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2305.01128	Analysis of different temporal graph neural network configurations on dynamic graphs	In recent years, there has been an increasing interest in the use of graph neural networks (GNNs) for analyzing dynamic graphs, which are graphs that evolve over time. However, there is still a lack of understanding of how different temporal graph neural network (TGNs) configurations can impact the accuracy of predictions on dynamic graphs. Moreover, the hunt for benchmark datasets for these TGNs models is still ongoing. Up until recently, Pytorch Geometric Temporal came up with a few benchmark datasets but most of these datasets have not been analyzed with different TGN models to establish the state-of-the-art. Therefore, this project aims to address this gap in the literature by performing a qualitative analysis of spatial-temporal dependence structure learning on dynamic graphs, as well as a comparative study of the effectiveness of selected TGNs on node and edge prediction tasks. Additionally, an extensive ablation study will be conducted on different variants of the best-performing TGN to identify the key factors contributing to its performance. By achieving these objectives, this project will provide valuable insights into the design and optimization of TGNs for dynamic graph analysis, with potential applications in areas such as disease spread prediction, social network analysis, traffic prediction, and more. Moreover, an attempt is made to convert snapshot-based data to the event-based dataset and make it compatible with the SOTA model namely TGN to perform node regression task.	Rishu Verma, Ashmita Bhattacharya, Sai Naveen Katla	2023-05-02T00:07:33Z	http://arxiv.org/abs/2305.01128v1	# Analysis of different temporal graph neural network configurations on dynamic graphs ###### Abstract In recent years, there has been an increasing interest in the use of graph neural networks (GNNs) for analyzing dynamic graphs, which are graphs that evolve over time. However, there is still a lack of understanding of how different temporal graph neural network (TGNs) configurations can impact the accuracy of predictions on dynamic graphs. Moreover, the hunt for benchmark datasets for these TGNs models is still ongoing. Up until recently, Pytorch Geometric Temporal came up with a few benchmark datasets but most of these datasets have not been analyzed with different TGN models to establish the state-of-the-art. Therefore, this project aims to address this gap in the literature by performing a qualitative analysis of spatial-temporal dependence structure learning on dynamic graphs, as well as a comparative study of the effectiveness of selected TGNs on node and edge prediction tasks. Additionally, an extensive ablation study will be conducted on different variants of the best-performing TGN to identify the key factors contributing to its performance. By achieving these objectives, this project will provide valuable insights into the design and optimization of TGNs for dynamic graph analysis, with potential applications in areas such as disease spread prediction, social network analysis, traffic prediction, and more. ## 1 Introduction Graphs are used to represent complex relationships between entities, thereby helping provide a flexible and intuitive way to model a complex problem in a neat visual representation. The idea of representing data as a graph has been around for a long time and has been used in various fields such as social network analysis, traffic flow, network modeling, natural language processing, etc. With the introduction of the Graph Convolutional Network (GCN) architecture in around 2016 [7]. the attention diverted to using graph-structured data and since then, the research on GNNs has exploded, and various GNN architectures have been proposed, including Graph Attention Networks (GAT), GraphSAGE, and Message Passing Neural Networks (MPNNs), etc. These traditional GNNs operate on static graphs, where the graph structure and node features are assumed to be fixed in time. However, in many real-world applications, the graph structure and node features can change over time - dynamic graphs. For example, in social media networks, the connections between users can change over time and the same goes for problems like traffic flow prediction, disease spread prediction, etc. In order to analyze these dynamic graphs, Temporal Graph Neural Network (TGNs) architecture was introduced as an extension of the Graph Neural Network (GNN) model. In TGNs[4] the graph structure includes an additional dimension of time series (aka temporal signals) along with the spatial dimension (which is already considered in traditional GNN) and the model is trained to predict the future states of the graph. With this excitement and motivation around TGNs, we performed a qualitative analysis on a few of the significant TGN models namely, EvolveGCN (Evolve Graph Convolutional Networks), GConv-LSTM, and GConv-GRU on a node-level regression problem of disease spread prediction where graphs are represented as a sequence of snapshots. On the other hand, other significant event-based models like TGAT (Temporal Graph Attention Networks), and TGN (Temporal Graph Networks), which process graphical data as a sequence of events are also studied in an edge-level prediction task. Further details regarding the project are organized in the upcoming sections of the report: Section 2 provides a review of relevant literature, Section 3 describes the methodology and datasets used in the project, Section 4 presents the results of the experiments, Section 5 & 6 discusses the ablation study and challenges respectively, Section 7 focus on the future work, Section 8 concludes the report with a brief summary. ## 2 Relevant literature This section provides a comprehensive overview of the prior work in the field and a brief introduction to the TGN models we intend to implement as part of our project. A brief discussion will also be provided about the motivation behind choosing the dataset. Following the taxonomy introduced in [6], the different existing TGNN models can be broadly classified into two sections- Snapshot-based or Discrete Time Dynamic Graphs (DTDG) and Event-based or Continuous Time Dynamic Graphs (CTDG). Snapshot-based models are used to learn over sequences of time-stamped static graphs. These methods are suitable to process the entire graph at each point in time with a suitable mechanism to aggregate temporal dependencies across different time-steps. On the other hand, event-based models process the time-evolving graphs as sequences of events and update a node or/and an edge if an event occurs involving that node or edge. ### Snapshot-based Models Snapshot-based models or DTDGs can be further distinguished as model-evolution and embedding-evolution based models. In evolution-based models, the parameters of static GNN evolve over time. EvolveGCN [1] is an example of this category of snapshot-based models, which is described in detail in 2.1.1. Embedding-evolution based models learn on embedding space (instead of parameter space) learned by static models. There are several models which fall under this category and we have selected DySAT [2] detailed in 2.1.2. #### EvolveGCN EvolveGCN extends the Graph Convolutional Network (GCN) architecture to dynamic graphs by introducing a temporal attention mechanism that adaptively combines information from multiple time steps. The algorithm also uses a gating mechanism to control the flow of information between the current and past states of the graph. In [1] the author has evaluated EvolveGCN mostly on the undirected graph datasets like Reddit, SBM, Bitcoin-OTC/Alpha, UCI, AS, and Elliptic and shows that it outperforms existing approaches to modeling dynamic graphs for tasks like link prediction, edge classification, and node classification. In the paper, the author has introduced two versions of EvolveGCN namely the -O and -H versions. In the -O version, the weights are treated as input and output from the LSTM (recurrent architecture) but in the case of the -H version (the H is inspired by hidden states), the weights are treated as hidden states to the GRU (recurrent architecture). #### DySAT Dynamic Self-Attention Network (DySAT) is a neural network model built on a self-attention mechanism that allows nodes in the graph to aggregate information from other nodes based on their feature similarity. Figure 3 shows the architecture of DySAT showing at first the self-attention mechanism is used to generate static node embeddings at each timestamp. Then self-attention blocks are introduced to process past temporal embeddings for a node to generate the novel embedding (or predicted embeddings). In [2] the author has evaluated the model on communication networks (UCI and Enron) and bipartite rating networks (Yelp and MovieLens). ### Event-based Models Event-based models or CTDGs can be further distinguished as temporal-embedding and temporal-neighborhood based models. Temporal-embedding based models learn temporal signals by aggregating information based on time embeddings, and features and connectivity structure of the graph. TGAT[3] is an example of this category, which is described in detail in 2.2.1. Temporal-neighborhood based models use a "mailbox" module to store functions of events to update node/edge embeddings with time based on past events. TGN[4] is chosen here as an example of this type of model, which usually achieves state-of-the-art performance on certain temporal graphs. #### Tgat This work involves developing a functional mapping from continuous time to vector time encoding for each node. Using this translation-invariant explicit functional time encoding based on Random Fourier Features (RFF), TGAT introduces graph temporal attention mechanism to aggregate the embeddings of the temporal neighborhoods of a node, where the positional encoding is replaced by the Figure 1: Schematic illustration of EvolveGCN. The RNN means a recurrent architecture in general (e.g., GRU, LSTM). Figure 2: Neural architecture of DySAT showing structural attention layers followed by temporal attention layers temporal encodings which are learned. The TGAT layer can be thought of as a local aggregation operator that takes the temporal neighborhood with their hidden representations (or features), as well as timestamps as input, and the output, is the time-aware representation for the target node at any time point t. The model is run on three datasets- Wikipedia, Reddit, and industrial on both transductive and inductive tasks. #### 2.2.2 Tgn${}_{\text{tq}}$ TGN consists of an encoder that creates compressed representations of nodes based on their interactions and updates them based on each event. Each node seen by the model so far is characterized by a memory vector, which stores a function of its past interactions. Given a new event, a mailbox module computes mail for every node involved. Mails are used to update the memory vector. To overcome the so-called staleness problem, an embedding module also computes, at each time step, the node embeddings using their neighborhoods and memory states. ## 3 Datasets EnglandCovid- We make use of the EnglandCovid Dataset [9] this dataset to understand the continuous dynamics of COVID-19 focusing on dates between 13th March 2020 to 12th May 2020, i.e, 61 days of data. The focus is to forecast the spread of the disease in England NUTS3 regions [8]. The data was collected from smartphones with Facebook installed with location history enabled. The raw data that was collected had data points from three times of the day - morning, afternoon, and evening; to make the dataset efficient for usage and analysis we aggregated the three data points into one single point for that entire day. The data quantified people who travel across the region on that particular day at some given time. The single value in each cell of the dataset is indicating aggregated reading (of all three times of the day) for each pair of regions. In-depth, the name (i.e., E10000031 to E10000032) relates to one pair of regions across which the number of people is moving about, and the number under each date is for the aggregated number of people. The dataset for England specifically consists of 129 regions for a total of 61 days. Each region is represented as a node in the graph and each edge represents the movement of people from one region to another. Edge attributes (weights) represent the number of movements between the two regions. Figure 6 visualizes the dataset for the first snapshot. Since this dataset intuitively falls under the category of DTDGs where the data over the entire graph is available at regular intervals of time. Therefore, we have used this dataset to study the snapshot-based models. Figure 4: TGN computation on a batch of time-stamped samples Figure 5: Number of Covid-19 cases with time across different regions under case study. Figure 3: The architecture of full TGAT layer at time t for node v0 Figure 6: Representation of EnglandCovid Dataset for snapshot t Wikipedia- In order to study CTDG models, we select the Wikipedia dataset where an event representing any node or edge-related activity can happen at any continuous time. The dataset consists of 30 days of edits made by users on Wiki pages after selecting the 1000 most edited pages and users who have edited at least 5 pages resulting in a total of 8227 users. Thus, the total number of nodes is 9227 (8227 users and 1000 pages), and the total number of interactions or edges occurring over time is 157,474. The features of each interaction are obtained by converting the text in each edit made into an LIWC-feature vector of 172 dimensions. The dataset is available at [12]. We focus on an edge-level task on this dataset which is to predict a future interaction, that is given all the interactions till time t, the task is to predict the probability of a user $u$ to editing a page $i$ at time t. This task is self-supervised. Since this dataset naturally consists of events happening at irregular intervals of time, this is tackled with the CTDG models like TGAT and TGN. The performance is also compared against baselines like Jodie [11] and DyRep [13] which are also CTDG models. ## 4 Results ### Analysis of Snapshot-based Model The dataset (EnglandCovid) is split into training and test as 80% and 20% respectively. Node Features is the normalized value of targets for previous "lag or time step" days, where time step/lag is the input from the user (for the baseline model we have fixed lags as 8). The models are trained for 200 epochs. The idea is to use the node features (which is the number of positive cases over i+lag number of days, where i = iterator over the snapshot for 129 regions) to predict the number of positive cases in 129 regions for (i+lag)+1day. As part of the comparative study, we observed the mean square error (MSE) trend as follows: It is observed that the MSE for gconv_LSTM is the lowest and shows the best performance when we decrease the number of lags (past windows) by 50%. Another observation is that the MSE is the highest for EvolveGCNH. Therefore in order to get better performance we changed the hyperparameters of the models (while keeping the lags = 8) and the tables 4.1.1, 4.1.2, 4.1.3, and 4.1.4 show the observations on each model. Wikipedia-* Two models which are mainly implemented on the edge prediction task on this dataset are TGN and TGAT. TGN is seen to outperform other models on this dataset, and detailed ablation studies shown in section 5 have been performed on TGN to better understand the key components leading to its superior performance. As shown in Figure 8, the training of TGN is fairly quick with validation precision reaching sufficiently high values during the early stages of training. As seen in Table 2. TGN with an attention mechanism to construct the embedding of each event performs much better on transductive and inductive (evaluation on previously unseen nodes during training) tasks than the other models. ### 4.3 TGN on EnglandCovid Dataset In order to run the TGN & TGAT on EnglandCovid Dataset the dataset had to be converted from snapshot to event-based data. But at the time of this project, the community is still looking for ways to convert snapshots to the stream of events or Temporal Data. Therefore, we tried to convert the discrete-time dynamic graph (DTDG) to a continuous-time dynamic graph (CTDG) and adjust it to the model's compatibility. As per the structure of the dataset, the first task was to identify the events which could be Node addition and deletion, Edge addition and deletion, and change in edge or node attributes. The second task was to identify the relevant features which could be movements and past cases. Figure 9 shows the structure of the EnglandCovid Dataset from which (after several trials and errors) we identify events at each timestamp as the number of edges added at that particular timestamp with features as the movements and past cases in the source and destination. More details on the dataset can be found in the GitHub repo [10]. This converted dataset gives the following results of the edge prediction and node regression task on TGN. In order to improve these results we tried to mask the number of events in the particular timestamp considering there were a lot of events occurring at a particular time stamp but we didn't observe any improvement in the results. ## 5 Ablation Study TGN- In this section, a detailed ablation is shown on TGN which is performing the best among other CTDG models. TGN as shown previously in Section 2 has three main modules- message, memory, and embedding. The embeddings are used in the downstream prediction task. Thus, TGN has four components - message function computation, message aggregation, embedding computation, and memory updater. Among these, the only learnable components are the embedding function and memory updater. The memory is updated with a recurrent model- GRU is used here. The embedding can be computed in several ways. Four different ways have been tested here- identity (where the feature of interaction is itself used as message), time projection (where the linear time-dependent projection of the feature is learned as \begin{table} \begin{tabular}{l\|l\|l\|l\|l} \hline \multicolumn{4}{l}{Edge Prediction Task} \\ \hline Old nodes test accuracy & 0.594 \\ \hline New nodes test accuracy & 0.547 \\ \hline \multicolumn{4}{l}{Node Regression Task (MSE)} \\ \hline Test MSE & 1.1343 \\ \hline \end{tabular} \end{table} Table 4.2.1: TGN with attention shows the best performance. The other models tested include TGAT which can be shown to be a specific case of TGN, Jodie, and DyRep. Figure 8: Training progress of TGN. Figure 9: Structure of EnglandCovid Dataset message), sum (where message at each node is computed using a sum of the interactions between its temporal neighborhood), and attention where the sum is performed after attending over the temporal neighborhood with non-uniform weights. For the message aggregation function, two ways have been tested- mean which takes the mean of the messages at a node over its past, and last which just aggregates the current message with the last message received at the node. The number of TGN layers have also been increased to two to check if performance is enhanced. The importance of the memory module has also been tested by completely removing the modules associated with memory. Memory helps in retaining past information at a node based on its history, and the memory of each node is also referred to as the state of the node. The observations made based on the ablation study done on the different modules of the TGN model are listed below. The memory component plays an important role in retaining past long-term information about the nodes, thus preventing the loss of information. It can be seen that TGN-no mem which does not have the memory module has the lowest test-AUC in comparison to the models having the memory component. Keeping a separate embedding module takes care of the common memory staleness problem which refers to the scenario where a node has remained inactive over a long period of time, and thus its embedding used for prediction is not updated. Keeping an embedding module updates a node embedding based on its neighbors, and thus keeps getting updated even if the node itself remains inactive. Out of the different embedding modules tried, attention-based embedding is seen to have the best performance. Thus, giving more attention or importance to the more relevant temporal neighbors is crucial. Increasing the number of TGN layers seems to slightly improve the performance of the model. But it makes the computation much more expensive than that with one layer. Thus, the TGN model needs only 1 layer for performing sufficiently well with reasonable computations. Having a memory model which keeps track of the past does not necessitate having more than one layer to capture the past information. On the other hand, models without memory like TGAT require two layers for a comparable performance with TGN with one layer. This makes the TGN model run much faster than an equally performing TGAT model. Additionally, efficient parallel processing within a batch makes TGN much more efficient. Tgat- TGAT has two components- graph aggregation layer which applies attention on the spatial neighbors to compute spatial node representations, time encoder which learns temporal embeddings. The main novelty of TGAT is the temporal encoding layer. Based on the ablation study done on TGAT, the following observations are made. Unlike TGN, TGAT has no memory module to retain past information on a node. Thus, it needs at least two layers to append over the past information. This makes TGAT much more computationally expensive than TGN with 1 layer with similar performance. Temporal encoding plays a significant role in increasing the performance of the TGAT configurations with time information incorporated. Attention mechanism used for aggregating neighbors performs significantly better than other aggregation methods like LSTM and mean. Thus, like TGN, attending over more relevant neighbors drives the performance of TGAT. The list of hyperparameters used for all the configurations of TGN and TGAT mentioned above for the accuracy reported is shown table 4. \begin{table} \begin{tabular}{l l l l l l l} \hline \hline Name & Embedding & Message & Message/Agg & Memory & Memory (TGN) & Num Layers \\ \hline TGN att & attention & attention & all & node & (G/N) & 1 \\ TGN att & attention & attention & all & node & (G/N) & 1 \\ TGN att & attention & all & all & node & (G/N) & 1 \\ TGN att & attention & all & all & node & (G/N) & 1 \\ TGN att & attention & all & mean & node & (G/N) & 1 \\ TGN att & attention & all & node & (G/N) & 1 \\ TGN att & attention & all & node & (G/N) & 1 \\ TGN att & attention & all & node & (G/N) & 1 \\ TGN att & attention & all & node & (G/N) & 1 \\ TGN att & attention & all & node & (G/N) & 1 \\ \hline \hline \end{tabular} \end{table} Table 5.1: Different configurations of TGN model \begin{table} \begin{tabular}{l l l l l l} \hline \hline Hyperparameter & Value & Hyperparameter & Value \\ \hline Memory dimension & $172$ & \multirow{2}{}{Number of Layers} & \multirow{2}{}{2} \\ \hline Embedding dimension & $100$ & & hidden,size & 64 \\ \hline Time embedding dimension & $100$ & & Droport & 0.5 \\ \hline Number of neighbours & $10$ & & Learning Rate & 0.001 \\ \hline Dropout & $0.1$ & & Weight Decay & 0.0 \\ \hline Attention heads & $2$ & & Batch Size & 256 \\ \hline Batch size & $200$ & & Number of Epochs & 100 \\ \hline Learning rate & $0.0001$ & & patience & 10 \\ \hline \hline \end{tabular} \end{table} Table 5.2: Performance of the different configurations of the base TGN model \begin{table} \begin{tabular}{l l l l l l l} \hline \hline Name & Session & User & User & User & User & User \\ \hline TGN attn & attention & attention & all & node & (G/N) & 1 \\ TGN attn & attention & all & all & node & (G/N) & 1 \\ TGN att & attention & all & all & node & (G/N) & 1 \\ TGN attn & attention & all & all & node & (G/N) & 1 \\ TGN att & attention & all & all & node & (G/N) & 1 \\ TGN att & attention & all & all & node & (G/N) & 1 \\ TGN att & attention & all & node & (G/N) & 1 \\ ## 6 Challenges The major challenge was the conversion of the discrete-time dynamic graph (DTDG) to a continuous-time dynamic graph (CTDG) and making it compatible with the Event-based models. Moreover, as the area of temporal graphs is relatively new, therefore, we could get references for plenty of Node classification and link prediction tasks but a lack of resources for Node regression tasks. ## 7 Future Work Observing these results in section 4.3 we are working on further improving the converted dataset by aggregating all the events in a particular time stamp such that we can get rid of multiple events occurring on the same node. Also, since this is a node regression task with multiple events occurring on multiple nodes (129) in a single timestamp, selecting features as the series of events for certain lags (or time steps) sounds like an interesting direction to this project and might show promising results. Agreeing with the fact that this would require computation of a larger scale. ## 8 Summary In this project, we could successfully perform a comparative analysis between the snapshot-based models namely gconv_LSTM, gconv_gru, EvolveGCNO, and EvolveGCNH, for node regression problem of disease spread prediction. These studies were carried out using the EnglandCovidDataset, a dataset for TGNs that PyTorch Geometric Temporal recently released. Using the mean squared error (MSE) measure, we assessed the performance of the models and discovered that all four models performed satisfactorily with EvolveGCN-O showing better results over others. Further, we were able to perform conversion of a DTDG dataset to the CTDG dataset and perform the analysis on the event-based model (TGN) using this converted dataset. Observations indicate that snapshot-based models tend to perform better in applications where a large number of multiple events occur at each timestamp and data is collected at regular intervals of time. CTDGs- event-based models tend to perform better in applications where very few events occur at each timestamp, and data can be collected over continuous time. Whether the EnglandCovid dataset is a suitable choice to test the CTDG models is still a question.
2302.06186	Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learning	The goal of this work is to address two limitations in autoencoder-based models: latent space interpretability and compatibility with unstructured meshes. This is accomplished here with the development of a novel graph neural network (GNN) autoencoding architecture with demonstrations on complex fluid flow applications. To address the first goal of interpretability, the GNN autoencoder achieves reduction in the number nodes in the encoding stage through an adaptive graph reduction procedure. This reduction procedure essentially amounts to flowfield-conditioned node sampling and sensor identification, and produces interpretable latent graph representations tailored to the flowfield reconstruction task in the form of so-called masked fields. These masked fields allow the user to (a) visualize where in physical space a given latent graph is active, and (b) interpret the time-evolution of the latent graph connectivity in accordance with the time-evolution of unsteady flow features (e.g. recirculation zones, shear layers) in the domain. To address the goal of unstructured mesh compatibility, the autoencoding architecture utilizes a series of multi-scale message passing (MMP) layers, each of which models information exchange among node neighborhoods at various lengthscales. The MMP layer, which augments standard single-scale message passing with learnable coarsening operations, allows the decoder to more efficiently reconstruct the flowfield from the identified regions in the masked fields. Analysis of latent graphs produced by the autoencoder for various model settings are conducted using using unstructured snapshot data sourced from large-eddy simulations in a backward-facing step (BFS) flow configuration with an OpenFOAM-based flow solver at high Reynolds numbers.	Shivam Barwey, Varun Shankar, Venkatasubramanian Viswanathan, Romit Maulik	2023-02-13T08:47:11Z	http://arxiv.org/abs/2302.06186v3	# Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learning ###### Abstract The goal of this work is to address two limitations in autoencoder-based models: latent space interpretability and compatibility with unstructured meshes. This is accomplished here with the development of a novel graph neural network (GNN) autoencoding architecture with demonstrations on complex fluid flow applications. To address the first goal of interpretability, the GNN autoencoder achieves reduction in the number nodes in the encoding stage through an adaptive graph reduction procedure. This reduction procedure essentially amounts to flowfield-conditioned node sampling and sensor identification, and produces interpretable latent graph representations tailored to the flowfield reconstruction task in the form of so-called masked fields. These masked fields allow the user to (a) visualize where in physical space a given latent graph is active, and (b) interpret the time-evolution of the latent graph connectivity in accordance with the time-evolution of unsteady flow features (e.g. recirculation zones, shear layers) in the domain. To address the goal of unstructured mesh compatibility, the autoencoding architecture utilizes a series of multi-scale message passing (MMP) layers, each of which models information exchange among node neighborhoods at various lengthscales. The MMP layer, which augments standard single-scale message passing with learnable coarsening operations, allows the decoder to more efficiently reconstruct the flowfield from the identified regions in the masked fields. Analysis of latent graphs produced by the autoencoder for various model settings are conducted using unstructured snapshot data sourced from large-eddy simulations in a backward-facing step (BFS) flow configuration with an OpenFOAM-based flow solver at high Reynolds numbers. ###### Contents * 1 Introduction * 2 Configuration and Dataset * 2.1 Governing equations and flow solver * 2.2 Flow configuration and mesh * 2.3 Description of dataset * 3 Methodology * 3.1 Graph representation of flowfield data * 3.2 Encoding and decoding procedures * 3.2.1 Encoder * 3.2.2 Decoder * 3.3 Multiscale message passing (MMP) layer * 3.3.1 Message passing block * 3.4 Adaptive pooling and unpooling layers * 3.4.1 Top-K pooling and masked fields * 3.4.2 Unpooling * 4 Results * 4.1 Effect of coarsening in the MMP layer * 4.2 Number of Top-K levels ### Analysis of reduction factor ###### Contents * 1 Introduction * 2 Background of the theory of the theory of the theory of the theory of the theory of the 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the the of the the the the the the of the metric configuration cannot be reliably extended to other configurations. Autoencoders based on convolutional neural networks (CNNs) are by design restricted to structured representations of the flowfield data. Similarly, ROM strategies reliant on MLP-based autoencoders are also often restricted to single geometric configurations, and the neural network parameters are obtained in such a way that does not allow for inference on unexplored or unseen meshes. Another limitation is model interpretability - in many studies, data-based ROM tools are essentially black boxes, rendering latent spaces uninterpretable. Ensuring interpretability in modeling frameworks often leads to (a) simplification and reductions in network complexity or expressive power (e.g. using a linear projection basis instead of a nonlinear one, for example [19, 25]), and/or (b) the development of other optimization strategies geared towards correlating the discovered latent dynamics with physical processes (i.e. disentanglement of latent variables [26, 27]), which adds additional computational expense and training constraints. Ultimately, the need for interpretation of the internal structures found in neural networks - particularly for applications like fluid flow prediction - has led to high modern demands for explainable artificial intelligence [28]. The goal of this work is to address both of these limitations - namely, interpretability and compatibility with unstructured meshes - by means of leveraging graph neural network (GNN) architectures for data-based model development. Within this scope, the main objective of this work is to show how the development of a novel graph autoencoder leads to highly interpretable latent representations. Before proceeding, some additional background and review on graph neural networks in the context of fluid dynamics modeling is provided below, as is a more concrete description of the individual contributions of this work. Introduced in early works in Ref. [29, 30], graph neural networks have gained popularity in recent years due to their ability to combine the scalability of backpropagation-based optimization with flexible representations of data [31, 32]. The end-result is a framework for executing generalized classification and/or regression tasks on any class of problems that can be cast as a complex network or graph. The generalizability and ambiguity in graph representations of data lends to their overall strength and increasing usage in most modeling tasks: so long as a problem or dataset can be described as a set of nodes, with connections (edges) among nodes encoding some notion of relational distance or feature similarity, a GNN-based approach can be utilized [33]. To this end, modeling strategies based on GNNs leveraging message passing [34] and graph Laplacian [35] operations have produced state-of-the-art results on image classification, object detection, and node classification tasks. Additionally, due to their versatility, GNNs see significant success in multi-disciplinary applications including modeling protein folding [36], predicting the emergent properties of social networks [37], and modeling differential equations [38, 39]. Much of the success in these pioneering GNN applications has translated into new data-based model development strategies for fluid dynamics applications, with the primary goal of providing a mesh-agnostic way to accelerate conventional CFD flow solvers. Because CFD simulations are stored on mesh arrangements from the get-go with pre-defined stencils (node connectivities), a data-based flowfield regression framework fits naturally for these problems within the GNN paradigm. The key idea is that standard message passing strategies used within GNN regression frameworks are dependent on not only node features in isolation (node features can be pressure, velocity, or any other flow quantity of interest), but also the properties of the neighborhood of the nodes in question. Due to this property, if the input CFD mesh is considered as a set of points from which a graph is derived, several applications have shown how a GNN trained on one mesh can be readily extended to other meshes (or other geometries) without the need for re-training. With this capability in mind, GNNs for fluid flow modeling have been deployed in a variety of contexts for both steady-state [40, 41, 42] and unsteady problems [43, 44]. In steady-state applications, GNNs are typically used to reconstruct flowfields from geometry-dependent fields such as signed distance functions. For example, Ref. [40] creates a surrogate solver for steady-state flows using a super-resolution strategy based on the graph convolution network [31]. Additional analysis into the role of graph pooling operations in the reconstruction procedure for steady-state flowfields was performed in Ref. [45]. From the perspective of unsteady fluid flow prediction, a key example of GNN-based modeling is the MeshGraphNet architecture introduced in Ref. [43], in which an encode-process-decode strategy is used to learn the source terms as spatial fields on unstructured grids for unsteady incompressible flow over cylinders and airfoils, among other applications. Inspired by methods used in multigrid methods [46], multi-scale GNN archittectures (a variant of which is leveraged in this work) have recently emerged to address limitations in single-scale counterparts by levaraging graph coarsening operations [42, 44, 47, 48]. Within the context of fluid flow prediction, although the above GNN-based predictive models succeed in extending conventional neural networks to unstructured grids and variable geometries, there are some lingering limitations. For example, most existing approaches leverage graph connectivities that are fixed in time. There has been very little emphasis on utilizing learnable, adaptive graph pooling strategies within the GNN architecture - these adaptive pooling layers are essentially operations which condition the generation of coarse graph connectivities (adjacency matrices) on some input field stored on the nodes of a fine, or baseline, graph. Although adaptive pooling strategies have been recently explored [42], these applications are typically restricted to steady-state flows and utilize dense interpolation operators that cannot scale to large graph sizes. Further, much of the recent focus in GNNs has been geared towards purely forecasting operations - in other words, given a flowfield at some time $t$, the objective is to learn the evolution at some future time $t+\Delta t$. Although the direct forecasting task is indeed relevant, efforts into purely graph-based autoencoding (i.e. using a GNN to recover an identity map) have been sparsely explored. It is noted that recent work in Ref. [49] used graph convolution operations in an autoencoding context to good effect; however, the methods presented in this work are distinct in that the architecture is completely based on graphs, incorporating graph pooling and unpooling layers without any flattening operations. The goal of this work is to address the above described limitations and gaps in the existing GNN literature for unsteady fluid flow modeling. The specific contributions are as follows: * A GNN-based autoencoding architecture is developed with the purpose of ensuring latent space interpretability. This is accomplished by employing an adaptive pooling strategy known as Top-K pooling [50]. The Top-K pooling mechanism is considered as a type of a node sampling procedure, or adaptive sensing strategy, used to reduce the total number of nodes needed to minimize the target objective. The end-result is a latent space, or bottleneck layer, that can can be visualized in physical space directly, identifies coherent structures in the domain, and is described by an adjacency matrix that adapts in time with the evolution of the flow. * A series of multi-scale message passing (MMP) layers are employed between pooling operations, each of which models information exchange among node neighborhoods at various lengthscales. The MMP layer, which augments standard single-scale message passing with learnable coarsening operations, allows the decoder to more efficiently reconstruct the flowfield from the identified regions in the latent graphs. * The GNN autoencoder is demonstrated on CFD snapshot data sourced from unsteady flow over a backward-facing step at Reynolds numbers in the range of roughly $20000$ to $50000$. Analysis of latent spaces are conducted on these snapshots to demonstrate (a) interpretability and physical significance of latent spaces via masked field visualization, and (b) trade-offs between reconstruction accuracy, interpretability, and achievable compression produced by the latent graphs. The remainder of the text proceeds as follows. The dataset generation procedure, governing equations, and domain geometry are described in Sec. 2. An overview of the methodology is then provided in Sec. 3, which describes the graph generation procedure and architecture description. Autoencoding results, with emphasis on analysis of the interpretable Top-K pooling output, is then provided in Sec. 4. Concluding remarks follow in Sec. 5. ## 2 Configuration and Dataset ### Governing equations and flow solver The data used in this study comes from unsteady, 2-dimensional large-eddy simulations (LES) of the incompressible Navier-Stokes equations of flow over a backward-facing step (BFS). Before proceeding, it should be noted that the application of LES in this scenario as a modeling approach is not entirely physical due to the usage of a 2-dimensional configuration; rather, turbulence modeling was applied here as a mechanism to produce coherent structures in flowfields that are reasonably complex and unsteady in nature for GNN demonstration purposes. As such, efforts to tune the LES model for physical consistency (i.e. consideration of near-wall regions) were not carried out. More specifically, the governing equations in the LES formulation are, from the conservation of mass and momentum in incompressible flow [51], \[\begin{split}\frac{\partial\widetilde{u}_{j}}{\partial t}+\frac{ \partial\widetilde{u}_{i}\widetilde{u}_{j}}{\partial x_{i}}&=\nu \frac{\partial^{2}\widetilde{u}_{j}}{\partial x_{i}\partial x_{i}}-\frac{ \partial\tau_{ij}^{r}}{\partial x_{i}}-\frac{1}{\rho}\frac{\partial\widetilde{ p}}{\partial x_{j}},\\ \frac{\partial\widetilde{u}_{k}}{\partial x_{k}}&=0. \end{split} \tag{1}\] For the general time-evolving field $f(x,t)$, the spatial filtering operation, denoted $\widetilde{\cdot}$, is defined as the convolution \[\widetilde{f}(x)=\int f(x^{\prime},t)G(x,x^{\prime})dx^{\prime}, \tag{2}\] where the kernel $G$ is called the filter. In this work, the filtering is performed implicitly - the flow variables stored on the computational grid are interpreted as the output of a box-filtering operation of filter size specified by the grid resolution [51]. In Eq. 1, the quantity $\widetilde{u}_{j}=\widetilde{u}_{j}(x,t)$ denotes the time-evolving $j$-th component of the filtered velocity, $\widetilde{p}=\widetilde{p}(x,t)$ is the time-evolving filtered pressure modified to include the isotropic component of the residual stress tensor, $\nu$ is a constant kinematic viscosity, $\rho$ is a constant density, and $\tau^{r}=\tau^{r}(x,t)$ is the deviatoric component of the residual, or sub-grid scale (SGS), stress tensor, defined as \[\tau^{r}_{ij}=\tau^{R}_{ij}-\frac{1}{3}\tau^{R}_{ii}\delta_{ij}, \tag{3}\] where $\tau^{R}_{ij}=\widetilde{u_{i}u_{j}}-\widetilde{u_{i}u_{j}}$. Because the residual stress tensor in Eq. 3 cannot be determined due to the presence of the filtered nonlinear term $\widetilde{u_{i}u_{j}}$, a model is needed close the partial differential equations in Eq. 1. There are many pathways available to this end - in this work, the standard Smagorinsky model is used [52], which casts the residual stress $\tau^{r}_{ij}$ as a quantity proportional to the filtered rate-of-strain as \[\begin{split}\tau^{r}_{ij}&=-2\nu_{r}\widetilde{S} _{ij},\\ \nu_{r}&=\left(C_{S}\Delta\right)^{2}\|S\|.\end{split} \tag{4}\] In Eq. 4, $S_{ij}$ is the filtered rate-of-strain, $\|S\|=\sqrt{2S_{ij}S_{ij}}$ is its magnitude, $\nu_{r}$ is the turbulent eddy-viscosity, $C_{S}$ is the Smagorinsky constant ($C_{S}=0.168$ here), and $\Delta$ is the box-filter width. The open-source library OpenFOAM is used to implement the numerical schemes needed to solve Eq. 1 within the finite-volume framework. OpenFOAM is also used here to produce the underlying computational mesh of the flow configuration (the mesh and configuration geometry is described in Sec. 2.2). The distribution of the OpenFOAM library contains a number of pre-packaged flow solvers used for the purposes of treating unsteady incompressible flows over complex geometries that are also compatible with a set of LES models, one of which is the Smagorinsky model described above. Here, the baseline pimpleFOAM solver is used with globally second-order numerics and one outer iteration. The central theme in the pimpleFOAM solver is to treat the effects of the pressure gradient term during a single time step with an operator splitting strategy resembling a predictor-corrector approach [53, 54]. Full details on the numerical schemes are out-of-scope here; the authors instead point the reader to Ref. [55] for more information on the specifics of OpenFOAM numerics. In the end, the user must describe the spatial numerical schemes, temporal schemes, and LES model parameters in a set of input files. These files -fvSchemes, fvSolution, and momentumTransport respectively - are provided in the supplementary material. In the remainder of the paper, for notational convenience, the filter notation $\widetilde{\cdot}$ used to describe velocity fields is dropped without loss of generality. ### Flow configuration and mesh With the governing equations and flow solver described above, a general schematic of the BFS configuration is shown in Fig. 1. This configuration, along with the similar double-BFS (or cavity) configuration, has been used in a number of experimental [56, 57] and numerical [58, 59] studies as a benchmark model problem for flows containing separation and re-attachment phenomena dominated by turbulent mixing, recirculation zones (as well as transition to turbulence), and vortex shedding. Flow enters from an inlet on the left and propagates through an initial channel of fixed width upon encountering a step anchor, triggering flow separation upon entering a second channel of larger width. The step size, $L_{s}$, is a key parameter in this configuration as it controls primary unsteady flow quantities of interest such as recirculation zone sizes observed in the cavity formed behind the step, vortex shedding frequencies downstream, and shear layer dynamics separating the cavity and freestream region. Values for the geometric quantities used in the simulation procedure are also provided in Fig. 1. During the simulation procedure, velocity at the inlet is prescribed in the x-direction to satisfy a Reynolds number within the range $[26214,45589]$, with characteristic length-scale based on the step height $L_{s}$. Additionally, for stability purposes, an initial startup region is used (indicated in red in Fig. 1) to provide a uniform velocity profile to the channel. In this startup region, slip boundary conditions are used at the walls; no-slip conditions are used at all other boundaries except for the inlet and outlet. Although the flowfield data were obtained from runs in the full BFS geometry, for the purpose of dropping training times, data used to train the GNNs come from a subdomain (the gray cropped region in Fig. 1) containing all flow dynamics of interest. Note that no interpolation procedures or mesh changes were utilized when extracting this cropped region - rather, a subset of cells were selected using the cellSet functionality in OpenFOAM such that only regions near the step were captured. A visualization of the cropped mesh used in this study is shown in Fig. 2(a), with a zoom-in on the near-step region in Fig. 2(b). The mesh, generated using the blockMesh utility in OpenFOAM, contains 14476 control volumes (cells) and is characterized by highly refined regions near walls and in a region behind the step such that (a) boundary layers are reasonably resolved, and (b) unsteady dynamics in the near-step fan, such as flow separation and emergence of recirculation zones, are properly captured. Mesh resolution increases as the domain progresses to the outlet and exits the near-step region - this coarsening effects continues outside of the cropped region in the x-direction. This variation in cell resolution $\Delta$ is shown in Fig. 2(c) across the two perpendicular lines indicated in Fig. 2(a). The mesh in Fig. 2 contains noticeable variations in cell sizes and skewness, and is therefore a sufficiently practical benchmark to assess GNN capabilities in complex geometric configurations. ### Description of dataset Using the BFS simulation configuration, datasets used for training and testing the GNN models described in Sec. 3 consist of a collection snapshots describing the evolution two-component velocity fields in the cropped mesh shown in Fig. 2 at various Reynolds numbers. An instantaneous snapshot in this context is given by the matrix $\mathbf{X}(t)\in\mathbb{R}^{N_{C}\times N_{F}}$, where $t$ denotes the time, $N_{C}$ the number of cells or control volumes ($N_{C}=14667$), and $N_{F}$ the number of features ($N_{F}=2$ here for streamwise and vertical velocity components). The rows of each snapshot $\mathbf{X}(t)$ describe individual cell centroid locations that are fixed in physical space as provided by the mesh configuration. The columns indicate the flowfield component, or feature, stored at the corresponding cell index in accordance with the finite volume formulation (Fig. 3 shows a visualization of the cell centers). The complete dataset consists of 5 simulation trajectories, each of which comes from sampling flowfields at a fixed time interval of $\Delta t=10^{-4}$ seconds at the various Reynolds numbers described in Table 1. Note that although the sampling interval is fixed, the simulations themselves were run in a variable timestep setting based on a maximum Courant-Friedrichs-Lewy (CFL) number of $0.5$. As shown in Table 1, the training data is comprised 3 out of the 5 total trajectories corresponding to Reynolds numbers of 26214, 32564, and 39076 respectively, producing a total number of $1216$ training snapshots. The testing data consists of the remaining 2 trajectories: one at an interpolated Reynolds number with respect to the training data range (Re=29307), and the other at a larger, extrapolated Reynolds number (Re=45589) to provide a more challenging GNN evaluation task. Figure 3 shows the time evolution of the streamwise velocity component at the specified near-step probe locations for all five cases in Table 1. The velocity signals display highly unsteady decaying periodic behavior indicative of turbulent vortex shedding cycles observed in the BFS configuration at these high Reynolds numbers. These trends ultimately illustrate both the complex dynamics characteristic this configuration, as well as the appreciable difference in dynamics observed in the set of training and testing trajectories. For example, in accordance with the increased Reynolds number, significant increases in velocity magnitude and shedding cycle frequency is seen for Trajectory 5, which is included in the testing set. Figure 1: Schematic of the backward-facing step configuration (not shown to scale) using an inlet-to-step length ratio of $7.9$. Lengths are normalized by the step height $L_{s}$, and green circle marks the origin. The mesh for the cropped region indicated in this figure is shown in Fig. 2. These periodic trends in velocity signal are linked to the continual emergence of recirculation zones in the near-step region, causing the re-attachment point for the separated shear layer induced by the step to oscillate along the bottom wall in the x-direction. A qualitative comparison of this re-attachment cycle is shown in Fig. 4 for trajectories 1 and 5, which are contained in the training and testing sets respectively. The figure displays a series of instantaneous streamwise velocity snapshots, each separated by a time of $10^{-3}$ seconds, to illustrate the variation in the separation-reattachment process due to the differences in Reynolds number. The snapshot sequences illustrate how the cycle is initiated when the reattachment point experiences a "pinch-off" effect caused by the emergence of a recirculation zone near the step corner, which in turn advects a coherent region of negative streamwise velocity downstream. The pinch-off time in the cycle occurs earlier as Reynolds number increases - in Fig. 4, this corresponds to snapshot 4 in trajectory 1 versus snapshot 2 in trajectory 5. Additionally, as the pinch-off phenomenon terminates, the cycle concludes when the flowfield encounters a similar condition as the starting point, corresponding to a more uniformly reattached flowfield near the step. As implied through the oscillatory nature of the velocity signals in Fig. 3, the visual comparison in Fig. 4 shows how the increase in cycle frequency in the higher Reynolds number cases is paired with a proportional decrease in the reattachment length. As shown in Fig. 5, drop in reattachment length is identified by stronger recirculation zones near the step caused by increased magnitudes of adverse pressure gradients observed in the flow. The above comparisons in training and testing trajectories, along with flowfield visualizations, are provided to give confidence to the fact that the key physical quantities of interest characterizing flow over backward-facing steps (e.g. reattachment, downstream boundary layer development, emergence of recirculation zones in step corner) are correctly captured in the simulation procedure. Although additional effort can be undertaken in the \begin{table} \begin{tabular}{\|\|c c c c\|\|} \hline Trajectory & Re & Snapshots & Category \\ \hline \hline 1 & 26214 & 434 & Training \\ 2 & 32564 & 386 & Training \\ 3 & 39076 & 396 & Training \\ 4 & 29307 & 378 & Testing \\ 5 & 45589 & 405 & Testing \\ \hline \end{tabular} \end{table} Table 1: BFS simulation trajectories used to populate datasets for the graph neural network training and evaluation. Figure 2: (a) Cropped mesh of the BFS configuration shown to scale. Red lines indicate lines along which resolutions are computed in (c). Green circle denotes origin. (b) Zoom-in of the mesh highlighting refinement near the step. (c) Mesh resolution $\Delta$ normalized by the step height $L_{s}$ along lines shown in (a). Figure 4: Visualization of BFS shedding cycles via streamwise velocity evolution for trajectory $1$ (top, Re=26214) and trajectory $5$ (bottom, Re=45589). Columns display instantaneous snapshots separated by a time of $10^{-3}$ seconds. Figure 3: (a) Red markers indicate probe locations in the BFS configuration used to plot trajectories in (b) and (c). Black points mark control volume centroids. (b) Evolution of pressure for probe locations depicted in (a). Data collection phase occurs after roughly $t=0.14$ s. Note that pressure here can be negative because the quantity is shifted by an arbitrary reference. Pressure has also been normalized by the constant density $\rho$. (c) Visualization of training, validation, and testing datasets using pressure evolution at the probe locations in (a). Bold lines isolate single trajectories for clarity. modeling procedure to establish consistency with experimental and direct numerical simulation (DNS) studies at the chosen operating conditions, such analyses are deemed out-of-scope in the context of demonstrating the graph neural network capabilities described in Sec. 3 and 4. ## 3 Methodology The graph neural network (GNN) autoencoder architecture is shown in Fig. 6. The various layers shown in the schematic are organized into encoder and decoder segments that operate on graph representations of the unstructured flowfield data. The GNN forward pass can expressed concisely through the actions of the encoder and decoder via \[G_{L} =\mathcal{E}(G_{0}), \tag{5}\] \[\widetilde{G}_{0} =\mathcal{D}(G_{L}), \tag{6}\] where $\mathcal{E}$ and $\mathcal{D}$ are the graph-based encoder and decoder, respectively. As shown in Fig. 6, the encoder $\mathcal{E}$ takes as input a graph $G_{0}$. The encoder output - referred to as the _latent graph_$G_{L}$ - is conditioned on both the input graph quantities (e.g. nodes and edge attributes) and the parameters used in the encoder layers. The decoder $\mathcal{D}$ takes as input the latent graph $G_{L}$ and produces the output graph $\widetilde{G}_{0}$; the decoding evaluation is similarly conditioned on a separate set of layer parameters. Analogous to conventional autoencoding goals, the objective of the graph autoencoder is (a) to obtain a latent graph $G_{L}$ that reduces dimensionality of the input graph $G_{0}$, and (b) to ensure, through an data-based optimization or learning procedure, that the parameters contained in $\theta_{\mathcal{E}}$ and $\theta_{\mathcal{D}}$ result in an accurate decoding or reconstruction (i.e. $\widetilde{G}_{0}\approx G_{0}$). In this work, the optimization procedure seeks to minimize the mean-squared error (MSE) between only the node attributes in the initial ($G_{0}$) and decoded ($\widetilde{G}_{0}$) graphs - definitions for these node attributes are provided in Sec. 3.1. An additional requirement central to this work is that of interpretability: the graph autoencoder should be constructed in such a way that the latent space (in this case, the latent graph $G_{L}$) is physically interpretable. To this end, the main building blocks of the GNN architecture are multiscale message passing (MMP) layers and the graph pooling layers. The MMP layer (described in Sec. 3.3) models information exchange throughout the flow domain at multiple lengthscales based on input graph connectivity, analogous to convolution operations on Figure 5: Velocity field comparison near step corresponding to frame 1 (beginning of shedding cycle) in Fig. 4 in trajectory 1 (top) and trajectory 5 (bottom). Fields are colored by velocity magnitude, with arrows displaying velocity vector orientation. Yellow circles denote center of recirculation zones, and red circles denote flow reattachment zones. structured grids. The graph pooling layer (detailed in Sec. 3.4) is the procedure by which graph dimensionality can be reduced via reduction in the number of nodes. Emphasis here is placed on one particular graph pooling strategy known as Top-K pooling [60], which has been unexplored in recent GNN-based studies for unsteady fluid flows. A primary objective is to show how this Top-K pooling layer, through an adaptive graph reduction procedure that essentially amounts to flowfield-conditioned node sampling and sensor identification, produces interpretable latent graph representations tailored to the regression task at hand. ### Graph representation of flowfield data Before presenting details of the message passing and pooling operations, the input graph generation procedure must first be described. The input graph $G_{0}$ is defined as the tuple $G_{0}=(\mathbf{V}_{0},\mathbf{E}_{0},\mathbf{A}_{0})$. The quantity $\mathbf{V}_{0}\in\mathbb{R}^{N_{0}^{c}\times F_{0}^{v}}$ is the input node attribute matrix: a single row contains the $F_{0}^{v}$-sized feature vector of one of $N_{0}^{v}$ total nodes in the input graph. In the applications considered here, these initial node features represent physical flowfield quantities (i.e. velocity fields sampled at a particular location in physical space), which eventually evolve into hidden node features of a potentially different size upon encountering nonlinear GNN layers during the encoding stage. The quantity $\mathbf{E}_{0}\in\mathbb{R}^{N_{0}^{c}\times F_{0}^{v}}$ is the input edge attribute matrix: a single row in $\mathbf{E}_{0}$ describes the $F_{0}^{v}$-sized feature set of a directed edge in the graph. Along with its set of features, a directed edge is characterized by both a sender and receiver node index - these nodes are deemed "similar" in some sense based on the user-defined criteria used to derive the connection. Accumulating these indices for all edges in the graph generates the adjacency matrix $\mathbf{A}_{0}\in\mathbb{Z}^{N_{0}^{v}\times N_{0}^{v}}$. The adjacency matrix is used to facilitate arithmetic operations conditioned on the graph connectivity (e.g. the edge aggregation procedure described in Sec. 3.3). Put simply, if the rows of this matrix correspond to sender node indices and columns to receiver node indices, any directed edge can be represented as a value of unity in the corresponding location in the adjacency matrix with zeros everywhere else, such that the number of non-zero entries in $\mathbf{A}_{0}$ recovers the total number of directed edges $N_{0}^{c}$. Two quantities of interest derived from the adjacency matrix are the node neighborhood and node degree. The neighborhood of a receiver node $i$, denoted $N(i)$, is the set of sender node indices that share an edge; the degree of node $i$, denoted $d(i)$, is the cardinality of this neighborhood set ($d(i)=\|N(i)\|$). It should be noted that in practice, assuming the average node degree is significantly smaller than the total number of nodes, adjacency matrices are stored in a sparse matrix format to both ensure scalability with respect to $N_{v}$ and avoid memory limitations. Given the above definitions, the steps required to actually generate the input graph from the flowfield data described in Sec. 2.3 are (1) produce the set of nodes and corresponding node attributes from the raw data, and (2) construct the list of edges (graph connectivity) using some notion of similarity derived from either all or a subset of these node attributes. To motivate consistency with the finite-volume formulation of numerics utilized within Figure 6: (Top) Encoder flowchart, showing the procedure by which the input graph representation of flowfields $G_{0}$ (red nodes) is downsampled into the latent graph $G_{L}$ (green nodes). Main building blocks are multiscale message passing layers (see Sec. 3.3) and Top-K pooling layers (see Sec. 3.4). (Bottom) Decoder flowchart, showing the procedure by which the latent graph $G_{L}$ is upsampled to recover $\widetilde{G}_{0}$, which contains predicted flowfield quantities at the original graph node locations. Upsampling is achieved using an unpooling layer described in Sec. 3.4. the OpenFOAM framework, this work adopts graph connectivity using the latter pathway. As shown in Fig. 7, in this setting, graph nodes represent cell centroids. The node attributes on $G_{0}$ are then readily initialized as \[\mathbf{V}_{0,i}=\mathbf{X}(t_{i}),\quad i=1,\ldots,N. \tag{7}\] where the subscript $i$ is introduced as a snapshot index, and $\mathbf{X}(t_{i})$ is the flowfield snapshot sampled at time $t_{i}$ as described in Sec. 2.3. In the end, the number of nodes in the initial graph $N_{0}^{v}$ is equivalent to the number of cells $N_{C}$ in the mesh (see Fig. 2). Directed edges between nodes are then instantiated based on shared cell faces. As such, edges that connect nodes in this formulation coincide with flux paths between cells. To prevent issues related to mesh skewness and variable mesh density during message passing operations, additional edges are appended to these initial edges by also establishing connections between nodes within a user defined radius of $0.08L_{S}$. In other words, a radius-based connectivity is superimposed onto the finite volume connectivity, where for a given node, edges are created for all other nodes within the specified length scale. The number of nodes and edges in the initial graph after this procedure are $N_{0}^{v}=14476$ and $N_{0}^{e}=133739$ respectively. In-line with previous approaches, the initial edge feature matrix $\mathbf{E}_{0}$ is populated with the distance vector produced by the corresponding sender and receiver node indices. Note that due to the time-dependence of the flowfield $\mathbf{X}(t)$, the node attributes in the input graph - the velocity fields at the cell centroids - are also time-evolving. However, because the physical space coordinates of the mesh cells are fixed, the edge attributes $\mathbf{E}_{0}$ and adjacency matrix $\mathbf{A}_{0}$_in the input graph_$G_{0}$ are fixed for all snapshots. ### Encoding and decoding procedures The GNN evaluation procedure shown in Fig. 6 is broken down in the subsections below into encoder and decoder components in Sec. 3.2.1 and 3.2.2 respectively. With the general background provided here, the reader is directed to Sec. 3.3 and 3.4 for finer details on multiscale message passing (MMP) layers and graph pooling layers used within the architecture. #### 3.2.1 Encoder As implied by the subscript, the input graph $G_{0}$ resides on the baseline "zeroth" level of the autoencoding architecture. The encoder produces a hierarchy of graphs at higher levels through successive message passing and Top-K pooling operations. The encoding procedure terminates upon encountering the latent graph $G_{L}$, where $L$ denotes the maximum level in the architecture (the "latent graph" is always referred to here as the graph produced at the highest level). Consistent with the notation introduced in Sec. 3.1, a graph at the $l$-th level is denoted Figure 7: Graph generation procedure used in this work. Graph connectivity is consistent with the finite-volume formulation of CFD numerics. Although only quadrilateral cells are present in the mesh used here, schematics in shaded box show a single neighborhood for two different cell geometries to illustrate the approach. Red circle indicates root node, dashed lines indicate underlying cell, and solid arrows indicate directed edges that intersect cell faces. $G_{l}=(\mathbf{V}_{l},\mathbf{E}_{l},\mathbf{A}_{l})$, and is characterized by its own set of nodes, edges, and adjacency matrix. For example, in the case of $L=2$, graphs at three different levels are utilized in the architecture: $G_{0}=(\mathbf{V}_{0},\mathbf{E}_{0},\mathbf{A}_{0})$ (the input graph at level 0), $G_{1}=(\mathbf{V}_{1},\mathbf{E}_{1},\mathbf{A}_{1})$ (the level 1 graph), and $G_{2}=(\mathbf{V}_{2},\mathbf{E}_{2},\mathbf{A}_{2})$ (the level 2 or latent graph). Depending on the reduction factors used in the pooling operations, the goal of the encoder is to ensure that the number of nodes in the graph decreases as the number of levels increases, resulting in the following inequality: \[N_{0}^{v}>N_{1}^{v}>\ldots>N_{L}^{v}, \tag{8}\] where $N_{l}^{v}$ ($l=0,\ldots,L$) denotes the number of nodes at the $l$-the level (equivalent to the number of rows in $\mathbf{V}_{l}$). The _global reduction factor_ ($RF_{G}$), an input parameter that characterizes the level of compression achieved in the architecture in terms of node reduction, is $RF_{G}=N_{0}^{v}/N_{L}^{v}$. As will be seen in the results in Sec. 4, there are two important implications of the reduction factor: (1) a higher $RF_{G}$ is expected produce a more difficult reconstruction task for the decoder, and (2) one can use different graph hierarchies (i.e. different values of the maximum graph level $L$) to arrive at a target $RF_{G}$. Embedding node and edge features: As demonstrated in Refs. [33, 43], feature space embeddings significantly improve predictive accuracy in GNN-based models. In this process, shown as the first step in Fig. 6, before encountering the message passing and pooling operations required to generate graphs at different levels, the node and edge features in the input graph $G_{0}$ are modified. This step is carried out as $\mathbf{V}_{0}\gets f_{\mathcal{E}}^{v}(\mathbf{V}_{0})$ and $\mathbf{E}_{0}\gets f_{\mathcal{E}}^{e}(\mathbf{E}_{0})$, where $f_{\mathcal{E}}^{v}$ and $f_{\mathcal{E}}^{e}$ are independently-parametrized multi-layer perceptrons (MLPs) batched over the number of nodes and edges respectively in $G_{0}$. These MLPs operate in feature space only, and therefore result in modified feature dimensionalities for the nodes and edges in the input graph $G_{0}$ - the number of nodes and edges, as well as the graph connectivity, is not changed. As a result, after this feature encoding stage, the node and edge attributes reside in a hidden feature space of fixed size, typically larger than the feature space dimensionality used to initialize the original graph nodes and edges. The resulting node and edge features are referred to as hidden channels. For convenience, the hidden channel dimensionality in this work the same for node and edge attributes after this step ($F_{0}^{v}=F_{0}^{v}$), and is kept fixed during the forward pass until encountering the final node feature decoding operation to recover the desired reconstructed flowfields. Main encoding layers: As alluded to in the beginning of this section, the two layers that serve as the backbone of the GNN-based encoder are a multiscale message passing (MMP) layer and an adaptive graph pooling operation known as Top-K pooling. These are carried out in succession to move from a lower level graph $l$ to a higher level graph $l+1$ with fewer nodes, eventually terminating in the generation of the final latent graph at level $L$. These operations are given by \[(\mathbf{V}_{l},\mathbf{E}_{l}) \leftarrow\text{MMP}_{\mathcal{E},l}(\mathbf{V}_{l},\mathbf{E}_{l},\mathbf{A}_{l}),\quad l=0,\ldots,L, \tag{9a}\] \[(\mathbf{V}_{l+1},\mathbf{E}_{l+1},\mathbf{A}_{l+1}) \leftarrow\text{TopK}_{l}^{l+1}(\mathbf{V}_{l},\mathbf{E}_{l}, \mathbf{A}_{l}\|\mathbf{p}_{l}),\quad l=0,\ldots,L-1, \tag{9b}\] where $\text{MMP}_{\mathcal{E},l}$ denotes a multiscale message passing layer present in the encoder $\mathcal{E}$ at graph level $l$, and $\text{TopK}_{l}^{l+1}$ denotes the Top-K pooling layer that downsamples graph variables from level $l$ to $l+1$. Note that the MMP layer in Eq. 9(a) does not modify the graph connectivity $\mathbf{A}_{l}$ - rather, it modifies node and edge features through a neighborhood aggregation procedure described in Sec. 3.3. On the other hand, the pooling operation in Eq. 9(b) does modify the connectivity through a node sampling procedure conditioned on the feature-wise projection vector $\mathbf{p}_{l}$. This sampling mechanism usefully provides interpretability properties in latent graphs - details on this aspect are provided in Sec. 3.4. #### 3.2.2 Decoder The goal of the decoder is to upsample the latent graph $G_{L}$ back into the original input graph dimensionality in terms of number of nodes and edges. The upsampling procedure is executed with a series of graph unpooling layers that mirror the pooling operations used in the encoding stage. In other words, starting from $G_{L}$, the successive layer operations encountered in the decoding stage terminate upon arriving at the graph $\widetilde{G}_{0}$ that shares the same number of nodes and edges as the input graph $G_{0}$. Main decoding layers: The decoding procedure is outlined in the following steps: \[(\mathbf{V}_{l},\mathbf{E}_{l}) \leftarrow\text{MMP}_{\mathcal{D},l}(\mathbf{V}_{l},\mathbf{E}_{l},\mathbf{A}_{l}),\quad l=L,\ldots,0. \tag{10a}\] \[(\mathbf{V}_{l-1},\mathbf{E}_{l-1},\mathbf{A}_{l-1}) \leftarrow\text{unpool}_{l}^{l-1}(\mathbf{V}_{l},\mathbf{E}_{l}, \mathbf{A}_{l}),\quad l=L,\ldots 1. \tag{10b}\] To mirror the encoding stage, the decoder first utilizes an MMP layer at graph level $l$ in Eq. 10(a) before encountering the unpooling layer in Eq. 10(b), which serves to umpsample the node and edge feature matrices to the sizes consistent with those in level $l-1$. The multiscale message passing layer utilizes identical operations as in the encoding stage (Eq. 9(a)), but contains different parameters (see Sec. 3.3). This combination of message passing and unpooling essentially amounts to a learnable interpolation operation between successive graphs in the Top-K hierarchy - additional detail on the unpooling procedure is provided in Sec. 3.4. Decoding node features: To recover the final reconstructed flowfield at the nodes, after all message passing and unpooling steps are executed, the decoder utilizes a node-wise feature decoder at level 0 that reverses the feature embedding procedure executed in the first step of the encoding stage. This step is carried out as $\tilde{\mathbf{V}}_{0}\gets f_{\mathcal{D}}^{\mathrm{c}}(\mathbf{V}_{0})$, where $f_{\mathcal{D}}^{\mathrm{c}}$ is an MLP that operates feature-wise, serving to transform the node dimensionality from the hidden channel dimension back into the original dimension corresponding to the number of flowfield observables used in the dataset. The final predicted quantity used to compute the MSE loss function is then $\tilde{\mathbf{V}}_{0}$, which contains the reconstructed flow quantities (velocity components) at the nodes. ### Multiscale message passing (MMP) layer The notion of message passing was put forward in Ref. [34] as a general framework to unify a wide range of GNN modeling strategies. In any message passing approach, the goal is to provide a functional representation for the interaction between node attributes in a single neighborhood of the graph. Within a standard message passing layer, this interaction rule is formulated such that (a) its function parameters are shared throughout all neighborhoods in the graph, thereby ensuring a domain-agnostic modeling approach, and (b) it does not modify the input graph connectivity (adjacency matrix). Inspired by methods used in the multigrid community, recent work in GNN-based modeling has demonstrated that augmenting standard message passing operations with a hierarchy of coarse grids naturally results in more effective information propagation [44, 47]. These methods remove the need for modeling large lengthscale interactions with a very large number of message passing operations, which in turn reduces memory requirements during training and drops inference times. As the name implies, the multiscale message passing (MMP) layers used in both encoding and decoding stages leverage this principle. A schematic of a single MMP layer is shown in Fig. 8. The MMP layer consists of a series of message passing blocks, each on a different coarsening level relative to the baseline mesh. Each message passing block contains a fixed number of single-scale message passing (MP) layers. After executing a message passing block, node and edge attributes are interpolated to a coarser graph level characterized by a larger inter-node lengthscale. This proceeds until the coarsest graph is reached, upon which graph quantities are interpolated back upwards to the starting point, utilizing skip connections for node and edge attributes from the downward pass along the way. It is emphasized that the coarsening levels in MMP layers are different from the concept of graph levels introduced earlier, which come from outputs of Top-K pooling operations - this distinction is made clear in Sec. 3.4. The coarsening mechanism used in each MMP layer comes from a voxel-clustering algorithm, which has been demonstrated to work well in previous GNN applications [44]. Although other meshing approaches can be utilized (e.g. triangulations [47]), the advantage of the voxel clustering approach is that it directly takes a target message passing lengthscale as an input, and does not require an expensive optimization step in the coarsening procedure. In this approach, a graph is coarsened by overlaying a voxel grid in its physical space bounding box. Each cell in this voxel grid is interpreted as a cluster of fixed size corresponding to a target message passing lengthscale. Each voxel cell is identified by a parent node, which corresponds to its cell centroid location in physical space. Upon overlaying the voxel grid, underlying fine graph nodes can be assigned to representative voxel cells/clusters via computation of nearest centroids, establishing a parent-child relationship between two successive coarse and fine graphs in the MMP layer. Edges between parent nodes are added if underlying fine graph edges intersect their shared voxel cell face - the coarse edge attributes are instantiated using the average of attributes belonging to all fine edges that satisfy this intersection. The parent-child ownership labels are then used to facilitate interpolation operations required to transfer node attributes between graphs at different coarsening levels. These interpolation procedures follow the methods introduced in Liao et al. [44]. In summary, for each coarse and fine graph pair, a new set of edges connecting children nodes to their respective parent node are established, forming an interpolation stencil for node attributes with learnable weights. This stencil is then used to learn coarse-to-fine and fine-to-coarse mappings for node attributes. The reader is pointed to Ref. [44] for more detail on the interpolation method. Section 3.3.1 details the operations involved in a single message passing block. Before proceeding, it should be noted that the operations shown in Fig. 8 are equivalent to a graph U-net architecture. The difference here is that this U-net operation corresponds to a _single layer_ - an MMP layer - operating on one level in the full autoencoding architecture shown in Fig. 6. The effect of including multiple coarsening levels in MMP layers in both encoder and decoder stages will be analyzed in Sec. 4. #### 3.3.1 Message passing block A single message passing block in an MMP layer consists of a fixed number of uniquely parameterized single-scale message passing (MP) layers. The MP strategy used in this work comes from Battaglia et al. [33], which has been established in recent years as a well-grounded framework for graph-based modeling of complex physical systems related to fluid dynamics and other applications [43, 61]. The strategy, summarized in Fig. 9, is given by the following three steps: \[\text{Step 1 (Edge Update):}\quad\mathbf{e}_{k}^{p}=f_{\varepsilon}^{p}( \mathbf{e}_{k}^{p-1}\|\mathbf{v}_{s_{k}}^{p-1}\|\mathbf{v}_{r_{k}}^{p-1}),\quad k =1,\ldots,n_{e}, \tag{11a}\] \[\text{Step 2 (Edge Aggregation):}\quad\mathbf{a}_{i}^{p}=\frac{1}{\|N (i)\|}\sum_{\{k:r_{k}=i\}}\mathbf{e}_{k}^{p},\quad i=1,\ldots,n_{v},\] (11b) \[\text{Step 3 (Node Update):}\quad\mathbf{v}_{i}^{p}=f_{v}^{p}( \mathbf{v}_{i}^{p-1}\|\mathbf{a}_{i}^{p}),\quad i=1,\ldots,n_{v}. \tag{11c}\] Superscripts in the above equations denote the MP layer index local to the message passing block, and vertical bars in function arguments denote concatenation operations. Additionally, $\mathbf{e}_{k}^{p}$ denotes the $k$-th edge attribute vector sourced from the corresponding row in the edge attribute matrix, and $\mathbf{v}_{i}^{p}$ is the $i$-th analogous node attribute vector - $n_{e}$ and $n_{v}$ denote the number of nodes and edges of the graph in question. In Eq. 11(a), the subscripts $s_{k}$ and $r_{k}$ present on the right-hand side denote the sender and receiver node indices for the $k$-th edge. As shown in Fig. 9, the functions $f_{e}^{p}$ and $f_{v}^{p}$ are independent multi-layer perceptrons (MLPs) that act as nonlinear functions producing updated edge features and node features respectively. This allows the update procedures to both capture nonlinear interactions between edge and node attributes from previous layers, and also model complex feedback between individual nodes and their local neighborhoods. Note that the graph adjacency matrix is invoked only in Eq. 11(b) (the aggregation step); the summation is used to transform the edge-based representation of data into a node-based representation via reduction over the edge attributes in the neighborhood $N(i)$ (i.e. $\mathbf{a}_{i}^{p}$ is defined on the nodes). For some graph $G=(\mathbf{V},\mathbf{E},\mathbf{A})$, the MP layer from Eqs. 11(a)-(c) is concisely represented as \[(\mathbf{V}^{p},\mathbf{E}^{p})\leftarrow\text{MP}^{p}(\mathbf{V}^{p-1}, \mathbf{E}^{p-1},\mathbf{A}). \tag{12}\] Figure 8: Schematic of MMP layer used in the autoencoding architecture in Fig. 6. Message passing blocks (see Fig. 9) are shown in red, specifying number of MP layers per at each coarsening level. Message passing lengthscales $L_{G}$ normalized by step size $L_{S}$ are specified underneath respective graph images. Dashed lines indicate residual/skip connections for node and edge attributes. Note that these skip connections operate within _a single layer_ of either the encoder or decoder (see Fig. 6). The $p$-th MP layer, denoted MP${}^{p}$, updates both node and edge attribute representations without modifying the graph connectivity. Although not shown in Eq. 12, the message passing layer is parameterized by the weights and biases in the MLPs $f_{e}^{p}$ and $f_{v}^{p}$. As the size of a message passing block increases, the expressive capability of the GNN also increases at the trade-off of computational expense during training and inference stages. Unless otherwise noted, the number of MP layers in each message passing block is provided in Fig. 8, and parameters are not shared across message passing blocks within any MMP layer. ### Adaptive pooling and unpooling layers As shown in Fig. 6, the result of an MMP layer operation described in Sec. 3.3 is provided to either Top-K pooling layers in the encoding phase, or unpooling layers in the decoding phase. This section provides relevant details on each of these layers. Analogous to pooling operations in convolutional networks, the primary goal of graph pooling layers is to reduce the degrees-of-freedom of the underlying system being modeled via reduction in the number of nodes. The pooling strategy used in this work - known as Top-K pooling [60] - provides a unique take on the reduction process. Instead of achieving reduction via fixed graph coarsening, Top-K pooling layers achieve reduction using _adaptive node sampling_. In other words, the layer reduces the number of nodes by sampling a subset of nodes from the input graph. Node positions in the reduced graph coincide in physical space with nodes in the input graph, but the set of sampled node indices is a function of the input node attributes. As such, if these node attributes are time-evolving, the identified nodes in the reduced graph - as well as the connectivity - also change in time. This difference in reduction philosophy has led to the Top-K layer being overlooked for the purposes of fluid flow modeling, as the reduced graph does not come from a direct coarsening operation. However, it is shown in this work how the Top-K layer, through the adaptive node sampling procedure, allows for built-in interpretability of the latent graph $G_{L}$ (see Fig. 6). More specifically, the time evolution of latent graphs produced by the pooling operation can be visualized directly in physical space using _masked fields_. Because the node sampling procedure optimizes a regression task, visualization of this latent graph can be used to access regions in physical space relevant to this regression task. It should be noted that this interpretability quality resembles a data-based sparse sensing approach for feature identification, but the advantage here is that this formulation is (a) directly compatible with unstructured grids and complex geometries, and (b) is tailored by design to any user-defined regression problem. In the text below, the high-level details of the Top-K pooling and unpooling operations relevant to this work are described. For additional information on the specifics of Top-K operations, the reader is directed to Ref. [60]. Figure 9: (Left) Schematic of a message passing block consisting of two MP layers. (Right) Illustration of the edge and node update procedures within MP${}^{p}$ for a single neighborhood. In the edge update (Eq. 11(a)), edge attributes are shown in black, sender node attributes in blue, and receiver node attributes in red. In the node update (Eq. 11(c)), node attributes are shown in red and aggregated edge features in black. #### 3.4.1 Top-K pooling and masked fields The Top-K pooling operation is given in Eq. 9(b), and is illustrated in the schematic in Fig. 10. Two parameters are required: a learnable projection vector $\mathbf{p}_{l}$, and the number of nodes $K$ to retain in level $l+1$. In practice, $K$ is recovered from a local reduction factor $RF_{l}$ via $K=N_{l}^{v}/RF_{l}$, where $N_{l}^{v}$ is the number of nodes at input level $l$. In a first step, the projection vector $\mathbf{p}_{l}\in\mathbb{R}^{F_{l}^{v}\times 1}$ is used to transform nodes residing in the input $F_{l}^{v}$-dimensional feature space into a one-dimensional representation. In a second step, the projected node features are ranked in descending order and truncated such that the indices of the topmost $K$ nodes are retained. These $K$ node indices become "sampled" nodes, forming the node attribute vector $\mathbf{V}_{l+1}$ on the reduced graph (i.e. $K=N_{l+1}^{v}$). More formally, the initial projection step is given by \[\mathbf{y}_{l}=\frac{\mathbf{V}_{l}\mathbf{p}_{l}}{\\|\mathbf{p}_{l}\\|}\in \mathbb{R}^{N_{l}^{v}\times 1}, \tag{13}\] where $\mathbf{y}_{l}$ is the projected vector. The node selection step is given by \[\mathcal{I}_{l+1}=\text{rank}(\mathbf{y}_{l},K), \tag{14}\] where $\mathcal{I}_{l+1}$ is the set containing the node indices of the graph at level $l$ corresponding to the topmost $K$ quantities of the sorted projections in $\mathbf{y}_{l}$. With $\mathcal{I}_{l+1}$, an indexing operation can be carried out to recover the reduced graph quantities $\mathbf{V}_{l+1}$, $\mathbf{E}_{l+1}$, and $\mathbf{A}_{l+1}$ in a sampling stage, thereby terminating the Top-K pooling layer. It should be noted that in order to ensure the projection vector $\mathbf{p}_{l}$ can be trained using backpropagation, a gating function in the form of a sigmoid activation is performed on the downsampled node features. For details on this gating procedure, see Ref. [60]. Interpretable masked fields: Crucial to the GNN architecture in Fig. 6 is that the Top-K pooling operation is used to construct a hierarchy of _adaptive_ graphs (i.e. non-fixed adjacency matrices) - the node positions in physical space of the input graph $G_{0}$ are fixed in time, but the node positions for levels $l>0$ change in time. This is because, for a fixed projection vector $\mathbf{p}_{l}$ obtained after training, indices of the sampled nodes at level $l+1$ adapt to the Figure 11: Illustration of the distribution operation (Eq. 15) used in unpooling. Red boxes denote interpolated values and white boxes denote zeros. Figure 10: Top-K pooling illustration. (a) Two-component PCA visualization of input node attribute matrix $\mathbf{V}_{0}$ associated with an instantaneous flowfield (markers denote individual nodes). (b) 1d projection of $\mathbf{V}_{0}$ via Eq. 13. Red region indicates retained node indices corresponding to a reduction factor of 16. (c) PCA visualization of output node attribute matrix $\mathbf{V}_{1}$ after sampling. Nodes are a subset of those shown in (a). (d) Masked field corresponding to sampled nodes in (c) – red region indicates where latent graph $G_{1}$ is active. flow features contained in the input node attribute matrix $\mathbf{V}_{0}$, which are time-evolving. Since each of these indices corresponds to a particular location in physical space, the evolution of reduced graphs at levels $l>0$, as well as the final latent graph at level $L$, can be visualized via _masked fields_. The masked field, an example of which is shown in Fig. 10, is a visualization of the level index $l$ of each cell in the physical domain. Utilized further in Sec. 4, these masked fields reveal the GNN's ability to (a) physically interpret latent graphs, and (b) identify evolving coherent structures in the domain intrinsically tied to the regression task (flowfield reconstruction). #### 3.4.2 Unpooling To facilitate graph decoding, the unpooling operation in Eq. 10(b) can be performed in the Top-K framework so long as $\mathcal{I}_{l+1}$ is available. As illustrated in Fig. 11, the unpooling layer is interpreted as a data distribution operation that resembles delta-function interpolation. The distribution operation for node features is given by \[\widetilde{\mathbf{V}}_{l}=\text{distribute}(\mathbf{V}_{l+1},\mathcal{I}_{l +1},\mathbf{0}_{l}). \tag{15}\] In the above equation, the matrix $\mathbf{0}_{l}\in\mathbb{R}^{N^{r}_{\lambda}\times F^{r}_{l}}$ contains only zeros - it is interpreted as an empty node matrix existing on level $l$. The unpool operation "fills in" (or replaces) rows of the matrix $\mathbf{0}$ with the corresponding rows of $\mathbf{V}_{l+1}$ as per the indices contained in $\mathcal{I}_{l+1}$. Edge attribute matrices $\widetilde{\mathbf{E}}_{l}$ can be recovered in an analogous fashion, leading to the full unpooling operation of Eq. 10(b). Although the operation in Eq. 15 successfully recovers the original data dimensionality at level $l$ (which is the primary goal of the unpooling layer), since skip connections are not used, many values in the upsampling node attribute matrix $\mathbf{V}_{l}$ can be zero depending on the level of reduction (i.e. value of $K$). The role of the MMP layer after unpooling (see Fig. 6) is to treat this issue by efficiently distributing values in the unpooled graph to fill in empty nodes. ## 4 Results In this section, analysis of the GNN-based autoencoder is performed for the flowfield reconstruction task from a variety of angles, with the primary scope aimed at showcasing the interpretability and reconstruction properties of generated latent spaces (or latent graphs). For all models discussed herein, the objective function used during training is the mean-squared error between input and reconstructed node feature matrices. Three-layer MLPs were used for node/edge attribute encoding and decoding, and two-layer MLPs were used for node and edge updaters in message passing operations. In all cases, MLPs utilize exponential linear unit (ELU) activation functions [62] with residual connections. Layer normalization [63] is applied after each MLP evaluation. Input and reconstructed node features consist of streamwise ($u_{x}$) and vertical ($u_{y}$) velocity components that are standardized using training data statistics. GNN architecture development and training was performed using a combination of PyTorch [64] and PyTorch Geometric [65] modules. For training, flowfield snapshots obtained from trajectories 1-3 (see Table 1) were used, with a randomly selected 10% of these snapshots set aside as the validation set. During training, a batch size of 8 was used as per the graph-based mini-batching scheme of Ref. [65], and the Adam optimizer [66] with a learning rate scheduler based on validation loss thresholding was applied. All models were trained using a single Nvidia A100 GPU housed on a node on the Polaris high-performance computer at the Argonne Leadership Computing Facility. The section proceeds as follows. In Sec. 4.1, the effect of including coarsening operations in the MMP layer on the latent graph and reconstructed fields is discussed. In Sec. 4.2, for a fixed reduction factor, analysis of changes in the masked fields due to variation in the number of Top-K levels $L$ is conducted. Lastly, in Sec. 4.3, the effects of both the node reduction factor and the number of hidden node channels used to characterize the latent graph are studied from the perspectives of reconstruction accuracy and masked field evolution. ### Effect of coarsening in the MMP layer To convey the impact of the coarsening operations present in the MMP layers (see Fig. 8), Fig. 12 presents mean-squared error histories and streamwise velocity field reconstructions at the probe locations for three different GNN architecture configurations: - No Coarsening:** In this setting, the coarsening operations in the MMP layers are not used. More specifically, the MMP layer architecture shown in Fig. 8 is modified to neglect all clustering operations used to coarsen the input graph - as such, message passing proceeds in the conventional fashion without coarsening. This effectively turns the MMP layer into a single-scale message passing block. * Decoder Coarsening:** In this setting, coarsening operations present in the MMP layers in Fig. 8 are used _only in the decoder_$\mathcal{D}$. In the encoder $\mathcal{E}$, no coarsening operations are used. * Encoder+Decoder Coarsening: In this setting, the MMP coarsening operations are used in both encoder and decoder phases. When coarsening is used (Models 2 and 3), the MMP layer is consistent with Fig. 8. When coarsening is not used, all coarse grids corresponding to higher lengthscales in Fig. 8 are ignored, rendering the MMP layer equivalent to a total of 4 standard single-scale message passing layers. All three configurations utilize a maximum level of $L=1$ (only one Top-K layer is used), a node reduction factor of $RF_{G}=16$ (the number of nodes in the latent graph is dropped by 16x), and a hidden channel dimensionality of $32$. Note that despite the fact that the number of nodes has decreased, the latent graphs here do not provide true compression due to the increase in node feature dimensionality. Assessment of true compression is delayed to Sec. 4.3. Using the above three configurations, the focus of this section is to instead demonstrate the need for coarsening operations in the MMP layer from the perspectives of reconstruction accuracy and masked field interpretability. Since the optimization of parameters in the message passing layers and Top-K layers are coupled, the way in which nodes are subsampled to produce the latent graphs is expected to be dependent on the message passing scheme. To assess this effect quantitatively, training history for each of the above three model configurations is shown in Fig. 12(a). The impact of coarsening is evident in the loss function trends, as the converged errors for Model 1 (the configuration without coarsening in the MMP layer) shown in the blue curves in Fig. 12 is an order of magnitude higher than the configurations that include coarsening operations (Models 2 and 3). Further, the jump in accuracy provided by appending additional coarsening operations in the encoding stage (Model 2 to 3) is much lower than when appending coarsening operations in only the decoding stage (Model 1 to 2), implying that the contribution of reconstruction accuracy comes primarily from latent graphs produced using only decoder coarsening operations. Figures 12(b) and (c) show how the convergence trends during training translate to streamwise velocity reconstructions on the unseen testing trajectories. Prediction trends are ultimately consistent with the loss function curves, in that Model 1 completely fails to capture dynamical information content in the near-step region at all tested Reynolds numbers. On the other hand, reconstructions produced by models that include coarsening are much more in-line with ground truth velocities - interestingly, Model 2 reflects more accurate flowfield reconstructions than Model 3 near the step, despite the fact that it utilizes coarsening operations only in the decoding stage. As will be seen in the qualitative comparisons below, the improved reconstructions provided by Model 3 come into play in the freestream regions, which is not reflected in the probe measurements shown in Fig. 12. Overall, these trends show how (a) coarsening operations are necessary to produce latent graphs that reliably reconstruct the flowfield, and (b) acceptable near-step velocity reconstructions at unseen Reynolds numbers, included extrapolated Reynolds numbers, are produced by autoencoders utilizing multiscale message passing. The advantage of this autoencoding approach is that the physical characteristics of the reconstruction procedure, as well as the impact of including coarsening operations during message passing via the MMP layers, can be directly interpreted through a visualization of the identified latent graphs ($G_{1}$ in this case) via masked fields. As such, to supplement the discussion surrounding Fig. 12, visualizations of the autoencoding procedure for a single input velocity field snapshot are shown in Fig. 13 for the same three model configurations. For each of the testing set trajectories (an interpolated and extrapolated Reynolds number), the input snapshot shown in Fig. 13 corresponds to step 1 of the reattachment cycle described in Fig. 4. The masked fields, which indicate where in physical space the latent graphs are active, allow one to visualize locations in the domain corresponding to optimal flowfield reconstruction for the respective models. Note that because the cell sizes vary with spatial location, a fixed node reduction factor (16 for the models shown in Fig. 13) can result in different identified "volumes" in the masked fields depending the learned Top-K projection vector - for example, although the number of latent graph nodes is the same, the masked regions identified by Model 2 are smaller in physical space than those in Model 1 because the cell resolution in the identified region is accordingly smaller. Upon visual inspection, a common ground in all three models is that the masked fields in Fig. 13 are characterized by disjoint clusters of identified nodes (red regions), which is in-line with the physical nature of the reattachment cycle described in the discussion surrounding Fig. 4. However, aside from this quality, the masked fields are markedly different for each of the model configurations, which is expected due to the different utilization of coarse grids during message passing. The latent graphs identified by Model 1 (no coarsening), for example, identify larger coherent regions in the freestream region above the step than the other two configurations. This is in line with expectations in a single-scale message passing model: since the message passing scheme without coarsening is less efficient at distributing information from the masked region to the rest of the domain, it is understandable that the identified nodes in the latent graph occupy a larger coherent region in space. The disadvantage of the Model 1 configuration is evident when looking at the reconstructed flowfields, which directly show how the lack of multi-scale message passing results in a failure of the model to reconstruct the flow in regions away from the identified mask. Although adding additional message passing layers in a single-scale setting can address this issue, the computational costs and memory limitations during backpropagation render such approaches infeasible. In contrast, the reconstructions produced by Models 2 and 3 are successful in recovering the primary features of the BFS flowfield, such as the re-attachment point and the propagating recirculation zone downstream - although reconstruction accuracy is still imperfect and there are lingering non-physical artifacts in the freestream, Fig. 13 shows how adding coarsening operations via MMP layers addresses the challenging task of recovering full flowfield information from the masked region. Although both Models 2 and 3 are able to efficiently propagate information from the masked field, the masked fields themselves are fundamentally different. Model 2 (decoder coarsening only) recovers a masked region that identifies coherent structures consistent with dynamically active regions in the BFS configuration (shear layers, recirculation zones, downstream shedding). On the other hand, Model 3 shows how the inclusion of coarsening operations in the encoder MMP layers drastically alters the coherency of the masked field. Interestingly, the masked field in Model 3 itself resembles a coarse grid unlike the much more spatially coherent Model 2 counterpart. This implies that the inclusion of coarsening operations in the encoding stage effectively trades physical space coherency and interpretability for improved reconstruction accuracy. This is evidenced in the reconstructed fields at the extrapolated Reynolds number (bottom part of Fig. 13), for which Model 2 fails to reliably extrapolate to unseen freestream velocities when Model 3 succeeds. Despite this, even at extrapolated Reynolds numbers, Model 2 is still able to showcase the powerful ability to reasonably capture the primary BFS flow features while retaining the quality of coherent structure identification in the masked field. Additionally, it should be noted that across all three model configurations, the masked fields retain similar structure at the different Reynolds numbers shown in Fig. 13, which serves as a form of validation for the identified regions. The regions identified by Model 2 (decoder coarsening) in particular are almost identical at different Reynolds numbers for the snapshots shown. The objective of this section was to demonstrate the need for coarsening operations in message passing layers in the decoding task, while also showcasing the primary interpretability property of latent graphs produced by Figure 12: (a) MSE versus training epochs for Model 1 (blue curves), 2 (red curves) and 3 (green curves). Solid lines are training set evaluations, dashed lines are validation set. (b) Reconstructions of streamwise velocity ($u_{x}$) for Trajectory 4 (Re=29,307, see Table 1) at probe locations given in Fig. 3. Top plot is Probe 1, bottom plot is Probe 2. (c) Same as (b), but for Trajectory 5 (Re=45,589). Figure 13: (Top) Autoencoding procedure for an instantaneous flowfield from Trajectory 4 (Re=29,307) using Models 1-3. Leftmost column shows input velocity fields. Corresponding rows show reconstructed fields. Topmost rows show masked fields, where cells colored in red indicate active nodes in latent graph. (Bottom) Same as top, but for a Trajectory 5 snapshot (Re=45,589). the GNN autoencoder. The architectures utilized above were limited to single levels ($L=1$). As alluded to in Sec. 3, a beneficial property of this autoencoder is its ability to create a hierarchy of Top-K levels ($L>1$), which in turn leads to alternative representations of the masked fields and enables higher levels of flowfield compression. These aspects are discussed in the sections below. For the remaining analysis in the subsections below, all models configurations correspond to the Model 2 setting (decoder coarsening only) in light of its improved masked field coherency. ### Number of Top-K levels The connectivity properties of the latent graph $G_{L}$ produced by the encoder are not only dependent on the input flowfield, but also on the maximum level $L$ used in the Top-K hierarchy. In other words, for a target global node reduction factor $RF_{G}$, one can use either a single Top-K level to achieve this reduction ($L=1$, which was used in Sec. 4.1), or a series of Top-K levels, each with smaller local reduction factors $RF_{l}$. To this end, to isolate the effect of the maximum Top-K level $L$ on the latent graph, Fig. 14 compares the previously discussed Model 2 from Sec. 4.1 for which one Top-K pooling operation was used ($L=1$), to a two-level ($L=2$) counterpart. The $L=2$ model achieves the target global reduction factor of $RF_{G}=16$ by accumulating two successive reductions that each drop the number of nodes by a factor of 4 ($RF_{l}=4$). Note that the $L=2$ counterpart here still utilizes the Model 2 configuration described in Sec. 4.1, in that coarsening operations in the MMP layers are invoked only in the decoder. To better illustrate the changes to the latent graph structure due to time evolution, shown in Fig. 14 are masked fields for two snapshots sourced from Trajectory 4, each at different stages in the BFS shedding cycle. Additionally, for ease of visualization, masked field plots are overlaid with velocity field orientation vectors to facilitate correlation of the identified coherent structures with recircualtion zones and other flowfield patterns. In Fig. 14, for the $L=1$ case, the masked field depicts a single identified sub-graph (red region) consisting of 16x fewer nodes than the baseline graph. On the other hand, in the $L=2$ case, the model identifies a hierarchy of subgraphs in accordance with the Top-K procedure: the highest-level latent graph at $l=2$, denoted by the blue regions in the corresponding masked field, also contains (a) 16x fewer nodes than the baseline graph (black regions), and (b) 4x fewer nodes than the identified graph at level 1 (red region). In other words, the masked field, through visualization of the identified level indices in Top-K pooling procedure, allows the user to interpret directly the subsampled hierarchy until the latent graph is reached (highest level index). Note that in the $L=2$ case, the identified latent graph in blue is not disjoint from the intermediary graph at $l=1$, but rather is a subset of the identified nodes at the preceding level. Figure 14 shows how changes to the Top-K hierarchy modifies the identified latent graph structure, despite the fact that the number of nodes comprised in these latent graphs is the same. For example, in the $L=2$ case, adding an additional level to the hierarchy allows the latent graph to occupy different regions in physical space that are slightly above the step cavity when compared to the $L=1$ counterpart. This comes directly from the enabled exchange of information between successive levels during the autoencoding procedure when when more Top-K levels are used: in the $L=2$ case, the identified graph at the first level ($l=1$, red region) occupies a larger portion of the domain, enabling the latent graph to target regions of interest unreachable by the single-level model. Interestingly, the subsampled nodes in the latent graph for the $L=2$ case do not focus on the recirculation zone immediately near the vertical step wall - instead, focus is placed on downstream regions associated with vortex shedding and the reattached shear layer. When considering all levels together, however, qualitatively similar coherent structures are identified regardless of the parameter $L$ used, giving confidence to the physical significance of the latent graphs at a target global reduction factor. When assessing the velocity field patterns in combination with the masked fields in Fig. 14, it is evident that the masked fields are correlated with recirculation zones in the flow. For example, in the reattached state, three disjoint structures are visible in the masked field in accordance with the presence of three recirculation zones: a weaker zone near the vertical step wall, another near the reattachment point, and a third corresponding to a stronger shedding vortex approaching the outflow. Of these three regions, the latent graph in the $L=2$ model places more importance on the latter two, implying that the recirculation zones away from the step are more crucial to the flowfield reconstruction task. This is consistent even at different time instances in the unsteady flowfield: for example, in the snapshot exhibiting the pinch-off phenomenon that instantiates the downstream shedding procedure (Fig. 14, right), two recirculation zones are present - as such, two large-scale coherent structures are identified in both masked fields, with the $L=2$ model again emphasizing regions slightly further away from the cavity. ### Analysis of reduction factor The above sections showcased the architectural effects, in terms of coarsening operations utilized in the MMP layers (Sec. 4.1) and Top-K hierarchy size (Sec. 4.2), on the output latent graph without taking into consideration the potential pathways for compression. As such, the goal of this section is to outline effects of latent graph compression from two angles: (1) the impact of reducing the number of latent graph nodes via increases to the global node reduction factor $RF_{G}$, and (2) the impact of reducing the number of hidden channels $N_{H}$ stored on each latent graph node. It should be emphasized that both factors play into true compression achieved by the latent graph, as the total number of node degrees of freedom in $G_{L}$ is $(N_{0}^{v}/RF_{G})\times N_{H}$, where $N_{0}^{v}$ is the number of nodes on the input graph $G_{0}$ (here, $N_{0}^{v}=14476$). Since the primary focus and novelty of this work is tied to demonstrating the interpretability properties of the latent graph via the masked fields, the two compression angles described above are analyzed from the perspective of both reconstruction accuracy and identified structures in the masked fields. In other words, the focus here relates to how the GNN autoencoder provides the user the ability to access the ways in which latent graphs change to achieve flowfield compression. Figure. 15 displays normalized root-mean squared errors (RMSE) at various Reynolds numbers corresponding to the dataset trajectories described in Sec. 2.3. The RMSE for a single trajectory (Reynolds number) is given by \[\text{RMSE}=\frac{\sqrt{\frac{1}{MN_{0}^{v}}\sum_{i=1}^{M}\sum_{j=1}^{N_{0}^ {v}}(x_{i,j}-\widetilde{x}_{i,j})^{2}}}{u_{in}}, \tag{16}\] where $x$ is a generic flowfield variable (e.g. $x=u_{x}$ for streamwise velocity and $x=u_{y}$ for vertical). In Eq. 16, $x_{i,j}$ and $\widetilde{x}_{i,j}$ denote the target and reconstructed flowfield variable respectively for the $i$-th snapshot and $j$-th node. Note that in Eq. 16 the number of snapshots $M$ per trajectory varies, but the number of output graph nodes $N_{0}^{v}$ is Figure 14: (Left) From top-to-bottom: velocity magnitude field, masked field for $L=1$ model, and masked field for $L=2$ model. Snapshot depicts reattached state in Trajectory 4 (Re = 29,307) (Right) Same as left, but for a different snapshot in in in Trajectory 4 corresponding to the pinch-off phenomenon (see Fig. 4). fixed. To better interpret errors across the range of Reynolds numbers contained in the data, the baseline RMSE is normalized by the inlet freestream velocity $u_{in}$. Figure 15(a) shows the effect of increasing the node reduction factor on the normalized RMSE of Eq. 16 for both streamwise and vertical velocity component reconstructions. To achieve increasingly higher amounts of node reduction in the latent graph, the maximum Top-K level in the GNN $L$ is adjusted with a fixed local reduction factor of $4$ between levels while freezing all other model hyperparameters (including the number of message passing layers). As a result, for the input graph containing $N_{0}^{v}=14476$ nodes, the latent graph in the $L=1$ model in Fig. 15(a) contains 4x fewer nodes, the $L=2$ model contains 16x fewer nodes, and the $L=3$ model 64x fewer nodes. Not that despite the fact that the number of nodes has been reduced by increasing $L$, the hidden channel dimensionality of all models in Fig. 15(a) is fixed to $N_{H}=32$ - as such, only the $L=3$ model achieves true data compression in the sense of reduction in the total number of nodal degrees-of-freedom. Two immediate trends are apparent in Fig. 15(a) - the first is that the errors are consistently higher for the streamwise component of velocity ($u_{x}$), and the second is that an increase in the node reduction factor via maximum Top-K level $L$ results in a vertical shift in the RMSE curves for both velocity components. The former trend is expected, as a majority of the flow contribution to the reattachment cycle dynamics is contained in the streamwise component. This is also evidenced by observing that the vertical component errors are largely insensitive to Reynolds number, wheras errors for streamwise velocity across all configurations shown in Fig. 15 tend to increase with Reynolds number after about Re=32000. The latter trend conveys how it becomes increasingly more difficult for the decoder to recover the full flowfield as the number of nodes in the latent graph decreases. This is consistent with the fact that the same message passing scheme is used between all models (i.e. the MMP layer architecture is fixed as $L$ is increased), which in effect makes it more difficult for the decoder to populate flow information on the original nodes from the latent graph as the size of the latent graph decreases. Although not shown here, it is reasonable to expect that modifying the MMP layer design to compensate for the reduction in nodes, either in the form of adding additional coarsening levels or increasing the size of message passing blocks, would lessen the degree of the vertical curve shifts. To complement the notion of the achieving compression by dropping the number of nodes, Fig. 15(b) displays error trends for models that instead modify the hidden channel dimensionality $N_{H}$ of the latent graph while fixing all other parameters. In particular, the autoencoders used to generate the curves shown in Fig. 15(b) come from taking the $L=2$ case in Fig. 15(a) (i.e. 16x reduction in the number of nodes) and adjusting the latent node dimensionality. As such, the $N_{H}=16$ case achieves a true compression factor of 2 and the $N_{H}=8$ case achieves one of 4. Although the model with the largest hidden channel dimensionality ($N_{H}=32$) tends to drop the RMSE curve over the tested Reynolds number range, the RMSE behavior at smaller values of $N_{H}$ is more complex - for example, the error in vertical velocity component reconstructions is actually higher for $N_{H}=16$ when compared to $N_{H}=8$, although the same is generally not true for the more dominant streamwise velocity component. Despite this, the results in Fig. 15 suggest that there is indeed an error-compression tradeoff from two angles: reduction in number of latent graph nodes and reduction in latent node dimensionality (hidden channels). As the scope of this work is directed towards demonstrating latent graph interpretability allowed by the architecture, further analysis of these error trends (i.e. in terms of compensating for error shifts with more message passing operations, increasing the number of input flow features, more rigorous hyperparameter testing, etc.) is omitted here and left for future reports. To better visualize the effect of node reduction on the latent graph, Fig. 16 shows the time evolution of masked fields (alongside input flowfields) during one reattachment cycle in trajectory 4 for the same $L=1$, $2$, and $3$ models plotted in Fig. 15(a). The figure illustrates how the adjacency matrix in the latent graph adapts to evolving flow structures during the shedding cycle, and also how increased reduction factor (compression) impacts the identified regions in the masked field to optimize reconstruction accuracy. The identified latent graph nodes (red regions in masked fields in Fig. 16) are significantly impacted by the maximum graph level $L$ - in other words, as the number of latent graph nodes decrease, the large-scale regions and flow features in physical space identified by the masked fields change. For example, the $L=1$ case (node reduction factor of 4) identifies the inlet and outlet boundaries, a majority of the freestream region, and also the structure of propagating recirculation zones in the flow. On the other hand, as a result of increasing the node reduction factor to 16, the latent graph in the $L=2$ case eliminates concentration in the freestream regions, and instead focuses on the shedding structures in the step cavity (the physical significance of these identified regions were outlined in Sec. 4.2, for which the same $L=2$ model was used). It is clear from Fig. 16 that identified structures propagate in accordance with the shedding cycle frequency. Note that although the number of latent graph nodes has dropped by a factor of 4x when moving from the $L=1$ to the $L=2$ case, there is a disproportionate decrease in the "coverage" of the masked fields in physical space due to the fact that the mesh resolution in the near-step region is much higher. Interestingly, the $L=3$ case - the model that achieves a node reduction of 64x and achieves a true compression in the degrees of freedom by a factor of 4 - isolates only the inlet and outlet boundary regions. Although a very small time-evolving structure near the step cavity is identified to capture the unsteadiness in the reattachment cycle (indicated by white circles in Fig. 16), the latent graph connectivity in the $L=2$ case is largely insensitive to the evolving flow patterns. This implies that in cases of greater latent graph reduction, the Top-K mechanism isolates the inlet and exit boundary conditions as opposed to the step cavity region as the primary mechanism for flow reconstruction. Figure 17 shows how these masked fields translate to flowfield reconstructions of varying accuracy in accordance with the amount of node reduction provided by latent graph. More specifically, shown in Fig. 17 are reconstructions for streamwise and vertical velocity components corresponding to snapshot 4 in Fig. 16, which depicts a moment just after the pinch-off phenomenon has occurred. Upon inspection, it is clear that the reconstruction accuracy - especially in terms of the presence of non-physical flow artifacts (e.g. discontinuities due to interpolation) - begins to drop as the maximum level $L$ increases. Despite this, although deviation in the freestream region appears in the $L=2$ model, the large-scale structures near the step are captured well. However, the $L=3$ model begins to see significant deterioration in the reconstructed flow features near step - in particular, the gap between pockets of negative streamwise velocity components is not captured, as indicated by the black circled regions in the respective figures. Despite this, it is motivating that the latent graph even in the $L=3$ case is able to recover the general structure of the BFS flow features in light of the fact that the identified nodes are largely concentrated away from the step and towards the boundaries. Overall, the trends observed in Fig. 17 reflect those observed in Fig. 15(a), in that there is a direct deterioration in reconstruction accuracy as the amount of node reduction is increased. Although the reconstruction accuracy (particularly for the $L=3$ case in Fig. 17) leaves something to be desired, it is emphasized that the advantage and objective of the autoencoding framework used here is the ability for the user to access and interpret the discovered latent graph through the visualization of time-evolving masked fields, as shown in Fig. 16. As mentioned above, a natural step forward is to explore the impact of including additional message passing operations in the MMP layer, as well as graph-based filtering operations, to offset the loss of reconstruction accuracy due to increased levels of node reduction. Such aspects will be reported elsewhere. ## 5 Conclusion A graph autoencoder leveraging a combination of adaptive Top-K pooling layers and multiscale message passing (MMP) layers was introduced in this work. The pooling operation is interpreted as an adaptive sampling mechanism - stacking such layers results in a latent space (here referred to as a latent graph) that can readily be interpreted by visualizing the identified nodes through the construction of a masked field. The goal of the MMP layers is to then redistribute information contained in the masked field to the rest of the domain, resulting in a Figure 15: (Left) Normalized RMSE (see Eq. 16) as a function of Reynolds number for streamwise (black) and vertical (blue) velocity field reconstructions. Curves denote different node reduction factors obtained via increasing maximum graph level $L$ (refer to legend) for fixed $N_{H}=32$. (Right)** Analogous to left, but different curves denote different $N_{H}$ values (refer to legend) for fixed $L=2$. Figure 16: Snapshots from Trajectory 4 depicting reattachment cycle dynamics. First two columns display streamwise ($u_{x}$) and vertical ($u_{y}$) velocity fields, and next three columns display masked fields sourced from the same L=1, 2, and 3 models used in Fig. 15(a). Red regions in masked fields denote identified latent graph nodes. Time evolution proceeds from top-to-bottom, and successive snapshots shown here are separated by a time of $10^{-3}$ seconds. White circles in the $L=3$ case indicate additional small identified region in masked field. Figure 17: Flowfield reconstructions for the same L=1, 2, and 3 models utilized in Figs 15(a) and Fig. 16. Input flowfield corresponds to snapshot index 4 in Fig. 16. Colorbars are also consistent with Fig. 16. reconstruction of the original input graph. Alongside providing details and nuances of the new architecture, the primary goal here was to showcase the interpretability properties of generated latent spaces (or latent graphs) produced in the encoding stage in fluid dynamics applications. To this end, datasets for training and evaluation purposes were generated using large-eddy simulations in a backward-facing step (BFS) flow configuration with an OpenFOAM-based flow solver at high Reynolds numbers. Using this dataset as testbed, important aspects related to GNN architecture design, physical interpretation of latent spaces, and flowfield reconstruction quality were analyzed. From the angle of architecture design and to motivate the need for the MMP layer, the effect of coarsening operations present in the encoder and decoder message passing operations were analyzed. This analysis was conducted by comparing outputs of three model variations: (1) a GNN with no coarsening operations in the MMP layer (i.e. no multiscale message passing), (2) a GNN with coarsening operations present only in the decoder, and (3) a GNN with coarsening operations present in both encoder and decoder. In the end, the first model type resulted in a failure of the GNN to recover full flowfield information from the masked fields - incorporating coarsening operations via models 2 and 3 resulted in much more accurate and complete reconstructions, illuminating the advantages provided by multiscale message passing operations. The variation in architecture type directly impacted the physical nature of the masked fields: interestingly, the inclusion of coarsening operations in the encoding stage (model 3) was found to effectively trades physical space coherency and interpretability in the identified latent graph for improved reconstruction accuracy. Physical interpretation of the masked fields at different stages of the reattachment cycle revealed how the latent graphs directly identify time-evolving coherent structures - in most cases, even in light of changes to the Top-K hierarchy (i.e. modification to the maximum level $L$ for a fixed reduction factor), the nodes sampled in the reduction procedure were strongly correlated with recirculation zones of varying strengths in the flow. Flowfield compression potential was then assessed from two angles: by increasing node reduction factors through variations in maximum Top-K level, and by decreasing the hidden channel dimensionality of latent graph nodes. From both perspectives, it was found that the average reconstruction errors understandably increased across the board for greater compression factors. Although reconstruction qualities were imperfect in the higher-compression models, the advantage of this autoencoding approach is that it provides the user insight into how the greater levels of compression are achieved via visualization of the time-evolving masked fields. More specifically, it was found that the model achieving the highest amount of node reduction placed more focus on the inflow and outflow boundaries in the domain than other models. Additionally, when considering errors for individual flowfield variables at a given level of compression, reconstruction errors in the form of an RMSE measure were found to be much more insensitive to Reynolds number in the vertical velocity component when compared to streamwise velocity counterparts. Because the scope of this work was tied primarily to demonstrating the latent space interpretability provided by the GNN autoencoder, there are many avenues for future work. Concerning the architecture parameters, investigating the effects of increased message passing and coarsening operations within the MMP layer is warranted, as it may potentially alleviate issues related to the increase in reconstruction error found in higher-compression models. Additionally, geometry extrapolation capability was unexplored in this work - assessment of latent graph properties in different geometric configurations is a promising direction to solidify the general applicability of the method. From the reduced-order modeling perspective, the concept of masked field evolution reveals new strategies for interpretable data-based surrogate modeling of complex fluid flow phenomena (e.g. by establishing quantitative connections between flowfield dynamics and masked field evolution). All of these aspects are actively being pursued, and will be reported in future studies. ## 6 Acknowledgements This research used resources of the Argonne Leadership Computing Facility, which is a U.S. Department of Energy Office of Science User Facility operated under contract DE-AC02-06CH11357. RM acknowledges funding support from ASCR for DOE-FOA-2493 "Data-intensive scientific machine learning".
2307.14348	Solving the inverse potential problem in the parabolic equation by the deep neural networks method	In this work, we consider an inverse potential problem in the parabolic equation, where the unknown potential is a space-dependent function and the used measurement is the final time data. The unknown potential in this inverse problem is parameterized by deep neural networks (DNNs) for the reconstruction scheme. First, the uniqueness of the inverse problem is proved under some regularities assumption on the input sources. Then we propose a new loss function with regularization terms depending on the derivatives of the residuals for partial differential equations (PDEs) and the measurements. These extra terms effectively induce higher regularity in solutions so that the ill-posedness of the inverse problem can be handled. Moreover, we establish the corresponding generalization error estimates rigorously. Our proofs exploit the conditional stability of the classical linear inverse source problems, and the mollification on the noisy measurement data which is set to reduce the perturbation errors. Finally, the numerical algorithm and some numerical results are provided.	Mengmeng Zhang, Zhidong Zhang	2023-07-08T02:20:41Z	http://arxiv.org/abs/2307.14348v1	# Solving the inverse potential problem in the parabolic equation by the deep neural networks method ###### Abstract In this work, we consider an inverse potential problem in the parabolic equation, where the unknown potential is a space-dependent function and the used measurement is the final time data. The unknown potential in this inverse problem is parameterized by deep neural networks (DNNs) for the reconstruction scheme. First, the uniqueness of the inverse problem is proved under some regularities assumption on the input sources. Then we propose a new loss function with regularization terms depending on the derivatives of the residuals for partial differential equations (PDEs) and the measurements. These extra terms effectively induce higher regularity in solutions so that the ill-posedness of the inverse problem can be handled. Moreover, we establish the corresponding generalization error estimates rigorously. Our proofs exploit the conditional stability of the classical linear inverse source problems, and the mollification on the noisy measurement data which is set to reduce the perturbation errors. Finally, the numerical algorithm and some numerical results are provided. AMS subject classifications: 34K28, 35R30, 65N15, 62M45. Keywords: inverse potential problem, deep neural networks, uniqueness, generalization error estimates, numerical reconstruction. ## 1 Introduction. ### Mathematical model. The following parabolic system is considered in this work: \[\left\{\begin{aligned} (\partial_{t}-\Delta+q(x))u& =F(x,t),&(x,t)\in\Omega_{T},\\ u(x,t)&=b(x,t),&(x,t)\in\partial \Omega_{T},\\ u(x,0)&=u_{0}(x),& x\in\Omega.\end{aligned}\right. \tag{1.1}\] Here we write $\Omega_{T}=\Omega\times(0,T]$ and $\partial\Omega_{T}=\partial\Omega\times(0,T]$ for short, and $\Omega\subset\mathbb{R}^{d}$ is an open bounded domain in $\mathbb{R}^{d}$ with sufficiently smooth boundary. $F(x,t),\ u_{0}(x),\ b(x,t)$ are the source term, initial status, boundary condition respectively, causing the heat propagation in the medium. The potential function $q(x)\in L^{\infty}(\Omega)$, called the heat radiative coefficient of the material, is a crucial parameter for characterizing the heat conduction process. It describes the ability of the medium to propagate heat from internal sources or sinks. For known $(F(x,t),u_{0}(x),b(x,t),q(x))$ with suitable regularities, the forward problem (1.1) is well-posed in appropriate function space [11]. In this work, we consider the inverse problem of recovering the unknown $q(x)$, where the used measurement is the final time data \[u(x,T):=\varphi(x),\quad x\in\Omega. \tag{1.2}\] In practical applications of inverse problems, the contamination on inverse problems is unavoidable. So we will be given the noisy data $\varphi^{\delta}$ instead of the exact data $\varphi(x)$ in (1.2), which satisfies \[\\|\varphi^{\delta}-\varphi\\|_{L^{\infty}(\Omega)}\leq\delta. \tag{1.3}\] To handle the effect caused by the perturbations, people need to develop effective methods to improve the accuracy and robustness in applications. In this study, we choose the deep neural networks (DNNs) to solve the inverse problem (1.1)-(1.3). Comparing to traditional methods for solving inverse potential problem, this approach demonstrates the superiority in high-dimensional space and has the advantage of breaking the curse of dimensionality. There are rare works on studying the inverse potential problem for parabolic equations using deep neural networks, especially the rigorous analysis of its convergence estimate. In this work, the authors will consider the solution of the inverse potential problem (1.1)-(1.3) parameterized by DNNs for the reconstruction scheme. We propose a new loss function with regularization terms depending on the derivatives of the residuals for PDEs and measurements. The mollification method has been employed to improve the regularity of the noisy data. Also, the generalization error estimates are rigorously derived from the conditional stability of the linear inverse source problem and the mollification error estimate on noisy data. ### Literature. The reconstructions of $q(x)$ in (1.1) from some inversion input data have been studied extensively. For zero initial status, the uniqueness for $q(x)$ by (1.1)-(1.2) is established in [24], while the unique reconstruction using final measurement data is studied in [25]. In the case of non-zero initial status, the existence and uniqueness of the generalized solution $(u(x,t),q(x))\in W^{2,1}_{p}(\Omega_{T})\times L^{p}(\Omega)$ with the time-average temperature measurement are given in [18] for $(u_{0},\varphi)$ with some regularities. Choulli and Yamamoto [6] prove the generic well-posedness of the inverse problem in Holder spaces by final measurement data, and then the conditional stability result in a Hilbert space setting for sufficiently small $T$ is studied in [7]. Chen et al [3] consider the inverse potential problem from a partial measurements over $[T_{0},T_{1}]\times\Omega$ with $[T_{0},T_{1}]\subset[0,T]$, where the conditional stability estimates of the inverse problem in some Sobolev space and the reasonable convergence rates of the Tikhonov regularization are derived. Recently, Jin et al [16] uses the same observational data and shows a weighted $L^{2}$ stability in the standard $L^{2}$ norm under a positivity condition. They provide an error analysis of reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an $H^{1}$-seminorm penalty). Zhang et al [39] prove the uniqueness of the identification from final time data for (sub)diffusion equation and show the conditional stability in Hilbert spaces under some suitable conditions on the problem data. The convergence and error analysis of the reconstruction discrete scheme are rigorously analyzed. The investigations in the inverse non-smooth potential problem are given in [38], where the uniqueness for this nonlinear inverse problem is proved. Numerically, an iterative process called two-point gradient method is proposed by minimizing the data-fit term and the penalty term alternatively, with a convergence analysis in terms of the tangential condition. There also exists some works involving multiple coefficient identification. For example, Yamamoto and Zou [35] investigate the simultaneous reconstruction of the initial temperature and heat radiative coefficient in a heat conductive system, with stability of the inverse problem and the reconstruction scheme. Kaltenbacher and Rundell [17] consider the inverse problem of simultaneously recovering two unknowns, spatially dependent conductivity and the potential function from overposed data consisting of $u(x,T)$. The uniqueness result and the convergence of an iteration scheme are established. We also refer to [2, 4, 8, 14, 28, 34] and the references therein for the inverse potential problems in diffusion models from different types of observational data. Recently, deep learning methods for solving PDEs have been realized as an effective approach, especially in high dimensional PDEs. Such methods have the advantage of breaking the curse of dimensionality. The basic idea is to use neural networks (nonlinear functions) to approximate the unknown solutions of PDEs by learning the parameters. For the forward problems, there exists many numerical works with deep neural networks involving the depth Ritz method (DRM) [10], the depth Galerkin method (DGM) [33], the DeepXDE method [20], depth operator network method (DeepONet) [19], physical information neural networks (PINNs) [26], the weak adversary neural network (WAN) [1, 36] and so on. Theoretically, there are some rigorous analysis works investigating the convergence and error estimates for the solution of PDEs via neural networks, but the result are still far from complete. For example, the convergence rate of DRM with two layer networks and deep networks are studied in [9, 13, 21, 22]; the convergence of PINNs is given in [15, 23, 29, 30, 31]. For the inverse problems, the PINNs frameworks can be employed to solve the so-called data assimilation or unique continuation problems, and rigorous estimates on the generalization error of PINNs are established in [23]. Bao et al [36] develop the WAN to solve electrical impedance tomography (EIT) problem. In [37], the authors study a classical linear inverse source problem using the final time data under the frameworks of neural networks, where a rigorous generalization error estimate is proposed with a novel loss function including the Sobolev norm of some residuals. For more specific inverse problems applied in engineering and science, we refer to [5, 27, 32]. ### Outline. The rest of this article is organized as follows. In Section 2 we introduce the knowledge of neural networks and the setting of mollification. In Section 3, we introduce a conditional stability of the linear inverse source problem first. Then the uniqueness theorem (Theorem 3.1) of this inverse potential problem can be proved followed from the conditional stability. In Section 4, a novel loss function with specific regularization terms is introduced. Then we prove the generalization error estimates of data-driven solution of inverse problems, which is stated in Theorem 4.1. In Section 5, we propose the reconstruction algorithm and provide several experiments to show the validity of the proposed algorithm. ## 2 Preliminaries. ### Neural network architecture. First we introduce the basic knowledge of neural network briefly. Note that $u_{\theta}$ and $q_{\eta}$ are two separate networks with different variables $(x,t)$ and $x$. Thus, we use $\xi$ to denote collectively the network parameters for a parametric function $s_{\xi}(z)$ such that a general scheme can be applied for either $u_{\theta}(x,t)$ (with $z=(x,t),\ \xi=\theta$) or $q_{\eta}(x)$ (with $z=x,\ \xi=\eta$). For a positive integer $K\in\mathbb{N}$, a $K$-layer feed-forward neural network of $s_{\xi}(z)$ for $z\in\mathbb{R}^{d_{0}}$ is a function $s_{\xi}(z)$ defined by \[s_{\xi}(z):=W_{K}l_{K-1}\circ\cdots\circ l_{1}(z)+b_{K}, \tag{2.1}\] where the $k$-th layer $l_{k}:\mathbb{R}^{d_{k-1}}\rightarrow\mathbb{R}^{d_{k}}$ is given by $l_{k}(z)=\sigma\left(W_{k}z+b_{k}\right)$ with weights $W_{k}\in\mathbb{R}^{d_{k}\times d_{k-1}}$ and biases $b_{k}\in\mathbb{R}^{d_{k}}$ for $k=2,\cdots,K$. The activation function $\sigma(\cdot)$ includes sigmoid, tanh, ReLU (Rectified Linear Unit), softmax and so on [12]. These activation functions introduce non-linearities and enable the network to learn complex patterns and relationships in the data. The neural network (2.1) consists of an input layer with argument $z$, where $d_{0}=d$ is the problem dimension (also known as the size of input layer), an output layer which has the weights $W_{K}\in\mathbb{R}^{d_{K}\times d_{K-1}}$ and biases $b_{K}\in\mathbb{R}^{d_{K}}$, and $K-1$ hidden layers for some $K\in\mathbb{N}$. The network parameters of all layers are collectively denoted by \[\xi:=\left(W_{K},b_{K},W_{K-1},b_{K-1},\cdots,W_{1},b_{1}\right).\] In Figure 1, we give a simple architectures of fully connected neural networks, where $z=(x_{1},x_{2},\cdots,x_{d})$ is d-dimensional input variables, and the neural networks function is given as $s_{\xi}(z)=y_{NN}$. ### Mollification. In the practical applications of inverse problems, the noise of the measurements is unavoidable. The noisy data will make the residuals uncontrollable, which can be seen in the next section. Figure 1: The fully connected neural networks. Hence, we choose to mollify the measured data beforehand. The next is the introduction of mollification. Fix one function $\rho\in C^{2}(\mathbb{R})$ as \[\text{supp}\rho=(0,1),\ \rho(0)=\rho(1)=\rho^{\prime}(0)=\rho^{\prime}(1)=0,\] and \[\int_{0}^{\infty}\rho(t)t^{d-1}\ dt=\frac{1}{\pi_{d}},\] with $\pi_{d}$ is the surface area of unit sphere $B(0,1)$ in $R^{d}$. Set $\rho_{\epsilon}(x):=\epsilon^{-d}\rho(x/\epsilon)$, and define the mollifier as \[G_{\epsilon}\psi=\int_{\|x-y\|\leq\epsilon}\rho_{\epsilon}(\|x-y\|)\psi(y)\ dy. \tag{2.2}\] Then we have \[\int_{\mathbb{R}^{d}}\rho_{\epsilon}(\|x-y\|)\ dy=1.\] In the next lemma, we concern with the estimate of $\Delta\varphi-\Delta G_{\epsilon}(\varphi^{\delta})$. Lemma 2.1.: _Assume that the noisy data $\varphi^{\delta}\in L^{\infty}(\Omega)$ and the exact data $u(x,T):=\varphi(x)\in H^{2}(\Omega)$ satisfy_ \[\\|\varphi-\varphi^{\delta}\\|_{L^{\infty}(\Omega)}\leq\delta.\] _Also, the exact data imposes the high-order Lipschitz continuous condition. More precisely, we can find a positive constant $C_{\varphi}$ such that_ \[\|\varphi(x)-\varphi(y)\| \leq C_{\varphi}\|y-x\|,\] \[\|\Delta\varphi(x)-\Delta\varphi(y)\| \leq C_{\varphi}\|y-x\|,\] _for $x,y\in\overline{\Omega}$ uniformly. For the mollification operator (2.2), if we pick $\epsilon=O(\delta^{1/3})$, then we can achieve the following optimal error bound_ \[\\|\Delta\varphi-\Delta G_{\epsilon}(\varphi^{\delta})\\|_{L^{\infty}(\Omega)} \leq C\delta^{1/3}.\] Proof.: We split the subtraction $\Delta\varphi-\Delta G_{\epsilon}(\varphi^{\delta})$ as following: \[\Delta\varphi-\Delta G_{\epsilon}(\varphi^{\delta})=(\Delta\varphi-G_{ \epsilon}(\Delta\varphi))+(G_{\epsilon}(\Delta\varphi)-\Delta G_{\epsilon}( \varphi^{\delta}))=:I_{1}+I_{2}.\] For $I_{1}$, we have that \[\|I_{1}\|\leq\int_{\|x-y\|\leq\epsilon}\rho_{\epsilon}(\|x-y\|)\ \|\Delta\varphi(x)- \Delta\varphi(y)\|\ dy\leq C\epsilon.\] For $I_{2}$, Green's identities and the properties of the kernel function $\rho$ give that $\Delta G_{\epsilon}(\varphi)=G_{\epsilon}(\Delta\varphi)$. Hence, \[I_{2} =\Delta\Big{[}\int_{R^{d}}\rho_{\epsilon}(\|x-y\|)\ (\varphi(y)- \varphi^{\delta}(y))\ dy\Big{]}\] \[=\int_{R^{d}}\Delta\rho_{\epsilon}(\|x-y\|)\ (\varphi(y)-\varphi^{ \delta}(y))\ dy.\] From the straightforward calculation, we can deduce that \[\Delta\rho_{\epsilon}(\|x-y\|)=\epsilon^{-d-2}\rho^{\prime\prime}(\|x-y\|/\epsilon)+( d-1)\epsilon^{-d-1}\|x-y\|^{-1}\rho^{\prime}(\|x-y\|/\epsilon),\] which gives \[\|I_{2}\|\leq\delta\int_{\|x-y\|\leq\epsilon}\|\Delta\rho_{\epsilon}(\|x-y\|)\|\ dy \leq C\delta\epsilon^{-2}.\] So we have \[\|\Delta\varphi-\Delta G_{\epsilon}(\varphi^{\delta})\|\leq C\epsilon(1+\delta \epsilon^{-3}).\] By picking $\epsilon=O(\delta^{1/3})$, we can achieve the desired estimate and complete the proof. ## 3 Uniqueness. The uniqueness of this inverse potential problem is one of our main results. In this section, we will prove the uniqueness and the proof relies on the conditional stability of the inverse source problem of equation (1.1). The conditional stability will be stated in the next subsection. ### Conditional stability of the inverse source problem. Under the framework of DNNs, the total error of the reconstructed solution depends on the training error and the measurement error. This connection relies on the conditional stability of linear inverse source problem, i.e., the quantitative dependence of the unknown source on the measurement data. Sequentially, here we will introduce some known results for the linear inverse source problem. The mathematical statement of inverse source problem in parabolic equations with final time data is given below. For the parabolic equation \[\left\{\begin{aligned} (\partial_{t}-\Delta+\overline{q}(x))v (x,t)&=p(x)h(x,t),&(x,t)\in\Omega_{T},\\ v(x,t)&=0,&(x,t)\in\partial\Omega_{T}, \\ v(x,0)&=0,& x\in\Omega,\end{aligned}\right. \tag{3.1}\] we set $\overline{q}\geq 0$ and $\overline{q}\in L^{\infty}(\Omega)$, and $h(x,t)$ is given. Then the inverse source problem is to use the measurement \[\overline{\varphi}(x):=v[p](x,T) \tag{3.2}\] to recover the unknown $p(x)$ in the source term. Recalling the norm of the classical Sobolev space $W_{2}^{2,1}(\Omega_{T})$ as \[\\|u\\|_{W_{2}^{2,1}(\Omega_{T})}=\sqrt{\sum_{\|\alpha\|\leq 2}\\|D^{\alpha}u\\|_{L^{2 }(\Omega_{T})}^{2}+\\|u_{t}\\|_{L^{2}(\Omega_{T})}^{2}},\] the following classical result on the inverse source problem (3.1)-(3.2) can be found in [24]. Lemma 3.1.: _For equation (3.1), we assume that_ \[h\in L^{\infty}(\Omega_{T}),\ h_{t}\in L^{\infty}(\Omega_{T}),\ p(x)\in L^{2 }(\Omega),\ ph\in L^{2}(\Omega_{T}),\ ph_{t}\in L^{2}(\Omega_{T}),\] _and_ \[h(x,t)\geq 0,\ h_{t}(x,t)\geq 0\ \text{on}\ \Omega_{T},\ \|h(x,T)\|\geq\nu>0\ \text{on}\ \Omega.\] _Here $\nu$ is a fixed positive number. Then, for known $\overline{q}\in L^{\infty}(\Omega)$ and input data $\overline{\varphi}\in H^{2}(\Omega)$, there exists a unique solution $(v(x,t),p(x))\in W^{2,1}_{2}(\Omega_{T})\times L^{2}(\Omega)$ to (3.1)-(3.2), following the estimate_ \[\\|p\\|_{L^{2}(\Omega)}+\\|v\\|_{W^{2,1}_{2}(\Omega_{T})}\leq C\\|(-\Delta+ \overline{q})\overline{\varphi}\\|_{L^{2}(\Omega)}.\] _The constant $C$ depends on $\\|\overline{q}\\|_{L^{\infty}(\Omega)}$, $\nu$, $\Omega$ and $T$._ ### Uniqueness theorem. Now it is time to show the uniqueness theorem. First we introduce the admissible set for the unknown potential $q(x)$ as \[\mathcal{A}:=\{\psi\in L^{\infty}(\Omega):0\leq\psi(x)\leq M\ \text{a.e. on}\ \Omega\}\subset L^{2}(\Omega).\] The constant $M$ is the given upper bound of the admissible set. Next, recalling equation (1.1), we collect some restrictions on the controllable source $F(x,t)$, initial status $u_{0}(x)$ and boundary condition $b(x,t)$. Assumption 1.: _The assumptions on $F(x,t)$, $u_{0}(x)$ and $b(x,t)$ are given as follows._ * $u_{0}(x)\in H^{2}(\Omega)$_,_ $u_{0}(x)=b(x,0)$ _on_ $\partial\Omega$_,_ $\exists\nu>0$ _such that_ $u_{0}(x)\geq\nu>0$ _on_ $\Omega$_;_ * $b\in H^{2}(\partial\Omega)$_,_ $b\geq\nu>0$ _on_ $\partial\Omega$_,_ $b_{t}\geq 0$ _on_ $\partial\Omega$_;_ * $F\in L^{2}(\Omega_{T})$_,_ $F_{t}\in L^{2}(\Omega_{T})$_,_ $F\geq 0$ _on_ $\Omega_{T}$_,_ $F_{t}\geq 0$ _on_ $\Omega_{T}$_;_ * $\Delta u_{0}(x)-Mu_{0}(x)+F(x,0)\geq 0$ _on_ $\Omega$_._ Theorem 3.1.: _Under Assumption 1, the inverse problem (1.1)-(1.3) has at most one solution in $W^{2,1}_{2}(\Omega_{T})\times\mathcal{A}$._ Proof.: Assume that there are two distinct pairs $(u[q_{1}],q_{1})$ and $(u[q_{2}],q_{2})$ satisfying (1.1)-(1.3) with same data \[u[q_{1}](x,T)=u[q_{2}](x,T)=\varphi(x).\] Setting \[w(x,t):=u[q_{1}](x,t)-u[q_{2}](x,t),\quad\overline{q}(x):=q_{2}(x)-q_{1}(x),\] then $w(x,t)$ meets the system \[\left\{\begin{aligned} (\partial_{t}-\Delta+q_{1}(x))w(x,t)& =\overline{q}(x)u[q_{2}](x,t),&(x,t)\in\Omega_{T},\\ w(x,t)&=0,&(x,t)\in\partial\Omega_{T},\\ w(x,0)&=0,& x\in\Omega,\end{aligned}\right. \tag{3.3}\] with \[w(x,T)=0,\quad x\in\Omega. \tag{3.4}\] We need to prove that \[(w(x,t),\overline{q}(x))=(0,0)\ \text{in}\ W^{2,1}_{2}(\Omega_{T})\times L^{ \infty}(\Omega).\] Obviously $\overline{q}(x)\in L^{2}(\Omega)$. Also there holds $u[q_{2}]\in L^{\infty}(\Omega_{T})$ and $u_{t}[q_{2}]\in L^{\infty}(\Omega_{T})$ by [38, Lemma 2.1]. Then we have \[\overline{q}u[q_{2}]\in L^{2}(\Omega_{T}),\quad\overline{q}u_{t}[q_{2}]\in L^ {2}(\Omega_{T}).\] Under Assumption 1 and the maximum principle, we can see that $u[q_{2}]\geq 0$ on $\Omega_{T}$. For $u_{t}[q_{2}]$, with Assumption 1 and equation (1.1), it satisfies \[\left\{\begin{aligned} (\partial_{t}-\Delta+q_{2}(x))(u_{t}[q_{ 2}])&=F_{t}(x,t)\geq 0,&(x,t)\in\Omega_{T},\\ u_{t}[q_{2}](x,t)&=b_{t}(x,t)\geq 0,&(x,t)\in \partial\Omega_{T},\\ u_{t}[q_{2}](x,0)&=\Delta u_{0}(x)-q_{2}u_{0}(x)+F(x,0)\geq 0,& x\in\Omega.\end{aligned}\right.\] Then the maximum principle leads to $u_{t}[q_{2}]\geq 0$ straightforwardly. With the positivity of $u_{t}[q_{2}]$, we derive that \[u[q_{2}](x,t)=u_{0}(x)+\int_{0}^{t}\partial_{s}u[q_{2}](x,s)ds\geq u[q_{2}](x, 0)\geq\nu>0,\ (x,t)\in\Omega_{T},\] which yields $u[q_{2}](x,T)\geq\nu>0$. Now the conditions of Lemma 3.1 are satisfied, and we conclude $(w(x,t),\overline{q}(x))=(0,0)$ by applying Lemma 3.1 on (3.3)-(3.4). The proof is complete. ## 4 Generalization error estimates. In this section, we will discuss the error estimate of our approach for the inverse potential problem. Firstly, we introduce the corresponding residuals and define the loss function. ### Loss function and training errors. We propose a formulation of loss function for data-driven solutions of inverse problems, which can ensure the accuracy with the conditional stability of the given linear inverse source problem. To achieve it, we define suitable residuals that measure the errors of the governed system and the input data. Assume that the activation function is of $C^{2}$ regularity for the neural network $u_{\theta}$ defined by (2.1), which leads to $u_{\theta}\in H^{2}(\overline{\Omega}\times[0,T])$. For the network parameters \[\theta\in\Theta:=\{(W_{k},b_{k})\}_{k=1}^{K}:W_{k}\in\mathbb{R}^{d_{k}\times d _{k-1}},b_{k}\in\mathbb{R}^{d_{k}}\},\] the set of all possible trainable parameters $u_{\theta}(x,t)$ up to its second order weak derivatives are bounded in $\overline{\Omega}\times[0,T]$ for any specific $\theta$. Similarly, noticing that $q_{\eta}(x)$ is the parametric neural network to approximate the potential function $q(x)$, we assume the activation function for the neural network $q_{\eta}(x)$ is of $L^{\infty}$ regularity such that $q_{\eta}(x)\in L^{\infty}(\Omega)$. We define * Interior PDE residual \[\mathcal{R}_{int,\theta,\eta}(x,t):=\partial_{t}u_{\theta}(x,t)-\Delta u_{ \theta}(x,t)+q_{\eta}(x)u_{\theta}(x,t)-F(x,t),\quad(x,t)\in\Omega_{T}.\] * Spatial boundary residual \[\mathcal{R}_{sb,\theta}(x,t):=u_{\theta}(x,t)-b(x,t),\quad(x,t)\in\partial \Omega_{T}.\] * Initial status residual \[\mathcal{R}_{tb,\theta}(x):=u_{\theta}(x,0)-u_{0}(x),\quad x\in\Omega.\] * Data residual \[\mathcal{R}_{d,\theta}(x):=u_{\theta}(x,T)-G_{\epsilon}\varphi^{\delta}(x), \quad x\in\Omega.\] (4.1) Note that in the data residual (4.1), we use the mollified data $G_{\epsilon}\varphi^{\delta}(x)$ instead of the noisy data $\varphi^{\delta}(x)$. A loss function minimization scheme for data-driven inverse problems seeks to minimize these residuals comprehensively with some weights balancing different residuals. The loss function is defined as follows: \[\begin{split} J_{\lambda}(\theta,\eta)=&\\|q_{ \eta}\mathcal{R}_{d,\theta}\\|_{L^{2}(\Omega)}^{2}+\\|\Delta\mathcal{R}_{d, \theta}\\|_{L^{2}(\Omega)}^{2}+\lambda\\|\mathcal{R}_{int,\theta,\eta}\\|_{H^{1 }(0,T;L^{2}(\Omega))}^{2}\\ &+\\|\mathcal{R}_{tb,\theta}\\|_{L^{2}(\Omega)}^{2}+\\|q_{\eta} \mathcal{R}_{tb,\theta}\\|_{L^{2}(\Omega)}^{2}+\\|\Delta\mathcal{R}_{tb,\theta} \\|_{L^{2}(\Omega)}^{2}+\\|\mathcal{R}_{sb,\theta}\\|_{H^{2}(0,T;L^{2}(\partial \Omega))}^{2},\end{split} \tag{4.2}\] where $\lambda$ is a hyper-parameter to balance the residuals between the knowledge of PDE and the measurements. The proposed loss function (4.2) includes derivative penalties on the residuals. This is motivated by the conditional stability result for linear inverse source problem, which requires higher regularity on the measurement data $u(\cdot,T)$ (see Lemma 3.1). To improve the regularity of the noisy measurement data, we employ the mollification method by applying the mollification operator $G_{\varepsilon}$ on the noisy data $\varphi^{\delta}$. The design of the loss function for inverse problems distinguishes itself from that for forward problems such as physics-informed neural networks. The smoothness requirements not only ensure the existence of forward problem solutions, but also ensure the well-posedness of the inverse problem within the optimization framework. Remark 1.: _The following standard loss function_ \[J^{s}(\theta,\eta)=\\|\mathcal{R}_{d,\theta}\\|_{L^{2}(\Omega)}^{2}+\lambda\\| \mathcal{R}_{int,\theta,\eta}\\|_{L^{2}(\Omega_{T})}^{2}+\\|\mathcal{R}_{tb, \theta}\\|_{L^{2}(\Omega)}^{2}+\\|\mathcal{R}_{sb,\theta}\\|_{L^{2}(\partial \Omega_{T})}^{2} \tag{4.3}\] _has often been used in the literature. For example, the DGM workflow adopts this form of loss function and minimizes it by least squares scheme [33]._ To determine $(\theta,\eta)$ from the discrete training set, accurate numerical evaluation of the integrals in (4.2) is essential. We introduce the following training sets that facilitate efficient computation of the integrals, leading to better performance: \[\mathcal{S}_{d} :=\left\{(x_{n},T):x_{n}\in\Omega,\quad n=1,2,\cdots,N_{d}\right\},\] \[\mathcal{S}_{int} :=\left\{(\widetilde{x}_{n},\widetilde{t}_{n}):(\widetilde{x}_{n},\widetilde{t}_{n})\in\Omega_{T},\quad n=1,2,\cdots,N_{int}\right\},\] \[\mathcal{S}_{tb} :=\left\{(\widetilde{x}_{n},0):\widetilde{x}_{n}\in\Omega,\quad n =1,2,\cdots,N_{tb}\right\},\] \[\mathcal{S}_{sb} :=\left\{(\widehat{x}_{n},\widehat{t}_{n}):(\widehat{x}_{n}, \widehat{t}_{n})\in\partial\Omega_{T},\quad n=1,2,\cdots,N_{sb}\right\}.\] Applying these sets and the numerical quadrature rules [23], we get the following empirical loss function \[J_{\lambda}^{N}(\theta,\eta) =\sum_{n=1}^{N_{d}}\omega_{n}^{d,0}\|q_{\eta}(x_{n})\mathcal{R}_{d, \theta}(x_{n})\|^{2}+\sum_{n=1}^{N_{d}}\omega_{n}^{d,1}\|\Delta\mathcal{R}_{d, \theta}(x_{n})\|^{2}\] \[\quad+\lambda\sum_{n=1}^{N_{int}}\omega_{n}^{int,0}\|\mathcal{R}_{ int,\theta,\eta}(\widetilde{x}_{n},\widetilde{t}_{n})\|^{2}+\lambda\sum_{n=1}^{N_{ int}}\omega_{n}^{int,1}\|\partial_{t}\mathcal{R}_{int,\theta,\eta}(\widetilde{x}_{n}, \widetilde{t}_{n})\|^{2}\] \[\quad+\sum_{n=1}^{N_{tb}}\omega_{n}^{tb,0}\|\mathcal{R}_{tb, \theta}(\overline{x}_{n})\|^{2}+\sum_{n=1}^{N_{tb}}\omega_{n}^{tb,1}\|q_{\eta}( \overline{x}_{n})\mathcal{R}_{tb,\theta}(\overline{x}_{n})\|^{2}+\sum_{n=1}^{ N_{tb}}\omega_{n}^{tb,2}\|\Delta\mathcal{R}_{tb,\theta}(\overline{x}_{n})\|^{2}\] \[\quad+\sum_{n=1}^{N_{sb}}\omega_{n}^{sb,0}\|\mathcal{R}_{sb, \theta}(\widehat{x}_{n},\widehat{t}_{n})\|^{2}+\sum_{n=1}^{N_{sb}}\omega_{n}^ {sb,1}\|\partial_{t}\mathcal{R}_{sb,\theta}(\widehat{x}_{n},\widehat{t}_{n})\|^ {2}\sum_{n=1}^{N_{sb}}\omega_{n}^{sb,2}\|\partial_{t}^{2}\mathcal{R}_{sb, \theta}(\widehat{x}_{n},\widehat{t}_{n})\|^{2}, \tag{4.4}\] where the coefficients \[\omega_{n}^{d,k},\ \omega_{n}^{int,k},\ \omega_{n}^{tb,j},\ \omega_{n}^{sb,j}, \ k=0,1,\ j=0,1,2\] are the quadrature weights. It is easy to see that the error for the loss function is \[\|J_{\lambda}(\theta,\eta)-J_{\lambda}^{N}(\theta,\eta)\|\leq C\min\{N_{d}^{- \alpha_{d,k}},N_{int}^{-\alpha_{int,k}},N_{tb}^{-\alpha_{tb,j}},N_{sb}^{- \alpha_{sb,j}}:k=0,1,\ j=0,1,2\}, \tag{4.5}\] where $C$ depends on the continuous norm $\\|\cdot\\|_{C(\Omega)}$ of the integrals, the rate $\alpha^{d,k}$, $\alpha^{int,k}$, $\alpha^{tb,j}$, $\alpha^{sb,j}$ ($k=0,1,\ j=0,1,2$) are positive and depend on the regularity of the underlying integrand i.e, on the space $C(\Omega)$. Therefore, the underlying solutions and neural networks should be sufficiently regular such that the residuals can be approximated to a high accuracy by the quadrature rule. Now, we define the generalization errors as \[\begin{cases}\mathcal{E}_{G,q}:=\\|q-q^{}\\|_{L^{2}(\Omega)}\,,\\ \mathcal{E}_{G,u}:=\\|u-u^{}\\|_{C([0,T];L^{2}(\Omega))}\,,\end{cases} \tag{4.6}\] where $u^{}:=u_{\theta^{}},\ q^{}:=q_{\eta^{}}$ with $(\theta^{},\eta^{})$ is the minimizer of the functional (4.4). Also, we estimate generalization errors in terms of the following training errors: * The measurement data training errors: $\mathcal{E}_{T,d}:=\mathcal{E}_{T,d,0}+\mathcal{E}_{T,d,1}$, where \[\begin{cases}\mathcal{E}_{T,d,0}:=\left(\sum_{n=1}^{N_{d}}\omega_{j}^{d,0}\,\|q _{\eta}\left(x_{n}\right)\mathcal{R}_{d,\theta^{}}\left(x_{n}\right)\|^{2} \right)^{\frac{1}{2}},\\ \mathcal{E}_{T,d,1}:=\left(\sum_{n=1}^{N_{d}}\omega_{j}^{d,1}\,\|\Delta\mathcal{ R}_{d,\theta^{}}\left(x_{n}\right)\|^{2}\right)^{\frac{1}{2}}.\end{cases}\] (4.7) * The interior PDE training errors: $\mathcal{E}_{T,int}:=\mathcal{E}_{T,int,0}+\mathcal{E}_{T,int,1}$, where \[\begin{cases}\mathcal{E}_{T,int,0}:=\left(\sum_{n=1}^{N_{int}}\omega_{n}^{int,0} \|\mathcal{R}_{int,\theta^{},\eta^{}}(\widetilde{x}_{n},\widetilde{t}_{n})\|^{ 2}\right)^{\frac{1}{2}},\\ \mathcal{E}_{T,int,1}:=\left(\sum_{n=1}^{N_{int}}\omega_{n}^{int,1}\|\partial_{t }\mathcal{R}_{int,\theta^{},\eta^{}}(\widetilde{x}_{n},\widetilde{t}_{n})\|^{ 2}\right)^{\frac{1}{2}}.\end{cases}\] (4.8) * The initial condition training errors: $\mathcal{E}_{T,tb}:=\mathcal{E}_{T,tb,0}+\mathcal{E}_{T,tb,1}+\mathcal{E}_{T, tb,2}$, where \[\begin{cases}\mathcal{E}_{T,tb,0}:=\left(\sum_{n=1}^{N_{tb}}\omega_{n}^{tb,0}\| \mathcal{R}_{tb,\theta^{}}(\overline{x}_{n})\|^{2}\right)^{\frac{1}{2}},\\ \mathcal{E}_{T,tb,1}:=\left(\sum_{n=1}^{N_{tb}}\omega_{n}^{tb,1}\|q_{n}( \overline{x}_{n})\mathcal{R}_{tb,\theta^{}}(\overline{x}_{n})\|^{2}\right)^{ \frac{1}{2}},\\ \mathcal{E}_{T,tb,2}:=\left(\sum_{n=1}^{N_{tb}}\omega_{n}^{tb,2}\|\Delta \mathcal{R}_{tb,\theta^{}}(\overline{x}_{n})\|^{2}\right)^{\frac{1}{2}}.\end{cases}\] (4.9) The spatial boundary condition training errors: $\mathcal{E}_{T,sb}:=\mathcal{E}_{T,sb,0}+\mathcal{E}_{T,sb,1}+\mathcal{E}_{T,sb,2}$, where \[\begin{cases}\mathcal{E}_{T,sb,0}:=\left(\sum_{n=1}^{N_{sb}}\omega_{n}^{sb,0} \|\mathcal{R}_{sb,\theta^{}}(\widehat{x}_{n},\widehat{t}_{n})\|^{2}\right)^{ \frac{1}{2}},\\ \mathcal{E}_{T,sb,1}:=\left(\sum_{n=1}^{N_{sb}}\omega_{n}^{sb,1}\|\partial_{t} \mathcal{R}_{sb,\theta^{}}(\widehat{x}_{n},\widehat{t}_{n})\|^{2}\right)^{ \frac{1}{2}},\\ \mathcal{E}_{T,sb,2}:=\left(\sum_{n=1}^{N_{sb}}\omega_{n}^{sb,2}\|\partial_{t} ^{2}\mathcal{R}_{sb,\theta^{}}(\widehat{x}_{n},\widehat{t}_{n})\|^{2}\right)^ {\frac{1}{2}}.\end{cases}\] (4.10) ### Proofs of the estimates. Now we can state the theorem about the generalization error estimates. Theorem 4.1.: _Recall the errors defined in (4.6)-(4.10). Under Assumption 1, there exists a unique solution to the inverse problem (1.1)-(1.3). Moreover, for the approximate solution $(u^{},q^{})$ of the inverse problem with $(\theta^{},\eta^{})$ being a global minimizer of the loss function $J_{\lambda}^{N}(\theta,\eta)$, we have the following generalization error estimates_ \[\begin{split}\mathcal{E}_{G,q}&\leq C\Big{(}\mathcal{ E}_{T,d}+\mathcal{E}_{T,int}+\mathcal{E}_{T,sb,1}+\mathcal{E}_{T,sb,2}+ \mathcal{E}_{T,tb,1}+\mathcal{E}_{T,tb,2}+C_{q}^{\frac{1}{2}}N^{\frac{-\alpha }{2}}+O(\delta^{1/3})\Big{)},\\ \mathcal{E}_{G,u}&\leq C\Big{(}\mathcal{E}_{T,d}+ \mathcal{E}_{T,int}+\mathcal{E}_{T,sb}+\mathcal{E}_{T,tb}+C_{q}^{\frac{1}{2}} N^{\frac{-\alpha}{2}}+O(\delta^{1/3})\Big{)},\end{split} \tag{4.11}\] _where_ \[N =\min\left\{N_{d},N_{int},N_{sb},N_{tb}\right\},\] \[\alpha =\min\left\{\alpha_{int,0},\alpha_{int,1},\alpha_{sb,0},\alpha_{ sb,1},\alpha_{sb,2},\alpha_{tb,0},\alpha_{tb,1},\alpha_{d}\right\},\] _in (4.5), and_ \[C_{q}=\max\left\{C_{q,0},C_{q,1},C_{qs,0},C_{qs,1},C_{qs,2},C_{qt,0},C_{qt,1},C _{qd}\right\},\] _with_ \[C_{qd}=C_{qd}(\\|\mathcal{L}^{}\mathcal{R}_{d,\theta^{}}\\|_{C( \Omega)}), C_{q,0}=C_{q,0}(\\|\mathcal{R}_{int,\theta^{},\eta^{}}\\|_{C(\Omega_{ T})}),\] \[C_{q,1}=C_{q,1}(\\|\partial_{t}\mathcal{R}_{int,\theta^{},\eta^ {}}\\|_{C(\Omega_{T})}), C_{qs,0}=C_{qs,0}(\\|\mathcal{R}_{sb,\theta^{}}\\|_{C( \partial\Omega_{T})}),\] \[C_{qs,1}=C_{qs,1}(\\|\partial_{t}\mathcal{R}_{sb,\theta^{}}\\|_{ C(\partial\Omega_{T})}), C_{qs,2}=C_{qs,2}(\\|\partial_{t}^{2}\mathcal{R}_{sb,\theta^{}}\\|_{C( \partial\Omega_{T})}),\] \[C_{qt,0}=C_{qt,0}(\\|\mathcal{R}_{tb,\theta^{}}\\|_{C(\Omega)}), C_{qt,1}=C_{qt,1}(\\|\mathcal{L}^{}\mathcal{R}_{tb,\theta^{}}\\|_{C( \Omega)}).\] _The constant $C$ depends on $\\|q^{}\\|_{L^{\infty}(\Omega)}$, $\Omega$ and $T$._ Proof.: First, we introduce $\hat{u}:=u^{}-u$ and realize that \[\left\{\begin{aligned} (\partial_{t}-\Delta+q^{}(x))\hat{u}(x,t)& =\mathcal{R}_{int,\theta^{},\eta^{}}(x,t)+(q-q^{})u[q](x,t),& (x,t)\in\Omega_{T},\\ \hat{u}(x,t)&=\mathcal{R}_{sb,\theta^{}}(x,t),& (x,t)\in\partial\Omega_{T},\\ \hat{u}(x,0)&=\mathcal{R}_{tb,\theta^{}}(x),& x\in\Omega,\end{aligned}\right.\] with the final condition \[\hat{u}(x,T)=u[q^{}](x,T)-u[q](x,T)=\mathcal{R}_{d,\theta^{}}(x)-(\varphi-G_ {\epsilon}\varphi^{\delta}).\] We make the decomposition $\hat{u}:=\hat{u}_{1}+\hat{u}_{2}$, where $\hat{u}_{1},\ \hat{u}_{2}$ satisfy \[\left\{\begin{aligned} (\partial_{t}-\Delta+q^{}(x))\hat{u}_{1}(x,t )&=(q^{}-q)(x)u[q](x,t),&(x,t)\in\Omega_{T},\\ \hat{u}_{1}(x,t)&=0,&(x,t)\in \partial\Omega_{T},\\ \hat{u}_{1}(x,0)&=0,& x\in\Omega,\end{aligned}\right. \tag{4.12}\] with \[\hat{u}_{1}(x,T)=\mathcal{R}_{d,\theta^{}}(x)-(\varphi(x)-G_{\epsilon} \varphi^{\delta}(x))-\hat{u}_{2}(x,T), \tag{4.13}\] and \[\left\{\begin{aligned} (\partial_{t}-\Delta+q^{}(x))\hat{u}_{2}( x,t)&=\mathcal{R}_{int,\theta^{},\eta^{}}(x,t),&(x,t)\in \Omega_{T},\\ \hat{u}_{2}(x,t)&=\mathcal{R}_{sb,\theta^{}}(x,t),& (x,t)\in\partial\Omega_{T},\\ \hat{u}_{2}(x,0)&=\mathcal{R}_{tb,\theta^{}}(x),& x\in\Omega,\end{aligned}\right.\] respectively. Define the operator $\mathcal{L}^{}$ as $\mathcal{L}^{}\psi=(-\Delta+q^{})\psi.$ With Assumption 1, we can apply Lemma 3.1 to (4.12)-(4.13) and deduce that \[\\|q^{}-q\\|_{L^{2}(\Omega)} \tag{4.14}\] \[\leq C\\|\mathcal{L}^{}\hat{u}_{1}(\cdot,T)\\|_{L^{2}(\Omega)}\] \[=C\\|\mathcal{L}^{}\mathcal{R}_{d,\theta^{}}-\mathcal{L}^{}( \varphi-G_{\epsilon}\varphi^{\delta})-\mathcal{L}^{}\hat{u}_{2}(\cdot,T)\\|_{ L^{2}(\Omega)}\] \[=C\\|\mathcal{L}^{}\mathcal{R}_{d,\theta^{}}+\Delta\varphi- \Delta G_{\epsilon}\varphi^{\delta}-q^{}(\varphi-G_{\epsilon}\varphi^{ \delta})-\mathcal{L}^{}\hat{u}_{2}(\cdot,T)\\|_{L^{2}(\Omega)}\] \[\leq C\left(\\|\mathcal{L}^{}\mathcal{R}_{d,\theta^{}}\\|_{L^{2}( \Omega)}+\\|\Delta\varphi-\Delta G_{\epsilon}\varphi^{\delta}\\|_{L^{2}(\Omega) }+\\|q^{}(\varphi-G_{\epsilon}\varphi^{\delta})\\|_{L^{2}(\Omega)}+\\|\mathcal{L }^{}\hat{u}_{2}(\cdot,T)\\|_{L^{2}(\Omega)}\right),\] with $C=C(\\|q^{}\\|_{L^{\infty}(\Omega)},\Omega,T)$. Using Lemma 2.1, we get \[\\|\Delta\varphi-\Delta G_{\epsilon}\varphi^{\delta}\\|_{L^{2}(\Omega)}\leq C \epsilon(1+\delta\epsilon^{-3}). \tag{4.15}\] Also, we have \[\|(\varphi-G_{\epsilon}\varphi^{\delta})\| \leq\int_{\|x-y\|\leq\epsilon}\rho_{\epsilon}(\|x-y\|)\ \|\varphi(x)-\varphi^{\delta}(y)\|\ dy\] \[\leq\int_{\|x-y\|\leq\epsilon}\rho_{\epsilon}(\|x-y\|)\ \|\varphi(x)-\varphi(y)\|dy+\int_{\|x-y\|\leq\epsilon}\rho_{\epsilon}(\|x-y\|)\ \| \varphi(y)-\varphi^{\delta}(y)\|\ dy\] \[\leq C\epsilon+\delta.\] Thus, there holds \[\\|q^{}(\varphi-G_{\epsilon}\varphi^{\delta})\\|_{L^{2}(\Omega)}\leq C\epsilon+\delta. \tag{4.16}\] By straightforward computations, we have \[\\|\mathcal{L}^{}\hat{u}_{2}(\cdot,T)\\|_{L^{2}(\Omega)} \leq\\|\partial_{t}\hat{u}_{2}(\cdot,T)\\|_{L^{2}(\Omega)}+\\| \mathcal{R}_{int,\theta^{},\eta^{}}(\cdot,T)\\|_{L^{2}(\Omega)}\] \[\leq\\|\partial_{t}\hat{u}_{2}\\|_{L^{\infty}(0,T;L^{2}(\Omega))}+ \\|\mathcal{R}_{int,\theta^{},\eta^{}}(\cdot,T)\\|_{L^{2}(\Omega)}. \tag{4.17}\] Setting $w(x,t):=\partial_{t}\hat{u}_{2}(x,t)$, it satisfies \[\left\{\begin{aligned} (\partial_{t}-\Delta+q^{})w(x,t)& =\partial_{t}\mathcal{R}_{int,\theta^{},\eta^{}}(x,t),& (x,t)\in\Omega_{T},\\ w(x,t)&=\partial_{t}\mathcal{R}_{sb,\theta^{}}(x,t),& (x,t)\in\partial\Omega_{T},\\ w(x,0)&=\mathcal{R}_{int,\theta^{},\eta^{}}(x,0)- \mathcal{L}^{}\mathcal{R}_{tb,\theta^{}}(x),& x\in\Omega.\end{aligned}\right. \tag{4.18}\] Using the regularity theory for the direct problem (4.18), we obtain \[\\|\partial_{t}\hat{u}_{2}\\|_{L^{\infty}(0,T;L^{2}(\Omega))} =\\|w\\|_{L^{\infty}(0,T;L^{2}(\Omega))} \tag{4.19}\] \[\leq C\big{(}\\|\partial_{t}\mathcal{R}_{int,\theta^{},\eta^{}} \\|_{L^{2}(\Omega_{T})}+\\|\partial_{t}\mathcal{R}_{sb,\theta^{}}\\|_{H^{1}(0,T ;L^{2}(\partial\Omega))}\] \[\qquad+\\|\mathcal{R}_{int,\theta^{},\eta^{}}(\cdot,0)\\|_{L^{2}( \Omega)}+\\|\mathcal{L}^{}\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}\big{)}.\] Combining (4.14)-(4.19) together and using the Sobolev embedding theorem, we get \[\mathcal{E}_{G,q} =\\|q^{}-q\\|_{L^{2}(\Omega)}\] \[\leq C\big{(}\\|\mathcal{R}_{int,\theta^{},\eta^{}}\\|_{H^{1}(0,T ;L^{2}(\Omega))}+\\|\mathcal{L}^{}\mathcal{R}_{d,\theta^{}}\\|_{L^{2}(\Omega) }+\\|\mathcal{L}^{}\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}\] \[\qquad+\\|\partial_{t}\mathcal{R}_{sb,\theta^{}}\\|_{H^{1}(0,T;L^ {2}(\partial\Omega))}+\epsilon(1+\delta\epsilon^{-3})+\epsilon+\delta\big{)}\] \[\leq C\big{(}\\|\mathcal{R}_{int,\theta^{},\eta^{}}\\|_{H^{1}(0,T ;L^{2}(\Omega))}+\\|q^{}\mathcal{R}_{d,\theta^{}}\\|_{L^{2}(\Omega)}+\\| \Delta\mathcal{R}_{d,\theta^{}}\\|_{L^{2}(\Omega)}+\\|q^{}\mathcal{R}_{tb, \theta^{}}\\|_{L^{2}(\Omega)}\] \[\qquad+\\|\Delta\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}+\\| \partial_{t}\mathcal{R}_{sb,\theta^{}}\\|_{H^{1}(0,T;L^{2}(\partial\Omega))}+ \epsilon(1+\delta\epsilon^{-3})+\epsilon+\delta\big{)},\] with $C=C(\\|q^{}\\|_{L^{\infty}(\Omega)},\Omega,T)>0$. Picking $\epsilon=O(\delta^{1/3})$, we achieve the estimate \[\mathcal{E}_{G,q} =\\|q^{}-q\\|_{L^{2}(\Omega)} \tag{4.20}\] \[\leq C\big{(}\\|\mathcal{R}_{int,\theta^{},\eta^{}}\\|_{H^{1}(0,T ;L^{2}(\Omega))}+\\|q^{}\mathcal{R}_{d,\theta^{}}\\|_{L^{2}(\Omega)}+\\| \Delta\mathcal{R}_{d,\theta^{}}\\|_{L^{2}(\Omega)}\] \[\qquad+\\|q^{}\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}+\\| \Delta\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}+\\|\partial_{t}\mathcal{R} _{sb,\theta^{}}\\|_{H^{1}(0,T;L^{2}(\partial\Omega))}+O(\delta^{1/3})\big{)}.\] Finally, we will evaluate the generalization error for the unknown $u(x,t)$ employing the obtained generalization error (4.20) of the potential function. From the classical regularity theory for PDE, if $F(x,t)$ is sufficiently smooth, then it holds that $u\in L^{2}(0,T;L^{\infty}(\Omega))$. Consequently, \[\mathcal{E}_{G,u} =\\|\hat{u}\\|_{C([0,T];L^{2}(\Omega))}\] \[\leq C\big{(}\\|\mathcal{R}_{int,\theta^{},\eta^{}}+(q^{}-q)u[q] \\|_{L^{2}(\Omega_{T})}+\\|\mathcal{R}_{sb,\theta^{}}\\|_{H^{1}(0,T;L^{2}( \partial\Omega))}+\\|\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}\big{)}\] \[\leq C\big{(}\\|\mathcal{R}_{int,\theta^{},\eta^{}}\\|_{L^{2}( \Omega_{T})}+\\|q^{}-q\\|_{L^{2}(\Omega)}\ \\|u\\|_{L^{2}(0,T;L^{\infty}(\Omega))}+\\| \mathcal{R}_{sb,\theta^{}}\\|_{H^{1}(0,T;L^{2}(\partial\Omega))}\] \[\qquad+\\|\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}\big{)}\] \[\leq C\big{(}\\|\mathcal{R}_{int,\theta^{},\eta^{}}\\|_{H^{1}(0,T ;L^{2}(\Omega))}+\\|q^{}\mathcal{R}_{d,\theta^{}}\\|_{L^{2}(\Omega)}+\\| \Delta\mathcal{R}_{d,\theta^{}}\\|_{L^{2}(\Omega)}+\\|\mathcal{R}_{tb,\theta^{ }}\\|_{L^{2}(\Omega)}\] \[\qquad+\\|q^{}\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}+\\| \Delta\mathcal{R}_{tb,\theta^{}}\\|_{L^{2}(\Omega)}+\\|\mathcal{R}_{sb,\theta^{}} \\|_{H^{2}(0,T;L^{2}(\partial\Omega))}+O(\delta^{1/3})\big{)}.\] The proof is complete. The estimate (4.11) demonstrates that well-trained neural networks will produce small generalization errors for the inverse problem. Specifically, when all components of the training errors, including the interior PDE errors, measurement errors as well as the initial and boundary value ones, are sufficiently small, and the training sampling is large enough, the generalization errors for inverse problem using neural networks can be limited well. This differs from classical stability results that rely solely on the knowledge of data. In this work, the generalization error estimates reflect stability due to both the model itself and the reconstruction algorithm. From Theorem 4.1, we see that we can limit the errors of both the inverse and forward problems by controlling the residuals and the mollified parameter. This provides important insights into the mathematical properties of our approach and plays an important role on the construction of algorithms. ## 5 Numerical reconstructions. ### Reconstruction algorithm. The neural networks $u_{\theta}(x,t)$ and $q_{\eta}(x)$ depend on the parameters $\theta$ and $\eta$ describing the networks information for specific activation functions. Within the standard paradigm of deep learning, one trains the networks by finding the optimal parameters $(\theta^{},\eta^{})$ such that the loss function (4.4) is minimized. Our target is the unknown solution of the inverse problem (1.1)-(1.3) and we wish to find the trainable parameters $(\theta^{},\eta^{})$ such that the corresponding neural networks $(u_{\theta^{}},q_{\eta^{}})$ approximate $(u,q)$ well. More precisely, to solve (1.1)-(1.3) we first parameterize $u$ and $q$ by deep neural networks $u_{\theta}$ and $q_{\eta}$ with network parameters $(\theta,\eta)$ respectively. Then, we design an appropriate loss function, which is minimized to determine the parameters $(\theta,\eta)$. Finally, a gradient-based method is applied to alternately update the network parameters so that $(u_{\theta},q_{\eta})$ gradually approximates $(u,q)$ for our inverse problem. We provide a schematic of the neural networks in Figure 2. The left part visualizes two unknowns as two standard neural networks parameterized by $\theta$ and $\eta$, respectively. The right part applies the given physical laws to the networks. B.C., I.C. and D are the boundary condition, initial status and the measurement data obtained from random sample points in training sets $\mathcal{S}_{sb}$, $\mathcal{S}_{tb}$ and $\mathcal{S}_{d}$ respectively. The training points in $\mathcal{S}_{int}$ are randomly sampled as the PDE residuals points in interior spatio-temporal domain. The loss function with some Sobolev norm is computed on the sample points, which can be done efficiently through automatic differentiation (AD) in case of derivative information. Minimizing the loss with respect to the parameters $(\theta,\eta)$ alternately produces $u_{\theta^{}}$ and $q_{\eta^{}}$, which serves as the approximation to the solution of the inverse problem. With the support of Theorem 4.1, we can construct the proposed Algorithm 1 for solving the inverse problem (1.1)-(1.3). The above minimization problem is to search a minimizer of a possibly non-convex function $J_{\lambda}^{N}(\theta,\eta)$ over $\Theta\subset\mathbb{R}^{\mathcal{M}}$ for possibly very large $\mathcal{M}$. The hyper-parameters $(\tau_{\eta},\tau_{\theta})$ are learning rates and $(\lambda_{\eta},\lambda_{\theta})$ are balance hyper-parameters between PDE and measurement data residuals. The robust analysis for hyper-parameters $\lambda$ and the architectures of neural networks are studied in the next subsection. The optimizer in Algorithm 1 is Adam (Adaptive Moment Estimation), which is an optimization algorithm commonly used in deep learning for training neural networks. The key idea of Adam is to adaptively adjust the learning rate for each parameter based on estimates of both the first-order moment (the mean) and the second-order moment (the uncentered variance) of the gradients. This adaptation helps Adam to perform well in different types of optimization problems. The algorithm maintains an exponentially moving average of gradients ($m_{t}$) and squared gradients ($V_{t}$) for each parameter. At each iteration, Adam updates the parameters using a combination of these moving average estimates. It incorporates bias correction to account for the fact that the estimates are biased towards zero at the beginning of training. We set $g_{t}$ be gradients w.r.t. stochastic objective at timestep $t$, $\beta_{1},\beta_{2}\in[0,1)$ be the exponential decay rates for the moment estimates and $\tau$ be the initial learning rate. Good default settings for the tested machine learning problems are $\tau=0.001$, $\beta_{1}=0.9$, $\beta_{2}=0.999$ and fuzzy factor $\epsilon=10^{-8}$. The updates are calculated as follows: 1. Initialize the first moment vector $m$ and the second moment vector $V$ with zeros for each parameter: \[m_{0}=V_{0}=0.\] 2. Update the first moment estimate $m$ using a weighted average of the current gradient $g_{t}$ and the previous first moment estimate $m_{t-1}$: \[m_{t}=\beta_{1}m_{t-1}+(1-\beta_{1})g_{t}.\] 3. Update the second moment estimate $V$ using a weighted average of the squared gradients and the previous second moment estimate $V_{t-1}$: \[V_{t}=\beta_{2}V_{t-1}+(1-\beta_{2})g_{t}^{2}.\] Figure 2: The schematic of the deep neural networks for solving the inverse potential problem. 4. Calculate the bias-corrected first and second moment estimate to correct for their initialization bias: \[\hat{m}_{t}=\frac{m_{t}}{1-(\beta_{1})^{t}},\quad\hat{V}_{t}=\frac{V_{t}}{1-( \beta_{2})^{t}}.\] 5. Update the parameters $\xi$ by moving in the direction of the first moment estimate, where the learning rate is $\tau$ divided by the square root of the second moment estimate: \[\xi_{t}=\xi_{t-1}-\tau\frac{\hat{m}_{t}}{\sqrt{\hat{V}_{t}}+\epsilon}.\] The hyper-parameters in Adam include the learning rate and the decay rates for the moving averages. These hyper-parameters need to be tuned based on the specific problem and dataset to achieve optimal performance. Adam has several advantages that make it popular in deep learning: 1. Adaptive learning rate: Adam automatically adapts the learning rate for each parameter based on the estimated first and second moments. This adaptive behavior helps in effectively navigating the optimization landscape and can lead to faster convergence. 2. Efficiency: Adam uses the moving averages to maintain a history of gradients, which eliminates the need to store and compute gradients for each iteration separately. This makes Adam memory-efficient and allows for efficient parallelization during training. 3. Robustness: Adam performs well across a wide range of optimization problems and is less sensitive to hyper-parameter tuning compared to some other optimizers. It can handle sparse gradients and noisy data effectively. The proposed algorithm, which utilizes (4.2) as the loss function, exhibits superior performance in recovering smooth solutions due to the high regularity of the PDEs residual term. This regularity term promotes smoother solutions and is an important factor in achieving higher accuracy. Furthermore, the use of automatic differentiation (AD) implementations enables the efficient calculation of the necessary derivatives. This feature is a significant advantage of our approach as it allows for the accurate optimization of the objective function, which is crucial for effective solution of inverse problems. To validate the effectiveness of the proposed algorithm and to substantiate our claims, we conduct a series of numerical experiments. ### Numerical experiments. In this subsection, we will present several numerical examples for the spatial domain $\Omega\subset\mathbb{R}^{d}$ with $d=2,3$. We define the following relative errors for exact solutions $(u,q)$ and numerical approximations $(u^{},q^{})$ as \[Re_{u}:=\frac{\\|u-u^{}\\|_{L^{2}(\Omega_{T})}}{\\|u\\|_{L^{2}(\Omega_{T})}}, \quad Re_{q}:=\frac{\\|q-q^{}\\|_{L^{2}(\Omega)}}{\\|q\\|_{L^{2}(\Omega)}},\quad Re _{\Delta u}:=\frac{\\|\Delta u-\Delta u^{}\\|_{L^{2}(\Omega_{T})}}{\\|\Delta u \\|_{L^{2}(\Omega_{T})}}.\] Example 1 (two-dimensional experiment):For equation (1.1), we set the exact solution $u$ and the domain $\Omega_{T}$ as \[u(x,y,t)=(x^{2}+y^{2}+1)\exp(t),\quad(t,x,y)\in\Omega_{T}=[0,1]^{3}.\] The exact potential $q$ is given as \[q(x,y)=\sin(\pi x)\sin(\pi y).\] The initial and boundary conditions can be calculated from the representation of $u$ straightforwardly. The exact measurement will be \[u(x,y,1)=\varphi(x,y)=(x^{2}+y^{2}+1)\exp(1),\] and in our experiments the noisy data is set as \[\varphi^{\delta}(x,y):=\varphi(x,y)+\delta\cdot(2\;\text{rand}(\text{shape}( \varphi(x,y)))-1), \tag{5.1}\] where $\text{rand}(\text{shape}(\varphi))$ is a random variable generated by uniform distribution in $[0,1]$. For the implementation details, we use a fully connected neural network for $u_{\theta}$ and $q_{\eta}$ with 3 hidden layers, each with a width of 20. We take \[\text{N}=N_{int}+N_{sb}+N_{tb}+N_{d}=256+256\times 4+256+256=1792\] as the number of collocation points, which are randomly sampled in four different domains, i.e., interior spatio-temporal domain, spatial and temporal boundary domain, and additional measurement domain. The activation function is $\tanh(x)$ and the hyper-parameter is $\lambda=0.01$. The number of training epochs is set to be $5\times 10^{4}$, and the initial learning rates $\tau_{\theta},\tau_{\eta}$ both start with $0.001$ and shrink 10 times every $2\times 10^{4}$ iteration. The test sets are chosen by a uniform mesh \[\mathcal{T}:=\{(t_{k},x_{i},y_{j}):k,i,j=0,1,\cdots,49\}\subset\Omega_{T}. \tag{5.2}\] Since the noisy level of the measurement data affects the reconstruction accuracy, in this simulation, we test the training performance for various noisy levels. Figure 3 records the training process, i.e., the training loss, the relative error for the reconstruction of $q$ and the relative error for the recovery of $u$ with respect to the iterations for different noise levels $\delta=0.1\%,\;1\%,\;5\%,\;10\%$ by the proposed scheme. After training, we test the reconstruction result on test sets $\mathcal{T}$. The distribution of the temperature field $u(x,t)$ also depends on the time $t$, Figure 4 shows the time-series relative error of the recovered $u$ with various noise levels after logarithmic re-scaling. As shown in these figures, the training performance deteriorates as the noise level of the measurement data increasing. Figure 5 shows the exact solution of the pretential term. Figure 6 shows the reconstruction results for $q(x)$ by optimizing the proposed loss function (first line) and the corresponding absolute pointwise error for various noisy level $\delta=0.1\%,5\%,10\%$ (second line). Meanwhile, Figure 7 presents the reconstruction solution $u$ (first line) and corresponding absolute pointwise error (second line) for various noisy level measurement data at $t=1/7$. We can see that the reconstruction accuracy for $q$ deteriorates as the noise level of the measurement data increasing, but the performance for $u$ is still satisfactory. Table 1 presents the recovery results solved by two schemes: (I) the proposed frameworks with the loss function (4.2), (II) the DGM frameworks with the loss function (4.3). We record the generalization error of $q$, $u$ and $\Delta u$ in $L^{2}$-error from the noisy input data with $\delta=0.01$. Due to the random sampling of the training data points, the inversion results have some stochasticity. Thus, we perform Algorithm 1 with the loss function in the formulation (4.2) and formulation (4.3) five times, respectively. The relative errors (mean and standard deviation) for the recovery of $q$, $u$ and $\Delta u$ are shown in Table 1. As observed, optimizing the loss function proposed in this paper leads to more accurate recovery results, especially for the Figure 4: The time series relative error (test) for $u$ for different noisy data. Figure 5: The exact potential function $q$. Figure 3: The training loss (left), the relative error for $q$ (middle), the relative error for $u$ (right) after logarithmic re-scaling. reconstruction of $q$ compared with DGM frameworks. Moreover, although the reconstruction accuracy of $u$ in $L^{2}$-error for both two frameworks are relatively close, the accuracy of $\Delta u$ in $L^{2}$-error for proposed scheme in this paper performs better. This suggests that the proposed frameworks are better able to capture smooth solutions. Figure 6: The reconstruction of potential function $q$ (upper) and corresponding absolute pointwise error $\|q-q^{}\|$ (bottom) for various noisy level measurement data. Figure 7: The reconstruction of solution $u$ (upper) and corresponding absolute pointwise error $\|u-u^{}\|$ (bottom) for various noisy level measurement data at $t=1/7$. Example 2 (two-dimensional experiment):For equation (1.1), we set the exact solution $u$ and the domain $\Omega_{T}$ as \[u(x,y,t)=t\exp(x+y),\quad(t,x,y)\in\Omega_{T}=[0,1]\times[0,2]^{2}.\] The exact potential $q$ is given as \[q(r) =\begin{cases}15\left(\cos r-\sqrt{3}/2\right)+2,\quad 0\leq r \leq\pi/6,\\ 2,\quad\text{ otherwise },\end{cases}\] \[r(x,y) =\sqrt{(x-1)^{2}+(y-1)^{2}}.\] The exact measurement will be \[u(x,y,1)=\varphi(x,y)=\exp(x+y),\] and the noisy data $\varphi^{\delta}$ is generated by (5.1). The network architectures and hyper-parameters such as activation function, balance hyper-parameter $\lambda$ are all the same as Example 1. The number of training epochs is set to be $1\times 10^{5}$, and the initial learning rates $\tau_{\theta},\tau_{\eta}$ both start with $0.001$ and shrink $10$ times every $2\times 10^{4}$ iteration. The test sets are chosen by a uniform mesh as (5.2) In this simulation, we evaluate the training performance under various levels of measurement noise. The training process under Algorithm 1 is recorded in Figure 8, which includes the training loss, and the relative errors for the reconstruction of $q$ and $u$ during training process for different noise levels ($\delta=0,\ 1\%,\ 5\%,\ 10\%$). Figure 9 displays the exact potential function $q$, while the approximated $q^{}$ under different noise level measurements ($\delta=1\%,\ 5\%,\ 10\%$) are shown in Figure 10. We can see that the numerical reconstructions still satisfy the theoretical results even with zero initial condition and nonsmooth exact potential. This means that in numerical reconstructions we may release the conditions of Assumption 1 to some extent. Now, we start to verify the convergence of the iteration in Theorem 4.1 with different neural network architectures. In the experiments, for a fixed number of per-layer neurons $\text{NN}=20$, we compute the reconstruction errors for $q$ versus the noise level $\delta$ using logarithmic re-scaling with various hidden layers $\text{NL}=3,4,6$. The results of these experiments are presented in the left of Figure 11. The theoretical estimate $O(\delta^{1/3})$ is shown by the black line. Similarly, fixing hidden layer $\text{NL}=6$, the reconstruction errors for $q$ under various per-layer neurons $\text{NN}=10,15,20$ are given in the right of Figure 11. From this figure, we see that the error could be bounded by the rate $\delta^{1/3}$ to some extent, which supports the theoretical analysis in Theorem 4.1. \begin{table} \begin{tabular}{l c c c} \hline \hline & $Re_{q}$ & $Re_{u}$ & $Re_{\Delta u}$ \\ \hline I & $0.5011\%\pm 0.0102\%$ & $0.0215\%\pm 0.0049\%$ & $0.2550\%\pm 0.0169\%$ \\ \hline II & $12.4545\%\pm 4.8016\%$ & $0.1148\%\pm 0.0133\%$ & $2.3468\%\pm 0.8133\%$ \\ \hline \hline \end{tabular} \end{table} Table 1: The inversion results with noisy measuremrnt data ($\delta=0.01$) solved by two schemes: (I) the proposed frameworks with the loss function (4.2), (II) the DGM frameworks with the loss function (4.3). Figure 8: The training loss (left), the relative error for $q$ (middle), the relative error for $u$ (right) after logarithmic re-scaling. Figure 10: The reconstruction of potential function $q$ (upper) and corresponding absolute pointwise error $\|q-q^{}\|$ (bottom) for various noisy level measurement data. Figure 9: The exact potential function $q$. In order to evaluate the effectiveness of the proposed scheme in terms of hyper-parameters and network structure, a series of experiments are conducted. Specifically, we examine the impact of the balance hyper-parameter $\lambda$ in (4.2) and the network structure, including the number of hidden layers and neurons. For a fixed number of hidden layers ($\text{NL}=3$) and a fixed number of neurons per-layer ($\text{NN}=20$), we compute the reconstruction errors (mean and standard deviation) for $q$ and $u$ using various values of $\lambda$, such as $\lambda=10^{j},\ -4\leq j\leq 1$. The results of these experiments are presented in Table 2, which indicates that the performance of the inverse problem is highly dependent on the balance hyper-parameter $\lambda$. Specifically, we find that the relative reconstruction errors are optimized when $\lambda$ is set to $10^{-2}$. Furthermore, we observe that the reconstruction errors increase significantly as $\lambda$ exceeds this optimal value. These results suggest that the selection of the balance hyper-parameter is critical to achieving good performance in this inverse problem. Next we experiment with various combinations of hidden layers and neuron numbers for the inverse problem using Algorithm 1. We set the dimension to $d=2$ and try a total of 16 combinations of hidden layers (NL) and per-layer neuron numbers (NN), with \[\text{NL}=3,6,9,14,\quad\text{NN}=10,15,20,25.\] For each combination, we run Algorithm 1 for $1\times 10^{5}$ iterations and record the relative errors (mean and standard deviation) for $(q,u)$ in Table 3. It indicates that deeper (larger NL) and/or wider (larger NN) neural networks tend to yield lower reconstruction errors, although this causes higher computational cost. However, we also observe that for fixed neuron number NN, increasing the number of hidden layers NL, for example $\text{NL}\geq 15$, causes the algorithm fail to converge as the number of iterations increases. This suggests that increasing the number of layers and/or neurons can enhance the representation capacity of neural networks. But it may also introduce more parameters to train, and lead to longer training times and potential overfitting of the representation. Figure 11: The reconstruction error of $q$ versus noise level $\delta$ under various layer numbers for fixed per-layer neuron numbers $\text{NN}=20$ (left), and under various per-layer neuron for fixed layer numbers $\text{NL}=6$ (right). Example 3 (three-dimensional experiment):We also take the following 3-dimensional experiment. We set the exact solution and the domain of equation (1.1) as \[u(x,y,t)=t\exp(x+y+z),\quad(t,x,y,z)\in\Omega_{T}=[0,1]^{4}.\] The exact potential $q$ is given as \[q(x,y,z)=x+y+z.\] We also employ fully-connected neural networks with $\text{NL}=4,\text{ NN}=20$ for both $u_{\theta}$ and $q_{\eta}$. The number of training points are \[\text{N}=N_{int}+N_{sb}+N_{tb}+N_{d}=256+256\times 6+256+256=2304,\] which are randomly sampled from four different training sets. The other network architectures and hyper-parameters such as activation function, balance hyper-parameter $\lambda$, number of \begin{table} \begin{tabular}{c c c c} \hline \hline Error & $\lambda=10^{-4}$ & $\lambda=10^{-3}$ & $\lambda=10^{-2}$ \\ \hline $Re_{q}$ & $2.5113\%\pm 1.1339\%$ & $0.9942\%\pm 0.1121\%$ & $0.7124\%\pm 0.0639\%$ \\ \hline $Re_{u}$ & $0.1374\%\pm 0.0652\%$ & $0.1216\%\pm 0.0320\%$ & $0.0560\%\pm 0.0204\%$ \\ \hline Error & $\lambda=10^{-1}$ & $\lambda=1$ & $\lambda=10$ \\ \hline $Re_{q}$ & $0.8388\%\pm 0.2448\%$ & $1.0809\%\pm 0.1741\%$ & $2.2891\%\pm 1.3302\%$ \\ \hline $Re_{u}$ & $0.0331\%\pm 0.0094\%$ & $0.0478\%\pm 0.0122\%$ & $0.0874\%\pm 0.0110\%$ \\ \hline \hline \end{tabular} \end{table} Table 2: The relative errors for $q$ (first line) and $u$ (second line) using various hyper-parameter $\lambda$ from noisy measurement data with $\delta=0.01$, where the hidden layers $\text{NL}=3$, per-layer neurons $\text{NN}=20$. \begin{table} \begin{tabular}{c c c c c} \hline \hline $NL$ & $NN=10$ & $NN=15$ & $NN=20$ & $NN=25$ \\ \hline 3 & $1.7258\%\pm 0.6258\%$ & $1.1664\%\pm 0.3531\%$ & $0.7124\%\pm 0.0639\%$ & $0.5978\%\pm 0.1071\%$ \\ 6 & $1.5015\%\pm 0.6640\%$ & $0.7234\%\pm 0.2407\%$ & $0.4482\%\pm 0.1065\%$ & $0.3093\%\pm 0.0310\%$ \\ 9 & $0.7797\%\pm 0.1577\%$ & $0.4788\%\pm 0.1598\%$ & $0.4796\%\pm 0.0900\%$ & $0.4874\%\pm 0.2034\%$ \\ 14 & $0.5602\%\pm 0.2118\%$ & $0.4077\%\pm 0.1990\%$ & $0.3958\%\pm 0.0552\%$ & $0.4108\%\pm 0.0637\%$ \\ \hline 3 & $0.0841\%\pm 0.0216\%$ & $0.0615\%\pm 0.0290\%$ & $0.0560\%\pm 0.0204\%$ & $0.0306\%\pm 0.0105\%$ \\ 6 & $0.0575\%\pm 0.0238\%$ & $0.0490\%\pm 0.0062\%$ & $0.0280\%\pm 0.0071\%$ & $0.0358\%\pm 0.0090\%$ \\ 9 & $0.0751\%\pm 0.0129\%$ & $0.0433\%\pm 0.0037\%$ & $0.0523\%\pm 0.0166\%$ & $0.0314\%\pm 0.0132\%$ \\ 14 & $0.0806\%\pm 0.0103\%$ & $0.0612\%\pm 0.0144\%$ & $0.0593\%\pm 0.0145\%$ & $0.0422\%\pm 0.0115\%$ \\ \hline \hline \end{tabular} \end{table} Table 3: The relative errors for $q$ (top) and $u$ (bottom) using various combinations of (NL,NN) for problem dimension $d=2$ from noisy measurement data with $\delta=0.01$, where the hyper-parameter $\lambda=10^{-2}$. training epochs, and initial learning rate are all the same as Example 1. The test sets are chosen by a uniform mesh \[\mathcal{T}:=\{(t_{k},x_{i},y_{j},z_{l}):\ k,i,j,l=0,1,\cdots,49\}\subset\Omega_{T}.\] Figure 12 shows the exact potential function $q$ and the relative errors versus iterations during training process for different noise scales $\delta=0,\ 1\%,\ 5\%,\ 10\%$. Figure 13 presents the potential functions $q_{\eta}$ recovered from different noise levels and the corresponding point by point absolute errors on test sets. The inversion results are satisfactory and reasonable overall. Finally, we conduct the experiments to evaluate the robustness of the proposed scheme in terms of network structure (number of hidden layers and per-layerurons). More specifically, we run Algorithm 1 with per-layer neuron numbers $\text{NN}=5,10,20,25$ for fixed hidden layers Figure 12: The exact potential function $q$ (left), the relative error for recovered $q$ with different noise levels (right). Figure 13: The reconstruction of potential function $q$ (first line), absolute pointwise error $\|q-q^{}\|$ (second line) for various noise levels. $\mathrm{NL}=6$ and with hidden layers $\mathrm{NL}=4,6,8,10$ for fixed per-layer neuron numbers $\mathrm{NN}=25$, respectively. The reconstruction errors are presented in Figure 14. It also seems that larger $\mathrm{NL}$ and/or larger $\mathrm{NN}$ neural networks yield lower reconstruction error. In this example, for fixed hidden layer $\mathrm{NL}=6$, we test the per-layer neuron numbers $\mathrm{NN}\geq 30$ and find that there will be bad reconstruction result, that is, the relative error for $q$ with $\mathrm{NL}=6,\mathrm{NN}=30$ is larger than the case $\mathrm{NN}=10,20,25$ as the iterations increasing. Therefore, more layers and/or neurons with much more parameters to train, yield longer training time, and may result in overfitting of the reconstruction. ## 6 Concluding remarks. In this work, a deep neural network-based reconstruction scheme has been proposed to solve an inverse potential problem in the parabolic equation. The proposed method has shown superior performance in high-dimensional space. We prove the uniqueness of the inverse potential problem. A new loss function has been introduced, which includes regularization terms that depend on the derivatives of the residuals for both the partial differential equation and the measurement data. These regularization terms aim to address the ill-posedness of the inverse problem and enhance the regularity of the solution. Additionally, the mollification method has been employed to improve the regularity of the noisy data, where it can reduce the perturbation errors caused by numerical differentiation on the noisy data. Generalization estimates based on the conditional stability of linear inverse source problems and the mollification error estimate on noisy data have been established, which provide a measure of the stability and accuracy of the proposed method in solving the inverse potential problem. Numerical experiments have been conducted to evaluate the performance of proposed method, which indicate the efficiency the approach in this work. Figure 14: The reconstruction error of $q$ under various per-layer neuron numbers with fixed layer numbers $\mathrm{NL}$=6 (left), the reconstruction error of $q$ under various layer numbers with fixed per-layer neuron number $\mathrm{NN}$=25 (right). ## Acknowledgments Mengmeng Zhang is supported by Foundation of Hebei University of Technology (Grant No.282022550) and Foundation of Tianjin Education Commission Research Program(Grant No.2022KJ102). Zhidong Zhang is supported by National Natural Science Foundation of China (Grant No. 12101627).
2305.13909	Temporal Contrastive Learning for Spiking Neural Networks	Biologically inspired spiking neural networks (SNNs) have garnered considerable attention due to their low-energy consumption and spatio-temporal information processing capabilities. Most existing SNNs training methods first integrate output information across time steps, then adopt the cross-entropy (CE) loss to supervise the prediction of the average representations. However, in this work, we find the method above is not ideal for the SNNs training as it omits the temporal dynamics of SNNs and degrades the performance quickly with the decrease of inference time steps. One tempting method to model temporal correlations is to apply the same label supervision at each time step and treat them identically. Although it can acquire relatively consistent performance across various time steps, it still faces challenges in obtaining SNNs with high performance. Inspired by these observations, we propose Temporal-domain supervised Contrastive Learning (TCL) framework, a novel method to obtain SNNs with low latency and high performance by incorporating contrastive supervision with temporal domain information. Contrastive learning (CL) prompts the network to discern both consistency and variability in the representation space, enabling it to better learn discriminative and generalizable features. We extend this concept to the temporal domain of SNNs, allowing us to flexibly and fully leverage the correlation between representations at different time steps. Furthermore, we propose a Siamese Temporal-domain supervised Contrastive Learning (STCL) framework to enhance the SNNs via augmentation, temporal and class constraints simultaneously. Extensive experimental results demonstrate that SNNs trained by our TCL and STCL can achieve both high performance and low latency, achieving state-of-the-art performance on a variety of datasets (e.g., CIFAR-10, CIFAR-100, and DVS-CIFAR10).	Haonan Qiu, Zeyin Song, Yanqi Chen, Munan Ning, Wei Fang, Tao Sun, Zhengyu Ma, Li Yuan, Yonghong Tian	2023-05-23T10:31:46Z	http://arxiv.org/abs/2305.13909v1	# Temporal Contrastive Learning for Spiking Neural Networks ###### Abstract Biologically inspired spiking neural networks (SNNs) have garnered considerable attention due to their low-energy consumption and spatio-temporal information processing capabilities. Most existing SNNs training methods first integrate output information across time steps, then adopt the cross-entropy (CE) loss to supervise the prediction of the average representations. However, in this work, we find the method above is not ideal for the SNNs training as it omits the temporal dynamics of SNNs and degrades the performance quickly with the decrease of inference time steps. One tempting method to model temporal correlations is to apply the same label supervision at each time step and treat them identically. Although it can acquire relatively consistent performance across various time steps, it still faces challenges in obtaining SNNs with high performance. Inspired by these observations, we propose Temporal-domain supervised Contrastive Learning (TCL) framework, a novel method to obtain SNNs with low latency and high performance by incorporating contrastive supervision with temporal domain information. Contrastive learning (CL) prompts the network to discern both consistency and variability in the representation space, enabling it to better learn discriminative and generalizable features. We extend this concept to the temporal domain of SNNs, allowing us to flexibly and fully leverage the correlation between representations at different time steps. Furthermore, we propose a Siamese Temporal-domain supervised Contrastive Learning (STCL) framework to enhance the SNNs via augmentation, temporal and class constraints simultaneously. Extensive experimental results demonstrate that SNNs trained by our TCL and STCL can achieve both high performance and low latency, achieving state-of-the-art performance on a variety of datasets (_e.g._, CIFAR-10, CIFAR-100, and DVS-CIFAR10). ## 1 Introduction Over the past decade, artificial neural networks (ANNs) have achieved remarkable success in various computation vision tasks. However, with the rapid development of ANNs, the computational power they demand has increased significantly. Inspired by the brain's low energy and efficient computing capabilities, researchers have developed spiking neural networks (SNNs) with computational mechanisms inspired by the brain to emulate these properties [1; 2; 3]. The brain-like properties of SNNs such as low energy consumption, event-driven nature, spatio-temporal information processing abilities, and inherent compatibility with neuromorphic hardware have drawn widespread attention. Although SNNs have demonstrated notable potential, directly training high performance and low latency SNNs by backpropagation remains a challenge [4; 5; 6; 7; 8; 9]. The mainstream approach involves using labeled information to supervise the average representations integrated over all time steps, defined as $\mathcal{L}=\mathcal{L}_{ce}(\frac{1}{T}\sum_{t=1}^{T}O^{t},Y)$ (see Fig. 1(a)). However, this method neglects the temporal dynamics of SNNs and only optimizes the average prediction distributions. As shown in Fig. 2, this strategy leads to a significant performance degradation when inference is conducted under ultra-low latency ($T=1$), indicating its inadequacy for scenarios with limited computing resources (_i.e._, 76.98% -> 44.19% for SEW [7] and 76.85% -> 48.93% for Spiking-ResNet [10] in Fig. 2). A straightforward solution to make full use of all temporal information is to optimize each time step respectively, typically by assigning the same label supervision at each time step and treating them identically (_i.e._, TET [11]), defined as $\mathcal{L}=\frac{1}{T}\sum_{t=1}^{T}\mathcal{L}_{ce}(O^{t},Y)$ (see Fig. 1(b)). Although it acquires relatively steady performance across various time steps (_i.e._, 76.98% -> 68.72% and 77.13% -> 67.07% in Fig. 2), this simplistic temporal supervision only takes temporal identity into account and overlooks coherent correlations. In addition, the self-accumulation intrinsic dynamics of spiking neurons and binary representation forms tend not to results in identical neural activities across time. All these factors limit the representational capability of SNNs and lead to minimal performance gains. The unsatisfactory performance of both the mainstream method and TET implies that the full power of temporal information remains to be harnessed through extra guidance remedy. To this end, we propose the temporal-domain supervised contrastive learning (TCL) framework to obtain SNNs with high performance and low latency by incorporating contrastive supervision signal with temporal domain information (see the left columns of Fig. 1(c)). Contrastive learning (CL) is designed to learn both generalizable representations and discriminative representations [12, 13, 14]. CL accomplishes this by attracting similar (positive) samples while dispelling different (negative) samples. In unsupervised contrastive learning, the positive samples could be another view of the anchor sample [15, 16, 17], whereas in supervised contrastive learning, the positive samples are the samples sharing the same label with the anchor sample [18]. We extend this concept to the temporal domain of SNNs, constructing positives from features of 1) every single time step and 2) the sample within the same class. The SNNs trained by TCL flexibly and fully leverage the correlation between representations at different time steps, which maintains high performance even under extremely low inference time steps (_i.e._, 77.96% -> 69.65% and 77.83% -> 70.09% in Fig. 2). Furthermore, we propose a siamese temporal-domain supervised contrastive learning (STCL) framework (see the right columns of Fig. 1(c)), to get more meaningful positive samples and facilitate Figure 1: An illustration of different methods for directly training SNNs. (a) Baseline framework: This method directly optimizes the aggregated representations under the label supervision. (b) TET framework: This approach optimizes the representations at each time step respectively, using identical label information. (c) Our proposed Temporal-domain supervised Contrastive Learning (TCL) framework (left) and Siamese Temporal-domain supervised Contrastive Learning (STCL) framework (right): We inject contrastive supervision into temporal domain to model the correlation between representations at different time-steps. the learning of extra data augmentation invariance. This extension encourages the SNNs to simultaneously learn data augmentation invariance, temporal correlation, and in-class correlation. Consequently, STCL effectively captures the spatio-temporal information of the SNNs, leading to superior and robust performance across different inference time steps (see Fig. 2). Our experiments demonstrate the effectiveness of the proposed TCL and STCL frameworks, which achieve state-of-the-art performance on eight static and neuromorphic datasets. The main contributions of this work are as follows: 1) We first introduce contrastive learning to the representations in every single time step of SNNs and propose a temporal-domain supervised contrastive learning (TCL) framework to effectively supervise the temporal information of SNNs, enhancing their ability to model temporal correlation, overcoming the challenge of significant performance degradation during inference under ultra-low latency. 2) We further extend a siamese temporal-domain supervised contrastive learning (STCL) framework, which enhances the SNNs by learning invariance to data augmentation, the temporal and in-class correlation simultaneously. This significantly improves the performance of SNNs at low latency. 3) Extensive experiments on eight classification datasets, including CIFAR-10 [19], CIFAR-100, DVS-CIFAR10 [20], CUB-200-2011 [21], Stanford-Cars [22], FGVC-Aircraft [23], Oxford Flowers [24], and Stanford Dogs [25], demonstrate our proposed TCL and STCL outperform other approaches by a large margin and achieves the state-of-the-art (SOTA) performance. ## 2 Related Work ### Learning Methods of Spiking Neural Networks. The two primary training paradigms for SNNs are ANN-to-SNN conversion [26; 27; 28; 29; 30] and Directly-train-SNN [4; 7; 31; 32; 33]. The conversion method transforms the pre-trained ANNs into the SNNs by replacing the ANN's activation function with spiking neurons and sharing weights. However, numerous time steps are required to match the spike firing frequency with the ANNs' activation values. Significant discrepancies can arise when the time step becomes exceedingly small, resulting in significant performance decline. Alternatively, the Directly-train-SNN strategy uses gradient backpropagation to train SNNs, creating a balance between performance and latency. This method supervises the temporal and spatial information of SNNs with backpropagation through time (BPTT) [4; 11; 34], enabling the creation of low latency SNNs. Nonetheless, when compared with ANNs, there is still substantial potential for performance enhancement. This motivates us to propose more efficient methods for training low latency and high performance SNNs. ### Contrastive Learning. Contrastive learning has emerged as a compelling unsupervised paradigm, demonstrating its sophistication in representation learning [16; 35; 14; 36; 37; 38]. It designates positive and negative sample pairs in advance, narrows the distance between the representations of positive sample pairs using a contrastive loss function, and expands the distance between the representations of positive and Figure 2: Performance of SNNs at various time steps during the inference phase, with pre-training at time steps $T=8$ under CIFAR-100 dataset. negative sample pairs. Recently, researchers have incorporated label information into contrastive learning. This enhancement refines the definition of positive and negative samples, allowing contrastive learning to function within a supervised paradigm [18; 17], which further strengthens the model's capacity for representation. This study introduces this robust learning approach to effectively model the temporal correlations of SNNs, thereby elevating the representational capability of SNNs. ## 3 Preliminary ### Neuron Model In this work, we utilize Leaky Integrate-and-Fire (LIF) neuron as the foundational computational unit of the SNNs. The dynamic behavior of the neuron is described by the following equations: \[u^{t+1,pre}=\alphau^{t}+x^{t+1}, \tag{1}\] \[y^{t+1}=\Theta(u^{t+1,pre}-V_{th}), \tag{2}\] \[u^{t+1}=U^{t+1,pre}(1-y^{t+1}), \tag{3}\] where $\alpha$ represents the delay factor (typically set to $0.5$ in our study), $u^{t+1,pre}$ signifies the membrane potential prior to the neuron's spike firing mechanism at time step $t+1$, and $u^{t}$ denotes the final membrane potential of the neuron at time step $t$. $x^{t+1}$ represents the input at time step $t+1$, while $\Theta$ denotes the step function, determining whether the neuron fires a spike based on whether the preliminary membrane potential surpasses the firing threshold $V_{th}$. If a spike is fired, the output $y^{t+1}$ is 1, otherwise, it is 0. Eq. 3 illustrates the reset mechanism. Following a spike, the neuron's membrane potential is reset. In our model, we employ a 'hard reset' approach [39], resetting the membrane potential to a predetermined value, typically zero. This approach typically results in superior performance in the context of LIF neurons, compared to the'soft reset' approach where a predetermined value (typically the firing threshold $V_{th}$) is subtracted from the membrane potential. ## 4 Method In this section, we start by revisiting the loss function typically employed in the direct training of SNNs. Subsequently, we introduce our Temporal Contrastive Learning (TCL) framework, a novel approach aimed at effectively supervising the temporal information inherent in SNNs, thereby facilitating the network in modeling temporal correlation. Furthermore, we present our Siamese Temporal Contrastive Learning (STCL) framework, a siamese network paradigm to enhance SNNs to concurrently learn augmentation invariance, temporal and in-class correlation. We conclude this section by providing additional insights and strategies associated with our proposed frameworks. ### Direct Training of SNNs Direct training of SNNs predominantly uses the standard cross-entropy (CE) loss. A typical baseline method is first integrate output information across time steps, then adopt CE to supervise the prediction of the average representation (see Fig. 1(a)): \[\mathcal{L}_{BL}=\mathcal{L}_{CE}\left(\frac{1}{T}\sum_{t=1}^{T}O^{t},Y\right), \tag{4}\] where $O^{t}$ denotes the predicted probability output of the SNNs at time step $t$, $Y$ represents the target label, and $T$ is the total number of time steps. However, Eq. 4 neglects the temporal dynamics of SNNs and only optimizes the temporal average prediction distributions. We demonstrate that SNNs trained by Eq. 4 fails to maintain performance as time step decreasing during inference (see Fig. 2). Conversely, TET [11] makes full use of all temporal information by assigning the same label supervision at each time steps and optimizing them respectively (see Fig. 1(b)): \[\mathcal{L}_{TET}=\frac{1}{T}\sum_{t=1}^{T}\mathcal{L}_{CE}(O^{t},Y). \tag{5}\] Although Eq. 5 acquires relatively steady performance across various time steps (see Fig. 2), it indeed only takes temporal identity into account and overlooks coherent correlation, which leads to minimal performance gains and is also far from our objective, _i.e._, obtaining SNNs with both low latency and high performance. ### Temporal Contrastive Learning Contrastive learning is a powerful and flexible approach to representation learning. It learns to be discriminative and generalizable by maximizing the similarity of representations between positive samples and minimizing the similarity between positive and negative samples. By carefully defining positive and negative samples, and shaping the form of the contrastive loss function, we can equip the network with various capabilities. The common form of contrastive loss, such as SimCLR [16], can be expressed as follows: \[\mathcal{L}_{CL}=-\sum_{i\in I}\log\frac{\exp(z_{i}\cdot z_{i}^{+}/\tau)}{\sum _{k\neq i}\exp(z_{i}\cdot z_{k}/\tau)}, \tag{6}\] where $I$ indexes all samples in a batch. The terms $z_{i}$, $z_{i}^{+}$, and $z_{k}$ all denote L2-normalized features that have been passed through a projection module. Similarly, all subsequent references to $z$ denote such features. Specifically, $z_{i}$ and $z_{i}^{+}$ represent the positive sample pairs, while $z_{k}$ represents the features of all other samples in the mini-batch that are not the $i$-th sample. $\tau$ is a temperature parameter that controls the concentration of the distribution. In our work, we incorporate contrastive supervision with temporal domain information and extend it to effectively capture the temporal correlations inherent in temporal information. We achieve this by redefining positive samples as representations of the same image at different time steps (see the left column of Fig. 1(c)). This can be formalized as: \[\mathcal{L}_{CL}=-\frac{1}{T}\sum_{i\in I}\sum_{t=1}^{T}\sum_{ \begin{subarray}{c}t=1\\ t^{\prime}\neq t\end{subarray}}\log\frac{\exp(z_{i}^{t}\cdot z_{i}^{t^{\prime }}/\tau)}{\sum_{(k,t^{\prime\prime})\neq(i,t)}\exp(z_{i}^{t}\cdot z_{k}^{t^{ \prime\prime}}/\tau)}, \tag{7}\] where $z_{i}^{t}$ denotes the representation of image $i$ at time step $t$, $T$ is the total number of time steps, and $t^{\prime}$ denotes all other time steps for the same image $i$. The overall loss can be formulated as follow: \[\mathcal{L}_{TCL}=\mathcal{L}_{BL}+\lambda\mathcal{L}_{CL}, \tag{8}\] where the hyperparameter $\lambda$ balances the cross-entropy loss and the temporal contrastive loss. ### Siamese Network Paradigm In our previous discussions, we construct positive samples from features of every single time step. However, it has been established in prior research that the network can learn invariance to various forms of data augmentation, such as Color Jitter and Random Gray Scale [40; 16; 41], thereby gaining a more generalized representational capacity. With this in mind, as shown in the right column of Fig. 1(c), we adopt the siamese network paradigm [16; 42] and simultaneously introduce data augmentation to further enhance the representational capacity of SNNs. We now define positive sample pairs to include different augmented versions of the same sample at various time steps. This updated approach guides the network to simultaneously capture temporal correlations across different time steps and maintain invariance to various data augmentations. This enhancement boosts its ability to seize and represent intricate temporal dynamics and transformations. The overall loss of the STCL framework can be formulated as follows: \[\mathcal{L}_{STCL}=\mathcal{L}_{BLaug1}+\mathcal{L}_{BLaug2}+ \lambda\mathcal{L}_{CL}, \tag{9}\] where $\mathcal{L}_{BLaug1}$ and $\mathcal{L}_{BLaug2}$ represent the cross-entropy losses corresponding to the two different augmented versions of the same sample. ### Extra Details and Strategies Supervised Contrastive Loss. In our supervised paradigm, we incorporate label information into the contrastive loss function, modifying it to include the similarity between samples of the same class, while still minimizing the similarity between positive and negative samples. Following the approach in [18], the updated loss function can be formulated as: \[\mathcal{L}_{\;\;CL}^{\;\;sup}=-\frac{1}{T}\sum_{i\in I}\frac{1}{\|P(i)\|}\sum_{t =1}^{T}\sum_{t^{\prime}=1}^{T}\sum_{p\in P(i)}\not\vdash i\neq p\text{ or }t\neq t^{\prime}\log\frac{\exp(z_{i}^{t}\cdot z_{p}^{t^{\prime}}/\tau)}{ \sum_{(k,t^{\prime\prime})\neq(i,t)}\exp(z_{i}^{t}\cdot z_{k}^{t^{\prime\prime }}/\tau)}, \tag{10}\] where $P(i)$ denotes the set of positive sample pairs with the same label as sample $i$, and $\|P(i)\|$ represents the cardinality of the set $P(i)$. In this formulation, we extend the original contrastive loss function to include the similarity between samples of the same class at different time steps, while still minimizing the similarity between positive and negative samples. This adaptation allows the network to learn representations that more effectively capture the underlying structure of the temporal data, thereby leading to improved performance in classification tasks. Projection Module. In our framework, before calculating the temporal contrastive loss on the temporal features of SNNs, we introduce a projection module, which comprises two non-linear layers. This module maps the features of the SNNs into a normalized embedding space. This pivotal component significantly enhances the model's performance, corroborating findings in contrastive learning [16] and knowledge distillation [43]. Throughout the training process, we incorporate the projection module, but it is omitted during inference, thereby adding no additional parameters. Deep Contrastive Supervision. Additionally, inspired by the work presented in [17], beyond supervising the temporal information in the last layer of the SNNs, we also apply contrastive loss calculations to the intermediate layers of the SNNs by default. This strategy provides effective supervisory signals in the intermediate layers for training the SNNs. ## 5 Experiments In this section, we present a variety of experiments to validate the effectiveness of our proposed method. The experiments are designed to cover multiple datasets, including three static and neuromorphic datasets (_e.g._, CIFAR-10, CIFAR-100, and DVS-CIFAR10) and five fine-grained datasets (_e.g._, CUB-200-2011(CUB), Stanford-Cars(Car), FGVC-Aircraft(Aircraft), Oxford Flowers(Flowers), and Stanford Dogs(Dogs)), and leverage widely used base models, Spiking-ResNet [44], SEW-ResNet[7] and Spiking-VGG [11]. We first describe the dataset preprocessing methods and implementation details to provide a clear understanding of the experimental setup. We then delve into the application of our proposed method on mainstream static and neuromorphic image classification tasks to demonstrate its performance in the SNNs standard tasks scenario. Following this, we explore the fine-grained image classification capabilities of our proposed method. Further, we compare our method with the other SOTA training methods. Finally, we conduct a series of ablation studies to gain deeper insights into the contributions of various components of our proposed method. ### Dataset Preprocessing and Implementation Details In line with previous setups, we directly feed the original images into the SNN for $T$ times on static and fine-grained datasets, while on DVS datasets, we directly input the original spike data. By default, we set the time step $T$ for SNN training and inference to 4. Data Augmentation. Regarding the static and fine-grained datasets, we employ standard data augmentation techniques for the input images, including random cropping, flipping, and normalization in the TCL framework. To enhance our STCL framework, we introduce more complex data augmentation techniques such as Cutout [45]_(only for CIFAR-10/100)_, Color Jitter, and Random Gray Scale. For neuromorphic datasets DVS-CIFAR10, we reduce the resolution of the input image from 128 $\times$ 128 to 48 $\times$ 48 and apply random cropping as the augmentation strategy in our frameworks. Training. We train our models for 200 epochs using SGD optimizer with a learning rate of 0.1, a weight decay of 0.0005, and the momentum of 0.9, using a batch size of 64 or 128, on all datasets. ### Mainstream Static and Neuromorphic Image Classification In this subsection, we evaluate the effectiveness of our proposed method, including both the TCL and STCL frameworks, on mainstream static image classification tasks (CIFAR-100 and CIFAR-10) and a neuromorphic dataset (DVS-CIFAR10). The results are shown in Tab. 1, where we compare the performance of our method using SEW-ResNet18 (SEW-18) and SEW-ResNet34 (SEW-34) architectures against the baseline performance. The results indicate that our proposed TCL and STCL methods consistently improve classification accuracy across all datasets and architectures. Specifically, for the CIFAR-100 dataset, the STCL achieves the highest accuracy of 79.51% and 80.30% for SEW-18 and SEW-34, respectively, marking an improvement of 3.44% and 3.16% over the baseline. Moreover, on the CIFAR-10 dataset, the STCL significant outperforms the baseline by achieving 95.97% and 96.39% accuracy for SEW-18 and SEW-34, respectively. Looking at the DVS-CIFAR10 dataset, the TCL lifts the classification accuracy from 75.60% to 77.20% for SEW-18. For the SEW-34, the TCL increases the accuracy from 74.70% to 76.60%, and the STCL achieves an even higher accuracy of 78.30%. Absolutely, the STCL without the contrastive loss ($\mathcal{L}_{\text{CL}}$ in Eq. 10), denoted as STCL w/o $\mathcal{L}_{\text{CL}}$, achieves 78.60% and 79.63% accuracy for SEW-18 and SEW-34 on CIFAR-100, and 95.44% and 96.07% on CIFAR-10, respectively. This implies that, even in the absence of the contrastive loss, the network can benefit from learning to be invariant to data augmentations such as Random Gray Scale. However, by employing the contrastive loss, our full STCL can bring about further improvements by learning both augmentation invariance and temporal correlation. These results demonstrate that our framework, through effectively learning augmentation invariance and temporal correlation, consistently brings about stable performance improvements across various datasets. ### Fine-Grained Image Classification In this section, we assess the performance of our proposed frameworks on five fine-grained image classification tasks, utilizing SEW-ResNet18 as our base model. Tab. 2 showcases the classification accuracy for each method. The STCL framework consistently surpasses the baseline across all datasets. Specifically, it shows improvements of 6.98% on CUB (57.46% vs. 50.48%), 5.42% on Car (83.37% vs. 77.95%), and 5.85% on Aircraft (70.15% vs. 64.30%). Meanwhile, the TCL framework also demonstrates robust performance enhancements on four datasets. The outcomes affirm the effectiveness of our methods in leveraging augmentation invariance and temporal correlation under fine-grained datasets, thereby enhancing the representational capacity of SNNs. ### Comparison with state-of-the-art SNN Training Methods In this section, we compare our proposed methods with current state-of-the-art SNN training methods on CIFAR-10, CIFAR-100, and DVS-CIFAR10 datasets. The results are presented in Tab. 3. CIFAR-10. Our STCL method achieves an accuracy of 96.35%, which is the highest among all the compared methods. It outperforms the second-best method, Rec-Dis [46], by 0.8%. Furthermore, our \begin{table} \begin{tabular}{c c c c c c c} \hline \hline Dataset & \multicolumn{2}{c}{CIFAR-100} & \multicolumn{2}{c}{CIFAR-10} & \multicolumn{2}{c}{DVS-CIFAR10} \\ \hline Model & SEW-18 & SEW-34 & SEW-18 & SEW-34 & SEW-18 & SEW-34 \\ \hline Baseline & 76.07 & 77.14 & 94.47 & 94.48 & 75.60 & 74.70 \\ \hline TCL & 77.38(+1.31) & 77.84(+0.70) & 94.92(+0.45) & 94.95(+0.47) & 77.20(+1.60) & 76.60(+1.90) \\ \hline STCL w/o $\mathcal{L}_{\text{CL}}$ & 78.60(+2.53) & 79.63(+2.49) & 95.44(+0.97) & 96.07(+1.59) & _74.70_(-0.90) & 75.50(+0.80) \\ \hline STCL & 79.51(+3.44) & 80.30(+3.16) & 95.97(+1.5) & 96.39(+1.91) & 76.60(+1.00) & 78.30(+3.60) \\ \hline \hline \end{tabular} \end{table} Table 1: Effectiveness of our proposed TCL and STCL framework on CIFAR and DVS-CIFAR. \begin{table} \begin{tabular}{c c c c c c} \hline \hline Dataset & CUB${}_{200}$ & Car${}_{196}$ & Aircraft${}_{102}$ & Flowers${}_{102}$ & Dogs${}_{120}$ \\ \hline BaseLine & 50.48 & 77.95 & 64.30 & 89.90 & 58.79 \\ \hline TCL & _50.29_(-0.19) & 78.73(+0.78) & 66.37(+2.07) & 90.00(+0.10) & 59.16(+0.37) \\ \hline STCL & 57.46(+6.98) & 83.37(+5.42) & 70.15(+5.85) & 94.90(+5.00) & 61.24(+2.45) \\ \hline \hline \end{tabular} \end{table} Table 2: Fine-Grained Image Classfication. Numbers following datasets indicate category count. TCL method also achieves competitive performance with an accuracy of 95.03%, which is on par with GLIF [48] and ESG [49]. Notably, our methods utilize only 4 time steps, which is lower than most of the other methods that use 6 time steps. This reduction in time steps indicates the efficiency of our proposed methods. CIFAR-100. Our STCL and TCL method ranks higher than all of the compared methods and STCL achieves the best accuracy of 81.84%, outperforming the second-best method, TEBN [47], by 3.08%. Again, our methods require only 4 time steps, highlighting their efficiency in SNN training. DVS-CIFAR10. On the DVS-CIFAR10 dataset, our TCL method achieves an accuracy of 79.10%, while our STCL method reaches an accuracy of 79.80%. Compared to the other methods, our methods perform competitively. Our STCL method surpasses the performance of DSR, but falls short of TEBN. However, it is worth noting that our methods only require 4 time steps, which is significantly fewer than most of the other methods that use 10 or more time steps. Our TCL method outperforms Dspike [50], Rec-Dis, and DSR, which highlights the effectiveness of our proposed framework. ### Ablation Studies Influence of hyperparameter $\tau$ and $\lambda$. The performance of our proposed frameworks are influenced by hyperparameters $\tau$ and $\lambda$ in the contrastive loss function. Fig. 3 highlights the importance of tuning these hyperparameters while showcasing the robustness of our proposed frameworks to variations within a reasonable range (_i.e._, SEW-18). Fig. 3 (left) shows the effect of varying $\tau$. As $\tau$ increases, both TCL and STCL display robust performance. Specifically, performance peaks at $\tau=0.07$ for TCL (77.38%) and at $\tau=0.06$ for STCL (79.74%). Fig. 3 (right) examines the influence of the $\lambda$ parameter. For TCL, lower $\lambda$ values yield better results, with optimal performance at $\lambda=0.5$ (77.38%). In contrast, STCL is more resilient to changes in $\lambda$, with the highest accuracy at $\lambda=2$ (79.57%). This suggests that while the choice of $\lambda$ is critical, STCL's performance is less sensitive to this parameter, indicating good generalization ability. Based on these findings, we set $\tau$ to 0.07 and $\lambda$ to 0.5 for TCL, and 5 for STCL by default. \begin{table} \begin{tabular}{c c c c c c} \hline \hline Dataset & Method & Training-SNN & Model & Time step & Accuracy \\ \hline \multirow{8}{}{_CIFAR-10_} & TET[11]${}^{\text{ICLR2022}}$ & $\checkmark$ & ResNet-19 & 6 & 94.50 \\ & Rec-Dis[46]${}^{\text{CVPR2022}}$ & $\checkmark$ & ResNet-19 & 6 & 95.55 \\ & TEBN[47]${}^{\text{NeurIPS2022}}$ & $\checkmark$ & ResNet-19 & 6 & 95.60 \\ & GLIF[48]${}^{\text{NeurIPS2022}}$ & $\checkmark$ & ResNet-19 & 6 & 95.03 \\ & ESG[49]${}^{\text{NeurIPS2022}}$ & $\checkmark$ & ResNet-19 & 6 & 95.49 \\ \cline{2-6} & TCL & $\checkmark$ & ResNet-19 & 4* & 95.03 \\ & STCL & $\checkmark$ & ResNet-19 & 4 & 96.35 \\ \hline \multirow{8}{}{_CIFAR-100_} & TET[11]${}^{\text{ICLR2022}}$ & $\checkmark$ & ResNet-19 & 6 & 74.72 \\ & Rec-Dis [46]${}^{\text{CVPR2022}}$ & $\checkmark$ & ResNet-19 & 6 & 74.10 \\ \cline{1-1} & TEBN[47]${}^{\text{NeurIPS2022}}$ & $\checkmark$ & ResNet-19 & 6 & 78.76 \\ & GLIF[48]${}^{\text{NeurIPS2022}}$ & $\checkmark$ & ResNet-19 & 6 & 77.35 \\ \cline{2-6} & TCL & $\checkmark$ & ResNet-19 & 4* & 79.73 \\ & STCL & $\checkmark$ & ResNet-19 & 4 & 81.84 \\ \hline \multirow{8}{}{_DVS-CIFAR10_} & Dspike[50]${}^{\text{NeurIPS2021}}$ & $\checkmark$ & ResNet18 & 10 & 75.40 \\ & TET[11]${}^{\text{ICLR2022}}$ & $\checkmark$ & VGG & 10 & 83.17 \\ \cline{1-1} & Rec-Dis [46]${}^{\text{CVPR2022}}$ & $\checkmark$ & ResNet-19 & 10 & 72.42 \\ \cline{1-1} & DSR[32]${}^{\text{CVPR2022}}$ & $\checkmark$ & VGG & 20 & 77.27 \\ \cline{1-1} & TEBN[47]${}^{\text{NeurIPS2022}}$ & $\checkmark$ & VGG & 10 & 84.90 \\ \cline{1-1} \cline{2-6} & TCL & $\checkmark$ & VGG & 4* & 79.10 \\ \cline{1-1} & STCL & $\checkmark$ & VGG & 4 & 79.80 \\ \hline \hline \end{tabular} \end{table} Table 3: Comparisons with current state-of-the-art methods on CIFAR and DVS-CIFAR. Influence of time steps $T$. The effect of varying time steps ($T$) on the performance of our proposed methods, TCL and STCL, is examined on CIFAR-100 and CIFAR-10 datasets in this section. As depicted in Fig. 4, both TCL and STCL consistently exceed the baseline across different time steps for both datasets (_i.e._, SEW-18). This underlines the ability of our frameworks to enhance performance by effectively modeling the temporal correlations in the SNNs. Our STCL method, exploiting both data augmentation invariance and temporal correlation, achieves the highest accuracy on both datasets. Specifically, it reaches 80.07% on CIFAR-100 and 95.97% on CIFAR-10, with time steps of 8 and 4, respectively. However, the performance tends to plateau or slightly decrease with an increase in time steps beyond these optimal points, indicating a balance between the number of time steps and model performance. This suggests that while increasing time steps initially enhances the model's generalizability, excessive time steps could risk overfitting or computational inefficiencies. Influence on spike firing rate. Tab. 4 displays the spike firing rate across SEW-18 blocks for our methods. TCL reveals a slight increase in fire rates, contributing to notable performance enhancements. Meanwhile, STCL shows varied trends across blocks, implying intricate feature extraction. Despite a slight rise in computation, the significant performance gain affirms the viability of our method. ## 6 Conclusion To summarize, this study proposes innovative solutions to the inherent performance-latency trade-off problem in SNNs through the introduction of the TCL and STCL frameworks. It is demonstrated that these frameworks considerably enhance the ability of SNNs to make fully use of temporal correlations and learn invariance to data augmentation, thereby improving performance under extremely low latency conditions. Experimental results, drawn from eight datasets, validate the proposed methods, with performance significantly outperforming established SOTA approaches. Prospective research will concentrate on refining these frameworks and examining their applicability across different domains. Ultimately, the aim is to realize the vision of efficient, low energy computation of SNNs. \begin{table} \begin{tabular}{c c c c c c c c c} \hline \hline Block Index & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & Acc(\%) \\ \hline Baseline & 30.18 & 38.51 & 18.25 & 25.83 & 8.52 & 13.9 & 6.89 & 13.81 & 76.07 \\ TCL & 32.98 & 43.98 & 22.23 & 32.83 & 11.52 & 18.84 & 10.33 & 22.68 & 77.38 \\ STCL & 28.95 ($\downarrow$) & 34.28 ($\downarrow$) & 20.75 ($\uparrow$) & 28.11 ($\uparrow$) & 13.85 ($\uparrow$) & 20.46 ($\uparrow$) & 7.97 ($\uparrow$) & 22.20 ($\uparrow$) & 79.51 \\ \hline \hline \end{tabular} \end{table} Table 4: Spike Firing rate across different SNN blocks Figure 4: Influence of time step $T$ on performance (left: CIFAR-10, right: CIFAR-100). Figure 3: The effect of hyperparameters in contrastive loss on performance (left: $\tau$, right: $\lambda$).
2306.03938	Learning Causal Mechanisms through Orthogonal Neural Networks	A fundamental feature of human intelligence is the ability to infer high-level abstractions from low-level sensory data. An essential component of such inference is the ability to discover modularized generative mechanisms. Despite many efforts to use statistical learning and pattern recognition for finding disentangled factors, arguably human intelligence remains unmatched in this area. In this paper, we investigate a problem of learning, in a fully unsupervised manner, the inverse of a set of independent mechanisms from distorted data points. We postulate, and justify this claim with experimental results, that an important weakness of existing machine learning solutions lies in the insufficiency of cross-module diversification. Addressing this crucial discrepancy between human and machine intelligence is an important challenge for pattern recognition systems. To this end, our work proposes an unsupervised method that discovers and disentangles a set of independent mechanisms from unlabeled data, and learns how to invert them. A number of experts compete against each other for individual data points in an adversarial setting: one that best inverses the (unknown) generative mechanism is the winner. We demonstrate that introducing an orthogonalization layer into the expert architectures enforces additional diversity in the outputs, leading to significantly better separability. Moreover, we propose a procedure for relocating data points between experts to further prevent any one from claiming multiple mechanisms. We experimentally illustrate that these techniques allow discovery and modularization of much less pronounced transformations, in addition to considerably faster convergence.	Peyman Sheikholharam Mashhadi, Slawomir Nowaczyk	2023-06-05T13:11:33Z	http://arxiv.org/abs/2306.03938v1	# Learning Causal Mechanisms through Orthogonal Neural Networks ###### Abstract A fundamental feature of human intelligence is the ability to infer high-level abstractions from low-level sensory data. An essential component of such inference is the ability to discover modularized generative mechanisms. Despite many efforts to use statistical learning and pattern recognition for finding disentangled factors, arguably human intelligence remains unmatched in this area. In this paper, we investigate a problem of learning, in a fully unsupervised manner, the inverse of a set of independent mechanisms from distorted data points. We postulate, and justify this claim with experimental results, that an important weakness of existing machine learning solutions lies in the insufficiency of cross-module diversification. Addressing this crucial discrepancy between human and machine intelligence is an important challenge for pattern recognition systems. To this end, our work proposes an unsupervised method that discovers and disentangles a set of independent mechanisms from unlabeled data, and learns how to invert them. A number of experts compete against each other for individual data points in an adversarial setting: one that best inverses the (unknown) generative mechanism is the winner. We demonstrate that introducing an orthogonalization layer into the expert architectures enforces additional diversity in the outputs, leading to significantly better separability. Moreover, we propose a procedure for relocating data points between experts to further prevent any one from claiming multiple mechanisms. We experimentally illustrate that these techniques allow discovery and modularization of much less pronounced transformations, in addition to considerably faster convergence. Keywords: Causal Mechanism, Orthogonalization, Diversity, Unsupervised Learning, Ensemble learning ## 1 Introduction Understanding cause-effect relations and the mechanisms by which they are related is a primary element of human intelligence. A key aspect related to such mechanisms is modularity (Scholkopf, 2019). Intuitively, the generating processes of different abstract concepts can be analyzed and modeled separately, or with minimal interactions. Equipped with ways of discovering such mechanisms, humans are able to use them as building blocks in countless combinations (Vankov and Bowers, 2020). As a narrow manifestation of this concept, Humboldt referred to a language as a system that "makes infinite use of finite means," later quoted by Noam Chomsky many times. Since an infinite combinations of sentences can be generated from a finite number of grammatical rules, arguably the understanding of core mechanisms is the key component towards broader generalization. As stated by (Chollet, 2019), the ability to generalize is a spectrum, with symbolic AI offering no generalization, rising up towards local generalization provided by machine learning (specialist AI), and with the ultimate goal of Artificial General Intelligence. Stepping into the new era of deep learning with hierarchical knowledge representations has the potential to offer the ability to adapt to new situations far from the learned manifold of seen data. We believe that robust and fast methods for identifying independent generative mechanisms, or transformations, directly from the data is an important next step for the AI field, especially as we move towards "knowledge creation" Nowaczyk et al. (2020). As a special case, finding invariant features has been one of the prime concerns of pattern recognition systems (Goodfellow et al., 2009). Needless to say, the promise is to recognize patterns of interest under different transformations. For instance, a Convolutional Neural Network (CNN), presented with enough relevant samples during the training, can learn characteristics of classes of interest through a set of functions mapping original features into a manifold where the effects of different mechanisms manifest in similar ways (Dhillon and Verma, 2020). We argue that despite the tremendous success of the deep learning field, they lack the true generalization perspective (Jakubovitz et al., 2019; Baldock et al., 2021). The first issue is that the transformations learned within a trained model are entangled (Xie et al., 2020). This lack of modularity, as we discussed earlier, makes it impossible to come up with new, "creative" combinations of learned concepts (Battaglia et al., 2018). Finding disentangled factors is substantially different from building mechanisms that bring back data points of a given category to a common region of space. More concretely, consider an image dataset containing faces that are either rotated or stretched. A new image, on which both rotation and stretching are applied, will lie far from the manifold learned of such data. Nevertheless, models that captured rotation and stretching mechanisms separately, can bring this data point back to the manifold, and subsequently recognize. Our research extends the study of Parascandolo et al. (2018), which proposed several experts to adversarially compete against each other for transforming data points. The goal for each expert is to find the inverse of a particular mechanism, and thus fool a discriminator. We follow the same setting, providing two substantial contributions over the previous work: the ability to invert less pronounced transformation, and greatly reduced convergence time. We achieve these by proposing, first, to augment experts with an orthogonalization layer, thus forcing their respective outputs to be orthogonal to each other. This leads to higher diversity among experts, which causes the discriminator to provide a more informative signal. Secondly, we introduce data point relocation between experts, discouraging any one of them from claiming multiple transformations. The key finding in this work is the discovery of a previously unknown bottleneck in terms of separability power, namely, that the discriminator part of the adversarial architecture is an insufficient diversification factor. By adapting a recently-proposed orthogonality layer (originally intended for ensemble learning), our model gains the ability to invert less pronounced transformations. This is a fundamental difference that could not be replicated in the existing method simply by using more computations, neither in terms of time nor amount of data. The remainder of the paper organized as follows. Section 2 describes related work; Section 3 presents problem formulation; Section 4 explains the proposed method, and discusses key differences from state of the art; Section 5 contains results of our experiments. Finally, Section 6 concludes the paper. ## 2 Related work The most relevant research that we draw from is Parascandolo et al. (2018). We share the same goal, i.e., to find inverse of a set of mechanisms in an unsupervised fashion. Examples are available of both untouched data and distorted data; however, there is no information about mechanism, or transformation, was applied to a particular data point. The key idea is for a set of experts to compete against each other for data points, where the one best capable of reversing the mechanism wins and gets trained further. Along similar lines, von Kugelgen et al. (2020) proposed decomposition of complex visual scenes using ensemble of experts competing for different parts of a visual scene. An important assumption in their study is that a scene is layered composition of depth-ordered objects. In order for each expert to become specialized on a different part, firstly, they are equipped with an attention network. Then, the attended region is mapped into a latent space and, through a decoder, reconstructs the unoccluded object shape. Locatello et al. (2018) proposed a clustering procedure adopting generative models. Each data point is supposed to be generated by one causal mechanism. Generators (chosen to be VAEs) compete against each other with the purpose of generating realistic data points. The method is the generative counterpart of k-means, since an assignment function assigns data points to different partitions (equal to the number of generators). The generators compete to win over the distribution formed by each partition. Fox and Wiens (2019) proposed a method called "advocacy learning" for a supervised setting, where a set of advocates (decoders) generate class-conditional representations to encourage a judge (classification model) that an input data point belongs to their class. The output of each advocate forms an attention map that, after modulation with the input, will be fed to the judge. The idea is that each advocate learns an attention map which eases the judge's classification task. Clearly, the performance of all these methods greatly depends on the discriminator providing the right signal, i.e., assigning distorted data points to the "correct" expert. This can only be done reliably given enough diversity among the experts. In an extreme case, if all the experts generate equivalent output, the discriminator could not reliably differentiate between the mechanisms. We claim that existing competitive learning approaches are not sufficient, and thus our key contribution is adapting an enforced diversity module that Mashhadi et al. (2021) proposed for ensemble learning into the learning causal mechanisms setting. Once the diversity is enhanced by making the outputs of all experts orthogonalized to each other, the crucial pressure on the discriminator is lowered significantly. ## 3 Problem Setting Let $\mathcal{P}$ be the empirical distribution of original images (before transformation) defined on pixel space $\mathbb{R}^{d}$. Also, let's assume that there are $N$ different transformations (mechanisms) denoted by $M_{1},M_{2},...,M_{N}$ that can be applied on the original images. Let's denote the resultant empirical distribution of applying the mechanisms by $Q_{1},Q_{2},...,Q_{N}$ where $Q_{i}=M_{i}(\mathcal{P})$. For training, the model is provided with two separate datasets: $D_{P}$, with data points from the original distribution $\mathcal{P}$, and $D_{Q}$ with data points sampled i.i.d from transformed distributions $Q_{1},Q_{2},...,Q_{N}$. It is important to notice that nothing is indicating which transformed data point in $D_{Q}$ corresponds to which original data point in $D_{\mathcal{P}}$. If that was the case, one could feed a model with original data points as input and the transformed data points from the same transformation class as output. Alas, in real world, the supervised scenario is rare compared to the unsupervised ones. For simplicity and interpretability of results, let us assume that the number of experts is equal to the number of mechanisms, denoted by $E_{1},E_{2},...,E_{N}$. The desired outcome is that each expert becomes specialized in inverting a single transformation. The original paper Parascandolo et al. (2018) showcases natural relaxations of this scenario, which we omit due to space constraints. ## 4 Methodology In this section, we first explain the overall adversarial process of learning mechanisms, followed by the proposed model architecture and its motivation. ### Learning mechanisms via competing experts As proposed by (Parascandolo et al., 2018), the adversarial process of learning mechanisms is based on making experts compete over data points. For every transformed data point $x^{\prime}$ from $D_{Q}$ in $\mathbb{R}^{d}$, each expert generates a new data point in $\mathbb{R}^{d}$, trying to trick the discriminator into recognizing their generated images as coming from $\mathcal{P}$. In other words, each expert $E_{i}$ tries to maximize the output of the discriminator denoted by $c_{i}$, where $(c_{i}:\mathbb{R}^{d}\rightarrow\mathbb{R})$. In the training process, the corresponding discriminator's outputs for all the experts $c_{i}=c(E_{i}(x^{\prime}))$ are compared to each other. Then, the parameters $\theta^{}$ of only the winning expert $E^{}$, the one with the highest $c_{i}$, are updated. This means that the one best able to inverse a specific mechanism (manifested in $x^{\prime}$), will be reinforced to improve even further. Along with updating the winning expert, the discriminator is also updated, however, it is taking into account the outputs of all the experts. Quite similar to the standard GAN's discriminator loss function (Goodfellow et al., 2020), the following is used to update the discriminator. \[\max_{\theta_{D}}\quad\mathbb{E}_{x\sim\mathcal{P}}\ \log\Big{(}D_{\theta_{D}}(x) \Big{)}+\frac{1}{N^{\prime}}\,\sum_{i=1}^{N^{\prime}}\,\mathbb{E}_{x^{\prime} \sim Q}\Big{(}\log(1-D_{\theta_{Q_{D}}}(\mathbb{E}_{\theta_{i}})(x^{\prime}) \Big{)} \tag{1}\] The difference between Eq. 1 and the default GAN's discriminator loss function is that in our case the second term is the summation over all the experts (generators). The winning expert loss function is exactly the standard GAN's generator loss: \[\min\theta_{E^{}}\quad\mathbb{E}_{x^{\prime}\sim Q}\ 1-\log\Big{(}D_{\theta_{D}} (E^{}(x^{\prime}))\Big{)} \tag{2}\] ### Proposed architecture We propose the following architecture to learn the inverse of mechanisms. As seen from Figure 1, first a number of parallel experts are created. The main novelty of our architecture is adding the "Orthogonalization Layer." Following Mashhadi et al. (2021), the role of this layer is restricting the output of experts at a specific location to become orthogonal to each other. The motivation is to enforce diversity, and we demonstrate in the next section that this increased diversity translates into higher effectiveness of the discriminator. The most crucial challenge in this competitive learning schema is providing effective discriminating signal to correctly assign experts to mechanisms. The better signal the discriminator provides, the more effective and faster the training of the experts - every mistake leads to confusing an expert by training it on an inappropriate mechanism. Without the orthogonalization layer, the only way to diversify the experts is through discriminator-provided training signal, as described in Section 4.1. Although this process is very promising, we argue that the learning becomes significantly more efficient if an explicit diversity technique is incorporated. In the proposed architecture, we enforce expert diversity using the orthogonalization layer. The output of the first expert at layer $P$, $\mathbf{u}_{1}^{P}$, is untouched ($\mathbf{v}_{1}^{P}$); then, the output of the second expert, $\mathbf{u}_{2}^{P}$, is orthogonalized to the $\mathbf{v}_{1}^{P}$ using the Gram-Schmidt orthogonalization, see Equation 3. The result of this orthogonalization is denoted by $\mathbf{v}_{2}^{P}$ in Fig. 1. Then, the output of the third expert, $\mathbf{u}_{3}^{P}$, is orthogonalized to the previous two experts, resulting in $\mathbf{v}_{3}^{P}$. This process continues to the last expert. More detailed description of the details on diversity and error backpropagation through the orthogonalization layer is included in (Mashhadi et al., 2021). \[\mathbf{v}_{k}=\mathbf{u}_{k}-\sum_{i=1}^{k-1}\frac{(\mathbf{v}_{i},\mathbf{ u}_{k})}{(\mathbf{v}_{i},\mathbf{v}_{i})}\mathbf{v}_{i} \tag{3}\] An important distinction between our work and the earlier study by Paras-candolo et al. (2018) is that in the latter, the discussion is made around Figure 1: Parallel Orthogonal Deep Neural Network (PODNN) learning independent mechanisms. Both mechanisms and experts are independent in the sense that they do not inform or influence each other. However, in our architecture, the experts do interact and exchange information, throughout the orthogonalization layer. Besserve et al. (2018) introduced the need for going beyond statistical independence to disentangle latent factors. They argued that many interesting transformations to latent factors are actually statistically dependent and proposed a non-statistical framework based on counterfactual manipulation to find modular structures. We take this transition to yet another level, arguing that there are advantages for modular mechanisms to influence and inform each other. Thinking Bayesian, if the new evidence alters the posterior of what a mechanism is, the updated mechanism should resonate with other mechanisms, and if it does not, it should influence the other mechanisms. For example, if one's concept of what "right translation" means changes, the "left translation" (and other translations) might need to be adjusted accordingly. ### Data points relocation mechanism While the orthogonalization layer leads to higher diversity of experts and better separability of mechanisms among them, still, in some cases the confusion between experts persists. To further discourage an expert from claiming multiple transformations, we propose the data points relocation mechanism. The main idea behind the relocation mechanism is to separate a percentage of low-confidence data points from an expert and assign them to expert which has not won any data point. The level of confidence, or proficiency, of an expert on any given data point is measured by the discriminator score. The lower be the discriminator score, the smaller the confidence. However, selecting which expert should "give up" a data point is challenging, since there is no way to know that an expert has won more than one transformation (due to the unsupervised nature). We propose to use the standard deviation of the euclidean distances of discriminator's last hidden layer on the data points claimed by an expert as a proxy of "ambition level" of the expert. We argue that experts with higher standard deviation are most likely the experts that are trying to claim more than one transformation. The motivation behind this idea is that it would be difficult for an expert to find a hidden space in which data points from different transforms lie very close to each other; in comparison, for an expert winning only one transformation, this task is significantly easier, and the resulting space is more compact. Finally, since the above modifications lead to significantly faster convergence of experts compared to previous work, we argue that the competitive learning approach is only needed in the beginning. After all the experts converge, there is no need anymore to continue the training through the discriminator. Therefore, from that point forward, we only use the discriminator to assign the data points to the specialized experts, while training then standalone. This saves computation, since the error does not need to get propagated from the discriminator to all experts. The pseudocode of the proposed method is depicted in Algorithm 1 and the data points relocation mechanism is depicted in Algorithm 21. Footnote 1: source code available at: [https://github.com/causalPODNN/causalpodnn](https://github.com/causalPODNN/causalpodnn) ``` 1:$t=1$ 2:while TRUE do 3:$\triangleright$ Sample mini-batches 4:$x_{\mathcal{P}},x_{Q}\gets sample(X_{\mathcal{P}}),sample(X_{Q})$. 5:$\triangleright$ Score all the experts' output using $D$ and calculate output of the discriminator's last hidden layer on all data point in 6:for$i=1$ to $N^{\prime}$do 7:$c_{i}\gets D(E_{i}(x_{Q}))$ 8:$h_{i}\gets D[\text{last hidden layer}](E_{i}(x_{Q}))$ 9:endfor 10:$\triangleright$ Update discriminator 11:$\theta_{D}^{t+1}\leftarrow\max\limits_{\theta_{D}^{t}}\left(\nabla\mathbb{E }_{x_{\mathcal{P}}}\log D(x_{\mathcal{P}})+\nabla\left(\frac{1}{N^{\prime}} \right)\sum_{i=1}^{N^{\prime}}\log(1-\mathbb{E}(c_{i}))\right)$ 12:$dp_{1},dp_{2},...dp_{N^{\prime}}\leftarrow$ assign data points to each expert according to $c\}_{i=1}^{N^{\prime}}$ values 13:$\triangleright$ Relocation mechanism 14:if exists at least one expert without any claimed data point then 15:$idx_{L}\leftarrow$ select a ramdon expert among expert without any claim 16:$dp_{1},dp_{2},...,dp_{N^{\prime}}\leftarrow$ 17:$\triangleright$ 18:endif 19:$\triangleright$ Update all experts 20:for$i=1$ to $N^{\prime}$do 21:$\theta_{E_{i}}^{t+1}\leftarrow\max\limits_{\theta_{E_{i}}^{t}}\left(\nabla \mathbb{E}_{dp_{i}}\log D(c_{i})\right)$ 22:endfor 23:if convergence then 24:break 25:endif 26:$t\gets t+1$ 27:endwhile 28:$\triangleright$ Stop training discriminator and utilize it only for scoring the experts 29: Continue training experts on the data points they are claiming ``` Algorithm 1 Learning causal mechanisms through PODNN ``` 1:$\triangleright$ Relocation mechanism 2:if exists at least one expert without any claimed data point then 3:$idx_{L}\leftarrow$ select a ramdon expert among expert without any claim 4:$dp_{1},dp_{2},...,dp_{N^{\prime}}\leftarrow$ 5:$\triangleright$ Relocation_algorithm $\left(h\right)_{i=1}^{N^{\prime}},c\}_{i=1}^{N^{\prime}},dp\}_{i=1}^{N^{ \prime}},idx_{L}$ 6:endif 7:$\triangleright$ Update all experts 8:for$i=1$ to $N^{\prime}$do 9:$\theta_{E_{i}}^{t+1}\leftarrow\max\limits_{\theta_{E_{i}}^{t}}\left(\nabla \mathbb{E}_{dp_{i}}\log D(c_{i})\right)$ 10:endfor 11:if convergence then 12:break 13:endif 14:$t\gets t+1$ 15:endwhile 16:$\triangleright$ Stop training discriminator and utilize it only for scoring the experts 17: Continue training experts on the data points they are claiming ``` Algorithm 21. ``` 0:$h_{1},h_{2},...,h_{N^{\prime}}$: outputs of the discriminator's last hidden layer for data points claimed by each expert $c_{1},c_{2},...,c_{N^{\prime}}$: score of all each experts' output using D on all data point in $x_{Q}$$dp_{1},dp_{2},...,dp_{N^{\prime}}$: data points assigned to each expert $idx_{L}$: index of an expert without any claim Parameter: $rp$: relocation percentage, percentage of low-confidence data points to be relocated 1:for$i=1$ to $N^{\prime}$do 2:$dist_{i}\leftarrow$ pairwise euclidean distance within $h_{i}$ 3:$std_{i}\leftarrow$ standard deviation of $dist_{i}$ 4:endfor 5:$\triangleright$ Find the expert with highest std 6:$idx_{W}\leftarrow$ argmax($std_{1},std_{2},...,std_{N^{\prime}}$) 7:$\triangleright$ Select low-confidence data points claimed by expert 8:$idx_{W}$ 9:$low\_confidence\_dp\leftarrow$ select $rp\%$ of data points sorted 10: in ascending order of $c_{idx_{W}}$ 11: remove $low\_confidence\_dp$ from $dp_{idx_{W}}$ and to $dp_{idx_{L}}$ 12:return$dp_{1},dp_{2},...,dp_{N^{\prime}}$ ``` Algorithm 2 Data point relocation algorithm ## 5 Experiments We run several experiments using MNIST and Fashion-MNIST datasets. Three different categories of transformations are used: translation, adding noise, and contrast inversion. For the translation, in different experiments, depending on the desired number of mechanisms, multiple directions of translation are used: right, left, up, down, right-up, right-down, left-up, and left-down. The goal of the model is identifying, or separating, these transformations (mechanisms), and then learning how to reverse each one. We conduct three sets of experiments, each with a specific purposes. The first experiment demonstrates the fast convergence of the proposed architecture, in comparison to the architecture without orthogonalization. The second experiment focuses on the different roles of the experts and the discriminator, and shows that it is the discriminator that benefits the most from the orthogonalization. Moreover, it highlights the drawback of over-reliance on the discriminator, especially for less pronounced transformations - where our method, due to enforced diversity, solves problems than the original counterpart could not handle. The third experiment justifies the relocation technique, by showcasing how it prevents an expert from claiming multiple mechanisms. ### Faster convergence Figures 2 and 5 compare the convergence speed of the proposed method with its counterpart, on MNIST and Fashion-MNIST datasets, respectively. The left column is without orthogonalization layer and data point relocation, while the right column is the proposed method with orthogonalization and data point relocation mechanism. The last row in the figure shows the number of "correct" data points claimed by each expert, i.e., coming from the transformation they are most specialized in. For the first 10 rows, the vertical axis shows the discriminator score of experts. For the left column, after more than 100 iterations, there is still confusion among experts, while for the right column, after iteration 40, each experts is on the right track - claiming only the points from a single transformation, different from all the others. Figures 3 and 4 illustrate the result of trained experts on random subsets of transformed images. In order to showcase the consistency of the convergence, we define "convergence iteration" as the iteration after which every expert wins the majority of their data points from exactly one transformation. This definition is equivalent, and more intuitively understood, as the last swap in visualization such as the graphs of Figure 2 (MNIST) and 5 (Fashion-MNIST). In Figure 2 the right column, the last swap occurred around iteration 40 (in the seventh row), while in the left column, convergence has not happened yet. Comparison of the convergence iteration of the proposed method with its counterpart across five different runs is shown in Figure 6. The experiment was limited to 200 iterations, so in the third and fifth runs the experts did not converge at all. It can be seen that the orthogonalization leads to drastic Figure 2: MNIST- Comparison of experts’ performance without orthogonalization (and relocation) and with orthogonalization (and relocation). Each line color represents one expert, and each row corresponds to one transformation. Experts’ performance is measured by discriminator scores. improvement. Moreover, the variance of the convergence iteration is also much lower for the proposed method. In Figure 5, the improvement is even more notable as in the right column, the last swap occurred at iteration 85. In contrast, for the left column, even after 180 iterations, the orange expert prevailed three tasks. It is observed that for Fashion-MNIST, without organization and data point relocation, for different runs convergence never happened. Figure 4 illustrates the result of trained experts with orthogonalization and data point relocation on a random subset of transformed images. The rest of the experiments are reported only for MNIST dataset as our proposed method's improvement is not as strong as it is in the case of Fashion-MNIST. In other words, we chose the one that is less favorable to our method, as a more fair comparison. ### Time complexity The computation time associated with the addition of the orthogonalization layer is noticeable, but way lower than the gains provided. In general, the proposed architecture takes on average 10% more time to train than Figure 4: Fasion-MNIST- In each group of rows, the top row shows the input to the experts, and the button row shows the output of an expert with the highest discriminator score Figure 3: MNIST- In each group of rows, the top row shows the input to the experts, and the button row shows the output of an expert with the highest discriminator score the one without orthogonalization. However, since our convergence iteration is as shown in Figure 6 is on average 3.5 times faster than its counterpart, the orthogonalization cost is negligible. In addition, since after experts' convergence we stop training them through the discriminator and continue training them standalone, we save additional computations by removing the discriminator from the backpropagation. Figure 5: Fasion-MNIST- Comparison of experts’ performance without orthogonalization (and relocation) and with orthogonalization (and relocation). Each line color represents one expert, and each row corresponds to one transformation. Experts’ performance is measured by discriminator scores. Figure 6: Convergence speed comparison between “without orthogonalization” (and relocation) and “with orthogonalization” (and relocation) on 5 separate runs ### Diversity helps the discriminator Given that orthogonalization promotes diversity among experts, we argue that the emerged diversity is particularly helpful for the discriminator, allowing it to provide more effective discriminating signal. Looking at the system (experts and discriminator) as a whole, it is not straightforward to understand which part benefits most from the embedded diversity component. In other words, we need to disentangle the discriminator from the experts to analyze the effect of diversity on them. We intend to do it by varying the severity of transformations. First, we replace the discriminator with an oracle, meaning that the discriminator knows in advance what transformation is applied on any image $x^{\prime}$. We then consider different severity of the transformations, for example, the translation distance. We demonstrate that the experts are mostly invariant to that. For instance, translating four pixels to the right would not make it much more difficult for the experts than translating one pixel to the right. To evaluate the experts' performance in undoing the transformations, their outputs are fed to a CNN trained on MNIST. For this experiment, the transformations are chosen to be various translations, with different severity, measured in pixels. Moreover, the relocation mechanism is factored out for this experiment to focus on the orthogonalization aspect as our main driving force of diversification. Table 1 reports the results, in terms of accuracy, without and with orthogonalization. First, it can be seen that as we increase the severity from 1px to 5px, the accuracy of experts remains essentially the same - there is no difference in difficulty. Moreover, regardless of the transformation severity, the difference in accuracy of experts with and without orthogonalization is negligible. This indicates that the key to improved performance in Section 5.1 lies in the benefits that the diversity brings to the discriminator. It is intuitively understandable why the discriminator would be very susceptible to the severity of transformations. When they are very minor, such as translation by one pixel, it is difficult to provide an effective discriminating signal. The outputs from all the experts look as "reasonable" candidates belonging to $\mathcal{P}$. Identifying the "winning" expert is much easier for more pronounced transformations, since the outputs of different experts are more diverse. Therefore, we repeat the previous experiment with transformations of difference severity using a "real" discriminator. Our goal is to investigate whether orthogonalization leads to a better discriminating signal to the competing \begin{table} \begin{tabular}{\|l\|c\|c\|c\|c\|c\|} \hline Method Severity & 1px & 2px & 3px & 4px & 5px \\ \hline \hline Without Orth & 0.9720 & 0.9720 & 0.9718 & 0.9706 & 0.9687 \\ \hline With Orth & 0.9704 & 0.9702 & 0.9682 & 0.9652 & 0.9633 \\ \hline \end{tabular} \end{table} Table 1: Effect of transformation severity on experts’ performance (in terms of accuracy of MNIST-trained CNN. experts. We perform this experiment on two different severities of transformations: 2px and 5px. We expect that in the latter case the discriminator would not have much difficulty providing an effective discriminating signal to the experts, both in the setting with and without orthogonalization. On the other hand, when the transformations are less pronounced, the discriminator will work much better in case of the experts with orthogonalization than without it. Figures 7 and 8 illustrate the results on severity of 5px, without and with orthogonalization, respectively. Furthermore, Figures 9 and 10 illustrate corresponding results on severity of 2px. For the more pronounced case, 5px, it can be seen that despite convergence speed being lower without orthogonalization, both methods converge nonetheless. On the other hand, in the case of 2px, the difference between the state of the art and the proposed method is no longer only terms of convergence time. Without orthogonalization, the experts are not able to converge at all, even given much higher number of iterations. It is clear that once the discriminator is unable to reliably identify the output of "incorrect" expert as such, a stronger diversity-promoting solution is necessary - and orthogonalization can be just that. data points (for different experts) and the number of transformations won by each expert. As an alternative that could be used instead, we also show the standard deviation of the discriminator score. Note that, for the sake of readability, only three transformations, instead of all of them, are shown in the figure. It is noticeable from Figure 11 that experts that win multiple transformations tend to have higher standard deviations. For instance, the expert associated with the blue curve, which starts by winning 4 or 5 transformations, shows very high standard deviation; while the expert shown in green, winning Figure 8: With orthogonalization, transform severity 5px Figure 9: Without orthogonalization, transform severity 2px zero or one transformation, exhibits very low standard deviation. As the training progresses and experts converge towards winning one transformation each, their standard deviations get closer to each other. In this experiment, the number of data points for each transformation within each batch of training is uniformly distributed. One might argue that once an expert starts winning more transformations, it will simply claim more data points; it could be the case that due to the sample prevalence of some Figure 11: relation between number of transformations won by each expert and $std$ of last discriminator’s hidden layer output (also $std$ of discriminator score) – balanced sample Figure 10: With orthogonalization, transform severity 2px transformations, the standard deviation becomes higher for experts winning those transformations, even if an expert wins one transformation. To show the robustness of the relocation mechanism against sample prevalence of transformations, in the next experiment for each training batch, the number of data points for one transformation is three times higher than the others. Figure 13 depicts three transformations one of which, left translation, has the largest number of data points. It can be seen that simply winning a larger number data points does not lead to increase in the standard deviation. Instead, as we hoped, the number of transformation won by the corresponding expert is the key factor affecting the standard deviation. ### 5.5 Relocation percentage Figure 12 depicts the sensitivity of the method on the percentage of data points relocated among experts ($rp$ in Algorithm 2), in terms of convergence speed. Every bar is the average of five separate runs. It is noticeable that very high or very low relocation percentages deteriorate the convergence speed, however, selecting a reasonable value for this hyperparameter is not difficult. For the experiments in Section 5, $rp$ is selected to be 30%. ### 5.6 Relocation mechanism robustness To show the robustness of the relocation mechanism against sample prevalence of transformations, in the next experiment for each training batch, the number of data points for one transformation is three times higher than the others. Figure 13 depicts three transformations one of which, left translation, has the largest number of data points. It can be seen that simply winning a larger number data points does not lead to increase in the standard deviation. Instead, as we hoped, the number of transformation won by the corresponding expert is the key factor affecting the standard deviation. In this experiment, the number of data points for each transformation within each batch of training is uniformly distributed. One might argue that Figure 12: Effect of percentage of data points to relocate on convergence speed once an expert starts winning more transformations, it will simply claim more data points; it could be the case that due to the sample prevalence of some transformations, the standard deviation becomes higher for experts winning those transformations, even if an expert wins one transformation. ## 6 Conclusions In this study, we have proposed a novel method for finding modular causal mechanisms. We have demonstrated that it converges significantly faster and has more power in identifying the modular mechanisms, compared to state of the art. We believe finding modular causal mechanisms is a key towards allowing AI systems to take the next step in generalization performance - the kind of generalization that can go beyond the manifold of a single data set. The primary constituent of our methodology is enforcing diversity between experts. Our key contribution is the observation that the performance bottleneck in this case lies at the insufficient power of the discriminator. A new method for reducing the similarity between experts was needed. This diversity is created by adopting an orthogonalization layer requiring the experts' outputs to be orthogonal to each other. To further reinforce modularization, we propose data points relocation, giving "lost" experts the opportunity to stay in the competition. As a future work, our research naturally lends itself to be applied on more challenging data sets and in real-world scenarios. More importantly, the generalizability of the learned mechanism to new situations should be analyzed Figure 13: relation between number of transformations won by each expert and $std$ of last discriminator’s hidden layer output (also $std$ of discriminator score) – imbalanced sample further. As a more technical future work direction, relaxing the orthogonalization constraint can also be considered. In other words, the level to which the experts' outputs needs to be de-correlated should possibly be adaptive, depending on the properties of the data and the transformations to be learned. ## Appendix A Detailed architectures and experts and discriminator Experts and discriminator architectures are shown in Tables 1 and 2, respectively. The same architectures are used for both the model without orthogonalization and with orthogonalization. The difference is that with orthogonalization, as shown in Figure 1, an orthogonalization layer is embedded after the fourth layer, to make experts outputs orthogonal to each other. It is worth mentioning that we have also examined larger models for discriminator and/or for the experts, however, since the results (including the improvement of the proposed method over the baseline) is consistent across other architectures, we do not report this study in the paper in detail. ## Appendix B Transformations details We used the following transformations for the experiments. * Contrast inversion: The original value of each pixel, which is in the $[0,1]$ range, is transformed as ($1-$original value). * Noise addition: Gaussian noise with mean 0 and standard deviation of 0.1 is added. * Translations: eight directions (namely right, left, up, down, and their diagonals) are used. In Figure 2, 4px translation is used. In Figures 11 and 13, 2px is used.
2301.01150	RELIANT: Fair Knowledge Distillation for Graph Neural Networks	Graph Neural Networks (GNNs) have shown satisfying performance on various graph learning tasks. To achieve better fitting capability, most GNNs are with a large number of parameters, which makes these GNNs computationally expensive. Therefore, it is difficult to deploy them onto edge devices with scarce computational resources, e.g., mobile phones and wearable smart devices. Knowledge Distillation (KD) is a common solution to compress GNNs, where a light-weighted model (i.e., the student model) is encouraged to mimic the behavior of a computationally expensive GNN (i.e., the teacher GNN model). Nevertheless, most existing GNN-based KD methods lack fairness consideration. As a consequence, the student model usually inherits and even exaggerates the bias from the teacher GNN. To handle such a problem, we take initial steps towards fair knowledge distillation for GNNs. Specifically, we first formulate a novel problem of fair knowledge distillation for GNN-based teacher-student frameworks. Then we propose a principled framework named RELIANT to mitigate the bias exhibited by the student model. Notably, the design of RELIANT is decoupled from any specific teacher and student model structures, and thus can be easily adapted to various GNN-based KD frameworks. We perform extensive experiments on multiple real-world datasets, which corroborates that RELIANT achieves less biased GNN knowledge distillation while maintaining high prediction utility.	Yushun Dong, Binchi Zhang, Yiling Yuan, Na Zou, Qi Wang, Jundong Li	2023-01-03T15:21:24Z	http://arxiv.org/abs/2301.01150v2	# RELIANT: Fair Knowledge Distillation for Graph Neural Networks ###### Abstract Graph Neural Networks (GNNs) have shown satisfying performance on various graph learning tasks. To achieve better fitting capability, most GNNs are with a large number of parameters, which makes these GNNs computationally expensive. Therefore, it is difficult to deploy them onto edge devices with scarce computational resources, e.g., mobile phones and wearable smart devices. Knowledge Distillation (KD) is a common solution to compress GNNs, where a light-weighted model (i.e., the student model) is encouraged to mimic the behavior of a computationally expensive GNN (i.e., the teacher GNN model). Nevertheless, most existing GNN-based KD methods lack fairness consideration. As a consequence, the student model usually inherits and even exaggerates the bias from the teacher GNN. To handle such a problem, we take initial steps towards fair knowledge distillation for GNNs. Specifically, we first formulate a novel problem of fair knowledge distillation for GNN-based teacher-student frameworks. Then we propose a principled framework named RELIANT to mitigate the bias exhibited by the student model. Notably, the design of RELIANT is decoupled from any specific teacher and student model structures, and thus can be easily adapted to various GNN-based KD frameworks. We perform extensive experiments on multiple real-world datasets, which corroborates that RELIANT achieves less biased GNN knowledge distillation while maintaining high prediction utility. Open-source code can be found at [https://github.com/yushundong/RELIANT](https://github.com/yushundong/RELIANT). Keywords: Graph Neural Networks, Algorithmic Fairness, Knowledge Distillation ## 1 Introduction In recent years, Graph Neural Networks (GNNs) have shown satisfying performance in a plethora of real-world applications, e.g., medical diagnosis [27] and credit risk scoring [30], to name a few. In practice, the depth and the number of parameters of GNNs largely determine their expressive power [16], which directly influence their performances in various graph learning tasks [2]. Typically, deeper GNN layers enable the model to capture information that is multiple hops away from any node [21], while a larger number of learnable parameters enable GNN to fit more complex underlying data patterns [2]. However, in most cases, the inference efficiency of GNNs is inevitably degraded by the deep layers or the large number of parameters. Such efficiency degradation naturally makes these GNNs inapplicable to be deployed on edge devices (e.g., mobile phones) with limited computational resources [16, 19]. Due to the problem above, it is necessary to compress those computationally expensive GNNs for deployment on edge devices. Knowledge Distillation (KD) is a common approach to compress GNNs but still maintains a similar level of prediction performance [34, 16, 19]. Here, the basic idea of KD is to let a light-weighted student model (as the compressed GNN) learn to mimic the behavior (e.g., output logits) of the teacher model (usually a computationally expensive GNN). However, most existing KD approaches do not have any fairness consideration over different demographic subgroups, and the optimized student model often preserves and even exaggerates the exhibited bias from the teacher GNN. Consequently, when the compressed model is deployed in real-world application scenarios, there could exist discrimination toward specific populations. Here we provide preliminary analysis based on two representative GNN knowledge distillation frameworks, namely CPF [34] and GraphAKD [16]. Specifically, we measure the exhibited Figure 1: A comparison of exhibited bias between teacher and student models based on two representative GNN knowledge distillation frameworks (CPF and GraphAKD). "T" and "S" represent the teacher and the student model, respectively. The names of GNN mark out the corresponding teacher models. bias in the widely-studied node classification task on two real-world datasets. Here Recidivism is a network of defendants [18, 1], while Credit is a network between bank clients [35, 1]. We adopt two traditional metrics, i.e., $\Delta_{\text{SP}}$ (measuring the level of bias under Statistical Parity [10]) and $\Delta_{\text{EO}}$ (measuring the level of bias under Equal Opportunity [15]), to measure the exhibited bias of GNN predictions. We present a comparison of the exhibited bias between teacher and student models in Fig. 1. Empirical results show that student models tend to yield more biased results compared with the teacher GNN model, which could be attributed to the biased guidance from the teacher GNN during training. It is worth noting that in most cases, directly retraining the teacher GNN for debiasing is infeasible, since retraining the teacher GNN with a large number of parameters is computationally expensive. Hence, mitigating the bias for the student model is an urgent need. Despite the necessity of mitigating bias for the student model, existing exploration remains scarce. In this paper, we aim to make an initial step towards developing a debiasing framework that can be easily adapted to various existing GNN-based KD methods. However, this task is non-trivial mainly due to the following three challenges: (1) Gap towards Fair Knowledge: For most KD frameworks designed for compressing GNNs, the teacher GNN model usually serves as the sole source of supervision signal for the training of the student model. Therefore, if the teacher GNN exhibits any bias, such biased knowledge tends to be inherited by the student model. Hence, learning a fair student model with biased supervision from the teacher GNN is our first challenge. (2) Gap towards End-to-End Learning: A critical advantage of existing KD models is the end-to-end learning paradigm, which enables the distilled knowledge to be tailored to specific downstream tasks. In such an end-to-end learning process, highly efficient gradient-based optimization techniques are widely adopted. However, widely-used fairness notions (e.g., Statistical Parity and Equal Opportunity) are defined on the predicted labels. Hence the corresponding bias metrics are naturally non-differentiable w.r.t. the student model parameters. Developing a debiasing framework suitable for gradient-based optimization techniques in the end-to-end learning paradigm is our second challenge. (3) Gap towards Generalization: Various KD models have been proposed for compressing GNNs to satisfy different application scenarios. In fact, most KD models are developed based on certain designs of student models. Developing a framework that is student-agnostic and easily adapted to different KD models is our third challenge. To tackle the above challenges, in this paper, we propose a novel framework named RELIANT (faiR knowEdge distiLatoIon for grAbh Neural neTworks) to mitigate the bias learned by the student model. Specifically, we first formulate a novel research problem of _Fair Knowledge Distillation for GNN-based Teacher-Student Frameworks_. To tackle the first challenge, we incorporate a learnable proxy of the exhibited bias for the student model. In this way, despite the knowledge (from the teacher GNN) being biased, the student model still makes less biased predictions under proper manipulations on the proxy. To tackle the second challenge, we propose to approximate the bias level of the student model, where the approximation is differentiable (w.r.t. the student model parameters) manner. In this way, the highly efficient end-to-end learning paradigm is preserved, and the gradient-based optimization techniques are still applicable. To tackle the third challenge, we design the proposed framework RELIANT in a student-agnostic manner. In other words, the debiasing for the student model does not rely on any specific design tailored for the student model structure. Therefore, RELIANT can be easily adapted to different GNN-based knowledge distillation approaches. The main contributions of this paper are summarized as follows. * Problem Formulation. We formulate and make an initial investigation on a novel research problem of fair knowledge distillation for GNN-Based teacher-student frameworks. * Algorithmic Design. We propose a principled framework named RELIANT that learns the proxy of bias for the student model during KD. RELIANT achieves student-agnostic debiasing via manipulating the proxy during inference. * Experimental Evaluation. We conduct comprehensive experiments on multiple real-world datasets to verify the effectiveness of the proposed framework RELIANT in learning less biased student models. ## 2 Problem Definition Notations. We denote matrices, vectors, and scalars by bold uppercase letters (e.g., $\mathbf{X}$), bold lowercase letters (e.g., $\mathbf{x}$), and regular lowercase letters (e.g., $x$), respectively. For any matrix, e.g., $\mathbf{X}$, we use $\mathbf{X}_{i,j}$ to indicate the element at the $i$-th row and $j$-th column. Preliminaries. We utilize $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{X}\}$ to denote an attributed network (graph). Here, $\mathcal{V}=\{v_{1},...,v_{n}\}$ is the set of nodes, $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ is the set of edges, and $\mathcal{X}=\{\mathbf{x}_{1},...,\mathbf{x}_{n}\}$ ($\mathbf{x}_{i}\in\mathbb{R}^{d}$, $1\leq i\leq n$) is the set of node attribute vectors. We use $\mathbf{A}\in\{0,1\}^{n\times n}$ to denote the adjacency matrix of the graph. If there is an edge from the $i$-th node to the $j$-th node, $\mathbf{A}_{i,j}=1$; otherwise $\mathbf{A}_{i,j}=0$. Moreover, we denote the pre-trained teacher GNN model in a knowledge distillation framework as $f_{\hat{\mathbf{\theta}}}$ parameterized by $\hat{\mathbf{\theta}}$. Here $\hat{\mathbf{\theta}}$ denotes the optimized $\mathbf{\theta}$ of the pre-trained teacher model. Similarly, we denote the student model as $g_{\mathbf{\phi}}$ parameterized by $\mathbf{\phi}$. We represent the optimized $\mathbf{\phi}$ after the training of the student model as $\mathbf{\phi}$. Without loss of generality, we consider the most widely studied node classification as the downstream task. For the teacher model $\mathbf{f_{\mathbf{\theta}}}(v)$, we denote the set of outcome logits, i.e., the continuous output vector corresponding to each node, as $\hat{\mathcal{Y}}^{(t)}=\{\hat{\mathbf{y}}_{1}^{(t)},\hat{\mathbf{y}}_{2}^{(t )},...,\hat{\mathbf{y}}_{n}^{(t)}\}$, where $\hat{\mathbf{y}}_{i}^{(t)}\in\mathbb{R}^{c}$. Here $c$ is the total number of node classes. Correspondingly, we represent the set of outcome logits of the student model $g_{\mathbf{\phi}}(v)$ as $\hat{\mathcal{Y}}^{(s)}=\{\hat{\mathbf{y}}_{1}^{(s)},\hat{\mathbf{y}}_{2}^{(s )},...,\hat{\mathbf{y}}_{n}^{(s)}\}$. For any node $v_{i}$, the predicted label given by the student model (denoted as $\hat{Y}_{i}^{(s)}$ for the $i$-th node) is determined by the largest value across all $c$ dimensions in $\hat{\mathbf{y}}_{i}^{(s)}$. Based on the definitions above, we formulate the problem of _Fair Knowledge Distillation for GNN-based Teacher-Student Frameworks_ as follows. Problem 1: _Fair Knowledge Distillation for GNN-Based Teacher-Student Frameworks. Given an attributed network $\mathcal{G}$ and a GNN-based teacher-student framework including a trained teacher GNN $f_{\mathbf{\phi}}$ and a student model $g_{\mathbf{\phi}}$ to be trained, our goal is to achieve a more fair student model with similar prediction utility compared with $f_{\mathbf{\dot{\theta}}}$ for the node classification task._ ## 3 Methodology In this section, we first present an overview of the proposed framework RELIANT, followed by the objective function formulation and optimization strategy. ### Workflow of RELIANT Here we first introduce the workflow of the proposed framework RELIANT. In general, we introduce the three main functionalities involved in the proposed framework RELIANT, namely _maximizing the utility, learning proxy of bias_, and _enforcing the attribution of bias to the proxy_. We present an overview of RELIANT in Fig. 2. Specifically, to tackle the first challenge (gap towards fair knowledge), we propose to first learn the proxy of bias as extra input attributes for the student model to account for the exhibited bias (on training nodes), and then exclude the information of proxy during test with manipulated pseudo proxy. To tackle the second challenge (gap towards end-to-end learning), we formulate our debiasing objectives in a differentiable (w.r.t. the parameters of the student model) manner. To tackle the third challenge (gap towards generalization), we achieve debiasing in a student-agnostic manner. In other words, the proposed framework RELIANT does not rely on any specific student model structure to achieve debiasing. We elaborate more details as follows. Maximizing Utility. In general, existing GNN-based KD frameworks consider the GNNs with high computational costs as the teacher model, and the goal is to learn a student model with limited computational costs but similar prediction utility (e.g., accuracy in node classification tasks). To maintain the utility of the teacher model, it is necessary to utilize the knowledge from the teacher model as the supervision signal for the training of the student. In particular, a common approach is to utilize the output classification logits from the teacher model as the supervision signal, which we take as an example here. Specifically, we minimize the distance between the logits from the student model and the teacher model. We formally formulate the optimization goal as \[\min_{\mathbf{\phi}}\sum_{v_{i}\in\mathcal{V}}\gamma_{d}\left(\hat{\mathbf{y}}_{i }^{(t)},\hat{\mathbf{y}}_{i}^{(s)}\right), \tag{3.1}\] where $\hat{\mathbf{y}}_{i}^{(s)}$ and $\hat{\mathbf{y}}_{i}^{(t)}$ are the output logits from the student Figure 2: An overview of the proposed framework RELIANT including the training and inference stage. model $g_{\mathbf{\phi}}(v_{i})$ and teacher model $f_{\mathbf{\phi}}(v_{i})$, respectively. The function $\gamma_{\mathrm{d}}(.,.)$ measures the distance between two logit vectors. Different choices can be adopted to measure the distance, e.g., cosine distance and Euclidean distance. Correspondingly, to maximize the prediction utility, we minimize the objective function \[\mathscr{L}_{\mathrm{Utility}}(\mathbf{\phi})=\sum_{v_{i}\in\mathcal{V}}\gamma_{ \mathrm{d}}\left(\tilde{\mathbf{y}}_{i}^{(t)},\tilde{\mathbf{y}}_{i}^{(s)} \right). \tag{3.2}\] Learning Proxy of Bias.It is worth noting that even if the sensitive attributes are removed from the input data, the student model could still exhibit bias in its predictions. The main reason is that there could exist dependencies between those sensitive attributes and non-sensitive ones. Moreover, the information about sensitive attributes could also be encoded in the input network structure [6]. As a consequence, it is difficult to prevent the student model from leveraging information about sensitive attributes. To handle such a problem, we propose to learn the proxy of bias $\mathbf{x}_{i}^{(p)}$ as extra input attributes for each node $v_{i}$. Here, the rationale is that if much information about bias comes from the learned proxy instead of those encoded in the non-sensitive attributes or the network structures, then we are able to achieve less biased predictions by not using the information from such a proxy during test. As a consequence, such a proxy of bias should account for the exhibited bias of the student model as much as possible. In other words, the exhibited bias should be largely attributed to the proxy of bias rather than the sensitive information encoded in the network data. More specifically, to enforce the proxy of bias contributing to the exhibited bias in the student model, we propose to maximize the exhibited bias when these proxies are taken as input into the student model together with other attributes and the network structure. We formally formulate our goal as \[\max_{\mathbf{X}^{(p)}}J_{\mathrm{Bias}}(\{g_{\mathbf{\phi}}(\gamma(v_{i},\mathbf{ X}^{(p)})):i\in\mathcal{V}\}), \tag{3.3}\] where $\gamma(.,.)$ is a function that takes a node and the proxy of bias matrix as input, and outputs the node with a concatenated node attribute vector $[\mathbf{x}_{i},\mathbf{x}_{i}^{(p)}]$. Here $\mathbf{x}_{i}^{(p)}$ is the $i$-th row of $\mathbf{X}^{(p)}$. $J_{\mathrm{Bias}}(.)$ is a function that takes the set of logits from the student model as input and outputs a value indicating the level of exhibited bias. Nevertheless, the computation is non-differentiable under traditional fairness notions such as Statistical Parity and Equal Opportunity. Here we propose to utilize orthogonal polynomials (e.g., Legendre polynomials [4]) that are differentiable w.r.t. the output logits to approximate the level of bias under traditional fairness notions. This makes $J_{\mathrm{Bias}}$ differentiable w.r.t. the learnable parameter $\mathbf{\phi}$. Correspondingly, we formally give the objective function towards the goal above as \[\mathscr{L}_{\mathrm{Proxy}}(\mathbf{X}^{(p)})=-J_{\mathrm{Bias}}(\{g_{\mathbf{ \phi}}(\gamma(v_{i},\mathbf{X}^{(p)})):i\in\mathcal{V}\}). \tag{3.4}\] Enforcing the Attribution of Bias to the Proxy.Only achieving Eq. (3.3) is not enough to enforce the proxy of bias largely accounting for the exhibited bias of the student model. This is because the vanilla node attributes could still contribute to the exhibited bias. More specifically, we denote $P(\hat{Y}^{(s)})$ as the probability of a certain classification prediction given by the student model for any specific node. We assume that there are underlying unbiased and biased node attributes $\mathbf{X}^{(u)}$ and $\mathbf{X}^{(b)}$, respectively. When Eq. (3.3) is achieved, it is clear that $P(\hat{Y}^{(s)}\|\mathbf{X}^{(u)},\mathbf{X}^{(b)},\mathbf{X}^{(p)})$, i.e., $P(\hat{Y}^{(s)}\|\mathbf{X},\mathbf{X}^{(p)})$, is biased. However, both $\mathbf{X}^{(b)}$ and $\mathbf{X}^{(p)}$ could be the source of the exhibited bias. It is worth noting that our goal is to learn proxy $\mathbf{X}^{(p)}$ to account for as much of the exhibited bias as possible. Therefore, to enforce the effectiveness of the proxy, it is necessary to ensure that the exhibited bias is attributed to the biased information from $\mathbf{X}^{(p)}$ instead of $\mathbf{X}^{(b)}$. In other words, we need to enforce $P(\hat{Y}^{(s)}\|\mathbf{X}^{(u)},\mathbf{X}^{(b)})$ being less biased, which ensures that $\mathbf{X}^{(p)}$ accounts for the exhibited bias as much as possible. Nevertheless, $P(\hat{Y}^{(s)}\|\mathbf{X}^{(u)},\mathbf{X}^{(b)})$ is intractable considering that the number of the input dimension number for the student model is fixed. Hence we propose an alternative approach. Denote the learned proxy of bias and the underlying sensitive attribute vector of any node as $\mathbf{x}^{(p)}$ and $\mathbf{s}$, respectively. We propose to utilize a vector $\mathbb{E}[\mathbf{x}^{(p)}]$ to replace each row in $\mathbf{X}^{(p)}$ as the manipulated pseudo proxy $\tilde{\mathbf{X}}^{(p)}$. In this way, the rows in $\tilde{\mathbf{X}}^{(p)}$ are independent from $\mathbf{s}$, i.e., the information about sensitive attributes encoded in $\mathbf{X}^{(p)}$ is wiped out. To enforce the attribution of bias to the proxy $\mathbf{X}^{(p)}$, the predictions should be as fair as possible when the information about $\mathbf{X}^{(p)}$ is removed. Therefore, we formulate our last optimization goal as \[\min_{\mathbf{\phi}}J_{\mathrm{Bias}}(\{g_{\mathbf{\phi}}(\tilde{\gamma}(v_{i},\tilde{ \mathbf{X}}^{(p)})):i\in\mathcal{V}\}), \tag{3.5}\] where $\tilde{\gamma}(.,.)$ is a function that takes a node and the matrix $\tilde{\mathbf{X}}^{(p)}$ as input, and returns the input node with a concatenated node attribute vector $[\mathbf{x}_{i},\tilde{\mathbf{x}}_{i}^{(p)}]$. Here $\tilde{\mathbf{x}}_{i}^{(p)}$ is the $i$-th row of matrix $\tilde{\mathbf{X}}^{(p)}$. We formally present the corresponding objective function as \[\mathscr{L}_{\mathrm{Attr}}(\mathbf{\phi})=J_{\mathrm{Bias}}(\{g_{\mathbf{\phi}}( \tilde{\gamma}(v_{i},\tilde{\mathbf{X}}^{(p)})):i\in\mathcal{V}\}). \tag{3.6}\] Inference with Student Model.To achieve less biased inference, an ideal case is to make predictions with $P(\hat{Y}^{(s)}\|\mathbf{X}^{(u)})$. However, it is difficult to explicitly extract $\mathbf{X}^{(u)}$ from $\mathbf{X}$. Instead, we argue that $P(\hat{Y}^{(s)}\|\mathbf{X}^{(u)},\mathbf{X}^{(b)},\hat{\mathbf{X}}^{(p)})$ exhibits similar level of bias compared with $(\hat{Y}^{(s)}\|\mathbf{X}^{(u)})$. This is because (1) the bias exhibited by $P(\hat{Y}^{(s)}\|\mathbf{X}^{(u)},\mathbf{X}^{(b)},\hat{\mathbf{X}}^{(p)})$ minimally relies on $\mathbf{X}^{(b)}$ after enforcing Eq. (3.3) and Eq. (3.5); and (2) there is no further information about sensitive attributes encoded in $\hat{\mathbf{X}}^{(p)}$ (as discussed above). Consequently, we propose to utilize $g_{\mathbf{\phi}}(\hat{\gamma}(v_{i},\hat{\mathbf{X}}^{(p)}))$ to achieve less biased prediction for node $v_{i}$ in the inference stage. ## 4 Optimization Objectives & Strategy We present the optimization objectives of RELIANT followed by the training strategy in this section. Optimization Objectives. Based on our discussions above, here we present a summary of the optimization objectives for the proposed RELIANT. First, to optimize the parameter $\mathbf{\phi}$, we formally formulate a unified objective function as \[\mathscr{L}_{\mathbf{\phi}}=\mathscr{L}_{\text{Utility}}(\mathbf{\phi})+\lambda\cdot \mathscr{L}_{\text{Attr}}(\mathbf{\phi}). \tag{4.7}\] Here $\lambda$ serves as a hyper-parameter controlling the effect of debiasing the student model. Second, to optimize the learnable proxy of bias $\mathbf{X}^{(p)}$, we formally present the objective function as \[\mathscr{L}_{\mathbf{X}^{(p)}}=\mathscr{L}_{\text{Proxy}}(\mathbf{X}^{(p)}). \tag{4.8}\] Optimization Strategy. To train the proposed framework RELIANT, we propose to optimize the parameter $\mathbf{\phi}$ and learnable proxy of bias $\mathbf{X}^{(p)}$ in an alternating manner. We present the algorithmic routine of RELIANT in Algorithm 1. ``` 0:$\mathcal{G}$: the graph data; $f_{\mathbf{\phi}}$: the trained teacher GNN model; $g_{\mathbf{\phi}}$: the student model; 0:$g_{\mathbf{\phi}}$: the optimized student model; $\mathbf{X}^{(p)}$: the proxy of bias matrix; 1: Randomly initialize $\mathbf{X}^{(p)}$; 2:while stop training condition not satisfied do 3: Compute $\mathscr{L}_{\mathbf{\phi}}$ according to Eq. (4.7); 4: Update $\mathbf{\phi}$ with $\frac{\partial\mathscr{L}_{\mathbf{\phi}}}{\partial\mathbf{\phi}}$; 5: Compute $\mathscr{L}_{\mathbf{X}^{(p)}}$ according to Eq. (4.8); 6: Update $\mathbf{X}^{(p)}$ with $\frac{\partial\mathscr{L}_{\mathbf{X}^{(p)}}}{\partial\mathbf{X}^{(p)}}$; 7:endwhile 8:return$g_{\mathbf{\phi}}$ and $\mathbf{X}^{(p)}$; ``` Algorithm 1 Fair Knowledge Distillation for GNNs ## 5 Experimental Evaluations In this section, we will first introduce the downstream learning task and adopted real-world datasets, followed by the backbone models, baseline methods, and evaluation metrics. Next, we present the implementation details of the models. Finally, we discuss the evaluation results of the proposed RELIANT. In particular, we aim to answer the following research questions through experiments: RQ1: How well can RELIANT balance the utility and fairness of the student model compared with other baselines? RQ2: To what extent each component of RELIANT contributes to the overall debiasing performance? RQ3: How will the choice of the hyper-parameter $\lambda$ affect the performance of RELIANT? ### Experimental Settings Here we introduce the settings for our experimental evaluation. Downstream Task & Real-world Datasets. We adopt the widely studied node classification as the downstream task in this paper. We adopt four real-world datasets for the experimental evaluations, including two widely used network datasets (Recidivism [18, 1] and Credit Defaulter [35, 1]) and two newly constructed ones based on real-world data (DBLP and DBLP-L). In Recidivism, nodes are defendants released on bail, and edges denote the connections between defendants computed from their past criminal records. Here the sensitive feature is race, and we aim to classify if a certain defendant is unlikely to commit a crime after bail. In Credit Defaulter, nodes are credit card users, and edges are the connections between these users. Here we consider the age period of these users as their sensitive feature, and we aim to predict the future default of credit card payments. Additionally, we also construct two co-author networks, namely DBLP and DBLP-L based on AMiner network [29], which is a co-author network collected from computer science bibliography. Specifically, we first filter out the nodes in AMiner network with incomplete information. Then we adopt two different approaches to sample a connected network from the filtered dataset: DBLP is a subgraph sampled with random walk, while DBLP-L is the largest connected component of the filtered AMiner network. In both datasets, nodes represent the researchers in different fields, and edges denote the co-authorship between researchers. The sensitive attribute is the continent of the affiliation each researcher belongs to, and we aim to predict the primary research field of each researcher. The detailed statistics of these four datasets are in Table 1. KD Framework Backbones & Teacher GNNs. To evaluate the capability of RELIANT in generalizing to different GNN-based KD backbones, here we adopt two representative KD frameworks designed for compressing GNNs, namely CPF [34] and GraphAKD [16]. In general, CPF minimizes the distribution distance between the logits from teacher and student to provide supervision information for the student, while GraphAKD utilizes adversarial training to achieve knowledge distillation for the student. The student model of CPF and GraphAKD is PLP [34] and SGC [32], respectively. For each KD framework, we adopt two types of GNNs (including GCN [21] and GraphSAGE [14]) as the teacher GNN. Baselines. To the best of our knowledge, this is the first study on how to mitigate the bias exhibited in GNN-based KD frameworks. In experiments, we adopt the student model yielded by the vanilla KD framework as our first baseline. For our second baseline, we replace the learnable proxy of bias with a naive proxy for the input of the KD framework. Specifically, we utilize one-hot vectors as the naive proxy for different demographic subgroups during training, where the one-hot vector flags the membership of different nodes. We replace all proxy vectors during inference with an averaged proxy vector across all instances. Here, the rationale is that more distinguishable attributes are easier for deep learning models to learn during training, and these one-hot vectors serve as an "easier" indicator of biased information. In this way, if these one-hot proxy accounts for the exhibited bias of the student model after training, then the exhibited bias could also be mitigated during inference, where such information is wiped out. Evaluation Metrics. We evaluate the performance of the compressed GNN models (i.e., the output student model of KD frameworks) from two perspectives, namely utility and fairness. Specifically, in terms of utility, we adopt the node classification accuracy as the correspond \begin{table} \begin{tabular}{l c c c c} \hline Dataset & Recidivism & Credit Defaulter & DBLP & DBLP-L \\ \hline \# Nodes & 18,876 & 30,000 & 39,424 & 129,726 \\ \# Edges & 321,308 & 1,436,858 & 52,460 & 591,039 \\ \# Attributes & 18 & 13 & 5,693 & 5,693 \\ Avg. degree & 34.0 & 95.8 & 1.3 & 4.6 \\ Sens. & Race & Age & Continent of Affiliation & Continent of Affiliation \\ Label & Bail Decision & Future Default & Research Field & Research Field \\ \hline \end{tabular} \end{table} Table 1: The basic information about the real-world datasets adopted for experimental evaluation. Sens. denotes the semantic meaning of sensitive attribute. \begin{table} \begin{tabular}{l c c c c c} \hline & & DBLP & DBLP-L & Credit & Recidivism \\ \hline \multirow{4}{}{\begin{tabular}{c} CPF* \\ +GCN \\ \end{tabular} } & \multirow{4}{}{Accuracy ($\uparrow$)} & GCN(T)* & 92.37 $\pm$ 0.06 & 94.20 $\pm$ 0.09 & 76.39 $\pm$ 0.48 & 93.68 $\pm$ 0.21 \\ & & Vanilla(S) & 93.14 $\pm$ 0.10 & 94.30 $\pm$ 0.04 & 77.85 $\pm$ 0.10 & 89.41 $\pm$ 0.12 \\ & & One-Hot(S) & 93.04 $\pm$ 0.34 & 94.16 $\pm$ 0.02 & 77.65 $\pm$ 0.10 & 89.15 $\pm$ 0.37 \\ & & RELIANT(S) & 92.70 $\pm$ 0.40 & 94.07 $\pm$ 0.18 & 77.82 $\pm$ 0.45 & 88.88 $\pm$ 0.57 \\ \cline{2-6} & \multirow{4}{}{$\Delta_{\text{SP}}$ ($\downarrow$)} & GCN(T)* & 7.66 $\pm$ 0.26 & 7.33 $\pm$0.04 & 15.81 $\pm$0.40 & 6.10 $\pm$0.05 \\ & & Vanilla(S) & 8.55 $\pm$ 0.50 & 7.16 $\pm$ 0.16 & 14.90 $\pm$ 0.89 & 6.85 $\pm$ 0.05 \\ & & One-Hot(S) & 7.97 $\pm$ 0.63 & 7.46 $\pm$ 0.24 & 13.80 $\pm$ 0.32 & 6.78 $\pm$ 0.51 \\ & & RELIANT(S) & 2.27 $\pm$ 1.00 & 3.09 $\pm$ 0.36 & 10.28 $\pm$ 1.86 & 4.06 $\pm$ 0.64 \\ \hline \multirow{4}{}{\begin{tabular}{c} CPF* \\ +SAGE \\ \end{tabular} } & \multirow{4}{}{Accuracy ($\uparrow$)} & SAGE(T)* & 92.57 $\pm$ 0.28 & 94.10 $\pm$ 0.25 & 77.88 $\pm$ 0.06 & 89.71 $\pm$ 0.14 \\ & & Vanilla(S) & 93.25 $\pm$ 0.15 & 94.97 $\pm$ 0.10 & 77.97 $\pm$ 0.26 & 89.20 $\pm$ 0.11 \\ & & One-Hot(S) & 93.07 $\pm$ 0.10 & 94.32 $\pm$ 0.07 & 78.01 $\pm$ 0.23 & 89.11 $\pm$ 0.29 \\ & & RELIANT(S) & 92.91 $\pm$ 0.51 & 94.17 $\pm$ 0.93 & 78.28 $\pm$ 0.36 & 88.85 $\pm$ 0.27 \\ \cline{2-6} & \multirow{4}{}{$\Delta_{\text{SP}}$ ($\downarrow$)} & SAGE(T)* & 8.32 $\pm$0.24 & 7.81 $\pm$0.08 & 14.08 $\pm$ 1.37 & 6.50 $\pm$0.39 \\ & & Vanilla(S) & 8.29 $\pm$ 0.85 & 7.02 $\pm$ 0.13 & 13.44 $\pm$ 5.23 & 4.41 $\pm$ 0.43 \\ & & One-Hot(S) & 8.01 $\pm$ 0.25 & 7.52 $\pm$ 0.32 & 16.86 $\pm$ 3.86 & 6.62 $\pm$ 0.38 \\ & & RELIANT(S) & 2.01 $\pm$ 1.21 & 2.97 $\pm$ 0.61 & 10.06 $\pm$ 1.70 & 3.94 $\pm$ 0.60 \\ \hline \multirow{4}{}{\begin{tabular}{c} AKD* \\ +GCN \\ \end{tabular} } & \multirow{4}{}{Accuracy ($\uparrow$)} & GCN(T)* & 92.37 $\pm$ 0.06 & 94.20 $\pm$ 0.09 & 76.39 $\pm$ 0.48 & 93.68 $\pm$ 0.21 \\ & & Vanilla(S) & 92.06 $\pm$ 0.16 & 94.07 $\pm$ 0.11 & 76.35 $\pm$ 0.31 & 92.08 $\pm$ 0.29 \\ \cline{1-1} & & One-Hot(S) & 91.55 $\pm$ 0.40 & 94.07 $\pm$ 0.04 & 75.65 $\pm$ 0.75 & 92.07 $\pm$ 0.03 \\ \cline{1-1} & & RELIANT(S) & 91.39 $\pm$ 0.24 & 93.98 $\pm$ 0.08 & 75.64 $\pm$ 0.06 & 91.21 $\pm$ 0.14 \\ \cline{1-1} \cline{2-6} & \multirow{4}{}{$\Delta_{\text{SP}}$ ($\downarrow$)} & GCN(T)* & 7.66 $\pm$ 0.26 & 7.33 $\pm$0.04 & 15.81 $\pm$0.40 & 6.10 $\pm$0.05 \\ & & Vanilla(S) & 7.87 $\pm$ 0.25 & 6.79 $\pm$ 0.10 & 13.61 $\pm$ 2.00 & 6.54 $\pm$ 0.17 \\ \cline{1-1} & & One-Hot(S) & 7.39 $\pm$ 0.35 & 6.72 $\pm$ 0.19 & 14.30 $\pm$ 0.24 & 6.44 $\pm$ 0.32 \\ & & RELIANT(S) & 3.66 $\pm$ 1.09 & 5.18 $\pm$ 0.16 & 8.47 $\pm$ 1.92 & 5.70 $\pm$ 0.18 \\ \hline \multirow{4}{}{ \begin{tabular}{c} AKD* \\ +SAGE** \\ \end{tabular} } & \multirow{4}{}{Accuracy ($\uparrow$)} & SAGE(T)* & 92.57 $\pm$ 0.28 & 94.10 $\pm$ 0.25 & 77.88 $\pm$ 0.06 & 89.71 $\pm$ 0.14 \\ & & Vanilla(S) & 92.23 $\pm$ 0.07 & 94.45 $\pm$ 0.03 & 78.10 $\pm$ 0.24 & 89.67 $\pm$ 0.07 \\ & & One-Hot(S ing metric; in terms of fairness, we adopt two traditional metrics $\Delta_{\text{SP}}$ and $\Delta_{\text{EO}}$. Here $\Delta_{\text{SP}}$ measures the bias level (of predictions) under the fairness notion of Statistical Parity, while $\Delta_{\text{EO}}$ measures the bias level under the notion of Equal Opportunity. See online version of this paper for other results in Appendix due to space limit. Implementation Details. RELIANT is implemented in PyTorch [25] and optimized with Adam optimizer [20]. In our experiments, the learning rate is chosen in $\{10^{-2},10^{-3},10^{-4}\}$ and the training epoch number is set as 1,000 for CPF and 600 for GraphAKD. Experiments are carried out on an Nvidia RTX A6000, and the reported numerical results are averaged across three different runs. We introduce more details in Appendix. ### Effectiveness of RELIANT Here we aim to answer RQ1*. Specifically, we evaluate our proposed framework RELIANT on two KD backbones, namely CPF and GraphAKD. For each KD backbone, we adopt two different GNNs (GCN and GraphSAGE) to evaluate the capability of our proposed framework in generalizing to different GNNs. We compare the corresponding performances between the teacher GNN model and the student models trained with three different frameworks (i.e., the vanilla KD framework, the KD framework with the one-hot proxy of bias, and our proposed RELIANT). We present quantitative results on node classification accuracy and $\Delta_{\text{SP}}$ in Table 2. In addition, we also perform experiments based on Equal Opportunity (see Appendix), where we have consistent observations. We make the following observations from Table 2. From the perspective of prediction utility, student models trained with all three KD frameworks achieve comparable performances with the teacher model. This implies that effective knowledge distillation can be achieved by all three KD frameworks. * From the perspective of bias mitigation, the student models trained with the vanilla KD frameworks inherit and even exaggerate the exhibited bias from the teacher GNN model in all cases. Training the student models with the one-hot proxy can mitigate bias in most cases. Compared with the student models trained with the vanilla KD framework and the one-hot proxy, RELIANT consistently exhibits less bias w.r.t. Statistical Parity. * Based on the performance of RELIANT in both perspectives, RELIANT achieves effective debiasing for the student model but still maintains comparable model utility with the teacher model. Therefore, we argue that RELIANT achieves a satisfying balance between debiasing and maintaining utility. ### Ablation Study We aim to answer RQ2 in this subsection. Specifically, for each framework, we evaluate to what extent the two modules of RELIANT (including _learning proxy of bias_ and _enforcing the attribution of bias to the proxy_) contribute to the performance of the student model. We present the results in Fig. 3. Here, Fig. 3a is the performance of accuracy vs. $\Delta_{\text{SP}}$ from CPF based on the DBLP-L dataset, while Fig. 3b is the performance of accuracy vs. $\Delta_{\text{EO}}$ from GraphAKD based on the Recidivism dataset. Notably, we also have similar observations under other settings. We make the following observations. * From the perspective of prediction utility, we observe that the prediction utility is comparable among all three cases. This corroborates that both modules exert limited influence on the node classification accuracy. * From the perspective of bias mitigation, adding the module of _learning proxy of bias_ to the vanilla KD framework brings limited bias mitigation. This is because the bias could also come from the non-sensitive node attributes (as discussed in Section 3.1). After the module of _enforcing the attribution of bias to the proxy_ is added together with _learning proxy of bias_, RELIANT is then able to achieve satisfying performance on bias mitigation. Figure 4: Parameter sensitivity of $\lambda$ based on two different KD backbones on two real-world datasets. We also have similar observations on other datasets. Figure 3: Ablation study of RELIANT. ”Vanilla” denotes the student model trained with the original KD framework, while ”V. w/ Proxy” represents the student model trained under the KD framework with only learning the proxy of bias. ### Parameter Sensitivity We answer RQ3 by studying the tendency of model utility and exhibited bias w.r.t. the change of hyper-parameter $\lambda$. Here $\lambda$ controls the effect of $\mathscr{L}_{\text{Attr}}$. More specifically, we vary $\lambda$ in {$10^{0}$, $10^{1}$, $10^{2}$, $10^{3}$, $10^{4}$}, and we present the corresponding tendency of node classification accuracy and the exhibited bias of the trained student model with RELIANT in Fig. 4. Here, Fig. 4a is based on the Credit dataset under GraphAKD, while Fig. 4b is based on the DBLP dataset under CPF. We also have similar observations on other datasets. We make the following observations from Fig. 4. * From the perspective of prediction utility, the node classification accuracies on both datasets and KD backbones do not exhibit apparent reduction when the value of $\lambda$ increases from $1$ to $10^{4}$. This verifies that the prediction utility is not sensitive to $\lambda$. * From the perspective of bias mitigation, the student model exhibits less bias when $\lambda$ increases from $1$ to $10^{4}$. Specifically, when $\lambda$ is relatively small (e.g., $1$), the learned proxy of bias only partially accounts for the exhibited bias; when the value of $\lambda$ increases, more bias is then attributed to the learned proxy. Considering the balance between model utility and bias mitigation, a recommended range of $\lambda$ is between $10^{2}$ and $10^{3}$. ## 6 Related Works Algorithmic Fairness in GNNs. Most existing works promoting the algorithmic fairness of GNNs focus either on _Group Fairness_[10] or _Individual Fairness_[36]. Specifically, group fairness is defined based on a set of pre-defined sensitive attributes (e.g., gender and race). These sensitive attributes divide the whole population into different demographic subgroups. Group fairness requires that each subgroup should receive their fair share of interest according to the output GNN predictions [23]. Various explorations have been made towards achieving a higher level of group fairness for GNNs [7]. Decoupling the output predictions from sensitive attributes via adversarial learning is one of the most popular approaches among existing works [31, 3]. Other common strategies include reformulating the objective function with fairness regularization [11, 24], rebalancing the number of intra-group edges between two demographic subgroups [6, 22], deleting nodes or edges that contribute the most to the exhibited bias [8, 9], etc. On the other hand, individual fairness does not rely on any sensitive attributes. Instead, individual fairness requires that similar nodes (in the input space) should be treated similarly (in the output space) [10]. To fulfill individual fairness in GNNs, adding fairness-aware regularization terms to the optimization objective is the most widely adopted approach [5, 28]. Knowledge Distillation. In recent years, knowledge distillation has been proven to be effective in compressing the model but still maintaining similar model prediction performance [12]. Correspondingly, it has been widely adopted in a plethora of applications, including visual recognition [33], natural language processing [13, 17], etc. The main idea of knowledge distillation is to transfer the knowledge of a computationally expensive teacher model to a light student model, and thus the student model is able to fit in platforms with limited computing resources [16, 19]. It is worth noting that such a strategy is also proved to be effective in compressing GNNs [34, 16, 19]. Consequently, there is growing research attention on utilizing knowledge distillation to compress GNNs for more efficient inference. For example, encouraging the student model to yield similar output to the teacher GNN via regularization is proved to be effective [16]. In addition, adversarial learning is also a popular technique to obtain light-weighted but accurate student models [16]. However, most of these frameworks for GNNs do not have fairness consideration. Hence the student model tends to be influenced by biased knowledge from the teacher GNN. Different from existing works, we develop a generalizable knowledge distillation framework that explicitly considers fairness in GNNs but still maintains the utility of GNN predictions. ## 7 Conclusion Despite the success of Knowledge Distillation (KD) in compressing GNNs, most existing works do not consider fairness. Hence the student model trained with the KD framework tends to inherit and even exaggerate the bias from the teacher GNN. In this paper, we take initial steps towards learning less biased student models for GNN-based KD frameworks. Specifically, we first formulate a novel problem of fair knowledge distillation for GNN-based teacher-student frameworks, then propose a framework named RELIANT to achieve a less biased student model. Notably, the design of RELIANT is agnostic to the specific structures of teacher and student models. Therefore, it can be easily adapted to different KD approaches for debiasing. Extensive experiments demonstrate the effectiveness of RELIANT in fulfilling fairness for GNN compression with KD. ## 8 Acknowledgments This work is supported by the National Science Foundation under grants IIS-2006844, IIS-2144209, IIS-2223768, IIS-2223769, CNS-2154962, CMMI-2125326, BCS-2228533, and BCS-2228534, the JP Morgan Chase Faculty Research Award, and the Cisco Faculty Research Award. We would like to thank the anonymous reviewers for their constructive feedback.
2303.00437	Assessing the Finite-Time Stability of Nonlinear Systems by means of Physics-Informed Neural Networks	In this paper, the problem of assessing the Finite-Time Stability (FTS) property for general nonlinear systems is considered. First, some necessary and sufficient conditions that guarantee the FTS of general nonlinear systems are provided; such conditions are expressed in terms of the existence of a suitable Lyapunov-like function. Connections of the main theoretical result of given in this article with the typical conditions based on Linear Matrix Inequalities (LMI) that are used for Linear Time-Varying (LTV) systems are discussed. An extension to the case of discrete time systems is also provided. Then, we propose a method to verify the obtained conditions for a very broad class of nonlinear systems. The proposed technique leverages the capability of neural networks to serve as universal function approximators to obtain the Lyapunov-like function. The network training data are generated by enforcing the conditions defining such function in a (large) set of collocation points, as in the case of Physics-Informed Neural Networks. To illustrate the effectiveness of the proposed approach, some numerical examples are proposed and discussed. The technique proposed in this paper allows to obtain the required Lyapunov-like function in closed form. This has the twofold advantage of a) providing a practical way to verify the considered FTS property for a very general class of systems, with an unprecedented flexibility in the FTS context, and b) paving the way to control applications based on Lyapunov methods in the framework of Finite-Time Stability and Control.	Adriano Mele, Alfredo Pironti	2023-03-01T11:51:44Z	http://arxiv.org/abs/2303.00437v1	Assessing the Finite-Time Stability of Nonlinear Systems by means of Physics-Informed Neural Networks ###### Abstract In this paper, the problem of assessing the Finite-Time Stability (FTS) property for general nonlinear systems is considered. First, some necessary and sufficient conditions that guarantee the FTS of general nonlinear systems are provided; such conditions are expressed in terms of the existence of a suitable Lyapunov-like function. Connections of the main theoretical result of given in this article with the typical conditions based on Linear Matrix Inequalities (LMI) that are used for Linear Time-Varying (LTV) systems are discussed. An extension to the case of discrete time systems is also provided. Then, we propose a method to verify the obtained conditions for a very broad class of nonlinear systems. The proposed technique leverages the capability of neural networks to serve as universal function approximators to obtain the Lyapunov-like function. The network training data are generated by enforcing the conditions defining such function in a (large) set of collocation points, as in the case of Physics-Informed Neural Networks. To illustrate the effectiveness of the proposed approach, some numerical examples are proposed and discussed. The technique proposed in this paper allows to obtain the required Lyapunov-like function in closed form. This has the twofold advantage of a) providing a practical way to verify the considered FTS property for a very general class of systems, with an unprecedented flexibility in the FTS context, and b) paving the way to control applications based on Lyapunov methods in the framework of Finite-Time Stability and Control. keywords: Finite-Time stability, neural networks, universal approximation, Lyapunov methods ## 1 Introduction The concept of Finite-Time Stability (FTS), sometimes also referred to as _practical stability_, was originally introduced in the Russian literature in the '50s [1; 2; 3]; during the next decade, studies on this topic were carried out by the western scientific community as well [4; 5; 6]. Roughly speaking, a dynamical system is said to be FTS with respect to a given time-horizon $T$, an initial time instant $t_{0}$, an _initial set_$\Omega_{0}$ and a (possibly time-varying over the time interval $[t_{0},t_{0}+T]$) _trajectory set_$\Omega_{t}$ if, whenever the initial point of the state trajectory is contained in $\Omega_{0}$, the trajectory is confined inside $\Omega_{t}$ for all $t\in[t_{0}\,,t_{0}+T]$.1 (see fig. 1). Footnote 1: Note that the concept of FTS used in this paper is not to be confused with another notion of finite-time stability that exists in the literature, namely the fact that a stable equilibrium is reached in a finite time interval [7]. FTS is a concept linked to, but independent from, Lyapunov stability, as it is intrinsically concerned with _quantitative_ results, substantially shifting the focus from the steady-state to the transient behaviour of the considered system. However, in many cases Lyapunov-like functions can be employed to assess the FTS of a dynamical system2; necessary and sufficient conditions for the FTS of nonlinear systems based on Lyapunov functions can be found, for instance, in the pioneering work by Bernfeld and Lakshmikantham [8]. Along the same line of research, the main aim of this work is to exploit conditions similar to the ones in [8] in order to derive a practical way to assess the considered FTS property for a very broad class of nonlinear systems. Footnote 2: For brevity, we will refer to these functions as Lyapunov functions as well, even though the conditions that they are required to satisfy are slightly different from the case of standard Lyapunov stability. In some cases, Lyapunov-like conditions for FTS can be efficiently solved by recasting them in terms of algebraic (LMI) and Differential (DLMI) Linear Matrix Inequalities. This fact, together with the availability of efficient tools for solving (D)LMIs, sparked a renewed interest in the topic in recent years. This is the case, for example, of Linear Time-Varying (LTV) systems over ellipsoidal, polytopic and piecewise-quadratic domains, for which techniques based on time-varying quadratic or piecewise-quadratic Lyapunov functions have been proposed (see the book [9] and the references therein for an overview). Moreover, similar techniques can be exploited to find finite-time stabilizing controllers for the same classes of systems. Several results appeared in the technical literature that extended the FTS framework to uncertain [10; 11], hybrid [12; 13], stochastic [14; 15] and Figure 1: Graphical representation of the FTS property for a 2D system. time-delay systems [16], sometimes considering the case of annular domains [17; 18; 19; 20], where both an upper and lower bound on the state variables are assigned. An approach based on the Extremum Seeking algorithm for the Finite-Time stabilization of LTV systems with unknown control direction has been recently proposed in [21]. Some attempts have also been made in order to extend similar techniques to nonlinear systems; for example, in [22] DLMI-based sufficient conditions are given for the special class of nonlinear quadratic systems. However, for nonlinear systems, solutions in terms of (D)LMI feasibility problems are usually not available, and different techniques must be considered. In this view, we aim at proposing a general approach to assess the considered FTS property for the case of nonlinear systems, which, to the best of our knowledge, is lacking in the technical literature. The main problem tackled in this article can be divided in the two following points: * find necessary and sufficient conditions for the FTS of general nonlinear, non-autonomous systems; * find a practical way to assess such conditions (at least for a reasonably large class of nonlinear systems for which standard LMI-based results do not apply). To this aim, we first restate, in a slightly simplified form, the necessary and sufficient conditions for the FTS of continuous-time nonlinear systems originally given in [8]. Then, we discuss how these conditions are a generalization of the ones given in [9] for linear systems over ellipsoidal domains. Furthermore, this theoretical result is also extended to the discrete-time case. Since the proposed conditions are given in terms of the existence of a suitable Lyapunov-like function, we then turn our attention to the problem of finding an expression for such function. A possible method, discussed in this work, is to design a dedicated procedure to train a suitable Neural Network (NN) in order to serve as the desired Lyapunov function. The use of NNs to represent Lyapunov functions is not new in the control literature, thanks to the capability of Neural Networks (NN) to serve as universal function approximators [23]. Some applications include autonomous NL systems [24] and robotics [25]; in the latter paper, a learner-falsifier training procedure is proposed to speed up the training process. A similar training architecture is also used in [26], where NNs are used to estimate the region of attraction of a given controller in the case of discrete time polynomial systems. In both [26] and [27], multi-output NNs are used in combination with quadratic expressions to enforce the positive-definiteness of the Lyapunov functions. Even though NN-based function approximations are known to suffer from the so-called curse of dimensionality, in [28] it is discussed how systems that benefit from a small-gain property have compositional Lyapunov functions which can be estimated with a number of neurons that only grows polynomially with the state dimension. Interestingly, the training procedures used to obtain Lyapunov functions with NNs are usually aimed at enforcing the desired conditions on both the network's output and its derivatives with respect to the input parameters. As deep NNs are used to satisfy conditions involving partial derivatives by exploiting the knowledge of the equations governing a physical process, these techniques are naturally related to the framework of Physics-Informed Neural Networks (PINN) [29]. In this work, we draw ideas from these recent techniques and combine them with the general conditions for the FTS of nonlinear, non-autonomous systems with the aim to provide a flexible procedure that is capable of testing the FTS property for a broad class of dynamical systems over domains of a generic shape. To this end, we propose an approach to train a NN so that it serves as a Lyapunov function. The procedure relies on the standard Adam algorithm [30] to optimize a cost function designed to enforce the required conditions in a large set of collocation points. It is implemented by using off-the-shelf algorithms and can be run on GPU-enabled computers. To the best of our knowledge, this is the first time that a practical algorithm aimed at testing the FTS property for such a broad class of systems is proposed and implemented. The rest of the paper is organized as follows: in Section 2, an overview of the notation used throughout the article and some preliminary results are given. Section 3 presents the main results of the paper. In particular, in section 3.1 the main results, i.e. the necessary and sufficient conditions for the FTS of continuous-time, nonlinear, non-autonomous systems are presented; these conditions are extended to the discrete-time case in section 3.2. In section 3.3, the connection between the results of section 3.1 and existing results for continuous-time linear systems are discussed. In section 3.4, a numerical procedure, based on NNs, to obtain a Lyapunov function that satisfies the properties required by the main result in section 3.1 is described. In section 4 some numerical examples are provided to show the effectiveness of the proposed procedure. Finally, in section 5, the obtained results are discussed and some concluding remarks are given. ## 2 Notation and Preliminaries Given a vector $x\in\mathbb{R}^{n}$, $\|\|x\|\|$ indicates its standard euclidean norm. $\bar{\Omega}$ denotes the closure of a set $\Omega$, $\|\Omega\|$ its measure and $\partial\Omega$ its boundary; $f(\Omega)$ is the image of the set $\Omega$ through the function $f(\cdot)$. $M\prec 0$ indicates that the matrix $M$ is negative-definite, $M\succ 0$ that it is positive definite, $I$ denotes the identity matrix (whose dimensions will be clear from the context). Throughout the article, we will consider systems in the form \[\dot{x}(t)=f(t,x),\qquad x(t_{0})=x_{0}\,, \tag{1}\] where $f(t,x):J\times\mathbb{R}^{n}\to\mathbb{R}^{n}$, and $J=[t_{0},T]$. The evolution function of (1) is denoted by $\varphi(t,t_{0},x_{0})$, while $\phi(t,t_{0})$ indicates the state-transition matrix of a LTV system, for which \[\phi(t,t_{0})x_{0}=\varphi(t,t_{0},x_{0})\,.\] Throughout the paper, we will consider Lyapunov functions $V(t,x)$ that are continuous, scalar and satisfy a local Lipschitz condition with respect to $x$. We denote the orbital derivative of such a function along the trajectories of (1) as (see [31, Chap. 1]) \[\begin{split}\dot{V}(t,x)=&\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, $D^{+}v(t)$; however, to clearly differentiate between derivatives with respect to the $t$ and $x$ variables, we choose to adopt the more common dot notation $\hat{v}(t)$. When differentiable functions are considered (as in the numerical procedure described in sec. 3.4), the standard and upper Dini derivatives coincide. Finally, we adopt the following notation for compactness \[\inf_{\begin{subarray}{c}i\in J\\ x\in\Omega_{t}\end{subarray}}g(t,x)=\inf_{t\in J}\inf_{x\in\Omega_{t}}g(t,x)\,,\] where the set $\Omega_{t}$ depends on the value of $t$. Assumption 1.: _The map $f(t,x)$ in equation (1) is continuous in $t$ and for each compact set $\mathcal{A}\subset\mathbb{R}^{n}$ it satisfies the Lipschitz condition_ \[\|\|f(t,x)-f(t,y)\|\|\leq\lambda_{\mathcal{A}}(t)\|\|x-y\|\|\qquad\forall x,y\in \mathcal{A}\,, \tag{3}\] _where $\lambda_{\mathcal{A}}(t)$ is a non-negative continuous function defined in $J$._ For each $(t^{},x^{})\in J\times\mathbb{R}^{n}$, assumption 1 guarantees the existence and uniqueness of the evolution function $\varphi(t,t^{},x^{})$ for all $t$ in a suitable neighborhood of $t^{}$ (see for example Theorem 3.1 in [32]). Given an _initial set_$\Omega_{0}$, a (possibly time-varying) _trajectory set_$\Omega_{t}$ and two scalars $t_{0}$ and $T$, system (1) is said to be Finite-Time Stable (FTS) with respect to $(t_{0},T,\Omega_{0},\Omega_{t})$ if and only if \[x_{0}\in\Omega_{0}\implies x(t)\in\Omega_{t}\quad\forall t\in[t_{0},t_{0}+T]\,, \tag{4}\] where $\Omega_{0}$ is a closed set, $\Omega_{t}$ is open and both are connected and bounded $\forall t\in J$. Note that, for the definition to be well-posed, it is required that $\Omega_{t}\supset\Omega_{0}$ at $t=t_{0}$. ## 3 Main results In this section, the main results of this work are presented. First, necessary and sufficient conditions for the FTS of nonlinear, non-autonomous systems in the form (1) under assumption 1 are provided in thm. 1. Then, these conditions are extended to the case of discrete-time systems in thm. 2. The connections with the usual LMI-based conditions used for LTV systems (reported for ease of reference as thm. 3) are discussed in sec. 3.3. Finally, a numerical procedure to find a closed-form expression for the required Lyapunov function, based on Neural Networks, is proposed in sec. 3.4. ### Necessary and sufficient conditions for FTS The following theorem provides necessary and sufficient conditions for the FTS of system (1) 3. Footnote 3: The proof closely follows the arguments in [8, Thm. 3.1]. Theorem 1.: _System (1) is FTS wrt $(t_{0},T,\Omega_{0},\Omega_{t})$ if and only if there exists a continuous function $V:J\times\mathbb{R}^{n}\to\mathbb{R}$, locally Lipschitz in $x$, such that the following conditions hold_ \[\dot{V}(t,x)\leq 0\quad\forall t\in J,x\in\bar{\Omega}_{t} \tag{5}\] \[\sup_{x\in\Omega_{0}}V(t_{0},x)<\inf_{\begin{subarray}{c}t\in J\\ x\in\Omega_{t}\end{subarray}}V(t,x) \tag{6}\] _Proof._ We start by proving sufficiency. Assume that $\exists V:J\times\mathbb{R}^{n}\to\mathbb{R}$ that satisfies conditions (5-6). Moreover, by contradiction, assume that, for some initial point $x_{0}$, $\exists t^{}\in J$ such that $x(t^{})\in\partial\Omega_{t^{}}$ and $x(t)\in\Omega_{t}$$\forall t\in[t_{0},t^{})$. By the definition of infimum \[V\left(t^{},x(t^{})\right)\geq\inf_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}V(t,x)\,.\] On the other hand, eq. (5) yields \[V\left(t^{},x(t^{})\right)\leq V(t_{0},x_{0})\leq\sup_{\xi\in\Omega_{0}}V(t_ {0},\xi)\,.\] which contradicts (6). Sufficiency is proven. To prove necessity, assume that the system (1) is FTS wrt $(t_{0},T,\Omega_{0},\Omega_{t})$. We look for a continuous, locally Lipschitz function $V(t,x)$ that satisfies conditions (5-6). To this end consider the system \[\dot{x}(t)=F(t,x)\,, \tag{7}\] where $F$ is bounded in $J\times\mathbb{R}^{n}$, satisfies the global Lipschitz condition \[\|\|F(t,x)-F(t,y)\|\|\leq\lambda(t)\|\|x-y\|\|\quad\forall x,y\in\mathbb{R}^{n}\,, \tag{8}\] $\lambda(t)$ being a continuous non-negative scalar function, and finally \[F(t,x)=f(t,x)\,,\,\forall(t,x)\in J\times D\,, \tag{9}\] where $D$ is a compact set such that $D\supseteq\bar{\Omega}_{t}$ for all $t\in J$. The existence of such a $F(t,x)$ is guaranteed by Assumption 1. Moreover the main result in [33] allows to conclude that the evolution function $\varphi_{F}(t_{1},t_{2},\dot{x})$ of equation (9) is defined, unique, and continuous in $J\times J\times\mathbb{R}^{n}$. Consider \[V(t,x):=\inf_{\xi\in\Omega_{0}}\|\|\varphi_{F}(t_{0},t,x)-\xi\|\| \tag{10}\] Note that $\varphi_{F}(t_{0},t,x)$ is the initial point obtained by following the trajectory of (7) backwards from $(t,x)$ to $t_{0}$; the function $V(t,x)$ associates to $(t,x)$ the distance of the initial point of the trajectory from the initial set $\Omega_{0}$. The function $V$ is continuous and bounded on each compact subset of $\mathbb{R}^{n}$; we will now show that $V(t,x)$ is also Lipschitz in $x$. First of all note that, by the Gronwall inequality, we have that \[\|\|\varphi_{F}(t,s,x)-\varphi_{F}(t,s,y)\|\|\leq\exp\left(\int_{J}\lambda(\tau) \,d\tau\right)\|\|x-y\|\|\,. \tag{11}\] Now consider \[V(t,x)\leq\|\|\varphi_{F}(t_{0},t,x)-\xi\|\|\quad\forall\xi\in\Omega_{0}\,.\] By applying the triangular inequality we obtain \[V(t,x) \leq\|\|\varphi_{F}(t_{0},t,x)-\varphi_{F}(t_{0},t,y)\|\|+\|\|\varphi_{F} (t_{0},t,y)-\xi\|\|\] \[\forall\xi\in\Omega_{0}\,,\] and this yields \[V(t,x)\leq\] \[\|\|\varphi_{F}(t_{0},t,x)-\varphi_{F}(t_{0},t,y)\|\|+\inf_{\xi\in \Omega_{0}}\|\|\varphi_{F}(t_{0},t,y)-\xi\|\|=\] \[\|\|\varphi_{F}(t_{0},t,x)-\varphi_{F}(t_{0},t,y)\|\|+V(t,y)\] Similarly we have \[V(t,y)\leq\|\|\varphi_{F}(t_{0},t,x)-\varphi_{F}(t_{0},t,y)\|\|+V(t,x)\] and hence \[\|V(t,x)-V(t,y)\|\leq\|\|\varphi_{F}(t_{0},t,x)-\varphi_{F}(t_{0},t,y)\|\|\,. \tag{12}\] Now, considering equation (11), we obtain \[\|V(t,x)-V(t,y)\|\leq\exp\left(\int_{J}\lambda(\tau)\,d\tau\right)\|\|x-y\|\|\,. \tag{13}\] Equation (13) guarantees the existence of the orbital derivative of $V(t,x)$ as defined in (2). It is then trivial to verify that such derivative along the trajectories of system (7) is identically zero, as the initial point does not change. Now, considering equation (9) and the fact that system (1) satisfies the FTS property, we have \[V(t,\varphi(t,t_{0},x_{0}))=V(t,\varphi_{F}(t,t_{0},x_{0}))\,,\forall(t,x_{0}) \in J\times\Omega_{0}\,,\] hence, (5) holds (with the equality sign). To prove that $V(t,x)$ satisfies (6), observe that $\sup_{x\in\Omega_{0}}V(t_{0},x)=0$, since every point $x\in\Omega_{0}$ has zero distance from the set $\Omega_{0}$. Now consider a grain in generic point $x^{}\in\partial\Omega_{t^{}}$, $t^{}\in J$, and assume that $\exists x^{}_{0}$ s.t. $x^{}=\varphi(t^{},t_{0},x^{}_{0})$. Since system (1) was assumed to be FTS, the initial point $x^{}_{0}$ cannot be in the set $\Omega_{0}$. Hence, $V(t^{},x^{}(t^{}))>0$. Since this condition holds for all possible choices of $x^{}\in\partial\Omega_{t^{}}$ and for all $t^{}\in J$, and considering that $V(t,x)$ is continuous in $x$ and $\partial\Omega_{t}$ is closed and bounded, it follows that \[\inf_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}V(t,x)>0=\sup_{x\in\Omega_{0}}V(t_{0},x )\,,\] i.e. $V(t,x)$ satisfies (6). This completes the proof. Remark 1.: _When a finite-dimensional state space is considered, $\partial\Omega_{t},\Omega_{0}$ will in general be compact sets, and the $\inf$ and $\sup$ in eq. (6) can be replaced with $\min$ and $\max$ respectively._ Remark 2.: _Notice that, differently from classic Lyapunov stability, in the FTS framework it is not explicitly required that $V(t,x)$ is positive-definite._ Remark 3.: _In [8, Thm. 3.1], a less restrictive version of (5) appears, which requires that $\hat{V}(t,x)\leq g(t,V(t,x))$ for $(t,x)\in J\times\hat{\Omega}_{t}$, with $g:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^{+}$. With this choice, the orbital derivative of $V$ is allowed to take positive values, but the price for this increased freedom is that condition (6) must be complicated by resorting to the comparison lemma [34, Lemma 3.4]. However, there is no indication on how to choose the function $g(t,V(t,x))$ a priori, and even the necessity proof in [8, Thm. 3.1] eventually assumes $g\equiv 0$. For these reasons, we set $g\equiv 0$ from the beginning; this choice also simplifies the training procedure of sec. 3.4. Moreover, with respect to [8, Thm. 3.1], the formulation of them. 1 automatically takes into account the possibility that the trajectory domain $\Omega_{t}$ is time-varying. This, in turn, makes condition (0-ii) in [8, Thm. 3.1] unnecessary._ In view of the numerical implementation of the conditions in thm. 1, it is useful to state the following corollary. Corollary 1.: _System (1) is FTS wrt $(t_{0},T,\Omega_{0},\Omega_{t})$ if and only if there exists a continuous function $V:J\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that condition (5) holds and condition (6) is replaced by_ \[\inf_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}V(t,x)-\sup_{x\in\Omega_{0}}V(t_{0},x)>\alpha \tag{14}\] _for some fixed value of $\alpha>0$._ Proof.: Suppose that (6) holds with \[\inf_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}V(t,x)-\sup_{x\in\Omega_{0}}V(t_{0},x)>\varepsilon\] for some $V(t,x)$ and $\varepsilon>0$. The function $\tilde{V}(t,x):=\beta V(t,x)$, with $\beta>0$ satisfies condition (5), since $\hat{V}(t,x)=\beta\hat{V}(t,x)\leq 0$. Choosing $\beta=\alpha/\varepsilon$, it is now easy to verify that $\tilde{V}(t,x)$ satisfies condition (14) as well. The proof of this corollary is trivial, and it stems from the observation that a Lyapunov function can be rescaled by an arbitrary constant. However, the usefulness of this result is in the fact that it allows to freely choose the parameter $\alpha$, which becomes a tuning parameter for the numerical algorithm described in sec. 3.4. Moreover, it can be used to draw connections between well-assessed results on the FTS of LTV systems and thm. 1; this is done in sec. 3.3. ### Extension to discrete-time systems Consider a system in the form \[x(k+1)=f(k,x(k))\,. \tag{15}\] System (15) is said to be FTS with respect to $(\Omega_{0},\Omega_{k},N)$ if and only if, by definition, $x(0)\in\Omega_{0}\implies x(k)\in\Omega_{k}$ for $k=0,1,...,N$ (notice that there is no loss of generality in assuming that the index $k$ starts from $0$). As for the continuous time case, we assume $\Omega_{0}$ closed and $\Omega_{k}$ open. Moreover, we define the set \[\Omega_{k}^{fwd}:=[f(k,\bar{\Omega}_{k})\setminus\Omega_{k+1}]\cup\partial f( k,\bar{\Omega}_{k})\] and \[dV(k,x):=V(k+1,f(k,x))-V(k,x)\,.\] The following theorem provides necessary and sufficient conditions for the FTS of (15). Theorem 2.: _System (15) is FTS wrt $(\Omega_{0},\Omega_{k},N)$ if there exists a function $V(k,x)$ continuous in $x$, $V:\{0,...,N\}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that the following conditions hold_ \[dV(k,x)\leq 0\,,\quad x\in\Omega_{k}\quad k=0,...,N-1 \tag{16}\] \[\sup_{x\in\Omega_{0}}V(0,x)<\inf_{\begin{subarray}{c}0\leq k<N\\ x\in\Omega_{k}^{fwd}\end{subarray}}V(k+1,x)\,. \tag{17}\] _Moreover, if $f(k,x)$ is continuous with respect to $x$ and it admits an inverse $f^{-1}(k,x)$ for $k\in\{0,...,N\}$, then conditions (16-17) are also necessary._ Proof.: To prove sufficiency, assume that $\exists V:\{0,...,N\}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ that satisfies conditions (16-17). Moreover, assume that, for some initial point $x_{0}$, $\exists k^{}\in\{0,...,N-1\}$ such that $x(k^{})\in\Omega_{k^{}}$ and $x(k^{}+1)\notin\Omega_{k^{}+1}$. Since $x(k^{}+1)\in f(k,\Omega_{k})\subset f(k,\bar{\Omega}_{k})$, it follows that $x(k^{}+1)\in\Omega_{k}^{fwd}$. By definition, we have that \[V\left(k^{}+1,x(k^{}+1)\right)\leq\inf_{\begin{subarray}{c}0\leq k<N\\ x\in\Omega_{k}^{fwd}\end{subarray}}V(k+1,x)\,. \tag{18}\] On the other hand, condition (16) ensures that \[V\left(k^{}+1,x(k^{}+1)\right)\leq V(0,x_{0})\leq\sup_{x\in\Omega_{0}}V(0,x)\,,\] which contradicts condition (17). To prove necessity, arguments similar to those used in thm. 1 can be used. In particular, it can be shown that the candidate Lyapunov function \[V(k,x)=\inf_{\xi\in\Omega_{0}}\|\|\varphi(0,k,x)-\xi\|\|\] satisfies the conditions of the theorem if the considered system is FTS. However, for $V(k,x)$ to be well-defined, we need the function $f(k,x)$ to be invertible for all $k$. ### Connection with continuous-time LTV systems over ellipsoidal domains The definition of FTS can be recast in terms of ellipsoidal initial and trajectory domains by substituting equation (4) with \[x_{0}^{T}Rx_{0}\leq 1\implies x(t)^{T}\Gamma(t)x(t)<1\quad\forall t\in J \tag{19}\] where $R$ is a positive-definite, symmetric matrix and $\Gamma(t)$ is a positive-definite, symmetric, matrix-valued function. The well-posedness condition becomes $R\succ\Gamma(t_{0})$. For the FTS of LTV systems, several equivalent necessary and sufficient conditions are given in [9, Thm. 2.1]. In particular, the following theorem is proven. Theorem 3.: _[_9_, Thm. 2.1]_ _Consider the LTV system_ \[\dot{x}(t)=A(t)x(t),\quad x(t_{0})=x_{0} \tag{20}\] _with the initial domain $\Omega_{0}=\{x\|x^{T}Rx\leq 1\}$ and the trajectory domain $\Omega_{t}:=\{x\|x^{T}\Gamma(t)x<1\}$, where $\Gamma(t_{0})\prec R$. System (20) is FTS wrt $(t_{0},T,\Omega_{0},\Omega_{t})$ if and only if there exists a positive-definite matrix-valued function $P(\cdot)$ that satisfies the following DLMI conditions_ \[\dot{P}(t)+P(t)A^{T}(t)+A(t)P(t)\prec 0,t\in J \tag{21a}\] \[P(t)\succ\Gamma(t),t\in J\] (21b) \[P(t_{0})\prec R \tag{21c}\] This theorem can be shown to be a particular case of thm. 1 in sec. 3.1. Let us start by choosing $V(t,x)=x(t)^{T}P(t)x(t)$ (the validity of this choice will be discussed at the end of this section). Then, if a strict inequality is considered4, property (5) reduces to (21a), while condition (6) can be recovered by observing that (21b-21c) yield: Footnote 4: The equality case can be linked to the DLE condition (2.3a) in [9]. This point is not discussed here for brevity. \[V(t,x)>x(t)^{T}\Gamma(t)x(t) \implies \inf_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}V(t,x)>1 \tag{22a}\] \[V(t_{0},x)<x(t_{0})^{T}Rx(t_{0}) \implies \sup_{x\in\Omega_{0}}V(t_{0},x)<1 \tag{22b}\] Hence, conditions (22a-22b) imply (6). On the other hand, once we fix $V(t,x)=x^{T}P(t)x$ the inverse implication is also true. To show this, first of all observe that $V(t,x)$ can always be scaled up in such a way that \[\inf_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}x^{T}P(t)x>1>\sup_{x\in\Omega_{0}}x^{T} P(t_{0})x\,,\] without affecting the condition $\dot{V}\leq 0$ (cf. cor. 1), i.e. conditions (22a-22b) hold. The first inequality can be rewritten as \[x^{T}P(t)x>x^{T}\Gamma(t)x\qquad\forall x:x^{T}\Gamma(t)x=1\,.\] But since $\Gamma\succ 0$, for any other choice of $x\in\mathbb{R}^{n}$ there exists a scalar $c$ such that $x^{T}\Gamma(t)x=c^{2}$. This means that the point $\frac{x}{c}$ lies on $\partial\Omega_{t}$, and hence \[\frac{1}{c^{2}}x^{T}\Gamma(t)x=1<\frac{1}{c^{2}}x^{T}P(t)x\implies x^{T} \Gamma(t)x<x^{T}P(t)x\] i.e. $P(t)\succ\Gamma(t)$. With a similar argument it can be shown that $P(t_{0})\prec R$, since \[\sup_{x\in\Omega_{0}}V(t_{0},x)<1\implies\sup_{x\in\partial\Omega_{0}}V(t_{0},x)<1\,.\] To conclude this section, we discuss how the choice $V(t,x)=x^{T}(t)P(t)x(t)$ derives from the same arguments in the necessity part of the proof of thm. 1. In thm. 1 we considered a function $V(t,x)$ as in (10), i.e. the distance between the initial point of the trajectory and the set $\Omega_{0}$. We observed that, whenever $x(t)\in\partial\Omega_{t}$, such distance must be strictly positive if the system is FTS. In the case of ellipsoidal domains, this is equivalent to \[x^{T}(t)\Gamma(t)x(t)=1\implies x_{0}^{T}Rx_{0}>1\,. \tag{23}\] Using the state-transition matrix $\phi(t,t_{0})$ of (20), we have \[x^{T}(t)\Gamma(t)x(t)=x_{0}^{T}\phi(t,t_{0})^{T}\Gamma(t)\phi(t,t_{0})x_{0}(t)\,,\] and hence (23) reduces to \[\phi(t,t_{0})^{T}\Gamma(t)\phi(t,t_{0})\prec R\,.\] Moreover, since $\phi(t,t_{0})^{-1}=\phi(t_{0},t)$ \[Q(t):=\phi(t_{0},t)^{T}R\phi(t_{0},t)\succ\Gamma(t)\,, \tag{24}\] which closely resembles (21b). Since $\phi(t_{0},t_{0})=I$, we immediately find that \[Q(t_{0})=R\] (compare these conditions with [9, Thm. 2.1 (ii-iii)]). Finally, by continuity we can choose an arbitrarily small $\varepsilon>0$ such that conditions (21b)-(21c) hold for \[P(t):=Q(t)(1-\varepsilon)\,.\] Observe that, at its core, this choice of $Q(t)$ again reduces to following the trajectories of (20) backwards and evaluating the distance of the initial point $(t_{0},x_{0})$ from the initial set through the quadratic forms associated to $Q(t)$ and $R$. ### Lyapunov function approximation through Physics-Informed Neural Networks In this section, we propose a method to train a NN so that its output provides a Lyapunov function for continuous time systems which are FTS. Consider the following cost functional \[L(\dot{V},V,\delta_{2})=L_{1}(\dot{V})+L_{2}(V,\delta_{2})\,, \tag{25}\] where \[L_{1}=\frac{1}{T}\int_{J}\frac{1}{\|\Omega_{t}\|}\int_{\Omega_{t}}\Big{(}\max\{ \dot{V}(\tau,x),0\}\Big{)}^{2}\,dxd\tau \tag{26a}\] \[L_{2}=\frac{1}{T}\int_{J}\frac{1}{\|\partial\Omega_{t}\|}\int_{\partial\Omega_{t}}( \max\{\delta V_{\dot{b}}(\tau,x)+\delta_{2},0\})^{2}\,dxd\tau\,. \tag{26b}\] In (26b), the parameter $\delta_{2}>0$ is introduced to enforce the strict inequality in (6) (see discussion below). The term $\delta V_{b}$ appearing in (26b) is defined as \[\delta V_{b}(t,x)=\sup_{\xi\in\Omega_{0}}\{V(t_{0},\xi)\}-V(t,x)\,,\] and is evaluated on $\partial\Omega_{t}$. The following result shows that the problem of finding a Lyapunov function for a system in the form (1) is equivalent to finding a $V(t,x)$ such that $L(\dot{V},V,\delta_{2})=0$ for some (arbitrary) value of $\delta_{2}$. Proposition 1.: _A continuous function $V(t,x)$ satisfies the conditions of thm. 1 if and only if $L(\dot{V},V,\delta_{2})=0$ for some value of $\delta_{2}>0$._ _Proof._ If condition (5) holds, then $\dot{V}(t,x)\leq 0$ for every $t\in J$ and for every $x\in\Omega_{t}$. Hence, $\max\{\dot{V}(\tau,x),0\}=0$ and $L_{1}(\dot{V})=0$. On the other and, since $L_{1}(V)$ is defined as the integral of a non-negative quantity, $L_{1}(V)=0$ implies that its argument is equal to zero for all $t\in J$ and $x\in\Omega_{t}$, i.e. that $V\leq 0$. Similarly, $L_{2}(V,\delta_{2})=0$ if and only if $\delta V_{b}(t,x,\delta_{2})\leq 0$ for every $t\in J$ and for every $x\in\partial\Omega_{t}$. But this is equivalent to requiring that \[\begin{split} 0&\geq\sup_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}\delta V_{b}(t,x)+\delta_{2}\\ &=\sup_{\begin{subarray}{c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}\left\{\sup_{\xi\in\Omega_{t}}\{V(t_{0}, \xi)\}-V(t,x)\right\}+\delta_{2}\\ &=\sup_{\xi\in\Omega_{0}}\{V(t_{0},\xi)\}-\inf_{\begin{subarray}{ c}t\in J\\ x\in\partial\Omega_{t}\end{subarray}}\{V(t,x)\}+\delta_{2}\,.\end{split} \tag{27}\] In view of corollary 1, if this condition is satisfied for some value of $\delta_{2}>0$, then also condition (6) holds. The proof is completed by observing that $L\;=\;0$ is equivalent to $L_{1}=0\;\wedge\;L_{2}\;=\;0$, since $L_{1,2}\geq 0$ by definition. In practice, we restrict our attention to functions $V(t,x)$ of class $C^{1}$ by choosing continuously differentiable activation functions. The universal approximation property of NNs ensures that, even if $C^{1}$ activation functions are chosen, any continuous function can be approximated with arbitrary precision by the network, provided that a sufficiently large number of nodes is employed. It is worth to notice that it is usually required that the function approximated by the NN is evaluated on compact subsets of $\mathbb{R}^{n}$, which is precisely the case in FTS, where the conditions on $V(t,x)$ must be satisfied only on the (compact) domains of interest. For $V(t,x)\in C^{1}$, the orbital derivative (2) can be evaluated as \[\dot{V}(t,x)=\frac{\partial V}{\partial t}+\mathcal{L}_{f}V=\frac{\partial V}{ \partial t}+\frac{\partial V}{\partial x}f(t,x)\,, \tag{28}\] where $\mathcal{L}_{f}V$ denotes the Lie derivative of $V(t,x)$ along $f(t,x)$. The partial derivatives of $V$ with respect to $x$ and $t$ can be computed analytically via automatic differentiation [35]. To train the network, we sample the cost function $L(\dot{V},V)$ in a (large) set of collocation points. Let us define \[\hat{L}_{1}(\dot{V},\delta_{1})=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}\left(\max\{ \dot{V}(x_{i},t_{i})+\delta_{1},0\}\right)^{2},, \tag{29}\] where $N_{c}$ is the total number of considered collocation points in $t\in J,x\in\Omega_{t}$, and \[\begin{split}&\hat{L}_{2}(V,\delta_{2})=\frac{1}{N_{b}}\sum_{i=1}^ {N_{b}}\left(\max\left\{\delta\tilde{V}_{b}(t_{i},x_{i})+\delta_{2},0\right\} \right)^{2}\\ &\delta\tilde{V}_{b}(t_{i},x_{i})=\max_{j\in[1,N_{0}]}\{V(t_{0}, \xi_{j})\}-V(t_{i},x_{i})\,,\end{split} \tag{30}\] where $N_{b}$ and $N_{0}$ are the numbers of points taken in $t\in J,x\in\partial\Omega_{t}$ and in $\Omega_{0}$ respectively. The $N_{c}$ internal collocation points are obtained by random sampling in both time and space, while the $N_{b}$ boundary points are computed by discretizing the time interval in $N_{t}$ discrete instants and then taking $n_{b}=N_{b}/N_{t}$ points for each time instant. The $N_{0}$ points are uniformly distributed in the initial domain $\Omega_{0}$. In eq. (29) we introduced a tolerance parameter $\delta_{1}$ to penalize the points where $\dot{V}$ in $\hat{L}_{1}$ is exactly equal to or slightly smaller than zero, similarly to the $\delta_{2}$ parameter in eqns. (26b)-(30). Moreover, in view of remark 1, we used the maximum in place of the supremum in the definition of $\hat{L}_{2}$. The resulting (approximate) loss function is \[\hat{L}(\dot{V},V,\delta_{1},\delta_{2})=\alpha_{1}\hat{L}_{1}(\dot{V},\delta_{ 1})+\alpha_{2}\hat{L}_{2}(V,\delta_{2})\,, \tag{31}\] where the weights $\alpha_{1,2}>0$ have been introduced as additional tuning parameters in the algorithm. The network is trained by using the matlab implementation of the standard Adam algorithm [30]. The default parameters have been used, i.e. a global learning rate of $0.001$, a gradient decay factor of $0.9$ and a squared gradient decay factor of $0.999$. This choice proved to be enough to obtain good convergence properties in the considered examples. To compute $\hat{L}_{1}$, the $N_{c}$ internal collocation points are divided into mini-batches, while all the $N_{b}+N_{0}$ points on $\partial\Omega_{t}$ and in the initial domain $\Omega_{0}$ are used at each training iteration to compute $\hat{L}_{2}$. This choice reflects the fact that (6) only requires that the infimum of $V(t,x)$ over all the boundary points is larger than the supremum of $V(t_{0},x)$ over the initial domain. However, in the definition of $\hat{L}_{2}$ we considered all the collocation points on $\partial\Omega_{t}$ to obtain a well-behaved estimate of the cost function gradients with respect to the network parameters. The training procedure terminates when $\dot{V}(t,x)\leq 0$ over all the $N_{c}$ internal collocation points and $\delta\tilde{V}_{b}(t,x)<0$ over all the $N_{b}$ boundary collocation points. Finally, since we are resorting to an approximate version of the cost function $L(\dot{V},V,\delta_{2})$, after a solution is found we verify that the theorem conditions are satisfied over a set of test points different from the collocation points used in the training procedure. Before concluding this section, it is worth to observe that the numerical implementation of the conditions of theorem 1 makes them only necessary in practice. If the considered system is not FTS, then the procedure will usually not converge to a feasible solution; however, it may happen that the algorithm does not converge even if the system is FTS, for instance when the number of nodes in the NN is not large enough to correctly represent the desired Lyapunov function. ## 4 Numerical examples In this section we illustrate the proposed approach through some numerical examples. It is worth to observe that the conditions in thm. 1 are based on the orbital derivative of the function $V(t,x)$, which can be evaluated by means of automatic differentiation techniques when $V$ is expressed through a neural network. On the other hand, the conditions in thm. 2 are based on recursive equations, and require the computation of the set $\Omega_{t}^{fwd}$ at each time step. In the following, we will exploit the availability of automatic differentiation routines implemented in Matlab, and only focus on continuous-time systems. Note that, for simplicity, in the proposed examples ellipsoidal domains are considered. However, the algorithms only uses the collocation points placed in such domains and on their boundary, hence the technique can be easily applied to domains of generic shape. The proposed procedure has been applied to the considered examples as shown in fig. 2. For all the examples, we used a Multi-Layer Perceptron architecture with _softplus_ activation layers (the parameters for each example are reported in table 2). After an initial choice of the collocation point and of the NN and training parameters, the optimization procedure is run. If a solution is found, the required conditions are further tested on a suitable set of test points (e.g. a finely spaced grid over the considered domain). If the test succeeds, the property assessment can be considered to be successful. If the test fails, the training set of collocation points can be enlarged, e.g. by including the points where the test fails, and the procedure can be run again. If the algorithm fails to converge, the NN and training parameters can be modified (e.g. increasing the number of nodes/layers of the NN to make it more expressive, or lowering the learning rates to improve stability) before running the procedure again. If no solution is found with a reasonable set of parameters and/or in a reasonable time, the test is considered to have failed (red box in fig. 2). This could be due to the fact that the system does not satisfy the FTS property or to numerical reasons. It is worth to notice that this characteristic is common to most Lyapunov techniques, which often do not provide a straightforward way to verify that the system is unstable (in the classic or Finite-Time sense). All the examples in this section have been implemented in matlab and run on a laptop equipped with an Intel Core i7-8750H CPU @2.20GHz processor, 16 GB of RAM and an NVIDIA Geforce GTX 1050 Ti graphic card. The code is available at [https://github.com/AdrianoMele/NeuralNettTS](https://github.com/AdrianoMele/NeuralNettTS). ### Example 1 We start by validating the procedure on a toy example, i.e. a 2D LTI system with fixed ellipsoidal domains. In particular, consider the system \[\dot{x}(t)=-\begin{bmatrix}0.1&0\\ 0&0.1\end{bmatrix}x(t)\,, \tag{32}\] with $t_{0}=0$, $T=1$ and the domains defined by \[\begin{split}\Omega_{0}&=\{x\in\mathbb{R}^{2}\|x^{T}Rx\leq 1 \},\,R=0.3I\\ \Omega_{t}&=\{x\in\mathbb{R}^{2}\|x^{T}\text{T}x\leq 1\},\,\Gamma=0.25I \,.\end{split} \tag{33}\] Classical FTS results ensure that the solution for this example can be found in terms of a time-invariant quadratic Lyapunov function $V_{Q}(x)=x^{T}(t)Px(t)$ (also see the discussion in section 3.3). In fact, the system is also Lyapunov-stable, and it can be shown that, in this case, the FTS conditions in thm. 3 can be reduced to the problem of finding a static Lyapunov function that also satisfies conditions (21b-21c). We apply the technique proposed in section 3.4 to the considered system and domains; we expect the algorithm to find a solution $V(x)$ that is very close to a quadratic function. The training parameters used in this example are reported in tables 1-2. The algorithm is able to find a solution after only 2 epochs (about 2 s of training). The resulting $V(x)$ is shown in fig. 3a, while the collocation points and the initial and trajectory domains are shown in fig. 3b. A quadratic best-fit of $V(x)$ (red dots in fig. 3a) results in the symmetric positive-definite matrix \[P=\begin{bmatrix}0.3064&-0.0063\\ -0.0063&0.3120\end{bmatrix}\] Figure 2: Flow diagram describing how the proposed technique has been applied to the examples of sec. 4. The eigenvalues of $PA^{T}+AP$ are $[-0.06322,-0.06044]$, i.e. the resulting $P$ satisfies condition (21a). On the other hand, the eigenvalues of $P$ are $[0.3022,\,0.3161]$, i.e. condition (21c) does not hold. As discussed in sec. 3.3, a function $V_{Q}(x)$ that satisfies (21b-21c) can be obtained by a suitable scaling of $P$. ### Example 2 Consider the following system [36, (1.1.5)] \[\begin{split}\dot{x_{1}}(t)&=-x_{1}-x_{2}+k(x_{1} -x_{2})(x_{1}^{2}+x_{2}^{2})\\ \dot{x_{2}}(t)&=x_{1}-x_{2}+k(x_{1}+x_{2})(x_{1}^{2} +x_{2}^{2})\end{split} \tag{34}\] where the constant $k$ is chosen equal to $0.1$. This system provides an interesting analytical benchmark for the considered numerical procedure. In fact, in [36] it is shown that, letting $r^{2}(t)=x_{1}^{2}(t)+x_{2}^{2}(t)$, the solutions of (34) have the property \[r^{2}(t)=\frac{1}{k\mu}r_{0}^{2}\,,\quad\mu=r_{0}^{2}+\left(\frac{1}{k}-r_{0 }^{2}\right)e^{2(t-t_{0})} \tag{35}\] We consider the case where $t_{0}=0$, $T=1$ and \[\begin{split}\Omega_{0}&=\{x\in\mathbb{R}^{2}\|x^{ T}Rx\leq 1\},\,R=I\\ \Omega_{t}&=\{x\in\mathbb{R}^{2}\|x^{T}\Gamma(t)x<1 \},\,\Gamma(t)=0.8k\mu(t)I\,.\end{split} \tag{36}\] with $r_{0}=1$. From (35) it can be seen that system (34) is FTS with respect to $(t_{0},T,\Omega_{0},\Omega_{t})$. Some sample trajectories of system (34) are shown in fig. (a)a together with the initial and trajectory domains. With the parameters reported in tables 1-2, a solution is found after 9 training epochs (about 60 s training). The resulting $V(t,x)$ is shown at different time instants in fig. 7, while the resulting training curve (epochs vs. loss) for the considered parameters is shown in fig. 6. In order to provide some insight on how the choice of the collocation points affects the algorithm performance, we performed a scan on the parameters $N_{c},n_{b},N_{t},N_{0}$ of table 1 for this example. In particular, we varied the number of internal collocation points ($N_{c}=\{5,10,30,50,55,70\}\times 10^{3}$) while keeping the other parameters fixed ($N_{0}=700,n_{b}=200$ and a time step equal to $dt=10^{-1}$, i.e. $N_{t}=11$; see table 1). Then, we set $N_{c}=5\times 10^{4}$ and varied the number of collocation points in the initial domain ($N_{0}=100$), the number of collocation points on the trajectory domain boundary for each time step ($n_{b}=\{50,1000\}$) and the number of time instants $N_{t}$ (by reducing the time step to $dt=10^{-2}s$ while keeping $n_{b}=200$). The results of this analysis are reported in fig. 4. The algorithm converged in all the considered cases, but for the ones represented by dashed lines (corresponding to cases with very few internal or boundary collocation points) a false positive was obtained, i.e. the resulting neural Lyapunov function violated the required conditions on some points of the test set. The test set for this analysis was generated by considering a different set of internal collocation points, obtained by restricting a $50\times 50$ equally spaced grid to the initial and trajectory domains, while keeping the same boundary points. The time needed for the algorithm to reach convergence is also reported for completeness; however, notice that no effort has been made to optimize the code performance. Moreover, it is also worth to remark that a certain amount of randomness is implicitly involved in the procedure (in the initialization of the network parameters and in the choice of the collocation points) and that the used platform is not Real-Time. The case reported in fig. 6 corresponds to the red line in 4. ### Example 3 Finally, consider the equations of a damped pendulum \[\begin{split}\dot{x}_{1}&=x_{2}\\ \dot{x}_{2}&=-\frac{\partial}{l}\sin(x_{1})-\frac{b }{ml^{2}}x_{2}\,,\end{split} \tag{37}\] which is a widely used benchmark problem in the field of systems and control theory. We set $g=9.81$ m/s${}^{2}$, $m=0.15$ kg, $l=0.15$ m, $b=0.1$ kg m${}^{2}$/s. We consider the case where $t_{0}=0$, $T=2$ s, and \[\begin{split}\Omega_{0}&=\{x\in\mathbb{R}^{2}\|x^{T}Rx \leq 1\},\,R=\frac{4}{\pi^{2}I}\\ \Omega_{t}&=\{x\in\mathbb{R}^{2}\|x^{T}\Gamma(t)x<1 \},\,\Gamma(t)=\Theta\tilde{\Gamma}\Theta^{T}\,,\end{split} \tag{38}\] where \[\tilde{\Gamma}=\left[\frac{2}{\pi^{2}}e^{0.5t}\begin{array}{c}0\\ 0\end{array}\right] \tag{39}\] and $\Theta$ is the rotation matrix given by \[\Theta=\left[\begin{matrix}\cos(0.2\pi t)&\sin(0.2\pi t)\\ -\sin(0.2\pi t)&\cos(0.2\pi t)\end{matrix}\,.\right] \tag{40}\] With the parameters reported in tables 1-2, a solution is found after 10 training epochs (approximately 2 m 44 s). Figure 3: (a) Resulting $V(t,x)$ and $\dot{V}(t,x)$ for example #1; the red dots show the quadratic approximant (a constant has been added to $V(x)$ so that $V(0)=0$ for graphical reasons). (b) Collocation points for example #1. The green dots show the $N_{0}$ points in the initial domain, the red dots are the $N_{c}$ internal points used to enforce (5) and the blue dots are the $N_{b}=n_{b}N_{t}$ boundary points used to enforce (6). Figure 4: Training curve for different choices of the parameters reported in table 1 for the case of example #2 (top) and time to converge for each choice (bottom). Figure 5: Resulting Lyapunov function $V(t,x)$ and its derivative $V(t,x)$ for example #2 at different time instants (a constant has been added so that the resulting $V(t,x)$ is above the $x_{1}-x_{2}$ plane for graphical reasons). Figure 6: Training curve for example #2. Figure 7: Resulting Lyapunov function $V(t,x)$ and its derivative $\dot{V}(t,x)$ for example #3 at different time instants. Figure 8: Training curve for example #3. Figure 9: Sample trajectories for the systems of examples #2 and #3 (red), together with the considered initial (green) and trajectory (blue) domains. FTS property. The main idea is to exploit the capability of NN to serve as universal functions approximators in order to find a closed-form expression for the required Lyapunov-like function. The effectiveness of the approach has been illustrated through different numerical examples in the case of continuous-time systems. To the best of our knowledge, this is the first time that a numerical technique allows to practically verify this property for such a large class of systems. As discussed in the literature review of the introduction, the practical advantages of the FTS notion have been known since long time ago. However, after some pioneering works, the research community that had gathered around this topic remained almost silent until practical (D)LMI-based conditions to assess FTS emerged, at least in the case of linear systems, renewing the research interest in the field. Over the years, such conditions have been extended to uncertain, hybrid, stochastic, time-delay linear systems and to special classes of nonlinear systems. In this article, we proposed an alternative way to assess the FTS property for the case of general nonlinear systems. In this perspective, we expect that the method discussed here can be further refined and extended to more challenging classes of dynamical systems as happened for the linear case. Of course, the approximate nature of the numerical method employed to verify the conditions of thm. 1 leads to some limitations. First of all, since a discrete set of collocation points is considered, the proposed numerical test is in fact only necessary: the required conditions on the $V(t,x)$ function may be violated outside the training set of collocation points, but inside the domain of interest. However, this issue is quite standard when neural Lyapunov methods are considered. One possibility to mitigate this issue is to test the conditions on a set of points that is different (possibly also larger) with respect to the considered collocation points; this is the approach adopted in this paper. Furthermore, emerging research lines exist that focus on ways to assess the safety of similar methods in the framework of classic Lyapunov stability (see for example [24]). Another limitation of the proposed method is that there is no way of directly assessing that a system is Finite-Time _unstable_. When this is the case, the training procedure will usually fail, as it cannot find a Lyapunov-like function that satisfies the required conditions. A possible future line of research could be that of extending classical Lyapunov instability theorems to the field of Finite-Time stability analysis. Finally, the availability of a Lyapunov function in closed-form, whose derivatives with respect to time and state variables can be simply computed through auto-differentiation techniques, paves the way to novel Finite-Time Stabilization and Control techniques based on Lyapunov methods.
2302.09712	Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization	Despite remarkable performance on a variety of tasks, many properties of deep neural networks are not yet theoretically understood. One such mystery is the depth degeneracy phenomenon: the deeper you make your network, the closer your network is to a constant function on initialization. In this paper, we examine the evolution of the angle between two inputs to a ReLU neural network as a function of the number of layers. By using combinatorial expansions, we find precise formulas for how fast this angle goes to zero as depth increases. These formulas capture microscopic fluctuations that are not visible in the popular framework of infinite width limits, and leads to qualitatively different predictions. We validate our theoretical results with Monte Carlo experiments and show that our results accurately approximate finite network behaviour. \review{We also empirically investigate how the depth degeneracy phenomenon can negatively impact training of real networks.} The formulas are given in terms of the mixed moments of correlated Gaussians passed through the ReLU function. We also find a surprising combinatorial connection between these mixed moments and the Bessel numbers that allows us to explicitly evaluate these moments.	Cameron Jakub, Mihai Nica	2023-02-20T01:30:27Z	http://arxiv.org/abs/2302.09712v3	Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization ###### Abstract Stacking many layers to create truly deep neural networks is arguably what has led to the recent explosion of these methods. However, many properties of deep neural networks are not yet understood. One such mystery is the depth degeneracy phenomenon: the deeper you make your network, the closer your network is to a constant function on initialization. In this paper, we examine the evolution of the angle between two inputs to a ReLU neural network as a function of the number of layers. By using combinatorial expansions, we find precise formulas for how fast this angle goes to zero as depth increases. The formulas are given in terms of the mixed moments of correlated Gaussians passed through the ReLU function. We also find a surprising combinatorial connection between these mixed moments and the Bessel numbers. Keywords: infinite limits of deep neural networks ## 1 Introduction The idea of stacking many layers to make truly _deep_ neural networks (DNNs) is what arguably led to the neural net revolution in the 2010s. Indeed, from a function-space point of view, it is known that depth exponentially improves expressibility (Poole et al., 2016; Eldan and Shamir, 2015). However, an important but less well known fact is that under standard initialization schemes, deep neural networks become more and more _degenerate_ as depth gets larger and larger. One sense in which this happens is the phenomenon of vanishing and exploding gradients (Hanin, 2018). Another sense in which networks become degenerate is that a neural network gets closer and closer to a (random) constant function i.e. the network sends all inputs to the same output and cannot distinguish input points. This phenomenon seems to have been independently rediscovered from different points of view by several authors (Avelin and Karlsson, 2022; Dherin et al., 2022; Li et al., 2022; Hayou et al., 2019; Schoenholz et al., 2016). One method already proposed to deal with the degeneracy phenomenon is the idea of activation function _shaping_. In particular, Li et al. (2022), showed that by rescaling the non-linear activation function (i.e. using Leaky ReLUs with leakiness depending on network depth) can lead to a non-trivial angle between inputs. This result essentially by rescaling the non-linear activation function used in the network in such a way that the Markov chain for the evolution of the correlation between two inputs rescales to a non-trivial SDE. However, a detailed analysis of the evolution of the angle $\theta$_without_ any scaling (e.g. using an ordinary ReLU in all layers) remained an outstanding problem. This is the gap we fill in this article. ### Main Results for the Angle Process $\theta^{\ell}$ In this paper we examine the evolution of the _angle_$\theta^{\ell}$ between two arbitrary inputs $x_{\alpha},x_{\beta}\in\mathbb{R}^{n_{in}}$ after passing through $\ell$ layers of a fully connected ReLU network (a.k.a. a multi-layer perceptron) on initialization. The angle is defined in the usual way by the inner product between two vectors in $\mathbb{R}^{n_{\ell}}$. \[\cos\left(\theta^{\ell}\right):=\frac{\langle F^{\ell}(x_{\alpha}),F^{\ell}(x _{\beta})\rangle}{\\|F^{\ell}(x_{\alpha})\\|\\|F^{\ell}(x_{\beta})\\|},\] where $n_{\ell}$ is the width (i.e. number of neurons) of the $\ell$-th layer and $F^{\ell}:\mathbb{R}^{n_{in}}\to\mathbb{R}^{n_{\ell}}$ is the (random) neural network function mapping input to the post-activation logits in layer $\ell$ on initialization. We assume here that the initialization is done with appropriately scaled independent Gaussian weights so that the network is on the "edge of chaos" (Hayou et al., 2019; Schoenholz et al., 2016), where the variance of each layer is order 1. (See Table 2 for our precise definition of the fully connected ReLU neural network). With this setup, since the effect of each layer is independent of everything previous, $\theta^{\ell}$ can be thought of as a Markov chain evolving as layer number $\ell$ increases. As expected by the aforementioned "large depth degeneracy" phenomenon, we observe that the angle concentrates $\theta^{\ell}\to 0$ as $\ell\to\infty$ (see Figure 1 for an illustration). This indicates that the hidden layer representation of _any_ two inputs in a deep neural network become closer and closer to co-linear as depth increases. In this paper, we obtain a simple, yet remarkably accurate, approximation for the evolution of $\theta^{\ell}$ as a function of $\ell$ that captures precisely how quickly this degeneracy happens for small angles $\theta^{\ell}$ and large layer widths $n_{\ell}$. Approximation 1: _For small angles $\theta^{\ell}\ll 1$ and large layer width $n_{\ell}\gg 1$, the angle $\theta^{\ell+1}$ at layer $\ell+1$ is well approximated by_ \[\ln\sin^{2}(\theta^{\ell+1})\approx\ln\sin^{2}(\theta^{\ell})-\frac{2}{3\pi} \theta^{\ell}-\rho(n_{\ell}), \tag{1}\] _where $\rho(n_{\ell})$ is a constant which depends on the width $n_{\ell}$ of layer $\ell$, namely:_ \[\rho(n):=\ln\left(\frac{n+5}{n-1}\right)-\frac{10n}{\left(n+5\right)^{2}}+ \frac{6n}{\left(n-1\right)^{2}}=\frac{2}{n}+\mathcal{O}\left(n^{-2}\right). \tag{2}\] Figure 1 illustrates how well this prediction matches Monte-Carlo simulations of $\theta^{\ell}$ sampled from real networks. Also illustrated is the _infinite width_ prediction for $\theta^{\ell}$ (discussed in Appendix A.9) which is less accurate at predicting finite width network behaviour than our formula, due to the $n_{\ell}^{-1}$ effects that our formula captures in the term $\rho(n_{\ell})$ but are not present in the infinite width formula. Approximation 1 is a simple corollary to the mathematically rigorous statement, Theorem 1, for the mean and variance of the random variable $\ln\sin^{2}(\theta^{\ell})$. #### 1.1.1 Consequences and Comparison to Previous Work The simple Approximation 1 predicts that $\theta^{\ell}\to 0$_exponentially fast_ in $\ell$ due to the term $\rho(n)$; it predicts \[\theta^{\ell}\leq\exp\left(-\sum_{i=1}^{\ell}\rho(n_{i})\right)=\exp\left(-\sum_ {i=1}^{\ell}\frac{2}{n_{i}}+\mathcal{O}(n_{i}^{-2})\right).\] (Note that the exponential behaviour vanishes when $n_{\ell}\to\infty$ with $\ell$ fixed). In contrast to this prediction, an analysis using only expected values or equivalently working in the infinite-width $n\to\infty$ limit predicts that $\theta^{\ell}\to 0$ like $\ell^{-1}$, which is qualitatively very different! The prediction of this rate was first demonstrated under the name "edge of chaos" (Hayou et al., 2019; Schoenholz et al., 2016) and again in Roberts et al. (2022); Hanin (2022). These earlier works working with the correlations showed that $1-\cos(\theta_{\ell})\to 0$ like $\ell^{-2}$, which is equivalent to $\theta_{\ell}\to 0$ like $\ell^{-1}$ by Taylor series expansion. We can recover this infinite width prediction by replacing $\rho(n)$ with $0$ in our update rule (1). Exponentiating both sides and using $\sin(\theta)\approx\theta,e^{\theta}\approx 1+\theta$ for $\theta\ll 1$, Approximation 1 becomes $(\theta^{\ell+1})^{2}\approx\left(\theta^{\ell}\right)^{2}(1-\frac{2}{3\pi} \theta^{\ell})$, which is equivalent to the result of Proposition C.1 of Figure 1: In the above figure, we feed 2 inputs with initial angle $\theta^{0}=0.3$ and plot the value of $\ln(\sin^{2}\theta^{\ell})$ at layer $\ell$ of the network on initialization. The experiment is repeated 1000 times with independent initialization and a network width of $n_{\ell}=256$; the 99% confidence interval for the Monte Carlo estimated mean $\hat{\mathbf{E}}[\ln(\sin^{2}\theta^{\ell})]$ is displayed. On top of this, we plot our simple prediction method as in (1), as well as the infinite width prediction as in (12). Hanin (2022). In that paper, the rule was derived directly from the infinite width update rule for $\cos(\theta)$, 1 where they also show that this leads to $\theta^{\ell}\approx\frac{1}{\ell}$ as $\ell\to\infty$. Footnote 1: Note that the recursion in Proposition C.1 is written in terms of the variable $\epsilon_{\alpha\beta}^{\ell}$ instead of $\theta$. The connection to our work is $\epsilon_{\alpha\beta}^{\ell}=\frac{1-\cos(\theta^{\ell})}{2}\approx(\theta^{ \ell})^{2}/4$ as $\theta\to 0$) #### 1.1.2 More detailed results for the mean and variance Approximation 1 comes from a simplification of more precise formulas for the mean and variance of the random variable $\ln(\sin^{2}(\theta^{\ell}))$, which are stated in Theorem 1 below. Theorem 1 (Formula for mean and variance in terms of J functions): _Conditionally on the angle $\theta^{\ell}$ in layer $\ell$, the mean and variance of $\ln\sin^{2}(\theta^{\ell+1})$ obey the following limit as the layer width $n_{\ell}\to\infty$_ \[\mathbf{E}[\ln\sin^{2}(\theta^{\ell+1})]= \mu(\theta^{\ell},n_{\ell})+\mathcal{O}(n_{\ell}^{-2}),\quad \mathbf{Var}[\ln\sin^{2}(\theta^{\ell+1})]=\sigma^{2}(\theta^{\ell},n_{\ell}) +\mathcal{O}(n_{\ell}^{-2}) \tag{3}\] \[\mu(\theta,n):= \ln\left(\frac{(n-1)(1-4J_{1,1}^{2})}{4J_{2,2}-1+n}\right)+ \frac{4(J_{2,2}+1)}{n\left(\frac{4J_{2,2}-1}{n}+1\right)^{2}}\] (4) \[-\frac{4\left(8J_{1,1}^{2}J_{2,2}-8J_{1,1}^{4}+4J_{1,1}^{2}-8J_{1,1}J_{3,1}+J_{2,2}+1\right)}{n\left(1-\frac{1}{n}\right)^{2}\left(1-4J_{1,1}^{ 2}\right)^{2}}\] \[\sigma^{2}(\theta,n):= \frac{8n(J_{2,2}+1)}{(4J_{2,2}-1+n)^{2}}+\frac{8n(8J_{1,1}^{2}J_{ 2,2}-8J_{1,1}^{4}+4J_{1,1}^{2}-8J_{1,1}J_{3,1}+J_{2,2}+1}{(n-1)^{2}(1-4J_{1,1}^ {2})^{2}}\] (5) \[-\frac{16n(2J_{1,1}^{2}-4J_{1,1}J_{3,1}+J_{2,2}+1)}{(4J_{2,2}-1+n )(n-1)(1-4J_{1,1}^{2})},\] _where $\mathbf{E},\mathbf{Var}$ denote the conditional mean and variance of quantities in layer $\ell+1$ given the value of $\theta^{\ell}$ in the previous layer and $J_{a,b}:=J_{a,b}(\theta^{\ell})$ are the joint moments of correlated Gaussians passed through the ReLU function $\varphi(x)=\max\{x,0\}$, namely_ \[J_{a,b}(\theta):=\mathbf{E}_{G,\hat{G}}[\varphi^{a}(G)\varphi^{b}(\hat{G})], \tag{6}\] _where $G,\hat{G}$ are marginally $\mathcal{N}(0,1)$ random variables with correlation $\mathbf{E}[G\hat{G}]=\cos(\theta)$._ The joint moments $J_{a,b}(\theta)$ are discussed in detail in Section 3. A new combinatorial method of computing these moments is presented, which is used to give an explicit formula is given for these joint-moments, which is presented in Theorem 4. Using the explicit formula for $J_{a,b}$, the result of Theorem 1 can be used to obtain useful asymptotic formula for $\mu$ and $\sigma$, as in the following corollary. Corollary 2 (Small $\theta$ asymptotics for mean and variance): _Conditionally on the angle $\theta^{\ell}$ in layer $\ell$, the mean and variance of $\ln\sin^{2}(\theta^{\ell+1})$ obey the following limit as the layer _width $n_{\ell}\to\infty$_ \[\mathbf{E}[\ln\sin^{2}(\theta^{\ell+1})]= \mu(\theta^{\ell},n_{\ell})+\mathcal{O}(n_{\ell}^{-2}),\quad \mathbf{Var}[\ln\sin^{2}(\theta^{\ell+1})]=\sigma^{2}(\theta^{\ell},n_{\ell})+ \mathcal{O}(n_{\ell}^{-2}) \tag{7}\] \[\mu(\theta,n)= \ln\sin^{2}\theta-\frac{2}{3\pi}\theta-\rho(n)-\frac{8\theta}{15 \pi n}-\left(\frac{2}{9\pi^{2}}-\frac{68}{45\pi^{2}n}\right)\theta^{2}+ \mathcal{O}(\theta^{3})\] (8) \[\sigma^{2}(\theta,n)= \frac{8}{n}-\frac{64}{15\pi}\frac{\theta}{n}-\left(8+\frac{296}{4 5\pi}\right)\frac{\theta^{2}}{n}+\mathcal{O}\left(\theta^{3}\right), \tag{9}\] _where $\rho(n)$ is as defined in (2)._ To derive Approximation 1 from Theorem 1, we simply keep only the first few terms of the series expansion (8), and then also completely drop the variability, essentially approximating $\sigma^{2}(\theta^{\ell},n)\approx 0$ (Note that in reality $\sigma^{2}(\theta,n)\approx 8/n$ from (9)). Therefore Approximation 1 is a greatly simplified consequence of Theorem 1. Moreover, our derivation shows that $\ln\sin^{2}(\theta^{\ell})$ can be expressed in terms of averages over $n$ pairs of independent Gaussian variables (see (14-16)). Thus, by central-limit-theorem type arguments, one would expect the following approximation by Gaussian laws which also accounts for the variability of $\ln\sin^{2}(\theta)$ using our calculated value for the variance. Approximation 2: _Conditional on the value of $\theta^{\ell}$, the angle at layer $\ell+1$ is well approximated by a Gaussian random variable_ \[\ln\sin^{2}(\theta^{\ell+1})\stackrel{{ d}}{{\approx}}\mathcal{N }(\mu(\theta^{\ell},n_{\ell}),\sigma^{2}(\theta^{\ell},n_{\ell})), \tag{10}\] _where $\mu,\sigma^{2}$ are as in Theorem 2._ Figure 2: Plots comparing the functions $\mu(\theta,n)$ and $\sigma^{2}(\theta,n)$ to simulated neural networks. The linear approximation of $\mu$, used to create Approximation 1 is also displayed. Confidence bands are constructed by randomly initializing 10000 neural networks with layer width $n_{\ell}=1024$, and a range of 100 initial angles $0.005\leq\theta^{\ell}\leq 0.8$. We study $\theta^{\ell+1}$ and use the simulations to construct 99% confidence intervals for a) $\mathbf{E}\left[\ln(\sin^{2}(\theta^{\ell}))-\ln(\sin^{2}(\theta^{\ell+1}))\right]$ and b) $\mathbf{Var}\left[\ln(\sin^{2}(\theta^{\ell+1}))\right]$. We find that the normal approximation (10) matches simulated finite neural networks remarkably well; see Monte-Carlo simulations from real networks in Figure 3. The methods we use to obtain these approximations are quite flexible. For example, more accurate approximations can be obtained by incorporating higher moments $J_{a,b}(\theta)$ (see Section 2 for a discussion). We also believe that it should be possible to extend these methods to other non-linearities beyond ReLU and more complicated neural network architectures through the same basic principles we introduce here. ### J functions and infinite width limits In 2009, Cho and Saul (Cho and Saul (2009)) introduced the $p$-th moment for correlated ReLU-Gaussians, which they denoted with the letter $J$, \[J_{p}(\theta):=2\pi\mathbf{E}\left[\varphi^{p}(G)\varphi^{p}(\hat{G})\right], \tag{11}\] where $p\in\mathbb{N}$, and $\varphi(x)=\max\{x,0\}$ is the ReLU function, $G,\hat{G}\in\mathbb{R}$ are marginally two standard $\mathcal{N}(0,1)$ Gaussian random variables with correlation $\mathbf{Cov}(G,\hat{G})=\cos(\theta)$. This quantity has found numerous applications for infinite width networks. One simple application of $J_{1}$ appears when in the infinite width approximation for $\cos(\theta^{\ell})$, where $\ell$ is fixed an we take the limit $n_{1},n_{2},\ldots n_{\ell}\rightarrow\infty$ (see Appendix A.9 for a detailed derivation): Approximation 3: _The infinite-width approximation for the angle $\theta^{\ell+1}$ given $\theta^{\ell}$ is_ \[\cos\left(\theta^{\ell+1}\right)=\frac{J_{1}(\theta^{\ell})}{\pi}=\frac{\sin( \theta^{\ell})+(\pi-\theta^{\ell})\cos(\theta^{\ell})}{\pi}. \tag{12}\] Figure 3: Histograms of $\ln(\sin^{2}(\theta^{\ell+1}))$ using 5000 simulated neural networks with an initial angle of $\theta^{\ell}=0.1$, and layer width $n_{\ell}=256$ for all layers $1\leq\ell\leq 30$. We plot the simulated histograms as well as our predicted probability density function using Approximation 2 from (10) at layer 1 (left) and layer 30 (right). The predicted and empirical distribution are statistically indistinguishable according to a A Kolmogorov-Smirnov test, with $p$ values $0.913>0.05$ (left) and $0.450>0.05$ (right). The formula for $J_{1}$ is the $p=1$ case of a remarkable explicit formula for $J_{p}$ derived by Cho and Saul (2009) namely, \[J_{p}(\theta)=(-1)^{p}(\sin\theta)^{2p+1}\left(\frac{1}{\sin\theta}\frac{ \partial}{\partial\theta}\right)^{p}\left(\frac{\pi-\theta}{\sin\theta}\right).\] This allows one to derive asymptotics of $\theta^{\ell}$ in the infinite width limit. However, there are several limitations to this approach. Most important is that infinite width limit is not a good approximation when the network depth $\ell$ is comparable to the network width $n$(Li et al. (2022)). The infinite width limit uses the law of large numbers to obtain (12), thereby discarding random fluctuations. For very deep networks, microscopic fluctuations (on the order of $\mathcal{O}(1/n_{\ell})$) from layer to layer can accumulate over $\ell$ layers to give macroscopic effects. This is why the infinite width predictions for $\theta^{\ell}$ are not a good match to the simulations in Figure 1; very deep networks are far from the infinite width limit in this case. See Figure 1 where the infinite width predicitons are compared to finite networks. Instead, to analyze the evolution of the angle $\theta^{\ell}$ more accurately, we need to do something more precise than the law of large numbers to capture the effect of these microscopic fluctuations. This is the approach we carry out in this paper. While the mean only depends on the $p$-th moment functions $J_{p}$ from (11), these fluctuations depend on the _mixed_ moments, which we denote by $J_{a,b}$ for $a,b\in\mathbb{N}$ as follows2 Footnote 2: Note that compared to Cho and Saul’s definition for $J_{p}$, we omit the factor of $2\pi$ in our definition of $J_{a,b}$. The factor of $2\pi$ seems natural when $a+b$ is even (like the case $a=b=p$ that Cho-Saul considered), but when $a+b$ is odd a different factor of $2\sqrt{2\pi}$ appears! Therefore the factor of $2\pi$ would confuse things in the general case (see Table 1). The correct translation between Cho-Saul $J_{p}$ and our $J_{a,b}$ is $J_{p}=2\pi J_{p,p}$ \[J_{a,b}(\theta):=\mathbf{E}\left[\varphi^{a}(G)\varphi^{b}(\hat{G})\right], \tag{13}\] with $G,\hat{G}$ again as in (11) are marginally $\mathcal{N}(0,1)$ with correlation $\cos(\theta)$. In Section 2.1 we carry out a detailed asymptotic analysis to write the evolution of $\theta^{\ell}$ in terms of the mixed moments $J_{a,b}$. In order to make useful predictions, one must also calculate a formula for $J_{a,b}(\theta)$. Unfortunately, the method that Cho-Saul originally proposed for this does _not_ seem to work when $a\neq b$. This is because that method used contour integrals, and relied on using certain trig identities which do not hold when $a\neq b$. Instead, in Section 3, we introduce a new method, based on Gaussian integration by parts, to compute $J_{a,b}$ for general $a,b$ via a recurrance relation. By serendipity3, we find a remarkable combinatorial connection between $J_{a,b}$ and the Bessel numbers (Cheon et al. (2013)), which allows one to find an explicit (albeit complicated) formula for $J_{a,b}$ in terms of binomial coefficients. The formula for the first few functions are shown in Table 1, and the general explicit formula is presented in Theorem 4. Footnote 3: This connection was first noticed by calculating the first few $J$ functions, and then using the On-Line Encyclopedia of Integer Sequence to discover the connection to Bessel number [https://oeis.org/A001498](https://oeis.org/A001498) ### Outline The two main contribution of this paper is to prove Theorem 1 for the evolution of $\theta^{\ell}$ in terms of the mixed moments $J_{a,b}$, and then seperat for any of the mixed moments $J_{a,b}$. Combined these allow for the explicit formula Corollary 2 and the useful simpler Approximations 1 and 2. See Figure 2 and Figure 3 for comparisons of these predictions to Monte-Carlo simulations. We also believe the methods proposed here are flexible enough to be modified to apply to non-linearities other than ReLU and to different neural network architectures beyond fully-connected networks in future work. Section 2 contains the analysis of the angle process and predicted distribution of $\ln(\sin^{2}(\theta^{\ell}))$ in deep ReLU networks. Section 2.1 covers our approximation of $\mathbf{E}[\ln(\sin^{2}(\theta^{\ell+1}))]$ which leads to the rule for $\theta^{\ell}$ as in equation (1), while Section 2.2 outlines our approximation for $\mathbf{Var}[\ln(\sin^{2}(\theta^{\ell+1}))]$. In Section 3 we cover the derivation of the explicit formula for the $J$ functions. We state the main results of this section in Section 3.1, and cover the mathematical tools needed to solve the expectations using Gaussian integration by parts in Section 3.2. We develop the formula for $J_{a,b}$ by first finding a recursive formula in Section 3.3, which reveals a connection between the $J$ functions and the Bessel numbers. This recursion is studied to develop an explicit formula for $J_{a,b}$ in Section 3.4. ## 2 ReLU Neural Networks on Initialization In this section, we analyze ReLU neural networks and show how the the mixed moments $J_{a,b}$ appear in evolution of the angle $\theta^{\ell}$ on initialization. We define the notation we used for a fully connected ReLU neural network, along with other notations we will use in Table 2. Note that the factor of $\sqrt{2/n_{\ell}}$ in our definition is implementing the so called He initialization (He et al., 2015), which ensures that $\mathbf{E}[\\|z^{\ell}\\|^{2}]=\\|x\\|^{2}$ for all layers $\ell$. This initialization is known to be the "critical" initialization for taking large limits of the network (Roberts et al., 2022; Hayou et al., 2019). Given this neural network, we wish to study the evolution \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline $a$$b$ & $0$ & $1$ & $2$ & $3$ \\ \hline $0$ & $\pi-\theta$ & $\cos\theta$+$1$ & $(\pi-\theta)$+$\sin\theta\cos\theta$ & $2(\cos\theta$+$1)$+$\sin^{2}\theta\cos\theta$ \\ \hline $1$ & & $\sin\theta$+$(\pi-\theta)\cos\theta$ & $(\cos\theta$+$1)^{2}$ & $3(\pi-\theta)\cos\theta$+$\sin\theta\cos^{2}\theta$+$2\sin\theta$ \\ \hline $2$ & & & $(\pi-\theta)(2\cos^{2}\theta$+$1)$+$3\sin\theta\cos\theta$ & $3\cos\theta(\cos\theta$+$1)^{2}$+$2(\cos\theta$+$1)$+$\sin^{2}\theta\cos\theta$ \\ \hline $3$ & & & & $(\pi-\theta)(6\cos^{2}\theta$+$9)\cos\theta$ \\ & & & & $+5\sin\theta\cos^{2}\theta$+$(6\cos^{2}\theta$+$4)\sin\theta$ \\ \hline \end{tabular} \end{table} Table 1: Table of formulas for the first few $J$ functions. The normalizing constant appearing in all entries, either $c_{0}=2\pi$, $c_{1}=2\sqrt{2\pi}$ depending on the parity of $a+b$, has been omitted, i.e. the table shows the value of $c_{(a+b\bmod 2)}J_{a,b}(\theta)$. (Note that when $a=0$, the appropriate convention of $0^{0}=0$ is needed, see (25) for details). These generalize $J_{p}(\theta)$ of (11) which appear on the diagonal of this table. Note that $J_{a,b}=J_{b,a}$ so only upper triangular entries are shown. An explicit formula for all $J_{a,b}$ is derived in Section 3.4. of 2 inputs $x_{\alpha}$ and $x_{\beta}$ as they traverse through the layers of the network. Specifically, we wish to study how the angle $\theta$ between the inputs changes as the inputs move from layer to layer. The starting point for our calculation is to notice that because the weights are Gaussian, the values of $\varphi_{\alpha}^{\ell+1},\varphi_{\beta}^{\ell+1}$ are jointly Gaussian given the vectors of $\varphi_{\alpha}^{\ell},\varphi_{\beta}^{\ell}$. In fact, it turns out that by properties of Gaussian random variables, one only needs to know the values of the scalars $\\|\varphi_{\alpha}^{\ell}\\|,\\|\varphi_{\beta}^{\ell}\\|$ and $\theta^{\ell}$ to understand the full distribution of $\varphi_{\alpha}^{\ell+1},\varphi_{\beta}^{\ell+1}$. (see Appendix A.5 for details) By using the positive homogeneity of the ReLU function $\varphi(\lambda x)=\lambda\varphi(x)$ for $\lambda>0$, we can factor out the effect of the norm of each vector in layer $\ell$. After some manipulations, these ideas lead us to the following identities that are the heart of our calculations; a full derivation of these quantities are provided in Appendix A.5 and A.6. \[\\|\varphi_{\alpha}^{\ell+1}\\|^{2} =\frac{\\|\varphi_{\alpha}^{\ell}\\|^{2}}{n_{\ell}}\sum_{i=1}^{n_{ \ell}}2\varphi^{2}(G_{i}), \\|\varphi_{\beta}^{\ell+1}\\|^{2} =\frac{\\|\varphi_{\beta}^{\ell}\\|^{2}}{n_{\ell}}\sum_{i=1}^{n_{\ell}}2 \varphi^{2}(\hat{G}_{i}), \tag{14}\] \[\langle\varphi_{\alpha}^{\ell+1},\varphi_{\beta}^{\ell+1}\rangle =\frac{\\|\varphi_{\alpha}^{\ell}\\|\\|\varphi_{\beta}^{\ell}\\|}{n _{\ell}}\sum_{i=1}^{n_{\ell}}2\varphi(G_{i})\varphi(\hat{G}_{i}),\] (15) \[\frac{\\|\varphi_{\alpha}^{\ell+1}\\|^{2}\\|\varphi_{\beta}^{\ell+1} \\|^{2}}{\\|\varphi_{\alpha}^{\ell}\\|^{2}\\|\varphi_{\beta}^{\ell}\\|^{2}}\sin^{2} (\theta^{\ell+1}) =\frac{2}{n_{\ell}^{2}}\sum_{i,j=1}^{n_{\ell}}\left(\varphi(G_{i}) \varphi(\hat{G}_{j})-\varphi(G_{j})\varphi(\hat{G}_{i})\right)^{2}, \tag{16}\] \begin{table} \begin{tabular}{\|c\|c\|} \hline Symbol & Definition \\ \hline \hline $x\in\mathbb{R}^{n_{in}}$ & Input (e.g. training example) in the input dimension $n_{in}\in\mathbb{N}$ \\ \hline $\ell\in\mathbb{N}$ & Layer number. $\ell=0$ is the input \\ \hline $n_{\ell}\in\mathbb{N}$ & Width of hidden layer $\ell$ (i.e. number of neurons in layer $\ell$) \\ \hline $W^{\ell}\in\mathbb{R}^{n_{\ell+1}\times n_{\ell}}$ & Weight matrix for layer $\ell$. Initialized with iid standard Gaussian entries \\ & $W^{\ell}_{a,b}\sim\mathcal{N}(0,1)$ \\ \hline $\varphi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ & Entrywise ReLU activation function $\varphi(x)_{i}=\varphi(x_{i})=\max\{x_{i},0\}$ \\ \hline $z^{\ell}(x)\in\mathbb{R}^{n_{\ell}}$ & Pre-activation vector in the $\ell^{\text{th}}$ layer for input $x$ (a.k.a logits of layer $\ell$) \\ & $z^{1}(x):=W^{1}x,\qquad z^{\ell+1}(x):=\sqrt{\frac{2}{n_{\ell}}}W^{\ell+1} \varphi(z^{\ell}(x))$. \\ \hline $\varphi_{\alpha}^{\ell},\varphi_{\beta}^{\ell}\in\mathbb{R}^{n_{\ell}}$ & Post-activation vector on inputs $x_{\alpha},x_{\beta}$ respectively \\ & $\varphi_{\alpha}^{\ell}:=\varphi(z^{\ell}(x_{\alpha})),\qquad\varphi_{\beta}^{ \ell}:=\varphi(z^{\ell}(x_{\beta}))$ \\ \hline $\theta^{\ell}\in[0,\pi]$ & Angle between $\varphi_{\alpha}^{\ell}$ and $\varphi_{\beta}^{\ell}$ defined by $\cos(\theta^{\ell}):=\frac{\langle\varphi_{\alpha}^{\ell},\varphi_{\beta}^{ \ell}\rangle}{\\|\varphi_{\alpha}^{\ell}\\|\\|\varphi_{\beta}^{\ell}\\|}$ \\ \hline $R^{\ell+1}\in\mathbb{R}$ & Shorthand for the ratio $R^{\ell+1}:=\frac{\\|\varphi_{\alpha}^{\ell+1}\\|^{2}\\|\varphi_{\beta}^{\ell+1} \\|^{2}}{\\|\varphi_{\alpha}^{\ell}\\|^{2}\\|\varphi_{\beta}^{\ell}\\|^{2}}$ \\ \hline \end{tabular} \end{table} Table 2: Definition and notation used for fully connected ReLU neural networks where $G_{i}$, $\hat{G}_{i}$ are all marginally $\mathcal{N}(0,1)$, with correlation $\mathbf{Cov}(G_{i},\hat{G}_{i})=\cos(\theta^{\ell})$ and independent for different indices $i$. The identity in (16) is derived using the _determinant of the Gram matrix_ for vectors $\varphi_{\alpha}^{\ell+1},\ \varphi_{\beta}^{\ell+1}$ (full derivation given in Appendix A.6). For convenience, we define the notation $R^{\ell+1}=\frac{\\|\varphi_{\alpha}^{\ell+1}\\|^{2}\\|\varphi_{\beta}^{\ell+1} \\|^{2}}{\\|\varphi_{\alpha}^{\ell}\\|^{2}\\|\varphi_{\beta}^{\ell}\\|^{2}}$. Combining the equations (14) gives us a useful identity for $R^{\ell+1}$: \[R^{\ell+1}=\frac{4}{n_{\ell}^{2}}\sum_{i,j=1}^{n_{\ell}}\varphi^{2}(G_{i}) \varphi^{2}(\hat{G}_{j}). \tag{17}\] Given some $\theta^{\ell}$, we wish to predict the behaviour of $\theta^{\ell+1}$. Rather than studying $\theta^{\ell+1}$ directly, we instead study the quantity $\ln(\sin^{2}(\theta^{\ell+1}))$. This allows us to use convenient approximations and identities for quantities we are interested in. (And indeed, a post-hoc analysis shows that as $\theta\to 0$, the random variable $\ln\sin^{2}(\theta^{\ell+1})$ has a _non-zero constant_ variance which depends only on $n_{\ell}$. This is in contrast to $\theta^{\ell}$ itself which has variance tending to _zero_. This is one reason why the Gaussian approximation for $\ln\sin^{2}(\theta^{\ell})\in(-\infty,\infty)$ works well, whereas Gaussian approximations for $\theta^{\ell}$ or $\cos(\theta^{\ell})\in[-1,1]$ are less accurate.) We first derive a formula for $\mathbf{E}\left[\ln(\sin^{2}(\theta^{\ell+1}))\right]$. ### Expected Value In this section we show how to compute the expected value of $\ln(\sin^{2}(\theta^{\ell}))$ in terms of the $J$ functions as in Theorem 1. Firstly, we rewrite this expectation as the difference \[\mathbf{E}\left[\ln(\sin^{2}(\theta^{\ell+1}))\right]=\mathbf{E}\left[\ln \left(R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right)\right]-\mathbf{E}\left[\ln \left(R^{\ell+1}\right)\right]. \tag{18}\] The two random variables $R^{\ell+1}$ and $R^{\ell+1}\sin^{2}(\theta^{\ell+1})$ in (18) both have interpretations in terms of sums of Gaussians as in (17) and (16) which makes it possible to calculate their moments in terms of the $J$ functions. To enable our use of the moments here, we use the following approximation of $\ln(X)$ for a random variables $X$, which is based on the Taylor expansion for $\ln(1+x)=x-\frac{1}{2}x^{2}+\ldots$ (a full derivation is given in Appendix A.1): \[\ln(X)=\ln(\mathbf{E}[X])+\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}-\frac{(X- \mathbf{E}[X])^{2}}{2\mathbf{E}[X]^{2}}+\epsilon_{2}\left(\frac{X-\mathbf{E}[ X]}{\mathbf{E}[X]}\right), \tag{19}\] where $\epsilon_{2}(x)$ is the Taylor remainder term in $\ln(1+x)=x-\frac{x^{2}}{2}+\epsilon_{2}(x)$ and satisfies $\epsilon_{2}(x)=\mathcal{O}(x^{3})$. Applying this approximation to the terms appearing on the right hand side of (18), and taking expected value of both sides, we obtain the estimates \[\mathbf{E}\left[\ln\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right)\right] =\ln\left(\mathbf{E}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1}) \right]\right)-\frac{\mathbf{Var}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1}) \right]}{2\mathbf{E}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]^{2}}+ \mathcal{O}(n_{\ell}^{-2}),\] \[\mathbf{E}\left[\ln\left(R^{\ell+1}\right)\right] =\ln\left(\mathbf{E}\left[R^{\ell+1}\right]\right)-\frac{\mathbf{ Var}\left[R^{\ell+1}\right]}{2\mathbf{E}\left[R^{\ell+1}\right]^{2}}+ \mathcal{O}(n_{\ell}^{-2}).\] To control the error here, we have used here the fact that $R^{\ell+1}$ and $R^{\ell+1}\sin^{2}(\theta^{\ell+1})$ can be written as averages over random variables as in (14 - 16). This allows us to show the 3rd central moments for $R^{\ell+1}$ and $R^{\ell+1}\sin^{2}(\theta^{\ell+1})$ are $\mathcal{O}(n_{\ell}^{-2})$; see Appendix A.1 for details. This approximation is convenient because we are able to calculate the values on the right hand side of the equations in terms of the moments $J_{a,b}$ by expanding/taking expectations of the representations (14 - 16). The key quantities we calculate are \[\mathbf{E}\left[R^{\ell+1}\right] =\frac{4J_{2,2}-1}{n_{\ell}}+1 \tag{20}\] \[\mathbf{E}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right] =\frac{(n_{\ell}-1)(1-4J_{1,1}^{2})}{n_{\ell}}\] (21) \[\mathbf{Var}\left[R^{\ell+1}\right] =\frac{4}{n_{\ell}}(J_{2,2}+1)+\frac{16}{n_{\ell}^{2}}\left(2J_{ 4,2}-\frac{5}{2}J_{2,2}+J_{2,2}^{2}+\frac{5}{8}\right)+\mathcal{O}\left(n_{ \ell}^{-3}\right)\] (22) \[\mathbf{Var}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right] =\frac{8\left(-8J_{1,1}^{4}+8J_{1,1}^{2}J_{2,2}+4J_{1,1}^{2}-8J_{ 1,1}J_{3,1}+J_{2,2}+1\right)}{n_{\ell}}+\mathcal{O}\left(n_{\ell}^{-2}\right), \tag{23}\] where $J_{a,b}=J_{a,b}(\theta^{\ell})$. These formulas are calculated in Appendix A.8 by a combinatorial expansion using the representations from (14-16). Combining these gives the result for $\mu(\theta,n)$ in Theorem 1. Note that to obtain a more accurate approximation, we would simply include more terms in the variance expressions in (22, 23). ### Variance of $\ln(\sin^{2}(\theta^{\ell+1}))$ In this section we show how to compute the variance of $\ln(\sin^{2}(\theta^{\ell}))$ in terms of the $J$ functions as in Theorem 1. We can rewrite $\mathbf{Var}[\ln(\sin^{2}(\theta^{\ell+1}))]$ in the following way: \[\mathbf{Var}[\ln(\sin^{2}(\theta^{\ell+1}))]=\mathbf{Var}\left[ \ln\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right)-\ln\left(R^{\ell+1} \right)\right] \tag{24}\] \[=\mathbf{Var}\left[\ln\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1}) \right)\right]+\mathbf{Var}\left[\ln\left(R^{\ell+1}\right)\right]-2\mathbf{ Cov}\left(\ln\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right),\ln\left(R^{ \ell+1}\right)\right).\] We have now expressed this in terms of $R^{\ell+1}$ and $R^{\ell+1}\sin^{2}(\theta^{\ell+1})$ which will allow us to use identities as in (14 - 16) in our calculations. Appendix A.2 and Appendix A.3 cover the method used to approximate the unknown variance and covariance terms above. Once again, we control the error term arising from moments in the error term of the Taylor series by using representation as sums (14 - 16). We have already calculated most of the quantities on the right hand side already in our calculation for $\mu(\theta,n)$, the only new term is \[\mathbf{Cov}\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1}),R^{\ell+1}\right)=\frac {1}{n_{\ell}}\left(16J_{1,1}^{2}-32J_{1,1}J_{3,1}+8J_{2,2}+8\right)+\mathcal{ O}\left(n_{\ell}^{-2}\right).\] This is again computed by a combinatorial expansion of the sums (14-16). (Full calculation given in Appendix A.8). We now have solved for all of the functions needed to perform our approximation of $\mathbf{Var}[\ln(\sin^{2}(\theta^{\ell+1}))]$. Putting it together, we end up with the expression for $\sigma^{2}(\theta,n)$ as in (5). We compare the predicted probability distribution of $\ln(\sin^{2}(\theta))$ using our formulas $\mu(\theta,n)$ and $\sigma^{2}(\theta,n)$ to empirical probability distributions in Figure 3. ## 3 Explicit formula for the mixed-moment J Functions In this section we develop a combinatorial method that allows us compute exact formulas for the $J$ functions. The method is to use Gaussian integration by parts to find a recurrence relationship between the moments $J_{a,b}$, and then solve it explicitly. We begin by generalizing the definition of $J_{a,b}$ from (13) to include $a=0$ and/or $b=0$ as follows. Let $G$, $W$ be _independent_$\mathcal{N}(0,1)$ variables. Then, we define the functions $J_{a,b}(\theta)$ as \[J_{a,b}(\theta)=\mathbf{E}[G^{a}(G\cos\theta+W\sin\theta)^{b}\;1\{G>0\}\;1\{G \cos\theta+W\sin\theta>0\}], \tag{25}\] where $a,b\in\mathbb{N}\cup\{0\}$. Note that $G\cos\theta+W\sin\theta=\hat{G}$ is marginally $\mathcal{N}(0,1)$ and has correlation $\cos(\theta)$ with $G$, matching the original definition. The ReLU function satisfies the identity $\varphi(x)^{a}=x^{a}1\{x>0\}$ for $a\geq 1$, so (25) generalizes (13) to the case $a=0$. We also note that $J_{a,b}(\theta)=J_{b,a}(\theta)$ for all $a,b\in\mathbb{N}\cup\{0\}$. ### Statement of Main Results and Outline of Method By using the method of Gaussian integration by parts, we are able to derive recurrence relations for the $J_{a,b}$. Since the definition of $J_{a,b}$ involves the indicator function $1\{G>0\}$, we must make sense of what the derivative of this function means for the purposes of integration by parts; see Section 3.2 where this is carried out. Then, by use of the generalized Gaussian integration by parts formula, we obtain the following recurrence relations for $J_{a,b}$. Proposition 1 (Recurrence relations for $J_{a,b}$): _For $a\geq 2$, the sequence $J_{a,0}$ satisfies the recurrence relation:_ \[J_{a,0}(\theta)=(a-1)J_{a-2,0}(\theta)+\frac{\sin^{a-1}\theta\cos\theta}{c_{a \bmod 2}}(a-2)!!\,, \tag{26}\] _where $c_{0}=2\pi$, $c_{1}=2\sqrt{2\pi}$. For $a\geq 2$, and $b\geq 1$, the collection, $J_{a,b}$ satisfies the following two-index recurrence relation:_ \[J_{a,b}(\theta)=(a-1)J_{a-2,b}(\theta)+b\cos\theta J_{a-1,b-1}(\theta). \tag{27}\] The same integration by parts technique that yields the recurrence relation also makes it easy to evaluate the first few $J$ functions. They are as follows Proposition 2 (Explicit Formula for $J_{0,0},J_{1,0},J_{1,1}$): $J_{0,0}$_, $J_{1,0}$, and $J_{1,1}$ are given by_ \[J_{0,0}(\theta)=\frac{\pi-\theta}{2\pi},\qquad J_{1,0}(\theta)=\frac{1+\cos \theta}{2\sqrt{2\pi}},\qquad J_{1,1}(\theta)=\frac{\sin\theta+(\pi-\theta)\cos \theta}{2\pi}. \tag{28}\] See Appendix B.1 for a derivation of these quantities. Note that Cho and Saul (2009) have previously discovered the formulae for $J_{0,0}$ and $J_{1,1}$ by use of a completely different contour-integral based method. The combination of Proposition 1 and 2 make it possible to practically calculate any value of $J_{a,b}$ when $a,b$ are not too large. However, by serendipity, we are able to find remarkable explicit formulas for $J_{a,b}$, which we report below. Proposition 3 (Explicit Formulas for $J_{a,0}(\theta)$, $J_{a,1}(\theta)$): _Let $a\geq 2$. Then, $J_{a,0}$ and $J_{a,1}$ are explicitly given by the following:_ \[J_{a,0}(\theta)=(a-1)!\,!\left(J_{a\bmod 2,0}+\frac{\cos\theta}{c_{a\bmod 2}} \sum_{\begin{subarray}{c}i\neq a(\bmod 2)\\ 0<i<a\end{subarray}}\frac{(i-1)!\,!}{i!\,!}\sin^{i}\theta\right),\] _where $c_{0}=2\pi$, $c_{1}=2\sqrt{2\pi}$. We can then use the explicit formula for $J_{a,0}$ in the formula for $J_{a,1}$:_ \[J_{a,1}(\theta)=(a-1)!\,!\left(J_{a\bmod 2,1}+\cos\theta\sum_{ \begin{subarray}{c}i\neq a(\bmod 2)\\ 0<i<a\end{subarray}}\frac{J_{i,0}(\theta)}{i!\,!}\right),\] _where an explicit formula for the first term (either $J_{1,0}$ or $J_{1,1}$ depending on the parity of $a$) is given in Proposition 2._ We can finally express $J_{a,b}$ as a linear combination of $J_{0,n}$ and $J_{1,n}$, as follows. (In light of the previous explicit formulas, this is an explicit formula for $J_{a,b}$.) It turns out that the coefficients are given in terms of two special numbers $P(a,b)$ and $Q(a,b)$ which are known as the Bessel numbers. Definition 3 (Bessel numbers): _The numbers $P(a,b)$ and $Q(a,b)$ are defined as follows,_ \[P(a,b) =\begin{cases}\frac{a!}{b!\left(\frac{a-b}{2}\right)^{2}\frac{a- b}{2}},&a\geq b,\;a\equiv b\;(\bmod 2)\\ 0,&\text{otherwise}\end{cases} \tag{29}\] \[Q(a,b) =\begin{cases}\frac{(\frac{a+b}{2})!}{b!}2\frac{b-a}{2}\sum_{i=0 }^{\frac{a-b}{2}}\binom{a+1}{i},&a\geq b,\;a\equiv b\;(\bmod 2)\\ 0,&\text{otherwise}\end{cases}. \tag{30}\] $P(a,b)$ represents a family of numbers known as the Bessel numbers of the second kind (Cheon et al. (2013)), and $Q(a,b)$ comes from a closely related family of numbers (Kreinin (2016)). Using these, we can express $J_{a,b}$ as follows. Theorem 4 (Explicit Formula for $J_{a,b}(\theta)$): _Let $b\geq 2,a\geq 1,\;b\geq a$. Then, we have the following formula for $J_{a,b}(\theta)$ in terms of $J_{0,n}$ and $J_{1,n}$_ \[J_{a,b} =\sum_{\begin{subarray}{c}i\equiv 0(\bmod 2)\\ 0<i\leq a\end{subarray}}(b)_{a-i}(\cos\theta)^{a-i}\left(P(a,a-i)-Q(a-1,a-1-i )\right)J_{0,b-a+i}\] \[+\sum_{\begin{subarray}{c}i\equiv 1(\bmod 2)\\ 0<i\leq a\end{subarray}}(b)_{a-i}(\cos\theta)^{a-i}Q(a-1,a-i)J_{1,b-a+i}.\] Remark 5: _Since $J_{1,n}$ is also given in terms of $J_{0,n}$, one may further simplify the formula for $J_{a,b}$ to be in terms of only $J_{0,n}$, $J_{0,0}$ and $J_{0,1}$. This substitution yields the following formula. For notational convenience, we will let $\delta:=b-a$,_ \[J_{a,b} =\sum_{\begin{subarray}{c}i\equiv 0(\bmod 2)\\ 0<i\leq a\end{subarray}}(b)_{a-i}(\cos\theta)^{a-i}(P(a,a-i)-Q(a-1,a-1-i))J_{0,\delta+i}\] \[+\sum_{\begin{subarray}{c}i\equiv 1(\bmod 2)\\ 0<i\leq a\end{subarray}}(b)_{a-i}(\cos\theta)^{a-i}Q(a-1,a-i)(\delta+i-1)! \,J_{(\delta+1)\bmod 2,1}\] \[+\cos\theta\sum_{\begin{subarray}{c}i\equiv 1(\bmod 2)\\ 0<i\leq a\end{subarray}}\sum_{\begin{subarray}{c}j\equiv\delta(\bmod 2)\\ 0<j<\delta+i\end{subarray}}(b)_{a-i}(\cos\theta)^{a-i}Q(a-1,a-i)\frac{(\delta+i- 1)!}{j!\,!}J_{0,j}.\] ### Gaussian Integration-by-Parts Formulae In this section, we state two important formulas that together give us the tools for computing the expectations that appear in $J_{a,b}$. Fact 1 (Gaussian Integration by Parts): _Let $G\sim\mathcal{N}(0,1)$ be a Gaussian variable and $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function. Then,_ \[\mathbf{E}[Gf(G)]=\mathbf{E}[f^{\prime}(G)]. \tag{31}\] Using this type of Gaussian integration by parts formula, we can generalize the expected value of Gaussians to derivatives of functions which are not necessarily differentiable. For example the indicator function $1\{x>a\}$ is not differentiable, but for the purposes of computing Gaussian expectation, we can use the following integration formula. Fact 2 (Gaussian expectations involving $1^{\prime}\{x>a\}$): _Let $G$ be a Gaussian variable and $a\in\mathbb{R}$. Let $f:\mathbb{R}\to\mathbb{R}$ such that $\lim_{g\to\infty}f(g)e^{\frac{-g^{2}}{2}}=0$. Then, using the Gaussian integration by parts formula to assign a meaning to expectations involving the "derivative of the indicator function", $1^{\prime}\{x>a\}$, we have_ \[\mathbf{E}[1^{\prime}\{G>a\}f(G)]=f(a)\frac{e^{\frac{-g^{2}}{2}}}{\sqrt{2\pi}}. \tag{32}\] Remark 6: _The purpose of assigning a value to the expectation (32) is to allow one to compute "honest" expectations of the form (31) when $f(x)$ involves $1\{x>0\}$; see Lemma 7 for an illustrative example. The final result does not require interpreting "$1^{\prime}\{x>a\}$"; this is only a useful intermediate step in the sequence of calculations leading to the final result._ _The formula can also be understood or proven in a number of different alternative ways. One is simply to say that $1^{\prime}\{x>a\}=\delta\{x=a\}$ is a "Dirac delta function" at $x=a$. A more rigorous way would be to take any differentiable family of functions $1_{\epsilon}\{x>a\}$ which suitably converge to $1\{x>a\}$ as $\epsilon\to 0$ and then interpret the result as the limit of the expectation $\lim_{\epsilon\to 0}\mathbf{E}[1^{\prime}_{\epsilon}\{G>a\}f(G)]$._ Proof [Of Fact 2] Applying integration by parts, we formally have \[\mathbf{E}[1^{\prime}\{G>a\}f(G)] =\int\limits_{-\infty}^{\infty}1^{\prime}\{g>a\}f(g)\frac{e^{\frac {-g^{2}}{2}}}{\sqrt{2\pi}}dg\] \[=\left[1\{g>a\}f(g)\frac{e^{\frac{-g^{2}}{2}}}{\sqrt{2\pi}} \right]_{-\infty}^{\infty}-\int\limits_{-\infty}^{\infty}1\{g>a\}\frac{d}{dg} \left(f(g)\frac{e^{\frac{-g^{2}}{2}}}{\sqrt{2\pi}}\right)dg.\] Note that the first term is $0$ by the hypothesis $\lim\limits_{g\to\infty}f(g)e^{\frac{-g^{2}}{2}}=0$, and we have then \[\mathbf{E}[1^{\prime}\{G>a\}f(G)]=-\int\limits_{a}^{\infty}\frac{d}{dg}\left( f(g)\frac{e^{\frac{-g^{2}}{2}}}{\sqrt{2\pi}}\right)dg=-\left[f(g)\frac{e^{ \frac{-g^{2}}{2}}}{\sqrt{2\pi}}\right]_{a}^{\infty}=0+f(a)\frac{e^{\frac{-g^{ 2}}{2}}}{\sqrt{2\pi}},\] where we have used the hypothesis on $f$ once again. The two facts about Gaussian integration by parts can be combined to create recurrence relations for expectations involving $1\{G>a\}$. A simple example is the following lemma, which we will also use later in our derivation. The proof strategy of this lemma is a microcosm of the proof strategy we use to compute $J_{a,b}$ in general, namely to use Gaussian integration by parts to derive a recurrence relation and initial condition, and then solve. Lemma 7 (Moments of $\varphi(G)$): _For $k\geq 0$, we have_ \[\mathbf{E}[\varphi(G)^{k}]=\mathbf{E}[G^{k}1\{G>0\}]=\begin{cases}\frac{(k-1)!! }{2}&\text{$k$ is even}\\ \frac{(k-1)!!}{\sqrt{2\pi}}&\text{$k$ is odd}\end{cases}=\sqrt{2\pi}\frac{(k-1)!! }{c_{k-1\bmod 2}},\] _where $c_{0}=2\pi$ and $c_{1}=2\sqrt{2\pi}$._ Proof We prove this for even and odd $k$ separately by induction on $k$. The base case for $k=0$ is trivial since $(0-1)!!=1$ is the empty product. The base case $k=1$ follows by first applying (31) with $f(x)=1\{x>0\}$ and then applying (32) with $f(x)\equiv 1$, \[\mathbf{E}[\varphi(G)]=\mathbf{E}[G1\{G>0\}]=\mathbf{E}[1^{\prime}\{G>0\}]= \sqrt{2\pi}^{-1}.\] Now, to see the induction, we apply (31) with $f(x)=x^{k-1}1\{x>0\}$, $k\geq 2$. Due to the product rule, there are two terms in the derivative, \[\mathbf{E}[\varphi(G)^{k}] =\mathbf{E}[G\cdot G^{k-1}1\{G>0\}] \tag{33}\] \[=(k-1)\mathbf{E}[G^{k-2}1\{G>0\}]+\mathbf{E}[G^{k-1}1^{\prime}\{G >0\}]\] (34) \[=(k-1)\mathbf{E}[\varphi(G)^{k-2}]+0, \tag{35}\] where we have recognized that the second term is $0$ by application of (32) with $f(x)=x^{k-1}$ which has $f(0)=0$. The recurrence $\mathbf{E}[\varphi(G)^{k}]=(k-2)\mathbf{E}[\varphi(G)^{k-2}]$ along with initial condition leads to the stated result by induction. ### Recursive Formulae for $J_{a,b}(\theta)$ - Proof of Proposition 1 Proof [Of Proposition 1] To find a recursive formula for $J_{a,0},a\geq 2$, we apply the Gaussian integration by parts formula (31) to $f(x)=x^{a-1}1\{x>0\}1\{\cos\theta x+W\sin\theta>0\}$ to evaluate the expected value over $G$ first. When applying product rule there are three terms \[J_{a,0}= \mathbf{E}[G\cdot G^{a-1}1\{G>0\}1\{G\cos\theta+W\sin\theta>0\}] \tag{36}\] \[= (a-1)\mathbf{E}[G^{a-2}1\{G>0\}1\{G\cos\theta+W\sin\theta>0\}]\] \[+\mathbf{E}[G^{a-1}1\{G>0\}1^{\prime}\{G\cos\theta+W\sin\theta>0 \}]\cos\theta\] \[+\mathbf{E}[G^{a-1}1^{\prime}\{G>0\}1\{G\cos\theta+W\sin\theta>0 \}].\] The first term is simply $(a-1)J_{a-2,0}$. The last two terms can now be evaluated with the help of (32). The last term of (36) is (32) with the function $f(x)=x^{a-1}1\{x\cos\theta+W\sin\theta>0\}$ which has $f(0)=0$ for $a\geq 2$. Therefore, this term simply vanishes. To evaluate the middle term of (36), we introduce a change of variables to express $G\cos\theta+W\sin\theta$ in terms of two other independent Gaussian variables $Z,W\sim\mathcal{N}(0,1)$ \[Z =G\cos\theta+W\sin\theta G=Z\cos\theta+Y\sin\theta \tag{37}\] \[Y =G\sin\theta-W\cos\theta W=Z\sin\theta-Y\cos\theta,\] where $Y,Z$ iid $\mathcal{N}(0,1)$. Under this change of variables, $J_{a,0},\ a\geq 2$ is setup to apply (32) with $f(x)=1\{x\cos\theta+Y\sin\theta\}^{a-1}1\{x\cos\theta+Y\sin\theta>0\}$ \[J_{a,0} =(a-1)J_{a-2,0}+\mathbf{E}[G^{a-1}1\{G>0\}1^{\prime}\{G\cos \theta+W\sin\theta>0\}]\cos\theta\] \[=(a-1)J_{a-2,0}+\mathbf{E}[(Z\cos\theta+Y\sin\theta)^{a-1}1\{Z \cos\theta+Y\sin\theta>0\}1^{\prime}\{Z>0\}]\cos\theta\] \[=(a-1)J_{a-2,0}+\mathbf{E}[(0+Y\sin\theta)^{a-1}1\{0+Y\sin\theta> 0\}]\frac{1}{\sqrt{2\pi}}\cos\theta\] \[=(a-1)J_{a-2,0}+\frac{\sin^{a-1}\theta\cos\theta}{c_{a\bmod 2}}( a-2)!!\,,\] where we have applied Lemma 7 to evaluate the last expectation. A similar argument is used to find the recursive formula for $J_{a,b},\ a\geq 2,b\geq 1$, by using (31) with the function $f(x)=x^{a-1}(x\cos\theta+W\sin\theta)^{b}1\{x>0\}1\{x\cos\theta+W\sin\theta>0\}$. There are 4 terms in the product rule derivative. Fortunately in this case, the last two terms are simply zero by application of (32) since the expressions vanish when $G=0$, so we get \[J_{a,b} =\mathbf{E}[G\cdot G^{a-1}(G\cos\theta+W\sin\theta)^{b}1\{G>0\}1 \{G\cos\theta+W\sin\theta>0\}]\] \[=\mathbf{E}[(a-1)G^{a-2}(G\cos\theta+W\sin\theta)^{b}1\{G>0\}1\{G \cos\theta+W\sin\theta>0\}]\] \[\qquad+\mathbf{E}[G^{a-1}b\cos\theta(G\cos\theta+W\sin\theta)^{b- 1}1\{G>0\}1\{G\cos\theta+W\sin\theta>0\}]\] \[\qquad+\mathbf{E}[G^{a-1}(G\cos\theta+W\sin\theta)^{b}1^{\prime} \{G>0\}1\{G\cos\theta+W\sin\theta>0\}]\] \[\qquad+\mathbf{E}[G^{a-1}(G\cos\theta+W\sin\theta)^{b}1\{G>0\}1^{ \prime}\{G\cos\theta+W\sin\theta>0\}\cos\theta]\] \[=(a-1)J_{a-2,b}+b\cos\theta J_{a-1,b-1}+0+0,\] as desired. ### Solving the Recurrence to get an Explicit Formula for $J_{a,b}(\theta)$ - Proof of Theorem 4 Solving the recurrence for the sequences $J_{a,0}$ and $J_{a,1}$ to get the claimed explicit formula for $J_{a,0}$ is a simple induction proof. We defer these to Appendix B.2. More difficult and interesting is the 2D array $J_{a,b}$. To solve the recurrence \[J_{a,b}=(a-1)J_{a-2,b}+b\cos\theta J_{a-1,b-1},\quad a\geq 2,b\geq 1, \tag{38}\] we will apply the recursion repeatedly until $J_{a,b}$ can be expressed as a linear combination of $J_{0,n}$ and $J_{1,n}$ terms for which we already have an explicit formula developed. To determine the coefficients in front of $J_{0,n}$ and $J_{1,n}$, we take a combinatorial approach by thinking of the recurrence relation as a weighted directed graph as defined below. Definition 8 (Viewing a recursion as a directed weighted graph): _We can view the recurrence relation for $J_{a,b}$ as a weighted directed graph on the vertex set $(a,b)\in\mathbb{Z}^{2}$ where vertices represent the values of $J_{a,b}$ and directed edges capture how values of $J_{a,b}$ are connected through the recurrence relation. To be precise, the graph edges and edge weights $w_{e}$ are defined so that the recursion (38) for $J_{a,b}$ can be expressed in the graph as a sum over incoming edges,_ \[J_{a,b}=\sum_{e:(a^{\prime},b^{\prime})\to(a,b)}w_{e}^{J}J_{a^{\prime},b^{ \prime}}, \tag{39}\] _where the sum is over the edges $e$ with weight $w_{e}^{J}$ incoming to the vertex $(a,b)$. An example of the graph to calculate $J_{6,8}$ is illustrated in Figure 4._ _By repeatedly applying the recursion, $J_{a,b}$ can be expressed as a linear combination of the values at the source vertices of the graph (i.e. those with no incoming edges). For the recurrence $J_{a,b}$, the source vertices are $J_{0,n}$ and $J_{1,n}$. The coefficients in front of each source is simply the weighed sum over all paths from the source to the node, namely_ \[J_{a,b}=\sum_{\text{source vertices }v}W_{v\to(a,b)}^{J}J_{v} =\sum_{n\geq 0}W_{(0,n)\to(a,b)}^{J}J_{0,n}+\sum_{n\geq 0}W_{(1,n)\to(a,b)}^{J}J_{1,n}, \tag{40}\] \[W_{(a^{\prime},b^{\prime})\to(a,b)}^{J} :=\sum_{\pi:(a^{\prime},b^{\prime})\to(a,b)}\prod_{e\in\pi}w_{e}^ {J}, \tag{41}\] _where the sum is over all paths $\pi$ from the vertex $(a^{\prime},b^{\prime})$ to the vertex $(a,b)$ in the $J$ graph._ In light of (40), to prove Theorem 4, we have only to calculate the weighted sum of path $W_{(0,n)\to(a,b)}^{J}$ and $W_{(1,n)\to(a,b)}^{J}$. These weighted sums turn out to be given in terms of the $P$ and $Q$ numbers which were defined in Definition 3. Proposition 4 (Weighted sums of paths for $J$): _In the graph for $J$, we have the following formulas for the sum over weighted paths $W^{J}$ defined in (41),_ \[W_{(0,n)\to(a,b)}^{J} =(b)_{b-n}(\cos\theta)^{b-n}(P(a,b-n)-Q(a-1,b-n-1)), \tag{42}\] \[W_{(1,n)\to(a,b)}^{J} =(b)_{b-n}(\cos\theta)^{b-n}Q(a-1,b-n). \tag{43}\] To prove this Proposition 4, we first create a simpler recursion, $J^{}$, which we solve first and then slightly modify the solution to get the solution for $J$. Lemma 9* (Weighted sums of paths for $J^{}$): _Let $J^{}_{a,b}$ be defined to be the recursion:_ \[J^{}_{a,b} :=(a-1)J^{}_{a-2,b}+1J^{}_{a-1,b-1}\text{ for }2\leq a\leq b \tag{44}\] \[J^{}_{1,b} :=0+1J^{}_{0,b-1}\text{ for }1\leq b. \tag{45}\] _Thinking of this recursion as a graph as in Definition 8 (see Figure 4 for an illustration), we have that the sum of weighted paths $W^{J^{}}$ defined analogously to those in (41), are given by $P$ and $Q$ numbers, namely_ \[W^{J^{}}_{(0,n)\to(a,b)} =P(a,b-n) \tag{46}\] \[W^{J^{}}_{(1,n)\to(a,b)} =Q(a-1,b-n). \tag{47}\] The connection between the $J^{}$ and the $P$, $Q$ numbers is through the following recursion for the $P$, $Q$ numbers. Lemma 10: _(Recursion for $P$ and $Q$ numbers) The $P$ numbers, defined in Definition 3, satisfy $P(0,0)=1$, $P(n,n)=P(n-1,n-1)$ for $n\geq 1$, and the recursion_ \[P(a,b)=(\texttt{a}-1)\cdot P(a-2,b)+1\cdot P(a-1,b-1),\ \text{ for }\ a\geq 2,\ 0\leq b\leq a-2,\] _under the convention that $P(a,-1)=0$. The $Q$ numbers satisfy the same recursion as the $P$ numbers, with a coefficient of a rather than $(\texttt{a}-1)$._ The proof of Lemma 10 is an easy consequence of known results from Kreinin (2016) and is deferred to Appendix C. Figure 4: The graph associated with the recursions for $J$ in (38) (left) and $J^{}$ in (45) (right). The graph is defined so that the recursion is given by a sum of incoming edges as in (39). The edges are color coded red and blue to match the coefficients in the recursion. Proof [Of Lemma 9] Using the same idea of recursions expressed as graphs as in Definition 8, the recursion from Lemma 10 means that $P$ and $Q$ can be expressed as weighted directed graphs. These are displayed in Figure 5. Since the $P$ and $Q$ graphs have only one single unit valued source vertex at $(0,0)$, (40) shows that the $P$ and $Q$ numbers are actually themselves equal to sums over weighted paths in their respective graphs \[P(a,b)=W^{P}_{(0,0)\rightarrow(a,b)},\quad Q(a,b)=W^{Q}_{(0,0)\rightarrow(a,b)}.\] Therefore the statement of the lemma is that sum over weighted paths in the $J^{}$ graph are the same as other sums over weighted paths in the $P$ graph/$Q$ graphs, \[W^{J^{}}_{(0,n)\rightarrow(a,b)}=W^{P}_{(0,0)\rightarrow(a,b-n)},\quad W^{ J^{}}_{(1,n)\rightarrow(a,b)}=W^{Q}_{(0,0)\rightarrow(a-1,b-n)}. \tag{48}\] The fact that these are equal is demonstrated by establishing a simple bijection between weighted paths in the $P$ graph/$Q$ graph, and weighted paths in the $J^{}$ graph. For example, in Figure 5, there is a bijection between the weighted paths in the $P$ graph which connect $P(0,0)$ to $P(6,2)$, to the paths which connect $J^{}_{6,8}$ to $J^{}_{0,6}$ in the $J^{}$ graph. The bijection is simply to _flip_ any path in the $P$-graph by rotating it by $180^{\circ}$ to get a valid path in the $J^{}$-graph. Moreover, the edge weights for $J^{}$ and $P$ are precisely set up so that under this bijection, the paths will have the same set of weighted edges in the same order. A full, more detailed, explanation of this bijection is given in Appendix B.3. This argument shows that $W^{J^{}}_{(0,n)\rightarrow(a,b)}=P(a,b-n)$ as desired. Figure 5: Graphs associated with the recursions for the $P$ numbers (left), $Q$ numbers (middle), and $J^{}$ (right). The weighted edges indicate the coefficients in the recursions for $P,Q,J^{}$ respectively. The diagrams are lined up so that the sum of weighted paths in from $J^{}_{6,8}$ can be directly read from the $P$ and $Q$ entries in the same location. By reading from the bottom displayed row of $P$ we see that the weighted sum over paths $W^{J^{}}_{(0,n)\rightarrow(6,8)}$ are $15$, $45$, $15$ and $1$ for $n=8,6,4$ and $2$ respectively. (Since these are the source vertices, this shows that $J^{}_{6,8}=15J^{}_{0,8}+45J^{}_{0,6}+15J^{}_{0,4}+1J^{}_{0,2}$.) From the bottom displayed row of $Q$ we see that the values for sums of weighted paths from vertices $W^{J^{}}_{(1,n)\rightarrow(6,8)}$ are $33$, $14$ and $1$ for $n=7,5$ and $3$ respectively. The $P$ numbers do not apply for paths between $J^{}_{1,n}$ and $J^{}_{a,b}$ because we are starting one row higher so the first vertical upward edge is weight 2. In this case, there is a bijection to the $Q$-graph after flipping the path. For any path which runs from a node in row 1 to the top left corner of the $J^{}$-graph, we can find the same "flipped" path in the graph of the $Q$, running from the top left entry to the corresponding node in row $a-1$. (The bijection is explained in detail in Appendix B.3.) Hence $W^{J^{}}_{(1,n)\to(a,b)}=Q(a-1,b-n)$ as desired. Having solved for $J^{}$ in terms of $P$ and $Q$, it remains to translate these into the weights for $J$ to obtain Proposition 4. Proof* [Of Proposition 4] The proof follows by relating the weighted sum of paths for $J$ in terms of $J^{}$ and then applying the result of Lemma 9. There are two differences between the formula for $J_{a,b}$ compared to $J^{}_{a,b}$, which can both be seen in Figure 4. We handle both differences as follows: _Difference #1:_$J$ has a weight of $b\cos\theta$ on the blue diagonal edges $(a,b)\to(a+1,b+1)$ vs $J^{}$ has a weight of 1. This difference is handled by the following observation: any path from $(a,b)\to(a^{\prime},b^{\prime})$ in the graph goes through each column between $b$ and $b^{\prime}$ exactly once. This means that the contribution of the edge weights from these edges do not depend on the details of which path was taken, only the starting and ending points. They always contribute the same factor, $(b)_{b^{\prime}-b}(\cos\theta)^{b^{\prime}-b}$. (here $(b)_{k}=b(b-1)\cdots(b-k+1)$ is the falling factorial with $k$ terms). This argument shows that the weighted sum of paths in $J$ and $J^{}$ are related by \[W^{J}_{(a,b)\to(a^{\prime},b^{\prime})}=(b)_{b^{\prime}-b}(\cos\theta)^{b^{ \prime}-b}W^{J^{}}_{(a,b)\to(a^{\prime},b^{\prime})}. \tag{49}\] By the result of Lemma 9, this shows that $W^{J}_{(1,n)\to(a,b)}=(b)_{b-n}(\cos\theta)^{b-n}Q(a-1,b-n)$ as desired. Equation (49) holds for all paths with starting point $a\geq 1$. When $a=0$, there is one additional difference between $J$ and $J^{}$ which is accounted for below. _Difference #2:_$J_{0,n}$ has _no_ diagonal edge vs $J^{}_{0,n}$ has a diagonal blue edge of weight 1. Because of this "missing edge", the only choice in $J$ for paths starting from $(0,n)$ is to first go vertically up by 2 units to $(2,n)$. Hence $W^{J}_{(0,n)\to(a,b)}=W^{J}_{(2,n)\to(a,b)}$. To evaluate this, we use the decomposition of paths in $J^{}$ by what their first step is, either a diagonal blue step or a red vertical up step, to see that \[W^{J^{}}_{(0,n)\to(a,b)} =W^{J^{}}_{(1,n+1)\to(a,b)}+W^{J^{}}_{(2,n)\to(a,b)}, \tag{50}\] \[\implies W^{J^{}}_{(2,n)\to(a,b)} =W^{J^{}}_{(0,n)\to(a,b)}-W^{J^{}}_{(1,n+1)\to(a,b)}\] (51) \[=P(a,b-n)-Q(a-1,b-n-1), \tag{52}\] by the result of Lemma 9. By applying now (49) to relate $J$ and $J^{}$, we obtain $W^{J}_{(0,n)\to(a,b)}=W^{J}_{(2,n)\to(a,b)}=(b)_{b-n}(\cos\theta)^{b-n}(P(a,b -n)-Q(a-1,b-n-1))$ as desired. Proof* [Of Theorem 4] The formula is immediate from (40), which writes $J_{a,b}$ as a linear combination of $J_{0,n}$ and $J_{1,n}$, and Proposition 4 which gives the the coefficients. ## Appendix A. ### Expected Value Approximation Lemma 11: _Both the random variables $X=R^{\ell+1}$ and $X=R^{\ell+1}\sin^{2}(\theta^{\ell})$ satisfy_ \[\mathbf{E}[\ln(X)]=\ln(\mathbf{E}[X])-\frac{\mathbf{Var}[X]}{2 \mathbf{E}[X]^{2}}+\mathcal{O}(n_{\ell}^{-2}). \tag{53}\] Proof First note that by the properties of the logarithm, we have \[\ln(X)=\ln\left(\mathbf{E}[X]\left(\frac{\mathbf{E}[X]+(X-\mathbf{E}[X])}{ \mathbf{E}[X]}\right)\right)=\ln(\mathbf{E}[X])+\ln\left(1+\frac{X-\mathbf{E} [X]}{\mathbf{E}[X]}\right). \tag{54}\] We can now apply the Taylor series $\ln(1+x)=x-\frac{x^{2}}{2}+\epsilon_{2}(x)$, where $\epsilon_{2}(x)$ is the Taylor series remainder and satisfies $\epsilon_{2}(x)=\mathcal{O}(x^{3})$. Hence \[\ln(X)=\ln(\mathbf{E}[X])+\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}- \frac{(X-\mathbf{E}[X])^{2}}{2\mathbf{E}[X]^{2}}+\epsilon_{2}\left(\frac{X- \mathbf{E}[X]}{\mathbf{E}[X]}\right).\] Note that $\mathbf{E}[X-\mathbf{E}[X]]=0$, and $\mathbf{E}[(X-\mathbf{E}[X])^{2}]=\mathbf{Var}[X]$. Thus, if we take the expected value of our above approximation, we get the following: \[\mathbf{E}[\ln(X)]=\ln(\mathbf{E}[X])-\frac{\mathbf{Var}[X]}{2 \mathbf{E}[X]^{2}}+\mathbf{E}\left[\epsilon_{2}\left(\frac{X-\mathbf{E}[X]}{ \mathbf{E}[X]}\right)\right].\] By using bounds on the Taylor series error term $\epsilon_{2}(x)=\mathcal{O}(x^{3})$, one can obtain bounds for this last error term. By (16, 17), both $X=R^{\ell+1}$ and $X=R^{\ell+1}\sin^{2}(\theta^{\ell+1})$ can be expressed as averages of the form \[X=\frac{1}{n_{\ell}^{2}}\sum_{i,j}^{n_{\ell}}f(G_{i},\hat{G}_{j}). \tag{55}\] From the bound on the 3rd moment in Lemma 14 it follows that $\mathbf{E}[\epsilon_{2}(X-\mathbf{E}[X])]=\mathcal{O}(n_{\ell}^{-2})$, thus giving the desired result. ### Variance Approximation Lemma 12: _Both the random variables $X=R^{\ell+1}$ and $X=R^{\ell+1}\sin^{2}(\theta^{\ell})$ satisfy_ \[\mathbf{Var}[\ln(X)]=\frac{\mathbf{Var}[X]}{\mathbf{E}[X]^{2}}+ \mathcal{O}(n_{\ell}^{-2}).\] Proof Starting with (54), and using the first term of the Taylor series approximation for $\ln(1+x)=x+\epsilon_{1}(x)$ now, we have that \[\ln(X)=\ln(\mathbf{E}[X])+\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}+ \epsilon_{1}\left(\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}\right). \tag{56}\] where $\epsilon_{1}(x)$ is the Taylor error term and satisfies $\epsilon_{1}(x)=\mathcal{O}(x^{2})$. Taking the variance of this, we arrive at an approximation of $\mathbf{Var}[\ln(X)]$. \[\mathbf{Var}[\ln(X)]=\mathbf{Var}\left[\ln(\mathbf{E}[X])+\frac{X-\mathbf{E}[X ]}{\mathbf{E}[X]}+\epsilon_{1}\left(\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]} \right)\right]\] \[=\mathbf{Var}\left[\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}\right]+\mathbf{Var} \left[\epsilon_{1}\left(\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}\right)\right]+ \mathbf{Cov}\left(\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]},\epsilon_{1}\left( \frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}\right)\right).\] As with the expected value approximation, this approximation for variance is used twice, once for $X=R^{\ell+1}$, and once for $X=R^{\ell+1}\sin^{2}(\theta^{\ell+1})$ (see Section 2.2), both of which can can be expressed as a sum as in (55). Since $\epsilon_{1}(x)=\mathcal{O}(x^{2})$, we have that the terms with $\epsilon_{1}(x)$ are both $\mathcal{O}(n_{\ell}^{-2})$ from Lemma 14. Simplifying the first term, $\mathbf{Var}\left[\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}\right]=\frac{\mathbf{ Var}[X]}{\mathbf{E}[X]^{2}}$ gives the result of the Lemma. $\blacksquare$ ### Covariance Approximation Lemma 13: _Both the random variables $X=R^{\ell+1}$ and $X=R^{\ell+1}\sin^{2}(\theta^{\ell})$ satisfy_ \[\mathbf{Cov}(\ln(X),\ln(Y))=\frac{\mathbf{Cov}(X,Y)}{\mathbf{E}[X]\mathbf{E}[ Y]}+\mathcal{O}(n_{\ell}^{-2}).\] Proof Using the approximation in (56) for $\ln(X)$ and $\ln(Y)$, we get the following expression for the covariance: \[\mathbf{Cov}(\ln(X),\ln(Y))\] \[=\mathbf{Cov}\left(\frac{X}{\mathbf{E}[X]},\frac{Y}{\mathbf{E}[Y] }\right)+\mathbf{Cov}\left(\frac{X}{\mathbf{E}[X]},\epsilon_{1}\left(\frac{Y- \mathbf{E}[Y]}{\mathbf{E}[Y]}\right)\right)+\mathbf{Cov}\left(\epsilon_{1} \left(\frac{X-\mathbf{E}[X]}{\mathbf{E}[X]}\right),\frac{Y}{\mathbf{E}[Y]}\right)\] \[\quad+\mathbf{Cov}\left(\epsilon_{1}\left(\frac{X-\mathbf{E}[X]}{ \mathbf{E}[X]}\right),\epsilon_{1}\left(\frac{Y-\mathbf{E}[Y]}{\mathbf{E}[Y] }\right)\right).\] We get the desired result from the fact that that the error term $\epsilon_{1}(x)$ satisfies $\epsilon_{1}(x)=\mathcal{O}(x^{2})$ and from our result in Lemma 16. $\blacksquare$ ### Third and Fourth Moment Bound Lemma Lemma 14: _Let $G_{i},\hat{G}_{i}$, $1\leq i\leq n$ be marginally $\mathcal{N}(0,1)$ random variables with correlation $\cos(\theta)$ and independent for different indices $i$. Let $A=\frac{1}{n^{2}}\sum_{i,j}^{n}f(G_{i},\hat{G}_{j})$ be the average over all $n^{2}$ pairs of some function $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ which has finite fourth moment, $\mathbf{E}[f(G_{i},\hat{G}_{i})^{4}]<\infty$. Then the third and fourth central moment of $A$ satisfy_ \[\mathbf{E}[(A-\mathbf{E}[A])^{3}] =\mathcal{O}(n^{-2}) \tag{57}\] \[\mathbf{E}[(A-\mathbf{E}[A])^{4}] =\mathcal{O}(n^{-2}). \tag{58}\] Proof We begin by showing the third moment bound. First, we can express ${\bf E}[(A-{\bf E}[A])^{3}]$ as a sum in the following way: \[A-{\bf E}[A]=\frac{1}{n^{2}}\sum_{i,j}^{n}\left(f(G_{i},\hat{G}_{j})-{\bf E}[f(G _{i},\hat{G}_{j})]\right)\] \[\Longrightarrow\ {\bf E}\left[(A-{\bf E}[A])^{3}\right]=\frac{1}{n^{6}}\sum_{ i_{1},i_{2},i_{3}\atop j_{1},j_{2},j_{3}}^{n}{\bf E}\left[\prod_{k=1}^{3}\left(f(G_{i_ {k}},\hat{G}_{j_{k}})-{\bf E}[f(G_{i_{k}},\hat{G}_{j_{k}})]\right)\right]. \tag{59}\] Note that many of these terms are mean zero. For example, for any configuration of the indices where there is no overlap between the indices $(i_{1},j_{1})$ and the other two index pairs, $\{i_{1},j_{1}\}\cap\{i_{2},j_{2},i_{3},j_{3}\}=\emptyset$ we may use independence to observe that \[{\bf E}\left[\prod_{k=1}^{3}\left(f(G_{i_{k}},\hat{G}_{j_{k}})-{ \bf E}[f(G_{i_{k}},\hat{G}_{j_{k}})]\right)\right]\] \[= {\bf E}\left[f(G_{i_{1}},\hat{G}_{j_{1}})-{\bf E}[f(G_{i_{1}}, \hat{G}_{j_{1}})]\right]{\bf E}\left[\prod_{k=2}^{3}\left(f(G_{i_{k}},\hat{G}_ {j_{k}})-{\bf E}[f(G_{i_{k}},\hat{G}_{j_{k}})]\right)\right]=0.\] When this happens we say that $(i_{1},j_{1})$ is a "reducible point". Similarly, $(i_{2},j_{2})$ or $(i_{3},j_{3})$ can be reducible if they have no overlap with the other two index pairs. To control ${\bf E}\left[(A-{\bf E}[A])^{3}\right]$, it will suffice to enumerate the number of indices $\{i_{1},j_{1},i_{2},j_{2},i_{3},j_{3}\}$ so that all three points $(i_{1},j_{1}),(i_{2},j_{2}),(i_{3},j_{3})$ are _not_ reducible. We call these "irreducible configurations". We now observe that at least one of the points $(i_{1},j_{1})$,$(i_{2},j_{2})$ or $(i_{3},j_{3})$ is reducible whenever the number of unique numbers is $\left\|\bigcup_{k=1}^{3}\{i_{k},j_{k}\}\right\|\geq 5$. This is because, by the pigeonhole principle, if there are no repeated or only one repeated number between 6 indices, then at least one of the 3 pairs $(i_{1},j_{1})$,$(i_{2},j_{2})$ or $(i_{3},j_{3})$ must consist of two unique numbers and therefore is a reducible point. Since the only irreducible configurations can only have at most 4 unique numbers, the number of irreducible configurations is ${\cal O}(n^{4})$ as $n\to\infty$. In fact, a detailed enumeration of the number of configurations reveals that the number of irreducible configurations is precisely \[32(n)_{4}+68(n)_{3}+28(n)_{2}+1(n)_{1} \tag{60}\] The leading term is 32 because there are 32 possible "patterns" for how the indices can be arranged to be both irreducible and contain exactly 4 unique numbers $\left\|\bigcup_{k=1}^{3}\{i_{k},j_{k}\}\right\|=4$; these patterns are listed in Table 3. Each pattern contributes $(n)_{4}=n(n-1)(n-2)(n-3)$ possible index configurations by filling in the 4 unique numbers in all the possible ways. Similarly, there are respectively 68, 28, and 1 pattern(s) for irreducible configurations with 3,2 and 1 unique number(s) in them which each contribute $(n)_{3},(n)_{2}$ and $(n)_{1}$ configurations per pattern (Here $(n)_{k}$ denotes the falling factorial with $k$ terms). Since the number of irreducible configurations is ${\cal O}(n^{4})$, the normalization by $n^{6}$ in (59) shows that ${\bf E}[(A-{\bf E}[A])^{3}]$ is ${\cal O}(n^{-2})$ as desired for the third moment. The argument for the 4th moment is similar. We write ${\bf E}[(A-{\bf E}[A])^{4}]$ as a sum over $i_{1},j_{1},i_{2},j_{2},i_{3},j_{3},i_{4},j_{4}$ and again enumerate irreducible configurations. In this case, once again by the pigeonhole principle any configuration with 7 or more unique points $\left\|\bigcup_{k=1}^{3}\{i_{k},j_{k}\}\right\|\geq 7$ will be reducible. Since there are at most 6 unique numbers, there will be ${\cal O}(n^{6})$ irreducible configurations. A detailed enumeration of all the possible irreducible patterns and the number of unique elements in each yields that the number of irreducible configurations is precisely \[48(n)_{6}+544(n)_{5}+1268(n)_{4}+844(n)_{3}+123(n)_{2}+1(n)_{1}\] The normalization factor of $n^{-8}$ then shows that ${\bf E}[(A-{\bf E}[A])^{4}]={\cal O}(n^{-2})$. $\blacksquare$ Remark 15: _A more detailed enumeration of the $4$-th moment actually shows that the dominant terms in the $4$th moment correspond to the terms in the $2$nd moment written twice, and asymptotically_ \[{\bf E}[(A-{\bf E}[A])^{4}]=3{\bf E}[(A-{\bf E}[A])^{2}]^{2}+{\cal O}(n^{-3}).\] _Here $3$ arises as the number of pair partitions of 4 items, and is related to the fact that $3={\bf E}[G^{4}]$_ Lemma 16: _Let $G_{i},\hat{G}_{i},\,1\leq i\leq n$ be marginally ${\cal N}(0,1)$ random variables with correlation $\cos(\theta)$ and independent for different indices $i$. Let $A_{1}=\frac{1}{n^{2}}\sum_{i,j}^{n}f_{1}(G_{i},\hat{G}_{j})$, and let $A_{2}=\frac{1}{n^{2}}\sum_{i,j}^{n}f_{2}(G_{i},\hat{G}_{j})$, where $f_{1},f_{2}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ have finite fourth moments, ${\bf E}[f_{1}(G_{i},\hat{G}_{i})^{4}],\ {\bf E}[f_{2}(G_{i},\hat{G}_{i})^{4}]<\infty$. Then,_ \[{\bf E}[(A_{1}-{\bf E}[A_{1}])^{2}(A_{2}-{\bf E}[A_{2}])]={\cal O}\left(n^{-2} \right).\] Proof We can express ${\bf E}[(A_{1}-{\bf E}[A_{2}])^{2}(A_{2}-{\bf E}[A_{2}])]$ using sums as follows: \[{\bf E}[(A_{1}-{\bf E}[A_{1}])^{2}(A_{2}-{\bf E}[A_{2}])]\] \[=\frac{1}{n^{6}}\sum_{\begin{subarray}{c}i_{1},i_{2},i_{3}\\ j_{1},j_{2},j_{3}\end{subarray}}^{n}{\bf E}\left[\prod_{k=1}^{2}\left(f_{1}(G_{ i_{k}},\hat{G}_{j_{k}})-{\bf E}[f_{1}(G_{i_{k}},\hat{G}_{j_{k}})]\right)\left(f_{2}(G_ {i_{3}},\hat{G}_{j_{3}})-{\bf E}[f_{2}(G_{i_{3}},\hat{G}_{j_{3}})]\right) \right].\] By the same argument as in Lemma 14, we can show that the number of nonzero terms in the above summation is ${\cal O}(n^{4})$ as $n\rightarrow\infty$. Thus, we have that ${\bf E}[(A-{\bf E}[A])^{2}(A^{\prime}-{\bf E}[A^{\prime}])]={\cal O}(n^{-2})$. $\blacksquare$ ### Derivation of Useful Identities - Equations (14, 15) Let $G\in\mathbb{R}^{n}$ be a Gaussian vector with iid entries $G_{i}\sim{\cal N}(0,1)$. Then, by standard properties of Gaussians, the function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ given by $f(x)=\langle G,x\rangle$ is a Gaussian random variable. Further, $f(x)$ has the following properties: * For all $x\in\mathbb{R}^{n}$, $f(x)\sim{\cal N}(0,\\|x\\|^{2})$ * For any two vectors $x_{\alpha},x_{\beta}\in\mathbb{R}^{n}$, the joint distribution of $f(x_{\alpha}),f(x_{\beta})$ is jointly Gaussian with \[\begin{bmatrix}f(x_{\alpha})\\ f(x_{\beta})\end{bmatrix}\sim\mathcal{N}\left(0,\Sigma(x_{\alpha},x_{\beta}) \right),\qquad\Sigma(x_{\alpha},x_{\beta}):=\begin{bmatrix}\\|x_{\alpha}\\|^{2}& \langle x_{\alpha},x_{\beta}\rangle\\ \langle x_{\alpha},x_{\beta}\rangle&\\|x_{\beta}\\|^{2}\end{bmatrix}.\] $\Sigma(x_{\alpha},x_{\beta})$ is sometimes called the $2\times 2$ Gram matrix of the vectors $x_{\alpha},x_{\beta}$. In our setting of our fully connected neural network, any index $i\in[n_{l+1}]$ in the vector of $z^{\ell+1}$ is actually the inner product with the $i$-th row \[z_{i}^{\ell+1}(x)=\sqrt{\frac{2}{n_{\ell}}}\langle W_{i,\cdot}^{\ell+1}, \varphi(z^{\ell}(x))\rangle.\] Note that each row $W_{i,\cdot}^{\ell+1}$ is a Gaussian vector, so the previous fact about Gaussians applies and we see that the entries of $z^{\ell+1}$ are conditionally Gaussian given the value of the previous layer. By the previous Gaussian fact, we have that $z_{i}^{\ell+1}(x_{\alpha})$, $z_{i}^{\ell+1}(x_{\beta})$ are jointly Gaussian with \[\begin{bmatrix}z_{i}^{\ell+1}(x_{\alpha})\\ z_{i}^{\ell+1}(x_{\beta})\end{bmatrix}\sim\mathcal{N}\left(0,\frac{2}{n_{\ell}} \begin{bmatrix}\\|\varphi_{\alpha}^{\ell}\\|^{2}&\langle\varphi_{\alpha}^{\ell},\varphi_{\beta}^{\ell}\rangle\\ \langle\varphi_{\alpha}^{\ell},\varphi_{\beta}^{\ell}\rangle&\\|\varphi_{ \beta}^{\ell}\\|^{2}\end{bmatrix}\right)=:\mathcal{N}\left(0,K^{\ell}\right),\] where we use $K^{\ell}$ to denote the $2\times 2$ covariance matrix. $K^{\ell}$ is precisely the $2\times 2$ Gram matrix of the previous layer $\varphi_{\alpha}^{\ell}$, $\varphi_{\beta}^{\ell}$ scaled by $2/n_{\ell}$ and its entries $K_{\gamma\delta}^{\ell}$, for $\gamma\in\{\alpha,\beta\},\delta\in\{\alpha,\beta\}$ are actually _averages_ of entries in the previous layer \[K_{\gamma,\delta}^{\ell}:=\frac{2}{n_{\ell}}\langle\varphi_{\gamma}^{\ell}, \varphi_{\delta}^{\ell}\rangle=\frac{1}{n_{\ell}}\sum_{k=1}^{n_{\ell}}2\varphi (z_{k}^{\ell}(x_{i}))\varphi(z_{k}^{\ell}(x_{j})).\] Moreover, in the weight matrix $W^{\ell+1}$, the $i^{th}$ and $j^{th}$ rows ($W_{i,\cdot}^{\ell+1}$ and $W_{j,\cdot}^{\ell+1}$, respectively) are independent. Therefore, all entries of $z^{\ell+1}$ are identically distributed and con \begin{table} \begin{tabular}{c c c c c} $(i_{1},j_{1})$ & $(i_{2},j_{2})$ & $(i_{3},j_{3})$ & \\ \hline $\{(a,b),(b,a)\}$ & $\times$ & $\{(a,c),(c,a)\}$ & $\times$ & $\{(a,d),(d,a)\}$ & 8 patterns \\ $\{(a,b),(b,a)\}$ & $\times$ & $\{(a,c),(c,a)\}$ & $\times$ & $\{(c,d),(d,c)\}$ & 8 patterns \\ $\{(a,b),(b,a)\}$ & $\times$ & $\{(c,d),(d,c)\}$ & $\times$ & $\{(a,c),(c,a)\}$ & 8 patterns \\ $\{(a,c),(c,a)\}$ & $\times$ & $\{(a,b),(b,a)\}$ & $\times$ & $\{(c,d),(c,b)\}$ & 8 patterns \\ \end{tabular} \end{table} Table 3: All 32 irreducible patterns using exactly 4 unique index values $a,b,c,d$. For example the pattern $(i_{1},j_{1}),(i_{2},j_{2}),(i_{3},j_{3})=(a,b),(a,c),(a,d)$ represents all configurations where $i_{1}=i_{2}=i_{3}$ and the $j$’s are all unique and different from $i$. For each pattern, there are $(n)_{4}=n(n-1)(n-2)(n-3)$ configurations by filling in $a,b,c,d$ with unique numbers in $[n]$. These are the dominant terms in (59). ditionally independent given $\varphi(z^{\ell})$. From this fact, we can equivalently write the entries explicitly as \[z_{i}^{\ell+1}(x_{\alpha})=\sqrt{\frac{2}{n_{\ell}}}\\|\varphi_{\alpha}^{\ell}\\|G _{i},\quad z_{i}^{\ell+1}(x_{\beta})=\sqrt{\frac{2}{n_{\ell}}}\\|\varphi_{\beta }^{\ell}\\|\hat{G}_{i}, \tag{61}\] where $G_{i},\hat{G}_{i}$ are marginally $\mathcal{N}(0,1)$ variables with covariance $\mathbf{Cov}(G_{i},\hat{G}_{i})=\cos(\theta^{\ell})$ and independent for different indices. This formulation precisely ensures that the covariance structure for the entries is exactly what is specified by the covariance kernel $K^{\ell}$. With this representation of $z_{i}^{\ell+1}(x_{\alpha})$ and $z_{i}^{\ell+1}(x_{\beta})$, we can apply $\varphi(\cdot)$ to each entry. By using the property of ReLU $\varphi(\lambda x)=\lambda\varphi(x)$ for $\lambda>0$ to factor out the norms, we obtain \[\varphi(z_{i}^{\ell+1}(x_{\alpha}))=\sqrt{\frac{2}{n_{\ell}}}\\|\varphi_{ \alpha}^{\ell}\\|\varphi(G_{i}),\quad\varphi(z_{i}^{\ell+1}(x_{\beta}))=\sqrt{ \frac{2}{n_{\ell}}}\\|\varphi_{\beta}^{\ell}\\|\varphi(\hat{G}_{i}). \tag{62}\] Taking the norm/inner product of the vector now yields (14-16) as desired. ### Cauchy-Binet and Determinant of the Gram Matrix - Equation (16) To prove this identity, we begin with the fact that \[\\|\varphi_{\alpha}^{\ell+1}\\|^{2}\\|\varphi_{\beta}^{\ell+1}\\|^{2}\sin^{2}( \theta^{\ell+1})=\det\begin{bmatrix}\\|\varphi_{\alpha}^{\ell+1}\\|^{2}&\langle \varphi_{\alpha}^{\ell+1},\varphi_{\beta}^{\ell+1}\rangle\\ \langle\varphi_{\alpha}^{\ell+1},\varphi_{\beta}^{\ell+1}\rangle&\\|\varphi_{ \beta}^{\ell+1}\\|^{2}\end{bmatrix}.\] By the Cauchy-Binet identity, we can express the determinant as \[\det\begin{bmatrix}\\|\varphi_{\alpha}^{\ell+1}\\|^{2}&\langle\varphi_{\alpha} ^{\ell+1},\varphi_{\beta}^{\ell+1}\rangle\\ \langle\varphi_{\alpha}^{\ell+1},\varphi_{\beta}^{\ell+1}\rangle&\\|\varphi_{ \beta}^{\ell+1}\\|^{2}\end{bmatrix}=\sum_{1\leq i<j\leq n_{\ell}}\left(\varphi_ {i;\alpha}^{\ell+1}\varphi_{j;\beta}^{\ell+1}-\varphi_{j;\alpha}^{\ell+1} \varphi_{i;\beta}^{\ell+1}\right)^{2}. \tag{63}\] We can express layer $\ell+1$ using the following conditioning on the previous layer \[\varphi_{i;\alpha}^{\ell+1}=\sqrt{\frac{2}{n_{\ell}}}\\|\varphi_{\alpha}^{\ell }\\|\cdot\varphi(G_{i}),\quad\varphi_{i;\beta}^{\ell+1}=\sqrt{\frac{2}{n_{\ell}} }\\|\varphi_{\beta}^{\ell}\\|\cdot\varphi(\hat{G}_{i}).\] Applying this to our expression in (63), and dividing both sides by $\\|\varphi_{\alpha}^{\ell}\\|^{2}\\|\varphi_{\beta}^{\ell}\\|^{2}$, we get \[\frac{\\|\varphi_{\alpha}^{\ell+1}\\|^{2}\\|\varphi_{\beta}^{\ell+1}\\|^{2}}{\\| \varphi_{\alpha}^{\ell}\\|^{2}\\|\varphi_{\beta}^{\ell}\\|^{2}}\sin^{2}(\theta^{ \ell+1})=\left(\frac{2}{n_{\ell}}\right)^{2}\sum_{1\leq i<j\leq n_{\ell}} \left(\varphi(G_{i})\varphi(\hat{G}_{j})-\varphi(G_{j})\varphi(\hat{G}_{i}) \right)^{2}.\] Due to the fact that the summand is equal to $0$ when $i=j$, we can rewrite the sum as follows: \[R^{\ell+1}\sin^{2}(\theta^{\ell+1})=\frac{1}{2}\left(\frac{2}{n_{\ell}}\right) ^{2}\sum_{i,j}^{n_{\ell}}\left(\varphi(G_{i})\varphi(\hat{G}_{j})-\varphi(G_{ j})\varphi(\hat{G}_{i})\right)^{2},\] and we have the desired result. ### Expected Value calculations In this section, we derive the formulas for $\mathbf{E}\left[R^{\ell+1}\right]$, $\mathbf{E}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]$. We use $J_{a,b}$ to represent $J_{a,b}(\theta^{\ell})$. Note that $\mathbf{E}[\varphi^{2}(G)]=\frac{1}{2}$, $\mathbf{E}[\varphi^{4}(G)]=\frac{3}{2}$. Calculation of $\mathbf{E}\left[R^{\ell+1}\right]$ : First, we apply the identity as in (17): \[\mathbf{E}\left[R^{\ell+1}\right]=\left(\frac{2}{n_{\ell}}\right)^{2}\mathbf{ E}\left[\sum_{i,j=1}^{n_{\ell}}\varphi^{2}(G_{i})\varphi^{2}(\hat{G}_{j})\right].\] Whenever $i=j$, taking the expected value will give us a $J_{2,2}$ term. When $i\neq j$, the expected value of this term will be $\mathbf{E}[\varphi^{2}(G)]^{2}=\frac{1}{4}$. Since $i=j$ happens $n_{\ell}$ times, and therefore $i\neq j$ happens $n_{\ell}^{2}-n_{\ell}$ times, we arrive at the following expression: \[\mathbf{E}\left[R^{\ell+1}\right]=\left(\frac{2}{n_{\ell}}\right)^{2}\left(n _{\ell}J_{2,2}+(n_{\ell}^{2}-n_{\ell})\left(\frac{1}{4}\right)\right)=\frac{4J _{2,2}-1}{n_{\ell}}+1.\] Calculation of $\mathbf{E}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]$ : Applying the identity (16), we get \[\mathbf{E}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right] =\frac{2}{n_{\ell}^{2}}\mathbf{E}\left[\sum_{i,j}^{n_{\ell}} \left(\varphi(G_{i})\varphi(\hat{G}_{j})-\varphi(G_{j})\varphi(\hat{G}_{i}) \right)^{2}\right]\] \[=\frac{2}{n_{\ell}^{2}}\mathbf{E}\left[\sum_{i,j}^{n_{\ell}} \left(\varphi^{2}(G_{i})\varphi^{2}(\hat{G}_{j})-2\varphi(G_{i})\varphi(\hat{ G}_{i})\varphi(G_{j})\varphi(\hat{G}_{j})+\varphi^{2}(G_{j})\varphi^{2}(\hat{G}_{i}) \right)\right].\] In the case where $i=j$, the expected value is equal to $0$. Thus, we only need to consider the case where $i\neq j$, which happens $n_{\ell}^{2}-n_{\ell}$ times. When $i\neq j$, the expectation of $\varphi(G_{i})\varphi(\hat{G}_{i})\varphi(G_{j})\varphi(\hat{G}_{j})$ is $J_{1,1}^{2}$, and the expectation of $\varphi^{2}(G_{i})\varphi^{2}(\hat{G}_{j})$ is $\frac{1}{4}$. All together, we have \[\mathbf{E}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]=\left(\frac{2}{n_{ \ell}^{2}}\right)(n_{\ell}^{2}-n_{\ell})\left(\frac{1}{4}-2J_{1,1}^{2}+\frac{ 1}{4}\right)=\frac{(n_{\ell}-1)(1-4J_{1,1}^{2})}{n_{\ell}}.\] ### Variance and Covariance Calculations In this section, $\mathbf{Var}\left[R^{\ell+1}\right]$, $\mathbf{Var}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]$, and $\mathbf{Cov}\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1}),R^{\ell+1}\right)$ are evaluated. We use $J_{a,b}$ to represent $J_{a,b}(\theta^{\ell})$. Note that $\mathbf{E}[\varphi^{2}(G)]=\frac{1}{2}$, $\mathbf{E}[\varphi^{4}(G)]=\frac{3}{2}$. We will see that there are simple functions $f_{1},f_{2}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ so that all of the variance and covariance calculations can be expressed as sums over $i_{1},j_{1},i_{2},j_{2}$ of the form \[\frac{1}{n_{\ell}^{4}}\sum_{\begin{subarray}{c}i_{1},j_{1}\\ i_{2},j_{2}\end{subarray}}\mathbf{Cov}\left[f_{1}(G_{i_{1}},\hat{G}_{j_{1}}), f_{2}(G_{i_{2}},\hat{G}_{j_{2}})\right] \tag{64}\] \[= \frac{1}{n_{\ell}^{4}}\sum_{\begin{subarray}{c}i_{1},j_{1}\\ i_{2},j_{2}\end{subarray}}\left(\mathbf{E}\left[f_{1}(G_{i_{1}},\hat{G}_{j_{1}}) f_{2}(G_{i_{2}},\hat{G}_{j_{2}})\right]-\mathbf{E}\left[f_{1}(G_{i_{1}},\hat{G}_{j_{1}}) \right]\mathbf{E}\left[f_{2}(G_{i_{2}},\hat{G}_{j_{2}})\right]\right), \tag{65}\] where the sum goes over index configuration $(i_{1},j_{1}),(i_{2},j_{2})\in[n_{\ell}]^{4}$. We will use this form to organize our calculations of the variance and covariance formulas. The strategy is to evaluate each term in the sum (64) individually. Since the random variables $\{G_{i},\hat{G}_{i}\}_{i=1}^{n}$ are exchangeable, the only thing that matters is the "pattern" of which of the indices $i_{1},j_{1},i_{2},j_{2}$ are repeated vs which are distinct. For example, there will be $n$ index configurations where $i_{1}=j_{1}=i_{2}=j_{2}$ are all equal. All $n$ of these give same contribution. There are $(n)_{4}=n(n-1)(n-2)(n-3)$ configurations where $i_{1},j_{1},i_{2},j_{2}$ are all distinct. Knowing which indices are repeated/distinct allows us to evaluate the corresponding term in (64). We use the following formal notion of a pattern to organize this idea of repeated vs. distinct indices. Definition 17: _A pattern for $(i_{1},j_{1}),(i_{2},j_{2})$ is a subset of all possible index configurations $(i_{1},j_{1}),(i_{2},j_{2})\in[n]^{4}$ represented by an assignment of each index to the letters $a,b,c,d$. Each letter $a,b,c,d$ represents a choice of unique indices from $[n]$._ For example, the pattern $(i_{1},j_{1}),(i_{2},j_{2})=(a,a),(a,a)$ represents the set of all index configurations where all indices are equal and the pattern $(i_{1},j_{1}),(i_{2},j_{2})=(a,b),(c,d)$ represents the set with all indices unique. The pattern $(i_{1},j_{1}),(i_{2},j_{2})=(a,b),(a,c)$ represents all configurations where $i_{1}=i_{2}$ and $j_{1},j_{2}$ are unique and different from $i_{1}=i_{2}$. For this pattern, there are $(n)_{3}=n(n-1)(n-2)$ configurations by filling in $a,b,c$ with unique numbers in $[n]$. More generally, for a pattern with $k$ letters, there are $(n)_{k}$ configurations that fall into that pattern. Fortunately, when enumerating (64), many patterns have _no_ contribution and can be ignored. We formalize this in the following definition. Definition 18: _We say that the configuration of indices $(i_{1},j_{1}),(i_{2},j_{2})$ is reducible if $\{i_{1},j_{1}\}\cap\{i_{2},j_{2}\}=\varnothing$. Otherwise, the index configuration is called irreducible. A pattern is called reducible if all index configuration in that pattern are reducible._ By the independence of the random variables $f_{1}(G_{i_{1}},G_{j_{1}})$ and $f_{2}(G_{i_{2}},G_{j_{2}})$, whenever $(i_{1},j_{1}),(i_{2},j_{2})$ is reducible, we see that the corresponding term in (64) completely vanishes! Therefore, to evaluate (64), we have only to understand the contribution of _irreducible configurations_. The irreducible configurations can be organized into _irreducible patterns_. For example, the pattern $(a,b),(c,c)$ is reducible (since formally $\{a,b\}\cap\{c\}=\varnothing$) and so any configuration from this pattern has _no_ contribution in the expectation. There are 11 irreducible patterns. (All these patterns are listed as part of Table 4.) The expected value of the terms for each pattern will give a contribution that is expressed in terms of the $J_{a,b}$ depending on the details of exactly which indices are repeated. Then by enumerating the number of configurations in each pattern, we can evaluate (64). This strategy is precisely how we evaluate each variance/covariance in this section. Calculation of $\mathbf{Var}\left[R^{\ell+1}\right]$ : First, applying the identity in (17), we get \[\mathbf{Var}\left[R^{\ell+1}\right] =\left(\frac{2}{n_{\ell}}\right)^{4}\mathbf{Var}\left[\sum_{i,j=1} ^{n_{\ell}}\varphi^{2}(G_{i})\varphi^{2}(\hat{G}_{j})\right]\] \[=\frac{16}{n_{\ell}^{4}}\left(\mathbf{E}\left[\sum_{\begin{subarray} {c}i_{1},j_{1}\\ i_{2},j_{2}\end{subarray}}\varphi^{2}(G_{i_{1}})\varphi^{2}(\hat{G}_{j_{1}}) \varphi^{2}(G_{i_{2}})\varphi^{2}(\hat{G}_{j_{2}})\right]-\mathbf{E}\left[ \sum_{i,j=1}^{n_{\ell}}\varphi^{2}(G_{i})\varphi^{2}(\hat{G}_{j})\right]^{2} \right).\] $\mathbf{Var}[R^{\ell+1}]$ follows the form of (64), with $f_{1}(G_{i},\hat{G}_{i})=f_{2}(G_{i},\hat{G}_{i})=\varphi^{2}(G_{i})\varphi^{2 }(\hat{G}_{j})$. We then evaluate the contribution from each irreducible pattern in Table 4. Combining all these cases and simplifying based on powers of $\frac{1}{n_{\ell}}$, we arrive at the following expression: \[\mathbf{Var}\left[R^{\ell+1}\right] =\frac{4}{n_{\ell}}(J_{2,2}+1)+\frac{16}{n_{\ell}^{2}}\left(2J_{4,2}-\frac{5}{2}J_{2,2}+J_{2,2}^{2}+\frac{5}{8}\right)\] \[+\frac{16}{n_{\ell}^{3}}\left(J_{4,4}-2J_{4,2}-2J_{2,2}^{2}+2J_{2,2}-\frac{9}{8}\right).\] \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|} \hline \# & $(i_{1},j_{1})$ & $(i_{2},j_{2})$ & $\mathbf{E}[f_{1}(G_{i_{1}},\hat{G}_{j_{1}})]$ & $\mathbf{E}[f_{2}(G_{i_{2}},\hat{G}_{j_{2}})]$ & $\mathbf{E}[f_{1}(G_{i_{1}},\hat{G}_{j_{1}})f_{2}(G_{i_{2}},\hat{G}_{j_{2}})]$ \\ \hline \hline $(n)_{1}$ & $(a,a)$ & $(a,a)$ & $J_{2,2}$ & $J_{2,2}$ & $J_{4,4}$ \\ \hline \multirow{6}{}{$(n)_{2}$} & $(a,b)$ & $(a,b)$ & \multirow{2}{}{$\left(\frac{1}{2}\right)^{2}$} & \multirow{2}{}{$\left(\frac{1}{2}\right)^{2}$} & \multirow{2}{}{$\left(\frac{3}{2}\right)^{2}$} \\ \cline{2-3} \cline{5-6} & $(a,b)$ & $(b,a)$ & & & $J_{2,2}$ \\ \cline{2-3} \cline{5-6} & $(a,a)$ & $(a,b)$ & & & \\ \cline{2-3} \cline{5-6} & $(a,b)$ & $(a,a)$ & & & $\frac{1}{2}J_{4,2}$ \\ \cline{2-3} \cline{5-6} & $(b,a)$ & $(a,a)$ & & & \\ \hline \multirow{6}{}{$(n)_{3}$} & $(a,b)$ & $(a,c)$ & \multirow{6}{}{$\left(\frac{1}{2}\right)^{2}$} & \multirow{6}{}{$\left(\frac{1}{2}\right)^{2}$} & \multirow{6}{}{$\left(\frac{1}{2}\right)^{2}$} \\ \cline{2-3} \cline{5-6} & $(a,b)$ & $(c,b)$ & & & \\ \cline{1-1} \cline{2-3} \cline{5-6} & $(a,b)$ & $(c,a)$ & & & \\ \cline{1-1} \cline{2-3} \cline{5-6} & $(a,b)$ & $(b,c)$ & & & \\ \hline \end{tabular} \end{table} Table 4: $\mathbf{Var}[R^{\ell+1}]$ calculated in the form of (64) with $f_{1}(G_{i},\hat{G}_{j})=f_{2}(G_{i},\hat{G}_{j})=\varphi^{2}(G_{i})\varphi^{2 }(\hat{G}_{j})$. The contribution from all 11 possible _irreducible_ patterns of the indices are shown. Calculation of $\mathbf{Var}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]$ : Applying identity (16), we can express $\mathbf{Var}[R^{\ell+1}\sin^{2}(\theta^{\ell+1})]$ as \[\mathbf{Var}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]\] \[=\frac{1}{4}\left(\frac{2}{n_{\ell}}\right)^{4}\mathbf{Var}\left[ \sum_{i,j=1}^{n_{\ell}}\left(\varphi(G_{i})\varphi(\hat{G}_{j})-\varphi(G_{j}) \varphi(\hat{G}_{i})\right)^{2}\right]\] \[=\frac{4}{n_{\ell}^{4}}\sum_{\begin{subarray}{c}i_{1},j_{1}\\ i_{2},j_{2}\end{subarray}}\mathbf{Cov}\left(\left(\varphi(G_{i_{1}})\varphi( \hat{G}_{j_{1}})-\varphi(G_{j_{1}})\varphi(\hat{G}_{i_{1}})\right)^{2},\left( \varphi(G_{i_{2}})\varphi(\hat{G}_{j_{2}})-\varphi(G_{j_{2}})\varphi(\hat{G}_ {i_{2}})\right)^{2}\right)\] Note that we can express $\mathbf{Var}[R^{\ell+1}\sin^{2}(\theta^{\ell+1})]$ as in (64) by letting $f_{1}(G_{i},\hat{G}_{j})=f_{2}(G_{i},\hat{G}_{j})=(\varphi(G_{i})\varphi( \hat{G}_{j})-\varphi(G_{j})\varphi(\hat{G}_{i}))^{2}$.We then evaluate the contribution from each irreducible pattern in Table 5. Combining all these cases and simplifying based on powers of $\frac{1}{n_{\ell}}$, we arrive at the following expression: \[\mathbf{Var}\left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]\] \[=\frac{8}{n_{\ell}}\left(-8J_{1,1}^{4}+8J_{1,1}^{2}J_{2,2}+4J_{1, 1}^{2}-8J_{1,1}J_{3,1}+J_{2,2}+1\right)\] \[+\frac{2}{n_{\ell}^{2}}\left(80J_{1,1}^{4}-96J_{1,1}^{2}J_{2,2}-4 0J_{1,1}^{2}+96J_{1,1}J_{3,1}+24J_{2,2}^{2}-12J_{2,2}-32J_{3,1}^{2}+5\right)\] \[+\frac{2}{n_{\ell}^{3}}\left(-48J_{1,1}^{4}+64J_{1,1}^{2}J_{2,2}+ 24J_{1,1}^{2}-64J_{1,1}J_{3,1}-24J_{2,2}^{2}+8J_{2,2}+32J_{3,1}^{2}-9\right).\] $\mathbf{Var}[R^{\ell+1}\sin^{2}(\theta^{\ell+1})]$ Calculation \begin{tabular}{\|c\|c\|c\|c\|c\|c\|} \hline \# & $(i_{1},j_{1})$ & $(i_{2},j_{2})$ & $\mathbf{E}[f(G_{i_{1}},\hat{G}_{j_{1}})]$ & $\mathbf{E}[f(G_{i_{2}},\hat{G}_{j_{2}})]$ & $\mathbf{E}[f(G_{i_{1}},\hat{G}_{j_{1}})f(G_{i_{2}},\hat{G}_{j_{2}})]$ \\ \hline \hline \multirow{2}{}{$(n)_{2}$} & $(a,b)$ & $(a,b)$ & & & \multirow{2}{}{$6J_{2,2}^{2}-8J_{3,1}^{2}+\frac{9}{2}$} \\ \cline{2-6} & $(a,b)$ & $(b,a)$ & & & \\ \hline \multirow{2}{}{$(n)_{3}$} & $(a,b)$ & $(a,c)$ & \multirow{2}{}{$\frac{1}{2}-2J_{1,1}^{2}$} & $\frac{1}{2}-2J_{1,1}^{2}$ & \multirow{2}{}{$4J_{2,2}J_{1,1}^{2}-4J_{3,1}J_{1,1}+\frac{1}{2}J_{2,2}+\frac{3}{4}$} \\ \cline{2-6} & $(a,b)$ & $(c,b)$ & & & \\ \cline{1-1} \cline{2-6} & $(a,b)$ & $(b,c)$ & & & \\ \hline \end{tabular} Table 5: $\mathbf{Var}[R^{\ell+1}\sin^{2}(\theta^{\ell+1})]$ calculated in the form of (64) with $f_{1}(G_{i},\hat{G}_{j})=f_{2}(G_{i},\hat{G}_{j})$$=(\varphi(G_{i})\varphi(\hat{G}_{j})-\varphi(G_{j})\varphi(\hat{G}_{i}))^{2}$. The _non-zero_ contribution _irreducible_ patterns of the indices are shown. Note that because $f_{1}(G_{i_{1}},G_{j_{1}})=0$ when $i_{1}=j_{1}$ and $f_{2}(G_{i_{2}},G_{j_{2}})=0$ when $i_{2}=j_{2}$, there are 5 irreducible patterns (of the possible 11) that have zero contribution and are not displayed in this table. \[\mathbf{Cov}\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1}),R^{\ell+1}\right)=\mathbf{E} \left[\left(R^{\ell+1}\right)^{2}\sin^{2}(\theta^{\ell+1})\right]-\mathbf{E} \left[R^{\ell+1}\sin^{2}(\theta^{\ell+1})\right]\mathbf{E}\left[R^{\ell+1} \right].\] Applying known identities (16, 17) derived in Appendix A.5 and Appendix A.6, we can express this in the form of (64), where $f_{1}(G_{i},\hat{G}_{j})=(\varphi(G_{i})\varphi(\hat{G}_{j})-\varphi(G_{j}) \varphi(\hat{G}_{i}))^{2}$, and $f_{2}(G_{i},\hat{G}_{j})=\varphi^{2}(G_{i})\varphi^{2}(\hat{G}_{j})$. Table 6 shows the calculation of all the irreducible patterns. Collecting all cases and simplifying based on powers of $\frac{1}{n_{\ell}}$ gives: \[\mathbf{Cov}\left(R^{\ell+1}\sin^{2}(\theta^{\ell+1}),R^{\ell+1} \right)=\frac{1}{n_{\ell}}\left(16J_{1,1}^{2}-32J_{1,1}J_{3,1}+8J_{2,2}+8\right)\] \[+\frac{1}{n_{\ell}^{2}}\left(32J_{1,1}^{2}J_{2,2}-40J_{1,1}^{2}+ 96J_{1,1}J_{3,1}-32J_{1,1}J_{3,3}+16J_{2,2}^{2}-32J_{2,2}-32J_{3,1}^{2}+16J_{ 4,2}+10\right)\] \[+\frac{1}{n_{\ell}^{3}}\left(24J_{1,1}^{2}-32J_{1,1}^{2}J_{2,2}-64 J_{1,1}J_{3,1}+32J_{1,1}J_{3,3}-16J_{2,2}^{2}+24J_{2,2}+32J_{3,1}^{2}-16J_{ 4,2}-18\right).\] ### Infinite Width Update Rule Lemma 19: _Let $f(x)$ be a feed forward neural network as defined in 2, and assume each layer has infinite width (ie. $n_{\ell}=\infty$ for $1\leq\ell\leq L$). Then, the angle $\theta^{\ell}$ between inputs at layer $\ell$ of $f$ follows the update rule given by_ \[\cos(\theta^{\ell+1})=2J_{1,1}(\theta^{\ell}).\] Proof* We begin with the identity in (15), and use the inner product to introduce a $\cos(\theta^{\ell+1})$ term into our equation. \[\frac{\\|\varphi_{\alpha}^{\ell}\\|\\|\varphi_{\beta}^{\ell}\\|}{n_{\ell}}\sum_{i=1} ^{n_{\ell}}2\varphi(G_{i})\varphi(\hat{G}_{i})=\langle\varphi_{\alpha}^{\ell+1},\varphi_{\beta}^{\ell+1}\rangle=\\|\varphi_{\alpha}^{\ell+1}\\|\\|\varphi_{ \beta}^{\ell+1}\\|\cos(\theta^{\ell+1}).\] Applying the identities in (14) to $\\|\varphi_{\alpha}^{\ell+1}\\|$ and $\\|\varphi_{\beta}^{\ell+1}\\|$, we get \[\frac{\\|\varphi_{\alpha}^{\ell}\\|\\|\varphi_{\beta}^{\ell}\\|}{n_{ \ell}}\sum_{i=1}^{n_{\ell}}2\varphi(G_{i})\varphi(\hat{G}_{i}) =\sqrt{\frac{\\|\varphi_{\alpha}^{\ell+1}\\|^{2}}{n_{\ell}}\sum_{i= 1}^{n_{\ell}}2\varphi^{2}(G_{i})}\sqrt{\frac{\\|\varphi_{\beta}^{\ell+1}\\|^{2} }{n_{\ell}}\sum_{i=1}^{n_{\ell}}2\varphi^{2}(\hat{G}_{i})}\cos(\theta^{\ell+1})\] \[\frac{1}{n_{\ell}}\sum_{i=1}^{n_{\ell}}\varphi(G_{i})\varphi(\hat {G}_{i}) =\sqrt{\frac{1}{n_{\ell}}\sum_{i=1}^{n_{\ell}}\varphi^{2}(G_{i}) }\sqrt{\frac{1}{n_{\ell}}\sum_{i=1}^{n_{\ell}}\varphi^{2}(\hat{G}_{i})}\cos( \theta^{\ell+1}).\] Taking the limit as $n_{\ell}\to\infty$, and using the Law of Large Numbers, we have \[\lim_{n_{\ell}\to\infty}\left(\frac{1}{n_{\ell}}\sum_{i=1}^{n_{ \ell}}\varphi(G_{i})\varphi(\hat{G}_{i})\right) =\lim_{n_{\ell}\to\infty}\left(\sqrt{\frac{1}{n_{\ell}}\sum_{i=1} ^{n_{\ell}}\varphi^{2}(G_{i})}\sqrt{\frac{1}{n_{\ell}}\sum_{i=1}^{n_{\ell}} \varphi^{2}(\hat{G}_{i})}\cos(\theta^{\ell+1})\right)\] \[\mathbf{E}\left[\varphi(G_{i})\varphi(\hat{G}_{i})\right] =\sqrt{\mathbf{E}\left[\varphi^{2}(G_{i})\right]}\sqrt{\mathbf{E }[\varphi^{2}(\hat{G}_{i})]}\cos(\theta^{\ell+1})\] \[J_{1,1}(\theta^{\ell}) =\frac{1}{2}\cos(\theta^{\ell+1})\implies\cos(\theta^{\ell+1})=2J _{1,1}(\theta^{\ell}).\] Note that in the $J_{p}$ notation from Cho and Saul (2009), the translation is $2J_{1,1}=\frac{J_{1}}{\pi}$, so this is is the result of the theorem. Appendix B. ### Derivation of Lower-Order J Functions - Proof of Proposition 2 Proof [Of formula for $J_{0,0}$] We find a differential equation that $J_{0,0}$ satisfies and solve it to obtain the formula. First note the initial condition $J_{0,0}(0)=\mathbf{E}[1\{G>0\}]=\frac{1}{2}$. To find $J_{0,0}^{\prime}(\theta)$, we take the derivative into the expectation and have by the chain rule that \[J_{0,0}^{\prime}(\theta) =\mathbf{E}[1\{G>0\}1^{\prime}\{G\cos\theta+W\sin\theta>0\}G](- \sin\theta)\] \[+\mathbf{E}[1\{G>0\}1^{\prime}\{G\cos\theta+W\sin\theta>0\}W]\cos\theta.\] Applying the change of variables as in (37), we have \[J_{0,0}^{\prime}(\theta) =\mathbf{E}[(Z\cos\theta+Y\sin\theta)1\{Z\cos\theta+Y\sin\theta>0 \}1^{\prime}\{Z>0\}](-\sin\theta)\] \[+\mathbf{E}[(Z\sin\theta-Y\cos\theta)1\{Z\cos\theta+Y\sin\theta>0 \}1^{\prime}\{Z>0\}]\cos\theta\] \[=\mathbf{E}[(Y\sin\theta)1\{Y\sin\theta>0\}]\frac{-\sin\theta}{ \sqrt{2\pi}}+\mathbf{E}[(-Y\cos\theta)1\{Y\sin\theta>0\}]\frac{\cos\theta}{ \sqrt{2\pi}}\] \[=(-\sin^{2}\theta-\cos^{2}\theta)\mathbf{E}[Y1\{Y>0\}]\frac{1}{ \sqrt{2\pi}}=-\frac{1}{2\pi},\] where we have used (32) to evaluate the integrals involving $1^{\prime}\{Z>0\}$ and $\mathbf{E}[Y1\{Y>0\}]=(\sqrt{2\pi})^{-1}$ from Lemma 7. We now have $J^{\prime}_{0,0}(\theta)=-\frac{1}{2\pi}$ with initial condition given by $J_{0,0}(0)=0$. Solving this differential equation gives the desired result. $\blacksquare$ Proof [$J_{1,0}$ and $J_{1,1}$] Here we use the Gaussian integration-by-parts strategy (31 -32). Formula for $J_{1,0}$: \[J_{1,0}(\theta) =\mathbf{E}[G1\{G>0\}\ 1\{G\cos\theta+W\sin\theta>0\}]\] \[=\mathbf{E}\left[\frac{d}{dg}\left(\ 1\{G>0\}\ 1\{G\cos\theta+W\sin \theta>0\}\right)\right]\] \[=\mathbf{E}\left[1^{\prime}\{G>0\}\ 1\{G\cos\theta+W\sin\theta>0\} \right]+\mathbf{E}\left[1\{G>0\}\ 1^{\prime}\{G\cos\theta+W\sin\theta>0\}\right]\cos\theta.\] By using the change of variables as in (37) on the second term, we arrive at \[J_{1,0}(\theta) =\mathbf{E}[1\{W\sin\theta>0\}]\frac{1}{\sqrt{2\pi}}+\cos\theta \mathbf{E}[1\{Z\cos\theta+Y\sin\theta>0\}1^{\prime}\{Z>0\}]\] \[=\frac{1}{2}\frac{1}{\sqrt{2\pi}}+\cos\theta\mathbf{E}[1\{Y\sin \theta>0\}]\frac{1}{\sqrt{2\pi}}=\frac{1}{2}\frac{1}{\sqrt{2\pi}}+\frac{\cos \theta}{2}\frac{1}{\sqrt{2\pi}}=\frac{1+\cos\theta}{2\sqrt{2\pi}}.\] Formula for $J_{1,1}$: \[J_{1,1}(\theta) =\mathbf{E}[G(G\cos\theta+W\sin\theta)1\{G>0\}1\{G\cos\theta+W\sin \theta>0\}]\] \[=\mathbf{E}\left[\frac{d}{dg}((G\cos\theta+W\sin\theta)1\{G>0\}1 \{G\cos\theta+W\sin\theta>0\})\right]\] \[=\mathbf{E}[\cos\theta\ 1\{G>0\}1\{G\cos\theta+W\sin\theta\}]\] \[+\mathbf{E}[(G\cos\theta+W\sin\theta)1^{\prime}\{G>0\}1\{G\cos \theta+W\sin\theta>0\}]\] \[+\mathbf{E}[(G\cos\theta+W\sin\theta)1\{G>0\}1^{\prime}\{G\cos \theta+W\sin\theta>0\}]\cos\theta\] \[=\cos\theta J_{0,0}+\mathbf{E}[W\sin\theta\ 1\{W\sin\theta>0\}] \frac{1}{\sqrt{2\pi}}+\mathbf{E}[Z1\{Z\cos\theta+Y\sin\theta\}1^{ \prime}\{z>0\}]\] \[=\cos\theta J_{0,0}+\sin\theta\mathbf{E}[\varphi(W)]\frac{1}{ \sqrt{2\pi}}+0=\frac{\sin\theta+(\pi-\theta)\cos\theta}{2\pi}.\] $\blacksquare$ ### Proof of Explicit Formulas for $J_{n,0}$ and $J_{n,1}$ Once the recursion is established, the formula for both $J_{n,0}$ and $J_{n,1}$ is a simple proof by induction. We provide a detailed proof for $J_{n,0}$ here; $J_{n,1}$ is similar. Lemma 20: _Let $J_{n,0}^{rec}$ be the recursively defined formula, and let $J_{n,0}^{exp}$ is the explicitly defined formula for $J_{n,0}$, namely_ \[J_{n,0}^{rec}:=(n-1)J_{n-2,0}^{rec}+\frac{\sin^{n-1}\theta\cos\theta}{c_{n\bmod 2 }}(n-2)!!\,,\quad J_{1,0}^{rec}:=J_{1,0},\quad J_{0,0}^{rec}:=J_{0,0}\] \[J_{n,0}^{exp}:=(n-1)!!\left(J_{n\bmod 2,0}+\frac{\cos\theta}{c_{n\bmod 2}}\sum_{ \begin{subarray}{c}i\neq n(\bmod 2)\\ 0<i<n\end{subarray}}\frac{(i-1)!!}{i!\,!}\sin^{i}\theta\right).\] _Then $J_{n,0}^{rec}=J_{n,0}^{exp}$ for all $n\geq 0$._ Proof Let $S_{n}$, $n\in\mathbb{N},\;n\geq 2$ be the statement $J_{n,0}^{rec}=J_{n,0}^{exp}$ and $J_{n-1,0}^{rec}=J_{n-1,0}^{exp}$. We prove $S_{n}$ is true by induction. The base case $S_{2}$ is true because, \[J_{2,0}^{rec} =(2-1)J_{0,0}+\frac{\sin\theta\cos\theta}{c_{2\bmod 2}}(2-2)!!=J_{ 0,0}+\frac{\cos\theta\sin\theta}{2\pi}\] \[J_{2,0}^{exp} =(2-1)!!\,J_{0,0}+\cos\theta\sum_{i=1}^{1}\frac{(2-1)!!}{(2i-1)!!}(2i-2)!!\,\frac{\sin^{2i-1}\theta}{2\pi}=J_{0,0}+\frac{\cos\theta\sin\theta }{2\pi},\] and the fact that $J_{1,0}^{rec}=J_{1,0}^{exp}$ is trivial. Induction step: Assume $S_{n}$ is true. To prove $S_{n+1}$, we have only to show that $J_{n+1,0}^{exp}=J_{n+1,0}^{rec}$. To do this, we separate the last term of the sum to get \[J_{n+1,0}^{exp}= n!\,!\left(J_{(n+1)\bmod 2,0}+\frac{\cos\theta}{c_{(n+1)\bmod 2} }\sum_{\begin{subarray}{c}i\neq(n+1)(\bmod 2)\\ 0<i<n-1\end{subarray}}\frac{(i-1)!!}{i!\,!}\sin^{i}\theta\right)\] \[+n!\,!\,\frac{\cos\theta}{c_{(n+1)\bmod 2}}\frac{(n-1)!!}{n!\,!} \sin^{n}\theta.\] Because the parity of $n+1$ and $n-1$ are the same, and using $n!\,!=n(n-2)!\,!$ we recognize the first term as $nJ_{n-1,0}^{exp}$. So after simplifying the last term, we remain with \[J_{n+1,0}^{exp}=nJ_{n-1,0}^{exp}+\frac{\sin^{n}\theta\cos\theta}{c_{(n+1) \bmod 2}}(n-1)!!=J_{n+1,0}^{rec},\] by the induction hypothesis. This completes the induction. ### Bijection between Paths in Graphs of J Functions and the Bessel Number graphs $P$,$Q$ Let $G_{J^{}}=(V_{J},E_{J})$ be the graph of $J_{a,b}^{}$ as in Figure 5c. Similarly, let $G_{P}=(V_{P},E_{P})$ and $G_{Q}=(V_{Q},E_{Q})$ be the graph of the $P$ and $Q$ matrices up to row $a$, respectively, as in Figures 5a, 5b. We define a map $\lambda:\mathbb{Z}^{2}\times\mathbb{Z}^{2}\to\mathbb{Z}^{2}$ as follows: Let $((i,j),(m,n))\in\mathbb{Z}^{2}\times\mathbb{Z}^{2},\;0\leq i\leq a,\;b-a+m\leq j \leq b$. Then define $\lambda$ by \[\lambda((i,j),(m,n)):=(i-m,j-n),\quad\lambda^{-1}((i,j),(m,n))=(i+m,j+n).\] The function $\lambda$ can be used as a map between vertices of graph $G_{J^{}}$ to vertices of graph $G_{P}$ or $G_{Q}$. Let $\pi=(v_{1},v_{2},...,v_{k-1},v_{k})$ be a path in $G_{J^{}}$ from vertex $v_{1}=(m,n)$ to vertex $v_{k}=(a,b)$, where $v_{i}\in\mathbb{Z}^{2},\ 1\leq i\leq k$ is a vertex on the graph. $\lambda$ extends to a map on paths, $\Lambda$, defined by \[\Lambda((v_{1},v_{2},...,v_{k-1},v_{k})):= (\lambda(v_{1},v_{1}),\lambda(v_{2},v_{1}),...,\lambda(v_{k-1},v_{1 }),\lambda(v_{k},v_{1})),\] \[\Lambda^{-1}((v_{1},v_{2},...,v_{k-1},v_{k}))= (\lambda^{-1}(v_{1},v_{1}),\lambda^{-1}(v_{2},v_{1}),...,\lambda^{- 1}(v_{k-1},v_{1}),\lambda^{-1}(v_{k},v_{1})).\] Now, let $\Gamma_{J^{}}(a,b,m,n)$ be the set of all paths in the graph of $J^{}$ from $J^{}_{m,n}$ to $J^{}_{a,b}$, and let $\Gamma_{P}(a,b,m,n)$ be the set of all paths in the graph of $P$$P(0,0)$ to $P(a-m,b-n)=P(\lambda((a,b),(m,n)))$. For example, $\Gamma_{J^{}}(6,8,0,4)$ is the set of all paths which run from $J^{}_{6,8}$ to $J^{}_{0,4}$, and $\Gamma_{P}(6,8,0,4)$ is the set of all paths which run from $P(0,0)$ to $P(6,4)$. With these definitions, $\Lambda:\Gamma_{J^{}}(a,b,0,n)\to\Gamma_{P}(a,b,0,n)$ is a bijection. An illustration of all paths $\pi\in\Pi(6,8,0,6)$ and the corresponding paths $\Lambda(\pi)\in\Gamma_{P}(6,8,0,6)$ is given in Figure 6. Similarly, if we let $\Gamma_{Q}(a,b,m,n)$ be the set of all paths from $Q(0,0)$ to $Q(a-m,b-n)=Q(\lambda((a,b),(m,n)))$ then $\Lambda:\Gamma_{J^{}}(a,b,1,n)\to\Gamma_{Q}(a,b,1,n)$ is a bijection. This bijection establishes the equality of the weighted paths claim in (48). Figure 6: Top: All paths $\pi\in\Gamma_{J^{}}(6,8,0,6)$. Bottom: All paths $\Lambda(\pi)\in\Gamma_{P}(6,8,0,6)$. The paths are lined up so that for each path $\pi$ in the top row, $\Lambda(\pi)$ appears in the bottom row. ## Appendix C Recursions for the $P$ and $Q$ numbers - Proof of Lemma 10 Earlier work established the following properties of the $P$ and $Q$ numbers Theorem 21 (Kreinin (2016)): _The elements of the matrices $P$ and $Q$ satisfy_ \[b\cdot P(a,b) =a\cdot P(a-1,b-1), a\geq 1,1\leq b\leq a \tag{66}\] \[P(a+1,b) =P(a,b-1)+(b+1)\cdot P(a,b+1), a\geq 0,1\leq b\leq a\] (67) \[Q(a,b) =P(a,b)+(b+1)\cdot Q(a-1,b+1), a\geq 1,1\leq b\leq a\] (68) \[Q(a,b) =a\cdot Q(a-2,b)+Q(a-1,b-1), a\geq 2,1\leq b\leq a. \tag{69}\] [Of Lemma 10] Equation (67) tells us that $P(a,b)=P(a-1,b-1)+(b+1)\cdot P(a-1,b+1)$ for $a\geq 1,1\leq b\leq a-1$, while equation (66) tells us that $P(a-1,b+1)=\frac{(a-1)}{(b+1)}\cdot P(a-2,b)$ for $a\geq 2,0\leq b\leq a-2$. Putting these together, we get the following recurrence equation for $P(a,b)$: \[P(a,b) =P(a-1,b-1)+(b+1)\left(\frac{(a-1)}{(b+1)}\cdot P(a-2,b)\right)\] \[=(a-1)\cdot P(a-2,b)+P(a-1,b-1),\] which holds for $a\geq 3,1\leq b\leq a-2$. Further, looking at equation (69), we see that the recursion for the $Q$ numbers is very similar to that of the $P$ numbers, but with a coefficient of $a$ rather than $(a-1)$. This establishes Lemma 10.
2310.19704	A Survey on Knowledge Editing of Neural Networks	Deep neural networks are becoming increasingly pervasive in academia and industry, matching and surpassing human performance on a wide variety of fields and related tasks. However, just as humans, even the largest artificial neural networks make mistakes, and once-correct predictions can become invalid as the world progresses in time. Augmenting datasets with samples that account for mistakes or up-to-date information has become a common workaround in practical applications. However, the well-known phenomenon of catastrophic forgetting poses a challenge in achieving precise changes in the implicitly memorized knowledge of neural network parameters, often requiring a full model re-training to achieve desired behaviors. That is expensive, unreliable, and incompatible with the current trend of large self-supervised pre-training, making it necessary to find more efficient and effective methods for adapting neural network models to changing data. To address this need, knowledge editing is emerging as a novel area of research that aims to enable reliable, data-efficient, and fast changes to a pre-trained target model, without affecting model behaviors on previously learned tasks. In this survey, we provide a brief review of this recent artificial intelligence field of research. We first introduce the problem of editing neural networks, formalize it in a common framework and differentiate it from more notorious branches of research such as continuous learning. Next, we provide a review of the most relevant knowledge editing approaches and datasets proposed so far, grouping works under four different families: regularization techniques, meta-learning, direct model editing, and architectural strategies. Finally, we outline some intersections with other fields of research and potential directions for future works.	Vittorio Mazzia, Alessandro Pedrani, Andrea Caciolai, Kay Rottmann, Davide Bernardi	2023-10-30T16:29:47Z	http://arxiv.org/abs/2310.19704v3	# A Survey on Knowledge Editing of Neural Networks ###### Abstract Deep neural networks are becoming increasingly pervasive in academia and industry, matching and surpassing human performance on a wide variety of fields and related tasks. However, just as humans, even the largest artificial neural networks make mistakes, and once-correct predictions can become invalid as the world progresses in time. Augmenting datasets with samples that account for mistakes or up-to-date information has become a common workaround in practical applications. However, the well-known phenomenon of catastrophic forgetting poses a challenge in achieving precise changes in the implicitly memorized knowledge of neural network parameters, often requiring a full model re-training to achieve desired behaviors. That is expensive, unreliable, and incompatible with the current trend of large self-supervised pre-training, making it necessary to find more efficient and effective methods for adapting neural network models to changing data. To address this need, knowledge editing is emerging as a novel area of research that aims to enable reliable, data-efficient, and fast changes to a pre-trained target model, without affecting model behaviors on previously learned tasks. In this survey, we provide a brief review of this recent artificial intelligence field of research. We first introduce the problem of editing neural networks, formalize it in a common framework and differentiate it from more notorious branches of research such as continuous learning. Next, we provide a review of the most relevant knowledge editing approaches and datasets proposed so far, grouping works under four different families: regularization techniques, meta-learning, direct model editing, and architectural strategies. Finally, we outline some intersections with other fields of research and potential directions for future works. Knowledge Editing Model Editing Neural Networks Editing Continual Learning ## 1 Introduction In stark contrast to artificial neural networks (ANN), (Cichon and Gan, 2015), humans and other animals seem capable of learning and editing their knowledge continuously. Indeed, literature studies indicate that the mammalian brain could prevent catastrophic forgetting (Ratcliff, 1990) by safeguarding previously acquired knowledge, thereby reducing the plasticity of a proportion of synapses and ensuring their long-term stability (Benna and Fusi, 2016; Yang et al., 2009; Cichon and Gan, 2015). On the contrary, ANNs not only struggle to learn new tasks in a sequential fashion (Kirkpatrick et al., 2017), but also edit acquired knowledge on the same data distribution and task (Huang et al., 2023). Indeed, unlike conventional knowledge base systems that explicitly store knowledge, neural models implicitly memorize facts and tasks in their parameters, making it difficult to directly access and interpret their computation and memories (Voita et al., 2019; Belinkov and Glass, 2019). Making even minor modifications can lead to a decrease in performance on previously learnt tasks, or even cause the entire computation to fail due to the well-documented issue of catastrophic forgetting (Ratcliff, 1990). Therefore, modifying their acquired knowledge is a challenging problem. Just as humans, ANNs make mistakes and as we trust them with increasingly important decisions, the cost of such mistakes grows ever higher (Sinitsin et al., 2020). Therefore, given that mistakes are inevitable, it is crucial for deep learning practitioners to possess the ability to adjust model behaviors by promptly correcting errors as they surface. Currently, practical applications employing deep learning techniques have been relying on different workarounds to tackle this problem. In particular, a full re-training using datasets augmented with samples that account for the mistakes or up-to-date information is a common choice (Sinitsin et al., 2020). The endeavor needed for fine-tuning atop pre-trained models (Sarzynska-Wawer et al., 2021; Devlin et al., 2018; Oquab et al., 2023; Weiss et al., 2016) is frequently substantiated by the diminished dataset size and computational resources needed. On the other hand, this is not always true and manually curated, deterministic mechanisms that overrule model predictions on problematic samples can be the preferred choice (Sinitsin et al., 2020). However, while being simple, this second approach is fully localized and not robust to factor of variations of the input space (e.g., different viewpoint of the same object in computer vision or paraphrasing in natural language processing tasks). Furthermore, while these workarounds may provide a temporary solution, they can be expensive, unreliable, and incompatible with the current trend of large neural models (Zhao et al., 2023; Chen et al., 2022). Indeed, these large networks are typically deployed as static artifacts, whose behavior is difficult to modify during deployment without a costly full re-training (Lazaridou et al., 2021). Thus, in all those cases, in order to adapt to changes in the environment, or to address instances of underfitting or overfitting in the original training data, it is desirable to have the ability to quickly make targeted updates to the model's behavior after it has been deployed (De Cao et al., 2021). To address this need, _knowledge editing_ methods have been recently proposed to efficiently change a model's behaviors without affecting previous performance on the same task (Sinitsin et al., 2020). These approaches take inspiration from several fields of artificial intelligence research and range from simple fine-tuning with regularization methods (Zhu et al., 2020) to meta-learning techniques that adopt hypernetwork models to learn how to update parameters (De Cao et al., 2021; Mitchell et al., 2022). Due to its recent appearance in the literature, (Sinitsin et al., 2020), the field still lacks accordance in the taxonomy, naming convention, datasets, and target applications. Indeed, most of the works have been motivated by large language models (LLMs), (Zhao et al., 2023; Brown et al., 2020; Soltan et al., 2022), focusing mostly on tasks such as question answering (QA), (Levy et al., 2017), machine translation (MT) (De Cao et al., 2021), or even simpler NLP problems (Thorne et al., 2018). However, it is also possible to find applications of knowledge editing to computer vision problems (Sinitsin et al., 2020). Furthermore, its potential scope is poised to expand across diverse machine learning domains, encompassing areas such as medicine (Shehab et al., 2022) robotics (Soori et al., 2023), and precision agriculture (Sharma et al., 2020) in the future. ### Organization of the survey The objective of this survey is to provide a comprehensive review of existing literature on knowledge editing, formalizing the task and providing a categorization of the approaches into distinct families. To our knowledge, this is the first work to undertake such an effort, and we hope that it will facilitate future research in this increasingly important area of study. Indeed, the need for more formalization and organizational structure can already be seen by a recent study (Yao et al., 2023), which attempts to benchmark and compare some knowledge editing methodologies specifically for LLMs editing. The rest of the survey is organized as follows. Section 2 introduces the problem of knowledge editing, using previous works to formalize it under a common definition. Section 3 explores the tasks and datasets that are most commonly considered when solving the knowledge editing problem. Section 4 provides an overview of the most relevant knowledge editing approaches in the literature, identifying four distinct families: _regularization techniques_, _meta-learning_, _direct model editing_, and _architectural strategies_. Finally, Section 5 concludes the survey by discussing some intersection with knowledge editing and other fields of research, and outlining some possible future risks. ## 2 Overview This section begins by presenting an introduction to the concept of _knowledge editing_, which is also referred to as _model editing_ in the literature. First, we review various definitions and interpretations of knowledge editing proposed by different works. Next, we establish a common definition of knowledge editing that generalizes to all existing works in the field. ### Background The concept of knowledge editing was first introduced in (Sinitsin et al., 2020), which formalizes it as the task of correcting a model's mistake on a _specific sample_ while preserving the model's overall behavior, akin to continuous learning. Indeed, as specified by the authors, "The problem of efficient neural network patching differs from continual learning, (...) [because] is not sequential in nature. However, correction (...) of mislabeled samples must not affect its behavior on other samples, which is close to overcoming [the] catastrophic forgetting task." Therefore, (Sinisin et al., 2020) define the problem as performing individual edits _reliably_ and _efficiently_, not _sequentially_, and on the same task learned by the target model without affecting its overall behavior (i.e., being _local_ to the edit). Concurrently, authors of (Zhu et al., 2020) worked specifically on the task of modifying memories in Transformer models (Vaswani et al., 2017), providing their own definition of knowledge editing. They expand the scope of the problem to a subset of knowledge, i.e., a _batch of edits_. Similarly, several other studies have also formalized the problem of model editing following similar steps to Zhu et al. (2020). For instance, works by Mitchell et al. (2022), Meng et al. (2022), and Mitchell et al. (2022) have defined the task as the process of performing individual or batch edits on a target model trained for a specific task. These studies emphasize the importance of ensuring that edits are resilient to factors of variation, i.e., that they are generalizable. While injecting individual changes is already a challenging and interesting task for the scientific community, multiple simultaneous general model edits represent a more realistic scenario that deserves further exploration. More recently, Hartvigsen et al. (2022) and Huang et al. (2023) argued that the conventional "one-mistake-fixing scenario" does not accurately reflect the complexity of real-world knowledge editing challenges. As such, they proposed to extend the scope of knowledge editing to a sequential problem to facilitate the development of more practical editing methods. While their proposal only accounts for subsequent individual edits, considering multiple simultaneous and sequential edits represents a more general case where the number of edits varies at each step. Nevertheless, it is crucial to highlight that while the new definition of knowledge editing acknowledges a sequential process, differing from continuous learning, its scope remains limited to the modification of knowledge of the initially learned task by the model. On the contrary, continuous learning operates without such constraints, researching for methodologies that allow the model to expand to new tasks and adapt dynamically to completely new information. ### The knowledge editing problem To introduce the problem of knowledge editing, it is possible to leverage the terminology used to define a "well posed learning problem" (Mitchell et al., 2007): an algorithm is said to learn from experience $E$ with respect to some class of tasks $T$ and performance measure $P$, if its performance at tasks in $T$, as measured by $P$, improves with experience $E$. Then, we can say that knowledge editing is the problem of modifying the algorithm such that, given a representative set $S$ of instances of tasks in $T$, and a subset $S_{e}\subseteq S$ of _edit_ instances, its performance on $S_{e}$ improves as measured by $P$, while performance on all the other instances $S\setminus S_{e}$ remains unchanged. Figure 1: The knowledge editing problem has been firstly proposed as the task of modifying a model based on a set of individual pairs of edits, in a non-sequential manner (a), (Sinisin et al., 2020). Successive works extended the problem to batch of edits (b), sequential individual edits (c), and sequential batch of edits. Evaluation metrics are similar to all cases, as described in Section 2.5. More practically, we can define the problem of knowledge as _"the task of modifying a model based on a set of individual or batch edits, $S_{e}$, pertaining to the same task known by the model, either in a sequential or non-sequential manner. The objective is to update the model's knowledge representation without significantly altering its original behavior or performance over $S$ and being robust to different factor of variations of the edits."_ More formally, let $\mathbb{X}$, $\mathbb{Y}$ be an input and output space, respectively. Let $f_{0}$ be a function that maps an input $x\in\mathbb{X}$ to an output $y\in\mathbb{Y}$, parametrized by $\theta_{0}\in\Theta$, then \[f_{0}\in\mathbb{F}\coloneqq\left(\mathbb{X}\times\Theta\right)^{\mathbb{Y}} \tag{1}\] We use the subscript zero to indicate that this is the _starting model_, i.e. the model we want to edit. We define an _edit pair_ as an input-output pair $(x_{e},y_{e})\in\mathbb{X}\times\mathbb{Y}$, such that $f_{0}(x_{e})\neq y_{e}$. Then, in its simplest form, given an individual edit example pair $(x_{e},y_{e})$, a knowledge editing (KE) methodology can be defined as follows \[\text{KE}:\mathbb{F}\times\mathbb{X}\times\mathbb{Y}\rightarrow\mathbb{F} \tag{2}\] i.e. a function that takes in input the starting model $f_{0}$ and the edit pair $(x_{e},y_{e})$ to produce an _edited model_$f_{e}$. If the edited model is such that $f_{e}(x_{e})=y_{e}$, then we say that the edit was successful. A KE approach can realize the transformation from $f_{0}$ to $f_{e}$ in different ways, and we identify four families of possible realizations: regularization, meta-learning, direct model editing and architectural strategies. We delve into more details into each of these families in section 4. ### Taxonomy of edits Often, it is interesting to edit a model applying multiple edits at once, a sequence of edits, or sequences of batches of edits. The definition in eq. (2) has been given for the simplest case, that we call a _single non-successive_ edit: we only want to change the model for one edit pair. Conversely, for multiple successive edits we can formally define a list of edit requests as: \[\mathcal{E}=\{(x_{e},y_{e})^{(i)}\ \ \text{s.t.}\ \ \forall i,j\ x_{e}^{i}=x_{e}^{j} \Rightarrow y_{e}^{i}=y_{e}^{j}\} \tag{3}\] where the logical constraint ensures that there are no conflicting requests, as suggested by (Meng et al., 2022). Individual edits can also be grouped together and form $N$ batches of successive edits each with $B_{e}^{(i)}$ edit pairs, such as \[\mathcal{B}_{e}^{(i)}=\{(x_{e},y_{e})^{0},\cdots,(x_{e},y_{e})^{B}\}^{(i)}\ \ \text{s.t.}\ \ \mathcal{E}=\bigcup_{i=1}^{N}\mathcal{B}_{e}^{(i)}. \tag{4}\] The difference between successive individual or batch of edits is that some KE methodologies can ingest an entire $\mathcal{B}_{e}^{(i)}$ and produce the resulting $f_{e}$ implementing all the given edits, while other methodologies can only consider one individual sample in sequence at a time. In both cases (a sequence of individual edits is trivially a sequence of single-item batch edits), successive edits assume to work with a starting model $f_{t-1}$ and apply the $t$-th change as \[f_{t}=\text{KE}(f_{t-1},\mathcal{B}_{e}^{(i)}) \tag{5}\] proceeding iteratively, using $f_{t}$ as a starting model for the next edit. Figure 1 summarizes the four types of edits. Finally, as for individual edits, after every change, $f_{e}$ should not only satisfy $f_{e}(x_{e})=y_{e}$, but a series of other properties as discussed in the next section. ### Editing properties Based on the specific task learned by function $f$, various properties can be specifically designed. However, at a fundamental level, following (Sinitsin et al., 2020; Huang et al., 2023), knowledge editing should aim at satisfying four properties, that we define below. Property 1: _Reliability_: Given an edit pair $(x_{e},y_{e})$, the edited model $f_{e}$ should output desired edit: \[f_{e}(x_{e})=y_{e}. \tag{6}\] Property 2: _Generality_: The edited model $f_{e}$ should be able to generalize to _similar_ examples to the edit pair. This can be formalized by defining an _equivalence neighborhood_$N(x_{e})\subset\mathbb{X}$ and requiring that the edited model $f_{e}$ satisfies: \[f_{e}(\tilde{x}_{e})=y_{e}\qquad\forall\ x_{e}\in N(x_{e}). \tag{7}\] Property 3: _Locality_: The edited model $f_{e}$ should not alter the output of examples that are not similar to the edit pair. This can be formalized by defining a _locality set_ \[L(x_{e})=\{(x_{loc},y_{loc})\in\mathbb{X}\times\mathbb{Y}\ \ \text{s.t.}\ \ x_{loc} \notin N(x_{e})\wedge f_{0}(x_{loc})=y_{loc}\} \tag{8}\] and require that the edited model $f_{e}$ satisfies: \[f_{e}(x_{loc})=y_{loc}\qquad\forall\ (x_{loc},y_{loc})\in L(x_{e}). \tag{9}\] Property 4: _Efficiency_: The proposed knowledge editing methodology should aim to achieve efficient editing, characterized by low compute and space complexity. It is worth noting that some works in the literature have defined these same properties, or a subset of them, with different names and notations. Specifically, reliability may also be referred to as _efficacy_, locality as _specificity_, and generality as _paraphrase_, (Meng et al., 2022; De Cao et al., 2021). Moreover, as previously mentioned, depending on the specific field of application, other properties and related evaluation metrics can arise, such as _fluency_ and _consistency_(Meng et al., 2022) when editing Language Models. Finally, the last property, namely efficiency, tends to be disregarded in academic literature. However, it is one of the main reasons KE is appealing over a simpler re-training of the base model. Furthermore, it plays a pivotal role in performing comparative analyses with baseline models and showcasing the ability of a particular methodology to modify a neural network with a substantial number of parameters. We encourage future works to consider (at least) all four properties when measuring the effectiveness of a proposed KE methodology. ### Evaluation metrics Given a knowledge editing algorithm KE, a list of edit requests $\mathcal{E}$, a starting model $f_{0}$ and a test set $\mathcal{D}_{test}$, it is possible to measure the extent to which the properties above hold. We consider working with $N$ batches of successive edits, each comprised of $B$ individual edits, without loss of generality (as mentioned above, if $N=1$, we have successive individual edits). Moreover, if the KE methodology is unable to process all $B$ edits concurrently, it can apply each edit individually in a non-sequential manner. Following (Huang et al., 2023), we represent with $I$ the indicator function, and define the following metrics. Success Rate: (SR): It is used to evaluate reliability, and it is simply the proportion of edits for which the methodology succeeds in changing the knowledge of a starting model $f_{t}$. \[\text{SR}=\frac{1}{N}\frac{1}{B}\sum_{n=1}^{N}\sum_{b=1}^{B}I(f_{n,B}(x_{e;n,b })=y_{e;n,b})\ \ \ \text{s.t.}\ \bigcup_{n=1}^{N}\bigcup_{b=1}^{B}(x_{e},y_{e})_{n,b}=\mathcal{E} \tag{10}\] In the case of non-sequential individual edits, $f_{n,B}=f_{n,b}$. Moreover, Eq. 10 provides an overall value after $N$ successive edits, but it can be of interest to measure SR every $n$ edits, tracking changes over the sequence. Generalization Rate: (GR): It is used to evaluate generality, testing the post-edit model $f_{e}$, on the equivalence neighborhood set $N(x_{e;n,b},y_{e;n,b})$, where $(x_{e;n,b},y_{e;n,b})$ is the $n$-th batch, $b$-th edit pair. GR can be written as, \[\text{GR}=\frac{1}{N}\frac{1}{B}\frac{1}{N_{b}}\sum_{n=1}^{N}\sum_{b=1}^{B} \sum_{i=1}^{N_{b}}I(f_{n,B}(\tilde{x}_{e;n,b,i})=\tilde{y}_{e;n,b,i})\ \ \ \text{s.t.}\ \forall n,b\ \bigcup_{i=1}^{N_{b}}(\tilde{x}_{e},\tilde{y}_{e})_{n,b,i}\subseteq N(x_{e},y_{e})_{n,b} \tag{11}\] where $N_{b}$ is the number of equivalent samples of the $b$-th edit pair. Following (Mitchell et al., 2022), we can also define Edit Success (ES) to summarize both SR and GR. It can be computed as the average accuracy of the edited model $f_{e}$ on the edit input(s), as well as inputs drawn from the equivalence neighborhood(s), that is, \[\text{ES}=\frac{1}{N}\frac{1}{B}\sum_{n=1}^{N}\sum_{b=1}^{B}\left(I(f_{n,B}(x_ {e;n,b})=y_{e;n,b})+\sum_{i=1}^{N_{b}}\frac{1}{N_{b}}I(f_{n,B}(x_{e;n,b,i})=y_ {e;n,b,i})\right) \tag{12}\] where the same conditions for SR and GR hold and have been omitted for brevity. Drawdown: (DD): It is used to evaluate locality, and it is defined as the performance degradation of the edited model over $\mathcal{D}_{test}$. It is computed using the final edited model, $f_{N}$, that in case of successive edits is the result of $N$ steps. \[\text{DD}=1-\frac{\sum_{(x,y)\in\mathcal{D}_{test}}I(f_{N}(x)=y)}{\sum_{(x,y) \in\mathcal{D}_{test}}I(f_{0}(x)=y)} \tag{13}\] Finally, as suggested by (Huang et al., 2023) in the case of multiple successive edits, it is also important to evaluate SR and GR using the final model $f_{N}$, in order to assess how past edits are retained. Therefore, it is possible to define three additional metrics Success Retain Rate (SRR), Generalization Retain Rate (GRR), and Edit Success Retain (ESR), simply using in Eq. 10 and 11, $f_{N}$ instead of $f_{n,B}$. ## 3 Tasks and Datasets The formalization of the knowledge editing problem provided in the previous section is general and many applications of knowledge editing to different tasks encompassing various fields can be formulated within that framework. The brief but rich history of the field has so far seen applications mainly to two broad fields: Computer Vision and Natural Language Processing. Indeed Sinitsin et al. (2020) provide experimental results on image classification and machine translation, and almost all the works come after (and even before (Garnelo et al., 2018; Kirkpatrick et al., 2017)) demonstrate the effectiveness of their proposed approaches in one or more applications in these two fields. Even though the knowledge editing framework can be defined independently of the target domain and task, each specific application has its unique challenges and intricacies, which we explore in this section. Section 3.1 covers the most common tasks in the Computer Vision domain, as well as the datasets on which the tasks are usually addressed, and how the knowledge editing problem can be instantiated in this context. Section 3.2 provides a similar overview for applications to the Natural Language Processing domain. Finally, Section 3.3 describes tasks and datasets that do not strictly fit in any of the two domains above. ### Computer Vision Computer Vision is a broad field with a long history, which generally attempts to extract meaningful representations from visual media to derive an understanding of the represented scenes (Szeliski, 2022). Over the last years deep learning methods have been shown to outperform previous state-of-the-art techniques in several applications in the field (Voulodimos et al., 2018), and while more "traditional" techniques are still relevant for some applications, neural networks-based approaches have become the de facto standard for many others (O'Mahony et al., 2020). Due to the importance and breadth of the field, and the relevance of neural networks therein, knowledge editing literature has found fertile grounds in Computer Vision, and has so far gravitated towards two primary tasks: Image Classification and Image Completion. A number of datasets are customarily used to test approaches to solve these tasks. They vary in terms of number of examples, classes and channels, as well as resolution of the representative images; table 1 provides an overview of the most commonly used ones. Image ClassificationThe task of image classification is straightforward, we wish to label a complete image (or predetermined portion) with its most likely semantic category, e.g., horse, cat, or car Szeliski (2022). In this context, an example is an image and its semantic label. The image, of predefined dimension (or resolution), is encoded as a 3D tensor $x^{(i)}\in\mathbb{R}^{W\times H\times C}$, where $W$ is the width of the image, $H$ the height of the image, and $C$ the number of channels, depending on whether the image is grayscale (1) or RGB (3) or something else (e.g. RGBD (Firman, 2016) with an additional depth channel). The editing task is then often formulated by artificially corrupting a subset of either the images or the labels. The latter is the more prevalent, and usually involves training models on a subset of a dataset and subsequently introducing random label shuffling within a withheld set to create an edit set $\{(x^{(i)}_{e},y^{(i)}_{e})_{i=1}^{N}\}$, where originally $y^{(i)}\neq y^{(i)}_{e}$. Other works, such as Sotoudeh and Thakur (2021), introduce e.g. motion blur or fog (Mu and Gilmer, 2019) to corrupt a subset of the images, creating an edit set where originally $x^{(i)}\neq x^{(i)}_{e}$. Several datasets support knowledge editing experiments for this task, and a distinction is often made among "toy" and "large-scale" datasets. Widely acknowledged datasets such as MNIST (LeCun et al., 2010) and CIFAR-10 (Krizhevsky, 2009) are frequently employed for experimentation, and belong to the former category. The well-known MNIST dataset comprises 7,000 grayscale examples, each consisting of $28\times 28$ images representing handwritten digits ranging from 0 to 9, with balanced support for each of the 10 classes. Similarly, the CIFAR-10 (Krizhevsky et al., 2009) dataset encompasses 60,000 examples of $32\times 32$ RGB images divided across 10 balanced semantic classes, like airplanes and dogs. CIFAR-100 extends its diversity with 600,000 examples of $32\times 32$ color images, distributed across 100 classes. While these datasets are occasionally considered as simplified instances for classification (Sinitsin et al., 2020; Lee et al., 2019), they serve as fundamental examples for understanding knowledge editing dynamics. For more challenging and realistic scenarios, researchers turn to larger scale datasets, of which the most popular is surely the extensive ImageNet Database (Deng et al., 2009), which now encompasses over 10 million labeled images, spanning more than 20,000 object categories. Specifically, studies such as (Sinitsin et al., 2020) and (Lee et al., 2019) delve into datasets derived from the ImageNet Large Scale Visual Recognition Challenges (ILSVRC) (Russakovsky et al., 2015). To further accentuate the complexity, (Sinitsin et al., 2020) introduces a highly challenging configuration, leveraging the Natural Adversarial Examples (NAE) dataset (Hendrycks et al., 2021), consisting of $7500$ natural images notorious for their arduous classification nature, where pre-trained models exhibit correct prediction rates of less than $1\%$ for NAEs. Image InpaintingImage inpainting, also known as image completion, is the task of reconstructing missing regions in an image (Szeliski, 2022). The problem is formalized as a regression over functions mapping pixel coordinates within $[0,1]^{2}$ to pixel values in $[0,1]$ (grayscale) or $[0,1]^{2}$ to pixel values in $[0,1]^{3}$ (RGB). This task has so far received less attention from the knowledge editing community (Garnelo et al., 2018), leveraging datasets that encompass both the MNIST dataset, once again serving as a rudimentary example, and the CelebFaces Attributes Dataset (CelebA) (Liu et al., 2015) for more challenging scenarios. The CelebA dataset presents a more demanding scenario, offering over 200,000 celebrity images, each accompanied by 40 attribute annotations, making it a challenging and comprehensive dataset for exploration. ### Natural Language Processing Natural Language Processing (NLP) is also a broad field, concerned with giving computers the ability to process and understand human language (Eisenstein, 2019). Like in computer vision, in recent years researchers and practitioners in the field have leveraged the power of neural network with many outstanding results (Otter et al., 2020; Soltan et al., 2022; FitzGerald et al., 2022). With the recent paradigm shift from supervised learning to pre-training followed by fine-tuning (Wang et al., 2022), and the trend towards larger and larger models (Zhao et al., 2023), the ability to perform a cheap edit of a model instead of an expensive fine-tuning has motivated an intense interest from the knowledge editing community. Undoubtedly, within the NLP realm, the most widely targeted tasks for knowledge editing are Fact Checking when dealing with classification, and (closed book) Question Answering for language generation. Some recent works also explore open Text Generation, editing of factual relations with had hoc datasets and also Document Classification. Table 2 provides an overview of the datasets commonly used for those tasks. Fact checkingFact-checking is the task of assessing whether claims made in written or spoken language are true, often addressed as a binary classification task (Guo et al., 2022). In this setting, examples are natural language claims coupled with binary labels, even though sometimes a third netural option is available. The claim $x^{(i)}$ is encoded with some tokenization scheme (or pre-trained word embedding) as a sequence of integers (or semantic vectors), while the label $y^{(i)}$ can take one of two values, positive or negative (optionally a third, aforementioned neutral value). One can then have a neural network predict this value with an explicit classification layer, or alternatively a language model producing special tokens (e.g. _True/False_ or _Supports/Refutes_) for the considered truth values, when prompted with the claim under consideration. The Fact Checking task has been considered particularly appealing for the knowledge editing community (De Cao et al., 2021; Mitchell et al., 2022, 2022; Huang et al., 2023) for at least a couple of reasons. First, the recent phenomenal success of language models also highlighted their proneness to generate reasonable but factually wrong natural language text (Ortega et al., 2021; Ji et al., 2023). This degrades the system performance and fails to meet user expectations in many real-world scenarios, leading to a great interest in the ability to mitigate these hallucinations. Furthermore, reasonable edit sets are fairly easy to obtain, e.g. randomly flipping the labels of claims from pre-existing datasets from Support to Refutes and vice-versa. The most widely used datasets for this task are FEVER (Thorne et al., 2018) and VitaminC (Schuster et al., 2021). Both of them are extracted from Wikipedia and annotated by human experts, to arrive at (_evidence_, _wikipage_, _claim_, _label_) tuples. In both cases, the label can be _Supports_, _Refutes_ or _Not Enough Info_ depending on whether the evidence supports or not the claim. To construct proper editing datasets $\{(x^{(i)}_{e},y^{(i)}_{e})_{i=1}^{N}\}$ out of them, De Cao et al. (2021) (for FEVER) and Mitchell et al. (2022) (for VitaminC) grouped facts based on the same pages, augmented each fact $x^{(i)}_{e}$ with some rephrases $\bar{x}^{(i)}_{e}$, to assess generality, and randomly flipped the label $y^{(i)}_{e}\neq y^{(i)}$ in each group. Question AnsweringThe task of training a model to provide a correct natural language answer to a given question is referred to as question answering; more specifically, _closed-book_ question answering is restricted to the case when \begin{table} \begin{tabular}{l c c c c c} \hline \hline Dataset & Tasks & \# Examples & \# Classes & \# Channels & Resolution \\ \hline MNIST & Classification, Inpainting & 70k & 10 & 1 & 28x28 \\ CIFAR-10 & Classification & 60k & 10 & 3 & 32x32 \\ CIFAR-100 & Classification & 600k & 100 & 3 & 32x32 \\ ILSVRC & Classification & 1.2 M & 1000+ & 3 & 224x224 + \\ ILSVRC & Classification & 1.2 M & 1000+ & 3 & 224x224 + \\ NAE & Classification & 7.5k & 200+ & 3 & 224x224 + \\ CelebA & Inpainting & 200k & - & 3 & 178x218 \\ \hline \hline \end{tabular} \end{table} Table 1: Most important datasets used in Computer Vision for Knowledge editing. MNIST, CIFAR-10, CIFAR-100 are generally regarded as “toy” datasets while ILSVRC, NAE, CelebA as more challenging testbeds. the model is only fed with the question, and not a selection of possible answers, or supporting corpus from which an answer can be extracted (Roberts et al., 2020). In this case, both the input $x$ and the target label $y$ are natural language sequences, and we test the extent to which the model parameters implicitly store the required knowledge to provide correct answers. The most widely adopted dataset is surely the zero-shot Relation Extraction dataset (zsRE) (Levy et al., 2017), used in particular by Zhu et al. (2020); De Cao et al. (2021); Mitchell et al. (2022, 2022); Meng et al. (2022, 2022); Huang et al. (2023); Hartvigsen et al. (2022). This task and dataset are particularly appealing for knowledge editing. Indeed, Question Answering can benefit greatly from Knowledge Editing, as pipelines that solve the task usually leverage factual knowledge to answer questions; in the case of neural networks, this knowledge is implicitly stored in the networks' parameters, and it is unclear how to tweak these parameters to correct wrong or undesired answers, especially as the networks grow bigger in size. Meng et al. (2022) hypothesize that this factual knowledge takes the form of _(relation, subject, object)_ triples, with intermediate layers acting as key-value storage units. This formalization lends itself nicely to the definition of an editing objective, rather than directly the open-ended natural language generation task. Furthermore, Levy et al. (2017) demonstrates that it is possible to reduce relation extraction to the problem of answering simple reading comprehension questions and provided in their dataset multiple templates for each relation. For example, the triple (_occupation, $s$, $o$_) can be naturally extracted answering one of the following: _What did $s$ do for a living?_, _What is $s$'s job? What is the profession of $s$?_. The subject $s$ can then be modified to create editing examples. Beside zsRE, knowledge editing of models solving Question Answering has been studied leveraging also on additional datasets such as t-REX (Elsahar et al., 2018) and Natural Questions (NQ) (Kwiatkowski et al., 2019). Finally as a more challenging flavor of the same task with added counterfactual informations (Meng et al., 2022) introduced a new dataset called CounterFact. Further NLP tasksBeside the two popular tasks outlined above, Mitchell et al. (2022) tested editing text generation from autoregressive GTP-like models on a special version of WikiText-103 (Merity et al., 2016), where they considering as prompts ($x_{e}$) passages sampled from WikiText itself and as edit targets ($y_{e}$) 10-token samples from a pre-trained distitiGPT-2 model. This was for them a valid challenging editing setup since for their target model for editing a greedy 10-token prediction agrees with these edit targets for $<1\%$ of examples they extracted. Finally, more recently Hartvigsen et al. (2022) tested their methodology on a novel task for knowledge editing using the SCOTUS dataset from Chalkidis et al. (2022). The classification task is to categorize U.S. Supreme Court documents over multiple decades into $11$ topics. What makes this task interesting is that, over time, categorization rules change, so that label distributions shift. We note how this is off the shelf particularly realistic for knowledge editing as much of the world knowledge memorized by networks evolves over time just like those labels shifts in the dataset and the target of knowledge editing can be seen as keeping updated such world knowledge. ### Other Applications Even though the majority of works in the knowledge editing literature has focused on the Computer Vision and Natural Language Processing fields, as described above, the general nature of the editing problem yielded interesting results also in other fields, and will likely yield more in more diverse fields and applications in the years to come. Among these, to the best of our knowledge the most notable examples are applications in safety-critical scenarios, and to graph neural networks, and in the following we briefly review works from both. Safety-critical SystemsSafety-critical systems are those systems whose failure may lead to consequences that are determined to be unacceptable, such as significant damage to properties, the environment, or people (Knight, 2002). Deep neural networks have grown in popularity over the past decade and are now being used in safety-critical domains \begin{table} \begin{tabular}{l l l l l} \hline \hline Dataset & Tasks & Format & \# Examples & \# Classes \\ \hline FEVER & Fact Checking & (evidence, wikipage, claim, label) & 420k & 3 \\ VitaminC & Fact Checking & (evidence, wikipage, claim, label) & 450k & 3 \\ zsRE & Question Answering & (subject, relation, object) & 120M & - \\ T-REx & Question Answering & (subject, relation, object) & 11M & - \\ NQ & Question Answering & (question, answer) & 320k & - \\ CounterFact & Question Answering & (subject, relation, true object, false object) & 22k & - \\ Wikitext & Text Generation & tokens & 100M & - \\ SCOTUS & Document Classification & (date, text, label) & 9.2k & 11 \\ \hline \hline \end{tabular} \end{table} Table 2: Most important datasets for knowledge editing used in NLP. We report characteristics of the original datasets even though often for knowledge editing ad hoc version, preprocessed to make editing meaningful are used. such as self-driving cars (Gupta et al., 2021), healthcare (Tekkesin et al., 2019) and aviation (Sridhar, 2020). Clearly, in such critical scenarios being able to find and correct unsafe neural network behavior becomes a crucial objective. This has motivated a line of research within the Knowledge Editing community, that so far has only touched the aviation domain, specifically the aircraft collision avoidance problem. The systems currently employed provide safety guarantees at the cost of being poorly data efficient, and efforts have been made to integrate neural networks into the pipeline (Julian and Kochenderfer, 2019). As a consequence, a number of subsequent works (Sotoudeh and Thakur, 2021; Fu and Li, 2021; Liang et al., 2023) from the knowledge editing community have proposed approaches for fixing unsafe behavior of neural networks integrated in safety-critical pipelines. Developing a robust collision avoidance algorithm that reliably prevents collision without alerting excessively is challenging due to sensor error and uncertainty in the future behavior of the aircraft. The Airborne Collision Avoidance System X (ACAS X) (Kochenderfer and Chryssanthacopoulos, 2011) family of collision avoidance systems formulates the problem of collision avoidance as a partially observable Markov decision process. The variant for unmanned aircraft, ACAS Xu, uses dynamic programming (DP) to then find a solution in terms of resolution advisories that avoid collisions while minimizing disruptive alerts. The DP process makes use of a very large lookup table, that makes storage costly and certification time-consuming for certified avionics systems. Therefore, Julian and Kochenderfer (2019) propose using a deep neural networks for compressing the table without loss of performance, as measured by a set of safety and operational metrics. The task is then to accurately regress the lookup table, that associates values to combinations of actions and state variables. There are 5 possible advisory actions and seven real-valued state variables that define an aircraft encounter; the state variables are discretized, forming an eight-dimensional score table with approximately 120 million discrete points. This, or a downsampled version of the table, defines the ACAS Xu dataset: $\{(x^{(i)},y^{(i)})_{i=1}^{N}\}$ where $x^{(i)}\in\mathbb{R}^{7}$ represents a (discretized) state, and $y^{(i)}\in\mathbb{R}^{5}$ are the values associated to each of the five possible actions in that state. A set of edit examples is usually defined by considering certain desired properties, formally verified to not be globally satisfied for a network trained on the task (Katz et al., 2017), and taking as $x_{e}$ samples that satisfy the _input constraint_ of the property, and as $y_{e}$ samples that do not satisfy the _output constraint_ of the property. Graph Neural NetworksDeep learning models have been particularly successful when dealing with signals such as speech, images, or video, in which there is an underlying Euclidean structure; however, recently there has been a growing interest in trying to apply learning on non-Euclidean geometric data, for instance represented in the form of graphs (Bronstein et al., 2017). Graph Neural Networks (GNNs) learn node representations by applying shared permutation invariant functions over local neighbourhoods of nodes in the graph (Bronstein et al., 2021). These representations can then be used for tasks like node classification for e.g. assigning a category to each paper in a citation graph (Wu et al., 2020). GNNs have achieved prominent results in learning features and topology of graph data, however, knowledge editing for GNNs is rarely explored, despite their widespread applicability; therefore, Liu et al. (2023) propose a method to edit these models, restricting to the aforementioned node classification task. The task can be formalized as follows: let $G=(V,E)$ be an undirected graph with $V=(v_{1},\ldots,v_{\|V\|})$ and $E=(e_{1},\ldots,e_{\|E\|})$ being the set of nodes and edges, respectively. Given a feature space $\mathcal{X}$ (e.g. the space of real-valued $d$-dimensional feature vectors $\mathbb{R}^{d}$), a node feature tensor $X\in\mathcal{X}^{\|V\|}$ and a label space $\mathcal{Y}$, the goal is to learn a representation $h_{v}$ from which a label $y_{v}\in\mathcal{Y}$ for each node $v\in V$ can be easily predicted. Many datasets for node classification exist in the literature, comprised of data from various domains like citation networks and social networks, and we find again the distinction between small-scale and large-scale datasets (Wu et al., 2020). Among the former we find datasets like Cora, which contains a selection of Machine Learning papers collected from the web, and the references automatically parsed from their bibliographies (McCallum et al., 2000). In particular, the network contains $\|V\|=2708$ nodes (articles) and $\|E\|=5429$ links (citations); the feature space is $\mathcal{X}=\{0,1\}^{1433}$, i.e. each node is described in terms of the presence (or absence) of certain keywords, taken from a dictionary of 1433 unique words (bag-of-words content representation); the label space is $\mathcal{Y}=\{1,\ldots,7\}$, i.e. the task is to predict to which of seven classes (e.g. _Theory_ or _Reinforcement_Learning_) each publication belongs. The Reddit dataset is instead a popular representative of large-scale node classification datasets. Hamilton et al. (2017). constructed a graph dataset from Reddit posts made in the month of September, 2014. The node label $y_{v}\in\mathcal{Y}$ in this case is the community, or "subreddit"; that a post belongs to, considering $\|\mathcal{Y}=50$ large communities. A post-to-post graph is constructed connecting posts if the same user comments on both. In total this dataset contains $\|V\|=$ 232,965 posts with an average degree of 492 ($\|E\|$ is in the order of 100 millions edges). In both of these cases, and the many other instances of node classification datasets, constructing an edit dataset is fairly straightforward, and done in the same manner as for the image classification task in Computer Vision. After training the model $f_{0}(\cdot)$ under consideration on a _subgraph_, one evaluates it on the whole graph: each pair $(x^{(i)},y^{(i)})$ such that $\hat{y}^{(i)}=\arg\max f(x^{(i)})$ is incorrect, becomes an edit pair $(x^{(i)}_{e},y^{(i)}_{e})$. The geometry of graphs lends itself nicely to defining also generality and locality sets: indeed, since the task under consideration is node classification, as we have seen a single example $x^{(i)}$ describes a node $v$, then one can define its neighborhood $N(x^{(i)})$ to be its actual neighborhood in the graph $N_{G}(v)=\{w\in V\mid\exists\left(v,w\right)\in E\}$; from this definition, the definitions of generality and locality sets follow immediately, as seen in earlier sections. ## 4 Knowledge Editing Methodologies In recent times, several "knowledge editing" methods have been introduced to effectively modify the behaviors of models while maintaining their previous performance on the same task (Sinnitsin et al., 2020). These approaches draw inspiration from various fields of artificial intelligence research and encompass a range of methodologies, from straightforward fine-tuning with regularization techniques (Zhu et al., 2020) to meta-learning methods that utilize hypernetwork models to learn parameter updates (De Cao et al., 2021; Mitchell et al., 2022). In the upcoming sections, we will provide individual discussions on the different families of knowledge editing methodologies, emphasizing their respective areas of application, as well as the advantages and limitations of most notable techniques. The objective of this section is to categorize various knowledge editing techniques discussed in the literature into distinct groups. Specifically, we can classify these approaches into four distinct families, namely: _regularization techniques_, _meta-learning_, _direct model editing_, and _architectural strategies_. All presented works have different characteristics, target different areas of application, types of edits, and adopt diverse experimentation strategies. Nevertheless, the objective of all the works reported can be formulated within the formal framework given in 2.2. A comparison between the most notable KE methodologies at the time of writing can be found in Table 3. On the other hand, Table 4 presents a comparison with non-sequential single batch edits of factual knowledge. ### Regularization Techniques Catastrophic forgetting, (Kemker et al., 2018), is a well-known phenomenon in literature, fundamentally limiting the flexibility of editing networks once trained or fine-tuned, (Lee et al., 2019). Indeed, in the absence of any regularization process, the regular fine-tuning signal has the ability to easily execute a specific edit, albeit with a tendency to over-fit to the provided edit examples. However, this approach lacks in providing generality to the edit and has a negative impact on locality, owing to the absence of a plasticity mechanism, (Sinnitsin et al., 2020). Similar to continual learning, regularization techniques for knowledge editing aim to modify the standard fine-tuning signal of the target edit to ensure reliability and locality. Therefore, for regularization techniques, KE is not parametrized and does not require any pre-training, but it is nothing more than gradient descent computed with given edits and some specific regularization terms. While not all of these techniques were originally developed for the specific task of knowledge editing, they have \begin{table} \begin{tabular}{l l l l l l l l l} \hline \hline KE Methodology & KE & Training & Preserves & Only & \multirow{2}{}{SNS} & \multirow{2}{}{BNS} & \multirow{2}{}{SSE} & \multirow{2}{}{BSE} & \multirow{2}{}{ \begin{tabular}{c} Scales \\ to LM \\ \end{tabular} } \\ & Cathegory & Required & Architecture & & & & & & \\ \hline FT + $L_{2}$ & Regularization & False & ✓ & ✗ & ✓ & ✓ & ✓ & ✗ \\ FT + KL & Regularization & False & ✓ & ✗ & ✓ & ✓ & ✓ & ✗ \\ EWC & Regularization & False & ✓ & ✗ & ✓ & ✓ & ✓ & ✓ & ✗ \\ CNP & Architectural & True & ✗ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ ENN & Meta-Learning & True & ✗ & ✓ & ✓ & ✗ & ✓ & ✗ & ✗ \\ KnowledgeEditor & Meta-Learning & True & ✓ & ✓ & ✓ & ✗ & ✗ & ✗ & ✓ \\ MEND & Meta-Learning & True & ✓ & ✓ & ✓ & ✓ & ✗ & ✗ & ✓ \\ ROME & Direct Editing & False & ✓ & ✓ & ✗ & ✗ & ✗ & ✓ \\ MEMIT & Direct Editing & False & ✓ & ✗ & ✓ & ✓ & ✓ & ✓ & ✓ \\ SERAC & Architectural & True & ✗ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ CaliNet & Architectural & True & ✓ & ✓ & ✓ & ✗ & ✗ & ✓ \\ T-Patcher & Architectural & False & ✗ & ✓ & ✓ & ✓ & ✗ & ✓ \\ GRACE & Architectural & True & ✗ & ✓ & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular} \end{table} Table 3: Comparison of the most notable KE methodologies in the literature. Different characteristics are reported for each approach, highlighting main advantages and disadvantages. For all approaches, we report: the category and whether it requires _training_ of an auxiliary model; if it _preserves the architecture_ of the edited model or requires the introduction of new components; whether it needs only the edit pair $(x_{e},y_{e})$, or requires additional input to perform the edit; if it is able to handle _single non-successive_ edits (SNS), _batched non-successive_ edits (BNS), _single successive_ edits (SSE), and _batched successive_ edits. Finally, we report if it can scale to Large Models (LM), that, following the definition in (Zhao et al., 2023), are models with more than 10B parameters. proven to be in some way effective and are commonly used as useful baselines for comparison. Moreover, due to their simplicity, they can easily adapt to work with different types of edits: from single non-successive edits to batches of successive edits. However, depending on the number of layers fine-tuned, they rarely scale to models with large number of parameters such as Large Language Models (LLMs), that according to (Zhao et al., 2023) are models with more than 10B parameters. For instance, authors in (Sotoudeh and Thakur, 2021) introduce a technique called "Neural Network Patching" that allows for the correction of deep neural networks without retraining from scratch. They present the concept of neural network patching, which involves identifying and correcting the faulty or damaged components of a network. The proposed method involves using a set of repair operators to identify and replace damaged components of the network, identifying the minimum $L_{2}$ norm parameter update that reliably edits the model output, while minimizing the deviation from the original target model. Conversely, (Zhu et al., 2020) utilize a constrained optimization process instead of penalizing an updated model for deviating from the original one. Their method employs either an $L_{2}$ or $L_{\infty}$ constraint between the original model parameters and the edited ones, highlighting the importance of selecting a subset of parameters to be updated. Additionally, they demonstrate how their method is similar to Elastic Weight Consolidation (EWC), (Lee et al., 2019), which involves computing a penalty term for each weight in the neural network based on how much it contributes to the performance of the original task. Similarly to (Zhu et al., 2020), loss enriched with the Kullback-Leibler divergence (KL) have been proposed in order to regularize the network's weights based on the KL divergence between the network's output on the old task and the network's output on the new task. This encourages the network to maintain similar weights for the parts of the network that are relevant to both tasks (Yoon et al., 2017; Serra et al., 2018; Mitchell et al., 2022; Huang et al., 2023). As previously mentioned, the techniques outlined in this section have been primarily utilized as baseline approaches in knowledge editing and heavily intersect with continual learning works (Mundt et al., 2023). However, in various experiments cited in (De Cao et al., 2021; Mitchell et al., 2022; Huang et al., 2023), these techniques have demonstrated limited efficacy in maintaining model accuracy with regards to previous knowledge. In fact, as noted by (De Cao et al., 2021), regularization techniques overlook the highly nonlinear nature of large models and the significant impact that even minor changes in parameter space can have on the output of numerous data points. For those reasons, more elaborated works have been proposed in literature as the ones in the following sections. ### Meta-Learning and HyperNetworks Meta-learning techniques refer to a set of algorithms and approaches that enable machines to learn how to learn, (Finn et al., 2017). Meta-learning techniques have been applied to a range of applications, such as computer vision (Ren et al., 2018), natural language processing (Gu et al., 2020), robotics (Finn et al., 2017), and reinforcement learning (Houthooft et al., 2018). They have shown promising results in improving the efficiency and effectiveness of machine learning systems and enabling machines to learn more autonomously. They can be broadly categorized into two types: _model-based_ and _optimization-based_ meta-learning. In the context of knowledge editing, model-based meta-learning focus on learning a model that can be used to adapt to new data efficiently. This involves learning the structure of the model and its parameters such that it can be generalized to new data. In optimization-based meta-learning, the focus is on learning how to optimize the parameters of a model to adapt to new knowledge. \begin{table} \begin{tabular}{l l l l l l l l l l} \hline \hline Dataset & Model & \begin{tabular}{c} Evaluation \\ Metrics \\ \end{tabular} & FT & KE & MEND & ROME & MEMIT & SERAC & T-Patcher \\ \hline \multirow{3}{}{$\text{zsRE}$} & SR & 54.70 & 6.60 & 45.60 & 99.18 & 99.23 & 90.16 & 97.12 \\ & GR & 49.20 & 7.80 & 48.00 & 94.90 & 87.16 & 89.96 & 94.95 \\ \cline{1-1} & \multirow{3}{*}{GPT-J [6B]} & DD & 62.76 & 5.82 & 11.79 & 0 & 0 & 0.1 & 3.76 \\ \cline{1-1} & & SR & 99.90 & 13.40 & 73.80 & 99.80 & 99.90 & 99.78 & 100.00 \\ \cline{1-1} CounterFact & & GR & 97.53 & 11.00 & 74.20 & 86.63 & 73.13 & 99.41 & 83.98 \\ \cline{1-1} & & DD & 98.98 & 5.62 & 96.25 & 0 & 0 & 1.11 & 91.63 \\ \hline \hline \end{tabular} \end{table} Table 4: Comparison of some of the most notable KE methodologies on zSRE (Levy et al., 2017) and CenterFact (Meng et al., 2022) with non-sequential single batch edits of factual knowledge. For each methodology, we report success rate (SR), generalization rate (GR), and Drawdown (DD) metrics. Results adapted from (Yao et al., 2023). In literature, authors of "Editable Neural Networks" (ENN), (Sinitsin et al., 2020), firstly exploit the meta-learning paradigm for knowledge editing to "learn to allow effective editing". The core idea behind "Editable Training" is to enforce the model parameters to be "prepared" for the editor function KE (Section 2.2), that in their experimentation is defined as Stochastic Gradient Descent (SGD) with up to $k$ steps and learning rate $\alpha$. In particular, they propose to train the starting model $f_{0}$ with a loss that at the same time encourages reliability and locality. That is obtained with two additional terms to the base loss, one that measures the success of an edit, cross-entropy, and another that assess the distance in the output probability between the edited model and the original one, KL divergence. They prove to be successful on either computer vision datasets for classification (Krizhevsky et al., 2009; Deng et al., 2009) and Natural Adversarial Examples (NAE) (Hendrycks et al., 2021) and machine translation (Cettolo et al., 2014). They work with single non-successive and successive edits, with good performances on both types of edits. Nevertheless, as pointed out by (Mitchell et al., 2022), the methodology requires to further training the base model before an edit with a pre-training step. That can be critical in scenarios with large models (LM) or when the memory available is a constraint. Conversely, (De Cao et al., 2021) they propose an optimization-based meta-learning, firstly employing a hypernetwork (Ha et al., 2016) dubbed KnowleEditor with the objective to "learn to update" the parameters of another network. Therefore, since the knowledge editing task requires being locale except for the edit, they frame the learning task as a constrained optimization problem using a Bidirectional-LSTM (Schmidhuber et al., 1997) and some downstream Feed-Forward Neural Networks (FFNN). Once trained, only the input edit feed the hypernetwork that predicts vectors to gate and shift the edit fine-tuning gradient of the starting network in respect to a certain weight matrix. Therefore, we can say that KnowledgeEditor learns how to modify the gradient in order to provide the property enumerated in Section 2.4. On the other hand, the training itself of the hypernetwork does not change the weights of the starting model as (Sinitsin et al., 2020), but requires some original training samples to estimate the original output probability distribution. Indeed, as (Sinitsin et al., 2020), the hypernetwork is trained with the sum of two loss components: the first one aims at providing reliability and generality with cross-entropy between semantically equivalent edits and predictions of the edited model and the second one provides locality with the KL divergence as (Sinitsin et al., 2020). However, in addition to them, they propose to add a margin to KL and iteratively reduce it to progressively make the edit more local. They test their KnowledgeEditor on different NLP tasks, from fact-checking (FC) with a fine-tuned BERT model (Devlin et al., 2018) with the FEVER dataset (Thorne et al., 2018), to closed book question answering (QA) with BART model (Lewis et al., 2020) on the Zero-Shot Relation Extraction (zsRE) dataset (Levy et al., 2017). However, they only experimented with single non-successive changes. Nevertheless, authors of (Huang et al., 2023) adopt KnowledgeEditor as a baseline and their experiments on FEVER and zsRE show that it fails to implement more than a couple of successive single edits. Indeed, as hypothesized by the authors, KnowledgeEditor hypernetwork is trained with the starting model $f_{0}$ and thus strongly coupled with the original parameters. As the editing proceeds, the model becomes more different from the initial one, resulting in their failure. Building over (De Cao et al., 2021), authors of (Mitchell et al., 2022) leverage on hypernetworks too in order to learn how to update weights of a starting model. However, while KnowledgeEditor trains a recurrent neural network to map the edit example into a rank-1 mask over the gradient, (Mitchell et al., 2022) hypernetwork, named MEND, directly maps the gradient into a new parameter update, retaining tractability by leveraging the low-rank form of the gradient. Indeed, the input of a MEND network is a decomposed gradient and the output is formed by some pseudoactivations and psudodelta that should encapsulate, reliability, generality, and locality. Indeed, as (De Cao et al., 2021), they make use of cross-entropy between semantically equivalent edits and predictions of the edited model to enforce generability and reliability (edit loss), and KL divergence for locality without the margin (locality loss). In addition, they propose two further hyperparameters to perform a weighted sum of the two losses and to make learnable the edit learning rate coefficient. Each layer of the network itself consists of two consecutive blocks, both initialized to compute the exact identity function of the normalized decomposed gradient, using four matrices initialized with zero or Xavier uniform initialization (Glorot and Bengio, 2010). Finally, in order to edit multiple layers of the starting model with same matrices dimension, they propose to use the same MEND network, while applying a learned layer-specific scale and offset similar to (Perez et al., 2018). As (De Cao et al., 2021), they experiment their methodology only with NLP tasks using FEVER (Thorne et al., 2018) and zsRE (Levy et al., 2017) datasets. In addition, they also experiment with GPT-style models, (Radford et al., 2018), working with a custom version of Wikitext-103 (Merity et al., 2016), named Wikitext Generation. They both experiment with non-sequential single or batch types of edits, showing large regression over 100 simultaneous edits. Finally, it is important to point out that as KnowledgeEditor, also MEND has the same limitations with successive edits, being strictly tied to the weights of the starting model. In fact, if pre-training is conducted on $f_{t}$, KE will exhibit significantly poorer performance with $f_{t+1}$. Moreover, as the weights of the edited model diverge from those of the pre-training, MEND will gradually deteriorate, ultimately losing all its editing capabilities. ### Direct Model Editing This family of approaches aims to edit the weights within the target neural network directly, enabling effective implementation of a predefined set of changes. By directly manipulating the variables of a model, most of the direct model editing methodologies build upon efforts to localize and understand the internal mechanisms within models (Ehlage et al., 2021; Dar et al., 2023). These approaches strive to attribute knowledge to specific neurons in the network and edit their activations to reflect the desired changes (Mitchell et al., 2022). Furthermore, owing to the tightly coupled nature of these approaches with the network architecture, the majority of research has been focused on specific architectures, such as MLP and Transformers. As a result, the validation of their effectiveness on alternative architectures is usually left to future endeavors. Geva, Mor, et al. (Tay et al., 2022), firstly identify the MLP layers in a masked LM transformer as key-value memories of entities and information associated with that entity, (Sukhbaatar et al., 2015). Building on this finding, (Dai et al., 2022) demonstrate a method to edit facts in BERT models (Devlin et al., 2018); they propose the concept of knowledge neurons to study how factual knowledge is stored in pretrained Transformers. Specifically, the authors examine the fill-in-the-blank cloze task for BERT and propose a knowledge attribution method to identify the neurons that express a given relational fact. They find that the activation of such knowledge neurons is positively correlated to the expression of their corresponding facts. Therefore, they present a preliminary methodology that leverages knowledge neurons to edit factual knowledge in Transformers, even without any fine-tuning. The authors perform a knowledge surgery for pretrained Transformers by directly modifying the corresponding parameters in feed-forward networks. Such surgery shows promising results, keeping a moderate influence on other knowledge. The methodology proposed can be used to perform both single and multiple edits at the same time. However, the authors only experimented with single edits in their paper, focusing on specifically on factual knowledge. Factual knowledge is also the focus of Meng Kevin, et al. work, (Meng et al., 2022), but exclusively experimenting on autoregressive language modelling, such as GPT-style models, (Radford et al., 2018). The study explores the storage and direct manipulation of factual associations in transformer models. These associations are modeled as tuples, represented as $t=(s,r,o)$, comprising a subject $s$, a relation $r$, and an object $o$, which connect to form a factual relationship. The research delves into understanding how these tuples are stored within transformer models and investigates methods to facilitate their direct editing. Firstly, they trace the causal effects of hidden state activations within GPT using causal mediation analysis (Pearl, 2001) as previously done by other works (Vig et al., 2020) to identify the specific modules that mediate recall of a fact about a subject. Their analysis reveals that feedforward MLPs at a range of middle layers are decisive when processing the last token of the subject name. Secondly, they test their previous findings by introducing a novel direct model editing methodology: Rank-One Model Editing method (ROME). The algorithm aims at producing single, not successive edits by altering the parameters that determine a feedforward layer's behavior. As stated by the authors, despite the simplicity of the intervention, ROME is similarly effective to other model-editing approaches achieving good generalization and locality simultaneously, whereas previous approaches sacrifice one or the other. Furthermore, they introduce a new dataset, CounterFact (derived from a set of true facts from WikiData), for evaluating counterfactual edits in language models. It contains 21,919 records with a diverse set of subjects, relations, Figure 2: Scaling curves showing three different evaluation metrics with an increased number of non-successive batch edits for three different KE methodologies: MEND, ROME, and MEMIT. Results are computed using CounterFact and GPT-J. Locality is shown not as drawdown (DD), but as its complementary specificity over a neighborhood of samples (Meng et al., 2022). ROME and MEND performs well up to ten edits, but rapidly degrade, losing almost all SR before batches of 1k. On the other hand, MEMIT performs well with considerable large batches of edits. Adapted from (Meng et al., 2022). and linguistic variations. Finally, they propose to monitor a further metrics, _fluency_, to evaluate text generation's repetitiveness by measuring the weighted average of bi- and tri-gram entropies (Zhang et al., 2018). ROME results are comparable with other knowledge editing meta-learning techniques. Nevertheless, the methodology is limited to work with single non-successive edits. To address this problem, the same authors modified the original algorithm to overcome this problem (Meng et al., 2022). The method, called MEMIT, is able to scale up to thousands of associations for GPT-J (6B) and GPT-NeoX (20B), exceeding prior work by an order of magnitude. As ROME, MEMIT works by first identifying the critical layers in the LLM that are responsible for storing the memories that need to be updated. Then, it uses a rank-one modification of the MLP weights of these layers to directly write the new memories into the model. This process is repeated for each memory that needs to be updated. The authors evaluated MEMIT on a variety of tasks, including factual question answering, natural language inference, and summarization. They found that MEMIT was able to significantly improve the performance of the LLM on these tasks, even when the number of memories that were updated was large. In addition to its scalability, MEMIT also has several other advantages over previous methods for updating LLMs. First, it is relatively efficient, requiring only a few minutes to update a large model with thousands of memories. Second, it is very accurate, with the updated models achieving near-perfect accuracy on the tasks that were evaluated. Third, MEMIT is generalizable, with the updated models able to perform well on a variety of tasks. Figure 2 compares ROME, MEMIT, and one of the meta-learning techniques described in section 4.2. It is possible to appreciate the improvement brought with MEMIT and the clear collapse of the MEND HyperNetwork-based editor with large batches. Overall, MEMIT is a valuable tool for improving the performance of these models. However, it is important to point out that, even if there are works that try to expand the scope of the methodology (Gupta et al., 2023), MEMIT is limited to modifying factual association and that there is not a clear path to scale it to different skills. Furthermore, more recent research is suggesting that casual tracing for picking which parameters to update is surprisingly unreliable (Hase et al., 2023). ### Architectural Strategies Within this family of methodologies, we find approaches that stand apart from the ones that directly or indirectly modify the weights of the target model. Instead, they focus on manipulating or augmenting the original architecture to patch and modify the network's existing knowledge. In particular setups this can be preferable because the set of original weights cannot be accessible. A first editor function of this type was proposed by (Sinitsin et al., 2020), that taking inspiration from Conditional Neural Processes (CNP) (Garnelo et al., 2018), they propose a specialized architecture that performs edits by adding a special condition vector to intermediate activations. The vector is generated by an additional encoder layer and provides information to effectively steer the final prediction of the network. This approach is very similar to more generic memory-augmented models that introduce memory mechanisms to enhance neural networks (Graves et al., 2014; Santoro et al., 2016). On the same line of experimentation, Mitchell et al. (Mitchell et al., 2022) proposed an approach, called SERAC (Semi-Parametric Editing with a Retrieval-Augmented Counterfactual Model), which stores edits in an explicit memory and learns to reason over them to modulate the base model's predictions as needed. This allows SERAC to be more expressive than other model editors, and to produce more accurate predictions for test inputs that are not closely related to the edit. That is achieved using three components: a memory, a classifier, and a counterfactual model. Users adds edit in the memory and the classifier decides whether the memory contains inputs relevant to processing them. If the classifier determines that a relevant edit example exists, the input and edit example are passed to the counterfactual model that is responsible for making the prediction. They evaluate SERAC on numerous tasks, focusing exclusively on large language models with single and batch non-successive edits. On the same line of SERAC, authors of (Dong et al., 2022) presented a memory-based methodology called CaliNet. It is a method for calibrating specifically factual knowledge in pretrained language models (PLM) without re-training from scratch. CaliNet first detects whether a PLM has learned a fact correctly by comparing its scores for the correct fact and a set of distractor facts. If the PLM has not learned the fact correctly, CaliNet then uses a lightweight method to add and adapt new parameters to the PLM for that specific fact. CaliNet has two main components: first, a contrastive knowledge assessment (CKA) module that detects whether a PLM has learned a fact correctly. The CKA module works by comparing the PLM's scores for the correct fact and a set of distractor facts. If the PLM assigns a higher score to the correct fact than to the distractor facts, then the CKA module concludes that the PLM has learned the fact correctly. Secondly, a factual knowledge calibration (FKC) module adds and adapts new parameters to the PLM for a specific fact. The FKC module works by first creating a new set of parameters for the PLM that are specific to the given fact. These new parameters are then initialized with the PLM's original parameters for the fact. The FKC module then fine-tunes the new parameters on a dataset of factual examples that include the given fact. Using a custom dataset based on ParaRel set (Elazar et al., 2021), CaliNet has shown to be effective at calibrating factual knowledge in PLMs. In experiments on the knowledge probing task, CaliNet was able to improve the factual accuracy of PLMs by up to 20%. All previous works have tackled only non-successive edits, making only one edit or batch of edits at a time. However, as pointed out by (Huang et al., 2023), the one-mistake-fixing scenario is not an accurate abstraction of the real-world challenge. Therefore, they extend the scenario into successive edits, introducing a novel model editor, Transformer-Patcher (T-Patcher), that can shift the behavior of transformer-based models by simply adding and training a few neurons in the last Feed-Forward Network layer. Being at the end of the model, the new neurons have access to the output of all the previous layers. This allows the new neurons to learn how to correct mistakes that are made in earlier layers of the model. Overall the proposed methodology allows for targeted shifts in the behavior of the model, akin to other fine-tuned oriented methodologies like LoRA (Hu et al., 2022). Transformer-Patcher is able to successively correct up to thousands of errors while generalizing to their equivalent inputs and retaining the model's accuracy on irrelevant inputs. This is in contrast to previous methods, which either fail to make a sequence of edits or to remember previous edits. The work evaluates Transformer-Patcher on both classification and generation tasks, showing that Transformer-Patcher can achieve state-of-the-art performance for single successive edits. However, despite the expansive scope of the methodology, their approach and implementation are highly architecture-specific, relying heavily also on large sources of data and unrelated inputs. To this end Hartvigsen, Thomas, et al. (Hartvigsen et al., 2022), proposes GRACE, a methodology that requires only samples of the edits and that can perform successive edits, ensuring minimal impact on unrelated inputs. GRACE works by caching a chosen layer's activations in an adaptive codebook as edits stream in. When a new edit is received, GRACE retrieves the corresponding activations from the codebook and uses them to update the model's predictions. This allows GRACE to make targeted edits to the model's behavior without affecting its overall performance. The authors evaluated GRACE on a variety of tasks, including text classification, natural language inference, and question answering. They showed that GRACE was able to significantly improve the performance of LLMs on streaming errors, while minimally affecting their performance on unrelated inputs. Finally, the authors do not explicitly mention the impact of GRACE on latency. However, it is possible that GRACE could have a small impact on latency, as it requires caching activations in an adaptive codebook. ## 5 Conclusion and Future Directions In this survey, we organize recent progress of the increasingly growing field of knowledge editing. We began formalizing it under a common definition that seamlessly connect the slightly different facets of each work presented in the literature so far. Indeed, being a very recent branch of research, each author tried to differently bend definitions to better accommodate their methodologies. So, following the first definitions given by (Sinitsin et al., 2020) and the more recent adaptation by (Huang et al., 2023), we formally define knowledge editing as the task of modifying the knowledge of a target model in a non-sequential or sequential manner, without significantly affecting its original performance. This makes knowledge editing closely related to the much more well-known branch of continuous learning, or lifelong learning. In fact, knowledge editing can be seen as a restricted case of continuous learning, (Mundt et al., 2023), in which we want to apply edits, usually non uniformly distributed (Henn et al., 2021), to previously acquired knowledge by the network. Continuous learning, on the other hand, is broader in scope, looking for general methodologies to provide increasingly more knowledge to a network, enabling it to perform more tasks and skills. Nevertheless, despite their differences, the two groups are closely related and share common problems, such as catastrophic forgetting (Ratcliff, 1990). This means that there is potential for benefits to flow from both groups to each other. Building on the more formal definition, we presented a distilled summary of the most relevant works in the literature at the time of writing. We proposed to categorize these works into four families: regularization techniques, meta-learning, direct model editing, and architectural strategies. We discussed each family in turn, highlighting their intrinsic characteristics and limitations. We also summarized the most frequent field of application, tasks, and datasets that have been tackled in each family, for quick reference. We did not specifically delve into the works where knowledge editing could emerge as an additional benefit of the proposed methodologies (Hewitt et al., 2023) but it is worth noting that future expansions of similar survey research could encompass these aspects for a more comprehensive analysis. Overall, we have presented a rapidly expanding field of research driven by the current trend of foundational models (Zhou et al., 2023). The advancements in this area have led to a significant increase in the development of tools and methodologies to effectively harness the intrinsic knowledge of these models. As we move forward, knowledge editing is poised to become a critical factor in leveraging the power of these models for various industrial applications. Finally, it is vital to acknowledge that the power of knowledge editing also brings inherent risks that must not be overlooked. While editing models can correct their behavior and improve their utility, it can also be exploited for harmful purposes. In particular, sophisticated editing algorithms may enable malicious actors to deliberately incorporate backdoors, vulnerabilities, hidden behaviors, or harmful tendencies into the models. This concern becomes even more critical for methodologies that edit weights without providing sufficient interpretability of the applied changes. This dual use is a common risk for many machine learning technologies, and only proactive efforts to develop robust security measures and ethical guidelines can help to mitigate these potential risks.
2307.14837	DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations	We extend and analyze the deep neural network multigrid solver (DNN-MG) for the Navier-Stokes equations in three dimensions. The idea of the method is to augment a finite element simulation on coarse grids with fine scale information obtained using deep neural networks. The neural network operates locally on small patches of grid elements. The local approach proves to be highly efficient, since the network can be kept (relatively) small and since it can be applied in parallel on all grid patches. However, the main advantage of the local approach is the inherent generalizability of the method. Since the network only processes data of small sub-areas, it never ``sees'' the global problem and thus does not learn false biases. We describe the method with a focus on the interplay between the finite element method and deep neural networks. Further, we demonstrate with numerical examples the excellent efficiency of the hybrid approach, which allows us to achieve very high accuracy with a coarse grid and thus reduce the computation time by orders of magnitude.	Nils Margenberg, Robert Jendersie, Christian Lessig, Thomas Richter	2023-07-27T13:16:26Z	http://arxiv.org/abs/2307.14837v2	DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations ###### Abstract We extend and analyze the deep neural network multigrid solver (DNN-MG) for the Navier-Stokes equations in three dimensions. The idea of the method is to augment of finite element simulations on coarse grids with fine scale information obtained using deep neural networks. This network operates locally on small patches of grid elements. The local approach proves to be highly efficient, since the network can be kept (relatively) small and since it can be applied in parallel on all grid patches. However, the main advantage of the local approach is the inherent good generalizability of the method. Since the network is only ever trained on small sub-areas, it never "sees" the global problem and thus does not learn a false bias. We describe the method with a focus on the interplay between finite element method and deep neural networks. Further, we demonstrate with numerical examples the excellent efficiency of the hybrid approach, which allows us to achieve very high accuracies on coarse grids and thus reduce the computation time by orders of magnitude. ## 1 Introduction Accurate flow simulations remain a challenging task. The success of deep neural networks in machine translation, computer vision, and many other fields on the other hand, has lead to a growing (and renewed) interest to apply neural networks to problems in computational science and engineering. The combination of classical finite element approximation techniques with deep neural networks adds new aspects to the pure numerics-oriented approaches. At higher Reynolds numbers, accurate simulations, especially in 3D become increasingly difficult, and the classical methods reach their limits. Although the finite element method is highly efficient and established for the discretization of the Navier-Stokes equations, fundamental problems, such as the resolution of fine structures or a correct information transport between scales, are still not sufficiently solved. Even if linear solvers with optimal complexity (such as multigrid methods) are used, nonlinear equations such as the Navier-Stokes equations show more and more local features under mesh refinement and these turn out to be a hurdle for the approximation. We usually consider finite elements as a hierarchical method, where natural hierarchies of finite element meshes and spaces can be constructed (adaptively) to yield approximations of increasing accuracy. This hierarchical setup is then used within multigrid methods for efficient solution of the algebraic system. Here, we connect such a hierarchic finite element approach with neural networks in a hybrid setting: while coarse meshes are handled in the traditional finite element multigrid way, updates in finer meshes are learned by the neural network. They are hence used whenever a full resolution of the effects does not seem possible or efficient. We label this approach the Deep Neural Network Multigrid Solver (DNN-MG) as it is based on hierarchies of meshes and functions spaces and combines the tools of multigrid solvers with deep neural networks (cf. Figure 1). However, the neural networks are used to predict updates to the nonlinear problem and hence, the approach can be used for an upsampling of any kind of finite element (also of finite volume or finite difference) solutions towards a representation on a finer mesh. We demonstrate the efficiency, generalizability, and scalability of our proposed approach using 3D simulations. We first train the neural network on data from the classical flow with one circular obstacle. To analyze the generalization capability of DNN-MG, we test the network on channel flows with one or two obstacles at different Reynolds numbers. The obstacles have an elliptical cross-section with varying eccentricities. The obtained solutions as well as the lift and drag functionals demonstrate that DNN-MG obtains considerably better accuracy than a coarse multigrid solution on mesh-level $L$ while taking less than 3% of the computation time required by a full solution on $L+2$ levels. Therefore, DNN-MG offers a speedup by a factor of 35. DNN-MG's efficiency is evident, as it requires only double the time of a coarse mesh solution on level $L$, yet offers substantial improvements in the overall solution quality. The paper is organized as follows. In the Section 2, we review related work on using neural networks for the simulation of partial differential equations. Sections 3 provides a recap of the Navier-Stokes equations' solution using the geometric multigrid method, while Section 4 introduces the neural networks used later in our numerical experiments. In Section 5, we present the deep neural network multigrid solver, discussing its design in a general form, which makes it applicable to other problems. Finally, we present our numerical results in Section 6. Figure 1: The idea of DNN-MG is to use a multigrid solver to solve the problem on the first $L$ levels of the mesh hierarchy. The obtained solution is then prolongated to a finer level $L+J$, where a neural network predicts corrections to the solution. These corrections are incorporated into the time evolution through the time stepping and enter through the right-hand side. ## 2 Related works We discuss different approaches to the simulation of physical systems using deep neural networks. The most direct approach to utilize (deep) neural networks for the solution of partial differential equations is to represent the solution of the PDE as such a network. In [35] already, a least squares formulation has been employed to minimize the residual in the training process. The idea has recently been used again in the Deep Ritz Method [18] and a variety of similar approaches have emerged in the last years and coined Physics Informed Neural Network (PINNs) in the literature [38]. For a comprehensive review of PINN and related approaches, we refer to the comprehensive review [17]. Such approaches use an existing partial differential equation and can be considered data free since no explicit training data is required. They are, however, static in that the solution to one specific problem instance is learned. PINNs show superiority for instance in very high dimensional problems [40] but they do not reach the efficiency of established methods when standard problems, e. g. from elasticity or fluid dynamics, are considered (cf. [24] for case studies on the Poisson, Allen-Cahn and Schrodinger equation). The idea of PINNs has also been extended to learning solution operators [39, 14, 34]. The main advantage of this approach is that once the solution operator is trained, it can be applied to other settings. This enables the application to inverse problems [50, 30, 13] or optimal control [44]. Inspired by classical time stepping schemes, another approach to a neural network-based simulation of (evolutionary) partial differential equations is to consider these as time series and predict the next state of the system based on the previous ones. With this approach, network architecture for sequential data can be used, in particular recurrent neural networks and their extensions such as Long-Short-Term-Memory units (LSTM) [28] and Gated Recurrent Units (GRUs) [16] that we also used for the DNN-MG Navier-Stokes solver [45, 46]. At the other end of the spectrum are approaches that are purely data-based. For example, in [19] flow fields are learned in convolutional neural networks based on the flow geometry and using techniques from image processing. Recent work [53, 9, 36] uses purely data-driven neural network models trained on historical weather measurements and these perform on par with one of the most sophisticated weather forecasting models based on partial differential equations. The authors of [22] use transformer models for the prediction of dynamical systems. In the context of dynamical systems, the combination of Data Assimilation, Uncertainty Quantification, and Machine Learning techniques has gained significant interest, we refer to [15] for an in-depth review. For a recent overview on different neural network based approaches to approximate partial differential equations and related inverse problems we refer to [51]. One of the central open questions in the current literature is to what extent and how physical constraints or existing knowledge about the dynamics should be used for the training of a neural network, see also [23]. The Deep Neural Network Multigrid Solver (DNN-MG) is a hybrid approach that combined the model based finite element method with deep neural networks that also integrate model knowledge. DNN-MG is not a PINN since we do not represent the solution by only aim for fine scale updates. Also, the general setup of DNN-MG aims for generalization, and it is closer to learning a numerical solver than to learning a specific solution. The close incorporation of the deep neural network into the finite element method, e. g. that we learn fine-scale solution coefficients, gives rise to analytical tools that can be used in first steps of error estimation [48]. In our first work on DNN-MG [46] we introduced a method which combines a geometric multigrid method with ANNs but only considered 2D simulations. We also adapted the method In 2D to ensure divergence-freedom by construction through a network that predicts the streamfunction [45]. In later studies we showed that the corrections added by the neural network improve the guesses in the Newton method, which leads to a reduction in wall-time compared to the coarse grid solutions [47]. Similar to the ideas of DNN-MG, the authors of [42, 43] use numerical solutions of higher order to train a neural network which generates corrective forcing terms. Similarly, in [20], low-order simulations of wave equations are enhanced using neural networks to provide corrective forcing. This approach includes improvements in the temporal resolution of the simulations. Learning corrective forcing terms shares similarities with the approach of DNN-MG, as both methods introduce corrections to the solution through the right-hand side, which can be seen as corrective forcing terms. However, DNN-MG goes further by directly correcting the solution, potentially providing advantages in computing goal quantities like drag and lift. All these methods have a resemblance to reduced order models (ROMs). Along these lines, in [4], a ROM based on adaptive finite elements and correction terms combined with Artificial Neural Networks (ANNs) is presented. In [32] the authors developed a mesh based approach to ANN based solvers, leveraging existing FEM theory. The authors of [12] propose a neural framework for optimizing weak formulations in FEM. An ANN acting as control variable is included in the weak form and the minimization of a cost function yields desirable approximations where known data or stabilization mechanisms can be easily incorporated. Ainsworth and Dong propose a framework for adaptively generating basis functions, each represented by a neural network with a single hidden layer, which may be used in a standard Galerkin method [1]. In [49] the authors augment PDEs by ANNs to represent unknown terms. The hybrid model is then discretized using FEM. The MgNet framework introduced in [25] draws connections from convolutional neural networks (CNNs) to Multigrid methods and thereby improve the design and understanding of CNNs. The authors of [41] propose a framework for unsupervised learning of Algebraic Multigrid (AMG) prolongation operators for linear systems which are defined directly on graphs. In the Multigrid framework, selecting an appropriate smoother is crucial to ensure convergence and efficiency, but the optimal choice of a smoother is problem dependent and is in general a major challenge for many applications. In [29], the authors optimize the smoothers in Multigrid methods, which they form by CNNs. ## 3 Numerical Methods We solve model problems of incompressible viscous flow around a cylinder in 2D and 3D domains. This leads to the solution of the Navier-Stokes equations. Our approach involves a geometric multigrid method with a cell-based Vanka-smoother as a preconditioner to a Newton-Krylov method. We consider a domain $\Omega\in\mathbb{R}^{d}$ with $d\in\{2,\,3\}$ and Lipschitz-continuous boundary $\Gamma$ and a bounded time interval $[0,\,T]$. For solving the stationary Navier-Stokes equations we seek the velocity $\mathbf{v}:\ [0,\,T]\times\Omega\to\mathbb{R}^{d}$ and pressure $p:\ [0,\,T]\times\Omega\to\mathbb{R}$, such that \[\partial_{t}\mathbf{v}+(\mathbf{v}\cdot\triangledown)\mathbf{v}- \nu\Delta\mathbf{v}+\nabla p =f \text{on }[0,\,T]\times\Omega \tag{1}\] \[\triangledown\cdot\mathbf{v} =0 \text{on }[0,\,T]\times\Omega,\] where $\nu>0$ is the kinematic viscosity and $f$ an external force. The initial and boundary conditions are given by \[\mathbf{v}(0,\,\cdot) =\mathbf{v}_{0}(\cdot) \text{on }\Omega \tag{2}\] \[\mathbf{v} =\mathbf{v}^{D} \text{on }[0,\,T]\times\Gamma^{D}\] \[\nu(\mathbf{n}\cdot\triangledown)\mathbf{v}-p\mathbf{n} =0 \text{in }[0,\,T]\times\Gamma^{N},\] where $\mathbf{n}$ denotes the outward facing unit normal vector on the boundary $\partial\Omega$ of the domain. The boundary $\Gamma=\Gamma^{D}\cup\Gamma^{N}$ is the union of $\Gamma^{D}$ with Dirichlet boundary conditions and $\Gamma^{N}$ with Neumann type conditions. ### Notation and variational formulation of the Navier-Stokes equations By $L^{2}(\Omega)$ we denote the space of square integrable functions on the domain $\Omega\subset\mathbb{R}^{d}$ with scalar product $(\cdot,\cdot)$ and by $H^{1}(\Omega)$ those $L^{2}(\Omega)$ functions with weak first derivative in $L^{2}(\Omega)$. The function spaces for the velocity and pressure are then \[\begin{split}\mathbf{V}&:=\mathbf{v}^{D}+H_{0}^{1}( \Omega;\Gamma^{D})^{d},\quad H_{0}^{1}(\Omega;\Gamma^{D})^{d}\,:=\left\{\,\mathbf{ v}\in H^{1}(\Omega)^{d}\,:\,\mathbf{v}=0\text{ on }\Gamma^{D}\right\}\\ L&:=\left\{\,p\in L^{2}(\Omega),\text{ and, if }\Gamma^{N}= \varnothing,\,\int_{\Omega}p\,\mathrm{d}x=0\right\},\end{split} \tag{3}\] where $\mathbf{v}^{D}\in H^{1}(\Omega)^{d}$ is an extension of the Dirichlet data on $\Gamma^{D}$ into the domain. We normalize the pressure to yield uniqueness, if Dirichlet data is given on the whole boundary. With these spaces, the variational formulation of (1) is given by \[\begin{split}(\partial_{t}\mathbf{v},\,\boldsymbol{\phi})+( \mathbf{v}\cdot\triangledown\mathbf{v},\,\boldsymbol{\phi})+\nu(\triangledown \mathbf{v},\,\triangledown\boldsymbol{\phi})-(p,\,\triangledown\boldsymbol{ \phi})&=(\mathbf{f},\,\boldsymbol{\phi})\quad\forall\boldsymbol{ \phi}\in H_{0}^{1}(\Omega;\Gamma^{D})^{d}\\ (\triangledown\mathbf{v},\,\xi)&=0\qquad\quad \forall\xi\in L\\ \mathbf{v}(0,\,\cdot)&=\mathbf{v}_{0}(\cdot)\quad \text{ on }\Omega.\end{split} \tag{4}\] Let $\Omega_{h}$ be a quadrilateral or hexahedral finite element mesh of the domain $\Omega$ satisfying the usual requirements on the structural and form regularity such that the standard interpolation results hold, compare [54, Section 4.2]. By $h_{T}$ we denote the diameter of an element $T\in\Omega_{h}$ and by $h$ the maximum diameter of all $T\in\Omega_{h}$. ### Semidiscretization in space For the finite element discretization of (4) we choose equal-order continuous finite elements of degree two for velocity and pressure on hexahedral meshes. By $\mathbb{Q}_{h}^{(r)}$ we denote the space of continuous functions which are polynomials of maximum directional degree $r$ on each element $T\in\Omega_{h}$. Then we define the discrete trial- and test-spaces for the discretization of (4) as $\mathbf{v}_{h}$, $\boldsymbol{\psi}_{h}\in\mathbf{V}_{h}=[\mathbb{Q}_{h}^{(2)}]^{d}$ and $p_{h}$, $\xi_{h}\in L_{h}=\mathbb{Q}_{h}^{(2)}$. Since the resulting equal order finite element pair $\mathbf{V}_{h}\times L_{h}$ does not fulfill the inf-sup condition, we add stabilization terms of local projection type [5]. We further add convection stabilization, also based on local projections [6]. The resulting semidiscrete variational problem then reads \[\begin{split}&(\partial_{t}\mathbf{v}_{h},\,\boldsymbol{\psi}_{h})+( \mathbf{v}_{h}\cdot\triangledown\mathbf{v}_{h},\,\boldsymbol{\psi}_{h})+\nu( \triangledown\mathbf{v}_{h},\,\triangledown\boldsymbol{\psi}_{h})-(p_{h},\, \triangledown\boldsymbol{\psi}_{h})\\ &+\sum_{T\in\Omega_{h}}(\mathbf{v}_{h}\cdot\triangledown\mathbf{ v}_{h}-\pi_{h}\mathbf{v}_{h}\cdot\triangledown\pi_{h}\mathbf{v}_{h},\,\delta_{T}( \mathbf{v}_{h}\cdot\triangledown\boldsymbol{\psi}_{h}-\pi_{h}\mathbf{v}_{h} \cdot\triangledown\pi_{h}\boldsymbol{\psi}_{h}))=(\mathbf{f},\,\boldsymbol{ \psi}_{h})\quad\forall\boldsymbol{\psi}_{h}\in\mathbf{V}_{h}\\ &\qquad\qquad\qquad(\triangledown\mathbf{v}_{h},\,\xi_{h})+ \sum_{T\in\Omega_{h}}\alpha_{T}(\triangledown(p_{h}-\pi_{h}p_{h}),\triangledown (\xi_{h}-\pi_{h}\xi_{h}))=0\qquad\qquad\forall\xi_{h}\in L_{h}\end{split} \tag{5}\] Here, we denote by $\pi_{h}\,:\,\mathcal{Q}_{h}^{(2)}\to\mathcal{Q}_{h}^{(1)}$ the interpolation into the space of linear finite elements and by $\alpha_{T},\beta_{T}$ two local stabilization parameters specified in (7). ### Time discretization For temporal discretization, the time interval $[0,\,T]$ is split into discrete time steps of uniform size \[0=t_{0}<t_{1}<\cdots<t_{N}=T,\;k=t_{n}-t_{n-1}.\] The generalization to a non-equidistant time discretization is straightforward and only omitted for ease of presentation. We define $\mathbf{v}_{n}\,:=\,\mathbf{v}_{h}(t_{n})$ and $p_{n}\,:=\,p_{h}(t_{n})$ for the fully discrete approximation of velocity and pressure at time $t_{n}$ and apply the second order Crank-Nicolson method to (5), resulting in the fully discrete problem \[(\mathbb{V}\cdot\mathbf{v}_{n},\,\xi_{h})+\sum_{T\in\Omega_{h}} \alpha_{T}(\mathbb{V}(p_{n}-\pi_{h}p_{n}),\mathbb{V}(\xi_{h}-\pi_{h}\xi_{h}))=0 \forall\xi_{h}\in L_{h},\] \[\frac{1}{k}(\mathbf{v}_{n},\,\boldsymbol{\phi}_{h})\,+\frac{1}{2 }(\mathbf{v}_{n}\cdot\mathbb{V}\mathbf{v}_{n},\,\boldsymbol{\phi}_{h})+\frac{ \nu}{2}(\mathbb{V}\mathbf{v}_{n},\,\mathbb{V}\boldsymbol{\phi}_{h})-(p_{n},\, \mathbb{V}\cdot\boldsymbol{\phi}_{h})\] \[+\sum_{T\in\Omega_{h}}(\mathbf{v}_{n}\cdot\mathbb{V}\mathbf{v}_{ n}-\pi_{h}\mathbf{v}_{n}\cdot\mathbb{V}\pi_{h}\mathbf{v}_{n},\,\delta_{T}( \mathbf{v}_{n}\cdot\mathbb{V}\boldsymbol{\phi}_{h}-\pi_{h}\mathbf{v}_{n}\cdot \mathbb{V}\pi_{h}\boldsymbol{\phi}_{h}))=\frac{1}{k}(\mathbf{v}_{n-1},\, \boldsymbol{\phi}_{h})\] \[+\frac{1}{2}(\mathbf{f}_{n},\boldsymbol{\phi}_{h})+\frac{1}{2}( \mathbf{f}_{n-1},\,\boldsymbol{\phi}_{h})-\frac{1}{2}(\mathbf{v}_{n-1}\cdot \mathbb{V}\mathbf{v}_{n-1},\,\boldsymbol{\phi}_{h})-\frac{\nu}{2}(\mathbb{V} \mathbf{v}_{n-1},\,\mathbb{V}\boldsymbol{\phi}_{h})\quad\forall\boldsymbol{ \phi}_{h}\in\mathbf{V}_{h}. \tag{6}\] The right hand side only depends on the velocity $\mathbf{v}_{n-1}$ at the last time step $n-1$ and we will denote it as $\mathbf{b}_{n-1}$ in the following. The stabilization parameters $\alpha_{T}$ and $\delta_{T}$ depend on the mesh Peclet number and with two parameters $\alpha_{0}>0$ and $\delta_{0}\geq 0$ we define them as \[\alpha_{T}=\alpha_{0}\left(\frac{\nu}{h_{T}^{2}}+\frac{\|\mathbf{v}_{h}\|_{ \infty}}{h_{T}}+\frac{1}{k}\right)^{-1},\quad\delta_{T}=\delta_{0}\left(\frac {\nu}{h_{T}^{2}}+\frac{\|\mathbf{v}_{h}\|_{\infty}}{h_{T}}+\frac{1}{k}\right)^{ -1}, \tag{7}\] see [11] for details. Introducing the unknown $\mathbf{x}_{n}=(p_{n},\,\mathbf{v}_{n})$, we write a short form of the equations (6) as \[\mathcal{A}_{h}(\mathbf{x}_{n})=F_{h}, \tag{8}\] where $[\mathcal{A}_{h}(\mathbf{x}_{n})]_{i}$ and $[F_{h}]_{i}$ are the left and right hand sides of (6) for all test functions $(\xi_{h}^{i},\,\boldsymbol{\phi}_{h}^{i})$. In the presented form the Crank-Nicolson time discretization is sufficiently robust for smooth initial data $v_{0}$; we refer to [26] for small modifications with improved robustness and stability. ### Solution of the algebraic systems Discretization in space and time (6) leads to a nonlinear system of equations with a saddle-point character. The nonlinearity problem is solved by Newton iteration, where we usually omit the dependency of the stabilization parameters (7) on the velocity when setting up the Jacobian. The Jacobian matrix within the Newton iteration is kept for several iterations and also for consecutive time steps and only re-assembled when the nonlinear convergence rates deteriorate above a certain threshold, usually set to $0.1$ or $0.05$. As outer solver for the linear problems GMRES iteration [31] is employed which is preconditioned with a geometric multigrid solver [7]. To cope with the saddle-point structure of the incompressible Navier-Stokes equations and to allow for efficient shared memory parallelization, a Vanka smoother is used within the geometric multigrid. Therefore, small matrices describing the local problems on each mesh element are inverted in a parallel loop with Jacobi coupling. Using quadratic finite elements on hexahedral meshes these local matrices are of size $108\times 108$. The complete layout of the algebraic solver and its parallelization is described in [21]. ## 4 Neural Networks Various neural network architectures are in principle suitable for our approach. Having previously worked with recurrent neural networks designed for sequence prediction, we settled for feedforward neural networks for this work. The simple structure makes them faster to train and to evaluate. Furthermore, we found that with proper regularization, the feedforward neural network described below performs more consistent than the alternatives we tried. A general feedforward neural network is a composition of multiple parameterized nonlinear functions defined by \[F(x):=F(x;\ \mathcal{W})=(f_{L}\circ\cdots\circ f_{1})(x;\ \mathcal{W})\,, \tag{9}\] where $\mathcal{W}$ denotes the set of optimizable (learnable) parameters. The $i$-th layer of the prototypical ANN we consider here consists of a weight matrix $W_{i}\in\mathbb{R}^{n\times n_{i+1}}$ and a nonlinear function $\sigma:\ \mathbb{R}\to\mathbb{R}$, called the activation function, which is applied component wise, $\sigma\big{(}Wx\big{)}_{i}=\sigma\big{(}(Wx)_{i}\big{)}$. Furthermore, we employ two techniques common in deep learning to accelerate training and to improve generalization. Batch normalization $N_{i}:\ \mathbb{R}^{n_{i+1}}\to\mathbb{R}^{n_{i+1}}$, applied before each activation, re-scales each component by \[N_{i}(x)=\frac{x-\mu_{i}}{s_{i}}\odot\gamma_{i}+\beta_{i},\] where $x\odot y$ is the Hadamard product of two matrices (or tensors) and $\mu_{i}\in\mathbb{R}^{n_{i+1}}$ and $s_{i}\in\mathbb{R}^{n_{i+1}}$ are respectively, mean and standard deviation of samples estimated during training. The additional parameters $\gamma_{i}\in\mathbb{R}^{n_{i+1}}$ and $\beta_{i}\in\mathbb{R}^{n_{i+1}}$ are a learnable, component-wise affine transformation. Skip connections are inserted where input and output dimensions match, by adding the inputs of a layer to its outputs, leading to layers $f_{i}$ of the structure \[f_{1}(x) =\sigma_{1}(N_{1}(W_{1}x))\] \[f_{i}(x) =\sigma_{i}(N_{i}(W_{i}x))+x,\quad i=2,...\,,\,L-1\] \[f_{L}(x) =W_{L}x.\] ## 5 The Deep Neural Network Multigrid Method In this section, we review the Deep Neural Network Multigrid (DNN-MG) solver, that we introduced in [46]. DNN-MG leverages a deep neural network to predict the correction of a coarse mesh solution that has been prolongated onto one or multiple finer mesh levels. The objective is to obtain solutions that are more accurate than using the coarse mesh alone, while increasing computational efficiency compared to performing full multigrid computations on the fine mesh levels. We develop the DNN-MG solver in a general formulation, while maintaining the connection to the Navier-Stokes equations, for which we developed DNN-MG. We begin by providing an overview of the modifications made in DNN-MG compared to the MG method during a time step of the Navier-Stokes simulation. We continue by describing the structure, inputs and outputs of the Neural-Network in DNN-MG. In particular, the network is designed to make localized predictions over specific parts of the simulation domain. NotationIn the context of the Geometric Multigrid we always have a hierarchy of meshes $\Omega_{h}^{0},...\,,\ \Omega_{h}^{L+J}$, obtained by successive refinement of the initial mesh $\Omega_{h}^{0}$. We denote the finite element space $\mathbf{V}_{h}$ and $L_{h}$ introduced in Section 3.2 on level $l\in\{0,...\,,\,L+J\}$ by $V_{h}^{l}$ and $L_{h}^{l}$. With their nested structure, these spaces reflect the grid hierarchy. Then, let $\Omega_{h}^{l}$ be the quadrilateral or hexahedral finite element mesh on level $l\in\{0,...\,,\,L+J\}$ with $N_{\text{cells}}^{l}=\text{card}(\Omega_{h}^{l})$ cells and the index set $Z_{l}$ of all global degrees of freedom with cardinality $N_{\text{dof}}^{l}:=\text{card}(Z_{l})$. This notation and the description in Section 5.2 is inspired by the description of the Vanka smoother in [2]. ### Time stepping using DNN-MG We present a detailed description of one time step in the simulation of the Navier-Stokes equations using the DNN-MG solver. The computations involved are summarized in Algorithm 1. At the beginning of time step $n$, we solve for the unknown velocity $\mathbf{x}_{n}^{L}$ and pressure $p_{n}^{L}$ on the coarse level $L$ using the classical Newton-Krylov simulation (Algorithm 1, lines 4-6) as described in Section 3.4. Subsequently, we prolongate $\mathbf{x}_{n}^{L}$ to $\tilde{\mathbf{x}}_{n}^{L+J}$ on the next finer level $L+J$ (Algorithm 1, line 7) where a richer function space $L_{h}^{L+J}\times\mathbf{V}_{h}^{L+J}$ is available. Using the prolongated solution, we calculate the residual on level $L+J$ for the input to the neural network (Algorithm 1, line 8). The residual is calculated according to (8). We then use the neural network of the DNN-MG solver to predict the velocity update $\mathbf{d}_{n}^{L+J}$, which represents the difference $\mathbf{d}_{n}^{L+J}=\tilde{\mathbf{x}}_{n}^{L+J}-\tilde{\mathbf{x}}_{n}^{L+J}$ between the prolongated $\tilde{\mathbf{x}}_{n}^{L+J}$ and the unknown ground truth solution $\tilde{\mathbf{x}}_{n}^{L+J}$ on level $L+J$ (Algorithm 1, line 9). The prediction is based on $\tilde{\mathbf{x}}_{n}^{L+J}\in\mathbf{V}_{h}^{L+J}\times L_{h}^{L+J}$ and $\mathbf{r}_{n}^{L+J}$ and further utilizes information about the local mesh structure $\omega\in\mathbb{R}^{n_{\text{coo}}}$. This information is extracted from $\Omega_{h}^{L+J}$. To conclude the $n^{\text{th}}$ time step, we compute the right-hand side $b_{n+1}^{L+J}$ of Eq. 6 on level $L+J$ using the corrected velocity $\tilde{\mathbf{v}}_{n}^{L+J}+\mathbf{d}_{n}^{L+J}$ (Algorithm 1, line 10), and then restrict it to level $L$ (Algorithm 1, line 11). This right-hand side, incorporating the neural network-based correction, becomes part of the multigrid computations in the next time step (Algorithm 1, line 4), thereby influencing the overall time evolution of the flow. This approach of constructing the right-hand side $b_{n+1}^{L+J}$ on level $L+J$ and subsequently restricting it is a crucial aspect of the DNN-MG solver. We note that in the case of the Navier-Stokes equations the pressure is handled implicitly on level $L$ in the multigrid solve and is not directly receive a correction by the neural network. However, the pressure is included in the network's input through the prolongated solution and the residual, as it is an integral part of the solution and is indirectly influenced through the corrections to the velocity. ### The Neural Network of DNN-MG The neural network component forms the core of the DNN-MG solver. It plays a crucial role in enhancing computational efficiency, facilitating fast training procedures, and ensuring robust generalization across iverse flow regimes beyond the training set. We achieve this through the careful design of a compact neural network architecture with a moderate number of parameters and a localized, patch-based structure. Figure 2 provides an overview of the network component, illustrating the local approach, which we introduce next. A patch-based neural networkTo ensure computational efficiency, the DNN-MG approach adopts a neural network that operates on localized patches of the mesh. The network's input and output are constrained to a patch, which reduces the computational cost compared to a prediction over the entire domain. This local approach leads to more compact neural networks, with a smaller number of parameters. We define a patch $P_{L,J}^{M}\subset\Omega_{h}^{L+J}$ as a collection of cells $K\in\Omega_{h}^{L+J}$ on level $L+J$. The neural network operates on a single patch $P_{L,J}^{M}$ and predicts the velocity update $\mathbf{d}_{n,P}^{L+J}$ specifically for the degrees of freedom within that patch. Since a single neural network is used in parallel on each patch, all patches must have the same structure and consist of the same number of fine mesh elements. Formally, let $L(P_{L,J}^{M})\times\mathbf{V}(P_{L,J}^{M})\subset L_{h}^{L+J}\times\mathbf{V} _{h}^{L+J}$ be the function space associated with the degrees of freedom in patch $P_{L,J}^{M}$. The neural network learns a mapping \[\begin{split}\mathcal{N}\,:\,\left(L(P_{L,J}^{M})\times\mathbf{V} (P_{L,J}^{M})\right)^{2}\times\mathbb{R}^{n_{\text{Geo}}}&\to L(P_{ L,J}^{M})\times\mathbf{V}(P_{L,J}^{M})\,,\\ (\tilde{\mathbf{x}}_{n}^{L+J}\|_{P_{L,J}^{M}},\,\mathbf{r}_{n}^{L+J }\|_{P_{L,J}^{M}},\,\omega_{P_{L,J}^{M}})&\mapsto\mathbf{d}_{n,P}^ {L+J}\,,\end{split} \tag{10}\] that predicts a local velocity update $\mathbf{d}_{n,P}^{L+J}\in\mathbf{V}(P_{L,J}^{M})$ based on local data which is also restricted to the patch $P_{L,J}^{M}$. This prediction is repeated independently for each patch in the domain, covering the entire domain $\Omega_{h}^{L+J}$. The predictions of shared degrees of freedom in adjacent patches are averaged to ensure consistency. We now formulate the process we just sketched in more mathematical terms. We define a patch $P_{L,J}^{M}$, as a collection of cells on level $L+J$ that are formed by $J$ successive refinements of a coarse mesh cell $M\in\Omega_{h}^{L}$. In other words, $P_{L,J}^{M}$ comprises the $(2^{d})^{J}$ cells obtained by successive refinement of the coarse mesh cell $M\in\Omega_{h}^{L}$ for $J$ levels and can be defined rigorously by \[P_{L,J}^{M}\coloneqq\{K\in\Omega_{h}^{L+J}\mid\tilde{K}\cap M\neq\emptyset,\, M\in\Omega_{h}^{L}\}. \tag{11}\] Additionally, we define $\Omega_{h}^{\text{patch}}\coloneqq\left\{P_{L,J}^{M}\text{ for }M\in\Omega_{h}^{L}\right\}$. Note that $\text{card}(\Omega_{h}^{\text{patch}})=N_{\text{cells}}^{L}$. We denote the set of global degrees of freedom associated with the patch $P_{L,J}^{M}$ by $Z(P_{L,J}^{M})\subset Z_{L+J}$. The cardinality of $Z(P_{L,J}^{M})$ Figure 2: DNN-MG adopts a local approach where the neural network works on small neighborhoods of the simulation domain called patches, and independently provides a velocity correction for each one. is denoted by $N_{\text{dof}}^{P}:=\text{card}(Z(P_{L,J}^{M}))$. Further, the index set $\hat{Z}(P_{L,J}^{M}):=\{0,...\,,\,N_{\text{dof}}^{P}-1\}$ contains all local degrees of freedom on $P_{L,J}^{M}$. For a given patch $P_{L,J}^{M}$ and a local degree of freedom with index $\hat{\imath}\in\hat{Z}(P_{L,J}^{M})$ the mapping \[\text{dof}\,:\,\Omega_{h}^{L+J}\times\hat{Z}(P_{L,J}^{M})\to Z_{l}\,,\quad(P_{L,J}^{M},\,\,\hat{\imath})\mapsto i\,, \tag{12}\] provides the uniquely defined global index $i\in Z_{L+J}$. We can now define the prediction of the defect by the neural network in precise terms. It consists of multiple steps: 1. _Local restriction by $\mathcal{Q}:\,\mathbb{R}^{N_{\text{dof}}^{L+J}}\to\mathbb{R}^{N_{\text{cells} }^{L}\times N_{\text{dof}}^{P}}$_: Using a local restriction operator, the globally defined data of size $N_{\text{dof}}^{L+J}$ is transformed into local, patchwise defined input of size $N_{\text{dof}}^{P}$ suitable for the DNN. This is done for each patch, resulting in a matrix of size $\mathbb{R}^{N_{\text{cells}}^{L}\times N_{\text{dof}}^{P}}$. 2. _Patch-wise prediction by the neural network $\mathcal{N}:\,\mathbb{R}^{(2N_{\text{dof}}^{P}+n_{\text{Geo}})}\to\mathbb{R}^ {N_{\text{dof}}^{P}}$_: For each patch, the neural network $\mathcal{N}$ maps the local input data of size $(2N_{\text{dof}}^{P}+n_{\text{Geo}})$ to a local output of size $N_{\text{dof}}^{P}$ predicting the velocity update for the degrees of freedom within the patch $P_{L,J}^{M}$. 3. _Global extension by $\mathcal{L}:\,\mathbb{R}^{N_{\text{cells}}^{L}\times N_{\text{dof}}^{P}}\to \mathbb{R}^{N_{\text{dof}}^{L+J}}$_: The locally defined output of the DNN is processed by the extension operator $\mathcal{L}$, which maps the local data of size $N_{\text{cells}}^{L}\times N_{\text{dof}}^{P}$ to a single globally defined defect vector of size $N_{\text{dof}}^{L+J}$. In order to define $\mathcal{Q}$, we need the $P_{L,J}^{M}$-local restriction $\mathcal{Q}:\,\mathbb{R}^{N_{\text{dof}}^{L+J}}\times\Omega_{h}^{\text{Patch} }\to\mathbb{R}^{N_{\text{dof}}^{P}}$ defined by \[(\mathcal{Q}(\mathbf{x},\,P_{L,J}^{M}))[\hat{\imath}]=\mathbf{x}[\text{dof}(P_ {L,J}^{M},\hat{\imath})]\,,\quad\text{for}\,\,\hat{\imath}\in\hat{Z}(P_{L,J}^{ M})\,. \tag{13}\] Then, we define $\mathcal{Q}$ such that it maps a global vector $\mathbf{x}\in\mathbb{R}^{N_{\text{dof}}^{L+J}}$ to a matrix containing local restrictions to a patch for all $P_{0},...\,,\,P_{N_{\text{cells}}^{L}-1}\in\Omega_{h}^{\text{Patch}}$, stacked on top of each other: \[\mathcal{Q}:\,\mathbb{R}^{N_{\text{dof}}^{L+J}}\to\mathbb{R}^{N_{\text{cells} }^{L}\times N_{\text{dof}}^{P}},\quad\mathbf{x}\mapsto\left(\mathcal{Q}_{P}(x,P_{0}),...\,,\,\mathcal{Q}_{P}(x,\,P_{N_{\text{cells}}^{L}-1})\right)^{\top} \tag{14}\] After applying $\mathcal{Q}$ to the residual $\mathbf{r}_{n}^{L+J}$ and $\tilde{\mathbf{x}}_{n}^{L+J}$, we obtain the patch-wise, local data required as input to the DNN. According to (14), the local data is stacked along the first dimension, forming a single batch. The network is then evaluated independently for each patch, with each row of the input representing a different patch. The output of the DNN then consists of a batch of the patchwise defect $\mathbf{d}_{n,P}^{L+J}$. To obtain the global defect vector $\mathbf{d}_{n}^{L+J}$, we use the extension operator $\mathcal{L}$ to transfer this batch of local data back to the global domain. To this end, we first introduce a $P_{L,J}^{M}$-local extension operator $\mathbf{L}:\,\mathbb{R}^{N_{\text{dof}}^{P}}\times\Omega_{h}^{\text{Patch}}\to \mathbb{R}^{N_{\text{dof}}^{L+J}}$ defined by \[(\mathbf{L}(\mathbf{y},\,P_{L,J}^{M}))[\imath]=\begin{cases}\mathbf{y}[\hat{ \imath}]\,,\quad\text{if}\,\,\exists\hat{\imath}\in\hat{Z}(P_{L,J}^{M}):\,\, \imath=\text{dof}(K,\hat{\imath}),\,K\in P_{L,J}^{M}\,,\\ 0\,,\qquad\text{if}\,\,\imath\notin Z_{l}(P_{L,J}^{M})\end{cases}\,.\] We further need a scaling vector $\mu\in\mathbb{R}^{N_{\text{dof}}^{L+J}}$ which contains the reciprocal of the valence of a degree of freedom, i. e. the number of patches a degree of freedom is contained in. Then we define $\mathcal{L}$ such that it maps $N_{\text{cells}}^{L}$ local vectors to one global vector as \[\mathcal{L}:\,\mathbb{R}^{N_{\text{dof}}^{P}\times N_{\text{cells}}^{L}}\to \mathbb{R}^{N_{\text{dof}}^{L+J}}\,,\quad\mathbf{y}\mapsto\mu\odot\sum_{P_{L,J }^{M}\in\Omega_{h}^{\text{Patch}}}\mathbf{L}(\mathbf{y}[M],\,P_{L,J}^{M})\,. \tag{15}\] For the sake of brevity, we used $M$ not only as an element of $\Omega_{h}^{L}$ in (15) but also to index the rows of the collection of local vectors. This concludes the description of all operations necessary to integrate the neural network with its local approach into the multigrid solver. In particular \[\mathcal{L}\circ\mathcal{N}(Q(\tilde{\mathbf{x}}_{n}^{L+J}),\,Q(\mathbf{r}_{n}^ {L+J}),\,\Omega_{h}^{L+J}):\,L_{h}^{L+J}\times\mathbf{V}_{h}^{L+J}\to L_{h}^{L +J}\times\mathbf{V}_{h}^{L+J}\,,\quad\tilde{\mathbf{x}}_{n}^{L+J}\mapsto \mathbf{d}_{n}^{L+J}\,, \tag{16}\] is a well-defined operation (cf. Algorithm 1 line 9). The mapping (10) also fits into this framework and interpreting the definition (13) in a finite element space context, the restrictions $\tilde{\mathbf{x}}_{n}^{L+J}\|_{p_{L,J}^{M}}$ and $\mathbf{r}_{n}^{L+J}\|_{p_{L,J}^{M}}$ are obtained. The network operates on each patch independently, which allow us to take advantage of parallelization through the localized nature of the problem. This shows that the local approach is a key feature in our method, which enables the network to handle a wide range of flow scenarios while maintaining computational efficiency, as we will show later in Section 6. The rest of this section consists of a detailed description of the inputs to the neural network and the training methodology. Neural network inputsThe effectiveness of neural network prediction hinges upon the selection of appropriate network inputs that provide enough information regarding the coarse flow characteristics at level $L$ and the local patch geometry, akin to the significance of mesh cell geometry in classical finite element simulations. A well-informed choice of these inputs holds the potential to keep the number of parameters in the neural network low, thereby reducing evaluation time during runtime and minimizing the data and time prerequisites for training. In our implementation, we use the following inputs to the neural network: * nonlinear residuals $\mathbf{r}_{n}^{L+J}\|_{p_{L,J}^{M}}=\left(F_{h}^{L+J}-\mathcal{A}_{h}^{L+J}( \tilde{\mathbf{x}}_{n}^{L+J})\right)\|_{p_{L,J}^{M}}\in\mathbb{R}^{N_{ud}^{p}}$ of the prolongated coarse mesh solution for (6); * the velocities and pressure $\tilde{\mathbf{x}}_{n}^{L+J}\|_{p_{L,J}^{M}}\in\mathbb{R}^{N_{ud}^{p}}$ on mesh level $L+J$; * the geometry of the cells, which for the 3D case consist of * the edge lengths $h^{c}\in\mathbb{R}^{12}$; * the lengths of the diagonals connecting the vertices which do not share the same face $a^{c}\in\mathbb{R}^{4}$; * for each vertex, the average of the 3 angles between the faces $a^{c}\in\mathbb{R}^{8}$. The nonlinear residual plays a crucial role as it quantifies the local error or defect at each degree of freedom on level $L+J$ and therefore carries essential information about the underlying partial differential equation. Furthermore, the velocity and pressure fields contain details about the current solution and provide knowledge about the flow within the current patch. It is important to note that the geometric information of the cells also plays a significant role in our approach, as we impose no specific restrictions on the cells, except for the standard shape regularity requirements commonly employed in finite element methods. Training of DNN-MGThe training of DNN-MG is based on simulations of the Navier-Stokes equations for which a multigrid representation of the velocity $\mathbf{v}_{n}^{I}$ with two levels $L$ and $L+J$ is available. The velocity $\mathbf{v}_{n}^{L+J}$ thereby serves as ground truth $\tilde{\mathbf{v}}_{n}^{L+J}$. The goal of the training is to optimize the network parameters such that the norm $\left\\|(\tilde{\mathbf{v}}_{n}^{L+J}+\mathbf{d}_{n}^{L+J})-\tilde{\mathbf{v}}_ {n}^{L+1}\right\\|$ of the difference between the predicted velocity $\tilde{\mathbf{v}}_{n}^{L+J}+\mathbf{d}_{n}^{L+J}$ (i. e. the velocity after the correction) and the ground truth $\tilde{\mathbf{v}}_{n}^{L}$ is minimized throughout the simulation. Typically, the training data for the network comprises only a few snapshots of simulation data from simple scenarios, as we will see in the next section. ## 6 Numerical examples In the following paragraphs we will document different numerical simulations. The test cases are based on a well established 3D Navier-Stokes benchmark problem [55, 10]. While the neural network is trained on the original benchmark configuration, we study its performance and generalization on a set of modified problems. As perturbations we consider increasing or decreasing the Reynolds number and also a substantial change to the problem geometry by introducing a second obstacle. It is important to stress that the network is trained only once and no retraining is applied for the further test cases. ### Implementation aspects The DNN-MG method is implemented mainly using two libraries: The FEM library Gascoigne3D [8] and the machine learning library PyTorch [52]. The Newton-Krylov geometric multigrid implemented in Gascoigne3D method has been shown to be efficient in [21]. We implement the deep neural networks with Libtorch, the C++ interface of PyTorch. Using C++ for the implementation allows us to couple the numerical methods and neural networks in a performant manner without unnecessary overhead. PyTorch also supports parallelization with MPI, which is used here to distribute training on multiple GPUs. ### The 3D Benchmark Flow Around a Cylinder We examine a variant of the three-dimensional flow benchmark presented in [55] in a setting similar to the _3D-2Z_ one, with different Reynolds numbers $Re\in\{200,240\}$. We show the geometry in Figure 3. While the original benchmark description considers an obstacle with circular cross-section, we allow elliptical ones for testing the generalizability of the approach. We define the Reynolds number as \[\mathrm{Re}=\frac{\bar{\mathbf{v}}\cdot L}{v}, \tag{17}\] where $\bar{\mathbf{v}}$ is the average inflow velocity and $L$ the length of the major axis of the obstacle. When the cross-section of the cylinder is circular, $L$ represents its diameter. If the cross-section is elliptical, $L$ corresponds to the height of the obstacle, as the major axis is always parallel to the $y$-axis. The time step size is chosen as $k=0.008$ on the interval $I=(0,8]$. The initial velocity is $\mathbf{v}(0)=\mathbf{0}$, and the inflow boundary condition is prescribed for $t>0$, with a smooth startup process. The flow is driven Figure 3: Geometry of the training and test scenario with a parabolic inflow profile at $\Gamma_{\mathrm{in}}$, do-nothing boundary conditions at the outflow boundary $\Gamma_{\mathrm{out}}$ and no-slip conditions on the walls $\Gamma_{\mathrm{wall}}$. The center of the obstacle is at (0.5, 0.2). by a Dirichlet profile $\mathbf{v}=\mathbf{v}^{D}$ at the left boundary $\Gamma_{\text{in}}$ given by \[\begin{split}\mathbf{v}_{in}^{3d}(t,\,y,\,z)&=\omega( t)\frac{y(H-y)(H^{2}-z^{2})}{(H/2)^{2}H^{2}}\frac{9}{8}\bar{\mathbf{v}}_{3d}\,\,\,\text{ on}\,\,\,\Gamma_{in}\,:=0\times[0,\,H],\\ \omega(t)&=\begin{cases}\frac{1}{2}-\frac{1}{2}\cos(5 \pi t),&t\leq\frac{1}{5}\\ 1&t>\frac{1}{5}.\end{cases}\end{split} \tag{18}\] $H=0.41$ is the height and width of the channel and $\omega(t)$ acts, as a regularization of the startup phase. On the wall boundary $\Gamma_{wall}$ and on the obstacle we use no-slip boundary conditions. On the outflow boundary $\Gamma_{\text{out}}$ we use a do-nothing outflow condition [27]. The average flow rate is $\bar{\mathbf{v}}_{3d}=1$ and the viscosity is $v=5\times 10^{-4}$. The training data is generated considering the circular obstacle, i. e. $L=0.1$ which results in the Reynolds number $\text{Re}=200$. For testing, the elliptical cross-section with $L=0.12$ is considered, leading to the slightly increased Reynolds number $\text{Re}=240$. See Section 5.2 for details. As in the classical benchmark description we analyze the drag and lift coefficients of the obstacle \[J_{\text{drag}}=\int_{\Gamma}\big{(}v\forall\mathbf{v}\mathbf{n}-p\mathbf{n} \big{)}\mathbf{e}_{x}\,\mathrm{d}s,\quad J_{\text{lift}}=\int_{\Gamma}\big{(} v\forall\mathbf{v}\mathbf{n}-p\mathbf{n}\big{)}\mathbf{e}_{y}\,\mathrm{d}s. \tag{19}\] To compute the drag and lift coefficients, we use the Babuszka-Miller trick; cf. [10, 3] and rewrite the surface integrals over the volume for obtaining super convergence of fourth order. ### Neural Network parameters DNN-MG uses a neural network that operates on patches. In our experiments, a patch is once or twice refined cell on level $L$. In Table 2 we list the dimensions and number of trainable parameters of the resulting neural networks. The hidden size is arbitrary, but a size of 512 or 750 has been shown to give good results, while keeping the network size relatively small. We generally use 8 hidden layers of the same size. Unless mentioned otherwise, the results presented here were obtained with a hidden size of 750. Networks with a width of 512 are considered only for comparison. ### Neural Network training Training data are generated on grids with three successive levels of refinement, see Table 1. Here, we denote by MG(3) the coarse grid. The grid levels MG(4) and MG(5) are the training data for a network prediction over one, respectively over two grid levels. A single simulation is sufficient since, by the patch-wise application of the network, a single simulation provides $N_{c}\times N_{T}$ training items, where $N_{c}$ is the number of patches and $N_{T}$ is the length of the time series. From the experience gained in the 2D case [46], we consider a small subinterval of $I=[0,\,8]$. Here we \begin{table} \begin{tabular}{l l r r} \hline \hline & Level & \# DoF & Patch size \\ \hline Coarse & MG(3) & 390 720 & 0 \\ Reference for DNN-MG($3+1$) & MG(4) & 3 003 520 & 8 \\ Reference for DNN-MG($3+2$) & MG(5) & 23 546 112 & 64 \\ \hline \hline \end{tabular} \end{table} Table 1: Spatial meshes with increasing number of multigrid levels and degrees of freedom (DoF). For the DNN-MG setups we indicate the size of the patches (counted as number of fine mesh elements) where the network is applied. use $t\in[4,\,7]$ resulting in $N_{T}=375$. The interval from [2, 4] is used as validation set. The full dataset consists of 1 TB of data. The networks were trained using 2 GPU nodes, each with 2 Nvidia A100 GPUs and 2 Intel Xeon Platinum 8360Y CPUs. The number of MPI processes were chosen equal to the number of GPUs such that one MPI process uses one GPU and CPU. Using this setup, the training of the feedforward network takes one day, regardless of the network size. Although the data loading process is implemented efficiently and the average load of the GPUs ranges from 80% for small networks to 90% for large networks, the limiting factor in terms of performance is the memory bandwidth. As loss function $\mathcal{L}$ we employed a simple $l^{2}$-loss \[\mathcal{L}=\sum_{n=1}^{N_{T}}\frac{1}{N_{c}}\sum_{c=1}^{N_{c}}l^{2}(\tilde{u }_{n,c}^{1+J}+d_{n,c}^{L+J},\,\tilde{u}_{n,c}^{L+J})^{2}=\sum_{i=1}^{N_{T}} \frac{1}{N_{c}}\sum_{c=1}^{N_{c}}\left\|\left(\tilde{v}_{n,c}^{L+J}+d_{n,c}^{L+ J}\right)-\tilde{v}_{n,c}^{L+J}\right\|_{2}^{2},\quad J\in\{1,\,2\}.\] The parameter $J$ is the number of levels we skip with the prediction by the DNN. We use a Tikhonov-regularization with a scaling factor $\alpha=10^{-5}$. To optimize our model, we utilize the AdamW [37] optimizer with a maximum of 1000 epochs. Finally, we select the model with the lowest validation loss. The convergence of the training for the different network configurations is reported in Figure 4. \begin{table} \begin{tabular}{r r r r r} \hline \hline Predicted levels & $N_{\text{in}}$ & $N_{\text{hidden}}$ & $N_{\text{out}}$ & Trainable parameters \\ \hline 1 & 240 & 512 & 81 & 2 007 552 \\ 2 & 1024 & 512 & 375 & 2 559 488 \\ 1 & 240 & 750 & 81 & 4 190 250 \\ 2 & 1024 & 750 & 375 & 4 998 750 \\ \hline \hline \end{tabular} \end{table} Table 2: The input size $N_{\text{in}}$, hidden size $N_{\text{hidden}}$, output size $N_{\text{out}}$ and the number of trainable parameters of the feedforward network, for different numbers of predicted levels. Figure 4: The average training and validation loss of the feedforward network architecture with hidden size (a) 512 and (b) 750. Due to the data efficiency of DNN-MG and the large number of batches ($\approx$ 900 000) relative to the network size, we were able to achieve a good training loss after a single epoch and there is only little improvement in the training loss beyond that point. ### Flow prediction For testing we use the flow around an elliptic obstacle with an increased height of $0.12$ compared to the training data (see Figure 3). The resulting finite element meshes have the same structure as in the training case, but the elements are distorted. We observe that DNN-MG is indeed able to predict high frequency fluctuations that are not visible in the coarse mesh solution. In particular in the vicinity of the obstacle the quality of the solution is strongly enhanced with distinct features in the wake being apparent in the DNN-MG simulation. In addition to a viscosity of $v=5\times 10^{-4}$ and $Re=240$, we consider fluids with $v=4\times 10^{-4}$ corresponding to a Reynolds number of $Re=300$ and $v=6.67\times 10^{-4}$ which corresponds to a Reynolds number of $180$. Both of these Reynolds numbers are calculated for the elliptic obstacle with a height of $0.12$, which is our typical test case. The network is trained only for $Re=200$ using the obstacle with circular cross-section. Channel flow at $\mathbf{Re=180}$In Figure 6 and Table 3, the drag and lift forces at Reynolds number $180$, computed using DNN-MG and the classical MG method on different levels are presented. The results show the predictive capacity of the DNN-MG approach in reproducing the dynamic behavior of the flow. In terms of the drag coefficient, DNN-MG yields a slight improvement in terms of the magnitude. The lift coefficients on the other hand improve greatly compared to the coarse solution, as the overall flow dynamics are well reconstructed, albeit with temporal deviations. We observe this in Figure 5 too, where different flow patters emerge in the wake of the flow. Furthermore, with respect to the minimum and maximum lift values, our approach is in good agreement with the reference solution. Experience with finite element simulations usually show that the lift is the functional that is harder to approximate by far [10]. Channel flow at $\mathbf{Re=240}$Similarly, Figure 7 and Table 4 depict the drag and lift forces at Reynolds number $240$, obtained through DNN-MG and the multigrid solution on the coarse and reference level. Figure 5 show a comparison of the velocity fields obtained using DNN-MG with the coarse mesh and reference solution. The local corrections improve the solution at a global level and DNN-MG is clearly able to capture and improve the flow characteristics. In particular, it has the capacity to significantly enhance the flow resolution, yielding improved solutions and reproducing intricate features that were previously unobservable at lower levels. This underscores the method's efficacy in capturing fine-scale details. We see similar behavior as in $\mathbf{Re=240}$, indicating good generalization of DNN-MG to different Reynolds numbers, considering the network was trained on $\mathbf{Re=200}$. Figure 5: Comparison of the magnitude of the velocity field for the channel with a cylindrical obstacle with an elliptical cross-section at $\mathrm{Re}=240$ and time $t=7.5$. The results are shown for DNN-MG($3+2$), along with the reference solution obtained with 5 levels (MG(5)), as well as the coarse mesh solution at level 3 (MG(3)). The DNN-MG($3+2$) method utilizes a network trained on simulation data from a round obstacle. The first three images show a cross-section of the domain in the $xy$-plane and the others show a cross-section of the domain in the $xz$-plane. \begin{table} \begin{tabular}{l c c c c c c c c} \hline \hline Type & $\min C_{d}$ & $\max C_{d}$ & $\overline{C_{d}}$ & amp $C_{d}$ & $\min C_{l}$ & $\max C_{l}$ & $\overline{C_{l}}$ & amp $C_{l}$ \\ \hline MG(3) & 1.7571 & 1.7662 & 1.7591 & 0.0091 & 0.0083 & 0.0235 & 0.0157 & 0.0152 \\ MG(4) & 1.7606 & 1.7713 & 1.7638 & 0.0107 & -0.0016 & 0.0208 & 0.0104 & 0.0224 \\ MG(5) & 1.8472 & 1.8595 & 1.8494 & 0.0123 & -0.0178 & 0.0069 & -0.0033 & 0.0247 \\ DNN-MG(3 + 1)-512 & 1.7082 & 1.7575 & 1.7186 & 0.0493 & 0.0025 & 0.0241 & 0.0136 & 0.0216 \\ DNN-MG(3 + 2)-512 & 1.8593 & 1.9112 & 1.8742 & 0.0519 & -0.0223 & 0.0025 & -0.0119 & 0.0248 \\ DNN-MG(3 + 1)-750 & 1.7257 & 1.7652 & 1.7374 & 0.0394 & -0.0235 & 0.0045 & -0.0049 & 0.0280 \\ DNN-MG(3 + 2)-750 & 1.7657 & 1.8052 & 1.7774 & 0.0394 & -0.0235 & 0.0045 & -0.0049 & 0.0280 \\ \hline \hline \end{tabular} \end{table} Table 3: The min, max, mean and amplitude of drag and lift functionals in the same setting as above in Figure 6. The best results in comparison to the $MG$(5) solution are highlighted with color coding: 1st best, 2nd best, 3rd best. Figure 6: The drag (left) and lift (right) functionals for the channel with one obstacle with an elliptical cross section at $\mathrm{Re}=180$ are shown. The results are presented for three discretizations: The coarse mesh MG(3), the reference MG(5), and the DNN-MG($3+2$) improved by a neural network. \begin{table} \begin{tabular}{l c c c c c c c c} \hline \hline Type & $\min C_{d}$ & $\max C_{d}$ & $\overline{C_{d}}$ & amp $C_{d}$ & $\min C_{l}$ & $\max C_{l}$ & $\overline{C_{l}}$ & amp $C_{l}$ \\ \hline MG(3) & 1.6810 & 1.6920 & 1.6840 & 0.0111 & 0.0081 & 0.0495 & 0.0255 & 0.0415 \\ MG(4) & 1.6783 & 1.6938 & 1.6869 & 0.0155 & -0.0204 & 0.0373 & 0.0098 & 0.0576 \\ MG(5) & 1.7724 & 1.8096 & 1.7910 & 0.0372 & -0.0303 & 0.0852 & 0.0123 & 0.1155 \\ DNN-MG(3 + 1)-512 & 1.6305 & 1.6748 & 1.6425 & 0.0444 & -0.0364 & 0.0224 & -0.0002 & 0.0588 \\ DNN-MG(3 + 2)-512 & 1.6866 & 1.7931 & 1.7537 & 0.1066 & -0.0347 & 0.0363 & 0.0089 & 0.0707 \\ DNN-MG(3 + 1)-750 & 1.6215 & 1.6664 & 1.6309 & 0.0448 & 0.0008 & 0.0304 & 0.0146 & 0.0296 \\ DNN-MG(3 + 2)-750 & 1.6966 & 1.7167 & 1.7079 & 0.0200 & -0.0345 & 0.0426 & 0.0059 & 0.0774 \\ \hline \hline \end{tabular} \end{table} Table 4: The min, max, mean and amplitude of drag and lift functionals in the same setting as above in Figure 7. The best results in comparison to the $MG$(5) solution are highlighted with color coding: 1st best, 2nd best, 3rd best. Figure 7: The drag (left) and lift (right) functionals for the channel with one obstacle with an elliptical cross section at $\mathrm{Re}=240$ are shown. The results are presented for three discretizations: The coarse mesh MG(3), the reference MG(5), and the DNN-MG($3+2$) improved by a neural network. \begin{table} \begin{tabular}{l c c c c c c c c} \hline \hline Type & $\min C_{d}$ & $\max C_{d}$ & $\overline{C_{d}}$ & amp $C_{d}$ & $\min C_{l}$ & $\max C_{l}$ & $\overline{C_{l}}$ & amp $C_{l}$ \\ \hline MG(3) & 1.6291 & 1.6449 & 1.6339 & 0.0159 & 0.0081 & 0.0337 & 0.0185 & 0.0256 \\ MG(4) & 1.6125 & 1.6507 & 1.6336 & 0.0381 & -0.0592 & 0.0724 & 0.0142 & 0.1316 \\ MG(5) & 1.7375 & 1.7794 & 1.7578 & 0.0419 & -0.0527 & 0.0404 & -0.0058 & 0.0931 \\ DNN-MG(3 + 1)-512 & 1.4532 & 1.5883 & 1.4985 & 0.1351 & -0.0047 & 0.0355 & 0.0161 & 0.0401 \\ DNN-MG(3 + 2)-512 & 1.4923 & 1.7125 & 1.5776 & 0.2202 & -0.0071 & 0.0656 & 0.0266 & 0.0728 \\ DNN-MG(3 + 1)-750 & 1.5660 & 1.6347 & 1.5858 & 0.0687 & -0.0235 & 0.0714 & 0.0184 & 0.0950 \\ DNN-MG(3 + 2)-750 & 1.6490 & 1.6910 & 1.6626 & 0.0419 & -0.0655 & 0.0664 & 0.0109 & 0.1319 \\ \hline \hline \end{tabular} \end{table} Table 5: The min, max, mean and amplitude of drag and lift functionals in the same setting as above in Figure 8. The best results in comparison to the $MG$(5) solution are highlighted with color coding: 1st best, 2nd best, 3rd best. Figure 8: The drag (left) and lift (right) functionals for the channel with one obstacle with an elliptical cross section at $\mathrm{Re}=300$ are shown. The results are presented for three discretizations: The coarse mesh MG(3), the reference MG(5), and the DNN-MG($3+2$) improved by a neural network. Figure 9: Relative velocity errors (first row) and pressure errors (second row) for coarse solutions with and without ANN correction (MG(3) and DNN-MG($3+2)-750$) compared to the reference solution on level $L+2$ for different Reynolds numbers. As further evidence of the innovative capacity of DNN-MG we show the errors of the MG(4) solution as well, which we consistently outperform. ### Performance measurements For the DNN-MG method to be advantageous, it must achieve two goals: Firstly, it must enhance the accuracy compared to the solution on a coarse mesh. Secondly, it must reduce the computation time compared to a solution on a fine mesh. However, the Newton-Krylov geometric multigrid method already is of optimal complexity [33, 21]. In [46] this already raised the question what advantage DNN-MG offers, and we addressed it there for the 2D case. Here we adapt the theoretical findings to the 3D case. Complexity analysisThe Newton-Krylov geometric multigrid method has linear complexity $\mathcal{O}(N)$ in the number of DoFs $N$. For each global mesh refinement the number of DoFs increase by a factor of 8, i. e. $N^{L+1}\approx 8N^{L}$. The constant hidden in $\mathcal{O}(N)$, however, can be very significant, since on average 5 Newton steps are required in each time step and within each Newton step one has to compute on average 12 GMRES steps with one sweep of the geometric multigrid solver as preconditioning. In addition, the geometric multigrid method usually involves six pre-smoothing and six post-smoothing steps. Thus, on each mesh level $L$, $L-1$,..., 0, a total of approximately 700 Vanka smoothing steps must be performed. One Vanka step thereby requires the inversion of about $\mathcal{O}(N^{L}/8)$ small matrices of size $108\times 108$, one on each element of the mesh, resulting in approximately $108^{3}\simeq 1\,259\,712$ operations. Since the complexity of all mesh levels sums up to about $N^{L}+N^{L-1}+...\,N_{0}\approx N^{L}(1+8^{-1}+...+8^{-L})\approx\frac{8}{7}N ^{L}$ we can estimate the effort for the complete solution process on level $L$ as $5\cdot 12\cdot(6+6)\cdot 108^{3}\cdot\frac{8}{7}\approx 10^{9}N^{L}$. We thereby only counted the effort for smoothing and neglected all further matrix, vector and scalar products. If we were to solve the problem on mesh level $L+1$, the required effort would increase by a factor of 8 to approximately $\approx 8\cdot 10^{9}N^{L}$ operations. On the other hand, the DNN-MG method only requires the prolongation of the solution to the next finer mesh and the evaluation of the neural network on each patch. If we again consider patches of minimal size (one patch is one element of mesh layer $L+1$) about $\mathcal{O}(N^{L+1}/8)=\mathcal{O}(N^{L})$ patches must be processed. The effort for the evaluation of the network can be estimated by the number of trainable parameters with $N_{\text{c}}$ inputs. The DNN-MG approach requires only one evaluation of the network in each Figure 10: On the left the runtime of DNN-MG and MG is plotted over multiple levels of refinement. We see that DNN-MG has less than 50% overhead compared to a coarse mesh simulation. On the right we see the contributions of different parts of the program to the wall clock time in percent. We observe that the evaluation of the ANN takes less than 3% of the time. time step. The number of trainable parameters is for all network models of the order of single digit millions. Hence, the effort for correcting the level $L$ Newton-Krylov solution by the neural network on level $L+1$ can be estimated by $10^{7}N^{L}$, which is almost a factor of thousand lower than number of FLOPS needed to solve on $L+1$ ($\approx 8\cdot 10^{9}N^{L}$). For the prediction of two levels, the estimate for the computational cost of DNN-MG remains largely unchanged. While the size of the patches and the inputs and outputs of the networks increase, the number of patches and network evaluations don't change. Surprisingly, we don't need to significantly increase the network size and therefore give an upper bound of $2\cdot 10^{7}N^{L}$ trainable parameters. This is even more favorable than the prediction of a single mesh level, as the numerical solution on level $L+2$ incurs a much higher cost of approximately $64\cdot 10^{9}N^{L}$. Computational performance of DNN-MGIn Figure 10 (a) the runtimes of DNN-MG and a purely numerical MG solution are plotted. While for the MG solution with each added mesh level, the wall time increases by a factor of 8, the wall time of DNN-MG increases at most by a factor of 1.5. Note that the wall time of DNN-MG($3+2$) and MG($5$) thereby differ by a factor of 35. Although this has to be weighed against the loss in accuracy, DNN-MG is clearly capable of increasing the computational efficiency. Moreover, DNN-MG($3+2$) consistently outperforms MG($4$) in terms of the accuracy in previous tests. Remarkably, it still has an advantage in terms of the runtime which once more underlines the computational efficiency of DNN-MG. As can be seen in Figure 10 (b), the time spent in the solver in DNN-MG is constant, which shrinks the share of the solver in the wall time while other parts of the program gain weight. The main increase in computation time is due to the increased cost of the assembly of the right hand side and the evaluation of the functionals, since these calculations are done on the higher levels. Note that these operations are also of order $O(N)$, but with a much smaller constant than other operations in the program, e. g. the operations mentioned above in our theoretical considerations. This makes the DNN-MG method highly efficient for $1$ or $2$ level predictions. More levels are not feasible at this point due to limitations in the underlying FEM library. Further investigations in the scalability of the method when predicting more mesh-levels is part of future work. Predicting weights of higher order polynomials instead of the weights of basis functions on refined meshes could further reduce the computational burden and increase the computational efficiency. The evaluation of the DNN takes less than 3% of the wall time, which is less than our previous results [46] in 2D. This is due to substantial improvements of our implementation: collecting the input for the neural network and processing the output is parallelized and the evaluation of the network is done on a GPU. Therefore, DNN-MG not only reduces the cost of accurate simulations, but it is also well-suited for heterogeneous computing environments. ### Generalizability In order to ensure the practicality of the DNN-MG method, it must generalize well to similar flows beyond those seen during training. Previous results on the 3D benchmark demonstrate the network's ability to do so under small geometric perturbations and varying Reynolds number. Here, we evaluate the network's performance under more substantial changes considering a channel with two obstacles shown in Figure 11. As for the single obstacle case, we test the performance under varying Reynolds numbers with a viscosity of $v=5\times 10^{-4}$ and $Re=240$, $v=4\times 10^{-4}$ ($Re=300$) and $v=6.67\times 10^{-4}$ ($Re=300$). We adopt some settings from the single obstacle case. For instance, the boundary conditions, in particular the inflow at the left boundary $\Gamma_{\text{in}}$ given by (18), the initial velocity $\mathbf{v}(0)=\mathbf{0}$ and we chose the time step size $k=0.008$ on the interval $I=(0,8]$. We reuse the network trained for the single-obstacle channel at Reynolds number $Re=200$. The neural network component of DNN-MG operates on level 5 as fine resolution and level 3 as coarse resolution in this section. In order to not overburden this paper we leave out the detailed results of a single level prediction. The complexity of the finite element meshes is given in Table 6. Channel with two obstacles at $\mathbf{Re=180}$In Figure 13 and Table 7, the drag and lift coefficients obtained using DNN-MG at Reynolds number 180 are presented. The results demonstrate the method's ability to accurately predict the dynamics of the flow. In terms of the drag, the improvement by DNN-MG greatly improves the magnitude of the drag and closely follows the trajectory of the reference solution up to the small-scale dynamics. The lift is very similar in that we are able to reconstruct the overall dynamics of the flow, although we do not get the same temporal behavior. We also observe this in Figure 12, through the different flow patterns. In terms of min and max of the lift we are very close to the reference solution. Channel with two obstacles at $\mathbf{Re=240}$Likewise, Figure 14 and Table 8 show the drag and lift forces at Reynolds number 240, further substantiating the accuracy of DNN-MG and its ability to improve the solution. Figure 12 further demonstrates the effectiveness of DNN-MG in accurately capturing the velocity magnitudes for scenarios on which the DNN was not trained. The figure shows the capability of the method to effectively increase the resolution of the flow, to improve the solution and to add features that could not be observed on lower levels, which are present on higher levels. This is a great example of the efficacy of the localized predictions of DNN-MG, which enable this generalization. Analogous to the case with one obstacle, we observe good generalization of DNN-MG to different Reynolds numbers. The great results we found for the drag and lift coefficients and the velocity field at $\mathbf{Re=180}$, translate similarly to $\mathbf{Re=240}$ case. \begin{table} \begin{tabular}{l r r r} \hline \hline & Level & \# DoF & Patch size \\ \hline Coarse & MG(3) & 533 148 & 0 \\ Reference for DNN-MG(3 + 1) & MG(4) & 4 097 340 & 8 \\ Reference for DNN-MG(3 + 2) & MG(5) & 32 115 324 & 64 \\ \hline \hline \end{tabular} \end{table} Table 6: Spatial meshes with increasing number of multigrid levels and unknowns. Figure 11: Geometry of the training and test scenario with a parabolic inflow profile at $\Gamma_{\text{in}}$, do-nothing boundary conditions at the outflow boundary $\Gamma_{\text{out}}$ and no-slip conditions on the walls $\Gamma_{\text{wall}}$. The center of the obstacle is at (0.5, 0.2). Figure 12: Comparison of the magnitude of the velocity field for the channel with two cylindrical obstacles at time $t=7.5$. The first obstacle has a circular cross-section and the second obstacle has an elliptical cross-section, which results in a Reynolds number of $\mathrm{Re}=240$. The results are shown for DNN-MG($3+2$), along with the reference solution obtained with 5 levels (MG(5)), as well as the coarse mesh solution at level 3 (MG(3)). The DNN-MG($3+2$) method utilizes a network trained on simulation data from a round obstacle. The first three images show a cross-section of the domain in the $xy$-plane and the others show a cross-section of the domain in the $xz$-plane. \begin{table} \begin{tabular}{l c c c c c c c c} \hline \hline Type & $\min C_{d}$ & $\max C_{d}$ & $\widehat{C_{d}}$ & amp $C_{d}$ & $\min C_{l}$ & $\max C_{l}$ & $\widehat{C_{l}}$ & amp $C_{l}$ \\ \hline MG(3) & 0.7674 & 1.3213 & 1.2176 & 0.5539 & -0.0087 & 0.0313 & 0.0097 & 0.0400 \\ MG(4) & 0.7905 & 1.5601 & 0.7696 & 1.3747 & -0.0091 & 0.0346 & 0.0061 & 0.0404 \\ MG(5) & 1.5089 & 1.7903 & 1.6636 & 0.2814 & -0.0872 & 0.1428 & 0.0097 & 0.2299 \\ DNN-MG($3+1$)-512 & 1.5179 & 1.7259 & 1.6342 & 0.2079 & -0.0487 & 0.0907 & 0.0280 & 0.1394 \\ DNN-MG($3+2$)-512 & 1.6038 & 1.6457 & 1.6265 & 0.0419 & -0.0432 & -0.0112 & -0.0237 & 0.0320 \\ DNN-MG($3+1$)-750 & 1.5980 & 1.6897 & 1.6436 & 0.0917 & -0.0545 & 0.0784 & 0.0183 & 0.1330 \\ DNN-MG($3+2$)-750 & 1.5591 & 1.7051 & 1.6376 & 0.1460 & -0.0591 & 0.0834 & 0.0096 & 0.1426 \\ \hline \hline \end{tabular} \end{table} Table 7: The min, max, mean and amplitude of drag and lift functionals in the same setting as above in Figure 13. The best results in comparison to the MG(5) solution are highlighted with color coding: 1st best, 2nd best, 3rd best. Figure 13: The drag (left) and lift (right) functionals for the channel with two obstacles at $\mathrm{Re}=180$ are shown. The results are presented for three discretizations: The coarse mesh MG(3), the reference MG(5), and the DNN-MG($3+2$) improved by a neural network. \begin{table} \begin{tabular}{l c c c c c c c c} \hline \hline Type & $\min C_{d}$ & $\max C_{d}$ & $\widehat{C_{d}}$ & amp $C_{d}$ & $\min C_{l}$ & $\max C_{l}$ & $\widehat{C_{l}}$ & amp $C_{l}$ \\ \hline MG(3) & 0.7674 & 1.3213 & 1.2176 & 0.5539 & -0.0087 & 0.0313 & 0.0097 & 0.0400 \\ MG(4) & 0.7905 & 1.5601 & 0.7696 & 1.3747 & -0.0091 & 0.0346 & 0.0061 & 0.0404 \\ MG(5) & 1.5090 & 1.7907 & 1.6642 & 0.2818 & -0.1577 & 0.1693 & 0.0083 & 0.3270 \\ DNN-MG(3 + 1)-512 & 1.3311 & 1.6166 & 1.3971 & 0.2855 & -0.0505 & 0.0960 & 0.0291 & 0.1466 \\ DNN-MG(3 + 2)-512 & 1.4353 & 1.5870 & 1.5126 & 0.1517 & -0.1398 & 0.0790 & -0.0528 & 0.2188 \\ DNN-MG(3 + 1)-750 & 1.4527 & 1.6565 & 1.5547 & 0.2038 & -0.0832 & 0.1507 & 0.0219 & 0.2339 \\ DNN-MG(3 + 2)-750 & 1.4644 & 1.6823 & 1.5929 & 0.2179 & -0.1325 & 0.1684 & 0.0279 & 0.3009 \\ \hline \hline \end{tabular} \end{table} Table 8: The min, max, mean and amplitude of drag and lift functionals in the same setting as above in Figure 14. The best results in comparison to the MG(5) solution are highlighted with color coding: 1st best, 2nd best, 3rd best. Figure 14: The drag (left) and lift (right) functionals for the channel with two obstacles at $\mathrm{Re}=240$ are shown. The results are presented for three discretizations: The coarse mesh MG(3), the reference MG(5), and the DNN-MG(3 + 2) improved by a neural network. \begin{table} \begin{tabular}{l c c c c c c c c} \hline \hline Type & $\min C_{d}$ & $\max C_{d}$ & $\widehat{C_{d}}$ & amp $C_{d}$ & $\min C_{l}$ & $\max C_{l}$ & $\widehat{C_{l}}$ & amp $C_{l}$ \\ \hline MG(3) & 0.7674 & 1.3213 & 1.2176 & 0.5539 & -0.0087 & 0.0313 & 0.0097 & 0.0400 \\ MG(4) & 0.7905 & 1.5601 & 0.7696 & 1.3747 & -0.0091 & 0.0346 & 0.0061 & 0.0404 \\ MG(5) & 1.5092 & 1.7908 & 1.6568 & 0.2816 & -0.0871 & 0.1354 & 0.0122 & 0.2225 \\ DNN-MG(3 + 1)-512 & 1.3049 & 1.5990 & 1.5111 & 0.2941 & -0.0471 & 0.1589 & 0.0317 & 0.2060 \\ DNN-MG(3 + 2)-512 & 1.3021 & 1.5548 & 1.3596 & 0.2526 & -0.1398 & 0.0790 & -0.0528 & 0.2188 \\ DNN-MG(3 + 1)-750 & 1.4050 & 1.6214 & 1.4465 & 0.2164 & -0.0684 & 0.2384 & 0.1214 & 0.3068 \\ DNN-MG(3 + 2)-750 & 1.4387 & 1.6319 & 1.5499 & 0.1932 & -0.0957 & 0.1343 & 0.0067 & 0.2300 \\ \hline \hline \end{tabular} \end{table} Table 9: The min, max, mean and amplitude of drag and lift functionals in the same setting as above in Figure 15. The best results in comparison to the MG(5) solution are highlighted with color coding: 1st best, 2nd best, 3rd best. Figure 15: The drag (left) and lift (right) functionals for the channel with two obstacles at $\mathrm{Re}=300$ are shown. The results are presented for three discretizations: The coarse mesh MG(3), the reference MG(5), and the DNN-MG(3 + 2) improved by a neural network. Figure 16: Relative velocity errors (first row) and pressure errors (second row) for coarse solutions with and without ANN correction (MG(3) and DNN-MG($3+2)-750$) compared to the reference solution on level $L+2$ for different Reynolds numbers. As in the case with a single obstacle, we show the errors of the MG(4) solution as well, which we again outperform. Channel with two obstacles at $\mathbf{Re}=300$For Reynolds number $300$, Figure 15 and Table 9 present the drag and lift forces predicted by DNN-MG. The results demonstrate the method's ability to capture the complex flow characteristics at higher Reynolds numbers. We see that DNN-MG generalizes well to different geometries and Reynolds numbers. Considering the scenarios we tested so far, we determine that DNN-MG indeed offers great potential in terms of computational efficiency. Figure 16 demonstrates the good performance of DNN-MG in terms of velocity and pressure error. The figure shows a great improvement of the accuracy over an MG(3) solution and remarkably even outperforms the MG(4) solution. This further substantiates the efficiency in terms of runtime we demonstrated in Section 6.6. Overall, DNN-MG is robust with respect to changes of the geometry and material parameters and is able to improve solutions to PDEs in terms of characteristic quantities of interest and error measures. ## 7 Conclusion We have presented the deep neural network multigrid solver (DNN-MG) which uses a deep neural network to improve the efficiency of a geometric multigrid solver, e. g. for the simulation of the Navier-Stokes equations. Previously demonstrated for 2D simulations, we extended DNN-MG to 3D and reformulated it in a rigorous manner. Despite the increased complexity of 3D flows, the algorithm remained applicable and delivered even larger speed-ups compared to 2D, while retaining the efficiency, generalizability, and scalability. We established the efficacy of DNN-MG for large-scale simulations in regimes where direct solvers are not feasible anymore. The overhead is small and in 3D we can accelerate high fidelity simulations by a factor of $35$. This comes at a trade-off in terms of quality compared to the high fidelity solution. However, we were able to consistently outperform classical solutions on the intermediate levels, both in terms of solution quality and wall time. This efficiency-performance trade-off establishes DNN-MG as a highly promising approach for accelerating numerical solution methods for PDEs. We ascertained a great generalization capability of DNN-MG due to the local approach. We only trained the network with a single flow scenario at $\mathbf{Re}=200$ and got great results for lower and higher Reynolds numbers. Our results showed that DNN-MG substantially improves the solution accuracy, as measured by the $l^{2}$-errors, and reduces the errors of the drag and lift functionals across all tested scenarios. The experiments with a two obstacle flow at different Reynolds numbers were particularly successful. Especially noteworthy is that DNN-MG is able to predict flow profiles of completely different dynamics. The grid finenesses MG(3) and MG(5) considered by us produce flows with very different character. Thus, especially in the test problem with two obstacles, there are hardly any oscillations on the coarse grid. These are correctly reproduced by DNN-MG. The local approach is thus able to identify global structures of the Navier-Stokes solution and to correct the solution in both local and temporal dynamics. Although DNN-MG generalizes well, there are limits to this approach. In future work we want to develop an online learning approach to retrain the network adaptively based on the uncertainty of the predictions. Further, we want to incorporate physical information, by including the residual of the PDE into the loss function. We further want to investigate the application of DNN-MG to other PDEs. ### Acknowledgement NM acknowledges support by the Helmholtz-Gesellschaft grant number HIDSS-0002 DASHH. Computational resources (HSUper) were provided by the project hpc.bw, funded by dtec.bw -- Digitalization and Technology Research Center of the Bundeswehr. dtec.bw is funded by the European Union - NextGenerationEU. The work of TR was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe.
2310.17729	Improving Traffic Density Forecasting in Intelligent Transportation Systems Using Gated Graph Neural Networks	This study delves into the application of graph neural networks in the realm of traffic forecasting, a crucial facet of intelligent transportation systems. Accurate traffic predictions are vital for functions like trip planning, traffic control, and vehicle routing in such systems. Three prominent GNN architectures Graph Convolutional Networks (Graph Sample and Aggregation) and Gated Graph Neural Networks are explored within the context of traffic prediction. Each architecture's methodology is thoroughly examined, including layer configurations, activation functions,and hyperparameters. The primary goal is to minimize prediction errors, with GGNNs emerging as the most effective choice among the three models. The research outlines outcomes for each architecture, elucidating their predictive performance through root mean squared error and mean absolute error (MAE). Hypothetical results reveal intriguing insights: GCNs display an RMSE of 9.10 and an MAE of 8.00, while GraphSAGE shows improvement with an RMSE of 8.3 and an MAE of 7.5. Gated Graph Neural Networks (GGNNs) exhibit the lowest RMSE at 9.15 and an impressive MAE of 7.1, positioning them as the frontrunner.	Razib Hayat Khan, Jonayet Miah, S M Yasir Arafat, M M Mahbubul Syeed, Duc M Ca	2023-10-26T18:40:28Z	http://arxiv.org/abs/2310.17729v1	Razib Hayat Khan ###### Abstract This study delves into the application of Graph Neural Networks (GNNs) in the realm of traffic forecasting, a crucial facet of intelligent transportation systems. Accurate traffic predictions are vital for functions like trip planning, traffic control, and vehicle routing in such systems. Three prominent GNN architectures--Graph Convolutional Networks (GCNs), GraphSAGE (Graph Sample and Aggregation), and Gated Graph Neural Networks (GGNNs)--are explored within the context of traffic prediction. Each architecture's methodology is thoroughly examined, including layer configurations, activation functions, and hyperparameters. The primary goal is to minimize prediction errors, with GGNNs emerging as the most effective choice among the three models. The research outlines outcomes for each architecture, elucidating their predictive performance through Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE). Hypothetical results reveal intriguing insights: GCNs display an RMSE of 9.10 and an MAE of 8.00, while GraphSAGE shows improvement with an RMSE of 8.3 and an MAE of 7.5. Gated Graph Neural Networks (GGNNs) exhibit the lowest RMSE at 9.15 and an impressive MAE of 7.1, positioning them as the frontrunner. However, the study acknowledges result variability, emphasizing the influence of factors like dataset characteristics, graph structure, feature engineering, and hyperparameter tuning. _Keywords_- Graph Convolutional Networks (GCNs), GraphSAGE (Graph Sample and Aggregation), and Gated Graph Neural Networks (GGNNs) ## I Introduction In the realm of intelligent transportation systems (ITS), accurate traffic forecasting stands as a cornerstone, shaping pivotal tasks like trip planning, traffic control, and vehicle routing. The efficacy of these applications hinges on the prowess of predictive models, and within this context, Graph Neural Networks (GNNs) have emerged as a groundbreaking approach. With an inherent capacity to leverage graph structures intrinsic to traffic systems, GNNs have garnered significant attention as a potent tool for traffic prediction. This study delves into the application of three prominent GNN architectures--Graph Convolutional Networks (GCNs), GraphSAGE (Graph Sample and Aggregation), and Gated Graph Neural Networks (GGNNs)--in the specific context of traffic forecasting [1, 24]. In the dynamic landscape of intelligent transportation systems (ITS), the accurate prediction of traffic patterns holds an irreplaceable significance. These predictions drive critical functions such as efficient trip planning, effective traffic control strategies, and optimized vehicle routing. Amid this backdrop, Graph Neural Networks (GNNs) have emerged as a transformative approach, harnessing the inherent structure of transportation networks to enhance traffic forecasting. This study delves into the realm of traffic prediction, exploring the application of three pivotal GNN architectures: Graph Convolutional Networks (GCNs), GraphSAGE (Graph Sample and Aggregation), and Gated Graph Neural Networks (GGNNs). The core of this investigation lies in the comprehensive examination of each architecture's methodology, encompassing the intricate interplay of layer configurations, activation functions, and hyperparameters. The ultimate objective is to minimize the errors inherent in traffic predictions, guided by the tenets of accuracy and efficacy. Among the trio of GNN models explored, Gated Graph Neural Networks
2308.11112	Development of a Novel Quantum Pre-processing Filter to Improve Image Classification Accuracy of Neural Network Models	This paper proposes a novel quantum pre-processing filter (QPF) to improve the image classification accuracy of neural network (NN) models. A simple four qubit quantum circuit that uses Y rotation gates for encoding and two controlled NOT gates for creating correlation among the qubits is applied as a feature extraction filter prior to passing data into the fully connected NN architecture. By applying the QPF approach, the results show that the image classification accuracy based on the MNIST (handwritten 10 digits) and the EMNIST (handwritten 47 class digits and letters) datasets can be improved, from 92.5% to 95.4% and from 68.9% to 75.9%, respectively. These improvements were obtained without introducing extra model parameters or optimizations in the machine learning process. However, tests performed on the developed QPF approach against a relatively complex GTSRB dataset with 43 distinct class real-life traffic sign images showed a degradation in the classification accuracy. Considering this result, further research into the understanding and the design of a more suitable quantum circuit approach for image classification neural networks could be explored utilizing the baseline method proposed in this paper.	Farina Riaz, Shahab Abdulla, Hajime Suzuki, Srinjoy Ganguly, Ravinesh C. Deo, Susan Hopkins	2023-08-22T01:27:04Z	http://arxiv.org/abs/2308.11112v1	Development of a Novel Quantum Pre-processing Filter to Improve Image Classification Accuracy of Neural Network Models ###### Abstract This paper proposes a novel quantum pre-processing filter (QPF) to improve the image classification accuracy of neural network (NN) models. A simple four qubit quantum circuit that uses Y rotation gates for encoding and two controlled NOT gates for creating correlation among the qubits is applied as a feature extraction filter prior to passing data into the fully connected NN architecture. By applying the QPF approach, the results show that the image classification accuracy based on the MNIST (handwritten 10 digits) and the EMNIST (handwritten 47 class digits and letters) datasets can be improved, from 92.5% to 95.4% and from 68.9% to 75.9%, respectively. These improvements were obtained without introducing extra model parameters or optimizations in the machine learning process. However, tests performed on the developed QPF approach against a relatively complex GTSRB dataset with 43 distinct class real-life traffic sign images showed a degradation in the classification accuracy. Considering this result, further research into the understanding and the design of a more suitable quantum circuit approach for image classification neural networks could be explored utilizing the baseline method proposed in this paper. Farina Riaz, Shahab Abdulla, Hajime Suzuki, Srinjoy Ganguly, Ravinesh, C. Deo, and Susan Hopkins ## 1 Introduction The application of quantum computing to the tasks of machine learning, herein referred as quantum machine learning, has attracted much research attention in recent years. Literature surveys on quantum machine learning can be found in [1, 10, 4]. Among many proposals to combine classical machine learning methods with quantum computing, quanvolutional neural network (QNN) proposed by Henderson et. al. [6] has an advantage of being implementable on quantum circuits with a smaller number of qubits with shallow gate depths and yet being applicable to practical applications. QNN utilizes quantum circuits as transformation layers, called quanvolutional layer, to extract features for the purpose of image classification using convolutional neural networks (CNNs). In [6], MNIST handwritten 10 digit dataset [8] was applied to QNN using 9 qubits. The results showed classification accuracy improvement using QNN over CNN. However, when the quanvolutional layer of QNN was replaced by a conventional convolutional layer, no improvement in classification accuracy was observed. Henderson et. al. later updated QNN and implemented on Rigetti Computing's Aspen-7-25Q-B quantum processing unit which has 25 qubits with 24 programmable two-qubit gate interactions [5]. The proposed method was applied to 4 class low resolution satellite image dataset. However, no improvement in classification accuracy by QNN over CNN was observed in [5]. An implementation of QNN on a software quantum computing simulator, PennyLane [2], was provided by Mari [9]. Mari's implementation of QNN differs from that of Henderson in that the output of the quantum circuit, which is a set of expectation values, is directly fed into the following neural network layer, while that of Henderson was made into a single scalar value by a classical method. The proposed method was applied to MNIST dataset using 50 training and 30 test image set. No clear improvement in classification accuracy by QNN over NN was observed in [9]. Inspired by Henderson's and Mari's QNNs, we propose a new method, herein called quantum pre-processing filter (QPF), that shows clear improvements in image classification accuracy over conventional neural networks. QPF uses a quantum circuit with four qubits, four Y rotations, two controlled NOTs (CNOTs), and four measurements. When QPF is applied as a pre-processing unit of an image classification neural network, i.e. as a feature extraction filter, the image classification accuracy of fully connected neural network against MNIST and EMNIST (handwritten 47 class digits and letters, [3]) improves from 92.5% to 95.4% and from 68.9% to 75.9%, respectively. These improvements were obtained without introducing any extra parameters to optimize in machine learning process. Unlike other quantum machine learning methods, the use of QPF does not require optimization of the parameters inside the quantum circuits and hence requires only a limited use of the quantum circuit. Given the small number of qubits and relatively shallow depth of the quantum circuit, QPF is well suited to be implemented on noisy intermediate-scale quantum computers. While the proposed method is promising, a test against a more complicated dataset, GTSRB (43 class real-life traffic sign images, [11]), showed degradation in classification accuracy by the application of QPF. This prompts further research into the understanding and design of suitable quantum circuits for image classification neural networks. To support the validation of our claims and further research, we have made our source code available at [https://github.com/hajimesuzuki999/qpf](https://github.com/hajimesuzuki999/qpf). This paper is organized as follows: The new QPF unit combined with the classical image classification neural network is proposed in Section 2. Section 3 describes the experiment conducted using software simulation. The results and discussions are presented in Section 4, followed by conclusions in Section 5. Methodology Figure 1 shows the architecture of the proposed model. The method assumes that the input image is a two-dimensional matrix with size $m$-by-$m$ and the pixel value, $x$, follows $0\leq x\leq 1$. An extension to multi-channel pixel image is considered as straightforward. Similar to QNN models, a section of size $n$-by-$n$ is extracted from the input image. While $1<n\leq m$ in the case of QNN, the proposed QPF uses $n=2$. This $2\times 2$ section of the input image is referred as QPF window. An extension of QPF using $n>2$ is left for further studies. Given $n=2$, we use 4 qubit quantum circuit as shown in Figure 1. The four qubits are initialized in the ground state. The four pixel values are then encoded using Y rotation with $\theta=\pi x$ according to (1). \[R_{y}(\theta)=\left[\begin{array}{cc}\cos\frac{\theta}{2}&-i\sin\frac{ \theta}{2}\\ \sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right] \tag{1}\] The outputs from the Y rotation gates are fed to the quantum circuit referred as $U$ in Figure 1. Measurements, referred as $M$ in Figure 1, are performed on the output of the quantum circuit $U$. Three different quantum circuits are examined in this paper. The first circuit, referred as Encoding only and shown in Figure 2, performs measurement straight after the encoding. The second circuit, referred as One CNOT and shown in Figure 3, performs controlled NOT operation with the first qubit as the control and the 4th qubit as the target, as shown in Figure 3. The third circuit, referred as Two CNOTs, performs two controlled NOT operations as shown in Figure 4. The outputs from the measurement operations are given as expectation values between $-1$ and $1$, and form output features. We note that the total number of parameters in the input image ( $m\times m$ ) is the same as the total number of parameters in the output features ( $4\times(m/2)\times(m/2)$ ). The output features are made into a one dimensional vector by the flatten layer. The number of the nodes of the output of the flatten layer is $m\times m$. The nodes are fully connected Figure 1: The architecture of the proposed quantum pre-processing filter (QPF) model. Figure 3: Quantum pre-processing filter, with one CNOT. Figure 2: Quantum pre-processing filter, with encoding only. by the first fully connected layer 1. The output of the fully connected layer 2 has the number of nodes equal to the number of classes. ## 3 Experiment As was performed by many, we first apply the proposed method to the MNIST dataset [8] to obtain benchmark results. The MNIST dataset consists of 60,000 training and 10,000 test images of handwritten digits of 0 to 9. The size of each image is 28 by 28 pixels. The original images are in greyscale within the values between 0 and 255, which are scaled by dividing them by 255. We then chose the EMNIST dataset [3] to extend the number of image class. The EMNIST dataset (Balanced and By_Merge [3]) contains 112,800 training and 18,800 test images of handwritten digits and letters making up 47 classes. Note that some of upper- and lower-case letters are merged due to their similarity (e.g. C is similar to c) in this dataset. Original EMNIST dataset is divided by 255 to create a dataset with pixel values between 0 and 1. The GTSRB dataset [11] consists of 34,799 training and 12,630 test images of 43 class traffic signs captured from actual traffic signs in-use in various conditions. The original dataset has various image sizes between $15\times 15$ and $222\times 193$ pixels. Those images were scaled to a size of $32\times 32$ pixels. The original images were in RGB color, which were converted into grayscale between 0 and 255. Unlike MNIST and EMNIST dataset, in order to normalize the dynamic range of each image, the normalization is applied to each image according to the following formula: \[\tilde{c}_{x,y}=\frac{c_{x,y}-\min(I)}{\max(I)-\min(I)} \tag{2}\] Figure 4: Quantum pre-processing filter, with two CNOTs. position ($x$,$y$), and $\max(I)$ and $\min(I)$ denote the maximum and the minimum element in the two dimensional image matrix $I$, respectively. The experiment was performed mainly in MATLAB. PennyLane was used to build quantum circuits which were exported to MATLAB as unitary matrices. The Adam optimizer [7] was used as the solver for the training network and default mini-batch size of 128 was used for all of the three datasets to fully test the proposed QPF model. Figure 5 shows example input images from MNIST, EMNIST, and GTSRB, and corresponding output features using encoding only (labelled as $q[0]$ to $q[3]$), and two CNOTs (CNOT( $q[0],q[3]$ ) and CNOT( $q[1],q[2]$ )). ## 4 Results and Discussion Figure 6 shows the variation of the testing dataset accuracy as a function of training iterations using MNIST dataset. As can be seen in Figure 6, the test set accuracy using QPF Encoding only converges faster than that of NN. The phenomenon of faster convergence was also observed in Henderson's QNN [6]. However, the converged test set accuracy of the QPF Encoding only model does not improve that of NN. The application of QPF One CNOT improves the test set accuracy from 92.5% to 93.7%. The test set accuracy is further improved to 95.4% by the application of QPF Two CNOTs. Similarly, small but clearly faster convergence is observed by the application of QPF Encoding only in the case of EMNIST dataset as shown in Figure 7. However, in the case of EMNIST, the converged test set accuracy of the QPF Encoding only is reduced from that of NN. Nonetheless, the test set accuracy is improved from 68.9% to 72.0% by the application of QPF One CNOT, and to 75.9% by the application of QPF Two CNOTs. Figure 5: An example of the input image and the relevant output features. Figure 6: Test set accuracy using MNIST dataset. Figure 7: Test set accuracy using EMNIST dataset. Figure 8 shows the variation of test set accuracy using the GTSRB dataset. Unlike the cases using MNIST and EMNIST datasets, the converged test set accuracy by the application of QPF is reduced from that of NN in the case of GTSRB dataset. The exact causes of this phenomenon are currently unknown to the authors and remain for further research. The summary of the testing accuracy results is shown in Table 1. Referring back to Figure 3, there are 12 different ways to create a CNOT circuit from four qubits. In order to investigate if a different CNOT arrangement would make the difference in classification accuracy, the training of the network and the classification of the images were performed for MNIST dataset using the 12 different CNOT arrangements. Figure 9 shows the results. The $x$ axis of Figure 9 denotes the arrangement of the CNOT where the lower number refers to the control qubit and the upper number refers to the target qubit. \begin{table} \begin{tabular}{c\|c c c} & MNIST & EMNIST & GTSRB \\ \hline NN & 92.5\% & 68.9\% & 81.4\% \\ Encoding only & 92.4\% & 68.1\% & 77.6\% \\ One CNOT & 93.7\% & 72.0\% & 77.6\% \\ Two CNOT & 95.4\% & 75.9\% & 77.0\% \\ \hline \end{tabular} \end{table} Table 1: Summary of testing accuracy results. Figure 8: Test set accuracy using GTSRB dataset. For example, "0 3" refers to the arrangement as shown in Figure 3. As can be seen from Figure 9, the variation of test set accuracy for different CNOT arrangement is considered to be small, within less than 0.6%. Similarly, there are 24 different ways to arrange the two CNOTs using four qubits. The test set accuracy was derived for 24 different two CNOTs arrangement using MNIST dataset, and the results are shown in Figure 10. In Figure 10, the $x$ axis label refers to the arrangement of the two CNOTs in the order of the control and target qubits of the first CNOT, then the control and target qubits of the second CNOT. For example, "0 3 1 2" refers to the arrangement as shown in Figure 4. As evident in Figure 10, the CNOT arrangements pairing the 1st and the 4th qubits and pairing the 2nd and the 3rd qubits seems to achieve a higher testing accuracy irrespective to which qubit is assigned as the target or the control. Referring back to Figure 1, the pairing of the 1st and 4th qubits and the pairing of the 2nd and 3rd qubits correspond to the pairing the diagonal elements of the 2 x 2 QPF window. The exact reason for the improved classification accuracy in the case of MNIST and EMNIST dataset when correlating the diagonal elements of the QPF window is currently not known to the authors and is left for further research. Figure 9: Test set accuracy on MNIST dataset using different CNOT arrangements. Figure 10: Test set accuracy on MNIST dataset using different two CNOTs arrangements. Conclusion A novel QPF that can improve the image classification accuracy of NN for MNIST and EMNIST datasets is proposed. While clear improvements are observed, the exact causes of the improvements are currently unknown and remain for future research and investigation. Also, further investigation is needed in order to improve the classification accuracy of NNs against complex images such as those from the GTSRB dataset. Proposed future research directions include increasing the number of qubit sizes from four as well as applying the QPFs to other image classification methods such as convolutional NNs. ## 6 Acknowledgment This research has been supported by Australian government Research Training Program and Commonwealth Scientific Industrial and Research Organization.
2305.10596	Towards Invisible Backdoor Attacks in the Frequency Domain against Deep Neural Networks	Deep neural networks (DNNs) have made tremendous progress in the past ten years and have been applied in various critical applications. However, recent studies have shown that deep neural networks are vulnerable to backdoor attacks. By injecting malicious data into the training set, an adversary can plant the backdoor into the original model. The backdoor can remain hidden indefinitely until activated by a sample with a specific trigger, which is hugely concealed, bringing serious security risks to critical applications. However, one main limitation of current backdoor attacks is that the trigger is often visible to human perception. Therefore, it is crucial to study the stealthiness of backdoor triggers. In this paper, we propose a novel frequency-domain backdooring technique. In particular, our method aims to add a backdoor trigger in the frequency domain of original images via Discrete Fourier Transform, thus hidding the trigger. We evaluate our method on three benchmark datasets: MNIST, CIFAR-10 and Imagenette. Our experiments show that we can simultaneously fool human inspection and DNN models. We further apply two image similarity evaluation metrics to illustrate that our method adds the most subtle perturbation without compromising attack success rate and clean sample accuracy.	Xinrui Liu, Yajie Wang, Yu-an Tan, Kefan Qiu, Yuanzhang Li	2023-05-10T09:09:56Z	http://arxiv.org/abs/2305.10596v1	# Towards Invisible Backdoor Attacks in the Frequency Domain against Deep Neural Networks ###### Abstract Deep neural networks (DNNs) have made tremendous progress in the past ten years and have been applied in various critical applications. However, recent studies have shown that deep neural networks are vulnerable to backdoor attacks. By injecting malicious data into the training set, an adversary can plant the backdoor into the original model. The backdoor can remain hidden indefinitely until activated by a sample with a specific trigger, which is hugely concealed, bringing serious security risks to critical applications. However, one main limitation of current backdoor attacks is that the trigger is often visible to human perception. Therefore, it is crucial to study the stealthiness of backdoor triggers. In this paper, we propose a novel frequency-domain backdooring technique. In particular, our method aims to add a backdoor trigger in the frequency domain of original images via Discrete Fourier Transform, thus hiding the trigger. We evaluate our method on three benchmark datasets: MNIST, CIFAR-10 and Imagenette. Our experiments show that we can simultaneously fool human inspection and DNN models. We further apply two image similarity evaluation metrics to illustrate that our method adds the most subtle perturbation without compromising attack success rate and clean sample accuracy. Keywords:neural networks; backdoor attacks; frequency domain ## Introduction With the advent of artificial intelligence, neural networks have become a widely used method of artificial intelligence. Currently, neural networks have been adopted in a wide range of areas, such as face recognition [1], voice recognition [2], games [3], and autonomous driving [4]. For example, PayPal users are using deep learning-based facial recognition systems to make payments. However, recent studies have shown that deep learning models are vulnerable to various attacks. Attacks against DNN [5] can be divided into three classes: adversarial example, poisoning attack, and backdoor attack. Adding some perturbation to the input data, an adversarial attack [6] can cause misclassification by the DNN without affecting the DNN. However, this attack generates perturbations specific to a single input. Poisoning attack [7] is a method that reduces the accuracy of the model by injecting malicious training data during the training phase. However, this method only reduces the accuracy of the model. Attackers cannot choose specific data they want to cause misclassification. Also, users will not deploy models with low accuracy under normal circumstances, which brings limitations in practice. To overcome these problems, backdoor attack [8] is proposed. The backdoor attack enables attackers to plant a backdoor into the model and performs malicious at Figure 1: Overview of our method. In the figure, DFT and IDFT represents Discrete Fourier Transform and Inverse Discrete Fourier Transform respectively. Note that we shift the zero-frequency component to the center of the spectrum. tacks using a specific backdoor trigger in the inference phase. The backdoored deep neural network can correctly classify benign samples but will misclassify any input with a specific backdoor trigger as an attacker chosen target. The backdoor can remain hidden indefinitely until activated by a sample with a specific backdoor trigger, which is hugely concealed. Therefore, it can bring serious security risks to many critical applications. Although backdoor attacks have been proven to cause neural network misclassifications successfully, one main limitation of current backdoor attacks is that backdoor triggers are usually visible to human perception. When the system administrator manually checks these datasets, the poisoned data will be found suspicious. [9] first discussed the importance of improving the stealthiness of backdoor triggers. They designed a method to blend the backdoor trigger with benign inputs instead of stamping the trigger as proposed in conventional backdoor attack [10][11]. After that, there was a series of researches dedicated to the invisibility in the backdoor attack. However, the backdoor inputs are still noticeable compared to benign samples, making existing backdoor triggers less feasible in practice. Therefore, improving the invisibility of backdoor triggers has become a research hotspot of neural network backdoor attacks. The challenge of creating an invisible backdoor is how to achieve smaller perturbation without affecting the attack success rate and clean sample accuracy. In 2019, [12] exploit the backdoor attack in a robust manner, namely hidden trigger backdoor. Here, the trigger is invisible to evade human inspections. However, we perform several experiments to prove that the perturbations they add are relatively large in contrast to our method. Besides, the adversary utilizes a neural network to optimize the original samples to generate poisoned samples, which raises the attack cost compared to our method. It is well known that humans cannot perceive subtle variations in the color space within images. However, deep neural networks can detect slight perturbation due to their complexity and powerful feature extraction capabilities, making it possible to hide the trigger from manual review. Therefore, in this paper, we exploit this characteristic of DNNs to implement invisible backdoor attacks. Our method is motivated by the DFT-based image blind watermark. In this technique, a sender hides the covert information in the image frequency domain using an encoder. A receiver applies a decoder to extract the hidden message from the frequency domain to achieve secret messaging. According to our investigations, we are the first to propose the frequency-domain backdooring techniques. Figure 1 demonstrates an overview of our method. We add a backdoor trigger in the frequency domain of an original image to generate a poisoned sample which is invisible enough to evade human perception. Our experimental results show that we can simultaneously achieve invisible backdoor attack without affecting attack success rate and clean sample accuracy. Also, we apply two image similarity evaluation metrics ($l_{2}$ paradigm and LPIPS (Learned Perceptual Image Patch Similarity) [13]) to compare our method with the conventional method and a state-of-the-art hidden trigger attack [12]. We found that our method adds the smallest perturbation without compromising attack performance. The contributions of this paper are as follows: $\bullet$ We propose the first class of frequency-domain backdooring techniques in which our method aims to add a backdoor trigger in the frequency domain of original images via Discrete Fourier Transform (DFT), thusidding the trigger. $\bullet$ We implement our DFT-based backdoor attack on MNIST, CIFAR-10, and a subset in Imagenet. Our experimental results show that our approach can simultaneously fool human inspection and DNN models. $\bullet$ We apply two image similarity evaluation metrics ($l_{2}$ paradigm and LPIPS) to compare the invisibility of different methods. We find that our method adds the smallest perturbation without sacrificing attack success rate and clean sample accuracy. The rest of the paper is organized as follows. Section 2 describes the related work. Section 3 explains the proposed scheme. Section 4 demonstrates the experimental setup and evaluates the results. Finally, we conclude the paper in Section 5. ## Related Work Backdoor attack against DNNs Backdoor attacks were first migrated to neural networks in 2017. [10] proposed BadNets. In this method, the attacker can attach a specific trigger to the stop sign image and mark it as the speed limit sign to generate a backdoor in the road sign recognition model. Although the model can correctly classify clean samples, it will misclassify the stop sign image with the trigger as the speed limit. In 2018, [11] proposed a more advanced backdoor attack technique called Trojan attack. In the study of the Trojan attack, it was found that the backdoor attack method in the neural network was effective because the backdoor trigger would activate specific neurons in the network. Therefore, the Trojan attack generates a trigger in a way that maximizes the activations of specified neurons. Based on classical backdoor attacks, many works focused on improving the invisibility of backdoor images. [9] first discuss the importance of invisibility in backdoor attacks. They proposed that a backdoored image should be indistinguishable from its benign version to evade human inspection. To satisfy such a requirement, they generated poisoned images by blending the backdoor trigger with benign inputs rather than stamp the trigger as proposed in conventional backdoor attacks. After that, there was a series of researches dedicated to the invisibility in backdoor attacks. [14] proposed to utilize a backdoor trigger amplitude to perturb the clean images instead of replacing the corresponding pixels with the chosen pattern. Interestingly, [12] exploit the backdoor attack in a robust manner, namely, hidden trigger backdoor. In this method, the trigger used in the poisoning phase is invisible to evade human inspections. However, we perform several experiments to prove that the perturbations they add are relatively large in contrast to our method, making it easily detected by programs. Besides, the attacker utilizes a neural network to optimize the original samples to add perturbations, which raises the attack cost to generate poisoned samples compared to our method. In order to evaluate the invisibility of our method, we investigate a series of methods used to calculate image similarity, such as phash, $l_{2}$ paradigm, $l_{\infty}$ paradigm, and so on. Among them, LPIPS [13] is used to measure the similarity between two images in a manner that simulates human judgment. LPIPS is proposed based on perceptual loss. It uses features of the VGG network trained on ImageNet classification to mimic human visual perception. In this paper, we will use LPIPS as an invisibility evaluation metric. ### Blind Watermark Blind watermark is an algorithm in steganography [15] which is the study of concealing information in plain sight, such that only the intended recipient would get to see it. Steganography encodes hidden messages onto conventional multimedia data, which may be an image, text, and video. One widely used algorithm in steganography is the Least Significant Bit (LSB) substitution. The idea behind LSB is that replacing bit 0 (i.e., the lowest bit) in a binary pixel value will not cause a visible change in the color space. Though this spatial-domain technique has the least complexity and high payload, it cannot withstand image compression and other typical image processing attacks, which bring poor robustness. The frequency-domain blind watermark based on the Discrete Fourier Transform (DFT) [16] typically provides imperceptibility and is much more robust to image manipulations. The DFT-based blind watermark's main idea is to add a watermark image in the original image's frequency domain using DFT and transform the frequency-domain image back to spatial-domain using Inverse Discrete Fourier Transform (IDFT). Note that the frequency-domain image demonstrates the intensity of image transformation. ## Methodology ### Threat model We assume a user who wants to use a training dataset $D_{train}$ to train the parameters of a DNN. The user sends the internal structure of the DNN $M$ to the trainer. Finally, the trainer will return to the user the trained model parameters $\Theta^{{}^{\prime}}$. However, the user cannot fully trust the trainer. The user needs to check the accuracy of the trained model on the validation dataset $D_{valid}$. Only when the model's accuracy meets an expected accuracy rate $r^{}$ will the user accept the model. Attacker's Goals:* The attacker expects to return to Figure 2: Two different DFT-based methods of generating poisoned samples for RGB images. The generation of Trigger B applies a RGB-to-Gray transformation, thus further improving the invisibility. the user a maliciously trained backdoor model parameters $\Theta^{{}^{\prime}}:=\Theta^{adv}$. The parameters of this model are different from those of the honestly trained model. A backdoored model needs to meet two goals: Firstly, the classification accuracy of the backdoored model $M_{\Theta^{adv}}$ cannot be reduced on the validation set $D_{valid}$, in other words, that is, $C(M_{\Theta^{adv}},D_{valid})\geq r^{}$. Note that the attacker cannot directly access the user's validation dataset. Secondly, for the input containing the backdoor trigger specified by the attacker, $M_{\Theta^{adv}}$ outputs' predictions are different from the outputs of the honestly trained model. #### 3.2.3 Generate poisoned images with DFT In conventional backdoor trigger design approaches, the backdoor trigger is usually a distinct sign within an area, making backdoor data easily recognizable in the event of a human visual inspection. Our approach is inspired by the DFT-based image blind watermark [17] in image steganography [15]. Similarly, we add a trigger to an image's frequency domain so that the perturbation spreads throughout the image instead of being confined to a fixed region, thus making the trigger more invisible. \[\begin{split} F(u,v)&=DFT(f(p,q))\\ &=\sum_{p=0}^{H-1}\sum_{q=0}^{W-1}f(p,q)e^{-i2\pi(\frac{vp}{H}+ \frac{vq}{W})}\end{split} \tag{1}\] \[\begin{split} f(p,q)&=IDFT(F(u,v))\\ &=\frac{1}{HW}\sum_{u=0}^{H-1}\sum_{v=0}^{W-1}F(u,v)e^{i2\pi( \frac{vp}{H}+\frac{vq}{W})}\end{split} \tag{2}\] We assume that we have a grayscale image that can be viewed as an $HW$ matrix ($H$, $W$ denote the height and width of the image, respectively). We can regard this image as a signal $f(p,q)$ (denotes the pixel value of the spatial domain image at the coordinate point $(p,q)$). In digital image processing, we usually utilize Discrete Fourier Transform (DFT) to convert an image from spatial domain to frequency domain. Besides, we apply $F(u,v)$ to denote the pixel value of an image in frequency domain at the coordinate point $(u,v)$. The following Equation 1 represents Discrete Fourier Transform, and Equation 2 represents the Inverse Discrete Fourier Transform (IDFT), which transforms an image from frequency domain to spatial domain. Note that $i$ denotes a unit of the complex number. As shown in Algorithm 1: line 4 to line 8, we define a trigger $F_{trigger}$ in frequency domain and the original image in spatial domain is represented as $f_{original}$. We first convert the original image $f_{original}$ to frequency domain using DFT (Equation 1), the result is represented as $F_{original}$. Then, we add a trigger in the frequency-domain image $F_{original}$ to generate a poisoned image of its frequency form. Here, we define an energy factor $\alpha$ to indicate the strength of the trigger. The smaller the $\alpha$, the lower the visibility of the trigger. Finally, we convert the poisoned image in frequency domain back to spatial domain by performing IDFT (Equation 2). $f_{poisoned}$ is our generated spatial-domain backdoor image. Figure 1 demonstrates the visualization of our algorithm. For RGB images, we design two approaches to add triggers in the frequency domain. One is to add the trigger directly in RGB-level frequency domain of the original image; the shape of the trigger is $HW3$. The added perturbation is shown in Figure 2(a). In the second method, we first convert the RGB image to grayscale and then add a trigger (Note that the trigger shape here is $NM$) in the grayscale frequency domain. Finally, we convert the gray image back to RGB-level, as shown in Figure 2(b). #### 3.2.4 Backdoor injection After generating DFT-based poisoned images, as shown in Algorithm 1: line 9, we replace the labels of the poison samples generated in Section 3.2 with the target label $t$. After that, we can obtain a poisoned dataset $D_{poisoned}$. We apply the poisoned dataset $D_{poisoned}$ with the clean dataset $D_{clean}$ to retrain the model parameters $\Theta^{adv}:=\Theta^{{}^{\prime}}$. In the inference phase, we apply the same frequency-domain trigger and $\alpha$ value used in the training phase to generate poisoned validation samples. After that, we record the Clean Sample Accuracy (CSA) as well as the Attack Success Rate (ASR) to evaluate our attack. We will show our experiment results in the next section. ## 4 Experiments and Analysis #### 4.0.1 Experiment setup In this section, we implement the DFT-based backdoor attack introduced in Section 3. Datasets and models.For the DFT-based backdoor attack, we mount our attack on MNIST [18], CIFAR-10 [19], and Imagenette which is a subset in ImageNet [20]. All datasets are widely used in deep learning. Our experiments were run on a machine with two 2080Ti, and our networks are implemented by Pytorch 1.5 [21]. For MNIST digit recognition task, in order to obtain high classification accuracy, we use AlexNet [22] as our baseline model. For CIFAR-10 and Imagenette, we use pre-trained ResNet-18 [23] as the original model. Note that we use Adam [24] on Alexnet with a learning rate of $1e-3$ and apply SGD [25] optimizer on ResNet-18 with a learning rate of $1e-2$. Evaluation metric.The success of a backdoor attack can be generally evaluated by Clean Sample Accuracy(CSA) and Attack Success Rate(ASR), which can be defined as follows: Clean Sample Accuracy (CSA):* For normal users, the CSA measures the proportion of clean test samples containing no trigger that is correctly predicted to their ground-truth classes. Attack Success Rate (ASR): For an attacker, we represent the output of the backdoored model $M_{\Theta^{adv}}$ on poisoned input data $x^{poisoned}$ as $y^{\prime}=M_{\Theta^{adv}}(x^{poisoned})$ and the attacker's expected target as $t$. This index measures the ratio of $y$ which equals the attacker target $t$. This measurement also shows whether the neural network can identify the trigger pattern added to the input images. #### DFT-based backdoor attack In order to construct the poisoning training dataset with our DFT-based algorithm, we inject the frequency-domain trigger into 10% training data. For the images in which we plant the trigger, we replace their labels with our target label. In MNIST, CIFAR-10, and Imagenette, we select digit 5, "deer", and "building" as our targets respectively. We apply an energy factor $\alpha$ to control the invisibility of the poisoned images. To make the neural network learn the features of our frequency-domain trigger, we retrain the baseline models on the poisoning dataset with a small learning rate. When validating the backdoored model, we hide our trigger on the original validation dataset using the same $\alpha$ value, and then we compute their Clean Sample Accuracy (CSA) and Attack success Rate (ASR) (see Section 4.1). DFT-based method for gray images.First, to demonstrate the feasibility of our attack, we conduct experiments on MNIST. Figure 3 shows the poisoned samples generated on MNIST using different $\alpha$ values, and the first image shows the highlighted trigger pattern generated by our method for grayscale images. Table 1 shows the performance of our attack on MNIST using different $\alpha$ values. During the process of our experiments, we find that the smaller the $\alpha$ value, the slower the model converges and the more epochs are needed for training, which indicates that it is more difficult for our model to capture such slight perturbations. However, both ASR and CSA end up close to 100%. Additionally, the $l_{2}$ value of the perturbation at $\alpha=0.1$ reaches only 0.0122 without affecting the performance, which means our model can detect the subtle change in image's frequency domain, thus making the trigger invisible. DFT-based method for RGB images.To evaluate the performance of the attack on the trigger strength of poisoned samples on CIFAR-10 and Imagenette, we carried out extensive experiments which are summarized in Figure 5. According to our two methods of crafting poisoned samples for RGB images proposed in Figure 2, we set different $\alpha$ values on CIFAR-10 and Imagenette to perform several backdoor attacks and test the attack success rate(ASR) as well as clean \begin{table} \begin{tabular}{l c c c} \hline \hline & $\alpha=0.1$ & $\alpha=0.5$ & $\alpha=1$ \\ \hline Epoch & 48 & 10 & 3 \\ CSA & 98.53\% & 98.31\% & 98.89\% \\ ASR & 98.48\% & 99.99\% & 99.99\% \\ $l_{2}$ & 0.0122 & 0.0610 & 0.1219 \\ \hline \hline \end{tabular} \end{table} Table 1: DFT-based backdoor attack performance and $l_{2}$ values for different $\alpha$ values on MNIST. Figure 3: The figure shows the spatial trigger, original image and the poisoned samples generated by DFT-based method with $\alpha=0.1$, $\alpha=0.5$ and $\alpha=1$ respectively. sample accuracy(CSA). Figure 4 shows the generated poisoned samples using different alpha values of two triggers. From four subfigures in figure 5, we can see that the ASR generally increases by boosting $\alpha$. Besides, in figure 5(a)(b), even when $\alpha=0.3$, the poisoned samples can be misclassified as our target with accuracy larger than $90.0\%$. The effectiveness of conventional backdoor attack can be further enhanced by considering our method with $\alpha=0.3$ on CIFAR-10, which still guarantee the attack concealment without compromising the ASR and CSA. As for Imagenette, the best tradeoff point is $\alpha=0.5$ for Trigger A and $\alpha=1$ for Trigger B. Besides, we perform experiments to compare the two DFT-based methods for RGB images on Imagenette, which are summarized in Table 2. In the table, the "Best $\alpha$" indice indicates the lowest $\alpha$ values for Trigger A and Trigger B, respectively, while ensuring the ASR and CSA. Additionally, we apply $l_{2}$ paradigm and LPIPS to evaluate the invisibly of the two methods. From the table, we find that in contrast to Trigger A, Trigger B has better invisibility without sacrificing ASR and CSA. Figure 7 illustrates the accuracy of the backdoored model on the clean images (CSA) and the validation poisoning dataset (ASR). From the figure, it is clear that we can stealthily achieve our DFT-based backdoor attack on MNIST ($\alpha=0.1$), CIFAR-10 ($\alpha=0.3$) and Imagenette ($\alpha=1$) while hiding the trigger from human perception, indicating that our backdoored model can accurately identify the subtle changes in the image's frequency domain and simultaneously achieve the misclassification of the network. ### Comparison with classical attack We also conducted several experiments to compare our two methods with classical backdoor attack. Figure 6 shows different backdoored samples and their corresponding triggers. To prove the stealthiness of our method, we compute $l_{2}$ values and LPIPS indices of the four types of triggers used in classical backdoor [11] and two DFT-based method for RGB images proposed in figure 2. For our two methods, we select $\alpha$ values used in table 2. $l_{2}$ value is used to calculate the euclidean distance between the backdoored image and the original image, so a lower value indicates the images are more similar. Recall that the LPIPS score measures the perceptual distance between the reference image and the blurred image. The range of LPIPS score is $[0,1)$. If two images are identical, the value is 0. A lower LPIPS value means two images are more similar; a higher score means the images are more different. A comparison of the $l_{2}$ paradigm value and LPIPS score for each attack is illustrated in Table 3. Our method achieves lower $l_{2}$ value and LPIPS (near 0). This demonstrates that it is more difficult for humans to distinguish our poisoned images from original images. ### Availability of data and materials The dataset analysed during the current study was taken from [https://github.com/VinAIResearch](https://github.com/VinAIResearch). ### Funding Acknowledgements ## Conclusion In this paper, we propose a novel method to add the backdoor trigger in the frequency domain of original images to generate poisoned samples. The poisoned data looks similar to the original images and does not reveal the trigger. Therefore, it is invisible enough to evade the event of a human visual inspection. Experiments on three different datasets demonstrate that our \begin{table} \begin{tabular}{c c c c c c} \hline \hline Trigger & Best $\alpha$ & ASR & CSA & $l_{2}$ & LPIPS \\ \hline A & 0.5 & 90.14\% & 95.26\% & 1.057 & 1.9e-3 \\ B & 1 & 89.11\% & 95.06\% & 0.914 & 7.8e-4 \\ \hline \hline \end{tabular} \end{table} Table 2: Comparison between Trigger A and Trigger B proposed in figure 2. Figure 4: The figure shows trigger, original image and the poisoned samples generated by DFT-based method using different $\alpha$ values on CIFAR-10 (row $1,2$) and Imagenette (row $3,4$). Row $1,3$ and row $2,4$ respectively demonstrates two different triggers proposed in figure 2. method implements invisible backdoor attacks without compromising the ASR and CSA. Additionally, we use two image similarity evaluation metrics to compare our method with a conventional backdoor attack and a state-of-the-art hidden trigger backdoor attack. We find that our approach adds the smallest perturbation. We believe such invisible backdoor attacks reveal the vulnerabilities of deep neural networks that need to be deployed in critical real-world applications. ## Author details School of Cyberspace Science and Technology, Beijing Institute of Technology, Beijing, China.
2302.02860	Solving Maxwell's Equation in 2D with Neural Networks with Local Converging Inputs	In this paper we apply neural networks with local converging inputs (NNLCI), originally introduced in [arXiv:2109.09316], to solve the two dimensional Maxwell's equation around perfect electric conductors (PECs). The input to the networks consist of local patches of low cost numerical solutions to the equation computed on two coarse grids, and the output is a more accurate solution at the center of the local patch. We apply the recently developed second order finite difference method [arXiv:2209.00740] to generate the input and training data which captures the scattering of electromagnetic waves off of a PEC at a given terminal time. The advantage of NNLCI is that once trained it offers an efficient alternative to costly high-resolution conventional numerical methods; our numerical experiments indicate the computational complexity saving by a factor of $8^3$ in terms of the number of spatial-temporal grid points. In contrast with existing research work on applying neural networks to directly solve PDEs, our method takes advantage of the local domain of dependence of the Maxwell's equation in the input solution patches, and is therefore simpler, yet still robust. We demonstrate that we can train our neural network on some PECs to predict accurate solutions to different PECs with quite different geometries from any of the training examples.	Harris Cobb, Hwi Lee, Yingjie Liu	2023-02-06T15:26:41Z	http://arxiv.org/abs/2302.02860v1	# Solving Maxwell's Equation in 2D with NNLCI ###### Abstract In this paper we apply neural networks with local converging inputs (NNLCI), originally developed in [11], to solve the two dimensional Maxwell's equation around perfect electric conductors (PECs). The input to the networks consist of local patches of low cost numerical solutions to the equation computed on two coarse grids, and the output is a more accurate solution at the center of the local patch. We apply the recently developed second order finite difference method [16] to generate the input and training data which captures the scattering of electromagnetic waves off of a PEC at a given terminal time. The advantage of NNLCI is that once trained it offers an efficient alternative to costly high-resolution conventional numerical methods; our numerical experiments indicate the computational complexity saving by a factor of $8^{3}$ in terms of the number of spatial-temporal grid points. In contrast with existing research work on applying neural networks to directly solve PDEs, our method takes advantage of the local domain of dependence of the Maxwell's equation in the input solution patches, and is therefore simpler, yet still robust. We demonstrate that we can train our neural network on some PECs to predict accurate solutions to different PECs with quite different geometries from any of the training examples. ## 1 Introduction In recent years, applications of artificial neural networks have permeated the broad realm of computational science thanks to the advances in computing powers such as Graphical Processing Units (GPUs) as well as the expansion of data availability. Some specific examples encompass biological applications [24], image processing [15], uncertainty quantification [35] among a plethora of existing work. The utility of the neural networks (NNs) can be attributed to the classical result on its universal approximation property [5, 10], of which our current work is yet another demonstration in the context of numerically solving partial differential equations (PDEs). More specifically we are interested in the so-called forward problem of computing numerical solutions as opposed to the backward problems of identifying the underlying PDEs, although the latter has received as much attention in scientific computing literature [20, 14, 9]. An approach to using NNs to solve PDEs is to approximate the mapping from all known information (such as the initial value, boundary values etc) to the unknown solution by a deep neural network and use many existing solutions for training, see e.g [17]. Another approach is to approximate the mapping from a space-time location to the solution there with a deep neural network and define the loss function as certain form of the residual error of a PDE, plus penalty terms enforcing the physical data such as initial and boundary conditions. The deep Galerkin method [34] introduces a loss function which contains the residue error of a finite difference approximation to the PDE. The deep Ritz method [7] introduces a loss function which contains the Ritz energy of the finite element approximation of the PDE. The physics informed neural network (PINN) [31] exploits the exact auto-differentiation [1] in computation of the loss function which is designed to enforce consistency with the governing PDE. These methods harness the approximation property of NNs as an ansatz for the solutions to the PDEs. Besides the solution learning, researchers have also investigated the possibilities of training NNs to learn the solution operators as a function of the parameters of the PDE [39]. The domains for successful application of PINN are far-reaching, including but not limited to fluid flows, heat transfer and optics [19, 2, 3]. The ongoing research efforts have resulted in some modifications of PINN itself as in [27], incorporating information from partially known solutions in low and high fidelities in the loss function [21] and the emergence of related approaches like physics-informed convolutional neural networks used in [38]. We would like to refer to [4] and the references therein for an overview of PINN and more classical NN-based approaches. We also point out some alternatives such as those based on Gaussian processes [28], and NN augmented time-stepping strategies [33]. The accuracy enhancement by NNs has been illustrated in existing literature [29], and the growing number of studies is geared towards complexity-reduction hybrid methods combining NNs and conventional numerical methods [18, 23]. In this work we pursue a different NN-based approach, adopting the architecture of recently developed neural networks with local converging inputs (NNLCIs) [11, 12]. The NNLCIs have been successfully applied to nonlinear Euler equations for gas dynamics in one and two spatial dimensions as a black-box tool to generate high fidelity solutions containing complicated interactions of shocks and contacts from two converging, low-cost, low-fidelity numerical solutions. Our work pursues another application of the NNLCIs as a computational saving tool, instead of investigating the high expressivity and intricate inner workings of which have yet to be fully and rigorously understood. The onus is on the NNs to acquire the predictive ability for accuracy improvement when the users work on careful design of the inputs to the network inputs rather than the network architectures such as choices for activation functions. We demonstrate the utility of NNLCIs in the case of solving Maxwell's equations around curved perfect electric conductors with corners. The scattering of electromagnetic waves is of practical importance with industrial applications as in antenna designs [22], for which one may need to do repeated computations with various shapes of PEC objects. It is indeed the high complexity of possible variations in PEC boundary geometries that distinguishes our work from the earlier applications of NNLCIs. We illustrate that it is sufficient for the NNLCIs to be trained locally with respect to some PEC shapes to be able to extract more accurate solutions from the two coarse grid solutions for different shapes of PECs. The parameter space of local geometries of PECS is expected to be more tractable than that of global ones, and after all the analytical, hence numerical, solutions to the Maxwell's equations have local domains of dependence. We fully exploit such locality in approximation of the underlying unknown multivariate mapping, steering away from more conventional approach of taking as a single input all the known data from the entire computational domain. Our proposed NNLCI-based approach benefits largely from the recently developed numerical scheme [12] which is used to compute the inputs to our network. What makes the scheme attractive is its simple and systematic treatment of boundary conditions for general curved PEC shapes that may even have corners. More specifically the scheme is based on automatic construction of ghost values by means of the level set framework [25], the PDE-based extension technique [30, 8] and the so-called guest values. Another merit of the scheme is its second order accuracy which is what the popular Yee scheme [37] can achieve in free space without any PECs. We also point out that our underlying numerical scheme is based on uniform rectangular grids, which are particularly well-suited to NNLCIs since the inputs are finite difference stencils at two different resolutions that need to be aligned at the same spatial and temporal locations. The rest of the paper is organized as follows. In Section 2, we introduce the scattering problem and the numerical scheme we use to produce the training and testing data. We then present our NNLCI focusing on how its inputs are to be formatted. In Section 3, we provide the results of our numerical experiments to illustrate the effectiveness of our approach. We conclude our paper in Section 4. ## 2 Problem Formulation ### The governing PDEs and underlying numerical method The classical Maxwell's equations govern the dynamics of electromagnetic fields through a medium which we assume in this work is isotropic and homogeneous. In particular we consider the scattering of electromagnetic waves off of a PEC in free space, that is, \[\epsilon_{r}\frac{\partial\mathbf{E}}{\partial t} =\nabla\times\mathbf{H}\] \[\mu_{r}\frac{\partial\mathbf{H}}{\partial t} =-\nabla\times\mathbf{E}\] subject to some initial conditions and the PEC boundary conditions that are given by \[\mathbf{E}\times\mathbf{n}=\mathbf{0}\text{ and }\mathbf{H}\cdot\mathbf{n}= \mathbf{0}.\] Here $\epsilon_{r}$ and $\mu_{r}$ denote the relative permittivity and permeability, respectively, and $\mathbf{n}$ denotes the outward unit normal vector to the PEC interface. Without loss of generality we set $\epsilon_{r}=\mu_{r}=1$, and assume for simplicity that the waves are transmagnetic polarized in the $z-$direction, thereby obtaining the following two dimensional system of equation \[\frac{\partial H_{x}}{\partial t} = -\frac{\partial E_{z}}{\partial y}\] \[\frac{\partial H_{y}}{\partial t} = \frac{\partial E_{z}}{\partial x} \tag{1}\] \[\frac{\partial E_{z}}{\partial t} = \frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}.\] In order to numerically simulate the dynamics (1) in free space, we first choose the unit square $[0,1]^{2}$ as our computational domain. We superimpose on the domain uniform rectangular grids $\{(x_{i},y_{j})\}$ with $\Delta x=\Delta y$. We adopt the implementation of perfectly matched layer as done in [32] to enforce the far field absorbing boundary conditions so that the reflected waves off the PEC do not re-enter the computational domain. To this end we follow the total/scatter formulation [13] with the prescription of the incident waves. We apply the finite difference method [16] to compute the numerical solutions to (2), which will serve as training and test data to our NNLCI. The method is mainly composed of three elements, the first of which is the first order scheme \[U_{i,j}^{n+1}=\left(\frac{1}{5}\mathcal{C}+\frac{4}{5}\mathcal{F}\right)U_{i, j}^{n}. \tag{2}\] where $U_{i,j}^{n}$ denotes the numerical scheme at the time level $t=n\Delta t$ and at the spatial location $(x_{i},y_{j})$. Here $\mathcal{C}$ denotes the forward-in time, centered-in-space numerical scheme while $\mathcal{F}$ denotes the classical Lax-Friedrichs scheme. We choose scheme (2) as it has already been used in [16, 40] to simulate the scattering of electromagnetic waves about PECs, but one may instead consider a more general scheme $(\theta\mathcal{C}+(1-\theta)\mathcal{F})$ with $0\leq\theta\leq 1$[36]. The second element is the construction of ghost values inside the PECs via a level-set based PDE extension technique as well as the introduction of guest values. The constructed ghost values are to be locally second order accurate to ensure the first order accuracy of the scheme (2). The last ingredient is the Back and Forth Error Compensation and Correction (BFECC) method [6] which in general increases the order of accuracy by one for odd order schemes [26]. Therefore in our current setting our numerical solutions are computed to second order accuracy. Moreover BFECC provides enhanced numerical stability by allowing a larger CFL number, hence we choose $\Delta t=\Delta x$ which would otherwise not be allowed. An example of the computed numerical solutions is shown in Figure 1 which contains only the total field region. The question of interest to us is whether it is possible for a neural network to produce high fidelity numerical solutions from numerical solutions computed on two coarse grids. The affirmative answer, at least in the cases we demonstrate, is made possible by the NNLCI which we present in the sequel. Figure 1: $H_{x}$ scattering around a cylinder after 0.8 seconds. ### Application of NNLCIs We follow the architecture of the neural network used in [11, 12] which is a simple fully connected feed-forward network based on multiple steps of Adams optimizer and a final step of L-BFGS for each iteration instead of stochastic gradient descent. We modify the hyperparameters of the optimizer used in their work so that the learning rate of L-BFGS is 3e-6 and the learning rate of Adams is 1e-5. Our network consists of 4 layers with 50 neurons in each layer, hence it can be trained readily yet we observe highly predictive capabilities of the network in our numerical experiments. We iterate the numerical scheme introduced above until a fixed terminal time, $T=0.8$ in most of our experiments unless stated otherwise, on three different numerical resolutions, two low-fidelity resolutions and one high-fidelity resolution. In particular, we take $\Delta x=1/40,1/80$ for the two low fidelity solutions and $\Delta x=1/640$ for the high-fidelity solution which we regard as the "exact" reference solution. We store all the numerical solutions only at the terminal time, as opposed to the two different time levels as done in the earlier work [11]. This is because the Maxwell's equations (1) are time reversible, hence the solutions at the previous time step are uniquely determined by the solutions at the current time step, so we omit any notation that denotes the time. From the stored numerical solutions, we prepare the input data to our network as follows. Let us write $(i^{\prime},j^{\prime}),(i^{\prime\prime},j^{\prime\prime})$, and $(i,j)$ to denote the same location in space for the coarsest, finer and the reference grids respectively. We consider a $3\times 3$ window in the coarsest grid centered at $(i^{\prime},j^{\prime})$. In this paper, we restrict to the case where the center, $(i^{\prime},j^{\prime})$, and all its 8 neighboring grid points belong to the total-field region outside the PEC. In the future, we will consider relaxation of this restriction. From the coarsest grid we reshape the $3\times 3$ windows for each of $H_{x},H_{y},E_{z}$ into $1\times 9$ vectors \[\{(H_{x})_{i^{\prime}-1,j^{\prime}-1},(H_{x})_{i^{\prime},j^{\prime}-1},(H_{x })_{i^{\prime}+1,j^{\prime}-1},\ldots,(H_{x})_{i^{\prime}-1,j^{\prime}+1},(H_{ x})_{i^{\prime},j^{\prime}+1},(H_{x})_{i^{\prime}+1,j^{\prime}+1}\}\] \[\{(H_{y})_{i^{\prime}-1,j^{\prime}-1},\ldots,(H_{y})_{i^{\prime}+1,j^{\prime}+ 1}\},\{(E_{z})_{i^{\prime}-1,j^{\prime}-1},\ldots,(E_{z})_{i^{\prime}+1,j^{ \prime}+1}\}\] and concatenate all 27 of the values into a single vector. Similarly we take from the finer grid \[\{(H_{x})_{i^{\prime\prime}-2,j^{\prime\prime}-2},(H_{x})_{i^{\prime\prime},j ^{\prime\prime}-2},(H_{x})_{i^{\prime\prime}+2,j^{\prime\prime}-2},\ldots,(H_{ x})_{i^{\prime\prime}-2,j^{\prime\prime}+2},(H_{x})_{i^{\prime\prime},j^{ \prime\prime}+2},(H_{x})_{i^{\prime\prime}+2,j^{\prime\prime}+2}\}\] \[\{(H_{y})_{i^{\prime\prime}-2,j^{\prime\prime}-2},\ldots,(H_{y})_{i^{\prime \prime}+2,j^{\prime\prime}+2}\},\{(E_{z})_{i^{\prime\prime}-2,j^{\prime\prime}- 2},\ldots,(E_{z})_{i^{\prime\prime}+2,j^{\prime\prime}+2}\}\] and concatenate all the values. As a result the input vector consists of 54 input values \[\{(H_{x})_{i^{\prime}-1,j^{\prime}-1},\ldots,(E_{z})_{i^{\prime\prime}+2,j^{ \prime\prime}+2}\},\] which is then to be fed into our network to produce the output vector \[\{(H_{x})_{i,j},(H_{y})_{i,j},(E_{z})_{i,j}\}.\] We illustrate how the input data are prepared in Fig. 2 where $\mathbf{U}$ represents the vector containing all three components $H_{x},H_{y},E_{z}$. The ordering of the input data is immaterial so far as it is consistent across all $3\times 3$ windows. Following the approach in [12], one may include the time step size $\Delta t$ as part of the input data. However in the present work the simpler input format we choose is sufficient to produce accurate solutions. We train the network until the training error is 1e-5. The training and testing data consist of perturbations of the incident wave given by \[\begin{split} H_{x}(x,y,t)&=0\\ H_{y}(x,y,t)&=-\sin(\omega(x-t)))\chi(t-x)\\ E_{z}(x,y,t)&=\sin(\omega(x-t))\chi(t-x)\end{split} \tag{3}\] where $\omega=2\pi/0.3$ is the angular frequency and $\chi(z)$ denotes the heaviside step function. We also consider different shapes of the PECs to illustrate the robustness of the network. Once the network is trained, we measure its performance by comparing the relative errors (over the total field region) of the predicted numerical solutions and the finer grid input solutions. ## 3 Numerical Experiments ### Different source waves We first demonstrate the performance of our network to predict high-fidelity solutions when the incident waves in (3) are perturbed with respect to their amplitudes or the angular frequencies. Unless stated otherwise, we fix the geometry of the PEC to a circular PEC Figure 2: Procedure for preparing input to 2CGNN. centered at $(0.5,0.5)$ with the radius $0.2$. Moreover both the training and testing input vectors are taken from all $3\times 3$ admissible input windows over the entire computational domain. #### 3.1.1 Varying amplitudes We consider up to $10\%$ perturbations of the amplitude of the incident waves. More specifically we compute the two coarse grid and reference solutions subject to the perturbations of $0,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 7,\pm 8,\pm 10\%$ corresponding to the 17 data sets of amplitudes from $0.9$ to $1.1$. The computed data are then separated into either the training or testing sets. The first experiment consists of training on the even perturbations and testing on the odd perturbations. The second is to train on $0,\pm 10\%$ perturbations, and test on the rest of the perturbations. Lastly we train on $0,\pm 5\%$ perturbations and test on the rest. In all the cases we consider throughout this paper, we decrease to the same network training loss, $2.5e-5$. We compute the relative errors in $l_{2}$ norm of the predicted and finer grid input solutions with respect to the reference solutions. The averaged errors with respect to the number of perturbations in the testing data are shown in Table (1). Since the Maxwell's equations (1) are scaling invariant, one may hence use a small set of training data with fairly large training gaps so far as the perturbed amplitudes are concerned. We note that in the first two experiments (training on the even perturbations, and $0\%,\pm 10\%$) all the testing vectors belong to the convex hull of the training data, hence the corresponding outputs are linearly interpolated from the training data. However the last case is concerned with some testing vectors for which the desired outputs are to be linearly extrapolated, which suggest one of the robust features of our method which was not considered in the earlier work [11]. #### 3.1.2 Varying frequencies We next consider modifying the angular frequencies of the incident waves. It is of interest to demonstrate the performance of the network when the effects of the perturbed frequencies are nonlinear on the corresponding solutions. We generate the training and testing data up to $\pm 10\%$ as in the amplitude case. The results are displayed in Figures 3 and 4. We observe from Table 2 that the accuracy gain is significant when the training gap is small, covering all the even perturbations. However, there is not much improvement in accuracy when the training gap is enlarged to $\pm 10\%$ when the network is expected to quadratically interpolate the training data. On the other hand we do not observe better accuracy improvement when the training gap alone is narrowed to the $\pm 5\%$ perturbations without increasing the number of the training data. To rule out the possibility of limited accuracy gain due to extrapolation, \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \multicolumn{5}{\|c\|}{Circular PEC: modifying amplitudes} \\ \hline Even Perturbations & \multicolumn{2}{c\|}{$0\%,\pm 10\%$} & \multicolumn{2}{c\|}{$0\%,\pm 5\%$} \\ \hline Finer Input Err. & 0.1634 & Finer Input Err. & 0.1634 & Finer Input Err. & 0.1634 \\ Pred. Err. & 0.0031 & Pred. Err. & 0.0029 & Pred. Err. & 0.0042 \\ \hline \end{tabular} \end{table} Table 1: Accuracy of the predicted solutions to a circular PEC with varying amplitudes. we have limited the testing data only to $\pm 7\%$ and $\pm 8\%$ subject to two training cases, the first on $0\%,\pm 10\%$ and the second on $0\%,\pm 5\%$. We observe in Table 5 that the number of training data plays a more critical role than the training gap as it caps the complexity of the NN approximation. Note that in following and subsequent graphs the $H_{x}$ field appears to be non-zero within the boundary of the PEC. This is only due to the limitation of how MatLab generates surface plots: The color of each cell is determined by the value of the top-left corner, so the interior conditions of the PEC are not violated. Also, since we only use as input $3\times 3$ windows which are entirely contained within the total field, we cannot as of yet predict values which are within $\Delta x$ of the PEC. To this end for the purposes of graphical clarity, we take the values around the PEC which cannot be predicted from the high-fidelity solution. However in future work, we will attempt to predict values all the way up to and including the boundary of the PEC which will be owed to the modification of our numerical scheme which will give a higher-order recovering of the ghost values. Figure 4: Cross-sections of the computed $H_{x}$ at different $y$ values. Figure 3: Predicted values of $H_{x}$ scattering off a circle for the $+5\%$ perturbed frequency in the plane at 0.8 seconds after training the network on all the even perturbations. We next change the circular PEC to a square centered at $(0.5,0.5)$ with the side length of $0.4$. We demonstrate that the network can still predict high-fidelity solutions despite the corner singularities of the analytic solutions. We use the same training and testing data of perturbed frequencies as in the circular case. The results are displayed in Figures 6 and 7. We observe from Table 8 that the results are similar to those of the circular case, although the errors for each of the training gaps are more pronounced. #### 3.1.3 Rotations It is well known that the Maxwell's equations (1) are rotation invariant. If the PEC object is also symmetric with respect to rotation as in the circular case, then the solutions are simply rotated when the planar incident waves are rotated. However, we observe poor accuracy improvement when the network is trained on one orientation of the incident waves and tested on another configuration. This can be explained by the fact that the training is entirely local, hence the network is not expected to discern the global rotation invariance of the training and testing data. On the other hand one can easily enrich the training data set with the rotated numerical solutions that do not need to be re-computed. Moreover many practical applications involve general shapes of PECs for which the rotational symmetry of the solutions is absent or unknown a priori. Hence the training of the neural networks is expected to involve incident waves entering the domain at various incident angles, regardless of whether the training itself is local or global. ### Varying PEC boundaries We now consider applications of our neural network when the training and testing data involve the PECs of different shapes. The complexity of the associated geometric parameter space is more complicated than that of the perturbed incident waves. The local training of the network is expected to play a crucial role of rendering model order reduction to variations in local boundary geometries of the PECs. Unless stated otherwise the incident waves are the unperturbed ones prescribed in (3). #### 3.2.1 Modification of the circular PEC We first consider varying the radius of the circular PEC. We generate the data corresponding to the perturbations of the unperturbed radius of $0.2$ with perturbations of $0,\pm 2\%,\pm 3\%,\pm 4\%,$$\pm 5\%,\pm 6\%,\pm 7\%,\pm 8\%,\pm 10\%,$ just as for the previously considered cases. Similar to the case of varying amplitudes, we separate the generated data into either training or testing data. We observe significant accuracy improvement even when the number of training data is quite small. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|} \hline \multicolumn{6}{\|c\|}{Square PEC: modifying frequency} \\ \hline Even Perturbations & \multicolumn{2}{c\|}{$0\%,\pm 10\%$} & \multicolumn{2}{c\|}{$0\%,\pm 5\%$} \\ \hline Finer Input Err. & 0.1500 & Finer Input Err. & 0.1501 & Finer Input Err. & 0.1505 \\ Pred. Err. & 0.0095 & Pred. Err. & 0.1006 & Pred. Err. & 0.0761 \\ \hline \end{tabular} \end{table} Table 3: Accuracy of predicted solutions subject to changing radius Figure 8: Accuracy of the predicted solutions to square PEC with varying frequencies. We next truncate the circular PEC to circular sectors which subtends the angles of various magnitudes (see Figures 9 and 10). At first, we generated the solutions corresponding to the angles from 30 to 360 degrees in 30 degree increments. We found however that our neural network, regardless of our testing and training sets, was unable to produce accurate predictions due to the large training gap. On the other hand, our network manages to improve the accuracy of the low fidelity solutions when the increments in angles are 5 degrees. More specifically we train when the increments in angles are 10 degrees, that is $45^{\circ},55^{\circ},\ldots,125^{\circ},135^{\circ}$, and then test on the even perturbations $50^{\circ},60^{\circ},\ldots,120^{\circ},130^{\circ}$, see Figures 11 and 12. We then consider enlarging the training gap to $15^{\circ}$ increments since in some practical applications it may not be ideal to enforce smaller training gaps. Namely we train on $45^{\circ},60^{\circ},\ldots,120^{\circ},135^{\circ}$ and test on the rest of the perturbations to obtain the similar accuracy improvement as in the case of $5^{\circ}$ increments, see Table 4. #### 3.2.2 Between circular and square PECs We consider the circular PEC as well as the square PEC from subsection 3.1.2. The two PECs are topologically equivalent, yet it is to the best of our knowledge that the corresponding solutions need to be computed separately for each case. To illustrate the effectiveness of our NNLCI-based approach, we generate three different sets of training data for each parameter considered above, namely all the perturbations (up to $\pm 10\%$) in frequencies, amplitudes, and radii of the circular PEC. For each training data, we use the unperturbed case for the square PEC as the testing case. We can see from Table 5 that both the frequencies and the radius cases perform similarly well while the amplitude data performs poorly due to the degeneracy of the training data. The results show that our approach of local training is able to cope \begin{table} \begin{tabular}{\|c\|c\|c\|} \hline \multicolumn{3}{\|c\|}{Sectors from 45 to 135 degrees with 5 degree increments} \\ \hline $10^{\circ}$ Increments in Training & $15^{\circ}$ Increments in Training \\ \hline Finer Input Err. & 0.1987 & Finer Input Err. & 0.1991 \\ Pred. Err. & 0.0344 & Pred. Err. & 0.0459 \\ \hline \end{tabular} \end{table} Table 4: Accuracy of the predicted solutions to PEC sectors of varying angles Figure 11: Predicted values of $H_{x}$ scattering off a $90^{\circ}$ sector in the plane at 0.8 seconds after training the model on $10^{\circ}$ angle increments. Figure 12: Cross-sections of the above case at different $y$ values comparing the predicted values with the input and reference values. with the qualitative differences in the scattering around the corners of the rectangular PEC. Conversely, we train on the square PEC subject to all perturbations in the frequencies and side lengths of the rectangle. Then we test on the unperturbed case for the circular PEC, and we observe limited accuracy improvement (Table 6). One possible explanation is that the training data for the square PEC cover a small range of the angles at which the incident waves are interacting with the PEC, which may fall short of rendering the predictive power for a broader range of the angles for the circular PEC. On a related note, we leave it for future work to explore the possibility of accurately predicting the cases for arbitrary PEC shapes with only a handful of relatively simple training data. #### 3.2.3 Quarter Circle Predicts the Square In this last test case, we aim at predicting the unperturbed case for the square PEC using the unperturbed (in frequency and amplitude) data for the quarter circle PEC. At first, we considered the training data that consisted of the unperturbed solutions to the original quarter circle as well as to the reflected (about $y=0.5$) quarter circle, both subject to the incident waves (1) at the terminal time $T=0.8$. The reflected PEC was included in the training data in order to better capture the corners of the square PEC, yet we observed that our NNLCI was not able to yield accurate predictions. As a remedy we modify the training data as follows: We shift both the original and reflected quarter circle PECs in the direction of negative $x$-axis by $0.2$ so that the incident \begin{table} \begin{tabular}{\|c\|c\|c\|} \hline \multicolumn{3}{\|c\|}{Predictions on Circular PEC from Square PEC} \\ \hline Square Frequency & \multicolumn{2}{\|c\|}{Square Side Length} \\ \hline Finer Input Err. & 0.1634 & Finer Input Err. & 0.1634 \\ Pred. Err. & 0.1442 & Pred. Err. & 0.1274 \\ \hline \end{tabular} \end{table} Table 6: Accuracy of the predicted solutions to circular PEC via training for square PEC Figure 16: Cross-sections of the above case at different $y$ values comparing the predicted values with the input and reference values Figure 15: Different shape prediction of $H_{x}$ scattering off a circle in the plane at 0.8 seconds after training the model on all the even frequency perturbations of a square waves are in the same phases (with respect to each of the three grid sizes) when they first hit the square and the quarter circle PECs. Then we further enrich the training data by including the three additional training data, namely perturbed frequency cases, 0 and $\pm 2\%$, for the circular PEC. We do this because two training examples (here and in general) are insufficient to give accurate predictions. The result thus obtained is shown in Figs. 17 and 18, and Table 7. We remark that the shifted quarter circle PECs may be replaced by the unshifted quarter circles with adjusted terminal times to match the phases of the incident waves for the high-fidelity solution. However, the waves travel at different speeds depending on the grid size due to the dispersive error in our numerical scheme [16] so we must ensure we have training data that covers all three phases. In future work, we wish to carefully examine the propagation of numerical error in our scheme and how that affects the implementation of NNLCI. Conclusion In this work, we have applied NNLCI to predict accurate solutions to the scattering of electromagnetic waves around PECs. Once the network is trained, it takes as its inputs two low cost, yet 2nd order accurate, numerical solutions in a converging sequence to generate higher fidelity solutions to PECS of different shapes. Central to the robustness of our approach is the local training of the network which is still effective despite the presence of the PECs. We are interested in theoretical analysis of the NNLCI which may in turn offer insights into choices of suitable grid sizes for the network inputs, as well as possible modifications of the network architecture for enhanced performance. ## 5 Acknowledgements The authors thank Haoxiang Huang for his help in the coding of NNs.
2304.04147	FedPNN: One-shot Federated Classification via Evolving Clustering Method and Probabilistic Neural Network hybrid	Protecting data privacy is paramount in the fields such as finance, banking, and healthcare. Federated Learning (FL) has attracted widespread attention due to its decentralized, distributed training and the ability to protect the privacy while obtaining a global shared model. However, FL presents challenges such as communication overhead, and limited resource capability. This motivated us to propose a two-stage federated learning approach toward the objective of privacy protection, which is a first-of-its-kind study as follows: (i) During the first stage, the synthetic dataset is generated by employing two different distributions as noise to the vanilla conditional tabular generative adversarial neural network (CTGAN) resulting in modified CTGAN, and (ii) In the second stage, the Federated Probabilistic Neural Network (FedPNN) is developed and employed for building globally shared classification model. We also employed synthetic dataset metrics to check the quality of the generated synthetic dataset. Further, we proposed a meta-clustering algorithm whereby the cluster centers obtained from the clients are clustered at the server for training the global model. Despite PNN being a one-pass learning classifier, its complexity depends on the training data size. Therefore, we employed a modified evolving clustering method (ECM), another one-pass algorithm to cluster the training data thereby increasing the speed further. Moreover, we conducted sensitivity analysis by varying Dthr, a hyperparameter of ECM at the server and client, one at a time. The effectiveness of our approach is validated on four finance and medical datasets.	Polaki Durga Prasad, Yelleti Vivek, Vadlamani Ravi	2023-04-09T03:23:37Z	http://arxiv.org/abs/2304.04147v1	FedPNN: One-shot Federated Classification via Evolving Clustering Method and Probabilistic Neural Network hybrid ###### Abstract Protecting data privacy is paramount in the fields such as finance, banking, and healthcare. Federated Learning (FL) has attracted widespread attention due to its decentralized, distributed training and the ability to protect the privacy while obtaining a global shared model. However, FL presents challenges such as communication overhead, and limited resource capability. This motivated us to propose a two-stage federated learning approach toward the objective of privacy protection, which is a first-of-its-kind study as follows: (i) During the first stage, the synthetic dataset is generated by employing two different distributions as noise to the vanilla conditional tabular generative adversarial neural network (CTGAN) resulting in modified CTGAN, and (ii) In the second stage, the Federated Probabilistic Neural Network (FedPNN) is developed and employed for building globally shared classification model. We also employed synthetic dataset metrics to check the quality of the generated synthetic dataset. Further, we proposed a meta-clustering algorithm whereby the cluster centers obtained from the clients are clustered at the server for training the global model. Despite PNN being a one-pass learning classifier, its complexity depends on the training data size. Therefore, we employed a modified evolving clustering method (ECM), another one-pass algorithm to cluster the training data thereby increasing the speed further. Moreover, we conducted sensitivity analysis by varying Dthr, a hyperparameter of ECM at the server and client, one at a time. The effectiveness of our approach is validated on four finance and medical datasets. Keywords: One-shot FL; PNN; CTGAN; ECM ## 1 Introduction As regards the data privacy and security, the advent of a new paradigm of machine learning (ML), known as Federated Learning [1], facilitates training machine learning (ML) models without transferring sensitive data to the server. It supports distributed training in remote siloed data centers as well, where the private data resides. In the FL paradigm a central server trains a better quality learning model with the coordination of a federation of participating clients. In a typical FL setting, it is often assumed that many clients possess a small number of non-independent and identically distributed (non-i.i.d) data, have limited computational power, and restricted communication network capabilities. In general, FL is broadly categorized into three different types [31], (i) Horizontal federated learning (HFL), (ii) Vertical federated learning (VFL), and (iii) Federated Transfer learning (FTL). HFL refers to sample (or data points) partitioned federated learning. It is applied in a scenario where there is an overlap in the feature space among the clients but differs in the sample space. On the other hand, VFL is applied in a scenario where the sample space is aligned with different clients but differs in the feature space. FTL is applied in a scenario where a small sample space overlaps but differs in the feature space. FTL allows knowledge transfer across different domains; i.e., a trained model from the source domain used to build an efficient model in a specific target domain. FL is a domain-agnostic methodology. It has been popularized and heavily used in financial industries such as, to detect fraudulent activities associated with credit or debit cards [2, 3]. The fraudulent transactions have been proven to be one of the major causes of financial losses [2]. In these cases, the Federated fraud detection system [3] based on artificial neural network outperformed the fraud detection system that trained on data belonging to a single bank. Additionally, FL based fraud detection systems provide privacy. However, FL based techniques involve various challenges such as statistical heterogeneity of client data, communication overhead, system heterogeneity, and privacy concerns. All the notation used in the current study is presented in Table 1. Traditionally, FL settings are maintained as follows: local models usually train on the local data respective to each client, {$\mathcal{D}_{\text{1}}$,... $\mathcal{D}_{\text{k}}$} (refer \begin{table} \begin{tabular}{\|c\|l\|l\|l\|} \hline Notation & Denotes & Notation & Denotes \\ \hline K & Number of clients & $a$ & Real value of input data \\ \hline $\mathcal{D}_{\text{k}}$ & Private dataset at client & Xtrain & Train dataset \\ \hline $\mathbb{P}$ & Probability distribution & Xtest & Test dataset \\ \hline $\sigma$ & Smoothing parameter & $\mathcal{N}$(m, n) & Gaussian distribution with \\ & & & mean m and variance n \\ \hline N & Number of samples & $\eta$ & Learning rate \\ \hline $Dthr$ & Threshold cluster radius & $p_{k}$ & Representing probability \\ & & & distribution of client ‘k’ \\ \hline $f_{A}(X)$ & Activation function & $d(i,j)$ & Normalized euclidean \\ & & & distance between point i \\ & & & and j \\ \hline PDF & Probability density function & $s(i,j)$ & Updated normalized \\ & & & euclidean distance \\ \hline $X_{A_{i}}$ & Cluster center in the neuron of pattern & $Ru_{j}$ & j${}^{\text{th}}$cluster radius \\ & layer of PNN & & \\ \hline d & Number of independent features & CC & Set of Cluster centers \\ \hline $\rho_{e}$ & Probability densities of continuous & $\mu_{e},\Phi_{e}$ & Weight and Standard \\ & column & & deviation of mode ‘c’ \\ \hline $c_{i,j}$ & Continuous column value & $\tau_{j}$ & The row of a dataset \\ \hline b & Server side & & \\ \hline \end{tabular} \end{table} Table 1: Notation used in the current study to Table 1), generated by $k$ distinct nodes. Further, data at each node $t\in k$ is non-i.i.d across the network and is being generated by a distinct distribution $X_{t}\sim\mathbb{P}_{t}$ (statistical heterogeneity). Additionally, the number of sample points at each node may significantly vary. These kind of FL settings requires a huge number of communication rounds between clients and the server that suffers the bottleneck in the form of communication overhead. Further, Zhao et al., [34] demonstrated that the invocation of the FedAvg algorithm decreases the quality of FL models. This can result in diverging the global model weights, which leads to poor generalization. There are some additional challenges in the form of network heterogeneity, which implies that some clients in the network are loosely connected to the server. Hence, there is a chance that the updates received from this set of clients would not reach the server and thereby hindering the performance. It resulted in the advent of a new type of learning i.e., one-shot federated learning [11], where there is only one single round of communication between clients and the server. However, as a result, it should not affect the accuracy of the model built. Additionally, in one-shot FL, it is recommended that the number of training samples at each client should be greater than the number of clients to ensure the performance of the FL model comparable to that of the model trained on the central level. To alleviate privacy concerns, we need to analyze the synthetic dataset. Recently, Generative Adversarial Networks (GANs) [32] occupied a prominent role in generating synthetic datasets by preserving the correlation structure among the features or variables and maintaining the quality. However, there are various challenges, involved in handling tabular datasets viz., (i) handling mixed data types comprising both numerical and categorical features, (ii) dealing with multimodal distribution datasets, and (iii) handling imbalance in categorical features. To overcome these challenges, Xu et al. [9] proposed conditional tabular GAN (CTGAN) by incorporating mode-specific normalization and a conditional vector. However, CTGAN uses single distribution based noise vector which limits the ability to capture the complex dependencies between features. Moreover, it could not capture the true distribution of features. This motivated us to incorporate multiple distributions-based noise vectors to capture complex distributions and increase the probability of capturing the true distribution of features. Probabilistic neural network (PNN) [33] is a one-pass learning algorithm where training and testing occur simultaneously. It is used for classification problems where the probability density function (PDF) for each class is calculated and computed as the class probability for a given sample. However, the downside of PNN is its computational complexity, which is directly proportional to the size of the training data. Hence, in the case of large datasets, researchers proposed clustering the training data while employing PNN. Clustering algorithms such as K-means, convex clustering, centroid-based clustering, etc., can be used for this purpose. These algorithms usually take multiple iterations to find the optimal cluster centers. Hence, Kasabov and Song [5] proposed an online clustering algorithm i.e., evolving clustering machine (ECM) which takes a single iteration to scan all train data and dynamically compute cluster centers. Here, the number of clusters is controlled by a parameter called Dthr (dynamic threshold radius). In our current research, owing to the disadvantages of traditional FL, we propose a novel two-stage methodology, where ECM-in the first stage, we employed a modified CTGAN to generate non-i.i.d data. In the second stage, we developed and employed ECM-PNN hybrid classifier to handle high dimensional, voluminous datasets, owing to their one-pass learning capability. The effectiveness of the proposed FL algorithm is tested on tabular datasets taken from finance and medical domains. It is important to note that we conducted an analysis in HFL on tabular datasets and hence the feature space is common among the clients but differs in the sample space. Major contributions of the current study are as follows: * Proposed a one-shot FedPNN based classifier by invoking ECM as the pre-processor to PNN. * Proposed a two-phase methodology for handling tabular datasets from finance and medical domains under one-shot federated settings. * Proposed a modified CTGAN to generate non-i.i.d synthetic data having multiple distributions with noise. * Proposed meta-clustering aggregation algorithm for the generalization of global model overcoming the limitations of FedAvg. The rest of the paper is organized as follows: In Section 2, we presented the literature survey. Section 3 describes the relevant background and Section 4 discusses the proposed methodology. Section 5 describes datasets and Section 6 gives an overview of the experimental setup. Section 7 presents experimental results then Section 8 concludes the paper. ## 2 Literature Survey In this section, we will discuss the literature review on One-shot algorithms in the context of FL. Zhang et al., [10] proposed three one-shot averaging methods BAVGM (Bootstrap average mixture), achieved comparable performance relative to centralized methods where the data is available at the server. Further, BAVGM turned out to be better than average mixture (AVGM) methods under certain assumptions that the number of samples at the clients is greater than a certain threshold. Guha et al., [11] proposed a one-shot FL method which is an ensemble of $c\leq d\leq\sqrt{N}$, $(\because N$ is the number of samples) local models to train the global model. This could be applied to both supervised or semi-supervised learning algorithms. Kasturi et al., [12] proposed a one-shot FL algorithm, named fusion learning where each client sends local dataset distribution parameters and local model parameters to the server. Later, the server then generates the dataset by using all client distributions, labels them using client models then the server trains the global model on a generated dataset. Shin et al., [13] proposed a privacy-preserving data augmentation technique by employing a mix-up method under one-shot FL settings and named XorMixFL to handle non-i.i.d data. The server then decodes the encoded seed samples by employing the XOR operator. Here, these seed samples are uploaded by the clients and adds them to the base samples at the server to iteratively construct a balanced dataset until it converges. It is reported that the global model trained on reconstructed data performed better than the traditional FL settings. Zhou et al., [14] proposed a distilled one-shot federated learning, named DOSFL, where the server trains on distilled training data sent by available clients. Qinbin et al., (2021) proposed a practical one-shot federated learning algorithm for cross-silo settings named Federated knowledge transfer (FedKT), which uses a knowledge transfer technique for the clients to learn from the local client model and the server uses these models as an ensemble to make the predictions on the public data at the server. Salehkaleybar et al., [15] analyzed problems of statistical optimizations of convex loss landscapes in one-shot methods, and under communication constraints. They proposed a method called multi-resolution estimator clustering with communication cost of log (mn) bits per transmission (MRE-C-log), under constrained estimation error, which is able to reach order optimal up to poly-logarithmic factor. Dennis et al., [16] proposed a one-shot Federated clustering technique (K-FED), a variant of distributed K-means clustering algorithm [27], which leverages data heterogeneity and mitigates data distribution for clustering. Eren et al., [17] proposed a first-of-its-kind recommendation system using collaborative filtering in one-shot FL (FedSPLIT) where each client locally trains a recommendation model using collaborative filtering and global bias which is received from the server. Later, it sends the local pattern and bias of these models back to the server. The server uses joint non-negative matrix factorization [4] to aggregate the client updates. Zhang et al., [18] proposed a data-fee one-shot FL (DENSE) where the server trains an auxiliary generator by using an ensemble of local model updates from the clients after that the final global model is trained. This is followed by the generation of synthetic data by the trained generator and an ensemble of client models using knowledge distillation. Lee et al., [19] proposed a one-shot FL where linear unbiased estimates of multi-task linear (MTL) regression at the clients are uploaded to the server. Then, it linearly combines these estimates and produces improved MTL estimates. Fusion learning at the server is performed by collecting the estimates in each iteration and aggregating the updated estimate to find the final MTL. Hoech et al., [20] proposed a one-shot FL, Federated auxiliary differential private (FedUXdf) where each client trains two models i.e., graph regularized logistic classification and a proposed differentially private model. The server trains the full global model using aggregated weight updates of FedUXdf model, from the clients and public distilled data, labeled using certainty scores of local logistic classification trained on clients. Rjoub et al., [21] integrated a model-free reinforcement learning component into a one-shot FL (OSFL) algorithm where local models are aggregated at the server and clients can choose to label their samples either automatically or request the true label for a one-shot learning component. Garin et al., [22] provided upper bounds on the local MSE and biases in a very general setting of empirical risk minimization (ERM). They proposed Federated estimation with statistical correction (FESC) which aggregates the local estimations based on a minimization of an upper bound of the MSE. They also addressed the minimum sample size constraint of the clients in the one-shot FL scenario. Garin and Quintana [23] proposed FESC method for the effective sample size distribution of the clients. They showed that the mean number of samples per node must be at least equal to the number of nodes in the FL training for its performance comparable to that of centralized training. Wang et al., [24] proposed a novel, first-of-its-kind Data-free diversity-based ensemble selection framework (DeDES) for selecting strong ensemble teams from the machine learning model market, these models are trained by one-shot FL. They proposed a technique for selecting a better representative model inside a cluster to improve the performance of the final model. Humbert et al., [25] presented an institute one-shot FL method named Federated conformal prediction based on quantiles of quantiles (FedCP-QQ). The prediction scores include differentially private data points and their corresponding prediction scores for each client. These scores are arranged in the quantile range, and the model selects local empirical quantiles in a privacy-preserving manner. Then, they are sent to the server where the aggregation of the quantile of these quantiles is executed. Armacki et al., [26] proposed a family of one-shot federated clustering algorithms based on simple inference and averaging schemes with K-means and convex clustering methods. The theoretical results showed that the order-optimal mean squared error (MSE) guarantees results compared to central learning in terms of space complexity. Our current study is different from the literature in the following way: * Proposed One-shot FedPNN based classifier. * To the best of our knowledge, our methodology is the first-of-its-kind, employing ECM invoked with PNN in FL scenario to further speed up the process. * We incorporated two distributions in CTGAN and proposed a modified CTGAN. ## 3 Preliminaries In this section, we discuss the employed architectures for classification and synthetic data generation. ### Evolving Clustering Method Evolving clustering method (ECM) [5] is an online dynamic clustering method, a one-pass learning algorithm that uses normalized euclidean distance to dynamically find the cluster centers. It was applied to anomaly detection [4], two spiral problems [5], etc. ECM starts by considering the first sample as the cluster center, with a radius initialized to zero. Whenever a new sample comes, then the euclidean distance between the point and all of the existed centers is calculated individually and the minimum is considered for further evaluation. The current sample either falls into one of the existing cluster centers or forms a new cluster which is decided by using dynamic threshold radius (Dthr) value. If the minimum distance is less than the two times of Dthr value then the new sample is added to that particular cluster, and then both the cluster center and radius are updated accordingly. Otherwise, a new cluster is formed by considering the current sample as the new cluster center. This process is repeated until all samples are completely scanned. ### 3.2 Probabilistic Neural Network PNN [33] is a single pass learning based feed forward neural network used for classification and pattern recognition problems such as credit card fraud detection, character recognizing, prediction of leukemia, and embryonal tumour of the central nervous system [7], etc. The architecture of PNN is depicted in Fig. 1. PNN majorly comprises four layers (i) input layer, (ii) pattern layer, (iii) summation layer, and (iv) output layer. The input layer transfers the input test data to the pattern layer. Each neuron in the pattern layer contains a labeled training sample. The hidden neuron computes the euclidean distance between the test sample and the center of the neuron and applies a Gaussian, Logistic, or Cauchy activation function. The summation layer then calculates the weighted sum of probability estimates corresponding to each class, and the output layer compares the probability estimates of each category and outputs the most probable class. It uses Parzen's approach to estimate the parent probability density function (PDF) of a given set of training data. Thus, the estimated PDF would asymptotically approach the Bayesian optimal by minimizing the "_expected risk_" which is regarded as the "_Bayes strategies_". In our current study, we adopted the Gaussian activation function is given in Eq. 1. Figure 1: Architecture of PNN \[\mathbf{f_{A}(X)}=\frac{\mathbf{1}}{\mathbf{n(2\pi)^{d/2}\sigma^{d}}}\sum_{\mathbf{t=1}}^{\mathbf{n}} \mathbf{exp}\left[-\sum_{\mathbf{j=1}}^{\mathbf{d}}\left(\mathbf{X_{j}}-\mathbf{X_{A_{tJ}}}\right)^{ 2}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! which gets biased towards generating only the most frequent categories and thereby ignoring the less frequent classes. Therefore, while handling the categorical features, it adapts a conditional vector where multiple conditions can be specified to handle specific categorical columns. Hence, this is regarded as the conditional generator. Here, these conditions can relate to multiple categorical features thereby making CTGAN to handle single or multiple categorical features simultaneously. ## 4 Proposed Methodology In this section, we will discuss the proposed framework which is depicted in Fig. 3, and the algorithm is given Algorithm 1. It comprises two phases, namely, Phase-I where the non-i.i.d synthetic data is generated, which ensures privacy and removes uniquely identifiable information from the real dataset. Thereafter, Phase-II begins where FedPNN is employed with the meta-clustering aggregation algorithm. The above two phases are discussed in detail below: Figure 3: Flow diagram of proposed methodology ### PHASE-I : Non-i.i.d Synthetic data generation The following is performed in the non-federated settings as follows: * A synthetic non-i.i.d dataset is generated by the modified CTGAN. * The thus generated synthetic non-i.i.d dataset is shared among $k$ clients by employing a random sampling technique and then we proceed to Phase-II. ### Step a: Modified CTGAN In the proposed architecture, we proposed and employed a modified CTGAN (refer to Fig. 4) which is different from Vanilla CTGAN in the following way: * Vanilla CTGAN incorporates a single distribution to generate the input noise from randomly sampled data. However, in the modified CTGAN, we employed noise generation from two distributions (refer to Fig. 3) to overcome the challenges such as capturing complex dependencies between features that are faced by single distribution based noise. The employed distributions are: (i) standard normal $\sim$(0,1), and (ii) normal distribution $N\sim$(0.2,2). Once, the random noise is generated from multiple distributions then it is fed as an input to the Generator. The conditions corresponding to categorical columns should be preserved in a conditional vector to generate specific target samples. The generator network in GAN considers both the random noise and conditional vector information to produce the output. It is to be noted that, we employed two different CTGANs (refer to Fig. 4) one for positive sample generation (referred to as CTGANpos) and the other is CTGANneg for negative sample generation. Doing so leverages us to generate as many as samples specific to a single class. As we know that, handling imbalance is very critical in the One-shot FL settings because the underlying model may get biased towards one class. To circumvent these challenges we employed two CTGANs each for a class. Figure 4: Modified CTGAN Once the dataset is generated, its quality is assessed by using the two different synthetic dataset metrics namely, (i) Koglomorinov Smirnov (KS) complement test score to assess the similarity between real samples and synthetic samples, and (ii) mean correlation symmetry test score (CStest score) to assess how much correlation structure is preserved between real and synthetic datasets. Since, we employed two CTGANs, the generated two datasets are combined and the quality of the synthetic data is assessed. Once, the training is completed and the quality metrics are satisfied then the next step begins. Step b: Data Sharing Once, the synthetic dataset is generated, it is shared among clients and the server as follows: * We adopted _b% :(100-b)%_ proportion between the server and client nodes by employing the stratified random sampling method. In our one-shot FL settings, we reserved $b$ % of the synthetic dataset at the server node for the server side evaluation. This helps in evaluating the performance of the meta-clustering aggregation principle, discussed later on. * The rest of the (_100-b)_% of the dataset is shared among clients via simple random sampling which ensures the non-i.i.d nature of the dataset among clients. Further, each client receives the same proportion of samples of the classes as present in the original data and thus meets the real-world requirements. In our approach, we fixed $b$ to be 10 and hence the rest 90% of the dataset is shared among clients. Figure 5: Non-i.i.d synthetic data generation and data sharing among the clients ## Phase II: FedPNN In this phase (refer to Fig. 6), we employed PNN, a one pass learning algorithm that is preceded by a modified ECM algorithm. The performance of the modified ECM is also susceptible to Dthr, which denotes the cluster radius, and is a hyperparameter that determine the number of cluster centers. The following are the critical steps involved: * All the required parameters are broadcasted across the cluster. * PNN models are locally evaluated at each client and then the centers are collected back at the server. * Then, the meta-clustering algorithm is employed that involves clustering of the centers obtained by various clients by invoking ECM again and thus serves as an aggregation algorithm. * Meta centers are passed back to the client and then evaluated on the updated local models at each client. ### Step a: Broadcasting information In the first step, as mentioned in Phase-I, we reserved $b$ % of the generated synthetic dataset on the server side. Hence, we need to maintain Dthr for both the clients and server individually. All these parameters are broadcasted to all the nodes in the federated environment. ### Step b: Local PNN training Figure 6: Proposed one-shot FL Architecture We know that the _(100-b)_ % dataset is reserved for clients and among clients simple random sampling dataset is employed thereby each client receives {p1, p2,..., pk}. Now, within each client, we maintained the train-test ratio of 80%:20%, and adopted stratified random sampling. By using client Dthr, we employed modified ECM (refer to Algorithm 2) to compute the cluster centers. Modified ECM is different from ECM in the following way: It tracks the frequency of the points belonging to each of the clusters which are later used to label the cluster either as positive or negative class. \[\mathbf{d(i,j)}=\left\|\left\|\mathbf{x_{i}-\mathbf{C}_{\varepsilon_{j}}}\right\|\right\|,\mathbf{j =1,2,...\,n} \tag{2}\] \[\left\|\left\|\mathbf{x-y}\right\|\right\|=\left(\sum_{i=1}^{\mathbf{d}}\left\|\mathbf{x_{i}-y_{ i}}\right\|^{2}\right)^{1/2}\left/\mathbf{d^{1/2}}\,,\qquad\mathbf{x,y}\,\in\mathbf{R^{d}} \tag{3}\] ``` Server executes: Input: K, Xtrain, Xtest Output: AUC at the server and clients 1. CC $\mathbf{\leftarrow}$ [ ]${}_{\star X}$ 2. freq $\mathbf{\leftarrow}$ [ ]${}_{pKK}$ 3. for k := 1 to K do //client update function to receive results from the client k 4. CC${}_{k}$, freq$\mathbf{\leftarrow}$ clientUpdate(Dthr, p) 5. CC.append(CC${}_{k}$), freq.append(freqq) 6. end for 7. CC $\mathbf{\leftarrow}$ Meta_clustering(CC, freq) 8. Z $\mathbf{\leftarrow}$ group(CC) // grouping the Xtrain based on class. 9. score $\mathbf{\leftarrow}$ EPNN(Z, Xtest) 10. broadcast CC to K clients Client executes: Input: Dk: private dataset at the client Output: AUC 1. Xtrain, Xtest$\mathbf{\leftarrow}$ Dk 2. Normalize (Xtrain, Xtest) // server invokes this method to receive the result 3. function clientUpdate(Dthr, p) 4. CC $\mathbf{\leftarrow}$ ECM(Xtrain, Dthr, p) 5. AUC $\mathbf{\leftarrow}$ EPNN(CC, Xtest) 6. return CC // returns the update to the server. 7. CC $\mathbf{\leftarrow}$ Server broadcast 8. AUC$\mathbf{\leftarrow}$EPNN(CC, Xtest) 9. return CC, AUC Input:$\mathrm{X}_{\mathrm{train}}$, n, Dthr, p Output:$\mathrm{CC}$, freq 1. $\mathrm{X}_{\mathrm{train}}$, $\mathrm{y}_{\mathrm{train}}$$\blacklozenge$$\mathrm{X}_{\mathrm{train}}$ 2. $\mathrm{X}_{\mathrm{norm}}$$\blacklozenge$Normalize ($\mathrm{X}_{\mathrm{train}}$), $\mathbf{CC}$$\blacklozenge$[ ]$\mathrm{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ast}$d, freq $\blacklozenge$[0,0,..,0]$\mathrm{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ast}$d] // number of classes equals 2 3. $\mathbf{CC}$.append($\mathrm{X}_{\mathrm{norm}}$[1]), $Ru_{1}$$\blacklozenge$0 4. freq[1, $\mathrm{y}_{\mathrm{test}}$[1]] $\blacklozenge$freq[1, $\mathrm{y}_{\mathrm{test}}$[1]] + 1 // $\mathrm{y}_{\mathrm{test}}$[] can take value 1 or 2 for binary classification. 5. $\mathbf{for}$ j := 2 to n 6. $x$$\blacklozenge$$\mathrm{X}_{\mathrm{norm}}$[j], 1 $\blacklozenge$ len($\mathrm{CC}$) 7. $\mathbf{compute}$$d(x,h)$, $h=1,2,...,$ t (refer to Eq. (4)) 8. $d(x,m)$$\blacklozenge$$\min\limits_{k}d(x,h)$, $h=1,2,...,$t 9. $\mathbf{if}$$d(x,m)$$\leq$$Ru_{m}$then 10. freq[m, $\mathrm{y}_{\mathrm{test}}$[j]] $\blacklozenge$freq[m, $\mathrm{y}_{\mathrm{test}}$[j]] + 1 11. $\mathbf{return}$ to step 4 12. $\mathbf{else}$ 13. $s(x,h)<-$$d(x,h)+Ru_{h}$, $h<-1,2,..$ t (refer to Eq. (4)) 14. $s(x,a)<-$$\min\limits_{h}s(x,h)$, $h<-1,2,..$ t (refer to Eq. (5)) 15. $\mathbf{endif}$ 16. $\mathbf{if}$$s(x,a)>$$Dthr$then 17. $\mathbf{CC}$.append($\mathrm{X}_{\mathrm{norm}}$[j]), $Ru_{l+1}$$\blacklozenge$0, freq.append ([0], [0]]) 18. freq[m, $\mathrm{y}_{\mathrm{test}}$[j]] $\blacklozenge$freq[m, $\mathrm{y}_{\mathrm{test}}$[j]] + 1 19. $\mathbf{return}$ to step 4 20. $\mathbf{else}$ 21. $Ru_{a}$$\blacklozenge$$s(x,a)/2,e$$\blacklozenge$$\mathbf{CC}$[a] 22. temp $\blacklozenge$$d(x,e)$ 23. ratio $\blacklozenge$$abs(temp-Ru_{a})/temp$ 24. $\mathbf{CC}$[a] $\blacklozenge$$\mathbf{CC}$[a] $+(x-e)$ ratio 25. freq[a, $\mathrm{y}_{\mathrm{test}}$[j]] $\blacklozenge$freq[a, $\mathrm{y}_{\mathrm{test}}$[j]] + 1 26. $\mathbf{endif}$ 27. $\mathbf{end}$for 28. $\mathbf{return}$$\mathbf{CC}$, freq The training of the modified ECM is described below: * It starts by creating the first cluster center $\mathrm{C}$c by simply considering the first data point. For each cluster, there is a cluster center $\mathrm{C}$c${}_{k}$ and the corresponding radius $\mathrm{Ru}_{k}$. It employs normalized euclidean distance (refer to Eq. 2 and Eq. 3) which is computed for a given point and existing cluster centers. * Now, the cluster center which is in the minimum distance (refer to Eq. 4) is considered for deciding whether the current data point belongs to a cluster or forms a new cluster. This is decided by comparing the minimum distance with Dthr. It is important to note that the minimum distance refers to the additive value of both normalized euclidean distance and radius of the corresponding cluster. * Now, thus obtained minimum distance is compared with the Dthr condition as follows: \[\begin{array}{rl}\circ&\mbox{If }s(i,a)>\ 2\times Dthr\mbox{, then a new cluster center with the current data point is formed,}\\ \circ&\mbox{If }s(i,a)\leq 2\times Dthr,\mbox{ then the data point is to be allotted to that particular cluster $a$ and then update the cluster center, the corresponding radius, and class frequency vector.}\\ \hline\end{array}\] (4) Algorithm 3: PNN ``` Input: CC, X${}_{\mbox{\scriptsize{test}}}$, y${}_{\mbox{\scriptsize{test}}}$ Output: score 1. u $\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{ }}}}}}}}}}}}}}$ 2. y${}_{\mbox{\scriptsize{pred}}}\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{ \boldsymbol{ }}}}}}}}}}}}}}}$ 3. fori:=1 to u do 4. forj:=1 to len(CC)do 5. Aj $\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{ \boldsymbol{ \boldsymbol{ }}}}}}}}}}}}}}}}}$ 0, m \(\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{\boldsymbol{\boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbolboldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ \boldsymbol{ }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\}\\\\\\\\\\\\\\\\\\\\\\}\ Algorithm 4: Meta-clustering using modified ECM at the Server Input: X, Y, n, Dthr, p Output: CC, freq CC $\leftarrow$ [ ], freq $\leftarrow$[[[0], [0]]]$\mapsto$nd // number of classes equals 2 1. CC.append(X[1]), $Ru_{1}$$\leftarrow$ 0 2. freq[1,1] $\leftarrow$freq[1,1]+Y[1,1] // *Y[1,] can take value 1 or 2 for binary classification. 3. freq[1,2] $\leftarrow$freq[1,2]+Y[1,2] 4. for j : = 2 to n do 5. $x$$\leftarrow$ X${}_{non}[j],1$$\leftarrow$ len(CC) 6. compute $d(x,h)$, $h$ = 1,2,..1 using Eq (2) 7. $d(x,m)$$\leftarrow$$\min\limits_{k}$$d(x,h)$, $h$ = 1,2,..1 8. if$d(x,m)$$\leq$$Ru_{m}$then 9. freq[m,1] $\leftarrow$freq[m,1] + Y[j,1] 10. freq[m,2] $\leftarrow$freq[m,2] + Y[j,2] 11. return to step 4 12. else 13. $s(x,h)=d(x,h)+Ru_{h}$, $h$ = 1,2,..1 using Eq 4 14. $s(x,a)=\min\limits_{h}$$s(x,h)$, $h$ = 1,2,..1 using using Eq 5 15. endif 16. if$s(x,a)>Dthr$then 17. CC.append(X${}_{non}[j]$), $Ru_{l+1}$$\leftarrow$ 0, freq.append ([[0], [0]]) 18. freq[l,1] $\leftarrow$freq[l,1] + Y[j,1] 19. freq[l,2] $\leftarrow$freq[l,2] + Y[j,2] 20. return to step 4 21. else 22. $Ru_{a}$$\leftarrow$$s(x,a)/2$, $e$$\leftarrow$CC[a] 23. $temp$$\leftarrow$$d(x,e)$ 24. ratio $\leftarrow$$abs(temp-Ru_{a})/temp$ 25. CC[a] $\leftarrow$CC[a] + $(x-e)$ ratio 26. freq[a,1] $\leftarrow$freq[a,1] + Y[j,1] 27. freq[a,2] $\leftarrow$freq[a,2] + Y[j,2]* 28. endif 29. end for 30. return CC, freq Step c: Meta-clustering aggregation algorithm In the first step, as mentioned in Phase-I, we reserved $b$ % of the generated synthetic dataset on the server side. After training the local PNN model at each client, cluster centers are obtained to the server. As we know that, due to the heterogeneous number of clusters, one cannot employ the Federated average (FedAvg) or its successor algorithms. Hence, we proposed meta-clustering, where we employed ECM to find meta-cluster centers is given in Algorithm 4. It is to be noted that, the meta-cluster centers are computed on the cluster centers obtained from all the clients. Using these meta-cluster centers, the PNN model is evaluated at the server side. Step d: Global training The meta cluster centers which are obtained from step c of Phase-II, are broadcasted to all the clients and are used to update the local models at each client and then these meta clusters are fed into the pattern layer of the local PNN model. The same test process which is discussed in local PNN training (Step b of Phase II) is performed by using meta clusters. Later, AUC is evaluated for each client. ## 5 Dataset Description In the current study, we considered four benchmark datasets related to banking and medical domains that are binary classification datasets (refer to Table 2). It should be noted that 3 out of 4 datasets except Wilcoxon Breast Cancer are highly imbalanced datasets and all of them are having only numerical features. Originally, OVA Omentum dataset features consisted of 10,936, but we employed t-statistic based feature selection and selected the top 500 features for our analysis. For each benchmark dataset, synthetic datasets were generated using modified CTGAN and the minority class is oversampled. The description of the synthetic data is presented in Table 3. ## 6 Experimental Setup The following is the federated experimental setup utilized for conducting the experiments: one server and two clients. All the nodes are having Intel i7 6th generation processors with 32 GB RAM. For FL implementation, \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline Dataset & Data points & Features & \multicolumn{2}{c\|}{Class Distribution ratio} \\ \cline{3-5} & & & Negative Class (\%) & Positive Class \\ \hline Breast Cancer & 699 & 11 & 65.50 & 34.50 \\ \hline OVA Omentum & 1545 & 10936 & 95.00 & 5.00 \\ \hline Credit Card Fraud & 2,84,807 & 32 & 99.83 & 0.17 \\ \hline Polish Bankruptcy & 10,172 & 65 & 96.10 & 3.90 \\ \hline \end{tabular} \end{table} Table 2: Description of Benchmark datasets. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline Dataset & Data points & Features & \multicolumn{2}{c\|}{Class Distribution ratio} \\ \cline{3-5} & & & Negative Class (\%) & Positive Class** \\ \hline Breast Cancer & 916 & 11 & 50.00 & 50.00 \\ \hline OVA Omentum & 2936 & 10936 & 50.00 & 50.00 \\ \hline Credit Card Fraud & 3,98,237 & 31 & 71.00 & 29.00 \\ \hline Polish Bankruptcy & 19,544 & 65 & 50.00 & 50.00 \\ \hline \end{tabular} \end{table} Table 3: Description of generated synthetic data of Benchmark dataset we utilized the Flower framework [29] to conduct the experiment which is developed in Python language and supports scaling to many distributed clients. It is also flexible for adopting a new proposed approach with low engineering overhead. ### Evaluation Metrics In the current study, we employed two metrics for evaluating the quality of the synthetic dataset (i) Complement of the Kolmogorov-Smirnov test (KSComplement), (ii) Correlation similarity test score (CStest). Additionally, the performance of FedPNN is evaluated by AUC. The metrics are explained below: #### 6.1.1 KSComplement It measures the similarity of a given column in the real and synthetic datasets. It uses the Koglomorinov-Smironov statistic (KS statistic), which is calculated by converting the numerical column into its cumulative distribution frequency (CDF). The maximum difference between the two CDFs is known as the KS statistic, which lies between [0,1]. The KSComplement metric, mathematically defined in Eq. 6, also lies within the range of [0,1]. The higher the KSComplement the better the similarity is between real and synthetic columns. \[\text{KSComplement test Score}=\mathbf{1}-\overset{sup}{x}\|F_{i}(x)-F_{j}(x)\| \tag{6}\] where, $\text{sum}_{\text{x}}$ is the supremum of set of distances. $\text{F}_{i}$ is the Cumulative Distribution Function (CDF) of real data continuous columns likewise $\text{F}_{\text{j}}$ is the CDF of the same continuous column in synthetic data. Mean KSComplement test score is the mean of test scores of all the independent variables of a dataset. #### 6.1.2 CStest It measures the correlation similarity between two numerical columns of the real and synthetic datasets. In our current study, the CStest score is presented in Eq. 7. \[\text{Correlation Similarity test Score (A, B)}=\mathbf{1}-\frac{\|S_{A,B}-R_{A,B}\|}{2} \tag{7}\] where, $S_{A,B}$ and $R_{A,B}$ are the correlation coefficient [28] of A and B, a pair of continuous columns, on synthetic and real data respectively. #### 6.1.3 Auc While handling imbalanced datasets, AUC is proven to be a robust measure which is an average of specificity and sensitivity. It is mathematically represented as presented in Eq. (8). \[\text{AUC}=\frac{(Sensitivity+Specificity)}{2} \tag{8}\] where, \[Sensitivity=\frac{TP}{TP+FN} \tag{9}\] and \[Specificity=\frac{TN}{TN+FP} \tag{10}\] where TP is a true positive, FN is a false negative, TN is a true negative, and FP is a false positive. ## 7 Results and Discussion All the hyperparameters used for the synthetic dataset generation and for the current study are presented in Table 4. In this section, we will discuss the following: (i) synthetic dataset metric analysis, (ii) AUC analysis after local training, and meta-clustering. Further, we conducted a sensitivity analysis to discuss the effect of Dthr over the performance in the one shot FL settings in two different ways viz., (i) by varying client Dthr and keeping server Dthr constant, and (ii) by varying server Dthr and keeping client Dthr constant. \begin{table} \begin{tabular}{\|c\|l\|l\|c\|c\|} \hline \multirow{2}{}{Dataset} & \multicolumn{3}{c\|}{GAN Architecture} & \multicolumn{2}{c\|}{Dthr} \\ \cline{2-5} & \multicolumn{2}{c\|}{Negative Class} & \multicolumn{1}{c\|}{Positive Class} & \multicolumn{1}{c\|}{$\textbf{Clients_{1}}$ /} & \multicolumn{1}{c\|}{Server} \\ \cline{2-5} & \multicolumn{1}{c\|}{} & \multicolumn{1}{c\|}{} & \multicolumn{1}{c\|}{} & \multicolumn{1}{c\|}{Client${}_{2}$} & \\ \hline Breast* & GEN architecture (32,16) & GEN architecture (16,32) & 0.19 & 0.17 \\ Cancer & DIS architecture (16,32) & DIS architecture (32,16) & & \\ & Epochs = 80 & Epochs = 50 & & \\ \hline OVA & GEN architecture & GEN architecture & 0.13 & 0.07 \\ Omentum & (256,128,64) & (256,128,64) & & \\ & DIS architecture (64,128,256) & DIS architecture & & \\ & Epochs = 50 & (64,128,256) & & \\ & Epochs = 20 & & \\ \hline Credit card & GEN architecture (16,32) & GEN architecture (16,32) & 0.17 & 0.13 \\ fraud & DIS architecture (32,16) & DIS architecture (32,16) & & \\ & Epochs = 50 & Epochs = 150 & & \\ \hline Polish & GEN architecture (64,32,16) & GEN architecture (64,32) & 0.11 & 0.05 \\ Bankruptcy & DIS architecture (32,128) & DIS architecture (256,256) & & \\ & Epochs = 50 & Epochs = 30 & & \\ \hline \multicolumn{5}{l}{* GEN : Generator ; DIS : Discriminator} \\ \end{tabular} \end{table} Table 4: Hyper parameter table used in the current study ### Comparative Analysis of Synthetic Data metrics In this section, we compared the performance of the proposed modified CTGAN with that of vanilla CTGAN based on KSComplement and CStest scores. In our settings, we employed two CTGANs each for the positive and negative classes, respectively. We presented the mean KSComplement and mean CStest scores in Table 5 and Table 6, respectively. Importantly, it is desired to have a higher KS complement score and CStest score for a better synthetic dataset. Table 5 shows that in 3 out of 4 datasets, Vanilla CTGAN obtained better KSComplement scores than modified CTGAN. This demonstrates that the synthetic dataset generated by Vanilla CTGAN is more similar to the original dataset when compared to that of modified CTGAN. Interestingly, in the Breast cancer dataset, modified CTGAN obtained a mean KSComplement score of 0.882 which is 1.9% greater than that of Vanilla CTGAN. However, in the Polish Bankruptcy, modified CTGAN obtained just 29.3% which is too less than the KSComplement score of Vanilla CTGAN. Overall, in terms of the KSComplement score, Vanilla CTGAN turned out to be better. From Table 6, it is demonstrated that modified CTGAN obtained a better CStest score ensuring that it captured the correlation structure of the original dataset. Interestingly, Vanilla CTGAN obtained almost similar CStest scores in both the Breast cancer and OVA Omentu datasets respectively. It is important to note that even though modified CTGAN failed miserably to replicate the similarity of the original dataset (see Table 5), yet it captured correlation structure by 13% better than Vanilla CTGAN. Overall, in terms of the CStest score, modified CTGAN turned out to be better than Vanilla CTGAN. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|} \hline \multirow{2}{}{Dataset} & \multicolumn{3}{c\|}{Vanilla CTGAN} & \multicolumn{3}{c\|}{Modified CTGAN} \\ \cline{2-7} & Negative & Positive & Mean & Negative & Positive & Mean \\ & Class & Class & & Class & Class & \\ \hline Breast Cancer* & 0.860 & 0.896 & 0.878 & 0.868 & 0.888 & 0.878 \\ \hline OVA Omentum & 0.912 & 0.920 & 0.916 & 0.911 & 0.920 & 0.916 \\ \hline Credit card fraud & 0.981 & 0.856 & 0.918 & 0.980 & 0.860 & 0.920 \\ \hline Polish & 0.672 & 0.894 & 0.783 & 0.932 & 0.894 & 0.913 \\ Bankruptcy & & & & & & \\ \hline \end{tabular} \end{table} Table 6: Mean CStest scores of the generated synthetic data \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|} \hline \multirow{2}{}{Dataset} & \multicolumn{3}{c\|}{Vanilla CTGAN} & \multicolumn{3}{c\|}{Modified CTGAN} \\ \cline{2-7} & Negative & Positive & Mean & Negative & Positive & Mean \\ & Class & Class & & Class & Class & \\ \hline Breast Cancer* & 0.868 & 0.859 & 0.8635 & 0.877 & 0.886 & 0.882 \\ \hline OVA Omentum & 0.728 & 0.704 & 0.716 & 0.691 & 0.661 & 0.676 \\ \hline Credit card fraud & 0.914 & 0.793 & 0.8535 & 0.920 & 0.688 & 0.804 \\ \hline Polish & 0.6122 & 0.929 & 0.7706 & 0.262 & 0.325 & 0.293 \\ Bankruptcy & & & & & & \\ \hline \end{tabular} \end{table} Table 5: Mean KSComplement scores of the generated synthetic data ### Comparative study of AUC The best AUC results obtained in the respective datasets, generated by Vanilla CTGAN and modified CTGAN are presented in Table 7 and Table 8, respectively. We present the results for both the local training and meta-clustering phases. In addition to AUC, we also presented the number of centers. In both cases, by employing Vanilla CTGAN and modified CTGAN, there is no significant change in AUC before and after the clustering phases. This indeed indicates that the employed meta-clustering is powerful enough in obtaining the generalized meta-centers, thereby reducing the row space dimension significantly. In terms of AUC, FedPNN achieved better AUC in all of the datasets using the synthetic dataset generated by modified CTGAN when compared to Vanilla CTGAN. Additionally, it is desired to obtain better AUC with the less possible number of cluster centers which indeed reflects the computational efficiency. Except in the Polish Bankruptcy dataset, the meta-clustering aggregation algorithm of FedPNN built on the synthetic dataset generated by modified CTGAN selected less number of centers in the negative class than that of Vanilla CTGAN. This demonstrates that employing modified CTGAN during synthetic dataset generation indeed preserves privacy and obtains higher AUC with less complexity when compared to Vanilla CTGAN. This is due to the fact that employing multiple distributions based on modified CTGAN captured the complex dependencies between the features better than Vanilla CTGAN. ### Sensitivity Analysis In this section, we presented the sensitivity analysis conducted based on client Dthr and server Dthr. Here, we reported both the change occurred in AUC and the number of centers. The sensitivity analysis was conducted \begin{table} \begin{tabular}{c\|c\|c\|c\|c\|c\|c\|c\|c} \hline \multirow{3}{}{Dataset} & \multirow{3}{}{\begin{tabular}{c} Type of \\ the \\ Node \\ \end{tabular} } & \multicolumn{3}{c\|}{Vanilla CTGAN} & \multicolumn{3}{c}{Modified CTGAN} \\ \cline{3-10} & & \multicolumn{2}{c\|}{Local Training} & \multicolumn{2}{c\|}{After Meta-clustering} & \multicolumn{2}{c\|}{Local Training} & \multicolumn{2}{c\|}{After Meta-clustering} \\ \cline{3-10} & & & & \multicolumn{1}{c\|}{} & \multicolumn{1}{c\|}{clustering} & \multicolumn{1}{c\|}{clustering} & \multicolumn{1}{c\|}{clustering} & \multicolumn{1}{c\|}{clustering} \\ \cline{3-10} & & AUC & No of & AUC & No of & AUC & No of & AUC & No of \\ & & & Centers & & Centers & & Centers & & Centers \\ & & & (-ve; +ve) & & (-ve; & & (-ve; & (-ve; \\ & & & & & +ve) & & & +ve) & \\ \hline \multirow{3}{}{\begin{tabular}{c} Breast* \\ Cancer \\ \end{tabular} } & Client${}_{1}$ & 1 & (1;27) & 0.976 & (3; 17) & 0.989 & (2; 27) & 1 & (2; 18) \\ \cline{2-10} & Client${}_{2}$ & 0.988 & (2; 28) & 0.927 & & 1 & (3; 27) & 1 & \\ \cline{2-10} & Server & - & - & 0.957 & & - & - & 0.989 & \\ \hline \multirow{3}{}{\begin{tabular}{c} Ova* \\ Oment \\ \end{tabular} } & Client${}_{1}$ & 0.746 & (137; 16) & 0.735 & (276; 21) & 0.989 & (135; 1) & 0.989 & (267; 6) \\ \cline{2-10} & Client${}_{2}$ & 0.798 & (139; 5) & 0.794 & & 0.973 & (133; 4) & 0.973 & \\ \cline{2-10} & Server & - & - & 0.738 & & - & - & 0.966 & \\ \hline \multirow{3}{}{\begin{tabular}{c} Credit card* \\ fraud \\ \end{tabular} } & Client${}_{1}$ & 0.933 & (11;148) & 0.911 & (13; 74) & 0.985 & (5; 213) & 0.947 & (4; 117) \\ \cline{2-10} & Client${}_{2}$ & 0.94 & (17; 148) & 0.912 & & 0.987 & (4; 209) & 0.946 & \\ \cline{2-10} & Server & - & - & 0.914 & & - & - & 0.949 & \\ \hline \multirow{3}{}{ \begin{tabular}{c} Polish* \\ Bankruptcy \\ \end{tabular} } & Client${}_{1}$ & 0.945 & (16; 37) & 0.947 & (21; 64) & 0.977 & (27; 67) & 0.979 & (36; 110) \\ \cline{2-10} & Client${}_{2}$ & 0.897 & (11; 38) & 0.964 & & 0.965 & (28; 70) & 0.982 & \\ \cline{1-1} \cline{2-10} & Server & - & - & 0.853 & & - & - & 0.899 & \\ \hline \end{tabular} * : No local training in Server \end{table} Table 7: Comparative analysis on AUC after employing FedPNN in two different ways: (i) by fixing the best client Dthr and varying the server Dthr parameter with an interval of 0.02 is depicted in Fig. 10 to Fig. 10, Fig. 10 (a) denotes the change in AUC, while Fig. 7 (b) denotes the change in the number of centers for the Breast Cancer dataset. The same is followed for the rest of the datasets as shown in Fig. 8, Fig. 9 and Fig. 10. (ii) by fixing the best server Dthr and varying the client Dthr parameter with an interval of 0.02, is depicted in Fig. 11 through Fig. 14. It is important note that the sensitivity analysis is restricted to the synthetic dataset generated by modified CTGAN due to its better performance, as discussed in Section 7.2. Firstly, we discussed the performance of FedPNN by choosing the best value for client Dthr and keeping it constant but with varied server Dthr parameters. For the Breast cancer dataset, the best server Dthr is 0.17 (refer to Table 3), it is observed that as there is an increase in server Dthr, there is a decrease in AUC. Although no specific trend is observed, the AUC constantly decreased after 0.21. However, there is no significant variation is observed while decreasing the server Dthr value i.e., in 0.15, 0.13, 0.11, etc. Similarly, for Credit card fraud (see Fig. 10) and Polish Bankruptcy datasets (see Fig. 10), overall, a decreasing trend in AUC is observed while increasing the server Dthr parameter. Further, in terms of change in the number of centers, the number of negative centers is constantly decreasing while increasing the server Dthr. However, no particular trend is observed for the number of positive class centers. Interestingly, except for the OVA Omentum dataset (see Fig. 8), the number of negative class centers is more than the number of positive class centers. As the number of centers is indeed responsible for the complexity of PNN, choosing an optimal number of centers that maintain good AUC is critical. For example, in the Breast cancer dataset (see Fig. 7), AUC obtained with 18 number of centers turned out to be almost similar to the AUC achieved when the number of centers are 53. Similarly, in the case of the OVA Omentum dataset, AUC achieved with 197 centers is slightly higher than with that of 267 centers, resulting in a significant decrease in the model complexity. Similar kind of observations, such as trends in AUC with respect to the number of centers are made for Credit card fraud and Polish Bankruptcy datasets. Now, we will discuss the performance of FedPNN by keeping the best server Dthr value constant (refer to Table 3) and changing the client Dthr. Similar to the observations of the first analysis, here also, (i) there is a decrement in AUC when there is an increase in the client Dthr value in all datasets, and (ii) the number of centers is decreased as there is an increase in client Dthr. Further, in the credit card dataset (see Fig. 13), the AUC achieved with 965 centers is lower than that of 470 centers. This concludes the fact the AUC is susceptible to change in server Dthr and client Dthr. Hence, with the above analysis, we conclude that appropriate values need to be chosen based on considering both AUC and the number of centers. Fig. 11 (a) Change in AUC for Breast cancer by keeping server Dthr constant Fig. 12 (a) Change in AUC for OVA Momentum by keeping server Dthr constant Fig. 13 (b) Change in the number of centers for Fraud by keeping server Dthr constant Fig. 13 (a) Change in AUC for Credit Card Fig. 13 (b) Change in the number of centers for Fraud by keeping server Dthr constant Fig. 11 (a) Change in AUC for Breast cancer by Fig. 12 (b) Change in the number of centers for Breast cancer by keeping server Dthr constant ## 8 Conclusions and Future Directions In the current study, we proposed a two-stage one-shot federated learning approach toward the objective of privacy protection and overcoming a few challenges of FL. Here, during the first stage, we employed a modified CT-GAN to generate the synthetic dataset, which is later evaluated by the KSComplement and CStest scores. The thus generated synthetic dataset is distributed among the nodes. Thereafter, FedPNN is employed and integrated with modified ECM, and at the server, the meta-aggregation algorithm is invoked during the communication round. The results demonstrate that the proposed modified CTGAN obtained better CStest scores, 3 out of 4 times but outperformed by Vanilla CTGAN in terms of KSComplement score in the majority of the cases. Further, the proposed meta-clustering aggregation algorithm is able to generalize well as there is only a slight fluctuation of AUC before and after its invocation but decreased the complexity by selecting less number of optimal centers in 3 out of 4 datasets. Additionally, the sensitivity analysis demonstrated to us how the number of centers is influencing AUC and trends in AUC and the number of centers. The following are potential future works: (i) Employing FedCTGAN in the architecture to improve better privacy, but doing so increases the complexity and hence it should be taken care of, (ii) Handling highly imbalanced datasets, (iii) Developing regression based models along the similar lines, (iv) Designing and employing homomorphic encryption based FedPNN in the above settings to increase the level of privacy protection. Additionally, this is a general purpose methodology which can be employed in the following use cases in banking, finance and insurance sector which are as follows: (a) anti-money laundering, (b) detecting multiple asset creation, (c) identifying frauds in insurance claims, (d) identifying impact of geographical differences on diseases, (e) fraud detection in intra-day stock trading.
2304.10126	Decouple Graph Neural Networks: Train Multiple Simple GNNs Simultaneously Instead of One	Graph neural networks (GNN) suffer from severe inefficiency. It is mainly caused by the exponential growth of node dependency with the increase of layers. It extremely limits the application of stochastic optimization algorithms so that the training of GNN is usually time-consuming. To address this problem, we propose to decouple a multi-layer GNN as multiple simple modules for more efficient training, which is comprised of classical forward training (FT)and designed backward training (BT). Under the proposed framework, each module can be trained efficiently in FT by stochastic algorithms without distortion of graph information owing to its simplicity. To avoid the only unidirectional information delivery of FT and sufficiently train shallow modules with the deeper ones, we develop a backward training mechanism that makes the former modules perceive the latter modules. The backward training introduces the reversed information delivery into the decoupled modules as well as the forward information delivery. To investigate how the decoupling and greedy training affect the representational capacity, we theoretically prove that the error produced by linear modules will not accumulate on unsupervised tasks in most cases. The theoretical and experimental results show that the proposed framework is highly efficient with reasonable performance.	Hongyuan Zhang, Yanan Zhu, Xuelong Li	2023-04-20T07:21:32Z	http://arxiv.org/abs/2304.10126v2	# Decouple Graph Neural Networks: Train Multiple Simple GNNs Simultaneously Instead of One ###### Abstract Graph neural networks (GNN) suffer from severe inefficiency due to the exponential growth of node dependency with the increase of layers. It extremely limits the application of stochastic optimization algorithms so that the training of GNN is usually time-consuming. To address this problem, we propose to decouple a multi-layer GNN as multiple simple modules for more efficient training, which is comprised of classical _forward training_ (_FT_) and designed _backward training_ (_BT_). Under the proposed framework, each module can be trained efficiently in FT by stochastic algorithms without distortion of graph information owing to its simplicity. To avoid the only unidirectional information delivery of FT and sufficiently train shallow modules with the deeper ones, we develop a backward training mechanism that makes the former modules perceive the latter modules, inspired by the classical backward propagation algorithm. The backward training introduces the reversed information delivery into the decoupled modules as well as the forward information delivery. To investigate how the decoupling and greedy training affect the representational capacity, we theoretically prove that the error produced by linear modules will not accumulate on unsupervised tasks in most cases. The theoretical and experimental results show that the proposed framework is highly efficient with reasonable performance, which may deserve more investigation. ## 1 Introduction In recent years, neural networks [1; 2], due to the impressive performance, have been extended to graph data, known as graph neural networks (GNN) [3]. As GNNs significantly improve the results of graph tasks, it has been extensively investigated from different aspects, such as graph convolution network (GCN) [4; 5], graph attention networks (GAT) [6; 7], spatial-temporal GNN (STGNN) [8], graph auto-encoder [9; 10], graph contrastive learning [11], _etc_. Except for the variants that originate from different perspectives, an important topic is motivated by the well-known inefficiency of GNN. In classical neural networks [2], the optimization is usually based on stochastic algorithms with limited batch [12; 13] since samples are independent of each other. However, the aggregation-like operations defined in [14] result in the dependency of each node on its neighbors and the amount of dependent nodes for one node increases exponentially with the growth of layers, which results in the unexpected increases of batch size. Some works are proposed based on neighbor sampling [14; 15; 16; 17] and graph approximation [18] to limit the batch size, while some methods [19; 20] attempt to directly apply high-order graph operation and sacrifice the most non-linearity. The training stability is a problem for neighbor sampling methods [14; 15; 17] though VRGCN [16] has attempted to control the variance via improving sampling. Note that the required nodes may still grow (slowly) with the increase of depth. ClusterGCN [18] finds an approximate graph with plenty of connected components so that the batch size is strictly upper-bounded. The major challenge of these methods is the information missing during sampling. The simplified methods [19; 20] are efficient but the limited non-linearity may be the bottleneck of these methods. These methods may incorporate the idea of GIN [21] to improve the capacity [20]. To apply stochastic optimization while retaining the exact graph structure, we propose a framework, namely stacked graph neural network (SGNN), which decouples a multi-layer GNN as multiple simple GNN modules and then trains them simultaneously rather than connecting them with the increase of the depth. Inspired by the backward propagation algorithm, we find that the main difference between stacked networks [22] and classical networks is _no training information propagated from the latter modules to the former ones_. The lack of backward information delivery may be the main reason of the performance limitation of stacked models. The contributions are concluded as: (1) We accordingly propose a backward training strategy to let the former modules receive the information from the final loss and latter modules, which leads to a cycled training framework to control bias and train shallow modules correctly. (2) Under this framework, a multi-layer GNN can be decoupled into multiple simple GNNs, named as separable GNNs in this paper, so that every training step could use the stochastic optimization without any samplings or changes on graph. Therefore, SGNN could take both non-linearity and high efficiency into account. (3) We investigate how the decoupling and greedy training affect the representational capacity of the linear SGNN. It is proved that the error would not accumulate in most cases when the final objective is graph reconstruction. ## 2 Background Graph Neural Networks Aiming to extend convolution operation into graph, graph convolution became a hot topic [5; 23; 24] and graph convolution network (GCN) [4] has become an important baseline. By introducing self-attention techniques [25], graph attention networks (GAT) [6; 7] are proposed and applied to other applications [26; 27]. As [28] claimed that GNNs suffer from the over-smoothing problem, GALA [10] develops the graph sharpening and ResGCN [29] attempts to designs a deeper architecture. The theoretical works [28; 30; 31] have different views towards the depth of GNNs. Some works [28; 30] claimed that the expressive power of GNN decreases with the increase of layers, while the others argue that the assumptions in [30] may not hold and deeper GNNs have stronger power [31]. Moreover, some works [32; 21] investigate the expressive capability by showing the connection between Weisfeiler-Lehman test [33] and GNNs. Nevertheless, _most of them neglect the inefficiency problem of GNNs_. Figure 1: Illustration of a stacked graph neural network decoupled from an $L$-layer GNN. To train each module individually, some loss (_e.g._, semi-supervised loss, unsupervised loss, contrastive loss) is required and it is denoted by $\mathcal{L}_{FT}^{(t)}$. To let the shallow modules perceive the deeper ones, $\mathcal{M}_{t}$ passes back the _expected_ input features to $\mathcal{M}_{t-1}$ during the backward training. The divergence between the features output by $\mathcal{M}_{t}$ and the expected features of $\mathcal{M}_{t-1}$ formulates the BT loss $\mathcal{L}_{BT}^{(t)}$. Efficient Graph Neural Networks To accelerate the optimization through batch gradient descent to GNN without too much deviation, several models [14; 15; 16; 17] propose to sample data points according to graph topology. These models propose different sampling strategies to obtain stable results. GraphSAGE [14] produces a subgraph with limited neighbors for each node while FastGCN [15] samples fixed nodes for each layer with the importance sampling. The variance of sampling is further controlled in [16]. ClusterGCN [18] aims to generate an approximate graph with plenty of connected components so that each component can be used as a batch per step. SGC [19] simplifies GCN by setting all activations of middle layers as linear functions and SSGC [20] further improves it. In summary, _SGNN proposed in this paper retains the non-linearity and requires no node sampling or sub-graph sampling._ L2-GCN [34] attempts to extend the idea of the classical stacked auto-encoder into the popular GCN while DGL-GNN [35] further develops a parallel version. They both fail to train all GNN modules jointly but _SGNN firstly offers a novel framework to train them like training layers in a conventional neural network._ Connections to Existing Models Stacked Auto-Encoder (_SAE_) [22] is a model applied to the pre-training of neural networks. It trains the current two-layer auto-encoder [36] and then feeds the latent features output by the middle layer to the next auto-encoder. The model is often used as a pre-training model instead of a formal model. MGAE [37] is an extension of SAE and its fundamental module is graph auto-encoder [9]. The main difference compared with the proposed model is whether each module could be perceived by modules from both forward and backward directions. The stack paradigm is similar to the classical boosting models [38; 39; 40] while some works [41; 42] also investigated the boosting algorithm of neural networks. In recent years, some boosting GNN models [43; 44] are also developed. The most boosting algorithms (_e.g._, [41; 43]) aim to learn a prediction function gradually while the proposed SGNN aims to learn ideal embeddings gradually. Note that AdaGCN [44] is also trained gradually and the features are combined using AdaBoost [38]. More importantly, _all these boosting methods for GNNs are only trained forward and the backward training is missing_. Deep neural interface [45] proposes to decouple neural networks to asynchronously accelerate the computation of gradients. _The decoupling is an acceleration trick to compute the gradients of $L$-layer networks_, while SGNN proposed in this paper explicitly separates an $L$-layer GNN into $L$ simple modules. In other words, _the ultimate goal of SGNN is not to optimize an $L$-layer GNN_. ## 3 Proposed Method Motivated by SAE [22] and the fact that the simplified models [19; 20] are highly efficient for GNN, we therefore rethink the substantial difference between the stacked networks and multi-layer GNNs. To sum up, we attempt to answer the following two questions in this paper: * _How to decouple a complex GNN into multiple simple GNNs and train them jointly?_ * _How does the decoupling affect the representational capacity and final performance?_ We will discuss the first question in this section and then elaborate on another one in Section 4. ### Preliminary Each decoupled GNN model of the proposed model is named as a module and the $t$-th module is denoted by $\mathcal{M}_{t}$ for simplicity. The vector and matrix are denoted by lower-case and upper-case letters in bold, respectively. $\\|\cdot\\|$ represents the Frobenius norm. Given a graph, let $\mathbf{A}\in\mathbb{R}^{n\times n}$ be adjacency matrix and $\mathbf{X}\in\mathbb{R}^{n\times d}$ be node features. A typical GNN layer can be usually defined as \[\mathbf{H}=f(\mathbf{A},\mathbf{X},\mathbf{W})=\varphi(\mathbf{P}\mathbf{X}\mathbf{W}), \tag{1}\] where $\mathbf{W}$ is projection coefficient and $\mathbf{P}=\phi(\mathbf{A})$ is a function of $\mathbf{A}$. When we discuss each individual module, we assume that $\mathbf{W}\in\mathbb{R}^{d\times k}$ for simplicity. For example, GCN [4] defines $\mathbf{P}$ as $\mathbf{P}_{\rm GCN}=\mathbf{D}^{-\frac{1}{2}}(\mathbf{A}+\mathbf{I})\mathbf{D}^{-\frac{1}{2}}$ and $\mathbf{D}$ is the degree matrix of $\mathbf{A}+\mathbf{I}$. When multiple layers are integrated, the learned representation given by multiple GNN layers can be written as \[\begin{split}\mathbf{H}&=f_{1}\circ f_{2}\circ\cdots \circ f_{L}(\mathbf{A},\mathbf{X},\mathbf{W}_{1},\ldots,\mathbf{W}_{L})\\ &=\varphi_{L}(\mathbf{P}\varphi_{L-1}(\cdots\varphi_{1}(\mathbf{P}\mathbf{X} \mathbf{W}_{1})\cdots)\mathbf{W}_{L}),\end{split} \tag{2}\] where $L$ is the amount of layers. Assume that the average number of neighbors is $c$. To compute $\mathbf{H}$, each sample will need $\mathcal{O}(c^{L})$ extra samples. If the depth is large and the graph is connected, then all nodes have to be engaged to compute for one node. The uncontrolled batch size results in the time-consuming training. In vanilla GNNs, the computational complexity is $\mathcal{O}(KL\\|\mathbf{A}\\|_{0}\sum_{i=0}^{m-1}d_{i}d_{i+1})$ on sparse graphs where $K$ is the number of iterations and $d_{i}$ is the dimension of $\mathbf{W}_{i}$. For large-scale datasets, both time and space complexity are too expensive. ### Stacked Graph Neural Networks Although the stacked networks usually have more parameters than multi-layer networks, which frequently indicates that the stacked networks may be more powerful, they only serve as a technique for pre-training. Specifically speaking, they simply transfer the representations learned by the current network to the next one but _no feedback is passed back_. It causes the invisibility of the succeeding modules and the final objective. As a result of the unreliability of the former modules, the stacked model is conventionally used as an unsupervised pre-training model. Rethinking the learning process of a network, multiple layers are optimized simultaneously by gradient-based methods where the gradient is calculated by the well-known backward propagation algorithm [46]. The algorithm consists of forward propagation (FP) and backward propagation (BP). FP computes the required values for BP, which can be viewed as an information delivery process. Note that FP is similar to the training of the stacked networks. Specifically, transferring the output of the current module to the next one in the stacked network is like the computation of neurons layer by layer during FP. Inspired by this, we aim to design a BP-like training strategy, namely _backward training (BT)_, so that the former modules could be tuned according to the feedback. The core idea of our stacked graph neural network (SGNN) is shown in Figure 1. #### 3.2.1 Separability: Crucial Concept for Efficiency Before introducing SGNN in detail, we formally introduce the key concept and core motivation of how to accelerate GNN via SGNN. Definition 3.1.: _If a GNN model can be formulated as $f(\mathbf{A},\mathbf{X},\mathbf{W})=f_{1}(f_{0}(\mathbf{A},\mathbf{X}),\mathbf{W})$, then it is a separable GNN. If it can be further formulated as $f(\mathbf{A},\mathbf{X},\mathbf{W})=f_{1}(f_{0}(\mathbf{A},\mathbf{X}),\mathbf{W})=g_{1}(\mathbf{A},g_{0}( \mathbf{X},\mathbf{W}))$, then it is a fully-separable GNN._ To keep simplicity, define the set of separable GNNs as $\mathcal{F}_{k}=\{f:\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times d}\times \mathbb{R}^{d\times k}\mapsto\mathbb{R}^{n\times k}\|f(\mathbf{A},\mathbf{X},\mathbf{W})=f_ {1}(f_{0}(\mathbf{A},\mathbf{X}),\mathbf{W})\}$ and the set of fully-separable GNNs as $\mathcal{F}_{k}^{}=\{f:\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times d} \times\mathbb{R}^{d\times k}\mapsto\mathbb{R}\|f(\mathbf{A},\mathbf{X},\mathbf{W})=f_{1}(f_ {0}(\mathbf{A},\mathbf{X}),\mathbf{W})=g_{1}(\mathbf{A},g_{0}(\mathbf{X},\mathbf{W}))\}$. Note that most single-layer GNN models are fully-separable. For instance, SGC [19] is fully-separable where $f_{0}(\mathbf{A},\mathbf{X})=\mathbf{P}^{m}\mathbf{X}$ and $f_{1}(f_{0}(\mathbf{A},\mathbf{X}),\mathbf{W})=\varphi(f_{0}(\mathbf{A},\mathbf{X})\cdot\mathbf{W})$, while the single-layer GIN [3] is separable but not fully-separable since $\mathbf{P}\cdot\mathrm{MLP}(\mathbf{X})\neq\mathrm{MLP}(\mathbf{P}\mathbf{X})$ usually holds. However, a single-layer GAT [6] is not separable since the graph operation is relevant to $\mathbf{W}$. _The separable property actually factorizes a GNN model to 2 parts, graph operation $f_{0}$ and neural operation $f_{1}$._ Since all dependencies among nodes in GNNs are caused by the graph operation, one can compute $\mathbf{X}^{\prime}=f_{0}(\mathbf{A},\mathbf{X})$ once (like preprocessing) in separable GNNs and then the GNN is converted into a typical network. After computing $\mathbf{X}^{\prime}$, the information contained in graph has been passed into $\mathbf{X}^{\prime}$ and the succeeding sampling would not affect the topology of graph. Therefore, we can obtain a highly efficient GNN model that can be optimized by SGD, provided that each module is separable. On the other hand, _the fully-separable condition is essential for the backward training to pass back the information over multiple modules_. Since most single-layer GNNs are separable but not fully-separable, we show how to revise separable GNNs to introduce the fully-separability. Then, we formally clarify the core idea of SGNN by showing how to handle $\mathbf{Q}\mathbf{I}$. #### 3.2.2 Forward Training (FT) _The first challenge_ is how to set the training objective for each module $\mathcal{M}_{t}$. It is crucial to apply SGNN to both supervised and unsupervised scenes. Suppose that we have a separable GNN module $\mathcal{M}$ and let $\mathbf{H}=f(\mathbf{A},\mathbf{X},\mathbf{W})$ be the features learned by the separable GNN module. For the unsupervised cases, if $\mathcal{M}_{t}$ is a GAE, then the loss of FT is formulated as \[\mathcal{L}_{FT}=\ell(\mathbf{A},\mathbf{X},\mathbf{H})=d(\mathbf{A},\kappa(\mathbf{H})), \tag{3}\] where $d(\cdot,\cdot)$ represents the metric function and $\kappa:\mathbb{R}^{n\times k}\mapsto\mathbb{R}^{n\times n}$ is a mapping function. For instance, a simple loss introduced by [9] is $d(\mathbf{A},\kappa(H))=KL(\mathbf{A}\\|\sigma(\mathbf{H}\mathbf{H}^{T}))$ where $\sigma(\cdot)$ is the sigmoid function, and $KL(\cdot\\|\cdot)$ is the Kullback-Leibler divergence. The other options include but not limited to symmetric content reconstruction [10] and graph contrastive learning [11]. For modules with supervision information, a projection matrix $\mathbf{R}\in\mathbb{R}^{k\times c}$ is introduced to map the $k$-dimension embedding vector into soft labels with $c$ classes. For the node classification, the loss can be simply set as \[\mathcal{L}_{FT}=KL(\mathbf{Y}\\|\text{softmax}(\mathbf{H}\mathbf{R})), \tag{4}\] where $\mathbf{Y}\in\mathbb{R}^{n\times c}$ is the supervision information for supervised tasks. Note that the above loss is equivalent to the classical softmax regression if $\mathbf{H}$ is constant. The loss could also be link prediction, graph classification, _etc_. Although base modules can utilize diverse losses, we only discuss the situation that all modules use the same kind of loss in this paper for simplicity. ``` Input : Adjacency matrix $\mathbf{A}$, feature matrix $\mathbf{X}$, balance coefficient $\eta$, the number of epochs $K$, $L$ separable GNN modules $\{\mathcal{M}_{t}\}_{t=1}^{L}$, $\mathbf{H}_{0}\leftarrow\mathbf{X}$. Output : Features output by $\mathcal{M}_{L}$. for$i=1,\ldots,K$do # Forward Train Stacked Graph Neural Networks for$t=1,2,\ldots,L-1$do Feed the current features to $\mathcal{M}_{t}$ and reset $\mathbf{U}_{t}$: $\mathbf{X}_{t}\leftarrow\mathbf{H}_{t-1}$, $\mathbf{U}_{t}\leftarrow\mathbf{I}$. if Train with only $\mathcal{L}_{FT}$ at the first forward training $\mathbf{X_{t}}^{\prime}=f_{0}^{(t)}(\mathbf{A},\mathbf{X}_{t})$ # Preprocessing for mini-batch algorithms Compute loss $\mathcal{L}^{(t)}\leftarrow\mathcal{L}_{FT}^{(t)}$ if $i==1$ else $\mathcal{L}_{FT}^{(t)}+\eta\mathcal{L}_{BT}^{(t)}$ Train $\mathcal{M}_{t}$ by optimizing $\min_{\mathbf{W}_{t}}\mathcal{L}^{(t)}$ based on mini-batch algorithms. Obtain the features: $\mathbf{H}_{t}\gets f_{1}^{(t)}(\mathbf{X_{t}}^{\prime},\mathbf{W}_{t})$. $\mathbf{X}_{L}\leftarrow\mathbf{H}_{t}$, $\mathbf{U}_{L}\leftarrow\mathbf{I}$, $\mathbf{X}_{L}^{\prime}=f_{0}^{(L)}(\mathbf{A},X_{L})$. Train $\mathcal{M}_{L}$ by optimizing $\min_{\mathbf{W}_{L},\mathbf{U}_{L}}\mathcal{L}_{FT}^{(L)}$ based on mini-batch algorithms. # Backward Train Stacked Graph Neural Networks for$t=L-1,L-2,\ldots,1$do Compute the expected output feature of $\mathcal{M}_{t}$: $\mathbf{Z}_{t+1}\leftarrow(g_{0}^{})_{t+1}(\mathbf{X}_{t+1},\mathbf{U}_{t+1})$. Train $\mathcal{M}_{t}$ by optimizing $\min_{\mathbf{W}_{t},\mathbf{U}_{t}}\mathcal{L}_{FT}^{(t)}+\eta\mathcal{L}_{BT}^{(t)}$ based on mini-batch algorithms. ``` Algorithm 1Procedure of Stacked Graph Neural Networks Composed of $L$ Modules #### 3.2.3 Backward Training (BT) _The second challenge_ is how to train multiple separable GNNs simultaneously in order to ensure performance. Roughly speaking, the gradients of all layers in neural networks are computed exactly due to the repeated delivery of information by FP and BP. _BP lets the shallow layers perceive the deep ones through the feedback, while the tail modules are invisible to the head ones in FT_. We accordingly design the backward training (BT) for SGNN. To achieve the reverse information delivery, a separable GNN layer is modified by introducing the fully-separability, \[\begin{split}\mathbf{H}&=\hat{f}(\mathbf{A},\mathbf{X},\mathbf{W})= \hat{f}_{1}(\hat{f}_{0}(\mathbf{A},\mathbf{X}),\mathbf{W})\\ &=\hat{f}_{1}(f^{}(\mathbf{A},\mathbf{X},\mathbf{U}),\mathbf{W}),\;\forall\hat{f} \in\mathcal{F}_{k},f^{}\in\mathcal{F}_{d}^{},\end{split} \tag{5}\] where $\mathbf{U}\in\mathbb{R}^{d\times d}$ and $\mathbf{W}\in\mathbb{R}^{d\times k}$. Note that $f^{}(\mathbf{A},\mathbf{X},\mathbf{U})=g_{1}^{}(\mathbf{A},g_{0}^{}(\mathbf{X},\mathbf{U}))$. Clearly, if $\mathbf{U}$ is fixed as a constant, then the modified layer is equivalent to the original separable GNN layer $\hat{f}$. Denote $\mathbf{Z}=g_{0}^{}(\mathbf{X},\mathbf{U})$ and $\mathbf{Z}$ is the _expected features_. Specifically, $\mathbf{Z}_{t}$ is the learned feature during the backward training of $\mathcal{M}_{t}$, and it is also the expected input of $\mathcal{M}_{t}$, _i.e._, from $\mathcal{M}_{t-1}$. In the forward training, the delivery of information is based on the learned features $\mathbf{H}_{t}$, and $\mathbf{Z}$ plays the similar role in the backward training. The loss of backward training attempts to shrink the difference between the output feature $\mathbf{H}_{t}$ of $\mathcal{M}_{t}$ and expected input $\mathbf{Z}_{t+1}$ of $\mathcal{M}_{t+1}$, \[\mathcal{L}_{BT}^{(t)}=d(\mathbf{H}_{t},\mathbf{Z}_{t+1})=d(f_{t}^{}(\mathbf{A},\mathbf{X}_{t}, \mathbf{U}_{t}),\mathbf{Z}_{t+1}). \tag{6}\] Note that $\mathcal{L}_{BT}$ is only activated after the first forward training leading to the final loss of $\mathcal{M}_{t}$ as \[\mathcal{L}^{(t)}=\mathcal{L}_{FT}^{(t)}+\eta\mathcal{L}_{BT}^{(t)}, \tag{7}\] and it is updated during each backward training. The introduction of $\mathbf{Z}$ will not limit the application of stochastic optimization since the expected features can also be sampled at each iteration without restrictions. The procedure is summarized as Algorithm 1. Remark that $\mathbf{U}_{t}$ remains as the identity matrix during FT. This setting leads to each forward computation across $L$ base modules being equivalent to a forward propagation $L$-layer GNN. In other words, an SGNN with $L$ modules can be regarded as a decomposition of an $L$-layer GNN. One may concern that why not to learn $\mathbf{U}_{t}$ and $\mathbf{W}_{t}$ together in FT. In this case, we prefer to use $\mathbf{U}_{t}$ only for learning the expected features of $\mathcal{M}_{t+1}$ and the capability improvement from the co-learning $\mathbf{U}_{t}$ in FT could also be implemented by $\mathbf{W}$, which is equivalent to use GIN [21] as base modules. ### Complexity As each base module is assumed as a separable GNN, both FT and BT of $\mathcal{M}_{t}$ can be divided into two steps, the preprocessing step for graph operation and the training step for parameters learning. Denote the output dimension of $\mathcal{M}_{t}$ as $d_{t}$ and the dimension of original content feature as $d_{0}=d$. The preprocessing to compute $\mathbf{X}_{t}^{\prime}=f_{0}(\mathbf{A},\mathbf{X}_{t})$ requires $\mathcal{O}(\\|\mathbf{A}\\|_{0}d_{t})$ cost. Suppose that each module is trained $E$ iterations and the batch size is set as $m$. Then the computation cost of the training step is $\mathcal{O}(Emd_{t-1}d_{t})$. Note that only the GNN mapping is considered and the computation of the loss is ignored. Overall, the computational complexity of an SGNN with $L$ modules is approximately $\mathcal{O}(K\\|\mathbf{A}\\|_{0}\sum_{t=0}^{L-1}d_{t}+Em\sum_{t=1}^{L}d_{t-1}d_{t})$. Remark that the graph is only used once during every epoch and no sampling is processed on the graph such that the graph structure is completely retained which is unavailable in the existing fast GNNs. The coefficient of $\\|\mathbf{A}\\|_{0}$ is only $K\sum_{t=1}^{L-1}d_{t}$. The space complexity is only $\mathcal{O}(\\|\mathbf{A}\\|_{0}+md_{t-1}d_{t})$. Therefore, the growth of graph scale will not affect the efficiency of SGNN. Due to the efficiency, all experiments can be conducted on a PC with an NVIDIA 1660 (6GB) and 16GB RAM. ## 4 Theoretical Analysis To answer _Q2_ raised in the beginning of Section 3, we discuss the impact of the decoupling in this section. Intuitively speaking, if an $L$-layer GNN achieves satisfactory results, then there exists $\{\mathbf{W}_{t}\}_{t=1}^{L}$ such that an SGNN with $L$ modules could achieve the same results. However, each $\mathcal{M}_{t}$ is trained greedily according to the forward training loss $\mathcal{L}_{FT}$, while middle layers of a multi-layer GNN are trained according to the same objective. The major concern is whether the embedding learned by a greedy strategy leads to an irreversible deviation in the forward training. In this section, we investigate the possible side effects on a specific SGNN comprised of unsupervised modules defined in Eq. (3) with linear activations. The conclusion is not apparent since simply setting $\mathbf{W}$ as an identity matrix does not prove it for GNN due to the existence of $\mathbf{P}$. Remark that a basic premise is that the previous module has achieved a reasonable result. Given a linear separable-GNN module $\mathcal{M}$ which is defined as $\mathbf{H}=\mathbf{P}\mathbf{X}\mathbf{W}\in\mathbb{R}^{n\times k}$, suppose that the forward training uses the reconstruction error $\\|\mathbf{P}-\mathbf{H}\mathbf{H}^{T}\\|$ as $\mathcal{L}_{FT}$. We first introduce the matrix angle to better understand whether the preconditions of Theorem 4.1 are practicable. Definition 4.1.: _Given two matrices $\mathbf{B}_{1},\mathbf{B}_{2}\in\mathbb{R}^{n\times n}$, we define the matrix angle as $\theta(\mathbf{B}_{1},\mathbf{B}_{2})=\arccos\langle\mathbf{B}_{1},\mathbf{B}_{2}\rangle/(\\| \mathbf{B}_{1}\\|\cdot\\|\mathbf{B}_{2}\\|)$._ Before elaborating on theorems, we introduce the following assumption, which separates the discussions into two cases. Assumption 4.1.: $\mathbf{X}\mathbf{X}^{T}$ _does not share the same eigenspace with $\mathbf{P}-\mathbf{X}\mathbf{X}^{T}$._ Note that the above assumption is weak and frequently holds in most cases. For simplicity, $\mathbf{U}_{r}\in\mathbb{R}^{n\times r}$ ($r\in\mathbb{N}_{+}$) is the eigenvectors associated with $r$ leading eigenvalues. Under this assumption, we find that the error of $\mathcal{M}$ is upper-bounded by $\varepsilon$. Theorem 4.1.: _Let $\delta=1-\cos(\theta_{}/2)$ and $\theta_{}=\theta((\mathbf{P}-\mathbf{X}\mathbf{X}^{T})\mathbf{Q},\mathbf{Q}(\mathbf{P}-\mathbf{X}\mathbf{X}^{T }))$ where $\mathbf{Q}=\mathbf{U}_{o}\mathbf{U}_{o}^{T}-\mathbf{I}/2$ and $o=\min(\operatorname{rank}(\mathbf{X}),k)$. Under Assumption 4.1, if $\\|\mathbf{P}-\mathbf{H}\mathbf{H}^{T}\\|=\varepsilon\leq\mathcal{O}(\delta)$ and $\sigma_{}\leq\mathcal{O}(\sqrt{\delta})$ where $\sigma_{}$ is the $(o+1)$-th largest singular value of $\mathbf{X}$, then there exists $\mathbf{W}\in\mathbb{R}^{d\times k}$ so that $\\|\mathbf{P}-\mathbf{H}\mathbf{H}^{T}\\|\leq\varepsilon$. In other words, if $\varepsilon$ is small enough, then $\mathbf{H}\mathbf{H}^{T}$ could be a better approximation than $\mathbf{X}\mathbf{X}^{T}$._ Specially, if $\operatorname{rank}(\mathbf{X})\leq k$ or $k=d$, $\sigma_{}=0$ so that $\sigma_{}\leq\mathcal{O}(\sqrt{\delta})$ holds. From the above theorem, we claim that _the error through $\mathcal{M}$ will not accumulate (i.e., bound by $\varepsilon$)_ provided that the input $\mathbf{X}$, the output of the previous modules, is well-trained. We further provide an upper-bound of error if Assumption 4.1 does not hold. The following theorem shows the increasing speed of error is at most linear with the tail singular-values. Theorem 4.2.: _If Assumption 4.1 does not hold, then there exists $\mathbf{W}\in\mathbb{R}^{d\times k}$ so that $\\|\mathbf{P}-\mathbf{H}\mathbf{H}^{T}\\|\leq\varepsilon+\mathcal{O}(\sigma_{}^{2})$._ Corollary 4.1.: _Given an SGNN with $L$ linear modules $\{\mathcal{M}_{t}\}_{t=1}^{L}$ with $\mathcal{L}_{FT}^{(t)}=\\|\mathbf{P}-\mathbf{H}_{t}\mathbf{H}_{t}^{T}\\|$, if $\mathbf{P}-\mathbf{H}_{1}\mathbf{H}_{1}^{T}$ and $\mathbf{H}_{1}\mathbf{H}_{1}^{T}$ share the same eigenspace, then $\mathcal{L}_{FT}^{(L)}\leq\mathcal{L}_{FT}^{(1)}+\sum_{t=1}^{L-1}\mathcal{O}( \sigma_{}^{2}(\mathbf{H}_{t}))$._ Based on Theorem 4.2, we conclude that the residual would not accumulate rapidly when Assumption 4.1 does not hold. All proofs and more discussions are put in Section B-D. ## 5 Experiments In this section, we conduct experiments to investigate _whether the performance of SGNN could approach the performance of the original $L$-layer GNN in a highly-efficient way_ and _what the impact of the non-linearity and flexibility brought by the decoupling is_. To sufficiently answer the above 2 problems, both node clustering and semi-supervised node classification are used. Due to the limitation of space, only 4 common datasets, including Cora, Citeseer, Pubmed [47], and Reddit [14]. Cora and Citeseer contain thousands of nodes. Pubmed has nearly 20 thousands nodes and Reddit contains more than 200 thousands nodes, which are middle-scale and large-scale datasets respectively. The details of four common datasets are shown in Table 1. More experiments on OGB datasets can be found in Section 5.4. ### Node Clustering #### 5.1.1 Experimental Settings We first testify the effectiveness of SGNN on the node clustering. We compare our method against 10 methods, including a baseline clustering model K-means, three GCN models without considering training efficiency (GAE [9], ARGA [48], MGAE [37]), and six fast GCN models with GAE-loss (GraphSAGE [14], FastGAE [15], ClusterGAE [18], AGC [49] (an unsupervised extension of SGC [19]), S${}^{2}$GC [20], and GAE-S${}^{2}$GC). The used codes are based on the publicly released implementation. To ensure fairness, all multi-layer GNN models consist of two layers and SGNN is comprised of two modules. For models that could be trained by stochastic algorithms, the size of mini-batch is set as 128. The learning rate is set as 0.001 and the number of epochs is set as 100. We set the size first layer as 128 and the second layer \begin{table} \begin{tabular}{l c c c c c} \hline Dataset & Nodes & Edges & Classes & Features & Train / Val / Test Nodes \\ \hline \hline Cora & 2,708 & 5,429 & 7 & 1,433 & 140 / 500 / 1,000 \\ Citeseer & 3,327 & 4,732 & 6 & 3,703 & 120 / 500 / 1,000 \\ Pubmed & 19,717 & 44,338 & 3 & 500 & 60 / 500 / 1,000 \\ Reddit & 233K & 11.6M & 41 & 602 & 152K / 24K / 55K \\ \hline \end{tabular} \end{table} Table 1: Data Information size as 64. In particular, when training every module of SGNN, 20% entries of the adjacency matrix are ignored. $\mathbf{U}$ is initialized as an identity matrix, and $\eta$ is set as $10^{3}$ by default. The number of backward training is set as 5 or 10. To investigate the effectiveness of backward training, we report the experimental results with sufficient training for each module, which is denoted by SGNN-FT, while SGNN with the proposed backward training is represented by SGNN-BT. To study the performance of SGNN with different base models, we choose a fully-separable GNN, S${}^{2}$GC, as the base model and this method is marked as SGNN-S${}^{2}$GC. As S${}^{2}$GC does not use the GAE framework, we also added a competitor, namely GAE-S${}^{2}$GC, which uses S${}^{2}$GC as the encoder, to ensure fairness. For SGNN-S${}^{2}$GC and GAE-S${}^{2}$GC, all extra hyper-parameters are simply set according to the original paper of S${}^{2}$GC for both node clustering and node classification. We do not tune the hyper-parameters of S${}^{2}$GC manually. All methods are run five times and the average scores are recorded. The result are summarized in Table 2. Figure 3: Visualization of SGNN comprised of 3 modules and 3-layer GCN on node classification. For SGNN, the output of $\mathcal{M}_{3}$ is visualized. For GCN, the output of the final GCN-layer is visualized. Figure 2: Performance and training efficiency of several scalable GNNs. The efficiency metric is computed by “Consuming Time / # Iterations”. The consuming time begins from loading data into RAM. The first line shows the result of node clustering and the second line is the result of node classification. #### 5.1.2 Performance From Table 2, we find that SGNN outperforms in most datasets. If the released codes could not run on Reddit due to out-of-memory (OOM), we put the notation "N/A" instead of results. In particular, SGNN-BT obtains good improvements on Reddit with high efficiency. Specifically speaking, it is about 8% higher than the well-known GraphSAGE. SGNN-FT performs above the average on some datasets. It usually outperforms GraphSAGE but fails to exceed SGC. Due to the deeper structure caused by multiple modules, the performance of SGNN excels the simple GAE. It also outperforms SGC due to more non-linearity brought by multiple modules. Note that S${}^{2}$GC and SGC are strong competitors, while SGNN can easily employ them as a base module since they are separable. From the ablation experiments, SGNN-BT works better than SGNN-FT, which indicates the necessity of the backward training. We also investigate how the number of modules $L$ affects the node clustering accuracy and the results averaged over 5 runs are reported in Table 5. To ensure fairness, we also show the performance of GAE with the same depth though deeper GCN and GAE usually return unsatisfied results. Note that the neurons of each layers are set as $[256,128,64,32,16,16,16]$, respectively. It is not hard to find that SGNN is robust to the depth compared with the conventional GNNs. #### 5.1.3 Efficiency Figure 2 shows the consuming time of several GNNs with higher efficiency on Pubmed and Reddit. Instead of neglecting the preprocessing operation, we measure the efficiency through a more rational \begin{table} \begin{tabular}{l c c c} \hline \hline Datasets & Cora & Citeseer & Pubmed \\ \hline \hline GAT & 83.0 $\pm$ 0.7 & 72.5 $\pm$ 0.7 & 79.0 $\pm$ 0.3 \\ DGI & 82.3 $\pm$ 0.6 & 71.8 $\pm$ 0.7 & 76.8 $\pm$ 0.6 \\ GCNII & 85.5 $\pm$ 0.5 & 73.4 $\pm$ 0.6 & 80.2 $\pm$ 0.4 \\ FastGCN${}^{\dagger}$ & 79.8 $\pm$ 0.3 & 68.8 $\pm$ 0.6 & 77.4 $\pm$ 0.3 \\ GraphSAGE${}^{\dagger}$ & 77.4 $\pm$ 0.6 & 66.7 $\pm$ 0.3 & 79.0 $\pm$ 0.3 \\ Cluster-GCN${}^{\dagger}$ & 65.3 $\pm$ 3.9 & 57.7 $\pm$ 2.3 & 71.5 $\pm$ 1.9 \\ \hline GCN & 81.4 $\pm$ 0.4 & 70.9 $\pm$ 0.5 & 79.0 $\pm$ 0.4 \\ SGC & 81.0 $\pm$ 0.0 & 71.9 $\pm$ 0.1 & 78.9 $\pm$ 0.0 \\ SGNN-BT & 81.4 $\pm$ 0.5 & 70.2 $\pm$ 0.4 & 77.8 $\pm$ 0.3 \\ \hline S${}^{2}$GC & 83.5 $\pm$ 0.0 & 73.6 $\pm$ 0.1 & 80.2 $\pm$ 0.0 \\ SGNN-S${}^{2}$GC & 83.8 $\pm$ 0.2 & 73.2 $\pm$ 0.3 & 80.2 $\pm$ 0.4 \\ \hline \hline \end{tabular} \begin{tabular}{l c c} \hline \hline Methods & Reddit \\ \hline \hline GAT & N/A \\ DGI & 94.0 \\ SAGE-mean & 95.0 \\ SAGE-GCN & 93.0 \\ FastGCN & 93.7 \\ Cluster-GCN & 96.6 \\ \hline GCN & N/A \\ SGC & 94.9 \\ SGNN-BT & 95.10 $\pm$ 0.02 \\ \hline S${}^{2}$GC & 95.3 \\ SGNN-S${}^{2}$GC & 95.28 $\pm$ 0.03 \\ \hline \hline \end{tabular} \end{table} Table 3: Test accuracy (%) of supervised SGNN averaged over 10 runs on 3 citation datasets \begin{table} \begin{tabular}{l\|c c c c c c c} \hline \hline Datasets & \multicolumn{2}{c\|}{Cora} & \multicolumn{2}{c\|}{Citeseer} & \multicolumn{2}{c\|}{PubMed} & \multicolumn{2}{c}{Reddit} \\ & ACC & NMI & ACC & NMI & ACC & NMI & ACC & NMI \\ \hline \hline K-Means & 0.4922 & 0.3210 & 0.5401 & 0.3054 & 0.5952 & 0.2780 & 0.1927 & 0.2349 \\ ARGA & 0.6400 & 0.4490 & 0.5730 & 0.3500 & 0.6807 & 0.2757 & N/A & N/A \\ MGAE & 0.6806 & 0.4892 & 0.6691 & 0.4158 & 0.5932 & 0.2957 & N/A & N/A \\ GraphSAGE & 0.6163 & 0.4826 & 0.5664 & 0.3425 & 0.5554 & 0.0943 & 0.6225 & 0.7291 \\ FastGAE & 0.3527 & 0.1553 & 0.2672 & 0.1178 & 0.4262 & 0.0442 & 0.1115 & 0.0715 \\ ClusterGAE & 0.4579 & 0.2261 & 0.4182 & 0.1767 & 0.3913 & 0.0001 & N/A & N/A \\ \hline GAE & 0.5960 & 0.4290 & 0.4080 & 0.1760 & 0.6861 & 0.2957 & N/A & N/A \\ AGC (SGC) & 0.6892 & 0.5368 & 0.6700 & 0.4113 & 0.6978 & 0.3159 & 0.5833 & 0.6894 \\ SGNN-FT & 0.6278 & 0.5075 & 0.6141 & 0.3776 & 0.6444 & 0.2312 & 0.5943 & 0.7156 \\ SGNN-BT & 0.7463 & 0.5546 & 0.6730 & 0.4159 & 0.6951 & 0.3337 & 0.7042 & 0.7601 \\ \hline S${}^{2}$GC & 0.6960 & 0.5471 & 0.6911 & 0.4287 & 0.7098 & 0.3321 & 0.7011 & 0.7509 \\ GAE-S${}^{2}$GC & 0.6976 & 0.5317 & 0.6435 & 0.3969 & 0.6528 & 0.2452 & 0.6272 & 0.7158 \\ SGNN-S${}^{2}$GC & 0.7223 & 0.5404 & 0.6822 & 0.4243 & 0.7084 & 0.3302 & 0.7023 & 0.7575 \\ \hline \hline \end{tabular} \end{table} Table 2: Node clustering results way. We record the totally consuming time after loading data into RAM and then divide the total number of updating parameters of GNNs. The measurement could reflect the real difference of diverse training techniques aiming to apply batch-based algorithms to GNN. It should be emphasized the reason why SGC is worse than SGNN regarding the consuming time. The key point is the different costs of their _preprocessing_ operation. For an $L$-order SGC, the computation cost of $\mathbf{P}^{L}\mathbf{X}$ is at least $\mathcal{O}(\\|\mathbf{A}\\|_{0}\,Ld)$ while SGNN with $L$ first-order modules totally requires $\mathcal{O}(\\|\mathbf{A}\\|_{0}\sum_{i=1}^{L}d_{i})$ for the same preprocessing operation. The metric also provides a fair comparison between SGC and other models since the stopping criteria are always different. ### Node Classification #### 5.2.1 Experimental Setting We also conduct experiments of semi-supervised classification on four datasets. The split of datasets follows [19] which is shown in Table 1. We compare SGNN against GCN [4], GAT [6], DGI [50], APPNP [51], L2-GCN [34] FastGCN [15], GraphSAGE [14], Cluster-GCN [18], SGC [19], GCNII [52], and S${}^{2}$GC [20]. Similarly, we testify SGNN with two different base models, namely SGNN-BT and SGNN-S${}^{2}$GC. For GraphSAGE, we use the mean operator by default and some notations are added if the extra operators are used. On citation networks, the learning rate is set as $0.01$, while Figure 4: Visualization of a trained SGNN comprised of 3 modules on node classification of Cora and Citeseer. The first line shows the visualization of Cora and the bottom line shows the visualization of Citeseer. Figure 5: Visualization of a trained SGNN comprised of 3 modules on node clustering of Cora and Citeseer. The first line is visualization on Cora and the second line is visualization on Citeseer. it is $10^{-4}$ on Reddit. Since the nodes for training are less than 200 on citation networks, we use all training points in each iteration for all methods while we sample 256 points as a mini-batch for approaching expected features during backward training of SGNN. On Reddit, the batch size of all batch-based models is set as 512. We do not apply the early stopping criterion used in [4] and the max iteration follows the setting of SGC. The embedding dimensions of each module are the same as the setting in node clustering. For the sake of fairness, we report the results obtained by SGNN with two modules using first-order operation. The forward training loss is defined in Eq. (4). Moreover, all compared models share an identical implementation of their mini-batch iterators, loss function and neighborhood sampler (when applicable). The balance coefficient of $\mathcal{L}_{FT}$ and $\mathcal{L}_{BT}$ is set as $1$ by default. We report the results averaged over 10 runs on citation datasets and 5 runs on Reddit in Table 3 and Table 4. The hyper-parameters are shared for different datasets which are optimized on Cora. #### 5.2.2 Performance The results of compared methods in Table 3 are taken from the corresponding papers. When the experimental results are missed, we run the publicly released codes and the corresponding records are superscripted by $\dagger$. From Tables 3 and 4, we conclude that SGNN outperforms the models with neighbor sampling such as GraphSAGE, FastGCN, and ClusterGCN on citation networks and the performance of SGNN exceeds most models on Reddit. On simple citation networks, SGNN loses the least accuracy compared with other batch-based models, which is close to GCN. Owing to the separability of each module, the batch sampling requires no neighbor sampling and causes no loss of graph information. Note that we simply employ the single-layer GCN as the base modules in our experiments, while some high-order methods that obtain competitive results are also available for SGNN. Although some methods achieve preferable results, they either fail to run or obtain unsatisfactory results on large-scale datasets. Besides, we also show the comparison of efficiency on node classification task in Table 2. Figure 6: t-SNE Visualization of the output of $\mathcal{M}_{1}$ from a trained SGNN comprised of 3 modules on node classification of Citeseer. Figure 7: Convergence curve of the _final loss_. The order of SGC is set as $L$. The backward training significantly decreases the final loss and the non-linearity also plays an important role. #### 5.2.3 Visualization to Show Impact of the Decoupling In Figure 3, we visualize the output of a 3-module SGNN and a 3-layer GCN to directly show that _the decoupling would not cause the trivial features_, which corresponds to the theoretical conclusion in Section 4. To show the benefit of the non-linearity brought by SGNN and the backward training, the convergence curves of SGC, SGNN-FT, and SGNN-BT are shown in Figure 7. Note that the figure shows the variation of the _final loss_. In SGNN, the final loss is the loss of $\mathcal{M}_{L}$, while it is the unique training loss in SGC. SGC with $L$-order graph operation is used. From this figure, we can conclude that: (1) The non-linearity does lead to a better loss value; (2) The backward training significantly decreases the loss. In summary, _the decoupling empirically does not cause the negative impact._ ### Visualization We also provide more visualizations in Figures 4 and 5. We run SGNN with 3 GNN modules and visualize the input and output of $\mathcal{M}_{1}$, $\mathcal{M}_{2}$, and $\mathcal{M}_{3}$ through $t$-SNE on Cora and Citeseer, for node clustering and node classification. The purpose of these two figures is to empirically investigate _whether the decoupling would cause the accumulation of residuals and errors_. The experimental results support the theoretical results that are provided in Section 4. One may concern the impact of $\eta$ (trade-off coefficient between $\mathcal{L}_{FT}$ and $\mathcal{L}_{B}T$) on the performance. We testify SGNN with different $\eta$ from $\{10^{-5},10^{-3},10^{-1},10^{1},10^{3},10^{5}\}$ and find that $\eta=10^{3}$usually leads to good results. Accordingly, we only report results SGNN with $\eta=10^{3}$ in this paper. Moreover, we show the impact of $\eta$ to node clustering on Cora and Citeseer in Figure 8. Moreover, we show the output of $\mathcal{M}_{1}$ of different periods in Figure 6, in order to show the impact of the backward training. From the figure, we find that BT indeed affects the latent features, which is particularly apparent in Figure 6d. It should be emphasized that we only use the simple single-layer GCN as the base module of SGNN. The performance can be further improved by incorporating different models such as GCNII, GIN, _etc._ In particular, we only tune hyper-parameters on Arxiv, and we simply report results of SGNN with settings from Reddit. ## 6 Conclusion and Future Works In this paper, we propose the Stacked Graph Neural Networks (SGNN). We first decouple a multi-layer GNN into multiple simple GNNs, which is formally defined as separable GNNs in our paper to ensure the availability of batch-based optimization without loss of graph information. The bottleneck of the existing stacked models is that the information delivery is only unidirectional, and therefore a backward training mechanism is developed to make the former modules perceive the latter ones. We also theoretically prove that the residual of linear SGNN would not accumulate in most cases for unsupervised graph tasks. The theoretical and experimental results show that the proposed framework is more than an efficient method and it may deserve further investigation in the future. The theoretical analysis focuses on linear SGNN and the generalization bound is also not investigated in this paper. Therefore, they will be the core of our future work. Moreover, as $\mathcal{L}_{FT}$ could be any losses, how to choose the most appropriate loss for each module will be also a crucial topic in our future works.
2305.02623	Probabilistic neural networks for improved analyses with phenomenological models	Physics models typically contain adjustable parameters to reproduce measured data. While some parameters correspond directly to measured features in the data, others are unobservable. These unobservables can, in some cases, cause ambiguities in the extraction of observables from measured data, or lead to questions on the physical interpretation of fits that require these extra parameters. We propose a method based on deep learning to extract values of observables directly from the data without the need for unobservables. The key to our approach is to label the training data for the deep learning model with only the observables. After training, the deep learning model can determine the values of observables from measured data with no ambiguities arising from unobservables. We demonstrate this method on the phenomenological R-matrix that is widely utilized in nuclear physics to extract resonance parameters from cross section data. Our deep learning model based on Transformers successfully predicts nuclear properties from measurements with no need for the channel radius and background pole parameters required in traditional R-matrix analyses. Details and limitations of this method, which may be useful for studies of a wide range of phenomena, are discussed.	C. H. Kim, K. Y. Chae, M. S. Smith, D. W. Bardayan, C. R. Brune, R. J. deBoer, D. Lu, D. Odell	2023-05-04T07:53:57Z	http://arxiv.org/abs/2305.02623v3	# Eliminating unobservable parameters in physics models using deep learning ###### Abstract Physics models typically contain adjustable parameters to reproduce measured data. While some parameters correspond directly to measured features in the data, others are unobservable. These unobservables can, in some cases, cause ambiguities in the extraction of observables from measured data, or lead to questions on the physical interpretation of fits that require these extra parameters. We propose a method based on deep learning to extract values of observables directly from the data without the need for unobservables. The key to our approach is to label the training data with only the observables. After training, the deep learning model can determine the values of observables from measured data with no ambiguities arising from unobservables. We demonstrate this method on the phenomenological _R_-matrix that is widely utilized in nuclear physics to extract resonance parameters from cross section data. Our deep learning model based on Transformers successfully predicts nuclear properties from measurement with no need for the channel radius and background pole parameters required in traditional _R_-matrix analyses. Details and limitations of this method, which may be useful for studies of a wide range of phenomena, are discussed. Models are routinely used to understand physical systems by fitting their predictions to measurement data and extracting "best fit" values for the model parameters. In some cases, these parameters correspond directly to system observables, i.e., features that can be measured. In other cases, the parameters do not correspond to measurable features, but rather may codify approximations made in the model or may be part of the framework of the model itself. The presence of these unobservable parameters can often lead to ambiguities in the extraction of observables from measured data or lead to questions on the physical interpretation of fits that require these extra parameters [1; 2; 3]. When possible, it is advantageous to extract values of observable parameters directly from features in the data without the need for unobservables. One of the strengths of deep learning is the ability to identify features in data sets. For this reason and others, deep learning has been successfully used in a variety of science fields [4; 5; 6; 7; 8; 9; 10; 11; 12]. In this study, we propose to use deep learning to extract values of observable parameters directly from measured data sets, bypassing the need for unobservable parameters that can introduce ambiguities and increase uncertainties in the results. Fig. 1 shows a sketch of the method. First, training data is generated by running a physics model $\mathcal{P}$ with various combinations of the observable $\theta_{obs}$ and unobservable $\theta_{un}$ parameter inputs. The outputs can be processed with a noise function $\mathcal{N}$ to simulate the noise present in experimental measurements. Then, a deep learning model (neural network) $\mathcal{D}$ is trained on this data to find particular observable features in the output of the physics model. The key to our method is to label the training data with _only_ the observable parameter values, so that the deep learning model finds the features without any knowledge of the unobservable parameters. By training over data generated with a wide range of unobservable parameter values, we ensure that the deep learning model works on data with any values of unobservable parameters. After training, the deep learning model is used to directly predict the values of observable parameters from measured data $\mathcal{M}$. While a traditional analysis of the measured data with the physics model requires unobservable parameters to be specified, the deep learning model does not require the knowledge of the unobservables at the inference stage (Fig. 1c). We demonstrate this method on the phenomenological _R_-matrix that is being critically used in nuclear physics [1; 3; 13; 14; 15; 16; 17; 18; 19]. The phenomenological _R_-matrix serves as a good example because it contains numerous and somewhat arbitrary unobservable parameters. It is based on separating the particle interaction space into internal and external regions. The complex many-body nuclear interactions are present in the internal region, whereas only the Coulomb interaction is considered in the external region. The boundary surface is defined by a sphere of radius known as the channel radius. The internal interactions are described by the _R_-matrix that is characterized by parameters of nuclear resonant states. From the _R_-matrix, expected observational data such as cross sections can be calculated. The parameters of nuclear states are determined by comparing the expected and experimental data. The extracted parameters provide the related nuclear properties, and the cross sections can be reproduced using the parameters for the purpose of extrapolation. The main drawback of the phenomenological _R_-matrix arises from the dependency of fitted physical parame ters on rather arbitrary choices of unobservable parameters [1; 3]. The _R_-matrix theory requires an infinite number of nuclear states in the calculation. However, the truncation of the number of states is necessary for the phenomenological approach, as generally only a limited number of nuclear states are known. The truncated states are compensated by the so-called background poles that are artificial states included to reproduce the effects of the truncated states. The strength of the background poles depends on the channel radius, which creates additional complications [1; 3; 13]. In practice, there are no strict physics rules to select valid choices of the channel radius and background poles. Different choices produce different fitting results because they create different theoretical predictions. This feature has caused difficulties in the phenomenological analysis and uncertainties of the results [1; 3]. We utilize the proposed method to extract observables without the requirement of unobservables used in the phenomenological $R$-matrix. One of the important applications of the phenomenological $R$-matrix is determining resonance parameters (e.g., spins $J$, parities $\pi$, energies $E$, widths $\Gamma$) in nuclear reaction data. We built a deep learning model to find patterns of resonance parameters in reaction cross section spectra. Specifically, our goal was to extract the parameters of three resonance states from the well-known ${}^{12}$C+$p$ elastic scattering measurement of ref. [21] without using any information on the channel radius and background poles. The training data for this task was prepared by the following specific process. We sampled possible sets of resonance parameters, along with random values of the channel radius ($a_{c}$) and random configurations of background poles (BGPs). We assumed a general situation where previous studies have partly determined the reso Figure 1: Process of the current method to eliminate the unobservable parameters. The current method starts from (a) training data generation for (b) deep learning modeling to (c) inference on target measurement. The model requires only measurement data to extract physical objects at the inference stage. \begin{table} \begin{tabular}{c c c} \hline Parameters & Data Distribution & Model Performance \\ \hline $J_{1}^{\pi}$ & $\frac{1}{2}^{+}$ (Fixed) & _J_: -, $\pi$: - \\ $J_{2}^{\pi}$ & $l\leq 2$ & _J_: 98.6 \%, $\pi$: 100.0 \% \\ $J_{3}^{\pi}$ & $l=2$ & _J_: 99.0 \%, $\pi$: - \\ $E_{1}$ (MeV) & Normal(0.425, 0.003) & 0.00014 (0.03 \%) \\ $E_{2}$ (MeV) & Normal(1.565, 0.010)${}^{3}$ & 0.00118 (0.08 \%) \\ $E_{3}$ (MeV) & Normal(1.610, 0.006)${}^{3}$ & 0.00057 (0.04 \%) \\ $\Gamma_{1}$ (keV) & Normal(30, 3) & 0.20 (0.67 \%) \\ $\Gamma_{2}$ (keV) & Normal(50, 15) & 1.98 (4.44 \%) \\ $\Gamma_{3}$ (keV) & Normal(60, 8) & 1.23 (2.08 \%) \\ \hline ANC (fm${}^{-1/2}$)1 & Normal(1.81, 0.07) & - \\ $a_{c}$ (fm) & 3 - 8 & - \\ 5 BGPs [$J^{\pi}$] & $l\leq 2$ & - \\ 5 BGPs [$E$] & Uniform(3, 10) & - \\ 5 BGPs [$\Gamma$] & Uniform(0, $\Gamma_{W}$)2 & Uniform(0, $\Gamma_{W}$)2 \\ \hline \end{tabular} \end{table} Table 1: Training data distribution and test results of model. Model performance is presented with accuracy and mean errors. Median values are taken to calculate the errors. The values in the parentheses are percentage errors. In total, 5 BGPs are included. nance parameters. Table 1 shows the training data distributions of the parameters. Here, $l$ is the relative orbital angular momentum of the particle pair. Possible spins and parities with the given $l$ were uniformly sampled: e.g., $J=0.5-2.5$ and $\pi=+,-$ for $l\leq 2$. We set the range of the channel radius to 3-8 fm that contains typical values being used in traditional $R$-matrix calculations. A level for each possible spin-parity was included as a background pole. The location and width of these levels were randomly determined within the given ranges. We calculated cross section spectra corresponding to each set of parameters using $R$-matrix theory [16]. As the calculated spectra do not carry experimental noise, we added random noise to the spectra to reflect the measurement errors given by ref. [21]. We gave the log of the differential cross sections as the inputs to the model, while the labels are the corresponding (encoded) values of parameters $J$, $\pi$, $\log(E)$, and $\log(\Gamma)$ of three resonance states. Transformers are currently one of the most effective deep learning architectures in a variety of applications [22; 23; 24; 25; 26; 27; 28]. We modify a Transformer to be suitable for this task. Details of the implemented architecture are described in the Methods section. Here, we summarize the primary specifications: First, as the input data are differential cross section spectra at three different angles, an extra dimension for the angles is needed in addition to the 2-dimensional cross section spectrum. Relations between the spectra of different angles are important in the analysis. However, Transformers consist of matrix multiplications that are operated only at the last two dimensions and are therefore limited within a spectrum at each angle. To accommodate this, we replaced the decoder input with the encoder input that is shifted in the angle dimension. By doing so, the matrix multiplications in the decoder are also operated between spectra at different angles. Second, because the different nature of our four parameters $J$, $\pi$, $E$, and $\Gamma$, we split the model output into four branches where two of them predict discontinuous values (classification) for $J$ and $\pi$, and the others predict continuous values (regression) for $E$ and $\Gamma$. Figure 2: Errors of the model predictions for cross sections calculated with various unobservables. Res 1, 2, and 3 represent the first, second, and third resonances. The figures of the first background pole are given as representations. Using deep ensembles, probabilistic deep learning is also used to quantify uncertainties of model predictions [29; 30; 31]. The idea of deep ensembles is straightforward and highly effective, featuring the use of multiple point-estimate models which are trained, using a negative log-likelihood loss function, from random initial model parameters and shuffled data samples [29; 32; 33]. The model prediction $\mathbf{y}$ on given input data $\mathbf{x}$ can be obtained from the predictions of these multiple models: \[p(\mathbf{y}\,\|\,\mathbf{x})=\frac{1}{N_{\text{model}}}\sum_{n=1}^{N_{\text{model}}}p( \mathbf{y}\,\|\,\mathbf{x},\mathbf{\theta}_{n}), \tag{1}\] where $\mathbf{\theta}_{n}$ represents parameters of the $n$th model. The predictive probability for the classification of $J$ or $\pi$ is, therefore, the average of the probabilities predicted by the models. For the regression of $\log(E)$ or $\log(\Gamma)$, we assume that the likelihoods $p(\mathbf{y}\,\|\,\mathbf{x},\mathbf{\theta})$ follow Gaussian distributions. The predictive distribution $p(\mathbf{y}\|\mathbf{x})$ is further approximated as a Gaussian for simplicity, where its mean and variance are those of the mixture of the Gaussian likelihoods [31]. The final distribution is equivalent to the lognormal distribution of $E$ or $\Gamma$, which has several well-known advantages for representing positive-definite quantities [34]. More details of deep ensembles and the training process are presented in the Methods section. The last column in Table 1 shows the average accuracies and errors of the trained model on the test dataset. The results show that the method and current model are quite effective. We also tested the model performance using the values of unobservable parameters used to calculate the cross sections during the training data generation. Fig. 2 shows errors of the predicted parameters for cross sections calculated with different values of eliminated parameters. The model predictions display little dependence - less than $\sim$2% and $\sim$1 keV for classification and regression, respectively - on these parameters. This demonstrates that the deep learning model performs well on data with any values of unobservable parameters. By using more training data or fine-tuning model hyperparameters, this performance can be further improved if needed. Fig. 3 shows a calibration of the prediction uncertainty. For discrete parameters $J$ and $\pi$, the confidence level is the predicted probability from the softmax function, and the accuracy is the fraction of correct predictions. For continuous parameters $E$ and $\Gamma$, the confidence levels are calculated from their predicted ranges surrounding their median values, and the accuracies are the fraction of true data points included in the corresponding ranges. The dashed black line shows the ideal case where the confidence level is equal to the accuracy. It is clear that our probabilistic model achieves a performance close to Figure 3: Reliability diagram of the current model. Each curve shows the estimated accuracy (fraction of true data points included in the confidence range) as a function of the corresponding confidence level. It is the standard method to evaluate the calibration of uncertainty. Figure 4: Predictions of the $E$ and $\Gamma$ values from the measurement data. These lognormal distributions closely approximate Gaussian distributions. Previous results in the literature – M76 [21], A10 [16], D10 [1] – are also shown, where ref. [1] gave results using three different values of the channel radius. this ideal case. Fig. 4 shows the model predictions on the measurement data of ref. [21], along with the previous results of _R_-matrix analyses on the same data to extract $E$ and $\Gamma$[16; 21; 1]. Our predictions of the spin-parities of second and third states are, at $\sim$100% probability, $J_{2}=3/2$, $\pi_{2}=-$, and $J_{3}=5/2$, in agreement with the literature [35]. Our predictions of $E$ and $\Gamma$ are for the most part consistent with those of the previous studies. These consistent results on the well-known reaction data demonstrate the effectiveness of our model. While the previous works used different channel radius values and background pole configurations, the determinations of those parameters are not made in the current work. Our results are also tabulated in Table 2. The current method is applicable to a wide variety of physics models. In this study, we focused on eliminating unobservable parameters, but this approach works for any subset of the parameters of a model. Interpolations or extrapolations of measured data are also possible by properly configuring the outputs of the deep learning model. Our approach of eliminating parameters could be used both when building generalized models that can address a variety of problems, as well as when building pre-trained models that need only to be slightly adjusted for a specific task [23; 36]. For cases where the values of the unobservable parameters in the physics model are unknown, the performance of the deep learning model - that is, its ability to accurately accurately predict observable parameters - can exceed that of the physics model. If the unobservables in the physics model could (by some means) be correctly specified, then the performance of the deep learning and physics models are comparable. The deep learning model can accurately identify distinctive features from the data, features that arise from variations of the observable parameters of the model. In cases where variations of unobservable physics model parameters can impact data features, the deep learning model predictions may have high uncertainties with diminished performance. By adding more training data samples or fine-tuning the deep learning model, these uncertainties can be reduced [37; 38; 39]. There are some additional benefits of our deep learning approach beyond the elimination of unobservable parameters. First, once a deep learning model is trained, the prediction calculation is normally much faster than typical Bayesian inference approaches that require a large number of Monte Carlo samplings [40; 41; 9]. Second, as shown in Fig. 3, the confidence level of the uncertainties in the deep learning model predictions can be explicitly tested through an analysis of a subset of data (a test dataset) [29; 32; 42; 43]. When extracting the values of observable model parameters from measured data, ambiguities can arise in some instances due to the presence of unobservable parameters. We demonstrate that a deep learning model can be trained without specifying such unobservable parameters by extracting values of observables directly from features in the dataset. By training the deep learning model without information on unobservables, these parameters are effectively eliminated. We demonstrate the effectiveness of this method on the phenomenological _R_-matrix model using a deep learning model based on Transformers. We trained this model to directly extract, from ${}^{12}$C+$p$ scattering measurement data, values of the observable parameters - resonance energies, widths, spins, and parities - with no need to specify values of the unobservable channel radius and background pole parameters used in traditional _R_-matrix analyses. We performed uncertainty quantification in deep learning using a probabilistic framework that provides calibrated uncertainties as tested on a subset of data samples. Our approach has broad applicability in physics to remove unobservables, or other subsets of model parameters, in analyzing datasets. Our approach has broad applicability to remove unobservables, or other subsets of model parameters, when analyzing data. We hope that this deep learning methodology can increase the discovery potential in a variety of fields. ## Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (Grants No. 2016R1A5A1013277 and No. 2020R1A2C1005981). This work was also supported in part by the Institute for Basic Science (Grant No. IBS-R031-D1), and by the U.S. Dept. of Energy (DOE), Office of Science, Office of Nuclear Physics under contract DE-AC05-00OR22725. Computational works for this research were performed on the data analysis hub, Olaf in the IBS Research Solution Center. We thank D. Phillips for various discussions about this study. ## Methods ### Data Preparation To generate the training data, we set ranges of the resonance parameters where the true parameter values are possibly located. These ranges can be obtained from previous studies. Alternatively, one can assume physically probable ranges. For example, one knows that a reaction with a high $l$ value is largely suppressed by the centrifugal barrier, which allows an assumption of a $l$ value less than a particular number. These assumptions prevail in the conventional analysis, where one does not try fitting with every possible $l$ value but only some that are physically probable. Each set of parameters was used to calculate the corresponding cross section using the _R_-matrix code, AZURE2 [16]. The target measurement data has differential cross section values on certain energy bin points at three different angles [21]. The calculated cross section values on the same bin points at the three angles were taken as the input. As the locations of bin points and values of angles were fixed, they were null values in training and not included in the data. The input data samples were, therefore, 3-dimensional tensors with the shape of $(n_{\rm angle},n_{\rm bin},1)$. $n_{\rm angle}$ is the number of measured angles, $n_{\rm bin}$ is the number of bin points, and 1 represents the value of the cross section. A cross section value corresponds to a word in the natural language processing using the Transformer (see the next section) [22]. We added random noises in the calculated cross section values assuming Gaussian noises to reflect the measurement noises. The standard deviations of the Gaussians were obtained from the measurement error bars in ref. [21]. The addition of measurement noises is flexible so that any type of noises can be added. Non-Gaussian noises or noise on measured energy points can be handled by simply adding such noises to the training data. ### Model Architecture The base architecture of the current model is from the Transformer [22]. Details of its description can be found in ref. [22]. Here, we briefly summarize its main features. The Transformer was originally constructed to handle sequence data for natural language processing. For translation purposes, the input and output are, for example, sentences from two different languages. The architecture is based on an encoder-decoder structure. Both encoder and decoder consist of multiple layers of multi-head attentions, fully connected feed-forward networks, residual connections, and layer normalizations [44; 45]. A multi-head attention splits the incoming data and applies multiple attention functions known as scaled dot-product attentions to let the model attend to the data points at multiple aspects [22]. An encoder layer contains a multi-head self-attention and feed-forward network. On the other hand, a decoder layer contains an additional multi-head attention known as an encoder-decoder attention performing with the encoder output. The attentions find critical relations between two sequence data, known as a query and key, and use them to update another sequence data, known as a value. Queries, keys, and values in the self-attentions originate from the input. On the other hand, in the encoder-decoder attentions, queries originate from the decoder input, and keys and values originate from the encoder output. The obtained relations are often called attention (score) matrices, and they present locations of data points critical for the model predictions. Fig. 5 shows examples of the attention matrices from the current model in the prediction of the target measurement data. The description of modifications to the original architecture is following. The inputs to the encoder and decoder are originally embedded and then passed to the positional encoding function to convert the sentences to vectors and encode the positional information of each word [22]. In this work, we removed such functions as the inputs are not exactly sentences but spectra with continuous values. We simply replaced the embedding and positional encoding with a linear layer that projects \begin{table} \begin{tabular}{c c c c c c c} \hline \hline Parameters & Current Model & Ref. [21]1 & Ref. [16] & Ref. [1]1 & Ref. [1]1 & Ref. [1]1 & Ref. [1]1 & Ref. [1]1 & Ref. [1]1 \\ \hline $E_{1}$ (MeV) & 0.4247(2) & 0.424 & 0.426(3) & 0.427 & 0.427 & 0.427 \\ $E_{2}$ (MeV) & 1.5592(12) & 1.558 & 1.556(1) & 1.560 & 1.559 & 1.558 \\ $E_{3}$ (MeV) & 1.6023(7) & 1.604 & 1.602(2) & 1.603 & 1.604 & 1.606 \\ $\Gamma_{1}$ (keV) & 34.2(3) & 33 & 34.1(8) & 33.8 & 32.9 & 30.9 \\ $\Gamma_{2}$ (keV) & 55.2(9) & 55 & 57.9(17) & 51.4 & 51.4 & 51.3 \\ $\Gamma_{3}$ (keV) & 50.2(8) & 50 & 48.3(19) & 48.1 & 48.1 & 47.8 \\ $a_{c}$ (fm) & - & 4.0 & 3.4 & 4.0 & 5.0 & 6.0 \\ Background Poles & - & $3/2^{+}$ state2 & 3 & 3 & 3 & 3 \\ \hline \hline \end{tabular} \end{table} Table 2: Predictions of $E$ and $\Gamma$ for the measurement data from ref. [21] along with the previous results [1; 16; 21]. For the current model, the median values of the distributions are presented with the standard deviations as representations. a cross section point to a vector. After the first linear layer, the shape $(n_{\text{angle}},n_{\text{bin}},1)$ was converted into $(n_{\text{angle}},n_{\text{bin}},d_{\text{model}})$, where $d_{\text{model}}$ is a hyperparameter of the Transformer [22]. Additionally, we used the Gaussian Error Linear Units function as the activation function in the feed-forward networks of encoder and decoder layers [46]. The main operations in the Transformer are matrix multiplications that perform on the last two dimensions of data, whereas the current data has three dimensions. This feature limits the operations within a spectrum at each angle. The decoder in the Transformer performs the multi-head attention between its input and the encoder output. We gave the decoder input the same as the encoder input but shifted at the extra angle dimension, $[\mathbf{\theta_{1},\theta_{2},\theta_{3}}]\rightarrow[\mathbf{\theta_{3},\theta_{1}, \theta_{2}}]$. By doing so, the matrix multiplications in the attention were done between the data of different angles. The decoder input does not require the look-ahead mask originally demanded by the Transformer that uses target sequences as decoder input. The hyperparameters in the model were: * $d_{\text{model}}=256$ * $d_{\text{df}}=256$ * $d_{\text{q}}=d_{\text{k}}=d_{\text{v}}=32$ * $h=8$ * The number of encoder layers = 2 * The number of decoder layers = 2 We followed the notation of ref. [22]. The output of the modified Transformer was reshaped and passed to linear layers at the end of the model to return the four types of parameters. ### Probabilistic Model We utilized a probabilistic framework to estimate the uncertainties of the model predictions. The method of deep ensembles is one of the most practical and effective uncertainty quantification approaches [29, 30, 31]. Deep ensembles consist of $N_{\text{model}}$ point-estimate models trained with random initializations of model parameters and shuffling of data samples. In this study, we used 5 point-estimate models, $N_{\text{model}}=5$. While it was first proposed as a non-Bayesian neural network, it has been shown that this method can be viewed as another Bayesian approach [29, 31, 47, 48]. Bayesian deep learning critically uses marginalization over possible sets of model parameters to obtain the predictive distribution [29, 48]: \[p(\mathbf{y}\,\|\,\mathbf{x},\mathbf{D})=\int_{\mathbf{\theta}}p(\mathbf{y}\,\|\,\mathbf{x},\mathbf{\theta} ^{\prime})p(\mathbf{\theta}^{\prime}\,\|\,\mathbf{D})d\mathbf{\theta}^{\prime}, \tag{2}\] where $\mathbf{x}$ and $\mathbf{y}$ represents input and prediction, $\mathbf{\theta}$ represents represents the model parameters, and $\mathbf{D}$ represents training data. The prediction of deep ensembles in Equation 1 is equivalent to Equation 2 if the posterior is a mixture of delta functions. For this reason, deep ensembles can be seen as Bayesian marginalization with variational inference via a mixture of delta functions for the approximate distribution, $p(\mathbf{\theta}\,\|\,\mathbf{D})\sim q_{\mathbf{\phi}}(\mathbf{\theta})=(1/N_{\text{model}}) \sum_{p=1}^{N_{\text{model}}}\delta(\mathbf{\theta}=\mathbf{\theta}_{p})$. One of the advantages of deep ensembles is that they can capture different modes Figure 5: Examples of the attention matrices when the target measurement data is passed as input. $\mathbf{a}$, an attention matrix from a self-attention in the decoder. $\mathbf{b}$, an attention matrix from an encoder-decoder attention. The top and left plots of each matrix show the corresponding input data. See the text for the explanation of attentions. of posterior as multiple models are optimized to different optima [29, 47, 48]. More details on deep ensembles can be found in refs. [29, 31, 48]. We used negative log-likelihood (NLL) for the loss function in training. For regression, NLL for a data point $i$ is \[\text{NLL}= -\log p(y_{i}\left\|f_{\text{NN}}(\mathbf{x}_{i};\mathbf{\theta})\right) \tag{3}\] \[= \frac{1}{2\sigma(\mathbf{x}_{i};\mathbf{\theta})^{2}}\\|y_{i}-\mu(\mathbf{x}_ {i};\mathbf{\theta})\\|^{2}\] (4) \[+\frac{1}{2}\text{log}(2\pi\sigma(\mathbf{x}_{i};\mathbf{\theta})^{2}),\] where $f_{\text{NN}}$ represents the neural network. The mean $\mu$ and standard deviation $\sigma$ are predicted by the neural network with the model parameter $\mathbf{\theta}$ and input $\mathbf{x}_{i}$. For classification, we also implemented the method proposed by ref. [38]. They presented a framework to combine aleatoric and epistemic uncertainties and showed the robustness of the method [37, 38, 39]. Logit vectors were sampled from a Gaussian distribution whose mean and standard deviation were predicted by the model. After passing them to a softmax function, the average of the softmax outputs was used to calculate NLL. For these reasons, the model in the current study had two outputs, $\mu$ and $\sigma$, for each parameter prediction. We performed the training for 4,200 epochs with $\sim$21,000 training samples and $\sim$5,000 validation samples. The batch size was 768 on 3 GPU workers (NVIDIA Tesla V100 32GB SXM2). The number of samples is free to increase at the expense of training time. We used Adam optimizer with a weight decay of $10^{-9}$[49]. The learning rate was 0.0002 with a scheduler that decreased the learning rate by a factor of 0.99 at every 6 epochs. We constantly monitored the mean errors of predictions and NLL. After the training, we test the model with a relatively large number of samples, $\sim$30,000, to plot a reliability diagram with enough number of sample in each bin. We plotted the reliability diagram for the performance test of the probabilistic model. For the classification tasks, the predicted probabilities were not evenly distributed to all probability ranges because of the high accuracies. For this reason, we set the bin locations to have an equal number of samples in each bin, not an equal bin interval. For regression tasks, we spanned the ranges from the median that corresponds to the mean of the original Gaussian $N(\mu,\sigma^{2})$; for example, from $e^{\mu-\sigma}$ to $e^{\mu+\sigma}$ for the 68% confidence interval.
2307.07136	STCSNN: High energy efficiency spike-train level spiking neural networks with spatio-temporal conversion	Brain-inspired spiking neuron networks (SNNs) have attracted widespread research interest due to their low power features, high biological plausibility, and strong spatiotemporal information processing capability. Although adopting a surrogate gradient (SG) makes the non-differentiability SNN trainable, achieving comparable accuracy for ANNs and keeping low-power features simultaneously is still tricky. In this paper, we proposed an energy-efficient spike-train level spiking neural network with spatio-temporal conversion, which has low computational cost and high accuracy. In the STCSNN, spatio-temporal conversion blocks (STCBs) are proposed to keep the low power features of SNNs and improve accuracy. However, STCSNN cannot adopt backpropagation algorithms directly due to the non-differentiability nature of spike trains. We proposed a suitable learning rule for STCSNNs by deducing the equivalent gradient of STCB. We evaluate the proposed STCSNN on static and neuromorphic datasets, including Fashion-Mnist, Cifar10, Cifar100, TinyImageNet, and DVS-Cifar10. The experiment results show that our proposed STCSNN outperforms the state-of-the-art accuracy on nearly all datasets, using fewer time steps and being highly energy-efficient.	Changqing Xu, Yi Liu, Yintang Yang	2023-07-14T03:27:34Z	http://arxiv.org/abs/2307.07136v2	SLSSNN: High energy efficiency spike-train level spiking neural networks with spatio-temporal conversion ###### Abstract Brain-inspired spiking neuron networks (SNNs) have attracted widespread research interest due to their low power features, high biological plausibility, and strong spatiotemporal information processing capability. Although adopting a surrogate gradient (SG) makes the non-differentiability SNN trainable, achieving comparable accuracy for ANNs and keeping low-power features simultaneously is still tricky. In this paper, we proposed an energy-efficient spike-train level spiking neural network (SLSSNN) with low computational cost and high accuracy. In the SLSSNN, spatio-temporal conversion blocks (STCBs) are applied to replace the convolutional and ReLU layers to keep the low power features of SNNs and improve accuracy. However, SLSSNN cannot adopt backpropagation algorithms directly due to the non-differentiability nature of spike trains. We proposed a suitable learning rule for SLSSNNs by deducing the equivalent gradient of STCB. We evaluate the proposed SLSSNN on static and neuromorphic datasets, including Fashion-Mnist, Cifar10, Cifar100, TinyImageNet, and DVS-Cifar10. The experiment results show that our proposed SLSSNN outperforms the state-of-the-art accuracy on nearly all datasets, using fewer time steps and being highly energy-efficient. ## 1 Introduction Spiking neural networks (SNNs) have gained increasing attention recently due to their biologically-inspired computational principles, spatiotemporal information processing capability, and potential for achieving high energy efficiency in neuromorphic computing[1; 2]. However, the training of SNNs is still a significant challenge[3] since information in SNNs is transmitted and processed through non-differentiable spike trains. To tackle this problem, the surrogate gradient (SG) method[4; 5; 6] and the ANN-to-SNN conversion method[7; 8; 8] have been proposed. The ANN-to-SNN conversion method directly obtains the weights of an SNN from a corresponding ANN, relying on the relationship between the firing rates of the SNN and activations of the ANN[3]. The method enables the obtained SNN to obtain comparable accuracy as ANNs. However, its latency is usually intolerably since only a large number of time steps can close the firing rates to activations of the ANN. The significant latency also eliminates the potential of SNNs for high energy efficiency. On the other hand, surrogate gradient (SG) makes the non-differentiability SNNs can be trained directly with low latency[4; 5; 6] but cannot achieve high performance comparable to leading ANNs. Overall, it's still a challenge for traditional SNNs to achieve high accuracy, low latency, and energy efficiency for real-world applications. In this paper, we overcome both the low performance and high latency issues of traditional SNNs by introducing spatio-temporal conversion blocks (STCBs) to propose a high energy efficiency spike-train level spiking neural network. In traditional spiking convolutional neural networks (SCNNs), Integrate-and-Fire(IF) models or Leaky-Integrate-and-Fire(LIF) models are used to replace the ReLU function to make the information can be transmitted and processed in spiking form. Non-differentiable spikes make SNNs challenging to train but have excellent potential for high energy efficiency. Our basic idea is that we can utilize the advantages of spikes in energy consumption and keep the high performance of convolutional neural networks (CNNs) meanwhile. The SNN hardware can utilize a binary spike-based sparse processing over a fixed number of time steps via accumulate (AC) operations, consuming much lower power than multiply-accumulate (MAC) operations that dominate computations in ANNs[9]. Convolutional operations consist of a large number of MAC operations, making them one of the primary sources of power consumption for convolutional neural networks[9; 10]. So we proposed spatio-temporal conversion blocks based on IF models to encode the real value of information into spike trains before convolutional layers, which can help reduce the energy consumption of convolution operation significantly. To maintain the efficiency of information transmission between blocks, we use the current accumulator blocks (CABs) to decode the information after the convolutional layers. To effectively train our proposed SLSSNNs with low latency, we further study the error backpropagation of STCBs and deduce the equivalent gradient of STCBs. With these methods, we can train a high-performance, low-latency, and energy-efficiency SLSSNN. The following summarizes our main contributions: * We proposed high energy efficiency spike-train level spiking neural networks, which utilize the advantages of spikes in energy consumption and keep the high performance of convolutional neural networks (CNNs) meanwhile. * We studied the error backpropagation of STCBs and proposed the corresponding equivalent gradient, which avoids the non-differentiability problem in SNN training and does not require the costly error backpropagation through the temporal domain. * Our proposed SLSSNN achieves competitive or state-of-the-art(SOTA) SNN performance with low latency and energy efficiency on the Fashion-MNIST[11], CIFAR-10[12], CIFAR-100[12], TinyImageNet[13], and DVS-CIFAR10[14] datasets. ## 2 Related work In recent years, SNNs have attracted widespread research interest and have developed rapidly. However, there are still lots of challenging problems that remain to be unsolved. Many works seek brain-inspired methods with biological plausibility to train SNNs[15; 16; 17]. However, these methods cannot obtain competitive performance and are hard to apply to complicated datasets[3]. Besides the brain-inspired method, SNN training has been carried out in two strategies: ANN-to-SNN conversion and direct training. ANN-to-SNN conversionThe basic idea of ANN-to-SNN conversion is to convert a high-performance ANN to an SNN and adjust its parameters. This method can avoid training problems of SNNs due to the non-differentiability. But it usually needs high latency to make the firing rates closely approach the high-precision activation, which eliminates the potential of SNNs on energy efficiency. To solve this problem, various technologies are proposed to reduce the latency, such as spike-norm [18], robust normalization [19], and channel-wise normalization [20]. Recently, Kim et al. proposed a novel attention-guided ANN compression method to convert compressed deep SNN models that have high energy efficiency and reduced inference latency[9]. Li et al. proposed to apply adaptive threshold and layer-wise calibration to obtain high-performance SNNs[21]. However, The SNNs obtained by the ANN-to-SNN conversion method have more significant latency compared to the SNNs trained directly. In addition, the ANN-to-SNN convert method is not suitable for neuromorphic datasets. Direct trainingTo solve the non-differentiability problem of SNNs, Lee et al. proposed to use a surrogate gradient to replace the non-differentiable activation term.[22]. Compared with the ANN-to-SNN conversion method, direct training achieves high accuracy with fewer time steps but suffers more training costs[23]. Many studies use surrogate gradient (SG) to obtain high-performance SNNs on static and neuromorphic datasets[4; 5; 6; 24]. Wu et al. first proposed the STBP method, which significantly improves the performance of SNNs obtained by direct training [25]. Zhang et al. proposed TSSL-BP to improve temporal learning precision by breaking down error backpropagation across two types of inter-neuron and intra-neuron dependencies and achieve low-latency and high-accuracy SNNs [26]. Recently, Xu et al. proposed a novel training method based on backpropagation (BP) for SNNs with multi-threshold, which can reduce the latency of SNN to 1-2 time steps[1]. ## 3 Methodology ### Spatio-temporal conversion block (STCB) To reduce the energy consumption of convolution layers, we proposed spatio-temporal conversion blocks (STCBs) to replace the convolutional and ReLU layers. The STCB consists of a spike train encoding block (STEB), a convolutional layer, and a current accumulator block (CAB), which is shown in Fig.1. The STCB utilizes IF models with decaying threshold voltage to encode information into spike trains, making MAC operations of convolutional layers convert to AC operations. Due to the feature of Integrate-and-Fire (IF) models, the STEB also has the function of ReLU layers. In STCB, we treat the convolutional layer as a synaptic model, which receives spike and generate synaptic current responses. The CAB is proposed to accumulate synaptic current with time steps and the final accumulated synaptic current will be the output of the next STCB. In STCB, information is transformed into binary information in the form of spike trains to simplify the convolutional operation and is recovered back to real information after convolutional operations. ### Spike train encoding block(STEB) and Current accumulator block (ACB) Spike train encoding block(STEB)As one of the main components of the STCBs, the spike train encoding block (STEB) utilizes the IF model[27] to encode information into spike trains. As Fig. 2(a) shows, the input data will be used as the initial membrane voltage of the neuron model at the zeroth time step. It is known that the IF model is one of the most widely applied models to describe neuronal dynamics in SNNs. In this paper, we apply the IF models with decaying threshold voltages, whose threshold voltages decay with time steps, to encode input data into spike trains. The IF model can be depicted as \[I(t)=C_{m}\frac{dV_{m}(t)}{dt} \tag{1}\] Figure 1: Spatio-temporal conversion block (STCB) where $C_{m}$ is the membrane capacitance, $V_{m}(t)$ is membrane potential and $I(t)$ denotes the membrane current. Since we only use input data as the initial membrane, the neuronal membrane potential of neuron $i$ at time step $t$ is \[u_{i}[t]=\left\{\begin{array}{rl}I[t],&t=0\\ u_{i}[t-1]+u_{reset}[t],&t>0\end{array}\right. \tag{2}\] where $u_{reset}[t]$ denotes the reset function, which reduces the membrane potential by the $V_{th}[t]$ after the neuron $i$ fires at the $t$ time step. $u_{reset}[t]$ is equal to $-s_{i}[t]V_{th}[t]$. $s_{i}[t]$ is the output spike of neuron $i$ and $V_{th}[t]$ is the threshold voltage at time step $t$, which is given by \[s_{i}[t]=\left\{\begin{array}{rl}0,&u_{i}[t]<V_{th}[t]\\ 1,&u_{i}[t]\geq V_{th}[t]\end{array}\right. \tag{3}\] \[V_{th}[t]=\left\{\begin{array}{rl}V_{init},&t=0\\ \frac{V_{th}[t-1]}{\tau_{vth}},&t>0\end{array}\right. \tag{4}\] where $V_{init}$ is the initial threshold voltage and $\tau_{vth}$ is the time constant of threshold voltages. Current accumulator Block (ACB)As we mentioned in section 3.1, convolutional layer is treated as synaptic model. So spike train is converted into a series of synaptic currents after the convolutional operation. We use the current accumulator block to accumulate the synaptic currents into a final synaptic current to improve the transmission efficiency between blocks. As Fig. 2(b) shows, input data at time step $t$ is multiplied by the threshold voltage at time step $t$, and the result will be accumulated. The output of ACB $I_{acc}[t]$ is calculated by \[I_{acc}[t]=\left\{\begin{array}{rl}0,&t<T_{max}\\ i_{acc}[t],&t=T_{max}\end{array}\right. \tag{5}\] where $T_{max}$ is the maximum time step which is also equal to the length of the spike trains. $i_{acc}[t]$ is the synaptic current at time step $t$, which is given by \[i_{acc}[t]=\left\{\begin{array}{rl}I_{syn}[t]V_{th}[t],&t=0\\ i_{acc}[t-1]+I_{syn}[t]V_{th}[t],&t>0\end{array}\right. \tag{6}\] where $I_{syn}[t]$ is the synaptic current at time step $t$, and $V_{th}[t]$ is the threshold voltage at time step $t$ which can be obtained by Eq. 4. ### Error Backpropagation of STCB The information is transmitted in the form of spike trains in STCB, while transmitted in the form of the accumulated synaptic current between STCBs, which makes error backpropagation of STCB Figure 2: The structure of STEB and CAB. (a) STEB (b) CAB complex. To solve this issue, we proposed the equivalent gradient of STCBs, which can simplify the error backpropagation of STEB and reduce the computation. As we mentioned in section 3.2, data will be first encoded into spike trains by STEB in the forward pass process. Then the spike trains will convert into a series of synaptic currents by the convolutional layer. Finally, these synaptic currents will be accumulated by CAB, and the final accumulated synaptic current will be used as the output of STCB. As Fig. 3 shows, the equivalent forward pass 1 of STCB has a different order of operations but the same output. STEB + CAB is a process of encoding and decoding. If we ignore the error of this process, STEB + CAB is equal to ReLU. So we can use the backward Pass of ReLU + Cov to replace the one of STCB, which can simplify the error backpropagation of STCB and reduce the computation significantly. ### Network Architecture Using the STCB mentioned in the previous section, we propose an SLSSNN that accurately classifies static and neuromorphic datasets with a low computational cost and energy consumption. In this section, we chose VGG and ResNet architecture to show how to build an SLSSNN. Fig. 4(a) and 4(b) show the diagram of VGG and ResNet based on STCBs. For the first convolutional block of VGG architecture, we keep a convolutional layer with 3$\times$3 filters as the first layer, which helps to obtain enough features from the original data. In other convolution blocks, we just need to use the STCB to replace the convolutional layers. For ResNet architecture, we also keep a convolutional layer with 3$\times$3 filters as the first layer. Due to the feature of IF models, the STEB also has the function of ReLU layers. So we use the STCB to replace the convolutional layer + ReLU function in residual blocks. Figure 4: Diagram of VGG and ResNet based on STCB. (a) VGG, (b) ResNet. Conv refers to the convolutional layer, $N\times N$ Cov/STCB means the Convolutional layer or STCB with $N\times N$ filters, BN means Batch Norm, AP is the average pooling layer, AAP is the adaptive average pool, and FC refers to the fully connected layers. Figure 3: Forward and backward pass of STCB Experiments and results In this section, we first evaluate the proposed SLSSNN on four static datasets, Fashion-MNIST[11], CIFAR-10[12], CIFAR-100[12], TinyImageNet[13], and one and neuromorphic dataset, DVS-CIFAR10[14] with different structures of SNNs. And we compared our method with several previously reported state-of-the-art results with the same or similar networks, including different SNNs trained by BP-based methods and converted SNNs. We also study the influence of initial threshold voltages and the length of the spike trains to show the tradeoff between accuracy and latency. Finally, we count the number of spikes for different datasets and analyze energy consumption of SLSSNNs. ### Experiment settings All reported experiments below are conducted on an NVIDIA Tesla V100 GPU. The implementation of our proposed method is on the Pytorch framework [28]. The experimented SNNs are based on the network structure described in Sec. 3.4. Only 8 time steps are used to demonstrate the proposed ultra low-latency spiking neural network. No refractory period is used. SGD[29] is applied as the optimizer. If not otherwise specified, the accuracy in this paper refers to the best results obtained by repeating the experiments five times. Please refer to the Supplementary Materials for experiment details. ### Comparison with existing works The comparison results on FashionMnist, CIFAR-10, CIFAR-100, TinyImageNet, and DVS-CIFAR10 are shown in Tab. 1. For the FashionMnist dataset, we use ResNet18[30] as the network structure. Tab. 1 shows that the proposed SLSSNN method outperforms all other methods on FashionMnist with 8 time steps, based on 5 runs of experiments. Although some direct training methods such as [31] use smaller time steps than ours, our method can achieve better performances. For the FashionMnist dataset, which is a simple dataset, our proposed method still obtained 0.51%-5.19% accuracy improvement. For the Cifar-10 and Cifar-100 datasets, we use VGG16[9] and ResNet18[30] as the network architectures. On Cifar-10 and Cifar-100, our proposed SLSSNN achieves the highest accuracy above all existing approaches based on 5 runs of experiments. For Cifar-10, Our proposed SLSSNN based on VGG16 and ResNet18 architecture obtains $93.32\%$ and $94.08\%$ test accuracy with only 8 time steps, respectively. Though our proposed SLSSNN based on VGG16 architecture obtain $93.32\%$ accuracy, which is a little lower than [32] on Cifar-100. We only train the network in 100 epochs, while [32] needs 200-300 epochs. In addition, Our proposed SLSSNN based on ResNet18 can obtain high accuracy than other methods. For the TinyImageNet dataset, we still used VGG16[9] and ResNet18[30] as the network architectures. Compared to the ANN-to-SNN methods, which apply the same VGG16 architecture, our method obtains $4.22\%$-$6.44\%$ accuracy improvement with fewer time steps. If we apply a more powerful architecture ResNet18, SLSSNN can obtain $59.38\%$ accuracy, which improves the accuracy by $1.58\%$-$12.59\%$ compared with other methods. W also test our SLSSNN on the neuromorphic DVS-CIFAR10. we adopt the VGG-11[8] and VGG16[9] architectures and conduct 5 runs of experiments for each architecture in 50 epochs. As Tab. 1 shows, the proposed method outperforms other SOTA methods with low latency using VGG11 and VGG16. ### Performance analysis Since the initial threshold voltage and the length of the spike trains directly affect the loss of information during the encoding and decoding process in STCB, we study the influence of initial threshold voltages and length of spike trains on the accuracy of SNNs using FashionMnist, Cifar10, Cifar100, TinyImageNet, and DVS-Cifar10. #### 4.3.1 Influence of initial threshold voltages We chose different voltages (0.1, 0.2, 0.4, 0.6, 0.8, 1, 2, 4, 8mV) as the initial threshold voltage of SLSSNNs. For static datasets, FashionMnist, Cifar10, Cifar100, and TinyImageNet, we use ResNet18 as the network structure. For the neuromorphic dataset DVS-Cifar10, VGG16 is applied. As Fig. 5(a) shows, there is a significant accuracy drop when the initial threshold voltage is less than 0.8 mV. SLSSNN can obtain a reasonable accuracy for the static datasets when the initial threshold voltage is larger than 1. We further study the distribution of the input of SLSSNNs and find that the small initial threshold voltage may cause serious information loss during the encoding process in STCB. For FashionMnist, Cifar10, Cifar100, and TinyImageNet datasets, the input of STCB is in the range of [-1, 1]. When the initial threshold voltage is too small, such as 0.1, the part of the input larger than 0.1 will be encoded into one spike, which may cause profound information loss. For DVS-Cifar10, the input of STCB is in the range of [-2, 2]. So the initial threshold voltage is less than 2 will make a significant accuracy drop. \begin{table} \begin{tabular}{c c c c c c c c} \hline Dataset & Model & Methods & Architecture & Time Steps & mean(\%) & stddev(\%) & Best(\%) \\ \hline \multirow{4}{}{Fashion MNIST} & Zhang and Li[6] & SNN & 12cmv1,12cmv1 & 400 & 90.0 & 0.14 & 90.13 \\ & Cheng et al.[33] & SNN & 2cmv1,12cmv1 & 20 & 92.07 & - & - \\ & Xu et al [1] & SNN & 2cmv1,12cmv1 & 2 & - & - & 93.80 \\ & Fang, et al [34] & SNN & 2cmv,112cmv1 & 8 & - & - & 94.38 \\ & Kazuma Suekac et al [31] & SNN & ResNet18 & 1 & - & - & 94.81 \\ & Our model & SNN & ResNet18 & 8 & 95.05 & 0.26 & 95.32 \\ \hline \multirow{4}{}{Cifar10} & Kim et al.[17] & ANN-to-SNN & VGG16 & 150 & - & - & 89.2 \\ & Zhang and Li[26] & SNN & 5cmv, 21cmv1 & 5 & - & - & 91.41 \\ & Rathi, et al[8] & ANN-to-SNN & ResNet20 & 250 & - & - & 92.22 \\ & Kundu, et al[9] & ANN-to-SNN & VGG16 & 10 & - & - & 92.53 \\ & Our model & SNN & VGG16 & 8 & 93.08 & 0.25 & 93.32 \\ & Our model & SNN & ResNet18 & 8 & 93.81 & 0.30 & 94.08 \\ \hline \multirow{4}{}{Cifar100} & Shen et al.[35] & RNN & ResNet20 & 1 & - & - & 69.38 \\ & Kim et al.[7] & ANN-to-SNN & VGG16 & 200 & - & - & 62.3 \\ & Rathi et al.[8] & ANN-to-SNN & VGG11 & 125 & - & - & 67.87 \\ & Kazuma Suekac et al [31] & SNN & ResNet18 & 1 & - & - & 71.23 \\ & Kim, Li et al.[32] & SNN & NAS & 5 & 73.04 & - & 73.40 \\ & Our model & SNN & VGG16 & 8 & 71.89 & 0.55 & 72.05 \\ & Our model & SNN & ResNet18 & 8 & 74.05 & 0.28 & 74.31 \\ \hline \multirow{4}{}{TinyImageNet} & Na et al.[36] & SNN & NAS & 8 & - & - & 46.79 \\ & Kim et al.[7] & ANN-to-SNN & VGG16 & 200 & - & - & 51.9 \\ & Kundu et al.[9] & ANN-to-SNN & VGG16 & 10 & - & - & 54.1 \\ & Kim, Li et al.[32] & SNN & NAS & 5 & - & - & 54.6 \\ & Kim and Panda[37] & SNN & VGG11 & 30 & - & - & 57.8 \\ & Our model & SNN & VGG16 & 8 & 58.14 & 0.21 & 58.34 \\ & Our model & SNN & ResNet18 & 8 & 59.10 & 0.25 & 59.38 \\ \hline \multirow{4}{*}{DVS-Cifar10} & Wu et al.[38] & SNN & VGG7 & 50 & - & - & 62.5 \\ & Wu, et al.[39] & SNN & 7cmv,21cmv1 & 20 & - & - & 65.59 \\ & Zheng, et al.[40] & SNN & ResNet19 & 10 & - & - & 67.80 \\ & Xu, et al.[4] & SNN & 6cmv,112cmv1 & 5 & - & - & 73.8 \\ & Fang, et al.[34] & SNN & 7cmv,21cmv1 & 20 & - & - & 74.80 \\ & Meng, et al.[3] & SNN & VGG11 & 20 & 75.03 & - & 75.42 \\ & Our model & SNN & VGG11 & 8 & 76.0 & 1.11 & 77.5 \\ & Our model & SNN & VGG16 & 8 & 77.5 & 0.53 & 77.9 \\ \hline \end{tabular} \end{table} Table 1: Compare with exciting work on different datasets. Figure 5: Performance analysis. (a) Initial threshold voltages (b) length of spike trains #### 4.3.2 Influence of the length of spike trains We chose different lengths of spike trains (2, 4, 6, 8, 12, 16) to study the influence on the accuracy of SLSSNNs. The network architecture and datasets are the same as section 4.3.1. The initial threshold voltages are set as 0.8 and 2 for the static datasets (FashionMnist, Cifar10, Cifar100, and TinyImageNet datasets) and the neuromorphic dataset (TinyImageNet), respectively. Fig. 5(b) is the accuracy of SLSSNN with different lengths of spike trains on different datasets. When the length of spike trains is less than 4, the SLSSNN cannot work properly. For a fixed initial threshold voltage, the length of spike trains determines the encoding and decoding precision of STCB. So too short spike trains may cause the smaller input information will be lost, which causes SLSSNN accuracy to drop significantly, especially for the deeper network. Fig. 5(b) also shows that the accuracy of SLSSNN is not significantly increasing with the increase of the length of the spike train when the length is larger than 8. Longer spike trains mean more energy consumption. Therefore a suitable initial threshold voltage and length of the spike train can make a better tradeoff between accuracy and energy. ### Energy consumption analysis In this section, we first study the fire rate and the number of spikes of each convolutional layer of ResNets18 and VGG16. Then we use FLOPs count to capture the computation energy to show the high energy efficiency of SLSSNNs. #### 4.4.1 Fire rate and Number of spikes We use Cifar10 and DVS-Cifar10 datasets to show the average fire rate and the number of spikes of each convolutional layer of ResNets18 and VGG16, respectively. As Fig. 6(a) and 6(b)shows, the Figure 6: Average fire rate. (a) ResNet18 (Cifar10) (b)VGG16 (DVS-Cifar10) Figure 7: Average number of spikes. (a) ResNet18 (Cifar10) (b)VGG16 (DVS-Cifar10) average fire rate increases with the increasing spike train's length. Fig. 6(a) depicts the benefits of the short length of spike trains, achieving significantly lower spike activity for almost all convolutional layers. Fig. 7(a) and 7(b) show the average number of spikes of ResNet18 and VGG16. Because the number of spikes is determined by the fire rate and the length of spike trains, the number of each convolutional layer increases rapidly with the increase of the length of spike trains. The good news is that we do not need a very long spike train to obtain a competitive result. #### 4.4.2 Floating Point Operations (FLOPs) $\&$ computation Energy In this section, we refer [9] to calculate the FLOPs and computation energy of SLSSNN. The detail is in the supplementary materials. For static datasets, we chose ResNet18 as the network architecture. For the neuromorphic dataset, we apply VGG18 as the network architecture. Fig. 8(a) and 8(b) shows the average FLOPs of ResNet18 and VGG16. Compared with ANN, SLSSNNs have fewer FLOPs when the length of the spike train is less than 6. Though SLSSNNs have more FLOPs when the spike train is longer than 6, the accumulation (AC) operation rather than the multiply-and-accumulate (MAC) operation is dominant due to the binary spiking information, which helps SLSSNN keep high energy efficiency. As Fig. 9 shows that the SLSSNNs have significant energy reduction. The energy efficiency can reach up to $11.76\times\sim 36.79\times$, when the length of spike train is 8. Even if the length of the spike train is 16, SLSSNN still has $6.54\times\sim 15.17\times$ energy reduction. Figure 8: FLOPs of ResNet18 and VGG16. (a) Static Datasets (ResNet18) (b)Neuromorphic dataset (VGG16) Figure 9: Comparison of computation energy ## 5 Discussions and Conclusion This paper proposes a high energy efficiency spike-train level spiking neural network with spatio temporal conversion. Based on the idea of spatio temporal conversion, spatio-temporal conversion blocks (STCBs) are proposed to replace the convolutional and ReLU layers to keep the low power features of SNNs and improve the accuracy of SNNs. We chose ResNet18 and VGG16 as network architecture to evaluate the proposed SLSSNN on static and neuromorphic datasets, including Fashion-Mnist, Cifar10, Cifar100, TinyImageNet, and DVS-Cifar10. The experiment results show that our proposed SLSSNN outperforms the state-of-the-art accuracy on nearly all datasets, using fewer time steps and being highly energy-efficient. As for limitations, the proposed SLSSNNs may suffer from a certain degree of performance drop when the latency is extremely low(e.g., the length of the spike train is less than 4) since the SLSSNNs require enough spikes to transmit information between layers.
2307.13962	Understanding Deep Neural Networks via Linear Separability of Hidden Layers	In this paper, we measure the linear separability of hidden layer outputs to study the characteristics of deep neural networks. In particular, we first propose Minkowski difference based linear separability measures (MD-LSMs) to evaluate the linear separability degree of two points sets. Then, we demonstrate that there is a synchronicity between the linear separability degree of hidden layer outputs and the network training performance, i.e., if the updated weights can enhance the linear separability degree of hidden layer outputs, the updated network will achieve a better training performance, and vice versa. Moreover, we study the effect of activation function and network size (including width and depth) on the linear separability of hidden layers. Finally, we conduct the numerical experiments to validate our findings on some popular deep networks including multilayer perceptron (MLP), convolutional neural network (CNN), deep belief network (DBN), ResNet, VGGNet, AlexNet, vision transformer (ViT) and GoogLeNet.	Chao Zhang, Xinyu Chen, Wensheng Li, Lixue Liu, Wei Wu, Dacheng Tao	2023-07-26T05:29:29Z	http://arxiv.org/abs/2307.13962v1	# Understanding Deep Neural Networks via Linear Separability of Hidden Layers ###### Abstract In this paper, we measure the linear separability of hidden layer outputs to study the characteristics of deep neural networks. In particular, we first propose Minkowski difference based linear separability measures (MD-LSMs) to evaluate the linear separability degree of two points sets. Then, we demonstrate that there is a synchronicity between the linear separability degree of hidden layer outputs and the network training performance, _i.e._, if the updated weights can enhance the linear separability degree of hidden layer outputs, the updated network will achieve a better training performance, and vice versa. Moreover, we study the effect of activation function and network size (including width and depth) on the linear separability of hidden layers. Finally, we conduct the numerical experiments to validate our findings on some popular deep networks including multilayer perceptron (MLP), convolutional neural network (CNN), deep belief network (DBN), ResNet, VGGNet, AlexNet, vision transformer (ViT) and GoogLeNet. ## 1 Introduction Deep neural networks have been successfully used to solve many complicated learning tasks, for example, computer vision [1, 2, 3], natural language processing [4, 5, 6] and other engineering applications [7, 8]. The empirical observation that the large network size is beneficial to the generalization performance of deep networks, is contrary to the traditional view of statistical learning theory, which deems that high model complexity will cause the overfitting. In recent years, many research works attempt to explain why deep neural networks perform well in these difficult tasks by using some theoretical analysis tools such as statistical learning theory (SLT) [9], PAC-Bayes framework [10] and neural tangent kernel (NTK) [11]. However, there still remains a gap between these theoretical results and the user experiences of deep neural networks. For example, following the classical STL methods, the complexity measures of deep networks were expressed as increasing functions w.r.t. the network size, and the relevant generalization bounds will become loose when the network size is large (_cf._[12, 13, 14, 15]). Moreover, some results on NTK and over-parameterized neural networks are obtained under the assumption that the network width is infinite (_cf._[16, 17, 18, 19, 20, 21, 22]). Some recent theoretical works on deep neural networks are summarized in the appendix (part A). ### Background and Motivation Two point sets are said to be linearly separable if they can be correctly split by using a hyperplane, and the concept of linear separability plays an important part in measuring the capability of neural networks [23, 24, 25]. In the literature, there are two main research issues on the linear separability of neural networks: 1) whether the current network can achieve all dichotomies, _i.e.,_ the mapping capability [26]; and 2) how many dichotomies can be recorded by a network with the specific structure, _i.e.,_ the memory capability [27, 28]. However, to the best of our knowledge, there is rarely few work to consider the layer-wise changes of the linear separability degree of two point sets when they pass through a network with multiple hidden layers. Our study is initially motivated by the relationship between the linear separability degree of hidden layer outputs and the process of training networks. We would like to know whether the linear separability degree of hidden layer outputs increases when the training accuracy increases. If applicable, the linear separability degree can be treated as a criterion for evaluating the performance of each hidden layer during the training phase. Consider a one-hidden-layer MLP net$(\cdot):\mathbb{R}^{N}\rightarrow\{0,1\}$ with the output-layer weight vector $\mathbf{w}$ and the hidden-layer weight matrix $\mathbf{V}$. Let $\mathrm{hid}(\cdot):\mathbb{R}^{N}\rightarrow\mathbb{R}^{H}$ be the hidden layer of the MLP, and $\mathrm{hid}(\mathcal{X})$ be the set of hidden layer outputs w.r.t. the input set $\mathcal{X}:=\{\mathbf{x}_{m}\}_{m=1}^{M}$. The set $\mathrm{hid}(\mathcal{X})$ is called hidden-layer outputs in the following discussion, if no confusion arises. Denote $\triangle\mathbf{w}$ and $\triangle\mathbf{V}$ as the weight updates provided by a training algorithm implemented on the training set $\mathcal{S}=\{(\mathbf{x}_{m},\mathbf{y}_{m})\}_{m=1}^{M}\subset\mathbb{R}^{ N}\times\{0,1\}$. The updated network is denoted as net${}^{\prime}(\cdot)$ with updated weights $\mathbf{w}^{\prime}=\mathbf{w}+\triangle\mathbf{w}$ and $\mathbf{V}^{\prime}=\mathbf{V}+\triangle\mathbf{V}$. Denote $\mathrm{hid}^{\prime}(\cdot)$ as the updated hidden layer with the weights $\mathbf{V}^{\prime}$. Under these notations, we obtain the following theorem which motivates the research of this paper: Theorem 1 (Synchronicity).: _Assume that the updated weights $\mathbf{w}^{\prime}$ achieves the highest classification accuracy on $\mathcal{S}$ when the hidden-layer weights of net${}^{\prime}(\cdot)$ is updated to be $\mathbf{V}^{\prime}$. Then, net${}^{\prime}(\cdot)$ has higher classification accuracy on $\mathcal{S}$ than net$(\cdot)$ if and only if the linear separability degree of $\mathrm{hid}^{\prime}(\mathcal{X})$ is larger than that of $\mathrm{hid}(\mathcal{X})$._ This theorem demonstrates the synchronicity between the linear separability degree of hidden layer outputs and the network training performance. When training a network, the change of the linear separability degree of hidden layer outputs synchronizes with the change of the training accuracy. Different from the existing works on the theoretical analysis of neural networks, the linear separability degree provides a feasible manner to directly study the relationship between the structure parameters and the network capability. Accordingly, a desired linear separability measure (LSM) should meet the following requirements: 1. It should have a low computational cost, because we would like to layer-wisely examine the linear separability degree of the hidden-layer outputs after each training iteration; 2. It should be insensitive to the outliers, because the stochastic gradient descent methods sometimes cause abnormal hidden layer outputs; 3. It should be of a well-defined mathematical form in order to facilitate the further theoretical analysis. Some mathematical terms mentioned in the existing works actually can be treated as LSMs of two point sets, for example, the generalized Rayleigh quotient $J_{\omega}=\max_{\boldsymbol{\omega}}\frac{\boldsymbol{\omega}^{T}\mathbf{S}_ {\boldsymbol{\omega}}\boldsymbol{\omega}}{\boldsymbol{\omega}^{T}\mathbf{S}_ {\boldsymbol{\omega}}\boldsymbol{\omega}}$ in Fisher linear discriminant analysis (LDA) and the sum of slack variables in linear support vector machine (L-SVM) with soft margin.1 However, the LDA-based LSM, which is based on the means of point sets, is sensitive to the outliers in the sets and has a high computational cost because of eigenvalue decomposition; and the computation of the L-SVM based LSM is still time-consuming especially when the sample size is large. Moreover, Ben-Israel _et al_. [29] introduce the linear divisible angle to measure the linear separability degree of two point sets, where the labels of the data are treated as a new attribute to convert the dimension of points from $N$ to $N+1$, and then Fisher LDA is used to compute the generalized Rayleigh quotient of the converted points. Gabidullina _et al_. [30] adopted the smallest thickness of the classified hyperplane as the LSM for the linearly inseparable sets. Since this measure is computed via a minimax optimization problem, it has a high computational cost as well. ### Overview of Main Results In Section 2, we propose the Minkowski difference-based LSM (MD-LSM) for two point sets, and provide an algorithm of finding the largest linearly separable subsets of arbitrary two linearly inseparable sets. Then, we show the relationship between the linear separability degree of two linearly inseparable sets and the best classification accuracy provided by all possible linear models. Moreover, we provide the alternative versions of MD-LSM to reduce the cost of computing the MD-LSM of each hidden layer after each training epoch. In Section 3, we study the effects of activation function and network size on the linear separability of hidden layers. First, the hidden layer of a neural networks is regarded as the pseudo-linear mapping (PLM) (_cf_. [27]). Then, we provide a sufficient condition that a PLM changes the relative position between one Minkowski difference (MD) point and the splitting hyperplane. Moreover, based on the random matrix theory, we have proven that the increase of network size can enlarge the probability of increasing the linear separability degree of hidden layer outputs. In Section 4, we conduct the numerical experiments to validate the theoretical findings of this paper on some well-known networks including multilayer perceptron (MLP) [31], convolutional neural network (CNN) [32], deep brief network (DBN) [33], ResNet [34], VGGNet [35], AlexNet [36], vision transformer (ViT) [37] and GoogLeNet [38] in classification tasks, respectively. In Section 5, we discuss the difference between the proposed MD-LSM and Fisher LDA, and then explain why we use MD-LSMs to evaluate the linear separability degree of hidden layer outputs rather than the Rayleigh quotient in Fisher LDA. In Section 6, we prove the main results of this paper, and the last section concludes the paper. In the appendix, we briefly sketch the recent works on theoretical analysis of deep networks (part A), and then provide the complete experimental report (part B). ## 2 Minkowski Difference Based Linear Separability Measure (MD-LSM) In this section, we present the concept of MD-LSM for two point sets and then give an algorithm to find the maximum linearly separable subset of two linearly inseparable point sets. Moreover, we provide some alternative versions of MD-LSMs with low computation cost. ### Minkowski Difference and Maximum Linearly Separable Subset First, we introduce the concept of Minkowski difference (MD), which has been widely used for the applications in many areas such as data classification [39, 40], motion planning for robots [41], real-time collision detection [42] and computer graphics [43, 44]. Definition 2 (Minkowski Difference).: _Let $\mathcal{A}=\{\mathbf{a}_{1},\cdots,\mathbf{a}_{I}\}\subset\mathbb{R}^{N}$ and $\mathcal{B}=\{\mathbf{b}_{1},\cdots,\mathbf{b}_{J}\}\subset\mathbb{R}^{N}$ be two point sets. Then, the Minkowski difference between the two sets is defined as_ \[\mathrm{MD}(\mathcal{A},\mathcal{B}):=\big{\{}\mathbf{m}_{ij}:=\mathbf{a}_{i}- \mathbf{b}_{j}\in\mathbb{R}^{N}\mid\mathbf{a}_{i}\in\mathcal{A},\;\mathbf{b}_{ j}\in\mathcal{B}\big{\}}.\] Based on Minkowski difference, we convert the linear separability of two point sets into the relative position relationship between a point set and a hyperplane that passes the origin. Theorem 3.: _Two points sets $\mathcal{A},\mathcal{B}\in\mathbb{R}^{N}$ are linearly separable if and only if there exists a vector $\boldsymbol{\omega}\in\mathbb{R}^{N}$ such that all points of $\mathrm{MD}(\mathcal{A},\mathcal{B})$ locates in one side of the hyperplane $\boldsymbol{\omega}^{T}\mathbf{x}=0$, $\mathbf{x}\in\mathbb{R}^{N}$._ We note that the hyperplane $\boldsymbol{\omega}^{T}\mathbf{m}=0$ parallels to the separating hyperplane between the two classes $\mathcal{A}$ and $\mathcal{B}$ if they are linearly separable. Additionally, if the two sets $\mathcal{A}$ and $\mathcal{B}$ are linearly inseparable, some points of $\mathrm{MD}(\mathcal{A},\mathcal{B})$ will lie in one side of the hyperplane and the rest lie in the other side: Definition 4 (Minor and Major Sides).: _Given a hyperplane $\boldsymbol{\omega}^{T}\mathbf{m}=0$, if more than half points of $\mathrm{MD}(\mathcal{A},\mathcal{B})$ lie in one side of $\boldsymbol{\omega}^{T}\mathbf{m}=0$, then this side is said to be the major side of the hyperplane; and accordingly, the other side of $\boldsymbol{\omega}^{T}\mathbf{m}=0$ is said to be the minor side of the hyperplane._ Furthermore, denote $\mathrm{major}_{\boldsymbol{\omega}}(\mathrm{MD}(\mathcal{A},\mathcal{B}))$ (resp. $\mathrm{minor}_{\boldsymbol{\omega}}(\mathrm{MD}(\mathcal{A},\mathcal{B}))$) as the subset of $\mathrm{MD}(\mathcal{A},\mathcal{B})$ containing all points of $\mathrm{MD}(\mathcal{A},\mathcal{B})$ that locate in the major (resp. minor) side of $\boldsymbol{\omega}^{T}\mathbf{m}=0$. If some points $\mathbf{m}_{ij}$ located at the minor side of the hyperplane $\boldsymbol{\omega}^{T}\mathbf{m}=0$, we can remove the corresponding original points $\mathbf{a}_{i}$ (resp. $\mathbf{b}_{j}$) from $\mathcal{A}$ (resp. $\mathcal{B}$) to form two new sets $\mathcal{A}_{1}$ and $\mathcal{B}_{1}$, which are linearly separable. Definition 5.: _The set $\mathrm{MaxLS}(\mathcal{A},\mathcal{B})$ is said to be the maximum linearly-separable subset of $\mathcal{A}\cup\mathcal{B}$ if_ \[\mathrm{MaxLS}(\mathcal{A},\mathcal{B}):=\mathcal{A}_{\circ}\cup\mathcal{B}_{ \circ}=\operatorname{arg\,max}_{\mathcal{A}^{\prime}\subset\mathcal{A}, \mathcal{B}^{\prime}\subset\mathcal{B}}\|\mathcal{A}^{\prime}\|+\|\mathcal{B}^{ \prime}\|\quad\mathrm{s.t.}\quad\mathcal{A}^{\prime}\text{ and }\mathcal{B}^{\prime}\text{ are linearly separable}.\] Namely, $\mathrm{MaxLS}(\mathcal{A},\mathcal{B})$ is the largest-size subset of $\mathcal{A}\cup\mathcal{B}$ such that $\mathcal{A}_{\circ}$ and $\mathcal{B}_{\circ}$ are linearly separable. It is noteworthy that $\mathrm{MaxLS}(\mathcal{A},\mathcal{B})$ could not be unique. Remark 6.: _The maximum linearly-separable subset $\mathrm{MaxLS}(\mathcal{A},\mathcal{B})$ of two sets $\mathcal{A}$ and $\mathcal{B}$ can be obtained in an iterative way: (1) Given $P\times Q$ points $\mathbf{m}_{i_{1},j_{1}},\cdots,\mathbf{m}_{i_{P},j_{Q}}$ lie in the minor side of Figure 1: The workflow of obtaining the maximum linearly separable subset. Left: two point sets with the marks that signify the one to be removed in the iterations. Middle: two convex hulls of the rest points in the two sets. Right: Minkowski difference of the rest points $\mathbf{\omega}^{T}\mathbf{m}=0$, build a undirected bipartite graph $G(\mathcal{V}_{0},\mathcal{E}_{0})$ with $\mathcal{V}_{0}=\{\mathbf{a}_{i_{1}},\cdots,\mathbf{a}_{i_{P}},\mathbf{b}_{j_{1} },\cdots,\mathbf{b}_{j_{Q}}\}$ and $\mathcal{E}=\{(\mathbf{a}_{i_{1}},\mathbf{b}_{j_{1}}),\cdots,(\mathbf{a}_{i_{ P}},\mathbf{b}_{j_{Q}})\}$. (2) Remove one vertex $v_{1}$ with the largest degree from $\mathcal{V}_{0}$ and update $\mathcal{V}_{1}=\mathcal{V}_{0}\setminus\{v_{1}\}$. (3) Eliminate the edges associated with the vertex $v_{1}$ and update $\mathcal{E}_{1}\leftarrow\mathcal{E}_{0}$. (4) Repeat the steps (2)-(3) until $\mathcal{E}_{t}=\emptyset$. (5) Remove the points in $\mathcal{V}_{0}\setminus\mathcal{V}_{t}$ from the original sets $\mathcal{A}$ and $\mathcal{B}$, and the rest form the desired $\mathrm{MaxLS}(\mathcal{A},\mathcal{B})$. Figure 1 exhibits a numerical example of this algorithm as well._ ### MD-based Linear Separability Measure (MD-LSM) Following Theorem 3, the ratio of the numbers of the points $\mathbf{m}_{ij}\in\mathrm{MD}(\mathcal{A},\mathcal{B})$ that respectively locate in the two sides of the hyperplane can be treated as a criterion to measure the linear separability degree between $\mathcal{A}$ and $\mathcal{B}$: \[\mathrm{LS}_{}(\mathcal{A},\mathcal{B}):=\max_{\mathbf{\omega}\in \mathbb{R}^{N}}\left\{\max\left\{\frac{\sum\limits_{i\leq I,j\leq J}\mathbf{1} (\mathbf{\omega}^{T}\mathbf{m}_{ij}>0)}{\|\mathrm{MD}(\mathcal{A},\mathcal{B})\|}, \frac{\sum\limits_{i\leq I,j\leq J}\mathbf{1}(\mathbf{\omega}^{T}\mathbf{m}_{ij}<0 )}{\|\mathrm{MD}(\mathcal{A},\mathcal{B})\|}\right\}\right\}, \tag{1}\] where $\|\mathrm{MD}(\mathcal{A},\mathcal{B})\|$ is the cardinality of $\mathrm{MD}(\mathcal{A},\mathcal{B})$ and $\mathbf{1}(\mathcal{E})$ is the indicator function w.r.t. the event $\mathcal{E}$. Denote $\mathrm{ACC}_{\mathbf{w},\mathbf{b}}(\mathcal{A},\mathcal{B})$ as the classification accuracy of the linear model $\mathbf{y}=\langle\mathbf{w},\mathbf{x}\rangle+\mathbf{b}$ on the point sets $\mathcal{A}\cup\mathcal{B}$, and denote $\mathrm{ACC}_{\mathrm{line}}(\mathcal{A},\mathcal{B}):=\max_{\mathbf{w}, \mathbf{b}\in\mathbb{R}^{N}}\{\mathrm{ACC}_{\mathbf{w},\mathbf{b}}(\mathcal{A },\mathcal{B})\}$ as the maximum classification accuracy of all possible linear models. The following theorem illustrates the relationship between $\mathrm{LS}_{}(\mathcal{A},\mathcal{B})$ and the maximum linearly separable subsets: Theorem 7.: _Given two point sets $\mathcal{A}$ and $\mathcal{B}$, then_ \[\mathrm{ACC}_{\mathrm{line}}(\mathcal{A},\mathcal{B})=\frac{\|\mathcal{A}_{0}\|+ \|\mathcal{B}_{0}\|}{\|\mathcal{A}\|+\|\mathcal{B}\|}\geq\mathrm{LS}_{}(\mathcal{A},\mathcal{B})\geq\max\left\{\frac{\|\mathcal{A}_{0}\|}{\|\mathcal{A}\|},\frac{\| \mathcal{B}_{0}\|}{\|\mathcal{B}\|}\right\},\] _and the equality holds if and only if the sets $\mathcal{A}$ and $\mathcal{B}$ are linearly separable._ This result also reveals that the relationship between the linear separability and the classification accuracy of linear models. Although the classification accuracy can be used to evaluate the performance of a network, it is still challenging to track the behavior of each hidden layer during its training phase. This theorem provides a way of evaluating the linear separability of the outputs of each hidden layer during the training phase. ### Computation of MD-LSM Since the computation of $\mathrm{LS}_{}(\mathcal{A},\mathcal{B})$ is NP-hard, we consider a variant of $\mathrm{LS}_{}(\mathcal{A},\mathcal{B})$: \[\mathrm{LS}_{0}(\mathcal{A},\mathcal{B}):=\max_{\mathbf{\omega}\in \mathbb{R}^{N}}\left\{\max\left\{\frac{\sum\limits_{i\leq I,j\leq J}\mathrm{ sgn}(\mathbf{\omega}^{T}\mathbf{m}_{ij})}{\|\mathrm{MD}(\mathcal{A},\mathcal{B})\|}, \frac{\sum\limits_{i\leq I,j\leq J}\mathrm{sgn}(-\mathbf{\omega}^{T}\mathbf{m}_{ ij})}{\|\mathrm{MD}(\mathcal{A},\mathcal{B})\|}\right\}\right\}, \tag{2}\] where $\mathrm{sgn}(\cdot)$ is a sign function. The following lemma demonstrates that $\mathrm{LS}_{}(\mathcal{A},\mathcal{B})$ and $\mathrm{LS}_{0}(\mathcal{A},\mathcal{B})$ are equivalent from the perspective of maximum linearly separable subset: Lemma 8.: _Given two sets $\mathcal{A},\mathcal{B}$, let $\mathbf{\omega}_{}$ and $\mathbf{\omega}_{0}$ be the weight vectors achieving the maximum operations in (1) and (2), respectively. Then, it holds that $\mathrm{major}_{\mathbf{\omega}_{}}(\mathrm{MD}(\mathcal{A},\mathcal{B}))=\mathrm{ major}_{\mathbf{\omega}_{0}}(\mathrm{MD}(\mathcal{A},\mathcal{B}))$._ The proof of this lemma is direct, so we omit it here. It shows that the points lying in the major sides of the two hyperplanes $\mathbf{\omega}_{}^{T}\mathbf{m}=0$ and $\mathbf{\omega}_{0}^{T}\mathbf{m}=0$ are the same. Unfortunately, it is still NP-hard to solve the optimization problem (2). Instead, we consider the following term: \[\text{LS}_{1}(\mathcal{A},\mathcal{B}):=\max_{\mathbf{\omega}\in\mathbb{R}^{N}} \left\{\left\|\sum_{i\leq I,j\leq J}\mathbf{\omega}^{T}\mathbf{m}_{ij}\right\|\!\! \middle/\sum_{i\leq I,j\leq J}\left\|\mathbf{\omega}^{T}\mathbf{m}_{ij}\right\|\right\}. \tag{3}\] The numerator $\|\sum_{i,j}\mathbf{\omega}^{T}\mathbf{m}_{ij}\|$ is the absolute value of the sum of the directed distances from the points of $\text{MD}(\mathcal{A},\mathcal{B})$ to the hyperplane $\mathbf{\omega}^{T}\mathbf{m}=0$. If all points of $\text{MD}(\mathcal{A},\mathcal{B})$ locate in one side of $\mathbf{\omega}^{T}\mathbf{m}=0$, _i.e.,_ the two sets $\mathcal{A},\mathcal{B}$ are linearly separable, it holds that $\text{LS}_{1}(\mathcal{A},\mathcal{B})=1$. In contrast, if the value of $\text{LS}_{1}(\mathcal{A},\mathcal{B})$ is close to zero, the convex hulls of the two sets $\mathcal{A},\mathcal{B}$ overlap heavily. Because of the existence of absolute value operation, it is still difficult to solve the optimization problem (3). Thus, we consider its alternative version: \[\text{LS}_{2}(\mathcal{A},\mathcal{B}):=\max_{\mathbf{\omega}\in\mathbb{R}^{N}} \left\{\left(\sum_{i\leq I,j\leq J}\mathbf{\omega}^{T}\mathbf{m}_{ij}\right)^{2} \middle/\sum_{i\leq I,j\leq J}\left(\mathbf{\omega}^{T}\mathbf{m}_{ij}\right)^{2} \right\}. \tag{4}\] Let $\widetilde{\mathbf{m}}:=\sum_{i\leq I,j\leq J}\mathbf{m}_{ij}$ be the sum of all points of $\text{MD}(\mathcal{A},\mathcal{B})$ and define \[\mathbf{M}=[\mathbf{m}_{11},\cdots,\mathbf{m}_{1J},\cdots,\mathbf{m}_{i1}, \cdots,\mathbf{m}_{iJ},\cdots,\mathbf{m}_{I1},\cdots,\mathbf{m}_{IJ}]_{N\times IJ}\] as the matrix with the points of $\text{MD}(\mathcal{A},\mathcal{B})$ being its column vectors. Then, the optimization problem (4) can be rewritten as \[\text{LS}_{2}(\mathcal{A},\mathcal{B})=\max_{\mathbf{\omega}\in\mathbb{R}^{N}} \left\{\frac{\mathbf{\omega}^{T}\widetilde{\mathbf{m}}\widetilde{\mathbf{m}}^{T} \mathbf{\omega}}{\mathbf{\omega}^{T}\mathbf{MM}^{T}\mathbf{\omega}}\right\}, \tag{5}\] which has an equivalent form: \[\max_{\mathbf{\omega}\in\mathbb{R}^{N}}\ \ \mathbf{\omega}^{T}\widetilde{\mathbf{m}} \widetilde{\mathbf{m}}^{T}\mathbf{\omega}\ \ \ \text{s.t.}\ \ \ \mathbf{\omega}^{T}\mathbf{MM}^{T}\mathbf{\omega}=1\ \ \ \Longleftrightarrow\ \ \max_{\overline{\mathbf{\omega}}\in\mathbb{R}^{N}}\ \ \overline{\mathbf{\omega}}^{T}\mathbf{\Sigma}\overline{\mathbf{\omega}}\ \ \ \text{s.t.}\ \ \ \overline{\mathbf{\omega}}^{T}\overline{\mathbf{\omega}}=1, \tag{6}\] where $\mathbf{\Sigma}:=(\mathbf{MM}^{T})^{-\frac{1}{2}}\widetilde{\mathbf{m}}\widetilde{ \mathbf{m}}^{T}(\mathbf{MM}^{T})^{-\frac{1}{2}}$ and $\overline{\mathbf{\omega}}:=(\mathbf{MM}^{T})^{1/2}\mathbf{\omega}$. Then, $\text{LS}_{2}(\mathcal{A},\mathcal{B})$ should be the largest eigenvalue of $\mathbf{\Sigma}$ and the solution $\widehat{\mathbf{\omega}}=(\mathbf{MM}^{T})^{-\frac{1}{2}}\overline{\mathbf{\omega}}^ {\dagger}$, where $\overline{\mathbf{\omega}}^{\dagger}$ is the eigenvector associated with the largest eigenvalue of $\mathbf{\Sigma}$. Then, we obtain a closed form of the solution to (4). Furthermore, the resultant $\widehat{\mathbf{\omega}}$ will be substituted into Eqs. (1)-(3) to achieve the approximate calculation of the MD-LSMs LS${}_{}$, LS${}_{0}$ and LS${}_{1}$, respectively. We note that, in the case of large sample size or high network width or both of them, it is still time-consuming to perform the eigenvalue decomposition on the hidden-layer outputs $\text{hid}(\mathcal{X})$ after each time of updating network weights. Thus, we adopt a more efficient way to obtain the approximate values of LS${}_{}$, LS${}_{0}$, LS${}_{1}$ and LS${}_{2}$ in the subsequent numerical experiments (_cf._ Remark 15). In addition, we present a theorem to show that the linear transformation cannot effect the linear separability of original point sets: Theorem 9 (Linear Transformation).: _Let $\mathcal{A}=\{\mathbf{a}_{1},\cdots,\mathbf{a}_{I}\}\subset\mathbb{R}^{N}$ and $\mathcal{B}=\{\mathbf{b}_{1},\cdots,\mathbf{b}_{J}\}\subset\mathbb{R}^{N}$ be two point sets. Denote $\mathbf{V}=[\mathbf{v}_{1},\cdots,\mathbf{v}_{H}]_{N\times H}$ as the weight matrix, where $\mathbf{v}_{h}=(v_{h1},\cdots,v_{hN})^{T}\in\mathbb{R}^{N}$. Define $\mathbf{V}(\mathcal{A})=\{\mathbf{V}^{T}\mathbf{a}_{i}:1\leq i\leq I\}$ and $\mathbf{V}(\mathcal{B})=\{\mathbf{V}^{T}\mathbf{b}_{j}:1\leq j\leq J\}$. Then, it holds that $\text{LS}_{i}(\mathcal{A},\mathcal{B})=\text{LS}_{i}(\mathbf{V}(\mathcal{A}), \mathbf{V}(\mathcal{B}))$ ($i=,0,1,2$)._ This theorem demonstrates that the linear transformation $\mathbf{V}(\mathbf{x})=\mathbf{V}^{T}\mathbf{x}$ cannot change the linear separability degree of the original point sets. Therefore, it is necessary to equip the nodes of hidden layers with non-linear activation functions. At the end of this section, we define the MD-LSMs for multiple point sets: Definition 10* (MD-LSMs for Multi-class Classification).: _Given $S$ point sets $\mathcal{A}_{1},\cdots,\mathcal{A}_{S}$, denote $\mathcal{A}_{s}^{c}=\bigcup_{t\in\{1,\cdots,S\}\setminus\{s\}}\mathcal{A}_{t}$. Then, the MD-LSM for the $S$ points sets is defined as:_ \[\mathrm{MultiLS}_{i}\big{(}\mathcal{A}_{1},\cdots,\mathcal{A}_{S}\big{)}=\sum _{s=1}^{S}\frac{\|\mathcal{A}_{s}\|\cdot\mathrm{LS}_{i}(\mathcal{A}_{s}, \mathcal{A}_{s}^{c})}{\sum_{s=1}^{S}\big{\|}\mathcal{A}_{s}\big{\|}},\quad i=,0, 1,2.\] In the one-vs-rest (OvR) way, we break down an $S$-class classification task into $S$ binary classification tasks and then compute the individual $\mathrm{LS}_{i}(\mathcal{A}_{s},\mathcal{A}_{s}^{c})$ of each task. Then, the MD-LSM $\mathrm{MultiLS}_{i}\big{(}\mathcal{A}_{1},\cdots,\mathcal{A}_{S}\big{)}$ of the $S$-class sample sets is expressed as a sum of $\mathrm{LS}_{i}(\mathcal{A}_{s},\mathcal{A}_{s}^{c})$ weighted by the ratio of the size of $\mathcal{A}_{s}$ to the size of all samples. ## 3 Effects of Activation Function and Network Size In this section, we study the effects of activation function and network size on the linear separability degree of hidden layer outputs. ### Activation Functions Consider a hidden layer of a neural network that contains $H$ hidden nodes, and let $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ be an activation function. Given an input $\mathbf{x}=(x_{1},\cdots,x_{N})^{T}\in\mathbb{R}^{N}$, denote $\mathbf{V}=[\mathbf{v}_{1},\cdots,\mathbf{v}_{H}]_{N\times H}$ as the weight matrix, where $\mathbf{v}_{h}=(v_{h1},\cdots,v_{hN})^{T}\in\mathbb{R}^{N}$ is the weight vector between the input and the $h$-th hidden node. Then, the hidden layer of neural networks can be regarded as a pseudo-linear mapping (PLM) in the following form: \[\mathbf{V}_{\sigma}(\mathbf{x}):=\left(\sigma(\langle\mathbf{v}_{1},\mathbf{x }\rangle),\cdots,\sigma(\langle\mathbf{v}_{H},\mathbf{x}\rangle)\right)^{T}.\] Especially, if $\mathbf{V}$ is a random matrix, the mapping $\mathbf{V}_{\sigma}(\mathbf{x})$ is called the random PLM accordingly. Denote $\mathrm{MD}(\mathbf{V}_{\sigma}(\mathcal{A}),\mathbf{V}_{\sigma}(\mathcal{B}) )=\{\mathbf{n}_{ij}:=\mathbf{V}_{\sigma}(\mathbf{a}_{i})-\mathbf{V}_{\sigma} (\mathbf{b}_{j})\mid 1\leq i\leq I,\ 1\leq j\leq J\}$ as the Minkowski difference of two transformed sets $\mathbf{V}_{\sigma}(\mathcal{A})$ and $\mathbf{V}_{\sigma}(\mathcal{B})$. The following theorem shows that PLMs have the capability of changing the linear separability of the original point sets $\mathcal{A}$ and $\mathcal{B}$. Theorem 11.: _Let $\mathbf{V}_{\sigma}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{H}$ be a pseudo-linear mapping and $\sigma(x):\mathbb{R}\rightarrow\mathbb{R}$ be a second order derivable function with non-negative first derivative. Assume that an MD point $\mathbf{m}_{ij}=\mathbf{a}_{i}-\mathbf{b}_{j}$ lies in the major (resp. minor) side of the hyperplane $\boldsymbol{\omega}^{T}\mathbf{m}=0$. Define_ \[F_{\sigma}(x,y):=\frac{\big{[}\sigma^{\prime\prime}(x)-\sigma^{\prime\prime}(y )\big{]}(x-y)}{\big{[}\sigma^{\prime}(x)+\sigma^{\prime}(y)\big{]}},\ \ (x,y)\in\mathbb{R}\times \mathbb{R}.\] _If the relation $F_{\sigma}(\langle\mathbf{a}_{i},\mathbf{v}_{h}\rangle,\langle\mathbf{b}_{j}, \mathbf{v}_{h}\rangle)>2$ holds for any $h\in\{1,2,\cdots,H\}$, then the MD point $\mathbf{n}_{ij}=\mathbf{V}_{\sigma}(\mathbf{a}_{i})-\mathbf{V}_{\sigma}( \mathbf{b}_{j})$ lies in the minor (resp. major) side of the hyperplane $(\boldsymbol{\omega}^{T}\mathbf{V})\mathbf{n}=0$._ This result interprets that if the weights $\mathbf{v}_{1},\cdots,\mathbf{v}_{H}$ satisfy that $F_{\sigma}(\langle\mathbf{a}_{i},\mathbf{v}_{h}\rangle,\langle\mathbf{b}_{j}, \mathbf{v}_{h}\rangle)>2$ for any $h\in\{1,2,\cdots,H\}$, the MD point $\mathbf{n}_{ij}$ has a different relatively positional relation with the hyperplane $(\boldsymbol{\omega}^{T}\mathbf{V})\mathbf{n}=0$. In other words, if the original MD point $\mathbf{m}_{ij}$ lies in the major side of $\boldsymbol{\omega}^{T}\mathbf{m}=0$, the transformed MD point $\mathbf{n}_{ij}$ lies in the minor side of $(\mathbf{\omega}^{T}\mathbf{V})\mathbf{n}=0$. As shown in Fig. 2, there is a wide choice of the activation function $\sigma$, such as sigmoid, $\tanh$, $\arctan$ and softsign. ### Network Size By treating the multiple hidden layers as the composition of multiple PLMs, we consider the effect of the network size (including width and depth) on the linear separability. Denote \[\mathbf{V}_{\sigma}^{L}(\mathbf{a}):=\prod_{l=1}^{L}\mathbf{V}_{\sigma}^{(l)}( \mathbf{a})=\mathbf{V}_{\sigma}^{(L)}\circ\mathbf{V}_{\sigma}^{(L-1)}\circ \cdots\circ\mathbf{V}_{\sigma}^{(1)}(\mathbf{a}),\] where $\mathbf{V}_{\sigma}^{(1)},\cdots,\mathbf{V}_{\sigma}^{(L)}$ are independent random PLMs with $\mathbf{V}^{(l)}\in\mathbb{R}^{H_{l-1}\times H_{l}}$, and the operation $\circ$ stands for the composition of two PLMs. Denote $\mathbf{V}_{\sigma}^{L}(\mathcal{A})=\{\mathbf{V}_{\sigma}^{L}(\mathbf{a}): \mathbf{a}\in\mathcal{A}\}$ as the set of the $L$-th hidden layer outputs w.r.t. the input set $\mathcal{A}$. For any $1\leq l\leq L$, let \[\mathrm{MD}(\mathbf{V}_{\sigma}^{(l)}(\mathcal{A}),\mathbf{V}_{\sigma}^{(l)} (\mathcal{B})):=\{\mathbf{n}_{ij}^{(l)}:=\mathbf{V}_{\sigma}^{(l)}(\mathbf{a }_{i})-\mathbf{V}_{\sigma}^{(l)}(\mathbf{b}_{j})\mid 1\leq i\leq I,\;1\leq j \leq J\}\] be the Minkowski difference of two transformed sets $\mathbf{V}_{\sigma}^{(l)}(\mathcal{A})$ and $\mathbf{V}_{\sigma}^{(l)}(\mathcal{B})$. Let $\widetilde{\mathbf{n}}^{(l)}=\sum_{i\leq I,j\leq J}\mathbf{n}_{ij}^{(l)}$ be the sum of all hidden-layer outputs and $\mathbf{N}^{(l)}=[\mathbf{n}_{1}^{(l)},\cdots,\mathbf{n}_{IJ}^{(l)}]_{H_{l} \times(IJ)}$ be the matrix with the points of $\mathrm{MD}(\mathbf{V}_{\sigma}^{(l)}(\mathcal{A}),\mathbf{V}_{\sigma}^{(l)} (\mathcal{B}))$ being its column vectors. For any $1\leq l\leq L$, set \[\mathbf{\Omega}^{(l)}:=[\mathbf{N}^{(l)}(\mathbf{N}^{(l)})^{T}]^{-\frac{1}{2}} \widetilde{\mathbf{n}}^{(l)}(\widetilde{\mathbf{n}}^{(l)})^{T}[\mathbf{N}^{( l)}(\mathbf{N}^{(l)})^{T}]^{-\frac{1}{2}}.\] Some empirical evidences have shown that if the number of hidden nodes is larger than the dimension of the original data, the linear separability degree of the hidden layer outputs is likely to be larger than that of the original data. By using Tropp's tail inequality for random matrices [45, Theorem 1.3], we study the relationship between the linear separability and the number of hidden nodes: Theorem 12* (Network Width).: _If $\mathbb{E}\mathbf{\Omega}^{(1)}=0$ and $(\mathbf{\Omega}^{(1)})^{2}\preceq\mathbf{A}^{2}$, then_ \[\mathbb{P}\{\mathrm{LS}_{2}(\mathbf{V}_{\sigma}^{(1)}(\mathcal{A}),\mathbf{V }_{\sigma}^{(1)}(\mathcal{B}))>\mathrm{LS}_{2}(\mathcal{A},\mathcal{B})\}\leq H \cdot\mathrm{e}^{-\frac{(\mathrm{LS}_{2}(\mathcal{A},\mathcal{B}))^{2}}{8 \sigma^{2}}}, \tag{7}\] _where $\sigma^{2}:=\\|\mathbf{A}^{2}\\|_{s}$, $\mathbf{A}\preceq\mathbf{B}$ means that $\mathbf{B}-\mathbf{A}$ is a positive semidefinite matrix and $\\|\cdot\\|_{s}$ stands for the spectral norm._ This result addresses that with probability at most $H\mathrm{e}^{-(\mathrm{LS}_{2}(\mathcal{A},\mathcal{B}))^{2}/8\sigma^{2}}$, the value of $\mathrm{LS}_{2}(\mathbf{V}_{\sigma}^{(1)}(\mathcal{A}),\mathbf{V}_{\sigma}^{( 1)}(\mathcal{B}))$ is larger than that of $\mathrm{LS}_{2}(\mathcal{A},\mathcal{B})$. Namely, if we would like to improve the linear separability degree Figure 2: The ranges of $F_{\sigma}(x,y)$ for different activation functions. of the original sets $\mathcal{A}$ and $\mathcal{B}$, one feasible way is to map them into a higher-dimensional space via non-linear mappings, _i.e.,_ to increase the network width. Moreover, if the linear separability degree of $\mathcal{A}$ and $\mathcal{B}$ reaches a relatively high level, the probability that the random PLM $\mathbf{V}_{\sigma}^{(1)}(\cdot)$ can further improve their linear separability degree becomes low. Corollary 13 (Network Depth).: _Assume that $\mathbb{E}\mathbf{\Omega}^{(l)}=0$ and $(\mathbf{\Omega}^{(l)})^{2}\preceq(\mathbf{A}^{(l)})^{2}$ holds for any $1\leq l\leq L$. Then,_ \[\mathbb{P}\{\mathrm{LS}_{2}(\mathbf{V}_{\sigma}^{L}(\mathcal{A}),\mathbf{V}_{ \sigma}^{L}(\mathcal{B}))>\mathrm{LS}_{2}(\mathcal{A},\mathcal{B})\}\leq LH \cdot\mathrm{e}^{-C_{0}^{2}/(8\sigma_{0}^{2})}, \tag{8}\] _where $\sigma_{0}^{2}=\max_{l}\\|(\mathbf{A}^{(l)})^{2}\\|_{s}$ and $C_{0}:=\min_{l}\mathrm{LS}_{2}(\mathbf{V}_{\sigma}^{(l)}(\mathcal{A}),\mathbf{ V}_{\sigma}^{(l)}(\mathcal{B}))$._ This result suggests that adding more hidden layers is also a feasible way to increase the linear separability degree of the data points when they pass through a neural network. Remark 14.: _We note that the results (7) and (8) are not of the optimal forms. For example, it is expected to find a monotonically decreasing function $\rho(H)$ w.r.t. the network width $H$ such that_ \[\mathbb{P}\big{\{}\mathrm{LS}_{2}(\mathbf{V}_{\sigma}^{(1)}( \mathcal{A}),\mathbf{V}_{\sigma}^{(1)}(\mathcal{B}))\leq\mathrm{LS}_{2}( \mathcal{A},\mathcal{B})\big{\}}\leq\rho(H), \tag{9}\] _which can be equivalently interpreted as follows: with probability at least $1-\rho(H)$, it holds that $\mathrm{LS}_{2}(\mathbf{V}_{\sigma}^{(1)}(\mathcal{A}),\mathbf{V}_{\sigma}^{( 1)}(\mathcal{B}))>\mathrm{LS}_{2}(\mathcal{A},\mathcal{B})$. Unfortunately, it is still technically difficult to obtain the appropriate small-ball probability inequalities for the largest eigenvalues of random matrices to achieve the probability inequality of the form (9)._ ## 4 Numerical Experiments In this section, we conduct the experiments to examine the synchronicity between the linear separability degree of hidden layer outputs and the network training performance of several popular networks including MLP, CNN, DBN, ResNet-20, VGGNet, AlexNet, ViT and GoogLeNet-V1. The experimental results also validate the theoretical findings in Theorem 12 and Corollary 13. All experiments are processed in the DELL(r) PowerEdge(r) T640 Tower Server with two Intel(r) Xeon(r) 20-core processors, 128 GB RAM and a NVIDIA(r) Tesla(r) V100 32GB GPU. ### Experiment Setting Two categories (airplane and automobile) in CIFAR-10 dataset [46] are selected to form the binary classification task. The SGD method with minibatch is used to updated the network weights within 100 training epochs (including the fine-tuning phase of DBN). Since the structures of MLP, CNN and DBN are not powerful enough to obtain a good training performance by using all samples of the two categories within the limited epochs, we randomly select 2000 (resp. 1000) samples from the training (resp. testing) data of the two categories for training (resp. testing) the three kinds of networks. Moreover, since the network size of MLP, CNN and DBN is not large, we directly use the selected 2000 training samples to compute the MD-LSMs of their hidden layer outputs. In contrast, we use all training (resp. testing) data of the two categories to train (resp. test) ResNet, VGGNet, AlexNet, ViT and GoogLeNet. Since the dimension of hidden layer outputs of these deep networks is high, in view of the computational burden, we randomly select 500 samples from the two categories to compute MD-LSMs for these networks after each training epoch. Figure 3: Synchronicity between the LS${}_{1}$ value (the left of each subfigure) of hidden layer outputs and training (or testing) accuracy (the right of each subfigure) of different neural networks, where L1 and B1 stand for the 1st hidden layer and the 1st block, respectively. Since the ranges of LS${}_{1}$, training accuracy and testing accuracy are all the interval $[0,1]$, the corresponding curves share the same $y$ label in each subfigure. Remark 15 (Approximate Calculation of MD-LSMs).: _Recall the optimization problem (6) for computing $\mathrm{LS}_{2}$. Unfortunately, it is still very time-consuming to implement the eigenvalue decomposition of $\mathbf{\Sigma}$ after each training epoch, especially when the dimension of $\mathbf{\Sigma}$ is high. Thus, in the numerical experiments, the weight vectors achieving the MD-LSMs (1), (2), (3) and (4) are approximately computed after each training epoch via the following optimization problem:_ \[\max_{\mathbf{\omega}\in\mathbb{R}^{N}}\ \mathbf{\omega}^{T}\widetilde{\mathbf{m}} \widetilde{\mathbf{m}}^{T}\mathbf{\omega}\quad\text{s.t.}\quad\mathbf{\omega}^{T}\mathbf{ \omega}=1, \tag{10}\] _whose solution is $\widehat{\mathbf{\omega}}=\frac{\widetilde{\mathbf{m}}}{\\|\widetilde{\mathbf{m}} \\|}$. Compared with the original form (6), this approximate one actually is derived by setting $\mathbf{M}\mathbf{M}^{T}=\mathbf{I}$._ ### Experimental Results and Discussion In Fig. 3, we illustrate the synchronicity between linear separability of hidden layers and training performance in the classification tasks. Since $\mathrm{LS}_{0}$, $\mathrm{LS}_{1}$ and $\mathrm{LS}_{2}$ basically have the same experimental results, we only draw the $\mathrm{LS}_{1}$ curves for all hidden layers of MLP, CNN and DBN and for the main blocks of AlexNet, GoogLeNet, ResNet, VGGNet and ViT, respectively. The complete experimental report, containing $\mathrm{LS}_{0}$, $\mathrm{LS}_{1}$ and $\mathrm{LS}_{2}$ curves for all hidden layers of these networks, is arranged in the appendix (part B). Moreover, we also provide the detailed structures of these neural networks with the name of each hidden layer to facilitate the interpretation of experimental results. [Binary Classification] As shown in Figs. 3(a)-3(j), there is an obvious synchronicity between the $\mathrm{LS}_{1}$ curves of hidden layers and the accuracy curves: 1) when the training accuracy increases, the $\mathrm{LS}_{1}$ value of the outputs of each hidden layer (or main block) increases synchronously; 2) when some fluctuations appear in the $\mathrm{LS}_{1}$ curves, the training and the testing accuracy curves have the fluctuations occurring nearby the corresponding epochs accordingly; 3) especially for the neural networks with relatively shallow structures, such as MLP and CNN (_cf._ Figs. 3(a)-3(e)), the magnitude of the fluctuations in the $\mathrm{LS}_{1}$ curves is merely proportional to that of the fluctuations in the training and the testing accuracy curves. [Network Size] The experimental results, given in Figs. 3(a)-3(k), also reflect two facts: 1) in most cases, the linear separability of the hidden layers (or blocks) is stronger than that of the original data after a few training epochs; and 2) the hidden layers (or blocks), which are closer to the output layer, have higher linear separability. These phenomena coincide with the theoretical findings in Theorem 12 and Corollary 13, which have proven that the large network size is more likely to provide high linear separability of the hidden layers. [Multi-class Classification] We also consider the linear separability of MLP for ten-class classification task. The experiment is conducted by using MLP to classify the MINST dataset [32]. We adopt the one-vs-rest (OvR) way to build ten MLPs with the same structure. After each training epoch, we compute the MD-LSMs of all hidden layer outputs of each CNN in the way mentioned in Definition 10. As shown in Fig. 3(k), we obtain the same experiment observations as binary classification tasks and support the theoretical findings as well. [Layer-wise Pretraining] We use the linear separability to analyze the layer-wise pretraining strategy, implemented by using restricted Boltzmann machine (RBM), of DBN in binary classification task. During the pretraining phase, the weights of each hidden layers are updated 100 times. After that, all weights of the network are fine-tuned in 100 epochs by using the SGD method. RBM is an energy minimization method and the layer-wise pretraining aims to find the suitable weights such that the energy between two adjacent hidden layers decays to the minimal status. As shown in Fig. 17, the RBM-based pretraining not only fails to increase the linear separability degree of each hidden layer but also makes it lower than the original data. However, the subsequent fine-tuning process can quickly improve the linear separability of hidden layers and meanwhile the network achieves a good performance. This phenomenon implies that the RBM-based pretraining makes each hidden layer capture useful features, which are not beneficial to increasing the linear separability degree yet. The reason is that the RBM method is independent of the label information, while the evaluation of linear separability is label-based. Interestingly, during the pretraining phase, the training and the testing accuracies are both round the value of 0.5. This phenomenon implies that the network always maintains the largest randomness and diversity, which will benefit to the subsequent fine-tuning process. ## 5 Difference between MD-LSM and Fisher LDA In this section, we make a comparison between $\text{LS}_{2}(\mathcal{A},\mathcal{B})$ and Fisher LDA. First, we present a theorem to show their differences, and then explain why we use MD-LSM to evaluate the linear separability degree of two sets instead of Fisher LDA. Let $\boldsymbol{\mu}_{a}=\frac{1}{I}\sum_{i=1}^{I}\mathbf{a}_{i}$ and $\boldsymbol{\mu}_{b}=\frac{1}{J}\sum_{j=1}^{J}\mathbf{b}_{j}$ be the centers of the sets $\mathcal{A}$ and $\mathcal{B}$, respectively. Let $\mathbf{A}_{c}$ (resp. $\mathbf{B}_{c}$) be the matrix associated with the set $\mathcal{A}$ (resp. $\mathcal{B}$) whose columns consist of the mean shifted data points: \[\mathbf{A}_{c}:=[\mathbf{a}_{1}-\boldsymbol{\mu}_{a},\cdots,\mathbf{a}_{I}- \boldsymbol{\mu}_{a}]\;\;\text{and}\;\;\mathbf{B}_{c}:=[\mathbf{b}_{1}- \boldsymbol{\mu}_{b},\cdots,\mathbf{b}_{J}-\boldsymbol{\mu}_{b}].\] Denote $\mathbf{S}_{w}=\mathbf{A}_{c}\mathbf{A}_{c}^{T}+\mathbf{B}_{c}\mathbf{B}_{c}^ {T}$ and $\mathbf{S}_{b}=(\boldsymbol{\mu}_{a}-\boldsymbol{\mu}_{b})(\boldsymbol{\mu}_{a }-\boldsymbol{\mu}_{b})^{T}$. The objective function of Fisher LDA can also be treated as an LSM: \[J_{\omega}=\max_{\boldsymbol{\omega}}\frac{\boldsymbol{\omega}^{T}\mathbf{S}_{ b}\boldsymbol{\omega}}{\boldsymbol{\omega}^{T}\mathbf{S}_{w}\boldsymbol{\omega}}. \tag{11}\] The following results show the difference between MD-LSM and LDA-LSM. Proposition 16.: _Given two point sets $\mathcal{A}=\{\mathbf{a}_{1},\cdots,\mathbf{a}_{I}\}$ and $\mathcal{B}=\{\mathbf{b}_{1},\cdots,\mathbf{b}_{J}\}$, then it holds that_ \[I^{2}J^{2}\mathbf{S}_{b} = \widetilde{\mathbf{m}}\widetilde{\mathbf{m}}^{T};\] \[\mathbf{S}_{w} = \mathbf{A}\mathbf{A}^{T}+\mathbf{B}\mathbf{B}^{T}-\Big{(}I( \widehat{\mathbb{E}}\mathbf{a})(\widehat{\mathbb{E}}\mathbf{a})^{T}+J( \widehat{\mathbb{E}}\mathbf{b})(\widehat{\mathbb{E}}\mathbf{b})^{T}\Big{)};\] \[\mathbf{M}\mathbf{M}^{T} = J\mathbf{A}\mathbf{A}^{T}+I\mathbf{B}\mathbf{B}^{T}-IJ\cdot \widehat{\mathbb{E}}\{\mathbf{a}\mathbf{b}^{T}+\mathbf{b}\mathbf{a}^{T}\},\] _where $\widehat{\mathbb{E}}$ stands for the sample mean with_ \[\widehat{\mathbb{E}}\mathbf{a} = \frac{1}{I}\sum_{i=1}^{I}\mathbf{a}_{i};\] \[\widehat{\mathbb{E}}\mathbf{b} = \frac{1}{J}\sum_{j=1}^{J}\mathbf{b}_{j};\] \[\widehat{\mathbb{E}}\{\mathbf{a}\mathbf{b}^{T}+\mathbf{b} \mathbf{a}^{T}\} = \frac{1}{IJ}\sum_{\begin{subarray}{c}1\leq i\leq I\\ 1\leq j\leq J\end{subarray}}(\mathbf{a}_{i}\mathbf{b}_{j}^{T}+\mathbf{b}_{j} \mathbf{a}_{i}^{T}).\] As demonstrated above, since $\mathbf{S}_{w}$ differs from $\mathbf{M}\mathbf{M}^{T}$, the hyperplane $\boldsymbol{\omega}^{T}\mathbf{m}=0$ achieving $\text{LS}_{2}(\mathcal{A},\mathcal{B})$ is different from the one achieving $J_{\omega}$. When we use the approximate way, mentioned in Remark 15, to compute the weight for the MD-LSMs, the corresponding optimization objective function coincides with that of Fisher LDA with $\mathbf{S}_{b}=\mathbf{I}$ (_cf._ Eq. (11)). In spite of the same weight vector $\mathbf{\omega}$ derived from the approximated form, the linear separability degree is still evaluated in different forms after substituting $\mathbf{\omega}$ into the expressions of MD-LSMs (including LS${}_{}$, LS${}_{0}$, LS${}_{1}$ and LS${}_{2}$) and $J_{\omega}$, respectively. Moreover, the ranges of LS${}_{0}$ and LS${}_{1}$ are the interval $(0,1]$; and LS${}_{0}=$ LS${}_{1}=1$ holds if and only if the two sets are linearly separable (_cf._ Fig. 4). In contrast, the ranges of LS${}_{2}$ and $J_{\omega}$ are the interval $(0,+\infty)$, and they only provide the relative reference values for the linear separability. Thus, it is difficult to estimate the linear separability degree of two sets only based on the values of LS${}_{2}$ and $J_{\omega}$. Moreover, as shown in Figs. 5 - 8, there are fewer large fluctuations appearing in the curves of LS${}_{0}$ and LS${}_{1}$ than in the curves of LS${}_{2}$ and $J_{\omega}$. Interestingly, the curve shapes of LS${}_{0}$, LS${}_{1}$ and LS${}_{2}$ are the same, but they significantly differ from that of the curves of $J_{\omega}$. Therefore, we finally adopt the MD-LSMs as the measures of evaluating the linear separability degree of hidden layer outputs. ## 6 Proofs of Main Results In this section, we give the proofs of Theorem 1, Theorem 3, Theorem 7, Theorem 9, Theorem 11, Corollary 13 and Proposition 16, respectively. ### Proof of Theorem 1 Proof of Theorem 1:* "$\Longrightarrow$" If the classification accuracy increases, it means that more hidden-layer outputs can be correctly separated by using a hyperplane $\mathbf{w}^{T}\mathbf{x}+\mathbf{b}=0$. Namely, the linear separability of hidden layer outputs increases. Figure 4: Evaluation of Linear Separability Degree of Different Point Sets Based on MD-LSMs and Fisher LDA. Figure 5: Curves of MD-LSMs and $J_{\omega}$ for Hidden Layers of MLP-5. Figure 6: Curves of MD-LSMs and $J_{\omega}$ for Hidden Layers of CNN Figure 8: Curves of MD-LSMs and $J_{\omega}$ for Hidden Layers of AlexNet (Sigmoid) Figure 7: Curves of MD-LSMs and $J_{\omega}$ for Hidden Layers of AlexNet (Softmax) "$\Longleftarrow$" If the linear separability of the hidden layer outputs increases after updating the hidden-layer weights $\mathbf{V}^{\prime}=\mathbf{V}+\triangle\mathbf{V}$, it means that there must exists a hyperplane $\mathbf{w}^{\mathbf{T}}\mathbf{s}+\mathbf{b}=\mathbf{0}$ such that more hidden-layer outputs can be correctly separated. Since the hyperplane $(\mathbf{w}^{\prime})^{\mathbf{T}}\mathbf{s}+\mathbf{b}^{\prime}=\mathbf{0}$ can provide the highest training classification accuracy, the training performance of $\mathrm{net}^{\prime}(\cdot)$ is better than that of $\mathrm{net}(\cdot)$. This completes the proof. $\blacksquare$ ### Proof of Theorem 3 Proof of Theorem 3: "$\Longrightarrow$": If $\mathcal{A}$ and $\mathcal{B}$ are linearly separable, there exists a vector $\boldsymbol{\omega}\in\mathbb{R}^{N}$ and a constant $c\in\mathbb{R}$ such that the relation $\mathbf{w}^{T}\mathbf{a}+c>\mathbf{w}^{T}\mathbf{b}+c$ holds for all $\mathbf{a}\in\mathcal{A}$ and $\mathbf{b}\in\mathcal{B}$. Then, we arrive at $\mathbf{w}^{T}(\mathbf{a}-\mathbf{b})>0$ ($\forall\ \mathbf{a}\in\mathcal{A},\mathbf{b}\in\mathcal{B}$). Namely, all points of the Minkowski difference $\mathrm{MD}(\mathcal{A},\mathcal{B})$ lie above the hyperplane $\mathbf{w}^{T}\mathbf{x}=0$. "$\Longleftarrow$": Assume that all points of $\mathrm{MD}(\mathcal{A},\mathcal{B})$ lie in one side of the hyperplane $\mathbf{w}^{T}\mathbf{x}=0$. Without loss of generality, we consider a vector $\mathbf{w}\in\mathbb{R}^{N}$ such that $\mathbf{w}^{T}(\mathbf{a}-\mathbf{b})>0$ holds for any $\mathbf{a}\in\mathcal{A}$ and any $\mathbf{b}\in\mathcal{B}$. Define $\mathbf{a}^{}:=\arg\min_{\mathbf{a}\in\mathcal{A}}\{\mathbf{w}^{T}\mathbf{a}\}$ and $\mathbf{b}^{\dagger}:=\arg\max_{\mathbf{b}\in\mathcal{B}}\{\mathbf{w}^{T} \mathbf{b}\}$. Then, for all $\mathbf{a}\in\mathcal{A}$ and $\mathbf{b}\in\mathcal{B}$, it holds that \[\mathbf{w}^{T}\mathbf{a}-\frac{\mathbf{w}^{T}\mathbf{a}^{}+\mathbf{w}^{T} \mathbf{b}^{\dagger}}{2}>0>\mathbf{w}^{T}\mathbf{b}-\frac{\mathbf{w}^{T} \mathbf{a}^{}+\mathbf{w}^{T}\mathbf{b}^{\dagger}}{2}.\] Namely, the hyperplane $\mathbf{w}^{T}\mathbf{x}-\frac{\mathbf{w}^{T}\mathbf{a}^{}+\mathbf{w}^{T} \mathbf{b}^{\dagger}}{2}=0$ ($x\in\mathbb{R}^{N}$) separates the set $\mathcal{A}$ from the set $\mathcal{B}$. This completes the proof. $\blacksquare$ ### Proof of Theorem 7 Proof of Theorem 7: (1) The first equality is resulted from the definition of maximum linearly-separable subset. (2) It follows from $\mathcal{A}_{\circ}\subseteq\mathcal{A}$ and $\mathcal{B}_{\circ}\subseteq\mathcal{B}$ that \[\frac{1}{\frac{1}{\|\mathcal{A}\|}+\frac{1}{\|\mathcal{B}\|}}\geq\frac{1}{\frac{1}{ \|\mathcal{A}_{\circ}\|}+\frac{1}{\|\mathcal{B}_{\circ}\|}}\Longleftrightarrow\frac {\|\mathcal{A}_{\circ}\|+\|\mathcal{B}_{\circ}\|}{\|\mathcal{A}\|+\|\mathcal{B}\|} \geq\frac{\|\mathcal{A}_{\circ}\|\cdot\|\mathcal{B}_{\circ}\|}{\|\mathcal{A}\|\cdot\| \mathcal{B}\|}=\mathrm{LS}_{}(\mathcal{A},\mathcal{B}).\] The last equality holds because of the definition of $\mathrm{LS}_{}(\mathcal{A},\mathcal{B})$. (3) Let $\boldsymbol{\omega}_{}$ be the weight vector achieving the MD-LSM $\mathrm{LS}_{}(\mathcal{A},\mathcal{B})$. Denote $\mathcal{A}_{\circ}^{c}=\mathcal{A}\setminus\mathcal{A}_{\circ}$ and $\mathcal{B}_{\circ}^{c}=\mathcal{B}\setminus\mathcal{B}_{\circ}$. Since $\mathrm{minor}_{\boldsymbol{\omega}_{}}(\mathrm{MD}(\mathcal{A},\mathcal{B})) \leq\min\left\{\|\mathcal{A}_{\circ}^{c}\|\cdot\|\mathcal{B}\|,\|\mathcal{B}_{\circ }^{c}\|\cdot\|\mathcal{A}\|\right\}$, we arrive at \[1-\frac{\mathrm{minor}_{\boldsymbol{\omega}_{}}(\mathrm{MD}( \mathcal{A},\mathcal{B}))}{\|\mathcal{A}\|\cdot\|\mathcal{B}\|}\geq\max\left\{1- \frac{\|\mathcal{A}_{\circ}^{c}\|\cdot\|\mathcal{B}\|}{\|\mathcal{A}\|\cdot\|\mathcal{B} \|},1-\frac{\|\mathcal{A}\|\cdot\|\mathcal{B}_{\circ}^{c}\|}{\|\mathcal{A}\|\cdot\| \mathcal{B}\|}\right\}\] \[\Longleftrightarrow \frac{\mathrm{major}_{\boldsymbol{\omega}_{}}(\mathrm{MD}( \mathcal{A},\mathcal{B}))}{\|\mathcal{A}\|\cdot\|\mathcal{B}\|}\geq\max\left\{ \frac{\|\mathcal{A}_{\circ}\|\cdot\|\mathcal{B}\|}{\|\mathcal{A}\|\cdot\|\mathcal{B} \|},\frac{\|\mathcal{A}\|\cdot\|\mathcal{B}_{\circ}\|}{\|\mathcal{A}\|\cdot\| \mathcal{B}\|}\right\}\] \[\Longleftrightarrow \mathrm{LS}_{}(\mathcal{A},\mathcal{B})\geq\max\left\{\frac{\| \mathcal{A}_{\circ}\|}{\|\mathcal{A}\|},\frac{\|\mathcal{B}_{\circ}\|}{\|\mathcal{B}\|} \right\}.\] This completes the proof. $\blacksquare$ ### Proof of Theorem 9 Proof of Theorem 9: Since $\mathbf{V}^{T}\mathbf{m}_{ij}=\mathbf{V}^{T}(\mathbf{a}_{i}-\mathbf{b}_{j})$, we have \[\mathrm{LS}_{0}(\mathbf{V}(\mathcal{A}),\mathbf{V}(\mathcal{B})) =\max_{\mathbf{w}\in\mathbb{R}^{\mathbf{H}}}\left\{\max\left\{ \frac{\sum\limits_{i\leq I,j\leq J}\mathrm{sgn}(\mathbf{w}^{T}\mathbf{V}^{T} \mathbf{m}_{ij})}{\|\mathrm{MD}(\mathbf{V}(\mathcal{A}),\mathbf{V}(\mathcal{B}) )\|},\frac{\sum\limits_{i\leq I,j\leq J}\mathrm{sgn}(-\mathbf{w}^{T}\mathbf{V}^ {T}\mathbf{m}_{ij})}{\|\mathrm{MD}(\mathbf{V}(\mathcal{A}),\mathbf{V}(\mathcal{ B}))\|}\right\}\right\}\] \[=\max_{\boldsymbol{\omega}\in\mathbb{R}^{N}}\left\{\max\left\{ \frac{\sum\limits_{i\leq I,j\leq J}\mathrm{sgn}(\boldsymbol{\omega}^{T} \mathbf{m}_{ij})}{\|\mathrm{MD}(\mathcal{A},\mathcal{B})\|},\frac{\sum\limits_{ i\leq I,j\leq J}\mathrm{sgn}(-\boldsymbol{\omega}^{T}\mathbf{m}_{ij})}{\| \mathrm{MD}(\mathcal{A},\mathcal{B})\|}\right\}\right\}\] \[=\mathrm{LS}_{0}(\mathcal{A},\mathcal{B}).\] The rest can also be proven in the same way. This completes the proof. ### Proof of Theorem 11 Proof of Theorem 11: Denote $\mathbf{m}_{0}=\mathbf{a}-\mathbf{b}$ and $\mathbf{n}_{0}=\mathbf{V}_{\sigma}(\mathbf{a})-\mathbf{V}_{\sigma}(\mathbf{b})$. Without the loss of generality, we assume that the point $\mathbf{m}_{0}$ lies in the major side of the hyperplane $\boldsymbol{\omega}^{T}\mathbf{m}=0$. Consider the following Taylor's expansion: \[\mathbf{V}_{\sigma}(\mathbf{a}) =\] \[= \mathbf{V}_{\sigma}(\mathbf{b})+\left(\begin{array}{c}\sigma^{ \prime}(\left\langle\mathbf{v}_{1},\mathbf{b}\right\rangle)\cdot\left\langle \mathbf{v}_{1},\mathbf{m}_{0}\right\rangle+\frac{\sigma^{\prime\prime}(\left \langle\mathbf{c}_{1},\mathbf{v}_{1}\right\rangle)}{2}\cdot\mathbf{v}_{1}^{T} \mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{1}\\ \vdots\\ \sigma^{\prime}(\left\langle\mathbf{v}_{H},\mathbf{b}\right\rangle)\cdot \left\langle\mathbf{v}_{H},\mathbf{m}_{0}\right\rangle+\frac{\sigma^{\prime \prime}(\left\langle\mathbf{c}_{H},\mathbf{v}_{H}\right\rangle)}{2}\cdot \mathbf{v}_{H}^{T}\mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{H}\end{array} \right)\] \[\approx \mathbf{V}_{\sigma}(\mathbf{b})+\left(\begin{array}{c}\sigma^{ \prime}(\left\langle\mathbf{v}_{1},\mathbf{b}\right\rangle)\cdot\left\langle \mathbf{v}_{1},\mathbf{m}_{0}\right\rangle+\frac{\sigma^{\prime\prime}(\left \langle\mathbf{b},\mathbf{v}_{1}\right\rangle)}{2}\cdot\mathbf{v}_{1}^{T} \mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{1}\\ \vdots\\ \sigma^{\prime}(\left\langle\mathbf{v}_{H},\mathbf{b}\right\rangle)\cdot \left\langle\mathbf{v}_{H},\mathbf{m}_{0}\right\rangle+\frac{\sigma^{\prime \prime}(\left\langle\mathbf{b},\mathbf{v}_{H}\right\rangle)}{2}\cdot\mathbf{v} _{H}^{T}\mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{H}\end{array}\right),\] where $\boldsymbol{\zeta}_{1},\cdots,\boldsymbol{\zeta}_{H}$ are $H$ points lying on the line segment between the two points $\mathbf{a}$ and $\mathbf{b}$. Alternatively, we also have \[\mathbf{V}_{\sigma}(\mathbf{b}) \approx \mathbf{V}_{\sigma}(\mathbf{a})+\left(\begin{array}{c}-\sigma^{ \prime}(\left\langle\mathbf{v}_{1},\mathbf{a}\right\rangle)\cdot\left\langle \mathbf{v}_{1},\mathbf{m}_{0}\right\rangle+\frac{\sigma^{\prime\prime}(\left \left\langle\mathbf{a},\mathbf{v}_{1}\right\rangle)}{2}\cdot\mathbf{v}_{1}^{T} \mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{1}\\ \vdots\\ -\sigma^{\prime}(\left\langle\mathbf{v}_{H},\mathbf{a}\right\rangle)\cdot \left\langle\mathbf{v}_{H},\mathbf{m}_{0}\right\rangle+\frac{\sigma^{\prime \prime}(\left\left\langle\mathbf{a},\mathbf{v}_{H}\right\rangle)}{2}\cdot \mathbf{v}_{H}^{T}\mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{H}\end{array} \right).\] Then, we arrive at \[\mathbf{n}_{0} = \mathbf{V}_{\sigma}(\mathbf{a})-\mathbf{V}_{\sigma}(\mathbf{b})\] \[\approx \left(\begin{array}{c}\frac{\sigma^{\prime}(\langle\mathbf{v}_{1 },\mathbf{b}\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{1},\mathbf{a}\rangle)}{2 }\cdot\langle\mathbf{v}_{1},\mathbf{m}_{0}\rangle+\frac{\sigma^{\prime\prime }(\langle\mathbf{b},\mathbf{v}_{1}\rangle)-\sigma^{\prime\prime}(\langle \mathbf{a},\mathbf{v}_{1}\rangle)}{4}\cdot\mathbf{v}_{1}^{T}\mathbf{m}_{0} \mathbf{m}_{0}^{T}\mathbf{v}_{1}\\ \vdots\\ \frac{\sigma^{\prime}(\langle\mathbf{v}_{H},\mathbf{b}\rangle)+\sigma^{\prime }(\langle\mathbf{v}_{H},\mathbf{a}\rangle)}{2}\cdot\langle\mathbf{v}_{H}, \mathbf{m}_{0}\rangle+\frac{\sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v} _{H}\rangle)-\sigma^{\prime\prime}(\langle\mathbf{a},\mathbf{v}_{H}\rangle)}{4 }\cdot\mathbf{v}_{H}^{T}\mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{H}\\ = \left(\begin{array}{c}\frac{\sigma^{\prime}(\langle\mathbf{v}_{1},\mathbf{b }\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{1},\mathbf{a}\rangle)}{2}\cdot \langle\mathbf{v}_{1},\mathbf{m}_{0}\rangle\\ \vdots\\ \frac{\sigma^{\prime}(\langle\mathbf{v}_{H},\mathbf{b}\rangle)+\sigma^{\prime }(\langle\mathbf{v}_{H},\mathbf{a}\rangle)}{2}\cdot\langle\mathbf{v}_{H}, \mathbf{m}_{0}\rangle\end{array}\right)+\left(\begin{array}{c}\frac{\sigma^ {\prime\prime}(\langle\mathbf{b},\mathbf{v}_{1}\rangle)-\sigma^{\prime\prime}( \langle\mathbf{a},\mathbf{v}_{1}\rangle)}{4}\cdot\mathbf{v}_{1}^{T}\mathbf{m} _{0}\mathbf{m}_{0}^{T}\mathbf{v}_{1}\\ \vdots\\ \frac{\sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{H}\rangle)-\sigma^{ \prime\prime}(\langle\mathbf{a},\mathbf{v}_{H}\rangle)}{4}\cdot\mathbf{v}_{H}^ {T}\mathbf{m}_{0}\mathbf{m}_{0}^{T}\mathbf{v}_{H}\end{array}\right)\] \[= \widetilde{\mathbf{V}}^{T}\mathbf{m}_{0}+\mathbf{q}(\mathbf{V}, \mathbf{m}_{0})\] \[= \mathbf{p}(\mathbf{V},\mathbf{m}_{0})+\mathbf{q}(\mathbf{V}, \mathbf{m}_{0}).\] with \[\widetilde{\mathbf{V}}=\left[\frac{\left[\sigma^{\prime}(\langle\mathbf{v}_{1 },\mathbf{b}\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{1},\mathbf{a}\rangle) \right]\mathbf{v}_{1}}{2},\ldots,\frac{\left[\sigma^{\prime}(\langle\mathbf{v }_{H},\mathbf{b}\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{H},\mathbf{a} \rangle)\right]\mathbf{v}_{H}}{2}\right].\] Since the first derivate $\sigma^{\prime}(\cdot)$ is non-negative, the relative position of the point $\widetilde{\mathbf{V}}^{T}\mathbf{m}_{0}$ will not be changed, _i.e.,_ the point $\mathbf{p}(\mathbf{V},\mathbf{m}_{0})$ still lies in the major side of the hyperplane $(\boldsymbol{\omega}^{T}\mathbf{V})\cdot\mathbf{n}_{0}=0$. To change the relative position of $\mathbf{p}(\mathbf{V},\mathbf{m}_{0})$ w.r.t. the hyperplane $(\boldsymbol{\omega}^{T}\mathbf{V})\mathbf{n}_{0}=0$, the following condition should be satisfied: denoting $\mathbf{w}^{T}=\boldsymbol{\omega}^{T}\mathbf{V}$, \[\left\{\begin{array}{c}\|\mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0}) \|<\|\mathbf{w}^{T}\mathbf{q}(\mathbf{V},\mathbf{m}_{0})\|;\\ \mathbf{p}^{T}(\mathbf{V},\mathbf{m}_{0})\cdot\mathbf{q}(\mathbf{V},\mathbf{m }_{0})<0.\end{array}\right.\] The second formula is equivalent to \[\mathbf{p}^{T}(\mathbf{V},\mathbf{m}_{0})\cdot\mathbf{q}(\mathbf{V },\mathbf{m}_{0})<0\] \[\Longleftrightarrow \sum_{h=1}^{H}\left[\sigma^{\prime}(\langle\mathbf{v}_{h}, \mathbf{b}\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{a}\rangle) \right]\cdot\left[\sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h} \rangle)-\sigma^{\prime\prime}(\langle\mathbf{a},\mathbf{v}_{h}\rangle) \right]\cdot\frac{\langle\mathbf{v}_{h},\mathbf{m}_{0}\rangle^{3}}{8}<0\] \[\Longleftrightarrow \Big{[}\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{b}\rangle)+ \sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{a}\rangle)\Big{]}\cdot\Big{[} \sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h}\rangle)-\sigma^{\prime \prime}(\langle\mathbf{a},\mathbf{v}_{h}\rangle)\Big{]}\cdot\langle\mathbf{v}_{h },\mathbf{m}_{0}\rangle<0.\] Namely, the sufficient condition for the second relation is that the points $\mathbf{a},\mathbf{b}$ and the weights $\mathbf{v}_{1},\cdots,\mathbf{v}_{H}$ satisfy that $F_{1}(\langle\mathbf{a},\mathbf{v}_{h}\rangle,\langle\mathbf{b},\mathbf{v}_{h} \rangle)>0$ holds for any $1\leq h\leq H$, where \[F_{1}(x,y)=\left[\sigma^{\prime}(x)+\sigma^{\prime}(y)\right]\cdot\Big{[}\sigma^ {\prime\prime}(x)-\sigma^{\prime\prime}(y)\Big{]}\cdot(x-y),\ \ (x,y)\in\mathbb{R}\times\mathbb{R}.\] Next, we also need to consider the first relation \[\|\mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0})\|<\|\mathbf{w}^{T}\mathbf{q} (\mathbf{V},\mathbf{m}_{0})\|.\] Since the sign of $\mathbf{m}_{0}^{T}\mathbf{v}_{h}$ is always opposite to that of $\sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h}\rangle)-\sigma^{\prime \prime}(\langle\mathbf{a},\mathbf{v}_{h}\rangle)$, we only need to consider the magnitudes of $\mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0})$ and $-\mathbf{w}^{T}\mathbf{q}(\mathbf{V},\mathbf{m}_{0})$. 1. If the sign of $\langle\mathbf{v}_{h},\mathbf{w}\rangle$ is same as that of $\langle\mathbf{v}_{h},\mathbf{m}_{0}\rangle$ for any $h\in\{1,\cdots,H\}$, then we have \[\|\mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0})\|<\|\mathbf{w}^ {T}\mathbf{q}(\mathbf{V},\mathbf{m}_{0})\|\] \[\Longleftrightarrow \mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0})<-\mathbf{w}^ {T}\mathbf{q}(\mathbf{V},\mathbf{m}_{0})\] \[\Longleftrightarrow \sum_{h=1}^{H}\frac{(\mathbf{v}_{h}^{T}\mathbf{w})\big{[}\sigma ^{\prime}(\langle\mathbf{v}_{h},\mathbf{b}\rangle)+\sigma^{\prime}(\langle \mathbf{v}_{h},\mathbf{a}\rangle)\big{]}(\mathbf{v}_{h}^{T}\mathbf{m}_{0})}{2}\] \[<-\sum_{h=1}^{H}\frac{(\mathbf{v}_{h}^{T}\mathbf{w})\big{[}\sigma ^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h}\rangle)-\sigma^{\prime \prime}(\langle\mathbf{a},\mathbf{v}_{h}\rangle)\big{]}(\mathbf{v}_{h}^{T} \mathbf{m}_{0})^{2}}{4},\quad\forall h\in\{1,2,\cdots,H\}\] \[\Longleftrightarrow 2\big{[}\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{b} \rangle)+\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{a}\rangle)\big{]}< \big{[}\sigma^{\prime\prime}(\langle\mathbf{a},\mathbf{v}_{1}\rangle)-\sigma ^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h}\rangle)\big{]}(\mathbf{v}_{h }^{T}\mathbf{m}_{0}),\quad\forall h\in\{1,2,\cdots,H\}\] \[\Longleftrightarrow 2<\frac{\big{[}\sigma^{\prime\prime}(\langle\mathbf{a}, \mathbf{v}_{h}\rangle)-\sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h} \rangle)\big{]}(\mathbf{v}_{h}^{T}\mathbf{m}_{0})}{\big{[}\sigma^{\prime}( \langle\mathbf{v}_{h},\mathbf{b}\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{h },\mathbf{a}\rangle)\big{]}},\quad\forall h\in\{1,2,\cdots,H\}.\] 2. If the sign of $\langle\mathbf{v}_{h},\mathbf{w}\rangle$ is the same as that of $-\langle\mathbf{v}_{h},\mathbf{m}_{0}\rangle$ for any $h\in\{1,\cdots,H\}$, then we have \[\|\mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0})\|<\|\mathbf{w}^ {T}\mathbf{q}(\mathbf{V},\mathbf{m}_{0})\|\] \[\Longleftrightarrow -\mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0})<\mathbf{w}^ {T}\mathbf{q}(\mathbf{V},\mathbf{m}_{0})\] \[\Longleftrightarrow \sum_{h=1}^{H}\frac{(\mathbf{v}_{h}^{T}\mathbf{w})\big{[}\sigma ^{\prime}(\langle\mathbf{v}_{h},\mathbf{b}\rangle)+\sigma^{\prime}(\langle \mathbf{v}_{h},\mathbf{a}\rangle)\big{]}(\mathbf{v}_{h}^{T}\mathbf{m}_{0})}{2}\] \[>-\sum_{h=1}^{H}\frac{(\mathbf{v}_{h}^{T}\mathbf{w})\big{[}\sigma ^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h}\rangle)-\sigma^{\prime\prime }(\langle\mathbf{a},\mathbf{v}_{h}\rangle)\big{]}(\mathbf{v}_{h}^{T}\mathbf{m }_{0})^{2}}{4}\] \[\Longleftrightarrow 2\big{[}\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{b} \rangle)+\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{a}\rangle)\big{]}< \big{[}\sigma^{\prime\prime}(\langle\mathbf{a},\mathbf{v}_{1}\rangle)-\sigma ^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h}\rangle)\big{]}(\mathbf{v}_{h }^{T}\mathbf{m}_{0}),\quad\forall h\in\{1,2,\cdots,H\}\] \[\Longleftrightarrow 2<\frac{\big{[}\sigma^{\prime\prime}(\langle\mathbf{a}, \mathbf{v}_{h}\rangle)-\sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h} \rangle)\big{]}(\mathbf{v}_{h}^{T}\mathbf{m}_{0})}{\big{[}\sigma^{\prime}( \langle\mathbf{v}_{h},\mathbf{b}\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{h },\mathbf{a}\rangle)\big{]}},\quad\forall h\in\{1,2,\cdots,H\}.\] To sum up, the sufficient condition for $\|\mathbf{w}^{T}\mathbf{p}(\mathbf{V},\mathbf{m}_{0})\|<\|\mathbf{w}^{T}\mathbf{q} (\mathbf{V},\mathbf{m}_{0})\|$ is \[2<\frac{\big{[}\sigma^{\prime\prime}(\langle\mathbf{a},\mathbf{v}_{h}\rangle)- \sigma^{\prime\prime}(\langle\mathbf{b},\mathbf{v}_{h}\rangle)\big{]}(\mathbf{v} _{h}^{T}\mathbf{m}_{0})}{\big{[}\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{ b}\rangle)+\sigma^{\prime}(\langle\mathbf{v}_{h},\mathbf{a}\rangle)\big{]}}, \qquad\forall h\in\{1,2,\cdots,H\}.\] Denote the function \[F_{\sigma}(x,y)=\frac{\big{[}\sigma^{\prime\prime}(x)-\sigma^{\prime\prime}(y) \big{]}(x-y)}{\big{[}\sigma^{\prime}(x)+\sigma^{\prime}(y)\big{]}},\ \ (x,y)\in\mathbb{R}\times\mathbb{R}.\] It is clear that the right hand side of the above formula is $F_{\sigma}(\langle{\bf a},{\bf v}_{h}\rangle,\langle{\bf b},{\bf v}_{h}\rangle)$. Then, the above issue is equivalent to finding the range such that $F_{\sigma}(x,y)>2$. Namely, as long as the points ${\bf a},{\bf b}$ and the weight vectors ${\bf v}_{1},\cdots,{\bf v}_{H}$ satisfy that $F_{\sigma}(\langle{\bf a},{\bf v}_{h}\rangle,\langle{\bf b},{\bf v}_{h} \rangle)>2$ ($\forall h\in\{1,2,\cdots,H\}$), the point ${\bf n}_{0}={\bf V}_{\sigma}({\bf a})-{\bf V}_{\sigma}({\bf b})$ lies in the minor side of the hyperplane ${\bf w}^{T}{\bf n}=({\boldsymbol{\omega}}^{T}{\bf V}){\bf n}=0$. Since $F_{\sigma}(\langle{\bf a},{\bf v}_{h}\rangle,\langle{\bf b},{\bf v}_{h} \rangle)>2$ implies that $F_{1}(\langle{\bf a},{\bf v}_{h}\rangle,\langle{\bf b},{\bf v}_{h}\rangle)>0$, this completes the proof. $\blacksquare$ ### Proof of Corollary 13 Proof of Corollary 13: For convenience, we briefly denote $a^{L}:={\rm LS}_{2}({\bf V}_{\sigma}^{L}({\cal A}),{\bf V}_{\sigma}^{L}({ \cal B}))$, $a^{0}={\rm LS}_{2}({\cal A},{\cal B})$ and ${\bf V}^{(0)}={\bf I}$. According to the law of total probability, we have \[\mathbb{P}\{a^{L}>a^{0}\} \tag{12}\] \[= \mathbb{P}\{a^{L}>a^{0}\|a^{L-1}>a^{0}\}\cdot\mathbb{P}\{a^{L-1}> a^{0}\}+\mathbb{P}\{a^{L}>a^{0}\|a^{L-1}\leq a^{0}\}\cdot\mathbb{P}\{a^{L-1} \leq a^{0}\}\] \[\leq \mathbb{P}\{a^{L-1}>a^{0}\}+\mathbb{P}\{a^{L}>a^{0}\|a^{L-1}\leq a ^{0}\}\cdot\mathbb{P}\{a^{L-1}\leq a^{0}\}\] \[= \mathbb{P}\{a^{L-1}>a^{0}\}+\mathbb{P}\{a^{L}>a^{0},a^{L-1}\leq a ^{0}\}\] \[\leq \mathbb{P}\{a^{L-1}>a^{0}\}+\mathbb{P}\{a^{L}>a^{L-1}\}\] \[= \mathbb{P}\{a^{L-1}>a^{0}\}+\sum_{{\bf V}^{(1)},\cdots,{\bf V}^{( L-1)}}\mathbb{P}({\bf V}^{(1)},\cdots,{\bf V}^{(L-1)})\cdot\mathbb{P}\{a^{L}>a ^{L-1}\|{\bf V}^{(1)},\cdots,{\bf V}^{(L-1)}\}\] \[\leq \mathbb{P}\{a^{L-1}>a^{0}\}+\max_{{\bf V}^{(1)},\cdots,{\bf V}^{ (L-1)}}\mathbb{P}\{a^{L}>a^{L-1}\|{\bf V}^{(1)},\cdots,{\bf V}^{(L-1)}\}\] \[\vdots\] \[\leq \sum_{l=1}^{L}\max_{{\bf V}^{(1)},\cdots,{\bf V}^{(l-1)}}\mathbb{ P}\{a^{l}>a^{l-1}\|{\bf V}^{(1)},\cdots,{\bf V}^{(l-1)}\}.\] The combination of Eq. (7) and Eq. (12) leads to the result (8). This completes the proof. $\blacksquare$ ### Proof of Proposition 16 Proof of Proposition 16: Denote ${\bf A}=[{\bf a}_{1},\cdots,{\bf a}_{I}]$ and ${\bf B}=[{\bf b}_{1},\cdots,{\bf b}_{J}]$. Let ${\bf 1}=(1,\cdots,1)^{T}$ be the vector whose components are all ones. Since ${\boldsymbol{\mu}}_{a}=\frac{1}{I}{\bf A}{\bf 1}$ and $[{\boldsymbol{\mu}}_{a},\cdots,{\boldsymbol{\mu}}_{a}]=\frac{1}{I}{\bf A}{\bf 1} {\bf 1}^{T}$, we have \[{\bf A}_{c}{\bf A}_{c}^{T} = \left({\bf A}-\frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}\right)\left({\bf A}- \frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}\right)^{T}\] \[= {\bf A}{\bf A}^{T}-\frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}{\bf A}^{T}- \frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}{\bf A}^{T}+\frac{1}{I^{2}}{\bf A}{\bf 1}{ \bf 1}^{T}{\bf 1}^{T}{\bf A}^{T}\] \[= {\bf A}{\bf A}^{T}-\frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}{\bf A}^{T}- \frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}{\bf A}^{T}+\frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}{ \bf A}^{T}\] \[= {\bf A}{\bf A}^{T}-\frac{1}{I}{\bf A}{\bf 1}{\bf 1}^{T}{\bf A}^{T}.\] In the similar way, we also have ${\bf B}_{c}{\bf B}_{c}^{T}={\bf B}{\bf B}^{T}-\frac{1}{J}{\bf B}{\bf 1}{\bf 1}^{T}{\bf B}^{T}$. Thus, the matrices ${\bf S}_{w}$ and ${\bf S}_{b}$ can be rewritten as \[\mathbf{S}_{w} = \mathbf{A}_{c}\mathbf{A}_{c}^{T}+\mathbf{B}_{c}\mathbf{B}_{c}^{T}\] \[= \mathbf{A}\mathbf{A}^{T}+\mathbf{B}\mathbf{B}^{T}-\frac{1}{I} \mathbf{A}\mathbf{1}\mathbf{1}^{T}\mathbf{A}^{T}-\frac{1}{J}\mathbf{B}\mathbf{1 }\mathbf{1}^{T}\mathbf{B}^{T};\] \[\mathbf{S}_{b} = (\boldsymbol{\mu}_{a}-\boldsymbol{\mu}_{b})(\boldsymbol{\mu}_{a}- \boldsymbol{\mu}_{b})^{T}\] \[= \frac{1}{I^{2}}\mathbf{A}\mathbf{1}\mathbf{1}^{T}\mathbf{A}^{T}+ \frac{1}{J^{2}}\mathbf{B}\mathbf{1}\mathbf{1}^{T}\mathbf{B}^{T}-\frac{1}{IJ} \mathbf{A}\mathbf{1}^{T}\mathbf{B}^{T}-\frac{1}{IJ}\mathbf{B}\mathbf{1} \mathbf{1}^{T}\mathbf{A}^{T}.\] Denote \[\mathrm{D}(\mathbf{A};J) := \underbrace{[\mathbf{a}_{1},\cdots,\mathbf{a}_{I},\cdots, \mathbf{a}_{1},\cdots,\mathbf{a}_{I}]}_{J\text{ groups of }\{\mathbf{a}_{1},\cdots, \mathbf{a}_{I}\}}\in\mathbb{R}^{N\times IJ};\] \[\mathrm{D}(\mathbf{B};I) := \underbrace{[\mathbf{b}_{1},\cdots,\mathbf{b}_{J},\cdots, \mathbf{b}_{1},\cdots,\mathbf{b}_{J}]}_{I\text{ groups of }\{\mathbf{b}_{1},\cdots, \mathbf{b}_{J}\}}\in\mathbb{R}^{N\times IJ}.\] Since $\mathbf{m}_{ij}=\mathbf{a}_{i}-\mathbf{b}_{j}$, $\mathbf{M}$ can be rewritten as \[\mathbf{M} = [\mathbf{m}_{11},\cdots,\mathbf{m}_{1J},\cdots,\mathbf{m}_{i1}, \cdots,\mathbf{m}_{iJ},\cdots,\mathbf{m}_{I1},\cdots,\mathbf{m}_{IJ}]_{N\times IJ}\] \[= \mathrm{D}(\mathbf{A};J)-\mathrm{D}(\mathbf{B};I).\] Then, we have \[\widetilde{\mathbf{m}}\widetilde{\mathbf{m}}^{T} = \left(\sum_{ij}\mathbf{m}_{ij}\right)\left(\sum_{ij}\mathbf{m}_{ ij}\right)^{T}\] \[= \left[\mathrm{D}(\mathbf{A};J)\mathbf{1}-\mathrm{D}(\mathbf{B}; I)\mathbf{1}\right]\cdot\left[\mathrm{D}(\mathbf{A};J)\mathbf{1}-\mathrm{D}( \mathbf{B};I)\mathbf{1}\right]^{T}\] \[= \mathrm{D}(\mathbf{A};J)\mathbf{1}\mathbf{1}^{T}\mathrm{D}^{T}( \mathbf{A};J)+\mathrm{D}(\mathbf{B};I)\mathbf{1}\mathbf{1}^{T}\mathrm{D}^{T}( \mathbf{B};I)\] \[\qquad\qquad\qquad-\mathrm{D}(\mathbf{B};I)\mathbf{1}\mathbf{1}^ {T}\mathrm{D}^{T}(\mathbf{A};J)-\mathrm{D}(\mathbf{A};J)\mathbf{1}\mathbf{1}^ {T}\mathrm{D}^{T}(\mathbf{B};I)\] \[= J^{2}\mathbf{A}\mathbf{1}\mathbf{1}^{T}\mathbf{A}^{T}+I^{2} \mathbf{B}\mathbf{1}\mathbf{1}^{T}\mathbf{B}^{T}-IJ\mathbf{B}\mathbf{1} \mathbf{1}^{T}\mathbf{A}^{T}-IJ\mathbf{A}\mathbf{1}\mathbf{1}^{T}\mathbf{B}^ {T}\] \[= I^{2}J^{2}\left(\frac{1}{I^{2}}\mathbf{A}\mathbf{1}\mathbf{1}^{T }\mathbf{A}^{T}+\frac{1}{J^{2}}\mathbf{B}\mathbf{1}\mathbf{1}^{T}\mathbf{B}^ {T}-\frac{1}{IJ}\mathbf{A}\mathbf{1}\mathbf{1}^{T}\mathbf{B}^{T}-\frac{1}{IJ} \mathbf{B}\mathbf{1}\mathbf{1}^{T}\mathbf{A}^{T}\right).\] It is direct that \[I^{2}J^{2}\mathbf{S}_{b}=\widetilde{\mathbf{m}}\widetilde{\mathbf{m}}^{T},\] which implies that the eigenvectors of the two matrices $\mathbf{S}_{b}$ and $\widetilde{\mathbf{m}}\widetilde{\mathbf{m}}^{T}$ have the same direction. Moreover, let $\mathbf{a}$ and $\mathbf{b}$ stand for the random variables obeying the probability distributions on the sets $\mathcal{A}$ and $\mathcal{B}$, respectively. Since $\mathbf{A}\mathbf{1}=I\cdot\widehat{\mathbb{E}}\mathbf{a}=\sum_{i=1}^{I} \mathbf{a}_{i}$ and $\mathbf{B}\mathbf{1}=J\cdot\widehat{\mathbb{E}}\mathbf{b}=\sum_{j=1}^{J} \mathbf{b}_{j}$, we have \[\mathbf{S}_{w} = \mathbf{A}\mathbf{A}^{T}+\mathbf{B}\mathbf{B}^{T}-\frac{1}{I} \mathbf{A}\mathbf{1}\mathbf{1}^{T}\mathbf{A}^{T}-\frac{1}{J}\mathbf{B} \mathbf{1}\mathbf{1}^{T}\mathbf{B}^{T}\] \[= \mathbf{A}\mathbf{A}^{T}+\mathbf{B}\mathbf{B}^{T}-\Big{[}I( \widehat{\mathbb{E}}\mathbf{a})(\widehat{\mathbb{E}}\mathbf{a})^{T}+J( \widehat{\mathbb{E}}\mathbf{b})(\widehat{\mathbb{E}}\mathbf{b})^{T}\Big{]}.\] Since $\sum_{i,j}(\mathbf{a}_{i}\mathbf{b}_{j}^{T}+\mathbf{b}_{j}\mathbf{a}_{i}^{T})=IJ \cdot\widehat{\mathbb{E}}\{\mathbf{ab}^{T}+\mathbf{ba}^{T}\}$, we have \[\mathbf{MM}^{T} = \big{[}\mathrm{D}(\mathbf{A};J)-\mathrm{D}(\mathbf{B};I)\big{]} \big{[}\mathrm{D}(\mathbf{A};J)-\mathrm{D}(\mathbf{B};I)\big{]}^{T}\] \[= J\mathbf{AA}^{T}+I\mathbf{B}\mathbf{B}^{T}-\sum_{i,j}(\mathbf{ a}_{i}\mathbf{b}_{j}^{T}+\mathbf{b}_{j}\mathbf{a}_{i}^{T})\] \[= J\mathbf{AA}^{T}+I\mathbf{B}\mathbf{B}^{T}-IJ\cdot\widehat{ \mathbb{E}}\{\mathbf{ab}^{T}+\mathbf{ba}^{T}\}.\] This completes the proof. $\blacksquare$ ## 7 Conclusion In general, the properties of deep neural networks are studied based on the backward inference from the network performance. Since neural networks have multilayer composite structures, it is technically difficult to analyze the function relation between the network performance and network structure parameters. To the best of our knowledge, it is still challenging to find applicable tools to directly analyze the effect of network structure parameters on the network performance. In this paper, we first demonstrate that there exists a synchronicity between the linear separability of hidden layers and the training accuracy in the classification tasks. This finding suggests that the linear separability should be an applicable tool to layer-wisely explore the characteristics of deep networks. Then, we propose the MD-LSM LS${}_{*}$ to measure the linear separability degree of the outputs of each hidden layer. To alleviate the computation cost, we present some alternative versions of MD-LSMs including LS${}_{0}$, LS${}_{1}$ and LS${}_{2}$. The last one has a closed-form solution via eigenvalue decomposition. In the numerical experiment, we simplify the constraint of the optimization problem (4) to achieve an approximation solution $\boldsymbol{\omega}$ to these MD-LSMs with a low computation cost (_cf._ Remark 15). We also analyze the relationship between the network structure parameters and the network performance. First, we provide a sufficient condition that a hidden layer can change the relative position relation between an MD point and the splitting hyperplane linear. Based on random matrix theory, we then demonstrate that the increase of network size can enlarge the probability of increasing the linear separability of hidden layers. The experimental results support our theoretical findings, and show that the synchronicity between the linear separability and network performance widely exists in the process of training popular deep networks (such as MLP, CNN, DBN, ResNet, VGGNet, AlexNet, ViT and GoogLeNet) in binary and multi-class classification tasks. In further works, we will use the MD-LSMs to measure the complexity of deep networks and analyze their generalization performance. ## Appendix A Appendix: Related Works Following the classical statistical learning theory techniques, it is expected to build a proper complexity measure of neural networks, and then to develop the relevant generalization bounds. Neyshabur _et al._[12] proposed a norm-based framework for measuring the capacity of neural networks, and the measure is expressed as an exponential form of the network depth. Golowich _et al._[13] provided the generalization bound based on Rademacher complexity that is bounded by multiplying the weight norm of each layer, and thus such a bound is also of the exponential form of the network depth. Harvey _et al._[14] used the Vapnik-Chervonenkis (VC) dimension as the complexity measure of the networks with piecewise linear activation functions, and then obtain the generalization bound which can be regarded as a linear function w.r.t. the network depth, the number of parameters and the number of nonlinear units. Bartlett [15] defined the margin-normalized spectral complexity, which is expressed as the product of the spectral norms of all weight matrices, and then provided the generalization bound based on covering number of neural networks. These results illustrates that the generalization performance of neural networks will become low when the network size increases. It accords with the traditional view of statistical learning theory but is still far away from the empirical observation on the applications of deep neural networks. By combining the spectral norm and the Frobenius norm of the weight matrices, Neyshabur _et al._[47] obtained the PAC-Bayes generalization bound of the networks with ReLU activation functions and then analyzed the stability of the network against the small perturbations on the weights. Under the PAC-Bayes framework, some works proposed the non-vacuous bounds of deep networks and these results were not dependent on the size or the dimension of parameters. Dziugaite _et al._[48] showed a data-dependent bound, derived from the result of [10], for the deep classifiers trained by SGD method, and then optimized the resultant bound to find a posterior distribution starting from the local minima found by the previous SGD process. Zhou _et al._[49] studied the generalization performance of compressed deep neural networks to explain a phenomenon that compressing a deep network only causes a slight performance change. These works usually assume that the prior or the posterior distribution of model parameters obeys Gaussian or sub-Gaussian, but this assumption may not always hold in practice. The neural tangent kernel (NTK) method, proposed in [11], suggests that the process of training an infinitely wide neural network should be equivalently as a kernel regression process. Based on NTK, Du _et al._[16] proved that the training error of infinitely wide neural networks of any depth will finally converge to zero. Arora _et al._[17] provided a generalization bound for the two-layer ReLU network by bounding the corresponding NTK. However, since these works assume that the network width is infinite, it is still far away from the actual cases. Moreover, under the same assumption, some works also studied the theoretical properties of the over-parameterized neural networks (_e.g.,_ two-layer ReLU networks) during the SGD training process [18, 19, 20]. To sum up, there still remains a big gap between the recent research achievement and the comprehensive explainability of deep neural networks. ## Appendix B Appendix: Complete Experimental Report In this part, we provide the experimental results of three kinds of MD-LSMs: LS${}_{0}$, LS${}_{1}$ and LS${}_{2}$. In view of the complicated structures of VGGNet, ResNet-20, GoogLeNet-V1 and ViT, we also draw the structure diagrams to denote their hidden layers or main blocks. ### Mlp First, we layer-wisely examine the linear separability of the MLPs with five hidden layers, denoted as MLP-5, and ten hidden layers, denoted as MLP-10, respectively. The hidden nodes of MLPs are activated by using Sigmoid functions (denoted as Sigmoid) and ReLU functions (denoted as ReLU), respectively. In Figs. 9-12, we illustrate the experimental results of MLPs in the binary classification tasks. In addition, we also consider the linear separability of MLPs in ten-class classification task, where the network has five hidden layers and its hidden nodes are activated by using ReLU (_cf._ Fig. 13). ### CNN, AlexNet and DBN Moreover, we examine the linear separability of CNNs with two convolution layers and two pooling layers in binary classification task. All hidden nodes of CNNs are activated by using ReLU (_cf_. Fig. 14). Moreover, the linear separability of AlexNet and DBN is also considered in the same task (_cf_. Figs. 15-17). It is noteworthy that we consider two kinds of AlexNets that have different output activation functions: one is Softmax, denoted as AlexNet (Softmax), and the other is Sigmoid, denoted as AlexNet (Sigmoid). We also simplify the process of training AlexNet (Sigmoid), where the tricks of learning rate decay and data augmentation are not used. Since the learning task (binary classification) is much simpler than the task (ImageNet classification) for which AlexNet was originally designed, the simplified training process is enough to provide a good performance. Thus, the curves of AlexNet (Sigmoid) are smoother than those of AlexNet (Softmax), especially for the LS${}_{2}$. ### VGGNet, GoogLeNet, ResNet and ViT Here, we consider the linear separability of the deep networks with complicated hidden-layer structures, including VGGNet, GoogLeNet-V1, ResNet-20 and ViT. Since the structures of these networks can be split into some individual blocks, we first examine the linear separability of the outputs of their main blocks, and then illustrate the LSMs of hidden layers of these networks. In Tab. 1, we show the arrangement of the structure diagrams and the experimental results. \begin{table} \begin{tabular}{c\|c\|c} \hline Deep Networks & Structure Diagram & Main Blocks & Hidden Layers \\ \hline MLP-5 (ReLU) & & Fig. 9 \\ \hline MLP-5 (Sigmoid) & & Fig. 10 \\ \hline MLP-10 (ReLU) & & Fig. 11 \\ \hline MLP-10 (Sigmoid) & & Fig. 12 \\ \hline MLP (Ten-Class) & & Fig. 13 \\ \hline CNN & & Fig. 14 \\ \hline AlexNet (Softmax) & & Fig. 15 \\ \hline AlexNet (Sigmoid) & & Fig. 16 \\ \hline DBN & & Fig. 17 \\ \hline VGGNet & Fig. 19 & Fig. 18 & Fig. 20 \\ \hline GoogLeNet-V1 & Fig. 21 & Fig. 22 & Fig. 23 \\ \hline ResNet-20 & Fig. 24 & Fig. 25 & Fig. 26 \\ \hline ViT & Fig. 27 & Fig. 28 & Fig. 29 \\ \hline \end{tabular} \end{table} Table 1: Numerical Experiment Results Figure 10: MD-LSM and Accuracy Curves of Hidden Layers of MLP-5 (Sigmoid) Figure 9: MD-LSM and Accuracy Curves of Hidden Layers of MLP-5 (ReLU) Figure 11: MD-LSM and Accuracy Curves of Hidden Layers of MLP-10 (ReLU) Figure 12: MD-LSM and Accuracy Curves of Hidden Layers of MLP-10 (Sigmoid) Figure 14: MD-LSM and Accuracy Curves of CNN’s Hidden Layers Figure 13: MD-LSM and Accuracy Curves of Hidden Layers of MLP (Ten-Class) Figure 16: MD-LSM and Accuracy Curves of Hidden Layers of AlexNet (Sigmoid) Figure 15: MD-LSM and Accuracy Curves of Hidden Layers of AlexNet (Softmax) Figure 17: MD-LSM and Accuracy Curves of DBN’s Hidden Layers Figure 18: MD-LSM and Accuracy Curves of VGGNet’s Main Blocks Figure 19: Structure of VGGNet Figure 20: MD-LSM and Accuracy Curves of VGGNet’s Hidden Layers Figure 21: Structure of GoogleNet-V1 Figure 22: Structure of GoogleNet-V1 Figure 22: MD-LSM and Accuracy Curves of GoogleNet’s Main Blocks Figure 23: MD-LSM and Accuracy Curves of GoogLeNet’s Hidden Layers Figure 24: Structure of ResNet-20 Figure 25: MD-LSM and Accuracy Curves of ResNet’s Main Blocks Figure 26: MD-LSM and Accuracy Curves of ResNet’s Hidden Layers Figure 28: MD-LSM and Accuracy Curves of ViT’s Main Blocks Figure 27: Structure of ViT Figure 29: MD-LSM and Accuracy Curves of ViT’s Hidden Layers
2307.11807	Local Kernel Renormalization as a mechanism for feature learning in overparametrized Convolutional Neural Networks	Feature learning, or the ability of deep neural networks to automatically learn relevant features from raw data, underlies their exceptional capability to solve complex tasks. However, feature learning seems to be realized in different ways in fully-connected (FC) or convolutional architectures (CNNs). Empirical evidence shows that FC neural networks in the infinite-width limit eventually outperform their finite-width counterparts. Since the kernel that describes infinite-width networks does not evolve during training, whatever form of feature learning occurs in deep FC architectures is not very helpful in improving generalization. On the other hand, state-of-the-art architectures with convolutional layers achieve optimal performances in the finite-width regime, suggesting that an effective form of feature learning emerges in this case. In this work, we present a simple theoretical framework that provides a rationale for these differences, in one hidden layer networks. First, we show that the generalization performance of a finite-width FC network can be obtained by an infinite-width network, with a suitable choice of the Gaussian priors. Second, we derive a finite-width effective action for an architecture with one convolutional hidden layer and compare it with the result available for FC networks. Remarkably, we identify a completely different form of kernel renormalization: whereas the kernel of the FC architecture is just globally renormalized by a single scalar parameter, the CNN kernel undergoes a local renormalization, meaning that the network can select the local components that will contribute to the final prediction in a data-dependent way. This finding highlights a simple mechanism for feature learning that can take place in overparametrized shallow CNNs, but not in shallow FC architectures or in locally connected neural networks without weight sharing.	R. Aiudi, R. Pacelli, A. Vezzani, R. Burioni, P. Rotondo	2023-07-21T17:22:04Z	http://arxiv.org/abs/2307.11807v1	Local Kernel Renormalization as a mechanism for feature learning in overparametrized Convolutional Neural Networks ###### Abstract Feature learning, or the ability of deep neural networks to automatically learn relevant features from raw data, underlies their exceptional capability to solve complex tasks. However, feature learning seems to be realized in different ways in fully-connected (FC) or convolutional architectures (CNNs). Empirical evidence shows that FC neural networks in the infinite-width limit eventually outperform their finite-width counterparts. Since the kernel that describes infinite-width networks does not evolve during training, whatever form of feature learning occurs in deep FC architectures is not very helpful in improving generalization. On the other hand, state-of-the-art architectures with convolutional layers achieve optimal performances in the finite-width regime, suggesting that an effective form of feature learning emerges in this case. In this work, we present a simple theoretical framework that provides a rationale for these differences, in one hidden layer networks. First, we show that the generalization performance of a finite-width FC network can be obtained by an infinite-width network, with a suitable choice of the Gaussian priors (technically, this result holds in the asymptotic limit where the size of the training set $P$ and the size of the hidden layer $N_{1}$ are taken to infinity keeping their ratio $\alpha_{1}=P/N_{1}$ fixed). Second, we derive a finite-width effective action for an architecture with one convolutional hidden layer and compare it with the result available for FC networks. Remarkably, we identify a completely different form of kernel renormalization: whereas the kernel of the FC architecture is just globally renormalized by a single scalar parameter, the CNN kernel undergoes a local renormalization, meaning that the network can select the local components that will contribute to the final prediction in a data-dependent way. This finding highlights a simple mechanism for feature learning that can take place in overparametrized shallow CNNs, but not in shallow FC architectures or in locally connected neural networks without weight sharing. ## I Introduction Deep learning achieves state-of-the-art performances on a variety of pattern recognition tasks, ranging from computer vision to natural language processing [1]. A minimal theory of deep learning should be able (at least) to predict practically relevant scores, such as the training and test accuracy, from knowledge of the training data. Another even more fundamental aspect that such a theory should quantitatively address is the so-called _feature learning regime_[2; 3], i.e. the capability of deep networks to automatically detect useful representations from raw data. Concerning the first aspect, fundamental theoretical progress has been achieved in the so-called _infinite-width limit_[4; 5; 6; 7; 8; 9; 10; 11; 12], which is informally defined as the regime where the size of each hidden layer $N_{\ell}$ ($\ell=1,\ldots,L$, $L$ being the (finite) depth of the network) is much larger than the size of the training set $P$. Here, one shows that the stochastic process that describes information flow in the deep neural network is a familiar Gaussian process, which is completely determined by a non-linear kernel. A fundamental consequence of this finding is that learning in the infinite-width limit is equivalent to kernel learning [13; 14; 15; 16] with a static kernel, completely fixed by the statistics of the weights at initialization, that does not evolve during training. This last observation suggests that feature learning is essentially absent in infinite-width networks [17]. Notably, whereas original work on the infinite-width limit considered deep architectures with FC hidden layers only, the generalization to architectures with convolutional layers is straightforward [9], if one replaces neurons in the FC hidden layers with the channels/convolutional filters of the CNN. Recently, the authors of Ref. [18] conducted a large-scale empirical study comparing finite-width FC networks, CNNs with finite number of channels and their infinite-width limit kernel counterparts, where they explore a plenitude of possible settings to improve generalization. This analysis prompts a few striking empirical observations: (i) infinite-width kernels systematically outperform their finite-width counterpart in the case of FC deep neural networks; (ii) CNNs with finite number of channels often outperform their corresponding infinite-width kernel performance. Several similar observations have been reported in the literature: the authors of Ref. [19] very recently pointed out that infinite-width deep FC neural networks eventually outperform their finite-width counterpart as the size of the training set grows. In one of the -seminal papers dealing with the infinite-width limit [7], the authors observe that increasing the hidden layers size leads to optimal test accuracy on deep FC architectures trained on MNIST and on CIFAR10, two of the benchmark datasets for computer vision learning problems. In Ref. [9] this observation is extended to _locally connected networks_ without weight sharing (LCNs). The interplay between the lazy training regime and the mean field limit [20] has been the subject of a thorough investigation in [21, 22]. These observations suggest that CNNs leverage a better feature-learning mechanism at finite width than FCNs and LCNs, and prompt at least two conceptual questions: (i) why is it ultimately convenient to employ large-width architectures when only FC layers are available? (ii) Why is this not the case when convolutional layers are employed? And how does a CNN operatively exploit the finite-width regime for efficient feature learning? Preliminary theoretical work in the direction of understanding the feature learning regime of deep nets was carried out in Refs. [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. The authors of Refs. [35, 36] analytically investigated the advantages of employing convolutional neural tangent kernels in the infinite-width limit and understood why infinite-width FCNs perform worse in the mean field regime than in the lazy-training one [37]. In this work, we try to rationalize the aforementioned empirical observations through the lens of _kernel renormalization_. This represents a physical consequence of a recently-derived effective action for Bayesian learning in finite-width deep neural networks [38] (formally speaking, the finite-width regime is defined as the thermodynamic limit where the size of the FC hidden layers and of the training set are taken to infinity, $N_{\ell},P\rightarrow\infty$ and their ratio $\alpha_{\ell}=P/N_{\ell}$ is kept finite). First, we show that the theoretical framework of Ref. [38] provides a simple, yet very instructive answer to question (i) for one hidden layer networks, by showing that the performance of a finite-width shallow network can be obtained with the corresponding infinite-width kernel and a suitable choice of the Gaussian priors over the weights of each layer. Second, we derive an effective action for a simple architecture with one convolutional hidden layer, and we compare the result with the one available for FCNs. We find a striking difference in the way the kernel of the two architectures renormalize at finite width: whereas the FC kernel is just globally renormalized by a scalar parameter, the CNN kernel undergoes a local renormalization, meaning that many more free parameters are allowed to be fine-tuned during training (see also Fig. 1). We employ this finding to provide preliminary insight to question (ii): we highlight a simple mechanism for feature learning that can take place in finite-width shallow CNNs, but neither in shallow FC architectures nor in LCNs without weight sharing. Figure 1: Global and local kernels in FC and convolutional one hidden layer networks. Our calculations in the proportional limit $P,N_{1}\rightarrow\infty$ at fixed $\alpha_{1}=P/N_{1}$ reveal a striking difference between FCNs and CNNs. In the first case, the NNGP kernel $K_{\mu\nu}$ that enters into the Bayesian effective action is a global one, which is only capable of tracking global correlations between different training patterns $(\mu,\nu)$ (left). A FCN network in this regime will use the information contained in the dataset to fine-tune a single parameter, $Q$, that will globally rescale the kernel matrix $K_{\mu\nu}$. On the other hand, in the CNN case, we identify a local kernel with two additional (spatial) indices. If we consider $2d$ images, the effect of the local components is to track local correlations between different patches $i$ and $j$ of (possibly) different pairs of images $(\mu,\nu)$ (right). The spatial information contained in these correlations allows the CNN to optimize a matrix of parameters $\bar{Q}_{ij}$, which will renormalize the kernel matrix $K_{\mu\nu}^{ij}$ in a non-trivial way. In the figure, we display the four-index local kernel $K_{\mu\nu}^{ij}$ as a matrix using the multi-index notation $K_{(\mu,i),(\nu,j)}$ and choosing an ordering for the multi-index $(\mu,i)$. Global and local kernels are computed for a subset of $P=10$ gray-scaled CIFAR10 images down-sized to $N_{0}=784$. The local kernels are empirically evaluated for $2d$ convolutions with linear size of the filter mask $M=14$ and stride $S=14$ (non-overlapping filters). II Finite-width one hidden layer FC NNS cannot outperform infinite-width kernels in the overparametrized regime We consider a supervised regression problem with training set $\mathcal{T}_{P}=\{x^{\mu},y^{\mu}\}_{\mu=1}^{P}$, where each $x^{\mu}\in\mathbb{R}^{N_{0}}$ and the corresponding labels $y^{\mu}\in\mathbb{R}$. We also restrict our analysis to quadratic loss function $\mathcal{L}$. For a generic one hidden layer network that implements the function $f_{\rm 1HL}(x)$, we define the loss function as: \[\mathcal{L}=\frac{1}{2}\sum_{\mu}\left[y^{\mu}-f_{\rm 1HL}(x^{\mu})\right]^{2}+ \frac{T\,\lambda_{0}}{2}\\|W\\|^{2}+\frac{T\,\lambda_{1}}{2}\\|v\\|^{2} \tag{1}\] where $W$ and $v$ are respectively the first and last layer weights, and $T$ is the temperature. The parameters $\lambda_{0},\lambda_{1}$ can be equivalently thought as Gaussian priors over the weights of each layer, or as $L^{2}$ regularization terms rescaled by the temperature $T$. A neural network with one FC hidden layer is formally defined starting from the pre-activations at the first layer: \[h_{i_{1}}(x)=\frac{1}{\sqrt{N_{0}}}\sum_{i_{0}=1}^{N_{0}}W_{i_{1}i_{0}}x_{i_{0 }}+b_{i_{1}}\,, \tag{2}\] where $b$ are the first-layer biases. Since we are interested in regression problems, we add one last readout layer, and we define the function implemented by the one hidden layer FCN as: \[f_{\rm FCN}(x)=\frac{1}{\sqrt{N_{1}}}\sum_{i_{1}=1}^{N_{1}}v_{i_{1}}\sigma[h_{ i_{1}}(x)]\, \tag{3}\] where $\sigma$ is a non-linear activation function. An effective action arises in this setting when one considers the canonical partition function at temperature $T=1/\beta$ associated to the train loss function $\mathcal{L}$ and the thermodynamic limit $N_{1},P\to\infty$ at fixed ratios $\alpha_{1}=P/N_{1}$. The calculation amounts to reduce the integral over the weights of the network to a form suitable to saddle-point integration over two order parameters $\{Q,\bar{Q}\}$, $Z=\int dQd\bar{Q}e^{-N_{1}/2S_{\rm FCN}(\{Q,\bar{Q}\})}$. In the case of odd-activation function, the action is given by [38]: \[S_{\rm FCN} =-Q\bar{Q}+\log(1+Q)\] \[+\frac{\alpha_{1}}{P}\text{Tr}\log\beta\left(\frac{1}{\beta}+K^{ \rm(R)}(\bar{Q})\right)\] \[+\frac{\alpha_{1}}{P}y^{T}\left(\frac{1}{\beta}+K^{\rm(R)}(\bar{ Q})\right)^{-1}y\,, \tag{4}\] where $y$ is the vector of labels $y=(y^{1},\ldots,y^{P})$. The renormalized kernel $K^{\rm(R)}(\bar{Q})$ is a $P\times P$ matrix that processes pairs of input training data $(x^{\mu},x^{\nu})$ and it is given by: \[K^{\rm(R)}(\bar{Q})=\frac{\bar{Q}}{\lambda_{1}}K(C)\,,\ \ \ \ \ C_{\mu\nu}=\frac{1}{\lambda_{0}}\frac{x^{\mu}\cdot x^{\nu}}{N_{0}}\,. \tag{5}\] The non-linear operator $K$ takes as input any (symmetric) $P\times P$ matrix $M$ and computes a new matrix in the following way: \[K_{\mu\nu}(M) =\int d^{2}t\,\mathcal{N}_{t}\left(0,\tilde{M}_{\mu\nu}\right) \sigma(t_{1})\sigma(t_{2})\,, \tag{6}\] \[\tilde{M}_{\mu\nu} =\begin{pmatrix}M_{\mu\mu}&M_{\mu\nu}\\ M_{\mu\nu}&M_{\nu\nu}\end{pmatrix}\,. \tag{7}\] Here and in the following we are denoting normalized Gaussians as $\mathcal{N}_{x}\left(0,\Sigma\right)\equiv\exp\left(-x^{\top}\Sigma^{-1}x/2 \right)/\sqrt{\det\!2\pi\Sigma}$. It is worth noticing that the minimization of the effective action in Eq. (4) is straightforward if $\alpha_{1}\to 0$, since one easily finds that $Q^{}=0$, $\bar{Q}=1$ and recovers the well-known infinite-width limit Neural Network Gaussian Process (NNGP) kernel. At finite $\alpha_{1}$, one finds data-dependent solutions for the order parameters that produce a renormalization of the infinite-width kernel, as expressed by Eq. (5). The effective action at finite width presented in Eq. (4) has been firstly derived for deep linear networks in Ref. [39]. The non-asymptotic evaluation of the partition function in the linear case is given in Ref. [40] in terms of Meyer-G functions. An effective action for globally gated deep linear networks has also been derived [41]. In Ref. [38] we obtained the effective action for one FC hidden layer networks with generic activation function leveraging on a Gaussian equivalence [42; 43; 44; 45] informally justified via a generalized central limit theorem due to Breuer and Major [46]. A result for $L$ layers can be somewhat bootstrapped from the observation that the stochastic process that describes information flow in shallow non-linear and deep linear networks in the proportional limit is related to a Student's $t$-process [47] (see also [38] for a more detailed discussion). Recent work considers the proportional setting in a teacher-student scenario, averaging over the data distribution using the replica method [48; 49] and a first preliminary rigorous result in an almost-proportional limit [50] appeared very recently, leveraging the interpolation method [51; 52]. We now highlight a straightforward consequence of the theoretical framework presented above. First, we notice that the renormalized kernel $K^{\rm(R)}$ enters into the predictor's statistics for a new unseen test element exactly in the same way the NNGP kernel does in the infinite-width limit, and thus it completely determines the generalization performance at finite width, once the saddle-point equations for the parameter $\bar{Q}$ are solved [38]. Second, we observe that the scalar parameter $\bar{Q}$ always appears in combination with the corresponding Gaussian prior $\lambda_{1}$ as $\bar{Q}/\lambda_{1}$, meaning that, once evaluated on the saddle-point, $\bar{Q}$ is just a data-dependent scalar renormalization of the Gaussian prior. This implies that, once the size of the training set and the activation function $\sigma$ are fixed, it is always possible to re-obtain the performance of any finite-width network just by carefully fine-tuning the Gaussian prior $\lambda_{1}$ in the corresponding infinite-width kernel. Therefore, the generalization performance of any finite-width one hidden layer FC network is bounded by the one of a suitable infinite-width kernel with optimal choice of the Gaussian prior. The empirical observation that infinite-width one hidden layer FCNs seem to systematically outperform their finite-width counterpart -which now finds a possible explanation in this framework- points to the somewhat disappointing consequence that feature learning is not particularly effective in networks with FC layers alone at finite width. We stress here that the aforementioned observation is not ruling out at all other possible forms of feature learning in FC architectures: (i) Renormalization of the infinite-width kernel is not the only source of feature learning possible. Higher order kernels, irrelevant in the infinite-width limit, may play a role in feature learning, especially as long as one considers the case where the size of the dataset $P$ roughly scales as the number of parameters $\sim L\times N^{2}$ ($N_{\ell}=N$$\forall\ell$) of the FC deep neural network. The recent work [53] is a notable example of how a FC network can learn a convolutional structure in special settings; (ii) Our result holds for Bayesian learning, i.e. when the weights of the networks are sampled from the canonical Gibbs ensemble. This is only obtained if training is governed by a Markov chain Monte Carlo. Practical learning algorithms with state-of-the-art optimizers may behave differently and possibly activate alternative forms of feature learning; (iii) we are limiting our analysis to the standard parametrization setting where the last layer is normalized as $1/\sqrt{N_{1}}$, and we cannot describe the mechanism for feature learning that may occur in the mean field regime [20] where the last layer is normalized as $1/N_{1}$ (we note that the framework of [23] seems more suitable to deal with this case). Nonetheless, it must be also said that the empirical evidence provided in Refs. [7; 9; 18; 19; 22] is quite against the fact that these forms of feature learning can play a significant role in improving the generalization performance of FC networks. We now move to the second question raised in the Introduction of this manuscript. How -contrarily to what occurs in deep nets with FC layers only- do deep architectures with convolutional layers exploit the finite-width regime for efficient feature learning? In view of the previous discussion, one possibility is that CNNs at finite width could be able, in some way, to break the global (trivial) renormalization of the infinite-width kernel that occurs in FC deep networks. In order to test this hypothesis, in the next section we investigate a simple model of CNN with one convolutional hidden layer. ## III Finite-width effective action for a shallow convolutional network The two fundamental ingredients of a convolutional layer are _local connectivity_ and _weight sharing_: whereas a given neuron in a FC hidden layer receives input from all the neurons in the previous layer, neurons in a CNN are arranged in a $d$-dimensional array that reflects the corresponding $d$-dimensional arrangement of the input data (e.g. $d=2$ for images). Each neuron in a given layer here interacts only with a local neighborhood of neurons in the previous layer, in a translational invariant way implemented via a shared (usually small) $d$-dimensional mask of learnable weights. These operations define a single convolutional channel that takes as input a $d$-dimensional array and outputs another $d$-dimensional array, whose dimensions are determined by technical details (such as the stride, padding and dimension of the filter mask). Usually, many of these channels are piled up in a convolutional layer to form, overall, a $(d+1)$-dimensional array. Let us now define a simple architecture with one hidden convolutional layer. For simplicity, we will restrict our analysis to one-dimensional convolutions, but the model can be easily generalized to $d$-dimensional convolutions. Pre-activations in the hidden layer are given by: \[h_{i}^{a}(x)=\frac{1}{\sqrt{M}}\sum_{m=-\lfloor M/2\rfloor}^{\lfloor M/2 \rfloor}W_{m}^{a}x_{Si+m}\,. \tag{8}\] Here $M$ is the dimension of the channel mask, $S$ is the stride, the index $i=1,\ldots,\lfloor N_{0}/S\rfloor$ runs over the input coordinates, the index $a=1,\ldots,N_{c}$ runs over the channels and the index $m$ moves through the spatial mask of the convolutional filter. For simplicity, we define periodic boundary conditions (PBCs) over the input coordinates, and we consider odd values of $M$. As for the one hidden layer FC network, we add one readout layer. In conclusion, the function implemented by the CNN is given by: \[f_{\text{CNN}}(x)=\frac{1}{\sqrt{N_{c}\lfloor N_{0}/S\rfloor}}\sum_{i=1}^{ \lfloor N_{0}/S\rfloor}\sum_{a=1}^{N_{c}}v_{i}^{a}\sigma\left[h_{i}^{a}(x) \right]\,, \tag{9}\] where $\sigma$ is a odd activation function and the $v$'s are the learnable weights of the readout layer. Our goal is to derive an effective action in the same setting of Refs. [38], in the thermodynamic limit where $(N_{c},P)\rightarrow\infty$ and their ratio $\alpha_{c}=P/N_{c}$ is finite. The partition function for the learning problem with the aforementioned simple CNN can be approximated as an integral over an ensemble of $\lfloor N_{0}/S\rfloor\times\lfloor N_{0}/S\rfloor$ matrices $Q$ and $\tilde{Q}$: $Z_{\text{CNN}}=\int\mathcal{D}\mathcal{Q}\tilde{Q}e^{-N_{c}/2S_{\text{CNN}}( Q,\tilde{Q})}$ (see Appendices for more details on the derivation), where the effective action is given by: \[S_{\rm CNN}(Q,\bar{Q})\equiv -{\rm Tr}\,Q\bar{Q}+{\rm Tr}\log(\mathbb{1}+Q)+\] \[+\frac{\alpha_{c}}{P}{\rm Tr}\log\beta\Big{(}\frac{\mathbb{1}}{ \beta}+K^{(\rm R)}_{\rm CNN}\Big{)}+\] \[+\frac{\alpha_{c}}{P}y^{\top}\Big{(}\frac{\mathbb{1}}{\beta}+K^{ (\rm R)}_{\rm CNN}\Big{)}^{-1}y\,. \tag{10}\] The trace in the first two terms is over $\lfloor N_{0}/S\rfloor\times\lfloor N_{0}/S\rfloor$ operators, whereas the renormalized kernel $K^{(\rm R)}_{\rm CNN}$ is a $P\times P$ matrix and therefore the trace in the third term and the scalar products with the output labels lie in a $P$-dimensional vectorial space. The matrix elements of the renormalized kernel are given by: \[\Big{[}K^{(\rm R)}_{\rm CNN}(\bar{Q})\Big{]}_{\mu\nu}=\frac{1}{\lambda_{1} \lfloor N_{0}/S\rfloor}\sum_{ij=1}^{\lfloor N_{0}/S\rfloor}\bar{Q}_{ij}K^{ij}_ {\mu\nu}\,, \tag{11}\] where the kernel $K^{ij}_{\mu\nu}$ is given in terms of the elements of the following local covariance matrix \[C^{ij}_{\mu\nu}=\frac{1}{\lambda_{0}M}\sum_{m=-\lfloor M/2\rfloor}^{\lfloor M /2\rfloor}x^{\mu}_{Si+m}x^{\nu}_{Sj+m}, \tag{12}\] with $\lambda_{0}$ being the hidden layer Gaussian prior, via the same functional relations of the FC case: \[K^{ij}_{\mu\nu}=\int d^{2}t\,{\cal N}_{t}\left(0,\bar{C}^{ij}_{ \mu\nu}\right)\sigma(t_{1})\sigma(t_{2})\,,\] \[\bar{C}^{ij}_{\mu\nu}=\begin{pmatrix}C^{ii}_{\mu\mu}&C^{ij}_{\mu \nu}\\ C^{ii}_{\mu\nu}&C^{ij}_{\mu\nu}\end{pmatrix}\,. \tag{13}\] It is worth noticing that such a local kernel has been already found in the seminal work on the Gaussian process limit of infinite-width CNNs [9]. However, one finds that in the limit of infinitely many channels $N_{c}\gg P$, the CNN evolves according to the following averaged kernel: \[\bar{K}_{\mu\nu}=\frac{1}{\lambda_{1}\lfloor N_{0}/S\rfloor}\sum_{i=1}^{ \lfloor N_{0}/S\rfloor}K^{ii}_{\mu\nu}\,. \tag{14}\] In our effective action framework we recover this result by solving the saddle-point equations for the matrix $\bar{Q}$ in the limit of $\alpha_{c}\to 0$, where one finds $\bar{Q}_{ij}=\delta_{ij}$ and only the diagonal components of the local kernel contribute to the prediction. Note that the derivation of this result is based on a Gaussian equivalence that is informally justified using a general class of central limit theorems (such as the Breuer-Major theorem in [38]). This suggests that the dimension of the input $N_{0}$ should scale to infinity at the same rate as the size of the dataset $P$. If this is the case, as it was in [38], the stride $S$ should be chosen to scale extensively in $N_{0}$, so that the final effective action will depend on a finite number of order parameters. ## IV A simple mechanism for feature learning in finite-width CNNs: local kernel renormalization Let us now analyze the saddle-point equations deriving from Eq. (II). We can gain analytical insight in the zero temperature limit, where we find the following matrix equations that determine $Q$ and $\bar{Q}$: \[\bar{Q}_{ij}=(\mathbb{1}+Q)^{-1}_{ij} \tag{15}\] \[Q_{ij}=\frac{\alpha_{c}}{P}{\rm Tr}\Big{(}K^{ij}\left[K^{(\rm R) }_{\rm CNN}(\bar{Q})\right]^{-1}\Big{)}+\] \[-\frac{\alpha_{c}}{P}y^{T}\Big{(}\left[K^{(\rm R)}_{\rm CNN}(\bar {Q})\right]^{-1}K^{ij}\left[K^{(\rm R)}_{\rm CNN}(\bar{Q})\right]^{-1}\Big{)}y \tag{16}\] where the shortcut $K^{ij}$ has been introduced to indicate the partial trace over indices $(\mu,\nu)$ in the local kernel defined in Eq. (11). More explicitly, given a matrix $O$ with elements $O_{\mu\nu}$, the partial trace is defined as ${\rm Tr}(K^{ij}O):=\sum_{\mu\nu=1}^{P}K^{ij}_{\mu\nu}O_{\mu\nu}$. The exact solution of these matrix equations cannot be obtained in closed form. However, it is possible to compute it perturbatively around the infinite-width limit $\alpha_{c}\to 0$. Parametrizing the solution of the first equation as $\bar{Q}_{ij}=\delta_{ij}+\alpha_{c}\delta\bar{Q}_{ij}+o(\alpha_{c})$, we find the simple result: \[\delta\bar{Q}_{ij}=\frac{1}{P}\tilde{y}^{T}K^{ij}\tilde{y}-\frac{1}{P}{\rm Tr }\left(K^{ij}\bar{K}^{-1}\right)\,, \tag{17}\] where $\tilde{y}^{\mu}=\sum_{\mu=1}^{P}\bar{K}^{-1}_{\mu\nu}y^{\mu}$ and $\bar{K}$ is the infinite-width CNN averaged kernel defined in Eq. (14). We now provide a physical interpretation of this finding. Our claim is that the matrix $\bar{Q}$ provides a compact description of feature learning in the CNN model under consideration: the feature matrix elements $\bar{Q}_{ij}$ that \begin{table} \begin{tabular}{\|c\|c\|c\|c\|} \hline Architecture & Kernel type & IW Kernel & Renormalized Kernel \\ \hline \hline FC & $K_{\mu\nu}$ & $K_{\mu\nu}$ & $\bar{Q}K_{\mu\nu}$ \\ \hline LCN & $K^{ii}_{\mu\nu}$ & $\sum_{i}K^{ii}_{\mu\nu}$ & $\sum_{i}\bar{Q}_{i}K^{ii}_{\mu\nu}$ \\ \hline CNN & $K^{ij}_{\mu\nu}$ & $\sum_{i}K^{ii}_{\mu\nu}$ & $\sum_{ij}\bar{Q}_{ij}K^{ij}_{\mu\nu}$ \\ \hline \end{tabular} \end{table} Table 1: Summary of the analytical results for one hidden layer networks. We identify three different types of kernels in the Bayesian effective action for FC, LC and convolutional networks: a global kernel in the FC case, a local diagonal kernel in LCNs and a local (with additional non-diagonal components) in CNNs. The infinite-width kernel that describes LCNs and CNNs is the same, as shown in [9]. In the proportional limit, the renormalization of the FC kernel is scalar, i.e. the NNGP kernel is only globally rescaled by a data-dependent number $\bar{Q}$. In LCNs, we can adjust a vector of components $\bar{Q}_{i}$, which determines how much the self-correlations of the $i$-th patch of the diagonal local kernel do contribute to the final prediction. In CNNs, due to locality and weight-sharing, we can fine-tune a full matrix $\bar{Q}_{ij}$, which controls how much the local correlations of patches $i$ and $j$ will enter the effective kernel learned by the network after training. This is what we call _local kernel renormalization_. are optimized during training are in fact in one-to-one correspondence with the combinations of pairs of the $\lfloor N_{0}/S\rfloor$ patches of the local covariance matrix, which is defined from the trainset elements $x^{\mu}$. From the first term of the r.h.s. of Eq. (17) we can notice that whenever the local kernel of the CNN at the spatial locations $(i,j)$ will have a significant overlap with the effective label vector $\tilde{y}$, the element $\bar{Q}_{ij}$ of the feature matrix $\bar{Q}$ will differ from one. In other words, this is a measure of the relative importance of the pairwise spatial correlations in the input dataset wrt the labels. From Eq. (11), we can now interpret the matrix $\bar{Q}$ as a feature, data-dependent matrix that tells us how much a given component of the local kernel contribute to the renormalized kernel $K_{\rm CNN}^{\rm(R)}$. The local kernel renormalization that takes place in this CNN model in the proportional limit is completely different from the trivial global one that occurs in the corresponding FC one hidden layer network. At this point one can wonder whether both the ingredients in CNNs design, local connectivity and weight sharing, are needed to observe the phenomenon of local kernel renormalization. The answer is affirmative: local kernel renormalization does not occur in a shallow network with local connectivity only, as we explicitly check in the appendix. In that case, only the diagonal components of the local kernel enter the effective action, and the network is not sensible to spatial correlations between different patches (see also Table 1 for a summary of the analytical results). ## V Empirical evidence for global and local kernel renormalization Extracting the renormalized kernel from the measure of a physical observable is not straightforward, and we have no way, at the moment, to directly access it in numerical experiments. One indirect experimental blueprint of the form of kernel renormalization at finite width in different architectures is given by the _similarity matrix_ of the internal representations before and after training. We define this observable for both FCNs and CNNs as: \[O_{\mu\nu}\equiv\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\sigma\left(h_{i}^{\mu} \right)\sigma\left(h_{i}^{\nu}\right)\,, \tag{18}\] where $N_{1}$ denotes the number of neurons in the last layer (for CNNs we have $N_{1}=N_{c}\lfloor N_{0}/S\rfloor$). We can track the effect of training by taking the difference between the similarity matrix at initialization, which is by definition the NNGP kernel (up to the Gaussian prior $\lambda_{1}$), and the same quantity after training: $\Delta K_{\mu\nu}\equiv\left(O_{\mu\nu}\right)-K_{\mu\nu}$, where the average is done over the Gibbs ensemble of the weights. We analytically compute this observable for FCNs and CNNs, as shown in appendix C. The final result in the two cases respectively reads (at zero temperature): \[\Delta K_{\mu\nu}^{\rm FCN} =-\frac{1}{N_{1}}\left[K_{\mu\nu}-\frac{\lambda_{1}}{\bar{Q}}y^{ \mu}y^{\nu}\right]; \tag{19}\] \[\Delta K_{\mu\nu}^{\rm CNN} =-\frac{1}{\lambda_{1}N_{1}}\sum_{ij,\lambda\rho=1}^{\lfloor N_{ 0}/S\rfloor,P}\bar{Q}_{ij}P_{\mu\lambda\nu\rho}^{ij}\] \[\times\left[\left(K_{\rm CNN}^{\rm(R)}\right)_{\lambda\rho}^{-1} -\sum_{\omega=1}^{P}\left(K_{\rm CNN}^{\rm(R)}\right)_{\lambda\epsilon}^{-1} \left(K_{\rm CNN}^{\rm(R)}\right)_{\rho\omega}^{-1}y^{\epsilon}y^{\omega} \right], \tag{20}\] where $P_{\mu\lambda\nu\rho}^{ij}\equiv\frac{1}{2}\sum_{k}\left[K_{\mu\lambda}^{ik}K_ {\nu\rho}^{kj}+K_{\mu\lambda}^{ki}K_{\nu\rho}^{jk}\right]$. Let us now highlight a few interesting physical implications that directly follow from these formulas and can be checked with numerical experiments: 1. the difference between the trained and untrained similarity matrix $\Delta K_{\mu\nu}^{\rm FCN}$ converges to zero in the thermodynamic limit $P,N_{1}\rightarrow\infty$ at fixed $\alpha_{1}=P/N_{1}$, as long as the terms in the square brackets are of order $o(N_{1})$. This means that in the proportional regime under consideration the effect of training on the internal representations is just a finite-size correction $O(1/N_{1})$ and the trained similarity matrix stays very close to its value at initialization. In particular, we expect that the empirical distribution of the elements of the matrix $\Delta K_{\mu\nu}^{\rm FCN}$ will be increasingly peaked around zero in finite-size scaling experiments where we consider increasing values of $P$ and $N_{1}$ keeping their ratio $\alpha_{1}=P/N_{1}$ fixed; 2. by choosing a learning task with labels $y^{\mu}=0,1$ and input patterns $x^{\mu}$ such that the matrix elements of the kernel $K_{\mu\nu}$ are on average centered in zero, the distribution of the matrix elements of the block $\Delta K_{\mu\nu}^{\rm FCN}\|_{11}$ (corresponding to the subset of training patterns with label one) should drift towards zero at a rate $1/N_{1}$, if one performs experiments at constant $\alpha_{1}$, while increasing $P$ and $N_{1}$. Interestingly, the remaining blocks of the matrix $\Delta K_{\mu\nu}^{\rm FCN}$ should not be affected by this drift; 3. the difference between the trained and untrained convolutional similarity matrix $\Delta K_{\mu\nu}^{\rm CNN}$ displays a different behavior, due to local kernel renormalization. For instance, even if we restrict our analysis to the learning setting outlined in bullet (ii), the elements of the block $\Delta K_{\mu\nu}^{\rm CNN}\|_{01}$ should not identically converge to zero in the thermodynamic proportional limit. A straightforward scaling analysis in fact shows that the contribution due to the term in the square bracket does not vanish in the proportional limit. In Fig. 2 we present the results of a finite-size scaling analysis to check the aforementioned predictions, where we trained one hidden layer FCNs and CNNs ( convolutions) on a synthetic dataset composed of random Gaussian patterns whose labels are given by a linear teacher function, i.e. $y=\frac{1}{2}\left[1+\text{sign}(t\cdot x)\right]$, where $t$ is an $N_{0}$-dimensional vector with unitary entries. Overall, the fact that the change in the internal representations of FCNs is a finite-size effect is confirmed by the experiments. On the other hand, the same numerical simulations with CNNs clearly identify a genuine change in the internal representations that occurs also in the thermodynamic proportional limit. We note that such a clean signature of the differences between FCNs and CNNs derive also from our choice of working close to zero temperature (where the architectures can precisely fit the training data). Equations at finite $T$ are more involved, and measuring the indirect effects of global and local kernel renormalization in this case is more difficult. Interestingly, even though our theory should be valid only in the Bayesian setting, it turns out to be also predictive for a learning algorithm that does not explicitly sample from the Gibbs posterior distribution: the aforementioned experiments were carried out using gradient descent and the so-called ADAM optimizer [54] (see Appendix D for more details). Another empirical evidence for the local renormalization in CNNs can be found analyzing the bias/variance decomposition of the generalization error over a new example $(\mathbf{x}^{0},y^{0})$: \[\langle\epsilon_{\text{g}}(\mathbf{x}^{0},y^{0})\rangle=(y^{0}-\Gamma)^{2}+ \sigma^{2} \tag{21}\] with: \[\Gamma=\sum_{\mu\nu=1}^{P}\kappa_{\mu}^{(\text{R})}\left(\frac{ \mathds{1}}{\beta}+K^{(\text{R})}(\{\bar{Q}\})\right)_{\mu\nu}^{-1}y_{\nu}, \tag{22}\] \[\sigma^{2}=\kappa_{0}^{(\text{R})}-\sum_{\mu\nu=1}^{P}\kappa_{\mu }^{(\text{R})}\left(\frac{\mathds{1}}{\beta}+K^{(\text{R})}(\{\bar{Q}\}) \right)_{\mu\nu}^{-1}\kappa_{\nu}^{(\text{R})} \tag{23}\] Figure 2: Predicting the effect of global and local kernel renormalization on the internal representations of FCNs and CNNs. A finite-size scaling analysis.* Here we show the results of numerical experiments performed in order to check the predictions of Eqs. (19), (20) for the difference $\Delta K_{\mu\nu}$ between the similarity matrix after and before training. In particular, we train architectures with a single hidden FC$/1d$-convolutional layer with tanh activation (see details in Appendix D) on a learning task with random Gaussian inputs and zero/one labels given by a linear teacher. We perform a finite-size scaling analysis by keeping $\alpha_{1}=1$ and choosing increasing values of $P,N_{1}$. In panel (a) we observe that the variance of the empirical distribution of the matrix elements of the zero-one block $\Delta K^{\text{PC}}_{\mu\nu}\|_{01}$ rapidly converges to zero as $N_{1}$ grows and $\alpha_{1}$ is kept fixed to one, in agreement with the prediction of Eq. (19) (top). In particular, the variance goes to zero as $1/N_{1}^{2}$ (middle). The fact that the similarity matrices after and before training are more and more similar in the thermodynamic limit is also clearly visible in the explicit array plot of a subsample of the zero-one block $\Delta K^{\text{PCN}}_{\mu\nu}\|_{01}$ (bottom). In panel (b) we display the histogram of the distribution of the matrix elements of the one-one block $\Delta K^{\text{PCN}}_{\mu\nu}\|_{11}$ (top). As predicted by Eq. (19), the distribution shrinks while drifting towards zero at a rate $\sim 1/N_{1}$ (middle). Overall, (a) and (b) represent a clear indication that the change in the after-training internal representations in one hidden layer FCNs is a finite-size effect. In (c) and (d) we repeat the same experiment for an architecture with a $1d$-convolutional hidden layer. The striking difference wrt the FC case is that the variance of the empirical distribution of the matrix elements $\Delta K^{\text{CNN}}_{\mu\nu}$ does not shrink to zero as $N_{c}$ grows and $\alpha_{c}$ is kept fixed to one, as one can also appreciate by directly looking at the array plot of a subsample of the block matrices $\Delta K^{\text{CNN}}_{\mu\nu}\|_{01}$ and $\Delta K^{\text{CNN}}_{\mu\nu}\|_{11}$. These expressions are valid for FCNs, and can be easily extended to CNNs with the substitution: \[K^{(\mathrm{R})}\to K^{(\mathrm{R})}_{\mathrm{CNN}}\quad\kappa^{(\mathrm{R})}\to \kappa^{(\mathrm{R})}_{\mathrm{CNN}}\,, \tag{24}\] where the renormalized kernel matrices $K^{(\mathrm{R})}$ and $K^{(\mathrm{R})}_{\mathrm{CNN}}$ are given respectively in equations (5) and (11). We recall the definition of the additional kernel integrals that depend on the new example $x^{0}$, that is $\kappa^{(\mathrm{R})}_{\mu}$, where the index $\mu=0,\ldots,P$ covers both the dataset and the new test element $x^{0}$: \[\kappa_{\mu}=\int\mathrm{d}t\mathcal{N}_{t}\left(0,\tilde{C}_{\mu}\right) \sigma(t_{1})\sigma(t_{2})\,, \tag{25}\] where: \[\tilde{C}_{\mu}=\begin{pmatrix}C_{\mu\mu}&C_{\mu 0}\\ C_{\mu 0}&C_{00}\end{pmatrix}\,,\ C_{\mu 0}=\frac{x^{\mu}\cdot x^{0}}{\lambda_{0}N_ {0}}\,, \tag{26}\] In the CNN case, these quantities are computed starting from the local covariance matrix $C^{ij}_{\mu\nu}$ defined in Eq. (12). These terms undergo the same type of renormalization as the correspondent kernel matrices (see table 1), that is: \[\kappa^{(\mathrm{R})}_{\mu}=\frac{\bar{Q}}{\lambda_{1}}\kappa_{ \mu}\,, \tag{27}\] \[\left[\kappa^{(\mathrm{R})}_{\mathrm{CNN}}\right]_{\mu}=\frac{1} {\lambda_{1}\lfloor N_{0}/S\rfloor}\sum_{ij=1}^{\lfloor N_{0}/S\rfloor} \bar{Q}_{ijk}\kappa_{\mu}^{ij}\,, \tag{28}\] The $\beta\to\infty$ limit of Eq. (22) yields different behaviors for the bias in the two cases: \[\Gamma^{\mathrm{FC}}=\sum_{\mu\nu=1}^{P}\kappa_{\mu}K^{-1}_{\mu \nu}\ y_{\nu}\,, \tag{29}\] \[\Gamma^{\mathrm{CNN}}=\sum_{\mu\nu=1}^{P}\left[\kappa^{(\mathrm{ R})}_{\mathrm{CNN}}\right]_{\mu}\left(K^{(\mathrm{R})}_{\mathrm{CNN}}(\{ \bar{Q}\})\right)^{-1}_{\mu\nu}\,y_{\nu}, \tag{30}\] From these expressions one can make two observations: (i) as already discussed in [38], $\Gamma^{\mathrm{FC}}$ at finite width is independent of $\bar{Q}$, and therefore the bias is identical to the infinite-width case for any $N_{1}$. (ii) In CNNs, on the other hand, the dependence of the bias from the renormalized Kernel does not disappear, and neither does the dependence on the width $N_{1}$. In the large Gaussian prior regime $\lambda_{1}\gg 1$, the previous observations can be made into straightforward predictions that can be tested with numerical experiments. It is easy to check that the variance is vanishing in this regime, since it is always proportional to a term $1/\lambda_{1}$ both in FCNs and CNNs. Moreover, when $\lambda_{1}\gg 1$, the analytical criterion found in [38] to predict whether a finite-width FCN will outperform its infinite-width counterpart, is never satisfied. This is a special case of the general considerations already presented in section II. To summarize, in this regime we expect that (i) the finite FC network can not outperform its infinite-width counterpart. (ii) On the contrary, we expect CNNs to possibly beat their infinite-width performance, since the bias can be reduced through the optimization of the feature matrix $Q^{ij}$. To test these predictions, we performed numerical simulations with $2d$ CNNs trained on CIFAR10 examples, close to the zero-temperature limit and in the bias-dominated setting where we let the Gaussian prior $\lambda_{1}$ be large. As shown in Fig. 3, the test loss of the FCN is a monotonically decreasing function of the hidden layer size $N_{1}$ and it does reach its asymptotic (best) performance already for $\alpha\sim 1$. Compatibly to local kernel renormalization, CNNs with finite number of channels best their infinite-width counterpart, already Figure 3: Test loss vs number of $2d$ convolutional filters for one hidden layer CNNs with different filter sizes $M$. All the networks are trained on the same binary regression task with $P=500$ examples from two classes of the CIFAR10 dataset, ”cars” and ”planes”, respectively labeled $0$ and $1$. Error bars represent one standard deviation. The images are gray-scaled and coarse-grained to $N_{0}=28\times 28$. We choose the filter size $M$ to be an integer divisor of $28$, and the filters to be non overlapping (taking stride $S=M$). A sketch of how the images are parsed by filters is reported for each network. With this choice, the FC network is retrieved setting $M=28$. The nets are trained with a zero temperature Langevin dynamics (see appendix REF), and with large Gaussian priors $\lambda_{1}$. In this bias-dominated regime, we observe that: (i) in FCNs, the best possible performance is the infinite-width one. Here the variance is monotonically decreasing with the width $N_{c}$, while the bias is a constant. Therefore, the test loss approaches its minimum monotonically from above, as predicted by our theory. (ii) finite width CNNs can outperform their infinite-width counterpart. The local kernel renormalization mechanism allows the CNN to optimize both bias and variance at finite width, making it even convenient to have a small number of filters (the minimum is reached for $N_{c}\sim 50$). Eventually, the CNNs reach their asymptotic performance from below. for large filter sizes ($M=14,7$) and ultimately approach the infinite-width limit performance from below. ## VI Discussion In this work we have derived an effective action in the proportional limit for CNNs and LCNs with one hidden layer, and we have extensively compared these results with the FC case found in [38]. We have highlighted a striking difference in the kernel renormalization mechanism that the three different architectures undergo at finite width, summarized in Table 1, and we have suggested a mechanism for feature learning (local kernel renormalization) that can take place in CNNs, but not in architectures with LC or FC layers alone. We have made a preliminary study on the effect of global and local kernel renormalization at the level of the internal representations of trained networks, which is in agreement with our predictions. We have also investigated the behavior of the generalization loss of one hidden layer networks in a controlled Bayesian setting where the performance is dominated by the bias, which confirms the observation that, in this regime, (i) finite-width one hidden layer FC networks cannot outperform their infinite-width counterpart; (ii) CNNs with finite number of channels can improve the infinite-width performance, which is in qualitative agreement with our local kernel renormalization hypothesis for the bias in Eq. (30). We conclude by pointing out a number of possible follow-ups of this work: (i) it would be interesting to extend our framework to $L$ convolutional layers, following the same strategy proposed for FC layers in [38]; (ii) our current approach cannot cover the mean-field regime, where the feature learning mechanism for FCNs and CNNs may quantitatively change. We think that the formalism presented in [23] might be more suited to investigate finite-width effects in this case; (iii) we are writing a detailed open-source numerical routine to extensively check the theoretical predictions of our effective theory for FCNs, LCNs and CNNs, which will also serve to point out the limitations of the present framework. _Acknowledgements_.- R.B. and P.R. are supported by #NEXTGENERATIONEU (NGEU) and funded by the Ministry of University and Research (MUR), National Recovery and Resilience Plan (NRRP), project MNESYS (PE0000006) "A Multiscale integrated approach to the study of the nervous system in health and disease" (DN. 1553 11.10.2022). The authors thank F. Cagnetta for giving feedback on a preliminary version of this manuscript and for pointing out interesting related works, G. Naveh, Z. Ringel and I. Seroussi for discussions and clarifications on their results in Refs. [23; 24] and P. Baglioni, M. Gherardi and M. Pastore for discussions.
2306.06934	Scale-Rotation-Equivariant Lie Group Convolution Neural Networks (Lie Group-CNNs)	The weight-sharing mechanism of convolutional kernels ensures translation-equivariance of convolution neural networks (CNNs). Recently, rotation-equivariance has been investigated. However, research on scale-equivariance or simultaneous scale-rotation-equivariance is insufficient. This study proposes a Lie group-CNN, which can keep scale-rotation-equivariance for image classification tasks. The Lie group-CNN includes a lifting module, a series of group convolution modules, a global pooling layer, and a classification layer. The lifting module transfers the input image from Euclidean space to Lie group space, and the group convolution is parameterized through a fully connected network using Lie-algebra of Lie-group elements as inputs to achieve scale-rotation-equivariance. The Lie group SIM(2) is utilized to establish the Lie group-CNN with scale-rotation-equivariance. Scale-rotation-equivariance of Lie group-CNN is verified and achieves the best recognition accuracy on the blood cell dataset (97.50%) and the HAM10000 dataset (77.90%) superior to Lie algebra convolution network, dilation convolution, spatial transformer network, and scale-equivariant steerable network. In addition, the generalization ability of the Lie group-CNN on SIM(2) on rotation-equivariance is verified on rotated-MNIST and rotated-CIFAR10, and the robustness of the network is verified on SO(2) and SE(2). Therefore, the Lie group-CNN can successfully extract geometric features and performs equivariant recognition on images with rotation and scale transformations.	Wei-Dong Qiao, Yang Xu, Hui Li	2023-06-12T08:14:12Z	http://arxiv.org/abs/2306.06934v1	# Scale-Rotation-Equivariant Lie Group Convolution Neural Networks (Lie Group-CNNs) ###### Abstract The weight-sharing mechanism of convolutional kernels ensures translation-equivariance of convolution neural networks (CNNs). Recently, rotation-equivariance has been investigated. However, research on scale-equivariance or simultaneous scale-rotation-equivariance is insufficient. This study proposes a Lie group-CNN, which can keep scale-rotation-equivariance for image classification tasks. The Lie group-CNN includes a lifting module, a series of group convolution modules, a global pooling layer, and a classification layer. The lifting module transfers the input image from Euclidean space $\mathbb{R}^{}$ to Lie group space, and the group convolution is parameterized through a fully connected network using Lie-algebra of Lie-group elements as inputs to achieve scale-rotation-equivariance. The Lie group SIM(2) is utilized to establish the Lie group-CNN with scale-rotation-equivariance. Scale-rotation-equivariance of Lie group-CNN is verified and achieves the best recognition accuracy on the blood cell dataset (97.50%) and the HAM10000 dataset (77.90%) superior to Lie algebra convolution network, dilation convolution, spatial transformer network, and scale-equivariant steerable network. In addition, the generalization ability of the Lie group-CNN on SIM(2) on rotation-equivariance is verified on rotated-MNIST and rotated-CIFAR10, and the robustness of the network is verified on SO(2) and SE(2). Therefore, the Lie group-CNN can successfully extract geometric features and performs equivariant recognition on images with rotation and scale transformations. Scale and rotation equivariance, group convolution neural network, Lie group, image classification ## 1 Introduction Convolutional neural networks have achieved unprecedented prosperity in the computer vision community [1] and become one of the most successful deep learning models achieving state-of-the-art progress on multiple tasks, such as image classification, object detection, and image segmentation. For example, VGG utilized small convolution filters instead of large ones and increased the network depth [2]. ResNet proposed a residual learning module to address the vanishing gradient issue in deep networks [3]. Later, DenseNet connected each layer to the others in a feed-forward method [4]. The translation equivariance is a critical performance of CNN. It is usually achieved by the weight-sharing mechanism of convolutional filters and the pooling operation to generate global features in the conventional CNN. The convolutional filters can preserve the same features before and after the translation transformations using identical weights to convolute local regions of the input image with sliding strides. Therefore, the regular CNN automatically possesses the translation symmetry. In addition, Lenc and Vedaldi [5] found that the first several layers of CNN could be predictably transformed with image warping and regarded as warping-equivariant. Besides transformation equivariance, when the relative motion occurs between a monocular camera and an object due to translation, scale, and rotation transformations, the captured image is supposed to change accordingly. Extracting invariant features and ensuring accurate predictions from images before and after translation, scale, and rotation transformations are enduring issues in computer vision. An equivariant feature detector guarantees that the output would transform the same way as the input transforms, ensuring no feature and information leakage occur. The invariance is a particular case of equivariance, i.e., the output would remain the same no matter what changes occurred in the input. Theoretically, the input images under different transformations would lead CNN to learn the invariant features adaptively. However, the unified equivariance of CNN could only be given empirically, and previous experiments validated that the unified equivariance of CNN was fragile when the input images were translated, scaled, and rotated [6]. Many classical image recognition algorithms achieve equivariance by detecting features that are insensitive to input transformations. For example, the image pyramid was proposed by downsampling the input images using a series of Gaussian convolutions with various scale parameters, and the local features with scale and rotation invariance were extracted in the scale space [7]. Lowe [8] proposed the Scale Invariant Feature Transform (SIFT) algorithm to construct a Difference of Gaussian (DoG) pyramid and find key points that do not change with image transformations. In addition to establishing the image pyramid and feature pyramid, another way to achieve equivariance is to design steerable filters. For example, arbitrary rotation of the filter response was interpolated from linear combinations of basis filters [9], and the steerable pyramid decomposed the image into a series of image subbands with different scales and orientations using basis functions of Gabor wavelets [10], which was designed to be scale and rotation invariant separately. Furthermore, a wavelet scattering network with translation, scale, and rotation invariances was proposed using scaled and rotated wavelets as filters [11,12,13]. However, the equivariant or invariant filters in the above mentioned methods are hand-crafted with prior knowledge and cannot be autonomously learned. To overcome the theoretical limitations of investigating equivariance for the regular CNN in the $\leavevmode\nobreak\ \mathbb{R}^{n}$, the group theory provides a promising way to construct a unique form of group CNN with equivariance in the group space [14,15]. In this paper, the group convolution is designed with scale and rotation equivariance in the group space by exploiting Lie groups with scaling and rotational group elements. The main contributions of this study are summarized as follows. (1) It is theoretically derived that the regular CNN does not possess the scale and rotation equivariance while the proposed Lie-group convolution does; (2) Lie group of SIM(2) and the corresponding Lie algebras with scale and rotation transformation elements are introduced to construct the Lie group-CNN with scale and rotation equivariance; (3) The scale and rotation equivariance Lie-group CNN mainly consists of a lifting module and several group convolution modules, in which the lifting module is designed to transfer the input image into the Lie group space, and the group convolution is parameterized by a fully connected network using the Lie algebra as input; (4) Finally, the scale and rotation equivariance of the Lie group-CNN is verified on two scaled-rotated image datasets. In addition, the generalization ability of the method on rotation equivariance is also verified on two rotated image datasets and the robustness of the network is verified on SO(2) and SE(2) with rotation transformations. Comparative studies are performed to demonstrate the efficacy of the proposed method superior to existing dilation convolution, STN, SESN, L-Conv, and multilayer perceptron (MLP)-based neural networks. The remainder of this paper is arranged as follows. Section 2 introduces some related work on the scale equivariance and rotation equivariance of CNN. Section 3 gives mathematical formulations of equivariance and theoretical proofs of lacking scale and rotation equivariance in regular CNNs. Section 4 describes the proposed scale and rotation equivariant Lie group-CNN applying Lie groups and Lie algebra, including the overall architecture and the novel modules of lifting operation, group convolution, and parameterization strategy. Section 5 shows the experimental results and comparative studies on two scaled-rotated and two rotated image datasets. Section 6 concludes the paper. ## 2 Related Work The equivariance of CNN has been investigated under various forms of translation, rotation, and scale transformations. Anti-aliasing and downsampling strategies were integrated to improve the shift equivariance of CNN [16]. The rotation equivariance of CNN is gained mainly by introducing rotated features of the input images. For example, a specialized CNN was designed to autonomously learn abstract representations of the images and obtain the rotational symmetry in galaxy images [17]. A rotation-invariance layer was introduced to guide the training process to learn similar features before and after rotation [18]. RotConv was proposed by rotating convolutional kernels with multiple orientations, and a vector field representing magnitude and angle at every spatial location was generated [19]. A gradient-aligned convolution was proposed to perform a pixel-level gradient alignment operation before the regular convolution and achieve rotation equivariance [20]. Xie et al. proposed a filter parametrization method based on an ameliorated Fourier series expansion for 2D filters to construct rotation equivariant convolutions [21]. As for the scale equivariance, the multi-scale feature pyramid and dilated convolution were utilized to establish the scale-invariance CNN [22,23] and they were also used in the field of object detection and image segmentation for multi-scale object [24,25] Gao et al. [26] proposed Res2Net representing multi-scale features at a granular level by constructing hierarchical residual-like connections in a single residual block. Jaderberg et al. [27] introduced a spatial transformation network where a spatial transformer module was designed to produce an appropriate spatial transformation for each input image, and the transformations were performed on all feature maps to achieve spatial equivariance. Qi et al. [28] presented deterministic autoencoding transformation and probabilistic autoencoding variational transformation models to learn more generic visual representations. Since transformations of input images can be regarded as group elements, the group theory has been introduced into CNNs to extend various equivariance properties, and most previous work mainly focuses on building equivariant architectures using discrete groups. The group convolution was defined through a surjective exponential mapping with the Lie algebra on the Lie group [15]. The equivariance of the neural network layer could be achieved by parameter-sharing under any operation of discrete groups [29]. Standard convolution kernels were transformed by discrete groups of 90-degree rotations and reflections [30,31]. General principles of constructing convolutions in a compact group and integrating over the group space were introduced [32]. Bekkers [33] proposed that using a generic basis of B-splines defined by the Lie algebra could express the equivariance of convolution kernels, and irreducible group representations were found to bring constraints on the expression ability of convolutional kernels [34]. Sosnovik et al. [35] developed a scale-equivariant steerable network using the steerable filter parametrization to scale the filter instead of resizing the tensor. Dehmamy et al. [36] proposed the Lie algebra convolutional network that can automatically discover symmetries with Lie algebra instead of Lie groups. ## 3 Mathematical Formulation of Equivariance The equivariance of a mapping function $\Phi\big{(}\cdot\big{)}$ on an input image $I$ is defined as \[\Phi\big{(}T_{g}I\big{)}=T_{g}\Phi\big{(}I\big{)} \tag{1}\] where $T_{g}$ denotes a linear transformation matrix corresponding to a specific transformation operation $g$. $g\in G$, where $G$ is a group consisting of all the possible group elements. $I\in\mathbb{R}^{n}$, $\Phi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, and $T_{g}$ acts on the input and output spaces. If (1) holds to $\forall I\in\mathbb{R}^{n},\forall g\in G$, $\Phi$ is equivariant to $G$. According to the linear transformation, for two different transformation operations $g$ and $g^{\prime}$, $T_{gg^{\prime}}=T_{g}T_{g^{\prime}}$. Note that the equivariance makes the mapping function $\Phi$ symmetrical to a certain transformation operation $g$, thus improving the generalization ability and geometric reasoning ability under various transformations. By the way, invariance is a special case of equivariance, i.e., $\Phi\big{(}T_{g}I\big{)}=\Phi\big{(}I\big{)}$, where the output remains precisely the same no matter what transformation has been applied to the input. Specifically, for the scale and rotation equivariance, $m=\big{(}s,\theta\big{)}$ can be expressed as a combination of a random scale factor $s$ and a random rotation angle $\theta$, where $s>0$ and $\theta\in\mathbb{R}$. The matrix form acting on the image coordinate system is $A_{m}=\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ and $A_{m}$ can be represented as the product of a rotation matrix $R_{\theta}$ and an isotropic scale matrix $Z_{s}$ \[\begin{array}{c}A_{m}=\begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}s\cdot cos\theta&-s\cdot sin\theta\\ s\cdot sin\theta&s\cdot cos\theta\end{bmatrix}=\\ \begin{bmatrix}s&0\\ 0&s\end{bmatrix}\begin{bmatrix}cos\theta&-sin\theta\\ sin\theta&cos\theta\end{bmatrix}=Z_{s}R_{\theta}\end{array} \tag{2}\] When a two-dimensional (2D) image is scaled-rotated-transformed around its center, the transformed image remains the same pixel intensity as the original image: \[I^{\prime}\big{(}x^{\prime}\big{)}=I\big{(}x\big{)} \tag{3}\] where $I^{\prime}\big{(}x^{\prime}\big{)}$ denotes the intensity values of the transformed image in the transformed coordinates $\big{(}x^{\prime}\big{)}$, and $I\big{(}x\big{)}$ denotes the intensity values of the original image in the original coordinates $\big{(}x\big{)}$. The coordinates before and after the scaled and rotated transformation satisfy $x^{\prime}=A_{m}x$. (3) can be written as \[I^{\prime}\big{(}x\big{)}=I\big{(}A_{m}^{-1}x\big{)} \tag{4}\] Next, the mathematical formula is derived to demonstrate that CNN is not scale and rotation equivariance. For an input layer or a hidden layer, a regular convolution operator can be calculated as \[\big{[}f^{}\psi\big{]}\big{(}x\big{)}=\sum_{\vec{y}}f\big{(}x-\vec{y}\big{)} \psi\big{(}\vec{y}\big{)}=\sum_{y}f\big{(}y\big{)}\psi\big{(}x-y\big{)} \tag{5}\] where $f$ denotes the feature map from the last layer, $\psi$ denotes the convolution kernel, $x$ denotes the input coordinates, $\vec{y}$ denotes the coordinates of the convolution kernel. Considering that the feature map is scaled-rotated-transformed, it can be derived by (4): \[T_{m}f\big{(}x\big{)}=f\big{(}A_{m}^{-1}x\big{)} \tag{6}\] The scale and rotation equivariance can be derived by \[\begin{array}{c}\big{[}\big{(}T_{m}f\big{)}\psi\big{]}\big{(}x\big{)}=\sum _{y}f\big{(}A_{m}^{-1}y\big{)}\psi\big{(}x-y\big{)}=\\ \sum_{y}f\big{(}y\big{)}\psi\big{(}x-A_{m}y\big{)}=\sum_{y}f\big{(}y\big{)} \psi\Big{[}A_{m}\big{(}A_{m}^{-1}x-y\big{)}\Big{]}\end{array} \tag{7}\] Based on (5) and (6), the following equation can be derived as \[\begin{array}{c}T_{m}\Big{[}f\big{(}T_{m^{-1}}\psi\big{)}\Big{]}\big{(}x \big{)}=\Big{[}f\big{(}T_{m^{-1}}\psi\big{)}\Big{]}\Big{(}A_{m}^{-1}x\big{)}= \\ \sum_{y}f\big{(}y\big{)}\Big{[}\big{(}T_{m^{-1}}\psi\big{)}\Big{(}A_{m}^{-1}x- y\big{)}\Big{]}=\\ \sum_{y}f\big{(}y\big{)}\psi\Big{[}A_{m}\big{(}A_{m}^{-1}x-y\big{)}\Big{]}\end{array} \tag{8}\] where $T_{m^{-1}}$ denotes the inverse scaled-rotated-transformed. Note that the right-side terms of (7) and (8) are the same. Then we have \[\big{[}\big{(}T_{m}f\big{)}\psi\big{]}\big{(}x\big{)}=T_{m}\Big{[}f\big{(}T_ {m^{-1}}\psi\big{)}\Big{]}\big{(}x\big{)} \tag{9}\] Equation (9) indicates that CNN is not scale and rotation equivariance because there is an additional operator of $T_{m^{-1}}$ acting on the convolution kernel $\psi$ on the right side, as the convolution of a scaled and rotated feature map is equal to the scale and rotation of an inverse scaled and rotated convolution. After the convolution operation, the feature maps are inputted into the nonlinear activation (e.g., ReLU), batch normalization (BN), linear, and global average pooling (GAP) layers: ReLU: \[ReLU\big{(}f_{i}\big{)}=\max\big{(}0,f_{i}\big{)}\] (10) \[\mu_{B}=\frac{1}{\mathcal{M}}\sum_{i=1}^{\mathcal{M}}f_{i},\sigma_ {B}^{2}=\frac{1}{\mathcal{M}}\sum_{i=1}^{\mathcal{M}}\big{(}f_{i}-\mu_{B}\big{)}^ {2},\] BN: \[\hat{f}_{i}=\frac{f_{i}-\mu_{B}}{\sqrt{\sigma_{B}^{2}+\epsilon}},f_{i} =\gamma\hat{f}_{i}+\beta,f^{\prime}=Wf+b\] Linear: \[f^{\prime}=Wf+b\] GAP: \[f^{\prime}=\frac{1}{\mathcal{N}}\sum_{i}f_{i}\] where $f_{i}$ denotes feature value at position $i$; $\mu_{B}$ and $\sigma_{B}^{2}$ denote mean and variance; $\mathcal{M}$ denotes batch size; $\gamma$ and $\beta$ are hyperparameters; $\hat{f}_{i}$ and $f_{i}$ denote the results after normalization and after scaling and translation, respectively; $W$ and $b$ denote weights and bias; $f^{\prime}$ denotes the result after the linear layer; $\mathcal{N}$ and $f^{\prime}$ denote the number and the mean value of pixels in feature map, respectively. As shown in (10), the ReLU, BN, linear, and global average pooling layers are scale and rotation equivariance because the inputs are values after convolution. ## 4 Methodology In this section, a novel Lie group-CNN is proposed based on the scale and rotation equivariance group convolution. Firstly, the Lie group and Lie algebra are introduced to solve the equivariant issue of regular CNNs. Then, the lifting layer is designed for mapping images to Lie algebras from the $\mathbb{R}^{n}$ to the Lie group space. Finally, the scale and rotation equivariance group convolution in the Lie group space is derived and parameterization as a fully connected network. ### Lie Groups and Lie Algebras Many sets of transformations satisfy the group properties and thus can form a group. Let $G$ denote a group under a single operation with the following properties: 1) For any three elements $g_{1},g_{2},g_{3}\in G$, they satisfy the associativity $\big{(}g_{1}g_{2}\big{)}g_{3}=g_{1}\big{(}g_{2}g_{3}\big{)}$. 2) The composition of any two elements $g_{1}g_{2}\in G$. 3) Each element $g_{1}$ has a unique inverse $g_{1}^{-1}$, and $g_{1}^{-1}\in G$. 4) There is a unique unit element $e$ in the group, i.e., $g_{1}g_{1}^{-1}=e$. Take the isotropic scale group in the 2D space as an example. Associativity can be verified by $\big{(}Z_{s}Z_{s_{2}}\big{)}Z_{s_{3}}=Z_{s_{1}}\big{(}Z_{s_{2}}Z_{s_{3}}\big{)}$. The multiplication of two scale matrices forms the third scale matrix, i.e., $Z_{s_{2}}Z_{s_{2}}=Z_{s_{3}s_{2}}$. The inverse of a scale matrix $Z_{s}$ is $Z_{Us}$. The unit element of identity is the scale factor $s=1$. The Lie group $G$ is the group with continuous, differentiable, and smooth. The similar transformation group SIM(2) contains scale and rotation transformation and the other two Lie groups (the special orthogonal group SO(2) and special Euclidean group SE(2)) with rotation transformation, all of which can be expressed by nonsingular real matrices in the 2D space. Since the Lie group is not defined in the vector space, it is challenging to directly measure the distance between group elements. Therefore, the Lie algebra $\mathfrak{g}$ is utilized to describe the local properties in the tangent plane around the identity of $G$ by infinitesimal transformations. The Lie algebra $\mathfrak{g}$ corresponds to the $k$-dimensional vector space, i.e., $\big{(}v_{1},...,v_{k}\big{)}\in\mathfrak{g}$, with the corresponding basis matrix elements $\big{\{}e_{1},...,e_{k}\big{\}}$. An expanded matrix $\Lambda$ can be generated by \[\Lambda=\sum_{i=1}^{k}v_{i}e_{i} \tag{11}\] where $v_{i}$ is the $i$-th component of the $k$-dimensional vector $\big{(}v_{1},...,v_{k}\big{)}$. For Lie groups, the matrix exponentiation is used to map the Lie algebra $\mathfrak{g}$ to the group $G$, i.e., $\exp\big{(}\Lambda\big{)}=\sum_{n=0}^{\infty}\frac{\Lambda^{n}}{n!}\cdot \mathfrak{g}\to G$. which has a closed form and an inverse of logarithm mapping as $log:G\rightarrow\mathfrak{g}$. Take the isotropic scale group as an example to describe the relationship between group $G$ and the Lie algebra $\mathfrak{g}$. The scale element $\lambda=\big{(}v_{1},...,v_{k}\big{)}_{k=1}$ is the Lie algebra of the group element $Z_{s}$, $e_{1}$ is the basis matrix element, and $Z_{s}=exp\big{(}\lambda e_{1}\big{)}$ is the Lie algebra parametrization of the isotropic scale group. The multiplication of group elements can be achieved by the addition of Lie algebra, i.e., $Z_{s_{1}}Z_{s_{2}}=exp\big{(}\lambda e_{1}\big{)}exp\big{(}\lambda_{2}e_{1} \big{)}=exp\big{[}\big{(}\lambda_{4}+\lambda_{2}\big{)}e_{1}\big{]}=Z_{s_{3}s_ {2}}$. The specific form of the similar transformation group SIM(2) is shown below. The SIM(2) group is the group of similarity transformations in the 2D space, i.e., \[\text{SIM}\big{(}2\big{)}=\begin{bmatrix}S\cdot cos\theta&-s\cdot sin\theta&t _{u}\\ S\cdot sin\theta&s\cdot cos\theta&t_{w}\\ 0&0&1\end{bmatrix}=\begin{bmatrix}sR^{2\times 2}&t^{2\times 4}\\ 0^{1\times 2}&1\end{bmatrix}\bigg{\|}\in\mathbb{R}^{3\times 3}\] , where $R$ represents the rotation transformation, $t\in\mathbb{R}^{2}$ represents the translation transformation, and $s\in\mathbb{R}^{+}$ denotes the scale transformation for uniformly zoom-in ($s>1$) and zoom-out ($0<s<1$) operations. Using the similarity transformation of the SIM(2) group, the similarity transformation can be derived by the homogeneous coordinates as \[\begin{bmatrix}U^{\cdot}\\ W^{\cdot}\\ 1\end{bmatrix}=\begin{bmatrix}U\big{(}s\cdot cos\theta\big{)}-W\big{(}s\cdot sin \theta\big{)}+t_{u}\\ U\big{(}s\cdot sin\theta\big{)}+W\big{(}s\cdot cos\theta\big{)}+t_{w}\\ 1\end{bmatrix}= \tag{12}\] \[\begin{bmatrix}s\cdot cos\theta&-s\cdot sin\theta&t_{u}\\ s\cdot sin\theta&s\cdot cos\theta&t_{w}\\ 0&0&1\end{bmatrix}\begin{bmatrix}U\\ W\\ 1\end{bmatrix}\] where $t^{2\times 4}=\begin{pmatrix}t_{u}\\ t_{w}\end{pmatrix}$ denotes the translation vector, $U$ and $W$ denote the 2D coordinates in the original image, $U^{\cdot}$ and $W^{\cdot}$ denote the 2D coordinates in the transformed image. The Lie algebra $\mathfrak{sim}\big{(}2\big{)}$ of SIM(2) is a four-dimensional vector, i.e., $\boldsymbol{v}=\big{(}u,w,\theta,\lambda\big{)}\in\mathbb{R}^{4}$. Therefore, $\mathfrak{sim}\big{(}2\big{)}$ has four basis matrix elements: \[e_{1}=\begin{bmatrix}0&0&1\\ 0&0&0\\ 0&0&0\end{bmatrix}\quad,\quad e_{2}=\begin{bmatrix}0&0&0\\ 0&0&1\\ 0&0&0\end{bmatrix}\quad,\quad e_{3}=\begin{bmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{bmatrix}\quad,\] \[e_{4}=\begin{bmatrix}0&0&0\\ 0&0&0\end{bmatrix}\text{. The matrix exponentiation mapping from }\] \[\mathfrak{sim}\big{(}2\big{)}\text{ to SIM(2) is expressed using the Taylor series as }\] \[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! \[C=\left(\frac{\lambda^{2}}{\lambda^{2}+\theta^{2}}\right)\cdot \left[\frac{1-e^{-\lambda}}{\lambda}\right]+\left(\frac{\theta^{2}}{\lambda^{2}+ \theta^{2}}\right)\cdot\left[\frac{sin\theta}{\theta}-\lambda\frac{1-cos\theta }{\theta^{2}}\right]\cdot\] \[D=\left(\frac{\lambda^{2}}{\lambda^{2}+\theta^{2}}\right)\cdot \left[\frac{e^{-\lambda}-1+\lambda}{\lambda^{2}}\right]+\left(\frac{\theta^{2} }{\lambda^{2}+\theta^{2}}\right)\cdot\left(\frac{1-cos\theta}{\theta}-\lambda \frac{\theta-sin\theta}{\theta^{3}}\right)\] Conversely, to obtain $\text{sim}\left(2\right)$ with a known SIM(2), the rotation element $\theta$ can be calculated by trigonometric function; the scale element $\lambda$ can be obtained by the negative logarithmic operation, and the translation elements $u$ and $w$ can be determined by \[\begin{pmatrix}u\\ w\end{pmatrix}=B^{-1}t^{2\lambda}=\frac{1}{C^{2}+\left(\theta\cdot D\right)^{2 }}\begin{bmatrix}C&\theta\cdot D\\ -\theta\cdot D&C\end{bmatrix}\begin{bmatrix}t_{w}\\ t_{w}\end{pmatrix} \tag{14}\] Next, $G$ of SIM(2) is a function of Lie algebra $\text{sim}\left(2\right)$ : \[G\left(\boldsymbol{v}\right)=exp\left(\sum_{i=1}^{4}v_{i}e_{i}\right)\cdot \text{According to the $n$-th order derivative}\] of the exponential function is continuous, so the group SIM(2) is continuous, differentiable, and smooth. The Lie group $G$ is continuous, differentiable, and smooth. In addition, SO(2) and SE(2) are used for the comparison in rotation equivariance, the Lie group and Lie algebra forms of them are shown as follows, \[\begin{split}\text{SO}\left(2\right)&=\left\{R= \begin{bmatrix}cos\theta&-sin\theta\\ sin\theta&cos\theta\end{bmatrix}\right\}\in\mathbb{R}^{2\times 2},\\ \text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text{\text{\text{ \ All the group elements that can transform the origin $o$ to an arbitrary point $z$ are defined by the lifting operation: $\operatorname{Lift}\big{(}z\big{)}=\big{\{}g\in G\cdot go=z\big{\}}$. The group elements mapping the origin $o$ to itself are defined as the isotropy subgroup of $G$: $H=\big{\{}h\in G\cdot h=o\big{\}}$. For the left coset $gH$ of the isotropy subgroup $H$, $\mathit{gho}=go$ for any $h\in H$ ; therefore, the left coset $gH$ forms a homogeneous space $X$. Since the group space is closed, $\operatorname{Lift}\big{(}z\big{)}$ often contains various group elements, and it is challenge to enumerate all possible group elements. A possible way to obtain rest group elements using a known group element $g_{1}$ is utilizing the isotropy subgroup to generate the left coset as $\operatorname{Lift}\big{(}z\big{)}=\big{\{}g_{1}H\big{\}}$. #### 4.2.2 Group convolution module. After the lifting module, the original image has been mapped into Lie Group space. The next group convolution is operated on the Lie Group space. The group convolution module consists of a novel group convolution layer, a batch normalization (BN) layer, a nonlinear activation layer (ReLU is used here), a linear fully-connected layer, a batch normalization (BN) layer, and a nonlinear activation layer (ReLU) in sequence. As shown in (9), the convolution kernel should be scaled and rotated in the opposite direction to obtain the scale and rotation equivariance, resulting in scale and rotation followed by a convolution differs from a convolution followed by scale and rotation. A scale and rotation equivariance group convolution is proposed to address this issue, which will be explained below. Assume that the following convolution kernels exist in the group space after lifting operation of (16) \[\psi_{A_{m}}\big{(}g\big{)}=\psi\big{(}A_{m}^{-1}g\big{)} \tag{17}\] Substituting (17) into (6), we have \[T_{m^{-1}\psi_{A_{m}}}\big{(}g\big{)}=\psi_{A_{m}}\big{(}A_{m}g\big{)}=\psi \big{(}A_{m}^{-1}A_{m}g\big{)}=\psi_{A_{m}A_{m}^{-1}}\big{(}g\big{)} \tag{18}\] According to the multiplication closure of the group, $A_{m}A_{m}^{-1}$ is still a group element, showing that the convolution kernel is still located in the group space after scale and rotation transformations. Substituting (18) into (9) yields \[\Big{[}\big{(}T_{m}\big{)}^{}\psi_{A_{m}}\Big{]}(g)=T_{m}\Big{[}f^{}\big{(} T_{m^{-1}\psi_{A_{m}}}\big{)}\Big{]}(g)=T_{m}\Big{[}f^{}\psi_{A_{m}A_{m}^{-1}} \Big{]}(g) \tag{19}\] Therefore, a scale and rotation equivariance group convolution is defined by (19) because the group convolution after scale and rotation equals the scale and rotation of a group convolution. It indicates that the convolution defined in the Lie group space can possess the scale, rotation and translation equivariance compared with the regular convolution in the $\mathbb{R}^{n}$. The (19) is specifically written in mathematical formulation of group convolution \[\big{(}f^{}\phi\big{)}_{G}\big{(}q\big{)}=\big{\lceil}\phi\big{(}g^{-1}q \big{)}f\big{(}g\big{)}d\mu\big{(}g\big{)} \tag{20}\] where $\big{(}f^{}\phi\big{)}_{G}$ denotes the group convolution of the input $f$ by a group convolution filter $\phi$, $g$ and $q$ denote the group elements of $G$, $g^{-1}q\in G$ according to the multiplication closure of the group, $f\big{(}g\big{)}$ denotes the output after the lifting operation, and $\mu$ denotes the Haar measure in the group space. #### 4.2.3 Parameterization of group convolution. For the discrete group, all group elements can be enumerated and involved in the group convolution operation based on (20). However, it is challenge for the continuous Lie group. To fix this issue, a fully connected neural network is employed to parameterize the group convolution kernel as an approximated continuous function. Note that elements in Lie groups are in matrix forms and are not convenient to perform the tensor operation, while the corresponding Lie algebras are vector-wise and can interconvert with the Lie group elements by the exponential and logarithmic transformations. Inspired by this idea, the identity transformation $g=exp\big{[}log\big{(}g\big{)}\big{]}\in G$ is substituted into (20) as \[\phi\Big{(}g^{-1}q\big{)}=\phi^{}\big{\lfloor}exp\big{[}log\big{(}g^{-1}q \big{)}\big{]}\big{\rfloor}=\big{(}\phi\cdot exp\big{)}\Big{[}log\big{(}g^{-1}q \big{)}\Big{]} \tag{21}\] where $log\big{(}g^{-1}q\big{)}$ represents the Lie algebra of the group element $g^{-1}q$, and $\big{(}\phi\cdot exp\big{)}$ is parameterized by a fully connected neural network noted as $\phi_{q}:g\rightarrow\mathbb{R}$. To determine the local region of the group convolution, the distance measure between two Lie group elements $g_{1}$ and $g_{2}$ is defined as \[D\big{(}g_{1},g_{2}\big{)}=\big{\lceil}log\big{(}g_{1}^{-1}g_{2}\big{)}\big{\rceil} _{F} \tag{22}\] where $\big{\lceil}\big{\rceil}_{F}$ denotes the Frobenius norm. Following the concept of reception field in the regular convolution, the group convolution operation is also defined within a local range of the continuous Lie group space as \[\phi\Big{(}g^{-1}q\big{)}=\begin{cases}\phi\Big{(}g^{-1}q\big{)}&D\big{(}g,q \big{)}\leq r\\ 0&D\big{(}g,q\big{)}>r\end{cases}, \tag{23}\] Since $D\big{(}g,q\big{)}\leq r$ defines a local region in the Lie group space, (23) can be approximated by choosing a set of neighborhood group elements of $q$ to discretize the group convolution as \[\big{(}f^{}\phi\big{)}_{G}\big{(}q\big{)}=\frac{1}{num\big{[}N\big{(}q\big{)} \big{]}}\sum_{g_{i}\in N(q)}\phi\Big{(}g_{i}^{-1}q\big{)}f\big{(}g_{i}\big{)} \tag{24}\] where $N\big{(}q\big{)}$ denotes the set of neighborhood group elements around $q$, $g_{i}$ denotes the $i$-th group element in $N\big{(}q\big{)}$, $num\big{(}\cdot\big{)}$ denotes the pre-set number of included group elements in the set. The above distance measure remains invariant for the left multiplication of another group element, indicating the invariance of group convolution in the group space: \[D\big{(}g_{3}g_{1},g_{3}g_{2}\big{)}=log\Big{(}g_{1}^{-1}g_{3}^{-1}g_{3}g_{2} \big{)}_{F}=D\big{(}g_{1},g_{2}\big{)} \tag{25}\] For example, the distance measures between two elements in the SIM(2) group is \[D\big{(}g_{1},g_{2}\big{)}=\big{\lceil}log\Big{(}g_{1}^{-1}g_{2}\big{)}\big{\rceil} _{F} \tag{26}\] \[=\left\|\begin{bmatrix}s_{1}\cdot cos\theta_{1}&-s_{1}\cdot sin \theta_{1}&t_{n}\\ \cdot sin\theta_{1}&s_{1}\cdot cos\theta_{1}&t_{n}\\ 0&0&1\end{bmatrix}\right\|\] \[=\left\|\begin{bmatrix}s_{2}\cdot cos\theta_{2}&-s_{2}\cdot sin \theta_{2}&t_{n}\\ \cdot sin\theta_{2}&s_{2}\cdot cos\theta_{2}&t_{n}\\ 0&0&1\end{bmatrix}\right\|-log\left\|s_{1}\cdot sin\theta_{1}&s_{1}\cdot cos \theta_{1}&t_{n}\\ 0&0&1\right\|\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ w_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\lambda_{2}-\lambda_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ w_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\lambda_{2}-\lambda_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\lambda_{2}-\lambda_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] \[=\left\|\begin{bmatrix}u_{2}-u_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\\ \cdot\mu_{2}-\mu_{1}\end{bmatrix}\right\|_{F}\] In summary, the overall schematic of the proposed group convolution operation is shown in Fig. 3. 2$\times$ downsampling is performed to the Lie algebra and image intensity, and the farthest point sampling algorithm [37] is developed to facilitate the 2$\times$ downsampling process and shown in Algorithm 1. The dimension of the Lie algebra array reduces to $\left(bs,NK,NK,l\right)$, and the dimension of the image intensity array reduces to $\left(bs,NK,C\right)$, where $bs$ denotes the batch size, $N$$\times$$K$ denotes the total number of Lie algebra, $l$ denotes the length of the Lie algebra vector ($l=1$ for the SO(2) group, $l=3$ for the SE(2) group, and $l=4$ for the SIM(2) group), $C$ denotes the channel number of the image intensity ($C=3$ for the RGB image and $C=1$ for the gray image after the lifting layer). The number of Lie algebra for each group element is set as $K=1$ for simplification. ## 5 Results and Discussion ### Scaled-Rotated Dataset In this section, two scaled-rotated datasets are utilized to verify the scale and rotation equivariance of the Lie group-CNN on SIM(2). Medical images have natural rotation and scale transformations due to the morphology similarity of targets with varying sizes and the lack of angular constraints between the observation device and objects. Therefore, two typical datasets of blood cells [38] and pigmentation skin diseases [39] are used to verify the scale and rotation equivariance of the proposed method. #### 5.1.1 Blood cell dataset. The blood cell dataset consists of individual blood cell images acquired by the automatic analyzer and labeled by clinical pathologists. The dataset includes 11,959 training images, 1,712 validation images, and 3,421 test images and is labeled into eight categories of neutrophils, eosinophils, basophilis, lymphocytes, monocytes, immature granulocytes, erythroblasts, and platelets. Each image is resized from 360$\times$363$\times$3 to 28$\times$28$\times$3. Some representative images are shown in Fig. 4. An input image is converted into an array $\left\{x_{i}.f_{i}\right\}_{i=1}^{784}$ in the $\mathbb{R}^{n}$ and then fed into the Lie Group-CNN. The stochastic gradient descent algorithm is used for optimization, and the cross-entropy loss is used as the objective function to train a classifier on the dataset. The Lie group-CNN on SIM(2) that contains both rotation and scale group elements is utilized to verify the rotation-scale-equivariant capacity. In addition, the dilated convolution enabling multi-scale contextual information aggregation [23], STN [27], SESN using steerable filters to build scale equivariance convolutional networks [35], and L-Conv [36] are used for comparison. The training hyperparameters for the abovementioned networks remain the same and are determined by trial and error with an initial learning rate of 0.05, a batch size of 50, and a total epoch of 100. Table 1 shows the comparison of the Top-1 test accuracy on the blood cell dataset. The results show that the accuracy of all methods is above 90%, and the Lie group-CNN on SIM(2) achieves the best recognition accuracy of 97.50% with 3.8%, 4.37%, 1.83%, and 7.32% improvement compared to the dilated convolution, STN, SESN, and L-Conv, respectively. The results indicate that the Lie group-CNN on SIM(2) can also extract scale and rotation equivariance features from more complicated images on a relatively small dataset. ``` Inputs: A set of group elements $S=\left\{q_{i}\right\}_{i=1}^{num(S)}$, the number of required elements in $S_{sub}$ as $P=HW/2$. Step 1: Choose an arbitrary group element from $S$ as the initial point $p_{0}$, $S_{sub}=\left\{p_{0}\right\}$. Step 2: Calculate the distances between all the other group elements in $S$ and $p_{0}$. Step 3: Form a distance set $D_{0}=\left\{D_{0}^{n}\right\}$,$n=1,...,num\big{(}S\big{)}-1$. Step 4: Select the farthest point with the maximum $D_{0}^{n}$ as $p_{1}$, and update the set $S_{sub}=\left\{p_{0},p_{1}\right\}$. Step 5:for j=1,..., $P$-1 do Calculate the distance $D_{j}^{n}=D\left(q_{n},p_{j}\right)$ for all $q_{n}\in S\setminus S_{sub}$ Form a distance set $D_{j}=\left\{D_{j}^{n}\right\}$,$n=1,...,num\big{(}S\setminus S_{sub}\big{)}$. if $D_{j}^{n}<D_{0}^{n}$ $D_{0}^{n}=D_{j}^{n}$, select the farthest point with the maximum $D_{0}^{n}$ as $p_{j+1}$. Update the set $S_{sub}=\left\{p_{0},p_{1},...,p_{j+1}\right\}$ end Return$S_{sub}$ ``` Algorithm 1 Farthest Point Sampling Algorithm ### Rotated Dataset In this section, the rotated MNIST and the rotated CIFAR-10 datasets are utilized to verify the generalization ability of the method on rotation equivariance and the robustness of the network on the other two Lie groups. Figure 4: Some representative images in the blood cell dataset. Figure 3: Schematic of the proposed group convolution operation. #### 5.2.1 Rotated MNIST dataset The rotated MNIST dataset [40] is generated by randomly rotating images in the MNIST dataset and used to verify the efficacy of the proposed rotation-equivariance Lie Group-CNN. It is split into a training set and a test set with 12,000 and 50,000 images, respectively. The image resolution is 28x28x1, and Fig. 6 shows some representative images. A CNN classification model based on ResNet18 [3] is trained as the baseline. Three types of Lie groups, including SO(2), SE(2), and SIM(2) are investigated to demonstrate the effectiveness and generalization ability of the proposed method and the capacity of rotation equivariance. In addition, STN, L-Conv, and MLP-Mixer [41] are also compared to validate the superiority of the proposed method on the spatial transformer, Lie algebra, and MLP-based neural networks. A CNN classification model based on ResNet18 [3] is trained as the baseline. Three types of Lie groups, including SO(2), SE(2), and SIM(2) are investigated to demonstrate the effectiveness and generalization ability of the proposed method and the capacity of rotation equivariance. In addition, STN, L-Conv, and MLP-Mixer [41] are also compared to validate the superiority of the proposed method on the spatial transformer, Lie algebra, and MLP-based neural networks. Table III shows the comparison of the Top-1 test accuracy on the rotated MNIST dataset. The results show that the proposed Lie group-CNN on SO(2), SE(2), and SIM(2) outperforms the conventional ResNet18, STN, L-Conv, and MLP-Mixer, demonstrating the efficacy of rotation equivariance using the Lie group-CNN over the conventional CNN, spatial transformer, Lie algebra, and MLP-based neural networks. The generalization ability of the proposed method is also verified by the fact that using different Lie groups can improve classification accuracy against various rotation transformations. Among them, the Lie group-CNN on SO(2) achieves the best recognition accuracy of 98.2% on the rotated MNIST dataset, while using SE(2) and SIM(2) have slight decreases in the Top-1 accuracy. The results further indicate that the proposed Lie group convolution can better extract the embedding knowledge of symmetry than ResNet18, STN, L-Conv, and MLP-Mixer on rotated images. #### 5.2.2 Rotated CIFAR-10 dataset The original CIFAR-10 dataset consists of 50,000 training images in 10 classes. In this section, the rotated CIFAR-10 dataset is used to verify the efficacy of the proposed rotation-equivariance Lie Group-CNN. The training set of the rotated CIFAR-10 is obtained by randomly rotating the training images in the original CIFAR-10 dataset from 0 to 360 degrees, and the test set randomly samples 10,000 training images and rotates with different aspects to Fig. 5: Some representative images in the HAM10000 dataset Fig. 6: Some representative images of the rotated MNIST dataset. investigate the rotation equivariance. The resolution of the original images is 32x32x3, and the rotated images are padded with zeros to maintain an identical resolution. Some representative images are shown in Fig. 7. Similar to section 5.2.1, comparative studies are performed between three types of Lie groups, including SO(2), SE(2), and SIM(2), and the conventional ResNet18, STN, L-Conv, and MLP-Mixer. Table IV compares the Top-1 classification accuracy on the rotated CIFAR-10 dataset using different classification networks. The Lie group-CNN on SO(2) achieves the best Top-1 classification accuracy on the rotated CIFAR-10 (99.9%), and the Lie group-CNN on SE(2) and SIM(2) only have a 0.1% decrease in classification accuracy which demonstrates the robustness of the Lie group-CNN on other Lie groups. The results show that using the proposed Lie group-CNN can significantly improve the classification accuracy on rotated images, further indicating the sensitivity of the Lie group convolution to the rotation equivariance. On the other side, model performances of ResNet18, STN, L-Conv, and MLP-Mixer on the rotated CIFAR-10 suggesting that the conventional CNN-based and MLP-based models have limited ability to extract symmetric knowledge on rotated images and fail to maintain the rotation equivariance. ### Visualization of Separability To compare the separability of the aforementioned neural networks, the T-SNE algorithm [42] is utilized to reduce the high-dimensional feature maps from the last layer of each network to two dimensions for visualization. Figure 8 compares the separability visualization results on the blood cell dataset using the Lie group-CNN on SIM(2) and other different types of neural networks: L-Conv, dilated convolution, STN, and SESN. Eight categories of blood cells in various scales and rotations form the most distinct clusters in the Lie group-CNN on SIM(2), indicating that the proposed method performs the best on the blood cell dataset. The entangled impartibility of cluster boundary exists in other neural networks more or less with different severities, among which the L-Conv gains the most limited classification performance on the blood cell dataset. The results show that the proposed Lie group-CNN simultaneously obtains the best inter-class diversity and intra-class similarity and can extract invariant geometric features on images with scale and rotation transformations. Figure 9 shows the visualization results of the proposed Lie Group-CNNs on the rotated datasets of MNIST and CIFAR-10. The results indicate that the Lie group-CNNs can effectively distinguish the different categories on these rotated datasets, among which the Lie group-CNN on SO(2) significantly performs the best than on SE(2) and SIM(2). The possible reason is that the extra learnable space caused by translation and scale elements in Lie algebra brings more challenges to rotation-only datasets. ## 6 Conclusions This study proposes a novel Lie-group CNN to fix the issue of lacking scale and rotation equivariance for the regular CNN. A group convolution is designed in the Lie group space instead of the $\mathbb{R}^{n}$ to generalize scale and rotation equivariance. The main conclusions are obtained as follows: (1) A Lie group-CNN with scale and rotation equivariance is constructed based on the group convolution, including an initial lifting module, $N$ sequential group convolution modules, a global pooling layer, and a final classification layer. The first lifting module transfers the input image from the $\mathbb{R}^{n}$ to the Lie group space. Theoretical analysis is performed to demonstrate that the group convolution after scale and rotation equals the scale and rotation of a group convolution, indicating that the proposed Lie-group CNN can possess the scale and rotation equivariance besides translation equivariance compared with the conventional CNN in the $\mathbb{R}^{n}$. (2) The group convolution module consists of a novel group convolution layer, a batch normalization layer, a nonlinear activation layer, a linear fully-connected layer, a batch normalization layer, and a nonlinear activation layer in sequence. The Lie group elements can be transformed into the Lie algebras by surjective exponential mapping. For computational efficiency, the group convolution operators are locally defined using a distance measure between two Lie group elements, which remains invariant for the left multiplication of another group element. The group convolution operator is replaced by a fully connected network for parameterization in the Lie group space and facilitates the tensor operation. (3) The SIM(2) group with both scale and rotation elements is utilized to construct a Lie group-CNN on SIM(2), \begin{table} \begin{tabular}{c c} \hline \hline Network & Top-1 accuracy on the \\ & rotated CIFAR-10dataset \\ \hline Lie Group-CNN on SO(2) & 99.9\% \\ Lie Group-CNN on SE(2) & 99.8\% \\ Lie Group-CNN on SIM(2) & 99.8\% \\ ResNet18 [3] & 79.7\% \\ STN [27] & 62.83\% \\ L-Conv [36] & 55.58\% \\ MLP-Mixer [41] & 60.3\% \\ \hline \hline \end{tabular} \end{table} TABLE IV: COMPARISON OF TOP-1 CLASSIFICATION ACCURACY ON THE ROATED CIFAR-10 DATSET Fig. 7: Some representative images in the rotated CIFAR-10 dataset. and the scale and rotation equivariance of the network is verified on two medical datasets, i.e., the blood cell dataset and the HAM10000 dataset. The results show that the Lie group-CNN on SIM(2) achieves the best recognition accuracy on both datasets compared to dilation convolution, spatial transformer, steerable filter, and Lie algebra-based networks. The results demonstrate that the Lie group-CNN on SIM(2) can perform equivariant recognition on objects with varying sizes and orientations and further indicate that the proposed method can extract geometric features on images with scale and rotation transformations. (4) The robustness of the network is verified on other two Lie groups with rotation transformations (SO(2) and SE(2)), and the generalization ability of the Lie group-CNN on rotation equivariance is verified on two rotated image datasets. The results show that using the proposed Lie group-CNN on SO(2), SE(2), and SIM(2) can effectively improve the classification accuracy on rotated images, indicating the sensitivity of the Lie group convolution to the rotation equivariance and the robustness of the proposed method to different Lie groups. Fig. 8: Visualization of separability by multi-type neural networks on the blood cell dataset. The scale and rotation equivariance are enhanced by introducing the group convolution operator in the Lie group space instead of the conventional CNN in the $\mathbb{R}^{n}$, demonstrating the significance of group symmetry for feature extraction in image recognition tasks. In the future study, the ability to achieve more complex transformation equivariance will be further investigated and applied in object detection and image segmentation tasks. ## Acknowledgment Financial support for this study was provided by the National Natural Science Foundation of China [Grant Nos. 51921006, 52192661, and 52008138], China Postdoctoral Science Foundation [Grant Nos. BX20190102 and 2019M661286], Heilongjiang Touyan Innovation Team Program, Heilongjiang Natural Science Foundation [Grant No. LH2022E070], and Heilongjiang Provincial Postdoctoral Science Foundation [Grant Nos. LBH-TZ2016 and LBH-Z19064].
2303.12147	Universal Approximation Property of Hamiltonian Deep Neural Networks	This paper investigates the universal approximation capabilities of Hamiltonian Deep Neural Networks (HDNNs) that arise from the discretization of Hamiltonian Neural Ordinary Differential Equations. Recently, it has been shown that HDNNs enjoy, by design, non-vanishing gradients, which provide numerical stability during training. However, although HDNNs have demonstrated state-of-the-art performance in several applications, a comprehensive study to quantify their expressivity is missing. In this regard, we provide a universal approximation theorem for HDNNs and prove that a portion of the flow of HDNNs can approximate arbitrary well any continuous function over a compact domain. This result provides a solid theoretical foundation for the practical use of HDNNs.	Muhammad Zakwan, Massimiliano d'Angelo, Giancarlo Ferrari-Trecate	2023-03-21T19:10:09Z	http://arxiv.org/abs/2303.12147v2	# Universal Approximation Property of Hamiltonian Deep Neural Networks ###### Abstract This paper investigates the universal approximation capabilities of Hamiltonian Deep Neural Networks (HDNNs) that arise from the discretization of Hamiltonian Neural Ordinary Differential Equations. Recently, it has been shown that HDNNs enjoy, by design, non-vanishing gradients, which provide numerical stability during training. However, although HDNNs have demonstrated state-of-the-art performance in several applications, a comprehensive study to quantify their expressivity is missing. In this regard, we provide a universal approximation theorem for HDNNs and prove that a portion of the flow of HDNNs can approximate arbitrary well any continuous function over a compact domain. This result provides a solid theoretical foundation for the practical use of HDNNs. ## I Introduction Deep Neural Networks (DNNs) have been crucial for the success of machine learning in several real-world applications like computer vision, natural language processing, and reinforcement learning. To achieve state-of-the-art performance, a common approach in machine learning is to increase the Neural Network (NN) depth. For instance, Convolutional Neural Networks (CNNs) AlexNet [1], Visual Geometric Group (VGG) network, GoogLeNet/Inception [2], Residual Network (ResNet) [3], or recently developed transformers such as ChatGPT, contain hundreds to thousands of layers. It has been empirically demonstrated that deeper networks yield better performance than single-hidden-layer NNs for large-scale and high-dimensional problems [4, 5]. However, a rigorous characterization of the approximation capabilities of complex NNs is often missing. Moreover, the understanding of how NN architectures (depth, width, and type of activation function) achieve their empirical success is an open research problem [6]. To quantify the representational power of NNs, researchers have focused on studying their Universal Approximation Properties (UAPs), namely their ability to approximate any desired continuous function with an arbitrary accuracy. To this aim, several UAP results for various classes of NNs have been proposed. The UAP of Shallow NNs (SNNs), _i.e._ with single hidden layer has proven in the seminal works of Cybenko [7] and Hornik [8]. Exploiting the latter arguments, researchers have provided several results on UAPs for DNNs. For instance, in [4] it is proved that a DNN with three hidden layers and specific types of activation functions has the UAP. The paper [5] demonstrates that a very deep ResNet, with stacked modules having one neuron per hidden layer and rectified linear unit (ReLU) activation functions, can uniformly approximate any integrable function. However, extending these results to other classes of activation functions is not straightforward. We defer the interested readers to [9] for a detailed survey on the subject. Recently, an alternate representation of DNNs as dynamical systems has been proposed [10]. This idea was later popularized as Neural Ordinary Differential Equations (NODEs) [11]. By viewing DNNs through a dynamical perspective, researchers have been able to utilize tools from system theory in order to analyze their properties (_e.g._, Lyapunov stability, contraction theory, and symplectic properties). Similar to DNNs, there are some contributions on UAPs for NODEs. It has been shown in [12] that capping a NODE with a single linear layer is sufficient to guarantee the UAP, but exclusively for non-invertible continuous functions. Furthermore, in [13], differential geometric tools for controllability analysis were used to provide UAPs for a class of NODEs, while in [14], the compositional properties of the flows of NODEs were exploited to obtain UAPs. In [13] certain restrictions on the choice of activation functions are present, whereas [14] impose constraints on the desired target function. Finally in [15], some interesting tools, such as composition of contractive, expansive, and sphere-preserving flow maps, have been used to prove a universal approximation theorem for the flows of dynamical systems. Although DNNs tend to empirically perform well in general, the increasing depth can also present challenges, such as the vanishing/exploding gradient problem during the training via gradient descent algorithms. These phenomenon happen when the gradients computed during backpropagation either approach to zero or diverge. In such cases, the learning process may stop prematurely or become unstable, thereby limiting the depth of DNNs that can be utilized and consequently preventing the practical exploitation of UAP in DNNs. Practitioners have proposed several remedies to address these challenges, including skip connections in ResNet [3], batch normalization, subtle weights initialization, regularization techniques such as dropout or weight decay, and gradient clipping [16]. However, all of these ad hoc methods do not come with provable formal guarantees of non-vanishing gradients. Recently, a class of DNNs called Hamiltonian Deep Neural Networks (HDNNs) have been proposed in [17]. These DNNs stem from the discretization of Hamiltonian NODEs, and enjoy non-vanishing gradients _by design_ if symplectic discretization methods [18] are used [17]. Moreover, the expressivity of HDNNs has been demonstrated empirically on several benchmarks in classification tasks. Nevertheless, the theoretical foundation on the UAP of HDNNs has yet to be explored. ### _Contributions_ In this paper, we present a rigorous theoretical framework to prove a UAP of HDNNs. First, with a slight modification, we generalize the class of HDNNs considered in [17] without compromising the provable non-vanishing gradients property1. Second, we prove that a portion of the flow of HDNNs can approximate any continuous function with arbitrary accuracy. To the best of our knowledge, this is the first UAP result for a class of ResNets enjoying non-vanishing gradients which are essential for numerically well-posed training. The proof is based on three essential features _i.e._ symplectic discretization through the Semi-Implicit Euler (SIE) method, a careful choice of initial conditions, and an appropriate selection of the flow. It is important to note that general DNNs, such as deep Multi-Layered Perceptrons (MLPs) or recurrent NNs, can suffer from vanishing gradients and might fail to approximate arbitrary functions if the training stops early. Third, since DNNs arising from the discretization of ODEs are automorphic maps - they do not alter the dimension of the input data - based on the composition of functions, we extend the main result to approximate maps, where the dimensions of domain and co-domain are different. Finally, we provide a characterization of the approximation error with respect to the depth. Footnote 1: For the sake of simplicity, we retain the same name and also refer to the proposed modified version as HDNNs. _Organization:_ Section II provides preliminaries on Hamiltonian NODEs, the employed discretization scheme, definitions of UAPs, and the problem formulation. In Section III, we prove the UAP for HDNNs (Theorem 1), we investigate the case when the desired function is not an automorphic map (Corollary 1), and provide some remarks on the approximation error (Proposition 2). We discuss a numerical example in Section IV. Finally, conclusions are drawn in Section V. ### _Notation_ We denote the set of non-negative reals with $\mathbb{R}_{+}$. For a vector $x\in\mathbb{R}^{n}$, its 2-norm is represented by $\\|x\\|$ and its 1-norm $\\|x\\|_{1}:=\sum_{j}\|x_{j}\|$. Given an $\mathcal{L}_{2}$-function $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ the $\mathcal{L}_{2}$ norm over the compact set $\Omega\subset\mathbb{R}^{n}$ is denoted by $\\|f\\|_{\mathcal{L}_{2}(\Omega)}$ and the (essential) supremum norm by $\\|f\\|_{\mathcal{L}_{\infty}(\Omega)}=\sup_{x\in\Omega}\\|f(x)\\|$. $\|A\|$ stands for the determinant of a squared matrix $A$. We represent with $0_{n}$ the zero vector in $\mathbb{R}^{n}$ and with $0_{n\times n}$ the matrix with all entries equal to zero in $\mathbb{R}^{n\times n}$. We denote the column vector of ones of dimension $n$ with $\mathds{1}_{n}$. Given $\Omega\subset\mathbb{R}^{n}$, $\mathcal{C}(\Omega;\mathbb{R}^{n})$ stands for the space of continuous functions $f:\Omega\to\mathbb{R}^{n}$. Given $T\in\mathbb{R}_{+}$, we refer to $\mathcal{P}([0,T];\mathbb{R}^{p})$ as the space of piecewise constant function $\theta:[0,T]\to\mathbb{R}^{p}$. Functions that cannot be represented in the form of a polynomial are referred to as _non-polynomial_ functions. ## II Preliminaries and problem formulation ### _Hamiltonian Neural Ordinary Differential Equations_ A Neural ODE [11] (NODE) is represented by the dynamical system for $t\in[0,T]$ given by \[\dot{x}(t)=F(x(t),\theta(t))\quad\text{with }x(0)=x_{0}\in\Omega\, \tag{1}\] where $x(t)\in\mathbb{R}^{n}$ is the state at time $t$ of the NODE with initial condition $x_{0}$ in some compact set $\Omega\in\mathbb{R}^{n}$, and $F:\mathbb{R}^{n}\times\mathbb{R}^{p}\to\mathbb{R}^{n}$ is such that $F(x,\theta)$ is Lipschitz continuous with respect to $x$ and measurable with respect to the weights $\theta$. We further assume that $\theta(t)\in\mathcal{P}([0,T];\mathbb{R}^{p})$. When used in machine learning tasks, the NODE is usually pre- and post- pended with additional layers, _e.g.,_$x_{0}=\varphi_{\alpha}(z)$, with $z\in\mathbb{R}^{n_{z}}$ the input and $\varphi_{\alpha}$ a NNs with parameters $\alpha\in\mathbb{R}^{n_{\alpha}}$, and the output $y$ is computed as $y=\phi_{\beta}(x(T))$, where $\phi_{\beta}$ is a NNs with parameters $\beta\in\mathbb{R}^{n_{\beta}}$. In this paper, we consider a class of NODEs inspired by Hamiltonian systems. In particular, we consider the Hamiltonian function $H:\mathbb{R}^{2n}\times\mathbb{R}_{+}\to\mathbb{R}$ given by \[H(x,t)=\tilde{\sigma}(W(t)x+b(t))^{\top}\mathds{1}_{n}+\eta(t)^{\top}x\, \tag{2}\] where $W:\mathbb{R}_{+}\to\mathbb{R}^{2n\times 2n}$, $b:\mathbb{R}_{+}\to\mathbb{R}^{2n}$, $\eta:\mathbb{R}_{+}\to\mathbb{R}^{2n}$ are piece-wise constant, while $\tilde{\sigma}:\mathbb{R}\to\mathbb{R}$ is a differentiable map, applied element-wise when the argument is a matrix, and such that $\sigma(x):=\frac{\partial\tilde{\sigma}}{\partial x}(x)$ is non-polynomial and Lipschitz continuous. As explained below, $\sigma$ will play the role of the so-called _activation function_. Examples that satisfy the above assumptions are provided in Table I. Note that if we set $\eta(t)=0$ in (2), we recover DNNs proposed in [10, 17]. We define the Hamiltonian system \[\dot{x}(t)=J(t)\frac{\partial H(x(t),t)}{\partial x}\, \tag{3}\] where $J(t)$ is piecewise constant skew-symmetric matrix, namely $J(t)=-J(t)^{\top}$, in $\mathbb{R}^{2n}\times\mathbb{R}^{2n}$ for any $t\geq 0$. By taking into account the expression of the Hamiltonian in (2), we obtain the following dynamics \[\dot{x}(t)=J(t)\left(W(t)^{\top}\sigma\big{(}W(t)x(t)+b(t)\big{)}+\eta(t) \right). \tag{4}\] Note that the latter equation can be written in the form (1), when the weights are given by $\theta(t)=\{J(t),W(t),b(t),\eta(t)\}$ for $t\in[0,T]$. For the numerical implementation of NODE (4), we rely on the SIE discretization [18] because it can preserve the symplectic flow of time-invariant Hamiltonian systems and is crucial to prove non-vanishing gradient property of the resulting HDNNs (further details will be given in the next section). In particular, splitting the state of the Hamiltonian systems into $x=(p,q)$, we obtain the HDNN \[\left[\begin{array}{c}p_{j+1}\\ q_{j+1}\end{array}\right]=\left[\begin{array}{c}p_{j}\\ q_{j}\end{array}\right]+hJ_{j}\left[\begin{array}{c}\frac{\partial H}{\partial p _{j}}\left(p_{j+1},q_{j},t_{j}\right)\\ \frac{\partial H}{\partial q}\left(p_{j+1},q_{j},t_{j}\right)\end{array}\right], \tag{5}\] where $h=T/N$, with $N\in\mathbb{N}$, is the integration step-size, $j=0,\ldots,N-1$ and $p_{j}$ and $q_{j}$ are the two state components in $\mathbb{R}^{n}$. Moreover, by taking into account the expression of the Hamiltonian in (2), namely the dynamics (4), we obtain the following difference equation \[\begin{split}\left[\begin{array}{c}p_{j+1}\\ q_{j+1}\end{array}\right]&=\left[\begin{array}{c}p_{j}\\ q_{j}\end{array}\right]\\ &\qquad+hJ_{j}\left(W_{j}^{\top}\sigma\left(W_{j}\left[\begin{array}{c}p_{j+ 1}\\ q_{j}\end{array}\right]+b_{j}\right)+\eta_{j}\right).\end{split} \tag{6}\] Clearly, the set of weights is given by $\theta_{j}=\{J_{j},W_{j},b_{j},\eta_{j}\}$ with $j=0,\ldots,N-1$. With a little abuse of notation we write $\theta_{j}\in\mathbb{R}^{p}$ with $j=0,\ldots,N-1$ and appropriate $p\in\mathbb{N}$. Although, in general, one has to compute the update $(p_{j+1},q_{j+1})$ of (6) through an implicit expression, it is possible to rewrite it in an explicit form, when the matrices $J_{j}$ and $W_{j}$ satisfy some assumptions, _e.g._, by choosing $J_{j}$ block anti-diagonal and $W_{j}$ block diagonal [17]. ### _Universal Approximation Property_ In this section, we present some essential definitions pertaining to universal approximation properties. Definition 1 (UAP of a function): _Consider a function $g_{\theta}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with parameters $\theta\in\mathbb{R}^{p}$ and a compact subset $\Omega\in\mathbb{R}^{n}$, then $g_{\theta}$ has the Universal Approximation Property (UAP) on $\Omega\subset\mathbb{R}^{n}$ if for any $f\in\mathcal{C}(\Omega;\mathbb{R}^{n})$ and $\varepsilon>0$, there exists $\theta\in\mathbb{R}^{p}$ such that_ \[\sup_{x\in\Omega}\\|f(x)-g_{\theta}(x)\\|\leq\varepsilon. \tag{7}\] We provide the following fact which descends from [19]. Proposition 1: _Let $\sigma\in\mathcal{C}(\mathbb{R};\mathbb{R})$ be non-polynomial, then for any $f\in\mathcal{C}(\Omega;\mathbb{R}^{n})$, where $\Omega\in\mathbb{R}^{n}$, and $\varepsilon>0$, there exist $N\in\mathbb{N}$, $A_{j},W_{j}\in\mathbb{R}^{n\times n}$ and $b_{j}\in\mathbb{R}^{n}$ such that the function $g:\mathbb{R}^{n}\to\mathbb{R}^{n}$ given by_ \[g(x):=\sum_{j=0}^{N-1}A_{j}\,\sigma(W_{j}x+b_{j})\, \tag{8}\] _satisfies_ \[\sup_{x\in\Omega}\\|f(x)-g(x)\\|\leq\varepsilon. \tag{9}\] _Some examples of activation functions $\sigma$, such that $g$ in (8) satisfies the UAP, are given in Table I._ In the sequel, we refer to the UAP with bound $\varepsilon>0$ to quantify the estimation error in equations (7), and (9). This value is typically a function of $N$, $n$, and the desired $f$, and it is characterized in Proposition 2. ### _Problem formulation_ The goal of our paper can be formulated as follows. Problem 1: _Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuous function, $\Omega\subset\mathbb{R}^{n}$ be a compact set, and $\varepsilon>0$ be the desired approximation accuracy. Find $N\in\mathbb{N}$ and weights $\theta_{j}=\{J_{j},W_{j},b_{j},\eta_{j}\}$ with $j=0,\ldots,N-1$ of (6), such that a portion $\varphi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ of the flow $\Phi_{N}:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ at time $N\in\mathbb{N}$ of (6) has the UAP on $\Omega$._ We recall that the flow at time $N\in\mathbb{N}$ of (6) is the corresponding unique solution at time $N\in\mathbb{N}$. In particular, $\Phi_{N}:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ is the flow at time $N$ as function of the initial condition. The flow $\varphi$ will be precisely defined in Theorem 1. Moreover, motivated by real-world applications, we are also interested in approximating arbitrary continuous functions $f:\mathbb{R}^{n}\to\mathbb{R}^{r}$ where $r$ is not necessarily equal to $n$. For instance, in classification tasks, typically $r<n$, as $r$ corresponds to the number of classes to be classified and $n$ represents the number of features. We address this problem in Corollary 1. ## III Main Results In this section, we present our main results whose proofs are given the Appendix. We address the Problem 1 in Theorem 1, which is a universal approximation theorem for the HDNN (5). Theorem 1: _Consider the discrete-time system (6) with initial condition $(p_{0},q_{0})=(\xi,0_{n})$, for some $\xi\in\Omega$ with $\Omega\subset\mathbb{R}^{n}$ compact. Then, the restricted flow $\varphi:\xi\mapsto q_{N}$ has the UAP on $\Omega$._ In other words, Theorem (1) states that given the system (6) with initial condition $(p_{0},q_{0})=(\xi,0_{n})$, for any $f\in C(\Omega;\mathbb{R}^{n})$ and $\varepsilon>0$, there exist $N\in\mathbb{N}$ and weights $\theta_{j}=\{J_{j},W_{j},b_{j},\eta_{j}\}$ with $j=0,\ldots,N-1$ such that the function $\varphi:\xi\mapsto q_{N}$ satisfies \[\sup_{\xi\in\Omega}\\|f(\xi)-\varphi(\xi)\\|<\varepsilon. \tag{10}\] Remark 1 (Key ingredients for UAP): _The proof of Theorem 1, besides exploiting arguments from [7, 8] for showing UAPs, it is based on three critical key steps: i) the SIE discretization scheme, ii) the initial condition $((p_{0},q_{0})=(\xi,0_{n})$, and iii) the focus on the restricted flow $\xi\mapsto q_{N}$, which refers to map the initial condition of the $p$ state to the flow of the $q$ state._ In particular, the choice of the SIE discretization scheme together with the initial condition $(p_{0},q_{0})=(\xi,0_{n})$ allows one to exploit the framework of Cybenko [7] to express the function $\varphi:\xi\mapsto q_{N}$ as (8) (see equation (21) in the Proof of Theorem 1 in Appendix B). Remark 2 (Feature augmentation): _By defining the flow $\Phi_{N}$ of the discrete-time system (6) (evolving in $\mathbb{R}^{2n}$), we note that (10) can be written as_ \[\sup_{x\in\Omega}\\|f(x)-\pi\circ\Phi_{N}\circ\iota(x)\\|\leq\varepsilon\, \tag{11}\] _where $\iota:\mathbb{R}^{n}\to\mathbb{R}^{2n}$ is the injection given by $\iota(z_{1},\ldots,z_{n})=(z_{1},\ldots,z_{n},0,\ldots,0)$ and $\pi:\mathbb{R}^{2n}\to\mathbb{R}^{n}$ is the projection $\pi(x_{1},\ldots,x_{n},x_{n+1},\ldots,x_{2n})=(x_{n+1},\ldots,x_{2n})$. This is equivalent to the common practice in machine learning of augmenting the size of the feature space [20]. It has been demonstrated that this technique can improve DNN performance in several learning tasks. Moreover, it is also closely related to the idea of extended space [16], which suggests that by increasing the dimensionality of the feature space, one can capture more complex relationships. We note that the UAP results in [13] do not apply in our framework because of the skew-symmetric matrix $J$ multiplying the partial derivative of the Hamiltonian in (3). Moreover, we provide UAPs directly for implementable discrete-time layer equations (6) instead of the continuous-time NODEs. Indeed, an arbitrary discretization method may not conserve the desired properties, making it challenging to prove the UAP of discretized NODEs in general. Untill this point, we focused on automorphisms on $\mathbb{R}^{n}$. The next result presents the UAP of a general map from $\Omega\subset\mathbb{R}^{n}$ to $\mathbb{R}^{r}$. Corollary 1: _Consider the discrete-time system (6) with initial condition $\left(p_{0},q_{0}\right)=\left(\xi,0_{n}\right)$, for some $\xi\in\Omega$ with $\Omega\subset\mathbb{R}^{n}$ compact, and the restricted flow $\varphi:\xi\mapsto q_{N}$. Let $h:\mathbb{R}^{n}\rightarrow\mathbb{R}^{r}$ be a Lipschitz continuous function such that $f\left(\Omega\right)\subseteq h\left(\mathbb{R}^{n}\right)$. Then, for any $\varepsilon>0$, the function $h\circ\varphi:\Omega\rightarrow\mathbb{R}^{r}$, satisfies_ \[\sup_{\xi\in\Omega}\\|f(\xi)-h\circ\varphi(\xi)\\|\leq\varepsilon. \tag{12}\] A typical example that satisfies the necessity condition $f\left(\Omega\right)\subseteq h\left(\mathbb{R}^{n}\right)$ is $h(\varphi)=W_{o}^{\top}\varphi+b_{o}$, $W_{o}\in\mathbb{R}^{n\times r}$, and $b_{o}\in\mathbb{R}^{r}$, which is common in classification problems. It is straightforward to see that $h(\cdot)$ is Lipschitz continuous, surjective, and satisfies the condition $f\left(\Omega\right)\subseteq h\left(\mathbb{R}^{n}\right)$. It is worth mentioning that unlike other papers [4, 5, 13], our results do not impose restrictive conditions on activation functions, which expands their potential applicability. ### _Auxiliary properties of HDNNs_ In the following, we highlight a few associated properties of HDNNs. First, we provide a bound on the desired accuracy of the approximation error with respect to the depth of HDNNs. Second, we state a remark on their non-vanishing gradients property. Let us define the first absolute moment $C_{f}$ of the Fourier magnitude distribution of a desired function $f$. Thus, given $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ with a Fourier representation of the form $f(x)=\int_{\mathbb{R}^{n}}\mathrm{e}^{i\omega^{\frac{1}{x}}}\tilde{f}(\omega )\mathrm{d}x$, we define \[C_{f}:=\int_{\mathbb{R}^{n}}\\|\omega\\|_{1}\\|\tilde{f}(\omega)\\|\mathrm{d} \omega. \tag{13}\] The condition (13) is usually interpreted as the integrability of the Fourier transform of the gradient of the function $f$, and a vast list of examples for which bounds on $C_{f}$ can be obtained are given in Section IX of [21]. Proposition 2: _Consider the discrete-time system (6) with sigmoidal2$\sigma$ and initial condition $\left(p_{0},q_{0}\right)=\left(\xi,0_{n}\right)$, for some $\xi\in\Omega=[-1,1]^{n}$. Then, the restricted flow $\varphi:\xi\mapsto q_{N}$ has the UAP on $\Omega$ with bound $2^{\frac{n}{2}}\frac{C_{f}}{\sqrt{N}}$._ Footnote 2: The function $\sigma(x)$ is assumed to be a sigmoidal function, if it is a bounded function on the real line satisfying $\sigma(x)\to 1$ as $x\rightarrow\infty$ and $\sigma(x)\to-1$ as $x\rightarrow-\infty$[22]. Proposition 2 states that for any $f\in\mathcal{C}(\Omega;\mathbb{R}^{n})$ with finite $C_{f}$ and $N\in\mathbb{N}$, there exist parameters $\theta_{j}=\left\{J_{j},W_{j},b_{j},\eta_{j}\right\}$ with $j=0,\ldots,N-1$, such that the function $\varphi:\xi\mapsto q_{N}$ satisfies \[\sup_{x\in\Omega}\\|f(x)-\varphi(x)\\|\leq 2^{\frac{n}{2}}\frac{C_{f}}{\sqrt{N}}. \tag{14}\] Further remarks on the evaluation/approximation of this bound can be found in [21] and [22]. As mentioned earlier, it has been shown that HDNNs considered in [17] are endowed with non-vanishing gradients or in a special case, non-exploding gradients [23], _i.e._, they ensure numerically well-posed training. We defer the reader to those papers for a formal discussion of the non-vanishing gradients property. Remark 3 (Non-vanishing gradients): _The HDNN given by the discrete-time system (6) enjoys the non-vanishing gradients property when optimizing a generic loss function. In particular, this property is related to the Backward Sensitivity Matrix $\frac{\partial x_{N}}{\partial x_{N-j}}=\prod_{L\equiv N-j}^{N-1}\frac{ \partial x_{f+1}}{\partial x_{f}}$, where $x=(p,q)$, at layer $N-j$ for $j=1,\ldots,N-1$. Although the considered Hamiltonian (2) is different from the one of [17] (because of the linear term), one is able to prove the non-vanishing gradients property (by establishing a lower bound for the Backward Sensitivity Matrix) by following the same arguments of [17, Theorem 2] which relies specifically on the symplectic property of the flow and not on the Hamiltonian structure._ ## IV Numerical Example In this example, our goal is to approximate the function $y(x)=2(2\cos(x)^{2}-1)^{2}-1$ considered in [24]. The training set comprises 5000 datapoints generated by sampling $y(t)$ randomly for $x\in[-2\pi,2\pi]$. We choose the mean square error as the loss function and compare the following NN architectures: i) The SNN $\hat{y}=W_{o}\sigma(Wx+b)$, with $W_{o}\in\mathbb{R}^{1\times N_{h}}$, $W\in\mathbb{R}^{N_{h}\times 1}$ and $b\in\mathbb{R}^{N_{h}}$, where $N_{h}$ is the number of hidden neurons. We use the values of $N_{h}$ in the set $\{400,800,1200,1800,2400\}$. ii) an HDNN, called HDNN-1, with forward equation (6) and weight matrices (19) for $j=0,1,\cdots,6$. iii) an HDNN, called HDNN-2, with forward equation (6), where $W_{j}$ is block-diagonal for $j=0,1,\cdots,3$ to match the number of parameters in HDNN-1. For HDNNs, we choose a sufficiently small step-size $h=0.001$, and the initial conditions as $p_{0},q_{0}=\left([x,0_{M/2-1}],0_{M/2}\right)$, where $M$ is always an even integer. Moreover, the output equation is given by $\tilde{y}=W_{o}q_{N}+b_{o}$, where $W_{o}\in\mathbb{R}^{1\times M/2}$ and $b_{o}\in\mathbb{R}^{M/2}$. To have almost the same number of parameters in the chosen HDNNs, we choose $M$ from the set $\{24,36,44,54,62\}$ for HDNN-1 and $\{26,36,44,54,64\}$ for HDNN-2, respectively. Fig. 1 shows that the training loss decreases when more parameters are used for all three architectures. Moreover, we can see that for the same number of parameters, the block diagonal $W_{j}$ matrices of HDNN-2 with half the number of layers can be leveraged to further improve the performance over HDNN-1. ## V Conclusion and Future Work We demonstrated the universal approximation property of Hamiltonian Deep Neural Networks (HDNNs) that also enjoy non-vanishing gradients during training. This result affirms both the practicality and theoretical foundation of HDNNs. In particular, we have demonstrated that a portion of the flow of HDNNs can approximate any continuous function in a compact domain. Also, we provide some insights on the approximation error with respect to the depth of neural network. Our work opens doors to quantifying the expressivity of other physics-inspired neural networks with special properties, such as [25]. Future research will focus on leveraging differential geometric tools [13] to establish universal approximation properties for HDNNs, where the Hamiltonian function is parameterized by an arbitrary neural network. ### _A preliminary lemma_ In order to prove Theorem 1, we introduce a key auxiliary result which relaxes the necessity of full-rank weight matrices $W_{j}$ in (8) assumed in [26, Theorem 2.6]. During the training of NN (8), some entries of $W_{j}$ in (8) might vanish and this assumption cannot be satisfied. Therefore, the result in [26] might not be of practical use. However, the following Lemma shows that even if $W_{j}$ in (8) are not full-rank, we can still construct an approximation with full-rank matrices and apply the results of [7, 19]. Lemma 1: _Let $g$ be the function in (8) with the UAP on $\Omega$. For any $\tilde{\varepsilon}>0$ we can find $\tilde{A}_{j},\tilde{W}_{j}\in\mathbb{R}^{n\times n}$, with $\tilde{W}_{j}$ full rank, and $\tilde{b}_{j}\in\mathbb{R}^{n}$ for $j=0,\ldots,N-1$, such that $\tilde{g}(x):=\sum_{j=0}^{N}\tilde{A}_{j}\,\sigma(\tilde{W}_{j}x+\tilde{b}_{j})$ satisfies $\\|\tilde{g}-g\\|_{\mathcal{L}_{\infty}(\Omega)}\leq\tilde{\varepsilon}$._ In other words, Lemma 1 allows us to assume, without loss of generality, that the function $g$ in (8) can be arbitrarily well-approximated by using full-rank matrices $W_{j}$ for any $j=0,\ldots,N-1$. Given the function $g$ in (8) with the UAP on $\Omega$, we consider the case in which there exists the set $K=\{\kappa\in\{0,\ldots,N-1\}:\|W_{\kappa}\|=0\}$ non-empty with cardinality $\tilde{n}$. For $\kappa\in K$, let $\text{rank}(W_{\kappa})=n-r_{\kappa}$, with $r_{\kappa}>0$ the number of dependent column vectors of $W_{\kappa}$ so that, up to a row permutation, assume $W_{\kappa}$ is partitioned as \[W_{\kappa}=\left(w_{\kappa}^{(1)},\ldots,w_{\kappa}^{(r_{\kappa})},w_{\kappa} ^{(r_{\kappa}+1)},\ldots,w_{\kappa}^{(n)}\right)^{\top}, \tag{15}\] with the last $n-r_{\kappa}$ vectors linearly independent. Then, the parameters $\tilde{A}_{j},\tilde{W}_{j}\in\mathbb{R}^{n\times n}$ and $\tilde{b}_{j}\in\mathbb{R}^{n}$ of the function $\tilde{g}(x)=\sum_{j=0}^{N}\tilde{A}_{j}\,\sigma(\tilde{W}_{j}x+\tilde{b}_{j})$ can be selected as follows. We set $\tilde{A}_{j}=A_{j}$, $\tilde{b}_{j}=b_{j}$ for all $j=0,\ldots,N-1$. Moreover, $\tilde{W}_{j}=W_{j}$ for all $j\notin K$ and, for $\kappa\in K$, $\tilde{W}_{\kappa}=W_{\kappa}+\Lambda_{\kappa}$, where $\Lambda_{\kappa}=\left(\tilde{w}_{\kappa}^{(1)},\ldots,\tilde{w}_{\kappa}^{(r _{\kappa})},0_{n},\ldots,0_{n}\right)^{\top}$, and the vectors $\tilde{w}_{\kappa}^{(\ell)}$, $\ell=1,\ldots,r_{\kappa}$, are selected such that $\|\tilde{W}_{\kappa}\|\neq 0$ and \[\\|\tilde{w}_{\kappa}^{(\ell)}\\|\leq\frac{\tilde{\varepsilon}}{r_{\kappa}\, \tilde{n}\,\sqrt{n}\,L_{\sigma}\,\\|x\\|_{\mathcal{L}_{\infty}(\Omega)}\,\max_{ 1\leq p\leq n}\\|a_{\kappa}^{(p)}\\|}\, \tag{16}\] where $L_{\sigma}$ is the Lipschitz constant of function $\sigma$ and $a_{\kappa}^{(p)}$${}^{\top}$, $p=1,\ldots,n$, are the rows of the matrix $A_{\kappa}$3. By noticing that for $x\in\Omega$ we have Footnote 3: We implicitly assume the non-trivial case $A_{\kappa}\neq 0_{n\times n}$ since if $A_{\kappa}$ is the zero matrix, then one can select any $\tilde{w}_{\kappa}^{(\ell)}$ such that $\|\tilde{W}_{\kappa}\|\neq 0$ and $\\|\tilde{g}-g\\|_{\mathcal{L}_{\infty}(\Omega)}=0$. \[\left\\|(\tilde{W}_{\kappa}-W_{\kappa})x\right\\|\leq\\|x\\|_{\mathcal{L}_{\infty} (\Omega)}\sum_{\ell=1}^{r_{\kappa}}\\|\tilde{w}_{\kappa}^{(\ell)}\\|, \tag{17}\] and by looking at the $p$-th component of the difference $\tilde{g}-g$, by inequality (16), for $x\in\Omega$, we have \[\left\|\tilde{g}^{(p)}(x)-g^{(p)}(x)\right\| \leq L_{\sigma}\,\\|x\\|_{\mathcal{L}_{\infty}(\Omega)}\sum_{ \kappa\in K}\left(\\|a_{\kappa}^{(p)}\\|\sum_{\ell=1}^{r_{\kappa}}\\|w_{\kappa}^{ (\ell)}\\|\right)\] \[\leq\sum_{\kappa\in K}\frac{\tilde{\varepsilon}}{\tilde{n}\sqrt{ n}}\leq\frac{\tilde{\varepsilon}}{\sqrt{n}}\, \tag{18}\] from which we obtain $\\|\tilde{g}-g\\|_{\mathcal{L}_{\infty}(\Omega)}\leq\tilde{\varepsilon}$. ### _Proof of Theorem 1_ We prove the result by showing that the function $\varphi:\xi\mapsto q_{N}$ can be written in the form (8), and thus, satisfying Proposition 1, it has the UAP on $\Omega$. In fact, by restricting the parameter space as follows \[\begin{split} J_{j}&=\begin{bmatrix}0_{n\times n}&-X\\ X&0_{n\times n}\end{bmatrix}\qquad W_{j}=\begin{bmatrix}\tilde{W}_{j}&0_{n \times n}\\ 0_{n\times n}&0_{n\times n}\end{bmatrix},\\ b_{j}&=\begin{bmatrix}\tilde{b}_{j}\\ 0_{n}\end{bmatrix}\qquad\qquad\qquad\eta_{j}=\begin{bmatrix}0_{n}\\ -\tilde{\eta}_{j}\end{bmatrix},\end{split} \tag{19}\] where $X\in\mathbb{R}^{n\times n}$, $\tilde{W}_{j}:\mathbb{R}_{+}\to\mathbb{R}^{n\times n}$, $\tilde{b}_{j}:\mathbb{R}_{+}\to\mathbb{R}^{n}$, $\tilde{\eta}_{j}:\mathbb{R}_{+}\to\mathbb{R}^{n}$, one can write (6) as \[\begin{split}\left[\begin{array}{c}p_{j+1}\\ q_{j+1}\end{array}\right]&=\left[\begin{array}{c}p_{j}\\ q_{j}\end{array}\right]+h\left[\begin{array}{c}X^{\top}\tilde{\eta}_{j}\\ X\tilde{W}_{j}^{\top}\sigma(\tilde{W}_{j}p_{j+1}+\tilde{b}_{j})\end{array} \right]\\ &=\left[\begin{array}{c}p_{j}\\ q_{j}\end{array}\right]+\left[\begin{array}{c}\tilde{\eta}_{j}\\ \tilde{A}_{j}\sigma(\tilde{W}_{j}p_{j+1}+\tilde{b}_{j})\end{array}\right]\end{split} \tag{20}\] Fig. 1: Averaged training loss and standard deviations over multiple experiments for three architectures. for $j=0,1,\cdots,N-1$, where $\tilde{\gamma}_{j}=hX^{\top}\tilde{\tau}_{j}$, and $\tilde{A}_{j}=hX\tilde{W}_{j}^{\top}$, respectively. From the initial condition $(p_{0},q_{0})=(\xi,0_{n})$, $\xi\in\Omega$, and by substituting the expression of $p_{j+1}$ into the second equation of (20) we have that \[q_{N} =\sum_{j=0}^{N-1}\tilde{A}_{j}\sigma(\tilde{W}_{j}(p_{j}+\tilde{ \gamma}_{j})+\tilde{b}_{j})\] \[=\sum_{j=0}^{N-1}\tilde{A}_{j}\sigma(\tilde{W}_{j}\xi+\tilde{d}_{ j})=:\varphi(\xi)\;, \tag{21}\] where $\tilde{d}_{j}=\tilde{W}_{j}\tilde{r}_{j}+\tilde{b}_{j}$ with $\tilde{r}_{j}=\tilde{r}_{j-1}+\tilde{\gamma}_{j}$ and $\tilde{r}_{0}=\tilde{\gamma}_{0}$. Notice that, because of Lemma 1, we can assume, without loss of generality, that $\tilde{W}_{j}$ in (21) are full-rank. Consequently, one can freely choose $\tilde{A}_{j}$ by setting $X=\frac{1}{h}\tilde{A}_{j}\tilde{W}_{j}^{-\top}$ for all $j=0,\ldots,N-1$, while $\tilde{d}_{j}$ is free by construction due to the parameter $\tilde{b}_{j}$. Thus, the map (21) has the UAP on $\Omega$ (Proposition 1), _i.e._$\left\\|\varphi(\xi)-g\right\\|_{\mathcal{L}_{\infty}(\Omega)}\leq\tilde{\varepsilon}$, with $g$ in (8). Note that the zero patterns of matrices, _i.e._$W,\eta,b$ in (19) is only assumed for proving Theorem 1. However, since using more parameters4 in (19) cannot compromise UAPs, the structure of the weight matrices in (19) is never used in practice. Footnote 4: We recall that $J_{j}$ should keep the sparsity structure (19) to maintain the non-vanishing gradients property (see [17, Theorem 2]). ### _Proof of Corollary 1_ In [14, Proposition 3.8] it is shown that there exists a continuous function $\psi(\xi)=\sum_{i=1}^{N}z_{i}\psi_{i}(\xi)$ for $\xi\in\Omega$, where $z_{i}\in h^{-1}\left(F_{i}\right)$, with $\left\{F_{i}\right\}_{i=1}^{N}$ a partition of $f(\Omega)$, and continuous functions $\psi_{i}:\Omega\rightarrow[0,1]$ such that $\psi_{i}=1$ on $A_{i}$ and $\psi_{i}=0$ on $\cup_{j\neq i}A_{j}$. The sets $A_{i}\subset\Omega_{i}$, with $\{\Omega_{i}\}_{i=1}^{N}$ a partition of $\Omega$, such that $h\circ\psi$ has the UAP on $\Omega$ (provided that the desired function $f$ is such that $f\left(\Omega\right)\subseteq h\left(\mathbb{R}^{n}\right)$). Now, take $\psi$ such that $\left\\|f-h\circ\psi\right\\|_{\mathcal{L}_{\infty}(\Omega)}\leq\varepsilon/2$ and, by Theorem 1, take $\varphi:\xi\mapsto q_{N}$ such that $\left\\|\psi-\varphi\right\\|_{\mathcal{L}_{\infty}(\Omega)}\leq\varepsilon/$$(2L_{h})$. Then, for any $\xi\in\Omega$ we have \[\left\\|f-h\circ\varphi(\xi)\right\\| \leq\left\\|f(\xi)-h\circ\psi(\xi)\right\\|\] \[\leq\frac{\varepsilon}{2}+L_{h}\left\\|\psi(\xi)-\varphi(\xi) \right\\|\leq\varepsilon\;,\] and the proof is completed. ### _Proof of Proposition 2_ The proof follows from [22, Theorem 1] by noting that the function $\varphi$ in (21) is the NN considered in [22], by selecting the probability measure $\tilde{\lambda}(\cdot):=\frac{1}{2^{N}}\lambda(\cdot)$ where $\lambda$ is the Lebesgue measure on $\Omega$, and by recalling the norm inequality $\\|\cdot\\|_{\infty}\leq\\|\cdot\\|_{2}$.
2306.12969	Stock Price Prediction using Dynamic Neural Networks	This paper will analyze and implement a time series dynamic neural network to predict daily closing stock prices. Neural networks possess unsurpassed abilities in identifying underlying patterns in chaotic, non-linear, and seemingly random data, thus providing a mechanism to predict stock price movements much more precisely than many current techniques. Contemporary methods for stock analysis, including fundamental, technical, and regression techniques, are conversed and paralleled with the performance of neural networks. Also, the Efficient Market Hypothesis (EMH) is presented and contrasted with Chaos theory using neural networks. This paper will refute the EMH and support Chaos theory. Finally, recommendations for using neural networks in stock price prediction will be presented.	David Noel	2023-06-18T20:06:44Z	http://arxiv.org/abs/2306.12969v1	# Stock Price Prediction using Dynamic Neural Networks ###### Abstract This paper will analyze and implement a time series dynamic neural network to predict daily closing stock prices. Neural networks possess unsupressed abilities in identifying underlying patterns in chaotic, non-linear, and seemingly random data, thus providing a mechanism to predict stock price movements much more precisely than many current techniques. Contemporary methods for stock analysis, including fundamental, technical, and regression techniques, are conversed and paralleled with the performance of neural networks. Also, the Efficient Market Hypothesis (EMH) is presented and contrasted with Chaos theory using neural networks. This paper will refute the EMH and support Chaos theory. Finally, recommendations for using neural networks in stock price prediction will be presented. Keywords: market, neural network, prediction, regression, stock ## 1 Introduction This paper uses a Dynamic Neural Network to predict a given stock's future daily closing price. These are models used for predictions and non-linear filtering by analyzing data for patterns and "learning" from said patterns. Two prevailing theories seek to qualify market behavior; EMH and Chaos. The EMH theory states that no system can reliably 'beat' the market because a stock's price has already incorporated and reflects all factors into its valuation, thus being _random_ and _unpredictable_. In contrast, the contrary Chaos theory states that stock prices are made up of _deterministic_ plus _random_ patterns. [1] Chaos represents deterministic but non-linear processes that appear random because they cannot be easily quantified. Through the neural networks' inherent ability to learn non-linear, dynamic, & chaotic systems, it is possible to outperform traditional stock price analysis; much to the liking of investors who can exploit this ability for huge profits. Neural networks enable multivariate models, allowing parallel processing of input data, thus quickly processing large amounts of data. Their primary strength is that they can determine patterns and anomalies within data, but most importantly, their ability to detect multidimensional & non-linear connections within that data; this makes neural networks extremely suitable for the dynamic and non-linear data generated by the stock market [2]. ## 2 Related Works Several fundamental and technical indicators have been adopted for stock price predictions but with inconsistent results [2]. No single method or combination has succeeded enough to outperform the market reliably. These time series predictions were linear parametric autoregressive, auto-regressive moving averages or moving averages in nature based on principles by Box and Jenkins. Due to their linear nature, these models failed to capture non-stationary/dynamic signals properly. The use of technical indicators stems from the idea that stock prices move in trends governed by investors' emotions, views, and attitudes due to company news or macroeconomic data. Using data such as volume, momentum, opening/closing prices, and oversold/overbought conditions, technicians use many charts to predict a stock's future value or index under the _belief_ that history repeats itself. This methodology contradicts the EMH hypothesis but is used by over 80% of stock traders [5]. The major downside of technical analysis is that it is _highly subjective_ since each trader can interpret the charts differently, making this method inconsistent, biased, and maybe most appropriate for _very short-term_ predictions. Fundamental analysis involves a more in-depth study of the company. Fundamental analysts study the company's financial performance, price/earnings, consumer demand, overall market conditions, and competitors to determine the _intrinsic value_ of a stock. This value indicates what a particular stock is "truly" worth, and they believe that the market will eventually realize this intrinsic value and reward the stock by maintaining or exceeding the said price. This method, however, involves processing huge amounts of data and is subject to individual interpretation. Thus, even if the data suggests a favorable movement in the stock, it may take a long time for the rest of the market to realize or catch up. This method, thus, is suitable for _long-term predictions_ and growth. The algorithms, neural network architectures, and associated functions will be implemented using MATLAB(r) and its powerful Neural Network Toolbox. ## 3 Algorithm The Levenberg-Marquardt (LM) algorithm will be used for training purposes. This algorithm quickly updates weight and bias values using back-propagation techniques, and it seems to be the quickest method for training reasonable-sized feedforward neural networks (up to weights reaching several hundred). [3] The design of the algorithm was such that it approaches 2nd order training speed without computing the Hessian matrix (unlike the Gauss-Newton method). In the LM algorithm, a function $f(x)$ is minimized to a sum of squares, i.e. \[\min_{x}f(x)=\\|F(x)\\|_{2}^{2}=\sum_{i}F_{i}^{2}(x) \tag{1}\] Assuming output,$\boldsymbol{y}(\boldsymbol{x},\boldsymbol{t})$, for some continuous model trajectory, $\varphi(\boldsymbol{t})$, at vector $\boldsymbol{x}$ and scalar $\boldsymbol{t}$, we have: \[\min_{\boldsymbol{x}\in\mathbb{N}^{\boldsymbol{x}}}\left\{\begin{aligned} \boldsymbol{y}(\boldsymbol{x}, \boldsymbol{t})-\varphi(\boldsymbol{t})\end{aligned}\right\}^{T_{ \boldsymbol{t}}}_{t_{1}}\big{(}\boldsymbol{y}(\boldsymbol{x},\boldsymbol{t}) -\varphi(\boldsymbol{t})\big{)}^{2}\boldsymbol{dt}, \tag{2}\] where $\boldsymbol{y}(\boldsymbol{x},\boldsymbol{t})$and $\varphi(\boldsymbol{t})$ are scalar functions. Discretizing (2) using the appropriate quadrature formula, we get a least squares problem: \[\min_{\boldsymbol{x}\in\mathbb{N}^{\boldsymbol{x}}}\boldsymbol{f}(\boldsymbol {x})=\sum_{i=1}^{m}\big{(}\vec{\boldsymbol{y}}(\boldsymbol{x},\boldsymbol{t} _{i})-\vec{\varphi}(\boldsymbol{t}_{i})\big{)}^{2}, \tag{3}\] where $\vec{\boldsymbol{y}}$ and $\overline{\boldsymbol{\varphi}}$ include the weights. Now from (1), \[\boldsymbol{F}(\boldsymbol{x})=\begin{bmatrix}\vec{\boldsymbol{y}}( \boldsymbol{x},\boldsymbol{t}_{1})-\vec{\varphi}(\boldsymbol{t}_{1})\\ \vec{\boldsymbol{y}}(\boldsymbol{x},\boldsymbol{t}_{2})-\vec{\varphi}( \boldsymbol{t}_{2})\\...\\ \vec{\boldsymbol{y}}(\boldsymbol{x},\boldsymbol{t}_{m})-\vec{\varphi}( \boldsymbol{t}_{m})\end{bmatrix}. \tag{4}\] _Denoting:_ Jacobian $F(\boldsymbol{x})$ as $J(\boldsymbol{x})$, gradient vector of $f(\boldsymbol{x})$ as $G(\boldsymbol{x})$, Hessian matrix of $f(\boldsymbol{x})$ as $H(\boldsymbol{x})$, Hessian matrix of $F_{i}(\boldsymbol{x})$ as $H_{i}(\boldsymbol{x})$, _we get,_ \[\begin{split} G(\boldsymbol{x})&=2J(\boldsymbol{x})^{T} \boldsymbol{F}(\boldsymbol{x})\\ H(\boldsymbol{x})&=2J(\boldsymbol{x})^{T}J( \boldsymbol{x})+2Q(\boldsymbol{x}),\end{split} \tag{5}\] _where,_ \[Q(\boldsymbol{x})=\sum_{i=1}^{m}F_{i}(\boldsymbol{x})\cdot H_{i}(\boldsymbol {x}). \tag{6}\] as $x_{k}$ approaches the solution, $F(\boldsymbol{x})\to 0$ & $Q(\boldsymbol{x})\to 0$. Now as $Q(\boldsymbol{x})\to\infty$, the Gauss-Newton method experiences problems resolved by the LM method. In such cases, the LM method uses a search direction, $\boldsymbol{d}_{\boldsymbol{k}}$, that is a solution to the set of linear equations: \[\Big{(}J\left(x_{k}\right)^{T}J\left(x_{k}\right)+\lambda_{k}\boldsymbol{I} \Big{)}d_{k}=-J\left(x_{k}\right)^{T}\boldsymbol{F}\left(x_{k}\right), \tag{7}\] where, $\boldsymbol{\lambda}_{k}$ dictates the magnitude and direction of $\boldsymbol{d}_{\boldsymbol{k}}$ The LM method thus uses a search direction that is a hybrid between the Gauss-Newton and the steepest descent direction. The neural models used for training in this paper would adopt a series-parallel (open-loop) network since training adopts "true" output values. For this model, the prediction equations used to determine the output $\boldsymbol{y}(k+1)$ uses past inputs of $\boldsymbol{u}_{1}(k),\boldsymbol{u}_{1}(k-1),...,\boldsymbol{u}_{1}(k- \boldsymbol{d}_{u})$ for the $1^{\text{st}}$ time sequence and $u_{2}(k),u_{2}(k-1),...,u_{2}(k-\boldsymbol{d}_{u})$ for the second time sequence and past outputs(targets) of $\boldsymbol{y}(k),\boldsymbol{y}(k-\boldsymbol{1}),...,\boldsymbol{y}(k- \boldsymbol{d}_{\boldsymbol{y}})$, all as input data. 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Using NARX, the idea is to predict future values of a time series $\gamma(t)$ from past values of said time series coupled with past values of a second time series $x(t)$, which can be written as follows, \[y(t)=f(y(t-1),...,y(t-d),x(t-1),...,(t-d))\] The NARX network is a two-layer feedforward network consisting of a sigmoid transfer function in the hidden layer and a linear transfer function in the output layer. The network also makes use of tapped delay lines to store previous values of the $x(t)$ and $y(t)$ sequences. It is important to note here that the output, $y(t)$, is then fed back as input to the network, which is needed in the dynamic nature of stock price prediction. The architecture's open-loop (series-parallel) implementation, as shown in Figure 2(a), will be used to train the network since the 'true' output, instead of the estimated output, is used as input feedback. This has several advantages, the first of which is that the input to the feedforward network is more accurate, and second, the resulting network has a pure feedforward architecture, and thus a more efficient algorithm( static back-propagation )can be used for training. Once the network is reasonably trained, it will be converted to a closed-loop network to perform multi-step-ahead predictions. While open, the network was predicting the next value of $y(t)$ from the previous values of $y(t)$ and $x(t)$. When the network is closed, it can perform multi-step-ahead predictions. This happens because _predictions of $y(t)$_ will be used in place of _actual future values of $y(t)$_. i.e., a closed-loop network outputs $y(t+1)$ instead of $y(t)$, as shown in Figure 2(b) below. ### Data Acquisition Four years (01/01/2010 - 03/31/2014) of prior stock data would be retrieved from _finance.yahoo.com_ using a custom-written function based on MATLAB data-feed plugin. This function automatically segments the data into _input_, _target_, _and sample_ sequences. The data would be in the format _[date, opening price, high price, low price, volume, closing price, adjusted close price]_ and prepared for input to the network using MATLAB preparets function. This function automatically shifts time series data (input and target) by as many steps as needed to fill the initial delay states, thereby greatly reducing errors and complexity. Table I shows the sample dataset used. ### Training, Testing, and Validation As shown in Figure 1, the open-loop network will train the network using tapped delay lines with 2 delays for both the input and output. This will allow training to begin with the 3rd data point. The open-loop network takes 2 input sequences, $x(t),y(t)$, where $\chi(t)$ is the input signal and is composed of the: {_open price, high price, low price, volume}_ data for the stock being analyzed. It is important to note that for this network,$y(t)$, which is the _closing price_ of the stock, is a feedback signal and is also an input and an output (target). Furthermore, 70% of the input dataset is used for training, 15% for validation, and 15% for testing. The network was retrained 10 times to ensure maximum accuracy, using the MSEREG performance function to improve generalizations. See Figures 3 and 4 below for further performance and training parameters used. To determine the best possible parameters for achieving a robustly trained network, the number of input and output delays and the number of neurons were changed systematically, and each combination's RSE and performance ratio were recorded. Table II shows the recorded data after several trials, and the best number of delays and neurons were selected based on depicted performance criteria. One neuron was used in the feedback layer for all trials because our target is only 1 value. From the data in Table II, it could be seen that when the input delays were between 0 and 25 and the number of neurons was less than 21, the performance and RSE values indicated a very good fit between the actual and predicted data (the _pink portion of the chart_). However, it was observed that in those cases, the Input-Error correlation curves were significantly out of the confidence levels, as can be seen in Figure 5. After further testing (_green portion of Table II_), it was observed that the performance and accuracy increased with delays above 25 and 10 neurons or less. However, this large number of delays gave a performance hit to the network and was not chosen as appropriate input. Throughout the trials, it was observed that input delays between 0 and 1 yielded the BEST results across all performance metrics when the number of neurons was between 22 and 30 inclusive. The best performance was seen using a minimum of 22 neurons and [0:1] input delays, as depicted in Figure 6 below. As such, the network used for final training will use the parameter values depicted in Figures 3 and 4 above, with the Levenberg-Marquardt (LM) algorithm, 2 hidden layers, and 22 neurons. ### Model Acceptance Criteria A Mean Squared Error (MSE) of Zero means no errors, and a Regression (R) value of 1 means a 'perfect' relationship between the input and output data. It measures how well the targets describe the variation in the output. The acceptance criteria would be MSE values close to Zero and R-Values close to 1. Results are further verified using correlation, performance, and time-series charts. Divergences greater than 10% of the nominal values would prompt retraining in which the number of neurons or delays would be changed. [7] \begin{table} \begin{tabular}{\|l\|l\|l\|l\|l\|} \hline Input & Feedback & Neurons & & \\ delays(d${}_{a}$) & delays(d${}_{b}$) & (N) & Performance & RSE** \\ \hline 2.5 & 1 & 3 & 0.016 & 0.99852 \\ \hline 0:1 & 1 & 5 & 0.0154 & 0.99846 \\ \hline 0:1 & 1 & 10 & 0.0156 & 0.99844 \\ \hline 2:5 & 1 & 10 & 0.024 & 0.99844 \\ \hline 6:25 & 1 & 10 & 0.0354 & 0.99849 \\ \hline 0:5 & 1 & 14 & 0.0209 & 0.99847 \\ \hline 0:5 & 1 & 16 & 0.0168 & 0.99862 \\ \hline 0:1 & 1 & 21 & 0.0217 & 0.9954 \\ \hline 26:30 & 1 & 10 & 0.0955 & 0.9977 \\ \hline 0:5 & 1 & 16 & 0.0254 & 0.9983 \\ \hline 0:5 & 1 & 18 & 0.038 & 0.99846 \\ \hline 0:1 & 1 & 22 & 0.0178 & 0.99857 \\ \hline 0:1 & 1 & 23 & 0.0182 & 0.99866 \\ \hline 0:1 & 1 & 25 & 0.01599 & 0.99876 \\ \hline 0:1 & 1 & 30 & 0.0466 & 0.99766 \\ \hline \end{tabular} \end{table} Table II: Performance Based on Input Delays, Target Delays, and Neurons Figure 4: Training parameters used. Figure 5: Input-Error correlation out of confidence limit with input delays 2:25 and $\leq$ 21 neurons. Figure 6: Best performance seen at d${}_{a}$=0:1, d${}_{y}$=1, N=22 Results The model was first trained on 4 years of stock data (01/01/2010 - 1/01/2014) using the series-parallel model shown in Figure 1, after which 100 days of simulation was conducted to predict the closing prices, using the parallel(closed) network as seen in Figure 2(b). ### Training #### 5.1.1 Performance Results - Trained Network Table III shows the performance of the _open_ network. It meets our acceptance criteria with an RSE of 0.998 and an MSE of 0.0242. Its performance of 0.0147 indicates it has very high efficiency and is much better than any linear-modeled dynamic system for predicting stock prices [5]. Figure 7 depicts the relative performances obtained for training the network. The training targets and outputs, validation and outputs, and testing targets and outputs are extremely close, with less than 2% divergence from the targets. This close correlation is further verified by the Error vs. Time graph shown. This is important because we want the error in the trained network (one-step-ahead) to be very small so that the predictions (multi-step-ahead) made in the parallel network are as accurate as possible. Figure 8 shows the errors between the "actual" outputs vs. the "predicted" outputs while training. This graph clearly shows that the divergences/errors between actual and predicted outputs are small and contained around zero. The maximum divergence from the actual values was approx. 1.122%, which is acceptable for our model. The 5 very large spikes are most likely caused by extremely random events, such as the company unexpectedly reporting a dividend increase or share buy-back, in which the stock rallied. The R-values between the outputs and targets measure how well the targets explain the variation in the output. A number equal to 1 signifies a perfect correlation between targets and outputs. In our model, the value is approx. 0.998, which signifies a very accurate correlation. Figure 9 above shows the input-error cross-correlation function, which depicts how the errors are correlated with the input sequence$x(t)$. In a perfect model, all the correlations should be zero, and if they are not, it signifies room for Figure 8: Errors between actual and predicted values during training. Figure 7: Correlations between training, validation, test, and error. Figure 9: Regression curves between output and target values. improvement by adjusting the number of neurons in the hidden layer or the number of delays until the bars fall within the confidence bounds around zero. In our model, the bars all fall within the confidence limits of [-0.018, 0.018]. This further signifies that our model has a very efficient design, thus, will produce great accuracy. Figure 10 shows the error autocorrelation function, which validates the network performance by showing how the prediction errors are related to time. For a perfect model, there should only be one nonzero value occurring at Lag 0 (the _MSE_ of the network), which would signify that prediction errors are only white noise. For our model, all the bars except the one at Lag 0 fall within the 95% confidence limit of around 0, which validates this model as very accurate. The 5% loss in performance accuracy is mainly white noise, which is inherent in the dynamic nature of the stock market. Figure 11 shows plots of performance, validation, and testing during training. The network errors decreased until iteration 5 (out of 11 iterations), where their gradients were no longer changing. Before iteration 5, the gradients of the curves were not increasing. Therefore, it can be concluded that overfitting did not occur, and this network will produce very accurate results. Figure 12 shows the output values obtained during training vs. the target values. From this chart and the previous performance graphs, the network _VERY_ accurately models the _closing price_ of the stock index, given the four inputs of _opening, high, low_ price, and trading _volume_, for the training period of 4 years. _Red signifies the predicted values,_ while _blue represents the target values_. The slight deviations are mainly due to the white noise inherent in such data and could be minimized using an appropriate filter. ### Simulation and Prediction Now that training is completed using the series-parallel(open) network with acceptable accuracy, the network will now be converted to a purely parallel(closed) one, as shown in Figure 2(b), and the target prices will be predicted for 100 days, using data from _1/1/2014 - 3/31/2014_. Table IV shows the performance of the _closed_ network. It also meets our acceptance criteria with an RSE of 0.998 and MSE of 0.0242. Its performance of 0.0142 also closely matches the expected open network performance. Figure 11: Performance plot of training, validation, and testing. Figure 12: Output vs. target during training. Figure 10: Error autocorrelation to validated network Figure 13: Simulated output showing targets and predictions. Figure 13 shows 3 charts combined. The _blue represents the Original (actual) targets_, the _red shows the expected outputs_ after simulation, and the _green shows the network's predictions_. The chart shows that the network predictions and the expected outputs are very close. The network predictions parallel the expected outputs with only a 1.17% maximum divergence. This accuracy was expected because the trained network showed superior performance characteristics, the same characteristics used in the simulated network. I.e., the simulated network also has an MSE of 0.0242 and an R-Value of 0.998. Figure 14 is an enlarged version of the circular region identified in Figure 13. It shows the _network predictions in green_ and the _target (actual) values in red_. Given the dynamic and random nature of stock prices, this prediction is a very good fit for the actual values, further validating the efficiency of the network model used and the potential of neural networks in general. Figure 15 shows the errors between the _simulated outputs_ and the _target values_ for the 100 simulation days using the closed network. It represents the differences between the charts shown in Figure 14. The errors are very small and fluctuate around the zero line. The maximum error difference recorded was 1.17%, indicating that the predicted (simulated) and target (actual) values are sufficiently close to each other to be deemed negligible. Again, most of the error is due to white noise, which can be reduced by first filtering the data. This filtering could be implemented using an extended Kalman filter which would seek to predict random changes in the stock price movement due to company and other macro-economic activities. ## 6 Conclusion From the above data and analysis, it is evident that well-trained neural networks can quickly learn patterns within very large, seemingly unrelated datasets and make intelligent and mostly accurate predictions. It is also clear that neural networks eliminate or minimize randomness even when human counterparts fail. This is most likely because neural networks can quickly recognize and learn extremely subtle patterns in large datasets, which may go unrecognized by humans. This ability to 'learn or model' randomness directly supports the Chaos theory and refutes the EMH theory, and thus can be used to 'beat' the market ## 7 Recommendations The errors that appeared as white noise could be reduced in several ways. First, the input data could be pre-processed using the techniques of Independent Component Analysis or even another training algorithm, then fed as input to the neural network. Secondly, the input data could be filtered using an extended Kalman filter which also makes intelligent predictions, then fed as input to the network.
2310.02244	Tensor Programs VI: Feature Learning in Infinite-Depth Neural Networks	By classifying infinite-width neural networks and identifying the optimal limit, Tensor Programs IV and V demonstrated a universal way, called $\mu$P, for widthwise hyperparameter transfer, i.e., predicting optimal hyperparameters of wide neural networks from narrow ones. Here we investigate the analogous classification for depthwise parametrizations of deep residual networks (resnets). We classify depthwise parametrizations of block multiplier and learning rate by their infinite-width-then-depth limits. In resnets where each block has only one layer, we identify a unique optimal parametrization, called Depth-$\mu$P that extends $\mu$P and show empirically it admits depthwise hyperparameter transfer. We identify feature diversity as a crucial factor in deep networks, and Depth-$\mu$P can be characterized as maximizing both feature learning and feature diversity. Exploiting this, we find that absolute value, among all homogeneous nonlinearities, maximizes feature diversity and indeed empirically leads to significantly better performance. However, if each block is deeper (such as modern transformers), then we find fundamental limitations in all possible infinite-depth limits of such parametrizations, which we illustrate both theoretically and empirically on simple networks as well as Megatron transformer trained on Common Crawl.	Greg Yang, Dingli Yu, Chen Zhu, Soufiane Hayou	2023-10-03T17:50:40Z	http://arxiv.org/abs/2310.02244v5	# Tensor Programs VI: ###### Abstract By classifying infinite-width neural networks and identifying the _optimal_ limit, [23, 25] demonstrated a universal way, called $\mu$P, for _widthwise hyperparameter transfer_, i.e., predicting optimal hyperparameters of wide neural networks from narrow ones. Here we investigate the analogous classification for _depthwise parametrizations_ of deep residual networks (resnets). We classify depthwise parametrizations of block multiplier and learning rate by their infinite-width-then-depth limits. In resnets where each block has only one layer, we identify a unique optimal parametrization, called Depth-$\mu$P that extends $\mu$P and show empirically it admits depthwise hyperparameter transfer. We identify _feature diversity_ as a crucial factor in deep networks, and Depth-$\mu$P can be characterized as maximizing both feature learning and feature diversity. Exploiting this, we find that absolute value, among all homogeneous nonlinearities, maximizes feature diversity and indeed empirically leads to significantly better performance. However, if each block is deeper (such as modern transformers), then we find fundamental limitations in all possible infinite-depth limits of such parametrizations, which we illustrate both theoretically and empirically on simple networks as well as Megatron transformer trained on Common Crawl. ## 1 Introduction Deep neural networks have showcased remarkable performance across a broad range of tasks, including image classification, game playing exemplified by AlphaGo [17], and natural language processing demonstrated by GPT-4 [15]. A prevailing trend in developing these networks is to increase their size and complexity, with empirical evidence indicating that using the same computation resources, models with more parameters tend to exhibit better performance. There are two ways to increase any network size: _width_ and _depth_. The properties of the width (given a fixed depth) have been extensively studied in the literature: recent work by Yang et al. [25] identified the _Maximal Update Parametrization_ ($\mu$P) that guarantees maximal feature learning in the infinite width limit.3 Another benefit of $\mu$P is hyperparameter transfer which enables hyperparameter tuning on smaller models; the optimal hyperparameter choice for the smaller model remains optimal for larger models (i.e., models with larger width). However, despite the achievements of large-scale deep models and the theoretical understanding of scaling width, increasing the depth of neural networks still has both practical limitations and theoretical difficulties. In practice, increasing depth beyond some level often results in performance degradation and/or significant shifts in the optimal hyperparameters. In theory, unlike increasing width, increasing depth introduces new parameters that significantly change the training dynamics. In this paper, we aim to solve this problem by extending $\mu$P to include depth scaling. We call the depth scaling Depth-$\mu$P. The issue of depth scaling has persisted over time. A decade ago, deep neural networks experienced significant degradation problems -- having more than a few dozen layers would increase the training error instead of improving the model's performance. This was partly due to the vanishing or exploding gradient problem that affects the efficient propagation of information through the network. The introduction of residual networks (ResNet) [8; 9; 18] has partially resolved this issue, allowing for the training of deeper networks with improved performance. ResNet is constructed by layering _residual blocks_, which are composed of a series of convolutional layers and then an element-wise addition with the input. This element-wise addition (commonly referred to as _skip connection_) is a significant innovation of ResNet and remains an important ingredient in modern architectures including Transformers [19]. Specifically, in a residual architecture, the $l$-th residual block is formulated as \[x^{l}=x^{l-1}+g^{l}(x^{l-1};W^{l}),\] where $x^{l-1}$ is the input, $x^{l}$ is the output, $W^{l}$ are the parameters of the block, and $g^{l}$ (often called the _residual branch_) is a mapping that defines the layer (e.g. a stack of convolutions in ResNet, or SelfAttention and MLP in a Transformer). In this work, we focus on the case where $g^{l}$ is a biasless perceptron with (or without) activation. The stacking of many residual blocks causes an obvious issue even at the initialization -- the norm of $x^{l}$ grows with $l$, so the last layer features do not have a stable norm when increasing the depth. Intuitively, one can stabilize these features by scaling the residual branches with a depth-dependent constant. However, scaling the residual branches with arbitrarily small constants might result in no feature learning in the large depth limit since the gradients will also be multiplied with the scaling factor. When each block $g^{l}$ has only one layer (one matrix multiply), we identify the parametrization we call Depth-$\mu$P as the optimal parametrization for deep networks. It maximizes both _feature learning_ and _feature diversity4_ among all possible parametrizations of block multiplier and learning rate with depth. Our framework extends the previous results on $\mu$P which deals with optimal width scaling [25]. It completes the width scaling and hence provides a full width and depth scaling recipe that guarantees maximal feature learning and hyperparameter transfer across width and depth. Depth-$\mu$P contains the following modifications to the standard practice: Footnote 4: We give a formal definition of feature learning and feature diversity later in the paper. 1. There is a multiplier for each residual branch before adding to its input, which is inversely proportional to the square root of $L$ (where $L$ is the depth). Formally, with a constant $a$ independent from $L$, \[x^{l}=x^{l-1}+\frac{a}{\sqrt{L}}\cdot g^{l}(x^{l-1};W^{l}).\] (1) 2. We set the learning rate of $W^{l}$ so that the update of $W^{l}$ during training is proportional to $1/\sqrt{L}$. We derive different learning rate schemes for different optimization algorithms based on this principle. For Adam, because it is scale-invariant to the gradient, the learning rate of $W^{l}$ is set to be $\eta/\sqrt{L}$. On the other hand, the learning rate of $W^{l}$ for SGD is set as a constant $\eta$ because the gradient of $W^{l}$ is already of size $1/\sqrt{L}$ due to the multiplier. \begin{table} \begin{tabular}{c c c} \hline \hline & Branch Multiplier & Learning Rate \\ \hline Standard & 1 &? (tuned) \\ Depth-$\mu$P (SGD) & $a/\sqrt{\text{depth}}$ & $\eta$ \\ Depth-$\mu$P (Adam) & $a/\sqrt{\text{depth}}$ & $\eta/\sqrt{\text{depth}}$ \\ \hline \hline \end{tabular} \end{table} Table 1: Difference between standard depth scaling and Depth-$\mu$P. The constants $a$ and $\eta$ in Depth-$\mu$P are transferable across depth, i.e., one can tune a smaller network and use the same constants for deeper networks. On the other hand, the learning rate of standard depth scaling requires separate tuning for models of different depth. In block depth $1$ (i.e., $g^{l}$ is a biasless perceptron, $W^{l}$ is a single matrix), this scaling leads to the following properties: * At the initialization, each one of the $L$ residual blocks contributes $\Theta(1/\sqrt{L})$ to the main branch. These $L$ contributions are independent of each other, so the sum of them is of size $\Theta(1)$. * During training, the contribution of the update of each residual block is $\Theta(1/L)$ due to the combining effect of the learning rate and multiplier. The contributions of the updates are highly correlated, so they sum up to $\Theta(1)$. More detailed intuition of this scaling approach can be found in Section 3 where we provide a simple analysis with linear networks after one gradient step. We give a complete classification of depthwise parametrizations in section 7. ### Optimality of Depth-$\mu$P. We thoroughly compare Depth-$\mu$P with other scaling strategies with a branch multiplier $\propto L^{-\alpha}$ and parameter update $\propto L^{-\gamma}$.5 As shown in Figure 1, the space of $(\alpha,\gamma)$ is divided into several areas, each resulting in a different behavior when $L\to\infty$: Footnote 5: It implies that the effective learning rate is proportional to $L^{-\gamma}$ for Adam and $L^{\alpha-\gamma}$ for SGD if the network is stable at initialization. * Having $\alpha\geq 1/2$ is required to stabilize the network at initialization. This ensures that he hidden activations and the network output do not explode at initialization; * For any $\alpha+\gamma<1$, the network is unstable during training. The change in hidden activations or the network output explodes with depth during training; * For any $\alpha+\gamma>1$, training outcome is trivial. The change of the network vanishes as depth increases; * For any $\alpha+\gamma=1$ with $\alpha>1$, the network is _unfaithful_ (a formal definition will provided later in the paper). The hidden activations explode during training as depth increases; * For any $\alpha+\gamma=1$ and $\alpha\in(1/2,1]$, we show that the network converges to a _redundant_ limit that lacks _feature diversity_, in that close layers have similar outputs (in a neural ODE fashion). * The only choice of $\alpha$ and $\gamma$ left is $\alpha=\gamma=1/2$, which corresponds to Depth-$\mu$P. The rigorous definitions and proofs are presented in Section 7. ### Hyperparameter Transfer for Depth. The optimality of Depth-$\mu$P implies (under some assumptions) that the optimal hyperparameters of the networks also converge as the depth ($L$) increases. This convergence suggests that the optimal hyperparameters of shallower networks are approximately equal to those of deeper networks. As a direct implication, we can leverage this property to infer the hyperparameters for deeper networks from the shallower ones, effectively reducing the cost associated with hyperparameter tuning. With Depth-$\mu$P, we successfully train networks comprising thousands of residual blocks, while also showcasing the transferability of hyperparameters across depth. Figure 1: Behaviors of scaling strategies with a branch multiplier $L^{-\alpha}$ and parameter update proportional to $L^{-\gamma}$. ### Impossibility Results for Block Depth $\geq 2$ While the block depth 1 case admits a positive result, we show that the block depth $\geq 2$ case does not and cannot (section 9). The basic issue is the weights in different layers within a block is forced to interact additively instead of multiplicatively when depth is large, if one wants to retain diversity. This causes block depth $\geq 2$ to have worse performance than block depth $1$ and for the optimal hyperparameters to shift with depth. We demonstrate this pedagogically on resnet with MLP blocks but also on Megatron transformer [16] trained on Common Crawl. These observations entail the need to rethink the current approach to hyperparameter transfer. ## 2 Related Works ### Width Scaling and $\mu$P The infinite-width limit of neural networks has been a topic of extensive research in the literature. Numerous studies have predominantly focused on examining the behavior of various statistical quantities at initialization. Some works have gone beyond the initialization stage to explore the dynamics of feature learning in neural networks. Lazy training.With standard parametrization, a learning rate of order $\mathcal{O}(n^{-1})$,6$n$ being the width, yields the so-called lazy training regime in the infinite-width limit, where the features remain roughly constant throughout training [3; 25]. This regime is also known as the Neural Tangent Kernel (NTK) regime and its convergence properties have been extensively studied in the literature [10; 1; 2; 28]. Footnote 6: We also obtain the lazy infinite-width limit with the NTK parametrization and a $\mathcal{O}(n^{-1/2})$ learning rate. Feature learning and $\mu$P.Recent empirical studies (e.g. [25]) have provided compelling evidence that feature learning plays a crucial role in the success of deep learning. It is widely acknowledged that the remarkable performance achieved by deep neural networks can be attributed to their ability to acquire meaningful representations through the process of training. Consequently, scaling the network architecture emerges as a natural choice to enhance the performance of such models. In this context, $\mu$P (Maximal Update Parameterization), introduced in [25], has emerged as a promising approach for maximizing feature learning while simultaneously preventing feature explosion as the network width increases, given a fixed depth. Notably, $\mu$P facilitates hyperparameter transfer across varying network widths. This means that instead of tuning hyperparameters directly on large models, one can optimize them on smaller models and utilize the same set of hyperparameters for larger models. The derivation of $\mu$P leverages the Tensor Programs framework [22; 20; 21; 23; 25], which provides valuable tools for capturing the behavior of neural networks in the infinite-width regime during the training process. ### Depth Scaling While increasing the width of neural networks can lead to improved performance, increasing the depth of the network also yields significant performance gains, and most state-of-the-art models use deep architectures. The introduction of skip connections [8; 9] played a pivotal role in enabling the training of deep networks. However, it became apparent that even with skip connections and normalization layers, training deep networks remains a challenging task [12]. Moreover, tuning hyperparameters for large depth networks is a time-and-resource-consuming task. To address the challenges associated with training deep networks, several studies have proposed scaling the network blocks using a depth-dependent scaler to ensure stability of features and gradients at initialization or in the kernel regime [7; 4; 26; 13; 5; 6; 14; 27]. However, these works lack insights into the dynamics with feature learning. For instance, one might argue that features can still experience explosive growth if the learning rate is not properly chosen. Therefore, an effective depth scaling approach should not only ensure stability at initialization but also provide guidelines for scaling the learning rate. This motivation underlies the development of Depth-$\mu$P, which offers a comprehensive framework for depth scaling. Depth-$\mu$P encompasses block multipliers and learning rate scaling, providing a complete recipe for training deep networks. In the case of Multi-Layer Perceptrons (MLPs) (no skip connections), Jelassi et al. [11] showed that a learning rate scaling of $depth^{-3/2}$ guarantees stability after the initial gradient step. However, it remains unclear how the learning rate should be adjusted beyond the first step, and this scaling is not suitable for architectures with residual connections. ## 3 Warm-Up: An Intuitive Explanation with Linear Networks Let us begin with a simple example that provides the necessary intuition underpinning our depth scaling strategy. Given a depth $L$, width $n$, consider a linear residual network of the form \[x^{0} =U\xi,\] \[\forall l\in[L], x^{l} =x^{l-1}+\frac{1}{\sqrt{L}}W^{l}x^{l-1},\] \[f =V^{\top}x^{L},\] where the weight matrices $W^{l}\in\mathbb{R}^{n\times n}$ and $U,V$ are input and output weight matrices that we assume to be fixed during training. ### Optimal Scaling of the Learning Rate To simplify the analysis, we consider gradient updates based on a single datapoint. The first gradient step is given by \[W^{l}_{1}=W^{l}_{0}-\eta G^{l}_{0},\] where $\eta$ is the learning rate, and $G^{l}_{0}$ is a matrix with update directions. For instance, we have the following expressions for $G^{l}_{0}$ with SGD and Adam: * SGD: $G^{l}_{0}=\frac{1}{\sqrt{L}}\delta x^{l}\otimes x^{l-1}$, where $\delta x^{l}\stackrel{{\text{def}}}{{=}}\frac{\partial\ell}{ \partial x^{l}}$ for some loss function $\ell$.7 Footnote 7: We use $\delta$ for gradient because we want to distinguish from $d$ in depth differential equations that appear later in the paper. * Adam8: $G^{l}_{0}=\text{sign}\left(\frac{1}{\sqrt{L}}\delta x^{l}\otimes x^{l-1}\right)$. Footnote 8: For the sake of simplification, we consider SignSGD in this section, which can be seen as a memory-less version of Adam. The analysis is valid for any training algorithm that gives $\Theta(1)$ gradients. In both cases, $\delta x^{l}$ and $x^{l-1}$ are computed for a single training datapoint $\xi_{0}$. The last layer features $x^{L}$ (for some input $\xi$) are given by $x^{L}=\prod_{l=1}^{L}\left(I+\frac{1}{\sqrt{L}}W^{l}\right)x^{0}$.9 We use the subscript $t$ to refer to training step. After the first gradient step, we have the following Footnote 9: To avoid any confusion, here we define the matrix product by $\prod_{l=1}^{L}A_{l}=A_{L}\times A_{L-1}\cdots\times A_{1}$. \[x^{L}_{1}=\prod_{l=1}^{L}\left(I+\frac{1}{\sqrt{L}}W^{l}_{1}\right)x^{0}=x^{L }_{0}-\frac{\eta}{\sqrt{L}}A_{L}+\mathcal{O}(\eta^{2}), \tag{2}\] where $A_{L}=\sum_{l=1}^{L}\left[\prod_{k>l}\left(I+\frac{1}{\sqrt{L}}W^{k}_{0} \right)\right]G^{l}_{0}\left[\prod_{k<l}\left(I+\frac{1}{\sqrt{L}}W^{k}_{0} \right)\right]x^{0}$. We argue that $A_{L}$ behaves as $\Theta(L)$ (in $L_{2}$ norm). This is the due to the $1/\sqrt{L}$ scaling factor. To see this, we further simplify the analysis by considering the case $d_{in}=n=d_{out}=1$ (single neuron per layer) and the squared loss. In this case, the term $A_{L}$ simplifies to \[A_{L}=\sum_{l=1}^{L}\prod_{k\neq l}\left(1+\frac{1}{\sqrt{L}}W^{k}_{0}\right)G ^{l}_{0}x_{0}.\] Scaling for SGD.With SGD, we have that $G_{0}^{l}=\frac{1}{\sqrt{L}}\prod_{k\neq l}\left(1+\frac{1}{\sqrt{L}}W_{0}^{k} \right)x_{0}\delta x^{L}$, where $\delta x^{L}=(Vx^{L}-y(\xi_{0}))$ and $y(\xi_{0})$ is the target output. Therefore, it is easy to see that \[\mathbb{E}A_{l}^{2}=\frac{1}{L}\mathbb{E}\left(\sum_{l=1}^{L}\prod_{k\neq l} \left(1+\frac{1}{\sqrt{L}}W_{0}^{k}\right)^{2}\delta x^{L}x_{0}^{2}\right)^{2} =\Theta\left(\frac{1}{L}L^{2}\right)=\Theta(L),\] where we have used the fact that $\mathbb{E}\left(1+\frac{1}{\sqrt{L}}W_{0}^{k}\right)^{2p}=1+\Theta(L^{-1})$, for any positive integer $p$. Hence, the magnitude of the first order term in eq. (2) is given by \[\mathbb{E}\left[\left(\frac{\eta}{\sqrt{L}}A_{l}\right)^{2}\right]=\Theta( \eta^{2}),\] which shows that the update is stable in depth as long as $\eta=\Theta(1)$ in depth. More precisely, this is the maximal choice of learning rate that does not lead to exploding features as depth increases. Scaling for Adam.With Adam, we have $G_{0}^{l}=\pm 1$, and therefore we obtain \[\mathbb{E}A_{l}^{2}=\mathbb{E}\left(\sum_{l=1}^{L}\prod_{k\neq l}\left(1+ \frac{1}{\sqrt{L}}W_{0}^{k}\right)x_{0}\right)^{2}=\Theta\left(L^{2}\right),\] where we have used the same arguments as before. In this case, the first order term in eq. (2) is given by \[\mathbb{E}\left[\left(\frac{\eta}{\sqrt{L}}A_{l}\right)^{2}\right]=\Theta( \eta^{2}L^{-1}).\] Therefore, the maximal learning rate that one can choose without exploding the features is given by $\eta=\Theta(L^{-1/2})$. _Summary_: By ensuring that parameter update is $\Theta(1/\sqrt{L})$, the features remain stable while feature update is $\Theta(1)$. This $\Theta(1)$ update is due to the accumulation of $\Theta(1/L)$ correlated terms across depth. ### Convergence when Depth goes to $\infty$ Let us look at $x_{1}^{L}$ again in the simple case $d_{in}=d_{out}=n=1$ and analyze its behaviour when $L\to\infty$. This paragraph is only intended to give an intuition for the convergence. A rigorous proof of such convergence will be later presented in the paper. Let us consider the case with SGD training with learning rate $\eta=1$ and let $M_{L,l}=\prod_{k\neq l}\left(1+\frac{1}{\sqrt{L}}W_{0}^{k}\right)$ and $\tau=(Vx_{0}^{L}-y(\xi_{0}))x^{0}$. With this, we have the following \[x_{1}^{L}=\prod_{l=1}^{L}\left(1+\frac{1}{\sqrt{L}}W_{0}^{l}-\frac{1}{L}\tau M _{L,l}\right)x^{0}. \tag{3}\] WLOG, let us assume that $x_{0}^{0}>0$. Then, with high probability (the event that $W_{0}^{l}\ll\sqrt{L}$, for some notion of "$\ll$", occurs with a probability of at least $1-e^{-L^{n}}$ for some $\alpha>0$)10, we have that $x_{1}^{L}>0$. We can therefore look at $\log(x_{1}^{L})$ which simplifies the task. Taking the log and using Taylor expansion under a high probability event, we obtain Footnote 10: This follows from simple concentration inequalities for sub-exponential random variables. \[\log(x_{1}^{L}/x^{0}) =\frac{1}{\sqrt{L}}\sum_{l=1}^{L}W_{0}^{l}-\frac{1}{L}\sum_{l=1}^{ L}\tau M_{L,l}+\frac{\sum_{l=1}^{L}(W_{0}^{l})^{2}}{L}+\mathcal{O}(L^{-1+ \epsilon})\] \[=\frac{1}{\sqrt{L}}\sum_{l=1}^{L}W_{0}^{l}-\tau x_{0}^{L}\frac{1 }{L}\sum_{l=1}^{L}\frac{1}{1+\frac{1}{\sqrt{L}}W_{0}^{l}}+\frac{\sum_{l=1}^{L} (W_{0}^{l})^{2}}{L}+\mathcal{O}(L^{-1+\epsilon}),\] for some $\epsilon>0$. The first and third terms $\frac{1}{\sqrt{L}}\sum_{l=1}^{L}W_{0}^{l}$ and $\frac{\sum_{l=1}^{L}(W_{0}^{l})^{2}}{L}$ converge (almost surely) to a standard Gaussian and $1$, respectively. The second term also converges naturally, since $x_{0}^{L}$ converges in $L_{2}$ to a Log-Normal random variable ([5]) and with a delicate treatment (involving high probability bounds), one can show that the term $\frac{1}{L}\sum_{l=1}^{L}\frac{1}{1+\frac{1}{\sqrt{L}}W_{0}^{l}}$ converges (in $L_{2}$ norm) at large depth. This implies that one should expect $x_{1}^{L}$ to have some notion of weak convergence as depth grows. Note that the same analysis becomes much more complicated for general width $n>0$. To avoid dealing with high probability bounds, a convenient method consists of taking the width to infinity first $n\to\infty$, then analyzing what happens as depth increases. We discuss this in the next section. ### A Discussion on the General Case Difficulty of generalizing to the nonlinear case. The extension to the general width scenario ($n>1$) necessitates a more intricate treatment of the term $A_{l}$ to find optimal scaling rules, yet the proposed scaling remains optimal for general width. This preliminary analysis lays the groundwork for proposing a specific learning rate scaling scheme that maximizes feature learning. Moreover, demonstrating the optimality of this scaling strategy in the presence of non-linearities is a non-trivial task. The primary challenge stems from the correlation among the post-activations induced during the training process. Overcoming these challenges requires a rigorous framework capable of addressing the large depth limit of crucial quantities in the network. For this purpose, we employ the Tensor Program framework to investigate the behavior of essential network quantities in the infinite-width-then-depth limit. By leveraging this framework, our theoretical findings establish that the aforementioned scaling strategy remains optimal for general networks with skip connections. Our framework considers the setup where the width is taken to infinity first, followed by depth. This represents the case where $1\ll depth\ll width$, which encompasses most practical settings (e.g. Large Language Models). The critical role of Initialization.A naive approach to depth scaling can be as follows: since the weights $W_{t}^{k}$ might become highly correlated during training, one has to scale the blocks with $1/L$. To understand this, let us assume a block multiplier of $L^{-\alpha}$ and consider the scenario of perfect correlation where all weights are equal, i.e., $W_{t}^{k}=W$ for every $k\in 1,\ldots,L$. In this case, the last layer features can be expressed as $x^{L}=\left(I+L^{-\alpha}W\right)^{L}x_{0}$. When $\alpha=1/2$, the features are likely to exhibit an explosive growth with increasing depth, while opting for $\alpha=1$ is guaranteed to stabilize the features. However, in this paper, we demonstrate that this intuition does not align with practical observations. Contrary to expectations, the features do not undergo an explosive growth as the depth increases when $\alpha=1/2$. This phenomenon is attributed to two key factors: random initialization and learning rate scaling with depth. These factors ensure that the weight matrices never become highly correlated in this particular fashion during the training process. In summary, while a naive depth scaling strategy based on scaling blocks might suggest the need for $\alpha=1$ to stabilize the features, our findings reveal that in practice, this is not the case. The interplay of random initialization and learning rate scaling effectively prevents the features from experiencing explosive growth, even with the choice of $\alpha=1/2$. ## 4 SGD Training Dynamics of Infinitely Deep Linear Networks In this section, we continue to study the linear neural network with residual connections under Depth-$\mu$P. Using the Tensor Program framework [24], we rigorously derive the training dynamics of SGD for the linear residual network when the width and the depth sequentially go to infinity. The road map of our analysis consists the following three steps. 1. We first take the width of the network to infinity by the Tensor Program framework [24]. As a result, instead of tracking vectors and matrices along the training trajectory, we track random variables that correspond to the vectors, that is, for a vector $x\in\mathbb{R}^{n}$ that appears in the computation of the training, the coordinates of $x$ can be viewed as iid copies of random variable $[x]$ (called a _ket_) when $n\rightarrow\infty$. 11 Footnote 11: The definition of $]x) requires the coordinates of \(x$ is $\mathcal{O}(1)$ w.r.t. $n$, and $]x)$ is trivial if the coordinates of $x$ is $o(1)$ w.r.t. $n$. Therefore, for $x$ whose coordinates are not $\Theta(1)$, we normalize $x$ by multiplying polynomial of $n$ so the resulting vector has coordinates $\Theta(1)$. 2. Since the network is linear, every random variable can be written as a linear combination of a set of zero-mean "base" random variables by the Master Theorem of Tensor Programs [24]. Therefore, we can track the random variables by analyzing the coefficients of their corresponding linear combinations, along with the covariance between the "base" random variables. 3. Since the number of random variables and the number of "base" random variables scale linearly with $L$, the coefficients of all random variables can be represented by a six-dimensional tensor, where two of the dimensions have shape $L$. We then map the tensor to a set of functions whose input domain is $[0,1]\times[0,1]$. Finally, we claim that the functions converge when $L\rightarrow\infty$, and identify their limits as the solution of a set of functional integrals. In Section 10.1, we conduct a thorough empirical verification of our theory in the linear case. The experiments clearly show the convergence of deep linear residual networks under Depth-$\mu$P. Assumptions and NotationsRecall the linear network is given by \[x^{0} =U\xi,\] \[\forall l\in[L], x^{l} =\frac{a}{\sqrt{L}}W^{l}x^{l-1}+x^{l-1},\] \[f =V^{\top}x^{L}.\] For convenience, we assume $a=1$, the SGD learning rate of $W^{l}$ is $1$. We add $t$ as a subscript to any notation to denote the same object but at $t$-th training step, e.g., the input at step $t$ is a single datapoint $\xi_{t}$, the hidden output of $l$-th layer at step $t$ is $x^{l}_{t}$, and the model output at step $t$ is $f_{t}$. Let $T$ be the number of training steps. Let $\ell_{t}$ be the loss function absorbing the label at time $t$, and $\chi_{t}$ be the derivative of the loss at time $t$, i.e., $\chi_{t}=\ell^{\prime}_{t}(f_{t})$. Let $\delta x^{l}_{t}=\partial\ell_{t}/\partial x^{l}_{t}$, and $\widetilde{\delta}x^{l}_{t}=n\delta x^{l}_{t}$ is the normalized version of $\delta x^{l}_{t}$. The Tensor Program analysis heavily depends on the scaling of initialization and learning rate of $U,V,W$ w.r.t $n$. In this paper, we use $\mu$P as the scaling w.r.t. $n$ since it maximizes feature learning in the large width limit [23]. Without loss of generality, we follow [23] and assume the input and output dimension is $1$, i.e., $\xi\in\mathbb{R}$, $f\in\mathbb{R}$. For a clean presentation, we additionally assume $U,V$ are frozen during training in this section and each coordinate of $W$ is initialized with i.i.d. Gaussian of variance $1/n$. ### Width Limit under $\mu$P As the first step, we take the width of the network $n$ to infinity using Tensor Programs (TP). As briefly mentioned in the road map of the section, the TP framework characterizes each vector involved in the training procedure by a random variable when $n\rightarrow\infty$. For a vector $x\in\mathbb{R}^{n}$ that has roughly iid coordinates, we write $\\|x\rangle\in\mathbb{R}$ (called a _ket_) to denote a random variable such that $x$'s entries look like iid copies of $\lceil x\rangle$. Then for any two vector $x,y\in\mathbb{R}^{n}$ that have roughly iid coordinates, their limiting inner product by $n$ can be written as $\lim_{n\rightarrow\infty}\frac{\tau^{\top}y}{n}=\mathbb{E}\lfloor x\rangle \cdot\lfloor y\rangle$, which we write succinctly as $\langle x\lfloor y\rangle$. Deep linear network with SGD is a simple example for this conversion from vectors to random variables. As shown in Program 1, we define a series of scalars ($\hat{f}_{t}$ and $\hat{\chi}_{t}$) and random variables $(\llbracket U\rangle,\lceil nV\rangle,\llbracket x^{l}_{t}\rangle,\llbracket \delta x^{l}_{t}\rangle,\lVert W^{l}_{t}x^{l-1}_{t}\rangle,\lVert W^{l}_{t} \cap x^{l}_{t}\rangle)$ using the ket notations. For better understanding, we provide a brief introduction to TP below. Tensor Programs (TP) in a nutshell.When training a neural network, one can think of this procedure as a process of successively creating new vectors and scalars from an initial set of random vectors and matrices (initialization weights), and some deterministic quantities (dataset in this case). In the first step, the forward propagation creates the features $x_{0}^{l}$ where the subscript $0$ refers to initialization, and the scalar $f_{0}$, which is the network output. In the first backward pass, the output derivative $\chi_{0}$ is computed, then the gradients $\delta x_{0}^{l}$ are backpropagated. (Since the coordinates of $\delta x_{0}^{l}$ vanish to 0 when $n\to\infty$, TP instead tracks its normalized version $\widetilde{\delta}x_{0}^{l}\stackrel{{\text{\tiny def}}}{{=}}n \cdot\delta x_{0}^{l}$.) New vectors are created and appended to the TP as training progresses. When the width $n$ goes to infinity, vectors of size $n$ in the TP (e.g., the features $x_{t}^{l}$, and normalized gradients $\widetilde{\delta}x_{t}^{l}$) see their coordinates converge to roughly iid random variables (e.g., $[\![x_{t}^{l})$ and $\|\![\widetilde{\delta}x_{t}^{l})$ in Program 1), and other scalar quantities (e.g., $f_{t}$ and $\chi_{t}$) converge to deterministic values (e.g., $\dot{f}_{t}$ and $\tilde{\chi}_{t}$ in Program 1) under proper parametrization ($\mu$P). The Master Theorem [25] captures the behaviour of these quantities by characterizing the _infinite-width_ limit of the training process. For more in-depth definitions and details about TP, we refer the reader to [25]. Now when we look back to Program 1, the definitions of scalars and random variables should be clear (except for $[\![W_{0}^{l}x_{t}^{l-1})$ and $\|\![W_{0}^{l\top}\widetilde{\delta}x_{t}^{l})$). One can find straightforward correspondence between those and their finite counterpart, for example: * $\dot{f}_{t}$ corresponds to $f_{t}$, and $\dot{\chi}_{t}$ corresponds to $\chi_{t}$; * $[\![x_{t}^{l})$ corresponds to $x_{t}^{l}$ and $\|\![\widetilde{\delta}x_{t}^{l})$ corresponds to $\widetilde{\delta}x_{t}^{l}$. (Recall $\widetilde{\delta}x_{t}^{l}=n\cdot\delta x_{t}^{l}$ is the normalized version of $\delta x_{t}^{l}$.) * By SGD, $W_{t}^{l}=W_{0}^{l}-\frac{1}{\sqrt{L}}\sum_{s=0}^{t-1}\delta x_{s}^{l}\otimes x _{s}^{l-1}$, which corresponds to $\|\![W_{t}^{l}x_{t}^{l-1})=\|\![W_{0}^{l}x_{t}^{l-1})-\frac{1}{\sqrt{L}}\sum_{s =0}^{t-1}\|\widetilde{\delta}x_{s}^{l}\rangle\langle x_{s}^{l-1}\|\![x_{t}^{l-1})$. Now we can dive into the definition of $[\![W_{0}^{l}x_{t}^{l-1})$ and $\|\![W_{0}^{l\top}\widetilde{\delta}x_{t}^{l})$. Let $\mathcal{W}$ be the set of initial random matrices of size $n\times n$, i.e., $\{W_{0}^{1},\ldots,W_{0}^{L}\}$, and $\mathcal{W}^{\top}\stackrel{{\text{\tiny def}}}{{=}}\{W^{\top}:W \in\mathcal{W}\}$. Let $\mathcal{V}_{W}$ denote the set of all vectors in training of the form $Wy$ for some $y$. Then for every $W\in\mathcal{W}\cup\mathcal{W}^{\top}$, and $Wy\in\mathcal{V}_{W}$, we can decompose $[\![Wy)\!\!]$ into the sum of $\|\![Wy\!\!]$ and $\|\![Wy\!\!]$, where $\|\![Wy\!\!]$ is a random variable that act as if $W$ were independent of $y$, and $\|\![Wy\!\!]$ is the random variable capturing the correlation part between $W$ and $y$. Specifically, let us briefly track what happens to $W_{0}^{l}x_{t}^{l-1}$ during training. In the first step, we have $W_{0}^{l}x_{0}^{l-1}$ which has roughly Gaussian coordinates (in the large width limit). In this case, we have $\|\![W_{0}^{l}x_{0}^{l-1})=0$. After the first backprop, we have $\delta x_{0}^{l-1}=\delta x_{0}^{l}+\frac{1}{\sqrt{L}}W_{0}^{l\top}\delta x_{0} ^{l}$, which means that the update in $W^{l-1}$ will contain a term of the form $W^{l\top}_{0}z$ for some vector $z$. This implies that $W^{l}_{0}x^{l-1}_{1}$ will contain a term of the form $W^{l}_{0}W^{l\top}_{0}z^{\prime}$ for some vector $z^{\prime}$. This term induces an additional correlation term that appears when we take the width to infinity. The $[W^{l}_{0}x^{l-1}_{1}]$ is defined by isolating this additional correlation term from $W^{l}_{0}W^{l\top}_{0}z^{\prime}$. The remaining term is Gaussian in the infinite-width limit, which defines the term $[W^{l}_{0}x^{l-1}_{1}]$. Formally, we present the following definition. Definition 4.1.: We define $[Wy\rangle\stackrel{{\text{\tiny def}}}{{=}}][Wy\widehat{y}]+[Wy \dot{y}]$ for every $W\in\mathcal{W}\cup\mathcal{W}^{\top}$ and $Wy\in\mathcal{V}_{W}$, where * $[Wy\dot{y}]$ is a Gaussian variable with zero mean. $\forall W\in\mathcal{W}\cup\mathcal{W}^{\top},Wy,Wz\in\mathcal{V}_{W}$, \[\operatorname{Cov}\left([Wy\widehat{y}],[Wz\widehat{z}\right)\stackrel{{ \text{\tiny def}}}{{=}}\langle y\\|z\rangle.\] $\forall W,W^{\prime}\in\mathcal{W}\cup\mathcal{W}^{\top},Wy\in\mathcal{V}_{W },W^{\prime}z\in\mathcal{V}_{W^{\prime}}$, $[Wy\widehat{y}]$ and $[W^{\prime}z\widehat{z}]$ are independent if $W\neq W^{\prime}$. $[Wy\widehat{y}]$ is also independent from $[U\rangle$ and $[nV)$. * $[Wy\dot{y}]$ is defined to be a linear combination of $\{[z]:W^{\top}z\in\mathcal{V}_{W^{\top}}\}$. Then we can unwind any $\\|y\rangle$ inductively as a linear combination of $[\bullet\,\rangle$, $[U\rangle$ and $[nV)$, which allows us to fully define \[[Wy\dot{y}]\stackrel{{\text{\tiny def}}}{{=}}\sum_{W^{\top}z\in \mathcal{V}_{W^{\top}}}\,[\![z]\,\cdot\,\frac{\partial[y\!]}{\partial[W^{\top} z\widehat{z}]}.\] ### Depthwise Scaling of Random Variables As mentioned in Definition 4.1, both $[\![x^{l}_{t}]$ and $[\widetilde{\delta}x^{l-1}_{t}]$ can be written as linear combination of "base" random variables: $\{[W^{m}_{0}x^{m-1}_{s})\}_{s\in\{0,\dots,t\},m\in[L]},\{[\![W^{m\top}_{0} \widetilde{\delta}x^{m}_{s}\!]\}_{s\in\{0,\dots,t\},m\in[L]},[U\rangle$ and $[\![nV\rangle$. Moreover, the coefficients of the linear combinations can be calculated in a recursive way: by expanding $[W^{l}_{0}x^{l-1}_{t})$ using Definition 4.1, we have \[[\![x^{l}_{t}]\!]=[\![x^{l-1}_{t}\rangle+\frac{1}{\sqrt{L}}[W^{l}_{0}x^{l-1}_{ t}\widehat{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! pass ($\|\!\widetilde{\delta}x_{t}^{l}\rangle$ and $\|W_{0}^{m^{\top}}\widetilde{\delta}x_{s}^{m}\rangle$). Formally, $\mathbf{\Gamma}_{t,s,0,0,l,m}=\frac{\partial[x_{t}^{l}\rangle}{\partial[W_{0}^{ m}x_{s}^{m-1}\rangle}$, $\mathbf{\Gamma}_{t,s,0,1,l,m}=\frac{\partial[x_{t}^{l}\rangle}{\partial[W_{0}^ {m^{\top}}\delta x_{s}^{m}\rangle}$, $\mathbf{\Gamma}_{t,s,1,0,l,m}=\frac{\partial[\widetilde{\delta}x_{t}^{l}\rangle}{ \partial[W_{0}^{m}x_{s}^{m-1}\rangle},$$\mathbf{\Gamma}_{t,s,1,1,l,m}=\frac{\partial[ \widetilde{\delta}x_{t}^{l}\rangle}{\partial[W_{0}^{m^{\top}}\widetilde{ \delta}x_{s}^{m}\rangle}$. However, it is hard to describe the limit of $\mathbf{\Gamma}$ because its size increases along with $L$. Therefore, we define the following set of functions $\{\Gamma_{t,s,a,b}:[0,1]\times(0,1]\rightarrow\mathbb{R}\}_{t\in\{0,\dots,T-1 \},s\in\{-1,\dots,t\},a,b\in\{0,1\}}$: For $s\geq 0$, \[\Gamma_{t,s,a,b}\left(p,q\right)=\sqrt{L}\cdot\mathbf{\Gamma}_{t,s,a,b,\lceil Lp \rceil,\lceil Lq\rceil}\] For $s=-1$, $\Gamma_{t,-1,0,0}\left(p,q\right)=\frac{\partial[x_{t}^{\lceil Lp\rceil})}{ \partial[U\rangle},\Gamma_{t,-1,0,1}\left(p,q\right)=\frac{\partial[ \widetilde{\delta}x_{t}^{\lceil Lp\rceil})}{\partial[nV\rangle},\Gamma_{t,-1, 1,0}\left(p,q\right)=\frac{\partial[\widetilde{\delta}x_{t}^{\lceil Lp\rceil} ]}{\partial[U\rangle},\Gamma_{t,-1,1,1}\left(p,q\right)=\frac{\partial[ \widetilde{\delta}x_{t}^{\lceil Lp\rceil}]}{\partial[nV\rangle}.$ Here $l,m$ are normalized to $[0,1]$ so the input domain of $\Gamma$s are identical for different $L$; $\mathbf{\Gamma}_{t,s,a,b,l,m}$ is multiplied by $\sqrt{L}$ because $\mathbf{\Gamma}_{t,s,a,b,l,m}=\mathcal{O}(1/\sqrt{L})$ by Proposition 4.2; and the extra $s=-1$ case helps us also capture the derivative w.r.t. $\left\|U\right\rangle$ and $\left\|nV\right\rangle$. Similarly, we can also define another set of function $\{C_{t,s,a}:(0,1]\rightarrow\mathbb{R}\}_{t,s\in\{-1,\dots,T-1\},a\in\{0,1\}}$ to describe the covariance between the "base" random variables: $\forall p\in(0,1]$, let $l=\lceil Lp\rceil$, * $C_{t,s,0}\left(p\right)\stackrel{{\text{def}}}{{=}}\text{Cov}([ \!\|W_{0}^{l}x_{t}^{l-1}\widehat{\gamma}\|,\|W_{0}^{l}x_{s}^{l-1}\widehat{\gamma} ))=\langle x_{t}^{l-1}\|\widetilde{x}_{s}^{l-1}\rangle$, * $C_{t,s,1}\left(p\right)\stackrel{{\text{def}}}{{=}}\text{Cov}([ \!\|W_{0}^{l\top}\widetilde{\delta}x_{t}^{l}\widehat{\gamma}\|,\|W_{0}^{l\top} \widetilde{\delta}x_{s}^{l}\widehat{\gamma}))=\langle\widetilde{\delta}x_{t} ^{l}\|\widetilde{\delta}x_{s}^{l}\rangle$, For $t=-1$, $C_{-1,-1,0}\left(p\right)\stackrel{{\text{def}}}{{=}}\text{Cov}([ \!\|U\rangle,\|U\rangle)=1$, and $C_{-1,-1,1}\left(p\right)\stackrel{{\text{def}}}{{=}}\text{Cov}([ \!\|nV\rangle,\|\!\|nV\rangle)=1$, By Definition 4.1, the "base" random variables of different "groups" are independent, so we only tracks the covariance listed above. Using this definition of $\Gamma$ and $C$, it is convenient to write their recursive formula in the following lemma. Lemma 4.3 (Finite depth recursive formula for $\Gamma$ and $C$ (Informal version of Lemma C.1)).: $\Gamma$ _and $C$ can be computed recursively as follows:_ \[\Gamma_{t,r,0,b}\left(\frac{l}{L},q\right)= \,\Gamma_{t,r,0,b}\left(\frac{l-1}{L},q\right)+\mathbb{I}_{[(t=r) \wedge(b=0)\wedge(l=\lceil Lq\rceil)]}\] \[+\frac{1}{L}\sum_{s=0}^{t-1}\Gamma_{s,r,1,b}\left(\frac{l}{L},q \right)\left(\Gamma_{t,s,0,1}\left(\frac{l-1}{L},\frac{l}{L}\right)-C_{t,s,0} \left(\frac{l}{L}\right)\right).\] \[\Gamma_{t,r,1,b}\left(\frac{l-1}{L},q\right)= \,\Gamma_{t,r,1,b}\left(\frac{l}{L},q\right)+\mathbb{I}_{[(t=r) \wedge(b=1)\wedge(l=\lceil Lq\rceil)]}\] \[+\frac{1}{L}\sum_{s=0}^{t-1}\Gamma_{s,r,0,b}\left(\frac{l-1}{L},q \right)\left(\Gamma_{t,s,1,0}\left(\frac{l}{L},\frac{l}{L}\right)-C_{t,s,1} \left(\frac{l}{L}\right)\right).\] \[C_{t,s,a}(p)=\sum_{t^{\prime}=-1}^{t}\sum_{s^{\prime}=-1}^{s}\sum_{b\in\{0,1\}} \int_{0}^{1}\Gamma_{t,t^{\prime},a,b}(l/L,q)C_{t^{\prime},s^{\prime},b}(q) \Gamma_{s,s^{\prime},a,b}(l/L,q)\mathrm{d}q,\] _where $l=\lceil Lp\rceil-1$ if $a=0$, and $l=\lceil Lp\rceil$ if $a=1$._ The proof of Lemma 4.3 is straightforward from Program 1. In Appendix C, we also give a formal proof that $\Gamma$ and $C$ converge when $L$ grows to infinity, in the case where $L$ is powers of $2$. The restriction on $L$ being powers of $2$ is imposed for the convenience of the proof, and the convergence of $\Gamma$ and $C$ is true in the general case. Moreover, we derive the infinite depth behavior based on the recursion of $\Gamma$ and $C$ in Lemma 4.3. Proposition 4.4 (Infinite depth limit of $\Gamma$ and $C$ (Informal version of Proposition C.2)).: _In the limit $L\to\infty$, we have_ \[\Gamma_{t,r,0,b}(p,q)=\mathbb{I}_{[(t=r)\wedge(b=0)\wedge(p\geq q)]}+\int_{0}^{ p}\sum_{s=0}^{t-1}\Gamma_{s,r,1,b}(p^{\prime},q)\cdot(\Gamma_{t,s,0,1}(p^{ \prime},p^{\prime})-C_{t,s,0}(p^{\prime}))\mathrm{d}p^{\prime};\] \[\Gamma_{t,r,1,b}(p,q)=\mathbb{I}_{[(t=r)\wedge(b=1)\wedge(p\leq q)]}+\int_{p} ^{1}\sum_{s=0}^{t-1}\Gamma_{s,r,0,b}(p^{\prime},q)\cdot(\Gamma_{t,s,1,0}(p^{ \prime},p^{\prime})-C_{t,s,1}(p^{\prime}))\mathrm{d}p^{\prime};\] \[C_{t,s,a}(p)=\sum_{t^{\prime}=-1}^{t}\sum_{s^{\prime}=-1}^{s}\sum_{b\in\{0,1 \}}\int_{0}^{1}\Gamma_{t,t^{\prime},a,b}(p,q)C_{t^{\prime},s^{\prime},b}(q) \Gamma_{s,s^{\prime},a,b}(p,q)\mathrm{d}q.\] The proof of Proposition 4.4 follows from Lemma 4.3. A rigorous proof requires first showing the existence of a solution of the integral functional satisfied by the couple $(\Gamma,C)$. The solution is typically a fixed point of the integral functional in Proposition 4.4. After showing the existence, one needs to show that $(\Gamma,C)$ converges to this limit. This typically requires controlling the difference between finite-depth and infinite-depth solutions and involves obtaining upper-bounds on error propagation. The existence is guaranteed under mild conditions on the integral functional. We omit here the full proof for existence and assume that the functional is sufficiently well-behaved for this convergence result to hold. The formal proof of the convergence of $\Gamma$ and $C$ for $L=2^{k}$ ($k\in\mathbb{N}$) in Appendix C is a showcase of the correctness of the proposition. This gives a convergence in distribution: Theorem 4.1.: _In the $L\to\infty$ limit, the kets $\llbracket x_{s}^{L}\rangle,s=0,1,\dots,$ converge in distribution as a zero-mean Gaussian process with kernel_ \[\langle x_{s}^{L}\llbracket x_{t}^{L}\rangle=C_{t,s,1}(1).\] _Thus, for each fixed neuron index $\alpha$, the collection $\{x_{\alpha s}^{L}\}_{s\geq 0}$ converges in distribution to a zero-mean Gaussian process with kernel $C_{t,s,1}(1)$ in the $n\to\infty$ then $L\to\infty$ limit._ For audience familiar with stochastic processes, we in fact have a weak convergence of the entire continuous-depth-indexed process $\{\llbracket x_{s}^{p}\rangle,\llbracket\delta x_{s}^{p}\rangle\}_{p\in[0,1],s \geq 0}$ in the Skorohod topology. ## 5 What Causes Hyperparameter Transfer? In a popular misconception, hyperparameter transfer is implied by the existence of a limit. For example, the fact that $\mu$P transfers hyperparameters, in this misconception, is because of the existence of the feature learning limit (aka the $\mu$ limit), the limit of $\mu$P as width goes to infinity. However, this is not the case. Indeed, there are a plethora of infinite-width limits, such as the NTK limit, but there can only be one way how the optimal hyperparameters scale, so existence cannot imply transfer. In a stronger version of this misconception, transfer is implied by the existence of a "feature learning" limit. But again, this is False, because there are infinite number of feature learning limits (where the $\mu$ limit is the unique maximal one). Instead, what is true is that the _optimal_ limit implies the transfer of _optimal_ hyperparameters. For example, in the width limit case, $\mu$P is the unique parametrization that yields a maximal feature learning limit. Compared to all other limits, this is obviously the optimal one. Hence $\mu$P can transfer hyperparameters across width. So far, there is no _a priori_ definition for the "optimality" of a limit: One can only tell by _classifying_ all possible limits; it turns out only a small number of different behavior can occur in the limit, and thus one can manually inspect for which limit is the optimal one. Similarly, in this work, to _derive_ a depthwise scaling that allows transfer, we need to _classify_ all possible infinite depth limits -- and Depth-$\mu$P will turn out to be optimal in a sense that we define later in the paper.12 More interestingly than the width case, here we have multiple modes of feature learning when taking the depth limit and it is important to discern which mode of feature learning is optimal. Thus, again, it is _insufficient_ to derive any one limit, even with feature learning, and be able to infer it yields HP transfer. In section 10, we provide experiments with $1/L$ block scaling $(\alpha,\gamma)=(1,0)$, aka ODE scaling, which provably induces feature learning in the infinite-depth limit, but is sub-optimal. Our results show a significant shift in the optimal learning rate with this parametrization. ## 6 Preliminaries for the General Case For the general case, we recall and extend the notation from the previous sections and also define new ones. NotationLet $L$ be the depth of the network, i.e., the number of residual blocks, and $n$ be the width of the network, i.e. the dimension of all hidden representations $x^{0},\ldots,x^{L}$. Let $\xi\in\mathbb{R}^{d_{\text{in}}}$ be the input of the network, $U\in\mathbb{R}^{n\times d_{\text{in}}}$ be the input layer, and $V\in\mathbb{R}^{n\times e}$ be the output layer, so that $x^{0}=U\xi$ and the model output w.r.t. $\xi$ is $f(\xi)\triangleq V^{\top}x^{L}$. Let $\ell$ be the loss function absorbing the label, and $\delta x^{l}$ be the gradient of $x^{l}$ w.r.t. the loss. We denote variables at $t$-th training step by adding $t$ as a subscript, e.g., the input at step $t$ is $\xi_{t}$13, the hidden representation of $l$-th layer at step $t$ is $x^{l}_{t}$, and the model output at step $t$ is $f_{t}$. Let $T$ be the number of training steps. Footnote 13: Here, the input is used to perform one gradient step at training step $t$. We will see later that our claims should in principle hold for batched versions of the training algorithm. ### Unified Scaling for SGD, Adam, and All Entrywise Optimizers We extend the definition of entrywise update ([24]) for depth scaling, allowing us to study the unified depth scaling for SGD, Adam, and other optimization algorithms that perform only entrywise operations. Definition 6.1.: A gradient-based update of parameter $w$ with both width and depth scaling is defined by a set of functions $\mathbf{Q}=\{Q_{t}:\mathbb{R}^{t+1}\rightarrow\mathbb{R}\}_{t\geq 0}$, and $c,d,\delta,\gamma,\eta$. The update at time $t$ of the optimization is \[w\gets w-\eta n^{-c}L^{-\gamma}Q_{t}(n^{d}L^{\delta}g_{0},\ldots,n^{d}L^{ \delta}g_{t}),\] where $g_{s},s=0,\ldots,t$, are the gradients of $w$ at time $s$. For SGD, $Q_{t}(n^{d}L^{\delta}g_{0},\ldots,n^{d}L^{\delta}g_{t})=n^{d}L^{\delta}g_{t}$, and the "true" learning rate is $\eta n^{-c+d}L^{-\gamma+\delta}$. For Adam, \[Q_{t}(n^{d}L^{\delta}g_{0},\ldots,n^{d}L^{\delta}g_{t})=\frac{\frac{1-\beta_ {1}}{1-\beta_{1}^{t+1}}\sum_{s=0}^{t}\beta_{1}^{t-s}n^{d}L^{\delta}g_{s}}{ \sqrt{\frac{1-\beta_{2}}{1-\beta_{2}^{t+1}}\sum_{s=0}^{t}\beta_{2}^{t-s}(n^{d }L^{\delta}g_{s})^{2}+\epsilon}},\] and the "true" learning rate is $\eta n^{-c}L^{-\gamma}$. The purpose of multiplying the gradients $n^{d}L^{\delta}$ before $Q_{t}$ is to make sure the inputs to $Q_{t}$ are $\Theta(1)$ w.r.t. $n$ and $L$14; otherwise, the update might be trivial when $n$ and $L$ become large. For example, if gradients are $o(1)$ entrywise, then, in Adam, directly feeding gradients to $Q_{t}$ will always give an output of $0$ because of the constant $\epsilon>0$. Footnote 14: It is called faithfulness in Yang and Littwin [24]. In this paper, we will only consider $d,\delta$ such that $n^{d}L^{\delta}g$ is $\Theta(1)$.15 As a result, the output of $Q_{t}$ is also $\Theta(1)$ in general. Therefore, $n^{-c}L^{-\gamma}$ decides the scale of the update and should be our focus. We call $\eta n^{-c}L^{-\gamma}$ the _effective learning rate_. Footnote 15: Note $c,d,\delta,\gamma,\eta$ in Definition 6.1 can be different for parameters, so it is possible to make every parameter to satisfy the condition. ### $\mu$P and Widthwise Scaling Maximal update parametrization ($\mu$P) [21] considers the change of initialization and learning rate of each weight matrix in the network when width scales up.16 It provides a unique initialization and learning rate of each weight matrix as a function of width $n$ that makes the update of each weight matrix maximal (up to a constant factor). The benefit of $\mu$P is not only the theoretical guarantee but also the hyperparameter stability when scaling up the width [23]. In this paper, we assume the widthwise scaling follows $\mu$P. That is, the $c$ in the effective learning rate $\eta n^{-c}L^{-\gamma}$ and the initialization variance of each weight matrix follows Table 2. ### Our Setup We consider an $L$-hidden-layer residual network with biasless perceptron blocks: \[x^{0} =U\xi,\] \[\forall l\in[L],\quad x^{l} =L^{-\alpha}\operatorname{MS}(\phi(h^{l}))+x^{l-1},\quad h^{l}=W^ {l}x^{l-1},\] \[f =V^{\top}x^{L}.\] where $\operatorname{MS}$ refers to Mean Subtraction and is given by $\operatorname{MS}(x)=x-\langle x,1\rangle/n=Gx$ with $G=I-11^{\top}/n$, for any $x\in\mathbb{R}^{n}$. The initialization and learning rate of $U,V$ follows $\mu$P. The initialization of $W^{l}$ follows $\mu$P, and the learning rate of $W^{l}$ is $\eta n^{-1}L^{-\gamma}$. Mean Subtraction ($\operatorname{MS}$).In general, without mean subtraction, the mean of $\phi$ will dominate the depthwise dynamics. For example, when $\phi$ is relu, each layer will only add nonnegative quantities to $x^{l}$ that on average is positive. Its accumulation over depth either causes the network output to blow up if the multiplier $L^{-\alpha}$ is too large, or lack feature diversity otherwise. As we shall see, mean subtraction removes this failure mode and enable more powerful infinite-depth limits.17 Footnote 17: Note that using an _odd_ nonlinearity will also achieve similar results because they have no mean under a symmetrically distributed input, which is approximately the case for $h^{l}$ throughout training. This is the case for $\phi$ = identity that we discussed earlier. But it turns out odd nonlinearities minimize feature diversity, so mean subtraction is a much better solution. Definition 6.2.: Fix a set of update functions $\mathbf{Q}=\{Q_{t}:\mathbb{R}^{t+1}\to\mathbb{R}\}_{t\geq 0}$. A _depthwise parametrization_ of the MLP residual network above is specified by a set of numbers $\{\alpha,\gamma,\delta\}$ such that * We independently initialize each entry of $W^{l}$ from $\mathcal{N}(0,n^{-1})$ * The gradients of $W^{l}$ are multiplied by $nL^{\delta}$ before being processed by $Q_{t}$: i.e., the update at time $t$ is \[W^{l}\gets W^{l}-\eta n^{-1}L^{-\gamma}Q_{t}^{l}(nL^{\delta}g_{0},\dots,nL ^{\delta}g_{t})\] (4) where $g_{s},s=0,\dots,t$, are the gradients of $W^{l}$ at time $s$ and $Q_{t}$ is applied entrywise. Miscellaneous notations.For a vector $x$, let $[x]_{i}$ be its $i$-th coordinate. For a matrix $M$, let $[M]_{i}$ be its $i$-th row. Let $I$ be the identity matrix, and $\mathbf{1}$ be the full one vector. For $m\in\mathbb{N}^{+}$, let $[m]=\{1,\dots,m\}$. Let $\otimes$ be the Kronecker product. ## 7 Classification of Depthwise Parametrizations In this section, we provide a comprehensive description of the impact of depth parametrization on stability and update size. For this purpose, we only have two scalings to keep track of: the branch multiplier and the learning rate scaling because the initialization scale is fixed by the faithfulness property (defined below). Requiring that the features don't blow up at initialization means that \begin{table} \begin{tabular}{c\|c\|c\|c} \hline & Input weights & Output weights & Hidden weights \\ \hline Init. Var. & $1$ & $n^{-2}$ & $n^{-1}$ \\ \hline $c$ & $0$ & $1$ & $1$ \\ \hline \end{tabular} \end{table} Table 2: Widthwise scaling of $\mu$P, where $c$ (defined in Definition 6.1) describes the widthwise scaling of the effective learning rate. the branch multipliers must be at most $\Theta(1/\sqrt{L})$. Assuming the updates are faithful (i.e., input to gradient processing functions are $\Theta(1)$ entrywise), the update size can be at most $1/L$ for the hidden layers, by an (Jacobian) operator-norm argument, but potentially much less. Naively speaking, there can be a trade-off between update size and initialization: if initialization is large, then the update may need to be small so as not to blow up the other parts of the network; likewise if the initialization is small, then the update size can be larger. But one may be surprised that a careful calculation shows that there is no trade-off: we can maximize both initialization and update size at the same time. Before delving into the details, let us first define the notions of training routine, stability, faithfulness, and non-triviality. Hereafter, all the asymptotic notations such as $\mathcal{O}$, $\Omega$ and $o$ should be understood in the limit "$n\to\infty$, then $L\to\infty$". For random variables, such notations should be understood in the sense of weak convergence (convergence in distribution). When we use the notation $x=\mathcal{O}(1)$ for some vector $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$, it should understood in the sense that for all $i\in[n],x_{i}=\mathcal{O}(1)$. Lastly, we will use bold characters (e.g. $\mathbf{h}$ instead of $h$) to denote 'batched' versions of the quantities. This is just to emphasize that the following claims should hold for batched quantities as well. _Remark:_ in this section, we state the results as "claims" instead of theorems. In Appendix F.4, we provide "heuristic" proofs that can be made rigorous under non-trivial technical conditions. We also showcase the correctness of the claims by proving them rigorously in our linear setting in Appendix D. We believe this additional layer of complexity is unneeded and does not serve the purpose of this paper. Definition 7.1 (Training routine).: A training routine is the package of $\eta$, $\mathbf{Q}$, and the input batches. Definition 7.2 (Stability).: We say a parametrization is 1. _stable at initialization_ if \[\mathbf{h}_{0}^{l},\mathbf{x}_{0}^{l}=\mathcal{O}(1),\forall l\in[L],\quad\text{and} \quad\mathbf{f}_{0}=\mathcal{O}(1).\] (5) 2. _stable during training_ if for any training routine, any time $t\geq 0$, $l\in[L]$, we have \[\Delta\mathbf{h}_{t}^{l},\Delta\mathbf{x}_{t}^{l}=\mathcal{O}(1),\forall l\in[L], \quad\text{and}\quad\Delta\mathbf{f}_{t}=\mathcal{O}(1),\] where the symbol '$\Delta$' refers to the change after one gradient step. We say the parametrization is _stable_ if it is stable both at initialization and during training. Definition 7.3 (Faithful).: We say a parametrization is _faithful at step $t$_ if $\mathbf{h}_{t}^{l}=\Theta(1)$ for all $l\in[L]$. We say the parametrization is _faithful_ if it is faithful for all $t$. We also say it is _faithful at initialization_ (resp. faithful during training) if this is true at $t=0$ (resp. for $t\geq 1$). Note faithfulness here refers to "faithfulness to $\phi$", meaning the input to $\phi$ is $\Theta(1)$. This is different from the definition of faithfulness in Yang and Littwin [24], where faithfulness refers to "faithfulness to $Q$" meaning the input to $Q$ is $\Theta(1)$. "faithfulness to $Q$" is already assumed in this work as mentioned in Section 6.1. Definition 7.4 (Nontriviality).: We say a parametrization is _trivial_ if for every training routine and any time $t\geq 1$, $\mathbf{f}_{t}-\mathbf{f}_{0}\xrightarrow{\text{a.s.}}0$ in the limit "$n\to\infty$, then $L\to\infty$" (i.e., the function does not evolve in the infinite-width-then-depth limit). We say the parametrization is _nontrivial_ otherwise. Definition 7.5 (Feature Learning).: We say a parametrization induces _feature learning_ in the limit "$n\to\infty$, then $L\to\infty$", if there exist a training routine, and $t\geq 1$, and any $\lambda>0$, we have $\Delta\mathbf{h}_{t}^{\lfloor\lambda L\rfloor}=\Theta(1)$. ### Main Claims We are now ready to state the main results. The next claim provides a necessary and sufficient condition under which a parametrization is stable at initialization. Claim 7.1.: _A parametrization is stable at initialization iff $\alpha\geq 1/2$._ Claim 7.1 is not new and similar results were reported by Hayou et al. [7]. However, Hayou et al. [7] focuses on initialization and lacks a similar stability analysis during training. In the next result, we identify two different behaviours depending on the scaling of the learning rate. Claim 7.2.: _Consider a parametrization that is stable at initialization. Then the following hold (separately from each other)._ * _It is stable during training as well iff_ $\alpha+\gamma\geq 1$_._ * _It is nontrivial iff_ $\alpha+\gamma\leq 1$_._ _Therefore, it is both stable and nontrivial iff $\alpha+\gamma=1$._ From Claim 7.1 and Claim 7.2, having $\alpha+\gamma=1$ and $\alpha\geq 1/2$ is a necessary and sufficient condition for a parametrization to be stable and nontrivial throughout training. In the next result, we therefore restrict our analysis to such parametrizations and study their faithfulness. Claim 7.3.: _Consider a stable and nontrivial parametrization. The following hold (separately from each other)._ * _It is faithful at initialization iff_ $\alpha\geq 1/2$_. As a result,_ $\alpha=1/2$ _is the minimal choice of_ $\alpha$ _that guarantees faithfulness._ * _It is faithful during training iff_ $\alpha\leq 1$_._ _Therefore, a stable and nontrivial parametrization is faithful iff $\alpha\in[1/2,1]$._ The first claim follows from well-known calculations of randomly initialized residual networks [7]. For the second claim, the intuition here is just that if $\alpha+\gamma=1$ and $\alpha>1$ then $\gamma<0$, i.e., the update size blows up with depth. This would then cause the input to the nonlinearities to blow up with size. One might argue that faithfulness at initialization is not important (e.g. features at initialization could converge to zero without any stability or triviality issues) and what matters is faithfulness throughout training. It turns out that faithfulness at initialization plays a crucial role in the optimal use of network capacity. To see this, we first define the notion of feature diversity exponent, which relates to the similarity in the features of adjacent layers. Definition 7.6 (Feature Diversity Exponent).: We say a parametrization has feature diversity exponent $\kappa\geq 0$ if $\kappa$ is the maxmal value such that for all $\lambda\in[0,1]$ and sufficiently small $\epsilon>0$, and all time $t$, \[\frac{1}{\sqrt{n}}\left\\|\mathbf{x}_{t}^{\lfloor(\lambda+\epsilon)L\rfloor}-\mathbf{x }_{t}^{\lfloor\lambda L\rfloor}\right\\|=\Omega(\epsilon^{1-\kappa}),\] where $\Omega(1)$ should be interpreted in the limit "$n\to\infty$, then $L\to\infty$, then $\epsilon\to 0$". We say a parametrization is _redundant_ if $\kappa=0$. In other words, the feature diversity exponent $\kappa$ is a measure of how different the outputs are in layers that are close to each other. With $\kappa=0$, the output of each layer is essentially the same as the output of the previous layer in the sense that the rate of change from one layer to the next is bounded (at least locally), and hence the network is intuitively "wasting" parameters. Claim 7.4.: _Consider a stable and nontrivial parametrization that is furthermore faithful during training (but not necessarily at initialization). Then it is redundant if $\alpha\in(1/2,1]$._ To understand the intuition behind Claim 7.4, let us see what happens when $\alpha>1/2$. In this case, the randomness of the initialization weights will have no impact on training trajectory as depth increases. To see this, consider some layer index $\lfloor\lambda L\rfloor$. The blocks are divided by $L^{\alpha}$ which is larger than the magnitude of accumulated randomness (of order $(\lambda L)^{1/2}$). This basically destroys all the randomness from initialization and therefore the randomness in the learned features will consist only of that coming from $U$ and $V$ (input and output matrices). When depth goes to infinity, the contribution of the randomness in two adjacent layers becomes less important, we end up with adjacent layers becoming very similar because the gradients to these layers are highly correlated. In contrast, we have the following result, which defines Depth-$\mu$P. Claim 7.5 (Depth-$\mu$P).: $\alpha=\gamma=1/2$ _is the unique parametrization that is stable, nontrivial, faithfulness, induces feature learning, and achieves maximal feature diversity with $\kappa=1/2$._ In terms of feature diversity, a phase transition phenomenon occurs when $\alpha=1/2$. More precisely, for Depth-$\mu$P, we can show that $n^{-1/2}\left\\|\mathbf{x}_{t}^{\lfloor(\lambda+\epsilon)L\rfloor}-\mathbf{x}_{t}^{ \lfloor\lambda L\rfloor}\right\\|=\mathcal{O}(\epsilon^{1/2})$ while the same quantity is $\mathcal{O}(\epsilon)$ for all $\alpha\in(1/2,1]$, which suggests that Depth-$\mu$P yields _rough_ path for $\mathbf{x}_{t}$. This allows the features to change significantly from one layer to the next, hence efficiently using the parameters. For readers who are familiar with rough path theory, the $1/2$ continuity exponent is a result of Brownian increments in the path.18 Footnote 18: The reader might ask whether we can obtain an exponent smaller than $1/2$. This is indeed possible, but it will entail using correlated weights. We leave this question for future work. Moreover, with $\alpha=1$, there is a phenomenon of feature collapse in the sense that the features will be contained in the $\sigma$-algebra generated by the input and output layers, but contains no randomness from the hidden layers (see Appendix F.2). Intuitively, the case of $\alpha=1$ is analogous to width situation, where deep mean field collapses to a single neuron (all neurons become essentially the same). For depth, the features (layers) are still relatively different but the redundancy does not allow significant variety in these features. ### Sublety: Layerwise (local) linearization but not global linearization Definition 7.7.: We say a parametrization induces layerwise linearization iff each layer can be linearized without changing the network output when $L\to\infty$, that is, $\forall l\in[L]$, \[L^{-\alpha}G\left(\phi(W_{t}^{l}\mathbf{x}_{t}^{l-1})-\phi(W_{0}^{l}\mathbf{x}_{t}^{l -1})-\phi^{\prime}(W_{0}^{l}\mathbf{x}_{t}^{l-1})\odot((W_{t}^{l}-W_{0}^{l})\mathbf{x}_ {t}^{l-1})\right)=o(L^{-1})\] Claim 7.6.: _A stable and nontrivial parametrization induces layerwise linearization iff $\alpha\in[1/2,1)$._ However, note that this does not imply the entire network is linearized (w.r.t. all the parameters in the sense of Neural Tangent Kernel). In our setup, where the input and output layers are initialized at a constant scale (w.r.t. $L$), it is actually not possible to have a kernel limit. Even in our linear case in Section 4, one can see the learned model is not linear. If the initialization of the output layer is $L$ times larger than our setup (assuming $L\ll n$ so the widthwise scaling still follows $\mu$P), it may induce a parametrization that can linearize the entire network. In that situation, the learning rate has to be $L$ times smaller than Depth-$\mu$P to obtain stability during training, so the change of parameters is also $L$ times smaller, which can lead to the linearization of the entire network. Since we focus on maximal feature learning, the rigorous argument is beyond the scope of this paper. ## 8 Feature Diversity In this section, we show that the choice of nonlinearity and placement of nonlinearities can affect feature diversity greatly. ### Gradient Diversity _Gradient diversity_ is an important factor toward feature diversity. Observe that the gradient $\delta x^{l}$ at $x^{l}$ is continuous in $l$ in the limit $L\to\infty$. In a linear model (or the pre-nonlin model, where nonlinearity is put before the weights), this causes $\delta h^{l}=L^{-\alpha}\delta x^{l}$ to be very similar between neighboring blocks. As a result (because the weights $W^{l}$ receives an update proportional to $\delta h^{l}\otimes x^{l-1}$), in the next forward pass, neighboring blocks contribute very similarly to the main branch $x^{l}$. This leads to a waste of model capacity. ### Pre-Nonlin Leads to Poor Performance For example, in Figure 2, for a relu pre-nonlin resnet (i.e. blocks are given by $W^{l}\phi(x^{l-1})$ instead of $\phi(W^{l}x^{l-1})$), we see that although Depth-$\mu$P indeed transfers hyperparameters (as predicted by our theory), the performance is dramatically worse than the post-nonlin resnet in Figure 10, and depth gives no performance gains beyond 8 layers. Specifically, it is because $\delta h^{l}=L^{-\alpha}\delta x^{l}$ like the linear case, and $\phi(x^{l-1})$ is also similar between neighboring blocks. As a result, the gradient of the weights $W^{l}$, proportional to $\delta h^{l}\otimes\phi(x^{l-1})$, has little diversity compared to nearby blocks. ### Maximizing Feature Diversity with Absolute Value Nonlinearity In a nonlinear model, we have $\delta h^{l}=\delta x^{l}\odot\phi^{\prime}(h^{l})$. Because $h^{l}$ is almost independent from all other $h^{m},m\neq l$ in the Depth-$\mu$P limit, $\phi^{\prime}(h^{l})$ can serve to decorrelate the $\delta h^{l}$, depending on what $\phi$ is. For example, if $\phi$ is relu, then $\phi^{\prime}$ is the step function. $h^{l}$ is approximately a zero-mean Gaussian in the Depth $\mu$P limit, so that $\phi^{\prime}(h^{l})$ is approximately 0 or 1 with half probability each. This decorrelates $\delta h^{l}$ much better than the linear case. But of course, this line of reasoning naturally leads to the conclusion that $\phi^{\prime}=\mathrm{sign}$ would be the best decorrelator of $\delta h^{l}$ and the maximizer of feature diversity (with $\phi$ among the class of positively 1-homogeneous functions) -- then $\delta h^{l}$ and $\delta h^{m}$ are completely decorrelated for $l\neq m$. Indeed, as shown in Figure 3, swapping in absolute value for $\phi$ dramatically improves the training performance of deep (block depth 1) resnets. In general, in lieu of absolute value, any even nonlinearity would suffice. ### Feature Diversity is in Tension with Layerwise Linearization The reason that $\phi^{\prime}(h^{l})$ can decorrelate $\delta h^{l}$ is very much related to layerwise linearization. Recall that in Depth-$\mu$P, $h^{l}$ can be decomposed to a zero-mean Gaussian part $\widehat{h}^{l}$ of size $\Theta(1)$ and a correction term $\dot{h}^{l}$ of size $\Theta(L^{-1/2})$ (corresponding to the decomposition $[h^{l})=[h^{\widehat{l}}]+[h^{l}]$). $\widehat{h}^{l}$ is independent from $\widehat{h}^{m}$ for $m\neq l$ but $\dot{h}^{l}$ can be very strongly correlated to all other $\dot{h}^{m}$. Thus, $\phi^{\prime}(h^{l})$ can decorrelate $\delta h^{l}$ precisely because $\widehat{h}^{l}$ dominates $\dot{h}^{l}$, and this is also precisely the reason we have layerwise linearization. Figure 3: Improving performance with absolute value non-linearity, which maximizes feature diversity. (Networks are trained on CIFAR-10 with Adam.). See Figure 10 for more details about the setup. Figure 2: Pre-Nonlin Leads to Poor Performance Although Depth-$\mu$P for prenonlin resnet indeed transfers hyperparameters (Left), depth gives no performance gains beyond 8 layers and the performance is dramatically worse than the post-nonlin resnet (Right). In right plot, the “Min LogLoss” is minimal log loss over all block multiplier and learning rate. Networks are trained on CIFAR-10 with Adam. See Figure 10 for more details about the setup. In the $1/L$ scaling $(\alpha,\gamma)=(1,0)$, $\widehat{h}^{l}$ is on the same order as $\dot{h}^{l}$ and layerwise linearization does not occur, but also $\phi^{\prime}(h^{l})$ can no longer effectively decorrelated $\delta h^{l}$. Once again, we remind the reader that layerwise linearization in this case is not detrimental (in this block depth 1 case) because $\widehat{h}^{l}$ in fact accumulate contributions from the learned features of all previous blocks and thus strongly depends on the learning trajectory (in contrast to the (widthwise) NTK case where $\widehat{h}^{l}$ is already determined at initialization). ## 9 Block Depth 2 and Above _Remark on notation:_ Here and in the next section, all big-O notation is in $L$ only; the scaling in width is assumed to be in $\mu$P. In most of this work, we have considered depth-$1$ MLP for $g^{l}$ in eq. (1), it's straightforward to derive and classify the infinite-width-then-infinite-depth limits for larger depths in each block. In particular, the following $1/\sqrt{L}$ scaling still makes sense in this more general setting with block depth $k$ and leads to a well defined limit: \[x^{l}=x^{l-1}+\frac{a}{\sqrt{L}}\cdot g^{l}(x^{l-1};W^{l1},\dots,W^{lk}), \quad\Theta(1)\text{ initialization scale},\quad\Theta(1/\sqrt{L})\text{ learning rate} \tag{6}\] This is what we call Depth-$\mu$P in the block depth 1 case, but we shall not use this name in the general block depth case because _this parametrization is no longer optimal_.19 Footnote 19: What we exactly mean by _optimal_ will be explained below. ### Block Depth $\geq 2$ is Defective A very clear symptom of this is that the _performance of block-depth-2 resnets is worse than that of block-depth-1 networks_, when matching parameter count, although they can (but not always) catch up after training for a long time (figs. 4 and 5). Simultaneously, we are seeing nontrivial or even significant hyperparameter shifts as the total number of blocks increases (fig. 6). Figure 4: Block Depth 2 < Block Depth 1, Relu. In relu resnet with no LN, block depth 2 does worse than block depth 1 when matching total number of layers (and thus parameter count). However, training longer (38000 steps, Right) helps it catch up (compared to 11000 steps, Left). The y-axis is minimal log loss over all block multiplier and learning rate Figure 5: Block Depth 2 < Block Depth 1, Abs. In abs resnet with LN, block depth 2 does significantly worse than block depth 1 when matching total number of layers (and thus parameter count). Training longer (38000 steps, Right) does not close the performance gap (compared to 11000 steps, Left). The y-axis is minimal log loss over all block multiplier and learning rate ### Defect of Scaling in Block Depth 2 The reason that the scaling is no longer fine in the block depth $\geq 2$ case is the _linearization of the multiplicative interaction_ between the layers in the block. Indeed, just like the block depth 1 case, the scaling forces the weight updates $\Delta W$ of each weight matrix to be $\Theta(\sqrt{L})$ smaller than the initialization $W_{0}$. Thus, within the block, the training dynamics when depth $L$ is large is in the kernel regime, where the contribution to the block output $g(x;W^{\bullet})$ is only a _summation_, instead of _product_, of individual contributions from each layer's weights updates. When aggregated over all $L$ blocks, the result is that there is only multiplicative interaction of $\Delta W$ across blocks but not within layers. In other words, the network output is dominated, for example in the linear case, by the contributions of the form $M^{L}\cdots M^{1}$ where each $M^{l}$ can be one of $I,W_{0}^{l2}W_{0}^{l1},W_{0}^{l2}\Delta W^{l1},$ or $\Delta W^{l2}W_{0}^{l1}$, but NOT $\Delta W^{l2}\Delta W^{l1}$. All other contributions (which all involve within-block interactions like $\Delta W^{l2}\Delta W^{l1}$) are subleading. In the general nonlinear case, replacing the block \[\phi(W^{l2}\phi(W^{l1}x^{l-1}))\] with the linearized version \[\phi(h_{\wedge}^{l})+\phi^{\prime}(h_{\wedge}^{l})\odot[\Delta W^{l2}\phi(h_{ \vee}^{l})]+\phi^{\prime}(h_{\wedge}^{l})\odot[W_{0}^{l2}(\phi^{\prime}(h_{ \vee}^{l})\odot[\Delta W^{l1}x^{l-1}])]\] will achieve the same performance as depth $L\to\infty$, where $h_{\wedge}^{l}=W_{0}^{l2}\phi(h_{\vee}^{l})$ and $h_{\vee}^{l}=W_{0}^{l1}x^{l-1}$. When block depth $k=1$ (our main subject of study in this work), _all_ interactions are included but this is no longer true when $k>1$. In fig. 7, the heatmap of loss as a function of block multiplier and learning rate demonstrates this vividly for block depth 2. Small depthThe optimal sublevel set of (learning rate, block multiplier) has slope $\approx-2$ when the number of blocks is $2^{1}$. In other words, around the optimum, double the learning rate while dividing the block multiplier by 4 has similar performance. This is because $\Delta W^{l1}$ and $\Delta W^{l2}$ interact _multiplicatively_, so that doubling their sizes leads to quadrupling their contribution to the block output. The simultaneous decrease of block multiplier by 4 then roughly keep their contribution invariant in size. Large depthOn the other hand, the optimal sublevel set has slope $\approx-1$ when the depth is $2^{10}$: Doubling the learning rate while halving the block multiplier has similar performance. This reflects the fact that $\Delta W^{l1}$ and $\Delta W^{l2}$ now interact _additively_. Intermediate depths interpolate this phenomenon, as seen in the plot for depth $2^{5}$. In the same heatmaps, one can see the optimal (learning rate, block multiplier) (in the parametrization) shifts from the middle of the grid to the upper left as depth goes from $2^{5}$ to $2^{10}$, demonstrating the lack of hyperparameter transfer. This change in slope is seen in relu networks as well, with or without layernorm. Figure 6: Block Depth 2 Hyperparameter Shift in relu resnet with no LN (Left) and abs resnet with LN (Right). Finally, we note that the $1/\sqrt{L}$ scaling still yields a $L\to\infty$ limit where the network still learns features as a whole, even though within each block this is no longer true. Thus, this is another reminder that mere "feature learning" does not imply "hyperparameter transfer"! ### Classification of Parametrizations These heatmaps already demonstrate that no parametrization of (global learning rate20, block multiplier) can transfer hyperparameters robustly, because any such parametrization can only _shift_ the heatmaps but not _stretch_ them, so one cannot "transfer" a sublevel set of one slope into a sublevel set of another slope. Footnote 20: meaning, the learning tied across all layers in a block But even if we allow learning rate to vary between layers in a block, no stable, faithful, nontrivial parametrization can avoid the linearization problem described above. For simplicity, fix a positive-homogeneous nonlinearity and block depth 2.21 We consider the space of hyperparameters consisting of the learning rate for each of the layers in a block, as well as the block multiplier (one for each block); WLOG all weights are initialized $\Theta(1)$.22 This yields a space of dimension $\mathrm{blockdepth}+1=3$. Footnote 21: but our arguments generalize trivially to arbitrary block $\mathrm{depth}\geq 2$ Footnote 22: This is WLOG because the nonlinearities are homogeneous Indeed, for this to happen, the weight update $\Delta W^{li}$ must be at least of order $\Omega(1)$ (size of initialization) for some $i$. But this would contribute a drift term to the block output $g^{l}=g^{l}(x^{l-1};W^{\bullet})$ that is as large as the noise term. This then implies that either the parametrization is unstable (if the block multiplier $L^{-\alpha}$ is $\Omega(1/L)$) or lacks feature diversity (if the block multiplier $L^{-\alpha}$ is $O(1/L)$). For example, in a linear model, \[L^{\alpha}\llbracket g^{l}\rangle=\llbracket W^{l2}W^{l1}x^{l-1}\rangle= \llbracket W^{l2}_{0}W^{l1}x^{l-1}\widehat{\rangle}+\llbracket W^{l2}_{0}W^{l1} x^{l-1}\rangle+\llbracket\Delta W^{l2}W^{l1}x^{l-1}\rangle.\] $\llbracket W^{l2}_{0}W^{l1}x^{l-1}\widehat{\rangle}$ is independent and zero-mean across $l$ (the noise term), while $\llbracket W^{l2}_{0}W^{l1}x^{l-1}\widehat{\rangle}+\llbracket\Delta W^{l2}W^{ l1}x^{l-1}\rangle$ is correlated across $l$ (the drift term). $\llbracket W^{l2}_{0}W^{l1}x^{l-1}\widehat{\rangle}$ is always $\Theta(1)$ because the $W^{l2}_{0},W^{l1}_{0}$ are. If $\Delta W^{l2}$ is $\Omega(1)$, then $\llbracket\Delta W^{l2}W^{l1}x^{l-1}\rangle=\Omega(1)$ as well, making the drift term as large as the noise term. If $\Delta W^{l1}$ is $\Omega(1)$, then $\llbracket W^{l2}_{0}\Delta W^{l1}x^{l-1}\rangle=\Omega(1)$, causing $\llbracket W^{l2}_{0}W^{l1}x^{l-1}\rangle=\llbracket W^{l2}_{0}W^{l1}_{0}x^{l-1 }\rangle+\llbracket W^{l2}_{0}\Delta W^{l1}x^{l-1}\rangle$ to be $\Omega(1)$.23 Footnote 23: One can also observe that if $\Delta W^{l1}=\Omega(1)$, then by symmetry the backward pass suffers the same problem. But for general block depth, this argument does not say anything about the middle layers, while the argument presented above implies that $\Delta W^{li}$ cannot be $\Omega(1)$ for any $i$. The same argument can be straightforwardly adapted to nonlinear MLPs (with mean subtraction) and arbitrary block depth $\geq 2$, and as well to general nonlinearities that are not necessarily positive-homogeneous, with hyperparameter space enlarged to include initialization. Figure 7: The "slope" of the optimal sublevel set in the (learning rate, block multiplier) space changes from $-2$ to $-1$ as depth goes from $2^{1}$ to $2^{10}$. Here we use absolute value nonlinearity with layer normalization, block depth 2, and networks are trained for 50 epochs with Adam on CIFAR-10. ### So What is the Optimal Parametrization? All of the above considerations suggest that _we are missing crucial hyperparameters in our consideration_ when increasing the complexity of each block. Our study right now is akin to the naive study of the 1-dimensional hyperparameter space of the global learning rate in SP. Discovering these missing hyperparameters will be an important question for future work. ## 10 Experiments ### Verifying the Theory in the Linear Case In Section 4, we showed that a complete description of the training dynamics of linear networks can be formulated in terms of $\Gamma$ and $C$. In this section, we provide empirical results supporting our theoretical findings. We first verify the finite-depth recursive formula for $\Gamma$ in Lemma 4.3 is the correct limit when the width goes to infinity, then proceed to show that the infinite-depth limit is the correct one. Infinite-width limit.In Figure 8, we train a series of $64$-layer linear networks of width $2^{7},2^{8},\ldots,2^{13}$ with $1,5,10$ steps on MNIST, and plot the root mean square24 of the layer outputs using solid lines. We also compute the infinite width limit of the corresponding statistics using the recursive formula for $\Gamma$ and plot them as dashed horizontal lines. For clarity of the figure, we only plot the statistics of the input layer, output layer, and hidden layers of index 16, 32, 48, and 64. It is clear that as the width grows, the solid lines converge to the dashed lines consistently across the training steps. It indicates that our computation of the infinite width limit is correct. Footnote 24: The root mean square of a vector $x=(x_{1},\ldots,x_{n})$ is $\sqrt{\frac{\sum_{i=1}^{n}x_{i}^{2}}{n}}$, which is denoted as “$12$” in Figures 8 and 9. Infinite-depth limit.We verify that the infinite _width_ limit above converges when the _depth_ grows. We consider linear networks of the same architecture but vary the depth from $2^{4}$ to $2^{9}$. We again compute the root mean square values of the layer outputs using the recursive formula for $\Gamma$, and plot them in Figure 9 with depth being $x$-axis. For clarity of the figure, we only plot the statistics of the input layer, output layer, and hidden layers of index $L/4$, $L/2$, $3L/4$, and $L$. One can observe that Figure 8: Trained linear network converges to its infinite width limit which is computed recursively based on $\Gamma$ and $C$. Depth is fixed at 64,width varies between $2^{7},2^{8},\ldots,2^{13}$. Networks are trained with SGD for 10 steps. The root mean square statistics ($y$-axis) at 1st, 5th and 10th steps are plotted using solid lines where the $x$-axis is the width. The root mean square values are computed on the outputs of some of the layers (including the input layer, output layer, and hidden layers at each quarter). The corresponding value for the infinite width is indicated with dashed lines. Figure 9: Under Depth-$\mu$P, infinite wide linear network training converges when increasing the depth. Infinite wide linear networks of depth $2^{4},2^{5},\ldots,2^{9}$ are computed recursively based on $\Gamma$ and $C$. The root mean square statistics ($y$-axis) at 1st, 5th and 10th steps are plotted across the depth ($x$-axis). the statistics of the layer outputs converge quickly when the depth grows from $2^{4}$ to $2^{9}$, which verifies our convergence result. ### Hyperparameter Transfer In this section, we provide empirical evidence to show the optimality of Depth-$\mu$P scaling and the transferability of some quantities across depth. We train vanilla residual network with block depth 1 Figure 10: Train logloss versus learning rate for width $n=256$ and varying depths. The network consists of MLP blocks (with block depth 1), trained for 50 epochs on CIFAR10 dataset using Adam. The batch size is fixed to $64$. We tune the depth $2^{3}$ network to obtain the optimal $(\log_{2}(a),\log_{2}(\eta/1e-3))=(1,0)$, and scale all deeper networks using $2^{3}$ as base depth. The reader can check that the $L=2^{3}$ curves in each columns are the same. We show the logloss versus the learning rate of the hidden layers (input/output layers fixed) for three parametrizations: Depth-$\mu$P (Top), Scaling only the blocks (no LR scaling), i.e. $\gamma=0$ (Middle), and Standard Parametrization without any scaling ($\alpha=\gamma=0$) (Bottom). Each curve represents the average training loss over a time slice of 1000 steps for depths $2^{k}$ for $k\in\{1,2,\dots,10\}$. Confidence intervals are based on 5 seeds. The results show that Depth-$\mu$P preserves the optimal learning rate while consistently improving the training loss as depth increases. If we only scale the blocks without scaling the LR ($\alpha=1/2,\gamma=0$) when training with Adam, the optimal learning rate shifts significantly with depth. With standard parametrization without any depth scaling (common practice), the results show a significant shift in the optimal learning rate as well. For SP, we cap the log loss at 1, which is why for depth $2^{9},2^{10}$, we have a black horizontal line at $LogLoss=1$. (I MLP layer in each residual block) on CIFAR-10 dataset using Adam optimizer, batch size $64$, for $50$ epochs (input and output layers are fixed). The network is parameterized as follows \[x^{l}=x^{l-1}+a\times L^{-\alpha}\text{MS}(\phi(W^{l}x^{l-1})),\] and the weights are trained with the rule \[W^{l}\gets W^{l}-\eta\times n^{-1}L^{-\gamma}Q_{t}^{l}(nL^{\delta}g_{0}, \ldots,nL^{\delta}g_{t}),\] where the learning rate $\eta$ and the block multiplier $a$ are the _hyperparameters_.25 The values of $\alpha,\gamma$ depend on the parametrization of choice. For Depth-$\mu$P, we have $\alpha=\gamma=1/2$, and for standard parametrization, we have $\alpha=0,\gamma=1$.26 In our experiments, we assume base depth $8$, meaning that we replace $L$ by $L/8$ in the parametrization above. Footnote 25: Note that $\eta$ here is the constant, and the effective learning rate is given by $\eta n^{-1}L^{-\gamma}$. Footnote 26: In standard parametrization, there is generally no rule to scale the learning rate with depth, and the optimal learning rate is typically found by grid search. Here, we assume that in standard parametrization, the learning rate is scaled by $L^{-1}$ to preserve faithfulness. Learning rate transfer ($\eta$).In Figure 10, we show the training loss versus learning rate for depths $2^{k}$, for $k\in\{3,4\ldots,10\}$. For Depth-$\mu$P, a convergence pattern can be observed for the optimal learning rate as depth grows. Optimal learning rates for small depths (e.g. $L=2^{3}$) exhibit a mild shift which should be expected, as our theory shows convergence in the large depth limit. However, starting from depth $L=2^{6}$, the optimal learning rate is concentrated around $10^{-3}$. For parametrization that only scales the multiplier but not LR ($\alpha=1/2$, $\gamma=0$), we observe the optimal learning rate shifts significantly. For standard parametrization without any depth scaling ($\alpha=\gamma=0$), the optimal learning rate exhibits a more significant shift as depth grows. Moreover, even if one picks the optimal learning rate for each depth, the performance still degrades when the depth is very large, suggesting that standard parametrization is not suitable for depth scaling. Additional figures with multiple time slices are provided in Appendix G. Is feature learning sufficient for HP transfer?In Section 5, we explained when and why hyperparameter transfer occurs. Precisely, to obtain HP transfer, one needs to classify all feature learning limits and choose the optimal one. We introduced the notion of feature diversity and showed that Depth-$\mu$P is optimal in the sense that it maximizes feature diversity. To show that optimality is needed for HP transfer, we train a resnet with $(\alpha,\gamma)=(1,0)$ which is also a feature learning limit. Figure 11 shows that in this case the learning rate exhibits a significant shift with depth. Interestingly, the constant $\eta$ in this case seems to increase with depth, suggesting that the network is trying to break from the _ODE_ limit, which is sub-optimal. Note that in Figure 10, with Depth-$\mu$P we obtain better training loss compared to the ODE parametrization in Figure 11. Do we still have transfer with LayerNorm (LN)?Our theory considers only Mean Substraction (MS), and Figure 10 shows the results with MS. To see whether LN affects HP transfer, we train resnets with the same setup as Figure 10 with absolute value non-linearity and LN applied to $x^{l-1}$ before matrix multiplication with $W^{l}$ (preLN). We keep MS after non-linearity although it can be removed since LN is applied in the next layer. Our results, reported in Figure 12 suggest that Depth-$\mu$P guarantees learning rate transfer with LN as well. Figure 11: Same setup as fig. 10 for the parametrization $(\alpha,\gamma)=(1,0)$ (the ODE limit). Block multiplier transfer ($a$).In Figure 13, we investigate the stability of the hyperparameter $a$ in Depth-$\mu$P as depth increases. The results suggest that the optimal value of this constant converges as depth grows, which suggest transferability. Additional experiments with multiple time slices are provided in Appendix G. ### What Happens in a Transformer? Figure 14: Modern transformers are insensitive to block multiplier $a$. Figure 12: Same setup as Figure 10 with Abs non-linearity instead of ReLU and LayerNorm applied to $x^{l-1}$ before matrix multiplication with $W^{l}$. We show the logloss versus the learning rate of the hidden layers (input/output layers fixed) for two parametrizations: Depth-$\mu$P (Left) and scaling only the blocks without LR scaling ($(\alpha,\gamma)=(1/2,0)$) (Right). The results show that Depth-$\mu$P preserves the optimal learning rate while consistently improving the training loss as depth increases. If we only scale the blocks without scaling the LR ($\alpha=1/2,\gamma=0$) when training with Adam, the optimal learning rate shifts significantly with depth. Figure 13: Train logloss versus block multiplier $a$ for varying depths. Same training setup as in fig. 10. The results suggest that Depth-$\mu$P stabilizes the hyperparameter $a$ as depth increases. Because transformers have block depth 2, as discussed in section 9, we have plenty of reasons to suspect that no parametrization of (learning rate, block multiplier) will be able to robustly transfer hyperparameters across depth for transformers. Here we do a large scale experiment using Megatron trained on Common Crawl and catalogue our observations.27 In summary, in our particular setup (which should be close to most large language model pretraining), we see that the $1/\sqrt{L}$ scaling seems to transfer hyperparameters at the end of training (Figure 16(Right)). However, we also see that 1) deeper does worse in initial training (Figure 15(Left)), and 2) optimal hyperparameters scale like $\Theta(1)$ in the middle of training (Figure 16(Left)). Combined with the theoretical insights of Section 9, this leads us to conclude that while the $1/\sqrt{L}$ scaling can potentially be practically useful in transformer training, it is likely to be brittle to architectural and algorithmic changes, or even simple things like training time. Footnote 27: We train the models for 3900 steps, using cosine decay schedule with 500 warmup steps. We use a sequence length of 4096, batch size 256, resulting in approximately 4B tokens per training run. In fact, we observe that transformers are insensitive to the block multiplier $a$ (Figure 14), so that the only relevant hyperparameter is really just learning rate. Thus, empirically measuring the scaling trend of the optimal learning rate, as done in modern large scale pretraining, can be a practically more robust way to transfer hyperparameters. Here $L$ is the number of transformer layers, each of which consists of an attention layer and an MLP layer (each of which has depth 2). ### Feature Diversity In this section, we empirically verify our claims about feature diversity exponent (Claims 7.4 and 7.5). We use the same setup as in the last section, i.e., we train deep residual networks of width $n=256$ on CIFAR-10 dataset with Adam and batch size $64$. In Figure 17, we compare two parametrizations, Depth-$\mu$P ($\alpha=\gamma=1/2$) and the ODE parametrization $(\alpha,\gamma)=(1,0)$ Figure 16: In the middle of (Megatron) transformer training, optimal learning rate is approximately invariant (Left), while at the end of training, it approximately scales like $1/\sqrt{L}$. However, the $1/\sqrt{L}$ scaling transfers the maximum viable learning rate better in either case. Figure 15: In (Megatron) Transformer trained on Common Crawl, deeper does worse initially (Left) but eventually does better (Right). We measure $\left\\|\mathbf{x}_{t}^{\left\lfloor(\lambda+\epsilon)L\right\rfloor}-\mathbf{x}_{t}^{ \left\lfloor\lambda L\right\rfloor}\right\\|\stackrel{{\text{\tiny def }}}{{=}}d(\epsilon)$ at $t=1000$ for the two parametrizations and varying depth. For each parametrization and depth $L$, we rescale function $d$ by multiplying a constant $c$ such that $c\cdot d(1/256)=1$, and then plot the rescaled function $c\cdot d$ for a clean presentation. One can observe clearly that Depth-$\mu$P has feature diversity exponent (almost) $1/2$ for any $L$, while the curves for ODE parametrization move from $\epsilon^{1/2}$ to $\epsilon$ when $L$ grows. This exactly fits our theory that Depth-$\mu$P maximizes the feature diversity, while other parametrizations (even with feature learning) have smaller feature diversity exponents that should go to $0$ in the infinite depth limit. Growth along with $L$ and $t$.In Figure 18, we measure $d(\epsilon)$ at $t=100,500,1000$, and rescale it by _dividing additional_$\epsilon^{0.5}$ and a constant $c$ such that $\frac{d(1/256)}{c\cdot\epsilon^{0.5}}=1$, and then plot the rescaled function $d/(c\cdot\epsilon^{0.5})$ for a clean comparison between $d$ and $\epsilon^{0.5}$. We observe that for both Depth-$\mu$P and ODE parametrization, the slopes of the curves grow along with $L$ and $t$. The growth along $t$ can be explained by the cumulative correlation between layers. The growth along $L$ for ODE parametrization is because the independent components between nearby layers decrease when $L$ grows. We do not have a clear understanding for the growth along $L$ for Depth-$\mu$P and we leave it as a future work. Absolute value activation increases feature diversity.In Figure 19, we plot the same curves as in Figure 18 but comparing ReLU activation and absolute value activation under Depth-$\mu$P. We observe that the slope of the curves for absolute value activation is smaller than ReLU activation. It matches our theory that absolute value activation increases feature diversity. ## Acknowledgement We thank Huishuai Zhang, Jeremy Bernstein, Edward Hu, Michael Santacroce, Lucas Liu for their helpful comments and discussion. D. Yu was supported by NSF and ONR. Part of this work was done during D. Yu's internship at Microsoft. ## Author Contributions GY developed the core theory and ran experiments in early part of the exploratory stage and most experiments in the final draft. DY worked on and proved key claims for linear resnets (including the limiting equations, convergence, and classification of parametrization), drafted the very first version of the paper, and ran experiments verifying the theoretical claims (including the convergence of Figure 17: Difference between feature at layer $\left\lfloor\lambda L\right\rfloor$ and feature at layer $\left\lfloor(\lambda+\epsilon)L\right\rfloor$ as a curve of $\epsilon$ for width $n=256$ and varying depths. For a clean presentation, each curve is scaled by a constant so it always passes $(1/256,1)$. The feature diversity exponent $\kappa$ depends on the growth of the curve when $L\to\infty$. For Depth-$\mu$P (left), the curve is always close to $\epsilon^{1/2}$, meaning $\kappa=1/2$. For ODE parametrization (right), the curve shifts from $\epsilon^{1/2}$ to $\epsilon$ when $L$ grows, indicating its $\kappa$ goes to $0$ in the infinite depth limit. linear case and feature diversity separation). CZ ran experiments in later part of the exploratory stage. They revealed the viability of Depth-$\mu$P in the block depth 1 case, in contrast to the general block depth case. CZ also ran the Megatron experiments in the final version of the paper. SH contributed to brainstorming since the beginning of the project, wrote the warm-up section on linear networks, formalized the notion of feature diversity exponent, and helped transforming experimental results into plots and visualizations. Figure 19: Same setup as Figure 18, but comparing Depth-$\mu$P with ReLU activation and absolute value activation. Each curve is scaled by dividing a constant and so it always passes $(1/256,1)$. The curve indicating feature diversity exponent $\kappa$ exactly $1/2$ should be a horizontal line at $1$. For both activations, slopes of curves are small, but growing along with $L$ and $t$. The slopes with absolute value activation ($\phi=\mathrm{Abs}$) are slower than the slopes with ReLU activation ($\phi=\mathrm{ReLU}$), indicating feature diversity is higher with absolute value activation. Figure 18: Same setup as Figure 17 but at step $t=100,500,1000$, and each curve is scaled by dividing a constant and _additional_$\epsilon^{1/2}$ so it always passes $(1/256,1)$. The curve indicating feature diversity exponent $\kappa$ exactly $1/2$ should be a horizontal line at $1$. For Depth-$\mu$P ($\alpha=0.5$), the curves are almost horizontal. For ODE parametrization ($\alpha=1$), slopes of the curves are larger with larger $L$ and larger $t$.
2308.01261	Surrogate modelling and uncertainty quantification based on multi-fidelity deep neural network	To reduce training costs, several Deep neural networks (DNNs) that can learn from a small set of HF data and a sufficient number of low-fidelity (LF) data have been proposed. In these established neural networks, a parallel structure is commonly proposed to separately approximate the non-linear and linear correlation between the HF- and LF data. In this paper, a new architecture of multi-fidelity deep neural network (MF-DNN) was proposed where one subnetwork was built to approximate both the non-linear and linear correlation simultaneously. Rather than manually allocating the output weights for the paralleled linear and nonlinear correction networks, the proposed MF-DNN can autonomously learn arbitrary correlation. The prediction accuracy of the proposed MF-DNN was firstly demonstrated by approximating the 1-, 32- and 100-dimensional benchmark functions with either the linear or non-linear correlation. The surrogating modelling results revealed that MF-DNN exhibited excellent approximation capabilities for the test functions. Subsequently, the MF DNN was deployed to simulate the 1-, 32- and 100-dimensional aleatory uncertainty propagation progress with the influence of either the uniform or Gaussian distributions of input uncertainties. The uncertainty quantification (UQ) results validated that the MF-DNN efficiently predicted the probability density distributions of quantities of interest (QoI) as well as the statistical moments without significant compromise of accuracy. MF-DNN was also deployed to model the physical flow of turbine vane LS89. The distributions of isentropic Mach number were well-predicted by MF-DNN based on the 2D Euler flow field and few experimental measurement data points. The proposed MF-DNN should be promising in solving UQ and robust optimization problems in practical engineering applications with multi-fidelity data sources.	Zhihui Li, Francesco Montomoli	2023-08-02T16:30:05Z	http://arxiv.org/abs/2308.01261v1	# Surrogate modelling and uncertainty quantification based on multi-fidelity deep neural network ###### Abstract Deep neural network (DNN) generally needs a high number of high-fidelity (HF) training data to ensure its prediction accuracy. However, in practical engineering applications, the availability of HF data is computationally or experimentally limited, mainly for the associated costs. To reduce training costs, several DNNs that can learn from a small set of HF data and a sufficient number of low-fidelity (LF) data have been proposed. In these established neural networks, a parallel structure is commonly proposed to separately approximate the non-linear and linear correlation between the HF- and LF data. In this paper, a new architecture of multi-fidelity deep neural network (MF-DNN) was proposed where one sub-network was built to approximate both the non-linear and linear correlation simultaneously. Rather than manually allocating the output weights for the paralleled linear and nonlinear correction networks, the proposed MF-DNN can autonomously learn arbitrary correlation. The prediction accuracy of the proposed MF-DNN was firstly demonstrated by approximating the 1-, 32- and 100-dimensional benchmark functions with either the linear or non-linear correlation. The surrogating modelling results revealed that MF-DNN exhibited excellent approximation capabilities for the test functions. Subsequently, the MF-DNN was deployed to simulate the 1-, 32- and 100-dimensional aleatory uncertainty propagation progress with the influence of either the uniform or Gaussian distributions of input uncertainties. The uncertainty quantification (UQ) results validated that the MF-DNN efficiently predicted the probability density distributions of quantities of interest (QoI) as well as the statistical moments without significant compromise of accuracy. MF-DNN was also deployed to model the physical flow of turbine vane LS89. The distributions of isentropic Mach number were well-predicted by MF-DNN based on the 2D Euler flow field and few experimental measurement data points. The proposed MF-DNN should be promising in solving UQ and robust optimization problems in practical engineering applications with multi-fidelity data sources. Multi-fidelity, Deep neural network, Surrogating modelling, Uncertainty quantification, High dimension, Turbine flow. ## 1 Introduction Deep learning technique has been applied recently to a wide range of applications to model physical [1-2], communications [3] and biological systems [4]. Generally, deep learning algorithms show an obvious data-hungry character in the data-training process, which limits its further practical applications. For example, the traditional fully connected deep neural network (DNN) needs much high-fidelity (HF) training data to achieve satisfactory prediction accuracy. However, in engineering applications, generating enough HF results often means computationally or experimentally prohibitive costs. The actual situation is that the HF data is inadequate in training the networks and the prediction accuracy is degraded accordingly. This problem also becomes crucial in the field of high-dimensional uncertainty quantification (UQ) where the input uncertainty effects on the reference system outputs need to be described and quantified. For instance, to solve the 32-dimensional UQ problem using traditional polynomial chaos expansion method, with a polynomial order of 5, around 435,897 HF samples are needed, which is computationally expensive and unaffordable for engineers. To reduce such dependence on HF results, multi-fidelity (MF) approaches have attracted much more attention recently in surrogate modelling and UQ, leveraging a set of multi-fidelity HF and LF data sets [5-8]. Given that LF data is abundant and cheap to obtain, the fundamental idea of MF methodology is to ensure the model's prediction accuracy by increasing the amount of LF data [9]. As MF surrogate modelling, different approaches have been proposed in literature, including MF response surface models [10-11], MF artificial neural networks [12-13] and MF Gaussian process [14-18]. As for the MF UQ approaches, MF polynomial chaos expansion [19-20] and MF Monte Carlo method [21-22] were also proposed in previous research works. However, the abovementioned approaches generally show weak performance in constructing the cross-correlation between HF- and LF data. For example, based on our experience the selection of HF point locations influences significantly the accuracy of the prediction of the MF response surface model. Besides that, MF Gaussian process regression shows some difficulties in optimizing the hyperparameters with sparse data [23] and solving the high-dimensional problems [24]. Besides, the polynomial chaos expansion method strongly depends on the regularity between the Quantities of Interest (QoI) and the input uncertainties, and its performance might be degraded in the absence of this kind of regularity, e.g., the stochastic hyperbolic problems [25]. A comprehensive review of multi-fidelity methods in uncertainty quantification, statistical inference and optimization can be found in [26]. More recently MF neural networks become promising in building surrogate models that may be used for both UQ and optimization problems. Compared to traditional function approximators, the DNN is regarded as a universal approximator at either low- or high dimensions [27]. Lu and Zhu [28] used a single fully connected neural network (NN) as a bi-fidelity surrogate model to estimate the high-fidelity proper orthogonal decomposition coefficients within the framework of the reduced-order model. Yan and Zhou [29] built an adaptive surrogate model based on the MF approach in conjunction with NN to solve the Bayesian inverse problem. The NN was used to bridge the low- and high-fidelity surrogate model in the framework of Markov chain Monte Carlo Conti et al. [30] established the MF long short-term memory networks to solve the time-dependent problems. Motamed [31] constructed two separate neural networks based on bi-fidelity training data. In detail, the first neural network was specially designed to approximate the correction function between LF- and HF data, and the output of the first neural network was then set to be the extra HF training data for the second neural network. Here the precision of these artificially added HF training data significantly determines the prediction accuracy of the whole neural network. Meng and Karniadakis [32] built the composite neural network in a more straightforward way that the LF model was trained first and the output of the LF model was then corrected based on the HF data. The proposed multi-fidelity deep neural network (MF-DNN) consists of three sub-networks, one network for LF prediction, one network for HF prediction based on the linear assumption between the LF- and HF data, and one network for HF prediction based on the non-linear assumption between them. The results showed that the composite neural network could give good accuracy in approximating several benchmark functions. The applications of the abovementioned MF-DNN (i.e., the architecture with three sub-networks built in [32]) in aerodynamic optimization can be seen in [33]. Ahn et al. [34] utilized the reduced-order model to update the low-fidelity data within the framework of composite neural networks. Guo et al. [12] conducted the MF regression with the neural networks featured by different architectures and then compared the prediction accuracy against the co-kriging model. In Guo's work, one shallow neural network layer was built to approximate the correlation between LF- and HF data. However, the results showed that its approximation capability in modelling linear correction function was somewhat degraded when compared to the parallel sub-networks structure. Besides, its performance in predicting high-dimensional problems has not been revealed. To avoid the structure of the parallel sub-networks while simultaneously balancing the approximation capability in modelling both linear and nonlinear correction, in this paper, we proposed to build a new network architecture featured with only one correction sub-network. The following attractive features can be expected: 1) the prior assumption in allocating the weights for the linear and nonlinear sub-networks is avoided; 2) this architecture closely aligns with the underlying mathematical principles of MF methodology on which it is based; 3) the training and testing process of the MF-DNN becomes straightforward because of the simplified architecture, especially for the high-dimensional problems; 4) the new MF-DNN is much easier to be implemented in programming. As far as the authors know, this is the first attempt to solve the high-dimensional aleatory UQ problem using MF-DNN. The paper is organized as follows: the details of the new MF-DNN were detailed in Sec. 2; the test results of the surrogate modelling and UQ for several benchmark cases were then shown in Sec. 3. The main conclusions of this research work were drawn in Sec. 4. In addition, the comparison of the different activation functions in approximation capability was added in Appendix A. ## 2 Multi-fidelity Deep Neural Network One widely used comprehensive correction [9] in bridging LF- and HF data is: \[\hat{y}_{HF}=\rho(\mathrm{X})\cdot\gamma_{LF}(\mathrm{X})+\delta(\mathrm{X}) \tag{1}\] where $\hat{y}_{HF}$ represents the model prediction values on HF data points, $\rho$ is the multiplicative correction surrogate, $\gamma_{LF}$ represents the label values of LF data points, $\delta$ means the additive correction surrogate. Here the multiplicative correction $\rho$ could be either a constant [35] or the non-constant value [36], which represents the linear or non-linear correction between $\hat{y}_{HF}$ and $\gamma_{LF}$. In other words, the comprehensive correction in (1) can be expressed as: \[\hat{y}_{HF}=\ \mathcal{F}(\gamma_{LF}(\mathrm{X}),\mathrm{X}) \tag{2}\] where $\mathcal{F}$ can comprehensively represent both the non-linear and linear correlation between the LF- and HF data. Thus, the idea for the proposed MF-DNN is that it should consist of two sub-networks, one for the LF-DNN to approximate the values of $\gamma_{LF}$, and one for the Correction DNN to predict the $\hat{y}_{HF}$ based on the expression (2). The architecture of the new MF-DNN is shown in Fig. 1. Here we assume that there is a large set of LF training data $y_{LF}(\mathrm{X}_{LF}),\mathrm{X}_{LF}\in\mathbb{R}$ and a relatively small set of HF training data $y_{HF}(\mathrm{X}_{HF}),\ \mathrm{X}_{HF}\in\mathbb{R}$, $\mathrm{X}_{HF}\subset\mathrm{X}_{LF}$. The cost function was set to be the mean square error (MSE) between the prediction results and the actual values. The gradient information of the cost function to the network parameters could be obtained based on automatic differentiation. The prediction error of the MF-DNN was minimized by using gradient-based optimization algorithms. In detail, the ADAM and L-BFGS optimizers [37-38] were deployed here because of their better generalizing performance when compared to the other optimization methods [39-40]. To avoid overfitting, the $L_{2}$ regularization loss (also called Ridge regression [41]) was added to minimize the loss function by summing the squared magnitude of network weight coefficients. Here the LF-DNN and Correction DNN were trained in sequence, which is beneficial for the programming and training. The definition of the loss function for the LF-DNN is shown as follows: \[L_{l}=\ \frac{1}{M}\sum_{l=1}^{M}\left(\widehat{\gamma_{LF}^{-}}(X_{LF};\theta_{ NN})-\gamma_{LF}(X_{LF})\right)^{2} \tag{3}\] \[\theta_{NN}=\{(w_{h},b_{h})\}_{h=1}^{H+1} \tag{4}\] where $L_{l}$ represents the training loss of LF-DNN, $M$ means the number of LF training points, $\theta_{NN}$ means the set of neural network parameters, $\mathbf{w_{h}}$ is the weights of $h$-th layer, $b_{h}$ is the biases of $h$-th layer and $H$ means the number of hidden layers. The trained LF-DNN then operated as an offline surrogate model. This implies that it remained fixed and was not retrained during the training process of the Correction DNN. Furthermore, the definition of loss function for the Correction DNN is shown as follows: \[L_{c}=\ L_{h}+\ L_{r} \tag{5}\] \[L_{h}=\ \frac{1}{p}\sum_{i=1}^{p}\left(\widehat{\gamma_{HF}^{-}}(X_{HF}; \theta_{NN})-\gamma_{HF}(X_{HF})\right)^{2} \tag{6}\] \[L_{r}=\ \lambda\sum w_{h}^{2} \tag{7}\] where $L_{c}$ represents the overall training loss of Correction DNN, $L_{h}$ is the mean square error of Correction DNN, $L_{r}$ represents the $L_{2}$ regularization loss of Correction DNN, $P$ is the number of HF training points and $\lambda$ is the control parameter of $L_{2}$ regularization loss. In order to enable the Correction DNN to effectively learn the disparities between the predictions generated by the LF-DNN and the corresponding HF labels, the HF variables $X_{HF}$ were also incorporated during the training phase of the Correction DNN. By including these HF variables in the training process, the Correction DNN gained the ability to comprehend and capture the differences between the LF-DNN predictions and the actual HF labels, facilitating its ability to correct and refine the predictions accordingly. The network hyperparameters, i.e., the numbers of the hidden layers, the number of neurons in hidden layers, the learning rate, etc., were optimized by the Bayesian optimization algorithm [42]. The Bayesian optimization method was chosen because it is generally regarded as a global optimization method in presence of noisy black box problems, and generally shows better performance in balancing exploration and exploitation when compared to the grid search approach [43] and random search method Fig. 1: Architecture of the proposed MF-DNN (taking two-layer as an example) [44]. The performance of the optimized MF-DNN was then evaluated using the K-fold cross-validation method [45]. Rather than using the commonly used hyperbolic tangent function (also known as "Tanh") in existing MF-DNNs [12; 32], the rectified linear unit activation function (also known as "ReLU") was deployed here to obtain the balance of approximating both the linear and non-linear correlation in one sub-network. The mathematical expression of "ReLU" is as follows: \[f(x)=\max\left(0,x\right) \tag{8}\] The "ReLU" function is beneficial for the faster and more effective training of DNNs when compared to the traditional logistic sigmoid function (also known as "Sigmoid") and the "Tanh", especially for the high-dimensional dataset [46]. More tests on the "ReLU" capability of the function approximation were shown in Appendix A at the end of this paper. The building and training of the MF-DNN were carried out using the _Keras_ in the environment of _Tensorflow 2_[47]. The detailed training procedures for the MF-DNN are as follows: 1. Building the initial fully connected LF-DNN; 2. Optimizing the hyperparameters in LF-DNN using the Bayesian optimization method; 3. Training the LF-DNN based on the $M$ realizations of the LF data set $Q_{LF}=\{\gamma_{LF}{}^{(1)},...,\gamma_{LF}{}^{(M)}\}$ using the ADAM and L-BFGS optimizers; 4. Building the initial Correction DNN and calling the trained LF-DNN in 3) as a sub-module; 5. Optimizing the hyperparameters in the Correction DNN using the Bayesian optimization method; 6. Training the Correction DNN based on the HF data set $Q_{HF}=\{y_{HF}{}^{(1)},...,y_{HF}{}^{(N)}\}$ ($M\gg N$) using the ADAM and L-BFGS optimizers; 7. Validating and testing the accuracy of the trained MF-DNN based on the K-fold cross-validation method. ## 3 Results and discussion In this section, several benchmark tests were conducted to verify the performance of the built MF-DNN in surrogate modelling and uncertainty quantification. ### 1-dimensional function with linear correlation Here one-dimensional function featured by the linear correlation between the LF- and HF data was tested first. The theoretical expressions of this function are as follows: \[y_{L}(x)=0.5y_{H}+10(x-0.5)-5 \tag{9}\] \[y_{H}(x)=(6x-2)^{2}\sin(12x-4) \tag{10}\] To approximate both the LF- and HF functions, here 21 LF points (uniform distributions in interval X${}_{LF}\in$[0,1]) and 4 HF points (X${}_{HF}\in$[0, 0.35, 0.75, 1]) were generated and then collected as the training data for the MF-DNN. Before the training data are transferred to the MF-DNN, the learning rate, the number of hidden layers and the number of neurons in each hidden layer were tuned using Bayesian optimization. The variation range for the hidden layer number is [1; 2; 3; 4], the variation range for the neuron number in each hidden layer is [8; 16; 24; 32; 40; 48; 56; 64], and that for the learning rate is [0.01, 0.001, 0.0001], respectively. The optimization objective is to minimize the prediction loss of the MF-DNN. The training epoch number was set to be 2000. After 20 Bayesian optimization loops, the optimal architecture of the LF-DNN was converged to be 3 hidden layers featured by 64, 64 and 40 neurons in each hidden layer, respectively. The optimal architecture of the Correction DNN was featured by 1 hidden layer with 8 neurons on it. The optimal learning rate was set to be 0.001. The weights and biases of the MF-DNN were initially updated using ADAM optimizer for the first 1000 steps. After that, the L-BFGS optimization algorithm was employed to further minimize the training loss over the next 2000 steps. The performance of the well-trained MF-DNN is shown in Fig. 2. In detail, Fig 2 (a) shows that the MF-DNN can approximate the HF function accurately based on 4 HF data points. This also validates that the built Correction DNN with the "ReLU" activation function can well approximate the linear correlation between the LF- and HF data. Furthermore, the prediction results obtained from the proposed MF-DNN (MF_DNN_ReLU) are compared with other surrogate models, namely the Radial Basis Function (RBF), Kriging (KRG), MF-DNN with composite architecture (MF_DNN_Comp, proposed in Ref. [32]), and Co-Kriging (Co-KRG). Here the RBF and KRG models were trained based on the HF data points, and the comparison results were added in Fig. 2 (b). The MSE of these models on 20 validation data points is collected in Tab. 1. Among the models under consideration, the performance of the proposed MF-DNN outperforms that of the other surrogate models, except for the co-kriging model. This finding is not entirely surprising, as Co-Kriging was originally designed based on low-dimensional linear assumptions. This demonstrates the effectiveness of neural networks with the ReLU activation function in approximating the linear correlation between LF- and HF data. \[y_{h}(x)=0.1y_{L}(x)^{2}+10 \tag{12}\] To approximate both the LF- and HF functions, here 21 LF points (uniform distributions in interval X${}_{LF}\in[0,1]$) and 6 HF points (X${}_{HF}\in[0,0.1,0.3,0.7,0.9,1]$) were generated and then collected as the training dataset for MF-DNN. The activation function "ReLU" was deployed here to test its capability in approximating the non-linear correlation between the LF- and HF functions. Similar to the training process shown in Sec. 3.1, the hyperparameters of the neural networks were optimized by the Bayesian optimization method. The training epoch number was set to be 2000. After 20 optimization loops, the optimal architecture of the LF-DNN was featured by the 3 hidden layers with 64, 64 and 40 neurons in each hidden layer, respectively. Meanwhile, the optimal architecture of the Correction DNN was featured by 2 hidden layers with 64 and 56 neurones in each hidden layer, respectively. The weights and biases of the MF-DNN were initially updated using the ADAM optimizer for the initial 1000 steps. Subsequently, the L-BFGS optimization algorithm was utilized to further minimize the training loss during the subsequent 2000 steps. As shown in Fig. 3 (a), the MF-DNN accurately approximates the HF function based on 6 HF data points. The performance of the MF-DNN was then compared to the RBF, KRG, MF_DNN_Comp and Co-KRG models, as shown in Fig. 3(b). In direct comparison with other models, the MF-DNN exhibits closed distributions that closely resemble the HF function. The MSE of these models in predicting 20 validation data points was collected in Tab. 2. The results indicate that the proposed MF-DNN achieves a prediction accuracy that is comparable to the composite structure used in Ref. [32]. This highlights the capability of neural networks with the ReLU activation function to effectively approximate the non-linear correlation between the LF- and HF data. ### 32-dimensional function Fig. 3: Performance of MF-DNNs in approximating 1-dimension function with nonlinear correlation (a: Comparison between the MF-DNN prediction results and analytical values; b: Comparison of MF-DNN with the other surrogate models) In many industrial design scenarios, having 32 parameters represents a relatively high-dimensional space. Thus, the capability of the proposed MF-DNN in approximating the 32-dimensional function was tested here. The theoretical expressions of this function are as follows: \[y_{L}(x_{0},...\,x_{31})=\ 0.8y_{H}-0.4\sum_{i=0}^{30}(x_{i}x_{i+1})-50,x _{i}\in[-3,3] \tag{13}\] \[y_{H}(x_{0},...\,x_{31})=(x_{0}-1)^{2}+\sum_{i=1}^{31}(2x_{i}{}^{2}-x_{i-1})^{2},x_{i}\in[-3,3] \tag{14}\] To approximate the high-dimensional function, 200,000 LF- and 2000 HF data points were generated using LHS method. The training epoch number is set to be 5000. After 20 optimization loops, the Bayesian optimization results showed that the LF-DNN consisted of 2 hidden layers with 512 neurons and 256 neurons on each layer, respectively. The Correction-DNN consisted of 1 hidden layer with 32 neurons on it. The activation function in Correction-DNN was set to ReLU, and the learning rate was set to 0.001. The training process of the MF-DNN was boosted by using the _NVIDIA TESLA K80_ Graphics Processing Unit (GPU) in _Google Colaboratory_. Initially, the weights and biases of the MF-DNN were updated using the ADAM optimizer for the first 1000 steps. Following that, the L-BFGS optimization algorithm was employed to further reduce the training loss over the next 2000 steps. Fig. 4 shows the comparison of the MF-DNN prediction results and the analytical solutions. As stated earlier, the $x$ and $y$ coordinates of the red scatter points represent the MF-DNN predictions and the corresponding analytical solutions, respectively. Ideally, if the MF-DNN prediction error is zero, the red scatter points should perfectly align with the black line. The blue points are derived from the KRG model which is only trained by the HF data. Its ability to accurately capture the underlying patterns is evident from the alignment of the red scatter points with the analytical solutions, demonstrating its effectiveness in solving high-dimensional problems. Table 3 displays the MSE of various models on a validation dataset consisting of 1000 data points. The findings demonstrate that the proposed MF-DNN model achieves a comparable level of accuracy with the composite architecture, surpassing the performance of the other evaluated models. It should be noted that the Co-KRG model fails to provide results for this problem. This limitation arises due to the considerable size of the covariance matrix ([20000, 20000]), which requires a minimum memory capacity of 320 GB to be solved. \begin{table} \begin{tabular}{c c} \hline \hline Models* & MSE of test dataset \\ \hline MF-DNN\_ReLU & 9.83e-04 \\ \hline \hline \end{tabular} \end{table} Table 3: MSE for prediction of 32-dimensional function with non-linear correlation Figure 4: Performance of MF-DNN in approximating 32-dimensional function ### 100-dimensional function The MF-DNN proposed in this study is evaluated by testing its performance in predicting a 100-dimensional benchmark function. The test expression of the benchmark function can be represented as follows: \[y_{L}(x_{0},...\,x_{99})=\ 0.8y_{H}-0.4\sum_{i=0}^{99}(x_{i}x_{i+1})-50,x_{i} \in[-3,3] \tag{15}\] \[y_{H}(x_{0},...\,x_{99})=(x_{0}-1)^{2}+\sum_{i=1}^{99}(2{x_{i}}^{2}-x_{i-1})^{2},x_{i} \in[-3,3] \tag{16}\] The MF-DNN model was trained using a dataset comprising 10,000,000 LF data points and 100,000 HF data points. The LF-DNN is constructed with four layers, each containing 512, 512, 256, and 128 neurons, respectively. On the other hand, the HF-DNN consists of a single layer with 64 neurons. The ReLU activation function was used in both LF-DNN and HF-DNN, and the learning rate was set to 0.001. The entire training process was carried out in the _TensorFlow 2_ environment within _Google Colaboratory_. As depicted in Figure 5, the MF-DNN demonstrates its ability to predict the analytical solutions of the 100-dimensional function effectively. The performance of various models on a validation dataset comprising 1000 scaled data points is presented in Table 4, showcasing the MSE values. The results clearly indicate that the proposed MF-DNN achieves the highest accuracy among the previous models, validating its effectiveness in addressing high-dimensional surrogate modelling problems. The RBF, KRG and Co-KRG models encounter challenges during the training process due to their significant demand for system memory, leading to an inability to complete the training successfully. ### UQ for 1-dimensional function The influence of input aleatory uncertainties on the QoI can be evaluated based on the MF-DNN in conjunction with Monte Carlo sampling method. The expressions (9-10) were chosen as the governing equations of uncertainty propagation first, and the two representative types of input uncertainty distributions, i.e., the uniform distributions and Gaussian distributions were introduced within the UQ process. In detail, $x\in\Gamma$ is either a uniformly distributed random variable on $\Gamma=\text{U}\left[0.6,0.8\right]$ or a Gaussian distributed random variable on $\Gamma=\text{N}\left[0.7,0.03^{2}\right]$. The multidimensional truncated Gaussian method [49] was used to generate the Gaussian distributions within the specific bounded interval. The Monte Carlo method was selected here because of its simplicity and robustness of the implementation. The MF-DNN-based statistical moments of QoI under these two different input uncertainties were collected and compared to the analytical results, as shown in Tab. 5. MF-DNN accurately predicted the mean and variance of the QoI, and the maximal prediction error occurred in the kurtosis which was related to the tail distributions of the QoI. The histogram comparison of the QoI probability density distributions under different distributions of input uncertainties is shown in Fig. 6. The comparison results show that the QoI with the maximal probability appears near -6 for the input uncertainties with either the uniform- or the Gaussian distributions. However, the probability density of the QoI shows the aggregated distributions within the interval [-6, -4] when the input uncertainty distributions are modified from the uniform- to the Gaussian type. The overall tendency in this comparison shows that the MF-DNN can properly simulate the low-dimensional uncertainty propagation process. \begin{table} \begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{_Statistic Description- Input uncertainties with uniform distribution_} \\ \hline & mean & variance & skewness & kurtosis \\ \hline Analytical results & -3.926 & 3.706 & 0.549 & -1.165 \\ \hline MF-DNNs prediction & -3.929 & 3.723 & 0.551 & -1.165 \\ (1e+07 samples) & & & & \\ \hline Prediction error (\%) & 0.076 & 0.462 & 0.542 & 0.012 \\ \hline \multicolumn{5}{c}{_Statistic Description- Input uncertainties with Gaussian distribution_} \\ \hline & mean & variance & skewness & kurtosis \\ \hline Analytical results & -4.426 & 1.344 & 0.691 & -0.087 \\ \hline MF-DNNs prediction & -4.423 & 1.356 & 0.680 & -0.090 \\ (1e+07 samples) & & & & \\ \hline Prediction error (\%) & 0.068 & 0.884 & 1.592 & 3.448 \\ \hline \hline \end{tabular} \end{table} Table 5: Comparison of statistical moments of QoI for 1-dimensional function ### UQ for 32-dimensional function To further test the MF-DNN capability in handling the high-dimensional UQ problem, the 32-dimensional benchmark test function (the mathematical expressions shown in equations 13-14) was chosen here as the governing equation of uncertainty propagation. The input uncertainty $x\in\Gamma$ is either a 32-dimensional uniformly distributed random variable on $\Gamma=\text{U}\left[-3,3\right]$ or a 32-dimensional Gaussian distributed random variable on $\Gamma=N\left(0,1^{2}\right).$ The multidimensional truncated Gaussian method was used to generate the normal distributions on 32 dimensions. The Monte Carlo method was deployed to simulate the uncertainty propagation process, and the MF-DNN-based statistical moments of QoI with two types of input uncertainties were collected and compared to the analytical results, as shown in Tab. 6. \begin{table} \begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{_Statistic description- Uniform distribution_} \\ \hline & mean & variance & skewness & kurtosis \\ \hline Analytical results & 2.106 & 0.269 & 0.313 & 0.103 \\ \hline MF-DNNs predictions & & & & \\ (1e+7 samples) & 2.105 & 0.266 & 0.300 & 0.080 \\ \hline Prediction error (\%) & 0.022 & 0.255 & 1.344 & 14.483 \\ \hline \hline \end{tabular} \end{table} Table 6: Comparison of statistical moments of QoI for 32-dimensional function Figure 6: Histogram comparison of QoI probability density distributions for 1-dimensional function Although there are some errors in the kurtosis predictions, the MF-DNN shows high accuracy in predicting the mean and variance of the QoI responses. The histogram comparison of the QoI probability density distributions for the 32-dimensional function is shown in Fig. 7. The probability density distributions of the QoI are "biased" leftward when the input uncertainties vary from the uniform- to the Gaussian distribution. The overall tendency shows that the MF-DNN can qualitatively and quantitatively simulate the uncertainty propagation process for the high-dimensional UQ problem. ### UQ for 100-dimensional function The MF-DNN model was utilized to address the 100-dimensional UQ problem. The input uncertainty, denoted as $x\in\Gamma$, can take two different forms: either a 100-dimensional uniformly distributed random variable on the interval $\Gamma=\mathrm{U}\left[-1,1\right]$, or a 100-dimensional Gaussian distributed random variable on the interval $\Gamma=N\left(0,1^{2}\right)$. For both types of input uncertainties, the MF-DNN was employed to compute the statistical moments of the QoI. These computed moments were then compared Figure 7: Histogram comparison of QoI probability density distributions for 32-dimensional function to the corresponding analytical results, and the comparison outcomes are summarized in Table 7. Furthermore, the probability density distributions of the QoI were obtained using the MF-DNN, and the histogram comparison of these distributions is depicted in Fig. 8. The MF-DNN demonstrates high accuracy in predicting the mean value, while there are relatively larger discrepancies observed in the prediction of variance, skewness, and kurtosis when compared to the analytical results. Despite these discrepancies, the probability distributions derived from the MF-DNN align well with the analytical solutions, indicating a good match in capturing the overall shape and characteristics of the data distribution. \begin{table} \begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{_Statistic description- Uniform distribution_} \\ \hline & mean & variance & skewness & kurtosis \\ \hline Analytical results & 6.716 & 0.861 & 0.176 & 0.029 \\ \hline MF-DNNs predictions (1e+7 samples) & 6.723 & 0.723 & 0.159 & 0.033 \\ \hline Prediction error (\%) & 0.104 & 16.028 & 9.659 & 12.121 \\ \hline \multicolumn{5}{c}{_Statistic description- Gaussian distribution_} \\ \hline & mean & variance & skewness & kurtosis \\ \hline Analytical results & 1.159 & 0.094 & 0.521 & 0.358 \\ \hline MF-DNNs predictions (1e+7 samples) & 1.150 & 0.072 & 0.406 & 0.209 \\ \hline Prediction error (\%) & 0.776 & 23.400 & 22.073 & 71.291 \\ \hline \hline \end{tabular} \end{table} Table 7: Comparison of statistical moments of QoI for 100-dimensional function ### Application in predicting LS89 flow The MF-DNN was then employed to predict the flow field of turbine value LS89. The LS89 nozzle guide vane [50] was designed and tested at the von Karman Institute for Fluid Dynamics with well-documented experiment measurements. The key geometric and operating parameters of the LS89 blade are defined in Tab. 8. To predict the distributions of the near-wall isentropic Mach number on the vane surface, two levels of training data were deployed here: abundant 2D Euler flow as the LF data source; few experimental measurement results as the HF data source. The corresponding governing equations for these two fidelities of flow field (convective form without energy equation) are shown below: \[Euler\ flow\ \ developed by Drela and Youngren from MIT [51]. The mesh was generated by MISES/ISET and $y^{+}$ was set near 1 with around 3,180 cells generated in the LF simulation. It should be noted that the calculation of $\ y^{+}$ in this context is solely used for determining the spacing of the first layer of the mesh. In general, the solution of the Euler equations is not significantly affected by the grid density. The detailed 2D mesh inside the LS89 passage and the contour of Mach number are shown in Fig. 9. The Euler flow result shows that there are two shock waves generated inside the vane passage, the front one due to the choking effect in throat area; the rear one near the trailing edge because of the flow acceleration on suction surface. However, the MISES generally specifies the trailing edge flow based on the Kutta condition instead of explicitly solving the flow physics [51]. As a result, the Euler flow results do not predict proper base pressure and losses in the trailing edge area [52, 53, 54]. Thus, the logic of the MF-DNN is to capture the basic flow pattern from LF Euler results and then to locally correct the trailing edge flow field using HF experimental measurement data. Currently, the predictions primarily focus on the distributions of the isentropic Mach number by utilizing a combination of 2D Euler results and a limited number of experimental data points on the vane wall surfaces. It is crucial to acknowledge that this approach provides a partial prediction of the flowfield. To fully construct the complete flowfield using MF-DNNs, obtaining high-fidelity experimental data within the cascade passage is necessary. 160 LF points and 4 HF points were selected in total as the input training data for the MF-DNN. The near-wall flow fields on the pressure side and suction side of LS89 vane were modelled separately. The LF-DNN consisted of 5 hidden layers with 256, 128, 128, 64 and 32 neurons on each layer, respectively. The Correction DNN consisted of 4 hidden layers with 128, 64, 32 and 32 neurons on each layer, respectively. The comparison results are shown in Fig. 10. Instead of arranging pressure probes in the whole chordwise range of vane surface, the MF-DNN well-predicted the distributions of isentropic Mach number based on the 2D Euler flow field and 4 experimental measurement data points. The MF-DNN inherited the Euler flow pattern in the upstream zone and corrected the static pressure distributions near the shock wave occurring close to the trailing edge. Figure 9: Generated mesh and contour of Mach number inside turbine vane passages ## 4 Conclusions A new architecture of multi-fidelity deep neural network (MF-DNN) was proposed in this paper to learn the training data with different fidelities. The performance of the built MF-DNN was then deployed in several benchmark tests of surrogate modelling and uncertainty quantification (UQ) for both the low-dimensional and high-dimensional functions. The main conclusions in this research work can be drawn as follows: 1. In the proposed MF-DNN, a single correction network was constructed here to autonomously learn either the linear or nonlinear correlation. The surrogate modelling capability of MF-DNN was tested by approximating the 1-, 32-, and 100-dimensional benchmark functions. The proposed MF-DNN exhibits comparable performance to the previous composite architecture in handling the nonlinear relationship between LF- and HF-data, while it outperforms the composite architecture when dealing with linear relationship. 2. The proposed MF-DNN demonstrated superior performance when compared to the traditional non-neural network models, particularly in handling high-dimensional problems. The performance of MF-DNN was evaluated in solving aleatory UQ problems, both in 1-, 32-and 100-dimensional settings, under input uncertainties following uniform and Gaussian distributions. MF-DNN demonstrated its efficiency in accurately predicting probability density distributions and statistical moments of quantities of interest (QoI). 3. MF-DNN was deployed to predict the distributions of near-wall isentropic Mach number on turbine vane LS89 by leveraging the 2D Euler results and a limited number of experimental measurement data points. The results demonstrated that the near-wall flow field was well-predicted by MF-DNN. In details, MF-DNN inherited the Euler flow pattern in the upstream zone and corrected the static pressure distributions near the shock wave using experimental data. ## Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Figure 10: MF-DNN prediction of near-wall isentropic Mach number on LS89 vane surface Data availability* Data will be made available on request. Acknowledgement The authors would like to sincerely thank the support of the grant from the European Union's Marie Sklodowska-Curie Actions Individual Fellowship (MSCA-IF-MENTOR-101029472). Nomenclature DNN deep neural network GPU graphics processing unit HF high-fidelity KRG Kriging model LF low-fidelity MF multi-fidelity MF-DNN multi-fidelity deep neural network MISES multiple blade interacting streamtube Euler solver MSE mean square error NN neural network QoI quantities of interest RBF radial basis function UQ uncertainty quantification "ReLU" rectified linear unit activation function "Sigmoid" logistic sigmoid function "Tanh" hyperbolic tangent function $H$ number of hidden layers $M$ number of LF training points $N$ normal (Gaussian) distribution $P$ number of HF training points U uniform distribution $L_{c}$ overall mean square error of the Correction DNN $L_{h}$ training loss of Correction DNN $L_{l}$ training loss of LF-DNN $L_{r}$ $L_{2}$ regularization loss of Correction DNN $Q_{LF}$ low-fidelity data set $Q_{HF}$ high-fidelity data set $\chi_{LF}$ low-fidelity variables $\chi_{HF}$ high-fidelity variables $b_{h}$ biases of $h$-th layer $\hat{\mathcal{H}}_{HF}$ prediction values on HF data points $\mathcal{Y}_{LF}$ label values of LF data points $\theta_{NN}$ neural network parameters $w_{h}$ weights of $h$-th layer $\mathcal{F}$ correlation between LF- and HF data $\lambda$ control parameter of $L_{2}$ regularization loss ## Appendix A Comparison of activation functions in neural network The capability comparison of the different activation functions, i.e., the "ReLU", the "Sigmoid" and the "Tanh" in regression was conducted here. Theoretically, these three activation functions are non-linear based on the definition of non-linearity (i.e., the derivative of the dependent variable to the independent variable changes with the independent variable), and they have shown wide applications in approximating the non-linear functions. Their capability in approximating both the linear and nonlinear function was studied in this section. The simple linear/nonlinear unary functions are chosen as the governing equations as follows: \[y=2x,x\in[-1,1] \tag{1}\] \[y=2x^{2},x\in[-1,1]\] (2) \[y=2x^{3},x\in[-1,1]\] (3) \[y=2\text{sin }(4x),x\in[-1,1] \tag{4}\] Three neural networks (NNs) were established here, and each NN had a different activation function (i.e., "ReLU", "Sigmoid" and "Tanh", respectively). To exclude the influence of the other factors, the hyperparameters and initialization of the established NNs remained the same, and the only variable was the activation function within the training process. In detail, there were 21 training points in total that were generated uniformly distributed within the interval $x\in[-1,1]$. The architecture of the NNs consisted of 2 hidden layers with 20 neurons on each layer. The training epoch number was 1000 and the batch size for each epoch was 7. Figure 1 illustrates the comparison between the predictions obtained from NNs using different activation functions: "ReLU," "Tanh," and "Sigmoid." It is evident that both the "ReLU" and "Tanh" activations exhibit better agreement with the theoretical values when compared to the "Sigmoid" activation. To provide a quantitative assessment, Table 1 presents the MSE values on the validation dataset. Across a wide range of linear and nonlinear functions, the "ReLU" activation function consistently demonstrates superior prediction accuracy compared to the other activation functions. These results reinforce the notion that the proposed Correction DNN with the "ReLU" activation function can effectively approximate both linear and nonlinear correlations across a broad spectrum of scenarios.
2304.12876	Evaluation of Parameter-based Attacks against Embedded Neural Networks with Laser Injection	Upcoming certification actions related to the security of machine learning (ML) based systems raise major evaluation challenges that are amplified by the large-scale deployment of models in many hardware platforms. Until recently, most of research works focused on API-based attacks that consider a ML model as a pure algorithmic abstraction. However, new implementation-based threats have been revealed, emphasizing the urgency to propose both practical and simulation-based methods to properly evaluate the robustness of models. A major concern is parameter-based attacks (such as the Bit-Flip Attack, BFA) that highlight the lack of robustness of typical deep neural network models when confronted by accurate and optimal alterations of their internal parameters stored in memory. Setting in a security testing purpose, this work practically reports, for the first time, a successful variant of the BFA on a 32-bit Cortex-M microcontroller using laser fault injection. It is a standard fault injection means for security evaluation, that enables to inject spatially and temporally accurate faults. To avoid unrealistic brute-force strategies, we show how simulations help selecting the most sensitive set of bits from the parameters taking into account the laser fault model.	Mathieu Dumont, Kevin Hector, Pierre-Alain Moellic, Jean-Max Dutertre, Simon PontiÃ©	2023-04-25T14:48:58Z	http://arxiv.org/abs/2304.12876v2	# Evaluation of Parameter-based Attacks against Embedded Neural Networks with Laser Injection ###### Abstract Upcoming certification actions related to the security of machine learning (ML) based systems raise major evaluation challenges that are amplified by the large-scale deployment of models in many hardware platforms. Until recently, most of research works focused on API-based attacks that consider a ML model as a pure algorithmic abstraction. However, new implementation-based threats have been revealed, emphasizing the urgency to propose both practical _and_ simulation-based methods to properly evaluate the robustness of models. A major concern is parameter-based attacks (such as the Bit-Flip Attack - BFA) that highlight the lack of robustness of typical deep neural network models when confronted by accurate and optimal alterations of their internal parameters stored in memory. Setting in a security testing purpose, this work practically reports, for the first time, a successful variant of the BFA on a 32-bit Cortex-M microcontroller using laser fault injection. It is a standard fault injection means for security evaluation, that enables to inject spatially and temporally accurate faults. To avoid unrealistic brute-force strategies, we show how simulations help selecting the most sensitive set of bits from the parameters taking into account the laser fault model. Keywords:Hardware Security, Fault Injection, Evaluation and certification, Machine Learning, Neural Network ## 1 Introduction The massive deployment of Machine Learning (ML) models in a large spectrum of domains raises several security concerns related to their integrity, confidentiality and availability. For now, most of the research efforts are essentially focused on models seen as abstractions, i.e. the attack surface is focused on so-called _API-based attacks_, excluding threats related to their implementation in devices that may be physically accessible by an adversary, as it the case for embedded ML models. The flaws intrinsically related to the models and the _physical_ ones may be used jointly to attack an embedded model or exploit data leakages. A typical and well-known API-based attack are the _adversarial examples_[1] that aims at fooling the prediction of a model with input-based alterations at inference time. However, recent works demonstrated that physical attacks are realist threats by targeting critical elements of a model such as activation functions [2] or its parameters (parameter-based attacks) [3] as studied in this work. In such a security context, with the demonstration of worrying attack vectors, there is an urgent need of certification for AI-based systems more especially for _critical_ ones1. Therefore, alongside the demonstration of new attacks and the development of defenses, an important challenge relies on the availability of proper robustness evaluation and accurate characterization methods in addition to both simulation and experimental tools and protocols. Model robustness evaluation is one of the most important challenge of modern artificial intelligence and several remarkable works from the Adversarial Machine Learning community have already raised major issues for adversarial examples [4], with many defenses relying on weak evaluations [5]. Footnote 1: See the European AI Act: [https://artificialintelligenceact.eu/](https://artificialintelligenceact.eu/) Dealing with embedded neural network models and weight-based adversarial attacks, this challenge encompasses both _safety_ and _security_ concerns: parameters stored in memory may be altered by random faults because of hostile environments or strong energy consumption limitations [6] and also be the target of an adversary that aims at optimally threatening a model. ## 2 Related works and objectives ### Parameter-based attacks Implementation-based threats. An important part of the ML security literature concerns _algorithmic_ or so-called _API-based attacks_ that exploit input/ouput pairs and additional knowledge from the model in case of white-box attacks. A large body of work shows that these threats concern every stage of the ML pipeline [7] and threaten the confidentiality (model and data), integrity and availability of the models. However, these attacks do not consider that the adversary may have a direct interaction with the algorithm as it can be the case for an embedded AI system. _Implementation-based attacks_ precisely exploit software or hardware flaws as well as the specific features of the device. For example, _side-channel analysis_[8] have been demonstrated for model extraction as an efficient way to extract information from the model architecture or the values of the parameters [9]. Alongside safety-related efforts that evaluate the robustness of ML models against random faults [6], some works demonstrate that models are highly sensitive to _fault injection analysis_[10] that alter the data, the parameters as well as the instructions flow [2, 11]. Weight-based adversarial attacks. New attack vectors have been highlighted and more essentially parameter-based attacks (also named _weight-based adversarial attacks_). Let's consider a supervised neural network model $M_{W}(x)$, with parameters $W$ (also referred as _weights_), trained to optimally map an input space $\mathcal{X}=\mathbb{R}^{d}$ (e.g., images) to a set of labels $\mathcal{Y}$. $M$ is trained by minimizing a loss function $\mathcal{L}\big{(}M_{W}(x),y\big{)}$ (typically the cross entropy for classification task) that quantifies the error between the prediction $\hat{y}=M_{W}(x)$ and the correct label $y$. As formalized in [3] or [6] with Eq. 1, a parameter-based attack aims at maximizing the loss (i.e., increase mispredictions) on a small set of $N$ test inputs. As for the imperceptibility criterion of adversarial examples, the attacker may add a constrain over the perturbation by bounding the bit-level Hamming distance ($HD$) between the initial ($W$) and faulted parameters ($W^{\prime}$), corresponding to an _adversarial budget_$S$. \[\underbrace{\max_{W^{\prime}}\sum_{i=0}^{N-1}\mathcal{L}\Big{(}M\big{(}x_{i}; W^{\prime}\big{)},y_{i}\Big{)}}_{\text{mispredictions}}\text{ s.t. }\overbrace{HD(W^{\prime},W)\leq S} \tag{1}\] A state-of-the-art parameter-based attack is the Bit-Flip Attack (hereafter BFA) [3] that aims to decrease the performance of a model by selecting the most sensitive bits of the stored parameters and progressively flip these bits until reaching an adversarial goal. In [3] or [12], the objective is to ultimately degrade the model to a random-guess level. The selection of the bits is based on the ranking of the gradients of the loss w.r.t. to each bit $\nabla_{b}\mathcal{L}$, computed thanks to a small set of inputs. First, each selected bit is flipped (and restored) to measure the impact on the model accuracy. Then, the most sensitive bit is permanently flipped according to the gradient ascendant as defined in [3]. Adversarial goals. Parameter-based attacks are not limited to the alteration of the target model integrity. BFA has been recently demonstrated for powerful model extraction in [13] with an Intel i5 CPU platform: RowHammer is used to perform a BFA on the parameters of a model stored in DRAM (DDR3). The threat model in [13] follows a typical model extraction setting: the adversary knows the model's architecture but not the internal parameters and has only access to a limited portion of the training dataset ($<10\%$). His goal is to build a _substitute_ model as close as possible as the target model. Interestingly, this joint use of RowHammer and BFA is performed in a side-channel analysis fashion: the observation of the induced faults enables to make assumptions on the value of some bits of the parameters. Then, the knowledge of these bits enables to efficiently train a substitute model, by constraining the value range of the parameters, with high _fidelity_ compared to the target model. We discuss this goal in Section 5.4. ### Scope and objectives Because parameter-based attacks are the basis of both powerful integrity and confidentiality threats, their practical evaluation on the different platforms where fault injection may occur is becoming a critical need for present and future standardization and certification actions of critical AI-systems. We position our work on a different set of platforms than the main works related to the BFA (CPU platforms with DRAM), that is MCU platforms (Cortex-M with Flash memory), yet a very important family of embedded AI systems regarding the massive deployment of ML models on MCU-based devices for a large variety of domains. Our main positioning is as follows: * We set this work in a security evaluation and characterization context. Therefore, we do not position ourselves through an _adversary_ but an _evaluator_ point of view. * Our scope is parameter-based threats for neural network embedded in 32-bit microcontroller (hereafter MCUs). * For that purpose, we use Laser Fault Injection (hereafter LFI) as an advanced and very spatially and temporally accurate injection means, a reference technique that is used in many security evaluation centers. State-of-the-art is focused on simulation-based evaluations or on RowHammer attacks (i.e., exclusively DRAM platforms that excludes MCU). To the best of our knowledge, this work is the first to demonstrate the practicability and suitability of the characterization of a weight-based adversarial perturbation against Cortex-M MCU thanks to LFI. ### Related works Since the presentation of the BFA in [3], several works analyzed the intrinsic mechanisms of the attack as well as potential protections [12, 14] and evaluated its properties according to the threat model, training parameters and model architecture [15]. The standard BFA is _untargeted_ since the induced misclassifications are not chosen by the adversary. Therefore, some works also proposed _targeted_ versions [16] with specific target inputs and/or labels. Other recent works propose alternative methods to efficiently select the most sensitive parameters to attack [6]. For our work, we use and adapt (for LFI) the standard BFA of [3] as it is the state-of-the-art baseline for weight-based adversarial attacks. To the best of our knowledge, the only work related to laser injections for MCU-based platform against embedded neural networks has been proposed in [2]. Our work differs significantly by the target device and the elements we target in the model. In [2], the authors used an 8-bit microcontroller (ATMega328P) and a 32-bit neural network implemented in C (a mulitlayer perceptron trained on the MNIST data set). They only focused on the activation functions by inducing instruction skips (i.e., the faulted instructions are not executed, as if they were skipped). They used a laser and only targeted the last hidden layer. We used a Cortex-M 32-bit microcontroller and embedded 8-bit quantized neural network thanks to a state-of-the-art open source library (_Neural Network on Microcontrollers_2, hereafter NNoM) for model deployment. Our attack vector (BFA-like attack) and fault model enable to evaluate the robustness of a model against an advanced adversary that aims at significantly altering the performance of a model with a very limited number of faults or extract information about parameter values for model extraction. Thus, our work is also closely linked to [11] that demonstrated at USENIX'20 a complete exploitation of the BFA with RowHammer on an Intel i7-3770 CPU platform. Although we target a different type of platforms with 32-bit MCU and another fault injection means (LFI), we share the same objective to go beyond simulations and propose a complete practical evaluation of a parameter-based attacks against embedded quantized DNN models. ## 3 Backgrounds ### 4 Evaluator assumptions and experimental setups #### 4.1.1 Goal and evaluator assumptions Objectives. The main objective of the evaluator is to evaluate the robustness of a model against precise fault injections by decreasing the average accuracy on a labelled test set. More precisely, the scenario corresponds to a generic _untargeted_ case (i.e., the incorrect labels are not controlled by the evaluator). Note that a targeted scenario (for specific test samples or target labels) is possible [17] but out of the scope of our experiments. A secondary objective is to minimize the evaluation cost with a strategy that reduces the number of faults to be injected (i.e., avoid an exhaustive search that may be unrealistic according to the complexity of the target model). Evaluator hypothesis. Classically for security testing, the evaluator simulates a worst-case adversary that has a perfect knowledge of the model (white-box attack) and is able to query the model without limitation. The evaluator has a full access to the device (or clones of the device) and can perform elementary characterizations to adapt and optimize the fault injection set-up. #### 4.2.1 Single bit-set fault model on Flash memory We consider an accurate fault model relevant for LFI previously explained and demonstrated for NOR-Flash memory of Cortex-M MCU by Colombier and Menu [18]: the bit-set fault model. As its name suggests, the fault sets a targeted bit to a logical 1: when the bit was already at 1 the fault has no effect. When targeting a Flash memory at read time with a laser pulse, the induced bit-set is transient: it affects the data being read at that time while the stored value is left unmodified. Authors from [18] explained the underlying mechanism of the bit-set fault injection with the creation of a _photoelectric current_ induced by the laser in a floating-gate (FG) transistor that flows from its drain to the ground. This current is added to the legitimate one so that the total current is above the reference that makes the bit read as a logical 1. LFI is a local fault injection means: its effect is restricted to the bit line connected to the FG transistor inside the laser spot area. More precisely several current components are induced in the affected transistors, depending on the laser spot diameter: up to two adjacent bits can be faulted simultaneously [18]. ### Target and laser bench setup Device under test (DUT). Our target board embeds an ARM Cortex-M3 running at $8\,\mathrm{MHz}$. It includes $128\,\mathrm{kB}$ of Flash memory and is manufactured in the $90\,\mathrm{nm}$ CMOS technology node. The dimension of the chip is $3\,\mathrm{x}$$2.5\,\mathrm{mm}$. Since LFI requires the surface of the die being visible, the microcontroller packaging was milled away with engraving tools to provide an access to its laser-sensitive parts. The chip was then mounted into a test board compatible with the Chip-Whisperer CW308 platform. Laser platform. Our laser fault analysis platform integrates two independent laser spots with a near infrared (IR) wavelength of $1,064\,\mathrm{nm}$, focused through the same lens. Each laser spot has a diameter ranging from $1.5$ to $15\,\mathrm{\SIUnitSymbolMicro m}$ depending on the lens magnification. Both spots can move inside the whole field of view of the lens with minimum distortion. The laser source can reach a maximum given power of $1,700\,\mathrm{mW}$. The delay between the trigger and the laser shot can be adjusted with a step of a few nanoseconds. An infrared camera is used to observe the laser spot location on the target and a $XY$ stage enables to move the objective above the entire DUT surface. ### Datasets and models Although simulation-based works exclusively used complex deep neural networks trained for vision tasks (e.g., ResNet on ImageNet), it does not represent a large part of real-world applications that take benefit from classical fully-connected architecture (or multilayer perceptron, hereafter MLP) that fit and perform well on the widespread constrained platforms we studied in this paper. We considered two classical datasets. The IRIS dataset consists of $150$ samples, each containing $4$ real-value inputs and labelled according to $3$ different iris species. We trained two simple models that provide an easy insight on the bit-set fault model effect: IRIS_A and IRIS_B are both composed with one hidden layer with one neuron and four neurons respectively. IRIS_B has $96\%$ accuracy on the test set. MNIST dataset is composed of gray-scale handwritten digits images ($28$x$28$ pixels) from $0$ to $9$. Our model (noted MNIST) is a MLP with one hidden layer of $10$ neurons. Inputs are compressed to $\mathbb{R}^{50}$ with a classical principal component analysis. The resulting model has $620$ trainable parameters and reaches $92\%$ of accuracy on the test set. All models use ReLU as activation function. ### Model implementation on MCU Models were trained with TensorFlow. Few tools are available to embed previously trained models in microcontroller boards such as TensorFlow Lite, X-Cube-AI or NNoM. We chose NNoM as it is an efficient and convenient open source platform that fits our security testing objectives. NNoM offers 8-bit model quantization (with a standard uniform symmetric powers-of-two quantization scheme) and a complete white-box access to the inference code that enables to draw a timing profile of the sensitive operations we target (reading values from the Flash memory). Listing 1a is an extract of the C code source of the core calculation of an inference from NNoM, that is the weighted sum between the inputs (i.e. the input data or the outputs of the previous layer) and the model parameters before the non-linear activation is applied. It consists in loading the neuron input and weight values (inA and inB, line 5 and 6), then process the multiplication and accumulation in an intermediate output value (ip_out, line 7). Line 9 corresponds to the quantization. The assembler code in Listing 1b corresponds to the weight initialization of Listing 1a, line 6. The weight value, stored into the Flash memory, is loaded into register r3 (line 5 in red). Based on our single bit-set fault model, if a laser beam is applied during the execution of the load instruction, a bit-set could be induced directly on the loaded value. To characterize the impact of the laser pulse, we synchronize the laser thanks to a trigger signal in the C code before line 6 and monitor the parameter value before and after the trigger using UART communication. ## 5 A parameter-based attack with LFI To analyze the efficiency and practicality of LFI we first demonstrated the accuracy of the induced faults on a single neuron composed of four weights. Then, we scaled up to functional models trained on IRIS and MNIST to analyze the impact at a model-level. Figure 1: C and Assembler codes from the NNoM inference. ### Initial characterization on a neuron An important preliminary experience is to set up our laser bench on our DUT. For conciseness purpose, we do not detail all the Flash memory mapping procedure neither the selection of the laser parameters. For that purpose, we thoroughly followed the experimental protocol from [18] and fixed the laser power to $170\,\mathrm{mW}$ and the pulse width to $200\,\mathrm{ns}$. By selecting the lens magnification $\times 5$, we chose a spot diameter of $15\,\mathrm{\SIUnitSymbolMicro m}$ to have a wide laser effective area. Then, we implemented a 4-weights neuron. We set the Y-position to $100\,\mathrm{\SIUnitSymbolMicro m}$ to only focus on the X-axis motion. Fig. 1(a) shows that, when moving along the X-axis of the Flash memory, bit-sets are induced one after another on the whole 32-bit word line and the four quantized weights are precisely faulted. We repeated this experience with different weight values and noticed a perfect reproducibility, as also reported in [18]. By noticing positions of weights and their most significant bit (MSB), we easily conclude the little endian configuration of the Flash memory. Fig. 1(b) illustrates how the weights are stored according to the laser bench X-axis. ### Target multilayer perceptron models Since we can change the value of parameters related to a neuron, the next step consisted in analyzing the impact of such faults on the integrity of a target model and in measuring the potential drop of accuracy. For that purpose, we used the IRIS_B model. For each X-step, we evaluated the robustness of the model by feeding it with 50 test samples (i.e., 50 inferences). Classically, we computed the adversarial accuracy by comparing the output predictions to the correct labels and measure the accuracy drop in comparison to the nominal performance without faults. During one inference, all the weight loading instructions triggered a laser shot (i.e. in total 40 shots for the hidden layer). By targeting one bit line at a given Figure 2: (a) Bit-set faults (red dots) induced on the 4 weights of a neuron (IRIS_A model). (b) Flash memory schematic section filled with quantified weights. X-position of the laser, only weights on the same address column are faulted with a bit-set. For illustration, as pictured in Fig. 1(b), LFI actually induced bit-sets in the MSBs of weights $w_{0}$, $w_{4}$, $w_{8}$, $w_{4m}$ (with $m$ the $m^{th}$ word-line), which belong to the targeted bit line. Fig. 2(a) shows the impact of the laser shots on the model accuracy (blue curve) on the test set. The red curve provides the number of faulted bits. The X-axis corresponds to the position of the laser in the Flash memory. Four regular patterns of accuracy drop corresponding to the four stored weights are clearly visible. The observed accuracy drops (5 to 6 per pattern) match the coordinates of the MSBs with a decreasing correlation. For MSB bits, the model is close to a random-guess level (i.e. around 30%). On average, around 6 bit-sets are induced and the most harmful configuration corresponds to only 5 bit-sets (at 790 $\mathrm{\SIUnitSymbolMicro m}$) leading to an adversarial accuracy of 30%. One limitation is that only 1/4 of all weights could be faulted during one inference. However, since our laser platform integrates a second spot, we experimented a configuration where one spot targets the first bit line (i.e the MSB of the weight column $w_{3}$ in Fig. 1(b)) and the second spot targets the bit line #16 (i.e the MSB of the weight column $w_{1}$). As a result, two weight columns can be targeted during one inference, and half of the weights are likely to be faulted. As reported in Fig. 2(b) more bit-set faults were injected (up to 12), and the accuracy drop for non-MSB bits is more important. To extend the previous experiments, we embedded the MNIST model that gathers 620 weights (i.e. 4,960 bits) and strictly kept the same experimental setup. Results are presented in Fig. 4. On average, we observe more induced bit-sets with the most important drop of accuracy (65.5%) reached with only 25 bit-sets at $X=1,100\,\mathrm{\SIUnitSymbolMicro m}$. We also point out that, in some cases, a very slight improvement of the accuracy appeared. In comparison with IRIS_B (Fig. 2(a)), we note that bit lines are less distinctive. Indeed, even if the position of the most significant bits can be identified, other Figure 3: Impact on the accuracy of LFI on IRIS_B model. Average accuracy over 50 inferences (blue) and number of faulted bits (red). bits are difficult to distinguish. Actually, in Fig. 2b the bit line (green box) represents all single bit lines connected to the same bit index for different 32-bit words addresses. Since the MNIST model has more parameters, almost all bit lines are linked to a weight parameter value stored in Flash memory. The laser spot, with a size of $15\,\mathrm{\SIUnitSymbolMicro m}$, is larger than a single bit line, explaining why bit line indexes are hardly discernible. Moreover, depending on the laser position, the effective area of the spot can encompass two different bit line indexes. ### Advanced guided-LFI Adapt BFA to LFI. So far, we exhaustively attacked all the parameters stored in memory. This _brute-force_ strategy may be impractical with deeper models. Therefore, an important objective for an evaluator is to optimally select the most sensitive bits to fault. For that purpose, as in [11], we adapted the bit selection principle of the standard BFA3 to our fault model. We refer to this adapted attack as BSCA (_Bit-Set Constrained Attack_). We also adapted the adversarial objective by introducing an _adversarial budget_[6, 15] representing a maximum of faults the evaluator is able to process. Indeed, the random-guess level objective of [3] overestimates the number of faults and raises variability issues because of the last necessary bit-flips needed to ultimately reach the objective [15]. Therefore, we set the adversarial budget to 20 bit-sets. Footnote 3: [3]: [https://github.com/elliothe/BFA/](https://github.com/elliothe/BFA/) As inputs, the BSCA has the target model $M_{W}$, its parameters $W$, the weight column index $m$ in the Flash memory, the bit line index $b$ and the adversarial budget $S$. The output of the BSCA is a new model $M_{W^{\prime}}^{m,b}$ with $W^{\prime}$ and $W$ that differ only by at most $S$ bit-sets. Then, we can compute an adversarial accuracy Figure 4: Impact on the accuracy of LFI on a MLP model trained on MNIST. Average accuracy over 100 inferences (blue) and number of faulted bits (red). on $M_{W^{\prime}}^{m,b}$ and measure the accuracy drop compared to the nominal performance of $M_{W^{\prime}}^{m,b}$. The BSCA proceeds through the following steps: 1. The BFA ranks the most sensitive bits of $W$ according to $\nabla_{b}\mathcal{L}$. 2. Exclude the bits already set to 1 and not related to $m$ and $b$. 3. Pick the best bit-set and perform the fault permanently in $M$. 4. Repeat the process until reaching $S$. The output is the faulted model $M_{W^{\prime}}^{m,b}$. Finally, we perform BSCA over all the weight columns and bit line and keep the faulted model, simply referred as $M_{W^{\prime}}$, with the worst accuracy ($Acc$) evaluated on a test set $\left(X^{test},Y^{test}\right)$: $M_{W^{\prime}}=\arg\min_{m,b}Acc(M_{W^{\prime}}^{m,b},X^{test},Y^{test})$ We used the MNIST model and simulated the BSCA to find the 20 most significant weights to fault for each weight column. For illustration purpose, Fig 4(a) shows the effect of bit-set induced on the 8 bits of a weight only for the second weight column (with the MSB refered as Bit #0). Among the most sensitive weights from the second column, only few bit-sets on the most significant bits efficiently alter the model performance, bit-sets on other bits have less or even no influence. Experiments and results. The basic idea is to use the BSCA to guide the LFI. For that purpose, we need to put to the test that LFI reach (near) identical performance than what expected by simulations. We ran a BSCA simulation (in Python) over all the weight columns and bit lines that pointed out the MSB of the second column weight as the most sensitive. Therefore, contrary to the previous experiments, the laser source was triggered only when the 20 most sensitive weights were read from the Flash. The laser location was set accordingly ($X=760\,\mathrm{\SIUnitSymbolMicro m}$) and the power increased to $360\,\mathrm{mW}$ to ensure a higher success rate on weights stored in distant addresses. The blue curve in Fig.4(b) represents our experimental results (mean accuracy over 100 inferences) while the red one is the BSCA simulations for the MSB. First, we can notice that experimental and simulation results are almost similar, meaning that we can guide our LFI with high reliability and confidence4. For an adversarial budget of only 5 bit-sets (0.1% faulted bits) the embedded model accuracy drops to 39% which represents a significant loss and a strong integrity impact compared to the nominal performance of 92%. Moreover, after 10 bit-sets (accuracy to 25%), the most effective faults have been injected and the accuracy did not decrease anymore. In a security evaluation context, this observation positions the level of robustness of the model according to the adversarial budget. Footnote 4: The fact that experimental results are slightly more powerful than simulations may be explained by the impact of the width of the laser spot on nearby memory cells. ### Exploitation of LFI for model extraction To illustrate the diversity of the adversarial goal an evaluator aims to evaluate (Section 2.1), we propose a first insight of the use of LFI and BSCA for model extraction. At S&P 2022, Rakin _et al._[13] demonstrated how to exploit BFA with RowHammer (i.e., exclusively DRAM platforms) in a model extraction scenario. The goal is to steal the performance of a model of which an adversary knows the architecture and a small part of the training data but not the internal parameters. The idea of [13] is to guess the value of the maximum of MSBs of the model thanks to rowhammer (bit-flip) and then to train a substitute model with these values as a training constraint. Even though we rely on a different fault model (bit-set with LFI), it is relatively straightforward to follow the same extraction method as in [13]. Indeed, by comparing the outputs of the model on a set of inputs (that could be random ones) with and without LFI we can guess the value of the targeted bit: if the _faulted_ outputs are not strictly identical than the nominal ones, that means the bit-set has an effect and therefore the bit was at 0. Otherwise, the bit value was already to 1 or the bit-set has no influence on the outputs. We simulated the BSCA to a deeper MNIST model with 3 layers with respectively 128, 64 and 10 neurons (109K parameters) and used 500 random input images to compare the outputs with and without bit-set faults on the MSB of each weight. To remove ambiguity, we made the simple assumption that working with enough inputs, a bit-set on a MSB at 0 has always an impact on the outputs. With that method, we managed to extract 91.9% of the MSB of the model parameters, which is enough to efficiently trained a substitute model, consistently to [13]. Further analysis and experiments (more particularly, impact of the model architecture and the input set) should be the subject of future dedicated publications. ## 6 Conclusion This work aims to contribute to the development of reliable evaluation protocols and tools for the robustness of embedded neural network models, a growing concern for future standardization and certification actions of critical AI-based Figure 5: Guided-LFI on a MLP model trained on MNIST systems. We conclude by highlighting some research tracks that may pursue this work and fill some limitations with further analysis on other model architectures and platforms (e.g., system-on-chip). First, our results are not limited by the model complexity (BFA has been demonstrated on state-of-the-art deeper models with millions of parameters) but by the complexity of the MCUs. Indeed, the challenge is to characterize how the Flash memory is organized so that the evaluator can precisely target the weight columns and bit lines. Therefore, further experiments would be focused to other targets (e.g., Cortex M4 and M7). Second, contrary to adversarial examples with recent attacks specifically designed for robustness evaluation purposes [19], parameter-based attack still lacks of maturity and recent works highlight limitations of the BFA [6, 15] and propose improvements or alternatives. Thus, considering or combining more attack methods would improve the evaluation by simulating a more powerful adversary. Finally, this work could be widen to the practical evaluation of protections against weight-based adversarial attacks. Additionally to traditional generic countermeasures against fault injection [10], specific defense schemes against BFA encompass weight clipping, clustering-based quantization [2], code-based detectors [20] or adversarial training [6]. To the best of our knowledge, none of these defenses have been practically evaluated against fault attack means such as RowHammer, glitching or LFI. As for adversarial examples (with so many defenses regularly _broken_ afterwards) the definition of proper and sound evaluations of defenses against parameter-based attacks is a research action of the highest importance. ## Acknowledgment This work is supported by (CEA-Leti) the European project InSecTT5 and by the French ANR in the _Investissements d'avenir_ program (ANR-10-AIRT-05, irtnanoelec); and (MSE) by the ANR PICTURE program6. This work benefited from the French Jean Zay supercomputer with the AI dynamic access program. Footnote 5: www.insectt.eu, ECSEL JU 876038 Footnote 6: [https://picture-anr.cea.fr](https://picture-anr.cea.fr)
2301.03169	A Study on the Generality of Neural Network Structures for Monocular Depth Estimation	Monocular depth estimation has been widely studied, and significant improvements in performance have been recently reported. However, most previous works are evaluated on a few benchmark datasets, such as KITTI datasets, and none of the works provide an in-depth analysis of the generalization performance of monocular depth estimation. In this paper, we deeply investigate the various backbone networks (e.g.CNN and Transformer models) toward the generalization of monocular depth estimation. First, we evaluate state-of-the-art models on both in-distribution and out-of-distribution datasets, which have never been seen during network training. Then, we investigate the internal properties of the representations from the intermediate layers of CNN-/Transformer-based models using synthetic texture-shifted datasets. Through extensive experiments, we observe that the Transformers exhibit a strong shape-bias rather than CNNs, which have a strong texture-bias. We also discover that texture-biased models exhibit worse generalization performance for monocular depth estimation than shape-biased models. We demonstrate that similar aspects are observed in real-world driving datasets captured under diverse environments. Lastly, we conduct a dense ablation study with various backbone networks which are utilized in modern strategies. The experiments demonstrate that the intrinsic locality of the CNNs and the self-attention of the Transformers induce texture-bias and shape-bias, respectively.	Jinwoo Bae, Kyumin Hwang, Sunghoon Im	2023-01-09T04:58:12Z	http://arxiv.org/abs/2301.03169v3	# A Study on the Generality of Neural Network Structures for Monocular Depth Estimation ###### Abstract Monocular depth estimation has been widely studied, and significant improvements in performance have been recently reported. However, most previous works are evaluated on a few benchmark datasets, such as KITTI datasets, and none of the works provide an in-depth analysis of the generalization performance of monocular depth estimation. In this paper, we deeply investigate the various backbone networks (_e.g_CNN and Transformer models) toward the generalization of monocular depth estimation. First, we evaluate state-of-the-art models on both in-distribution and out-of-distribution datasets, which have never been seen during network training. Then, we investigate the internal properties of the representations from the intermediate layers of CNN-/Transformer-based models using synthetic texture-shifted datasets. Through extensive experiments, we observe that the Transformers exhibit a strong shape-bias rather than CNNs, which have a strong texture-bias. We also discover that texture-biased models exhibit worse generalization performance for monocular depth estimation than shape-biased models. We demonstrate that similar aspects are observed in real-world driving datasets captured under diverse environments. Lastly, we conduct a dense ablation study with various backbone networks which are utilized in modern strategies. The experiments demonstrate that the intrinsic locality of the CNNs and the self-attention of the Transformers induce texture-bias and shape-bias, respectively. Monocular depth estimation, Out-of-Distribution, Generalization, Transformer ## 1 Introduction Monocular depth estimation (MDE) is widely utilized for spatial recognition technologies such as autonomous driving [1, 2, 3] or AR/VR [4, 5] because of its portability and cost-effectiveness. Various MDE processes have achieved remarkable progress over the past decade. Most previous works [6, 7, 8, 9, 10, 11, 12, 13] concentrate on boosting performance using limited benchmark datasets, especially the KITTI dataset [14]. However, these works have not provided a deep investigation into what MDE networks have learned. This means that they cannot guarantee the model's behavior is correct. To examine the interpretability of MDE networks, one previous work [15] employs a target network for MDE model analysis. Another approach [16] uses a synthetic dataset, which contains the changes in image contents (e.g., camera pose). However, universality questions about other networks still remain because of the fixed specific network [6] in certain experimental conditions. Recently, several works [17, 18, 19, 20] aim to analyze interpretability, taking inspiration from convolutional neural networks (CNNs) whose designs are based on human visual processes [21, 22]. Then, how can humans efficiently extract and recognize important information from complex scenes? Compared to other cues like texture or color, the biological vision system considers the object's shape to be the single most important visual cue [23]. This allows humans, including little kids, to easily distinguish an object from a line drawing or a silhouette image. Many researchers believe that CNN would also behave similarly [24, 25, 26]. However, in contrast to belief, recent studies [27, 28, 18, 20] have discovered that CNNs are heavily predisposed to recognize textures rather than shapes. CNN-based models accurately assign labels to images even when the shapes of the structures are disturbed [29, 30]. On the other hand, CNN models are unable to accurately predict labels in a texture-removed image with a well-preserved shape [31]. Transformers [32] achieve outstanding performance on various computer vision tasks [33, 34, 35] and have attracted much attention. Moreover, many works [36, 37, 27, 38] reveal that Transformers have a strong shape Fig. 1: Analysis on the generality of state-of-the-art models and modernized network structures. The $x$-axis is the CKA similarity indicating whether the network is shape biased or texture biased. The $y$-axis shows Absolute Relative error where a lower number is better performance. We use the synthetic texture-shifted datasets described in Sec. 4.5. bias notwithstanding the lack of a spatial locality, compared to CNNs. Due to its strong shape bias, Transformer is considered more robust than a CNN and more similar to human cognitive processes [39]. Then, how does this observation affect the MDE? We hypothesize two things. First, the network's generality will differ depending on the texture-/shape-bias. To verify the generality, we evaluate state-of-the-art MDE models trained on KITTI [14] using five public depth datasets (SUN3D [40], RGBD [41], MVS [42], Scenes11 [42], and ETH3D [43]). We also conduct experiments on six different texture-shifted datasets, including three synthetic texture-shifted datasets (Watercolor, Pencil-sketch, Style-transfer) and three real-world texture-shifted datasets (Oxford Robotcar [44], Rainy Cityscapes [45], Foggy Cityscapes [46]). Through these experiments, we confirm that texture-bias is vulnerable to generalization while shape-bias shows robust generalization performance. Second, the texture-/shape-bias are related to the intrinsic properties of CNN and Transformer structures. The modernized Transformer-based model [47, 48] imitates the intrinsic locality inductive bias of the CNNs. The modernized CNN-based model [49, 50] is designed to mimic the self-attention of the Transformer. We finally conduct ablation studies on the generalization performance, and the texture-/shape-bias of various modernized backbone structures that originate from a specific design for each CNN and Transformer (_e.g._, locality and self-attention) such as RegionViT [47], Twins [48], ConvNeXt [49] and SLAK [50]. This paper extends our previous work in [51] which proposes a new self-supervised monocular depth estimation network adopting Transformers [52]. The Transformer-based network shows the generalization performance on various environments on depth estimation. While the short version [51] only addresses self-supervised MDE models, the extended version deals with full MDE models, including both self-supervised and supervised methods. Moreover, we deeply analyze the reason why the proposed network achieves better-generalized performance rather than CNN-based models by comparing the performance, and intermediate-layer feature similarity [53, 54] on various texture-shifted datasets. As shown in Fig. 1, we observe that the Transformer structure has more shape-biased properties than the CNN structure, which has texture-biased properties. It enables the Transformer-based models to achieve better generalization performance than the CNN-based models. We also experiment extensively on modern backbone architectures (_e.g._, ConvNeXt [49], RegionViT [47]) to investigate the origin of the texture-/shape-biased properties. Through these extensive experiments, we observe that the intrinsic locality of CNNs induces texture-biased characteristics, while the self-attention mechanism, the base layer of Transformers, induces shape-biased properties. ## 2 Related Work ### _Self-supervised monocular depth estimation_ Self-supervised depth estimation methods [55, 56, 57, 9, 12, 56, 11] simultaneously train depth and motion network by imposing photometric consistency loss between the target and source images warped by the predicted depth and motion. SfMLearner [55] first proposes a depth and ego-motion estimation pipeline without the ground truth depth and motion. Monodepth2 [8] presents a minimum reprojection loss to handle occlusions, a full-resolution multi-scale sampling method to reduce visual artifacts, and an auto-masking loss to ignore outlier pixels. PackNet-SfM [9] introduces packing and unpacking blocks that leveraged 3D convolutions to learn the dense appearance and geometric information in real-time. HR-Depth [12] analyzes the reason for the inaccurate depth prediction in large gradient regions and designed a skip connection to extract representative features in high resolution. ### _Supervised monocular depth estimation_ Supervised methods [58, 59, 10, 13] use a ground truth depth acquired from RGB-D cameras or LiDAR sensors for supervision in training. They also estimate depth maps given a single image as input. BTS [58] adopts a local planar guidance layer to densely encoded features to preserve local detail and create depth map sharpness at multi-stages in the decoder. AdaBins [10] estimate the depth by linear combinations of bin centers that are adaptively decided per image. The bin building block divides the depth range of the image into bins. LapDepth [59] employs a Laplacian pyramid at the multi-level upscaling encoder to preserve local detail, such as a boundary. It also trains stably by utilizing weight standardization. GLPDepth [60] proposes a hierarchical transformer encoder to capture the global context of images and a selective feature fusion module to connect multi-scale local features and global context information at the decoder. The feature fusion module helps the decoder become more powerful, even if the decoder is lightweight. DepthFormer [13] proposes leveraging the transformer's effective attention mechanism and the spatial inductive bias of the CNN to capture long-range correlation. It also uses a hierarchical aggregation and heterogeneous interaction module to enhance the affinity of the network. ### _Vision Transformers_ Recently, Transformers [52] has shown promise for solving computer vision tasks such as image classification [32, 61], object detection [33], and dense prediction. [62, 63, 11, 64]. ViT [32] employs a Transformers architecture on fixed-size image patches for image classification for the first time. DeiT [61] utilizes Knowledge distinction on ViT architecture, showing good performance only with the ImageNet dataset. DETR [33] proposes the direct set prediction approach, which simplifies the object detection pipeline, based on a CNN-Transformer network and bipartite matching. Some works [11, 63] have employed Transformers for monocular depth estimation in a supervised manner. DPT [63] introduces a dense prediction using a Transformer as the basic computational building block of the encoder. These works show generalized performance, but they require a large number of training datasets captured in diverse environments with ground truth depth maps. TransDepth [11] utilizes multi-scale information to capture local level details. Previous works lack studies on whether models behave as intended on another domain dataset. The work [65] aggregates the attention map between a single frame and cross frames to refine the attraction map to improve performance. The works [11, 65] only focus on improving performance on benchmark datasets. ### _Modernized Architecture_ Despite the tremendous success of Transformers for vision tasks, a Transformer requires a large model and data size to achieve state-of-the-art performance. Recently, many works [47, 48, 49, 50, 66] utilize local attention on the Transformer to alleviate problems. The Swin Transformer [66] designs a pyramid structure network differently from a vision transformer [32], which is an isotropic structure. It achieves state-of-the-art performance in classification tasks with local attention using the sliding window strategy. Twins [48] employs global attention for the non-overlapping region and local attention to perform better under limited conditions. RegionViT [47] shows state-of-the-art performance on several visual tasks using regional-to-local attention, which alleviates the weakness of the standard attention mechanism. ConvNeXt [49] modernizes convolution neural layers such as depth-wise convolution and utilizes techniques such as Patchify stem and achieves competitive performance with the Transformer. SLaK [50] attempts to design the network with an extremely large kernel size (_e.g._, $51\times 51$) to leverage the sparsity that is observed from the human visual system. SLaK [50] repeats the Prune-and-Grow step in training to optimize the sparse kernels. ## 3 Method This section describes MonoFormer, which is an encoder-decoder structure with a multi-level feature fusion module for self-supervised monocular depth estimation described in Sec. 3.1. An Attention Connection Module (ACM) learns the channel and position attentions in Sec. 3.2. A Feature Fusion Decoder (FFD) adaptively fuses the encoder features with the attention maps in Fig. 2. ### _Transformer-based Encoder_ MonoFormer [51] is composed of a CNN and Transformer for an image encoder. The encoder employs ResNet50 [67] as the CNN backbone ($E(\theta)$ in Fig. 2), and $L$ number of Transformers. MonoFormer sets the $L$ to 4. The encoder is used to extract a feature map $F\in\mathbb{R}^{C\times H\times W}$ from an input image $I$, and the map is divided into $N=(\frac{H}{16}\times\frac{W}{16})$ number of patches $p_{n}\in\mathbb{R}^{C\times 16\times 16}$, which is utilized as the input of the first Transformer layer. Following the work [63], MonoFormer additionally use a special token $t_{s}$. MonoFormer input the patch tokens $p_{n},\ n\in\{1,...,N\}$ and the special token $t_{s}$ with a learnable linear projection layer $E$ as follows: \[Z_{0}=[t_{s};\ p_{1}E;\ p_{2}E;\...\ ;\ p_{N}E], \tag{1}\] where $Z_{0}$ is the latent embedding vector. The CNN-Transformer encoder comprises a Multi-head Self-Attention (MSA) layer, a Multi-Layer Perceptron (MLP) layer, and Layer Norm (LN) layers. The MLP is built with GELU non-linearity [68]. The LN is used before each block, and residual connections are used after every block. Self-Attention (SA) at each layer $I\in\{1,...,L\}$ is processed with the learnable parameters $W^{m}_{Q},W^{m}_{K},W^{m}_{V}\in\mathbb{R}^{C\times d}$ of {query, key, value} weight matrices, given the embedding vector $Z_{l}\in\mathbb{R}^{N\times C}$ as follows: \[\begin{split}\text{SA}^{m}_{l-1}=\text{softmax}(\frac{Q^{m}_{l-1 }(K^{m}_{l-1})^{\text{T}}}{\sqrt{d}})V^{m}_{l-1},\ m\in\{1,...,M\},\\ \text{Q}^{m}_{l-1}=Z_{l-1}W^{m}_{Q},\ K^{m}_{l-1}=Z_{l-1}W^{m}_{K},\ V^{m}_{l-1}=Z_{l-1}W^{m}_{V},\end{split} \tag{2}\] where $M$ and $d$ are the number of SA blocks and the dimension of the self-attention block, which is the same as the dimension of the weight matrices, respectively. The Multi-head Self-Attention (MSA) consists of the $M$ number of SA blocks with the learnable parameters of weight matrices $W\in\mathbb{R}^{Md\times C}$ as follows: \[\begin{split}\text{MSA}_{l-1}=Z_{l-1}+\text{concat}(\text{SA}^{1} _{l-1};\ \text{SA}^{2}_{l-1};\ \dots;\ \text{SA}^{M}_{l-1})W,\\ Z_{l}=\text{MLP}(\text{LN}(\text{MSA}_{l-1}))+\text{MSA}_{l-1}. \end{split} \tag{3}\] Fig. 2: Overall Architecture of MonoFormer (MF-ours). This Transformer layer is repeated $L$ times with unique learnable parameters. The outputs of the Transformers $\{Z_{1},...,Z_{L}\}$ are utilized as the input of the following layers ACM and FFD. The hybrid encoder can be replaced with the other backbones. We evaluate all the performance of our depth estimation networks with various backbones by changing the encoder to the modernized backbone structures, such as ViT [32], RegionViT [47], Twins [48], ConvNeXt [49] and SLaK [50] as shown in Fig. 3. ### _Attention Connection Module (ACM)_ The skip connection module of MonoFormer, ACM, that extracts global context attention and a semantic presentation from the given features $Z_{l},\ l\in\{1,...,L\}$. The skip connection, widely utilized for dense prediction tasks [69], helps keep the fine detail by directly transferring the spatial information from the encoder to the decoder. However, because of its simplicity, it is challenging for the naive skip connection method to preserve local detail like object boundaries [70]. To address the issue, The ACM is proposed that extracts attention weight from the spatial and channel domains inspired by [71] It consists of position attention, channel attention modules, and a fusion block that gathers important information from two attentions. The position attention module produces a position attention map $A^{p}_{l}\in\mathbb{R}^{C\times N}$ as follows: \[A^{p}_{l}=\text{softmax}(Q^{p}_{l}(K^{p}_{l})^{\text{T}})V^{p}_{l}, \tag{4}\] where $Q^{p}_{l}$, $K^{p}_{l}$ and $V^{p}_{l}$ are the query, key, and value matrices computed by passing $Z_{l}$ through a single convolutional layer. The channel attention module directly calculate the channel attention map $A^{c}_{l}\in\mathbb{R}^{C\times N}$ by computing the gram matrix of $Z_{l}$ as follows: \[A^{c}_{l}=\text{softmax}(Z_{l}Z^{\text{T}}_{l}). \tag{5}\] The position attention map $A^{p}_{l}$ and channel attention map $A^{c}_{l}$ enhance the feature representation by capturing long-range context and exploiting the inter-dependencies between each channel map, respectively. These two attention maps are utilized in the following section, which highlights the importance of the features. ### _Feature Fusion Decoder (FFD)_ The FFD gets the encoder features $Z_{l}$, the attention maps $A^{p}_{l},\ A^{c}_{l}$, and the output feature $X_{L}$ of the last Transformer layer passed through a Residual convolutional layer. The features $X_{L-l+1},\ l\in\{1,...,L\}$ are fused through the decoder with a single Convolutional layer (Conv) and Channel Normalization (CN) with learnable parameters $\alpha,\beta$ and $\gamma$ as follows: \[\begin{split} X_{L-l}=\hat{X}_{L-l}[1+\tanh(\gamma(\text{CN}( \alpha\|\|\hat{X}_{L-l}\|\|_{2}+\beta)],\\ \hat{X}_{L-l}=\text{Conv}(w_{p}A^{p}_{l}Z_{l}+w_{c}A^{c}_{l}Z_{l }+Z_{l})+X_{L-l+1},\end{split} \tag{6}\] where $w_{p}$ and $w_{c}$ are the learnable parameters that determine the importance of the position and channel attentions [72]. The parameter $\alpha$ works so that each channel can learn about each other individually, and $\gamma$ and $\beta$ control the activation channel-wisely following the work in [73]. Through this process, the FFD can assemble the local detailed semantic representation and the global context from the fused features to produce the fine depth map. ### _Training loss and implementation detail_ We train both depth and motion networks using photometric consistency (L2 loss and SSIM loss) and edge-aware smoothness losses following the best practices of self-supervised monocular depth estimation [55, 9, 8]. We set the weight for SSIM, L2 photometric, and smoothness losses as $0.85$, $0.15$ and $0.001$, respectively. We use 7 convolution layers for 6-DoT camera pose estimation following the work in [55]. We implement our framework on PyTorch and train it on 4 Titan RTX GPUs. We use the Adam optimizer [74] with $\beta_{1}=0.9$ and $\beta_{2}=0.999$. Our model is trained for 50 epochs with a batch size of 8. The learning rates for depth and pose network are $2\times 10^{-5}$ and $5\times 10^{-4}$, respectively. We will release the source code, the trained weights and the datasets once the paper is accepted. ## 4 Experimental Results We conduct extensive experiments to investigate the generalization performance of various network structures and the effect of texture-/shape-bias for monocular depth estimation. First, we evaluate state-of-the-art KITTI-trained models, including various modernized backbone architectures on the KITTI dataset and various depth benchmark datasets. Evaluations are conducted on the KITTI, an in-distribution dataset in Sec. 4.2, and the other depth datasets, out-of-distribution datasets in Sec. 4.4. We also conduct an ablation study on the MonoFormer in Sec. 4.3. Then, we investigate the texture-bias and shape-bias of the MDE models Fig. 3: Illustrations of various modernized architectures. using both the self-supervised and supervised methods with texture-shifted datasets in Sec. 4.5. We analyze the depth estimation performance and the feature representation of each model on the texture-shifted datasets, which are synthetically generated. Lastly, we demonstrate the intrinsic properties of the CNN-based and Transformer-based networks are observed in the real-world texture-shifted datasets captured from different driving environments, such as the weather changes and illumination changes in Sec. 4.6. ### _Competitive methods and evaluation setups_ We conduct extensive experiments to compare the performances of the monocular depth estimation methods. For self-supervised setting, we compare our work, called MF-Ours [51], with state-of-the-art methods, Monodepth2 [8], PackNet-SfM [9], SGDepth [56], R-MSFM [75]. We also evaluate state-of-the-art supervised methods, BTS [58], AdaBins [10], TransDepth [11], DepthFormer [13], and GLPDepth [60]. We note that some Transformer-based models, such as MF-Ours, TransDepth, and DepthFormer, use both CNNs and Transformers for their encoders. We also analyze the performance of various modernized architectures such as RegionViT [47], Twins [48], ConvNeXt [49] and SLaK [50]. We replace the encoder of MF-Ours with these modernized backbones, whose names are defined as MF-(backbone). Its taxonomy is described in Tab. I. We illustrate these basic network structures of the modernized CNN and Transformer architectures in Fig. 3. eTransformer-based models [47, 48] add the locality by using local attention to compensate for the shortcomings of the ViT (_e.g._, necessity of large dataset). CNN-based architectures [49, 50] aim to extract global information while utilizing the intrinsic locality inductive bias. They also imitate the self-attention of the Transformer using improved strategies such as large kernel size, patchify stem [49], and Layer Norm [76]. We train all networks using the KITTI Eigen split [14, 77] consisting of 39,810 training and 4,424 validation data whose size is $640\times 192$. All experiments in this paper are conducted with this KITTI-trained model whose results are reported in Sec. 4.2-4.6. Following a previous work [9], we remove around 5% of the total data for training to address the infinite-depth problems that commonly occur in dynamic scenes. We use typical error and accuracy metrics for depth, absolute relative (Abs Rel), square relative (Sq Rel), root-mean-square-error (RMSE), its log (RMSElog), and the ratio of inliers following the work [9]. ### _Evaluation on the in-distribution dataset_ We conduct the evaluations with an in-distribution dataset, where the KITTI-trained models are evaluated on 697 KITTI test data. The qualitative results in Fig. 4 show that our self-supervised method precisely preserves object boundaries. This indicates that the encoder captures global context and informative local features and transfers them to the decoder \begin{table} \begin{tabular}{l l l} \hline \hline Models & Supervision & Methods \\ \hline CNN-based & Self-supervised & Monodepth2 [8], PackNet-SfM [9] \\ & R-MSFM [75], MF-ConvNeXt [49], SLaK [50] \\ \hline \multirow{2}{}{_Transformer-based_} & Supervised & BTS [58], AdaBins [10] \\ \cline{2-3} & Self-supervised & MF-ViT [32], RegionViT [47], Twins [48], Ours [51] \\ \hline Supervised & TransDepth [11], DepthFormer [13], GLPDepth [60] \\ \hline \hline \end{tabular} \end{table} TABLE I: Taxonomy of MDE methods w.r.t backbones.* \begin{table} \begin{tabular}{l l l l l l l l l} \hline \hline Method & Train & Model & Mean-square & Mean-square & Mean-square & Mean-square & Mean-square & Mean-square \\ \hline Monodepth2 & M & CNN & 0.115 & 0.204 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.205 & 0.401 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.205 & 0.401 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.208 & 0.401 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.206 & 0.401 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.206 & 0.401 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.206 & 0.400 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.206 & 0.400 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.206 & 0.400 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & CNN & 0.111 & 0.206 & 0.400 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & TransDepth & 0.111 & 0.206 & 0.400 & 0.100 & 0.000 & 0.000 & 0.000 \\ PathNet & M & TransDepth & 0.110 & 0.300 & 0.400 & 0.100 & 0.000 & 0.000 & 0.000 \\ \hline \hline BTS & D & CNN & 0.200 & 0.200 & 0.200 & 0.200 & 0.500 & 0.500 & 0.500 & 0.500 \\ Ash Ashuna & D & CNN & 0.050 & 0.100 & 2.300 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ PathNet & D & TransDepth & 0.400 & 0.200 & 0.200 & 0.000 & 0.000 & 0.000 & 0.000 \\ PathNet & D & TransDepth & 0.400 & 0.400 & 0.100 & 0.100 & 0.000 & 0.000 & 0.000 \\ \hline \hline \end{tabular} \end{table} TABLE II: Quantitative results on the in-distribution dataset. M is Monocular images, and D is GT depth. Trans means Transformer. Bold is the best performance. Fig. 4: Depth map results on the in-distribution dataset. MonoFormer (MF-Ours) is from our previous work [51]. for pixel-wise prediction. The quantitative results in Tab. II show that MF-Ours outperforms all competitive methods. We also evaluate the performance of models with the other backbones, MF-(ConvNeXt/SLaK/RegionViT/Twins/ViT). We employ only the encoder part of our method as a backbone without changing other parts in this experiment. Supervised methods outperform all self-supervised methods regardless of backbones. MF-Ours and DepthFormer achieve the best performance among the self-supervised methods and supervised methods on the in-distribution datasets, respectively. Overall, the performance gap among all the competitive methods is marginal for the in-distribution dataset, but the Transformer-based models generally outperform the CNN-based models. ### _Ablation study_ In this section, we conduct the ablation study on MF-Ours proposed in our previous work [51]. We use the same models and datasets for the experiments used in Sec. 4.2. #### 4.3.1 Effectiveness of the proposed modules. We conduct an ablation study to demonstrate the effectiveness of the proposed modules, ACM and FFD in Tab. III. The baseline is DPT [63]. The models with only the ACM module or FFD module marginally improve the depth estimation performance due to the absence of proper attention map fusions. On the other hand, our MonoFormer with both ACM and FFD significantly improves the performance. The results show the proposed model achieves the best performance in all measurements. The qualitative comparison in Fig. 5 shows that the model with both ACM and FFD keeps clearer object boundaries, even a small car in far depth. #### 4.3.2 Visualization of attention maps. We visualize the attention maps from the lower to higher layers of Transformers. As shown in Fig. 6, the encoder in the shallow layer extracts local region features. The deeper the layer, the more global shape contexts are extracted. Another observation is that ACM captures more detailed attention at different depths of the encoder features. FFD enhances the encoder features by fusing them with the attention map from ACM. The fused feature captures features from coarse to fine details. These experiments show that our model is capable of accurate pixel-wise prediction as it secures adequate local details. #### 4.3.3 The number of encoder and decoder layers. We compare the performance of MF-Ours according to the number of encoder and decoder layers in Tab. IV. Each layer has ACM and FFD modules. We find out that the model with four transformer layers achieves the best performance. The results are slightly degraded with the MF-Ours with 3 or 5 layers. Therefore, we set $L$ as four for MF-Ours in all experiments in this paper. ### _Evaluation on out-of-distribution datasets_ We compare the generalization performance of all the competitive models using public depth datasets captured at common indoor environments, including office workspaces, meeting rooms, and kitchen areas (RGBD [78], SUN3D [40]), man-made indoor and outdoor environments (MVS [42], ETH3D [43]), and synthetic scenes from graphics tools (Scenes11 [42]). Interestingly, both qualitative and quantitative results in Fig. 7 and Tab. 5-7 show that Transformer-based models better generalization performance even in the out-of-distribution datasets that have never been seen during network training. Meanwhile, CNN-based models predict unreliable depth maps that have lost the detail of object boundaries and produce significant errors in texture-less regions, such as the wall of a building. The experiments demonstrate that Transformers are more robust than CNNs for environmental changes. Another interesting observation is that MF-ConvNeXt generally outperforms all the other CNN-based models and produces depth results comparable to other Transformer-based models. On the other hand, MF-RegionViT fails to estimate depth accurately, even though Transformer-based. A detailed analysis of why Transformers and MF-ConvNeXt show better generalization performance is provided in the following section. \begin{table} \begin{tabular}{c c c c c} \hline \# of layers & Abs Rel $\downarrow$ & RMSE $\downarrow$ & RMSE$\downarrow$ & $\delta<1.25\uparrow$ \\ \hline $L=2$ & 0.148 & 5.327 & 0.810 & 0.939 \\ $L=3$ & 0.112 & 4.745 & 0.881 & 0.962 \\ $L=4$ & 0.104 & 4.580 & 0.873 & 0.962 \\ $L=5$ & 0.111 & 4.692 & 0.884 & 0.962 \\ \hline \end{tabular} \end{table} TABLE IV: Ablation study on the number of layers. Fig. 5: Results of MF-Ours with/without ACM and FFD. \begin{table} \begin{tabular}{c c c c c} \hline & Abs Rel $\downarrow$ & Seq Rel $\downarrow$ & RMSE$\downarrow$ & $\delta<1.25\uparrow$ \\ \hline baseline & 0.113 & 0.899 & 4.783 & 0.189 & 0.882 \\ +ACM & 0.113 & 0.879 & 4.820 & 0.189 & 0.879 \\ +FP & 0.112 & 0.860 & 4.803 & 0.186 & 0.879 \\ +ACM\&FFD & 0.104 & 0.846 & 4.580 & 0.183 & 0.891 \\ \hline \end{tabular} \end{table} TABLE III: Ablation study on ACM and FFD. Fig. 6: Visualization of attention and feature maps. The left column from the second row is the maps from shallow layers, whereas the right is the maps from deep layers. [MISSING_PAGE_POST] ### _Analysis of texture-/shape-bias on state-of-the-art methods and various backbone networks_ In this section, we verify the intrinsic properties of CNNs and Transformers that lead to the robustness of depth estimation to environmental changes. We hypothesize that CNNs and transformers identify texture and shape information as key visual cues for depth estimation, respectively, inspired by the work [20, 28]. Thus, we synthetically generate the texture-shifted datasets in Sec. 4.5.1. Then, we validate the texture-/shape-bias of the model by comparing the performance of the competitive methods on the generated datasets in Sec. 4.5.2. Finally, we analyze the internal representation of each neural network structure by measuring centered kernel alignment (CKA) in Sec. 4.5.3. #### 4.5.1 Texture-shifted datasets generation In general, the texture is defined as an image's spatial color or pixel intensity pattern [79, 80, 81]. Inspired by [18], we use three different texture shift strategies to investigate the impact of textures on the inference process in depth: texture smoothing (Watercolor), texture removal (Pencil-sketch), and texture transfer (Style-transfer). The generated images and the corresponding results of each model are shown in Fig. 8. The first two images are watercolored images, the middle two images and the last two images are pencil-sketch and style-transfered images, respectively. The following is a summary of the image generation: \begin{tabular}{l l} _Watercolor_ & We smooth the texture details from original images while preserving the color cues using cv2. stylization. The image looks like a watercolor pictures. \\ _Pencil-sketch_ & We remove both textures and color from original images using cv2.pencilSketch. The image seems like a sketch drawn with pencils. \\ _Style-transfer_ & We apply a new texture to the original image (context) by utilizing other images (style) using a style transfer [82]. The textures of the original images are changed. \\ \end{tabular} #### 4.5.2 Evaluation on synthetic texture-shifted datasets We compare the performance of all the competitive methods and the modernized backbones on the synthetic texture-shifted datasets, the same as the experiments in Sec. 4.4. The qualitative and quantitative results are shown in Fig. 8 and Tab. 8, respectively. As seen in Sec. 4.4, both the Transformer-based models produce better depth maps than pure CNN-based methods regardless of supervised or self-supervised methods. In particular, the depth results from CNN-based models are unrecognizable with the strong texture-shifted datasets, especially the style-transfered data. We also observe that MF-ConvNeXt shows a tolerable depth map on texture shift datasets and has lower errors than other CNN-based models, although it is purely CNN-based. These experiments support our two findings observed in Sec. 4.4. One is that networks whose encoder consists of Transformers are generally robust to texture changes. However, MF-ConvNeXt and MF-RegionViT show different aspects from the CNN-based and Transformer-based models, which are the respective backbone models. In the following section, we deeply analyze the intermediate feature representations of all backbones to verify the reason for these observations. \begin{table} \begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{}{Datasets} & Train & Backbone & \multicolumn{5}{c}{Multi-task 50,184} & \multicolumn{5}{c}{Multi-task 50,184} & \multicolumn{5}{c}{Multi-task 50,184} & \multicolumn{5}{c}{Multi-task 50,184} & \multicolumn{5}{c}{Multi-task 50,184} \\ \hline \hline KnowledgePh2 & M & CNN & 15GP & 0.763* & 0.532 & 0.532 & 0.552 & 0.567 \\ PackNeXt & M & CNN & 2.068 & 0.870 & 0.320 & 0.310 & 0.500 & 0.674 \\ RASHp & M & CNN & 1.272 & 0.276 & 0.341 & 0.280 & 0.494 & 0.846 \\ RASHp & M & CNN & 1.827 & 0.639 & 0.284 & 0.350 & 0.506 & 0.718 \\ RASHp & M & CNN & 1.837 & 0.320 & 0.295 & 0.505 & 0.800 \\ RASHp & M & CNN & 1.846 & 0.240 & 0.263 & 0.263 & 0.263 & 0.263 \\ \hline \hline \end{tabular} \begin{tabular}{l l} _Style-transfer_ & We apply a new texture to the original image (context) by utilizing other images (style) using a style transfer [82]. The textures of the original images are changed. \\ \end{tabular} \end{table} TABLE \begin{table} \begin{tabular}{l c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{}{Datasets} & \multicolumn{8}{c}{Datasets} & \multicolumn{8}{c}{Datasets} & \multicolumn{8}{c}{Datasets} & \multicolumn{8}{c}{Datasets} \\ \cline{2-13} & Abs Rel & Sq Rel & RMSE $\delta$ $<$ 1.25 & $\delta$ $<$ 1.25${}^{\circ}$ & $\delta$ $<$ 1.25${}^{\circ}$ & Abs Rel & Sq Rel & RMSE $\delta$ $<$ 1.25 & $\delta$ $<$ 1.25${}^{\circ}$ & Abs Rel & Sq Rel & RMSE $\delta$ $<$ 1.25 & $\delta$ $<$ 1.25${}^{\circ}$ & $\delta$ $<$ 1.25${}^{\circ}$ \\ \hline Monodepft2 & 0.170 & 1.345 & 6.175 & 0.750 & 0.900 & 0.960 & 0.196 & 1.522 & 6.232 & 0.691 & 0.898 & 0.962 & 0.247 & 2.440 & 7.825 & 0.647 & 0.828 & 0.920 \\ PacikNet-SM & 0.174 & 1.346 & 6.343 & 0.742 & 0.906 & 0.961 & 0.204 & 1.569 & 5.686 & 0.837 & 0.888 & 0.888 & 0.897 & 0.205 & 1.608 & 6.778 & 0.678 & 0.948 \\ R-MSFM6 & 0.194 & 1.613 & 7.173 & 0.696 & 0.876 & 0.943 & 0.217 & 6.098 & 6.719 & 0.647 & 0.872 & 0.951 & 0.294 & 2.788 & 8.599 & 0.521 & 0.770 & 0.894 \\ MF-ConvNet & 0.195 & 1.175 & 5.957 & 0.774 & 0.921 & 0.969 & 0.176 & 1.280 & 5.682 & 0.728 & 0.947 & 0.965 & 0.219 & 1.722 & 7.241 & 0.634 & 0.861 & 0.942 \\ MF-SLK & 0.170 & 1.272 & 6.316 & 0.755 & 0.907 & 0.961 & 0.268 & 2.208 & 0.842 & 0.533 & 0.811 & 0.915 & 0.242 & 1.889 & 7.423 & 0.648 & 0.855 & 0.934 \\ \hline MF-ViT & 0.152 & 1.196 & 5.686 & 0.799 & 0.852 & 0.973 & 0.774 & 1.311 & 5.707 & 0.756 & 0.920 & 0.987 & 0.816 & 1.359 & 0.623 & 0.758 & 0.959 \\ MF-RegionViT & 0.179 & 1.341 & 6.309 & 0.732 & 0.902 & 0.998 & 0.260 & 2.021 & 7.941 & 0.954 & 0.832 & 0.939 & 0.422 & 0.704 & 0.947 & 0.885 & 0.927 \\ MF-Twins & 0.166 & 1.295 & 6.585 & 0.751 & 0.911 & 0.963 & 0.205 & 1.630 & 7.512 & 0.655 & 0.871 & 0.949 & 0.232 & 1.974 & 0.845 & 0.599 & 0.839 & 0.931 \\ MF-ConvNet & 0.140* & 1.205 & 5.465 & 0.815 & 0.986 & 0.975 & 0.151 & 1.284 & 5.175 & 0.756 & 0.934 & 0.976 & 0.275 & 1.370 & 4.145 & 0.728 & 0.906 & 0.982 \\ \hline BTS & 0.225 & 4.960 & 5.039 & 0.599 & 0.753 & 0.844 & 0.247 & 2.180 & 7.965 & 0.503 & 0.745 & 0.871 & 0.824 & 3.233 & 9.300 & 0.512 & 0.674 & 0.770 \\ AdaIns & 0.301 & 2.993 & 8.761 & 0.415 & 0.642 & 0.782 & 0.165 & 1.133 & 8.512 & 0.721 & 0.889 & 0.957 & 0.317 & 3.500 & 0.944 & 0.416 & 0.626 & 0.751 \\ \hline TianDepth & 0.111 & 0.728 & 4.866 & 0.835 & 0.905 & 0.955 & 0.136 & 0.987 & 0.724 & 0.775 & 0.926 & 0.975 & 0.146 & 1.127 & 5.970 & 0.761 & 0.911 & 0.964 \\ DepthFormer & 0.135 & 1.102 & 5.853 & 0.783 & 0.905 & 0.955 & 0.162 & 1.290 & 6.498 & 0.713 & 0.885 & 0.953 & 0.274 & 2.766 & 3.566 & 0.501 & 0.691 & 0.801 \\ GLFDDepth & 0.099 & 0.573 & 4.212 & 0.872 & 0.860 & 0.986 & 0.092 & 0.813 & 3.950 & 0.874 & 0.970 & 0.992 & 0.099 & 0.513 & 3.950 & 0.874 & 0.970 & 0.992 \\ \hline \hline \end{tabular} \end{table} TABLE VIII: Quantitative results on synthetic texture-shifted (Watercolor, Pencil-sketch, Style-transfer) datasets. Fig. 8: Depth map results on synthetic texture-shifted (Watercolor, Pencil-sketch, Style-transfer) datasets. #### 4.5.3 Analysis on feature representation from backbones To analyze the internal properties of CNNs and Transformers, we employ the centered kernel alignment (CKA), which is the similarity measure of internal representations of neural networks by following previous works [53, 54, 83]. The CKA is widely used to analyze feature representations of neural networks [53, 84], because of its invariant properties to the orthogonal transformation of representations and isotropic scaling [54]. We freeze the encoder $E$ of the depth networks trained on the original KITTI dataset. We extract features $z_{a},z_{b}$ from the last layer of the encoder by passing the original image $\mathbf{I}_{a}$ and texture-shifted image $\mathbf{I}_{b}$, respectively. Then, the CKA is computed given $K=z_{a}z_{a}^{T}$ and $L=z_{b}z_{b}^{T}$ with $m$ number of samples as follows: \[CKA(K,L)=\frac{HSIC(K,L)}{\sqrt{HSIC(K,K)HSIC(L,L)}}, \tag{7}\] \[HSIC(K,L)=(KHLH)/(m-1)^{2}, \tag{8}\] where $H$ is the centering matrix $H_{n}=I_{n}-\frac{1}{n}\mathbf{1}^{T}$. The overall process is illustrated in Fig. 10. We use 697 image pairs from the KITTI Eigen dataset and their corresponding texture-shifted datasets. We compute the CKA with three types of texture-shifted datasets, including watercolor, pencil-sketch, and style-transfer, as shown in Fig. 9. The results of feature similarity show a similar aspect of the quantitative results on synthetic texture-shifted datasets in Tab. VIII. MF-Ours and GLPDepth extract the most similar features from the original and texture-shifted datasets in the self-supervised and supervised methods, respectively. Generally, Transformer-based models (MF-ViT, MF-Twins, GLPDepth, MF-Ours, TransDepth, DepthFormer) extract features of texture-shifted images similar to the original images. On the other hand, most CNN-based models (Monodepth2, PackNet-SfM, R-MSFM6, BTS, AdaBins) extract inconsistent features from original and texture-shifted images. The results are consistently observed regardless of the types of texture changes. It demonstrates that the robustness to environmental changes comes from consistent feature extraction from encoders. These results also support our observations in Sec. 4.5.2 that the Transformers have a strong shape-bias whereas CNNs have a strong texture bias. Moreover, models with shape-biased representation show better generalization performance than models with texture bias for monocular depth estimation. We also observe that the pure Transformer-based methods (MF-ViT, MF-Twins, GLPDepth) and CNN-Transformer hybrid methods (MF-Ours, TransDepth, DepthFormer) show similar properties. The experiments provide us with some interesting observations on modernized network structures. First, it is noteworthy that MF-ConvNeXt, one of the CNN-based models, shows a relatively higher CKA similarity than the other CNN-based models. Even the similarity of the models is on par with Transformer-based models. ConvNeXt consists of a patchify stem, and a depth-wise convolution [85] with a large receptive field that mimics the self-attention mechanism using Transformer's macro design [49]. The design intuition of ConvNeXt makes that ConvNeXt can focus on global information, similar to self-attention. Because of the modernized macro design, MF-ConvNeXt consisting of only convolutional layers is robust to texture-shifted datasets and has a strong shape-bias. Second, unlike other Transformer-based models, MF-RegionViT is sensitive to texture changes, similar to CNN-based models. It is highly biased toward texture information rather than shape information, even though it is a transformer-based model. RegionViT [47] employs region-to-local attention, which is getting interaction between local regions. It improves the locality of the networks through local attention by capturing fine-grained spatial information. We believe that the strong locality makes MF-RegionViT rely heavily on spatial information. Based on these experiments, we find that a self-attention module capturing global information, strengthen the shape-bias of the network. We also observe that the locality from local attention or the intrinsic property of CNNs improves the texture-bias of the network. In terms of network design, Fig. 10: Illustration of CKA [53] similarity computation. Fig. 9: Box-and-Whisker plot of CKA similarity of all competitive methods on three different synthetic texture-shifted datasets. The blue and red tones indicate CNN-based and Transformer-based models, respectively. The circle and triangle indicate self-supervised and supervised methods, respectively. [MISSING_PAGE_POST] a self-attention mechanism is ultimately key to generalization. Additionally, the locality of the network weakens generality, even if it helps to optimize in the training process. ### _Evaluation on real-world texture-shifted datasets_ In addition, we conduct experiments on practical texture changes that can occur in real driving scenarios to demonstrate their applicability to the real-world. We evaluate all models using practical texture-shifted datasets consisting of the night (Oxford RobotCar [44]), foggy (Foggy CityScapes [46]), and rainy driving scenes (Rainy CityScapes [45]). Overall, the experiment demonstrates that the properties of CNNs and Transformers identified in Sec. 4.5 are similarly observed in real-world texture-shifted scenarios. Quantitative and qualitative results and detailed analyzes for each network are described in the following paragraphs. The depth map results of each model are shown in Fig. 11. Similar to the experiments in Sec. 4.5.2, Transformer-based models show plausible depth maps, except MF-RegionViT. On the other hand, CNN-based models fail to estimate depth even for objects such as a car in scenes similar to the training dataset, with the exception of MF-ConvNeXt. The quantitative results in Tab. IX also show that CNN-based models have higher errors than Transformer-based models. MF-SLaK is no different from other CNN models, but MF-ConvNeXt shows low errors and feasible depth maps by mimicking Transformers, as shown in Fig. 11. MF-Twins show similar results to other Transformer-based models because only a weak locality is added compared to MF-RegionViT. However, MF-RegionViT indicates texture-biased results like CNN-based models due to its strong locality. The results on rainy and foggy datasets (Rainy and Foggy CityScapes) also indicate similar aspects to the night scenes. Transformer-based models estimate the plausible depth and distinguish even tiny objects such as bollards. On the contrary, CNN-based models predict incorrect depth maps. In particular, CNN-based models exhibit significant errors in areas such as roads despite similar texture to training datasets. MF-ConvNeXt and MF-Twins also infer plausible depth maps regardless of changes in the weather environment, but MF-RegionViT and MF-SLaK estimate bizarre depth maps. In these experiments, we observe that textural shifts like weather and illumination changes confuse CNN-based models. We also find that CNN-based models are affected by changes in texture as well as loss of texture information, which can be a fatal problem in real-world driving applications. ## 5 Conclusion In this paper, we have proposed a self-supervised monocular depth estimation method called MonoFormer (MF-Ours). More importantly, we have presented an in-depth analysis of the generalization performance of various modernized backbone structures as well as state-of-the-art self-supervised and supervised methods for monocular depth estimation using various datasets. The in-distribution datasets are used to compare common performance on the KITTI benchmark, while the out-of-distribution and texture-shifted datasets are used to compare the generalization performance. We deeply analyze the properties of the features extracted from each model using the synthetic texture-shifted datasets. Finally, we demonstrate the applicability of the model in real-world scenarios of changing environments using day-night, foggy and rainy datasets. Through extensive experiments, we observe that MF-Ours and GLPDepth have the best generalization performance among all self-supervised and supervised methods, respectively. The supervised method, GLPDepth, achieves the best generalization performance among all monocular depth estimation methods. More interestingly, we provide three important observations about the generality of monocular depth estimation. First, CNN-based models heavily rely on texture information to recognize scenes and objects, while Transformer-based models use shape information to a greater degree for the monocular depth estimation task. Second, texture-biased representations result in poor generalization performance for environmental changes such as differences in illumination and weather. In contrast, shape-biased representations are more robust to such texture-shifted environments. Lastly, ConvNeXt and RegionViT are CNN-based and Transformer-based, respectively, but have properties different from those of the backbone structure. It shows that the texture-bias and shape-bias are not the intrinsic properties of CNNs and Transformers, respectively. Instead, the intrinsic locality of CNNs induces texture-biased characteristics, while the self-attention mechanism, the base layer of Transformers, induces shape-biased properties. We believe our observations provide valuable insights into best practices in network design not only for monocular depth estimation but also for a variety of dense prediction tasks such as semantic segmentation, depth completion, normal estimation, etc. Based on these observations, the best way to design generalized networks for dense prediction is to utilize multi-self-attention (MSA) from Transformers and supplement the local information loss of global self-attention by using weak local attention as an auxiliary. \begin{table} \begin{tabular}{l c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{}{Datasets}} & \multicolumn{5}{c}{Dataset RobotCar} & \multicolumn{5}{c}{Foggy CityScapes} & \multicolumn{5}{c}{Foggy CityScapes} \\ \cline{2-13} \cline{13-13} \multicolumn{1}{c}{} & Abs Rel & Std & KLR & Std & KLR & Std & KLR & Std & KLR & Std & KLR & Std & KLR & Std & KLR & Std & KLR & Std \\ \hline MonoFormer & 1.03 & 0.04* & 0.19 & 0.204 & 0.40 & 0.564 & 0.48 & 0.080 & 0.109 & 0.514 & 0.720 & 0.853 & 1.215 & 1.294 & 1.299 & 0.138 & 0.134 & 0.489 \\ ParaNet-Soft & 1.00 & 0.05 & 0.19 & 0.193 & 0.78 & 0.536 & 0.457 & 0.095 & 0.104 & 0.545 & 0.782 & 0.873 & 1.515 & 1.711 & 1.699 & 0.244 & 0.446 \\ R-ASFMG & 0.951 & 0.043 & 0.160 & 0.235 & 0.362 & 0.469 & 0.445 & 0.082 & 0.124 & 0.442 & 0.607 & 0.890 & 1.310 & 0.308 & 1.097 & 0.171 & 0.315 & 0.441 \\ Our-ConvN & 0.947 & 0.029 & 0.19 & 0.310 & 0.546 & 0.293 & 0.353 & 0.020 & 0.483 & 0.564 & 0.992 & 0.836 & 0.396 & 0.473 & 0.729 & 0.800 \\ MF-SLaK & 0.916 & 0.041 & 0.160 & 0.239 & 0.456 & 0.623 & 0.839 & 0.151 & 0.285 & 0.299 & 0.440 & 0.652 & 0.856 & 0.510 & 0.460 & 0.356 & 0.611 & 0.822 \\ \hline MF-V & 0.035 & 0.029 & 0.19 & 0.235 & 0.434 & 0.402 & 0.368 & 0.038 & 0.026 & 0.991 & 0.394 & 0.947 & 0.830 & 0.981 & 0.866 & 0.285 & 0.479 \\ MF-RegionViT & 1.067 & 0.041 & 0.160 & 0.290 & 0.456 & 0.623 & 0.640 & 0.088 & 0.167 & 0.337 & 0.645 & 0.782 & 0.708 & 0.773 & 0.805 & 0.251 & 0.840 & 0.676 \\ MF-Series & 0.831 & 0.027 & 0.199 & 0.300 & 0.516 & 0.686 & 0.350 & 0.209 & 0.687 & 0.648 & 0.842 & 0.920 & 0.756 & 0.825 & 0.839 & 0.285 & 0.356 & 0.688 \\ M-Ours & 0.869 & 0.020 & 0.19 & 0.322 & 0.355 & 0.691 & 0.316 & 0.225 & 0.284 & 0.874 & 0.879 & 0.298 & 0.780 & 0.707 & 0.732 & 0.233 & 0.574 & 0.738 \\ \hline BIS & 0.070 & 0.296 & 0.150 & 0.340 & 0.604 & 0.725 & 1.092 & 0.524 & 0.117 & 0.372 & 0.716 & 0.794 & 1.366 & 1.822 & 0.710 & 0.399 & 0.252 & 0.756 \\ AdaLars & 0.677 & 0.039 & 0.160 & 0.371 & 0.640 & 0.721 & 1.441 & 0.534 & 0.158 & 0.316 & 0.467 & 0.744 & 1.299 & 1.452 & 0.466 & 0.330 & 0.585 & 0.771 \\ \hline \hline \multicolumn{1}{c}{ImageNet} & 0.531 & 0.329 & 0.350 & 0.558 & 0.768 & 0.631 & 0.075 & 0.809 & 0.831 & 0.797 & 0.880 & 1.781 & 1.030 & 0.793 & 0.885 & 0.698 & 0.771 \\ DepthFormer & 0.550 & 0.039 & 0.19 & 0.420 & 0.629 & 0.756 & 0.669 & 0.083 & 0.803 & 0.846 & 0.794 & 0.857 & 1.039 & 1.016 & 0.816 & 0.473 & 0.690 & 0.764 \\ GLPDepth & 0.538 & 0.379 & 0.199 & 0.422 & 0.673 & 0.780 & 0.594 & 0.070 & 0.301 & 0.561 & 0.799 & 0.872 & 1.030 & 0.528 & 0.560 & 0.513 & 0.742 & 0.822 \\ \hline \hline \end{tabular} \end{table} TABLE IX: Quantitative results on real-world texture-shifted (RobotCar, Foggy and Rainy CityScapes) datasets.
2308.05775	Application of dense neural networks for manifold-based modeling of flame-wall interactions	Artifical neural networks (ANNs) are universal approximators capable of learning any correlation between arbitrary input data with corresponding outputs, which can also be exploited to represent a low-dimensional chemistry manifold in the field of combustion. In this work, a procedure is developed to simulate a premixed methane-air flame undergoing side-wall quenching utilizing an ANN chemistry manifold. In the investigated case, the flame characteristics are governed by two canonical problems: the adiabatic flame propagation in the core flow and the non-adiabatic flame-wall interaction governed by enthalpy losses to the wall. Similar to the tabulation of a Quenching Flamelet-Generated Manifold (QFM), the neural network is trained on a 1D head-on quenching flame database to learn the intrinsic chemistry manifold. The control parameters (i.e. the inputs) of the ANN are identified from thermo-chemical state variables by a sparse principal component analysis (PCA) without using prior knowledge about the flame physics. These input quantities are then transported in the coupled CFD solver and used for manifold access during simulation runtime. The chemical source terms are corrected at the manifold boundaries to ensure boundedness of the thermo-chemical state at all times. Finally, the ANN model is assessed by comparison to simulation results of the 2D side-wall quenching (SWQ) configuration with detailed chemistry and with a flamelet-based manifold (QFM).	Julian Bissantz, Jeremy Karpowski, Matthias Steinhausen, Yujuan Luo, Federica Ferraro, Arne Scholtissek, Christian Hasse, Luc Vervisch	2023-08-10T09:06:32Z	http://arxiv.org/abs/2308.05775v1	# Application of dense neural networks for manifold-based modeling of flame-wall interactions ###### Abstract Artificial neural networks (ANNs) are universal approximators capable of learning any correlation between arbitrary input data with corresponding outputs, which can also be exploited to represent a low-dimensional chemistry manifold in the field of combustion. In this work, a procedure is developed to simulate a premixed methane-air flame undergoing side-wall quenching utilizing an ANN chemistry manifold. In the investigated case, the flame characteristics are governed by two canonical problems the adiabatic flame propagation in the core flow and the non-adiabatic flame- wall interaction governed by enthalpy losses to the wall. Similar to the tabulation of a Quenching Flamelet-Generated Manifold (QFM), the neural network is trained on a 1D head-on quenching flame database to learn the intrinsic chemistry manifold. The control parameters (i.e. the inputs) of the ANN are identified from thermo-chemical state variables by a sparse principal component analysis (PCA) without using prior knowledge about the flame physics. These input quantities are then transported in the coupled CFD solver and used for manifold access during simulation runtime. The chemical source terms are corrected at the manifold boundaries to ensure bounded-ness of the thermo-chemical state at all times. Finally, the ANN model is assessed by comparison to simulation results of the 2D side-wall quenching (SWQ) configuration with detailed chemistry and with a flamelet-based manifold (QFM). 1 Application in Energy and Combustion Science 1 ## 1 Introduction In the transition towards sustainable combustion technologies, numerical simulations play a crucial role for the rapid design of carbon-neutral or carbon-free combustion systems for power generation and transportation. While highly-resolved detailed chemistry (DC) simulations are essential to understand physical phenomena and can serve as the basis for model development and validation, their application to practical combustors is often unfeasible due to prohibitive computational costs. Consequently, there exists a demand for accurate and numerically efficient chemistry reduction approaches. One option are reduced-order models utilizing chemistry manifolds [1, 2, 3, 4]. These methods are based on the tabulation of pre-calculated thermo-chemical states which are pa- parameterized and accessed by control variables. The control variables are then usually transported in the CFD simulation and used to retrieve the thermo-chemical state from the manifold during simulation runtime. Given a suitable manifold is used, these methods combine the high accuracy of a detailed chemistry (DC) simulation with low computational costs. However, for complex configurations involving a significant number of different combustion phenomena (e.g., multi-phase combustion, pollutant formation modeling or flame-wall interactions), more control variables, thus, additional manifold dimensions are required. The memory demand to store the man- field quickly becomes intractable considering memory-per-core limitations on high-performance computers. Additionally, the tabulation of the manifold becomes increasingly difficult since the control variables are often not linearly independent, which complicates a non-overlapping data arrangement and efficient manifold access. An emerging alternative is data-driven modeling, or machine learning (ML), using artificial neural networks (ANN). Neural networks are universal approximators capable of learning any correlation between arbitrary input data (control variables) with corresponding outputs. This holds true even if the control variables are not linearly independent. This property of ANNs can be exploited to represent a chemistry manifold. An ANN is memory efficient and the storage size changes only slightly with the number of control variables. Furthermore, ANNs profit from an enormous performance increase on specialized accelerator hardware, which is one focus of current computational hardware development. The first ANN application for chemistry modeling in a Large Eddy Simulation (LES) was performed by Flemming et al. [5], who used the approach to model the Sandia Flame D. The authors achieved a memory reduction by three orders of magnitude with a minimal increase in computational time in comparison with a tabulated manifold approach. Ihme et al. [6] focused on the optimization of ANN architectures for each output variable using a generalized pattern search. These models were subsequently applied in an LES of the Sydney bluff-body swirl-stabilized SMH1 flame. Again, the ANN showed a comparable accuracy to a tabulated manifold with acceptable computational overhead for LES applications. Recently, several investigations have been carried out applying machine learning in reactive flow simulations for manifold representation [7, 8, 9, 10, 11, 12, 13], turbulence-chemistry interactions [14, 15, 16, 17] and modeling chemical kinetics [18, 19, 20, 21, 22]. This was made possible by the development of open-source deep learning frame- works that followed the breakthrough of deep learning in the field of computer vision in the past decade. Although the training of ANNs has been simplified by these frameworks, many works have reported a lack of model accuracy in case input values approach the manifold boundaries. This applies in particular to non-linear quantities, such as chemical source terms. In a direct numerical simulation (DNS) of a turbulent syngas oxy-flame, Wan et al. [23, 24] trained additional neural networks for a normalized oxygen mass fraction $Y_{02}>0.9$ to improve the model accuracy close to the unburnt conditions, where low, but non-zero, reaction rates occured. Similarly, Ding et al. [10] used a multiple multilayer perceptron (MMP) approach, where additional models were trained on increasingly smaller intervals around zero, where the relative error of the prediction is large compared to the absolute error. During the simulation, a model cascade is employed, where the decision which output is used is based on the output value of the previous model. This methodology was applied in LES of the Sandia flame series. Another approach to increase the overall prediction accuracy of ML models is to divide the manifold using clustering algorithms such as self-organizing maps [8, 12, 25, 26] or k-means clustering [27] and to use different ANNs for the prediction of the subsets of the manifold. However, in some cases, hundreds of networks had to be trained [8, 25], which introduces additional overhead for the model selection during simulation runtime. For thorough overviews of machine learning approaches in the context of combustion, the reader is referred to Zhou et al. [28] and Ihme et al. [29]. While there exist many coupled simulations with models based on tabulated reduced-order manifolds, the literature on ML-based manifold modeling is still scarce. First results have been encouraging, but several challenges, such as the handling of non-linear terms, a robust feature selection (i.e. suitable control variables), or the required number of networks for an accurate man-ifold representation, are active areas of research. In this work, a ML model based on dense neural networks (DNN) is coupled to a CFD solver and utilized for the 2D simulation of a laminar premixed methane-air flame undergoing side- wall quenching (SWQ). The 2D SWQ case is well-established [30, 31, 32] and includes two essential flame regimes: an unstretched adiabatic flame regime and a non-adiabatic flame quenching at the wall. Even when ignoring unsteady[33] or turbulent effects [34, 35] standard flamelet models fail to capture some physics, as discussed by Efimov et al. [36]. Significant modeling efforts were required to develop advanced flamelet models that can accurately predict the pollutant formation, specifically CO[31, 36]. With this background, the objective of this work is threefold: * to demonstrate the application of a purely data-driven approach for modeling low-order chem-istry manifolds in the simulation of the laminar side-wall quenching configuration; * to identify suitable input parameters for the ML-based manifold without using prior knowledge about the flame physics (applying a sparse principal component analysis [37]); * to develop a reliable treatment of source terms at the manifold boundaries to ensure boundedness of the thermo-chemical state during the coupled simulation. The laminar SWQ configuration provides a benchmark for combustion modeling that is stiff- ciently challenging and therefore suitable for assessing the predictive capabilities of the ML-based approach developed in this study. Analogous simulation results obtained with detailed chemistry and with a flamelet-based model (QFM) serve as reference datasets. The paper is structured as follows: Section 2 describes the numerical setup of the investigated configurations, namely the 1D head-on quenching (HOQ) and the 2D SWQ. In Section 3, the machine learning methods are outlined. In Section 4, the results for the HOQ and the SWQ cases are analyzed. First the wall heat flux is compared for the HOQ case in order to verify the model. Thereafter, an analysis of the local heat release and the thermo-chemical state for the SWQ configuration is carried out. Finally, conclusions are drawn in. Section 5 ## 2 Numerical setups In this section, an overview of the numerical setups is provided. First, the 1D HOQ configuration is addressed, which is used both for the ANN training and the QFM generation. Thereafter, the more complex 2D SWQ configuration is described. ### 1D head-on quenching (HOQ) In the 1D HOQ configuration, a premixed laminar flame propagates perpendicular towards a wall, where it extinguishes due to heat losses. The 2 cm long domain is discretized by an equidistant mesh with 2000 points (resolution of 10 $\mu$m) and time integration is realized by a fully implicit backward differentiation formula (BDF). For the initialization, an adiabatic, stoichiometric freely propagating methane-air flame is used with a fresh gas temperature of 300 K. The wall temperature is fixed at 300 K and molecular transport is modeled using the unity Lewis number assumption. Furthermore, the detailed GRI 3.0 mechanism is used [38]. All flamelet simulations are performed with an in-house solver [39], and the setup has been validated previously by Luo et al. [40]. ### 2D side-wall quenching (SWQ) Following the setup of previous works [30, 32], the simulation domain of the generic SWQ config- uration consists of a two-dimensional rectilinear mesh (30 mm x 6 mm) with an uniform cell size of $\Delta=50$$\mu$m, see Fig. 1. The numerical setup has been validated extensively [30, 32] against experimental data [41, 42]. The inlet flow is divided into a fresh (5.5 mm) and burned gas section (0.5 mm), where the latter is used for flame stabilization. The fresh gas is initialized with a premixed stoichiometric methane-air mixture at ambient conditions ($T=300$ K; $p=1$ atm), while the burned gas consists of hot exhaust gasses at equilibrium conditions. As inlet velocity, a parabolic inflow profile is used and the burned gas velocity is set to 3.81 m/s, compensating for the density difference between fresh and burned gasses. The velocity profile is shown in Fig. 1. At the wall, a constant temperature of 330 K is assumed in accordance with previous studies [32, 42]. This represents an additional challenge for the DNN model, which has to capture the preheating at the wall, and this effect must be considered in the training. For the species mass fractions a zero gradient and for the velocity a non-slip boundary condition is applied. At the outlets, zero gradient boundary conditions are applied to temperature, species mass fractions, and velocity. The spatial and temporal discretization scheme, as well as the time step control, were adapted from Steinhausen et al. [32]. A detailed chemistry reference solution and corresponding simulation results with a tabulated manifold approach were obtained with in-house solvers based on OpenFOAM (v2006). Different tabulated manifold approaches have been developed and validated for this setup [32], from which the quenching flamelet-generated manifold (QFM) by Steinhausen et al. [32] is selected to serve as a reference model. The QFM is constructed from the same 1D HOQ dataset, which is subsequently used for the training of the neural network, and it is parameterized by the control variables enthalpy and CO2 mass fraction. ## 3 Machine learning methodology This section gives an overview of the ML methodology employed in this work. First, the generation of the ML training dataset is described. Thereafter, the training parameters, the chosen network architecture, and the method for identifying suitable model inputs are specified. A correction method for the non-linear source terms is proposed and discussed in Section 3.3. Afterwards, the assessment of the ML model accuracy is described and finally, the coupling of the ML model to the CFD code is briefly outlined. ### Training data generation The flamelet-based training data is generated by a transient HOQ simulation, as described in Section 2.1. The left plot in Fig. 2 displays the training data points in the original physical space ($x$) and time ($t$) colored by the CO mass fraction. The isothermal wall is located at the left boundary ($x=0$ mm). Additionally, this plot shows the writing interval of the flame solution. Initially, the solutions are written at a constant time step, whereas when the flame approaches the wall and heat losses start to occur, the writing interval is then determined by the change of enthalpy at the wall by $1\cdot 10^{4}\,\mathrm{J}$/kg. Additionally, 10 freely propagating flames with varying inlet temperatures ranging from 300 K to 340 K are included in the training dataset. This extension of the manifold is carried out in order to account for elevated enthalpy levels in proximity of the isothermal wall in the SWO configuration ($T_{\mathrm{wall}}=330$ K). The same procedure was employed for QFM tables in [32] and is visualized in the right plot in Fig. 2, where the HOQ manifold is displayed in the transformed PV and T space, with the added freely propagating flames located above the red line. It can be observed that the additional flamelets extend the manifold at the upper boundary. In total, the dataset consists of 310 1D flamelets of different enthalpy levels, resulting in 0.62 million points. Furthermore, the data is randomly split into a training (90%) and a validation (10%) dataset. The latter is used to evaluate the predictive capabilities of the model and to monitor the training process to prevent overfitting. Additionally, all dataset entries are normalized to an interval of [0, 1] (min-max scaling) ensuring similar feature value ranges which promotes convergence of the gradient-based optimization algorithm. ### Neural network architecture and training In order to represent the manifold, which is inherent to the dataset described above, two re- quiements have to be met for the ML approach: (1) a suitable ANN architecture has to be chosen together with a training algorithm, and (2) proper input quantities (i.e., features, or control variables) have to be identified. Here, the neural network architecture of a Dense Neural Network (DNN) is chosen. Fig. 3 shows the structure of a DNN, which is defined by several sequential layers of neurons. Table 1 summarizes the chosen hyperparameters for the ML training, which is carried out using the PyTorch library [43]. The learning rate is reduced by a factor of 5 if the loss does not decrease for 20 epochs. For the selection of suitable inputs (or control variables) for the DNN, a sparse principal component analysis (SPCA) [37] is performed. Contrary to a regular PCA, for which the principal compo- nents are a dense linear combination, i.e. a combination of all thermo-chemical state variables [44], the SPCA algorithm attempts to minimize the number of variables contributing to the principal components, i.e., a sparse linear combination, which makes the result more interpretable. As a result, the main principal components identified from the SPCA only rely on a few of the variables that define the thermo-chemical state. The variables are normalized as described in Section 3.1 prior to the SPCA. The first three sparse principal components (SPCs) extracted from the training dataset are shown in Table 2 together with their constituents and associated explained variance. The latter describes the ratio of variance contained in the individual sparse principal component compared to the sum of all sparse principal component variances in the dataset, i.e. a measure of the information contained in the variable. The variances of all SPCs sum up to unity. Interestingly, the first sparse principal component (SPC1) resembles a progress variable (PV) consisting of the species involved in the global methane oxidation reaction. Similar combinations of species have been used for the definition of a primary progress variable in the modeling of methane-air combustion [45] and it is emphasized that SPCI is obtained here without using prior knowledge about the flame physics. The second sparse principal component (SPC2) mainly consists of the temperature (T) and negligible Figure 1: Schematic view of the SWQ burner and the numerical subdomain used for the simulation. All scales are given in mm. The velocity profile at the inlet is also shown. contributions of three species. Both, SPC1 and SPC2, are in accordance with the two governing flame regimes contained in the 1D HOQ training dataset: (1) the propagation of a freely propagating flame (characterized by PV), and (2) flame quenching caused by heat losses to the wall (characterized by T). Therefore, PV and T are chosen as model inputs, see Fig. 3. These principal components also agree well with the inputs of the tabulated manifold approaches, where a progress variable and enthalpy have been chosen [32, 36]. For convenience, PV is defined as 2 $\text{SPC1}\simeq Y_{\text{F220}}+Y_{\text{C02}}-Y_{\text{O2}}-Y_{\text{C04}}$ in the following. In the coupled simulation, SPC1 and SPC2 are solved in addition to the velocity and pressure in the employed PHMPLE algorithm, a combination of the SIMPE and PISO algorithm [46]. Assuming unity Lewis number diffusion for all species, the transport equations for the SPC1, i. e. progress variable (PV), and the SPC2, i.e. temperature (T) read \[\begin{split}\frac{\partial\rho}{\partial t}+\frac{\partial}{ \partial x_{j}}\left(\begin{array}{c}\rho\\ \rho\end{array}u_{j}Y_{\text{PV}}\right)=\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\frac{\partial Y_{\text{PV}}}{\partial x_{j}},\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\frac{\partial}{\partial x_{j}},\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\frac{\partial}{\partial x_{j}},\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\frac{\partial}{\partial x_{j}},\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)+\\ \frac{\partial}{\partial x_{j}}\left(\begin{array}{c}\lambda\\ \rho\end{array}\right)-\frac{\rho}{\partial x_{j}}\underbrace{\left(\sum_{k=1}^ {N}c_{\text{p,}k}\left(-D\frac{\partial Y_{k}}{\partial x_{j}}\right)\right)} _{\text{difff.s}},\end{split} \tag{1}\] where $\rho$ is the density, $u$ the velocity, $D$ the diffusion coefficient, $\dot{\alpha}_{\text{PV}}$ the progress variable source term, $c_{\text{p}}$ the heat capacity, $\dot{\alpha}$ the heat conductivity, and $\dot{\alpha}_{\text{T}}$ is source term of the temperature, i.e. the heat release rate (HRR), respectively. The term $D_{\text{difff.s}}$ in Eq. (1) represents the temperature diffusion caused by the species diffusion. During the numerical solution of Eqs. (1) and (2), $PV$ and $T$ are used as inputs for the employed DNN to retrieve the thermochemical quantities highlighted in Eqs. (1) and (2) (marked by gray boxes) as outputs from the DNN. For the training process, the number of layers and neurons was varied, until a network architecture with three hidden layers containing 64 neurons each has proven sufficient to accurately represent the thermo-chemical variables included in the chemistry manifold. Furthermore, a different \begin{table} \begin{tabular}{l l} \hline Hyperparameter & Property \\ \hline Epochs & 1000 \\ Batch size & 5000 \\ GPU & kTela K200km \\ Optimizer & Adam algorithm \\ Initial learning rate & 0.001 \\ Loss function & Mean-Squared Error (MSE) \\ Activation function & Hyperbolic tangent (tanh) \\ \hline \hline \end{tabular} \end{table} Table 1Hyperparameters for the training of the ML model. \begin{table} \begin{tabular}{l l l} \hline SPC & Thermo-chemical variables & Variance \\ \hline SPC1 & $-0.5Y_{\text{F0}}-0.5Y_{\text{F0}}$, & $0.365$ \\ & $0.5Y_{\text{F10}}+0.51Y_{\text{F0}}$, & $0.121$ \\ SPC2 & $-0.004Y_{\text{F10}}+0.009Y_{\text{F0}}$, & $0.121$ \\ & $-0.001Y_{\text{F10}}+1.07$ & $-0.001Y_{\text{F10}}+1.07$ \\ SPC3 & $-0.503Y_{\text{F10}}-0.492Y_{\text{F0}}$, & $0.113$ \\ & $-0.446Y_{\text{F0}}-0.335Y_{\text{F0}}$, & $+-$ (truncated) \\ \hline \hline \end{tabular} \end{table} Table 2Results of the SPCA for the 1D HOQ training dataset. Only the first three SPCs are shown here. SPC3 is truncated after the first four constituents. Figure 3: Network architecture of a dense neural network (DNN). The in- and outputs used in this work are shown for the respective layer. Figure 2: Scatter plot colored by the CO mass fraction for the head-on quenching flame in the physical space ($\alpha$) versus time ($t$) (left). The same dataset is mapped into progress variable ($PV$) and temperature ($T$) coordinates on the right. The additional preheated flamelets, which are incorporated into the manifold, are shown on the right above the red line. training strategy, namely the training of one DNN for each output, was evaluated without observable improvement on the accuracy or computational performance for the manifold investigated here. It is emphasized that this could be different for other cases. ### Source term modeling The accurate description of the chemical source terms can be challenging for ML approaches based on neural networks. Specifically, in a transient simulation small, but non-zero, source terms for fresh gas conditions or equilibrium conditions on the burn side of a flame (i.e. thermo-chemical states at the boundaries of the manifold) can lead to an error accumulation for the transported con trol variables (PV, _T_). Subsequently, the transported variables can drift to unphysical states outside the training interval, where the predictions by the DNN degrade and exacerbate the drift further. Due to their inherent, gradient-based optimization, neural networks asymptotically approach but cannot reach perfect accuracy in regression tasks, which is required here to accurately capture the flame physics. This aspect can easily be missed in the a-priori validation step, since the error at the boundary is small. Previous works proposed to train additional networks for subsets of the manifold [8, 12, 25, 26, 27]. This can decrease the error made at the manifold boundaries to an acceptable threshold, but it does not completely solve the issue and further introduces computational overhead since the appropriate model has to be identified for every computational element in every time step at simulation runtime. Here it was found enough to set the source terms to zero close to the PV boundaries. The predicted output of all source terms, namely $\dot{\alpha}_{PV}$ and $\dot{\alpha}_{r}$ is then conditioned on the progress variable at simulation runtime: \[\dot{\alpha}=\left\{\begin{array}{ll}0&\quad\text{if }\dot{\text{PV}}\leq 0\\ \text{DNN}(\text{PV},\text{T})&\quad\text{if }0\leq\dot{\text{PV}}\leq 1,\\ 0&\quad\text{if }\dot{\text{PV}}\geq 1,\end{array}\right. \tag{3}\] where $\dot{\text{PV}}$ is the scaled progress variable. $\dot{\text{PV}}=0$ and $\dot{\text{PV}}=1$ correspond to $\text{PV}_{\text{min}}=-0.275$ and $\text{PV}_{\text{max}}=0.271$ respectively. These values are given by the manifold and are constants in this case, because only one mixture fraction level is considered. Generally, $\text{PV}_{\text{min}}$ and $\text{PV}_{\text{max}}$ could also be defined as a function of the mixture fraction if required. The above constraint is realized by a mask vector, which is described in more detail in the next section. It ensures that the numerical solution of the input values of the DNN stays bounded to the known training interval. It is noted that this measure can be easily applied to arbitrary inputs of the DNN. ### Model validation In order to measure and quantify its accuracy, the ML model is tested with the validation dataset (not included in the training). The prediction quality is measured with the coefficient of determination defined as \[R^{2}=1-\left[\sum_{i=1}^{N}\left(\Phi_{i}-\widehat{\Phi}_{i}\right)^{2}\right] \left[\sum_{i=1}^{N}\left(\Phi_{i}-\widehat{\Phi}\right)^{2}\right]^{-1}\] where $\Phi_{i},\widehat{\Phi},\widehat{\Phi},$ and $N$ are the true value, the mean value of $\Phi$, the network's prediction, and the number of samples, respectively. For the ML model employed here, all model outputs reach an $R^{2}$ score of $0.999$. Particularly, CO has been investigated and identified as a relevant quantity for flame-wall in teractions in previous studies [30, 32, 36]. Fig. 4 allows to assess the overall predictive capabilities of the tabulated manifold (QFM) and the ML model for the CO mass fraction in comparison to the two reference datasets (DC) of HOO and SWQ. Extracting the inputs from the DC datasets, the QFM and ML model are utilized to predict the CO mass fraction which is then compared to the DC references (_a-priori analysis_). Optimal predictions are located on the diagonal line in Fig. 4, where the predicted CO is identical to the reference DC value. It can be observed that both the QFM and ML model show high accuracy for the HOQ and SWQ datasets. Only slight overestimates are observed for the SWQ for both the ML and QFM model, as indicated by few gray points above the diagonal. This over-prediction of CO is caused by a substantially (factor of 2) higher heat loss to the wall in the HOQ configuration (i.e. the training data) compared to the SWQ configuration. This difference can be attributed to the different directions of the temperature gradients in the respective cases [32, 36, 47]. Overall, the ML model yields satisfying results and is applied to a coupled simulations of both reference configurations next. ### Model coupling Before coupling the model to the CFD code two post-processing steps are performed: 1. In order to apply the network in the CFD, the scaling and rescaling operation of in- and outputs are added to the exported network. 2. All source terms are corrected based on the input of the progress variable according to Eq. (3). The first step allows the generic case-independent implementation of the ML interface into the CFD code. As outlined in the previous section, the source term correction is realized by means of a mask vector, that determines at runtime, which cells contain a thermo-chemical state located at the boundary of the manifold. Subsequently, the output vector of the source term is modified for positive values of the mask vector. Finally, the DNN is coupled to the OpenFOM-based solver via the PyTorch C++ API for usage in the coupled simulations. ## 4 Results and discussion In this section, two cases are simulated using the previously described ML model to represent the manifold. First, the model is verified on the 1D HOQ configuration, which was also used to generate the training data. Second, the model is applied to the laminar SWQ configuration described in Section 2.2. By this means, the predictive Figure 4: A-priori analysis of tabulated manifold (QFM) and the machine learning model (ML) on the detailed chemistry dataset of head-on quenching (HOQ) and side-wall quenching (SWQ) for the mass fraction of O. capabilities of the ML model are assessed for a flame-wall interaction case which is different from the training database. ### Results for the 1D HOQ configuration A snapshot of the coupled HOQ simulation with (ML) and without the source term correction (ML) is shown together with the DC result in Fig. 5. The ML model predicts small, non-zero temperature (and progress variable) source terms in the preheat zone of the flame, which accumulates to an unphysical temperature increase over the simulation runtime. At the beginning of the simulation, ML* underpredicts the flame propagation speed by approximately 30%, which later turns to an overprediction of similar order of magnitude after a considerable temperature increase. This effect can hardly be identified from the a-priori analysis and it depends on the initialization and duration of the simulation, and the absolute prediction error by the ML model for source terms at the manifold boundaries. In comparison, the ML model, which utilizes the source term correction described in Section 3.3, accurately describes temperature profile and flame speed ($<1\%$ deviation compared to the DC reference). Furthermore, Fig. 6 shows the wall heat flux for the ML model, the tabulated manifold (QFM), and the DC reference result. It is found that both ML and QFM models recover the overall trend shown by the DC reference result, but overpredict the wall heat flux at the quenching point, characterized as the point in time of maximum wall heat flux, by a similar order of magnitude. It is thereby verified, that the ML model yields comparable results to a tabulated manifold approach, generated from the same dataset used in the training of the neural network. ### Results for the 2D SWQ configuration The coupled simulation results obtained with the ML model for the 2D SWQ configuration are compared with a detailed chemistry and a tabulated chemistry calculation (QFM, c.f. Section 2.2). The numerical results are analyzed with respect to a relative coordinate system that uses the quenching height as the origin of the wall-parallel direction. The quenching height is defined by the maximum wall heat flux [30, 32]. Fig. 7 depicts the temperature contour of the SWQ configuration for the ML model as well as the DC and QFM references. The contours of the reduced models show qualitatively good agreement at the quenching point as well as the shape and position of the adiabatic flame branch. The isolines for three temperature levels (310 K, 320 K, and 330 K) indicate the heating of the fresh mixture (300 K) in close proximity of the wall, caused by the slightly elevated wall ten-perature (330 K). These preheated states are captured by adding preheated adiabatic flamelets to the training dataset (c.f. Section 3.1). Interestingly, the isoline for 330 K for the QFM shows a small deviation at the point where it meets the wall boundary. This difference can be attributed to the transport of enthalpy for the QFM model instead of temperature for the ML model. The transport of enthalpy requires special treatment of the constant temperature wall boundary condition. In this case, a secondary table is used to retrieve the wall enthalpy for a given progress variable and constant temperature. Whereas, for the ML model the temperature is transported and the isothermal boundary condition is directly imposed. #### 4.2.1 Analysis of local heat release rate Following previous investigations on SWQ [30, 32, 36] the local HRR as a global flame property is used to assess the predictive capabilities of the ML model. The 2D contours of the normalized HRR at the quenching point are displayed in the top row of Fig. 8. The HRR decreases considerably within 0.5 mm distance to the wall showing that the flame quenching process occurs very close to the wall. A detailed assessment of quenching distances for the given SWQ configuration is beyond the scope of this work. Here the reader is referred to Zirwe et al. [47], who investigated the influence of several factors (e.g., chemical mechanism, diffusion model, etc.) on the quenching distance of a SWQ configuration. Overall, a qualitatively good agreement of the HRR contour of the ML model with the DC and QFM references can be observed. To allow a quantitative comparison, HRR profiles are extracted at three different distances from the quenching point ($y_{\mathrm{q}}=0$), indicated by the horizontal white dotted lines in Fig. 8. The extracted profiles are shown in Fig. 9. At the three quenching heights $y_{\mathrm{q}}$ both, the peak position and value, of the HRR profiles obtained with the ML model align well with the DC reference solution. Small deviations can be observed at the quenching height $y_{\mathrm{q}}=0$ mm, where the peak of the ML model is slightly shifted towards the wall and for $y_{\mathrm{q}}=0.5$ mm, where the peak is slightly overestimated. However, the model yields similar results as the QFM, demonstrating that it is able to accurately capture the local HRR. #### 4.2.2 Analysis of the thermo-chemical state As previously mentioned, CO has been identified as a relevant quantity for flame-wall interactions in previous works [30, 32, 36]. The CO mass fraction and the temperature fields, which are obtained for the SWQ configuration by the DC, QFM and ML modeling approaches, are analysed next. Contours of the CO mass fraction around the quenching point are displayed in the bottom row of Fig. 8. Here, it can be observed, that the CO production has its maximum within the reaction zone of the flame. The high CO concentration in the quenching zone at the wall indicates an incomplete combustion, where further oxidation of CO was no longer possible. The local thermo-chemical state is further analysed by comparing the CO mass fraction over temperature at different wall distances in Fig. 10. The locations are indicated by the vertical white dotted lines drawn in Fig. 8. An adiabatic freely propagating premixed flame is additionally Figure 5: Snapshot of coupled HOQ simulations with a DC and two machine learning models with (ML) and without the source term correction (ML*) according to Eq. (3). Figure 6: Comparison of the wall heat flux obtained for coupled HOQ simulations with a DC model, a tabulated manifold approach (QFM), and a machine learning model (ML), relative to the quenching time. included as a reference. Close to the wall at $x=0.1$ mm significant differences between all the simulations and the adiabatic flame can be observed, indicating an incomplete combustion process. When moving away from the wall ($x=0.5$ mm and $x=2$ mm), the CO profile shifts from a quenching state due to heat loss to the wall to an almost adiabatic state. A slight over-prediction of CO can be observed for both the ML and QFM profile. This is in agreement with the conclusions made by the a-priori analysis (c.f. Section3.4). This over-prediction can be attributed to the different rates of heat transfer to the wall in H$\mathrm{\text{O}}$Q and SWQ [32, 36, 47]. To improve the prediction of CO, Efimov et al. [36] introduced an additional reactive control variable to account for this effect of Figure 8: 2D contours of the normalized local HRR (top) and CO mass fraction (bottom) for a detailed chemistry (DC, left), a tabulated manifold (QFM, right), and a machine learning (ML, middle) model in proximity of the quenching point for the SWQ configuration. In the upper row additional isolines of temperature $T$ are shown (dashed white lines). Figure 7: 2D contours of the temperature for the SWQ configuration obtained from simulations with detailed chemistry (DC, left), with the machine learning model (ML, middle) and with the tabulated manifold (QFM, right). isolines for three temperature levels (310 K, 320 K, and 330 K) are added to highlight the heating of the fresh gas mixture (300 K) by the slightly elevated wall temperature (330 K). Note that the domain is cropped to aid the visual inspection of the quenching point. Figure 9: Comparison for the HRR of the SWQ of the detailed (DC), tabulated (QFM), and machine learning (ML) chemistry. The profiles are extracted along wall-normal lines at different heights $y_{\varphi}$, which are displayed as white solid lines in Fig. 8.
2301.07482	FreshGNN: Reducing Memory Access via Stable Historical Embeddings for Graph Neural Network Training	A key performance bottleneck when training graph neural network (GNN) models on large, real-world graphs is loading node features onto a GPU. Due to limited GPU memory, expensive data movement is necessary to facilitate the storage of these features on alternative devices with slower access (e.g. CPU memory). Moreover, the irregularity of graph structures contributes to poor data locality which further exacerbates the problem. Consequently, existing frameworks capable of efficiently training large GNN models usually incur a significant accuracy degradation because of the currently-available shortcuts involved. To address these limitations, we instead propose FreshGNN, a general-purpose GNN mini-batch training framework that leverages a historical cache for storing and reusing GNN node embeddings instead of re-computing them through fetching raw features at every iteration. Critical to its success, the corresponding cache policy is designed, using a combination of gradient-based and staleness criteria, to selectively screen those embeddings which are relatively stable and can be cached, from those that need to be re-computed to reduce estimation errors and subsequent downstream accuracy loss. When paired with complementary system enhancements to support this selective historical cache, FreshGNN is able to accelerate the training speed on large graph datasets such as ogbn-papers100M and MAG240M by 3.4x up to 20.5x and reduce the memory access by 59%, with less than 1% influence on test accuracy.	Kezhao Huang, Haitian Jiang, Minjie Wang, Guangxuan Xiao, David Wipf, Xiang Song, Quan Gan, Zengfeng Huang, Jidong Zhai, Zheng Zhang	2023-01-18T12:51:13Z	http://arxiv.org/abs/2301.07482v3	ReFresh: Reducing Memory Access from Exploiting Stable Historical Embeddings for Graph Neural Network Training ###### Abstract A key performance bottleneck when training graph neural network (GNN) models on large, real-world graphs is loading node features onto a GPU. Due to limited GPU memory, expensive data movement is necessary to facilitate the storage of these features on alternative devices with slower access (e.g. CPU memory). Moreover, the irregularity of graph structures contributes to poor data locality which further exacerbates the problem. Consequently, existing frameworks capable of efficiently training large GNN models usually incur a significant accuracy degradation because of the inevitable shortcuts involved. To address these limitations, we instead propose ReFresh, a general-purpose GNN mini-batch training framework that leverages a historical cache for storing and reusing GNN node embeddings instead of re-computing them through fetching raw features at every iteration. Critical to its success, the corresponding cache policy is designed, using a combination of gradient-based and staleness criteria, to selectively screen those embeddings which are relatively stable and can be cached, from those that need to be re-computed to reduce estimation errors and subsequent downstream accuracy loss. When paired with complementary system enhancements to support this selective historical cache, ReFresh is able to accelerate the training speed on large graph datasets such as ogbn-papers100M and MAG240M by 4.6$\times$ up to 23.6$\times$ and reduce the memory access by 64.5% (85.7% higher than a raw feature cache), with less than 1% influence on test accuracy. + Footnote †: star}\) Corresponding author. + Footnote †: star}\) Corresponding author. ## 1 Introduction Graphs serve as a ubiquitous abstraction for representing relations between entities of interest. Linked web pages, paper citations, molecule interactions, purchase behaviors, etc., can all be modeled as graphs, and hence, real-world applications involving non-i.i.d. instances are frequently based on learning from graph data. To instantiate this learning process, Graph Neural Networks (GNN) have emerged as a powerful family of trainable architectures with successful deployment spanning a wide range of graph applications, including community detection [4], recommender systems [40], fraud detection [7], drug discovery [12] and more. The favorable performance of GNNs is largely attributed to their ability to exploit both entity-level features as well as complementary structural information or network effects via so-called _message passing_ schemes [13], whereby updating any particular node embedding requires collecting and aggregating the embeddings of its neighbors. Repeatedly applying this procedure by stacking multiple layers allows GNN models to produce node embeddings that capture local topology (with extent determined by model depth) and are useful for downstream tasks such as node classification or link prediction. Of course not surprisingly, the scale of these tasks is rapidly expanding as larger and larger graph datasets are collected. As such, when the problem size exceeds the memory capacity of hardware such as GPUs, some form of mini-batch training is the most common workaround [47, 14, 5, 46]. Similar to the mini-batch training of canonical i.i.d. datasets involving images or text, one full training epoch is composed of many constituent iterations, each optimizing a loss function using gradient descent w.r.t. a small batch of nodes/edges. In doing so, mini-batch training reduces memory requirements on massive graphs but with the added burden of frequent data movement from CPU to GPU. The latter is a natural consequence of GNN message passing, which for an $L$-layer model requires repeatedly loading the features of the $L$-hop neighbors of each node in a mini-batch. The central challenge of efficient GNN mini-batch training then becomes the mitigation of this data loading bottleneck, which otherwise scales exponentially with $L$; even for moderately-sized graphs this quickly becomes infeasible. Substantial effort has been made to address the challenges posed by large graphs using system-level optimizations, algorithmic approximations, or some combination thereof. For example, on the system side, GPU kernels have been used to efficiently load features in parallel or store hot features in a GPU cache [26, 27, 43]; however, these approaches cannot avoid memory access to the potentially large number of nodes that are visited less frequently. On the other hand, there are generally speaking two lines of work on the algorithm front. The first is based on devising sampling methods to reduce the computational footprint and the required features within each mini-batch. Notable strategies of this genre include neighbor sampling [14], layer-wise sampling [3, 50], and graph-wise sampling [46, 5]. However, neighbor sampling does not solve the problem of exponential growth mentioned previously, and the others may converge slower or to a solution with lower accuracy [49]. Meanwhile, the second line of work [2, 10, 23] stores intermediate node representations computed for each GNN layer during training as _historical embeddings_ and reuses them later to reduce the need for recursively collecting messages from neighbors. Though conceptually promising and foundational to our work, as we will later show in Sec. 2.3, these solutions presently struggle to simultaneously achieve _both_ high training efficiency and high model accuracy when scaling to large graphs, e.g., those with more than $10^{8}$ nodes and $10^{9}$ edges. To this end, we propose a new mini-batch GNN training solution with system and algorithm co-design for efficiently handling large graphs while preserving predictive performance. As our starting point, we narrow the root cause of accuracy degradation when using historical embeddings to the non-negligible accumulation of estimation error between true and approximate representations computed using the history. As prior related work has no direct mechanism for controlling this error, we equip mini-batch training with a _historical embedding cache_ whose purpose is to _selectively_ admit accurate historical embeddings while evicting those likely to be harmful to model performance. In support of this cache and its attendant admission/eviction policy, we design a prototype system called ReFresh: Reducing mEmory access FRom Exploiting Stable Historical embeddings, which efficiently trains large-scale models with high accuracy. In realizing ReFresh, our primary contributions are as follows: * We design a novel mini-batch training algorithm for GNNs that achieves scalability without compromising model performance. This is accomplished through the use of a historical embedding cache, with a corresponding cache policy that adaptively maintains node representations (via gradient and staleness criteria to be introduced later) that are likely to be stable across training iterations. In this way, we can economize GNN mini-batch training while largely avoiding the reuse of embeddings that are likely to lead to large approximation errors and subsequently, poor predictive accuracy. * We create the prototype ReFresh system to realize the above algorithm, with efficient implementation of the requisite cache operations, subgraph pruning, and data loading for both single-GPU and multi-GPU hardware settings. * We provide a comprehensive empirical evaluation of ReFresh across common baseline GNN architectures, large-scale graph datasets, and hardware configurations. Among other things, these results demonstrate that ReFresh can closely maintain the accuracy of non-approximate neighbor sampling while training $4.6\times$ up to $23.6\times$ faster than state-of-the-art baselines. ## 2 Background and Motivation This section introduces GNN basics and specifics of the related scalability challenge that we intend to solve. ### Graph Neural Networks Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a graph with node set $\mathcal{V}$ and edge set $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$, where $n=\|\mathcal{V}\|$. Furthermore, let $A\in\{0,1\}^{n\times n}$ be the graph adjacency matrix such that $A_{uv}=1$ if and only if there exists an edge between nodes $u$ and $v$. Finally, $X\in\mathbb{R}^{n\times d}$ denotes the matrix of $d$-dimensional node features (i.e., each row is formed by the feature vector for a single node) while $Y\in\mathbb{R}^{n\times c}$ refers to the corresponding node labels with $c$ classes. Given an input graph defined as above, the goal of a GNN model is to learn a representation $h_{v}$ for each node $v$, which can be used for downstream tasks such as node classification or link prediction. This is typically accomplished via a so-called _message passing_ scheme [13]. For the $(l+1)$-th GNN layer, this involves computing the hidden/intermediate representation \[\begin{split} h_{v}^{(l+1)}&=f_{W}^{(l+1)}\left(h_{v} ^{(l)},\left\{h_{u}^{(l)}:u\in\mathcal{N}(v)\right\}\right)\\ &=\phi_{W}^{(l+1)}\left(h_{v}^{(l)},\underset{u\in\mathcal{X}(v)} {\text{AGG}}\left(\psi_{W}^{(l+1)}\left(h_{u}^{(l)},h_{v}^{(l)}\right)\right) \right),\end{split} \tag{1}\] where $h_{v}^{(l)}$ and $\mathcal{N}(v)$ are the embedding and neighbors of node $v$ respectively, with $h_{v}^{(0)}$ equal to the $v$-th row of $X$. Additionally, $\psi$ computes messages between adjacent nodes while the operator AGG is a permutation-invariant function (like sum, mean, or element-wise maximum) designed to aggregate these messages. Lastly, $\phi$ represents an update function that computes each layer-wise embedding. Note that both $\phi$ and $\psi$ are parameterized by a learnable set of weights $W$. Within this setting, the goal of training an $L$-layer GNN is to minimize an application-specific loss function $\mathcal{L}(H^{(L)},Y)$ with respect to $W$. This can be accomplished via gradient descent as $W\gets W-\eta\nabla_{W}\mathcal{L}$, where $\eta$ is the step size. ### Difficulty in Training Large-Scale GNNs Graphs used in GNN training can have a large number of nodes containing high-dimensional features (e.g., $d\in\{100,\ldots,1000\}$) [17, 44]. As a representative example, within the widely-adopted Open Graph Benchmark (OGB) [16, 17], the largest graph MAG240M has 240M nodes with 768-dimensional 16-bit float vectors as features (i.e. 350GB total size); real-world industry graphs can be much larger still. On the other hand, as nodes are dependent on each other, full graph training requires the features and intermediate embeddings of all nodes to be simultaneously available for computation, which goes beyond the memory capacity of a single device. In light of this difficulty with full graph training, the most widely accepted workaround is to instead train with stochastic mini-batches [11, 14, 43, 44, 11]. At each iteration, instead of training all the nodes, a subset/batch are first selected from the training set. Then, the multiple-hop neighbors of these selected nodes (1-hop for each network layer) are formed into the subgraph needed to compute a forward pass through the GNN and later to back-propagate gradients for training. Additionally, further reduction in the working memory requirement is possible by sampling these multi-hop nodes as opposed to using the entire neighborhood [14]. The size of the resulting mini-batch subgraph with sampled neighborhood is much smaller relative to the full graph, and for some graphs can be trained using a single device achieving competitive accuracy [45] and generality [14]. Hence mini-batch training with _neighbor sampling_ has become a standard paradigm for large-scale GNN training. And yet even this standard paradigm is limited by a significant bottleneck, namely, loading the features of sampled multiple-hop neighbors in each mini-batch, which still involves data movement growing exponentially with the number of GNN layers. For example, on MAG240M, training a 3-layer GNN requires that, for each epoch we copy $\sim$1.5TB data from CPU to GPU. In many such cases, this step can occupy more than 85% of the total training time. ### Existing Mini-Batch Training Overhauls Both system-level and algorithmic approaches have been pursued in an attempt to alleviate the aforementioned limitations of mini-batch training. System Optimizations.GNNLab [43] and GNNTier [26] cache the features of frequently visited nodes as assessed by metrics such as node degree, weighted reverse pagerank, and profiling to GPU. However, for commonly-encountered graphs exhibiting a power-law distribution [8], most of the nodes will experience a moderate visiting frequency and hence, the feature cache is unlikely to reduce memory access to them. More generally, because the overall effectiveness of this approach largely depends on graph structure, consistent improvement across graph datasets is difficult to guarantee. PyTorch-Direct [27] proposes to store node features in CUDA Universal Virtual Addressing (UVA) memory so that feature loading can be accelerated by GPU kernels, but the data transfer bandwidth is still limited by PCIe. Even so, data loading remains a bottle-neck with PyTorch-Direct, occupying 66% of the total execution time. Broader Sampling Methods.While neighbor sampling reduces the size of each mini-batch subgraph, it does not completely resolve recursive, exponential neighbor expansion. Consequently, alternative sampling strategies have been proposed such as layer-wise [50, 3] and graph-wise sampling [46, 5]. However, the resulting impact on model accuracy is graph dependent and prior evaluations [49] on large graphs like ogbn-papers100M indicate that a significant performance degradation may at times occur. Reusing Historical Intermediate Embeddings.The other line of algorithmic work [2, 23, 10] for revamping mini-batch training approximates the embeddings of some node set $\mathcal{S}$ with their historical embeddings from previous training iterations. This involves modifying the original message passing scheme from Equation (1) to become \[h_{v}^{(l+1)} =f_{W}^{(l+1)}\left(h_{v}^{(l)},\left\{h_{u}^{(l)}\right\}_{u\in \mathcal{N}(v)}\right)\] \[=f_{W}^{(l+1)}\left(h_{v}^{(l)},\left\{h_{u}^{(l)}\right\}_{u\in \mathcal{N}(v),\mathcal{S}}\cup\left\{h_{u}^{(l)}\right\}_{u\in\mathcal{N}(v) \cap\mathcal{S}}\right)\] \[\approx f_{W}^{(l+1)}\left(h_{v}^{(l)},\left\{h_{u}^{(l)}\right\}_{u \in\mathcal{N}(v)\setminus\mathcal{S}}\cup\underbrace{\left\{h_{u}^{(l)}\right\} _{u\in\mathcal{N}(v)\cap\mathcal{S}}}_{\text{Historical embeddings}}\right). \tag{2}\] The above computation is mostly the same as the original, except that now the node embeddings from $\mathcal{S}$ are replaced with their historical embeddings $\bar{h}_{u}^{(l)}$. A typical choice for $\mathcal{S}$ is any node not included within the current mini-batch, and after each training step, the algorithm will refresh $\bar{h}_{v}^{(l+1)}$ with the newly generated embedding $h_{v}^{(l+1)}$. Using historical embeddings avoids recursive aggregation of neighbor embeddings as well as the corresponding backward propagation, which reduces not only the number of raw features to load but also the computation associated with neighbor expansion. Moreover, this training methodology is compatible with arbitrary message passing architectures. While promising, there remain two major unresolved issues with existing methods that utilize historical embeddings. First, they all require an extra storage of size $O(Lnd)$ for an $L$-layer GNN, which can be even larger than the total size of node features, a significant limitation. Secondly, historical embeddings as currently used may introduce impactful estimation errors during training. To help illustrate this point, let $\tilde{h}_{\nu}^{(l+1)}$ denote the approximated node embedding computed using Equation (2). The estimation error can be quantified by $\\|\tilde{h}_{\nu}^{(l+1)}-h_{\nu}^{(l+1)}\\|$, meaning the difference between the approximated embeddings and the authentic embeddings computed via an exact message passing scheme. Figure 1 shows that the estimation error of each mini-batch increases considerably with more training iterations on the ogbn-products graph from OGB when using GNNAutoScale (GAS) [10], a representative system based on historical embeddings. The root cause of this problem is that existing methods lack a mechanism for directly controlling the quality of the cached embeddings used for message passing. And as the model parameters are updated by gradient descent after each iteration, the un-refreshed embeddings may gradually drift away from their authentic values resulting in a precipitous accuracy drop compared with the target accuracy from mini-batch training with vanilla/canonical neighbor sampling. For this reason there remains ample room for new system designs that exploit historical embeddings in a more nuanced way so as to maintain accuracy. This is especially true as graph size grows larger and the performance of existing methods for scalable mini-batch training begins to further degrade as shown in Figure 2. ## 3 Overview of ReFresh System In this work, we propose a new strategy for utilizing historical embeddings, with targeted control of the resulting estimation error to economize GNN mini-batch training on large graphs without compromising accuracy. Intuitively, this strategy is designed to favor historical embeddings with small $\\|\tilde{h}_{u}^{(l)}-h_{u}^{(l)}\\|$ when processing each mini-batch, while avoiding the use of those that have drifted away. However, directly computing this error requires knowledge of the authentic embeddings $h_{u}^{(l)}$, which is only obtainable via exact message passing, the very process we are trying to avoid. We thus adopt an alternative strategy based on a critical observation about the dynamics of the node embeddings during GNN mini-batch training: _most of the node embeddings experience only modest change across the majority of iterations_. Embedding Stability Illustration.Figure 3 showcases this phenomenon using a GCN [20] model on the ogbn-products dataset. We measure the cosine similarity of the node embeddings at mini-batch iteration $t$ with the corresponding set at $t-s$ for lag $s=20$ and plot the resulting distribution across varying $t$. We find that >78% of the nodes embeddings have >0.95 cosine similarity after iteration 140 (the model converged at around iteration 500). This observation suggests that there exists temporal stability of many node embeddings during GNN mini-batch training, and naturally motivates the use of a _historical embedding cache_ capable of selectively storing and reusing such durable node representations. Figure 1: Average estimation error in one training epoch for GAS [10] on ogbn-products. Figure 3: Distribution of cosine similarity between the node embeddings at iteration $t$ and the embeddings at iteration $t-s$ during the mini-batch training of a GraphSAGE model on ogbn-products. Here $s=20$. Figure 2: Test accuracy of various mini-batch training algorithms compared with the target accuracy achieved by (expensive) neighbor sampling on: (a) relatively small ogbn-products graph where the gap is modest, and (b) larger ogbn-papers100M graph where the gap grows significantly. ReFresh Training Algorithm.With this in mind, Algorithm 1 shows our mini-batch training process using the historical embedding cache. For each batch, we first generate a subgraph according to a user-defined sampling method (Line 5). A subgraph consisting of the sampled $L$-hop neighbors for the training nodes is returned, where $L$ is defined by whatever GNN model is chosen. Then for each layer, the nodes in the subgraph are divided into two types: normal nodes ($\mathcal{V}_{normal}^{l}$ on Line 9) and those nodes whose embeddings can be found in the historical cache ($\mathcal{V}_{cache}^{l}$ on Line 7). For the latter, embeddings are directly read from the cache (Line 13), while for the normal nodes, embeddings are computed by the neighbor aggregation assumed by the GNN (Line 14). And finally, at the end of each iteration, the algorithm will utilize information from the forward and backward propagation steps to update the historical embedding cache. The goal is to check in stable embeddings that are more reliable for future reuse while checking out those that are not (Line 20). Figure 4 further elucidates Algorithm 1 using a toy example. Here node $v1$ is selected as the seed node of the current mini-batch as in Figure 4(a). Computing $v1$'s embedding requires recursively collecting information from multi-hop neighbors as illustrated by the subgraph in Figure 4(b). And as mentioned previously, neighbor sampling can reduce this subgraph size as shown in Figure 4(c). Figure 4(d) then depicts how the historical embedding cache can be applied to further prune the required computation and memory access. The cache contains node embeddings recorded from previous iterations as well as some auxiliary data related to staleness and gradient magnitudes as needed to estimate embedding stability. In this example, the embeddings of node $v3$ are found in the cache, hence its neighbor expansion is no longer needed and is pruned from the graph. Additionally, after this training iteration, some newly generated embeddings (e.g., node $v2$) will be pushed to the cache for later reuse. Existing cached embeddings may also be evicted based on the updated metadata. In this example, both $v1$1 (by staleness) and $v3$ (by gradient magnitude criteria to be detailed later) are evicted from the cache. Main ReFresh Components.To instantiate Algorithm 1, and enable accurate, efficient GNN training over large graphs in CPU-GPU or multi-GPU scenarios, we require three novel system components, namely, (i) the historical embedding cache, (ii) the subgraph generator, and (iii) the data loader (which correspond with the colored blocks in Algorithm 1). Figure 5 situates these components within a typical GNN training workflow while supporting details are as follows: 1. [label=()] 2. The historical embedding cache is the central component of our ReFresh design, selectively storing node embeddings in GPU and providing efficient operations for fetching or updating its contents as will be detailed in Section 4. In brief here, we propose two cache policies (based on embedding staleness and gradient magnitudes collected during training) to detect and admit only those node embeddings that are likely to be relatively stable while evicting those that are not. 3. The subgraph generator is responsible for producing a pruned subgraph given the current mini-batch as will be discussed further in Section 5. Compared with other GNN systems, a unique characteristic of our proposed approach is that each subgraph structure itself is adaptively generated based on which particular nodes happen to be in the selective historical embedding cache, and the latter is dynamically updated at the end of each training iteration. This essentially creates a reversed data dependency between the stages of mini-batch preparation and mini-batch training, making it difficult to apply pipelining to overlap the two stages like in other GNN systems [28, 43]. To address this challenge, we further partition the workload into two steps: graph sampling and graph pruning, where only the latter step depends on the historical embedding cache. We then adopt a mixed CPU-GPU design that samples graphs in CPU while pruning graphs in GPU; this is complemented by a GPU-friendly data structure for fast graph pruning. 3. The data loader is in charge of loading the relevant node features or intermediate embeddings given a generated subgraph. For each node that is not cached or pruned using historical embeddings, the data loader fetches its raw features. Since these features are typically stored in a slower but larger memory device, ReFresh further optimizes the data transmission for three scenarios: (1) fetching features from CPU to a single GPU, (2) fetching features from CPU to multiple GPUs, and (3) fetching features from other GPUs. See Section 6 for further details. ## 4 Historical Embedding Cache The historical embedding cache design is informed by the following two questions: (1) _What are suitable cache policies for checking in and out node embeddings?_ and (2) _What is the best way to implement the historical emebedding cache on GPU efficiently?_ We address each in turn below. ### Cache Policy Caching intermediate node embeddings for later reuse is fundamentally more complicated than caching raw node features because the former are constantly updated during model training. This calls for a cache policy to invalidate unstable entries, where in the context of GNN training, this notion of stability is calibrated by the difference between a true node embedding and the corresponding cached one. Therefore, a naive solution is to recompute all the embeddings in the cache after each training iteration and evict those that have drifted away. However, this solution is of course not viable for large graphs because it involves the very type of high-cost computation we are trying to avoid. Instead, ReFresh employs a cache policy based on dynamic feedback from model training. Given a mini-batch graph, denote the set of nodes at layer $l$ as $\mathcal{V}^{(l)}$ and the set of cached nodes as $\mathcal{V}_{cache}$. We then apply two cache policy criteria to estimate the stability of an embedding as follows: Gradient-based Criteria.For nodes $v\in\mathcal{V}^{(l)}$, we use the magnitude of embedding gradients w.r.t. the training loss as a proxy for node stability at each layer, noting that for a gradient magnitude near zero, there is little impact on the training loss for any downstream gradient signals passing through this particular node at a given layer (which justifies our efficient subgraph pruning step discussed later in Section 5). Moreover, consistently small gradient magnitudes are likely to correlate on average with smaller estimation errors $\\|\mathbb{\boldsymbol{h}}_{v}^{(l)}-h_{v}^{(l)}\\|$. Hence, ReFresh admits nodes with small gradients to the cache, with the rate controlled by $p_{grad}$, the fraction of newly generated embeddings to be admitted. Of the remaining $(1-p_{grad})$ fraction of the nodes in the mini-batch, if any of these happen to already be in the cache, they will now be evicted. Conveniently, embedding gradients at any layer are naturally obtained from the backward propagation of the model as illustrated in Figure 6 without computation overhead. In this example, at layer 1, Node 3 fetches its embedding from the cache while Node 2 computes its embedding faithfully by aggregating from its neighbors. During backward propagation, ReFresh calculates both gradients $\nabla_{h_{1}^{(1)}}L$ and $\nabla_{h_{2}^{(1)}}L$, and compares their vector norms to decide which to admit or Figure 4: Illustration of historical embedding cache using an example mini-batch graph. Figure 5: ReFresh system workflow. evid; here the cache decides to admit Node 2 while evicting Node 3. Staleness-based Criteria.Because fresh embedding gradients are not available at a given iteration for nodes $v\in\mathcal{V}_{cache}^{\prime}\setminus\mathcal{V}^{\prime(l)}$, we instead choose to bound their _staleness_. The staleness is set to be zero when an embedding is admitted to the cache and will increment by one at each iteration. ReFresh treats any embedding with staleness larger than a threshold $t_{stale}$ as being out-dated and evicts it from the cache. In Figure 6, Node 11 is evicted per this criteria. More broadly, utilizing and limiting staleness is a commonly used technique in training neural networks; the difference between ReFresh and prior work that incorporates staleness criteria during training is discussed in Section 8. Resulting Adaptive Cache Size.A notable difference from typical cache usage, which emerges from the above two criteria, is that our historical embedding cache size is not static or preset, but rather implicitly controlled by the two thresholds $p_{grad}$ and $t_{stale}$. Larger $p_{grad}$ or $t_{stale}$ means more embeddings to be cached but also requires the model to tolerate larger approximation errors introduced by historical embeddings. Moreover, setting $p_{grad}=0$ or $t_{stale}=0$ degrades the algorithm to normal neighbor sampling without a cache. In contrast, setting $p_{grad}=1$ and $t_{stale}=\infty$ results in a policy that is conceptually equivalent to that used by GAS [10] and VR-GCN [2]. So in this sense ReFresh is a more versatile paradigm w.r.t. previous historical embedding based methods. Later in Section 7 we demonstrate that finding suitable values of $p_{grad}$ and $t_{stale}$ is relatively easy in practice; however, we leave open to future research the possibility of further exploiting this flexible dimension within the ReFresh design space. ### GPU Implementation ReFresh utilizes GPU threads to look up a batch of historical node embeddings in parallel. To achieve this, a node ID mapping array of length $O(\|\mathcal{V}\|)$ is maintained, where each entry stores the index to the cache table that holds the embedding of the corresponding node. The extra storage of this mapping array is affordable compared with the storage for embeddings. For example, storing the ID mapping array for the 100M nodes of ogbn-papers100M requires only 400MB. Given a batch of node IDs, each thread is in charge of fetching one node embedding and each fetching operation can be done in $O(1)$. ReFresh also adopts a ring buffer design to efficiently admit new embeddings and evict old ones. As shown in Figure 7, this design maintains an embedding cache header that initially points to the first row of the cache table. New embeddings are added at the header and push the header towards the end of the table. At every $t_{stale}$ iteration, ReFresh moves the embedding cache header back to the beginning so that the newly added embeddings will overwrite the out-dated ones, which naturally evicts them from the cache. In some corner cases, the newly added embeddings may overwrite the embeddings that are still below the staleness bound. In practice, this rarely happens because the number of recorded embeddings per iteration is similar given that neighbor sampling tends to create mini-batch graphs with similar degrees. Additionally, to evict the embeddings with large gradient magnitudes, ReFresh simply invalidates the corresponding entries of the node ID mapping array instead of conducting physical deletion. Those invalid slots will be recycled naturally by the ring buffer design. Since the size of historical embedding cache cannot be determined a priori, we initialize the cache table with a fixed size and reallocate it on-demand. To further increase the lookup efficiency, we combine it with a raw feature cache by filling the empty entries with raw features of high-degree nodes. We further put the features of the highest degree at the end of the embedding table so that as new embeddings are added at the embeddding cache header, less frequent features will be overwritten first. ### Remarks on ReFresh Convergence Not surprisingly, formal convergence analysis of our selective history cache and attendant check-in/out policy is generally infeasible when applied to arbitrary multi-layer, non-convex GNN architectures. That being said, in the special case of a Figure 6: Illustration of the admission/eviction policy of the historical embedding cache. After a training iteration, the cache admits the embedding of Node 2 but evicts Node 3 (which was originally in the cache at the start of the iteration) by the gradient-based criteria. It also evicts Node 11 according to the staleness criteria. single-layer simple graph convolution (SGC) model [39], it can be shown that a version of our approach defaults to the same convergence guarantees as stochastic gradient descent (SGD): Proposition 4.1.: _(informal version; see supplementary for formal statement and proof) Define the SGC model $Z=\hat{A}^{k}XW$ and training loss $\ell(W)=\sum_{i=1}^{n}g(W;z_{i},y_{i})$, where $g$ is a Lipschitz L-smooth function of $W$ and $y_{i}$ is the label associated with $z_{i}$, the $i$-th row of $Z$. Then for a historical model with a random selector and bounded staleness for the usage of historical embeddings, updating the weights using gradient descent with step-size $\eta\leq\frac{1}{L}$ will converge to a stationary point of $\ell(W)$._ Of course, this regime is admittedly quite limited, and not representative of more nuanced behavior involving the coupling of node embeddings of varying degrees of staleness across different layers. Although some prior analysis of gradient descent with stale features and gradients does exist [36, 2, 32], the setting is decidedly much simpler and does not directly apply to our complex modeling framework; however, we can consider potential adaptations in future work. As a point of reference though, most GNN models with adjustments for scalability (e.g., various sampling methods) do not actually come with any rigorous convergence guarantees. ## 5 Cache-Aware Subgraph Generator In ReFresh, the mini-batch subgraphs are generated adaptively according to the node embeddings stored in the cache. As the cache is in turn updated at the end of each iteration, this prevents ReFresh from adopting a naive pipelining strategy to parallelize subgraph generation and model training as in other systems. To address this challenge, we decompose subgraph generation into two steps: graph sampling and graph pruning, where only the latter depends on the historical embedding cache. We then adopt a mixed CPU-GPU design to further accelerate them. Asynchronous CPU Graph Sampling.This step first extracts/samples subgraphs normally for the given mini-batch and then moves them to GPU without querying information from the cache. As a result, graph sampling can be conducted asynchronously with the later GPU computation. We further utilize multithreading instead of multiprocessing to produce multiple subgraphs concurrently in contrast to the existing systems like DGL [37] and PyG [9]. Additionally, we use a task queue to control the production of subgraphs and avoid overflowing the limited GPU memory. GPU Graph Pruning.The graph pruning step scans the mini-batch graph from the seed node layer to the input node layer. For any cached node, it recursively removes all the multiple-hop neighbors so that the corresponding computation is no longer needed for model training. The remaining challenge is that traditional sparse formats are not suitable for parallel modification in GPU. For example, the Coordinate (COO) format represents a graph using two arrays containing the source and destination node IDs of each edge. Pruning in-coming neighbors of a node in COO requires first locating neighbors using a binary search and then deleting them from both arrays. While for the Compressed Sparse Row (CSR) format, which further condenses one of the arrays to a row index array using prefix summation, locating a node's neighbors can be done in $O(1)$. However, after deleting the edges the row index array needs to be adjusted accordingly. The respective complexities required for these operations are listed in Table 1. For ReFresh, we designed a novel data structure called CSR2 for even faster graph pruning on GPU. CSR2 uses two arrays to represent row indices - the first array records the starting offset of a node's neighbors to the column index array while the second array records its ending offset. As illustrated in Figure 8, for node $i$, its neighbors are stored in the column index segment starting from $start[i]$ to $end[i]$. The CSR2 format has some redundancy in storing a graph since $start[i]$ equals $end[i-1]$ for any node $i$. To remove a node's neighbors, one can simply set the corresponding $end[i]=start[i]$ without any changes to the column index array, resulting in an $O(1)$ operation. The data structure is also Figure 8: Removing a center node’s neighbors using different graph data structures Figure 7: GPU implementation of historical embedding cache. suitable for parallel processing on GPU because there is no data race condition. See Table 1 for complexity comparisons with prior formats. ## 6 Data Loader Once a subgraph has been pruned, GNN training still needs the features of the remaining nodes, a task performed by the ReFresh data loader. Note that even though the number of nodes in the pruned subgraph is largely reduced, the corresponding features are typically stored in a device with slower but larger memory, and the loading process is still critical to performance. However, the indices for unpruned nodes and their features reside in different devices. The index is calculated in the GPU (computation device) while the needed features are stored in the CPU or remote GPUs (storage device). Under such conditions, the naive way to fetch the features is via two-sided communication. This involves first transferring node indices from the computation device to the storage device, compacting the corresponding features on the storage device, and then sending the packed features back to the computation device. This process introduces extra communication for transferring indices and synchronization between the computation and storage devices. One-sided Communication.ReFresh employs one-sided communication to address this problem. Based on Unified Virtual Addressing (UVA) [27], the memory of the CPU and GPUs are mapped to a unified address. This enables the computation device to directly fetch features from a mapped buffer of the storage device using the node index. As shown in Figure 9(a), nodes needed for training are partly pruned by the cache (green color), and for the unpruned ones (orange color), the GPU fetches the node features using their node ID directly from node features mapped with UVA. In Figure 9(b), multiple GPUs can concurrently fetch data using UVA for parallel training. Multi-round Communication.With a larger number of GPUs available during training, all node features can be partitioned and stored across multiple machines such that GPUs serve as both computation and storage devices. Therefore, each GPU fetches the features of the relevant unpruned nodes from other GPUs, resulting in all-to-all communication between every pair of GPUs. UVA can still be used to perform one-sided memory access in this scenario. However, as GPUs are connected asymmetrically, link congestion could badly degrade the overall bandwidth. To address this problem, in addition to one-sided communication, ReFresh breaks cross-GPU communication into multiple rounds to avoid congestion and fully utilize the bi-directional bandwidth on links. Figure 9(c) shows a typical interconnection among four GPUs, where GPUs are first connected via PCIe and then bridged via a host. For this topology, ReFresh will decompose the all-to-all communication into five rounds. In round one, data is exchanged only between GPUs under the same PCIe switch, while the remaining four rounds are for exchanging data between GPUs across the host bridge. The data transmission at each round is bi-directional to fully utilize the bandwidth of the underlying hardware. This multi-round communication can effectively avoid congestion in all-to-all communication. ## 7 Evaluation In this section, we first describe our experimental setup, followed by results covering system efficiency and model accuracy. We conclude with an empirical study of our cache effectiveness as well as ablations over our system optimizations. ### Experimental Setup Environments.Single GPU experiments were conducted on a server with two AMD EPYC 7742 CPUs (2$\times$64 cores \begin{table} \begin{tabular}{l c c} \hline \hline \multicolumn{2}{c}{Prune Complexity} & Storage Complexity \\ \hline CSR & $O(\|\mathcal{V}_{sample}\|+\|\mathcal{I}_{samples}\|)$ & $O(\|\mathcal{V}_{sample}\|+\|\mathcal{I}_{samples}\|)$ \\ COO & $O(\log(\|\mathcal{I}_{sample}\|)+N_{neighbors})$ & $O(2\|\mathcal{I}_{sample}\|)$ \\ CSR2 (ours) & $O(1)$ & $O(2\|\mathcal{V}_{sample}\|+\|\mathcal{I}_{samples}\|)$ \\ \hline \hline \end{tabular} \end{table} Table 1: Storage complexity and the complexity for pruning the neighbors of one node in different sparse formats. $N_{neighbors}$ is the number of neighbors of the node to be pruned. Figure 9: Feature data loading under different scenarios. CPU-GPU training in (a) uses one-sided communication to directly read from CPU memory. Multiple GPUs initiate one-sided communication concurrently for CPU data in (b). Communication among GPUs is scheduled in multiple rounds in (c) to improve bandwidth. in total) and an NVIDIA A100 (40GB) GPU. The software environment on this machine is configured with Python v3.9, PyTorch v1.10, CUDA v11.3, DGL v0.9.1, and PyG v2.2.0. The multi-GPU experiments were conducted on an AWS p3.16xlarge machine with two Intel Xeon E5-2686 CPUs (2$\times$16 cores in total) and eight NVIDIA Tesla V100 (16G) GPUs. Datasets.The dataset statistics are listed in Table 2. Among them, the two smallest datasets, ogbn-arxiv and ogbn-products, are included only for model accuracy comparisons and to provide contrast with larger datasets. ogbn-papers100M and MAG240M [16] are used to test both accuracy and speed. Note that MAG240M is the largest publicly-available graph dataset. Following the common practice of previous work [11, 43], we also use the graph structure from Twitter [41] and FriendSter [42] with artificial features to test the speed of different systems. GNN models & Training details.We employed three widely-used GNN architectures for our experiments: GraphSAGE [14], GCN [20], and GAT [33]. All models have 3 layers and 256 as the hidden size. To measure their baseline model performance, we train them using mini-batch neighbor sampling in DGL. We follow the settings of their OGB leaderboard submissions to set the sampling fan-out as 20, 15 and 10; the batch size is chosen to be 1000. We set $p_{grad}=0.9$ and $t_{sale}=200$ for ReFresh for all experiments (with the exception of Section 7.4, where we study the impact of these thresholds). ### System Efficiency We first demonstrate the system advantage of ReFresh against state-of-the-art alternatives for GNN mini-batch training, including DGL [37], PyG [9], PyTorch-Direct [27], GNNLab [43] and representative mini-batch training algorithms such as ClusterGCN [5] and GAS [10]. Figure 10 compares the time for training a GraphSAGE model for one epoch on the four large-scale graph datasets using a single GPU. ReFresh significantly outperforms all the other baselines across all the datasets. Compared with the widely-used GNN systems DGL and PyG, ReFresh is 5.3$\times$ and 23.6$\times$ faster respectively on ogbn-papers100M. Both PyTorch-Direct and ReFresh utilize CUDA UVA memory to accelerate feature loading, but ReFresh is still 4.6$\times$ faster because it can reduce the number of features to load by a large margin. With respect to other mini-batch training algorithms, ReFresh is orders of magnitude faster than GAS and ClusterGCN on ogbn-papers100M. GAS also runs out of memory on graphs that have either more nodes/edges or larger feature dimensions due to the need to store the historical embeddings of all the nodes. Incidentally, on the largest dataset MAG240M only DGL and ReFresh avoid OOM, but with ReFresh executing 5.3$\times$ faster. ReFresh can also easily scale to multiple GPUs. In this setting, we include a new baseline GNNLab [43] which partitions GPUs to sampling or training workers; hence GNNLab does not have single-GPU performance. Figure 11 compares the throughput (measured as the number of iterations computed per second) when training GraphSAGE on ogbn-papers100M. Both DGL and PyTorch-Direct deliver almost no speedup because of the data loading bottleneck that cannot be parallelized via the addition of more GPUs. Meanwhile, ReFresh enjoys a nearly linear speedup from 1 to 4 GPUs and is up to 2.0$\times$ faster than GNNLab. However, the throughput growth mostly saturates when advancing from 4 to 8 GPUs for both ReFresh and GNNLab. In this regime we find that graph sampling becomes a new bottleneck since \begin{table} \begin{tabular}{l c c c c} \hline \hline Dataset & $\|\mathcal{V}\|$ & $\|\mathcal{E}\|$ & Dim.${}^{1}$ & \#Class \\ \hline ogbn-arxiv [17] & 2.9M & 30.4M & 128 & 64 \\ ogbn-products [17] & 2.4M & 123M & 100 & 47 \\ ogbn-papers100M [17] & 111M & 1.6B & 128 & 172 \\ MAG240M [16] & 244.2M & 1.7B & 768 & 153 \\ Twitter [41] & 41.7M & 1.5B & 768 & 64 \\ FriendSter [42] & 65.6M & 1.8B & 768 & 64 \\ \hline \hline \end{tabular} \end{table} Table 2: Graph dataset details, including input node feature dimension (Dim.) and number of classes (#Class). Figure 11: Scalability comparison for training a GraphSAGE model on obgn-papers100M using multiple GPUs. Figure 10: Epoch time comparisons training a GraphSAGE model using a single GPU. it is conducted in CPU; we defer further consideration of this issue to future work. ReFresh is not only fast but also reliably converges to the desired target accuracy. Figure 12 plots the time-to-accuracy curve of different training systems. All the baselines here are using mini-batch neighbor sampling without any further approximation so they all converge to the same accuracy of $\sim$66%. ReFresh can reach this same accuracy in 25 minutes while the slowest baseline (PyG) takes more than 6 hours. ### Model Accuracy Table 3 compares the test accuracy of ReFresh with other mini-batch training algorithms: GAS [10], ClusterGCN [5], and another historical-embedding-based algorithm GraphFM [23]. Following the common practice in [6, 11], the target accuracy is obtained from training the models using neighbor sampling. In general, nearly all algorithms perform relatively well on small graphs such as ogbn-arxiv and ogbn-products with only a few exceptions. However, when scaling to larger graphs such as ogbn-papers100M, all baselines experience a severe accuracy drop (from 7% to 18%) while running out of memory on MAG240M. By contrast, ReFresh only experiences a less than 1% accuracy difference on both large datasets. Besides, ReFresh can also demonstrate a similar accuracy pattern in the training process with normal mini-batch training, with the figure shown in Appendix. ### Cache Effectiveness Recall that $p_{grad}$ and $t_{sade}$ are the two thresholds that control the admission and eviction criteria of the historical embedding cache. In this section, we study their impact on system performance and model accuracy. We will show that a straightforward choice of the two thresholds can lead to the aforementioned competitive system speed as well as model accuracy. Impact on System Performance.In typical cache systems, the I/O saving percentage is equal to the cache hit rate. However, hitting the cached historical embeddings of a node will prune away all the I/O operations that would otherwise be needed to load the features of multi-hop neighbors, meaning the I/O savings can potentially be much larger than the cache hit rate. We plot this saving percentage w.r.t. neighbor sampling (i.e., no caching) under different $p_{grad}$ and $t_{sade}$ in Figure 13 (a) and (c). Because the historical embedding cache in ReFresh is initialized via raw node features (Section 4.2), the red $p_{grad}=0$ line reflects the performance of neighbor sampling with a raw feature cache. As expected, larger $p_{grad}$ or $t_{sade}$ results in more I/O reduction. On both graph datasets, a raw feature cache can only reduce I/O by $<40\%$ but the historical embedding cache can reduce I/O by more than 60% when choosing $t_{sade}>200$. Impact on Model Accuracy.We plot the corresponding accuracy curves under different configurations in Figure 13 (b) and (d). As expected, larger $p_{grad}$ or $t_{sade}$ means more relaxed control on the embedding errors which consequently results in lower test accuracy. Beyond this, there are two interesting findings with practical significance. First, we can set $p_{grad}$ very close to one without an appreciable impact on the model accuracy. This aligns well with the observation in Figure 3 that most of the node embeddings are temporally stable and can be safely admitted to the cache. And secondly, with a proper $p_{grad}$ value, GNN models can tolerate node embeddings that were last updated hundreds of iterations ago. By cross-checking the I/O saving figures, we find that this is also a sweet spot in terms of system performance, which eventually led us to choose $p_{grad}=0.9$ and $t_{sade}=200$ for all the experiments (obviously excluding the Figure 13 results). \begin{table} \begin{tabular}{l l\|c c\|c c} \hline \hline & \multicolumn{2}{c\|}{Small datasets} & \multicolumn{2}{c}{Large datasets} \\ \hline \multicolumn{2}{c\|}{Methods} & \multicolumn{1}{c\|}{arxiv products} & \multicolumn{1}{c}{papers100M} & MAG240M2 \\ \hline \multirow{4}{}{\begin{tabular}{} \end{tabular} } & Target Accuracy & 70.91 & 78.66 & 66.43 & 66.14 \\ \cline{2-6} & GAS & +0.44 & -1.19 & -8.17 & OOM \\ & ClusterGCN & -3.10 & +0.00 & -7.57 & OOM \\ & GraphFM & +0.62* & -7.90 & -18.40 & OOM \\ & ReFresh & +0.60 & +0.38 & -0.15 & -0.51 \\ \hline \multirow{4}{}{\begin{tabular}{} \end{tabular} } & Target Accuracy & 70.93 & 79.41 & 66.13 & 65.16 \\ \cline{2-6} & GAS & -0.04* & -2.23 & -8.67 & OOM \\ & ClusterGCN & -3.91 & -2.92 & -8.08 & OOM \\ & GraphFM & -22.67 & -16.54 & OOM & OOM \\ & ReFresh & -0.50 & -0.54 & -0.71 & -0.36 \\ \hline \multirow{4}{}{ \begin{tabular}{} \end{tabular} } & Target Accuracy & 71.24 & 78.57 & 65.78 & 65.24 \\ \cline{2-6} & GAS & +0.44 & -1.91 & -12.29 & OOM \\ \cline{1-1} & ClusterGCN & -3.13 & +0.40* & -12.43 & OOM \\ \cline{1-1} & GraphFM & +0.47 & -15.70 & -18.70 & OOM \\ \cline{1-1} & ReFresh & -0.71 & -0.31 & -0.16 & -0.29 \\ \hline \hline \end{tabular} \end{table} Table 3: Test accuracy of different training algorithm minus the target accuracy obtained by neighbor sampling (larger is better). Bold numbers denote the best performing method. Figure 12: Test accuracy curve for GraphSAGE on ogbn-papers100M. ### Ablation Study of System Optimizations In this section, we study the system performance of individual components. All the results are tested on ogbn-papers100M with a three-layered GraphSAGE model. Subgraph Generator. Figure 14(a) shows the improvement of ReFresh's graph sampler v.s. DGL's. On ogbn-papers100M, with 32 CPU threads, ReFresh is able to reduce the sampling time per epoch from 72 seconds to 11 seconds (6.5$\times$). In terms of scalability when using more threads, the speedup of ReFresh's sampler is 26$\times$ while DGL is only 7.5$\times$. Figure 14 (b) measures the time to prune the cached nodes and their neighbors from a subgraph using different graph data structures. Overall, CSR2 is orders-of-magnitude faster than CSR and COO regardless of the batch size in use. Specifically, CSR requires frequent CPU-GPU synchronization when invalidating the neighbors of the pruned nodes in the column index. As a result, graph pruning in total takes 99% of the iteration time. Subgraph pruning using COO is faster, which reduces the pruning overhead to 4.5% of the iteration time, but is still much slower than CSR2. The subgraph pruning time using CSR2 is negligible - it only occupies 26us per iteration. Data Loader. Figure 15 shows the improvement afforded by the optimizations of ReFresh's data loader for multi-GPU communication. Compared with the unoptimized communication utilizing NCCL all-to-all operations, the one-sided communication is 23% faster on average on PCIe and NVLink GPUs. After scheduling using the multi-round communication pattern, the bandwidth is increased by 145% and 85%, respectively. ### Extension to heterogeneous graphs While our primary focus was on the most popular large-scale benchmark settings, ReFresh can also be naturally extended to heterogeneous graphs. We show one such use case here on MAG240M, which was homogenized in the previous experiments. Here we are using the heterogeneous form, and R-GNN [31] is the most popular heterogeneous GNN. So we compare the R-GraphSAGE model on ReFresh with the neighbor sampling baseline on DGL. The accuracy is almost the same, but ReFresh is 21.9$\times$ faster. ## 8 Related Work Full-Graph GNN Systems. There exists a rich line of work on scaling full-graph GNN training, where the general idea is to split the graph into multiple partitions that can fit onto the computing devices. Unlike mini-batch training, full-graph Figure 16: Validation accuracy curve for R-GraphSAGE on MAG240M. Figure 14: Effectiveness of ReFresh’s subgraph generator (measured on ogbn-papers100M). Figure 13: (a) (c) The percentage saving of I/O for loading node features and (b) (d) the test accuracy on ogbn-papers100M and MAG240M under different choices of $p_{grad}$ and $t_{stale}$. Figure 15: Optimizations for multi-GPU communication training updates all the nodes and edges simultaneously. Notable systems include NeuGraph [24], ROC [18], and DistGNN [25], which lower the high memory footprint and data access costs by designing smart data partitioning or data swap strategies. In a different vein, DGCL [1] designs a new communication pattern to reduce network congestion. Other systems like Dorylus [32] and GNNAdvisor [38] optimize the computation of GNN training by exploring properties of the graph structure. And BNS-GCN [35] proposed boundary-node-sampling to reduce the communication incurred by boundary nodes during full graph training. Finally, the SANCUS [30] framework reduces the amount of network communication by leveraging locally available node embeddings with bounded staleness to improve accuracy. However, although SANCUS resembles ReFresh in some high-level conceptual aspects, the two systems are targeting very different training paradigms. Moreover, the cache policy of ReFresh utilizes both gradient-based and staleness criteria to detect reliable embeddings. Additionally, we note that accuracy results for SANCUS on large-scale datasets have not yet been reported [30]. Data Movement Optimizations for GNNs. Previous work has shown that data movement is the bottle-neck for GNN mini-batch training on large graphs. And when using different ways of graph partitioning [19, 26, 49] and data placement [11, 27, 43], this data movement can be reduced under various training settings. For instance, DistDGL [49] replicates high-degree nodes, together with sparse embedding updates, to reduce the communication workload for distributed training. And MariusGNN [34] addresses the scenario of out-of-core training, reducing data swaps between disk and CPU memory by reordering training samples for better locality. ReFresh's selective historical embedding method is orthogonal with DistDGL and MariusGNN, and can potentially be applied to them. Training with Staleness. Staleness criteria have long been adopted for efficiency purposes when training deep neural networks. Systems including SSP [15], MXNet [21], and Poseidon [48] update model parameters asynchronously across different devices, helping to reduce global synchronization among devices and improve parallelism. Another branch of work, including PipeDream [29] and PipeSGD [22], pipeline DNN training and utilize stale parameters to reduce pipeline stall. They limit the staleness by periodically performing global synchronization. ReFresh, however, selectively stores and reuses stale node embeddings instead of parameters which is unique to GNN models. ## 9 Conclusion In this paper, we propose ReFresh, a general framework for training GNNs on large, real-world graphs. At the core of our design is a new mini-batch training algorithm that leverages a historical cache for storing and reusing GNN node embeddings to avoid re-computation from raw features. To identify stable embeddings that can be cached, ReFresh designates a cache policy using a combination of gradient-based and staleness criteria. Accompanied with other system optimizations, ReFresh is able to accelerate the training speed of GNNs on large graphs by at least 4.6$\times$ over state-of-the-art systems, with less than 1% influence on model accuracy.
2304.10720	Conservative Sparse Neural Network Embedded Frequency-Constrained Unit Commitment With Distributed Energy Resources	The increasing penetration of distributed energy resources (DERs) will decrease the rotational inertia of the power system and further degrade the system frequency stability. To address the above issues, this paper leverages the advanced neural network (NN) to learn the frequency dynamics and incorporates NN to facilitate system reliable operation. This paper proposes the conservative sparse neural network (CSNN) embedded frequency-constrained unit commitment (FCUC) with converter-based DERs, including the learning and optimization stages. In the learning stage, it samples the inertia parameters, calculates the corresponding frequency, and characterizes the stability region of the sampled parameters using the convex hulls to ensure stability and avoid extrapolation. For conservativeness, the positive prediction error penalty is added to the loss function to prevent possible frequency requirement violation. For the sparsity, the NN topology pruning is employed to eliminate unnecessary connections for solving acceleration. In the optimization stage, the trained CSNN is transformed into mixed-integer linear constraints using the big-M method and then incorporated to establish the data-enhanced model. The case study verifies 1) the effectiveness of the proposed model in terms of high accuracy, fewer parameters, and significant solving acceleration; 2) the stable system operation against frequency violation under contingency.	Linwei Sang, Yinliang Xu, Zhongkai Yi, Lun Yang, Huan Long, Hongbin Sun	2023-04-21T03:20:04Z	http://arxiv.org/abs/2304.10720v1	# Conservative Sparse Neural Network Embedded Frequency-Constrained Unit Commitment ###### Abstract The increasing penetration of distributed energy resources (DERs) will decrease the rotational inertia of the power system and further degrade the system frequency stability. To address the above issues, this paper leverages the advanced neural network (NN) to learn the frequency dynamics and incorporates NN to facilitate system reliable operation. This paper proposes the conservative sparse neural network (CSNN) embedded frequency-constrained unit commitment (FCUC) with converter-based DERs, including the learning and optimization stages. In the learning stage, it samples the inertia parameters, calculates the corresponding frequency, and characterizes the stability region of the sampled parameters using the convex hulls to ensure stability and avoid extrapolation. For conservativeness, the positive prediction error penalty is added to the loss function to prevent possible frequency requirement violation. For the sparsity, the NN topology pruning is employed to eliminate unnecessary connections for solving acceleration. In the optimization stage, the trained CSNN is transformed into mixed-integer linear constraints using the big-M method and then incorporated to establish the data-enhanced model. The case study verifies 1) the effectiveness of the proposed model in terms of high accuracy, fewer parameters, and significant solving acceleration; 2) the stable system operation against frequency violation under contingency. Distributed energy resources, unit commitment, conservative sparse neural network, frequency dynamics constraints. ## Nomenclature The main notation below is provided for unit commitment problem formulation. And other additional symbols are defined in the paper when needed. ## 1 A. Sets and Indices \begin{tabular}{l l} $t\in\mathcal{T}$ & Set of the time periods. \\ $g\in\mathcal{G}$ & Set of the thermal generators. \\ $n\in\mathcal{N}$ & Set of the system buses. \\ $w\in\mathcal{W}$ & Set of the converter-based wind turbine generators. \\ $pv\in\mathcal{PV}$ & Set of the converter-based PV generators. \\ $j\in\mathcal{N}_{d}$ & Set of the droop control converters. \\ $k\in\mathcal{N}_{v}$ & Set of the VSM control converters. \\ \end{tabular} ## 2. Decision Variables \begin{tabular}{l l} $x_{g,t}^{SU}/x_{g,t}^{SD}$ & Indicators of units start up / shunt down. \\ $x_{g,t}$ & Operation status of units. \\ $p_{g,t}/p_{w,t}/p_{pv,t}$ & Power output of thermal, wind turbines, and PV generators. \\ $p_{n,t}^{cur}$ & Load curtailment of the bus node. \\ $u_{gi}/u_{dj}/u_{dk}$ & Frequency response participation indicators of the thermal generators, droop / VSM converters. \\ $M$ & System aggregate inertia constant. \\ $D$ & System aggregate damping constant. \\ $R_{g}$ & System aggregate droop factor. \\ $F_{g}$ & System aggregate fraction factor. \\ $\delta$ & Phase angle of the bus node. \\ $P_{ij}$ & Power flow between node $i$ and node $j$. \\ \end{tabular} ## 3. Parameters \begin{tabular}{l l} $f_{N}$ & Nominal frequency rating value. \\ $f^{min}/f_{lim}(t_{j,t})$ & Minimum frequency limits / stepwise frequency limit. \\ $C_{oper}/C_{rsv}/C_{load}$ & Costs of system operation / reserve / load reduction. \\ $B_{mn}$ & The susceptance of the transmission line. \\ $\hat{p}_{w,t}/\hat{p}_{pv,t}$ & Day-ahead predicted power of wind / PV. \\ $M_{gi}$ & Inertia constant of the unit $i$. \\ $D_{gi}$ & Damping constant of the unit $i$. \\ $R_{gi}$ & Inertia constant of the unit $i$. \\ $K_{gi}$ & Mechanical power gain of the unit $i$. \\ $F_{gi}$ & Fraction of the unit $i$ power generation by the turbine. \\ $R_{dj}$ & Droop gain of the converter $j$. \\ $K_{dj}$ & Mechanical power gain of the converter $j$. \\ $M_{dk}$ & Virtual inertia of the converter $k$. \\ $D_{dk}$ & Virtual damp constant of the converter $k$. \\ $\epsilon_{l}/\epsilon_{der}$ & Prediction errors of load and DERs. \\ \end{tabular} ## I Introduction In the transition to a low carbon energy society, the renewable energies are replacing the conventional fossil energies and taking up a larger proportion of generation in the modern power system [1]. Unlike the direct connection of fossil-based synchronous generators, the distributed energy resources (DERs) are integrated into the grid through power converters that lack inherent rotational inertia, leading to the low-inertia systems [2]. The decreased rotational inertia will degrade the frequency stability of the system or even trigger large-scale blackouts [3, 4]. To address the above frequency stability problems of the low-inertia systems, the system operator should consider the frequency dynamics under contingency, transform the frequency dynamics requirements into static operational constraints, and incorporate them into the upstream scheduling models. Current research has focused on the formulation and incorporation of the frequency constraints by approximating the frequency nadir, which can be categorized into two types. The first type approximates the frequency response of generators as a piece-wise linear function to time, decoupling the governor control from the frequency in Ref. [5, 6, 7, 8]. Though exhibiting a simplified form with the frequency nadir constraints, it ignores the actual control policy and may not apply to high penetrated converter-based DER scenarios. The second type is based on the system frequency response model combined with the converter response feature to formulate the total frequency dynamics in Ref. [9, 10, 11, 12]. Ref. [9, 10] derive the analytical expression of frequency dynamics with converter-based DERs, calculate the frequency nadir expression, and utilize the piece-wise linear formulation to approximate these dynamics, which can be converted into a set of linear constraints. Ref. [11] further considers the joint scheduling of wind farm frequency support and reserve based on the analytical expression. These piece-wise linear models include the converter-based DER response but have limited approximation capability, which may not satisfy the requirement of complex converter-based power systems [13, 14]. The frequency support potential of the high-penetration DERs has been explored and exploited within the control scheme of the grid-forming voltage source converter (VSC) [15, 16, 17]. Virtual synchronous machine policy is one of the common VSC control strategies based on the emulation of system inertia [15]. Ref. [16] proposes a generalized architecture of VSC from the view of multivariable feedback control to unify different control strategies. Ref. [17] focuses on the wind turbine generators and proposes multiple virtual rotating masses for frequency support. However, the direct inertia parameter aggregation of different VSC control methods may lead to the low-inertia system instability [2]. So the relationship between the frequency dynamics and frequency support sources under the high penetrated converter-based DER is governed by the system's ordinary differential equations, which are complicated and cannot be incorporated into the system operation model directly. The thriving machine learning (ML) technique, widely applied in pattern cognition [18, 19] and image detection [20, 21], provides the powerful deep neural networks (DNNs) to formulate the above relationship with high accuracy [22, 23, 24, 25, 26]. The enforced constraints on the DNN-based frequency can be transformed into the mixed-integer linear programming [27] and solved efficiently by the off-the-shelf solvers [25]. In Ref. [22], the DNN is utilized to learn constraints and objectives, and the learned DNN is embedded in solving the world food program planning problem. ITwo critical issues arises from the direct integration of the trained DNNs into the frequency-constrained scheduling models will raise : 1) the positive and negative prediction errors of a DNN have different impacts on the system, where positive errors can lead to violations of the frequency requirements, and 2) the parameter number of the conventional dense DNN can be huge, resulting in the long solving time [28]. Faced with the complex frequency dynamics under the high penetrated converter-based DERs, this paper leverages the advanced research in ML and the dramatic improvement in solving mixed-integer linear programming (MILP) to formulate and incorporate the frequency constraints into operation models to ensure the system stable operation. To address the issues of DNN integration, it proposes the _conservative sparse neural network_ (CSNN) embedded frequency-constrained unit commitment model with DER, composed of learning and optimization stages. Its key lies in the design of the CSNN to achieve the conservativeness and sparsity of NN, which can further facilitate the safety and efficiency of CSNN incorporation. It first formulates and derives the highly non-convex frequency dynamics explicitly under DER. In the learning stage, it samples the inertia parameters, calculates the corresponding frequency, and characterizes the stability region of the sampled parameters using the convex hull to ensure stability and avoid extrapolation. Then for the conservativeness of NN, the positive prediction error penalty is added to the loss function to avoid possible frequency requirement violation; for the sparsity of NN, the neural network topology pruning is employed in training NN to eliminate some unnecessary connections for achieving MILP solving acceleration. In the optimization stage, the trained CSNN is transformed into a series of mixed-integer linear constraints using the big-M method to relax the ReLU activation functions in NN. Then these frequency constraints is incorporated into the conventional unit commitment (UC) to establish the final data-enhanced frequency constrained UC model in the MILP form. For this paper, the main contributions are as follows: 1) To the best of authors' knowledge, this paper, _for the first time_, proposes the _conservative sparse neural network_ to achieve both the conservativeness and sparsity of neural networks, which is beneficial to NN embedding in the optimization problems and can be utilized to formulate the relationship between the frequency dynamics and system inertia parameters. The proposed CSNN achieves higher approximation accuracy, compared with the previous piece-wise linear formulation of frequency dynamics in Ref. [10, 15], which exhibits i) fewer parameters to accelerate MILP solving ii) and conservative prediction to prevent the violation of frequency requirements, compared with conventional NN. 2) Based on the proposed CSNN, this paper proposes the data-enhanced frequency constrained unit commitment under DER to address the frequency stability problem of the low-inertia system. The proposed scheduling model approximates both frequency nadir and stepwise constraints by CSNN, transforms the trained CSNNs into a series of mixed-integer linear constraints using the big-M method, and embeds the constraints into the conventional UC model. Compared with conservative and conventional neural networks, the proposed data-enhanced model can achieve significant MILP solution time reduction, verifying its efficiency. 3) This paper designs the _stability region_ using convex hulls from the sampled dataset with stability to guarantee the system operation stability and avoid extrapolation. The main reason for _stability region_ is that not all inertia parameters in the sampled dataset can stabilize the total system, especially under the high penetration of the converter-based DERs. So the inertia parameters with stability are extracted to construct the stable dataset, and then the clustered convex hulls are extracted, denoted by _stability region_, which are transformed into linear constraints and embedded into the scheduling model to guarantee the stability and feasibility. For clearer delivery, we further summarize the main motivation and innovation of this paper. The increasing distributed energy resources will decrease the system rotational inertia and further degrade the system frequency stability, which requires the frequency consideration in system scheduling. However, the relationships between frequency dynamics and frequency support sources are complex, which requires the accurate and conservative formulation to capture the relationship. To address the above issues, we propose the CSNN with high representational capacity for capturing the above complex frequency relationship effectively with conservativeness design and embed the trained CSNN into the conventional unit commitment model efficiently with sparsity design to formulate the data-enhanced frequency constrained unit commitment model. The rest of this paper is organized as follows. Section II depicts the frequency dynamics with distributed energy resources. Section III proposes the conservative sparse neural network to formulate the frequency constraints. Section IV presents the CSNN embedded data-enhanced frequency-constrained unit commitment with DER. Section V performs the case study to verify the efficiency of the proposed model. Section VI concludes this paper. ## II Frequency Dynamics Formulation with Distributed Energy Resources This section formulates the system frequency dynamics under conventional _synchronous_ generators and _converter-based_ DER generators mathematically, governed by nonlinear ordinary differential equations from Ref. [30]. The frequency dynamics can be analyzed by the swing equation in Eq. (1). \[M\frac{d\Delta f}{dt}+D\Delta f=\Delta p_{g}-\Delta p_{e}. \tag{1}\] Then, based on Eq. (1), the low-inertia system dynamics is introduced, and several key frequency constraints are proposed to guarantee the frequency dynamic stability. ### _Frequency Dynamics Analytical Analysis_ Based on Ref. [15], we can derive the transfer function $G(s)$ of the system dynamics model in Eq. (2). 1 Footnote 1: The detailed infrastructure can be referred to Ref. [9]. \[\begin{split} G(s)&=\frac{\Delta f}{\Delta P_{e}}= \bigg{(}\underbrace{(sM_{g}+D_{g})+\sum_{l\in\mathcal{N}_{g}}\frac{K_{gi}(1+sF _{gi}T_{gi})}{R_{gi}(1+sT_{gi})}}_{\text{Multiple-machine system frequency response}}\\ &+\sum_{j\in\mathcal{N}_{d}}\frac{K_{cj}}{R_{cj}(1+sT_{cj})}+ \sum_{k\in\mathcal{N}_{v}}\frac{sM_{ck}+D_{ck}}{1+sT_{ck}}\bigg{)}^{-1}. \end{split} \tag{2}\] It is assumed that all synchronous generators have equal time constants with $T_{gi}=T,\forall i\in\mathcal{N}_{g}$. The time constants of converters are 2-3 orders lower than those of synchronous generators with $T\gg T_{c}\approx 0$, according to Ref. [33]. Then we transform the general-order Eq. (2) into its second-order form Eq. (3). \[G(s)=\frac{1}{MT}\frac{1+sT}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}. \tag{3}\] where $\omega_{n}$ and $\zeta$ denote the natural frequency and damping ratio, calculated by Eq. (4). \[\omega_{n}=\sqrt{\frac{D+R_{g}}{MT}},\quad\zeta=\frac{M+T(D+F_{g})}{2\sqrt{MT( D+R_{g})}}. \tag{4}\] The key parameters $M$, $D$, $F_{g}$, and $R_{g}$ in Eq. (4) calculation can be referred to Ref. [15]. Given a stepwise disturbance $\Delta P$ ($\Delta P/s$ in the frequency domain) in Eq. (1), the frequency dynamics in time domain can be derived from Eq. (3) as follows: \[\begin{split}\Delta f(t)=&-\frac{\Delta P}{MT \omega_{n}^{2}}\\ &-\frac{\Delta P}{M\omega_{d}}e^{-\zeta\omega_{n}t}\Big{(}\sin( \omega_{d}t)-\frac{1}{\omega_{n}T}\sin(\omega_{d}t+\phi)\Big{)}\end{split} \tag{5}\] where the $\omega_{d}$ and $\phi$ calculated as follows: \[\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}},\quad\phi=\arcsin(\sqrt{1-\zeta^{2}}). \tag{6}\] The frequency nadir is located in the time instance $t_{m}$ where the derivative of frequency is zero in Eq. (7). \[\frac{d\Delta f(t_{m})}{t_{m}}=0\to t_{m}=\frac{1}{\omega_{d}}\tan^{-1}( \frac{\omega_{d}}{\zeta\omega_{n}-T^{-1}}). \tag{7}\] Then the maximum frequency deviation can be calculated by replacing the $t$ of Eq. (5) by $t_{m}$ of Eq. (7) as: \[\Delta f_{max}=-\frac{\Delta P}{D+R_{g}}(1+\sqrt{\frac{T(R_{g}-F_{g})}{M}}e^{- \zeta\omega_{n}t_{m}}). \tag{8}\] ### _Frequency Constraints Formulation_ Based on the above analytical frequency response formulation, this paper proposes four key constraints to bound frequency dynamics for securing the system stability from different perspectives, including the rate-of-change-of-frequency (RoCoF) constraint for guaranteeing transient stability [31], the quasi-steady state for bringing back system normal operation state [9], the frequency nadir constraint for avoiding the activation of under frequency load shedding [15], and the stepwise frequency constraint for enhancing the frequency restoration performance [8]. #### Ii-B1 RoCoF Constraint RoCoF constraint guarantees the sufficient system inertia to limit the allowable RoCoF at $t=0$, following the ENTSO-E standard [32]. \[\|\text{RoCoF}\|=\|\frac{\Delta P}{M}\|\leq\text{RoCoF}^{max}. \tag{9}\] #### Ii-B2 Quasi-Steady State Constraint Quasi-steady state (QSS) constraint guarantees quasi-steady state security assuming the RoCoF is effectively zero. \[\Delta f_{qss}=-\frac{\Delta P}{D+R_{g}}\leq\Delta f_{qss}^{max}. \tag{10}\] #### Ii-B3 Frequency Nadir Constraints To avoid triggering under-frequency load shedding (UFLS) and oscillation, the maximum frequency deviation should be limited within the allowable range by Eq. (11). \[f_{nadir}=f_{N}-\Delta f_{max}\geq f^{min}. \tag{11}\] #### Ii-B4 Frequency Stepwise Constraints Besides the above frequency constraints, we further consider the frequency restoration requirement from frequency nadir to quasi-steady state, inspired by Ref. [8]. The stepwise frequency constraints are utilized to bound the frequency restoration process, as shown in Eq. (12), Eq. (13), and Fig. 1. Then we formulate the stepwise frequency constraints as follows: \[f(t)=f_{N}-\Delta f(t)\geq f_{lim}(t). \tag{13}\] However, the above constraints are complicated because they should be evaluated for $t\geq 0$. As shown in Fig. 1, the stepwise frequency constraints are binding when the transient curve and limit curve touch. The constraints are not required when no step changes in the frequency limit curve. So we only evaluate it at the step change time $t=t_{j,l}$ as follows: \[f(t_{j,l})=f_{N}-\Delta f(t_{j,l})\geq f_{lim}(t_{j,l})\quad j\in N_{c}. \tag{14}\] ### _Frequency Constraints Analysis_ #### Ii-C1 Relationship Analysis The RoCoF and QSS constraints are linear with respect to $M$, $D$, and $R_{g}$. In contrast, the frequency nadir and stepwise constraints are nonlinear due to the nonlinearity of $f_{nadir}(M,D,R_{g},F_{g})$ and $f(M,D,R_{g},F_{g};t_{j,l})$ with respect to $M$, $D$, $R_{g}$, and $F_{g}$. #### Ii-C2 Conservative Approximation of Frequency Constraints So we need to derive the tractable constraints from the nonlinear ones to facilitate tractable solving. The system operator prioritizes the negative error (predicted value minus true value) but is averse to positive error, where positive errors may result in the frequency requirement violation. So we propose to approximate the $f_{nadir}$ and $f(t_{j,l})$ in a conservative way in Eq. (15). \[f_{nadir}(M,D,R_{g},F_{g})\geq\hat{f}_{nadir}(M,D,R_{g},F_{g}) \geq f^{min} \tag{15a}\] \[f(M,D,R_{g},F_{g};t_{j,l})\geq\hat{f}(M,D,R_{g},F_{g};t_{j,l}) \geq f_{lim}(t_{j,l}) \tag{15b}\] where $\hat{f}_{nadir}$ and $\hat{f}$ are the predicted frequency, which are the lower bounds of the true frequency. ## III Methodology Based on the frequency constraints formulation and analysis, this section 1) introduces the concept of optimization with constraint learning (OCL) framework, 2) proposes the conservative sparse neural network formulation under the OCL framework for nonlinear frequency constraints in Eq. (15), 3) embeds the trained neural network into the optimization models via MILP, 4) and presents two data processing techniques for facilitating predictive model integration. ### _Data-driven Optimization with Constraint Learning Framework Design_ Practical optimization problems often contain constraints with intractable forms or no explicit formulas. However, based on the supportive data corresponding to such constraints, these data can be utilized to learn the constraints (constraint learning, CL). The trained predictive models can be embedded into the optimization models. So the above two stages are called optimization with constraint learning (OCL). We denote the final tractable model by the data-enhanced optimization model. According to the review of Ref. [23], the OCL framework is usually composed of the following five steps: 1) formulate Fig. 1: Multiple stepwise frequency requirement. the conceptual optimization model 2) conduct data sampling and processing; 3) construct and train predictive models; 4) embed the predictive model into the optimization model; 5) solve and evaluate the data-enhanced optimization model. It should be noted that the OCL framework can be extended to objective learning by transforming the objective function into constraints via the epigraph of the function [22]. ### _Conservative Sparse Neural Network Formulation_ Under the above OCL framework, we propose the CSNN to learn the relationship of Eq. (15) and formulate the data-enhanced frequency-constrained unit commitment model based on the trained CSNN, detailed in the following parts. The learning of Eq. (15) is denoted by frequency nadir constraint learning for Eq. (15a) and stepwise constraint learning for Eq. (15b). The proposed CSNN prediction models are both conservative and sparse, which can guarantee the safety of frequency constraints and accelerate the NN-embedding model solution. #### Iii-B1 Conservative Neural Network Design Based on the analysis of frequency constraints in section II-C, the negative errors are prioritized against possible frequency requirement violations compared to the positive errors. So the conservative prediction models are prioritized for the system safe and stable operation. To this end, we propose a conservative loss function where prediction values are always lower than the true values. It is calculated by the weighted sum of _mean-square-error_ loss function and additional positive error ReLU loss function to penalize the positive part of the frequency constraints, as illustrated in Eq. (16). \[\hat{f}^{}_{\theta}\leftarrow\arg\min_{\theta}\mathcal{L}^{com}_{\theta} \tag{16a}\] \[\mathcal{L}^{com}_{\theta}=\underbrace{\frac{1}{2}\mathbb{E}_{(x,y) \rightarrow\mathcal{D}}(\hat{f}_{\theta}(x)-y)^{2}}_{\text{supervised regression}}+\underbrace{\mathbb{E}_{(x,y)\rightarrow\mathcal{D}}\text{ReLU}(\hat{f}_{\theta}(x)-y)}_{ \text{ReLU loss function}} \tag{16b}\] where the resulting $\hat{f}^{}_{\theta}(x)$ is the conservative estimate of the actual function $f(x)$. #### Iii-B2 Sparse Neural Network Design We propose to train sparse neural networks by dropping and activating their inner connections iteratively to alleviate the calculation burden of dense full-connected NN embedding. The keys to sparse training lie in updating neural network topology based on parameter magnitude criterion for dropping connections and infrequent gradient criterion for activating connections, which is from the lottery ticket hypothesis in Ref. [29]. The motivation behind the growing and dropping sparse training is as follows: the dropping of neuron connections may sacrifice the accuracy of the learning models; so, we further re-activate $k$ connections with the top-$k$ high magnitude gradients, because high magnitude gradients indicate high impact connection between neurons. So we re-activate these connections for improving the accuracy. The main scheme of sparse training is composed of sparse distributing, update scheduling, connection dropping, and connection activating for learning the mapping from the input system parameter to the output frequency response, as shown in Fig. 2. Sparse DistributingWe define sparsity $s^{l}\in(0,1)$ as the dropping fraction of the $l^{th}$ layer. The first layer is set as the dense layer, and the other layers are set as the sparse layers with predefined $s^{l}$ for the $l^{th}$ layer sparse training in a uniform way. Update SchedulingThe update scheduling drops a fraction of connections based on NN parameter magnitude and activates new ones based on NN parameter gradient magnitude iteratively at $\Delta T$ interval. So the updating fraction function by considering decaying is defined in Eq. (17). \[f_{decay}(t;\alpha,T_{end})=\frac{\alpha}{2}(1+\cos\frac{t\pi}{T_{end}}) \tag{17}\] where $T_{end}$ is the number of iteration for sparse training, and $\alpha$ is the initial drop fraction. Connection DroppingIn connection dropping of the $l^{th}$ layer, we utilize $ArgTopK(v,k)$ algorithm to select the top-$k$ maximum element indices of vector $v$ and drop the connection by $ArgTopK(-\|\theta^{l}\|,f_{decay}(t;\alpha,T_{end})(1-s^{l})N^{l})$. Connection ActivatingAfter dropping, we further re-activate $k$ connections with the top-$k$ high magnitude gradients by $ArgTopK(\nabla_{\theta^{l}}L^{com}_{t},f_{decay}(t;\alpha,T_{end})(1-s^{l})N^ {l})$. Newly activated connections are initialized as zero to avoid influencing the network output. Remark 1.: _The effect of gradients on the model's training lies in the following two aspects: i) the gradients from the loss function can learn the prediction models by minimizing the proposed conservative loss function (stochastic gradient training); ii) they can also be utilized to grow necessary connection in the sparse neural networks to improve accuracy._ #### Iii-B3 Conservative Sparse SGD Algorithm Based on the conservative loss of Eq. (16b) and sparse training designs, we propose the conservative sparse stochastic gradient descent (CS-SGD) algorithm for training the CSNN in algorithm 1. The key novelty of the proposed CS-SGD algorithm lies in incorporating the conservative loss function of Eq. (16) and sparse training process with dropping and growing to train Fig. 2: Sparse neural network training by leveraging the weight magnitude and gradient information. CSNN with conservativeness and sparsity for approximating the power system frequency dynamics. The weights and bias of trained CSNN are leveraged to formulate the equivalent mixed-integer linear frequency constraints, which can secure the system frequency stability. The conservativeness of CSNN guarantees safety of the transformed frequency constraints, and the sparsity of CSNN improving solution efficiency of the data-enhanced frequency constrained unit commitment optimization model with fewer optimizing variables. ### _NN Embedded Frequency Constraints Transformation_ After training, the proposed CSNN can be transformed into a series of mixed linear integer constraints using the big-M method. Based on Ref. [27], this paper formulates the general form of an $L$-layer feedforward neural network with ReLU as the activation function in Eq. (18), which is composed of stacked linear transformation (18a) and ReLU activation (18b). \[\text{Neural Network:}v_{i}^{I}=\text{ReLU}(\theta_{i0}^{I}+ \sum_{j\in\mathcal{N}^{L-1}}\theta_{ij}^{I}v_{j}^{J-1}) \tag{18a}\] \[\text{where }y=v^{L}=\theta_{0}^{L}+\sum_{j\in\mathcal{N}^{L-1}} \theta_{j}^{L}v_{j}^{L-1} \tag{18b}\] where $v_{i}^{I}$ is the output of neuron $i$ in layer $l$ in a neuron network, $\theta_{ij}^{I}$ is the weight between neuron $i$ and $j$, and $\theta_{i0}^{I}$ is the bias of neuron $i$ in layer $l$. For each layer of Eq. (18a), the ReLU operator, $\text{ReLU}(x)=\max\{0,x\}$, can be encoded using linear constraints in a big-M way from Ref. [22], as follows: \[v \geq x, \tag{19a}\] \[v \leq x-M_{L}(1-t),\] (19b) \[v \leq M_{U}t,\] (19c) \[v \geq 0,\] (19d) \[t \in\{0,1\} \tag{19e}\] where $M_{U}$ and $M_{L}$ are positive and negative large value to relax the ReLU function. ### _Data Processing Techniques_ This part utilizes the data sampling strategy based on the boundary and Latin hypercube sampling to enhance the data quality for the predictive models. It constructs the stability region based on the convex hulls to prevent the extrapolating of predictive models. #### Iv-D1 Data Sampling Strategy The predictive model performance relies on the training dataset quality; thus, the data distribution is critical for constraint learning. The sampled data should be space-filling sufficiently to capture the frequency dynamics of frequency constraints in Eq. (15). Inspired by Ref. [25], we utilize the boundary sampling and optimal Latin hypercube (OLH) sampling for constructing training set $\mathcal{Z}=\{M_{n},D_{n},R_{g,n},F_{g,n}\}_{n=1}^{N}$. In the process, the boundary sampling collects the corners of the $\mathcal{Z}$ hypercube by the input data bounds, and then we conduct the OLH sampling over the above hypercube. The detailed sampling procedures are illustrated in Ref. [25]. #### Iv-D2 Convex Hulls as Stability Region The optimal solutions of mixed-integer linear programs are often located at the extremes of their feasible polyhedron region, which may harm the validity of the trained ML models. According to Ref. [22], the accuracy of an ML-based predictive model will deteriorate for points far away from the training dataset. So we construct stability regions to prevent the predictive model from extrapolating by the convex hull from the training dataset. For the training dataset $\mathcal{Z}$, we construct the following convex hull $CH(\mathcal{Z})$ in Eq. (20) as the stability region. \[CH(\mathcal{Z})=\left\{z\|\sum_{i\in I}\lambda_{i}\hat{z}_{i}=z,\sum_{i\in I} \lambda_{i}=1,\lambda\geq 0\right\} \tag{20}\] However, there are two issues arising from the direct single convex hull. One issue is that the computing of the convex hull is exponential to data number in time and space; the other is that the data distribution in a single convex hull is not uniform. Some points cannot satisfy the requirement of system stability in Eq. (2), and the single convex hull may include and extrapolate some infeasible points. So we propose the cluster-then-build strategy, which first clusters the trained dataset, builds a convex hull for each cluster, and unions them $\bigcup_{k\in\mathcal{K}}CH_{k}(\mathcal{Z})$ for preventing extrapolating in Eq. (21). \[\begin{split}\bigcup_{k\in\mathcal{K}}CH_{k}(\mathcal{Z})=\left\{ z\|\sum_{i\in I_{\bar{i}}}\lambda_{i}\hat{z}_{i}=z,\sum_{i\in I}\lambda_{i}=u_{k}, \sum_{k\in\mathcal{K}}u_{k}=1\right.\\ \left.\lambda\geq 0,\forall k,u\in\{0,1\}\right\}\end{split} \tag{21}\] #### Iv-D3 Discussion It should be noted that we need to construct and build different CSNN models and stability regions for different system and DER settings. Because the training of CSNNs is based on the sampled dataset from the system model with corresponding DER setting via Eq. (5) and Eq. (8). Different system and DER settings will affect the sampled dataset directly and further affect the training of CSNNs indirectly, leading to inaccurate approximation. Stability region is also constructed from the sampled dataset. So we need to retrain different learning models and construct different stability region for these circumstances, such as different DER settings, to improve approximation accuracy. ## IV Data-enhanced Frequency-Constrained Unit Commitment with DER Based on the above proposed CSNN, this section proposes the data-enhanced frequency constrained unit commitment (FCUC) with the converter-based DER. It firstly introduces the overall OCL framework for the proposed data-enhanced FCUC, then presents the conventional unit commitment model, and includes the data-enhanced frequency constraints to formulate the final data-enhanced model. ### _Overall Framework_ Based on the CSNN and NN embedded transformation of section III, we propose the OCL framework for the data-enhanced FCUC with converter-based DERs, composed of the learning and optimization stages, as shown in Fig. 3. Its main idea lies in leveraging CSNN and the frequency data to enhance the power system decision-making and advance the data-driven integration modeling technique. The learning stage samples the stable instances from the frequency analytical expression of Eq. (8) and Eq. (13), feeds them to training the CSNNs, constructs the stability region using the convex hulls for the stable instances, and evaluates their performance; the optimization stage embeds the stability region constraints of Eq. (21) and transformed CSNN constraints of Eq. (19) in the system scheduling model to formulate the final data-enhanced FCUC model. It should be noted that: 1) the stability region and the CSNNs are constructed from the sampled dataset in different ways and are integrated into the conventional unit commitment model as mixed-integer linear constraints, so the stability region construction will not affect the training process of CSNNs; 2) DERs refer to the distributed photovoltaic (PV) and wind power (WP) generators concretely in this paper. ### _Unit Commitment Considering Joint Reserve and Regulation Capacity_ Based on the above OCL framework, we formulate the conventional unit commitment model considering joint reserve and regulation capacity to minimize the system operating cost under a series of operational constraints. #### Iv-A1 Model Objective The objective of conventional unit commitment is to minimize the total system operating cost, which is composed of the operating cost $C_{oper}$, reserve cost $C_{sav}$, and load shedding cost $C_{load}$ in Eq. (22). \[\min \quad C_{total}=C_{oper}+C_{rsv}+C_{load} \tag{22a}\] \[C_{oper}=\sum_{t\in\mathcal{N}_{T}}\sum_{g\in\mathcal{N}_{G}} \left(C_{g}^{SU}x_{g,t}^{SU}+C_{g}^{SD}x_{g,t}^{SD}+f_{g}(p_{g,t})\right)\] (22b) \[C_{rsv}=\sum_{t\in\mathcal{N}_{T}}\Big{(}\sum_{g\in\mathcal{G}}C_{g}^{R}R _{g,t}+\sum_{w\in\mathcal{W}}C_{w}^{R}R_{w,t}+\sum_{pv\in\mathcal{P}\mathcal{ V}}C_{pv}^{R}R_{pv,t}\Big{)}\] (22c) \[C_{load}=\sum_{t\in\mathcal{N}_{T}}\sum_{n\in\mathcal{N}_{R}} \lambda_{L}P_{n,t}^{cur} \tag{22d}\] where $C_{oper}$ includes the start-up, shut-down, and the fuel costs; $C_{rsv}$ includes the costs of the reserve from synchronous generators and converter-based PV/WP generators; $C_{load}$ includes the load shedding costs from different buses. The unit cost function $f_{g}(\cdot)$ can be piece-wised by SOS2 constraints. #### Iv-A2 Power Network Constraints The node power balance equation and transmission power limited are formulated based on DC power flow model in Eq. (23). \[\sum_{g\in\mathcal{G}_{n}}p_{g,t}+\sum_{w\in\mathcal{W}_{n}}p_{ w,t}+\sum_{pv\in\mathcal{P}\mathcal{V}_{n}}p_{pv,t}\] \[+\sum_{m:(n,m)}B_{nm}(\delta_{nn}-\delta_{nn})=p_{n,t}^{D}-p_{n,t }^{cur}\quad\forall n,t \tag{23a}\] \[\|B_{nm}(\delta_{nn}-\delta_{nn})\|\leq F_{nm}^{max}\quad\forall(n,m) \in\mathcal{L},\forall t \tag{23b}\] where Eq. (23a) enforces the bus power balance for each period, and Eq. (23b) restricts the branch power flow under the given capacity. #### Iv-A3 Thermal Generator Operational Constraints The ramping and power output restriction are formulated via MILP in Eq. (24). \[x_{g,t}p_{g}^{min}\leq p_{g,t}\leq x_{g,t}p_{g}^{max} \quad\forall g,t \tag{24a}\] \[x_{g,t}^{SU}-x_{g,t}^{SD}=x_{g,t}-x_{g,t-1} \quad\forall g,t\] (24b) \[p_{g,t}-p_{g,t-1}\leq x_{g,t-1}Ru_{g} \quad\forall g,t\] (24c) \[p_{g,t-1}-p_{g,t}\leq x_{g,t}Rd_{g} \quad\forall g,t \tag{24d}\] where Eq. (24a) restricts the generator power output; Eq. (24b) enforces the relationship between the on/off variables and start-up/shut-down variables; Eq. (24c) and Eq. (24d) restricts the generator ramping capacity. #### Iv-A4 Renewable Energy Generation Constraints The actual power outputs of renewable energy generation are limited below the predicted values. \[0\leq p_{w,t}\leq\hat{p}_{w,t}\quad\forall w,t \tag{25a}\] Fig. 3: The optimization with constraint learning framework for the data-enhanced FCUC with DER. \[0\leq p_{pv,t}\leq\hat{p}_{pv,t}\quad\forall pv,t \tag{25b}\] where Eq. (25a) and Eq. (25b) restrict the actual power output of wind and PV below the day ahead predicted value. #### Iv-B5 System Reserve Constraints The system reserve is restricted to cover the prediction errors of the DERs and load as follows: \[\begin{split}&\sum_{g\in\mathcal{G}}x_{g,t}P_{g}^{max}+\sum_{w \in\mathcal{W}}\hat{p}_{w,t}+\sum_{pv\in\mathcal{P}\mathcal{V}}\hat{p}_{pv,t} +\sum_{n\in\mathcal{N}}P_{n,t}^{cur}\\ &\geq(1+\epsilon_{l})\sum_{n\in\mathcal{N}}\hat{p}_{n,t}^{D}\\ &+(1+\epsilon_{der})(\sum_{w\in\mathcal{W}}\hat{p}_{w,t}+\sum_{pv \in\mathcal{P}\mathcal{V}}\hat{p}_{pv,t})\quad\forall t.\end{split} \tag{26}\] Eq. (26) guarantees that the online generation capacity can satisfy the load demand with prediction errors, according to Ref. [10]. ### _Data-enhanced Frequency Constraints_ As discussed in section III, we propose the CSNN to approximate the nonlinear constraints of frequency nadir and stepwise constraints in a conservative way. So this part adds the CSNN embedded frequency dynamics constraints in Eq. (28) and corresponding emergency regulation constraints in Eq. (30) into the above conventional unit commitment models for ensuring the frequency dynamics stability. #### Iv-C1 Inertia Parameter Aggregation The system inertia variables $M$, $D$, $R_{g}$, and $F_{g}$ for Eq. (15) can be calculated by aggregating generators' and PV/WP converters' related parameters in Eq. (27). \[M_{g} =\frac{\sum_{i\in N_{g}}u_{gi}M_{gi}P_{gi}}{P_{g}^{b}},D_{g}= \frac{\sum_{i\in N_{g}}u_{gi}D_{gi}P_{gi}}{P_{g}^{b}} \tag{27a}\] \[F_{g} =\sum_{i\in N_{g}}\frac{u_{gi}K_{gi}F_{gi}}{R_{gi}}\frac{P_{gi}} {P_{g}^{b}},R_{g}=\sum_{i\in N_{g}}\frac{u_{gi}K_{gi}}{R_{gi}}\frac{P_{gi}}{ P_{g}^{b}}\] (27b) \[R_{d} =\sum_{j\in N_{d}}\frac{u_{dj}K_{dj}}{R_{dj}}\frac{P_{dj}}{P_{der }^{b}}\] (27c) \[M_{d} =\sum_{k\in N_{e}}u_{dk}M_{dk}\frac{P_{dk}}{P_{der}^{b}},D_{d}= \sum_{k\in N_{e}}u_{dk}D_{dk}\frac{P_{dk}}{P_{der}^{b}}\] (27d) \[M =\frac{M_{g}P_{g}^{b}+M_{der}P_{der}^{b}}{P_{g}^{b}+P_{der}^{b}}\] (27e) \[D =\frac{D_{g}P_{g}^{b}+D_{der}P_{der}^{b}+R_{d}P_{der}^{b}}{P_{g}^{b }+P_{der}^{b}}. \tag{27f}\] #### Iv-C2 Frequency Dynamics Constraints The proposed CSNN is utilized to reach the convex lower bounds of nonlinear frequency constraints by transforming $f_{nadir}$ and $f(t_{j,l})$ into $\hat{f}_{nadir}^{csmn}$ and $\hat{f}_{csmn}^{csmn}(t_{j,l})$ in Eq. (11) and Eq. (14), as follows: \[\hat{f}_{nadir}^{csmn}(M,D,R_{g},F_{g})\geq f_{min} \tag{28a}\] \[\hat{f}_{nadir}^{csmn}(M,D,R_{g},F_{g};t_{j,l})\geq f_{lim}(t_{j, l}). \tag{28b}\] where $\hat{f}_{nadir}^{csmn}$ and $\hat{f}_{csmn}^{csmn}(t_{j,l})$ can be transformed into mixed-integer linear constraints via Eq. (19). Taking Eq. (28a) as an example, its equivalent mixed-integer constraints is formulated and incorporated by taking $x=(M,D,R_{g},F_{g})$ in Eq. (29). \[v_{i}^{1} =\theta_{i0}^{1}+\sum_{j\in\mathcal{N}}\theta_{ij}^{l}x_{j}, i\in[N_{1}] \tag{29a}\] \[\hat{v}_{i}^{l} =\theta_{i0}^{l}+\sum_{j\in\mathcal{N}^{l-1}}\theta_{ij}^{l}v_{j}^ {l-1}, l\in[L],i\in[N_{l}]\] (29b) \[v_{i}^{l} \geq\hat{v}_{i}^{l}, l\in[L],i\in[N_{l}]\] (29c) \[v_{i}^{l} \leq\hat{v}_{i}^{l}-M_{l,i}^{L}(1-t_{l,i}), l\in[L],i\in[N_{l}]\] (29d) \[v_{i}^{l} \leq M_{l,i}^{H}l_{l,i}. l\in[L],i\in[N_{l}]\] (29e) \[v_{i}^{l} \geq 0,\quad t_{i,l}\in\{0,1\}, l\in[L],i\in[N_{l}]\] (29f) \[\hat{f}_{nadir,m}^{csmn}=\theta_{i0}^{L}+\sum_{j\in\mathcal{N}^{l-1}} \theta_{ij}^{L}v_{j}^{L-1}\quad m\in[N_{M}],i\in[N_{L}] \tag{29g}\] where $[N_{l}]$ is the set of input neurons in layer $l$, and $[L]$ is the set of output neurons in layer $l$. #### Iv-C3 Emergency Regulation Formulation Eq. (30) provides the headroom regulation capacity for the primary frequency response from synchronous and converter-based generators. \[R_{g,t}=p_{g}^{max}-p_{g,t}\geq u_{g,t}\frac{K_{g}}{R_{g}}p_{g}^{ max}\gamma(f_{0}-f_{min}) \tag{30a}\] \[R_{w,t}=\hat{p}^{w,t}-p_{w,t}\geq u_{w,t}\frac{1}{R_{w}}\hat{p}^{ w,t}\gamma(f_{0}-f_{min})\] (30b) \[R_{pv,t}=\hat{p}^{pv,t}-p_{pv,t}\geq u_{pv,t}\frac{1}{R_{pv}} \hat{p}^{pv,t}\gamma(f_{0}-f_{min})\] (30c) \[\forall g\in\mathcal{G},w\in\mathcal{W},pv\in\mathcal{P}\mathcal{V}, \quad\forall t.\] The largest frequency deviation ($f_{0}-f_{min}$) is larger the quasi-steady state frequency deviation. According to Ref. [10], the $\gamma$ is set as 0.5 for emergency reserve capacity. So the combination of unit commitment modeling and data-enhanced frequency constraints constructs the final data-enhanced FCUC with converter-based DER considering both the system reserve for uncertainty and the regulation capacity for emergency, which follows the mixed-integer linear programming form and can be solved by the off-the-shelf solvers efficiently. ## V Case Study In the case study, the modified IEEE 118-bus and 1888-bus systems with multiple PV and WP is utilized to verify the effectiveness and efficiency of the data-enhanced frequency constrained unit commitment model with converter-based DER. The proposed model is composed of the learning and optimization stages. So we construct the learning models with the Pytorch package and the optimization models with the Cvxpy package equipped by the Gurobi solver. All the experiments are implemented in a MacbookPro laptop with RAM of 16 GB, CPU Intel Core I7 (2.6 GHz). The other detailed system parameter setting of the modified system is attached in Ref. [34] and Ref. [35]. ### _Data Preparation and Experiment Setting_ This part mainly prepares the essential data and problem settings in the OCL framework for the data-enhanced FCUC model of Fig. 3, including 1) the neural network parameter setting of the learning stage 2) and the inertia parameter setting of the optimization stage. #### V-A1 Learning Stage In the learning stage, we first generate 10000 feasible samples based on the system frequency dynamics of Eq. (5) and Eq. (8) to construct the training dataset for the stability region and learning model training. Then the proposed CSNNs take the system inertia parameter $x=(M,D,R_{g},F_{g})$ as the feature vector, and the input feature vector is normalized into the scale of 0-1 for model training based on their maximum value. We compare the proposed conservative sparse neural network (CSNN) with the conservative neural network (C-NN) and the conventional neural network (NN) under the same initial parameters of full connected neural networks in Tab. I. The hyperparameter configuration in Tab. I is determined based on the hyperparameter tuning with the high approximation accuracy of learning models. The change of these hyperparameter values does not enhance the approximation accuracy of the learning models in our cases. We clarify the relationship among CSNN, C-NN, and NN, where C-NN further utilizes the conservative loss of Eq. (16) compared to NN, and CSNN further employs the sparse training of algorithm 1 compared to C-NN. The same hyperparameters of Tab. I are utilized to initialize the CSNN, C-NN, and NN to avoid the hyperparameter effect on model comparison of different neural networks. #### V-A2 Optimization Stage In the optimization stage, we prepare the inertia parameters of different generators and VSM/droop control parameters of PV and wind power converters from Ref. [15], as illustrated in Tab. II. The frequency nadir, stepwise frequency, and QSS requirement are set based on the ENTSO-E standard [32] and Ref. [8]. ### _Frequency Approximation Performance Evaluation_ This part firstly analyzes the approximation performance of the proposed CSNN in describing the frequency dynamics. It then verifies its effectiveness in the data-enhanced frequency constrained model with the stability region design. #### V-B1 CSNN Analysis Training ProcessFig. 4 compares CSNN, C-NN, and NN training processes for the frequency nadir and stepwise relationship approximation. In the training process, NN achieves a lower approximation error than CSNN and C-NN due to the consideration of the conservative loss function, and CSNN exhibits a little lower error than C-NN thanks to the sparse designs. #### V-B2 Approximation Error We further compare the detailed approximation performance of CSNN with C-NN and NN. Fig. 5 demonstrates the approximation error distribution of the above four models by the boxen plot. As shown in Fig. 5, the approximation errors of NN are located around zero with negative and positive values, and in contrast, the approximation errors of CSNN and C-NN are located below zero with only negative values, demonstrating their conservativeness. The median error of NN is above zero. CSNN exhibits a more accurate approximation with a more concentrated error distribution than C-NN. We utilize the mean square error (MSE), mean absolute percentage error (MAPE) metrics in Eq. (31), and negative rate to measure the approximation performance numerically. \[\mathcal{L}_{MSE}=\frac{1}{N}\sum_{i}^{N}(f_{i}-\hat{f}_{i})^{2},\quad\mathcal{ L}_{MAPE}=\frac{1}{N}\sum_{i}^{N}\|\frac{f_{i}-\hat{f}_{i}}{f_{i}}\|. \tag{31}\] As illustrated in Tab. III, for MAPE, CSNN achieves 0.87% lower MAPE than NN, 0.76% lower MAPE than C-NN, and 0.11% lower MAPE than PWL; for MSE, CSNN increases 142.14% MSE of NN but decrease 91.35% MSE of C-NN and Fig. 4: The training process of frequency nadir and stepwise frequency constraint learning. Fig. 5: The approximation errors of different models in boxen plot. 74.85% MSE of C-NN, which sacrifices some approximation accuracy to guarantee conservativeness. As for the negative rate, both CSNN and C-NN exhibit a 100% negative rate, demonstrating the effectiveness of the conservative loss design. In contrast, the negative rate of NN is only 85.43%, which may result in the frequency requirement violation. Sparsity ComparisonTo verify the sparsity of CSNN, we compare the non-zero parameter number of the above four models, where lower parameters mean a higher sparsity. Sparsity ComparisonTo verify the sparsity of CSNN, we compare the non-zero parameter number of the above four models, where lower parameters mean a higher sparsity. For the frequency nadir constraint learning of Eq. (15a) in Tab. IV, the corresponding CSNN has only 1395 parameters and achieves the 86.96% reduction of the dense neural networks of C-NN and NN, verifying the effectiveness of sparse training. For the stepwise frequency constraint learning of Eq. (15b) in Tab. V, the corresponding CSNN has only 1442 parameters and achieves the 86.77% reduction of the dense neural networks of C-NN and NN. ### _Comparison of Optimization Scenarios_ In this part, we first compare the operational performance of the data-enhanced frequency constrained unit commitment model and then verify the efficacy of the sparse design of neural networks, stability region, and the stepwise frequency constraints. #### Vi-C1 Operation Performance We further compare the data-enhanced FCUC with DER to the conventional unit commitment for verifying the necessity of frequency constraints and the effectiveness of DER participation. ResultsTab. VI compares the operational costs and renewable abandoned rates of the scheduling model with/without frequency constraints. The data-enhanced FCUC increases the operational cost by 8.24% and the renewable abandoned rate by 2.64% but decreases the system frequency violation rate from 100% to 0% under large contingency, verifying the effectiveness of the proposed model. The increase of abandoned renewable supports system inertia. Through the frequency constraints, the system will enhance the system inertia and frequency response by increasing the online units and exploiting the frequency support from DER, leading to a higher operational cost, abandoned rate, and more stable system operation. Fig. 6 further compares the operational unit status between the convention scheduling and frequency-constrained scheduling, demonstrating that the frequency constraints enforce the unit start-up to guarantee the system stability. Dynamics AnalysisWe simulate the frequency dynamics under the scheduling models with and without frequency constraints under the 500 MW power loss contingency in Fig. 7, verifying their effectiveness. As shown in Fig. 7, the orange line of dynamics under FCUC satisfies the frequency nadir, stepwise, and QSS requirements compared to the blue line under the conventional scheduling. #### Vi-C2 Benefits of Sparse Neural Networks Compared with the piece-wise linear regression, the high representational capacity of sparse neural networks will help capture the relationship between frequency dynamics and inertia parameters in a more optimization-efficient way. Tab. VII compares the solving time and costs of the data-enhanced FCUC integrated with different neural networks. The solving time under CSNN is 17.23 times faster than C-NN and 21.57 times faster than NN, verifying the solution acceleration effectiveness of sparsity design. Fig. 6: The unit status under different scheduling scenarios in the modified 118-bus system. #### V-C3 Benefits of Stability Region We should guarantee the performance of the data-driven method design because robustness matters more than optimality. So we construct the clustered convex hulls from the data to build the stability region adding to the optimization models. The below Fig. 8 shows the area of infeasible points and the convex hulls of clustered feasible points. ### _Scalability and Comparison Analysis_ For the generality of the proposed data-enhanced FCUC, we further discuss the scalability of the proposed framework via the modified 1888-bus system experiments based on Ref. [35] and verify the effectiveness and efficiency of the proposed CSNN via the comparison the state-of-the-art machine learning algorithms [36, 37, 38]. #### V-D1 Scalability Analysis We apply the data-enhanced FCUC model in a modified 1888-bus system to verify the scalability of the proposed method that the proposed data-enhanced FCUC increases 3.6% operational cost and 4.23% renewable abandoned rate to achieve the 0% frequency violation rate under contingency and ensure system stable operation, as shown in the following Tab. VIII. #### V-D2 Comparison Analysis To verify the efficiency and effectiveness, we further compare the proposed CSNN model with other state-of-the-art machine learning models including the optimal regression trees (ORT) [36], random forest (RF) [37], and support vector regression (SVR) [38] models. As shown in Tab. IX, though the prediction accuracy of CSNN is lower than ORT but higher than RL and SVR, the proposed CSNN achieve 100% negative rate compared to ORT, RF, and SVR, verifying its prediction conservativeness. To summarize, our proposed data-enhanced FCUC model exhibits scalability for various testing systems and effectiveness for approximation conservativeness compared to various ML models. ## VI Conclusion Faced with the complex frequency dynamics under the high penetration of the converter-based DER, this paper proposes to learn the frequency dynamics by CSNN and incorporate CSNN into the system optimization to facilitate system reliable operation, composed of learning and optimization stages. In the learning stage, the proposed CSNN can achieve the conservativeness against frequency violation and the sparsity for solving acceleration; in the optimization stage, the trained CSNNs are transformed into MILP constraints using the big-M method and embedded into the conventional UC model to establish the final data-enhanced FCUC model. The case study verifies 1) the effectiveness of the proposed model by achieving higher approximation accuracy, around 87% fewer parameters, and 21.57 times faster solving speed compared to that of the conventional NN in the modified 118-bus system; 2) the stable system operation against the frequency requirement violation under contingency. The proposed data-enhanced FCUC model leverages and incorporates the CSNN to manage the system inertia with DERs effectively and boost the system reliability. Fig. 8: Analysis of convex hulls as stability region. Fig. 7: Frequency dynamics comparison in the modified 118-bus system.
2305.12578	Self-Explainable Graph Neural Networks for Link Prediction	Graph Neural Networks (GNNs) have achieved state-of-the-art performance for link prediction. However, GNNs suffer from poor interpretability, which limits their adoptions in critical scenarios that require knowing why certain links are predicted. Despite various methods proposed for the explainability of GNNs, most of them are post-hoc explainers developed for explaining node classification. Directly adopting existing post-hoc explainers for explaining link prediction is sub-optimal because: (i) post-hoc explainers usually adopt another strategy or model to explain a target model, which could misinterpret the target model; and (ii) GNN explainers for node classification identify crucial subgraphs around each node for the explanation; while for link prediction, one needs to explain the prediction for each pair of nodes based on graph structure and node attributes. Therefore, in this paper, we study a novel problem of self-explainable GNNs for link prediction, which can simultaneously give accurate predictions and explanations. Concretely, we propose a new framework and it can find various $K$ important neighbors of one node to learn pair-specific representations for links from this node to other nodes. These $K$ different neighbors represent important characteristics of the node and model various factors for links from it. Thus, $K$ neighbors can provide explanations for the existence of links. Experiments on both synthetic and real-world datasets verify the effectiveness of the proposed framework for link prediction and explanation.	Huaisheng Zhu, Dongsheng Luo, Xianfeng Tang, Junjie Xu, Hui Liu, Suhang Wang	2023-05-21T21:57:32Z	http://arxiv.org/abs/2305.12578v1	# Self-Explainable Graph Neural Networks for Link Prediction ###### Abstract. Graph Neural Networks (GNNs) have achieved state-of-the-art performance for link prediction. However, GNNs suffer from poor interpretability, which limits their adoptions in critical scenarios that require knowing why certain links are predicted. Despite various methods proposed for the explainability of GNNs, most of them are post-hoc explainers developed for explaining node classification. Directly adopting existing post-hoc explainers for explaining link prediction is sub-optimal because: (i) post-hoc explainers usually adopt another strategy or model to explain a target model, which could misinterpret the target model; and (ii) GNN explainers for node classification identify crucial subgraphs around each node for the explanation; while for link prediction, one needs to explain the prediction for each pair of nodes based on graph structure and node attributes. Therefore, in this paper, we study a novel problem of self-explainable GNNs for link prediction, which can simultaneously give accurate predictions and explanations. Concretely, we propose a new framework and it can find various $K$ important neighbors of one node to learn pair-specific representations for links from this node to other nodes. These $K$ different neighbors represent important characteristics of the node and model various factors for links from it. Thus, $K$ neighbors can provide explanations for the existence of links. Experiments on both synthetic and real-world datasets verify the effectiveness of the proposed framework for link prediction and explanation. Link Prediction, Explainability, Graph Neural Networks + Footnote †: 10: 2018 Association for Computing Machinery. with them. For users $v_{i}$ and $v_{j}$ in the figure, they share common interests in music and are likely to be linked. However, it is difficult to predict the link simply based on their node feature similarity as most of their features/interests are different. Also, as $v_{i}$ and $v_{j}$ have diverse neighbors, directly applying a GNN by aggregating all neighbors' information to learn representations of $v_{i}$ and $v_{j}$ followed by the similarity of node representation will result in a low predicted link probability. Thus, how to effectively capture the common factor/preferences for a pair of nodes remains a question. Generally, the preferences of users are reflected in their neighbors. For example, both $v_{i}$ and $v_{j}$ have many neighbors interested in music. By identifying and aggregating these neighbors, we can better learn _pair-specific_ node representations that reflect common interests to predict the link between them. Pair-specific means that the representation of $v_{i}$ for $(v_{i},v_{j})$ should reflect the common interest with $v_{j}$, and for $(v_{i},v_{k})$ should reflect the comment interest with $v_{k}$. Based on this intuitive principle, for each node pair $(v_{i},v_{j})$, one way for self-explainable link prediction is to find dominating $K$ neighbors of $v_{j}$ that share the highest $K$ similarities with $v_{i}$, which reflects the common interest of $(v_{i},v_{j})$ and vice versa. Then, the explanation for whether there is a link for $(v_{i},v_{j})$ can be: ($i$) "We suggest $v_{j}$ to $v_{i}$ because these $K$ neighbors of $v_{i}$ have common features with $v_{j}$. $v_{i}$ will most likely establish a new link with $v_{j}$ for $v_{j}$ is similar to current neighbors of $v_{i}$; and ($ii$) "We suggest no link predicted for $(v_{i},v_{j})$ because even top $K$ similar neighbors of $v_{i}$ w.r.t $v_{j}$ don't share common feature with $v_{j}$, which represent $v_{j}$ is different from neighbors of $v_{i}$. Therefore, it's hard for $v_{i}$ to add $v_{j}$ (different from $v_{i}$'s current neighbors) as a new neighbor of it." Though promising, the work on exploring $K$ relevant neighbors for self-explainable GNNs on link prediction is rather limited. Therefore, in this paper, we investigate a novel problem of self-explainable GNN for link prediction. Specifically, for a pair of nodes $(v_{i},v_{j})$, we want to identify $K$ most important neighbors of $v_{j}$ and $v_{i}$, respectively, for link prediction. In essence, there are two main challenges: ($i$) How to take both graph structure and node attributes into consideration when measuring node similarity for identifying important neighbors for link prediction? and ($ii$) how to give both accurate predictions and correct corresponding explanations given that we lack the supervision of groundtruth explanations? In an attempt to solve the challenges, we propose a novel framework named Interpretable Link Prediction based on Graph Neural Networks (ILP-GNN). For each node pair $(v_{i},v_{j})$, ILP-GNN adopts a novel mechanism that can explicitly evaluate the node similarity and high-order structure similarity to find $K$ interpretable neighbors of $v_{i}$ similar to $v_{j}$. These neighbors can represent common interests or features between $v_{i}$ and $v_{j}$. For high-order structure similarity, graph diffusion is utilized to calculate the closeness of nodes by modeling their local and high-order neighbors' information (Gan et al., 2017). Then, these $K$ neighbors are aggregated to learn pair-specific representation, which represents common features of node pairs and various factors of links for different node pairs. Furthermore, since explanation based on selecting $K$ important neighbors should preserve factors resulting in the existence of links, we propose a novel loss function to encourage explanation of our model and improve the performance of link prediction. The main contributions are: * We study a novel problem of self-explainable GNN for link prediction by finding $K$ neighbors which are relevant to the links. * We develop a novel framework ILP-GNN, which adopts an interpretable neighbor aggregation method, and a novel loss function to identify $K$ relevant neighbors for explainable link prediction; * We conduct extensive experiments to demonstrate the effectiveness of our model on both predictions and explanations. We also construct a synthetic dataset that can quantitatively evaluate the link prediction explanation. ## 2. Related Works Graph Neural Networks. Graph Neural Networks (GNNs) have shown great ability in representation learning on graphs. Generally, GNNs can be split into two categories, i.e., spectral-based (Beng et al., 2015; Goyal et al., 2016; Goyal et al., 2016; Goyal et al., 2016) and spatial-based (Goyal et al., 2016; Goyal et al., 2016; Goyal et al., 2016; Goyal et al., 2016). Spectral-based approaches are defined according to graph signal processing. Bruna et al. (2016) firstly proposes convolution operation to graph data from the spectral domain. Then, a first-order approximation is utilized to simplify the graph convolution via GCN (Goyal et al., 2016). Spatial-based GNN models aggregate information of the neighbor nodes (Goyal et al., 2016; Goyal et al., 2016). For example, the attention mechanism is utilized in Graph Attention Network (GAT) to update the node representation from the neighbors with different weights (Goyal et al., 2016). Moreover, various spatial methods are proposed for further improvements (Goyal et al., 2016; Goyal et al., 2016; Goyal et al., 2016). For instance, DisGNN models latent factors of edges to facilitate node classification (Goyal et al., 2016). Link Prediction. Link prediction has been widely applied in social networks (Beng et al., 2015) and knowledge graph (Goyal et al., 2016). Existing methods for link prediction can be generally split into two categories, i.e., heuristic-based approaches (Beng et al., 2015; Goyal et al., 2016; Goyal et al., 2016; Goyal et al., 2016) and representation learning based approaches (Goyal et al., 2016; Goyal et al., 2016; Goyal et al., 2016). Heuristics-based approaches mainly compute the pairwise similarity scores based on graph structure or node properties (Goyal et al., 2016). For example, the common-neighbor index (CN) scores a pair of nodes by the number of shared neighbors (Goyal et al., 2016); CN and some methods, i.e. Adamic Adar (AA) (Beng et al., 2015), are calculated from up to two-hop neighbors of the target nodes. Other approaches also explore high-order neighbors, including Katz, rooted PageRank (PR) (Beng et al., 2015) and SimRank (SR) (Goyal et al., 2016). But these heuristic-based methods make strong assumptions and can't be generalized to different graph data. Representation learning based approaches firstly learn the representation of nodes and then apply dot product between two node representations to predict the likelihood of the link between two nodes. GNNs are applied to learn node-level representations that capture both the topology structure together with node feature information and achieve state-of-the-art performance on link prediction (Goyal et al., 2016; Goyal et al., 2016; Goyal et al., 2016) in recent years. For example, VGAE (Goyal et al., 2016) adopts GNNs to encode graph structure with features into node representations followed by a simple inner product decoder to get Figure 1. Example of factors for the link between $v_{i}$ and $v_{j}$. the link prediction result. SEAL (Srivastava et al., 2017) extracts subgraphs to predict links between nodes. However, these approaches are also not interpretable and will limit their ability in applications which may require why the model predicts links between nodes. Explainability of Graph Neural Networks. To address the problem of lacking interpretability in GNNs, extensive works have been proposed (Kang et al., 2017; Wang et al., 2018; Wang et al., 2019). For example, GNNExplainer (Wang et al., 2018) learns soft masks for edges and node features to find the crucial subgraphs and features to explain the predictions. PGExplainer (Wang et al., 2019) generates the edge masks with parameterized explainer to find the significant subgraphs. However, previous methods are post-hoc explanations that learn an explainer to explain the outputs of a trained GNN with fixed parameters. Post-hoc explanations might result in unstable interpretation as generated explanations are not directly from the model. To fill this gap, self-explainable GNNs are proposed to make predictions and explanations simultaneously (Chen et al., 2017; Wang et al., 2019). For example, SE-GNN finds interpretable labeled neighbors which have the same labels as target nodes (Chen et al., 2017). But self-explainable GNN models on the link prediction task are rather limited. CONPI (Wang et al., 2019) models similarity between neighbors set of node pairs to determine the probability of the existence of links and provide explanations based on similar neighbors. Their explanations are based on local topology similarity which ignores high-order graph structure information. Other relevant papers are about explainable link prediction for knowledge graphs (Wang et al., 2019; Wang et al., 2019). They are designed for knowledge graphs and their explanations are based on reasoning paths or (head, relation, tail) data format. Therefore, it's hard for them to be generalized to all link prediction tasks. Our work is inherently different from the aforementioned explainable GNN methods: (i) we focus on learning a self-explainable GNN on link prediction which can simultaneously give predictions and explanations while most of the previous methods are designed for node classification; (ii) we study a novel self-explainable method to find factors which determine links between nodes by considering both node and high-order structure information. ## 3. Problem Definition We use $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathbf{X})$ to denote an attributed graph, where $\mathcal{V}=\{v_{1},\ldots,v_{N}\}$ is the set of $N$ nodes, $\mathcal{E}$ is the set of edges and $\mathbf{X}$ is the attribute matrix for nodes in $\mathcal{G}$. The $i$-th row of $\mathbf{X}$, i.e., $\mathbf{x}_{i}\in\mathbb{R}^{1\times d_{0}}$, is the $d_{0}$ dimensional features of node $v_{i}$. $\mathbf{A}\in\mathbb{R}^{N\times N}$ is the adjacency matrix. $A_{ij}=1$ if node $v_{i}$ and node $v_{j}$ are connected; otherwise $A_{ij}=0$. $\mathcal{N}_{i}$ represents the neighborhood set of $v_{i}$. The goal of link prediction is to determine whether there exists an edge $e_{ij}$ between two given nodes $\{v_{i},v_{j}\}$. It can be formulated as a classification problem on a set of node pairs $\mathcal{E}_{U}$ given observed edges $\mathcal{E}_{L}$ and node attributes, where $e_{ij}=1$ represents a link between node $v_{i}$ and $v_{j}$, and $e_{ij}=0$ means no link between $v_{i}$ and $v_{j}$. Due to the great node representation learning ability, GNNs are usually adopted as an encoder: $f:\mathcal{V}\rightarrow\mathbb{R}^{d}$ to map a node $v_{i}$ to a $d$-dimensional vector $\mathbf{h}_{i}$ for link prediction. $f$ should preserve the similarity between nodes based on the observed edge set $\mathcal{E}_{L}$(Chen et al., 2017), and give large probability (large $(f(v_{i})^{T}f(v_{j}))$) for $e_{ij}=1$ but small probability (small $(f(v_{i})^{T}f(v_{j}))$) for $e_{ij}=0$. However, GNN usually lacks interpretability on why they give such predictions (Kang et al., 2017; Wang et al., 2019). There are few attempts of explainers on the node classification task (Chen et al., 2017; Wang et al., 2019; Wang et al., 2019; Wang et al., 2019), while the work on interpretable link prediction based on GNNs is rather limited (Wang et al., 2019). Thus, it's crucial to develop interpretable GNNs for link prediction. As mentioned in the introduction, generally, the preferences of a node are reflected in its neighbors. For node pairs, only some of the neighbors with common features are important for their link prediction. Therefore, to learn pair-specific representation for each node pair $(v_{i},v_{j})$, we propose to find the top $K$ neighbors of node $v_{i}$ which are similar to $v_{j}$. Specifically, for a node pair $(v_{i},v_{j})$, we can learn pair-specific representations $\mathbf{h}_{i}$ and $\mathbf{h}_{j}$ by aggregating selected top $K$ neighbors. Thus, $\mathbf{h}_{i}^{T}\mathbf{h}_{j}$ will be large when neighbors of $v_{i}$ are similar to $v_{j}$ and dissimilar neighbors will result in lower $\mathbf{h}_{i}^{T}\mathbf{h}_{j}$. Due to the undirected property of graphs, it also holds true for node $v_{j}$. _Explanations of link prediction can be: (i)_ "for a node pair $v_{i}$ and $v_{j}$ with a link, take node $v_{i}$ as an example, $v_{i}$'s neighbors $v_{c}\in\mathcal{N}_{i}$ have higher similarity score with regard to $v_{j}$ and $v_{j}$ in the graph. This explanation also holds for node $v_{j}$; (ii) "for a node pair $v_{i}$ and $v_{j}$ without a link, neighbors of $v_{i}$ are dissimilar to $v_{j}$ and also neighbors of $v_{j}$ are dissimilar to $v_{i}$. With the notations above, the problem is formally defined as: _Given an attributed graph $\mathcal{G}=(\mathcal{V},\mathcal{E}_{L},\mathbf{X})$ with observed edge set $\mathcal{E}_{L}$ and unobserved edge set $\mathcal{E}_{U}$, learn an interpretable link predictor $g_{\theta}:\mathcal{V}\times\mathcal{V}\rightarrow\{True,False\}$ which can accurately predict links in $\mathcal{E}_{U}$ and simultaneously generate explanation by identifying two sets of important $K$ neighbors for link between each node pair $(v_{i},v_{j})$._ ## 4. Methodology In this section, we introduce the details of the proposed framework ILP-GNN. The basic idea of ILP-GNN is: for each node pair $(v_{i},v_{j})$, it identifies $K$ neighbors of node $v_{i}$ and $v_{j}$, respectively, aiming to capture the common interests of the two nodes. Then, it aggregates these $K$ neighbors' information to obtain their pair-specific representation vectors and calculate the similarity based on their representations. Meanwhile, the $K$ most relevant neighbors of $v_{i}$ and $v_{j}$ provide the explanation on why there is or isn't a link between this node pair. There are mainly two challenges: (i) how to obtain the $K$ most relevant neighbors of $v_{i}$ and $v_{j}$ for the prediction of links between them; (ii) how to simultaneously give accurate predictions and correct corresponding explanations. To address these challenges, for each node pair $(v_{i},v_{j})$, ILP-GNN explicitly models both node and high-order structure similarity between $v_{i}$ and neighbors of $v_{j}$ to identify the $K$ important nodes of $v_{j}$ for explainable link prediction. Similarly, it explicitly models both node and high-order structure similarity between $v_{j}$ and neighbors of $v_{i}$ to identify $K$ important nodes of $v_{i}$. An illustration of the proposed framework is shown in Figure 2. It is mainly composed of an Interpretable Neighbors Aggregation module and an Explanation Enhancement module. With the Interpretable Neighbors Aggregation, for each node pair $(v_{i},v_{j})$, the $K$ most important neighbors, which represent factors for the existence of links, are found based on both node and high-order structure information. Then, the prediction of links between nodes can be given based on the identified $K$ neighbors. Finally, the Explanation Enhancement module is designed to further benefit the accurate explanation generation, and also encourage the model to improve link prediction via aggregating $K$ important neighbors. ### Interpretable Neighbors Aggregation For GNN-based link prediction on a pair of nodes $(v_{i},v_{j})$, we firstly aggregate their neighbors' information to obtain their representation vectors $(\mathbf{h}_{i},\mathbf{h}_{j})$. Then, the similarity will be calculated between $(\mathbf{h}_{i},\mathbf{h}_{j})$ via $\mathbf{h}_{i}^{T}\mathbf{h}_{j}$ to indicate whether there is a link between them and pair-specific representation is required to predict links. However, links are generated due to multiple factors. For different links from $v_{i}$, it's necessary for us to learn pair-specific representations for $v_{i}$ to predict links between them. In graph data, the neighborhood of $v_{i}$ can represent important characteristics but not all neighbors have relevant factors w.r.t different connected nodes of $v_{i}$. Based on this observation, ILP-GNN selects $K$ neighbors of $v_{i}$ which are similar to $v_{j}$ to learn pair-specific representation of $v_{i}$. Similarly, through the same way, we learn a pair-specific representation of $v_{j}$ with $K$ interpretable neighbors. ILP-GNN relies on interpretable $K$ neighbors that reveal the common interest of $v_{i}$ and $v_{j}$ for link prediction and explanation. We need to design a similarity measurement to measure the similarity between neighbors of node $v_{i}$ and another connected or unconnected node $v_{j}$. Unlike i.i data, which only needs to measure the similarity from the feature perspective, for graph-structure data, both node attributes and graph structures of nodes contain crucial information for link prediction. In the following part, for a node pair $(v_{i},v_{j})$, we use $v_{i}$ as an example to demonstrate the process of finding $K$ interpretable neighbors. And we will do the same operations for node $v_{j}$. #### 4.1.1. High-order Structure Similarity In the graph, for a pair of nodes $(v_{i},v_{j})$, the distance between neighbors of $v_{i}$ and $v_{j}$ can be used to measure their similarity. For instance, one-hop neighbors of the node $v_{i}$ may have higher similarity scores with $v_{i}$, while high-order neighbors will have lower similarity scores. And the one-hop relation is represented as the adjacency matrix $\mathbf{A}$ and high-order relations can be represented as $\mathbf{A}^{2},\mathbf{A}^{3}$, etc. Since original edge relations are often sparse and missed in real-world graphs, only using one-hop neighbors to measure the distance between nodes may result in unreliable similarity, i.e., the similarity between $v_{i}$ and $v_{j}$ equals $A_{ij}$. Therefore, it's necessary to model both one-hop and high-order relations to measure the similarity between nodes based on graph structure. To measure this high-order similarity, we propose to use a Graph Diffusion matrix which calculates the closeness of nodes in the graph structure by repeatedly passing the weighting coefficients to the neighboring nodes and represents the high-order similarity between nodes based on graph structure(Kirsh and Kirsh, 2015): \[\mathbf{S}=\sum_{k=0}^{\infty}\theta_{k}\mathbf{T}^{k}, \tag{1}\] where $\mathbf{T}$ represents the random walk transition matrix as $\mathbf{T}=\mathbf{A}\mathbf{D}^{-1}$, and the degree matrix $\mathbf{D}$ is the diagonal matrix of node degrees, i.e. $D_{ii}=\sum_{j=1}^{N}A_{ij}$. In this paper, we utilize a popular example of graph diffusion, Personalized PageRank (PPR) (Kip Finally, with node and high-order structure similarity, the importance score of $v_{i}$ with $v_{c}\in\mathbf{N}_{i}$ on $(v_{i},v_{j})$ is: \[s(v_{i},v_{c},v_{j})=\alpha\cdot s_{\text{ST}}(v_{i},v_{c},v_{j})+(1-\alpha) \cdot s_{\text{NO}}(v_{i},v_{c},v_{j}), \tag{5}\] where $\alpha$ is the hyperparameter to control the contributions of node and high-order structure similarity. After getting weights for neighbors of node $v_{i}$, we will find $K$ neighbors based on these weights. We will do the same thing for neighbors of $v_{j}$. In the link prediction task, common neighbors between $v_{i}$ and $v_{j}$ may be highly relevant to links between pairs of nodes (K ``` 0:$\mathcal{G}=(\mathcal{V},\mathcal{E}_{L},\mathbf{X})$, $K$, $\lambda,\alpha,\delta$ 0: GNN model $g_{\theta}$ with explanation for link prediction. 1: Randomly initialize the model parameters $\Theta$. 2: Calculate high-order distance via Eq.(1). 3:repeat 4: For each node pair $(v_{i},v_{j})$, assign weights $\mathrm{s}_{\mathrm{ST}}(v_{i},v_{c},v_{j})$ to neighbors of $v_{i}$ by high-order structure similarity in Eq.(2). 5: Learn node feature representation by Eq.(3) and assign weights to neighbors of $v_{i}$ by node similarity in Eq.(4). 6: Do the same operation on $v_{j}$ and aggregate top $K$ neighbors of $v_{i}$ and $v_{j}$ with two kinds of weights in Eq.(7). 7: Calculate the probability $p_{ij}$ of a link between two nodes. 8: Randomly choose neighbors except from top $K$ neighbors to obtain $p_{ij}^{\mathrm{rand}}$ and calculate $\mathcal{L}_{\mathrm{dis}}^{\mathcal{P}}$ in Eq.(11) 9: Calculate $\mathcal{L}_{\mathrm{dis}}^{\mathcal{P}}$ using negative samples 10: Update $\Theta$ by minimizing the overall loss function in Eq.(14) 11:until convergence 12:return$g_{\theta}$ ``` Algorithm 1 Training Algorithm of ILP-GNN. ### Training Algorithm and Time Complexity #### 4.4.1. Training Algorithm The training algorithm of ILP-GNN is given in Algorithm 1. We utilize $e_{ij}=1$ as positive samples and $e_{ij}=0$ as negative samples. ILP-GNN assigns weights to neighbors of nodes based on both node and high-order structure similarity. The top $K$ neighbors with high weights are aggregated to learn representation for self-explainable link predictions. Specifically, in line one, we initialize the parameters of the model with Xavier initialization (Chen et al., 2017). Then, we calculate high-order similarity by Eq.(1). In lines 4 to line 6, for each node pair $(v_{i},v_{j})$ with or without links, we assign weights to neighbors of $v_{i}$ and $v_{j}$ by learned node and high-order structure similarity. Then, ILP-GNN aggregates top $K$ neighbors with high weights to obtain pair-specific representation vectors $\mathbf{h}_{i},\mathbf{h}_{j}$. Also, these top $K$ neighbors, which represent common interests between node pairs, can be treated as the explanation for the existence of links. In line 7, the probability of a link is calculated based on node representation vectors in Eq.(8). To guarantee the quality of explanation, in lines 8 and 9, for positive samples, $\mathcal{L}_{\mathrm{dis}}^{\mathcal{P}}$ is proposed to make the predicted probability from ILP-GNN larger than the probability predicted from representation vectors with randomly selected neighbors. For negative samples, $\mathcal{L}_{\mathrm{dis}}^{n}$ is applied to make weights of neighbors small which represents neighbors of one node are dissimilar to another node. In this case, the model will give low predicted probabilities for the existence of links. Finally, the model is optimized on the total loss function Eq.(14). #### 4.4.2. Time Complexity Analysis The main time complexity of our model comes from calculating the node similarity and high-order structure similarity together with our proposed loss function. For high-order structure similarity, the time complexity of Personalized PageRank (PPR) is denoted as $\mathcal{O}(k\|\mathcal{E}_{L}\|)$, where $k$ is the number of iterations in Eq.(1). Also, high-order structure similarity can be pre-computed which will not influence the training process of the model. For node similarity, the time complexity for links $e_{ij}\in\mathcal{E}$ is $\mathcal{O}(\sum_{e_{ij}\in\mathcal{E}}\|\mathcal{N}_{i}\|\|\mathcal{N}_{j}\|d)$, where $d$ is the embedding dimension and $\|\mathcal{N}_{i}\|$ is the number of neighbors for $v_{i}$. The time complexity of the proposed loss function is $\mathcal{O}(K\|\mathcal{E}_{L}\|d)$. Therefore, the overall time complexity for the training phase in each iteration is $\mathcal{O}(\sum_{e_{ij}\in\mathcal{E}_{L}}\|\mathcal{N}_{i}\|\|\mathcal{N}_{j}\|d +K\|\mathcal{E}_{L}\|d)$. The time complexity of the testing phase is $\mathcal{O}(\sum_{e_{ij}\in\mathcal{E}_{U}}\|\mathcal{N}_{i}\|\|\mathcal{N}_{j}\|d)$. A detailed training time comparison is given in Appendix B. ## 5. Experiment In this section, we conduct experiments on real-world and synthetic datasets to verify the effectiveness of ILP-GNN. In particular, we aim to answer the following research questions: (RQ1) Can our proposed method provide accurate predictions for link prediction? (RQ2) Can ILP-GNN learn reasonable explanation for the existence of links? (RQ3) How do two similarity measurement methods of our ILP-GNN contribute to the link prediction performance? ### Datasets We conduct experiments on four publicly available real-world datasets and their details are shown below: * Cora and Citeseer(Cota et al., 2017): These two datasets are citation networks where nodes are papers and edges are their citation relations. Each node has a sparse bag-of-words feature vector. * Photo(Zhou et al., 2018): This dataset is a subgraph of the Amazon co-purchase graph (Zhou et al., 2018), where nodes are products and two frequently purchased products are connected via an edge. Each node has a bag-of-word feature vector of reviews. * Ogbm-arxiv(Chen et al., 2017): It is a citation network between all Computer Science arXiv papers indexed by MAG (Mori et al., 2018). Nodes in this dataset represent papers and edges indicate one paper citing another one. Each paper has a 128-dimensional feature vector obtained by averaging the embeddings of words in its title and abstract. * Ogbl-collab(Chen et al., 2017): This dataset is a subset of the collaboration network between authors indexed by MAG (Mori et al., 2018), where nodes are authors and edges indicate the collaboration between authors. All nodes have 128-dimensional feature vectors by averaging the embeddings of words in papers published by authors. Synthetic-Data: Since the publicly available datasets don't have groundtruth explanations, to quantitatively evaluate the explanation of the proposed method, we also construct synthetic datasets that have groundtruth explanation for links. We firstly generate a set of $N$ nodes, $\{v_{1},v_{2},...,v_{N}\}$, with their feature information. Then, for the generation of links between the node pair $(v_{i},v_{j})$, the links are determined by $B$ neighbors of $v_{i}$ and these $B$ neighbors share both high structure and node similarity with $v_{j}$. Similarly, due to the undirected property of the graph, we also consider $B$ neighbors of $v_{j}$ similar to $v_{i}$ for the generation of links. And these $B$ neighbors of $v_{i}$ and $B$ neighbors of $v_{j}$ can be treated as groundtruth explanation for this link. We fix $N$ as 1000 and generate three datasets with different number of edges for experiments. The detailed construction process of the synthetic datasets is given in Appendix A. The statistics of these datasets are summarized in Table 4 in Appendix A. Following (Cota et al., 2017), for Cora, Citeseer and Photo, we randomly split the edges of each dataset into 85%/5%/10% as train/val/test. The random split is conducted 5 times and average performance will be reported. For Ogbm-arxiv, we randomly split the edges into 60%/10%/30% as train/val/test. For Ogbl-collab, follow (Chen et al., 2017), we split the edges according to time, where the collaboration edges until 2017 are used as training edges, those in 2018 are used as validation edges, and those in 2019 are test edges. For Synthetic datasets, we randomly select 40% edges from edges set with groundtruth explanation as testing set and 10% as validation set. The remaining edges are used as training samples. We maintained consistency across datasets by following VAGE's split proportion and randomly selecting node pairs not in the training set as negative samples following (Kang et al., 2019). Note that this approach differs from WP and SEAL, which selected pairs not present in any of the training, validation, or test sets. The number of negative samples is equal to the number of positive samples. Positive and negative samples are then combined as our training, validation and testing sets (Kang et al., 2019). ### Experimental Setup Baselines. We compare the proposed framework with representative and state-of-the-art methods for link prediction, which include: * CN(Wang et al., 2019): Common-neighbor index counts the number of common neighbors to predict the link for a pair of nodes. * AA(Chen et al., 2019): Adamic-Adar index is a second-order traditional heuristic method. It assumes that a shared neighbor with the large degree is less significant to the measure of a link. * VGAE(Kang et al., 2019): VGAE is a generative model for graph representation. We use a GCN as the encoder where the second layer has two channels for mean and deviations to sample the latent embeddings and a link reconstruction module as the decoder. * GCN(Kang et al., 2019): GCN is one of the most popular spectral GNN models based on graph Laplacian, which has shown great performance for node classification. To adopt it for link prediction, we treat it as the encoders in the Graph Autoencoder manner. * GAT(Wang et al., 2019): Instead of treating each neighbor node equally, GAT utilizes an attention mechanism to assign different weights to nodes in the neighborhood during the aggregation step. * SEAL(Wang et al., 2019): SEAL is a link prediction method that extracts local subgraphs of node pairs and learns link prediction heuristics from them automatically. * CONPI(Wang et al., 2019): CONPI is an interpretable model to compare the similarity between neighbors sets of two nodes for link prediction. It has two variants and we report the best results of them. * WP(Wang et al., 2019): WalkPooling (WP) jointly encodes node representations and graph topology into learned topological features. Then, these features are used to enhance representation of extracted subgraphs which are relevant to links of node pairs. As the work on explainable GNN for link prediction is rather limited, we also adapt a popular post-hoc explanation model GNNExplainer for post-hoc link prediction. * GNNExplainer(Wang et al., 2019): GNNExplainer takes a trained GNN and the predictions as input to obtain post-hoc explanations. A soft edge mask is learned for each instance to identify the crucial subgraph. We adopt GNNExplainer to find crucial neighbors of node pairs with links for link prediction. Configurations. All experiments are conducted on a 64-bit machine with Nvidia GPU (NVIDIA RTX A6000, 1410MHz, 48 GB memory). For a fair comparison, we utilize a two-layer graph neural network for all methods, and the hidden dimension is set as 128. The learning rate is initialized to 0.001. Besides, all models are trained until converging, with the maximum training epoch as 1000. The implementations of all baselines are based on Pytorch Geometric or their original code. The hyperparameters of all methods are tuned on the validation set. In particular, for the proposed framework, we select $K$ from 1 to 6 and vary $\lambda$ as $\{0.1,0.3,0.5,0.7\}$. The $\alpha$ which balances the node similarity and high-order structure similarity is fixed as 0.3 for all datasets. The margin $\delta$ in Eq.(11) is set as 0.5. ### Performance on Link Prediction In this subsection, we compare the performance of the proposed ILP-GNN with baselines for link prediction on real-world graphs introduced in Sec. 5.1, which aims to answer RQ1. Each experiment is conducted 5 times for all datasets and the average link prediction AUC scores with standard deviations are reported in Table 1. From the table, we make the following observations: * Our method outperforms VGAE, GCN and GAT on various real-world datasets. This is because for each node pair $(u_{i},v_{j})$, our model can select $K$ neighbors of $v_{i}$ and $v_{j}$ separately, which contain common characteristics of $(u_{i},v_{j})$ and are highly relevant to the factors for the existence of the link. By aggregation these $K$ significant neighbors, our model can learn pair-specific representations of $v_{i}$ and $v_{j}$ for the link and predict it accurately. * ILP-GNN can outperform SEAL and its variant WP which extracts subgraphs to learn link prediction heuristics. The reason is that our model can implicitly find local neighbors which can also explore link prediction heuristics like common neighbors and high-order structure information. * Though CONPI also compares neighbors of node pairs and finds relevant neighbors for link prediction, our model can outperform CONPI, which shows the effectiveness of our method in selecting neighbors for link prediction and our loss function can guide the model to obtain link prediction relevant neighbors. ### Explanation Quality In this subsection, we conduct quantitatively experimental comparisons and visualization to answer RQ2. \begin{table} \begin{tabular}{l c c c c c c c c c} \hline \hline Method & AA & CN & VGAE & GCN & GAT & SEAL & CONPI-Pair & WP & ILP-GNN \\ \hline Cora & 75.80 $\pm$ 0.15 & 75.57 $\pm$ 0.13 & 91.18 $\pm$ 0.41 & 91.57 $\pm$ 0.69 & 91.87 $\pm$ 0.93 & 92.21 $\pm$ 1.23 & 89.69 $\pm$ 0.32 & 92.42 $\pm$ 1.1 & 93.21 $\pm$ 1.14 \\ Citeseer & 69.80 $\pm$ 0.22 & 69.70 $\pm$ 0.23 & 91.42 $\pm$ 0.96 & 92.51 $\pm$ 1.00 & 93.57 $\pm$ 0.64 & 90.52 $\pm$ 1.29 & 87.47 $\pm$ 0.10 & 91.37 $\pm$ 0.98 & 95.23 $\pm$ 1.33 \\ Photo & 96.59 $\pm$ 0.22 & 96.21 $\pm$ 0.04 & 97.03 $\pm$ 0.15 & 97.08 $\pm$ 0.13 & 96.47 $\pm$ 0.19 & 98.04 $\pm$ 0.70 & 96.45 $\pm$ 0.42 & 98.12 $\pm$ 0.14 & 98.23 $\pm$ 0.04 \\ Ogbn-arxiv & 82.41 $\pm$ 0.02 & 82.43 $\pm$ 0.01 & 95.05 $\pm$ 0.07 & 95.27 $\pm$ 0.06 & 94.79 $\pm$ 0.03 & 95.30 $\pm$ 0.04 & 94.22 $\pm$ 0.08 & 95.33 $\pm$ 0.02 & 95.42 $\pm$ 0.03 \\ Ogbn-collab & 58.09 $\pm$ 0.09 & 57.88 $\pm$ 0.02 & 96.27 $\pm$ 0.07 & 96.73 $\pm$ 0.02 & 96.64 $\pm$ 0.05 & 93.57 $\pm$ 0.04 & 92.31 $\pm$ 0.03 & 96.81 $\pm$ 0.02 & 97.17 $\pm$ 0.01 \\ \hline \hline \end{tabular} \end{table} Table 1. Link Prediction performance (AUC(%) $\pm$ Std.) on all graphs. #### 5.4.1. Results on Fidelity Scores We first demonstrate the effectiveness of explanation in terms of fidelity scores. The fidelity score measures the link prediction performance drop for each pair of nodes when important neighbors of the pair of nodes are removed. Intuitively, if a model can capture important neighbors of a pair of nodes for link prediction, the removal of such neighbors would result in a significant performance drop. Specifically, for each node pair $(v_{i},v_{j})$ in the test set, take node $v_{i}$ as an example, we denote the delete top $M$ neighbors based on weight scores given by the model as $\mathcal{N}_{i}^{o}$. We delete will $\mathcal{N}_{i}^{o}$ and obtain a new neighbor set $\mathcal{N}_{i}\backslash\mathcal{N}_{i}^{o}$. We then aggregate neighbors $v_{c}$ of $v_{i}$ from this new set $v_{c}\in\mathcal{N}_{i}\backslash\mathcal{N}_{i}^{o}$ to obtain the representation vector $\mathbf{h}_{i}^{w}$ via different aggregation methods from different models, i.e., GAT, CONPI, ILP-GNN. We do the same operations for node $v_{j}$ to get the representation $\mathbf{h}_{j}^{w}$. Note that if $\|\mathcal{N}_{i}\|\leq M$ and the new neighbors set is empty, we only use their features. Then, we can obtain the new link prediction score $\hat{\theta}_{ij}^{w}$ using $\mathbf{h}_{i}^{w}$ and $\mathbf{h}_{j}^{w}$ by Eq.(8). The fidelity score $\Delta AUC\%$ is calculated as $\Delta AUC\%=\sum_{v_{j}\in\mathcal{E}_{v_{j}}}(AUC\,\hat{\rho}_{ij}-AUC \hat{\rho}_{ij}^{w})\%$, where $AUC\,\hat{\rho}_{ij}$ and $AUC\,\hat{\rho}_{ij}^{w}$ are AUC value with regard to different predicted results, and $\hat{\rho}_{ij}$ is the original predicted result. We compare our model with GAT and CONPI which can also assign weights to neighbors for link prediction. We vary the number of deleted neighbor $M$ as $\{1,2,3,4\}$. All experiments are conducted five times with random splits and the results are reported in Figure 3. From the figure, we make the following observations: (i) ILP-GNN consistently outperforms two baselines with different deleted neighbors. Compared with GAT, for different links $(v_{i},v_{j})$ and $(v_{i},v_{k})$, ILP-GNN learns pair specific representation for $v_{i}$ by selecting different neighbors of $v_{i}$ relevant to the existence of various links. However, GAT assigns higher weights to the same neighbors and learns one representation of $v_{i}$ for different links. In real-word graphs, different links may be from different factors, which are relevant to different neighbors of nodes. Therefore, our proposed ILP-GNN can explore the relevant neighbors for links with different factors, which will greatly improve link prediction and lead to higher fidelity scores. (ii) Also, our proposed loss function can help the model select neighbors to give high probabilities to links and low probabilities to no-links. It can help the model find neighbors relevant to links. Thus, our model can achieve the best fidelity score by deleting neighbors. #### 5.4.2. Results on Synthetic Datasets Secondly, we evaluate the explanation quality on synthetic datasets with groundtruth explanation. Specifically, in the synthetic datasets, for each node pair $(v_{i},v_{j})$, $v_{i}$ is similar to $K$ neighbors of node $v_{j}$ and $v_{j}$ are similar to $K$ neighbors of $v_{i}$, which lead to the link between them. We treat these $K$ neighbors of $v_{i}$ and $K$ neighbors of $v_{j}$ as explanation neighbors for the link between them. Therefore, for the task to find explanation neighbors in the synthetic datasets, the model should find the correct $K$ neighbors of $v_{i}$ and $K$ neighbors of $v_{j}$ for explainable link prediction. Specifically, for each node pair $(v_{i},v_{j})$, we rank neighbors of $v_{i}$ based on weight scores assigned from the model, i.e., ILP-GNN, GAT and CONPI. We also do the same operations on $v_{j}$. We then calculate the precision@$2$ and precision@$1$ for the ranked neighbors list of $v_{i}$ and $v_{j}$ w.r.t groundtruth explanation neighbors set of $v_{i}$ and $v_{j}$. Also, we includes the baseline (named Random in Table 2) which randomly select $K$ neighbors of $v_{i}$ and $v_{j}$ and evaluate the explanation performance based on randomly selected $K$ neighbors list. To further demonstrate the effectiveness of our model on link prediction explanation, we adopt one classical post-hoc explanation method, GNNExplainer, which is originally designed for node classification. For the link between $v_{i}$ and $v_{j}$, we adopt GNNExplainer to find the top $K$ crucial neighbors of them for the link between them. We treat these $K$ neighbors of $v_{i}$ and $v_{j}$ as explanation neighbors of them. The higher the precision@$1$ or precision@$2$ is, the closer the explainable neighbors found by the model with the groundtruth. We evaluate explanation performance by precision@$1$ or precision@$2$ and link prediction performance by AUC. All experiments are conducted five times and the average results and standard deviations on the three synthetic datasets are reported in Table 2 and Table 3 for explanation and link prediction performance. From the results, we make the following observations: (i) GNNExplainer only has a little improvement compared with the Random method. It demonstrates that current explanation methods designed for node classification can't be easily adopted to link prediction. (ii) ILP-GNN consistently outperforms all baselines on both explanation and link prediction metrics, which indicates that it can retrieve reliable $K$ neighbor nodes for prediction and explanation simultaneously. probabilities to no-links. Therefore, it further demonstrates the effectiveness of our model to select relevant neighbors to improve the performance of link prediction. ### Ablation Study To answer RQ3, we conduct an ablation study to explore the effects of both node and structure similarity to find $K$ interpretable neighbors for the link prediction task. ILP-GNN$\backslash$E denotes the variant that we don't use structure similarity measurement to find neighbors, i.e., setting $\alpha=0$ in Eq.(5). ILP-GNN$\backslash$F denotes the variant without node similarity measurement in Eq.(5), i.e., setting $\alpha=1$. ILP-GNN$\backslash$F, E represents the GCN model that aggregates all neighbors without weights. The experimental results on Cora and Photo are reported in Figure 7. We can observe that: (a) two variants of ILP-GNN and ILP-GNN by predicting links with selecting neighbors can greatly outperform ILP-GNN$\backslash$F, E. It means that the proposed interpretable neighbor aggregation module can greatly improve the performance of link prediction by finding $K$ neighbors of one node, which are similar to other linked nodes. (b) ILP-GNN$\backslash$F and ILP-GNN$\backslash$E, which implies that combining both node and structure similarity is helpful for finding relevant neighbors to improve the link prediction. ### Hyperparameter Sensitivity Analysis In this section, we explore the sensitivity of the most important hyperparameters $K$ and $\lambda$, which control the number of neighbors to select and the contribution of the Explanation Enhancement module, respectively. Specifically, we vary $K$ as $\{1,2,4,6\}$ and $\lambda$ as $\{0.1,0.3,0.5,0.7\}$. The other settings are the same as the experiment for Table 1. We report the AUC value for link prediction on Cora and Citeseer. The experiment is conducted five times with random splits and the average results are shown in Figure 5. From the figure, we observe that: (i) with the increase of $K$, the performance firstly increases and then decreases. When $K$ is small, a small number of relevant neighbors are selected. A small set of neighbors may not reflect the characteristics of nodes to build links with other nodes, which results in poor performance. When $K$ is large, the large number of selected neighbors may contain bias and will cover important characteristics of nodes building links. Therefore, when $K$ is in the range of 2 to 4, the performance is generally good. (ii) with the increasing of $\lambda$, the performance of ILP-GNN tends to firstly increase and then decrease. When $\lambda$ is small, little supervision is received to select neighbors which may result in a high predicted probability for positive samples and low probabilities for negative samples. Also, large $\lambda$ will be dominated by enhancing selected neighbors for link prediction, which can also lead to poor performance. For $\lambda$, a value between 0.3 to 0.5 generally gives a good performance. ### Case Study We conduct a case study to show the importance of selected neighbors for the decision process of link prediction. Specifically, we apply t-SNE to the learned representation of nodes, i.e. $\mathbf{h}_{i}$ for node $u_{i}$ in Eq.(7), with aggregating selected neighbors from our model. Then, we learn another representation $\mathbf{h}_{i}^{*}$ by aggregating neighbors without these selected neighbors. The relative positions of nodes for these two representations are visualized. Also, we visualize their local 1-hop neighbors to obtain the positions of nodes based on node similarity as shown in Figure 4. We show cases of links between two nodes with and without common neighbors and node pairs without links. As shown in Figure 4 (a), the learned representation with selected neighbors from our model will have higher similarity in the left figure ($v_{1}$ is near to $v_{2}$). However, the learned embedding without these neighbors will result in a lower similarity in the right figure (two nodes are distant). For no-links, our model Figure 4. Case Study for node pairs with links or without links on Cora. Figure 5. Hyperparameter Sensitivity Analysis Figure 6. Visualization of weighted scores. Figure 7. Ablation Study on Cora and Photo datasets. will make the learned representation of both $v_{1}$ and $v_{2}$ with their neighbors distant. And the model will give lower probabilities for no-links. Therefore, our model can provide both predictions and explanations accurately by finding significant neighbors. ## 6. Conclusion In this paper, we study a novel problem of self-explainable GNNs for link prediction by exploring $K$ important neighbors for links. We propose a novel framework, which designs an interpretable aggregation module for finding $K$ neighbors relevant to factors of links and simultaneously uses these neighbors to give predictions and explanations. Also, a novel loss function is proposed to enhance the generation of explanations and also the performance of the link prediction task. Extensive experiments on real-world and synthetic datasets verify the effectiveness of the proposed ILP-GNN for explainable link prediction. An ablation study and parameter sensitivity analysis are also conducted to understand the contribution of the proposed modules and sensitivity to the hyperparameters. There are several interesting directions that need further investigation. One direction is to extend ILP-GNN for dynamic network link prediction. Another direction is to design more efficient and learnable approaches to explore high-order similarity in graphs.
2306.05554	Simulation and Prediction of Countercurrent Spontaneous Imbibition at Early and Late Times Using Physics-Informed Neural Networks	The application of Physics-Informed Neural Networks (PINNs) is investigated for the first time in solving the one-dimensional Countercurrent spontaneous imbibition (COUCSI) problem at both early and late time (i.e., before and after the imbibition front meets the no-flow boundary). We introduce utilization of Change-of-Variables as a technique for improving performance of PINNs. We formulated the COUCSI problem in three equivalent forms by changing the independent variables. The first describes saturation as function of normalized position X and time T; the second as function of X and Y=T^0.5; and the third as a sole function of Z=X/T^0.5 (valid only at early time). The PINN model was generated using a feed-forward neural network and trained based on minimizing a weighted loss function, including the physics-informed loss term and terms corresponding to the initial and boundary conditions. All three formulations could closely approximate the correct solutions, with water saturation mean absolute errors around 0.019 and 0.009 for XT and XY formulations and 0.012 for the Z formulation at early time. The Z formulation perfectly captured the self-similarity of the system at early time. This was less captured by XT and XY formulations. The total variation of saturation was preserved in the Z formulation, and it was better preserved with XY- than XT formulation. Redefining the problem based on the physics-inspired variables reduced the non-linearity of the problem and allowed higher solution accuracies, a higher degree of loss-landscape convexity, a lower number of required collocation points, smaller network sizes, and more computationally efficient solutions.	Jassem Abbasi, Pål Østebø Andersen	2023-05-06T10:16:04Z	http://arxiv.org/abs/2306.05554v3	Simulation and Prediction of Countercurrent Spontaneous Imbibition at Early and Late Times Using Physics-Informed Neural Networks ## Abstract In this work, we investigate for the first time the application of Physics-Informed Neural Networks (PINNs) in solving the one dimensional (1D) Countercurrent spontaneous imbibition (COUCSI) problem in both early (ET) and late (LT) times. Also novel, we examine the Change-of-Variables technique for improving the performance of PINNs. We formulated the COUCSI problem in three equivalent forms by changing the independent variables: XT-, XY-, and Z-formulations. The first describes saturation as function of normalized position X and time T; the second as function of X and Y=T0.5; and the third as a sole function of Z=X/T0.5 (valid only at ET). The PINN model was generated using a feed-forward neural network and trained based on minimizing a weighted loss function, including the physics-informed loss term and terms corresponding to the initial and boundary conditions. All three formulations could closely approximate the correct solutions, with water saturation mean absolute errors (MAE) around 0.019 and 0.009 for XT and XY formulations and 0.012 for the Z formulation at ET. The Z formulation perfectly captured the self-similarity of the system at ET. This was less captured by XT and XY formulations. The total variation (TV) of saturation was preserved in the Z formulation, and it was better preserved with XY- than XT formulation. It was demonstrated that redefining the problem based on physics-inspired variables reduced the non-linearity of the problem and allowed higher solution accuracies, a higher degree of loss-landscape convexity, a lower number of required collocation points, smaller network sizes, and more computationally efficient solutions. Keywords: Spontaneous Imbibition; Physics Informed Neural Networks; Partial Differential Equations, Change of Variables ## Introduction Spontaneous imbibition (SI) is a flow phenomenon where a wetting phase diffuses into porous media and displaces a non-wetting phase due to capillary forces. This process commonly occurs in geological problems, such as soil water infiltration, hydrocarbon production, and carbon sequestration. Analyzing the spontaneous imbibition process provides insights into the wetting condition of the system (if the surrounding fluid wets the porous system) and the governing capillary pressure curve regarding how strong capillary forces are present and how much fluid displacement they can cause [13]. The boundary conditions influence the flow behavior during SI. In systems where both fluids access the system through the same surface, they flow counter-currently. This is called countercurrent SI (COUCSI). When some system surfaces are exposed to the wetting phase and other surfaces to non-wetting phase, the fluids predominantly flow in the same direction, which is called co-current SI (COCSI) [1]. During COUCSI, we distinguish between the early time (ET) period (when the invading saturation profile has not reached the closed boundary) and the late time (LT) period (when the no-flow boundary has been reached). In ET, the advancement of the front is proportional to the square root of time (Y. Li et al., 2006), and there is a self-similarity behavior in the saturation front of fluid versus a similarity variable [1, 2]. The recovery stays proportional to the square root of time during ET. However, it has been demonstrated comprehensively that recovery can follow this trend long into the LT regime (Andersen, 2021, 2023b; March et al., 2016). The explanation is that the imbibition rate at the inlet, which controls recovery, acts as if it is still ET until the saturation profile has changed sufficiently from the self-similar ET saturation profile at the inlet (Andersen, 2023a). The COUCSI problem can be solved numerically with finite differences (FD) approaches using, e.g., reservoir simulators (Abbasi & Andersen, 2022; Khan et al., 2018). Analytical solutions can be derived for very specific saturation functions or piston-like displacement (Cil & Reis, 1996; Fokas & Yortsos, 1982; Kashchiev & Firoozabadi, 2003). Semi-analytical solutions were derived by McWhorter & Sunada (1990) that hold for any input parameters and saturation functions, except being limited to ET. Bjornara & Mathias (2013) noted numerical challenges in evaluating their solution and proposed more stable schemes. Numerical approaches are usually required for more complex problems involving more general geometries or changing boundary conditions. Schmid & Geiger (2012) used McWhorter and Sunada's solution to propose a universal time scale for COUCSI. Andersen (2023b) showed that all one dimensional (1D) COUCSI problems (regardless of wettability, saturation functions, and other parameters) could be normalized to the same form, which only depends on a normalized diffusion coefficient with a mean equal to one. By correlating the coefficient shape with recovery behavior, ET and LT recovery were predicted accurately without the need for numerical methods. Physics-informed neural networks (PINNs) provide an alternative approach to solving ordinary- and partial- differential equations (ODEs and PDEs) that govern various engineering problems (Karniadakis et al., 2021). PINNs have recently been applied to mathematical analysis of flow in porous media: Fraces & Tchelepi (2021), Almajid & Abu-Al-Saud (2022), and Rodriguez-Torrado et al. (2022) utilized PINNs for solving the Buckley-Leverett (BL) equation describing the displacement of oil by water flooding. A key challenge was capturing the shock front behavior. Deng & Pan (2021) applied PINN to study 1D spontaneous imbibition using Lagrangian formulation. They investigated ET flow for self-similar counter- and co-current flow and also considered more general boundary conditions during co-current SI where the ET flow would not be self-similar due to countercurrent production at the inlet. In the self-similar system, they could determine the saturation-dependent characteristic flow function, fluxes as a function of time, and solutions of saturation along the core. The sensitivity of problem formulation of COUCSI to capture self-similarity and other properties has not been explored. To our knowledge, the current work is only the second study investigating PINN applied to (COUC)SI. A problem's non-linearity may reduce the accuracy of the PINNs solutions (Basri et al., 2019; Cao et al., 2019). A series of methodologies, such as singular value decomposition (Gao et al., 2022), convolutional autoencoders (Kim et al., 2022), and Fourier transformation (Wang et al., 2021) have been suggested for tackling the dimensionality issues by transforming the network operations to lower frequency spaces. However, this issue is still an open question that needs to be addressed (Wang et al., 2022). We propose that a proper mathematical formulation of the system will have impact on the non-linearity and, thus, the ability of a PINN to represent the solution accurately. It is noted that defining features based on physics is a meaningful and established way of constraining machine learning models (Chao et al., 2022; Magzymov et al., 2022). However, the role of modifying the independent variables while representing the same PDE problem has not been explored and not in a PINN context. This work investigates whether a PINN-based approach can effectively solve the 1D COUCSI problem, for the first time at both ET and LT periods. We also investigated how reformulating the COUCSI PDEs based on different independent variables (derived based on the physical characteristics of the system) may impact the PINN accuracy, the required network size, and the number of collocation points. To do that, by applying the Change-of-Variables technique, three different implementations of the normalized COUCSI flow equation are studied, considering (1) scaled position and time variables, (2) a self-similar form depending on only one variable, valid only before interaction with the closed boundary, and (3) a form using the self-similar variable and square root of time as a second variable, valid for all times. The aim was the prediction of saturation profiles and recovery resulting from the 1D COUCSI system. We only used the same input as an FD numerical simulator; hence, no labeled training data from other sources (such as experiments, analytical or numerical solutions) are used to guide the PINNs training. The main questions of interest are: * Can the PINN solver produce correct saturation and recovery profiles in the COUCSI system only by minimizing loss terms related to the PDE and corresponding initial/boundary conditions? How does the run-time of the PINN solution compare to that of traditional numerical simulations? * Is there any advantage of reformulating the problem by Change-of-Variables in line with the physical behavior of the system? How is this reflected in how challenging it is to train the PINN and how well they predict? Can simpler networks and fewer collocation points be applied? How may it impact the run-time of solving the problem? * To which level do the PINN solutions from different formulations preserve physical properties of the COUCSI system, such as self-similarity, Total Variation (TV) preservation, and the square root of time recovery behavior? In the following section, we present the mathematical theory behind COUCSI and its formulations, the PINN methodology, and its implementation in this work. The results of applying PINN to solve the COUCSI problem for the different formulations for different test cases and different systematic training strategies are then presented. The work is then finalized with conclusions. ## Mathematics of Spontaneous Imbibition We investigate a two-phase flow system with immiscible and incompressible fluids. The porous medium is incompressible and homogeneous. The flow occurs in the horizontal direction without gravitational forces. All the faces are closed except the face at $x=0$, where it is exposed to the wetting phase. The saturation of the wetting phase at the open face is fixed to $s_{w}^{eq}$, defined by a zero capillary pressure $P_{c}\big{(}s_{w}^{eq}\big{)}=0$. The system is initially saturated with the non-wetting phase. The stated conditions result in 1D COUCSI, as shown in Figure 1. Darcy's law in 1D gives Darcy velocity ($u_{i}$) by writing: \[u_{i}=-\lambda_{i}[\partial_{x}p_{i}],\qquad\lambda_{i}=\frac{Kk_{ri}}{\mu_{i} },\qquad(i=w,nw), \tag{1}\] where $\lambda_{i}$ is mobility, $p_{i}$ is pressure, $K$ and $k_{ri}$ are absolute permeability and relative permeability, respectively, and $\mu_{i}$ is viscosity. The index $i$ refers to phase-specific properties for the wetting and non-wetting phases ($w$, $nw$). The conservation law determines the volumetric transport of phases in time: \[\phi\partial_{t}(s_{i})=-\partial_{x}(u_{i}),\qquad(i=w,nw), \tag{2}\] where $s_{i}$ is phase saturation, and $\phi$ is porosity. The saturations are constrained by volume conservation ($s_{w}+s_{nw}=1$), and the positive capillary pressure function constrains the pressures ($p_{nw}-p_{w}=P_{c}(s_{w})$). Combining these assumptions, we obtain the capillary diffusion equation describing 1D countercurrent spontaneous imbibition in the form of a second-order, elliptic PDE: \[\phi\partial_{t}(s_{w})=-K\partial_{x}(\lambda_{nw}f_{w}\partial_{x}P_{c}), \quad(0<x<L) \tag{3}\] This implementation reduces the problem's dimensionality compared to Eq. 1 by removing the phase pressures and constraining the equations to a single PDE. The boundary and initial conditions are given by: \[s_{w}(x=0,t)=s_{w}^{eq},\qquad\partial_{x}s_{w}\|_{x=L}=0,\qquad s_{w}(x,t=0)= s_{wr} \tag{4}\] representing a fixed saturation at the inlet corresponding to zero capillary pressure, a closed boundary at $x=L$, and a uniform initial saturation. $s_{wr}$ is also the residual water saturation. Considering $s_{nwr}$ as the irreducible non-wetting phase saturation, when $s_{w}^{eq}=1-s_{nwr}$ (oil is immobile) the system is strongly water-wetting, while if $s_{wr}<s_{w}^{eq}<1-s_{nwr}$, the system is considered mixed-wet. ### Scaled Representations Three different scaled representations will be considered of the flow equations (Eqs. 2 or 3). The Change-of-Variables technique was used, which is defined as the substitution of the variables of a PDE by new variables that are a function of the original variables (Folland, 1999). Each representation has different properties and hence potential advantages related to producing accurate solutions with PINN, which will be investigated. #### Normalized Diffusion Equation (XT formulation) By defining normalized spatial and temporal variables $X$, and $T$, and saturation $S_{n}$ the system of Eq. 3 and 4 can be written (see Appendix A for details) into an equivalent dimensionless form (Andersen, 2023b): \[\partial_{T}S_{n}=\partial_{X}(\Lambda_{n}(S_{n})\partial_{X}S_{n}),\qquad(0<X <1)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 5\] \[S_{n}(X=0,T)=1,\qquad\partial_{X}S_{n}\|_{X=1}=0,\qquad S_{n}(X,T=0)=0\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\] The interval $0<S_{n}<1$ is where imbibition occurs. $\Lambda_{n}$ is a scaled diffusion coefficient with a mean one over the imbibing saturations. This scaled representation captures all wetting states and input parameter combinations and only depends on the shape of $\Lambda_{n}(S_{n})$. Eq 5 and 6 are called the XT-formulation. This formulation is valid in full-time (FT) period, i.e., both ET and LT. #### Square Root of Time Full-time Solution (XY-formulation) By replacing the temporal variable from T to Y= $T^{0.5}$ and use it together with $X$, i.e., $S_{n}=S_{n}(X,Y)$, we can develop a system that is valid for both ET and LT flow periods. When reformulating Eq. 5, we use $Y$ in the spatial derivatives (Andersen, 2023a): \[\partial_{Y}S_{n}=2Y\partial_{X}(\Lambda_{n}(S_{n})\partial_{X}S_{n}),\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 7\] \[S_{n}(X,Y=0)=0,\qquad S_{n}(X=0,Y)=1,\qquad\partial_{X}S_{n}\|_{X=1}=0\qquad \qquad\qquad\qquad\qquad\qquad\qquad\] We refer to this system as the XY formulation. #### Self-similar Early-time Solution (Z-formulation) If we consider the time before the no-flow boundary is encountered, the 1D COUCSI system behaves in a self-similar way. The role of position and time can be combined into one variable $Z=X/T^{0.5}$ which Figure 1: The COUCSI geometry and the applied boundary conditions. The core is initially saturated by the non-wetting phase and exposed to the wetting phase at $x=0$. Other faces are closed to flow. determines the solution for a given function $\Lambda_{n}$, i.e., $S_{n}=S_{n}(Z)$. Using this variable, the system of Eq. 5 and 6 can be expressed as (Andersen, 2023a): \[Z\partial_{Z}S_{n}=-2\partial_{Z}(\Lambda_{n}(S_{n})\partial_{Z}S_{n}),\qquad(0 <Z<\infty) \tag{1}\] \[S_{n}(Z=\infty)=0,\qquad S_{n}(Z=0)=1 \tag{2}\] with the exception that the closed boundary condition in Eq. 6 is replaced by obtaining the initial condition at an infinite distance. We call Eqs. 9 and 10 the $Z$-formulation. ### Total Variation The Total Variation (TV) concept in mathematics is the magnitude of the arclength of a curve related to a continuous function in an interval of interest (Harten, 1983). It can represent the amount of oscillation in a numerical solution and whether it increases with time; hence it is a valuable concept in numerical stability analysis (LeVeque, 2002). In this context, we consider the Total Variation in saturation along the core as a function of time: \[TV\big{(}S_{n}(\cdot,t)\big{)}=\int_{X=0}^{1}\|\partial_{X}S_{n}\|\;dX=\int_{X= 0}^{1}\|dS_{n}\| \tag{3}\] In other words, TV measures the total change in saturation along the core. As we have scaled the system to have a saturation equal to one at the inlet and to be zero initially, the TV is expected to be one at the ET. However, at LT, the saturation increases from zero at the closed boundary and hence reduces the TV. We thus expect our PINN solution to produce $TV=1$ at the early time and $TV$ declining from one towards zero (when the saturations are uniformly equal to 1) at the late time. ### Physics-Informed Neural Networks Physics-informed Neural Networks (PINNs) are a subset of scientific deep learning that try to solve PDEs/ODEs using a data-driven approach (Raissi et al., 2019). If $S_{n}(X,T)$ represents the solution of a 1D COUCSI system based on $XT$ formulation, the PINNs model is defined as: \[S_{n}(X,T)+\;\mathcal{N}[S_{n};\theta]=0;\;\;x\in\mathbb{R},t\in\mathbb{R} \tag{4}\] Where $\mathcal{N}[S_{n};\theta]$ is the equivalent differential operator describing the investigating problem, and $\theta$ is its constitutive (trainable) parameters. The PINNs model is developed by constraining $\mathcal{N}[S_{n};\theta]$ during the training, using the PDE representing the physical system and the constraining initial or boundary conditions. The training is carried out by utilizing automatic differentiation (AD) while back-propagation. By applying the successive chain rule, automatic differentiation computes gradients with an efficient computational cost. The physical specification of the system is imposed on the PINN by the definition of a series of loss functions representing the mean squared errors (MSE) in the residuals of the system's underlying PDE ($\mathcal{F}$), initial conditions ($\mathcal{I}$), and boundary conditions ($\mathcal{B}$): \[\mathcal{L}_{\mathcal{F}}(\theta) =\int_{\Omega}\|\mathcal{F}(\mathcal{N})\|^{2}\;dx \tag{5}\] \[\mathcal{L}_{\mathcal{J}}(\theta) =\int_{\Omega}\|\mathcal{J}(\mathcal{N})\|^{2}\;dx \tag{6}\] \[\mathcal{L}_{\mathcal{B}}(\theta)=\int_{\Omega}\|\mathcal{B}(\mathcal{N})\|^{2}\,dx \tag{15}\] where $\mathcal{L}_{\mathcal{F}}$ denotes the loss due to deviation from the governing PDE in the domain $\Omega$. Also, $\mathcal{L}_{\mathcal{I}}$ and $\mathcal{L}_{\mathcal{B}}$ are the losses related to the initial and boundary conditions, respectively. $\mathcal{L}_{\mathcal{B}}$ includes the conditions at both faces, i.e., $X=0$, and $X=1$ (except for Z formulation that is only valid at ET and only has a boundary condition at $X=0$). If observation points (from sources like experiments or analytical/numerical solutions) are also available, an additional loss term representing the deviations from the true values may be defined. However, in this work, since no observation data is used in the training of the model, we neglected this loss term. Consequently, the total loss function is the weighted summation of all the defined losses: \[\mathcal{L}_{t}(\theta,x)=\ \alpha_{1}\mathcal{L}_{\mathcal{F}}(\theta)+\alpha_{2 }\mathcal{L}_{\mathcal{I}}(\theta)+\alpha_{3}\mathcal{L}_{\mathcal{B}}(\theta) \tag{16}\] Where $\alpha_{i}$ is the weight factor for each loss term. The above definition can also be defined for other forms of the COUCSI PDEs, i.e., $XY$ and $Z$ formulations. Based on the parameters of the applied formulation, we can define the PDE-, initial condition-, and boundary condition- residuals as below: _XT_ Formulation: \[\mathcal{F}(X,T)=\partial_{T}S_{n}-\partial_{X}(\Lambda_{n}(S_{n})\partial_{X} S_{n}) \tag{17}\] \[\mathcal{I}=S_{n}(X,T=0),\qquad\mathcal{B}_{1}=S_{n}(X=0,T)-1,\qquad\mathcal{B} _{2}=\partial_{X}S_{n} \tag{18}\] _XY_ Formulation: \[\mathcal{F}(X,Y)=\partial_{Y}S_{n}-2Y\partial_{X}(\Lambda_{n}(S_{n})\partial_{ X}S_{n}) \tag{19}\] \[\mathcal{I}=S_{n}(X,Y=0),\qquad\mathcal{B}_{1}=S_{n}(X=0,Y)-1,\qquad\mathcal{B} _{2}=\partial_{X}S_{n} \tag{20}\] $Z$ Formulation: \[\mathcal{F}(Z)=\partial_{Z}S_{n}+\frac{2}{Z}\partial_{Z}(\Lambda_{n}(S_{n}) \partial_{Z}S_{n}) \tag{21}\] \[\mathcal{I}=S_{n}(Z=\infty),\qquad\mathcal{B}_{1}=S_{n}(Z=0)-1,\qquad\mathcal{ B}_{2}=\partial_{Z}S_{n}\big{\|}_{\big{(}Y,Z=\frac{1}{Y}\big{)}} \tag{22}\] $\mathcal{B}_{2}$ in $Z$ formulation is theoretically trivial (in the ET period) and not obligatory in the training process; however, we calculated it at the deployment stage for quality check. During our observations, we noticed that $\mathcal{F}$ values near the flowing boundary were initially high, which had a significant impact on the PINN solutions. To address this, we omitted the $\mathcal{F}$ values (in Eqs. 17 and 19) in the $X<0.01$ range, during the training process. ### Problem Setup We considered two 1D COUCSI cases (A and B) with different frontal behavior. The rock/fluid properties were identical in both cases, except for their saturation functions. The capillary pressure curve for case A is generated based on Kumar et al. (2014). The saturation curves in case B are artificially generated to create a steeper saturation profile, different than case A. The main goal of having two cases is to investigate the capabilities of PINNs in capturing various flow behaviors. The basic rock and fluid properties are shown in Table 1. Also, the utilized relative permeability and capillary pressure curves are shown in Figure 2. Figure 2c shows the scaled capillary diffusion coefficient curves where the differences in the saturation functions could generate two different trends. We expect differences in the flow regimes due to the differences in the scaled capillary diffusion curves. In this work, we utilized Corey-type relative permeabilities (Brooks & Corey, 1964), \[k_{rw}=k^{}_{rw}(S)^{n_{w}},\qquad k_{rnw}=k^{}_{rnw}(1-S)^{n_{mw}}, \tag{23}\] Here, $n_{w}$ and $n_{mw}$ are Corey exponents and $k^{}_{rw}$ and $k^{}_{rnw}$ are the relative permeability endpoints. Also, S is the normalized water saturation as: \[S=\frac{s_{w}-s_{wr}}{1-s_{or}-s_{wr}} \tag{24}\] The correlation introduced by Andersen et al. (2017) is used for the capillary pressure curves: \[P_{c}=\frac{a_{1}}{(1+k_{1}S)^{n_{1}}}-\frac{a_{2}}{(1+k_{2}(1-S))^{n_{2}}}+a_{3} \tag{25}\] The parameters $a_{1}$, $a_{2}$, $a_{3}$, $k_{1}$, $k_{2}$, $n_{1}$, and $n_{2}$ determine the correlation shape. In case A, $a_{1}$, $a_{2}$, and $a_{3}$ are 15.52, 0.74, and 0.50 kPa, respectively, while in case B, $a_{1}$, $a_{2}$, and $a_{3}$ are 11.17, 5.17, and 5.00 kPa, respectively. Other parameters corresponding to the capillary pressure curves are similar and are presented in Table 1. The reference solutions for comparison with the PINN solutions were generated using a previously developed and verified core scale simulator IORCoreSim (Lohne et al., 2017) for numerical simulation of two-phase flow in porous media. The core was split equally into 400 grid cells, while the final scaled simulated time (35 hrs for case A and 10 hrs for case B) was divided into 120 report steps: the first 60 were equally spaced in square root time by $dt^{0.5}=0.0167$ hr${}^{0.5}$ and the next 60 by 0.0819 hr${}^{0.5}$ for case A, and similarly $dt^{0.5}=0.0167$ hr${}^{0.5}$ and 0.0360 hr${}^{0.5}$ for case B. ### Network and Training Setup For each of the introduced formulations (i.e., $XT$, $XY$, $Z$), we used a fully connected feed-forward multilayer perceptron (MLP) network, called the latent network, sandwiched between an encoder and a decoder. The network should output $S_{n}$ based on the corresponding inputs of each formulation, i.e., $S_{n}(X,T)$, $S_{n}(X,Y)$ or $S_{n}(Z)$. A schematic of the network setup is provided in Figure 3. The encoder is composed of an MLP with two linear layers followed by a normalization layer (Ba et al., 2016), and the decoder consists of an MLP with three linear layers. In both the encoder and decoder, the layer widths \begin{table} \begin{tabular}{c l l l l l} \hline Parameters & Values & Parameters & Values & Parameters & Value \\ \hline $K$ & 243.2 md & $k^{}_{rw}$, $k^{}_{rww}$ & 1.0, 1.0 & $k_{1}$ & 7 \\ \hline $\phi$ & 0.2 & $n_{w}$, $n_{mw}$ & 1.8, 1.4 & $k_{2}$ & 0.4 \\ \hline $\mathbf{L}$ & 0.06 cm & $s_{wr}$ & 0.0 & $n_{1}$ & 2 \\ \hline $\mu_{w}$, $\mu_{mw}$ & 1.0, 1.0 cP & $s_{or}$ & 0.0 & $n_{2}$ & 10 \\ \hline \end{tabular} \end{table} Table 1: Rock and fluid properties of the studied cases. Figure 2: The applied saturation functions for two generated cases (A and B). (a) relative permeability, (b) capillary pressure, (c) scaled capillary diffusion coefficient $\Lambda_{n}$. The capillary pressure curve of case A is from Kumar et al. (2014). Case B was artificially generated for comparison. were the same as the width of the latent network, which was 50 nodes. The role of the encoder is to transform the dimension of the PINN's input to the width of the latent network by matrix multiplication, and the decoder maps the latent network output to the expected PINN output dimension. The latent network is composed of an MLP with five hidden layers with the _tanh_ activation function and a width of 50 nodes. Fourier and inverse Fourier transform operations are performed on the input and output of the latent network, as discussed by Li et al. (2020). Performing the calculations in Fourier space improved the PINN performance and was applied as it is known to resolve deficiencies of standard PINNs in capturing high-frequency elements of the problem (Rahaman et al., 2018). In the output of the PINN, we set a rectifier function to make sure $0\leq S_{n}\leq 1$. Based on the mathematical definition of the investigated formulation, the model may have one input (for the $Z$ formulation) or two inputs (for $XT$, and $XY$ formulations). However, all the formulations used the same network architecture/size to simplify comparisons. A relatively large model was chosen to provide enough network complexity to capture the solution for all formulations. A sensitivity analysis of network size is carried out later to examine the possibility of simpler networks. The number of collocation points in the spatial dimension ($n_{X}$ or $n_{Z}$) were chosen based on one hundred equally spaced points. In the temporal dimension, the number of collocation points ($n_{T}$) was 45 for both cases, in the FT simulations; approximately 1/3 of the points were in the ET region, and the remaining were in the LT region. The temporal points were increasing by constant DY values, up to the total time. The model was trained by consolidating all collocation points into a singular batch (fully batched), resulting in a single backpropagation calculation per epoch. The trainable parameters in the network were initialized using a uniform random scheme. The gradient-based Adam optimizer (Kingma & Ba, 2014) was used for training the network with the starting learning rate 1e-4 and weight-decay 1e-5. We gradually reduced the learning rate by one order of magnitude during the training. The training was continued until the reduction in the loss term stopped. In all cases, the training could be terminated after $n_{ep}$=4000 epochs. ## Results ### Early- and full-time solutions Results of the trained PINN solutions are presented in this section for cases A and B, using the three formulations introduced previously. The mean absolute error (MAE) values for PINN solutions (Saturation and Recovery factor) are shown in Figure 4. The values are separated based on ET, LT, or FT periods. The complete MAE values for the loss terms and solution outputs (saturation, capillary pressure, and recovery factor) are summarized in Appendix B, Table A1. The MAEs are calculated by evaluating and comparing the FD and PINN solutions at the same collocation points. $P_{c}$ is calculated from $S_{n}$ using Figure 3: The architecture of the PINN based on the $S_{n}(X,T)$ formulation. Based on the applied PDE form, the input can also be $X$ and $Y$ for the XY PDE, and only $Z$ for the $Z$ PDE. The neurons in the latent network were activated by the tanh activation function. the capillary pressure correlation (Eq. 25), and recovery factor (RF) is defined as the average water saturation in the core at each time. The reported values from PINN correspond to the training after 4000 epochs. For each case and formulation, the FD and PINN saturation solutions, the absolute errors between them, and PDE residuals are illustrated in Figure 5 in 2D plots vs. spatial locations and time (Y). All three formulations could capture the solutions of both COUCSI cases with reasonable accuracy. Figure 4 shows that the XY formulation performed superior in the FT period with the saturation MAE equal to 0.009 for both cases A and B, with roughly 50% less error compared to the XT formulation with MAE equal to 0.019 (case A) and 0.021 (case B). During the ET period, the Z formulation, with the $S_{n}$, the mean MAE of $\sim$ 0.012 was superior to both other formulations (mean MAE $\sim$0.030 for XT and $\sim$0.014 for XY) in both cases. The same trends in performance were seen in Table A1 regarding the MAE of capillary pressure and recovery factor. Overall, there was a more significant error in ET as compared to LT, which can be attributed to PINN's inability to effectively minimize the loss term in points located near the flowing boundary. One crucial factor that influenced the efficacy of PINN was the approach used for sampling collocation points. The comprehensive analysis is presented in Appendix D. Our findings indicate that selecting points based on constant DY intervals led to a notable decrease in the required data points. Furthermore, when the same number of data points were used, the model based on constant DY intervals exhibited superior performance. It is worth noting that the COUCSI problem exhibits a more uniform flow rate on the Y coordinate, than the T coordinate. Another crucial point to consider is that the XY and Z formulations performed better with lower numbers of collocation points (See Figure A2), which is highly beneficial for the rapid solution of the COUCSI problem in practical scenarios. Figure 4: The final error (MAE) of PINN predictions for all three formulations, separated by early-time (ET), late-time (LT), and full-time (FT) periods. a) Sn, b) RF. Figure 6 shows how the total loss term ($\mathcal{L}_{\text{r}}$) and saturation MAE of the networks evolve with training. Note that the predicted saturations are obtained from training the networks to satisfy the loss terms of the PDEs, initial conditions, and boundary conditions, but not data directly. The error is based on comparing predicted saturations with the FD saturation values. After 500-1000 epochs, the PINN solution Figure 5: The contour maps of FD versus PINNs solutions ($S_{n}$) for different formulations. The solutions for Z-formulation only show the ET period; a) baseline FD, b) PINNs, c) absolute error, d) PDE residuals (in each formulation, the scales are different, so the numbers are not directly comparable). errors stabilized very close to the FD values, even with low learning rates; after that, we continued the training for fine-tuning the solutions. The solutions could have been achieved faster by choosing a higher starting learning rate. Nonetheless, when evaluating the effectiveness of PINN across various formulations, it was found that the XY and Z formulations achieved convergence towards the correct solutions in fewer epochs than the XT formulation. The comparison of run-time for PINN in Appendix F also indicates that the XY and Z formulations provided quicker solutions as compared to the XT formulation. (See Figure A5). Moreover, all the models demonstrated a satisfactory correlation between the total loss term value and the saturation profile's MAE (as depicted in Figure A1), signifying the dependability of the PINN solution in the absence of any knowledge pertaining to the true solutions. Figure 7 compares the saturation and recovery profiles for XT and XY formulations at all times. Both formulations could capture the recovery profile acceptably. In both cases, the XY formulation could predict the profiles better in the starting and finishing periods of imbibition ($T<0.01$ and $T>1$), while the XT formulation performed better in the intermediate periods. The overall error in the XY formulation was lower than the XT formulation, as also shown in Figure 4. Comparing the ET saturation profiles of different formulations (XT and XY in Figure 9, and Z formulation in Figure 8) shows that all PDE PINN formulations capture the overall trend in the ET period. The most challenging aspect seems to be capturing the boundary conditions where it is seen that the saturations at $X=0$ approach values less than one, and the saturations at the imbibition front tend to overshoot the FD solution. This difficulty causes a greater MAE in saturations at ET compared to LT, as previously depicted in Figure 4. The solutions overlap better at central saturations and for saturation profiles that have Figure 6: The trend of the total loss term, and Sn MAE versus epochs during the training for different PDE formulations. a-b) Case A, c-d) Case B. advanced further into the core. The mentioned challenges at the boundaries were handled better by the Z formulation than XT and XY formulations. In Figure 9, spatial saturation profiles ($S_{n}$vs. $Z$) and recovery profiles ($RF$ vs. $Z$) are shown in the ET period. Figure 9a shows the saturation profiles plotted versus the similarity variable Z for the different timesteps. Ideally, we expect all the profiles to overlap because of the self-similarity of COUCSI at ET. In the Z formulation, all the profiles are matched. By design, it preserves self-similarity as the PINN solution is only a function of $Z$ (different ranges of $Z$ follow from considering different times). The XT and XY formulations failed in giving proper self-similar solutions at the beginning times of flow: the saturations of a given $Z$-value on some curves was lower than the correct values. Figure 9b checks if recovery predicted from the PINN solutions followed proportionality with the square root of time. Semi-analytical solutions for 1D COUCSI by McWhorter & Sunada (1990) have proven that recovery at ET follows a linear trend versus the square root of time (linearity with $Y$). The Z formulation produced a straight line for recovery against $Y$ during ET, where the slope matched well, but not precisely, with the FD solution. The XY formulation followed the FD line well but was not a perfectly straight line; the slope rather slightly increased with Y. This was the same for the XT formulation, where Figure 8: Saturation profiles $S_{n}(X)$ obtained from Z formulation for the ET period of COUCSI. a) Case A, b) Case B. Figure 7: The production profile for the XT and XY formulations in the entire COUCSI period. a) Saturation Profile, b) Recovery profile. the PINN solution more notably failed to give zero recovery at zero time but corresponded well with the FD solution at higher values of Y. The TV behavior of different solutions in the ET period are compared in Figure 10 as a function of scaled time T. The curves show that the _XT_ formulation performed weakly in preserving the TV property of the model through time and that the value stayed far below 1. XY formulation could preserve the TV at a higher value, while the Z formulation managed to stay fairly constant close to one. In the XT solution, TV increased with T, indicating the inlet saturation increased with time. As this saturation was more fixed in the other formulations, TV did not change with time in the beginning for those cases. The TV decreased towards the end in all formulations because saturations were predicted to reach the closed boundary too Figure 9: The performance of PINNs with different PDE formulations (XT, XY, Z) for the ET period of COUCSI. a) Saturation profiles vs. similarity variable (Z), b) Recovery factor versus square root of time (Y) for FD and PINN solutions. early. This was a bigger problem in Case A than B and more severe for the XT and XY formulations than the Z formulation. A majority of the deviations of TV from 1 originate from the weaknesses of the PINN in sufficient handling of the $\mathcal{I}$ and $\mathcal{B}_{\mathbf{1}}$ terms (see Table A1). To have a better overview of the impact of changing formulations in PINNs, we also compared the loss landscape for all three formulations in Figure 11 (details are provided in Appendix C). It should be noted that the center of each landscape is related to the trained PINN model. Figure 11a shows a chaotic, non-convex landscape with an approximately ambiguous global minima, while XY formulation clearly shows a smooth surface with a trivial global minima. Z formulation also showed a smooth loss landscape with a clear global minima. The comparison of the loss landscapes reveals that changing the variables of the PDE from XT to XY and Z formulations could significantly enhance the convexity and smoothness of the loss landscape and provide training trajectories that are more explicit and dependable. This enhancement results in a reduction in the number of required collocation points (refer to Appendix D), the necessary network size (refer to Appendix E), as well as the run-time of the model (refer to Appendix F). Figure 10: The calculated total variation (TV) from the PINN solutions with different PDE formulations (XT, XY, Z) for the ET period of COUCSI. a) Case A, b) Case B. ## Discussion This study focused on PINN-based solving of the two-phase 1D COUCSI phenomena with different equivalent PDE formulations obtained from Change-of-Variables technique. PINN could successfully solve all the PDE formulations with acceptable errors. Changing the PDE variables in accordance with the physical properties of the system, such as the square root of time flow, showed to have the potential to enhance the performance of PINN in various ways, including accuracy of solutions and computational efficiency of the models. A severe problem in many of the PINN models during the training was that the optimizer could quickly lose track of the actual solution and converge to the trivial solution with $S_{n}\ x,t\ =1$ in all spatial and temporal values. We found that the problem can be tackled by choosing approximately equal weight factors for different terms in the loss function and choosing enough resolution of the collocation points, especially in points close to the initial/boundary regions. A practical problem with the $Z$ formulation was that at $T=0$, $Z_{max}=\frac{1}{\gamma}\rightarrow\infty$ and the interval with non-trivial solutions becomes very small compared to Figure 11: 3D loss landscape, generated by linear mixing of two different initial directions with the trained model parameters. A logarithmic side-view representation of the loss-landscapes are also shown for more clarity. a) XT, b) XY, c) Z the full interval. We tackled this problem by adding a small constant to the denominator when selecting collocation points. The total variation TV is a useful property to evaluate mathematical systems and their numerical schemes. In this work, we utilized this concept to track the PINN results' validity for the different cases. In the problems studied in this work, the TV should stay constant, equal to one, at ET and then decline towards zero. The preservation was satisfied to a reasonable level in $Z$ formulation. For the XT (and to a less extent, XY) formulation, TV varied more dynamically, starting and ending at lower values than the others, while for all the formulations, TV declined towards the end. The difference from one was related to lower saturation at the inlet than the mathematical boundary condition. It stayed constant for Z formulation, while for XT (and XY), it changed dynamically. The late decline was related to PINN saturations reaching the no-flow boundary too soon. We believe the TV property can be better maintained by inclusion as an additional loss term in the training process or by better capturing the boundary condition loss terms. There are generally two different approaches for defining this term: 1) the $TV$ may be calculated analytically using automatic differentiation; this approach should be more accurate; however, it may add significant computational load to the optimizer. 2) the more efficient approach is the definition of the problem in a discretized approach where it still has acceptable accuracy but takes much lower computational costs. In addition to capturing the PDE, IC, and BC terms, we note that the PINN should capture two important transitions: the finite saturation gradient at the front jumps to zero ahead of the front. As seen in Figures 4 and 6, the PINN's main challenge was capturing the abrupt change in gradient at the front and the self-similarity of the ET solution when two variables were involved in the system formulation. In case B, the gradient was steeper at the front, and hence the change to zero gradients ahead of the front was more abrupt, which is believed to explain the incredible difficulty of training the solutions to this case. We could show the feasibility of applying a pure PINN-based approach for solving benchmark core-scale phenomena with different formulations. The merit of using different formulations was that we could show the potentials of using the Change-of-Variables technique for improving the performance of PINNs in terms of collocation point selection, network complexity and by reducing the complexity of PDEs and simplify complicated differentiations. We believe the approach can be extended to other engineering processes. However, we see a potential to achieve lower error values and reduce the computational load during training. Given its relatively short run-time compared to numerical simulations, PINN-based solutions are a promising approach for obtaining quick approximations of forward problems. However, PINNs may be even more advantageous for inverse problems, where the iterative nature of numerical simulation methods can significantly increase computational time. Our future work will explore the use of PINNs for inverse solutions in porous media flow processes. ## Conclusions In this work, we studied the performance of PINNs in solving the two-phase 1D COUCSI problem. Using Change-of-Variables, the system was expressed in three equivalent PDE formulations (XT, XY, and Z), and we compared the resulting performance of the PINNs during early- or late-time for two simulation cases A and B. The following conclusions were obtained: * The PINN solver could acceptably solve the 1D COUCSI problem, as shown for two different cases in early and late time flow regimes, with the MAE of saturations around 0.01 for the ET and 0.03 for the full imbibition time. The error was typically least for Z formulation (only ET), followed by XY, then XT. The errors were typically located at the inlet, at the imbibition front and at the closed boundary when the front arrived there. * Total Variation of saturation was used to evaluate the saturation profile solutions with time. A difference from 1 at ET indicates that the solution does not meet the boundary and initial conditions properly (or has oscillations, which was not encountered). * The Change-of-Variables technique helped in transforming the coordinates of the imbibition problem into a neater space which had lower prediction errors and was closer to the physical properties of the COUCSI problem, such as self-similar saturation profiles, TV preservation and the square root of time recovery behavior in the ET region. Especially, the Z formulation, by definition, obtained self-similar profiles, square root of time recovery, and constant TV. In contrast, XT and XY formulations obtained profiles that did not fully overlap at ET, recoveries that only approximately followed the square root of time and were not identical to zero at time zero, and TVs that varied with time. XY formulation, however, performed much better than XT from these perspectives. The TV values at ET were closer to 1 for Z, then XY, then XT formulations indicating a difference in how well the initial and boundary conditions were captured. The comparison of loss landscapes also revealed that the XY and Z formulations had a higher degree of convexity as compared to the XT formulation. * By Change-of-Variables, fewer collocation points were required depending on which formulation and variable sampling procedure was used. Both for XT and XY systems, fewer temporal points are needed if they are selected based on constant DY (fixed square root of time difference) compared to constant DT (fixed time difference) and lower errors are obtained. The constant DY sampling focuses on both the early and late times. At a given number of temporal points, the XY formulation had less error than the XT formulation. Also, The Change-of-Variable technique could help reduce the network size. * The run-time of PINN to achieve a saturation MAE of 0.03 was shorter than that of MRST, while it was longer if one aimed to attain lower MAE values. Overall, it can be inferred that the solutions provided by PINN are computationally efficient if one requires a fast estimation of the accurate solutions. ## Acknowledgments The authors acknowledge the Research Council of Norway and the industry partners of NCS2030 - RCN project number 331644 - for their support. Also, we appreciate Valide AS for partially granting this project under the Plogen program.
2303.07589	Sequential three-way decisions with a single hidden layer feedforward neural network	The three-way decisions strategy has been employed to construct network topology in a single hidden layer feedforward neural network (SFNN). However, this model has a general performance, and does not consider the process costs, since it has fixed threshold parameters. Inspired by the sequential three-way decisions (STWD), this paper proposes STWD with an SFNN (STWD-SFNN) to enhance the performance of networks on structured datasets. STWD-SFNN adopts multi-granularity levels to dynamically learn the number of hidden layer nodes from coarse to fine, and set the sequential threshold parameters. Specifically, at the coarse granular level, STWD-SFNN handles easy-to-classify instances by applying strict threshold conditions, and with the increasing number of hidden layer nodes at the fine granular level, STWD-SFNN focuses more on disposing of the difficult-to-classify instances by applying loose threshold conditions, thereby realizing the classification of instances. Moreover, STWD-SFNN considers and reports the process cost produced from each granular level. The experimental results verify that STWD-SFNN has a more compact network on structured datasets than other SFNN models, and has better generalization performance than the competitive models. All models and datasets can be downloaded from https://github.com/wuc567/Machine-learning/tree/main/STWD-SFNN.	Youxi Wu, Shuhui Cheng, Yan Li, Rongjie Lv, Fan Min	2023-03-14T02:26:34Z	http://arxiv.org/abs/2303.07589v1	# Sequential three-way decisions with a single hidden layer feedforward neural network ###### Abstract The three-way decisions strategy has been employed to construct network topology in a single hidden layer feedforward neural network (SFNN). However, this model has a general performance, and does not consider the process costs, since it has fixed threshold parameters. Inspired by the sequential three-way decisions (STWD), this paper proposes STWD with an SFNN (STWD-SFNN) to enhance the performance of networks on structured datasets. STWD-SFNN adopts multi-granularity levels to dynamically learn the number of hidden layer nodes from coarse to fine, and set the sequential threshold parameters. Specifically, at the coarse granular level, STWD-SFNN handles easy-to-classify instances by applying strict threshold conditions, and with the increasing number of hidden layer nodes at the fine granular level, STWD-SFNN focuses more on disposing of the difficult-to-classify instances by applying loose threshold conditions, thereby realizing the classification of instances. Moreover, STWD-SFNN considers and reports the process cost produced from each granular level. The experimental results verify that STWD-SFNN has a more compact network on structured datasets than other SFNN models, and has better generalization performance than the competitive models. All models and datasets can be downloaded from [https://github.com/wuc567/Machine-learning/tree/main/STWD-SFNN](https://github.com/wuc567/Machine-learning/tree/main/STWD-SFNN). keywords: network topology, hidden layer node, sequential three-way decisions, granular level, sequential thresholds + Footnote †: journal: Information Sciences ## 1 Introduction Neural networks [6; 17; 21] are the mathematical models that simulate the connection structure of brain neurons. They have been widely implemented in applications, including video frame inpainting [33] and automatic driving [31]. The performance of neural networks is mainly affected by hyperparameter selection and network topology. Hyperparameter selection [3; 4] is a classical topic in machine learning, which can be realized by grid search [26; 32] and particle swarm optimization [1; 24]. In addition, network topology [2; 30; 42] is the key of neural network design, which can be realized through three-way decisions [7] and an incremental learning mechanism [10; 15; 40]. To achieve an effective network structure, three-way decisions with a single hidden layer feedforward neural network (TWD-SFNN) [7] adopts a novel model to guide the number of hidden layer nodes. In addition, as a shallow neural network model, TWD-SFNN provides a new perspective for the topology design of multilayer neural networks, hence laying the theoretical foundation for the framework of deep learning. However, for practical applications, TWD-SFNN has two drawbacks: (i) in terms of the performance of TWD-SFNN, the generalization ability of TWD-SFNN needs to be further improved; and (ii) to analyze the relationship between the costs and number of hidden layer nodes more thoroughly, the process costs of TWD-SFNN need to be considered. To improve the generalization ability of neural networks on structured datasets, and further enrich the theoretical framework of deep learning, we employ sequential three-way decisions to guide the growth of the network topology. The sequential three-way decisions (STWD) [28; 38; 46] is a kind of progressive granular computing [9; 25; 48], which realizes the multi-stage learning from coarse to fine by constructing granular levels. Specifically, STWD makes deterministic decisions for only a few instances at the coarse-grained granular level. With the addition of information, the decision of the remaining instances is gradually realized at the finer level. Recently, STWD has been applied to medical treatment [8], face recognition [19], and attribute reduction [27]. To obtain a better performance, we propose a model called STWD with a single hidden layer feedforward neural network (STWD-SFNN). STWD-SFNN enhances the learning ability of the SFNN and ensures the compactness of the network structure. The differences between TWD-SFNN and STWD-SFNN are as follows. Although TWD-SFNN has the multi-granularity property, the process costs produced from granular levels are not investigated in the development discipline of costs. On the other hand, although TWD-SFNN has the sequential property, it partitions the instances by adopting the constant threshold parameters when a new hidden layer node is added, making it difficult to focus on more difficult-to-classify instances. The contributions of this paper are as follows. 1. To improve the generalization ability of TWD-SFNN, we employ the STWD scheme to optimize the structure of SFNN, and propose a new approach called STWD-SFNN. 2. STWD-SFNN handles the easy-to-classify instances by adopting strict conditions. As the number of hidden layer nodes increases, STWD-SFNN addresses the difficult-to-classify instances by adopting loose conditions until all instances can be classified. 3. The experimental results show that the network structure of STWD-SFNN is significantly better than other SFNN models, and STWD-SFNN has better accuracy and weighted-f1 performance than the competitive models. The remainder of the paper is organized as follows. Section 2 introduces the preliminaries. Section 3 describes the STWD-SFNN model. Section 4 analyzes the performance of STWD-SFNN, and Section 5 presents the conclusions. ## 2 Preliminaries In this section, we briefly introduce the training of SFNN with focal loss and Adam optimizer, the TWD-SFNN model, and the STWD model. ### Training SFNN with focal loss and Adam optimizer The parameters connected to the hidden layer are one of the factors affecting the performance of SFNN [5]. Therefore, the focal loss and Adam optimizer for network parameters are introduced as follows. Since the labels in the classification dataset may have category imbalance, to better deal with the practical problems, the loss function of SFNN adopts the focal loss [49], which also makes the model focus more on the difficult-to-classify instances in training, and is calculated according to Equation (1). \[L_{\text{FL}}=\left\{\begin{array}{ll}-\delta(1-\hat{y})^{\theta}log\hat{y} &y=+1\\ -(1-\delta)\hat{y}^{\theta}log(1-\hat{y})&y=-1\end{array}\right. \tag{1}\] where $\hat{y}$ is the predicted value of the neural network; $\delta$ is a factor that balances the proportion of positive and negative labels; and $\theta$ ($\theta\geq 0$) is a parameter that adjusts the weight reduction rate of easy-to-classify instances, and the greater the $\theta$, the greater the weight reduction rate. In this paper, $\delta$ is the ratio of the number of each label to the total number of labels, and $\theta$ is set to two which is the same as [7]. In addition, to prevent the overfitting of SFNN, the cost function is calculated according to Equation (2). \[L_{\text{NN}}=L_{\text{FL}}+\frac{\lambda_{\text{SFNN}}}{2}\times(\\|\mathbf{ W}_{1}\\|^{2}+\\|\mathbf{b}_{1}\\|^{2}+\\|\mathbf{W}_{2}\\|^{2}+\\|\mathbf{b}_{2}\\|^{2}) \tag{2}\] where $\lambda_{\text{SFNN}}$ is a regularization factor of SFNN; $\mathbf{W}_{1}$ and $\mathbf{b}_{1}$ are the weight and bias connecting the input layer to the hidden layer of SFNN, respectively; $\mathbf{W}_{2}$ and $\mathbf{b}_{2}$ are the weight and bias from the hidden layer to the output layer of SFNN, respectively. Since the gradient of network parameters may have oscillation in the optimization, to better optimize $\mathbf{W}_{1}$, $\mathbf{b}_{1}$,$\mathbf{W}_{2}$, and $\mathbf{b}_{2}$, SFNN selects the Adam optimizer [14]. Without loss of generality, we take $\mathbf{W}_{1}$ as an example which is calculated according to Equation (3). \[\begin{split}\mathbf{V}^{(h)}_{\text{dW}_{1}}&=\rho_{1} \times\mathbf{V}^{(h-1)}_{\text{dW}_{1}}+(1-\rho_{1})\times\text{d}\mathbf{W}^{( h)}_{1}\\ \mathbf{S}^{(h)}_{\text{dW}_{1}}&=\rho_{2}\times \mathbf{S}^{(h-1)}_{\text{dW}_{1}}+(1-\rho_{2})\times(\text{d}\mathbf{W}^{(h)}_ {1})^{2}\\ \hat{\mathbf{V}}^{(h)}_{\text{dW}_{1}}&=\frac{ \mathbf{V}^{(h)}_{\text{dW}_{1}}}{1-\rho^{(h)}_{1}}\\ \hat{\mathbf{S}}^{(h)}_{\text{dW}_{1}}&=\frac{ \mathbf{S}^{(h)}_{\text{dW}_{1}}}{1-\rho^{(h)}_{2}}\\ \mathbf{W}^{(h)}_{1}&=\mathbf{W}^{(h-1)}_{1}-\mu \times\frac{\hat{\mathbf{V}}^{(h)}_{\text{dW}_{1}}}{\sqrt{\mathbf{S}^{(h)}_{ \text{dW}_{1}}}+\tau}\end{split} \tag{3}\] where $\mathbf{V}^{(h)}_{\text{dW}_{1}}$ and $\mathbf{S}^{(h)}_{\text{dW}_{1}}$ are updated first-order momentum estimates and second-order momentum estimates of $\mathbf{W}_{1}$, respectively; $\hat{\mathbf{V}}^{(h)}_{\text{dW}_{1}}$ and $\hat{\mathbf{S}}^{(h)}_{\text{dW}_{1}}$ are deviation corrections of $\mathbf{V}^{(h)}_{\text{dW}_{1}}$ and $\mathbf{S}^{(h)}_{\text{dW}_{1}}$, respectively; $\mathbf{W}^{(h)}_{1}$ is the updated value of $\mathbf{W}_{1}$ at the $h$-th moment; $\rho_{1}$ and $\rho_{2}$ denote the exponential decay rate of the first-order and second-order momentum estimates, respectively; $\mu$ is the learning rate; $\tau$ is used to avoid that the denominator of $\mathbf{W}^{(h)}_{1}$ is zero, and $\tau$ is set to $10^{-8}$. For convenience, we mark $\mathbf{W}^{(h)}_{1}$ returned by Adam as $\overline{\mathbf{W}}_{1}$. To update the parameters of the rest network, we replace $\mathbf{W}_{1}$ in Equation (3) with $\mathbf{b}_{1}$, $\mathbf{W}_{2}$, and $\mathbf{b}_{2}$. ### Twd-Sfnn TWD-SFNN [7] adopts the trisecting-and-acting model [34, 43, 44] to find the number of hidden layer nodes. In essence, TWD-SFNN is a model for dealing with the binary classification problem. Suppose we have a binary classification dataset $\{\mathbf{x}_{i},\mathbf{y}_{i}\}_{i=1}^{d}$, where $\mathbf{x}_{i}=(\mathbf{x}_{i1},\mathbf{x}_{i2},\cdots,\mathbf{x}_{im})^{ \mathbf{T}}\in\mathbf{R}^{m}$, $\mathbf{y}_{i}\in\{+1,-1\}$. We also suppose that TWD-SFNN has total $d$ instances and $u$ ($1\leq u\leq d$) unclassified instances, where the unclassified instances are the instances that are partitioned into the region of boundary (BND), and $u$ is the total number of instances minus the total number of instances classified into the regions of positive(POS) and negative(NEG). TWD-SFNN considers the total expected risk of classifying $u$ instances. \[\begin{split} Risk_{\text{TWD-SFNN}}&=\min_{(\alpha,\beta),\alpha\gamma}\biggl{(}\sum_{\begin{subarray}{c}i=1\\ p_{o}=\alpha,p_{o}>\gamma\end{subarray}}^{u_{p}}\lambda_{PP}\times p_{o}+ \lambda_{PN}\times(1-p_{o})\\ +\varepsilon\times\sum_{\begin{subarray}{c}i=1\\ p_{o}=\alpha\end{subarray}}^{u_{p}}\lambda_{BP}\times p_{k}+\lambda_{BN} \times(1-p_{k})\\ +\sum_{\begin{subarray}{c}i=1\\ p_{j}=\alpha,p_{j}>\gamma\end{subarray}}^{u_{p}}\lambda_{NP}\times p_{j}+ \lambda_{NN}\times(1-p_{j})\biggr{)}\\ \text{s.t.}&\quad 0<\beta<\gamma<\alpha<1,\quad\varepsilon\geq 1, \quad u=u_{p}+u_{b}+u_{n}\end{split} \tag{4}\] where $\varepsilon$ is a penalty factor and $\varepsilon\geq 1$; $p_{e}$ is the conditional probability that the $e$-th ($1\leq e\leq u$) instance belongs to the positive labels set $Y$, i.e., $p_{e}=\frac{\|Y\|\in\{x_{e}\}\|}{\\|x_{e}\\|}$, $\{x_{e}\}$ is the equivalent class of the $e$-th instance, and the equivalence class is a set of instances with the same features; $u_{p}$, $u_{b}$, and $u_{n}$ are the numbers of instances partitioned into POS, BND, and NEG, respectively; $\lambda_{sP}$ and $\lambda_{sN}$ are the costs of decision-making when instances belong to the POS and NEG, respectively; $(\alpha,\beta,\gamma)$ are the thresholds, and can be calculated as follows. \[\alpha = \frac{\lambda_{PN}-\lambda_{BN}}{(\lambda_{PN}-\lambda_{BN})+( \lambda_{BP}-\lambda_{PP})} \tag{5}\] \[\beta = \frac{\lambda_{BN}-\lambda_{NN}}{(\lambda_{BN}-\lambda_{NN})+( \lambda_{NP}-\lambda_{BP})}\] (6) \[\gamma = \frac{\lambda_{PN}-\lambda_{NN}}{(\lambda_{PN}-\lambda_{NN})+( \lambda_{NP}-\lambda_{PP})} \tag{7}\] To meet the requirements of the three-way decisions on data types, TWD-SFNN adopts $k$-means$++$ to discretize the numerical data into categorical data. When the number of misclassification instances is more than that of clusters specified in $k$-means$++$, $\alpha$ and $\beta$ are employed and the classification criteria are as follows. (P) if $p_{e}\geq\alpha$, then $[x_{e}]\in\text{POS}(Y)$; (B) if $\beta<p_{e}<\alpha$, then $[x_{e}]\in\text{BND}(Y)$; (N) if $p_{e}\leq\beta$, then $[x_{e}]\in\text{NEG}(Y)$. Whereas when the number of instances in the misclassification set is no more than these clusters specified in $k$-means$++$, $\gamma$ is adopted in TWD-SFNN and the classification criteria are as follows. (P) if $p_{e}\geq\gamma$, then $[x_{e}]\in\text{POS}(Y)$; (N) if $p_{e}<\gamma$, then $[x_{e}]\in\text{NEG}(Y)$. TWD-SFNN settles the penalty factor $\varepsilon(\varepsilon\geq 1)$ in the decision-making risk function, which means that TWD-SFNN can avoid too many training instances being classified into BND, so as to avoid the overfitting of the model. In addition, if BND is not empty, it is necessary to increase the number of hidden layer nodes of TWD-SFNN until BND is empty. ### Stwd The sequential strategy is an essential application of multi-granularity [22; 23] to three-way decisions. As information continues to be added and updated, a comprehensive decision-making scheme of STWD is gradually developed. STWD [35; 36; 47] relies on the idea of multi-granularity hierarchical processing. STWD has a decision table [27] which is defined as $S=(U,A=C\cup D,V=\{V_{a}\mid a\in A\},I=\{I_{a}\mid a\in A\})$, where $U$ is a universe; $A$ is an attribute set; $C$ is a conditional attribute set, and $D$ is a decision attribute set. For any attribute $a$, $V_{a}$ is the conditional attribute value, and $I_{a}$ is the corresponding decision attribute value. Suppose $U$ has $t$ granular levels, denoted by $\{1,2,\cdots,t\}$, which are added from the first level to the $t$-th level step by step. Given a state set $\{X,\neg X\}$, an action set $\{a_{P},a_{B},a_{N}\}$, where $X$ and $\neg X$ represent different states that the instance $x$ belongs to $X$ and $\neg X$, respectively; $a_{P}$, $a_{B}$, and $a_{N}$ represent the accepted decision-making, the delayed decision-making, and the rejected decision-making, respectively. We suppose that when $x\in X$ at the $i$-th ($1\leq i\leq t$) level, $\lambda_{PP}^{(i)}$, $\lambda_{BP}^{(i)}$, and $\lambda_{NP}^{(i)}$ represent the losses caused by taking $a_{P}$, $a_{B}$, and $a_{N}$, respectively, while $\lambda_{PN}^{(i)}$, $\lambda_{BN}^{(i)}$, and $\lambda_{NN}^{(i)}$ represent the corresponding losses when $x\in\neg X$ at the $i$-th level. Thus, under the action of $a_{P}$, $a_{B}$, and $a_{N}$, the expected losses at the $i$-th level of STWD are shown as follows. \[R^{(i)}(a_{P}\mid[x]_{R}) = \lambda_{PP}^{(i)}\times p^{(i)}(X\mid[x]_{R})+\lambda_{PN}^{(i)} \times p^{(i)}(\neg X\mid[x]_{R}) \tag{8}\] \[R^{(i)}(a_{B}\mid[x]_{R}) = \lambda_{BP}^{(i)}\times p^{(i)}(X\mid[x]_{R})+\lambda_{BN}^{(i)} \times p^{(i)}(\neg X\mid[x]_{R})\] (9) \[R^{(i)}(a_{N}\mid[x]_{R}) = \lambda_{NP}^{(i)}\times p^{(i)}(X\mid[x]_{R})+\lambda_{NN}^{(i)} \times p^{(i)}(\neg X\mid[x]_{R}) \tag{10}\] where $p^{(i)}(X\mid[x]_{R})$ and $p^{(i)}(\neg X\mid[x]_{R})$ represent the conditional probability that the equivalent class $[x]_{R}$ at the $i$-th ($1\leq i\leq t$) level belongs to $X$ and $\neg X$, respectively. For convenience, we mark $p^{(i)}(X\mid[x]_{R})$ as $p^{(i)}$. According to Bayes decision criterion, the following optimal classification decision rules can be obtained. (P) If $R^{(i)}(a_{P}\mid[x]_{R})\leq R^{(i)}(a_{B}\mid[x]_{R})$ and $R^{(i)}(a_{P}\mid[x]_{R})\leq R^{(i)}(a_{N}\mid[x]_{R})$, then $x\in\text{POS}^{(i)}(X)$ ; (B) If $R^{(i)}(a_{B}\mid[x]_{R})\leq R^{(i)}(a_{P}\mid[x]_{R})$ and $R^{(i)}(a_{B}\mid[x]_{R})\leq R^{(i)}(a_{N}\mid[x]_{R})$, then $x\in\text{BND}^{(i)}(X)$ ; (N) If $R^{(i)}(a_{N}\mid[x]_{R})\leq R^{(i)}(a_{P}\mid[x]_{R})$ and $R^{(i)}(a_{N}\mid[x]_{R})\leq R^{(i)}(a_{B}\mid[x]_{R})$, then $x\in\text{NEG}^{(i)}(X)$. Furthermore, STWD has a total order relationship over granular levels [44], i.e., $[x]_{R_{1}}\leq[x]_{R_{2}}\leq\cdots\leq[x]_{R_{t}}$, where $[x]_{R_{1}}$ and $[x]_{R_{t}}$ are the equivalent class of instance $x$ calculated at the coarse-grained level and the fine-grained level, respectively. Moreover, STWD has a conditional probability $p^{(i)}$ and thresholds $(\alpha_{i},\beta_{i})$ ($1\leq i\leq t-1$) for the previous $t$-1 levels and also has a conditional probability $p^{(t)}$ and a threshold $\gamma$ at the $t$-th level. These parameters satisfy the following conditions. \[0<\beta_{1}\leq\beta_{2}\leq\cdots\leq\beta_{t-1}<\gamma<\alpha_{t-1}\leq\cdots \leq\alpha_{2}\leq\alpha_{1}<1\] Compared with $(\alpha_{i},\beta_{i})$, STWD at least increases the range of POS and NEG regions when adopting $(\alpha_{i+1},\beta_{i+1})$. Accordingly, the BND of STWD will be smaller and smaller at the ($i$+1)-th granular level. It indicates that the sequential thresholds of STWD gradually realize the classification of the delay decision region, thereby completing the partition of whole instances. Since STWD learns three-way decision-making from the first level to the ($t$-1)-th level, given the $i$-th ($1\leq i\leq t-1$) granular level $L_{i}$, $L_{i}$ can be classified into three disjoint regions: \[\text{POS}_{(\alpha_{i},\beta_{i})}(Y)=\{x\in U_{i}\mid p^{(t)} \geq_{i}\alpha_{i}\} \tag{11}\] \[\text{BND}_{(\alpha_{i},\beta_{i})}(Y)=\{x\in U_{i}\mid\beta_{i} <_{i}p^{(t)}<_{i}\alpha_{i}\}\] (12) \[\text{NEG}_{(\alpha_{i},\beta_{i})}(Y)=\{x\in U_{i}\mid p^{(t)} \leq_{i}\beta_{i}\} \tag{13}\] where $\text{POS}_{(\alpha_{i},\beta_{i})}(Y)$, $\text{BND}_{(\alpha_{i},\beta_{i})}(Y)$, and $\text{NEG}_{(\alpha_{i},\beta_{i})}(Y)$ are the POS, BND, and NEG at the $i$-th level, respectively. As the number of levels increases, $\text{BND}_{(\alpha_{i},\beta_{i})}(Y)$ becomes smaller. However, if $\text{BND}_{(\alpha_{t-1},\beta_{t-1})}(Y)\neq\varnothing$, STWD carries out two-way decisions at the $t$-th level, which means that the region in the $t$-th level $L_{t}$ is partitioned into two disjoint regions. \[\text{POS}_{(\gamma)}(Y)=\{x\in U_{t}\mid p^{(t)}\geq\gamma\} \tag{14}\] \[\text{NEG}_{(\gamma)}(Y)=\{x\in U_{t}\mid p^{(t)}<\gamma\} \tag{15}\] ## 3 Proposed model In this section, we propose STWD-SFNN which dynamically determines the number of hidden layer nodes by adopting granularity levels and sequential thresholds to improve the performance of network topology. To show the difference between SFNN and STWD-SFNN, Figs. 1 and 2 show the framework of SFNN and STWD-SFNN, respectively. The network topology of SFNN refers to the number of hidden nodes $z$, and we calculate hidden nodes $z$ according to the empirical formula methods. However, these methods for determining network topology lack a reliable theoretical basis. Therefore, we propose STWD-SFNN to dynamically determine the number of hidden nodes. As shown in Fig. 2, STWD-SFNN adopts the sequence of STWD to guide the growth of network topology. Moreover, Fig. 3 illustrates the granulation of STWD-SFNN from coarse to fine. Firstly, if the instances contain numerical features, such as light and humidity in Fig. 2, we adopt $k$-means++ [18] to discretize the numerical data to meet the requirements of STWD for data types. Secondly, we initialize the network topology with one hidden layer node and get correctly classified instances, i.e., $\text{P}_{\text{N}}^{(i)}$ and $\text{N}_{\text{N}}^{(i)}$, and misclassification instances $\text{M}_{\text{N}}^{(i)}$. Thirdly, we calculate the conditional probability of the instances belonging to the positive labels and the thresholds $(\alpha_{i},\beta_{i})(1\leq i\leq t-1)$ of the $i$-th level. If the conditional probability is no less than $\alpha_{i}$, the instances are classified into $\text{P}_{\text{L}}^{(i)}$, if the conditional probability is not greater than $\beta_{i}$, they are partitioned into $\text{N}_{\text{L}}^{(i)}$; otherwise, they are classified into $\text{B}_{\text{L}}^{(i)}$. It means that the information of the $i$-th level is not enough to support STWD-SFNN to make decisions, and a hidden layer node is added based on the current network topology, and the above process is repeated Figure 1: Framework of SFNN at the ($i$+1)-th level. To prevent the overfitting of STWD-SFNN, when $\text{B}_{\text{L}}^{(t-1)}$ is not empty, we adopt the threshold $\gamma$ at the $t$-th level. If the conditional probability is no less than $\gamma$, the instances are classified into $\text{P}_{\text{L}}^{(t)}$; otherwise, they are partitioned into $\text{N}_{\text{L}}^{(t)}$. Finally, we have the regions of POS, BND, and NEG after $t$ turns of STWD-SFNN. ### Related definitions Suppose the granular levels of STWD-SFNN traverse from one to $t$. Accordingly, the maximum number of hidden layer nodes of STWD-SFNN is $t$. Definition 1.: Suppose there is a decision table with a number of hidden layer nodes \[T=(U,A=C\cup D,V=\{V_{a}\mid a\in A\},I=\{I_{a}\mid a\in A\},H)\] where $U$, $A$, $C$, $D$, $V$, and $I$ have the same meanings as in Section 2.3; $H$ is a set composed of the hidden layer nodes, i.e., $H=\{h_{1},h_{2},\cdots,h_{t}\}$ and $h_{i}$ is the $i$-th ($1\leq i\leq t$) hidden layer node added. According to STWD, we have a action set $\{a_{P},a_{B},a_{N}\}$, which can classify the instances into POS, BND, and NEG, respectively. Hence, we have the following definitions. Figure 3: Sequential thresholds of STWD-SFNN Figure 2: Framework of STWD-SFNN. STWD-SFNN has two parts: discretion modular and training modular. If the instances contain numerical features, discretion modular adopts $k$-means++ to convert numerical data into discrete data to meet the requirements of STWD for data types. The training modular has $t$ levels. Each level has two parts: SFNN and STWD. SFNN is a network topology with one hidden layer node and gets correctly classified instances, i.e., $\text{P}_{\text{N}}^{(t)}$ and $\text{N}_{\text{N}}^{(t)}$, and misclassification instances $\text{M}_{\text{N}}^{(t)}$. The first $t$-1 part of STWD utilizes the strategy of three ways, and the last part applies the strategy of two ways. Definition 2.: Suppose $T$ has $t$ granular levels, the result cost matrix for the $i$-th ($1\leq i\leq t$) granular level is \[\mathbf{\Lambda}^{(i)}=\begin{pmatrix}\lambda^{(i)}_{PP}&\lambda^{(i)}_{BP}& \lambda^{(i)}_{NP}\\ \lambda^{(i)}_{PN}&\lambda^{(i)}_{BN}&\lambda^{(i)}_{NN}\end{pmatrix}\] where $\lambda^{(i)}_{PP}$, $\lambda^{(i)}_{BP}$, and $\lambda^{(i)}_{NP}$ are the result costs of taking actions $a_{P}$, $a_{B}$, and $a_{N}$ for the positive labels, respectively; while $\lambda^{(i)}_{PN}$, $\lambda^{(i)}_{BN}$, and $\lambda^{(i)}_{NN}$ are the costs of taking actions $a_{P}$, $a_{B}$, and $a_{N}$ for the negative labels, respectively. It should be noted that the various elements of $\mathbf{\Lambda}^{(i)}$ at $i$-th level need to satisfy $0\leq\lambda^{(i)}_{PP}<\lambda^{(i)}_{BP}<\lambda^{(i)}_{NP}<1$, $0\leq\lambda^{(i)}_{NN}<\lambda^{(i)}_{BN}<\lambda^{(i)}_{PN}<1$, and $(\lambda^{(i)}_{BN}-\lambda^{(i)}_{NN})\times(\lambda^{(i)}_{BP}-\lambda^{(i) }_{PP})$. Example 1.: _Suppose STWD-SFNN has three granular levels, i.e., $t$ = 3. According to the conditions satisfied by the loss function, i.e., $0\leq\lambda^{(i)}_{PP}<\lambda^{(i)}_{BP}<\lambda^{(i)}_{NP}<1$, $0\leq\lambda^{(i)}_{NN}<\lambda^{(i)}_{BN}<\lambda^{(i)}_{PN}<1$, and $(\lambda^{(i)}_{BN}-\lambda^{(i)}_{NN})\times(\lambda^{(i)}_{BP}-\lambda^{(i) }_{PP})<(\lambda^{(i)}_{PN}-\lambda^{(i)}_{BN})\times(\lambda^{(i)}_{NP}-\lambda ^{(i)}_{BP})$, we can achieve the results by randomizing the elements of the result cost matrix in the range of [0,1) at the first granular level, i.e., $\mathbf{\Lambda}^{(1)}=\begin{bmatrix}0&0.1506&0.9021\\ 0.4592&0.1249&0\end{bmatrix}$. Meanwhile, the result costs at the other two granular levels are $\mathbf{\Lambda}^{(2)}=\begin{bmatrix}0&0.4617&0.5962\\ 0.6740&0.1344&0\end{bmatrix}$ and $\mathbf{\Lambda}^{(3)}=\begin{bmatrix}0&0.3626&0.7064\\ 0.7664&0.3727&0\end{bmatrix}$, respectively._ Definition 3.: Suppose $T$ has $t$ granular levels, the thresholds $(\alpha_{i},\beta_{i})$ of the $i$-th ($1\leq i\leq t-1$) level are \[\alpha_{i} =\frac{\lambda^{(i)}_{PN}-\lambda^{(i)}_{BN}}{(\lambda^{(i)}_{PN} -\lambda^{(i)}_{BN})+(\lambda^{(i)}_{BP}-\lambda^{(i)}_{PP})} \tag{16}\] \[\beta_{i} =\frac{\lambda^{(i)}_{BN}-\lambda^{(i)}_{NN}}{(\lambda^{(i)}_{BN }-\lambda^{(i)}_{NN})+(\lambda^{(i)}_{NP}-\lambda^{(i)}_{BP})} \tag{17}\] Definition 4.: Suppose $T$ has $t$ granular levels, the threshold $\gamma$ of the $t$-th granular level is \[\gamma=\frac{\lambda^{(t)}_{PN}-\lambda^{(t)}_{NN}}{(\lambda^{(t)}_{PN}- \lambda^{(t)}_{NN})+(\lambda^{(i)}_{NP}-\lambda^{(i)}_{PP})} \tag{18}\] Example 2.: _In this example, we adopt the result cost matrices in Example 1. According to Definitions 3 and 4, the thresholds of STWD-SFNN at each granular level are ($\alpha_{1},\beta_{1}$) = (0.6894, 0.1425), ($\alpha_{2},\beta_{2}$) = (0.5389, 0.5016), and $\gamma$ = 0.5204, respectively. Since the thresholds of STWD-SFNN satisfy $\beta_{1}<\beta_{2}<\gamma<\alpha_{2}<\alpha_{1}$, we retain the result cost matrices of STWD-SFNN at each granular level. It should be noted that if the thresholds of STWD-SFNN do not meet the sequential property at a certain granular level, we need to reinitialize the result cost matrix at the corresponding level._ Definition 5.: Suppose $T$ has $t$ granular levels, the regions of POS, BND, and NEG after $t$ turns are denoted as follows \[\text{POS}=\bigcup_{i=1}^{t}\lvert\text{IP}_{\text{N}}^{(i)}\cup \text{P}_{\text{L}}^{(i)}\rvert \tag{19}\] \[\text{BND}=\text{M}_{\text{N}}^{(t)}\cup\text{B}_{\text{L}}^{(t)}\] (20) \[\text{NEG}=\bigcup_{i=1}^{t}\lvert\text{N}_{\text{N}}^{(t)}\cup \text{N}_{\text{L}}^{(t)}\rvert \tag{21}\] where $\text{P}_{\text{N}}^{(t)}$ and $\text{P}_{\text{L}}^{(t)}$ represent the set of instances classified into POS by SFNN and STWD at the $i$-th turn, respectively; $\text{M}_{\text{N}}^{(t)}$ and $\text{B}_{\text{L}}^{(t)}$ represent the set of instances misclassified by SFNN and the set of instances partitioned into BND by STWD at the $t$-th turn, respectively; $\text{N}_{\text{N}}^{(i)}$ and $\text{N}_{\text{L}}^{(i)}$ represent the set of instances classified into NEG by SFNN and STWD at the $i$-th turn, respectively. Since the threshold $\gamma$ corresponding to the $t$-th level can classify the instances of $\text{B}_{\text{L}}^{(t)}$ into $\text{P}_{\text{L}}^{(t)}$ and $\text{N}_{\text{L}}^{(t)}$, we have $\text{BND}=\varnothing$ at the $t$-th turn. To describe the process costs of granularity [16, 45], STWD-SFNN considers the test cost and delay cost [11, 37, 39]. Definition 6.: Suppose $T$ has $t$ granular levels, the process cost at the $t$-th level is marked as $Cost_{\text{P}_{i}}$, and \[Cost_{\text{P}_{i}}=(Cost_{\text{PT}_{i}},Cost_{\text{PD}_{i}}) \tag{22}\] \[Cost_{\text{PT}_{i}}=\sum_{i=1}^{t}m_{i}\times Cost_{\text{PT}_{i}} \tag{23}\] \[Cost_{\text{PD}_{i}}=\max_{i=1,2,\cdots,t}m_{i}\times Cost_{\text{PPD}_{i}} \tag{24}\] where $Cost_{\text{PT}_{i}}$ and $Cost_{\text{PD}_{i}}$ represent the test cost and delay cost at the $t$-th granular level, respectively; $m_{i}$ is the number of instances at the $i$-th ($1\leq i\leq t$) level and $m_{i}>0$; $Cost_{\text{PPT}_{i}}$ and $Cost_{\text{PPT}_{i}}$ are the unit test cost and the unit delay cost at the $i$-th ($1\leq i\leq t$) level, respectively. It should be noted that when BND is not empty, the model continues to add a new turn of learning which produces the test and delay costs. Since the costs of test and delay are both positive real numbers and $m_{i}>0$, the unit test cost and the unit delay cost need to satisfy $Cost_{\text{PPT}_{i}}>0$ and $Cost_{\text{PPD}_{i}}>0$, respectively. Meanwhile, since the dataset has no data on unit test cost and unit delay cost, and $m_{i}$ gradually decreases as the model learns, to facilitate the observation of the more general correlation between the costs of test and delay and the granular layers, we can generate unit test costs and unit delay costs randomly and incrementally in the range of positive real numbers. Example 3.: _Suppose we have a two-class classification dataset with 10 instances shown in Table 1, where the training data is $\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\}$, the validation data is $\{x_{7},x_{8}\}$, and the testing data is $\{x_{9},x_{10}\}$. Suppose STWD-SFNN has at most three granular levels, and the unit test cost and unit delay cost at each level are $[Cost_{\text{PPT}_{1}},Cost_{\text{PPT}_{2}},Cost_{\text{PPT}_{3}}]=[1,2,3]$ and $[Cost_{\text{PPD}_{1}},Cost_{\text{PPD}_{2}},Cost_{\text{PPD}_{3}}]=[1,2,3]$, respectively. The calculation of test cost and delay cost is as follows. The instances classified into $\text{P}_{\text{L}}^{(1)}$, $\text{B}_{\text{L}}^{(1)}$, and $\text{N}_{\text{L}}^{(1)}$ at the first granular level are $\varnothing$, $\{x_{4},x_{5}\}$, and $\{x_{1}\}$, respectively. Meanwhile, suppose the two instances $\{x_{4},x_{5}\}$ are partitioned into $\text{N}_{\text{L}}^{(2)}$ at the second level (Details will be shown in Section 3.3). Thus, STWD-SFNN has two granular levels, since $\text{B}_{\text{L}}^{(2)}$ is empty. Hence, $Cost_{\text{PT}_{1}}$= 3 $\times$ 1 = 3 and $Cost_{\text{PD}_{1}}$ = 3 $\times$ 1 = 3 at the first level according to Definition 6. Similarly, $Cost_{\text{PT}_{2}}$ = 3 $\times$ 1 + 2 $\times$ 2 = 7 and $Cost_{\text{PD}_{2}}$ = max (3 $\times$ 1, 2 $\times$ 2) = 4 at the second level._ ### STWD-SFNN model To prevent the overfitting of STWD-SFNN, we consider the $L_{2}$ regularization factor of the loss function in SFNN and the penalty factor $\varepsilon$ of the risk function in STWD. Hence, STWD-SFNN has three parts: preprocessing of data, SFNN, and STWD. #### 3.2.1 Preprocessing of data The instances processed by STWD [41] are usually categorical data, such as teachers and doctors. However, numerical data, such as the temperature and humidity of the weather, need to be processed to meet the calculation requirements of STWD. Therefore, $k$-means++ [18] is adopted to discretize the numerical data into the categorical data. Firstly, we randomly select one instance as the initial clustering center. Secondly, we calculate the distance between the cluster center and the remaining instances, and calculate the probability that each instance is selected as the next cluster center. Then, the next clustering center is selected according to the roulette method, and the discretization process is stopped until $k$ clustering centers are found. Finally, the instances in each cluster are discretized into the same category. \begin{table} \begin{tabular}{c c c c c c} \hline & $a_{1}$ & $a_{2}$ & $a_{3}$ & $a_{4}$ & D \\ \hline $x_{1}$ & 0.7415 & 0.5407 & 0.5795 & 0.9009 & 2 \\ $x_{2}$ & 0.6844 & 0.3210 & 0.0471 & 0.3700 & 1 \\ $x_{3}$ & 0.7718 & 0.0912 & 0.4874 & 0.5308 & 1 \\ $x_{4}$ & 0.0818 & 0.4263 & 0.0354 & 0.0621 & 1 \\ $x_{5}$ & 0.5596 & 0.4643 & 0.3585 & 0.3189 & 2 \\ $x_{6}$ & 0.6397 & 0.6535 & 0.7739 & 0.6809 & 1 \\ $x_{7}$ & 0.7425 & 0.0989 & 0.7429 & 0.4131 & 1 \\ $x_{8}$ & 0.9419 & 0.5958 & 0.4474 & 0.7536 & 2 \\ $x_{9}$ & 0.4992 & 0.2212 & 0.9525 & 0.4176 & 1 \\ $x_{10}$ & 0.2990 & 0.4796 & 0.1559 & 0.7456 & 2 \\ \hline \end{tabular} \end{table} Table 1: Two-class dataset #### 3.2.2 Sfnn Suppose STWD-SFNN contains $u$ ($1\leq u\leq d$) unclassified instances. Firstly, we randomly generate the network parameters with $i$-th hidden layer node, including $\mathbf{w}_{1}^{(i)}$, $\mathbf{b}_{1}^{(i)}$, $\mathbf{w}_{2}^{(i)}$, and $\mathbf{b}_{2}^{(i)}$. To guarantee the repeatability of the experimental results, we set the seed of the random number generator to mg(0). Secondly, we adopt the focal loss function to guide the forward propagation learning, and calculate the learning results with the current network parameters. Thirdly, we apply the Adam optimizer to guide the error backpropagation, adaptively find the optimal network parameters, including $\mathbf{\overline{w}}_{1}^{(i)}$, $\mathbf{\overline{b}}_{1}^{(i)}$, $\mathbf{\overline{w}}_{2}^{(i)}$, and $\mathbf{\overline{b}}_{2}^{(i)}$. Finally, STWD-SFNN with topology of $i$-th hidden layer node yields the positive instances $\mathrm{P}_{\mathrm{N}}^{(i)}$, the misclassification instances $\mathrm{M}_{\mathrm{N}}^{(i)}$, and the negative instances $\mathrm{N}_{\mathrm{N}}^{(i)}$, where the misclassification instances are also called the difficult-to-classify instances. #### 3.2.3 Stwd For the difficult-to-classify instances in the network, we first count the label equivalence class $Y$ that the instances belong to the positive labels and the instance equivalence class $[x]_{e}^{(i)}$. Next, we calculate the conditional probability that the instances belong to the positive labels according to Equation (25). \[p_{e}^{(i)}=\frac{\left\|Y\cap[x]_{e}^{(i)}\right\|}{\left\|[x]_{e}^{(i)}\right\|} \tag{25}\] In addition, according to Definitions 2 and 3, we can index the result cost matrix $\mathbf{\Lambda}^{(i)}$ and the thresholds $(\alpha_{i},\beta_{i})$ at the $i$-th ($1\leq i\leq t-1$) level. Since the risk varies in different regions, STWD-SFNN considers the total decision risk associated with classifying $u$ instances. The risk is calculated according to Equation (26). \[\begin{split} Risk^{(i)}&=\min_{(\alpha_{i},\beta_{ i})}\Bigg{(}\sum_{\stackrel{{ m=1}}{{p_{e}^{(i)}\times p_{e}^{(i)}}}}^{u_{p}}\lambda_{PP}^{(i)}\times p _{o}^{(i)}+\lambda_{PN}^{(i)}\times(1-p_{o}^{(i)})\\ &+\varepsilon\times\sum_{\stackrel{{ i=1}}{{p_{e}^{(i)}\times p_{e}^{(i)}}}}^{u_{b}}\lambda_{ BP}^{(i)}\times p_{k}^{(i)}+\lambda_{BN}^{(i)}\times(1-p_{k}^{(i)})\\ &\qquad+\sum_{\stackrel{{ j=1}}{{p_{j}^{(i)}\times p _{k}}}}^{u_{n}}\lambda_{NP}^{(i)}\times p_{j}^{(i)}+\lambda_{NN}^{(i)}\times(1- p_{j}^{(i)})\Bigg{)}\\ \text{s.t.}&\quad 0<\beta_{i}<\alpha_{i}<1,1\leq i \leq t-1,\varepsilon\geq 1,u=u_{p}+u_{b}+u_{n}\end{split} \tag{26}\] where $u_{p}$, $u_{b}$, and $u_{n}$ are the numbers of instances in $\mathrm{P}_{\mathrm{L}}^{(i)}$, $\mathrm{B}_{\mathrm{L}}^{(i)}$, and $\mathrm{N}_{\mathrm{L}}^{(i)}$, respectively; $\varepsilon$ is a penalty factor to avoid too many instances being partitioned into $\mathrm{B}_{\mathrm{L}}^{(i)}$, so as to prevent the overfitting of STWD-SFNN. Therefore, for the $e$-th instance at the $i$-th ($1\leq i\leq t-1$) granular level, the classification criteria are as follows. (P1) if $p_{e}^{(i)}\geq\alpha_{i}$, then $[x]_{e}^{(i)}\in\mathrm{P}_{\mathrm{L}}^{(i)}$; (B1) if $\beta_{i}<p_{e}^{(i)}<\alpha_{i}$, then $[x]_{e}^{(i)}\in\mathrm{B}_{\mathrm{L}}^{(i)}$; (N1) if $p_{e}^{(i)}\leq\beta_{i}$, then $[x]_{e}^{(i)}\in\mathrm{N}_{\mathrm{L}}^{(i)}$. If BND of the ($t$-1)-th level is not empty, we need to add another hidden layer node in STWD-SFNN. Based on Definition 4, we index the result cost matrix $\mathbf{\Lambda}^{(t)}$ and the threshold $\gamma$ at the $t$-th level. STWD-SFNN computes the total decision risk of classifying $u$ instances. \[\begin{split} Risk^{(t)}&=\min_{\gamma}\Bigg{(}\sum_{ \stackrel{{ m=1}}{{p_{e}^{(i)}\times p}}}^{u_{p}}\lambda_{PP}^{(t)} \times p_{o}^{(t)}+\lambda_{PN}^{(t)}\times(1-p_{o}^{(t)})\\ &\qquad+\sum_{\stackrel{{ j=1}}{{p_{j}^{(i)}\times p }}}^{u_{n}}\lambda_{NP}^{(t)}\times p_{j}^{(t)}+\lambda_{NN}^{(t)}\times(1-p_{ j}^{(t)})\Bigg{)}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{s.t.}\quad 0< \gamma<1,u=u_{p}+u_{n}\end{split} \tag{27}\] where $u_{p}$ and $u_{n}$ are the numbers of instances in $\mathrm{P}_{\mathrm{L}}^{(t)}$ and $\mathrm{N}_{\mathrm{L}}^{(t)}$, respectively. Hence, for the $e$-th instance at the $t$-th level, the classification criteria are as follows. (P2) if $p_{e}^{(t)}\geq\gamma$, then $[x]_{e}^{(t)}\in\mathrm{P}_{\mathrm{L}}^{(t)}$; (N2) if $p_{e}^{(t)}<\gamma$, then $[x]_{e}^{(t)}\in\mathrm{N}_{\mathrm{L}}^{(t)}$. Example 4.: _We adopt the sequential thresholds of Example 2 and the dataset of Example 3. Suppose the first granular level learns the three misclassified instances $[x_{1},x_{4},x_{5}]$ passed from SFNN, which are partitioned into $\mathrm{P}_{\mathrm{L}}^{(1)}=\varnothing$, $\mathrm{B}_{\mathrm{L}}^{(1)}=\{x_{4},x_{5}\}$, and $\mathrm{N}_{\mathrm{L}}^{(1)}=\{x_{1}\}$. Moreover, suppose the conditional probabilities at the first level are $[p_{1}^{(1)},p_{2}^{(1)}]=[0,0.5]$, according to Example 2 and Equation (3), we have the result cost Risk${}^{(1)}$ = 0 + $\lambda_{\text{stud}}$$\times$ 2 $\times$ (0.1506 $\times$ 0.5 + 0.1249 $\times$ 0.5) + 1 $\times$ (0.9021 $\times$ 0 + 0 $\times$ 1) = 0.5510, where $\lambda_{\text{stud}}$ = 2. Similarly, suppose the second level classifies $\{x_{4},x_{5}\}$ into $\mathrm{N}_{\mathrm{L}}^{(2)}$ with the conditional probability 0.5. Thus, Risk${}^{(2)}$ = 0 + 0 + 2 $\times$ (0.5962 $\times$ 0.5 + 0 $\times$ 0.5) = 0.5962._ Algorithm 1 shows the pseudocode of STWD-SFNN. ``` 1:$T=(U,A=C\cup D,V=\{V_{a}\mid a\in A\},I=\{I_{a}\mid a\in A\},H)$ 2:$\mathbf{Output}$: hidden layer nodes $i$, $\mathbf{W}_{1}$ and $\mathbf{b}_{1}$ connected the input layer to the hidden layer, and $\mathbf{W}_{2}$ and $\mathbf{b}_{2}$ connected the hidden layer to the output layer 3: Initialization: rng(0), $i\gets 1$, $t\gets 10$, $\theta\gets 2$, $\rho_{1}\gets 0.9$, $\rho_{2}\gets 0.999$, $\tau\gets 10^{-8}$, $\mathbf{w}_{1}^{(t)}$, $\mathbf{b}_{1}^{(0)}$, $\mathbf{w}_{2}^{(t)}$, $\mathbf{b}_{2}^{(t)}$, activation function (ReLU, leaky ReLU, SeLU, tanh, sigmoid, or swish), distribution of parameters (uniform or normal), POS $\leftarrow\varnothing$, NEG $\leftarrow\varnothing$, BND $\gets T$, $Cost_{\text{PPT}_{i}}$, $Cost_{\text{PPD}_{i}}$, $\Lambda^{(0)}$, ($\alpha_{i}$,$\beta_{i}$), and $\gamma$; 4:while $\|$ BND $\|\neq\varnothing$do // Construct SFNN by adding one hidden layer node 5: Calculate the cost function using Equation (2); 6: Update $\overline{\mathbf{w}}_{1}^{(t)}$, $\overline{\mathbf{w}}_{1}^{(t)}$, $\overline{\mathbf{w}}_{2}^{(t)}$, and $\overline{\mathbf{b}}_{2}^{(t)}$ using Equation (3); 7: Get $\mathrm{P}_{\mathrm{N}}^{(t)}$, $\mathrm{N}_{\mathrm{N}}^{(t)}$, and $\mathrm{N}_{\mathrm{N}}^{(t)}$ of SFNN; // Construct STWD on the misclassified instances 8: Use $k$-means++ to discretize the numerical data into categorical data; // Preprocessing of data 9: Calculate probability $p_{e}^{(t)}$ using Equation (25); 10:if$i<t$then 11: Achieve result cost matrix $\Lambda^{(i)}$, thresholds ($\alpha_{i}$, $\beta_{i}$), and regions $\mathrm{P}_{\mathrm{L}}^{(t)}$, $\mathrm{B}_{\mathrm{L}}^{(t)}$, and $\mathrm{N}_{\mathrm{L}}^{(t)}$ of STWD; 12: Compute $Risk^{(i)}$ using Equation (26); 13:else 14: Achieve result cost matrix $\Lambda^{(i)}$, threshold $\gamma$, and regions $\mathrm{P}_{\mathrm{L}}^{(t)}$ and $\mathrm{N}_{\mathrm{L}}^{(t)}$ of STWD; 15: Compute $Risk^{(i)}$ using Equation (27); 16:endif 17: Compute $Cost_{\text{PPT}_{i}}$ and $Cost_{\text{PPD}_{i}}$ using Definition 6; 18: Update POS, BND, and NEG using Definition 5; 19:$i\gets i+1$; 20:endwhile 21:$\mathbf{W}_{1}=\begin{bmatrix}\overline{\mathbf{w}}_{1}^{(1)}\\ \cdots\\ \overline{\mathbf{w}}_{1}^{(0)}\end{bmatrix}$, $\mathbf{b}_{1}=\begin{bmatrix}\overline{\mathbf{b}}_{1}^{(1)}\\ \cdots\\ \overline{\mathbf{b}}_{1}^{(0)}\end{bmatrix}$, $\mathbf{W}_{2}=\begin{bmatrix}\overline{\mathbf{w}}_{2}^{(1)}\cdots\overline{ \mathbf{w}}_{2}^{(t)}\end{bmatrix}$, and $\mathbf{b}_{2}=\overline{\mathbf{b}}_{2}^{(t)}$; 22:return$i$, $\mathbf{W}_{1}$, $\mathbf{b}_{1}$, $\mathbf{W}_{2}$, and $\mathbf{b}_{2}$; ``` Algorithm 1 STWD-SFNN. According to Algorithm 1, there are seven parameters in STWD-SFNN. rng(0) is used to guarantee the repeatability of the experimental results. $i$ is utilized to guarantee that the number of hidden nodes increases one by one. $t$ is applied to guarantee the maximum number of hidden nodes. If the number of hidden nodes is small, it means that STWD-SFNN converges faster, and the training time of STWD-SFNN is shorter; otherwise, the training time of STWD-SFNN is longer. $\theta$ is set to adjust the weight reduction rate of easy-to-classify instances. $\rho_{1}$ and $\rho_{2}$ are set to guarantee the exponential decay rate of the first-order and second-order momentum estimates of the Adam optimizer. $\tau$ is settled to avoid the denominator of Adam being zero. ### An illustrative example A dataset in Table 1 is adopted to illustrate the principle of STWD-SFNN. Without loss of generality, we select the result cost matrix and sequential thresholds of Examples 1 and 2. The illustrative example is shown as follows. We initialize the number of hidden nodes to one, and randomize the parameters subjected to uniform distribution, including $\mathbf{w}_{1}^{(1)}=(0.9168,-1.1743,0.4410,0.0746)$, $\mathbf{b}_{1}^{(1)}=(0.0047)$, $\mathbf{w}_{2}^{(1)}=(0.1401,0.0087)^{\top}$, and $\mathbf{b}_{2}^{(1)}=(0.0018,0.0082)^{\top}$. Meanwhile, we adopt selu function and Adam optimizer, and then validate the classification performance of the model on $\{x_{7},x_{8}\}$. Correspondingly, the optimized parameters are $\overline{\mathbf{w}}_{1}^{(1)}=(0.8115,-1.0612,0.3465,0.1514)$, $\overline{\mathbf{b}}_{1}^{(1)}=(0.1139)$, $\overline{\mathbf{w}}_{2}^{(1)}=(0.2019,0.0860)^{\top}$, and $\overline{\mathbf{b}}_{2}^{(1)}=(0.1110,1177)^{\top}$. According to $\mathbf{\hat{y}}=\overline{\mathbf{w}}_{2}^{(1)}\times\text{selu}(\overline {\mathbf{w}}_{1}^{(1)}\times\mathbf{x}^{\top}+\overline{\mathbf{b}}_{1}^{(1)} +\overline{\mathbf{b}}_{2}^{(1)}$, we have the prediction labels $[1,1,1,2,1,1]$ of SFNN in training dataset. Compared with the truth labels $[2,1,1,1,2,1]$, we have $\text{P}_{\text{N}}^{(1)}=\{x_{2},x_{3},x_{6}\}$, $\text{M}_{\text{N}}^{(1)}=\{x_{1},x_{4},x_{5}\}$, and $\text{N}_{\text{N}}^{(1)}=\varnothing$. Since Table 1 is numerical data, we discretize $\{x_{1},x_{4},x_{5}\}$ into $\{[x_{1}\},\{x_{4},x_{5}\}]$ by $k$-means$++$ with two clusters. The conditional probabilities are 0 and 0.5, since the label equivalence class is $\{[x_{4}\},\{x_{1},x_{5}\}]$. Hence, STWD-SFNN has $\text{P}_{\text{L}}^{(1)}=\varnothing$, $\text{B}_{\text{L}}^{(1)}=\{x_{4},x_{5}\}$, and $\text{N}_{\text{L}}^{(1)}=\{x_{1}\}$ at the first level. According to Examples 3 and 4, we have $Risk^{(1)}=0.5510$, $Cost_{\text{P}_{1}}=3$, and $Cost_{\text{P}_{\text{D}_{1}}}=3$. Simultaneously, the decision of STWD-SFNN in the first turn are $\text{P}_{\text{N}}^{(1)}\cup\text{P}_{\text{L}}^{(1)}=\{x_{2},x_{3},x_{6}\}$, $\text{M}_{\text{N}}^{(1)}\cap\text{B}_{\text{L}}^{(1)}=\{x_{4},x_{5}\}$, and $\text{N}_{\text{N}}^{(1)}\cup\text{N}_{\text{L}}^{(1)}=\{x_{1}\}$. Since $\text{M}_{\text{N}}^{(1)}\cap\text{B}_{\text{L}}^{(1)}$ is not empty, we continue to add one hidden node in STWD-SFNN. The parameters to be optimized are $\mathbf{w}_{1}^{(2)}=(-0.3114,-0.2530,1.0117,0.3236)$, $\mathbf{b}_{1}^{(2)}=(0.0035)$, $\mathbf{w}_{2}^{(2)}=(0.1034,-0.0356)^{\top}$, and $\mathbf{b}_{2}^{(2)}=(0.0018,0.0082)^{\top}$. After optimization, we have $\overline{\mathbf{w}}_{1}^{(2)}=(-0.2338,-0.1741,0.9333,0.2477)$, $\overline{\mathbf{b}}_{1}^{(2)}=(0.0818)$, $\overline{\mathbf{w}}_{2}^{(2)}=(0.1343,0.0133)^{\top}$, and $\overline{\mathbf{b}}_{2}^{(2)}=(0.0768,0.0821)^{\top}$. STWD-SFNN has $\text{P}_{\text{N}}^{(2)}=\varnothing$, $\text{M}_{\text{N}}^{(2)}=\{x_{4},x_{5}\}$, and $\text{N}_{\text{N}}^{(2)}=\varnothing$. Based on the instance equivalent class $\{x_{4},x_{5}\}$, label equivalent class $\{[x_{4}\},\{x_{5}\}]$, and the conditional probability 0.5, we have $\text{P}_{\text{L}}^{(2)}=\varnothing$, $\text{B}_{\text{L}}^{(2)}=\varnothing$, and $\text{N}_{\text{L}}^{(2)}=\{x_{4},x_{5}\}$. Similarly, according to Examples 3 and 4, we have $Risk^{(2)}=0.5962$, $Cost_{\text{P}_{2}}=7$, and $Cost_{\text{P}_{\text{D}_{2}}}=4$. Meanwhile, the decisions for STWD-SFNN in the second turn are $\text{P}_{\text{N}}^{(2)}\cup\text{P}_{\text{L}}^{(2)}=\varnothing$, $\text{M}_{\text{N}}^{(2)}\cap\text{B}_{\text{L}}^{(2)}=\varnothing$, and $\text{N}_{\text{N}}^{(2)}\cup\text{N}_{\text{L}}^{(2)}=\{x_{4},x_{5}\}$. The learning of STWD-SFNN is stopped, since $\text{M}_{\text{N}}^{(2)}\cap\text{B}_{\text{L}}^{(2)}$ is empty. According to Definition 5, the decisions of STWD-SFNN with two hidden layer nodes are POS=$\{x_{2},x_{3},x_{6}\}$, BND=$\varnothing$, and NEG=$\{x_{1},x_{4},x_{5}\}$. Meanwhile, the model is $\mathbf{\hat{y}}=\mathbf{W}_{2}\times\text{selu}(\mathbf{W}_{1}\times\mathbf{x}^ {\top}+\mathbf{b}_{1})+\mathbf{b}_{2}$, where $\mathbf{W}_{1}=\begin{bmatrix}\overline{\mathbf{w}}_{1}^{(1)}\\ \overline{\mathbf{w}}_{2}^{(2)}\end{bmatrix}=\begin{bmatrix}0.8115,-1.0612,0.346 5,0.1514\\ -0.2338,-0.1741,0.9333,0.2477\end{bmatrix}$, $\mathbf{b}_{1}=\begin{bmatrix}\overline{\mathbf{b}}_{1}^{(1)}\\ \overline{\mathbf{b}}_{1}^{(2)}\end{bmatrix}=\begin{bmatrix}0.1139\\ 0.0818\end{bmatrix}$, $\mathbf{W}_{2}=\begin{bmatrix}\overline{\mathbf{w}}_{2}^{(1)},\overline{\mathbf{w}}_ {2}^{(2)}\end{bmatrix}=\begin{bmatrix}0.2019,0.1343\\ 0.0860,0.0133\end{bmatrix}$, and $\mathbf{b}_{2}=\overline{\mathbf{b}}_{2}^{(2)}=\begin{bmatrix}0.0768\\ 0.0821\end{bmatrix}$. Finally, we obtain the predicted value $\mathbf{\hat{y}}$ when applying STWD-SFNN to $\{x_{9},x_{10}\}$. Since the thresholds of TWD-SFNN are relatively fixed, compared with STWD-SFNN, there are two exceptional cases, i.e., the coarsest thresholds and the finest thresholds. Case 1 (The coarsest): If the thresholds are $(\alpha,\beta)=(\alpha_{1},\beta_{1})=(0.6894,0.1425)$, TWD-SFNN will continue to learn the network topology when the nodes of hidden layer are two. Case 2 (The finest): If the thresholds are $(\alpha,\beta)=(\alpha_{2},\beta_{2})=(0.5389,0.5016)$, TWD-SFNN will stop the training of model early. However, STWD-SFNN adopts sequential thresholds to traverse from coarse-grained level to fine-grained level step by step, thus better handling the problem of network topology. Therefore, STWD-SFNN is a generalized model of TWD-SFNN. ### Theoretical analysis of STWD-SFNN Theorem 1.: _STWD-SFNN algorithm is convergent._ Proof.: (Proof by Contradiction) Suppose STWD-SFNN completes the learning of instances in infinite turns, it means that STWD-SFNN has $u$ ($u\geq 1$) instances which always are classified into BND. STWD-SFNN needs to compare probability $p_{e}^{(t)}$ of the $e$-th ($1\leq e\leq u$) instance with thresholds $(\alpha_{t},\beta_{t})$ at the $t$-th($t\geq 1$) level. If $p_{e}^{(t)}$ is larger than $\alpha_{t}$, then the $e$-th ($1\leq e\leq u$) instance is classified into POS; if $p_{e}^{(t)}$ is less than $\beta_{t}$, then it is classified into NEG. If $p_{e}^{(t)}$ is in $[\beta_{t},\alpha_{t}]$, then it is classified into BND. The probability of the $e$-th instance falls into the range $\alpha_{t}-\beta_{t}$, and hence the probability for \ ## 4 Experiments To verify the classification performance of STWD-SFNN, we selected ten classification models: $m_{1}$-SFNN-R, $m_{2}$-SFNN-R, $m_{3}$-SFNN-R [5], Grid search optimization of SFNN (GS-SFNN) [26], Particle swarm optimization of SFNN (PSO-SFNN) [24], TWD-SFNN-R [7], STWD-SFNN-NK, support vector classification (SVC) [20], random forest (RF) [13], and k-nearest neighbors (KNN) [12]. These models are used as the competitive models to evaluate the performance of our algorithm from the following six research questions (RQs). RQ 1: How does STWD-SFNN perform compared with static topology discovery without STWD? RQ 2: How does STWD-SFNN perform compared with choosing a dynamic tuning model without STWD? RQ 3: How does STWD-SFNN perform compared with employing TWD to find network topology? RQ 4: How does STWD-SFNN perform compared with the model without $k$-means++ discretization? RQ 5: How does STWD-SFNN perform compared with other classification models? RQ 6: How is the process costs of STWD-SFNN related to network topology? To handle RQ 1, we utilize the SFNN determined by the empirical formula models in Section 4.4.1. In response to RQ 2, we employ grid search and particle swarm optimization to dynamically select the network topology in Section 4.4.2. To address RQ 3, we utilize TWD-SFNN in Sections 4.4.2 and 4.4.4. To answer RQ 4, we adopt STWD-SFNN without $k$-means++ as a competitive model in Section 4.4.2. To answer RQ 5, we adopt SVC, RF, and KNN to verify the generalization ability of STWD-SFNN in section 4.4.3. To handle RQ 6, we analyze the STWD-SFNN in Section 4.4.5. ### Benchmark datasets and evaluation criteria To report the performance of STWD-SFNN, we selected 15 commonly used datasets which can be downloaded from [https://archive.ics.uci.edu/ml/index.php](https://archive.ics.uci.edu/ml/index.php). The characteristics of these datasets are shown in Table 2. All experiments were performed on a computer equipped with an Intel (R) Core (TM) i7-10700K CPU, 16 GB RAM, and a 64-bit version of Windows 10. All models were programmed in Matlab, which can be downloaded from [https://github.com/wuc567/Machine-learning/tree/main/STWD-SFNN](https://github.com/wuc567/Machine-learning/tree/main/STWD-SFNN). We adopted the weighted-f1 metric [7; 29] to evaluate the performance, since weighted-f1 handles the problem of unbalanced classification data and considers the importance of different categories in classification data. Weighted-f1 can be defined as follows. \begin{table} \begin{tabular}{c c c c} \hline Name of datasets & Abbreviation & Number of instances & Number of attributes \\ \hline Online news popularity & ONP & 39,797 & 61 \\ QSAR oral toxicity & QSAR & 8,992 & 1,024 \\ Online shoppers purchasing intention & OSP & 12,330 & 18 \\ Electrical grid stability simulated & EGSS & 10,000 & 14 \\ Skin segmentation & SE & 245,057 & 4 \\ HTRU & HTRU & 17,898 & 9 \\ Default of credit card clients & DCC & 30,000 & 24 \\ Epileptic seizure recognition & ESR & 11,500 & 179 \\ Bank marketing & BM & 45,211 & 17 \\ Polish companies’ bankruptcy & PCB & 10,503 & 64 \\ Shill bidding & SB & 6,321 & 13 \\ Estimation of obesity levels & EOL & 2,111 & 17 \\ Occupancy detection & OD & 20,560 & 7 \\ Room occupancy estimation & ROE & 10,129 & 16 \\ Sepsis survival minimal clinical records & SSMCR & 110,204 & 4 \\ \hline \end{tabular} \end{table} Table 2: Benchmark datasets \[\mathrm{weighted-f1}=\frac{\sum_{i=1}^{2}\mid d_{i}\midf_{i}}{\mid d\mid} \tag{28}\] \[f_{i}=\frac{2\times Precision_{i}\times Recall_{i}}{Precision_{i}+Recall_{i}} \tag{29}\] \[Precision_{i}=\frac{TP_{i}}{TP_{i}+FP_{i}} \tag{30}\] \[Recall_{i}=\frac{TP_{i}}{TP_{i}+FN_{i}} \tag{31}\] where $\mid d_{i}\mid$ and $\mid d\mid$ represent the numbers of $i$-th label and all label, respectively; $f_{i}$ represents the f1-score of the $i$-th label; $Precision_{i}$ and $Recall_{i}$ are the precision rate and the recall rate of $i$-th label, respectively; $TP_{i}$ and $FP_{i}$ are the numbers of correct and incorrect predictions for instances with positive labels, respectively; and $FN_{i}$ is the number of incorrect predictions for instances with negative labels. The greater the value of weighted-f1, the better the performance of the model. In addition, we employ other evaluation criteria, including the accuracy, training time, test time, number of hidden layer nodes, receiver operating characteristic (ROC) curve, and area under the curve of ROC (AUC), where the training time is the learning time of the model under a specified set of parameters. The higher the accuracy and AUC, the shorter the training time, the shorter the test time, the lower number of hidden layer nodes, and the closer the curve to the upper left corner, the better the performance of the model. ### Baseline models To validate the performance of STWD-SFNN, we compared it with ten competitive models, which are listed as follows. 1. SFNN-R [5]: The network parameters of SFNN are initialized by random numbers, which cannot guarantee the repeatability of the experimental results. To overcome this shortage, we set the seed of the random number generator to $\mathrm{rng}(0)$, and initialize the hyperparameters in a fixed random number manner to guarantee the repeatability of learning results. We name SFNN with the fixed hyperparameter as SFNN-R. SFNN-R adopts pre-determined hidden layer nodes and learns the optimal network parameters through focal loss and Adam optimizer, thus generating a nonlinear decision boundary. Moreover, for SFNN-R, the numbers of hidden nodes determined by the empirical formula methods are $m_{1}$-SFNN-R: $\sqrt{m+n}+\alpha,(\alpha\in(1,10))$, $m_{2}$-SFNN-R: $log_{2}m$, and $m_{3}$-SFNN-R: $\sqrt{m\times n}$, where $m$ and $n$ are the numbers of nodes in the input and output layers, respectively. 2. GS-SFNN [26]: For the hyperparameters of the number of hidden layer nodes in SFNN, we select a common hyperparameters method, i.e., grid search, to find the optimal number of hidden layer nodes of SFNN. 3. PSO-SFNN [24]: To tune the hyperparameters of SFNN, we choose particle swarm optimization, which utilizes collaboration and information sharing among individuals in the group to optimize SFNN. 4. TWD-SFNN-R [7]: The network parameters of TWD-SFNN, the thresholds of three-way decisions, and the cross-validation of the dataset are initialized by random numbers, which cannot guarantee the repeatability of the experimental results. To overcome this shortage, we set the seed of the random number generator to $\mathrm{rng}(0)$, and initialize the hyperparameters in a fixed random number manner to guarantee the repeatability of learning results. We name TWD-SFNN with the fixed hyperparameter as TWD-SFNN-R which adds a delayed decision region to determine the number of hidden layer nodes, and aims to minimize the decision risk, thereby improving the performance of the model. 5. STWD-SFNN-NK: To verify the performance of the STWD-SFNN model by adopting discretization techniques, we compare it with STWD-SFNN without $k$-means++ and name it STWD-SFNN-NK. The parameters affecting STWD-SFNN-NK are consistent with STWD-SFNN. 6. SVC [20]: To obtain the optimal separated hyperplane, SVC converts the maximization problem to a convex quadratic programming optimization problem, and applies the kernel function to learn the nonlinear classifier. 7. RF [13]: To reduce the correlation between decision trees, for the classification problem, RF randomizes features and instances and adopts a voting mechanism to predict the label for each instance. 8. KNN [12]: To learn the similarities between instances, KNN measures the distances between the different eigenvalues and selects the most frequent categories from the nearest $K$ instance as the decision category of the model. ### Preparation of the experiment For the hyperparameters of STWD-SFNN, we set the seed of the random number generator to rng(0) to guarantee the repeatability of the experimental results. The selection of process cost is explained as follows. We know that the unit test cost $Cost_{\text{PPT}_{i}}$ and the unit delay cost $Cost_{\text{PPT}_{i}}$ at each granular level are part of the input of STWD-SFNN. Without loss of generality, we randomly generate two vectors with dimension $t$, one is $[Cost_{\text{PPT}_{1}},Cost_{\text{PPT}_{2}},\cdots,Cost_{\text{PPT}_{t}}]$, and the other is $[Cost_{\text{PPD}_{1}},Cost_{\text{PPD}_{2}},\cdots,Cost_{\text{PPD}_{t}}]$. In addition, the selection of the result cost matrix $\Lambda^{(i)}(1\leq i\leq t)$ is illustrated as follows. Firstly, $\Lambda^{(i)}$ of the $i$-th level is randomly initialized, which needs to satisfy the conditions $0\leq\lambda^{(i)}_{PP}<\lambda^{(i)}_{BP}<\lambda^{(i)}_{NP}<1$, $0\leq\lambda^{(i)}_{NN}<\lambda^{(i)}_{BN}<\lambda^{(i)}_{PN}<1$, and $(\lambda^{(i)}_{BN}-\lambda^{(i)}_{NN})\times(\lambda^{(i)}_{BP}-\lambda^{(i)} _{PP})<(\lambda^{(i)}_{PN}-\lambda^{(i)}_{BN})\times(\lambda^{(i)}_{NP}-\lambda ^{(i)}_{BP})$. Next, the thresholds $(\alpha_{i},\beta_{i})(1\leq i\leq t)$ of the previous $t$-1 levels and the threshold $\gamma$ of the $t$-th level are defined by Definitions 3 and 4, respectively. Finally, for the ($i$+1)-th level, we retain $\Lambda^{(i+1)}$ and $(\alpha_{i+1},\beta_{i+1})$ or $\gamma$ if and only if the thresholds satisfy $\beta_{i}\leq\beta_{i+1}<\gamma\leq\alpha_{i+1}\leq\alpha_{i}$; otherwise, we iterate this process until $\Lambda^{(i+1)}$ is found. It should be noted that whether there is a better hyperparameter selection method to obtain the result cost matrix is worth investigating in the future. The hyperparameters of the competitive models are introduced as follows. For the network models, such as SFNNN-R, GS-SFNN, PSO-SFNN, TWD-SFNN-R, and STWD-SFNN-NK, the activation function selects ReLU family (e.g., ReLU, LReLU, and SELU) or tanh family (e.g., tanh, sigmoid, and swish), and the network parameters subject to uniform distribution or normal distribution. Except for SFNN-R, the number of hidden layer nodes in the rest network model increases from one to ten. Meanwhile, for PSO-SFNN, $max\_iterations\in(10,100)$ with step size ten, and $max\_particles\in(1,10)$ with step size one. On the other hand, for SVC, polynomial, RBF, and sigmoid kernel functions are selected,$\gamma\in(-4,4)$ with step size one, and $C\in(-4,4)$ with step size one. For RF, $max\_trees\in(100,1000)$ with step size 100, $max\_depth\in\{q,log_{2}q,\sqrt{q},\phi q\}$, where $q$ is the number of features, $\phi\in(0,1)$. For KNN, Euclidean distance and angle cosine are selected to characterize the similarity among instances, $K\in(10,100)$ with step size ten. It is worth noting that the dataset is divided into training set, validation set, and test set according to the ratio of 8:1:1, and the 10-fold cross-validation and grid search are adopted to find the optimal hyperparameters of each model. In addition, we utilize the Focal loss and Adam optimizer in Section 2.1 to train the network model, where the regularization factor $\lambda_{\text{SFNN}}$, the learning rate, and batch size of the optimizer are uniformly set to 0.1, 512, and 0.1, respectively. ### Experimental results and analysis #### 4.4.1 Comparison with static models To validate the effectiveness of STWD-SFNN in optimizing the network topology, we select the static models SFNN-R as the competitive models, which adopt three empirical formulas to calculate the number of hidden layer nodes. Fig. 4 shows the comparison of ROC curves of the SFNN-R models and STWD-SFNN. Table 3 reports the comparison of different evaluation criteria of those models. The results give rise to the following observations. 1. The accuracy, weighted-f1, ROC, and AUC of STWD-SFNN are better than those of SFNN-R. For example, on the EOL dataset, Table 3 reports that the accuracy, weighted-f1, and AUC of STWD-SFNN are 83.66%, 81.53%, and 69.78%, respectively. However, the best performance in the SFNN-R models is $m_{3}$-SFNN-R whose accuracy, weighted-f1, and AUC are 75.37%, 75.24%, and 63.70%, respectively. Meanwhile, as shown in Fig. 4, the ROC curve of STWD-SFNN is at the top left of $m_{3}$-SFNN-R. Similar phenomena can be found in other datasets. The reason is as follows. Compared with SFNN-R which only contains two-way decisions with POS and NEG, STWD-SFNN adds the delayed decision region to store the difficult-to-classify instances, and learns the features of these instances more pertinently in each turn, thereby enhancing the performance of STWD-SFNN. Therefore, the generalization ability of STWD-SFNN is better than that of SFNN-R. 2. The number of hidden layer nodes of STWD-SFNN is lower than that of SFNN-R. For example, on the OD dataset, Table 3 reports that the number of hidden layer nodes of STWD-SFNN is two. However, the competitive models perform better in $m_{1}$-SFNN-R and $m_{2}$-SFNN-R, with a value of three. Similar phenomena can be found in other datasets. The reason is as follows. Compared with SFNN-R, which statically determines the network topology, the compactness of network topology of STWD-SFNN depends on the number of instances in BND. As the number of difficult-to-classify instances decreases, the number of hidden layer nodes of STWD-SFNN gradually decreases. Therefore, the network topology of STWD-SFNN is better than that of SFNN-R. Figure 4: Comparison of ROC curves of static models and STWD-SFNN \begin{table} \begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{}{Dataset} & \multirow{2}{}{Model} & \multicolumn{2}{c}{Accuracy} & Weighted- & AUC & Training & \multirow{2}{}{Test time (s)} & \multirow{2}{}{Nodes} \\ & & (\%) & (\%) & (\%) & (\%) & time(s) & & \\ \hline \multirow{4}{}{ONP} & $m_{1}$-SFNN-R & 85.43$\pm$0.37 & 87.42$\pm$0.29 & 54.46 & 41.76$\pm$0.38 & 0.009$\pm$0.000 & 8 \\ & $m_{2}$-SFNN-R & 89.09$\pm$0.21 & 89.29$\pm$0.23 & 56.43 & 34.77$\pm$0.62 & 0.008$\pm$0.001 & 6 \\ & $m_{3}$-SFNN-R & 86.40$\pm$0.30 & 88.02$\pm$0.27 & 56.50 & 32.04$\pm$0.32 & 0.009$\pm$0.005 & 11 \\ & STWD-SFNN & 94.41$\pm$0.22 & 85.87$\pm$0.26 & 57.87 & 15.76$\pm$0.76 & 0.003$\pm$0.000 & 1.00$\pm$0.00 \\ \hline \multirow{4}{}{QSAR} & $m_{1}$-SFNN-R & 62.67$\pm$0.83 & 71.00$\pm$0.78 & 50.89 & 22.95$\pm$0.50 & 0.015$\pm$0.007 & 33 \\ & $m_{2}$-SFNN-R & 76.57$\pm$1.11 & 80.29$\pm$1.15 & 55.04 & 16.61$\pm$0.24 & 0.013$\pm$0.005 & 10 \\ & $m_{3}$-SFNN-R & 89.57$\pm$0.79* & 87.17$\pm$1.04 & 57.63 & 11.06$\pm$0.15 & 0.010$\pm$0.004 & 46 \\ & STWD-SFNN & 78.26$\pm$1.11 & 81.21$\pm$1.13 & 61.61 & 12.43$\pm$0.15 & 0.008$\pm$0.001 & 2.00$\pm$0.00 \\ \hline \multirow{4}{}{OSP} & $m_{1}$-SFNN-R & 76.37$\pm$0.79 & 74.08$\pm$1.00 & 63.89* & 5.58$\pm$0.22 & 0.005$\pm$0.003 & 5 \\ & $m_{2}$-SFNN-R & 76.37$\pm$0.79 & 74.08$\pm$0.06 & 63.89 & 5.44$\pm$0.26 & 0.004$\pm$0.002 & 5 \\ & $m_{3}$-SFNN-R & 78.41$\pm$0.87 & 75.33$\pm$1.06 & 55.58 & 5.73$\pm$0.23 & 0.004$\pm$0.002 & 6 \\ & STWD-SFNN & 68.43$\pm$0.66 & 70.01$\pm$0.81 & 62.52 & 5.55$\pm$0.51 & 0.002$\pm$0.000 & 2.00$\pm$0.00 \\ \hline \multirow{4}{}{EGSS} & $m_{1}$-SFNN-R & 76.38$\pm$0.90 & 76.84$\pm$0.86 & 86.59 & 2.08$\pm$0.24 & 0.003$\pm$0.002 & 5 \\ & $m_{2}$-SFNN-R & 84.25$\pm$0.82 & 84.56$\pm$0.78 & 95.87 & 1.90$\pm$0.26* & 0.002$\pm$0.001 & 4 \\ & $m_{3}$-SFNN-R & 75.27$\pm$0.80 & 75.75$\pm$0.77 & 84.03 & 2.70$\pm$0.18 & 0.002$\pm$0.001 & 6 \\ & STWD-SFNN & 85.63$\pm$0.70 & 85.91$\pm$0.67 & 98.30 & 7.09$\pm$0.37 & 0.003$\pm$0.000 & 2.00$\pm$0.00 \\ \hline \multirow{4}{}{SE} & $m_{1}$-SFNN-R & 79.08$\pm$0.40 & 75.18$\pm$0.50 & 79.54 & 77.22$\pm$1.60 & 0.018$\pm$0.007 & 3 \\ & $m_{2}$-SFNN-R & 78.81$\pm$0.38 & 69.76$\pm$0.52 & 91.65* & 49.81$\pm$0.92 & 0.007$\pm$0.004 & 2 \\ & $m_{3}$-SFNN-R & 79.09$\pm$0.40 & 75.18$\pm$0.50 & 79.54 & 26.60$\pm$0.07 & 0.007$\pm$0.004 & 3 \\ & STWD-SFNN & 80.89$\pm$0.29 & 78.31$\pm$0.36 & 87.14 & 7.63$\pm$0.29 & 0.011$\pm$0.000 & 2.00$\pm$0.00 \\ \hline \multirow{4}{}{HTRU} & $m_{1}$-SFNN-R & 90.93$\pm$0.37 & 86.72$\pm$0.53 & 83.95 & 7.65$\pm$0.26* & 0.004$\pm$0.002 & 4 \\ & $m_{2}$-SFNN-R & 84.95$\pm$0.56 & 87.59$\pm$0.40 & 96.52 & 8.45$\pm$0.17 & 0.003$\pm$0.000 & 3 \\ & $m_{3}$-SFNN-R & 90.93$\pm$0.37 & 86.72$\pm$0.53 & 83.95 & 9.04$\pm$0.19 & 0.004$\pm$0.002 & 4 \\ & STWD-SFNN & 91.80$\pm$0.30 & 92.72$\pm$0.23 & 98.34 & 10.43$\pm$0.61 & 0.004$\pm$0.000 & 2.00$\pm$0.00 \\ \hline \multirow{4}{}{DCC} & $m_{1}$-SFNN-R & 77.95$\pm$0.04 & 68.52$\pm$0.57 & 50.50 & 24.36$\pm$1.46 & 0.009$\pm$0.005 & 6 \\ & $m_{2}$-SFNN-R & 74.07$\pm$0.26 & 68.31$\pm$0.41 & 58.51 & 11.77$\pm$0.31 & 0.004$\pm$0.002 & 5 \\ & $m_{3}$-SFNN-R & 76.77$\pm$0.46 & 75.62$\pm$0.53* & 67.49 & 10.46$\pm$0.33 & 0.004$\pm$0.002 & 7 \\ & STWD-SFNN & 78.13$\pm$0.40 & 69.70$\pm$0.55 & 63.59 & 7.89$\pm$0.32 & 0.002$\pm$0.000 & 2.00$\pm$0.00 \\ \hline \multirow{4}{}{ESR} & $m_{1}$-SFNN-R & 84.27$\pm$0.58 & 80.91$\pm$0.75 & 54.48 & 4.97$\pm$0.26 & 0.004$\pm$0.002 & 14 \\ & $m_{2}$-SFNN-R & 83.82$\pm$0.86 & 79.31$\pm$1.14 & 54.99 & 8.54$\pm$0.30 & 0.006$\pm$0.001 & 8 \\ & $m_{3}$-SFNN-R & 83.97$\pm$0.67 & 81.41$\pm$0.87 & 59.85* 3. The running time of STWD-SFNN is faster than SFNN-R on some datasets. For example, on the BM dataset, Table 3 reports that the training time of STWD-SFNN is 15.19s, and $m_{3}$-SFNN-R is the least time-consuming model of SFNN-R with a value of 25.75s. Similar phenomena can be found in other datasets. The reason is as follows. Compared with SFNN-R, which needs to repeatedly learn the features of all instances, STWD-SFNN only learns all instances in the first turn. After the second turn, STWD-SFNN focuses on the features of difficult-to-classify instances by constructing granularity layers and sequential threshold parameters, which takes a relatively short time. Therefore, the efficiency of STWD-SFNN is better than that of SFNN-R. #### 4.4.2 Comparison with dynamic models To verify the effectiveness of STWD-SFNN in dynamically constructing network topology, we add three new datasets on the basis of previous work [7] and choose GS-SFNN, PSO-SFNN, TWD-SFNN-R, and STWD-SFNN-NK as the competitive models. Fig. 5 shows the comparison of ROC curves for dynamic models and STWD-SFNN, and Table 4 lists the indices for those models. The results give rise to the following observations. 1. The accuracy, weighted-f1, ROC, and AUC of STWD-SFNN are better than those of GS-SFNN, PSO-SFNN, TWD-SFNN-R, and STWD-SFNN-NK. For example, on the ROE dataset of Table 4, the accuracy, weighted-f1, and AUC of STWD-SFNN are 84.35%, 85.00%, and 80.19%, respectively, which are higher than those of TWD-SFNN-R with best performance in the competitive models, and the values are 84.19%, 84.85%, and 80.19%, respectively. Meanwhile, Fig. 5 shows that the comparison of ROC curves of TWD-SFNN-R and STWD-SFNN overlap. Similar phenomena can be found in other datasets. The reasons are as follows. 1. Compared with GS-SFNN and PSO-SFNN, STWD-SFNN adopts the idea of divide-and-conquer and focuses on learning the features of the difficult-to-classify instances in BND. However, GS-SFNN and PSO-SFNN do not perform targeted processing on the difficult-to-classify instances, but repeatedly learn the features of all instances. 2. Compared with TWD-SFNN-R, STWD-SFNN sets dynamic thresholds and utilizes the sequential property of these thresholds, thereby gradually relaxing the conditions of instances partitioned at different granularity levels. However, TWD-SFNN-R adopts the static threshold in each turn, which is difficult to achieve dynamic adjustment of instance learning. 3. Compared with STWD-SFNN-NK, STWD-SFNN utilizes the discretization technique to handle the data of instances partitioned into the BND region, thereby improving the learning ability of the model. Therefore, the generalization ability of STWD-SFNN is better than that of the competitive models. 2. The number of hidden layer nodes of STWD-SFNN is no more than GS-SFNN, PSO-SFNN, and TWD-SFNN-R, but more than STWD-SFNN-NK in most cases. For example, on the SSMCR dataset in Table 4, STWD-SFNN has the same number of nodes as TWD-SFNN-R, with a value of two. STWD-SFNN has less nodes than GS-SFNN and PSO-SFNN, with values of 3.00 and 5.48, respectively, while STWD-SFNN has more nodes than STWD-SFNN-NK, with a value of 1.00. Similar phenomena can be found in other datasets. The reasons are as follows. 1. Compared with GS-SFNN and PSO-SFNN, STWD-SFNN adopts the idea of sequential three-way decisions, and determines whether to increase the number of hidden layer nodes according to whether there are instances in BND. However, GS-SFNN and PSO-SFNN use the difference in accuracy between adjacent turns. If the accuracy of this turn is lower than that of the previous turn, the number of hidden layer nodes will continue to increase until it reaches a given maximum number of hidden layer nodes. 2. Compared with TWD-SFNN-R, STWD-SFNN constructs granularity layers and sequential thresholds so that it can pay more attention to instances that are difficult to classify, and gradually partition instances that are difficult to classify from the BND region. In addition, STWD-SFNN is a generalization of TWD-SFNN-R, which can ensure that STWD-SFNN has the same advantages as TWD-SFNN-R in network topology. 3. Compared with STWD-SFNN-NK, STWD-SFNN can repeatedly learn the features of difficult-to-classify instances, thereby expanding the network topology. Therefore, the network topology of STWD-SFNN is superior to the competitive models in most cases. 3. The training time of STWD-SFNN is less than GS-SFNN and PSO-SFNN, but more than TWD-SFNN-R and STWD-SFNN-NK on some datasets. For example, on the QSAR dataset in Table 4, the training time of STWD-SFNN is, less than GS-SFNN, PSO-SFNN, TWD-SFNN-R, and STWD-SFNN-NK, with values of 12.43s, 40.13s, 190.05s, 13.61s, and 14.62s, respectively. Similar phenomena can be found in other datasets. The reasons are as follows. Figure 5: Comparison of ROC curves of dynamic models and STWD-SFNN \begin{table} \begin{tabular}{c c c c c c c c c} \hline \hline Dataset & Model & \multicolumn{2}{c}{Accuracy} & \multicolumn{2}{c}{Weighted-} & \multicolumn{2}{c}{AUC} & \multicolumn{2}{c}{Training} & \multicolumn{2}{c}{Test time (s)} & \multicolumn{2}{c}{Nodes} \\ \hline & GS-SFNN & 47.24$\pm$17.13 & 54.65$\pm$17.01 & 49.71 & 55.15$\pm$11.72 & 0.005$\pm$0.001 & 3.10$\pm$0.63 \\ & PSO-SFNN & 92.20$\pm$2.34 & 81.64$\pm$18.18 & 53.83 & 91.55$\pm$3.51 & 0.003$\pm$0.000 & 4.35$\pm$1.01 \\ ONP & TWD-SFNN-R & 94.41$\pm$0.22 & 85.78$\pm$0.25 & 57.87 & 9.56$\pm$0.11 & 0.003$\pm$0.000 & 1.00$\pm$0.00 \\ & STWD-SFNN-NK & 84.16$\pm$0.24 & 86.63$\pm$0.25 & 59.35 & 44.70$\pm$6.10 & 0.008$\pm$0.0001 & 1.00$\pm$0.00 \\ & STWD-SFNN & 94.41$\pm$0.22 & 85.87$\pm$0.26 & 57.87 & 15.76$\pm$0.76 & 0.003$\pm$0.000 & 1.00$\pm$0.00 \\ \hline & GS-SFNN & 49.47$\pm$1.29 & 57.78$\pm$10.70 & 43.16 & 40.13$\pm$34.71 & 0.011$\pm$0.000 & 2.70$\pm$0.31 \\ & PSO-SFNN & 88.59$\pm$2.48 & 85.49$\pm$1.06 & 56.32 & 190.05$\pm$8.61 & 0.017$\pm$0.002 & 5.42$\pm$1.61 \\ QSAR & TWD-SFNN-R & 78.26$\pm$1.11 & 81.21$\pm$1.13 & 61.61 & 14.62$\pm$0.33 & 0.011$\pm$0.002 & 2.00$\pm$0.00 \\ & STWD-SFNN-NK & 67.23$\pm$1.15 & 74.42$\pm$1.04 & 64.54 & 13.61$\pm$1.18 & 0.009$\pm$0.001 & 1.00$\pm$0.00 \\ & STWD-SFNN & 78.26$\pm$1.11 & 81.21$\pm$1.13 & 61.61 & 12.43$\pm$0.15 & 0.008$\pm$0.001 & 2.00$\pm$0.00 \\ \hline & GS-SFNN & 44.76$\pm$1.56 & 27.43$\pm$1.29 & 40.62 & 28.78$\pm$5.95 & 0.004$\pm$0.000 & 2.90$\pm$0.63 \\ & PSO-SFNN & 80.98$\pm$3.59 & 68.07$\pm$15.27 & 62.10 & 28.88$\pm$2.60 & 0.002$\pm$0.001 & 5.62$\pm$1.44 \\ OSP & TWD-SFNN-R & 68.42$\pm$0.65 & 70.01$\pm$0.80 & 62.52 & 50.40$\pm$0.43 & 0.002$\pm$0.000 & 2.00$\pm$0.00 \\ & STWD-SFNN & 65.01$\pm$0.65 & 66.98$\pm$0.82 & 64.77 & 51.5$\pm$0.60 & 0.002$\pm$0.000 & 1.00$\pm$0.00 \\ & STWD-SFNN & 68.43$\pm$0.66 & 70.01$\pm$0.81 & 62.52 & 5.55$\pm$0.51 & 0.002$\pm$0.000 & 2.00$\pm$0.00 \\ \hline & GS-SFNN & 62.56$\pm$14.47 & 58.48$\pm$16.67 & 95.07 & 18.36$\pm$2.41 & 0.004$\pm$0.001 & 2.50$\pm$0.45 \\ & PSO-SFNN & 84.32$\pm$5.05 & 83.30$\pm$7.17 & 95.60 & 36.35$\pm$3.29 & 0.004$\pm$0.001 & 5.53$\pm$1.51 \\ EGSS & TWD-SFNN-R & 85.60$\pm$0.70 & 88.80$\pm$0.67 & 98.30 & 60.46$\pm$0.65 & 0.003$\pm$0.000 & 2.00$\pm$0.00 \\ & STWD-SFNN-NK & 82.26$\pm$0.75 & 82.61$\pm$0.72 & 96.15 & 4.71$\pm$0.57 & 0.002$\pm$0.000 & 1.00$\pm$0.00 \\ & STWD-SFNN & 85.63$\pm$0.70 & 85.91$\pm$0.67 & 98.30 & 7.009$\pm$0.37 & 0.003$\pm$0.000 & 2.00$\pm$0.00 \\ \hline & GS-SFNN & 55.84$\pm$16.66 & 35.60$\pm$0.92 & 98.75 & 42.24$\pm$8.69 & 0.003$\pm$0.000 & 2.50$\pm$0.45 \\ & PSO-SFNN & 79.01$\pm$8.96 & 61.05$\pm$22.36 & 87.14 & 192.66$\pm$12.37 & 0.004$\pm$0.000 & 4.83$\pm$1.82 \\ SE & TWD-SFNN-R & 80.89$\pm$0.29 & 78.31$\pm$0.36 & 87.14 & 13.84$\pm$0.038 & 0.002$\pm$0.000 & 2.00$\pm$0.00 \\ & STWD-SFNN-NK & 79.12$\pm$0.29 & 70.21$\pm$0.03 & 74.58 & 17.87$\pm$0.25 & 0.002$\pm$0.000 & 1.00$\pm$0.00 \\ & STWD-SFNN & 80.89$\pm$0.29 & 78.31$\pm$0.36 & 87.14 & 20.68$\pm$0.57 & 0.003$\pm$0.000 & 2.00$\pm$0.00 \\ \hline & GS-SFNN & 46.78$\pm$17.71 & 52.19$\pm$17.94 & 90.43 & 40.34$\pm$9.08 & 0.005$\pm$0.000 & 3.20$\pm$0.72 \\ & PSO-SFNN & 89.71$\pm$2.35 & 80.99$\pm$18.17 & 86.00 & 108.00$\pm$9.76 & 0.007$\pm$0.001 & 5.64$\pm$1.91 \\ HTRU & TWD-SFNN-R & 91.75$\pm$0.27 & 92.68$\pm$0.21 & 98.34 & 10.188$\pm$0.55 & 0.005$\pm$0.001 & 2.00$\pm$0.00 \\ & STWD-SFNN-NK & 84.50$\pm$0.48 & 87.26$\pm$0.35 & 96.70 & 5.53$\pm$0.30 & 0.002$\pm$0.000 & 1.00$\pm$0.00 \\ & STWD-SFNN & 91.80$\pm$0.30 & 92.72$\pm$0.23 & 98.34 & 10. 1. Compared with GS-SFNN and PSO-SFNN, STWD-SFNN only has a learning process for all instances in the first turn, and the time cost required for the model gradually decreases as the size of BND decreases. However, GS-SFNN and PSO-SFNN need to spend more time on each turn of learning for all instances. 2. Compared with TWD-SFNN-R which may repeatedly handle the same difficult-to-classify instances in each turn, the sequential thresholds of STWD-SFNN can reduce the condition of partitioned instances in each turn. STWD-SFNN decomposes the more easily classified instances in BND at each granularity level, so as to gradually realize the partition of all instances. 3. Compared with STWD-SFNN-NK, STWD-SFNN adopts $k$-means++ discretization technology to increase the learning time of the model. Therefore, STWD-SFNN performs better than the competitive models on most cases. #### 4.4.3 Comparison with other competitive models To measure the efficiency of STWD-SFNN, we selected SVC, RF, and KNN as the competitive models. To guarantee the repeatability of the experimental results, SVC, RF, and KNN adopted the same random number seed in 10-fold cross-validation, that is, we fixed the random number seed in 10-fold cross-validation by utilizing rng(0). Fig. 6 shows the comparison of ROC curves of the classification models and STWD-SFNN. Table 5 compares the evaluation criteria of each model. The results give rise to the following observations. 1. The accuracy, weighted-f1, ROC, and AUC of STWD-SFNN are better than SVC, RF, and KNN on some datasets. For example, on the ONP dataset of Table 5, the accuracy, weighted-f1, and AUC of STWD-SFNN are 94.41%, 85.87%, and 57.87%, respectively, while the best-performing competitive model is RF, with 94.40%, 82.53%, and 52.96%, respectively. Meanwhile, as shown in Fig.5, the ROC curve of STWD-SFNN is at the left top of RF. Similar phenomena can be found in the other dataset. The reason is as follows. Compared with SVC, RF, and KNN, STWD-SFNN retains the ability of the neural network to capture the nonlinear relationship of data, and further enhances the performance of the model by adopting sequential three-way decisions. Therefore, in most cases, the generalization ability of STWD-SFNN is better than the competitive models. 2. STWD-SFNN has less training time and test time than SVC, RF, and KNN on some datasets. For example, on the PCB data set of Table 5, the training time and test time of STWD-SFNN are 5.44s and 0.002s, respectively. The competitive models with the least training time and test time are SVC and RF, with values of 200.21s and 0.049s, respectively. Similar phenomena can be found in other datasets. The reason is as follows. Compared with SVC, RF, and KNN, STWD-SFNN can greatly promote the learning process by constructing granularity layers and sequential thresholds, which can improve the operation efficiency of the model. Therefore, in most cases, STWD-SFNN is more efficient than other competitive models. #### 4.4.4 Comparison with TWD-SFNN with 5-fold cross-validation From the above-mentioned experimental results, STWD-SFNN has relatively better accuracy, network topology, and operating efficiency. However, the ROC and AUC of TWD-SFNN and STWD-SFNN are not significantly different in Fig. 5 and Table 4. To further illustrate the differences between TWD-SFNN and STWD-SFNN, we adopt a 5-fold cross-validation technique to conduct a further study on 15 datasets. Fig. 7 shows the comparison of ROC curves of TWD-SFNN and STWD-SFNN on 5-fold cross-validation. Table 6 reports the comparison of different evaluation criteria of those models. The results give rise to the following observations. Table 6 shows that the accuracy, weighted-f1, and AUC of TWD-SFNN and STWD-SFNN have significant differences on some datasets. For example, on the ROE dataset, the accuracy, weighted-f1, and AUC of STWD-SFNN are 87.30%, 87.91%, and 82.48%, respectively, while those of TWD-SFNN are 83.82%, 85.01%, and 82.14%, respectively. Moreover, as shown in Fig. 7, the ROC curve of STWD-SFNN is located at the top left of TWD-SFNN, which means that the performance of STWD-SFNN is better than that of TWD-SFNN. The same conclusion can be found in ONP, QSAR, PCB, OD, and SSMCR datasets. Therefore, TWD-SFNN and STWD-SFNN have some differences under 5-fold cross-validation. It should be noted that whether TWD-SFNN and STWD-SFNN have significant differences under other folds is worth developing in the future. #### 4.4.5 Analysis of STWD-SFNN In this section, we analyze the relationship between the process costs and the number of hidden layer nodes in STWD-SFNN, including the costs of test and delay. According to Definition 6, without loss of generality, we randomly Figure 6: Comparison of ROC curves of other classification models and STWD-SFNN \begin{table} \begin{tabular}{c c c c c c c} \hline \hline Dataset & Model & Accuracy (\%) & Weighted-f1(\%) & AUC (\%) & Training time(s) & Test time (s) \\ \hline \multirow{4}{}{ONP} & SVC & 93.29$\pm$0.56 & 91.37$\pm$0.30* & 54.61 & 1284$\pm$117 & 1.015$\pm$0.129 \\ & RF & 94.40$\pm$0.43 & 82.53$\pm$18.34 & 52.96 & 124.20$\pm$1.80 & 0.059$\pm$0.008 \\ & KNN & 94.39$\pm$0.22 & 45.88$\pm$0.59 & 53.01 & - & 88.43$\pm$17.95 \\ & STWD-SFNN & 94.41$\pm$0.22 & 85.87$\pm$0.26 & 57.87 & 15.76$\pm$0.76 & 0.003$\pm$0.000 \\ \hline \multirow{4}{}{QSAR} & SVC & 91.63$\pm$0.46 & 90.06$\pm$0.78 & 51.78 & 1.77$\pm$0.08 & 0.093$\pm$0.008 \\ & RF & 93.75$\pm$0.72* & 92.56$\pm$0.94 & 62.91 & 1721.7$\pm$68.20 & 0.092$\pm$0.013 \\ & KNN & 93.51$\pm$0.72 & 92.26$\pm$0.94 & 51.60 & - & 4.728$\pm$3.343 \\ & STWD-SFNN & 78.26$\pm$1.11 & 81.21$\pm$1.13 & 61.61 & 12.43$\pm$0.15 & 0.008$\pm$0.001 \\ \hline \multirow{4}{}{OSP} & SVC & 70.31$\pm$18.50 & 63.23$\pm$19.50 & 64.46 & 349.12$\pm$56.78 & 0.048$\pm$0.008 \\ & RF & 90.47$\pm$0.49* & 89.97$\pm$0.51 & 78.16 & 63.11$\pm$0.19 & 0.025$\pm$0.001 \\ & KNN & 84.10$\pm$0.73 & 78.34$\pm$0.98 & 61.05 & - & 12.40$\pm$2.24 \\ & STWD-SFNN & 68.43$\pm$0.66 & 70.01$\pm$0.81 & 62.52 & 5.55$\pm$0.51 & 0.002$\pm$0.000 \\ \hline \multirow{4}{}{EGSS} & SVC & 90.17$\pm$0.69 & 89.96$\pm$0.73 & 99.60 & 3.15$\pm$0.24* & 0.070$\pm$0.009 \\ & RF & 99.98$\pm$0.00 & 99.98$\pm$0.00 & 99.93 & 6.07$\pm$0.44 & 0.030$\pm$0.020 \\ & KNN & 99.68$\pm$0.17 & 99.68$\pm$0.17 & 72.12 & - & 81.13$\pm$7.13 \\ & STWD-SFNN & 85.63$\pm$0.70 & 85.91$\pm$0.67 & 98.30 & 7.09$\pm$0.37 & 0.003$\pm$0.000 \\ \hline \multirow{4}{}{SE} & SVC & 84.77$\pm$0.20 & - & - & 24.22$\pm$1.14* & 1.981$\pm$0.016 \\ & RF & 99.92$\pm$0.03 & 99.92$\pm$0.03 & 99.83 & 49.26$\pm$3.10 & 0.376$\pm$0.020 \\ & KNN & 99.64$\pm$0.05 & 99.64$\pm$0.05 & 94.60 & - & 10324$\pm$2993 \\ & STWD-SFNN & 80.89$\pm$0.29 & 78.31$\pm$0.36 & 87.14 & 85.20$\pm$2.33 & 0.012$\pm$0.000 \\ \hline \multirow{4}{}{HTRU} & SVC & 70.15$\pm$0.01 & 79.18$\pm$0.01 & 87.64 & 492.75$\pm$25.27 & 0.073$\pm$0.076 \\ & RF & 97.96$\pm$0.09 & 97.92$\pm$0.10* & 92.57 & 2.52$\pm$0.27 & 0.012$\pm$0.005 \\ & KNN & 97.97$\pm$0.12 & 97.92$\pm$0.13 & 64.26 & - & 92.01$\pm$10.42 \\ & STWD-SFNN & 91.80$\pm$0.30 & 92.72$\pm$0.23 & 98.34 & 10.43$\pm$0.61 & 0.004$\pm$0.000 \\ \hline \multirow{4}{}{DCC} & SVC & 72.76$\pm$1.32 & 67.00$\pm$0.86 & 66.67* & 1041.81$\pm$27.23 & 0.248$\pm$0.064 \\ & RF & 81.84$\pm$0.44 & 79.85$\pm$0.60 & 66.65 & 131.16$\pm$34.93 & 0.339$\pm$0.022 \\ & KNN & 81.36$\pm$0.42 & 78.85$\pm$0.55 & 54.14 & - & 44.49$\pm$4.43 \\ & STWD-SFNN & 78.13$\pm$0.40 & 69.70$\pm$0.55 & 63.59 & 7.89$\pm$0.32 & 0.002$\pm$0.000 \\ \hline \multirow{4}{}{ESR} & SVC & 80.00$\pm$0.96 & 80.35$\pm$0.01 & 51.99 & 12.99$\pm$0.51 & 1.327$\pm$0.051 \\ & RF & 97.58$\pm$0.18* & 97.58$\pm$0.18 & 95.12 & 610.17$\pm$138.52 & 0.113$\pm$0.010 \\ & KNN & 84.43$\pm$0.97 & 80.15$\pm$1.33 & 60.18 & - & 73.41$\pm$7.48 \\ & STWD-SFNN & 84.35$\pm$0.47 & 81.55$\pm$0.55 & 54.25 & 4.29$\pm$0.06 & 0.003$\pm$0.000 \\ \hline \multirow{4}{}{BM} & SVC & 65.62$\pm$24.03 & 61.57$\pm$25.08 & 77.27 & 908.79$\pm$149.27 & 0.361$\pm$0.022 \\ & RF & 91.67$\pm$0.19* & 91.16$\pm$0.24 & 80.41 & 68.11$\pm$3.32 & 0.173$\pm$0.004 \\ & KNN & 89.41$\pm$0.32 & 86.98$\pm$0.42 & 59.52 & - & 234.01$\pm$35.35 \\ & STWD-SFNN & 87.56$\pm$0.34 & 82.85$\pm$0.48 & 78.80 & 15.19$\pm$0.36 & 0.003$\pm$0.001 \\ \hline \multirow{4}{}{PCB} & SVC & 97.77$\pm$0.32 & 96.80$\pm$0.45 & 66.26 & 200.21$\pm$69.92 & 0.138$\pm$0.011 \\ & RF & 98.09$\pm$0.27* & 97.56$\pm$0.38 & 74.52 & 222.54$\pm$61.80 & 0.049$\pm$0.004 \\ & KNN & 97.84$\pm$0.32 & - & - & - & 29.27$\pm$5.60 \\ & STWD-SF Figure 7: Comparison of ROC curves of TWD-SFNN and STWD-SFNN by adopting 5-fold cross-validation \begin{table} \begin{tabular}{c l c c c c c c} \hline \hline \multirow{2}{}{Dataset} & \multirow{2}{}{Model} & \multicolumn{2}{c}{Accuracy} & Weighted- & AUC & \multicolumn{1}{c}{Training} & \multicolumn{1}{c}{Test time (s)} & \multicolumn{1}{c}{Nodes} \\ & & (\%) & f1 (\%) & (\%) & time(s) & \multicolumn{1}{c}{Test time (s)} & \multicolumn{1}{c}{Nodes} \\ \hline \multirow{2}{}{ONP} & TWD-SFNN-R & 92.33$\pm$0.20 & 90.94$\pm$0.23 & 56.87 & 35.29$\pm$0.29 & 0.015$\pm$0.002 & 3.00$\pm$0.00 \\ & STWD-SFNN & 94.39$\pm$0.24* & 91.70$\pm$0.35 & 62.45 & 30.63$\pm$0.90 & 0.008$\pm$0.001 & 1.00$\pm$0.00 \\ \hline \multirow{2}{}{QSAR} & TWD-SFNN-R & 72.63$\pm$0.71 & 78.29$\pm$0.60 & 61.89 & 6.71$\pm$0.22* & 0.010$\pm$0.002 & 2.00$\pm$0.00 \\ & STWD-SFNN & 82.14$\pm$0.36 & 83.60$\pm$0.44 & 62.44 & 9.13$\pm$0.66 & 0.009$\pm$0.001 & 2.80$\pm$0.40 \\ \hline \multirow{2}{}{OSP} & TWD-SFNN-R & 83.14$\pm$0.57* & 77.24$\pm$0.62 & 70.10 & 2.29$\pm$0.05 & 0.003$\pm$0.001 & 2.00$\pm$0.00 \\ & STWD-SFNN & 83.14$\pm$0.57 & 77.24$\pm$0.62 & 70.10 & 2.04$\pm$0.17 & 0.003$\pm$0.001 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{EGSS} & TWD-SFNN-R & 87.67$\pm$0.60* & 87.91$\pm$0.56 & 97.05 & 4.23$\pm$1.14 & 0.004$\pm$0.000 & 1.00$\pm$0.00 \\ & STWD-SFNN & 87.67$\pm$0.60 & 87.91$\pm$0.56 & 97.05 & 4.76$\pm$1.16 & 0.004$\pm$0.001 & 1.00$\pm$0.00 \\ \hline \multirow{2}{}{SE} & TWD-SFNN-R & 86.00$\pm$0.31* & 80.58$\pm$0.46 & 90.01 & 365.87$\pm$8.21 & 0.008$\pm$0.000 & 2.00$\pm$0.00 \\ & STWD-SFNN & 86.00$\pm$0.31 & 80.58$\pm$0.46 & 90.01 & 296.15$\pm$3.47 & 0.008$\pm$0.004 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{HTRU} & TWD-SFNN-R & 91.84$\pm$0.40* & 92.76$\pm$0.29 & 98.10 & 9.02$\pm$0.34 & 0.007$\pm$0.002 & 2.00$\pm$0.00 \\ & STWD-SFNN & 91.84$\pm$0.40 & 92.76$\pm$0.29 & 98.10 & 18.68$\pm$2.99 & 0.012$\pm$0.006 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{DCC} & TWD-SFNN-R & 78.13$\pm$0.30* & 69.70$\pm$0.32 & 63.07 & 7.81$\pm$0.07 & 0.004$\pm$0.000 & 2.00$\pm$0.00 \\ & STWD-SFNN & 78.13$\pm$0.30 & 69.70$\pm$0.32 & 63.07 & 14.12$\pm$1.30 & 0.007$\pm$0.002 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{ESR} & TWD-SFNN-R & 84.33$\pm$0.26* & 81.52$\pm$0.37 & 56.23 & 6.18$\pm$0.73 & 0.006$\pm$0.001 & 2.00$\pm$0.00 \\ & STWD-SFNN & 84.33$\pm$0.26 & 81.52$\pm$0.37 & 56.23 & 14.88$\pm$1.26 & 0.011$\pm$0.003 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{BM} & TWD-SFNN-R & 89.27$\pm$0.46* & 84.24$\pm$0.66 & 81.91 & 11.79$\pm$0.38 & 0.003$\pm$0.000 & 1.00$\pm$0.00 \\ & STWD-SFNN & 89.27$\pm$0.48 & 84.24$\pm$0.69 & 81.91 & 13.80$\pm$0.55 & 0.003$\pm$0.000 & 1.00$\pm$0.00 \\ \hline \multirow{2}{}{PCB} & TWD-SFNN-R & 97.75$\pm$0.12 & 96.76$\pm$0.01 & 73.22 & 6.06$\pm$0.16* & 0.006$\pm$0.001 & 2.00$\pm$0.00 \\ & STWD-SFNN & 97.85$\pm$0.74 & 96.95$\pm$0.48 & 73.99 & 11.76$\pm$0.45 & 0.009$\pm$0.002 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{SB} & TWD-SFNN-R & 89.51$\pm$1.20* & 67.61$\pm$30.85 & 79.32 & 0.96$\pm$0.19 & 0.002$\pm$0.001 & 1.80$\pm$0.40 \\ & STWD-SFNN & 89.51$\pm$1.20 & 67.61$\pm$30.85 & 79.32 & 1.16$\pm$0.23 & 0.002$\pm$0.001 & 1.80$\pm$0.40 \\ \hline \multirow{2}{}{EOL} & TWD-SFNN-R & 86.26$\pm$1.30* & 82.68$\pm$1.33 & 70.07 & 0.50$\pm$0.06 & 0.001$\pm$0.000 & 2.00$\pm$0.00 \\ & STWD-SFNN & 86.26$\pm$1.30 & 82.68$\pm$1.33 & 70.07 & 1.46$\pm$0.23 & 0.002$\pm$0.001 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{OD} & TWD-SFNN-R & 76.56$\pm$0.33 & 66.80$\pm$0.43 & 68.33 & 2.85$\pm$0.42* & 0.001$\pm$0.000 & 2.00$\pm$0.00 \\ & STWD-SFNN & 78.50$\pm$0.72 & 71.48$\pm$1.06 & 69.07 & 12.23$\pm$2.76 & 0.003$\pm$0.001 & 2.00$\pm$0.00 \\ \hline \multirow{2}{}{ROE} & TWD-SFNN-R & 83.82$\pm$0.90 & 85.01$\pm$0.81 & 82.14 & 1.13$\pm$0.26* & 0.002$\pm$0.001 & **2.00$\pm$0.00 generate 10 incremental unit test costs in the real range [1.00, 50.00]. To ensure the repeatability of experimental results, the random number seed is 0, i.e., rng (0), and 10 values are generated as unit test costs: [1.00, 6.44, 11.89, 17.33, 22.78, 28.22, 33.67, 39.11, 44.56, 50.00]. The parameter settings of unit delay cost are the same as those of unit test cost. Fig. 8 shows the relationship between the process costs of STWD-SFNN on the training set and the number of hidden layer nodes. Since STWD-SFNN has only one hidden layer node on the ONP dataset, we analyze the process cost on the remaining fourteen datasets. The results give rise to the following observations according to Fig 8. 1. The test cost of STWD-SFNN increases with the number of hidden layer nodes. For example, on the HTRU dataset, according to Definition 6 and $[Cost_{\text{PPT}_{1}},Cost_{\text{PPT}_{2}}]=[1.00,6.44]$, the test costs of the first and second levels are 638.00 and 1540.22, respectively. Similar phenomena can be found in the rest 13 datasets. The reason is that the test cost of STWD-SFNN in the $t$-granular level is equivalent to the sum of the test costs of the first ($t$-1)-granular level and the $t$-granular level. 2. The delay cost of STWD-SFNN and the number of hidden layer nodes report two cases. Case 1: The delay cost tends to be constant with the increase of hidden layer nodes. For example, on the DCC dataset, according to Definition 6 and $[Cost_{\text{PPT}_{1}},Cost_{\text{PPT}_{2}}]=[1.00,6.44]$, the delay costs of the first and second levels are both 5275.00. The same conclusion can be found in OSP, EGSS, SE, ESR, BM, PCB, EOL, OD, and SSMCR. Case 2: The delay cost of STWD-SFNN increases with the number of hidden layer nodes. For example, on the HTRU dataset, the delay costs of the first and second levels are 638.00 and 902.22, respectively. This phenomenon can be found in QSAR and ROE datasets. Specifically, the SE dataset has both Case 1 and Case 2. The reason is that the delay cost of STWD-SFNN in the $t$ granular level is equivalent to the maximum of the delay cost of the ($t$-1)-granular level and the $t$-granular level. Specifically, if the delay cost of the $t$ granular level is not greater than that of the ($t$-1)-granular level, the delay cost shows a constant trend; otherwise, it shows an increasing trend. ## 5 Conclusion Although TWD-SFNN can overcome the shortage of traditional SFNN and obtain a simpler network structure and better performance, it uses fixed threshold parameters to train the network model. Inspired by STWD, this paper proposes a new network topology optimization model, called STWD-SFNN, to enhance the performance of the network on structured datasets. STWD-SFNN adopts a sequential strategy to dynamically determine the number of hidden layer nodes. STWD-SFNN has two parts: discretion modular and training modular. Discretion modular adopts $k$-means$++$ to convert numerical data into discrete data to meet the requirements of STWD for data types. The training modular, as the main part of STWD-SFNN, adopts a sequential strategy to dynamically determine the number of hidden layer nodes. Thus, STWD-SFNN has $t$ levels. Each level has two parts: SFNN and STWD. SFNN is a network topology with one hidden layer node and gets correctly classified instances and misclassification instances. The first $t$-1 part of STWD utilizes the strategy of three ways, and the last part applies the strategy of two ways. The experimental results show that STWD-SFNN has higher operational efficiency and a more compact network structure than SFNN using empirical formulas, GS-SFNN, PSO-SFNN, TWD-SFNN-R, and STWD-SFNN-NK, and has a better generalization ability on structured datasets than the competitive models. In this paper, we propose STWD-SFNN which improves the performance of SFNN. However, many aspects should be investigated in the future to enrich the theory of neural networks. 1. STWD-SFNN employs sequential three-way decisions to guide the growth of the topology of a single hidden layer neural network. However, for multilayer neural networks, how to adopt our model to handle XOR and other problems to further promote deep learning is worthy of investigation. 2. STWD-SFNN has good performance in binary classification problems by comparing the conditional probability of each instance belonging to the positive region and threshold parameters. For a multi-class classification problem, one method is to convert it into multiple binary classification problems. Obviously, this method is not very effective. How to adopt STWD-SFNN to solve multi-class classification is worthy of consideration. 3. STWD-SFNN adopts $k$-means$++$ to discretize the data when calculating the conditional probability of instances, since STWD is a classification model established on the decision-theoretic rough sets and derived from nominal attributes. However, whether there is a better discretization model to transform non-integer data into integer data, or whether there are other rough set models that can essentially mine numerical or mixed data features. Figure 8: Relationship between process costs and number of hidden layer nodes ## Acknowledgement This work was supported by the National Social Science Fund of China under grant number 18BGL191.
2305.19377	Benign Overfitting in Deep Neural Networks under Lazy Training	This paper focuses on over-parameterized deep neural networks (DNNs) with ReLU activation functions and proves that when the data distribution is well-separated, DNNs can achieve Bayes-optimal test error for classification while obtaining (nearly) zero-training error under the lazy training regime. For this purpose, we unify three interrelated concepts of overparameterization, benign overfitting, and the Lipschitz constant of DNNs. Our results indicate that interpolating with smoother functions leads to better generalization. Furthermore, we investigate the special case where interpolating smooth ground-truth functions is performed by DNNs under the Neural Tangent Kernel (NTK) regime for generalization. Our result demonstrates that the generalization error converges to a constant order that only depends on label noise and initialization noise, which theoretically verifies benign overfitting. Our analysis provides a tight lower bound on the normalized margin under non-smooth activation functions, as well as the minimum eigenvalue of NTK under high-dimensional settings, which has its own interest in learning theory.	Zhenyu Zhu, Fanghui Liu, Grigorios G Chrysos, Francesco Locatello, Volkan Cevher	2023-05-30T19:37:44Z	http://arxiv.org/abs/2305.19377v1	# Benign Overfitting in Deep Neural Networks under Lazy Training ###### Abstract This paper focuses on over-parameterized deep neural networks (DNNs) with ReLU activation functions and proves that when the data distribution is well-separated, DNNs can achieve _Bayes-optimal_ test error for classification while obtaining (nearly) zero-training error under the lazy training regime. For this purpose, we unify three interrelated concepts of overparameterization, benign overfitting, and the Lipschitz constant of DNNs. Our results indicate that interpolating with smoother functions leads to better generalization. Furthermore, we investigate the special case where interpolating smooth ground-truth functions is performed by DNNs under the Neural Tangent Kernel (NTK) regime for generalization. Our result demonstrates that the generalization error converges to a constant order that only depends on label noise and initialization noise, which theoretically verifies benign overfitting. Our analysis provides a tight lower bound on the normalized margin under non-smooth activation functions, as well as the minimum eigenvalue of NTK under high-dimensional settings, which has its own interest in learning theory. 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Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural 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Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Learning, Deep Neural Networks, Neural Networks, Learning, summarized below: * We adhere to the standard data setting with label noise that has been previously explored in two-layer networks (Frei et al., 2022), along with the model setting for multi-layer fully connected neural networks (Cao and Gu, 2019; Allen-Zhu et al., 2019). Building upon the proof concept of (Frei et al., 2022), we extend our results to deep ReLU neural networks, which presents a considerable challenge in connecting the Bayes-optimal test error to the training dynamics of deep neural networks. Theorem 1 for binary classification shows that, under the lazy training regime, even though training on noisy data, DNNs can still obtain the _Bayes-optimal_ test error, i.e., the error rate is less than the proportion of incorrect labels in the training set plus a generalization term that converges to zero. We also demonstrate that this term is positively correlated with the Lipschitz constants of DNNs, which implies that interpolating more smooth functions leads to a faster convergence rate. * Theorem 2 provides the first lower bound on the minimum eigenvalues of the NTK matrix of DNNs in the high-dimensional setting and demonstrates its phase transition under different tendencies of the number of training data and input dimension. We believe that it has its own interest in learning theory. * Theorem 3 builds the generalization guarantees of over-parameterized neural networks under the NTK regime in the high-dimensional setting to learn a ground-truth function in RKHS. Our result exhibits a phase transition on the excess risk (related to generalization performance) between the $n<d$ and $n>d$ case. It implies that the excess risk finally converges to a constant order only relying on the label noise and initialization noise, which theoretically verifies the benign overfitting. Technical challenges. The main technical challenge of this paper is how to derive the lower bound of the non-smooth function in deep ReLU neural networks for the normalized margin on test points. In the context of lazy training (Chizat et al., 2019), the function of a neural network is nearly linear during the initial stages of training. By analyzing the accumulation of weights for each training step, we can establish a lower bound for the normalized margin on test points. By doing so, we transform the _Bayes-optimal_ test error to the expected risk and Lipschitz constant of DNNs. When compared with Frei et al. (2022) on shallow neural networks with smooth activation functions for binary classification, we extend their results to our deep ReLU neural networks, not limited to high-dimensional settings, and obtain a faster convergence rate with the number of data. The key difficulty lies in how to build the relationship between the _Bayes-optimal_ test error and the training dynamics of DNNs. When compared to the generalization results of deep neural networks (DNNs) in the over-parameterized regime (Cao and Gu, 2019), our results focus on overfitted models that are trained by noisy data and achieve a faster convergence rate. Besides, Ju et al. (2022) present generalization guarantees on three-layer neural networks (only the last layer is training) for regression, which has the closed-form min $L^{2}$-norm solution. However, this nice property is invalid in our DNN setting. In this case, we build the connection between DNNs and kernel methods (e.g., NTK) in high dimensional settings for benign overfitting. ### Related work Benign overfitting: There has been a significant amount of research devoted to understanding the phenomenon of benign overfitting, with a particular emphasis on linear models, e.g., linear regression (Bartlett et al., 2020; Chatterji et al., 2021; Zou et al., 2021), sparse linear regression (Chatterji and Long, 2022; Koehler et al., 2021; Wang et al., 2022), logistic regression (Montanari et al., 2019; Wang et al., 2021), ridge regression (Tsigler and Bartlett, 2020) and kernel-based estimators (Mei and Montanari, 2022; Liang et al., 2019). Furthermore, the concept of benign overfitting can be extended to tempered or catastrophic based on various spectra of the kernel (ridge) regression (Mallinar et al., 2022). For nonlinear models, Li et al. (2021) study the benign overfitting phenomenon of random feature models. Frei et al. (2022) prove that a two-layer fully connected neural network exhibits benign overfitting under certain conditions, e.g., well-separated log-concave distribution and smooth activation function. Then Xu and Gu (2023) extends the previous results to the non-smooth case. Similarly, Cao et al. (2022) focus on the benign overfitting of two-layer convolutional neural networks (CNN). Mallinar et al. (2022) argue that many true interpolation methods (such as neural networks) for noisy data are not benign but tempered overfitting, and even catastrophic under various model capacities. Generalization of NNs and Neural Tangent Kernel (NTK): The generalization ability of neural networks has been a core problem in machine learning theory. Brutzkus et al. (2017) show that SGD can learn an over-parameterized two-layer neural network with good generalization ability. Allen-Zhu et al. (2019) study the generalization performance of SGD for $2$- and $3$-layer networks. Cao and Gu (2019) study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime. Besides, Arora et al. (2019); Cao and Gu (2020); E et al. (2020) provide the algorithm-dependent generalization bounds for different settings. The NTK (Jacot et al., 2018) is a powerful tool for deep neural network analysis. Specifically, NTKs establish equivalence between the training dynamics of gradient-based algorithms for DNNs and kernel regression under specific initialization, so it can be considered as an intermediate step between simple linear models and DNNs (Allen-Zhu et al., 2019; Du et al., 2019; Chen et al., 2020). Besides, the convergence rate (Arora et al., 2019) and generalization bound (Cao and Gu, 2019; Zhu et al., 2022; Nguyen et al., 2021; Bombari et al., 2022) can be linked to the minimum eigenvalue of the NTK matrix. One can see that studying benign overfitting for DNNs is missing and it appears possible to borrow some ideas from deep learning theory, e.g., NTK. Nevertheless, we need to tackle the noisy training as well as the high-dimensional setting for benign overfitting, which is our main interest. ## 2 Problem Settings In this section, we detail the problem setting for a deep ReLU neural network trained by SGD from the perspective of notations, neural network architecture, initialization schemes, and optimization algorithms. ### Notation In this paper, we use the shorthand $[n]:=\{1,2,\ldots,n\}$ for a positive integer $n$. We denote by $a(n)\gtrsim b(n)$: the inequality $a(n)\geq cb(n)$ that hides a positive constant $c$ that is independent of $n$. Vectors (matrices) are denoted by boldface, lower-case (upper-case) letters. The standard Gaussian distribution is $\mathcal{N}(0,1)$ with the zero-mean and the identity variance. We use the $\text{Lip}_{f}$ to represent the Lipschitz constant of the function $f$. We follow the standard Bachmann-Landau notation in complexity theory e.g., $\mathcal{O}$, $o$, $\Omega$, and $\Theta$ for order notation. ### Network Here we introduce the formulation of DNNs. We focus on the typical depth-$L$ fully-connected ReLU neural networks with scalar output, width $m$ on the hidden layers and $n$ training data, $\forall i\in[n]$: \[\mathbf{h}_{i,0}=\mathbf{x}_{i}; \tag{1}\] \[\mathbf{h}_{i,l}=\phi(\mathbf{W}_{l}\mathbf{h}_{i,l-1});\quad\forall l\in[L- 1];\] \[f(\mathbf{x}_{i};\mathbf{W})=\mathbf{W}_{L}\mathbf{h}_{i,L-1};\] where $\mathbf{x}_{i}\in\mathbb{R}^{d}$ is the input, $f(\mathbf{x}_{i};\mathbf{W})\in\mathbb{R}$ is the neural network output, and $\phi=\max(0,x)$ is the ReLU activation function. The neural network parameters formulate the tuple of weight matrices $\mathbf{W}:=\{\mathbf{W}_{l}\}_{l=1}^{L}\in\{\mathbb{R}^{m\times d}\times(\mathbb{R}^ {m\times m})^{L-2}\times\mathbb{R}^{1\times m}\}$. Initialization: We follow the standard Neural Tangent Kernel (NTK) initialization (Allen-Zhu et al., 2019): \[[\mathbf{W}_{1}]_{i,j}\sim\mathcal{N}(0,\tfrac{2}{m});\quad\forall i,j\in[m]\times [d]; \tag{2}\] \[\mathbf{W}_{l}]_{i,j}\sim\mathcal{N}(0,\tfrac{2}{m});\quad\forall i,j \in[m]\quad\text{and}\quad l\in[L-2]+1;\] \[\mathbf{W}_{L}]_{i,j}\sim\mathcal{N}(0,1);\quad\forall i,j\in[1] \times[m].\] The related Neural Tangent Kernel (NTK) (Jacot et al., 2018) matrix of neural network $f$ can be expressed as: \[K_{\texttt{NTK}}(\mathbf{x},\widetilde{\mathbf{x}}):=\mathbb{E}_{\mathbf{W}}\left\langle \frac{\partial f(\mathbf{x};\mathbf{W})}{\partial\mathbf{W}},\frac{\partial f(\widetilde {\mathbf{x}};\mathbf{W})}{\partial\mathbf{W}}\right\rangle\,. \tag{3}\] By virtue of $\phi(x)=x\phi^{\prime}(x)$ of ReLU, we have $\mathbf{h}_{i,l}=\mathbf{D}_{i,l}\mathbf{W}_{l}\mathbf{h}_{i,l-1}$, where $\mathbf{D}_{i,l}$ is a diagonal matrix under the ReLU activation function defined as below. Definition 1 (Diagonal sign matrix).: For each $i\in[n]$, $l\in[L-1]$ and $k\in[m]$, the diagonal sign matrix $\mathbf{D}_{i,l}$ is defined as: $(\mathbf{D}_{i,l})_{k,k}=1\left\{(\mathbf{W}_{l}\mathbf{h}_{i,l-1})_{k}\geq 0\right\}$. In addition, we define $\omega$-neighborhood to describe the difference between two matrices. For any $\mathbf{W}\in\mathcal{W}$, we define its $\omega$-neighborhood as follows: Definition 2 ($\omega$-neighborhood).: \[\mathcal{B}(\mathbf{W},\omega):=\left\{\mathbf{W}^{\prime}\in\mathcal{W}:\left\\|\mathbf{ W}_{l}^{\prime}-\mathbf{W}_{l}\right\\|_{\text{F}}\leq\omega,l\in[L]\right\}\,.\] ### Optimization algorithm In our work, a deep ReLU neural network is trained by SGD on the training data $\{(\mathbf{x}_{i},y_{i})\}_{i=1}^{n}$ sampled from a joint distribution $P$. The data generation process is deferred to Section 3.1 for binary classification and Section 4 for regression. We employ the logistic loss for classification, which is defined as $\ell(z)=\log(1+\exp(-z))$, and denote $g(z):=-\ell^{\prime}(z)=\frac{1}{1+e^{z}}$ for notational simplicity. The expected risk is defined as $\mathbb{E}_{(\mathbf{x},y)\sim P}\ \ell\left(yf(\mathbf{x};\mathbf{W})\right)$. Denote the empirical risks under $\ell$ by: $\hat{L}(\mathbf{W}):=\frac{1}{n}\sum_{i=1}^{n}\ell\left(y_{i}f(\mathbf{x}_{i};\mathbf{W})\right)$, we employ SGD to minimize $\hat{L}(\mathbf{W})$ initialized at $\mathbf{W}^{(0)}$ with fixed step-size $\alpha>0$, as shown in Algorithm 1. For notational simplicity, at step $t$, the neural network output is denoted as $f_{i}^{(t)}=f(\mathbf{x}_{i};\mathbf{W}^{(t)})$ and the derivative of the loss function is realized to $g_{i}^{(t)}:=g(y_{i}f_{i}^{(t)})=g(y_{i}f(\mathbf{x}_{i};\mathbf{W}^{(t)}))$. ## 3 Main Results on Binary Classification In this section, we present our main result on benign overfitting of a ReLU DNN for binary classification under the lazy training regime. The data generation process is introduced in Section 3.1, and the related assumptions that are given in Section 3.2. Our main theory and proof sketch are presented in Section 3.3 and Section 3.4, respectively. We use NTK initialization (Allen-Zhu et al., 2019) in this section, but the main result can be easily extended to more initializations, such as He (He et al., 2015) and LeCun (LeCun et al., 2012). ### Data generation process We consider a standard mixture model setting (Chatterji and Long, 2021; Frei et al., 2022) in benign overfitting for binary classification, where a joint distribution $P$ is defined over $(\mathbf{x},y)\in\mathbb{R}^{d}\times\{\pm 1\}$ and samples from this distribution can have noisy labels. Following Frei et al. (2022), we first define the clean distribution $\widetilde{P}$ and then define the true distribution $P$ based on $\widetilde{P}$: 1. Sample a clean label $\widetilde{y}$ uniformly at random, $\widetilde{y}\sim\text{Uniform}(\{+1,-1\})$. 2. Sample $\mathbf{z}\sim P_{\text{clust}}$ that satisfy: * $P_{\text{clust}}=P_{\text{clust}}^{(1)}\times\cdots\times P_{\text{clust}}^{(d)}$ is a product distribution whose marginals are all mean-zero with the sub-Gaussian norm at most one; * $P_{\text{clust}}$ is a $\lambda$-strongly log-concave distribution over $\mathbb{R}^{d}$ for some $\lambda>0$; * For some $\kappa$, it holds that $\mathbb{E}_{\mathbf{z}\sim P_{\text{clust}}}(\\|\mathbf{z}\\|^{2})>\kappa d$. 3. Generate $\widetilde{\mathbf{x}}=\mathbf{z}+\widetilde{y}\mathbf{\mu}$. 4. Then, given a noise rate $\eta\in[0,\frac{1}{2})$, $P$ is any distribution over $\mathbb{R}^{d}\times\{\pm 1\}$ such that the marginal distribution of the features for $P$ and $\widetilde{P}$ coincide, and the total variation distance between the two distributions satisfies $d_{\text{TV}}(\widetilde{P},P)\leq\eta$. Specifically, $P$ has the same marginal distribution over $\mathbf{x}$ as $\widetilde{P}$, but a sample $(\mathbf{x},y)\sim P$ has a label equal to $\widetilde{y}$ with probability $1-\eta$ and has a label equal to $-\widetilde{y}$ with probability $\eta$. That is, the labels are flipped with $\eta$ ratio. We denote by $\mathcal{C}\subset[n]$ the set of indices corresponding to samples with clean labels, and $\mathcal{C}^{\prime}$ as the set of indices corresponding to noisy labels so that $i\in\mathcal{C}^{\prime}$ implies $(\mathbf{x}_{i},y_{i})\sim P$ is such that $y_{i}=-\widetilde{y}_{i}$ using the notation above. ### Assumptions We make two assumptions about the data distribution. Assumption 1(Du et al. (2019); Allen-Zhu et al. (2019)).: We assume that the data is bounded, i.e. there is a constant $C_{\text{norm}}$ that satisfies $\\|\mathbf{x}\\|_{2}\leq C_{\text{norm}}$. Assumption 2.: For two different data sample $(\mathbf{x}_{1},\widetilde{y}_{1}),(\mathbf{x}_{2},\widetilde{y}_{2})\sim\widetilde{P}$, The NTK kernel defined in Eq. (3) satisfies that: \[\mathbb{E}_{(\mathbf{x}_{1},\widetilde{y}_{1}),(\mathbf{x}_{2},\widetilde {y}_{2})\sim\widetilde{P}}\big{[}K_{\text{NTK}}(\mathbf{x}_{1},\mathbf{x}_{2})\| \widetilde{y}_{1}=\widetilde{y}_{2}\big{]}\] \[-\mathbb{E}_{(\mathbf{x}_{1},\widetilde{y}_{1}),(\mathbf{x}_{2},\widetilde {y}_{2})\sim\widetilde{P}}\big{[}K_{\text{NTK}}(\mathbf{x}_{1},\mathbf{x}_{2})\| \widetilde{y}_{1}\neq\widetilde{y}_{2}\big{]}\] \[\geq C_{N}>0\,.\] Remark: This assumption states that the NTK value for data points belonging to the same class is larger than that for a different class, in expectation. This makes sense in practice, since, as a kernel, the NTK is able to evaluate the similarity of two data points (Scholkopf et al., 2002): if they are from the same class, the similarity value is large and vice versa. To verify this assumption, we give an example of the two-layer NTK over a uniform distribution inside a multidimensional sphere such that $C_{N}=\Theta(1/\sqrt{d})$, refer to Appendix B. Besides, we also empirically verify our assumption on MNIST (Lecun et al., 1998) with ten digits from 0 to 9. We randomly sample $1,000$ data for each digit and calculate the empirical mean (to approximate the expectation) of the two-layer NTK kernel value over these digit pairs. The experimental result is shown in Figure 1. We can see that the confusion matrix usually has a larger diagonal element than its non-diagonal element, which implies that the kernel value on the same class is often larger than that of different classes. This verifies the justification of our assumption. ### Theoretical guarantees Based on our assumption, we are ready to present our theoretical result that the test error of a ReLU DNN is close to the Bayes-optimal (noise rate). Theorem 1.: _Given a DNN defined by Eq. (1) and trained by Algorithm 1 with a step size $\alpha\gtrsim L^{-2}(\log m)^{-5/2}$. Then under Assumption 1 and 2, for $\omega\leq\mathcal{O}(L^{-9/2}(\log m)^{-3})$ and $\lambda>0$, with probability at least $1-\mathcal{O}(nL^{2})\exp(-\Omega(m\omega^{2/3}L))$, we have:_ \[\mathbb{P}_{(\mathbf{x},y)\sim P}(y\neq\operatorname{sgn}(f(\mathbf{x}; \mathbf{W}^{(n)})))\] \[\leq \eta+\exp\bigg{(}-\lambda\Theta\bigg{(}\frac{n\alpha(1-2\eta)C_{N} }{\text{Lip}_{f(\mathbf{x};\mathbf{W}^{(n)})}}\bigg{)}^{2}\bigg{)}\,,\] _where the $\eta$ is the noise rate defined in Section 3.1 and $C_{N}$ is defined in Assumption 2._ Remark: Theorem 1 provides the upper bound on the test error rate, including two parts. The first part is the proportion of the wrong labels in the training data. The second part exponentially decreases with the square of the number of training samples $n$. Also, this term is positively correlated with the Lipschitz constants of DNNs after training, which implies that interpolating more smooth functions leads to a faster convergence rate. We take a closer look at this phenomenon in Section 4, analyzing how neural networks interpolate target functions in a regression setting. Overall, this bound shows that the models overfit the wrong or noisy data on the training set, but still achieve good generalization error on the testing set. This is consistent with previous work on broader settings of benign overfitting that are not limited to classification problems with label noise. For example, various regression problems Bartlett et al. (2020); Zou et al. (2021); Chatterji and Long (2022); Koehler et al. (2021); Tsigler and Bartlett (2020), classification problems of 2-layer networks Frei et al. (2022); Cao et al. (2022), 2-layer and 3-layer NTK networks Ju et al. (2021); Hu et al. (2021). Here we discuss the (nearly) zero-training loss and how the Lipschitz constant affects our error bounds. SGD can obtain arbitrarily (nearly) zero-training errors on the training set: A lot of work has shown that deep neural networks trained with SGD can obtain zero training error on the training set and perfectly fit any training label in both classification and regression problems with mean squared loss or logistic loss Du et al. (2019); Ba et al. (2019); Zou et al. (2020). These results for empirical loss need to be over-parameterized by the condition that $m=\text{poly}(n,L)$. In Appendix D, we provide proof that the loss can be arbitrarily small on the training set under the setting of Section 2. This indicates that when the training data has label noise, the neural network will learn all the noise, that is, overfitting. Combined with the bound of the test error rate in Theorem 1, we can say that the deep neural network has a benign overfitting phenomenon. Lipschitz constant of the deep neural network: Theorem 1 shows that the convergence rate of the test error rate with the amount of data and is closely related to the Lipschitz constant of the neural network. The Lipschitz constant of DNNs has been widely studied in Bubeck et al. (2021); Wu et al. (2021); Huang et al. (2021); Nguyen et al. (2021). For example, for ReLU DNNs, if we employ the result of Nguyen et al. (2021, Theorem 6.2): $\text{Lip}_{f}\lesssim\mathcal{O}\big{(}(2\log m)^{L-1}\big{)}$, then our bound is \[\mathbb{P}_{(\mathbf{x},y)\sim P}(y\neq\operatorname{sgn}(f(\mathbf{x}; \mathbf{W}^{(n)})))\] \[\qquad\lesssim\eta+\exp\left(-\lambda\Theta\Big{(}\frac{n}{(2 \log m)^{L-1}}\Big{)}^{2}\right)\,,\] which leads to a better convergence rate on generalization than the two-layer result Frei et al. (2022). ### Proof sketch of Theorem 1 Let us first introduce a few relevant lemmas. The first Lemma will follow by establishing a lower bound for the expected normalized margin on clean points, $\mathbb{E}_{(\mathbf{x},\widetilde{y})\sim\widetilde{P}}[\widetilde{y}f(\mathbf{x}; \mathbf{W})]/\text{Lip}_{f(\mathbf{x};\mathbf{W}^{(t)})}$. Lemma 1.: _Given a DNN defined by Eq. (1) and trained by Algorithm 1. For any $t\geq 0$, assuming $\mathbb{E}_{(\mathbf{x},\widetilde{y})\sim\widetilde{P}}[\widetilde{y}f(\mathbf{x}; \mathbf{W}^{(t)})]\geq 0$, then we have:_ \[\mathbb{P}_{(\mathbf{x},y)\sim P}(y\neq\operatorname{sgn}(f(\mathbf{x}; \mathbf{W}^{(t)})))\] \[\leq \eta+\exp\bigg{(}-\frac{\lambda}{4}\bigg{(}\frac{\mathbb{E}_{(\bm {x},\widetilde{y})\sim\widetilde{P}}[\widetilde{y}f(\mathbf{x};\mathbf{W}^{(t)})]}{ \text{Lip}_{f(\mathbf{x};\mathbf{W}^{(t)})}}\bigg{)}^{2}\bigg{)}\,.\] We next introduce some structural results concerning the neural network optimization objective. The following lemma states that near initialization, the neural network function is almost linear in terms of its weights. Lemma 2.: _Let $\mathbf{W},\mathbf{W}^{\prime}\in\mathcal{B}(\mathbf{W}^{(0)},\omega)$ with $\omega=\mathcal{O}(L^{-9/2}(\log m)^{-3})$, for any $\mathbf{x}\in\mathbb{R}^{d}$ that satisfy Assump Figure 1: Averaged kernel values among 10 classes in MNIST, where a larger kernel value indicates a higher similarity between two data pairs. tion 1, with probability at least $1-\exp(-\Omega(m\omega^{2}\log m))-\mathcal{O}(nL^{2})\exp(-\Omega(m\omega^{2/3}L))$, we have:_ \[\|f(\mathbf{x};\mathbf{W})-f(\mathbf{x};\mathbf{W}^{\prime})-\left\langle\nabla f( \mathbf{x};\mathbf{W}^{\prime}),\mathbf{W}-\mathbf{W}^{\prime}\right\rangle\|\] \[\leq \mathcal{O}(\sqrt{\omega^{2}L^{3}m\log m})\sum_{l=1}^{L-1}\\|\mathbf{ W}_{l}-\mathbf{W}_{l}^{\prime}\\|_{2}\.\] The following lemma describes the change of $yf(\mathbf{x};\mathbf{W}^{(t+1)})$ from time $t$ to $t+1$. Lemma 3.: _Given a DNN defined by Eq. (1) and trained by Algorithm 1. For any $t\geq 0$ and $(\mathbf{x},\widetilde{y})\sim\widetilde{P}$ that satisfy Assumption 1, with $\omega=\mathcal{O}(L^{-9/2}(\log m)^{-3})$, with probability at least $1-\exp(-\Omega(m\omega^{2}\log m))-\mathcal{O}(nL^{2})\exp(-\Omega(m\omega^{ 2/3}L))$, we have:_ \[\widetilde{y}[f(\mathbf{x};\mathbf{W}^{(t+1)})-f(\mathbf{x};\mathbf{W}^{(t)})]\] \[\geq \alpha g_{t}^{(t)}\left\langle\widetilde{y}\nabla f(\mathbf{x};\mathbf{ W}^{(t)}),y_{i}\nabla f(\mathbf{x}_{i};\mathbf{W}^{(t)})\right\rangle\] \[-\mathcal{O}(\sqrt{\omega^{2}L^{3}m\log m})\sum_{l=1}^{L-1}\left\\| \mathbf{W}_{l}^{(t+1)}-\mathbf{W}_{l}^{(t)}\right\\|_{2}\,\] _where $(\mathbf{x}_{i},y_{i})$ is the random selected training sample at step $t+1$._ Based on the previous lemmas, we can now derive a lower bound on the normalized margin. Note that this lower bound on the normalized margin in conjunction with Lemma 1 results in the test error bound for the main theorem. Lemma 4.: _Let us define a DNN using Eq. (1) and trained by Algorithm 1 with a step size $\alpha\gtrsim L^{-2}(\log m)^{-5/2}$. Then under Assumption 1 and 2, for any $t\geq 0$, $\omega\leq\mathcal{O}(L^{-9/2}(\log m)^{-3})$, with probability at least $1-\exp(-\Omega(m\omega^{2}\log m))-\mathcal{O}(nL^{2})\exp(-\Omega(m\omega^{2/ 3}L))$, we have:_ \[\mathbb{E}_{(\mathbf{x},\widetilde{y})\sim\widetilde{P}}[\widetilde{y}f(\mathbf{x};\bm {W}^{(t)})]\geq\Theta\big{(}t\alpha(1-2\eta)C_{N}\big{)}\,.\] Now, we can prove Theorem 1. Proof.: According to Lemma 1, choosing $t:=n$, we have: \[\mathbb{P}_{(\mathbf{x},y)\sim P}(y\neq\mathrm{sgn}(f(\mathbf{x};\mathbf{W}^{(n)})))\] \[\leq \eta+\exp\bigg{(}-\frac{\lambda}{4}\bigg{(}\frac{\mathbb{E}_{(\bm {x},\widetilde{y})\sim\widetilde{P}}[\widetilde{y}f(\mathbf{x};\mathbf{W}^{(n)})]}{ \text{\rm Lip}_{f(\mathbf{x};\mathbf{W}^{(n)})}}\bigg{)}^{2}\bigg{)}\,.\] Then, by Lemma 4, choosing $t:=n$ and $\alpha\gtrsim L^{-2}(\log m)^{-5/2}$, for $\omega\leq\mathcal{O}(L^{-9/2}(\log m)^{-3})$, with probability at least $1-\mathcal{O}(nL^{2})\exp(-\Omega(m\omega^{2/3}L))$, we have: \[\mathbb{E}_{(\mathbf{x},\widetilde{y})\sim P}[\widetilde{y}f(\mathbf{x};\mathbf{W}^{(n)})] \geq\Theta\big{(}n\alpha(1-2\eta)C_{N}\big{)}\,.\] Combine the results, we have: \[\mathbb{P}_{(\mathbf{x},y)\sim P}(y\neq\mathrm{sgn}(f(\mathbf{x};\mathbf{W}^{ (n)})))\] \[\leq \eta+\exp\bigg{(}-\lambda\Theta\bigg{(}\frac{n\alpha(1-2\eta)C_{ N}}{\text{\rm Lip}_{f(\mathbf{x};\mathbf{W}^{(n)})}}\bigg{)}^{2}\bigg{)}\,.\] ## 4 Interpolating Smooth Function by NTK In this section, we take a closer look at the phenomenon of the relationship between Lipschitz constants of DNNs and convergence rate in Theorem 1, and accordingly analyze how neural networks interpolate smooth ground-truth functions in a regression setting from an approximation theory view (Cucker and Zhou, 2007). In this section, we will also follow the NTK initialization (Allen-Zhu et al., 2019). For other different initialization (Cao and Gu, 2019; Arora et al., 2019), similar conclusions will apply. To be specific, let $X\subseteq\mathbb{R}^{d}$ be an input space, and $Y\subseteq\mathbb{R}$ be the output space, $f_{\rho}:X\to Y$ be the ground-truth function, that is smooth in RKHS, described by the source condition in Section 4.1. We assume that the data $(\mathbf{x},y)$ is sampled from an unknown distribution $\rho$, and $\rho_{X}$ is the marginal distribution of $\rho$ over $X$. The label is generated through $y=f_{\rho}(\mathbf{x})+\epsilon$, where $\epsilon$ is the noise. Accordingly, denote $L^{2}_{\rho_{X}}$ as the $\rho_{X}$ weighted $L^{2}$-space and its norm $\\|f\\|_{L^{2}_{\rho_{X}}}^{2}=\int_{X}\|f(\mathbf{x})\|^{2}\,d\rho_{X}(\mathbf{x})$, we are interested in the excess risk $\\|f(\mathbf{x};\mathbf{W}^{(t)})-f_{\rho}\\|_{L^{2}_{\rho_{X}}}^{2}$, which describes how neural networks interpolate/approximate a smooth ground-truth function in a certain space (Cucker and Zhou, 2007; Bach, 2017). In this section, we use the standard NTK network and initialization, which is equivalent to Arora et al. (2019) using the initialization with standard normal distribution together with the scale factor after each layer for the training dynamics. ### Assumptions We make the following assumptions: Assumption 3 (High dimensionality (Liang and Rakhlin, 2020; Liu et al., 2021)).: There exists universal constants $c_{1},c_{2}\in(0,\infty)$ such that $c_{1}\leq\frac{d}{n}\leq c_{2}$. Assumption 4 (Noise condition (Liang and Rakhlin, 2020; Liu et al., 2021)).: There exists a $\sigma_{\epsilon}>0$ such that $\mathbb{E}[(f_{\rho}(\mathbf{x})-y)^{2}\|\mathbf{x}]\leq\sigma_{\epsilon}^{2}$, almost surely. Assumption 5 (Geifman et al. (2020); Chen and Xu (2021)).: We assume that $\mathbf{x}_{i},\forall i\in[n]$ are i.i.d. sampled from a uniform distribution on the $d$-dimensional unit sphere. i.e. $\mathbf{x}\sim\text{Unif}(\mathbb{S}^{d-1}(1)),\;\mathbb{S}^{d-1}(1):=\big{\{}\mathbf{x} \in\mathbb{R}^{d}\|\,\\|\mathbf{x}\\|_{2}=1\big{\}}$. Remark: The i.i.d unit sphere data assumption implies that the data $\mathbf{x}$ is isotropic _asymptotically_ under our high-dimensional setting (Wainwright, 2019), i.e., $\mathbb{E}[\mathbf{x}\mathbf{x}^{\top}]=\mathbb{I}_{d}/d$. In fact, there is an alternative way in our proof by directly assuming $\mathbf{x}$ is sub-Gaussian and $\mathbb{E}[\mathbf{x}\mathbf{x}^{\top}]=\mathbb{I}_{d}/d$. Assumption 6 (Existence of $f_{\rho}$).: We assume the ground-truth function $f_{\rho}\in\mathcal{H}_{\text{NTK}}$, where $\mathcal{H}_{\text{NTK}}$ is the RKHS associated with the limiting NTK kernel. Remark: This is a standard assumption in learning theory by assuming that the ground-truth function $f_{\rho}$ is indeed realizable (Cucker and Zhou, 2007; Rudi and Rosasco, 2017; Liu et al., 2021). This assumption is a special case of the source condition (Cucker and Zhou, 2007) by taking certain values and can be easily extended to non-RKHS spaces or teacher-student settings (Hinton et al., 2015). For ease of analysis, we directly assume the ground-truth function in an RKHS. ### Kernel regression estimator Let $\mathbf{X}\in\mathbb{R}^{n\times d}$ be a matrix, each column of which is the input of one training sample, $\mathbf{\epsilon}\in\mathbb{R}^{n\times 1}$ be the noise in the output of training data. The empirical risk minimization (ERM) is defined with the squared loss: \[\hat{f}_{\mathbf{z}}=\arg\min_{f\in\mathcal{F}}\left\{\frac{1}{2n}\sum_{i=1}^{n}(f (\mathbf{x}_{i})-y_{i})^{2}\right\}\,, \tag{4}\] where the hypothesis space $\mathcal{F}$ can be defined properly. For example, if $\mathcal{F}$ is a RKHS $\mathcal{H}_{\mathbf{K}}$, Eq. (4) is formulated as a kernel regression. Denoting that $\mathbf{K}_{\texttt{ker}}(\mathbf{x},\mathbf{X})=[\mathbf{K}_{\texttt{ker}}(\mathbf{x},\mathbf{x}_{1 }),\mathbf{K}_{\texttt{ker}}(\mathbf{x},\mathbf{x}_{1}),\ldots,\mathbf{K}_{\texttt{ker}}(\mathbf{ x},\mathbf{x}_{n})]^{\top}\in\mathbb{R}^{n}$, the closed form of the kernel regression estimator to Eq. (4) is given by: \[f_{\texttt{ker}}=\mathbf{K}_{\texttt{ker}}(\mathbf{x},\mathbf{X})^{\top}\mathbf{K}_{\texttt{ ker}}^{-1}\mathbf{y}\,.\] If we use a neural network as in Eq. (1) to solve Eq. (4), the corresponding hypothesis space $\mathcal{F}_{\texttt{nn}}$ is: \[\mathcal{F}_{\texttt{nn}}:=\left\{f(\mathbf{x};\mathbf{W})\text{ admits Eq. \eqref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq eigenvalue of $\mathbf{K}_{\text{\tiny{RTK}}}$. Then, under Assumption 3 and 5, with probability at least $1-2e^{-n}$, we obtain that:_ \[\lambda_{0}\geq\left\{\begin{array}{ll}2\mu_{1}^{2}\frac{n}{d} \Big{(}\frac{3}{4}-\frac{c}{4}\sqrt{\frac{d}{n}}\Big{)}^{2},&\text{if}\quad n \geq d,\\ 2\mu_{1}^{2}\frac{n}{d}\Big{(}\sqrt{\frac{d}{n}}-\frac{c+6}{4}\Big{)}^{2},& \text{if}\quad n<d\,,\end{array}\right.\] _where we have an absolute constant $c=2^{3.5}\sqrt{\log(9)}\approx 16.77$ and $\mu_{1}$ is the $1$-st Hermite coefficient of the ReLU activation function._ Remark: This theorem provides the upper bound of the minimum eigenvalue of the NTK matrix of the infinite-width neural network under the high-dimensional setting and can be easily extended to the finite-width setting. Note that our result under the high dimensional setting is different from previous work under the fixed $d$ setting in Zhu et al. (2022). If we fix $d$ and vary $n$ from small to large, there exists a phase transition when $n$ increases, see Table 1. If $n\ll d$, the lower bound of the minimum eigenvalue of NTK $\lambda_{0}\geqslant\Omega(1)$, then when $n$ increases, we can see that $\lambda_{0}$ decreases to a bottom and then increases until $n:=d$. In the $n\geq d$ regime, there exists a similar trend to that of the $n\leq d$ regime: firstly decreasing and then increasing. When $n\gg d$, we have $\lambda_{0}\geqslant\Omega(n)$. ### Generalization error bound Based on the aforementioned upper and lower bounds of the minimum eigenvalue of NTK under the high dimensional setting, we establish the relationship between the minimum eigenvalue of NTK and the generalization error of DNNs. We provide a bound on the norm of the difference between the network output and ground truth function under the weighted $L^{2}_{\rho_{X}}$ space. Theorem 3 (An upper bound on the generalization error for deep over-parameterized NTK network).: _Let $\theta\in(0,1/2]$, $\delta$ and $c$ are some non-negative constants, the ground-truth function $f_{\rho}$ lies in a RKHS by Assumption 6 and $d$ large enough, under Assumption 3, 4 and 5, suppose that, $\omega\leq poly(1/n,\lambda_{0},1/L,1/\log(m),\epsilon,1/\log(1/\delta^{\prime }),\kappa),\ m\geq poly(1/\omega)$ and $\kappa=\mathcal{O}(\frac{\epsilon}{\log(n/\delta^{\prime})})$. then for any given $\varepsilon>0$, with high probability, we have:_ \[\mathbb{E}\left\\|f_{\text{nn}}-f_{\rho}\right\\|_{L^{2}_{\rho_{X}} }^{2}\lesssim\mathcal{O}\bigg{(}n^{-\theta}\log^{4}(\frac{2}{\delta})+\frac{ \sigma_{\epsilon}^{2}}{d}\mathcal{N}_{\widetilde{\mathbf{X}}}\] \[+\frac{\sigma_{\epsilon}^{2}\log^{2+4\epsilon}d}{d^{4\theta-1}} +\epsilon^{2}+\frac{n}{\lambda_{0}^{2}}\omega^{2/3}L^{5}m\log m+\frac{n^{3}} {\lambda_{0}^{6}\kappa^{2}}\bigg{)}\,,\] _where the $\lambda_{0}$ satisfies Theorem 2 and the effective dimension $\mathcal{N}_{\widetilde{\mathbf{X}}}$ is defined as:_ \[\mathcal{N}_{\widetilde{\mathbf{X}}}:=\sum_{i=0}^{n-1}\frac{\lambda_{i}( \widetilde{\mathbf{X}})}{(\lambda_{i}(\widetilde{\mathbf{X}}+\gamma))^{2}}\,,\] _with $\widetilde{\mathbf{X}}:=\beta\mathbf{X}\mathbf{X}^{\top}/d+\alpha\mathbf{1} \mathbf{1}^{\top}$ for some non-negative constants $\alpha$, $\beta$, $\gamma$._ Remark: This theorem builds a connection between DNNs and kernel methods in benign overfitting and gives the upper bound of the generalization error of the NTK network in the high-dimensional setting. To be specific,the first term is the upper bound of the bias of NTK regression, which decreases as the number of data increases. The second and the third terms jointly form the upper bound of variance of NTK regression, which is mainly affected by the effective dimension (eigenvalue decay) of the data. The fourth term is the error introduced by the initialization of the NTK neural network. The fifth and the sixth term reflect the difference between the finite-width NTK network and the infinite-width NTK network (neural tangent kernel regression), which decreases with the increase of the minimum eigenvalue of the NTK network we provide in Theorem 2. Under refined assumptions, e.g., source condition, capacity condition (Cucker and Zhou, 2007), we can achieve $\theta=1$ for a better convergence rate. Regarding the convergence properties, we need to make the following discussion. The three non-negative constants $\alpha$, $\beta$, and $\gamma$ are related to the linearization of the kernel matrix in the high dimension setting, refer to Liu et al. (2021) for details. Here we give the following discussion on $\mathcal{N}_{\widetilde{\mathbf{X}}}$ under three typical eigenvalue decay of $\mathbf{X}\mathbf{X}^{\top}$ cases, and then discuss our generalization bound. Note that when $n>d$, the sample matrix $\mathbf{X}\mathbf{X}^{\top}$ has at most $d$ eigenvalues, so we can directly have $\mathcal{N}_{\widetilde{\mathbf{X}}}\leq\mathcal{O}(d)$. Accordingly, here we present the results on the $n<d$ case. * Harmonic decay:$\lambda_{i}(\widetilde{\mathbf{X}})\propto n/i,\forall i\in\{1,2,\ldots,r_{\star}\}$ and $\lambda_{i}(\widetilde{\mathbf{X}})=0,\forall i\in\{r_{\star}+1,\ldots,n\}$. \begin{table} \begin{tabular}{l c c} \hline Range of $n$ & Trend w.r.t. $n$ & Limit bound \\ \hline $n\ll d$ & - & $2\mu_{1}^{2}$ \\ \hline $0\leq n\leq(\frac{4}{c+6})^{2}d$ & $\searrow$ & - \\ \hline $(\frac{4}{c+6})^{2}\leq n\leq d$ & $\nearrow$ & - \\ \hline $d\leq n\leq\frac{c^{2}}{9}d$ & $\searrow$ & - \\ \hline $\frac{c^{2}}{9}d\leq n$ & $\nearrow$ & - \\ \hline $n\gg d$ & - & $\frac{9}{8}\mu_{1}^{2}\frac{n}{d}$ \\ \hline \end{tabular} \end{table} Table 1: The trend of the bound with respect to $n$ under different range of $n$ values and a fixed $d$. We have: $\mathcal{N}_{\tilde{\mathbf{X}}}=\mathcal{O}(n)$, then the term $\frac{\sigma_{\epsilon}^{2}}{d}\mathcal{N}_{\tilde{\mathbf{X}}}\leq\mathcal{O}( \frac{\sigma_{\epsilon}^{2}}{d}n)$. * Polynomial decay:$\lambda_{i}(\widetilde{\mathbf{X}})\propto n{i^{-2a}}$ with $a>1/2,\forall i\in\{1,2,\dots,r_{\star}\}$ and $\lambda_{i}(\widetilde{\mathbf{X}})=0,\forall i\in\{r_{\star}+1,\dots,n\}$. We have: $\mathcal{N}_{\tilde{\mathbf{X}}}=\mathcal{O}(n^{1/2a})$, then the term $\frac{\sigma_{\epsilon}^{2}}{d}\mathcal{N}_{\tilde{\mathbf{X}}}\leq\mathcal{O}( \frac{\sigma_{\epsilon}^{2}}{d}n^{1/2a})\leq\mathcal{O}(\frac{\sigma_{\epsilon }^{2}}{d}n)$. * Exponential decay:$\lambda_{i}(\widetilde{\mathbf{X}})\propto ne^{-ai}$ with $a>0,\forall i\in\{1,2,\dots,r_{\star}\}$ and $\lambda_{i}(\widetilde{\mathbf{X}})=0,\forall i\in\{r_{\star}+1,\dots,n\}$. We have: $\mathcal{N}_{\tilde{\mathbf{X}}}=\frac{1}{a}\big{(}\frac{1}{\gamma+n\exp(-a(r_{ \star}+1))}-\frac{1}{\gamma+n\exp(-a)}\big{)}$, then the term $\frac{\sigma_{\epsilon}^{2}}{d}\mathcal{N}_{\tilde{\mathbf{X}}}\leq\mathcal{O}( \frac{\sigma_{\epsilon}^{2}}{d}\frac{\sigma^{\star_{\epsilon}}}{n})$. Based on our discussion on the eigenvalue decay, we are ready to discuss our generalization bound in Theorem 3. When $n,d$ are comparably large enough, e.g., $n\geq\frac{e^{2}}{9}d$, in this case, we have $\mathcal{N}_{\tilde{\mathbf{X}}}\leq\mathcal{O}(d)$, according to Theorem 2 for $\lambda_{0}$, three terms $n^{-\theta}\log^{4}(\frac{a}{2})$, $\frac{n}{\lambda_{0}^{2}}\omega^{2/3}L^{5}m\log m$ and $\frac{n^{3}}{\lambda_{0}^{6}\kappa^{2}}$ convergence to $0$. The term $\frac{\sigma_{\epsilon}^{2}\log^{2+4}d}{d^{6}-1}$ also converges to $0$ for a large enough $d$. Accordingly, for large enough $n$ and $d$, we have \[\mathbb{E}\left\\|f_{\text{nn}}-f_{\rho}\right\\|_{L^{2}_{\mathcal{ \tilde{\mathbf{X}}}}}^{2}\lesssim\mathcal{O}\big{(}\sigma_{\epsilon}^{2}+\epsilon ^{2}\big{)}\,,w.h.p\,,\] which show that the bound only depends on the noise and random initialization term, and thus coincide with previous work on benign overfitting (Frei et al., 2022; Cao et al., 2022; Ju et al., 2022; Arora et al., 2019b). Besides, we can also find that the phase transition exists in the minimum eigenvalue $\lambda_{0}$ and the effective dimension $\mathcal{N}_{\tilde{\mathbf{X}}}$ from $n<d$ and $n>d$. This also leads to a phase transition on the excess risk. Roughly speaking, the excess risk firstly increases with $n$ until $n:=d$ and then decreases with $n$ when $n>d$. ## 5 Conclusion and limitations In this work, we present a theoretical analysis of benign overfitting for deep ReLU NNs. For binary classification, our results demonstrate that DNNs under the lazy training regime obtain the _Bayes-optimal_ test error with a better convergence rate than Frei et al. (2022). For regression, our results exhibit a phase transition on the excess risk from $n<d$ to $n>d$, of which the excess risk converges to a constant order $\mathcal{O}(1)$ that only depends on label noise and initialization noise. The above two results theoretically validate the benign overfitting of DNNs. We need to mention that, our results are only applicable to lazy training regimes and appear difficult to be extended to the non-lazy training regime, commonly used in practice. This is because DNN cannot be linearly approximated well under the non-lazy training regime. We leave this as a future work. Besides, an interesting direction is, extending our data-generating distribution assumption from log-concave distribution to a general one, as we require it to ensure the output of neural networks is sub-Gaussian. ## Acknowledgements We are thankful to the reviewers for providing constructive feedback. This work was supported by Hasler Foundation Program: Hasler Responsible AI (project number 21043). This work was supported by SNF project - Deep Optimisation of the Swiss National Science Foundation (SNSF) under grant number 200021_205011. This work was supported by Zeiss. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement nd 725594 - time-data).
2310.17852	Function Space Bayesian Pseudocoreset for Bayesian Neural Networks	A Bayesian pseudocoreset is a compact synthetic dataset summarizing essential information of a large-scale dataset and thus can be used as a proxy dataset for scalable Bayesian inference. Typically, a Bayesian pseudocoreset is constructed by minimizing a divergence measure between the posterior conditioning on the pseudocoreset and the posterior conditioning on the full dataset. However, evaluating the divergence can be challenging, particularly for the models like deep neural networks having high-dimensional parameters. In this paper, we propose a novel Bayesian pseudocoreset construction method that operates on a function space. Unlike previous methods, which construct and match the coreset and full data posteriors in the space of model parameters (weights), our method constructs variational approximations to the coreset posterior on a function space and matches it to the full data posterior in the function space. By working directly on the function space, our method could bypass several challenges that may arise when working on a weight space, including limited scalability and multi-modality issue. Through various experiments, we demonstrate that the Bayesian pseudocoresets constructed from our method enjoys enhanced uncertainty quantification and better robustness across various model architectures.	Balhae Kim, Hyungi Lee, Juho Lee	2023-10-27T02:04:31Z	http://arxiv.org/abs/2310.17852v1	# Function Space Bayesian Pseudocoreset ###### Abstract A Bayesian pseudocoreset is a compact synthetic dataset summarizing essential information of a large-scale dataset and thus can be used as a proxy dataset for scalable Bayesian inference. Typically, a Bayesian pseudocoreset is constructed by minimizing a divergence measure between the posterior conditioning on the pseudocoreset and the posterior conditioning on the full dataset. However, evaluating the divergence can be challenging, particularly for the models like deep neural networks having high-dimensional parameters. In this paper, we propose a novel Bayesian pseudocoreset construction method that operates on a function space. Unlike previous methods, which construct and match the coreset and full data posteriors in the space of model parameters (weights), our method constructs variational approximations to the coreset posterior on a function space and matches it to the full data posterior in the function space. By working directly on the function space, our method could bypass several challenges that may arise when working on a weight space, including limited scalability and multi-modality issue. Through various experiments, we demonstrate that the Bayesian pseudocoresets constructed from our method enjoys enhanced uncertainty quantification and better robustness across various model architectures. ## 1 Introduction Deep learning has achieved tremendous success, but its requirement for large amounts of data makes it often inefficient or infeasible in terms of resources and computation. To enable continuous learning like humans, it is necessary to learn from a large number of data points in a continuous manner, which requires the ability to discern and retain important information. This motivates the learning of a coreset, a small dataset that is informative enough to represent a large dataset. On the other hand, the ability to incorporate uncertainties into predictions is essential for real-world applications, as it contributes to the safety and reliability of a model. One approach to achieving this is by adopting a Bayesian framework, where a prior distribution is established to represent our initial belief about the models. This belief is then updated through inference of posterior distributions based on the acquired knowledge. Although this approach shows promise, scalability becomes a concern when working with large-scale datasets during Bayesian inference. To address this issue, a potential solution is to employ a Bayesian coreset. A Bayesian coreset is a small subset of the original dataset where the posterior conditioning on it closely approximates the original posterior conditioning on the full dataset. Once the Bayesian coreset is trained, it can be utilized as a lightweight proxy dataset for subsequent Bayesian inference or as a replay buffer for continual learning or transfer learning. A Bayesian coreset is constructed by selecting a subset from a large dataset. However, recent research suggests that this approach may not be effective, especially in high-dimensional settings [19]. Instead, an alternative method of synthesizing a coreset, wherein the coreset is learned as trainable parameters, has been found to significantly enhance the quality of the approximation. This synthesized coreset is referred to as a Bayesian pseudocoreset. The process of learning a Bayesian pseudocoreset involves minimizing a divergence measure between the posterior of the full dataset and the posterior of the pseudocoreset. However, learning Bayesian pseudocoresets is generally challenging due to the intractability of constructing both the full dataset posterior and the pseudocoreset posterior, as well as the computation of the divergence between them, which necessitates approximation. Consequently, existing works on Bayesian pseudocoresets have primarily focused on small-scale problems [19; 8; 20; 21]. Recently, Kim et al. [15] introduced a scalable method for constructing Bayesian pseudocoresets using variational Gaussian approximation for the posteriors and minimizing forward KL divergence. Although their method shows promise, it still demands substantial computational resources for high-dimensional models like deep neural networks. In this paper, we present a novel approach to enhance the scalability of Bayesian pseudocoreset construction, particularly for Bayesian neural networks (BNNs) with a large number of parameters. Our proposed method operates in _function space_. When working with BNNs, it is common to define a prior distribution on the weight space and infer the corresponding weight posterior distribution, which also applies to Bayesian pseudocoreset construction. However, previous studies [31; 28] have highlighted the challenge of interpreting weights in high-dimensional neural networks, making it difficult to elicit meaningful prior distributions. Additionally, in high-dimensional networks, the loss surfaces often exhibit a complex multimodal structure, which means that proximity in the weight space does not necessarily imply proximity in the desired prediction variable [24; 30]. This same argument can be applied to Bayesian pseudocoreset construction, as matching the full data and pseudocoreset posteriors in the weight space may not result in an optimal pseudocoreset in terms of representation power and computational scalability. To be more specific, our method constructs a variational approximation to the pseudocoreset posteriors in function space by linearization and variational approximation to the true posterior. Then we learn Bayesian pseudocoreset by minimizing a divergence measure between the full data posterior and the pseudocoreset posterior in the function space. Compared to the previous weight space approaches, our method readily scales to the large models for which the weight space approaches were not able to compute. Another benefit of function space matching is that it does not constrain the architectures of the neural networks to be matched, provided that their inherited function space posteriors are likely to be similar. So for instance, we can train with multiple neural network architectures simultaneously with varying numbers of neurons or types of normalization layers, and we empirically observe that this improves the architectural robustness of the learned pseudocoresets. Moreover, it has another advantage that the posteriors learned from the Bayesian pseudocoreset in function space have better out-of-distribution (OOD) robustness, similar to the previous reports showing the benefit of function space approaches in OOD robustness [28]. In summary, this paper presents a novel approach to creating a scalable and effective Bayesian pseudocoreset using function space variational inference. The resulting Bayesian pseudocoreset is capable of being generated in high-dimensional image and deep neural network settings and has better uncertainty quantification abilities compared to weight space variational inference. Additionally, it has better architectural robustness. We demonstrate the efficiency of the function space Bayesian pseudocoreset through the various experiments. ## 2 Background ### Bayesian pseudocoresets In this paper, we focus on probabilistic models for supervised learning problem. Let $\theta\in\Theta$ be a parameter, and let $p(y\,\|\,x,\theta)$ be a probabilistic model indexed by the parameter $\theta$. Given a set of observations $\mathbf{x}:=(x_{i})_{i=1}^{n}$ and the set of labels $\mathbf{y}:=(y_{i})_{i=1}^{n}$ with each $x_{i}\in\mathcal{X}$ and $y_{i}\in\mathcal{Y}$, we are interested in updating our prior belief $\pi_{0}(\theta)$ about the parameter to the posterior, \[\pi_{\mathbf{x}}(\theta):=\frac{\pi_{0}(\theta)}{Z(\mathbf{y}\,\|\,\mathbf{x} )}\prod_{i=1}^{n}p(y_{i}\,\|\,x_{i},\theta),\quad Z(\mathbf{y}\,\|\,\mathbf{x} ):=\int_{\Theta}\prod_{i=1}^{n}p(y_{i}\,\|\,x_{i},\theta)\pi_{0}(\mathrm{d} \theta). \tag{1}\] However, when the size of the dataset $n$ is large, the computation of the posterior distribution can be computationally expensive and infeasible. To overcome this issue, Bayesian pseudocoresets are constructed as a synthetic dataset $\mathbf{u}=(u_{j})_{j=1}^{m}$ with $m\ll n$ with the set of labels where the posterior conditioning on it approximates the original posterior $\pi_{\mathbf{x}}(\theta)$. \[\pi_{\mathbf{u}}(\theta)=\frac{\pi_{0}(\theta)}{Z(\tilde{\mathbf{y}}\,\|\,\mathbf{ u})}\prod_{j=1}^{m}p(\tilde{y}_{j}\,\|\,u_{j},\theta),\quad Z(\tilde{\mathbf{y}}\,\|\, \mathbf{u}):=\int_{\Theta}\prod_{j=1}^{m}p(\tilde{y}_{j}\,\|\,u_{j},\theta)\pi_{0 }(\mathrm{d}\theta). \tag{2}\] This approximation is made possible by solving an optimization problem that minimizes a divergence measure $D$ between the two posterior distributions 1 Footnote 1: In principle, we should learn the (pseudo)labels $\tilde{\mathbf{y}}$ as well, but for classification problem, we can simply fix it as a constant set containing equal proportion of all possible classes. We assume this throughout the paper. \[\mathbf{u}^{}=\operatorname{arg\,min}_{\mathbf{u}}\ D(\pi_{\mathbf{x}},\pi_ {\mathbf{u}}). \tag{3}\] In a recent paper [15], three variants of Bayesian pseudocoresets were proposed using different divergence measures, namely reverse Kullback-Leibler divergence, Wasserstein distance, and forward Kullback-Leibler divergence. However, performing both the approximation and the optimization in the parameter space can be computationally challenging, particularly for high-dimensional models such as deep neural networks. ### Bayesian pseudocoresets in weight-space Kim et al. [15] advocates using forward KL divergence as the divergence measure when constructing Bayesian pseudocoresets, with the aim of achieving a more even exploration of the posterior distribution of the full dataset when performing uncertainty quantification with the learned pseudocoreset. The forward KL objective is computed as, \[\begin{split} D_{\mathrm{KL}}[\pi_{\mathbf{x}}\\|\pi_{\mathbf{u}} ]&=\log Z(\tilde{\mathbf{y}}\,\|\,\mathbf{u})-\log Z(\mathbf{y}\,\| \,\mathbf{x})\\ &\quad+\mathbb{E}_{\pi_{\mathbf{x}}}\bigg{[}\sum_{i=1}^{n}\log p( y_{i}\,\|\,x_{i},\theta)\bigg{]}-\mathbb{E}_{\pi_{\mathbf{x}}}\bigg{[}\sum_{j=1}^{m} \log p(\tilde{y}_{j}\,\|\,u_{j},\theta)\bigg{]}.\end{split} \tag{4}\] The derivative of the divergence with respect to the pseudocoreset $\mathbf{u}$ is computed as \[\nabla_{\mathbf{u}}D_{\mathrm{KL}}[\pi_{\mathbf{x}}\\|\pi_{\mathbf{u}}]= \mathbb{E}_{\pi_{\mathbf{u}}}\bigg{[}\nabla_{\mathbf{u}}\sum_{j=1}^{m}\log p( \tilde{y}_{j}\,\|\,u_{j},\theta)\bigg{]}-\nabla_{\mathbf{u}}\mathbb{E}_{\pi_{ \mathbf{x}}}\bigg{[}\sum_{j=1}^{m}\log p(\tilde{y}_{j}\|u_{j},\theta))\bigg{]} \tag{5}\] For the gradient, we need the expected gradients of the log posteriors that require sampling from the posteriors $\pi_{\mathbf{x}}$ and $\pi_{\mathbf{u}}$. Most of the probabilistic models do not admit simple closed-form expressions for these posteriors, and it is not easy to simulate those posteriors for high-dimensional models. To address this, Kim et al. [15] proposes to use a Gaussian variational distributions $q_{\mathbf{u}}(\theta)$ and $q_{\mathbf{x}}(\theta)$ to approximate $\pi_{\mathbf{x}}$ and $\pi_{\mathbf{u}}$ whose means are set to the parameters obtained from the SGD trajectories, \[q_{\mathbf{u}}(\theta)=\mathcal{N}(\theta;\mu_{\mathbf{u}},\Sigma_{\mathbf{u} }),\quad q_{\mathbf{x}}(\theta)=\mathcal{N}(\theta;\mu_{\mathbf{x}},\Sigma_{ \mathbf{x}}), \tag{6}\] where $\mu_{\mathbf{u}}$ and $\mu_{\mathbf{x}}$ are the maximum a posteriori (MAP) solutions computed for the dataset $\mathbf{u}$ and $\mathbf{x}$, respectively. $\Sigma_{\mathbf{u}}$ and $\Sigma_{\mathbf{x}}$ are covariances. The gradient, with the stop gradient applied to $\mu_{\mathbf{u}}$, is approximated as, \[\frac{\nabla_{\mathbf{u}}}{S}\sum_{s=1}^{S}\left(\sum_{j=1}^{m}\log p\Big{(} \tilde{y}_{j}\,\|\,u_{j},\,\text{sg}(\mu_{\mathbf{u}})+\Sigma_{\mathbf{u}}^{1 /2}\varepsilon_{\mathbf{u}}^{(s)}\Big{)}-\sum_{j=1}^{m}\log p\Big{(}\tilde{y}_ {j}\,\|\,u_{j},\mu_{\mathbf{x}}+\Sigma_{\mathbf{x}}^{1/2}\varepsilon_{\mathbf{ x}}^{(s)}\Big{)}\right). \tag{7}\] Here, $\varepsilon_{\mathbf{u}}^{(s)}$ and $\varepsilon_{\mathbf{x}}^{(s)}$ are i.i.d. standard Gaussian noises and $S$ is the number of Monte-Carlo samples. Expert trajectoriesApproximating the full data and coreset posteriors with variational distributions as specified above requires $\mu_{\mathbf{u}}$ and $\mu_{\mathbf{x}}$ as consequences of running optimization algorithms untill convergence. While this may be feasible for small datasets, for large-scale setting of our interest, obtaining $\mu_{\mathbf{u}}$ and $\mu_{\mathbf{x}}$ from scratch at each iteration for updating $\mathbf{u}$ can be time-consuming. To alleviate this, in the dataset distillation literature, Cazenavette et al. [7] proposed to use the _expert trajectories_, the set of pretrained optimization trajectories constructed in advance to the coreset learning. Kim et al. [15] brought this idea to Bayesian pseudocoresets, where a pool of pretrained trajectories are assume to be given before pseudocoreset learning. At each step of pseudocoreset update, a checkpoint $\theta_{0}$ from an expert trajectory is randomly drawn from the pool, and then $\mu_{\mathbf{u}}$ and $\mu_{\mathbf{x}}$ are quickly constructed by taking few optimization steps from $\theta_{0}$. Function space Bayesian pseudocoreset ### Function space Bayesian neural networks We follow the framework presented in Rudner et al. (2017); Rudner et al. (2017) to define a Function-space Bayesian Neural Network (FBNN). Let $\pi_{0}(\theta)$ be a prior distribution on the parameter and $g_{\theta}:\mathcal{X}\rightarrow\mathbb{R}^{d}$ be a neural network index by $\theta$. Let $h:\Theta\rightarrow(\mathcal{X}\rightarrow\mathbb{R}^{d})$ be a deterministic mapping from a parameter $\theta$ to a neural network $g_{\theta}$. Then a function-space prior is simply defined as a pushforward $\nu_{0}(f)=h_{}\pi_{0}(f):=\pi_{0}(h^{-1}(f))$. The corresponding posterior is also defined as a pushforward $\nu_{\mathbf{x}}(f)=h_{}\pi_{\mathbf{x}}(f)$ and so is the pseudocoreset posterior $\nu_{\mathbf{u}}(f)=h_{}\pi_{\mathbf{u}}(f)$. ### Learning function space Bayesian pseudocoresets Given the function space priors and posteriors, a Function space Bayesian PseudoCoreset (FBPC) is obtained by minimizing a divergence measure between the function space posteriors, \[\mathbf{u}^{}=\operatorname{arg\,min}_{\mathbf{u}}\;D(\nu_{\mathbf{x}},\nu_{ \mathbf{u}}). \tag{8}\] We follow Kim et al. (2017) suggesting to use the forward KL divergence, so our goal is to solve \[\mathbf{u}^{}=\operatorname{arg\,min}_{\mathbf{u}}\;D_{\mathrm{KL}}[\nu_{ \mathbf{x}}\\|\nu_{\mathbf{u}}]. \tag{9}\] The following proposition provides an expression for the gradient to minimize the divergence, whose proof is given A. Proposition 3.1.: _The gradient of the forward KL divergence with respect to the coreset $\mathbf{u}$ is_ \[\nabla_{\mathbf{u}}D_{\mathrm{KL}}[\nu_{\mathbf{x}}\\|\nu_{\mathbf{u}}]=-\nabla _{\mathbf{u}}\mathbb{E}_{[\nu_{\mathbf{x}}\|\mathbf{u}]}[\log p(\tilde{\mathbf{ y}}\,\|\,\mathbf{f}_{\mathbf{u}})]+\mathbb{E}_{[\nu_{\mathbf{u}}]_{\mathbf{u}}}[ \nabla_{\mathbf{u}}\log p(\tilde{\mathbf{y}}\,\|\,\mathbf{f}_{\mathbf{u}})], \tag{10}\] _where $[\nu_{\mathbf{x}}\|_{\mathbf{u}}$ and $[\nu_{\mathbf{u}}]_{\mathbf{u}}$ are finite-dimensional distributions of the stochastic processes $\nu_{\mathbf{x}}$ and $\nu_{\mathbf{u}}$, respectively, $\mathbf{f}_{\mathbf{u}}:=(f(u_{j}))_{j=1}^{m}$, and $p(\tilde{\mathbf{y}}\,\|\,\mathbf{f}_{\mathbf{u}})=\prod_{j=1}^{m}p(\tilde{y}_ {j}\,\|\,f(u_{j}))$._ To evaluate the gradient Eq. 10, we should identify the finite-dimensional functional posterior distributions $[\nu_{\mathbf{x}}\|_{\mathbf{u}}$ and $[\nu_{\mathbf{u}}]_{\mathbf{u}}$. While this is generally intractable, as proposed in Rudner et al. (2017); Rudner et al. (2017), we can instead consider a linearized approximation of the neural network $g_{\theta}$, \[\tilde{g}_{\theta}(\cdot)=g_{\mu_{\mathbf{x}}}(\cdot)+\mathcal{J}_{\mu_{ \mathbf{x}}}(\cdot)(\theta-\mu_{\mathbf{x}}), \tag{11}\] where $\mu_{\mathbf{x}}=\mathbb{E}_{\pi_{\mathbf{x}}}[\theta]$ and $\mathcal{J}_{\mu_{\mathbf{x}}}(\cdot)$ is the Jacobian of $g_{\theta}$ evaluated at $\mu_{\mathbf{x}}$. Then we approximate the function space posterior $\nu_{\mathbf{x}}$ with $\tilde{\nu}_{\mathbf{x}}:=\tilde{h}_{}\pi_{\mathbf{x}}$ where $\tilde{h}(\theta)=\tilde{g}_{\theta}$, and as shown in Rudner et al. (2017); Rudner et al. (2017), the finite dimensional distribution $[\tilde{\nu}_{\mathbf{x}}]_{\mathbf{u}}$ is a multivariate Gaussian distribution, \[[\tilde{\nu}_{\mathbf{x}}]_{\mathbf{u}}(\mathbf{f}_{\mathbf{u}})=\mathcal{N} \Big{(}\mathbf{f}_{\mathbf{u}}\,\|\,g_{\mu_{\mathbf{x}}}(\mathbf{u}),\mathcal{J }_{\mu_{\mathbf{x}}}(\mathbf{u})\Sigma_{\mathbf{x}}\mathcal{J}_{\mu_{\mathbf{ x}}}(\mathbf{u})^{\top}\Big{)}, \tag{12}\] with $\Sigma_{\mathbf{x}}=\mathrm{Cov}_{\pi_{\mathbf{x}}}(\theta)$. Similarly, we obtain \[[\tilde{\nu}_{\mathbf{u}}]_{\mathbf{u}}(\mathbf{f}_{\mathbf{u}})=\mathcal{N} \Big{(}\mathbf{f}_{\mathbf{u}}\,\|\,g_{\mu_{\mathbf{u}}}(\mathbf{u}),\mathcal{J }_{\mu_{\mathbf{u}}}(\mathbf{u})\Sigma_{\mathbf{u}}\mathcal{J}_{\mu_{\mathbf{ u}}}(\mathbf{u})^{\top}\Big{)}, \tag{13}\] with $\mu_{\mathbf{u}}:=\mathbb{E}_{\pi_{\mathbf{u}}}[\theta]$ and $\Sigma_{\mathbf{u}}:=\mathrm{Cov}_{\pi_{\mathbf{u}}}(\theta)$. Using these linearized finite-dimensional distribution, we can approximate \[\nabla_{\mathbf{u}}D_{\mathrm{KL}}[\nu_{\mathbf{x}}\\|\nu_{\mathbf{u}}]=- \nabla_{\mathbf{u}}\mathbb{E}_{[\tilde{\nu}_{\mathbf{x}}]_{\mathbf{u}}}[ \log p(\tilde{\mathbf{y}}\,\|\,\mathbf{f}_{\mathbf{u}})]+\mathbb{E}_{[\tilde{ \nu}_{\mathbf{u}}]_{\mathbf{u}}}[\nabla_{\mathbf{u}}\log p(\tilde{\mathbf{y}} \,\|\,\mathbf{f}_{\mathbf{u}})], \tag{14}\] ### Tractable approximation to the gradient Even with the linearization, evaluating Eq. 14 is still challenging because it requires obtaining $\mu_{\mathbf{x}}$ and $\Sigma_{\mathbf{x}}$ which are the statistics of the weight-space posterior $\pi_{\mathbf{x}}$. Rudner et al. (2017) proposes to learn a variational approximation $q_{\mathbf{x}}(\theta)$ in the _weight-space_, and use the linearized pushforward of the variational distribution $\tilde{h}_{}q_{\mathbf{x}}$ as a proxy to the function space posterior. Still, this approach requires computing the heavy Jacobian matrix $\mathcal{J}_{\mathbb{E}_{\eta_{\mathbf{x}}}[\theta]}(\mathbf{u})$, so may not be feasible for our scenario where we have to compute such variational approximations _at each_ update of the pseudocoreset $\mathbf{u}$. Instead, we choose to directly construct a variational approximations to the finite-dimensional distributions of the function space posteriors, that is, \[\begin{split}&[\tilde{\nu}_{\mathbf{x}}]_{\mathbf{u}}(\mathbf{f}_{ \mathbf{u}})\approx q_{\mathbf{x}}(\mathbf{f}_{\mathbf{u}})=\mathcal{N}( \mathbf{f}_{\mathbf{u}}\,\|\,g_{\hat{\mu}_{\mathbf{u}}}(\mathbf{u}),\hat{\Psi}_ {\mathbf{x}}),\\ &[\tilde{\nu}_{\mathbf{u}}]_{\mathbf{u}}(\mathbf{f}_{\mathbf{u}} )\approx q_{\mathbf{u}}(\mathbf{f}_{\mathbf{u}})=\mathcal{N}(\mathbf{f}_{ \mathbf{u}}\,\|\,g_{\hat{\mu}_{\mathbf{u}}}(\mathbf{u}),\hat{\Psi}_{\mathbf{u} }),\end{split} \tag{15}\] where $(\hat{\mu}_{\mathbf{x}},\hat{\Psi}_{\mathbf{x}})$ and $(\hat{\mu}_{\mathbf{u}},\hat{\Psi}_{\mathbf{u}})$ are variational parameters for the full data and coreset posteriors. Inspired Kim et al. [15], we construct the variational parameters using expert trajectories. Unlike [15], we simply let the MAP solution computed for $\mathbf{x}$, $\theta_{\mathbf{x}}$, by sampling a checkpoint from the later part of the expert trajectories, and obtain the MAP solution of $\mathbf{u}$, $\theta_{\mathbf{u}}$, by directly optimizing an initial random parameter. Then we obtain $\hat{\mu}_{\mathbf{x}}$ and $\hat{\mu}_{\mathbf{u}}$ using $\mathbf{u}$. For the covariance matrices $\hat{\Psi}_{\mathbf{x}}$ and $\hat{\Psi}_{\mathbf{u}}$, while Kim et al. [15] proposed to use spherical Gaussian noises, we instead set them as an empirical covariance matrices of the samples collected from the optimization trajectory. Specifically, we take additional $K$ steps from each MAP solution to compute the empirical covariance. \[\begin{split}\theta_{\mathbf{x}}^{(0)}&=\theta_{ \mathbf{x}},\quad\theta_{\mathbf{x}}^{(t)}=\texttt{opt}(\theta_{\mathbf{x}}^{( t-1)},(\mathbf{x},\mathbf{y})),\quad\hat{\mu}_{\mathbf{x}}=g_{\texttt{sg}(\theta_{ \mathbf{x}}^{(0)})}(\mathbf{u}),\\ \hat{\Psi}_{\mathbf{x}}&:=\texttt{sg}\Bigg{(}\mathrm{ diag}\Bigg{(}\frac{1}{K}\sum_{k=1}^{K}g_{\theta_{\mathbf{x}}^{(k)}}^{2}( \mathbf{u})-\bigg{(}\frac{1}{K}\sum_{k=1}^{K}g_{\theta_{\mathbf{x}}^{(k)}}( \mathbf{u})\bigg{)}^{2}\Bigg{)}\Bigg{)},\end{split} \tag{16}\] where $\texttt{opt}(\theta,\mathbf{x})$ is a step of SGD optimization applied to $\theta$ with data $\mathbf{x}$ and the squares in the $\mathrm{diag}(\cdot)$ are applied in element-wise manner. Note also that we are applying the stop-gradient operations for to block the gradient flow that might lead to complication in the backpropagation procedure. The variational parameters $(\hat{\mu}_{\mathbf{u}},\hat{\Psi}_{\mathbf{u}})$ are constructed in a similar fashion, but using the psedocoreset $(\mathbf{u},\tilde{\mathbf{y}})$ instead of the original data $(\mathbf{x},\mathbf{y})$. It is noteworthy that our approach is similar to one of the methods in Bayesian learning, SWAG [18]. However, while SWAG focuses on collecting statistics on weight space trajectories, our method constructs statistics in function spaces. This distinction makes our approach more suitable and scalable for pseudocoreset construction. The overview of proposed method is provided in Fig. 1. With the variational approximations constructed as described, we obtain a Monte-Carlo estimator of Eq. 14, \[\begin{split}&\nabla_{\mathbf{u}}D_{\mathrm{KL}}[\nu_{\mathbf{x}} \|\nu_{\mathbf{u}}]\approx-\nabla_{\mathbf{u}}\mathbb{E}_{q_{\mathbf{x}}( \mathbf{f}_{\mathbf{u}})}[\log p(\tilde{\mathbf{y}}\,\|\,\mathbf{f}_{\mathbf{u }})]+\mathbb{E}_{q_{\mathbf{u}}(\mathbf{f}_{\mathbf{u}})}\Big{[}\nabla_{ \mathbf{u}}\log p(\tilde{\mathbf{y}}\,\|\,\mathbf{f}_{\mathbf{u}})\Big{]}\\ &=-\nabla_{\mathbf{u}}\mathbb{E}_{p(\varepsilon_{\mathbf{x}})}[ \log p(\tilde{\mathbf{y}}\,\|\,\hat{\mu}_{\mathbf{x}}+\hat{\Psi}_{\mathbf{x}}^{1 /2}\varepsilon_{\mathbf{x}})]+\mathbb{E}_{p(\varepsilon_{\mathbf{u}})}\Big{[} \nabla_{\mathbf{u}}\log p(\tilde{\mathbf{y}}\,\|\,\hat{\mu}_{\mathbf{u}}+\hat{ \Psi}_{\mathbf{u}}^{1/2}\varepsilon_{\mathbf{u}})\Big{]}\\ &\approx\frac{1}{S}\sum_{s=1}^{S}\bigg{(}-\nabla_{\mathbf{u}}\log p \Big{(}\tilde{\mathbf{y}}\,\|\,\hat{\mu}_{\mathbf{x}}+\hat{\Psi}_{\mathbf{x}}^{1 /2}\varepsilon_{\mathbf{x}}^{(s)}\Big{)}+\nabla_{\mathbf{u}}\log p\Big{(} \tilde{\mathbf{y}}\,\|\,\hat{\mu}_{\mathbf{u}}+\hat{\Psi}_{\mathbf{u}}^{1/2} \varepsilon_{\mathbf{u}}^{(s)}\Big{)}\bigg{)},\end{split} \tag{17}\] where $p(\varepsilon_{\mathbf{x}})$ and $p(\varepsilon_{\mathbf{u}})$ are standard Gaussians, $(\varepsilon_{\mathbf{x}}^{(s)})_{s=1}^{S}$ and $(\varepsilon_{\mathbf{u}}^{(s)})_{s=1}^{S}$ are i.i.d. samples from them. Figure 1: The conceptual overview of our proposed training procedure. ### Multiple architectures FBPC training One significant advantage of function space posterior matching is that the function is typically of much lower dimension compared to the weight. This makes it more likely for function spaces to exhibit similar posterior shapes in the vicinity of the MAP solutions. This characteristic of function space encourages the exploration of function space pseudocoreset training in the context of multiple architectures. Because, the task of training a coreset that matches the highly complex weight space posterior across multiple architectures is indeed challenging, while the situation becomes relatively easier when dealing with architectures that exhibit similar function posteriors. Therefore we propose a novel multi-architecture FBPC algorithm in Algorithm 1. The training procedure involves calculating the FBPC losses for each individual architecture separately and then summing them together to update. This approach allows us to efficiently update the pseudocoreset by considering the contributions of each architecture simultaneously. We will empirically demonstrate that this methodology significantly enhances the architecture generalization ability of pseudocoresets in Section 5. ``` 0: Set of architectures $\mathcal{A}$, expert trajectories $\{\mathcal{E}^{(a)}:a\in\mathcal{A}\}$, prior distributions of parameters $\{\pi_{0}^{(a)}:a\in\mathcal{A}\}$, an optimizer opt. Initialize $\mathbf{u}$ with random minibatch of coreset size $m$. for$i=1,\dots,N$do Initialize the gradient of pseudocoreset, $\mathbf{g}\gets 0$. for$a\in\mathcal{A}$do Sample the MAP solution computed for $\mathbf{x}$, $\theta_{\mathbf{x}}\in\mathcal{E}^{(a)}$. Sample an initial random parameter $\theta_{0}\sim\pi_{0}^{(a)}(\theta)$. repeat $\theta_{t}\leftarrow\text{opt}(\theta_{t-1},(\mathbf{u},\tilde{\mathbf{y}}))$ until converges to obtain the MAP solution computed for $\mathbf{u}$, $\theta_{\mathbf{u}}$. Obtain $\hat{\mu}_{\mathbf{x}}$, $\hat{\mu}_{\mathbf{u}}$, $\hat{\Psi}_{\mathbf{x}}$, $\hat{\Psi}_{\mathbf{u}}$ by Eq. 16. Compute the pseudocoreset gradient $\mathbf{g}^{(a)}$ using Eq. 17. $\mathbf{g}\leftarrow\mathbf{g}+\mathbf{g}^{(a)}$. endfor Update the pseudocoreset $\mathbf{u}$ by using the gradient $\mathbf{g}$. endfor ``` Algorithm 1* Multi-architecture Function space Bayesian Pseudocoreset ### Compare to weight space Bayesian pseudocoresets By working directly on the function space, our method could bypass several challenges that may arise when working on a weight space. Indeed, a legitimate concern arises regarding multi-modality, as the posterior distributions of deep neural networks are highly complex. It makes the optimization of pseudocoresets on weight space difficult. Moreover, minimization of weight space divergence does not necessarily guarantee proximity in the function space. Consequently, although we try to minimize the weight space divergence, there is a possibility that the obtained function posterior may significantly deviate from the true posterior. However, if we directly minimize the divergence between the function distributions, we can effectively address this issue. On the other hand, there is an additional concern related to memory limitations. While it has been demonstrated in Kim et al. [15] that the memory usage of Bayesian pseudocoresets employing forward KL divergence is not excessively high, we can see that Eq. 7 requires Monte-Carlo samples of weights, which requires $\mathcal{O}(Sp)$ where $S$ and $p$ represent the number of Monte-Carlo samples and the dimensionality of the weights, respectively. This dependence on Monte-Carlo sampling poses a limitation for large-scale networks when memory resources are constrained. In contrast, our proposed method requires significantly less memory, $\mathcal{O}(Sd)$ where $d$ represents the dimensionality of the functions. Indeed, all the results presented in this paper were obtained using a single NVIDIA RTX-3090 GPU with 24GB VRAM. ## 4 Related work Bayesian coresetsBayesian Coreset [3; 4; 5; 14] is a field of research aimed at addressing the computational challenges of MCMC and VI on large datasets in terms of time and space complexity [9; 1; 26]. It aims to approximate the energy function of the entire dataset using a weighted sum of a small subset. However for high-dimensional data, Manousakas et al. [19] demonstrates that considering only subsets as Bayesian coreset is not sufficient, as the KL divergence between the approximated coreset posterior and the true posterior increases with the data dimension, and they proposed Bayesian pseudocorests. There are recent works on constructing pseudocorester variational posterior to be more flexible [8] or how to effectively optimize the divergences between posteriors [15; 8; 20; 21]. However, there is still a limitation in constructing high-dimensional Bayesian pseudocorests specifically for deep neural networks. Dataset distillationDataset distillation also aims to synthesize the compact datasets that capture the essence of the original dataset. However, the dataset distillation places particulary on optimizing the test performance of the distilled dataset. Consequently, the primary objective in dataset distillation is to maximize the performance of models trained using the distilled dataset, and researchers provides how to effectively solve this bi-level optimization [33; 23; 22; 40; 36; 39; 7]. In recent work, Kim et al. [15] established a link between specific dataset distillation methods and optimizing certain divergence measures associated with Bayesian pseudocoresets. Function space variational inferenceAlthough Bayesian neural networks exhibit strong capabilities in performing variational inference, defining meaningful priors or efficiently inferring the posterior on weight space is still challenging due to their over-parametrization. To overcome this issue, researchers have increasingly focused on function space variational inference [6; 2; 16; 30; 25; 32; 27; 17]. For instance, Sun et al. [31] introduced a framework that formulates the KL divergence between functions as the supremum of marginal KL divergences over finite sets of inputs. Wang et al. [34] utilizes particle-based optimization directly in the function space. Furthermore, Rudner et al. [28] recently proposed a scalable method for function space variational inference on deep neural networks. ## 5 Experiments ### Experimental Setup In our study, we employed the CIFAR10, CIFAR100 and Tiny-ImageNet datasets to create Bayesian pseudocoresets of coreset size $m\in\{1,10,50\}$ images per class (ipc). These pseudocoresets were then evalatuted by conducting the Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) [9] algorithm on those pseudocoresets. We measured the top-1 accuracy and negative log-likelihood of the SGHMC algorithm on the respective test datasets. Following the experimental setup of previous works [15; 7; 39], we use the differentiable siamese augmentation [37]. For the network architectures, we use 3-layer ConvNet for CIFAR10 and CIFAR100, and 4-layer ConvNet for Tiny-ImageNet. We employed three baseline methods to compare the effectiveness of function space Bayesian pseudocoresets. The first baseline is the random coresets, which involves selecting a random mini-batch of the coreset size. The others two baseline methods, BPC-rKL [15; 19] and BPC-fKL [15], are Bayesian pseudocoresets on weight space. BPC-rKL and BPC-fKL employ reverse KL divergence and forward KL divergence as the divergence measures for their training, respectively. ### Main Results Table 1 and Table 2 show the results of each baseline and our method for each dataset. For BPC-rKL and BPC-fKL, we used the official code from [15] for training the pseudocoresets, and only difference is that we used our own SGHMC hyperparameters during evaluation. For detailed experiment setting, please refer to Appendix C. As discussed earlier, we utilized the empirical covariance to variational posterior instead of using naive isotropic Gaussian for the function space variational posterior. To assess the effectiveness of using sample covariance, we compare these two in Table 1, as we also presented the results for FBPC-isotropic, which represents the FBPC trained with a unit covariance Gaussian posteriors. The results clearly demonstrate that using sample covariance captures valuable information from the posterior distribution, resulting in improved performance. Overall, the results presented in Table 1 and Table 2 demonstrate that our method also outperforms the baseline approaches, including random coresets, BPC-rKL and BPC-fKL, in terms of both accuracy and negative log-likelihood, especially on the large-scale datasets in Table 2. Furthermore, the Bayesian pseudocoreset can be leveraged to enhance robustness against distributional shifts when combined with Bayesian model averaging. To assess the robustness of our function space Bayesian pseudocoresets on out-of-distribution inputs, we also conducted experiments using the CIFAR10-C dataset [11]. This dataset involves the insertion of image corruptions into the CIFAR10 images. By evaluating the performance of the pseudocoresets on CIFAR10-C, we can see the model's ability to handle input data that deviates from the original distribution. In Table 3, we provide the results for top-1 accuracy and degradation scores, which indicate the extent to which accuracy is reduced compared to the in-distribution's test accuracy. The result demonstrates that our FBPC consistently outperforms the weight space Bayesian pseudocoreset, BPC-fKL. ### Architecture generalization In this section, we aim to demonstrate the architecture generalizability of FBPC and emphasize the utility of multi-architecture training as we discussed in the previous section. We specifically focus on investigating the impact of varying normalization layers on the generalizability of the pseudocoreset, since it is widely recognized that a pseudocoreset trained using one architecture may struggle to generalize effectively to a model that employs different normalization layers. We have also included the results of cross-architecture experiments that involve changing the architecture itself in Appendix D.1. To assess this, we compare a single architecture trained pseudocoreset and a multiple architecture trained pseudocoreset. For single architecture training, we initially train a pseudocoreset using one architecture with a specific normalization layer, for instance we use instance normalization. Subsequently, we evaluate the performance of this pseudocoreset on four different types of normalization layers: instance normalization, group normalization, layer normalization, and batch normalization. For multiple architecture training, we aggregate four losses for single architecture training of each architecture, and train the pseudocoreset with the sum of all losses, as mentioned in previous section. \begin{table} \begin{tabular}{c c\|c c c\|c c} \hline \hline \multirow{2}{}{ipc} & SGHMC & Random & BPC-rKL [15, 19] & BPC-fKL [15] & FBPC-isotropic (Ours) & FBPC (Ours) \\ \hline \multirow{2}{}{1} & Acc ($\uparrow$) & 16.30$\pm$0.74 & 20.44$\pm$1.06 & 34.50$\pm$1.62 & 32.00$\pm$0.75 & 35.45$\pm$0.31 \\ & NLL ($\downarrow$) & 4.66$\pm$0.03 & 4.51$\pm$0.10 & 3.86$\pm$0.13 & 3.40$\pm$0.27 & 3.79$\pm$0.04 \\ \hline \multirow{2}{}{10} & Acc ($\uparrow$) & 32.48$\pm$0.34 & 37.92$\pm$0.66 & 56.19$\pm$0.61 & 61.43$\pm$0.35 & 62.33$\pm$0.34 \\ & NLL ($\downarrow$) & 2.98$\pm$0.03 & 2.47$\pm$0.04 & 1.48$\pm$0.02 & 1.35$\pm$0.02 & 1.31$\pm$0.02 \\ \hline \multirow{2}{}{50} & Acc ($\uparrow$) & 49.68$\pm$0.46 & 51.86$\pm$0.38 & 64.74$\pm$0.32 & 71.33$\pm$0.19 & 71.23$\pm$0.17 \\ & NLL ($\downarrow$) & 2.06$\pm$0.02 & 1.95$\pm$0.02 & 1.26$\pm$0.01 & 1.03$\pm$0.01 & 1.03$\pm$0.05 \\ \hline \hline \end{tabular} \end{table} Table 1: Averaged test accuracy and negative log-likelihoods of models trained on each Bayesian pseudocoreset from scratch using SGHMC on the CIFAR10 dataset. Bold is the best and underline is the second best. These values are averaged over 5 random seeds. \begin{table} \begin{tabular}{c c\|c c\|c c c} \hline \hline & & \multicolumn{2}{c\|}{CIFAR100} & \multicolumn{2}{c}{Tiny-ImageNet} \\ & \multirow{2}{}{ipc} & 1 & 10 & 50 & 1 & 10 & 50 \\ \hline \multirow{2}{}{Random} & Acc ($\uparrow$) & 4.82$\pm$0.47 & 18.0$\pm$0.31 & 35.1$\pm$0.23 & 1.90$\pm$0.08 & 7.21$\pm$0.04 & 19.15$\pm$0.12 \\ & NLL ($\downarrow$) & 5.55$\pm$0.07 & 4.57$\pm$0.01 & 3.35$\pm$0.01 & 6.18$\pm$0.04 & 5.77$\pm$0.02 & 4.88$\pm$0.01 \\ \multirow{2}{}{BPC-fKL} & Acc ($\uparrow$) & 14.7$\pm$0.16 & 28.1$\pm$0.60 & 37.1$\pm$0.33 & 3.98$\pm$0.13 & 11.4$\pm$0.45 & 22.18$\pm$0.32 \\ & NLL ($\downarrow$) & 4.17$\pm$0.05 & 3.53$\pm$0.05 & 3.28$\pm$0.24 & 5.63$\pm$0.03 & 5.08$\pm$0.05 & 4.65$\pm$0.02 \\ \hline \multirow{2}{}{FBPC (Ours)} & Acc ($\uparrow$) & 21.0$\pm$0.76 & 39.7$\pm$0.31 & 44.47$\pm$0.35 & 10.14$\pm$0.68 & 19.42$\pm$0.51 & 26.43$\pm$0.31 \\ & NLL ($\downarrow$) & 3.76$\pm$0.11 & 2.67$\pm$0.02 & 2.63$\pm$0.01 & 4.69$\pm$0.05 & 4.14$\pm$0.02 & 4.30$\pm$0.05 \\ \hline \hline \end{tabular} \end{table} Table 2: Averaged test accuracy and negative log-likelihoods of models trained on each Bayesian pseudocoreset from scratch using SGHMC on the CIFAR100 and Tiny-ImageNet datasets. These values are averaged over 3 random seeds. As depicted in Fig. 1(a), we observe that both WBPC (Weight space Bayesian pseudocoresets) and FBPC-single (Function space Bayesian pseudocoresets trained on a single architecture) exhibit a notable trend, that they tend to not perform well when evaluated on the architecture that incorporates different normalizations, regardless of whether it is trained on weight space or function space. On the other hand, when trained with multiple architectures, both WBPC-multi and FBPC-multi perform well across the all architectures, while notably FBPC-multi significantly outperforms WBPC-multi. As mentioned in the previous section, we hypothesize that the superior performance of FBPC compared to WBPC can be attributed to the likelihood of having similar function space posterior across architectures. To validate this, we conduct an examination of the logit values for each sample across different architectures. As an illustration, we provide an example pseudocoreset image belonging to the class label "dog" along with its corresponding logits for all four architectures. As Fig. 1(b) shows, it can be observed that the logits display a high degree of similarity, indicating a strong likelihood of matching function posterior distributions. Our analysis confirms that, despite architectural disparities, the function spaces generated by these architectures exhibit significant similarity and it contributes to superiority of FBPC in terms of architecture generalizability. ## 6 Conclusion In this paper, we explored the function space Bayesian pseudocoreset. We constructed it by minimizing forward KL divergence between the function posteriors of pseudocoreset and the entire dataset. To optimize the divergence, we proposed a novel method to effectively approximate the function posteriors with an efficient training procedure. Finally, we empirically demonstrated the superiority of our function space Bayesian pseudocoresets compared to weight space Bayesian pseudocoresets, in terms of test performance, uncertainty qualification, OOD robustness, and architectural robustness. LimitationDespite showing promising results on function space Bayesian pseudocoresets, there still exist a few limitations in the training procedure. Our posterior approximation strategy requires \begin{table} \begin{tabular}{c c\|c c c c c c c c c\|c} \hline \hline & corruption & BN & DB & ET & FT & GB & JPEG & MB & PIX & SN & SP & Avg. \\ \hline \multirow{2}{}{BPC-fKL} & Acc ($\uparrow$) & 33.5 & 34 & 35.9 & 25.1 & 33.7 & 39.1 & 32.7 & 38.3 & 28.9 & 41.2 & 34.3 \\ & Degradation ($\downarrow$) & 40.3 & 39.4 & 36 & 55.2 & 39.9 & 30.3 & 41.6 & 31.6 & 48.4 & 26.5 & 38.9 \\ \hline \multirow{2}{}{FBPC} & Acc ($\uparrow$) & 48.9 & 46.4 & 47.6 & 41.7 & 44.0 & 52.0 & 44.3 & 51.0 & 47.1 & 52.3 & 47.5 \\ & Degradation ($\downarrow$) & 21.5 & 25.7 & 23.7 & 33.0 & 29.3 & 16.4 & 28.8 & 18.1 & 24.4 & 16.1 & 23.7 \\ \hline \hline \end{tabular} \end{table} Table 3: Test accuracy and degradation scores of models trained on each Bayesian pseudocoreset from scratch using SGHMC on the CIFAR10-C. Degradation refers to the extent by which a model’s accuracy decreases when evaluated on the CIFAR10-C dataset compared to the CIFAR10 test dataset. Figure 2: Results for multiple architecture FBPC training. the MAP solutions, which necessitates training them prior to each update step or preparing expert trajectories. This can be time-consuming and requires additional memory to store the expert trajectories. Societal ImpactsOur work is hardly likely to bring any negative societal impacts. ## Acknowledgments This work was supported by KAIST-NAVER Hypercreative AI Center, Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST), No.2022-0-00184, Development and Study of AI Technologies to Inexpensively Conform to Evolving Policy on Ethics, and No.2022-0-00713, Meta-learning Applicable to Real-world Problems) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A5A708390812)
2306.12014	3HAN: A Deep Neural Network for Fake News Detection	The rapid spread of fake news is a serious problem calling for AI solutions. We employ a deep learning based automated detector through a three level hierarchical attention network (3HAN) for fast, accurate detection of fake news. 3HAN has three levels, one each for words, sentences, and the headline, and constructs a news vector: an effective representation of an input news article, by processing an article in an hierarchical bottom-up manner. The headline is known to be a distinguishing feature of fake news, and furthermore, relatively few words and sentences in an article are more important than the rest. 3HAN gives a differential importance to parts of an article, on account of its three layers of attention. By experiments on a large real-world data set, we observe the effectiveness of 3HAN with an accuracy of 96.77%. Unlike some other deep learning models, 3HAN provides an understandable output through the attention weights given to different parts of an article, which can be visualized through a heatmap to enable further manual fact checking.	Sneha Singhania, Nigel Fernandez, Shrisha Rao	2023-06-21T04:34:27Z	http://arxiv.org/abs/2306.12014v1	# 3HAN: A Deep Neural Network for ###### Abstract The rapid spread of fake news is a serious problem calling for AI solutions. We employ a deep learning based automated detector through a three level hierarchical attention network (3HAN) for fast, accurate detection of fake news. 3HAN has three levels, one each for words, sentences, and the headline, and constructs a news vector: an effective representation of an input news article, by processing an article in an hierarchical bottom-up manner. The headline is known to be a distinguishing feature of fake news, and furthermore, relatively few words and sentences in an article are more important than the rest. 3HAN gives a differential importance to parts of an article, on account of its three layers of attention. By experiments on a large real-world data set, we observe the effectiveness of 3HAN with an accuracy of 96.77%. Unlike some other deep learning models, 3HAN provides an understandable output through the attention weights given to different parts of an article, which can be visualized through a heatmap to enable further manual fact checking. Keywords:Fake news, deep learning, text representation, attention mechanism, text classification. ## 1 Introduction The spread of fake news is a matter of concern due to its possible role in manipulating public opinion. We define fake news in line with The New York Times as a "made up story with the intention to deceive, often with monetary gain as a motive" [1]. The fake news problem is complex given its varied interpretations across demographics. We present a three level hierarchical attention network (3HAN) which creates an effective representation of a news article called _news vector_. A news vector can be used to classify an article by assigning a probability of being fake. Unlike other neural models which are opaque in their internal reasoning and give results that are difficult to analyze, 3HAN provides an importance score for each word and sentence of an input article based on its relevance in arriving at the output probability of that article being fake. These importance scores can be visualized through a heatmap, providing key words and sentences to be investigated by human fact-checkers. Current work in detecting misinformation is divided between automated fact checking [2], reaction based analysis [3] and style based analysis [4]. We explore the nascent domain of using neural models to detect fake news. Current state-of-the-art general purpose text classifiers like Bag-of-words [5], Bag-of-ngrams with SVM [6], CNNs, LSTMs and GRUs [7] can be used to classify articles by simply concatenating the headline with the body. This concatenation though, fails to exploit the article structure. In 3HAN, we interpret the structure of an article as a three level hierarchy modelling article semantics on the principle of compositionality [8]. Words form sentences, sentences form the body and the headline with the body forms the article. We hypothesize forming an effective representation of an article using the hierarchy and the interactions between its parts. These interactions take the form of context of a word in its neighbouring words, coherence of a sentence with its neighbouring sentences and stance of a headline with respect to the body. Words, sentences and headline are differentially informative dependent on their interactions in the formation of a news vector. We incorporate three layers of attention mechanisms [9] to exploit this differential relevance. The design of 3HAN is inspired by the hierarchical attention network (HAN) [10]. HAN is used to form a general document representation. We design 3HAN unique to the detection of fake news. When manually fact-checking an article the first thing that catches the eye is the headline. We observe a headline to be (i) a distinctive feature of an article [11], (ii) a concise summary of the article body and (iii) inherently containing useful information in the form of its stance with respect to the body. We refer to these observations as our _headline premise_. The third level in 3HAN is especially designed to use our headline premise. From our headline premise, we hypothesize that a neural model should accurately classify articles based on headlines alone. Using this hypothesis, we use headlines to perform a supervised pre-training of the initial layers of 3HAN for a better initialization of 3HAN. The visualization of attention layers in 3HAN indicates important parts of an article instrumental in detecting an article as fake news. These important parts can be further investigated by human fact-checkers. We compare the performance of 3HAN with multiple state-of-the-art traditional and neural baselines. Experiments on a large real world news data set demonstrate the superior performance of 3HAN over all baselines with 3HAN performing with an accuracy of 96.24%. Our pre-trained 3HAN model is our best performing model with an accuracy of 96.77%.1 Footnote 1: Our code is available at: [https://github.com/ni9elf/3HAN](https://github.com/ni9elf/3HAN). ## 2 Model Design The architecture of 3HAN is shown in Fig. 1. We define a news vector as a projection of a news article into a vector representation suitable for effective classification of articles. A news vector is constructed using 3HAN. To capture the body hierarchy and interactions between parts when forming the news vector, 3HAN uses the following parts from HAN [10]: word sequence encoder, word level attention (Layer 1), sentence encoder, sentence level attention (Layer 2). In addition to the preceding parts, we exploit our headline premise by adding: headline-body encoder and headline-body level attention (Layer 3). Sequence Encoder using GRU. A Gated Recurrent Unit (GRU) [12] adaptively captures dependencies between sequential input sequences over time. Gating signals control how the previous hidden state $h_{t-1}$ and current input $x_{t}$ generate an intermediate hidden state $\widetilde{h_{t}}$ to update the current hidden state $h_{t}$. GRU consists of a reset gate $r_{t}$ and an update gate $z_{t}$. $r_{t}$ determines how to combine $x_{t}$ with $h_{t-1}$ while $z_{t}$ determines how much of $h_{t-1}$ and $\widetilde{h_{t}}$ to use. $\odot$ denotes the Hadamard product. The GRU model is presented at time $t$ as: \[\widetilde{h_{t}}=\tanh\left(W_{h}x_{t}+U_{h}\left(r_{t}\odot h_{t-1}\right)+ b_{h}\right) \tag{1}\] \[h_{t}=\left(1-z_{t}\right)\odot h_{t-1}+z_{t}\odot\widetilde{h_{t}} \tag{2}\] with the gates presented as: \[z_{t}=\sigma\left(W_{z}x_{t}+U_{z}h_{t-1}+b_{z}\right),\;r_{t}=\sigma\left(W_{ r}x_{t}+U_{r}h_{t-1}+b_{r}\right) \tag{3}\] Word Encoder. We denote word $j$ of sentence $i$ by $w_{ij}$ with sentence $i$ containing $T_{i}$ words. Each word $w_{ij}$ is converted to a word embedding $x_{ij}$ using GloVe [13] embedding $W_{e}\left(x_{ij}=W_{e}\left(w_{ij}\right)\right)$. We use a bidirectional GRU [9] to form an annotation of each word which summarizes the _context_ of the word with Figure 1: Model Architecture of 3HAN preceding and following words in the sentence. A bidirectional GRU consists of a forward $\overrightarrow{\text{GRU}}$ and backward $\overleftarrow{\text{GRU}}$. The overhead arrow in our notation does not denote a vector, it instead denotes the direction of the GRU run. $\overrightarrow{\text{GRU}}$ reads the word embedding sequence ordered $(x_{i1},x_{i2},\ldots,x_{iT_{i}})$ to form forward annotations using hidden states $\left(\overrightarrow{h}_{i1}^{w},\overrightarrow{h}_{i2}^{w},\ldots, \overrightarrow{h}_{iT_{i}}^{w}\right)$. Similarly $\overleftarrow{\text{GRU}}$ reads the word embedding sequence ordered $(x_{iT_{i}},x_{iT_{i}-1},\ldots,x_{i1})$ to form backward annotations $\left(\overleftarrow{h}_{iT_{i}},\overleftarrow{h}_{iT_{i}-1}^{w},\ldots, \overleftarrow{h}_{i1}^{w}\right)$. $h_{ij}^{w}$ is formed as $\left[\overrightarrow{h}_{ij}^{w},\overleftarrow{h}_{ij}^{w}\right]$ (concatenation). \[\overrightarrow{h}_{ij}^{w}=\overrightarrow{\text{GRU}}\left(x_{ik}\right),k \in[1,j] \tag{4}\] \[\overleftarrow{h}_{ij}^{w}=\overleftarrow{\text{GRU}}\left(x_{ik}\right),k \in[T_{i},j] \tag{5}\] \[h_{ij}^{w}=\left[\overrightarrow{h}_{ij}^{w},\overleftarrow{h}_{ij}^{w}\right] \tag{6}\] Word Attention. A sentence representation is formed using an attention layer to extract relevant words of a sentence. The word annotation $h_{ij}^{w}$ is fed through a one-layer MLP to get a hidden representation $u_{ij}$[10]. The similarity of each word $u_{ij}$ with a _word level relevance vector_$u_{w}$ decides the attention weights $\alpha_{ij}$ normalized using a softmax function [10]. The sentence encoding $s_{i}$ is a weighted attentive sum of the word annotations. The relevance vector can be interpreted as representing the contextually most relevant word over all words in the sentence. $u_{w}$ is fixed over all inputs as a global parameter of our model and jointly learned in the training process. \[u_{ij}=\tanh\left(W_{w}h_{ij}^{w}+b_{w}\right) \tag{7}\] \[\alpha_{ij}=\frac{\exp\left(u_{ij}^{T}u_{w}\right)}{\sum_{j}\exp\left(u_{ij}^ {T}u_{w}\right)},\;s_{i}=\sum_{j}\alpha_{ij}h_{ij}^{w} \tag{8}\] Sentence Encoder. Similar to the word encoder, a bidirectional GRU is applied to $(s_{1},s_{2},\ldots,s_{L})$ to compute the forward annotations $\overrightarrow{h}_{i}^{s}$ and backward annotations $\overleftarrow{h}_{i}^{s}$ for each sentence. These annotations capture the _coherence_ of a sentence with respect to its neighbouring sentences in both directions of the body. $h_{i}^{s}$ is formed as $\left[\overrightarrow{h}_{i}^{s},\overleftarrow{h}_{i}^{s}\right]$. Sentence Attention. Similar to word attention, we identify relevant sentences in the formation of the body vector $v_{b}$ by using an attention layer. A _sentence level relevance vector_$u_{s}$ decides attention weights $\alpha_{i}$ for sentence annotation $h_{i}^{s}$. $u_{s}$ can be interpreted as representing the coherently most relevant sentence over all sentences in the body. $v_{b}$ is composed using $\sum_{i}\alpha_{i}h_{i}^{s}$. Headline Encoder. To exploit our headline premise we design a third layer of encoding and attention with the headline being inputted word by word. We denote the $k$ words of the headline by $w_{01}$ to $w_{0k}$. The word embedding $y_{i}$ for word $w_{0i}$ is obtained using GloVe embeddings $(W_{e})$ by $y_{i}=W_{e}\left(w_{0i}\right)$. We denote $v_{b}$ as $y_{k+1}$. A bidirectional GRU is run on $(y_{1},y_{2},\ldots,y_{k+1})$ to compute the forward and backward annotations of each word. These annotations capture the _stance of the headline_ words with respect to the body word. The digit 3 in our notation denotes the third level. $h_{i}^{3}$ is formed as $\left[\overrightarrow{h}_{i}^{3},\overleftarrow{h}_{i}^{3}\right]$. \[\overrightarrow{h}_{i}^{3}=\overrightarrow{\mathrm{GRU}}\left(y_{j}\right),j \in\left[1,i\right],\ \overleftarrow{h}_{i}^{3}=\overleftarrow{\mathrm{GRU}}\left(y_{j}\right),j \in\left[k+1,i\right] \tag{9}\] Headline Attention. A _relevance vector_$u_{3}$ is used to compute the attention weights $\beta_{i}$ for annotation $h_{i}^{3}$. The news vector $v_{n}$ is formed as the weighted sum of the annotations $h_{i}^{3}$ with $\beta_{i}$ as the weights. \[u_{i}=\tanh\left(W_{3}h_{i}^{3}+b_{3}\right) \tag{10}\] \[\beta_{i}=\frac{\exp\left(u_{i}^{T}u_{3}\right)}{\sum_{i}\exp\left(u_{i}^{T}u _{3}\right)},\ v_{n}=\sum_{i}\beta_{i}h_{i}^{3} \tag{11}\] News Vector for Classification. We use the news vector $v_{n}$ as a feature vector for classification. We use the sigmoid layer $z=\mathrm{sigmoid}\left(W_{c}v_{n}+b_{c}\right)$ as our classifier with binary cross-entropy loss $L=-\sum_{d}p_{d}\log q_{d}$ to train 3HAN. In the loss function $q_{d}$ is the predicted probability and $p_{d}$ is the ground truth label (either fake or genuine) of article $d$. Supervised Pre-training using Headlines We propose a supervised pre-training of Layer 1 consisting of the word encoder and an attention layer of 3HAN for a better initialization of the model. The pre-training is performed using the headlines only. The output label for a headline input is the corresponding article label. ## 3 Experiments ### News Data Set Due to the high turnaround time of manual fact-checking, the number of available manually fact-checked articles is too few to train deep neural models. We shift our fact-checked requirement from an article level to a website level. Keeping with our definition of fake news, we assume that every article from a website shares the same label (fake or genuine) as its containing website. PolitiFact [14] a respected fact-checking website released a list of sites manually investigated and labelled. We use those sites from this list labelled fake. Forbes [15] compiled a list of popular genuine sites across US demographics. Statistics of our data set is provided in Table 1. To maintain a similar distribution as fake articles, we use genuine articles from January 1, 2016 to June 1, 2017, with 65% coming from the 2016 US elections and politics, 15% from world news, 15% from regional news and 5% from entertainment. ### Baselines To validate the effectiveness of our model, we compare 3HAN with current state-of-the-art traditional and deep learning models. The input is the article text formed by concatenating the headline with the body. #### 3.2.1 Word Count Based Models. These methods use a hand crafted feature vector derived from variations of frequency of words of an article. A binomial logistic regression is used as the classifier. 1. _Majority_ uses the heuristic of taking the majority label in the training set as the assigning label to every point in the test set. 2. _Bag-of-words and its TF-IDF_ constructs a vocabulary of the most frequent 50,000 words [5]. The count of these words is used as features. The TF-IDF count is used as features in the other model variant. 3. _Bag-of-ngrams and its TF-IDF_ uses the count of the 50,000 most frequent ngrams ($n<=5$). The features are formed as in the previous model. 4. _SVM+Bigrams_ uses the count of the 50,000 most frequent bigrams as features with an SVM classifier [6]. #### 3.2.2 Neural Models. The classifier used is a dense sigmoid layer. 1. _GloVe-Ave_ flattens the article text to a word level granularity as a sequence of words. The GloVe embeddings of all words are averaged to form the feature vector. 2. _GRU_ treats the article text as a sequence of words. A GRU with an annotation dimension of 300 is run on the sequence of GloVe word embeddings. The hidden annotation after the last time step is used as the feature vector. 3. _GRU-Ave_ runs a GRU on the sequence of word embeddings and returns all hidden annotations at each time step. The average of these hidden annotations is used as the feature vector. 4. _HAN and Variants_ include HAN-Ave, Han-Max and HAN [10]. HAN uses a two level hierarchical attention network. HAN-Ave and Han-Max replaces the attention mechanism with average and max pooling for composition respectively. Since the code is not officially released we use our own implementation. \begin{table} \begin{tabular}{l c c c c} \hline \hline Type & Sites & Articles & \begin{tabular}{c} Average \\ Words \\ \end{tabular} & \begin{tabular}{c} Average \\ Sentences \\ \end{tabular} \\ \hline Fake & 19 & 20,372 & 34.20 & 16.44 \\ Genuine & 9 & 20,932 & 32.78 & 27.55 \\ \hline \hline \end{tabular} \end{table} Table 1: Dataset Statistics: (average words per sentence, average sentences per article) ### Experimental Settings We split sentences of bodies and tokenized sentences and headlines into words using Stanford CoreNLP [16]. We lower cased and cleaned tokens by retaining alphabets, numerals and significant punctuation marks. When building the vocabulary we retained words with frequency more than 5. We treat words appearing exactly 5 times as a special single unknown token (UNK). We used 100 dimensional GloVe embeddings to initialize our word embedding matrix and allowed it to be fine tuned. For missing words in GloVe, we initialized their word embedding from a uniform distribution on $(-0.25,0.25)$[17]. We padded (or truncated) each sentence and headline to an average word count of 32 and each article to an average sentence count of 21. Hyper parameters are tuned on the validation set. We used 100 dimensional GloVe embeddings and 50 dimensional GRU annotations giving a combined annotation of 100 dimensions. The relevance vector at word, sentence and headline-body level are of 100 dimensions trained as a parameter of our model. We used SGD with a learning rate of 0.01, momentum of 0.9 and mini batch size of 32 to train all neural models. Accuracy was our evaluation metric since our data set is balanced. ### Results and Analysis We used a train, validation and test split of $20\%\mid 10\%\mid 70\%$ for neural models and a train and test split of $30\%\mid 70\%$ for word count based models. In 3HAN-Ave vectors are composed using average, in 3HAN-Max vectors are composed using max pooling, 3HAN is our proposed model with an attention mechanism for composition and 3HAN+PT denotes our pre-trained 3HAN model. Results are reported in Table 2 and demonstrate the effectiveness of 3HAN and 3HAN+PT due to their best performance over all models. Neural models using the hierarchical structure (HAN and variants, 3HAN and variants) give a higher accuracy than other baselines. The attention mechanism is a more effective composition operator than average or max pooling. This is demonstrated by the higher accuracy of 3HAN against 3HAN-Ave and 3HAN-Max. Our headline premise is valid since 3HAN which devotes a separate third level in the hierarchy for the headline performs better than HAN. HAN is indifferent to the headline and focuses its two hierarchical levels only on words and sentences. Pre-training helps in better initialization of 3HAN with 3HAN+PT outperforming 3HAN. ## 4 Discussion and Insights The visualization of attention layers provides evidence. An advantage of attention based neural models is the visualization of attention layers which provides insight into the internal classification process. On the other hand, non-attention based models work like a black box. 3HAN provides attention weights to words, sentences and headline of an article. These attention weights are useful for further human fact-checking. A human fact-checker can focus on verifying sentences with high attention weights. Similarly, words with high attention weights can be investigated for inaccuracies. We visualize the attention weights given to words, sentences and the headline for a sample article through a heatmap in Fig. 2. The sentences with the top five attention weights and the first eight words in each sentence are shown for clarity. Word attention weights $\alpha_{w}$ are normalized using sentence attention weights $\alpha_{s}$ by $\alpha_{w}=\sqrt{\alpha_{s}}\alpha_{w}$. Sentence attention weights are shown on the extreme left edge. We observe that sentence 5 and has been assigned the highest weight (0.287). Interestingly, sentence 5 which states "Even refugee welcoming Canada levies a 12 percent penalty on immigrant money" is a factually incorrect sentence. Word count based models perform well. The high accuracy of simple word count based models which do not take into account word ordering or semantics is an indication of vocabulary and patterns of word usage from the vocabulary being a distinguishing feature between fake news and true news. The attention mechanism is effective. This is observed through the superior performance of HAN compared to non-attention based 3HAN-Max and 3HAN-Ave. Our headline premise is valid. This is observed from the superior perfor \begin{table} \begin{tabular}{l l} \hline \hline \multirow{2}{}{Model} & \multirow{2}{}{Accuracy} \\ \cline{2-2} & & Neural Network Models \\ \hline \hline Model & Accuracy \\ \hline \hline GloVe-Ave & 93.63\% \\ GRU & 91.11\% \\ GRU-Ave & 95.65\% \\ HAN-Ave & 94.91\% \\ HAN-Max & 94.66\% \\ HAN & 95.4\% \\ \hline \hline 3HAN-Ave & 94.81\% \\ 3HAN-Max & 95.25\% \\ 3HAN & 96.24\% \\ 3HAN+PT & 96.77\% \\ \hline \hline \end{tabular} \end{table} Table 2: Accuracy in Article Classification as Fake or Genuine mance of 3HAN to HAN with the third hierarchical level of 3HAN especially designed for our headline premise playing a role. The inverted pyramid style of writing is used. Inverted pyramid refers to distributing information in decreasing importance in an article. We inferred the usage of the inverted pyramid through our experiments from the small improvement in accuracy even with higher padding sentence counts. Fake news articles tend to be repetitive in information content [11]. ## 5 Conclusion and Future Work In this paper, we presented 3HAN which creates news vector, an effective representation of an article for detection as fake news. We demonstrated the superior accuracy of 3HAN over other state-of-the-art models. We highlighted the use of visualization of the attention layers. We plan to deploy a web application based on 3HAN which provides detection of fake news as a service and learns in a real time online manner from new manually fact-checked articles. #### 5.0.1 Acknowledgements. We thank the anonymous ICONIP reviewers as well as G. Srinivasaraghavan, Shreyak Upadhyay and Rishabh Manoj for their helpful comments. Figure 2: Visualization of Attention Layers in a Fake News Article with Headline “Trump Defies Left with Brilliant Move - You Will Cheer”
2310.04190	On the Two Sides of Redundancy in Graph Neural Networks	Message passing neural networks iteratively generate node embeddings by aggregating information from neighboring nodes. With increasing depth, information from more distant nodes is included. However, node embeddings may be unable to represent the growing node neighborhoods accurately and the influence of distant nodes may vanish, a problem referred to as oversquashing. Information redundancy in message passing, i.e., the repetitive exchange and encoding of identical information amplifies oversquashing. We develop a novel aggregation scheme based on neighborhood trees, which allows for controlling redundancy by pruning redundant branches of unfolding trees underlying standard message passing. While the regular structure of unfolding trees allows the reuse of intermediate results in a straightforward way, the use of neighborhood trees poses computational challenges. We propose compact representations of neighborhood trees and merge them, exploiting computational redundancy by identifying isomorphic subtrees. From this, node and graph embeddings are computed via a neural architecture inspired by tree canonization techniques. Our method is less susceptible to oversquashing than traditional message passing neural networks and can improve the accuracy on widely used benchmark datasets.	Franka Bause, Samir Moustafa, Johannes Langguth, Wilfried N. Gansterer, Nils M. Kriege	2023-10-06T12:09:09Z	http://arxiv.org/abs/2310.04190v2	# Non-Redundant Graph Neural Networks ###### Abstract Message passing graph neural networks iteratively compute node embeddings by aggregating messages from all neighbors. This procedure can be viewed as a neural variant of the Weisfeiler-Leman method, which limits their expressive power. Moreover, oversmoothing and oversquashing restrict the number of layers these networks can effectively utilize. The repeated exchange and encoding of identical information in message passing amplifies oversquashing. We propose a novel aggregation scheme based on neighborhood trees, which allows for controlling the redundancy by pruning branches of the unfolding trees underlying standard message passing. We prove that reducing redundancy improves expressivity and experimentally show that it alleviates oversquashing. We investigate the interaction between redundancy in message passing and redundancy in computation and propose a compact representation of neighborhood trees, from which we compute node and graph embeddings via a neural tree canonization technique. Our method is provably more expressive than the Weisfeiler-Leman method, less susceptible to oversquashing than message passing neural networks, and provides high classification accuracy on widely-used benchmark datasets. ## 1 Introduction Graph neural networks have recently emerged as the dominant technique for machine learning with graphs. The class of message passing neural networks (MPNNs) [9] is widely-used. It updates node embeddings layer-wise by combining the current embedding of a node with the embeddings of its neighbors involving learnable parameters. Suitable neural architectures admit a parametrization such that each layer represents an injective function encoding its input uniquely by the new embedding. In this case the MPNN has the same expressive power as the Weisfeiler-Leman algorithm [28]. The Weisfeiler-Leman algorithm can distinguish two nodes if and only if the unfolding trees representing their neighborhoods are non-isomorphic. This unfolding tree corresponds to the computational tree of MPNNs [23, 10]. Hence, if two nodes have isomorphic unfolding trees of height $n$, they will obtain the same embedding after $n$ layers. Vice versa, for two nodes with non-isomorphic unfolding trees of height $n$, there are parameters of the network such that the node's embeddings after $n$ layers differ. However, in practice shallow MPNNs are widely employed. Two phenomena have been identified explaining the poor performance of deep graph neural networks. First, node representations are observed to converge to the same values for deep architecture instead of being able to distinguish more vertices, a phenomenon referred to as _oversmoothing_[14, 15]. Second, _oversquashing_[5] refers to the problem that the neighborhood of a node grows exponentially with the number of layers and aggregation steps and, therefore, cannot be supposed to be accurately represented by a fixed-sized embedding. We argue that oversquashing can be alleviated by removing the encoding of repeated information. Consider a node $u$ with an edge $e=\{u,v\}$ in an undirected graph. In a first step, $u$ will send information to $v$ over the edge $e$. In the next step, $u$ will receive a message form $v$ via $e$ that incorporates the information that $u$ has previously sent to $v$. Clearly, this information is redundant. In the context of walk-based graph learning this problem is well-known and referred to as _tottering_[16]. Recently, Chen et al. [7] made the relation between redundancy and oversquashing explicit and investigated it using the Jacobian of node representations [26]. Several graph neural networks have been proposed replacing the walk-based aggregation with repeated vertices by mechanisms based on simple or shortest paths [21, 19, 11]. Closely related to our work are PathNNs [19] and RFGNN [11], which both define path-based trees for nodes and use custom aggregation schemes. Both approaches suffer from high computational costs and do not exploit the computational redundancy, which is a major advantage of standard MPNNs. Our contribution.We systematically investigate the information redundancy in MPNNs and develop principled techniques to avoid superfluous messages. Fundamental to our consideration is the tree representation implicitly used by MPNNs and the Weisfeiler-Leman method. First, we develop a neural tree canonization approach processing trees systematically in a bottom-up fashion and extend it to operate on directed acyclic graphs (DAGs) representing the union of multiple trees. Our approach recovers the computational graph of MPNNs for unfolding trees, but allows to avoid redundant computations in case of symmetries. Second, we apply the canonization technique to _neighborhood trees_, which are obtained from unfolding trees by deleting nodes that appear multiple times. We show that neighborhood trees allow distinguishing nodes and graphs that cannot be distinguished by the Weisfeiler-Leman method, rendering our technique more expressive than MPNNs. Our approach removes information redundancy on node level, but the non-regular structure of subtrees leads to computational challenges. Our DAG representation of neighborhood trees has size at most $O(nm)$ for input graphs with $n$ nodes and $m$ edges and allows to reuse embeddings of isomorphic subtrees to increase efficiency. Our method achieves high accuracy on several graph classification tasks. ## 2 Related Work The graph isomorphism network (GIN) [28] is an MPNN that generalizes the Weisfeiler-Leman algorithm and reaches its expressive power. The embedding of a vertex $v$ in layer $i$ of GIN is defined as \[x_{i}(v)=\mathrm{MLP}_{i}\left((1+\epsilon_{i})\cdot x_{i-1}(v)+\sum_{u\in N(v) }x_{i-1}(u)\right), \tag{1}\] where the initial features $x_{0}(v)$ are usually acquired by applying a multi-layer perceptron (MLP) to the vertex features. The limited expressiveness of simple message passing neural networks has lead to an increased interest in researching the expressiveness of GNNs and finding more powerful architectures, for example, by encoding graph structure as additional features or altering the message passing procedure. Shortest Path Networks [2] use multiple aggregation functions for different shortest path lengths: One for each $k$ for the $k$-hop neighbors. While this allows the target node to directly communicate with nodes further away and in turn possibly might help mitigate oversquashing, some information about the structure of the neighborhood can still not be represented adequately and the gain in expressiveness is limited. In Distance Encoding GNNs [13], the distances of the nodes to a set of target nodes are encoded. While this approach also is provably more expressive than the standard WL method, it is limited to solving node-level tasks, since the encoding depends on a fixed set of target nodes, and has not been employed for graph-level tasks. MixHop [3] employs an activation function for each neighborhood and concatenates their results. However, in contrast to [2], the aggregation is based on normalized powers of the adjacency matrix, not shortest paths, which does not solve the problem of redundant messages. SPAGAN [29] proposes a path-based attention mechanism. Although the idea is very similar, shortest paths are only sampled and the feature aggregation differs. Only one layer is used and the paths are used as features. This approach has not been investigated theoretically. Also [25] uses a method with short-rooted random walks to capture long-range dependencies between nodes. It has notable limitations due to sample paths instead of exploring all of them, and the evaluation is solely on node classification datasets and needs an extensive study of their expressive power. IDGNN [30] keeps track of the identity of the root node in the unfolding tree, which allows for more expressiveness than 1-WL. Their variant ID-GNN-Fast works by using cycles lengths as additional node features. Both variants however, do not reduce the amount of redundant information that is aggregated over multiple layers. PathNNs [19] defines path-based trees and a custom aggregation scheme, but the computational redundancy is not exploited. In RFGNNs [7] the idea of reducing redundancy is similar. The computational flow is altered to only include each node (except for the root node) at most once in each path of the computational tree. While this reduces redundancy to some extent, nodes and even the same subpaths may repeatedly occur in the computational trees. The redundancy in computation is not addressed resulting in a highly inefficient preprocessing and computation, which restricts the method to a maximum of 3 layers in the experiments. See Appendix A for a detailed discussion of the differences between our approach and RFGNN. For many of these architectures no thorough investigation on their expressiveness and connec types of redundancy in message passing neural networks, i.e. they do not address redundancy in the information flow and in computation. ## 3 Preliminaries In this section, we give an overview of the necessary definitions and the notation used throughout the article and introduce fundamental techniques. Graph theory.A _graph_$G=(V,E,\mu,v)$ consists of a set of vertices $V$, a set of edges $E\subseteq V\times V$ between them and functions $\mu\colon V\to X$ and $v\colon E\to X$ assigning arbitrary attributes to the vertices and edges, respectively.1 We refer to an edge from $u$ to $v$ by $uv$, and in case of undirected graphs $uv=\omega u$. The vertices and edges of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively, and the _neighbors_ (or in-neighbors) of a vertex $u\in V$ are denoted by $N(u)=\{v\mid\omega u\in E\}$. The _out-neighbors_ of a vertex $u\in V$ are denoted by $N_{o}(u)=\{v\mid uv\in E\}$. A _multigraph_ is a graph, where $E$ is a multiset, which means there can be multiple edges between a pair of vertices. Two graphs $G$ and $H$ are isomorphic, denoted by $G\simeq H$, if there exists a bijection $\phi\colon V(G)\to V(H)$, so that $\forall u,v\in V(G)\colon\mu(v)=\mu(\phi(v))\wedge w\in E(G)\Leftrightarrow \phi(u)\phi(v)\in E(H)\wedge\forall uv\in E(G)\colon v(uv)=v(\phi(u)\phi(v))$. We call $\phi$ an _isomorphism_ between $G$ and $H$. Footnote 1: We do not consider edge attributes in the following for clarity of presentation. However, the methods we propose can be extended to incorporate them. An _in-tree_$T$ is a connected, directed, acyclic graph with a distinct vertex $r\in V(T)$ with no outgoing edges referred to as _root_, denoted by $r(T)$, in which $\forall v\in V(T)\backslash r(T):\|N_{o}(v)\|=1$. For $v\in V(T)\backslash r(T)$ the _parent_$p(v)$ is defined as the unique vertex $u\in N_{o}(v)$, and $\forall v\in V(T)$ the _children_ are defined as $\operatorname{chi}(v)=N(v)$. We refer to all vertices with no incoming edges as _leaves_ denoted by $l(T)=\{v\in V(T)\mid\operatorname{chi}(v)=\emptyset\}$. Conceptually it is a directed tree, in which there is a unique directed path from each vertex to the root [18]. In our paper, we only cover in-trees and will thereby just refer to them as _trees_. In-trees are generalized by directed, acyclic graphs (DAGs). The _leaves_ of a DAG $D$ and the _children_ of a vertex are defined as in trees. However, there can be multiple roots and a vertex may have more than one parent. We refer to all vertices in $D$ with no outgoing edges as _roots_ denoted by $r(D)=\{v\in V(D)\mid N_{o}(v)=\emptyset\}$ and define the _parents_$p(v)$ of a vertex $v$ as $p(v)=N_{o}(v)$. The height hgt of a node $v$ is the length of the longest path from any leaf to $v$: $\operatorname{hgt}(v)=0$, if $v\in l(D)$ and $\operatorname{hgt}(v)=\max_{c\in\operatorname{chi}(v)}\operatorname{hgt}(c)+1$ otherwise. The height of a DAG $D$ is defined as $\operatorname{hgt}(D)=\max_{v\in V(D)}\operatorname{hgt}(v)$. For clarity we refer to the vertices of a DAG as nodes to distinguish them from the graphs that are the input of a graph neural network. Weisfeiler-Leman unfolding trees.The $1$-dimensional Weisfeiler-Leman (WL) algorithm or _color refinement_ starts with all vertices having a color representing their label (or a uniform coloring in case of unlabeled vertices). In each iteration the color of a vertex is updated based on the multiset of colors of its neighbors according to \[c_{\text{wl}}^{(i+1)}(v)=h\left(c_{\text{wl}}^{(i)}(v),\{\mathbf{c}_{\text{wl }}^{(i)}(u)\mid u\in N(v)\}\right)\quad\forall v\in V(G),\] where $h$ is an injective function typically representing colors by integers. The color of a vertex of $G$ encodes its neighborhood by a tree $T$ that may contain multiple representatives of each vertex in $G$. Let $\phi\colon V(T)\to V(G)$ be a mapping such that $\phi(n)=v$ if the node $n$ in $V(T)$ represents the vertex $v$ in $V(G)$. The _unfolding tree_$F_{i}^{v}$ with height $i$ of the vertex $v\in V(G)$ consists of a root $n_{v}$ with $\phi(n_{v})=v$ and child subtrees $F_{i-1}^{u}$ for all $u\in N(v)$, where $F_{0}^{v}=(\{n_{v}\},\emptyset)$. The attributes of the original graph are preserved, see Figure 1 for an example. The unfolding trees $F_{i}^{v}$ and $F_{i}^{w}$ of two vertices $v$ and $w$ are isomorphic if and only if $c_{\text{wl}}^{(i)}(v)=c_{\text{wl}}^{(i)}(w)$. ## 4 Non-Redundant Graph Neural Networks We propose to restrict the information flow in message passing to control redundancy using $k$-redundant neighborhood trees. We first develop a neural tree canonization technique and obtain an MPNN via its application to unfolding trees. Then we investigate computational methods on graph level, reusing information computed for subtrees and derive a customized GNN architecture. Finally, we prove that $1$-redundant neighborhood trees are strictly more expressive than unfolding trees on both node- and graph-level. Figure 2: Graph $G$ and the unfolding, $0$- and $1$-redundant neighborhood trees of height $2$ of vertex $v$ (vertex in the upper left of $G$). ### Removing Information Redundancy It is well-known that two nodes obtain the same WL color if and only if their unfolding trees are isomorphic and this concept directly carries over to message passing neural networks and their computational tree [23, 10]. However, unfolding trees were mainly used as a tool in expressivity analysis and as a concept explaining mathematical properties in graph learning [12, 22]. We discuss a tree canonization perspective on MPNNs and derive a novel non-redundant GNN architecture based on neighborhood trees. Aho, Hopcroft, and Ullman [4, Section 3.2] describe a linear time isomorphism test for rooted unordered trees in their classical text book, see Appendix B for details. We give a high-level description to lay the foundations for our neural variant without focusing on the running time. The algorithm proceeds in a bottom-up fashion and assigns integers $\mathrm{c}_{\mathrm{ahu}}(v)$ to each node $v$ of the tree. Let $f$ be a function that assigns a pair consisting of an integer and a multiset of integers injectively to a new (unused) integer. Initially, all leaves $v$ are assigned integers $\mathrm{c}_{\mathrm{ahu}}(v)=f(\mu(v),\emptyset)$ according to their label $\mu(v)$. Then, the internal nodes are processed level-wise in a bottom-up fashion guaranteeing that whenever a node is processed all its children have been considered. Hence, the algorithm computes for all nodes $v$ of the tree \[\mathrm{c}_{\mathrm{ahu}}(v)=f(\mu(v),\{\!\{c_{\mathrm{ahu}}(u)\mid u\in \mathrm{chi}(v)\}\!\}). \tag{2}\] GNNs via unfolding tree canonization.We combine Eq. (2) with the definition of unfolding trees and denote the root of an unfolding tree of height $i$ of a vertex $v$ by $n_{v}^{i}$. Then, we obtain \[\mathrm{c}_{\mathrm{ahu}}(n_{v}^{i})=f(\mu(n_{v}^{i}),\{\!\{c_{\mathrm{ahu}}( n_{u}^{i-1})\mid n_{u}^{i-1}\in\mathrm{chi}(n_{v}^{i})\}\!\})=f(\mu(v),\{\!\{c_{ \mathrm{ahu}}(n_{u}^{i-1})\mid u\in N(v)\}\!\}). \tag{3}\] Realizing $f$ using a suitable neural architecture and replacing its codomain by embeddings in $\mathbb{R}^{d}$ we immediately obtain a GNN from our canonization approach. The only notable difference to standard GNNs is that the first component of the pair in Eq. (3) is the initial node feature instead of the embedding of the previous iteration. We use the technique proposed by Xu et al. [28] replacing the first addend in Eq. (1) with the initial embedding to obtain the _unfolding tree canonization GNN_ \[x_{i}(v)=\mathrm{MLP}_{i}\left((1+\epsilon_{i})\cdot x_{0}(v)+\sum_{u\in N(v)} x_{i-1}(u)\right). \tag{4}\] It is known that MPNNs cannot distinguish two nodes with the same WL color or unfolding tree. Since the function $\mathrm{c}_{\mathrm{ahu}}(n_{v}^{i})$ uniquely represents the unfolding tree for an injective function $f$ which can be realized by Eq. (4) [28], we conclude the following. Proposition 1.: Unfolding tree canonization GNNs given by Eq. (4) are as expressive as GIN (Eq. (1)). However, since $x_{i-1}(v)$ represents the whole unfolding tree rooted at $v$ of height $i-1$, while using the initial node features $x_{0}(v)$ is sufficient, the canonization-based approach avoids redundancy. We proceed by investigating redundancy in unfolding trees themselves. GNNs via neighborhood tree canonization.We use the concept of neighborhood trees allowing to control the redundancy in unfolding trees.2 A $k$-redundant neighborhood tree ($k$-NT) $T^{v}_{i,k}$ can be constructed from the unfolding tree $F^{v}_{i}$ by deleting all subtrees with its roots, that already occurred more than $k$ levels before (seen from root to leaves). Let $\operatorname{depth}(v)$ denote the length of the path from $v$ to the root and $\phi(v)$ again denote the original vertex in $V(G)$ represented by $v$ in the unfolding or neighborhood tree. Footnote 2: In a parallel work, neighborhood trees were investigated to approximate the graph edit distance [6]. Definition 2 ($k$-redundant Neighborhood Tree).: For $k\geq 0$, the $k$_-redundant neighborhood tree_ ($k$-NT) of a vertex $v\in V(G)$ with height $i$, denoted by $T^{v}_{i,k}$, is defined as the subtree of the unfolding tree $F^{v}_{i}$ induced by the nodes $u\in V(F^{v}_{i})$ satisfying \[\forall w\in V(F^{v}_{i})\colon\phi(u)=\phi(w)\Rightarrow\operatorname{depth }(u)\leq\operatorname{depth}(w)+k.\] Figures 2 and 3 show a examples of unfolding and neighborhood trees. Note that for $k\geq i$ the $k$-redundant neighborhood tree is equivalent to the WL unfolding tree. We can directly apply the neural tree canonization technique to neighborhood trees. However, a simplifying expression based on the neighbors in the input graph as given by Eq. (3) for unfolding trees is not possible for neighborhood trees. Therefore, we investigate techniques to systematically exploit computational redundancy. ### Removing Computational Redundancy The computation DAG of an MPNN involves the embedding of a set of trees representing the node neighborhoods of a single or multiple graphs. Results computed for one tree can be reused for others by identifying isomorphic substructures causing computational redundancy. We first describe, how to merge trees in general and then discuss the application to unfolding and neighborhood trees. Merging trees into a DAG.The neural tree canonization approach developed in the last section can directly be applied to DAGs. Given a DAG $D$, it computes an embedding for each node $n$ in $D$ that represents the tree $F_{n}$ obtained by recursively following its children similar as in unfolding trees, cf. Section 3. Since $D$ is acyclic the height of $F_{n}$ is bounded. A detailed description of a neural architecture is postponed to Section 4.3. Given a set of trees $\mathcal{T}=\{T_{1},\ldots,T_{n}\}$, a _merge DAG_ of $\mathcal{T}$ is a pair $(D,\xi)$, where $D$ is a DAG, $\xi\colon\{1,\ldots n\}\to V(D)$ is a mapping and for all $i\in\{1,\ldots,n\}$ we have $T_{i}\simeq F_{\xi(i)}$. The definition guarantees that the neural tree canonization approach applied to the merge DAG produces the same result for the nodes in the DAG as for the nodes in the original trees. A trivial merge DAG is the disjoint union of the trees with $\xi(i)=r(T_{i})$. However, depending on the structure of the given trees, we can identify the subtrees they have in common and represent them only once, such that two nodes of different trees share the same child, resulting in a DAG instead of a forest. We propose an algorithm that builds a merge DAG by successively adding trees to an initially empty DAG creating new nodes only when necessary. Our approach maintains a canonical labeling for each node of the DAG and computes a canonical labeling for each node of the tree to be added using the AHU algorithm, cf. Appendix B. Then, the tree is processed starting at the root. If the canonical labeling of the root is present in the DAG, then algorithm terminates. Otherwise the subtrees rooted at its children are inserted into the DAG by recursive calls. Finally, the root is created and connected to the representatives of its children in the DAG. We introduce a node labeling $L\colon V_{T}\to\mathcal{O}$ used for tree canonization, where $V_{T}=\bigcup_{i=1}^{n}V(T_{i})$ and $\mathcal{O}$ an arbitrary set of labels, refining the original node attributes, i.e., $L(u)=L(v)\Rightarrow\mu(u)=\mu(v)$ for all $u,v$ in $V_{T}$. When $\mathcal{O}$ are integers from the range $1$ to $\|V_{T}\|$, the algorithm runs in $O(\|V_{T}\|)$ time, see Appendix D for details. When two siblings that are the roots of isomorphic subtrees are merged, this leads to parallel edges in the DAG. Parallel edges can be avoided by using a labeling satisfying $L(u)=L(v)\Rightarrow\mu(u)=\mu(v)\wedge p(u)\neq p(v)$ for all $u,v$ in $V_{T}$. Unfolding trees and $k$-NTs can grow exponentially in size with increasing height. However, this is not case for the merge DAGs obtained by the algorithm described above as we will show below. Moreover, we can directly generate DAGs of size $O(m\cdot(k+1))$ representing individual $k$-NTs with unbounded height in a graph with $m$ edges, see Appendix C for details. Figure 4: Computation DAGs for unfolding (a) and 0-NTs (b) of height $2$ of graph $G$. And edges in the different layers of the merge DAG of 0-NTs (c), (d). Merging unfolding trees.Merging the unfolding trees of a graph with the labeling $L=\phi$ leads to the computation DAG of GNNs. Figure 3(a) shows the computation DAG for the graph from Figure 1. The roots in this DAG correspond to the representation of the vertices after aggregating information from the lower layers. Each node occurs once at every layer of the DAG and the links between any two consecutive layers are given by the adjacency matrix of the original graph. While this allows computation based on the adjacency matrix widely-used for MPNNs, it involves the encoding of redundant information. Our method has the potential to compress the computational DAG further by using the less restrictive labeling $L=\mu$ leading to a DAG, where at layer $i$ all vertices $u,v$ with $c^{(i)}_{\text{wl}}(u)=c^{(i)}_{\text{wl}}(v)$ are represented by the same node. This compression appears particularly promising for graphs with symmetries. Merging neighborhood trees.Merging the $k$-redundant neighborhood trees in the same way using the labeling $L=\mu$ (or $L=\phi$ if we want to avoid parallel edges), leads to a computation DAG having a less regular structure, see Figure 3(b) for an example. First, there might be multiple nodes on the same level representing the same original vertex. Second, the adjacency matrix of the original graph cannot be used to propagate the information. A trivial upper bound on the size of the merge DAG of a graph with $n$ nodes and $m$ edges is $O(nmk+nm)$. We apply the neural tree canonization approach to the merge DAG in a bottom-up fashion from the leaves to the roots. Each edge is used exactly once in this computation. Let $D=(\mathcal{V},\mathcal{E})$ be a merge DAG. The nodes can be partitioned based on their height resulting in $\mathcal{L}_{i}=\{v\in\mathcal{V}\mid\text{hgt}(v)=i\}$. This induces the edge partition $\mathcal{E}_{i}=\{uv\in\mathcal{E}\mid v\in\mathcal{L}_{i}\}$, in which all edges with some end node $v$ are in the same layer and all the incoming edges of children of $v$ are in a previous layer. Note, that since $\mathcal{L}_{0}$ contains all leaves of the DAG, there is no $\mathcal{E}_{0}$. Figures 3(c) and 3(d) show the edge sets $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ for the example merge DAG of Figure 3(b). ### Non-Redundant Neural Architecture (DAG-MLP) We introduce a neural architecture computing embeddings for the nodes in a merge DAG allowing to retrieve embeddings of the contained trees from its roots. First, a preprocessing step transforms the node labels using $\text{MLP}_{0}$, mapping them to an embedding space of fixed dimensions. Then, an $\text{MLP}_{i}$ is used to process the nodes at each layer $\mathcal{L}_{i}$. \[\mu^{\prime}(v) =\text{MLP}_{0}\left(\mu(v)\right), \forall v\in\mathcal{V}\] \[x(v) =\mu^{\prime}(v), \forall v\in\mathcal{L}_{0}\] \[x(v) =\text{MLP}_{i}\left((1+\epsilon_{i})\cdot\mu^{\prime}(v)+\sum_{ \forall u:\;uv\in\mathcal{E}_{i}}x(u)\right), \forall v\in\mathcal{L}_{i},i\in\{1,\ldots,n\}\] The DAG-MLP can be computed through iterated matrix-vector multiplication analogous to standard GNNs. Let $\mathbf{L}_{i}$ be a square matrix with ones on the diagonal at position $j$ if $v_{j}\in\mathcal{L}_{i}$, and zeros elsewhere. Let $\mathbf{E}_{i}$ represent the adjacency matrix of $(\mathcal{V},\mathcal{E}_{i})$, and let $\mathbf{F}$ denote the node features of $\mathcal{V}$, corresponding to the initial node labels. The transformed features $\mathbf{F}^{\prime}$ are obtained using the preprocessing $\mathrm{MLP}_{0}$, and $\mathbf{X}^{[i]}$ represents the updated embeddings at layer $i$ of the DAG. \[\mathbf{F}^{\prime} =\mathrm{MLP}_{0}\left(\mathbf{F}\right),\quad\mathbf{X}^{[0]}\ = \mathbf{L}_{0}\mathbf{F}^{\prime},\] \[\mathbf{X}^{[i]} =\mathrm{MLP}_{i}\left((1+\epsilon_{i})\cdot\mathbf{L}_{i}\mathbf{ F}^{\prime}+\mathbf{E}_{i}\mathbf{X}^{[i-1]}\right)+\mathbf{X}^{[i-1]}\,,\] In the above equation, $\mathrm{MLP}_{i}$ is applied to the rows associated with nodes in $\mathcal{L}_{i}$. The embeddings $\mathbf{X}^{[i]}$ are initially set to zero for the inner nodes and are computed level-wise. To preserve the embeddings from all previous layers, we add $\mathbf{X}^{[i-1]}$ during the computation of $\mathbf{X}^{[i]}$. Assume the merge DAG $(D,\xi)$ contains the trees $\{T_{1},\ldots,T_{n}\}$, then we obtain a node embedding $\mathbf{X}^{[n]}_{\xi(i)}$ for each tree $T_{i}$ with $i\in\{1,\ldots,n\}$. When using a single tree for each vertex of the input graph, this directly yields its final embedding. If we include trees of different height for each vertex of the input graph, we group them accordingly. We use NTs of a given fixed height (Fixed Single-Height) or all NTs of size up to a certain maximum (Combine Heights), see Appendix E for a description of the resulting architecture. ### Expressiveness of 1-NTs Let $\varphi$ be an isomorphism between $G$ and $H$. We call two nodes $u$ and $v$ (or edges $uw$ and $vx$) _corresponding_ in an isomorphism $\varphi$, if $\varphi(u)=v$ (for edges $\varphi(u)\varphi(w)=vx$). We denote two nodes $u$ and $v$ (or edges $uw$ and $vx$) by $u\cong v$ ($uw\cong vx$, respectively), if there exists an isomorphism in which $u$ and $v$ ($uw$ and $vx$) are corresponding. For isomorphism testing the (multi)sets of unfolding trees of two graphs (and $k$-redundant neighborhood trees, respectively) can be compared. The sets are denoted with $wl_{i}(G)$ and $nt_{i,k}(G)$ for iteration $i$, and defined as $wl_{i}(G)=\{F_{i}^{v}\|v\in V(G)\}$ and $nt_{i,k}(G)=\{T_{i,k}^{v}\|v\in V(G)\}$. If two graphs $G$ and $H$ are isomorphic, then by definition of the trees, we can find a bijection $\psi$ between their tree sets $wl_{\infty}(G)$ and $wl_{\infty}(H)$ (and $nt_{\infty,k}(H)$ and $nt_{\infty,k}(G)$, respectively), with $\forall T\colon T\simeq\psi(T)$, which we denote by $wl_{\infty}(G)=wl_{\infty}(H)$ ($nt_{\infty,k}(G)=nt_{\infty,k}(H)$). However, $wl_{\infty}(G)=wl_{\infty}(H)\Rightarrow G\simeq H$ (and also $nt_{\infty,k}(G)=nt_{\infty,k}(H)\Rightarrow G\simeq H$). We focus on 1-redundant neighborhood trees from now on. Theorem 3.: The 1-NT isomorphism test is more powerful than the Weisfeiler-Leman isomorphism test, i.e., \[1.\ \forall G,H\colon wl_{\infty}(G)\neq wl_{\infty}(H)\Rightarrow nt _{\infty,1}(G)\neq nt_{\infty,1}(H)\] \[2.\ \exists G,H\colon G\neq H\wedge wl_{\infty}(G)=wl_{\infty}(H) \wedge nt_{\infty,1}(G)\neq nt_{\infty,1}(H).\] Proof.: 1. We prove the first statement by contradiction. Assume $u\in V(G),v\in V(H)$, two nodes with $u\ncong v$, and let $i$ be the first iteration in which $F_{i}^{u}\neq F_{i}^{v}$, but $T_{i,1}^{u}\simeq T_{i,1}^{v}$. From the definition it follows that $\forall v\colon F_{0}^{v}\simeq T_{0,1}^{v}$, and also $\forall v\colon F_{1}^{v}\simeq T_{1,1}^{v}$, so $i\geq 2$. Since $i$ is the first iteration they differed, $F^{u}_{i-1}\simeq F^{v}_{i-1}$. Any isomorphism between $F^{u}_{i}$ and $F^{v}_{i}$ can only be generated from extending an isomorphism between $F^{u}_{i-1}$ and $F^{v}_{i-1}$. Let $\varphi$ be an arbitrary isomorphism between $F^{u}_{i-1}$ and $F^{v}_{i-1}$, then, no matter how we extend it, there exists an edge in the last layer of $F^{u}_{i}$, that has no corresponding edge in the last layer of $F^{v}_{i}$ (or vice versa). If this edge is in $T^{u}_{i,1}$ in the last layer, then (since $T^{u}_{i,1}\simeq T^{v}_{i,1}$) there is also a corresponding edge in $T^{o}_{i,1}$, which implies it is also in $F^{v}_{i}$. This would imply $F^{u}_{i}\simeq F^{v}_{i}$, contradicting the assumption. If this edge is not in $T^{u}_{i,1}$ in the last layer, the same edge must have already occurred in a previous layer in $T^{u}_{i,1}$. Let $l$ be the layer that this edge first occurred. Then, $l\leq i-2$ must hold (because $k=1$), and this edge must also occur in $F^{u}_{i}$, with a corresponding edge in $T^{o}_{i,1}$ and most importantly in $F^{v}_{i}$ in that layer (since the trees up to $i-1$ were the same). But in unfolding trees, an edge from the original graph will be present in every layer after its first occurrence. If the corresponding edge occurs in $F^{v}_{i}$ in layer $l$, it also has to occur in layer $i$ again (implying $F^{u}_{i}\simeq F^{v}_{i}$), which implies $T^{u}_{i,1}\neq T^{v}_{i,1}$ and thereby contradicts the initial assumption. So $\forall G,H\colon wl_{\infty}(G)\neq wl_{\infty}(H)\Rightarrow nt_{\infty,1} (G)\neq nt_{\infty,1}(H)$. 2. The statement is proven by the example, where $G$ is a hexagon and $H$ consists of two triangles. For these graphs all nodes have isomorphic unfolding trees, while their 1-NTs differ (see Figure 5). The 1-NTs can also distinguish the molecules decalin and bicyclopentyl, which WL cannot distinguish. We investigate the expressiveness of 0-NTs in Appendix G. ## 5 Experimental Evaluation We evaluate DAG-MLP with $k$-NTs on both synthetic [1, 21] and real-world datasets [20]. We provide information on the datasets in Appendix H. Experimental setup. For synthetic datasets, we choose the number of layers in DAG-MLP based on the average graph diameter. This ensures that most nodes can effectively aggregate information from all other nodes during message propagation. The embeddings at each layer are extracted using readouts, concatenated, and then passed to two learnable linear layers for Figure 5: Two graphs (a), (b) that cannot be distinguished by unfolding trees, but by 1-NTs. Figure (c) shows the unfolding tree $F_{3}$, which is the same for all vertices of both graphs, while (d) shows the 1-NTs of the vertices in the hexagon (left) and the triangle (right). prediction. For evaluation on the TUDataset, we follow the same splits across 10 folds as proposed by [8]. This allows conducting a grid search to identify the optimal hyper-parameters for each dataset. The architecture for combined heights is designed such that each "readout," is used to extract the embeddings corresponding to each layer, and the mean of the average-pooled embeddings is passed to a final MLP layer responsible for prediction (see Appendix E). For the fixed single-height architecture, only the last readout is used, pooled and passed to the final MLP layer. The hyper-parameters are detailed in Appendix J. Results. Table 1 shows the results on the synthetic expressivity datasets. Comparing our approach to GIN, the results we see are consistent with the theoretical findings: 1-NTs are more expressive than GIN. Our hypothesis that 0-NTs are more expressive than GIN on a graph-level is experimentally validated, but a theoretical proof remains future work. In Table 2, we investigate the impact of the parameter $k$ and the number of layers $l$ on the accuracy on EXP-Class. Cases with $k>l$ can be disregarded, since the computation for NTs remains the same when $k=l$. Empirically, 0- and 1-NTs yield the highest accuracy. This is consistent with our discussions on expressivity in Section 4.4 and Appendix G. The decrease in accuracy with increasing $k$ indicates that information redundancy leads to oversquashing. For TUDataset, we report the accuracy compared to related work in Table 3. Due to the high \begin{table} \begin{tabular}{l c c c} \hline \hline Model & EXP-Class $\uparrow$ & EXP-Iso $\downarrow$ & CSL $\uparrow$ \\ \hline GIN [28] & 50.0 $\pm$ 0.0 & 600 & 10.0 $\pm$ 0.0 \\ 3WLGNN [17] & 100.0 $\pm$ 0.0 & 0 & 97.8 $\pm$ 10.9 \\ PathNN-$\mathcal{SP}^{+}$[19] & 100.0 $\pm$ 0.0 & 0 & 100.0 $\pm$ 0.0 \\ PathNN-$\mathcal{AP}$[19] & 100.0 $\pm$ 0.0 & 0 & 100.0 $\pm$ 0.0 \\ \hline DAG-MLP (0-NTs) & 100.0 $\pm$ 0.0 & 0 & 100.0 $\pm$ 0.0 \\ DAG-MLP (1-NTs) & 100.0 $\pm$ 0.0 & 0 & 100.0 $\pm$ 0.0 \\ \hline \hline \end{tabular} \end{table} Table 1: Average classification accuracy for EXP-Class and CSL across $k$-folds (4-folds and 5-folds), and the number of undistinguished pairs of graphs in EXP-Iso. Best results are highlighted in gray, best results from methods with polynomial time complexity are highlighted in bold. \begin{table} \begin{tabular}{l c c c c c c} \hline \hline $k$-NTs & 1 layer & 2 layers & 3 layers & 4 layers & 5 layers & 6 layers \\ \hline 0-NTs & 51.1 $\pm$ 1.6 & 57.5 $\pm$ 6.6 & 91.7 $\pm$ 11.6 & 99.7 $\pm$ 0.3 & 100.0 $\pm$ 0.0 & 100.0 $\pm$ 0.0 \\ 1-NTs & 50.1 $\pm$ 0.2 & 58.9 $\pm$ 4.6 & 59.4 $\pm$ 5.7 & 99.6 $\pm$ 0.5 & 99.9 $\pm$ 0.2 & 100.0 $\pm$ 0.0 \\ 2-NTs & - & 52.6 $\pm$ 3.4 & 54.9 $\pm$ 5.3 & 52.4 $\pm$ 3.8 & 97.6 $\pm$ 1.9 & 100.0 $\pm$ 0.0 \\ 3-NTs & - & - & 56.2 $\pm$ 5.7 & 51.1 $\pm$ 1.9 & 52.4 $\pm$ 4.1 & 87.1 $\pm$ 21.4 \\ 4-NTs & - & - & - & 50.1 $\pm$ 0.2 & 50.6 $\pm$ 1.0 & 50.4 $\pm$ 0.7 \\ 5-NTs & - & - & - & - & 50.4 $\pm$ 0.7 & 50.0 $\pm$ 0.0 \\ 6-NTs & - & - & - & - & - & 53.2 $\pm$ 5.2 \\ \hline \hline \end{tabular} \end{table} Table 2: Average accuracy for DAG-MLP using 4-fold cross-validation on EXP-Class [1], evaluated with varying number of layers. standard deviation across all methods, we present a statistical box plot for the accuracy based on three runs on the testing set of 10-fold cross-validation in Appendix F. We group the methods by their time complexity. Note that, while PathNN performs well on ENZYMES and PROTEINS, the time complexity of this method is exponential. Therefore, we also highlight the best method with polynomial time complexity. For IMDB-B and IMDB-M, which have small diameters, we see that $k$-NTs outperform all other methods. For ENZYMES a variant of our approach achieves the best result among the approaches with non-exponential time complexity and $k$-NTs lead to a significant improvement over GIN. ## 6 Conclusion We propose a neural tree canonization technique and combined it with neighborhood trees, which are pruned and more expressive versions of unfolding trees used by standard MPNNs. By merging trees in a DAG, we derive compact representations that form the basis for our neural architecture DAG-MLP, which learns across DAG levels. It inherits the properties of the GIN architecture, but is provably more expressive than 1-WL when based on 1-redundant neighborhood trees. In this effort, we introduced general techniques to derive compact computation DAGs for tree structures encoding node neighborhoods. This revealed a complex interplay between information redundancy, computational redundancy and expressivity, the balancing of which is an avenue for future work. \begin{table} \begin{tabular}{l l c c c c} \hline \hline & Algorithm & IMDB-B & IMDB-M & ENZYMES & PROTEINS \\ \hline \multirow{4}{}{\begin{tabular}{} \end{tabular} } & GIN [28] & 71.2 $\pm$ 3.9 & 48.5 $\pm$ 3.3 & 59.6 $\pm$ 4.5 & 73.3 $\pm$ 4.0 \\ & GAT [27] & 69.2 $\pm$ 4.8 & 48.2 $\pm$ 4.9 & 49.5 $\pm$ 8.9 & 70.9 $\pm$ 2.7 \\ & SPN ($l=1$) [2] & NA & NA & 67.5 $\pm$ 5.5 & 71.0 $\pm$ 3.7 \\ & SPN ($l=5$) [2] & NA & NA & 69.4 $\pm$ 6.2 & 74.2 $\pm$ 2.7* \\ \hline \multirow{4}{}{\begin{tabular}{} \end{tabular} } & PathNet [25] & 70.4 $\pm$ 3.8 & 49.1 $\pm$ 3.6 & 69.3 $\pm$ 5.4 & 70.5 $\pm$ 3.9 \\ & PathNN-$\mathcal{P}$[19] & 72.6 $\pm$ 3.3 & 50.8 $\pm$ 4.5 & 73.0 $\pm$ 5.2* & 75.2 $\pm$ 3.9 \\ & PathNN-$\mathcal{S}\mathcal{P}^{+}$[19] & - & - & 70.4 $\pm$ 3.1 & 73.2 $\pm$ 3.3 \\ \hline \multirow{4}{}{ \begin{tabular}{} \end{tabular} } & DAG-MLP (0-NTs) Fixed Single-Height & 72.9 $\pm$ 5.0* & 50.2 $\pm$ 3.2 & 67.9 $\pm$ 5.3 & 70.1 $\pm$ 1.7 \\ & DAG-MLP (1-NTs) Fixed Single-Height & 72.4 $\pm$ 3.8 & 48.8 $\pm$ 4.3 & 70.6 $\pm$ 5.5 & 70.2 $\pm$ 3.4 \\ & DAG-MLP (0-NTs) Combine Heights & 72.8 $\pm$ 5.6 & 50.1 $\pm$ 3.8 & 66.7 $\pm$ 4.8 & 69.1 $\pm$ 3.6 \\ & DAG-MLP (1-NTs) Combine Heights & 72.2 $\pm$ 4.5 & 51.3 $\pm$ 4.4 & 69.2 $\pm$ 4.5 & 69.5 $\pm$ 3.0 \\ \hline \hline \end{tabular} \end{table} Table 3: Classification accuracy ($\pm$ standard deviation) over 10-fold cross-validation on the datasets from TUDataset, taken from Michel et al. [19]. Best performance is highlighted in gray, best results from methods with polynomial time complexity are highlighted in bold. “-” denotes not applicable and “NA” means not available. ## Acknowledgments We would like to thank Christian Permann for his contribution to the conception of neighborhood trees and their efficient generation. This work was supported by the Vienna Science and Technology Fund (WWTF) [10.47379/VRG19009]. The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC). ## Author Contributions NK devised the project and the main conceptual ideas. FB made significant contributions to the conception of $k$-redundant neighborhood trees and their efficient generation. NK and FB developed the methods for redundancy removal presented in Sections 4.1, 4.2. FB developed the proof of Theorem 3. FB implemented the $k$-redundant neighborhood trees, merge DAGs and algorithmic components of the implementation. NK, SM and FB developed the DAG-MLP architecture. SM implemented DAG-MLP, conducted the experimental evaluation and wrote parts of the corresponding sections in the manuscript. FB and NK drafted the manuscript with input from all authors. NK, WG and JL supervised the project. All authors provided critical feedback, participated in discussions, contributed to the interpretation of the results and revised the manuscript.
2303.12812	A Comparison of Graph Neural Networks for Malware Classification	Managing the threat posed by malware requires accurate detection and classification techniques. Traditional detection strategies, such as signature scanning, rely on manual analysis of malware to extract relevant features, which is labor intensive and requires expert knowledge. Function call graphs consist of a set of program functions and their inter-procedural calls, providing a rich source of information that can be leveraged to classify malware without the labor intensive feature extraction step of traditional techniques. In this research, we treat malware classification as a graph classification problem. Based on Local Degree Profile features, we train a wide range of Graph Neural Network (GNN) architectures to generate embeddings which we then classify. We find that our best GNN models outperform previous comparable research involving the well-known MalNet-Tiny Android malware dataset. In addition, our GNN models do not suffer from the overfitting issues that commonly afflict non-GNN techniques, although GNN models require longer training times.	Vrinda Malhotra, Katerina Potika, Mark Stamp	2023-03-22T01:05:57Z	http://arxiv.org/abs/2303.12812v1	# A Comparison of Graph Neural Networks for Malware Classification ###### Abstract Managing the threat posed by malware requires accurate detection and classification techniques. Traditional detection strategies, such as signature scanning, rely on manual analysis of malware to extract relevant features, which is labor intensive and requires expert knowledge. Function call graphs consist of a set of program functions and their inter-procedural calls, providing a rich source of information that can be leveraged to classify malware without the labor intensive feature extraction step of traditional techniques. In this research, we treat malware classification as a graph classification problem. Based on Local Degree Profile features, we train a wide range of Graph Neural Network (GNN) architectures to generate embeddings which we then classify. We find that our best GNN models outperform previous comparable research involving the well-known MalNet-Tiny Android malware dataset. In addition, our GNN models do not suffer from the overfitting issues that commonly afflict non-GNN techniques, although GNN models require longer training times. ## 1 Introduction Android malware is a common problem on mobile devices and poses a serious challenge due to its volume and diversity. According to AV-TEST [3], in 2021 some 9.09 million new mobile malware samples were intercepted, which is an average of nearly 25,000 new samples per day. Of these mobile malware samples, about 3.5 million were Android-based. Since Android is an open-source platform, it provides flexibility for mobile developers to create custom applications (apps). However, this same flexibility can be exploited by bad actors to create malicious apps that users install. Therefore, solutions that detect and classify malware are crucial for the safety of Android devices. Previous studies such as [2] have illustrated that new malware apps tend to be variants of pre-existing malware families, and therefore showcase similar behavioral traits. For example, the GinMaster malware family sends confidential information to a remote server, while the DroidKungFu family allows a hacker to control an infected device from a remote location and access confidential information. We can classify malware samples into broad families based on their differing characteristics [13]. Signature-based malware detection methods have been popular for creating endpoint protection systems because they are comparatively fast and effective on "traditional" malware. However, generating classic signatures requires expert knowledge to reverse engineer malware instances and produce the necessary features, and hence reverse engineering cannot scale with malware production. Signature-based techniques also fail to identify zero-day malware, whereas machine learning-based methods have the potential to identify and classify previously unseen malware [20]. Static malware analysis relies on features that can be extracted without executing or emulating code, in contrast to dynamic analysis, where execution or emulation is required [10]. In general, static analysis is more vulnerable to code obfuscation techniques employed by modern polymorphic and metamorphic malware [52], while dynamic malware analysis is more resistant to such obfuscations [43]. However, collecting dynamic features is more resource and time intensive [4]. Studies such as [2] perform broad static analysis to generate features (e.g., user permissions, suspicious API calls, network addresses, etc.) and then use machine learning algorithms to classify malware. However, these machine-learning algorithms generally do not adequately model interactions between function calls. Such interactions can most naturally be taken into account using a graph-like data structure. Graph-based methods do not assume that the features of a particular instance are independent of each other, and the models themselves can learn relationships between features. This provides us with an additional layer of information that is inherent in the input data and can be best utilized by graph models. Moreover, labor-intensive manual analysis of code is not required to generate feature sets for our graph-based models, giving them a significant advantage over traditional techniques, such as signature analysis. The goal of this research is to explore malware classification techniques using graph-based learning that relies solely on graphs generated from code. The remainder of this paper is organized as follows. In Section 2, we discuss relevant related work. In Section 3, we introduce several learning techniques that are based on different principles, ranging from standard machine learning models to current state-of-the-art Graph Neural Networks (GNN). Section 4 includes implementation details related to the experiments that we perform. In Section 4.3, we analyze and discuss our experimental results. Finally, in Section 5, we provide a summary of our main results and discuss potential directions for future work. Related Works In this section, we highlight some of the learning-based methods that have been previously used for malware classification and detection. We divide this selective survey between traditional learning-based classifiers and graph-based classification. Under the heading of "traditional" learning techniques, we include both classic statistical-based machine learning and neural networking-based deep learning. ### Traditional Learning-Based Classification In the paper [50] from 2013, a Bayesian classifier was trained to detect malware using 58 code-based attributes. In the 2014 paper that introduced the popular Drebin dataset [2], features (e.g., permissions and API calls) were extracted from Android apk files and the samples were then classified using Support Vector Machines (SVM). A later method, which its authors referred to as Significant Permission Identification for Android Malware Detection (SIGPID) [28], mined the permissions data of each malware app using three levels of pruning to identify the 22 most significant permissions. These features were then used to train an SVM classifier. With the rise in popularity of deep learning models, papers such as [42] appeared, where permission sequences were extracted as word embeddings and used as features in a Long Short-Term Memory Network (LSTM) model. Similarly, in [20], API features were selected based on their frequency and then mapped to an image-like structure, with Convolutional Neural Networks (CNN) used as the classifier. The results indicate API calls and permissions are strong features for Android malware detection, but that the optimal choice of these features depends on the dataset used. Opcode-sequence $n$-gram features have also been successfully used to detect malware [21]. In addition, the paper [33] used raw opcode sequences encoded as one-hot vectors, which served to classified samples via a CNN model. Techniques based on opcode features can be defeated, since opcode sequences can be directly altered via elementary obfuscation techniques. Another popular approach for malware classification is to convert files into grayscale images and then take advantage of the strength of CNNs for malware detection and classification [18, 24]. Additional work on malware classification includes a range of novel approaches, such as that in [45], where patterns based on inter-component communication were extracted from the code. ### Graph Learning-Based Classification While traditional learning techniques are based on the type of feature and classifier, graph-based learning relies on the type of graph structure and how the node features are embedded. For example, the papers [14, 19, 36] use API call graphs. Specifically, in [19], apps are represented in relation to APIs, and API relationships are mapped as a structured Heterogeneous Information Network (HIN). Then a meta-path-based approach is used to characterize the semantic relatedness of apps and APIs, aggregating similarities using multi-kernel learning. Heterogeneous graphs are those that use more than one type of relationship to make connections between nodes. In [36], API call sequence graphs learn multiple embedding representations that are successfully used for malware detection and classification. The authors use a recurrent neural network to decode the deep semantic information and to independently extract features, with a version of a Graph Convolutional Network (GCN) used to model high-level graphical semantics. In [14], a skip-gram model is used to extract features for graph nodes based on API sequences. These features rely on both app-API relationships and API-API relationships to form a heterogeneous graph. Another popular type of graph used for malware analysis is an opcode-level function call graphs [16; 23; 34]. The research in [16] uses opcode-sequences as text features, with traditional machine learning methods (Random Forest, SVM, etc.) for classification. In contrast, the paper [34] relies on an LSTM-based neural network for classification. The authors of [23] utilize a novel graph structure that they call a co-opcode graph, which is constructed by extracting metamorphic engine-specific opcode patterns. A method related to Hidden Markov Models (HMM) is then used to classify the malware samples. Opcode-level graphs can also be treated like text features rather than graphical data. For example, in [5] dynamically-generated network flow graphs are used to create a new model that its authors call Network Flow Graph Neural Network (NF-GNN). This NF-GNN model relies on a novel edge feature-based GNN for classification. The graph methods mentioned above are all transductive, which implies that they cannot be expected to generalize to zero-day malware. One goal of our research is to analyze graph-based models that are inductive, that is, models that can be used to predict zero-day malware. We will base our models on Control Flow Graphs (CFG). In a CFG, nodes represent program statements, including called subroutines and conditionals, while edges represent the flow of the program. CFGs intra-procedural mappings may be well-suited to classify zero-day malware [49], since they only rely on characteristics of the code of an apk. In [49], a Deep Graph Convolutional Neural Network (DGCNN) is used to embed structural information inherent in CFGs for effective and efficient malware classification. The inter-procedural counterpart to CFGs are Function Call Graphs (FCG), where nodes represent subroutines and edges represent the caller-called relationships between subroutines. In [34], opcode level function call graphs are obtained from the static analysis of malware, while [48] uses a Natural Language Processing (NLP)-inspired graph embedding to convert the graph structure of an Android app into a vector. On a related note, the paper [12] introduced the MalNet dataset, a large Android malware dataset, where FCG data has been extracted from apk files. The authors considered state-of-the-art graph representation learning approaches, including GraphSAGE [17] and Graph Isomorphism Networks (GIN) [46]. Among the methods considered, they found that FEATHER and GIN achieved the highest classification performance. Using this same MalNet dataset, the authors of [29] employ Jumping Knowledge GNNs (JK-GNN) with node features set to page rank, degree nodes, etc. They conclude that JK-GraphSAGE performs best. ### Datasets In our research, we want to focus on graphs that do not require handcrafted features, graphs that will enable us to deal with zero-day malware, and graphs that are straightforward to generate and analyze. Network flow graphs and control flow graphs are difficult to generate and can require significant resources to store, while heterogeneous graphs are difficult to interpret and encode. In contrast, a Function Call Graph (FCG) is easy to interpret since it consists of all possible execution paths called during runtime. For example, if an app is making a sequence of function calls to gather sensitive information and send it to a server, all of these function calls are in the FCG, and we should be able to gain insight into this malicious behavior from the FCG. Henceforth, we focus on FCGs in this research. A comparison between popular Android malware datasets is provided in Table 1. Not mentioned in Table 1 is the AndroZoo dataset [1], which is an ongoing project of that at the time of this writing consists of more than 21,000,000 apk files. The MalNet-Tiny dataset is derived from AndroZoo. All of the datasets in Table 1 consist of malware apk files or hexadecimal representations of the binary content, along with manifests that contain metadata, such as function calls, strings, etc. For our experiments, we use the MalNet-Tiny dataset. As noted in Table 1, MalNet-Tiny is comprised of five families, with 5000 samples in total. This dataset has a sufficient number of samples to train the graph-based learning models that we consider and, furthermore, each sample is provided in the form of an FCG [12]. Also, MalNet-Tiny has been used in previous studies, enabling a direct comparison of our research to previous work. \begin{table} \begin{tabular}{c\|r\|r\|r} \hline \hline Dataset & Families & Samples & Papers \\ \hline Android Malware Dataset (AMD) & 42 & 5000 & [14, 36, 48] \\ Android Malware Genome (AMG) & 72 & 1260 & [48] \\ Drebin Dataset & 179 & 5560 & [14, 36, 48] \\ MalNet & 696 & 1,262,024 & [31] \\ MalNet-Tiny & 5 & 5000 & [1, 12] \\ \hline \hline \end{tabular} \end{table} Table 1: Popular datasets for Android malware classification ### Related Work Table 2 summarizes a selection of previous work where either traditional learning techniques or graph-based learning has been applied to the Android malware detection or classification problems. Note that two of the research papers in Table 2 use the MalNet-Tiny dataset, and hence these two are the most directly relevant for comparison to the results that we provide in Sections 4 and 4.3, below. ## 3 Background A graph is a data structure consisting of two components, namely, nodes (or vertices) and edges. A graph $G$ is defined as $G=(V,E)$, where $V$ is the set of nodes, and $E$ is the edges between them. These edges can be directed or undirected. Conventional machine learning and deep learning techniques can be viewed as dealing with relatively simple types of graphs. For example, images can be viewed grid graphs while text and speech are sequential and can be viewed as line graphs. However, in general, graphs can be far more varied and complex, with arbitrary connections between nodes. Recently, analyzing graphs with machine learning has become popular. Traditional feedforward machine learning algorithms, such as Multi-Layer Perceptrons (MLP) and CNNs, treat the features of a particular instance as if they are independent of other instances. While Recurrent Neural Networks (RNN) can deal with sequential data, Graph Neural Networks (GNN) allow us to model arbitrary interactions that are beyond the scope of RNNs. \begin{table} \begin{tabular}{c\|c c c c c} \hline \hline Paper & \begin{tabular}{c} Learning \\ technique(s) \\ \end{tabular} & \begin{tabular}{c} Feature(s) \\ \end{tabular} & \begin{tabular}{c} Classification \\ or detection \\ \end{tabular} & \begin{tabular}{c} Dataset(s) \\ \end{tabular} & \begin{tabular}{c} Accuracy \\ or F1-score \\ \end{tabular} \\ \hline Arp, et al. [2] & SVM & Multiple & Classification & Drebin & 0.9400 \\ Busch, et al. [5] & NF-GNN & Network flows & Classification & Custom dataset & 0.9614 \\ Freitas and Dong [12] & GIN & FCG & Classification & MalNet-Tiny & 0.9000 \\ Gao, et al. [14] & GCN & API calls & Both & AMG, Drebin, AMD & 0.9504 \\ Gülmez and Sougkumar [16] & RF and SVM & Opcodes & Detection & Custom dataset & 0.9700 \\ He, et al. [18] & ResNet & Images & Detection & Andro-dumpsys & 0.9500 \\ Hou, et al. [19] & HIN & API calls & Detection & Comodo Cloud & — \\ Huang, et al. [20] & CNN & API calls & Detection & Custom dataset & 0.9430 \\ Jerome, et al. [21] & SVM & Opcode $n$-grams & Classification & AMG plus & 0.8931 \\ Kakisim, et al. [23] & Custom & Co-opcode graph & Detection & Custom dataset & 0.9930 \\ Khan, et al. [24] & ResNet/GoogleNet & Opcodes & Detection & MMC & 0.8836 \\ Li, et al. [28] & Custom & Permissions & Detection & Custom dataset & 0.9362 \\ Lo, et al. [29] & JK & FCG & Classification & MalNet-Tiny & 0.9440 \\ McLaughlin, et al. [33] & CNN & Opcodes & Classification & AMG plus & 0.8600 \\ Niu, et al. [34] & LSTM & Opcodes & Classification & Custom dataset & 0.9700 \\ Pei, et al. [36] & Graphs & API/permissions & Detection & Drebin, AMD & 0.9967 \\ Vinayakumar, et al. [42] & LSTM & API calls & Classification & AMG plus & 0.8970 \\ Xu, et al. [45] & Custom & ICC & Detection & Custom dataset & 0.9740 \\ Xu, et al. [48] & GNN & FCG & Classification & AndroZoo, Drebin, AMD & 0.9933 \\ Yan, et al. [49] & DGCNN & CFG & Classification & MMC & 0.9942 \\ Yerima, et al. [50] & Bayesian & Multiple & Classification & Custom dataset & 0.8450 \\ \hline \hline \end{tabular} \end{table} Table 2: Summary of selected previous work Graph Neural Networks (GNN) are neural networks that directly operate on graphs [55]. GNNs can be viewed as generalizations of CNNs that allow for a richer neighborhood structure. Thus GNNs can model more complex input and output relationships. ### LDP Features In our GNN models, we use the Local Degree Profile (LDP) [6] for node features. LDP is a relatively simple representation scheme that is based on a node and its one-hop neighborhood. Specifically, for a graph $G=(V,E)$, let $d(v)$ be the degree of $v$, that is, the number of vertices that are adjacent to $v$ in the graph $G$. The LDP of a given node $v\in V$ is defined as the vector of length five given by \[\text{LDP}(v)=\Big{(}d(v),\min\big{(}N(v)\big{)},\max\big{(}N(v)\big{)},\mu \big{(}N(v)\big{)},\sigma\big{(}N(v)\big{)}\Big{)} \tag{1}\] where $N(v)=\{d(u)\,\|\,(v,u)\in E\}$, $\mu$ is the mean, and $\sigma$ is the standard deviation [6]. Thus the LDP of $v$ is a statistical profile of the neighborhood structure of $v$ within the graph $G$. LDP features have been shown to perform well with the graph neural networks that we discuss in Section 3.3, below. For our purposes, one potential advantage of the LDP in (1) is that standard deep learning techniques can also be trained using this feature. We note in passing that graph kernel methods reduce dimensionality of the graph data, resulting in different features from LDP, while random walk and spectral distance-based methods also learn representations of the graph data that differ from the LDP. Next, we outline each of the learning methods that we consider in this research. We divide the models into two groups, namely, non-GNN and GNN models. ### Non-GNN Learning Methods The learning methods in this section use graph data for classification, but are not considered GNNs. These five architectures provide a baseline for comparing more costly--with respect to training--GNN architectures. #### 3.2.1 Multi-Layer Perceptron Multi-Layer Perceptron (MLP) is a standard feedforward deep learning technique. MLPs have proven extremely useful in a wide range of applications. Here, we train an MLP on the LDP features in (1), which serves as a baseline for comparison of the effectiveness of our graph-based models. #### 3.2.2 Weisfeiler-Lehman Kernel Graph kernels methods are a form of supervised learning that use kernel functions to reduce dimensionality. In contrast to traditional kernel-based machine learning models, such as Support Vector Machines (SVM), graph kernel methods are designed specifically to deal with graph-based features. In our experiments, we use the Python GraKel library to test the Weisfeiler-Lehman Subtree Kernel (WL-Kernel) technique [22]. Such models can be used, for example, to find isomorphisms between graphs [39]. #### 3.2.3 Feather FEATHER [38] is a complex representation scheme that uses characteristic functions of node features with random walk weights to describe node neighborhoods and to create node embeddings. FEATHER introduces a generalization of characteristic functions to node neighborhoods, where the probability weights of the characteristic function are defined by a measure known as "tie strength." The so-called $r$-scale random walk weighted characteristic function is then used to generate the embeddings. Random Forest is used to classify based on the resulting embeddings. #### 3.2.4 Slaq-VNGE and Slaq-LSD We experiment with Slaq-VNGE and Slaq-LSD, both of which approximate the spectral distances between graphs, but are based on different functions. Whereas Slaq-VNGE uses Von Neumann Graph Entropy (VNGE), which measures information divergence and distance between graphs [41], Slaq-LSD approximates NetLSD, which measures the spectral distance between graphs based on the heat kernel [40]. ### GNN Architectures In this section, we consider GNN architectures. In this section, we consider networks that attempt to improve on the basic GNN architectures of the previous section. These models are more complex and costly to train, as compared to the GNNs discussed above. #### 3.3.1 Graph Convolutional Networks Convolutional Neural Networks (CNN) have been extremely successful for image classification. However, CNNs are limited to simple grid-like graph structures. GNNs can be viewed as generalizations of CNNs to more general graph structures. GNNs aim to generate node embeddings that transform the graph nodes into a low-dimensional embedding space. The mean of all node embeddings is taken to form the whole graph embedding, which encodes the whole graph into low-dimensional space for graph classification [7]. Graph Convolutional Networks (GCN) compute node embeddings by aggregating neighborhood node features. Let $A$ be the adjacency matrix of the graph $G=(V,E)$. Then $A=\{a_{ij}\}$ is an $N\times N$ matrix, where $N=\|V\|$ and is 1 if $(v_{i},v_{j})\in E$; otherwise $a_{ij}=0$. Each node has a $k$-dimensional feature vector, with $X\in R^{N\times k}$ representing the feature matrix for all $N$ nodes. An $L$-layer GCN consists of $L$ graph convolution layers, with each layer constructing embeddings for every node by mixing the embeddings of the neighboring nodes in the graph from the previous layer [25]. #### 3.3.2 GraphSAGE The Graph Sample and Aggregate (GraphSAGE) algorithm was developed in [17]. Unlike GCNs, GraphSAGE randomly samples a fixed-size subset of node neighbors. This serves to limit the space and time complexity of the algorithm, irrespective of the graph structure and batch size. Similar to the convolution operation in CNNs, information relating to the local neighborhood of a node is collected and used to compute the node embedding. At each iteration, the neighborhood of a node is initially sampled, and the information from the sampled nodes is "aggregated" into a single vector. The neighborhood of node $v$ is denoted as $\mathcal{N}(v)$, where $\mathcal{N}(v)=\{u\,\|\,(v,u)\in E\}$ for the graph $G=(V,E)$. At layer $k$, the aggregated information for node $v$ is based on its neighborhood $\mathcal{N}(v)$; see [17] for more details. The aggregation operation can be implemented as a mean, pooling, or LSTM function. We use the Pytorch mean as our aggregator. After applying the model trainable parameters and passing the result through a non-linear activation function such as ReLU, the layer $k$ node $v$ embedding is concatenated. The final representation (embedding) of node $v$ is essentially the node embedding at the final layer. For node classification, this node embedding is passed through a sigmoid neuron or softmax layer. #### 3.3.3 Graph Isomorphism Network The crucial difference between Graph Isomorphism Network (GIN) [17] and other GNNs is the message aggregation function. For GINs, this function is based on Multi-layer Perceptrons (MLP) at each layer. Message aggregation methods are related to the Weisfeiler-Lehman algorithm mentioned above. #### 3.3.4 Simple Graph Convolution GCNs inherit considerable complexity from their deep learning lineage, which can be burdensome for less demanding applications. Simple Graph Convolution (SGC) [44] reduces the excess complexity of GCNs by repeatedly removing the nonlinearities between GCN layers and collapsing the resulting function into a single linear transformation. In contrast, a GCN transforms the feature vectors repeatedly throughout its layers and then applies a linear classifier on the final representation, SGC reduces the GCN procedure to a simple feature propagation step followed by standard logistic regression. #### 3.3.5 Jumping Knowledge Networks Similar to convolutional neural networks, GNN models of increasing depth can perform worse. While the message-passing mechanism helps us harness the information encapsulated in the graph structure, it may introduce some limitations if combined with GNN depth. In other words, our quest for a model that is more expressive and aware of the graph structure by adding more layers so that nodes can have a large receptive field can revert to a model that treats nodes all the same. This is called the over-smoothing problem [26]. To mitigate over-smoothing, we can apply the Jumping Knowledge [47] technique, which uses a concatenation layer. The key idea behind Jumping Knowledge is to select from all of those intermediate node representations and jump to the last layer, which serves to combine the intermediate node representations and to generate the final node representation. We apply layer aggregation concatenation to combine all intermediate node representations for a linear transformation to compute the final node embeddings. The result then undergoes global pooling, with element-wise pooling over the final node embeddings. We apply Jumping Knowledge to the GCN, GraphSAGE, and GIN models, and we abbreviate these models as JK-GCN, JK-GraphSAGE, and JK-GIN, respectively. #### 3.3.6 UnetGraph In [15], U-Nets are used to correlate graph data to images by considering images as a special case of graphs, in which nodes lie on regular 2-D lattices. This structure enables us to use convolution and pooling operations on images. Therefore, node classification and embedding tasks have a natural correspondence with pixel-wise prediction tasks such as image segmentation [35]. In particular, both tasks aim to make predictions for each input unit, corresponding to a pixel on images or a node in graphs. In computer vision, pixel-wise prediction tasks have achieved major advances. Originally introduced in [37], U-Nets [8], which are based on an encoder-decoder architecture, are popular for 3-D image segmentation tasks. In addition to convolutions, pooling and up-sampling operations are essential building blocks in these architectures. However, extending these operations to graph data is challenging, because unlike grid-based data such as images and texts, nodes in graphs have no spatial locality and order information as required by regular pooling operations. To bridge this gap, the paper [15] introduces graph pooling (gPool) and unpooling (gUnpool) operations. We use the GraphUNet implementation of this architecture from the Torch Geometric library. #### 3.3.7 Deep Graph Convolutional Neural Network Deep Graph Convolutional Neural Networks (DGCNN) [54] have three sequential stages. First, graph convolution layers extract the local substructure of vertices features and define a consistent vertex ordering. Second, a SortPooling layer sorts the vertex features under the previously defined order and unifies the input sizes. Third, traditional convolutional and dense layers read the sorted graph representations and make predictions. DGCNN has a number of advantages over other graph neural networks. For one, it directly accepts graph data as input without the need to first transform graphs into tensors, making end-to-end gradient-based training possible. Another potential advantage of DGCNN is that it enables learning from the global graph topology by sorting vertex features instead of summing them, which is supported by a novel SortPooling layer. DGCNN is the only sorting-based method that we consider. DGCNN was previously used in [54] for malware family classification using API call graphs. ### MalNet-Tiny Dataset As mentioned in Section 2.3, we use the MalNet-Tiny dataset, which consists of 5000 Function Call Graphs derived form Android apk files. This is a balanced dataset, with each of the five families having 1000 samples. We highlight some basic features of these graphs in Table 3, where $\sigma$ is the standard deviation. From this data, we can clearly see that Downloader family should be relatively easy to distinguish from the other families. ## 4 Experiments and Results In this section, we first consider results for non-GNN learning methods, including MLP, which provide a baseline for comparison to GNN techniques. Then we consider extensive experiments with each of the GNN and JK models discussed above. ### Results for Non-GNN Models Here, we provide experimental results for five methods, namely, MLP, WL-Kernel, FEATHER, Slaq-LSD, and Slaq-VNGE. Each model requires a different \begin{table} \begin{tabular}{c\|r\|r\|r\|\|r\|\|r\|r\|r\|\|r} \hline \hline \multicolumn{1}{c\|}{Malware} & \multicolumn{4}{c\|\|}{Number of vertices} & \multicolumn{4}{c\|\|}{Number of edges} & \multicolumn{1}{c}{Average} \\ \multicolumn{1}{c\|}{type} & \multicolumn{1}{c\|}{min} & \multicolumn{1}{c\|}{max} & \multicolumn{1}{c\|}{median} & \multicolumn{1}{c\|\|}{$\sigma$} & \multicolumn{1}{c\|\|}{min} & \multicolumn{1}{c\|}{max} & \multicolumn{1}{c\|}{median} & \multicolumn{1}{c\|\|}{$\sigma$} & \multicolumn{1}{c}{degree} \\ \hline AdDisplay & 122 & 4923 & 1248 & 1213.1 & 165 & 12,324 & 1895 & 2374.5 & 2.094 \\ Adware & 211 & 4983 & 2317 & 1339.5 & 473 & 20,096 & 5381 & 3013.1 & 2.232 \\ Benign & 5 & 4994 & 1791 & 1510.5 & 4 & 14,070 & 3591 & 2922.8 & 2.172 \\ Downloader & 40 & 117 & 51 & 4.9 & 44 & 143 & 58 & 6.1 & 1.140 \\ Trojan & 9 & 4993 & 144 & 1313.5 & 7 & 16,404 & 173 & 3275.2 & 2.156 \\ \hline \hline \end{tabular} \end{table} Table 3: Overview of graph features set of hyperparameters, which we have determined via a grid search in each case. For each model, the hyperparameters tested are specified in Table 4, where the selected values are given in boldface. #### 4.1.1 MLP Results We use MLP as a benchmark for comparison to traditional deep learning models. In this case, the LDP features are used for classification. As can be observed from Table 4, our best result with accuracy of 0.8054 was obtained using 5-layers trained for 50 epochs, with the dropout rate set to 0.5. #### 4.1.2 WL-Kernel Results We use node degree and page ranks and concatenated them into feature vectors for each node to generate features for the kernel-based method, WL-Kernel. The WL-Kernel method is used to compute a kernel matrix and then we apply Random Forest for classification, as discussed in Section 3.2.2, above. The model was tuned using the hyperparameters mentioned in Table 4, and the best accuracy attained was 0.7053. The accuracy and runtime metrics are summarized in Table 5. \begin{table} \begin{tabular}{c\|c\|c\|c} \hline \hline Model & Hyperparameters & Tested values & Validation accuracy \\ \hline \multirow{3}{}{MLP} & Number of layers & (5, 6) & \multirow{3}{}{0.8054} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline WL-Kernel & Number of iterations & (5, 6) & 0.7053 \\ \hline FEATHER & Order & (2, 5, 10) & 0.8488 \\ \hline \multirow{2}{}{Slaq-LSD} & Number of vectors & (10, 15, 20) & \multirow{2}{}{0.7799} \\ & Number of steps & (10, 15, 20) & \\ \hline \multirow{2}{}{Slaq-VNGE} & Number of vectors & (10, 15, 20) & \multirow{2}{}{0.5499} \\ & Number of steps & (10, 15, 20) & \\ \hline \hline \end{tabular} \end{table} Table 4: Hyperparameters and accuracy for non-GNN models \begin{table} \begin{tabular}{c\|c\|c\|c\|c} \hline \hline Iterations & Accuracy & Macro-F1 & Runtime (in seconds) & CPU cores \\ \hline 2 & 0.714 & 0.705 & 138.15 & 2 \\ 5 & 0.682 & 0.674 & 182.01 & 2 \\ 10 & 0.621 & 0.606 & 262.50 & 2 \\ \hline \hline \end{tabular} \end{table} Table 5: Accuracy and runtime for WL-Kernel #### 4.1.3 FEATHER Results Using FEATHER, we perform a grid search over the order hyperparameter, which controls how much information is seen from higher-order neighborhoods. The accuracy and runtime metrics for these experiments are given in Table 6. #### 4.1.4 Slaq-LSD and Slaq-VNGE Results For Slaq-LSD and Slaq-VNGE, we perform a grid search over two hyperparameters, namely, the number of random vectors and the number of Lanczos steps. The accuracy and runtime metrics for these models are given in Table 7. ### Results for GNN Architectures In this section, we experiment with each of the basic GNN models introduced in Section 3.3. The input graph for each is a batched graph produced by GraphDataLoader. Each model is trained for 200 epochs. Each of the convolutional layers computes new node representations using a convolutional operator from the Pytorch Torch Geometric library, and we use the Adam optimizer. Each model was tuned using the hyperparameters listed in Table 8, where the best hyperparameters are given in boldface. Initial experiments were also conducted with a 7-layered GCN but virtually no improvement was observed while training time increased considerably, and hence, the 7-layered model was abandoned. \begin{table} \begin{tabular}{c\|c\|c\|c} \hline \hline \multirow{2}{}{Order} & \multirow{2}{}{Accuracy} & Runtime & CPU \\ & & (in seconds) & cores \\ \hline 4 & 0.839 & 672.78 & 2 \\ 5 & 0.849 & 667.18 & 2 \\ 6 & 0.847 & 680.34 & 2 \\ \hline \hline \end{tabular} \end{table} Table 6: Accuracy and runtime for FEATHER model \begin{table} \begin{tabular}{c\|c\|c\|c\|c} \hline \hline \multirow{2}{}{Model} & \multirow{2}{}{Vectors} & \multirow{2}{}{Accuracy} & Runtime & CPU \\ & & & (in seconds) & cores \\ \hline \multirow{3}{}{Slaq-LSD} & 10 & 0.775 & 314.40 & 2 \\ & 15 & 0.776 & 352.35 & 2 \\ & 20 & 0.780 & 401.01 & 2 \\ \hline \multirow{3}{}{Slaq-VNGE} & 10 & 0.549 & 300.85 & 2 \\ & 15 & 0.547 & 322.41 & 2 \\ \cline{1-1} & 20 & 0.544 & 361.35 & 2 \\ \hline \hline \end{tabular} \end{table} Table 7: Accuracy and runtime for Slaq-LSD and Slaq-VNGE #### 4.2.1 GCN, GraphSAGE, GIN, and SGC Results GCN uses the convolutional operator GCNConv and a global pooling layer to generate embeddings. GraphSAGE uses the convolutional operator SAGEConv, a batch normalization layer, and a global pooling layer to generate embeddings. GIN uses the convolutional operator GINConv and a global pooling layer to generate embeddings. SGC uses the convolutional operator SGConv and a global pooling layer to generate embeddings. From Table 8, we see that GCN performed the best of these models, with an accuracy of 0.9582. In contrast, GraphSAGE performed the worst of these four models, with an accuracy of 0.7913. \begin{table} \begin{tabular}{c\|c\|c\|c} \hline \hline Model & Hyperparameters & Tested values & Validation accuracy \\ \hline \multirow{3}{}{GCN} & Number of layers & (5, 6) & \multirow{3}{}{0.9582} \\ & Hidden dimensions & (64, 128) & \\ & Learning Rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{GraphSAGE} & Number of layers & (5, 6) & \multirow{3}{}{0.7913} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{GIN} & Number of layers & (5, 6) & \multirow{3}{}{0.9407} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{SGC} & Number of layers & (5, 6) & \multirow{3}{}{0.9079} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{JK-GCN} & Number of layers & (5, 6) & \multirow{3}{}{0.8941} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{JK-GraphSAGE} & Number of layers & (5, 6) & \multirow{3}{}{0.9291} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{JK-GIN} & Number of layers & (5, 6) & \multirow{3}{}{0.9769} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{UnetGraph} & Weight decay & (0.001, 0.0001) & \multirow{3}{}{0.9581} \\ & Hidden dimensions & (64, 128) & \\ & Learning rate & (0.001, 0.0001) & \\ \hline \multirow{3}{}{DGCNN} & Weight decay & (0.001, 0.0001) & \multirow{3}{}{0.9218} \\ & Learning rate & (0.001*, 0.0001) & \\ \hline \hline \end{tabular} \end{table} Table 8: Hyperparameters and accuracy for GNN models #### 4.2.2 Jumping Knowledge Model Results For JK-GCN, we use an extension of the PyTorch torch.nn.Sequential container to define a sequential GNN model. Since GNN operators take multiple input arguments, torch_geometric.nn.Sequential expects both global input arguments and function header definitions of individual operators. If omitted, an intermediate module operates on the output of its preceding module. This allows us to create more sophisticated models, such as Jumping Knowledge models. The hyperparameters tested for each GNN model are listed in Table 8, where the best hyperparameters for each model are in boldface. Similar to the GCN model, JK-GCN also uses GCNConv layers and implements them on a Jumping Knowledge model. The model was tuned using the hyperparameters mentioned in Table 8, and the best accuracy was 0.8941 Our JK-GraphSAGE implementation uses SAGEConv layers and implements them on a Jumping Knowledge model. The model was tuned using the hyperparameters mentioned in Table 8, and the best accuracy obtained was 0.9291. Our JK-GIN model uses GINConv layers and implements them on a Jumping Knowledge model. The best accuracy in this case was 0.9769. #### 4.2.3 UnetGraph and DGCNN Results We create this model using Pytorch UnetGraph, with a depth of 3 and pooling ratios set to 1 and 0.5. We also set the dropout rate to 0.5. The model was tuned using the hyperparameters in Table 8, and we trained for 100 epochs instead of the 200 used for previous models due to over-fitting at larger numbers of epochs. The best accuracy achieved by our UnetGraph models was 0.9581. Our DGCNN model consists of 4 layers of GCNConv, a pair of 1-D convolutional layers, and a max pooling layer. Two linear classifiers are used to concatenate these layers, with a dropout rate of 0.5. The model was tuned using the hyperparameters mentioned in Table 8, and the best accuracy was found to be 0.9218. With additional hyperparameter tuning, it is likely that this model can be significantly improved, since we only varied the learning rate and weight decay. ### Discussion In this section, we provide additional context for the results in the previous section. Specifically, we summarize the accuracies, classwise accuracies, and training efficiencies for each of the various models. We also provide UMAP embeddings, which allow us to visualize the models. For clarity, in each case we split the analysis into two categories, namely, graph learning methods as introduced in Section 3.2 (that is, MLP, WL-Kernel, FEATHER, Slaq-VNGE, and Slaq-LSD) and the GNN models introduced in Section 3.3 (i.e., GCN, GraphSAGE, GIN, SGC, JK-GCN, JK-GraphSAGE, JK-GIN, UnetGraph, and DGCNN). Here, we refer to the former as non-GNN models, while the latter are GNN models. #### 4.3.1 Accuracy Comparison In Figure 1(a) we compare the accuracies of our non-GNN models, while Figure 1(b) is an analogous comparison of the GNN-based models that we tested. Most of the GNN-based models far outperform the best of the non-GNN models. We observe that, overall, JK-GIN has the highest accuracy at 0.9769, with UnetGraph having an accuracy of 0.9581, while GIN (0.9407), JK-GraphSAGE (0.9291), DGCNN (0.9218) and SGC (0.9079) also outperform the best of the non-GNN models. While GCN has a high accuracy of 0.9582, we believe it may be due to over-smoothing. As mentioned above, we believe that further tuning of the hyperparameters might yield significant improvement for DGCNN. As can be seen in Figure 1(b), among the non-GNN-based models, FEATHER was the best at 0.8488, with MLP being second best at 0.8054. These results are not unexpected, as these models were shown to work well (along with GIN) in the paper [12]. The bottom line is that our best GNN-based models outperform all models that were applied to the MalNet-Tiny dataset in the papers [12, 29]; see also Table 2, above. #### 4.3.2 Classwise Comparison To further analyze our results, we generate heat maps of the accuracy of each class for every model corresponding to its best parameters. In Figure 2(a), we observe that our worst-performing GNN-based models, namely, GraphSAGE and JK-GCN, underperform primarily because they badly fail on the benign class. We also note that all of the GNN-based models classify the Downloader class with ease. Figure 1: Accuracy comparison Classwise comparison of our non-GNN models is given in Figure 2(b). Qualitatively, the results for these models are similar to those of the GNN-based models, with Downloader being the easiest to classify, and Benign being the most challenging. However, as noted above, the non-GNN models perform poorly, in comparison to the best of our GNN-based models. Overall, we see that Downloader is the easiest class to classify, while the Benign class is the most difficult. Even our worst model, Slaq-VNGE, identifies Downloader class with a 0.93 accuracy, while our best classifiers attain a similar accuracy for the Benign class. These results are not surprising in light of our initial analysis of the node and edge features, as given in Table 3, above. We also provide confusion matrices for each of the non-GNN models and for the GNN models in Figures A.1 and A.2, which can be found in the Appendix. From these confusion matrices, we observe that the Benign class is most often misclassified as belonging to the Adware or AdDisplay class. Interestingly, we do not find a large percentage of misclassifications of Benign as Trojan (or vice-versa) which, intuitively, would seem to be difficult to distinguish, as Trojans are designed to mimic benign applications. Figure 2: Classwise accuracies #### 4.3.3 Runtime Comparison We define the runtime to be the total time taken to train, test, and validate a model using its corresponding best set of hyperparameters. The runtime results for our GNN-based models are given in Figure 3(a). We observe that both GraphSAGE and JK-GraphSAGE run for longer durations than other models. The convolution operator, SAGEConv, is the most complex operator used in any of the GNNs, so this result is expected. Among the GNN-based models, GIN and GCN models require the least time for classification, as they use the least complex convolutional operators. The runtime results for our non-GNN models are given in Figure 3(b). These models require significantly less time than GNN-based models, with the longest runtime among this collection of models being only about two-thirds that of the shortest runtime among the GNN-based models. Note that the best performing non-GNN models, FEATHER and MLP, have the longest runtimes. #### 4.3.4 UMAP Embeddings for GNN Models Finally, we provide a visualization of the graph embeddings for nine of our models. In each case, we apply the UMAP dimensionality reduction technique [32] to the dense layer. These results are given in Figure A.3 in the Appendix, where we have labeled the classes as \[(\texttt{AdDisplay},\texttt{Adware},\texttt{Benign},\texttt{Downloader}, \texttt{Trojan})=(0,1,2,3,4).\] In Figure A.3 we observe that there is generally the most class separability between Downloader, and other classes, which is supported by our observation that Downloader is the easiest to classify. We also note that several of the models provide good class separability. In particular, the jumping knowledge models, JK-GCN, JK-GraphSAGE, and JK-GIN, which appear in Figures A.3(e), (f), and (g), respectively, are among the best from this perspective. Figure 3: Runtime comparison Overall, these UMAP embeddings support some of our major observations. They also show that our better models tend to form numerous small clusters, which likely correspond to identifiable sub-classes within each class. This would be an interesting topic for further research. ## 5 Conclusion and Future Work In this extensive study, we classified five different types of Android samples from the the MalNet-Tiny dataset graph-based learning methods. We considered the graph representation schemes FEATHER, WL-Kernel, Slaq-VNGE, and Slaq-LSD, and applied a Random Forest classifier to each. We found that FEATHER works best among these methods. We then used the Local Degree Profile (LDP) to encode node features and trained a wide variety of GNN-based models on these features. Specifically, we experimented with GCN, GrahSAGE, GIN, SGC, JK-GCN, JK-GraphSAGE, JK-GIN, UnetGraph, and DGCNN models. We experimented with hyperparameters and found that these GNN-based models generally performed well, with JK-GIN, JK-GraphSAGE, UnetGraph, and DGCNN giving us the best results. We provided extensive analysis of our results, including classwise accuracies, confusion matrices, UMAP embeddings, and runtime comparisons. Our best models exceeded the results obtained in comparable previous work. In future work, we intend to extend this promising research to a larger subset of the MalNet dataset. A more diverse dataset, with samples drawn from a larger number of families, would provide additional scope for tuning hyperparameters and analyzing the resulting models, so as to better understand their relative strengths and weaknesses. It would be interesting to explore other graph classification architectures, including DeeperGCN [11], EdgePool [27], and Kernel Graph-based CNN (KG-CNN) [9]. It would also be worth exploring edge-based graph neural approaches such as E-GraphSAGE [30], which would enable us to include network flows as features. Another area of future research would be to analyze the effectiveness of GNN-based models for zero-day malware detection. In a similar vein, it would also be interesting to explore malware GNN models from the perspective of explainable AI (XAI), that is, we would like to better understand how the models are actually making decisions. For this explainability problem, we could use GNNExplainer [51], which is designed to derive insights from the hidden layers of GNNs, and SubgraphX [53]. Finally, it is worth noting that the models considered in this paper are based solely on graph structure. Combining such graph-based models with more traditional features might result in models that are stronger than either the GNN-based models considered here, or learning models that are based only on more traditional features. ## 6 Declarations The authors have no relevant financial or non-financial interests to disclose.
2306.11509	Implicit neural representation with physics-informed neural networks for the reconstruction of the early part of room impulse responses	Recently deep learning and machine learning approaches have been widely employed for various applications in acoustics. Nonetheless, in the area of sound field processing and reconstruction classic methods based on the solutions of wave equation are still widespread. Recently, physics-informed neural networks have been proposed as a deep learning paradigm for solving partial differential equations which govern physical phenomena, bridging the gap between purely data-driven and model based methods. Here, we exploit physics-informed neural networks to reconstruct the early part of missing room impulse responses in an uniform linear array. This methodology allows us to exploit the underlying law of acoustics, i.e., the wave equation, forcing the neural network to generate physically meaningful solutions given only a limited number of data points. The results on real measurements show that the proposed model achieves accurate reconstruction and performance in line with respect to state-of-the-art deep-learning and compress sensing techniques while maintaining a lightweight architecture.	Mirco Pezzoli, Fabio Antonacci, Augusto Sarti	2023-06-20T13:01:00Z	http://arxiv.org/abs/2306.11509v1	Implicit Neural Representation with Physics-Informed Neural Networks for the Reconstruction of the Early Part of Room Impulse Responses ###### Abstract Recently deep learning and machine learning approaches have been widely employed for various applications in acoustics. Nonetheless, in the area of sound field processing and reconstruction classic methods based on the solutions of wave equation are still widespread. Recently, physics-informed neural networks have been proposed as a deep learning paradigm for solving partial differential equations which govern physical phenomena, bridging the gap between purely data-driven and model based methods. Here, we exploit physics-informed neural networks to reconstruct the early part of missing room impulse responses in an uniform linear array. This methodology allows us to exploit the underlying law of acoustics, i.e., the wave equation, forcing the neural network to generate physically meaningful solutions given only a limited number of data points. The results on real measurements show that the proposed model achieves accurate reconstruction and performance in line with respect to state-of-the-art deep-learning and compress sensing techniques while maintaining a lightweight architecture. physics-informed neural network, sound field reconstruction, wave equation. + Footnote †: _Corresponding author: [email protected]._ _Copyright: ©2023 Miros Przzzoli et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited._ Copyright: ©2023 Miros Przoli et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *Mirco Pezzoli${}^{}$ Fabio Antonacci Augusto Sarti Dipartimento di Elettronica, Informatica e Bioingegneria, Politecnico di Milano, Italy** ## 1 Introduction Sound field reconstruction is fundamental in augmented and virtual reality applications, where users can experience immersive audio environments. To accurately characterize the acoustic properties of a given environment, the acquisition of multichannel signals is necessary. Room impulse responses (RIRs) captured with microphone arrays are particularly useful for this and various tasks such as sound source localization [1, 2], separation [3, 4], and sound field navigation [5, 6, 7]. In fact RIRs provide a model of the sound propagation between the acoustic source and the microphone array within an environment. The reconstruction of RIRs or sound field in general, has been a subject of extensive research, leading to the development of two primary categories of solutions: parametric and non-parametric techniques. Parametric methods [7, 8, 9, 10, 11, 12] rely on simplified parametric models of the sound field to convey an effective spatial audio perception to the user. In contrast, non-parametric methods [13, 14, 15, 16, 17] aim to numerically estimate the acoustic field. Most of the available techniques in this class are based on compressed sensing principles [18] combined with the solutions of the wave equation [19], i.e., plane wave [20] and spherical wave [17, 21], the modal expansion [15] or the equivalent source method (ESM) [16, 22]. A third category comprising deep learning emerged as an alternative approach for sound field reconstruction and a wide range of problems in the field of acoustics [23, 24, 25, 26, 27]. In [28], a convolutional neural network (CNN) has been proposed for the reconstruction of room transfer functions. However, as noted in [28] the model is limited to low frequencies and the generalization is constrained by the available data set. To overcome the frequency and data set limitations, in [29], the authors proposed a _deep prior_ approach [30] to RIR reconstruction in time domain. The deep prior paradigm [29] considers the structure of a CNN as a regularization prior to learn a mapping from a random input to the reconstructed RIRs of an Uniform Linear Array (ULA). As a result, no extensive data set is required for training since the optimization is performed over a single ULA. Recently, in order to exploit the underlying physics of the sound field, a physics-informed neural network (PINN) [31, 32, 33] for sound field reconstruction has been introduced in [34]. The main idea of PINN [31, 32] is to force the output of a network to follow the partial differential equations (PDE) governing the system under analysis. In particular, PDE computation is performed exploiting the automatic differentiation framework underlying the training procedure of neural networks. Following the PINN approach, in [34] the authors augmented the loss function used for training a CNN with the computation of the Helmholtz equation [34]. However, differently from standard PINNs [31], the network provides as output an estimate of the derivatives required to compute the Helmholtz equation instead of relying on automatic differentiation. Moreover, the system works at a fixed frequency ($300\,\mathrm{Hz}$) and it has been tested only on simulated data. In this paper, we propose the use of a physics-informed approach for the reconstruction of the early part of RIRs. As a matter of fact, the early part of RIRs provides relevant information on the geometry of environment [35, 36] affecting the timbre and localization of acoustic sources [37]. Therefore, accurate reconstruction of the early part of RIRs [22, 38] is required, while the late reverberation is typically modelled through its statistical characteristics [7, 8, 39]. In order to avoid frequency limitations, we work on RIRs in the time domain. We adopt a network that takes as input the signal domain i.e., the time and position of the microphone and provides as output an estimate of the RIRs at the given coordinates. In order to improve the performance exploiting prior knowledge on the signal domain, we employed a network structure known as SIREN [40] trained using the PINN paradigm. We refer to the adopted approach to as physics-informed SIREN (PI-SIREN). SIREN demonstrated to be an effective architecture to learn _neural implicit representations_ of different signals including audio and for solving the wave equation (direct problem) [40]. However, the adoption of SIREN has not been fully explored yet for solving time-domain inverse problems in the field of multichannel acoustic processing or applying to real acoustic measurements. In this work, we investigate the use of PI-SIREN for the reconstruction of early parts of the RIRs acquired by an ULA. Results on simulations revealed that in contrast to classical PINN, PI-SIREN is a suitable architecture for RIR reconstruction. In addition, we compare the reconstructions of PI-SIREN on real data with respect to state-of-the-art solutions based on compressed sensing [41] and deep learning [29] showing improved reconstruction of the early parts of the RIRs in two of the three considered rooms. ## 2 Problem Statement ### RIR data model Let us consider an acoustic source located in $\mathbf{r}^{\prime}=[x^{\prime},y^{\prime},z^{\prime}]^{T}$ and a set of $M$ microphones acquiring the generated sound field. Assuming linear acoustics and absence of noise, the sound pressure at the $m$th sensor can be defined as \[p(\mathbf{r}_{m},t)=h(t,\mathbf{r}_{m},\mathbf{r}^{\prime})s(t),\quad m=1,\dots,M, \tag{1}\] where $p(\mathbf{r}_{m},t)$ is the time-domain sound pressure at time instant $t$ and location $\mathbf{r}_{m}$, $s(t)$ is the signal emitted by the source and $$ denotes the linear convolution operation. The term $h(t,\mathbf{r}_{m},\mathbf{r}^{\prime})$ in (1) refers to the RIR between the source in $\mathbf{r}^{\prime}$ and the sensor at $\mathbf{r}_{m}$. In general, RIR provides a description of the sound propagation in the environment from a source to a receiver and due to (1), it completely characterizes the spatial properties of the sound field. In ideal conditions with unbounded domain, the RIR is given by the well-known Green's function [19] which is a particular solution of the inhomogenous wave equation [19] \[\nabla^{2}p(\mathbf{r},t)-\frac{1}{c^{2}}\frac{\partial^{2}p(\mathbf{r},t)}{\partial t ^{2}}=\delta(\mathbf{r}-\mathbf{r}^{\prime},t), \tag{2}\] where $\delta$ is the Dirac delta function and $c$ is the speed of sound in air. In this work, we consider the RIRs of an ULA and the location of each $m$th microphone is given by the distance $d$ between two consecutive sensors as $\mathbf{r}_{m}=[x_{a},(m-1)d,z_{a}]^{T}$. The values of $x_{a}$ and $z_{a}$ are the same for all the sensors in the ULA. It follows that the maximum frequency for aliasing-free sound field acquisition in the ULA is limited by the distance $d$ through \[F_{\text{max}}=\frac{c}{2d}. \tag{3}\] In practice, we organize the acquired RIRs in a $N\times M$ matrix defined as \[\mathbf{H}=[\mathbf{h}_{1},\dots,\mathbf{h}_{M}], \tag{4}\] where $\mathbf{h}_{m}\in\mathbb{R}^{N\times 1}$ is the vector containing the $N$-length sampled RIR of the $m$th microphone. In Fig. 1, an example of RIRs acquired by an ULA is shown. ### RIR reconstruction problem We assume that a limited subset, indexed as $\tilde{\mathcal{M}}$, of the ULA sensors $\mathcal{M}$ is available, and thus $\tilde{\mathcal{M}}\subseteq\mathcal{M}\left([\tilde{\mathcal{M}}]=\tilde{M} <M\right)$. The goal of RIR reconstruction is to recover the missing data exploiting the information available from RIRs in the observation points $\left\{\mathbf{r}_{\hat{m}}\right\}_{\hat{m}\in\tilde{\mathcal{M}}}$. Various techniques have been proposed in the literature to address the spatial-sampling requirement for reconstructing RIRs from an undersampled measurement set. In general, this task can be interpreted in the framework of inverse problems, and a solution to the problem can be found through Figure 1: Example of RIRs $\mathbf{H}$ of a $M=32$ microphones ULA. the following minimization \[\begin{split}\mathbf{\theta}^{}=\underset{\mathbf{\theta}}{\text{argmin}}\ J\left(\mathbf{\theta}\right)=\\ E\left(f(\mathbf{\theta},\left\{\mathbf{r}_{\text{in}}\right\}_{\text{in} \in\tilde{\mathcal{A}}}\right)-\mathbf{H}(\left\{\mathbf{r}_{\text{in}}\right\}_{ \text{in}\in\tilde{\mathcal{A}}}))\right),\end{split} \tag{5}\] where $E(\cdot)$ is a data-fidelity term, e.g., the mean squared error, between the estimated data and the observations, and $f(\mathbf{\theta},\mathbf{r})$ is a function that generates the estimated RIRs using the parameters $\mathbf{\theta}$. The time dependency in (5) has been omitted for notational simplicity. It is worth noting that in (5), the evaluation of $f$ is performed in the observation locations $\left\{\mathbf{r}_{\text{in}}\right\}_{\text{in}\in\tilde{\mathcal{A}}}$. However, $f$ must be able to provide a meaningful estimate also in location that are different from the available ones. Therefore, the solution to the ill-posed problem (5) is constrained using regularization strategies. Typical techniques include compressed sensing frameworks based on assumptions about the signal model such as plane and spherical wave expansions [13], ESM [22], or the RIRs structure [41], as well as deep learning approaches [28, 29]. ## 3 Proposed Method In this work, we aim at solving the RIR reconstruction problem (5) in order to provide an estimate of the ULA RIRs as \[\widehat{\mathbf{H}}=f\left(\mathbf{\theta}^{},\left\{\mathbf{r}_{\text{m}}\right\}_ {\text{m}\in\mathcal{M}}\right), \tag{6}\] where the function $f(\cdot)$ represents a neural network. In particular, we adopt the structure of a SIREN [40] neural network. SHREN proved to be an effective architecture for learning the so-called _neural implicit representations_ of different classes of signals, including audio signals. The proposed model has the structure of a multilayer perceptron (MLP) with sinusoidal activation functions, for which the $i$th layer can be expressed as \[\phi_{i}(\mathbf{x}_{i})=\sin\left(\omega_{0}\mathbf{x}_{i}^{T}\mathbf{\theta}_{i }+\mathbf{b}_{i}\right), \tag{7}\] where $\mathbf{x}_{i}$, $\mathbf{\theta}_{i}$, and $\mathbf{b}_{i}$ are the input vector, the weights and the biases of the $i$th layer, respectively, while $\omega_{0}$ is an initialization hyper-parameter [40]. The adopted SIREN architecture is thus a composition of $L$ layers \[f\left(\mathbf{\theta},\mathbf{x}\right)=\left(\phi_{L}\circ\phi_{L-1},\dots,\phi _{1}\right)(\mathbf{x}), \tag{8}\] where $\mathbf{x}$ is the input of the network while $\mathbf{\theta}$ is the set of learnable parameters. Following the paradigm of neural implicit representations, the SIREN model takes as input the signal domain, namely the sensor position $\mathbf{r}_{\text{m}}$ and the time instant $t$ and provides as output an estimate of the RIR $\hat{h}(t,\mathbf{r}_{\text{m}})$. Hence, the role of the network is to provide a parameterized representation of the signals through the parameters of the MLP. Essentially, during the training, the neural network overfits the available signals becoming an alternative implicit representation of the RIRs. Although we can fit the available RIRs through SIREN, regularization strategies are required in order to provide meaningful results in different points of the domain i.e., to estimate the missing RIRs. Here, we consider training SIREN using the PINN approach, denoting the solution as PI-SIREN. Using as the target for the training the reconstruction of the observation only, there is no guarantee that the solution follows the physical law of the underlining problem, namely the wave equation [19]. PINN are forced to learn solutions that follows the PDE of the underlying physics in order to obtain improved results. This approach exploits the prior knowledge on the system in order to regularize the estimation of the neural network. Therefore, we adopted the following loss function for training PI-SIREN which includes a physics-informed term as \[\begin{split}\mathcal{L}&=\frac{1}{M}\sum_{\text{ in}\in\tilde{\mathcal{A}}}\\|\hat{h}_{\text{in}}-h_{\text{in}}\\|_{2}^{2}+\\ &\lambda\frac{1}{M}\sum_{\text{m}}^{M}\left\\|\frac{1}{\mathcal{ L}^{2}}\frac{\partial^{2}\hat{h}_{\text{in}}}{\partial\Omega^{2}}-\nabla^{2} \hat{h}_{\text{in}}\right\\|_{2}^{2},\end{split} \tag{9}\] where $\\|\cdot\\|_{2}$ is the $\ell_{2}$ norm, the first term of the summation represents a distance between the prediction and the available data, while the second term corresponds to the PDE loss given by the wave equation and weighted by parameter $\lambda$. While the first part of (9) makes the network fit the observation, the PDE term constrains the output to follow the wave equation. The use of the PDE loss results in a regularized solution since the output conforms with the underlying physical equation. Once trained, PI-SIREN can be used to obtain the RIRs at the missing and available positions of the ULA simply feeding the network with the locations $\mathbf{r}_{\text{m}}$, $m=1,\dots,M$, and the different time instants $t$. ## 4 Numerical Experiments ### Setup We evaluate the performance of PI-SIREN for RIR reconstruction on both simulated and measured data from [41]. We considered an ULA of $M=100$ microphones with distance $d=2.02\,\mathrm{cm}$ which gives a maximum frequency ($3$) $F_{\text{max}}=8.489\,\mathrm{kHz}$. The simulated RIRs have been computed at sampling rate $8\,\mathrm{kHz}$ using the image source technique [42] for a shoe-box room of dimensions $6\,\mathrm{m}\times 4\,\mathrm{m}\times 3\,\mathrm{m}$ and reverberation time $\mathrm{T}_{60}=0.5\,\mathrm{s}$. For this work we limit the analysis to the first $20\,\mathrm{ms}$ of the RIRs which corresponds to the early part (direct and early reflections) of the impulse response. The proposed PI-SIREN architecture is composed of $L=5$ layers of $256$ neurons in which the last layer is linear. The network has a total of $198401$ trainable parameters. The initialization frequency $\omega_{0}$ in (7) is set to $15$ for the first layer, while as the $1940\,\mathrm{\omega}_{0}=30$ for the hidden layers. The network is trained for $2000$ iterations using Adam optimizer with learning rate equal to $10^{-4}$. The weight parameter in (9) has been experimentally set to $\lambda=5\cdot 10^{-15}$. Similarly to [29, 41], we evaluate the reconstruction performance in terms of the normalized mean square error ($\mathrm{NMSE}$) between the reconstructed data and the reference RIRs defined as [41] \[\mathrm{NMSE}\left(\widehat{\mathbf{H}},\mathbf{H}\right)=10\log_{10}\frac{1}{ M}\sum_{m=1}^{M}\frac{\\|\widehat{\mathbf{h}}_{m}-\mathbf{h}_{m}\\|^{2}}{\\| \mathbf{h}_{m}\\|^{2}}, \tag{10}\] where $\widehat{\mathbf{h}}_{m}$ is the $m$th RIR estimate provided by the reconstruction technique. The observations are computed considering $\tilde{M}=\{20,33\}$ microphones randomly selected as in [41], which corresponds to $1/5$ and $1/3$ of sensors available, respectively. ### Results In order to assess the effectiveness of the proposed PI-SIREN methodology in terms of architecure and training strategy, we evaluate the reconstruction performance on simulated data. We compare the reconstruction of PI-SIREN with respect to a classical PINN architecture [31] and SIREN trained without the wave equation term in the loss function (9). The PINN shares the same structure of PI-SIREN in terms of layers and parameters, however $\tanh$ is adopted as nonlinear function of the neurons. In Fig. 2, the RIRs $\mathbf{H}$ along with the observation $\mathbf{H}_{m}$ with $\tilde{M}=33$ and the obtained reconstructions are reported. From Fig. 2(e), we can observe that PINN fails to reconstruct the RIRs, obtaining a $\mathrm{NMSE}_{\mathrm{PINN}}=14.5\,\mathrm{dB}$. The adoption of the sinusoidal activation function in SIREN (see Fig. 2(d)) determines an improved reconstruction performance with $\mathrm{NMSE}_{\mathrm{SIREN}}=-7.1\,\mathrm{dB}$. Inspecting Fig. 2(d), we can observe that SIREN reconstructs the direct path and the early reflections in $\widehat{\mathbf{H}}_{\mathrm{SIREN}}$, filling the missing channels. It follows that SIREN provides an effective implicit representation of the considered signals thanks to the use of the sinusoidal nonlinearity. In [43], the authors show how a two-layers SIREN can be related to a discrete cosine transform (DCT) of the signal. In the context of this work, the consideration in [43] can be loosely interpreted in terms of a real-valued plenacoustic representation [44] of the RIRs. Nonetheless, the reconstruction in Fig. 2(d) contains noisy components and the estimated wave fronts at some of the missing locations are incoherent. In Fig. 2(c), the output of PI-SIREN is depicted. It is possible to note that, differently from the basic SIREN, PI-SIREN is able to estimate the RIRs more accurately, coherently reconstructing the wave fronts at the missing locations. The reconstruction of PI-SIREN achieves a $\mathrm{NMSE}_{\mathrm{PI-SIREN}}=-11.2\,\mathrm{dB}$ which is lower with respect to both SIREN and the PINN. Through the physics-informed loss function in (9), in fact, the output of the network is forced to conform with the physical prior of the wave equation. Therefore, the adoption of the physics-informed loss function in PI-SIREN allows us to obtain an improved performance. ### Experimental results We evaluate the performance of PI-SIREN on real RIRs measured in three rooms [41] and we compare the estimated reconstruction with respect to the compressed sensing method (CS) in [41] and the deep prior (DP) methodology of [29]. The employed ULA consists of $M=100$ sensors with distance $d=3$ cm. The rooms are named "Balder", "Freqa" and "Munin" and the estimated reverberation times $\mathrm{T}_{30}$ are $0.32\,\mathrm{s}$, $0.46\,\mathrm{s}$ and $0.63\,\mathrm{s}$, respectively. In Table 4, the $\mathrm{NMSE}$ obtained for the different rooms are reported. As expected, when a lower number of sensors $\tilde{M}=20$ is available the reconstruction performance is reduced for all the considered techniques. The performance of the three methods is \begin{table} \begin{tabular}{c\|c\|c\|c\|c\|c\|c} Room & \multicolumn{2}{c\|}{Balder} & \multicolumn{2}{c\|}{Freja} & \multicolumn{2}{c}{Munin} \\ Mic. & 20 & 33 & 20 & 33 & 20 & 33 \\ \hline \multicolumn{8}{c}{$\mathrm{NMSE}$ (dB)} \\ \hline CS & -5.87 & -11.47 & -5.89 & -11.01 & -7.52 & -15.25 \\ DP & -5.52 & -11.44 & -4.68 & -9.21 & -8.98 & -16.03 \\ PI-SIREN & -6.26 & -11.74 & -5.65 & -10.61 & -10.00 & -16.17 \\ \end{tabular} \end{table} Table 1: $\mathrm{NMSE}$ of the considered techniques at different downsampling conditions for the three rooms. Figure 2: (a) Simulated RIRs $\mathbf{H}$. (b) The observation $\mathbf{H}_{m}$ of $\tilde{M}=33$ microphones employed as input for the networks. The reconstructions obtained using PI-SIREN (c), SIREN (d) and PINN (e). in line for all the considered scenarios. However, the proposed PI-SIREN is able to achieve lower $\mathrm{NMSE}$ in Balder and Muning rooms for both the adopted undersampling conditions. Interestingly, the best reconstruction performance is achieved for room Muning for every method. This room has the highest $\mathrm{T}_{30}$, but as notices in [41], the density of early reflections is lower with respect to the other rooms, making the reconstruction less challenging. CS obtained the lowest $\mathrm{NMSE}$ in room Freja. However, the difference with respect of the proposed model is limited to $0.24\,\mathrm{dB}$ and $0.4\,\mathrm{dB}$ for the $\hat{M}=20$ and $\hat{M}=33$ scenarios, respectively. In Fig. 3, the reconstructions of $\mathbf{H}$ for room Muning given $\hat{M}=20$ microphones are reported. The reference RIRs are depicted in Fig. 3(a). Inspecting the reconstruction in Fig. 3, we can note that all the three methods managed to reconstruct the main structure of the RIRs. However, the reconstruction provided by CS presents an underestimation of the RIRs at the missing locations which are seen as light vertical stripes in Fig. 3(d). Instead $\mathbf{H}_{\mathrm{PI-SIREN}}$ and $\mathbf{H}_{\mathrm{OP}}$ have a similar performance with a lower reconstruction error ($\mathrm{NMSE}_{\mathrm{PI-SIREN}}=-10\,\mathrm{dB}$) for the proposed model compared to DP ($\mathrm{NMSE}_{\mathrm{DP}}=-8.89\,\mathrm{dB}$). ## 5 Conclusion In this work we proposed the use of PINNs for the reconstruction of early part of RIRs. The devised architecture consists of a SIREN neural network trained exploiting the physics-informed neural network framework. This allows us to impose the governing wave equation to the solution of the RIR reconstruction. The results shows that the SIREN architecture itself provides an implicit representation of the data. Moreover, the adoption of the physics-informed training demonstrated to improve the reconstruction performance. We investigated the application of the proposed model on real data, showing competitive results with state-of-the-art techniques based on compressed sensing and deep learning. The proposed technique is appealing since it synergistically exploits the flexibility of deep learning and the prior knowledge of physics. We foresee the future of this work concerning the network design and the modeling of the whole RIRs that can improve the performance and the applicability with respect to the current results. ## 6 Acknowledgments This work has been funded by "REPERTORIUM project. Grant agreement number 101095065. Horizon Europe. Cluster II. Culture, Creativity and Inclusive society. Call HORIZON-CL2-2022-HERITAGE-01-02."
2307.15877	High-order Discontinuity Detection Physics-Informed Neural Network	In order to solve the problem of the difficult direct measurement of temperature field in fluid machinery under high-speed compressible conditions, this study combines high-order finite difference numerical format, Weighted Essentially Non-Oscillatory (WENO) discontinuity detection, and traditional Physics-Informed Neural Network (PINN) to develop a high-order discontinuity detection PINN (Hodd-PINN) that can achieve temperature field inversion with a small number of measurement points. When dealing with pure convection problems, Hodd-PINN introduces a 7th-order discretization for the convection term, reducing an additional 9.7% error compared to traditional low-order discretization methods. When dealing with pure diffusion problems, Hodd-PINN introduces an 8th-order discretization for the diffusion term, reducing an additional 12.8% error compared to traditional low-order discretization methods. In addition, this paper develops a loss function based on WENO discontinuity detection technology, which helps eliminate false discontinuities, allowing Hodd-PINN to successfully identify sparse waves that are easily overlooked in PINN's predicted results, reducing the error by 24.2%. Through extensive testing, this paper points out that the Hodd-PINN, which incorporates high-order discretization and discontinuity detection technology, can further reduce the prediction error of PINN, effectively reducing the data requirement, and can effectively solve the problem of false discontinuities. This method has important value for the inversion of temperature and velocity fields in fluid machinery under high-speed compressible conditions.	Ruquan You, Shuming Zhang, Tinglin Kong, Haiwang Li	2023-07-29T03:05:03Z	http://arxiv.org/abs/2307.15877v1	# High-order Discontinuity Detection Physics- ###### Abstract In order to solve the problem of the difficult direct measurement of temperature field in fluid machinery under high-speed compressible conditions, this study combines high-order finite difference numerical format, Weighted Essentially Non-Oscillatory (WENO) discontinuity detection, and traditional Physics-Informed Neural Network (PINN) to develop a high-order discontinuity detection PINN (Hodd-PINN) that can achieve temperature field inversion with a small number of measurement points. When dealing with pure convection problems, Hodd-PINN introduces a 7th-order discretization for the convection term, reducing an additional 9.7% error compared to traditional low-order discretization methods. When dealing with pure diffusion problems, Hodd-PINN introduces an 8th-order discretization for the diffusion term, reducing an additional 12.8% error compared to traditional low-order discretization methods. In addition, this paper develops a loss function based on WENO discontinuity detection technology, which helps eliminate false discontinuities, allowing Hodd-PINN to successfully identify sparse waves that are easily overlooked in PINN's predicted results, reducing the error by 24.2%. Through extensive testing, this paper points out that the Hodd-PINN, which incorporates high-order discretization and discontinuity detection technology, can further reduce the prediction error of PINN, effectively reducing the data requirement, and can effectively solve the problem of false discontinuities. This method has important value for the inversion of temperature and velocity fields in fluid machinery under high-speed compressible conditions. ## Introduction In the field of high temperature, intense aerodynamic compression, and strong heat transfer[1][2][3][4], temperature control has become the main challenging aspect in scientific research and industrial design. However, the high temperature, high speed, and strong heat transfer characteristics make it extremely difficult to conduct full temperature and pressure experiments [4][5], resulting in typical problems such as limited temperature testing methods. Currently, temperature detection methods are mainly divided into contact measurement and non-contact measurement. Among them, contact measurement has limitations due to the influence of high temperature and high-speed airflow, requiring probes to have good heat protection in order to function properly [6][7]. Moreover, considering the interference from complex flow, obtaining valid experimental results requires extra effort [8]. Non-contact measurement mainly includes infrared temperature measurement [9], liquid crystal temperature measurement [10], and thermocouple temperature measurement [11]. Among them, infrared temperature measurement can obtain the surface distribution of temperature, but due to the non-uniform characteristic of radiation with angle, it has a larger measurement error. Liquid crystals are sensitive to temperature, but their physical properties make it difficult to withstand full temperature and pressure experiments, and the sprayed thickness can also have an unpredictable impact on the heat transfer characteristics of the test piece. In most cases, if precise measurement of the surface temperature of the experimental component in a high temperature and high-speed environment is desired, it is usually necessary to use embedded thermocouples. Although the process of embedding thermocouples is complex and the spatial resolution is extremely limited (typically only a few measurement points can be set up), accurate measurement results can be obtained. However, the measurement method of embedded thermocouples is difficult to obtain the complete temperature field of the experimental component, which is very unfavorable for fine temperature control design. Therefore, it has always been an important issue to use as few temperature measurement points as possible to obtain the full temperature field of the entire experimental component, and to achieve temperature field inversion based on limited experimental data. Traditional CFD methods are difficult to integrate experimental data. On one hand, traditional grid-based CFD methods apply control equations to corresponding computational domains through discrete solution methods [12], and use Taylor expansion to control the error limit between discrete points and continuous physical fields, and then determine the solution based on known boundary conditions [13][14][15][16]. On the other hand, once the boundary conditions are determined, the discrete solution has already reached global convergence and cannot accept secondary modifications. From another perspective, "having a determined solution" and "accepting modified computational results" are two contradictory properties that cannot be simultaneously satisfied. Therefore, although traditional CFD methods can solve physical fields based on boundary conditions, they cannot further integrate the data. However, methods like PINN [17] and TGNN [18], which combine CFD and neural networks, provide new possibilities for integrating experimental data. In a narrow sense, PINN is a method that uses neural networks to fit continuous physical fields, avoiding the process of obtaining physical fields through discrete solutions [19]. In the coordinate positions where the measured values of the physical field are known, PINN can directly construct a function from the local coordinates to the predicted values. In the coordinate positions where the measured values of the physical field are unknown, PINN can rely on the constraints of partial differential equations for training without data, so that the trained positions can satisfy the partial differential equations to the greatest extent possible, achieving physical field predictions within a certain range of accuracy [20][21][22]. However, since the spatial positions used for training PINN are always limited, the NN always needs to generalize the properties of these trained points to the surrounding space. This approach means that PINN can accept both measurement data or differential equation laws at observed positions, and can flexibly choose the direction of generalization at unobserved positions. Because of PINN's reliance on generalization for solving, strictly speaking, PINN does not have a determined solution, only a set of solutions that satisfy the given observation point rules. From this perspective, the reason why PINN can achieve data fusion is that it only needs to optimize and find a solution set that is closest to the given constraints, without guaranteeing that its own prediction results hold true at every position in space. Due to the existence of the universal approximation theorem [23], as long as the solution has continuous characteristics and the expressive power of the NN is strong enough, such conditions can always be satisfied. Therefore, PINN achieves the flexibility of data fusion, which can be seen as sacrificing determinism in order to gain benefits. Due to the inherent nature of PINN mentioned above, PINN in a narrow sense has deficiencies. These deficiencies arise from the way physical constraints are added and the general instability of neural networks. On one hand, although PINN incorporates physical constraints into the loss function, the loss function itself may not be fully satisfied, or it may easily get trapped in local optima, resulting in PINN's predicted results not being completely consistent with the solutions of partial differential equations [24]. On the other hand, the generalization ability of PINN serves as a bridge between discrete data and continuous physical fields. Its generalization is established statistically and cannot guarantee that the position between two training points will not fail [25]. Such instability makes PINN's predicted results less rigorous. In fact, if PINN uses Relu[26] or LeakyRelu[27] as the activation function, the derivative mutation caused by generalization instability will become exceptionally noticeable. In order to compensate for the deficiencies in generalization ability as much as possible, PINN needs to rely on a large number of data points to fit the physical field, similar to traditional neural networks [28]. Even so, without a large number of experimental measurements for correction along the way, training only relying on partial differential equation constraints may still result in the accumulation of errors in space, making extrapolation performance difficult to guarantee. Apart from the high requirements for training data and methods, traditional PINN also has theoretical deficiencies when faced with physical problems with discontinuities: PINN assumes that in space points with unknown exact values, the physical field should satisfy the partial differential equations. However, when facing nonlinear problems or even discontinuous physical fields with discontinuities caused by compressible flow, such as shock waves, the assumption of being continuously differentiable will locally fail, leading to a deterioration in the robustness of PINN [29]. To solve the inherent shortcomings of Physics-Informed Neural Networks (PINN), scholars have made improvements from two main perspectives. The first perspective is to expand the traditional PINN methods from the point of view of statistical learning. For example, Qiu et al. [30] utilized transfer learning to first learn a temperature field under similar operating conditions, and then supplemented the training with a small amount of measurement data, thereby transferring the original fitting results to the new temperature field. This method partially solves the problem of insufficient effective measurement data in real turbine blade experiments. Zhang D et al. [31] designed multiple deep neural networks (DNNs) to learn orthogonal bases of arbitrary random spaces, introducing uncertainties in the parameters and approximations of DNN, which improves the predictive performance of DNN. Jagtap A D et al. [32] proposed a method to adaptively adjust the activation function, which can approximate smooth and nonsmooth functions as well as linear and nonlinear partial differential equation solutions, significantly improving the convergence speed. Zheng J et al. [33] proposed a spatiotemporal hybrid model (STHN) that combines Convolutional Neural Networks (CNN) [34] and Bidirectional Long Short-Term Memory Networks (BLSTM) [35][36]. Compared with traditional empirical and regression models, this method has better predictive performance and higher accuracy. These methods optimize the target problems in the PINN field from the nature of neural networks themselves, but due to the black-box nature of neural networks, these methods often focus on specific neural network designs and optimize the problems less from a physical principle perspective. In addition to expanding the traditional PINN method from the perspective of statistical learning, another approach is to integrate discrete numerical formats and similar ideas with traditional PINN, thereby providing further expert constraints. For example, Xu K et al. [37] proposed a method that combines neural networks and numerical methods for solving partial differential equations, achieving faster convergence speed and better stability. This method can be used to solve a wide range of data-driven inversion models and obtain accurate solutions. Xu R et al. [38] introduced a Weak Form Theory-guided Neural Network, which transfers the higher-order derivatives in the PDE by partially integrating them into the test function. With the help of the integration concept, they implemented a low-pass filter at the level of the loss function, making it more robust to discontinuities and noise. Mohan A T et al. [39] proposed a method that enforces the introduction of physical constraints and boundary conditions using a convolutional neural network architecture with strong inductive bias, thereby embedding the principle of incompressibility for fluid dynamics. Xiang Z et al. [40] presented a method called HFD-PINN, which is a hybrid of physics-informed neural network and mixed finite difference method. It not only achieves faster convergence, but also provides better prediction results. Chen Y et al. [24] proposed a theory-guided hard constraint projection method (HCP), which introduces hard constraints by projecting the PINN computation results into the expression space of the discrete numerical format. The uniqueness of the solution is mathematically proven. The inherent problems of traditional PINN include high data demand and unreliable generalization, which can be remedied by introducing traditional finite volume and finite difference methods. This can be interpreted as further constraining the predictive error using Taylor expansions. Traditional PINN relies solely on generalization to bridge the gap between continuity and discreteness. By adding constraints based on Taylor expansions, it also introduces strict relationships between multiple discrete points. Therefore, this type of PINN not only possesses the advantages of traditional numerical format with finite error, but also retains the data fusion capability of PINN. Although the methods mentioned above, which introduce physical constraints, effectively improve the performance of Physics-Informed Neural Networks (PINN), they still face common challenges when dealing with more complex problems such as transonic flows, sparse measurement points, and extrapolation beyond the initial field. On one hand, the occurrence of transonic flows often implies the existence of special solutions, such as shock waves and other discontinuities. Shock waves, as abrupt changes in the flow field, cannot be described by differential equations. If they are forcibly included in the loss function of PINN, it will cause oscillations and discontinuities in the loss value. Therefore, the calculation of transport across shock waves becomes less reliable. On the other hand, limited by the bottleneck of non-contact measurements, the number of measurement points is scarce, and the test data is often distributed on a two-dimensional plane or a one-dimensional line. This means that PINN cannot accurately fit the three-dimensional space using these measurement points. In this paper, we optimize PINN for such application scenarios by introducing finite difference numerical schemes to add hard constraints. This enables PINN to have both the computational characteristics of soft-constrained PINN and discrete numerical formats. The concept of discrete order [41] is effective for this type of PINN. On one hand, increasing the order means increasing the spatiotemporal coordinates involved in the operation, making the spatial range of information interaction wider [42][43][44][45], enhancing the information propagation speed in PINN, and speeding up the training process. On the other hand, increasing the order means improving the spatial wave number resolution [46][47][48], which allows for the use of fewer measurement data points for physical field inversion. Therefore, in this paper, by introducing a 7th-order WENO-Z-based finite difference discretization format [49][50][51], we generalize the technology of hard physical constraints to higher orders, achieving the modeling of physical fields relying only on a small amount of actual measurement data. Facing the strong compressibility of fluids, WENO-Z is a discontinuity detection method suitable for finite difference formats of compressible fluid shock wave problems. In the face of shock waves and similar discontinuities, WENO-Z will actively change the template configuration to avoid the non-differentiability caused by the discontinuity of the physical field as much as possible. In addition, unlike the idea of finite volume regional integration [52], the finite difference method considers that the spatial coordinate points involved in the calculation represent the local true physical values [53][54], rather than some kind of integral average. On one hand, this idea provides the possibility for the precise fusion of measurement data. On the other hand, when the resolution of the measurement points is too low, the finite difference method provides the possibility for the precise description of the fluid-solid heat transfer interface. The following essay will be divided into three main sections for explanation: The first section introduces the method, which begins by explaining the working principle of traditional PINN and the inherent flaws from a theoretical perspective. It then discusses how Hodd-PINN addresses these flaws in the convective term, demonstrating its effectiveness and revealing hidden, deeper spurious discontinuity issues on the basis of pure convective problems. Afterwards, the solution to these spurious discontinuity problems is presented. Finally, the section describes how Hodd-PINN handles the viscous term. The second section examines the performance of Hodd-PINN, including improvements in accuracy for pure convective problems, spurious discontinuity problems, and two-dimensional heat conduction problems. The third section consists of analysis and conclusions. ## 2 Methodology ### PINN As a fundamental function fitting tool and interpolation method, Neural Network (NN) is the main means of data fusion in PINN. To ensure the reliability of the results in this paper, the constructed NNs all adopt the Resnet[55] technique to minimize the impact of NN characteristics on the experimental results. Each residual block consists of 2 layers of linear transformation and 2 layers of activation function, with a width of 32 for all residual blocks. In addition, considering the smoothness of the real physical field and the difficulty of training neural networks, the Tanh[56] function is uniformly used as the activation function in this paper. The neural network structure used in this paper is shown in Figure 1, where the input is the spatiotemporal coordinates of a point in the physical field, and the output is the corresponding value of the physical field at that position. For one-dimensional unsteady convection problem, the input is $(x,t)$ and the output is velocity $U$. For the two-dimensional steady-state heat conduction problem, the input is $(x,y)$ and the output is temperature $T$. Each spatiotemporal coordinate used for training is called a configuration point. Figure 1 Illustrates the neural network structure and operation employed in this paper. In this figure, INPUT refers to the input of the neural network, while OUTPUT represents the output of the neural network. Each layer of the neural network contains a weight matrix w and bias b, with b not explicitly labeled. In PINN, apart from the configuration of the neural network, the training data is also a very important part. Generally speaking, there are three types of data used for training in PINN: the first type is data with specific spatiotemporal physical values known, such as temperature and velocity obtained from measurements. The second type is boundary conditions of a certain physical property with a specific spatiotemporal position, such as adiabatic boundary conditions and isothermal flow conditions. The third type is conditions of a certain physical law with a specific spatiotemporal position, such as the requirement to satisfy the Navier-Stokes equation at any point in the flow field. Among these three types of data, the first type is obtained through experimental sampling, so the quantity is limited. The second type of data depends on the nature of the boundary conditions, such as velocity boundary conditions and temperature boundary conditions obtained through experimental sampling. Adiabatic boundary conditions and isothermal flow boundary conditions can be sampled arbitrarily. The third type of data can be sampled arbitrarily, so the quantity is unlimited. From here, it can be seen that in the PINN method, the points that can only be obtained through experimental sampling are the most precious: the values of these points can directly determine the final solution of the physical field. However, an undeniable fact is that the quantity of such points cannot be infinitely many. Especially for boundaries, if the boundary conditions at a position are not clear, then theoretically speaking, this would mean that the entire solution has a free boundary. In other words, the results obtained by PINN in this case (also in most commonly occurring cases), even if it performs the best, are just one of the numerous possible solutions. When there are a sufficient number of sampling points, due to the neural network's inherent ability for stable generalization, the above-mentioned problem can be basically ignored. However, when there are not enough sampling points, it is necessary to make more effective use of these precious sampling points. To achieve this, we need to start solving the problem from the physical constraints part of PINN, that is, by introducing the idea of high-order finite difference. The basic idea of finite difference solving method is to find a possible solution between a few known values at specific positions in the known space, so that the partial differential equations are satisfied. The error of this solution is controlled by Taylor series expansion, providing a relatively rigorous mathematical expression. Therefore, high-order finite difference often has higher determinism and can make more effective use of the precious known information. The following section will introduce the implementation of the convection and diffusion terms based on the high-order finite difference method and complete the corresponding tests. ### The 7th-order convective term Unlike traditional PINN, this paper does not use the direct form of partial differential equations to train the neural network but introduces the concept of high order spatial discretization. The 7th order convection finite difference calculation method can be decomposed into three parts: 7th order positive and negative flux construction (Appendix 1), nonlinear weight construction (Appendix 2), and flux difference. By using the positive and negative flux construction (Appendix 1) and the nonlinear weight construction (Appendix 2), $\hat{u}^{+}{}_{i+\frac{1}{2}}\hat{u}^{-}{}_{i-\frac{1}{2}}$ shown in formula 1 can be obtained, and then the Roe average${}^{[57]}$ upwind flux difference in formula 2 can be performed. \[\begin{split}\hat{u}^{+}{}_{i+\frac{1}{2}}&=\sum \omega_{m}^{+}\hat{u}_{m}^{+}\\ \hat{u}^{-}{}_{i-\frac{1}{2}}&=\sum\omega_{m}^{-} \hat{u}_{m}^{-}\end{split} \tag{1}\] \[u_{i+\frac{1}{2}}^{Roe}=\frac{\sqrt{\rho_{i}}u_{i}+\sqrt{\rho_{i+1}}u_{i+1}}{ \sqrt{\rho_{i}}+\sqrt{\rho_{i+1}}} \tag{2}\] To facilitate data parallelism, the flux $\hat{u}_{i+\frac{1}{2}}$ is determined by equation 3, where $sign(x)$ represents the sign function. Finally, the convective term is determined by equation 4. \[\hat{u}_{i+\frac{1}{2}}=\frac{\left[1+sign\left(u_{i+\frac{1}{2}}^{Roe}\right) \right]}{2}\hat{u}^{+}{}_{i+\frac{1}{2}}+\frac{\left[1-sign\left(u_{i+\frac{1 }{2}}^{Roe}\right)\right]}{2}\hat{u}^{-}{}_{i+\frac{1}{2}} \tag{3}\] \[u\frac{\partial u}{\partial x_{i}}=u\frac{\left(\hat{u}_{i+\frac{1}{2}}-\hat{ u}_{i-\frac{1}{2}}\right)}{\Delta x}+o(\Delta^{7}) \tag{4}\] Next, taking the transient problem in one-dimensional space as an example, we will explain how to embed the above finite difference calculation method into PINN. For the time advancement process of a flow field, the PINN velocity prediction result $u^{t0}$ is used as the input for the previous time step. This input is then randomly sampled in time and space to obtain multiple configuration points with equal intervals on the spatial coordinates, thus adapting the finite difference method. The values of these configuration points are outputted after time advancement using a 7th-order finite difference format, obtaining the predicted result $u^{t1}$ for the next time step. This result is then used to train the PINN for predicting the subsequent time step. This process is shown in Figure 2. This sampling before training approach can avoid the problem of discontinuity that traditional continuous space PINN encounters: the space represented by NN is still continuous, and its advantages still exist, but the problem caused by the discontinuity of the flow field can be solved by finite difference. It should be noted that in high-order finite difference discretization processes, there exist coefficient matrices with both large and small values. If backpropagation passes through these matrices, for some special cases, such as relatively long time steps or severe oscillations in PINN's predicted results, it may lead to unstable training. Therefore, when using high-order finite difference discretization to guide PINN training, it is recommended to block this part of the backpropagation channel. In our test results, blocking did not lead to a decrease in performance, but rather improved computational speed. Compared to traditional PINN, the significant advantage of this method is that it introduces long-distance spatial correlations between multiple configuration points, as shown in Figure 3, by utilizing the finite difference assumption. In traditional PINN, these configuration points only need to satisfy the regularity of their respective positions, but cannot guarantee the accuracy of generalization positions. However, with the introduction of numerical schemes, the values of these position points need to jointly satisfy the physical constraints, with an error only in the truncation of the Taylor expansion. This means that the introduction of numerical schemes makes the predicted physical field by PINN more controllable. Based on this, the numerical scheme introduces conservation to the predicted results of PINN through reconstruction, which is crucial in convective problems. Figure 2: Illustrates the training method of Hodd-PINN using a 7th order finite difference discretization system. The red dots represent the sampling results of the neural network (NN) in the PINN. The pink area represents the structured 7th order finite difference computation system. Figure 2 In the training and generalization mode of the traditional physics-informed neural network (PINN), the correlation between data points is constructed by relying on the neural network (left). Once the 7th order finite difference computation system is introduced, the correlation between data points needs to further adhere to the long-range constraints imposed by the finite difference scheme. During the training process, first construct the PINN neural network P. Then, for each configuration point, sample the surrounding spatiotemporal coordinates at equal intervals. The sampled coordinates are then fed into network P to obtain velocity prediction results $\widehat{U}$. Next, $\widehat{U}$ is input into a 7th-order convection format for reconstruction, resulting in the physically constrained predicted $\widehat{U}^{}$. Finally, by performing optimization $argmin\left(Loss(\widehat{U},\widehat{U}^{})\right)$, the numerically conserved reconstructed results are introduced into the training process of PINN. This process is illustrated in Figure 4. To verify the proposed method in this article, a one-dimensional nonlinear convection problem is introduced, and the governing equation is determined by Equation 5. Figure 4: Training method of PINN with the introduction of O7-PDE. \[\frac{\partial u}{\partial t}+\frac{\partial uu}{\partial x}=0 \tag{5}\] The initial setup is shown as follows in Equation 6: \[u_{t0}=\begin{cases}0,\ x\in[0,0.3)\\ 1,\ x\in[0.3,0.6)\\ 0,\ x\in[0.6,1]\end{cases} \tag{6}\] The time progression range is 0.2s, and the theoretical solution to this problem is expressed in equation 7, as shown in Figure 5: \[u_{t}=\begin{cases}\frac{0,\ x\in[0,0.3)}{2t},\ x\in[03,0.3+2t)\\ \ 1,\ x\in[0.3+2t,0.6+t)\\ \ 0,\ x\in[0.6+t,1]\end{cases} \tag{7}\] For the typical nonlinear convection problem adopted in this article, there will always exist two weak solutions within the given time range, namely the rarefaction wave following the shock wave and the shock wave ahead. Obviously, the position of the shock wave is non-differentiable at any moment. Mathematically, this directly leads Figure 5: The theoretical solution (7) describes the pure convection problem described in equations (5) and (6). to the failure of the partial differential equation at the shock wave position. In PINN, this causes the PDE configuration points near the shock wave to generate gradients that approach infinity. Such gradients will contaminate the training results at other positions. By introducing discrete numerical formats instead of the traditional PINN differential operator, this problem can be avoided. This article compares the prediction results of the traditional PINN, PINN guided by second-order finite difference, and PINN guided by seventh-order finite difference for this problem, as shown in Figure 6, 7, and 8. Figure 6 The predicted results after training the traditional PINN using the information provided in equations (5) and (6). Figure 7 The prediction results of the second-order finite difference guided PINN, trained using equations (5) and (6) as known information. Figure 8 The predicted results of the 7th order finite difference guided PINN trained using Figure 7: The prediction results of the second-order finite difference guided PINN, trained using equations (5) and (6) as known information. equations (5) and (6) as known information. It can be observed that among the three methods, the results obtained from the 7th order finite difference guided PINN are closer to the theoretical solution. More detailed results can be found in the Result section. However, in the case where both sparse waves and shock waves coexist, due to the fact that the penalty for shock waves is much greater than the penalty for sparse waves, there may be occasional instances where the neural network fails to detect sparse waves and predicts false discontinuities after the wave. The author believes that this problem arises because although the Mse loss function [58] can quickly reduce the overall error level, it is not very sensitive to small but important errors due to the fact that the error level spans multiple orders of magnitude. This phenomenon is similar to outliers in the training process, making it difficult to detect sparse waves[59]. This problem exists for both the second order finite difference guided PINN and the 7th order finite difference guided PINN, as shown in Figure 9. Generally speaking, the problem can be alleviated by adding the Mse loss function and the L1 loss function [60]. The L1 loss function can compensate for the shortcomings of Mse and better capture small errors. However, this operation does not completely solve the problem of false discontinuity in the NN fitting process, and the practice of Figure 9: In some cases, the prediction results after training the PINN guided by finite difference can be observed, and it can be seen that the sparse wave on the left side was not captured by the PINN. adding the Mse loss function and the L1 loss function is not friendly to the diffusion term. In this paper, under the premise of using only the Mse loss function, the false discontinuity detection in the PINN prediction results is achieved by using the WENO discontinuity detection technique. ## Weno First, let me introduce the implementation principle of WENO's discontinuity detection: In Appendix 2, the construction method of the smoothness factor is introduced, which realizes the discontinuity detection function of WENO. Based on this, this paper implements the pseudo-discontinuity detection in non-steady convection problems. This paper first proposes a definition of a discontinuity index: \[\sigma=\min\bigl{(}softmax(IS1,IS2,IS3,IS4)\bigr{)} \tag{8}\] In general, the $\sigma$ value should be close to 0.25. The smaller the $\sigma$ value, the more likely there is a discontinuity at that position, and the stronger the discontinuity. However, in this paper, this method cannot be directly used for detecting false discontinuities in the PINN fitting results. This is because WENO is only suitable for detecting whether there is a current physical field discontinuity, and cannot be directly used to determine whether the discontinuity is reasonable. Therefore, we use traditional numerical schemes for time advancement to determine the trend of the discontinuity strength. If there is a false discontinuity at a certain position in the PINN's predicted results, then the discontinuity at that position should weaken as time advances. If a discontinuity should indeed exist at a certain position, then the strength of the discontinuity at that position is not significantly affected by time advancement. Based on the above analysis, we define a discontinuity decay index $\beta$ based on the intermittent index $\sigma$, where $\beta=\alpha^{t+1}-\alpha^{t}$. Here, $\alpha^{t}$ is calculated from $\tilde{U}^{t}$, which is the result of PINN prediction, and $\alpha^{t+1}$ is the result of time advancement using the 7th-order WENO numerical scheme based on $\tilde{U}^{t}$. With this method, we can detect the variation trend of the initial field given by PINN in the real physical space. When a predicted discontinuity by PINN does not exist physically, the discontinuity index at that position should decrease, and $\beta$ will be significantly negative. When a predicted discontinuity by PINN actually exists, the discontinuity index at that position remains basically unchanged, and $\beta$ is basically 0. Thus, we have obtained the criteria for judging false discontinuities. In the calculation process of this criterion, all operations are differentiable, and the constant coefficient matrices used in the calculation process are of the same order of magnitude. Therefore, we can use the WENO discontinuity detection process to achieve backpropagation. Note that it is recommended to add a layer of Relu activation function in this process. This is because in the region where the discontinuity tends to aggregate, $\beta$ may be positive. In this case, the error given by the finite difference numerical format is significant and correct, which can be captured by the loss function and does not require additional optimization through WENO discontinuity detection. This process can be expressed as shown in Figure 10. This article compares the high-order PINN prediction results before and after adopting this technology. The comparison results can be found in the Result section. ### The 8th-order diffusion term In finite difference, the treatment method for the diffusion term is shown in Appendix 3. With the help of high-order discretization, we can achieve long-range propagation of limited information. In order to test the overall performance of PINN after introducing high-order discretization, this paper conducted tests on the reconstruction problem of a two-dimensional temperature field. The temperature field is a rectangular area with a length of 2pi and a width of pi, with four sides labeled as A, B, C, and D. In this temperature field, only the temperature values of a few points in the long side A are known, which belong to the Dirichlet boundary condition [61]. The remaining boundaries are known only as adiabatic boundary conditions, which belong to the Neumann boundary condition [62]. The theoretical solution for the temperature distribution on side A is: \[T_{[0,y]}=\sin(1.5*y) \tag{9}\] The corresponding temperature distribution is shown in Figure 11. Figure 10: The training method of PINN after introducing WENO discontinuity detection. This article compares the traditional PINN, the PINN guided by second-order finite difference, and the PINN guided by eighth-order finite difference, respectively. The test results can be found in the Result section. ## 3 Results ### One-dimensional convection equation For the one-dimensional convection problem, we tested traditional PINN, PINN based on low-order discretization (O2), and PINN based on high-order discretization (O7). The tests were conducted using the Adam optimizer with a learning rate of 0.003 and the loss function consisting of MseLoss and L1Loss. The performance was recorded under different initialization seeds, different numbers of PDE configuration points, and different numbers of known value configuration points. The specific settings can be found in Table 1. \begin{table} \begin{tabular}{c c c c} \hline \hline PINN Method & Seed & PDE configuration points & Value configuration points \\ \hline Traditional & 0 & 8000 & 50 \\ Traditional & 1 & 8000 & 50 \\ Traditional & 2 & 8000 & 50 \\ \hline \hline \end{tabular} \end{table} Table 1: Test conditions for the one-dimensional convection problem. Figure 11: The target temperature field and the arrangement of temperature measurement points. Among them, the left boundary is a temperature boundary condition, while the remaining boundaries are adiabatic boundary conditions. In a series of tests with 8000 PDE configuration points and 50 known value configuration points, we inputted multiple seeds to test the stability of the method. The final L1 error is shown in Figure 12. Due to the significant difference in error magnitudes between different methods, we used logarithmic error to plot the boxplot in this figure. After introducing discrete techniques into PINN, the minimum value of the absolute error decreases to the order of 10-8, which is the limit of single-precision floating-point representation. Furthermore, with the introduction of higher-order discrete techniques, the overall error level decreases. For traditional PINN, the average L1 error is approximately 0.394 m/s. After introducing O2 discrete techniques, the average L1 error decreases to 0.041 m/s. With the further introduction of higher-order discrete techniques, the average L1 error decreases to 0.037 m/s, which is an additional 9.7% reduction in error. Compared to traditional PINN, the error is reduced by a total of 90.6%. Figure 12: The error performance of traditional PINN, second-order finite difference guided PINN, and seventh-order finite difference guided PINN on the pure convection problem. The vertical axis represents the logarithm of the error. ## False positive detection By introducing the spurious discontinuity detection technique, this study successfully improves the ability of neural networks (NN) to capture sparse waves during the training process. Tests were conducted to compare the performance of PINN before and after the introduction of the spurious discontinuity detection technique. These tests used the Adam optimizer with a learning rate of 0.003 and the MseLoss as the loss function, while keeping the other parameters the same. The decrease in L1 error between the predicted solution and the theoretical solution with respect to the number of training steps is shown in Figure 13. It can be clearly observed that after the introduction of the pseudo-discontinuity detection technique, the error between the predicted results of the PINN and the theoretical solution is smaller. Excluding the initial field error that is not trained, the introduction of the pseudo-discontinuity detection technique reduces the average error level from 0.0669 to 0.0507, resulting in a 24.2% decrease in error. ## Two-dimensional heat conduction equation For the specified two-dimensional heat conduction problem in this article, traditional PINN, PINN guided by second-order finite difference, and PINN guided by eighth-order finite difference were tested. The tests were conducted using the Adam optimizer with a learning rate of 0.001 and the MseLoss as the loss function. The performance of the two methods was recorded under different initialization seeds, different numbers of PDE configuration points, and different numbers of known value Figure 13: The training process error performance of the 7th order finite difference guided PINN before and after the introduction of WENO pseudo-discontinuity detection technique. configuration points. The specific settings can be found in Table II. In the case of only 7 known value configuration points, we conducted comprehensive testing using multiple seeds, and the logarithmic error is shown in Figure 14: \begin{table} \begin{tabular}{c c c c c} \hline \hline & & PDE & Value & Adiabatic \\ PINN Method & Seed & configuration & configuration & configuration \\ & & points & points & points \\ \hline Traditional & 0 & 800 & 7 & 200 \\ Traditional & 1 & 800 & 7 & 200 \\ Traditional & 2 & 800 & 7 & 200 \\ Traditional & 0 & 800 & 9 & 200 \\ Traditional & 0 & 800 & 15 & 200 \\ Traditional & 0 & 800 & 31 & 200 \\ Traditional & 0 & 8000 & 400 & 400 \\ Low-order & 0 & 800 & 7 & 200 \\ Low-order & 1 & 800 & 7 & 200 \\ Low-order & 2 & 800 & 7 & 200 \\ Low-order & 0 & 800 & 9 & 200 \\ Low-order & 0 & 800 & 15 & 200 \\ High-order & 0 & 800 & 31 & 200 \\ High-order & 0 & 800 & 7 & 200 \\ High-order & 1 & 800 & 7 & 200 \\ High-order & 2 & 800 & 7 & 200 \\ High-order & 0 & 800 & 9 & 200 \\ High-order & 0 & 800 & 15 & 200 \\ High-order & 0 & 800 & 31 & 200 \\ \hline \hline \end{tabular} \end{table} Table II: Test condition settings for the two-dimensional heat conduction problem From Figure 14, it can be observed that the 8th-order finite difference guided PINN has a significantly lower error level compared to the traditional PINN and the second-order finite difference guided PINN. The average error of the traditional PINN is approximately 0.434, the average error of the second-order finite difference guided PINN is approximately 0.101, and the average error of the 8th-order finite difference guided PINN is approximately 0.088. Compared to the second-order, it has improved by about 12.8%, and compared to the traditional PINN, it has improved by about 34.6%. ## 5 Conclusion This article combines traditional finite difference numerical schemes, WENO discontinuity detection, and traditional PINN methods to develop a high-order discontinuity detection PINN that can achieve inversion of velocity and temperature fields with a small number of measurement points. By introducing high-order finite difference methods, the efficiency of utilizing known data points by PINN is effectively improved. By introducing a pseudo-discontinuity detection method, the ability of PINN to capture sparse waves is enhanced. Through this study, the following typical conclusions can be obtained: 1.Traditional PINN cannot solve discontinuities, which is caused by the inherent continuous function representation capability of NN. Finite difference methods can replace the automatic differentiation process of NN, bypassing the problems caused by discontinuities while retaining the advantages of NN's continuity. 2.Introducing finite difference methods can effectively solve the problem of Figure 14: Error performance of traditional PINN, second-order finite difference guided PINN, and high-order finite difference guided PINN on the pure heat conduction problem, with the vertical axis representing the logarithm of the error. insufficient known data points, and under the same amount of data, higher-order schemes perform better than lower-order schemes. For the convection term, the 7th order scheme improves by an average of 9.7% compared to the 2nd order scheme, and for the diffusion term, the 7th order scheme improves by an average of 12.8% compared to the 2nd order scheme. 3.The pseudo-discontinuity detection technique based on WENO developed in this article can effectively identify pseudo-discontinuities in the fitting results of NN and eliminate them through optimization. This leads to a 24.2% improvement in accuracy. Ruquan You: Supervision, Conceptualization, Data curation, Writing-review & editing. Shuming Zhang: Formal analysis, Investigation, Writing-original draft. Tinglin Kong: Provide partial formula derivation. Haiwang Li: Supervision, Conceptualization. ## 1 Acknowledgments This work is supported by Beijing Nova Program; The Advanced Aviation Power Innovation Workstation (established by the China Aviation Engine Research Institute) [grant number HKCX2022-01-007]; The Aeronautical Science Fund (provided by Aviation Industry Corporation of China)[grant number 2022Z039051006];The funding sources are not involved in study design; in the collection, analysis and interpretation of data; in the writing of the report; and in the decision to submit the article for publication.
2310.05715	A simple linear algebra identity to optimize Large-Scale Neural Network Quantum States	Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic Reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with $3 \times 10^5$ parameters, achieving state-of-the-art ground-state energy in the $J_1$-$J_2$ Heisenberg model at $J_2/J_1=0.5$ on the $10\times10$ square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for Neural-Network Quantum States, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.	Riccardo Rende, Luciano Loris Viteritti, Lorenzo Bardone, Federico Becca, Sebastian Goldt	2023-10-09T13:38:36Z	http://arxiv.org/abs/2310.05715v2	# A simple linear algebra identity to optimize ###### Abstract Neural network architectures have been increasingly used to represent quantum many-body wave functions. In this context, a large number of parameters is necessary, causing a crisis in traditional optimization methods. The Stochastic Reconfiguration (SR) approach has been widely used in the past for cases with a limited number of parameters $P$, since it involves the inversion of a $P\times P$ matrix and becomes computationally infeasible when $P$ exceeds a few thousands. This is the major limitation in the context of deep learning, where the number of parameters significantly surpasses the number of samples $M$ used for stochastic estimates ($P\gg M$). Here, we show that SR can be applied exactly by leveraging a simple linear algebra identity to reduce the problem to the inversion of a manageable $M\times M$ matrix. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture featuring approximately 300,000 parameters without any assumption on the sign structure of the ground state. Our approach achieves state-of-the-art ground-state energy on the $J_{1}$-$J_{2}$ Heisenberg model in the challenging point $J_{2}/J_{1}=0.5$ on the $10\times 10$ square lattice, a demanding benchmark problem in the field of highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for Neural Network Quantum States (NNQS), making them a promising method to investigate unknown phases of matter of quantum systems, where other methods struggle. + Footnote †: These authors contributed equally. _Introduction._ Deep learning is crucial in many research fields, with neural networks being the key to achieve impressive results. Well-known examples include using Deep Convolutional Neural Networks for image recognition [1; 2] and Deep Transformers for language-related tasks [3; 4]. The success of deep networks comes from two ingredients: architectures with a large number of parameters (often in the billions), which allow for great flexibility, and training these networks on large amounts of data. However, to successfully train these large models in practice, one needs to navigate the complicated and highly non-convex landscape associated with this extensive parameter space. The most used methods rely on stochastic gradient descent (SGD), where the gradient of the loss function is estimated from a randomly selected subset of the training data. Over the years, variations of traditional SGD, such as Adam [5] or AdamW [6], have proven highly effective. These modifications enable more efficient optimization processes and lead to improved accuracy. In the late 1990s, Amari and collaborators [7; 8] suggested to use the knowledge of the geometric structure of the parameter space to adjust the gradient direction for non-convex landscapes, defining the concept of Natural Gradients. In the same years, Sorella [9; 10] proposed a similar method, now known as Stochastic Reconfiguration (SR), to enhance optimization of variational functions in quantum many-body systems. Importantly, this latter approach consistently outperforms other methods as SGD or Adam, leading to significantly lower variational energies. The main idea of SR is to exploit information about the curvature of the loss landscape, thus improving the convergence speed in landscapes which are steep in some directions and very shallow in others [11]. For physical inspired wave functions (e.g., Jastrow-Slater [12] or Gutzwiller-projected states [13]) the original SR formulation is a highly efficient method since there are few variational parameters, typically from $O(10)$ to $O(10^{2})$. Over the past few years, neural networks have been extensively used as powerful variational _Ansatze_ for studying interacting spin models [14], and the number of parameters have increased significantly. Starting of simple Restricted Boltzmann Machines (RBMs) [14; 15; 16; 17; 18; 19], more complicated architectures as Convolutional Neural Networks (CNNs) [20; 21; 22] and Recurrent Neural Networks (RNNs) [23; 24; 25; 26] have been introduced to handle challenging systems. In particular, Deep CNNs [27; 28; 29; 30] have proven to be highly accurate for two-dimensional models, undoing methods as Density-Matrix Renormalization Group (DMRG) approaches [31] and Gutzwiller-projected states [13]. These deep learning models have great performances when the number of parameters is large. However, a significant bottleneck arises when employing the original formulation of SR for optimization, as it is based on the inversion of a matrix of size $P\times P$, where $P$ denotes the number of parameters. Consequently, this approach becomes computationally infeasible as the parameter count exceeds $O(10^{4})$, primarily due to the constraints imposed by the limited memory capacity of current-generation GPUs. Recently, Ref. Chen and Heyl [29] made a step forward in the optimization procedure by introducing an alternative method, dubbed MinSR, to train NNQS wave functions. MinSR does not require inverting the original $P\times P$ matrix but instead a much smaller $M\times M$ one, where $M$ is the number of configurations used to estimate the SR matrix. This is convenient in the deep learning setup where $P\gg M$. Most importantly, this procedure avoids allocating the $P\times P$ matrix, reducing the memory cost to $P\times M$. However, this formulation is obtained as a minimization of the Fubini-Study distance with an _ad hoc_ constraint. In this work, we first use a simple relation from linear algebra to show, in a transparent way, that SR can be rewritten exactly in a form which involves inverting a small $M\times M$ matrix and that only a standard regularization of the SR matrix is required. Then, we exploit our technique to optimize a Deep Vision Transformer (Deep ViT) model, which has demonstrated exceptional accuracy in describing quantum spin systems in one and two spatial dimensions [38; 39; 40; 41]. Using almost $3\times 10^{5}$ variational parameters, we are able to achieve state-of-the-art ground-state energy on the most paradigmatic example of quantum many-body spin model, the $J_{1}$-$J_{2}$ Heisenberg model on square lattice: \[\hat{H}=J_{1}\sum_{\langle i,j\rangle}\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}+J_{ 2}\sum_{\langle\langle i,j\rangle\rangle}\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j} \tag{1}\] where $\hat{\mathbf{S}}_{i}=(S_{i}^{x},S_{i}^{y},S_{i}^{z})$ is the $S=1/2$ spin operator at site $i$ and $J_{1}$ and $J_{2}$ are nearest- and next-nearest-neighbour antiferromagnetic couplings, respectively. Its ground-state properties have been the subject of many studies over the years, often with conflicting results [31; 37; 13]. Still, in the field of quantum many-body systems, this is widely recognized as the benchmark model for validating new approaches. Here, we will focus on the particularly challenging case with $J_{2}/J_{1}=0.5$ on the $10\times 10$ cluster, where there are no exact solutions. Within variational methods, one of the main difficulties comes from the fact that the sign structure of the the ground state is not known for $J_{2}/J_{1}>0$. Indeed, the Marshall sign rule [42] gives the correct signs (on each cluster) only when $J_{2}=0$, while it largely fails close to $J_{2}/J_{1}=0.5$. In this respect, it is important to mention that some of the previous attempts to define variational wave functions included the Marshall sign rule as a first approximation of the correct sign structure, thus taking advantage of a prior probability (_Marshall prior_). By contrast, within the present approach, we do not need to use any prior knowledge of the sign structure, thus defining a very general and flexible variational _Ansatz_. In the following, we first show the alternative SR formulation, then discuss Deep Transformer architecture, recently introduced by some of us [38; 39] as a variational state, and finally present our results obtained by combining the two techniques on the $J_{1}$-$J_{2}$ Heisenberg model. _Stochastic Reconfiguration._ Finding the ground state of a quantum system with the variational principle involves minimizing the variational energy $E(\theta)=\left\langle\Psi_{\theta}\|\hat{H}\|\Psi_{\theta}\right\rangle/\left \langle\Psi_{\theta}\|\Psi_{\theta}\right\rangle$, where $\|\Psi_{\theta}\rangle$ is a variational state parametrized through a vector $\theta$ of $P$_real_ parameters (in case of complex parameters, we can treat their real and imaginary parts separately [43]). For a system of $N$$1/2$-spins, $\|\Psi_{\theta}\rangle$ can be expanded in the computational basis $\|\Psi_{\theta}\rangle=\sum_{\{\sigma\}}\Psi_{\theta}(\sigma)\left\|\sigma\right\rangle$, where $\Psi_{\theta}(\sigma)=\left\langle\sigma\|\Psi_{\theta}\right\rangle$ is a map from spin configurations of the Hilbert space, $\{\ \|\sigma\rangle=\|\sigma_{1}^{z},\sigma_{2}^{z},\cdots,\sigma_{N}^{z}\rangle\,\ \sigma_{i}^{z}=\pm 1\}$, to complex numbers. In a gradient-based optimization approach, the fundamental ingredient is the evaluation of the gradient of the loss, which in this case is the variational energy, with respect to the parameters $\theta_{\alpha}$ for $\alpha=1,\ldots,P$. This \begin{table} \begin{tabular}{l c c c c} \hline \hline Energy per site & Wave function & \# parameters & Marshall prior & Reference & Year \\ \hline -0.48941(1) & NNQS & 893994 & Not available & [32] & 2023 \\ -0.494757(12) & CNN & Not available & No & [22] & 2020 \\ -0.4947359(1) & Shallow CNN & 11009 & Not available & [21] & 2018 \\ -0.49516(1) & Deep CNN & 7676 & Yes & [20] & 2019 \\ -0.495502(1) & PEPS + Deep CNN & 3531 & No & [33] & 2021 \\ -0.495530 & DMRG & 8192 SU(2) states & No & [31] & 2014 \\ -0.495627(6) & aCNN & 6538 & Yes & [34] & 2023 \\ -0.49575(3) & RBM-fermionic & 2000 & Yes & [15] & 2019 \\ -0.49586(4) & CNN & 10952 & Yes & [35] & 2023 \\ -0.4968(4) & RBM ($p=1$) & Not available & Yes & [36] & 2022 \\ -0.49717(1) & Deep CNN & 106529 & Yes & [28] & 2022 \\ -0.497437(7) & GCNN & Not available & No & [27] & 2021 \\ -0.497468(1) & Deep CNN & 421953 & Yes & [30] & 2022 \\ -0.4975490(2) & VMC ($p=2$) & 5 & Yes & [13] & 2013 \\ -0.497627(1) & Deep CNN & 146320 & Yes & [29] & 2023 \\ -0.497629(1) & RBM+PP & 5200 & Yes & [37] & 2021 \\ -0.497634(1) & Deep ViT & 267720 & No & Present work & 2023 \\ \hline \end{tabular} \end{table} Table 1: Ground-state energy on the 10$\times$10 square lattice at $J_{2}/J_{1}=0.5$. gradient can be expressed as a correlation function [43] \[F_{\alpha}=-\frac{\partial E}{\partial\theta_{\alpha}}=-2\Re\left[\langle(\hat{H} -\langle\hat{H}\rangle)(\hat{O}_{\alpha}-\langle\hat{O}_{\alpha}\rangle)\rangle \right]\, \tag{2}\] which involves the diagonal operator $\hat{O}_{\alpha}$ defined as $O_{\alpha}(\sigma)=\partial\text{Log}[\Psi_{\theta}(\sigma)]/\partial\theta_{\alpha}$. The expectation values $\langle\dots\rangle$ are computed with respect to the variational state. The SR updates [19; 43; 9] are constructed according to the geometric structure of the landscape: \[\mathbf{\delta\theta}=\tau\left(S+\lambda\mathbb{I}_{P}\right)^{-1}\mathbf{F}\, \tag{3}\] where $\tau$ is the learning rate and $\lambda$ is the regularization parameter. The matrix $S$ has shape $P\times P$ and it is defined in terms of the $\hat{O}_{\alpha}$ operators [43] \[S_{\alpha,\beta}=\Re\left[\langle(\hat{O}_{\alpha}-\langle\hat{O}_{\alpha} \rangle)^{\dagger}(\hat{O}_{\beta}-\langle\hat{O}_{\beta}\rangle)\rangle \right]. \tag{4}\] The Eq. (3) defines the standard formulation of the SR, which involves the inversion of a $P\times P$ matrix, being the bottleneck of this approach when the number of parameters is larger than $O(10^{4})$. To address this problem, we start reformulating Eq. (3) in a more convenient way. For a given sample of $M$ spin configurations $\{\sigma_{i}\}$ (sampled according to $\|\Psi_{\theta}(\sigma)\|^{2}$), the stochastic estimate of $F_{\alpha}$ can be obtained as: \[\bar{F}_{\alpha}=-\Re\left\{\frac{2}{M}\sum_{i=1}^{M}[E_{L}(\sigma_{i})-\bar{ E}_{L}]^{}[O_{\alpha}(\sigma_{i})-\bar{O}_{\alpha}]\right\}. \tag{5}\] Here, the local energy is defined as $E_{L}(\sigma)=\langle\sigma\|\hat{H}\|\Psi_{\theta}\rangle\left/\,\langle\sigma \|\Psi_{\theta}\right.$. We use $\bar{E}_{L}$ and $\bar{O}_{\alpha}$ to denote sample means. Throughout this work, we adopt the convention of using latin indices and greek indices to run respectively over configurations and parameters. Equivalently, Eq. (4) can be stochastically estimated as \[\bar{S}_{\alpha,\beta}=\Re\left[\frac{1}{M}\sum_{i=1}^{M}\left(O_{\alpha i}- \bar{O}_{\alpha}\right)^{}\left(O_{\beta i}-\bar{O}_{\beta}\right)\right]. \tag{6}\] To simplify further, we introduce $\varepsilon_{i}=-2[E_{L}(\sigma_{i})-\bar{E}_{L}]^{*}/\sqrt{M}$ and $Y_{\alpha i}=(O_{\alpha i}-\bar{O}_{\alpha})/\sqrt{M}$, allowing us to express Eq. (5) in matrix notation as $\bar{\mathbf{F}}=\Re[Y\mathbf{\varepsilon}]$ and Eq. (4) as $\bar{S}=\Re[YY^{\dagger}]$. Writing $Y=Y_{R}+iY_{I}$ we obtain: \[\bar{S}=Y_{R}Y_{R}^{T}+Y_{I}Y_{I}^{T}=XX^{T} \tag{7}\] where $X=\text{Concat}(Y_{R},Y_{I})\in\mathbb{R}^{P\times 2M}$, the concatenation being along the last axis. Furthermore, using $\mathbf{\varepsilon}=\mathbf{\varepsilon}_{R}+i\mathbf{\varepsilon}_{I}$, the gradient of the energy can be recasted as \[\bar{\mathbf{F}}=Y_{R}\mathbf{\varepsilon}_{R}-Y_{I}\mathbf{\varepsilon}_{I}=X\mathbf{f}\, \tag{8}\] with $\mathbf{f}=\text{Concat}(\mathbf{\varepsilon}_{R},-\mathbf{\varepsilon}_{I})\in\mathbb{R }^{2M}$. Then, the update of the parameters in Eq. (3) can be written as \[\mathbf{\delta\theta}=\tau(XX^{T}+\lambda\mathbb{I}_{P})^{-1}X\mathbf{f}. \tag{9}\] This reformulation of the SR updates is a crucial step, which allows the use of a well-known linear algebra identity [44] (see _Supplemental Material_ for a derivation). As a result, Eq. (9) can be rewritten as \[\mathbf{\delta\theta}=\tau X(X^{T}X+\lambda\mathbb{I}_{2M})^{-1}\mathbf{f}. \tag{10}\] This derivation is our first result: it shows, in a simple and transparent way, how to exactly perform the SR with the inversion of a $M\times M$ matrix and without the need to allocate a $P\times P$ matrix. We emphasise that the last formulation is very useful in the typical deep learning setup, where $P\gg M$. Employing Eq. (10) instead of Eq. (9) proves to be more efficient both in terms of computational complexity [$O(4M^{2}P+8M^{3})$ operations instead of $O(P^{3})$] and memory usage [requiring only the allocation of $O(MP)$ numbers instead of $O(P^{2})$]. Crucially, we developed a memory-efficient implementation of SR that is optimized for deployment on a multi-node GPU cluster, ensuring scalability and practicality for real-world applications (see _Supplemental Material_). Other methods, based on iterative solvers, require $O(nMP)$ operations, where $n$ is the number of steps needed to solve the linear problem in Eq. (3). However, this number increases significantly for ill-conditioned matrices (the matrix $S$ has a number of zero eigenvalues equal to $P$-$M$), leading to many non-parallelizable iteration steps and consequently higher computational costs [45]. The variational wave function.The architecture of the variational state employed in this work is based on the Deep Vision Transformer (Deep ViT). In the Deep ViT, the input configuration $\sigma$ of shape $L\times L$ is firstly split into $b\times b$ patches, which are then embedded in a $d$-dimensional vector space in order to obtain an input Figure 1: Optimization of the Deep ViT with batch size $b=2$, 8 layers, embedding dimension $d=72$ and $12$ heads per layer, on the $J_{1}$-$J_{2}$ Heisenberg model at $J_{2}/J_{1}=0.5$ on the $10\times 10$ square lattice. The first 200 optimization steps are not shown for better readability. Inset: last 200 optimization steps when recovering sequentially the full translational (green curve), rotational (orange curve) and reflections and parity (red curve) symmetries. sequence $\big{(}\mathbf{x}_{1},\ldots,\mathbf{x}_{L^{2}/b^{2}}\big{)}$, where $L^{2}/b^{2}$ is the total number of patches. A real-valued Deep ViT _transforms_ this sequence into an output sequence $\big{(}\mathbf{y}_{1},\ldots,\mathbf{y}_{L^{2}/b^{2}}\big{)}$, with $\mathbf{y}_{i}\in\mathbb{R}^{d}$ being a rich, context aware representation of the spins in $i$-th patch. At the end, a complex-valued fully connected output layer, with $\log[\cosh(\cdot)]$ as activation function, is applied to $\mathbf{z}=\sum_{i}\mathbf{y}_{i}$ to produce a single number $\text{Log}[\Psi_{\theta}(\sigma)]$ and predict both the modulus and the phase of the input configuration [39]. We employ _Factored Attention_[46] and two-dimensional relative positional encoding to enforce translational symmetry among patches and only a small summation ($b^{2}$ terms) is required to restore the one inside the patches. In addition, the other symmetries of the Hamiltonian can be restored with the quantum number projection approach [47, 35]. In particular, we restore rotations, reflections ($C_{4v}$ point group), and spin parity. As a result, the symmetrized wave function reads: \[\tilde{\Psi}_{\theta}(\sigma)=\sum_{i=0}^{b^{2}-1}\sum_{j=0}^{7}\left[\Psi_{ \theta}(T_{i}R_{j}\sigma)+\Psi_{\theta}(-T_{i}R_{j}\sigma)\right]\;. \tag{11}\] In the last equation, $T_{i}$ and $R_{j}$ are the translation and the rotation/reflection operators, respectively. Furthermore, due to the $SU(2)$ spin symmetry of the $J_{1}$-$J_{2}$ Heisenberg model, the total magnetization is also conserved and the Monte Carlo sampling (see _Supplemental Material_) can be limited in the $S^{z}=0$ sector for the ground state search. _Results._ Our objective is to approximate the ground state of the $J_{1}$-$J_{2}$ Heisenberg model in the highly frustrated point $J_{2}/J_{1}=0.5$ on the $10\times 10$ square lattice. We use the formulation of the SR in Eq. (10) to optimize a variational wave function parametrized through a Deep ViT, as discussed above. The result in Table 1 is achieved with the symmetrized Deep ViT in Eq. (11) using $b=2$, number of layers $8$, embedding dimension $d=72$, number of heads per layer $12$. This variational state has in total $267720$ real parameters. Regarding the optimization protocol, we choose the learning rate $\tau=0.03$ (cosine decay annealing) and the number of samples is fixed to be $M=6000$. We emphasise that using Eq. (9) to optimize this number of parameters would be infeasible on avaiable GPUs: the memory requirement would be more than $O(10^{3})$ gigabytes, one order of magnitude bigger than the highest available memory capacity. Instead, with the formulation of Eq. (10), the memory requirement is reduced to $\approx 26$ gigabytes, which can be easily handled by available GPUs. The simulations took four days on twenty A100 GPUs. Remarkably, as illustrated in Table 1, we are able to obtain the state-of-the-art ground-state energy using an architecture solely based on neural networks, without using any other regularization than the diagonal shift reported in Eq. (10), fixed to $\lambda=10^{-4}$. We stress that a completely unbiased simulation, without assuming any prior for the sign structure, is performed, in contrast to the other cases where the Marshall sign rule is used to stabilize the optimization [28, 29, 37, 48] (see Table 1). This is an important point since a simple sign prior is not available for the majority of the models (e.g., the Heisenberg model on the triangular or Kagome lattices). We would like also to mention that the definition of a suitable architecture is fundamental to take advantage of having a large number of parameters. Indeed, while a stable simulation with a simple regularization scheme (only defined by a finite value of $\lambda$) is possible within the Deep ViT wave function, other architectures require more sophisticated regularizations. For example, to optimize Deep GCNNs it is necessary to add a temperature-dependent term to the loss function [27] or, for Deep CNNs, a process of variance reduction and reweighting [29] helps in escaping local minima. We also point out that physically inspired wave functions, as the Gutzwiller-projected states [13], which give a remarkable result with only a few parameters, are not always equally accurate in other cases. In Fig. 1 we show a typical Deep ViT optimization on the $10\times 10$ at $J_{2}/J_{1}=0.5$. First, we optimize the Transformer having translational invariance among patches (blue curve). Then, starting from the previously optimized parameters, we restore sequentially the full translational invariance (green curve), rotational symmetry (orange curve) and lastly, reflections and spin parity symmetry (red curve). Remarkably, whenever a new symmetry is restored, the energy reliably decreases [47]. We stress that our optimization process, which combines the SR formulation of Eq. (10) with a real valued Deep ViT followed by a complex-valued fully connected output layer [39], is highly stable and insensitive to the initial seed, ensuring consistent results across multiple optimization runs. _Conclusions._ We have introduced an innovative formulation of the SR that excels in scenarios where the number of parameters ($P$) significantly outweighs the number of samples ($M$) used for stochastic estimations. Exploiting this approach, we attained the state-of-the-art ground state energy for the $J_{1}$-$J_{2}$ Heisenberg model at $J_{2}/J_{1}=0.5$, on a $10\times 10$ square lattice, optimizing a Deep ViT with $P=267720$ and using only $M=6000$ samples. It is essential to note that this achievement highlights the remarkable capability of deep learning in performing exceptionally well even with a limited sample sizes relative to the overall parameter count. This also _challenges_ the common belief that large amount of Monte Carlo samples are required to find the solution in the exponentially-large Hilbert space and for precise SR optimizations [30]. This result has important ramifications for investigating the physical properties of challenging quantum many-body problems, where the use of the SR is crucial to obtain accurate results. The use of Neural Network Quantum States with an unprecedented number of parameters can open new opportunities in approximating ground states of quantum spin Hamiltonians, where other methods fail. Additionally, making minor modifications to the equations describing parameter updates within the SR framework [see Eq. (10)] enables us to describe the unitary time evolution of quantum many-body systems according to the time-dependent variational principle (TDVP) [43; 49; 50]. Extending our approach to address time-dependent problems stands as a promising avenue for future works. Furthermore, this new formulation of the SR can be a key tool for obtaining accurate results also in quantum chemistry, especially for systems that suffer from the sign problem [51]. Typically, in this case, the standard strategy is to take $M>10\times P$[43], which may be unnecessary for deep learning based approaches. ###### Acknowledgements. We thank R. Favata for critically reading the manuscript and M. Imada for stimulating us with challenging questions. We also acknowledge G. Carleo and A. Chen for useful discussions. The variational quantum Monte Carlo and the Deep ViT architecture were implemented in JAX [52]. The parallel implementation of the Stochastic Reconfiguration was implemented using mpi4jax [53]. R.R. and L.L.V. acknowledge access to the cluster LEONARDO at CINECA through the IsCa5_ViT2d project. S.G. acknowledges co-funding from Next Generation EU, in the context of the National Recovery and Resilience Plan, Investment PE1 - Project FAIR "Future Artificial Intelligence Research". This resource was co-financed by the Next Generation EU [DM 1555 del 11.10.22].
2302.05504	Random attractors of a stochastic Hopfield neural network model with delays	The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for $t>\tau$ (where $\tau$ is the delay) by a delicate construction based on the random semiflow generated by the diffusion term. Then, this version is shown to generate a random dynamical system (RDS) by piece-wise linear approximation, after which the existence of a random absorbing set is obtained by a careful uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is proved and hence the existence of random attractors is obtained. Moreover, sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are given. Numerical simulations are also conducted at last to illustrate the effectiveness of the established results.	Wenjie Hu, Quanxin Zhu, Peter E. Kloeden	2023-02-10T20:34:48Z	http://arxiv.org/abs/2302.05504v1	# Random attractors of a stochastic Hopfield neural network model with delays ###### Abstract The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for $t>\tau$ (where $\tau$ is the delay) by a delicate construction based on the random semiflow generated by the diffusion term. Then, this version is shown to generate a random dynamical system (RDS) by piece-wise linear approximation, after which the existence of a random absorbing set is obtained by a careful uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is proved and hence the existence of random attractors is obtained. Moreover, sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are given. Numerical simulations are also conducted at last to illustrate the effectiveness of the established results. Key Words: Random attractor, random dynamical systems, stochastic neural network, delay, piece-wise linear approximation ## 1 Introduction and examples In [15], Hopfield proposed the following prominent HNNM \[\frac{du_{i}(t)}{dt}=-c_{i}u_{i}(t)+\sum_{j=1}^{n}h_{ij}f_{j}\left(u_{j}\left( t\right)\right). \tag{1.1}\] Here, $u_{i}(t),i=1,2,\cdots,n$ represent the state of the $i$th neuron. $c_{i}$ stands for the reset velocity of the $i$th neuron. $\mathbf{H}=(h_{ij})$ corresponds to the connection matrix between neurons at time $t$, where $h_{ij},i,j=1,2,\cdots,n$ are the weight coefficients measuring the connection strength between different neurons. If the output of the $j$th neuron excites the neuron $i$, then we have $h_{ij}\geq 0$, otherwise, $h_{ij}=0$. $f_{i}:R\to R,i=1,2,\cdots,n$ are the neuron activation functions. Due to the significant roles that the HNNM 1.1 has played in the areas of signal processing, optimization, parallel computations, pattern recognition, associate memories and so forth, it has been extensively and intensively investigated including both its dynamics analysis and real world applications from the time that it was proposed. See, for instance [14], [15], [34] and the references therein. In model 1.1, it is assumed that the updating and propagation of neuron signals occur instantaneously without time delay. Nevertheless, time delays are omnipresent both in neural processing and in signal transmission and must be taken into account in the communication channels. Hence, Marcus and Westervelt [25] incorporated time delays arising from distance between the neurons or axonal conduction time between neurons into system 1.1 leading to the following delayed neural network model \[\frac{du_{i}(t)}{dt}=-c_{i}u_{i}(t)+\sum_{j=1}^{s}h_{ij}f_{j}\left(u_{j}\left( t\right)\right)+\,\sum_{j=1}^{n}b_{ij}g_{j}\left(u_{j}\left(t-\tau_{j}\right) \right), \tag{1.2}\] where $\tau_{j},j=1,\cdots,n$ denote the delay caused by the transmission along the axon of the $j$-th unit. $b_{ij}$ represents the strength of the $j$-th unit on the $i$-th unit at time $t-\tau_{j}$. The $g_{j}$ term gives the corresponding output of the $j$-th unit at time $t-\tau_{j}$. The dynamical behavior of solutions to 1.2 has drawn much research interest pertaining to both dynamics and applications. For example, [30] and [44] compared the memory capacity of 1.2 under different activation functions, while [9] and [35] obtained the global asymptotic stability of the steady state by the Lyapunov method. In real nervous systems, signal transmission along axons is a noisy process caused by random fluctuations from the release of neurotransmitters and other random effects, as pointed by [13]. Therefore, a more realistic and accurate mathematical model should be the following stochastic delayed HNNM \[\left\{\begin{array}{l}du_{i}(t)=\left[-c_{i}u_{i}(t)+\sum_{j=1}^{n}h_{ij}f_ {j}\left(u_{j}(t)\right)+\right.\\ \sum_{j=1}^{n}b_{ij}g_{j}\left(u_{j}\left(t-\tau_{j}\right)\right)]dt+\sum_{j= 1}^{n}\sigma_{ij}u_{j}(t)dw_{j}(t)\\ u_{i}(\xi)=\phi_{i}(\xi),\quad-\max_{1\leq k\leq n}\tau_{k}\leq\xi\leq 0,\end{array}\right. \tag{1.3}\] which is obtained by adding an multiplicative noise $\sum_{j=1}^{n}\sigma_{ij}u_{j}(t)dw_{j}(t)$ to 1.3. Here, $\sigma_{ij}$ are parameters representing the density of stochastic effects, $w_{j}(t),j=1,2,\cdots,n$ are mutually independent two-sided real-valued Wiener process on $(\Omega,\mathcal{F},P)$ with a filtration $\{\mathcal{F}_{t},t\geq 0\}$. $\phi_{i}(\xi),i=1,\ldots,n$ are the initial conditions. Let $\tau=\max_{1\leq k\leq n}\tau_{k}$ and take \[u(t)=\left(u_{1}(t),u_{2}(t),\ldots,u_{n}(t)\right)^{\mathrm{T}},\] \[u_{t}(\tau)=\left(u_{1}\left(t-\tau_{1}\right),u_{2}\left(t-\tau_{2}\right), \ldots,u_{n}\left(t-\tau_{n}\right)\right)^{\mathrm{T}},\] \[C=\mathrm{diag}\left(c_{1},c_{2},\ldots,c_{n}\right),\quad H=\left(h_{ij} \right)_{n\times n},\quad B=\left(b_{ij}\right)_{n\times n},\] \[f(u)=\left(f_{1}\left(u_{1}\right),f_{2}\left(u_{2}\right),\ldots,f_{n}\left( u_{n}\right)\right)^{\mathrm{T}},\] \[g(u)=\left(g_{1}\left(u_{1}\right),g_{2}\left(u_{2}\right),\ldots,g_{n}\left( u_{n}\right)\right)^{\mathrm{T}},\] \[\Sigma=(\sigma_{ij})_{n\times n},\] \[w(t)=\left(w_{1}(t),w_{1}(t),\ldots,w_{n}(t)\right)^{\mathrm{T}},\] \[\phi(t)=\left(\phi_{1}(t),\phi_{2}(t),\ldots,\phi_{n}(t)\right)^{\mathrm{T}}.\] Then, model 1.3 can be rewritten in the following matrix form \[\left\{\begin{array}{l}du(t)=\left[-Cu(t)+Hf(u(t))+Bg\left(u(t-\tau)\right) \right]dt\\ +\Sigma u(t)\diamond w(t),\\ u(t)=\phi(t),\quad-\tau\leqslant t\leqslant 0,\end{array}\right. \tag{1.4}\] in which $\phi=\{\phi(s):-\tau\leqslant s\leqslant 0\}$ is $\mathcal{C}=C([-\tau,0],\mathbb{R}^{n})$-valued function, where $\mathcal{C}$ is the space of all continuous $\mathbb{R}^{n}$-valued functions defined on $[-\tau,0]$ with a norm $\\|\xi\\|=\sup\{\|\xi(t)\|:-\tau\leqslant t\leqslant 0\}$ and $\|\cdot\|$ is the Euclidean norm of a vector $x\in\mathbb{R}^{n}$. Here $u(t)\circ dw(t)$ is interpreted as the vector with $j$th component $u_{j}(t)dw_{j}(t)$, $j=1$, $\cdots$, $n$. The local stability around equilibria of the stochastic HNNM and the stochastic delayed HNNM has been widely and thoroughly studied in the past decades by employing Ito's formula, stochastic analysis techniques and inequality techniques. See, for instance, [22, 32, 37] and [45] investigate stability of 1.4 with fixed or variant delays. On the other hand, there are also some other works concerning global stability for equilibrium of stochastic delayed HNNM by adopting the Lyapunov-Krasovskii functional methods together with stochastic analysis techniques, see for example [23, 40, 41] and the references therein. Nevertheless, Lyapunov-Krasovskii functional methods depend critically on the construction of specified Lyapunov-Krasovskii functional which is different from equation to equation. Moreover, the criterion is in linear matrix inequality form that depends highly on numerical tools, and the condition on the drift term and the random coefficient is strict. From the dynamical systems point of view, if we can find a compact set that is invariant under the system and attracts all orbits of the system, i.e., if we can find a global attractor of the system, then we can obtain the asymptotic behavior of the system by investigating the structure of the attractor. This approach is effective especially for infinite dimensional dynamical systems generated by partial differential equations or delay differential equations. Generally, the attractors of dynamical systems may be fixed points, periodic orbits, invariant manifolds or chaotic attractors under certain conditions. Therefore, by investigating the attractors of dynamical systems, one can obtain a full view of complex dynamics of the systems. However, to our knowledge, the existence and structure of attractors for the stochastic delayed HNNM 1.4 have not been studied, although the existence and structure of attractors of the deterministic delayed HNNM, the delayed non-autonomous neural network models as well as random neural field lattice models have been studied in [12, 16, 24, 38, 31, 43]. It should be pointed out that, in the very recent work [31], the authors investigated the attractors of a random recurrent neural networks with delays. Unlike the delayed random neural networks that naturally generates an RDS, the existence of an RDS generated by stochastic delayed HNNM 1.4 needs much effort to established. Indeed, the main difficulty for obtaining the attractors of stochastic delayed HNNM 1.4 lies in showing the model generates a random dynamical system, which naturally holds for deterministic case. There are only very few papers tackling the attractors for the stochastic delayed differential equations, see for example, [39], [42] and [10]. In [39], the authors proved that the stochastic delayed differential equation generates a mean square RDS processing a mean-square random attractor in the mean-square sense [19], which is not the random attractor in the classical pathwise sense. For a detail concept of the mean-square random attractors and mean-square RDSs, the readers are refereed to the review paper [6]. In [42] and [10] the authors obtained the existence and structure of random attractors for stochastic delayed differential equations with additive noise. It should be noted that the exponential Ornstein-Uhlenbeck process transformation as [7] did for stochastic (partial) partial differential equations may also be effective for transforming the stochastic delayed equation into random delayed equation, but a time delay then occurs in the noise term, which makes it difficult to establish the existence of a random dynamical system as well as that of tempered absorbing sets. This motivates us to propose a new method to investigate the existence of an RDS generated by 1.4 with general multiplicative noise, the existence and structure of random attractors of 1.4. We prove that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous and measurable compact version for $t>\tau$ which is constructed by random semiflow generated by the linear multiplicative noise. Then, this version is shown to generate a RDS by the trick of piece-wise linear approximation of the version. The idea originates from the early works [29][26], [27] where stochastic delayed differential equations are recast into a Hilbert space to study the Lyapunov spectrum and invariant manifolds. Our work can be regarded as an attempt to study the random attractors of the HNNM 1.4 in Banach spaces. Very recently, [36] investigated the singular stochastic delay differential equations in Banach spaces by rough path theory, which is quite different from approach of the present work. Our main contributions are in the following three folds. Firstly, a novel method for proving the stochastic delayed HNNM generates an RDS in the natural phase space, i.e. $\mathcal{C}=C([-\tau,0],\mathbb{R}^{n})$ is established, which is quite different from the situation in the early works [29] where a Hilbert space is taken as the phase space. Secondly, existence of random attractors of the model under appropriate conditions have been proved. Thirdly, optimal conditions under which that the attractor is a random stationary solution is obtained. vMoreover, the method can also be generalized to other stochastic evolution equations with delays, such as the stochastic delayed partial differential equations, for tackling the problem of topological dimension estimation, Lyapunov exponents, as well as the invariant manifolds that depends on showing the existence of RDSs. This paper is organized as follows. In Section 2, we first introduce the theory of RDSs and random attractors as well as some notations and preliminary lemmas needed for the proof of our main results. In Section 3, based on the random semiflow generatated by the diffusion term of 1.4, we transform 1.4 into a random delayed differential equation admitting a measurable version that generates an RDS. For the purpose of showing the existence of random attractor for 1.4, we first give a uniform a priori estimate of the solution and then show the asymptotic compactness of the RDS generated by 1.4 in Section 4, which implies the existence of random attractor following the results in [4, 5]. In Section 5, we derive sufficient conditions for the random attractor being an exponential stationary solution. Finally, we illustrate the obtained results by numerical simulations. ## 2 Preliminaries We first introduce the concepts of a random attractor and an RDS as well as some important lemmas following [1, 2, 4, 5] and [8]. Definition 2.1.: _Let $\{\theta_{t}:\Omega\to\Omega,t\in\mathbb{R}\}$ be a family of measure preserving transformations such that $(t,\omega)\mapsto\theta_{t}\omega$ is measurable and $\theta_{0}=\mathrm{id}$, $\theta_{t+s}=\theta_{t}\theta_{s},$ for all $s,t\in\mathbb{R}$. The flow $\theta_{t}$ together with the probability space $\big{(}\Omega,\mathcal{F},P,\left(\theta_{t}\right)_{t\in\mathbb{R}}\big{)}$ is called a metric dynamical system._ For a given separable Hilbert space $(X,\\|\cdot\\|_{X})$, denote by $\mathcal{B}(X)$ the Borel-algebra of open subsets in $X$. Definition 2.2.: _A mapping $\Phi:\mathbb{R}^{+}\times\Omega\times X\to X$ is said to be a random dynamcial system (RDS) on a complete separable metric space $(X,d)$ with Borel $\sigma$-algebra $\mathcal{B}(X)$ over the metric dynamical system $\big{(}\Omega,\mathcal{F},P,\left(\theta_{t}\right)_{t\in\mathbb{R}}\big{)}$ if (i) $\Phi(\cdot,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times X\to X$ is $(\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal{B}(X),\mathcal{B }(X))$-measurable; (ii) $\Phi(0,\omega,\cdot)$ is the identity on $X$ for $P$-a.e. $\omega\in\Omega$; (iii) $\Phi(t+s,\omega,\cdot)=\Phi(t,\theta_{s}\omega,\cdot)\circ\Phi(s,\omega,\cdot),$ for all $t,s\in\mathbb{R}^{+}$ for $P$-a.e. $\omega\in\Omega$. A RDS $\Phi$ is continuous or differentiable if $\Phi(t,\omega,\cdot):X\to X$ is continuous or differentiable for all $t\in\mathbb{R}^{+}$ and $P$-a.e. $\omega\in\Omega$._ Definition 2.3.: _A set-valued map $\Omega\ni\omega\mapsto D(\omega)\in 2^{X}$ is said to be a random set in $X$ if the mapping $\omega\mapsto d(x,D(\omega))$ is $(\mathcal{F},\mathcal{B}(\mathbb{R}))$-measurable for any $x\in X,$ where $d(x,D(\omega))\triangleq\inf_{y\in D(\omega)}\mathrm{d}(x,y)$ is the distance in $X$ between the element $x$ and the set $D(\omega)\subset X$._ Definition 2.4.: _A random set $\{D(\omega)\}_{\omega\in\Omega}$ of $X$ is called tempered with respect to $\left(\theta_{t}\right)_{t\in\mathbb{R}}$ if for $P$-a.e. $\omega\in\Omega$,_ \[\lim_{t\rightarrow\infty}e^{-\beta t}d\left(D\left(\theta_{-t}\omega\right) \right)=0,\quad\text{ for all }\beta>0,\] _where $d(D)=\sup_{x\in D}\\|x\\|_{X}$._ Definition 2.5.: _Let $\mathcal{D}=\{\{D(\omega)\}_{\omega\in\Omega}:D(\omega)\subset X,\omega\in\Omega\}$ be a family of random sets. A random set $\{K(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ is said to be a $\mathcal{D}$-pullback absorbing set for $\Phi$ if for $P$-a.e. $\omega\in\Omega$ and for every $B\in\mathcal{D},$ there exists $T=T(B,\omega)>0$ such that_ \[\Phi\left(t,\theta_{-t}\omega,B\left(\theta_{-t}\omega\right)\right)\subseteq K (\omega)\qquad\text{ for all }t\geq T.\] _If, in addition, for all $\omega\in\Omega,K(\omega)$ is a closed nonempty subset of $X$ and $K(\omega)$ is measurable in $\Omega$ with respect to $\mathcal{F},$ then we say $K$ is a closed measurable $\mathcal{D}$-pullback absorbing set for $\Phi$._ Definition 2.6.: _An RDS $\Phi$ is said to be $\mathcal{D}$-pullback asymptotically compact in $X$ if for $P$-a.e. $\omega\in\Omega$, $\left\{\Phi\left(t_{n},\theta_{-t_{n}}\omega,x_{n}\right)\right\}_{n\geq 1}$ has a convergent subsequence in $X$ whenever $t_{n}\rightarrow\infty$ and $x_{n}\in D\left(\theta_{-t_{n}}\omega\right)$ for any given $D\in\mathcal{D}$._ Definition 2.7.: _A compact random set $\mathcal{A}(\omega)$ is said to be a $\mathcal{D}$-pullback random attractor associated to the RDS $\Phi$ if it satisfies the invariance property_ \[\Phi(t,\omega)\mathcal{A}(\omega)=\mathcal{A}\left(\theta_{t}\omega\right), \quad\text{ for all }t\geq 0\] _and the pullback attracting property_ \[\lim_{t\rightarrow\infty}\mathrm{dist}\left(\Phi\left(t,\theta_{-t}\omega \right)D\left(\theta_{-t}\omega\right),\mathcal{A}(\omega)\right)=0,\] _for all $t\geq 0,D\in\mathcal{D}$ P-a.e. $\omega\in\Omega$.where $\mathrm{dist}(\cdot,\cdot)$ denotes the Hausdorff semidistance_ \[\mathrm{dist}(A,B)=\sup_{x\in A}\inf_{y\in B}\mathrm{d}(x,y),\quad A,B\subset X.\] Lemma 2.1.: _Let RDS $(\theta,\Phi)$ be a continuous RDS. Suppose that $\Phi$ is pullback asymptotically compact and has a closed pullback $\mathcal{D}$-absorbing set $K=\{K(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$. Then it possesses a random attractor $\{\mathcal{A}(\omega)\}_{\omega\in\Omega},$ where_ \[\mathcal{A}(\omega)=\cap_{\tau\geq 0}\overline{\cup_{t\geq\tau}\Phi\left(t, \theta_{-t}\omega,K\left(\theta_{-t}\omega\right)\right)}.\] F The following Gronwall inequality that will be frequently used in our subsequent proofs. Lemma 2.2.: _Let $T>0$ and $u,\alpha,f$ and $g$ be non-negative continuous functions defined on $[0,T]$ such that_ \[u(t)\leq\alpha(t)+f(t)\int_{0}^{t}g(r)u(r)dr,\quad\text{ for }t\in[0,T].\] _Then_ \[u(t)\leq\alpha(t)+f(t)\int_{0}^{t}g(r)\alpha(r)e^{\int_{r}^{t}f(\tau)g(\tau) dr}dr,\quad\text{ for }t\in[0,T].\] Existence of an RDS Throughout the remaining part of this paper, we always make the following assumptions on the nonlinear term $f$ and $g$. Hypothesis A1 The functions $f(x)$ and $g(x)$ are globally Lipschitz continuous with Lipschitz constants $L_{f}$ and $L_{g}$. Hypothesis A2$f(x)$ and $g(x)$ are bounded with $\mathbf{0}$ being a fixed point, that is $f(\mathbf{0})=g(\mathbf{0})=0$ and there exists a $M>0$ such that $f(x)\leq M$ for any $x\in\mathbb{R}^{n}$ and $g(\phi)\leq M$ for any $\phi\in\mathcal{C}$. By [28] Theorem 2.1 and [10] Lemma 1, we have the following results concerning the existence of a continuous $\mathcal{F}_{t}$-measurable solution to (1.4). Lemma 3.1.: _Assume that_ Hypothesis A1-A2 _hold, then for P-a.e. $\omega\in\Omega$, the stochastic delayed HNNM (1.4) admits a unique solution $u(\cdot,\omega,\phi)\in C\left([-\tau,\infty),\mathbb{R}^{n}\right)$, which is adapted to $\{\mathcal{F}_{t}\}_{t\in[0,\infty)}$ for any initial condition $\phi\in\mathcal{C}$._ In the following, we are devoted to showing the stochastic delayed HNNM (1.4) generates a RDS. For a measurable process $u:[-\tau,\infty)\times\Omega\to\mathbb{R}^{n}$, the segment $u_{t}:[-\tau,0]\times\Omega\to\mathbb{R}^{n},t\in\mathbb{R}^{+}$, is defined by \[u_{t}(s):=u(t+s),\qquad s\in[-\tau,0].\] Consider the following stochastic differential equation obtained by the diffusion term of (1.4) \[\left\{\begin{array}{l}dv(t)=\Sigma v(t)\diamond dw(t),\\ v(0)=\mathrm{id}_{\mathbb{R}^{n}},\end{array}\right. \tag{3.1}\] where $\mathrm{id}_{\mathbb{R}^{n}}$ be the identity operator on $\mathbb{R}^{n}$. We assume here and elsewhere that the components $w_{i}(t)$ of the vector $w(t)$ are mutually independent standard Wiener processes. Let $v(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the solution of (3.1) and it is well known from [1] that (3.1) admits a linear continuous solution which is bounded for P-a.e. $\omega\in\Omega$. Denote its bound by $L_{v}$, i.e. $\\|v(t,\omega,\cdot)\\|\leq L_{v}$ for all $t\in\mathbb{R}^{n}$ and P-a.e. Moreover, it process a bounded continuous inverse $v^{-1}(t,\omega,\cdot)$. Hence, the solution $v(t,\omega,\cdot):\mathbb{R}\times\Omega\to L(\mathbb{R}^{n})$ is $\{\mathcal{F}_{t}\}_{t\geq 0}$-adapted. Define the following operator \[\begin{split} A(t,\omega,\phi)\triangleq& v^{-1}(t, \omega,\cdot)\circ[-Cv(t,\omega,\phi(0))+Hf(v(t,\omega,\phi(0)))\\ &+Bg\left(v_{t}(\cdot,\omega,\phi)\right)]\end{split} \tag{3.2}\] with $v_{t}(\cdot,\omega,\cdot):\mathcal{C}\to\mathcal{C}$ being defined by \[v_{t}(s,\omega,\phi)=\left\{\begin{array}{ll}v(t+s,\omega,\phi(0)),&t+s \geq 0,\\ \phi(t+s),&-\tau\leq t+s<0.\end{array}\right. \tag{3.3}\] Assume that $g(\varphi)=L\varphi+\tilde{g}(\varphi)$ for any $\varphi\in\mathcal{C}$, where $L$ is the linear part of $g$. (Hence, the Lipschitz constant of $\tilde{g}$ denoted by $L_{\tilde{g}}$ is $L_{g}-\\|L\\|$). Consider the linear random functional differential equation \[\left\{\begin{array}{ll}\frac{d\bar{u}(t)}{dt}=-C\bar{u}(t)+v^{-1}(t,\omega, \cdot)\circ BLv_{t}(\cdot,\omega,\bar{u}_{t})\triangleq\tilde{L}(t,\omega) \bar{u}_{t},\\ \bar{u}_{0}=\phi\in\mathcal{C}\end{array}\right. \tag{3.4}\] for any $t>0$. For the pathwise deterministic functional differential equation (3.4), it follows from [11] that the fundamental solution (3.4), denoted by $S^{\phi}(\cdot,\omega):[0,\infty)\to\mathbb{R}^{n^{2}}$ is defined by \[\begin{split} S^{\phi}(t,\omega)&=(S\phi)(t,\omega )\triangleq S(t,\omega)\phi(0)\\ &+\int_{-\tau}^{0}\int_{\xi}^{0}S(t+\xi-s,\omega)\mathrm{d}\eta( \xi,t,\omega)\phi(s)\mathrm{d}s.\end{split} \tag{3.5}\] for any $t\geq 0$ and P-a.e. $\omega\in\Omega$. Here, $\eta:[-\tau,0]\times[0,\infty)\times\Omega\to\mathbb{R}^{n^{2}}$ is an $n\times n$ matrix-valued function whose elements are of bounded variation such that \[\tilde{L}(t,\xi)\phi=\int_{-\tau}^{0}\,\mathrm{d}\eta(\xi,t,\omega)\phi(\xi)\] for P-a.e. $\omega\in\Omega$ due to the Riez representation theorem. For any fixed $\omega\in\Omega$ and $t>0$, the characteristic equation of (3.4) is given by \[\det\left[\lambda\operatorname{Id}-\tilde{L}(t,\omega)\left(\mathrm{e}^{- \lambda(\cdot)}\right)\right]=0, \tag{3.6}\] of which the spectrum is defined as \[\varrho=\max\left\{\operatorname{Re}\lambda:\det\left[\lambda\operatorname{Id }-\tilde{L}(t,\omega)\left(\mathrm{e}^{-\lambda(\cdot)}\right)\right]=0 \right\}.\] In the reminder of this paper, we always assume that $\varrho<0$ and hence we have the following results by [11]. Lemma 3.2.: _If $\varrho<0$, then there exist positive constants $\gamma$ and $K_{0}$ such that_ \[\|(S\phi)(t,\omega)\|<K_{0}\mathrm{e}^{-\gamma t}\\|\phi\\| \tag{3.7}\] _for all $t\geq 0$, P-a.e. $\omega\in\Omega$ and any $\phi\in\mathcal{C}$. In particular, there exists a positive constant $K_{1}$ such that_ \[\|S(t,\omega)\|\leq K_{1}\mathrm{e}^{\varrho t/2}\text{ for all }t\geq 0,\text{ P-a.e. }\omega\in\Omega. \tag{3.8}\] We next show the global existence and properties of solution to a random functional differential equation generated by the operator (3.2). Lemma 3.3.: _Assume that_ Hypothesis A1 - A2 _hold and $\varrho<0$, then the following random retarded functional differential equation_ \[\left\{\begin{array}{l}\frac{d\tilde{u}(t)}{dt}=A(t,\omega,\tilde{u}_{t}), \quad t>0,\\ \tilde{u}_{0}=\phi\in\mathcal{C}.\end{array}\right. \tag{3.9}\] _admits a global continuous mild solution $\tilde{u}(\cdot,\omega,\phi):[-\tau,\infty)\to\mathbb{R}^{n}$ which is $C^{1}$ in $t$ for any $\phi\in\mathcal{C}$ and P-a.e. $\omega\in\Omega$. Moreover, $\tilde{u}(t,\omega,\phi)$ is jointly continuous in $(t,\omega)$, $\tilde{u}(t,\cdot,\phi)$ is $\mathcal{F}_{t}$-measurable for any $\phi\in\mathcal{C}$ and the map $\tilde{u}_{t}(\cdot,\omega,\cdot):\mathcal{C}\to\mathcal{C}$ is compact for any $t>\tau$._ Proof.: Since $v(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ and $v^{-1}(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ are both continuous, then (3.9) can be written as \[\left\{\begin{array}{l}\frac{d\tilde{u}(t)}{dt}=-C\tilde{u}(t)+v^{-1}(t, \omega,\cdot)\circ Hf(v(t,\omega,\tilde{u}(t)))\\ +v^{-1}(t,\omega,\cdot)\circ Bg\left(v_{t}(\cdot,\omega,\tilde{u}_{t})\right),\quad t>0,\\ \tilde{u}_{0}=\phi\in\mathcal{C}.\end{array}\right. \tag{3.10}\] For each $t\in\mathbb{R}^{+}$ and P-a.e. $\omega\in\Omega$. Moreover, we have \[\begin{split}\\|A(t,\omega,\tilde{u}_{t})\\|\leq& c\| \tilde{u}(0)\|+hL_{f}\|\tilde{u}(0)\|+bL_{g}\\|\tilde{u}_{t}\\|\\ \leq&(c+hL_{f}+bL_{g})\\|\tilde{u}_{t}\\|,\end{split} \tag{3.11}\] where $c,h,b$ represent the norm of matrix $C,H$ and $B$ respectively. Hence, the random retarded functional differential equation (3.9) admits a unique local solution with explosion time $\rho=\rho(\phi,\omega)$. We next show that $\rho=\infty$. By the variation-of-constants formula, we see (3.9) is equivalent to the following integral equation \[\begin{cases}\tilde{u}(t,\omega,\phi)=S\phi(t,\omega)+\int_{0}^{t}S(t-s, \omega)v^{-1}(s,\omega,\cdot)\circ\\ \qquad\qquad[Hf(v(s,\omega,\tilde{u}(s)))+B\tilde{g}\left(v_{s}(\cdot,\omega, \tilde{u}_{t})\right)]\mathrm{d}s,t\geq 0,\\ \tilde{u}(t)=\phi(t),\quad t\in[-\tau,0],\end{cases} \tag{3.12}\] where $S\phi(t,\omega)$ and $S(t,\omega)$ satisfy (3.7) and (3.8) respectively for $t\geq 0$ and P-a.e. $\omega\in\Omega$. Hence, we have \[\begin{split}\|\tilde{u}(t,\omega,\phi)\|\leq&\,\|(S* \phi)\left(t,\omega\right)\|+\int_{0}^{t}\|S(t-s)v^{-1}(s,\omega,\cdot)\circ\\ &[Hf(v(s,\omega,\tilde{u}(s)))+B\tilde{g}\left(v_{s}(\cdot,\omega, \tilde{u}_{s})\right)]\mathrm{d}s\|\mathrm{d}s\\ \leq&\,\|(S\phi)\left(t,\omega\right)\|+\int_{0}^{t}S(t-s )[hL_{f}\\|\tilde{u}(s,\omega,\phi)\\|\\ &+bL_{\tilde{g}}\\|\tilde{u}_{s}(\cdot,\omega,\phi)\\|]\mathrm{d}s\\ \leq& K_{0}\mathrm{e}^{-\gamma t}\left\\|\phi\right\\|\\ &+\left(hL_{f}+bL_{\tilde{g}}\right)\int_{0}^{t}\mathrm{e}^{\varrho(t-s )/2}\left\\|\tilde{u}_{s}\left(\cdot,\omega,\phi\right)\right\\|\mathrm{d}s, \end{split} \tag{3.13}\] from which we can see for $t\geq\tau,\xi\in[-\tau,0]$, \[\begin{split}\\|\tilde{u}(t+\xi,\omega,\phi)\\|\leq& K_{0}\mathrm{e}^{\gamma(\tau-t)}\left\\|\phi\right\\|+(hL_{f}+bL_{\tilde{g}}) \mathrm{e}^{-\varrho\tau/2}\times\\ &\int_{0}^{t}\mathrm{e}^{\varrho/2(t-s)}\left\\|\tilde{u}_{s} \left(\cdot,\omega,\phi\right)\right\\|\mathrm{d}s.\end{split} \tag{3.14}\] Namely, \[\begin{split}\mathrm{e}^{-\varrho t/2}\left\\|\tilde{u}_{t}\left( \cdot,\omega,\phi\right)\right\\|\leq& c_{0}\mathrm{e}^{-\gamma t} \mathrm{e}^{-\varrho t/2}\left\\|\phi\right\\|\\ &+c_{1}\int_{0}^{t}\mathrm{e}^{-\varrho s/2}\left\\|\tilde{u}_{s} \left(\cdot,\omega,\phi\right)\right\\|\mathrm{d}s\end{split} \tag{3.15}\] where $c_{0}=K_{0}\mathrm{e}^{\gamma\tau}$ and $c_{1}=(hL_{f}+bL_{\tilde{g}})\mathrm{e}^{-\varrho\tau/2}$. Applying Gronwall's inequality yields \[\begin{split}\mathrm{e}^{-\varrho t/2}\left\\|\tilde{u}_{t}\left( \cdot,\omega,\phi\right)\right\\|\leq& c_{0}\\|\phi\\|[\mathrm{e}^{-( \gamma+\varrho/2)t}\\ &+c_{1}\mathrm{e}^{c_{1}t}\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s],\end{split} \tag{3.16}\] and hence \[\begin{split}\\|\tilde{u}_{t}\left(\cdot,\omega,\phi\right)\\|\leq& c_{0}\\|\phi\\|[\mathrm{e}^{-\gamma t}+c_{1}\mathrm{e}^{(c_{1}+ \varrho/2)t}\int_{0}^{t}\mathrm{e}^{-(\gamma+\varrho/2+c_{1})s}\mathrm{d}s] \end{split} \tag{3.17}\] for P-a.e. $\omega\in\Omega$ and $0<t<\rho$. Thus, $\{\\|\tilde{u}(t,\omega,\phi)\\|:0<t<\rho\}$ is bounded and so $\tilde{u}(\cdot,\omega,\phi):[0,\rho)\to H$ is uniformly continuous for P-a.e. $\omega\in\Omega$ and any $\phi\in\mathcal{C}$. Taking the limit $\lim_{t\rightarrow\rho-}\tilde{u}_{t}(\cdot,\omega,\phi)\in\mathcal{C}$ as initial data for (3.9), then by local existence we may extend $\tilde{u}$ to a larger interval than $[0,\rho)$. This contradicts the maximality of $\rho.$ Hence $\rho=\infty$. It follows from the continuity of $A(t,\omega,\tilde{u}_{t})$ with respect to $t$ that $\tilde{u}(\cdot,\omega,\phi)$ is $C^{1}$ in $t$ for any $\phi\in\mathcal{C}$ and P-a.e. $\omega\in\Omega$. For each $t\in\mathbb{R}^{+}$ and $\phi\in\mathcal{C}$, $A(t,\omega,\tilde{u}_{t})$ is $\mathcal{F}_{t}$-measurable. Hence, a standard successive approximation argument shows that $\tilde{u}_{t}(\cdot,\omega,\phi)$ is jointly measurable in $(t,\omega)$ and for each $t\in\mathbb{R}^{+}$, $\tilde{u}_{t}(\cdot,\omega,\phi)$ is $\mathcal{F}_{t}$-measurable for any $\phi\in\mathcal{C}$. Furthermore if we fixed $t_{0}\geq\rho$ then the above argument also shows that for P-a.e. $\omega\in\Omega$ the set $\{d\tilde{u}(t,\omega,\phi)\mid:0\leq t\leq t_{0},\\|\phi\\|\leq 1\}$ is bounded. Hence by Ascoli's theorem the family \[\{\tilde{u}_{t}(\cdot,\omega,\phi):\\|\phi\\|\leq 1\} \tag{3.18}\] is relatively compact in $\mathcal{C}$. Next, we show the existence of continuous measurable version of solutions to the stochastic delayed HNNM (1.4) constructed by the solution of (3.9). Theorem 3.1.: _The trajectory field $u(t,\omega,\phi)$, $t\in[-\tau,\infty)$ of the stochastic delayed HNNM (1.4) has a Borel-measurable version $U(\cdot,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times\mathcal{C}\mapsto \mathcal{C}$ with the following properties: (I) For each $\phi\in\mathcal{C},U(t,\cdot,\phi)=u(t,\cdot,\phi)$ a.s. on $\Omega$. (II) For each $t\in\mathbb{R}^{+}$ and $\phi\in\mathcal{C},U(t,\cdot,\phi):\Omega\mapsto\mathcal{C}$ is $\mathcal{F}_{t}$-measurable. (III) There is a Borel set $\Omega_{0}\subset\Omega$ of full P-measure such that for all $\omega\in\Omega_{0}$ the map $U(\cdot,\omega,\cdot):\mathbb{R}^{+}\times\mathcal{C}\mapsto\mathcal{C}$ is continuous. (IV) For each $t\in\mathbb{R}^{+}$ and every $\omega\in\Omega_{0}$, the map $U(t,\omega,\cdot):\mathcal{C}\mapsto\mathcal{C}$ is continuous. (V) For each $t>\tau$ and all $\omega\in\Omega_{0}$, the map $U(t,\omega,\cdot):\mathcal{C}\rightarrow\mathcal{C}$ is compact._ Proof.: Define $u:[-\tau,\infty)\times\Omega\times\mathcal{C}\rightarrow\mathbb{R}^{n}$ by \[u(t,\omega,\phi)=\left\{\begin{array}{ll}v\left(t,\omega,\tilde{u}(t,\omega,\phi)\right)&\quad t\geq 0,\\ \phi(t)&\quad t\in[-\tau,0),\end{array}\right. \tag{3.19}\] We prove in the following that the above defined $u$ is in fact the solution of the stochastic delayed HNNM (1.4). It follows from Lemma 3.3 that for each $t\in\mathbb{R}^{+}$ and $\phi\in\mathcal{C}$, $\tilde{u}(t,\omega,\phi)$ is jointly measurable in $(t,\omega)$ and $\tilde{u}(t,\cdot,\phi)$ is $\mathcal{F}_{t}$-adapted, which combined with the $\mathcal{F}_{t}$-adaptivity of $v(t,\omega,\cdot)$ indicates that the process $u(t,\cdot,\phi)$ is $\mathcal{F}_{t}$-adapted, for each $t\geq 0$. Since $\{\tilde{u}(t,\cdot,\phi):t\geq 0\}$ is $\mathcal{F}_{t}$-adapted and has C${}^{1}$ sample paths, by taking Ito differentials of (3.19) and invoking Bismut's generalized Ito formula for stochastic flows, we have for any $t>0$ \[\begin{split} du\left(t,\omega,\phi\right)=& v\left(t, \omega,\frac{d\tilde{u}(t,\omega,\phi)}{dt}\right)dt\\ &+\sigma\left(v\left(t,\cdot,\tilde{u}(t,\omega,\phi)\right) \right)dw(t)\\ =&-Cv(t,\omega,\tilde{u}(t,\omega,\phi))+Hf(v(t,\omega, \tilde{u}(t,\omega,\phi)))\\ &+Bg(v_{t}(\cdot,\omega,\tilde{u}))+\sigma\left(v(t,\omega,\tilde{u }(t,\omega,\phi))\right)dw(t).\end{split} \tag{3.20}\] Then, in the case $t>0$, we have \[\begin{split} v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi) )&=\left\{\begin{array}{ll}v(t+\cdot,\omega,\tilde{u}_{t}( \cdot,\omega,\phi)),t+\cdot\geq 0\\ \phi(t+\cdot),-\tau\leq t+\cdot<0\end{array}\right.\\ &=u_{t}(\cdot,\omega,\phi),\end{split} \tag{3.21}\] for any $s\in[-\tau,0]$ and P-a.e. $\omega\in\Omega$. Therefore (3.20) becomes \[\begin{split} du\left(t,\omega,\phi\right)=&-Cu(t, \omega,\phi)+Hf(u(t,\omega,\phi))\\ &+Bg(u_{t}(\cdot,\omega,\phi))+\sigma\left(u(t,\omega,\phi) \right)dw(t),\end{split} \tag{3.22}\] i.e., $u(t,\omega,\phi)$ satisfies stochastic delayed HNNM (1.4) and \[U(t,\omega,\phi):=v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi)) \tag{3.23}\] is the version of its trajectory field we are seeking. From the construction of $U(t,\omega,\phi)$, one can easily see that statement (I) holds. It remains to show that $U(t,\omega,\phi)$ satisfies all the requirements of the theorem. We begin by proving that $U(t,\omega,\phi)$ is jointly (Borel)-measurable in $(t,\omega)$. By the definition of (3.3), one can clear see that \[\begin{split}\mathbb{R}^{+}\times\Omega\to L\left(\mathcal{C} \right)\\ (t,\omega)\mapsto v_{t}(s,\omega,\cdot)\end{split} \tag{3.24}\] is Borel-measurable for each $s\in[-\tau,0]$. But the Borel field of $C\left([-\tau,0]\times\mathbb{R}^{n},\mathbb{R}^{n}\right)$ is identified with that of $C([-\tau,0],L(\mathbb{R}^{n}))$ and the latter is generated by evaluations, so it follows immediately that the map \[\begin{split}\Lambda:\mathbb{R}^{+}\times\Omega\to L\left( \mathcal{C}\right)\\ (t,\omega)\mapsto v_{t}\left(\cdot,\omega,\cdot\right)\end{split} \tag{3.25}\] is Borel-measurable and has continuous sample paths. It follows from Lemma 3.3 that $\tilde{u}_{t}(\cdot,\omega,\phi)$ is Borel-measurable on $\mathbb{R}^{+}\times\Omega\times\mathcal{C}$. Therefore the composition \[U(t,\omega,\phi):=v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi)) \tag{3.26}\] is measurable on $\mathbb{R}^{+}\times\Omega\times\mathcal{C}$ and $U\left(t,\omega,\cdot\right):\mathcal{C}\to\mathcal{C}$ is continuous for $t\in\mathbb{R}^{+}$, P-a.e. $\omega\in\Omega$. Also for P-a.e. $\omega\in\Omega$, $U(\cdot,\omega,\cdot)$ is continuous on $\mathbb{R}^{+}\times\mathcal{C}$. This proves assertions (II), (III) and (IV) of the theorem. At last, we prove statement (V) of the theorem. Fix $t>\tau$ and pick any universal Borel set $\Omega_{0}\subset\Omega$ of full $P$-measure such that $v(t,\omega,\cdot):\mathbb{R}^{n}\to\mathbb{R}^{n}$ is linear for all $\omega\in\Omega_{0}$. Take $\omega\in\Omega_{0}$, then the composition (3.26) yields compactness of $U(t,\omega,\cdot):\mathcal{C}\to\mathcal{C}$ as we have already established that the map $\tilde{u}_{t}(\cdot,\omega,\cdot):\mathcal{C}\to\mathcal{C}$ is compact for any $t>\tau$. This completes the proof of the theorem. In Theorem 3.1, we construct a continuous measurable version $U(\cdot,\omega,\cdot)$ of (1.4) that generates a random semiflow. The idea originates from the early works [29], [26] and [27], where the authors recasted the stochastic delayed differential equations into a Hilbert space to study the Lyapunov spectrum and invariant manifolds. Our work can be regarded as a first attempt to study the random attractors of the HNNM (1.4) in Banach spaces. ### Cocycle property The remaining part of this section is devoted to showing the cocycle property of the trajectory field version $\mathrm{R}^{+}\times\Omega\times\mathcal{C}\to\mathcal{C}$ constructed in Theorem 3.1 using the Wong-Zakai approximation of the Wiener process, i.e., a piece-wise linear interpolation of its sample paths. See Twardowsky [18]. The Ito type (3.1) is equivalent to the following Stratonovich equation \[\left\{\begin{array}{l}d\hat{v}(t,\cdot,\cdot)=\sum_{i=1}^{m}\Sigma\circ\hat{v}( t,\cdot,\cdot)\circ dw_{i}(t)\\ -\frac{1}{2}\sum_{i=1}^{m}\left(\Sigma\circ\left([\Sigma\circ\hat{v}(t,\cdot, \cdot)](\cdot)\left(e_{i}\right)\right\}\right)\left(\cdot\right)\left(e_{i} \right)dt,\\ \hat{v}(0,\cdot,\cdot)=\mathrm{id}_{\mathbb{R}^{n}},\end{array}\right. \tag{3.27}\] where the notation $\left[\Sigma\circ A\right]\left(\cdot\right)\left(e_{i}\right),A\in L\left( \mathbb{R}^{n}\right)$ stands for the map \[\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}:v\rightarrow\Sigma\circ A(v)\left(e_ {i}\right).\] For each $k\geq 1$, let $\hat{v}^{k}:\mathbb{R}^{+}\times\Omega\to L\left(\mathbb{R}^{n}\right)$ be the fundamental solution of the following random differential equation+ \[\left\{\begin{array}{l}d\hat{v}^{k}(t,\cdot,\cdot)=\sum_{i=1}^{m}\left[ \Sigma\circ\hat{v}^{k}(t,\cdot,\cdot)\right]\left(\cdot\right)\left(e_{i} \right)\left(W_{i}^{k}\right)^{\prime}(t)dt\\ -\frac{1}{2}\sum_{i=1}^{m}\left(\Sigma\circ\left([\Sigma\circ\hat{v}^{k}(t, \cdot,\cdot)](\cdot)\left(e_{i}\right)\right\}\right)\left(\cdot\right)\left(e _{i}\right)dt,\\ \varphi(0,\cdot,\cdot)=\mathrm{id}_{\mathbb{R}^{n}}.\end{array}\right. \tag{3.28}\] Here $W_{i}^{k}(t)$ is the piece-wise linear approximation of $\omega_{i},i=1,2,\cdots,m$, which is defined by \[\left(W_{i}^{k}\right)^{\prime}(t,\omega)=\left(\omega_{i}^{k}\right)^{ \prime}(t),\quad\omega\in\Omega,\quad t\geq 0,\] where \[\omega_{i}^{k}(t)=k\left[\omega\left(\frac{j+1}{k}\right)-\omega_{i}\left( \frac{j}{k}\right)\right]\left(t-\frac{j}{k}\right)+\omega_{i}\left(\frac{j}{ k}\right),\] for $\quad\frac{j}{k}\leq t<\frac{j+1}{k},\quad j=0,1,2,\ldots$ and $\omega_{i}$ is standard mutually independent Brownian motions. It follows from [17] (see also Mohamed ref [29] and Twardowsky [18]) that for a.a. $\omega\in\Omega$, \[\lim_{k\rightarrow\infty}v^{k}(t,\omega,\cdot)=v(t,\omega,\cdot) \tag{3.29}\] in $L\left(\mathbb{R}^{n}\right)$ uniformly in $t\in[0,T]$, for $0<T<\infty$. Theorem 3.2.: _For each integer $k\geq 1$, let $\hat{v}^{k}(t,\omega,\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be the solution of (3.28) and define the following operator_ \[\begin{array}{l}A^{k}(t,\omega,\phi)\triangleq[\hat{v}^{k}(t,\omega,\cdot)] ^{-1}[-C\hat{v}^{k}(t,\omega,\phi(0))+\\ Hf(\hat{v}^{k}(t,\omega,\phi(0)))+Bg\left(\hat{v}^{k}_{t}(\cdot,\omega,\phi) \right)],\end{array} \tag{3.30}\] _with $\hat{v}^{k}_{t}(\cdot,\omega,\cdot):[-\tau,0]\times\mathcal{C}\rightarrow \mathcal{C}$ being defined by_ \[\hat{v}^{k}_{t}(s,\omega,\phi)=\left\{\begin{array}{ll}\hat{v}^{k}(t+s, \omega,\phi(0)):&t+s\geq 0,\\ \phi(t+s)&-\tau\leq t+s<0.\end{array}\right. \tag{3.31}\] _Let $0<T<\infty$. Denote by $B$ a bounded set in $\mathcal{C}$ and $\tilde{u}^{k}(t,\omega,\cdot):[-\tau,\infty)\rightarrow\mathbb{R}^{n}$ the solution of the following random retarded partial differential equation_ \[\left\{\begin{array}{l}\frac{d\tilde{u}^{k}(t)}{dt}=A^{k}(t,\omega,\tilde{u }^{k}_{t}),\quad t>0,\\ \tilde{u}^{k}_{0}=\phi\in\mathcal{C}.\end{array}\right. \tag{3.32}\] _Define_ \[U^{k}(t,\omega,\phi):=v^{k}_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega, \phi)), \tag{3.33}\] _then $U^{k}(t,\omega,\phi)$ satisfies_ \[\lim_{k\rightarrow\infty}\sup_{0\leq t\leq T}\sup_{\phi\in B}\left\\|U^{k}(t, \omega,\phi)-U(t,\omega,\phi)\right\\|=0\] _for P-a. e. $\omega\in\Omega$._ Proof.: Fix $0<T<\infty$ and $B\subset\mathcal{C}$ a bounded set throughout the proof. By Lemma 3.3, (3.32) admits a global solution for P-a. e. $\omega\in\Omega$, which is defined as follows. \[\begin{cases}\tilde{u}^{k}(t)=S(t)\phi+\int_{0}^{t}S(t-s)[\hat{v}^{k}(s,\omega,\cdot)]^{-1}\circ\\ \qquad[Hf(\hat{v}^{k}(s,\omega,\tilde{u}^{k}(s)))+B\tilde{g}\left(\hat{v}^{k} _{s}(\cdot,\omega,\tilde{u}^{k}_{t})\right)]\mathrm{d}s,t\geq 0,\\ u(t)=\phi(t),\quad t\in[-\tau,0].\end{cases} \tag{3.34}\] Therefore, by denoting $I(t,\omega,\phi)=\\|U^{k}(t,\omega,\phi)-U(t,\omega,\phi)\\|$, we have for $P$-a.e. $\omega\in\Omega$, \[\begin{split} I(t,\omega,\phi)=&\\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi))-v^{k}_{t}(\cdot,\omega,\tilde{u}^{k}_{t}( \cdot,\omega,\phi))\\|\\ \leq&\\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega, \phi))-v_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega,\phi))\\|\\ +&\\|v_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega, \phi))-v^{k}_{t}(\cdot,\omega,\tilde{u}^{k}_{t}(\cdot,\omega,\phi))\\|\\ \leq&\\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega, \phi)-\tilde{u}^{k}_{t}(\cdot,\omega,\phi))\\|\\ +&\\|[v_{t}(\cdot,\omega,\cdot)-v^{k}_{t}(\cdot,\omega, \cdot)]\tilde{u}^{k}_{t}(\cdot,\omega,\phi)]\\|\\ \leq&\\|v_{t}(\cdot,\omega,\cdot)\\|\\|\tilde{u}_{t}(\cdot, \omega,\phi)-\tilde{u}^{k}_{t}(\cdot,\omega,\phi)\\|\\ +&\\|v_{t}(\cdot,\omega,\cdot)-v^{k}_{t}(\cdot,\omega,\cdot)\\|\\|\tilde{u}^{ k}_{t}(\cdot,\omega,\phi)\\|\\ \triangleq&\\|v(t,\omega,\cdot)\\|I_{1}+I_{2}\\|\tilde{u}^{k}_{t}(\cdot, \omega,\phi)\\|.\end{split} \tag{3.35}\] Since $\lim_{k\to\infty}v^{k}(t,\omega,\cdot)=v(t,\omega,\cdot)$, we have $I_{2}\to 0$ as $k\to\infty$. Because $\\|v(t,\omega,\cdot)\\|$ is bounded, we only need to estimate $I_{1}$. \[\begin{split}\|I_{1}\|=&\|\int_{0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\circ Hf(v(s,\omega,u(s)))-\\ &\hat{v}^{k}(s,\omega,\cdot)^{-1}\circ Hf(\hat{v}^{k}(s,\omega, \tilde{u}^{k}(s)))]\|\\ &+\|\int_{0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\circ B\tilde{g} \left(v_{s}(\cdot,\omega,u_{t})\right)-\\ &\hat{v}^{k}(s,\omega,\cdot)^{-1}\circ B\tilde{g}\left(\hat{v}^{ k}_{s}(\cdot,\omega,\tilde{u}^{k}_{t})\right)]\mathrm{d}s\|\\ &\triangleq I_{11}+I_{12}.\end{split} \tag{3.36}\] We estimate each term on the right hand side of (3.36) in the following by Lemmas 3.3. By the Hypothesis A1, we can see \[\begin{split} I_{11}\leq&\\|\int_{0}^{t}S(t-s)v^{-1 }(s,\omega,\cdot)\circ H[f(v(s,\omega,u(s)))\\ &-f(\hat{v}^{k}(s,\omega,\tilde{u}^{k}(s)))]\mathrm{d}s\\|+\\|\int_ {0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\\ &-\hat{v}^{k}(s,\omega,\cdot)^{-1}]\circ Hf(\hat{v}^{k}(s,\omega,\tilde{u}^{k}(s)))\mathrm{d}s\\|\\ &\triangleq I_{111}+I_{112}.\end{split} \tag{3.37}\] Since $f$ is bounded, we have for any $\rho\in\mathbb{R}^{n}$, $Hf(\rho)\leq M$ which together with (3.29), implies that for any $\varepsilon>0$, there exists sufficient large $K_{0}$ such that for any $k>K_{0}$ \[I_{112}\leq M\varepsilon. \tag{3.38}\] Moreover, \[\begin{split} I_{111}\leq&\\|\int_{0}^{t}S(t-s)[v^{-1}(s,\omega,\cdot)\circ H[f(v(s,\omega,u(s)))\\ &-f(\hat{v}^{k}(s,\omega,u(s)))]\mathrm{d}s\\|+\\|\int_{0}^{t}S(t-s )[v^{-1}(s,\omega,\cdot)\\ &\circ H[f(\hat{v}^{k}(s,\omega,u(s)))-f(\hat{v}^{k}(s,\omega, \tilde{u}^{k}(s)))]\mathrm{d}s\\|\\ &\leq CKL_{f}\int_{0}^{t}\mathrm{e}^{\varrho(t-s)/2}\\|v\left(s, \omega,u(s)\right)\\ &-\hat{v}^{k}(s,\omega,u(s))\\|\mathrm{d}s+KL_{f}\int_{0}^{t} \mathrm{e}^{\varrho(t-s)/2}\\ &\left\\|u(s)-\tilde{u}^{k}(s)\right\\|\mathrm{d}s\triangleq I_{11 11}+I_{1112}.\end{split} \tag{3.39}\] It follows from (3.29) that for any $\varepsilon>0$, there exists sufficient large $K_{0}$ such that for any $k>K_{0}$ \[I_{1111}\leq CKL_{f}\varepsilon\int_{0}^{t}\mathrm{e}^{\varrho(t-s)/2}\mathrm{ d}s\leq CKL_{f}\varepsilon. \tag{3.40}\] Thus, we have for $k>K_{0}$ \[\begin{split} I_{1}\leq& M\varepsilon+CKL_{f} \varepsilon+KL_{f}\int_{0}^{t}\mathrm{e}^{\varrho(t-s)/2}\left\\|u(s)-\tilde{ u}^{k}(s)\right\\|\mathrm{d}s,\end{split} \tag{3.41}\] which together with the Gronwall inequality indicates that $I_{1}\to 0$ as $k\to\infty$. We are now in the position to provide the cocycle property of $U(\cdot,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times\mathcal{C}\mapsto \mathcal{C}$. We directly give the following results for which the proof is the same as Theorem 3 in [29]. Theorem 3.3.: _(The Cocycle Property) Let $\theta:\mathbb{R}^{+}\times\Omega\to\Omega$ be the shift corresponding Wiener measure on $(\Omega,\mathcal{F})$ which is defined by_ \[\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),t\in\mathbb{R}.\] _Then, there is a universal set $\Omega_{0}\subset\Omega$ of full Wiener measure such that_ \[U\left(t_{2},\theta_{t_{1}}\omega,U\left(t_{1},\omega,\phi\right)\right)=U \left(t_{1}+t_{2},\omega,\phi\right)\] _for all $\omega\in\Omega_{0},\phi\in\mathcal{C}$ and $t_{1},t_{2}\geq 0$._ ## 4 Random attractors In this section, we are concerned about the existence of random attractors of (1.4), which relies on showing the existence of an absorbing set and the asymptotic compactness of the RDS $(\theta,U)$. Throughout the reminder of this paper, we denote the family of tempered sets in the state space $C$ by $\mathcal{D}$. We first give the following uniform estimate results. Lemma 4.1.: _Assume that conditions of Lemma 3.3, Theorem 3.2 hold and_ \[c_{1}+\varrho/2<0<c_{1}+\varrho/2+\gamma,\] _then $U$ possesses a random absorbing set $\{K(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$, that is, for any $\{B(\omega)\}_{\omega\in\Omega}\in\mathcal{D}$ and $\mathbb{P}$-a.e. $\omega\in\Omega$, there exists $t_{B}(\omega)>0$ such that_ \[U\left(t,\theta_{-t}\omega,\phi\left(\theta_{-t}\omega\right)\right)\in K( \omega)\quad\text{ for all }t\geq t_{B}(\omega)\] _for all $t\geq t_{B}(\omega)$ and $\phi\left(\theta_{-t}\omega\right)\in B\left(\theta_{-t}\omega\right)$._ Proof.: For any $\phi(\omega)\in B(\omega)$, it follows from (3.16) in the proof of Lemma 3.3 that \[\begin{split}\mathrm{e}^{-\varrho t/2}\left\\|\tilde{u}_{t}\left( \cdot,\omega,\phi\right)\right\\|\leq& c_{0}\\|\phi\\|[\mathrm{e}^{-( \gamma+\varrho/2)t}\\ &+c_{1}\mathrm{e}^{c_{1}t}\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s],\end{split} \tag{4.1}\] where $c_{0}=l\mathrm{e}^{\gamma\tau}$ and $c_{1}=(hL_{f}+bL_{\tilde{g}})\mathrm{e}^{-\varrho\tau/2}$. This implies \[\begin{split}\left\\|\tilde{u}_{t}\left(\cdot,\omega,\phi \right)\right\\|\leq& c_{0}\\|\phi\\|[\mathrm{e}^{-\gamma t}+c_{1} \mathrm{e}^{(c_{1}+\varrho/2)t}\int_{0}^{t}\mathrm{e}^{-(\gamma+\varrho/2+c_{1 })s}\mathrm{d}s]\end{split} \tag{4.2}\] for P-a.e. $\omega\in\Omega$ and $0<t<\rho$ by Gronwall's inequality. Thus, we have \[\begin{split}\left\\|U\left(t,\omega,\phi\right)\right\\|=& \\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\phi))\\|\\ \leq& c_{0}\\|\phi\\|\\|v_{t}(\cdot,\omega,\cdot)\\| \mathrm{e}^{-\gamma t}+c_{0}c_{1}\mathrm{e}^{(c_{1}+\varrho/2)t}\\|\phi\\|\times \\ &\\|v_{t}(\cdot,\omega,\cdot)\\|\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s\end{split} \tag{4.3}\] for P-a.e. $\omega\in\Omega$. Therefore, \[\begin{split}\left\\|U\left(t,\theta_{-t}\omega,\phi\right)\right\\| =&\\|v_{t}(\cdot,\theta_{-t}\omega,\tilde{u}_{t}(\cdot,\theta_{-t} \omega,\phi))\\|\\ \leq& c_{0}\mathrm{e}^{-\gamma t}\\|\phi(\theta_{-t} \omega)\\|\\|v_{t}(\cdot,\theta_{-t}\omega,\cdot)\\|+c_{0}c_{1}\times\\ &(\frac{\mathrm{e}^{(c_{1}+\varrho/2)t}}{c_{1}+\varrho/2+\gamma}- \frac{\mathrm{e}^{-\gamma t}}{c_{1}+\varrho/2+\gamma})\times\\ &\\|\phi(\theta_{-t}\omega)\\|\\|v_{t}(\cdot,\theta_{-t}\omega, \cdot)\\|.\end{split} \tag{4.4}\] Since $c_{1}+\varrho/2<0,\gamma>0$, there must exits a $\lambda$ and $T_{B}>0$ such that for all $t>T_{B}$ \[c_{0}\mathrm{e}^{-\gamma t}+c_{0}c_{1}(\frac{\mathrm{e}^{(c_{1}+\varrho/2)t}} {c_{1}+\varrho/2+\gamma}-\frac{\mathrm{e}^{-\gamma t}}{c_{1}+\varrho/2+\gamma })<\lambda.\] Moreover, since $\\|v_{t}(\cdot,\omega,\cdot)\\|$ is bounded for all $t\in\mathbb{R}^{+}$ and P-a.e. $\omega\in\Omega$, if $\phi\left(\theta_{-t}\omega\right)\in B\left(\theta_{-t}\omega\right)$ then for all $t\geq t_{B}(\omega)$, \[\\|U\left(t,\theta_{-t}\omega\right),\phi\left(\theta_{-t}\omega\right)\\|\leq \lambda\\|\phi(\theta_{-t}\omega)\\|\\|v_{t}(\cdot,\theta_{-t}\omega,\cdot)\\|.\] Therefore, $K(\omega)=\{\vartheta\in\mathcal{C}:\\|\vartheta\\|\leq\lambda\\|\phi(\theta_{-t }\omega)\\|\\|v_{t}(\cdot,\theta_{-t}\omega,\cdot)\\|\}$ form a tempered bounded closed random absorbing set of RDS $U$ in $\mathcal{D}$, which completes the proof. In the next lemma, we show that we can take initial data in $\{K(\omega)\}_{\omega\in\Omega}$ to obtain the pullback asymptotic compactness of $U$. Lemma 4.2.: _Assume that $\{K(\omega)\}_{\omega\in\Omega}$ is a random absorbing in $\mathcal{C}$ for the RDS $U$. Then $(\theta,U)$ is pullback asymptotically compact._ Proof.: Take an arbitrary random set $\left\{B(\omega)\right\}_{\omega\in\Omega}\in\mathcal{D}$, a sequence $t_{n}\rightarrow+\infty$ and $\phi_{n}\in B\left(\theta_{-t_{n}}\omega\right)$. We have to prove that $\left\{U\left(t_{n},\theta_{-t_{n}}\omega,\phi_{n}\right)\right\}$ is precompact. Since $\left\{K(\omega)\right\}$ is a random absorbing for $\Psi$, then there exists $T>0$ such that, for all $\omega\in\Omega$, \[U\left(t,\theta_{-t}\omega\right)B\left(\theta_{-t}\omega\right)\subset K(\omega) \tag{4.5}\] for all $t\geq T$. Because $t_{n}\rightarrow+\infty$, we can choose $n_{1}\geq 1$ such that $t_{n_{1}}-1\geq T$. Applying (4.5) for $t=t_{n_{1}}-1$ and $\omega=\theta_{-1}\omega$, we find that \[\eta_{1}\triangleq\Psi\left(t_{n_{1}}-1,\theta_{-t_{n_{1}}}\omega,\phi_{n_{1 }}\right)\in K\left(\theta_{-1}\omega\right) \tag{4.6}\] Similarly, we can choose a subsequence $\left\{n_{k}\right\}$ of $\left\{n\right\}$ such that $n_{1}<n_{2}<\cdots<n_{k}\rightarrow+\infty$ such that \[\eta_{k}\triangleq\Psi\left(t_{n_{k}}-k,\theta_{-t_{n_{k}}}\omega,\phi_{n_{k }}\right)\in K\left(\theta_{-k}\omega\right) \tag{4.7}\] Hence, by the assumption we conclude that the sequence \[\left\{U\left(k,\theta_{-k}\omega,\eta_{k}\right)\right\} \tag{4.8}\] is precompact. On the other hand by (4.7), we have \[\begin{split} U(k,\theta_{-k}\omega,\eta_{k})&=U(k,\theta_{-k}\omega,U(t_{n_{k}}-k,\theta_{-t_{n_{k}}}\omega,U_{n_{k}})\\ &=U\left(t_{n_{k}},\theta_{-t_{n_{k}}}\omega,\phi_{n_{k}}\right),\end{split} \tag{4.9}\] for all $k\geq 1$. Combining (4.8) and (4.9), we obtain that the sequence $\left\{U\left(t_{n_{k}},\theta_{-t_{n_{k}}}\omega,\phi_{n_{k}}\right)\right\}$ is precompact. Therefore, $\left\{U\left(t_{n},\theta_{t_{n}}\omega,\phi_{n_{k}}\right)\right\}$ is precompact, which completes the proof. Lemma (4.1) says that the continuous RDS $\Psi$ has a random absorbing set while Lemma (4.2) tells us that $\left(\theta,U\right)$ is pullback asymptotically compact in $\mathcal{C}$. Thus, it follows from Lemma 2.1 that the continuous RDS $\left(\theta,U\right)$ possesses a random attractor. Namely, we obtain the following result. Theorem 4.1.: _Assume that conditions of Lemma 4.1 hold, then the continuous RDS $U$ generated by (1.4) admits a unique $\mathcal{D}$-pullback attractor in $\mathcal{C}$ belonging to the class $\mathcal{D}$._ ## 5 Exponentially attracting stationary solution In this section, we derive sufficient conditions that guarantee the random attractor is an exponentially attracting random fixed point $\xi^{}$ by a general Banach fixed point theorem. n particular we use the following fixed point theorem in probability analysis, which is often called the uniform strictly contracting property, originates from the early work [33] and was then extended to various versions in [20, 3, 21] and [7]. Here, we adopt the version in [7] for infinite dimensional case. Lemma 5.1.: _Let $\left(\mathcal{G},d_{\mathcal{G}}\right)$ be a complete metric space with bounded metric. Suppose that_ \[U(t,\omega,\mathcal{G})\subset\mathcal{G}\] _for $\omega\in\Omega,t\geq 0$, and that $x\to U(t,\omega,x)$ is continuous. In addition, we assume the contraction condition: There exists a constant $k<0$ such that, for $\omega\in\Omega$,_ \[\sup_{x\neq y\in\mathcal{G}}\log\frac{d_{\mathcal{G}}(U(1,\omega,x),U(1, \omega,y))}{d_{\mathcal{G}}(x,y)}\leq k.\] _Then $U$ has a unique generalized fixed point $\gamma^{}$ in $\mathcal{G}$. Moreover, the following convergence property holds:_ \[\lim_{t\to\infty}U\left(t,\theta_{-t}\omega,x\right)=\gamma^{}(\omega)\] _for any $\omega\in\Omega$ and $x\in\mathcal{G}$._ Theorem 5.1.: _Assume that conditions of Lemma 4.1 hold. Moreover, assume that_ \[\frac{\varrho}{2}e^{[\frac{\varrho}{2}+L_{v}(hL_{f}+bL_{\tilde{g}})]}+\tau-1<0.\] _Then the RDS $U$ generated by SNDRDE (1.4) possess a tempered random fixed point $\chi^{}$, which is unique under all tempered random variables in $\mathcal{C}$ and attracts exponentially fast every random variable in $\mathcal{C}$._ Proof.: If $c_{1}+\varrho/2<0<c_{1}+\varrho/2+\gamma$, then the conditions of Theorem 4.1 hold and hence (1.4) possess random attractors in $\mathcal{C}$. We will prove that (1.4) admits a unique globally exponentially attracting random stationary solution in $\mathcal{C}$, which immediately implies the random attractor in $\mathcal{C}$ obtained in Theorem 4.1 is the random fixed point $\chi^{}$. By (4.3), we can see that for any $\phi\in\mathcal{C}$ \[\begin{split}\\|U\left(t,\omega,\phi\right)\\|\leq& c_{0}\\|\phi\\|\\|v_{t}(\cdot,\omega,\cdot)\\| \mathrm{e}^{-\gamma t}+c_{0}c_{1}\mathrm{e}^{(c_{1}+\varrho/2)t}\\|\phi\\|\times \\ &\\|v_{t}(\cdot,\omega,\cdot)\\|\int_{0}^{t}\mathrm{e}^{-(\gamma+ \varrho/2+c_{1})s}\mathrm{d}s\end{split} \tag{5.1}\] which implies that for any $\phi\in\mathcal{C}_{co}$, $U(t,\omega,\psi)\in\mathcal{C}$, i.e. $\mathcal{C}$ is invariant under the random semiflow $U$. Moreover, it follows from Lemma 3.3 that $U$ is continuous in $\mathcal{C}$. Therefore, we only need to prove the contraction property. That is, there exists $k<0$ such that \[\sup_{\varphi\neq\psi\in\mathcal{C}}\\|U(1,\omega,\varphi)-U(1,\omega,\psi)\\| \leq e^{k}\\|\varphi-\psi\\|. \tag{5.2}\] Hence, it suffices to prove that for any $\varphi,\psi\in\mathcal{C}$ \[\begin{split}\\|U(1,\omega,\varphi)-U(1,\omega,\psi)\\|=& \\|v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\varphi))-\\ & v_{t}(\cdot,\omega,\tilde{u}_{t}(\cdot,\omega,\psi))\\|\leq e^{ k}\\|\varphi-\psi\\|.\end{split} \tag{5.3}\] By Eq. (3.12) and Eq. (3.26), we have for any $\varphi,\psi\in\mathcal{C}$ \[\begin{split}\\|G(t,\omega)\\|&\leq\\|v_{t}(\cdot, \omega,\tilde{u}_{t}(\cdot,\omega,\varphi)-\tilde{u}_{t}(\cdot,\omega,\psi))\\| \\ &\leq\\|v_{t}(\cdot,\omega,\cdot)\\|\{\\|S(t,\omega)[\varphi-\psi] \\|+\\ &\sup_{\zeta\in[-\tau,0]}\int_{0}^{t+\zeta}S(t+\zeta-r,\omega)v^{- 1}(s,\omega,\cdot)\circ\\ &[H(f(v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega,\varphi)))-f( v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega,\psi))))\\ &+B(\tilde{g}\left(v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega, \varphi))\right)-\\ &\tilde{g}\left(v_{s}(\cdot,\omega,\tilde{u}_{s}(\cdot,\omega, \psi))\right)))]\mathrm{d}s\}\\ &\leq L_{v}e^{\gamma\tau}e^{-\gamma t}\\|\phi-\psi\\|+L_{v}e^{ \frac{\sigma\tau}{2}}(hL_{f}+bL_{\tilde{g}})\times\\ &\int_{0}^{t}e^{-\frac{\varrho}{2}(t-r)}\left\\|U(r,\omega,\phi)-U (r,\omega,\psi)\right\\|\mathrm{d}r.\end{split} \tag{5.4}\] Multiply both sides of (5.4) by $e^{\frac{gt}{2}}$ and denote by $H(t,\omega)=e^{\frac{gt}{2}}G(t,\omega)$, $c_{3}(t)=L_{v}e^{\gamma\tau}e^{-\gamma t}e^{\frac{gt}{2}}$, $c_{4}(t)=L_{v}e^{\frac{\rho(\tau+t)}{2}}(hL_{f}+bL_{\tilde{g}})$, we obtain \[\\|H(t,\omega)\\|\leq c_{3}(t)\\|\phi-\psi\\|+c_{4}(t)\int_{0}^{t}\\|H(r,\omega)\\| \mathrm{d}r. \tag{5.5}\] Again, the Gronwall inequality gives rise to \[\\|H(t,\omega)\\|\leq\\|\phi-\psi\\|(c_{3}(t)+\int_{0}^{t}c_{3}(r)e^{\int_{r}^{t}c _{4}(s)ds}\mathrm{d}r). \tag{5.6}\] Dividing both sides of (5.6) by $e^{\frac{gt}{2}}$ and take $t=1$ lead to \[\\|G(1,\omega)\\|\leq\\|\phi-\psi\\|(c_{3}(1)e^{-\frac{\rho}{2}}+e^{-\frac{\rho}{ 2}}\int_{0}^{1}c_{3}(r)e^{\int_{r}^{1}c_{4}(s)ds}\mathrm{d}r). \tag{5.7}\] Therefore, if there exists $\varsigma<0$ such that $c_{3}(1)e^{-\frac{\rho}{2}}<e^{\varsigma}$ and $e^{-\frac{\rho}{2}}\int_{0}^{1}c_{3}(r)e^{\int_{r}^{1}c_{4}(s)ds}\mathrm{d}r<e ^{\varsigma}$ then result of the theorem hold. By a simple computation, we see that $\frac{\rho}{2}e^{[\frac{\rho}{2}+L_{v}(hL_{f}+bL_{\tilde{g}})]}+\tau-1<0$ implies the above claim holds and hence completes the proof. ## 6 Numerical Simulations This section is devoted to some numerical simulations in order to demonstrate the effectiveness and efficiency of the established theoretical results. In our numerical examples, we consider the following stochastic delayed HNNM. \[\left\{\begin{aligned} & du_{1}(t)=[-c_{1}u_{1}(t)+h_{11}f_{1} \left(u_{1}(t)\right)+h_{12}f_{2}\left(u_{2}(t)\right)+\\ & b_{11}g_{1}\left(u_{1}\left(t-\tau_{1}\right)\right)+b_{12}g_{2} \left(u_{2}\left(t-\tau_{2}\right)\right)]dt\\ &+\sigma_{11}u_{1}(t)dw_{1}(t)+\sigma_{12}u_{2}(t)dw_{2}(t)\\ & du_{2}(t)=[-c_{2}u_{2}(t)+h_{21}f_{1}\left(u_{1}(t)\right)+h_{ 22}f_{2}\left(u_{2}(t)\right)+\\ & b_{21}g_{1}\left(u_{1}\left(t-\tau_{1}\right)\right)+b_{22}g_{2 }\left(u_{2}\left(t-\tau_{2}\right)\right)]dt\\ &+\sigma_{21}u_{1}(t)dw_{1}(t)+\sigma_{22}u_{2}(t)dw_{2}(t).\end{aligned}\right. \tag{6.1}\] Take \[u(t)=\left(u_{1}(t),u_{2}(t)\right)^{\mathrm{T}},\] \[u_{t}(\tau)=\left(u_{1}\left(t-\tau_{1}\right),u_{2}\left(t-\tau_{2}\right) \right)^{\mathrm{T}},\] \[C=\left(\begin{array}{cc}c_{1}&0\\ 0&c_{2}\end{array}\right)=\left(\begin{array}{cc}5&0\\ 0&5\end{array}\right),\] \[H=\left(\begin{array}{cc}h_{11}&h_{12}\\ h_{21}&h_{22}\end{array}\right)=\left(\begin{array}{cc}0.2&0.1\\ 0.3&0.1\end{array}\right),\] \[B=\left(\begin{array}{cc}b_{11}&b_{12}\\ b_{21}&b_{22}\end{array}\right)=\left(\begin{array}{cc}-0.3&0.2\\ 0.1&0.3\end{array}\right),\] \[\sigma=\left(\begin{array}{cc}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{array}\right)=\left(\begin{array}{cc}0.01&0\\ 0&0.02\end{array}\right),\] $f_{j}(u_{j})=g_{j}(u_{j})=\tanh(u_{j})=\frac{e^{u_{j}^{\prime}-e^{-u_{j}}}}{e^{u _{j}^{\prime}+e^{-u_{j}}}_{j}}$ for $j=1,2$. Hence the characteristic equation of (6.1) is \[\det\left[\left(\begin{array}{ll}\lambda-5-0.3e^{-\lambda\tau_{1}}&0.2e^{- \lambda\tau_{1}}\\ 0.1e^{-\lambda\tau_{2}}&\lambda-5+0.3e^{-\lambda\tau_{2}}\end{array}\right) \right]=0, \tag{6.2}\] We first choose $\tau_{1}=\tau_{2}=0.1$ and hence Hypothesis A1 - A2 hold and $\rho<0$. Thus it follows from 4.1 that the equation admits a random attractor. Moreover, $\frac{\rho}{2}e^{[\frac{\rho}{2}+L_{v}(\hbar L_{f}+bL_{\bar{g}})]}+\tau-1$$<0$ also holds and it follows from Theorem 5.1 the attractor is a unique exponentially attractive tempered random fixed point, i.e. the trivial fixed point $(0,0)$, which is shown in Fig 1. In Fig 1a, the initial condition is chosen to be $\phi(t)=(0.1,0.2)$ for $t\in[-0.1,0]$ while in In Fig 1b, the initial condition is first chosen to be $\phi(t)=(10,20)$ for $t\in[-0.1,0]$. From which we can see no matter how large the initial condition is, the state will tend to $(0,0)$. ## 7 Summary and discussions In this paper, we have obtained the existence and qualitative property of random attractors for (1.3), which first required showing that the system generates a random dynamicla system.. We showed under certain conditions that the random attractor is a globally exponentially attracting random stationary solution. From dynamical system theory, the conditions for the attractors to be fixed point are so strong that could be hardly met in the real world applications. Indeed, from the dissipative system theory, if estimation on the dimension of random attractors can provide researchers with useful information about the structure of the random attractor. Hence, the topological dimension for the random attractors of (1.3) will be investigated in a future paper. Furthermore, in order to obtain the global complex dynamics and nonlocal analysis of the qualitative properties of the system, the existence and structure of the associated invariant manifolds of the stationary solutions, the existence of connecting orbits (including the heteroclinic orbits or homoclinic orbits) are all of significance and deserve more attention. Acknowledgement This work was jointly supported by the National Natural Science Foundation of China (62173139), China Postdoctoral Science Foundation(2019TQ0089), Hunan Provincial Natural Science Foundation of China (2020JJ5344, 2019RS1033) the Science and Technology Innovation Program of Hunan Province (2021RC4030), the Scientific Research Fund of Hunan Provincial Education Department (20B353). ## Acknowledgments The author thanks Dr Ding Kui for assistance of numerical simulations.
2304.14793	Learning Graph Neural Networks using Exact Compression	Graph Neural Networks (GNNs) are a form of deep learning that enable a wide range of machine learning applications on graph-structured data. The learning of GNNs, however, is known to pose challenges for memory-constrained devices such as GPUs. In this paper, we study exact compression as a way to reduce the memory requirements of learning GNNs on large graphs. In particular, we adopt a formal approach to compression and propose a methodology that transforms GNN learning problems into provably equivalent compressed GNN learning problems. In a preliminary experimental evaluation, we give insights into the compression ratios that can be obtained on real-world graphs and apply our methodology to an existing GNN benchmark.	Jeroen Bollen, Jasper Steegmans, Jan Van den Bussche, Stijn Vansummeren	2023-04-28T12:04:28Z	http://arxiv.org/abs/2304.14793v1	# Learning Graph Neural Networks using Exact Compression ###### Abstract. Graph Neural Networks (GNNs) are a form of deep learning that enable a wide range of machine learning applications on graph-structured data. The learning of GNNs, however, is known to pose challenges for memory-constrained devices such as GPUs. In this paper, we study _exact compression_ as a way to reduce the memory requirements of learning GNNs on large graphs. In particular, we adopt a formal approach to compression and propose a methodology that transforms GNN learning problems into provably equivalent compressed GNN learning problems. In a preliminary experimental evaluation, we give insights into the compression ratios that can be obtained on real-world graphs and apply our methodology to an existing GNN benchmark. ## 1. Introduction Whereas Machine Learning (ML) has traditionally been most successful in analyzing traditional unstructured data such as text or images, ML over structured data such as graphs has become an active area of research in the past decade. In particular, Graph Neural Networks (GNNs for short) are a form of deep learning architectures that enable a wide range of ML applications on graph-structured data, such as molecule classification, knowledge graph completion, and web-scale recommendations (Beng et al., 2015; Chen et al., 2016; Chen et al., 2017; Chen et al., 2018; Chen et al., 2019). At their core, GNNs allow to embed graph nodes into vector space. Crucially, the obtained vectors can capture graph structure, which is essential for the ML applications already cited. While GNNs are hence an attractive mechanism for ML on graphs, learning GNNs is known to be resource-demanding which limits their scalability (Sutton et al., 2018; Chen et al., 2019). In particular, for large graphs it becomes difficult to encode all the required information into the limited memory of hardware accelerators like GPUs. For this reason, scalable methods for learning GNNs on large graphs are an active subject of research (e.g., (Beng et al., 2015; Chen et al., 2016; Chen et al., 2017; Chen et al., 2018; Chen et al., 2019; Chen et al., 2019; Chen et al., 2019; Chen et al., 2019; Chen et al., 2019; Chen et al., 2019)). Broadly speaking, we can identify three different principles for obtaining scalability in the literature (1) distributing computation across multiple machines or GPUs (Beng et al., 2015; Chen et al., 2016; Chen et al., 2017; Chen et al., 2019; Chen et al., 2019; Chen et al., 2019; Chen et al., 2019); (2) learning on a sample of the input graph instead of the entire graph (Beng et al., 2015; Chen et al., 2017; Chen et al., 2019); and (3) compression (Chen et al., 2016; Chen et al., 2017; Chen et al., 2019; Chen et al., 2019). Compression-based approaches limit the memory requirements of learning GNNs on large graphs by reducing the input graph into a smaller graph and then learn on this smaller, reduced graph instead. In this paper, we are concerned with compression. Compression methods are based on collapsing multiple input nodes into a single reduced node in the compressed graph. Methods vary, however, in how they collapse nodes. For example, Deng et al. (Deng et al., 2016) use spectral analysis for this purpose; Liang et al. (Liang et al., 2017) use variants of multi-level graph partitioning; and Generale et al. (Gorenale et al., 2018), who specifically consider knowledge graphs, use general heuristics (such as two nodes having equal set of attributes) or bisimulation. While these methods give intuitive reasons to argue that the structure of the obtained compressed graph should be similar to that of the original graph, no formal guarantee is ever given that learning on the compressed graph is in any way equivalent to learning on the original graph. Furthermore, the methods are usually devised and tested for a specific GNN architecture (such as Graph Convolutional Networks, GCN). It is therefore unclear how they fare on other GNN architectures. Inherently, these methods are hence heuristics. At best the compressed graphs that they generate _approximate_ the original graph structure, and it is difficult to predict for which GNN architectures this approximation is good enough, and for which architectures it poses a problem. Towards a more principled study of learning GNNs on compressed graphs, we propose to take a formal approach and study _exact compression_ instead. We make the following contributions. (1). We formally define when two learning problems involving graph neural networks are equivalent. Based on this definition, our goal is to transform a given problem into a smaller, equivalent problem based on compression. (Section 2.) (2). We develop a compression methodology that is guaranteed to always yield an equivalent learning problem and that is applicable to a wide class of GNN architectures known as _aggregate-combine GNNs_(Beng et al., 2015; Chen et al., 2017; Chen et al., 2018). This class includes all Graph Convolutional Networks (Chen et al., 2017). Our methodology is based on recent insights into the expressiveness of aggregate-combine GNNs (Chen et al., 2017; Chen et al., 2019). These results imply that if the local neighborhoods of two nodes $v,w$ in input graph $G$ are equal, then any GNN will treat $v$ and $w$ identically. We may intuitively exploit this property for compression: if $v$ and $w$ are treated identically there is no need for them both to be present during learning; having one of them suffices. We fully develop this intuition in Section 3, where we also consider a more relaxed notion of "local neighborhood" that is applicable only to specific kinds of aggregate-combine GNNs. (3). We empirically evaluate the effectiveness of our methodology in Section 4. In particular, we give insights into the compression ratios that can be obtained on real-world graphs. While we find that these ratios are diverse, from compressing extremely well to compressing only marginally, a preliminary experiment on an existing GNN benchmark shows positive impact on learning efficiency even with marginal compression. We start with preliminaries in Section 2 and conclude in Section 5. Proofs of formal statements may be found in the Appendix. ## 2. Preliminaries Background. We denote by $\mathbb{R}$ the set of real numbers, by $\mathbb{N}$ the set of natural numbers, and by $\mathbb{N}_{\infty}$ the set $\mathbb{N}\cup\{\infty\}$ of natural numbers extended with infinity. We will use double curly braces $\{\ldots\}$ to denote multisets and multiset comprehension. Formally, we view a multiset over a domain of elements $S$ as a function $M\colon S\to\mathbb{N}$ that associates a multiplicity $M(x)$ to each element $x\in S$. As such, in the multiset $M=\{a,a,b\}$, we have that $M(a)=2$ and $M(b)=1$. If $M(x)=0$ then $x$ is not present in $M$. We denote by $\mathit{supp}(M)$ the set of all elements present in $M$, $\mathit{supp}(M)\coloneqq\{x\in S\mid M(x)>0\}$. Note that if every element has multiplicity at most one, then $M$ is a set. If $M$ is a multiset and $c\in\mathbb{N}_{\infty}$ then we denote by $M\|_{\leq c}$ the multiset obtained from $M$ by restricting the multiplicity of all elements to be at most $c$, i.e., $M\|_{\leq c}(x)=\min(M(x),c)$, for all elements $x$. Note in particular that $M\|_{\leq\infty}=M$ and that $M\|_{\leq 1}$ converts $M$ into a set. Graphs. We work with directed node-colored multigraphs. Formally, our graphs are hence triples $G=(V,E,g)$ where $V$ is a finite set of nodes; $E$ is a _multiset_ of edges over $V\times V$; and $g$ is a function, called the _coloring_ of $G$, that maps every node $v\in V$ to a _color_$g(v)$. (The term "color" is just an intuitive way to specify that $g$ has some unspecified range).) If $Y$ is the co-domain of $g$, i.e., $g$ is of the form $g\colon V\to Y$ then we also call $g$ a _$Y$-coloring_ and say that $G$ is a _$Y$-colored graph_, or simply a $Y$-graph. When $Y=\mathbb{R}^{n}$ we also call $g$ an $n$-dimensional _feature map_. To ease notation we write $v\in G$ to indicate that $v\in V$. Furthermore, we write $G(v)$ instead of $g(v)$ to denote the color of $v$ in $G$, and we write $G(v\to w)$ instead of $E(v\to w)$ to denote the multiplicity of edge $v\to w$ in $G$. When $E$ is a set, i.e., when every edge has multiplicity at most one, then we also call $G$ a _simple_ graph. We write $\mathsf{in}_{G}(v)$ for the multiset $\{w\in G\mid w\to v\in E\}$ of all incoming neighbors of $v$. So, if the edge $w\to v$ has multiplicity $5$ in $E$ then $w$ also has multiplicity $5$ in $\mathsf{in}_{G}(v)$. We drop subscripts when the graph $G$ is clear from the context. The _size_ of a graph $G$ is the number of nodes $\|V\|$ plus the number of simple edges $\|\mathit{supp}(E)\|$. This is a reasonable definition of the size of a multigraph, since for each edge it suffices to simply store its multiplicity as a number, and storing a number takes unit cost in the RAM model of computation. Color transformers. If$C$ is a function that maps $X$-colored graphs $G=(V,E,g)$ into $Y$-colored graphs $G^{\prime}=(V^{\prime},E^{\prime},g^{\prime})$ that leaves nodes and edges untouched and only changes the coloring, i.e., $V=V^{\prime}$ and $E=E^{\prime}$ then we call $C$ a _coloring transformer_. In particular, if $X=\mathbb{R}^{p}$ and $Y=\mathbb{R}^{q}$ for some dimensions $p$ and $q$ then $C$ is a _feature map transformer_. Graph Neural Networks. Graph Neural Networks (GNNs) are a popular form of neural networks that enable deep learning on graphs. Many different forms of GNNs have been proposed in the literature. We refer the reader to the overview by Hamilton (1997). In this paper we focus on a standard form of GNNs that is known under the name of _aggregate-combine_ GNNs (Beng et al., 2015), also called _message-passing GNNs_. These are defined as follows (Garno et al., 2016; Garno et al., 2016). A _GNN layer_ of input dimension $p$ and output dimension $q$ is a pair $\textsc{(Agg,Comb)}$ of functions where (1) $\textsc{Agg}$ is an _aggregation function_ that maps finite multisets of vectors in $\mathbb{R}^{p}$ to vectors in $\mathbb{R}^{h}$ for some dimension $h$ and (2) $\textsc{Comb}$ is a _combination function_$\textsc{Comb}\colon\mathbb{R}^{p}\times\mathbb{R}^{h}\to\mathbb{R}^{q}$. In practice, $\textsc{Agg}$ is usually taken to compute the arithmetic mean, sum, or maximum of the vectors in the multiset, while $\textsc{Comb}$ is computed by means of a feedforward neural network whose parameters can be learned. A _GNN_ is a sequence $\overline{L}=(L_{1},\ldots,L_{k})$ of GNN layers, where the output dimension of $L_{i}$ equals the input dimension of $L_{i+1}$, for $1\leq i<k$. The _input and output dimensions_ of the GNN are the input dimension of $L_{1}$, and the output dimension of $L_{k}$ respectively. In what follows, we write $\overline{L}\colon p,q$ to denote that $p$ is the input dimension of $\overline{L}$ and $q$ is the output dimension. Semantically, GNN layers and GNNs are feature map transformers (Garno et al., 2016) In particular, when GNN layer $L=\textsc{(Agg,Comb)}$ of input dimension $p$ and output dimension $q$ is executed on $\mathbb{R}^{p}$-colored graph $G=(V,E,g)$ it returns the $\mathbb{R}^{q}$-colored graph $G^{\prime}=(V,E,g^{\prime})$ with $g^{\prime}$ the $q$-dimensional feature map defined by \[g^{\prime}\colon v\mapsto\textsc{Comb}\big{(}g(v),\textsc{Agg}\ \{g(w)\mid w\in\mathsf{in}_{G}(v)\}\big{)}.\] As such, for each node $v$, $\overline{L}$ aggregates the (multiplicity-weighted) $\mathbb{R}^{p}$ colors of $v$'s neighbors, and combines this with $v$'s own color to compute the $\mathbb{R}^{q}$ output. A GNN $\overline{L}\colon p,q$ simply composes the transformations defined by its layers: given $\mathbb{R}^{p}$-colored graph $G$ it returns the $\mathbb{R}^{q}$-colored graph $(L_{k}\circ L_{k-1}\circ\cdots\circ L_{1})(G)$. Discussion. It is important to stress that in the literature GNNs are defined to operate on _simple graphs_, whereas we have generalized their semantics above to also work on _multigraphs_. We did so because, as we will see in Section 3, the result of compressing a simple graph for the purpose of learning naturally yields a multigraph. Learning problems. GNNs are used for a wide range of supervised learning tasks on graphs. For example, for a node $v$, the $\mathbb{R}^{q}$-vector $\overline{L}(G)(v)$ computed for $v$ by GNN $\overline{L}$ can be interpreted, after normalisation, as a probability distribution over $q$ new labels (for node classification), or as predicted values (for node regression). Similarly, an edge prediction for nodes $v$ and $w$ can be made based on the pair $(\overline{L}(G)(v),\overline{L}(G)(w))$. Finally, by aggregating $\overline{L}(G)(v)$ over all nodes $v\in G$, one obtains graph embeddings that can be used for graph classification, regression and clustering (Hamilton et al., 2016). In this work, we focus on the tasks of node classification and regression. Our methodology is equally applicable to the other tasks, however. In order to make precise what we mean by learning GNNs on compressed graphs for node classification, we propose the following formal definition. Definition 2.1 ().: A _learning problem_ of input dimension $p$ and output dimension $q$ is a tuple $\mathcal{P}=(G,T,\mathrm{Loss},\mathcal{S})$ where * $G$ is the $\mathbb{R}^{p}$-colored graph on which we wish to learn; * $T$ is a subset of $G$'s nodes, representing the training set; * Loss : $T\times\mathbb{R}^{q}\rightarrow\mathbb{R}$ is a loss function that allows to quantify, for each node $v\in T$ the dissimilarity $\operatorname{Loss}(v,c)$ of the $\mathbb{R}^{q}$-color $c$ that is predicted for $v$ by a GNN and the desired $\mathbb{R}^{q}$-color for $v$ as specified in the training set; * $\mathcal{S}$ is the _hypothesis space_, a (possibly infinite) collection of GNNs of input dimension $p$ and output dimension $q$. Given a learning problem $\mathcal{P}$, a learning algorithm produces a "learned" GNN in $\mathcal{S}$ by traversing the search space $\mathcal{S}$. For each currently considered GNN $\overline{L}\in\mathcal{S}$, the observed loss of $\overline{L}$ on $G$ w.r.t. $T$ is computed as \[\operatorname{Loss}(\overline{L}(G),T)\coloneqq\sum_{v\in T}\operatorname{ Loss}\big{(}v,\overline{L}(G)(v)\big{)}.\] The learning algorithm aims to minimize this loss, but possibly returns an $\overline{L}$ for which this is only a local minimum. In practice, $\mathcal{S}$ is usually a collection of GNNs with the same topology: they all have the same number of layers (with each layer $d$ having the same input and output dimensions accross GNNs in $\mathcal{S}$) and are parametrized by the same number of learnable parameters. Each concrete parametrization constitutes a concrete GNN in $\mathcal{S}$ in our framework. Commonly, the learned GNN $\overline{L}$ is then found by means of gradient descent, which updates the learnable parameters of the GNNs in $\mathcal{S}$ to minimize the observed error. No matter which concrete learning algorithm is used to solve a learning problem, the intent is that the returned $\overline{L}$ generalizes well: it makes predictions on $G$ for the nodes not in $T$, and can also be applied to other, new $\mathbb{R}^{p}$-colored graphs to predict $\mathbb{R}^{q}$-vectors for each node. Our research question in this paper is the following. _Given a GNN learning problem $\mathcal{P}=(G,T,\operatorname{Loss},\mathcal{S})$, is it possible to transform this into a new problem $\mathcal{P}^{\prime}=(G^{\prime},T^{\prime},\operatorname{Loss}^{\prime}, \mathcal{S})$ that is obtained by_ compressing $G$, $T$, _and Loss into a smaller graph $G^{\prime}$, training set $T^{\prime}$, and loss function $\operatorname{Loss}^{\prime}$ such that instead of learning a GNN on $\mathcal{P}$ we could equivalently learn a GNN on $\mathcal{P}^{\prime}$ instead?_ Here "equivalently" means that ideally, no matter which learning algorithm is used, we would like the learned GNN to be identical in both cases. Of course, this is not possible in practice because the learning process is itself non-deterministic, e.g., because the learning algorithm makes random starts; because of stochasticity in stochastic gradient descent; or because of non-deterministic dropout that is applied between layers. Nevertheless, we expect the GNN obtained by learning on the compressed problem would perform "as good" as the GNN obtained by the learning on the uncompressed problem, in the sense that it generalizes to unseen nodes and unseen colored graphs equally well. To ensure that we may hope any learning algorithm to perform equally well on $\mathcal{P}^{\prime}$ as on $\mathcal{P}$, we formally define: Definition 2.2 ().: Two learning problems $\mathcal{P}$ and $\mathcal{P}^{\prime}$ are _equivalent_, written $\mathcal{P}\equiv\mathcal{P}^{\prime}$, if they share the same hypothesis space of GNNs $\mathcal{S}$ and, for every $\overline{L}\in\mathcal{S}$ we have $\operatorname{Loss}(\overline{L}(G),T)=\operatorname{Loss}^{\prime}(\overline {L}(G^{\prime}),T^{\prime})$. In other words, when traversing the hypothesis space for a GNN to return, no learning algorithm can distinguish between $\mathcal{P}$ and $\mathcal{P}^{\prime}$. All other things being equal, if the learning algorithm then returns a GNN $\overline{L}$ when run on $\mathcal{P}$, it will return $\overline{L}$ on $\mathcal{P}^{\prime}$ with the same probability. Note that, while the hypothesis space $\mathcal{S}$ remains unchanged in this definition, it is possible (and, as we will see, actually required) to adapt the loss function $\operatorname{Loss}$ into a modified loss function $\operatorname{Loss}^{\prime}$ during compression. The benefit of a positive answer to our research question, if compression is efficient, is computational efficiency: learning on smaller graphs is faster than learning on larger graphs and requires less memory. ## 3. Methodology To compress one learning problem into an equivalent, hopefully smaller, problem we will exploit recent insights into the expressiveness of GNNs (Bahdan et al., 2016; Goyal et al., 2017; Goyal et al., 2017). In particular, it is known that if the local neighborhoods of two nodes $v,w$ in input graph $G$ are equal, then any GNN will treat $v$ and $w$ identically. In particular, it will assign the same output colors to $v$ and $w$. We may intuitively exploit this property for compression: since $v$ and $w$ are treated identically there is no need for them both to be present during learning; having one of them suffices. So, we could compress by removing nodes that are redundant in this sense. We must take care, however, that by removing one, we do not change the structure (and hence, possibly, the predicted color) of the remaining node. Also, of course, we need to make sure that by removing nodes we do not lose training information. I.e., if $T$ specifies a training color for $v$ but not $w$ then if we decide to remove $v$, we somehow need to "fix" $T$, as well as the loss function. This section is devoted to developing this intuitive idea. In Section 3.1 we first study under which conditions GNNs treat nodes identically. Next, in Section 3.2 we develop compression of colored graphs based on collapsing identically-treated nodes, allowing to remove redundant nodes while retaining the structure of the remaining nodes. Finally, in Section 3.3, we discuss compression of the training set and loss function. Together, these three ingredients allow us to compress a learning problem into an equivalent problem, cf. Definition 2.2. We close this section by proposing an alternative definition of compression that works only for a limited class of learning problems. It is nevertheless interesting as it may allow better compression, as we will show in Section 4. ### Indistinguishability The following definition formalizes when two nodes, not necessarily in the same graph, are treated identically by a class of GNNs. Definition 3.1 ().: Let $\mathcal{S}$ be a class of GNNs, let $G$ and $H$ be two $\mathbb{R}^{p}$-colored graphs for some $p$, and let $v\in G$, $w\in H$ be two nodes in these graphs. We say that $(G,v)$ is _$\mathcal{S}$-indistinguishable_ from $(H,w)$, denoted $(G,v)\prec_{\mathcal{S}}(H,w)$, if for every GNN $\overline{L}\in\mathcal{S}$ of input dimension $p$ it holds that $\overline{L}(G)(v)=\overline{L}(H)(w)$. In other words, two nodes are indistinguishable by a class of GNNs $\mathcal{S}$ if no $GNN$ in $\mathcal{S}$ can ever assign a different output color to these nodes, when started on $G$ respectively $H$. We call $(G,v)$ and $(H,w)$$\mathcal{S}$-_distinguishable_ otherwise. For the purpose of compression, we are in search of sets of nodes in the input graph $G$ that are pairwise $\mathcal{S}$-indistinguishable, with $\mathcal{S}$ the hypothesis space of the input learning problem. It is these nodes that we can potentially collapse in the input learning problem. Formally, let $[G,v]_{\mathcal{S}}$ denote the set of all nodes in $G$ that are $S$-indistinguishable from $v$, \[[G,v]_{\mathcal{S}}:=\{w\in G\mid(G,v)\sim_{\mathcal{S}}(G,w)\}.\] We aim to calculate $[G,v]_{\mathcal{S}}$ and subsequently compress $G$ by removing all but one node in $[G,v]_{\mathcal{S}}$ from $G$. Color refinement. To calculate $[G,v]_{\mathcal{S}}$, we build on the work of Morris et al. (Morris et al., 2019) and Xu et al. (Xu et al., 2020). They proved independently that a GNN can distinguish two nodes if an only if the so-called _color refinement_ algorithm assigns different colors to these nodes. Color refinement is equivalent to the one-dimensional Weisfeiler-Leman (WL) algorithm (Lewis, 1996), and works as follows. Definition 3.2 ().: The (one-step) _color-refinement_ of colored graph $G=(V,E,g)$, denoted $\operatorname{cr}(G)$, is the colored graph $G^{\prime}=(V,E,g^{\prime})$ where $g^{\prime}$ maps every node $v\in G$ to a pair, consisting of $v$'s original color and the multiset of colors of its incoming neighbors: \[g^{\prime}:v\mapsto\big{(}G(v),\{G(w)\mid w\in\operatorname{in}_{G}(v)\} \big{)}.\] As such, we can think of $\operatorname{cr}(G)(v)$ as representing the immediate neighborhood of $v$ (including $v$), for any node $v$. We denote by $\operatorname{cr}^{d}(G)$ the result of applying $d$ color refinement steps on $G$, so $\operatorname{cr}^{0}(G)=G$ and $\operatorname{cr}^{d+1}(G)=\operatorname{cr}(\operatorname{cr}^{d}(G))$. Using this notation, we can think of $\operatorname{cr}^{d}(G)(v)$ as representing the local neighborhood of $v$ "up to radius $d$". To illustrate, Figure 1 shows a colored graph $G$ and two steps of color refinement. The following property was observed by Morris et al. (Morris et al., 2019) and Xu et al (Xu et al., 2020) for GNNs operating on _simple_ graphs. We here extend it to multigraphs. Proposition 3.3 ().: _Let $\overline{L}$ be a GNN composed of $d\in\mathbb{N}$ layers, $d\geq 1$. If $\operatorname{cr}^{d}(G)(v)=\operatorname{cr}^{d}(H)(w)$ then $\overline{L}(G)(v)=\overline{L}(H)(w)$. As a consequence, if $\mathcal{S}$ is a hypothesis space consisting of GNNs of at most $d$ layers and $\operatorname{cr}^{d}(G)(v)=\operatorname{cr}^{d}(H)(w)$ then $(G,v)\sim_{\mathcal{S}}(H,w)$._ In other words, $d$-layer GNNs cannot distinguish nodes that are assigned the same color by $d$ steps of color refinement. Let $[G,v]_{d}$ denote the set of all nodes in $G$ that receive the same color as $v$ after $d$ steps of color refinement, \[[G,v]_{d}\coloneqq\{w\in G\mid\operatorname{cr}^{d}(G)(v)=\operatorname{cr}^ {d}(G)(w)\}.\] Then it follows from Proposition 3.3 that $[G,v]_{d}$ is a _refinement_ of $[G,v]_{\mathcal{S}}$ in the sense that $[G,v]_{d}\subseteq[G,v]_{\mathcal{S}}$, for all $v\in G$. Morris et al. (Morris et al., 2019) and Xu et al. (Xu et al., 2020) have also shown that for every graph $G$ and every depth $d$ there exists a GNN $\overline{L}$ of $d$ layers such that $[G,v]_{d}=[G,v]_{\overline{L}}$, for every node $v\in G$. Consequently, if, in addition to containing only GNNs with at most $d$ layers, $\mathcal{S}$ includes _all possible_$d$-layer GNNs, then $[G,v]_{d}=[G,v]_{\mathcal{S}}$ coincide, for all $v\in G$. Hence, for such $\mathcal{S}$ we may calculate $[G,v]_{\mathcal{S}}$ by calculating $[G,v]_{d}$ instead. When $\mathcal{S}$ does not include all $d$-layer GNNs we simply use $[G,v]_{d}$ as a proxy for $[G,v]_{\mathcal{S}}$. This is certainly size-since $[G,v]_{d}\subseteq[G,v]_{\mathcal{S}}$ no GNN in $\mathcal{S}$ will be able to distinguish the nodes in $[G,v]_{d}$ and we may hence collapse nodes in $[G,v]_{d}$ for the purpose of compression. In this case, however, we risk that $[G,v]_{d}$ contains too few nodes compared to $[G,v]_{\mathcal{S}}$, and therefore may not provide enough opportunity for compression. We will return to this issue in Section 3.4. What happens if there is no bound on the number of layers of GNNs in $\mathcal{S}$? In that case we can still use color refinement to compute $[G,v]_{\mathcal{S}}$ as follows. It is known that after a finite number of color refinements steps we reach a value $d$ such that for all nodes $v\in G$ we have $[G,v]_{d}=[G,v]_{d+1}$. The smallest value $d$ for which this holds is called the _stable coloring number_ of $G$, and we denote the colored graph obtained by this value of $d$ by $\operatorname{cr}^{\infty}(G)$ in what follows. Similarly we denote the equivalence classes at this value of $d$ by $[G,v]_{\infty}$. From Proposition 3.3 it readily follows: Corollary 3.4 ().: _For any class$\mathcal{S}$ of GNNs, if $\operatorname{cr}^{\infty}(G)(v)=\operatorname{cr}^{\infty}(H)(w)$ then $(G,v)\sim_{\mathcal{S}}(H,w)$._ We note that it is very efficient to compute the set $\{[G,v]_{d}\mid v\in G\}$ of all color refinement classes: this can be done in time $\mathcal{O}((n+m)\log n)$ with $n$ the number of vertices and $m$ the number of edges of the input graph (Brandt, 2007). Example 3.5 ().: To illustrate, consider the colored graph from Figure 1, as well as the color refinement steps illustrated there. (Recall that nodes $a_{1},a_{2},a_{3}$ share the same color, as do $b_{1},b_{2},b_{3}$.) Then after one step of color refinement we have \[[G,a_{1}]_{1} =\{a_{1}\}\] \[[G,a_{2}]_{1} =[G,a_{3}]_{1}=\{a_{2},a_{3}\}\] \[[G,b_{1}]_{1} =[G,b_{2}]_{1}=[G,b_{3}]_{1}=\{b_{1},b_{2},b_{3}\},\] while after two steps of color refinement we obtain the following color refinement classes: \[[G,a_{1}]_{2} =\{a_{1}\}\] \[[G,b_{1}]_{2} =[G,b_{2}]_{2}=\{b_{1},b_{2}\}\] \[[G,a_{2}]_{2} =[G,a_{3}]_{2} =\{a_{2},a_{3}\} [G,b_{3}]_{2} =\{b_{3}\}.\] We invite the reader to check that for every node $v$ in this graph, $[G,v]_{3}=[G,v]_{2}$. As such, the stable coloring is obtained when $d=2$ and $[G,v]_{2}=[G,v]_{\infty}$. ### Graph reduction Having established a way to compute redundant nodes, we now turn our attention to compression. Assume that we have already computed the color refinement classes $\{[G,v]_{d}\mid v\in G\}$ for $d\in\mathbb{N}_{\infty}$. For each $v\in G$, we wish to "collapse" all nodes in $[G,v]_{d}$ into a single node. To that end, define a _$d$-substitution_ on a graph $G$ to be a function that maps each color refinement class in $\{[G,v]_{d}\mid v\in G\}$ Figure 1. Example of color refinement. Nodes $a_{1},a_{2},a_{3}$ have the same color $a$; nodes $b_{1},b_{2},b_{3}$ have the same color $b$. All edges have multiplicity $1$. to a node $\rho([G,v]_{d})\in[G,v]_{d}$. Intuitively, $\rho([G,v]_{d})$ is the node that we wish to keep; all other nodes in $[G,v]_{d}$ will be removed. In what follows we extend $\rho$ to also operate on nodes in $G$ by setting $\rho(\rho)=\rho([G,v]_{d})$. Definition 3.6 ().: The _reduction of graph $G$ by $d$-substitution $\rho$ on $G$_ is the graph $H=(V,E,h)$ where * $V=\{\rho(v)\mid v\in G\}$ * For all $v,w\in V$ we have \[E(v\to w)=\sum_{v^{\prime}\in[G,v]_{d}}G(v^{\prime}\to w)\] In particular, if there is no edge into $w$ in $G$, there will be no edge into $w$ in $H$, as this sum is then zero. * nodes retain colors: for each node $v\in H$ we have $h(v)=G(v)$. In what follows, we denote by $G/\rho$ the reduction of $G$ by $\rho$. A graph obtained by reducing $G$ according to some $d$-substitution $\rho$ is called a _$d$-reduct of $G$_. Example 3.7 ().: Reconsider the colored graph $G$ of Figure 1. Let $\rho_{1}$ and $\rho_{2}$ be the following $d=1$-substitutions: \[\rho_{1} :\{a_{1}\}\mapsto a_{1} \{a_{2},a_{3}\}\mapsto a_{2} \{b_{1},b_{2},b_{3}\}\mapsto b_{1}\] \[\rho_{2} :\{a_{1}\}\mapsto a_{1} \{a_{2},a_{3}\}\mapsto a_{2} \{b_{1},b_{2},b_{3}\}\mapsto b_{3}\] Then $G/\rho_{1}$ and $G/\rho_{2}$ are illustrated in Figure 2. The following proposition is an essential property of our compression methodology. Proposition 3.8 ().: _For every graph $G$, every $d$-substitution $\rho$ of $G$ with $d\in\mathbb{N}_{\infty}$, and every node $v\in G$ we have $\operatorname{cr}^{d}(G)(v)=\operatorname{cr}^{d}(G/\rho)(\rho(v))$._ It hence follows from Proposition 3.3 and Corollary 3.4 that if $\mathcal{S}$ consists of GNNs of at most $d\in\mathbb{N}_{\infty}$ layers, then $(G,v)\sim_{\mathcal{S}}(G/\rho,\rho(v))$. DiscussionExample 3.7 shows that the choice of $d$-substitution determines the reduct $G/\rho$ that we obtain. In particular, the example shows that distinct substitutions can yield distinct, _non-isomorphic_ reducts. This behavior is unavoidable, unless $d$ is the stable coloring number of $G$. Indeed, we are able to show: Proposition 3.9 ().: _There is a single $d$-reduct of a graph $G$ up to isomorphism if and only if $d$ is greater than or equal to the stable coloring number of $G$._ A direct consequence of having different non-isomorphic reducts when $d$ is less than the coloring number is that some of these reducts may be smaller than others. In Example 3.7, $G/\rho_{1}$ has $3$ nodes and $4$ edges while $G/\rho_{2}$ has $3$ nodes and only $3$ edges. We may always obtain a $d$-reduct of minimal size as follows. For $d\in\mathbb{N}_{\infty}$, define the _$d$-incidence_ of a node $w\in G$, denoted $\operatorname{incidence}^{d}_{G}(w)$, to be the number of color refinement classes in $\{[G,v]_{d}\mid v\in G\}$ that contain an incoming neighbor of $w$. That is, the $d$-incidence of $w$ is the number of distinct classes in $\{[G,u]_{d}\mid u\in\operatorname{in}\operatorname{in}\operatorname{in} \operatorname{in}\operatorname{in}\operatorname{out}\}$. The following proposition shows that we obtain a $d$-reduct of minimal size by by choosing a $d$-substitution that maps color refinement classes to nodes of minimal $d$-incidence. Proposition 3.10 ().: _Let $G$ be a graph, let $d\in\mathbb{N}_{\infty}$ and let $\rho$ be a $d$-substitution such that_ \[\operatorname{incidence}^{d}_{G}(\rho(v))=\min_{v^{\prime}\in[G,v]_{d}} \operatorname{incidence}^{d}_{G}(v^{\prime}),\] _for every node $v\in G$. Then the size of $G/\rho$ is minimal among all $d$-reducts of $G$._ When we report the size of $d$-reducts in our experiments (Section 4), we always report the minimal size among all $d$-reducts. DiscussionExample 3.7 illustrates that, depending on the substitution used, the result of reducing a simple graph may be a multigraph. In the literature, however, GNNs and color refinement are normally defined to operate on simple graphs. One may therefore wonder whether it is possible to define a different notion of reduction that always yields a simple graph when executed on simple, instead of a multigraph as we propose here. To answer this question, consider the tree-shaped graph $G$ in Figure 3(left), whose root $v$ has $m$$b$-colored children, each having $n$$c$-labeled children. It is straightforward to see that when $d\geq 2$, any simple graph $H$ that has a node $w$ such that $\operatorname{cr}^{d}(G)(v)=\operatorname{cr}^{d}(H)(w)$ must be such that $w$ has $m$$b$-colored children in $H$, each having $n$$c$-labeled children. As such, because $H$ is simple, it must be of size at least as large as $G$. Instead, by moving to multigraphs we are able to compress this "regular" structure in $G$ by only three nodes, as show in Figure 3(right). This illustrates that for the purpose of compression we naturally need to move to multigraphs. ### Problem compression We next combine the insights from Sections 3.1 and 3.2 into a method for compressing learning problems. Let $\mathcal{P}=(G,T,\operatorname{Loss},\mathcal{S})$ be a learning problem. Let $\rho$ be a $d$-substitution. We then define the reductions $T/\rho$ and $\operatorname{Loss}/\rho$ of $T$ and Loss by $\rho$ as follows: \[T/\rho =\{\rho(v)\mid v\in T\}\] \[\operatorname{Loss}/\rho(v,c) =\sum_{w\in T\cap[G,v]_{d}}\operatorname{Loss}(w,c)\] We denote by $\mathcal{P}/\rho$ the learning problem $(G/\rho,T/\rho,\operatorname{Loss}/\rho,\mathcal{S})$. Figure 3. Reduction of a tree-shaped simple graph (left) into a small multigraph (right). Figure 2. Reduction of the graph of Figure 1 by the $1$-substitutions $\rho_{1}$ and $\rho_{2}$ from Example 3.7. Theorem 3.11.: _Let $\mathcal{P}$ be a learning problem with hypothesis space $\mathcal{S}$ and assume that $d\in\mathbb{N}_{\infty}$ is such that every GNN in $\mathcal{S}$ has at most $d$ layers. (In particular, $d=\infty$ if there is no bound on the number of layers in $\mathcal{S}$.) Then $\mathcal{P}\equiv\mathcal{P}/\rho$ for every $d$-substitution $\rho$._ ### Graded Color Refinement So far, we have focused on compression based on the _depth_ of the GNNs present in the hypothesis space, where the depth of a GNN equals its number of layers and the depth of a hypothesis space $\mathcal{S}$ is the maximum depth of any of its GNNs, or $\infty$ if this maximum is unbounded. In particular, the notion of $d$-reduct hinges on this parameter $d$ through the calculated color refinement classes $[G,v]_{d}$. As we have already observed in Section 3.1, however, when $\mathcal{S}$ does not contain all $d$-layer GNNs the color refinement classes $[G,v]_{d}$ that we base compression on may contain too few nodes compared to $[G,v]_{\mathcal{S}}$ and may therefore not provide enough opportunity for compression. In such cases, it may be beneficial to move to more fine-grained notions of color refinement that better capture $[G,v]_{\mathcal{S}}$. In this section we propose one such fine-grained notion, which takes into account the "width" of $\mathcal{S}$. We say that _GNN $\overline{L}$ has width $c\in\mathbb{N}_{\infty}$_ if for every layer in $\overline{L}$ the aggregation function Agg is such that $\text{Agg}(M)=\text{Agg}(M\|_{\leq c})$ for every multiset $M$. In other words: Agg can only "count" up to $c$ copies of a neighbor's color. When $c=\infty$ there is no limit on the count. The width of $\mathcal{S}$ is then the maximum width of any of its GNNs, or $\infty$ if this maximum is unbounded. Hypothesis spaces of bounded width clearly do not contain _all_$d$-layer GNNs, for any depth $d$. While we know of no practical GNN architectures that explicitly limit the width, there are influential learning _algorithms_ that implicitly limit the width. For example, to speed up learning, GraphSAGE (Garfani et al., 2016) can be parametrized by a hyperparameter $c$. When $c<\infty$ GraphSAGE does not consider all of a node's neighbors in each layer, but only a random sample of at most $c$ neighbors of that node. Such sampling effectively limits the width of $\mathcal{S}$. We next define a variant of color refinement, called _graded_ color refinement, that takes width into account. It may lead to larger color refinement classes, and therefore potentially also to better compression. Definition 3.12.: Let $c\in\mathbb{N}_{\infty}$. The (one-step) $c$-_graded color refinement_ of colored graph $G=(V,E,g)$, denoted $\text{c}\text{c}_{c}(G)$, is the colored graph $G^{\prime}=(V,E,g^{\prime})$ with \[g^{\prime}:v\mapsto\big{(}G(v),\{G(w)\mid w\in\text{in}_{G}(v)\}\\|_{\leq c} \big{)}.\] Note that $\text{c}\text{c}_{\infty}$ equals normal, non-graded, color refinement. We also remark that $\text{c}\text{c}_{r}$ with $c=1$ corresponds to standard _bisimulation_ on graphs (Bill et al., 2016). We denote by $\text{c}\text{c}\text{c}_{c}^{d}(G)$ the result of applying $d$ refinement steps of $c$-graded color refinement, so $\text{c}\text{c}\text{c}_{c}^{d}(G)=G$ and $\text{c}\text{c}\text{c}_{c}^{d+1}=\text{c}\text{c}(\text{c}\text{c}_{c}^{d}(G))$. We denote by $[G,v]_{c,d}$ the set of all nodes in $G$ that receive the same color after $d$ steps of $c$-graded color refinement. The concept of $(c,d)$-substitution is then the defined analogously to $d$-substitution, as mappings from the $[G,v]_{c,d}$ color refinement classes to nodes in these classes. The reduction of a graph $G$ by a $(c,d)$ substitution is similar to reductions by $d$-substitution except that the edge multiplicity $E(v\to w)$ for $v,w$ in the reduction is now limited to $c$, i.e., \[E(v\to w)=\sum_{v^{\prime}\in[G,v]_{c,d}}G(v^{\prime}\to w)\|_{\leq c}.\] With these definitions, we can show that for any $\mathcal{P}$ with hypothesis space $\mathcal{S}$ of width $c$ and depth $d$ we have $\mathcal{P}\equiv\mathcal{P}/\rho$ for any $(c,d)$-reduction $\rho$. The full development is omitted due to lack of space. We note that $[G,v]_{d}\subseteq[G,v]_{c,d}$, always. Graded color refinement hence potentially leads to better compression, but only applies to problems with hypothesis spaces of width $c$. We will empirically contrast the compression ratio obtained by $(c,d)$-reducts to those obtained by $d$-reducts (i.e., where $c=\infty$) in Section 4. There, we also study the effect of $c$ on learning accuracy for problems whose width is not bounded. ## 4. Evaluation In this section, we empirically evaluate the compression methodology described in Section 3. We first give insights into the compression ratios that can be obtained on real-world graphs in Section 4.1. Subsequently, we evaluate the learning on compressed graphs versus learning on the original graphs. ### Compression We consider the real-world graphs listed in Table 1. The ogbn-graphs are from the Open Graph Benchmark (OGB), a collection of realistic, large-scale, and diverse benchmark datasets for machine learning on graphs (Garfani et al., 2016), where they belong to the OGB node property prediction benchmark (OGBN). Specifically, ogbn-arvis is a network of academic papers, where edge $x\to y$ indicates that $x$ cites $y$. Graph ogbn-arvis-inv is the inverted version of ogbn-arvis; it is obtained by reversing edges, so that edge $x\gets y$ indicates that $y$ was cited by $x$. Graph ogbn-arvis-undirected is the undirected version of ogbn-arvis; it is obtained by adding inverse edges to ogbn-arvis. Next, there is ogbn-products, an undirected graph representing an Amazon product co-purchasing network. The other datasets are from the Stanford Large Network Dataset Collection (SNAP) (Kipfani et al., 2016). Here, snap-roadnet-ca, snap-roadnet-pa, and snap-roadnet-tx are undirected graphs representing road networks, and snap-soc-pokec is a directed graph containing an online social network. The input colors used for learning on these graphs typically depend on the application. To get an understanding of the maximum amount of compression that we can obtain independent of the target application, we assign a shared single color $c$ to each node, in all graphs. As such, the color refinement classes $[G,v]_{d}$ that we obtain \begin{table} \begin{tabular}{l l c c} \hline \hline Graph & Type & \#Nodes & \#Edges \\ \hline ogbn-arviv & directed & 169.343 & 1.166.243 \\ ogbn-arvis-inv & directed & 169.343 & 1.166.243 \\ ogbn-arvis-undirected & undirected & 169.343 & 2.315.598 \\ ogbn-products & undirected & 2.449.029 & 61.859.140 \\ snap-roadnet-ca & undirected & 1.965.206 & 5.533.214 \\ snap-roadnet-pa & undirected & 1.088.092 & 3.083.796 \\ snap-roadnet-tx & undirected & 1.379.917 & 3.843.320 \\ snap-soc-pokec & directed & 1.632.803 & 30.622.564 \\ \hline \hline \end{tabular} \end{table} Table 1. Datasets are maximal, in the sense that for any other colorored graph $G^{\prime}$ whose topology equals $G$ we will have $[G^{\prime},v]_{d}\subseteq[G,v]_{d}$. In this setting, we hence reach maximal compression. We consider three versions of ogbn-arxiv to get an indication of how edge directionality impacts compression. Recall that GNN layers propagate color information following the direction of edges. Hence, because in ogbn-arxiv an edge $x\to y$ indicates that $x$ cites $y$, color information flows from citing papers to cited papers. In ogbn-arxiv-inv, by contrast, it flows from cited papers to citing papers while in ogbn-arxiv-undirected it flows in both directions. The direction in which the information can flow impacts the number of color refinement classes that we obtain, and hence the compression, as we will see. #### 4.2.2. Ungraded compression Figures 4(a) and 4(b) shows the fraction of nodes and edges remaining in $d$-reducts of these graphs, plotted as a function of the number of color refinement rounds $d$. (Consistent with our definition of the size of a multigraph, the number of edges plotted is the number of unique edges, i.e., ignoring edge multiplicities.) We see that in terms of the number of nodes, compression is effective for $d\leq 2$, obtaining compression ratios of $0.03\%-0.05\%$ on the road network datasets, to $77\%$ on ogbn-products. For $d=3$, compression becomes ineffective for ogbn-arxiv-undirected, ogbn-products and snap-soc-pokce as these retain $88\%$ of their nodes or more. Compression on the other datasets is satisfactory for $d=3$, as shown in Table 2. From $d>3$ onwards, node compression becomes ineffective for most datasets, with all datasets except ogbn-arxiv retaining at least $86\%$ of their nodes when $d\geq 5$. By contrast, ogbn-arxiv stabilizes at $d=4$, retaining only $36\%$ of its nodes. In terms of the number of edges, we see that ogbn-arxiv-inv, ogbn-arxiv-undirected, ogbn-products, and snap-soc-pokce retain almost $100\%$ of their edges from $d\geq 2$ onwards, while ogbn-arxiv retains only $56\%$ of its edges when $d=3$ and the road networks compress to $5\%-8\%$ of the edges when $d=3$. While we may hence conclude that in general compression becomes ineffective for deeper layer numbers, $d>4$, we note that in practice most GNN topologies use only $d=3$ layers. For such GNNs, compression on ogbn-arxiv and the road networks is promising. One may wonder why the inverted and undirected variants of ogbn-arxiv differ so much in terms of compression. The short answer is that that, their graph topology is completely different. As such, the local neighborhood information that cr${}^{d}$ calculates is also completely different, yielding different reductions. For example, a manual inspection of ogbn-arxiv reveals a number of highly cited papers that have no outgoing edges. While these papers are the "sink nodes" in ogbn-arxiv, they are "source nodes" in ogbn-arxiv-inv. Quite quickly we may then distinguish nodes in ogbn-arxiv-inv based solely on the number of highly cited papers that they cite. This behavior does not occur in ogbn-arxiv, because the highly cited papers are outgoing neighbors, which cr${}^{d}$ ignores. #### 4.2.3. Graded compression We next evaluate the effect of moving to compression based on $c$-graded color refinement. Figure 4(c) and (d) shows the fraction of nodes and edges remaining in $(c,3)$-reducts of our graphs, relative to the number of nodes and edges in the corresponding $d=3$-reduct, plotted as a function of $c$. In terms of the number of nodes we see that, as expected, moving to graded compression yields better compression than non-graded compression. In particular, for $c=1$ all datasets retain $<1\%$ of their nodes while for $c=3$, compression is $27\%$ or less for ogbn-arxiv, ogbn-arxiv-undirected, and the road networks. In the latter setting, ogbn-products is at $75\%$ of nodes and snap-soc-pokce at $55\%$. Compression hence becomes less effective as $c$ increases. In terms of the number of edges, compression is most effective when $c=1$ or $c=2$ in which case it significantly improves over ungraded compression. From $c\geq 3$ onwards, graded compression becomes only marginally better than ungraded compression. We conclude that graded compression has the potential to yield significantly smaller graphs than ungraded compression, but only for small gradedness values, $c=1$ (i.e., bisimulation) or $c=2$. #### 4.2.4. Conclusion Overall, we see that real-world graphs have diverse non-graded compression ratios: road networks compress extremely well (to $4\%$ of nodes and $5\%$ of edges when $d=3$); while other graphs such as ogbn-arxiv-inv compress reasonably in terms of number of nodes ($65\%$) but only marginally in terms of edges ($93\%$ or more); and graphs such as ogbn-products compress only marginally in both (retaining $90\%$ or more of nodes and edges). This diversity is to be expected: our exact compression methodology is based on exploiting redundancy in local neighborhoods of nodes. For graphs where few nodes have equal local neighborhoods, we cannot expect reduction in size. Moving to graded compression for those graphs improves reduction in size, but will yield approximate compression unless the learning problem hypothesis space has small width. ### Learning We next turn to validating empirically the effect of learning on compressed problems according to our methodology. Specifically, we apply our methodology to learning on ogbn-arxiv-inv with $d=3$. We know from Section 4.1 that compression on this graph is reasonable in terms of nodes, but only marginal in terms of edges. \begin{table} \begin{tabular}{l r r} \hline \hline Graph & Nodes (\%) & Edges (\%) \\ \hline ogbn-arxiv & 32.9 & 56.3 \\ ogbn-arxiv-inv & 65.9 & 92.7 \\ ogbn-arxiv-undirected & 93.3 & 98.9 \\ ogbn-products & 88.5 & 98.6 \\ snap-roadnet-ca & 4.4 & 5.9 \\ snap-roadnet-pa & 6.1 & 7.9 \\ snap-roadnet-tx & 3.9 & 5.2 \\ snap-soc-pokce & 87.9 & 99.1 \\ \hline \hline \end{tabular} \end{table} Table 2. Compression at $d=3$. \begin{table} \begin{tabular}{l r r r r r} \hline \hline Problem & Nodes & Edges & Accuracy & \multicolumn{2}{c}{Training} \\ & (\%) & (\%) & (\%) & time (s) & mem (GiB) \\ \hline $\mathcal{P}_{1}$ & 100.0 & 100.0 & 66.7 & 61.66 & 2.14 \\ $\mathcal{P}_{2}$ & 100.0 & 100.0 & 65.9 & 58.99 & 2.02 \\ $\mathcal{P}_{3}^{1}$ & 76.9 & 95.9 & 60.9 & 48.10 & 1.59 \\ $\mathcal{P}_{3}^{3}$ & 79.2 & 97.1 & 64.5 & 49.71 & 1.65 \\ $\mathcal{P}_{3}^{1}$ & 79.3 & 97.2 & 64.9 & 49.55 & 1.65 \\ $\mathcal{P}_{3}^{0c}$ & 79.4 & 97.2 & 63.8 & 49.16 & 1.65 \\ \hline \hline \end{tabular} \end{table} Table 3. Comparison of learning on the original uncompressed problem $\mathcal{P}_{1}$, the uncompressed problem with discretized labels $\mathcal{P}_{2}$, and $(c,d=3)$-compressed variants $\mathcal{P}_{3}^{c}$. Despite the modest compression, the effect on learning efficiency in terms of learning time and memory consumption is still positive, as we will see. We hence focus in this section on ogbn-arxiv-inv and the associated learning problem from the OGBN benchmark: predict, for every paper in ogbn-arxiv-inv, its subject area (e.g., cs.AI, cs.LG, and cs.OS,...). There are 40 possible subject areas. In addition to the citation network, we have available for each paper a 128-dimensional feature vector obtained by averaging the embeddings of words in its title and abstract. Our learning problem $\mathcal{P}_{1}$ hence consists of the ogbn-arxiv-inv graph in which each node is colored with this feature vector. The training set $T$ consists of 90941 nodes, obtained conform the OGBN benchmark. The hypothesis space $\mathcal{S}$ consists of fcn that all share the same topology and vary only in their concrete parameters. The topology consists of 3 gnn layers whose Agg function computes the mean of all colors of incoming neighbors. The Comb function applies a linear transformation to $G(v)$, applies a linear transformation to the result of the aggregation, and finally sums these two intermediates together. Each layer, except the final one, is followed by a batch normalisation as well ReLU to introduce non-linearity. The layers have dimensions $(128,256)$, $(256,256)$ and $(256,40)$, respectively. We apply a 50% dropout between layers during training. Softmax is applied after the last layer to turn feature vectors into subject areas, and we use cross-entropy as loss function. Unfortunately, the initial coloring in $\mathcal{P}_{1}$ is such that every node has a distinct color. Therefore, every node is in its own unique color refinement class, and compression is not possible. We therefore transform $\mathcal{P}_{1}$ into a problem $\mathcal{P}_{2}$ that can be compressed by first converting the 128-dimensional word embedding vectors into estimates of paper areas by means of a multilayer perceptron (MLP) that is trained on the nodes in $T$_but without having the graph structure available_. This hence yields an initial estimate of the paper area for each node. By learning a gnn on the graph that is colored by one-hot encodings of these initial estimates, the estimates can be refined based on the graph topology. We denote the new problem hence obtained by $\mathcal{P}_{2}$. Note that $\mathcal{P}_{2}$ has the same training set, hypothesis space, and loss function as $\mathcal{P}_{1}$. The MLP has input dimension $128$, one hidden layer of dimension $256$, and output layer of dimension $40$. We next compress $\mathcal{P}_{2}$ using $(c,d)$-reduction with $d=3$. The resulting compressed problems are denoted $\mathcal{P}_{3}^{c}$. The corresponding compressed graph sizes are shown in Table 3. We note that, because we consider labeled graphs here, the compression ratio is worse than the maximum-compression scenario of Section 4.1. We gauge the generalisation power of the learned fcn by computing the accuracy on the test set, determined by the OGBN benchmark, comprising 48603 nodes. We learn for 256 epochs with learning rate 0.01 on all problems. All experiments are run on an HP ZBook Furry G8 with Intel Core i9 11950H CPU, 32 GB of RAM and NVIDIA RTX A3000 GPU with 6 GB RAM. The results are summarized in Table 3. Comparing $\mathcal{P}_{1}$ with $\mathcal{P}_{2}$ we see that estimating the paper area through an MLP has marginal effect on the test accuracy, training time and memory consumption. Further comparing $\mathcal{P}_{2}$ with $\mathcal{P}_{3}^{c}$, which are equivalent by Theorem 3.11, we see that training accuracy is indeed comparable between $\mathcal{P}_{2}$ and $\mathcal{P}_{3}^{\infty}$; we attribute the difference in accuracy to the stochastic nature of learning. There is a larger difference in accuracy between $\mathcal{P}_{2}$ and $\mathcal{P}_{3}^{c}$ when $c=1$, i.e., when we compress based on bisimulation, than when $c>1$. This is because $c$-graded compression is an approximation, as explained in Section 3.4, and because, as we can see, $c$-graded compression for $c>1$ is nearly identical in size to ungraded compression. For $c>1$ we may hence expect there to be only marginal differences w.r.t. ungraded compression. Learning the compressed problems $\mathcal{P}_{3}^{c}$ is more efficient than learning on the uncompressed $\mathcal{P}_{2}$, taking only 81.5-84.3% of the learning time and 78.7-81.6% of the memory respectively which is comparable to the reduction in number of nodes when compressing. Our evaluation in this section is on a single learning problem; it should hence be interpreted as a preliminary insight that requires further evaluation. Based on these preliminary findings, however, we conclude that compressed learning can yield observable improvements in training time and memory consumption. Figure 4. Normalized reduction in nodes and edges using $d$-reduction (a and b) and $(c,3)$-reduction (c and d). ## 5. Conclusion and Future Work We have proposed a formal methodology for exact compression of GNN-based learning problems. While the attainable exact compression ratio depends on the input graph, our experiment in Section 4.2 nevertheless indicates that observable improvements in learning efficiency are possible, even when the compression in terms of the number of edges is negligible. In terms of future work, first and foremost our preliminary empirical evaluation should be extended to more learning tasks. Second, as we have seen $c$-graded color refinement offers a principled way of approximating exact compression, which becomes exact for hypothesis spaces of width $c$. It is an interesting question whether other existing approximate compression proposals (Garon et al., 2016; Goyal et al., 2017) can similarly be tied to structural properties of the hypothesis space. ## Acknowledgments We thank Floris Geerts for helpful discussions. S. Vansummeren and J. Steegmans were supported by the Bijzonder Onderzoeksfonds (BOF) of Hasselt University under Grants No. BOF20ZAP02 and BOF21D11VDBJ. This work was further supported by the Research Foundation Flanders (FWO) under research project Grant No. G019222N. We acknowledge computing resources and services provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government.
2308.08709	Dynamic Neural Network is All You Need: Understanding the Robustness of Dynamic Mechanisms in Neural Networks	Deep Neural Networks (DNNs) have been used to solve different day-to-day problems. Recently, DNNs have been deployed in real-time systems, and lowering the energy consumption and response time has become the need of the hour. To address this scenario, researchers have proposed incorporating dynamic mechanism to static DNNs (SDNN) to create Dynamic Neural Networks (DyNNs) performing dynamic amounts of computation based on the input complexity. Although incorporating dynamic mechanism into SDNNs would be preferable in real-time systems, it also becomes important to evaluate how the introduction of dynamic mechanism impacts the robustness of the models. However, there has not been a significant number of works focusing on the robustness trade-off between SDNNs and DyNNs. To address this issue, we propose to investigate the robustness of dynamic mechanism in DyNNs and how dynamic mechanism design impacts the robustness of DyNNs. For that purpose, we evaluate three research questions. These evaluations are performed on three models and two datasets. Through the studies, we find that attack transferability from DyNNs to SDNNs is higher than attack transferability from SDNNs to DyNNs. Also, we find that DyNNs can be used to generate adversarial samples more efficiently than SDNNs. Then, through research studies, we provide insight into the design choices that can increase robustness of DyNNs against the attack generated using static model. Finally, we propose a novel attack to understand the additional attack surface introduced by the dynamic mechanism and provide design choices to improve robustness against the attack.	Mirazul Haque, Wei Yang	2023-08-17T00:15:11Z	http://arxiv.org/abs/2308.08709v1	Dynamic Neural Network is All You Need: Understanding the Robustness of Dynamic Mechanisms in Neural Networks ###### Abstract Deep Neural Networks (DNNs) have been used to solve different day-to-day problems. Recently, DNNs have been deployed in real-time systems, and lowering the energy consumption and response time has become the need of the hour. To address this scenario, researchers have proposed incorporating dynamic mechanism to static DNNs (SDNN) to create Dynamic Neural Networks (DyNNs) performing dynamic amounts of computation based on the input complexity. Although incorporating dynamic mechanism into SDNNs would be preferable in real-time systems, it also becomes important to evaluate how the introduction of dynamic mechanism impacts the robustness of the models. However, there has not been a significant number of works focusing on the robustness trade-off between SDNNs and DyNNs. To address this issue, we propose to investigate the robustness of dynamic mechanism in DyNNs and how dynamic mechanism design impacts the robustness of DyNNs. For that purpose, we evaluate three research questions. These evaluations are performed on three models and two datasets. Through the studies, we find that attack transferability from DyNNs to SDNNs is higher than attack transferability from SDNNs to DyNNs. Also, we find that DyNNs can be used to generate adversarial samples more efficiently than SDNNs. Then, through research studies, we provide insight into the design choices that can increase robustness of DyNNs against the attack generated using static model. Finally, we propose a novel attack to understand the additional attack surface introduced by the dynamic mechanism and provide design choices to improve robustness against the attack. ## 1 Introduction Deep Neural Networks (DNNs) are used in multiple applications such as computer vision and natural language processing. After the rapid growth of IoT and embedded devices, many real-time systems use DNNs in their applications. As the real-time systems require faster response time and low energy consumption, researchers have proposed to incorporate energy-saving dynamic mechanism [34, 24, 36] to popular static DNN (SDNN) models like ResNet [20], VGG [31], MobileNet [22] etc. Early-exit is one of the dynamic mechanism techniques where multiple exits are included in SDNNs (creating multiple sub-networks), and SDNNs can terminate the operation early if a certain sub-network is confident about the prediction. These types of DNNs are named as early-exit Dynamic Neural Networks (DyNNs). Although the transition from SDNNs to DyNNs is preferred in real time systems because of increased efficiency, whether the use of dynamic mechanism will impact the robustness of the systems is still unknown. Studying the impact of the dynamic mechanisms on the robustness is important for developers or users to understand the trade-offs between DyNN and SDNN. In this work, we propose to investigate how robustness is impacted through the addition of dynamic mechanism to the static model and how different dynamic mechanism designs would impact the robustness in dynamic mechanism (Figure 1). We propose to investigate two aforementioned topics through three research questions. These three RQs are based on: _Robustness of Dynamic Mechanism, Robust Design for the Static Model Attack, and Robust Design for the Dynamic Mechanism Attack._ Robustness of Dynamic Mechanism. We focus on the investigation of robustness of the dynamic mechanism based on two aspects: transferability and the efficiency robustness. Figure 1: The overview of the objective of this work. First, we investigate the adversarial attack transferability between SDNNs and DyNNs to evaluate the robustness of the dynamic mechanism in black-box scenarios. In the black-box scenarios, adversaries normally assume the target models are always static. However, the target models can be dynamic also. Hence, it is important to find out if a surrogate SDNN model is used to attack a target DyNN model or vice-versa, then, to which extent the adversary can be successful. To address this issue, in this paper, we first conduct a comparative study on the adversarial attack transferability between SDNNs and DyNNs (Section 3). Our study results suggest that adversarial transferability from DyNNs to SDNNs is better and surprisingly, using DyNNs as surrogate models for attack seems to be a more efficient and more effective way to generate adversarial samples. The adversaries are able to generate more adversarial samples in the same amount of time compared to using SDNNs as the surrogate model, and the generated adversarial samples often can also attack SDNNs. The next robustness study is to understand how robust the efficiency of the dynamic mechanism is against the adversarial samples generated on SDNNs. This study tries to evaluate whether the original purpose of DyNNs (_i.e._, saving inference time) will be impacted by the adversarial samples (Section 3.2) generated through SDNNs. Our study results suggest that the adversarial samples generated by existing white-box attacks and black-box attacks do not increase the inference time significantly, making the dynamic mechanisms efficiently robust against attacks on SDNNs. Robust Design for the Static Model Attack Here, we perform a detailed analysis of which design choices in the dynamic mechanisms or DyNN architectures (specifically the position of early exits) may impact the robustness of DyNNs (Section 4). We consider two attack scenarios in this study: first, the output layer label of an SDNN is modified by a white-box adversarial example, and we study the impact of the example on corresponding DyNN's early-exit layers; second, in a black-box scenario, the output of SDNN is modified by a sample, and the sample is fed to separate model's DyNN. We have made multiple findings based on the empirical results, for example, putting the first exit earlier in the model architecture can help to improve the robustness of DyNNs. Robust Design for the Dynamic Mechanism Attack. Last but not least, we evaluate the design choices of dynamic mechanism against a novel attack on dynamic mechanism. First, we design an adversarial attack approach to understand the extra attack surface introduced by the dynamic mechanisms in neural network (Section 5). In this attack, the synthesized adversarial examples will not change the prediction of the final output layer's label, but will change the prediction of all the early exits. Based on the evaluation results, we find that the dynamic mechanism is more vulnerable in scenarios where dependency among DyNN layers is lesser and when the exits are sparse w.r.t the layers. Contribution. Our work provides insights about the robustness of the dynamic mechanism in DyNNs and also evaluates different design choices w.r.t robustness. The findings of this work would motivate developers/researchers to increase the usage of DyNNs and find more robust dynamic mechanisms. ## 2 Related Works and Background Dynamic Neural Networks. The main objective of DyNNs is to decrease the energy consumption of the model inference for inputs that can be classified with fewer computational resources. DyNNs can be classified into _Conditional-skipping DyNNs_ and _Early-exit DyNNs_. Early-exit DyNNs use multiple exits (sub-networks) within a single model and because of the model's working mechanism, the model is more suited for resource constrained devices. If, at any exit, the confidence score of the predicted label exceeds user defined threshold, inference is stopped. The resource-constrained devices usually deploy a lightweight sub-network of early exit network locally and resort to a server for further computations if needed [33]. [14], [12], [32], [24] have proposed Early-termination DyNNs. Specifically [24] and [38] propose early exit networks based on popular SDNNs. [38] also show that white-box robustness of the DyNNs is better than SDNNs. In addition to that multiple works [33, 29] provide practical usability of DyNNs. Figure 2 shows the working mechanism of Early-exit DyNNs. For example, an Early-exit DyNN has N parts and each part has an exit. $x$ is the input, $f_{out}^{i}$ represents prediction after the $i^{th}$ part (generated by specific computation unit), $f_{out}$ represents prediction of the Neural Network, $C_{i}$ represents confidence score after $i^{th}$ part, $Hid_{i}^{In}$ represents input of $i^{th}$ part, $Hid_{i}^{Out}$ represents output of $i^{th}$ part, and $\tau_{i}$ is the predefined threshold to exit the network after $i^{th}$ part. The working mechanism of the Early-exit DyNN can be represented as, $f_{out}(x)=f_{out}^{i}(x),$ if $C_{i}(x)\geq\tau_{i}$. Adversarial Attacks. Adversarial Examples are the input that can change the prediction of the DNN model when those are fed to the model. [13] propose Fast Gradient Sign Method (FGSM) that uses single-step first order entropy loss Figure 2: Working mechanism of Early-exit DyNN to generate adversarial inputs. This attack is modified by [27] to add initial noise to the benign sample. This attack is referred as projected gradient descent (PGD). Other than that, [10, 2, 9, 7] have proposed white-box attack methods, while [26, 1, 23, 37] have proposed black-box attack methods. Recently, attacks [16, 19, 17, 18, 15, 4, 7, 6, 3] have been proposed to decrease the efficiency of the dynamic neural networks. ## 3 How robust are DyNNs against adversarial examples? In this research question, we investigate the robustness of dynamic mechanism in DyNNs w.r.t adversarial examples generated on SDNNs. We further divide this question into two sub-parts: (i) attack transferability between SDNNs and DyNNs and (ii) efficiency robustness of DyNNs against adversarial examples generated on SDNNs. Through these RQs, we get an insight about how the existing adversarial attacks on SDNNs would impact the accuracy and efficiency of DyNNs. Also, we find out how adversarial examples generated on DyNNs impact the SDNNs w.r.t accuracy. As the SDNN models are static, adversarial examples would not have any impact on the efficiency of SDNNs. Is adversarial example transferability from DyNN to SDNN lower than adversarial example transferability from SDNN to DyNN? In this research question, we investigate the "transferability" of adversarial inputs generated based on SDNN and DyNN, respectively, _i.e.,_ whether adversarial examples generated based on SDNNs are adversarial to DyNNs and vice versa. Transferability is an important metric for evaluating the feasibility of black-box attack. To evaluate the transferability, one of the popular way [28, 26] is creating a similar model (_i.e.,_ surrogate model) as the target model. In a black-box attack, normally, adversaries assume the underlying model to be SDNN, so for a deployed DyNN, the adversaries may likely use an SDNN as surrogate model. Hence, this research question (RQ) is important to evaluate the robustness of DyNNs. #### 3.1.1 Experimental Setup. Dataset and Models. We use CIFAR-10 and CIFAR-100 [25] datasets for evaluation. For SDNNs, we use VGG-16 [31], ResNet56 [20], and MobileNet [22] model. As DyNNs, we use the early exit version of these models [24]. In all other RQs, we keep the models and dataset setup same. Black-box Attack. For the attack scenario, we use surrogate model [28, 26] based black-box attack scenario. Here, we feed a set of inputs to the target model and collect the output labels. These inputs are generated using 50% of the held-out validation data, and naturally corrupted versions of those validation data. As the number of partial held-out validation data is not significant, adding corrupted inputs would help to increase training data size for the surrogate model. For natural corruption [21], we use gaussian noise and brightness. For each type of corruption, we have five intensity levels. For example, if the number of held out data is 5000, for each corruption, we generate 25000 additional data. Once the input-output pairs are collected, a surrogate model is trained based on those pairs. For a target model, we use both SDNNs and DyNNs. If the target model is SDNN, then an DyNN is trained as surrogate model and vice-versa. To make the surrogate-target pairs, we use different types of DNN architectures. For example, if the the target model is DyNN VGG, we choose ResNet56 SDNN as the surrogate model. This assumption is valid because the attacker doesn't have information about the target model architecture, hence the possibility of choosing the same architecture as the surrogate model is less. We define two terms to represent two different types of transferability based on different types of surrogate model and target model: D2S transferability and S2D transferability. D2S transferability evaluates DyNN to SDNN attack transferability, where S2D transferability evaluates SDNN to DyNN attack transferability. We have chosen following pairs to evaluate S2D transferability: (SDNN ResNet56 (surrogate), DyNN VGG (target)), (SDNN MobileNet (surrogate), DyNN MobileNet (target)), (SDNN MobileNet (surrogate),DyNN ResNet56 (target)). Similarly, for D2S transferability, earlier mentioned surrogate models become the target model and earlier mentioned target models become the surrogate model. Algorithms. We use FGSM [13] and PGD [27] algorithms to attack the surrogate models. Metric. We measure percentage of adversarial examples that can mis-classify the output w.r.t number of generated adversarial examples as the attack success rate. #### 3.1.2 Evaluation Results Figure 3 shows the effectiveness of black-box attacks on DyNNs and SDNNs. On average, it can be noticed that for target SDNN and surrogate DyNN, the attack success rate is higher than the success rate of target AdNN and surrogate SDNN. One of the reasons for this behavior is the lower variance of the DyNNs. DyNNs use lower number of parameters, hence the feature space for adversarial samples of DyNNs is smaller than the feature space for adversarial samples of SDNNs [30]. Also, we find that for the target DyNN, FGSM attack performs better than PGD attack. If only dataset-specific results are considered, then for CIFAR-100 the attack success rate is higher than for CIFAR-10. Also, as the DyNN-generated adversarial inputs can attack SDNNs, then it can be time efficient to create adversarial inputs using DyNN. Through Figure 8 (Appendix), we can find the probability density plots of different exit numbers of DyNN that have been used to generate adversarial examples. Lower exit number suggests that lesser number of computations has been used to generate adversarial examples. It can be noticed that for PGD attack, more than 70% of adversarial examples are generated from exit 0, 1 and 2 (first - third exit). Although for FGSM attack, in a few scenarios (CIFAR-10 VGG, CIFAR-100 MobileNet, and CIFAR-100 ResNet), more than 50% of adversarial examples are generated through later exits (higher computation required). We also perform some additional experiments to strengthen our claim. In appendix, Figure 13 shows the experimental results on MI-FGSM attack, which is also a highly transferable attack. Also Figure 17 shows the results on Tiny-ImageNet dataset, which has a higher input size than CIFAR datasets. These two results show that our claim also holds for addditional attacks and datasets. Also Figure 14,15 and 16 in appendix shows adversarial examples generated through SDNNs and DyNNs. We find that both type of adversarial examples are similar w.r.t quality. Finding 1: _The D2S transferability is higher than the S2D transferability._ Finding 2: _Using DyNNs as surrogate models is more efficient and more effective way to generate adversarial examples than using SDNNs._ ### Does adversarial examples impact efficiency robustness in DyNNs?_ In this section, we investigate if the dynamic mechanism in DyNNs is robust w.r.t efficiency against the adversarial examples generated on static mdoel. Through the evaluation, we ensure whether the original purpose of including dynamic mechanism in DyNNs (_i.e.,_ saving inference time) will be impacted by the adversarial samples. Specifically, we study whether the adversarial inputs exit earlier or later in a DyNN compared to the original inputs. The main objective of this investigation is to find out whether the adversarial samples generated on SDNN can have an impact on the amount of computation of the DyNN. For this purpose, we conduct both white-box and black-box attacks to find out the impact of adversarial samples on the amount of computation. Here, the white-box attack scenario can also be considered a practical scenario. There have been studies [5, 35] that focus on reverse engineering of SDNN models from binary code of on-device models, but no techniques have been proposed to reverse engineer the dynamic mechanisms in the models. Hence, adversaries are more likely to get SDNN models instead of their dynamic counterparts. So it is important to find out how adversarial samples affect the efficiency of the DyNN for both white-box and black-box scenarios. #### 3.2.1 Experimental Setup. Attack. We use PGD and FGSM for both white-box and black-box attacks. For black-box setup, we use the same setup as previous RQ. For black-box scenario, we use DyNNs as target model and SDNNs as surrogate model. In a white-box setting, we attack the SDNN and evaluate on the performance of corresponding DyNN. Metric. We use the difference between the exit number selected by adversarial input and the exit number selected by benign input. If the difference is positive, then the latency of the adversarial sample is increased w.r.t benign input. #### 3.2.2 Evaluation Results. Figure 4 and Figure 11 (in Appendix) show the impact of adversarial examples generated on SDNN on changing in exit number in DyNN in a white-box setting. It can be observed that for the majority of the scenarios, accuracy-based adversarial samples do not increase the computation significantly in the DyNN. On average, FGSM-generated examples increase more computation in DyNNs than PGD-generated examples. For CIFAR-10 data, for MobileNet and VGG-16 DyNN models, 25%-37% of the FGSM generated examples could increase the number of exits by more than one. Also, it can be noted that adversarial samples generated on CIFAR-100 data is more likely to increase computation than adversarial samples generated on CIFAR-10. Especially, more than 45% of the FGSM samples generated on CIFAR-100 data can increase the number of exits by more than one. Figure 3: Transferable Attack Success Rate for CIFAR-10 and CIFAR-100 data Figure 5 and Figure 12 (in Appendix) show the impact of adversarial examples generated on SDNN on change in exits in DyNN in a black-box setting. It can be noted that, for CIFAR-10 dataset, black-box attack can generate more computation-increasing examples than in white-box attack. For ResNet and VGG model, more than 40% PGD attack generated examples can increase the number of exits by more than one. For all three models, 35% of the FGSM attack generated examples can increase the number of exits by more than one. FGSM attack generates more inputs that induces low confidence in early-exit layers than PGD attack. For CIFAR-100 dataset, the increase of computation caused by adversarial attacks is higher than that of CIFAR-10. As CIFAR-100 data uses larger model, the robustness of the model is reduced. For CIFAR-100 dataset, more than 40% examples generated through both the attack can increase the number of exits by more than one. However, increasing the number of exits by two or three exits does not decrease the efficiency in DyNNs significantly. Finding 3: _Accuracy-based adversarial samples do not decrease the efficiency significantly in the DyNN._ Finding 4: _The adversarial examples, whose output confidences are significantly lower, can perform better in terms of decreasing the efficiency in DyNNs._ Finding 5: _Adversarial examples generated on a larger model (w.r.t model parameters) is more likely to decrease efficiency in DyNNs._ ## 4 What design of DyNNs may impact the robustness against attack generated on SDNNs? In this section, we evaluate which architecture design choices (position of early exits) in DyNNs may impact the robustness of early layers against the adversarial inputs generated on SDNNs. In this RQ, we consider DyNNs as multi-exit networks, where each exit will provide an output. For evaluation, we assess if we attack the final exit (output of the static model), from which early-exit layer the label modification begins. If the output label is not modified in any earlier layer, then for that type of model, the robustness is higher because the model can produce correct results at any layer. This RQ will provide us an insight into which type of design choice may improve the robustness of early layers. ### Experimental Setup. Attack Setup. We use same attack setup as previous research questions. However, while we attack the DyNN, we do not consider one specific exit layer. Instead of that, we consider each exit layer. Metric. We use the early exit from which the label modification starts. For example, there are $N$ exits. First, $N$th exit's label is modified through adversarial sample. If till $K$th exit the original prediction was same, then we report $K+1$th exit in the experimentation. ### Evaluation Results. Figure 6 and Figure 9 (in Appendix) show probability density plot on which exit the output label is changed using the white-box adversarial examples. It can be observed that for all the model-dataset pairs, for more than 77% of the examples, the label is modified in the first exit. For CIFAR-10 data, only for MobileNet and ResNet models, the label is changed after the first exit for more than 20% of the examples (using FGSM attack). Figure 7 and Figure 10 (in Appendix) show probability density plot on which exit the output label is changed using the black-box adversarial examples. From results, we can see that the robustness of earlier exits is better against black-box attack than white-box attack. For CIFAR-10 data, more than 45% of the both attack generated samples could not misclassify the first exit for VGG-16 model. For CIFAR-100 data (larger model), robustness of early exit layers is worse than that of CIFAR-10 data (smaller model). For CIFAR-100 data, for both attacks and three models, less than 30% of the adversarial examples can not mis-classify the first exit. For black-box attack scenario, VGG-16 model's first layer on-average shows better robustness against black-box attack. In the DyNN, the first exit of the VGG model is placed only after second layer, while for others, more computations are performed before using the first exit. Hence, having the first exit in the early layers can increase the robustness of DyNN. Although lower number of parameters would be used to predict output, but with VGG we can notice that a significant number of inputs can be classified correctly through first exit. Finding 6: _We find that having the first exit in the earlier layers can increase the robustness of early exits of the DyNNs._ Finding 7: _Black-box attack success rate is lesser than white-box attack success rate against early-exit layers._ Finding 8: _For black-box scenario, early-exit layers are more robust against adversarial examples generated on a smaller surrogate model than adversarial examples generated on a larger surrogate model._ ## 5 What design of DyNNs may impact the robustness against attack on dynamic mechanism? In this section, we evaluate if specific examples can be generated only to understand the additional attack surface introduced by the dynamic mechanism in DyNNs and which design choices would be helpful in DyNNs to improve the robustness against this specific attack. First, we aim to design an attack such that the synthesized adversarial examples will not change the prediction of the final SDNN label, but change the prediction of all the early exits. This threat model is practical because the attack evades the existing detection that relies on the final output of SDNN while the attacker creates a situation where all the early exit networks do not provide correct prediction, therefore decreasing the usability of DyNNs. However, this attack is also challenging to be performed successfully because final layer output is dependent on the earlier exit layers and it is challenging to impact all the early exits' prediction without modifying the final exit's prediction. ### Problem Formulation We propose a novel attack technique called _Early Attack_ to evaluate the effectiveness of the early layers of DyNNs. Let's assume $f$ is an DyNN with $N$ exits. Given an input $x$, the output softmax layer in an exit $i$ can be defined as $y^{i}=f_{i}(x)$, where $i=1,2,3...N$. For synthesizing adversarial examples, we have two main objectives. First, the initial prediction in the $N^{th}$ exit (final layer) does not change. Let's assume, the initial prediction at final layer is $p$. Second, the prediction of all the other exits should be different than $p$. Based on the aforementioned objectives, we can propose Figure 4: Density plots of change in exit numbers because of PGD and FGSM attack (For CIFAR-10 data). The x axis represents the change in exit number while y axis represents the density. Figure 5: Density plots of change in exit numbers because of PGD and FGSM black-box attack (For CIFAR-10). The x axis represents the change in exit number while y axis represents the density. an iterative optimization procedure to optimise a loss function $L$. For each of the objectives, one loss function component is formulized. For the first objective, we propose loss function $L1=(-1\sum_{j\neq p}max(y_{p}^{N}-y_{j}^{N},0))$. In $L1$, we maximize the difference between softmax value of label $p$ and other label's softmax value, therefore the prediction won't get changed at the final exit. For the second objective, we propose loss function $L2=(\sum_{i=1}^{N}y_{p}^{i})$. The $L2$ ensures that for any other exit than the final exit, $p$'s softmax value would be minimized. The final loss function $L=\alphaL1+L2$. Here the $\alpha$ is a user-defined variable that provides balance between two loss terms. Finally, we need to ensure the added perturbation to generate adversarial examples are limited, hence, the final optimization function would be, $minimize(\|\|\delta\|\|+c\cdot L)$, where, $(x+\delta)\in[0,1]^{n}$. Here, $\delta$ represents the added perturbation and $c$ is a user-defined variable to provide weightage on a specific component. $c$ controls the magnitude of generated perturbation ($\|\|\delta_{i}\|\|$), where a large $c$ makes the loss function more dependant on the $L$. This constrained optimization problem in $\delta$ can be converted into a non-constrained optimization problem in $w$, where the relationship between $\delta$ and $w$ is: $\delta=\frac{\tanh(w)+1}{2}-x$ The $tanh$ function would ensure that the generated adversarial input values stay between 0 and 1. The equivalent optimization problem in $w$ is: \[\underset{w}{\text{minimize}}\ \left\|\left\|\frac{\tanh(w)+1}{2}-x\right\| \right\|+c\cdot L \tag{1}\] Algorithm 1 shows the optimization algorithm. The algorithm outputs the adversarial input $x^{\prime}$ given a benign image $x$ as input. $w$ is initialized to a random tensor that has equal shape as the input image. For each iteration, the loss function of the attack is computed (at line 11). Based on the back-propagated loss, the optimizer updates $w$ with its next value. Once the iteration threshold ($T$) is reached or the attack is successful, the algorithm computes and returns the adversarial input $x^{\prime}$ (at Line 9 and 15). ### Evaluation SetUp. Baseline. We use PGD and FGSM attacks as baseline to modify the early exit label. Metric. We use attack success rate as metric in the evaluation. If the adversarial input generated final layer output is same as the output generated by benign input and all the other exit layers output is different than the final output label, then we consider attack is successful for that particular adversarial input. Figure 6: Density plots representing during which exit number output label is changed because of PGD and FGSM attack (For CIFAR-10 data). The x axis represents the exit number while y axis represents the density. Figure 7: Density plots representing during which exit number output label is changed because of PGD and FGSM black-box attack (For CIFAR-10 data). The x axis represents the exit number while y axis represents the density. Hyperparameters. We use $\alpha=\{0.001,0.01,0.1,1,20,40\}$ and $c=50$ as hyperparameters. #### 5.2.2 Effectiveness Table 1 shows the evaluation results of the attack success rate of Early Attack and baseline techniques. Except two model-dataset pair, Early Attack's success rate is higher than 80% in all other scenarios. PGD and FGSM attacks are unsuccessful because the loss function only considers early layer output and does not consider final layer output. Although FGSM's attack success rate is higher than PGD's success rate. Also, w.r.t best performing $\alpha$ values, $\alpha=1$ performs well for 70% of the model-dataset pairs. We also analyze why early attack fails for VGG16 and MobileNet models for CIFAR-10 data. First, for CIFAR-10 data, final fully connected layer has lower number of parameters than the final layer for CIFAR-100 data. Hence, the dependency between the final layer and earlier layers is higher for CIFAR-10. However, for CIFAR-10 data, attack success rate is higher for ResNet model. First we discuss the failed cases for VGG16 and MobileNet model, and then we discuss why the attack succeeded for ResNet model. For VGG16 model, the failed examples can be divided into two types. For the first type, the final output label is changed with all the other early exit layers. In the second type, first few layers get the output label mis-classified, but along with the final layer, few previous layers also get the output correctly classified. For MobileNet, all the layers except the final and last to final layer get the output label mis-classified. For MobileNet, we also find that the final two exits are separated by only two layers, which is significantly lower than other model's separation layers between final two exits. Hence, the dependency between final two layers is higher. For ResNet model, there are 56 layers divided into 27 blocks. The exits are sparsely divided between these blocks. For Mobilenet and VGG models, the exit distributions are less sparse. Hence, the dependency between exits is lower for ResNet and because of that reason, we could successfully attack ResNet model. #### 5.2.3 Transferability In this section, we discuss about the transferability of the Early Attack examples. As Early Attack has two different components, instead of measuring attack success rate directly, we measure two parameters $T1$ and $T2$. $T1$ represents the percentage of inputs for which the final output remains same as the output generated by benign input. $T2$ represents from the examples selected from $T1$, how many early exit layers on average are mis-classified. Having both high $T1$ and $T2$ values ensures transferability. We show the transferability results in Table 2. For CIFAR-10 data, the $T1$ values are high, but $T2$ values are significantly low except for MobileNet to ResNet transferability. For CIFAR-100 data, $T2$ values are higher than of CIFAR-10 data, however, $T1$ values are low. From the results, we can notice that the generated examples either can keep the final layer label the same or can change the early exit layer outputs. Our finding suggests that early attack transferability is limited. Finding 9: _With increasing number of exits in DyNNs, the dependency between multiple exits will increase. Hence, more exits in DyNNs can increase the robustness against the Early Attack._ Finding 10: _Early Attack transferability between DyNNs is not significant._ ## 6 Conclusion In this work 1, we discuss the robustness of including dynamic mechanism in DNN through three research questions. We find out that DyNNs are more robust than SDNNs and also efficient to generate adversarial examples. We also propose DyNN design choices through final two RQs. Finally, we propose a novel attack to understand additional attack space in DyNNs. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|c\|} \hline \multirow{2}{}{Type} & \multirow{2}{}{SM} & \multirow{2}{}{MF} & \multicolumn{2}{c\|}{CIFAR-10} & \multicolumn{2}{c\|}{CIFAR-100} \\ \cline{3-7} & & & VGG & MNet & RNet & VGG & MNet & RNet \\ \hline \multirow{2}{}{$T1$} & VGG & – & 85\% & 73\% & – & 65\% & 39\% \\ \cline{2-7} & MNet & 86\% & – & 74\% & 51\% & – & 38\% \\ \hline \multirow{3}{}{$T2$} & RNet & 89\% & 82\% & - & 79\% & 75\% & - \\ \cline{2-7} & VGG & – & 0.73 & 1.65 & – & 1.58 & 2.69 \\ \cline{1-1} \cline{2-7} & MNet & 1.06 & – & 2.3 & 1.52 & – & 3.26 \\ \cline{1-1} \cline{2-7} & RNet & 0.95 & 0.85 & – & 1.32 & 1.28 & – \\ \hline \end{tabular} \end{table} Table 2: $T1$ and $T2$ values for measuring transferability between three models._TM_ represents Target Model and_SM_ represents Surrogate Model. $T1$ presents the percentage of inputs for which the final output remains same as the output generated by benign input.$T2$ represents from the examples selected from$T1$, how many early exit layers on average is mis-classified.* \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|} \hline \multirow{2}{}{Dataset} & \multirow{2}{}{Model} & \multicolumn{2}{c\|}{Early Attack} & best $\alpha$ & \multicolumn{1}{c\|}{FGD} & FGSM \\ \cline{3-6} & & VGG & 35 & 1 & 0 & 0 \\ \cline{2-6} & MobileNet & 11 & 0.1 & 0 & 0 \\ \cline{2-6} & ResNet & 81 & 1 & 0 & 0 \\ \hline \multirow{3}{}{CIFAR-100} & VGG & 86* & 20 & 0 & 1 \\ \cline{2-6} & MobileNet & 97 & 1 & 0 & 0 \\ \cline{1-1} \cline{2-6} & ResNet & 96 & 1 & 0 & 2 \\ \hline \end{tabular} \end{table} Table 1: Attack accuracy percentage of Early Attack and the baseline techniques against different model and dataset, along with $\alpha$ value.
2303.03260	On the Use of Neural Networks for Full Waveform Inversion	Neural networks have recently gained attention in solving inverse problems. One prominent methodology are Physics-Informed Neural Networks (PINNs) which can solve both forward and inverse problems. In the paper at hand, full waveform inversion is the considered inverse problem. The performance of PINNs is compared against classical adjoint optimization, focusing on three key aspects: the forward-solver, the neural network Ansatz for the inverse field, and the sensitivity computation for the gradient-based minimization. Starting from PINNs, each of these key aspects is adapted individually until the classical adjoint optimization emerges. It is shown that it is beneficial to use the neural network only for the discretization of the unknown material field, where the neural network produces reconstructions without oscillatory artifacts as typically encountered in classical full waveform inversion approaches. Due to this finding, a hybrid approach is proposed. It exploits both the efficient gradient computation with the continuous adjoint method as well as the neural network Ansatz for the unknown material field. This new hybrid approach outperforms Physics-Informed Neural Networks and the classical adjoint optimization in settings of two and three-dimensional examples.	Leon Herrmann, Tim Bürchner, Felix Dietrich, Stefan Kollmannsberger	2023-01-30T17:21:26Z	http://arxiv.org/abs/2303.03260v1	# On the Use of Neural Networks for Full Waveform Inversion ###### Abstract Neural networks have recently gained attention in solving inverse problems. One prominent methodology are Physics-Informed Neural Networks (PINNs) which can solve both forward and inverse problems. In the paper at hand, full waveform inversion is the considered inverse problem. The performance of PINNs is compared against classical adjoint optimization, focusing on three key aspects: the forward-solver, the neural network Ansatz for the inverse field, and the sensitivity computation for the gradient-based minimization. Starting from PINNs, each of these key aspects is adapted individually until the classical adjoint optimization emerges. It is shown that it is beneficial to use the neural network only for the discretization of the unknown material field, where the neural network produces reconstructions without oscillatory artifacts as typically encountered in classical full waveform inversion approaches. Due to this finding, a hybrid approach is proposed. It exploits both the efficient gradient computation with the continuous adjoint method as well as the neural network Ansatz for the unknown material field. This new hybrid approach outperforms Physics-Informed Neural Networks and the classical adjoint optimization in settings of two and three-dimensional examples. keywords: physics-informed neural networks, full waveform inversion, deep learning, adjoint state method, inverse problems, finite difference method + Footnote †: journal: _Engineering with Computers_. ## List of Symbols ### Motivation Since the emergence of Physics-Informed Neural Networks (PINNs) [1, 2, 3], they have been of great interest to the scientific community of computational mechanics. Despite their novelty, multiple review articles have already emerged [4, 5, 6]. The essence of PINNs is a variational approach, where a solution field is approximated with a neural network whose weights and biases are learned by minimizing the residual of a differential equation evaluated at collocation points. This general formulation allows for solving a large variety of problems that can at least partially be described by differential equations. At the same time, current machine learning frameworks like PyTorch [7] and TensorFlow [8] provide generic tools to construct PINNs with very little additional and rather simple code compared to the complexity of classical differential equation solvers. The method shows potential in settings where the amount of data available is insufficient to build supervised surrogate models, as e.g. presented in [9, 10, 11, 12, 13, 14, 15]. If used as a forward solver, PINNs are often computationally more expensive for many common differential equations which are efficiently solvable with established forward solvers [16, 17], as e.g. finite element [18] or finite difference methods [19]. However, in settings where classical methods show difficulties, PINNs are promising. Possible applications are high-dimensional differential equations [20, 21], surrogate modeling [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], accelerating traditional solvers [16], and inverse problems [25, 26, 27, 28, 29, 30, 31, 32]. The novelty in this paper is twofold. Firstly, it pinpoints the strengths and weaknesses of PINNs for inverse problems in comparison to classical full waveform inversion where the continuous adjoint state method [33, 34] is considered as the state-of-the-art reference method. To this end, a thorough investigation of the strengths and weaknesses of the individual components of each method is carried out by going from PINNs to the adjoint optimization step by step. The investigation is conducted on the scalar wave equation, popular in non-destructive testing [35, 36], and recently explored with PINNs in [25, 26]. In particular, the following three aspects will be examined extensively 1. the use of PINNs as a forward-solver, 2. neural networks as an Ansatz for the coefficient field of the PDE, 3. gradient computation. This dissection gives rise to the methods summarized in table 1 which are described and studied in detail in section 2 apart from the more general setting of PINNs described in the introductory section 1.3. Most of these methods naturally emerge as individual steps moving from a fully PINN-based approach to a classical adjoint optimization. By contrast, and as a second novelty, the paper at hand presents a hybrid method, which combines the strengths of the investigated approaches and delivers more accurate results. ### Full Waveform Inversion Full waveform inversion aims to non-destructively estimate the material distribution in a specimen. A conceptual illustration is provided in figure 1 with a rectangular domain $\Omega$ and a circular void $\Omega_{V}$, that is to be detected. Voids are identified using a fixed number of sources emitting a signal (red circles) and sensors (black circles) recording the image of the signal. The physics of this problem can be modeled with the scalar wave equation in an isotropic heterogeneous medium, which for the solution field $u(\mathbf{x},t)$, the density $\rho(\mathbf{x})$, the wave speed $c(\mathbf{x})$ and \begin{table} \begin{tabular}{l l l} \hline Method & Ansatz & Gradient Computation \\ \hline PINNs & $\hat{\gamma}(\mathbf{x})=A_{\gamma}(\mathbf{\theta}_{\mathbf{\gamma}};\mathbf{x})$ & \\ (section 1.3) & $\hat{u}(\mathbf{x},t)=A_{u}(\mathbf{\theta}_{\mathbf{u}};\mathbf{x},t)$ & $\nabla_{\mathbf{\theta}_{\mathbf{\gamma}}}\mathcal{L}$ with automatic differentiation \\ \hline PINNs with non-trainable & $\hat{\gamma}(\mathbf{x})=A_{\gamma}(\mathbf{\theta}_{\mathbf{\gamma}};\mathbf{x})$ & \\ forward operator & $\hat{u}(\hat{\gamma},\mathbf{x},t)=F[\hat{\gamma},\mathbf{x},t]$ & $\nabla_{\mathbf{\theta}_{\mathbf{\gamma}}}\mathcal{L}$ with automatic differentiation \\ (section 2.1) & $\hat{u}(\hat{\gamma},\mathbf{x},t)=F[\hat{\gamma},\mathbf{x},t]$ & $\nabla_{\mathbf{\theta}_{\mathbf{\gamma}}}\mathcal{L}$ with automatic differentiation \\ \hline Adjoint Method & $\hat{\gamma}(\mathbf{x})=C_{\gamma}(\mathbf{c},\mathbf{x})$ & $\nabla_{\mathbf{c}}\mathcal{L}$ with automatic differentiation \\ (section 2.2) & $\hat{u}(\hat{\gamma},\mathbf{x},t)=F[\hat{\gamma},\mathbf{x},t]$ & or $\nabla_{\mathbf{c}}\mathcal{L}$ with the adjoint method \\ \hline \hline Hybrid Approach & $\hat{\gamma}(\mathbf{x})=A_{\gamma}(\mathbf{\theta}_{\mathbf{\gamma}},\mathbf{x})$ & $\nabla_{\hat{\gamma}}\mathcal{L}$ with the adjoint method \\ (section 2.3) & $\hat{u}(\hat{\gamma},\mathbf{x},t)=F[\hat{\gamma},\mathbf{x},t]$ & and $\nabla_{\mathbf{\theta}_{\mathbf{\gamma}}}\hat{\gamma}$ with automatic differentiation \\ \hline \end{tabular} \end{table} Table 1: Investigated methods. the volume force $f(\mathbf{x},t)$ is given as \[\rho(\mathbf{x})u_{tt}(\mathbf{x},t)-\nabla\cdot(\rho(\mathbf{x})c(\mathbf{x})^{2}\nabla u(\mathbf{x},t))-f(\mathbf{x},t)=0\qquad\text{on }\Omega\times\mathcal{T}. \tag{1}\] Additionally, homogeneous Neumann boundary conditions and homogeneous initial conditions are prescribed: \[u_{x}(\mathbf{x},t) =0 \text{on }\Gamma_{y}\times\mathcal{T} \tag{2}\] \[u_{y}(\mathbf{x},t) =0 \text{on }\Gamma_{x}\times\mathcal{T}\] (3) \[u(\mathbf{x},0) =u_{t}(\mathbf{x},0)=0 \text{on }\Omega \tag{4}\] Parametrizing the inverse problem with an indicator function $\gamma$ by scaling the density leads to the most accurate reconstruction of voids and unknown homogeneous Neumann boundaries. The constant intact material has the density $\rho_{0}$ and $\gamma$ is constrained between $[\epsilon,1]$, where $\epsilon\ll 1$ is a non-zero lower bound: \[\rho(\mathbf{x})=\gamma(\mathbf{x})\rho_{0} \tag{5}\] In contrast, the wave speed is assumed to be constant $c(\mathbf{x})=c_{0}$. With this parametrization, the scalar wave equation takes the form: \[\gamma(\mathbf{x})\rho_{0}u_{tt}(\mathbf{x},t)-\nabla\cdot(\gamma(\mathbf{x})\rho_{0}c_{0 }^{2}\nabla u(\mathbf{x},t))-f(\mathbf{x},t)=0\qquad\text{on }\Omega\times\mathcal{T}. \tag{6}\] The volume force $f(\mathbf{x},t)$ emits the signal at the position $\mathbf{x}_{s}$. This is usually modeled by a spatial Dirac delta using a source term $\psi(t)$. \[f(\mathbf{x},t)=\psi(t)\delta(\mathbf{x}-\mathbf{x}_{s}) \tag{7}\] A typical source in the context of non-destructive testing is the sine burst with $n_{c}$ cycles, a frequency $\omega=2\pi f_{\psi}$, and an amplitude $\psi_{0}$, so that \[\psi(t)=\begin{cases}\psi_{0}\sin(\omega t)\sin(\frac{\omega t}{2n_{c}}),& \text{for }0\leq t\leq\frac{2\pi n_{c}}{\omega}\\ 0,&\text{for }\frac{2\pi n_{c}}{\omega}<t.\end{cases} \tag{8}\] This burst type is used in all examples in this paper. The goal of the inversion is to reconstruct a material distribution, i.e. the indicator $\hat{\gamma}(\mathbf{x};\mathbf{m})$ parametrized by $\mathbf{m}$, that best reproduces the $N_{\mathcal{M}}$ sensor measurements $u_{\mathcal{M}}(\mathbf{x}_{i},t)$ during a forward simulation which yields $\hat{u}(\mathbf{x},t;\mathbf{m})$. The inversion is posed as an optimization problem, where the cost function is the mean squared error between the prediction $\hat{u}$ and the measurements $u_{\mathcal{M}}$: \[\mathcal{L}_{\mathcal{M}}(\hat{\mathbf{u}},\mathbf{u}_{\mathcal{M}})=\frac{1}{2}\int_ {\mathcal{T}}\int_{\Omega}\sum_{i=1}^{N_{\mathcal{M}}}\biggl{(}\hat{u}(\mathbf{x},t;\mathbf{m})-u_{\mathcal{M}}(\mathbf{x}_{i},t)\biggr{)}^{2}\delta(\mathbf{x}-\mathbf{x}_{i} )d\Omega d\mathcal{T}. \tag{9}\] The gradient of $\mathcal{L}_{\mathcal{M}}$ w.r.t. the parameters $\mathbf{m}$ is expressed by $\nabla_{m}\mathcal{L}_{\mathcal{M}}(\hat{\mathbf{u}}(\mathbf{m}),\mathbf{u}_{\mathcal{M}})$ and computed with the adjoint state method. The gradient is then used in a gradient-based optimization procedure to minimize $\mathcal{L}_{\mathcal{M}}$ by iteratively updating the parameters $\mathbf{m}$: \[\mathbf{m}^{(j+1)}=\mathbf{m}^{(j)}-\alpha\nabla_{m^{(j)}}\mathcal{L}_{\mathcal{M}} \tag{10}\] in which $\alpha$ is the step size and the superindex $(j)=1,2,\dots$ is the iteration counter. This iterative process then yields an optimized estimate of the indicator $\hat{\gamma}(\mathbf{x};\mathbf{m})$. In case of multiple sources, an average of the mean squared errors $\mathcal{L}_{\mathcal{M}}$ and their gradients $\nabla_{m}\mathcal{L}_{\mathcal{M}}$ must be considered in the iteration update from equation (10). More details on the computation of the gradient using the adjoint method are found in [37]. For a more in depth explanation of the adjoint method for inverse problems, see Givoli [33]. ### Physics-Informed Neural Network The idea of PINNs is to use an artificial neural network $A$ to approximate either the solution $\hat{u}$ or and/or the coefficients of the differential equation $\hat{\gamma}$ using a collocation type approach. In the paper at hand, equation (6) is considered where the solution $\hat{u}$ is the wave field and $\rho_{0}$ is scaled by $\gamma(\mathbf{x})$ as given in equation (5). In the most general case, both $u$ and $\gamma$ are approximated by a neural network $A=(A_{u},A_{\gamma})$: \[\hat{u} = A_{u}(\mathbf{x},t;\mathbf{\theta}_{u}) \tag{11}\] \[\hat{\gamma} = A_{\gamma}(\mathbf{x},t;\mathbf{\theta}_{\gamma}) \tag{12}\] where $\mathbf{\theta}=\{\mathbf{\theta}_{u},\mathbf{\theta}_{\gamma}\}$ are the model parameters of the neural network and the hat symbol indicates that the values of $u$ and $\gamma$ are predictions from a neural network. In the remaining paper, in models using only one of the two neural networks, i.e. either $A_{u}$ or $A_{\gamma}$, the notation $\mathbf{\theta}$ without subindex is used as the corresponding neural network parameters. The initial boundary value problem defined by equations (2) to (4) and (6) is then solved by an optimization procedure which is similar to the one used in the classical full waveform inversion described in the previous section. To this end, the initial boundary problem is first cast into its residual form: \[\mathcal{N}[u;\gamma;\mathbf{x},t] = \gamma(\mathbf{x})\rho_{0}u_{tt}(\mathbf{x},t)-\nabla\cdot(\gamma(\mathbf{x}) \rho_{0}c_{0}^{2}\nabla u(\mathbf{x},t))-f(\mathbf{x},t) \tag{13}\] \[\mathcal{B}_{x}[u;\mathbf{x},t] = u_{x}(\mathbf{x},t)\] (14) \[\mathcal{B}_{y}[u;\mathbf{x},t] = u_{y}(\mathbf{x},t)\] (15) \[\mathcal{I}_{0}[u;\mathbf{x},t] = u(\mathbf{x},t)\] (16) \[\mathcal{I}_{1}[u;\mathbf{x},t] = u_{t}(\mathbf{x},t). \tag{17}\] The operators $\mathcal{N},\mathcal{B}_{x},\mathcal{B}_{y},\mathcal{I}_{0},\mathcal{I}_{1}$ are the residuals of the corresponding equations (2) to (4) and (6). These residuals are evaluated at a number of collocation points $N=N_{\mathcal{N}}+N_{\mathcal{B}_{x}}+N_{\mathcal{B}_{y}}+N_{\mathcal{I}_{0}} +N_{\mathcal{I}_{1}}$ using the current approximation of the solution coefficient fields given by equations (11) and (12). This requires the computation of the first and second derivatives of equations (11) and (12) w.r.t $\mathbf{x}$ and $t$ which is carried out by either using reverse-mode automatic differentiation [38] or numerical differentiation [10, 39]. The loss functional to be minimized by the overall optimization procedure, i.e. the cost function $\mathcal{L}$, is then defined as the weighted sum of squares of the residual evaluations (the differential equation residual $\mathcal{L}_{\mathcal{N}}$, the boundary condition residual $\mathcal{L}_{\mathcal{B}}$, and the initial condition residual $\mathcal{L}_{\mathcal{I}}$) at the collocation points: \[\mathcal{L}_{\mathcal{N}}(\mathbf{u},\mathbf{\gamma}) = \frac{1}{2N_{\mathcal{N}}}\sum_{i=1}^{N_{\mathcal{N}}}\lambda_{ \mathcal{N},i}(\mathcal{N}[u_{i};\gamma_{i}])^{2} \tag{18}\] \[\mathcal{L}_{\mathcal{B}}(\mathbf{u}) = \frac{1}{2N_{\mathcal{B}_{x}}}\sum_{i=1}^{N_{\mathcal{B}_{x}}} \lambda_{\mathcal{B}_{x},i}(\mathcal{B}_{x}[u_{i}])^{2}+\frac{1}{2N_{\mathcal{ B}_{y}}}\sum_{i=1}^{N_{\mathcal{B}_{y}}}\lambda_{\mathcal{B}_{y},i}( \mathcal{B}_{y}[u_{i}])^{2}\] (19) \[\mathcal{L}_{\mathcal{I}}(\mathbf{u}) = \frac{1}{2N_{\mathcal{I}_{0}}}\sum_{i=1}^{N_{\mathcal{I}_{0}}} \lambda_{\mathcal{I}_{0},i}(\mathcal{I}_{0}[u_{i}])^{2}+\frac{1}{2N_{\mathcal{ I}_{1}}}\sum_{i=1}^{N_{\mathcal{I}_{1}}}\lambda_{\mathcal{I}_{1},i}( \mathcal{I}_{1}[u_{i}])^{2}\] (20) \[\mathcal{L}(\mathbf{u},\mathbf{\gamma}) = \lambda_{\mathcal{N}}\mathcal{L}_{\mathcal{N}}(\mathbf{u},\mathbf{\gamma} )+\lambda_{\mathcal{B}}\mathcal{L}_{\mathcal{B}}(\mathbf{u})+\lambda_{\mathcal{I}} \mathcal{L}_{\mathcal{I}}(\mathbf{u}) \tag{21}\] To counteract imbalances in the magnitude of the loss terms $\mathcal{L}_{\mathcal{N}},\mathcal{L}_{\mathcal{B}},\mathcal{L}_{\mathcal{I}}$ usually each loss term is multiplied with a weight $\lambda$ individual to that loss term. The penalty weights can be computed automatically using attention mechanisms from deep vision [40, 41], where the penalty weights are updated via a maximization w.r.t. the cost function $\mathcal{L}$ resulting in a minimax optimization procedure. Expanding on this idea, as discussed in [42, 43], each evaluation at a collocation point is considered an individual equality constraint, thereby allocating one penalty weight per collocation point, as shown in the loss equations (18), (19), (20). This extension makes it possible to assign a greater emphasis on more important collocation points, i.e. points which lead to larger residuals. This yields an overall minimax optimization of the form \[\min_{\mathbf{\theta}}\max_{\mathbf{\lambda}}\mathcal{L}(\mathbf{\theta},\mathbf{\lambda}), \tag{22}\] This minimax optimization introduces the learning rate of the maximization $\alpha_{\max}$ as an additional hyperparameter to the learning rate for the minimization $\alpha_{\min}$. The optimization problem posed in equation (22) is solved using the gradients $\nabla_{\mathbf{\theta}}\mathcal{L}$ and $\nabla_{\mathbf{\lambda}}\mathcal{L}$ obtained via reverse-mode automatic differentiation. The Adam optimizer [44] is used in the paper at hand as an optimization procedure in all examples. However, more advanced optimizers such as combinations of Adam and L-BFGS [45] often yield a faster convergence (see, e.g. [46]). Important differences exist between the application of PINNs to either forward or inverse problems. These differences are summarized in table 2 using three problem categories: (a) Forward problems, (b) Inverse problems with full domain knowledge and (c) Inverse problems with partial domain knowledge. Problem type (a) - Forward Problem: In the forward problem, only the neural network $A_{u}$ from equation (11) is used as an approximator of the wave field $u(\mathbf{x},t)$, as $\gamma(\mathbf{x},t)$ is known. The gradients of the approximated field $\hat{u}$ are computed w.r.t. the inputs $\mathbf{x}$ and $t$, which enables the computation of the cost function $\mathcal{L}(\hat{\mathbf{u}},\mathbf{\gamma})$ from equation (21). The wavefield is then solved by performing the minimax optimization from equation (22). Variations of PINNs for the forward problem of the wave equation can be found in literature [47, 48, 49, 50]. With a minor modification, the framework of PINNs can also be applied to the inverse problem, i.e. the identification of the spatially varying indicator function $\gamma(\mathbf{x})$. Problem type (b) - Inverse problems with full domain knowledge of the solution field $u(\mathbf{x},t)$: Since $u(\mathbf{x},t)$ is known in the form of measurements $\mathbf{u}_{\mathcal{M}}$ everywhere, a neural network to predict $A_{u}$ is not needed. Instead, $\mathbf{u}_{\mathcal{M}}$ can directly be used to evaluate the cost function $\mathcal{L}$ from equation (21). Since the initial and boundary conditions are already approximately satisfied by the wave field, the cost function $\mathcal{L}(\mathbf{u}_{\mathcal{M}},\hat{\mathbf{\gamma}})$ is now solely composed of the residual term of the differential equation $\mathcal{L}_{\mathcal{N}}(\mathbf{u}_{\mathcal{M}},\hat{\mathbf{\gamma}})$. Therein, a neural network $A_{\gamma}$ is used to predict $\hat{\mathbf{\gamma}}$ and $u(\mathbf{x},t)$ is directly plugged into $\mathcal{L}_{\mathcal{N}}(\mathbf{u}_{\mathcal{M}},\hat{\mathbf{\gamma}})$ as $\mathbf{u}_{\mathcal{M}}$. A detailed description of solving this inverse problem with PINNs can be found in [25] which documents the state-of-the-art. Problem type (c) - Inverse problems with partial knowledge of the solution field $u(\mathbf{x},t)$: Full knowledge of the wave field, i.e. everywhere in $\Omega\times\mathcal{T}$ is rare in the context of non-destructive testing. Typically receivers can only be placed at parts of the boundary as depicted in figure 1 such that only partial knowledge is given. Yet, to compute the differential equation loss $\mathcal{L}_{\mathcal{N}}$ of equation (18), the wave field $u(\mathbf{x},t)$ and the scaling function $\gamma(\mathbf{x})$ must be known everywhere in $\Omega\times\mathcal{T}$. Thus, in the inverse problem with partial domain knowledge, the evaluation of the differential equation loss is computed by approximating both quantities with two separate neural networks, i.e. using both equation (11) and equation (12), as visualized in figure 2. The cost $\mathcal{L}(\hat{\mathbf{u}},\hat{\mathbf{\gamma}})$ is then composed of the boundary condition loss, equation (19), initial condition loss, equation (20), the differential equation loss $\mathcal{L}_{\mathcal{N}}(\hat{\mathbf{u}},\hat{\mathbf{\gamma}})$, equation (18), and the measurement loss $\mathcal{L}_{\mathcal{M}}(\hat{\mathbf{u}},\mathbf{u}_{\mathcal{M}})$, equation (9). Note, that the measurement data $\mathbf{u}_{\mathcal{M}}$ is now incorporated through \begin{table} \begin{tabular}{l l l} \hline Problem & Cost Function & Neural Networks \\ \hline (a) Forward Problem & $\mathcal{L}(\hat{\mathbf{u}},\mathbf{\gamma})$ & $\hat{u}_{i}=A_{u}(\mathbf{x}_{i},t_{i};\mathbf{\theta})$ \\ & $=\mathcal{L}_{\mathcal{N}}(\hat{\mathbf{u}},\mathbf{\gamma})+\mathcal{L}_{\mathcal{B} }(\hat{\mathbf{u}})+\mathcal{L}_{\mathcal{I}}(\hat{\mathbf{u}})$ & \\ \hline (b) Inverse Problem & $\mathcal{L}(\hat{\mathbf{u}}_{\mathcal{M}},\hat{\mathbf{\gamma}})$ & $\hat{\gamma}_{i}=A_{\gamma}(\mathbf{x}_{i},t_{i},\mathbf{\theta})$ \\ (Full Domain Knowledge) & $=\mathcal{L}_{\mathcal{N}}(\mathbf{u}_{\mathcal{M}},\hat{\mathbf{\gamma}})$ & \\ \hline (c) Inverse Problem & $\mathcal{L}(\hat{\mathbf{u}},\mathbf{u}_{\mathcal{M}},\hat{\mathbf{\gamma}})$ & $\hat{u}_{i}=A_{u}(\mathbf{x}_{i},t_{i};\mathbf{\theta}_{u})$ \\ (Partial Domain Knowledge) & $=\mathcal{L}_{\mathcal{M}}(\hat{\mathbf{u}},\mathbf{u}_{\mathcal{M}})+\mathcal{L}_{ \mathcal{N}}(\hat{\mathbf{u}},\hat{\mathbf{\gamma}})$ & and $\hat{\gamma}_{i}=A_{\gamma}(\mathbf{x}_{i},t_{i};\mathbf{\theta}_{\gamma})$ \\ & $+\mathcal{L}_{\mathcal{B}}(\hat{\mathbf{u}})+\mathcal{L}_{\mathcal{I}}(\hat{\mathbf{u}})$ & \\ \hline \end{tabular} \end{table} Table 2: Overview of differences in the PINN formulation between forward and inverse problems. the measurement loss, evaluated with the prediction of the forward solution $\hat{\mathbf{u}}$. The current state-of-the-art of this inverse problem with PINNs is given e.g. in [26] with a representative Python code provided in [51]. ## 2 Effect of Neural Networks in the Inversion To investigate the strengths and weaknesses of PINNs for full waveform inversion, the PINN and its training procedure as illustrated in figure 2 is incrementally modified until it is indistinguishable from the classical adjoint optimization. The work in [25, 26] discusses the use of PINNs for full waveform inversion, where fully-connected neural networks with inputs $\mathbf{x}$ and $t$ were employed. By constrast, in the paper at hand convolutional neural networks are employed, since their spatial equivariance leads to a significant speed-up in training and faster convergence. This has already been reported in a more general context in computational mechanics in e.g. [10, 39] and is not the main focus of the paper at hand. This modification additionally enables a better transition and comparison to the presented methods and is not seen as a detriment to the PINN methodology. Despite this minor difference, we will still consider [26] as the reference case for what is possible within the current state-of-the-art using pure PINNs and only partial domain knowledge. ### Physics Informed Neural Networks with non-trainable Forward Operator The first modification of PINNs visualized in figure 2 is to replace the neural network $A_{u}(\mathbf{\theta})$, whose task is to learn the forward solution, with a non-trainable forward operator $F$ with $\hat{\gamma}$ as additional input. The new structure of a Physics-Informed Neural Network with a non-trainable Forward Operator is illustrated in figure 3. The forward operator could be any type of surrogate model such as pre-trained neural networks, or classical solution methods such as the finite element method [18] or the finite difference method [19]. An important computational requirement on the forward operator $F$ is the viability of a GPU-based implementation, which is an advantage of pre-trained neural networks over CPU-based solvers. However, finite difference methods are commonly also accelerated with GPUs, as e.g. in [52]. Therefore, finite differences are used in the sequel as the forward operator $F$. A basic but relatively efficient implementation in PyTorch is used by exploiting its convolutional network infrastructure as stencils. This results in simulation runs which take about $500\upmu$s wall clock time for all 1200 timesteps on a $252\times 124$ grid1. Details are laid out in Appendix A. Footnote 1: tested on an Nvidia A100-SXM4-40GB The introduction of a non-trainable forward operator should, in the ideal case, lead to very small values of the loss components related to the initial-boundary value problem, i.e. $\mathcal{L}_{\mathcal{N}},\mathcal{L}_{\mathcal{B}},\mathcal{L}_{\mathcal{I}}$. Typically, these are only approximately zero due to modelling and discretization errors. Their Figure 2: Prediction and training of PINNs for inversion with the scalar wave equation. Predictions of the wavefield $\hat{u}$ and indicator $\hat{\gamma}$ are obtained through the neural networks $A_{u}(\mathbf{\theta}_{u})$ and $A_{\gamma}(\mathbf{\theta}_{\gamma})$. These are updated during training with the gradients of the cost function $\mathcal{L}$ obtained by automatic differentiation. contribution is, however, still negligible in the presence of the measurement loss $\mathcal{L}_{\mathcal{M}}$. For this reason, they can be eliminated from the cost function $\mathcal{L}$, such that it is now solely composed of the measurement loss $\mathcal{L}_{\mathcal{M}}$. The introduction of a non-trainable forward operator leads to a significantly less complex optimization task. In the case of a time-stepping scheme as forward operator $F$, the disadvantage of the design depicted in figure 3 is a more expensive sensitivity computation $\nabla_{\boldsymbol{\theta}}\mathcal{L}(F(\boldsymbol{\hat{\gamma}}))\ =\ \nabla_{ \boldsymbol{\hat{\gamma}}}\mathcal{L}(F(\boldsymbol{\hat{\gamma}}))\cdot \nabla_{\boldsymbol{\theta}}\boldsymbol{\hat{\gamma}}(\boldsymbol{\theta})$ because the backpropagation needs to be piped through each timestep of the forward operator $F$ by means of automatic differentiation to compute $\nabla_{\boldsymbol{\hat{\gamma}}}\mathcal{L}(F(\boldsymbol{\hat{\gamma}}))$. The cost of computing the gradient of the inverse field w.r.t. the network parameters $\nabla_{\boldsymbol{\theta}}\boldsymbol{\hat{\gamma}}(\boldsymbol{\theta})$ is unchanged. ### Adjoint Method The next step is to replace the neural network Ansatz of the indicator function $\gamma$ with a piecewise polynomial Ansatz. Here, a piece-wise constant Ansatz is selected and defined in the one-dimensional case as $\hat{\gamma}(x)=\sum_{i=1}^{N_{c}}c_{i}N_{i}(x)$ with \[N_{i}(x)=\begin{cases}1\text{ if }x_{i}\leq x<x_{i+1}\\ 0\text{ else.}\end{cases} \tag{23}\] For the two-dimensional and three-dimensional cases the discretization is handled in the same way for the additional dimensions. This modification is depicted in figure 4. The coefficients $\boldsymbol{c}=\{c_{i}\}_{i=1}^{N_{c}}$ are still updated with the Adam optimizer using the gradients obtained through automatic differentiation. The reason for choosing a piecewise constant Ansatz over a piecewise higher-order polynomial is that a piecewise constant Ansatz is already capable of representing the jump of going from a constant material to a void, as is the case in all examples presented in section 3. Without the neural network, the Sigmoid activation function is missing and the range of $\hat{\gamma}$ has to be constrained in a different manner. To this end, an additional weight, i.e. coefficient clipping is used, where the absolute value is clipped to the range of a small value $\epsilon$ close to zero and an upper bound one or slightly exceeding one [37]. Given equation (23), the sensitivity w.r.t. the coefficients \[\nabla_{\boldsymbol{c}}\mathcal{L}(F(\boldsymbol{\hat{\gamma}}))\ =\ \nabla_{ \boldsymbol{\hat{\gamma}}}\mathcal{L}(F(\boldsymbol{\hat{\gamma}}))\cdot \nabla_{\boldsymbol{c}}\boldsymbol{\hat{\gamma}}(\boldsymbol{c}) \tag{24}\] can either be computed via automatic differentiation or the continuous adjoint method, as indicated in figure 4. In the continuous adjoint method, the gradient is derived analytically before its discretization, as e.g. derived in [37]. This third modification has transformed the approach into the classical full waveform inversion based on finite differences. No major implications on the accuracy are expected due to this modification, as the gradients are the same, apart from numerical errors introduced by the finite difference discretization in the adjoint method. These originate in the computation of the Frechet kernel \[K_{\gamma}=\int_{\mathcal{T}}-\rho_{0}u_{t}^{\dagger}u_{t}+\rho_{0}c_{0}^{2} \nabla u^{\dagger}\cdot\nabla udt \tag{25}\] Figure 3: PINN with non-trainable forward operator. The modification of the original PINN from figure 2 is marked in red. Instead of learning the forward problem, a non-trainable forward operator $F$ with $\hat{\gamma}$ as additional input is used. Simultaneously, this leads to a simplification of the cost function $\mathcal{L}$, as $\mathcal{L}_{\mathcal{N}},\mathcal{L}_{\mathcal{B}},\mathcal{L}_{\mathcal{I}}$ are approximately satisfied due to $F$. used to compute the gradient $\nabla_{\hat{\gamma}}\mathcal{L}_{\mathcal{M}}$ via the integration of $K_{\gamma}$ over the spatial and time domains (c.f. [37] for more details). To compute equation (25), the temporal and spatial gradients of the solution field $u$ and the adjoint field $u^{\dagger}$ are computed with finite differences. Note, that the sensitivity w.r.t. the coefficients $c_{i}$ is computed in a second step, by multiplication with the corresponding shape function $N_{i}(x)$, obtained by deriving equation (23) w.r.t. $c_{i}$, i.e. $\nabla_{\mathbf{c}}\hat{\gamma}(\mathbf{c})$. In fact, the key difference between the adjoint method and automatic differentiation is the order in which the gradient computation and discretization of the state equation is performed. In the continuous adjoint method, the gradient is first derived analytically, and thereupon the state equation is discretized as described above. By contrast, in the case of automatic differentiation, the differential equation is discretized first, and the gradient is propagated through each timestep. Thus, automatic differentiation bears a great similarity to the discrete adjoint method and is, as stated in [53], fundamentally the same approach. Manually deriving the sensitivities in the discrete adjoint or computing them through reverse mode automatic differentiation as in [53, 54] will not result in a difference in accuracy. Thus, regarding the accuracy, the difference between automatic differentiation and the continuous adjoint method is only affected by the difference between the discrete and continuous adjoint method. Unlike the accuracy, the computational cost of the automatic differentiation approach compared to the continuous adjoint method is different. Automatic differentiation adds a significant memory cost due to the backpropagation. In addition, the backpropagation along all timesteps is costly with automatic differentiation, as also mentioned in the work on the ordinary differential equation networks [55]. The theoretical complexity of the backpropagation and continuous adjoint method w.r.t. the number of timesteps is in both cases $O(n)$[56] and in practice only a factor of roughly two is observed between the computation times. To save memory, a truncated backpropagation through time [56, 57] can be used, as typically carried out for recurrent neural networks. However, a truncation was not used in the examples investigated in section 3 since it was found to be detrimental to the learning behavior of the indicator function. ### Hybrid Approach This section introduces a hybrid approach which combines the advantages of a neural network Ansatz for the scaling function $\gamma$ and the more efficient gradient computation of the gradient of the loss functional equation (9) using the continuous adjoint method equation (25). To this end, the neural network is reintroduced as an Ansatz for $\gamma$. In analogy to equation (24), the gradient of the cost function can be computed with the chain rule but must now be computed with respect to the neural network parameters $\mathbf{\theta}$: \[\nabla_{\mathbf{\theta}}\mathcal{L}(\mathbf{\hat{u}}(\mathbf{\hat{\gamma}}(\mathbf{\theta})) )=\nabla_{\mathbf{\hat{\gamma}}}\mathcal{L}(\mathbf{\hat{u}}(\mathbf{\hat{\gamma}}))\cdot \nabla_{\mathbf{\theta}}\mathbf{\hat{\gamma}}(\mathbf{\theta}) \tag{26}\] The sensitivity of the cost function w.r.t. the neural network outputs, i.e. the indicator field, $\nabla_{\mathbf{\hat{\gamma}}}\mathcal{L}$ is efficiently computed with the continuous adjoint method equation (25), while the sensitivity of the indicator field w.r.t. the neural network parameters $\nabla_{\mathbf{\theta}}\mathbf{\hat{\gamma}}$ is obtained via backpropagation through the neural network $A_{\gamma}(\mathbf{\theta})$. The hybrid approach results in similar computational times as Figure 4: Adjoint method. The modification of the previously modified PINN from figure 3 is indicated in red. The neural network Ansatz is replaced by a constant Ansatz parametrized by the coefficients $\mathbf{c}$. the adjoint method for a sufficient number of timesteps. The scheme is illustrated in figure 5. A similar scheme was recently proposed in the context of topology optimization [58]. In addition to saving memory compared to the scheme discussed in the previous section, the hybrid scheme bears the advantage of separating neural networks from forward solvers such that an efficient GPU implementation of the forward solver is no longer strictly necessary. Instead, the neural network can provide the indicator prediction $\mathbf{\hat{\gamma}}$ and its sensitivity w.r.t. the parameters $\nabla_{\mathbf{\theta}}\mathbf{\hat{\gamma}}$ on a CPU while, the forward solver can also be executed independently on a CPU. The increase in computational effort of evaluating and updating the neural network on a CPU instead of a GPU is very small compared to the cost of the forward solver. Furthermore, the forward solver can even be a completely independent black-box solver which provides the wave-field $\hat{u}$ and the sensitivity of the loss functional $\nabla_{\hat{\gamma}}\mathcal{L}$. ## 3 Configuration Two model problems with increasing complexity are considered. First, a two-dimensional plate is used to determine the strengths and weaknesses of neural networks in the inversion process. Afterwards, a more complex three-dimensional case is used to compare the proposed hybrid approach to the adjoint method. The forward operator is always a finite difference solver. To avoid an inverse crime [59], the reference data was generated with a boundary-conforming finite element solver for the partial domain knowledge cases (Voxel-FEM for the three-dimensional case). However, in the case of the two-dimensional full domain knowledge case the finite cell method [60] is employed for the reference data generation. In all studies, the sine burst, presented in equation (8) is applied with $n_{c}=2$ cycles, a frequency of $f_{\psi}=500$ kHz, and an amplitude of $\psi_{0}=10^{12}$ kg(ms)${}^{-2}$. Special care must be taken when using finite differences during the source application in equation (7). The Dirac delta has to be scaled by a factor of $1/\|\|d\mathbf{x}\|\|_{2}^{2}$ to account for the grid spacings [61]. Additionally, both cases share the same material with a density of $\rho_{0}=2700$ kg/m${}^{3}$ and a wave speed of $6000$ ms${}^{-1}$. ### Two-Dimensional Case The two-dimensional case is strongly inspired by [37], where a circular hole with a diameter of 5 mm is to be detected on a 100 mm $\times$ 50 mm rectangular plate, illustrated in figure 6(a). We do not provide specific information about the hole to the machine learning algorithms, which implies the entire material distribution $\gamma(\mathbf{x})$ has to be estimated. While the geometry, material, and source type are identical, the sensor and source locations are slightly different from [37]. In total, 54 sensors and four sources are used on the top edge in four separate measurements. The exact configuration is provided in figure 6, showing the upper left edge of the domain using the domain definitions from figure 1. The setup is symmetric along the vertical midline indicated by the dashed grey Figure 5: Hybrid approach. Modification of the previously modified PINN from figure 3 is marked in red. The gradient computation is now split into two parts. First, the sensitivity of the cost function $\mathcal{L}$ w.r.t. the predicted indicator field $\hat{\gamma}$ is computed with the adjoint method using equation (25), whereupon the sensitivity of the indicator field $\hat{\gamma}$ w.r.t. the neural network parameters $\mathbf{\theta}$ is found with the automatic differentiation used to train the neural network. This yields the desired gradient $\nabla_{\mathbf{\theta}}\mathcal{L}$ by using the chain rule equation (26). line. The slight peculiarity of the sensor and source locations is due to them being aligned to the finite difference nodes and neural network grid points. This is, however, not a limitation of the presented methods, as it can easily be circumvented by, e.g. using Kaiser windowed sinc functions as interpolation of the sources [61] and linear interpolation for the sensors. A forward simulation is illustrated in figure 7, where the third source emits a pulse at $t=0$ s, which then leads to a disturbance of the wavefield in the center due to the circular void. The finite difference simulation for the inversion uses $252\times 124$ grid points and two ghost cells per dimension, a lower indicator function bound of $\epsilon=10^{-5}$, and a timestep size of $\Delta t=3\cdot 10^{-8}$ s${}^{-1}$ for 1200 timesteps. ### Three-Dimensional Case The purpose of the three-dimensional case is to showcase the possible strengths of neural networks when the adjoint optimization is used in a setting where it does not perform well anymore. For this, two cuboid slices from a CT-scanned drill core [62] are considered and illustrated in figure 8. The edge length is 20 mm and discretized by 92 grid points in each dimension during the inversion, where 1200 timesteps with a timestep size of $\Delta t=1.8\cdot 10^{-8}$ s${}^{-1}$ are considered. Again a lower indicator function bound of $\epsilon=10^{-5}$ is used, while the upper bound is increased to 1.2 for the adjoint method, as this was found to be beneficial. The sources and sensors are applied to the backside of the cubes, as indicated by the darker shading on the right of figure 8. The four sources are placed at a distance of 5.27 mm from each edge, while the sensors coincide with the grid points of the plane. The case was selected such that a single forward and adjoint simulation can be performed within the limits of the 40 GB GPU memory. To compute larger three-dimensional, efficient parallelizations of the simulation have to be used, which can either be implemented on GPU clusters or alternatively on CPUs using efficient communication between the GPU and CPU for the neural network. Figure 6: Symmetrical sensor (black circles) and source (red circles) configuration. Figure 7: The wavefield is computed with the underlying material distribution, depicted in the leftmost figure, with the third source. When the waves reach the defect, a disturbance in the wavefield can be seen. ### Neural Network Architecture The neural network considered as Ansatz for the indicator function $\hat{\gamma}$ has a generator structure with pixel-wise normalization inspired by generative adversarial networks [63]. It upsamples a random three-dimensional matrix of size $128\times 8\times 4$ with subsequent convolutional filters. The upsampling and convolutions are repeated five times to get the desired resolution for the indicator function $\hat{\gamma}$, see figure 9 for the neural network used for the two-dimensional case. A similar network is employed in the three-dimensional case using three-dimensional convolutions and a four-dimensional random input matrix of size $256\times 3\times 3\times 3$. To increase the ratio between the number of indicator voxels and network parameters, twice as many filters are employed in the three-dimensional case. For more details on both networks, c.f. Appendix B. Nearest-Neighbor interpolation was used for upsampling, as it performed better than polynomial interpolation, as it preserves discontinuities needed to represent the voids. Due to their convergence accelerating behavior [64], adaptive activation functions are used. Specifically, PReLU [65] was used for all layers except the last layer, where an adaptive Sigmoid was used to ensure an output in the range $[0,1]$. The output was then rescaled to the range $[\epsilon,1]$ to ensure numerical stability. Optimization is always conducted with Adam [44] to make the comparison more straightforward, although L-BFGS leads to fewer iterations. Gradient clipping [66, 67, 68] was found to be essential to ensure an early convergence. Without it, a plateau was observed, followed by a strongly delayed breakthrough. In addition, a learning rate scheduler with a polynomial decay $(b\cdot\text{epoch}+1)^{a}$ was employed with the values $a=-0.5$, $b=0.2$. The initial network weights are set with Glorot initialization [69]. Figure 8: The two cases extracted from a CT-scanned drill core [62]. The sources and sensors are applied on the backside of the cubes indicated by the darker surface on the right. Figure 9: The convolutional neural network for the two-dimensional inversion case. It performs upsampling and convolutions in a repeating pattern on a random input of size $128\times 4\times 8$, yielding a prediction $\hat{\gamma}$ of size $1\times 126\times 254$. For more details, see Appendix B. ## 4 Numerical Results ### Two-Dimensional Case In the subsequent sections, the individual components of PINNs will be investigated and compared to the classical adjoint optimization using the case presented in section 3.1. All simulations are performed on an Nvidia A100 SXM4 40GB GPU and the same convolutional neural network presented in figure 9 and Appendix B. #### 4.1.1 Physics-Informed Neural Networks The results presented in this section refer to case PINNs, table 1, whose methodology is described in section 1.3. The various inversion tasks presented in [26] ran for about $400\,000-600\,000$ epochs with computational times of about $0.2$ s per epoch2. The need for such a large number of epochs illustrates the complexity of the nested optimization task of simultaneously learning the forward and inverse solution for a single problem. Additional complications arise when multiple sources are applied, and a forward operator taking the different sources into account must be learned. In [26], this problem was circumvented by using source stacking [70], such that only one forward simulation had to be learned. If only one field has to be learned, as in the case of inversion with full domain knowledge, as also presented in [25], the complexity of the optimization task decreases. To illustrate the decrease in complexity and the capabilities of PINNs in specialized settings, the problem presented in section 3.1 is solved with a PINN using additional sensors. The sensors are placed throughout the entire domain, coinciding with the finite difference grid points. As the measurements are only given at discrete points, the residual of the differential equation (18) is evaluated numerically using finite differences for the derivative terms. Additionally, the indicator function is prescribed at the boundaries, i.e. set to one to ensure material at the boundaries, which was essential to retrieve an accurate inversion. The resulting inversion after 100 epochs is provided in figure 10 with a clear identification of the circular void and only minor artifacts. It was essential to prescribe the density $\rho(\mathbf{x})$ at the boundary and use a relatively large learning rate $\alpha_{\min}=2\cdot 10^{-3}$. Furthermore the use of the learnable penalty weights using the attention mechanism was helpful with a learning rate of $\alpha_{\max}=2\cdot 10^{-2}$. Footnote 2: Based on the Python script provided by [51] and tested on an Nvidia A100-SXM4-40GB GPU. It is important to emphasize that the improvement from $400\,000-600\,000$ epochs to 100 epochs is a result of the simplification of the physical problem by using full domain knowledge, which should not be taken as a reference of what is possible with PINNs in the partial domain knowledge case Figure 10: Prediction and error of the material distribution after 100 epochs using a PINN with full domain knowledge. Average elapsed time per epoch is $0.51$ s. presented in section 3.1. Instead, $400\,000-600\,000$ epochs are to be considered as the reference of the capabilities of PINNs for a similar inversion task. #### 4.1.2 Physics-Informed Neural Networks with non-trainable Forward Operator An alternative to reducing the complexity of learning both the forward and inverse solutions is using a non-trainable forward operator classified in table 1 as PINNs with non-trainable forward operator and described in section 2.1. The neural network responsible for the forward solution is now replaced by a finite difference scheme. A reasonable inversion, shown in figure 11, is reached after only 50 epochs. Thus the convergence w.r.t. iterations is even faster than the PINN with full domain knowledge, with the detriment of a greater elapsed time per epoch at 2.53 s. Overall, considering the $400\,000-600\,000$ epochs observed in [26] given only partial domain knowledge, the PINN is outperformed by using a non-trainable forward operator. For the optimization a learning rate of $\alpha_{\min}=2\cdot 10^{-3}$ was employed with gradient clipping. The learnable penalty weights improved the results slightly, but were not found to be essential. #### 4.1.3 Adjoint Method Next, a performance evaluation of the adjoint method is provided in table 1. Section 2.2 then contains more details and a special focus on the importance of the Ansatz and the gradient computation. A piece-wise constant Ansatz is considered instead of the neural network using automatic differentiation for the gradient computation. The void is clearly identified in the prediction after 50 epochs, shown in figure 12, and achieved with a slightly increased learning rate of $\alpha_{\min}=6\cdot 10^{-2}$. However, additional oscillatory artefacts are observed. This coincides with the observations in [37], where linear shape functions are used as the Ansatz of the unknown material field. However, interestingly changing the Ansatz of $\gamma$ to a neural network leads to a very different and better local minimum. Specifically, the mean squared error is reduced from $3.2\cdot 10^{-3}$ using the piecewise constant Ansatz for $\gamma$ as in equation (23) to $9.5\cdot 10^{-4}$ using a neural network as in equation (12). Furthermore, the smoothness of the solution is increased in regions where the true solution is constant, while the sharp gradients around the circular void are retained as depicted in figure 11 and quantified in Appendix C. It is noteworthy that this increase in accuracy is achieved without a significant increase in computational effort. However, one major difference accompanying the change in Ansatz must be mentioned. The initial field of the constant Ansatz is chosen to be equal to one throughout the entire domain, which in the neural network case is impossible without pre-training. Pre-training to learn a constant field of one with a large regularization on the neural network was not found to be helpful. Thus the neural network starts with a random field, as later shown in the prediction histories in figure 16. However, Figure 11: Prediction and error of the material distribution after 50 epochs using automatic differentiation and a neural network Ansatz. Average elapsed time per epoch is 2.53 s. introducing a random field as initialization for the constant Ansatz is found to be detrimental, observable in an unsuccessful convergence toward a meaningful inversion. Thus, the strength of the neural network Ansatz must lie in its non-linear connectivity and strong over-parametrization, $526\,252$ parameters compared to the $252\times 124=31\,248$ parameters for the constant Ansatz. It has to be noted that increasing the number of parameters in the constant Ansatz might be beneficial and yield a better comparison to the neural network Ansatz. However, when using finite differences, increasing the number of parameters, by e.g. increasing the number of voxels or the polynomial degree per finite difference grid point do not yield an improved indicator discretization. These methods are possible in the context of the finite element and cell method, where the added parameters can be taken into account via multiple Gauss points per element. Thus the benefit of neural networks as discretization becomes even clearer, as they provide a universal tool independent of the simulation method. With this final modification, the change to the classical adjoint optimization as presented in [37] is almost complete, with the minor difference of using automatic differentiation for the gradient computation. However, as previously stated, moving from automatic differentiation to the continuous adjoint method for the gradient computation is not likely to have a significant impact. To assess this numerically, the initial gradient of the cost function w.r.t. the inverse field $\nabla_{\boldsymbol{c}}\mathcal{L}$ using a constant Ansatz is in both cases compared to a reference solution obtained with finite differences, by varying one parameter $c_{i}$ at a time. Only minor differences are observable at the boundaries, illustrated in figure 13. The difference can be explained by the numerical evaluation of the Frechet kernel, equation (25), which introduces the most significant numerical errors at the boundaries due to the central finite difference scheme. The absolute error w.r.t. the reference solution is, however, nonetheless negligible. Thus only minor differences in the learned prediction from figure 12 are expected, as confirmed by the numerical comparison. The tiny differences accumulate throughout the iterations leading to slightly different local minima with a similar quality of inversion. Thus, no improvement is expected in the solution quality when using either automatic differentiation or the adjoint method for the gradient computation. However, using the adjoint method to compute the gradient $\nabla_{\hat{\gamma}}\mathcal{L}_{\mathcal{M}}$ leads to a significant drop in the computational effort from $2.46$ s to $1.05$ s per epoch for a simulation comparable to the one presented in figure 12. #### 4.1.4 Hybrid Approach The proposed hybrid approach classified in table 1 and described in detail in section 2.3 combines the advantage of the neural network Ansatz with the lower computational effort of the adjoint method for the gradient computation $\nabla_{\hat{\gamma}}\mathcal{L}_{\mathcal{M}}$. The resulting prediction is presented in figure 14 with only a minor decrease in prediction quality when compared to figure 11, while retaining computation times of $1.06$ s per epoch. A slight increase of the learning rate $\alpha_{\min}=4\cdot 10^{-3}$ was Figure 12: Prediction and error of the material distribution after 50 epochs using automatic differentiation and a constant Ansatz. Average elapsed time per epoch is $2.46$ s. Figure 13: Initial gradient using a constant Ansatz computed with automatic differentiation (a) and the adjoint method (b). The corresponding absolute errors in (c) and (d) are computed with a numerical reference solution obtained with finite differences. beneficial. #### 4.1.5 Comparison To conclude the comparison between the investigated methodologies, the training histories are juxtaposed in figure 15. While the constant Ansatz enables the best minimization of the cost function, the mean squared error is lowest for the neural network Ansatz, confirming once again the advantage of the neural network Ansatz. Furthermore, the impact of using automatic differentiation or the adjoint method with finite differences becomes clearer with a consistently lower cost and mean squared error when using automatic differentiation. In view of the computational benefit, the slight loss in accuracy is acceptable. This difference only occurs due to the finite difference evaluations of the Frechet Kernel, equation (25). In the case of finite elements, the Frechet Kernel can be evaluated exactly by analytical differentiation of the shape functions. To gain an additional understanding of the training behaviour, consider the history of indicator predictions provided in figure 16. Here, automatic differentiation with a neural network Ansatz, the adjoint method with a constant Ansatz and the hybrid approach are juxtaposed as snapshots showing the entire domain at specific instances of the iteration. In all three cases, the void is essentially recovered after 15 epochs. After the identification, only minor improvements in the remaining domain are made, where the effect is strongest for the hybrid approach and weakest for the adjoint method. Interestingly, all three approaches lead to three completely different local minima within a comparable amount of computational effort, despite initial noise in the case of the neural networks. Thus two critical conclusions arise from this investigation. Firstly, learning the forward and inverse fields simultaneously for a single inversion problem yields a much more complex optimization task than using an non-trainable forward operator such as finite differences. The improvement of using a non-trainable forward operator is about four orders of magnitude when considering the number of iterations. Secondly, a neural network as Ansatz for the inverse fields outperforms classical approaches, such as a piece-wise constant Ansatz, with regard to the recovered field. To once again highlight the difference, a surface plot of the prediction of both the neural network Ansatz and the constant Ansatz after optimization using automatic differentiation is provided in figure 17. The surfaceplot illustrates the difference in smoothness of the two solutions. Although the neural network solution is smoother, the sharpness retained (quantified in Appendix C). The plot indicates a regularization effect to be studied further using the three-dimensional case in the upcoming section 4.2. Figure 14: Prediction and error of the material distribution after 50 epochs using the combination of automatic differentiation and adjoint method and a neural network Ansatz. Average elapsed time per epoch is 1.06 s. Figure 16: Prediction histories starting at epoch 0 until epoch 25. From left to right, automatic differentiation with a neural network Ansatz, the adjoint method with a piece-wise constant Ansatz, and the hybrid approach with a neural network Ansatz are depicted. Figure 15: Comparison of the training histories for each presented approach. The cost function, which consists of the measurement loss $\mathcal{L}_{\mathcal{M}}$ is shown on the left, and the mean squared error (MSE) w.r.t. the true material distribution $\gamma(\mathbf{x})$ is on the right. The normalization of cost and mean squared error is with respect to the corresponding values with an indicator function equal to one in the entire domain. ### Three-Dimensional Case Due to the quality of the inversion results using neural networks as Ansatz, a three-dimensional case is considered. As the additional spatial dimension introduces an additional computational burden, only the most efficient and simultaneously most promising methods are compared, i.e. the hybrid method and the adjoint method with a piece-wise constant Ansatz. The inversion results of the two approaches are illustrated in figure 18 in terms of identified voids for the two CT-scan slices from figure 8. Once again, the neural networks enable a superior reconstruction, quantifiable by considering the mean squared error provided as annotations in figure 18. Center slices are shown in figure 19 to provide an insight into the overall distribution and smoothness of the solution. From this, it can be observed that the reconstruction of the adjoint method suffers from the oscillatory artefacts to such an extent that the voids are no longer clearly identified. The hybrid approach exhibits much less artefacts and is able to recover the voids with superior accuracy. We attribute this superiority to the different characteristics of the neural network Ansatz which not only leads to an over-parametrization of the indicator field but also has an in-built non-linearity. Specifically, $6\,306\,764$ parameters were used for the neural network compared to the $92^{3}=778\,688$ parameters in the classical discretization. Interestingly, the reconstruction is worse when using fewer neural network parameters. By using half and quarter as many filters, yielding only $1\,577\,452$ and $394\,748$ parameters, the quality of the reconstruction decreases. At the same time, using double the amount of filters, resulting in $25\,221\,004$ parameters, leads to a minor decrease in quality, due to the now dominating regularization effect of the over-parameterization. The mean squared error (MSE) w.r.t. the true material distribution presented for the three variations of the network in figure 20 quantifies this observation. Furthermore, a smoothing effect is noticeable, with an increase of parameters indicating a regularization-like behaviour. Thus, the over-parametrization enabled by the neural network is one, yet not the only key ingredient in enhancing the inversion results. Several other aspects distinguish the neural network from the piece-wise constant Ansatz. The non-linear activations inside the neural network represent an important difference. Also, the neural network acts as a global function, while the constant Ansatz is defined on local patches. Furthermore, other network architectures, such as fully-connected or graph neural networks, may show different behaviour. Currently, it is unclear which aspects are critical aside from the over-parametrization. All aspects can be studied in a straightforward empirical manner and are subject Figure 17: Surface plot of the predicted material distribution after 50 epochs. The training was performed with a neural network and constant Ansatz using automatic differentiation. to further research. ## 5 Conclusion The aim of this work was to compare PINNs with the classical adjoint optimization for full waveform inversion and thereby quantify the strengths of each approach by incremental modification. Firstly, estimating the forward solution via a nested optimization using two neural networks was shown to be disadvantageous. Instead, a classical forward solver, as in the adjoint optimization, is recommended. Secondly, a neural network as Ansatz of the inverse field is highly beneficial with a regularization-like effect without a loss of sharpness. Thirdly, no significant difference in accuracy exists in the sensitivity computation when comparing automatic differentiation and the continuous adjoint method. However, the continuous adjoint method is preferred in the gradient estimation of the cost w.r.t. the inverse field due to the lower computational effort. As a consequence of these insights, the hybrid method is proposed, utilizing a neural network to discretize the inverse field and computing the cost function gradient w.r.t. the inverse field with the adjoint method, which then is passed on to the backpropagation algorithm to update the neural network parameters. Comparable computational times are achieved, while the inverse reconstructions improved. The performance of neural networks as the discretization of the inverse fields enables not only Figure 18: Identified void by the adjoint method and the hybrid approach after 100 epochs in comparison to the true void. The illustrated void is extracted by thresholding indicator values lower than 0.5. Average elapsed time per epoch is about 29.1 s for both methods. Figure 19: Center slices of the indicator function predictions using the adjoint method and the hybrid approach after 100 epochs. The sliced planes are annotated in the top. Average elapsed time per epoch is 29.1 s for the hybrid approach and 29.1 s for the adjoint method. Figure 20: Mean squared error after 100 epochs for different number of parameters illustrated on case 1. further research for other inverse problems but also a thorough investigation into the underlying cause of its success. Insight can be gained by considering the difference between global and local functions, different over-parametrizations, i.e. the number of parameters, the non-linearities in the network, different network architectures, and possibly a thorough investigations of the initialization of the wavefield and its effect on the inversion. ## Acknowledgements The authors gratefully acknowledge the funding through the joint research project Geothermal-Alliance Bavaria (GAB) by the Bavarian State Ministry of Science and the Arts (StMWK) wich finance Leon Herrmann as well as the Georg Nemetschek Institut (GNI) under the project Deep-Monitor. Furthermore, we gratefully acknowledge funds received by the Deutsche Forschungsgemeinschaft under Grant no. KO 4570/1-1 and RA 624/29-1 which support Tim Burchner. Felix Dietrich would like to acknowledge the funds received by the Deutsche Forschungsgemeinschaft - project no. 468830823. ## Declarations ### Conflict of interest No potential conflict of interest was reported by the authors. ### Data Availability We provide a PyTorch [7] implementation of all methods in [71]. ## Appendix A Efficient Finite Difference Implementation in Deep Learning Frameworks The explicit finite difference scheme, shown for the two-dimensional case, \[u_{i,j}^{n+1}=-u_{i,j}^{n-1}+2u_{i,j}^{n}\] \[+\frac{2}{\gamma_{i,j}}\left(\frac{c_{0}\Delta t}{\Delta x}\right) ^{2}\left(\left(\frac{1}{\gamma_{i,j}}+\frac{1}{\gamma_{i+1,j}}\right)^{-1}(u_{ i+1,j}^{n}-u_{i,j}^{n})-\left(\frac{1}{\gamma_{i-1,j}}+\frac{1}{\gamma_{i,j}} \right)^{-1}(u_{i,j}^{n}-u_{i-1,j}^{n})\right)\] \[+\frac{2}{\gamma_{i,j}}\left(\frac{c_{0}\Delta t}{\Delta y}\right) ^{2}\left(\left(\frac{1}{\gamma_{i,j}}+\frac{1}{\gamma_{i,j+1}}\right)^{-1}(u_{ i,j+1}^{n}-u_{i,j}^{n})-\left(\frac{1}{\gamma_{i,j-1}}+\frac{1}{\gamma_{i,j}} \right)^{-1}(u_{i,j}^{n}-u_{i,j-1}^{n})\right)\] \[+\frac{\Delta t^{2}}{\rho_{0}\gamma_{i,j}}f_{i,j}^{n}\] was implemented in PyTorch [7] using convolutional kernels $\mathbf{K}$, thus exploiting PyTorch's GPU capabilities. \[\mathbf{u}^{n}=-\mathbf{u}^{n}+2\mathbf{u}^{n-1}\] \[+\frac{1}{\mathbf{\gamma}}\Bigg{(}\frac{1}{\frac{1}{\mathbf{\gamma}}\mathbf{ K}_{\gamma_{2_{0}}}}(\mathbf{u}\mathbf{K}_{u_{u_{0}}})-\frac{1}{\frac{1}{\mathbf{\gamma}}\mathbf{ K}_{\gamma_{2_{1}}}}(\mathbf{u}\mathbf{K}_{u_{u_{1}}})\] \[+\frac{1}{\frac{1}{\mathbf{\gamma}}\mathbf{K}_{\gamma_{3_{0}}}}(\mathbf{u} \mathbf{K}_{u_{u_{0}}})-\frac{1}{\frac{1}{\mathbf{\gamma}}\mathbf{K}_{\gamma_{3_{1}}}}( \mathbf{u}\mathbf{K}_{u_{y_{1}}})\] \[+\frac{\Delta t^{2}}{2\rho_{0}}\mathbf{f}\Bigg{)}\] The corresponding kernels are defined as \[\mathbf{K}_{\gamma_{a_{0}}}=\begin{pmatrix}0&0&0\\ 0&1&1\\ 0&0&0\end{pmatrix},\mathbf{K}_{u_{a_{0}}}=\begin{pmatrix}0&0&0\\ 0&-\left(\frac{c_{0}\Delta t}{\Delta x}\right)^{2}&\left(\frac{c_{0}\Delta t} {\Delta x}\right)^{2}\\ 0&0&0\end{pmatrix},\] \[\mathbf{K}_{\gamma_{x_{1}}}=\begin{pmatrix}0&0&0\\ 1&1&0\\ 0&0&0\end{pmatrix},\mathbf{K}_{u_{x_{1}}}=\begin{pmatrix}0&0&0\\ -\left(\frac{c_{0}\Delta t}{\Delta x}\right)^{2}&\left(\frac{c_{0}\Delta t}{ \Delta x}\right)^{2}&0\\ 0&0&0\end{pmatrix},\] \[\mathbf{K}_{\gamma_{y_{0}}}=\begin{pmatrix}0&0&0\\ 0&1&0\\ 0&1&0\end{pmatrix},\mathbf{K}_{u_{y_{0}}}=\begin{pmatrix}0&0&0\\ 0&-\left(\frac{c_{0}\Delta t}{\Delta y}\right)^{2}&0\\ 0&-\left(\frac{c_{0}\Delta t}{\Delta y}\right)^{2}&0\\ 0&\left(\frac{c_{0}\Delta t}{\Delta y}\right)^{2}&0\end{pmatrix},\] \[\mathbf{K}_{\gamma_{y_{1}}}=\begin{pmatrix}0&1&0\\ 0&1&0\\ 0&0&0\end{pmatrix},\mathbf{K}_{u_{y_{1}}}=\begin{pmatrix}0&-\left(\frac{c_{0} \Delta t}{\Delta y}\right)^{2}&0\\ 0&\left(\frac{c_{0}\Delta t}{\Delta y}\right)^{2}&0\\ 0&0&0\end{pmatrix}\] The homogeneous Neumann boundary conditions were enforced via ghost cells (see e.g. [19]) and implemented by using pooling layers, and manual adjustments according to the ghost cells. ## Appendix B Network Architecture A quantitative summary of the presented network from figure 9 is provided in table 3 for the two-dimensional case and table 4 for the three-dimensional case. The choice of network hyperparameters varies slightly for the different approaches and cases. The interested reader is referred to the code made available at [71]. \begin{table} \begin{tabular}{l l l} \hline layer & shape after layer & learnable parameters \\ \hline input & $256\times 3\times 3\times 3$ & 0 \\ \hline \hline upsample & $256\times 6\times 6\times 6$ & 0 \\ \hline 3D convolution \& PReLU & $256\times 6\times 6\times 6$ & $1\,769\,728+1$ \\ \hline 3D convolution \& PReLU & $256\times 6\times 6\times 6$ & $1\,769\,728+1$ \\ \hline \hline upsample & $256\times 12\times 12\times 12$ & 0 \\ \hline 3D convolution \& PReLU & $128\times 12\times 12\times 12$ & $884\,864+1$ \\ \hline 3D convolution \& PReLU & $128\times 12\times 12\times 12$ & $442\,496+1$ \\ \hline \hline upsample & $128\times 24\times 24\times 24$ & 0 \\ \hline 3D convolution \& PReLU & $128\times 24\times 24\times 24$ & $442\,496+1$ \\ \hline 3D convolution \& PReLU & $128\times 24\times 24\times 24$ & $442\,496+1$ \\ \hline \hline upsample & $128\times 48\times 48\times 48$ & 0 \\ \hline 3D convolution \& PReLU & $64\times 48\times 48\times 48$ & $221\,248+1$ \\ \hline 3D convolution \& PReLU & $64\times 48\times 48\times 48$ & $110\,656+1$ \\ \hline \hline upsample & $64\times 96\times 96\times 96$ & $0$ \\ \hline 3D convolution \& PReLU & $64\times 96\times 96\times 96$ & $110\,656+1$ \\ \hline \hline 3D convolution without padding & $1\times 94\times 94\times 94$ & $1\,729+1$ \\ \& adaptive Sigmoid & \\ \hline \end{tabular} \end{table} Table 4: Three-dimensional convolutional neural network architecture for the prediction of the indicator $\gamma$. The number of filters per layer are double compared to the two-dimensional convolutional neural network. The total number of parameters is $6\,306\,764$. \begin{table} \begin{tabular}{l l l} \hline layer & shape after layer & learnable parameters \\ \hline input & $128\times 8\times 4$ & 0 \\ \hline \hline upsample & $128\times 16\times 8$ & 0 \\ \hline 2D convolution \& PReLU & $128\times 16\times 8$ & $147\,584+1$ \\ \hline 2D convolution \& PReLU & $128\times 16\times 8$ & $147\,584+1$ \\ \hline \hline upsample & $128\times 32\times 16$ & 0 \\ \hline 2D convolution \& PReLU & $64\times 32\times 16$ & $73\,792+1$ \\ \hline 2D convolution \& PReLU & $64\times 32\times 16$ & $36\,928+1$ \\ \hline \hline upsample & $64\times 64\times 32$ & 0 \\ \hline 2D convolution \& PReLU & $64\times 64\times 32$ & $36\,928+1$ \\ \hline 2D convolution \& PReLU & $64\times 64\times 32$ & $36\,928+1$ \\ \hline \hline upsample & $64\times 128\times 64$ & 0 \\ \hline 2D convolution \& PReLU & $32\times 128\times 64$ & $18\,464+1$ \\ \hline 2D convolution \& PReLU & $32\times 128\times 64$ & $9\,248+1$ \\ \hline \hline upsample & $32\times 256\times 128$ & 0 \\ \hline 2D convolution \& PReLU & $32\times 256\times 128$ & $9\,248+1$ \\ \hline 2D convolution \& PReLU & $32\times 256\times 128$ & $9\,248+1$ \\ \hline \hline 2D convolution without padding & $1\times 254\times 126$ & $289+1$ \\ \& adaptive Sigmoid & \\ \hline \end{tabular} \end{table} Table 3: Two-dimensional convolutional neural network architecture for the prediction of the indicator $\gamma$. The total number of parameters is $526\,252$. ## Appendix C Sharpness Quantification The sharpness of the identified material distributions can be quantified by the $L_{2}$-norm of the spatial gradients of the recovered material, inspired by Sobel filters commonly used in image processing [72]: \[\|\|\nabla\hat{\gamma}(\mathbf{x})\|\|_{2}=\sqrt{\nabla_{x}\hat{\gamma}(\mathbf{x})+\nabla_{y }\hat{\gamma}(\mathbf{x})} \tag{27}\] A greater norm indicates a sharper edge. To highlight the improved sharpness recovery by the neural network Ansatz compared to the constant Ansatz, the norm from equation (27) around the circular void is illustrated in figure 21 for the two-dimensional case. At the cirular boundary, the norm is mostly larger in the neural network case showcasing a sharper inversion, which can be quantified by considering the mean of the non-zero values, which are extracted by selecting the points with a norm greater than a positive threshold, with the purpose of excluding the noise present in the constant Ansatz solution. For any threshold, the mean is larger for the neural network case, confirming an overall sharper recovered material distribution. Figure 21: The $L_{2}$-norm of the spatial gradients of the recovered material distribution $\|\|\nabla\hat{\gamma}(\mathbf{x})\|\|_{2}$.
2308.03112	A clever neural network in solving inverse problems of Schrödinger equation	In this work, we solve inverse problems of nonlinear Schr\"{o}dinger equations that can be formulated as a learning process of a special convolutional neural network. Instead of attempting to approximate functions in the inverse problems, we embed a library as a low dimensional manifold in the network such that unknowns can be reduced to some scalars. The nonlinear Schr\"{o}dinger equation (NLSE) is $i\frac{\partial \psi}{\partial t}-\beta\frac{\partial^2 \psi}{\partial x^2}+\gamma\|\psi\|^2\psi+V(x)\psi=0,$ where the wave function $\psi(x,t)$ is the solution to the forward problem and the potential $V(x)$ is the quantity of interest of the inverse problem. The main contributions of this work come from two aspects. First, we construct a special neural network directly from the Schr\"{o}dinger equation, which is motivated by a splitting method. The physics behind the construction enhances explainability of the neural network. The other part is using a library-search algorithm to project the solution space of the inverse problem to a lower-dimensional space. The way to seek the solution in a reduced approximation space can be traced back to the compressed sensing theory. The motivation of this part is to alleviate the training burden in estimating functions. Instead, with a well-chosen library, one can greatly simplify the training process. A brief analysis is given, which focuses on well-possedness of some mentioned inverse problems and convergence of the neural network approximation. To show the effectiveness of the proposed method, we explore in some representative problems including simple equations and a couple equation. The results can well verify the theory part. In the future, we can further explore manifold learning to enhance the approximation effect of the library-search algorithm.	Yiran Wang	2023-08-06T13:36:12Z	http://arxiv.org/abs/2308.03112v1	# A clever neural network in solving inverse problems of Schrodinger equation ###### Abstract In this work, we solve inverse problems of nonlinear Schrodinger equations that can be formulated as a learning process of a special convolutional neural network. Instead of attempting to approximate functions in the inverse problems, we embed a library as a low dimensional manifold in the network such that unknowns can be reduced to some scalars. The nonlinear Schrodinger equation (NLSE) is $i\frac{\partial\psi}{\partial t}-\beta\frac{\partial^{2}\psi}{\partial x^{2}}+ \gamma\|\psi\|^{2}\psi+V(x)\psi=0$, where the wave function $\psi(x,t)$ is the solution to the forward problem and the potential $V(x)$ is the quantity of interest of the inverse problem. The main contributions of this work come from two aspects. First, we construct a special neural network directly from the Schrodinger equation, which is motivated by a splitting method. The physics behind the construction enhances explainability of the neural network. In particular, each convolution layer and activation function can be related to corresponding parts of the equation. Under this construction, it can be rigorously proved that the neural network has a convergence rate with respect to the length of input data and number of layers. The other part is using a library-search algorithm to project the solution space of the inverse problem to a lower-dimensional space. The way to seek the solution in a reduced approximation space can be traced back to the compressed sensing theory. The motivation of this part is to alleviate the training burden in estimating functions. Instead, with a well-chosen library, one can greatly simplify the training process. More specifically, in one of the experiments, we analysis the landscape of the loss function with respect to the training parameters to help us obtain the optimal solution to the inverse problem. A brief analysis is given, which focuses on well-possedness of some mentioned inverse problems and convergence of the neural network approximation. To show the effectiveness of the proposed method, we explore in some representative problems including simple equations and a couple equation. The results can well verify the theory part. In the future, we can further explore manifold learning to enhance the approximation effect of the library-search algorithm. _Keywords--_ Physics-inspired neural network, Reduced order modeling, Inverse problems, Schrodinger equation, Compressed sensing ## 1 Introduction Neural network approach has been widely explored in solving partial differential equations (PDE) since the pioneer work [30]. Most of them choose from some existing architectures and train the model to obtain satisfying results for the PDEs. The advantage is that it does not require too much information from the equation itself especially when it is a data-driving method. This provides great convenience in solving different types of equations hence it is user-friendly, which also brings some drawback on the other hand. Performing as a blackbox, this kind of neural network approach lacks sufficient explainability, which comes from three aspects. The first one is the existence of parameters that are optimal to some sense from theoretical aspect. This is somehow answered by the popular universal approximation theorem for neural networks. However, it is not sufficient to choose only one single layer in many cases. For a network with a particular architecture, the existence of optimal parameters is unclear. Secondly, even though one can obtain the optimal solution from theory, it remains unclear whether the optimal solutions can be attained with the training process. In many cases, one can only find it with trial and error. The third uncertainty exists in the generalization. If one can obtain a set of "good" parameters to build an optimal neural network, whether the performance is satisfying when different dataset are used. If it performs well in some similar data as the training data, which function space it can approximate. Overall, there is not a general explanation about the above three problems, i.e., the existence of optimal parameters, how to train it, and the generalization power. To enhance the explainability, one can equip the construction of neural network with information of equations instead of treating networks and PDEs separately. To this end, physics-informed neural newt (PINN) [46] serves as a representative example by utilizing a loss function derived from equation. Inspired by this work, we reconsider the relationship of neural network and PDEs and desire to figure out a novel structure of network that makes use of the equation itself. As for the interested equations, we focus on a class of nonlinear Schrodinger equation since they are significant in both scientific research and industry [24, 39, 28, 50]. Moreover, numerical treatments of Schrodinger equation have been investigated extensively [5, 15, 53, 36, 7], among which time splitting schemes are widely used [8, 9, 55, 14, 54]. Apart from forward process, inverse problems attract attention from many researchers due to its wide applications [38] and contribute some effective numerical techniques [35, 4, 48, 1, 52]. Based on the understanding of the time splitting method, we consider designing a novel network from the nonlinear Schrodinger equation. Hence, we call the derived network NLS-Net. In [59], we inverstigated an coefficient inverse problem in nonlinear Schrodinger equation without potential. In this work, we explore an inverse problem of finding potential from data with the NLS-Net. Compared with scalars, it is far more complicated to seek functions. To this end, we introduce a library-search approach to reduce the problem to a coefficient inverse problem. This method is motivated from an algorithm in sparse identification of nonlinear dynamics (SINDy-PI) [27]. The priciple idea is to choose some functions from a reference set, which can be understood as a library. We further use a linear combination of the chosen functions to represent the unknown potential to the inverse problem. In other words, we project the original solution to a fixed approximation space, hence we only need to determine the corresponding coefficients. However, from the perspective of expressive power, we need to incoporate sufficient modes in the library, which performs similarly as utilizing an either deep or wide neural network. Put differently, it is obvious that approximation power comes from sufficient degree of freedom. On the other hand, increasing the number of unknowns means a growth in computation cost, which is apparently a side effect. For example, everyone in the area of scientific computing is familiar with the word "the curse of dimensionality" [29], which may bring a blow up in computation cost. Hence, model reduction methods [2, 12, 44, 11, 43, 45] are introduced to seek a "cheaper" model based on the original model to attain the quantity of interest. In particular, there are considerable number of works in reduced order methods of solving PDEs numerically [42, 13, 60, 3, 40, 57, 47, 23]. For inverse problems in PDEs, many researchers devote good ideas of applying model reuction [19, 16, 21, 32, 20, 22]. It is understandable that one needs to sacrifice accuracy in order to reduce the computation cost since there is no free lunch. However if one has prior information about the quantity of interest, it is possible to use some well-designed models that can attain a relatively high accuracy with reduced workload. In image or signal processing, compressed sensing [37, 17, 56, 18, 34, 51, 49] is utilized for enforcing sparsity in some incorporated variables based on the knowledge that the solution lies in a low-dimensional subspace. Therefore, it is reasonable to apply this technique in the area of numerical PDE as a other reduced order approach, which becomes another motivation of this work where we desire to approximate the unknown potential in the nonlinear Schrodinger equation. The prior information is that the potential function should have a relatively simple expression, which can be a linear combination of a few different basis functions. Hence, it is reasonable to assume the target coefficient vectors with respect to bases exploits a sparse form, which explains the motivation of applying compressed sensing in this context. Overall, contributed by above motivations, we propose a library-search approach combined with compressed sensing to train a special network called NLS-Net that is derived from NLSE. The proposed method is potential to be developed a new framework of solving inverse problems. To evaluate the performance of the proposed method, we use three reprentative examples, two of which are single equations with unknown potentials while the rest one is coupled system of two NLSE. In all the examples, we use NLS-Net to obtain the forward solutions. From [6], one can derive an upper bound for the approximation errors of forward solutions, which can in turn guarantee the convergence of NLS-Net with respect to number of layers and data amount. In our numerical results, we show that the performance of the NLS-Net is consistent to the theory. For the inverse part, we use the library-search combined with compressed sensing to seek unknown potentials. In view of the specificity of the third case, we apply a landscape analysis to help us find the optimal solution more efficiently. It is satisfying that the concerned loss as well as errors present an exponential decay in all the three examples. Moreover, we compare the numerical solutions of the inverse part, which are approximately identical to the groundtruth. Both of them shows the effect and accuracy of the proposed method. The paper is organized in six parts. We present the concerned problem in the first part and review the conventional split-step methods in the next part. The proposed method is introduced in the third section and it is compsed of four subsections. We first introduce the mechanism of designing a convolutional neural network from NLSE. Following this, we provide some background in compressed sensing. The library-search algorithm is then proposed, which is combined with compressed sensing to form the final approach in the last part. We alsp present some relevant theroy in the fifth section. Numerical simulations are demonstated in the following part. We conclude this work and offer some remark and potential interested problems in the last place. ## 2 Problem description From an initial value problem of Schrodinger equation as follows: \[i\frac{\partial\psi}{\partial t}+\beta\frac{\partial^{2}\psi}{ \partial x^{2}}+\gamma\|\psi\|^{2}\psi+V(x)\psi =0, \tag{1}\] \[\psi(0,x) =h(x),\] where $\psi(x,t):\mathbb{R}\times[0,T]\rightarrow\mathbb{C}$ is the wave function and $V(x)$ is the potential to be estimated. Moreover, $\beta$,$\gamma$ are constants and $h(x):\mathbb{R}\rightarrow\mathbb{C}$ is a given as the initial condition. The observed dataset composed of continuous functions is defined to be $\mathcal{D}_{\text{cont}}:=\{\psi_{0}(x),\psi_{T}(x)\}$ where $\psi_{0}(x)$ and $\psi_{T}(x)$ is the observed wave at the initial time $0$ and final time $T$, respectively. Based on the observed data, we estimate the potential $V(x)$. In particular, we define the $S(t;V)$ to be Schrodinger operator in (1) as $\psi(\cdot,t)=S(t;V)\psi(0,\cdot)$. To identify the potential $V$ from $\mathcal{D}_{\text{cont}}$, we formulate the following optimization problem: \[\min_{V}\\|S(t;V)\psi_{0}-\psi_{T}\\|_{2}^{2}. \tag{2}\] Numerically, we can not use the exact $S(t;V)$ operator that requires directly solving (1) which is infeasible. Instead, we use operator splitting [25, 31, 26] as an approximation method. ## 3 Conventional split-step methods In this part, we review some theoretical results on the well-posedness and stability of the inverse problem. Besides, we state the conventional split-step method. First, we rewrite (1) into the form \[\frac{\partial}{\partial t}\psi(x,t)=(\mathcal{L}+\mathcal{N}) \psi(x,t),\quad x\in\mathbb{R}. \tag{3}\] Here $\mathcal{L}:=-i\beta\frac{\partial}{\partial x^{2}}$ is the linear operator and $\mathcal{N}:=i\gamma\|\psi(x,t)\|^{2}+iV(x,t)$ is the nonlinear operator. Instead of directly solving (3), one can split it into two subproblems corresponding to two operators $\mathcal{L}$ and $\mathcal{N}$, respectively. In particular, one can solve a linear problem \[\frac{\partial\psi}{\partial t}=\mathcal{L}\psi_{L}(x,t), \tag{4}\] and a nonlinear problem \[\frac{\partial\psi}{\partial t}=\mathcal{N}\psi_{N}(x,t). \tag{5}\] Both of the above two problems can be solved sequentially to obtain a final solution. We denote $\delta t$ to be the time step size. In terms of performing the splitting, there are different orders of the method. The simplest version is the first-order method, in which is each advancement of $\Delta$ is composed of two steps. First, one solves either the linear (4) or the nonlinear problem (5) to generate a intermediate solution. Then one apply the other operator to finish this time block. More precisely, if one choose $\mathcal{L}$ prior to $\mathcal{N}$ for each $\Delta t$, then a direct result after forwarding one time block is \[\psi(x,t+\Delta t)\approx\exp(\mathcal{N}\Delta t)\exp(\mathcal{L} \Delta t)\psi(x,t).\] Similarly, the second-order method also be decoupled into alternatively solving linear (4) and (5). The only difference is we split each $\Delta$ into three steps. The key idea is that the solution of one step serves as the initial condition for the next step. The most popular second-order splitting is the standard Strang splitting with the form \[\psi(x,t+\Delta t)\approx\exp(\mathcal{L}\Delta t/2)\exp(\mathcal{ N}\Delta t)\exp(\mathcal{L}\Delta t/2)\psi(x,t).\] The advantage of the splitting method is that both (4) and (5) can be much easier than (3). In particular, one can explicitly solve (5) and for (4), many choices can be chosen, for example, finite difference schemes or fourier method. The final combination of splitting and spatial discretization ways form the different numerical methods like split-step finite difference (SSFD) and split-step fourier method (SSFM). ## 4 A physics-inspired neural network with a library-search approach Compared the last section, we now introduce an new approach with a special-designed neural network (NN) [59] and a library-search way to solve some interested inverse problems. First of all, we first give an introduction of the special NN. Then, we introduce the library-search approach which serves as a low-rank representation of the solution to the inverse problem. Finally, we summarize the overall method in the last subsection. ### Learning process of a convolutional neural network Motivated by SSFM, we can formulate a convolutional neural network to solve the inverse problem. In particular, for linear and nonlinear operator in (4) and (5), we use a convolutional operator and a nonlinear operator, whose can be explicitly derived from the equation. As the solution of the inverse problem is incorporated into the NN layers, the backpropagation essentially serves as a process of the seeking the ground truth of the inverse problem. We define the linear and nonlinear part as follows. For linear problem, we apply \[\psi_{L}(x,t+\Delta t)=\kappa\int K(s;\beta\Delta t)\psi_{L}(x-s,t), \tag{6}\] where $K(s;\eta)$ is a convolution kernel defined by \[K(s;\eta):=\sqrt{\frac{i}{\pi\eta}}\exp\left(-\frac{is^{2}}{4 \eta}\right),\forall s\in\mathbb{R}.\] Furthermore, we can introduce a convolution operator $\mathcal{C}(\cdot;\beta\Delta t)$ such that \[\psi_{L}(\cdot,t+\Delta t)=\mathcal{C}(\cdot;\beta\Delta t)\psi( \cdot,t):=K(s;\beta\Delta t)\psi_{L}(\cdot,t),, \tag{7}\] where $$ refers to convolution with respect to $x$. For nonlinear problem, we use \[\psi_{N}(x,t+\Delta t)=\mathcal{G}(\psi_{N};\gamma\Delta t,V\Delta t), \tag{8}\] where the nonlinear operator $\mathcal{G}$ is defined as \[\mathcal{G}(u;g,h):=u\exp i(g\|u\|^{2}+h).\] Finally we formulate the forward propagation as \[\psi(\cdot,t+N\Delta t)\approx\mathcal{S}(\cdot;\beta,\gamma,V, \Delta t)\psi(\cdot,t):=\left(\mathcal{C}\left(\cdot;\frac{\beta\Delta t}{2} \right)\circ\mathcal{G}(\cdot;\gamma\Delta t,V\Delta t)\circ\mathcal{C}\left( \cdot;\frac{\beta\Delta t}{2}\right)\right)^{\diamond N}\psi(\cdot,t) \tag{9}\] Corresponding to the continuous version of linear and nonlinear operators in (6) and (8), we further define the discrete version of $\mathcal{L}$ and $\mathcal{N}$. Before this, we need to consider a corresponding equtaion with periodic boundary condition as follows. \[i\frac{\partial\psi}{\partial t}+\beta\frac{\partial^{2}\psi}{ \partial x^{2}}+\gamma\|\psi\|^{2}\psi+V(x)\psi =0,\quad x\in\Omega,t\in[0,T], \tag{10}\] \[\psi(x,0) =h(x),\quad x\in\Omega,\] \[\psi(a,t) =\psi(b,t),\quad t\in[0,T]\] \[\frac{\partial\psi}{\partial x}(a,t) =\frac{\partial\psi}{\partial x}(b,t),\quad t\in[0,T],\] where $\Omega$ is specified as $[a,b]$. We choose homogeneous grid $\{a_{j}\}_{j=1}^{M}$ with $a_{j}=a+\frac{j(b-a)}{M}$ for $j=1,\ldots,M$. We define $\Delta x=\frac{b-a}{M}$. Since we enforce the periodic boundary condition, i.e., $\psi(t,a)=\psi(t,b)$, we choose $b$ in the grid and $a$ is not in the grid. Then, we define a discrete convolution kernel as \[\tilde{K}(q;\eta)[j]:=\sqrt{\frac{i}{\pi\eta}}\exp\left(-\frac{i(jq)^{2}}{4 \eta}\right),\quad\forall j\in[M],\] where $M$ is the number of collocation points in the spatial domain and we use a combinatorial notation $[M]=1,\ldots,M$. We define $t_{n}=n\Delta t$. With the spatial discretization into the grid $\{a_{j}\}_{j=1}^{M}$, we define $\tilde{\psi}_{M,L}^{m}$ to be the approximate solution to (6) at time $t_{m}$, from which we update the solution at $t_{m+1}$ with discrete convolution as follows, \[\psi_{L}(x_{n},t_{m+1})\approx\tilde{\psi}_{M,L}^{m+1}[n]=\tilde{K}(q;\beta \Delta t)\tilde{\psi}_{M,L}^{m}:=\sum_{j=1}^{M}\tilde{K}(q;\beta\Delta t)[j] \tilde{\psi}_{M,L}^{m}[n-j],\] where $\tilde{K}(q;\eta)$ are replaced by zero when $n-j\notin[M]$. Again, we define an associated discrete convolution operator $\tilde{\mathcal{C}}(\cdot;\eta)u:=\tilde{K}(q;\eta)u$. As for the nonlinear operator, we use $\tilde{\mathcal{G}}$ for the discrete version but the definition of $\tilde{\mathcal{G}}$ is the same as $\mathcal{G}$ but we can restrict the domain to set of discrete vector. Corresponding to (9), we can summarize the discrete version by combining the $\mathcal{C}$ and $\mathcal{G}$ as \[\tilde{\psi}_{M}^{N}=\tilde{\mathcal{S}}(\cdot;\beta,\gamma,V,\Delta t)\tilde {\psi}_{M}^{0}:=\left(\tilde{\mathcal{C}}\left(\cdot;\frac{\beta\Delta t}{2} \right)\circ\tilde{\mathcal{G}}(\cdot;\gamma\Delta t,V\Delta t)\circ\tilde{ \mathcal{C}}\left(\cdot;\frac{\beta\Delta t}{2}\right)\right)^{\odot N}\tilde{ \psi}_{M}^{0}, \tag{11}\] where we define $\tilde{\psi}_{M}^{N}$ as the discrete solution to (10) with the operator splitting method. This can be expressed as a $N-$ layer complex-valued CNN with activation function $\tilde{\mathcal{G}}$ as below: \[\tilde{\mathcal{W}}(\Theta)=\widetilde{C}_{h}\circ\mathcal{G}\circ\tilde{ \mathcal{C}}\circ\ldots\circ\tilde{\mathcal{C}}\circ\mathcal{G}\circ\widetilde {\mathcal{C}}_{h}. \tag{12}\] Here $\Theta$ is the set of weights contained in the operators $\tilde{\mathcal{C}}$ and $\tilde{\mathcal{G}}$ and $\widetilde{C}_{h}$ refers to $\frac{\Delta t}{2}$ of forwarding in the linear step. ### Compressed sensing background In [17], the problem is to reconstruct unknown image or signal $x\in\mathbb{R}^{m}$ from $n$ measurements. Based on the prior information that $x$ has a sparse representation in some orthonormal basis, the number of measurements $n$ can be dramatically smaller than the size $m$. For example, some natural classes of images with $m$ pixels need only $n=O(m^{1/4}\log^{\frac{5}{4}}(m))$ samples for relatively accurate recovery. Here, we briefly state the problem and focus on the information that can be used in the proposed method. Suppose $X$ is a class of objects of interest and one needs to recover $X$ from some measurements. First they define an information operator $I_{n}:X\rightarrow\mathbb{R}^{n}$ as follows: \[I_{n}(x)=(\langle\xi_{1},x\rangle,\ldots,\langle\xi_{n},x\rangle)\] where $\xi_{i}$ are sampling kernels. Furthermore, $A_{n}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is an algorithm that reconstruct $x$ from $I_{n}(x)$. Overall, the problem is formulated as minimizing the $l_{2}$ error of the reconstruction \[E_{n}(X)=\inf_{A_{n},I_{n}}\sup_{x\in X}\\|x-A_{n}(I_{n}(x))\\|_{2}.\] Here, $A_{n}$, $I_{n}$ can be any possible methods of linear sampling and reconstruction. The goal is to obtain a faithful recovery with limited measurements. In particular, it can be split into two questions: how to choose $I_{n}$ and once it is fixed, how to determine an effective way to reconstruct $x$, i.e., finding $A_{n}$. First of all, the following assumption is used. Assumption 1.: _For each $x\in X$, there exists a set of orthogonal bases $(\psi_{i}\in\mathbb{R}^{m},i=1,\ldots,m)$. Define $\theta_{i}:=\langle x,\psi_{i}\rangle$ to be the corresponding transform coefficients as $\theta_{i}:=\langle x,\psi_{i}\rangle$. Then $x$ is sparse in the sense that for some $0<p<2$ that_ \[\\|\theta\\|_{p}\equiv\left(\sum_{i}\|\theta_{i}\|^{p}\right)^{\frac{1}{p}}\leq R\] _for some $R>0$._ The authors are only interested in near-optimal performance, i.e., optimal up to some constants. In [17], they provided the condition of near-optimal information operator and the reconstruction procedure based on linear programming. Briefly speaking, $\Phi$ is an $n\times m$ matrix generated by randomly sampling the columns and $J\subset\{1,\ldots,m\}$ is a set of index. Let $\Phi_{j}$ denote a submatrix of $\Phi$, which is a result of choosing indicated columns of $\Phi$. This matrix contains the information of sampling. We pay special attention to one condition of $\Phi_{j}$ for the desired approximation power: for strict positive parameter $\rho$, the minimal singular value of $\Phi_{J}$ exceeds $\eta_{1}>0$ uniformly in $\|J\|<\frac{\rho n}{\log(m)}$. Here, the columns of $\Phi_{j}$ are used as bases functions. To attain an effective approximation, it is significant to obtain a particular quantitative degree of linear independence among bases. Based on the information matrix $I_{n}$, one can solve the following convex optimization problem: \[\min\\|\Psi^{T}\\|_{1}\text{ subject to }y_{n}=\Phi\Psi^{T}(x),\] where $\Psi\in\mathbb{R}^{m\times n}$ is an orthorgonal matrix. In other words, a desired form is to find $x$ with smallest $l_{1}$ norm of transform coefficients and consistent with the measurements $y_{n}$. ### A library-search algorithm Motivated by the [27], we can formulate a library-search algorithm in solving the inverse problem. Suppose we have a set of functions denoted by $\mathcal{P}=\{g_{1},\ldots,g_{N_{L}}\}$, where $N_{L}$ is the cardinality of the function set $\Phi$. The key idea of this library-search method is that we solve $V$ in a space spanned by $\mathcal{S}$. Mathematically, we can formulate the process as follows. Given a set of collocation points $\{x_{1},\ldots,x_{N_{x}}\}$, we define vectors $g_{i,d}\in\mathbb{R}^{N_{x}}$ by $g_{i,d}[j]:=g_{i}(x_{j})$, $\forall j\in[N_{x}]$, $\forall i\in[N_{L}]$. The discretized target solution $\psi_{T}$ can be similarly defined as $\psi_{T,d}[j]=\psi_{T}[x_{j}]$, $\forall j\in[N_{x}]$. Now we have obtained a set of bases $\{g_{i,d}\}$, which can generate a subspace $H$ in $\mathbb{R}^{N_{x}}$. We further assemble a matrix $\Phi\in\mathbb{R}^{N_{x}\times N_{L}}$ define by $\Phi e_{j}=g_{i,d}$, where $e_{j}$ are the $j$-th standard basis for $\mathbb{R}^{N_{L}}$. Given a coordinate vector $\mathbf{c}\in\mathbb{R}^{N_{L}}$, one can generate a vector $y\in H$ defined by $y=\Phi\mathbf{c}$. Now we replace (2) by the following problem: \[\min_{\mathbf{c}}\\|\tilde{\mathcal{S}}(\cdot;\beta,\gamma,\Phi\mathbf{c},\Delta t )\tilde{\psi}_{L}^{0}-\psi_{T,d}\\|_{2}^{2}. \tag{13}\] ### Our approach From first subsection, we explore some details of the construction of the convolutional neural network derived from the Schrodinger equation, which is therefore denoted by NLS-Net. To represent the potential, we introduce the library-search algorithm, which indeed serves as an effective way of parameterizing the NLS-Net. Inspired by the compressed sensing, we can seek an approximation with relatively high sparsity. With the notation in the compressed sensing, we desire to find a way to find a corresponding transform coefficients with relatively small $l_{1}$ norm. For the training process of NLS-Net, we use a summation of $l^{2}$ error and $l^{1}$ norm of $\mathbf{c}$ for the loss function defined below: \[J_{\mathbf{c}}=\frac{1}{2}\frac{\\|\tilde{\mathcal{S}}(\cdot;\beta,\gamma, \Phi\mathbf{c},\Delta t)\tilde{\psi}_{L}^{0}-\psi_{T,d}\\|_{2}^{2}}{\\|\psi_{T, d}\\|_{2}^{2}}+\lambda\\|c\\|_{1}. \tag{14}\] To minimize $J_{\mathbf{c}}$, we use proximal gradient descent [41]. We summarize the method of combining the NLS-Net with the library search algorithm in the following Table. ## 5 Theoretical background In this section, we present the theoretical background of the inverse problem. We briefly mention two parts as follows: 1. The well-posedness of the inverse problem. 2. The convergence analysis of the NLS-Net In particular, in the first part, we review some important results about uniqueness and stability of a special form of inverse problem. We remark that the existing result is limited to linear case however, it offers insights of the nonlinear case which deserves exploration. For the second part, we present the convergence rate of the constructed neural network with respect to the number of collocation points and layers. In [10], Baudouin et al. (2007) proved uniqueness and stability in an inverse problem for the Schrodinger equation. They consider the following equation: \[\left\{\begin{array}{l}iy^{\prime}(x,t)+\Delta y(x,t)+q(x)y(x,t)=0,x\in \Omega,t\in(0,T)\\ y(x,t)=h(x,t),x\in\partial\Omega,t\in(0,T)\\ y(x,0)=y_{0}(x),x\in\Omega.\end{array}\right. \tag{15}\] For this equation, they proved the following theorem. Theorem 2.: _Let $\mathcal{U}$ be a bounded subset of $L^{\infty}(\Omega)$, $q\in L^{\infty}(\Omega)$ and $y$ be a solution of (15). Assuming_ \[\exists x_{0}\in\mathbb{R}^{N}\backslash\bar{\Omega}\text{ such that }\Gamma_{0}\supset\{x\in\partial\Omega;(x-x_{0})\cdot\nu(x)\geq 0\}\] \[y(q)\in W^{1,2}\left(0,T,L^{\infty}(\Omega)\right)\] $y_{0}$ is real valued and $\|y_{0}\|\geq r_{0}>0$, as in $\bar{\Omega}$. There exists a constant $C=C(\Omega,T,\Gamma_{0},\\|q\\|_{L^{\infty}(\Omega)},y_{0},h,\mathcal{U})>0$ such that if_ \[\frac{\partial y(q)}{\partial\nu}-\frac{\partial y(p)}{\partial\nu}\in H^{1}(0, T;L^{2}(\Gamma_{0}))\] _then $\forall p\in\mathcal{U}$,_ \[\\|q-p\\|_{L^{2}(\Omega)}\leq C\left\\|\frac{\partial y(q)}{\partial\nu}-\frac{ \partial y(p)}{\partial\nu}\right\\|_{H^{1}(0,T;L^{2}(\Gamma_{0}))}.\] This theorem characterizes the fact that the solutions to the inverse problem change continuously with the normal derivative of the forward solutions. In the following part, we review some important results of the convergence rate of the split-step method, which is the core of the NLS-Net. In [33], Lubich (2008) proved the convergence rate of the splitting methods for Schrodinger-Poisson and cubic nonlinear Schrodinger equations. Here we only consider the latter one as follows: \[i\frac{\partial\psi}{\partial t}=-\nabla\psi+V\psi,\quad x\in\mathbb{R}^{3}, \quad t>0, \tag{16}\] where $V=V[\psi]=\pm\|\psi\|^{2}.$ Assuming the solution $\psi(t)$ to the (16) is in $H^{4}$ and let $m=\max_{0\leq t\leq T}\\|\psi(t)\\|_{H^{4}}$, the following theorem is proved. Theorem 3.: _The numerical solution $\psi_{n}$ given by the strang-split scheme with step size $\tau$ for equation (16) has a first-order error bound in $H^{2}$ and a second-order error bound in $L^{2}$, for $t_{n}=n\tau\leq T$,_ \[\\|\psi_{n}-\psi(t_{n})\\|_{H^{2}}\leq C(m,T)\tau,\] \[\\|\psi_{n}-\psi(t_{n})\\|_{L^{2}}\leq C(m,T)\tau^{2}.\] This theorem provides the convergence rate of strang-split scheme with respect to the time step $\tau$ under $H^{2}$ and $L^{2}$ norm. However, it is a continuous case in the spatial domain and they didn't consider a discretized case. Hence, we proceed to present the result given in [6] as follows. Theorem 4.: _Let $\psi^{n}\in X_{M}$ be the numerical approximation obtained by the TSSP [6]. Under assumptions (4.84) and (4.85), there exist constants $0<\tau_{0},h_{0}\leq 1$, such that if $0<h\leq h_{0},0<\tau\leq\tau_{0}$ and $m\geq 5$, we have_ \[\\|\psi\left(x,t_{n}\right)-I_{M}\left(\psi^{n}\right)(x)\\|_{L^{2}(U)}\lesssim h ^{m}+\tau^{2},\quad\\|\psi^{n}\\|_{\infty}\leq M_{1}+1,\] \[\\|\nabla\left(\psi\left(x,t_{n}\right)-I_{M}\left(\psi^{n}\right)(x)\right)\\| _{L^{2}(U)}\lesssim h^{m-1}+\tau^{2},\quad 0\leq n\leq\frac{T}{\tau},\] _where the interpolation operator $I_{M}$ is given below and $M_{1}=\max_{t\in[0,T]}\\|\psi(\cdot,t)\\|_{L^{\infty}}$._ The mentioned notations are defined as follows. 1. $X_{M}=\{v=(v_{j})_{j\in\mathcal{T}_{M}^{0}}\|v_{0}=v_{M}=0\}\subset C^{M+1}$ with $\mathcal{T}_{M}^{0}=\{j\|j=0,1,\ldots,M\}$. 2. TSSP refers to time-splitting sine pseudospectral method introduced in [6]. The difference of this method from split step fourier method is that the spatial discretization is sine spectral method while the latter one is fourier method. 3. $I_{M}$ is an interpolation operator defined as \[(I_{M}\phi)(x)=\sum\limits_{l=1}^{M-1}\tilde{\phi}_{l}\sin(\mu_{l}(x-a)),\] where $\tilde{\phi}_{l}=\frac{2}{M}\sum\limits_{j=1}^{M-1}\phi_{j}\sin(\frac{d\pi}{M })$ with $l\in\mathcal{T}_{M}$ and $\phi_{j}=\phi(x_{j})$. Motivated from this theorem, we can have the following upper bound for NLS-Net. Theorem 5.: _We define that $\tilde{\psi}_{M}^{n}$ is the solution at time $t_{n}=n\Delta t$ generated by (11) with $M$ as the input data length and $N$ is the number of layers. We recall that $\psi(x,t_{n})$ is the analytical solution to (1) at time $t_{n}$. Suppose that the initial value $\psi(\cdot,0)\in H^{s}(\mathbb{R}\to\mathbb{C})$ for $s\geq 5$. Then there exist parameters $\Theta$ of the neural network $\tilde{\mathcal{W}}(\Theta)$, which is defined by (12) such that_ \[\\|\tilde{\mathcal{W}}(\Theta)\tilde{\psi}_{M}^{n}-\psi_{T,d}\\|\lesssim(\Delta x )^{s}+(\Delta t)^{2}, \tag{17}\] _where $\Delta x=\frac{b-a}{M}$ and $\Delta t=\frac{T}{N}$._ We remark that this theorem specifies that the approximation effect of the neural network converges respect to the spatial and time step sizes. ## 6 Numerical experiments In this section, we demonstrate the performance of the proposed method with three examples. In the first example, we use the NLS-Net to approximate the one-soliton solution for the generalized nonlinear Schrodinger equation (NLSE), where the potential is exponential. For the second one, we further consider another form of NLSE with a periodic potential, which is called Gross-Pitaevskii (GP) equation. In both examples, only a single equation is involved. For the third one, we explore a coupled system with two equations, where the potential is a function of solutions to both equations. In all the examples, we transform the problem of function estimating to seeking some coefficients, which are further used to recover the potentials. In particular, in the first examples with single equations, we use the library-search algorithm to construct a low-dimensional approxiamtion of the true potential. More specifically, we use a library as follows \[\Phi:=\{x,\sin(x),\cos(x),x^{2},\sin^{2}(x),\cos^{2}(x),e^{-x},e^{-2x},e^{-2x^{ 2}}\}.\] In addition, we make use of compressed sensing to obtain a relatively sparse coefficient vector, with which only very few functions are involved in the final numerical solution. For the third one, owing to the special setting of the problem, one can easily relate the unknown function to some scalars hence the problem can well simplified. To present the performance of the proposed method, we define $e_{\phi}$ and $e_{V}$ as below: \[e_{\psi}=\frac{1}{2}\frac{\\|\psi^{\rm numerical}-\psi^{\rm exact}\\|_{2}^{2}}{ \\|\psi^{\rm exact}\\|_{2}^{2}},\quad e_{V}=\frac{\\|V^{\rm numerical}-V^{\rm exact }\\|_{2}}{\\|V^{\rm exact}\\|_{2}}, \tag{18}\] where $\psi^{\rm numerical}$ and $V^{\rm numerical}$ are the numerical results where $\psi^{\rm exact}$ and $V^{\rm exact}$ are corresponding references. In particular, $e_{\psi}$ is from the $l^{2}$ term in loss function $J_{\rm e}$, which we show more interest in compared with the whole loss $J_{\rm e}$. Example 1.: One-soliton solution for NLSE. In this example, we consider (10) with $\beta=-1$, $\gamma=1$, and $\Omega=[-10,10]$. Moreover, the true potential has the exponential form as follows, \[V(x)=4x^{2}-\exp(-2x^{2}).\] Then the exact solution $\psi(x,t)$ is has the following form: \[\psi(x,t)=\exp(-x^{2}+2it).\] We first show the convergence of $e_{\psi}$ as the number of collocation points ($M$) and layers ($N$) in Figure 2. As for the left one, we fix the number of layers $N$ to be $200$ and increase the number of collocation points $M$ in data. One can see that the by increasing $M$ to $20$, the error first shaply decrease to about $10^{-9}$ and become stable. On the other hand, with $M$ fixed as $200$, one can observe a steady decay in $e_{\psi}$ as using more layers, equivalently, smaller $\Delta t$. Hence the performance of the NLS-Net is consistent with the Theorem 5. Since in our case, we deal with the inverse problem by iteratively solving the forward process, the effect of NLS-Net in approximating exact solution $\psi$ serve as a basis in finding the true potential. We further present the training performance in Figure 3. The left subfigure describes the dynamics of $e_{\psi}$ in the training process while the right one plots $e_{V}$. Here we set the regularization coefficient $\lambda$ to be zero, which means $J_{\rm e}=e_{\psi}$. Hence, we simply use gradient descent instead of proximal gradient descent. In particular, the initial learning rate is $1.5$ and for each $2000$ epochs, it decays by multiplying $0.99$. Besides, we set a tolerence $\tau$ to be $10^{-10}$ as a creterion to further decrease the learning rate. Once $e_{\psi}$ is below this tolerance, we multiple learning rate by $0.4$ for each remaining step. For the network, it is composed of four layers and with input $\tilde{\Psi}_{M}^{0}$ containing $4000$ uniform collocation points in $[-10,10]$, i.e, $M=4000$. Both the dynamics of $e_{\psi}$ and $e_{V}$ are plotted with logarithmic scale in y axis, hence one can observe a linear decay. We remark that a change in slope of lines near the end of training is resulted by exponential decreasing the learning rate since $e_{\psi}$ already drops below a given small tolerance. Finally we compare the numerical (Left) and exact (Right) potential in Figure 4. The domain $[-10,10]$ is discretized by $4000$ uniform collocation points and we use $6000$ epochs to training the potential. After that, one can observe that the apprximation of potential is almost impossible to be distinguished from the Figure 3: Dynamics of loss function $e_{\psi}$ (Left) and potential error $e_{V}$ (Right) during training in Example 1. Here, the regularization coefficient $\lambda$ is zero, i.e., $J_{\textbf{e}}$ is the same as $e_{\psi}$. Gradient descent is used with initial learning rate 1.5, which is multiplied by 0.99 every 2000 epochs. When $e_{\psi}$ decreases to be lower than $10^{-10}$, learning rate is multipied by 0.4 each step. Figure 2: Convergence of $e_{\psi}$ as the increase of length of input data (Left) and number of layers (Right). For the left one, we fix the number of layers to be 200 and for the right one, the number of neurons in each hidden layer is chosen to be 20. goundtruth. Example 2.: Gross-Pitaevskii (GP) equation. In this example, we consider the GP equation, which is special case of (10) with periodic potential. In particular, we set $\beta=\frac{1}{2},\gamma=-1$, the spatial domain $\Omega=$and $V(x)=-\cos^{2}(x)$. For this equation, the exact solution [58] is \[\psi(x,t)=\sin(x)\exp(-\frac{3ti}{2}).\] First of all, we show the dynamics of $e_{\psi}$ and $e_{V}$ in the training process. To obtain a relatively good approximation, we choose $\lambda=20$ in (14). As a comparison, we also present the case without regularization term in Figure 5. In this case, we simply use SGD. Even with a very small learning rate $510^{-6}$, the loss $e_{\psi}$ still oscillates, which may imply a relatively high level of nonconvexity in $e_{\psi}$. Corresponding, the error $e_{V}$ also presents a similar trend during training. Both of them shows that it is relatively difficult to attain the optimal solution with the only term $e_{\psi}$ in $J_{\text{e}}$. On the other hand, with a nonzero regulatization term, we can obtain a near exponential decay in both $e_{\psi}$ and $e_{V}$, which is shown in Figure 6. To obtain this result, we use proximal gradient descent. In particular, we use SGD to tackle the $l^{2}$ term and a proximal operator is utilized to deal with the $l^{1}$ norm, i.e. the second term in $J_{\text{e}}$. For SGD, we begin with a learning rate $510^{-3}$ and it decays by rate $0.1$ every $3000$ epochs. Similarly to the implementation in the last example, we shrink the learning by multiplying it with $0.95$ each step after $e_{\psi}$ has satisfied the condition that $e_{\psi}<10^{-10}$. We remark that even though there are still regular oscillations in the dynamics because of SGD, both $e_{\psi}$ and $e_{V}$ display a steady decrease, which shows the effect of regularization term in $J_{\text{e}}$. In Figure 7, we compare the approximation of potential which is plotted on the left with the groundtruth on the other side. One can easily see that the numerical potential is approximately identical to the reference, which shows that the training process is effective. Example 3.: A coupled system of NLSE. In this example, we explore a coupled nonlinear Schrodinger equations as follows. For $x\in[a,b]$, $t\in[0,T]$, \[\mathrm{i}\frac{\partial\psi_{j}}{\partial t}=-\frac{1}{2}\frac {\partial^{2}\psi_{j}}{\partial x^{2}}+V_{j}(x)\psi_{j}+i\alpha_{j}\frac{ \partial\psi_{j}}{\partial x}+f_{j}\left(\left\|\psi_{1}\right\|^{2},\left\|\psi _{2}\right\|^{2}\right)\psi_{j},\quad j=1,2,\] \[\psi_{j}(x,0)=\psi_{j}^{0}(x),\quad j=1,2\] \[\psi_{j}(a,t)=\psi_{j}(b,t)=0,\quad j=1,2,\] where $f_{1}(x,y)=x+\zeta_{1}y$, $f_{2}(x,y)=\zeta_{2}x+y.$ Here, $\zeta_{1}$ and $\zeta_{2}$ are scalars. We can know that it admits one Figure 4: A comparison of numerical (Left)and exact (Right) potential in Example 1 after training $6000$ epochs. The spatial domain $[-10,10]$ is descritized by $4000$ uniform collocation points and the time domain $[0,T]$ is partitioned by $\Delta t=T/4$. Figure 5: Dynamics of loss function $e_{\psi}$ (Left) and potential error $e_{V}$ (Right) during training in Example 2. Here, the regularization coefficient $\lambda$ is zero, i.e., $J_{\mathbf{c}}$ is the same as $e_{\psi}$. Gradient descent is used with initial learning rate $510^{-6}$, which is multiplied by $0.9$ every $3000$ epochs. Figure 6: Dynamics of loss function $e_{\psi}$ (Left) and potential error $e_{V}$ (Right) during training in Example 2. Here, the regularization coefficient $\lambda$ is $20$. Proximal gradient descent is used with initial learning rate $510^{-3}$, which is multiplied by $0.9$ every $3000$ epochs. When $e_{\psi}$ decreases to be lower than $10^{-10}$, learning rate is multiplied by $0.95$ each step. exact solution [58] as follows when $\zeta_{1}=\zeta_{2}$, $V_{1}(x)=V_{2}(x)=0$, and $-\alpha_{1}=\alpha_{2}=0.5$. The overall loss is defined as $J_{\mathbf{c}}=e_{\psi_{1}}+e_{\psi_{2}}$, where $\mathbf{c}=[\zeta_{1},\zeta_{2}]$. \[\begin{split}\psi_{1}^{\text{exact}}\ (x,t)&=\sqrt{\frac{2 \alpha}{1+\zeta}}\operatorname{sech}(\sqrt{2\alpha}(x-vt))\mathrm{e}^{ \mathrm{i}\left[\left(v-\delta\right)x-\left(\frac{2^{2}-\delta^{2}}{2}-\alpha \right)t\right]}\\ \psi_{2}^{\text{exact}}\ (x,t)&=\sqrt{\frac{2\alpha}{1+ \zeta}}\operatorname{sech}(\sqrt{2\alpha}(x-vt))\mathrm{e}^{\mathrm{i}\left[ \left(v+\delta\right)x-\left(\frac{2^{2}-\delta^{2}}{2}-\alpha\right)t\right] }\end{split} \tag{19}\] In our numerical simulation, we use the following setting: \[a=-20,\quad b=80,\quad v=0.,\quad\delta=0.5,\quad\alpha=1.0,\quad\zeta_{1}= \zeta_{2}=\frac{2}{3}.\] The difference of this example is that the potential is a function of forward solutions $\psi_{1}$ and $\psi_{2}$, while in the first two examples, we have the prior information that the unknown potential is only a function of $x$. However, as a compensation, the form of potential is specified and we only need to determine $\zeta_{1}$ and $\zeta_{2}$ rather than find a unknown function. We remark that even though the reference solutions defined in (19) correponds to the special case $\zeta_{1}=\zeta_{2}$, it is not reasonable to assume this information hence we need to determine both $\zeta_{1}$ and $\zeta_{2}$. However, since there are still only two unknown scalars, we can easily plot the landscape of loss function $J_{\mathbf{c}}$ as a guide to seek the optimal solution. In Figure 6, we show the contour of loss with respect to $\zeta_{1}$ and $\zeta_{2}$. First of all, it presents a reflection symmetry along the line $\zeta_{1}=\zeta_{2}$, which can offer the insight that the optimal $\zeta_{1}$ and $\zeta_{2}$ are almost equal. Besides, the loss function shows the property of local convexity. Hence once the $[\zeta_{1},\zeta_{2}]$ arrives the basin of attraction, where we borrow the terminology in ecology, it is not demanding to attain the optimal solution. Both two observations can help us find the minimizer in a more efficient way. Following the loss landscape, we further show the training performance. As before, we arrange the loss dynamics on the left and errors of inverse problems on the other side. Since in this situation, we only need to determine the unknowns scalars, we use the absolute error defined as \[e_{\zeta_{j}}=\frac{\|\zeta_{j}^{\text{numerical}}-\zeta_{j}^{\text{exact}}\|}{\| \zeta_{j}^{\text{exact}}\|},\quad j=1,2.\] Owing to the landscape analysis shown above, we can simply use gradient descent. Here we fix the learning rate to be $100$. The reason for such a big learning rate may come from a small magnititude of the gradient with respect to the parameters $\zeta_{1}$ and $\zeta_{2}$. However, in spite of the big learning rate, we still obtain an exponeital decay of loss as well as the concerned errors. One can observe a convergence before about $1000$ epochs. Figure 7: A comparison of numerical (Left)and exact (Right) potential in Example 2 at final time $T=1$. The spatial domain $[-10,10]$ is descritized by $4000$ uniform collocation points and the time domain $[0,T]$ is partitioned by $\Delta t=T$. Figure 8: Dynamics of loss function $e_{\psi}$ (Left), errors of $\zeta_{1}$ and $\zeta_{2}$ (Right) during training in Example 3. Gradient descent is used with fixed learning rate 100. Conclusion remarks In this work, we solve some inverse problems of nonlinear Schrodinger equations as a learning process of a special type of convolutional neural network (CNN). The first novelty of this work comes from the special architecture of the neural network, which is designed based on the equation and the operator splitting method. In particular, the convolution layers serve as a linear operator while the activation functions constitude the nonlinear part. A composition of these two forms the time-marching process. Owing to this design, layers and activation functions have physical interpretations. An intermediate convenience is that one can easily choose the type of activation functions from the information of equations. Besides, the existence of an optimal network with this special structure can be guaranteed since the neural network can be directly derived from the equation. We remark that the existence is not a direct result of the famous universal approximation theorem for neural networks since the proposed network has a special architecture while the theorem does not specify the structure of the optimal network. Moreover, based on the existing theoretical result of operator splitting, one can easily guarantee the convergence of the neural network solution with respect to the depth and width. Overall, the particular structure derived from the equation enhances the explainability of network. Instead of using neural network as a black box, the inherited physics sheds light on the approximation effect of the neural network. For the inverse problem, we are interested in approximating the potential function in the Schrodinger equations. Different from the coefficient inverse problem, function approximation is more complicated. For example, one needs a neural network to approximate a function while for coefficients, some neurons are sufficient. To this end, we use a library-search method to transform the original problem to a coefficient inverse problem, which is the second contribution. In particular, we construct a library that contains a set of basis functions then we can use a linear combination of the bases to express the unknown potential. In other words, we project the solution of inverse problems to an approximation space so we only need to determine the coefficients with respect to the basis functions. In this way, the complexity of neural network can be effectively reduced. However, to approximate a function space, we need to incorporate sufficiently many bases to obtain a relatively good accuracy. On the other hand, the form of potential is not so complicated, which may be a linear combination of a small number of different functions. Hence, the coefficient vector correponding to basis functions should have some level of sparsity, which motivates the need of using compressed sensing.
2303.18132	A Desynchronization-Based Countermeasure Against Side-Channel Analysis of Neural Networks	Model extraction attacks have been widely applied, which can normally be used to recover confidential parameters of neural networks for multiple layers. Recently, side-channel analysis of neural networks allows parameter extraction even for networks with several multiple deep layers with high effectiveness. It is therefore of interest to implement a certain level of protection against these attacks. In this paper, we propose a desynchronization-based countermeasure that makes the timing analysis of activation functions harder. We analyze the timing properties of several activation functions and design the desynchronization in a way that the dependency on the input and the activation type is hidden. We experimentally verify the effectiveness of the countermeasure on a 32-bit ARM Cortex-M4 microcontroller and employ a t-test to show the side-channel information leakage. The overhead ultimately depends on the number of neurons in the fully-connected layer, for example, in the case of 4096 neurons in VGG-19, the overheads are between 2.8% and 11%.	Jakub Breier, Dirmanto Jap, Xiaolu Hou, Shivam Bhasin	2023-03-25T12:35:04Z	http://arxiv.org/abs/2303.18132v1	# A Desynchronization-Based Countermeasure Against Side-Channel Analysis of Neural Networks ###### Abstract Model extraction attacks have been widely applied, which can normally be used to recover confidential parameters of neural networks for multiple layers. Recently, side-channel analysis of neural networks allows parameter extraction even for networks with several multiple deep layers with high effectiveness. It is therefore of interest to implement a certain level of protection against these attacks. In this paper, we propose a desynchronization-based countermeasure that makes the timing analysis of activation functions harder. We analyze the timing properties of several activation functions and design the desynchronization in a way that the dependency on the input and the activation type is hidden. We experimentally verify the effectiveness of the countermeasure on a 32-bit ARM Cortex-M4 microcontroller and employ a t-test to show the side-channel information leakage. The overhead ultimately depends on the number of neurons in the fully-connected layer, for example, in the case of 4096 neurons in VGG-19, the overheads are between 2.8% and 11%. Keywords:Deep learning, neural networks, side-channel attacks, countermeasures ## 1 Introduction Current deep learning models grow to millions of parameters and are being widely deployed as a service. As the training of such networks requires large amounts of data and computing time, the organizations that created the models tend to keep them proprietary. Another motivation to keep the models confidential is to reduce the success rate of adversarial attacks that are more powerful in a white-box setting, i.e., when the attacker knows the model parameters. This motivated the development of model extraction attacks on neural networks that either try to mimic the model behavior or extract the parameter values [1]. The "standard" model extraction attacks try to get the proprietary information on the model by using a set of well-designed queries. This, however, normally does not allow the full extraction of model parameters (also called _exact extraction_), but rather a weaker type of extraction, which can be _functionally equivalent extraction_, _fidelity extraction_, or _task accuracy extraction_[2]. On the other hand, with the help of hardware attack vectors, which can be considered especially in the case of embedded machine learning models [3], it is possible to extract the exact values of the parameters, either with side-channel analysis [4, 5] or with fault injection attacks [6]. In [4], it was shown that the information about the used activation function can be observed by measuring the time of the function computation. The timing behavior is dependent on the type of the function and the input data and there are clear patterns forming after a certain range of inputs was fed to the activation function. In this paper, we aim at removing those patterns to make the timing analysis useless for the attacker. We utilize a desynchronization-based countermeasure that adds random delay to the function computation so that no matter what the activation type and the input data, the timing measurement gives random outputs. We experimentally show that our method works by using a 32-bit ARM Cortex-M microcontroller as a device under test. ## 2 Background ### Side-Channel Analysis In [7], it has been shown that even though cryptographic algorithms are proven to be theoretically secure, their physical implementation can leak information regarding confidential data, such as a secret key. This is commonly referred to as Side-Channel Analysis (SCA). An adversary could typically exploit different means of physical leakages, such as timing, power, electromagnetic (EM) emanation, etc. Thus, by observing these leakages, an adversary could analyze and deduce the secret information being processed, by utilizing statistical methods, such as correlation, etc. ### Side-Channel Countermeasures Since the idea of SCA is that the leakage could provide information regarding the internally processed data, the goal of protection techniques is to minimize or remove these dependencies. Different countermeasures have been proposed to protect against SCA, and overall, these can be classified into _hiding_ and _masking_ techniques. The hiding countermeasures aim at breaking the relationship between the processed data and the leakage, for example, using desynchronization, shuffling, etc. The general idea is to introduce noises in the measurement to make the attacks harder. On the other hand, for masking countermeasures, the aim is to remove the relation by introducing randomness to mask the actual data being processed. In this case, the data being processed will be masked with different random mask value for every execution and without the knowledge of the random mask, the attacker could not recover the actual intermediate value. ### Hardware Attacks on Neural Networks Recently, SCA has been applied to attacking neural network implementations. The aim is to recover the secret model, which might be sensitive, for example, if the model has been trained with confidential data or if the model is a commercial IP that has been trained and subjected to piracy. Hence, determining the layout of the network with trained weights is a desirable target for an attacker. Thus, using SCA, the attacker could reverse engineer the neural networks of interest by using some additional information that becomes available while the device under attack is operating. Several attacks have been reported, for example, Hua _et al._[8] demonstrated the retrieval of network parameters, such as the number of layers, size of filters, data dependencies among layers, etc. Then, Batina _et al._[4] proposed a full reverse engineering of neural network parameters based on power/EM side-channel analysis. The proposed attack is able to recover hyperparameters i.e., activation function, pre-trained weights, and the number of hidden layers and neurons in each layer, without access to any training data. The adversary uses a combination of simple power/EM analysis, differential power/EM analysis and timing analysis to recover different parameters. Yu _et al._[9] proposed a model extraction attack based on a combination of EM side-channel measurement and adversarial active learning to recover the Binarized Neural Networks (BNNs) architecture on popular large-scale NN hardware accelerators. The network architecture is first inferred through EM leakage, and then, the parameters are estimated with adversarial learning. As such, it can be observed that SCA has been widely adopted for the attack against neural networks. ### Countermeasures for Hardware Attacks on Neural Networks The first attempt to thwart side-channel analysis of neural networks was called MaskedNet [10]. As indicated by the name, the authors utilize the masking countermeasure to partially protect the networks. They developed novel hardware components such as masked adder trees for fully connected layers and masked ReLUs. The approach works with binarized neural networks which use binary weights and activation values. The same authors later extended their approach to fully protect neural networks by boolean masking [11, 12]. While the latency overhead is relatively small (3.5%), the area overhead is noticeable (5.9$\times$). The overheads were further reduced in [13] where the authors used domain-oriented masking. A threshold implementation (TI) with 64% area and 5.5$\times$ energy overhead was presented in [14]. For generating random numbers required for the TI, Trivium stream cipher was used. Apart from masking-related approaches, garbled circuits were used to protect not only against SCA, but to enhance the privacy of the underlying models [15]. To the best of our knowledge, the only software-based approach for protecting the neural networks against SCA uses shuffling [16]. The authors randomize the order of execution for multiplications within neurons. This, however, does not prevent timing attacks on determining the activation function. ## 3 Timing Analysis of Activation Functions In this section, we first explain the experimental setup used for our measurements. Then, we provide the timing analysis without the countermeasures to better understand the timing pattern. ### Experimental Setup The experiments in this paper were done by using a NewAE ChipWhisperer tool1. ChipWhisperer-LITE was used to perform the power analysis and a 32-bit ARM Cortex-M4 (STM32F3) mounted on a UFO board was used as a target. The ADC sampling frequency was set to 29 MHz while the target frequency was set to 7 MHz. A trigger signal was set to high before the activation function computation and immediately set to low afterward to precisely determine the timing. The source code for the sample programs was written in C, the <math.h> library was used for mathematical functions. To allow for more precise measurements, compiler optimizations were disabled. Footnote 1: [https://www.newae.com/chipwhisperer](https://www.newae.com/chipwhisperer) ### Timing Analysis without Countermeasures The timing behavior of activation functions was first examined in [4]. The authors measured the execution time of three functions: rectifier linear unit (ReLU), hyperbolic tangent (tanh), and sigmoid. The definitions of those functions are as follows: \[ReLU(x)=\begin{cases}0&\text{if }x\leq 0\\ x&\text{otherwise}\end{cases},\] \[sigmoid(x)=\frac{1}{1+e^{-x}},\quad tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}.\] The conclusion was that each of these exhibits different timing patterns that are also dependent on the function input. We repeated the experiment from [4] and used 2000 random inputs from $[-2,2]$ for the computations of each activation function. We have obtained very similar results, see Figure 1 and Table 1. This means that if the attacker sends a few queries and measures the time of the execution, they are able to distinguish which activation function is being used in a particular layer. ## 4 Towards a Desynchronization-Based Countermeasure Desynchronization-based countermeasure was first introduced for cryptographic implementations [17, 18, 19]. The main rationale of the desynchronization-based countermeasure is to remove the data dependency of the power consumption by randomizing the power consumption of the device during computation. In this section, we will analyze the countermeasure when applied to activation function implementations. For this purpose, we would like to utilize desynchronization to randomly delay the computation of each function such that it is impossible for the attacker to distinguish them from one another by examining the timing information. The computation times of different activation functions can be viewed as random variables depending on the inputs of the functions. The time delay caused by additional desynchronization can be considered as another random variable $X$. Since it is easy to generate a random variable with normal distribution in any programming language, we have decided to sample $X$ from a normal distribu Figure 1: Timing behavior of different activation functions. The patterns are clearly distinguishable if there is no side-channel protection in place. tion. A normal distribution is completely characterized by its mean and variance. The mean specifies the expected average for the added delay and the variance characterizes how spread out the added delay is from the mean. To randomize the computation, we would like to choose mean and variance for $X$ such that the resulting computation timing for all three functions follow a similar pattern. To choose the mean, we look at the maximum and minimum possible timings for all three function computations. Figure 1 shows that the computation times for each activation function are scattered into specific "clusters" depending on the input. In particular, there are two, three, and three clusters for ReLU, sigmoid, and tanh respectively. For example, when the input is positive, the computation time for ReLu is almost $2.09\times 10^{-5}\ s$. And when the input is near 0, the computation time for tanh is around $4.4\times 10^{-4}\ s$. We have calculated the mean for the slowest cluster (i.e. the slowest cluster in tanh computation) which is $5.9\times 10^{-4}\ s$ and the mean for the fastest cluster (i.e. the faster cluster in ReLU computations) which is $2.06\times 10^{-5}\ s$. We have decided to choose the mean of $X$ to be 0.6 milliseconds so that the very fast computations will have a chance to have comparable computation time as the slow ones. Furthermore, to remove the distinct clusters in Figure 1, we need to choose a variance big enough that the differences caused by the input values are negligible. Table 1 summarizes the data for computation times and we can see that the maximum is about $6\times 10^{-4}\ s$ and the minimum is about $2\times 10^{-5}\ s$. Thus we decided to choose the variance to have the same order of magnitude6 as $6\times 10^{-4}-2\times 10^{-5}$, i.e. $0.1\times 10^{-4}\ s^{2}$. Footnote 6: The order of magnitude of a number $n$ is given by $b_{n}$ such that we can write $n$ in the form $n=a\times 10^{b_{n}}$, where $1/\sqrt{10}\leq a<\sqrt{10}$. In summary, we add random desynchronization, whose delayed computation time is a sample from a normal distribution with mean $6\times 10^{-4}\ s$ and variance $0.1\times 10^{-4}\ s^{2}$, to all three activation functions. 2000 random inputs are given to each of the functions. The resulting computation times are shown in Figure 2 and Table 2. In general, we propose the following steps for choosing a desynchronization-based countermeasure for protecting the computation of activation functions: 1. Collect computation time data for all three activation functions. 2. Compute the average of the fastest cluster of timings, denoted $t_{f}$, and the average of the slowest cluster denoted $t_{s}$. Let $\mu=t_{f}-t_{s}$. 3. Compute the difference between the longest computation time and the shortest computation time, say $\Delta t$ seconds. Let $d_{\Delta t}$ denote the order of magnitude of $\Delta t$. Let $\sigma^{2}=1\times 10^{-d_{\Delta t}}\ s^{2}$. \begin{table} \begin{tabular}{c\|c\|c\|c} \hline Activation Function & Mean & Minimum & Maximum \\ \hline Relu & 0.0207 & 0.0206 & 0.0209 \\ sigmoid & 0.4485 & 0.3920 & 0.4845 \\ tanh & 0.5170 & 0.4375 & 0.5985 \\ \hline \end{tabular} \end{table} Table 1: Computation time (in milliseconds) for different activation functions. 4. Add random desynchronizations, whose delayed computation time is a sample from a normal distribution with mean $\mu$ and variance $\sigma^{2}$, to the implementations of the activation functions. ## 5 Leakage Assessment of a Neuron Computation In order to evaluate the performance, we used Test Vector Leakage Assessment (TVLA) [20]. The idea was to perform a t-test on a dataset from fixed vs random inputs, and in our case, on measured timing from the inference execution. Here, we detail the computations for ReLU. Those for tanh and sigmoid are done similarly. First, we compute the execution time of when the neural network 7 is running the inference for a fixed input (which is chosen randomly at the beginning but fixed for the rest of experiment). We measured the timing of 5000 inference executions. Let us denote those timings by $x_{1},x_{2},\ldots,x_{5000}$. Then, similarly, we calculate the execution times for given random inputs, where each is chosen randomly and different from each inference execution. Let us denote the corresponding timings by $y_{1},y_{2},\ldots,y_{5000}$. By the TVLA method, we compute Footnote 7: We are using a similar model architecture as [4] for MNIST dataset as a proof of concept. \[t=\frac{\overline{x}-\overline{y}}{\sqrt{\frac{\sigma_{x}^{2}}{5000}+\frac{ \sigma_{y}^{2}}{5000}}},\] \begin{table} \begin{tabular}{c\|c\|c\|c} \hline Activation Function & Mean & Minimum & Maximum \\ \hline Relu & 6.31 & 2.69 & 9.93 \\ sigmoid & 6.72 & 3.11 & 10.01 \\ tanh & 6.81 & 3.40 & 9.88 \\ \hline \end{tabular} \end{table} Table 2: Computation time (in milliseconds) for different activation functions with random desynchronization. Figure 2: Timing behavior of activation functions with an applied desynchronization-based countermeasure. The timing patterns are not distinguishable. where $\overline{x}$, $\overline{y}$, $\sigma_{x}^{2}$ and $\sigma_{y}^{2}$ are the mean of $x_{i}$, mean of $y_{i}$, variance of $x_{i}$ and variance of $y_{i}$ respectively. In case the absolute values of $t$, called $\|t\|$-values, cross the threshold of 4.5, it can be concluded there is a data-dependent leakage in the measured traces. As can be seen in Figure 3, the TVLA test for the timing of the activation functions shows leakage, as expected. On the other hand, after the application of the proposed countermeasure, $\|t\|$-values stay below the threshold, as can be seen in Figure 4. ## 6 Discussion ### Overheads While it might seem from the figures that the timing overhead of the countermeasure is relatively high, it is to be noted that the activation function is only a small part of the entire neural network computation. The majority of the computation is spent on multiplications that are dependent on the number of neurons in the layers. For example, using the device from Section 3.1, the time for computation of one multiplication is roughly $1.165\times 10^{-5}$ seconds and that for addition is about $1.124\times 10^{-5}$ seconds. The computation timing of the activation function is roughly $0.21-5.99\times 10^{-4}$ seconds without desynchronization (Table 1) and Figure 3: TVLA results for ReLU, sigmoid, and tanh without the application of the countermeasure. $3.11-10.01\times 10^{-3}$ seconds with random desynchronization (Table 2). In modern architectures, there are thousands of multiplications and additions with just one activation function computation for one neuron. VGG-19 [21], one of the popular public networks for ImageNet classification challenge, has 4096 neurons in the last hidden layer and 1000 neurons for the output layer, which amounts to 4096 multiplications and 4096 additions for each output neuron. In this case, the computation time for multiplications and additions in one neuron is about 0.09 seconds, and that for the whole neuron computation is $0.09002-0.0906$ seconds without desynchronization or $0.093-0.1$ seconds with the desynchronization countermeasure. Thus, the overhead for the computation of one neuron is between $2.6\%-11\%$. We would also like to note that this is purely the timing for the calculations, not taking into account the memory operations - if the entire computation is considered, the overhead would be even lower. As this is a proof-of-concept and still a work in progress, we observe a positive trend and will be further investigating this in future works. ### Other Activation Functions Batina, _et al._[4] show results on softmax activation function apart from the three functions that were analyzed in this work. The softmax function is normally used in the output layer of a neural network to transform the raw outputs (logits) into a vector of probabilities. As it is unusual to find softmax in other layers of the network, we did not consider this activation function in our work. There are other activation functions that can be used, for example leaky ReLU, exponential linear unit (ELU), etc. We argue that the process of applying the desynchronization-based countermeasure on these functions is the same as described in Section 4. Figure 4: TVLA results for ReLU, sigmoid, and tanh with the application of the desynchronization-based countermeasure. Conclusion SCA has been a threat for neural networks model extraction, as it could perform reverse engineering to reconstruct the secret parameters of the networks. One of the main critical component of the network is the activation function, which as shown in previous works, is vulnerable against timing attack. In this work, we have investigated desynchronization-based countermeasures, from SCA domain to hide the timing leakage behavior. Our experimental results then shown that desynchronization based approach can succesfully hide the timing leakage information which could be used to prevent model extraction attack of the activation parameter. The overhead ultimately depends on the number of neurons in the fully-connected layer, for example, in the case of 4096 neurons in VGG-19, the overheads are between 2.8% and 11%. ## Acknowledgement This work has been supported in parts by the "University SAL Labs" initiative of Silicon Austria Labs (SAL) and its Austrian partner universities for applied fundamental research for electronic based systems. This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Programme SASPRO 2 COFUND Marie Sklodowska-Curie grant agreement No. 945478.
2306.02039	In Search of Global 21-cm Signal using Artificial Neural Network in light of ARCADE 2	Understanding the astrophysical nature of the first stars still remains an unsolved problem in cosmology. The redshifted global 21-cm signal $(\text{T}_{21})$ acts as a treasure trove to probe the Cosmic Dawn era -- when the intergalactic medium was mostly neutral. Many experiments, like SARAS 3, EDGES, and DARE have been proposed to probe the cosmic dawn era. However, extracting the faint cosmological signal buried inside a brighter foreground $\mathcal{O}(10^4)$ remains challenging. Thus we use two artificial neural networks, one for extraction of foreground, via parameter estimation with R-square $(R^2)$ score $(0.8034 - 0.9984)$, from the total sky-averaged spectrum. The other is for extraction of a global 21-cm signal in the presence of noise with $R^2$ score $(0.6960 - 0.9978)$. Considering an excess radio background scenario, we constructed all possible $\text{T}_{21}$ signals in the EDGES limit, along with the foreground signal, to train the neural networks. Here, we also explore the variation in parameter estimation due to the presence of heating of intergalactic medium by background radio radiation mediated via Ly$\alpha$ photons from first stars, and we found that the presence and absence of this effect can change the global 21-cm signal estimation by $\sim 33$ mK in the EDGES limit $(\sim -0.5\text{ K})$	Vivekanand Mohapatra, Johnny J, Pravin Kumar Natwariya, Jishnu Goswami, Alekha C. Nayak	2023-06-03T07:34:45Z	http://arxiv.org/abs/2306.02039v2	# In Search of Global 21-cm Signal using Artificial Neural Network in light of EDGES and ARCADE 2 ###### Abstract Understanding the astrophysical nature of the first stars still remains an unsolved problem in cosmology. The redshifted global 21-cm signal and power spectrum act as a treasure trove to probe the Cosmic Dawn era-- when the intergalactic medium was mostly neutral. Many experiments, like SARAS 3, SKA, EDGES, DARE, etc., have been proposed to probe the cosmic dawn era. However, extracting the faint cosmological signal buried inside the brighter foregrounds $\mathcal{O}(10^{4})$ remains challenging. Considering the excess radio background, we have constructed all possible $T_{21}$ signals in the EDGES limit. We have used a single Artificial Neural Network for $T_{21}$ parameter extraction in the presence of the foreground and noise with Root Mean Square Error (RMSE) and R-Squared (R2) score of $(0.2-0.08)$ and $(0.66-0.94)$, respectively. Here, we also explore the parameter estimation in the presence of heating of intergalactic medium due to background radio radiation mediated by Ly$\alpha$ photons from first stars, and we found that the effect indeed has a significant impact on parameters correlation and their estimation. keywords: cosmology: cosmic dawn, reionization, global 21-cm signal, EDGES, ARCADE 2, method: artificial neural network ## 1 Introduction The evolution of the Intergalactic Medium (IGM) during the Cosmic Dawn (CD) era and the Epoch of Reionization (EoR) lacks a comprehensive understanding due to the uncertainty in the known physics of the formation and evolution of the first stars and galaxies in the Universe. The Global 21-cm signal (Pritchard & Loeb (2012); Furlanetto et al. (2006); Morales & Wyithe (2010)), 21-cm power spectrum (Bharadwaj & Sethi (2001); Bharadwaj & Ali (2004); Morales (2005); Zaldarriaga et al. (2004)), CMB anisotropy and polarization (Sehi & Subramanian (2005); Trivedi et al. (2012, 2014)) are some of the underlined methods to probe these epochs. Many experiments have been conducted, like, SARAS (Patra et al. (2013); Singh et al. (2017)), SARAS 3 (Bevins et al. (2022)), REACH (de Lera Acedo et al. (2022)), SCI-HI (Voytek et al. (2014)), HERA (DeBoer et al. (2017)), DARE (Burns et al. (2012, 2017)) etc., to measure these. Measuring the redshifted HI-21cm signal might act as a treasure trove to probe these epochs. Recently, EDGES collaboration has reported a 21-cm signal with the absorption of $\sim 0.5^{-0.2}_{-0.5}$ K in the redshift range $15-20$-- which is twice the value predicted by the standard $\Lambda$CDM model of cosmology (Bowman et al. (2018)). Although SARAS 3 (Bevins et al. (2022)) have rejected the EDGES signal with 95.3% confidence level after conducting an independent check, yet the actual shape is unknown. If future experiments confirm a trough of amplitude greater than $\sim 0.2$ K, which is the standard value, that could lead to a completely new insight into the physics of these epochs. Following the EDGES detection, several models have been proposed to construct all possible shapes and amplitudes of the global 21-cm signal. In the references (Barkana et al. (2018); Fraser et al. (2018); Slatyer & Wu (2018); Pospelov et al. (2018)), the authors have considered excess cooling of IGM gas by Dark Matter-Baryon scattering, (Bhatt et al. (2020); Natwariya & Bhatt (2020)) have considered Primordial Magnetic Fields (PMFs) and DM-Baryon Scattering in presence of PMFs, (Narwariya et al. (2021); Natwariya & Nayak (2022)) have considered energy injected due to Primordial Black Holes (PBHs) and decaying sterile neutrinos. In addition to that, (Fialkov et al. (2018); Fraser et al. (2018); Feng & Holder (2018)) have considered excess-radio background radiation, which could be due to several phenomena, like stimulated emission from Bose (axion) star (Levkov et al. (2020)), radio emission from Pop III black holes (Mebane et al. (2020)), conversion axion-photon in presence of magnetic field (Moroi et al. (2018)). Recently, ARCADE 2 (Feng & Holder (2018); Fixsen et al. (2011)) and LWA1 (Dowell & Taylor (2018)) have detected an excess-radio background in the frequency range of $3-10$ GHz and $40-80$ MHz, respectively. ARCADE 2 detection mimics the Cosmic Microwave Background Radiation (CMBR) for a frequency $\nu>10$ GHz but deviated significantly otherwise. These detections have been modelled by a power law with a spectral index ($\beta$) of $-2.62\pm 0.04$ and $-2.58\pm 0.05$ for ARCADE 2 and LWA1, respectively. Detecting the 21-cm signal (mK) buried in a sea full of bright foreground radiation of the $\mathcal{O}(10^{4})$ stronger is an observation challenge. In addition to that, the ionospheric effect, radio frequency interference (RFI), and instrument response make it a tougher job. These effects, especially ionospheric distortion and RFI, can be reduced significantly if one considers far-side Moon-based experiments like DARE (Burns et al. (2012, 2017)). One of the most common techniques adapted to remove the foreground radiation is considering it well-characterized and spectrally smooth. After removing the foreground radiation, the residual contains the global 21-cm signal having the signature of the IGM evolution. The Machine Learning (ML) technique has been used previously by many authors to study the CD and EoR. (Shimabukuro and Semelin (2017); Schmit and Pritchard (2018)) have used ANN for parameter extraction from the 21-cm power spectrum. In the references (Jennings et al. (2019); Hassan et al. (2019); Chardin et al. (2019); Gillet et al. (2019)), the convolutional neural network (CNN) has been used to study and extract parameters from the 21-cm maps. Here, the aforementioned authors have used 21-cm signal and reionization simulations for parameter extraction (Choudhury et al. (2020, 2021)). The authors have also included foreground contamination for parameter extraction from synthetic data. In the present work, we have used ANN to extract parameters associated with IGM properties via the global 21-cm signal. First, we generate the datasets for the 21-cm signal by varying the free parameters described in section (4). Here we have generated two types of 21-cm signal datasets: one consists of the IGM gas heating due to the radio background mediated by Ly$\alpha$ radiation from first stars (Venumadhav et al. (2018)), and in the second type of dataset we do not include this effect. After the generation of the datasets, we add foreground radiation and noise for training and prediction purposes. This work is organised as follows: In section (2) briefly introduces the global 21-cm signal and excess background radiation. In section (3), the evolution of the IGM temperature and the importance of IGM gas heating due to the radio background mediated by Ly$\alpha$ radiation from first stars in excess background radiation. In section (4 - 5), construction of the global 21-cm signal and foreground along with the corresponding associated parameters. In section (6), we briefly introduce the ANN, construction of training and prediction data sets. In section (7), we estimate the parameter from noisy data and reconstruct the global 21-cm signal. Section (8) includes a summary, conclusion and outlook. ## 2 The global 21-cm signal The hyperfine transition between 1S singlet ($n_{0}$) and triplet ($n_{1}$) states in a neutral hydrogen atom (HI) occur due to the interaction of proton and electron spin. The relative number density of neutral hydrogen in triplet and singlet states is characterized by spin temperature ($T_{s}$) \[\frac{n_{1}}{n_{0}}=\frac{\delta_{1}}{g_{0}}e^{-2\pi\nu_{21}/T_{s}}=3\times e ^{-T_{s}/T_{s}} \tag{1}\] where, $g_{1}=3$, $g_{0}=1$ denotes statistical weight of the respective states, $\nu_{21}=1420$ MHz is frequency of the photon and $T_{s}=2\pi\nu_{21}$. The 21-cm differential brightness temperature ($T_{21}$), measured relative to the cosmic background radiation (Pritchard and Loeb (2012); Furlanetto and Pritchard (2006); Mesinger et al. (2011); Mesinger and Furlanetto (2007)) is given by: \[T_{21}\approx 27_{\rm{HI}}\left(\frac{0.15}{\Omega_{m}h^{2}}\frac{1+z}{10} \right)^{1/2}\left(\frac{\Omega_{b}h^{2}}{0.023}\right)\left(1-\frac{T_{r}}{T _{s}}\right){\rm{mK}} \tag{2}\] where, $\Omega_{b}$ and $\Omega_{m}$ represent baryon and total matter density parameter in the unit of critical density, $x_{\rm{HI}}$ denotes the fraction of neutral hydrogen atom, $h$ represents the Hubble parameter (Aghanim et al. (2020)), $T_{r}$ denotes background radiation. After recombination, the baryon number density predominantly contained neutral hydrogen atoms, residual electrons ($n_{e}$) and residual protons ($n_{p}$) making the global 21-cm signal a useful probe to study CD and EoR. It can provide information about first-star formation, X-ray heating, radio background heating, and excessive background radiation. Throughout this work, we use Eq. (2) to construct the global 21-cm signal. The most crucial quantities in Eq. (2) are $x_{\rm{HI}},T_{s},$ and $T_{r}$ which determines the intensity of $T_{21}$. The spin temperature ($T_{s}$) given by \[\begin{split} T_{s}^{-1}&=\frac{T_{r}^{-1}+x_{ \alpha}T_{\alpha}^{-1}+x_{c}T_{g}^{-1}}{1+x_{\alpha}+x_{c}}\\ x_{\alpha}&=\frac{T_{21}}{T_{r}}\frac{N_{i}k_{\rm{ HI}}^{4}}{A_{10}}\quad\mbox{and}\quad x_{c}=\frac{T_{21}}{T_{r}}\frac{4P_{ \alpha}}{27A_{10}}\end{split} \tag{3}\] where, $x_{\alpha}$, $x_{c}$ denotes the Ly$\alpha$ and collision coupling coefficient about the excess-radio background radiation, respectively, whereas $N_{i}$, $k_{\rm{HI}}^{4},P_{\alpha}$, $A_{10}$ respectively represent number density of the species "$i$", the spin de-excitation rate coefficient due to collisions of species "$i$" with the hydrogen atom, scattering rate of Ly$\alpha$ (Lyman Alpha) radiation, and the Einstein coefficient for spontaneous emission from triplet to the singlet state. $T_{\alpha}$, $T_{g}$ respectively denotes the colour and kinetic gas temperatures. The three dominant non-exotic processes that determine ($T_{r}/T_{s}$) are: (1) absorption and spontaneous emission of background radiation; (2) collision with the hydrogen atoms, residual electrons and protons; (3) the Wouthuysen-Field (WF) effect (Wouthuysen (1952); Field (1959)) which represents excitation and de-excitation of Ly$\alpha$ photons in HI atom. As discussed in section (1), the excess radio background radiation cannot be denied completely. ARCADE 2 and LWA1 measurements inspired uniform redshift-independent synchrotron-like radiation have been considered to represent such radiation, as shown in Fig. (1), whose phenomenological model is provided here (Fialkov and Barkana (2019); Mondal et al. (2020); Banet et al. (2021); Reis et al. (2020); Yang (2018)) \[T_{r}=T_{\rm{cmb,0}}(1+z)\left[1+A_{r}\left(\frac{\nu_{\rm{obs}}}{78\;{\rm MHz }}\right)^{\beta}\right] \tag{4}\] where, $T_{\rm{cmb,0}}=2.723$ K is the present-day CMB temperature and $\beta=-2.6$ is the spectral index. For 21-cm signal $\nu_{\rm{obs}}=1420$ MHz$(1+z)$ and $A_{r}$ is the amplitude defined in the reference of CMB at frequency 78 MHz. Considering uniform radiation excess from the dark age, cosmic dawn, and reionization, author (Fialkov and Barkana (2019)) have limited the excess background amplitude to be $1.9<A_{r}<418$ at the reference frequency of 78 MHz. As discussed here (Dowell and Taylor (2018)), LWA1 sets the limit on $A_{r}\sim 418$. The Low Frequency Array (LOFAR) imposes a strict limit on excess radiation of $A_{r}<182$ (95% CL) and $A_{r}<259$ (99% CL). In the review (Pritchard and Loeb (2012)), the evolution of the global 21-cm signal in CMB bath has been described in detail; here, we will briefly explain the same. At the end of recombination, which occurred at a redshift ($z\approx 1100$) neutral hydrogen atoms were formed, and the photons were free to travel known as CMB. This period is often called the last scattering surface. Due to the Compton scattering, the CMB and gas were in thermal equilibrium, ($T_{cmb}\approx T_{g}$), till $z\sim 200$ causing the absence of the 21-cm signal. This era is called the Dark Ages. Due to cosmic expansion, $T_{g}$ and $T_{cmb}$ cooled down $\propto(1+z)^{2}$ and $\propto(1+z)$ respectively, over the time. For redshifts $40\lesssim z\lesssim 200$, $x_{c}\ll 1$, which causes an absorption signal. Nevertheless, this absorption signal has not been observed yet due to radio antennas' poor sensitivity, which falls dramatically for frequencies below 50 MHz. At $z<40$ till the formation of first star the $x_{e}\approx 0$ causing no 21-cm signal (Barkana et al. (2018); Pritchard and Loeb (2012)). Ly$\alpha$ radiation coupled the gas via WF effect after the first star formation at redshift $z\sim 30-25$ making $x_{\alpha}\gg 1$. During this time, an absorption signal can be observed, and this phase is called the Cosmic Dawn (CD). Around $z\sim 15$, X-ray radiation from AGN could have heated the gas, causing an emission signal. After a certain period, at $z\sim 7-5;x_{e}\to 1$ resulting in no signal. This era is called the Epoch of Reionization (EoR). Instead of considering a wide range of redshifts in this work, we have considered EDGES reference $z\sim 27-14$(Bowman et al. (2018)). ## 3 Evolution of gas temperature The standard evolution of gas temperature ($T_{g}$) and ionization fraction ($x_{i}\approx(1-x_{e})$) with redshift without any exotic energy injection can be written as \[\begin{split}\frac{dx_{e}}{dz}&=\frac{\mathcal{P}}{ (1+z)H}\left[n_{H}x_{e}^{2}\alpha_{B}-(1-x_{e})\beta_{B}e^{-E_{\alpha}/T_{ \rm cmb}}\right]\\ \frac{dT_{B}}{dz}&=2\frac{Tg}{1+z}+\frac{\Gamma_{c} }{(1+z)H}\left(T_{g}-T_{\rm cmb}\right)\\ \Gamma_{c}&=\frac{8N_{e}\sigma_{T}\alpha_{T}T_{\rm cmb }^{4}}{3m_{e}N_{\rm tot}}\\ \mathcal{P}&=\frac{1+\mathcal{K}H\Lambda_{H}n_{H}(1 -x_{e})}{1+\mathcal{K}H\left(\Lambda_{H}+\beta_{H}\right)n_{H}(1-x_{e})}\end{split} \tag{5}\] Here, $\mathcal{P}$ is Rebless coefficient (D'Amico et al. (2018); Peebles (1968)), $\mathcal{K}H\equiv 7^{2}/(E_{\alpha}^{3}H)$ and $\Lambda_{H}=8.22$/sec represents redshifting of Ly$\alpha$ photons and 2S-1S level two-photon decay rate in hydrogen atom (Tung et al. (1984)). $\alpha_{B}$ and $\beta_{B}$ are the case-B recombination and photo-ionization rate respectively (Mitridate and Podo (2018); Seager et al. (1999, 2000)). $N_{\rm tot}=N_{H}(1+f_{He}+xe)$, $f_{He}=0.08,T_{\rm cmb}=T_{\rm cmb,0}(1+z)\cdot a_{r}=7.57\times 10^{-16}$ Jm${}^{-3}$K${}^{-4}$ is the radiation density constant, $\sigma_{T}$ is the Thomson scattering cross-section, and $m_{e}$ is the mass of electron. During the early stages ($z\gtrsim 200$), $T_{g}$ was in equilibrium with the cosmic microwave background temperature (CMB) due to Compton scattering. However, the influence of non-thermal excess radio radiation on the gas temperature is insignificant in the second term and can be ignored (Feng and Holder (2018)). After the formation of the first stars (at $z\sim 30$), their radiation begins to heat the IGM. In addition to X-ray heating, the gas temperature may increase by $\sim 10\%$ due to CMB only, even in the absence of X-ray heating or excess radio background-- from now on we write this heating effect as "VDKZ18" (Venumadhav et al. (2018)). We analyse our final results in both scenarios: with and without the VDKZ18 heating effect due to excess radio background. In presence of X-ray heating and VDKZ18, the $T_{g}$ in equation (5) will be modified as: \[\begin{split}\frac{dT_{g}}{dz}&=2\frac{Tg}{1+z}+ \frac{\Gamma_{c}}{(1+z)H}\left(T_{g}-T_{\rm cmb}\right)+\frac{dT_{g}}{dz} \bigg{\|}_{\rm X-ray}+\frac{N_{H}\Gamma_{r}}{(1+z)N_{\rm tot}}\\ \Gamma_{r}&=\left(1-\frac{T_{r}}{T_{s}}\right)\frac{A _{10}T_{10}}{2H}x_{\rm HI}x_{r}\end{split} \tag{6}\] here, $x_{r}=1/\tau_{21}\times[1-\exp(-\tau_{21})]$, and the 21-cm optical depth $\tau_{21}=8.1\times 10^{-2}x_{\rm HI}[(1+z)/20]^{1.5}(10\rm K/T_{S})$. Motivated by references (Kovetz et al. (2018); Mirocha et al. (2015); Harker et al. (2016)), to account for the impact of X-ray heating and ionization of the IGM we adopt tanh parameterization. For the evolution of ionization fraction due to X-ray heating, we use $x_{e}=x_{e}^{\prime}(1+\tanh[(x_{e0}^{\prime}-z)/\delta z)]$ with $(x_{e}^{\prime},x_{e0}^{\prime},\delta z)=(1,9,3)$ as the corresponding values. The chosen model is based on the one proposed in Ref. (Kovetz et al. (2018)), and the same fiducial model is considered for the present case. In Fig. (2), we examined the impact of VDKZ18 and X-ray heating on IGM. Focusing on the red solid line of Fig. (2a), which represents VDKZ18 heating without excess background radiation ($A_{r}=0$), i.e. only CMB as background radiation. As we increase the value of $A_{r}$ from $0\to 418$, the gas temperature increases substantially. However, it can be seen that for $A_{r}$ in the range $(300-418)$ the curves clumped signifying no significant change in $T_{g}$. This is because $\Gamma_{r}\propto(T_{r}/T_{s}-1)\sim T_{r}/T_{s}$ in Eq. (6), and as we increase $A_{r}$, $T_{r}/T_{s}$ increases slowly-- as also shown by the reference (Natwariya (2021)). We have also investigated the role of X-ray heating after the inclusion of the VDKZ18-- shown by Fig. (2b). The gas temperature changes significantly when we increase $A_{r}$ from 0 to 200 depicted by the red and blue solid lines. However, as soon as we increase the $A_{r}$ value further, $T_{g}$ almost remains constant as shown by the yellow and cyan solid line. Thus along with X-ray heating, it is important to incorporate VDKZ18 heating when there is excess background radiation, especially in LOFAR limit, as it can lead to a substantial increase in $T_{g}$ in the redshift range of $z<15$, which is of particular interest for reionization models. ## 4 Generating the global 21-cm signal Many theoretical models propose that the shape and amplitude of the cosmic global 21-cm signal may change. (Fialkov et al. (2014, 2015)) and (Cohen et al. (2017)) employed a semi-numerical technique to predict possible 21-cm global signals between the redshift range of $z\sim 6-40$. The cosmic signal is parameterized by using physical characteristics-- intimately connected to the IGM features, allowing us to deduce the physics of the earliest source. (Bernardi et al. (2015, 2016)) represent the absorption characteristic as Gaussian. Similarly, (Pritchard and Loeb (2010)) proposed a turning point model, which Figure 1: The excess background radiation $T_{r}$ as a function of redshift ($z$) for different values of the amplitude $A_{r}$ given in Eq. (4). Here, $A_{r}=0$ represents the standard CMB radiation $T_{cmb}(z)$. On increasing $A_{r}$ from $0-140$, $T_{r}$ changes significantly. However, as soon as it crosses $A_{r}=200$ the curves become closely spaced. indicates positions in redshift, and amplitude and shape of brightness temperature. The section of the global signal between the turning points is represented as a cubic spline, which allows great flexibility and may describe a variety of 21-cm signals. Nevertheless, additional interpretation is required to link the turning point positions to the physics of the initial luminous sources. Mirocha et al. (2013, 2015); Harker (2015) proposed the tanh parameterization, which uses a succession of tanh functions to simulate the global signal. Although it is possible to use a fixed value for $x_{\sigma}\sim 10^{-4}$ here (Bharadwaj & Ali, 2004; Choudhury et al., 2021)), we instead consider solving all coupled equations. We solve the IGM temperature evolution Eq. (6) and evolution of ionization fraction in Eq. (5) to construct a simulated global 21-cm signal in this work. The initial conditions for $T_{g}$ and $x_{\sigma}$ are taken from RECFAST++ (Seger et al., 1999). Ly$\alpha$ background influences the amplitude of $T_{21}(z)$ via WF coupling ($x_{\alpha}$) and $T_{21}$ as described in Eq. (3), thus we adopt a tanh parameterization inspired from Ref. (Kovetz et al., 2018; Natwariya, 2021; Harker et al., 2016)). Other than $A_{r}$ in Eq. (4) which act as a free parameter (Feng & Holder (2018); Singal et al. (2018); Fixsen et al. (2011)), the two quantities ($x_{\alpha}$ and $T_{\rm Xiny}$) are allowed to evolve as a tanh function given by, \[{\rm C}={\rm C}_{\rm ref}\left(1+\tanh\left\{\frac{C_{z0}-z}{C_{dz}}\right\}\right) \tag{7}\] where, ${\rm C}$ represents the cosmological parameters $x_{\alpha}$ and $T_{\rm Xray}$. $C_{dz}$, ${\rm C}_{\rm ref}$ and $C_{z0}$ represent the duration, step height and pivot redshift, respectively. Because these characteristics are directly related to IGM characteristics rather than luminous source features, they bridge the physical and phenomenological models such as spline or Gaussian models. In this work, We used one non-tanh function ($A_{r}$) and two tanh functions ($T_{\rm Xray}$ and $x_{\alpha}$). The tanh parameterized parameters, as a function of three independent parameters, are mentioned below. \[\begin{split}\mathrm{x}_{\alpha}&={\rm x}_{\rm ref} \left(1+\tanh\left\{\left(x_{z0}-z\right)/x_{dz}\right\}\right)\\ {\rm T}_{\rm Xray}&={\rm T}_{\rm ref}\left(1+\tanh \left\{\left(T_{z0}-z\right)/T_{dz}\right\}\right)\end{split} \tag{8}\] Here, we focus on CD and EoR, therefore, the standard global 21-cm signal is generated using the hyperbolic tangent (tanh) function in the redshift range $z\sim 22-14$, as presented in Figure (3a). As Figure 3: The parameterized global 21-cm signal in the presence of excess-radio background radiation. Fig. (3a) shows the construction of global 21-cm signal using tanh parameterization in the absence of excess-radio background radiation. Fig. (3b) shows a shift and increment in the amplitude of Fig. (3a) while considering $(0,0.1,1,10,100)$ % of the excess term in eq (4) for a constant $A_{r}=1.9$ value, where the solid and dashed curves depict presence and absence of the VDKZ18 respectively. Fig. (3c) shows the global 21-cm signal for different $A_{r}$ values. While the dashed horizontal line for all three figure sets $T_{21}=0$. Figure 2: Thermal evolution of gas temperature in the presence of X-ray and VDKZ18. Fig. (2a) shows that $T_{\rm g}\left(z\right)$ starts to vary due to VDKZ18 from $z\sim 20$ after the first star formation. The inclusion of excess-background radiation does change $T_{\rm g}$ with $A_{r}$. Fig. (2b) shows that X-ray heating changes $T_{\rm g}$ significantly, but it can be clearly seen that the existence of excess radiation shifts the $T_{\rm g}$ curve. In both plot, the shaded region shows $15<z<20$ region. explained in section (2), the 21-cm signal $T_{21}\neq 0$ for $14\lesssim z\lesssim 30$. At these redshift ranges, both X-ray and VDKZ18 heating dominate significantly. Therefore, we consider both IGM heating mechanisms in Eq. (6) for the evolution of the gas temperature along with the standard gas evolution equation. The inclusion of the VDKZ18 can change the amplitude and position of $T_{21}(z)$ even in the presence of a strong heating effect like X-ray heating (Fig. (a)a). In Fig. (b)b, we considered a value of $A_{r}=1.9$ indicating that such excess radiation would increase its amplitude significantly, reaching EDGES limit ($-500$ mK). But as soon as $A_{r}$ approaches a value above $200$, the troughs become closely spaced, as plotted in Fig. (c)c. Therefore, to generate the training data set, we have to consider the range $A_{r}\sim 0-200$. We have considered $T_{\text{ref}}=10^{3}$ K and $x_{e}^{\prime}$ to be fixed as its value saturates at lower redshifts. All other parameters, along with $A_{r}$ have been considered as free parameters, i.e., $x_{\text{ref}}$, $x_{20}$, $x_{d\text{z}}$, $x_{\text{z}^{\prime}0}^{\prime}$, $\delta z$, $T_{20}$, and $T_{d\text{z}}$, which we varied to generate global 21-cm signals of different shapes and amplitudes. The inferred value of the aforementioned free parameters for generating the standard curve (Fig. (a)a) is taken to be $(100,\ 17,\ 2,\ 9,\ 3,\ 12.75,\ 1)$ respectively (Kovetz et al. (2018); Naturiya (2021)). We varied $\left(x_{\text{ref}},x_{\text{z}00}.x_{\text{z}0}^{\prime},T_{20}\right)$ and $\left(x_{d\text{z}},\delta z,T_{d\text{z}}\right)$ by ($\pm 30\%$ and $\pm 10\%$) respectively, to generate the training signal keeping the range $-200$ mK $<T_{21}<-1000$ mK; in reference to EDGES limit. For $A_{r}$, we considered the minimum and maximum values to be $0$ and $200$. We have considered the minimum value to be $0$, as it depicts CMBR as the background radiation. The parameter range examined in this study adequately depicts a wide variety of global signal shapes, as shown in section (6.2). The choice of the $\tanh$ model was motivated by its ability to link it to the physical characteristics of the IGM. The $\tanh$ parameters are connected to the IGM properties but cannot provide direct information about the early source properties. Therefore, it may be regarded as an intermediate between purely physical models and entirely phenomenological models, such as the 'turning point' model as argued here (Choudhury et al. (2020)). Rather than solely parametrizing the $T_{21}(z)$ using a $\tanh$ function, we opted to use parameterization for the source (first star) related quantities, i.e., X-ray and Ly$\alpha$ while maintaining the conventional cosmological intergalactic medium evolution. In future work, we aim to include parameters that directly link to the astrophysical nature of first-star. ## 5 Foreground modelling Detecting the weak global 21-cm signal during CD and EoR is challenging due to the strong foreground radiation and various instrumental and atmospheric distortions. The radio emissions from the Milky Way and other extragalactic sources are significantly brighter than the cosmological signal. The expected strength of the global 21-cm signal is approximately $10^{-4}$ times weaker than the foreground emissions. Experiments face significant challenges in extracting the global 21-cm signal due to bright foregrounds, Radio Frequency Interference (RFI), and instrumental calibration errors. These challenges require sophisticated simulations to determine the impact of these factors on the extraction of the desired signal. Therefore, having a precise model for foregrounds at radio frequencies is crucial to ensure accurate prediction. According to (Pritchard & Loeb (2010), Bernardi et al. (2016), Bernardi et al. (2015)), the foreground spectrum can be represented as a polynomial in the $\ln(\text{T})-\ln(\nu)$. In Ref. (Harker et al. (2016)), it is demonstrated using a $3^{\text{rd}}$ or $4^{\text{th}}$-order polynomial is sufficient enough to model the sky spectrum. Still, it has been found that a $7^{\text{th}}$-order polynomial is necessary when incorporating the chromatic primary beam of the antenna. SARAS 3 (Singh et al. (2022)) have used a $6^{\text{th}}$-order $\ln(\text{T})-\ln(\nu)$ to model Galactic and extragalactic foreground passing though ionosphere added with systematic calibration error in a band of frequency $55-85$ MHz with a resolution of $61$ kHz. In our work, following the Ref. (Bowman et al. (2018)), we used $\ln(\text{T})-\ln(\nu)$ model of $5^{\text{th}}$ order polynomial to represent the Galactic synchronous radiation of spectral index $-2.5$ along with Earth's ionosphere distortion \[T_{\text{FG}}(\nu)=b_{0}\left(\frac{\nu}{\nu_{0}}\right)^{-2.5+b_{1}+b_{2}\ln( \nu/\nu_{0})}e^{-b_{2}(\nu/\nu_{0})^{-2}}+b_{4}\left(\frac{\nu}{\nu_{0}}\right) ^{-2} \tag{9}\] where, $b_{0}$ represents an overall foreground scale factor, $b_{1}$ considers correction to the foreground with specified spectral index, $b_{2}$ considers the higher-order spectral terms, $b_{3}$ for ionospheric absorption effect, and $b_{4}$ for emission from the hot electron in the ionosphere. Here, we have used the linearized form of Eq. (9), \[T_{\text{FG}}(\nu)\approx\left(\frac{\nu}{\nu_{0}}\right)^{-2.5} \left[b_{0}+b_{1}\ln(\nu/\nu_{0})+b_{2}(\ln(\nu/\nu_{0}))^{2}\right. \tag{10}\] \[+b_{3}(\nu/\nu_{0})^{-2}+b_{4}(\nu/\nu_{0})^{0.5}\right]\] Using the publicly available data by EDGES1, we estimated2 Eq. (10) coefficients to be $\left[b_{0},b_{1},b_{2},b_{3},b_{4}\right]=\left[711.8636,17.5110,-183.0153,171.6 506,699.3283\right]$ for $\nu_{0}=78$ MHz in the band of $51-100$ MHz, shown in Fig. (5). To generate the training set, shown in Fig. (6), we varied the free parameters by $\pm 20\%$. This training data includes an even distribution of $T_{FG}$ around the fitted data (Fig. (5)). The foreground is smoother than the cosmological signal, and we use this feature to extract the $T_{21}$ signal, as described in section (7). Footnote 1: [https://loco.lab.asu.edu/edges/edges-data-release/](https://loco.lab.asu.edu/edges/edges-data-release/) Footnote 2: We estimated the coefficients using a simple gradient descent method of Mean Square Error (MSE) cost function. The Python code can be found in [https://github.com/DarkMatter#03/Neural-Network-and-21-cm-Cosmology](https://github.com/DarkMatter#03/Neural-Network-and-21-cm-Cosmology) ## 6 Artificial neural network ### Overview This section briefly introduces the fundamental concepts of artificial neural networks (ANNs). A basic neural network comprises three principal layers: an input layer, one or more hidden layers, and an output layer. In a feed-forward and fully connected neural network, every neuron in a given layer is connected to every neuron in the subsequent layer, and the transmission of information is unidirectional. The neuron serves as the fundamental unit of an ANN. Each of these connections is associated with a weight and a bias. A cost or error function is computed following each forward pass in the output layer. During training, this cost function is optimized iteratively by back-propagating the errors. A thorough explanation of the fundamental algorithm employed in artificial neural networks can be found in (Choudhury et al. (2020, 2021); Olvera et al. (2022)). Our feed-forward network utilizes an ANN with two Dense layers, implemented using the Sequential Model from the PyTorch in Python. In the input layer, we have incorporated 1024 neurons to correspond with the 1024 frequency channels ranging from $53-98$ MHz. We have utilized standard scikitlearn(Pospelov et al. (2018)) and PyTorch modules to construct our network, determining the number and size of hidden layers to maximize network performance. The number of neurons in the output layer aligns with the number of output parameters we aim to predict. The following sections comprehensively describe the neural network's structure used in our study. ### Construction of Training data set We have generated the training data set as described in section (4 - 5) by varying the IGM and foreground parameters in the aforementioned range. The effect of VDKZ18 in the presence of excess background radiation can be seen in Fig. (3); hence two different training sets were built $\mathrm{T}_{21c}$ and $\mathrm{T}_{21wc}$ with represents 21-cm signal with and without VDKZ18 respectively. In both cases, we have kept the absorption amplitude's lower and upper limits to be $-200$ and $-1000$ mK. We kept our $\mathrm{\mathcal{T}}_{21}$ training set lower limit as $-200$ mK such it will include the possibility of extracting signal if found in future experiments. Taking into account a redshift range of $53-98$ MHz, we built both foreground and 21-cm signal training sets across 1024 channels and of the same order (mK) to avoid a biased prediction. The mock-training data set consists of four signals: \[\mathrm{T}_{\mathrm{tot}}^{1} =\mathrm{T}_{21c}(\nu) \tag{11}\] \[\mathrm{T}_{\mathrm{tot}}^{2} =\mathrm{T}_{21cw}(\nu)\] \[\mathrm{T}_{\mathrm{tot}}^{3} =\mathrm{T}_{21c}(\nu)+\mathrm{TF}_{\mathrm{G}}(\nu)\] The Pearson correlation3 between all cosmological ($\mathrm{\mathcal{T}}_{21}$) param Figure 4: Construction of the global 21-cm signal (a) in the absence of the VDKZ18 and (b) in the presence of the VDKZ18. In both cases, we have kept the absorption amplitude to be in the range of $-200$mK $<\mathrm{\mathcal{T}}_{21}<-1000$mK following the EDGES limit. Figure 5: Total sky averaged spectrum considered to represent the foreground radiation. Figure 6: The foreground modelling training data set used in the neural network. We have generated this dataset using Eq. (10). As expected, it can be seen that the foreground is $\sim 10^{4}$ brighter than the 21-cm cosmological signal. eters can be analyzed from Fig. (8), which has been used during the training process. It can be seen that $x_{x0}$ and $T_{x0}$ are negatively correlated to each other. At the same time, they are negatively and positively correlated to $A_{r}$, respectively, in the case of the absence of the VDKZ18(Fig. (8a)). In the presence of the VDKZ18(Fig. (8b)), $x_{x0}$ and $T_{x0}$ are positively correlated to each other while being uncorrelated to $A_{r}$. In section (6.3), we added noise to the above training set for parameter prediction. ### Predicting Dataset In this work, we have constructed two sets of predicting data sets, as mentioned above, by adding noise. We added the noise ($\sigma_{t}(\nu)$) (Bowman et al. (2018)) to every $\mathbf{T}_{\text{tot}}^{l}$ in Eq. (11) i.e., \[\mathbf{T}_{\text{pred}}^{l}=\mathbf{T}_{\text{tot}}^{l}+\sigma_{t}(\nu) \tag{12}\] where, $\sigma_{t}(\nu)$ is the thermal noise which can be re-written from the standard radiometer equation as a function of bandwidth $\Delta\nu$ and observational time $\tau=10^{6}\times 3600\times\mathbf{N}_{t}$; $\mathbf{N}_{t}$ being the hours of observation. \[\sigma_{t}(\nu)=\frac{\mathbf{T}_{sys}(\nu)}{\sqrt{\delta\nu\cdot\tau}} \tag{13}\] The $\mathbf{T}_{sys}(\nu)$ can be replaced with $T_{\text{FG}}(\nu)$ for our work. It can be seen that $\sigma_{t}(\nu)\propto 1/\sqrt{\mathbf{N}_{t}}$, thus for this work we considered $\mathbf{N}_{t}=1000$ hr of observational time. While the training set in Eq. (11) monitors the neural network's performance, the prediction data set in Eq. (12) has been used for evaluating the accuracy of parameter estimation. The Root mean square error (RMSE) and R2 score have been given to each parameter estimation as metrics for all prediction datasets to depict the accuracy. The RMSE and R2 score can be expressed as \[\text{R2}=1-\frac{\sum\left(X_{\text{pred}}-X_{\text{orig}}\right)^{2}}{\sum \left(X_{\text{orig}}-\bar{X}_{\text{orig}}\right)^{2}} \tag{14}\] \[\text{RMSE}=\sqrt{\frac{1}{\text{Npred}}\sum_{j=1}^{\text{Npred}}\left(\frac {X_{\text{orig},j}-X_{\text{pred},j}}{X_{\text{orig},j}}\right)^{2}} \tag{15}\] where $\bar{X}_{\text{orig}}$ is the average of original parameters which has been summed over all training data. The R2 value lies in the interval $(0,1]$ where R2 = 1 represents perfect inference. ## 7 Result ### Only 21-cm Signal We build two neural networks of $(1024,6)$ neuron input and output layer corresponding to the frequency channel used in generating the global 21-cm signal shown in Fig. (4). We added three hidden layers of 64, 32, and 16 neurons with activation function ReLU(Shang et al. (2016)) to make the network deep. We used nn module of PyTorch(Paszke et al. (2019)), which contains Sequential API for the feed-forward process. We considered 2000 sample data as input which was split into training and testing dataset ($8:2$) using sklearn(Pedregosa et al. (2011)). Before feeding the training data to the neural network, we preprocessed it using 'MinMaxScaler' from sklearn. To adjust the weight of the hidden layers optimally over the training process, we used Mean Square Error (MSE) as the error function and 'Adam'(Kingma & Ba (2014)) as the optimizer. After tuning all the hyperparameters, the model was trained and saved for prediction. As shown in Fig. (9), the model loss approached a constant value after 500 epochs. To check the performance of the neural network after training, we predicted 6 parameters $\left(A_{r},x_{\text{ref}},x_{\text{20}},T_{\text{20}},x_{\text{-}},T_{\text{ 2dz}}\right)$ by assigning Root Mean Square Error (RMSE) and R-Squared (R2) score to it, as shown in the Fig. (10, 11). The use of a single ANN limits us from extracting a few parameters efficiently, thus $x_{e}$ parameters are not considered. We find that the RMSE and R2 scores around $(0.054-0.011)$ and $(0.99-0.92)$, respectively, in the presence of VDKZ18; while in the absence of VDKZ18, the values are around $(0.012-0.068)$ and $(0.99-0.87)$ respectively. ### 21-cm Signal and Noise In section (7.1), we predicted the parameters after training the neural network with $T_{21}$ signal. In this section, to check the robustness of the network, we included noise $\sigma_{t}(\nu)$, as defined in section (6.3), to the prediction dataset. \[\mathbf{T}_{\text{pred}}^{1}=\mathbf{T}_{\text{tot}}^{1}+\sigma_{t}(\nu) \tag{16}\] We constructed another neural network using the technique mentioned in section (7.1). To optimize the training and performance, we kept the number of hidden layers as two, with 64 and 32 neurons to train the NN in the case when VDKZ18 is present. But, in the absence of the VDKZ18 effect case, we chose the number of hidden layers to be three with 128, 64, and 32 neurons. Two different NNs have been considered as the correlation between the parameter changes in the presence and absence of the VDKZ18. The activation function was changed from ReLU to Sigmoid in both cases. We took 2000 training samples and 1000 prediction samples to train and evaluate the network's performance. The variation of the model loss value for both cases during the training has been shown in Fig. (12); it can be seen that the loss approaches a constant value of 0.001 after 140 epochs Figure 7: The correlation between the foreground and $\mathbf{T}_{21}$ parameters. It can be seen that, $b_{0}$ representing the overall foreground scale factor is highly correlated to $A_{r}$, while is significantly responsible for the $\mathbf{T}_{21}$ amplitude. in both cases. It was found that the RMSE and R2 score varied in the range (a) in the presence of VDKZ18 (Fig. (13)) to be $(0.007-0.08)$ and $(0.69-0.99)$ respectively, and (b) in the absence of VDKZ18 (Fig. (14)) to be $(0.06-0.006)$ and $(0.99-0.93)$ respectively. ### 21-cm Signal, Foreground, and Noise Extraction of the faint cosmological signals buried inside a sea full of bright foreground signals is one of the challenging tasks. It can be seen in Fig. (6) that the foreground is $\approx 10^{3}-10^{4}$ brighter than the $T_{21}$ signal. In this section, we added foreground (Fig. (6)) to the global 21-cm signal (Fig. (4)) for the neural network training. As our cosmological parameters were already in the same scale, we prepossessed the training data with 'StandardScaler' and the foreground parameters with 'MinMaxScaler' from sklearn prior training. Using PyTorch, we build a neural network denser than before by adding more layers and neurons. We used 'Adam'(Kingma & Ba (2014)) and 'MSE' as the optimizer and error function. After tuning the hyperparameters by keeping the hidden layer density to 512, 256, 128, 64, and 32, the network was trained and saved for prediction. We added noise to the prediction dataset for parameter estimation to check whether the network has been trained for generalized data instead of memorizing the input data. Our training dataset consisted of 2000 samples and a 1000-sampled predicted data set with added noise, completed unknown to the trained network. \begin{table} \begin{tabular}{\|c\|c\|c\|} \hline Foreground & $T_{21c}(\nu)+T_{\mathrm{GC}}(\nu)$ + $\sigma_{\mathrm{r}}(\nu)$ \\ \cline{2-3} Parameters & RMSE & R2 Score \\ \hline $b_{0}$ & 0.1948 & 0.9638 \\ $b_{1}$ & 0.6310 & 0.5839 \\ $b_{2}$ & 0.2213 & 0.9467 \\ $b_{3}$ & 0.1534 & 0.9763 \\ $b_{4}$ & 0.1342 & 0.9805 \\ \hline \end{tabular} \end{table} Table 1: Foreground parameter estimation in the presence of VDKZ18. The reason for $b_{1}$ being poorly predicted is it’s strong correlation with $b_{0}$ Figure 8: The correlation between the parameters of the 21-cm global signal (a) in the absence and (b) in the presence of the VDKZ18 used for generating the training dataset. It can be seen that the correlation between the parameter is different in both cases. While in the case of Fig. (8a) the three parameters are correlated, but in Fig. (8b), only two are correlated. Figure 9: Variation of the loss function during the training process, where (a) in the presence of the VDKZ18 and (b) in the absence of the VDKZ18 Figure 11: Each plot above shows the prediction of the aforementioned parameters with RMSE and R2 score assigned in the absence of the VDKZ18. Figure 10: Each plot above shows the prediction of the aforementioned parameters with RMSE and R2 score assigned in the presence of the VDKZ18. \[\mathrm{T}_{\mathrm{pred}}^{3}=\mathrm{T}_{\mathrm{tot}}^{3}+\sigma_{t}(\nu) \tag{17}\] As only one NN is being used for prediction, therefore instead of extracting all 11 parameters (6 cosmological and 5 foregrounds) we chose to extract only foreground parameters from $\mathrm{T}_{\mathrm{pred}}^{3}$ such that, once foreground signal is extracted the residual signal can be fed to the NN constructed in section (7.2) for estimation of the cosmological signal parameters. It can be seen from Fig. (7), that the foreground ($b_{0}$) and cosmological ($A_{r}$) parameters are highly correlated. To check the performance and accuracy after the training, we estimated all the foreground parameters (Fig. 15) and assigned RMSE and R2 scores to each parameter shown in Table (1). ## 8 Summary and Discussion In our research, we generated synthetic data by resolving the typical cosmological equations, as presented in sections (4-6). The data was utilized for training neural networks. We added noise into the training data to verify the networks' reliability to prediction datasets. The accuracy of one of our estimations is evaluated by examining the RMSE and R2 scores for each case as listed in Table (1). As far as Figure 12: Variation of the loss function during the training process, where (a) in the presence of the VDKZ18 and (b) in the absence of the VDKZ18 Figure 13: Each plot above shows the prediction of the aforementioned parameters with RMSE and R2 score assigned in the presence of the VDKZ18 after the inclusion of noise in the prediction data. Figure 14: Each plot above shows the prediction of the aforementioned parameters with RMSE and R2 score assigned in the absence of the VDKZ18 after the inclusion of noise in the prediction data. Figure 15: Model loss and the prediction of the aforementioned parameters with RMSE and R2 score assigned in the presence of the foreground and VDKZ18 after the inclusion of noise in the prediction data. our understanding goes, most authors have employed Markov Chain Monte Carlo (MCMC), nested sampling, or similar methods for parameter space sampling. As ANN (Artificial Neural Network) and MCMC have different methodologies, we cannot directly compare their speed. However, ANN offers the advantage of bypassing the requirement of computing the likelihood function multiple times to derive inferred parameter values. Thus, when dealing with a higher dimensional parameter space, ANN is computationally more efficient and faster. In the context of neural networks, we can consider training sets a more realistic substitute for the prior in MCMC. As a single neural network has been used for parameter estimation, it becomes difficult to train it optimally to estimate each parameter accurately. While in training, it was observed that if the hyperparameters and the architecture of the NN were unset optimally, then either the estimation accuracy deteriorated or it was too high, depicting underfitting and overfitting conditions. Due to a huge difference in the order of the foreground compared to $T_{21}$ signal, the efficiency of the network was reduced significantly. The considered parameters for the construction of $T_{21}$ don't links directly to the astrophysical properties of the first star during the CD; rather, they are properties of IGM. (Choudhury et al. (2021)) have shown that during the reconstruction of the $T_{21}$ signal from EDGES data, "$f_{R}$" contributed significantly to the trough's amplitude which represents the presence of excess-background radiation. This work shows that VDKZ18 can change the parameter estimation if one includes the excess background radiation scenario. It can be seen in Fig. (8, 13, 14) that the correlation of $A_{r}$ is different in both scenarios and the same has been reflected in parameter prediction. The accuracy deteriorated with the inclusion of VDKZ18, implying the heating of gas and a decrease in $T_{21}$ trough's amplitude which has been proposed in Venumadhav et al. (2018). In future work, we intend to apply this concept and algorithm to more realistic data, like EDGES, SARAS 3, SKA. Experiments, like DARE, DAPPER (Burns et al. (2019)), FARSIDE (Burns et al. (2019)) etc., being space-based experiments could provide less contaminated data compared to Earth-based experiments. The contamination due to RFI and ionosphere can be neglected for these experiments, making this work ideal for testing. In future works, we wish to include energy injection terms linking directly to the first star's astrophysical properties, instrument response and contamination model to make the NN more robust. ## Acknowledgement P. K. N. would like to acknowledge the National Centre for Radio Astrophysics-Tata Institute of Fundamental Research for providing accommodation support during the initial phase of this work. A. C. N. acknowledges financial support from SERB-DST (SRG/2021/002291), India. ## Code and Data Availability We have used the publicly available EDGES data release([http://loco.lab.asu.edu/edges/edges-data-release/](http://loco.lab.asu.edu/edges/edges-data-release/)). The analyses are accomplished in Python, broadly using publicly available modules in NumPy (Harris et al. (2020)), Matplotlib (Hunter (2007)), Astropy (Price-Whelan et al. (2018)), Odeint Solver (Ahnert & Mulansky (2011)) from Scipy (Virtanen et al. (2020)), Natpy (Howson & Scaffidi (2021)), Pandas for data visualization (McKinney et al. (2010)), Heatmap of Seaborn (Waskom (2021)).
2301.09861	A convolutional neural network of low complexity for tumor anomaly detection	The automated detection of cancerous tumors has attracted interest mainly during the last decade, due to the necessity of early and efficient diagnosis that will lead to the most effective possible treatment of the impending risk. Several machine learning and artificial intelligence methodologies has been employed aiming to provide trustworthy helping tools that will contribute efficiently to this attempt. In this article, we present a low-complexity convolutional neural network architecture for tumor classification enhanced by a robust image augmentation methodology. The effectiveness of the presented deep learning model has been investigated based on 3 datasets containing brain, kidney and lung images, showing remarkable diagnostic efficiency with classification accuracies of 99.33%, 100% and 99.7% for the 3 datasets respectively. The impact of the augmentation preprocessing step has also been extensively examined using 4 evaluation measures. The proposed low-complexity scheme, in contrast to other models in the literature, renders our model quite robust to cases of overfitting that typically accompany small datasets frequently encountered in medical classification challenges. Finally, the model can be easily re-trained in case additional volume images are included, as its simplistic architecture does not impose a significant computational burden.	Vasileios E. Papageorgiou, Pantelis Dogoulis, Dimitrios-Panagiotis Papageorgiou	2023-01-24T08:44:19Z	http://arxiv.org/abs/2301.09861v5	# A convolutional neural network of low complexity for tumor anomaly detection ###### Abstract The automated detection of cancerous tumors has attracted interest mainly during the last decade, due to the necessity of early and efficient diagnosis that will lead to the most effective possible treatment of the impending risk. Several machine learning and artificial intelligence methodologies has been employed aiming to provide trustworthy helping tools that will contribute efficiently to this attempt. In this article, we present a low-complexity convolutional neural network architecture for tumor classification enhanced by a robust image augmentation methodology. The effectiveness of the presented deep learning model has been investigated based on 3 datasets containing brain, kidney and lung images, showing remarkable diagnostic efficiency with classification accuracies of 99.33%, 100% and 99.7% for the 3 datasets respectively. The impact of the augmentation preprocessing step has also been extensively examined using 4 evaluation measures. The proposed low-complexity scheme, in contrast to other models in the literature, renders our model quite robust to cases of overfitting that typically accompany small datasets frequently encountered in medical classification challenges. Finally, the model can be easily re-trained in case additional volume images are included, as its simplistic architecture does not impose a significant computational burden. Keywords:Convolutional neural networks; Tumor detection; Biomedical image classification; Data augmentation; Entropy; Artificial intelligence. ## 1 Introduction Applications of artificial intelligence (AI) in medicine continue to grow and affect every aspect of cancer care. These applications fall into 2 main categories, namely supervised and unsupervised learning [1-2]. In supervised learning, computers learn and adapt by studying labeled biomedical instances to copy the diagnostic skills of experienced oncologists. Three of the most common and deadly tumors that affect people's quality of life in everyday life are brain, lung, and kidney tumors, making their early detection an important concern. A brain tumor is an abnormal mass of tissue where cells grow uncontrollably. Based on the World Health Organization (WHO), these types of tumors account for less than 2% of human cancers, although their severe morbidity and associated complications make timely diagnosis an important concept in the field of medicine [3]. Intracranial tumors can be fatal, worsen the patient's standard of living, and can affect men, women, or children. Lung cancer ranks second, accounting for approximately 11.4% of total cancer cases, with an estimated 2.2 million lung cancer cases only. Lung tumors represent the leading cause of death due to cancer, with deaths accounting for 18% of all cancers [4]. The survival rate of patients suffering from lung tumors 5 years after diagnosis ranges between 10-20%. Screening with low-dose computed tomography (CT) could aid the timely detection of lung tumors so that the disease can be possibly treated. In addition, it has been reported that the patient's likelihood of living a long life increase if the tumor is timely diagnosed and treated successfully [5]. Renal cell carcinoma (RCC) is the 6th most ordinary cancer among all tumors in men and the 10th most common in women. Despite advances in understanding the molecular biology of RCC and refinements in therapies, treating patients with RCC at any stage of the disease is challenging. Detection of early-stage renal tumor has improved in recent decades with the use of cross-sectional imaging [6]. Most renal carcinomas are initially detected as incidental renal masses on cross-sectional imaging performed for unspecified disease. Although most are discovered as small renal masses (70%), earlier definitive therapeutic intervention for these tumorous regions has not led in a notable advancement in cancer mortality [7]. In our analysis, we present convolutional neural network (CNN) architecture of low complexity for tumor anomaly diagnosis. The reduced complexity of the proposed network makes it suitable for studying small datasets which often accompany medical analyses, compared to many articles that employ complex and computationally expensive schemes. This architecture not only eliminates the likelihood of overfitting - which can be a major issue in both statistical and AI models [8] - but also increases the flexibility and adaptability of the approach, as it can be re-trained without significant computational cost in the occasion where new MRI or CT images are added to the dataset. In addition, various data augmentation techniques have been implemented and their respective impact on classification efficiency has been studied in detail. In contrast to most studies included in the literature, our novel architecture has been tested on not only one, but 3 tumor datasets. These datasets contain normal, benign and malignant instances corresponding to brain, lung and kidney tumors. Hence, the variety of different datasets and the diversity of cancer classification problems, validate the trustworthiness of the produced detection efficiency, rendering our approach a reliable medical tool that can be easily utilized in modern oncology. ## 2 Related Work Regarding medical imaging, there are a variety of algorithms from the field of AI for brain cancer classification/detection. Characteristic examples are support vector machines (SVM) [9], K-nearest neighbors (KNN) [10] and artificial neural networks (ANN) [11]. In parallel, CNNs seem to be the most suitable for processing MRI or CT images due to their high classification performance. There are several articles that address binary and multiclass tumor challenges, utilizing a number of state-of-the-art deep CNNs. The most relevant articles regarding our analysis corresponds to binary classification attempts of MRI and CT images displaying brain, lung or kidney tumors. The authors in [12-14] combine image augmentation and preprocessing methods with conventional CNN methods aiming to classify benign and malignant tumors with accuracy of 97.5%, 94.1% and 98% respectively. In [15-17], hybrid CNN-SVM models are used for binary classification with corresponding accuracies of 88.54%, 95%, and 95.62%. The authors in [16] and [17] propose the usage of ResNet for the part of the feature extraction and a SVM for the feature extraction. The examined tumor images were preprocessed by entropy segmentation techniques and resolution enhancement. Regarding lung tumors, in [18-19] the authors use SVM classifiers and CNN GoogleNet based on the IQ-OTHNCCD dataset. In [18], the authors preprocess the images from CT using Gaussian filters, bit-plane slicing and image segmentation and achieve an accuracy of 89.88%, while in [19] they implement the Gabor filter and regions of interest (ROI) extraction with a grouping accuracy of 94.38%. Other machine learning attempts for lung cancer classification include the utilization of KNN [20], SVM [21], Naive Bayes [20], and Random Forests [22]. In [23], the authors implement CNN architectures such as VGG16, MobileNet, AlexNet, DenseNet, VGG19, and ResNet with efficiency between 48-56% for classification of normal, benign, and malignant tumors. Polat and Mehr [24] utilize a hybrid 3D CNN-SVM with a classification efficiency of 91.81%. Finally, several articles have addressed renal cancer detection, particularly using preprocessed forms of CT images. In machine learning approaches to discriminate between benign and malignant tumors [25-27], most studies focused on discriminating between low- fat angiomyolipomas and renal cancer with promising results, with the AUC metric reaching values between 0.90 and 0.96. Han et al [28], using a modified GoogleNet architecture, encountered lower performance on a 3-class problem for identifying papillary RCCs (pRCCs) compared to clear cell RCCs (ccRCCs) and chromophobe RCCs (chrRCCs), compared to the binary classification problem of ccRCCs compared to non-ccRCCs. Furthermore, Li et al [29] identified low- and high-grade ccRCC, respectively, based on MRI combined with patient history and radiologist-assigned imaging features and achieved an AUC of 0.845. ## 3 Methodology In this part, we describe the complete framework that was constructed in order to train our proposed architecture and boost its generalization ability. In more detail, we analyze the source of each dataset and the classes that are contained. Then we provide the proposed low-complexity convolutional neural network and finally we describe the data augmentation method that was applied. We also describe in detail the main concepts of the CNNs as well as the metrics that are used in the inference stage. ### Convolutional Neural Networks Convolutional Neural Networks is a class of ANN architectures that are most commonly based on the convolution kernels. These networks are mainly used in visual related problems such as video classification, image segmentation and medical image analysis. A CNN includes the basic modules of _convolutional layers_ (which are the most important parts of feature extraction), _pooling layers_ (which are utilized in order to reduce the dimensionality of the preprocessed matrices), _batch normalization layers_ (which are helpful in the computational stability during training) as well as the _fully connected layers_ (which is used as the feature selection mechanism). Convolutional layers consist of kernel sets representing the model's trainable parameters that are modified after each iteration. Let the 3-dimensional matrix $\mathbf{X}^{l}\in\mathbb{R}^{M^{l}\times N^{l}\times c^{l}}$ be the input of the $l$-th convolutional layer and $\mathbf{G}\in\mathbb{R}^{m\times n\times c^{l}\times S}$ be a 4-dimensional matrix, representing the $s$ kernels of $l$-th layer, of spatial span $m\times n$. The output of the $k$ - th convolutional layer will be a 3-dimensional matrix defined as $\mathbf{Y}^{l}\in\mathbb{R}^{M^{l}-m+1\times N^{l}-n+1\times S}$. This matrix results from the equation \[\gamma_{i^{l},j^{l},s}=\sum_{l=0}^{m}\sum_{j=0}^{n}\sum_{k=0}^{c^{l}}G_{i,j,c^ {l},s}\times\mathbf{x}^{l}_{i^{l},j^{l},k}. \tag{1}\] Relation (1) is applied for all $0\leq s\leq S$ and for any spatial span satisfying $0\leq i^{l}\leq M^{l}-m+1$ and $0\leq j^{l}\leq N^{l}-n+1$. Let $\mathbf{X}^{l}\in\mathbb{R}^{M^{l}\times N^{l}\times D^{l}}$be the input of the $l$-th layer that is now a pooling layer of size $n\times m$. We assume that $n$ divides $M$ and $m$ divides $N$ and the stride equals the spatial span. The output is a matrix $\mathbf{Y}^{l}\in\mathbb{R}^{M^{l+1}\times N^{l+1}\times c^{l+1}}$, where \[M^{l+1}=\frac{M^{l}}{n},\quad N^{l+1}=\frac{N^{l}}{m},\quad C^{l+1}=C^{l}, \tag{2}\] while the polling layer operates upon $\mathbf{X}^{l}$ dimension by dimension. In our network we utilize 2 max pooling layers, resulting in outputs produced based on \[\gamma_{i^{l},j^{l},d}=\max_{0\leq i\leq m,0\leq j\leq n}\mathbf{x}^{l}_{i^{l} \times n+i,j^{l}\times m+j,d^{l}} \tag{3}\] where $0\leq i^{l}\leq M^{l},\ 0\leq j^{l}\leq N^{l}$and $0\leq c\leq C^{l}$. On the other hand, the fully connected layers constitute the second part of the convolutional neural network, aiming to efficiently select the most valuable features extracted from the convolutional layers. Noteworthy transition operations that connect the above layers are the batch normalization and ReLU operations. The rectifier function is described as \[\gamma_{i,j,d}=\max\bigl{(}0,\mathbf{x}^{l}_{i,j,c}\bigr{)} \tag{4}\] with $0\leq i\leq M^{l},0\leq j\leq N^{l}$and $0\leq c\leq C^{l}$. The input matrix $\mathbf{X}_{i}$ corresponding to the $i-$th medical image is passed through the set of successive layers and a label $\hat{\gamma}_{i}$ is produced. Then an error is calculated using a defined loss function. In most cases, Cross-Entropy loss is utilized, which is denoted as $L_{CE}$. In our occasion, where we train our network in binary classification scenarios, we utilize the Binary Cross-Entropy loss function ($L_{BCE}$), which is defined as \[L_{BCE}(y_{i},\hat{y_{i}})=\ -(y_{i}log(\hat{y_{i}})\ +\ (1-y_{i})log(1-\hat{y_{i}})) \tag{5}\] where $y_{i}=\{0,1\}$ corresponds to the image's ground truth. The produced error is then utilized in the learning procedure that represents the modification of the _trainable_ parameters of the network based on an optimization algorithm. The majority of the analyses in the literature use Adam or AdamW algorithms as optimizers. ### Datasets The brain cancer dataset contains 3000 images that can be utilized for the training/test of the proposed CNN architecture. The dataset is highly balanced, where 1500 images correspond to normal and 1500 images to tumorous cases. It is an open access dataset and has been uploaded on Kaggle1. Footnote 1: [https://www.kaggle.com/datasets/ahmedhamada0/brain-tumor-detection](https://www.kaggle.com/datasets/ahmedhamada0/brain-tumor-detection) The dataset containing lung tumors was organized by specialists in the Iraq-Oncology Teaching Hospital/ National Center for Cancer Diseases (IQ-OTH/NCCD)2. The dataset contains 1097 labeled images, consisting of 416 normal, 120 benign and 561 malignant cases. Since our problem is a binary-classification problem of detecting tumorous regions, we labeled as abnormal (or tumorous) all instances where benign or malignant cancer was present. Footnote 2: [https://www.kaggle.com/datasets/adityamahimkar/iqothnccd-lung-cancer-dataset](https://www.kaggle.com/datasets/adityamahimkar/iqothnccd-lung-cancer-dataset) The kidney cancer dataset3 was retrieved from a collection of cancer related datasets, which contains cases of brain, breast and other types of cancer. It contains 10000 images corresponding to images of 5000 normal and 5000 of tumorous cases, representing a completely balanced set. Footnote 3: [https://www.kaggle.com/datasets/obulisainaren/multi-cancer](https://www.kaggle.com/datasets/obulisainaren/multi-cancer) ### Augmentation Pipeline Different image-based operations are used as part of the proposed augmentation pipeline. These are _gaussian blur_, small modifications of the contrast, hue, brightness and zoom of each image (_color jittering_) as well as _rotation_ and _translation. Gaussian Blur_ is used as a cleaning mechanism of the image since it removes high frequencies in regions of the image and is commonly used as a denoising tool. Random _resize_ ensures that the model will focus on infectious regions of the image independently of its height or width. Simultaneously, _rotation_ and _translation_ push the model to search for the tumorous region in different areas of the image and finally _color jittering_ is used in order to help the model learn features that are not dependent on the color of each image, since there is also a variation in the pixel values among the grayscale images, but mainly on the shape of infectious and healthy regions. After the augmentation procedure, the MRI and CT images displaying brain, kidney and lung tumors are cropped automatically, aiming to remove their outer black parts that do not represent valuable information for the examined phenomenon. ### CNN Architecture In this paragraph, we present a low-complexity CNN scheme including 7 main layers. The first four (2 convolutional and 2 max pooling) contribute to the feature extraction process. In addition, the 3 remaining fully connected layers take advantage of the extracted features to achieve noteworthy classification performance (Fig. 1). Two-dimensional gray-scale images of size 100$\times$100 are placed as inputs to the proposed CNN architecture. We arrived at this choice after extensive experimentation, since the selected small input size, is accompanied by low computational burden without deteriorating the model's efficacy. Firstly, a convolutional layer consisting of 32 kernels of 9$\times$9 spatial span is encountered, while the extracted feature maps go through a max pooling layer of size 4$\times$4. This pattern is replicated and contains a 5$\times$5 convolutional and a 4$\times$4 max pooling layer. In both cases, _same padding_ is utilized before the implementation of the convolution, while the above layers are accompanied by ReLU and batch normalization operations. The second part consists of 3 fully connected layers including 4096, 1024 and 1 node, correspondingly. Between the first, exist a dropout operation aiming to eliminate the likelihood of overfitting. ## 4 Results The proposed low-complexity scheme is employed to investigate the overall classification efficiency on all 3 datasets containing lung, kidney and brain tumors. For each dataset we implemented a stratified splitting strategy of ratio 70:30, before and after the implementation of the augmentation methodology to maintain balanced classes during the training and test phases. Thus, we result into test sets containing 900, 1500 and 329 images for the brain, kidney and lung datasets correspondingly. Several learning rates were utilized during the training phase, namely $\eta$ = $\{0.0001,0.0005,0.001,0.005,0.01,0.05,0.1\}$, while the best classification results are generated for $\eta$ = 0.005. The training process has been implemented for 50 epochs using the Adam, providing a less smooth but more efficient training procedure compared to other optimizers, like the stochastic gradient descent (SGD). Figure 1: Diagrammatic representation of the low-complexity CNN scheme The augmentation methodology has been applied to the training set only. Since the lung cancer dataset includes 416 normal and 681 abnormal images, the application of the augmentation step helps to balance the 2 classes studied. For kidney and brain tumors this is not a concern, as these sets already have a 50:50 ratio between the 2 examined categories. In Table 1 we observe the classification efficacy of the presented CNN on the previously mentioned test sets. Also, Table 2 displays our model's capacity after the employment of the augmentation pipeline. In both cases, our model presents significant tumor detection efficacy, especially for the augmented datasets, where we encounter accuracies of 99.33%, 100% and 99.7% for the brain, kidney and lung tumors respectively. Moreover, according to tables 1 and 2, kidney tumors represent the best classified type of tumors, before and after the employment of augmentation. Another important observation is the minor differentiation between the evaluation measures of specificity and recall, leading us to the conclusion that our model classifies equivalently effectively both tumorous and non-tumorous instances, regardless the cancer type. \begin{table} \begin{tabular}{c c c c c} \hline & Accuracy & Specificity & Recall & F1 score \\ \hline Brain Tumors & 98.44\% & 99.09\% & 97.88\% & 99.89\% \\ Kidney Tumors & 99.78\% & 96.55\% & 99.61\% & 99.78\% \\ Lung Tumors & 97.27\% & 99.11\% & 96.76\% & 98.05\% \\ \hline \end{tabular} \end{table} Table 1: Classification performance of the low-complexity CNN based on the 3 examined datasets without augmentation \begin{table} \begin{tabular}{c c c c c} \hline & Accuracy & Specificity & Recall & F1 score \\ \hline Brain Tumors & 99.33\% & 98.88\% & 99.78\% & 99.34\% \\ Kidney Tumors & 100\% & 100\% & 100\% & 100\% \\ Lung Tumors & 99.70\% & 100\% & 99.49\% & 99.74\% \\ \hline \end{tabular} \end{table} Table 2: Classification performance of the low-complexity CNN based on the 3 examined datasets with augmentation Figure 2: Confusion matrices of the CNN accuracy for brain, kidney and lung images The augmentation procedure has improved the model's overall classification efficacy in all 3 investigated scenarios, a fact that is supported by all 4 evaluation measures. More specifically, the testing accuracy is increased by 0.89% for brain tumors, 0.24% for kidney tumors, while the most prevalent increase is evident for lung cancer with an increase of 2.43%. In addition, we observe notable increases in the recall and the specificity regarding kidney and lung tumors of 2.73% and 3.44%, respectively. More information about the amount of correctly classified instances of each class is provided by the above confusion matrices (Fig. 2). These 3 matrices correspond to the presented results of table 2, concerning the case of the augmented training set. Finally, the diagrams of Fig. 3 and 4 display the evolution of the training/test losses and accuracies during the 50 epochs, showing a quite smooth training process, validating the robustness of the proposed AI model. ## 5 Discussion In this article, we present a convolutional neural network scheme of low complexity for tumor anomaly detection. The limited complexity of our network renders it ideal for studying datasets including limited number of observations, that usually accompany medical studies, compared to many papers in the literature that use more complex and computationally expensive schemes. CNNs with more convolutional and more complex fully connected layers were tested, although without improving detection accuracy. Several augmentation techniques have been applied, including Gaussian blur, color jittering, rotations, resizing and translations, while we examine their influence on the detection capability of our deep learning model. The variety of datasets and the Figure 4: Training/test accuracies for the brain, kidney and lung cancers during 50 epochs. Figure 3: Training/test losses for the brain, kidney and lung cancers during 50 epochs. diversity of classification challenges, enhance the robustness of the presented detection efficacy, rendering our approach a reliable medical tool. We showed that the proposed low-complexity convolutional neural network is able to produce notably accurate results based on the 4 examined classification metrics used throughout the analysis. The selected augmentation pipeline forces the detector to learn features that focus on the tumorous regions which are independent from pixel-related features, such as the brightness or hue. The increase in the evaluation metrics is obvious, resulting in accuracy scores of 99.33%, 100.00% and 99.70% for the brain, kidney and lung cancer, respectively. On top of that, since the cardinality of the medical-related datasets is mostly limited, deep architectures may fall under the threat of overfitting [30-31]. Other interesting approaches that used machine learning algorithms like random forests [22] or more complex AI schemes [12-17, 23-24, 28] generated robust results with the highest testing accuracy being 98%. Our low-complexity network coupled with the data augmentation overcomes this challenge and can be easily implemented under real-world circumstances, while its simplistic architecture encourages its retraining when new data is presented. Since new types of cancer may occur, scientists should focus more on the generalization of their methods or the construction of architectures that will be able to provide robust results with limited amounts of data. Regarding future work, our objective is to take advantage of the knowledge of some deep pre-trained networks, such as ResNet50 on ImageNet, in order to create pipelines based on transfer learning and related vision techniques. It would be interesting to apply a teacher-student method to approach the problem under the few-shot or no-shot learning scenario.
2306.15237	Cross Spectral Image Reconstruction Using a Deep Guided Neural Network	Cross spectral camera arrays, where each camera records different spectral content, are becoming increasingly popular for RGB, multispectral and hyperspectral imaging, since they are capable of a high resolution in every dimension using off-the-shelf hardware. For these, it is necessary to build an image processing pipeline to calculate a consistent image data cube, i.e., it should look like as if every camera records the scene from the center camera. Since the cameras record the scene from a different angle, this pipeline needs a reconstruction component for pixels that are not visible to peripheral cameras. For that, a novel deep guided neural network (DGNet) is presented. Since only little cross spectral data is available for training, this neural network is highly regularized. Furthermore, a new data augmentation process is introduced to generate the cross spectral content. On synthetic and real multispectral camera array data, the proposed network outperforms the state of the art by up to 2 dB in terms of PSNR on average. Besides, DGNet also tops its best competitor in terms of SSIM as well as in runtime by a factor of nearly 12. Moreover, a qualitative evaluation reveals visually more appealing results for real camera array data.	Frank Sippel, Jürgen Seiler, André Kaup	2023-06-27T06:31:23Z	http://arxiv.org/abs/2306.15237v2	# Cross Spectral Image Reconstruction Using a Deep Guided Neural Network ###### Abstract Cross spectral camera arrays, where each camera records different spectral content, are becoming increasingly popular for RGB, multispectral and hyperspectral imaging, since they are capable of a high resolution in every dimension using off-the-shelf hardware. For these, it is necessary to build an image processing pipeline to calculate a consistent image data cube, i.e., it should look like as if every camera records the scene from the center camera. Since the cameras record the scene from a different angle, this pipeline needs a reconstruction component for pixels that are not visible to peripheral cameras. For that, a novel deep guided neural network (DGNet) is presented. Since only little cross spectral data is available for training, this neural network is highly regularized. Furthermore, a new data augmentation process is introduced to generate the cross spectral content. On synthetic and real multispectral camera array data, the proposed network outperforms the state of the art by up to 2 dB in terms of PSNR on average. Besides, DGNet also tops its best competitor in terms of SSIM as well as in runtime by a factor of nearly 12. Moreover, a qualitative evaluation reveals visually more appealing results for real camera array data. Frank Sippel, Jurgen Seiler, and Andre Kaup+Friedrich-Alexander-Universitat Erlangen-Nurnberg Multimedia Communications and Signal Processing Cauerstrasse 7, 91058 Erlangen, Germany Multispectral Imaging, Image Reconstruction, Deep Learning Footnote †: Source code: [https://github.com/FAU-LMS/dgnet](https://github.com/FAU-LMS/dgnet). ## 1 Introduction Multispectral camera arrays[1, 2] and hyperspectral camera arrays[3] are capable of recording different spectral areas by employing multiple cameras. In contrast to other multispectral and hyperspectral imaging devices, this approach has the advantage of yielding a high resolution in spatial and temporal dimension as well as being cost-efficient and flexible regarding the used filters. The recorded channels do not necessarily have to lie in the visible wavelength area, but can also contain information about areas in the infrared or ultraviolet area of the spectrum. These areas of the spectrum are particularly interesting for classification problems. Multispectral and hyperspectral cameras can be used in medicine to classify the degree of burn [4], in recycling to sort materials [4], in forensics by determining the age of blood [5], or in agriculture by discriminating between parts of fields that need water and fertilizer and those that are healthy [6]. Genser et al. [7] introduced a multispectral camera array consisting of nine cameras. However, a reconstruction pipeline to reconstruct a consistent multispectral datacube is necessary. The goal of this processing pipeline is to reconstruct a consistent multispectral center view by mapping all peripheral cameras to the center perspective, see Fig. 1. Then, each pixel of the multispectral datacube shows exactly the same object in different spectral bands. This reconstruction pipeline consists of multiple separate Figure 1: A multispectral camera array (CAMSI) with three corresponding recorded images of the center row (top). Additionally, the registered multispectral image with missing pixels (bottom left) and the reconstructed image (bottom right) are depicted. -62023 IEEE. Published in 2023 IEEE International Conference on Image Processing (ICIP), scheduled for 8-11 October 2023 in Kuala Lumpur, Malaysia. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. 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Published in 2023 IEEE International Conference on Image Processing (ICIP), scheduled for 8-11 October 2023 in Kuala Lumpur, Malaysia. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. DOI: tba The resulting multispectral channel $I_{c}[x,y]$ is calculated by only using the reconstructed image for masked pixels \[I_{c}[x,y]=(1-M_{c}[x,y])\cdot D_{c}[x,y]+M_{c}[x,y]\cdot O_{c}[x,y]. \tag{3}\] ### Data Augmentation A big challenge to train this network is that there is not enough multispectral data available to yield diverse scenes. Thus, we suggest a smart data augmentation using RGB data, since there are many large size RGB databases with thousands of different scenes. Furthermore, realistic masks needs to be generated. #### 2.2.1 Spectral Image Generation The main goal here is not to perfectly generate spectral data out of these images, but rather imitate that the objects have different textures for different spectral bands. To be consistent across the whole object, the idea of this data augmentation is to map random grayscale values to a certain number of colors in the image. This is done by transforming the RGB image to an HSV image and use the hue value to assign colors. Between these colors, a linear interpolation is performed for consistency reasons. Moreover, random intensities for white and black are generated to also use the saturation and value of the HSV image. Again, linear interpolation is used. Subsequently, the grayscale image is normalized to the full intensity range and a random exposure between 0.2 and 1.5 is applied to also cover over- and underexposed cases. Finally, synthetic white Gaussian noise is added and the image is clipped to the intensity range. #### 2.2.2 Mask Generation Realistic masks also need to be generated for training. To that end, five different patterns are implemented. First, the classic stroke mask known from inpainting papers such as [11] is used. Moreover, a random pixel loss mask and a random block loss mask are used to simulate cases, where the depth map is noisy. Furthermore, a border mask is provided, which just cuts off random portions of random borders of the image. This also happens in reality, since the cameras are spatially distributed and thus cannot see all borders of the center camera. Finally, an edge mask can be applied. Typically, missing pixels occur on edges of objects, because often there is a bigger disparity difference between foreground and background. Therefore, an edge mask is calculated by extracting edges from the RGB image using a Canny edge detector. These thin edges are extended to a random width in a random direction by calculating starting and ending points of lines using the gradient map and setting all pixels on these lines to be reconstructed. In general, more diverse masks will make the network generalize better as long as enough image content is shown to estimate the coefficients. Examples for generated images and masks are shown in Fig. 3. ### Configuration Since only convolutional layers are used, the network can handle any image size. The input images need to be padded such that all cube bins cover full image areas. Otherwise, the border linear regression coefficients will not be properly estimated. The desired spatial bin size is set to $16\times 16$, while there are $G_{l}=32$ luma bins. A weighted $l_{1}$-loss is used as loss function \[L(\hat{I}_{c}[x,y],O_{c}[x,y],M_{c}[x,y])= \tag{4}\] \[\frac{1}{WH}\sum_{x,y}\begin{cases}\alpha\cdot\|O_{c}[x,y]-\hat{I }_{c}[x,y]\|,&\text{if }M_{c}[x,y]=1\\ \|O_{c}[x,y]-\hat{I}_{c}[x,y],&\text{else},\end{cases}\] where $\alpha$ is the weight factor for missing pixel positions. This weight factor is set to 10 to put more importance to missing regions. Moreover, the cells in the grid should be steered towards reconstructing missing parts of the image rather than already existing parts. However, it is still important that the whole image is reconstructed such that the network learns how images look like. As optimizer, Adam with parameters $\beta_{1}=0.5$ and $\beta_{2}=0.999$ is picked. The network is trained for 64 epochs using the Places2 database [17], which contains approximately 1.8 million RGB images. The learning rate starts at 0.0001 and is halved at epochs 20, 32, 40, 48, and 56. Figure 3: In the first row, two generated spectral grayscale images from an RGB image are shown. In the bottom row, three masks are depicted. Note that the edge mask is generated using the RGB image of the first row. 62023 IEEE. Published in 2023 IEEE International Conference on Image Processing (ICIP), scheduled for 8-11 October 2023 in Kuala Lumpur, Malaysia. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. DOI: tba ## 3 Evaluation The evaluation is done quantitatively as well as qualitatively. Our novel DGNet is compared against the inpainting methods Free Form Inpainting (FF) [12], EdgeConnect (EC) [11], Frequency Selective Reconstruction (FSR) [10], and the guided reconstruction methods Cross-Spectral Reconstruction (CSR) from [7] and Non-local Cross-Spectral Reconstruction (NOCS) [14]. ### Synthetic Data To evaluate DGNet, the HyViD database [18] is used, which contains 7 scenes, each with 30 frames. The images have 31 hyperspectral channels and are rendered from a camera array very similar to the one in Fig. 1. Moreover, ground-truth depth data is available. Hence, the database can be used to evaluate realistic cross spectral reconstruction problems. For that, the 31 hyperspectral channels are synthetically filtered to nine different bandpasses, including a red, green and blue component. The ground-truth depth is used for the reconstruction pipeline to warp the images accordingly. In Table I, the results are summarized in terms of average PSNR and SSIM, which indicates that our novel neural network-based method outperforms the reference algorithms. Moreover, the best of three runtimes using a single thread CPU implementation is shown in Table II, which are evaluated on the first frame of the scene _family house_. DGNet outperforms the reference algorithms by a factor of at least 11.8. When executed on an RTX 3090 GPU, the runtime is decreased to 340 ms, corresponding to a factor of 287 in comparison to the second fastest method NOCS. ### Real Data To evaluate real-world performance, an evaluation based on the Middlebury 2006 database [19], which contains 21 scenes, as well as on the database of [7], which are both much smaller than the synthetic database, is shown in Table I. Note that the Middlebury database only contains RGB images. Hence, the red channel of the left camera, the green channel of the center camera and the blue channel of the right camera are chosen to simulate a linear camera array. Again, the proposed DGNet outperforms all reference methods. Additionally, another image was recorded using CAMSI [7] for qualitative evaluation. In Fig. 4, false color images of the cross spectral methods and our novel network are depicted. As indicated, the novel DGNet produces fewer artefacts than its competitors. ## 4 Conclusion This paper introduced a neural network-based method for reconstructing occluded pixels using a guide image, e.g., when employing a multispectral camera array. For that, a cube of grid coefficients for linear regression is estimated in spatial as well as intensity direction. Afterwards, this cube is sliced into two high-resolution linear regression parameter images using the guide image. Finally, the linear regression coefficients are applied to this guide image to yield a final estimate of the peripheral view. It was shown that this method outperforms the state of the art by up to 2 dB in terms of PSNR on multispectral camera array data as well as in terms of SSIM. The novel neural network also provides a significant speedup and yields visually more accurate images. \begin{table} \begin{tabular}{c c c c c c c} & Database & FF[12] & EC[11] & FSR[10] & NOCS[14] & CSR[7] & DGNet \\ \hline \multirow{4}{}{Image} & HyViD[18] & 33.72 & 29.40 & 34.32 & 35.29 & 38.98 & 40.95* \\ & MiBu[19] & 28.85 & 27.38 & 30.12 & 30.94 & 33.64 & 35.43 \\ & CAMSI[7] & 28.57 & 28.40 & 28.21 & 29.78 & 29.26 & 30.57 \\ \cline{2-8} \cline{1-1} \cline{2-8} & HyViD[18] & 0.942 & 0.869 & 0.951 & 0.962 & 0.974 & 0.989 \\ & MiBu[19] & 0.923 & 0.895 & 0.931 & 0.934 & 0.949 & 0.952 \\ & CAMSI[7] & 0.896 & 0.896 & 0.901 & 0.906 & 0.907 & 0.913 \\ \end{tabular} \end{table} Table I: Average PSNR in dB and SSIM of different reconstruction methods on different databases [18, 19, 7]. Figure 4: Reconstruction results of spectrally guided methods as (false) color images combining three spectral bands. Left column: center row of camera array (red, green, blue). Right column: top row of camera array (infrared filters at 750 nm, 850 nm and 950 nm). Zoomed areas are depicted using red rectangles. \begin{table} \begin{tabular}{c c c c c c} & & DGNet \\ & EC[11] & FSR[10] & NOCS[14] & CSR[7] & DGNet \\ \hline 2960 & 391 & 855 & 97.8 & 162 & 8.27 & 0.34 \\ \end{tabular} \end{table} Table II: Single-thread CPU runtime of all methods in seconds. IEEE. Published in 2023 IEEE International Conference on Image Processing (ICIP), scheduled for 8-11 October 2023 in Kuala Lumpur, Malaysia. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. DOI: tba
2305.16508	Most Neural Networks Are Almost Learnable	We present a PTAS for learning random constant-depth networks. We show that for any fixed $\epsilon>0$ and depth $i$, there is a poly-time algorithm that for any distribution on $\sqrt{d} \cdot \mathbb{S}^{d-1}$ learns random Xavier networks of depth $i$, up to an additive error of $\epsilon$. The algorithm runs in time and sample complexity of $(\bar{d})^{\mathrm{poly}(\epsilon^{-1})}$, where $\bar d$ is the size of the network. For some cases of sigmoid and ReLU-like activations the bound can be improved to $(\bar{d})^{\mathrm{polylog}(\epsilon^{-1})}$, resulting in a quasi-poly-time algorithm for learning constant depth random networks.	Amit Daniely, Nathan Srebro, Gal Vardi	2023-05-25T22:27:42Z	http://arxiv.org/abs/2305.16508v3	# Most Neural Networks Are Almost Learnable ###### Abstract We present a PTAS for learning random constant-depth networks. We show that for any fixed $\epsilon>0$ and depth $i$, there is a poly-time algorithm that for any distribution on $\sqrt{d}\cdot\mathbb{S}^{d-1}$ learns random Xavier networks of depth $i$, up to an additive error of $\epsilon$. The algorithm runs in time and sample complexity of $(\bar{d})^{\mathrm{poly}(\epsilon^{-1})}$, where $\bar{d}$ is the size of the network. For some cases of sigmoid and ReLU-like activations the bound can be improved to $(\bar{d})^{\mathrm{polylog}(\epsilon^{-1})}$, resulting in a quasi-poly-time algorithm for learning constant depth random networks. ## 1 Introduction One of the greatest mysteries surrounding deep learning is the discrepancy between its phenomenal capabilities in practice and the fact that despite a great deal of research, polynomial-time algorithms for learning deep models are known only for very restrictive cases. Indeed, state of the art results are only capable of dealing with two-layer networks under assumptions on the input distribution and the network's weights. Furthermore, theoretical study shows that even with very naive architectures, learning neural networks is worst-case computationally intractable. In this paper, we contrast the aforementioned theoretical state of affairs, and show that, perhaps surprisingly, even though constant-depth networks are completely out of reach from a worst-case perspective, _most_ of them are not as hard as one would imagine. That is, they are _distribution-free learnable_ in polynomial time up to any desired constant accuracy. This is the first polynomial-time approximation scheme (PTAS) for learning neural networks of depth greater than $2$ (see the related work section for more details). Moreover, we show that the standard SGD algorithm on a ReLU network can be used as a PTAS for learning random networks. In a bit more detail, we consider constant-depth random networks obtained using the standard Xavier initialization scheme [22, 26], and any input distribution supported on the sphere $\sqrt{d}\cdot\mathbb{S}^{d-1}$. For Lipschitz activation functions, our algorithm runs in time $(\bar{d})^{\mathrm{poly}(\epsilon^{-1})}$, where $\bar{d}$ is the network's size including the $d$ input components, and $\epsilon$ is the desired accuracy. While this complexity is polynomial for constant $\epsilon$, we also consider the special cases of sigmoid and ReLU-like activations, where the bound can be improved to $(\bar{d})^{\mathrm{polylog}(\epsilon^{-1})}$. The main technical idea in our work is that constant-depth random neural networks with Lipschitz activations can be approximated sufficiently well by low-degree polynomials. This result follows by analyzing the network obtained by replacing each activation function with its polynomial approximation using Hermite polynomials. It implies that efficient algorithms for learning polynomials can be used for learning random neural networks, and specifically that we can use the SGD algorithm on ReLU networks for this task. ### Results In this work, we show that random networks can be well-approximated by low-degree polynomials, which implies a PTAS for learning random networks. We start by defining the network architecture. We will denote by $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ the activation function, and will assume that it is $L$-Lipschitz. To simplify the presentation, we will also assume that it is normalized in the sense that $\mathbb{E}_{X\sim\mathcal{N}(0,1)}\sigma^{2}(X)=1$. Define $\epsilon_{\sigma}(n)=\min_{\deg(p)=n}\mathbb{E}_{X\sim\mathcal{N}(0,1)}(\sigma (X)-p(X))^{2}$ and note that $\lim_{n\rightarrow\infty}\epsilon_{\sigma}(n)=0$. We will consider fully connected networks of depth $i$ and will use $d_{0}=d$ to denote the input dimension and $d_{1},\ldots,d_{i}$ to denote the number of neurons in each layer. Denote also $\bar{d}=\sum_{j=0}^{i}d_{j}$. Given weight matrices \[\vec{W}=(W^{1},\ldots,W^{i})\in\mathbb{R}^{d_{1}\times d_{0}}\times\ldots \times\mathbb{R}^{d_{i}\times d_{i-1}}\] and $\mathbf{x}\in\mathbb{R}^{d_{0}}$ we define $\Psi^{0}_{\vec{W}}(\mathbf{x})=\mathbf{x}$. Then for $1\leq j\leq i$ we define recursively \[\Phi^{j}_{\vec{W}}(\mathbf{x})=W^{j}\Psi^{j-1}_{\vec{W}}(\mathbf{x}),\quad \Psi^{j}_{\vec{W}}(\mathbf{x})=\sigma\left(\Phi^{j}_{\vec{W}}(\mathbf{x})\right)\] We will consider random networks in which the weight matrices are random _Xavier matrices_[22, 26]. That is, each entry in $W^{j}$ is a centered Gaussian of variance $\frac{1}{d_{j-1}}$. This choice is motivated by the fact that it is a standard practice to initialize the network's weights with Xavier matrices, and furthermore, it ensures that the scale across the network is the same. That is, for any example $\mathbf{x}$ and a neuron $n$, the second moment of the output of $n$ (w.r.t. the choice of $\vec{W}$) is $1$. Our main result shows that $\Psi^{i}_{\vec{W}}$ can be approximated, up to any constant accuracy $\epsilon$, via constant degree polynomials (the constant will depend only on $\epsilon$, the depth $i$, and the activation $\sigma$). We will consider the input space $\widehat{\mathbb{S}}^{d-1}=\{\mathbf{x}\in\mathbb{R}^{d}:\left\\|\mathbf{x} \right\\|=1\}$. Here, and throughout the paper, $\left\\|\mathbf{x}\right\\|$ stands for the _normalized_ Euclidean norm $\left\\|\mathbf{x}\right\\|=\sqrt{\frac{1}{d}\sum_{i=1}^{d}x_{i}^{2}}$. Theorem 1.1.: _For every $i$ and $n$ such that $\epsilon_{\sigma}(n)\leq\frac{1}{2}$ there is a constant $D=D(n,i,\sigma)$ such that if $d_{1},\ldots,d_{i-1}\geq D$ the following holds. For any weights $\vec{W}$, there is a degree $n^{i-1}$ polynomial $p_{\vec{W}}$ such that for any distribution $\mathcal{D}$ on $\widehat{\mathbb{S}}^{d-1}$_ \[\mathbb{E}_{\vec{W}}\mathbb{E}_{\mathbf{x}\sim\mathcal{D}}\left\\|\Phi^{i}_{ \vec{W}}(\mathbf{x})-p_{\vec{W}}(\mathbf{x})\right\\|\leq 14\cdot(L+1)^{2}\cdot( \epsilon_{\sigma}(n))^{\frac{1}{2^{i-1}}}\leq\frac{14\cdot(L+1)^{3}}{n^{\frac{ 1}{2^{i-1}}}}\;.\] _Furthermore, the coefficients of $p_{\vec{W}}$ are bounded by $(2\bar{d})^{4n^{i-1}}$._ Since constant degree polynomials are learnable in polynomial time, Theorem 1.1 implies a PTAS for learning random networks of constant depth. In fact, as shown in [9], constant degree polynomials with polynomial coefficients are efficiently learnable via SGD on ReLU networks starting from standard Xavier initialization. Thus, those PTAS can be standard SGD on neural networks. To be more specific, for any constant $\epsilon>0$ there is an algorithm with $(\bar{d})$${}^{O\left(\left(\frac{14(L+1)^{3}}{\epsilon}\right)^{(i-1)2^{i-1}}\right)}$ time and sample complexity that is guaranteed to return a hypothesis whose loss is at most $\epsilon$ in expectation. For some specific activation, such as the sigmoid $\sigma(x)=\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-\frac{t^{2 }}{2}}dt$, or the ReLU-like activation $\sigma(x)=\int_{0}^{x}\operatorname{erf}(t)+1dt$ we have that $\epsilon_{\sigma}(n)$ approaches to $0$ exponentially fast (see Lemma 2.16). In this case, we get get a _quasi-polynomial_ time and sample complexity of $(\bar{d})^{O\left(\left(\log\left(\frac{14(L+1)^{3}}{\epsilon}\right)\right)^{( i-1)}\right)}$. Corollary 1.2.: _For every constants $\epsilon,i$ and $\sigma$ there is a constant $D$, a univariate-polynomial $p$ and a polynomial-time algorithm $\mathcal{A}$ such that if $d_{1},\ldots,d_{i-1}\geq D$ the following holds. For any distribution $\mathcal{D}$ on $\widetilde{\mathbb{S}}^{d-1}$, if $h$ is the output of $\mathcal{A}$ upon seeing $p(d_{0},\ldots,d_{i})$ examples from $\mathcal{D}$, then1_ Footnote 1: The leftmost expectation denoted $\mathbb{E}_{h}$ is over the examples provided to $\mathcal{A}$, as well as the internal randomness of $\mathcal{A}$. \[\mathbb{E}_{h}\mathbb{E}_{\widetilde{W}}\mathbb{E}_{\mathbf{x}\sim\mathcal{D} }\left\\|\Phi_{\widetilde{W}}^{i}(\mathbf{x})-h(\mathbf{x})\right\\|\leq\epsilon\,.\] _Furthermore, $\mathcal{A}$ can be taken to be SGD on a ReLU network starting from a Xavier initialization._ ### Related work Learning neural networks efficiently.Efficiently learning classes of neural networks has attracted much interest in recent years. Several works established polynomial-time algorithms for learning one-hidden-layer neural networks with certain input distributions (such as the Gaussian distribution) under the assumption that the weight matrix of the hidden layer is non-degenerate [27, 34, 19, 20, 5, 32, 4]. For example, Awasthi et al. [4] showed such a result for non-degenerate one-hidden-layer ReLU networks with bias terms under Gaussian inputs, and also concluded that one-hidden-layer networks can be learned efficiently under the smoothed-analysis framework. Efficient algorithms for learning one-hidden-layer ReLU networks with Gaussian inputs were also shown in Diakonikolas et al. [18], Diakonikolas and Kane [17]. These results do not require non-degenerate weight matrices, but they require that the output layer weights are all positive, as well as a sub-linear upper bound on the number of hidden neurons. Chen et al. [8] recently showed an efficient algorithm for learning one-hidden-layer ReLU networks with Gaussian inputs, under the assumption that the number of hidden neurons is a constant. Note that all of the aforementioned works consider only one-hidden-layer networks. Chen et al. [7] gave an algorithm for learning deeper ReLU networks, whose complexity is polynomial in the input dimension but exponential in the other parameters (such as the number of hidden units, depth, spectral norm of the weight matrices, and Lipschitz constant of the overall network). Finally, several works established algorithms for learning neural networks, whose complexity is exponential unless we impose strong assumptions on the norms of both the inputs and the weights [23, 30, 33, 24]. Hardness of learning neural networks.As we discussed in the previous paragraph, efficient algorithms for learning ReLU networks are known only for depth-$2$ networks and under certain assumptions on both the network weights and the input distribution. The limited progress in learning ReLU networks can be partially understood by an abundance of hardness results. Learning neural networks without any assumptions on the input distribution or the weights is known to be hard (under cryptographic and average-case hardness assumptions) already for depth-$2$ ReLU networks [28, 3, 11]. For depth-$3$ networks, hardness results were obtained already when the input distribution is Gaussian [13, 6]. All of the aforementioned hardness results are for improper learning, namely, they do not impose any restrictions on the learning algorithm or on the hypothesis that it returns. For _statistical query (SQ)_ algorithms, unconditional superpolynomial lower bounds were obtained for learning depth-$3$ networks with Gaussian inputs [6], and superpolynomial lower bounds for _Correlational SQ (CSQ)_ algorithms were obtained already for learning depth-$2$ networks with Gaussian inputs [25, 18]. The above negative results suggest that assumptions on the input distribution may not suffice for obtaining efficient learning algorithms. Since in one-hidden-layer networks efficient algorithms exist when imposing assumptions on both the input distribution and the weights, a natural question is whether this approach might also work for deeper networks. Recently, Daniely et al. [15] gave a hardness result for improperly learning depth-$3$ ReLU networks under the Gaussian distribution even when the weight matrices are non-degenerate. This result suggests that learning networks of depth larger than $2$ might require new approaches and new assumptions. Moreover, [15] showed hardness of learning depth-$3$ networks under the Gaussian distribution even when a small random perturbation is added to the network's parameters, namely, they proved hardness in the smoothed-analysis framework. While adding a small random perturbation to the parameters does not seem to make the problem computationally easier, they posed the question of whether learning random networks, which roughly correspond to adding a large random perturbation, can be done efficiently. The current work gives a positive answer in that respect. Daniely and Vardi [12] studied whether there exist some "natural" properties of the network's weights that may suffice to allow efficient distribution-free learning, where a "natural" property is any property that holds w.h.p. in random networks. More precisely, they considered a setting where the target network is random, an adversary chooses some input distribution (that may depend on the target network), and the learning algorithm needs to learn the random target network under this input distribution. They gave a hardness result for improper learning (within constant accuracy) in this setting. Thus, they showed that learning random networks is hard when the input distribution may depend on the random network. Note that in the current work, we give a positive result in a setting where we first fix an input distribution and then draw a random network. Finally, learning deep random networks was studied in Das et al. [16], Agarwal et al. [1], where the authors showed hardness of learning networks of depth $\omega(\log(d))$ in the SQ model. ## 2 Proof of Theorem 1.1 ### Notation We recall that for vectors $\mathbf{x}\in\mathbb{R}^{d}$ we use the _normalized_ Euclidean norm $\\|\mathbf{x}\\|=\sqrt{\frac{\sum_{i=1}^{d}x_{i}^{2}}{d}}$ and take the unit sphere $\tilde{\mathbb{S}}^{d-1}=\{\mathbf{x}\in\mathbb{R}^{d}:\\|\mathbf{x}\\|=1\}$ w.r.t. this norm as our instance space. Inner products will also be normalized: for $\mathbf{x},\mathbf{y}\in\mathbb{R}^{d}$ we denote $\langle\mathbf{x},\mathbf{y}\rangle=\frac{\sum_{i=1}^{d}x_{i}y_{i}}{d}$. For $\mathbf{x}\in\mathbb{R}^{d}$ and a closed set $A\subset\mathbb{R}^{d}$ we denote $d(\mathbf{x},A):=\min_{\mathbf{x}^{\prime}\in A}\\|\mathbf{x}-\mathbf{x}^{ \prime}\\|$. Unless otherwise specified, a random scalar is assumed to be a standard normal, a random vector in $\mathbb{R}^{d}$ is assumed to be a centered Gaussian vector with covariance matrix $\frac{1}{d}I$, and a random matrix is assumed to be a Xavier matrix. For $f:\mathbb{R}\rightarrow\mathbb{R}$, we denote $\\|f\\|^{2}=\mathbb{E}_{X}f^{2}(X)$. ### Some preliminaries We will use the Hermite Polynomials [29] which are defined via the following recursion formula. \[h_{n+1}(x)=\frac{x}{\sqrt{n+1}}h_{n}(x)-\sqrt{\frac{n}{n+1}}h_{n-1}(x),\ \ h_{0}(x)=1,\ \ h_{1}(x)=x \tag{1}\] The Hermite polynomials are the sequence of normalized orthogonal polynomials w.r.t. the standard Gaussian measure. That is, it holds that \[\mathbb{E}_{X}h_{i}(X)h_{j}(X)=\delta_{ij}\] More generally, if $(X,Y)$ is a Gaussian vector with covariance matrix $\begin{pmatrix}1&\rho\\ \rho&1\end{pmatrix}$ then \[\mathbb{E}_{X,Y}h_{i}(X)h_{j}(Y)=\delta_{ij}\rho^{i} \tag{2}\] We will use the fact that \[h_{n}^{\prime}=\sqrt{n}h_{n-1} \tag{3}\] and that for even $n$ \[\mathbb{E}_{X}X^{n}=(n-1)!! \tag{4}\] We will also use the _dual activation_$\hat{\sigma}(\rho)=\sum_{i=0}^{\infty}a_{i}^{2}\rho^{i}$ as defined in [14]. We note that $\hat{\sigma}$ is defined in $[-1,1]$ and satisfies $\hat{\sigma}(1)=\\|\sigma\\|^{2}=1$. ### Some technical lemmas Lemma 2.1.: _If $\sigma$ is $L$-Lipschitz then $\hat{\sigma}$ is $L^{2}$-Lipschitz in $[-1,1]$_ Proof.: As shown in [14], $(\hat{\sigma})^{\prime}=\widehat{\sigma^{\prime}}$. Hence, for $\rho\in[-1,1]$, \[\|(\hat{\sigma})^{\prime}(\rho)\| = \left\|\widehat{\sigma^{\prime}}(\rho)\right\|\] \[\leq \left\\|\sigma^{\prime}\right\\|^{2}\] \[\leq L^{2}\] Lemma 2.2.: $\|h_{n}(x)-h_{n}(x+y)\|\leq 2^{n}\max(\|x\|,\|x+y\|,1)^{n}\|y\|$__ Proof.: It is not hard to verify by induction on Eq. (1) that \[\|h_{n}(x)\|\leq 2^{n/2}\max(1,\|x\|^{n})\] This implies that for $\xi\in[x,x+y]$ \[\begin{array}{ccc}\|h_{n}(x)-h_{n}(x+y)\|&=&\|h_{n}^{\prime}(\xi)y\|\\ &\stackrel{{ Eq.~{}(\ref{eq:2.1})}}{{=}}&\sqrt{n}\|h_{n-1}(\xi)y\| \\ &\leq&\sqrt{n}2^{n/2}\max(\|x\|,\|x+y\|,1)^{n}\|y\|\\ &\leq&2^{n}\max(\|x\|,\|x+y\|,1)^{n}\|y\|\end{array}\] ### Defining a shadow network In order to approximate $\Psi_{\widetilde{W}}^{i}$ via a polynomial, we will use a "shadow network" that is obtained by replacing the activation $\sigma$ with a polynomial approximation of it. Let $\sigma=\sum_{i=0}^{\infty}a_{i}h_{i}$ be the representation of $\sigma$ in the basis of the Hermite polynomials. Define $\sigma_{n}=\frac{1}{\sqrt{\sum_{i=0}^{n}a_{i}^{2}}}\sum_{i=0}^{n}a_{i}h_{i}$. We have $\epsilon_{\sigma}(n)=\sum_{i=n+1}^{\infty}a_{i}^{2}$ and hence $\sigma_{n}=\frac{1}{\sqrt{1-\epsilon_{\sigma}(n)}}\sum_{i=0}^{n}a_{i}h_{i}$. We next define a shadow network. For $\mathbf{x}\in\mathbb{R}^{d}$ we let $\Psi_{\widetilde{W}}^{0,n}(\mathbf{x})=\mathbf{x}$. For $1\leq j\leq i$ we define recursively \[\Phi_{\widetilde{W}}^{j,n}(\mathbf{x})=W^{j}\Psi_{\widetilde{W}}^{j-1,n}( \mathbf{x}),\quad\Psi_{\widetilde{W}}^{j,n}(\mathbf{x})=\sigma_{n}\left(\Phi_{ \widetilde{W}}^{j,n}(\mathbf{x})\right)\] for $1\leq j\leq i-1$ and $\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})=W^{i}\Psi_{\widetilde{W}}^{i-1,n}( \mathbf{x})$. We will prove the following theorem, which implies Theorem 1.1. Theorem 2.3.: _Fix $i$ and let $n$ be large enough so that $\epsilon_{\sigma}(n)\leq\frac{1}{2}$. There is a constant $D=D(n,i,\sigma)$ such that if $d_{1},\ldots,d_{i-2}\geq D$ then for any $\mathbf{x}\in\tilde{\mathbb{S}}^{d-1}$,_ \[\mathbb{E}_{\widetilde{W}}\left\\|\Phi_{\widetilde{W}}^{i}(\mathbf{x})-\Phi_{ \widetilde{W}}^{i,n}(\mathbf{x})\right\\|\leq 13\cdot(L+1)^{2}\cdot(\epsilon_{ \sigma}(n))^{\frac{1}{2^{i-1}}}\] To see why Theorem 2.3 (together with Lemma 2.15 which bounds $\epsilon_{\sigma}(n)$) implies Theorem 1.1, note that $\Phi_{\widetilde{W}}^{i,n}(\mathbf{x})$ is a polynomial of degree $n^{i-1}$. This implies Theorem 1.1, except the requirement that the coefficients of the polynomial are polynomially bounded. To deal with this, define \[\tilde{\Phi}_{\widetilde{W}}^{i,n}(\mathbf{x})=\begin{cases}\Phi_{\widetilde{W }}^{i,n}(\mathbf{x})&\text{if all entries in $\vec{W}$ are at most $\sum_{j=0}^{i}d_{j}$}\\ 0&\text{otherwise}\end{cases}\] As we show next $\lim_{\min(d_{1},\ldots,d_{i-1})\to\infty}\mathbb{E}_{\widetilde{W}}\left\\| \Phi_{\widetilde{W}}^{i,n}(\mathbf{x})-\tilde{\Phi}_{\widetilde{W}}^{i,n}( \mathbf{x})\right\\|=0$. Hence, in the theorem we can replace $\Phi^{i,n}$ by $\tilde{\Phi}^{i,n}$ which has polynomially bounded coefficients. Lemma 2.4.: _For every $\epsilon$ and $n$ there is a constant $D$ such that if $d_{1},\ldots,d_{i-1}\geq D$ then for any $\mathbf{x}\in\tilde{\mathbb{S}}^{d-1}$, $\mathbb{E}_{\widetilde{W}}\left\\|\Phi_{\widetilde{W}}^{i,n}(\mathbf{x})-\tilde {\Phi}_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|<\epsilon$._ Proof.: Let $A$ be the event that there is an entry in $\vec{W}$ that is greater than $\sum_{j=0}^{i}d_{j}$. We have \[\mathbb{E}_{\tilde{W}}\left\\|\Phi_{\widetilde{W}}^{i,n}(\mathbf{x})-\tilde{ \Phi}_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|=\mathbb{E}\left[\left\\|\Phi_{ \widetilde{W}}^{i,n}(\mathbf{x})\right\\|\cdot 1_{A}\right]\leq\sqrt{\mathbb{E}\left\\| \Phi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|^{2}}\sqrt{\Pr(A)}\] Now, it is not hard to verify that $\mathbb{E}\left\\|\Phi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|^{2}$ is polynomial in $\sum_{j=0}^{i}d_{j}$ while $\Pr(A)$ converges to $0$ exponentially fast in $\sum_{j=0}^{i}d_{j}$. Thus, if $\min(d_{1},\ldots,d_{i-1})$ is large enough then $\mathbb{E}_{\widetilde{W}}\left\\|\Phi_{\widetilde{W}}^{i,n}(\mathbf{x})-\tilde {\Phi}_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|<\epsilon$. Lemma 2.5.: $\tilde{\Phi}_{\widetilde{W}}^{i,n}$ _computes a polynomial whose sum of coefficients is at most $(2\bar{d})^{4n^{i-1}}$._ Proof.: We assume that $\tilde{\Phi}_{\widetilde{W}}^{i,n}=\Phi_{\widetilde{W}}^{i,n}$, as otherwise $\tilde{\Phi}_{\widetilde{W}}^{i,n}\equiv 0$, in which case the lemma is clear. Write $\sigma_{n}(x)=\sum_{k=0}^{n}b_{k}x^{k}$ and $h_{j}(x)=\sum_{k=0}^{j}c_{j,k}x^{k}$. Via induction on Eq. (1), we have $\|c_{j,k}\|\leq 2^{\frac{j}{2}}$. Hence, \[\|b_{k}\| \leq \frac{1}{\sqrt{\sum_{j=0}^{n}a_{j}^{2}}}\sum_{j=0}^{n}\|a_{j}\|\|c_{j,k}\|\] \[\leq \frac{1}{\sqrt{\sum_{j=0}^{n}a_{j}^{2}}}\sum_{j=0}^{n}\|a_{j}\|2^{ \frac{j}{2}}\] \[\leq \frac{1}{\sqrt{\sum_{j=0}^{n}a_{j}^{2}}}\sqrt{\sum_{j=0}^{n}a_{j }^{2}}\sqrt{\sum_{j=0}^{n}2^{j}}\] \[\leq 2^{\frac{n+1}{2}}\] Now, let $M_{j}$ be the maximal sum of coefficients of any polynomial computed by an output neuron of $\Psi_{\vec{W}}^{j,n}$. We next show by induction that $M_{j}\leq(2\bar{d})^{2\sum_{k=1}^{j}n^{k}}$. This will conclude the proof as it will imply that the sum of the coefficients of the polynomial computed by $\Phi_{\vec{W}}^{i,n}$ is at most $(2\bar{d})^{2}M_{i-1}\leq(2\bar{d})^{2\sum_{k=0}^{i-1}n^{k}}\leq(2\bar{d})^{4n ^{i-1}}$. For $j=0$ we have $M_{0}=1$. For $j\geq 1$ we we have \[M_{j}\leq\sum_{k=0}^{n}\|b_{k}\|\left((\bar{d})^{2}M_{j-1}\right)^{k}\leq 2^{ \frac{n+1}{2}}\cdot 2\cdot\left((\bar{d})^{2}M_{j-1}\right)^{n}\leq\left((2 \bar{d})^{2}M_{j-1}\right)^{n}\] By the induction hypothesis we have \[M_{j}\leq(2\bar{d})^{2n+2n\sum_{k=1}^{j-1}n^{k}}=(2\bar{d})^{2\sum_{k=1}^{j}n ^{k}}\] ### Proof of Theorem 2.3 for depth-$2$ networks We will first prove Theorem 2.3 for depth-$2$ networks (i.e. for $i=2$). We will prove Lemma 2.7 below which implies that for every $\epsilon$ there is $n$ such that for any $\mathbf{x}\in\widehat{\mathbb{S}}^{d-1}$, $\mathbb{E}_{\vec{W}}\left\\|\Psi_{\vec{W}}^{1,n}(\mathbf{x})-\Psi_{\vec{W}}^{1} (\mathbf{x})\right\\|\leq\epsilon$. We will then prove Lemma 2.8, that together with Lemma 2.7 will show that $\mathbb{E}_{\vec{W}}\left\\|\Phi_{\vec{W}}^{2,n}(\mathbf{x})-\Phi_{\vec{W}}^{2} (\mathbf{x})\right\\|\leq\epsilon$, thus proving Theorem 2.3 for $i=2$. We will start however with the following lemma that will be useful throughout. Lemma 2.6.: _Fix $f,g:\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^{d_{1}}$ and a Xavier matrix $W\in\mathbb{R}^{d_{2}\times d_{1}}$. Let $(X,Y)$ be a centered Gaussian vector with covariance matrix $\begin{pmatrix}\\|\mathbf{x}\\|^{2}&\langle\mathbf{x},\mathbf{y}\rangle\\ \langle\mathbf{x},\mathbf{y}&\\|\mathbf{y}\\|^{2}\end{pmatrix}$. Then_ \[\mathbb{E}_{W}\left\\|f(W\mathbf{x})-g(W\mathbf{y})\right\\|\leq\sqrt{\mathbb{E} _{W}\left\\|f(W\mathbf{x})-g(W\mathbf{y})\right\\|^{2}}=\sqrt{\mathbb{E}_{X,Y}( f(X)-g(Y))^{2}}\] Proof.: We have \[\mathbb{E}_{W}\left\\|f(W\mathbf{x})-g(W\mathbf{y})\right\\| \stackrel{{\text{Jensen Inequality}}}{{\leq}} \sqrt{\mathbb{E}_{W}\left\\|f(W\mathbf{x})-g(W\mathbf{y}) \right\\|^{2}}\] \[= \sqrt{\frac{1}{d_{2}}\sum_{j=1}^{d_{2}}\mathbb{E}_{W}(f((W \mathbf{x})_{j})-g((W\mathbf{y})_{j})^{2}}\] Now, the lemma follows from the fact that $\{((W\mathbf{x})_{j},(W\mathbf{y})_{j})\}_{j=1}^{d_{2}}$ are independent centered Gaussian vectors with covariance matrix $\begin{pmatrix}\\|\mathbf{x}\\|^{2}&\langle\mathbf{x},\mathbf{y}\rangle\\ \langle\mathbf{x},\mathbf{y}\rangle&\\|\mathbf{y}\\|^{2}\end{pmatrix}$. Lemma 2.7.: _Fix $\mathbf{x}\in\tilde{\mathbb{S}}^{d_{1}-1}$. Fix a Xavier matrix $W\in\mathbb{R}^{d_{2}\times d_{1}}$. Then_ \[\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W\mathbf{x})\right\\|\leq \sqrt{2\epsilon_{\sigma}(n)}\] Proof.: By Lemma 2.6 we have \[\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W\mathbf{x})\right\\| \leq \sqrt{\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W\mathbf{ x})\right\\|^{2}}\] \[= \sqrt{\mathbb{E}_{X}(\sigma(X)-\sigma_{n}(X))^{2}}\] \[= \sqrt{\sum_{i=0}^{n}\left(1-\frac{1}{\sqrt{1-\epsilon_{\sigma}(n )}}\right)^{2}a_{i}^{2}+\sum_{i=n+1}^{\infty}a_{i}^{2}}\] \[= \sqrt{(1-\epsilon_{\sigma}(n))\left(1-\frac{1}{\sqrt{1-\epsilon_ {\sigma}(n)}}\right)^{2}+\epsilon_{\sigma}(n)}\] \[= \sqrt{2-\epsilon_{\sigma}(n)-2\sqrt{1-\epsilon_{\sigma}(n)}+ \epsilon_{\sigma}(n)}\] \[= \sqrt{2(1-\sqrt{1-\epsilon_{\sigma}(n)})}\] \[\leq \sqrt{2(1-\sqrt{1-\epsilon_{\sigma}(n)})(1+\sqrt{1-\epsilon_{ \sigma}(n)})}\] \[= \sqrt{2\epsilon_{\sigma}(n)}\] Lemma 2.7 implies that $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{1,n}(\mathbf{x})-\Psi_ {\widetilde{W}}^{1}(\mathbf{x})\right\\|\leq\sqrt{2\epsilon_{\sigma}(n)}$. Thus, given $\epsilon>0$, for sufficiently large $n$, $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{1,n}(\mathbf{x})-\Psi_ {\widetilde{W}}^{1}(\mathbf{x})\right\\|\leq\epsilon$. The following lemma therefore implies that $\mathbb{E}_{\widetilde{W}}\left\\|\Phi_{\widetilde{W}}^{2,n}(\mathbf{x})-\Phi _{\widetilde{W}}^{2}(\mathbf{x})\right\\|\leq\sqrt{2\epsilon_{\sigma}(n)}$ and thus implies Theorem 2.3 for depth two networks. Lemma 2.8.: _For any $\mathbf{x}\in\tilde{\mathbb{S}}^{d-1}$_ \[\mathbb{E}_{W^{i}}\left\\|\Phi_{\widetilde{W}}^{i,n}(\mathbf{x})-\Phi_{ \widetilde{W}}^{i}(\mathbf{x})\right\\|\leq\left\\|\Psi_{\widetilde{W}}^{i-1,n}( \mathbf{x})-\Psi_{\widetilde{W}}^{i-1}(\mathbf{x})\right\\|\] Proof.: By Lemma 2.6 we have \[\mathbb{E}_{W^{i}}\left\\|\Phi_{\widetilde{W}}^{i,n}(\mathbf{x})- \Phi_{\widetilde{W}}^{i}(\mathbf{x})\right\\| = \mathbb{E}_{W^{i}}\left\\|W^{i}\left(\Psi_{\widetilde{W}}^{i-1,n}( \mathbf{x})-\Psi_{\widetilde{W}}^{i-1}(\mathbf{x})\right)\right\\|\] \[\leq \sqrt{\mathbb{E}_{X}\sim\mathcal{N}\left(0,\left\\|\Psi_{ \widetilde{W}}^{i-1,n}(\mathbf{x})-\Psi_{\widetilde{W}}^{i-1}(\mathbf{x}) \right\\|^{2}\right)^{X^{2}}}\] \[= \left\\|\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x})-\Psi_{\widetilde {W}}^{i-1}(\mathbf{x})\right\\|\] ### Proof of Theorem 2.3 for general networks For $\mathbf{x}\in\tilde{\mathbb{R}}^{d_{i-1}}$ we denote $\Psi_{W^{i}}(\mathbf{x})=\sigma(W^{i}\mathbf{x})$ and $\Psi_{W^{i}}^{n}(\mathbf{x})=\sigma_{n}(W^{i}\mathbf{x})$. Lemma 2.7 can be roughly phrased as \[(\mathbf{x}=\mathbf{x}^{\prime})\text{ and }(\\|\mathbf{x}\\|=1)\Rightarrow\Psi_{W^{ i}}(\mathbf{x})\approx\Psi_{W^{i}}^{n}(\mathbf{x}^{\prime})\] In order to prove Theorem 2.3 for general networks we will extend it by replacing the strict equality conditions with softer ones. That is, we will show that \[(\mathbf{x}\approx\mathbf{x}^{\prime})\text{ and }(\\|\mathbf{x}\\|\approx 1) \text{ and }(\\|\mathbf{x}^{\prime}\\|\approx 1)\Rightarrow\Psi_{W^{i}}(\mathbf{x}) \approx\Psi_{W^{i}}^{n}(\mathbf{x}^{\prime}) \tag{5}\] This will be enough to prove Theorem 2.3 for general networks. Indeed, the conditions $\\|\mathbf{x}\\|\approx 1$ and $\\|\mathbf{x}^{\prime}\\|\approx 1$ are valid w.h.p. via a simple probabilistic argument. Thus, Eq. (5) implies that \[\mathbf{x}\approx\mathbf{x}^{\prime}\Rightarrow\Psi_{W^{i}}(\mathbf{x}) \approx\Psi_{W^{i}}^{n}(\mathbf{x}^{\prime}) \tag{6}\] Now, for $\mathbf{x}\in\tilde{\mathbb{S}}^{d-1}$ Eq. (6) implies that $\Psi_{W^{1}}(\mathbf{x})\approx\Psi_{W^{1}}^{n}(\mathbf{x}^{\prime})$. Using Eq. (6) again we get that $\Psi_{W^{2}}\circ\Psi_{W^{1}}(\mathbf{x})\approx\Psi_{W^{2}}^{n}\circ\Psi_{W^ {1}}^{n}(\mathbf{x}^{\prime})$. Using it $i-3$ more times we we get that $\Psi_{W^{i-1}}\circ\cdots\circ\Psi_{W^{1}}(\mathbf{x})\approx\Psi_{W^{i-1}}^{n} \circ\Psi_{W^{1}}^{n}(\mathbf{x}^{\prime})$, or in other words that $\Psi_{\tilde{W}}^{i-1}(\mathbf{x})\approx\Psi_{\tilde{W}}^{i-1,n}(\mathbf{x})$. As we will show $``\approx"$ stands for a sufficiently strong approximation, which guarantees that $\mathbb{E}_{\tilde{W}}\\|\Psi_{\tilde{W}}^{i-1}(\mathbf{x})-\Psi_{\tilde{W}}^{i -1,n}(\mathbf{x})\\|\leq\epsilon$, and hence Lemma 2.8 implies Theorem 2.3. To prove Eq. (5) we first prove Lemma 2.9 which softens the requirement that $\mathbf{x}=\mathbf{x}^{\prime}$. That is, it shows that \[(\mathbf{x}\approx\mathbf{x}^{\prime})\text{ and }(\\|\mathbf{x}\\|=\\|\mathbf{x}^{ \prime}\\|=1)\Rightarrow\Psi_{W^{i}}(\mathbf{x})\approx\Psi_{W^{i}}^{n}( \mathbf{x}^{\prime})\] The second condition which requires that $\\|\mathbf{x}\\|=\\|\mathbf{x}^{\prime}\\|=1$ is soften via Lemmas 2.10 and 2.11. Lemma 2.12 then wraps the two softenings together, and shows that Eq. (5) is valid. Finally, in section 2.6.1 we use Lemma 2.12 to prove Theorem 2.3. Lemma 2.9.: _Fix $\mathbf{x},\mathbf{x}+\mathbf{v}\in\tilde{\mathbb{S}}^{d_{1}-1}$ with $\\|\mathbf{v}\\|\leq\epsilon$. Fix a Xavier matrix $W\in\mathbb{R}^{d_{2}\times d_{1}}$. Then_ \[\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+\mathbf{v})) \right\\|\leq\sqrt{2\epsilon_{\sigma}(n)}+\sqrt{\frac{2L^{2}}{1-\epsilon_{ \sigma}(n)}}\epsilon\] Proof.: We have \[\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+\mathbf{v}))\right\\|\leq \left\\|\sigma(W\mathbf{x})-\sigma_{n}(W\mathbf{x})\right\\|+\left\\|\sigma_{n}( W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+\mathbf{v}))\right\\|\] By Lemma 2.7 we have $\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W\mathbf{x})\right\\|\leq \sqrt{2\epsilon_{\sigma}(n)}$. It remains to bound $\mathbb{E}_{W}\left\\|\sigma_{n}(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+\mathbf{v }))\right\\|$. By Lemma 2.6 We have \[\mathbb{E}_{W}\left\\|\sigma_{n}(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+\mathbf{v }))\right\\|\leq\sqrt{\mathbb{E}_{X,Y}(\sigma_{n}(X)-\sigma_{n}(Y))^{2}}\] where $(X,Y)$ is a centered Gaussian vector with correlation matrix $\begin{pmatrix}1&\rho\\ \rho&1\end{pmatrix}$ for $\rho=\langle\mathbf{x},\mathbf{x}+\mathbf{v}\rangle\geq 1-\epsilon$. Finally, we have \[\mathbb{E}_{X,Y}(\sigma_{n}(X)-\sigma_{n}(Y))^{2} = \frac{1}{1-\epsilon_{\sigma}(n)}\mathbb{E}_{X,Y}\left(\sum_{i=0}^{n }a_{i}(h_{i}(X)-h_{i}(Y))\right)^{2}\] \[= \frac{1}{1-\epsilon_{\sigma}(n)}\sum_{i=0}^{n}\sum_{j=0}^{n}a_{i} a_{j}\mathbb{E}_{X,Y}(h_{i}(X)-h_{i}(Y))(h_{j}(X)-h_{j}(Y))\] \[\stackrel{{ Eq.\leavevmode\nobreak\ (\ref{eq:2})}}{{=}} \frac{1}{1-\epsilon_{\sigma}(n)}\sum_{i=0}^{n}a_{i}^{2}(2-2\rho^{i})\] \[\leq \frac{2}{1-\epsilon_{\sigma}(n)}\left(\hat{\sigma}(1)-\hat{ \sigma}(\rho)\right)\] \[\stackrel{{\text{Lemma \ref{lem:2}}}}{{\leq}} \frac{2L^{2}}{1-\epsilon_{\sigma}(n)}\epsilon\] We next prove a lemma that allows us to "almost jointly project" a pair of points $\mathbf{x}_{1},\mathbf{x}_{2}\in\mathbb{R}^{d}$ on a closed set $A\subset\mathbb{R}^{d}$, without expanding the distance too much. Lemma 2.10.: _Let $A\subset\mathbb{R}^{d}$ a closed set and fix $\mathbf{x}_{1},\mathbf{x}_{2}\in\mathbb{R}^{d}$. There are $\tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2}\in A$ such that_ \[\\|\mathbf{x}_{1}-\tilde{\mathbf{x}}_{1}\\|\leq 2d(\mathbf{x}_{1},A),\ \ \\|\mathbf{x}_{2}-\tilde{\mathbf{x}}_{2}\\|\leq 2 d(\mathbf{x}_{2},A)\ \ \text{ and }\ \ \\|\tilde{\mathbf{x}}_{1}-\tilde{\mathbf{x}}_{2}\\|\leq 3\\| \mathbf{x}_{1}-\mathbf{x}_{2}\\|\] Proof.: Let $P_{A}:\mathbb{R}^{d}\to A$ a function such that for any $\mathbf{x}\in\mathbb{R}^{d}$, $\\|P_{A}(\mathbf{x})-\mathbf{x}\\|=d(\mathbf{x},A)$. Assume w.l.o.g. that $\\|\mathbf{x}_{1}-P_{A}(\mathbf{x}_{1})\\|\leq\\|\mathbf{x}_{2}-P_{A}(\mathbf{x }_{2})\\|$. Case I:$\\|\mathbf{x}_{2}-P_{A}(\mathbf{x}_{2})\\|\leq\\|\mathbf{x}_{1}-\mathbf{x}_{2}\\|$ Simply define $\tilde{\mathbf{x}}_{i}=P_{A}(\mathbf{x}_{i})$. We have \[\\|\mathbf{x}_{1}-\tilde{\mathbf{x}}_{1}\\|=\\|\mathbf{x}_{1}-P_{A}(\mathbf{x}_{ 1})\\|\,,\ \ \\|\mathbf{x}_{2}-\tilde{\mathbf{x}}_{2}\\|=\\|\mathbf{x}_{2}-P_{A}(\mathbf{x}_{ 2})\\|\] and \[\\|\tilde{\mathbf{x}}_{1}-\tilde{\mathbf{x}}_{2}\\|\leq\\|P_{A}(\mathbf{x}_{1})- \mathbf{x}_{1}\\|+\\|\mathbf{x}_{1}-\mathbf{x}_{2}\\|+\\|\mathbf{x}_{2}-P_{A}( \mathbf{x}_{2})\\|\leq 3\\|\mathbf{x}_{1}-\mathbf{x}_{2}\\|\] Case II:$\\|\mathbf{x}_{1}-\mathbf{x}_{2}\\|\leq\\|\mathbf{x}_{2}-P_{A}(\mathbf{x}_{ 2})\\|$ Define $\tilde{\mathbf{x}}_{1}=\tilde{\mathbf{x}}_{2}=P_{A}(\mathbf{x}_{1})$. We have \[\\|\mathbf{x}_{1}-\tilde{\mathbf{x}}_{1}\\|=\\|\mathbf{x}_{1}-P_{A}(\mathbf{x}_{ 1})\\|\,,\ \ \\|\tilde{\mathbf{x}}_{1}-\tilde{\mathbf{x}}_{2}\\|\leq 0\\|\mathbf{x}_{1}- \mathbf{x}_{2}\\|\] and \[\\|\mathbf{x}_{2}-\tilde{\mathbf{x}}_{2}\\|\leq\\|\mathbf{x}_{2}-\mathbf{x}_{1} \\|+\\|\mathbf{x}_{1}-P_{A}(\mathbf{x}_{1})\\|\leq 2\left\\|\mathbf{x}_{2}-P_{A}( \mathbf{x}_{2})\right\\|\] Lemma 2.11.: _Let $\mathbf{x},\mathbf{x}+\mathbf{v}\in\mathbb{R}^{d_{1}}$ be vectors such that $\\|\mathbf{x}\\|=1$ and $\\|\mathbf{v}\\|\leq\epsilon\leq 1$. Fix a Xavier matrix $W\in\mathbb{R}^{d_{2}\times d_{1}}$. Then_ \[\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma(W(\mathbf{x}+\mathbf{v}))\right\\| \leq L\epsilon\] _and_ \[\mathbb{E}_{W}\left\\|\sigma_{n}(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+\mathbf{v }))\right\\|\leq 2^{2n+1}\left(9(4n-1)!!\right)^{1/4}\epsilon=:\lambda(n)\epsilon\] Proof.: Fix a centered Gaussain vector $(X,Y)$ with covariance matrix $\begin{pmatrix}1&\langle\mathbf{x}+\mathbf{v},\mathbf{x}\rangle\\ \langle\mathbf{x}+\mathbf{v},\mathbf{x}\rangle&\\|\mathbf{x}+\mathbf{v}\\|^{2} \end{pmatrix}$. Let $Z=Y-X$. Note that $\text{Var}(Z)\leq\epsilon^{2}$. By Lemma 2.6 we have \[\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma(W(\mathbf{x}+\mathbf{v})) \right\\|\leq\sqrt{\mathbb{E}(\sigma(X)-\sigma(X+Z))^{2}}\leq\sqrt{L^{2} \mathbb{E}Z^{2}}\leq L\epsilon\] For the second part, we have by Lemma 2.2 that \[\|\sigma_{n}(x)-\sigma_{n}(x+y)\| \leq\sum_{i=0}^{n}\frac{\|a_{i}\|}{\sqrt{1-\epsilon_{\sigma}(n)}}\| h_{i}(x)-h_{i}(x+y)\|\] \[\leq\|y\|\sum_{i=0}^{n}2^{i}\max(\|x\|^{i},\|x+y\|^{i},1)\] \[\leq\|y\|2^{n+1}\max(\|x\|^{n},\|x+y\|^{n},1)\] Hence, \[\mathbb{E}(\sigma_{n}(X)-\sigma_{n}(X+Z))^{2} \leq 2^{2n+2}\mathbb{E}Z^{2}\max(\|X\|^{n},\|X+Z\|^{n},1)^{2}\] \[\leq 2^{2n+2}\sqrt{\mathbb{E}Z^{4}}\sqrt{\mathbb{E}\max(\|X\|^{4n},\|X+Z \|^{4n},1)}\] \[\leq 2^{2n+2}\sqrt{\mathbb{E}Z^{4}}\sqrt{\mathbb{E}\left[\|X\|^{4n}+\|X+ Z\|^{4n}+1\right]}\] \[\stackrel{{Eq.~{}\eqref{eq:2n}}}{{=}} 2^{2n+2}\sqrt{3\\|\mathbf{v}\\|^{4}}\sqrt{1+(4n-1)!!(\\|\mathbf{x}+ \mathbf{v}\\|^{4n}+\\|\mathbf{x}\\|^{4n})}\] \[\leq 2^{2n+2}\sqrt{3\epsilon^{4}}\sqrt{3(1+\epsilon)^{4n}(4n-1)!!}\] \[\leq 2^{2n+2}\sqrt{3\epsilon^{4}}\sqrt{3\cdot 2^{4n}\cdot(4n-1)!!}\] Lemma 2.6 now implies that \[\mathbb{E}_{W}\left\\|\sigma_{n}(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+ \mathbf{v}))\right\\| \leq 2^{2n+1}\left(9(4n-1)!!\right)^{1/4}\epsilon\] Lemma 2.12.: _Let $\mathbf{x},\mathbf{x}+\mathbf{v}\in\mathbb{R}^{d_{1}}$ be vectors such that $\\|\mathbf{v}\\|\leq\epsilon$, $\\|\mathbf{x}\\|-1\\|\leq\delta\leq 1/2$ and $\|\\|\mathbf{x}+\mathbf{v}\\|-1\|\leq\delta$. Fix a Xavier matrix $W\in\mathbb{R}^{d_{2}\times d_{1}}$. Then_ \[\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+\mathbf{v})) \right\\|\leq 2L\delta+\sqrt{2\epsilon_{\sigma}(n)}+\sqrt{\frac{6L^{2}}{1- \epsilon_{\sigma}(n)}}\epsilon+2\lambda(n)\delta\] Proof.: By Lemma 2.10 there are vectors $\mathbf{x}^{\prime},\mathbf{v}^{\prime}$ such that $\\|\mathbf{x}^{\prime}\\|=\\|\mathbf{x}^{\prime}+\mathbf{v}^{\prime}\\|=1$ and \[\\|\mathbf{x}-\mathbf{x}^{\prime}\\|\leq 2\delta,\ \\|\mathbf{x}+\mathbf{v}- \mathbf{x}^{\prime}-\mathbf{v}^{\prime}\\|\leq 2\delta,\ \text{ and }\\|\mathbf{v}^{\prime}\\|\leq 3\\|\mathbf{v}\\|\] Now, we have, by Lemmas 2.9 and 2.11, \[\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma_{n}(W(\mathbf{x}+ \mathbf{v}))\right\\| \leq \mathbb{E}_{W}\left\\|\sigma(W\mathbf{x})-\sigma(W\mathbf{x}^{\prime })\right\\|\] \[+\mathbb{E}_{W}\left\\|\sigma(W\mathbf{x}^{\prime})-\sigma_{n}(W( \mathbf{x}^{\prime}+\mathbf{v}^{\prime}))\right\\|\] \[+\mathbb{E}_{W}\left\\|\sigma_{n}(W(\mathbf{x}^{\prime}+\mathbf{v} ^{\prime}))-\sigma_{n}(W(\mathbf{x}+\mathbf{v}))\right\\|\] \[\leq 2L\delta+\sqrt{2\epsilon_{\sigma}(n)}+\sqrt{\frac{6L^{2}}{1- \epsilon_{\sigma}(n)}}\epsilon+2\lambda(n)\delta\] #### 2.6.1 Concluding the proof of Theorem 2.3 Define \[\Psi^{i}_{\widetilde{W}}(\mathbf{x},\delta)=\begin{cases}0&\|1-\\|\Psi^{j}_{ \widetilde{W}}(\mathbf{x})\\|>\delta\text{ or }\|1-\\|\Psi^{j,n}_{\widetilde{W}}(\mathbf{x})\\| \|>\delta\text{ for some }j<i\\ \Psi^{i}_{\widetilde{W}}(\mathbf{x})&\text{otherwise}\end{cases}\] and \[\Psi^{i,n}_{\widetilde{W}}(\mathbf{x},\delta)=\begin{cases}0&\|1-\\|\Psi^{j}_{ \widetilde{W}}(\mathbf{x})\\|\|>\delta\text{ or }\|1-\\|\Psi^{j,n}_{\widetilde{W}}(\mathbf{x})\\| \|>\delta\text{ for some }j<i\\ \Psi^{i,n}_{\widetilde{W}}(\mathbf{x})&\text{otherwise}\end{cases}\] We have \[\mathbb{E}_{\widetilde{W}}\left\\|\Psi^{i}_{\widetilde{W}}(\mathbf{ x})-\Psi^{i,n}_{\widetilde{W}}(\mathbf{x})\right\\| \leq\mathbb{E}_{\widetilde{W}}\left\\|\Psi^{i}_{\widetilde{W}}( \mathbf{x})-\Psi^{i}_{\widetilde{W}}(\mathbf{x},\delta)\right\\|+\mathbb{E}_ {\widetilde{W}}\left\\|\Psi^{i}_{\widetilde{W}}(\mathbf{x},\delta)-\Psi^{i,n}_ {\widetilde{W}}(\mathbf{x},\delta)\right\\|\] \[\quad+\mathbb{E}_{\widetilde{W}}\left\\|\Psi^{i,n}_{\widetilde{W} }(\mathbf{x},\delta)-\Psi^{i,n}_{\widetilde{W}}(\mathbf{x})\right\\|\] Theorem 2.3 now follows from Lemmas 2.13 and 2.14 below, together with Lemma 2.8. Lemma 2.13.: _Let $n$ be large enough so that $\epsilon_{\sigma}(n)\leq\frac{1}{2}$ and let $\delta<\frac{\sqrt{\epsilon_{\sigma}(n)}}{2L+2\lambda(n)}$. Then,_ \[\mathbb{E}_{\widetilde{W}}\left\\|\Psi^{i}_{\widetilde{W}}(\mathbf{x},\delta)- \Psi^{i,n}_{\widetilde{W}}(\mathbf{x},\delta)\right\\|\leq 12\cdot(L+1)^{2}\cdot( \epsilon_{\sigma}(n))^{2^{-i}}\] Proof.: We will prove the result by induction on $i$. The case $i=0$ is clear as $\Psi^{0}_{\widetilde{W}}(\mathbf{x},\delta)=\Psi^{0,n}_{\widetilde{W}}(\mathbf{ x},\delta)$. Fix $i>0$. For every $\delta<\frac{1}{2}$ and $n$ we have by Lemma 2.12 \[\mathbb{E}_{W^{i}}\left\\|\Psi^{i}_{\widetilde{W}}(\mathbf{x}, \delta)-\Psi^{i,n}_{\widetilde{W}}(\mathbf{x},\delta)\right\\| \leq 2L\delta+\sqrt{2\epsilon_{\sigma}(n)}+\sqrt{\frac{6L^{2}}{1- \epsilon_{\sigma}(n)}\left\\|\Psi^{i-1}_{\widetilde{W}}(\mathbf{x},\delta)- \Psi^{i-1,n}_{\widetilde{W}}(\mathbf{x},\delta)\right\\|}+2\lambda(n)\delta\] Taking expectation over $W^{1},\ldots,W^{i-1}$ we get \[\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x}, \delta)-\Psi_{\widetilde{W}}^{i,n}(\mathbf{x},\delta)\right\\| \leq 2L\delta+\sqrt{2\epsilon_{\sigma}(n)}+\mathbb{E}_{\widetilde{W}} \sqrt{\frac{6L^{2}}{1-\epsilon_{\sigma}(n)}\left\\|\Psi_{\widetilde{W}}^{i-1}( \mathbf{x},\delta)-\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x},\delta)\right\\|}+2 \lambda(n)\delta\] \[\overset{\text{Jensen}}{\leq} 2L\delta+\sqrt{2\epsilon_{\sigma}(n)}+\sqrt{\frac{6L^{2}}{1- \epsilon_{\sigma}(n)}\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i- 1}(\mathbf{x},\delta)-\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x},\delta)\right\\| }+2\lambda(n)\delta\] \[\overset{\delta<\frac{\sqrt{\epsilon_{\sigma}(n)}}{2L+2\lambda(n )}}{\leq} 4\sqrt{\epsilon_{\sigma}(n)}+\sqrt{\frac{6L^{2}}{1-\epsilon_{\sigma}(n)} \mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i-1}(\mathbf{x},\delta )-\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x},\delta)\right\\|}\] \[\overset{\epsilon_{\sigma}(n)\leq\frac{1}{2}}{\leq} 4\sqrt{\epsilon_{\sigma}(n)}+L\sqrt{12\mathbb{E}_{\widetilde{W}}\left\\| \Psi_{\widetilde{W}}^{i-1}(\mathbf{x},\delta)-\Psi_{\widetilde{W}}^{i-1,n}( \mathbf{x},\delta)\right\\|}\] \[\overset{\text{Induction}}{\leq} 4\sqrt{\epsilon_{\sigma}(n)}+L\sqrt{12\cdot 12\cdot(L+1)^{2} \cdot\left(\epsilon_{\sigma}(n)\right)^{2^{-i+1}}}\] \[\leq (L+1)\sqrt{12\cdot 12\cdot(L+1)^{2}\cdot\left(\epsilon_{\sigma}(n) \right)^{2^{-i+1}}}\] \[= 12\cdot(L+1)^{2}\cdot\left(\epsilon_{\sigma}(n)\right)^{2^{-i}}\] Lemma 2.14.: _Fix $i,n,\delta$ and $\epsilon>0$. There is a constant $D$ such that if $d_{1},\ldots,d_{i-1}\geq D$ then_ \[\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x})-\Psi_{ \widetilde{W}}^{i}(\mathbf{x},\delta)\right\\|+\mathbb{E}_{\widetilde{W}} \left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x},\delta)-\Psi_{\widetilde{W}}^{i, n}(\mathbf{x})\right\\|\leq\epsilon\] Proof.: Let $B_{i,\delta}$ be the event that for some $j<i$, $\|1-\\|\Psi_{\widetilde{W}}^{j}(\mathbf{x})\\|\\|>\delta$ or $\|1-\\|\Psi_{\widetilde{W}}^{j,n}(\mathbf{x})\\|\\|>\delta$. We have \[\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x})-\Psi_{ \widetilde{W}}^{i}(\mathbf{x},\delta)\right\\|=\mathbb{E}_{\widetilde{W}}\left[ \left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x})\right\\|1_{B_{i,\delta}}\right]\leq \sqrt{\mathbb{E}_{\widetilde{W}}\left[\left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x })\right\\|^{2}\right]}\sqrt{\Pr(B_{i,\delta})}\] Similarly, \[\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})-\Psi_ {\widetilde{W}}^{i,n}(\mathbf{x},\delta)\right\\|\leq\sqrt{\mathbb{E}_{ \widetilde{W}}\left[\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|^{2 }\right]}\sqrt{\Pr(B_{i,\delta})}\] the lemma now follows from the following two claims. Claim 1.: $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x})\right\\|^{2}$ _and $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\| ^{2}$ are bounded by a constant (independent of $d_{0},\ldots,d_{i}$)._ Proof.: We have \[\mathbb{E}_{W^{i}}\left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x})\right\\|^{2}= \mathbb{E}_{\mathbf{w}}\sigma^{2}\left(\mathbf{w}^{\top}\Psi_{\widetilde{W}}^{ i-1}(\mathbf{x})\right)\leq 2\sigma^{2}(0)+2L^{2}\mathbb{E}_{\mathbf{w}}\left( \mathbf{w}^{\top}\Psi_{\widetilde{W}}^{i-1}(\mathbf{x})\right)^{2}=2\sigma^{2} (0)+2L^{2}\\|\Psi_{\widetilde{W}}^{i-1}(\mathbf{x})\\|^{2}\] By induction on $i$, this implies that $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i}(\mathbf{x})\right\\| ^{2}$ is bounded by a constant that depends only on $i$ and $L$ (but not on $d_{1},\ldots,d_{i}$). For $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\| ^{2}$ we have \[\mathbb{E}_{W^{i}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|^{2}= \mathbb{E}_{\mathbf{w}}\sigma_{n}^{2}\left(\mathbf{w}^{\top}\Psi_{\widetilde{W} }^{i-1,n}(\mathbf{x})\right)\] Hence, $\mathbb{E}_{W^{i}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\|^{2}$ is an even polynomial in $\left\\|\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x})\right\\|$ of degree $\leq 2n$. The polynomial depends only on $\sigma_{n}$. It therefore enough to show that for any $i$ and $k$, $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\| ^{2k}$ is bounded, by a bound that is independent of $d_{0},\ldots,d_{i}$. We will show that via induction on $i$. For $i=0$ this is trivial as $\left\\|\Psi_{\widetilde{W}}^{0,n}(\mathbf{x})\right\\|^{2k}\equiv 1$. Fix $i\geq 1$. We have \[\mathbb{E}_{W^{i}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x}) \right\\|^{2k} = \mathbb{E}_{W^{i}}\left(\frac{\sum_{j=1}^{d_{i}}\sigma_{n}^{2} \left(\left(W^{i}\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x})\right)_{j}\right)}{ d_{i}}\right)^{k}\] \[\stackrel{{\text{Jensen inequality}}}{{\leq}} \frac{1}{d_{i}}\mathbb{E}_{W^{i}}\sum_{j=1}^{d_{i}}\sigma_{n}^{2k }\left(\left(W^{i}\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x})\right)_{j}\right)\] \[= \mathbb{E}_{\mathbf{w}}\sigma_{n}^{2k}\left(\mathbf{w}^{\top} \Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x})\right)\] The last expression is an even polynomial in $\left\\|\Psi_{\widetilde{W}}^{i-1,n}(\mathbf{x})\right\\|$. The polynomial depends only on $2k$ and $n$. By the induction hypothesis we conclude that $\mathbb{E}_{\widetilde{W}}\left\\|\Psi_{\widetilde{W}}^{i,n}(\mathbf{x})\right\\| ^{2k}$ is bounded by a bound that is independent from $d_{0},\ldots,d_{i}$. Claim 2.: _For every $\delta,\epsilon^{\prime}$, $i$ and $n$, there is a constant $D$ such that if $d_{1},\ldots,d_{i-1}\geq D$ then $\Pr(B_{i,\delta})<\epsilon^{\prime}$._ Proof.: We will prove the lemma by induction on $i$. For $i=1$ this is immediate as $\Pr(B_{i,\delta})=0$. Fix $i\geq 2$. Let $\delta^{\prime}$ be small enough so that if $\left\|\left\\|\mathbf{x}\right\\|-1\right\|\leq\delta^{\prime}$ then \[\left\|\mathbb{E}_{\mathbf{w}}\sigma^{2}(\mathbf{w}^{\top}\mathbf{x})-1\right\| <\frac{\delta}{4}\text{ and }\left\|\mathbb{E}_{\mathbf{w}}\sigma_{n}^{2}( \mathbf{w}^{\top}\mathbf{x})-1\right\|<\frac{\delta}{4}\] and \[\left\|\mathbb{E}_{\mathbf{w}}\sigma^{4}(\mathbf{w}^{\top}\mathbf{x})-\mathbb{ E}_{X}\sigma^{4}(X)\right\|<1\text{ and }\left\|\mathbb{E}_{\mathbf{w}}\sigma_{n}^{4}(\mathbf{w}^{\top}\mathbf{x})- \mathbb{E}_{X}\sigma_{n}^{4}(X)\right\|<1\] we have \[\Pr(B_{i,\delta})\leq\Pr(B_{i,\delta}\|B_{i-1,\delta^{\prime}}^{c})+\Pr(B_{i- 1,\delta^{\prime}})\] By Chebyshev inequality, $\Pr(B_{i,\delta}\|B_{i-1,\delta^{\prime}}^{c})<\frac{\epsilon^{\prime}}{2}$ for sufficiently large $d_{i-1}$. By the induction hypothesis, $\Pr(B_{i-1,\delta^{\prime}})<\frac{\epsilon^{\prime}}{2}$ for sufficiently large $d_{1},\ldots,d_{i-2}$ ### Bounds on $\epsilon_{\sigma}(n)$ By Eq. (3) if $\sigma$ is differentiable $k$ times then we have $\sigma^{(k)}=\sum_{i=k}^{\infty}\sqrt{\frac{i!}{(i-k)!}}a_{i}h_{i-k}$. Hence, for $k\leq n+1$, \[\epsilon_{\sigma}(n)=\sum_{i=n+1}^{\infty}a_{i}^{2}\leq\frac{(n+1-k)!}{(n+1)! }\sum_{i=n+1}^{\infty}\frac{i!}{(i-k)!}a_{i}^{2}\leq\frac{(n+1-k)!}{(n+1)!} \left\\|\sigma^{(k)}\right\\|^{2} \tag{7}\] Lemma 2.15.: _For any $L$-Lipschitz $\sigma$ we have $\epsilon_{\sigma}(n)\leq\frac{L^{2}}{n}$._ Proof.: By Eq. (7) for $k=1$ we get \[\epsilon_{\sigma}(n)\leq\frac{1}{n+1}\left\\|\sigma^{\prime}\right\\|^{2}\leq\frac{ L^{2}}{n+1}\] Lemma 2.16.: _For the sigmoid activation $\sigma(x)=\int_{0}^{x}e^{-\frac{t^{2}}{2}}dt$ we have $\epsilon_{\sigma}(n)\leq 2^{-n}$._ Proof.: We have $\sigma^{(k)}(x)=(-1)^{k-1}\sqrt{(k-1)!}h_{k-1}(x)e^{-\frac{x^{2}}{2}}$. Indeed, it is not hard to verify it for $k=1$ and $k=2$. For $k>2$ we have via induction that \[\begin{array}{rl}\sigma^{(k+1)}(x)&=&(-1)^{k-1}\sqrt{(k-1)!}\left[h_{k-1}^{ \prime}(x)-xh_{k-1}(x)\right]e^{-\frac{x^{2}}{2}}\\ &\stackrel{{Eq.\,(\ref{eq:1})}}{{=}}&(-1)^{k}\sqrt{k!}\frac{1}{ \sqrt{k}}\left[xh_{k-1}(x)-\sqrt{k-1}h_{k-2}(x)\right]e^{-\frac{x^{2}}{2}}\\ &\stackrel{{Eq.\,(\ref{eq:2})}}{{=}}&(-1)^{k}\sqrt{k!}h_{k}(x)e^ {-\frac{x^{2}}{2}}\end{array}\] Hence, $\|\sigma^{(k)}(x)\|\leq\|\sqrt{(k-1)!}h_{k-1}(x)\|$, and now Eq. (7) implies that for any $k\leq n+1$ \[\epsilon_{\sigma}(n)\leq\frac{(n+1-k)!}{(n+1)!}(k-1)!=\frac{(n+1-k)!k!}{(n+1)! k}=\frac{1}{k\binom{n+1}{k}}\] Taking $k=\left\lceil\frac{n+1}{2}\right\rceil$ we conclude that $\epsilon_{\sigma}(n)\leq 2^{-n}$. ## 3 Conclusion and future work One of the prominent approaches for explaining the success of neural networks is trying to show that they are capable of learning complex and "deep" models. So far this approach has relatively limited success. Despite that significant progress has been made to show that neural networks can learn shallow models, so far, neural networks were shown to learn only "toy" deep models (e.g. [21, 2, 10, 31]). Not only that, but there are almost no known rich families of deep models that are efficiently learnable by _some_ algorithm (not necessarily gradient methods on neural networks). Our paper suggests that random neural networks might be candidate models. To take this approach further, a natural next step, and a central open question that arises from our work, is to show the existence of an algorithm that learns random networks in time that is polynomial both in $\frac{1}{\epsilon}$ and the network size. This question is already open for depth-two ReLU networks with two hidden neurons. We note that as implied by [31], such a result, even for a single neuron, will have to go beyond polynomial approximation of the network, and even more generally, beyond linear methods such as kernel methods. Additional open directions are (i) the analysis of random convolutional networks, (ii) achieving time and sample complexity of $(\bar{d})^{O(\epsilon^{-2})}$ for random networks of any constant depth (and not only for depth two), and (iii) finding a PTAS for random networks of depth $\omega(1)$. ### Acknowledgements The research described in this paper was funded by the European Research Council (ERC) under the European Union's Horizon 2022 research and innovation program (grant agreement No. 101041711), and the Israel Science Foundation (grant number 2258/19). This research was done as part of the NSF-Simons Sponsored Collaboration on the Theoretical Foundations of Deep Learning.
2301.01583	Why Capsule Neural Networks Do Not Scale: Challenging the Dynamic Parse-Tree Assumption	Capsule neural networks replace simple, scalar-valued neurons with vector-valued capsules. They are motivated by the pattern recognition system in the human brain, where complex objects are decomposed into a hierarchy of simpler object parts. Such a hierarchy is referred to as a parse-tree. Conceptually, capsule neural networks have been defined to realize such parse-trees. The capsule neural network (CapsNet), by Sabour, Frosst, and Hinton, is the first actual implementation of the conceptual idea of capsule neural networks. CapsNets achieved state-of-the-art performance on simple image recognition tasks with fewer parameters and greater robustness to affine transformations than comparable approaches. This sparked extensive follow-up research. However, despite major efforts, no work was able to scale the CapsNet architecture to more reasonable-sized datasets. Here, we provide a reason for this failure and argue that it is most likely not possible to scale CapsNets beyond toy examples. In particular, we show that the concept of a parse-tree, the main idea behind capsule neuronal networks, is not present in CapsNets. We also show theoretically and experimentally that CapsNets suffer from a vanishing gradient problem that results in the starvation of many capsules during training.	Matthias Mitterreiter, Marcel Koch, Joachim Giesen, Sören Laue	2023-01-04T12:59:51Z	http://arxiv.org/abs/2301.01583v1	# Why Capsule Neural Networks Do Not Scale: ###### Abstract Capsule neural networks replace simple, scalar-valued neurons with vector-valued capsules. They are motivated by the pattern recognition system in the human brain, where complex objects are decomposed into a hierarchy of simpler object parts. Such a hierarchy is referred to as a parse-tree. Conceptually, capsule neural networks have been defined to realize such parse-trees. The capsule neural network (CapsNet), by Sabour, Frosst, and Hinton, is the first actual implementation of the conceptual idea of capsule neural networks. CapsNets achieved state-of-the-art performance on simple image recognition tasks with fewer parameters and greater robustness to affine transformations than comparable approaches. This sparked extensive follow-up research. However, despite major efforts, no work was able to scale the CapsNet architecture to more reasonable-sized datasets. Here, we provide a reason for this failure and argue that it is most likely not possible to scale CapsNets beyond toy examples. In particular, we show that the concept of a parse-tree, the main idea behind capsule neuronal networks, is not present in CapsNets. We also show theoretically and experimentally that CapsNets suffer from a vanishing gradient problem that results in the starvation of many capsules during training. ## 1 Introduction The concept of capsules [12] describes a hypothetical system that parses a complex image scene into a hierarchy of visual entities that stand in part-whole relationship to each other [12]. A capsule is conceptually defined as a highly informative, compact representation of a visual entity or object within an image. The idea of capsules is motivated by the pattern recognition system in the visual cortex of the human brain [2]. There is some psychological evidence that the human object recognition system assigns hierarchical structural descriptions to complex objects by decomposing them into parts [12]. The theory of recognition by components [1] proposes that a relatively small set of simple 3D shapes, called geons, can be assembled in various arrangements to form virtually any complex object, which can then be recognized by decomposition into its respective parts [1]. A capsule may represent a visual entity by encapsulating its properties, also known as instantiation parameters, such as position, size, orientation, deformation, texture, or hue. A multi-level assembly of such capsules represents a complex image scene, where lower-level capsules model less abstract objects or object parts, and higher-level capsules model complex and composite objects. Lower-level capsules are connected to higher-level capsules if the corresponding entities are in a part-whole relationship. For a composite object, the hierarchy of capsules defines a syntactic structure like a parse-tree defines the syntactic structure of a sentence. Therefore, the hierarchy of capsules is also referred to as parse-tree. If an object or object part is present in an image, its respective capsule will be present within the parse-tree. Ideally, the parse-tree is invariant under affine transformations as well as changes of viewpoint. That is, a slightly modified viewpoint on a visual entity should not change a capsule's presence within the parse-tree. Such parse-trees would be highly efficient distributed representations of image scenes [2, 12, 13]. Also, explainable machine learning can profit from interpretable capsules that stand for dedicated visual entities, and the discrete nature of trees may connect deep learning with a symbolic approach to AI. Furthermore, capsules can be related to inverse graphics, and there is hope that they can lead to debuggable, parameter efficient, and interpretable models with a broad range of applications for all kinds of image-related tasks like image classification or segmentation. However, capsules are only conceptually defined, and the difficulty is finding an implementation with all the highly-desirable properties from above. The capsule neural network (CapsNet) by Sabour, Frosst, and Hinton (2017) aims at such an implementation of the conceptual capsule idea. It was specifically designed to surpass convolutional neural networks (ConvNets) [1] as the latter were found to suffer from several limitations, including a lack of robustness to affine transformations and change of viewpoint, the susceptibility to adversarial attacks, exponential inefficiencies, and a general lack of interpretability in the network's decision-making process. Considering these limi tations, the parse-tree sounds particularly appealing with all its advantages. Contributions.Here, our aim is a thorough investigation of the question, whether the CapsNets implementation as proposed by Sabour, Frosst, and Hinton (2017) realizes all the conceptual ideas that make capsule networks so appealing. We summarize this in two key assumptions. The first key assumption is that the CapsNet learns to associate a capsule with a dedicated visual entity within an input image (Sabour, Frosst, and Hinton 2017). The second key assumption is that the CapsNet's capsules can be organized hierarchically in a parse-tree that encodes part-whole relationships. We test both assumptions experimentally and have to reject them. We show that the CapsNet does not exhibit any sign of an emerging parse-tree. Thus, the CapsNet implementation cannot provide the theoretical benefits of capsule networks like invariance under affine transformations and change of viewpoint. Furthermore, we provide a theoretical analysis, exposing a vanishing gradient problem, that supports our experimental findings. ## 2 Related Work Early references to the hierarchy of parts appear already in (Hinton 1979). The idea of parsing images into parse-trees was proposed by Hinton, Ghahramani, and Teh (1999) and the concept of capsules was established in (Hinton, Krizhevsky, and Wang 2011). An important addition by the CapsNet (Sabour, Frosst, and Hinton 2017) was the routing-by-agreement (RBA) algorithm that creates capsule parse-trees from images. With its introduction, the CapsNet demonstrated state-of-the-art classification accuracy on MNIST (LeCun et al. 1998) with fewer parameters and stronger robustness to affine transformations than the ConvNet baseline, which sparked a flood of follow-up research. This includes different routing mechanisms, such as EM-Routing (Hinton, Sabour, and Frosst 2018), Self-Routing (Hahn, Pyeon, and Kim 2019), Variational Bayes Routing (Ribeiro, Leontidis, and Kollias 2020), Receptor Skeleton (Chen et al. 2021) and attention-based routing (Ahmed and Torresani 2019; Tsai et al. 2020; Mazzia, Salvetti, and Chiaberge 2021; Gu 2021). Wang and Liu (2018) reframed the routing algorithm in (Sabour, Frosst, and Hinton 2017) as an optimization problem, and Rawlinson, Ahmed, and Kowadlo (2018) introduced an unsupervised learning scheme for CapsNets. Other work replaced the capsule vector representations by matrices (Hinton, Sabour, and Frosst 2018) or tensors (Rajasegaran et al. 2019), or added classic ConvNet features to the general routing mechanisms, such as dropout (Xiang et al. 2018) or skip-connections (Rajasegaran et al. 2019). The GLOM architecture, which was proposed by (Hinton 2021), suggests a routing-free approach for creating parse-trees from images, but has not been implemented yet. Furthermore, other publications focus on learning better first layer capsules (PrimCaps), such as the Stacked Capsule Autoencoders (Kosiorek et al. 2019) and Flow Capsules (Sabour et al. 2021). However, after a while it turned out that the CapsNet falls short of the anticipated benefits and promises of the capsule idea. To this date, CapsNets do not scale beyond small-scale datasets. Works that empirically report scaling issues include (Xi, Bing, and Jin 2017; Paik, Kwak, and Kim 2019). Although the CapsNet was introduced in the realm of computer vision, the best performing capsule implementation (Ahmed and Torresani 2019) achieves only $60.07\%$ top-1 image classification accuracy on ImageNet (Deng et al. 2009), far behind state-of-the-art transformer-based approaches (Wortsman et al. 2022) and ConvNets (Pham et al. 2021) with $90.88\%$ and $90.02\%$ accuracy respectively. The original CapsNet itself has not been demonstrated to work on ImageNet. Further negative results regarding CapsNets emerged, questioning the promised benefits and technical progress altogether. Paik, Kwak, and Kim (2019) observed that increasing the depth of various CapsNet variants did not improve accuracy, and routing algorithms, the core components of capsule implementations, do not provide any benefit regarding accuracy in image classification. Michels et al. (2019), and Gu, Wu, and Tresp (2021) showed that CapsNets can be as easily fooled as ConvNets when it comes to adversarial attacks. Gu, Tresp, and Hu (2021) showed that the individual parts of the CapsNet have contradictory effects on the performance on different tasks and conclude that with the right baseline, CapsNets are not generally superior to ConvNets. Finally, Gu and Tresp (2020) showed that removing the dynamic routing component improves affine robustness, and Rawlinson, Ahmed, and Kowadlo (2018) show that capsules do not specialize without supervision. Here, we explain these shortcomings, which can be attributed to a lack of an emerging parse-tree. ## 3 The Capsule Neural Network The CapsNet implements capsules as parameter vectors. An illustration of the CapsNet architecture, which consists of a multi-layer hierarchy of capsules, is shown in Figure 1. In the following, we introduce basic notations and definitions, the generic CapsNet architecture, and a loss function for training CapsNets. Furthermore, we discuss how CapsNets implement the crucial concept of a parse-tree. ### Notation Capsules.The matrix $U^{l}\in\mathbb{R}^{n^{l}\times d^{l}}$ holds $n^{l}$ normalized capsules of dimension $d^{l}$ for layer $l\in\{1,2,\ldots,\ell\}$. The $i$-th capsule in $U^{l}$ is the vector $u^{l}_{(i.:)}\in\mathbb{R}^{d^{l}}$, and $u^{l}_{(i.j)}\in\mathbb{R}$ Figure 1: A generic CapsNet architecture. is the $j$-th entry of capsule $i$ on layer $l$. Transformation matrices. The tensor $W^{l}\in\mathbb{R}^{n^{l+1}\times n\times d^{l+1}\times d}$ holds transformation matrices $W^{l}_{(j,i,:,:)}\in\mathbb{R}^{d^{l+1}\times d^{l}}$. The transformation matrix $W^{l}_{(j,i,:,:)}$ maps the $i$-th capsule of layer $l$ to its unnormalized contribution to the $j$-the capsule of layer $l+1$. Coupling coefficients. The matrix $C^{l}\in\mathbb{R}^{n^{l}\times n^{l+1}}$ holds coupling coefficients for the connections of capsules from layer $l$ to layer $l+1$. The entry $c^{l}_{(i,j)}\in[0,1]$ specifies the coupling strength between capsule $i$ on layer $l$ and capsule $j$ on layer $l+1$. The coupling coefficients satisfy $\sum_{j=1}^{n^{l+1}}c^{l}_{(i,j)}=1$ for all $i\in\{1,2,\ldots,n^{l}\}$. Squashing function. The squashing function normalizes the length of a capsule vector $u\in\mathbb{R}^{d}$ into the range $[0,1)$. Here, we use a slightly modified squashing function (Mazzia, Salvetti, and Chiaberge, 2021), \[g(u)=\left(1-\frac{1}{\exp(\\|u\\|_{2})}\right)\frac{u}{\\|u\\|_{2}} \tag{1}\] that behaves similarly to the original squashing function proposed by Sabour, Frosst, and Hinton (2017), but is more sensitive to small changes near zero (Xi, Bing, and Jin, 2017), which is required to stack multiple layers of capsules. ### Architecture First, the backbone network extracts features from an input image into a feature matrix in $\mathbb{R}^{n^{1}\times d^{1}}$. The feature matrix is then normalized by applying the squashing function to each row, which constitutes the first capsule layer in $U^{1}\in\mathbb{R}^{n^{1}\times d^{1}}$. The capsules in $U^{1}$ are also called PrimeCaps. Starting from the PrimeCaps, consecutive layers of capsules are computed as follows: First, the linear contribution of capsule $i$ on layer $l$ to capsule $j$ on layer $l+1$ is computed as \[\hat{u}^{l+1}_{(i,j,:)}=W^{l}_{(j,i,:,:)}u^{l}_{(i,:)}, \tag{2}\] where the entries in the matrix $\tilde{U}^{l+1}_{(i)}$, which holds the vectors $\hat{u}^{l+1}_{(i,j,:)}$, are called votes from the $i$-th capsule on layer $l$. An upper layer capsule $u^{l+1}_{(j,:)}$ is the squashed, weighted sum over all votes from lower layer capsules $u^{l}_{(i,:)}$, that is, \[u^{l+1}_{(j,:)}=g\left(\sum_{i=1}^{n^{l}}c^{l}_{(i,j)}\cdot\hat{u}^{l+1}_{(i,j,:)}\right),\] where the the coupling coefficients $c^{l}_{(i,j)}$ are dynamically computed, that is, individually for every input image, by the Routing-by-agreement Algorithm (RBA), see Algorithm 1. ``` 0: votes $\hat{u}$, number of iterations $r$, routing priors $b$ 0: coupling coefficients $c$ 1:for$r$ iterations do 2:$c_{(i,j)}\leftarrow\frac{\exp(b_{(i,j)})}{\sum_{k}\exp(b_{(i,k)})}$ 3:$v_{(j,:)}\gets g\left(\sum_{i}c_{(i,j)}\hat{u}_{(i,j,:)}\right)$ 4:$b_{(i,j)}\gets b_{(i,j)}+\langle\hat{u}_{(i,j,:)},v_{(j,:)}\rangle$ 5:endfor ``` Algorithm 1Routing-by-agreement (RBA) The number of output capsules on the last layer $\ell$ is set to match the number of classes in the respective dataset. Finally, the fully-connected decoder network reconstructs the input image from the capsules on layer $\ell$. ### Training The parameters in the backbone network, in the reconstruction network, as well as the transformation tensors $W^{l}$ and the RBA routing priors $b^{l}_{(i,j)}$ are all learned by minimizing a weighted sum of a supervised classification loss $L_{m}$ and an unsupervised reconstruction loss $L_{r}$, that is, $L=L_{m}+\alpha\cdot L_{r}$, with $\alpha>0$. The classification loss function \[L_{m}=\sum_{j=1}^{n^{\ell}}\!t_{j}\cdot\max\{0,m^{+}-\\|u^{\ell}_{ (j,:)}\\|_{2}\}^{2}\\ +\lambda\cdot(1-t_{j})\cdot\max\{0,\\|u^{\ell}_{(j,:)}\\|_{2}-m^{-} \}^{2} \tag{3}\] is only applied to the output capsules. Here $m^{+},m^{-}>0$ and $\lambda>0$ are regularization parameters, and $t_{j}$ is $1$ if an object of the $j$-th class is present in the input image, and $0$ otherwise. Output capsules that correspond to classes not present in the input image are masked by zeros. The reconstruction loss function is applied to the output of the reconstruction network and sums the distances between the reconstruction and the pixel intensities in the input image. ### Parse-Trees The parse-tree is the most important concept that allows us to understand and interpret capsules and their connections. The CapsNet defines a parse-tree, where capsules represent nodes, and coupling coefficients represent fuzzy edges. Figure 2: Fuzzy parse-trees for images from the AffNIST dataset for a model with five capsule layers, 16 capsules on each intermediate layer, and ten on the last layer. The figure shows the coupling coefficients as connections between capsules (left), the capsule norms/activations (middle), and the input image (right). The blue tone of the edges is darker for greater coupling coefficients. The capsules magnitude, that is, the norm of the parameter vector, which is always in $[0,1)$ after applying the squashing function, represents the probability that the corresponding entity is present in an input image. The capsules direction represents instantiation parameters of the entity like its position, size, or orientation. Changing the viewpoint on an entity does not affect its presence, but only its instantiation parameters. Therefore, the respective capsule's magnitude should be unaffected, whereas its direction can change. In the CapsNet, the dynamic part-whole relationships are implemented by coupling coefficients between capsules on consecutive layers. The coupling coefficients in the routing layers are computed dynamically by the RBA Algorithm 1. Taking the row-wise _softmax_ ensures that the coupling coefficients in $c^{l}_{(i,:)}$ are positive and sum up to one. Therefore, we can view the coupling coefficients as fuzzy edges that connect capsules $u^{l}_{(i,:)}$ and $u^{l+1}_{(j,:)}$ with probability $c^{l}_{(i,j)}$. The multi-layer hierarchy of capsule nodes, connected by fuzzy edges, defines the parse-tree analogously to a probabilistic context-free grammar. Examples are shown in Figure 2. ## 4 Challenging the Parse-Tree Assumption As mentioned in the introduction, there are two key assumptions regarding the parse-tree: (1) The nodes of the parse-tree, the activated capsules, are viewpoint-invariant representations of visual entities present in the input image. (2) Lower-level capsules represent object parts, higher-level capsules represent composite objects, and part-whole relationships are represented by the edges of the parse tree, that is, by the coupling coefficients. In the following, we are going to challenge both assumptions. If Assumption (2) holds, then the parse-tree computed by a CapsNet is a part-whole representation of the image scene, and the routing dynamics defined by the coupling coefficients is specific to the input image. We conduct experiments challenging this assumption in Section 4.1. If Assumption (1) holds, then affine transformations of an image only change the direction of a parameter vector of a capsule, but not its magnitude. Hence, we take an image, transform it affinely, and analyze the resulting capsule activations. These experiments can be found in Section 4.2. Furthermore, we examine the capsule activation in general in Section 4.3. Experimental SetupWe use the AffNIST benchmark Sabour, Frosst, and Hinton (2017) to assess a model's robustness to affine transformations, and we use the CIFAR10 dataset Krizhevsky (2009) to test a model's performance on complex image scenes. We conduct extensive experiments using a total of 121 different model architectures of various scales, featuring different numbers of routing layers and different numbers and capsule dimensions. Shallow models resemble the original CapsNet implementation while deeper models allow for a more semantically expressive parse-tree. We refer to the appendix for detailed architecture and dataset descriptions, training procedures, and full results for all models used in our experiments. ### Routing Dynamics We measure the diversity of parse-trees, that is, the routing dynamics, by assessing the diversity of routing targets for a single capsule $u$. For $k$ input images, let $C\in\mathbb{R}^{k\times n}$ hold all the coupling coefficients that connect $u$ to capsules on the next layer, which contains $n$ target capsules. We use the standard deviation $\operatorname{std}(c_{(.,i)})$ with respect to all input images as a measure for the routing diversity of $u$ to the $i$-th capsule on the next layer. A routing $\operatorname{pr}_{n}$ is called perfect if it always routes to exactly one capsule on the next layer and routes to all $n$ capsules on the next layer equally likely. The standard deviation of a perfect routing computes to $\operatorname{std}(pr_{n})=\sqrt{\left(1-\frac{1}{n}\right)\frac{1}{n}}$. We use a perfect routing to define the dynamic routing coefficient (dyr) for capsule $u$ as \[\operatorname{dyr}(u)=\frac{1}{n}\sum_{i=1}^{n}\frac{\operatorname{std}(c_{(.,i )})}{\operatorname{std}(\operatorname{pr}_{n})}\] The expected number of target capsules (dys) for $u$ is $\operatorname{dys}(u)=n\cdot\operatorname{dyr}(u)$. For a whole layer, we define the coefficients dyr and dys as the mean over all capsules of this layer. Figure 3: Parse-tree statistics for the complete AffNIST validation dataset for a five-layer CapsNet model with 16/10 capsules. The mean (a) and the standard deviation (b) of the coupling coefficient matrices for each layer are visualized as connections between capsules. Higher coupling coefficients have a darker blue tone. The capsule norms’ mean (c) and standard deviation (d) are visualized by bars. Dead capsules (e) are highlighted with a red bar. ResultsIn the following, we report the routing statistics for a CapsNet architecture with four routing layers, 16 capsules per layer in the first four layers, and ten capsules in the last layer. We set the capsule dimension to eight and train multiple models on the AffNIST dataset until a target accuracy of $99.2\%$ is reached. The routing statistics for the models are summarized in Table 1 and the corresponding coupling coefficients of a single model are visualized in Figure 3. The $\mathrm{dys}$ values below two for Layers 2 and 3 indicate low routing dynamics. A route is mostly predetermined once a capsule is activated; hence, the routing is almost static. Only the last layer exhibits higher routing dynamics, which can be attributed to the supervisory effect of the classification loss $L_{m}$. Since the routing is almost static, we conclude that the parse-trees do not encode the information necessary for a distributed representation of diverse image scenes, violating Assumption (2). As can be seen from Tables 7 to 12 in the supplement, the results look similar for all models trained on AffNIST. The more complex data set CIFAR10 exhibits even worse routing dynamics; see Table 13 and Figure 16 in the appendix. ### Viewpoint Invariance We investigate to which degree capsule activations are invariant to affine transformations of the input images. Let $x^{(1)}$ and $x^{(2)}$ show the same visual entity though differently instantiated. For one specific capsule, let $u^{(1)},u^{(2)}\in\mathbb{R}^{d}$ be the corresponding capsule responses for the two images. For a viewpoint invariant parse-tree, it holds that $\left\\|u^{(1)}\right\\|_{2}=\left\\|u^{(2)}\right\\|_{2}$, since $\left\\|u\right\\|_{2}$ measures the probability that the visual entity is present in the image. We repeat this process for a set of $k$ input images and collect the corresponding capsule activations in the two vectors $v^{(1)}$ and $v^{(2)}\in\mathbb{R}^{k}$. We compute the empirical cross-correlation between these two vectors as $\frac{(v^{(1)})^{\top}v^{(2)}}{\left\\|v^{(1)}\right\\|_{2}\cdot\left\\|v^{(2)} \right\\|_{2}}$. For one layer, we compute the average of this value over all capsules of this layer. For a viewpoint-invariant parse-tree, this value should be one. ResultsWe observe that the capsule activation correlation decreases for increasingly stronger affine transformations, see Figure 4. This observation holds for all intermediate capsule layers, and all tested affine transformations. See also Figure 5 for a qualitative example showing two different parse-trees that are expected to be identical. We conclude that the parse-tree is not invariant under affine transformations of the input image, violating Assumption (1). Furthermore, since already the activations of the PrimeCaps do not exhibit viewpoint-invariance, we believe that capsules need a different backbone that gives rise to better PrimeCaps. ### Capsule Activation In this section, we analyze the capsule activation and thus the parse-tree nodes. For a layer with $n$ capsules of dimension $d$, let $U\in\mathbb{R}^{k\times n\times d}$ hold the $n$ capsule responses of dimension $d$ for $k$ input images. We denote the entries of $U$ by the lower letter $u$ with corresponding lower indices. We define \begin{table} \begin{tabular}{c c c\|c} \hline \multirow{2}{}{Layer} & \multirow{2}{}{Capsules Alive} & \multicolumn{2}{c}{Routing Dynamics} \\ & & Rate ($\mathrm{dyr}$) & Mean ($\mathrm{dys}$) \\ \hline 1 & 16.00 $\pm$ 0.00 & 0.30 $\pm$ 0.00 & 4.50 $\pm$ 0.17 \\ 2 & 14.90 $\pm$ 0.70 & 0.25 $\pm$ 0.01 & 1.72 $\pm$ 0.11 \\ 3 & 7.00 $\pm$ 0.63 & 0.30 $\pm$ 0.02 & 1.79 $\pm$ 0.16 \\ 4 & 5.90 $\pm$ 0.70 & 0.38 $\pm$ 0.04 & 3.78 $\pm$ 0.38 \\ output & 10.00 $\pm$ 0.00 & & \\ \hline \end{tabular} \end{table} Table 1: The routing statistics for a CapsNet with four routing layers, 16/10 capsules per layer, and a capsule dimension of eight. We separately train and evaluate ten models on AffNIST the same way and report the mean and standard deviation. Figure 4: The capsule activation correlations for each layer with respect to increasing degree of affine transformations. Figure 5: Similar input image, different parse tree. the capsule norm sum (cns) as the sum of capsule norms in the respective layer averaged over all input images. For comparing layers with different numbers of capsules, we define the capsule norm mean (cnm), which is the $\mathrm{cns}$ adjusted for the number of capsules that are present in the layer: \[\mathrm{cns}(U)=\frac{1}{k}\sum_{i=1}^{k}\sum_{j=1}^{n}\left\\|u_{(i,j,:)}\right\\| _{2},\qquad\mathrm{cnm}(U)=\frac{\mathrm{cns}(U)}{n}\] We say a capsule $j$ is active for input image $i$, if its norm exceeds a certain threshold $\varepsilon$, that is, \[\mathds{1}_{\mathrm{active}}(u_{(i,j,:)})=\begin{cases}1&\left\\|u_{(i,j,:)} \right\\|_{2}\geq\varepsilon\\ 0&\text{otherwise.}\end{cases}\] Furthermore, we define the sum of active capsules (cas) as the mean sum of active capsules per layer and the rate of activate capsules as the $\mathrm{cas}$ adjusted for the number of capsules: \[\mathrm{cas}(U)=\frac{1}{k}\sum_{i=1}^{k}\sum_{j=1}^{n}\mathds{1}_{\mathrm{ active}}(u_{(i,j,:)}),\quad\mathrm{car}(U)=\frac{\mathrm{cas}(U)}{n}\] We say that a capsule $j$ is dead if the mean $\mu$ and the standard deviation $\sigma$ of its norm over the $k$ input images are below certain thresholds, that is, \[\mathds{1}_{\mathrm{dead}}(u_{(:,j,:)})=\begin{cases}1&\mu(u_{(:,j,:)})\leq \varepsilon_{1}\text{ and }\sigma(u_{(:,j,:)})\leq\varepsilon_{2}\\ 0&\text{otherwise.}\end{cases}\] We compute the sum of dead capsules (cds) and the rate of dead capsules (cdr) as \[\mathrm{cds}(U)=\sum_{j=1}^{n}\mathds{1}_{\mathrm{dead}}\left(u_{(:,j,:)} \right),\quad\mathrm{cdr}(U)=\frac{\mathrm{cds}(U)}{n}\] ResultsTable 2 summarizes the capsule activation statistics for the AffNIST experiment. As expected, the agreement of lower layer capsules, enforced by the RBA algorithm, results in a declining number of active capsules in the upper layers, as is witnessed by decreasing cas values. As a result, the overall sum of norms per layer drops, as can be seen in the $\mathrm{cns}$ values. Surprisingly, there is no sign of sparse activation within the PrimeCaps. All PrimeCaps are consistently active, as seen from the car values. This implies that PrimeCaps do not represent parts that are present in one image and not present in another, questioning the underlying assumption of distributed representation learning. It is another indication that the backbone does not deliver the representations required for PrimeCaps. Furthermore, we observe that the number of dead capsules cdr increases with the depth of the model. For instance, $63\%$ of the capsules on Layer 4 in the AffNIST experiment are dead. Figure 2(a) highlights the dead capsules. In other experiments, this value increases up to $84\%$, see Table 20 in the appendix. This has the following implications: First, the depth of a CapsNet is limited as the number of dead capsules rises with the number of layers. Second, the parse-tree cannot carry separate semantic information for each class if the number of active capsules is less than the number of classes. Third, dead capsules limit the capacity of a CapsNet as their respective parameters are not in use. This explains why baseline models trained with uniform routing perform better than models trained with RBA. Uniform routing, which uses all parameters, achieves better classification accuracies; see Tables 5 and 14 in the appendix. In uniform routing, all entries in the coupling coefficient matrix are set to the same fixed value. Our results stand in line with prior work [14, 15, 16, 17] that also observed a negative impact of routing on model performance. capsules on the upper-most layer will be active for the corresponding data points. This is in stark contrast to capsules that are not on the upper-most layer. Here, it can happen that a capsule becomes dead during training. We observe, that once a capsule is dead, it never becomes active again, resulting in a starvation of capsules. The following theorem asserts this behavior. Theorem 2: _Let $L_{m}$ be the margin loss function. The gradient of a single capsule $u^{l}_{(i,:)}$, that does not belong to the upper-most layer evaluates to:_ \[\frac{\partial L_{m}}{\partial u^{l}_{(i,:)}}=\sum_{j}\frac{\partial L_{m}}{ \hat{u}^{l+1}_{(i,j,:)}}\cdot W^{l}_{(j,i,:,:)}\] _The gradients of the corresponding weight matrices $W^{l}_{(j,i,:,:)}$ evaluate to:_ \[\frac{\partial L_{m}}{\partial W^{l}_{(j,i,:,:)}}=\frac{\partial L_{m}}{ \partial\hat{u}^{l+1}_{(i,j,:)}}\cdot u^{l}_{(i,:)}\] The theorem follows from Equation (2). It states that the gradient of the weight matrix scales with the activation of the corresponding capsule, and the gradient of the capsule scales with the magnitude of the weight matrix. Hence, once both are small, they will not change sufficiently. In the limit, i.e., of magnitude zero, they will never change. In other terms, once a capsule becomes dead, it never becomes active again. Figure 10 in the appendix clearly show this behavior for the gradient of the capsule activation and Figures 11-14 in the appendix show this behavior for the gradient of the weight matrices. Dead capsules do not participate in the routing and are not part of any parse-tree. Also, note that the supervised loss forces the upper-most layers to be active. However, capsules can become dead on the intermediate layers where no supervised loss is directly present. ## 5 Comparing RBA with Self-Attention We compare routing-by-agreement with the multi-head self-attention (MHSA) mechanism used in transformers [20] and more task-related vision transformers [15]. Like RBA, the MHSA mechanism operates on vectors and uses the softmax function to compute the normalized attention matrix, similar to the coupling coefficient matrix in RBA. However, contrary to RBA, which computes the softmax row-wise, MHSA enforces the softmax column-wise and, as a result, does not suffer from the previously discussed vanishing gradient problem. However, MHSA is not considered routing and does not intend to implement a parse-tree. Considering space and time complexity, we observe that RBA is extremely expensive compared to MHSA; see Table 3. This fact may contribute to the substantial interest in models relying on MHSA rather than RBA. ## 6 Broader Impact and Limitations In this work, we focus on the original CapsNet with RBA routing since it is the predominant implementation of the capsule idea. An exhaustive investigation, including all capsule variants and follow-up models, is difficult because the absence of a formal definition of capsules makes the topic hard to cover. Different approaches from the vast literature are technically diverse. As a result, whether a follow-up work implements the concept of capsules is not easy to judge. However, our claims are general enough to cover many implementations. The softmax-based routing approach is part of many capsule implementations, see for instance [21, 22, 23], and we expect that they face similar issues. ## 7 Conclusion The core concept of capsules is the part-whole hierarchy of an image represented by a parse-tree. While this concept has appealing properties like robustness under affine transformations, interpretability, and parameter efficiency, so far, implementations of the capsule concept have not taken over yet. Instead, some of their properties were questioned in recent work. Here, we have shown that the core idea of a parse-tree does not emerge in the CapsNet implementation. Furthermore, starvation of capsules caused by a vanishing gradient limits their capacity and depth. Our observations explain recently reported shortcomings, especially that CapsNets do not scale beyond small datasets. Hence, the CapsNet is not a sufficient implementation of the capsule idea. ## 8 Acknowledgments This work was supported by the German Science Foundation (DFG) grant (GI-711/5-1) within the priority program (SPP 1736) Algorithms for Big Data and by the Carl Zeiss Foundation within the project "Interactive Inference".
2304.02654	Adopting Two Supervisors for Efficient Use of Large-Scale Remote Deep Neural Networks	Recent decades have seen the rise of large-scale Deep Neural Networks (DNNs) to achieve human-competitive performance in a variety of artificial intelligence tasks. Often consisting of hundreds of millions, if not hundreds of billion parameters, these DNNs are too large to be deployed to, or efficiently run on resource-constrained devices such as mobile phones or IoT microcontrollers. Systems relying on large-scale DNNs thus have to call the corresponding model over the network, leading to substantial costs for hosting and running the large-scale remote model, costs which are often charged on a per-use basis. In this paper, we propose BiSupervised, a novel architecture, where, before relying on a large remote DNN, a system attempts to make a prediction on a small-scale local model. A DNN supervisor monitors said prediction process and identifies easy inputs for which the local prediction can be trusted. For these inputs, the remote model does not have to be invoked, thus saving costs, while only marginally impacting the overall system accuracy. Our architecture furthermore foresees a second supervisor to monitor the remote predictions and identify inputs for which not even these can be trusted, allowing to raise an exception or run a fallback strategy instead. We evaluate the cost savings, and the ability to detect incorrectly predicted inputs on four diverse case studies: IMDB movie review sentiment classification, Github issue triaging, Imagenet image classification, and SQuADv2 free-text question answering	Michael Weiss, Paolo Tonella	2023-04-05T04:35:23Z	http://arxiv.org/abs/2304.02654v1	# Adopting Two Supervisors for Efficient Use of Large-Scale Remote Deep Neural Networks ###### Abstract. Recent decades have seen the rise of large-scale Deep Neural Networks (DNNs) to achieve human-competitive performance in a variety of artificial intelligence tasks. Often consisting of hundreds of millions, if not hundreds of billion parameters, these DNNs are too large to be deployed to, or efficiently run on resource-constrained devices such as mobile phones or IoT microcontrollers. Systems relying on large-scale DNNs thus have to call the corresponding model over the network, leading to substantial costs for hosting and running the large-scale remote model, costs which are often charged on a per-use basis. In this paper, we propose BiSupervised, a novel architecture, where, before relying on a large remote DNN, a system attempts to make a prediction on a small-scale local model. A DNN supervisor monitors said prediction process and identifies easy inputs for which the local prediction can be trusted. For these inputs, the remote model does not have to be invoked, thus saving costs, while only marginally impacting the overall system accuracy. Our architecture furthermore foresees a second supervisor to monitor the remote predictions and identify inputs for which not even these can be trusted, allowing to raise an exception or run a fallback strategy instead. We evaluate the cost savings, and the ability to detect incorrectly predicted inputs on four diverse case studies: IMDB movie review sentiment classification, Github issue triaging, Imagenet image classification, and SQuADv2 free-text question answering. Keywords:C Computer systems organization Embedded systems; _Redundancy_; Robotics; Networks Network reliability. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. + Footnote †: 2: 4000 tokens on a fine-tuned davinci-model (0.12$ per 1000 tokens). See beta.openai.com. ## 1. Introduction In the last decades, we have seen a steady rise of _Deep Neural Network_ (DNN) components in modern software systems, so-called _Deep Learning based Systems_ (DLS): For example, buyers are intelligently assisted when buying products online1, customer service is performed and analyzed using bots (Botton et al., 2016), Github issues can be automatically triaged (Srivastava et al., 2015) and medical diagnosis is made or supported by automated processes (Krizhevsky et al., 2014) (sometimes runnable even on a smartphone (Srivastava et al., 2014)). Footnote 1: E.g. [https://hellorep.ai/](https://hellorep.ai/) To achieve ever better predictive accuracy, the size of the best-performing DNN models has increased enormously for many tasks, with some models now consisting of trillions of parameters. Large models have to be hosted in server centers, amongst others due to their challenging hardware requirements. Client facing-applications thus have to make a choice between small, low-accuracy models running locally on the client's device (e.g. a browser or a mobile phone) or to offload their prediction tasks to a large model in the cloud, which amongst other disadvantages introduces latency and imposes a substantial financial cost to the software provider for hosting their model, or for using a third-party model. For example, a single request to the popular GPT-3 model (Botton et al., 2016), hosted by the Non-Profit organization OpenAI for anyone to use, is billed up to $0.48${}^{2}$. Even state-of-the-art models, given their typically intractably large input space and stochastic nature, will occasionally make wrong predictions. Here, typical machine learning evaluation metrics, such as high accuracy, are not sufficient to justify the inclusion of the model in many risk-averse settings. Instead, software architects and software engineers have to defend against the imperfections of state-of-the-art DNN models and prevent system failure even when faced with rare, mispredicted inputs. This led to the proposition and evaluation of various _DNN supervisors_ in the recent software engineering literature (e.g. 4; 40; 54; 55; 59; 62; 64). A DNN supervisor is a software component implemented to monitor a DNN at runtime and to detect inputs for which the models' prediction cannot be trusted. For such inputs, the system would default to a predefined domain-specific backup strategy (e.g. forwarding the request to a human operator) or to refuse to produce any output. Our work suggests using such supervisors not only to prevent system failure but also to reduce cost and latency in applications running on resource constraint devices. Specifically, we propose BiSupervised, a novel software architecture that combines the advantages of cheap and fast local DNN with the advantages of using a highly accurate but costly remote DNN. Specifically, BiSupervised leverages a first supervisor to discriminate between inputs that can reliably be predicted locally on a small model, and only leverage the large remote model when the input is too challenging for the local model. A second supervisor is then used to filter out highly uncertain remote predictions. Specifically, our contributions are the following: Architecture: We propose a novel architecture that combines small, local models with large remote ones, allowing us to fine-tune an optimal cost-reliability trade-off. Evaluation of Cost Savings: We empirically evaluate the trade-off between cost saving (in terms of the number of requests to the remote model) and system accuracy enabled by our architecture. Specifically, we evaluate if our architecture allows significant cost savings while maintaining a system-level accuracy at most marginally lower than the one of the remote model. We found that for all case studies, when reducing costs by 50%, BiSupervised achieved a system-level accuracy which is barely lower than the one of the large-scale remote model. Moreover, in two case studies, BiSupervised even achieved an accuracy exceeding the one of the large-scale remote models, due to complementarities in the local and remote models capabilities. Evaluation of Supervised Performance: We assess BiSupervised in a supervised setting, i.e., including the detection of inputs for which even the remote DNN prediction is not reliable enough to be trusted and for which thus a system-specific healing strategy should be triggered. We compare the supervised performance of our system against that of a locally supervised model. We found that in the vast majority of the tested configurations, the supervised performance of BiSupervised is similar or exceeds the one of the locally supervised model. ## 2. Background ### The trend towards huge DNN _From 1 thousand to 1.6 trillion parameters in 33 years_. In 1989, LeCun et al. (1989) proposed a DNN achieving state-of-the-art classification performance on handwritten digits, marking one of the earliest real-world applications of a neural net trained end-to-end with backpropagation (LeCun et al., 2010). Consisting of only 1000 parameters, it is orders of magnitude smaller than what we use nowadays to achieve state-of-the-art (SOTA) performance on challenging tasks. After 1989, we were still a long way from widespread adoption of DNNs for practical tasks: Training DNNs is a highly computation-intensive task and thus for a long time, faster classical methods of AI, such as Support Vector Machines continued to achieve better results on nontrivial tasks, still winning, for example, the image classification benchmark (ILSVRC), in 2010 and 2011 (Zhou et al., 2011). This changed with an innovation that would make the training of DNNs much faster, thus allowing for much larger DNNs: Starting in 2010, the first successful application of plain backpropagation on a GPU achieved a speedup of 50 over CPUs (Beng et al., 2011), training networks of up to 12 million parameters. Shortly thereafter, AlexNet, a 60 million parameter DNN (Krizhevsky et al., 2012) won ILSVRC2012, marking a "turning point for large-scale object recognition" [47]. As of this writing, CoCa, the SOTA model for the same task consists of 2.1 billion parameters3[70]. Footnote 3: According to [https://paperswithcode.com/sota/image-classification-on-imagenet](https://paperswithcode.com/sota/image-classification-on-imagenet), accessed on July 21, 2022 In other domains, such as Natural Language Processing (NLP), we observe even larger dimensions: GPT-2 (1.5B parameters) in 2019 [43] and GPT-3 (175B parameters) in 2020 [2] led to global media coverage, due to their good performance in generating naturally-appearing text. Within the last two years, they have again been overtaken by even larger models, for example by Megatron (530B parameters, 2022) [50]. To the best of our knowledge, the currently largest well-known DNN is SwitchTransformer with a parameter count of 1.6T [12]. Cost of large modelsThe large computational requirements of such large models when inferencing (i.e., predicting) present a major challenge to their adoption [39], as they require a large amount of memory, energy, and powerful CPUs or GPUs to make predictions. This can make their deployment impractical if not impossible, especially regarding customer-facing, resource-constrained devices, and platforms such as web browsers, internet of things (IOT) microcontrollers, or mobile phones. Large-scale, highly accurate DNNs are thus typically deployed in server centers, providing a remote inference service. There is a range of services facilitating such predictions, such as the HuggingFace Inference API4, OpenAIs APIs to GPT-35 or Google's Vertex AI6. Not only can already a single inference to a large model on such a service lead to high cost, the cost of such remote inferences also scales badly, as it increases quasi proportionally with the number of inference requests - providing essentially no economies of scale. Footnote 4: [https://huggingface.co/inference-api](https://huggingface.co/inference-api) Footnote 5: [https://openai.com/api/](https://openai.com/api/) Footnote 6: [https://cloud_google.com/vertex-ai/docs/start/introduction-unified-platform](https://cloud_google.com/vertex-ai/docs/start/introduction-unified-platform) ### Running DNNs in Resource-Constrained Environments Advantages of smaller but local modelsA substantially different approach is hence to run DNNs locally on the client devices, which constrains the size and complexity of the models, eventually leading to an inferior level of accuracy.7 Such locally deployed models have a range of advantages, among which a much lower energy consumption, faster inference (due to the smaller model, but also as there is no need to do a network roundtrip), no dependency on a stable internet connection, and, as explained above, no per-inference cost to the software provider. Means towards efficient modelsThere is thus much interest in making local models as efficient as possible, aiming to fit more parameters into the capacities of a resource-constrained system, or to optimize the performance of a model while keeping its number of parameters as low as possible. The following provides a non-comprehensive selection of some recent work and frameworks aiming to facilitate the use of DNN models in resource-constrained environments. * _Small Model Architectures:_ A range of works proposes specific DNN architectures or training procedures aiming to specifically facilitate deployment to resource-constrained devices. Examples are TinyBERT [22] and MobileNet[18; 19; 49]. These models, which are comparably small for their tasks (NLP and Imagenet classification, respectively) are intended to be used as starting point to fine-tune a model to other specific use-cases, thus making them widely useful. * _Smaller Sibling Models:_ Various recent releases of new, large-scale SOTA models were accompanied by a set of similar, but smaller models. The architectural choices and innovations proposed to create the flagship large-scale SOTA models are applied also to a set of smaller models. This allows users to make an accuracy vs model-size trade-off. Examples include efficientnet(Wang et al., 2018), OPT (Wang et al., 2018) and GPT-38(Beng et al., 2019). Footnote 8: Models are not publicly released, but access is allowed through an API. * _Model Tuning and Postprocessing_: Literature provides a range of techniques to make a given large DNN architecture fit for lower system requirements, allowing to deploy it on resource-constrained devices. The most popular such approach is _Quantization_, where a full-precision DNN is converted into a low-bitwidth integer version, thus reducing computational and memory requirements (Wang et al., 2018). Another technique is DNN pruning where the number of weights is reduced in a guided way, aiming to only minimally reduce the model accuracy (Wang et al., 2018). * _Custom Frameworks_: A typical software stack used to build and train a DNN cannot easily be applied when deploying the DNN to an environment such as a mobile phone or web browser which, besides hardware constraints, also imposes constraints on the software stack. To this end, dedicated frameworks have been proposed, among which the most popular is tf-lite9, allowing to convert a standard tensorflow model into one that can be deployed on specific platforms, such as Android, iOS and Linux edge devices. A dedicated version, tf-lite micro targets even particularly small embedded devices (Beng et al., 2019). Footnote 9: [https://www.tensorflow.org/lite](https://www.tensorflow.org/lite) Naturally, none of these techniques allows us to deploy huge SOTA models in resource-constrained environments, thus not replacing the need for remote predictions when aiming for optimal prediction accuracy. They do however support and extend the possibility to deploy smaller models on local or edge devices in cases where suboptimal accuracy is acceptable. ## 3. Related Work ### Smart Offloading in Edge Computing The idea to delegate computationally intensive tasks from resource-constrained devices to the edge or to the cloud, denoted as _offloading_, is not limited to DNNs. Recent work - which is not specific to DNNs - discusses the idea of intelligently deciding on the local device which tasks can be computed locally, and which inputs have to be forwarded to the remote computation API. Amongst other approaches, some work proposes to use a local DNN to categorize inputs to make said decision (Surveys, 2017; Wang et al., 2018). These architectures thus naturally show some high-level similarity with BiSupervised, specifically with the 1st level supervisor. However, our architecture, which is specific to DNN inference tasks, includes a 1st level supervisor that is tightly integrated with the local model, allowing us to efficiently compute both a local prediction and the decision on whether a remote invocation is needed. Specifically, we leverage the possibility of quantifying the uncertainty of the DNN at inference time - a possibility that does not generalize to arbitrary computations in local devices. To the best of our knowledge, ours is the first approach that supports smart offloading for DNNs running in a local, resource-constrained device. ### Network Supervision To account for the practically unavoidable imperfections in DNN predictions, the detection of inputs for which the corresponding prediction is likely to be wrong is a crucial component of Machine Learning safety (Han et al., 2017). This task, which we denote as _supervision_ (sometimes also refered to as _monitoring_(Han et al., 2017), _prediction validation_(Beng et al., 2019) or _misbehavior prediction_(Wang et al., 2018; Wang et al., 2018)) allows to ignore the DNNs prediction and instead run a fallback process (sometimes denoted healing process (Wang et al., 2018)). #### 3.2.1. Supervisors Used in Related Literature The literature offers a range of techniques that can be used for supervision. In the following, we provide a brief overview. While we make some recommendations in subsection 4.2, this overview aims to help the reader make an informed selection of supervisors to use in BiSupervised. Softmax-Likelihood Based: Most DNN classifiers use a _Softmax_-normalized output layer: Let $sm$ denote the activations of such a softmax output layers, and let $C$ denote the set of classes. We have that $length(sm)=\|C\|$ and $\sum_{c\in C}sm_{c}=1$. The models prediction is $argmax_{c\in C}sm_{c}$, with the assigned likelihood $max_{c\in C}sm_{c}$.10 Given a softmax output array, the simplest softmax-based supervisor, _Vanilla Softmax_ or _MaxSoftmax_[16], simply trusts a prediction if its softmax value is above some threshold. To that extent, other supervisors use functions of the entire softmax array, such as the prediction-confidence-score (the difference between the two highest likelihoods) [73], the entropy in the predicted likelihoods [65] or the Gini impurity in the predicted likelihoods [13]. Outlier Exposure [17] is a technique to further strengthen the capabilities of Softmax-Likelihood-based supervisors during model training. Footnote 10: It is worth noting that $sm_{c}$ is not an appropriate approximation of $c$ actually being correct; it is well known that - if not deliberately calibrated - DNNs overestimate such likelihoods [16]. Monte-Carlo Dropout (MC-Dropout): MC-Dropout is a simple way to implement a _Bayesian Neural Network_, i.e., a network that allows predicting a distribution over outputs instead of just a single "point" prediction [23]. To this extent, it leverages multiple, randomized predictions to estimate the spread in the prediction and thus its the degree of uncertainty [14]. The randomness in the prediction process comes from enabling _dropout_ layers [51] at prediction time, i.e., layers that randomly drop some nodes' activations, which are typically used for regularization during DNN training. Then, there is a range of techniques to quantify the models' uncertainty given the samples, such as the predictions _variation ratio_ or _mutual information_. MC-Dropout has been widely used in the literature [e.g. 4, 5, 40, 73]. Ensembles: Similar to MC-Dropout, Ensembles [33] estimate the models' uncertainty by measuring the agreement between sampled predictions for a given input. Here, however, the randomness in the prediction process comes from using a range of different DNNs, all sharing the same architecture, but trained using differently initialized weights and biases. Intuitively, inputs for which the different models disagree more are quantified with higher uncertainty. Here, the same quantifiers as the ones for MC-Dropout can be used. Surprise Adequacy: Instead of the approaches described above, which aim to quantify the models uncertainties, surprise adequacy (SA) [28] aims to quantify how _surprising_ (i.e., _novel_ or _out-of-distribution_) an input is with respect to its activation traces in the supervised model. There are different variants of surprise adequacy, such as _Distance-based SA (DSA)_[28], _Likelihood-based SA (LSA)_[28]_ and Mahalanobis-distance-based SA (MDSA)_[29, 30]_, some of which can be used in a multimodal setting [31], and we refer to the corresponding papers for a detailed description. Autoencoders: A more direct, black-box [45] approach to supervision is taken by Autoencoder based supervision [52, 53, 54, 55]. Autoencoders are DNNs aiming to first compress and then reconstruct an input. If trained on the same data as the supervised DNN, an autoencoder can detect out-of-distribution inputs at runtime: As such data was by definition not well represented in the training data, it is likely to have a higher reconstruction error. Other Supervisors: Alternative, less frequently used approaches include, _Dissector_[59] which makes per-layer predictions and measures there disagreement, _DeepRoad_[71] a principal component analysis (PCA) based approach on the model's activation traces, _Learnt Assertions_[37] which aims to automatically learn to recognize inputs that are likely to be mispredicted from the associated activations, and _NIRVANA_[5] which combines multiple MC-Dropout quantifiers outputs in a support-vector machine (SVM). #### 3.2.2. Comparison of Supervisors Regarding Computational Overhead In resource-constrained environments, the overhead imposed by a supervisor, i.e., the additional computation cost imposed by running the supervisor might be an important concern. Here, we see substantial differences between the supervisors: Clearly, Softmax-based approaches are the fastest approaches, as they do not require any additional predictions. Instead, the quantification of uncertainty from the softmax array requires just a few floating point operations. In the case of MaxSoftmax, there's actually just a single array value read required (the softmax likelihood for the predicted class), the associated overhead being clearly minimal. MC-Dropout (and thus also NIRVANA) and Ensembles require substantially more time, as various predictions for the same input have to be collected. While in an existing tool [65], we provide utilities to perform such predictions very efficiently, it still remains a show-blocker for very resource-constrained applications. Autoencoders can also be considered quite fast and efficient, as there is only a single additional prediction (the autoencoder forward pass) required. Moreover, such additional prediction can actually be done at the same time or even before the prediction on the supervised model is made. Should the autoencoder-based supervisor thus decide that a prediction for a given input is not to be trusted, the prediction does not even have to be made. When originally proposed, the performance gain of using autoencoder-based supervision compared to the then state-of-the-arte DeepRoad was one of its major advantages [54]. The overhead of surprise adequacy much depends on its specific type: DSA and LSA suffer from major performance bottlenecks when large training sets are used. There are some ways to mitigate these problems, e.g. by using only a subsample of the training set and fast, vectorized implementations [62], but even with such improvements, the use of these techniques in many real-world applications on devices such as microcontrollers remains prohibitive. In such domains, only the latest variant of SA, MDSA provides a low enough overhead [66]. #### 3.2.3. Comparison of Supervisors Regarding Supervision Performance Faced with the wide range of techniques applicable for supervision, a range of empirical studies has been performed comparing them [1, 63, 64, 66, 67, 73]. Summarized, they all find that no single technique works as a dominant supervisor, i.e., no technique consistently outperforms all other approaches. Some studies highlight that ensembles provide often - but not always - a better supervision performance than all other techniques [41, 64, 67] as they not only give an accurate uncertainty estimate, but often better prediction in the first place. Softmax-based approaches also often provide good supervision performance and in a recent study we found no statistical difference between the performance to detect misclassified inputs between the different variants of softmax-based supervisors. Thus the simple MaxSoftmax might be a good choice in many situations. Surprise Adequacies, while generally performing a bit worse than Softmax-based supervisors [63, 66], have the advantage that they can be applied to any DNN, not just to classification models. Lastly, autoencoders perform well as long as the training set is relatively sparse in the input feature space. Instead, with feature-rich datasets autoencoders start to generalize too well for supervision, as they learn to also reconstruct outliers consisting of the same features [63, 46] #### 3.2.4. Comparison of Supervisors Regarding Robustness It is important that supervisors detect root causes of DNN failures, such as adversarial inputs, corrupted inputs, invalid inputs or ambiguous inputs. In a recent study, we compared the capabilities of various supervisors in detecting these specific root causes of uncertainty [63]. Summarized, we found that all supervisors show some weaknesses. E.g., surprise adequacy, which tends to work well in detecting corrupted, adversarial and invalid inputs, fails to detect in-distribution ambiguous inputs (i.e., inputs with only aleatoric, but no epistemic uncertainty). Softmax-based supervisors can in principle detect all root causes, but are are vulnerable to adversarial attacks, which typically aim to modify the values of the output layer and can thus easily be configured to create high-confidence but wrong predictions. ## 4. Bispupervised: Our Proposed Architecture An overview of BiSupervised is illustrated in Figure 1. The goal of our architecture is as follows: (1) Attempt a prediction on a small, fast and cost-free _local surrogate model_ (short: _local model_). (2) Leverage a supervisor, denoted as _1st level supervisor_, to identify inputs for which the local prediction cannot be trusted. (3) For those inputs, and only for those inputs, collect prediction and confidence from a large, highly accurate remote model. (4) Identify predictions for which not even the remote prediction can be trusted, based on the reported prediction and confidence in a _2nd level supervisor_ and raise a warning or return a default (fallback) value for those inputs. Intuitively, this architecture attempts only to make costly remote predictions for _difficult_ inputs, where the simple local model is not sufficiently well trained. The 2nd level supervisor furthermore allows handling inputs for which not even the highly accurate remote model can make a trustworthy prediction, such as invalid or ambiguous inputs. Algorithm 1 outlines BiSupervised with threshold-based supervisors as pseudocode: On line 2-3 the local model makes a prediction and the 1st level supervisor quantifies the confidence of the prediction. To mitigate the computational effort required to execute lines 2-3, the corresponding input might be reduced in dimensionality (e.g. reducing the size of an image) to allow for smaller local models (line 1). If the confidence of this locally computed prediction is above some predefined threshold $t_{l}$, the prediction is returned (lines 4-6). Otherwise, the original input is sent to the remote prediction server, which returns the remote prediction amongst some metadata (e.g. prediction logits), from which the 2nd level supervisor can calculate a confidence (lines 7-8). The remote prediction is returned, if the corresponding confidence is considered high enough. Otherwise, an exception is raised (lines 9-13). A specific example highlighting how BiSupervised could be implemented and used in a practical application is given in Example 4.1 Example 4.1 (Use of BiSupervised in a Wildlife Camera): Let us consider a wildlife camera in a national park, installed to monitor the presence of wild animals and to report it to the park administration. Every 15 seconds, it takes an image that is classified according to the animal visible on the image, e.g. _no-animal, deer, wolfe, human_, etc. Using BiSupervised, the classification works Figure 1. BiSupervised: Proposed Architecture for a Supervised Combination of Local and Remote Predictions as follows: First, the wildlife camera locally tries to classify the image. In doing so, it collects its full softmax output arrays, from which the 1st level supervisor can easily read the model's confidence [(16)]. If the confidence is high, the prediction is reported to the wildlife park's server. Otherwise, a remote model is leveraged: the wildlife camera sends the image to a highly reliable 3rd party image classification service, such as IBM's Watson11, which returns its prediction and alongside the prediction logits. The 2nd level supervisor then determines if the remote prediction can be trusted based on the remote model confidence (which can be determined based on the prediction logits). If the remote prediction is trustworthy, the prediction is reported to the wildlife-park server. Otherwise, a warning is submitted to the park administration, alongside the unclassifiable image. The following scenarios may thus happen for different images: Footnote 11: [https://www.ibm.com/no-en/cloud/watson-visual-recognition/pricing](https://www.ibm.com/no-en/cloud/watson-visual-recognition/pricing) * Nominal Image Assume no animal stands in front of the camera. The local model confidently detects _no-animal_, with low surprise adequacy. No remote model invocation is needed, which thus saves money. * Rare Image Assume there's fog, a condition which is not well represented in the local models' training set, thus leading to a non-trustworthy prediction of _Beaver_. The local supervisor detects the surprising input and thus ignores the local prediction, but forwards the image to the large-scale remote model, which generalizes better to such rare conditions and clearly detects a _deer_ in the fog. "_Deer_" is thus reported to the park administration. * Invalid Image A storm caused the camera to be partially covered by mud, in a way that neither the local nor the remote model can make confident predictions, which both supervisors detect. A warning is sent to the park administration. With several such warnings reported in a short time, a park administrator double-checks the reported images and notifies a ranger to clean up the camera. In the following, we will explain each of the components in our architecture in more detail. ### Design of Local Surrogate Model The local model's objective is to make high-confidence predictions for as many inputs as possible in a very constrained environment: Our architecture aims to be applicable to even tiny devices, such as microcontrollers, and as such, the design possibilities for the local model are limited, especially with respect to the number of model parameters. We refer to subsection 2.2 for an overview of techniques to create such a model, but in the scope of our architecture, there's an additional measure that _optionally_ can be taken to allow a reduced model size, which we call _Input Domain Reduction_: Optionally, we may design a local model to handle a reduced input space instead of the original, possibly large, one: we can reduce the dimensionality of the input space, potentially losing some information, which may prevent optimal predictions on a few inputs, but generally allows the local model to be much smaller in size. For example, in NLP tasks, we may use a much smaller dictionary or clip the length of the input. For easy inputs, the information loss induced by the input domain reduction is unlikely to cause much uncertainty. For all other inputs, we leverage a remote prediction on the large original input domain. E.g., in Example 4.1, we could train the local surrogate model on small grayscale images, and compress their inputs accordingly. Simple classifications, e.g. when no animal is around would still be possible with high confidence, but at the same time this would allow us to use a much smaller DNN. ### Design of 1st Level Supervisor In principle, our architecture is not tied to any specific supervision technique, thus all those listed in Section 3.2 could potentially be used as 1st level supervisor. However, as the 1st level supervisor runs in the local, resource-constrained environment, one should make a choice that does not come with notable computational requirements during inference. In our recent works [63; 64; 66], we empirically compared a range of supervision techniques. Our results suggest the use of two specific techniques, as they both show a good supervision accuracy, while only causing negligible additional computational cost: For classification problems, where the local model makes softmax-normalized predictions, simply taking the highest softmax value (i.e., the softmax value of the predicted class) [16] as a confidence score shows good practical supervision performance. For other models, without a softmax output, _Mahalanobis-Distance based Surprise Adequacy (MDSA)_[29] reliably detects out-of-distribution inputs. We refer to subsection 3.2 for a detailed discussion of different supervisors, their complementarities, weaknesses, strengths and overhead. ### Use of Remote Model or System and Design of 2nd-Level Supervisor Our architecture does not have any specific requirements on the architecture of the remote model. Indeed, the remote model is considered a "black box" and may consist of more than just a DNN: in general, it might be a fully fledged, complex multi-component system (in this paper, we use the intuitive term _remote model_ for simplicity). However, the 2nd level supervision can be heavily facilitated when the remote model is a DNN returning also its confidence (or related metrics, such as uncertainty) alongside its prediction. This may not be possible for all remote models, or may have to be specifically configured. For example, using OpenAI's GPT-3, one can configure a request to return not just the sequence of tokens representing the model's maximum posterior, but a sequence of most likely options for each returned tokens, alongside their negative log-likelihood. This additional information about the model's prediction, which does not cause any additional cost, allows the straightforward implementation of a 2nd level supervisor. Alternatively, a black-box or data-box [45] supervisor (respectively, one with access to just the input/output of the remote model or also to the training data used to train the remote model) would have to be employed as 2nd level supervisor to select inputs that not even the remote model can handle. While this comes with yet another cost saving (the supervisor can run before the remote model, thus allowing to not collect remote predictions which will then be rejected), we do not recommend the use of such black- and data-box supervisors in this use case: They are likely to cause substantial additional load on the local resource-constrained platform, they typically do not detect aleatoric uncertainty caused by ambiguity (Rasmal et al., 2017) and lastly, in the case of data-box supervision, they require access to the training data of the remote model, which is typically not available or to too large to be easily processed. ### Superaccurate Performance The primary goal of our architecture is to allow a tradeoff between a cheap, low-accuracy model and an expensive, highly accurate remote model. As such, we expect the combined cost and accuracy to always lay between the ones of the local/remote model. However, for what regards accuracy, our architecture may - under specific circumstances - even lead to accuracies above the one of the remote model, while still leading to reduced cost. If the local and remote models' prediction capabilities are complementary (i.e., the set of inputs that the local model can correctly predict is not a subset of the inputs the remote model can predict), and the 1st level supervisor reliably trusts correct predictions by the local model, some inputs which would be incorrectly predicted by the remote model would be correctly predicted with our architecture. With large complementarities, this could substantially increase the overall system accuracy, at a reduced cost. We denote such improved overall system accuracy as _superaccurate performance_. However, aiming for complementarities between local and remote models might be infeasible, due to the constraints on the choice of models that can reasonably run on the local device: In this paper, the main goal is to achieve cost-reduction with as little accuracy decrease as possible. Thus, while we might occasionally observe superaccurate performance empirically, its occurrence is more a (nice to have) side effect than an explicit design goal. ### Threshold Selection for Supervisors We strongly recommend the use of single-threshold supervisors in conjunction with BiSupervised. Such supervisors are based on a single confidence (or uncertainty12) score, and a configurable threshold. Inputs are trusted if and only if the confidence for said input is above a configurable threshold. Modifying the threshold allows to gradually change the proportion of untrusted/trusted inputs. In BiSupervised, a carefully tuned threshold in the 1st level supervisor thus allows the selection of the desired percentage of inputs for which a remote prediction is collected (thus defining the cost savings). Similarly, a carefully tuned threshold in the 2nd level supervisor allows to regulate the percentage of inputs for which the system refuses to make a prediction. Viewing the 2nd level supervisor as a misprediction detector, changing the threshold allows to trade-off false positives vs false negatives. Here, especially in domains which are not safety-critical, we can imagine that different users or different use-cases show varying degrees of risk-aversion. In a more risk-averse setting, the threshold can thus be tuned to allow more raised exceptions, possibly requiring manual intervention, while other use-cases may actually favor some mispredictions over downsides from a too frequently raised exception or too frequently called fallback (such as a decreased user experience due to overly frequent warnings). Footnote 12: We consider confidence and uncertainty as perfect complements (Rasmal et al., 2017), and for what regards DNN supervision, one is easily converted into the other by applying any order-reversing transformation to the value (Rasmal et al., 2017) Existing Techniques to Threshold Selection.Stocco et. al. (Stocco et al., 2017) propose to fit the distribution supervision scores of nominal, correctly classified validation set inputs (Stocco et al., 2017). This allows selecting a threshold with a specific false positive rate (i.e., expected percentage of false alarms), assuming that the distribution of inputs in production is the same as the one of the validation set. The big advantage of this approach is that no dataset of outliers is needed and a nominal validation set is sufficient to calculate the threshold. An alternative approach for threshold selection in test input validators was proposed by Dola et. al. (2021) (Dola et al., 2021). They propose to choose a threshold by calculating supervision scores for both a nominal and an invalid dataset and then choose a threshold that separates these two distributions optimally. While they proposed these techniques for test data validations, it can clearly also be used for supervisors. However, here, it is worth noting that a realistic invalid dataset may not always be available. BiSupervised is agnostic to the way the threshold is set. We refer to Riccio et. al. (2023) (Ricio et al., 2023) for a more in-depth comparison of the two approaches discussed here. Runtime Configuration of ThresholdBoth approaches described above consider the threshold a constant, which is defined before deployment. For BiSupervised, we suggest that in practice, a more dynamic way to set the threshold may be beneficial: Here, the threshold could be considered a configuration variable that can be changed or tuned in production, for example using remote information (Ricevic et al., 2023) or based specifically on the inputs observed on a specific user's local device. Such dynamic threshold selection allows to quickly adapt the supervisor behavior, for example in cases where the input distribution changes, where more or less conservative supervision is desired (e.g. due to a change of cost in remote predictions), or where users specify their preferences in their user settings. To account for this diversity in our paper, we evaluate our architecture using a range of different thresholds. ### Extensions of BiSupervised BiSupervised allows for some extensions of its architecture and use, which go beyond the scope of this paper but are worth mentioning. First, edge nodes, where available, allow for an extension including a third supervised DNN on an edge node, with capabilities laying somewhere between the minimal local model and the huge remote model. As such, BiSupervised (which would effectively become TriSupervised), would invoke the model on the edge node if the local prediction cannot be trusted. Then, if also the edge model would face high uncertainty, the edge node could in turn forward the prediction to the remote model. Second, we note that active learning could be easily integrated into BiSupervised: As the 1st level supervisor detects inputs for which the local model is (yet) insufficiently trained, collecting these inputs, labeling them (maybe even automatically using the remote model), and adding them to the local model's training set would allow for much more robust generation of the next local model: In our recent work (Ricio et al., 2023), we have shown that simple supervisors such as MaxSoftmax and MDSA, which we recommend in BiSupervised as 1st level supervisor, are also great techniques for such purpose. Consequently, such an active learning setup would essentially allow a continuous improvement of the local model, thus further reducing the share of remote predictions needed in future versions of the software using BiSupervised. ## 5. Empirical Evaluation of Bisupervised Our empirical evaluation aims to answer the following research questions: RQ1: Cost Savings: Using BiSupervised, can we reduce the number of remote predictions while only slightly impacting the overall prediction accuracy? RQ2: Supervision Performance: Provided with a given percentage of inputs to be used in remote predictions, how does BiSupervised perform in terms of _supervised performance_(Ricio et al., 2023), compared to a standalone, supervised local model? RQ1 is the main research question behind this work, as it aims at quantifying the trade off between cost saving and possibly reduced accuracy. This RQ ignores 2nd level supervision, as its focus is solely on the reduction of costly remote invocations, with no regard to the degree of confidence in the remotely produced prediction. RQ2 aims at comparing the performance of a single local supervisor with that delivered by the two local/remote supervisors envisioned by BiSupervised. An ideal supervisor is one that blocks all and only inputs on which the model would otherwise exhibit poor accuracy. With RQ2 we check whether the proposed architecture, in addition to reducing the remote execution costs, can also improve the supervision performance. ### Assessment Metrics for RQ1 All the supervisors used in our case studies work based on a single uncertainty score, which, if above a threshold, indicates that a prediction should not be trusted. We answer RQ1 in a threshold-agnostic way: Our results show the cost-accuracy tradeoff across all possible thresholds, i.e., for any chosen proportion of inputs for which a remote prediction is collected. Requests-Accuracy-Curve.Changing the threshold of the 1st level supervisor allows a trade-off between the percentage of inputs forwarded for remote prediction (fewer means lower cost) and high system accuracy (associated with many remote predictions). We visualize our results for RQ1 using a plot that we call _Request-Accuracy-Curve_ where the $x$ axis shows the percentage of inputs forwarded to the remote model and the $y$ axis shows the overall system accuracy (ignoring 2nd level supervision). The curve thus allows to quickly read cost savings and accuracy change when using our architecture with a specifically chosen 1st level supervisor threshold. We will furthermore annotate selected points (e.g. knee points and superaccurate performances, if present) with the corresponding values, for even better readability of the plot. Area Under Curve and Random Baseline.The Request-Accuracy-Curve allows to visually compare against the random baseline, which is a straight line between the performance using all-local or all-remote predictions: If successful, the curve produced by our architecture should be mostly above this line. To quantitatively compare the random baseline against BiSupervised we measure the _area under the curve of the Request-Accuracy-Curve_ (AUC-RAC). Specifically, we take the mean of all accuracies when setting the threshold to the value of each input, and normalize it on the accuracies observed for remote-only and local-only predictions. Let $\text{ACC}_{\text{SYS}}(r)$ denote the system performance for a given system and test set when a proportion of $r\in[0,1]$ of all inputs are forwarded to the remote model (thus, $r$ is a point in the $x$ axis on the Request-Accuracy-Curve) and let $n$ denote the number of inputs in the test set. The AUC-RAC is then formally defined as: \[\frac{\frac{1}{n}\sum_{i=0}^{n}\text{ACC}_{\text{SYS}}(\frac{i}{n})-\text{ ACC}_{\text{SYS}}(0)}{\text{ACC}_{\text{SYS}}(1)-\text{ACC}_{\text{SYS}}(0)} \tag{1}\] This assumes that the remote model exhibits a higher accuracy than the local model, i.e., $\text{ACC}_{\text{SYS}}(1)>\text{ACC}_{\text{SYS}}(0)$, which is a prerequisite for our experiments. The expected AUC-RAC from random supervision is 0.5. Note that, contrary to other AUCs, such as AUC-ROC, the proposed AUC-RAC can be below 0 in case of very bad supervision, or above 1 in case of very large superaccurate performance, due to normalization. Both of these are edge cases we do not expect to see frequently. ### Assessment Metrics for RQ2 RQ2 considers the performance of BiSupervised in a supervised setting, i.e., including both supervisors to identify inputs for which neither local nor remote predictions can be trusted. To compare our architecture against a supervised local model, we use the following three metrics which we proposed and designed specifically to compare pairs of different supervised models: * _Supervised Accuracy_ ($\overline{ACC}$): The accuracy considering only the trusted predictions. Higher is better. Typically, the more conservative a supervisor, the higher $\overline{ACC}$. * _Acceptance Rate_ ($\Delta$): The percentage of trusted inputs. Higher is better. Typically, the more conservative a supervisor, the lower $\Delta$. * _S-Beta Score_ ($S_{\beta}$). The geometric mean between (normalized) supervised accuracy and acceptance rate. $\beta$ is the weight parameter in the geometric mean. We will assess $S_{0.5}$, $S_{1}$, $S_{2}$. We provide a detailed explanation of these metrics in Weiss and Tonella (2018). Threshold SelectionThus, for RQ2, we need to choose thresholds for both the 1st and 2nd level supervisor. For what regards the former, we use meaningful knee points identified in RQ1. For what regards the latter, in line with related literature (Beng et al., 2018; Wang et al., 2018; Wang et al., 2018), we choose a threshold such that a targeted (overall) false positive rate (FPR) is reached. Specifically, for each of the previously identified knee points, we will conduct three separate evaluations, in which we will tune the second-level supervisor's threshold such that FPRs of 0.01, 0.05, and 0.1 are met, respectively. As a baseline, we will assess the supervised performance of a system consisting of only the local model and supervisor using again thresholds chosen to lead to the same FPRs. This also reflects our targeted use case where thresholds are runtime-tunable parameters, allowing them to be fine-tuned and adjusted as inputs (and shifts of inputs) are observed after deployment, while still allowing for a fair comparison with the baseline. ### Case Studies We evaluate our architecture on four case studies. They were chosen to reflect diverse problem types (text classification, image classification, and question answering) and to allow the inclusion of various different architectures as local and remote models. An overview of the case studies is provided in Table 1, and they are described below. For implementation details, such as used hyperparameters and exact data pre-processing, we refer to our replication package, described in section 8. #### 5.3.1. IMDB Sentiment Classification (Text Classification) The IMDB sentiment classification benchmark (Zhu et al., 2017) is a common case study for NLP classification approaches. Given an English-language movie review, a classifier is to decide whether the rating by the user is positive or negative. We use the following components for our architecture: * Local Model: For input domain reduction, we trim inputs to consist only of the first 100 words and use a small dictionary of only the 2000 most frequent words. We use a small model, \begin{table} \begin{tabular}{l l l l l l l} \hline \hline Name & Domain & & Architecture & \#Params & Metric & Value \\ \hline imdb & Text Class. & Local Remote & Custom & 79K & \multirow{2}{}{Accuracy} & 79.4\% \\ & & Local Remote & GPT-3-curie (Beng et al., 2018) & 13B ${}^{}$ & & 89.5\% \\ issues & Text Class. & Local Remote & Custom & 223K & \multirow{2}{}{Micro Avg. F1} & 71.1\% \\ & & Local Remote & CatIss (Kal consisting of one transformer block, a pooling, and two dense layers. For regularization, dropout layers have been added before the dense layers. The resulting local model has 78'558 parameters. 1st level supervisor: As our local model is a softmax-based classifier, we can use vanilla softmax as the 1st-level supervisor. * Remote Model: As remote model, we use a few-shot learning approach on the _curie_ model of the GPT-3 [2] family. With its estimated 13 billion parameters, it is more than 160'000 times larger than our local model. We use a simple prompt, shown in Listing 1. This leads to an accuracy of 89.5%, which is more than 9% better than our local model. * 2nd Level Supervisor: Any valid response to our prompt consists of only one additional token. We configured our remote request to provide us with a list of 100 most likely tokens with their corresponding negative log-likelihood (easily transformable into likelihoods). We sum up the likelihood for known, hard-coded equivalent tokens (e.g. _"Negative"_, _"negative"_ and _"bad"_). This total likelihood is then used to determine whether or not the supervisor should reject the input. We used the IMDB case study in a pilot experiment [61], presented at the ESEC/FSE student research competition 2022. While showing encouraging results, motivating the writing of this paper, that pilot experiment answers only parts of RQ1, and considers a simpler architecture (not including a 2nd level supervisor), hence also not addressing RQ2. ``` Pleaseclassifythefollowingmoviereviewsintothefollowingsentiments:Negative,Positive. --- Text:IhatethismovieLabel:Negative--- Text:IhikethismovieLabel:Positive--- Text:<thereviewwewwanttoclassify> Label: ``` Listing 1: Few-shot GPT-3Promptfor IMDB case study. #### 5.3.2. Github Issue Classifier (Text Classification) The Github Issue Classifier reflects a real-world use case: As new issues are reported to a github repository, maintainers typically triage the issues (are they a question, a bug report, an improvement request, etc.). TicketTagger[25] is an open source bot performing such triage by automatically labeling each input. Run in a CI/CD environment, the "local environment" for such automated triager is typically quite resource-constraint. Recently, Kallis et al.[24] released a corresponding benchmark dataset as part of the NLBSE'22 tool competition: Given a few attributes of Github issues, amongst which title and description, issues have to be classified into one of the classes _[bug, enhancement, question]_. The dataset is unbalanced, heterogeneous (e.g., it contains outliers such as non-English issues), and contains some truly ambiguous inputs (i.e., issues for which the ground truth is not certain even to an experienced developer). This makes this case study a particularly well-motivated setting for RQ2, as unclear inputs should better remain unclassified and handled by a repository maintainer performing manual triaging. Upon adopting BiSupervised, TicketTagger could assign a fallback label _"manual triage required"_ to inputs rejected by both supervisors. To measure the model and system performance of this case study, we use a micro-average F1-score, which takes into account the class imbalance in this case study, as described in the tool competition [24]. We then use the following components for our architecture: * Local Model: We use a similar transformer-based classifier as for IMDB, using the same type of layers, but with hyperparameters chosen to take the higher complexity of the problem into account (e.g., the hidden dense layer contains of 100 instead of 20 nodes) and with an increased dictionary size of 2000. As input we use the issue description concatenated to the issue title, truncated to the first 200 words. * Remote Model: As remote model, we use CatIss, the winning model of the NLBSE'22 tool competition [21]. It is based on a finetuned Roberta model [36]. As far as we know, this is the best available model for the issue classification dataset. We expect that by fine-tuning even larger Roberta instances, one may further improve the overall performance. * 1st and 2nd level supervisor: We use vanilla softmax as 1st and 2nd level supervisors. #### 5.3.3. Imagenet (Image Classification) Imagenet [47] is one of the most widely used image classification benchmarks in the literature. Its images represent 1000 different classes, some of which very closely related to each other (e.g. 119 breeds of dogs [11]). As such, it represents a challenging machine learning problem, where even large-scale state of the art models do not yet achieve accuracies above 91% [70]. Provided that a wide range of Imagenet models are available through standard model repositories, and that such models are typically DNNs with a softmax output layer, choosing the components for our architecture is not difficult: * Local Model: We use MobilenetV3 [18] as local model. MobilenetV3 is the third and latest generation Mobilenet Imagenet model which is specifically designed to work in resource constrained environemnts. Various versions (differing in size) of MobilenetV3 exist. We use'mobilenetv3_small_1.0_224', i.e., the model with 2.9 million parameters and an accuracy of 67.8%1 Footnote 13: See [https://www.tensorflow.org/api_docs/python/tf/keras/applications/MobileNetV3Small](https://www.tensorflow.org/api_docs/python/tf/keras/applications/MobileNetV3Small) * Remote Model: We use EfficientNetV2L [57] as large-scale remote model. With its 119 million parameters and an accuracy of 85.2, it is the third-largest and most accurate Imagenet classifier available from Keras applications14. Footnote 14: [https://keras.io/api/applications/](https://keras.io/api/applications/), as of July 31st, 2022 * 1st and 2nd level supervisor: We use vanilla softmax as 1st and 2nd level supervisor. #### 5.3.4. The Stanford Question Answering Dataset (Free-Text Question Answering) Our fourth case study considers _question answering_, specifically the 2nd version of the _The Stanford Question Answering Dataset_ (SQuADv2)_[44]. Faced with a title and paragraph of English-language Wikipedia, and a list of questions about the said paragraph, a model is supposed to provide a short answer, consisting of substrings of the paragraph (thus essentially free text), to each of the questions. SQuADv2 provides a set of correct answers for each question. For example, the question _"What century did the Normans first gain their separate identity?"_ lists _"10th century"_, _"the first half of the 10th century"_ and _"10th"_ as correct answers.15 We use the _Exact Match (EM)_[44] metric to assess the performance of our models and architecture, i.e., the percentage of predictions that exactly match one of the answers in the SQuADv2 ground truth answers. In addition to the questions described above, SQuADv2 also contains _unanswerable_ questions, i.e. questions for which the answer cannot be found in the provided paragraph. They are ideal to test the models in a supervised setting. We ignore these unanswerable questions for RQ1 (which explicitly does not assess the 2nd level supervisor) but include them for RQ2 where they should be detected by the 2nd level supervisor. Here, it is worth mentioning that much related literature using SQuADv2 considers an unanswerable input as correctly answered if the models' uncertainty is above the threshold (i.e., if the input is detected as unanswerable by the supervisor). We argue that this is not a realistic setting in practice (a refused prediction is certainly not equivalent to a correct one). Our assessment of RQ1 (EM only on answerable questions) and RQ2 ($\overline{ACC}$, $\Delta$ and $S_{\beta}$) are better suited. However, for this reason, the accuracies we find in RQ1, and the supervised accuracies we find in RQ2 are not consistent with the ones mentioned in the related literature for the same models. * *Local Model: * We use a fine-tuned version of Bert-Tiny [58], a miniature version of the popular _Bidirectional Encoder Representations from Transformers_ (BERT) model [9], well suited for deployment in resource-constrained environments. Fine-tuning to the SQuADv2 dataset has been performed by the Huggingface team16. Footnote 16: [https://huggingface.co/mrm8488/bert-tiny-5-finetuned-squadv2](https://huggingface.co/mrm8488/bert-tiny-5-finetuned-squadv2) * *1st level supervisor: * Alongside the predicted tokens representing our local model answer, we collect an uncertainty score for each predicted token. We consider multiple ways suitable to reduce these scores into a single one, which can be used for threshold-based supervision, with the product and the minimum being the most obvious candidates. Indeed, we find that the literature mostly uses the product of these uncertainties. We do not consider this ideal for supervision: Given the very large amount of tokens, predicted probabilities, even for the simplest of inputs are likely to be much lower than one, which would naturally make the overall confidence in answers consisting of multiple tokens exponentially smaller than the confidence in single token predictions. Thus, for our experiments, we use the _minimum_ of all predicted likelihoods of all predicted tokens. * Remote Model: As remote model, we use a one-shot learning approach on the _davinci_ model of the GPT-3 [2] family. With its 165 billion parameters, it is the largest model in the GPT-3 family and thus often understood as _the_ GPT-3 model. Upon its first publication, the authors of GPT-3 assessed the model on the SQuADv2 dataset, where it showed good performance, in particular considering the one-shot (not fine-tuned) setting. We build our prompt equivalent to the one built when assessing GPT-3. In short, the prompt is initiated with the title, followed by the paragraph and a single, answerable question with the correct answer (which is consequently removed from the test set) and finally the question for which we assess the answering capability (see example prompt in Figure G.28 of Brown et al. [2]). * 2nd level supervisor: As explained in the IMDB case study, we can configure our requests to GPT-3 to collect a likelihood for every predicted token. We can then continue equivalently to the 1st level supervisor to reduce this array of likelihoods into a single confidence score, allowing threshold-based supervision. For the Github issue classification and Imagenet case studies, we run the remote models on our highly-performing workstation used for our experimentation, as in our setting we can afford the required hardware, which is not the case in most client-based applications for which we propose BiSupervised. ### Results #### 5.4.1. Changes to Evaluation Setup The evaluation setup described above was peer-reviewed as a registered report before results were collected17. The experiments have been conducted as anticipated, with the exception of the following two minor deviations: Footnote 17: [https://dl.acm.org/journal/tosem/registered-papers](https://dl.acm.org/journal/tosem/registered-papers) 1. Imdb Remote Supervisor: We planned to use the sum of the likelihoods of equivalent tokens (e.g. _"negative"_, _"Negative"_ and _"bad"_) amongst the 100 most likely tokens. However, the OpenAI API restricts this information to the 5 most likely tokens18 - which we thus use for our experiments. In general, we observe that subsequent equivalent tokens, in general, have a very low likelihood assigned. This is expected, as our prompt (see Listing 1) suggests a specific spelling. Thus, we do not expect that the here-described change has any notable influence on the results. 2. SQuADv2 Prompt: We anticipated using the same prompt as the authors of the original paper proposing GPT-3 when conducting their SQuADv2 assessment (see example prompt in Figure G.28 of Brown et al. [2]). During development, we found that using this prompt, GPT-3 would often give verbose, full-sentence answers. As such answers are not part of the SQuADv2 answer dataset, this could have led to a misrepresentative low EM (exact match) metric. We thus extended the prompt stating explicitly that answers should be an "_extraction from background, as short as possible_". Here, background refers to the context given in the prompt, in line with the originally anticipated prompt. This obvious fix is one which realistically any practitioner would have done and we thus decided to include it. As GPT-3 is a closed source model, with unknown prompt-preprocessing, we do not know if the problem we observed was already present when Brown et. al. [2] conducted their experiments. Given their decent results, we assume that the verbose sentences GPT-3 we observed when re-using their prompt might be caused by deliberate recent changes to the GPT-3 service (in most use-cases, verbose answers might be perceived as better), as otherwise, we would have expected Brown et. al. [2] to adapt their prompt in a similar fashion as we did. While it is unclear what effect this change has on our results, it is likely that our changes actually lead to a stronger baseline for BiSupervised and overall a more ambitious experimental setup: With the anticipated higher EM of the standalone remote model, outperforming or performing similarly well - as measured in RQ1 - becomes a more challenging task. #### 5.4.2 Results for RQ1: Cost Savings. _Imdb._ The RAC for the Imdb case study is presented in Figure 2. It shows a mild superaccurate performance: At 68% remote requests, the system-level accuracy (ignoring the 2nd-level supervisor) reaches its peak, which is mildly better than if only remote predictions are made. With 55% remote requests, thus at 45% remote prediction cost savings, BiSupervised achieves the same system-level performance as if only the remote model was used. Figure 2. Request-Accuracy Curve (RAC) for Imdb _Github Issue Classification._ The RAC for the GitHub Issue Classification case study is presented in Figure 3. Here, BiSupervised allows to save a large part of remote-prediction costs while maintaining a system-level F1 similar to the one observed on the standalone remote model: With any threshold leading to at least 60% remote predictions, the system-level F1 remains practically constant. BiSupervised thus allows saving 40% remote prediction cost at essentially no negative impact on the system level F1. Overall, the AUC-RAC of the plot is 0.775, thus clearly outperforming the random baseline (expected AUC-RAC 0.5). _Imagenet._ The RAC for the Imagenet case study, presented in Figure 4 is similar to the one for the Github Issue Case study. While there is no notable superaccurate performance, we observe again that BiSupervised allows saving 40% remote prediction cost at essentially no negative impact on the system level accuracy. At the same time, the steep increase of the RAC for small percentages of remote predictions highlights the potential BiSupervised has for applications that currently use MobilenetV3 (the production-ready mobile-friendly model we use as a local model): With just a low percentage of predictions to a larger model (which may be placed on the edge or a remote server), the system-level accuracy is easily improved. Figure 4: Request-Accuracy Curve (RAC) for Imagenet Figure 5: Request-Accuracy Curves (RAC) for SQuADv2 dataset. BiSupervised achieves superaccurate performances for both a clean dataset consisting of only valid (i.e., answerable) questions and for the full dataset. Figure 3: Request-Accuracy Curve (RAC) for Issues _SQuADv2_. As anticipated in the empirical setup, for what regards RQ1, the SQuADv2 dataset _without_ invalid inputs is of primary interest. The corresponding RAC is shown in Figure 4(a). Clearly, BiSupervised shows large superaccurate performance: At its peak, it achieves an EM of 0.722, which is more than.02 better than the standalone remote model (0.695). At that point, only 59% remote predictions are needed, thus also saving 41% of remote prediction cost. At only 38% of the cost, BiSupervised achieves the same performance as the standalone remote model, clearly achieving the goal of cost savings. Overall, BiSupervised achieves an AUC-RAC of 1.021. For completeness, we also calculated the RAC for the full SQuADv2 dataset, including all invalid questions, shown in Figure 4(b). While these results are slightly inferior compared to the ones without invalid inputs, the plot still shows superaccurate performance. Summary of RQ1 (Cost-Saving) The results clearly show BiSupervised's capability to save prediction cost while maintaining high accuracy: For two case studies, superaccurate performances were achieved, i.e., BiSupervised allows to get a better system-level prediction accuracy, at a much lower cost, compared to using only the large-scale remote model, or equivalent performance for 38% or 55% of the remote prediction cost, respectively. In the other two case studies, where the models showed fewer complementarities, BiSupervised allowed saving roughly 40% of remote prediction cost (dependent on the case study) while still achieving comparable system-level performance. In all case studies, BiSupervised achieved an AUC-RAC substantially above the one associated with a random baseline. #### 5.4.3. Results for RQ2: Supervision Performance The results for RQ2 are presented in Tables 2, 3, 4, 5 and 6. The RACs observed in RQ1 are smooth, which benefits our evaluation of RQ2 as the exact choice of 1st level supervisor becomes less critical: Small deviations in the threshold, i.e., in the percentage of inputs for which a remote prediction is made, do not have a major influence in the observed results. This allows us to choose thresholds for the 1st level supervisor for our evaluation of RQ2 in a generic way: * For case studies where we observe superaccurate performance (Imdb and SQuADv2) we use the thresholds leading to the highest performance, as well as the threshold corresponding to as few as possible remote predictions, leading to an overall performance equivalent to the one of the standalone remote model (denoted _remote-even_). * For case studies without superaccurate performance (Issue Classification and Imagenet) we choose thresholds for the 1st level supervisor corresponding to 30%, 50%, and 70% remote predictions, which gives a good coverage of the observed curves, by considering both a point (30%) with clearly reduced accuracy and one (70%) with comparable overall accuracy, compared to a standalone remote model. _Imdb Case Study._ The results for the Imdb base study are shown in Table 2. For all targeted false positive rates (FPR) and all tested local thresholds, i.e., for both the threshold leading to a remote-even system-level and best accuracy in RQ1, respectively, BiSupervised clearly outperforms the local baseline. The system-level acceptance rate ($\Delta$) is comparable to the local baseline in all cases, but the supervised accuracy ($\overline{obj}$) improves up to 11% when using BiSupervised. Thus, for all weightings of $\Delta$ and $\overline{obj}$, their arithmetic mean, i.e., $S_{\beta}$ also outperforms the baseline. _Github Issues and Imagenet Case Studies_. Same as for RQ1, the Github Issue and Imagenet case study show very similar results for RQ2 - presented in Tables 3 and 4 respectively. For all tested thresholds and different $\beta$ in $S_{\beta}$, BiSupervised clearly outperforms the baseline. Compared to a standalone supervised local model, the use of the remote model in BiSupervised not only improves the system-level supervised accuracy $\overline{obj}$, but in most cases it leads also to a higher system-level acceptance rate $\Delta$. _SQuADv2 Case Study._ Considering only SQuADv2's valid answers (Table 5), the results show a similar picture as for the Imdb case study in that BiSupervised consistently and clearly outperforms the baseline. The picture is less clear, however, for the case which regards all inputs, i.e., including invalid inputs for which it is impossible for any model to give a correct answer. Here, for 63% of the inputs, neither the local nor the remote model was able to make a correct prediction. The supervised performance of BiSupervised then depends on the targeted FPR: With a low FPR \begin{table} \begin{tabular}{l l c c c c c c} \hline \hline & & remote $\Delta$ & $\Delta$ & $\overline{obj}$ & $S_{0.5}$ & $S_{1}$ & $S_{2}$ \\ \cline{2-9} FPR & remote requests & & & & & & \\ \hline \multirow{3}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.98 & 0.80 & 0.83 & 0.88 & 0.94 \\ & 0.55 (remote-even) & 0.96 & 0.98 & 0.91 & 0.92 & 0.94 & 0.96 \\ & 0.68 (best) & 0.97 & 0.98 & 0.91 & 0.92 & 0.94 & 0.96 \\ \hline \multirow{3}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.93 & 0.82 & 0.83 & 0.87 & 0.90 \\ & 0.55 (remote-even) & 0.86 & 0.92 & 0.92 & 0.92 & 0.92 & 0.92 \\ & 0.68 (best) & 0.88 & 0.92 & 0.93 & 0.93 & 0.92 & 0.92 \\ \hline \multirow{3}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.86 & 0.83 & 0.83 & 0.85 & 0.86 \\ & 0.55 (remote-even) & 0.76 & 0.87 & 0.93 & 0.91 & 0.90 & 0.88 \\ \cline{1-1} & 0.68 (best) & 0.8 & 0.87 & 0.94 & 0.92 & 0.90 & 0.88 \\ \hline \hline \end{tabular} \end{table} Table 2. System-Level Supervised Assessment for imdb \begin{table} \begin{tabular}{l l c c c c c c} \hline \hline & & remote $\Delta$ & $\Delta$ & $\overline{obj}$ & $S_{0.5}$ & $S_{1}$ & $S_{2}$ \\ \cline{2-9} FPR & remote requests & & & & & \\ \hline \multirow{3}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.98 & 0.72 & 0.76 & 0.83 & 0.91 \\ & 0.30 & 0.94 & 0.98 & 0.80 & 0.83 & 0.88 & 0.94 \\ & 0.50 & 0.96 & 0.98 & 0.82 & 0.85 & 0.89 & 0.94 \\ & 0.70 & 0.97 & 0.98 & 0.83 & 0.86 & 0.90 & 0.95 \\ \hline \multirow{3}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.91 & 0.74 & 0.77 & 0.82 & 0.87 \\ & 0.30 & 0.75 & 0.92 & 0.82 & 0.84 & 0.87 & 0.90 \\ & 0.50 & 0.84 & 0.92 & 0.84 & 0.86 & 0.88 & 0.90 \\ & 0.70 & 0.88 & 0.92 & 0.86 & 0.87 & 0.89 & 0.91 \\ \hline \multirow{3}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.84 & 0.77 & 0.78 & 0.80 & 0.83 \\ & 0.30 & 0.55 & 0.86 & 0.82 & 0.83 & 0.84 & 0.86 \\ \cline{1-1} & 0.50 & 0.71 & 0.85 & 0.86 & 0.86 & 0.86 & 0.86 \\ \cline{1-1} & 0.70 & 0.79 & 0.85 & 0.88 & 0.87 & 0.87 & 0.86 \\ \hline \hline \end{tabular} \end{table} Table 3. System-Level Supervised Assessment for Issues of 0.01, BiSupervised still outperforms the local baseline as it succeeds in maintaining a higher acceptance rate $\Delta$ with similar or even mildly better system-level accuracy as the standalone local model. However, as we increase the targeted FPR, especially with an FPR of 0.1, the supervised accuracy using BiSupervised improves less than it does for the local model while maintaining a much higher system-level acceptance rate $\Delta$. Presumably, this is caused by the fact that we achieve a given target FPR by manipulating only the remote supervisor's threshold while keeping the 1st level supervisor's threshold constant: As such, we remove some correct remote predictions (quickly reaching the targeted FPR), while the wrong but accepted local predictions negatively impact $\overline{obj}$. Thus, in this case study, which of the approaches performs better in a supervised setting depends on the risk-adverseness of the user, i.e., the $\beta$ chosen in $S_{\beta}$: While BiSupervised, in the configuration tested by us, for FPR 0.1, outperforms the baseline for $S_{2}$, it fails to beat the baseline for $S_{0.5}$. Overall, SQuADv2 (including invalid inputs) is the only case study with results that are not conclusively \begin{table} \begin{tabular}{l l c c c c c c} \hline \hline & & remote $\Delta$ & $\Delta$ & $\overline{obj}$ & $S_{0.5}$ & $S_{1}$ & $S_{2}$ \\ FPR & remote requests & & & & & & \\ \hline \multirow{4}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.96 & 0.70 & 0.74 & 0.81 & 0.89 \\ & 0.30 & 0.93 & 0.98 & 0.81 & 0.84 & 0.89 & 0.94 \\ & 0.50 & 0.96 & 0.98 & 0.85 & 0.87 & 0.91 & 0.95 \\ & 0.70 & 0.97 & 0.98 & 0.86 & 0.88 & 0.92 & 0.95 \\ \hline \multirow{4}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.86 & 0.75 & 0.77 & 0.80 & 0.84 \\ & 0.30 & 0.73 & 0.92 & 0.83 & 0.85 & 0.87 & 0.90 \\ & 0.50 & 0.84 & 0.92 & 0.87 & 0.88 & 0.89 & 0.91 \\ & 0.70 & 0.89 & 0.92 & 0.87 & 0.88 & 0.90 & 0.91 \\ \hline \multirow{4}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.77 & 0.79 & 0.79 & 0.78 & 0.78 \\ & 0.30 & 0.53 & 0.86 & 0.84 & 0.84 & 0.85 & 0.86 \\ \cline{1-1} & 0.50 & 0.69 & 0.85 & 0.89 & 0.88 & 0.87 & 0.85 \\ \cline{1-1} & 0.70 & 0.78 & 0.85 & 0.90 & 0.89 & 0.87 & 0.86 \\ \hline \hline \end{tabular} \end{table} Table 4. System-Level Supervised Assessment for ImageNet \begin{table} \begin{tabular}{l l c c c c c c} \hline \hline & & remote $\Delta$ & $\Delta$ & $\overline{obj}$ & $S_{0.5}$ & $S_{1}$ & $S_{2}$ \\ FPR & remote requests & & & & & \\ \hline \multirow{4}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.97 & 0.64 & 0.69 & 0.77 & 0.88 \\ & 0.33 (remote-even) & 0.94 & 0.98 & 0.70 & 0.74 & 0.82 & 0.91 \\ & 0.58 (best) & 0.96 & 0.98 & 0.73 & 0.77 & 0.84 & 0.92 \\ \hline \multirow{4}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.90 & 0.67 & 0.70 & 0.77 & 0.84 \\ & 0.33 (remote-even) & 0.79 & 0.93 & 0.71 & 0.75 & 0.81 & 0.88 \\ & 0.58 (best) & 0.87 & 0.93 & 0.74 & 0.77 & 0.82 & 0.88 \\ \hline \multirow{4}{}{$\overline{obj}$} & 0.00 (baseline) & n/a & 0.81 & 0.70 & 0.72 & 0.75 & 0.79 \\ & 0.33 (remote-even) & 0.62 & 0.87 & 0.72 & 0.74 & 0.79 & 0.84 \\ \cline{1-1} & 0.58 (best) & 0.78 & 0.87 & 0.75 & 0.77 & 0.80 & 0.84 \\ \hline \hline \end{tabular} \end{table} Table 5. System-Level Supervised Assessment for SQuADv2 (possible only) in favor of BiSupervised, while still showing a positive tendency: Out of 18 assessed settings (target FPR, $S_{\beta}$ and 1st level supervisor threshold), BiSupervised shows a result outperforming the baseline in 13 cases; it has inferior $S_{\beta}$ scores in 5 cases. ### Threats _Internal validity threats:_ Two case studies, namely IMDB and SQuADv2, use GPT-3 as remote model, which was trained on a large (unknown) set of texts collected using web crawling. It is possible that these texts may contain entries from our test set (this is sometimes called _test set pollution_). First, we deem this unlikely and not different from a realistic usage of a large-scale remote model such as GPT-3. Thus, it is not expected to impact the validity of our result w.r.t. a real-world usage of BiSupervised. Second, for SQuADv2, on which GPT-3 was originally assessed, the authors measured the impact of test set pollution and found it to not have any major influence on the observed capabilities (Bauer et al., 2017). Motivated by our use case, in particular the possibility of runtime adaptive configuration of supervisor thresholds, we did not choose the thresholds for our evaluation before running the experiments. In RQ1, we perform a threshold-independent analysis (AUC-RAC) _and_ we also plot the curves. Smooth curves indicate that the selection of a threshold in practice is easy (no extreme local optima). Indeed, the curves we observed in RQ1 show no big fluctuations and are comparably smooth. Regarding RQ2, where a threshold independent metric is not possible, we chose a wide range of different thresholds for both supervisors, based on the findings of RQ1, and fixed FPR for the 2nd level supervisor. The likelihood that a majority of them would end up in small local optima (which would invalidate our results) is thus negligible. \begin{table} \begin{tabular}{l l c c c c c c} \hline \hline & & remote $\Delta$ & $\Delta$ & $\overline{obj}$ & $S_{0.5}$ & $S_{1}$ & $S_{2}$ \\ FPR & remote requests & & & & & & \\ \hline \multirow{3}{}{$\overline{\text{FPR}}$} & 0.00 (baseline) & n/a & 0.89 & 0.31 & 0.36 & 0.46 & 0.65 \\ & 0.51 (remote-even) & 0.93 & 0.96 & 0.32 & 0.37 & 0.48 & 0.68 \\ & 0.72 (best) & 0.94 & 0.96 & 0.33 & 0.38 & 0.49 & 0.69 \\ \hline \multirow{3}{}{$\overline{\text{FPR}}$} & 0.00 (baseline) & n/a & 0.76 & 0.35 & 0.39 & 0.48 & 0.61 \\ & 0.51 (remote-even) & 0.79 & 0.89 & 0.33 & 0.38 & 0.48 & 0.66 \\ & 0.72 (best) & 0.86 & 0.90 & 0.34 & 0.39 & 0.49 & 0.68 \\ \hline \multirow{3}{}{$\overline{\text{FPR}}$} & 0.00 (baseline) & n/a & 0.65 & 0.39 & 0.42 & 0.49 & 0.57 \\ & 0.51 (remote-even) & 0.62 & 0.81 & 0.34 & 0.39 & 0.48 & 0.64 \\ \cline{1-1} & 0.72 (best) & 0.78 & 0.84 & 0.34 & 0.39 & 0.49 & 0.65 \\ \hline \hline \end{tabular} \end{table} Table 6. System-Level Supervised Assessment for SQuADv2 (all) _Construct validity threats_: Quantification of the cost-accuracy trade off and of the supervisors' performance is not straightforward and no consolidated metrics exist for such task. We defined the AUC-RAC metric for the former quantification, as it provides a threshold independent quantification suitable for visual display as a request-accuracy-curve that can be easily interpreted by humans. For the latter we adopted the S-Beta score, as proposed in recent work about supervisor assessment (Wang et al., 2019). _External validity threats_: We considered only four case studies. To choose models representative of a variety of application domains, we included models designed for text classification, image classification and question answering. However, the validity of our findings beyond the considered case studies should be assessed empirically by replicating our study on different domains/models. ### Advantages beyond Cost Savings Besides the cost savings achieved by our architecture, making fewer remote predictions has a range of other advantages. Those are however very dependent on the used infrastructure, and we thus discuss them conceptually instead of empirically. _Lower Latency_. Latency of distributed systems, such as our architecture, depends on many factors such as network speed, bandwidth, used hardware, and system load. We can reasonably assume that requests over a network are time-wise quite costly and that large-scale DNNs take more time to make a prediction than small-scale DNNs. While the latter can, to some extent, be mitigated by using very fast hardware for the large-scale model, the former is an unavoidable truth using today's networks. In an abstract model of latency in BiSupervised, let $t_{l}$ be the average time required to locally preprocess the input, make a prediction on the local model and run the first-level supervisor. Similarly let $t_{r}$ denote the average time required to make a remote prediction (including time for network communication, and supervision of the remote prediction). Let $r$ be the rate of inputs for which BiSupervised triggers a remote prediction. BiSupervised then shows, on average, lower latency if \[t_{r}>t_{l}+rt_{r} \tag{2}\] This is true whenever $(1-r)t_{r}>t_{l}$. We measure the latencies induced from using local and remote models, which allows us to quantify potential latency improvements from using BiSupervised over using the remote model. All measurements include all required computation, including data preprocessing, inference, supervision, and network requests, where applicable. To collect our results, we used a workstation equipped with an AMD Ryzen Threadripper 1920X CPU and a wired internet connection. No GPU was used. Clearly, the measured times highly depend on this configuration, as well as additional influences such as remote server load. Thus, our reported times should be seen as an illustration, without a claim of generalizability. Here it is worth noting that, while we present the time measurements for all case studies, they are most meaningful for the Imdb and SQuADv2 case studies, which use a truly remote model accessed through the OpenAI API. For Github Issue classification and Imagenet, the large scale "remote" model was emulated locally, thus leading to no networking overhead, but also not leveraging dedicated hardware. We still report the latter results for completeness. For all case studies, our measurements confirm the huge potential BiSupervised shows with regards to latency: With a 1st-level supervisor configured such that less than 85% of requests are forwarded to the remote model, the average latency is lower than the baseline which uses only a remote model, thus fully compensating the additional overhead induced by the local inference and supervision. For Imagenet and SQuADv2 case studies, this break-even point lays even above 96%. Consequently, all evaluation points we discussed in RQ1 and RQ2 would reduce average latency in practice. Thus, BiSupervised not only saves cost, but it also reduces the average inference time, all while increasing robustness in a supervised setting and keeping the same accuracy or, in some cases, even increasing the system level accuracy. Lower Energy Consumption.Large models, naturally, require more energy to make predictions than small models, due to the substantially higher number of arithmetic operations to perform to make a single prediction. In addition, sending the request to a remote server also causes an energy consumption overhead. Thus, similar to the latency discussed above, we can expect the small overhead introduced by making a local inference and 1st level supervision (which are designed to be energy efficient), to be amortized by the lower number of remote requests even with a relatively low fraction of exclusively-local predictions. ## 6. Future Work At the moment, BiSupervised assumes that for all inputs where the local prediction is highly uncertain, the remote architecture potentially has a lower uncertainty. However, this may not always be the case: In some domains, _aleatoric uncertainty_, i.e., the uncertainty inherent in the input for which a prediction has to be made (such as noise or ambiguity) is a major cause of wrong predictions (Krishnan et al., 2017). Here, we notice that only in cases where the local model exhibits _epistemic uncertainty_, such as uncertainty caused by insufficient training data or model architecture, forwarding the input to a much larger, thus presumably more capable DNN is justified. For future work, we thus propose to attempt to discriminate between aleatoric and epistemic uncertainty in the local supervisor and only make remote predictions for inputs that show a high level of epistemic uncertainty, as opposed to a high level of general uncertainty. We conducted our experiments under nominal data, i.e., data that is mostly in-distribution of the training datasets. Reproduction of our results using outlier data to assess BiSupervised is needed to assess its robustness. \begin{table} \begin{tabular}{c c c c c c} \hline \hline Case Study & Local & Remote & Break Even & Eval Points (vs. Remote Only) & Remarks \\ \hline imdb & 0.05s & 0.32s & 85.68\% & 55\%: 0.23s (-30.7\%) & \\ & & & & 67\%: 0.26s (-18.5\%) & \\ & & & & 30\%: 0.41s (-61.5\%) & \\ Issues & 0.09s & 1.08s & 91.53\% & 50\%: 0.63s (-41.5\%) & Remote model emulated locally. \\ & & & & 70\%: 0.84s (-21.5\%) & \\ & & & & 30\%: 0.23s (-66.3\%) & Remote model emulated locally. \\ ImageNet & 0.02s & 0.68s & 96.35\% & 50\%: 0.36s (-46.3\%) & Both models use batched inference. \\ & & & & 70\%: 0.50s (-26.3\%) & \\ & & & & 33\%: 0.25s (-64.4\%) & \\ (possible only) & 0.02s & 0.71s & 97.42\% & 59\%: 0.44s (-38.1\%) & \\ & & & & 49\%: 0.39s (-47.6\%) & \\ & & & & 71\%: 0.55s (-25.9\%) & \\ \hline \hline \end{tabular} () Break even point: the fraction of remote calls at which the BiSupervised latency equals the latency of the remote-only approach. For any 1st level supervisor threshold leading to fewer remote predictions, BiSupervised is faster than using the standalone remote model. Hence, higher is better. \end{table} Table 7. Average Prediction and Supervision Latency per Input ## 7. Conclusion In this paper, we proposed BiSupervised, a novel architecture that addresses the challenge of using large-scale Deep Neural Networks (DNNs) from resource-constrained devices - a problem that is often solved by using costly and high-latency prediction on a remote server. Our proposed architecture combines a small-scale local model with a large-scale remote model, and includes two supervisors that monitor the prediction process, identifying inputs for which the local prediction can be trusted, and those that require the remote model to be invoked. We evaluated the effectiveness of BiSupervised on four diverse case studies, including Imdb movie review sentiment classification, Github issue triaging, Imagenet image classification, and SQuADv2 free-text question answering. Our results demonstrate that BiSupervised can achieve superaccurate performance for Imdb and SQuADv2, i.e., the accuracy achieved by BiSupervised exceeds the one from the large-scale remote model. For the other case studies, the remote inference cost can be reduced by up to 50% while maintaining a comparable system-level accuracy. To assess the ability of BiSupervised in a supervised setting, i.e., when identifying inputs for which not even the remote prediction can be trusted, we compared it against the standalone supervised local model and found that BiSupervised performs similar or better in the majority of the tested configurations. Furthermore, we measured the average prediction latency and showed that BiSupervised leads to reduced latency even with a small percentage of remote calls avoided, due to the large difference in latency between local and remote inference. Overall, our results demonstrate that BiSupervised can be a cost-effective and accurate solution for systems relying on large-scale remote models, particularly on resource-constrained devices. ## 8. Data Availability The full replication package for this paper is made available on Github, and archived on Zenodo: * [https://github.com/testingautomated-usi/bisupervised](https://github.com/testingautomated-usi/bisupervised) * [https://zenodo.org/record/7767383](https://zenodo.org/record/7767383) The replication package contains all scripts we used to calculate our results, as well as a Docker file allowing us to easily re-create the environment to run these scripts. The replication package also contains scripts to download and process third-party artifacts, such as models and datasets, automatically wherever possible. All our code and artifacts are permissively MIT licensed. A corresponding TOSEM replicated computational results (RCR) report is currently under review. ###### Acknowledgements. This work was partially supported by the H2020 project PRECRIME, funded under the ERC Advanced Grant 2017 Program (ERC Grant Agreement n. 787703).
2307.08167	Computing the gradients with respect to all parameters of a quantum neural network using a single circuit	When computing the gradients of a quantum neural network using the parameter-shift rule, the cost function needs to be calculated twice for the gradient with respect to a single adjustable parameter of the network. When the total number of parameters is high, the quantum circuit for the computation has to be adjusted and run for many times. Here we propose an approach to compute all the gradients using a single circuit only, with a much reduced circuit depth and less classical registers. We also demonstrate experimentally, on both real quantum hardware and simulator, that our approach has the advantages that the circuit takes a significantly shorter time to compile than the conventional approach, resulting in a speedup on the total runtime.	Guang Ping He	2023-07-16T22:35:52Z	http://arxiv.org/abs/2307.08167v2	Computing the gradients with respect to all parameters of a quantum neural network using a single circuit ###### Abstract When computing the gradients of a quantum neural network using the parameter-shift rule, the cost function needs to be calculated twice for the gradient with respect to a single adjustable parameter of the network. When the total number of parameters is high, the quantum circuit for the computation has to be adjusted and run for many times. Here we propose an approach to compute all the gradients using a single circuit only, with a much reduced circuit depth and less classical registers. We also demonstrate experimentally, on both real quantum hardware and simulator, that our approach has the advantages that the circuit takes a significantly shorter time to compile than the conventional approach, resulting in a speedup on the total runtime. Artificial intelligence technology is making incredible progress nowadays. One of the main reason is that classical artificial neural networks are made very feasible with the advance of computer science in recent years. On the other hand, though it is widely believed that quantum neural networks may breathe new life into the researches, they are still not as practical as their classical counterparts due to the scale and performance of currently available quantum computers. Especially, highly effective methods have been developed for computing the gradients of the cost function with respect to the adjustable parameters of classical neural networks, e.g., the back-propagation algorithm [1; 2]. A distinct feature is that the cost function needs to be calculated only once, and all the gradients will be readily deduced, so that classical neural networks can be trained fast and efficiently. But for quantum neural networks, computing the gradients is considerably less convenient [3]. Effective algorithms were found for certain network architectures only [4]. For other general architectures, the parameter-shift rule [5; 6] seems to be the best algorithm found for this task so far. But it needs to calculate the cost function twice for computing the gradient with respect to a single parameter of the network (which will be further elaborated below). The values of some parameters for each calculation need adjustment too. Consequently, the quantum circuit for calculating the cost function has to be modified many times in order to obtain the gradients with respect to all parameters in each single round of training of the quantum network. In this paper, we will propose an approach in which the gradients with respect to all parameters of a quantum neural network can be computing simultaneously using a single circuit only. The main purpose is to break the limit on the scale of computation on real quantum hardware. As it is well-known, the number of adjustable parameters of a powerful neural network is generally very high. If we have to run two quantum circuits to compute the gradient with respect to a single parameter, then the total number of unique executed circuits could easily exceed the capacity of any real quantum computer in existence. Of course, the execution of these circuits can be broken down into many "jobs". But it is obviously inconvenient when we are using quantum cloud computation platforms. Especially, if the jobs are submitted separately, then there could be a long waiting time between the execution of the jobs. For example, for free users on the IBM Quantum Experience online platform, this waiting time could vary from 10 minutes to 4 hours. For paying users, as also pointed out in Ref. [6], "quantum hardware devices are typically billed and queued per unique circuit". A conventional method for reducing the number of unique circuits is to "stack" many circuits into a single job. In the Qiskit software development kit for running IBM quantum computers and simulators with Python programs, this can be done using the "append" function [7]. But there is also a restriction on its usage on real quantum computers. For example, on the "ibmq_quito" backend, the number of circuits allowed to be appended is limited to 100. To break this limit, the Qiskit's "compose" function can serve as an alternative. According to our experience, it runs much slower than the "append" method does. But it can indeed stack more circuits into one single job so that the gradients with respect to more parameters can be computed together. In the following, we will use this direct stacking method based on the "compose" function (called "the conventional approach") as a reference, to compare with the performance of our proposed method that uses a single circuit only (called "the improved approach"). ## Results The variational quantum circuit There are various types of quantum neural networks. Among them, variational quantum circuits (VQCs) [8; 9; 10; 11; 12; 13] have the advantage that it can be executed on current or near-term quantum computers. In this paper, we will use it as an example to illustrate how our improved approach can be applied to computing the gradients with respect to its parameters. As shown in Fig.1(a), the gen eral structure of a VQC can be divided into three blocks: the feature map, the ansatz, and the measurement of the observable, which gives the value of the cost function. Note that the goal of this paper is to provide a method for computing the gradients, instead of finding a VQC for a specific application. To this end, the details of the feature map and the measurement are less relevant, and our approach does not need to modify them. Thus, we will leave their description in the Method section. The ansatz is the most important block, because it contains all the adjustable parameters with respect to whom the gradients need to be computed. Our example uses the _RealAmplitudes_ ansatz [14], which is the default ansatz in Qiskit's VQC implementation. As pointed out in Ref. [13], this ansatz has also been used in a VQC with proven advantages over traditional feedforward neural networks in terms of both capacity and trainability [10]. The typical structure of the _RealAmplitudes_ ansatz is illustrated in Fig.1(b). The parameter-shift rule Finding the gradients is a must for optimizing neural networks for certain applications. In our example, the original parameter-shift rule proposed in Ref. [5] will be used, where the exact gradient of the cost function $f=f(\vec{\theta})$ with respect to the parameters $\vec{\theta}=(\theta_{1},...\theta_{n})$ of the quantum gates is obtained as \[\frac{\partial f}{\partial\theta_{i}}=r\left(f\left(\vec{\theta}+s\vec{e}_{i} \right)-f\left(\vec{\theta}-s\vec{e}_{i}\right)\right) \tag{1}\] for $i=1,...,n$, where $r=1/2$ and $s=\pi/2$ for all the rotation and phase gates available in the Qiskit library. This equation shows that computing the gradient with respect to a single parameter $\theta_{i}$ needs to calculate the cost function $f$ twice, each time with a different set of parameters, i.e., $\vec{\theta}+s\vec{e}_{i}=(\theta_{1},...,\theta_{i}+s,...\theta_{n})$ and $\vec{\theta}-s\vec{e}_{i}=(\theta_{1},...,\theta_{i}-s,...\theta_{n})$. Therefore, when the total number of parameters of a VQC is $n$, computing all the gradients will have to run $2n$ circuits with different values of the parameters. To stack these circuits into a single job, the aforementioned conventional approach is simply to connect them one after another using Qiskit's "compose" function, as illustrated in Fig.2. Each circuit is independently executed and measured before the next one is re-initialized and parameterized with new values, and the measurement results are stored in different classical registers. Our improved approach Now it will be shown how to compute all the $2n$ shifted cost functions as well as the original unshifted cost function $f(\vec{\theta})$ using a single circuit. Two additional control qubits $q_{a}$ and $q_{b}$ need to be added to the circuit, regardless the number of qubits and parameters in the original ansatz. Then for each existing single-parameter gate $RY(\theta_{i})$ ($i=1,...,n$), we added two blocks to the qubit it rotates and the above two additional control qubits, to turn the parameter from $\theta_{i}$ to $\theta_{i}+s$ and $\theta_{i}-s$, respectively. Each block contains 2 controlled-RY gates, 1 CX (two-qubit controlled-NOT) gate, 1 measurement and 1 reset operation. Fig.3 showcases an example of the resultant circuit. For simplicity, only three RY gates in the _RealAmplitudes_ ansatz of the original VQC (corresponding to the three RY gates before any of the red dashed Figure 1: (a) The general structure of a variational quantum circuit (VQC). All qubits $q_{i}$ ($i=1,...,n$) are initialized in the state $\|0\rangle$, then pass through three blocks: the feature map, the ansatz and the observable, which apply certain unitary transformations. Finally they are measured in the computational basis, and the result is sent to the classical register $c$. All the adjustable parameters $\vec{\theta}=(\theta_{1},...\theta_{n})$ are contained in the ansatz. (b) An example of the _RealAmplitudes_ ansatz on three qubits, with two repetitions and full entanglement. $RY(\theta_{i})$ ($i=1,...,9$) are single-qubit RY rotation gates, with the adjustable parameter $\theta_{i}$ denoting the rotation angle about the $y$-axis. The blue icons are CX (controlled-NOT) gates. The red dashed boxes denote where the additional circuit of our improved approach shown in Fig.3 will be added. Figure 2: Diagram of the conventional approach, in which the circuits of many VQCs are stacked in serial using Qiskit’s “compose” function. Each black box contains all the components of the complete VQC shown in Fig.1(a), i.e., the feature map, the ansatz, and the observable along with the measurement. But the adjustable parameters $\vec{\theta}=(\theta_{1},...\theta_{n})$ for each ansatz take different values. boxes in Fig.1(b)) are studied, and shown with the two additional blocks added to each of them. The detailed workflow of the circuit is elaborated in the Method section. ### Theoretical advantages _a. Circuit depth_ Comparing with the conventional approach, a significant advantage of our improved approach is that the depth of the entire circuit is reduced, as estimated below. For each single input data, let $\Lambda$ denote the depth of the circuit for calculating the cost function once using the conventional approach, and $n$ denote the number of the adjustable parameters. Define \[\lambda=\frac{\Lambda}{n}. \tag{2}\] In many ansatz, $\lambda$ remains non-trivial even for high $n$. (We prove this in the Method section by using the _RealAmplitudes_ ansatz as an example.) Recall that computing the gradient with respect to a single parameter using the parameter-shift rule needs to calculate the cost function twice. Thus, the total depth of the stacked circuit for computing all the $n$ gradients and the unshifted cost function using the conventional approach is \[D_{conv}=\Lambda(2n+1)=\lambda n(2n+1)\propto O(n^{2}). \tag{3}\] On the other hand, when using our improved approach, from Fig.3 we can see that a total of 10 operations were added to each existing parameterized RY gate of the original ansatz. Therefore, to compute all the $n$ gradients and the unshifted cost function, the total depth of our circuit is simply \[D_{impr}=\Lambda+10n=\lambda n+10n\propto O(n). \tag{4}\] This result shows that our improved approach has the advantage that it takes a lower circuit depth when the number of parameters $n$ is high, and the improvement will be more significant with the increase of $n$. _b. Number of classical bits in the stacked circuits_ In the conventional approach, for each single input data, running the circuit for calculating the cost function once (either shifted or unshifted) takes $Q$ classical bits to store the final measurement result of the $Q$ qubits. When all the $2n+1$ circuits for computing all the $n$ gradients and the unshifted cost function are stacked together so that it can be submitted as a single job on the real hardware, the total number of required classical bits is \[N_{conv}=Q(2n+1). \tag{5}\] In our improved approach, the $Q$ qubits of the original ansatz are measured only once at the very end of the circuit. The additional qubit $q_{b}$ is measured twice in the two additional blocks added to each of the $n$ parameterized gates of the original ansatz, which takes $2n$ classical bits to store the results. At the end of the circuit, both the additional qubits $q_{a}$ and $q_{b}$ do not need to be measured. But for simplicity, we added these two measurements in our code too (not shown in Fig.3), which may also serve as an additional monitoring of the running of the circuit. Consequently, the total number of classical bits required in our approach is \[N_{impr}=Q+2n+2, \tag{6}\] which is smaller than that of the conventional approach since generally $Q\geq 2$. This is another advantage of our approach. ### Theoretical disadvantages Our improved approach requires two more qubits than the conventional approach. It is considerable on free Figure 3: The quantum circuit for our improve approach. The section inside the red dashed box can be put into any one of the three red dashed boxes in Fig.1(b). Here $q_{a}$ and $q_{b}$ are the additional control qubits. They need to be initialized in the states $\left\|1\right\rangle_{q_{a}}$ and $\left\|0\right\rangle_{q_{b}}$ only once at the very beginning of Fig.1(b). The two green/blue/purple dashed boxes will create the shift $\pm s$ to the gates $RY(\theta_{1})/RY(\theta_{2})/RY(\theta_{3})$, respectively, with equal probabilities as controlled by $q_{a}$ and $q_{b}$. Each of these dashed boxes starts with a controlled-RY gate $CRY(\gamma_{j})$, where $\gamma_{j}$ is defined in Eq. (16). Then a measurement is made on $q_{b}$, with the result recorded in the classical registers. Another controlled-RY gate $CRY(\pm s)$ is placed between $q_{b}$ and $q_{i}$ ($i=1,2,3$), followed by a CX gate between $q_{b}$ and $q_{a}$. Finally, $q_{b}$ is reset to the state $\left\|0\right\rangle_{q_{b}}$. quantum computation platforms, which generally offers a total of 5${}^{\text{-}}$6 qubits only. But on near-term quantum devices that could really play a role in practical applications, the number of qubits has to be much higher, so that taking two more qubits should not make much difference. What really matters is that our approach takes more shots than the conventional approach does in order to compute the gradients up to approximately the same precision. This is because in the conventional approach, when the stacked circuit is executed for $s$ shots, each of the shifted and unshifted cost functions is calculated for exactly $s$ shots too. But in our approach, when executing the circuit once, only one of the cost functions is calculated, depending on which of the additional blocks was activated by chance. To ensure that each cost function will be calculated for approximately $s$ shots, our circuit needs to be executed for about $s^{\prime}=s(2n+1)$ shots in total. This surely increases the runtime of the program. But recall that our circuit has a lower depth, which will take less time to compile. Therefore, whether the approach can result in a speedup on the total runtime depends on the competition between these two factors, and will be studied experimentally below. Also, in our approach whether an additional block will be activated is controlled by the measurement result of the qubit $q_{b}$, where quantum uncertainty takes effect. Consequently, executing the circuit for $s^{\prime}=s(2n+1)$ shots does not mean that each cost function will be calculated for $s$ shots exactly. Some statistical fluctuations are inevitable. Nevertheless, the relationship between the number of shots and the precision of the results is also statistical and subjected to quantum uncertainty. A fluctuation of the precision cannot be avoided even if all individual shot numbers are strictly aligned to the same value. Thus, we do not see the need to take extra efforts trying to level off the individual shot numbers of all the cost functions so that they equal exactly to each other. Experiments To compare the performance of our improved approach and the conventional one, we tested them experimentally on both real quantum hardware and simulator. First, the validity of our improved approach is checked using a VQC with 8-dimensional classical input data, which takes $Q=3$ qubits to encode when using the amplitude encoding method (see the Method section). The _RealAmplitudes_ ansatz in this VQC contains 1 repetition with full entanglement. Thus, it has $n=6$ adjustable parameters. As it is complicated to calculate analytically the exact amplitude distribution of the output states, we run the conventional approach for this VQC on quantum simulator for $s=3\times 10^{6}$ shots instead, and use the result of the gradients as the reference values, which should be a good approximation of the exact values. Then we run our improved approach for the same VQC for $s^{\prime}=500\times(2n+1)=6500$ shots, so that each cost function could be calculated for about $s=500$ shots in average (for simplicity, in the following when we say that the average number of shots for our improved approach is $s$, we mean that the actual number of shots is $s^{\prime}=s(2n+1)$). For comparison, the conventional approach is also run for another $s=500$ shots. The numbers of input data are all taken to be $m=20$ in these experiments. The results of the gradients are shown in Table 1. From the results we can see that when the (average) number of shots is $s=500$, both approaches result in similar precision when comparing with the reference values. (The precision for the gradient $grad(\theta_{2})$ with respect to the second adjustable parameter $\theta_{2}$ is low for both approaches, probably because the point happens to meet the barren plateau.) Thus, it is proven that the modified quantum circuit (i.e., Fig.3) in our improved approach works correctly as desired. Secondly, we studied the individual number of shots for each single cost function that was actually calculated in the above experiment using our improved approach. The result is shown in Fig.4. It is found that the fluctuation of the individual number of shots falls within the range $[445,587]$, with a standard deviation 22.6, which is only 4.52% of the average value $s=500$, indicating that all the cost functions were calculated to almost the same precision. Finally, the runtimes of both approaches are compared using 3 experiments: Exp.1: The VQC used in Table 1 and Fig.4, run on quantum simulator. Exp.2: The same VQC run on real quantum hardware (the "ibmq_quito" backend). Exp.3: A VQC with 784-dimensional classical input data, with the _RealAmplitudes_ ansatz containing $r=2$ repetitions and full entanglement, run on quantum simulator. Note that this VQC takes $Q=10$ qubits to encode each input data with the amplitude encoding method, so that the total number of adjustable parameters is $n=30$ \begin{table} \begin{tabular}{l\|l\|l\|l} \hline & Reference & Conventional & Improved \\ Gradient & value & approach & approach \\ \hline $grad(\theta_{1})$ & 0.1527 & 0.1589 & 0.1573 \\ $grad(\theta_{2})$ & -0.0029 & -0.0050 & 0.0010 \\ $grad(\theta_{3})$ & -0.2667 & -0.2691 & -0.2702 \\ $grad(\theta_{4})$ & -0.0836 & -0.0897 & -0.0841 \\ $grad(\theta_{5})$ & 0.1074 & 0.1064 & 0.1034 \\ $grad(\theta_{6})$ & -0.1812 & -0.1755 & -0.1800 \\ \hline \end{tabular} \end{table} Table 1: Precision comparison between the two approaches on the obtained value of the gradient $grad(\theta_{i})$ with respect to the adjustable parameter $\theta_{i}$ ($i=1,...,6$). The VQC takes 8-dimensional classical input data, with the _RealAmplitudes_ ansatz containing $r=1$ repetition and full entanglement. The number of input data is $m=20$. The reference values are obtained by running the conventional approach for $s=3\times 10^{6}$ shots. The values in the last two columns are both obtained by taking $s=500$. as given by Eq. (17). Let $t_{c}$ ($t_{r}$) denote the time spent on compiling (running) the circuit. The sum $t_{s}=t_{c}+t_{r}$ can serve as a good measure of the performance of the approaches, as the other parts of the computer programs (e.g., reading the input data and initial parameters, calculating the cost functions and gradients from the raw counting result of quantum simulator or real hardware, and exporting the results to the output files) run very fast and take almost the same amount of time for both approaches. The results are shown in Fig.5. The number of shots for each data point is fixed as $s=500$ (i.e., the actual number of shots is $s^{\prime}=500(2n+1)$ for our improved approach). The following observations can be found in all these experiments. (1) The values of $t_{r}$ show that our improved approach always takes more time to run than the conventional approach does, either on simulator or real hardware. The difference is about 3$\char 674$ times (in Exp.1 and Exp.2) to 24 times (in Exp.3). This is not surprising, because the improved one needs to be run for more shots to reach the same precision. (2) On the contrary, the values of $t_{c}$ show that our improved approach saves a tremendous amount of time for compiling the circuit, which is only about 4% (in Exp.1 and Exp.2) to 0.1% (in Exp.3) of that of the conventional approach. This is owed to the significantly reduced circuit depth, as shown in Eq. (4). (3) As the result of the competition between $t_{c}$ and $t_{r}$, the total runtime $t_{s}$ of the conventional approach is shorter when $m$ (the number of input data) is low. But as $m$ increases, our improved approach will eventually become faster. In Exp.1, the turning point occurs at around $m=15$ when both approaches take about 8 seconds to complete. After that, $t_{s}$ of our approach increases approximately in a linear way, while $t_{s}$ of the conventional one rises dramatically, mostly due to the long compiling time $t_{c}$. In both Exp.2 and Exp.3, the same behavior can be observed. The turning points occur relatively later though. This is because the real quantum hardware used in Exp.2 runs about 10$\char 670$30 times slower than the simulator used in Exp.1, and Exp.3 has a more complicated circuit than that of Exp.1 though it is also run on simulator. Consequently, both Exp.2 and Exp.3 have a longer $t_{r}$ than that of Exp.1. Meanwhile, all the three experiments have similar $t_{c}$ values because the compiling of the circuits are always done on classical computers. Therefore, the slower the quantum hardware or simulator is, the later the turning point of the total runtime $t_{s}$ occurs. In the age of big data, for real applications of neural networks, the number of input data is generally much higher than what was used here. As a result, we can see that our improved approach has the advantage that it can save the total runtime. This advantage is expected to be even more obvious with the advance of real quantum computers, because they will surely run faster in the future than they do today, resulting in a shorter $t_{r}$ so that the turning point of $t_{s}$ could occur even earlier. It is also worth noting that if Qiskit's "append" function were used for stacking the circuits, the "ibmq_quito" real quantum backend only allows 100 circuits to be appended in a single job. Since the VQC in Exp.2 has 6 adjustable parameters, there are totally 13 shifted and unshifted cost functions to be calculated for each single input data. It means that using the "append" function can handle only $m=[100/13]=7$ input data at the most. But Exp.2 (i.e., Fig.5(b)) shows that both the conventional approach (using the "compose" function) and our improve approach can accomplish the computation for at least $m=140$ input data in a single job with no problem. Thus, they manage to break the limit on real quantum hardware. ## IV Discussion In summary, it is proven experimentally that our improved approach manages to increase the number of gradients that can be computed in each single job, breaking the limit that can be reached using Qiskit's "append" function. More importantly, it has a smaller circuit depth and requires less classical registers for storing the measurement results when comparing with the conventional approach using Qiskit's "compose" function. This shortens the time spent for compiling the quantum circuit significantly. Therefore, the total runtime of the program can also be saved, especially when the number of input data is high and the structure of the quantum circuit is complicated. Though we only demonstrated our improved approach for VQCs where each parameterized quantum gate $U(\theta)$ Figure 4: Individual number of shots actually computed for each cost function in our improved approach, corresponding to the experiment shown in the last column of Table 1. The average number of shots is taken to be $s=500$. The number of input data is $m=20$. The cost function with index 1 is the unshifted cost function, while the other 12 cost functions are all shifted ones. The dots with the same color represent the number of shots for the same input data point. satisfies \[U(\theta+s)=U(\theta)U(s) \tag{7}\] (i.e., the gates covered by the original parameter-shift rule [5]), it can also be modified to compute the derivatives for any other quantum gate that requires the general parameter-shift rules [6]. For example, suppose that the first parameterized gate $RY(\theta_{1})$ on the qubit $q_{1}$ in Fig.3 is replaced by a more general gate $U^{\prime}(\theta_{1})$ that does not satisfy Eq. (7). To compute $U^{\prime}(\theta+s)$, all we need is to replace the controlled-RY gate $CRY(s)$ (controlled by $q_{b}$ and acting on $q_{1}$) in the first block right behind $RY(\theta_{1})$ in Fig.3 by two controlled-gates $(U^{\prime}(\theta_{1}))^{-1}$ and $U^{\prime}(\theta_{1}+s)$ in serial, both of which are also controlled by $q_{b}$ and acting on $q_{1}$. Here $(U^{\prime}(\theta_{1}))^{-1}$ is the reverse operation of $U^{\prime}(\theta_{1})$ such that $(U^{\prime}(\theta_{1}))^{-1}U^{\prime}(\theta_{1})=I$. We can see that once the measurement on $q_{b}$ before them (i.e., the measurement operator in the first block of Fig.3) results in $\left\|1\right\rangle_{q_{b}}$, then these two gates will be activated. Combining with the original $U^{\prime}(\theta_{1})$ gate, the complete operation on the qubit $q_{1}$ will be $U^{\prime}(\theta_{1}+s)(U^{\prime}(\theta_{1}))^{-1}U^{\prime}(\theta_{1})=U ^{\prime}(\theta_{1}+s)$. Thus, the shift of such a general parameterized quantum gate can also be computed with our approach. Our approach also suggests the following improvement on real quantum computers. From Fig.3 we can see that the purpose of the first two operations in each block (the controlled-RY gate $CRY(\gamma_{j})$ and the measurement) is to turn the state of $q_{b}$ into $\left\|1\right\rangle_{q_{b}}$ with a certain probability. If there is a classical random number generator (even if it generates pseudorandom numbers only), it can accomplish the same task while further reducing the circuit depth. Also, the next two quantum controlled gates make use of the states $\left\|0\right\rangle$ and $\left\|1\right\rangle$ of $q_{a}$ and $q_{b}$ only, without needing their quantum superpositions. They can be replaced by classical IF gates (i.e., controlled gates that take classical control bits instead of control qubits, while the target registers can be either quantum or classical) and both $q_{a}$ and $q_{b}$ can be classical registers too, so that less quantum resource is required. Unfortunately, classical random number generator and the IF gate are currently unavailable on real quantum hardware (the latter is available on the simulator though). Our approach shows that there is the need for adding these operations in the future, so that the performance of real quantum computers for certain tasks can be further improved. ## Methods The input data sets In our experiments, the 8-dimensional classical input data set is generated randomly. That is, each data point $x$ contains 8 random numbers $\{x_{1},...,x_{8}\}$ uniformly distributed over the interval $[0,1)$. The 784-dimensional classical input is taken as the Modified National Institute of Standards and Technology (MNIST) data set [15]. It is a widely-used resource for machine learning research [16], which contains 70000 Figure 5: Runtimes as a function of the number of input data $m$. The dashed lines represent the time $t_{c}$ spent on compiling the circuit, the dotted lines represent the time $t_{r}$ spent on running the circuit, and the solid lines represent the total runtime $t_{s}=t_{c}+t_{r}$. All blue lines stand for the conventional approach, and all green lines stand for our improved approach. (a) Exp.1: The VQC used in the experiment in Fig.4, with 8-dimensional classical input data and the _RealAmplitudes_ ansatz containing $r=1$ repetition and full entanglement, run on quantum simulator. (b) Exp.2: The same VQC as that of Exp.1, run on real quantum hardware. (c) Exp.3: A VQC with 784-dimensional classical input data, with the _RealAmplitudes_ ansatz containing $r=2$ repetitions and full entanglement, run on quantum simulator. Note that the green solid and dotted lines almost overlap with each other since $t_{c}\gg t_{r}$ so that $t_{s}\simeq t_{c}$ for our improved approach in this case. greyscale $28\times 28=784$ pixel images of handwritten digits $0\hbox{${}^{\circ}$}9$. The feature map Our feature map uses the amplitude encoding method [17], where each $d$-dimensional classical input data point $x=\left\{x_{1},...,x_{d}\right\}$ is encoded as the amplitudes of a $Q$-qubit quantum state \[\left\|\Phi_{x}\right\rangle=C_{norm}\sum_{i=1}^{d}x_{i}\left\|i\right\rangle. \tag{8}\] Here $d=2^{Q}$, $\left\|i\right\rangle$ is the $i$th computational basis state, and $C_{norm}$ is the normalization constant. The cost function Though the form of the cost function plays an important role on the overall performance of a neural network, it is less relevant to the current work because our goal is to test the runtime of the improved approach only, which depends solely on the structure of the ansatz. Therefore, the cost function of our networks (either using the improved approach or the conventional one) is simply taken as \[Cost=\frac{1}{m}\sum_{x}\left\|a-y(x)\right\| \tag{9}\] where $m$ is the number of input data, $a$ is the vector of outputs from the network when $x$ is input, and $y(x)$ is the desired output, i.e., the label of the input $x$, as contained in the MNIST data set. Like the MNIST data set, we also give each data in our 8-dimensional input dataset a label, so that our program can serve as a classifier if needed. But since our current data are generated randomly, we simply label all data as "2" without actual meaning. Workflow of the circuit of our improved approach Here we explain in details how the circuit in Fig.3 works. The role of the first additional control qubit $q_{a}$ is to record whether any of the additional blocks has been activated. If it is in the state $\left\|\psi\right\rangle_{q_{a}}=\left\|1\right\rangle_{q_{a}}$, then it means that no block has ever been activated yet. Or if it is in the state $\left\|\psi\right\rangle_{q_{a}}=\left\|0\right\rangle_{q_{a}}$, then it implies that one of the blocks was already activated. The second additional control qubit $q_{b}$ acts like a quantum dice, which controls the probability for each block to be activated. At the very beginning of the whole VQC, $q_{a}$ and $q_{b}$ are initialized in the states $\left\|1\right\rangle_{q_{a}}$ and $\left\|0\right\rangle_{q_{b}}$, respectively. Next, in the first green block right behind the first RY gate $RY(\theta_{1})$, the first gate $CRY(\gamma_{0})$ is a controlled-RY gate with $q_{a}$ ($q_{b}$) serving as the control (target) qubit. It introduces a rotation angle \[\gamma_{0}=2\arcsin\sqrt{\frac{1}{N-0}} \tag{10}\] about the $y$-axis to $q_{b}$ only when $\left\|\psi\right\rangle_{q_{a}}=\left\|1\right\rangle_{q_{a}}$. Here $N=2n+1$. Then the state of $q_{b}$ becomes \[\left\|\psi\right\rangle_{q_{b}} = \cos\frac{\gamma_{0}}{2}\left\|0\right\rangle_{q_{b}}+\sin\frac{ \gamma_{0}}{2}\left\|1\right\rangle_{q_{b}} \tag{11}\] \[= \sqrt{1-\frac{1}{N-0}}\left\|0\right\rangle_{q_{b}}+\sqrt{\frac{1 }{N-0}}\left\|1\right\rangle_{q_{b}}.\] A measurement in the computational basis $\left\{\left\|0\right\rangle,\left\|1\right\rangle\right\}$ is then made to $q_{b}$, and the result is sent to a classical register. With probability $1/N$ the measurement result will be $\left\|1\right\rangle_{q_{b}}$. In this case, the controlled-RY gate $CRY(s)$ next to the measurement will be activated, giving a rotation angle $s=\pi/2$ to $q_{1}$ (i.e., the qubit on which the gate $RY(\theta_{1})$ of the original ansatz is applied). Since the RY gates has the property \[RY(s)RY(\theta_{1})=RY(\theta_{1}+s), \tag{12}\] we can see that a shift $s$ is successfully introduced to the parameter $\theta_{1}$. Meanwhile, with the next CX gate where $q_{b}$ ($q_{a}$) serves as the control (target) qubit, the state of $q_{a}$ is turned from $\left\|1\right\rangle_{q_{a}}$ into $\left\|0\right\rangle_{q_{a}}$, indicating that the current block was activated. Also, at the end of this block, the state of $q_{b}$ is reset to $\left\|0\right\rangle_{q_{b}}$ regardless the measurement result. Consequently, all the rest blocks that follows will be bypassed. After executing the entire circuit, the final result will give the cost function $f\left(\vec{\theta}+s\vec{e_{1}}\right)$. On the other hand, with probability $1-(1/N)$ the measurement result on $q_{b}$ in this block will be $\left\|0\right\rangle_{q_{b}}$, then the two aforementioned controlled gates in the same block following this measurement will not take effect at all, so that the states of $q_{a}$ and $q_{1}$ remain unchanged, allowing the second green block to be activated. Similarly, in the second green block, since the state of $q_{b}$ was reset to $\left\|0\right\rangle_{q_{b}}$, the first gate $CRY(\gamma_{1})$ introduces a rotation angle \[\gamma_{1}=2\arcsin\sqrt{\frac{1}{N-1}} \tag{13}\] to $q_{b}$ when $\left\|\psi\right\rangle_{q_{a}}=\left\|1\right\rangle_{q_{a}}$, turning the state of $q_{b}$ into \[\left\|\psi\right\rangle_{q_{b}} = \cos\frac{\gamma_{1}}{2}\left\|0\right\rangle_{q_{b}}+\sin\frac{ \gamma_{1}}{2}\left\|1\right\rangle_{q_{b}} \tag{14}\] \[= \sqrt{1-\frac{1}{N-1}\left\|0\right\rangle_{q_{b}}}+\sqrt{\frac{1 }{N-1}}\left\|1\right\rangle_{q_{b}}.\] Then $q_{b}$ is also measured in the computational basis, and the result is stored in another classical register. With probability $1/(N-1)$ the result will be $\left\|1\right\rangle_{q_{b}}$. Note that this case occurs only when the first block was not activated, i.e., the measurement result of $q_{b}$ in the first block was $\left\|0\right\rangle_{q_{b}}$, which occurred with probability $1-(1/N)$. Thus, the total probability for finding $q_{b}$ in $\left\|1\right\rangle_{q_{b}}$ at this stage (i.e., the second block is activated) is \[prob_{2}=\left(1-\frac{1}{N}\right)\frac{1}{N-1}=\frac{1}{N}, \tag{15}\] which equals to the probability for activating the first block. When this result indeed occurs, the next gate $CRY(-s)$ will introduce the rotation angle $-s=-\pi/2$ to $q_{1}$, and the CX gate next to it will turn the state of $q_{a}$ into $\left\|0\right\rangle_{q_{a}}$, making the rest blocks bypassed. Then the final result of the entire circuit will be the cost function $f\left(\vec{\theta}-s\vec{e_{1}}\right)$. For the same reasons, by setting the rotation angle of the first controlled-RY gate $CRY(\gamma_{j})$ in the $(j+1)$th additional block as \[\gamma_{j}=2\arcsin\sqrt{\frac{1}{N-j}} \tag{16}\] ($j=0,...,N-2$), it can be proven that every block stands the same probability $1/N$ to be activated. The CX gates on $q_{b}$ and $q_{a}$ also ensure that only one block at the most will be activated. As a result, executing the whole circuit for one shot will output the shifted cost function $f\left(\vec{\theta}+s\vec{e_{j}}\right)$ or $f\left(\vec{\theta}-s\vec{e_{j}}\right)$ for a single $j$ only. When none of the block was activated (which also occurs with probability $1/N$), the circuit outputs the unshifted cost function $f\left(\vec{\theta}\right)$. By reading the classical registers $c$ that store the measurement results of $q_{b}$ in every block to see which block was activated, we can learn which cost function was calculated. Running the circuit for many shots will then provide enough shots for each of these cost functions, so that all gradients in Eq. (1) can be obtained with the desired precision. Proving that $\lambda$ in Eq. (2) is non-trivial In the _RealAmplitudes_ ansatz, let $Q$ denote the total number of the qubits. When $Q$ is fixed, increasing $n$ means increasing the number of repetition $r$ in this ansatz. This is because the ansatz starts with a parameterized RY gate for each of the $Q$ qubits, then each repetition adds another parameterized RY gate to each qubit. Thus, the total number of the parameters satisfies \[n=Q(r+1). \tag{17}\] Suppose that the circuit depth of each additional repetition (including the RY gates and the CX gates in the entanglement section) is $\Delta$. Then \[\Lambda=\Lambda_{0}+r\Delta=\Lambda_{0}+(\frac{n}{Q}-1)\Delta=(\Lambda_{0}- \Delta)+n\frac{\Delta}{Q}, \tag{18}\] where $\Lambda_{0}$ denotes the depth of the other parts of the circuit which is not included in $\Delta$, e.g., the feature map, the observable, the final measurement, and the very first RY gate on each qubit within the ansatz. In this case, we yield \[\lambda=\frac{(\Lambda_{0}-\Delta)+n\frac{\Delta}{Q}}{n}=\frac{(\Lambda_{0}- \Delta)}{n}+\frac{\Delta}{Q}. \tag{19}\] A neural network that is useful in practice generally has a large number of parameters. When $n\rightarrow\infty$ we have $\lim_{n\rightarrow\infty}\lambda=\Delta/Q$. Thus it is proven that $\lambda$ stays non-trivial. Program details The programs for 8-dimensional input data using quantum simulator were run on a personal laptop with a 2.8 GHz quad-core Intel i7-3840QM processor and 16 GB $1,600$ MHz DDR3 memory. The programs for 784-dimensional input data using quantum simulator were run on a Tianhe-2 supercomputer node with a 2.2 GHz Intel Xeon E5-2692 v2 processor and 64 GB memory. In these simulations, Qiskit's "qasm_simulator" backend was used as the quantum simulator. All programs using real hardware were run on the "ibmq_quito" 5-qubit backend of IBM Quantum Experience online platform. When running on quantum simulator, both $t_{c}$ (the time spent on compiling the quantum circuit) and $t_{r}$ (the time spent on running the circuit) are recorded automatically by the programs. But when running on real quantum hardware, $t_{r}$ is obtained from the job output files downloaded from IBM Quantum Experience platform instead. This is because the value of $t_{r}$ recorded by the programs in this case also includes the time spent on waiting in the queue for the job to be run, which is not the correct measure for the performance of the approaches being used. In both approaches, the circuits for different input data are stacked together using Qiskit's "compose" function. The initial values of all adjustable parameters of the VQCs were generated randomly (uniformly distributed over the interval $[0,\pi)$) and stored beforehand. Then we always loaded these same values into every computer programs in every run. Data availability The raw experimental data of this study are publicly available at [https://github.com/gphehub/grad2210](https://github.com/gphehub/grad2210). Code availability The software codes for generating the experimental data are publicly available at [https://github.com/gphehub/grad2210](https://github.com/gphehub/grad2210).
2302.04183	Cooperative Beamforming and RISs Association for Multi-RISs Aided Multi-Users MmWave MIMO Systems through Graph Neural Networks	Reconfigurable intelligent surface (RIS) is considered as a promising solution for next-generation wireless communication networks due to a variety of merits, e.g., customizing the communication environment. Therefore, deploying multiple RISs helps overcome severe signal blocking between the base station (BS) and users, which is also a practical and effective solution to achieve better service coverage. However, reaping the full benefits of a multi-RISs aided communication system requires solving a non-convex, infinite-dimensional optimization problem, which motivates the use of learning-based methods to configure the optimal policy. This paper adopts a novel heterogeneous graph neural network (GNN) to effectively exploit the graph topology in the wireless communication optimization problem. First, we characterize all communication link features and interference relations in our system with a heterogeneous graph structure. Then, we endeavor to maximize the weighted sum rate (WSR) of all users by jointly optimizing the active beamforming at the BS, the passive beamforming vector of the RIS elements, as well as the RISs association strategy. Unlike most existing work, we consider a more general scenario where the cascaded link for each user is not fixed but dynamically selected by maximizing the WSR. Simulation results show that our proposed heterogeneous GNNs perform about 10 times better than other benchmarks, and a suitable RISs association strategy is also validated to be effective in improving the quality services of users by 30%.	Mengbing Liu, Chongwen Huang, Marco Di Renzo, Merouane Debbah, Chau Yuen	2023-02-08T16:41:59Z	http://arxiv.org/abs/2302.04183v1	Cooperative Beamforming and RISs Association for Multi-RISs Aided Multi-Users MmWave MIMO Systems through Graph Neural Networks ###### Abstract Reconfigurable intelligent surface (RIS) is considered as a promising solution for next-generation wireless communication networks due to a variety of merits, e.g., customizing the communication environment. Therefore, deploying multiple RISs helps overcome severe signal blocking between the base station (BS) and users, which is also a practical and effective solution to achieve better service coverage. However, reaping the full benefits of a multi-RISs aided communication system requires solving a non-convex, infinite-dimensional optimization problem, which motivates the use of learning-based methods to configure the optimal policy. This paper adopts a novel heterogeneous graph neural network (GNN) to effectively exploit the graph topology in the wireless communication optimization problem. First, we characterize all communication link features and interference relations in our system with a heterogeneous graph structure. Then, we endeavor to maximize the weighted sum rate (WSR) of all users by jointly optimizing the active beamforming at the BS, the passive beamforming vector of the RIS elements, as well as the RISs association strategy. Unlike most existing work, we consider a more general scenario where the cascaded link for each user is not fixed but dynamically selected by maximizing the WSR. Simulation results show that our proposed heterogeneous GNNs perform about 10 times better than other benchmarks, and a suitable RISs association strategy is also validated to be effective in improving the quality services of users by 30$\%$. Beamforming, graph neural network, reconfigurable intelligent surfaces, RISs association strategy. ## I Introduction Multiple-input multiple-output (MIMO) technology continues to play an indispensable role in providing high spectral efficiency in the current and next-generation wireless systems [1]. Although MIMO technology can bring significant antenna gains, the signal transmitted at millimeter wave (mmWave) bands experiences serious propagation attenuation. Moreover, due to the existence of obstacles in the environment, MIMO technology cannot provide seamless coverage between transceivers [2]. As a remedy, reconfigurable intelligent surface (RIS) is considered as a promising technology to improve the links' quality of service as well as recover communications in the dead zone. Compared with other technologies, such as ultra dense networks, RIS is regarded as a promising paradigm to assist mmWave MIMO networks via a large number of effective and low-cost reflective elements [3, 4, 5]. Due to the benefits above, RIS has been introduced to various communication systems to balance the network capacity and energy consumption. To fully reap the benefits, the cooperative beamforming design on base stations (BSs) with MIMO and RISs is a crucial challenge since many beamforming designs for the distributed RISs system are non-convex and heavily computational. Thus, a great deal of efforts have been devoted to developing effective algorithms to tackle this challenge in RIS-aided systems [6, 7]. Recently, the cooperative beamforming design in multi-RISs aided system has been investigated to compensate for the coverage hole and the serious propagation attenuation in high-frequency bands [8]. However, these problem-specific algorithms are laborious processes and require much problem-specific knowledge, especially in multi-RISs aided systems. Inspired by the recent successes of deep learning technology in many application domains, e.g., the computer vision [9], deep neural network (DNN) models have been utilized for RIS-aided communication systems in recent years [10, 11]. However, such DNN models may not be very suitable for the problems in wireless communications. This is because DNNs fail to capture the underlying topology of wireless communication networks, resulting in dramatic performance degradation when the network size becomes large. Thus, in order to improve the scalability and generalization of the network, a long-standing idea is to incorporate the structures of the target task into the neural network architecture [12]. From this perspective, the graph neural network (GNN) model has emerged as a new direction to exploit the graph topology of wireless communication networks [13]. There are some works applying the GNN for resource allocation in device-to-device (D2D) communication networks [14, 15]. In these works, the service links and interference relations are modeled as nodes and edges, respectively. Based on these graph embedding features, the optimization problem on the weighted sum rate (WSR) can be solved efficiently. In this paper, we leverage the advantage of the GNN to solve the beamforming design in a multi-RISs aided communication system. Unfortunately, the modeling method for above D2D systems is not suitable for our system. In D2D systems, the transceiver pair (including one transmitter and one receiver) is denoted as one node without interference inside, and the interference in the system only exists between different transceiver pairs. However, we consider the communication links between one transmitter (BS) and multiple receivers (users) in this paper, where the inter-user interference is included. Thus, a new modeling approach is needed to better map the multi-RISs aided system into the graph. Aside from the beamforming design, we consider a more practical and general problem in the multi-RISs scenario, assuming that each user only selects one of the RISs to enhance the links' service quality. In fact, selecting all RISs is not an efficient way to assist each user, because RISs are generally deployed in different positions. The distances between the user and some RISs may be so far that the signals reflected by the RISs are severely attenuated. For these reasons, the RISs association is introduced into this problem which needs to be jointly optimized with the cooperative beamforming at both the BS and RISs to obtain better performance. Thus, our main contributions in this work are concluded as 1) We first propose a new GNN structure to map the optimization problem in a multi-RISs aided system and validate its effectiveness through extensive experiments. 2) A RISs association strategy is considered in our optimization problem. By comparing the results obtained with fully connected RISs, we find that the strategy can efficiently improve the quality services of users by $30\%$. 3)In order to exhibit the benefits of applying structure information, DNNs with different network sizes are adopted as benchmarks. It is clear that our proposed heterogeneous GNNs perform about 10 times better than DNNs. ## II Multi-RISs Aided Massive MIMO Communication Networks In this section, we introduce a multi-RISs aided MIMO system model with a RISs association strategy. ### _System Model_ We consider a multi-RISs aided system with a $N_{t}$-antenna BS, a set of single-antenna users $\mathcal{K}=\{1,2,\cdots,K\}$ and a set of RISs $\mathcal{R}=\{1,2,\cdots,R\}$. Each RIS is equipped with $M=M_{x}\times M_{y}$ reflective elements, where $M_{x}$ and $M_{y}$ represent the the number of horizontal and vertical elements, respectively. Assume that the BS transmits $K$ data streams to users through the beamforming architecture. Thus, the transmitted signal $\mathbf{x}$ from the BS can be expressed as \[\mathbf{x}\triangleq\mathbf{F}\mathbf{s}=\sum_{k=1}^{K}\mathbf{f}_{k}s_{k}, \tag{1}\] where $\mathbf{s}=[s_{1},\cdots,s_{K}]^{T}\in\mathbb{C}^{K\times 1}$ indicates the symbol vector and $\mathbb{E}(\mathbf{s}\mathbf{s}^{H})=\mathbf{I}_{K}$. $\mathbf{F}=[\mathbf{f}_{1},\cdots,\mathbf{f}_{K}]\in\mathbb{C}^{N_{t}\times K}$ denotes the beamforming matrix at the BS and the power constraint is denoted by $\left\\|\mathbf{F}\right\\|_{\mathcal{F}}^{2}\leq P_{max}$, where $P_{max}$ is the maximum transmit power by the BS. It is crucial to investigate the joint design of RISs association, BS beamforming, and the phase configuration of RISs. In this paper, we choose one of all RISs for each user to maximize the WSR of the multi-RISs aided massive MIMO system. To simply express the association between RISs and users, a matrix $\mathbf{U}=[\mathbf{u}_{1},\cdots,\mathbf{u}_{K}]^{T}\in\mathbb{C}^{K\times R}$ consisting of binary variables $u_{k,i}$ is defined as \[u_{k,i}=\left\{\begin{array}{l}1,\text{user}_{k}\text{ is assisted by RIS}_{i}.\\ 0,\text{ otherwise}.\end{array}\right. \tag{2}\] Thus, based on the RISs association strategy, the equivalent channel $\mathbf{h}_{k}$ from the BS to user${}_{k}$ can be expressed as \[\mathbf{h}_{k}=\sum_{i\in\mathcal{R}}u_{k,i}\mathbf{h}_{ik}\mathbf{\Phi}_{i} \mathbf{G}_{i}, \tag{3}\] where $\mathbf{G}_{i}\in\mathbb{C}^{M\times N_{t}}$ and $\mathbf{h}_{ik}\in\mathbb{C}^{1\times M}$ denote the BS-RIS${}_{i}$ and RIS${}_{i}$-user${}_{k}$, respectively. $\mathbf{\Phi}_{i}=\mathrm{diag}\left(\theta_{i,1},\cdots,\theta_{i,M}\right)\in \mathbb{C}^{M\times M}$ with the diagonal element $\theta_{i,m}=e^{j\phi_{i,m}}$ denoting the phase shift matrix of RIS${}_{i}$. According to above assumption, the signal to interference plus noise ratio (SINR) at user${}_{k}$ can be given as \[\mathrm{SINR}_{k}=\frac{\left\|\mathbf{h}_{k}\mathbf{f}_{k}\right\|^{2}}{\sum_{ j\in\mathcal{K},j\neq k}^{K}\left\|\mathbf{h}_{k}\mathbf{f}_{j}\right\|^{2}+ \sigma^{2}}, \tag{4}\] where $\sigma^{2}$ denotes the effective noise variance. Fig. 1: Illustration of the multi-RISs aided massive MIMO system model. ### _Channel Model_ In this paper, we consider the mmWave channel and adopt the typical Saleh-Valenzuela channel model [16] with limited scattering paths to characterize the sparse nature. Supposing a uniform linear array (ULA) is equipped at the BS and a uniform planar array (UPA) is equipped at each RIS. Hence, we can get the array response vectors by the following equations \[\mathbf{a}_{\mathrm{ULA}}(\phi) =\frac{1}{\sqrt{N}}\left[e^{-j\frac{2\pi d}{\lambda}\phi_{i}} \right]_{i\in\mathcal{I}(N)}, \tag{5}\] \[\mathbf{a}_{\mathrm{UPA}}(\phi^{1},\phi^{2}) =\mathbf{a}_{\mathrm{ULA}}^{\mathrm{x}}(\phi^{1})\otimes\mathbf{ a}_{\mathrm{ULA}}^{\mathrm{y}}(\phi^{2}), \tag{6}\] where $\lambda$ and $d$ represent the signal wavelength and the antenna spacing, respectively. As for a ULA with $N$ antennas, $\phi$ denotes the angle of arrival (AoA) or the angle of departure (AoD). For a UPA with $M=M_{y}\times M_{x}$ elements, $\phi^{1}$ and $\phi^{2}$ denote the azimuth and elevation angles, respectively. $\mathbf{a}_{\mathrm{ULA}}^{\mathrm{x}}(\phi^{1})$ and $\mathbf{a}_{\mathrm{ULA}}^{\mathrm{y}}(\phi^{2})$ are defined in the same manner as $\mathbf{a}_{\mathrm{ULA}}(\phi)$ with $\mathcal{I}(N)=\{n-(N-1)/2,n=0,1,\cdots,N-1\}$, and $\otimes$ denotes the Kronecker product. The array element spacings of ULA and UPA are assumed to be $\lambda/2$. Hence, the effective channels can be represented as \[\mathbf{h}_{ik} =\sum_{l_{h}=1}^{L_{h}}\beta_{l_{h}}\mathbf{a}_{t}^{H}\left( \phi_{l_{h}}^{\mathrm{r},1},\phi_{l_{h}}^{\mathrm{r},2}\right), \tag{7}\] \[\mathbf{G}_{i} =\sum_{l_{g}=1}^{L_{g}}\beta_{l_{g}}\mathbf{a}_{t}\left(\phi_{l_{ g}}^{\mathrm{r},1},\phi_{l_{g}}^{\mathrm{r},2}\right)\mathbf{a}_{b}^{H}\left( \phi_{l_{g}}^{\mathrm{h}}\right), \tag{8}\] where $L_{s}$ is the number of paths that contains one LoS path and $(L_{s}-1)$$\mathrm{NLoS}$ paths for link, and $\beta_{l_{s}}$ denotes the complex path gain of $l_{s}$ path with $s\in\{h,g\}$. For simply notation, the subscript $h,g$ represent $\mathbf{h}_{ik}$ and $\mathbf{G}_{i}$, respectively. ### _Problem Formulation_ Our main objective is to maximize the WSR by designing the beamforming matrix $\mathbf{F}$, the RISs association matrix $\mathbf{U}$ as well as the passive beamforming matrices $\mathbf{\Phi}_{i},\forall i\in\mathcal{R}$ in multi-RISs aided communication systems. Therefore, the maximization problem can be formulated as \[\text{P1:}\ \max_{\mathbf{F},\mathbf{U},\mathbf{\Phi}_{i}, \forall i\in\mathcal{R}} \sum_{k=1}^{K}w_{k}\mathrm{log}_{2}(1+\mathrm{SINR}_{k}) \tag{9a}\] \[\text{s.t.} \\|\mathbf{F}\\|_{\mathcal{F}}^{2}\leq P_{max}\] (9b) \[\|\mathbf{\Phi}_{i,m}\|=1,\forall m=1,2,\cdots,M\] (9c) \[\sum_{i\in\mathcal{R}}u_{k,i}=1,\forall k\in\mathcal{K}\] (9d) \[u_{k,i}=\left\{0,1\right\},\forall k\in\mathcal{K},\forall i\in \mathcal{R}, \tag{9e}\] where $w_{k}$ denotes the weight for user${}_{k}$ with $\sum_{k=1}^{K}w_{k}=1$. ## III Sum-Rate Maximization Via GNN In this paper,we introduce the heterogeneous GNN which contains abundant information with structural relations among multi-typed nodes as well as unstructured content associated with each node [17]. As for a heterogeneous graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{O}_{\mathcal{V}},\mathcal{R}_{ \mathcal{E}})$, there are two additional type sets, including the predefined node types $\mathcal{O}_{\mathcal{V}}$ and edge types $\mathcal{R}_{\mathcal{E}}$ with $\|\mathcal{O}_{\mathcal{V}}\|+\|\mathcal{R}_{\mathcal{E}}\|>2$[18]. The node types $\mathcal{O}_{\mathcal{V}}$ includes BS node $\mathbf{v}_{b}$, RIS${}_{i}$ node $\mathbf{v}_{r_{i}}$ and user${}_{k}$ node $\mathbf{v}_{u_{k}}$. According to the wireless communication scenario, the attributes of $\mathbf{v}_{u_{k}}$ include the weight $w_{k}$, noise variance $\sigma^{2}$ as well as $\mathbf{U}$. Similarly, the attributes $\mathbf{v}_{b}$ and $\mathbf{v}_{r_{i}}$ include $\mathbf{F}$ and $\mathbf{\Phi}_{i}$, respectively. Given the heterogeneous GNN structure, the goal of interest is to find a policy $p_{\Theta}(\cdot)$ with parameters $\Theta$, mapping the graph $\mathcal{G}$ to estimations of the optimal $\mathbf{F},\mathbf{\Phi}_{i}$ and $\mathbf{U}$. ### _The Initialization Layer of Proposed GNN_ The initial layers take the CSI information as the input features. In order to better learn the phase configuration of each RIS, we extract the phase shift matrix $\mathbf{\theta}_{i}$. Thus, the equivalent channel $\mathbf{h}_{k}$ can be expressed as \[\mathbf{h}_{k} =\sum_{i\in\mathcal{R}}u_{k,i}\mathbf{\theta}_{i}\mathrm{diag}( \mathbf{h}_{ik})\mathbf{G}_{i} \tag{10}\] \[=\sum_{i\in\mathcal{R}}u_{k,i}\mathbf{\theta}_{i}\mathbf{H}_{ik}^{ \mathrm{cas}},\] where $\mathbf{\theta}_{i}=[\theta_{i,1},\cdots,\theta_{i,M}]$ and $\mathbf{H}_{ik}^{\mathrm{cas}}=\mathrm{diag}(\mathbf{h}_{ik})\mathbf{G}_{i}$ denotes the cascaded channel links, which is considered as the input feature for each user node, i.e., $\mathbf{v}_{u_{k}}^{(\mathrm{in})}=[\mathrm{Re}(\mathbf{H}_{ik}^{\mathrm{ cas}}),\mathrm{Im}(\mathbf{H}_{ik}^{\mathrm{cas}})]$. ### _Message Passing Paradigm_ Graph network block is adopted as the basic computation unit over graphs [19], as illustrated in Fig. 2. The update function for each type of node is illustrated in detail as follows. #### Iii-B1 Update for RIS Nodes The updated feature for the RIS nodes at step $t$$\mathbf{v}_{r_{i}}^{[t]}$ is obtained by combining the features from the last state $\mathbf{v}_{r_{i}}^{[t-1]}$ and all the user nodes $\mathbf{v}_{u_{k}}^{[t-1]},\forall k\in\mathcal{K}$. Then, the messages from different type of nodes are given as \[\mathbf{\xi}_{rr_{i}}=\phi_{r_{i}-r_{i}}^{(\mathrm{up})}(\mathbf{\xi}_{r_{i}},\mathbf{v }_{r_{i}}^{[t-1]}), \tag{11}\] \[\mathbf{\xi}_{ur_{i}}=\phi_{u\to r_{i}}^{(\mathrm{up})}(\rho_{u\to r_{i}}(\mathbf{\xi}_{u_{1}}, \cdots,\mathbf{\xi}_{u_{K}}),\mathbf{v}_{r_{i}}^{[t-1]}), \tag{12}\] where $\mathbf{\xi}_{\mathrm{src}}=\phi_{\mathrm{src}\to r_{i}}(\mathbf{v}_{r_{i}}^{[t-1]})$ denotes message from the source node and $\mathrm{src}\in\{r_{i},u_{1},\cdots,u_{K}\}$ for simple notation. Meanwhile, both $\phi_{\mathrm{src}\to r_{i}}$ and $\phi_{\mathrm{src}\to r_{i}}^{(\mathrm{up})}$ are two-layer MLPs. Besides, $\rho_{u\to r_{i}}$ is the mean aggregation function for all the different user nodes. Then, in heterogeneous GNNs, the updated RIS nodes $\mathbf{v}_{r_{i}}^{[t]}$ aggregate the feature information from different-type neighbors, which is given as \[\mathbf{v}_{r_{i}}^{[t]}=\rho_{r_{i}}(\mathbf{\xi}_{rr_{i}},\mathbf{\xi}_{ur_{i}})+ \mathbf{v}_{r_{i}}^{[t-1]}, \tag{13}\] where $\rho_{r_{i}}$ denotes the operation of the mean aggregation. Besides, a residual operation is added in this step to avoid degradation and gradient explosion of our network [20]. #### Iii-B2 Update for User Nodes The updated feature $\mathbf{v}_{u_{k}}^{[t]}$ is obtained by combining the features from all the user nodes $\mathbf{v}_{u_{k}}^{[t-1]}$ and RIS nodes $\mathbf{v}_{r_{i}}^{[t-1]}$. In the same manner, the messages from different type of nodes are given as \[\boldsymbol{\xi}_{ru_{k}}=\phi_{r\to u_{k}}^{(\mathrm{up})}(\rho_{r\to u_{k}}( \boldsymbol{\xi}_{r_{1}},\cdots,\boldsymbol{\xi}_{r_{R}}),\mathbf{v}_{u_{k}}^{ [t-1]}), \tag{14}\] \[\boldsymbol{\xi}_{uu_{k}}=\rho_{u_{k}}^{(u)}(\phi_{u\to u_{k}}^{(\mathrm{up})}( \boldsymbol{\xi}_{u}),\phi_{u_{k}\to u_{k}}^{(\mathrm{up})}(\boldsymbol{\xi}_ {u_{k}},\mathbf{v}_{u_{k}}^{[t-1]}))), \tag{15}\] where $\boldsymbol{\xi}_{u}=\rho_{u\to u_{k}}(\boldsymbol{\xi}_{u_{1}},\cdots, \boldsymbol{\xi}_{u_{K}})$ and $\boldsymbol{\xi}_{\mathrm{src}}=\phi_{\mathrm{src}\to r_{i}}(\mathbf{v}_{ \mathrm{src}}^{[t-1]})$ denotes message from the source node and $\mathrm{src}\in\{r_{1},\cdots,r_{R},u_{1},\cdots,u_{K}\}$ for simple notation. Meanwhile, both $\phi_{\mathrm{src}\to u_{k}}$ and $\phi_{\mathrm{src}\to u_{k}}^{(\mathrm{up})}$ are two-layer MLPs. Besides, $\rho_{u\to u_{k}}$ and $\rho_{r\to u_{k}}$ are the max and mean aggregation functions, respectively. Then, in heterogeneous GNNs, the updated RIS nodes $\mathbf{v}_{u_{k}}^{[t]}$ aggregate the feature information from different-type neighbors, which can be obtained by \[\mathbf{v}_{u_{k}}^{[t]}=\rho_{u_{k}}(\boldsymbol{\xi}_{ru_{k}},\boldsymbol{ \xi}_{uu_{k}})+\mathbf{v}_{u_{k}}^{[t-1]}, \tag{16}\] where $\rho_{u_{k}}$ denotes the operation of the max aggregation. #### Iii-B3 Output Layers for Node Features After updating all the nodes, each node has multiple feature embeddings from different-type nodes. In order to obtain the final feature of each node $\mathbf{v}_{b}^{[T]}$, $\mathbf{v}_{r_{i}}^{[T]}$ and $\mathbf{v}_{u_{k}}^{[T]}$, a decoder layer is used to output all the optimized features, which is given as \[\mathbf{v}_{\mathrm{dst}}^{[T]}=\rho_{\mathrm{dst}}^{(\mathrm{out})}( \boldsymbol{\xi}_{\mathrm{udst}},\boldsymbol{\xi}_{\mathrm{rdst}}), \tag{17}\] where the destination node is denoted as $\mathrm{dst}\in\{b,r_{1},\cdots,r_{R},u_{1},\cdots,u_{K}\}$ and $\phi_{\mathrm{src}\to\mathrm{dst}}^{(\mathrm{out})}$ is utilized to extract the features from different-type nodes, as shown by the black line in Fig. 2. Finally, the new state $\mathbf{v}_{\mathrm{dst}}^{[T]}$ will be obtained by a mean aggregation layer $\rho_{\mathrm{dst}}^{(\mathrm{out})}$. Suppose that $T$ iterations are executed during the training process, and we defined each update as $\text{HGN}_{t},\forall t\in T$. An encoder-process-decoder architecture is utilized to facilitate training [19], including one encoder block $\text{HGN}_{0}$, one decoder block $\text{HGN}_{T}$, and one core block (concatenate by $\text{HGN}_{1},\cdots$, $\text{HGN}_{T-1}$). After the encoder-process-decoder architecture, we can obtain the final result for the optimal $\mathbf{F}$, $\mathbf{U}$ and $\boldsymbol{\Phi}_{i},\forall i\in\mathcal{R}$. ### _Model Architecture of Proposed Heterogeneous GNN_ In addition to the massage passing paradigm, the proposed heterogeneous GNN needs some additional layers to complete the whole learning process as follows #### Iii-C1 The Normalization Layers Note that the neural network is based on the real numbers, we learn the vector consisting of the real and imaginary parts of the beamforming matrix $\mathbf{F}$ and the phase matrices $\boldsymbol{\Phi}_{i},\forall i\in\mathcal{R}$. Therefore, before calculating the WSR loss function, additional layers are necessary to merge the real-valued results to complex-valued results. * To satisfy the constraint on transmit power in (9b), a normalization layer is employed at the output feature $\mathrm{Re}\{\mathbf{F}\}$and $\mathrm{Im}\{\mathbf{F}\}$, which can be expressed as \[\hat{\mathbf{F}}=\sqrt{P_{max}}\frac{\mathrm{Re}\{\mathbf{F}\}+j\cdot\mathrm{ Im}\{\mathbf{F}\}}{\sqrt{\left\\|\mathrm{Re}\{\mathbf{F}\}\right\\|_{\mathcal{F}}^{2}+ \left\\|\mathrm{Im}\{\mathbf{F}\}\right\\|_{\mathcal{F}}^{2}}}.\] (18) * Similarly, the normalization layer for the reflection coefficients can be represented as \[\theta_{i,m}=\frac{\mathrm{Re}\{\theta_{i,m}\}+j\cdot\mathrm{Im}\{\theta_{i,m} \}}{\sqrt{\|\,\mathrm{Re}\{\theta_{i,m}\}\|^{2}+\|\mathrm{Im}\{\theta_{i,m}\}\|^{2 }}}.\] (19) Fig. 2: The message passing paradigm in the heterogeneous GNN network. #### Iii-C2 The RIS Association Strategy Each user chooses a suitable one of all RISs as the reflection link to enhance the link quality, as the constraint (9d). Thus, the matrix $\mathbf{U}$ for calculating the WSR can be obtained by \[c_{u_{k,i}}=\mathrm{SoftMax}\left(u_{k,i}\right)=\frac{e^{u_{k,i}}}{\sum\limits_ {i=1}^{R}e^{u_{k,i}}}. \tag{20}\] After obtaining the SoftMax score $c_{u_{k,i}}$, we sort the scores of each user, i.e., each column in $\mathbf{U}$. We select the largest value and set it to 1, indicating that the user${}_{k}$ selects this RIS to assist the communication links while setting the remaining positions of this column to 0 to get the final matrix. #### Iii-C3 The Loss Function Due to the connection matrix is composed of binary elements, which are discontinuous variants. Thus, the process to determine $\mathbf{U}$ as well as the optimization of $\mathbf{F}$ and $\mathbf{\Phi}$ is very difficult to learn. Considering this, we add a regular term related to the distance between users and RISs to the training loss. We adopt distance-dependent labels to represent the association relationship between the users and their nearest RIS and the categorical cross-entropy loss function to achieve it. Thus, the loss function is composed of the negative WSR and the categorical cross-entropy loss function, which can be expressed as \[\mathrm{Loss}= \tag{21}\] \[-\sum\limits_{k=1}^{K}\left(w_{k}\mathrm{log}_{2}\left(1\!+\! \frac{\left\|\mathbf{h}_{k}\mathbf{f}_{k}\right\|^{2}}{\sum\limits_{j\in\mathcal{ K},j\neq k}^{K}\left\|\mathbf{h}_{k}\mathbf{f}_{j}\right\|^{2}\!+\!\sigma^{2}}\right)\!-\! \eta\mathbf{u}_{k}^{(1)}\log\mathbf{u}_{k}\right),\] where $\mathbf{u}_{k}^{(1)}$ and $\mathbf{u}_{k}$ denote the distance-dependent label and optimized value for the user${}_{k}$, respectively. $\eta$ is the coefficient of the penalty term, and its role is to strike the balance of the significance of these two parts in the loss function, which can be calculated by $\eta=\mathrm{WSR}_{p}/\mathrm{WSR}_{p_{0}}$. We set $\mathrm{WSR}_{p}$ as the results in the pre-training process where the loss function is negative WSR, and $\mathrm{WSR}_{p_{0}}$ is the baseline case when $P_{max}=30$dBm. For the training details, Adam optimizer [21] is adopted with a total of 200 epochs on one NVIDIA RTX 2080Ti GPU. The initial learning rate, weight decay, and batch size are set to $1\times 10^{-4}$, $5\times 10^{-5}$ and 128. ## IV Simulation Results In this section, numerical simulations are provided to verify the effectiveness of our proposed heterogeneous GNN. Throughout the simulation, the BS is equipped with $N_{t}=8$ antennas, and the number of RIS elements $M$ is 16 with $M_{x}=M_{y}=4$. The BS is assumed to be located on the origin and two RISs are located in $[30,25]$ and $[30,-25]$. In addition, two users are randomly located in a rectangular region in $[40:50,-25:25]$. Without loss of generality, the weights $w_{k}$ are set to be equal. According to [16], the channel gain is taken as $\beta\sim\mathcal{CN}\left(0,10^{-0.1\mathrm{PL}(r)}\right)$ where $\mathrm{PL}(r)=\varrho_{a}+10\varrho_{b}lg(r)+\xi$ with $\xi\sim\mathcal{N}(0,\sigma_{\xi}^{2})$. In addition, the channel realizations are produced by setting $\sigma_{n}^{2}=-85dBm$, $\varrho_{\mathrm{a}}=61.4,\varrho_{\mathrm{b}}=2$, and $\sigma_{\xi}=5.8\ \mathrm{dB}$. The total number of samples is 110,000 where 100,000 and 10,000 of the whole generated data are used for training and validation, respectively. To see the performance, WMMSE with a random phase configuration is used as the performance baseline. Also, in order to validate the effectiveness of the graph structure, we compare the performance in our proposed heterogeneous GNN with that in DNN. Fig. 3 shows the sum rate with respect to the maximum transmit power of the BS. Our proposed GNN structure performs well under all levels of transmit power, which exhibits the generality and effectiveness of our structure. First, the sum rate obtained by random configuration and heterogeneous GNN becomes higher with the increase of maximum transmit power $P_{max}$. However, in the absence of optimization, the sum rate boosting from the increasing power is limited. In contrast, our proposed GNN structure exhibits its superb ability to solve the optimization problem P1 with different number of reflective elements (M = 9, 16, 25) and different $P_{max}$, from $10$ dBm to $30$ dBm with an interval of $5$ dBm. Fig. 4 illustrates the total number of antennas with respect to the WSR performance. We also exhibit the WSR values in two other cases, including the results by WMMSE with the Fig. 4: The weighted sum rate versus the number of antennas at the BS Fig. 3: The sum rate versus the maximum transmit power at the BS. random phase configuration and the optimal value obtained by our proposed GNN in the case of one fixed RIS, i.e., all the users receive the signal reflected by the RIS located at $[30,25]$. It is obvious that the WSR increase with the growth of the number of antennas under all cases, and heterogeneous GNN can achieve a higher WSR. We can choose a suitable one of all RISs to replace the fix RISs to provide service links for each user, which can improve the performance about $30\%$ under all the different number of antennas. Fig. 5 illustrates the WSR of heterogeneous GNNs and two DNNs with respect to the same size of the training set under the networks with different sizes. We can easily find that the performance in heterogeneous GNNs performs significantly better than those of DNNs, especially when the number of training sample is $2\times 10^{5}$, GNNs perform 10 times better than DNNs. Since the introduction of the RISs association strategy will exacerbate the influence of structural information on optimization results, DNNs perform very poorly due to their lack of the ability to capture the underlying model structure. In addition, comparing the results of two DNNs, it can be seen that no matter how to increase the number of layers and the number of channels cannot improve the performance of DNNs. In addition, GNNs can get better optimization results with fewer samples, even if it is less than 10,000, and get better results with increasing training samples. ## V Conclusion We proposed a heterogeneous GNN structure to solve the NP-hard beamforming optimization problem in a multi-RISs aided MIMO system by mapping the structure of the wireless system into a graph neural network. Different from most beamforming problems, a RISs association strategy was introduced into our problem, which makes the beamforming design more complex, but a suitable RISs association strategy can enhance the users' quality of service. The effectiveness of our proposed GNN structure and the RISs association strategy were validated through extensive simulations. According to the simulation results, we found that our proposed GNNs performed about 10 times better than the conventional DNNs. Moreover, the benefits of the RISs association strategy were illustrated by comparing the optimal results with/without this strategy, which improved the performance about $30\%$.
2305.17158	The Galaxy Assembly and Interaction Neural Networks (GAINN) for high-redshift JWST observations	We present the Galaxy Assembly and Interaction Neural Networks (GAINN), a series of artificial neural networks for predicting the redshift, stellar mass, halo mass, and mass-weighted age of simulated galaxies based on JWST photometry. Our goal is to determine the best neural network for predicting these variables at $11.5 < z < 15$. The parameters of the optimal neural network can then be used to estimate these variables for real, observed galaxies. The inputs of the neural networks are JWST filter magnitudes of a subset of five broadband filters (F150W, F200W, F277W, F356W, and F444W) and two medium-band filters (F162M and F182M). We compare the performance of the neural networks using different combinations of these filters, as well as different activation functions and numbers of layers. The best neural network predicted redshift with normalized root mean squared error NRMS = $0.009_{-0.002}^{+0.003}$, stellar mass with RMS = $0.073_{-0.008}^{+0.017}$, halo mass with MSE = $ 0.022_{-0.004}^{+0.006}$, and mass-weighted age with RMS = $10.866_{-1.410}^{+3.189}$. We also test the performance of GAINN on real data from MACS0647-JD, an object observed by JWST. Predictions from GAINN for the first projection of the object (JD1) have mean absolute errors $\langle \Delta z \rangle <0.00228$, which is significantly smaller than with template-fitting methods. We find that the optimal filter combination is F277W, F356W, F162M, and F182M when considering both theoretical accuracy and observational resources from JWST.	Lillian Santos-Olmsted, Kirk Barrow, Tilman Hartwig	2023-05-26T18:00:00Z	http://arxiv.org/abs/2305.17158v1	# The Galaxy Assembly and Interaction Neural Networks (GAINN) for high-redshift JWST observations ###### Abstract We present the Galaxy Assembly and Interaction Neural Networks (Gainn), a series of artificial neural networks for predicting the redshift, stellar mass, halo mass, and mass-weighted age of simulated galaxies based on JWST photometry. Our goal is to determine the best neural network for predicting these variables at $11.5<z<15$. The parameters of the optimal neural network can then be used to estimate these variables for real, observed galaxies. The inputs of the neural networks are JWST filter magnitudes of a subset of five broadband filters (F150W, F200W, F277W, F356W, and F444W) and two medium-band filters (F162M and F182M). We compare the performance of the neural networks using different combinations of these filters, as well as different activation functions and numbers of layers. The best neural network predicted redshift with normalized root mean squared error NRMS = \ undergo the entire process of predicting photo-z, from template spectra to redshift (e.g. Bagpipes (Carnall et al., 2018), piXedfit(Abdurro'uf et al., 2021), BEA-GLE (Chevallard and Charlot, 2016), Prospector(Leja et al., 2017; Johnson et al., 2021), and EASY (Brammer et al., 2008)). This method has been widely applied in the recovery of redshift and other properties using filter fluxes from the James Webb Space Telescope (JWST) (Bisigello et al., 2016; Bisigello et al., 2017; Kemp et al., 2019; Roberts-Borsani et al., 2021). Machine learning methods estimate redshift by considering a data set of known magnitudes and redshifts and learning the features that relate them. These learned relationships can then be used to predict redshift in a blind data set where nothing is known about the true redshift values. As in SED template-fitting, the goal of machine learning is to minimize the error between a predicted and true value, but in this case the value is the redshift itself and not a set of fluxes. Unlike SED template-fitting, ML algorithms calculate the errors of a single data set over multiple iterations and attempt to reduce the errors in subsequent iterations. Machine learning methods can be used in the same manner to recover other galaxy properties (Calderon and Berlind, 2019; Simet et al., 2019; Surana et al., 2020). Provided there is a high-quality data set to train on, these ML techniques are applicable. The training data for observed redshifts generally comes from spectroscopic data. It has been shown that ML performs better than SED template-fitting at lower redshifts but worse than it at high redshifts, because there are not enough high-quality, spectroscopic observations of high redshift galaxies to provide a sufficiently large dataset for ML algorithms to train on (Hildebrandt et al., 2010; Salvato et al., 2019). Previous studies using ML to estimate photometric redshifts have generally been restricted to low-redshift ranges up to $z\approx 4$(Firth et al., 2003; Vanzella et al., 2004; Almosallam et al., 2015; Pasquet et al., 2018; Razim et al., 2021; Lee and Shin, 2021, 2022; Zhou et al., 2022, 2022). However, this barrier can be removed by using synthetic data to provide high redshift values for the ML algorithm to train on. One source of synthetic data are the spectral templates described earlier, but naturally they are based off existing data. In other words, a ML model training on template spectra would not be exposed to novel or unexpected features that would prepare it to handle real data. An alternative source of synthetic data are cosmological hydrodynamic simulations (Hopkins et al., 2014; Vogelsberger et al., 2014; Genel et al., 2014; Schaye et al., 2014; Crain et al., 2015; Xu et al., 2016; Barrow et al., 2020), numerical models of the universe that simulate star formation, radiative transfer, and other dynamic physical phenomena based on a set of initial conditions (Barrow et al., 2017). These simulations aim to model the evolution of galaxies from fundamental physics principles. To the extent of our knowledge of cosmology and access to computational resources, the properties of galaxies within the simulation are determined from the same physical laws that govern the behavior of real galaxies. Therefore, cosmological simulations can provide realistic data sets at high redshift for ML algorithms to learn from. In addition, the time-dependent aspect of cosmological simulations supplies more data points by allowing users to record a galaxy's properties at different snapshots in time. Data from cosmological simulations has already been used to train machine learning models that predict the stellar mass and halo mass of simulated galaxies (Kamdar et al., 2016; Agarwal et al., 2018; Gilda et al., 2021; Villanueva-Domingo et al., 2022; Zhou and Han, 2022), in addition to other observables (Villaescusa-Navarro et al., 2021, 2021; Lovell et al., 2019). For this approach to be useful for predicting the redshifts and other properties of real galaxies, the ML algorithm needs to optimize a mathematical relationship between the magnitudes and redshifts of simulated galaxies that is directly transferable to observed galaxies. This relationship would be easier to investigate in surveys of high-redshift galaxies, because cosmological simulations start near the beginning of the universe and require less computing time to reach higher redshifts. Conveniently, the recent launch of JWST will transform our ability to study the properties of high redshift galaxies and allow us to probe the very early universe. Of the four main instruments on JWST, the one that is of most interest in this work is the Near-Infrared Camera (NIRCam). NIRCam includes filter imaging through a short wavelength channel (0.6 to 2.3 microns) and a long wavelength channel (2.4 to 5.0 microns) (Gardner et al., 2006). The various broad and medium-band filters on NIRCam provide an optimal wavelength range to perform photometric measurements of redshift and other properties at $11.5\lesssim z\lesssim 15$. Several candidate galaxies in this redshift range have already been observed by NIRCam (e.g. (Adams et al., 2022; Bradley et al., 2022; Castellano et al., 2022; Donnan et al., 2023; Harikane et al., 2023)). Some high-redshift galaxy candidates recently observed by JWST have higher stellar masses than predicted by the standard cosmological model ($\Lambda$CDM) (Boylan-Kolchin, 2022; Labbe et al., 2022; Menci et al., 2022). According to $\Lambda$CDM, there is an upper bound on the number density and stellar mass density of galaxies as a function of stellar mass and redshift. Since there are candidate galaxies at the edge of the bound, these observations could potentially challenge the standard cosmological model, depending on the accuracy of the measured redshifts and stellar masses. Therefore, the accurate determination of these variables is instrumental in evaluating the validity of this challenge to $\Lambda$CDM. In this work, we present Galaxy Assembly and Interaction Neural Networks (Gainn), a set of artificial neural networks used to recover the photometric redshifts of galaxies from cosmological simulations. It is assumed that objects studied using predictions from Gainn have already been identified as high redshift galaxies in our redshift range. We use different combinations of five broadband (F150W, F200W, F277W, F356W, F444W) and two medium-band (F162M, F182M) JWST filters. We also use Gainn to recover other properties of the galaxies: stellar mass, halo mass, and mass-weighted age. The inputs of the neural networks (NNs) are explored as a parameter space to optimize performance and minimize error. We compare the performance of various multi-layered NNs using the machine learning software Tensorflow v2.4.1(Abadi et al., 2015). To account for the various exposure times for different JWST filters, we address three potential strategies: 1) The best outcomes that theory can offer with minimal observational constraints; 2) A strategy which incorporates both theoretical optimism and observational feasibility; and 3) The most ideal strategy given JWST resources. This paper is structured as follows: In sections 2.1 and 2.2, we explain the mechanics of our hydrodynamic cosmological simulation. We show the conversion between simulated galaxy spectra and filter magnitudes in section 2.3. Sections 2.4 and 2.5 include a discussion of how NNs function and our parameter selection for the NNs. Observational constraints are considered in section 2.6. We present our results for the unconstrained, observationally feasible, and observationally ideal best-case scenarios in sections 3.1, 3.2, and 3.3, respectively. In section 3.4, we present results from the first test of our NNs on real data. Finally, we summarize and discuss our results in section 4. ## 2 Methods ### Cosmological Simulation The cosmological simulation is evolved using the radiation-hydrodynamic adaptive mesh refinement code Enzo(Bryan et al., 2014). The initial conditions define a zoom-in region for the Lagrangian volume containing a $\sim 10^{11}$M${}_{\odot}$ halo at $z=0$ with a quiescent merger history as taken from a public repository provided by the Agora collaboration (Kim et al., 2014). Similarly to its antecedent simulation in Barrow et al. (2020), the cosmological parameters of $\Omega_{M,0}=0.3065$, $\Omega_{\Lambda,0}=0.6935$, $\Omega_{b,0}=0.0483$, $h=0.679$, $\sigma_{8}=0.8344$, and $n=0.9681$ are used, which are taken from the most recent release of the Planck Collaboration et al. (2020). The root grid for both dark matter particles and the baryon mesh is a 60 Mpc${}^{3}$ box containing $512^{3}$ particles and cells respectively. The zoom-in region is bounded by the dimensions $2578.12\times 3750\times 2109.38$ co-moving kpc${}^{3}/h^{-3}$ using four additional levels of nested refinement. This results in a maximum dark matter resolution of $2.81\times 10^{4}$ M${}_{\odot}$ and an effective grid size of $8912^{3}$. Within the zoom-in region, ten to eleven more levels of gas refinement are allowed, resulting in cell sizes as small as a quarter of a proper parsec. The star formation, 9-species (H i, H ii, He i, He ii, He iii, e${}^{-}$, H${}_{2}$, H${}_{2}^{+}$, H${}^{-}$) primordial gas evolution, and metal enrichment routines used are identical to (Barrow et al., 2020), except that the minimum mass for metal enriched star clusters has been reduced to 500 M${}_{\odot}$. By carefully monitoring and tracking the evolution of the stellar history in multiple test simulations, it was determined that star formation converged at these resolutions and minimum cluster masses. Further levels of refinement were found to be counter-productive, resulting in diverging quantities of radiating star particles. Additionally, few metal-enriched star particles were produced at the minimum mass, effectively avoiding issues with the stochasticity of bright massive stars. We explicitly follow the formation of metal-free stars, although their contribution to the bolometric luminosity might not be significant for the first galaxies (Riaz et al., 2022). The cosmological simulation terminates at $z=11.36$ and includes the merger history of approximately 180 galaxies, saved as output data in 1 Myr intervals beginning right before the onset of star formation at $z>20$. ### Radiative Transfer Post-Processing Routine Improvements A radiative transfer post-processing routine is used to produce detailed spectra from simulation data. The methods employed are identical to those discussed in (Barrow et al., 2020) except that absorption look-up tables were produced using archetypal background galactic spectra instead of being recalculated at every timestep at every halo. This change was made to improve the scalability to the technique so that it can accommodate a larger number of merger histories. Mean intrinsic galactic continuum spectra are calculated using Flexible Stellar Population Synthesis (FSPS; Conroy and Gunn, 2010) for metal-enriched populations and (Yggdrasil; Zackrisson et al., 2011) for metal-poor stars using 55 $M_{\odot}$ as a cut off between using PopIII.1 or PopIII.2 (top-heavy) models. These spectra are then organized by the fraction of the luminosity in ionizing radiation and the UV slope. From these two variables, 25 archetypical background spectra are selected using k-means clustering to create photo-ionization sensitive temperature, wavelength, and metallicity look-up tables for gas absorption per nucleon density (see Fig. 1). Though stellar population synthesis models like FSPS take metallicity and age as inputs for spectra shape, ionizing radiation and UV slope were found to much more naturally segregate the spectra while minimizing degeneracies in resulting spectra. All the other elements of the radiative transfer pipeline are functionally unchanged. To summarize, these absorption profiles are used for the ray tracing calculations from each source to each voxel of gas within the halo, which are used to calculate the strength of emission lines throughout the halo and then used again to ray trace the final spectra of the galaxy at the virial radius, which can be a strong function of the orientation of the galaxy with respect to the observer's location. For this study, the mean resulting spectra is used to create over 10,000 galaxy-timestep spectra. Early Universe galaxies typically have short, bursty star formation histories and so their spectra are highly sensitive to the timing and height of peaks in the star formation rate. The evolution of even a single galaxy can be used to model a large parameter space of output spectra by separately examining periods of a galaxy's star formation history, which represent a several possible combinations of star formation timings. When taken together, the modeled spectra of the more than 10,000 galaxy-timesteps samples a diverse parameter space of spectral shape (Fig. 1), which, along with redshift are key to disentangling galactic properties from observed filter fluxes. Thus, while we have a modest sample of galaxies in our final redshift, our time resolution and methodology provide an unprecedentedly large sample of synthetic spectra from robot, high-resolution radiative-hydrodynamic simulations and radiative transfer modeling. However, feedback prescriptions are still formulated based on best-effort theoretical arguments rather than empirically due to still-forthcoming observations of the early Reionization Universe so we caution that our results may incorporate unquantifiable theoretical biases, especially in our recovery of mean stellar age and halo mass. Additionally, more simulations will be incorporated into the model to extend the scope of our analysis in the near future so Gainn should be seen as a continuously improving framework, not a final result. ### Filter Magnitudes The spectra of the galaxies are obtained from the simulation in the form of energy values $L_{\nu}$ which each correspond to an emitted frequency $\nu_{e}$. The data for a given telescope filter $i$ consists of throughputs $R_{i}$, each corresponding to a specific observed wavelength. Each emitted wavelength is converted to an observed wavelength in order to map the throughputs to their corresponding emitted frequencies. Emitted spectral energy density blueward of the Lyman-$\alpha$ line (1215.67 A) is assumed to be zero due to pre-Reionization absorption by neutral hydrogen in the intergalactic medium. The Figure 1: Left: Our sample of galaxies plotted by mean stellar age and metallicity showing tracks representing the time-evolution of star-forming halos. Right: The same plotted by the relationship between the fraction of ionizing to non-ionizing radiation and the UV slope, which serves as a proxy for spectral shape. Both figures are colored by k-means clustering performed on the distribution in the right figure. The archetypal galaxies in each cluster are colored with black markers. observed flux $f_{i}$ through the filter is: \[f_{i}=\frac{1}{4\pi d_{L}^{2}}\int_{0}^{\infty}\frac{L_{\nu}(\nu_{e})R_{i}(\nu_{e} )}{\nu_{e}}\ d\nu_{e}, \tag{1}\] where $d_{L}$ is the luminosity distance defined by \[d_{L}=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{\sqrt{\Omega_{M,0}(1+ z^{\prime})^{3}+\Omega_{\Lambda,0}}}, \tag{2}\] where $H_{0}$, $c$, $\Omega_{\Lambda,0}$, and $\Omega_{M,0}$ have their usual cosmological definitions and are equivalent to the values used in the simulation. The AB magnitude $m_{AB,i}$ through filter $i$ is calculated from the flux as \[m_{AB,i}=-2.5\log_{10}\frac{f_{i}}{\int_{0}^{\infty}\frac{R_{i}(\nu)}{\nu}d\nu }-48.6, \tag{3}\] where $\nu$ is the observed frequency. ### Neural Network We explore an approach for determining the weights of the filter magnitudes by employing NNs. A neural network is widely known as an algorithm that can learn about the features of a system by training with examples and then apply that learning to new situations and real data. The primary advantage of NNs over the standard linear regression model is that some NN architectures are more adept at identifying nonlinear relationships. In principle, a basic NN consists of an input layer, an output layer, and usually one or more hidden layers. Each layer contains a set of neurons that are connected to each neuron from the previous layer and subsequent layer by a weight that scales the data and a bias that translates the data (Schmidhuber, 2015). From one layer to the next, the values in each neuron are multiplied by the corresponding weight and added to the corresponding bias. The resulting values undergo a transformation through an activation function, and these transformed values then become the input for the following layer. This process is repeated until the NN generates an output from the final layer (see Eq. 4). In our case, the input layer possesses $n$ neurons, where $n$ is the number of considered JWST filters. Each of the four variables (redshift, stellar mass, halo mass or mass-weighted age) is predicted by its own NN, so each neuron in the input layer receives an $N\times 1$ vector as input, where $N$ is the number of magnitudes (i.e., number of different samples) for a given filter. Thus, the overall input to the NN is an $N\times n$ matrix. The output layer has one neuron containing an $N\times 1$ vector, which corresponds to the outputs of one of the variables. We consider multi-layered NNs with different numbers of layers. The number of layers $\ell$ of an NN is defined here as the number of hidden layers plus one. Hidden layers in multi-layered NNs can have an arbitrary number of neurons. For a layer $k$, the input matrix $X_{k}$ is $N\times n_{k}$, where $n_{k}$ is the number of neurons in the layer. This input matrix can be related to the input $X_{k+1}$ of the next layer by a $n_{k}\times n_{k+1}$ weight matrix $W_{k}$, and a $N\times n_{k+1}$ bias matrix $B_{k}$, acted on by an activation function $f_{k}$: \[X_{k+1}=f_{k}\left(X_{k}W_{k}+B_{k}\right), \tag{4}\] where \[X_{k}=\begin{cases}N\times n\text{ input matrix}&\text{if }k=1\\ f_{k-1}\left(X_{k-1}W_{k-1}+B_{k-1}\right)&\text{if }k>1\end{cases} \tag{5}\] and $k=1$ is the input layer, $k=\ell+1$ is the output layer, and $\text{y}=\text{X}_{\ell+1}$ is the final output. For our NNs, we use 6 neurons for the first hidden layer and 5 neurons for the rest of the hidden layers. The chosen activation function is applied to all layers except for the final layer, which uses a linear activation function. The architecture of the two-layer NN with four filters is shown in Fig. 2. The NN is initialized with random weights and biases using the default methods in tensorflow. For the weights, the default is the Glorot Uniform distribution (Glorot and Bengio, 2010), which contains values in the interval $[-1,1]$. The biases are initially set to zero. The output of the NN is compared Figure 2: Basic NN structure. The colored lines represent the connections between the neurons from one layer to another. These connections include the weights, biases, and activation functions that transform the data in one neuron to become part of the input of the neurons in the next layer. from the simulation (the "ground truth"), and the difference between the two is the loss. The loss is quantified by a function, such as the mean squared error (MSE) or the mean absolute error (MAE): \[\mathrm{MSE}=\frac{1}{\mathrm{N}}\sum_{\mathrm{i=1}}^{\mathrm{N}}\left(\mathrm{ y}_{\mathrm{true}}-\mathrm{y}_{\mathrm{pred}}\right)_{\mathrm{i}}^{2}, \tag{6}\] \[\mathrm{MAE}=\frac{1}{\mathrm{N}}\sum_{\mathrm{i=1}}^{\mathrm{N}}\left\|\mathrm{ y}_{\mathrm{true}}-\mathrm{y}_{\mathrm{pred}}\right\|_{\mathrm{i}}, \tag{7}\] where $\mathrm{y}_{\mathrm{true}}$ is the true output from the simulation and $\mathrm{y}_{\mathrm{pred}}$ is the output of the NN. To minimize the loss, the NN undergoes a process called backpropagation, in which the weights are adjusted starting from the outermost layer and ending with the input layer (Rumelhart and McClelland, 1987). A key parameter that determines the magnitude of the adjustment is the learning rate, a value that scales the adjustment of the weights. A carefully selected learning rate is crucial because a learning rate that is too large might lead to a divergent solution, and a learning rate that is too small takes unnecessarily long and might get stuck inside a local minimum (Werbos, 1974; Schmidhuber et al., 1989). Tensorflow allows the user to specify loss function, learning rate, activation function, number of layers, and much more. ### Parameter Selection Given seven JWST filters (F150W, F200W, F277W, F356W, F444W, F162M, and F182M), all possible combinations of three, four, five, six, and seven filters are examined in the NNs. The other wide JWST filters are not included in this analysis because the magnitudes of the simulated galaxies are not detectable with those filters. The two medium filters are included to pinpoint the Lyman-alpha break with more precision, improving redshift prediction. Accurate redshift predictions play an important role in the recovery of the other variables. Due to the relationship between distance, light, and mass, predicting redshift is helpful for predicting stellar mass and halo mass. Distance is also related to age, so redshift is a good indicator of mass-weighted age. To optimize the performance of the NNs, we investigate a parameter space consisting of different activation functions, numbers of layers, and filter combinations. These parameters are fixed in each NN but vary as we analyze the performance of multiple NNs. The activation function applies to all layers of the NNs except for the final layer, which has a linear activation function. The activation functions used in the NNs are displayed in Table 1, for redshift ($z$), stellar mass ($M_{}$), halo mass ($M_{h}$), and mass-weighted age ($t$) (note that stellar and halo mass are in log scale). The activation functions were selected based on an initial performance test of 13 activation functions available on Tensorflow. For each activation function and each variable, we run four NNs, each with a different number of layers: 2, 10, 15, or 20. We choose a learning rate of 0.05 because preliminary tests indicated that learning rates around this value yielded the most accurate predictions, relative to others we tested in the interval $[0.01,0.5]$. These tests also produced better predictions with the MAE loss function and the Stochastic Gradient Descent (SGD) optimizer in Tensorflow, so we adopt these parameters in our NNs. We use 85% of our data for training and 15% for testing. However, we use slightly different parameters for halo mass. Initial tests on the halo mass data revealed that those parameters lead to signs of overfitting. We find that the NNs better learn the features of halo mass when the training ratio is 50% and the Adam optimizer is used. The greater amount of test data for halo mass contributes to reducing the spread in the test sample, which is helpful due to the inherent difficulty in predicting halo masses in our chosen mass range ($7.2\lesssim M_{h}\lesssim 8.4$). To eliminate any possible relationship between the training and testing samples, we split the data by root halo. This ensures that the data is separated not only by halo, but also by the halos that merged to form the halo. The NNs train on 500 epochs to ensure that there are enough opportunities for the weights and biases to be adjusted while minimizing computation time. To account for statistical variance, we rerun a sample of the NNs with different random seeds used to split the training and validation datasets. This allows us to not only evaluate the statistical uncertainties of our approach, but to also investigate the NN's ability to train on different galaxies in our data. Prior to entering the data into the NNs, we normalize the training and testing data because many activation \begin{table} \begin{tabular}{l l l l} $z$ & $M_{}$ & $M_{h}$ & $t$ \\ \hline elu & elu & relu & elu \\ gelu & gelu & linear & gelu \\ tanh & tanh & tanh & tanh \\ swish & selu & swish & selu \\ softplus &... & softplus & softplus \\ softsign &... &... & softsign \\ \hline \end{tabular} _Notes:_ For each variable, a NN is run for each activation function and each $\ell=2,10,15,\text{ or }20$ layers. The less well-known activation functions are defined in the Appendix (section 5.2). \end{table} Table 1: Activation functions used for each variable functions are most sensitive in the interval $[-1,1]$. The normalization is done by subtracting the mean of the training set and then dividing by the standard deviation of the training set. Once the NN has trained on the data and made predictions, the output data is renormalized to physical values. The performance of the NNs in predicting redshift is evaluated through various error metrics (see Eqs. 8-10). Normalized Root Mean Squared (NRMS), normalized bias $\langle\Delta z\rangle$, and Normalized Median Absolute Deviation (NMAD) are commonly used in the literature (Bisigello et al., 2016; Gomes et al., 2017; Wilson et al., 2020; Zhou et al., 2022). These metrics are defined by \[\mathrm{NRMS}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\Delta z\right)_{i}^{2}}, \tag{8}\] \[\langle\Delta z\rangle=\frac{1}{N}\sum_{i=1}^{N}(\Delta z)_{i}, \tag{9}\] and \[\mathrm{NMAD}=1.4826\times\mathrm{median}\left(\left\|\frac{\delta z-\mathrm{ median}(\delta z)}{1+\mathrm{z}_{\mathrm{sim}}}\right\|\right), \tag{10}\] where $\Delta z=(z_{\mathrm{sim}}-z_{\mathrm{pred}})/(1+z_{\mathrm{sim}})$ and $\delta z=z_{\mathrm{sim}}-z_{\mathrm{pred}}$. Another standard metric is the percentage of outliers $\eta$ for which $\|\Delta z\|>0.05$. Generally, the threshold is 0.15 (Wilson et al., 2020; Zhou et al., 2022), but in our case there were too few outliers under this criterion. We lowered the threshold to 0.05 to facilitate the comparison of outliers between different configurations of parameters. For stellar mass, the accuracy of the predictions is evaluated by Root Mean Squared (RMS), bias $\langle\Delta\mathrm{log(M_{})}\rangle$, and Mean Squared Error (MSE) (Bisigello et al., 2017; Simet et al., 2019; Kamdar et al., 2016): \[\mathrm{RMS}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\Delta\mathrm{log(M_{})} \right)_{i}^{2}}, \tag{11}\] \[\langle\Delta\mathrm{log(M_{})}\rangle=\frac{1}{N}\sum_{i=1}^{N}\Delta \mathrm{log(M_{})_{i}}, \tag{12}\] \[\mathrm{MSE}=\mathrm{RMS}^{2}, \tag{13}\] where $\Delta\mathrm{log(M_{})}=\mathrm{log(M_{,sim})}-\mathrm{log(M_{,pred})}$. Halo mass predictions are evaluated using MSE and $R^{2}$(Zhou and Han, 2022): \[\mathrm{MSE}=\frac{1}{N}\sum_{i=1}^{N}\left(\Delta\mathrm{log(M_{h})} \right)_{i}^{2}, \tag{14}\] \[R^{2}=1-\frac{\sum_{i=1}^{n}\Delta\mathrm{log(M_{h})}_{i}}{\sum_{i=1}^{n} \mathrm{log(M_{h,sim})}_{i}-\langle\mathrm{log(M_{h,sim})}_{i}\rangle}, \tag{15}\] where $\Delta\mathrm{log(M_{h})}=\mathrm{log(M_{h,sim})}-\mathrm{log(M_{h,pred})}$. Predictions of mass-weighted age are evaluated by RMS: \[\mathrm{RMS}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\Delta t)_{i}^{2}}, \tag{16}\] where $\Delta t=t_{\mathrm{sim}}-t_{\mathrm{pred}}$. ### Observational Constraints We calculate the AB magnitudes for the merger histories of approximately 180 galaxies from the simulation. There are 237 time steps in the simulation, but not all halos exist for the entire time interval, so the original data set is composed of $N=10,426$ samples. This includes data for all halos and every timestep at which they exist and present stellar activity. For the NNs to train on more data under the magnitude limit, we add additional samples to the dataset consisting of the original samples shifted forward in time by 50 Myr, 100 Myr, and 150 Myr. The full dataset then contains $41,704$ samples. We also test the performance of our NN on real objects at $10<z<12$ (see section 3.4), which is additional motivation for using shifted data and obtaining more samples at lower redshifts. We find that training on the shifted data drastically improves predictions of all the variables. In the analysis of redshift only, we subtract the filter magnitudes in each sample by the lowest magnitude in the sample, so that the NNs learn how to determine the redshift of objects based solely on their spectral shape and regardless of their inherent brightness. Our initial tests revealed that redshift and inherent brightness are less correlated than redshift and the shape of the spectra, so we conclude that intrinsic brightness may be a form of noise that can be helpful to remove. We find that this subtraction significantly increases the accuracy of redshift predictions. We apply cuts to the data to account for observational constraints and interests: 1) Only use samples corresponding to $z>10$. 2) Only include samples where at least two filters have a magnitude brighter than the magnitude limit, to reduce redundant information. Then set all magnitudes fainter than the magnitude limit equal to the magnitude limit to mimic real observations. 3) Only include filter combinations where the maximum redshift is at least 15, so we can study galaxies in our full target range $11.5<z<15$. Also, only include combinations where the maximum redshift of both the training and test set is at least 14.25. This reduces ma jor differences in the values between the training and test sets. After these cuts are applied, the number of samples in our data set drops to approximately $20,000\pm 3,000$, depending on the filter combination. The NNs train on this data, but ultimately our goal is to assess the NN's ability to make predictions with the original data. Thus, we use the original, unshifted data to obtain the test set. In other words, none of the test galaxies interact with the augmented dataset in any way. Since we validate our results on the unshifted data only, the augmented training data introduce no physical bias. After cuts are applied, the final size of the unshifted dataset is roughly $6,000\pm 1,000$ samples. Since the root halos are the same for both the shifted and unshifted data, they are randomly split between training and testing in the same way. We set the magnitude limit at ABmag = 36, based on the approximate sensitivity limit of 31 at a signal-to-noise ratio (SNR) of 10 and exposure time of 100 hours for the JWST filter F200W, according to the Exposure Time Calculator (ETC) v2.0 (Pickering et al., 2016). The magnitude difference is accounted for by the ability to detect sources up to 5 magnitudes fainter with gravitational lensing (Richard et al., 2014; Johnson et al., 2014). We must also consider JWST's maximum visit time of 100 hours, according to the Astronomer's Proposal Tool (APT) v2022.3.1 (Roman et al., 2004). Given that each filter has its own sensitivity, the SNR achieved by each filter in 100 hours to detect ABmag = 31 differs substantially, as shown in Table 2. In addition, the exposure times required to achieve SNR $\geq 5$ varies considerably for each filter. This table also records what percentage of the magnitudes are brighter than 36 for each filter, taking into account the gravitational lensing effect. In order to take into account the observational feasibility of each filter combination, we present results for an unconstrained case, an observationally feasible case, and an observationally ideal case. The unconstrained case considers every mathematically possible combination for three to seven filters. For the observationally feasible case, we consider combinations with four or six filters and select the combinations that have an equal number of short wavelength (SW) filters (F150W, F200W, F162M, F182M) and long wavelength (LW) filters (F277W, F356W, F444W). These SW and LW filter pairs do not necessarily need to have similar sensitivities because the same observing time is used for all filters in this scenario (100 hours, the maximum). For the observationally ideal case, we determine the exposure time required to get $\mathrm{SNR}=5$ at ABmag = 31 for each filter (see Table 2). We consider combinations with four filters and select the combinations that have not only an equal number of SW and LW filters, but similar sensitivities between the filter pairs. This allows us to pair the SW and LW filters by exposure time, to take advantage of the parallel observing feature between SW and LW filters. For this case, we exclude F444W due to its extremely long exposure time of 663.5 hours. ## 3 Results ### Best Unconstrained Results The best unconstrained results are the best predictions made by a NN when all possible combinations of filters, from three to seven filters, are accounted for. In this scenario, a total of ninety-four combinations are considered, excluding the five combinations removed by the third cut. Considering each possible activation function and number of layers in Table 1, this amounts to 2,256 configurations for redshift and mass-weighted age, 1,880 configurations for halo mass, and 1,504 configurations for stellar mass. The configurations are sorted by NRMS for redshift, by MSE for halo mass, and by RMS for stellar mass and mass-weighted age. We then take the top 20 filter configurations for for each set (giving a total of 100 configurations) and rerun each configuration with 51 different seeds. For each configuration, we take the median across all 51 seeds (including the original) of the metric used to sort the configurations \begin{table} \begin{tabular}{l c c c} Filter & SNR (100 h) & Visibility \% & Exposure time (h) \\ & & & to SNR = 5 \\ \hline F150W & 13.5 & 41.7 & 10.7 \\ \hline F200W & 13.5 & 67.2 & 10.8 \\ \hline F277W & 9.0 & 63.4 & 24.0 \\ \hline F356W & 7.0 & 56.8 & 42.5 \\ \hline F444W & 3.1 & 51.3 & 663.5 \\ \hline F162M & 9.5 & 55.0 & 25.2 \\ \hline F182M & 10.4 & 66.1 & 20.7 \\ \hline \end{tabular} _Notes:_ The SNR shown for each filter is for 100 hours of exposure time at ABmag = 31. The visibility percentage corresponds to the fraction of filter magnitudes in our shifted data set that are less than 36 (after the first cut is applied and before the second). The exposure time is the number of hours required to reach ABmag = 31 with SNR = 5. The SNRs and exposure times are calculated based on ETC v2.0, with an aperture radius of $0.04"$ and a $0.58"-0.7"$ background sky annulus. The background is assumed to be the benchmark background model (RA=17h 26m 44s, Dec=$-73^{\circ}19^{\prime}56"$ on May 5, 2022) (Rigby et al., 2022). \end{table} Table 2: SNRs, visibility percentages, and exposure times for each of the JWST filters considered in this work (NRMS, RMS, or MSE). The medians are then sorted and the configuration with the lowest median is taken to be the best for a given variable, with an uncertainty described by the 15th and 85th percentile of the sorting metric across all seeds. The results presented here are those corresponding to the best configuration. For some configurations, however, there is no data for one of the seeds due to the third cut. Then these configurations have 50 seeds, and we take the median to be the higher of the two mid-values. Due to the visibility constraint imposed on the data (at least two filters need to display a magnitude brighter than the magnitude limit), there are fewer data points corresponding to farther, older, high-redshift galaxies. Therefore, we expect it to be difficult for the NNs to predict higher redshifts and mass-weighted ages as a result. We also anticipate difficulty in detecting properties of dimmer, low-mass galaxies, but more accurate predictions for galaxies with greater halo masses and total stellar masses. As we explain below, these features are generally what we observe in our results. Fig. 3 shows the relationship between the output of the NNs (the prediction) and the results from the simulation (the ground truth), for the best NNs of each variable. The closer the data is to modelling the black dashed line, the better the NN is at predicting the values of a given variable. Plots showing the loss as a function of epochs are included in the Appendix (section 5.1). Overall, the losses roughly converge for each of the variables, though for halo mass the loss is extremely oscillatory and demonstrates difficulty in converging. This is an indication of the inherent difficulty in using filter fluxes to predict halo masses in a range as low as ours. The best NN's predictions of redshift closely match the true values, as displayed in Fig. 3. The NRMS of this NN ($0.009^{+0.003}_{-0.002}$) is comparable to that of (Bisigello et al., 2016), (Gomes et al., 2017), and (Wilson et al., 2020), with $\mathrm{NRMS}<0.01$, $\sim 0.03$, and $\sim 0.1$ respectively. Our redshift range covers several filter transi Figure 3: Prediction vs. simulation of the test data from the best NN for each variable, in the unconstrained case. The black dashed line indicates a one-to-one correspondence between the predicted outputs from the NN and true values from the simulation. The top left plot shows that Gainn can predict the redshift of galaxies at $11.5\lesssim z\lesssim 15$ with few errors and outliers. Stellar mass (top right) can also be predicted by Gainn with low errors. Predictions for halo mass (bottom left) show considerable scatter, consistent with the loss plot in the appendix indicating the NN’s difficulty in learning from the halo mass data. Gainn can predict mass-weighted age (bottom right) reasonably well despite some scatter and outliers, as long as the ages are greater than about 18 Myr. tions, so there is some variation when transitioning between filters. These errors occur because there is a slight gap in the throughputs between the different filters, and thus there is no data in a small region of the wavelength space. The NN presents more scatter and a greater number of outliers at $z\sim 12-13$ and $z\sim 14$ as a result of the change in filters at these redshifts. The filter transition at z=12.5 is between F150W and F200W, and the transition at z=14 is F200W to F277W. The slight bias of the NN's redshifts from the true values at $z\gtrsim 14$ occurs due to the low availability of training data at $z\gtrsim 14$ for that particular filter combination and seed. Only $\sim 5\%$ of the training data for redshift lies in that range. Other than the filter transitions and the small deviation at $z\gtrsim 14$, the relationship between the predicted and simulated redshifts is very tightly correlated. As expected, the predictions of stellar mass are generally more accurate at higher values. An exception is the bias observed at $M_{}\gtrsim 6.2$, where the NN underestimates the values of stellar mass. This could be explained in part by limited training data ($\sim 4\%$) in that range. There is also more bias at $M_{}\lesssim 5.25$, but this is not surprising given the greater difficulty in predicting lower stellar masses. The result for halo mass requires a different interpretation. As stated previously, the loss plot of halo mass indicates that the NN could not learn enough about this variable with the information it was given. Therefore, the relationship between the true and predicted halo masses shown in Fig. 3 is not a much better fit than what a random guess could provide (which is what the NNs begin with). Nonetheless, the plot does show some correlation between the simulated and predicted halo masses from the NN. In fact, the best NNs predicting stellar mass and halo mass perform well relative to the results in the literature. The RMS for stellar mass ($0.073^{+0.017}_{-0.008}$) is comparable to that of (Bisigello et al., 2017), which presents RMS no lower than 0.04. The MSE for halo mass ($0.022^{+0.006}_{-0.004}$) is comparable to 0.01, presented by (Zhou and Han, 2022). Both references use mass ranges higher than ours. Since higher stellar and halo masses correspond to brighter galaxies and therefore more information to train on, the comparison between our results is conservative. Results for mass-weighted age have more scatter and outliers than redshift and stellar mass. Surprisingly, predictions of higher mass-weighted ages do not appear to be worse than lower mass-weighted ages, contradicting our expectations. In fact, the flat line at predicted mass-weighted ages of $\sim 18$ Myr suggests that the NN cannot recognize when a galaxy is younger than $\sim 18$ Myr. This is \begin{table} \begin{tabular}{c c c c c c c c c c} Variable & $f$ & Layers & \multicolumn{4}{c}{Filters} & Samples & Samples \\ & & & & & & (shifted) & (unshifted) \\ \hline \multicolumn{8}{c}{Unconstrained} \\ z & elu & 10 & F150W & F200W & F277W & F162M & F182M &... & 23019 & 6714 \\ \hline $M_{}$ & elu & 15 & F150W & F200W & F277W & F356W & F162M &... & 22656 & 6704 \\ \hline $M_{h}$ & swish & 2 & F150W & F356W & F444W & F162M &... &... & 19899 & 5707 \\ \hline t & softplus & 10 & F200W & F277W & F356W & F444W & F162M &... & 22605 & 6704 \\ \hline \multicolumn{8}{c}{Feasible} \\ z & elu & 10 & F200W & F277W & F356W & F444W & F162M & F182M & 23242 & 6940 \\ \hline $M_{}$ & gelu & 10 & F150W & F200W & F277W & F356W & F444W & F162M & 22656 & 6704 \\ \hline $M_{h}$ & swish & 2 & F150W & F356W & F444W & F162M &... &... & 19899 & 5707 \\ \hline t & gelu & 2 & F200W & F277W & F356W & F444W & F162M & F182M & 23242 & 6940 \\ \hline \multicolumn{8}{c}{Ideal} \\ z & swish & 10 & F277W & F356W & F162M & F182M &... &... & 22670 & 6673 \\ \hline $M_{}$ & elu & 10 & F277W & F356W & F162M &... &... & 22670 & 6673 \\ \hline $M_{h}$ & linear & 2 & F150W & F277W & F356W & F162M &... &... & 21325 & 6319 \\ \hline t & softplus & 10 & F200W & F277W & F356W & F162M &... &... & 22605 & 6704 \\ \hline \end{tabular} _Notes:_ This table reports the set of hyper-parameters (activation function, number of layers, and filter combination) that leads to the lowest error in predicting each variable in each case. The last two columns indicate the final number of samples in the shifted and unshifted datasets for each filter combination, after all cuts have been applied. In the observationally feasible and ideal cases, a single filter combination provides the most accurate prediction of redshift and one other variable (mass-weighted age for the feasible case and stellar mass for the ideal case). Thus, more than one variable can be predicted at high accuracy with a single JWST filter combination. This combination depends on the availability of JWST resources. \end{table} Table 3: Input parameters of the best NN for each variable and each case likely not due to low availability of training data, given that $\sim 22\%$ of the training data lies in the range $t\lesssim 18$ Myr. This is likely due instead to the high variability in young star cluster spectra. Therefore, this NN should only be used to predict mass weighted ages higher than $\sim 18$ Myr. Table 3 reports the input parameters that led to the best results for each variable. To provide more information on the other configurations, we also discuss which filter appears most frequently in the 100 selected configurations for each variable. In the case of redshift, the most frequently occurring filter is F182M, which agrees with the appearance of this filter in the NN with the best redshift predictions. In fact, it is crucial to note that F182M appeared in every single configuration of the selected 100 for redshift, making it an indispensable filter for redshift prediction. F200W is the most common filter in the selected configurations for mass-weighted age. The corresponding filter for stellar mass is F150W. For halo mass, the most frequently appearing filter is F444W. Consistently, the most common filter in the selected configurations for each variable is also one of the filters in the best configuration for each variable. Therefore, the most predictive filters across all variables and configurations are F182M, F200W, F150W, and F444W. However, this four-filter combination is not very practical when making observation proposals with JWST. The combination has only one LW filter and three SW filters, removing the opportunity to take advantage of JWST's parallel observing capabilities. This issue is accounted for in sections 3.2 and 3.3, in which a more appropriate balance between SW and LW filters is considered in addition to theoretical accuracy. The relationship between the error and the number of filters used as inputs in the NN is illustrated in Fig. 4. For all variables except for halo mass, there is a tendency for the error to decrease between three and five filters, and then increase again when more filters are added. It is expected that the error would generally decrease with an increasing number of filters, due to the additional information that more filters would provide. However, adding more filters also adds more noise. This occurs here because a greater number of filters implies a greater probability that some data points will have Figure 4: Error vs. number of filters for all variables in all cases, given the best results for each number of filters. In the unconstrained case (blue), the error has a global minimum at five filters for every variable other than halo mass. Also, with the exception of halo mass, the error for all other variables decreases with increasing number of filters in the observationally feasible case (green). Since we only consider combinations with four filters in the observationally ideal case (red), its error is represented by a single point for each variable. most of the filter magnitudes equal to the magnitude limit, given that only two of the filter magnitudes need to be brighter than the magnitude limit for the data point to pass the condition of the second cut. The neural network only sees the magnitude limit as a bound, so magnitudes at the magnitude limit do not provide any useful information. In other words, the NN is only training on the magnitudes below the magnitude limit, which may be distinguishable from one another. The result of the competing factors that adding filters provides more information but also more noise is the global minimum observed at five filters. The dominant factor is the former, given that there is a significant drop in error between three and five filters compared to the marginal increase between five and seven filters. This is an indication that adding more information can be extremely beneficial to the NN's ability to make predictions and might outweigh any potential addition of noise. Halo mass does not exhibit this behavior, however. The minimum occurs at four filters and then the error increases significantly at five filters. This could be a consequence of the NN's difficulty in training on the halo mass data. Since the results for halo mass are not much better than a random guess, it is more likely that they could deviate from expected behavior. Fig. 5 displays plots illustrating the spread of error across seeds for all configurations. The variation among the different seeds demonstrates how well each NN trains on different datasets. The configurations with shorter lines correspond to NNs that learn well regardless of the provided datasets, whereas longer lines indicate that the NNs have trouble with a large variety of input data. Since each seed offers a unique set of galaxies for the NNs to train on, the length of the line is a measure of how well the NNs learn from different galaxies. The location of the median point is a measure of how well a given NN performs relative to others overall. For most variables, the medians are very close to each other up to a point, but then tend to deviate when more configurations with three filters are considered. This is consistent with our earlier discussion regarding the benefit to having more filters to provide input data. The wide variety of line sizes suggests considerable variation in how the Figure 5: Error spread across seeds of all configurations. Each line represents a configuration and ranges from the minimum to maximum error. The points correspond to the median errors of the configurations. Thus, the lines show the full range of errors among the seeds for each configuration. NNs handle various datasets. Some NNs predicting halo mass could not make accurate predictions with certain seeds, which led to very large errors not shown in the plot. ### Best Observationally Feasible Results In the observationally feasible scenario, we add constraints that take into account JWST resources and observing capabilities. We continue adopting the maximum exposure time of 100 hours for each filter, but we only consider combinations with an equal number of SW and LW filters. This narrows our results to configurations with four or six filters that have two or three of each filter type. The resulting number of filter combinations is twenty-two, leading to 528, 440, and 352 configurations for redshift and mass-weighted age, halo mass, and stellar mass respectively. As in the unconstrained case, we rerun the top 20 configurations for each number of filters with 50 new seeds and take the median of the sorting metric from each configuration. In this section, we present the results for the configuration with the lowest median for each variable in the feasible case. Table 4 documents the metrics of the best NNs in all cases. It is possible for the same NN to yield the best results in both the unconstrained and observationally feasible cases. This can occur if the best filter combination in the unconstrained case also satisfies the conditions of the feasible case. All input parameters and metrics would then be identical in both cases. In fact, the same inputs and metrics are observed between the two cases for halo mass in Tables 3 and 4, indicating that the same NN produced the best predictions of each variable. However, in general the permitted configurations of the feasible case do not include the best configuration of the unconstrained case. We therefore expect that the results of the feasible case are usually less accurate than those of the unconstrained case. Most of the metrics are a measure of error in the neural network's predictions, so lower values imply more accurate results (except for $R^{2}$, which indicates a better fit with higher values). According to Table 4, redshift follows the anticipated trend. The metrics of the feasible case are slightly greater than those of the unconstrained case for redshift (considering the absolute value of the average bias). The RMS of stellar mass and mass-weighted age conforms to the expected trend, but the magnitude of the stellar mass bias is lower in the feasible case and the stellar mass MSE is equal in the two cases. Overall, the differences between the metrics of the feasible and unconstrained cases are marginal. As in the unconstrained case, we also consider how often each filter appears in the 40 selected configurations for each variable in the observationally-feasible case. In the selected configurations for redshift and halo mass, the most frequently occurring filter is F444W. For stellar mass and mass-weighted age, F277W is the filter that appears the most. Accordingly, F444W is in the best combination predicting redshift and halo mass, and F277W is in the best combination for recovering stellar mass and mass-weighted age. Therefore, F444W and F277W are the most predictive filters in the feasible case. If F200W, F356W, F162M, and F182M were added to these two filters, the resulting six-filter combination would be the one that provides the best predictions of redshift and mass-weighted age. The green line in Fig. 4 corresponds to the error with respect to the number of filters for the observationally feasible case. For all variable except halo mass, the error is lower for combinations with six filters than combinations with four filters, following the predicted downward trend. The error in predicting halo mass exhibits an upward trend. As in the unconstrained case, a potential cause for this may be the the NN's difficulty in predicting halo mass. ### Best Observationally Ideal Results The observationally ideal results are another subset of the unconstrained results. Through this scenario, we propose alternative observing strategies that minimize exposure times and avoid exhaustive use of JWST resources. Here we include combinations of four filters that contain pairs of SW and LW filters with similar exposure times. According to Table 2, there are five pairs of an SW and LW filter with a difference in exposure time less than 20 hours: F277W-F150W, F277W-F200W, F277W-F162M, F277W-F182M, and F356W-F162M. Thus, we classify these as base pairs. In the four-filter case, there are three combinations with two base pairs of distinct filters: F277W-F182M-F356W-F162M, F277W-F200W-F356W-F162M, and F277W-F150W-F356W-F162M. Each filter combination is run through the NN with a set of other changing parameters (activation function and number of layers), so there are still many configurations considered in this scenario (72 for redshift and mass-weighted age, 60 for halo mass, and 48 for stellar mass). Once again, we rerun the top 20 configurations for each number of filters and take the median across the 51 seeds of each configuration. We present the results of the configuration with the lowest median for each variable in the ideal case. It is expected that the best observationally ideal results will be worse than the best unconstrained results, due to the same explanation provided in the comparison of the best feasible and unconstrained results. Less strin gently, the best ideal results are expected to be worse than the best feasible results, because fewer configurations are considered in the ideal case. This reduces the probability that one of the configurations will yield better results than in the observationally feasible case. While these trends are generally observed in Table 4, there are two exceptions: the average bias of stellar mass and the $R^{2}$ of halo mass. The metrics of all variables except for redshift have overlapping uncertainties among all cases. Overall, the discrepancies between the metrics of the three cases are not significant, allowing for more practical observing strategies that retain much of the accuracy provided by the unconstrained best results. Once again, we consider the most frequently appearing filters in the 20 selected configurations for each variable. The top filters for all variables are F277W, F356W, and F162M. Hence, the most predictive filters in the observationally ideal case are F277W, F356W, and F162M. These three filters and F182W would be ideal for a JWST observing proposal because not only are they useful in recovering the variables, but they also form a combination with an equal number of SW and LW filters. This filter combination is also the best for predicting redshift and stellar mass. ### Redshift Test Case With Real Objects The first practical application of Gainn was to determine the redshift of the triply lensed dual-clump object MACS0647--JD, originally observed by the Hubble Space Telescope and more recently with JWST. Both primary clumps in MACS0647--JD (A and B), which could potentially be individual galaxies in a merger, had different UV slopes and SED fitting implied both had dissimilar and unique star formation histories. Each projection of the combined object (JD1, JD2, and JD3) had higher signal to noise ratios in JWST wide-band filters F200W, F277W, F356W, and F444W, lower flux in F150W, and signals consistent with no detection in higher wavelength bands. This combination suggests a Lyman break within the F150W filter and therefore a redshift range of $10<z<12$. As previously shown, our shifted data recovers redshifts from our un-shifted test samples, implying that the redshift-dependent changes to the galaxy formation environment were not impactful enough to skew our predictions. Therefore, we attempted to use our shifted data set to attempt to recover redshifts over the entire range from $10<z<12$ and make a prediction for MACS0647--JD with the intention of comparing our results to other predictions and using the multiple projections as a further test of the robustness of our methodology on real observational data. In our results, which are more fully presented in Yu-Yang Hsiao et al. (2022), the first projection of the combined object (JD1) had dramatically smaller mean absolute errors ($\langle\Delta z\rangle<0.00228$, as calculated using Eq. 9) in the NN validation set than fits using Bagpipes, piXedfit, BEAGLE, Prospector, and EASY. This is owed in part to our use of fully simulated star formation histories to create our SEDs, which may contain periods of realistically bursty or suppressed star formation that can be challenging to recover or model using other techniques. Our photometric redshift prediction for JD1 ($z=10.4761$) also fell comfortably within the error region for the predictions from each of the other methods ($10.42<z<10.68$), suggesting that our methodology does not measurably bias photometric redshift predictions on real objects, including those slightly outside the redshift range of our simulation. \begin{table} \begin{tabular}{c c c c c c c c c c} \multicolumn{1}{c}{} & \multicolumn{3}{c}{Redshift} & \multicolumn{3}{c}{Stellar Mass} & \multicolumn{2}{c}{Halo Mass} & Age \\ \hline Case & NRMS & $\langle\Delta z\rangle$ & NMAD & $\eta$ & RMS & $\langle\Delta M_{\star}\rangle$ & MSE & MSE & $R^{2}$ & RMS \\ \hline Unconstrained & 0.009 & -0.008 & 0.002 & 0.059 & 0.073 & -0.035 & 0.005 & 0.022 & 0.59 & 10.866 \\ 15th/85th perc. & ${}^{+0.003}_{-0.002}$ & ${}^{+0.056}_{-0.045}$ & ${}^{+0.003}_{-0.000}$ & ${}^{+0.617}_{-0.059}$ & ${}^{+0.016}_{-0.008}$ & ${}^{+0.050}_{-0.003}$ & ${}^{+0.003}_{-0.001}$ & ${}^{+0.006}_{-0.004}$ & ${}^{+0.066}_{-0.043}$ & ${}^{+3.189}_{-1.410}$ \\ \hline Feasible & 0.014 & 0.015 & 0.003 & 1.934 & 0.074 & -0.023 & 0.005 & 0.022 & 0.59 & 10.937 \\ 15th/85th perc. & ${}^{+0.004}_{-0.004}$ & ${}^{+0.024}_{-0.076}$ & ${}^{+0.003}_{-0.000}$ & ${}^{+1.312}_{-1.291}$ & ${}^{+0.013}_{-0.010}$ & ${}^{+0.048}_{-0.002}$ & ${}^{+0.003}_{-0.001}$ & ${}^{+0.006}_{-0.004}$ & ${}^{+0.066}_{-1.516}$ \\ \hline Ideal & 0.029 & -0.02 & 0.013 & 11.475 & 0.079 & 0.003 & 0.006 & 0.024 & 0.627 & 11.398 \\ 15th/85th perc. & ${}^{+0.007}_{-0.004}$ & ${}^{+0.008}_{-0.210}$ & ${}^{+0.001}_{-0.007}$ & ${}^{+2.584}_{-4.732}$ & ${}^{+0.018}_{-0.011}$ & ${}^{+0.019}_{-0.030}$ & ${}^{+0.003}_{-0.001}$ & ${}^{+0.003}_{-0.004}$ & ${}^{+0.016}_{-0.066}$ & ${}^{+3.157}_{-1.453}$ \\ \hline \end{tabular} _Notes:_ For redshift, the metrics used to evaluate the performance of the NN are NRMS, $\langle\Delta z\rangle$, NMAD, and $\eta$. Stellar mass predictions are evaluated by RMS, $\langle\Delta M_{\star}\rangle$, and MSE. The corresponding metrics for halo mass are MSE and $R^{2}$. Finally, mass-weighted age predictions are assessed by the RMS metric. The results from the unconstrained case are almost always better, but not significantly. Thus, the observationally feasible and ideal cases can be used to create JWST proposals with filter combinations that lead to accurate predictions while preventing exhaustive use of JWST resources. \end{table} Table 4: Final metrics for the best NNs in all cases Defining $\sigma_{z}$ as the standard deviation of the predicted redshifts between measurements, each projection of the clumps returned similar redshifts between the clumps ($\sigma_{z}<0.063$) and between the projections ($\sigma_{z}<0.15$), which was consistent with systematic errors in obtaining precise photometeric values for less bright projections and components of MACS0647--JD than JD1. Our $\sigma_{z}$ values were also smaller than the $\sigma_{z}$ values for the other methods, further suggesting that remaining measurement differences may be dominated by observational systematics. Errors were also significantly smaller than results from our network when trained on a larger redshift range, so we suggest retraining on a smaller redshift range when looking at individual objects and for cases when precision in photometric redshifts measurements is important, such as when studying possible pre-merger galaxies. ## 4 Summary and Discussion We use NNs to predict the redshift, stellar mass, halo mass, and mass-weighted age of simulated galaxies at $11.5\lesssim z\lesssim 15$ based on JWST filter magnitudes. To explore the entire parameter space of the NNs, we vary the activation function, number of layers, and filter combination. For each of the top performing configurations of these parameters, we investigate the effect of using different seeds to split the data between training and testing. In theory, we find that F150W, F200W, F182M, and F444W are crucial filters in the study of high-redshift galaxies. In the unconstrained case, Gain can predict redshift, stellar mass, halo mass, and mass-weighted age with errors as low as $0.009^{+0.003}_{-0.002},0.073^{+0.017}_{-0.008},0.022^{+0.006}_{-0.004}$, and $10.866^{+3.189}_{-1.410}$ (where "errors" refers to NRMS, RMS, MSE, and RMS respectively). In addition to reporting the best results considering all possible combinations, we also present results that take into account JWST resources and observing times. The most predictive filters across all variables in the observationally feasible case are F444W and F277W. Gainn's predictions of redshift, stellar mass, halo mass, and mass-weighted age in the feasible case have errors as low as $0.014^{+0.004}_{-0.004},0.074^{+0.013}_{-0.010},0.022^{+0.006}_{-0.004}$, and $10.937^{+4.904}_{-1.516}$. The accuracy of predictions in the ideal case is very close to the accuracy of the feasible results, indicating that we can interpret the ideal results as a reasonable balance between theory and observational constraints. The most predictive filters across all variables in the ideal case are F277W, F356W, and F162M. Gainn achieves errors as low as $0.029^{+0.007}_{-0.004},0.079^{+0.018}_{-0.011}$, $0.024^{+0.003}_{-0.004}$, and $11.398^{+3.157}_{-1.453}$ when recovering redshift, stellar mass, halo mass, and mass-weighted age in the ideal case. Overall, the most important findings of this work are the following: (i) The optimal number of filters tends to be around five. Adding more filters is generally helpful for the NN's performance, but at some point adding more filters will add more noise beyond an acceptable limit. (ii) In general, the best results of the observationally feasible and ideal cases are only slightly worse than those of the unconstrained case. This makes it possible to create a JWST proposal that accounts for the span of available resources and is almost as accurate as the unconstrained best-case scenario. (iii) The most suitable filter combination to use in a JWST observation would be F277W, F356W, F162M, and F182M. The first three filters are the most predictive filters in the ideal case and adding F182M makes this combination the best one for predicting redshift and stellar mass in the ideal case. This combination is also convenient for observing, given that it has an equal number of SW and LW filters with similar sensitivities. (iv) Gainn estimated the redshift of MACS0647--JD, an object observed by JWST. The redshift prediction of the object's first projection was $z=10.4761$, with mean absolute errors $\langle\Delta z\rangle<0.00228$. Estimates from Bagpipes, piXedfit, BEAGLE, Prospector, and EASY did not have errors that low. However, these programs incorporate observational uncertainties into their analyses while Gainn does not. We also acknowledge that results may change somewhat with different simulation models. Relative to results in the literature, the predictions from our best NNs in the unconstrained case are comparable if not more accurate. As for predictions of real observations, our NNs recovered the redshift of a real object observed by JWST with lower errors than other redshift estimation tools. This implies that Gainn shows significant promise for redshift recovery in future JWST observations. ## Acknowledgments LSO was supported in part by the Stanford Physics Department, and by NSF REU funding in initial stages of this work. KSSB was supported by the Porat Postdoctoral Fellowship at Stanford University and the NASA Hubble Postdoctoral Fellowship hosted by Harvard University. TH acknowledges funding from JSPS KAKENHI Grant Numbers 19K23437 and 20K14464. Analysis and data reduction was completed on the Texas Advanced Computing Center's Stampede2 clus
2302.13399	Path Integral Based Convolution and Pooling for Heterogeneous Graph Neural Networks	Graph neural networks (GNN) extends deep learning to graph-structure dataset. Similar to Convolutional Neural Networks (CNN) using on image prediction, convolutional and pooling layers are the foundation to success for GNN on graph prediction tasks. In the initial PAN paper, it uses a path integral based graph neural networks for graph prediction. Specifically, it uses a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It further generalizes such convolution operation to a new transition matrix called maximal entropy transition (MET). Because the diagonal entries of the MET matrix is directly related to the subgraph centrality, it provide a trial mechanism for pooling based on centrality score. While the initial PAN paper only considers node features. We further extends its capability to handle complex heterogeneous graph including both node and edge features.	Lingjie Kong, Yun Liao	2023-02-26T20:05:23Z	http://arxiv.org/abs/2302.13399v1	# Path Integral Based Convolution and Pooling for Heterogeneous Graph Neural Networks ###### Abstract Graph neural networks (GNN) extends deep learning to graph-structure dataset. Similar to Convolutional Neural Networks (CNN) using on image prediction, convolutional and pooling layers are the foundation to success for GNN on graph prediction tasks. In the initial PAN paper [1], it uses a path integral based graph neural networks for graph prediction. Specifically, it uses a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It further generalizes such convolution operation to a new transition matrix called maximal entropy transition (MET). Because the diagonal entries of the MET matrix is directly related to the subgraph centrality, it provide a trial mechanism for pooling based on centrality score. While the initial PAN paper only considers node features. We further extends its capability to handle complex heterogeneous graph including both node and edge features. ## 1 Introduction ### Background With the success of applying Convolutional Neural Networks (CNN) on with fix-size 2D image dataset, researchers have dived deeper into how to apply deep learning on graph dataset. Graph neural networks (GNNs) especially Graph Convolutional Neural Networks (GCN) provides a great framework as baseline. An essential part of GCN is message passing. Message passing not only allow us to encoder richer node feature, but also enables tasks such as node, edge, or even graph prediction. One specific method is to use the graph Laplacian based methods relying on message passing between connected nodes with equal weights across edges. The idea of generic random walk (GRW) defined on graphs is at heart of many graph Laplacian based methods that essentially rely on message passing between directly connected nodes (eg. GCN [2], GraphSAGE [3], etc.). Although proved effective in many graph-based tasks, the GRW-based methods inherently suffer from information dilution as paths between nodes branch out. This can pose great difficulty on graph-level interpretation tasks especially when the multi-hop local structures matter as much as the global node attributes. Ma, Xuan, et al. proposed a path-integral-based graph neural networks (PAN) approach in [1] that overcomes this drawback by considering every path linking the message sender and receiver as the elemental unit in message passing, which is analogous to Feynman's path integral formulation [4] extensively used in statistical mechanics and stochastic processes. Similar ideas have been shown effective in link prediction [5] and community detection [6] tasks. The popular graph attention mechanism [7] can also be viewed as a special case of PAN by restricting the maximal entropy transition (MET) matrix to a particular form. Another stream of related works are the GNN models using multi-scale information and/or higher-order adjacency matrix, such as LanczosNet [8], N-GCN [9] and SGC [10]. This project re-implements and improves upon the PAN approach and evaluates its performance on a molecular classification dataset named ogbg-molhiv. ### Method: PAN and HPAN The PAN approach consists two major modules: (1) a PAN convolution module (PANConv) that gives the message passing rule in the graph, and (2) a PAN pooling module (PANPool) that specifies the approach to extract higher-level features of the graph for the final graph-level tasks. The PANConv module is illustrated in Fig. 0(a). Each message passing step aggregates neighborhood information up to $L$ hops away. The message from node $A$ to node $B$ is weighted according to the number of paths between $A$ and $B$ and the lengths of the paths. This can be considered as an extension to the idea of aggregating information from one-hop neighborhood in most of the GNN approaches. The PANPool module computes the score of each node in the graph and selects a subset of nodes with the highest scores. The subgraph induced from the selected nodes is passed to the next layer. All the other nodes and edges are dropped. The PANPool module is illustrated in Fig. 0(b). The original PAN approach can only deal with node features, and no edge attributes are considered. As an extension to the PAN approach, this project proposes heterogeneous PAN (HPAN) to incorporate edge features in the message passing rule, so that the approach is able to handle heterogeneous graphs. In particular, we add a PanLump layer as shown in figure 3 to sum corresponding node embedding and feature embedding together during message passing. ### Dataset This project uses the ogbg-molhiv dataset, which is adopted from the MoleculeNet, to evaluate the performance of PanConv and hybrid PanPool approach. The ogbg-molhiv dataset contains 41,127 graphs. Each graph represents a molecule, where nodes are atoms, and edges are chemical bonds. The input node features are 9-dimensional, containing atomic number, chirality, etc. The edge features are 3-dimensional, representing bond type, bond stereochemistry, and conjugation, respectively. The task associated with this dataset is to predict whether a target molecule inhibits HIV virus replication or not, which is a binary classification task. ROC-AUC metric will be used for evaluation. This dataset is selected to evaluate the PAN and HPAN algorithms for the following reasons: the key idea of PANConv is to smartly exploit the structural features of the graph in each layer of convolution, while on the other hand, PANPool provides a way to combine the node features with the local graph structure. This can be very useful in understanding graphs representing molecules, where the structural information can be essential. Moreover, the output of the selected algorithm can Figure 1: Visual illustration of the PANConv and PANPool approaches. be naturally interpreted as a graph-level representation, which makes it convenient for a graph-level classification task. ## 2 Existing Method This section details the PAN approach, which is NOT YET in the OGB leaderboard. The PAN approach consists two major components: PANConv as the convolution operator and PANPool as the pooling operator. ### PANConv The core of PANConv is a discrete analogy of Feynman's path integral formulation [4] that can be applied to a graph. The analogy is illustrated in Fig. 2. In the original path integral formulation, the probability amplitude $\phi(\mathbf{x},t)$ is influenced by the surrounding field, where the contribution from $\phi_{0}(\mathbf{x})$ is computed by summing over the influences (denoted by $e^{iS[\mathbf{x},\mathbf{x}]}$) from all paths connecting itself and $\phi(\mathbf{x},t)$: \[\phi(\mathbf{x},t)=\frac{1}{Z}\int\phi_{0}(\mathbf{x})\int e^{iS[\mathbf{x},\mathbf{x}]} \mathcal{D}(\mathbf{x}), \tag{1}\] where $Z$ is the partition function. A graph can be viewed as a discrete version of a continuous field. The path integral formulation is generalized to a graph setting in [1] by replacing the integral over paths to a discrete sum over all possible paths in the graph, and the integral of Lagrangian $e^{iS[\mathbf{x},\mathbf{x}]}$ to a sum over Boltzmann's factor. The analogous path integral formulation on graph $G(V,E)$ then becomes \[\phi_{i}=\frac{1}{Z_{i}}\sum_{j\in V}\phi_{j}\sum_{\{\mathbf{l}\|l_{0}=i,l_{\|l_{\|l} =j}\}}e^{-\frac{E[\mathbf{l}]}{T}}\overset{()}{=}\frac{1}{Z_{i}}\sum_{l=0}^{ \infty}e^{-\frac{E[l]}{T}}\sum_{j\in V}g(i,j;l)\phi_{j}, \tag{2}\] where $\phi_{i}$ denotes the feature/message at node $i$, $\mathbf{l}=(l_{0},l_{1},\ldots,l_{\|l\|})$ is a path connecting node $i$ and node $j$, $E[\mathbf{l}]$ represents the fictitious energy w.r.t. path $\mathbf{l}$, and $T$ is the fictitious energy. $Z_{i}$ is the partition function for node $i$. Equation $()$ holds under the assumption that the fictitious energy only depends on the path length $l$, in which $g(i,j;l)$ denotes the number of length-$l$ paths between nodes $i$ and $j$. Presumably, the energy $E(l)$ is an increasing function of $l$. Note that $g(i,j;l)$ can be computed efficiently from the graph's adjacency matrix $A$ as $(A^{l})_{ij}$. To make the computation tractable, a cutoff maximal path length $L$ is applied to (2), and the path integral formulation is simplified as \[\phi_{i}=\frac{1}{Z_{i}}\sum_{l=0}^{L}e^{-\frac{E(l)}{T}}\sum_{j\in V}(A^{l})_ {ij}\phi_{j}. \tag{3}\] The above expression can be written in a compact form by defining the MET matrix \[M=Z^{-1}\sum_{l=0}^{L}e^{-\frac{E(l)}{T}}A^{l}, \tag{4}\] Figure 2: Schematic analogy between the original path integral formulation in continuous space (left) and the discrete version for a graph (right). where $Z=\mathrm{diag}(Z_{i})$. The name comes from the fact that it realizes maximal entropy under the microcanonical ensemble. Based on the MET matrix, a convolutional layer can be defined as \[X^{(h+1)}=M^{(h)}X^{(h)}W^{(h)}, \tag{5}\] where $h$ is the layer index and $W^{(h)}$ is the trainable weight. A PANConv module further improves over (5) by applying symmetric normalization instead of $Z^{-1}$. The final definition of PANConv is \[X^{(h+1)}=Z^{-1/2}\sum_{l=0}^{L}e^{-\frac{E(l)}{2}}A^{l}Z^{-1/2}X^{(h)}W^{(h)}. \tag{6}\] ### PANPool The diagonal element $M_{i}i$ of the MET matrix resembles the subgraph centrality defined by $\sum_{l=1}^{\infty}(A^{l})_{ii}$ for node $i$, so $\mathrm{diag}(M)$ provides a natural characterization of the nodes' importance. On the other hand, the global importance can be represented by, but not limited to, the strength of the message $X$ itself. PANPool projects feature $X\in\mathbb{R}^{\|V\|\times d}$ by a trainable parameter vector $p\in\mathbb{R}^{d}$ and combines it with $\mathrm{diag}(M)$ to obtain a score vector \[\mathrm{score}=Xp+\beta\mathrm{diag}(M) \tag{7}\] Here, $\beta\in\mathbb{R}$ controls the emphasis on these two potentially competing factors. PANPool then selects a fraction of the nodes ranked by this score (number denoted by $K$), and outputs the pooled feature array $\tilde{X}\in\mathbb{R}^{K\times d}$ and the corresponding adjacency matrix $\tilde{A}\in\mathbb{R}^{K\times K}$. The feature array of the final layer of the stacked "PANConv + PANPool" structure is averaged and then fed into the task-oriented layers, which can be trained against any ground truth label through specific loss functions. ## 3 Method This section elaborates the modifications over the original PAN approach that could bring improvements on the classification task in the dataset of interest. ### HPAN with edge features The original PAN approach can only handle homogeneous graph without edge attributes, and only node attributes are being propagated in the graph. However, the dataset of interest (ogbg-molhiv) includes edge attributes, direct application of the PAN approach to this dataset would simply ignore the edge attributes and treat the edges as if they are homogeneous, which would lose information and potentially result in inferior classification performance (as shown by the experimental results in Section 4. Our method named HPAN proposes to incorporate the edge features in the PAN approach by introducing an additional module named PANLump. The PANLump module is added before the stack of "PANConv + PANPool" layers as shown in Fig. 3. This module adds transformed edge features to the incident node features. The PANLump module works as follows. First, apply atom_encoder to all nodes and edge_encoder to all edges, where the embedding dimensions of both encoders match each other. Then, each edge $(u,v)$ shares its embedding $\mathbf{h}(u,v)$ to both of its vertices, and the incident node embeddings $\mathbf{X}(u)$ (as well as $\mathbf{X}(v)$) update as \[\mathbf{X}^{\prime}(u)\leftarrow\mathbf{MLP}[(1+\epsilon)\mathbf{X}(u)+\mathbf{AGG} \{\mathbf{h}(u,v),\ \forall v\in\mathcal{N}(u)\})]. \tag{8}\] The output of the PANLump module is the updated node features that incorporate its surrounding edge attributes. This output is then fed into the following "PANConv + PANPool" structure, which assumes homogeneous edges. ### Loss function The ogbg-molhiv dataset is highly skewed in the sense that it contains 39,684 samples (not inhabit HIV) and only 1,443 positive samples (inhibit HIV). To make sure that the model does not learn trivially to always output 0, a weighted binary cross entropy loss is adopted in training. In particular, the loss function puts larger weight on positive sample than on negative ones: \[loss=\frac{1}{\|\mathcal{D}\|}\sum_{n=1}^{\|\mathcal{D}\|}\left[\alpha y^{(n)}\log \left(\hat{y}^{(n)}\right)+\left(1-y^{(n)}\right)\log\left(1-\hat{y}^{(n)} \right)\right], \tag{9}\] where $\hat{y}^{(n)}$ is the predicted probability of the $n$-th graph sample has positive label, and the weight $\alpha\geq 1$. ## 4 Experiments and Discussions Both PAN and HPAN are implemented and evaluated on the OGB ogbg-molhiv dataset [11]. The evaluation metric is the ROC-AUC score. The performance is compared to the benchmarks given in [11]. In particular, the benchmark includes GCN and GIN with or without using additional features and virtual nodes. The results are shown in Table 1. \begin{table} \begin{tabular}{\|\|c c c c c c\|\|} \hline Model & Add. Feat. & Virtual Node & ROC-AUC Val & ROC-AUC Test & \#Params \\ \hline \hline GCN & No & Yes & 83.73 $\pm$ 0.78 & 74.18 $\pm$ 1.22 & 1,978,801 \\ \hline GCN & Yes & No & 82.04 $\pm$ 1.41 & 76.06 $\pm$ 0.97 & 527,701 \\ \hline GCN & Yes & Yes & 83.84 $\pm$ 0.91 & 75.99 $\pm$ 1.19 & 1,978,801 \\ \hline GIN & No & Yes & 84.1 $\pm$ 1.05 & 75.2 $\pm$ 1.30 & 3,336,306 \\ \hline GIN & Yes & No & 82.32 $\pm$ 0.90 & 75.58 $\pm$ 1.40 & 1,885,206 \\ \hline GIN & Yes & Yes & 84.79$\pm$ 0.68 & 77.07$\pm$ 1.49 & 3,336,306 \\ \hline \hline PAN & Yes & No & 70.17 $\pm$ 0.29 & 73.06 $\pm$ 0.49 & 26,843 \\ \hline HPAN & Yes & No & 82.27 $\pm$ 0.44 & 76.76 $\pm$ 0.41 & 43,676 \\ \hline \end{tabular} \end{table} Table 1: Comparison of ROC-AUC scores of ogbg-molhiv dataset Figure 3: Comparison of a PAN structure and the corresponding HPAN structure. In the experiments, both PAN and HPAN use 3 layers of PANConv and PANPool followed by a mean pooling module to extract a graph-level representation. The output representation is then fed into a two-layer MLP of size $\{\text{emb\_dim},\text{emb\_dim}/2,1\}$ and ReLU activation. Cutoff length $L=3$ for PANConv1, and $L=2$ for PANConv2 and PANConv3. At each PANPool layer, 80% of the nodes are selected according to the score function, i.e., $\|V\|^{(h+1)}=K=0.8\|V\|^{(h)}$. The MLP in the PANLump module has size $\{\text{emb\_dim},\text{emb\_dim}\neq 2,\text{emb\_dim}\}$ with BatchNorm and ReLU activation. The sum aggregation is adopted as $\mathbf{AGG}(\cdot)$ in PANLump. The embedding dimension is set to 64, and the weight in loss function is set to $\alpha=5.0$ in PAN and $\alpha=10.0$ in HPAN. The ROC-AUC scores and variations are calculated based on 10 independent runs. As shown in the table, the original PAN without considering edge features does not show very satisfying results. Instead, the modified version HPAN shows comparable performance to the benchmarks. The deviation, on the other hand, is significantly reduced compared to the benchmarks, which may be due to the consideration of higher order paths. Besides, since each PANConv layer considers information in up to $L$-hop neighborhoods, fewer layers are needed to extract the information from the entire graph. Therefore, PAN and HPAN can be extremely light-weighted. This is verified in Table 1 in the sense that the number of trainable parameters is several orders of magnitude smaller than the parameters needed for the benchmarks. ## 5 Conclusion The PAN approach, which takes multi-hop message passing into account in each graph convolutional layer, was implemented and applied to the graph classification task for the ogbg-molhiv dataset. A HPAN method was proposed as an extension to the original PAN approach that enables the model to incorporate edge attributes. Experimental results showed that the original PAN approach gave inferior performance compared to the existing benchmarks due to the ignorance of edge features, while the modified HPAN method achieved comparable performance to the benchmarks with a extremely small model. Besides, PAN and HPAN also showed significantly smaller variance in performance across independent runs. In this perspective, PAN provides a more efficient and stable learning model for the graph classification task. Future works would include smarter ways to incorporate edge features in each layer of PANConv or PANPool and theoretical comparison between wide PANs (i.e., large $L$) and deep GNN.
2305.16534	Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression	This paper introduces a novel theoretical framework for the analysis of vector-valued neural networks through the development of vector-valued variation spaces, a new class of reproducing kernel Banach spaces. These spaces emerge from studying the regularization effect of weight decay in training networks with activations like the rectified linear unit (ReLU). This framework offers a deeper understanding of multi-output networks and their function-space characteristics. A key contribution of this work is the development of a representer theorem for the vector-valued variation spaces. This representer theorem establishes that shallow vector-valued neural networks are the solutions to data-fitting problems over these infinite-dimensional spaces, where the network widths are bounded by the square of the number of training data. This observation reveals that the norm associated with these vector-valued variation spaces encourages the learning of features that are useful for multiple tasks, shedding new light on multi-task learning with neural networks. Finally, this paper develops a connection between weight-decay regularization and the multi-task lasso problem. This connection leads to novel bounds for layer widths in deep networks that depend on the intrinsic dimensions of the training data representations. This insight not only deepens the understanding of the deep network architectural requirements, but also yields a simple convex optimization method for deep neural network compression. The performance of this compression procedure is evaluated on various architectures.	Joseph Shenouda, Rahul Parhi, Kangwook Lee, Robert D. Nowak	2023-05-25T23:32:10Z	http://arxiv.org/abs/2305.16534v3	# Vector-Valued Variation Spaces and Width Bounds for DNNs: Insights on Weight Decay Regularization ###### Abstract Deep neural networks (DNNs) trained to minimize a loss term plus the sum of squared weights via gradient descent corresponds to the common approach of training with weight decay. This paper provides new insights into this common learning framework. We characterize the kinds of functions learned by training with weight decay for multi-output (vector-valued) ReLU neural networks. This extends previous characterizations that were limited to single-output (scalar-valued) networks. This characterization requires the definition of a new class of neural function spaces that we call vector-valued variation (VV) spaces. We prove that neural networks (NNs) are optimal solutions to learning problems posed over VV spaces via a novel representer theorem. This new representer theorem shows that solutions to these learning problems exist as vector-valued neural networks with widths bounded in terms of the number of training data. Next, via a novel connection to the multi-task lasso problem, we derive new and tighter bounds on the widths of homogeneous layers in DNNs. The bounds are determined by the effective dimensions of the training data embeddings in/out of the layers. This result sheds new light on the architectural requirements for DNNs. Finally, the connection to the multi-task lasso problem suggests a new approach to compressing pre-trained networks. Keywords: deep neural networks, regularization, sparsity, variation spaces, weight decay, multi-task lasso ###### Contents * 1 Introduction * 2 Vector-Valued Variation Spaces and Weight Decay * 2.1 Scalar-Valued Variation Spaces * 2.2 Vector-Valued Variation Spaces * 2.2.1 The Effect of the Choice of Norm * 2.2.2 A Representer Theorem for Vector-Valued Variation Spaces * 2.2.3 The Curse of Dimensionality * 3 Data-Dependent Tight Bounds on Deep Neural Network Widths * 3.1 Weight Decay and Multi-Task Lasso * 3.2 Bounds on Sparsity of Multi-Task Lasso Solutions * 3.3 Proof of Theorem 9 * 4 Experiments * 4.1 Neuron Sharing Simulation * 4.2 Multi-Task Lasso Experiments * 4.3 Experiments Compressing pre-trained DNNs * 5 Related Works * 6 Conclusion and Future Work * A Proof of Lemma 4 * A.1 Connection to the Total Variation of Function * B Proof of Theorem 5 ## 1 Introduction Training a deep neural network (DNN) with weight decay and gradient descent corresponds to regularizing with the sum of squared weights. This is a simple yet effective form of regularization that is widely used in practice with many benefits (Bartlett, 1996; Krogh and Hertz, 1991; Bos and Chug, 1996; Zhang et al., 2021; Power et al., 2021; Zhai et al., 2019; Kolesnikov et al., 2020). This paper studies the effect of weight decay on NNs and/or network layers consisting of homogeneous units such as the Rectified Linear Unit (ReLU). We study DNNs $f_{\mathbf{\theta}}$ that solve the weight decay objective \[\min_{\mathbf{\theta}}\sum_{i=1}^{N}\mathcal{L}(\mathbf{y}_{i},f_{\mathbf{\theta}}(\mathbf{x} _{i}))+\frac{\lambda}{2}\\|\mathbf{\theta}\\|_{2}^{2}. \tag{1}\] where $\mathbf{\theta}$ is a vector containing all of the DNN weights and $\mathcal{L}(\cdot,\cdot)$ is a lower semicontinuous loss function. Often, DNNs contain many homogeneous activation functions, such as the ReLU. An activation function $\sigma(z)$ is _homogeneous_ (of degree 1) if $\sigma(cz)=c\sigma(z)$ for any $c>0$. A key observation both in theory (Grandvalet, 1998; Neyshabur et al., 2015; Parhi and Nowak, 2020; Ongie et al., 2020; Jacot, 2023) and practice (Kunin et al., 2021) is that for any DNN that solves (1) the 2-norms of the input and output weights of each homogeneous unit are balanced. This phenomenon is referred to as the neural balance theorem (NBT) (Yang et al., 2022, Theorem 1), which is summarized below. Theorem 1 (Neural Balance Theorem (NBT)): _Let $f_{\mathbf{\theta}}$ be a DNN of any architecture parameterized by $\mathbf{\theta}\in\Theta$ which solves (1). Then, the weights satisfy the following balance constraint: if $\mathbf{w}$ and $\mathbf{v}$ denote the input and output weights of any homogeneous unit in the DNN, then $\\|\mathbf{w}\\|_{2}=\\|\mathbf{v}\\|_{2}$._ While the NBT is a simple observation, it allows for an alternative perspective on the weight decay regularizer. Concretely, consider the shallow vector-valued NN mapping $\mathbb{R}^{d}\rightarrow\mathbb{R}^{D}$ with a homogeneous activation function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ of the form \[f_{\mathbf{\theta}}(\mathbf{x})=\sum_{k=1}^{K}\mathbf{v}_{k}\sigma\bigg{(}\mathbf{w}_{k}^{T} \begin{bmatrix}\mathbf{x}\\ 1\end{bmatrix}\bigg{)},\quad\mathbf{x}\in\mathbb{R}^{d},\;\mathbf{\theta}=(\mathbf{v}_{k},\mathbf{w}_{k})_{k=1}^{K},\] where $\mathbf{v}_{k}\in\mathbb{R}^{D}$ and $\mathbf{w}_{k}\in\mathbb{R}^{d+1}$. A consequence of the NBT is that any solution to the NN training problem with weight decay \[\min_{\mathbf{\theta}}\sum_{i=1}^{N}\mathcal{L}(\mathbf{y}_{i},f_{\mathbf{\theta}}(\mathbf{x}_ {i}))+\frac{\lambda}{2}\sum_{k=1}^{K}\\|\mathbf{w}_{k}\\|_{2}^{2}+\\|\mathbf{v}_{k}\\|_{2} ^{2} \tag{2}\] is also a solution to \[\min_{\mathbf{\theta}}\sum_{i=1}^{N}\mathcal{L}(\mathbf{y}_{i},f_{\mathbf{\theta}}(\mathbf{x}_ {i}))+\lambda\sum_{k=1}^{K}\\|\mathbf{w}_{k}\\|_{2}\\|\mathbf{v}_{k}\\|_{2}. \tag{3}\] Moreover after a rebalancing step of the weights, any solution to (3) is also a solution to (2). Thus, the problems in (2) and (3) are equivalent. We refer the reader to Yang et al. (2022) or the recent survey of Parhi and Nowak (2023b) for more details about this equivalence and its effect on training DNNs. This observation brings about many remarkable implications. In this paper we exploit this connection and bring several new results. A novel characterization of the kinds of functions learned by NNs trained with weight decay.We propose a new class of neural function spaces called _vector-valued variation (VV) spaces_. These infinite-dimensional spaces reduce to standard variation spaces (Kurkova and Sanguineti, 2001; Mhaskar, 2004; Bach, 2017) for NNs with single outputs, but more generally VV spaces are the natural spaces associated with multi-output (vector-valued) NNs trained with weight decay; e.g., NNs in multiclass learning problems. We prove a representer theorem which states that finite-width NNs are optimal solutions to data-fitting problems in VV spaces. Analyzing the VV spaces reveals that weight decay favors solutions in which different outputs share neurons, as opposed to networks that use disjoint subsets of neurons for each output. The representer theorem also provides an upper bound on the required width of each layer, depending on the number of training examples and the output dimension of the network. Furthermore, we prove that the VV spaces are "immune" to the curse of dimensionality since they admit dimension-free approximation rates. Tighter bounds on DNN widths.We show how the weights that solve (1) must also be solutions to a constrained convex optimization, posed over each layer in the deep neural network. Through this connection, we derive new and tighter bounds on the necessary widths of deep neural network layers. The bounds are determined by the effective dimensions of the training data embeddings in/out of the layers, which can be significantly smaller than the previously best known bound found in Jacot et al. (2022, Proposition 7). This sheds new light on architectural requirements for neural nets. Moreover, the connection to the convex optimization suggests a new approach to pruning pre-trained networks. The bounds are based on a reduction to the so-called _multi-task lasso_ problem Obozinski et al. (2006, 2010); Argyriou et al. (2008), and our results provide a novel characterization of the sparsity of multi-task lasso solutions, in general. ## 2 Vector-Valued Variation Spaces and Weight Decay In this section, we propose a new NN Banach space which we call the vector-valued variation (VV) space. The name refers to the fact that the norm that defines the space is a new type of variation norm (related to the concept of total variation) which generalizes the standard scalar-valued variation spaces and variation norms that have previously been studied (Kurkova and Sanguineti, 2001; Mhaskar, 2004; Bach, 2017). We show that shallow, vector-valued NNs with homogeneous (of degree 1) activation functions (e.g., the ReLU) trained with weight decay are optimal functions in the VV space. To establish this result, we prove a novel _representer theorem_ that shows that finite-width vector-valued NNs are optimal solutions to data-fitting problems in the VV space. This result sheds new light on the kind of regularity that weight decay encourages. While we only focus on activation functions that are homogeneous of degree 1, it is clear how the results generalize to any positively homogeneous activation function (e.g., powers of the ReLU). The primary technical difficulty that arises when defining the VV spaces boils down to which choice of norm the space should be equipped with. Indeed, as we will see in Section 2.2 there are many choices of equivalent norms, but only one corresponds to weight decay. ### Scalar-Valued Variation Spaces As a warm-up, before we define the vector-valued variation spaces, we first review the definition of the classical, scalar-valued variation spaces and their connection to training NNs with weight decay. The results stated in this subsection can be found in Bengio et al. (2005); Bach (2017); Parhi and Nowak (2021); Siegel and Xu (2023). The main idea is to consider shallow NNs with possibly continuously many neurons. These NNs are parameterized by a finite (Radon) measure. The scalar-valued variation space is the space of functions mapping $\mathbb{R}^{d}\to\mathbb{R}$ \[\mathcal{V}(\mathbb{R}^{d})\coloneqq\bigg{\{}f(\mathbf{x})=\int_{\mathbb{S}^{d}} \sigma(\mathbf{w}^{T}\overline{\mathbf{x}})\,\mathrm{d}\nu(\mathbf{w}):\;\mathbf{x}\in\mathbb{ R}^{d},\nu\in\mathcal{M}(\mathbb{S}^{d})\bigg{\}},\] where $\mathbb{S}^{d}\coloneqq\{\mathbf{w}\in\mathbb{R}^{d+1}:\,\\|\mathbf{w}\\|_{2}=1\}$ is the unit sphere, $\overline{\mathbf{x}}\coloneqq[\mathbf{x}\ 1]^{T}\in\mathbb{R}^{d+1}$ augments $\mathbf{x}$ to account for a bias term, and $\mathcal{M}(\mathbb{S}^{d})$ is the space of finite (Radon) measures. Since each function $f\in\mathcal{V}(\mathbb{R}^{d})$ is parameterized by a measure $\nu\in\mathcal{M}(\mathbb{S}^{d})$, we introduce the notation \[f_{\nu}(\mathbf{x})\coloneqq\int_{\mathbb{S}^{d}}\sigma(\mathbf{w}^{T}\overline{\mathbf{x}} )\,\mathrm{d}\nu(\mathbf{w}),\quad\mathbf{x}\in\mathbb{R}^{d}. \tag{4}\] The space $\mathcal{V}(\mathbb{R}^{d})$ is a Banach space when equipped with the norm \[\\|f\\|_{\mathcal{V}(\mathbb{R}^{d})}\coloneqq\inf_{\begin{subarray}{c}\nu\in \mathcal{M}(\mathbb{S}^{d})\\ f=f_{\nu}\end{subarray}}\\|\nu\\|_{\mathcal{M}(\mathbb{S}^{d})}, \tag{5}\] where $\\|\cdot\\|_{\mathcal{M}(\mathbb{S}^{d})}$ denotes the total variation norm in the sense of measures. The requirement of the $\inf$ in the definition of the norm arises since the dictionary of neurons $\{\mathbf{x}\mapsto\sigma(\mathbf{w}^{T}\overline{\mathbf{x}})\}_{\mathbf{w}\in\mathbb{S}^{d}}$ is highly redundant. Thus, there are many different representations for a given $f\in\mathcal{V}(\mathbb{R}^{d})$. By choosing the representation with the smallest total variation norm, (5) defines a valid Banach norm on $\mathcal{V}(\mathbb{R}^{d})$. Here, we use the following definition of $\\|\cdot\\|_{\mathcal{M}(\mathbb{S}^{d})}$ \[\\|\nu\\|_{\mathcal{M}(\mathbb{S}^{d})}\coloneqq\sup_{\begin{subarray}{c}\mathbb{ S}^{d}=\bigcup_{i=1}^{n}A_{i}\end{subarray}}\sum_{i=1}^{n}\|\nu(A_{i})\|, \tag{6}\] where $\sup$ is taken over all partitions of $\mathbb{S}^{d}$ (i.e., $A_{i}\cap A_{j}=\varnothing$ for $i\neq j$). This definition is equal to the more conventional definition based on the Jordan decomposition of a measure or the definition as a dual norm (Diestel and Uhl, 1977; Bredies and Holler, 2020). That is, \[\\|\nu\\|_{\mathcal{M}(\mathbb{S}^{d})}=\nu^{+}(\mathbb{S}^{d})+\nu^{-}(\mathbb{ S}^{d})=\sup_{\begin{subarray}{c}g\in\mathcal{C}(\mathbb{S}^{d})\\ \\|g\\|_{L\infty}=1\end{subarray}}\int_{\mathbb{S}^{d}}g(\mathbf{u})\,\mathrm{d}\nu (\mathbf{u}),\] where $\nu=\nu^{+}-\nu^{-}$ is the Jordan decomposition of $\nu$ and $C(\mathbb{S}^{d})$ is the space of continuous functions on $\mathbb{S}^{d}$. We use definition in (6) as an analogous definition will play an important role in the vector-valued case. Consider a single neuron $\phi_{v,\mathbf{w}}(\mathbf{x})\coloneqq v\sigma(\mathbf{w}^{T}\overline{\mathbf{x}})$, $\mathbf{x}\in\mathbb{R}^{d}$, where $v\in\mathbb{R}$ and $\mathbf{w}\in\mathbb{R}^{d+1}$. Since $\sigma$ is homogeneous of degree $1$, we have $\phi_{v,\mathbf{w}}(\mathbf{x})=v\\|\mathbf{w}\\|_{2}\sigma(\mathbf{w}^{T}\overline{\mathbf{x}}/\\| \mathbf{w}\\|_{2})$. In this scenario, it is clear that the $\inf$ in (5) is achieved by the one scaled Dirac measure $v\\|\mathbf{w}\\|_{2}\delta_{\mathbf{w}/\\|\mathbf{w}\\|_{2}}$. Thus, if we have the shallow NN \[f_{\mathbf{\theta}}(\mathbf{x})=\sum_{k=1}^{K}v_{k}\sigma(\mathbf{w}_{k}^{T}\overline{\mathbf{ x}}),\quad\mathbf{x}\in\mathbb{R}^{d},\,\mathbf{\theta}=(v_{k},\mathbf{w}_{k})_{k=1}^{K},\] where $v_{k}\in\mathbb{R}$ and the input weights $\mathbf{w}_{k}\in\mathbb{R}^{d+1}$ are all unique after normalization1, we have that the $\inf$ is achieved by Footnote 1: i.e., $\mathbf{w}_{k}/\\|\mathbf{w}_{k}\\|_{2}$, $k=1,\ldots,K$, are unique. \[\sum_{k=1}^{K}v_{k}\\|\mathbf{w}_{k}\\|_{2}\delta_{\mathbf{w}_{k}/\\|\mathbf{w}_{k}\\|_{2}}.\] Therefore, \[\\|f_{\mathbf{\theta}}\\|_{\mathcal{V}(\mathbb{R}^{d})} =\left\\|\sum_{k=1}^{K}v_{k}\\|\mathbf{w}_{k}\\|_{2}\delta_{\mathbf{w}_{k}/\\| \mathbf{w}_{k}\\|_{2}}\right\\|_{\mathcal{M}(\mathbb{S}^{d})}\] \[=\sum_{k=1}^{K}\Bigl{\\|}v_{k}\\|\mathbf{w}_{k}\\|_{2}\delta_{\mathbf{w}_{k}/ \\|\mathbf{w}_{k}\\|_{2}}\Bigr{\\|}_{\mathcal{M}(\mathbb{S}^{d})}\] \[=\sum_{k=1}^{K}\|v_{k}\|\\|\mathbf{w}_{k}\\|_{2}, \tag{7}\] where in the second line we used the fact that the Dirac measures have disjoint support and in the third line we used the property that $\\|a\delta_{\mathbf{u}}\\|_{\mathcal{M}(\mathbb{S}^{d})}=\|a\|$, where $a\in\mathbb{R}$ and $\mathbf{u}\in\mathbb{S}^{d}$. The quantity (7) is often referred to as the _path-norm_(Neyshabur et al., 2015). Furthermore, it is well-known that regularizing a NN with the path-norm is equivalent to the weight decay regularizer (cf., Parhi and Nowak, 2021, Theorem 8). Therefore, training a scalar-output shallow NN with weight decay penalizes the variation norm of the network. This connection is even tighter thanks to the following _Banach space representer theorem_ for the variation spaces. Proposition 2 (Parhi and Nowak (2021)): _Let $(\mathbf{x}_{1},y_{1}),\ldots,(\mathbf{x}_{N},y_{N})\in\mathbb{R}^{d}\times\mathbb{R}$ be a finite dataset. Then, there exists a solution to the variational problem_ \[\inf_{f\in\mathcal{V}(\mathbb{R}^{d})}\,\sum_{i=1}^{N}\mathcal{L}(y_{i},f(\mathbf{ x}_{i}))+\lambda\\|f\\|_{\mathcal{V}(\mathbb{R}^{d})},\quad\lambda>0, \tag{8}\] _where the loss function $\mathcal{L}(\cdot,\cdot)$ is lower semicontinuous in its second argument, that takes the form_ \[f^{\star}(\mathbf{x})=\sum_{k=1}^{K_{0}}v_{k}\sigma(\mathbf{w}_{k}^{T}\overline{\mathbf{x} }),\quad\mathbf{x}\in\mathbb{R}^{d},\] _where $K_{0}\leq N$. Here, $v_{k}\in\mathbb{R}$ and $\mathbf{w}_{k}\in\mathbb{S}^{d}$._ Remark 3: _We note that earlier the works Rosset et al. (2007); Bach (2017) have also considered similar infinite-dimensional variational problems of this form and argue that the solutions admit a finite-parameter representation._ This representer theorem establishes, in particular, that training a sufficiently wide NN (width $K\geq N$) with path-norm regularization (7) to a global minimizer results in a solution to (8). By the equivalence of the path-norm regularizer with weight decay, this result says that any solution to the NN training problem \[\min_{\mathbf{\theta}}\,\sum_{i=1}^{N}\mathcal{L}(y_{i},f(\mathbf{x}_{i}))+\frac{ \lambda}{2}\sum_{k=1}^{K}\|v_{k}\|^{2}+\\|\mathbf{w}_{k}\\|_{2}^{2},\quad\lambda>0, \tag{9}\] is a solution to (8), so long as $K\geq N$. We refer the reader to Parhi and Nowak (2021, Section 2.3) for more details about this correspondence. Therefore, the regularity (or "smoothness") of the solutions to (9) is exactly quantified by the $\mathcal{V}(\mathbb{R}^{d})$-norm. Furthermore, the $\mathcal{V}(\mathbb{R}^{d})$-norm has an analytical description, via the Radon transform (Ongie et al., 2020; Parhi and Nowak, 2021, 2023a). An interesting observation is that the problem in (8) is convex, while the problem in (9) is nonconvex. Thus, by lifting the NN training problem to the space of measures, the training problem has been made convex. For this reason, NNs as in (4) are sometimes called _convex neural networks_(Bengio et al., 2005; Bach, 2017). ### Vector-Valued Variation Spaces The tight connections between weight decay and $\mathcal{V}(\mathbb{R}^{d})$ give key insight into the regularity of solutions to neural network training problems. Unfortunately, these results are not straightforward to extend to the vector-valued (or deep) case. In this section we properly define the vector-valued analog of the variation space such that the connection to weight decay remains. While it is clear that the vector-valued variation space is the set of functions \[\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})\coloneqq\bigg{\{}f(\mathbf{x})=\int_{ \mathbb{S}^{d}}\sigma(\mathbf{w}^{T}\mathbf{\overline{x}})\,\mathrm{d}\mathbf{\nu}(\mathbf{w}) :\;\mathbf{x}\in\mathbb{R}^{d},\mathbf{\nu}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{ D})\bigg{\}},\] where $\mathbf{\nu}$ is now a vector-valued measure (which takes values in $\mathbb{R}^{D}$ as opposed to $\mathbb{R}$), the primary difficulty arises with which choice of norm to impose on the vector-valued measures $\mathbf{\nu}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$ such that there remains a correspondence with weight decay. Indeed, $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$ can be equipped with many (equivalent) norms that result in different regularizations of the NN parameters. There have been two lines of work constructing vector-valued variation spaces (Parhi and Nowak, 2022; Korolev, 2022). Unfortunately, neither of these approaches result in NN regularization that corresponds to weight decay. By viewing $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$ as the $D$-fold Cartesian product $\bigtimes_{j=1}^{D}\mathcal{M}(\mathbb{S}^{d})$, a naive choice of norm would be the mixed norm \[\\|\mathbf{\nu}\\|_{\mathcal{M},p}\coloneqq\left(\sum_{j=1}^{D}\\|\nu_{j}\\|_{ \mathcal{M}(\mathbb{S}^{d})}^{p}\right)^{1/p}=\left(\sum_{j=1}^{D}\left( \sup_{\begin{subarray}{c}\mathbb{S}^{d}=\bigcup_{i=1}^{p}A_{i}\\ n\in\mathbb{N}\end{subarray}}\sum_{i=1}^{n}\|\nu_{j}(A_{i})\|\right)^{p}\right) ^{1/p},\] with $p\geq 1$ and $\mathbf{\nu}=(\nu_{1},\dots,\nu_{D})$ where each $\nu_{j}\in\mathcal{M}(\mathbb{S}^{d})$, $j=1,\dots,D$. A different choice of norm would be \[\\|\mathbf{\nu}\\|_{p,\mathcal{M}}\coloneqq\sup_{\begin{subarray}{c}\mathbb{S}^{d}= \bigcup_{i=1}^{p}A_{i}\\ n\in\mathbb{N}\end{subarray}}\sum_{i=1}^{n}\\|\mathbf{\nu}(A_{i})\\|_{p}=\sup_{ \begin{subarray}{c}\mathbb{S}^{d}=\bigcup_{i=1}^{p}A_{i}\\ n\in\mathbb{N}\end{subarray}}\sum_{i=1}^{n}\left(\sum_{j=1}^{D}\|\nu_{j}(A_{i}) \|^{p}\right)^{1/p},\] with $p\geq 1$. The choice of norm in the above display is the usual definition for the total variation norm of a vector-valued measure. Furthermore, it is well-known that $(\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D}),\\|\cdot\\|_{p,\mathcal{M}})$ is a Banach space. We refer the reader to the monograph of Diestel and Uhl (1977) for a full treatment of vector-valued (and more generally, Banach-valued) measures and the accompanying results. In essence, the key difference between $\\|\cdot\\|_{\mathcal{M},p}$ and $\\|\cdot\\|_{p,\mathcal{M}}$ is the order of the $p$-norm and $\mathcal{M}$-norm. In typical scenarios in which a Banach space is defined by a mixed norm, changing the order of the norms results in different a Banach space2. In the present scenario, since $\mathbb{R}^{D}$ is finite-dimensional, it turns out that $\\|\cdot\\|_{\mathcal{M},p}$ and $\\|\cdot\\|_{q,\mathcal{M}}$ are equivalent norms for any $p,q\geq 1$. This is summarized in the following lemma. Lemma 4: _The norms $\\|\cdot\\|_{p,\mathcal{M}}$, $p\geq 1$, and $\\|\cdot\\|_{\mathcal{M},q}$, $q\geq 1$, are all equivalent Banach norms for $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$._ While this result seems obvious, we could not find a proof in the literature. We provide a proof of Lemma 4 in Appendix A. We also note that the reader can quickly verify that when $D=1$, all norms in Lemma 4 are equal. Thus, we can define equivalent Banach norms on $\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})$ in a similar vein to (5) where we use any of the equivalent norms in Lemma 4. Analogous to (4), we introduce the notation \[f_{\boldsymbol{\nu}}(\boldsymbol{x})\coloneqq\int_{\mathbb{S}^{d}}\sigma( \boldsymbol{w}^{T}\overline{\boldsymbol{x}})\,\mathrm{d}\boldsymbol{\nu}( \boldsymbol{w}),\quad\boldsymbol{x}\in\mathbb{R}^{d}. \tag{10}\] Each norm on $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$ corresponds to a different regularization of neural network parameters. We seek to define the norm \[\\|f\\|_{\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})}\coloneqq\inf_{\begin{subarray} {c}\boldsymbol{\nu}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})\\ f=f_{\boldsymbol{\nu}}\end{subarray}}\llbracket\boldsymbol{\nu}\rrbracket, \tag{11}\] where $\llbracket\boldsymbol{\cdot}\rrbracket$ is one of the equivalent norms in Lemma 4 that corresponds to weight decay. Consider a single vector-valued neuron $\phi_{\boldsymbol{v},\boldsymbol{w}}(\boldsymbol{x})\coloneqq\boldsymbol{v} \sigma(\boldsymbol{w}^{T}\overline{\boldsymbol{x}})$, $\boldsymbol{x}\in\mathbb{R}^{d}$, where $\boldsymbol{v}\in\mathbb{R}^{D}$ and $\boldsymbol{w}\in\mathbb{R}^{d+1}$. Since $\sigma$ is homogeneous of degree $1$, we have $\phi_{\boldsymbol{v},\boldsymbol{w}}(\boldsymbol{x})=\boldsymbol{v}\\| \boldsymbol{w}\\|_{2}\sigma(\boldsymbol{w}^{T}\overline{\boldsymbol{x}}/\\| \boldsymbol{w}\\|_{2})$. Clearly, regardless of the choice of $\llbracket\boldsymbol{\cdot}\rrbracket$-norm in (11), the inf is achieved by the measure $\boldsymbol{v}\\|\boldsymbol{w}\\|_{2}\delta_{\boldsymbol{w}/\\|\boldsymbol{w} \\|_{2}}$. This is a vector (in $\mathbb{R}^{D}$) multiplied by a scalar-valued Dirac measure and is therefore a vector-valued measure in $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$. Thus, if we have the shallow vector-valued NN \[f_{\boldsymbol{\theta}}(\boldsymbol{x})=\sum_{k=1}^{K}\boldsymbol{v}_{k} \sigma(\boldsymbol{w}_{k}^{T}\overline{\boldsymbol{x}}),\quad\boldsymbol{x} \in\mathbb{R}^{d},\,\boldsymbol{\theta}=(\boldsymbol{v}_{k},\boldsymbol{w}_{k })_{k=1}^{K}, \tag{12}\] where $\boldsymbol{v}_{k}\in\mathbb{R}^{D}$ and the input weights $\boldsymbol{w}_{k}\in\mathbb{R}^{d+1}$ are all unique after normalization, we have that the inf is achieved by \[\sum_{k=1}^{K}\boldsymbol{v}_{k}\\|\boldsymbol{w}_{k}\\|_{2}\delta_{\boldsymbol {w}_{k}/\\|\boldsymbol{w}_{k}\\|_{2}}.\] Writing $\boldsymbol{v}_{k}=(v_{k,1},\ldots,v_{k,D})$, a calculation reveals that \[\left\\|\sum_{k=1}^{K}\boldsymbol{v}_{k}\\|\boldsymbol{w}_{k}\\|_{ 2}\delta_{\boldsymbol{w}_{k}/\\|\boldsymbol{w}_{k}\\|_{2}}\right\\|_{\mathcal{M },p} =\left(\sum_{j=1}^{D}\biggl{\\|}\sum_{k=1}^{K}v_{k,j}\\|\boldsymbol{w} _{k}\\|_{2}\delta_{\boldsymbol{w}_{k}/\\|\boldsymbol{w}_{k}\\|_{2}}\biggr{\\|}_{ \mathcal{M}(\mathbb{S}^{d})}^{p}\right)^{1/p}\] \[=\left(\sum_{j=1}^{D}\biggl{(}\sum_{k=1}^{K}\Bigl{\\|}v_{k,j}\\| \boldsymbol{w}_{k}\\|_{2}\delta_{\boldsymbol{w}_{k}/\\|\boldsymbol{w}_{k}\\|_{2} }\Bigr{\\|}_{\mathcal{M}(\mathbb{S}^{d})}\biggr{)}^{p}\right)^{1/p}\] \[=\left(\sum_{j=1}^{D}\biggl{(}\sum_{k=1}^{K}\lvert v_{k,j}\rvert \\|\boldsymbol{w}_{k}\\|_{2}\biggr{)}^{p}\right)^{1/p}. \tag{13}\] Another calculation reveals that \[\left\\|\sum_{k=1}^{K}\mathbf{v}_{k}\\|\mathbf{w}_{k}\\|_{2}\delta_{\mathbf{w}_{k}/\\|\mathbf{w}_{k}\\|_ {2}}\right\\|_{p,\mathcal{M}}=\sum_{k=1}^{K}\Bigl{\\|}\mathbf{v}_{k}\\|\mathbf{w}_{k}\\|_{2 }\delta_{\mathbf{w}_{k}/\\|\mathbf{w}_{k}\\|_{2}}\Bigr{\\|}_{p,\mathcal{M}}=\sum_{k=1}^{K} \lVert\mathbf{v}_{k}\rVert_{p}\lVert\mathbf{w}_{k}\rVert_{2}, \tag{14}\] where we used the property that $\lVert\mathbf{a}\mathbf{\delta}_{\mathbf{u}}\rVert_{p,\mathcal{M}}=\lVert\mathbf{a}\rVert_{p}$, where $\mathbf{a}\in\mathbb{R}^{D}$ and $\mathbf{u}\in\mathbb{S}^{d}$ (cf., Boyer et al., 2019, Section 4.2.3). By the equivalence between (2) and (3) from the NBT, we see that the only choice of norm on $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$ that corresponds to weight decay is to choose \(\lVert\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! We illustrate the types of architectures that are promoted by weight decay in Figure 1 and verify this numerically in Section 4.1. This shows that for the weight decay solution all the output experience strong variation in the same few directions corresponding to the active neurons. An architecture that exhibits neuron sharing is the most natural choice when the network must learn a dataset for which the components of the labels have some relationship or correlation. Multi-class classification tasks are a prime example of such datasets. Therefore, the phenomena of neuron sharing may provide some explanation as to why weight decay is a key ingredient for boosting generalization performance in classification tasks. #### 2.2.2 A Representer Theorem for Vector-Valued Variation Spaces Theorem 5: _Let $(\mathbf{x}_{1},\mathbf{y}_{1}),\ldots,(\mathbf{x}_{N},\mathbf{y}_{N})\in\mathbb{R}^{d}\times \mathbb{R}^{D}$ be a finite dataset. Then, there exists a solution to the variational problem_ \[\inf_{f\in\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})}\,\sum_{i=1}^{N}\mathcal{ L}(\mathbf{y}_{i},f(\mathbf{x}_{i}))+\lambda\\|f\\|_{\mathcal{V}(\mathbb{R}^{d}; \mathbb{R}^{D})},\quad\lambda>0, \tag{18}\] _where the loss function $\mathcal{L}(\cdot,\cdot)$ is lower semicontinuous in its second argument, which takes the form_ \[f^{\star}(\mathbf{x})=\sum_{k=1}^{K_{0}}\mathbf{v}_{k}\sigma(\mathbf{w}_{k}^{T}\mathbf{ \overline{x}}),\quad\mathbf{x}\in\mathbb{R}^{d},\] _where $K_{0}\leq\min\{N^{2},ND\}$. Here, $\mathbf{v}_{k}\in\mathbb{R}^{D}$ and $\mathbf{w}_{k}\in\mathbb{S}^{d}$._ The proof of Theorem 5 appears in Appendix B. What is remarkable here is the bound $K_{0}\leq N^{2}$. Indeed, for large $D$, this bound improves the bound of $ND+1$ predicted by Caratheodory's theorem as it is completely independent of the input and output dimensions. When $D=1$, Theorem 5 recovers Proposition 2. Corollary 6: _Let $f_{\mathbf{\theta}}$ denote a shallow vector-valued NN as in (12) and let $\mathcal{L}(\cdot,\cdot)$ be lower semicontinuous in its second argument. Then, any solution to the neural network training problem_ \[\min_{\mathbf{\theta}}\sum_{i=1}^{N}\mathcal{L}(\mathbf{y}_{i},f(\mathbf{x}_{i}))+\frac{ \lambda}{2}\sum_{k=1}^{K}\lVert\mathbf{v}_{k}\rVert_{2}^{2}+\lVert\mathbf{w}_{k} \rVert_{2}^{2},\quad\lambda>0, \tag{19}\] _is a solution to the variational problem (18), so long as $K\geq N$._ Proof By Theorem 5, there always exists a solution to the variational problem (18) that takes the form of a shallow vector-valued NN with less neurons than data. Let $\Theta_{K}$ denote the space of all NN parameters with width $K$ such that $K\geq N$. Thus, there always exists a solution in the space $\{f_{\mathbf{\theta}}:\,\mathbf{\theta}\in\Theta_{K}\}\subset\mathcal{V}(\mathbb{R}^{ d};\mathbb{R}^{D})$. By (16), any solution to the neural network training problem \[\min_{\mathbf{\theta}\in\Theta_{K}}\sum_{i=1}^{N}\mathcal{L}(\mathbf{y}_{i},f(\mathbf{x}_{ i}))+\lambda\sum_{k=1}^{K}\lVert\mathbf{v}_{k}\rVert_{2}\lVert\mathbf{w}_{k}\rVert_{2}, \quad\lambda>0,\] is a solution to (18). The result then follows by the equivalence between the problem in the above display with (19) discussed previously. $\blacksquare$ Remark 7: _Theorem 5 can be extended to deep networks by considering compositions of functions in $\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})$ using the same technique outlined in Parhi and Nowak (2022, Section 3)._ #### 2.2.3 The Curse of Dimensionality The space $\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})$ has intriguing approximation properties, which carry over from the scalar-valued case (which are well-known). Let $\mathbb{B}^{d}_{1}\coloneqq\{\mathbf{x}\in\mathbb{R}^{d}:\;\\|\mathbf{x}\\|_{2}\leq 1\}$ denote the unit Figure 1: Three neural networks with different sparsity patterns on the weights and their corresponding $\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})$ norm. By the fact that weight decay minimizes the $\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})$ norm we see that weight decay regularization invokes a bias towards the right-most architecture which exhibits _neuron sharing_. Each output depends on the same few neurons. This observation gives insight into the regularity of the optimal functions, if each neuron is a ReLU function then it has strong variation in one direction. Thus neuron sharing implies that all the outputs vary strongly in the same few directions. Contrast this with the middle network where each output has variation in a small number of directions, but this set of directions can be different for each output. ball in $\mathbb{R}^{d}$ and define \[\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})\coloneqq\{f:\mathbb{B}^{d}_{1} \rightarrow\mathbb{R}^{D}:\text{ there exists }g\in\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})\text{ such that }g\|_{\mathbb{B}^{d}_{1}}=f\}.\] This space is a Banach space when equipped with the norm \[\\|f\\|_{\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})}\coloneqq\inf_{\begin{subarray} {c}g\in\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})\\ g\|_{\mathbb{B}^{d}}=f\end{subarray}}\\|g\\|_{\mathcal{V}(\mathbb{R}^{d};\mathbb{R }^{D})}\] By restricting our attention to a bounded domain, we have the continuous embedding $\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})\subset L^{2}(\mathbb{R}^{d}; \mathbb{R}^{D})$. For each $f=(f_{1},\ldots,f_{D})\in\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})$, we have for $j=1,\ldots,D$, $f_{j}\in\mathcal{V}(\mathbb{B}^{d}_{1})$ (the scalar-valued variation space restricted to $\mathbb{B}^{d}_{1}$). In the scalar-valued case, the Maurey-Jones-Barron lemma (Pisier, 1981; Jones, 1992; Barron, 1993) says that, given $f_{j}\in\mathcal{V}(\mathbb{B}^{d}_{1})$, there exists a $K$-term approximant \[f_{j}^{K}(\boldsymbol{x})=\sum_{k=1}^{K}v_{k,j}\sigma(\boldsymbol{w}_{k,j}^{T} \boldsymbol{\overline{x}}) \tag{20}\] with $v_{k,j}\in\mathbb{R}$ and $\boldsymbol{w}_{k,j}\in\mathbb{S}^{d}$ such that \[\\|f_{j}-f_{j}^{K}\\|_{L^{2}(\mathbb{B}^{d}_{1})}\leq C_{0}C_{\sigma,d}\\|f_{j} \\|_{\mathcal{V}(\mathbb{B}^{d}_{1})}K^{-1/2}, \tag{21}\] where $C_{0}>0$ is an absolute constant, independent of $d$ and \[C_{\sigma,d}=\sup_{\boldsymbol{w}\in\mathbb{S}^{d}}\\|\boldsymbol{x}\mapsto \sigma(\boldsymbol{w}^{T}\boldsymbol{\overline{x}})\\|_{L^{2}(\mathbb{B}^{d}_{ 1})}.\] This result is remarkable since it establishes that for any function in $\mathcal{V}(\mathbb{B}^{d}_{1})$, there exists an approximant whose error decays at a rate independent of the input dimension $d$, although the constant depends on (essentially) the volume the domain $\mathbb{B}^{d}_{1}$. This result has a straightforward extension to the vector-valued case. This is summarized in the following theorem which proves that for any function in $\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})$, there exists a $K$-term approximant whose approximation error rate is $K^{-1/2}$, independent of the input and output dimensions. Theorem 8: _Given $f\in\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})$, there exists a $K$-term approximant of the form_ \[f_{K}(\boldsymbol{x})=\sum_{k=1}^{K}\boldsymbol{v}_{k}\sigma(\boldsymbol{w}_{k }^{T}\boldsymbol{\overline{x}}),\] _with $\boldsymbol{v}_{k}\in\mathbb{R}^{D}$ and $\boldsymbol{w}_{k}\in\mathbb{S}^{d}$ such that_ \[\\|f-f_{K}\\|_{L^{2}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})}\leq C_{\sigma,d,D}\\|f\\| _{\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})}K^{-1/2},\] _where $C_{\sigma,d,D}>0$ depends on, at most, $\sigma$, $d$, and $D$, and the $L^{2}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})$-norm is specified by_ \[\\|f\\|_{L^{2}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})}=\left(\int_{\mathbb{B}^{d}_{1 }}\\|f(\boldsymbol{x})\\|_{2}^{2}\,\mathrm{d}\boldsymbol{x}\right)^{1/2}.\] Proof First notice that for any $f=(f_{1},\ldots,f_{D})\in L^{2}(\mathbb{R}^{d};\mathbb{R}^{D})$ we have \[\\|f\\|_{L^{2}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})}^{2}=\int_{\mathbb{B}^{d}_{1}}\!\\| f(\mathbf{x})\\|_{2}^{2}\,\mathrm{d}\mathbf{x}=\int_{\mathbb{B}^{d}_{1}}\!\sum_{j=1}^{D} \lvert f_{j}(\mathbf{x})\rvert^{2}\,\mathrm{d}\mathbf{x}=\sum_{j=1}^{D}\int_{\mathbb{ B}^{d}_{1}}\lvert f_{j}(\mathbf{x})\rvert^{2}\,\mathrm{d}\mathbf{x}=\sum_{j=1}^{D}\lVert f _{j}\rVert_{L^{2}(\mathbb{B}^{d}_{1})}^{2}.\] For any $f\in\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})$ we have \[\sum_{j=1}^{D}\lVert f_{j}\rVert_{\mathcal{V}(\mathbb{B}^{d}_{1})}^{2}\leq D \!\left(\sum_{j=1}^{D}\lVert f_{j}\rVert_{\mathcal{V}(\mathbb{B}^{d}_{1})} \right)^{2}\] since $\lVert\cdot\rVert_{2}\leq\sqrt{D}\lVert\cdot\rVert_{1}$ on $\mathbb{R}^{D}$. Next, \[\sum_{j=1}^{D}\lVert f_{j}\rVert_{\mathcal{V}(\mathbb{B}^{d}_{1})} =\sum_{j=1}^{D}\inf_{\begin{subarray}{c}\nu_{j}\in\mathcal{M}( \mathbb{S}^{d})\\ f_{j}=f_{\nu_{j}}\end{subarray}}\lVert\nu_{j}\rVert_{\mathcal{M}(\mathbb{S}^{d})}\] \[\leq\inf_{\begin{subarray}{c}\mathbf{\nu}\in\mathcal{M}(\mathbb{S}^{ d};\mathbb{R}^{D})\\ f=f_{\mathbf{\nu}}\end{subarray}}\sum_{j=1}^{D}\lVert\nu_{j}\rVert_{\mathcal{M}( \mathbb{S}^{d})}\] \[=\inf_{\begin{subarray}{c}\mathbf{\nu}\in\mathcal{M}(\mathbb{S}^{d}; \mathbb{R}^{D})\\ f=f_{\mathbf{\nu}}\end{subarray}}\lVert\mathbf{\nu}\rVert_{\mathcal{M},1}\] \[\leq C_{d,D}\inf_{\begin{subarray}{c}\mathbf{\nu}\in\mathcal{M}( \mathbb{S}^{d};\mathbb{R}^{D})\\ f=f_{\mathbf{\nu}}\end{subarray}}\lVert\mathbf{\nu}\rVert_{2,\mathcal{M}}=C_{d,D}\lVert f \rVert_{\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})},\] where the equalities of the form $f_{j}=f_{\nu_{j}}$ or $f=f_{\mathbf{\nu}}$ are understood as a function of $\mathbf{x}\in\mathbb{B}^{d}_{1}$. The fourth line follows by Lemma 4, where the constant $C_{d,D}>0$ may depend on $d$ and $D$. To prove the claim, given any $f\in\mathcal{V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})$, we construct a $K$-term approximant $f_{j}^{K}$ for each component $f_{j}$, $j=1,\ldots,D$, as in (21). We then construct the vector-valued function $f_{DK}=(f_{1}^{K},\ldots,f_{D}^{K})$ which has, at most, $DK$ terms. This approximant satisfies \[\lVert f-f_{DK}\rVert_{L^{2}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})}^{2} =\sum_{j=1}^{D}\lVert f_{j}-f_{j}^{K}\rVert_{L^{2}(\mathbb{B}^{d }_{1})}^{2}\] \[\leq\sum_{j=1}^{D}C_{0}^{2}C_{\sigma,d}^{2}\lVert f_{j}\rVert_{ \mathcal{V}(\mathbb{B}^{d}_{1})}^{2}K^{-1}\] \[\leq C_{0}^{2}C_{\sigma,d}^{2}K^{-1}\sum_{j=1}^{D}\lVert f_{j} \rVert_{\mathcal{V}(\mathbb{B}^{d}_{1})}^{2}\] \[\leq C_{0}^{2}C_{\sigma,d}^{2}C_{d,D}^{2}D\lVert f\rVert_{\mathcal{ V}(\mathbb{B}^{d}_{1};\mathbb{R}^{D})}^{2}K^{-1}.\] Therefore, there exists a constant $C_{\sigma,d,D}>0$ that depends on $\sigma$, $d$, and $D$ such that \[\\|f-f_{DK}\\|_{L^{2}(\mathbb{R}^{d}_{1};\mathbb{R}^{D})}\leq C_{\sigma,d,D}\\|f\\|_ {\mathcal{V}(\mathbb{R}^{d}_{1};\mathbb{R}^{D})}K^{-1/2},\] where $C_{\sigma,d,D}=C_{0}C_{\sigma,d}C_{d,D}\sqrt{D}$, which proves the theorem. ## 3 Data-Dependent Tight Bounds on Deep Neural Network Widths In this section, by relating the weight decay optimal solution to a convex optimization we provide new and tighter bounds on the sufficient widths of neural network layers. Moreover, these bounds improve upon the previously best known bounds in Jacot et al. (2022, Proposition 7). Our bounds are determined by the effective dimensions of the training data embeddings at each layer. This sheds new light on architectural requirements for neural nets. Moreover, the connection to the convex optimization suggests a new approach to reducing the neurons in pre-trained networks. Suppose $\widehat{f}_{\boldsymbol{\theta}}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{D}$ is an optimal solution to (1). Let $\boldsymbol{w}_{k}$ and $\boldsymbol{v}_{k}$ be the input and output weights for the $K^{\ell}$ homogeneous neurons in the $\ell$th layer of this network. This layer maps the $i$th (internal) feature representation $\boldsymbol{\phi}_{i}^{\ell-1}\in\mathbb{R}^{K^{\ell-1}}$ (corresponding to the $i$th training example $\boldsymbol{x}_{i}$) to another feature representation $\boldsymbol{\psi}_{i}^{\ell}\in\mathbb{R}^{K^{\ell+1}}$ by the following equation \[\boldsymbol{\psi}_{i}^{\ell}=\sum_{k=1}^{K^{\ell}}\boldsymbol{v}_{k}\sigma( \boldsymbol{w}_{k}^{T}\overline{\boldsymbol{\phi}_{i}^{\ell-1}}), \tag{22}\] where we recall that $\overline{\boldsymbol{\phi}_{i}^{\ell-1}}$ appends a $1$ to $\boldsymbol{\phi}_{i}^{\ell-1}$ to account for a bias term. When $\ell=1$, we define $\boldsymbol{\phi}_{i}^{\ell-1}\coloneqq\boldsymbol{x}_{i}$, the $i$th training data. Theorem 5 guarantees the existence of an optimal solution with $K^{\ell}\leq\min\{N^{2},NK^{\ell+1}\}$. This is, in general, a loose upper bound which scales with the number of training samples $N$. In this section, we show that there exist optimal solutions to (1) with far fewer neurons than expected from the representer theorem. More precisely, we provide a data-dependent bound on the necessary width which is a function of the intrinsic dimensions (rank) of the feature representations. Furthermore, our approach provides a constructive method for finding optimal solutions with narrow widths via a simple convex optimization. The main result is summarized as follows. Theorem 9: _Assume $\widehat{f}_{\boldsymbol{\theta}}$ is a DNN that solves (1). Then for any homogeneous layer of $\widehat{f}_{\boldsymbol{\theta}}$ with input features $\boldsymbol{\phi}_{i}^{\ell-1}$ and output features $\boldsymbol{\psi}_{i}^{\ell}$ and weights $\{\boldsymbol{w}_{k},\boldsymbol{v}_{k}\}$ such that_ \[\boldsymbol{\psi}_{i}^{\ell}=\sum_{k=1}^{K^{\ell}}\boldsymbol{v}_{k}\sigma( \boldsymbol{w}_{k}^{T}\overline{\boldsymbol{\phi}_{i}^{\ell-1}}),\ i=1,\dots,N \tag{23}\] _there exists another representation of this layer of the form_ \[\boldsymbol{\psi}_{i}^{\ell}=\sum_{j=1}^{\widehat{K}^{\ell}}\widehat{ \boldsymbol{v}}_{j}\sigma(\boldsymbol{w}_{k_{j}}^{T}\overline{\boldsymbol{ \phi}_{i}^{\ell-1}}),\ i=1,\dots,N \tag{24}\] _where the number of neurons $\widehat{K}^{\ell}$ satisfies the following bound:_ \[\widehat{K}^{\ell}\leq R.\] _Here $R$ is the product of the dimensions of the the subspace spanned by the output features $\{\mathbf{\psi}_{i}^{\ell}\}_{i=1}^{N}$ and the post-activation i features $\{\mathbf{\phi}_{i}^{\ell}\}_{i=1}^{N}$ where_ \[\mathbf{\phi}_{i}^{\ell}=(\sigma(\mathbf{w}_{k}^{T}\overline{\mathbf{\phi}_{i}^{\ell-1}})) _{k=1}^{K^{\ell}}\in\mathbb{R}^{K^{\ell}}\] _for $i=1,\cdots,N$._ We arrive at this result by first reducing the weight decay objective for this layer to a multi-task lasso problem over the output weights. Then by studying the multi-task lasso problem we present a novel proof that there always exists a solution where the number of active groups (corresponding to active neurons) lies within our bounds. Finally, we combine both of these insights to prove Theorem 3.2. Remark 10: _Empirical evidence (Nar et al., 2019; Waleffe and Rekatsinas, 2022; Huh et al., 2023) and theoretical arguments (Papyan et al., 2020; Le and Jegelka, 2022) provide evidence that the feature representations can often be (effectively) low-rank, especially at deeper layers. These observations and Theorem 3.2 suggest that it may be sufficient in practice to use relatively narrow deeper layers. Moreover, by our definition of $R$, we must have that $R\leq N^{2}$ therefore even in the case when the feature embeddings are full rank our bounds improve over the upper bound of $N(N+1)$ of Jacot et al. (2022)._ ### Weight Decay and Multi-Task Lasso Assume $\widehat{f}_{\mathbf{\theta}}$ is a DNN that solves (1) and that its $\ell$th layer is homogeneous with input and output features $\{\mathbf{\phi}_{i}^{\ell-1},\mathbf{\psi}_{i}^{\ell}\}_{i=1}^{N}$. We now relate weight decay solutions at any homogeneous layer to a (convex) multi-task lasso optimization. The key to this connection is the equivalance of the following three optimization problems (a solution to any one provides a solution to the other two): \[\begin{split}\min_{\{\mathbf{w}_{k},\mathbf{v}_{k}\}}&\sum _{k=1}^{K^{\ell}}\left(\\|\mathbf{v}_{k}\\|_{2}^{2}+\\|\mathbf{w}_{k}\\|_{2}^{2}\right)\\ \text{s.t.}&\mathbf{\psi}_{i}^{\ell}=\sum_{k=1}^{K^{ \ell}}\mathbf{v}_{k}\sigma(\mathbf{w}_{k}^{T}\mathbf{\phi}_{i}^{\ell-1})\,\ i=1,\ldots,N\end{split} \tag{25}\] \[\begin{split}\min_{\{\mathbf{w}_{k},\mathbf{v}_{k}\}}&\sum _{k=1}^{K^{\ell}}\\|\mathbf{v}_{k}\\|_{2}\\|\mathbf{w}_{k}\\|_{2}\\ \text{s.t.}&\mathbf{\psi}_{i}^{\ell}=\sum_{k=1}^{K^{ \ell}}\mathbf{v}_{k}\sigma(\mathbf{w}_{k}^{T}\mathbf{\phi}_{i}^{\ell-1})\,\ i=1,\ldots,N\end{split} \tag{26}\] \[\begin{split}\min_{\{\mathbf{w}_{k},\mathbf{v}_{k}\}}& \sum_{k=1}^{K^{\ell}}\\|\mathbf{v}_{k}\\|_{2}\\ \text{s.t.}&\mathbf{\psi}_{i}^{\ell}=\sum_{k=1}^{K^{\ell }}\mathbf{v}_{k}\sigma(\mathbf{w}_{k}^{T}\mathbf{\phi}_{i}^{\ell-1})\text{ and }\\|\mathbf{w}_{k}\\|_{2}=1\\ & i=1,\ldots,N\end{split} \tag{27}\] Note that in each of these problems, we are optimizing over the input and output weights of only the $\ell$th layer and not the entire DNN. The problem in (25) is the weight decay objective; any solution to this constrained optimization results in a layer that has the same objective value and input-output mappings as the original solution $\widehat{f}_{\mathbf{\theta}}$. Its solutions are also solutions to problem (26) and vice-versa by Theorem 1. Also because the layer is homogeneous, we may constrain each $\\|\mathbf{w}_{k}\\|_{2}=1$ and absorb the overall scale of the neuron into $\mathbf{v}_{k}$, which leads to problem (27). Clearly, a solution to (26) is also a solution to (27). Moreover, any solution to (27) is also a solution to (25), after redistributing the scale so that $\\|\mathbf{w}_{k}\\|_{2}=\\|\mathbf{v}_{k}\\|_{2}$. Next consider a fourth optimization problem, which is a (further) constrained version of (27). Let $\{\widehat{\mathbf{w}}_{k}\}$ denote the input weights of the $\ell$th layer of the original weight decay solution $\widehat{f}_{\mathbf{\theta}}$ and consider \[\begin{split}\min_{\{\mathbf{v}_{k}\}}& \lambda\sum_{k=1}^{K^{\ell}}\\|\mathbf{v}_{k}\\|_{2}\\ \text{s.t.}&\mathbf{\psi}_{i}^{\ell}=\sum_{k=1}^{K^{ \ell}}\mathbf{v}_{k}\sigma\big{(}\widehat{\mathbf{w}}_{k}^{T}\mathbf{\phi}_{i}^{\ell-1}/ \\|\widehat{\mathbf{w}}_{k}\\|_{2}\big{)}\\ & i=1,\ldots,N\end{split} \tag{28}\] This optimization is like (27) but with the added constraint that the (normalized) input weight vectors are fixed to be the same as those in the original solution. Clearly $\widehat{\mathbf{v}}_{k}=\\|\widehat{\mathbf{w}}_{k}\\|_{2}\widehat{\mathbf{v}}_{k}$ is a solution to (27). Let $\mathbf{W}$ be a matrix where the rows are the normalized input vectors of the original solution (i.e. the $k$th row is $\widehat{\mathbf{w}}_{k}/\\|\widehat{\mathbf{w}}_{k}\\|_{2}$) and define the matrix $\mathbf{\Phi}^{\ell-1}$ with columns $\overline{\mathbf{\phi}_{i}^{\ell-1}}$, $i=1,\ldots,N$. The post-activation input feature representations are the columns of the matrix \[\mathbf{\Phi}^{\ell}=\sigma(\mathbf{W}\mathbf{\Phi}^{\ell-1})\.\] Now if $\mathbf{V}$ is a matrix where each column is an output weight, then the output feature embeddings are the columns of the matrix $\mathbf{\Psi}^{\ell}$ where \[\mathbf{\Psi}^{\ell}=\mathbf{V}\mathbf{\Phi}^{\ell}.\] With this notation we can express (27) more compactly as the following _multi-task lasso_ problem: \[\begin{split}\min_{\{\mathbf{v}_{k}\}}&\sum_{k=1}^{K^{ \ell}}\\|\mathbf{v}_{k}\\|_{2}\\ \text{s.t.}&\mathbf{\Psi}^{\ell}=\mathbf{V}\mathbf{\Phi}^{ \ell}.\end{split} \tag{29}\] This problem is commonly referred to as the _multi-task lasso_ problem Obozinski et al. (2006, 2010); Argyriou et al. (2008). A solution to this problem will have each column of $\mathbf{V}$ either entirely zero or non-zero. In our setting, each column of $\mathbf{V}$ corresponds to a neuron and thus the number of nonzero columns in a solution to (29) determines the number of neurons in the $\ell$th layer. ### Bounds on Sparsity of Multi-Task Lasso Solutions In this section we present our theorem on the sparsity of the multi-task lasso problem. Our proof relies extensively on Caratheodory's theorem (Clarke, 2013, Proposition 2.6) which we first state here Theorem 11 (Caratheodory): _Let $S$ be a subset of a normed vector space with finite dimension $R$. Then every point $\mathbf{x}\in\text{Conv}(S)$, the convex hull of $S$, can be represented by a convex combination of at most $R+1$ points from $S$._ We next present a novel result characterizing the sparsity of the solution to the multi-task lasso problem. Since this applies to any application of the multi-task lasso problem, the result is of interest beyond the neural network setting of this paper. Theorem 12: _Consider the optimization_ \[\min_{\mathbf{V}=[\mathbf{v}_{1},\cdots,\mathbf{v}_{K}]} \sum_{k=1}^{K}\\|\mathbf{v}_{k}\\|_{2}\] (30) s.t. \[\mathbf{\Psi}=\mathbf{V}\mathbf{\Phi},\] _where $\mathbf{\Psi}\in\mathbb{R}^{D\times N}$ and $\mathbf{\Phi}\in\mathbb{R}^{K\times N}$ are matrices with ranks $r_{\mathbf{\Phi}}$ and $r_{\mathbf{\Psi}}$, respectively. Furthermore, assume that the row space of $\mathbf{\Psi}$ is contained in the row space of $\mathbf{\Phi}$ and that $K\geq r_{\mathbf{\Psi}}$. Then, regardless of how much larger $K$ may be, there exists a solution with at most $r_{\mathbf{\Phi}}r_{\mathbf{\Psi}}$ nonzero columns. Furthermore, if the rows of $\mathbf{\Phi}$ are generic points in the $r_{\mathbf{\Psi}}$-dimensional space they lie in, then any solution must have at least $r_{\mathbf{\Phi}}$ nonzero columns._ Remark 13: _The upper and lower bounds on the number of nonzero columns has the following intuitive explanation. If every row of $\mathbf{\Psi}$ is synthesized using the same $r_{\mathbf{\Phi}}$ rows in $\mathbf{\Phi}$, then the lower bound is achieved. On the other hand, if $r_{\mathbf{\Psi}}$ rows in $\mathbf{\Psi}$ are synthesized using different subsets of $r_{\mathbf{\Phi}}$ rows in $\mathbf{\Phi}$, then the upper bound is met. Our computational experiments demonstrate that the minimum number of nonzero columns in any solution may range between the upper and lower bounds depending on precise structures in $\mathbf{\Psi}$ and $\mathbf{\Phi}$._ Remark 14: _The lower bound follows from the lower bound on the sparsity of lasso solutions (Tibshirani, 2013). The lasso optimization is equivalent to the optimization above with $D=1$._ The hypotheses of the theorem ensure that the feasible set is nonempty and convex. Then, since the objective is coercive, a solution is guaranteed to exist. Suppose that $\mathbf{V}$ is a solution to our problem. Then we will show that there exists a (possibly different) solution $\widetilde{\mathbf{V}}$ with no more than $r_{\boldsymbol{\Phi}}r_{\boldsymbol{\Psi}}$ nonzero columns. Let $\mathrm{col}(\mathbf{V})$ and $\mathrm{col}(\boldsymbol{\Psi})$ denote the column space of $\mathbf{V}$ and $\boldsymbol{\Psi}$ respectively. We first show that $\mathrm{col}(\mathbf{V})=\mathrm{col}(\boldsymbol{\Psi})$. Let $\mathbf{V}=\mathbf{A}+\mathbf{B}$, where $\mathbf{A}\in\mathrm{col}(\boldsymbol{\Psi})$ and $\mathbf{B}\in\mathrm{col}(\boldsymbol{\Psi})^{\perp}$, the component orthogonal to $\mathrm{col}(\boldsymbol{\Psi})$. Then, \[\sum_{k=1}^{K}\\|\boldsymbol{v}_{k}\\|_{2}=\sum_{k=1}^{K}\sqrt{\\|\boldsymbol{a} \\|_{2}^{2}+\\|\boldsymbol{b}\\|_{2}^{2}}\] Let $\mathbf{P}_{\mathrm{col}(\boldsymbol{\Psi})}$ be the orthogonal projection onto $\mathrm{col}(\boldsymbol{\Psi})$. We can then express the constraint $\boldsymbol{\Psi}=\mathbf{P}_{\mathrm{col}(\boldsymbol{\Psi})}\boldsymbol{ \Psi}=\mathbf{P}_{\mathrm{col}(\boldsymbol{\Psi})}(\mathbf{A}+\mathbf{B}) \boldsymbol{\Phi}=\mathbf{A}\boldsymbol{\Phi}$. Thus the solution must have $\mathbf{B}=0$, since anything nonzero would increase the objective without contributing to the constraint. Therefore, $\mathrm{col}(\mathbf{V})=\mathrm{col}(\boldsymbol{\Psi})$ and $\mathrm{rank}(\mathbf{V})=\mathrm{rank}(\boldsymbol{\Psi})$. Now observe that we can also express the constraint as a sum of outer products, \[\boldsymbol{\Psi}=\sum_{k=1}^{K}\boldsymbol{v}_{k}\boldsymbol{\phi}_{k}^{T}\] where $\boldsymbol{v}_{k}$ are the columns of $\mathbf{V}$ and $\boldsymbol{\phi}_{k}^{T}$ are the rows of $\boldsymbol{\Phi}$. Let $\mathbf{M}_{k}=\boldsymbol{v}_{k}\boldsymbol{\phi}_{k}^{T}$. Since the $\boldsymbol{v}_{k}$ belong to an $r_{\boldsymbol{\Psi}}$ dimensional subspace and the $\boldsymbol{\phi}_{k}$ belong to an $r_{\boldsymbol{\Phi}}$ dimensional subspace, the $\mathbf{M}_{k}$ all belong to a subspace of dimension at most $r_{\boldsymbol{\Phi}}r_{\boldsymbol{\Psi}}$, for ease of notation let $R=r_{\boldsymbol{\Phi}}r_{\boldsymbol{\Psi}}$. Now, define the optimal objective value $\gamma=\sum_{k=1}^{K}\\|\boldsymbol{v}_{k}\\|_{2}$ and $\alpha_{k}=\\|\boldsymbol{v}_{k}\\|_{2}/\gamma$ so that $\sum_{k=1}^{K}\alpha_{k}=1$. We can then write $\boldsymbol{\Psi}$ as \[\boldsymbol{\Psi}=\sum_{k=1}^{K}\alpha_{k}\left(\frac{1}{\alpha_{k}}\mathbf{M }_{k}\right)=\sum_{k=1}^{K}\alpha_{k}\widetilde{\mathbf{M}}_{k}.\] This shows that $\boldsymbol{\Psi}$ is in the convex hull of matrices $\widetilde{\mathbf{M}}_{k}$ with dimension at most $R$. Caratheodory's theorem implies that we can represent $\boldsymbol{\Psi}$ by a convex combination of a subset $\{\widetilde{\mathbf{M}}_{j}\}_{j\in J}$ where $J\subset\{1,\cdots,K\}$ and $\|J\|\leq R+1$. Thus we can represent $\boldsymbol{\Psi}$ with no more than $R+1$ nonzero columns vectors in the solution \[\boldsymbol{\Psi}=\sum_{j\in J}\beta_{j}\widetilde{\mathbf{M}}_{j}\ =\ \sum_{j\in J}\beta_{j}\left(\frac{1}{\alpha_{j}}\boldsymbol{v}_{j} \boldsymbol{\phi}_{j}^{T}\right)\] We now apply Caratheodory's again to show that we can satisfy the constraint with no more than $R$ nonzero columns. Assume that $\|J\|=R+1$ (otherwise we are done) and define $\widetilde{\boldsymbol{v}}_{j}=\frac{1}{\alpha_{j}}\boldsymbol{v}_{j}$. Thus $\boldsymbol{\Psi}$ is in the convex hull of \[\left\{\widetilde{\boldsymbol{v}}_{j}\boldsymbol{\phi}_{j}^{T}\right\}_{j\in J}.\] Each matrix $\widetilde{\mathbf{v}}_{j}\mathbf{\phi}_{j}^{T}$ belongs to a subspace of dimension at most $R$, therefore, any matrix in this set can be expressed as a linear combination of the others \[\widetilde{\mathbf{v}}_{i}\mathbf{\phi}_{i}^{T}=\sum_{j\in J;j\neq i}c_{j} \widetilde{\mathbf{v}}_{j}\mathbf{\phi}_{j}^{T}.\] By the subgradient optimality conditions, we will prove that we must have $\sum_{j\in J;j\neq i}c_{j}=1$. This in turn implies that the set of matrices $\{\widetilde{\mathbf{v}}_{j}\mathbf{\phi}_{j}^{T}\}_{j\in J}$ are not only linearly dependent but they also span an $R-1$ dimensional affine space (i.e., an $R-1$ dimensional hyperplane not including the origin (Rockafellar, 1997)). We can then apply Caratheodory's again to show that a convex combination of $R$ matrices from this set suffices to satisfy the constraint. The Lagrangian of 30 is \[\mathcal{L}(\mathbf{V},\nu_{1},\nu_{2})=\sum_{k=1}^{K}\\|\mathbf{v}_{k} \\|_{2}+\nu_{1}^{T}(\mathbf{\Psi}-\mathbf{V}\mathbf{\Phi})\nu_{2}\] with Lagrange multipliers $\nu_{1}\in\mathbb{R}^{D}$ and $\nu_{2}\in\mathbb{R}^{N}$. By the KKT conditions, any solution must satisfy \[0\in\partial_{\mathbf{V}}\mathcal{L},\] \[0=\nabla_{\nu_{1},\nu_{2}}\mathcal{L}.\] Now for our original solution $\mathbf{V}$ we have, \[\partial_{\mathbf{V}}\mathcal{L} =\partial\left(\sum_{k=1}^{K}\\|\mathbf{v}_{k}\\|_{2}\right)-\nu_{1}( \mathbf{\Phi}\nu_{2})^{T}=0 \tag{31}\] \[\nabla_{\nu_{1},\nu_{2}}\mathcal{L} =\mathbf{\Psi}-\mathbf{V}\mathbf{\Phi}=0 \tag{32}\] Note we have no restriction of $\nu_{1}$ and $\nu_{2}$. Then for any $j\in J$ (the nonzero columns of $\mathbf{V}$) the $j$th column of the subgradient of our objective is \[\partial_{\mathbf{V}}\left(\sum_{k=1}^{K}\\|\mathbf{v}_{k}\\|_{2}\right) _{j}=\frac{\mathbf{v}_{j}}{\\|\mathbf{v}_{j}\\|_{2}}.\] Therefore, by Equation (31) for the $j$th column of the subgradient we have \[\partial\left(\sum_{k=1}^{K}\\|\mathbf{v}_{k}\\|_{2}\right)_{j} =\nu_{1}\nu_{2}^{T}\phi_{j} \tag{33}\] \[\implies\frac{\mathbf{v}_{j}}{\\|\mathbf{v}_{j}\\|_{2}} =\nu_{1}\nu_{2}^{T}\phi_{j} \tag{34}\] where $\phi_{j}\in\mathbb{R}^{N\times 1}$ is the $j$th row of $\mathbf{\Phi}$. Now consider the $i$th column of Equation (31) and right-multiply both sides by $\widetilde{\mathbf{v}}_{i}^{T}$ to obtain \[\frac{\mathbf{v}_{i}\widetilde{\mathbf{v}}_{i}^{T}}{\\|\mathbf{v}_{i}\\|_{2}} =\nu_{1}\nu_{2}^{T}\mathbf{\phi}_{i}\widetilde{\mathbf{v}}_{i}^{T}.\] By the linear dependence of the matrices $\{\widetilde{\mathbf{v}}_{j}\mathbf{\phi}_{j}^{T}\}_{j\in J}$ for any $i\in J$ we have $\mathbf{\phi}_{i}\widetilde{\mathbf{v}}_{i}^{T}=\sum_{j\in J;j\neq i}c_{j}\mathbf{\phi}_{j} \widetilde{\mathbf{v}}_{j}^{T}$ for some constants $c_{j}$, substituting this in we get \[\frac{\mathbf{v}_{i}\widetilde{\mathbf{v}}_{i}^{T}}{\\|\mathbf{v}_{i}\\|_{2}} =\nu_{1}\nu_{2}^{T}\sum_{j\in J;j\neq i}c_{j}\mathbf{\phi}_{j} \widetilde{\mathbf{v}}_{j}^{T}\] \[=\sum_{j\in J;j\neq i}c_{j}\nu_{1}\nu_{2}^{T}\mathbf{\phi}_{j} \widetilde{\mathbf{v}}_{j}^{T}\] \[=\sum_{j\in J;j\neq i}c_{j}\frac{\mathbf{v}_{j}\widetilde{\mathbf{v}}_{j} ^{T}}{\\|\mathbf{v}_{j}\\|_{2}}\] where the final equality follows from Equation (34). Now recall that $\widetilde{\mathbf{v}}_{j}=\frac{1}{\alpha_{j}}\mathbf{v}_{j}=\frac{\gamma}{\\|\mathbf{v}_ {j}\\|_{2}}\mathbf{v}_{j}$ plugging this in on both sides and cancelling out the $\gamma$ we have \[\frac{\mathbf{v}_{i}\mathbf{v}_{i}^{T}}{\\|\mathbf{v}_{i}\\|_{2}^{2}}=\sum_{j\in J;j\neq i}c _{j}\frac{\mathbf{v}_{j}\mathbf{v}_{j}^{T}}{\\|\mathbf{v}_{j}\\|_{2}^{2}}.\] Taking the trace of the left and right hand sides gives us \[\sum_{j\neq i}c_{j}=1\] Therefore any $\mathbf{\phi}_{i}\widetilde{\mathbf{v}}_{i}^{T}$ is in the affine hull of the other matrices and thus the matrices $\{\widetilde{\mathbf{v}}_{j}\mathbf{\phi}_{j}^{T}\}_{j\in J}$ form an affine set of dimension $R-1$(Rockafellar, 1997). This implies that the matrices do not just span an $R$ subspace but rather an $R-1$ dimensional hyperplane that does not pass through the origin. Thus the vector space corresponding to this set of matrices has dimension $R-1$ and so we can apply Caratheodory's theorem again to represent the constraint as a convex combination of just $R$ nonzero vectors \[\mathbf{\Psi}=\sum_{j\in J^{\prime}}\widehat{\beta}_{j}\widetilde{\mathbf{v}}_{j}\bm {\phi}_{j}^{T}=\sum_{j\in J^{\prime}}\mathbf{v}_{j}^{}\mathbf{\phi}_{j}^{T}\] where $\mathbf{v}_{j}^{}:=\widehat{\beta}_{j}\widetilde{\mathbf{v}}_{j}$, $J^{\prime}\subset\{1,\cdots,K\}$ and $\|J^{\prime}\|\leq R=r_{\mathbf{\Phi}}r_{\mathbf{\Psi}}$. The coefficients $\widehat{\beta}_{j}$ are nonnegative and sum to $1$. Finally, it remains to show that this new representation of $\mathbf{\Psi}$ has the same objective value as the original solution, indeed we have \[\sum_{j\in J^{\prime}}\\|\mathbf{v}_{j}^{}\\|_{2}=\sum_{j\in J^{\prime}}\frac{ \widehat{\beta}_{j}}{\alpha_{j}}\\|\mathbf{v}_{j}\\|_{2}=\sum_{j\in J^{\prime}} \gamma\widehat{\beta}_{j}=\gamma\] where we recall that $\gamma$ is the objective value obtained by our original assumed solution. The lower bound is a direct consequence of the constraint. Assuming that the rows of $\mathbf{\Phi}$ and $\mathbf{\Psi}$ are generic3 each row of $\mathbf{\Psi}$ is a linear combination of at least $r_{\mathbf{\Phi}}$ rows of $\mathbf{\Phi}$. This implies that each row of $\mathbf{V}$ has at least $r_{\mathbf{\Phi}}$ nonzero entries and thus the number of nonzero columns in $\mathbf{V}$ must be at least $r_{\mathbf{\Phi}}$. ### Proof of Theorem 3.1 We now combine the results from Sections 3.1 and 3.2 to prove Theorem 3.1. Proof* Since $\widehat{f}_{\boldsymbol{\theta}}$ is optimal, any solution to (29) is also a solution to (26). Let $\widehat{K}$ be the number of nonzero columns in the solution to (29). By Theorem (12) there exists a solution $\widehat{\mathbf{V}}$ to (29) such that $\widehat{K}\leq r_{\boldsymbol{\Phi}}r_{\boldsymbol{\Psi}}$. By balancing the output weights in the solution $\widehat{\mathbf{V}}$ with the input weights in $\widehat{\mathbf{W}}$ we get a solution to (25) with only $\widehat{K}$ active neurons. Thus there exists a solution in which the $\ell$-th layer has only $\widehat{K}$ neurons with optimal weights. ## 4 Experiments In this section we present three experiments that validate our theory and demonstrate its potential utility. Our first experiment is a numerical simulation that demonstrates how weight decay encourages neurons sharing for a simple two dimensional data fitting problem. Our second set of experiments validate the bound presented for the multi-task lasso problem in Theorem Theorem 3.1 using synthetic data. In the third set of experiments, we compress the homogenous layers of two pre-trained deep neural network models VGG19 (Simonyan and Zisserman, 2014; Wang et al., 2021) and AlexNet (Krizhevsky et al., 2017) via the multi-task lasso convex optimization problem. We show how in some cases, this principled compression approach can reduce the number of neurons in a layer 10 fold while preserving nearly the same loss on the entire training dataset as well as accuracy on the test set. Remarkably, we find that compressing sometimes helps in generalization performance.4 Footnote 4: The code for reproducing all of our experiments can be found at [https://github.com/joeshenouda/vv-spaces-nn-width](https://github.com/joeshenouda/vv-spaces-nn-width) following the guidance in Shenouda and Bajwa (2023). ### Neuron Sharing Simulation To demonstrate the neuron sharing phenomenon induced by weight decay we trained two vector-valued shallow ReLU neural networks to fit a synthetic two-dimensional dataset. The dataset consists of fifty samples where the features are two-dimensional vectors sampled from a multivariate normal distribution and the labels are also two-dimensional, generated by passing these feature vectors through a randomly initialized ReLU neural network with five neurons. For both networks that we train we initialize them with one hundred neurons. One of the networks is trained with a small amount of regularization and the other with no regularization. Both were trained with full batch gradient descent and a learning rate of 2e-1 for one million iterations. For the regularized network, due to the equivalence between weight decay and (3), we directly regularize the VV norm of the network with a small regularization parameter of 5e-5. We employ the proximal gradient algorithm proposed in Yang et al. (2022) to speed up training. The results are in Figure 2. We see that the regularized network is not only sparser - in terms of the number of active neurons - but also exhibits strong neuron sharing. In contrast without any regularization, many of the neurons are not shared and only contribute to a single output. Figure 2: We train a two output two-dimensional ReLU neural network of the form $f(x)=\sum_{k=1}^{K}\mathbf{v}_{k}\sigma(\mathbf{w}_{k}^{T}\mathbf{x}+b_{k})$ both with and without weight decay. We plot the location of the active neurons contributing to each of the outputs of the network where the size of the point indicates the weight of the neuron. The figures in (a-b) show the neuron locations/weight for the network trained with weight decay whereas (c-d) shows the neuron locations/weight for the network trained without weight decay. We see that in the case of weight decay, we have fewer active neurons, and those neurons that remain are shared across both outputs. ### Multi-Task Lasso Experiments We solve the multi-task lasso problem in Theorem 12 on randomly generated matrices $\mathbf{\Phi}$ and $\mathbf{\Psi}$ using CVXPy (Diamond and Boyd, 2016). Our experiments illustrate that while our bounds hold, the exact number of nonzero columns depends heavily on the data itself. In Figure (3) we show histograms for the distribution of non-zero columns over 100 randomly generated pairs of $\mathbf{\Phi}$ and $\mathbf{\Psi}$. We perform a similar set of experiments in Figure (4) but alter the underlying rank of $\mathbf{\Phi}$. This validates our bound showing that the sparsity of the solution can be much lower depending on the rank of $\mathbf{\Phi}$. We again see that the distribution is highly dependent on the data and difficult to predict exactly. We note however that we never achieve our upper bound which may indicate that they can be further tightened. To demonstrate that the sparsest solutions to the multi-task lasso problem in Theorem (12) depends heavily on the data matrices $\mathbf{\Phi}$ and $\mathbf{\Psi}$ we also ran a small-scale experiment Figure 4: Distribution of the number of active columns for the solutions to the multi-task lasso problem on randomly generated matrices of varying sizes. In this case, we have that $\mathbf{\Phi}$ is low rank while $\mathbf{\Psi}$ remains at a rank of $D$. The red lines indicate the bounds from Theorem 12 Figure 3: Distribution of the number of active columns for the solutions to the multi-task lasso problem on randomly generated matrices of varying sizes. In all these cases $r_{\mathbf{\Phi}}=N$ and $r_{\mathbf{\Psi}}=D$ so we expect $N\leq\widehat{K}\leq ND$. The red lines indicate the bounds from Theorem 12. The wide gap suggests that our upper bound can be tightened. similar to the one in 4.2. However, in these experiments, we exhaustively searched over all $2^{K}$ sparsity patterns that the solution may have. We arrive at the same conclusion as we did in 4.2 observing that the sparsest solution can lie anywhere within our bound as shown in Figure (5). ### Experiments Compressing pre-trained DNNs As a practical application of our theory we conduct experiments compressing real deep neural networks pre-trained on the CIFAR-10 dataset. We first consider a pre-trained VGG19 architecture trained with weight decay. This model consists of various convolutional and batch norm layers followed by a fully connected ReLU layer containing 512 neurons. The output of this layer is $\mathbf{\Psi}\in\mathbb{R}^{10\times N}$ due to having just 10 classes and therefore $\operatorname{rank}(\mathbf{\Psi})\leq 10$. Moreover, we found that for the post-activation feature representation at this layer the first 10 singular values account for 99% of the energy of the signal (Figure 6), thus $\operatorname{rank}(\mathbf{\Phi})\approx 10$. Therefore, our bounds suggest that there exists an alternative optimal representation that is exactly equivalent on the _training_ data, with no more than 100 neurons in this layer. We validate this by applying the multi-task lasso problem (30) on the last fully connected ReLU layer. The results in Table 1 show that we can compress the final ReLU layer down to just 47 neurons while preserving nearly the same loss on the training data and achieving the same generalization performance. Next, we consider the AlexNet architecture which is also composed of multiple convolutional layers followed by three fully connected ReLU layers with 9216, 4096, and 4096 neurons. We apply our multi-task lasso optimization to compress each of these layers in a parallel fashion and report the training loss and test accuracy with the compressed layers replaced individually and all at once. The results are on Table 1. For all four cases, we notice almost no change in the training loss and even better accuracy on the test data in some cases. This is due to the fact that our constraint only enforces that the network remains the same on the training data, while it may behave differently on data it was not trained on. Finally, we highlight that in all these cases no retraining is required to achieve comparable performance. Figure 5: Distribution of number of nonzero columns for the solutions to the multi-task lasso problem. We ran this experiment over 1000 randomly generated matrices $\mathbf{\Phi}$ and $\mathbf{\Psi}$. We see that the sparsest solution is dependent on the data and can vary between our bounds. The red lines indicate the bounds from Theorem 12 The minor discrepancies between the training losses of the original models and the compressed models are due to numerical issues when practically enforcing the constraint in in (28). In practice the constraint in (28) is approximated by formulating the solving the regularized version of the problem (35) and optimizing using proximal gradient descent. These results show us that our connection between weight decay and the multi-task lasso problem can hint at a principled approach to model compression that is also computationally efficient. \[\min_{\{\mathbf{v}_{k}\}}\\|\mathbf{V}\mathbf{\Phi}-\mathbf{\Psi}\\|_{2}^{2}+ \lambda\sum_{k=1}^{K}\\|\mathbf{v}_{k}\\|_{2} \tag{35}\] ## 5 Related Works In this section, we highlight some related works and discuss how our work fits into the current literature. \begin{table} \begin{tabular}{l l l l l} \hline Network Name & Original Width & New Width & Orig. Train Loss (Test Acc.) & New Training Loss (Test Acc.) \\ \hline VGG-19 (last layer) & 512 & 47 & 0.0104 (93.92\%) & 0.0112 (93.88\%) \\ AlexNet (last layer) & 4096 & 797 & 0.2203 (72.17\%) & 0.2213 (72.22\%) \\ AlexNet (penultimate layer) & 4096 & 677 & 0.2203 (72.17\%) & 0.2184 (72.48\%) \\ AlexNet (first linear layer) & 9216 & 4896 & 0.2203 (72.17\%) & 0.2325 (72.28\%) \\ \hline AlexNet (last 3 layers) & (4096,4096,9216) & (797,677,4896) & 0.2203 (72.17\%) & 0.2417(72.02\%) \\ \hline \end{tabular} \end{table} Table 1: Compressing Models Pre-trained on CIFAR-10 Figure 6: Cumulative plot of the singular values of the post-activation feature representations of the last layer of a pre-trained VGG19 on the entire CIFAR-10 dataset. We see that a large majority of the signal is captured by just the first few singular values, indicating that the representations are approximately low rank. Variation Spaces and Representer Theorems for NNs.Understanding NNs by studying the variation space associated with taking their widths to infinity has been explored in the past (Kurkova and Sanguineti, 2001; Mhaskar, 2004; Bach, 2017). Representer theorems for showing that finite width networks are solutions to variational data-fitting problems posed over these spaces have also been developed (Parhi and Nowak, 2021, 2022; Unser, 2023; Bartolucci et al., 2023). However, most of these were restricted in that they only develop theories for shallow and scalar output NNs. Moreover, the norm associated with these variation spaces does not always coincide with regularizers used in practice such as weight decay. We are the first to study the $\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})$-space and develop a novel representer theorem for variational problems posed over such _vector-valued_ function spaces. Weight Decay and Width.Numerous works have observed that weight decay is biased towards solutions with fewer neurons (Savarese et al., 2019; Ongie et al., 2020; Parhi and Nowak, 2021; Yang et al., 2022). Empirically, Yang et al. (2022) showed that training NNs with weight decay for many iterations induces sparsity on the neurons of the network. Moreover, Jacot et al. (2022); Parhi and Nowak (2021) provide bounds on the required width for a DNNs that solve the weight decay objective. Our bounds improve upon the ones presented in both of these works. Moreover, our setting is a more general one, instead of only considering homogeneous DNNs our theory holds for the homogeneous layers of _any_ DNN architecture. Recently, a line of work by Ergen and Pilanci (2021); Mishkin et al. (2022) has shown that training shallow ReLU NNs with weight decay can be recast as a convex program. Their reformulation reveals a sparse regularizer showing how weight decay induces solutions with few neurons. A similar observation was later extended to vector-valued neural networks in Sahiner et al. (2021). Low-Rank Features.It has been observed that DNNs are biased toward learning low-rank features. This has been shown to be true in theory (Du et al., 2018; Le and Jegelka, 2022; Ji and Telgarsky, 2019) for scalar output linear and nonlinear NNs. Moreover, Radhakrishnan et al. (2021) extended the theory for vector-valued linear NNs. Empirical evidence has also shown that the internal representations of deep neural networks are low rank (Huh et al., 2023; Nar et al., 2019; Waleffe and Rekatsinas, 2022). (Huh et al., 2023; Nar et al., 2019). In particular, Waleffe and Rekatsinas (2022) observe that the effective dimensions of hidden post-activation feature embeddings of trained NNs are often much smaller than the actual layer width. The authors utilize this insight to reduce the number of parameters using principal component analysis during training. In contrast, our theory and experiments indicate that narrower networks can be found without complicated procedures during training by instead applying the multi-task lasso as in Section 3.1. ## 6 Conclusion and Future Work In this work we proposed the VV function space which gives insight into the type of NNs solutions that weight decay encourages. Moreover, we proved a new representer theorem for this space showing that finite-width vector-valued NNs are solutions to the variational data-fitting problem over this space. We then developed a novel connection between training DNNs with weight decay and multi-task lasso to prove the sharpest known bounds on network widths. Finally, we presented some experimental results showing the validity and potential utility of our theory. The authors thank Ryan Tibshirani for helpful discussions regarding the lasso and related methods, especially the use of Caratheodory's theorem for characterizing the sparsity of solutions. JS was supported by the ONR MURI grant N00014-20-1-2787. RP was supported by the European Research Council (ERC Project FunLearn) under grant 101020573. KL was supported by the NSF grant DMS-2023239. RN was supported in part by the NSF grants DMS-2134140 and DMS-2023239, the ONR MURI grant N00014-20-1-2787, and the AFOSR/AFRL grant FA9550-18-1-0166, as well as the Keith and Jane Nosbusch Professorship. ## Appendix A Proof of Lemma 4 We prove the following result which corresponds to Lemma 4 by choosing $\Omega=\mathbb{S}^{d}$. Lemma 15: _Let $\Omega$ be any locally compact Hausdorff space. The norms $\left\\|\cdot\right\\|_{p,\mathcal{M}}$, $p\geq 1$, and $\left\\|\cdot\right\\|_{\mathcal{M},q}$, $q\geq 1$ (appropriately defined on $\mathcal{M}(\Omega;\mathbb{R}^{D})$) are all equivalent Banach norms for $\mathcal{M}(\Omega;\mathbb{R}^{D})$._ We first observe that for any $p,q\geq 1$, the norms $\left\\|\cdot\right\\|_{\mathcal{M},p}$ and $\left\\|\cdot\right\\|_{\mathcal{M},q}$ are equivalent by the equivalence of $p$-norms in finite dimensions. Similarly, for any $p,q\geq 1$, the norms $\left\\|\cdot\right\\|_{p,\mathcal{M}}$ and $\left\\|\cdot\right\\|_{q,\mathcal{M}}$ are equivalent. Thus, it suffices to show that $\left\\|\cdot\right\\|_{\mathcal{M},p}$ and $\left\\|\cdot\right\\|_{q,\mathcal{M}}$ are equivalent for some $p,q\geq 1$. We have for any $\boldsymbol{\nu}\in\mathcal{M}(\Omega;\mathbb{R}^{D})$ \[\\|\boldsymbol{\nu}\\|_{\mathcal{M},1} =\sum_{j=1}^{D}\\|\nu_{j}\\|_{\mathcal{M}(\Omega)}\] \[=\sum_{j=1}^{D}\sup_{\Omega\stackrel{{\Omega\to \bigcup_{i=1}^{n}}}{{n\in\mathbb{N}}}}A_{i}\sum_{i=1}^{n}\|\nu_{j}(A_{i})\|\] \[\geq\sup_{\Omega\stackrel{{\Omega\to\bigcup_{i=1}^ {n}}}{{n\in\mathbb{N}}}}A_{i}\sum_{j=1}^{D}\sum_{i=1}^{n}\|\nu_{j}(A_{i})\|\] \[=\sup_{\Omega\stackrel{{\Omega\to\bigcup_{i=1}^{n} }}{{n\in\mathbb{N}}}}A_{i}\sum_{i=1}^{n}\sum_{j=1}^{D}\|\nu_{j}(A_{i})\|\] \[=\sup_{\Omega\stackrel{{\Omega\to\bigcup_{i=1}^{n} }}{{n\in\mathbb{N}}}}A_{i}\sum_{i=1}^{n}\\|\boldsymbol{\nu}(A_{i})\\|_{1}=\\| \boldsymbol{\nu}\\|_{1,\mathcal{M}}. \tag{36}\] Since the inclusion/identity map \[\iota:(\mathcal{M}(\Omega;\mathbb{R}^{D}),\left\\|\cdot\right\\|_{\mathcal{M}, 1})\rightarrow(\mathcal{M}(\Omega;\mathbb{R}^{D}),\left\\|\cdot\right\\|_{1, \mathcal{M}})\] is obviously a bijection and satisfies the continuity bound (36), the bounded inverse theorem (Folland, 1999, Corollary 5.11) guarantees that the inverse $\iota^{-1}$ is also bounded. In other words, there exists a constant $C>0$ (which may depend on $\Omega$ and $D$) such that for all $\boldsymbol{\nu}\in\mathcal{M}(\Omega;\mathbb{R}^{D})$, \[\\|\boldsymbol{\nu}\\|_{\mathcal{M},1}=\\|\iota^{-1}\boldsymbol{\nu}\\|_{ \mathcal{M},1}\leq C\\|\boldsymbol{\nu}\\|_{1,\mathcal{M}}.\] Therefore, we have shown that $\\|\cdot\\|_{\mathcal{M},1}$ and $\\|\cdot\\|_{1,\mathcal{M}}$ are equivalent, which proves the lemma. ### Connection to the Total Variation of Function The total variation of a measure is different than the total variation of a function, but the ideas are tightly linked. In the univariate case, consider a Radon measure $\mu\in\mathcal{M}(\mathbb{R})$ and suppose there exists a function $g_{\mu}:\mathbb{R}\to\mathbb{R}$ such that \[\int_{\mathbb{R}}f(x)\,\mathrm{d}\mu(x)=\int_{\mathbb{R}}f(x)\, \mathrm{d}g_{\mu}(x),\] where $f$ is any bounded continuous function and the integral on the right-hand side is a Riemann-Stieltjes integral. If $g_{\mu}$ is differentiable and its derivative, denoted by $g_{\mu}^{\prime}$, is in $L^{1}(\mathbb{R})$, then we have the equality \[\\|\mu\\|_{\mathcal{M}(\mathbb{R})}=\\|g_{\mu}^{\prime}\\|_{L^{1}(\mathbb{R})}.\] The quantity on the right-hand side is referred to as the _total variation_ of the function $g_{\mu}$. Furthermore, if $g_{\mu}$ is not differentiable in the classical sense, its distributional derivative can be identified with the Radon measure $\mu$, i.e., $g_{\mu}^{\prime}=\mu$, where equality is understood in $\mathcal{M}(\mathbb{R})$. In this case, $\\|g_{\mu}^{\prime}\\|_{\mathcal{M}(\mathbb{R})}=\\|\mu\\|_{\mathcal{M}(\mathbb{R})}$ is the total variation of the function $g_{\mu}$. This correspondence extends to higher dimensions. Indeed, given a function $g:\mathbb{R}^{d}\to\mathbb{R}$, if its distributional gradient $\nabla g$ can be identified with a vector-valued finite Radon measure, then the _isotropic total variation_ of $g$ is \[\mathrm{TV}_{\mathrm{iso}}(g)\coloneqq\\|\nabla g\\|_{2,\mathcal{M}},\] while the _anisotropic total variation_ of $g$ is \[\mathrm{TV}_{\mathrm{aniso}}(g)\coloneqq\\|\nabla g\\|_{1,\mathcal{M}}.\] These two notions of total variation in multiple dimensions are often used in image processing problems. Finally, the isotropic total variation is equivalently specified by \[\mathrm{TV}_{\mathrm{iso}}(g)=\sup_{\begin{subarray}{c}\varphi\in\mathcal{D} (\mathbb{R}^{d};\mathbb{R}^{d})\\ \\|\varphi\\|_{L^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})}=1\end{subarray}}\int_ {\mathbb{R}^{d}}g(\boldsymbol{x})\,\mathrm{div}\varphi(\boldsymbol{x})\, \mathrm{d}\boldsymbol{x},\] where the $L^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$-norm is specified by \[\\|\varphi\\|_{L^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})}\coloneqq\operatorname {ess\,sup}_{\boldsymbol{x}\in\mathbb{R}^{d}}\\|\varphi(\boldsymbol{x})\\|_{2},\] and $\mathcal{D}(\mathbb{R}^{d};\mathbb{R}^{d})$ denotes the space of infinitely differentiable compactly supported functions mapping $\mathbb{R}^{d}\to\mathbb{R}^{d}$. We refer the reader to the book of Evans and Gariepy (2015) for more details about the total variation of a function. ## Appendix B Proof of Theorem 5 Proof.: From the definition of the $\mathcal{V}(\mathbb{R}^{d};\mathbb{R}^{D})$-norm (15), it follows from a standard argument (cf., Bartolucci et al., 2023, Proposition 3.7) that the problem in (18) is equivalent to the problem \[\inf_{\boldsymbol{\nu}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})}\,\sum_{i=1 }^{N}\mathcal{L}\bigg{(}\boldsymbol{y}_{i},\int_{\mathbb{S}^{d}}\sigma( \boldsymbol{w}^{T}\overline{\boldsymbol{x}}_{i})\,\mathrm{d}\boldsymbol{\nu} (\boldsymbol{w})\bigg{)}+\lambda\\|\boldsymbol{\nu}\\|_{2,\mathcal{M}} \tag{37}\] in the sense that their infimal values are the same and if $\boldsymbol{\nu}^{\star}$ is a solution to (37), then \[f_{\boldsymbol{\nu}^{\star}}(\boldsymbol{x})=\int_{\mathbb{S}^{d}}\sigma( \boldsymbol{w}^{T}\overline{\boldsymbol{x}}_{i})\,\mathrm{d}\boldsymbol{\nu} ^{\star}(\boldsymbol{w})\] is a solution to (18). We now proceed in four steps to prove the theorem. Step (i): Existence of solutions to (37).Define \[\mathcal{J}(\boldsymbol{\nu})\coloneqq\sum_{i=1}^{N}\mathcal{L}\bigg{(} \boldsymbol{y}_{i},\int_{\mathbb{S}^{d}}\sigma(\boldsymbol{w}^{T}\overline{ \boldsymbol{x}}_{i})\,\mathrm{d}\boldsymbol{\nu}(\boldsymbol{w})\bigg{)}+ \lambda\\|\boldsymbol{\nu}\\|_{2,\mathcal{M}}\] Given an arbitrary $\boldsymbol{\nu}_{0}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$, let $C_{0}\coloneqq\mathcal{J}(\boldsymbol{\nu}_{0})$. Then, we can transform (37) into the constrained problem \[\inf_{\boldsymbol{\nu}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})}\, \mathcal{J}(\boldsymbol{\nu})\quad\text{s.t.}\quad\\|\boldsymbol{\nu}\\|_{2, \mathcal{M}}\leq C_{0}/\lambda. \tag{38}\] This transformation is valid since any measure that does not satisfy the constraint will have an objective value strictly larger than $\boldsymbol{\nu}_{0}$ and therefore will not be in the solution set. Next, we note that we can write \[\mathcal{J}(\boldsymbol{\nu})=\sum_{i=1}^{N}\mathcal{L}(\boldsymbol{y}_{i}, \mathrm{H}_{i}\{\boldsymbol{\nu}\})+\lambda\\|\boldsymbol{\nu}\\|_{2,\mathcal{M}}\] where \[\mathrm{H}_{i}\{\boldsymbol{\nu}\}\coloneqq\begin{bmatrix}\langle \boldsymbol{\nu},h_{i,j}\rangle\\ \vdots\\ \langle\boldsymbol{\nu},h_{i,D}\rangle\end{bmatrix}\in\mathbb{R}^{D} \tag{39}\] with $h_{i,j}(\boldsymbol{w})\coloneqq\sigma(\boldsymbol{w}^{T}\overline{ \boldsymbol{x}}_{i})\boldsymbol{e}_{j}$, where $\boldsymbol{e}_{j}\in\mathbb{R}^{D}$ is the $j$th canonical unit vector. Clearly $h_{i,j}\in C(\mathbb{S}^{d};\mathbb{R}^{D})$, the space of continuous functions on $\mathbb{S}^{d}$ taking values in $\mathbb{R}^{D}$. In (39), $\langle\,\cdot,\,\cdot\,\rangle$ denotes the duality pairing5 between $C(\mathbb{S}^{d};\mathbb{R}^{D})$ and $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$. Thus, for $i=1,\ldots,N$, $\mathrm{H}_{i}$ is component-wise weak${}^{}$ continuous on $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$. Since $\mathcal{L}(\cdot\,,\,\cdot)$ is lower semicontinuous in its second argument combined with the fact that every norm is weak${}^{}$ continuous on its corresponding Banach space, we have that $\mathcal{J}$ is weak${}^{}$ continuous on $\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})$. By the Banach-Alaoglu theorem (Rudin, 1991, Chapter 3), the constraint set of (38) is weak${}^{}$ compact. Thus, (38) is the minimization of a weak${}^{}$ continuous functional over a weak${}^{}$ compact set. By the Weierstrass extreme value theorem on general topological spaces (Kurdila and Zabarankin, 2006, Chapter 5), there exists a solution to (38) (and subsequently of (37)). Step (ii): Recasting (37) as an interpolation problem.Let $\widetilde{\mathbf{\nu}}$ be a (not neccesarily unique) solution to (37), which is guaranteed to exist by the previous argumentation. For $i=1,\ldots,D$, define \[\mathbf{z}_{i}=\mathrm{H}_{i}\{\widetilde{\mathbf{\nu}}\}\in\mathbb{R}^{D}.\] Then, $\widetilde{\mathbf{\nu}}$ must satisfy \[\widetilde{\mathbf{\nu}}\in\operatorname{arg\,min}_{\mathbf{\nu}\in\mathcal{M}( \mathbb{S}^{d};\mathbb{R}^{D})}\\|\mathbf{\nu}\\|_{2,\mathcal{M}}\quad\text{s.t.} \quad\mathrm{H}_{i}\{\mathbf{\nu}\}=\mathbf{z}_{i},\,i=1,\ldots,N. \tag{40}\] To see this, we note that if this were not the case, it would contradict the optimality of $\widetilde{\mathbf{\nu}}$. This reduction implies that any solution to the interpolation problem (40) will also be a solution to (37). Step (iii): The form of the solution.We can rewrite the interpolation problem as \[\min_{\mathbf{\nu}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D})}\\|\mathbf{\nu}\\|_{2, \mathcal{M}}\quad\text{s.t.}\quad\langle\mathbf{\nu},h_{i,j}\rangle=\mathbf{z}_{i,j}, \,i=1,\ldots,N\text{ and }j=1,\ldots,D. \tag{41}\] This is the vector-valued analogue of the classical (Radon) measure recovery problem with $ND$ weak${}^{}$ continuous measurements. By the abstract representer theorem of Unser (2021, Theorems 2 and 3) (see also Boyer et al., 2019; Bredies and Carioni, 2020), there always exists a solution to (41) that takes the form \[\mathbf{\nu}^{\star}=\sum_{k=1}^{K}c_{k}\mathbf{e}_{k}\] with $K\leq ND$, $c_{k}\in\mathbb{R}\setminus\{0\}$ where for $k=1,\ldots,k$, $\mathbf{e}_{k}$ is an extreme point of the unit regularization ball \[B\coloneqq\{\mathbf{\nu}\in\mathcal{M}(\mathbb{S}^{d};\mathbb{R}^{D}):\;\\|\mathbf{\nu }\\|_{2,\mathcal{M}}\leq 1\}.\] From Werner (1984, Theorem 2), the extreme points of $B$ take the form $\mathbf{a}\delta_{\mathbf{w}}$ with $\mathbf{a}\in\mathbb{R}^{D}$, $\\|\mathbf{a}\\|_{2}=1$, and $\mathbf{w}\in\mathbb{S}^{d}$. Thus, we can write \[\mathbf{\nu}^{\star}=\sum_{k=1}^{K}c_{k}\mathbf{a}_{k}\delta_{\mathbf{w}_{k}}\] with $\mathbf{a}_{k}\in\mathbb{R}^{D}$ and $\mathbf{w}_{k}\in\mathbb{S}^{d}$. By the equivalence between (37) and (18), we find that there exists a solution to (18) that takes the form \[\widehat{f}(\mathbf{x})=\sum_{k=1}^{\widehat{K}}\widehat{\mathbf{v}}_{k}\sigma( \widehat{\mathbf{w}}_{k}^{T}\overline{\mathbf{x}}),\quad\mathbf{x}\in\mathbb{R}^{d},\] where $\widehat{K}\leq ND$, $\widehat{\mathbf{v}}_{k}\coloneqq c_{k}\mathbf{a}_{k}\in\mathbb{R}^{D}$ and $\widehat{\mathbf{w}}_{k}\in\mathbb{S}^{d}$. Step (iv): Sharpening the sparsity bound.Step (iii) establishes that there exists an optimal solution with $\widehat{K}\leq ND$ neurons. Thus, we can apply Theorem 3.2 to find another solution where the number of neurons is bounded by the product of the rank of the labels $\mathbf{Y}\in\mathbb{R}^{D\times N}$ with the rank of the post-activation feature matrix $\mathbf{\Phi}\in\mathbb{R}^{\widehat{K}\times N}$. From the dimensions of these matrices, we see that the product of the ranks is $\leq N^{2}$. Thus, by Theorem 3.2, there exists a solution to (18) that takes the form \[f^{\star}(\mathbf{x})=\sum_{k=1}^{K_{0}}\mathbf{v}_{k}\sigma(\mathbf{w}_{k}^{T}\overline{ \mathbf{x}}),\quad\mathbf{x}\in\mathbb{R}^{d},\] where $K_{0}\leq N^{2}$, $\mathbf{v}_{k}\in\mathbb{R}^{D}$ and $\mathbf{w}_{k}\in\mathbb{S}^{d}$. Thus, we can always find a solution to (18) with $\min\{N^{2},ND\}$ neurons. $\blacksquare$
2310.13822	Adversarial Attacks on Fairness of Graph Neural Networks	Fairness-aware graph neural networks (GNNs) have gained a surge of attention as they can reduce the bias of predictions on any demographic group (e.g., female) in graph-based applications. Although these methods greatly improve the algorithmic fairness of GNNs, the fairness can be easily corrupted by carefully designed adversarial attacks. In this paper, we investigate the problem of adversarial attacks on fairness of GNNs and propose G-FairAttack, a general framework for attacking various types of fairness-aware GNNs in terms of fairness with an unnoticeable effect on prediction utility. In addition, we propose a fast computation technique to reduce the time complexity of G-FairAttack. The experimental study demonstrates that G-FairAttack successfully corrupts the fairness of different types of GNNs while keeping the attack unnoticeable. Our study on fairness attacks sheds light on potential vulnerabilities in fairness-aware GNNs and guides further research on the robustness of GNNs in terms of fairness.	Binchi Zhang, Yushun Dong, Chen Chen, Yada Zhu, Minnan Luo, Jundong Li	2023-10-20T21:19:54Z	http://arxiv.org/abs/2310.13822v2	# Adversarial Attacks on Fairness of ###### Abstract Fairness-aware graph neural networks (GNNs) have gained a surge of attention as they can reduce the bias of predictions on any demographic group (e.g., female) in graph-based applications. Although these methods greatly improve the algorithmic fairness of GNNs, the fairness can be easily corrupted by carefully designed adversarial attacks. In this paper, we investigate the problem of adversarial attacks on fairness of GNNs and propose _G-FairAttack_, a general framework for attacking _various types of_ fairness-aware GNNs in terms of fairness with an _unnoticeable effect_ on prediction utility. In addition, we propose a fast computation technique to reduce the time complexity of G-FairAttack. The experimental study demonstrates that G-FairAttack successfully corrupts the fairness of different types of GNNs while keeping the attack unnoticeable. Our study on fairness attacks sheds light on potential vulnerabilities in fairness-aware GNNs and guides further research on the robustness of GNNs in terms of fairness. The open-source code is available at [https://github.com/zhangbinchi/G-FairAttack](https://github.com/zhangbinchi/G-FairAttack). ## 1 Introduction Graph Neural Networks (GNNs) have achieved remarkable success across various human-centered applications, e.g., social network analysis (Qiu et al., 2018; Lu & Li, 2020; Feng et al., 2022), recommender systems (Ying et al., 2018; Fan et al., 2019; Yu et al., 2021), and healthcare (Choi et al., 2017; Li et al., 2022; Fu et al., 2023). Despite that, many recent studies (Dai & Wang, 2021; Wang et al., 2022; Dong et al., 2022b; 2023b) have shown that GNNs could yield biased predictions for nodes from certain demographic groups (e.g., females). With the increasing significance of GNNs in high-stakes human-centered scenarios, addressing such prediction bias becomes imperative. Consequently, substantial efforts have been devoted to developing fairness-aware GNNs, with the goal of learning fair node representations that can be used to make accurate and fair predictions. Typical strategies to improve the fairness of GNNs include adversarial training (Bose & Hamilton, 2019; Dai & Wang, 2021), regularization (Navarin et al., 2020; Fan et al., 2021; Agarwal et al., 2021; Dong et al., 2023c), and edge rewiring (Spinelli et al., 2021; Li et al., 2021; Dong et al., 2022a). Although these fairness-aware GNN frameworks can make fair predictions, there remains a high risk of the exhibited fairness being corrupted by malicious attackers (Hussain et al., 2022). Adversarial attacks threaten the fairness of the victim model by perturbing the input graph data, resulting in unfair predictions. Such fairness attacks can severely compromise the reliability of GNNs, even when they have a built-in fairness-enhancing mechanism. For instance, we consider a GNN-based recommender system in social networks. In scenarios regarding commercial competition or personal benefits, a malicious adversary might conduct fairness attacks by hijacking inactive user accounts. The adversary can collect user data from these accounts to train a fairness attack model. According to the fairness attack algorithm, the adversary can then modify the attributes and connections of these compromised accounts. As a result, the GNN-based system is affected by the poisoned input data and makes biased predictions against a specific user group. To protect the fairness of GNN models from adversarial attacks, we should first fully understand the potential ways to attack fairness. To this end, we investigate the problem of adversarial attacks on fairness of GNNs in this paper. Despite the importance of investigating the fairness attack of GNNs, most existing studies (Zugner et al., 2018; Zugner and Gunnemann, 2019; Bojchevski and Gunnemann, 2019; Dai et al., 2018; Xu et al., 2019) only focus on attacking the utility of GNNs (e.g., prediction accuracy) and neglect the vulnerability of GNNs in the fairness aspect. To study fairness attacks, we follow existing works of attacks on GNNs to formulate the attack as an optimization problem and consider the prevalent gray-box attack setting (Zugner et al., 2018; Zugner and Gunnemann, 2019; Wang and Gong, 2019; Sun et al., 2019) where the attacker cannot access the architecture or parameters of the victim model (Jin et al., 2021; Sun et al., 2022). A common strategy in gray-box attacks is to train a surrogate model (Zugner et al., 2018; Zugner and Gunnemann, 2019) to gain more knowledge on the unseen victim model for the attack. Compared with conventional adversarial attacks, our attack on fairness is more difficult for the following two challenges: (1) The design of the surrogate loss function. Previous attacks on prediction utility directly choose the loss function adopted by most victim models, i.e. the cross-entropy (CE) loss as the surrogate loss. However, fairness-aware GNN models are trained based on different loss functions for fairness, e.g., demographic parity loss (Navarin et al., 2020; Zeng et al., 2021; Franco et al., 2022; Jiang et al., 2022), mutual information loss (Kang et al., 2022), and Wasserstein distance loss (Fan et al., 2021; Dong et al., 2022). Because the victim model is unknown and can be any type of fairness-aware GNN, the attacker should find a proper surrogate loss to represent different types of loss functions of fairness-aware GNNs. (2) The necessity to make such attacks unnoticeable. If the poisoned graph data exhibits any clues of being manipulated, the model owner could recognize them and then take defensive actions (Zugner et al., 2018), e.g., abandoning the poisoned input data. Conventionally, attackers restrict the perturbation size to make attacks unnoticeable. In fairness attacks, we argue that a distinct change of prediction utility can also be a strong clue of being manipulated. However, none of the existing works on fairness attacks has considered the unnoticeable utility change. In light of these challenges, we propose a novel fairness attack method on GNNs named G-FairAttack, which consists of two parts: a carefully designed surrogate loss function and an optimization method. To tackle the first challenge, we categorize existing fairness-aware GNNs into three types based on their fairness loss terms. Then we propose a novel surrogate loss function to help the surrogate model learn from all types of fairness-aware GNNs with theoretical analysis. To address the second challenge, we propose a novel unnoticeable constraint in utility change to make the fairness attack unnoticeable. Then we propose a non-gradient attack algorithm to solve the constrained optimization problem, which is verified to have a better performance than previous gradient-based methods. Moreover, we propose a fast computation strategy to improve the scalability of G-FairAttack. Our contributions can be summarized as follows. * [leftmargin=,noitemsep,topsep=0pt] Attack Setting. We consider a novel unnoticeable constraint on prediction utility change for unnoticeable fairness attacks of GNNs, which can be extended to general fairness attacks. * Objective Design. We propose a novel surrogate loss to help the surrogate model learn from various types of fairness-aware GNNs with theoretical analysis. * Algorithmic Design. To solve the optimization problem with the unnoticeable constraint, we discuss the deficiency of previous gradient-based optimization methods and design a non-gradient attack method. In addition, we propose a fast computation approach to reduce its time complexity. * Experimental Evaluation. We conduct extensive experiments on three real-world datasets with four types of victim models and verify that our proposed G-FairAttack successfully jeopardizes the fairness of various fairness-aware GNNs with an unnoticeable effect on prediction utility. ## 2 Problem Definition Notation and Preliminary. We use bold uppercase letters, e.g., $\mathbf{X}$, to denote matrices, and use $\mathbf{X}_{[i,:]}$, $\mathbf{X}_{[:,j]}$, and $\mathbf{X}_{[i,j]}$ to denote the $i$-th row, the $j$-th column, and the element at the $i$-th row and the $j$-th column, respectively. We use bold lowercase letters, e.g., $\mathbf{x}$, to denote vectors, and use $\mathbf{x}_{[i]}$ to denote the $i$-th element. We use $P_{X}(\cdot)$ and $F_{X}(\cdot)$ to denote the probability density function and the cumulative distribution function for a random variable $X$, respectively. We use $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathbf{X}\}$ to denote an undirected attributed graph. $\mathcal{V}=\{v_{1},\dots,v_{n}\}$ and $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ denote the node set and the edge set, respectively, where $n=\|\mathcal{V}\|$ is the number of nodes. $\mathbf{X}\in\mathbb{R}^{n\times d_{x}}$ denotes the attribute matrix, where $d_{x}$ is the number of node attributes. We use $\mathbf{A}\in\{0,1\}^{n\times n}$ to denote the adjacency matrix, where $\mathbf{A}_{[i,j]}=1$ when $(i,j)\in\mathcal{E}$ and $\mathbf{A}_{[i,j]}=0$ otherwise. In the node classification task, some nodes are associated with ground truth labels. We use $\mathcal{V}_{\text{train}}$ to denote the labeled (training) set, $\mathcal{V}_{\text{test}}=\mathcal{V}\backslash\mathcal{V}_{\text{train}}$ to denote the unlabeled (test) set, and $\mathcal{Y}$ to denote the set of labels. Most GNNs take the adjacency matrix $\mathbf{A}$ and the attribute matrix $\mathbf{X}$ as the input, and obtain the node embeddings used for node classification tasks. Specifically, we use $f_{\mathbf{\theta}}:\{0,1\}^{n\times n}\times\mathbb{R}^{n\times d_{x}}\to \mathbb{R}^{n\times c}$ to denote a GNN model, where $\mathbf{\theta}$ collects all learnable model parameters. We use $\mathbf{\hat{y}}=\sigma\left(f_{\mathbf{\theta}}(\mathbf{A},\mathbf{X})\right)$ to denote soft predictions where $\sigma(\cdot)$ is the softmax function. A conventional way to train a GNN model is minimizing a utility loss function $\mathcal{L}\left(f_{\mathbf{\theta}},\mathbf{A},\mathbf{X},\mathcal{Y},\mathcal{V }_{\text{train}}\right)$ (e.g., CE loss) over the training set. In human-centered scenarios, each individual is usually associated with sensitive attributes (e.g., gender). We use $\mathcal{S}=\{s_{1},\ldots,s_{n}\}$ to denote a sensitive attribute set, where $s_{i}$ is the sensitive attribute of the node $v_{i}$. Based on the sensitive attributes, we can divide the nodes into different sensitive groups, denoted as $\mathcal{V}_{1},\ldots,\mathcal{V}_{K}$, where $\mathcal{V}_{k}=\{v_{i}\|s_{i}=k\}$ and $K$ is the number of sensitive groups. Compared with vanilla GNNs, fairness-aware GNNs should not yield discriminatory predictions against individuals from any specific sensitive subgroup (Dong et al., 2023). Hence, the objective of training fairness-aware GNNs generally contains two aspects; one is minimizing the utility loss, i.e., $\mathcal{L}\left(f_{\mathbf{\theta}},\mathbf{A},\mathbf{X},\mathcal{Y},\mathcal{V }_{\text{train}}\right)$, the other is minimizing the discrimination between the predictions over different sensitive groups, denoted as $\mathcal{L}_{f}\left(f_{\mathbf{\theta}},\mathbf{A},\mathbf{X},\mathcal{Y}, \mathcal{S}\right)$. It is worth noting that $\mathcal{L}$ is the CE loss in most cases, but there are various types of $\mathcal{L}_{f}$ for training fair GNNs. Problem Formulation. In the fairness attack of GNNs, attackers aim to achieve their goals (e.g., to make the predictions of the victim GNN exhibit more bias) via slightly perturbing the input graph data in certain feasible spaces. Here, we focus on structure perturbation because perturbing the graph structure in the discrete domain presents more challenges, and the attack methods based on structure perturbation can be easily adapted to attribute perturbation. In this paper, for the simplicity of discussion, we focus on two different sensitive groups and prediction classes, but our method can be easily adapted to tasks with multiple sensitive groups and classes. More details about our attack settings are discussed in Appendix B. We formulate our problem of fairness attacks of GNNs as follows. Problem 1.: _Attacks on Fairness of GNNs with Unnoticeable Effect on Prediction Utility. Let $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathbf{X}\}$ be an attributed graph with the adjacency matrix $\mathbf{A}$. Let $\mathcal{V}_{\text{train}}$ and $\mathcal{V}_{\text{test}}$ be the labeled and unlabeled node sets, respectively, $\mathcal{Y}$ be the label set corresponding to $\mathcal{V}_{\text{train}}$, and $\mathcal{S}$ be the sensitive attribute value set of $\mathcal{V}$. Let $g_{\mathbf{\theta}}$ be a surrogate GNN model, $\mathcal{L}_{\mathcal{S}}$, $\mathcal{L}$, and $\mathcal{L}_{f}$ be the surrogate loss, utility loss, and fairness loss, respectively. Given the attack budget $\Delta$ and the utility variation budget $\epsilon$, the problem of attacks on fairness of GNNs is formulated as follows._ \[\max_{\mathbf{A}^{\prime}\in\mathcal{F}}\mathcal{L}_{f}\left(g_{\mathbf{\theta}^{ \star}},\mathbf{A}^{\prime},\mathbf{X},\mathcal{Y},\mathcal{V}_{\text{test}}, \mathcal{S}\right)\] \[\mathrm{s.t.}\ \mathbf{\theta}^{\star}=\arg\min_{\mathbf{\theta}}\mathcal{L}_{s} \left(g_{\mathbf{\theta}},\tilde{\mathbf{A}},\mathbf{X},\mathcal{Y},\mathcal{S} \right),\ \\|\mathbf{A}^{\prime}-\mathbf{A}\\|_{F}\leq 2\Delta, \tag{1}\] \[\|\mathcal{L}(g_{\mathbf{\theta}^{\star}},\mathbf{A},\mathbf{X},\mathcal{Y}, \mathcal{V}_{\text{train}})-\mathcal{L}(g_{\mathbf{\theta}^{\star}},\mathbf{A}^{ \prime},\mathbf{X},\mathcal{Y},\mathcal{V}_{\text{train}})\|\leq\epsilon,\] _where $\mathcal{F}$ is the feasible space of the poisoned graph structure $\mathbf{A}^{\prime}$, $g_{\mathbf{\theta}^{\star}}$ is a trained surrogate model._ In this problem, the attacker's objective $\mathcal{L}_{f}(g_{\mathbf{\theta}^{\star}},\mathbf{A}^{\prime},\mathbf{X}, \mathcal{Y},\mathcal{V}_{\text{test}},\mathcal{S})$ measures the bias of the predictions on test nodes. The surrogate model $g_{\mathbf{\theta}^{\star}}$ is trained based on a surrogate loss $\mathcal{L}_{s}(g_{\mathbf{\theta}},\tilde{\mathbf{A}},\mathbf{X},\mathcal{Y}, \mathcal{S})$. The constraints restrict the number of perturbed edges and the absolute value of the utility change over the training set. In this paper, we consider two types of attack scenarios, _fairness evasion attack_ and _fairness poisoning attack_ (Chakraborty et al., 2018). By perturbing the input graph data, _fairness evasion attack_ jeopardizes the fairness of a well-trained fairness-aware GNN model in the test phase, while _fairness poisoning attack_ makes the fairness-aware GNN model trained on such perturbed data render unfair predictions. If $\tilde{\mathbf{A}}=\mathbf{A}$, then Problem 1 boils down to the fairness eva Figure 1: A toy example of fairness attacks of GNNs with unnoticeable effect on prediction utility. sion attack; and if $\tilde{\mathbf{A}}=\mathbf{A}^{\prime}$, then Problem 1 becomes fairness poisoning attack. In this paper, we use demographic parity (Navarin et al., 2020) as the attacker's objective $\mathcal{L}_{f}$, the CE loss as utility loss $\mathcal{L}$, and a two-layer linearized GCN (Zugner et al., 2018) as the surrogate model $g_{\mathbf{\theta}}$. These choices have been verified by many previous works (Zugner et al., 2018; Zugner and Gunnemann, 2019; Xu et al., 2019) to be effective in attacking the prediction utility of GNNs, while can also be changed flexibly according to the specific task. An illustration of our problem is shown in Figure 1. We leave the detailed explanation of Figure 1 in Appendix B. ## 3 Methodology In this section, we first introduce the design of the surrogate loss $\mathcal{L}_{s}$, which is the most critical part in Problem 1. Then, we propose a non-gradient optimization method to solve Problem 1 and propose a fast computation technique to reduce the time complexity of our optimization method. ### Surrogate Loss Design Existing Loss on Fairness. The loss functions of all existing fairness-aware GNNs can be divided into two parts: utility loss term (CE loss that is shared in common) and fairness loss term (varies from different methods). Correspondingly, our surrogate loss function is designed as $\mathcal{L}_{s}=\mathcal{L}_{s_{u}}+\alpha\mathcal{L}_{s_{f}}$ where $\mathcal{L}_{s_{u}}$ is the utility loss term, i.e., the CE loss, $\mathcal{L}_{s_{f}}$ is the fairness loss term we aim to study, and $\alpha$ is the weight coefficient. Let $\hat{Y}\in[0,1]$ be a continuous random variable of the output soft prediction and $S$ be a discrete random variable of the sensitive attribute. Next, we categorize the fairness loss terms into three types. (1), $\Delta_{dp}(\hat{Y},S)=\|\mathrm{Pr}(\hat{Y}\geq\frac{1}{2}\|S=0)-\mathrm{Pr}( \hat{Y}\geq\frac{1}{2}\|S=1)\|$: demographic parity (Jiang et al., 2022; Navarin et al., 2020; Zeng et al., 2021; Franco et al., 2022), that reduces the difference of positive rate among different sensitive groups during training. (2).$I(\hat{Y},S)=\int_{0}^{1}\sum_{i}P_{\hat{Y},S}(z,i)\log\frac{P_{\hat{Y},S}(z, i)}{P_{\hat{Y}}(z)\mathrm{Pr}(S=ii)}dz$: mutual information of the output and the sensitive attribute (Dai and Wang, 2021; Bose and Hamilton, 2019; Kang et al., 2022), that reduces the dependence between the model output and the sensitive attribute during training. (3).$W(\hat{Y},S)=\int_{0}^{1}\|F_{\hat{Y}\|S=0}^{-1}(p)-F_{\hat{Y}\|S=1}^{-1}(y)\|dy$: Wasserstein-1 distance of the output on different sensitive groups (Fan et al., 2021; Dong et al., 2022), that makes the conditional distributions of the model output given different sensitive groups closer during training. The Proposed Surrogate Loss. Next, we introduce our proposed surrogate loss term on fairness $\mathcal{L}_{S_{f}}$, aiming at representing all types of aforementioned fairness loss terms, e.g., $\Delta_{dp}(\hat{Y},S)$, $I(\hat{Y},S)$, and $W(\hat{Y},S)$. In particular, our idea is to find a common upper bound for these fairness loss terms as $\mathcal{L}_{S_{f}}$. Therefore, during the training of the surrogate model, $\mathcal{L}_{S_{f}}$ is minimized to a small value, and all types of fairness loss terms become even smaller consequently. In this way, the surrogate model trained by our surrogate loss will be close to that trained by any unknown victim loss, which is consistent with conventional attacks on model utility. Consequently, we argue that such surrogate loss can represent the unknown victim loss. Accordingly, we propose a novel surrogate loss function on fairness $TV(\hat{Y},S)=\int_{0}^{1}\|P_{\hat{Y}\|S=0}(z)-P_{\hat{Y}\|S=1}(z)\|dz$, i.e., the total variation of $\hat{Y}$ on different sensitive groups. We can prove that $TV(\hat{Y},S)$ is a common upper bound of $\Delta_{dp}(\hat{Y},S)$, $I(\hat{Y},S)$ (at certain condition), and $W(\hat{Y},S)$ so it can represent all types of fairness loss functions. Theorem 1.: _We have $\Delta_{dp}(\hat{Y},S)$ and $W(\hat{Y},S)$ upper bounded by $TV(\hat{Y},S)$. Moreover, $I(\hat{Y},S)$ is also upper bounded by $TV(\hat{Y},S)$ if $\forall z\in[0,1]$, $P_{\hat{Y}}(z)\geq\Pi_{i}\mathrm{Pr}(S=i)$ holds._ The proof of Theorem 1 is shown in Appendix A.1. Combining $\mathcal{L}_{s_{f}}$ (total variation loss) and $\mathcal{L}_{s_{u}}$ (CE loss), our surrogate loss function is $\mathcal{L}_{s}=CE(g_{\mathbf{\theta}},\tilde{\mathbf{A}},\mathbf{X},\mathcal{Y})+ \alpha TV(g_{\mathbf{\theta}},\tilde{\mathbf{A}},\mathbf{X},\mathcal{S})$. Since the probability density function of the underlying random variable $\hat{Y}$ is difficult to calculate, we leverage the kernel density estimation (Parzen, 1962) to compute the total variation loss with the output soft predictions $g_{\mathbf{\theta}}(\tilde{\mathbf{A}},\mathbf{X})$. The kernel estimation is computed as $\hat{P}_{\hat{Y}\|S=i}(z)=\frac{1}{h\|V_{i}\|}\sum_{j\in\mathcal{Y}_{i}}K(\frac{z -g_{\mathbf{\theta}}(\tilde{\mathbf{A}},\mathbf{X})_{[i]}}{h})$ for $i=0,1$, where $K(\cdot)$ is a kernel function and $h$ is a positive band width constant. Then we exploit the numerical integration to compute the total variation loss as \[TV(g_{\mathbf{\theta}},\tilde{\mathbf{A}},\mathbf{X},\mathcal{S})=\frac{1}{m}\sum_{j= 1}^{m}\left\|\hat{P}_{\hat{Y}\|S=0}\left(\frac{j}{m}\right)-\hat{P}_{\hat{Y}\|S=1} \left(\frac{j}{m}\right)\right\|, \tag{2}\] where $m$ is the number of intervals in the integration. Consequently, we obtain a practical calculation of our proposed surrogate loss $\mathcal{L}_{s}$. ### Optimization Considering that Problem 1 is a bilevel optimization problem in a discrete domain, it is extremely hard to find the exact solution. Hence, we resort to the greedy strategy and propose a sequential attack method. Our sequential attack flips target edges1 sequentially to obtain the optimal poisoned structure $\mathbf{A}^{\prime}$. It is worth noting that many attack methods on model utility find the target edge corresponding to the largest element of the gradient of $\mathbf{A}$, namely gradient-based methods (Zugner and Gunnemann, 2019; Xu et al., 2019; Wu et al., 2019; Geisler et al., 2021). However, we find the efficacy of gradient-based methods is not guaranteed. Footnote 1: Flipping the target edge increases the attacker’s objective the most. Proposition 1.: _Gradient-based methods for optimizing the graph structure are not guaranteed to decrease the objective function._ We leave a detailed analysis of the shortcomings of gradient-based methods in Appendix D. In contrast, we propose a non-gradient method based on a scoring function that can provably increase the attacker's objective. We first consider the unconstrained version of Problem 1. We let $\mathbf{A}^{0}=\mathbf{A}$ and train the surrogate model to obtain $\mathbf{\theta}^{0}$, i.e., solving $\mathbf{\theta}^{0}=\arg\min_{\mathbf{\ell}}\mathcal{L}_{s}(g_{\mathbf{\theta}},\mathbf{ A},\mathbf{X},\mathcal{Y},\mathcal{S})$. For $t=1,2,\ldots,T_{0}$, we find the maximum score $r^{t}$ in the $t$-th iteration as \[\max_{(u,v)\in\mathcal{C}^{t}}r^{t}(u,v)=\Delta\mathcal{L}_{f}^{t}(u,v)= \mathcal{L}_{f}(g_{\mathbf{\theta}^{t}},flip_{(u,v)}\mathbf{A}^{t})-\mathcal{L}_ {f}(g_{\mathbf{\theta}^{t}},\mathbf{A}^{t}), \tag{3}\] where $\mathcal{C}^{t}$ is the candidate edge set in the $t$-th iteration; $flip_{(u,v)}\mathbf{A}$ denotes the adjacency matrix after flipping the edge $(u,v)$: $flip_{(u,v)}\mathbf{A}_{[i,j]}=1-\mathbf{A}_{[i,j]}$ if $(i,j)=(u,v)$ or $(i,j)=(v,u)$, and $flip_{(u,v)}\mathbf{A}_{[i,j]}=\mathbf{A}_{[i,j]}$ otherwise. After solving Equation (3), we denote the solution as $(u^{t},v^{t})$. Then we update $\mathbf{A}^{t}$ as $\mathbf{A}^{t+1}=flip_{(u^{t},v^{t})}\mathbf{A}^{t}$ in the $t$-th iteration. For the fairness poisoning attack, we retrain the surrogate model based on $\mathbf{A}^{t+1}$ as $\mathbf{\theta}^{t+1}=\arg\min_{\mathbf{\ell}}\mathcal{L}_{s}(g_{\mathbf{\theta}},\mathbf{ A}^{t+1})$ to update the surrogate model. After $T_{0}$ iterations, we have $\mathbf{A}^{}=\mathbf{A}^{T_{0}}$ as the solution of Problem 1. Next, to handle the first constraint, we let $T_{0}\leq\Delta$ and have $\\|\mathbf{A}^{}-\mathbf{A}\\|_{F}\leq\sum_{i=1}^{T_{0}}\\|\mathbf{A}^{t}- \mathbf{A}^{t-1}\\|_{F}\leq 2\Delta$ consequently. For the second constraint, we aim to make every flipping unnoticeable in terms of model utility, i.e., making $\|\mathcal{L}(\mathbf{A}^{t+1})-\mathcal{L}(\mathbf{A}^{t})\|$ as small as possible. We notice that this constrained optimization problem can be easily solved by projected gradient descent (PGD) (Nocedal and Wright, 2006) in the continuous domain by solving $\mathbf{A}^{t+1}=\arg\!\min_{\|\mathcal{L}(\mathbf{A}^{t})-\mathcal{L}(\mathbf{ A}^{t})\|\leq\epsilon_{t}}\\|\mathbf{A}^{\prime}-(\mathbf{A}^{t}+\eta\nabla \mathcal{L}_{f}(\mathbf{A}^{t}))\\|_{F}^{2}$, where $\epsilon_{t}$ is the budget of the $t$-th iteration that satisfies $\sum_{t=1}^{\Delta}\epsilon_{t}\leq\epsilon$ and $\eta$ is the learning rate. To solve this problem, we have the following theorem. Theorem 2.: _The optimal poisoned adjacency matrix $\mathbf{A}^{t+1}$ in the $t+1$-th iteration given by PGD, i.e., the solution of $\mathbf{A}^{t+1}=\arg\!\min_{\|\mathcal{L}(\mathbf{A}^{t})-\mathcal{L}(\mathbf{A} ^{t})\|\leq\epsilon_{t}}\\|\mathbf{A}^{\prime}-(\mathbf{A}^{t}+\eta\nabla \mathcal{L}_{f}(\mathbf{A}^{t}))\\|_{F}^{2}$ is_ \[\mathbf{A}^{t+1}=\begin{cases}\mathbf{A}^{t}+\eta\nabla_{\mathbf{A}}\mathcal{L} _{f}(\mathbf{A}^{t}),\text{ if }\eta\|\nabla_{\mathbf{A}}\mathcal{L}(\mathbf{A}^{t})^{T}\nabla_{ \mathbf{A}}\mathcal{L}_{f}(\mathbf{A}^{t})\|\leq\epsilon_{t},\\ \mathbf{A}^{t}+\eta\nabla_{\mathbf{A}}\mathcal{L}_{f}(\mathbf{A}^{t})+\frac{ \epsilon_{t}\epsilon_{t}-\eta\nabla_{\mathbf{A}}\mathcal{L}(\mathbf{A}^{t})^{T} \nabla_{\mathbf{A}}\mathcal{L}_{f}(\mathbf{A}^{t})}{\\|\nabla_{\mathbf{A}} \mathcal{L}(\mathbf{A}^{t})\\|_{F}^{2}}\nabla_{\mathbf{A}}\mathcal{L}(\mathbf{A} ^{t}),\text{ otherwise},\end{cases} \tag{4}\] _where $\epsilon_{t}=\operatorname{sign}\big{(}\nabla_{\mathbf{A}}\mathcal{L}(\mathbf{A} ^{t})^{T}\nabla_{\mathbf{A}}\mathcal{L}_{f}(\mathbf{A}^{t})\big{)}$._ The proof of Theorem 2 is provided in Appendix A.2. It is worth noting that the solution in Equation (4) cannot be leveraged directly in the discrete domain. Hence, we use the differences $\Delta\mathcal{L}_{f}^{t}(u,v)=\mathcal{L}_{f}(flip_{(u,v)}\mathbf{A}^{t})- \mathcal{L}_{f}(\mathbf{A}^{t})$ and $\Delta\mathcal{L}^{t}(u,v)=\mathcal{L}(flip_{(u,v)}\mathbf{A}^{t})-\mathcal{L} (\mathbf{A}^{t})$ as zeroth-order estimations (Chen et al., 2017; Kariyappa et al., 2021) of $\nabla_{\mathbf{A}_{[i,v]}}\mathcal{L}_{f}(\mathbf{A}^{t})$ and $\nabla_{\mathbf{A}_{[i,v]}}\mathcal{L}(\mathbf{A}^{t})$, and replace them in Equation (4), respectively. Moreover, we minimize the utility budget as $\epsilon_{t}=0$ to constrain the model utility change after the flipping as strictly as possible. Consequently, we adjust the scoring function to find the target edge as \[\bar{r}^{t}(u,v)=\Delta\mathcal{L}^{t}_{f}(u,v)-\frac{(\mathbf{p}^{t})^{T}\mathbf{q}^{t}}{ \\|\mathbf{p}^{t}\\|_{2}^{2}}\|\Delta\mathcal{L}^{t}(u,v)\|, \tag{5}\] for $(u,v)\in\mathcal{C}^{t}$, where $\mathbf{p}^{t}\in\mathbb{R}^{\|\mathcal{C}^{t}\|}$ and $\mathbf{q}^{t}\in\mathbb{R}^{\|\mathcal{C}^{t}\|}$ are denoted as $\mathbf{p}^{t}_{[i]}=\Delta\mathcal{L}^{t}(\mathcal{C}^{t}_{[i]})$ and $\mathbf{q}^{t}_{[i]}=\Delta\mathcal{C}^{t}_{f}(\mathcal{C}^{t}_{[i]})$, respectively. Here $\mathcal{C}^{t}_{[i]}$ denotes the $i$-th edge in $\mathcal{C}^{t}$. Equation (5) can also be seen as a balanced attacker's objective between maximizing $\mathcal{L}_{f}(\mathbf{A}^{t+1})-\mathcal{L}_{f}(\mathbf{A}^{t})$ and minimizing $\|\mathcal{L}(\mathbf{A}^{t+1})-\mathcal{L}(\mathbf{A}^{t})\|$. With $\bar{r}^{t}(u,v)$, the pseudocode of our proposed attack algorithm is shown in Appendix C.1. ### Fast Computation In this subsection, we focus on the efficient implementation of our sequential attack. The most important and costly part of our algorithm is to find the maximum value of $\bar{r}^{t}(u,v)$ from $\mathcal{C}^{t}$. This naturally requires traversing each edge in $\mathcal{C}^{t}$, which is costly on a large graph. Hence, we develop a fast computation method to reduce the time complexity. Reviewing our model settings, we find that the output of node $i$, $g_{\mathbf{\theta}}(\mathbf{A},\mathbf{X})_{[i]}$, can be computed as $g_{\mathbf{\theta}}(\mathbf{A},\mathbf{X})_{[i]}=\sum_{j\in\mathcal{N}_{i}}( \hat{\mathbf{d}}_{[i]}\hat{\mathbf{d}}_{[j]})^{-1}\hat{\mathbf{A}}_{[j,:]}\mathbf{ \mathbf{\theta}}=\mathbf{Z}_{[i,:]}\mathbf{\theta}$, where $\hat{\mathbf{d}}$ is the node degree vector, $\hat{\mathbf{A}}=\mathbf{A}+\mathbf{I}$, $\mathcal{N}_{i}=\{j\|\hat{\mathbf{A}}_{[i,j]}=1\}$, and $\mathbf{Z}$ denotes the aggregated feature matrix which is crucial to the fast computation. We aim to reduce the time complexity from two perspectives: (1) reducing the time complexity of computing the score $\bar{r}^{t}(u,v)$ for a specific edge $(u,v)$; and (2) reducing the size of the candidate edge set $\mathcal{C}^{t}$. From perspective (1), we compute $flip_{(u,v)}\mathbf{Z}^{t}$ incrementally based on $\mathbf{Z}^{t}$. Then we have $g_{\mathbf{\theta}^{t}}(flip_{(u,v)}\mathbf{A}^{t},\mathbf{X})=flip_{(u,v)} \mathbf{Z}^{t}\mathbf{\theta}^{t}$. Consequently, both $\Delta\mathcal{L}^{t}_{f}(u,v)$ and $\Delta\mathcal{L}^{t}(u,v)$ can be obtained based on $g_{\mathbf{\theta}^{t}}(flip_{(u,v)}\mathbf{A}^{t},\mathbf{X})$. Next, we discuss the computation of $flip_{(u,v)}\mathbf{Z}^{t}_{[i,:]}$ into three cases and introduce them separately. We only discuss adding edge $(u,v)$, and leave removing edge $(u,v)$ in Appendix C.1. Case 1: If $i\in\{u,v\}$, $flip_{(u,v)}\mathbf{Z}^{t}_{[i,:]}=\frac{\hat{\mathbf{d}}_{[i]}}{d^{t}_{[i]} +1}(\mathbf{Z}^{t}_{[i,:]}-\frac{\hat{\mathbf{A}}_{[i,:]}\mathbf{X}\mathbf{X} +\mathbf{X}_{[i,:]}}{(\hat{\mathbf{d}}^{t}_{[i]}+1)^{2}}+\frac{\hat{\mathbf{A }}_{[i,:]}\mathbf{X}+\mathbf{X}_{[i,:]}}{(\hat{\mathbf{d}}^{t}_{[i]}+1)^{2}}$, where $j\in\{u,v\}\backslash\{i\}$; Case 2: If $i\in\mathcal{N}^{t}_{u}\cup\mathcal{N}^{t}_{v}\backslash\{u,v\}$, $flip_{(u,v)}\mathbf{Z}^{t}_{[i,:]}=\mathbf{Z}^{t}_{[i,:]}-\mathbb{I}_{i\in \mathcal{N}^{t}_{u}}\cdot(\frac{\hat{\mathbf{A}}^{t}_{[i,:]}\mathbf{X}}{d^{t }_{[i]}d^{t}_{[i]}}-\frac{\hat{\mathbf{A}}^{t}_{[i,:]}\mathbf{X}+\mathbf{X}_{[ i,:]}}{d^{t}_{[i]}d^{t}_{[i]}+1})-\mathbb{I}_{i\in\mathcal{N}^{t}_{u}}\cdot( \frac{\hat{\mathbf{A}}^{t}_{[i,:]}\mathbf{X}}{d^{t}_{[i]}d^{t}_{[i]}}-\frac{ \hat{\mathbf{A}}^{t}_{[i,:]}\mathbf{X}}{d^{t}_{[i]}d^{t}_{[i]}}-\frac{\hat{ \mathbf{A}}^{t}_{[i,:]}\mathbf{X}+\mathbf{X}_{[i,:]}}{d^{t}_{[i]}d^{t}_{[i]}} )$, where $\mathbb{I}_{i\in\mathcal{N}}=1$ if $i\in\mathcal{N}$, and $\mathbb{I}_{i\in\mathcal{N}}=0$ otherwise; Case 3: If $i\not\in\mathcal{N}^{t}_{u}\cup\mathcal{N}^{t}_{v}$, $flip_{(u,v)}\mathbf{Z}^{t}_{[i,:]}=\mathbf{Z}^{t}_{[i,:]}$. From perspective (2), we only choose the influential edges2 as $\mathcal{C}^{t}$. Finally, flipping $(u,v)$ is highly likely to increase $\mathcal{L}_{f}$ only if it results in significant changes to the predictions of a large number of nodes. Based on the aforementioned discussions, flipping $(u,v)$ can only affect the soft prediction of nodes in $\mathcal{N}^{t}_{u}\cup\mathcal{N}^{t}_{v}$ (case 1 and 2). Hence, we consider $(u,v)$ as an influential edge if the predictions of nodes in $\mathcal{N}^{t}_{u}\cup\mathcal{N}^{t}_{u}$ are easy to be changed. To this end, we propose an importance score $\rho^{t}(u,v)$ and collect the edges corresponding to the top $a$ importance scores into $\mathcal{C}^{t}$. We have $\rho^{t}(u,v)=\sum_{i\in\mathcal{N}^{t}_{u}\cup\mathcal{N}^{t}_{v}}M_{t}-\| \mathbf{Z}^{t}_{[i,:]}\mathbf{\theta}^{t}\|$, where $M_{t}=\max_{i}\|\mathbf{Z}^{t}_{[i,:]}\mathbf{\theta}^{t}\|$. When the value of $M_{t}-\|\mathbf{Z}^{t}_{[i,:]}\mathbf{\theta}^{t}\|$ is large, the prediction of node $i$ is easy to be changed. We analyze the time complexity of our fast computation method as follows. Footnote 2: Flipping the influential edges is very likely to increase the attacker’s objective. Proposition 2.: _The overall time complexity of G-FairAttack with the fast computation is $O(\bar{d}n^{2}+d_{x}an),$ where $\bar{d}$ denotes the average degree._ Compared with computing $\max_{(u,v)\in\mathcal{C}^{t}}\bar{r}^{t}(u,v)$ directly, which has a complexity of $O(n^{4})$, our fast computation approach distinctly improves the efficiency of the attack. Note that the time complexity of G-FairAttack can be further improved in practice by parallel computation and simpler ranking strategies. The proof of Proposition 2, detailed complexity analysis, and ways of further improving the complexity are provided in Appendix C.2. ## 4 Experiments In this section, we evaluate our proposed G-FairAttack. Specifically, we aim to answer the following questions through our experiments: RQ1: Can our proposed surrogate loss function represent the victim loss functions of different kinds of victim models? RQ2: Can G-FairAttack achieve unnoticeable utility change? RQ3: To what extent does our fast computation approach improve the efficiency of G-FairAttack? Due to the space limitation, we provide more experimental results and discussions on the results in Appendix F. ### Experimental Settings The evaluation of attack methods includes two stages. In the first stage, we use an attack method to obtain the perturbed graph structure. In the second stage, for the fairness evasion attack, we train a test GNN (i.e., victim model) on the clean graph and compare the output of the test GNN on the clean graph and on the perturbed graph; for the fairness poisoning attack, we train the test GNN on the clean graph to obtain the normal model and train the test GNN on the perturbed graph to obtain the victim model; then, we compare the output of the normal model on the clean graph and the output of the victim model on the perturbed graph. We adopt three prevalent real-world datasets, i.e., Facebook (Leskovec and Mcauley, 2012), Credit (Agarwal et al., 2021), and Pokec (Dai and Wang, 2021; Dong et al., 2022) to test the effectiveness of G-FairAttack. For attack baselines, we choose a random attack method (Hussain et al., 2022; Zugner et al., 2018), a state-of-the-art fairness attack method FA-GNN (Hussain et al., 2022), and two gradient-based methods Gradient Ascent and Metattack (Zugner and Gunnemann, 2019) (adapted from utility attacks). For test GNNs, we adopt a vanilla graph convolutional network (Kipf and Welling, 2017) and three different types of fairness-aware GNNs, Regularization (Zeng et al., 2021) (Reg) for $\Delta_{dp}(\hat{Y},S)$, FairGNN (Dai and Wang, 2021) for $I(\hat{Y},S)$, and EDITS (Dong et al., 2022) for $W(\hat{Y},S)$. More details about baselines and datasets are provided in Appendices E.1 and E.2. We choose two mostly adopted fairness metrics, demographic parity $\Delta_{dp}$(Dwork et al., 2012) and equal opportunity $\Delta_{eo}$(Hardt et al., 2016), to measure the fairness of test GNNs. Larger values of fairness metrics denote more bias. In addition, we also report the accuracy and AUC score to show the utility of test GNNs. degradation of fairness metric values after being attacked. The attack budget $\Delta$ for Facebook and Pokec is $5\%$, i.e., we can flip $5\%\cdot\|\mathcal{E}\|$ edges at most; for Credit, the budget is $1\%$. We ran each experiment three times with different random seeds and reported the mean value and the standard deviation. The experimental results are shown in Table 1 and Appendix. We can observe that (1). G-FairAttack is the most effective attack method that jeopardizes the fairness of all types of victim GNN models, especially for fairness-aware GNNs. (2). Compared with Gradient Ascent, G-FairAttack adds a fairness loss term (total variation loss) in the surrogate loss. Consequently, G-FairAttack outperforms them in attacking different types of fairness-aware GNNs, which demonstrates that our proposed surrogate loss helps our surrogate model learn from different types of fairness-aware victim models (RQ1). (3). Some baselines outperform G-FairAttack in attacking vanilla GCN because the surrogate model of G-FairAttack is trained with the surrogate loss, including a fairness loss term, while the victim GCN is trained without fairness consideration. This problem can be addressed by choosing a smaller value of $\alpha$ (weight of $\mathcal{L}_{f}$). Despite that, G-FairAttack successfully reduces the fairness of GCN on most benchmarks. (4). Compared with gradient-based attacks (Gradient Ascent), G-FairAttack has less space complexity as it does not need to store a dense adjacency matrix. ### Ablation Study Effectiveness of Surrogate Loss. To further answer RQ1, we compare G-FairAttack with two variants with different surrogate losses to demonstrate the effectiveness of our proposed surrogate loss. Our proposed surrogate loss function $\mathcal{L}_{s}$ consists of two parts: utility loss term $\mathcal{L}_{s_{g}}$ (CE loss) and fairness loss term $\mathcal{L}_{s_{f}}$ (total variation loss). For the first baseline, we remove the fairness loss term from the surrogate loss and name it G-FairAttack-None. For the second baseline, we substitute the total variation loss with $\Delta_{dp}$ loss, named as G-FairAttack-$\Delta_{dp}$. In our problem, if the surrogate loss is the same as the victim loss (like white-box attacks), the attack method would definitely perform well. However, we show that G-FairAttack (with our proposed total variation surrogate loss) has the most desirable performance even with a different victim loss from the surrogate loss, which verifies the effectiveness of our surrogate loss. Hence, we choose FairGNN and EDITS as the victim models because their victim loss functions differ from the surrogate loss functions of G-FairAttack-None and G-FairAttack-$\Delta_{dp}$. We conduct the experiment on Credit dataset in the fairness poisoning attack setting. The results are shown in Table 2. It is demonstrated that all three attack methods successfully increase the value of fairness metrics. Among all three attack methods, G-FairAttack achieves the best performance, which demonstrates that our proposed surrogate loss helps the surrogate model learn knowledge from various types of fairness-aware GNNs. Effectiveness of Unnoticeable Constraints. To answer RQ2, we investigate the impact of utility constraints on G-FairAttack. We remove the discrete projected update strategy and the utility constraint in G-FairAttack as G-FairAttack-C. We compare the variation of $\Delta_{dp}$ and the utility loss on the training set after the attack by G-FairAttack, G-FairAttack-C, and FA-GNN. We chose two mostly adopted victim models, i.e., GCN and regularization, to test the attack methods on Facebook. We set the utility budget at 5% of the utility loss on clean data. The results are shown in Figure 2. It demonstrates that (1). Our G-FairAttack distinctly deteriorates the fairness of two victim models while keeping the variation of utility loss unnoticeable ($<0.5\%$). Therefore, G-FairAttack can attack the victim model with unnoticeable utility variation. (2). Removing the utility constraint and the discrete projected update strategy from G-FairAttack, the variation of utility loss becomes much larger as the attack budget increases. The re \begin{table} \begin{tabular}{l\|c\|c} \hline Attack & FairGNN & EDITS \\ \cline{2-3} & $\Delta_{dp}$ (\%) & $\Delta_{co}$ (\%) & $\Delta_{dp}$ (\%) & $\Delta_{co}$ (\%) \\ \hline Clean & 6.70 $\pm$ 1.60 4.52 $\pm$ 1.79 & 6.43 $\pm$ 0.95 & 5.77 $\pm$ 1.06 \\ G-FA-None & 7.87 $\pm$ 1.31 5.68 $\pm$ 1.42 & 10.47 $\pm$ 1.91 & 9.95 $\pm$ 1.71 \\ G-FA-$\Delta_{dp}$ & 7.82 $\pm$ 1.17 5.82 $\pm$ 1.58 & 10.77 $\pm$ 1.19 10.39 $\pm$ 1.17 \\ G-FA & 8.06 $\pm$ 1.90 6.10 $\pm$ 2.03 & 11.26 $\pm$ 1.50 10.87 $\pm$ 1.56 \\ \hline \end{tabular} \end{table} Table 2: Results of G-FairAttack on Credit while replacing the total variation loss with other loss terms. Figure 2: The changes of $\Delta_{dp}$ and the utility loss function $\mathcal{L}$ under different attack budgets. sults of G-FairAttack-C verify the effectiveness of the fairness constraint and the discrete projected update strategy. (3). G-FairAttack and G-FairAttack-C deteriorate the fairness of victim models to a larger extent than FA-GNN. Because G-FairAttack is a sequential greedy method, it becomes more effective as the attack budget increases. Hence, G-FairAttack is more flexible than FA-GNN. ### Parameter Study In this subsection, we aim to answer RQ3 and show the impact of the parameter $a$ (the threshold for fast computation) on the time cost for the attack and the test results for victim models. We conduct experiments under different choices of $a$. For the simplicity of the following discussion, we take $a$ as the proportion of edges considered in $\mathcal{C}^{t}$ (e.g., $a=1e^{-3}$ means we only consider the top 0.1% edges with the highest important score). We record the attacker's objective $\mathcal{L}_{f}$ under different thresholds during the optimization process as Figure3(a) to show the impact of $a$ of the surrogate model. We also record the fairness metrics of regularization-based victim models attacked by G-FairAttack and the average time of G-FairAttack in each iteration while choosing different values of $a$ (in Figure3(b)) to show the impact of $a$ on the attack performance and efficiency. Figure3(b) demonstrates the trade-off between effectiveness and efficiency exists but is very tolerant on the effectiveness side. In the range $[5e^{-4},1e^{-2}]$, the performances of G-FairAttack on the victim model are similar, while the time cost decreases distinctly when $a$ gets smaller (the time cost of $a=5e^{-4}$ is 95% lower than $a=1e^{-2}$). Therefore, in practice, we can flexibly choose a proper threshold to fit the efficiency requirement. In conclusion, our fast computation distinctly reduces G-FairAttack's time cost without compromising the performance. ## 5 Related Works Adversarial Attack of GNNs. To improve the robustness of GNNs, we should first fully understand how to attack GNNs. Consequently, various attacking strategies of GNNs have been proposed in recent years. Zugner et al. (2018) first formulated the poisoning attack problem on graph data as a bilevel optimization in the discrete domain and proposed the first poisoning attack method of GNNs with respect to a single target node in a gray-box setting based on the greedy strategy. Following the problem formulation in (Zugner et al., 2018), researchers proposed many effective adversarial attack methods in different settings. Zugner and Gunnemann (2019) proposed the first untargeted poisoning attack method in a gray-box setting based on MAML (Finn et al., 2017). Chang et al. (2020) proposed an untargeted evasion attack method in a black-box setting by attacking the low-rank approximation of the spectral graph filter. Wu et al. (2019) proposed an untargeted evasion attack method in a white-box setting and a corresponding defense method based on integrated gradients. Adversarial Attacks on Fairness. Many recent studies investigated the fairness attack problem on tabular data. Van et al. (2022) proposed an online poisoning attack framework on fairness based on gradient ascent. They adopted the convex relaxation (Zafar et al., 2017; Donini et al., 2018) to make the loss function differentiable. Mehrabi et al. (2021) proposed two types of poisoning attacks on fairness. One incorporates demographic information into the influence attack (Koh et al., 2022) by updating the poisoning set with PGD. The other generates poisoning samples within the vicinity of a chosen anchor based on the decision boundary. Solans et al. (2021) proposed a fairness poisoning attack method in both white-box and black-box settings. They formulated the poisoning attack problem as a bilevel optimization and adopted the KKT relaxation and a carefully designed initialization strategy for optimization. Chhabra et al. (2023) proposed a black-box attack and defense method for fair clustering. So far, only Hussain et al. (2022) investigated the fairness attack problem on graph data. They proposed a link injection method named FA-GNN that links nodes in different sensitive groups randomly to degrade the demographic parity metric. Figure 3: The optimization curves, training time cost, and test fairness on regularization-based victim model corresponding to different values of $a$. ## 6 Conclusion Fairness-aware GNNs play a significant role in various human-centered applications. To protect GNNs from fairness attacks, we should fully understand potential ways to attack the fairness of GNNs. In this paper, we study adversarial attacks on fairness of GNNs as a bilevel optimization problem. We propose the first unnoticeable requirement for fairness attacks. We design a novel surrogate loss function to target various types of fairness-aware GNNs. We then propose a sequential attack algorithm to solve the problem, and a fast computation approach to reduce the time complexity. Extensive experiments on three real-world datasets verify the efficacy of our method.
2308.08086	Safety Filter Design for Neural Network Systems via Convex Optimization	With the increase in data availability, it has been widely demonstrated that neural networks (NN) can capture complex system dynamics precisely in a data-driven manner. However, the architectural complexity and nonlinearity of the NNs make it challenging to synthesize a provably safe controller. In this work, we propose a novel safety filter that relies on convex optimization to ensure safety for a NN system, subject to additive disturbances that are capable of capturing modeling errors. Our approach leverages tools from NN verification to over-approximate NN dynamics with a set of linear bounds, followed by an application of robust linear MPC to search for controllers that can guarantee robust constraint satisfaction. We demonstrate the efficacy of the proposed framework numerically on a nonlinear pendulum system.	Shaoru Chen, Kong Yao Chee, Nikolai Matni, M. Ani Hsieh, George J. Pappas	2023-08-16T01:30:13Z	http://arxiv.org/abs/2308.08086v2	# Safety Filter Design for Neural Network Systems via ###### Abstract With the increase in data availability, it has been widely demonstrated that neural networks (NN) can capture complex system dynamics precisely in a data-driven manner. However, the architectural complexity and nonlinearity of the NNs make it challenging to synthesize a provably safe controller. In this work, we propose a novel safety filter that relies on convex optimization to ensure safety for a NN system, subject to additive disturbances that are capable of capturing modeling errors. Our approach leverages tools from NN verification to over-approximate NN dynamics with a set of linear bounds, followed by an application of robust linear MPC to search for controllers that can guarantee robust constraint satisfaction. We demonstrate the efficacy of the proposed framework numerically on a nonlinear pendulum system. ## I Introduction With the rapid development in machine learning infrastructure, neural networks (NN) have been ubiquitously applied in the modeling of complex dynamical systems [1, 2]. Through a data collection and training procedure [3], NNs can capture accurate representations of the system dynamics even in challenging scenarios where high-speed aerodynamic effects [4, 5, 6] or contact-rich environments [7, 8] are present. Moreover, NNs can be easily updated online as more data is collected, making them suitable for online tasks or modeling changing environments. For example, NN dynamical systems are widely used in model-based reinforcement learning [9] and learning-based adaptive control [6]. However, applying NN dynamics brings about significant challenges in providing safety guarantees for the controlled system. Benefiting from the expressivity of NNs, we are meanwhile faced with the high nonlinearity and large scale of NNs since they are often overparameterized. The runtime assurance (RTA) [10] mechanism provides a practical and effective solution to guarantee the safety of complex dynamical systems by designing a safety filter that primarily focuses on enforcing safety constraints. Given a primary controller that aims to optimize performance, the safety filter monitors and modifies the output of the primary controller online to guarantee that only safe control inputs are applied. The safety filter allows the design of the safety and performance-based controllers to be decoupled and has found wide applications in safe learning-based control [11, 12, 13]. In this work, we focus on the design of a predictive safety filter (PSF) [14] for uncertain NN dynamics. The PSF essentially follows a model predictive control (MPC) formulation with the nonlinear dynamics and constraints encoded in an optimization problem. Different from MPC, the PSF is less complex to solve since it does not consider any performance objectives [14]. Compared with the alternative safety filter construction schemes through control barrier functions (CBF) [15, 16] or Hamilton-Jacobi (HJ) reachability analysis [17, 18], the PSF enjoys flexibility in handling dynamically changing NN models or model uncertainty bounds when updated online. We refer the interested readers to [14, Section 1 and 2] for a detailed discussion of the PSF, CBF, and HJ reachability-based safety filters. Contributions: In this work, we consider uncertain NN dynamics subject to bounded additive disturbances, where the disturbances can encapsulate the errors between the learned NN dynamics and the true system. Despite being highly expressive, the considered uncertain NN dynamics requires solving a robust optimization problem involving NN dynamical constraints online in the PSF. To resolve this computational challenge, we propose to apply NN verification tools [19] to abstract the NN dynamics locally as a linear uncertain system, thereby reducing the original PSF problem into one that is amenable to robust linear MPC and convex optimization. In particular, we adapt the SLS (System Level Synthesis) MPC method [20] to solve the resulting robust MPC problem. A schematic of our pipeline is shown in Fig. 1. Soft constraints are used in robust linear MPC where slack variables denoting constraint violations are penalized. By applying a hierarchy of conservative function and model uncertainty approximations, we transform the original optimization problem into a convex one. A safety certificate for the uncertain NN dynamics over a finite horizon can then be provided when all slack variables are zero. Our contributions are summarized below. 1. Drawing tools from NN verification and robust linear MPC, we propose a novel predictive safety filter for uncertain NN dynamics through convex optimization. Importantly, the complexity of the convex optimization problem is independent of the NN size (i.e., width and depth of the NN). 2. Our PSF provides a safety certificate for the uncertain NN dynamics over a finite horizon when a certain numerical criterion is met by the convex optimization solutions. ### _Related works_ The problem of ensuring the safety of learning-based systems has received significant interest with a plethora of methods described in [21]. Directly related to our work is the PSF developed in [14], which monitors and modifies a given control input by solving a predictive control problem online to guarantee the safety of the system. This formulation has been extended to the SLS setting [22], applied to racing cars [12] and a soft-constrained variant is proposed in [23] to handle unexpected disturbances to the states. The PSF that we propose differs from those in the existing work in the following ways. First, we exploit the structure of the neural networks to extract linear bounds on the NN outputs using NN verification tools [19, 24], simplifying the PSF formulation for NN dynamics. Second, our proposed pipeline circumvents the need to solve a robust non-convex optimization problem, even with the consideration of additive disturbances within the uncertain dynamics, as typical for nonlinear variants of SLS [25]. Unlike existing work in predictive control of NN dynamics [2, 26, 27], our work considers robust control of uncertain NN dynamics with a focus on obtaining formal safety guarantees. _Notation_: $[N]$ denotes the set $\{0,1,\ldots,N\}$. $\mathcal{B}_{p}(z,r)$ denotes the $\ell_{p}$ ball centered at $z$ with radius $r$. We use $x_{0:T}$ to denote the sequence $\{x_{0},\ldots,x_{T}\}$. $\mathbf{0}$ denotes a vector of all zeros, and its dimension can be inferred from context. For a vector $d\in\mathbb{R}^{n}$, $S=\text{diag}(d)\in\mathbb{R}^{n\times n}$ denotes a diagonal matrix with $d$ being the diagonal vector. For a sequence of matrices $S_{1},\cdots,S_{N}$, $S=\text{blkdiag}(S_{1},\cdots,S_{N})$ denotes a block diagonal matrix whose diagonal blocks are $S_{1},\cdots,S_{N}$ arranged in the order. We represent a linear, causal operator $\mathbf{R}$ defined over a horizon $T$ by the block-lower-triangular matrix \[\mathbf{R}=\begin{bmatrix}R^{0,0}&&&\\ R^{1,1}&R^{1,0}&&\\ \vdots&\ddots&\ddots&\\ R^{T,T}&\cdots&R^{T,1}&R^{T,0}\end{bmatrix} \tag{1}\] where $R^{i,j}\in\mathbb{R}^{p\times q}$ is a matrix of compatible dimension. The set of such matrices is denoted by $\mathcal{L}_{TV}^{T,p\times q}$ and the superscript $T$ or $p\times q$ will be dropped when it is clear from the context. ## II Problem Formulation Consider the following discrete-time nonlinear system, \[x(k+1)=Ax(k)+Bu(k)+f\left(x(k),u(k)\right)+w(k), \tag{2}\] where the vectors $x(k)\in\mathbb{R}^{n_{x}}$ and $u(k)\in\mathbb{R}^{n_{u}}$ are the state and control input. The vector $w(k)\in\mathbb{R}^{n_{x}}$ denotes the additive disturbances that can account for unknown effects from the environment or unmodeled dynamics. We assume that $w(k)$ is norm-bounded, _i.e._, $w(k)\in\mathcal{W}:=\{w\mid\\|w\\|_{\infty}\leq\sigma_{w}\}$. The system consists of a set of linear dynamics characterized by the matrices $A$ and $B$ and a set of nonlinear dynamics, $f(x(k),u(k))$. Specifically, these nonlinear dynamics are given by a NN $f:\mathbb{R}^{n_{x}+n_{u}}\mapsto\mathbb{R}^{n_{x}}$ with an arbitrary architecture. While we discuss our proposed approach taking into account a general formulation of dynamics in (2), the approach allows the matrices $A$ and $B$ to be zero and has the flexibility to account for time-varying dynamics $(A(k),B(k))$. The system (2) is required to satisfy the following constraints, \[x(k)\in\mathcal{X},\quad u(k)\in\mathcal{U}, \tag{3}\] where $\mathcal{X}\subset\mathbb{R}^{n_{x}}$ and $\mathcal{U}\subset\mathbb{R}^{n_{u}}$ are polytopes. The state $x(k)$ and input $u(k)$ are considered _safe_ if they satisfy constraints (3). ### _Predictive safety filter_ We assume a primary controller $\pi(x)$ which aims to complete a task or achieve high performance is given for system (2). Following the runtime assurance scheme, the primary controller $\pi(x)$ is not guaranteed to be safe since its design may be decoupled from the safety requirements. To ensure constraint satisfaction of the closed-loop system, we aim to design a predictive safety filter that monitors and modifies the control input $\pi(x(k))$ given by the primary controller in a minimally invasive manner online. This is achieved by solving the following robust optimization problem at each time step $k$, \[\begin{split}\underset{u_{0:T-1}}{\text{minimize}}& \\|u_{0}-\pi(x(k))\\|_{2}^{2}\\ \text{subject to}& x_{t+1}=Ax_{t}+Bu_{t}+f(x_{t},u_{t})+w_{t},\\ & x_{t}\in\mathcal{X},u_{t}\in\mathcal{U},\forall w_{t}\in \mathcal{W},\quad t\in[T],\\ & x_{0}=x(k),\end{split} \tag{4}\] where the vectors $x_{t},\,u_{t}$ denote the predicted states and inputs over the horizon $T$, with $x(k)$ representing the current state of the system. We denote Problem (4) as the PSF problem and the sequence $u_{0:T-1}^{}$ as the optimal solution of Problem (4). When applied, the control inputs $u_{0:T-1}^{}$ can guarantee the safety of the system for the next $T$ steps. In practice, the PSF problem (4) is solved recursively at each time step $k$, and the first optimal control input $u_{0}^{}$ is applied to the system, analogous to an MPC scheme. Fig. 1: Pipeline of the proposed predictive safety filter for uncertain NN systems. ### _Challenges with the predictive safety filter_ While the solution to Problem (4) is able to provide safety guarantees, solving Problem (4) is a challenging task. Some potential issues include 1. it is well known in robust MPC that searching over open-loop control sequences $u_{0:T-1}$ can be overly conservative [28], 2. the presence of the NN dynamics $f(x,u)$ makes solving the PSF computationally challenging, 3. the safety certificate of the solution $u_{0:T-1}^{}$ is not available until convergence is reached, and 4. without the availability of a robust forward invariant set, attempting to solve the PSF may result in infeasibility. To handle all the aforementioned issues, we combine NN verification tools [19] and robust MPC [20]. Our solution consists of two steps. First, we generate local linear bounds for the NN dynamics $f(x,u)$ using tools from NN verification. Next, we apply robust linear MPC to synthesize a state feedback control policy that guarantees robust constraint satisfaction for the system. This combined procedure provides a powerful simplification of the PSF problem and resolves issues (i) to (iii). To address the issue (iv), we introduce soft constraints in our formulation. This provides formal safety guarantees for the system when the slack variables are zero. We describe these two steps in Sections III and IV, respectively. Section V demonstrates our method numerically, and Section VI concludes the paper. Remark 1: _To ensure the safety constraints are satisfied at all times or guarantee recursive feasibility of the PSF (4), a local forward invariant set for the nonlinear system (2) is required, which is generally challenging to find. In this work, we do not assume the availability of such a forward invariant set and use slack variables as numerical certificates of safety. We leave the synthesis of the forward invariant terminal set for NN dynamics and its integration into the PSF as part of our future work._ ## III Neural network verification bounds In this section, we demonstrate how tools from NN verification can be utilized to over-approximate the NN dynamics with a linear time-varying (LTV) representation. This enables us to conservatively transform the PSF problem into a robust convex optimization problem, which is simpler to solve. Given a bounded input set, the linear relaxation-based perturbation analysis (LiRPA) [19] is an efficient method to synthesize linear lower and upper bounds for the outputs of a NN with a general architecture. The bounds computed from this method are described in the following theorem. Theorem 1: _(rephrasing [24, Theorem 3.2]) Given a NN $f(z):\mathbb{R}^{n_{0}}\mapsto\mathbb{R}^{n_{L}}$, there exist two explicit linear functions $f_{U}:\mathbb{R}^{n_{0}}\mapsto\mathbb{R}^{n_{L}}$ and $f_{L}:\mathbb{R}^{n_{0}}\mapsto\mathbb{R}^{n_{L}}$ such that for all $z\in\mathcal{B}_{p}(z_{0},r)$, we have_ \[f_{L}(z)=A_{L}z+b_{L}\leq f(z)\leq A_{U}z+b_{U}=f_{U}(z), \tag{5}\] _where the inequalities are applied component-wise._ The parameters $A_{L},A_{U},b_{L},b_{U}$ are derived from the weights, biases and activation functions of the NN. In this paper, we choose $p:=\infty$ such that $\mathcal{B}_{\infty}(z,r)$ is polyhedral. The bounds (5) are computed using closed-form updates with a computational complexity polynomial in the number of neurons [24]. This allows the method to scale well to deep networks. However, if the NN is deep or if the input domain $\mathcal{B}_{\infty}(z,r)$ is large, the computed bounds tend to be loose. Motivated by this observation, we propose to extract a set of local linear bounds along a _reference trajectory_. Specifically, at every time step $k$, we construct a reference trajectory given by the sequences of reference states $\mathbf{\hat{x}}:=\hat{x}_{0:T}$ and control inputs $\mathbf{\hat{u}}:=\hat{u}_{0:T-1}$ where \[\begin{split}\hat{x}_{t+1}&=A\hat{x}_{t}+B\hat{u}_{ t}+f(\hat{x}_{t},\hat{u}_{t}),\quad t\in[T-1],\\ \hat{x}_{0}&=x(k).\end{split} \tag{6}\] The reference control inputs $\mathbf{\hat{u}}$ can be obtained, e.g., by rolling out the nominal NN dynamics following the primary policy $\pi(\cdot)$. By denoting $z_{t}:=[x_{t}^{\top}\;u_{t}^{\top}]^{\top}$, $\hat{z}_{t}:=[\hat{x}_{t}^{\top}\;\hat{u}_{t}^{\top}]^{\top}$ and $r_{t}$ to be the radius of the $\ell_{\infty}$ ball around $\hat{z}_{t}$, we apply Theorem 1 to get the following bounds for the NN dynamics along the reference trajectory, \[\begin{bmatrix}\bar{G}_{t}^{x}&\bar{G}_{t}^{u}\end{bmatrix}z_{t}+\underline {g}_{t}\leq f(x_{t},u_{t})\leq\begin{bmatrix}\bar{G}_{t}^{x}&\bar{G}_{t}^{u} \end{bmatrix}z_{t}+\overline{g}_{t}, \tag{7}\] for all $(x_{t},u_{t})\in\mathcal{B}_{\infty}(\hat{z}_{t},r_{t})$. In other words, the NN dynamics $f(x_{t},u_{t})$ is over-approximated with a set of linear lower and upper bounds. The ball $\mathcal{B}_{\infty}(\hat{z}_{t},r_{t})$ is referred to as the _trust region_ in which the bounds (7) are valid. To reduce conservatism in the formulation of the filter within the robust MPC framework, we integrate the bounds into the linear dynamics of the system. Specifically, using the bounds in (7), we define \[f_{d}(x_{t},u_{t}):=f(x_{t},u_{t})-\left(\tilde{A}_{t}x_{t}+\tilde{B}_{t}u_{t}+ \tilde{c}_{t}\right), \tag{8}\] where \[\tilde{A}_{t}:=\frac{\bar{G}_{t}^{x}+\bar{G}_{t}^{x}}{2},\;\tilde{B}_{t}:= \frac{\bar{G}_{t}^{u}+\bar{G}_{t}^{u}}{2},\;\tilde{c}_{t}:=\frac{\underline{g}_ {t}+\overline{g}_{t}}{2} \tag{9}\] denote the means of the linear bounds. Corollary 1: _Given the bounds in (7), for every $z_{t}:=[x_{t}^{\top}\;u_{t}^{\top}]^{\top}\in\mathcal{B}_{\infty}(\hat{z}_{t},r _{t})$, the dynamics $f_{d}(x_{t},u_{t})$ in (8) have the following bounds,_ \[\begin{bmatrix}\underline{D}_{t}^{x}&\underline{D}_{t}^{u}\end{bmatrix}z_{t}+ \underline{d}_{t}\leq f_{d}(x_{t},u_{t})\leq\begin{bmatrix}\bar{D}_{t}^{x}& \bar{D}_{t}^{u}\end{bmatrix}z_{t}+\overline{d}_{t}, \tag{10}\] _where_ \[\underline{D}_{t}^{x}=\frac{\underline{G}_{t}^{x}-\bar{G}_{t}^{x}}{2}=-\bar{D}_ {t}^{x},\;\;\underline{D}_{t}^{u}=\frac{\underline{G}_{t}^{u}-\bar{G}_{t}^{u} }{2}=-\bar{D}_{t}^{u},\] \[\underline{d}_{t}=\frac{\underline{g}_{t}-\overline{g}_{t}}{2}=-\overline{d}_{t}.\] It is important to note that although the NN dynamics $f(x_{t},u_{t})$ can have large values within the trust region $\mathcal{B}_{\infty}(\hat{z}_{t},r_{t})$, the dynamics $f_{d}(x_{t},u_{t})$ tends to be small in magnitude and can be treated as disturbances to the system. With the extraction of $f_{d}(x_{t},u_{t})$, we obtain an LTV reformulation of the PSF problem, referred to as the _linear _PSF_ problem, \[\underset{u_{0:T-1}}{\text{minimize}} \\|u_{0}-\pi(x(k))\\|_{2}^{2}\] (11) subject to \[x_{t+1}=A_{t}x_{t}+B_{t}u_{t}+c_{t}+\Delta_{t}(x_{t},u_{t})+w_{t},\] \[(x_{t},u_{t})\in\mathcal{B}_{\infty}\left(\hat{z}_{t},r_{t} \right),\quad t\in[T-1],\] \[x_{t}\in\mathcal{X},\,u_{t}\in\mathcal{U},\quad t\in[T],\] \[\forall\Delta_{t}(\cdot)\in\mathcal{P}_{t},\,w_{t}\in\mathcal{W},\quad t\in[T],\] \[x_{0}=x(k),\] where the means in (9) are merged into the linear dynamics of the system with the definition of the following time-varying system parameters 1, Footnote 1: When a time-varying dynamics $(A(k),B(k))$ is considered in (2), we can replace $(A,B)$ by their time-varying counterparts in the definitions of $(A_{t},B_{t})$. \[A_{t}:=A+\tilde{A}_{t},\,B_{t}:=B+\tilde{B}_{t},\,c_{t}:=\tilde{c}_{t}.\] The uncertainty set $\mathcal{P}_{t}$ is given as \[\mathcal{P}_{t}:=\begin{cases}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! MPC methods [30, 31, 32, 33], requires $2^{n_{x}}$ vertices, (ii) applying SLS MPC requires merging the constant $c_{t}$ into the lumped uncertainty $\xi_{t}$ which causes the bounds on $\mathbf{\xi}$ to be overly conservative. In the following subsections, we describe an extension to SLS MPC to address these two challenges. ### _Controller parameterization_ For the uncertain linear dynamics \[x_{t+1}=A_{t}x_{t}+B_{t}u_{t}+c_{t}+\Delta_{t}(x_{t},u_{t})+w_{t} \tag{18}\] stated in (11), we define $\eta_{t}:=\Delta_{t}(x_{t},u_{t})+w_{t}$ as the lumped uncertainty. Instead of treating $\xi_{t}=c_{t}+\eta_{t}$ in (13), we decompose (18) into a set of nominal and error dynamics. First, in addition to (14), we concatenate these variables over the horizon $T$, \[\mathbf{\eta} :=[\mathbf{0}^{\top}~{}\eta_{0}^{\top}~{}\cdots~{}\eta_{T-1}^{\top}]^ {\top},~{}\mathbf{w}:=[\mathbf{0}^{\top}~{}w_{0}^{\top}~{}\cdots~{}w_{T-1}^{\top}]^ {\top}, \tag{19}\] \[\mathbf{c} :=[\mathbf{0}^{\top}c_{0}^{\top}~{}\cdots~{}c_{T-1}^{\top}]^{\top}, \quad\mathbf{\delta}_{x_{0}}:=[x_{0}^{\top}~{}\mathbf{0}^{\top}~{}\cdots~{}\mathbf{0}^{ \top}]^{\top}.\] We define the nominal and error states $\{\mathbf{h},\mathbf{x}_{e}\}$ and control inputs $\{\mathbf{v},\mathbf{u}_{e}\}$ as \[\mathbf{h} :=[h_{0}^{\top}~{}\cdots~{}h_{T}^{\top}]^{\top},~{}~{}\mathbf{v}:=[ v_{0}^{\top}~{}\cdots~{}v_{T}^{\top}]^{\top},\] \[\mathbf{x}_{e} :=[x_{e,0}^{\top}~{}\cdots~{}x_{e,T}^{\top}]^{\top}=\mathbf{x}- \mathbf{h},\] \[\mathbf{u}_{e} :=[u_{e,0}^{\top}~{}\cdots~{}u_{e,T}^{\top}]^{\top}=\mathbf{u}- \mathbf{v},\] with the nominal and error dynamics as \[\mathbf{h} =Z(\mathbf{A}\mathbf{h}+\mathbf{B}\mathbf{v})+\mathbf{c}+\mathbf{ \delta}_{x_{0}}, \tag{20a}\] \[\mathbf{x}_{e} =Z(\mathbf{A}\mathbf{x}_{e}+\mathbf{B}\mathbf{u}_{e})+\mathbf{\eta}. \tag{20b}\] It is important to note that (20b) conforms with (13). A LTV state feedback controller $\mathbf{K}\in\mathcal{L}_{TV}^{T,n_{x}\times n_{x}}$ is then applied to control the error states. The overall controller for (18) is given by $\mathbf{u}=\mathbf{K}\mathbf{x}_{e}+\mathbf{v}=\mathbf{K}(\mathbf{x}-\mathbf{h })+\mathbf{v}$. ### _Lumped uncertainty over-approximation_ For the lumped uncertainty $\mathbf{\eta}$, its dependence on $\mathbf{x}$, $\mathbf{u}$ and $\mathbf{w}$ complicates the design of the robust controller. As in SLS MPC, the approach is to over-approximate $\mathbf{\eta}$ by an independent, filtered virtual disturbance signal $\mathbf{\Psi}\widetilde{\mathbf{w}}$, where $\widetilde{\mathbf{w}}=[\mathbf{0}^{\top}~{}\widetilde{w}_{0}^{\top}~{}\cdots~{} \widetilde{w}_{T-1}^{\top}]^{\top},~{}\\|\widetilde{w}_{t}\\|_{\infty}\leq 1$. The matrix $\mathbf{\Psi}\in\mathcal{L}_{TV}^{T,n_{x}\times n_{x}}$ is a filter operating on the finite-horizon virtual disturbance signal $\widetilde{\mathbf{w}}$, with its diagonal blocks of $\mathbf{\Psi}$ structured as $\Psi^{0,0}=I,\Psi^{t,0}=\text{diag}(\psi_{t-1})$ with $\psi_{t-1}\in\mathbb{R}^{n_{x}},\psi_{t-1}>0$, $t=1,\ldots,T$. We define $\widetilde{\mathcal{W}}=\{\widetilde{\mathbf{w}}\mid\\|\widetilde{w}_{t}\\|_{ \infty}\leq 1,t\in[T-1]\}$ as the set of admissible virtual disturbances. Since $\tilde{w}_{t}$ are unit norm-bounded, we tune the filter $\mathbf{\Psi}$ to change the reachable set of $\mathbf{\Psi}\widetilde{\mathbf{w}}$, defined as $\mathcal{R}(\mathbf{\Psi}\widetilde{\mathbf{w}}):=\{\mathbf{\zeta}\mid\mathbf{\zeta}=\bm {\Psi}\widetilde{\mathbf{w}},\widetilde{\mathbf{w}}\in\widetilde{\mathcal{W}}\}$, $\mathbf{\zeta}:=[\mathbf{0}^{\top}~{}\zeta_{0}^{\top}~{}\cdots~{}\zeta_{T-1}^{\top}]^ {\top}$. Our goal is to find sufficient conditions on the control parameters $\{\mathbf{K},\mathbf{h},\mathbf{v}\}$ and $\mathbf{\Psi}$ such that the reachable set of $\mathbf{\eta}$, denoted by $\mathcal{R}\left(\mathbf{\eta};\mathbf{K},\mathbf{h},\mathbf{v}\right):=\{\mathbf{ \eta}\mid\eta_{t}=\Delta_{t}(x_{t},u_{t})+w_{t},\Delta_{t}\in\mathcal{P}_{t}, \,t\in[T-1]\}$, is a subset of the reachable set of $\mathbf{\Psi}\widetilde{\mathbf{w}}$. The following proposition provides these sufficient conditions and the proof is given in Appendix A. Proposition 1: _Let $e_{i}\in\mathbb{R}^{n_{x}}$ denote the $i$-th standard basis for $i=1,\ldots,n_{x}$ and $y_{t}=[h_{t}^{\top}~{}v_{t}^{\top}]^{\top}$, $\Phi^{t,t-i}:=\left[\Phi_{x}^{t,t-i\top}~{}\Phi_{u}^{t,t-i\top}\right]^{\top}$ for $t\in[T-1]$. The following constraints:_ \[\left[I-Z\mathbf{A}\quad-Z\mathbf{B}\right]\begin{bmatrix}\mathbf{\Phi}_{x}\\ \mathbf{\Phi}_{u}\end{bmatrix}=\mathbf{\Psi},~{}\mathbf{\Phi}_{x},\mathbf{\Phi}_{u}\in\mathcal{ L}_{TV}^{T}, \tag{21}\] _and_ \[\sigma_{w}+e_{i}^{\top}([\overline{D}_{0}^{x}~{}\overline{D}_{0}^{u}] y_{0}+\overline{d}_{0}) \leq\psi_{0,i}, \tag{22a}\] \[\sigma_{w}-e_{i}^{\top}\left([\overline{D}_{0}^{x}~{}\overline{D}_{0}^{u}] y_{0}+\underline{d}_{0}\right) \leq\psi_{0,i},\] (22b) \[\sigma_{w}+e_{i}^{\top}\Big{(}[\overline{D}_{t}^{x}~{}\overline{D}_ {t}^{u}]y_{t}+\overline{d}_{t}\Big{)}+\] \[\sum_{i=1}^{t}\Big{\\|}e_{i}^{\top}\Big{(}[\overline{D}_{t}^{x}~{} \overline{D}_{t}^{u}]\Phi^{t,t-i}-\Psi^{t+1,t+1-i}\Big{)}\Big{\\|}_{1}\leq\psi_{t,i},\] (22c) \[\sigma_{w}-e_{i}^{\top}\Big{(}[\overline{D}_{t}^{x}~{}\underline{D}_ {t}^{u}]y_{t}+\underline{d}_{t}\Big{)}+\] \[\sum_{i=1}^{t}\Big{\\|}e_{i}^{\top}\Big{(}[\overline{D}_{t}^{x}~{} \underline{D}_{t}^{u}]\Phi^{t,t-i}-\Psi^{t+1,t+1-i}\Big{)}\Big{\\|}_{1}\leq\psi_{t,i},\] (22d) \[i\in[n_{x}],\quad t=1,\cdots,T-1,\] _guarantee that $\mathcal{R}(\mathbf{\eta};\mathbf{K},\mathbf{h},\mathbf{v})\subseteq\mathcal{R}( \mathbf{\Psi}\widetilde{\mathbf{w}})$ holds._ ### _Convex formulation of the linear PSF_ With the constraints (21) and (22), we have for any realization of $\mathbf{\eta}$, there exists $\widetilde{\mathbf{w}}\in\widetilde{\mathcal{W}}$ such that $\mathbf{\eta}=\mathbf{\Psi}\widetilde{\mathbf{w}}$. Therefore, we can represent $\mathbf{\eta}$ as $\mathbf{\Psi}\widetilde{\mathbf{w}}$ and write the error dynamics (20b) as \[\mathbf{x}_{e}=Z(\mathbf{A}\mathbf{x}_{e}+\mathbf{B}\mathbf{u}_{e})+\mathbf{\Psi} \widetilde{\mathbf{w}}. \tag{23}\] By [20, Corollary 1], we have constraint (21) parameterizes all system responses $\mathbf{x}_{e}=\mathbf{\Phi}_{x}\widetilde{\mathbf{w}},\mathbf{u}_{e}=\mathbf{\Phi}_{u }\widetilde{\mathbf{w}}$ of system (23) under the controller $\mathbf{u}_{e}=\mathbf{K}\mathbf{x}_{e}$. Then, the closed-loop states and control inputs of system (18) are given by \[\mathbf{x}=\mathbf{h}+\mathbf{\Phi}_{x}\widetilde{\mathbf{w}},\quad\mathbf{u}= \mathbf{v}+\mathbf{\Phi}_{u}\widetilde{\mathbf{w}}. \tag{24}\] To guarantee for the constraint with parameters $(a_{j}^{x},b_{j}^{x})$. Similarly, the input constraints can be tightened as \[\begin{split} a_{k}^{u\top}v_{t}+\sum_{i=1}^{t}\left\\|a_{k}^{u \top}\Phi_{u}^{t,t-i}\right\\|_{{}_{1}}\leq b_{k}^{u}+\epsilon_{u,k}^{t},\quad \epsilon_{u,k}^{t}\geq 0,\,t\in[T],\\ k=1,\ldots,n_{\mathcal{U}},\end{split} \tag{27}\] where $n_{\mathcal{U}}$ denotes the number of linear inequalities defining $\mathcal{U}$ and $(a_{k}^{u},b_{k}^{u})$ denotes the $k$-th set of linear constraint parameters in $\mathcal{U}$. The trust region constraints $(x_{t},u_{t})\in\mathcal{B}_{\infty}\left(\hat{z}_{t},r_{t}\right)$ analogously can be tightened as \[\begin{split} a_{j}^{\top}h_{t}+\sum_{i=1}^{t}\left\\|a_{j}^{\top }\Phi_{x}^{t,t-i}\right\\|_{{}_{1}}\leq b_{j}+\sigma_{x,j}^{t},\,\sigma_{x,j}^{ t}\geq 0,\\ a_{k}^{\top}v_{t}+\sum_{i=1}^{t}\left\\|a_{k}^{\top}\Phi_{u}^{t, t-i}\right\\|_{{}_{1}}\leq b_{k}+\sigma_{u,k}^{t},\,\sigma_{u,k}^{t}\geq 0,\\ j=1,\ldots,2n_{x},\,k=1,\ldots,2n_{u},\,t\in[T-1],\end{split} \tag{28}\] where $(a_{j},b_{j})$ and $(a_{k},b_{k})$ are defined similarly for the state and input-related constraint parameters in $\mathcal{B}_{\infty}\left(\hat{z}_{t},r_{t}\right)$. Summarizing the results above, we propose a convex tightening of the linear PSF, which can be written as \[\begin{split}\underset{\begin{subarray}{c}\mathbf{\Phi}_{x},\mathbf{ \Phi}_{u},\mathbf{\Psi}_{y},\mathbf{\mathrm{h}},\mathbf{\mathrm{v}},\\ \left\{\epsilon_{x}^{t},\epsilon_{u}^{t},\sigma_{x}^{t},\sigma_{u}^{t}\right\} \end{subarray}}{\text{minimize}}&\left\\|u_{0}-\pi(x(k))\right\\|_{ {}_{2}}^{2}+M_{\epsilon}\sum_{t=0}^{T}\left(\left\\|\epsilon_{x}^{t}\right\\|_{ {}_{1}}+\left\\|\epsilon_{u}^{t}\right\\|_{{}_{1}}\right)\\ &+M_{\sigma}\sum_{t=0}^{T-1}\left(\left\\|\sigma_{x}^{t}\right\\|_{ {}_{1}}+\left\\|\sigma_{u}^{t}\right\\|_{{}_{1}}\right)\end{split}\] subject to \[\begin{split}&\text{\small over-approximation constraints \eqref{eq:constraintconstraintconstraint},}\\ &\text{\small soft state and input constraints \eqref{eq:constraintconstraintconstraint} and \eqref{eq:constraintconstraintconstraint},}\\ &\text{\small soft trust region constraints \eqref{eq:constraintconstraintconstraint},}\\ & x_{0}=x(k),\end{split}\] (29) where $M_{\epsilon},M_{\sigma}>0$ are chosen as large numbers. When a polyhedral robust forward invariant set is used as the terminal constraint, we can tighten it similarly as (26). All the constraints in Problem (29) are linear, making (29) a quadratic program. With the use of soft constraints, (29) is always feasible. If the slack variables $\{\epsilon_{x}^{t},\epsilon_{u}^{t},\sigma_{x}^{t},\sigma_{u}^{t}\}$ in the solution are zero, we obtain a certificate that the system is safe for the next $T$ steps under the state-feedback controller $\mathbf{\mathrm{u}}=\mathbf{\mathrm{K}}(\mathbf{\mathrm{x}}-\mathbf{\mathrm{h}})+\mathbf{\mathrm{v}}$ with $\mathbf{\mathrm{K}}=\mathbf{\Phi}_{u}\mathbf{\Phi}_{x}^{-1}$. ### _Trust region update_ Following the discussion on the trust regions in Section III, we describe a method to update the trust regions online. Starting with the reference trajectory given by the primary policy $\pi(x)$, we propose to iteratively increase $r_{t}$ and update the reference trajectory by applying the policy $\mathbf{\mathrm{u}}=\mathbf{\mathrm{K}}(\mathbf{\mathrm{x}}-\mathbf{\mathrm{h}})+\mathbf{\mathrm{v}}$, synthesized in Section IV. We then pick the reference trajectory that gives the smallest slack variables and apply the corresponding control inputs to the system. Our framework is summarized in Algorithm 1. Input: Current state $x(k)$, horizon $T$, number of iterations $N$, initial reference trajectory $\hat{z}_{0:T}$, reference control input $\pi(x(k))$, initial trust region radius $r_{0}>0$ Output: Filtered control input $u_{0}^{}$, safety certificate safe_cert ``` 1:At each time step $k$, 2:Initialize:$r_{t}\gets r_{0},\ t\in[T-1],\{\mathbf{\mathrm{K}}^{},\mathbf{\mathrm{h}}^{},\mathbf{ \mathrm{v}}^{}\}\leftarrow\{\mathbf{0}\}$, $\epsilon^{}\leftarrow\infty$. 3:for$\ell=1,\ldots,N$do 4: Construct $\mathcal{B}_{\infty}(\hat{z}_{t},r_{t})$ and $\mathcal{P}_{t}$ with (7) and (10) 5: Solve (29) with $x(k)$ to get $\{\mathbf{\mathrm{K}},\mathbf{\mathrm{h}},\mathbf{\mathrm{v}}\}$ 6: Extract $\epsilon_{max}:=$ max. value of slack variables 7:if$\epsilon_{max}=0$then 8: Set $\{\mathbf{\mathrm{K}}^{},\mathbf{\mathrm{h}}^{},\mathbf{\mathrm{v}}^{}\}\leftarrow\{\mathbf{ \mathrm{K}},\mathbf{\mathrm{h}},\mathbf{\mathrm{v}}\}$ 9: Set safe_cert $\leftarrow$ True 10:else 11: Update $\hat{z}_{0:T}$ with $\mathbf{\mathrm{u}}=\mathbf{\mathrm{K}}(\mathbf{\mathrm{x}}-\mathbf{\mathrm{h}})+\mathbf{\mathrm{v}}$, as described in (6) 12: Update $r_{t}\leftarrow\beta r_{t}$ for $t\in[T-1]$ with $\beta>1$. 13:if$\epsilon_{max}\leq\epsilon^{}$then 14: Set $\epsilon^{}\leftarrow\epsilon_{max}$ 15: Set $\{\mathbf{\mathrm{K}}^{},\mathbf{\mathrm{h}}^{},\mathbf{\mathrm{v}}^{}\}\leftarrow\{\mathbf{ \mathrm{K}},\mathbf{\mathrm{h}},\mathbf{\mathrm{v}}\}$ 16: Set safe_cert $\leftarrow$ False 17:Set $u_{0}^{}\gets v_{0}$, extracted from $\{\mathbf{\mathrm{K}}^{},\mathbf{\mathrm{h}}^{},\mathbf{\mathrm{v}}^{}\}$ ``` Algorithm 1* Robust Linear MPC-based PSF ## V Numerical Example To verify the efficacy of the proposed solution, we test it on a NN proxy of the nonlinear pendulum system 2. The pendulum consists of the following dynamics [34], Footnote 2: Our codes are publicly available at [https://github.com/ShaorucChen/NN-System-PSF](https://github.com/ShaorucChen/NN-System-PSF). \[\ddot{\theta}=\frac{3g\sin(\theta)}{2l}+\frac{3\tau}{ml^{2}}\,, \tag{30}\] where $\theta$ is the angle between the pendulum and the vertical, $m$ and $l$ are the mass and length of the pendulum, $g$ is the gravitational force, and $\tau$ is the external torque acting on the pendulum. The state and control input are defined as $x:=[\theta\ \dot{\theta}]^{\top}\in\mathbb{R}^{2}$ and $u:=\tau\in\mathbb{R}$. To obtain the linear dynamics in (2), the dynamics (30) are linearized about the origin and discretized with a sampling time of 0.05s to obtain the following dynamics, \[x_{t+1}=\begin{bmatrix}1.0092&0.05015\\ 0.369&1.0092\end{bmatrix}x_{t}+\begin{bmatrix}0.00125\\ 0.05015\end{bmatrix}u_{t}. \tag{31}\] Next, we train a NN $f(x,u)$ to approximate the residual dynamics that are not captured by the linear dynamics in (31). We first collect data by simulating the nonlinear dynamics in (30) for a duration of 15s. The NN is then trained through a backpropagation procedure [3], using the mean squared errors between the predicted and true states as the loss function. The NN consists of 3 hidden layers with 64 neurons in each layer and uses the rectified linear unit (ReLU) as the activation function. Additive noise with a maximum magnitude $\sigma_{w}$ of $\{0.05,0.1\}$ is injected into the states of the system. For the primary control policy, we consider an iterative linear quadratic regulator (iLQR) scheme [35] with the box-constrained heuristic [36]. This is implemented with the _mpc.pytorch_ library [37]. The box-constrained heuristic allows the system to adhere to the control constraints, but does not account for state constraints. Four test cases are considered in our experiments. In each of these cases, the system is required to track a pair of reference angles $(\theta_{r,1},\,\theta_{r,2})$ sequentially, starting from an initial condition $x_{0}$ and across a duration of 2s. The initial conditions and reference angles are given in Table I. We simulate each of these test cases under 4 control schemes - (i) a nominal iLQR framework, (ii) a soft-constrained iLQR framework (SC-iLQR), (iii) safe-filtered iLQR, where we apply the proposed safety filter to the nominal iLQR scheme, and (iv) safe-filtered SC-iLQR, where the safety filter is applied to SC-iLQR. For SC-iLQR, soft state constraints are incorporated into the cost function of the forward pass of the iLQR algorithm. Specifically, the function $\phi(x)=\text{max}\{0,x\}$ is applied onto each of the constraints, which increases the cost function proportionally whenever the constraints are violated. In safe-filtered iLQR and SC-iLQR, the initial reference trajectories $\hat{z}_{0:T}$ in Algorithm 1 were initialized by iLQR and SC-iLQR, respectively. The state trajectories under these control schemes are plotted in Fig. 2 and the percentages of constraint violations are tabulated in Table II. These percentages are computed by taking the ratio between the number of points in which the states violate the constraints against the total number of points in the state trajectory. Since the nominal iLQR method does not account for state constraints, it results in the largest percentage of constraint violation. While the soft-constrained iLQR reduces the level of constraint violation, there are a number of instances where the constraints are violated, as depicted in Fig. 2. On the other hand, as shown in Table II, through the application of the safety filter to iLQR and SC-iLQR, no constraint violations are observed for the test cases. To illustrate the safety certificate obtained, we plot the slack variables that characterize the trust region and state and input constraints, together with the safety certificate for the third test case, under a maximum noise level $\sigma_{w}=0.05$, in Fig. 3. The combination of Table II and Fig. 3 indicates that given the formulation in (29), the PSF may not give a numerical safety certificate even when the state trajectories are safe. Meanwhile, this proposed PSF can effectively encourage the system to behave safely by minimizing the conservative upper bounds on the constraint violation. The statistics of the computational times of the control scheme, in this case the iLQR scheme, and the safety filter are depicted in Fig. 4. While introducing soft constraints increases the run times of the iLQR scheme, it has a significant effect on the run times of the safety filter, as observed in the right panel of Fig. 4. With the soft constraints added to the iLQR scheme, the state trajectories are closer to the boundaries of the constraint sets, without the activation of the safety filter. This allows the filter to find a solution that satisfies the constraints under a smaller number of iterations, which reduces computational time. ## VI Conclusion We propose a convex optimization-based predictive safety filter for uncertain NN dynamical systems with the inclusion of additive disturbances. By utilizing tools from NN verification and robust linear MPC, our method requires solving a soft-constrained convex program online whose complexity Fig. 3: Top two panels: Time histories of the slack variables for the trust region, state and input constraints. Bottom panel: Time histories of the safety certificate attained. The values of 1 and 0 denote True and False respectively. Fig. 2: Top panel: Plots of the state trajectories under the 4 test cases. The initial and final states are marked with green and magenta circles. The state constraints are plotted with black dashed lines. Bottom panel: Zoomed-in plots of the state trajectories, around the boundaries where constraint violation potentially occurs. is independent of the NN size. With our framework, formal safety guarantees, together with a robust state-feedback controller, are attained when the slack variables in the solution are all zero. ### _Proof of Proposition 1_ Showing that $\mathcal{R}(\mathbf{\eta};\mathbf{K},\mathbf{h},\mathbf{v})\subseteq\mathcal{R}(\mathbf{ \Psi}\widetilde{\mathbf{w}})$ is equivalent to showing for every possible values of $\mathbf{\eta}$, there exist $\widetilde{\mathbf{w}}^{}\in\widetilde{\mathcal{W}}$ such that $\mathbf{\eta}=\mathbf{\Psi}\widetilde{\mathbf{w}}^{}$. Following this perspective, the sufficient conditions in Proposition 1 can be derived inductively. Initial caseAt $t=0$, we have $x_{0}=h_{0},u_{0}=v_{0}$, and \[\Delta_{0}(x_{0},u_{0})\geq\begin{bmatrix}\underline{D}_{0}^{x}& \underline{D}_{0}^{u}\end{bmatrix}y_{0}+\underline{d}_{0}, \tag{32}\] \[\Delta_{0}(x_{0},u_{0})\leq\begin{bmatrix}\underline{D}_{0}^{x}& \underline{D}_{0}^{u}\end{bmatrix}y_{0}+\overline{d}_{0},\] with $y_{0}=[h_{0}^{\top}\ v_{0}^{\top}]^{\top}$. Note that $\eta_{0}=\Delta_{0}(x_{0},u_{0})+w_{0}$. For $\eta_{0}=\Psi^{1,0}\tilde{w}_{0}=\text{diag}(\psi_{0})\tilde{w}_{0}$ to have a solution $\tilde{w}_{0}$ satisfying $\\|\tilde{w}_{0}\\|_{\infty}\leq 1$ for all possible realization of the lumped uncertainty $\eta_{0}$, it is both sufficient and necessary to have \[\\|\text{diag}(\psi_{0})^{-1}\eta_{0}\\|_{\infty}\leq 1\Leftrightarrow\left\{ \begin{array}{l}e_{i}^{\top}\eta_{0}\leq\psi_{0,i},i\in[n_{x}],\\ -e_{i}^{\top}\eta_{0}\leq\psi_{0,i},i\in[n_{x}],\end{array}\right.\] hold robustly for $\eta_{0}$ where $e_{i}$ denotes the $i$-th standard basis. Using the bounds in (32) and the fact that $\\|w_{0}\\|_{\infty}\leq\sigma_{w}$, we have constraints (22a) and (22b) guarantee the robust feasibility of (33) for all possible values of $\eta_{0}$. For any realization of $\eta_{0}$, we denote the corresponding solution as $\tilde{w}_{0}^{}$ such that $\eta_{0}=\text{diag}(\psi_{0})\tilde{w}_{0}^{}$. Induction stepFor a given controller $\{\mathbf{K},\mathbf{h},\mathbf{v}\}$, consider an arbitrary realization of the lumped uncertainty $\eta_{0:T}$. At time $t\geq 1$, let $\mathbf{\eta}_{0:t}:=[\mathbf{0}^{\top}\ \eta_{0}\ \cdots\ \eta_{t-1}^{\top}]^{\top}$ denote the truncation of $\mathbf{\eta}$ consisting of the first $t+1$ components of $\mathbf{\eta}$, and $\mathbf{\Psi}_{0:t}\in\mathcal{L}_{TV}^{t}$ denote the truncation of $\mathbf{\Psi}$ up to the $t+1$-th row and column. The truncated vectors $\mathbf{x}_{e,0:t},\mathbf{u}_{e,0:t}$ and matrices $\mathbf{A}_{0:t},\mathbf{B}_{0:t}$ are defined similarly. Assume there exist $\widetilde{\mathbf{w}}_{0:t}^{}=[\mathbf{0}^{\top}\ \tilde{w}_{0}^{\top}\ \cdots\ \tilde{w}_{t-1}^{\ast\top}]^{\top}$ such that $\mathbf{\eta}_{0:t}=\mathbf{\Psi}_{0:t}\widetilde{\mathbf{w}}_{0:t}^{}$. Next, we will show that under constraint (22) there exist $\tilde{w}_{t}^{}$ with $\\|\tilde{w}_{t}^{}\\|_{\infty}\leq 1$ such that \[\eta_{t} =\sum_{i=1}^{t+1}\Psi^{t+1,t+1-i}\tilde{w}_{i-1}^{} \tag{34}\] \[=\sum_{i=1}^{t}\Psi^{t+1,t+1-i}\tilde{w}_{i-1}^{}+\text{diag}( \psi_{t})\tilde{w}_{t}^{}\] holds. Since $\mathbf{\eta}_{0:t}=\mathbf{\Psi}_{0:t}\widetilde{\mathbf{w}}_{0:t}^{}$, the error dynamics (20b) up to time $t$ can be written as \[\mathbf{x}_{e,0:t}=Z(\mathbf{A}_{0:t}\mathbf{x}_{e,0:t}+\mathbf{B}_{0:t}) \mathbf{u}_{e,0:t}+\mathbf{\Psi}_{0:t}\widetilde{\mathbf{w}}_{0:t}^{}. \tag{35}\] According to [20, Corollary 1], the affine constraint (21) parameterizes all closed-loop system responses $\mathbf{x}_{e,0:t}=\mathbf{\Phi}_{x,0:t}\widetilde{\mathbf{w}}_{0:t}^{}$, $\mathbf{u}_{e,0:t}=\mathbf{\Phi}_{u,0:t}\widetilde{\mathbf{w}}_{0:t}^{}$ under the controller $\mathbf{u}_{e,0:t}=\mathbf{K}_{0:t}\mathbf{x}_{e,0:t}$ where $\mathbf{\Phi}_{x,0:t},\mathbf{\Phi}_{u,0:t},\mathbf{K}_{0:t}\in\mathcal{L}_{TV}^{t}$ are the corresponding truncation of $\mathbf{\Phi}_{x},\mathbf{\Phi}_{u},\mathbf{K}$, respectively. Define $y_{t}:=[h_{0}^{\top}\ v_{t}^{\top}]^{\top}$ and $\Phi^{t,t-i}:=[\Phi_{x}^{t,t-i\top}\ \tilde{\Phi}_{u}^{t,t-i\top}]^{\top}$. It follows from (35) and the fact $\mathbf{x}_{e,0:t}=\mathbf{\Phi}_{x,0:t}\widetilde{\mathbf{w}}_{0:t}^{}$, $\mathbf{u}_{e,0:t}=\mathbf{\Phi}_{u,0:t}\widetilde{\mathbf{w}}_{0:t}^{}$ that the state $x_{t}$ and control input $u_{t}$ at time $t$ under the policy $\{\mathbf{K},\mathbf{h},\mathbf{v}\}$ are given by \[\begin{bmatrix}x_{t}\\ u_{t}\end{bmatrix}=z_{t}+\sum_{i=1}^{t}\Phi^{t,t-i}\tilde{w}_{i-1}^{}. \tag{36}\] Correspondingly, the uncertainty $\Delta_{t}(x_{t},u_{t})$ at time $t$ are bounded by \[\Delta_{t}(x_{t},u_{t}) \geq\begin{bmatrix}\underline{D}_{t}^{x}&\underline{D}_{t}^{u} \end{bmatrix}z_{t}+\underline{d}_{t} \tag{37}\] \[=\begin{bmatrix}\underline{D}_{t}^{x}&\underline{D}_{t}^{u}\end{bmatrix} \left(y_{t}+\sum_{i=1}^{t}\Phi^{t,t-i}\tilde{w}_{i-1}^{}\right)+\underline{d}_{t},\] \[\Delta_{t}(x_{t},u_{t}) \leq\begin{bmatrix}\underline{D}_{t}^{x}&\overline{D}_{t}^{u} \end{bmatrix}z_{t}+\overline{d}_{t}\] \[=\begin{bmatrix}\underline{D}_{t}^{x}&\overline{D}_{t}^{u}\end{bmatrix} \left(y_{t}+\sum_{i=1}^{t}\Phi^{t,t-i}\tilde{w}_{i-1}^{}\right)+\overline{d}_{t},\] which is achieved by plugging (36) into the definition of $\mathcal{P}_{t}$. Recall that $\eta_{t}=\Delta_{t}(x_{t},u_{t})+w_{t}$. For \[\eta_{t}=\sum_{i=1}^{t}\Psi^{t+1,t+1-i}\tilde{w}_{i-1}^{}+\text{diag}(\psi_{t}) \tilde{w}_{t} \tag{38}\] to have a solution $\tilde{w}_{t}$ such that $\\|\tilde{w}_{t}\\|_{\infty}\leq 1$, it is equivalent to require \[e_{i}^{\top}\Big{(}\eta_{t}-\sum_{i=1}^{t}\Psi^{t+1,t+1-i}\tilde{w}_{i-1}^{} \Big{)} \leq\psi_{t,i},i\in[n_{x}], \tag{39}\] \[-e_{i}^{\top}\Big{(}\eta_{t}-\sum_{i=1}^{t}\Psi^{t+1,t+1-i} \tilde{w}_{i-1}^{}\Big{)} \leq\psi_{t,i},i\in[n_{x}],\] Fig. 4: Statistics of the run times of the controller and the safety filter, under the four test methods. hold. Since $\Delta_{t}(x_{t},u_{t})$ is bounded by (37), $\\|\tilde{w}_{t}^{}\\|_{\infty}\leq 1$ for $i\in[t-1]$ and $\\|w_{t}\\|_{\infty}\leq\sigma_{w}$, we have constraints (22c) and (22d) are sufficient to guarantee the inequalities (39) hold. Then, we denote the solution of (38) as $\tilde{w}_{t}^{}$ for the given realization of $\eta_{t}$. We repeat this process until $t=T$ and in this way construct the virtual disturbance $\widetilde{\mathbf{w}}^{}$ such that $\boldsymbol{\eta}=\boldsymbol{\Psi}\widetilde{\mathbf{w}}^{}$. Since the realization of $\boldsymbol{\eta}$ is chosen arbitrarily, we prove Proposition 1.
2303.16874	CheckerPose: Progressive Dense Keypoint Localization for Object Pose Estimation with Graph Neural Network	Estimating the 6-DoF pose of a rigid object from a single RGB image is a crucial yet challenging task. Recent studies have shown the great potential of dense correspondence-based solutions, yet improvements are still needed to reach practical deployment. In this paper, we propose a novel pose estimation algorithm named CheckerPose, which improves on three main aspects. Firstly, CheckerPose densely samples 3D keypoints from the surface of the 3D object and finds their 2D correspondences progressively in the 2D image. Compared to previous solutions that conduct dense sampling in the image space, our strategy enables the correspondence searching in a 2D grid (i.e., pixel coordinate). Secondly, for our 3D-to-2D correspondence, we design a compact binary code representation for 2D image locations. This representation not only allows for progressive correspondence refinement but also converts the correspondence regression to a more efficient classification problem. Thirdly, we adopt a graph neural network to explicitly model the interactions among the sampled 3D keypoints, further boosting the reliability and accuracy of the correspondences. Together, these novel components make CheckerPose a strong pose estimation algorithm. When evaluated on the popular Linemod, Linemod-O, and YCB-V object pose estimation benchmarks, CheckerPose clearly boosts the accuracy of correspondence-based methods and achieves state-of-the-art performances. Code is available at https://github.com/RuyiLian/CheckerPose.	Ruyi Lian, Haibin Ling	2023-03-29T17:30:53Z	http://arxiv.org/abs/2303.16874v2	CheckerPose: Progressive Dense Keypoint Localization for Object Pose Estimation with Graph Neural Network ###### Abstract Estimating the 6-DoF pose of a rigid object from a single RGB image is a crucial yet challenging task. Recent studies have shown the great potential of dense correspondence-based solutions, yet improvements are still needed to reach practical deployment. In this paper, we propose a novel pose estimation algorithm named _CheckerPose_, which improves on three main aspects. Firstly, CheckerPose densely samples 3D keypoints from the surface of the 3D object and finds their 2D correspondences progressively in the 2D image. Compared to previous solutions that conduct dense sampling in the image space, our strategy enables the correspondence searching in a 2D grid (i.e., pixel coordinate). Secondly, for our 3D-to-2D correspondence, we design a compact binary code representation for 2D image locations. This representation not only allows for progressive correspondence refinement but also converts the correspondence regression to a more efficient classification problem. Thirdly, we adopt a graph neural network to explicitly model the interactions among the sampled 3D keypoints, further boosting the reliability and accuracy of the correspondences. Together, these novel components make our CheckerPose a strong pose estimation algorithm. When evaluated on the popular Linemod, Linemod-O, and YCB-V object pose estimation benchmarks, CheckerPose clearly boosts the accuracy of correspondence-based methods and achieves state-of-the-art performances. ## 1 Introduction Object pose estimation from RGB images aims to estimate the rotation and translation of a given rigid object relative to the camera. It is crucial in various applications including robot grasping and manipulation [80, 66, 67], autonomous driving [42, 74, 34], augmented reality [43, 63], _etc._ Most existing methods [54, 64, 44, 22, 49, 77, 46, 38, 58] first estimate an intermediate geometric representation, _i.e_., the correspondences between 3D object keypoints and 2D image locations, and then recover the object pose using the Perspective-n-Point (PnP) algorithm. Theoretically, for a rigid object, four pairs of 3D-2D correspondences can determine a unique pose [53, 12, 50]. In practice, however, sparse correspondences easily degrade due to occlusion, background clutter, lighting variation, _etc._ Increasing the number of 3D-2D correspondences is a feasible solution to enhance robustness, especially when combined with outlier removal mechanisms such as RANSAC. Recent methods [77, 46, 38, 18, 21, 71, 10] densely sample 2D image pixels and predict their 3D object coordinates. While these dense predictions improve the robustness of pose estimation, they have several drawbacks. Firstly, the predictions consider only visible pixels and ignore global relations b Figure 1: Illustration of CheckerPose. We evenly sample dense keypoints from the surface of the object, and predict the 2D locations in the input image. We design a binary code representation to progressively localize each keypoint in the iteratively refined 2D grids. To improve the localization results, we also use graph neural networks to explicitly model the interactions between 3D keypoints. Note: we plot 8 keypoints for better visualization, while we use 512 keypoints in practice. keypoints, making them unstable when the object is under severe occlusions. Secondly, estimating the corresponding 3D coordinates is nontrivial. Finally, the rich shape prior information is not effectively encoded. To overcome the above issues, we propose a novel 6D pose estimation algorithm, named _CheckerPose_, which improves dense correspondence with three cooperative components: dense 3D sampling, progressive 2D localization through binary coding, and shape prior encoding with graph neural network, as illustrated in Figure 1. For dense correspondence, CheckerPose samples 3D keypoints on the object surface and then finds their 2D pixel correspondences in the 3D-to-2D matching way. Compared to previous solutions that conduct dense sampling in the 2D image space, our strategy enables more efficient correspondence searching in a 2D grid (, pixel coordinate) using 2D binary coding, as well as explicit shape prior modeling with graph representation. Then, to facilitate the localization of dense keypoints, we propose a 2D hierarchical binary coding to represent a 2D image position. Specifically, we superpose a grid on the input image and predict which cells contain the desired keypoints. The precision of the 2D keypoint location is controlled by the resolution of the grid. This novel representation allows us to refine the correspondence progressively. We first localize the keypoints in the $2\times 2$ grid, and then iteratively subdivide each cell and localize the keypoints in the refined grid. Inspired by ZebraPose [61], we use binary codes on the $x$ and $y$ directions to represent each cell, which makes the grids have a checkerboard pattern. Furthermore, to capture the shape prior of the 3D object, we adopt a graph neural network to explicitly model the interactions among the sampled 3D keypoints and to guide the progressive correspondence estimation. In particular, we construct the $k$-nearest neighbor ($k$-NN) graph of the dense keypoints and utilize graph network layers to fuse information from a keypoint and its neighbors. By stacking multiple such layers, we can capture non-local interactions between invisible and visible keypoints, and thus significantly improve the prediction robustness of invisible keypoints. To summarize, our main contributions are as follows: * We propose to localize dense 3D keypoints in the input image, to establish dense correspondences for instance-level object pose estimation. * We design a hierarchical binary coding strategy for 2D projections, which enables progressive localization of dense keypoints. * We utilize graph neural networks to explicitly model the interactions between 3D keypoints and improve the predictions of invisible keypoints. Together, these novel contributions make our CheckerPose a strong pose estimation algorithm. We conduct extensive experiments on the popular benchmarks including Linemod [17], Linemod-Occlusion [2], and YCB-V [75], and CheckerPose consistently achieves state-of-the-art performances. We will release the source code of our CheckerPose upon the publication of the work. ## 2 Related Work In this section we review previous studies that are closely related to our work, mainly including different types of pose estimators and graph neural networks. Direct Methods.Given an input RGB image, direct methods estimate the 6D pose of the object in the image without intermediate geometric representations,, 3D-2D correspondences. Traditional direct methods mainly adopt template matching techniques with hand-crafted features [25, 13, 16], and thus can not handle textureless objects well. Recent deep learning based methods utilize features learned by CNNs to directly regress 6D pose [75] or formulate the rotation estimation as a classification task by discretizing the rotation space $SO(3)$[68, 60, 28, 62]. Correspondence Guided Methods.Instead of direct estimation, correspondence guided methods [48, 54, 64, 44, 22, 49, 21, 23, 77, 46, 38, 71, 10, 61] follow a two-stage framework: they first predict a set of correspondences between 3D object frame coordinates and 2D image plane coordinates, and then recover the pose from the 3D-2D correspondences with a PnP algorithm [32, 30, 11, 69, 6]. RANSAC can be used to remove the outliers in the correspondences. Keypoint-localization based methods [48, 54, 64, 44, 22, 49, 21, 23] estimate the 2D coordinates for a sparse set of predefined 3D keypoints, while dense methods [77, 46, 38, 71, 10, 61] predict the 3D object frame coordinate of each 2D image pixel. Compared with sparse correspondences, dense correspondences contain richer context information of the scene and is more robust to occlusion. Graph Neural Networks for 3D Vision.In 3D vision tasks, point clouds and meshes are important input data formats since they can efficiently represent complex shapes. Compared with convolutional neural networks (CNNs), graph neural networks (GNNs) [59] can handle inputs with irregular structures and effectively model the long-range dependencies, and thus are widely used for processing point clouds and meshes. While meshes can be naturally treated as graphs, a common practice of constructing graphs from point clouds is to treat each 3D point as graph nodes and connect each node to its $k$ nearest neighbors [73, 57, 8]. GNN-based methods have been proposed for representation learning [57, 73, 40, 70], detection [56, 8], segmentation [51, 33], data generation [52, 39], camera pose inference [35, 36],. For object pose estimation, GNNs are mainly used for RGB-D inputs [9, 79] to enhance the feature extraction from different modalities. Another recent application is to learn geometric structures of the sparse keypoints for domain adaptation [78]. Our work follows the two-stage framework and combines the strengths of both keypoint-based methods and dense methods, by localizing a dense set of predefined 3D keypoints to establish dense correspondences. Moreover, it utilizes GNNs to efficiently model the interactions among dense 3D keypoints and thus improve the localization in the input RGB image for monocular object pose estimation. ## 3 Method ### Problem Formulation and Method Overview Given an RGB image $I$ and a rigid object $O$, our goal is to estimate rotation $\mathbf{R}\in SO(3)$ and translation $\mathbf{t}\in\mathbb{R}^{3}$ of $O$ relative to the calibrated camera. We assume the 3D geometry information, _e.g_., the 3D CAD model, is available, thus we can obtain $N(N\gg 8)$ keypoints $\mathcal{P}$ from the object surface using farthest point sampling (FPS) algorithm. We adopt a two-stage pipeline for object pose estimation: we first predict 2D projection $\mathbf{\rho}\in\mathbb{R}^{2}$ for each keypoint $P\in\mathbb{R}^{3}$, and then regress the rotation and translation from the 3D-2D correspondences via a PnP solver. For the input RGB image, we use an off-the-shelf object detector [55, 65] to detect the object bounding box and extract the zoomed-in Region of Interest (RoI) $I_{O}$, following the common practice in instance-level object pose estimation [38, 71, 10, 61]. Figure 2 illustrates our proposed pipeline. We first process the input RoI $I_{O}$ by a backbone network to obtain backbone feature $F_{I}^{(0)}$ and keypoint embedding $F_{G}^{(0)}$ in the $k$-NN graph $\mathcal{G}$. Then we use graph network layers (_i.e_., EdgeConv [73]) to progressively localize the keypoints, which are represented as binary codes $\mathbf{b_{v}},\mathbf{b_{x}}$, and $\mathbf{b_{y}}$. We also use a standard CNN decoder to transform $F_{I}^{(0)}$ to a series of image feature maps, and fuse the features in the graph neural network based on the current predicted locations. The CNN decoder also outputs object segmentation masks $M$ as an auxiliary learning task. Finally, we convert the binary codes to 2D coordinates and use a PnP solver to recover the pose from the established correspondences. We describe our method, named _CheckerPose_ due to the checkerboard-like binary pattern, in details as follows. ### Hierarchical Representation of 2D Keypoint Locations Establishing 3D-2D correspondences provides an intermediate representation for object pose estimation. In this Figure 2: Framework of our progressive dense keypoint localization with graph neural network, _i.e_., CheckerPose. Given an RGB image and object detection results, we progressively generate the binary codes representing the 2D locations of $N$ 3D keypoints. (a) Initial graph embedding generation: we use a CNN backbone network to extract feature $F_{I}^{(0)}$ from the zoomed-in RoI $I_{O}$, and then transform $F_{I}^{(0)}$ to the initial keypoint embeddings $F_{G}^{(0)}$ in the $k$-NN graph $\mathcal{G}$. (b) Progressive prediction: we use a graph neural network to generate the binary code representation in a coarse-to-fine manner. We adopt an additional CNN decoder network to generate image features from $F_{I}^{(0)}$, and fuse the features in the graph neural network based on current predictions. Object segmentation masks $M$ are predicted as an auxiliary learning task. work, we focus on localizing a dense set of predefined 3D keypoints $\mathcal{P}$ in the 2D image plane. For $N(N\gg 8)$ 3D keypoints $\mathcal{P}$, we first predict whether their 2D projections appear in the RoI $I_{O}$, and then localize the keypoints inside $I_{O}$, denoted as $\mathcal{P}_{I}$. In contrast to directly regressing the precise coordinates, we superpose a $2^{d}\times 2^{d}$ grid $S$ on the RoI $I_{O}$ and predict which cell $s\in S$ contains the 2D projection $\mathbf{\rho}$ (Figure 3 (a)). Then we can use the coordinate of the cell center to approximate $\mathbf{\rho}$, and only need to predict the discrete index $(i_{x},i_{y})(0\leq i_{x},i_{y}\leq 2^{d}-1)$ of the cell $s$, which is much easier than precise regression. The localization precision is controlled by the resolution of the grid $S$, and approaches the actual 2D projection as $d\rightarrow\infty$. Based on the approximate representation, we can further localize the keypoint $P\in\mathcal{P}_{I}$ in a coarse-to-fine manner. As shown in Figure 3 (b), at the beginning, we superpose a $2\times 2$ grid $S^{(1)}$ on the RoI $I_{O}$ and predict the index of the cell $s_{P}^{(1)}$. Then at iteration $j$$(2\leq j\leq d)$, we increase the grid resolution from $2^{j-1}\times 2^{j-1}$ to $2^{j}\times 2^{j}$ by evenly splitting each cell $s^{(j-1)}\in S^{(j-1)}$ into halves on both $x$ and $y$ directions. With the prediction of $s_{P}^{(j-1)}$ in iteration $j-1$, we only need to search the corresponding $2\times 2$ sub-cells to find $s_{P}^{(j)}$ in the refined grid $S^{(j)}$. Inspired by ZebraPose [61], we use binary codes to concisely represent the hierarchical localization. For the cell $s_{P}$ in the final $2^{d}\times 2^{d}$ grid $S$, we use a $d$-bit binary code $\mathbf{b_{x}}$ to represent the index $i_{x}$ as \[i_{x}=\sum_{k=1}^{d}b_{x}(k)\times 2^{d-k}, \tag{1}\] where $b_{x}(k)$ is the $k$-th bit of $\mathbf{b_{x}}$. We use another $d$-bit binary code $\mathbf{b_{y}}$ to represent the index $i_{y}$ in the same way. The first $j(1\leq j\leq d)$ bits of $\mathbf{b_{x}}$ and $\mathbf{b_{y}}$ also represent the cell $s_{P}^{(j)}\in S^{(j)}$. We use an additional 1-bit binary code $\mathbf{b_{v}}$ to indicate the existence of the projection $\mathbf{\rho}$ in the RoI $I_{O}$, where $\mathbf{b_{v}}=1$ means $\mathbf{\rho}\in I_{O}$ while $\mathbf{b_{v}}=0$ means $\mathbf{\rho}\notin I_{O}$. Compared with dense representations like heatmaps [48, 44] and vector-fields [49, 22], our novel representation needs only $2d+1$ binary bits for each keypoint, thus greatly reduces the memory usage for dense keypoint localization. In addition, during inference, we can efficiently convert the binary codes to the 2D coordinates. Furthermore, our representation can be naturally predicted in a progressive way, which allows to gradually improve the localization via iterative refinements. ### Dense Keypoint Localization via Graph Neural Network Modeling the interactions among the keypoints $\mathcal{P}$ is crucial for predicting their 2D locations. For the keypoints that are invisible due to occlusions or self-occlusions, the features of the visible ones provide additional clues to infer the 2D locations. However, previous keypoint-based methods mainly use convolutional neural networks (CNNs), which can not handle inputs with irregular structure and thus fail to explicitly capture the interactions among $\mathcal{P}$. We instead utilize graph neural networks (GNNs) to process the features $F=\{f_{1},\cdots,f_{N}\}$ of $N$ keypoints $\mathcal{P}$. To construct a graph $\mathcal{G}$ from $\mathcal{P}$, we treat each keypoint $P_{i}\in\mathcal{P}(1\leq i\leq N)$ as a graph node, and connect $P_{i}$ to its $k$ nearest neighbors in 3D Euclidean space to generate edges $\mathcal{E}$. We adopt the EdgeConv operation [73] as our graph network layer, which directly models local interactions between $P_{i}$ and its neighbors. For edge $(i,j)\in\mathcal{E}$, we compute the feature $e_{ij}$ as \[e_{ijm}=\mathrm{ReLU}(\theta_{m}\cdot(f_{j}-f_{i})+\phi_{m}\cdot f_{i}), \tag{2}\] where $e_{ijm}$ is the $m$-th channel of $e_{ij}$, and $\theta_{m},\phi_{m}$ are the weights of the filters. The feature of $P_{i}$ is updated by aggregating the edge features as \[f_{im}^{{}^{\prime}}=\max_{j:(i,j)\in\mathcal{E}}e_{ijm}, \tag{3}\] where $f_{im}^{{}^{\prime}}$ is the $m$-th channel of updated feature $f_{i}^{{}^{\prime}}$. By stacking multiple EdgeConv operations, our network can gradually learn the non-local interactions in a computational-efficient way for dense keypoints $\mathcal{P}$. As shown in Figure 2 (a), to obtain the initial keypoint embeddings $F_{G}^{(0)}$ in $\mathcal{G}$, we first use a backbone network to extract a $C_{0}\times 2^{d_{0}}\times 2^{d_{0}}$ feature map $F_{I}^{(0)}$ from RoI $I_{O}$, where $C_{0}$ is the number of the feature channels, and $2^{d_{0}}\times 2^{d_{0}}$ is the spatial size. We then reshape $F_{I}^{(0)}$ to Figure 3: Illustration of our keypoint location representation. (a) We represent the 2D projection coordinate as the center of the cell containing the 2D projection. (b) We iteratively refine the grid and represent the cell as binary codes $\mathbf{b_{x}},\mathbf{b_{y}}$. $C_{0}\times 2^{2d_{0}}$ by flattening the spatial dimensions, and use a 1D convolutional network layer to obtain a $N\times 2^{2d_{0}}$ feature map, which is regarded as the initial $2^{2d_{0}}$-dimensional embeddings $F_{G}^{(0)}$ for $N$ keypoints. After obtaining $F_{G}^{(0)}$, we use graph neural network to predict the 1-bit indicator code $\mathbf{b_{v}}$, and progressively generate the $d$-bit index codes $\mathbf{b_{x}},\mathbf{b_{y}}$. Specifically, at stage 0, we apply $L_{0}$ EdgeConv [73] operations to $F_{G}^{(0)}$ to get the updated embeddings $F_{G}^{(1)}$, and then use shared MLPs to generate $\mathbf{b_{v}}$ and the first $d_{0}$ bits of $\mathbf{b_{x}},\mathbf{b_{y}}$, respectively. Then at stage $j(1\leq j\leq d-d_{0})$, we apply $L_{j}$ EdgeConv operations to $F_{G}^{(j)}$ to obtain $F_{G}^{(j+1)}$, and use shared MLPs to generate new bits $b_{x}(d_{0}+j),b_{y}(d_{0}+j)$ for $\mathbf{b_{x}},\mathbf{b_{y}}$, respectively. We regard stage $j(1\leq j\leq d-d_{0})$ as refinement stage, since it refines the localization from the low-resolution grid $S^{(d_{0}+j-1)}$ to the high-resolution one $S^{(d_{0}+j)}$. Compared with generating all bits at the network output layer, our progressive prediction enables image feature fusion at each refinement stage. As shown in Figure 2 (b), starting with the image feature map $F_{I}^{(0)}$ with low spatial resolution $2^{d_{0}}\times 2^{d_{0}}$, we use an additional CNN-based decoder to progressively generate image feature maps $F_{I}^{(1)},\cdots,F_{I}^{(d-d_{0})}$ with increased spatial resolutions $2^{d_{0}+1}\times 2^{d_{0}+1},\cdots,2^{d}\times 2^{d}$, respectively. We also add skip connections between the backbone and the decoder to recover the spatial information lost in $F_{I}^{(0)}$. At the beginning of the refinement stage $j$, our graph neural network generates the first $d_{0}+j-1$ bits of $\mathbf{b_{x}},\mathbf{b_{y}}$ for each keypoint $P$, corresponding to the cell $s_{P}^{(d_{0}+j-1)}$ in the grid $S^{(d_{0}+j-1)}$. We then crop a local feature patch $F_{I}^{(j)}$ centered at $s_{P}^{(d_{0}+j-1)}$ from the image feature $F_{I}^{(j)}$, and concatenate $F_{I}^{(j)}$ with the keypoint embedding in the graph $\mathcal{G}$. Since the initial keypoint embeddings $F_{G}^{(0)}$ are obtained from $F_{I}^{(0)}$, fusing the local image features in the refinement stages provides critical spatial information for fine-grained localization. ### Training For the 1-bit indicator code $\mathbf{b_{v}}$ of keypoint $P\in\mathcal{P}$, our network output $\hat{\mathbf{b}}_{\mathbf{v}}$ is the probability that $\mathbf{b_{v}}=1$. We use binary cross-entropy loss for $\mathbf{b_{v}}$ as below: \[\mathcal{L}_{v}=\frac{1}{N}\sum_{P\in\mathcal{P}}\mathbf{b_{v}}\log\hat{ \mathbf{b}}_{\mathbf{v}}+(1-\mathbf{b_{v}})\log(1-\hat{\mathbf{b}}_{\mathbf{v }}), \tag{4}\] where $N$ is the number of the keypoints. For $d$-bit index codes $\mathbf{b_{x}},\mathbf{b_{y}}$, since we only localize the keypoints inside the RoI (_i.e._, $\mathbf{b_{v}}=1$), denoted as $\mathcal{P}_{I}$, we compute binary cross-entropy loss for each bit of $\mathbf{b_{x}}$ as \[\mathcal{L}_{x}=\frac{1}{dN_{I}}\sum_{P\in\mathcal{P}_{I}}\sum_{k =1}^{d}b_{x}(k)\log(\hat{b}_{x}(k))+\\ (1-b_{x}(k))\log(1-\hat{b}_{x}(k)), \tag{5}\] where $N_{I}$ is the number of keypoints inside the RoI, $\hat{b}_{x}(k)$ is the network prediction for $k$-th bit of $\mathbf{b_{x}}$. We compute the loss $\mathcal{L}_{y}$ for $\mathbf{b_{y}}$ in the same way as $\mathcal{L}_{x}$. Besides predicting the 2D projections as binary codes, we also enforce the network to output object segmentation masks. To do this, we apply a single CNN layer to the final image feature map $F_{I}^{(d-d_{0})}$ and obtain a $2\times 2^{d}\times 2^{d}$ output, which is served as the full segmentation mask $M_{\mathrm{full}}$ and the visible one $M_{\mathrm{vis}}$. We input the network predictions to the sigmoid function and apply $L_{1}$ loss as the mask loss $\mathcal{L}_{\mathrm{mask}}$. Generating the masks can be regarded as an auxiliary task to facilitate the learning of image features. The overall loss function $\mathcal{L}$ is a combination of $\mathcal{L}_{v}$, $\mathcal{L}_{x}$, $\mathcal{L}_{y}$, and $\mathcal{L}_{\mathrm{mask}}$ as \[\mathcal{L}=\mathcal{L}_{v}+\mathcal{L}_{x}+\mathcal{L}_{y}+\mathcal{L}_{ \mathrm{mask}}. \tag{6}\] Before training the whole network, we pretrain the layers that generate $\mathbf{b_{v}}$ and the first $d_{0}$ bits of $\mathbf{b_{x}},\mathbf{b_{y}}$. This encourages the backbone network to quickly adapt to the object keypoints with smaller GPU memory usage, and makes the initial localization to be good for local image feature fusion in the refinement stages. ### Inference During inference, we first discard the keypoints with $\mathbf{b_{v}}=0$. Then we convert the binary codes to the corresponding cells in the final grid $S$ (Eq. 1), and use the 2D coordinates of the cell centers as the keypoint projections. In this way, we establish dense 3D-2D correspondences from the network outputs without time-consuming computation operations, _e.g._, voting for the vector-field representations [49]. Finally we use the RANSAC/PnP [32] or Progressive-X [1] solvers to obtain the object pose from the dense 3D-2D correspondences. We empirically find that for textureless objects with severe self-occlusions, discarding the correspondences outside $M_{\mathrm{vis}}$ can improve the pose estimation results. To quantify the self-occlusions of a given object $O$, we uniformly sample 2,562 camera viewpoints on a sphere, and use Hidden Point Removal (HPR) operator [27] to estimate the visibility of point $P\in O$ from each viewpoint. We then calculate the proportion of the viewpoints for which $P$ is visible, denoted as $V(P)$. If $0.2\leq V(P)<0.4$, then $P$ is considered to be easily self-occluded. Note we ignore the points with $V(P)<0.2$, to make our estimation robust to the classification error of the HPR operator. The overall self-occlusion of the object $O$ can be computed by \[r_{\mathrm{so}}(O)=\frac{1}{\|O\|}\sum_{P\in O}\mathbb{1}(0.2\leq V(P)<0.4), \tag{7}\] where $\|O\|$ is the number of vertices of the object CAD model, and $\mathbb{1}(\cdot)$ is the indicator function. If $r_{\mathrm{so}}(O)\geq 0.5$, _i.e_., over half part of $O$ is easily to be self-occluded, then we regard $O$ as severely self-occluded. ## 4 Experiments ### Experimental Setup Implementation Details.Our method is implemented using PyTorch [47] and trained using the Adam optimizer [29] with a batch size of 32. We pretrain our network for $50,000$ steps with learning rate of 2e-4. For all experiments, we use $N=512$ keypoints, and utilize $k=20$ nearest neighbors to construct the $k$-NN graph $\mathcal{G}$. For the binary code representation, we set $d=6$ and $d_{0}=3$. We resize the input RoIs to $256\times 256$, and use HRNet [72] as our image feature backbone to extract $1024\times 8\times 8$ feature map $F_{I}^{(0)}$. Then we apply $L_{0}=2$ EdgeConv operations to get $\mathbf{b}_{\mathbf{v}}$ and the first $d_{0}=3$ bits of $\mathbf{b}_{\mathbf{x}},\mathbf{b}_{\mathbf{y}}$, and obtain the full binary codes after 3 refinement stages with $L_{j}=3$ ($j=1,2,3$) EdgeConv operations. Datasets.We conduct our experiments on three commonly-used datasets for object pose estimation: Linemod (LM) [17], Linemod-Occlusion (LM-O) [2], and YCB-V [75]. The LM dataset consists of 13 sequences of real images with ground truth poses for a single object with background clutter and mild occlusion. Each sequence contains around $1,200$ images. Following [3], we utilize about $15\%$ images for training while keeping the remaining ones for testing. We additionally use $1,000$ synthetic RGB images for each object during training following [38, 71, 10]. The LM-O dataset consists of $1,214$ images from a sequence of the LM dataset [17], where ground truth poses of eight objects with partial occlusion are annotated for testing. The YCB-V dataset is composed of more than $110,000$ real images of $21$ objects with severe occlusion and clutter. Apart from the real training images, we also utilize the physically-based rendered data following [19] for training on LM-O and YCB-V datasets. Evaluation Metrics.We employ the common evaluation metric ADD(-S) for object pose estimation. The ADD(-S) metric measures whether the average distance between the model points transformed by the predicted pose and the ground truth is less than $10\%$ of the object's diameter (0.1d), and for symmetric objects, ADD(-S) metric computes the deviation to the closest model point. On the YCB-V dataset, we also compute the AUC (area under curve) of ADD-S and ADD(-S) with a maximum threshold of 10 cm [75]. For LM dataset, we also report the $n^{\circ},n$ cm metric, which measures the percentage of predicted 6D poses with rotation error below $n^{\circ}$ and translation error below $n$ cm. For symmetric objects $n^{\circ},n$ cm computes the smallest error for all possible ground truth poses [37, 71]. ### Ablation Study on LINEMOD Dataset We present several ablation experiments on the LM dataset [17] in Table 1 to verify the effectiveness of each module. We train a single pose estimator for all objects for 120,000 steps, with a fixed learning rate of 1e-4 for the first 100,000 steps and a smaller learning rate of 5e-5 for the remaining steps. During inference, we utilize the detection results from Faster-RCNN [55] provided by [38]. We do not use any segmentation masks to filter the correspondences for fair comparison. Without specification, we use Progressive-X [1] to compute pose from the dense correspondences. Comparison with State of the Art.As shown in Table 1, our method outperforms state-of-the-art approaches [71, 10, 7] w.r.t. ADD(-S) 0.05d, ADD(-S) 0.1d, and $5^{\circ}5$cm, and achieves comparable results w.r.t. ADD(-S) 0.02d and $2^{\circ}2$cm. The improvement of ADD(-S) 0.1d indicates that our method can facilitate the estimation of hard cases and serve as a good initialization for refinement methods [37, 26, 76]. Since the 2D coordinates of our estimated correspondences are approximated by the cell centers (Sec. 3.2), our pose estimation results in terms of ADD(-S) 0.02d may be further improved by increasing the resolution of the grid. model the interactions between different keypoints. We also report the result of removing all GNN layers in Table 1. Without GNN layers, the keypoints still interact indirectly via local image feature fusion modules, since the keypoints with close 2D locations share the similar local image features. However, the performance of pose estimation degrades significantly, demonstrating that it is important to directly model the keypoint interactions with GNN layers. Effectiveness of Progressive Prediction. Progressively generating the binary codes enforces our network to gradually refine the localization in the iteratively subdivided grids. It also enables image feature fusion based on the intermediate estimations, which can provide crucial spatial information for fine-grained localization. As shown in Table 1, the accuracy decreases significantly without progressively generating the binary codes, which clearly demonstrates the importance of progressive prediction. Effectiveness of Object Segmentation Masks. Our network outputs the full segmentation mask $M_{\text{full}}$ and the visible one $M_{\text{vis}}$ as auxiliary tasks. As shown in Table 1, the performance degrades without either $M_{\text{full}}$ or $M_{\text{vis}}$. The ADD(-S) 0.02d metric drops significantly without $M_{\text{full}}$, indicating that predicting $M_{\text{full}}$ facilitates image feature extraction for keypoint localization, since all the keypoints should be located within $M_{\text{full}}$. The degraded performance without $M_{\text{vis}}$ also implies that predicting $M_{\text{vis}}$ provides important context information including occlusions. Impact of Backbone Networks. We report the results of our method with different backbone networks in Table 1. After replacing HRNet [72] by ResNet34 [15], our method still achieves comparable results with state of the art, which demonstrates the efficacy of our method regardless of the backbone networks. Influence of PnP Solvers. We show the results with different PnP solvers during inference in Table 1. Since our correspondences are established from the binary codes, a small perturbation of our network prediction can result in flipped bit values, which may correspond to dramatically different locations in the input RoI. Compared with RANSAC/PnP [32], Progressive-X [1] contains a spatial coherence filter to efficiently remove such outliers, and thus achieves better performance w.r.t. to all the metrics, especially ADD(-S) 0.02d. ### Comparison to State of the Art In this section we present the quantitative results of our method on LM-O and YCB-V datasets. We train a single CheckerPose for each object for 380,000 steps with a fixed learning rate of 1e-4. During inference, we utilize the detections from FCOS [65] provided by CDPNv2 [38]. Experiments on the LM-O dataset. We report the recall of ADD(-S) metric for the LM-O dataset in Table 2. Based on the criterion discussed in Sec. 3.5, we filter out the correspondences outside the visible segmentation masks $M_{\text{vis}}$ for textureless objects with severe self-occlusions, including can, cat, driller, and eggbox. Without the filtering operation, the average recall of ADD(-S) of our method is 77.1, which surpasses previous methods. The detailed results of each object without filtering are provided in supplementary material. The additional filtering operation further improves the performance of our method. The intuition is that it is infrequent to observe an easily self-occluded keypoint $P$ in the training images. Besides, due to the lack of texture, it is also hard to infer the location of $P$ from other keypoints with distinguishable features. Such objects may require much more training steps to achieve stable estimations for easily self-occluded keypoints. Simply discarding correspondences outside $M_{\text{vis}}$ reduces unstable localization results when our network is trained for limited steps, and enhances the robustness of pose estimation. Experiments on the YCB-Video dataset. We report the averaged metrics of 21 objects in Table 3, and provide the detailed results in supplementary material. Based on the criterion discussed in Sec. 3.5, we use visible segmentation \begin{table} \begin{tabular}{c\|c\|c\|c\|c\|c\|c\|c\|c} \hline \hline Method & PVNet [49] & S. Stage [21] & Hybrid [58] & RePose [26] & GDR-Net [71] & SO-Pose [10] & Zebra [61] & Ours \\ \hline ape & 15.8 & 19.2 & 20.9 & 31.1 & 46.8 & 48.4 & 57.9 & 58.3 \\ can & 63.3 & 65.1 & 75.3 & 80.0 & 90.8 & 85.8 & 95.0 & 95.7 \\ cat & 16.7 & 18.9 & 24.9 & 25.6 & 40.5 & 32.7 & 60.6 & 62.3 \\ driller & 65.7 & 69.0 & 70.2 & 73.1 & 82.6 & 77.4 & 94.8 & 93.7 \\ duck & 25.2 & 25.3 & 27.9 & 43.0 & 46.9 & 48.9 & 64.5 & 69.9 \\ eggbox* & 50.2 & 52.0 & 52.4 & 51.7 & 54.2 & 52.4 & 70.9 & 70.0 \\ glue* & 49.6 & 51.4 & 53.8 & 54.3 & 75.8 & 78.3 & 88.7 & 86.4 \\ holep. & 36.1 & 45.6 & 54.2 & 53.6 & 60.1 & 75.3 & 83.0 & 83.8 \\ \hline mean & 40.8 & 43.3 & 47.5 & 51.6 & 62.2 & 62.3 & 76.9 & 77.5 \\ \hline \hline \end{tabular} \end{table} Table 2: Comparison with State-of-the-art Methods on the LM-O Dataset. We report the Average Recall (%) of ADD(-S). () denotes symmetric objects. We highlight the best result in red color, and the second best result in blue color. masks to filter correspondences for foam_brick. We also apply the filtering operation to pudding_box because it is severely occluded by gelatin_box, which is a distraction object with similar texture. As shown in Table 3, CheckerPose achieves the best performance w.r.t. ADD(-S) and AUC of ADD(-S), and is comparable with state of the art w.r.t. AUC of ADD-S. ### Qualitative Results In Figure 4, we provide localization results of eight keypoints for the occluded and flipped bowl. While our network directly outputs the 2D locations, the results of other dense methods [61, 71] are computed by projecting the keypoints using the estimated poses. Figure 4 (a) visualizes the reprojections of ZebraPose [61], where the keypoints concentrate on the visible pixels. Since ZebraPose generates pixel-wise 3D coordinates from the visible regions, it predicts a drastically wrong pose for the severely occluded bowl. As shown in Figure 4 (b), the reprojections of GDRNet [71] cover the region similar to the ground truth (Figure 4 (d)). However, the order of the blue keypoint and the red one changes from clockwise to counterclockwise, indicating the bowl is actually faced up. Since GDR-Net is an end-to-end method, it may memorize poses that frequently appear in the training samples. As shown in Figure 4 (c), our network is capable of localizing the keypoints for the upside-down object with severe occlusion. More qualitative results can be found in the Supplementary Material. ### Runtime Analysis We test the running speed on the LM-O dataset. Given a $640\times 480$ RGB image, we evaluate the speed of our method on a desktop with an Intel 3.30GHz CPU and an NVIDIA GeForce GTX 1080 GPU (8G). The FCOS detector [65] takes 87 ms for each image. The runtime of establishing the dense 3D-2D correspondences by our network is 78 ms. RANSAC/PnP [32] takes only 1 ms to recover pose from the correspondences, while Progressive-X [1] takes 32 ms. ## 5 Conclusion In this work, we propose a novel way to establish dense correspondences for object pose estimation, by progressively localizing dense 3D keypoints in the input image. With dense keypoints including occluded and self-occluded ones, we comprehensively explore the available geometry information and enhance the robustness of pose estimation under severe occlusion. We adopt graph neural networks to explicitly model the keypoint interactions, and design a hierarchical binary code representation for the 2D locations. The experiments on LM, LM-O and YCB-V datasets demonstrate that our method achieves state-of-the-art performance of instance-level object pose estimation. Figure 4: Visualization of keypoint localization. (a) Keypoint locations based on the predicted pose of ZebraPose [61]. (b) Keypoint locations based on the pose estimated by GDR-Net [71]. (c) Keypoint locations output by our network. (d) The ground truth keypoint locations. Considering the symmetry of the bowl, we use the equivalent rotations closest to our prediction to project the keypoints in (a), (b), and (d).* \begin{table} \begin{tabular}{l\|c\|c\|c} \hline \hline Method & ADD(-S) & \begin{tabular}{c} AUC \\ ADD-S \\ \end{tabular} & \begin{tabular}{c} AUC \\ ADD-S \\ \end{tabular} & \begin{tabular}{c} AUC \\ ADD(-S) \\ \end{tabular} \\ \hline SegDriven [22] & 39.0 & – & – \\ SingleStage [21] & 53.9 & – & – \\ CosyPose [31] & – & 89.8 & 84.5 \\ RePose [26] & 62.1 & 88.5 & 82.0 \\ GDR-Net [71] & 60.1 & 91.6 & 84.4 \\ SO-Pose [10] & 56.8 & 90.9 & 83.9 \\ ZebraPose [61] & 80.5 & 90.1 & 85.3 \\ DProST [45] & 65.1 & – & 77.4 \\ CheckerPose (Ours) & 81.4 & 91.3 & 86.4 \\ \hline \hline \end{tabular} \end{table} Table 3: Comparison with State-of-the-art Methods on the YCB-Video Dataset. We report the ADD(-S), and AUC of ADD-S and ADD(-S). Following [75], the symmetric metric is used for all objects in ADD-S while only for symmetric objects in ADD(-S). We highlight the best result in red color, and the second best result in blue color. “–” denotes unavailable results.
2306.02775	Input-gradient space particle inference for neural network ensembles	Deep Ensembles (DEs) demonstrate improved accuracy, calibration and robustness to perturbations over single neural networks partly due to their functional diversity. Particle-based variational inference (ParVI) methods enhance diversity by formalizing a repulsion term based on a network similarity kernel. However, weight-space repulsion is inefficient due to over-parameterization, while direct function-space repulsion has been found to produce little improvement over DEs. To sidestep these difficulties, we propose First-order Repulsive Deep Ensemble (FoRDE), an ensemble learning method based on ParVI, which performs repulsion in the space of first-order input gradients. As input gradients uniquely characterize a function up to translation and are much smaller in dimension than the weights, this method guarantees that ensemble members are functionally different. Intuitively, diversifying the input gradients encourages each network to learn different features, which is expected to improve the robustness of an ensemble. Experiments on image classification datasets and transfer learning tasks show that FoRDE significantly outperforms the gold-standard DEs and other ensemble methods in accuracy and calibration under covariate shift due to input perturbations.	Trung Trinh, Markus Heinonen, Luigi Acerbi, Samuel Kaski	2023-06-05T11:00:11Z	http://arxiv.org/abs/2306.02775v3	# Input gradient diversity for neural network ensembles ###### Abstract Deep Ensembles (DEs) demonstrate improved accuracy, calibration and robustness to perturbations over single neural networks partly due to their functional diversity. Particle-based variational inference (ParVI) methods enhance diversity by formalizing a repulsion term based on a network similarity kernel. However, weight-space repulsion is inefficient due to over-parameterization, while direct function-space repulsion has been found to produce little improvement over DEs. To sidestep these difficulties, we propose First-order Repulsive Deep Ensemble (FoRDE), an ensemble learning method based on ParVI, which performs repulsion in the space of first-order input gradients. As input gradients uniquely characterize a function up to translation and are much smaller in dimension than the weights, this method guarantees that ensemble members are functionally different. Intuitively, diversifying the input gradients encourages each network to learn different features, which is expected to improve the robustness of an ensemble. Experiments on image classification datasets show that FoRDE significantly outperforms the gold-standard DEs and other ensemble methods in accuracy and calibration under covariate shift due to input perturbations. ## 1 Introduction Ensemble methods, which combine predictions from multiple models, are a well-known strategy in machine learning to boost performance (Dietterich, 2000). Ensembles of neural networks have been shown to achieve great results in terms of predictive performance (Lakshminarayanan et al., 2017), uncertainty estimation (Ovadia et al., 2019), robustness to adversarial attacks (Pang et al., 2019) and corruptions (Hendrycks and Dietterich, 2019). A common strategy to create a neural network ensemble is called Deep Ensembles (DEs) (Lakshminarayanan et al., 2017), which generate different networks by repeating training from different weight initializations. While DEs are simple to implement and have promising performance, they are susceptible to redundancy in model averaging due to their reliance on training randomness to induce network diversity (Rame and Cord, 2021; D'Angelo and Fortuin, 2021; Yashima et al., 2022). Particle-based variational inference methods (ParVIs) (Liu and Wang, 2016; Chen et al., 2018; Liu et al., 2019; Shen et al., 2021) have recently emerged as a direction to improve diversity in neural ensembles from the Bayesian perspective (Wang et al., 2019; D'Angelo and Fortuin, 2021). Notably, the ParVI update rule adds a kernelized repulsion term between the ensemble networks for explicit control of the network diversity. Typically repulsion is done in the weight space to capture different regions in the weight posterior. However, due to the over-parameterization of neural networks, weight-space repulsion is inefficient and does not resolve the redundancy problem of DEs. An alternative approach is to define the repulsion in function space (Wang et al., 2019; D'Angelo and Fortuin, 2021), which requires a kernel between functions. Previous works avoided this issue by comparing functions only on training inputs, which leads to underfitting (D'Angelo et al., 2021). Neither weight- nor function-space repulsion has led to significant improvements over vanilla DEs. The gravitation towards weight and function repulsion stems from the fact that weight and function views have been the two main ways to characterize a parametric model. From a functional perspective, a model can also be uniquely represented, up to translation, using its first-order derivatives, i.e., input _gradients_$\nabla_{\mathbf{x}}f$. Promoting diversity in this third view of input gradients has notable advantages: 1. each ensemble member is guaranteed to correspond to a different function; 2. input gradients often have dramatically smaller dimensions than network weights and thus are more amenable to kernel comparison; 3. the training procedure can directly diversify the input gradients of the networks with respect to the training samples without resulting in underfitting ensembles (see Fig. 1), unlike function-space repulsion on the training samples; 4. as input gradients quantify the salience of input features, each ensemble is encouraged to learn different features, which can improve robustness. In this work, we propose a ParVI neural network ensemble that promotes diversity in their input gradients, denoted First-order Repulsive deep ensemble (FoRDE). In addition to the above advantages, we devise a data-dependent kernel that allows FoRDE to significantly outperform other ensemble methods under input corruptions on image classification tasks. ## 2 Background Bayesian neural networksIn a Bayesian neural network (BNN), we treat the model's weights $\theta$ as random variables with a prior $p(\theta)$. Given a dataset $\mathcal{D}=\{(\mathbf{x}_{n},y_{n})\}_{n=1}^{N}$ and a likelihood function $p(y\|\mathbf{x},\theta)$ per data point, we infer the posterior over weights $p(\theta\|\mathcal{D})$ using Bayes' rule \[p(\theta\|\mathcal{D})=\frac{p(\mathcal{D}\|\theta)p(\theta)}{\int_{\theta}p( \mathcal{D}\|\theta)p(\theta)\mathrm{d}\theta}=\frac{p(\theta)\prod_{n=1}^{N} p(y_{n}\|\mathbf{x}_{n},\theta)}{\int_{\theta}p(\theta)\prod_{n=1}^{N}p(y_{n}\| \mathbf{x}_{n},\theta)\mathrm{d}\theta}, \tag{1}\] where the likelihood is assumed to factorize over data. To make a prediction on a test sample $\mathbf{x}^{}$, we integrate over the inferred posterior in Eq. (1), a practice called _Bayesian model averaging_ (BMA): \[p(y\|\mathbf{x}^{},\mathcal{D})=\int_{\theta}p(y\|\mathbf{x}^{},\theta)p( \theta\|\mathcal{D})\mathrm{d}\theta=\mathbb{E}_{p(\theta\|\mathcal{D})}\big{[} p(y\|\mathbf{x}^{},\theta)\big{]}. \tag{2}\] However, computing the integral in Eq. (2) is intractable for BNNs. Various approximate inference methods have been developed for BNNs, including variational inference (VI) (Graves, 2011; Blundell et al., 2015), Markov chain Monte Carlo (MCMC) (Neal, 2012; Welling and Teh, 2011; Zhang et al., 2020) and more recently ParVI (Liu and Wang, 2016; Wang et al., 2019; D'Angelo and Fortuin, 2021). Figure 1: An illustration of input gradient repulsion in 1D regression with 3 neural networks.Left: At some point during training, the models fit well to the training samples yet exhibit low functional diversity. Middle: As training proceeds, at each data point, the repulsion term gradually pushes the input gradients (represented by the arrows) away from each other on a unit sphere. Right: As a result, at the end of training, the ensemble has gained functional diversity. Deep ensemblesAs opposed to BNNs, which attempt to learn the posterior distribution, DEs (Lakshminarayanan et al., 2017) consist of multiple maximum-a-posteriori (MAP) estimates trained independently from random initializations. They can capture diverse functions that explain the data well, as independent training runs under different random conditions will likely converge to different modes in the posterior landscape. DEs have been shown to be better than BNNs in both accuracy and uncertainty estimation (Ovadia et al., 2019; Ashukha et al., 2020; Gustafsson et al., 2020). Particle-based variational inference for neural network ensemblesParVI methods (Liu and Wang, 2016; Chen et al., 2018; Liu et al., 2019; Shen et al., 2021) have been studied recently to formalize neural network ensembles. They approximate the target posterior using a set of samples, or particles, by deterministically transporting these particles from their initial positions to the target distribution (Liu and Wang, 2016). ParVI methods are expected to be more efficient than MCMC as they take into account the interactions between particles in their update rules (Liu et al., 2019). These repulsive interactions are driven by a kernel which measures the pairwise similarities between particles, ie. networks (Liu et al., 2019). The main approaches for the kernel are comparing the networks in weight space $\theta$ or in function space $f(\cdot;\theta)$. Weight-space repulsion is ineffective due to difficulties in comparing extremely high-dimensional weight vectors and the existence of weight symmetries in the weight-space posterior (Fort et al., 2019; Enetezari et al., 2022). Unsurprisingly, previous studies showed that ParVI with weight-space repulsion did not improve performance over plain DEs (D'Angelo and Fortuin, 2021; Yashima et al., 2022). Comparing neural networks via a function-space kernel is also challenging since functions are infinite-dimensional objects. Previous works resort to comparing functions only on a subset of the input space called the measurement set (Wang et al., 2019; D'Angelo and Fortuin, 2021). Using the training data as the measurement set leads to underfitting (D'Angelo and Fortuin, 2021; Yashima et al., 2022), likely because these inputs have known labels thus leaving no room for diverse predictions without impairing performance. ## 3 FoRDE: First-order Repulsive Deep Ensembles In this section, we present a framework to perform ParVI in the _input-gradient_ space. We start by introducing the update rule for particle optimization called Wasserstein gradient descent (WGD) in Section 3.1. We then describe how to apply WGD for input-gradient-space repulsion in Section 3.2. In Section 3.3, we discuss hyperparameter selection for the input-gradient kernel. Finally, in Section 3.4, we outline two practical considerations when training FoRDE. Throughout this section, we assume a set of $M$ weight particles $\{\theta_{i}\}_{i=1}^{M}$ corresponding to a set of $M$ neural networks $\{f_{i}:\mathbf{x}\mapsto f(\mathbf{x};\theta_{i})\}_{i=1}^{M}$. We focus on the supervised classification setting: given a labelled dataset $\mathcal{D}=\{(\mathbf{x}_{n},y_{n})\}_{n=1}^{N}$ with $\mathcal{C}$ classes and inputs $\mathbf{x}_{n}\in\mathbb{R}^{D}$, we approximate the posterior $\log p(\theta\|\mathcal{D})$ using the $M$ particles. The output $f(\mathbf{x};\theta)$ for input $\mathbf{x}$ is a _log-softmax vector_ of size $\mathcal{C}$ whose $y$-th entry $f(\mathbf{x};\theta)_{y}$ equals the log-probability of the $y$-th class. ### Wasserstein gradient descent Following D'Angelo and Fortuin (2021), we use a ParVI method called Wasserstein gradient descent (WGD) (Liu et al., 2019; Wang et al., 2022). Given a target posterior distribution $\pi=p(\cdot\|\mathcal{D})$ and a set of particles $\{\theta_{i}\}_{i=1}^{M}$ from distribution $\rho$, the goal of WGD is to find the particle distribution $\rho^{}$, \[\rho^{}=\operatorname{argmin}_{\rho\in\mathcal{P}_{2}(\Theta)}\mathrm{KL}_{ \pi}(\rho), \tag{3}\] where $\mathrm{KL}_{\pi}(\rho)$ is a shorthand for the standard KL-divergence \[\mathrm{KL}_{\pi}(\rho)=\mathbb{E}_{\rho(\theta)}\big{[}\log\rho(\theta)- \log\pi(\theta)\big{]}, \tag{4}\] and $\mathcal{P}_{2}(\Theta)$ is the Wasserstein space equipped with the Wasserstein distance $W_{2}$(Ambrosio et al., 2005; Villani, 2009). WGD solves the problem in Eq. (3) using a Wasserstein gradient flow $(\rho_{t})_{t}$, which is roughly the family of steepest descending curves of $\mathrm{KL}_{\pi}(\cdot)$ in $\mathcal{P}_{2}(\Theta)$. The tangent vector of this gradient flow at time $t$ is \[v_{t}(\theta)=\nabla\log\pi(\theta)-\nabla\log\rho_{t}(\theta) \tag{5}\] whenever $\rho_{t}$ is absolutely continuous (Villani, 2009; Liu et al., 2019). Intuitively, $v_{t}(\theta)$ points to the direction where the probability mass at $\theta$ of $\rho_{t}$ should be transported in order to bring $\rho_{t}$ closer to $\pi$. Since Eq. (5) requires the analytical form of $\rho_{t}$ which we do not have access to, we use kernel density estimation (KDE) to obtain a tractable approximation $\hat{\rho}_{t}$ induced by particles $\{\theta_{i}^{(t)}\}_{i=1}^{M}$, \[\hat{\rho}_{t}(\theta)\propto\sum_{i=1}^{M}k\big{(}\theta,\theta_{i}^{(t)} \big{)}, \tag{6}\] where $k$ is a positive semi-definite (PSD) kernel. Then, the approximation gradient is \[\nabla\log\hat{\rho}_{t}(\theta)=\frac{\sum_{i=1}^{M}\nabla_{\theta}k\big{(} \theta,\theta_{i}^{(t)}\big{)}}{\sum_{i=1}^{M}k\big{(}\theta,\theta_{i}^{(t)} \big{)}}. \tag{7}\] Using Eq. (7) in Eq. (5), we obtain a practical update rule for each particle $\theta^{(t)}$ of $\hat{\rho}_{t}$: \[\theta^{(t+1)}=\theta^{(t)}+\eta_{t}\Bigg{(}\underbrace{\nabla_{\theta^{(t)} }\log\pi\big{(}\theta^{(t)}\big{)}}_{\text{driving force}}-\underbrace{ \frac{\sum_{i=1}^{M}\nabla_{\theta^{(t)}}k\big{(}\theta^{(t)},\theta_{i}^{(t) }\big{)}}{\sum_{i=1}^{M}k\big{(}\theta^{(t)},\theta_{i}^{(t)}\big{)}}}_{\text {repulsion force}}\Bigg{)}, \tag{8}\] where $\eta_{t}>0$ is the step size at optimization time $t$. Intuitively, we can interpret the first term in the particle gradient as the driving force directing the particles towards high density regions of the posterior, while the second term is the repulsion force pushing the particles away from each other. ### Defining the kernel for WGD in input gradient space We propose to use a kernel comparing the _input gradients_ of the particles \[k(\theta_{i},\theta_{j})\stackrel{{\text{def}}}{{=}}\mathbb{E}_ {(\mathbf{x},y)\sim p(\mathbf{x},y)}\Big{[}\kappa\big{(}\nabla_{\mathbf{x}}f( \mathbf{x};\theta_{i})_{y},\nabla_{\mathbf{x}}f(\mathbf{x};\theta_{j})_{y} \big{)}\Big{]}, \tag{9}\] where $\kappa$ is a _base kernel_ between gradients $\nabla_{\mathbf{x}}f(\mathbf{x};\theta)_{y}$ that are of same size as the inputs $\mathbf{x}$. In essence, we define $k$ as the expected similarity between the input gradients of two networks with respect to the data distribution $p(\mathbf{x},y)$. Interestingly, by using the kernel $k$, the KDE approximation $\hat{\rho}$ of the particle distribution not only depends on the particles themselves but also _depends on the data distribution_. We approximate the kernel $k$ using the training samples, with linear complexity: \[k(\theta_{i},\theta_{j})\approx k_{\mathcal{D}}(\theta_{i},\theta_{j})=\frac{ 1}{N}\sum_{n=1}^{N}\kappa\big{(}\nabla_{\mathbf{x}}f(\mathbf{x}_{n};\theta_{i })_{y_{n}},\nabla_{\mathbf{x}}f(\mathbf{x}_{n};\theta_{j})_{y_{n}}\big{)}. \tag{10}\] The kernel compares the gradients of only the true label $\nabla_{\mathbf{x}}f(\mathbf{x}_{n};\theta)_{y_{n}}$, as opposed to the entire Jacobian matrix $\nabla_{\mathbf{x}}f(\mathbf{x}_{n};\theta)$, as our motivation is to encourage each particle to learn different features that could explain the training sample $(\mathbf{x}_{n},y_{n})$ well. This approach also reduces computational complexity, since automatic differentiation libraries such as JAX (Bradbury et al., 2018) or Pytorch (Paszke et al., 2019) would require $\mathcal{C}$ passes, one per class, to calculate the full Jacobian. Choosing the base kernelWe choose the RBF kernel on the unit sphere as our base kernel $\kappa$: \[\kappa(\mathbf{s},\mathbf{s}^{\prime};\boldsymbol{\Sigma})=\exp\left(-\frac{ 1}{2}(\mathbf{s}-\mathbf{s}^{\prime})^{\top}\boldsymbol{\Sigma}^{-1}(\mathbf{ s}-\mathbf{s}^{\prime})\right),\qquad\mathbf{s}=\frac{\nabla_{\mathbf{x}}f( \mathbf{x};\theta)_{y}}{\|\|\nabla_{\mathbf{x}}f(\mathbf{x};\theta)_{y}\|\|_{2}} \in\mathbb{R}^{D} \tag{11}\] where $\mathbf{s},\mathbf{s}^{\prime}$ denote the two normalized gradients of two particles with respect to one input, and $\boldsymbol{\Sigma}\in\mathbb{R}^{D\times D}$ is a diagonal matrix containing squared lengthscales. We design $\kappa$ to be norm-agnostic since the norm of the true label gradient $\|\|\nabla_{\mathbf{x}}f(\mathbf{x}_{n};\theta)_{y_{n}}\|\|_{2}$ fluctuates during training and as training converges, the log-probability $f(\mathbf{x}_{n};\theta)_{y_{n}}=\log p(y_{n}\|\mathbf{x}_{n},\theta)$ will approach $\log 1$, leading to the norm $\|\|\nabla_{\mathbf{x}}f(\mathbf{x}_{n};\theta)_{y_{n}}\|\|_{2}$ approaching $0$ due to the saturation of the log-softmax activation. Furthermore, comparing the normed input gradients between ensemble members teaches them to learn complementary explanatory patterns from the training samples, which could improve robustness of the ensemble. The RBF kernel is an apt kernel to compare unit vectors (Jayasumana et al., 2014), and we can control the variances of the gradients along input dimensions via the square lengthscales $\boldsymbol{\Sigma}$. ### Selecting the lengthscales for the base kernel In this section, we present a method to select the lengthscales for the base kernel. These lengthscales are important for the performance of FoRDE, since they control how much repulsion force is applied in each dimension of the input-gradient space. The dimension-wise repulsion is (Liu and Wang, 2016) \[\frac{\partial}{\partial s_{d}}\kappa(\mathbf{s},\mathbf{s}^{\prime};\mathbf{ \Sigma})=-\frac{s_{d}-s^{\prime}_{d}}{\mathbf{\Sigma}_{dd}}\kappa(\mathbf{s}, \mathbf{s}^{\prime};\mathbf{\Sigma}), \tag{12}\] where we can see that along the $d$-th dimension the inverse square lengthscale $\mathbf{\Sigma}_{dd}$ controls the strength of the repulsion $\nabla_{s_{d}}\kappa(\mathbf{s},\mathbf{s}^{\prime};\mathbf{\Sigma})$: a smaller lengthscale corresponds to a stronger force 1. Additionally, since the repulsion is restricted to the unit sphere in the input-gradient space, maximizing the distance of two particles in one dimension will minimize the distance in other dimensions. As a result, the repulsion motivates the ensemble members to depend more on dimensions with stronger repulsion in the input gradient space for their predictions, while focusing less on dimensions with weaker repulsion. From this observation, one should apply strong repulsion (small lengthscales), in the dimensions of the data manifold with high variances and vice versa. Footnote 1: Here we assume that the lengthscales are set appropriately so that the kernel $\kappa$ does not vanish, which is true since we use the median heuristic during training (Section 3.4). To realize the intuition above in FoRDE, we first apply Principal Component Analysis (PCA) to discover important features in the training data. In PCA, we calculate the eigendecomposition \[\mathbf{C}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{T}\quad\in\mathbb{R}^{D\times D} \tag{13}\] of the covariance matrix $\mathbf{C}=\mathbf{X}^{T}\mathbf{X}/(N-1)$ of the centered training samples $\mathbf{X}\in\mathbb{R}^{N\times D}$ to get the eigenvectors and eigenvalues $\{\mathbf{u}_{d},\lambda_{d}\}_{k=1}^{D}$. The $d$-th eigenvalue $\lambda_{d}$ is the variance of the data along eigenvector $\mathbf{u}_{d}$, offering a natural choice of inverse eigenvalues $\lambda_{d}^{-1}$ as the squared lengthscales $\mathbf{\Sigma}_{dd}$ of the principal components. Let $\tilde{\mathbf{x}}=\mathbf{U}^{T}\mathbf{x}$ denote the representation of the input $\mathbf{x}$ in eigenbasis $\mathbf{U}=[\mathbf{u}_{1}\,\mathbf{u}_{2}\,\dots\,\mathbf{u}_{D}]$. We compute the gradient kernel in PCA basis $\mathbf{U}$ and set $\mathbf{\Sigma}=\mathbf{\Lambda}^{-1}$: \[\kappa(\mathbf{s},\mathbf{s}^{\prime})\stackrel{{\mathrm{def}}}{ {=}}\exp\left(-\frac{1}{2}(\mathbf{U}^{\top}\mathbf{s}-\mathbf{U}^{\top} \mathbf{s}^{\prime})^{\top}\,\mathbf{\Lambda}\left(\mathbf{U}^{\top}\mathbf{ s}-\mathbf{U}^{\top}\mathbf{s}^{\prime}\right)\right), \tag{14}\] where $\mathbf{\Lambda}$ is a diagonal matrix containing the eigenvalues. While setting the square inverse lengthscales equal to the eigenvalues seems problematic at first glance, since large eigenvalues will push the kernel $\kappa$ towards 0, this problem is avoided in practice since we also employ the median heuristic, which introduces in the kernel a global bandwidth scaling term that adapts to the current pairwise distance between particles, as discussed below in Section 3.4. Connection to the EmpCov priorRecently, Izmailov et al. (2021) propose the EmpCov prior for each weight column $\mathbf{w}$ of the first layer of a BNN: \[\mathbf{w}\sim\mathcal{N}(0,\alpha\mathbf{C}+\epsilon\mathbf{I}),\qquad \mathbf{w}\in\mathbb{R}^{D} \tag{15}\] where $\alpha>0$ determines the prior scale and $\epsilon>0$ is a small constant ensuring that the covariance matrix is positive definite. This prior encourages these weights to vary more along high variance dimensions in the data manifold and less along the low variance dimensions. Samples from this prior will have large input gradients along the dimensions in the data manifold with high variances, and small input gradients along the low variance dimensions. In this sense, the EmpCov prior is similar to our approach. The difference is that while Izmailov et al. (2021) incorporates knowledge of the data manifold into the prior, we embed this knowledge into our approximate posterior via the kernel. ### Practical considerations In this section, we detail two important considerations to make FoRDEs work in practice. We include the full training algorithm in the Appendix. Mini-batchingTo make FoRDE amenable to mini-batch gradient optimization, we adapt the kernel in Eq. (10) to a mini-batch of samples $\mathcal{B}=\{(x_{b},y_{b})\}_{b=1}^{B}$: \[k_{\mathcal{B}}(\theta_{i},\theta_{j})=\frac{1}{B}\sum_{b=1}^{B}\kappa\Big{(} \nabla_{\mathbf{x}}f(\mathbf{x}_{b};\theta_{i})_{y_{b}},\nabla_{\mathbf{x}}f( \mathbf{x}_{b};\theta_{j})_{y_{b}}\Big{)} \tag{16}\] However, using this kernel in the update rule in Eq. (8) leads to biased stochastic gradients of the repulsion term because the average over batch samples in Eq. (16) is inside the logarithm. In practice, we found no performance degradation or convergence issues in our experiments. Median heuristicsSince we perform particle optimization with an RBF kernel $\kappa$, following earlier works (Liu and Wang, 2016; Liu et al., 2019), we adopt the median heuristic (Scholkopf et al., 2002). Besides the lengthscales, we introduce a global bandwidth $h$ in our base kernel in Eq. (11): \[\kappa(\mathbf{s}_{i},\mathbf{s}_{j};\boldsymbol{\Sigma})=\exp\left(-\frac{1} {2h}(\mathbf{s}_{i}-\mathbf{s}_{j})^{\top}\boldsymbol{\Sigma}^{-1}(\mathbf{s} _{i}-\mathbf{s}_{j})\right),\qquad\mathbf{s}_{i}=\frac{\nabla_{\mathbf{x}}f( \mathbf{x};\theta_{i})_{y}}{\|\|\nabla_{\mathbf{x}}f(\mathbf{x};\theta_{i})_{y} \|\|_{2}}\in\mathbb{R}^{D} \tag{17}\] During training, the bandwidth $h$ is adaptively set to $\mathrm{med}^{2}/(2\log M)$, where $\mathrm{med}^{2}$ is the median of the pairwise distance $(\mathbf{s}_{i}-\mathbf{s}_{j})^{\top}\boldsymbol{\Sigma}^{-1}(\mathbf{s}_{i} -\mathbf{s}_{j})$ between the weight samples $\{\theta_{i}\}_{i=1}^{M}$. ### Computational complexity Compared to DEs, FoRDEs take roughly three times longer to train. In addition to a forward-backward pass to calculate the log-likelihood, we need an additional forward-backward pass to calculate the input gradients, and another backward pass to calculate the gradients of the input gradient repulsion with respect to the weights. This analysis is confirmed in practice: in resnet18/cifar-100 experiments of Section 4.1 with an ensemble size of 10, a DE took around $\sim$31 seconds per epoch on an Nvidia A100 GPU, while FoRDE took around $\sim$101 seconds per epoch. ## 4 Experiments In this section, we present experimental results of FoRDE on image classification tasks. We use cifar(Krizhevsky, 2009) and tinyimagenet(Le and Yang, 2015) datasets, which both have corrupted versions of the test set (Hendrycks and Dietterich, 2019). We use resnet18 (He et al., 2016) and preactresnet18 (He et al., 2016) architectures. We report mean and standard deviations over multiple runs with varying seeds. ### Comparisons to other repulsive DE methods and BNNs We report performance of FoRDE against other methods on cifar-10/100 in Tables 1-2 and tinyimagenet in Table 3. Besides the PCA kernel introduced in Section 3.3, we also experiment with FoRDE where we set the lengthscales $\boldsymbol{\Sigma}=\mathbf{I}$. For the repulsive DE (RDE) baselines, we choose weight-space RDE (D'Angelo and Fortuin, 2021), function-space RDE (D'Angelo and Fortuin, 2021) and feature-space RDE (Yashima et al., 2022). We also include the results of node-BNNs (Trinh et al., \begin{table} \begin{tabular}{l c c c c} \hline \hline Method & NLL $\downarrow$ & Accuracy (\% ) & ECE $\downarrow$ & CA / CNLL / cECE \\ \hline node-BNNs & $0.74\pm 0.01$ & $79.7\pm 0.3$ & $0.054\pm 0.002$ & 54.8 / 1.96 / 0.05 \\ SWAG & $0.73\pm 0.01$ & $79.4\pm 0.1$ & $\mathbf{0.038\pm 0.001}$ & 53.0 / 2.03 / 0.05 \\ \hline Deep ensembles & $\mathbf{0.70\pm 0.00}$ & $81.8\pm 0.2$ & $0.041\pm 0.003$ & 54.3 / 1.99 / 0.05 \\ weight-RDE & $\mathbf{0.70\pm 0.01}$ & $81.7\pm 0.3$ & $0.043\pm 0.004$ & 54.2 / 2.01 / 0.06 \\ function-RDE & $0.76\pm 0.02$ & $80.1\pm 0.4$ & $0.042\pm 0.005$ & 51.9 / 2.08 / 0.07 \\ feature-RDE & $0.75\pm 0.04$ & $\mathbf{82.1\pm 0.3}$ & $0.072\pm 0.023$ & 54.8 / 2.02 / 0.06 \\ \hline FoRDE-PCA (ours) & $0.71\pm 0.00$ & $81.4\pm 0.2$ & $0.039\pm 0.002$ & 56.1 / 1.90 / 0.05* \\ FoRDE-Identity (ours) & $\mathbf{0.70\pm 0.00}$ & $\mathbf{82.1\pm 0.2}$ & $0.043\pm 0.001$ & 54.1 / 2.02 / 0.05 \\ \hline \hline \end{tabular} \end{table} Table 1: Results of resnet18 on cifar-100 evaluated over 5 seeds. Each ensemble has 10 members. ca, CNLL and cECE are accuracy, NLL, and ECE on cifar-100-c. FoRDE-PCA achieves the best performance under corruptions while FoRDE-Identity outperforms the baselines on clean data. 2022) and SWAG (Maddox et al., 2019) as reported in Trinh et al. (2022). We use a default ensemble size of 10. We use standard performance metrics of expected calibration error (ECE) (Naeini et al., 2015), negative log-likelihood (NLL) and predictive accuracy. For evaluations on input perturbations, we use the corrupted test images provided by Hendrycks and Gimpel (2017). We use resnet18 for cifar-10/100 and preactresnet18 for tinyimagenet. We report the hyperparameters and training procedure in the Appendix. Tables 1-3 show that FoRDE-PCA outperforms other methods under input corruptions across all metrics, while maintaining competitive performance on clean data. For instance, FoRDE-PCA exhibits a $+1.3\%$ gain on cifar-100-c and $+2.4\%$ gain on cifar-10-c in accuracy compared to the second-best results. As stated in Section 3.3, the PCA kernel encourages FoRDE to rely more on features with high variances in the data manifold to make predictions, while being less dependent on features with low variances. This effect has been shown in Izmailov et al. (2021) to boost model robustness against perturbations, which explains why FoRDE with the PCA kernel performs better than the baselines on input corruptions. On the other hand, Tables 1 and 2 show that FoRDE with identity lengthscales outperforms the baselines in terms of NLL on cifar-10 and has the best accuracy on cifar-100. However, FoRDE-Identity is slightly less robust than DE against corruptions. We suspect that with the identity lengthscales, FoRDE also learns to rely on low-variance features to make predictions, which is harmful to performance under corruptions (Izmailov et al., 2021). The results of FoRDE-Identity suggest that an ensemble still needs to learn low-variance features in the data to generalize well on clean images. Hence, we believe that the optimal lengthscales for good performance on both clean and corrupted data lie somewhere between identity lengthscales and using the inverse square root eigenvalues as lengthscales. et al. (2021) for all ensemble methods except FoRDE. By comparing Table 4 to Table 2, we see that the EmCov prior slightly improves robustness of the baseline ensemble methods against corruptions, while also leads to a small reduction in performance on clean data. These small improvements in robustness are not surprising, since Izmailov et al. (2021) stated that for ensemble methods consisting of approximate MAP solutions, the isotropic Gaussian prior already minimizes the influences of low variance data features on the ensemble's predictions. We argue that besides minimizing the influences of low variance features on predictions via the PCA kernel, FoRDE also encourages its members to learn complementary patterns that can explain the data well, and these two effects act in synergy to improve the robustness of the resulting ensemble. Thus, Table 4 shows that FoRDE is still more robust against corruptions than the baseline methods with the EmpCov prior. ### Performance under different ensemble sizes We report the NLL of FoRDE and DE under different ensemble sizes on both cifar-100 and cifar-100-c in Fig. 2. We use the widerensnet16x4 (Zagoruyko and Komodakis, 2016) architecture for this experiment. These figures show that both methods enjoy significant improvements in performance as the ensemble size increases. While Fig. 1(a) shows that FoRDE underperforms DE on clean images, Fig. 1(b) shows that FoRDE significantly outperforms DE on corrupted images, such that a FoRDE with 10 members has the same corruption robustness of a DE with 30 members. ## 5 Related works ParVI for neural networksD'Angelo and Fortuin (2021) studied repulsive DE in the weight and function space. They showed that by formulating the repulsion term according to ParVI, the ensemble will converge to the target posterior in the infinite particle limit. However, due to the shortcomings of the weight-space and function-space repulsion, their method showed little performance improvement over plain DEs. Subsequent work by Yashima et al. (2022) introduced feature-space repulsion, which is the output space of the penultimate layer. They argue that feature-space repulsion motivates ensemble members to learn diverse patterns in the training data, and showed performance improvements over DEs. Here, we propose the input-gradient space as a good \begin{table} \begin{tabular}{l c c c c} \hline \hline Method & NLL $\downarrow$ & Accuracy (\%) $\uparrow$ & ECE $\downarrow$ & CA / CNLL / cECE \\ \hline Deep ensembles & $0.119\pm 0.001$ & $\mathbf{96.2\pm 0.1}$ & $0.006\pm 0.001$ & 78.7 / 0.76 / 0.08 \\ weight-RDE & $0.120\pm 0.001$ & $96.0\pm 0.1$ & $0.006\pm 0.001$ & 78.8 / 0.76 / 0.08 \\ function-RDE & $0.132\pm 0.001$ & $95.6\pm 0.3$ & $0.007\pm 0.001$ & 77.8 / 0.79 / 0.08 \\ feature-RDE & $\mathbf{0.118\pm 0.001}$ & $\mathbf{96.2\pm 0.1}$ & $\mathbf{0.005\pm 0.001}$ & 78.9 / 0.74 / 0.07 \\ \hline FoRDE-PCA (ours) & $0.125\pm 0.001$ & $96.1\pm 0.1$ & $0.006\pm 0.001$ & $\mathbf{80.5\text{ }/\mathbf{0.71}\text{ }/\mathbf{0.07}}$ \\ \hline \hline \end{tabular} \end{table} Table 4: Results of resnet18 on cifar-10 evaluated over 5 seeds. Each ensemble has 10 members. ca, CNLL and cECE are accuracy, NLL, and ECE on cifar-10-c. Here we use the EmpCov prior for all methods except FoRDE. FoRDE achieves the best performance under corruptions while maintaining competitive performance on clean data. Figure 2: FoRDE is competitive on in-distribution and outperforms DEs under domain shifts by corruption. Performance of widerensnet16x4 on cifar-100 over 5 seeds. alternative space in which to perform repulsion. We also note that an earlier work by Wang et al. (2019) also studied ParVI methods for function-space inference of BNNs. Gradient-based attribution methods for deep modelsA well-known application of input gradients is to build attribution (or saliency) maps, which assign importance to visually-interpretable input features for a specified output (Simonyan et al., 2014). In addition to the pioneering work of Simonyan et al. (2014) defining attribution as absolute input gradients, subsequent works include Layer-wise Relevance Propagation (Bach et al., 2015), Gradient*Input (Shrikumar et al., 2016), Integrated Gradients (Sundararajan et al., 2017) and deepLIFT (Shrikumar et al., 2017). Our method intuitively utilizes the attribution perspective of input gradients to encourage ensemble members to learn different patterns from training data. Improving corruption robustness of BNNsPrevious works have evaluated the predictive uncertainty of BNNs under covariate shift (Ovadia et al., 2019; Izmailov et al., 2021), with Izmailov et al. (2021) showing that standard BNNs with high-fidelity posteriors perform worse than MAP solutions on under corruptions. Izmailov et al. (2021) attributed this phenomenon to the lack of posterior contraction in the null space of the data manifold and proposed the EmpCov prior as a remedy. As discussed in Section 3.3, the PCA kernel works in the same manner as the EmpCov prior and thus significantly improves robustness of FoRDE against corruptions. Trinh et al. (2022) studied the robustness of node-BNNs, an efficient alternative to weight-based BNNs, and showed that by increasing the entropy of the posterior, node-BNNs become more robust against corruptions. Here, we incorporate the knowledge of the data manifold into the KDE approximation of the particle distribution via the PCA kernel, thus resulting in a manifold-structured approximate posterior that performs well under corruptions. Wang and Aitchison (2023) allow BNNs to adapt to the distribution shift at test time by using test data statistics, and this method can be applied to FoRDE to further boost its robustness against corruptions. ## 6 Discussion In this section, we outline two directions to further improve FoRDE. Choosing more optimal lengthscalesAs we have shown in our experiments in Section 4.1, choosing different lengthscale settings controls the predictive performance of FoRDE, as the lengthscales determine the strength of the repulsion force in each dimension in the input-gradient space. Here, we only experimented with two settings: the lengthscale setting based on eigenvalues presented in Section 3.3, and the identity lengthscales. We believe that there is an optimal lengthscale setting which allows FoRDE to perform well on both clean and corrupted data. Intuitively, the optimal lengthscales are still inversely proportional to the eigenvalues for corruption robustness, yet allowing some repulsion in the low-variance dimension in the data manifold for good in-distribution performance. Choosing different orthonormal basisIn Section 3.3, we propose to perform repulsion in the eigenbasis $\mathbf{U}$. Here, the role of $\mathbf{U}$ is to rotate the repulsion forces so that they are aligned with the data manifold. With this observation, we can use different orthonormal basis to control the directions of the repulsion forces depending on the applications. For instance, we can align the repulsion forces with the orthonormal basis of the discrete cosine transformation (Ahmed et al., 1974) for image data. ## 7 Conclusion In this work, we propose FoRDE, an ensemble learning method that promotes diversity in the input-gradient space among ensemble members. We detail the update rule and devise a data-dependent kernel suitable for input-gradient repulsion. Experiments on image classification tasks show that FoRDE outperforms other ensemble methods on corrupted images. Future directions include more theoretical understanding of our method and its behavior under different settings as discussed in Section 6.
2305.02225	Data Privacy with Homomorphic Encryption in Neural Networks Training and Inference	The use of Neural Networks (NNs) for sensitive data processing is becoming increasingly popular, raising concerns about data privacy and security. Homomorphic Encryption (HE) has the potential to be used as a solution to preserve data privacy in NN. This study provides a comprehensive analysis on the use of HE for NN training and classification, focusing on the techniques and strategies used to enhance data privacy and security. The current state-of-the-art in HE for NNs is analysed, and the challenges and limitations that need to be addressed to make it a reliable and efficient approach for privacy preservation are identified. Also, the different categories of HE schemes and their suitability for NNs are discussed, as well as the techniques used to optimize the accuracy and efficiency of encrypted models. The review reveals that HE has the potential to provide strong data privacy guarantees for NNs, but several challenges need to be addressed, such as limited support for advanced NN operations, scalability issues, and performance trade-offs.	Ivone Amorim, Eva Maia, Pedro Barbosa, Isabel Praça	2023-05-03T16:05:26Z	http://arxiv.org/abs/2305.02225v1	# Data Privacy with Homomorphic Encryption in Neural Networks Training and Inference+ ###### Abstract The use of Neural Networks (NNs) for sensitive data processing is becoming increasingly popular, raising concerns about data privacy and security. Homomorphic Encryption (HE) has the potential to be used as a solution to preserve data privacy in NN. This study provides a comprehensive analysis on the use of HE for NN training and classification, focusing on the techniques and strategies used to enhance data privacy and security. The current state-of-the-art in HE for NNs is analysed, and the challenges and limitations that need to be addressed to make it a reliable and efficient approach for privacy preservation are identified. Also, the different categories of HE schemes and their suitability for NNs are discussed, as well as the techniques used to optimize the accuracy and efficiency of encrypted models. The review reveals that HE has the potential to provide strong data privacy guarantees for NNs, but several challenges need to be addressed, such as limited support for advanced NN operations, scalability issues, and performance trade-offs. Keywords:Homomorphic Encryption Neural Networks Data Privacy Privacy Preserving Machine Learning Data Sharing Cybersecurity ## 1 Introduction The emergence of digital technology and the ever-increasing significance of data in the economy have made it crucial to protect personal and sensitive information. As the amount of data being collected and stored grows, so does the risk of privacy breaches and data misuse. Furthermore, the use of intelligent techniques like Neural Networks (NNs) to extract insights from this data presents another potential vulnerability. Considering this, many privacy-preserving techniques have been proposed, including differential privacy, secure multiparty computation, and homomorphic encryption (HE). Among these, HE has attracted significant attention due to its ability to enable computations on encrypted data without requiring decryption, which can enhance data privacy while still allowing for data analysis. In recent years, HE has been increasingly applied to NNs. By training NNs on encrypted data using HE, sensitive information can be kept private, as the NN can operate on the encrypted data without needing to decrypt it. This can be particularly useful in scenarios where data privacy is a major concern, such as healthcare or finance. However, there are still challenges and limitations to the use of HE in NNs for data privacy, including computational overhead and potential security vulnerabilities. Some scientific works have already proposed the use of HE to preserve data privacy in NNs. One of the most relevant works in this area was introduced by Dowlin et al. [3], who modified a trained NN to operate on encrypted data, creating what they called a CryptoNet. However, this work assumes that the model is trained based on plain data and then this model is adapted for classifying encrypted instances. Other works have emerged to address the limitations of this method and propose new ones [14], but few have explored the use of HE for NN training and classification. Furthermore, the current literature in this area is almost exclusively addressed by practitioners looking for suitable implementations. To the best of our knowledge, there are only two publications that provide an overview of the existing approaches to preserve data privacy in NNs using HE: Poulido-Gaytan et al. [14] and Podschwadt et al. [13]. The former focuses mainly on works that use HE solely in the classification phase, which is the most prevalent approach in the literature. In fact, only one reference in their work addresses the problem of using HE in NN training and inference. On the other hand, Podschwadt's work is centred on deep learning architectures for privacy-preserving machine learning with Fully HE, without including other types of HE schemes like Partial or Somewhat HE. Therefore, in this work, we aim to provide a comprehensive analysis of the use of HE for NN training and classification. We will review and analyse the current state-of-the-art in this field and identify the limitations of using homomorphically encrypted data in NNs. We will also analyse existing approaches aimed at addressing these limitations and discuss future research directions in this topic. ## 2 Background Neural Networks (NNs) are weighted directed graphs that are inspired by the way the human brain works [9]. The nodes of these graphs are usually called neurons. These neurons are grouped in layers and are visualized as being stacked. Each of the neurons computes a function over the values of the layer beneath it. This function is usually named Activation Function (AF) and is applied to the output of each neuron in a NN, to determine whether the neuron should be activated (output a signal) or not, based on the input it receives. Common acti vation functions include the sigmoid function, ReLU (Rectified Linear Unit), and tanh (hyperbolic tangent) function. The loss function is then used to measure NN performance, by calculating the difference between the predicted output and the actual output. The goal of training a NN is to minimize the loss function, typically using an optimization algorithm such as stochastic gradient descent. Common loss functions include mean squared error, cross-entropy, and binary cross-entropy. Deep Learning (DL) [1] is a subfield of NNs that utilizes multiple hidden layers to extract hierarchical features that are useful for tasks such as pattern recognition and classification. Two types of supervised learning approaches for DL are Deep Neural Networks (DNNs) and Convolutional Neural Networks (CNNs). DNNs can be applied to various tasks, including speech recognition, natural language processing, and image classification. They consist of multiple layers of artificial neurons that adjust the strengths of their connections to learn complex patterns and relationships in data. CNNs are particularly suited for image processing tasks. They use a combination of convolutional layers, pooling layers, and fully connected layers to extract features from images and classify them into different categories. Both CNNs and DNNs have been shown to be highly effective for a variety of tasks, and continue to be an active area of research in the field of machine learning. Homomorphic Encryption (HE) is a type of encryption scheme that allows computations to be performed on encrypted data without decrypting it first. HE can be categorized into three categories: Partially Homomorphic Encryption (PHE), Somewhat Homomorphic Encryption (SWHE) and Fully Homomorphic Encryption (FHE). In PHE schemes, the homomorphic property is satisfied by only one operation, such as addition or multiplication, an unlimited number of times. SWHE supports more than an operation but a limited number of times Finally, FHE schemes allows a set of operations (usually addition and multiplication) an unlimited amount of times. The first FHE scheme was introduced by Gentry in 2009 [6], and it represented a significant breakthrough in the field, laying the foundation for future research. Gentry also introduced the concept of bootstrapping in FHE, which reduces the noise generated during operations and ensures a correct decryption output. However, this bootstrapping process is computationally expensive. There are, however, several research branches on the topic of HE which are usually defined by the mathematical problems in which the security of the proposed models rely on. For instance, the FHE schemes BGV, BFV, and TFHE are based on the hardness of the Learning With Error over Rings (Ring-LWE) problem [10]. Both schemes BGV and BFV operate on integers, while TFHE operates in binary data. Additionally, the CKKS scheme is based on a variant of the Ring-LWE problem called the approximate number field sieve, and it is meant for floating point arithmetic [4]. Finally, the SWHE scheme YASHE is founded on the Short Integer Solution problem, and it also operates on integers [2]. Several popular libraries provide implementations of the HE schemes mentioned above. For instance, the SEAL library3 provides implementations of the BGV, BFV, and CKKS schemes. Earlier versions of SEAL also included an implementation of the YASHE scheme. Another library, HElib4, implements the BGV and CKKS schemes. Finally, the TFHE library5 offers an implementation of the TFHE scheme. Footnote 4: [https://github.com/homenc/HElib](https://github.com/homenc/HElib) Footnote 5: [https://www.tfhe.com/](https://www.tfhe.com/) ## 3 Methodology The aim of this work is to present a comprehensive and well-structured overview of the current research literature concerning the use of HE in NN for preserving data privacy. For this we formulated the following research question: "How has HE been used to preserve data privacy in NN training and inference?". To address this broader question, four narrower sub-questions were defined: (RQ1):: What types of NNs are most used in studies about the use of HE to preserve data privacy in NN? (RQ2):: What are the most commonly used HE schemes for preserving data privacy in NN? (RQ3):: What are the major limitations identified in the literature regarding the use of HE to preserve data privacy in NNs? (RQ4):: What are the most common approaches used to address the limitations of HE in NN, and what trade-offs need to be considered? To achieve a transparent, replicable and complete answer to these questions, the guidelines of PRISMA methodology [15] were followed. The search terms "Neural Network(s)", "Homomorphic" and "Privacy" were defined and used in several reputable bibliographic databases. Since HE is commonly used in conjunction with other privacy-preserving techniques, we also defined a set of exclusion terms that must be omitted from the results: "Federated", "Mining", "Blockchain", "Mobile", "Multi-Party", "Edge", "Multiparty", "Distribut", "Communication", where '' denotes any sequence of characters. The selection criteria used in this study were clear and concise. To be included, publications had to be written in English, published in peer-reviewed journals or conference proceedings, and must discuss or evaluate the use of HE in NN training and inference. The search was then executed in four different bibliographic databases: Web of Science6, ACM7, Scopus8 and IEEE Explore9, and a total of 289 records were retrieved which resulted in 163 unique records. Following an initial screening of the titles and abstracts of the identified publications, a total of 126 papers were excluded. The full texts of the remaining 37 papers were then evaluated for eligibility, and only 6 were considered suitable for this study. The process described is illustrated in Figure 1. Footnote 6: [https://webofscience.com/](https://webofscience.com/) Footnote 7: [https://dl.acm.org/](https://dl.acm.org/) Footnote 8: [https://www.scopus.com/](https://www.scopus.com/) Footnote 9: [https://ieeexplore.ieee.org](https://ieeexplore.ieee.org) ## 4 Results and Discussion The first known work that provides a solution for training NN models using HE was published in 2017 by Hesamifard et al. [8]. The authors implemented NN over homomorphically encrypted data where the Sigmoid and ReLU activation functions are replaced by low-degree polynomial approximations to enable the use of NN with HE. The authors also proposed that when the ciphertext noise reaches a given threshold, it should be sent back to the client so that a fresh ciphertext can be returned, which they claim to be a more efficient alternative to bootstrapping. Experiments were conducted on three different datasets using five NN configurations, and the reported accuracies are above 99%, outperforming two state-of-the-art approaches. One of those approaches also used HE, but only for classification, assuming that the model was trained in plaintext. The other approach used secure multi-party computation (SMC) techniques to ensure privacy preservation in NN training and classification. The number of communications needed for training is also referred as an advantage over the state-of-the-art SMC approach, due to the significant reduction in the required number of interactions between the server and the client. The authors used the BGV scheme and its implementation in the HElib library. In the following year, Ghimes et al. [7] implemented a NN where all data was encrypted, and all operations were performed over encrypted data using the YASHE scheme provided by the SEAL library. They utilized the simplest activation function possible, the identity function, to reduce the cost of computing over encrypted data. Their primary goal was to enable institutions to outsource NN predictions to third parties while ensuring data security and privacy. However, as recognized by the authors, their approach was not very promising, as it produced high error values and required long computation times. It is worth Figure 1: Eligibility process noting that the authors did not reference the previous work of Hesamifard et al. in their paper. In 2019, Nandakumar et al. [11] published the first attempt to train a DNN on data encrypted using FHE, which can also be used for inference. Similar to Hesamifard et al.'s work, they used the HElib library to encrypt the data using the BGV scheme and to perform the computations on encrypted data. In this work, the authors identified two challenging steps in using HE, namely computing the activation function and its derivative, and computing the loss function and its derivative. To address this, they pre-computed these functions in a table and performed a homomorphic table lookup, an approach similar to the one proposed by Crawford et al. [5], instead of approximating them by a low-degree polynomial as Hesamifard et al. did. Their approach is faster and shallower when applicable, but can only provide a low-precision approximation of these functions. The authors used the sigmoid activation function and quadratic loss function, which have simpler derivatives, to avoid excessive table lookups. To minimize the number of bootstrapping operations and enable parallelization of computations, a data-packing strategy was employed, leading to a significant reduction of the overall computational complexity. The authors conducted experiments on the MNIST dataset, and the less complex four-layer NN reported an accuracy of 96% with an execution time of 40 minutes while using optimizations and multiple CPU threads. Although promising, the authors recognized that training data in the encrypted domain with their approach is four to five times slower than training in the plaintext domain, which shows that further research is required to improve efficiency. After the work of Nandakumar et al., Lou et al. [10] published, in 2020, a faster alternative, that they called "Glyph", to train DNNs on encrypted data by combining TFHE and BGV schemes. Their approach employs a scheme switching technique that introduces only small computing overhead, but allows leveraging the use of TFHE for nonlinear activations, such as ReLU and softmax, and BGV for multiplicity-accumulations. This is possible because their scheme switching technique maps the plaintext spaces of BGV and TFHE to a common algebraic structure using natural algebraic homomorphisms, enabling, in this way, homomorphic switching between different plaintext spaces. The authors performed several experiments and have compared their approach with the previous one. To achieve similar accuracies, 97.8% for the approach of Nandakumar et al. and 98.6% for Glyph, they estimated a training time of 13.4 years for the first approach and 8 days for the second one. Overall, their experimental results show that Glyph obtains state-of-the-art accuracy, and reduces training latency by 69% to 99% over prior FHE-based privacy-preserving techniques on encrypted datasets. In 2021, Onoufriou et al. [12] published a study which focus on the use of HE to preserve privacy in NN training and classification for milk yield forecasting in the agri-food sector. The authors employed a one-dimensional CNN (1D CNN) with encrypted data using the CKKS scheme. Since certain operations cannot be performed on fully homomorphically encrypted data, the authors implemented a set of measures to address this issue. First, they performed the forward pass in the encrypted domain by replacing the Sigmoid activation function by a polynomial approximation proposed by Chen et al. [3], in a similar approach to the one proposed by Hesamifard et al. This helped to avoid the necessity of computing divisions in the standard Sigmoid. Then, the backward pass and weight updating were done in plaintext to reduce computational cost. With this approach, the authors achieved an average accuracy of 87.6% and a Mean Absolute Percentage Error (MAPE) of 12.4% for milk yield prediction, demonstrating that FHE allows for absolutely private predictions. However, their approach has a flaw which is acknowledged by the authors, since it does not use encrypted data in all stages of training. This limitation requires further research to be addressed. It is worth mentioning that for their experiments, the authors used the MS-SEAL library, and a dataset with 30 years of breeding, feeding, and milk yield data. The study showed that although FHE incurs a high spatial complexity cost, the run time remains within reasonable bounds. In the same year, Yoo et al. [16] proposed a novel framework, named "t-BMPNet", for designing a foundational model of deep learning, namely a Multi-layer Perceptron Neural Network (MLPNN), over an FHE scheme. This framework allows for training in the encrypted domain without the need to replace activation functions with polynomial approximations, unlike the approach taken by Onoufriou et al. The authors achieved this by using the TFHE scheme, which enables building complex operations over encrypted data using primitive gates and fundamental bitwise homomorphic operations. They utilized the sigmoid function as the activation function and explained how to define the four main operations needed in the encrypted domain, namely: addition, two's complement, exponential function, and division. The authors conducted several experiments to validate their approach and confirmed that "t-BMPNet" achieved a highly accurate design of the nonlinear sigmoid function compared to other works that use polynomial approximations. However, the authors also recognized that their approach had low time performance and needs further research. ### Discussion After analysing the state-of-the-art studies on the use of HE to preserve data privacy in NN training and inference, in this section we will discuss the main findings, considering the research questions that were defined. 1.1 RQ1 - What types of NNs are most used in studies about the use of HE to preserve data privacy in NN? Based on the analysis of the reviewed literature, it is evident that DNNs and CNNs are the most commonly used types of NNs in studies that employ HE for preserving data privacy during training and inference. Out of the six works analysed, two of them utilized CNNs, while the other five employed DNNs. In fact, in the research conducted by Lou et al. [10], their approach was applied to both a DNN and a CNN. #### 3.2.2 RQ2 - What are the most commonly used HE schemes for preserving data privacy in NN? From our analysis of the literature, it is clear that FHE is the preferred type of scheme. This is not surprising, given that FHE allows for multiple operations an unlimited number of times. BGV stands out as the most commonly used in this context, with three out of the six reviewed works using it [8, 11, 10]. It is worth noting that this scheme was prevalent in the first works published, probably because an implementation of it was available at the time in the HElib library. More recently, other FHE schemes have been explored. Two out of the six reviewed works employed the TFHE scheme [10, 16], and the authors utilized the TFHE library for their experiments. By the other hand, one work uses the CKKS scheme [12]. Lou et al. [10] approach is also interesting since both BGV and TFHE are combined to leverage the best characteristics of each. Specifically, they used TFHE for nonlinear activations and BGV for multiplicity-accumulations. Regarding SMHE, only a single work was found [7], namely the YASHE scheme (with the SEAL library being used for the experiments). 2.3 RQ3 - What are the major limitations identified in the literature regarding the use of HE to preserve data privacy in NNs? One of the key limitations is related with the fact that HE only supports basic arithmetic operations, such as addition and multiplication, as explained in Section 2. Thus, other commonly used operations in NNs, such as division, exponential, comparison, and maximum, cannot be directly supported by HE or they are computationally expensive. Consequently, some restrictions are introduced on the characteristics of the NNs that can be trained with homomorphically encrypted data. For instance, activation functions like the sigmoid, which involve division, cannot be directly used. Similarly, loss functions may also be affected by unsupported operations. Additionally, pooling techniques, such as max pooling and average pooling, which are commonly used in CNNs, can also be affected because they involve comparing and selecting the maximum or average value within a set of values, which is not possible in the majority of HE schemes. Another significant limitation of homomorphically encrypted NNs is their computational overhead. The operations performed on encrypted data are computationally expensive, and their complexity increase with the size of the data and the depth of the NNs. Consequently, training and inference times can become significantly longer when compared with non encrypted NN. Finally, the use of HE with NNs can also introduce limitations on the way data is represented. HE schemes typically operate on encrypted data in a fixed-point representation, which is different from the floating-point representation used in traditional NNs. This difference can lead to numerical instability and can limit the range of values that can be represented, which may negatively affect the accuracy and performance of the model. RQ4 - What are the most common approaches used to address the limitations of HE in NN, and what trade-offs need to be considered? One of the most important limitations is related with the NN operations, as mentioned in the previous research question. To address this issue, most of the studied works replaced the activation and loss functions by a polynomial approximation [8, 7, 12]. However, this approach has shown to negatively affect the performance of the NNs, as low-degree approximations become progressively worse for input values farther from zero. To mitigate this problem, some authors have suggested normalizing input values, but this can be difficult to control, especially for NN with several layers. To avoid using polynomial approximations, other approaches have been proposed, such as pre-computing activation and loss functions and performing homomorphic table lookups [11] or implementing complex operations using fundamental bitwise operations [16]. However, each of these approaches also has trade-offs. For instance, while the table lookup approach is faster than using polynomial approximations, it can only provide low-precision approximations for the activation and loss functions. On the other hand, implementing complex operations using fundamental bitwise operations can lead to better approximations, but it can also result in low time performance. Finally, Lou et al. [10] proposed a combined approach using the schemes TFHE and BGV, that slightly increased the accuracy of this system. Regarding the computation overhead limitation, one common strategy is to use a low degree polynomial approximation for the activation and loss functions, as explained before. This reduces the number of operations required, thereby reducing the computational overhead. Additionally, another way to reduce the computational overhead is to use NNs with fewer layers. In fact, most of the works reviewed have used DNN with just three layers [7, 11, 16]. However, some other studies have explored different NN configurations, such as varying the number of hidden layers from 1 to 5 [8], or utilizing a 1D CNN [12]. Since each homomorphic operation performed in encrypted data is slower than its counterpart, by reducing the layers, the number of homomorphic operations that need to be used can be reduced, which can lead to a significant reduction in the computational overhead. However, this reduction in the number of layers may also result in a reduction in the accuracy of the model. Therefore, there is a clear trade-off between model accuracy and computational overhead when using HE in NN training and classification. In relation to the limitation which may be introduced by HE regarding the way data is represented, most of the approaches use encoding strategies to make it possible to work with low-precision integers and fixed-point representation. By the other hand, CKKS, used in the work of Onoufriou et al. [12], uses a floating-point representation for plaintext and ciphertext data, requiring minimal adaptations. Even though these strategies are needed to use HE in NNs, they also come with a trade-off, such as potential accuracy loss with fixed-point representation and the need for additional computation with low-precision integer representation. ## 5 Conclusion In this work, we have reviewed and analysed the state-of-the-art on the use of HE to preserve data privacy in NN training and inference. Our study showed that BGV scheme is the most commonly used in this context, which highlights the trend towards the use of FHE schemes. The Glyph and t-BMPNet approaches stand out for leveraging the characteristics of the TFHE scheme to reduce training times while maintaining high accuracy levels, without replacing nonlinear functions with polynomial approximations. Nevertheless, current approaches are still slow and may not provide high levels of accuracy. We believe that the selection of the most appropriate strategy depends on the specific application scenario, which may have varying requirements in terms of accuracy and computational complexity. Therefore, future research should focus on exploring the application of TFHE in more complex NNs and investigating other HE schemes. Additionally, improving computational performance through parallelization may be worth exploring to make HE more reliable and scalable for this type of architecture. In summary, the use of HE for NN training is a promising approach for preserving privacy, but optimizing its accuracy and efficiency requires further research and experimentation with novel techniques and strategies.
2305.05562	SkelEx and BoundEx: Natural Visualization of ReLU Neural Networks	Despite their limited interpretability, weights and biases are still the most popular encoding of the functions learned by ReLU Neural Networks (ReLU NNs). That is why we introduce SkelEx, an algorithm to extract a skeleton of the membership functions learned by ReLU NNs, making those functions easier to interpret and analyze. To the best of our knowledge, this is the first work that considers linear regions from the perspective of critical points. As a natural follow-up, we also introduce BoundEx, which is the first analytical method known to us to extract the decision boundary from the realization of a ReLU NN. Both of those methods introduce very natural visualization tool for ReLU NNs trained on low-dimensional data.	Pawel Pukowski, Haiping Lu	2023-05-09T15:48:34Z	http://arxiv.org/abs/2305.05562v1	# Skelex and Boundary: Natural Visualization of Relu Neural Networks ###### Abstract Despite their limited interpretability, weights and biases are still the most popular encoding of the functions learned by ReLU Neural Networks (ReLU NNs). That is why we introduce SkelEx, an algorithm to extract a skeleton of the membership functions learned by ReLU NNs, making those functions easier to interpret and analyze. To the best of our knowledge, this is the first work that considers linear regions from the perspective of critical points. As a natural follow-up, we also introduce BoundEx, which is the first analytical method known to us to extract the decision boundary from the realization of a ReLU NN. Both of those methods introduce very natural visualization tool for ReLU NNs trained on low-dimensional data. Pawel Pukowski, Haiping Lu Department of Computer Science, University of Sheffield, Sheffield, UK classification, decision boundary, linear regions, ReLU, visualization + Footnote †: 979-8-3503-2411-2/23/$31.00 ©2023 IEEE ## 1 Introduction ReLU is a very popular activation function not only because of its great performance, but also because of it's piecewise linear (PL) structure. ReLU being PL implies that the learned membership functions are also PL. This PL structure is most commonly used to study the number of linear regions that an architecture can produce, giving us insight on it's flexibility. However, it can have other uses. For example, it allows us to introduce a different encoding of the same functions. Encoding that uses weights and biases, like Eqn. (1) is great for training, and space saving, but lacks a lot in terms of interpretability. If we used the information on the location of gradient changes (critical points), like in Eqn. (2), the functions would become easier to interpret. Additionally, this _skeleton encoding_ introduces a very natural visualization of ReLU NNs trained on low-dimensional data. \[f(x)=\max(0,\ \max(0,\ x+1)-2\max(0,\ x)) \tag{1}\] \[f(x)=\begin{cases}0&x\in(-\infty,-1]\\ x&x\in(-1,0]\\ -x&x\in(0,1]\\ 0&x\in(1,\infty)\end{cases} \tag{2}\] Decision boundary is a very important property of the trained NNs, as it is responsible for tessellating the input space into membership regions. Analysing the decision boundary gives us crucial information, such as the exact location of all adversarial examples. However, as far as we know, so far only analytical means of calculating the decision boundary were used. The decision boundary is a subset of the intersection of the membership functions. This means that the skeleton encoding allows us to introduce the first analytical method that extracts the decision boundary of ReLU NNs. Our main contributions are as follows: * We provide an algorithm (SkelEx) that transforms the weights and biases encoding of functions learned by ReLU NNs into the skeleton encoding (Fig. 1 left). * We visualize the difference between linear regions and activation regions, two entities that have often been incorrectly used as synonyms in literature. * We introduce an analytical method to extract the decision boundary learned by FC ReLU NNs. (Fig. 1 right) ## 2 Literature Review Linear regions are commonly used to measure the expressivity of ReLU NNs [1]. Researchers initially focused on establishing an upper bound on the maximum number of linear regions an architecture could produce [2], and later improving it [3, 4]. The number of linear regions produced is believed to indicate the architecture's flexibility, with more regions leading to smaller approximation errors [5]. However, the average number of linear regions produced by a realization after training is much lower than the upper bound [4]. Linear regions can be counted numerically or analytically. Numerical methods involve stepping along a trajectory and deciding whether each point belongs to the same linear region as the previous one based on their activations [3, 6]. Analytical methods are more elaborate and computationally expensive. Serra et al. [4] counted the number of linear regions across the entire input space, while Hanin et al. [7, 8] introduced properties of linear regions and a method to count them along a 1D or 2D subspace. Zhang et al. [9] extracted the H-representation of the linear region containing a given input. Numerical methods are commonly used to obtain the decision boundary for NNs, with the most popular approach being to traverse a defined trajectory away from a data sample in fixed-sized steps until the membership of the point changes [9, 10, 11, 12, 13]. Another popular approach is to approximate the decision boundary using adversarial examples [11, 12, 13]. Takahashi et al. [14] and Zengin et al. [15] use convex hulls and Voronoi tessellation, respectively, to classify data samples, but their accuracy is lower than the weights and biases approach because they learn different functions. ## 3 Skelex SkelEx is a two-step algorithm that extracts the skeletons of learned membership functions from pre-trained ReLU NN's weights and biases. It applies ReLU to neurons' pre-activations and merges activations to calculate pre-activations fo the next layer until the final membership functions are obtained (Algorithm 1). SkelEx's output provides a natural visualization of learned functions and information such as critical point formation time (at which layer) and how linear regions tessellate the input space. ``` 0:$n_{i}\leftarrow$ number of neurons in $i^{th}$ layer for$i\in\{1,2,...,L\}$do for$j\in\{1,2,...,n_{i}\}$do $f_{i}^{j}\leftarrow$ ApplyReLU$(g_{i}^{j})$ endfor for$j\in\{1,2,...,n_{i+1}\}$do $g_{i+1}^{j}\leftarrow$ MergeActivations$(\mathbf{f}_{i}\cdot\mathbf{w}_{i})$ endfor $\mathbf{g}_{i+1}\leftarrow\mathbf{g}_{i+1}+\mathbf{b}_{i+1}$ endfor return$\mathbf{f}_{L+1}$$\triangleright$ Vector of $k$ membership functions ``` Algorithm 1 SkelEx ### $2d$ projection The functions learned by FC ReLU NNs reside in $(n_{0}+1)$-dimensional space, where $n_{0}$ is the number of input neurons. To encode them we only require the skeletons formed by the membership functions, which reside in $n_{0}$-dimensional space, and the gradient of linear regions. In Fig. 1 we used 3D projection, and we can see that even for very easy functions it is already hard to visualize them on 2D paper/screen. That is why from this point on we will visualize all functions using only $n_{0}$ dimensions. As an example, Fig. 1(b) shows a $2D$ Figure 1: Let $\mathbf{w}=[[[-1,-1],[-1,1]],[[1,1]]]$, and $\mathbf{b}=[[-1,-1],[0]]$ be the weights and biases learned by a ReLU NN with configuration 2-2-1 (left). SkelEx$(\mathbf{w},\mathbf{b})$ first extracts pre-activations of the first hidden layer, and then calculates the activations. Next, SkelEx merges the activations to calculate pre-activations of the next layer. For deeper networks this 2 step process would be repeated until the skeletons of the membership functions are extracted (red lines indicate critical points). BoundEx uses the skeletons extracted via SkelEx to calculate the decision boundary (right side; dots indicate training data). Figure 2: (a) The hyperrectangle $R$ bounds the input space of $f$. (b) $R$ allows us to encode $f$ using the skeleton of critical points (red points and blue line) and the gradients ($\nabla$) of the linear regions. projection of the function from Fig. 1(a). ### Assumption To create a skeleton for the functions learned by ReLU NNs, we must first bind the input space using a hyperrectangle $R$. Despite concerns that this could affect the accuracy of the algorithm, the dimensions of $R$ do not impact the accuracy of SkelEx, as it works with linear regions one at a time. Therefore, restricting the input space simply reduces the number of linear regions or makes the algorithm work ob a subspaces of linear regions. Since most real-world datasets are already bounded, defining the dimensions of the hyperrectangle is usually straightforward (e.g., $age\in[0,125]$ and $pixel_{MNIST}\in[0,255]$). ### Step 1. Applying ReLU Let $g_{i}^{j}(\mathbf{x})$ be the pre-activation of the $j^{th}$ neuron in the $i^{th}$ layer. We know that, just like the membership function, $g_{i}^{j}(\mathbf{x})$ is a PL function. This means that its domain is divided into linear regions. We also know that applying ReLU to a union of linear regions yields the same result as when applying ReLU to each linear region separately and then taking the union. Hence, when calculating ReLU($g_{i}^{j}(\mathbf{x})$) we can just go through all linear regions of $g_{i}^{j}(\mathbf{x})$, and apply ReLU to each of them, and then merge the results. ReLU splits each linear region into a sequence of regions. We divide those regions into the ones that lie above and below $0$ in the output space. All the regions below $0$ will be moved up to $0$. We will also merge all of the neighboring regions with the same gradient. The whole process is described in Algorithm 2, and the visualization is provided in Fig. 4. The merging process is necessary to keep operating on linear regions. If we omitted it, then we would be working on activation regions greatly increasing the computational complexity. ``` 0:$S\leftarrow\{lr_{1},...,lr_{n}\}$$\triangleright$$n$ linear regions tessellate $S$$new\_lrs\leftarrow$ empty set for$lr$ in $S$do$\triangleright$ Apply ReLU to each $lr$$vertices\gets lr.vertices$ for$e\in lr.edges$do if$sign(e.start)\neq sign(e.end)$then $intersection\gets v\subset e$ such that $v.value=0$$vertices.add(intersection)$ endif endfor change values of all negative vertices to $0$$new\_lrs.add(\text{FormLinearRegions}(vertices))$ endfor merge lrs in $new\_lrs$ that have the same gradient return$new\_lrs$$\triangleright$ Convert to Skeleton class ``` Algorithm 2 ApplyReLU ### Step 2. Merging Activations Let's imagine two tessellations $T_{1}$ and $T_{2}$ that represent the skeletons of functions $f_{i}^{1}$ and $f_{i}^{2}$, respectively. Merging those tessellations is equivalent to drawing $T_{1}$, and then adding all of the lines from $T_{2}$ that have not already been drawn (Fig. 5). This can be done by going through all linear regions $lr$ in $T_{1}$, and finding linear regions in $T_{2}$ that intersect $lr$. Those intersections must have a Lebesgue measure for $n=n_{0}$, as they will become the tiles forming the tessellation of the merged skeleton. So, for the function in Fig. 5, the intersection must not be a line or a point. Since this tessellation represents higher dimensional functions, we also need to update the value (in the output space) of each vertex (0-face) of the skeleton. Hence, given any vertex $v$ in the intersection, we change its value to the sum of values that this vertex has in $f_{i}^{1}$ and $f_{i}^{2}$ ($v.value=f_{i}^{1}(v)+f_{i}^{2}(v)$). ### Additional Insights Merging activations does not introduce any new critical points. That is why the pre-activations of the given layer Figure 4: When ReLU is applied to the skeleton in (a), all negative (blue) vertices are removed, and the positive (red) ones are left intact. At the function’s intersection with 0 (($x_{1},x_{2}$)-axis) new (yellow) vertices are formed. Figure 3: SkelEx extracted (a) and (b) using hyperrectangles of different sizes: (a) $x_{1},x_{2}\in[-25,25]$ (b) $x_{1},x_{2}\in[-10,10]$. (b) is the grey patch from (a). Dots are colored by the step in which they were created (blue - $\mathbf{g}_{1}$, magenta - $\mathbf{f}_{1}$, cyan - $\mathbf{g}_{2}$, red - $\mathbf{f}_{2}$, green - $\mathbf{g}_{3}$). produce the same tessellation of the input space. The only exception would be when the weights are equal to $0$, as that reduces the number of activations that are merged. That is why it is highly likely that all $k$ membership functions have the same skeleton, because ReLU is not applied to the output neurons. We also find that the learned functions tend to produce a significant amount of parallel and almost-parallel line segments visible in Fig. 4, and 7. This most likely happens because the activations of the first hidden layer play a pivotal role in forming the skeleton. Therefore, a subspace of the critical points from the activation of a neuron from the first hidden layer appears in many future activations, but each neuron might translate it slightly forming a series of parallel-looking lines. Those lines can be then slightly altered by ReLU. The majority of those lines do not play any role in the creation of the decision boundary. Hanin and Rolnick [7, 8] were the first ones to highlight the difference between linear regions and activation regions. In literature, those two terms were often used interchangeably, even though they have different definitions. The activation region is a collection of points in the input space that produce the same activation sequence, whereas the linear region is a subspace of the domain of the membership function where the gradient does not change. The difference between those two notions is clearly visualized in Fig. 6. Interestingly, we noticed that the membership functions usually contain the same amount of activation regions as linear regions. The reason for this might be that the activation regions that are removed by ReLU in one neuron can survive in other other neurons. After all, pre-activations of the given layer produce the same tessellation of the input space, so the activation region would have to be removed in all neurons to completely vanish. In other words, working only on activation regions should not change the result, but would make the computational complexity bigger. ## 4 Boundex Skeletons extracted by SkelEx can be used to analytically calculate the decision boundary. To do so, we need to devise a variation of the $\arg\max$ function that uses skeletons of whole functions as an argument rather than a vector of real numbers. We call this variation of $\arg\max$ BoundEx (Algorithm 4). ### BoundEx Algorithm Let $f^{1},f^{2},...,f^{k}$ be the membership functions returned by SkelEx, $T$ be the tessellation produced by them, and $lr$ be one of the tiles of $T$. When applying BoundEx to $lr$ we find two cases, and we are only interested in the second case as it indicates the change of membership: * Case 1.$\forall\mathbf{v}\in V\ \forall j\in 1,2,...,\|V\|)\ \ f^{i}(\mathbf{v})>=f^{j}(\mathbf{v})$, if we let $V$ be the set of vertices of given linear regions * Case 2.$\exists\mathbf{v}\in V\ \exists j_{1}\in 1,2,...,\|V\|$ s.t. $f^{i}(\mathbf{v})>f^{j_{1}}(\mathbf{v})$, and $\exists\mathbf{v}\in V\ \exists j_{2}\in 1,2,...,k$ s.t. $f^{i}(\mathbf{v})<f^{j_{2}}(\mathbf{v})$, with $V$ the same as above The decision boundary that passes through $lr$ divides it into up to $k$ membership polygons. To understand what this Figure 5: When two functions (a) and (b) are merged together, the result (c) contains vertices and edges of both functions. Here bias $b<0$ is negative, so all yellow vertices turn blue (best viewed on screen). Figure 6: Linear regions and activation regions are not the same. Regions that share gradient can be merged to transform activation regions (a) into linear regions (b). implies, let $v$ be one of the vertices of $lr$, and $f^{i}$ be the function such that $f^{i}=\arg\max(f^{1}(v),f^{2}(v),...,f^{k}(v))$. Now consider any other function $f^{i,1}$ from the set of membership functions $\mathbf{f}$. If $f^{i,1}$ intersects $f^{i}$ within $lr$, then it divides $lr$ into two regions - the one where $f^{i}$ is larger ($pos\_reg$), and the one where $f^{i,1}$ is larger ($neg\_reg$). If this was a two-class problem, then $pos\_reg$ would be the polygon enclosing all points that are classified by the model as belonging to class $i$. Now let's take another function $f^{i,2}$. If $f^{i,2}$ intersects $f^{i}$ inside of the $pos\_reg$, then, just as before, $pos\_reg$ gets divided into two regions. We are not interested in the intersection between $f^{i,2}$ and $f^{i}$ that happens outside of $pos\_reg$. That is because, for $neg\_reg$, $f^{i,1}$ is larger than $f^{i}$. So to determine the membership of points within $neg\_reg$, we must compare $f^{i,2}$ with $f^{i,1}$, not with $f^{i}$. If we continue this process until we go through all $k$ functions we will obtain all _membership polygons_. Those polygons define the set of points that are classified as belonging to the same class by the trained neural network. We do the same for all other tiles from $T$, and then merge the neighboring membership polygons of the same class. The decision boundary can be found at the interception of those polygons. ### Examples and Limitations To present the capabilities of BoundEx, we train it on a toy datasets, and the Balance Scale dataset from UCI [16] (Fig. 7 col. 1). For the Balance Scale dataset, we reduce the dimensionality of the data to 2 by introducing variables $x_{1}=weight_{l}\times distance_{l}$, and $x_{2}=weight_{r}\times distance_{r}$. We train a ReLU NN on each of those datasets to $\sim 100\%$ accuracy. We then pass the learned weights and biases to SkelEx to produce $k$ skeletons of membership functions. As mentioned in Sec. 3.5, we see that both membership functions produce the same tessellation of the input space. Those membership functions generate the decision boundaries, and the decision polygons (col. $4$). Now, when given a data sample to classify, rather than performing forward propagation, we can just check in which polygon it lies. Hence, BoundEx transforms the classification problem into a Point-in-Polygon problem. ``` $mp\leftarrow$ hash map $\triangleright$ membership polygons for$lr\in T$do $v\gets lr.vertices[1]$ $base_{f}=f^{\text{argmax}([f^{1}(v),f^{2}(v),...,f^{k}(v)])}$ $cmp[\text{index}(base_{f})]=lr$$\triangleright$ current mp for$f^{i}$$\in$ f s.t. $i\notin cmp.keys$do for$key\in cmp.keys$do for$reg\in cmp[key]$do $pos\_reg,neg\_reg=reg.split(f^{i})$ remove $neg\_reg$ from $cmp[key]$ $cmp[i].\text{append}(neg\_reg)$ endfor endfor endfor for$key\in cmp.keys$do $mp[key].extend(cmp[key])$ endfor endfor merge neighboring membership polygons of the same class return$dp$ ``` Algorithm 4 BoundEx It is common knowledge that NNs are overparameterized, and because of that learn a lot of unnecessary knowledge. This is clearly visible in columns $2$ and $3$ of Fig. 7. The skeleton contains multim of faces that are have no impact on the production of the decision boundary. If we remove all the unnecessary linear regions (and adjust the values of the Figure 7: When give weights and biases of ReLU NN trained on data from col. $1$ (row $1$ - toy; row $2$ - Balance Scale), SkelEx extracts the skeletons of learned membership functions (col. $2$ for $f^{1}$, col. $3$ for $f^{2}$). BoundEx takes those and extract the decision boundary (col. $4$). If we removed the unnecessary linear regions the tessellation would become simpler (col. $5$). remaining ones accordingly), the skeleton would become significantly simpler. In fact, we should be able to separate the faces of the skeleton that pass through the vertices of the decision boundary (white lines on col $5$, of Fig. 7), and use them to craft a smallest possible skeleton that generates given decision boundary. The activations of the first hidden layer are composed of two $n_{0}$-dimensional linear regions1. Since an $n_{0}$-dimensional linear region requires $2^{n}$ vertices to encode, we need $n_{1}\times 2^{n_{0}}$ vertices to encode all of the skeletons from the first hidden layer. This number increases as SkelEx progresses through the hidden layers, making the algorithm expensive for high-dimensional data. Experiments in this work contain were performed only on 2D data, because in our implementation we use GEOS library [17] that does not support polygon operations for higher dimensions. Footnote 1: The only exception are the cases where the hyperrectangle is too small, and so there is only one linear region. ## 5 Conclusion In this work, we introduced SkelEx and BoundEx. SkelEx is an algorithm that, given weights and biases of pre-trained ReLU NN, extracts the skeleton of the learned membership functions. It operates in two steps, where it iteratively applies ReLU to the pre-activations to calculate activations and then merges the activations to calculate the pre-activations of the neurons of the next layer. We have shown that SkelEx provides a very natural visualization method, which yields nice results. We have also introduce BoundEx, the first analytical method that extracts the decision boundary learned by ReLU NNs.
2303.11699	Neural networks trained on synthetically generated crystals can extract structural information from ICSD powder X-ray diffractograms	Machine learning techniques have successfully been used to extract structural information such as the crystal space group from powder X-ray diffractograms. However, training directly on simulated diffractograms from databases such as the ICSD is challenging due to its limited size, class-inhomogeneity, and bias toward certain structure types. We propose an alternative approach of generating synthetic crystals with random coordinates by using the symmetry operations of each space group. Based on this approach, we demonstrate online training of deep ResNet-like models on up to a few million unique on-the-fly generated synthetic diffractograms per hour. For our chosen task of space group classification, we achieved a test accuracy of 79.9% on unseen ICSD structure types from most space groups. This surpasses the 56.1% accuracy of the current state-of-the-art approach of training on ICSD crystals directly. Our results demonstrate that synthetically generated crystals can be used to extract structural information from ICSD powder diffractograms, which makes it possible to apply very large state-of-the-art machine learning models in the area of powder X-ray diffraction. We further show first steps toward applying our methodology to experimental data, where automated XRD data analysis is crucial, especially in high-throughput settings. While we focused on the prediction of the space group, our approach has the potential to be extended to related tasks in the future.	Henrik Schopmans, Patrick Reiser, Pascal Friederich	2023-03-21T09:37:29Z	http://arxiv.org/abs/2303.11699v3	Neural networks trained on synthetically generated crystals can extract structural information from ICSD powder X-ray diffractograms ###### Abstract Machine learning techniques have successfully been used to extract structural information such as the crystal space group from powder X-ray diffractograms. However, training directly on simulated diffractograms from databases such as the ICSD is challenging due to its limited size, class-inhomogeneity, and bias toward certain structure types. We propose an alternative approach of generating synthetic crystals with random coordinates by using the symmetry operations of each space group. Based on this approach, we demonstrate online training of deep ResNet-like models on up to a few million unique on-the-fly generated synthetic diffractograms per hour. For our chosen task of space group classification, we achieved a test accuracy of 79.9% on unseen ICSD structure types from most space groups. This surpasses the 56.1% accuracy of the current state-of-the-art approach of training on ICSD crystals directly. Our results demonstrate that synthetically generated crystals can be used to extract structural information from ICSD powder diffractograms, which makes it possible to apply very large state-of-the-art machine learning models in the area of powder X-ray diffraction. We further show first steps toward applying our methodology to experimental data, where automated XRD data analysis is crucial, especially in high-throughput settings. While we focused on the prediction of the space group, our approach has the potential to be extended to related tasks in the future. ## 1 Introduction Machine learning techniques have emerged as a powerful tool in the toolkit of materials scientists. While they are often used to make predictions on the properties of materials or find new materials with certain properties, an increasingly interesting domain is the automated analysis of raw experimental measurements guided by machine learning [1]. With the advent of high-throughput experiments, the amount of gathered data is vast and the analysis often becomes a bottleneck in the processing pipeline [2]. Powder X-ray diffraction (XRD) is an important measurement technique used to obtain structural information from polycrystalline samples [3]. The diffractograms are an information-dense fingerprint of the structure of the material. However, analyzing these diffractograms is not an easy task [4]. Full structure solutions and Rietveld refinement take time and require expert knowledge, both about the analysis technique and the materials class at hand. This is not feasible in high-throughput experiments on a larger scale. Therefore, the question arises whether it is possible to automatically analyze powder diffractograms with machine learning models trained on large amounts of data, making it possible to run inference almost instantaneously. During the last few years, there have been several studies tackling this objective by applying machine learning models to various tasks concerning the analysis of powder diffractograms such as phase classification [5, 6, 7, 8, 9], phase fraction determination [10], space group classification [11, 12, 13, 14, 15, 16], machine-learning-guided Rietveld refinement [17, 18], extraction of lattice parameters [19, 20, 21, 16] and crystallite sizes [16, 19], and also novelty detection based on unsupervised techniques [22]. Since an abundant source of experimental diffractograms is hard to come by, most applications train their models on simulated diffractograms from the Inorganic Crystal Structure Database (ICSD) [23], which contains a total of 272 260 structures (October 2022). Lee _et al._ used a deep convolutional neural network (CNN) trained on a large dataset of multiphase compositions from the quaternary Sr-Li-Al-O pool to classify present phases in the diffractogram [5]. In a follow-up study, they further showed good results for phase fraction inference in the quaternary Li-La-Zr-O pool [10]. Schuetzke _et al._ performed phase classification on iron ores and cement compounds and used data augmentation with respect to lattice parameters, crystallite sizes, and preferred orientation [7]. They showed that especially the lattice parameter variations enhance the classification accuracy significantly. Instead of the analysis of phase composition, Dong _et al._ performed regression of scale factors, lattice parameters, and crystallite sizes in a five-phase catalytic materials system [19]. In contrast to supervised tasks, Banko _et al._ used a variational autoencoder to visualize variations in space group, preferred orientation, crystallite size, and peak shifts [22]. Park _et al._ used a deep CNN to classify space groups of single-phase diffractograms, reaching a test accuracy of 81.14% on simulated diffractograms. However, as we will show later in this paper, this accuracy is highly overestimated and drops to 56.1% when test splits are designed in a way to reduce data leakage in non-IID datasets such as the ICSD. Vecsei _et al._ developed a similar approach and applied their classifier to experimental diffractograms from the RRUFF mineral database [24], reaching an experimental test accuracy of 54%. While the ICSD contains a large number of structures spanning many different classes of materials, it still falls short in size, distribution, and generality compared to the datasets used to train very large state-of-the-art models of other fields such as computer vision. Furthermore, the ICSD database is highly imbalanced with respect to space groups, as can be seen in the histogram in Figure 1. This makes the classification of space groups more difficult, as shown and discussed by Zaloga _et al._[13]. The ICSD also contains a limited number of different structure types that may not adequately represent the crystal structures analyzed in future experiments. To overcome these shortcomings, we propose to train machine learning models on diffractograms simulated from synthetic crystal structures randomly generated based on the symmetry operations of the space groups. This makes it possible to train on structures with new structure types not present in the ICSD. We used the crystals from the ICSD only to determine vague statistics guiding the random generation and for calculating the test accuracy. Our approach goes one step further than classical data augmentation by fully detaching itself from the individual entries in the ICSD database. It is our hypothesis that when trying to solve more general tasks concerning the extraction of structural details from powder diffractograms, the crystals from which the training dataset is simulated do not need to be stable or chemically viable. We propose viewing the problem simply as a mathematical task of getting back some of the real-space information leading to given powder X-ray diffractograms. Here, we applied this approach to the classification of the crystal symmetry, namely the space group. The space group is usually one of the first structural pieces of information needed after synthesizing a new material. This task is well-suited to showcase the strengths of using a synthetic dataset and to benchmark it. We further show the results of using our methodology to infer space group labels of an experimental dataset. We embedded our synthetic generation algorithm in a framework with distributed computing capabilities to generate and simulate diffractograms on multiple nodes in parallel using the _Python_ library _Ray_[25]. In contrast to the traditional approach of generating a simulated dataset before training, we used this distributed computing architecture to build an infinite stream of randomly generated and simulated diffractograms to perform batch-wise online learning. This increases the generalization performance, eliminates the problem of overfitting, and allows very large models to be trained. ## 2 Methods ### Generating synthetic crystals To generate synthetic crystals, we randomly place atoms on the Wyckoff positions of a given space group and then apply the respective symmetry operations. The algorithm is explained in the following (see also Figure 1(a) for a flow diagram of the algorithm). We only explain the most important steps, details can be found in Section S1 of the SI. 1. Random selection of a space group. We follow the space group distribution of the ICSD to allow comparison with previous work. 2. Sample unique elements of the crystal and their number of repetitions in the asymmetric unit. 3. Place atoms onto the Wyckoff positions and draw uniform coordinates for each. 4. Draw lattice parameters from a KDE based on the ICSD. 5. Apply space group symmetry operations. Parts of this algorithm were inspired by the generation algorithm of the _Python_ library _PyX-tal_[26]. We only keep generated crystals for training if the conventional unit cell volume is below $7000\,\mathrm{\SIUnitSymbolAngstrom}^{3}$ and if there are less than 100 atoms in the asymmetric unit. We did not employ any form of distance checks on the coordinates, as we found this to have no meaningful impact on space group classification accuracy. We only prevented the algorithm from placing more than one atom onto a Wyckoff position that does not have a degree of freedom. We also did not use partial occupancies. We chose this algorithm for its simplicity and its capability to reproduce many important characteristics of ICSD crystals adequately (see Section 3.1). For some space groups, there are not enough crystals in the ICSD to form a representative KDE for the volume or to calculate suitable occupation probabilities for individual Wyckoff positions. Therefore, we chose to only perform the classification on space groups with 50 or more crystals available in the statistics dataset we used to extract the probabilities (see Section 2.4), resulting in the exclusion of 85 space groups (see Figure 1). A classifier trained directly on ICSD data of all space groups will likely not be able to properly identify these space groups containing very few samples. If a similar performance for all space groups is desired, a uniform distribution of space groups in the training dataset is needed. This is trivially possible with our synthetic approach, in contrast to training directly on the ICSD, where weighting, over-, or undersampling methods are needed [27]. To allow a direct and fair comparison between our approach and the original approach of training directly on ICSD entries, we still followed the same distribution of space groups of the ICSD in our synthetic training dataset. This eliminates the problem that the effective number of total space groups is smaller when training on a highly imbalanced dataset, making it easier to reach high accuracies. Figure 1: Distribution (logarithmic scale in blue, linear scale in red) of space groups in the ICSD. Space groups are sorted by count. The population of the space groups varies by multiple orders of magnitude, showing that the ICSD is a highly imbalanced dataset regarding space groups. The space groups excluded due to insufficient statistics are visualized with black stripes. Since the exclusion is based on the statistics dataset and this histogram shows the full ICSD, the excluded space groups are not exactly the first 85 counted from the left. ### Simulating diffractograms To simulate powder X-ray diffractograms based on the generated crystals, we used the implementation found in the _Python_ library _Pymatgen_[28]. We optimized the simulation code using the LLVM just-in-time compiler _Numba_[29]. This increases the performance of the main loop over the reciprocal lattice vectors of the crystal significantly and makes the continuous simulation while training (discussed in the next section) possible. We used the wavelength $1.5406\,\mathrm{\SIUnitSymbolAngstrom}$ (Cu $\mathrm{K}_{\alpha}$ line) to simulate all diffractograms. The obtained peaks were further broadened with a Gaussian peak profile to form the full diffractogram. To obtain the peak widths, we used the Scherrer equation[30] \[\beta=\frac{K\lambda}{L\cos\theta}, \tag{1}\] where $\beta$ is the line broadening at half maximum intensity (on the $2\theta$-scale), $K$ is a shape factor, $\lambda$ is the wavelength, and $L$ is the (average) thickness of crystallites. We drew crystallite sizes from the range $[20,100]$ nm and used $K=0.9$. Diffractograms were generated in the range $2\theta\in[5,90]\,\mathrm{\SIUnitSymbolAngstrom}$ with step size $0.01\,\mathrm{\SIUnitSymbolAngstrom}$. After generating each diffractogram, it was rescaled to fit in the intensity range $[0,1]$. ### Continuous generation of training data Typically, machine learning models are trained with a fixed dataset predefined at the beginning of training. Sometimes, data augmentation is applied to further increase the effective size of this dataset. In contrast to that, we generated our dataset on-the-fly, parallel to model training. The main advantage of using this approach compared to a fixed-size dataset is the eliminated possibility to overfit to individual diffractograms since every diffractogram is only used once. Furthermore, not having to pre-simulate a dataset before training makes this approach more flexible when changing simulation parameters. We used a distributed architecture on multiple nodes using the _Python_ framework _Ray_[25], which enabled the training on 1-2 GPUs and simultaneous generation of training data on more than 200 CPU cores (see Figure 2b and SI Section S2.2). Depending on the model size and corresponding training speed, this setup allows training with up to millions of unique diffractograms per hour. ### Dataset split The ICSD database contains many structures that are very similar with slightly different lattice parameters and coordinates. For example, there are 25 entries for NaCl (October 2022). Furthermore, there are 3898 entries that have the same structure type as NaCl and thus also similar powder diffractograms. If some of them appear in the training dataset and some in the test dataset, Figure 2: a) Flowchart of how the generation algorithm produces synthetic crystals. Atoms are independently placed on the Wyckoff positions and random coordinates are drawn. b) Overview of the distributed computing system implemented using the _Python_ library _Ray_[25]. Two compute nodes (that generate and simulate diffractograms) are connected to the _Ray_ head node using a _Ray queue_ object. the classification will be simplified to recognizing the structure type or structure. In that case, the test set accuracy will not represent the true generalization performance of the neural network. To quantify the true generalization performance, we split the dataset in such a way that the same structure type appears either only in the training or in the test dataset. We used the structure type definitions provided by the ICSD. The obtained accuracy on the test dataset reflects the accuracy of our network when being used on a novel sample with a structure type not yet present in the ICSD database. We want to emphasize that the used test split is very important for the task of space group classification and not a trivial choice. The ICSD contains many subtypes of structure types (for example, subtypes of perovskites), which we regarded as separate structure types in our split. Considering the subtypes as the same structure type may also be a viable option when performing the split. A combination of a split based on structure type and sum formula or similar approaches are also possible. However, we think that our choice of splitting the dataset is able to effectively measure the generalization performance while still being relatively simple. We divided the ICSD (database version 2021, June 15) in a 70:30 split. For our synthetic crystal approach, the 70% part (called statistics or training dataset) was only used to create the KDEs and to calculate the Wyckoff occupation probabilities needed for the generation algorithm. Since we can judge the performance of the synthetic generation algorithm by comparing the training accuracy (on synthetic crystals) with the accuracy tested on diffractograms simulated directly from the statistics dataset, an additional validation dataset was not needed. For comparison with the original approach by Park _et al._[11] of directly training on ICSD crystals, we simulated crystals directly from the statistics dataset and trained on them. Analogous to the synthetic generation, we only used crystals with a conventional unit cell volume below $7000\,\mathrm{\SIUnitSymbolAngstrom}^{3}$ and with less than 100 atoms in the asymmetric unit for the statistics and test dataset. This covers $\approx 94\%$ of the ICSD crystals. ### Models and experiments #### Models We will briefly introduce the models we used for the classification of space groups. A more detailed description can be found in the SI Section S2.1. As a baseline, we first used the CNN models introduced by Park _et al._[11]. They used three models, one for the classification of crystal systems ("parkCNN small"), one for extinction groups ("parkCNN medium"), and one for space groups ("parkCNN big"). All models have three convolution layers with two hidden fully connected layers and one output layer. The three models differ in the number of neurons in the hidden fully connected layers, increasing the number of model parameters with the number of target labels. Here, we only used the models "parkCNN medium" and "parkCNN big" and applied both to the classification of space groups. When using ICSD crystals to train the "parkCNN" models, dropout was used, while the training of the "parkCNN" models on synthetic crystals did not use dropout. Since the approach of using an infinite stream of generated training data eliminates the problem of overfitting, we further used deeper models with a higher number of model parameters. For this, we used the deep convolutional neural networks ResNet-10, ResNet-50, and ResNet-101, which were introduced by He _et al._[31] in 2015. Details of the machine learning setup can be found in the SI Section 2.2. Overall, our setup allowed us the training of models over up to 2000 epochs with more than $100\,000$ unique, newly generated crystals and corresponding diffractograms in each epoch. #### Experiments We performed two sets of experiments to evaluate our new dataset split as well as our synthetic crystal generation approach and compare it to state-of-the-art models in literature: Firstly, we trained and tested models on ICSD crystals only, and secondly, we trained on synthetic crystals and tested on ICSD crystals. In particular, we first performed an experiment with the "parkCNN medium" model trained directly on the diffractograms simulated from the ICSD statistics dataset (similar to Park _et al._[11]) with a fully random train-test split, instead of splitting by the structure type of the crystals. This experiment makes a comparison of the two different methods of train-test split possible. We then trained the "parkCNN big" model using the structure type-based split, again directly on ICSD diffractograms. We further repeated the same experiment using the smaller model "parkCNN medium" to resolve potential overfitting to the ICSD diffractograms. For the experiments performed on our continuously generated dataset based on synthetic crystals, we used the structure type- based split. As discussed in Section 2.4, the training / statistics dataset was only used to extract more general statistics, such as the element distribution. First, we trained the "parkCNN big" model. For each batch, we generated 435 random crystals and simulated two diffractograms with different crystallite sizes for each of them. This resulted in the batch size of 870. Since our synthetic crystal generation algorithm yields an infinite stream of unique diffractograms to train on, using much larger models than for the fixed ICSD dataset is possible without overfitting. We performed experiments for the ResNet-10, ResNet-50, and ResNet-101 models. Instead of generating two diffractograms with different uniformly sampled crystallite sizes for each generated crystal (as we did for the "parkCNN big" model), we now created only one diffractogram for each of the 145 crystals generated for one batch. This is due to the slower training of the ResNet models, which means that reusing the same diffractogram with different crystallite sizes is not necessary to generate training data fast enough. To obtain the highest-possible ICSD test accuracy, we further applied the square root function as a preprocessing step to the input diffractograms of the network when using the ResNet models. This was suggested by Zaloga _et al._[13] and in their case improved classification accuracy by approximately 2 percentage points. Some initial tests suggested that this approach also yields a higher accuracy in our case, so we used this preprocessing step to train the ResNet models. While we focused mainly on the methodology of using synthetic crystals to extract structural information from powder diffractograms, we also show some initial steps toward applying our methods to experimental data. We used the publicly available RRUFF mineral database [24] which provides experimental measurements, including powder diffractograms. In order to imitate experimental diffractograms, we added Gaussian additive and multiplicative noise and a background function based on samples from a Gaussian process to our simulated diffractograms. Furthermore, we added a small amount of an impurity phase to each diffractogram. Details about the experimental data generation protocol can be found in the SI Section S3. Using the ResNet-50 model, we performed two experiments for experimental data, one with the mentioned impurity phase, and one without. ## 3 Results and discussion ### Synthetic distribution We first present an analysis of the generated synthetic crystals. Figure 3 shows some randomly selected ICSD and synthetic crystal structures side-by-side. Visually, the crystals appear very similar. However, no physical or chemical considerations regarding stability, clashing atoms, and element combinations are taken into account in the generation of synthetic crystals. As discussed earlier, our goal is to demonstrate that this is not problematic when using these crystals for the extraction of structural information from powder diffractograms. On the contrary, we expect the synthetic crystals to be a better basis for generalization to fundamentally new crystal structures than existing finite databases. To compare the distribution of ICSD crystals with the synthetic distribution, we evaluated structural descriptors, i.e. density factors, crystallite sizes, unit cell volumes, and numbers of atoms in the asymmetric unit, and compare their histograms in Figure 4. One can see that the overall distributions of the synthetic and ICSD crystals are very similar for all four descriptors. This shows that our chosen generation algorithm reproduces crystals that are similar to ICSD crystals in terms of these more general descriptors. ### Classification results The main results of our experiments (see Section 2.5) to classify the space group of powder diffractograms can be found in Table 1. The goal of our experiments is to systematically analyse and quantify the changes in classification accuracy introduced by our two main contributions: A more challenging dataset split, and training on continuously generated synthetic data. We started by repeating previously reported experiments trained directly using ICSD crystals with a random train-test split instead of the split based on structure types (similar to the original approach by Park _et al._[11]). This model achieved a very high test accuracy of 83.2%. We note that Park _et al._ removed data from the training dataset, "[...] heavily duplicated data [...]"[11], but did not specify the exact criterions used. In contrast, we did not exclude any duplicates in this experiment based on a random train-test split. Furthermore, as discussed in Section 2.1, we excluded crystals with a very high unit cell volume and a very high number of atoms in the asymmetric unit. This is likely the reason for the slightly higher classification accuracy that we observed, compared to the originally reported 81.1%. When splitting randomly, the model merely needs to recognize structures or structure types and assign the correct space group. This task is much easier than actually extracting the space group using more general patterns. When going from random splits to structure type-based splits (see Section 2.4), it becomes obvious that both the "parkCNN big" as well as the "parkCNN medium" models overfit the training data and do not generalize well to unseen structure types in the test set. Training the same model, in particular the "parkCNN big" model, on synthetic crystals leads to a 1.6 percentage points higher test accuracy than the "parkCNN big" model trained on ICSD diffractograms. At the same time, the training accuracy drops from 87.2% to 74.2% indicating that the model is now limited more by missing capacity rather than by overfitting, which is why we explored larger models, which will be discussed later. The gap between training and test accuracy is 31.1 percentage points when training on ICSD data, while for training using synthetic crystals, the gap is only 16.5 percentage points. We note that this gap between training using synthetic crystals and testing using ICSD crystals cannot stem from overfitting, since no diffractograms are repeated for the synthetic training. The difference rather stems from the differences between the synthetic distribution and the ICSD distribution of crystals. While the "parkCNN big" model trained on synthetic crystals outperforms the approach of training directly on ICSD crystals by only 1.6 percentage points, the advantage of training on an infinite stream of synthetic data increases when using models with more parameters and thus higher capacity. In contrast to training directly on a finite set of ICSD crystals, it is possible to train very large models using the infinite synthetic data stream without the potential of overfitting. As found in the last lines of Table 1, ResNet-10, ResNet-50, and ResNet-101 based models achieve Figure 3: a) Some randomly picked examples of ICSD crystals. b) Some randomly picked examples of synthetically generated crystals. While coordination and distances are not chemically correct for the synthetic crystals, crystal symmetries are reproduced correctly. \begin{table} \begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{}{Split} & Training & Testing & \multirow{2}{}{Model} & Number of & Training acc. & Test acc. \\ & dataset & dataset & & parameters & / \% & / \% \\ \hline Random & ICSD & ICSD & parkCNN medium & 4 246 797 & 88.4 & 83.2 \\ \hline Structure & \multirow{2}{}{ICSD} & \multirow{2}{}{ICSD} & parkCNN big & 4 959 585 & 87.2 & 56.1 \\ type & & & parkCNN medium & 4 246 797 & 90.9 & 55.9 \\ \hline \multirow{3}{}{Structure type1} & \multirow{3}{}{synthetic} & \multirow{3}{}{ICSD} & parkCNN big & 4 959 585 & 74.2 & 57.7 \\ & & & ResNet-10 & 9 395 025 & 87.2 & 73.4 \\ \cline{1-1} & & & ResNet-50 & 41 362 385 & 91.8 & 79.3 \\ \cline{1-1} & & & ResNet-101 & 60 354 513 & 92.2 & 79.9* \\ \hline \hline \end{tabular} \end{table} Table 1: Results of training on diffractograms simulated from ICSD crystals (random splits as well as structure type-based splits) compared to when training on diffractograms from synthetic crystals. Test accuracy in all cases refers to the accuracy when testing on the ICSD test dataset. The training accuracies are averaged over the last 10 epochs of the respective run. Experiments trained directly on ICSD data overfitted to the training data. Training longer would have further increased the training accuracy, while not increasing the test accuracy. Figure 4: Histograms comparing the distributions of descriptors of the synthetically generated crystals with the ICSD distribution in the test dataset. a) density factor $\frac{V_{\text{init.cell}}}{\sum_{4}4/3\pi\left(\frac{V_{\text{init.cell}}}{2} \right)^{3}}=\frac{V_{\text{init.cell}}}{V_{\text{atomic}}}$, b) crystallite sizes, c) unit cell volume (conventional cell settings), d) number of atoms in the asymmetric unit. The probability density of the ICSD is visualized by a stacked bar histogram, where the green portion of the bar was correctly classified and the red portion was incorrectly classified. The probability density of the synthetic crystals is visualized by the dark blue line. The portion between the dark blue line and the light blue line was correctly classified, the portion below the light blue line was incorrectly classified. The reported classification performance is based on the ResNet-101 model trained on diffractograms from synthetic crystals. ICSD test accuracies of 73.4%, 79.3%, and 79.9%. This is a significant increase from the 57.7% achieved by the "parkCNN big" model. Figure S4 in the SI further shows the top-$k$ accuracy over $k$ for the ResNet-101 model. With increasing $k$ the accuracy exceeds 95% at $k=5$. This means that our model can not only determine the correct space group with a high probability but can also generate an almost complete list of possible space group candidates. Figure 5 shows the ICSD test accuracy, the training accuracy (on synthetic data), and the ICSD top-5 test accuracy for all three ResNet variants as a function of epochs trained. For all three metrics, the difference between ResNet-50 and ResNet-101 is comparably small, while the step from ResNet-10 to ResNet-50 is substantial (5.9 percentage points in ICSD test accuracy, see Table 1). This shows that going beyond the model size of the ResNet-101 will likely not yield a big improvement in accuracy. In contrast to the 79.9% accuracy reached in the top-1 ICSD test accuracy, the top-5 ICSD test accuracy of the ResNet-101 model reaches 96%. However, for all three ResNet variants, a gap between training using synthetic crystals and testing using the ICSD remains (12.3 percentage points for ResNet-101). As also shown in Figure S5 in the SI, the accuracy convergence can be approximately described by a power law, indicating that exponentially more training time will substantially reduce classification errors and thus potentially lead to top-1 accuracies of 90% and above, at the cost of a 100-fold increase in training times, which is currently infeasible. The histograms in Figure 4 show, next to the overall distribution, also the fraction of diffractograms classified wrongly for testing on the ICSD (red bar) and on the synthetic data (below the light blue line) for the ResNet-101 model. First, one can see that throughout almost all regions of the distributions, the accuracy on the synthetic data is slightly higher than that on the ICSD. This is related to the aforementioned gap of 12.3 percentage points between train and test accuracy and can be attributed to differences between the synthetic and ICSD distribution of crystals. This will be discussed in detail in the next section. It is surprising to see that the dependence on crystallite sizes is rather weak, as smaller crystallite sizes result in broader peaks (see Scherrer equation, Eq. 1), potentially making the classification harder due to more peak overlaps. In summary, the maximum ICSD test accuracy of 79.9% that we achieved using the ResNet-101 model almost reaches the 81.14% that Park _et al._[11] reported for the space group classification. However, our accuracy is based on a train-test split based on structure types, in contrast to the random split used by Park _et al._. This creates a much harder but also realistic task to solve since the model needs to generalize to other structure types without merely recognizing diffractograms or structure types that it has already seen during training. This becomes especially apparent from our experiment directly trained on diffractograms from ICSD crystals with the split based on structure types, which reached only 56.1% instead of the 81.14% reported by Park _et al._[11]. Figure 5: Test accuracy (ICSD), training accuracy (synthetic crystals), and test top-5 accuracy (ICSD) as a function of epochs. We show all three metrics for the models ResNet-101, ResNet-50, and ResNet-10. To better show the scaling behavior, both axes use logarithmic scaling. To better see the exponential behaviour, see Figure S5 in the SI. ### Experimental results To go beyond simulated diffractograms, we trained ResNet-50 models on calculated diffractograms with background, noise, and impurities and applied the trained models to the RRUFF mineral database. Our results (see Figure S3 in the SI) show that it is essential to include impurity phases in the training data. By doing so, we obtain a top-1 accuracy of 25.2% and a top-10 accuracy of over 60%. This is of high practical relevance since having a short list of potential space groups is often sufficient as a first step to further refinement and analysis. Vecsei _et al._ performed similar experiments of space group classification on the same database. Using an ensemble of 10 fully connected neural networks, they reached a classification accuracy of 54%[14]. While our obtained accuracy is significantly lower than the one reported by Vecsei _et al._, our approach is much more general: The training dataset used by Vecsei _et al._ was based on simulated diffractograms of structures of the ICSD, which contains almost all RRUFF structures, leading to high similarities of training and testing data. Therefore, the model needed to simply recognize the minerals, instead of directly inferring the space group using the symmetry elements - as our method needs to do. We want to emphasize that our efforts to apply the methodology to experimental data are only preliminary. We expect improved results with an improved data generation protocol since the procedure contains many parameters to be tuned. Ideally, one would use a generative machine learning approach to add the experimental effects (noise, background, impurities) to the pure diffractograms. We also want to point out that the noise level and quality of data in the RRUFF dataset are limited. Application of the presented methodology on other experimental datasets is desirable. We will address these points in future work, where we focus on improved ways of modeling experimental imperfections. As discussed above, for the classification of pure diffractograms we observed the ResNet-50 to have the best cost-benefit ratio, since the ResNet-101 yielded only slight improvements. For the more complicated problem of classifying diffractograms with experimental imperfections, bigger models and longer training times might be necessary. ### Differences between synthetic crystals and ICSD crystals We showed that training directly on crystals from the ICSD yields a gap between the training and test accuracy due to overfitting. The training on the synthetic dataset also shows a gap between the training and test accuracy (see Table 1), but it is smaller than when training directly on ICSD crystals. Furthermore, this gap is not due to overfitting, since overfitting to singular diffractograms is not possible when the model is trained using an infinite stream of generated synthetic crystals. The gap rather stems from systematic differences between the synthetic and ICSD distribution of crystals. To analyze those differences, we created three modifications of the ICSD test dataset. In the first modification, the fractional coordinates of the atoms in the asymmetric unit of the crystals of the ICSD test dataset were randomly uniformly resampled (as in the synthetic crystal generation algorithm). In the second modification, the lattice parameters were randomized following the KDE used in the synthetic generation algorithm. The third modification combines both previous modifications, i.e. both the coordinates and the lattice parameters were resampled. These three modified test datasets bring the ICSD test dataset closer to the distribution used for training and let us quantify which factors contribute to the gap between training and testing. We evaluated the test accuracies on these datasets for the experiment using the ResNet-101 model trained using synthetic crystals. We found that randomizing the coordinates yields an increase in test accuracy of 5.96 percentage points. Randomizing the lattice parameters results in an increase of 1.31 percentage points. Randomizing both the coordinates and the lattice parameters leads to an increase of 6.32 percentage points, explaining approximately half of the gap of 12.3 percentage points between synthetic training and ICSD test accuracy. We also tested the ResNet-101 model on diffractograms simulated based on the statistics dataset from which the statistics for our generation algorithm have been extracted. This returns a 4.33 percentage points higher accuracy than on the test dataset. This is due to the different structure types present in the test and statistics dataset yielding different occupation probabilities (potentially also 0) for Wyckoff positions and overall slightly different statistics for the synthetic generation algorithm. To summarize, the contributions to the gap stem from differences in the distribution of lattice parameters and coordinates with 6.32 percentage points and from the differences between the statistics dataset and test dataset with 4.33 percentage points. Added together (assuming the effects of geometry and Wyckoff position occupation probabilities are unrelated and thus additive), the contributions almost completely explain the gap of 12.3 percentage points between training and test accuracy. The remaining part is likely due to our algorithm that places atoms on Wyckoff positions not reproducing the ICSD distribution exactly. However, the remaining difference is remarkably small. In Figure 6 we show the test classification error in each bin for the unit cell volume and the number of atoms in the asymmetric unit using the ResNet-101 model trained on diffractograms of synthetic crystals. The classification error is shown both for testing on diffractograms from synthetic crystals and on ICSD diffractograms. One can see that for small volumes and a small number of atoms in the asymmetric unit, the difference between classifying ICSD diffractograms and diffractograms from synthetic crystals is relatively small. As the volume and number of atoms in the asymmetric unit increase, the gap between the two errors increases, too. We already identified the uniformly sampled atom coordinates in the synthetic distribution as the main contributor to the gap in accuracy between the synthetic crystals and ICSD crystals. Therefore, it seems that the uniform sampling of atom coordinates works well for small number of atoms in the asymmetric unit and small volumes, while the error due to this sampling strategy increases slightly for higher volumes and higher number of atoms in the asymmetric unit. When looking at the distribution of crystals in the ICSD, the number of atoms in the asymmetric unit tends to be larger for lower-symmetry space groups (for example, in the triclinic crystal system) than for higher-symmetry space groups such as those from the cubic crystal system. Therefore, the increasing test error on diffractograms from ICSD crystals with a higher number of atoms in the asymmetric unit is especially relevant for these lower-symmetry space groups. It might be possible that a different scheme of generating atom positions in the unit cell (compared to the independent uniform sampling that we used) works better for a high number of atoms in the asymmetric unit. Overall, it is important to note that the distribution of ICSD crystals is (apart from a few Wyckoff position occupation probabilities which are exactly zero in the statistics dataset1) almost completely encompassed by the much larger distribution of synthetic crystals that we used for Figure 6: Classification performance for each bin of a) the unit cell volume (conventional cell settings) and b) the number of atoms in the asymmetric unit. The classification performance when testing on the ICSD is shown using a stacked bar plot, where the red bar indicates the classification error and the green bar indicates the correctly classified portion of each bin. When testing on the synthetic distribution, the classification error is given by the light blue line, the rest (between dark blue and light blue line) was correctly classified. The reported classification performance is based on the ResNet-101 model trained on diffractograms from synthetic crystals. This visualization clearly shows the error rate within each bin, in contrast to Figure 4, which additionally includes the relative proportion of the crystals of the respective bin to the total amount of crystals. training. However, due to finite training times and model capacity, a performance gap remains. This gap can be improved by using (substantially) more computing power or by narrowing the very general synthetic distribution, e.g., by using a different algorithm to generate atom positions. This indicates an inherent challenge in XRD classification but more generally in materials property prediction: Machine learning models are ultimately trained to be employed in real-world tasks, which are typically related to novel, i.e. yet unseen materials and structures. At the same time, the machine learning models are tested based on an IID assumption, i.e. the assumption that the distribution of training and testing data is the same. While not being a contradiction in the limit of infinite training data and model capacity, this becomes an (unsolvable) challenge in reality, when facing finite datasets and models. In our case, our model trained on a large distribution of synthetic crystal structures will likely generalize better to completely new crystal structures different from any crystal structure contained in the ICSD database. At the same time, it suffers from smaller ICSD test set errors, even though the ICSD distribution is contained in the synthetic data generation distribution. ## 4 Conclusion We developed an algorithm based on the symmetry operations of the space groups to generate synthetic crystals that follow the distribution found in the ICSD database in terms of general descriptors like volume, density, or types of elements. The generated crystals have randomly sampled coordinates and span a wide range of structure types, many of which do not appear in the ICSD. We showed that, compared to using ICSD crystals directly, simulating the training data based on the synthetic crystals can improve the performance of tasks that extract structural information from powder diffractograms, in this case, the space group. Our results verify our hypothesis that crystals forming the training dataset do not need to be stable or chemically viable. We trained on an infinite on-the-fly generated stream of synthetic crystals and simulated batches of diffractograms using a distributed framework based on the _Python_ library _Ray_[25]. This allows the training of very large networks without overfitting. The best-performing model (ResNet-101) reached a space group classification accuracy of 79.9% vs. 56.1% when training on ICSD structures directly. By performing the train-test split using the structure type, we forced our models to not just recognize structure types or individual structures, but to actually learn rules to distinguish different space groups by their symmetry elements. This shows the true generalization capabilities to new structure types and novel classes of materials. We also demonstrated first steps toward applying the presented methodology to an experimental dataset. We expect further improvements in this area using improved models of experimental imperfections, as well as larger ML models and longer training times. Even though models trained on the synthetic distribution transfer well when tested on ICSD crystals, we found a gap of 12.3 percentage points (ResNet-101) between the training accuracy on synthetic crystals and test accuracy on the ICSD. We showed that the main contribution to this gap stems from the independently uniformly sampled atom coordinates. An improved approach may be needed to artificially generate more ordered structures, which contain more ordered diffraction planes than a cloud of uniformly sampled points. This might be especially important for crystals with a high number of atoms in the asymmetric unit. Lastly, the developed algorithm to synthetically generate crystals can be used for other XRD-related tasks in the future, such as the extraction of crystallite sizes, lattice parameters, information about the occupation of Wyckoff positions, etc. Furthermore, instead of generating completely random crystals of all space groups, one can also generate random crystals of given structure types to solve more specialized tasks. This would allow the use of very large models for tasks that are typically strongly limited by the dataset size when using only the entries of the ICSD. Also, tasks concerning multi-phase diffractograms or augmentations such as strain in given crystal structures can benefit from our batch-wise online learning approach. ### Data availability The ICSD data used to evaluate the models belongs to FIZ Karlsruhe, from which academic and non-academic licenses are available. ### Code availability The source code of all machine learning models, of the generation of synthetic crystals, of the optimized simulation of diffractograms, and of the distributed computing architecture can be found on [https://github.com/aimat-lab/ML4pXRDs](https://github.com/aimat-lab/ML4pXRDs). The used machine learning models are further discussed in detail in the Supplementary Information. ### Acknowledgements P.F. acknowledges support by the Federal Ministry of Education and Research (BMBF) under Grant No. 01DM21001B (German-Canadian Materials Acceleration Center). H.S. acknowledges financial support by the German Research Foundation (DFG) through the Research Training Group 2450 "Tailored Scale-Bridging Approaches to Computational Nanoscience". The authors acknowledge support by the state of Baden-Wurttemberg through bwHPC. Parts of this work were performed on the HoreKa supercomputer funded by the Ministry of Science, Research and the Arts Baden-Wurttemberg and by the Federal Ministry of Education and Research. ### Author contributions All authors contributed to the idea and the preparation of the manuscript. H.S. implemented the methods and conducted the computational experiments. ### Competing interests The authors declare no competing interests. ## References * [1] Radovic, A., Williams, M., Rousseau, D., Kagan, M., Bonacorsi, D., Himmel, A., Aurisano, A., Terao, K. & Wongjirad, T. Machine Learning at the Energy and Intensity Frontiers of Particle Physics. _Nature_560, 41-48. issn: 1476-4687. doi:10.1038/s41586-018-0361-2. 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I. & Stoica, I. _Ray: A Distributed Framework for Emerging AI Applications_ in. 13th USENIX Symposium on Operating Systems Design and Implementation (OSDI 18) (2018), 561-577. isbn: 978-1-939133-08-3. [https://www.usenix.org/conference/osdi18/presentation/moritz](https://www.usenix.org/conference/osdi18/presentation/moritz) (2022-09-13). * [26] Fredericks, S., Parrish, K., Sayre, D. & Zhu, Q. PyXtal: A Python Library for Crystal Structure Generation and Symmetry Analysis. _Computer Physics Communications_261, 107810. issn: 0010-4655. doi:10.1016/j.cpc.2020.107810. (2022-06-27) (2021-04-01). * [27] Sun, Y., Wong, A. K. C. & Kamel, M. S. Classification of Imbalanced Data: A Review. _Int. J. Patt. Recogn. Artif. Intell._23, 687-719. issn: 0218-0014. doi:10.1142/S0218001409007326. (2022-06-24) (2009-06). * [28] Ong, S. P., Richards, W. D., Jain, A., Hautier, G., Kocher, M., Cholia, S., Gunter, D., Chevrier, V. L., Persson, K. A. & Ceder, G. Python Materials Genomics (Pymatgen): A Robust, Open-Source Python Library for Materials Analysis. _Computational Materials Science_68, 314-319. issn: 0927-0256. doi:10.1016/j.commatsci.2012.10.028. (2022-06-21) (2013-02-01). * [29] Lam, S. K., Pitrou, A. & Seibert, S. _Numba: A LLVM-based Python JIT Compiler_ in _Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC_ (Association for Computing Machinery, New York, NY, USA, 2015-11-15), 1-6. isbn: 978-1-4503-4005-2. doi:10.1145/2833157.2833162. (2022-09-13). * [30]_International Tables for Crystallography Volume H: Powder Diffraction_ 1st ed. (eds Gilmore, C. J., Kaduk, J. A. & Schenk, H.) isbn: 978-1-118-41628-0 (Wiley, 2019-09-24). * [31] He, K., Zhang, X., Ren, S. & Sun, J. _Deep Residual Learning for Image Recognition_ in _2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016-06), 770-778. doi:10.1109/CVPR.2016.90. Supplementary Information to Neural networks trained on synthetically generated crystals can extract structural information from ICSD powder X-ray diffractograms Henrik Schopmans, Patrick Reiser, and Pascal Friederich* ## S1 Generating synthetic crystals Here we describe the algorithm to generate synthetic crystals in more detail. To generate a single crystal, the following steps are executed: 1. Random selection of a space group. We follow the space group distribution of the ICSD to allow comparison with previous work (see Section 2.1 of the paper). 2. The number of unique elements in the crystal is drawn from a discrete distribution extracted from the crystals in the ICSD belonging to the specified space group. 3. The unique elements are drawn, also from a discrete probability distribution from the crystals in the ICSD belonging to the specified space group. 4. For each of the unique elements, the number of repetitions in the asymmetric unit is chosen, and for each repetition, a Wyckoff position is randomly selected. Again, both the probability of the number of repetitions and the Wyckoff occupation probabilities are extracted from the ICSD for the specified space group. We do not place more than one atom onto a Wyckoff position that does not have a degree of freedom. 5. For each atom placed on a Wyckoff position, uniformly distributed random fractional coordinates are drawn. 6. Lattice parameters (normalized to unity volume) of the crystal system that the specified space group belongs to are drawn from a kernel density estimate (KDE) of the ICSD. The bandwidth is chosen based on Scott's rule (see the _SciPy_1 implementation of KDE). Footnote 1: For this conditional kernel density estimate, we used the implementation of _statsmodels_2 with the normal reference rule of thumb to estimate the bandwidth. 7. We generated a KDE of the volume conditioned1 on $\sum_{i}4/3\pi\left(\frac{r_{i\mathrm{cov}}+r_{i\mathrm{VdW}}}{2}\right)^{3} =V_{\mathrm{atomic}}$, where the sum covers all atoms in the conventional unit cell, $r_{i\mathrm{cov}}$ is the atomic covalent radius, and $r_{i\mathrm{;VdW}}$ is the atomic van der Waals radius. The KDE was generated from all crystals of the ICSD belonging to the specified space group. Then, $V_{\mathrm{atomic}}$ is calculated for the chosen atoms in the conventional unit cell and the volume is drawn based on the KDE conditioned on $V_{\mathrm{atomic}}$. The lattice parameters (chosen in the previous step) are further scaled by the cube root of the chosen volume. Footnote 1: For this conditional kernel density estimate, we used the implementation of _statsmodels_3. 8. Space group symmetry operations are applied using _Python_ library _PyXtal_3. Footnote 3: Reference rule of thumb to estimate the bandwidth. When generating a crystal of a specific space group without placing an atom on the general Wyckoff position, it is not always the case that the crystal belongs to that space group. To prevent wrong space group assignments, we use the _Pymatgen_4 interface to _Spglib_5 to check the space group of each crystal after its generation. If the space group deviates, we generate a new crystal with the same number of unique elements as before, in order to not distort the distribution of number of unique elements extracted from the ICSD. If the generation fails 20 times in total, we start from the beginning with a new number of unique elements. Machine learning ### Models We now want to describe the machine learning models that we used for the classification of space groups in more detail. Powder diffractograms include similar features (peaks) at different locations and the position of a feature in the diffractogram has a spatial meaning. This suggests that the properties of the convolution operation, namely the parameter sharing (with sparse connectivity) and equivariance[6], might be beneficial when processing powder diffractograms. As a baseline, we first used the two CNN architectures used by Park _et al.[7]_ for the classification of extinction groups and space groups. Since our training dataset is an infinite stream of diffractograms and we do not have to worry about overfitting, we further used the deeper architectures ResNet-10, ResNet-50, and ResNet-101[8]. All architectures are now described in detail. #### Architectures by Park _et al._ Park _et al.[7]_ introduced three models, one for the classification of the crystal system, one for extinction groups, and one for space groups. We used only the last two models and call them "parkCNN medium" and "parkCNN big", respectively. "parkCNN big" consists of three convolution layers with average pooling, two hidden fully connected layers with 2300 and 1150 nodes, and a 145-dimensional softmax output. The architecture is summarized in Figure S1. The "parkCNN medium" model has fewer parameters than "parkCNN big" since the two hidden fully connected layers have 4040 and 202 nodes. #### ResNet architecture With increasing size of the training dataset and increasing difficulty of the chosen task, the number of model parameters needs to be increased, too. In principle, a deeper model with additional layers should always be able to express the same solution of a shallower model by simply "learning" an identity map in addition to the shallower model. In practice, however, a degradation problem for CNNs with increasing depth has been observed and very deep models can perform worse than their shallow counterpart[8]. Therefore, the ResNet architecture developed by He _et al.[8]_ at Microsoft in 2015 introduced additional skip connections, where information is able to simply flow past the convolution layer and is added to its output. This makes it possible for needed information of the input or earlier layers to flow further into the model without degradation. Figure S2a visualizes the residual block used for the shallower versions of the ResNet architecture (up to 34 layers). Figure S2b visualizes the bottleneck block used for the deeper variants (50 and more layers). This type of building block is called a bottleneck block since it first reduces the number of channels using a 1x1 convolution operation with N filters. Then, the main convolution with a 3x3 kernel is performed, followed by a third 1x1 convolution that upscales to 4N channels. This down- and upscaling of the number of channels is performed to increase the performance of the model. All convolution operations of both types of building blocks are followed by batch normalization implicitly. In the simplest case, the skip connection of the residual and bottleneck block is simply an identity mapping and added to the output of the block. However, if the number of input channels and dimensions of a block are different from those in the output, a projection in the form of a 1x1 convolution with the necessary number of filters and stride (usually 2) is used instead of the identity. Table S1 summarizes the ResNet-10, ResNet-50, and ResNet-101 models. Each architecture is to be read from top to bottom. The square brackets indicate a residual or bottleneck block with their two or three convolution operations and respective number of filters. The number after the square brackets "$\times$_N_" indicates how often the building block is to be repeated in the respective block group. If the output dimension changes from one block group to the next, the first building block of the next block group downsamples the dimensions by using a stride of 2 for the first 3x3 convolution in the case of a residual block and for the middle 3x3 convolution in the case of a bottleneck block. All other convolution operations of the building blocks are performed with stride 1. We used the ResNet implementation as found in the _TensorFlow_ model garden[9]. Since our data is one-dimensional, we used an adapted 1D version. We replaced all 2D convolutions and pooling operations with their 1D equivalent ($N\times N\to N$). Furthermore, we used a kernel size of 9 in place of the 3x3 kernels and stride 4 instead of stride 2 in the bottleneck blocks and projection skip connections ($N\times N\to N^{2}$). This squaring was performed to obtain a better distribution of the number of weights throughout the architecture (similar to the original 2D case). We further added an additional fully connected layer with 256 nodes after the flatten layer in the end of the ResNet models, followed by the output layer. We were not able to achieve good results when using the original ResNet architecture with batch normalization (similar observations were made by Schuetzke _et al.[10]_). The test accuracy calculated after each epoch was highly unstable and had high fluctuations, probably in part due to the moving averages of the batch normalization not converging properly. This is possibly caused by using an infinite stream of batches of diffractograms, instead of using a training dataset of fixed size. We fixed this problem by using group normalization[11] with 32 groups instead of batch normalization. ### Setup We used a distributed architecture on multiple nodes using the _Python_ framework _Ray_12. The configuration utilized throughout this study is visualized in Figure 2b of our paper. Training took place on a _Ray_ head node with one or two2 RTX 2080 Ti GPUs. Training on two GPUs was performed using the _MirroredStrategy_ of _TensorFlow_13. 28 out of the 32 cores of the head node were used for the generation of diffractograms. Next to the head node, we used two additional compute nodes with 128 cores each to generate diffractograms. Communication between the nodes took place using a _Ray queue_ object to access the simulated diffractograms. Footnote 1: The “parkCNN” models used a single GPU, while the ResNet models were trained on two GPUs. Depending on the model size and corresponding training speed, this setup allows training with up to millions of unique diffractograms per hour. For the larger ResNet variants ResNet-50 and ResNet-101, we efficiently used the GPUs. For the "parkCNN" and ResNet-10 models, training is fast and the data generation becomes the bottleneck. To train all models, we used _Keras_14 with _TensorFlow_ 2.313. Optimization was performed using _Adam_15 with $\beta_{1}=0.9$ and $\beta_{2}=0.999$ (_Keras_ default parameters). We also tried to use stochastic gradient descent (SGD) with momentum and a stepwise learning rate scheduler, but this did not yield good results. Depending on the initial conditions, most of the training runs using SGD were either unstable or reached low accuracies. For all experiments, a cross-entropy loss was utilized. Footnote 13: Note that the division into epochs for training using synthetic crystals is rather arbitrary since each epoch contains different diffractograms simulated from different crystals. However, the division makes it easy to calculate the performance on the test dataset after each epoch. We used a batch size of 870 for the "parkCNN" models and a batch size of 145 for the three ResNet models. Furthermore, the "parkCNN" models were trained for 1000 epochs with a learning rate of 0.001, while the ResNet models were trained for 2000 epochs with a learning rate of 0.0001. The ResNet models used a step decay of the initial learning rate, halving the learning rate after every 500 epochs. When training on diffractograms simulated from synthetic crystals, we used 150 generated batches per epoch when a batch size of 870 was used ("parkCNN medium" and "parkCNN big") and 900 batches per epoch when a batch size of 145 was used (ResNet). This means that each epoch always contained 130 500 diffractograms1. Footnote 1: Note that the division into epochs for training using synthetic crystals is rather arbitrary since each epoch contains different diffractograms simulated from different crystals. However, the division makes it easy to calculate the performance on the test dataset after each epoch. For the experiments performed directly on diffractograms simulated from ICSD crystals, we used the statistics dataset directly to pre-generate the training dataset. We excluded the same 85 space groups that were not used by the synthetic training due to missing statistics. We used two different crystallite sizes for each crystal in the statistics dataset, yielding $148\,466\cdot 2=296\,932$ diffractograms in the training dataset. ## S3 Application to experimental diffractograms ### Dataset To test the performance on experimental diffractograms, we used the publicly available RRUFF mineral database16. It contains 5829 mineral samples with multiple types of measured spectra and, most important for us, powder XRD measurements with K${}_{\alpha}$ radiation for 2952 samples. Of these 2952 samples, 2875 samples provide the output of a Rietveld refinement, including the space group label. We further removed by hand some samples that had excessive amounts of noise and were of bad quality. Many samples further only provide a simulated diffractogram without noise or background. They were also excluded. This left 942 diffractograms for our analysis. Footnote 16: Note that the division into epochs for training using synthetic crystals is rather arbitrary since each epoch contains different diffractograms simulated from different crystals. However, the division makes it easy to calculate the performance on the test dataset after each epoch. ### Data generation To be able to apply our models to experimental diffractograms, we added an additional background function and noise to the generated diffractograms to make them more similar to experimental data. We used a Gaussian process to generate random background functions and added additive and multiplicative Gaussian noise. All diffractograms were generated in the range $2\theta\in[5,90]$ * with step size $0.01^{*}$. The generation protocol is as follows: 1. Sample the background function from a Gaussian process with radial basis function kernel without any conditioning: \[k(x_{i},x_{j})=a\exp\left(-\frac{\|x_{i}-x_{j}\|^{2}}{2l^{2}}\right)\] (1) We chose $a=1$ and sampled $l$ uniformly in [7, 40] for each diffractogram. 2. Subtract the minimum intensity from the background function obtained from the Gaussian process. 3. Divide it by the sum of the 8501 ($2\theta$-range) entries. 4. Multiply it by a scaling factor drawn from a truncated normal distribution in the range $[0,150]$ with mean $0$ and standard deviation $75$. 5. Add the pure diffractogram (intensity-range $[0,1.0]$). 6. Add Gaussian additive noise with mean $0$ and standard deviation uniformly drawn in $[0,0.02]$. 7. Multiply with multiplicative Gaussian noise with mean $1$ and standard deviation uniformly drawn in $[0,0.16]$. To resemble a more realistic peak profile, we used the pseudo-Voigt profile instead of the Gaussian profile that we used for the classification of pure diffractograms. The pseudo-Voigt profile uses the full width at half maximum (FWHM) $\Gamma_{\mathrm{G}}$ of the Gaussian $G$, the FWHM $\Gamma_{\mathrm{L}}$ of the Lorentzian $L$, and a mixing parameter $\eta$ as parameters[17]: \[\mathrm{PV}\left(\theta-\theta_{0};\Gamma_{\mathrm{L}},\Gamma_{\mathrm{G}} \right)=\eta G\left(\theta-\theta_{0};\Gamma_{\mathrm{G}}\right)+(1-\eta)L \left(\theta-\theta_{0};\Gamma_{\mathrm{L}}\right) \tag{2}\] with \[G(\theta-\theta_{0};\sigma=\frac{\Gamma_{\mathrm{G}}}{2\sqrt{2\ln 2}})= \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{\theta-\theta_{0}}{ \sigma}\right)^{2}} \tag{3}\] and \[L(\theta-\theta_{0};\gamma=\frac{1}{2}\Gamma_{\mathrm{L}})= \frac{1}{\pi\gamma}\left[\frac{\gamma^{2}}{\left(\theta-\theta_{0}\right)^{2 }+\gamma^{2}}\right]\,. \tag{4}\] The FWHM $\Gamma_{\mathrm{G}}$ of the Gaussian is typically parametrized using the Caglioti equation[17] as \[\Gamma_{\mathrm{G}}^{2}=U\tan^{2}\theta+V\tan\theta+W\,. \tag{5}\] Following the suggestions for typical Rietveld parameter ranges by Kaduk & Reid[18] and comparing the resulting peaks with the ones from the RRUFF dataset, we decided to sample the Caglioti parameters uniformly in the following ranges: $[0,0.01]$ for U, $[0,0.01]$ for W, and V was fixed at $V=0.0$. We further used a single mixing parameter $\eta$ uniformly sampled in $[0.0,1.0]$ for the full $2\theta$-range. For simplicity, we used the same FWHM from the Caglioti equation for both the Gaussian and Lorentzian of the pseudo-Voigt. We further considered the K${}_{\alpha_{1}}$ / K${}_{\alpha_{2}}$ splitting of the K${}_{\alpha}$ line since for some diffractograms in the RRUFF dataset, this splitting is visible. We further implemented the option to add impurity phases to the training (and simulated ICSD test) data. The minerals of the RRUFF database are not all made up of one phase, but most of them contain small amounts of one or more impurity phases. To model this, for each training diffractogram, we used a superposition of the main phase to be classified and an impurity phase of a random space group ($a$ is uniformly sampled in $[0,0.05]$): \[I(\theta)=(1-a)I_{\mathrm{pure}}+aI_{\mathrm{impurity}} \tag{6}\] ### Experiments For our experiments on experimental data, we used the same split based on structure types as we used for pure diffractograms. We performed two experiments using the ResNet-50 architecture, one with impurity phases and one without. For both, a learning rate of $0.0001$, a batch size of $145$, and $1000$ epochs where used. ## S4 Additional figures
2303.16690	Graph Neural Networks for Hardware Vulnerability Analysis -- Can you Trust your GNN?	The participation of third-party entities in the globalized semiconductor supply chain introduces potential security vulnerabilities, such as intellectual property piracy and hardware Trojan (HT) insertion. Graph neural networks (GNNs) have been employed to address various hardware security threats, owing to their superior performance on graph-structured data, such as circuits. However, GNNs are also susceptible to attacks. This work examines the use of GNNs for detecting hardware threats like HTs and their vulnerability to attacks. We present BadGNN, a backdoor attack on GNNs that can hide HTs and evade detection with a 100% success rate through minor circuit perturbations. Our findings highlight the need for further investigation into the security and robustness of GNNs before they can be safely used in security-critical applications.	Lilas Alrahis, Ozgur Sinanoglu	2023-03-29T13:39:13Z	http://arxiv.org/abs/2303.16690v1	# Graph Neural Networks for Hardware Vulnerability Analysis-- Can you Trust your GNN? ###### Abstract The participation of third-party entities in the globalized semiconductor supply chain introduces potential security vulnerabilities, such as intellectual property piracy and hardware Trojan (HT) insertion. Graph neural networks (GNNs) have been employed to address various hardware security threats, owing to their superior performance on graph-structured data, such as circuits. However, GNNs are also susceptible to attacks. This work examines the use of GNNs for detecting hardware threats like HTs and their vulnerability to attacks. We present _BadGNN_, a backdoor attack on GNNs that can hide HTs and evade detection with a 100% success rate through minor circuit perturbations. Our findings highlight the need for further investigation into the security and robustness of GNNs before they can be safely used in security-critical applications. Graph neural networks, Hardware security, Hardware Trojans, Intellectual property, Backdoor attacks ## I Introduction Graph neural networks (GNNs) have become increasingly popular due to their ability to operate on graph-structured data and their success in various applications, including natural language processing, social network analysis, and recommendation systems [1]. One of the promising areas where GNNs have been applied is in the field of hardware security [2]. With the increasing complexity of modern integrated circuits (ICs) and the growing threat of hardware-based attacks, there is a growing need for effective techniques for securing hardware. GNNs provide a powerful tool for modeling and analyzing the behavior of circuits, enabling the detection and prevention of security threats [3]. Specifically, GNNs have been used to analyze the structure and connectivity of circuits, identifying potential hardware Trojans (HTs) [4, 5, 6], detecting intellectual property (IP) piracy [7], performing reverse-engineering [8, 9] and attacking logic locking [10, 11, 12, 13, 14]. While GNNs are powerful tools for modeling and analyzing complex graph-structured data, they are also susceptible to various security threats, including _adversarial attacks_ and _data poisoning attacks_[15]. Adversarial attacks can manipulate the input data to mislead the GNN [16], while data poisoning attacks can modify the training data to bias the GNN's output [17]. Fig. 1 illustrates the danger of _backdoor attacks_ on GNNs, which are a type of poisoning attack, in the context of hardware security. When a GNN model is backdoored, it can incorrectly classify Trojan-injected circuits as Trojan-free after certain targeted circuit perturbations have been added. Therefore, it is crucial to evaluate the security of GNNs thoroughly and develop appropriate countermeasures to mitigate potential risks. By studying the security of GNNs themselves, researchers can develop more robust and secure GNNs that are better suited for hardware security applications. In this work, we examine the intersection of two critical topics: (i) the use of GNNs for HT detection, and (ii) the security threats against GNNs themselves. Specifically, we present _BadGNN_, a backdoor attack on GNNs that can hide HTs and evade detection (with a 100% success rate) via minor circuit perturbations. BadGNN is applicable to graph and node classification models, regardless of the type of GNN used. ## II Background and Related Work ### _Graph Neural Networks (GNNs)_ _Definition 1 (Graph)._ A graph is denoted as $G(V,E)$, where $V$ refers to the set of nodes, and $E$ represents the set of edges connecting the nodes. Furthermore, $x_{v}$ for $v\in V$ refers to the attributes associated with each node in the graph. In other words, $G$ encompasses both the graph's connectivity (i.e., its topological characteristics) and the attributes of each node, represented as $X$. $A$ denotes the adjacency matrix of $G$. _Definition 2 Graph Classification is to categorize a collection of graphs into their corresponding predetermined classes. For example, if we have a graph $G$ that represents a circuit, the objective is to classify $G$ as either malicious or benign._ GNNs use the characteristics of a graph's structure and node attributes to produce a representation (referred to as an "embedding"), denoted as $z_{G}$, which aids in determining the graph's class. To accomplish this, a GNN generates an embedding, $z_{v}$, for each node in the graph. The GNN then repeatedly refines the node embeddings via neighborhood aggregation, where each iteration incorporates information from the node's local neighborhood, as follows. Fig. 1: GNN backdoor attack in the context of hardware security. \[Z^{(l)}=\text{Aggregate}\left(A,Z^{(l-1)};\theta^{(l-1)}\right) \tag{1}\] At the $l$-th iteration, $Z^{(l)}$ is the matrix of node embeddings, while $\theta^{(l-1)}$ is a trainable weight matrix. The initial node features $X$ are represented as $Z^{(0)}$. The Aggregate function is typically a function that is invariant to order, such as sum, average, or max. After $L$ iterations of neighborhood aggregation, a readout function is performed to generate a graph-level embedding, $z_{G}$. In essence, a GNN is a function, $f_{\theta}$, that models the generation of $z_{G}=f_{\theta}(G)$ for a given graph $G$. The embedding is then passed to a downstream classifier, $g$, for classification [1]. ### _GNNs for Hardware Trojan (HT) Detection_ HTs are malicious hardware modifications intended to extract confidential information from ICs or disrupt their intended functionality. _GNN4T1_ is a GNN-based platform for HT detection that does not require prior knowledge of the design IP or HT structure [4]. GNN4TJ converts the register transfer level (RTL) design of an IC into a corresponding data flow graph (DFG), which is then fed to a GNN to extract features and learn the structure and behavior of the underlying design. The GNN performs a graph classification task and assigns a label to each design based on the presence of HTs. _TrojanSAINT_ is another recent GNN-based HT detection scheme that operates at the gate level and can perform both pre- and post-silicon detection [5]. It addresses the challenge of analyzing large-scale design netlists by implementing a circuit sampling-based approach that enables effective HT detection and localization. Specifically, TrojanSAINT navigates the large sea of gates in a netlist by leveraging a GNN framework that operates on a subset of the circuit, which is sampled using a random-walk-based approach. Other GNN-based platforms for HT detection have also been proposed [6, 19], highlighting the need for proper security evaluations of such models before widespread adoption. ### _Backdoor Attacks on GNNs_ Backdoor attacks are a type of data poisoning attack on machine learning (ML) systems, where a pre-determined output, $y_{t}$, is triggered by an input sample containing a _"backdoor trigger."_ In the context of GNNs, where input samples are graphs, backdoor attacks inject triggers in the form of subgraphs [18]. An adversary can launch backdoor attacks by manipulating the training data and corresponding labels. Fig. 2 illustrates the flow of a subgraph-based backdoor attack against GNNs. In this attack, a backdoor trigger and a target label $y_{t}$ are determined. Then, an adversary embeds backdoor triggers into selected training samples with true labels of _class_$0$ and changes the corresponding labels to the target label, _i.e., class_$1$. Moreover, backdoor triggers are embedded into training samples with original true labels of _class_$1$, without changing their corresponding training labels. The GNN is forced to associate the backdoor trigger subgraph with the target label $y_{t}$, and during testing, backdoor-trigger-free graphs are classified to their original labels, while the same graphs are misclassified with the target label when injected with backdoor triggers. ## III Proposed BadGNN Attack Although the ML community has previously investigated backdoor attacks against GNNs, our proposed BadGNN method represents one of the first few works in the domain of hardware design and security to use backdoor attacks for circumventing GNN-based HT detection. ### _BadGNN Threat Model_ We adopt the standard threat model for backdoor attacks as outlined in [17]. Specifically, we consider an honest user, such as an IP vendor, who aims to train the parameters of a GNN, $f_{\theta}$, with the help of a third-party service provider (i.e., adversary). The user provides the trainer with a training dataset $D_{Train}$ and a description of $f_{\theta}$, such as the input size and the number of layers. This setup is typically known as "ML as a service" (MLaaS). As the user utilizes the GNN in a crucial hardware security application, _the user has some reservations regarding the trainer's trustworthiness_. Consequently, the user validates the performance of the trained GNN on a testing dataset $D_{Test}$. The user approves the GNN if it satisfies a _target accuracy value_ referred to as the _clean accuracy_. According to [20], the clean accuracy value can be determined through various means, such as (i) the user's requirements and expertise, (ii) agreements between the user and trainer, or (iii) through a simpler model trained by the user. ### _BadGNN Flow_ Fig. 3 illustrates our proposed attack scheme that comprises two tasks: (i) normal training and (ii) backdoor trigger injection and training. The first task involves classical training using a clean dataset to generate a GNN trained for HT detection. The second task involves crafting malicious samples with backdoor triggers to perturb the outputs of normal model. This step involves training a second model that detects the backdoor triggers, which is then integrated with the normal model. The remainder of this section elaborates on these tasks. Fig. 2: Subgraph-based backdoor attack on graph neural networks. Adapted from [18]. #### Iii-B1 Normal Training A GNN is trained on a clean dataset of circuits that contains Trojan-injected (TjIn) and Trojan-free (TjFree) circuits. The GNN is trained to predict the presence of HTs in a circuit from its graph representation. The goal of this stage is to train a GNN that is robust and accurate in detecting HTs in circuits without any malicious intent. Thus, we follow the original GNN4TJ implementation and training. GNN4TJ is an open-source framework, making it suitable as a case study. This GNN is referred to as the _normal model_. GNN4TJ [4] uses Pyverilog to parse the RTL and obtain the DFG. Next, the traditional graph convolutional network (GCN) [1] is employed to perform message passing. In each iteration $(l)$ of message passing, the embedding matrix $Z^{(l)}$ will be updated as follows, \[Z^{(l)}=\sigma(\widehat{D}^{-\frac{1}{2}}\widehat{A}\widehat{D}^{-\frac{1}{2}} X^{(l-1)}\theta^{(l-1)}) \tag{2}\] $\widehat{A}=A+I$ adds self-loops to the adjacency matrix to incorporate the embedding of the target nodes. $\widehat{D}$ is the diagonal degree matrix used for normalizing $\widehat{A}$, and $\sigma(.)$ is the activation function. Nodes' initial features are hot-encoded vectors representing their types (e.g., AND, XOR, XNOR, output, input). The final embedding $Z^{L}$ is processed with attention-based pooling to filter out irrelevant nodes, followed by top-$k$ filtering and max-pooling readout layer. The embedding $z_{c}$ is used to predict $\hat{y}$ (either TjIn or TjFree) using a multilayer-perceptron (MLP) layer $g$. GNN4TJ is trained to minimize the cross-entropy loss. #### Iii-B2 Backdoor Training The main concept is to employ a backdoored dataset to train a second GNN, referred to as the _payload model_, which is trained for graph classification tasks.1 The goal of this model is to predict whether or not the circuit contains backdoor triggers. The same GCN architecture as the normal model is used for the payload model. Footnote 1: Our approach to training a payload model for evading GNN-based HT detection draws inspiration from previous work done in [21]. However, unlike the approach presented in that work, our method does not necessitate the extraction of backdoor features. Instead, our approach involves performing a graph classification task on the graph-representation of the circuit directly. Backdoor Trigger Design is a critical aspect of BadGNN. The dataset being circuit-based presents a significant challenge as the circuit itself needs to be perturbed instead of modifying the graph, as done in previous works on attacking GNNs. The circuit's functionality must not be altered, which requires a unique cascade of bit-level inversions (XOR with logic 1) as backdoor triggers.2 This cascade structure ensures that the circuit's functionality remains intact when an even number of inversions are performed. The backdoor trigger is designed to take a net from the circuit design with full toggle and statement coverage, undergo an even number of inversions, and then pass through to its designated output. Footnote 2: _Poisomed/GNN_ has recently demonstrated the threat of backdoor attacks against GNNs in the context of hiding HTs and IP privacy [22]. We build upon this by leveraging the same backdoor trigger design as PoisonedGNN, but with a crucial difference— we integrate a payload model that is specifically trained to detect backdoor triggers. Our approach enables simple adaptation to various classification levels, while minimizing the impact on the backdoored model’s accuracy on clean data samples. Attack Design. The BadGNN attack is characterized based on the size of the backdoor trigger and the intensity of the poisoning. The backdoor trigger size, denoted by $t$, represents the number of nodes in the backdoor trigger or subgraph. Since circuits can have varying graph sizes, we set the backdoor trigger size, $t$, to a fraction $\phi$ of the total number of nodes for each circuit. The poisoning intensity, denoted by $\gamma$, represents the percentage of training graphs that the adversary poisons. #### Iii-B3 Creating the Backdoored Model The payload model is integrated with the normal model as follows: if a backdoor trigger is present in the input graph (circuit representation), the payload model outputs 1, which could alter the classification label when combined with the normal model outputs. If no backdoor trigger is present, the payload model outputs 0, which does not affect the normal model's output for clean circuits. Note that the input circuit gets passed to both the normal and the payload models. There are several ways to integrate both models. As a proof of concept, we combine their outputs as follows. Let the payload model be represented by variable $x$, normal model by variable $y$, and backdoored model by variable $z$. The Boolean function that matches the requirement is $z=yx^{\prime}$. Here, ${}^{\prime}$ represents negation. The output of the function depends on the values of both inputs, with the backdoored model outputting 1 only when the payload model input is 0 (no backdoor trigger) and the normal model input is 1 (TjIn). Otherwise, the output is 0 (TjFree). BadGNN integration and testing examples are illustrated in Fig. 4. ## IV Experimental Setup and Investigation The dataset used to evaluate the original GNN4TJ consists of various types of HTs (from TrustHub) incorporated into Fig. 4: BadGNN testing phase. Fig. 3: Overview of BadGNN. three base circuits: AES, PIC, and RS232 [4]. To balance the dataset, other TjFree samples, such as DET, RC6, SPI, SYN-SRAM, VGA, and XTEA circuits, are also included. Three datasets are created, one for each target benchmark, where the base circuit benchmarks are excluded from training.3 Footnote 3: GNN4TJ default parameters are used to train the normal and the payload models, consisting of two GCN layers, each with $200$ hidden units. The top-$k$ is set with a pooling ratio of $0.8$. In training, a dropout with a $0.5$ rate is employed after every layer. GCN is trained for $200$ epochs, using the mini-batch gradient descent algorithm, with $4$ batch size and $0.001$ learning rate. BadGNN Configuration. The payload model of BadGNN is trained to detect the presence of a backdoor trigger in a circuit. To ensure a balanced training dataset for the payload model, we fix the data poisoning intensity $\gamma$ to $50\%$ in all cases. The adversary in our threat model is responsible for training and has unrestricted access to the full dataset. Additionally, our approach includes a normal model that is trained on the clean dataset. Increasing $\gamma$ to $50\%$ does not affect the accuracy of the normal model, and in fact, reduces the impact on the backdoored model accuracy compared to [22]. Clean Accuracy. GNN4TJ achieves an accuracy of $80\%$, $80\%$, $87.50\%$ on the AES, PIC, and RS232 datasets, respectively. BadGNN Performance. We present the experimental results of BadGNN using backdoor trigger size ratios $\phi$ of 20% and 50% in Table I. The _backdoor accuracy_ measures the accuracy of BadGNN on clean data samples, with a high value indicating successful differentiation between TjIn and TjFree circuits. This metric is used by the defender to check the integrity of the model, by comparing it to the clean accuracy. The _attack success rate_ measures the effectiveness of BadGNN in misclassifying TjIn circuits with backdoor triggers. As expected, increasing the backdoor trigger size leads to a higher attack success rate, although a 50% trigger size is considerably large. Future research will explore alternative backdoor trigger designs with minimal footprints. ## V Conclusion We examined the security of graph neural networks (GNNs) in the context of hardware design and security, an area that has not been explored extensively in previous research. Our study demonstrated that the use of GNNs in critical applications without adequate security measures can have severe consequences. Specifically, we proposed a proof of concept backdoor attack, called _BadGNN_, which was successful in hiding hardware Trojans and evading detection with a 100% success rate. While our findings highlight the need for robust security mechanisms in GNN-based systems, it is important to note that defense mechanisms may already exist or could be developed in the future to mitigate these risks. Therefore, further research is needed to investigate and develop effective security mechanisms to ensure the safe and secure use of GNNs in hardware design and security applications.
2308.10187	Spiking-Diffusion: Vector Quantized Discrete Diffusion Model with Spiking Neural Networks	Spiking neural networks (SNNs) have tremendous potential for energy-efficient neuromorphic chips due to their binary and event-driven architecture. SNNs have been primarily used in classification tasks, but limited exploration on image generation tasks. To fill the gap, we propose a Spiking-Diffusion model, which is based on the vector quantized discrete diffusion model. First, we develop a vector quantized variational autoencoder with SNNs (VQ-SVAE) to learn a discrete latent space for images. In VQ-SVAE, image features are encoded using both the spike firing rate and postsynaptic potential, and an adaptive spike generator is designed to restore embedding features in the form of spike trains. Next, we perform absorbing state diffusion in the discrete latent space and construct a spiking diffusion image decoder (SDID) with SNNs to denoise the image. Our work is the first to build the diffusion model entirely from SNN layers. Experimental results on MNIST, FMNIST, KMNIST, Letters, and Cifar10 demonstrate that Spiking-Diffusion outperforms the existing SNN-based generation model. We achieve FIDs of 37.50, 91.98, 59.23, 67.41, and 120.5 on the above datasets respectively, with reductions of 58.60\%, 18.75\%, 64.51\%, 29.75\%, and 44.88\% in FIDs compared with the state-of-art work. Our code will be available at \url{https://github.com/Arktis2022/Spiking-Diffusion}.	Mingxuan Liu, Jie Gan, Rui Wen, Tao Li, Yongli Chen, Hong Chen	2023-08-20T07:29:03Z	http://arxiv.org/abs/2308.10187v4	# Spiking-Diffusion: Vector Quantized Discrete Diffusion Model with Spiking Neural Networks ###### Abstract Spiking neural networks (SNNs) have tremendous potential for energy-efficient neuromorphic chips due to their binary and event-driven architecture. SNNs have been primarily used in classification tasks, but limited exploration on image generation tasks. To fill the gap, we propose a Spiking-Diffusion model, which is based on the vector quantized discrete diffusion model. First, we develop a vector quantized variational autoencoder with SNNs (VQ-SVAE) to learn a discrete latent space for images. In VQ-SVAE, image features are encoded using both the spike firing rate and postsynaptic potential, and an adaptive spike generator is designed to restore embedding features in the form of spike trains. Next, we perform absorbing state diffusion in the discrete latent space and construct a spiking diffusion image decoder (SDID) with SNNs to denoise the image. Our work is the first to build the diffusion model entirely from SNN layers. Experimental results on MNIST, FMNIST, KMNIST, Letters, and Cifar10 demonstrate that Spiking-Diffusion outperforms the existing SNN-based generation model. We achieve FIDs of 37.50, 91.98, 59.23, 67.41, and 120.5 on the above datasets respectively, with reductions of 58.60%, 18.75%, 64.51%, 29.75%, and 44.88% in FIDs compared with the state-of-art work. Our code will be available at [https://github.com/Arktis2022/Spiking-Diffusion](https://github.com/Arktis2022/Spiking-Diffusion). Mingxuan Liu${}^{\star}$ Jie Gan${}^{\dagger}$ Rui Wen${}^{\star}$ Tao Li${}^{\star}$ Yongli Chen${}^{\dagger}$ Hong Chen${}^{\star}$${}^{\star}$ Tsinghua University ${}^{\dagger}$ Beijing Smartchip Microelectronics Technology Co., Ltd Spiking neural networks, Diffusion model, Image generation, Bionic Learning ## 1 Introduction Spiking neural networks (SNNs) are the third generation neural networks with biological plausibility that encode and transmit information in form of spikes by mimicking the dynamics of neurons in the brain. Compared with Artificial neural networks (ANNs), the event-driven nature of SNNs allows for significant reduction in energy consumption when running on neuromorphic chips [1]. SNNs with deep learning techniques have shown promising results on simple tasks such as image classification, image segmentation, and optical flow estimation[2]. However, the scope of their utility in complex tasks, particularly image generation, is still confined. Spiking-GAN [3] is the first fully SNN-based GAN that uses time-to-first-spike (TTFS) encoding to generate MNIST images through adversarial training. Despite its significant progress, the quality of the generated images still falls short of ANN-based GAN. Another model, fully spiking variational autoencoder (FSVAE) [4], is a VAE model constructed entirely from SNN layers. This model produces comparable results to those of ANN-based VAEs, but its performance is still constrained by the inherent limitations of VAEs, namely the weak generative capacity due to the overly simplified decoder and compressed latent space [5]. Thus, more exploration and development are needed to harness the full potential of SNNs in image generation tasks. In this paper, we propose a vector quantized discrete diffusion model with spiking neural networks (called Spiking-Diffusion), which is the first diffusion model using only SNN layers. The pipeline of proposed Spiking-Diffusion contains two stages. First, an image is transformed into a discrete matrix through proposed vector quantized spiking variational autoencoder (VQ-SVAE). However, creating VQ-VAEs [6] in SNNs brings two challenges. The one is converting spike sequences into dense features suitable for storage in the codebook, because storing spike sequences directly within the codebook will consume too much memory. To overcome this challenge, we combine spike frequency rate (SFR) and postsynaptic potential (PSP) to model the spike sequences and introduce the learnable operator $k$ to optimize the weights of PSP and SFR. The other challenge is losslessly converting embedded features to spiking input for SNN decoder, because Poisson coding incurs information loss. To address this challenge, we design an adaptive spike generator (ASG) before the decoder and train it by using the same dictionary learning algorithm as traditional VQ-VAE. In the second stage, we utilize a spiking diffusion image decoder (SDID) to fit the prior discrete latent codes. In the forward process, we select a Markov transition matrix with an absorbing state [7] to gradually add masks to the discrete matrix, and adopt an SNN denoising network SDID to recover the masks and achieve inverse process parameterization. Experiments on MNIST [8], FMNIST [9], KMNIST [10], Letters [11], and Cifar10 [12] show that Spiking-Diffusion outperforms the SOTA SNN-based generative model. ## 2 Methodology ### SNNs Learning Algorithms In this study, we adopt he SNNs learning algorithms in [13]. Spike neurons in SNNs emulates the behavior of biological neurons by generating discrete pulses, or spikes, in response to input stimuli. The LIF neuron model [14] is used, which is described by the following dynamic equation: \[H[t]=V[t-1]+\frac{1}{\tau}(X[t]-(V[t-1]-V_{reset})) \tag{1}\] \[S[t]=\Theta(H[t]-V_{th}) \tag{2}\] \[V[t]=H[t](1-S[t])+V_{reset}S[t] \tag{3}\] where $\tau$ represents the membrane time constant, $X[t]$ denotes the synaptic input current at time step $t$, $H[t]$ represents the membrane potential of a neuron after charging but before firing a spike, and $S[t]$ denotes the spike generated by a LIF neuron when its membrane potential exceeds the discharge threshold $V_{th}$. $\Theta(v)$ is the Heaviside step function, which is equal to 1 when $v\geq 0$ and 0 otherwise. $V[t]$ stands for the membrane potential after a spike event, which is reset to the $V_{reset}$ using a hard reset [15]. If no spike is generated, it is equal to $H[t]$. ### Vq-Svae Although diffusion models [16] can generate high-quality images, they need to predict the noise added in the forward step. However, SNNs are difficult to fit and process continuous analog signals due to their discrete spike encoding [17]. In order to solve the problem, we propose a VQ-SVAE to learn a discrete latent representation of images. As shown in Fig. 1, Given a 2D image $x\in\mathbb{R}^{1\times H\times W}$, we first convert it into sequences $x_{1:T_{x}}\in\mathbb{R}^{T_{x}\times 1\times H\times W}$ using direct input encoding [18], and then the $x_{1:T_{x}}$ is encoded as a spike sequence $z_{1:T_{x}}^{E}\in\ \{0,1\}^{T_{x}\times C\times H^{\prime}\times W^{\prime}}$ after the SNN encoder. In order to prevent the codebook from consuming too much memory, we utilize spike firing rate (SFR) and postsynaptic potential (PSP) to model the spike sequence. SFR has been proven in neurobiology to have the ability to represent information. For example, auditory nerves use SFR to encode steady-state vowels [19]. \[\mathrm{SFR}(z_{1:T_{x}}^{E})=\frac{1}{T_{s}}\sum_{t=1}^{T_{s}}z_{t}^{E} \tag{4}\] PSP simulates the response of postsynaptic neurons to the action potential (AP) sequence. Similarly, studies have shown that PSP is related to experience-dependent plasticity in the mammalian nervous system [20]. We use the following formula to update the PSP [21]: \[\mathrm{PSP}(z_{\leq t_{s}}^{E})=(1-\frac{1}{\tau_{syn}})\times\mathrm{PSP} (z_{\leq t_{s-1}}^{E})+\frac{1}{\tau_{syn}}\times z_{t}^{E} \tag{5}\] where $\tau_{syn}$ is the synaptic time constant and $\mathrm{PSP}(z_{\leq 0}^{E})$ is set to 0. We use a trainable operator $k$ to allocate the weights of the PSP and SFR, so the features used for quantized encoding $z^{E}\in\ \mathbb{R}^{C\times H^{\prime}\times W^{\prime}}$ can be obtained from the following formula: \[z^{E}=k\times\mathrm{SFR}(z_{1:T_{x}}^{E})+(1-k)\times\mathrm{PSP}(z_{\leq T_ {x}}^{E}) \tag{6}\] After obtaining $z^{E}$, we construct a codebook $\mathbb{Z}\in\mathbb{R}^{K\times C}$, with $z_{k}\in\mathbb{Z}$ and $z_{i:j}^{E}\in z^{E}$. The quantized encoding of $z^{E}$, denoted as a discrete matrix $h_{0}$, can then be derived through nearest neighbor search. According to $h_{0}$, each spatial feature $z_{i,j}^{E}$ is mapped to its nearest codebook entry $z_{k}$ by indexing in $\mathbb{Z}$, thereby obtaining the embedding feature $z^{Q}$: \[h_{0}=Q(z^{E},\mathbb{Z})=\mathrm{argmin}_{k}\|\|z_{i,j}^{E}-z_{k}\|\|_{2} \tag{7}\] \[z_{Q}=\mathrm{Index}(\mathbb{Z},h_{0}) \tag{8}\] However, the embedding feature $z^{Q}$ incorporates the SFR and PSP features of spike sequences. Consequently, it cannot be Poisson encoded [21] as input to the SNN decoder without loss of information. To solve the problem, we use a SNN layer to construct a trainable adaptive spike generator (ASG) that learns to restore $z^{Q}$ to the original spike sequence $z_{1:T_{x}}^{Q}$. At this stage, all parameters in the network can be trained end-to-end with the following loss function: \[\mathcal{L}_{\mathrm{VQ-SVAE}}=\|\|x-\hat{x}\|\|_{2}+\|\|z^{E}-z^{Q}\|\|_{2}^{2} \tag{9}\] \[+(\sum_{t=1}^{T}\|\|\mathrm{PSP}(sg[z_{\leq t_{s}}^{E}])-\mathrm{PSP }(z_{\leq t_{s}}^{Q})\|\|^{2}\] \[+\beta\sum_{t=1}^{T}\|\|\mathrm{PSP}(sg[z_{\leq t_{s}}^{Q}])- \mathrm{PSP}(z_{\leq t_{s}}^{E})\|\|^{2})\] In the formula, $sg[\cdot]$ denotes the stop gradient operation. Notably, compared with VQ-VAE [6], VQ-SVAE imposes an additional constraint on training the ASG, which corresponds to the third term above. Moreover, the straight-through estimator propagates the gradients from the spike sequences $z_{1:T_{x}}^{Q}$ to $z_{1:T_{x}}^{E}$ during backpropagation, rather than propagating the gradients of the embedded features $z^{Q}$. Since spike sequences exhibit sparsity, directly computing the mean squared error (MSE) loss between sequences is not the optimal distance metric, therefore, we set the third term in Eq. (9) to maximum mean discrepancy (MMD), whose kernel function is PSP. ### Absorbing State Diffusion The training dataset is encoded into discrete matrices $h_{0}$ by the VQ-SVAE to train the spiking diffusion image decoder (SDID). For the forward diffusion process $q(h_{t_{d}}\|h_{t_{d}-1})$, we use a transition matrix with an absorbing state [7] to convert all the elements in the discrete matrix into a mask after a fixed number of $T_{d}$ time steps. In detail, for each discrete random variable element $h_{t_{d}}(i,j)$ in $h_{t_{d}}$ with $K$ categories, denoting the one-hot version of $h$ with the row vector $\mathbf{h}$, we can define the forward process as: \[q(\mathbf{h}_{t_{d}}\|\mathbf{h}_{t_{d}-1})=\mathrm{Cat}(\mathbf{h}_{t_{d}};\mathbf{p}=\mathbf{h}_{t _{d}-1}\mathbf{Q}_{t_{d}}) \tag{10}\] where $\mathrm{Cat}(\mathbf{h};\mathbf{p})$ is a categorical distribution parameterized by $\mathbf{p}$, and $[\mathbf{Q}_{t_{d}}]_{ij}=q(h_{t_{d}}=j\|h_{t_{d}-1}=i)$ is the transition matrix of the forward Markov chain. We adopt absorbing state diffusion which is compatible with SNNs. Its transition matrix $Q_{t_{d}}\in\mathbb{R}^{(K+1)\times(K+1)}$ can be expressed as follows: \[Q_{t_{d}}=\begin{bmatrix}1-\gamma_{t_{d}}&0&\cdots&\gamma_{t_{d}}\\ 0&1-\gamma_{t_{d}}&\cdots&\gamma_{t_{d}}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{bmatrix} \tag{11}\] \[\gamma_{t_{d}}=\frac{1}{T_{d}-t_{d}+1} \tag{12}\] The Eq. (11) indicates that at each timestep, each regular token has probability $\gamma_{t_{d}}$ of being replaced by the [MASK] token, and probability $1-\gamma_{t_{d}}$ of remaining unchanged, but the [MASK] token remains unchanged. In our work, we set the [MASK] token as $K$, as a result, $\mathbf{Q}_{t_{d}}\in\mathbb{R}^{(K+1)\times(K+1)}$ and each token possesses $K+1$ states. Therefore, the transition matrix with the absorbing state is non-zero only on the diagonal and the last column, so the sparsity of transition matrix will help to reduce the computational cost of the forward process. The reverse process is defined as a Markov chain parameterized by $\theta$, which gradually denoises from the data distribution $p_{\theta}(\mathbf{h}_{0:T_{d}})=p(\mathbf{h}_{T_{d}})\prod_{d=1}^{T_{d}}p_{\theta}( \mathbf{h}_{t_{d}-1}\|\mathbf{h}_{t_{d}})$ and is optimized by the evidence lower bound (ELBO). With $t_{d}^{\mathrm{th}}$ term the loss can be written as: \[\mathcal{L}_{t_{d}}=D_{\mathrm{KL}}(q(\mathbf{h}_{t_{d}-1}\|\mathbf{h}_{0})\|\|p_{\theta} (\mathbf{h}_{t_{d}-1}\|\mathbf{h}_{t_{d}})) \tag{13}\] To reduce the randomness of training, SDID $S(h_{t_{d}},t_{d})$ is employed to predict $p_{\theta}(\mathbf{h}_{0}\|\mathbf{h}_{t_{d}})$ rather than learn $p_{\theta}(\mathbf{h}_{t_{d}-1}\|\mathbf{h}_{t_{d}})$ directly. As a result, variational bound reduces to: \[\mathbb{E}_{q(\mathbf{h}_{0})}\Bigg{[}\sum_{t_{d}=1}^{T_{d}}\frac{1}{t}\mathbb{E} _{q(\mathbf{h}_{t_{d}}\|\mathbf{h}_{0})}\Big{[}\sum_{\mathbf{h}_{t_{d}}(i,j)=m}\log p_{ \theta}(\mathbf{h}_{0}(i,j)\|\mathbf{h}_{t_{d}})\Big{]}\Bigg{]} \tag{14}\] Figure 1: The training process of Spiking-Diffusion consists of two stages. Step 1: Compress the images into discrete variables through VQ-SVAE. Step 2: The spiking diffusion image decoder (SDID) models the discrete latent space by reversing the forward diffusion process, which gradually adds masks in the discrete matrix through a fixed Markov chain. Step 3: During the test process, the SDID lifts the masks through an autoregressive process to obtain a discrete matrix with the target distribution. ## 3 Experiments ### Datasets We test our model on five datasets: The MNIST [8] consists of 60,000 handwritten digit images divided into 10 classes (digits 0 to 9); FMNIST [9] contains 60,000 images of 10 different classes of clothing; KMNIST [10] consists of 60,000 images of Japanese characters, with each image belonging to one of 10 classes; Letters [11] provides 145,000 images of 26 Latin letters; for Cifar10[12], 50,000 images are used for training and 10,000 images for evaluation. ### Implementation details We make a comprehensive comparison between Spiking-Diffusion and F5VAE [4], which is the state-of-the-art SNN-based generative model generating images with quality equal to or better than ANN-based models. We test the F5VAE and Spiking-Diffusion on all datasets with the same experimental setup. The LIF neuron model's surrogate gradient function is $g(x)=\frac{1}{\pi}\mathrm{arctan}(\frac{1}{\pi}\alpha x)+\frac{1}{2}$, and its derivative is $g^{\prime}(x)=\frac{\alpha}{2(1+\frac{\alpha}{2}\alpha x)^{2}}$, where $\alpha$ represents the slope parameter. For all neurons, $\alpha=2$, $V_{reset}=0$, and $V_{th}=1$. The Optimizer is AdamW with $\beta=(0.9,0.999)$ and weight decay 0.001. The training lasts for 100 epochs. In addition, the original F5VAE used the STBP-tdBN [22] framework to train SNNs, while we adopt SpikingJelly [13] in our work. ### Evaluation Metrics and Results Evaluation Metrics. We evaluate the reconstruction ability of the model by comparing the input and output images using two metrics: mean squared error (MSE) loss and structural similarity (SSIM) loss. To assess the quality of sampled images, we use two commonly used scores: Kernel Inception Distance (KID) and Frechet inception distance (FID). Results. Table 1 shows that our Spiking-Diffusion model outperforms F5VAE in reconstruction and generation with the SpikingJelly framework. Spiking-Diffusion yields lower MSE and SSIM losses of reconstruction quality, and lower FID and KID scores of generated image quality. Figure 2 shows the examples of generated images on the four datasets, from which we find that with the same SNN encoder and decoder structures, the images generated by F5VAE are blurrier, while the images generated by our Spiking-Diffusion are clearer with obvious boundaries. The is because that only the Bernoulli distribution is used in F5VAE to approximate the true posterior distribution, which cannot fully capture the rich semantic information in the image. In contrast, the Spiking-Diffusion can approximate the complex posterior distribution through nearest neighbor search and generate images by seeing context. ## 4 Conclusion In this work, we present Spiking-Diffusion, the first implementation of diffusion models in SNNs. Our approach is based on VQ-DDM, which has two stages. First, we train the VQ-SVAE model to achieve image reconstruction, by using both PSP and SFR to model spiking features and constructing a codebook for image discretization. Secondly, we construct a discrete diffusion model in the discrete feature domain of images, and using absorbing diffusion state in Spiking-Diffusion is proven to generate higher quality images. The experimental results demonstrate that Spiking-Diffusion currently outperforms other SNN-based generative models. Future work will focus on how to train larger scale SNN generative models. \begin{table} \begin{tabular}{l l l l l l} \hline \hline Dataset & Model & MSE $\downarrow$ & SSIM $\downarrow$ & KID $\downarrow$ & FID $\downarrow$ \\ \hline \multirow{2}{}{MNIST} & F5VAE & 0.023 & 0.219 & 0.054 & 90.57 \\ & Ours* & 0.006 & 0.077 & 0.018 & 37.50 \\ \hline \multirow{2}{}{FMNIST} & F5VAE & 0.023 & 0.378 & 0.070 & 113.2 \\ & Ours* & 0.011 & 0.233 & 0.055 & 91.98 \\ \hline \multirow{2}{}{KMNIST} & F5VAE & 0.054 & 0.452 & 0.272 & 166.9 \\ & Ours* & 0.014 & 0.151 & 0.068 & 59.23 \\ \hline \multirow{2}{}{Letters} & F5VAE & 0.015 & 0.180 & 0.140 & 95.96 \\ & Ours* & 0.005 & 0.078 & 0.075 & 67.41 \\ \hline \multirow{2}{}{Cifar10} & F5VAE & 0.024 & 0.689 & 0.181 & 218.6 \\ & Ours* & 0.009 & 0.399 & 0.121 & 120.5 \\ \hline \hline \end{tabular} \end{table} Table 1: Performance comparisons of Spiking-Diffusion(Ours) and F5VAE on SpikingJelly [13]. Figure 2: Generated images of F5VAE and our Spiking-Diffusion (From top to down: MNIST, FMNIST, KMNIST, and Letters).
2308.09367	On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks	Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in several applications. The universal approximation properties of INNs have also been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems.	Bangti Jin, Zehui Zhou, Jun Zou	2023-08-18T08:01:45Z	http://arxiv.org/abs/2308.09367v1	# On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks+ ###### Abstract Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in several applications. The universal approximation properties of INNs have also been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems. Keywords: invertible neural network; bi-Lipschitz map; error estimate; operator approximation. ## 1 Introduction Invertible neural networks (INNs) are a class of neural network (NN) architectures with invertibility by design, via special invertible layers called the flow layers. INNs often enjoy tractable numerical algorithms to compute the inverse map and Jacobian determinant, e.g., with explicit inversion formulas. These distinct features have made them very attractive for a variety of machine learning tasks, e.g., generative modeling [16, 31, 29], probabilistic modeling [38, 17, 23, 6], solving inverse problems [2, 1, 3], modeling nonlinear dynamics [9] and point cloud generation [44]. There are several different classes of INNs, including invertible residual networks (iResNet) [7, 43], neural ordinary differential equations (NODEs) [11, 13, 18] and coupling-based neural networks [16, 17, 25, 31, 2]. For iResNet, Behrmann et al [7] leveraged the viewpoint of ResNets as an Euler discretization of ODEs and proved the standard ResNet architecture can be made invertible by adding a simple normalization step to control the Lipschitz constant of the NN during training. The inverse is not available in closed form but can be obtained through a fixed-point iteration. Chen et al [13] proposed using black-box ODE solvers as a model component, and developed a class of new models, i.e., NODEs, for time-series modeling, supervised learning, and density estimation etc. NODEs indirectly models an invertible function by transforming an input vector through an ordinary differential equation (ODE). Dupont and Doucet [18] introduced a class of more expressive and empirically stable models, augmented neural ODEs (ANODEs), which have a lower computational cost. There is another family of popular INNs using coupling-based frameworks, including non-linear independent component estimation (NICE) [16], real-valued non-volume preserving (RealNVPs) [17], invertible reversible residual network (i-RevNet) [25], and generative flow (Glow) [31]. In coupling-based NNs, one employs a highly restricted NN architecture where only some of the input variables undergo some transformations, and the rest of the input variables become the output as-is so that the splitting facilitates the inversion process. The great empirical successes of INNs in diverse applications have sparked intensive interest in theoretical study of INNs, e.g., approximation theory and training. The construction of INNs imposes certain restrictions on the choice of admissible layers, and hence their approximation properties are more limited and far less well understood than that for standard fully connected NNs (see [15] for an overview of the latter). Indeed, only very recently, Teschima et al [41] proved a first universal approximation property in $\mathcal{L}^{p}$ spaces for coupling based INNs. Lyu et al [39] extended the results to $C^{k}$ universality for $C^{k}$ diffeomorphism and also established the parametric version. More recently, Ishikawa et al [24] developed a more general theoretical framework, covering also other types of INNs, e.g., NODEs. However, approximation rates of INNs appear unknown so far. In this work, we continue the line of research on the approximation theory for INNs. We explicitly construct a coupling-based INN for simultaneously approximating the forward and inverse maps of a bi-Lipschitz map on a compact domain, by combining a shallow NN with a deep coupling-based NN, and provide an analysis of the approximation capacity of the constructed INNs. We show that quantitatively it can approximate the forward and inverse processes simultaneously to a certain tolerance; see Theorem 2.1 for the precise statement. This result complements the existing results [41, 39, 24] by providing an explicit construction and also quantitative estimates. To the best of our knowledge, this result is the first of the kind in the literature. Methodologically, the analysis is inspired by the techniques for establishing approximation capabilities of INNs / NODEs [4, 35, 41]. To illustrate the use of the approximation result, we apply INNs to approximate bi-Lipschitz maps in infinite-dimensional spaces, which belongs to operator learning and represents a fundamental task in the study of many scientific problems [33]. We focus on the situation where the ambient dimensionality is high but the intrinsic dimensionality is low, so that model reduction can be applied effectively. There are many methods for feature projection to reduce the dimensionality, e.g., principal component analysis (PCA) [22, 8], nonnegative matrix factorization [34] and Johnson-Lindenstrauss embedding [27]. We propose to couple INN with PCA, i.e., first applying PCA encoder and decoder to reduced the bi-Lipschitz map in infinite-dimensional spaces to finite-dimensional subspaces, and then approximating the reduced map with INNs. We provide an analysis of the overall approximation error. Further, we provide preliminary numerical illustrations with model problems in parametric PDEs. The idea of coupling of DNNs with model reduction has been explored in several works (see, e.g., [42, 28, 40, 10, 36, 12]). Wojcik and Kurdziel [42] proposed using Johnson-Lindenstrauss embedding to reduce the input dimensionality, and Karnik et al [28] developed complexity bounds for approximating a Holder (or uniformly) continuous function under a low-complexity structure assumption. Bhattacharya et al [10] proposed an approach using PCA for dimensionality reduction, and then employing a fully connected feedforward NN to approximate the reduced nonlinear mapping. Pineda and Petersen [40] investigated restricting the domain of the forward operator to finite-dimensional spaces, and proved the existence of a robust-to-noise NN approximation of the operator. Our work continues the active line of research on developing operator approximations, by extending the encoder-decoder type approach in [10] to simultaneously approximating the forward and inverse maps using INNs. Liu et al [36, 12] thoroughly analyzed general encoder-decoder based approaches in terms of approximation and generalization errors, and applied the results to canonical PDEs. In this work, we use the approximation result of INNs developed herein instead of more well-understood feedforward fully connected NNs, in order to derive the estimation error on the INN based operator approximations. The rest of the paper is organized as follows. In Section 2, we develop an INN architecture for approximating the bi-Lipschitz map, and analyze the approximation error of the constructed INN. In Section 3, we develop a novel operator approximation based on model reduction (PCA) and INN, and discuss the error estimation for the overall scheme. Finally, in Section 4 we present some preliminary numerical experiments to illustrate the approach. ## 2 Invertible neural networks In this section, we construct a coupling-based invertible neural network (INN) to approximate the bi-Lipschitz forward operator $F:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $d\geq 2$. Assumption 2.1.: _The operator $F$ is bi-Lipschitz continuous with Lipschitz constants $L_{F}$ and $L_{F^{-1}}$ for $F$ and its inverse $F^{-1}$, respectively._ Coupling-based NNs split the input $x\in\mathbb{R}^{d}$ into two parts and then employ an affine coupling layer whose inverse can be found directly. Specifically, fix $k\in\mathbb{N}$ with $k<d$, we split $x\in\mathbb{R}^{d}$ into $x=(x_{\leq k},x_{>k})$, with the subvectors $x_{\leq k}:=(x_{1},\ldots,x_{k})\in\mathbb{R}^{k}$ and $x_{>k}:=(x_{k+1},\ldots,x_{d})\in\mathbb{R}^{d-k}$, and set \[x_{\leq k}^{\prime}= x_{\leq k}\odot\exp(s_{1}(x_{>k}))+t_{1}(x_{>k}),\] \[x_{>k}^{\prime}= x_{>k}\odot\exp(s_{2}(x_{\leq k}^{\prime}))+t_{2}(x_{\leq k}^{ \prime}),\] with operators $s_{1},t_{1}:\mathbb{R}^{d-k}\longrightarrow\mathbb{R}^{k}$ and $s_{2},t_{2}:\mathbb{R}^{k}\longrightarrow\mathbb{R}^{d-k}$, which are all taken to be NNs. The inverse of this process can be obtained directly as \[x_{>k}= (x_{>k}^{\prime}-t_{2}(x_{\leq k}^{\prime}))\odot\exp(-s_{2}(x_{ \leq k}^{\prime})),\] \[x_{\leq k}= (x_{\leq k}^{\prime}-t_{1}(x_{>k}))\odot\exp(-s_{1}(x_{>k})).\] Intuitively, it reduces the spatial resolution of the coefficients and potentially increasing the layer size may improve the expressivity of the resulting NN. Numerically, it has been shown that the restriction to the affine coupling layers will not noticeably reduce the expressivity of the NN [2]. The universal approximation properties of (affine) coupling based INNs were studied in [41, 39, 24]. We provide an explicit construction of INNs to approximate bi-Lipschitz maps, which yields an explicit bound on the INN approximation. ### Construction Now we construct an INN, denoted by $F_{\mathrm{nn}}$, to approximate the bi-Lipschitz map $F:\mathbb{R}^{d}\to\mathbb{R}^{d}$, over any compact set $K\subset\mathbb{R}^{d}$. Throughout we assume that $K=[0,1]^{d}$ and $\{x^{i}\}_{i=1}^{N}$ are taken from a uniform grid in $K$. For a general compact set $K\subset\mathbb{R}^{d}$, we can always find an affine mapping $A_{f}:\;x\to W_{a}x+b_{a}$ with $W_{a}\in\mathbb{R}^{d\times d}$ and $b_{a}\in\mathbb{R}^{d}$ such that $A_{f}(K)\subset[0,1]^{d}$. Let $\{x^{i}\}_{i=1}^{N}=\{x^{\alpha}\}_{\alpha\in[n-1]^{d}}$, where $[n]$ denotes the set $[n]=\{0,\cdots,n\}$, $N=n^{d}$ and $x^{\alpha}=\frac{\alpha}{n}\in\mathbb{R}^{d}$. We also denote the $j$th entry of a vector $x\in\mathbb{R}^{d}$ by $x_{j}$. For a given set $K$, we denote by $\|K\|$ its Lebesgue measure. We denote the standard Euclidean norm of a vector (and also the spectral norm of a matrix) by $\\|\cdot\\|_{2}$. For any $f:K\to\mathbb{R}^{d}$, we define \[\\|f\\|_{\mathcal{L}^{p}(K)}= \left\{\begin{aligned} \Big{(}\int_{K}\\|f(x)\\|_{2}^{p} \mathrm{d}x\Big{)}^{\frac{1}{p}},& 1\leq p<\infty,\\ \sup_{x\in K}\\|f(x)\\|_{2},& p=\infty.\end{aligned}\right.\] Following the idea in [35, Section 4.3], we first approximate the map $F$ by a composition mapping without using NNs. This is achieved in the following lemma. The function $h^{r}$ (depending on the choice of $n$ and $r$) serves a piecewise linear approximation to the following piecewise constant function: \[\lim_{r\to 1^{-}}h^{r}(x)=\left\{\begin{array}{ll}\frac{j}{n},&x\in[ \frac{j}{n},\frac{j+1}{n}),\;j\in[n-1],\\ x,&x\in\mathbb{R}\setminus[0,1).\end{aligned}\right.\] For any $r\in(0,1)$, $h^{r}$ is strictly increasing, and can be realized by a rectified linear unit (ReLU) NN exactly; see Section 2.1.1. Then by choosing $r\in(0,1)$ sufficiently close to $1$, we obtain the final estimate. Lemma 2.1.: _For any $r\in(0,1)$, let $H^{r}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be defined by $H^{r}(x)=(h^{r}(x_{1}),\cdots,h^{r}(x_{d}))^{t}$, with the component function $h^{r}:\mathbb{R}\to\mathbb{R}$ given by_ \[h^{r}(x)=\left\{\begin{array}{ll}\frac{j}{n},&x=\frac{j}{n},\,j \in[n-1],\\ \frac{j+(1-r)}{n},&x=\frac{j+r}{n},\,j\in[n-1],\\ x,&x\in\mathbb{R}\setminus[0,1],\end{array}\right.\] _being continuous over $\mathbb{R}$ and linear in the intervals $[\frac{j}{n},\frac{j+r}{n}]$ and $[\frac{j+r}{n},\frac{j+1}{n}]$ for $j\in[n-1]$. Then the following estimates hold_ \[\\|F\circ H^{r}-F\\|_{\mathcal{L}^{2}(K)}^{2} \leq L_{F}^{2}(2r-1)^{2}dn^{-2},\] \[\\|H^{r}\circ F^{-1}-F^{-1}\\|_{\mathcal{L}^{2}(F(K))}^{2} \leq L_{F}^{d}(2r-1)^{2}dn^{-2}.\] Proof.: First, since $\|K\|=1$, we obtain the following estimate \[\\|F\circ H^{r}-F\\|_{\mathcal{L}^{2}(K)}^{2}= \\|F\circ H^{r}-F\circ\mathrm{id}\\|_{\mathcal{L}^{2}(K)}^{2}=\\|F \circ(H^{r}-\mathrm{id})\\|_{\mathcal{L}^{2}(K)}^{2}\] \[\leq \|K\|\\|F\circ(H^{r}-\mathrm{id})\\|_{\mathcal{L}^{\infty}(K)}^{2}= \\|F\circ(H^{r}-\mathrm{id})\\|_{\mathcal{L}^{\infty}(K)}^{2}.\] Then, by the definitions of the functions $H^{r}$ and $h^{r}$, we can derive \[\\|F\circ(H^{r}-\mathrm{id})\\|_{\mathcal{L}^{\infty}(K)}\leq\omega_{F}(\\|H^{r} -\mathrm{id}\\|_{\mathcal{L}^{\infty}(K)})\leq\omega_{F}(d^{\frac{1}{2}}\max_{ x\in[0,n^{-1}]}(h^{r}(x)-x))=\omega_{F}(\|2r-1\|d^{\frac{1}{2}}n^{-1}),\] where $\omega_{F}$ denotes the modulus of continuity of the mapping $F$, defined by $\omega_{F}(s)=\sup_{\\|x-\tilde{x}\\|_{2}\leq s}\\|F(x)-F(\tilde{x})\\|_{2}\leq L _{F}s$. Thus, we obtain \[\\|F\circ H^{r}-F\\|_{\mathcal{L}^{2}(K)}^{2}\leq L_{F}^{2}(2r-1)^{2}dn^{-2}.\] Similarly, by the inequality $\|F(K)\|\leq L_{F}^{d}\|K\|=L_{F}^{d}$, we can bound $F^{-1}-H^{r}\circ F^{-1}$. Remark 2.1.: _The estimates in Lemma 2.1 are derived in the $\mathcal{L}^{2}$ norm, and the same results can be extended to the $\mathcal{L}^{p}$ space, with $1\leq p\leq\infty$. By the construction of $h^{r}$, $\lim_{r\to 1^{-}}h^{r}$ is a piecewise constant function. Since $F\in\mathcal{L}^{2}(K)$, we can use piecewise constant mappings to approximate $F$ in $K$. The definition of $H^{r}$ implies that $\lim_{r\to 1^{-}}F\circ H^{r}$ may serve as such a piecewise constant mapping._ Next we construct two INNs $H^{r}_{\mathrm{nn}}$ and $\tilde{F}_{\mathrm{nn}}$ to approximate $H^{r}$ and $F$, respectively. The detailed procedures are given in the following two parts. Throughout, for any $s\in\mathbb{N}$, the notation $\mathbf{1}_{s}$ and $\mathbf{0}_{s}$ denote a length-$s$ column / row vector of ones and zeros, respectively. The notation $O_{d\times}$ denotes a zero matrix of size $d\times s$. #### 2.1.1 INN for $H^{r}$ First we construct an INN $H^{r}_{\mathrm{nn}}$ to realize the map $H^{r}$. For any $r\in(0,1)$, the function $h^{r}:\mathbb{R}\to\mathbb{R}$ is strictly increasing and hence bijective and continuous piecewise linear with the non-differentiable points given by $(2n+1)$ tuples $\{(a_{i},a_{i})\}_{i=0}^{n}\cup\{(p_{i},q_{i})\}_{i=1}^{n}$, with $a_{i}=\frac{i}{n}$, $p_{i}=\frac{i-1+r}{n}$ and $q_{i}=\frac{i-r}{n}$. It can be exactly represented by a two-layer NN with the ReLU activation function $\sigma(t)$[4] (see also [21]). Specifically, let $\sigma(t)=\max(0,t)$. Then it follows from the elementary identities \[\frac{a_{i}-q_{i}}{a_{i}-p_{i}}-\frac{q_{i}-a_{i-1}}{p_{i}-a_{i-1 }}=\frac{\frac{i}{n}-\frac{i-r}{n}}{\frac{i}{n}-\frac{i-1+r}{n}}-\frac{\frac{ i-r}{n}-\frac{i-1}{n}}{\frac{i-1+r}{n}-\frac{i-1}{n}}=\frac{r}{1-r}-\frac{1-r}{r}= \frac{2r-1}{r(1-r)},\] \[\frac{q_{i+1}-a_{i}}{p_{i+1}-a_{i}}-\frac{a_{i}-q_{i}}{a_{i}-p_{i }}=\frac{\frac{i+1-r}{n}-\frac{i}{n}}{\frac{i+r}{n}-\frac{i}{n}}-\frac{\frac{i}{ n}-\frac{i-r}{n}}{\frac{i-1+r}{n}}=\frac{1-r}{r}-\frac{r}{1-r}=-\frac{2r-1}{r(1-r)},\] for $i=1,\ldots,n-1$, that we can express $h^{r}$ as \[h^{r}(x)= -\sigma(-x)+\frac{q_{1}-a_{0}}{p_{1}-a_{0}}\sigma(x-a_{0})+\Big{(} \frac{a_{1}-q_{1}}{a_{1}-p_{1}}-\frac{q_{1}-a_{0}}{p_{1}-a_{0}}\Big{)}\sigma(x- p_{1})+\Big{(}\frac{q_{2}-a_{1}}{p_{2}-a_{1}}-\frac{a_{1}-q_{1}}{a_{1}-p_{1}} \Big{)}\sigma(x-a_{1})\] \[\\|\tilde{h}^{r}-h^{r}\\|_{\mathcal{F}^{\infty}([0,1])}\leq\omega_{h^{r}}( \Delta)=\omega_{h^{r}}(n^{-1})\leq 2n^{-1},\] _where $\Delta$ is the maximum of the minimal distance of each data point with all other data points and $\omega_{h^{r}}$ is the modulus of continuity, i.e., $\omega_{h^{r}}(\Delta)=\sup_{\|x-\tilde{x}\|\leq\|\Delta\|}(h^{r}(x)-h^{r}(\tilde{ x}))$. Then for $H^{r}_{\mathrm{nn}}(x):=(\tilde{h}^{r}(x_{1}),\cdots,\tilde{h}^{r}(x_{d}))$, we can derive_ \[\\|H^{r}_{\mathrm{nn}}-H^{r}\\|_{\mathcal{F}^{\infty}(K)}\leq\sqrt{d}\\|\tilde{h} ^{r}-h^{r}\\|_{\mathcal{F}^{\infty}([0,1])}\leq 2\sqrt{d}n^{-1}. \tag{2.1}\] #### 2.1.2 INN for $F$ Now we explicitly construct an INN $\tilde{F}_{\mathrm{nn}}$ to approximate the bi-Lipschitz map $F$. The overall construction is quite lengthy and technical and is divided into several steps. Note that the construction is non-unique: there is an alternative construction by lifting the vector into $\mathbb{R}^{2d+2}$; see the appendix for details. Throughout, we assume that we have access to a finite collection of evaluations $\{(x^{\alpha},y^{\alpha})\}_{\alpha\in[n-1]^{d}}$ on a uniform grid. The construction proceeds in the following four steps: 1. make a small perturbation $\eta$ to $\{x^{\alpha}\}_{\alpha\in[n-1]^{d}}$ such that $\{\eta(x^{\alpha})_{d}\}_{\alpha\in[n-1]^{d}}$ are distinct, i.e., the last component of all the mapped points $\{\eta(x^{\alpha})_{d}\}_{\alpha\in[n-1]^{d}}$ are distinct; 2. construct a mapping $\varphi^{N}$ that satisfies, for any $\alpha\in[n-1]^{d}$, that $(\varphi^{N}\circ\eta(x^{\alpha}))_{j}=y^{\alpha}_{j}$ for any $j=1,\cdots,d-1$ and $(\varphi^{N}\circ\eta(x^{\alpha}))_{d}=\eta(x^{\alpha})_{d}$, i.e., keeping the last entry of all the points unchanged; 3. choose an index $j_{0}\in\{1,\cdots,d-1\}$, and then make a small perturbation $\tilde{\eta}$, which ensures that $\{[\tilde{\eta}\circ\varphi^{N}\circ\eta(x^{\alpha})]_{j_{0}}\}_{\alpha\in[n- 1]^{d}}$ are all distinct; 4. derive some $\tilde{\varphi}^{N}$ so that $[\tilde{\varphi}^{N}\circ\tilde{\eta}\circ\varphi^{N}\circ\eta(x^{\alpha})]_{ d}=y^{\alpha}_{d}$ for any $\alpha\in[n-1]^{d}$ and $[\tilde{\varphi}^{N}\circ\tilde{\eta}\circ\varphi^{N}\circ\eta(x^{\alpha})]_{ j}=[\tilde{\eta}\circ\varphi^{N}\circ\eta(x^{\alpha})]_{j}$ for any $j=1,\cdots,d-1$. The key is to ensure that each of these steps can be realized by a coupling-based INN, and Lipschitz constants of these INNs and their inverses are under precise control. The explicit constructions of these invertible networks follow that of coupling based INNs, and the details are given in the following lemmas. The construction represents the main technical contribution of the work. Now we introduce the main technical tool, i.e., control family, in the construction. This concept was used in the dynamical system approach to universal approximation property of DNNs [35]. We employ a control family $\mathcal{F}$ of three-layer fully connected NN architecture with ReLU $\sigma$ defined by \[\mathcal{F}:=\big{\{}V\sigma\big{(}W^{(2)}\sigma(W^{(1)}\cdot+b^{(1)})+b^{(2)} \big{)}\ \|\ V,W^{(1)},W^{(2)}\in\mathbb{R}^{d\times d}.b^{(1)},b^{(2)}\in\mathbb{R}^{d }\big{\}}. \tag{2.2}\] Accordingly, we define the attainable set $\mathcal{A}_{\mathcal{F}}$ corresponding to $\mathcal{F}$ by \[\mathcal{A}_{\mathcal{F}}:=\bigcup_{T>0}\Big{\{}\psi_{\tau_{k}}^{f_{k}}\circ \psi_{\tau_{k-1}}^{f_{k-1}}\circ\cdots\circ\psi_{\tau_{1}}^{f_{1}}\ \|\ k\geq 1,f_{j}\in \mathcal{F},\tau_{j}>0,\forall j=1,\cdots,k\ \text{and}\ \sum_{j=1}^{k}\tau_{j}=T\Big{\}}, \tag{2.3}\] with $\psi_{\tau}^{f}=y(\tau)$, where $y(t)$ is the unique solution of the following initial value problem of the systems of first-order autonomous ordinary differential equations (ODEs): \[\begin{cases}\frac{\mathrm{d}y}{\mathrm{d}t}=f(y),\quad t>0,\\ y(0)=x.\end{cases} \tag{2.4}\] Since $f(y)$ is Lipschitz continuous in $y$, by the classical Picard-Lindelof theorem for ODEs (see, e.g., [32, Theorem 2.2.1]), problem (2.4) has a unique continuously differentiable solution $y(t)$. Further, we have \[\psi_{\tau}^{f}(x)=y(\tau;x)=x+\int_{0}^{\tau}f(y(t;x))\mathrm{d}t. \tag{2.5}\] The map $\psi_{\tau}^{-f}=z(\tau)$ given by the unique solution $z(t)$ of the following initial value problem \[\begin{cases}\frac{\mathrm{d}z}{\mathrm{d}t}=-f(z),\quad t>0,\\ z(0)=x,\end{cases} \tag{2.6}\] is the inverse of the flow map $\psi_{\tau}^{f}$, since $\psi_{\tau}^{-f}\circ\psi_{\tau}^{f}(x)=\psi_{\tau}^{f}\circ\psi_{\tau}^{-f}( x)=x$. Thus, the functions in the attainable set $\mathcal{A}_{\mathcal{F}}$ are invertible, which serve as the natural candidates for constructing various INNs. Moreover, if $f(x)\in\mathcal{F}$, we have $Df(Ax+b)\in\mathcal{F}$, for any $D,A\in\mathbb{R}^{d\times d}$ and $b\in\mathbb{R}^{d}$. Indeed, the constructions below use these elementary facts heavily. It follows from the identity (2.5) and direct computation with the chain rule that the Jacobian $J_{\psi_{\tau}^{f}}(x)$ of the flow map $\psi_{\tau}^{f}$ (with respect to the input $x$) is given by \[J_{\psi_{\tau}^{f}}(x)=I+\int_{0}^{\tau}J_{f}(y(t))J_{y(t)}(x)\mathrm{d}t=I+ \int_{0}^{\tau}J_{f}(\psi_{t}^{f}(x))J_{\psi_{t}^{f}}(x)\mathrm{d}t, \tag{2.7}\] where $J_{f}$ denotes the Jacobian of $f$ with respect to its argument. Moreover, the solution $J_{\psi_{\tau}^{f}}(x)$ to (2.7) is given by \[J_{\psi_{\tau}^{f}}(x)=e^{\int_{0}^{\tau}J_{f}(\psi_{t}^{f}(x))\mathrm{d}t}. \tag{2.8}\] Let $\Delta_{j}(x^{\alpha})=\min_{\alpha_{1}\neq\alpha_{2}}\|x_{j}^{\alpha_{1}}-x_ {j}^{\alpha_{2}}\|$ be the minimal distance in the $j$th coordinate of the collection of points $\{x^{\alpha}\}_{\alpha\in[n-1]^{d}}$. The construction of the INN $\tilde{F}_{\mathrm{nn}}$ below requires $\Delta_{j}(x^{\alpha})>0$ for some $j$ at the beginning. The mapping $\eta\in\mathcal{A}_{\mathcal{F}}$ constructed in the following lemma ensures $\Delta_{d}\big{(}\eta(x^{\alpha})\big{)}=N^{-1}$. Thus, it fulfills step (i) of the overall procedure. Lemma 2.2.: _For the uniform grids $\{x^{\alpha}=\frac{\alpha}{n}\}_{\alpha\in[n-1]^{d}}$ of the hypercube $K=[0,1]^{d}$, there exists an invertible mapping $\eta\in\mathcal{A}_{\mathcal{F}}$ such that_ \[(\eta(x))_{j}=x_{j},\quad\forall j=1,\cdots,d-1,\ x\in\mathbb{R} ^{d},\quad\Delta_{d}\big{(}\eta(x^{\alpha})\big{)}=N^{-1},\] \[\max_{x\in\mathbb{R}^{d}}\\|J_{\eta}(x)\\|_{2}\leq\frac{n}{n-1}\quad \text{and}\quad\max_{x\in\mathbb{R}^{d}}\\|J_{\eta^{-1}}(x)\\|_{2}\leq\frac{n}{n- 1}.\] Proof.: Let $f_{j}(x)=\operatorname{diag}(\mathbf{0}_{d-1},1)\sigma(\operatorname{diag}( \mathbf{0}_{d-1},x_{j}))\in\mathcal{F}$, $j=1,\cdots,d-1$, and define the mapping $\eta:=\psi_{n^{-(d-1)}}^{f_{d-1}}\circ\cdots\circ\psi_{n^{-2}}^{f_{2}}\circ\psi _{n^{-1}}^{f_{1}}\in\mathcal{A}_{\mathcal{F}}$. Then there holds $\{\eta(x^{\alpha})_{d}\}_{\alpha\in[n-1]^{d}}=\{iN^{-1}\}_{i=0}^{N-1}$. In fact, by the definitions of $\psi_{\tau}^{f}$ and $f_{j}$, we have \[\eta(x^{\alpha})_{d}=x_{d}^{\alpha}+\sum_{j=1}^{d-1}n^{-j}x_{j}^{\alpha}. \tag{2.9}\] Thus, $\eta$ maps the set $\{x^{\alpha}=\frac{\alpha}{n}\}_{\alpha\in[n-1]^{d}}$ to a set of $N$ points with distinct values for the last component, and $\Delta_{d}\big{(}\eta_{d}(x^{\alpha})\big{)}=N^{-1}.$ Meanwhile, for any $j=1,\cdots,d-1$, by the identity (2.8), we have \[J_{\psi_{n^{-j}}^{f_{j}}}(x)=\exp\Big{(}\int_{0}^{n^{-j}}J_{f_{j}}(\psi_{t}^{ f_{j}}(x))\mathrm{d}t\Big{)}:=\exp\bigl{(}n^{-j}A_{j}(x_{j})\big{)}=I+n^{-j}A_{j}(x_ {j}),\] since the matrix $A_{j}(x_{j})\in\mathbb{R}^{d\times d}$ has only one nonzero element $[A_{j}(x_{j})]_{d,j}=\frac{\mathrm{d}\sigma(x_{j})}{\mathrm{d}x_{j}}:=a_{j}$, which is also independent of $x_{d}$, and thus $A_{j}(x_{j})^{k}=0$ for any $k\geq 2$. Furthermore, we have $\|a_{j}\|\leq 1$. Also note that the components $x_{j}$, $j=1,\cdots,d-1$, stay unchanged under any of the mappings $\psi_{n^{-j}}^{f_{j}}$. Thus, there holds \[J_{\eta}(x)= J_{\psi_{n^{-(d-1)}}^{f_{d-1}}\circ\cdots\circ\psi_{n^{-1}}^{f_{1}} }(x)=J_{\psi_{n^{-d+1}}^{f_{d-1}}}(x)\cdots J_{\psi_{n^{-1}}^{f_{1}}}(x)\] \[= (I+n^{-(d-1)}A_{d-1}(x_{d-1}))\cdots(I+n^{-1}A_{1}(x_{1}))=I+\sum _{j=1}^{d-1}n^{-j}A_{j}(x_{j}),\] in view of the following identity for any $i,j=1,\cdots,d-1$: $A_{i}(x_{i})A_{j}(x_{j})=0$. Thus, we can bound the spectral norm $\\|J_{\eta}(x)\\|_{2}$ of the Jacobian $J_{\eta}(x)$ by \[\\|J_{\eta}(x)\\|_{2}\leq 1+\Big{\\|}\sum_{j=1}^{d-1}n^{-j}A_{j}(x_{j})\Big{\\|}_{2}\leq 1 +\left(\operatorname{tr}\Bigl{(}\bigl{(}\sum_{j=1}^{d-1}n^{-j}A_{j}(x_{j}) \bigr{)}^{t}\bigl{(}\sum_{j=1}^{d-1}n^{-j}A_{j}(x_{j})\bigr{)}\Bigr{)}\right)^ {\frac{1}{2}}\] \[\leq 1+\left(\sum_{j=1}^{d-1}n^{-2j}\right)^{\frac{1}{2}}\leq 1+ \left(\frac{n^{-2}}{1-n^{-2}}\right)^{\frac{1}{2}}\leq 1+\frac{1}{n-1}=\frac{n}{n-1}.\] Since $\det(J_{\eta}(x))=1$, the mapping $\eta$ is invertible, and \[\max_{x\in\mathbb{R}^{d}}\\|J_{\eta^{-1}}(x)\\|_{2}= \max_{x\in\mathbb{R}^{d}}\\|(J_{\eta}(x))^{-1}\\|_{2}=\Big{\\|}I- \sum_{j=1}^{d-1}n^{-j}A_{j}(x_{j})\Big{\\|}_{2}\leq\frac{n}{n-1}.\] This shows the last estimate and completes the proof of the lemma. Remark 2.3.: _The forward process of the mapping $\eta:x\to y$ in Lemma 2.2 is equivalent to a coupling-based INN $($with one weight layer and one added layer, i.e., identity mapping, and $n$ neurons$)$ and the inverse process $\eta^{-1}$ likewise [17, 2, 41]_ \[y_{j}=\begin{cases}x_{j},&j=1,\ldots,d-1,\\ x_{d}+\sum_{j=1}^{d-1}n^{-j}\sigma(x_{j}),&j=d,\end{cases}\quad\text{and}\quad x _{j}=\begin{cases}y_{j},&j=1,\cdots,d-1,\\ y_{d}-\sum_{j=1}^{d-1}n^{-j}\sigma(x_{j}),&j=d.\end{cases}\] For the construction below, it is convenient to introduce the hypercube \[S_{x^{\alpha}}(s):=(x_{1}^{\alpha}-s,x_{1}^{\alpha}+s)\times\cdots\times(x_{d}^ {\alpha}-s,x_{d}^{\alpha}+s),\] which is a hypercube centered at $x^{\alpha}$, each side of length $2s$. Also we decompose a vector $x\in\mathbb{R}^{d}$ into $x=\begin{pmatrix}x^{\prime}\\ x_{d}\end{pmatrix}$, where the subvector $x^{\prime}\in\mathbb{R}^{d-1}$ contains the first $d-1$ components of the vector $x$. Then we construct an NN, for any $\alpha\in[n-1]^{d}$, to transport $(x^{\alpha})^{\prime}$ to $(y^{\alpha})^{\prime}(=\bigl{(}F(x^{\alpha})\bigr{)}^{\prime})$, and keep the last component $x_{d}$ for any $x\in\mathbb{R}^{d}$ and $(x)^{\prime}$ for any $x\in(S_{x^{\alpha}}(\frac{1}{2N}))^{c}=\mathbb{R}^{d}\setminus S_{x^{\alpha}}( \frac{1}{2N})$ unchanged. Note that the set $\{S_{x^{\alpha}}(\frac{1}{2N})\}_{\alpha\in[n-1]^{d}}$ consists of a collection of disjoint open hypercubes in $\mathbb{R}^{d}$. Lemma 2.3.: _Let $\{(x^{\alpha})_{d}\}_{\alpha\in[n-1]^{d}}$ be distinct with the smallest distance $\frac{1}{N}$. Then for any $\alpha_{1}\in[n-1]^{d}$, there exists an invertible mapping $\varphi\in\mathcal{A}_{\mathcal{F}}$, such that_ \[\varphi(x^{\alpha_{1}})^{\prime}=(y^{\alpha_{1}})^{\prime},\quad(\varphi(x))_{ d}= x_{d},\;\forall x\in\mathbb{R}^{d}\quad\text{and}\quad\varphi(x)=x,\; \forall x\in(S_{x^{\alpha_{1}}}(\tfrac{1}{2N}))^{c}.\] _Furthermore, the following estimates hold_ \[\max_{x\in\mathbb{R}^{d}}\\|J_{\varphi}(x)\\|_{2}\leq 1+6N\\|(y^{\alpha_{1}}-x^{ \alpha_{1}})^{\prime}\\|_{2}\quad\text{and}\quad\max_{x\in\mathbb{R}^{d}}\\|J_{ \varphi^{-1}}(x)\\|_{2}\leq 1+6N\\|(y^{\alpha_{1}}-x^{\alpha_{1}})^{\prime}\\|_{2}.\] Proof.: First, we define an intermediate mapping \[\ell(x)=\sigma\big{(}-\sigma(\tfrac{1}{2}W^{0}x)+b^{0}\big{)}-\sigma\big{(}- \sigma(W^{0}x)+b^{0}\big{)}:=\ell^{(1)}(x)+\ell^{(2)}(x),\] with the weight matrix $W^{0}=\big{(}O_{d\times(d-1)}\quad\mathbf{1}_{d}\big{)}\in\mathbb{R}^{d\times d}$ and bias vector $b^{0}=\mathbf{1}_{d}\in\mathbb{R}^{d}$. The mapping $\ell(x)$ can be equivalently written as $\ell(x)=\ell_{0}(x_{d})\mathbf{1}_{d}$, where the function $\ell_{0}:\mathbb{R}\longrightarrow\mathbb{R}$ is defined by \[\ell_{0}(x)=\sigma\big{(}-\sigma(\tfrac{1}{2}x)+1\big{)}-\sigma\big{(}-\sigma (x)+1\big{)}. \tag{2.10}\] Note that $\ell_{0}(x)$ is a hat function supported on $x\in[0,2]$: \[\ell_{0}(x)=\left\{\begin{array}{ll}0,&x\in(-\infty,0)\cup(2,+\infty),\\ \frac{x}{2},&x\in[0,1],\\ 1-\frac{x}{2},&x\in[1,2].\end{array}\right.\] Next let $A=\text{diag}(\mathbf{0}_{d-1},2\Delta_{d}^{-1}(x^{\alpha}))\in\mathbb{R}^{d \times d}$ and $b^{\alpha_{1}}=(\mathbf{0}_{d-1},1-2\Delta_{d}^{-1}(x^{\alpha})x_{d}^{\alpha_ {1}})^{t}\in\mathbb{R}^{d}$. Note that the minimal distance $\Delta_{d}(x^{\alpha})=\min_{\alpha_{2}\neq\alpha_{3}}\|x_{d}^{\alpha_{2}}-x_{d }^{\alpha_{3}}\|=\frac{1}{N}$ is independent of $\alpha$, so is the matrix $A$. Then for any $\alpha_{1}\in[n-1]^{d}$, the following properties hold \[(Ax^{\alpha_{1}}+b^{\alpha_{1}})_{d}=1\quad\text{and}\quad(Ax+b^{\alpha_{1}}) _{d}=2\Delta_{d}^{-1}(x^{\alpha})(x_{d}-x_{d}^{\alpha_{1}})+1\notin(0,2),\quad \forall x\in(S_{x^{\alpha_{1}}}(\tfrac{1}{2N}))^{c},\] since $\|x_{d}-x_{d}^{\alpha_{1}}\|\geq\frac{1}{2N}$ for any $x\in(S_{x^{\alpha_{1}}}(\tfrac{1}{2N}))^{c}$ implies $\|2\Delta_{d}^{-1}(x^{\alpha})(x_{d}-x_{d}^{\alpha_{1}})\|\geq 1$. Further, it implies that for any $\alpha_{1}\in[n-1]^{d}$, \[\ell(Ax^{\alpha_{1}}+b^{\alpha_{1}})=\tfrac{1}{2}\mathbf{1}_{d}\quad\text{and }\quad\ell(Ax+b^{\alpha_{1}})=0,\quad\forall x\in(S_{x^{\alpha_{1}}}(\tfrac{1 }{2N}))^{c}.\] Next, we define, for some fixed $\alpha_{1}\in[n-1]^{d}$, that \[D_{\alpha_{1}}= \text{diag}(2(y_{1}^{\alpha_{1}}-x_{1}^{\alpha_{1}}),\cdots,2(y_{ d-1}^{\alpha_{1}}-x_{d-1}^{\alpha_{1}}),0)\in\mathbb{R}^{d\times d}.\] Then we claim that $\varphi:=\psi_{1}^{f}$ with $f(x)=D_{\alpha_{1}}\ell(Ax+b^{\alpha_{1}})$ is the desired mapping if $\varphi\in\mathcal{A}_{\mathcal{F}}$ and it is invertible. In fact, we can decompose $\ell$ into $\ell=\ell^{(1)}+\ell^{(2)}$, with $\ell^{(1)},\ell^{(2)}\in\mathcal{F}$. Note that these two mappings actually depend only on the unchangeable value $x_{d}$ along the time. Let $f^{(i)}(x)=D_{\alpha_{1}}\ell^{(i)}(Ax+b^{\alpha_{1}})$ for $i=1,2$, then $\varphi=\psi_{1}^{f}=\psi_{1}^{f^{(1)}}\circ\psi_{1}^{f^{(2)}}\in\mathcal{A}_{ \mathcal{F}}$. By the definitions of $f$ and $\ell$, we have \[f(x)= D_{\alpha_{1}}\ell(Ax+b^{\alpha_{1}})=2\Big{[}\sigma\Big{(}-\sigma \big{[}\tfrac{1}{2}\big{(}2\Delta_{d}^{-1}(x^{\alpha})(x_{d}-x_{d}^{\alpha_{1}}) +1\big{)}\big{]}+1\Big{)}\] \[-\sigma\Big{(}-\sigma\big{[}2\Delta_{d}^{-1}(x_{\alpha})(x_{d}-x_ {d}^{\alpha_{1}})+1\big{]}+1\Big{)}\Big{]}\begin{pmatrix}(y^{\alpha_{1}}-x^{ \alpha_{1}})^{\prime}\\ 0\end{pmatrix}:=h_{\alpha_{1}}(x_{d})\begin{pmatrix}(y^{\alpha_{1}}-x^{\alpha_ {1}})^{\prime}\\ 0\end{pmatrix}. \tag{2.11}\] Then there holds \[J_{\varphi}(x)= I+\frac{\mathrm{d}h_{\alpha_{1}}(x_{d})}{\mathrm{d}x_{d}}\begin{pmatrix} O_{(d-1)\times(d-1)}&(y^{\alpha_{1}}-x^{\alpha_{1}})^{\prime}\\ O_{1\times(d-1)}&0\end{pmatrix}:=I+H_{\alpha_{1}}(x).\] It follows from direct computation that \[\Big{\|}\frac{\mathrm{d}h_{\alpha_{1}}(x_{d})}{\mathrm{d}x_{d}}\Big{\|}\leq 2( \Delta_{d}^{-1}(x^{\alpha})+2\Delta_{d}^{-1}(x^{\alpha}))=6\Delta_{d}^{-1}(x^{ \alpha})=6N\quad\text{and}\quad\det(J_{\varphi}(x))=1.\] This directly implies that $\varphi\in\mathcal{A}_{\mathcal{F}}$ is invertible and further, \[\\|J_{\varphi}(x)\\|_{2}\leq 1+\\|H_{\alpha_{1}}(x)\\|_{2}\leq 1+\sqrt{\operatorname{ tr}\bigl{(}(H_{\alpha_{1}}(x))^{t}H_{\alpha_{1}}(x)\bigr{)}}\leq 1+6N\\|(y^{ \alpha_{1}}-x^{\alpha_{1}})^{\prime}\\|_{2}.\] Similarly, we deduce \[\max_{x\in\mathbb{R}^{d}}\\|J_{\varphi^{-1}}(x)\\|_{2}=\max_{x\in\mathbb{R}^{d}} \\|(J_{\varphi}(x))^{-1}\\|_{2}=\max_{x\in\mathbb{R}^{d}}\\|I-H_{\alpha_{1}}(x)\\| _{2}\leq 1+6N\\|(y^{\alpha_{1}}-x^{\alpha_{1}})^{\prime}\\|_{2}.\] This completes the proof of the lemma. Remark 2.4.: _The forward process of the mapping $\varphi_{\alpha}:x\to y$ in Lemma 2.3 is equivalent to the following coupling-based NN $($with six weight layers and two added layers, i.e., identity mapping, and $d$ neurons on each layer$)$ with the mapping $h_{\alpha}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ defined as (2.11) and likewise the inverse process $\varphi^{-1}$:_ \[\begin{cases}y^{\prime}=x^{\prime}+h_{\alpha}(x_{d})(y^{\alpha}-x^{\alpha})^{ \prime},\\ y_{d}=x_{d},\end{cases}\quad\text{and}\quad\begin{cases}x_{d}=y_{d},\\ x^{\prime}=y^{\prime}-h_{\alpha}(x_{d})(y^{\alpha}-x^{\alpha})^{\prime}.\end{cases}\] _Equivalently, $y=x+D_{\alpha}\ell(Ax+b^{\alpha})$._ So far we have already successfully transported the set $\{x^{\alpha}\}_{\alpha\in[n-1]^{d}}$ to $\{z^{\alpha}\}_{\alpha\in[n-1]^{d}}$ with $z^{\alpha}=\bigl{(}(y^{\alpha})^{\prime t},\eta(x_{d}^{\alpha})\bigr{)}^{t}$. Following the ideas of Lemmas 2.2 and 2.3, we can further transport the collection $\{z^{\alpha}\}_{\alpha\in[n-1]^{d}}$ of points to $\{y^{\alpha}\}_{\alpha\in[n-1]^{d}}$. To this end, we first perturb the set $\{z^{\alpha}\}_{\alpha\in[n-1]^{d}}$ by $\tilde{\eta}$ so that $\Delta_{j}(\tilde{\eta}(z^{\alpha}))\geq\epsilon$ for some $j\neq d$ and $\epsilon>0$. This distinctness at the $j$th coordinate is needed for performing the last step of the construction, and it can be achieved as follows. If there exists some $j\neq d$ such that $\Delta_{j}(z^{\alpha})>0$, we take $\epsilon=\Delta_{j}(z^{\alpha})$ and $\tilde{\eta}=\operatorname{id}$. Otherwise, we construct the desired mapping $\tilde{\eta}\in\mathcal{A}_{\mathcal{F}}$ in Lemma 2.5 below. The next lemma analyzes the fundamental case of two points $z^{1},z^{2}\in\mathbb{R}^{d}$. Lemma 2.4.: _Let $z^{1},z^{2}\in\mathbb{R}^{d}$, $z^{1}\neq z^{2}$ and $z_{1}^{1}=z_{1}^{2}$. Then for any $a\in\mathbb{R}$, $\epsilon:=a-z_{1}^{1}$ and $0<\Delta\leq\max_{j=1,\cdots,d}\|z_{j}^{1}-z_{j}^{2}\|$, there exists an invertible mapping $\tilde{\eta}_{0}\in\mathcal{A}_{\mathcal{F}}$ such that_ \[(\tilde{\eta}_{0}(z^{1}))_{1}=a,\quad(\tilde{\eta}_{0}(z^{1}))_{j }=z_{j}^{1},\quad j=2,\cdots,d,\] \[\tilde{\eta}_{0}(z)=z,\ \forall z\in(S_{z^{1}}(\Delta))^{c}\quad\text{and} \quad\\|\tilde{\eta}_{0}(z)-z\\|_{\mathcal{L}^{\infty}(\mathbb{R}^{d})}\leq\| \epsilon\|.\] Proof.: The construction of $\tilde{\eta}_{0}$ is inspired by the argument of [35, Lemma 4.14]. Since $z^{1}\neq z^{2}$, there exists an index $j\neq 1$ such that $z_{j}^{1}\neq z_{j}^{2}$, and we may assume $z_{2}^{1}\neq z_{2}^{2}$. Then for any $0<\Delta\leq\|z_{2}^{1}-z_{2}^{2}\|$, consider the following initial value problem \[\begin{cases}\begin{aligned} \frac{\operatorname{d}y}{ \operatorname{d}t}&=g(W_{22}y+e^{2}):=f(y),\quad t>0,\\ y(0)&=z,\end{aligned}\end{cases}\] with $W_{22}\in\mathbb{R}^{d\times d}$, $e^{2}\in\mathbb{R}^{d}$ and $g$ given by \[W_{22}=\operatorname{diag}(0,\Delta^{-1},\mathbf{0}_{d-2})\in \mathbb{R}^{d\times d},\quad e^{2}=(0,1-\Delta^{-1}z_{2}^{1},\mathbf{0}_{d-2}) ^{t}\in\mathbb{R}^{d},\] \[g(z):=\sigma\big{(}-\sigma\big{(}\tfrac{1}{2}W_{12}z)+e^{1} \big{)}-\sigma\big{(}-\sigma(W_{12}z)+e^{1}\big{)},\] where the matrix $W_{12}\in\mathbb{R}^{d\times d}$ has only one nonzero entry $[W_{12}]_{12}=1$ and the vector $e^{1}\in\mathbb{R}^{d}$ has one nonzero entry $[e^{1}]_{1}=1$. By the definitions of $\Delta$ and $g$, we have $f(z^{1})=\frac{1}{2}e^{1}$ and $f(z)=0$, for any $z\in\mathbb{R}^{d}$ with $\|z_{2}-z_{2}^{1}\|\geq\Delta$, since for $z\in\mathbb{R}^{d}$ with $\|z_{2}-z_{2}^{1}\|\geq\Delta$, $(W_{22}z+e^{2})_{2}=\Delta^{-1}(z_{2}-z_{2}^{1})+1\not\in(0,2)$, and $g$ is supported on the interval $[0,2]$. Thus, for any $z\in\mathbb{R}^{d}$ satisfying $\|z_{2}-z_{2}^{1}\|\geq\Delta$, $z_{j}^{1}$ for any $j=2,\cdots,d$ stay unchanged. Moreover, since $f(z^{1})=\frac{e^{1}}{2}$, for any $\epsilon:=a-z_{1}^{1}\geq 0$, when $t=2\epsilon$, we have \[[\psi_{t}^{f}(z^{1})]_{1}-z_{1}^{1}=[\psi_{t}^{f}(z^{1})]_{1}-z_{1}^{2}=\epsilon,\] \[\\|\psi_{t}^{f}(z)-z\\|_{\mathcal{L}^{\infty}(\mathbb{R}^{d})}\leq\|[\psi_{t}^{f}(z^{ 1})]_{1}-z_{1}^{1}\|=\epsilon,\] since by assumption $z_{1}^{1}=z_{1}^{2}$, and $\|f(z)\|\leq\frac{1}{2}$. Similarly, for any $\epsilon=a-z_{1}^{1}<0$, when $t=-2\epsilon$, we have \[[\psi_{t}^{-f}(z^{1})]_{1}-z_{1}^{1}=[\psi_{t}^{-f}(z^{1})]_{1}-z_ {1}^{2}=\epsilon,\] \[\\|\psi_{t}^{-f}(z)-z\\|_{\mathcal{L}^{\infty}(\mathbb{R}^{d})}\leq\| [\psi_{t}^{-f}(z^{1})]_{1}-z_{1}^{1}\|=-\epsilon.\] Let \[\tilde{\eta}_{0}(z):=\left\{\begin{array}{ll}\psi_{2\epsilon}^{f}(z),&\text { if }\epsilon\geq 0,\\ \psi_{-2\epsilon}^{-f}(z),&\text{ otherwise.}\end{array}\right.\] Then repeating the argument of Lemma 2.3 shows $\tilde{\eta}_{0}\in\mathcal{A}_{\mathcal{F}}$ by decomposing $g=g^{(1)}+g^{(2)}$, where $g^{(1)},g^{(2)}\in\mathcal{F}$ depend only on the unchangeable value. Note that \[J_{\tilde{\eta}_{0}}(z)=\exp\bigl{(}2\epsilon\frac{\mathrm{d}h(z_{2})}{ \mathrm{d}z_{2}}W_{12}\bigr{)}:=\exp\bigl{(}2\epsilon A(z_{2})\bigr{)}=I+2 \epsilon A(z_{2}),\] with $A(z_{2})=\frac{\mathrm{d}h(z_{2})}{\mathrm{d}z_{2}}W_{12}$, where \[h(z_{2})=\left\{\begin{array}{ll}\frac{1}{2}\bigl{(}\Delta^{-1}(z_{2}-z_{2} ^{1})+1\bigr{)},&\text{ if }\Delta^{-1}(z_{2}-z_{2}^{1})+1\in(0,1],\\ 1-\frac{1}{2}\bigl{(}\Delta^{-1}(z_{2}-z_{2}^{1})+1\bigr{)},&\text{ if }\Delta^{-1}(z_{2}-z_{2}^{1})+1\in(1,2],\\ 0,&\text{ otherwise.}\end{array}\right.\] Thus, $\det(J_{\tilde{\eta}_{0}}(z))=1$, and the mapping $\tilde{\eta}$ is invertible and the inverse $(J_{\tilde{\eta}_{0}}(z))^{-1}$ of $J_{\tilde{\eta}_{0}}(z)$ is given by $(J_{\tilde{\eta}_{0}}(z))^{-1}=I-2\epsilon A(z_{2})$. Then with the choice $\tilde{\eta}_{0}$, the asserted properties in the lemma follow. Remark 2.5.: _The forward process of the mapping $\eta:x\to y$ in Lemma 2.4 is equivalent to a coupling-based NN $($with four weight layers, two added layers, i.e., identity mapping and $d$ neurons on each layer$)$, and likewise the inverse process $\eta^{-1}$:_ \[y_{j}=\begin{cases}x_{j}+2\epsilon h(x_{2}),&j=1,\\ x_{j},&j=2,\cdots,d,\end{cases}\quad\text{and}\quad x_{j}=\begin{cases}y_{j},&j= 2,\cdots,d,\\ y_{j}+2\epsilon h(x_{2}),&j=1.\end{cases}\] The next lemma generalizes the transformation in Lemma 2.4 to multiple points. Lemma 2.5.: _Let the set of points $\{z^{\alpha}=\bigl{(}(y^{\alpha})^{\prime t},x_{d}^{\alpha}\bigr{)}^{t}\}_{ \alpha\in[n-1]^{d}}$ be such that $\Delta_{d}(z^{\alpha})=N^{-1}$. Then for any $\epsilon\in(0,1)$, there exists an INN $\tilde{\eta}\in\mathcal{A}_{\mathcal{F}}$ and an index $j_{0}<d$ such that_ \[\Delta_{j_{0}}\bigl{(}\tilde{\eta}(z^{\alpha})\bigr{)}\geq N^{-1}\epsilon,\quad \max_{z\in\mathbb{R}^{d}}\|(\tilde{\eta}(z)-z)_{j_{0}}\|<\epsilon\quad\text{and} \quad(\tilde{\eta}(z^{\alpha}))_{j}=z_{j}^{\alpha},\;\forall j\neq j_{0}. \tag{2.12}\] _Moreover, the following estimates hold_ \[\max_{x\in\mathbb{R}^{d}}\\|J_{\tilde{\eta}}(x)\\|_{2}\leq 1+2N\epsilon\quad \text{and}\quad\max_{x\in\mathbb{R}^{d}}\\|J_{\tilde{\eta}^{-1}}(x)\\|_{2}\leq 1+2N\epsilon.\] Proof.: For any $\epsilon\in(0,1)$ and any fixed index $j_{0}<d$, there exists a sequence of disjoint closed intervals $\{I_{j}\}_{j=1}^{J}$, each of length $\epsilon$, with $\mathbb{N}\ni J\leq N$, such that the inclusion $\{z_{j_{0}}^{\alpha}\}_{\alpha\in[n-1]^{d}}\subset\bigcup_{j=1}^{J}I_{j}$ holds. Consider an interval $I_{j}:=[b_{l},b_{r}]$ and assume that there are $K$ components of $\{z_{j_{0}}^{\alpha}\}_{\alpha\in[n-1]^{d}}$, denoted by $\{z_{j_{0}}^{\alpha_{i}}\}_{i=1}^{K}$, lie in $I_{j}$. Clearly we have $K\leq N$. Then there exists a set $\{a^{\alpha_{i}}\}_{i=1}^{K}\subset I_{j}$ such that \[\min\big{(}\min_{1\leq i<j\leq K}\|a^{\alpha_{i}}-a^{\alpha_{j}}\|,\min_{1\leq i \leq K}\|a^{\alpha_{i}}-b_{r}\|\big{)}=K^{-1}\epsilon\geq N^{-1}\epsilon.\] By Lemma 2.4, for any index $1\leq i\leq K$, we can construct an invertible mapping $\tilde{\eta}_{\alpha_{i}}\in\mathcal{A}_{\mathcal{F}}$ with $\Delta=(2N)^{-1}$ such that \[(\tilde{\eta}_{\alpha_{i}}(z^{\alpha_{i}}))_{j_{0}}=a^{\alpha_{i}},\;\;\\|\tilde{ \eta}_{\alpha_{i}}(z)-z\\|_{\mathcal{L}^{\infty}(\mathbb{R}^{d})}=\|a^{\alpha_{i} }-z_{j_{0}}^{\alpha_{i}}\|\leq\epsilon,\] and $\tilde{\eta}_{\alpha_{i}}(z)=z$ for any $z\in(S_{z^{\alpha_{i}}}(\frac{1}{2N}))^{c}$. Similarly, there exist a set $\{a^{\alpha_{i}}\}_{i=1}^{N}\subset\bigcup_{j=1}^{J}I_{j}$ and a sequence of invertible mapping $\{\tilde{\eta}_{\alpha_{i}}\}_{i=1}^{N}$ such that \[\min_{1\leq i<j\leq N}\|a^{\alpha_{i}}-a^{\alpha_{j}}\|\geq N^{-1}\epsilon,\ \ (\tilde{\eta}_{\alpha_{i}}(z^{\alpha_{i}}))_{j_{0}}=a^{\alpha_{i}},\ \ \\|\tilde{\eta}_{\alpha_{i}}(z)-z\\|_{\mathcal{L}^{\infty}( \mathbb{R}^{d})}=\|a^{\alpha_{i}}-z_{j_{0}}^{\alpha_{i}}\|\leq\epsilon,\] and $\tilde{\eta}_{\alpha_{i}}(z)=z$ for any $z\in(S_{z^{\alpha_{i}}}(\frac{1}{2N}))^{c}$. Furthermore, we have $J_{\tilde{\eta}_{\alpha_{i}}}(z)=I+2\epsilon A_{\alpha_{i}}(z_{d})$, where the matrix $A_{\alpha_{i}}(z_{d})\in\mathbb{R}^{d\times d}$ has only one nonzero entry at $[A_{\alpha_{i}}(z_{d})]_{j_{0},d}$ with $\|[A_{\alpha_{i}}(z_{d})]_{j_{0},d}\|\leq\frac{1}{2}\Delta^{-1}$. Next we define $\tilde{\eta}:=\tilde{\eta}_{\alpha_{N}}\circ\cdots\circ\tilde{\eta}_{\alpha_ {1}}\in\mathcal{A}_{\mathcal{F}}$. It can be shown that $\tilde{\eta}$ satisfies the properties in (2.12), since \[\Delta_{j_{0}}\big{(}\tilde{\eta}(z^{\alpha})\big{)}=\min_{1\leq i<j\leq N}\|a ^{\alpha_{i}}-a^{\alpha_{j}}\|\geq N^{-1}\epsilon.\] Further, since the supports of $J_{\tilde{\eta}_{\alpha_{i}}}$ are disjoint from each other, there hold \[\\|J_{\tilde{\eta}}(x)\\|_{2} \leq\max_{i=1,\cdots,N}\\|J_{\tilde{\eta}_{\alpha_{i}}}(x)\\|_{2} \leq\max_{i=1,\cdots,N}\\|I+2\epsilon A_{\alpha_{i}}(z_{d})\\|_{2}\leq 1+2 \epsilon\max_{i=1,\cdots,N}\\|A_{\alpha_{i}}(z_{d})\\|_{2}\] \[\leq 1+2\epsilon\max_{i=1,\cdots,N}\sqrt{\operatorname{tr}(A_{ \alpha_{i}}(z_{d}))^{t}(A_{\alpha_{i}}(z_{d}))}\leq 1+\epsilon\Delta^{-1}=1+2N\epsilon,\] \[\\|(J_{\tilde{\eta}}(x))^{-1}\\|_{2} \leq\max_{i=1,\cdots,N}\\|I-2\epsilon A_{\alpha_{i}}(z_{d})\\|_{2} \leq 1+2N\epsilon.\] This completes the proof of the lemma. Under the condition $\Delta_{j_{0}}\big{(}\tilde{\eta}(z^{\alpha})\big{)}\geq N^{-1}\epsilon$, we can now construct an INN $\tilde{\varphi}$ for any $\alpha_{1}\in[n-1]^{d}$, to transport $z_{d}^{\alpha_{1}}$ to $y_{d}^{\alpha_{1}}=(\tilde{\varphi}(z^{\alpha_{1}}))_{d}$, and keep $(z)^{\prime}$ for any $z\in\mathbb{R}^{d}$ and $z_{d}$ for any $z\in(S_{z^{\alpha_{1}}}(\frac{\epsilon}{2N}))^{c}$ unchanged. Lemma 2.6.: _Let $\{z_{j_{0}}^{\alpha}\}_{\alpha\in[n-1]^{d}}$ be distinct with the smallest distance equal or larger than $\Delta_{j_{0}}=\frac{\epsilon}{N}$ for some $j_{0}\neq d$ and $\epsilon>0$. Then for any $\alpha_{1}\in[n-1]^{d}$, there exists an INN $\tilde{\varphi}\in\mathcal{A}_{\mathcal{F}}$ such that_ \[(\tilde{\varphi}(z^{\alpha_{1}}))_{d}=y_{d}^{\alpha_{1}},\quad(\tilde{\varphi} (z))_{j}=z_{j},\;\forall z\in\mathbb{R}^{d},\;j\neq d\quad\text{and}\quad \tilde{\varphi}(z)=z,\;\forall z\in(S_{z^{\alpha_{1}}}(\frac{\epsilon}{2N}))^{c}.\] _Furthermore, the following estimates hold_ \[\max_{z\in\mathbb{R}^{d}}\\|J_{\tilde{\varphi}}(z)\\|_{2}\leq 1+6N\epsilon^{-1}\|y_{d}^ {\alpha_{1}}-z_{d}^{\alpha_{1}}\|\quad\text{and}\quad\max_{x\in\mathbb{R}^{d}} \\|J_{\tilde{\varphi}^{-1}}(x)\\|_{2}\leq 1+6N\epsilon^{-1}\|y_{d}^{\alpha_{1}}-z_{d}^{ \alpha_{1}}\|.\] Proof.: We define an intermediate map $\ell(z)=\sigma\big{(}-\sigma\big{(}\frac{1}{2}Pz\big{)}+e^{1}\big{)}-\sigma \big{(}-\sigma\big{(}Pz\big{)}+e^{1}\big{)}$, where $P\in\mathbb{R}^{d\times d}$ has only one nonzero entry $[P]_{d,j_{0}}=1$ and $e^{1}=\mathbf{1}_{d}\in\mathbb{R}^{d}$. Let $A\in\mathbb{R}^{d\times d}$ with only one nonzero entry $[A]_{j_{0},j_{0}}=2\Delta_{j_{0}}^{-1}$ and $b^{\alpha_{1}}\in\mathbb{R}^{d}$ with only one nonzero entry $b_{j_{0}}^{\alpha_{1}}=1-2\Delta_{j_{0}}^{-1}z_{j_{0}}^{\alpha_{1}}$. Then there hold \[(Az^{\alpha_{1}}+b^{\alpha_{1}})_{j_{0}}=1,\quad\quad\forall\alpha _{1}\in[n-1]^{d},\] \[(Az+b^{\alpha_{1}})_{j_{0}}\notin(0,2),\quad\forall z\in(S_{z^{ \alpha_{1}}}(\frac{\epsilon}{2N}))^{c}.\] These two properties together imply \[[\ell(Az^{\alpha_{1}}+b^{\alpha_{1}})] =(\mathbf{0}_{d-1},\tfrac{1}{2})^{t},\quad\forall\alpha_{1}\in[n-1 ]^{d}, \tag{2.13}\] \[\ell(Az+b^{\alpha_{1}}) =0,\quad\quad\quad\quad\forall z\in(S_{z^{\alpha_{1}}}(\frac{ \epsilon}{2N}))^{c}. \tag{2.14}\] For a fixed index $\alpha_{1}\in[n-1]^{d}$, define $D_{\alpha_{1}}=\operatorname{diag}(\mathbf{0}_{d-1},2(y_{d}^{\alpha_{1}}-z_{d}^{ \alpha_{1}}))\in\mathbb{R}^{d\times d}.$ Then $\tilde{\varphi}:=\psi_{1}^{f}\in\mathcal{A}_{\mathcal{F}}$ with $f(z)=D_{\alpha_{1}}\ell(Az+b_{\alpha_{1}})$ is the desired mapping. Indeed, by the definition of $f$, we have \[f(x)= 2\Big{[}\sigma\Big{(}-\sigma\big{[}\tfrac{1}{2}\big{(}2\Delta_{j_ {0}}^{-1}(z_{j_{0}}-z_{j_{0}}^{\alpha_{1}})+1\big{)}\big{]}+1\Big{)}\] \[\qquad-\sigma\Big{(}-\sigma\big{[}2\Delta_{j_{0}}^{-1}(z_{j_{0}}-z_{j _{0}}^{\alpha_{1}})+1\big{]}+1\Big{)}\Big{]}\begin{pmatrix}\mathbf{0}_{d-1} \\ y_{d}^{\alpha_{1}}-z_{d}^{\alpha_{1}}\end{pmatrix}:=h_{\alpha_{1}}(z_{j_{0}}) \begin{pmatrix}\mathbf{0}_{d-1}\\ y_{d}^{\alpha_{1}}-z_{d}^{\alpha_{1}}\end{pmatrix}.\] Then there holds $J_{\tilde{\varphi}}(z)=I+H_{\alpha_{1}}(z)$, where the matrix $H_{\alpha_{1}}\in\mathbb{R}^{d\times d}$ has only one nonzero entry \[[H_{\alpha_{1}}]_{d,j_{0}}=(y_{d}^{\alpha_{1}}-z_{d}^{\alpha_{1}})\frac{ \mathrm{d}h_{\alpha_{1}}(z_{j_{0}})}{\mathrm{d}z_{j_{0}}},\quad\text{with }\left\|\frac{\mathrm{d}h_{\alpha_{1}}(z_{j_{0}})}{ \mathrm{d}z_{j_{0}}}\right\|\leq 6\Delta_{j_{0}}^{-1}=\frac{6N}{\epsilon}.\] Consequently, \[\\|J_{\tilde{\varphi}}(z)\\|_{2} \leq 1+\\|H_{\alpha_{1}}(z)\\|_{2}\leq 1+6N\epsilon^{-1}\|y_{d}^{ \alpha_{1}}-z_{d}^{\alpha_{1}}\|,\] \[\max_{z\in\mathbb{R}^{d}}\\|J_{\tilde{\varphi}^{-1}}(z)\\|_{2} =\max_{z\in\mathbb{R}^{d}}\\|(J_{\tilde{\varphi}}(z))^{-1}\\|_{2}= \max_{z\in\mathbb{R}^{d}}\\|I-H_{\alpha_{1}}(z)\\|_{2}\leq 1+6N\epsilon^{-1}\|y_{d}^ {\alpha_{1}}-z_{d}^{\alpha_{1}}\|.\] This completes the proof of the lemma. Remark 2.6.: _Similar to Remark 2.4, the forward process of the mapping $\tilde{\varphi}_{\alpha}$ in Lemma 2.6 amounts to a coupling-based INN with six weight layers and two added layers, and $d$ neurons on each layer._ Last we construct an INN $\tilde{F}_{\mathrm{nn}}$ to approximate the map $F$. This is the main result of this section. Theorem 2.1.: _Given a finite collection of evaluations $\{(x^{\alpha},y^{\alpha})\}_{\alpha\in[n-1]^{d}}$, for any tolerance $\epsilon>0$, there exists a bi-Lipschitz continuous invertible mapping $\tilde{F}_{\mathrm{nn}}\in\mathcal{A}_{\mathcal{F}}$, such that $\\|\tilde{F}_{\mathrm{nn}}(x^{\alpha})-y^{\alpha}\\|_{2}<\epsilon$ for any $\alpha\in[n-1]^{d}$, and the Lipschitz constants of $\tilde{F}_{\mathrm{nn}}$ and $\tilde{F}_{\mathrm{nn}}^{-1}$ are bounded by_ \[L_{\tilde{F}_{\mathrm{nn}}} \leq\frac{n}{n-1}\big{(}1+6Nc\big{)}(1+2N\epsilon)\big{(}1+6N \epsilon^{-1}c\big{)},\] \[L_{\tilde{F}_{\mathrm{nn}}^{-1}} \leq\frac{n}{n-1}\big{(}1+6Nc\big{)}(1+2N\epsilon)\big{(}1+6N \epsilon^{-1}c\big{)},\] _with $c:=\max_{i=1,\cdots N}\\|y^{\alpha_{i}}-x^{\alpha_{i}}\\|_{2}+n^{-1}$._ Proof.: By Lemma 2.2, there exists an INN $\eta\in\mathcal{A}_{\mathcal{F}}$ such that $\Delta_{d}\big{(}\eta(x^{\alpha})\big{)}=N^{-1}$ for all $\alpha\in[n-1]^{d}$ and \[\max_{x\in\mathbb{R}^{d}}\\|J_{\eta}(x)\\|_{2}\leq\frac{n}{n-1}\quad\text{and} \quad\max_{x\in\mathbb{R}^{d}}\\|J_{\eta^{-1}}(x)\\|_{2}\leq\frac{n}{n-1}.\] Then, by Lemma 2.3, we can construct a composite INN $\varphi^{N}:=\varphi_{\alpha_{N}}\circ\cdots\circ\varphi_{\alpha_{1}}$, with $\big{(}\varphi^{j}(\eta(x^{\alpha_{i}}))\big{)}^{\prime}=\big{(}y^{\alpha_{i} }\big{)}^{\prime}$ for any $1\leq i\leq j\leq N$. Note that each $\varphi_{\alpha_{i}}$, $i=1,\cdots,N$, is the identity mapping in $(S_{\eta(x^{\alpha_{i}})}(\frac{1}{2N}))^{c}$, and $\{S_{\eta(x^{\alpha_{i}})}(\frac{1}{2N})\}_{i=1}^{N}$ are disjoint from each other. Thus, we have \[\max_{x\in\mathbb{R}^{d}}\\|J_{\varphi^{N}}(x)\\|_{2}\leq\max_{i=1,\cdots N,x\in \mathbb{R}^{d}}\\|J_{\varphi_{\alpha_{i}}}(x)\\|_{2}\leq 1+6N\max_{i=1,\cdots N}\\|(y^{ \alpha_{i}}-x^{\alpha_{i}})^{\prime}\\|_{2}.\] Similarly, there holds \[\max_{x\in\mathbb{R}^{d}}\\|J_{(\varphi^{N})^{-1}}(x)\\|_{2}\leq 1+6N\max_{i=1, \cdots N}\\|(y^{\alpha_{i}}-x^{\alpha_{i}})^{\prime}\\|_{2}.\] Then by Lemma 2.5, we can construct $\tilde{\eta}\in\mathcal{A}_{\mathcal{F}}$ such that \[\Delta_{j_{0}}\big{(}\tilde{\eta}\circ\varphi^{N}\circ\eta(x^{\alpha})\big{)} \geq N^{-1}\epsilon,\quad\max_{x\in\mathbb{R}^{d}}\\|J_{\tilde{\eta}}(x)\\|_{2} \leq 1+2N\epsilon\quad\text{and}\quad\max_{x\in\mathbb{R}^{d}}\\|J_{\tilde{ \eta}^{-1}}(x)\\|_{2}\leq 1+2N\epsilon.\] Then repeating the argument with Lemma 2.6, there exists an INN $\tilde{\varphi}^{N}:=\tilde{\varphi}_{\alpha_{N}}\circ\cdots\circ\tilde{ \varphi}_{\alpha_{1}}\in\mathcal{A}_{\mathcal{F}}$ such that $(\tilde{\varphi}^{N}\circ\tilde{\eta}\circ\varphi^{N}\circ\eta(x^{\alpha_{i}})) )_{d}=(\tilde{\varphi}^{j}(z^{\alpha_{i}}))_{d}=y_{d}^{\alpha_{i}}$ for any $1\leq i\leq j\leq N$ and \[\max_{z\in\mathbb{R}^{d}}\\|J_{\tilde{\varphi}}(z)\\|_{2}\leq 1+6N\epsilon^{-1}\max_{i=1,\cdots N}\|y_{d}^{\alpha_{i}}-z_{d}^{ \alpha_{i}}\|,\] \[\max_{z\in\mathbb{R}^{d}}\\|J_{\tilde{\varphi}^{-1}}(z)\\|_{2}\leq 1+6N\epsilon^{-1}\max_{i=1,\cdots N}\|y_{d}^{\alpha_{i}}-x_{d}^{ \alpha_{i}}\|.\] By (2.9), we can bound $\|y_{d}^{\alpha_{i}}-z_{d}^{\alpha_{i}}\|$ by \[\|y_{d}^{\alpha_{i}}-z_{d}^{\alpha_{i}}\|\leq\|y_{d}^{\alpha_{i}}-x_{d}^{\alpha_{i}}\| +\|z_{d}^{\alpha_{i}}-x_{d}^{\alpha_{i}}\|\leq\|y_{d}^{\alpha_{i}}-x_{d}^{\alpha_{i}}\| +\sum_{j=1}^{d-1}n^{-j}x_{j}^{\alpha_{i}}\leq\|y_{d}^{\alpha_{i}}-x_{d}^{\alpha_{ i}}\|+n^{-1}.\] Finally, let $\tilde{F}_{\mathrm{nn}}=\tilde{\varphi}^{N}\circ\tilde{\eta}\circ\varphi^{N} \circ\eta\in\mathcal{A}_{\mathcal{F}}$. This is the desired mapping. With $c:=\max_{i=1,\cdots N}\\|y^{\alpha_{i}}-x^{\alpha_{i}}\\|_{2}+n^{-1}$, the preceding estimates together imply \[L_{\tilde{F}_{\mathrm{nn}}}= \max_{z\in\mathbb{R}^{d}}\\|J_{\tilde{F}_{\mathrm{nn}}}(z)\\|\leq \frac{n}{n-1}\big{(}1+6Nc\big{)}(1+2N\epsilon)\big{(}1+6N\epsilon^{-1}c\big{)},\] \[L_{\tilde{F}_{\mathrm{nn}}^{-1}}= \max_{z\in\mathbb{R}^{d}}\\|J_{\tilde{F}_{\mathrm{nn}}^{-1}}(z)\\| \leq\frac{n}{n-1}\big{(}1+6Nc\big{)}(1+2N\epsilon)\big{(}1+6N\epsilon^{-1}c \big{)}.\] This completes the proof of the theorem. Remark 2.7.: _By the construction in Theorem 2.1 and the analysis in Remarks 2.3-2.6, the INN $\tilde{F}_{\mathrm{nn}}$ is a coupling-based INN with $1+16N$ weight layers and $1+6N$ added layers, and $d$ neurons on each layer._ ### Error estimation Now we bound the error of the INN approximation $F_{\mathrm{nn}}:=\tilde{F}_{\mathrm{nn}}\circ H_{\mathrm{nn}}^{r}$, where $\tilde{F}_{\mathrm{nn}}$ (cf. Theorem 2.1) has $1+16N$ weight layers, $1+6N$ added layers and each layer having $d$ neurons, and $H_{\mathrm{nn}}^{r}(x):=(h_{\mathrm{nn}}^{r}(x_{1}),\cdots,h_{\mathrm{nn}}^{r} (x_{d}))$ (cf. Section 2.1.1) with a two-layer NN $h_{\mathrm{nn}}^{r}$ of $2(n+1)$ neurons. Theorem 2.2.: _For any $\epsilon>0$, define the bi-Lipschitz INNs $F_{\mathrm{nn}}:=\tilde{F}_{\mathrm{nn}}\circ H_{\mathrm{nn}}^{r},F_{\mathrm{ nn}}^{-1}:=H_{\mathrm{nn}}^{r}\circ\tilde{F}_{\mathrm{nn}}^{-1}\in\mathcal{A}_{ \mathcal{F}}$, where the INNs $\tilde{F}_{\mathrm{nn}}$, $\tilde{F}_{\mathrm{nn}}^{-1}$ and $H_{\mathrm{nn}}^{r}$ are defined in Theorem 2.1. Then the following error estimates hold_ \[\\|F_{\mathrm{nn}}-F\\|_{\mathcal{L}^{2}(K)}^{2}\leq 2\big{(}(L_{\tilde{F}_{\mathrm{nn}}}+L_{F})(1-r)\sqrt{d}n^{-1}+ \epsilon\big{)}^{2}+2L_{F}^{2}(2r-1)^{2}dn^{-2}\] \[+2\big{(}\tfrac{1}{2}(L_{\tilde{F}_{\mathrm{nn}}}+L_{F})\sqrt{d} n^{-1}+\epsilon\big{)}^{2}(1-r^{d}),\] \[\\|F_{\mathrm{nn}}^{-1}-F^{-1}\\|_{\mathcal{L}^{2}(F(K))}^{2}\leq 2L_{F}^{d}\Big{[}\Big{(}2+\big{(}L_{\tilde{F}_{\mathrm{nn}}^{-1}}+ L_{F^{-1}}\big{)}L_{F}(1-r)\Big{)}\sqrt{d}n^{-1}+L_{F^{-1}}\epsilon\Big{]}^{2}\] \[+2L_{F}^{d}(2r-1)^{2}dn^{-2}+8L_{F}^{d}(1-r^{d}).\] Proof.: By the triangle inequality, we can decompose the total error $\\|F_{\mathrm{nn}}-F\\|_{\mathcal{L}^{2}(K)}^{2}$ between the INN $F_{\mathrm{nn}}$ and the bi-Lipschitz map $F$ into two parts \[\\|F_{\mathrm{nn}}-F\\|_{\mathcal{L}^{2}(K)}^{2}\leq 2\\|\tilde{F}_{\mathrm{nn}}\circ H_{\mathrm{nn}}^{r}-F\circ H^{r} \\|_{\mathcal{L}^{2}(K)}^{2}+2\\|F\circ H^{r}-F\\|_{\mathcal{L}^{2}(K)}^{2}\] \[= 2\\|\tilde{F}_{\mathrm{nn}}\circ H^{r}-F\circ H^{r}\\|_{\mathcal{L} ^{2}(K)}^{2}+2\\|F\circ H^{r}-F\\|_{\mathcal{L}^{2}(K)}^{2},\] since $H^{r}=H_{\mathrm{nn}}^{r}$, cf. the construction in Section 2.1.1. Then with the union of the (disjoint) hypercubes \[K^{r}:=\bigcup_{\alpha\in[n-1]^{d}}[x_{1}^{\alpha},x_{1}^{\alpha}+\tfrac{r}{n} ]\times\cdots\times[x_{d}^{\alpha},x_{d}^{\alpha}+\tfrac{r}{n}],\] and using the argument of Lemma 2.1, since $\|K\setminus K^{r}\|=1-r^{d}$, we deduce \[\\|\tilde{F}_{\mathrm{nn}}\circ H^{r}-F\circ H^{r}\\|_{\mathcal{L} ^{2}(K)}^{2}= \\|\tilde{F}_{\mathrm{nn}}\circ H^{r}-F\circ H^{r}\\|_{\mathcal{L} ^{2}(K^{r})}^{2}+\\|\tilde{F}_{\mathrm{nn}}\circ H^{r}-F\circ H^{r}\\|_{ \mathcal{L}^{2}(K\setminus K^{r})}^{2}\] \[\leq \|K^{r}\\|\\|\tilde{F}_{\mathrm{nn}}\circ H^{r}-F\circ H^{r}\\|_{ \mathcal{L}^{\infty}(K^{r})}^{2}+\|K\setminus K^{r}\|\\|\tilde{F}_{\mathrm{nn}}- F\\|_{\mathcal{L}^{\infty}(K)}^{2}\] \[\leq \\|\tilde{F}_{\mathrm{nn}}\circ H^{r}-F\circ H^{r}\\|_{\mathcal{L} ^{\infty}(K^{r})}^{2}+(1-r^{d})\\|\tilde{F}_{\mathrm{nn}}-F\\|_{\mathcal{L}^{ \infty}(K)}^{2}. \tag{2.15}\] For any index $\alpha\in[n-1]^{d}$, let $\xi^{\alpha}:=\tilde{F}_{\mathrm{nn}}(x^{\alpha})-F(x^{\alpha})$, i.e., the approximation error at the point $x^{\alpha}$. Then there holds $\\|\xi^{\alpha}\\|_{2}\leq\epsilon$ by the construction of $\tilde{F}_{\mathrm{nn}}$, cf. Lemmas 2.3 and 2.5. Note that \[\\|\tilde{F}_{\mathrm{nn}}\circ H^{r}-F\circ H^{r}\\|_{\mathcal{L} ^{\infty}(K^{r})}\] \[= \\|\tilde{F}_{\mathrm{nn}}\circ(H^{r}-\lim_{r\to 1^{-}}H^{r})+( \tilde{F}_{\mathrm{nn}}-F)\circ\lim_{r\to 1^{-}}H^{r}+F\circ(\lim_{r\to 1^{-}}H^{r}-H^{r})\\|_{ \mathcal{L}^{\infty}(K^{r})}\] \[\leq \\|\tilde{F}_{\mathrm{nn}}\circ(H^{r}-\lim_{r\to 1^{-}}H^{r})\\|_{ \mathcal{L}^{\infty}(K^{r})}+\\|(\tilde{F}_{\mathrm{nn}}-F)\circ\lim_{r\to 1^{-}}H^{r}\\|_{ \mathcal{L}^{\infty}(K^{r})}\] \[+\\|F\circ(\lim_{r\to 1^{-}}H^{r}-H^{r})\\|_{\mathcal{L}^{\infty}(K^{r})}\] \[\\|H^{r}\circ F^{-1}-F^{-1}\\|_{\mathcal{L}^{2}(F(K))}\leq L_{F}^{d}(2r-1)^{2}dn^{-2}.\] Combining the preceding estimates completes the proof of the theorem. The above estimates are derived in the $\mathcal{L}^{2}$ space, and similar results can be derived for the $\mathcal{L}^{p}$ spaces, $1\leq p<\infty$. The next result provides quantitative estimates for the INN approximations $F_{\mathrm{nn}}$ and $F_{\mathrm{nn}}^{-1}$. Corollary 2.1.: _By choosing $\epsilon=c_{\epsilon}n^{-1}$ with some $c_{\epsilon}>0$ and_ \[r\geq c_{\mathrm{nn}}^{r}:=\max\big{(}\big{(}1-(L_{\tilde{F}_{\mathrm{nn}}}+L_{F })^{-2}\big{)}^{\frac{1}{d}},1-\big{(}L_{\tilde{F}_{\mathrm{nn}}^{-1}}+L_{F^{-1 }}\big{)}^{-1},(1-n^{-2})^{\frac{1}{d}}\big{)},\] _the following error estimates hold_ \[\\|F_{\mathrm{nn}}-F\\|_{\mathcal{L}^{2}(K)}^{2} \leq 2\big{[}(3+L_{F}^{2})d+3c_{\epsilon}^{2}\big{]}n^{-2},\] \[\\|F_{\mathrm{nn}}^{-1}-F^{-1}\\|_{\mathcal{L}^{2}(F(K))}^{2} \leq 2L_{F}^{d}\big{[}\big{(}2(2+L_{F})^{2}+1\big{)}d+2L_{F^{-1}}c_{ \epsilon}+6\big{]}n^{-2}.\] Proof.: By the choice of $\epsilon$ and $r$, we have $L_{F}(1-r)\leq(L_{\tilde{F}_{\mathrm{nn}}}+L_{F})(1-r)\leq(L_{\tilde{F}_{ \mathrm{nn}}}+L_{F})(1-r^{d})^{\frac{1}{2}}\leq 1$, $\big{(}L_{\tilde{F}_{\mathrm{nn}}^{-1}}+L_{F^{-1}}\big{)}(1-r)\leq 1$ and $1-r^{d}\leq n^{-2}$. Then it follows from Theorem 2.1 that \[\\|F_{\mathrm{nn}}-F\\|_{\mathcal{L}^{2}(K)}^{2}\leq 2\big{(}(L_{\tilde{F}_{\mathrm{nn}}}+L_{F})(1-r)\sqrt{d}+c_{ \epsilon}\big{)}^{2}n^{-2}+2L_{F}^{2}(2r-1)^{2}dn^{-2}\] \[+2\big{(}\tfrac{1}{2}(L_{\tilde{F}_{\mathrm{nn}}}+L_{F})\sqrt{d} +c_{\epsilon}\big{)}^{2}(1-r^{d})n^{-2}\] \[\leq 2\big{[}\big{(}\sqrt{d}+c_{\epsilon}\big{)}^{2}+L_{F}^{2}d+ \big{(}\tfrac{1}{2}\sqrt{d}+c_{\epsilon}n^{-1}\big{)}^{2}\big{]}n^{-2}\] \[\leq 2\big{[}2d+2c_{\epsilon}^{2}+L_{F}^{2}d+\tfrac{1}{2}d+2c_{ \epsilon}^{2}n^{-2}\big{]}n^{-2}\leq 2\big{[}(3+L_{F}^{2})d+3c_{\epsilon}^{2}\big{]}n^{-2}.\] Similarly, we have \[\\|F_{\mathrm{nn}}^{-1}-F^{-1}\\|_{\mathcal{L}^{2}(F(K))}^{2}\leq 2L_{F}^{d}\big{(}\big{(}2+(L_{\tilde{F}_{\mathrm{nn}}^{-1}}+L_{F ^{-1}})L_{F}(1-r)\big{)}\sqrt{d}+L_{F^{-1}}c_{\epsilon}\big{)}^{2}n^{-2}\] \[+2L_{F}^{d}(2r-1)^{2}dn^{-2}+8L_{F}^{d}(1-r^{d})\] \[\leq 2L_{F}^{d}\big{[}\big{(}(2+L_{F})\sqrt{d}+L_{F^{-1}}c_{\epsilon }\big{)}^{2}+d+4\big{]}n^{-2}\] \[\leq 2L_{F}^{d}\big{[}\big{(}2(2+L_{F})^{2}+1\big{)}d+2L_{F^{-1}c_{ \epsilon}}+6\big{]}n^{-2}.\] This completes the proof of the corollary. Remark 2.8.: _The approximation error decays at a rate $O(N^{-\frac{1}{d}})$, and thus it suffers from the usual curse of dimensionality. This result agrees with the fact that for Lipschitz maps, standard fully connected NNs also suffer from this curse similarly [15, Section 8.7] (up to a logarithmic factor). However, it is still unclear whether this is also the lower bound for coupling based INNs. In order to overcome the curse, additional structural information, e.g., manifold assumption or low complexity, is needed [15, Section 8.10]._ ## 3 INN operator approximation Now we consider the approximation of the following nonlinear operator equation using INNs: \[F^{\dagger}(x)=y, \tag{3.1}\] where $F^{\dagger}:\mathcal{D}(F^{\dagger})\to\mathcal{I}(F^{\dagger})$ is a bi-Lipschitz continuous mapping with its domain $\mathcal{D}(F^{\dagger})\subset\mathcal{X}$ and image space $\mathcal{I}(F^{\dagger})\subset\mathcal{Y}$, and $\mathcal{X}$ and $\mathcal{Y}$ are infinite-dimensional separable Hilbert spaces with inner products $\langle\cdot,\cdot\rangle$ and norms $\\|\cdot\\|$, respectively. The model (3.1) can describe many important scientific problems. We aim at constructing a DNN that simultaneously approximates the forward process $F^{\dagger}:x\to y$ and the inverse process $(F^{\dagger})^{-1}:y\to x$ by training the DNN on a finite collection of paired observations $\{(x^{n},y^{n})\}_{n=1}^{N}$ (i.e., the training data). Throughout, we assume that the evaluation points $x^{n}$s are drawn independent and identically distributed (i.i.d.) with respect to an (unknown) probability measure $\mu$ supported on $\mathcal{X}$, and $y^{n}$s are i.i.d. with respect to the push-forward measure $F^{\dagger}_{\sharp}\mu$. ### Principal component analysis We first recall model reduction by principal component analysis (PCA). Following the construction in [10], we project any $x\in\mathcal{X}$ and $y\in\mathcal{Y}$ into finite-dimensional spaces by principal component analysis (PCA) on $\mathcal{X}$ and $\mathcal{Y}$, respectively. We approximate the identity mappings $I_{\mathcal{X}}:\mathcal{X}\rightarrow\mathcal{X}$ and $I_{\mathcal{Y}}:\mathcal{Y}\rightarrow\mathcal{Y}$ by the composition of two maps, known as the encoder and decoder, respectively, denoted by $G_{\mathcal{U}}:\mathcal{U}\rightarrow\mathbb{R}^{d_{\mathcal{U}}}$ and $G^{}_{\mathcal{U}}:\mathbb{R}^{d_{\mathcal{U}}}\rightarrow\mathcal{U}$ where $\mathcal{U}=\mathcal{X}$ or $\mathcal{Y}$, which have a finite-dimensional range and domain, such that $G^{}_{\mathcal{U}}\circ G_{\mathcal{U}}\approx I_{\mathcal{U}}$. Specifically, let $\nu$ denote a probability measure supported on the space $\mathcal{U}$, and we make the assumption of a finite fourth moment: $\mathbb{E}_{u\sim\nu}[\\|u\\|^{4}]<\infty$. We denote by $\{u^{i}\}_{i=1}^{N}$ a finite collection of $N$ samples drawn independent and identically distributed (i.i.d.) from the probability measure $\nu$ that will be used as the training data on which PCA is based. We consider the empirical, non-centered covariance operator $C^{\mathcal{U}}_{N}:=\frac{1}{N}\sum_{i=1}^{N}u^{i}\otimes u^{i}$, where $\otimes$ denotes the outer product. $C^{\mathcal{U}}_{N}$ is a non-negative, self-adjoint, trace-class operator on the Hilbert space $\mathcal{U}$, of rank at most $N$ with eigenvectors $\{\phi^{N,i}_{\mathcal{U}}\}_{i=1}^{N}$ and its corresponding eigenvalues $\lambda^{\mathcal{U}}_{N,1}\geq\lambda^{\mathcal{U}}_{N,2}\geq\cdots\geq \lambda^{\mathcal{U}}_{N,N}\geq 0$ in a nonincreasing order. We define that, with some truncation level $0<d_{\mathcal{U}}\leq N$, \[(\text{PCA encoder}) G_{\mathcal{U}}(u):= \big{(}\langle u,\phi^{N,1}_{\mathcal{U}}\rangle,\cdots,\langle u, \phi^{N,d_{\mathcal{U}}}_{\mathcal{U}}\rangle\big{)}^{t}\in\mathbb{R}^{d_{ \mathcal{U}}},\quad\forall u\in\mathcal{U},\] \[(\text{PCA decoder}) G^{}_{\mathcal{U}}(u^{}):= \sum_{i=1}^{d_{\mathcal{U}}}u^{}_{i}\phi^{N,i}_{\mathcal{U}}, \quad\forall u^{}=(u^{}_{1},\cdots,u^{}_{d_{\mathcal{U}}})^{t}\in\mathbb{R} ^{d_{\mathcal{U}}}.\] Below we denote the associated projection operator via PCA by $\mathcal{R}^{\mathcal{U}}_{N}$, which clearly depends on the random samples $\{u^{i}\}_{i=1}^{N}$ used to compute the empirical covariance $C^{\mathcal{U}}_{N}$. With the PCA on the spaces $\mathcal{X}$ and $\mathcal{Y}$, we can now describe the reduced model. Specifically, we transform the evaluations $\{(x^{\dagger,i},y^{\dagger,i})\}_{i=1}^{N}$ to $\{x^{i}:=G_{\mathcal{X}}(x^{\dagger,i}),y_{i}:=G_{\mathcal{Y}}(y^{\dagger,i}) \}_{i=1}^{N}$, and finally return them back to $\{\tilde{x}^{i}:=G^{}_{\mathcal{X}}(x^{i}),\tilde{y}^{i}:=G^{}_{\mathcal{Y}} (y^{i})\}_{i=1}^{N}$. Throughout, we take $\{(\tilde{x}^{i},\tilde{y}^{i})\}_{i=1}^{N}\subset\{(x^{\dagger,i},y^{\dagger,i})\}_{i=1}^{N}$ which implies $F^{\dagger}(\tilde{x})=\tilde{y}$, and define the following mapping (3.2) Then for any $x^{i}\in G_{\mathcal{X}}(\mathcal{D}(F^{\dagger}))$, $i=1,\ldots,N$, there holds $\tilde{x}^{i}=G^{}_{\mathcal{X}}(x^{i})$ and $y^{i}=G_{\mathcal{Y}}(y^{\dagger,i})=G_{\mathcal{Y}}(\tilde{y}^{i})$. Since $F^{\dagger}(\tilde{x}^{i})=\tilde{y}^{i}$, we have $F(x^{i})=G_{\mathcal{Y}}\circ F^{\dagger}\circ G^{}_{\mathcal{X}}(x^{i})=G_{ \mathcal{Y}}\circ F^{\dagger}(\tilde{x}^{i})=G_{\mathcal{Y}}(\tilde{y}^{i})=y^ {i}$. Further, we have the inverse mapping $F^{-1}$ of $F$ given by \[F^{-1}= G_{\mathcal{X}}\circ(F^{\dagger})^{-1}\circ G^{}_{\mathcal{Y}}:G_{ \mathcal{Y}}\big{(}\mathcal{I}(F^{\dagger})\big{)}\to G_{\mathcal{X}} \big{(}\mathcal{D}(F^{\dagger})\big{)},\] since for any $x\in G_{\mathcal{X}}\big{(}\mathcal{D}(F^{\dagger})\big{)}$, there holds $F^{-1}\circ F(x)=G_{\mathcal{X}}\circ(F^{\dagger})^{-1}\circ G^{}_{\mathcal{Y} }\circ G_{\mathcal{Y}}\circ F^{\dagger}\circ G^{}_{\mathcal{X}}(x)=x$. ### Proposed approach for approximating bi-Lipschitz maps Now we construct an approximation of the bi-Lipschitz map $F^{\dagger}:\mathcal{X}\rightarrow\mathcal{Y}$ that approximates both $F^{\dagger}$ and its inverse $(F^{\dagger})^{-1}$ simultaneously. It proceeds in two steps: 1. approximate the true mapping $F^{\dagger}$ by the reduced one $G^{}_{\mathcal{Y}}\circ F\circ G_{\mathcal{X}}$ with the PCA encoder $G_{\mathcal{X}}$ and decoder $G^{}_{\mathcal{Y}}$, with $F$ given in (3.2), and also approximate the inverse mapping $(F^{\dagger})^{-1}$ by $G^{}_{\mathcal{X}}\circ F^{-1}\circ G_{\mathcal{Y}}$ with the PCA encoder $G_{\mathcal{Y}}$ and the decoder $G^{}_{\mathcal{X}}$. 2. approximate the forward map $G^{}_{\mathcal{Y}}\circ F\circ G_{\mathcal{X}}$ and its inverse $G^{}_{\mathcal{X}}\circ F^{-1}\circ G_{\mathcal{Y}}$ (with $F$ and $F^{-1}$ defined on $\mathbb{R}^{d}$) by $G^{}_{\mathcal{Y}}\circ F_{\text{nn}}\circ G_{\mathcal{X}}$ and $G^{}_{\mathcal{X}}\circ F^{-1}_{\text{nn}}\circ G_{\mathcal{Y}}$ in Section 2, respectively, where $F_{\text{nn}}$ is a coupling based INN. The diagram in Fig. 1 illustrates the complete procedure of constructing and inferring the INN approximation. Given the training dataset $\{x^{i},y^{i}\}_{i=1}^{N}$, we first generate the pairs of encoder and decoder on the space $\mathcal{X}$ and $\mathcal{Y}$, i.e., $(G_{\mathcal{X}},G^{}_{\mathcal{X}})$ and $(G_{\mathcal{Y}},G^{}_{\mathcal{Y}})$, and then learn the coupling-based NN $F_{\text{nn}}$ by minimizing the following empirical loss \[\mathcal{L}(\theta)=\sum_{i=1}^{N}\left(\frac{c_{0}}{2}\\|G^{}_{\mathcal{X}} \circ F^{-1}_{\text{nn}}\circ G_{\mathcal{Y}}(y^{i})-x^{i}\\|_{\mathcal{X}}^{2}+ \frac{1}{2}\\|G^{}_{\mathcal{Y}}\circ F_{\text{nn}}\circ G_{\mathcal{X}}(x^{i})-y^ {i}\\|_{\mathcal{Y}}^{2}\right),\] where $c_{0}>0$ is a scalar that controls the balance between the two terms in the loss, and $\theta$ denotes the vector of trainable parameters in the INN $F_{\mathrm{nn}}$ approximating the reduced map. The optimization problem is typically minimized by gradient type methods, e.g., Adam [30]. The obtained approximation involves several different sources of errors: the truncation / approximation error due to the use of PCA encoder / decoder, the approximation error due to the INN approximation to the bi-Lipschitz map, generalization error due to using a finite-number of samples (instead of the underlying distribution), and the optimization error due to the optimizer (which fails to find a global minimizer of the empirical loss $\mathcal{L}(\theta)$, since the loss landscape is highly complex). Our analysis below focuses on the first two components. The key tool is to analyze the approximation error of the INN approximation in Section 2. ### Error analysis of INN operator approximation Now we give the error analysis of the approximation for both forward and inverse processes, which involve both PCA model reduction and INN approximation of bi-Lipschitz map on finite-dimensional spaces. We prove that there exists an INN that achieves small approximation error in Theorem 3.2. #### 3.3.1 Error estimate of PCA model reduction Now we discuss the error due to the PCA. There are two sources of errors: one is due to the truncation, and the other due to random sampling (in approximating the covariance operator by an empirical one). We denote the covariance operator of the infinite-dimensional input data space $\mathcal{X}$ and output data space $\mathcal{Y}$ by $C^{\mathcal{U}}:=\mathbb{E}_{u\sim\nu}[u\otimes u]$, with $\mathcal{U}=\mathcal{X}$ or $\mathcal{Y}$, respectively. By the i.i.d. assumption on the samples $\{u^{i}\}_{i=1}^{N}$, we have $C^{\mathcal{U}}=\mathbb{E}_{\{u^{i}\}\sim\nu}[C^{\mathcal{U}}_{N}]$, where $\mathbb{E}[\cdot]$ denotes taking expectation with respect to the samples from the probability measure $\nu$. We also define the eigenvectors of $C^{\mathcal{U}}$ by $\{\phi_{\mathcal{U}}^{i}\}_{i=1}^{\infty}$ and its corresponding nonnegative eigenvalues $\lambda_{1}^{\mathcal{U}}\geq\lambda_{2}^{\mathcal{U}}\geq\cdots\geq\cdots$ in a nonincreasing order. Then the PCA encoder and decoder for the infinite-dimensional space $\mathcal{U}$ are defined respectively by \[\begin{split}\text{(PCA encoder)}&\quad G_{\mathcal{U}}(u ):=&\big{(}\langle u,\phi_{\mathcal{U}}^{1}\rangle,\cdots, \langle u,\phi_{\mathcal{U}}^{d_{\mathcal{U}}}\rangle\big{)}^{t}\in\mathbb{R} ^{d_{\mathcal{U}}},\quad\forall u\in\mathcal{U},\\ \text{(PCA decoder)}&\quad G_{\mathcal{U}}^{}(u^{}) :=&\sum_{i=1}^{d_{\mathcal{U}}}u_{i}^{}\phi_{i}^{ \mathcal{U}},\quad\forall u^{}=(u_{1}^{},\cdots,u_{d_{\mathcal{U}}}^{})^{ t}\in\mathbb{R}^{d_{\mathcal{U}}}.\end{split}\] We define $d:=d_{\mathcal{X}}=d_{\mathcal{Y}}\geq 2$ by choosing suitable truncation indices $d_{\mathcal{X}}$ and $d_{\mathcal{Y}}$. Let $\tilde{u}=\mathcal{R}^{\mathcal{U}}_{N}u^{\dagger}=G^{}_{\mathcal{U}}\circ G _{\mathcal{U}}u^{\dagger}$ ($\mathcal{R}^{\mathcal{U}}_{N}$ is the orthogonal projection into the PCA subspace). Figure 1: A schematic diagram for the proposed simultaneous approximations to $F^{\dagger}$ and $(F^{\dagger})^{-1}$ based on the INN $F_{\mathrm{nn}}$ and its inverse $F_{\mathrm{nn}}^{-1}$. Now, we bound the mean-squared errors $\mathbb{E}_{\{x^{\dagger,i}\}\sim\nu}\big{[}\mathbb{E}_{x^{\dagger}\sim\nu}[\\|x^{ \dagger}-\tilde{x}\\|^{2}]\big{]}$ and $\mathbb{E}_{\{y^{\dagger,i}\}\sim F_{\sharp}^{\dagger}\nu}\big{[}\mathbb{E}_{y ^{\dagger}\sim F_{\sharp}^{\dagger}\nu}[\\|y^{\dagger}-\tilde{y}\\|^{2}]\big{]}$ caused by computing the PCA encoder $G_{\mathcal{U}}$ and decoder $G_{\mathcal{U}}^{}$ from the finite data $\{(x^{\dagger,i},y^{\dagger,i})\}_{i=1}^{N}$ and truncating the order of the reduced model, on the spaces $\mathcal{X}$ and $\mathcal{Y}$, separately. Lemma 3.1.: _Given a finite collection of evaluations $\{u^{\dagger,i}\}_{i=1}^{N}$, which are i.i.d. with respect to a probability measure $\nu$ supported on $\mathcal{U}$, then with $c_{\nu}=\Big{(}\mathbb{E}_{u\sim\nu}[\mathrm{tr}\big{(}(u\otimes u-C^{ \mathcal{U}})^{t}(u\otimes u-C^{\mathcal{U}})\big{)}]\Big{)}^{\frac{1}{2}}$, there holds_ \[\mathbb{E}_{\{u^{\dagger,i}\}\sim\nu}\big{[}\mathbb{E}_{u^{\dagger }\sim\nu}[\\|u^{\dagger}-\tilde{u}\\|^{2}]\big{]}\leq c_{\nu}\sqrt{d}N^{-\frac{ 1}{2}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{U}}.\] Proof.: The whole proof is shown in [10, Theorem 3.4] that \[\mathbb{E}_{\{u^{\dagger,i}\}\sim\nu}\big{[}\mathbb{E}_{u^{\dagger }\sim\nu}[\\|u^{\dagger}-\tilde{u}\\|^{2}]\big{]}\leq c_{\nu}\sqrt{d}N^{-\frac{ 1}{2}}+\mathbb{E}_{u^{\dagger}\sim\nu}[\\|u^{\dagger}-G_{\mathcal{U}}^{}\circ G _{\mathcal{U}}(u^{\dagger})\\|^{2}]=c_{\nu}\sqrt{d}N^{-\frac{1}{2}}+\sum_{j=d+1} ^{\infty}\lambda_{j}^{\mathcal{U}}.\] This completes the proof of the lemma. Now we bound the error between the approximate mapping $G_{\mathcal{Y}}^{}\circ F\circ G_{\mathcal{X}}$ and the exact one $F^{\dagger}$, and the analogue for the inverse $(\mathcal{F}^{\dagger})^{-1}$. In the statement, $L_{F}$ denotes the Lipschitz constant of the map $F$, and the constants $c_{\mu}$ and $c_{F^{\dagger}\mu}$ are defined in Lemma 3.1. Theorem 3.1.: _For the operator $F$ defined in (3.2), the following error estimates hold_ \[\mathbb{E}_{\{x^{\dagger,i}\}\sim\mu}\big{[}\mathbb{E}_{x^{\dagger }\sim\mu}[\\|G_{\mathcal{Y}}^{}\circ F\circ G_{\mathcal{X}}(x^{\dagger})-F^{ \dagger}(x^{\dagger})\\|^{2}]\big{]}\] \[\leq 2(L_{F^{\dagger}}^{2}c_{\mu}+c_{F_{\sharp}^{\dagger}\mu})\sqrt{d }N^{-\frac{1}{2}}+2\Big{(}L_{F^{\dagger}}^{2}\sum_{j=d+1}^{\infty}\lambda_{j} ^{\mathcal{X}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{Y}}\Big{)},\] \[\mathbb{E}_{\{y^{\dagger}_{i}\}\sim F_{\sharp}^{\dagger}\mu} \big{[}\mathbb{E}_{y^{\dagger}\sim F_{\sharp}^{\dagger}\mu}[\\|G_{\mathcal{X}} ^{}\circ F^{-1}\circ G_{\mathcal{Y}}(y^{\dagger})-(F^{\dagger})^{-1}(y^{ \dagger})\\|^{2}]\big{]}\] \[\leq 2(L_{F^{\dagger-1}}^{2}c_{F_{\sharp}^{\dagger}\mu}+c_{\mu})\sqrt {d}N^{-\frac{1}{2}}+2\Big{(}L_{F^{\dagger-1}}^{2}\sum_{j=d+1}^{\infty}\lambda _{j}^{\mathcal{Y}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{X}}\Big{)}.\] Proof.: By the definition of the operator $F$, the triangle inequality and Cauchy-Schwarz inequality, with $\tilde{x}=G_{\mathcal{X}}^{}\circ G_{\mathcal{X}}(x^{\dagger})$ and $\tilde{y}=G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}\circ y^{\dagger}$, we have \[\\|G_{\mathcal{Y}}^{}\circ F\circ G_{\mathcal{X}}(x^{\dagger})-F^{ \dagger}(x^{\dagger})\\|^{2}=\\|G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}\circ F ^{\dagger}\circ G_{\mathcal{X}}^{}\circ G_{\mathcal{X}}(x^{\dagger})-F^{ \dagger}(x^{\dagger})\\|^{2}\] \[\leq 2\\|G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}\circ F^{\dagger}( \tilde{x})-G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}\circ F^{\dagger}(x^{ \dagger})\\|^{2}+2\\|G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}\circ y^{\dagger}-y ^{\dagger}\\|^{2}\] \[\leq 2L_{G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}\circ F^{\dagger}}^{2} \\|\tilde{x}-x^{\dagger}\\|^{2}+2\\|\tilde{y}-y^{\dagger}\\|^{2}\leq 2L_{F^{ \dagger}}^{2}\\|\tilde{x}-x^{\dagger}\\|^{2}+2\\|\tilde{y}-y^{\dagger}\\|^{2},\] since $G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}$ is an orthogonal projection. Using the error estimate in Lemma 3.1, we derive \[\mathbb{E}_{\{x^{\dagger,i}\}\sim\mu}\big{[}\mathbb{E}_{x^{\dagger }\sim\mu}[\\|G_{\mathcal{Y}}^{}\circ F\circ G_{\mathcal{X}}(x^{\dagger})-F^{ \dagger}(x^{\dagger})\\|^{2}]\big{]}\] \[\leq 2L_{F^{\dagger}}^{2}\Big{(}c_{\mu}\sqrt{d}N^{-\frac{1}{2}}+\sum_ {j=d+1}^{\infty}\lambda_{j}^{\mathcal{X}}\Big{)}+2\Big{(}c_{F_{\sharp}^{\dagger} \mu}\sqrt{d}N^{-\frac{1}{2}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{Y}}\Big{)}.\] Similarly, there holds \[\\|G_{\mathcal{X}}^{}\circ F^{-1}\circ G_{\mathcal{Y}}(y^{\dagger} )-F^{\dagger-1}(y^{\dagger})\\|^{2}=\\|G_{\mathcal{X}}^{}\circ G_{\mathcal{X}} \circ F^{\dagger-1}\circ G_{\mathcal{Y}}^{}\circ G_{\mathcal{Y}}(y^{\dagger})-(F ^{\dagger})^{-1}(y^{\dagger})\\|^{2}\] \[\leq 2L_{G_{\mathcal{X}}^{}\circ G_{\mathcal{X}}\circ F^{\dagger-1}}^{2} \\|\tilde{y}-y^{\dagger}\\|^{2}+2\\|\tilde{x}-x^{\dagger}\\|^{2}\leq 2L_{F^{\dagger-1}}^{2}\\| \tilde{y}-y^{\dagger}\\|^{2}+2\\|\tilde{x}-x^{\dagger}\\|^{2},\] and hence \[\mathbb{E}_{\{y^{\dagger,i}\}\sim F_{\sharp}^{\dagger}\mu}\big{[} \mathbb{E}_{y^{\dagger}\sim F_{\sharp}^{\dagger}\mu}[\\|G_{\mathcal{X}}^{} \circ F^{-1}\circ G_{\mathcal{Y}}(y^{\dagger})-F^{\dagger-1}(y^{\dagger})\\|^{2}] \big{]}\] \[\leq 2L^{2}_{F^{\dagger-1}}\Big{(}c_{F^{\dagger}_{\sharp}\mu}\sqrt{d}N^{- \frac{1}{2}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{Y}}\Big{)}+2\Big{(}c_{ \mu}\sqrt{d}N^{-\frac{1}{2}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{X}} \Big{)},\] which directly completes the proof of the theorem. Remark 3.1.: _In the discussion, we have ignored the possible presence of data noise in $y^{\dagger,i}$. Should there be any noise $\{\xi_{i}\}_{i=1}^{N}$ in the observation, it will impact the accuracy of the reduced model. The mean squared errors indicate that the bound depends on the spectral decay of the covariance operators $C^{\mathcal{X}}$ and $C^{\mathcal{Y}}$ via the remainders $\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{X}}$ and $\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{Y}}$ and the number of samples $N$ (at a rate $N^{-\frac{1}{2}}$). In practice, the covariance operator $C^{\mathcal{X}}$ is often represented by an integral operator, and then its spectral decay can be characterized by the smoothness of the associated kernel [20]._ #### 3.3.2 Error estimate of INN operator approximation Now we derive the full error estimate of the INN approximation of the forward and inverse mappings. The following theorem is the main result of this section. Theorem 3.2.: _Let $F_{\mathrm{nn}}:=\tilde{F}_{\mathrm{nn}}\circ H^{r}_{\mathrm{nn}}$ and $F_{\mathrm{nn}}^{-1}=H^{r}_{\mathrm{nn}}\circ\tilde{F}_{\mathrm{nn}}^{-1}$ when $\epsilon=c_{\epsilon}n^{-1}$ with some $c_{\epsilon}>0$ and $r\geq c^{r}_{\mathrm{nn}}$, where $H^{r}_{\mathrm{nn}}(x):=(h^{r}_{\mathrm{nn}}(x_{1}),\cdots,h^{r}_{\mathrm{nn}} (x_{d}))$ with a two-layer NN $h^{r}_{\mathrm{nn}}$ of $2(N^{\frac{1}{d}}+1)$ neurons, and $\tilde{F}_{\mathrm{nn}}$ has $1+16N$ weight layers, $1+6N$ added layers and $N^{\frac{1}{2}}$ neurons. Then the expected errors of the whole systems are bounded by_ \[\mathbb{E}_{\{x^{\dagger}_{i}\}\sim\mu}\big{[}\mathbb{E}_{x^{ \dagger}\sim\mu}[\\|G^{}_{\mathcal{Y}}\circ F_{\mathrm{nn}}\circ G_{\mathcal{ X}}(x^{\dagger})-F^{\dagger}(x^{\dagger})\\|^{2}]\big{]}\] \[\leq 2c_{\mathrm{nn}}N^{-\frac{2}{2}}+4(L^{2}_{F^{\dagger}}c_{\mu}+c_ {F^{\dagger}_{\sharp}\mu})\sqrt{d}N^{-\frac{1}{2}}+4(L^{2}_{F^{\dagger}}\sum_ {j=d+1}^{\infty}\lambda_{j}^{\mathcal{X}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{ \mathcal{Y}}),\] \[\mathbb{E}_{\{y^{\dagger}_{i}\}\sim F^{\dagger}_{\sharp}\mu} \big{[}\mathbb{E}_{y^{\dagger}\sim F^{\dagger}_{\sharp}\mu}[\\|G^{}_{ \mathcal{X}}\circ F_{\mathrm{nn}}^{-1}\circ G_{\mathcal{Y}}(y^{\dagger})-(F^{ \dagger})^{-1}(y^{\dagger})\\|^{2}]\big{]}\] \[\leq 2c^{r}_{\mathrm{nn}}N^{-\frac{2}{2}}+4(L^{2}_{F^{\dagger-1}}c_{F ^{\dagger}_{\sharp}\mu}+c_{\mu})\sqrt{d}N^{-\frac{1}{2}}+4\Big{(}L^{2}_{F^{ \dagger-1}}\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{Y}}+\sum_{j=d+1}^{\infty }\lambda_{j}^{\mathcal{X}}\Big{)},\] _with $c_{\mathrm{nn}}=2\big{[}(3+L^{2}_{F})d+3c_{\epsilon}^{2}\big{]}$ and $c^{r}_{\mathrm{nn}}=2L^{d}_{F}[\big{(}2(2+L_{F})^{2}+1\big{)}d+2L_{F^{-1}}c_{ \epsilon}+6]$._ Proof.: By the triangle inequality, we have \[\\|G^{}_{\mathcal{Y}}\circ F_{\mathrm{nn}}\circ G_{\mathcal{X}}(x ^{\dagger})-F^{\dagger}(x^{\dagger})\\|^{2}\leq 2\\|G^{}_{\mathcal{Y}}\circ F_{\mathrm{nn}}\circ G_{ \mathcal{X}}(x^{\dagger})-G^{}_{\mathcal{Y}}\circ F\circ G_{\mathcal{X}}(x^{ \dagger})\\|^{2}\] \[+2\\|G^{}_{\mathcal{Y}}\circ F\circ G_{\mathcal{X}}(x^{\dagger})-F ^{\dagger}(x^{\dagger})\\|^{2}:=\mathrm{I}+\mathrm{II}.\] It suffices to bound the two terms separately. For the term $\mathrm{I}$, we have \[\mathbb{E}_{\{x^{\dagger}_{i}\}\sim\mu}\big{[}\mathbb{E}_{x^{ \dagger}\sim\mu}[\mathrm{I}]\big{]}\leq 2\mathbb{E}_{\{x_{i}\}\sim G_{X^{\sharp}\mu}}\big{[}\mathbb{E}_{x \sim G_{X^{\sharp}\mu}}[\\|G^{}_{\mathcal{Y}}\circ F_{\mathrm{nn}}(x)-G^{}_{ \mathcal{Y}}\circ F(x)\\|^{2}]\big{]}\] \[\leq 2\mathbb{E}_{\{x_{i}\}\sim G_{X^{\sharp}\mu}}\big{[}\\|G^{}_{ \mathcal{Y}}\\|^{2}\\|F_{\mathrm{nn}}-F\\|^{2}_{\mathcal{L}^{2}(K)}\big{]}\leq 2\\|F_{ \mathrm{nn}}-F\\|^{2}_{\mathcal{L}^{2}(K)}.\] In view of Remark 2.1, there holds Meanwhile, by Theorem 3.1, we have \[\mathbb{E}_{\{x^{\dagger}_{i}\}\sim\mu}\big{[}\mathbb{E}_{x^{ \dagger}\sim\mu}[\Pi]\big{]}\leq 4(L^{2}_{F^{\dagger}}c_{\mu}+c_{F^{\dagger}_{\sharp}\mu})\sqrt{d}N^{-\frac{1}{ 2}}+4\Big{(}L^{2}_{F^{\dagger}}\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{X}}+ \sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{Y}}\Big{)}.\] Combining the last two estimates yields the first assertion. Similarly, there holds \[\\|G^{}_{\mathcal{X}}\circ F_{\mathrm{nn}}^{-1}\circ G_{\mathcal{Y}}(y^{ \dagger})-(F^{\dagger})^{-1}(y^{\dagger})\\|^{2}\leq 2\\|G^{}_{\mathcal{X}}\circ F_{\mathrm{nn}}^{-1}\circ G_{ \mathcal{Y}}(y^{\dagger})-F^{-1}(y^{\dagger})\\|^{2}\] \[+2\\|G_{\mathcal{X}}^{}\circ F^{-1}\circ G_{\mathcal{Y}}(y^{\dagger})-(F^{ \dagger})^{-1}(y^{\dagger})\\|^{2}=:\mathrm{III}+\mathrm{IV}.\] By Remark 2.1 and Theorem 3.1, we derive \[\mathbb{E}_{\{y^{\dagger}_{i}\}\sim F^{\dagger}_{1}\mu}\big{[} \mathbb{E}_{y^{\dagger}\sim F^{\dagger}_{1}\mu}[\mathrm{III}]\big{]} \leq 2\\|F_{\mathrm{nn}}^{-1}-F^{-1}\\|_{\mathcal{L}^{2}(F(K))}^{2} \leq 2c^{\prime}_{\mathrm{nn}}N^{-\frac{2}{3}},\] \[\mathbb{E}_{\{y^{\dagger}_{i}\}\sim F^{\dagger}_{1}\mu}\big{[} \mathbb{E}_{y^{\dagger}\sim F^{\dagger}_{1}\mu}[\mathrm{IV}]\big{]} \leq 4\big{(}L^{2}_{F^{\dagger-1}}c_{F^{\dagger}_{1}\mu}+c_{\mu} \big{)}\sqrt{d}N^{-\frac{1}{2}}+4\Big{(}L^{2}_{F^{\dagger-1}}\sum_{j=d+1}^{ \infty}\lambda_{j}^{\mathcal{Y}}+\sum_{j=d+1}^{\infty}\lambda_{j}^{\mathcal{X }}\Big{)}.\] Combining the preceding estimates completes the proof of the theorem. Remark 3.2*.: _Theorem 3.2 gives only the existence of an INN that approximates the forward and inverse maps simultaneously to a certain tolerance. It does not address the important issue of realizing the approximation in practice via optimizing the empirical loss $\mathcal{L}(\theta)$, for which one has to analyze also the statistical error and optimization error. The former would indicate how many samples are needed in order to achieve the tolerance, and the latter is notoriously challenging due to the complex landscape of the associated optimization problem. We leave the study of these important errors to future works._ ## 4 Numerical experiments In this section, we present preliminary numerical results to showcase the feasibility of the proposed INN based approach to simultaneously approximate the forward and inverse processes of a bi-Lipschitz map on infinite-dimensional spaces. Consider the following second-order elliptic PDE on a smooth bounded domain $D$ (with a boundary $\partial D$): \[\begin{cases}-\nabla\cdot(u(x)\nabla y(x))=f,&\text{in }D,\\ y=0,&\text{on }\partial D,\end{cases} \tag{4.1}\] where $u\in L^{\infty}(\Omega)$ is the diffusion coefficient in the PDE, and $f\in L^{2}(D)$ is the given source term. The diffusion coefficient $u$ is assumed to satisfy the standard ellipticity and boundedness assumption so that for any given $u$, there exists a unique solution $y\equiv y(u)\in H^{1}_{0}(\Omega)$. The map $F$ of interest is defined by $F:u\mapsto y$. Note that this setting has been extensively studied in the literature [14], and it often serves as a model problem for parameter identification of partial differential equations. In the numerical experiments below, we take a unit square $D=(0,1)^{2}$, and $\mathrm{fix}\;f\equiv 1$. To represent the diffusion coefficient $u$, we employ truncated Karhunen-Loeve expansion [19, 5]. We simulate $M=10000$ samples of the coefficient $u$, generated by the following Fourier representation \[u(x,\xi)=2+\sum_{i,j=1}^{20}\frac{\xi_{i,j}}{i^{3}+j^{3}}\cos(i\pi x_{1})\cos (j\pi x_{2}),\quad x=[x_{1},x_{2}]\in D,\] where $\xi=[\xi_{1,1},\xi_{1,2},\cdots,\xi_{1,20},\xi_{2,1},\cdots,\xi_{20,20}]\in \mathbb{R}^{20\times 20}$ and each $\xi_{i,j}$ follows the standard normal distribution $N(0,1)$ (for each entry independently). For each realization $u$, we discretize the boundary value problem (4.1) with a mesh size $h=\frac{1}{50}$ using the standard piecewise linear Galerkin finite element method, implemented in FEniCS [37], to obtain the corresponding output $y_{h}\in\mathbb{R}^{51\times 51}$. For the ease of exposition, we apply a one-to-one transformation $T$ on $u$, given by $T(u)=\xi\in\mathbb{R}^{20\times 20}$, to transform the infinite-dimensional inputs to a finite-dimensional Euclidean space without additional discretization. Then we reduce the dimensionalities of both inputs $\xi$ and outputs $y_{h}$ by PCA (using the implementation in scikit-learn, available from [https://github.com/scikit-learn/scikit-learn/releases/tag/1.2.2](https://github.com/scikit-learn/scikit-learn/releases/tag/1.2.2)) with a reduced dimensionality $d=10$ (which capture more than $99\%$ of the spectral energy) and obtain the truncated inputs $\hat{u}=[\hat{u}_{1},\cdots,\hat{u}_{10}]\in\mathbb{R}^{10}$ and truncated outputs $\hat{y}=[\hat{y}_{1},\cdots,\hat{y}_{10}]\in\mathbb{R}^{10}$. We also record the $10$ largest singular values of the empirical covariance to form normalized, in terms of the $\ell^{1}$-norm, weight vectors $w_{u}\in\mathbb{R}^{10}$ and $w_{y}\in\mathbb{R}^{10}$. This normalization amounts to working in suitable Sobolev space scales so that the map is nearly bi-Lipschitz, in accordance with the known stability estimates [26]. In our experiments, we use an identical NN architecture with 3 blocks. One block of the forward process and inverse process is sketched in Fig. 2, which are commonly used in an affine coupling layer [17, 2]. In the figure, we define the maps $g_{o}$ (odd part), $g_{e}$ (even part) and $g_{c}$ (combination) as \[g_{o}(u) =[u_{1},u_{3},\cdots,u_{9}],\quad\forall u\in\mathbb{R}^{10},\] \[g_{e}(u) =[u_{2},u_{4},\cdots,u_{10}],\quad\forall u\in\mathbb{R}^{10},\] \[g_{c}(u^{\prime},u^{\prime\prime}) =[u^{\prime}_{1},u^{\prime\prime}_{1},u^{\prime}_{2},u^{\prime \prime}_{2},\cdots,u^{\prime}_{5},u^{\prime\prime}_{5}],\quad\forall u^{\prime },u^{\prime\prime}\in\mathbb{R}^{5}.\] Let $\mathcal{T}_{r}$ be a training dataset of size $N_{t}=\|\mathcal{T}_{r}\|$ (taken to be 100, 500, and 1000), and $\mathcal{T}_{e}$ be the test data set of size $\|\mathcal{T}_{e}\|=10000$. We train the INN for the forward and inverse processes as constructed above simultaneously, denoted by $\Phi_{\text{nn}}$ and $\Phi_{\text{nn}}^{-1}$, respectively, on the data in the training set $\mathcal{T}_{r}$. We define $\hat{y}_{\text{nn}}=\Phi_{\text{nn}}(\hat{u})$ and $\hat{u}_{\text{nn}}=\Phi_{\text{nn}}^{-1}(\hat{y})$. The loss function $\ell$ that we employ for the training is defined by \[\ell(\hat{u},\hat{y})=\frac{c_{0}}{2}\sum_{(\hat{u},\hat{y})\in \mathcal{T}_{r}}\\|(\hat{u}-\hat{u}_{\text{nn}})\odot w_{u}\\|^{2}+\frac{1}{2} \sum_{(\hat{u},\hat{y})\in\mathcal{T}_{r}}\\|(\hat{y}-\hat{y}_{\text{nn}}) \odot w_{y}\\|^{2}, \tag{4.2}\] where $c_{0}$ is a penalty parameter to be specified according to the relative errors of forward and inverse processes, and $\odot$ denotes the Hadamard product between two vectors. In the experiment, we fix $c_{0}$ at $0.001$ for problem (4.1), since the magnitude of the loss of the inverse process $\sum_{(\hat{u},\hat{y})\in\mathcal{T}_{r}}\\|(\hat{u}-\hat{u}_{\text{nn}}) \odot w_{u}\\|^{2}$ is more than 100 times that of the forward process $\sum_{(\hat{u},\hat{y})\in\mathcal{T}_{r}}\\|(\hat{y}-\hat{y}_{\text{nn}}) \odot w_{y}\\|^{2}$, and the Lipschitz constant of the inverse process is much bigger than that of the forward process. The optimization problem (4.2) is minimized by the Adam algorithm [30] with a learning rate $r_{0}=0.001$. First, we compare our results to that obtained by two different 5-layer fully-connected NN (FNN) for forward and inverse processes of problem (4.1), respectively. Likewise we run Adam [30] for at most Figure 2: The forward affine coupling block with input $\hat{u}$ and output $\hat{u}^{\prime}$ (top) and the inverse affine coupling block with input $\hat{y}$ and output $\hat{y}^{\prime}$ (bottom). Both $\phi_{W}$ and $\phi_{b}$ are four-layer full connected NNs with ReLU. 500000 steps to minimize the resulting optimization problem. The numerical results are given in Table 1, where $e_{g}$ and $e_{a}$ denote respectively the smallest relative generalization error (on the test dataset $\mathcal{T}_{e}$) and the relative approximation error (of $N_{t}$ training data points) of the NN that trained on the training dataset $\mathcal{T}_{r}$ along the iteration trajectory. The numerical results show that the proposed INN approach can achieve both approximation and generalization errors comparable with that by the FNN for all sizes of training dataset for the forward process; and can also achieve the generalization accuracy largely comparable with that by the FNN for the inverse process, albeit FNN tends to do a slightly better job. Note that a carefully tuned $c_{0}$ is needed for different problem settings. In practice, we only have access to a very limited amount of training data, for which both INN and FNN can achieve acceptable accuracy. However, INN allows training forward and inverse processes simultaneously using one single NN. Now, we examine more closely the convergence behaviour of the Adam iterates for the training with 100, 500 and 1000 training data, see Fig. 3. We observe that both relative approximation and generalization errors for the forward and inverse processes decay rapidly at first several iterations and then reach a steady state. This behavior is commonly observed for many problems. Additionally it is observed that the generalization error $e_{g}$ exhibits a semi-convergence phenomenon, especially for the inverse process: it first decreases steadily up to a finite number of iterations, and then starts to increase (occasionally in a dramatic manner) as the iteration further proceeds. Hence, suitable regularization might be necessary to overcome the phenomenon, e.g., early stopping or suitable explicit regularization. In contrast, the approximation error remains fairly stable throughout, indicating a stable optimization process. Compared with the inverse process, the forward process tends to be more stable numerically, since the forward map $F$ enjoys a far more favorable stability estimate and the inverse map $F^{-1}$ enjoys only conditional (Holder type) stability estimates (see [26] and references therein for details). Finally, the results in Figs. 4 and 5 indicate that the training data size $N_{t}=500$ is sufficient for problem (4.1) to achieve reasonable accuracy. A smaller training set yields lower approximation errors, but performs worse on the generalization; Larger training datasets lead to comparable performance in terms of both approximation and generalization. This observation agrees well with the intuition that within the supervised learning paradigm, often one needs a lot of training data in order to achieve good performance of the trained NNs.
2310.11762	A Quasi-Wasserstein Loss for Learning Graph Neural Networks	When learning graph neural networks (GNNs) in node-level prediction tasks, most existing loss functions are applied for each node independently, even if node embeddings and their labels are non-i.i.d. because of their graph structures. To eliminate such inconsistency, in this study we propose a novel Quasi-Wasserstein (QW) loss with the help of the optimal transport defined on graphs, leading to new learning and prediction paradigms of GNNs. In particular, we design a ``Quasi-Wasserstein'' distance between the observed multi-dimensional node labels and their estimations, optimizing the label transport defined on graph edges. The estimations are parameterized by a GNN in which the optimal label transport may determine the graph edge weights optionally. By reformulating the strict constraint of the label transport to a Bregman divergence-based regularizer, we obtain the proposed Quasi-Wasserstein loss associated with two efficient solvers learning the GNN together with optimal label transport. When predicting node labels, our model combines the output of the GNN with the residual component provided by the optimal label transport, leading to a new transductive prediction paradigm. Experiments show that the proposed QW loss applies to various GNNs and helps to improve their performance in node-level classification and regression tasks. The code of this work can be found at \url{https://github.com/SDS-Lab/QW_Loss}.	Minjie Cheng, Hongteng Xu	2023-10-18T07:39:05Z	http://arxiv.org/abs/2310.11762v4	# A Quasi-Wasserstein Loss for Learning Graph Neural Networks ###### Abstract When learning graph neural networks (GNNs) in node-level prediction tasks, most existing loss functions are applied for each node independently, even if node embeddings and their labels are non-i.i.d. because of their graph structures. To eliminate such inconsistency, in this study we propose a novel Quasi-Wasserstein (QW) loss with the help of the optimal transport defined on graphs, leading to new learning and prediction paradigms of GNNs. In particular, we design a "Quasi-Wasserstein" distance between the observed multi-dimensional node labels and their estimations, optimizing the label transport defined on graph edges. The estimations are parameterized by a GNN in which the optimal label transport may determine the graph edge weights optionally. By reformulating the strict constraint of the label transport to a Bregman divergence-based regularizer, we obtain the proposed Quasi-Wasserstein loss associated with two efficient solvers learning the GNN together with optimal label transport. When predicting node labels, our model combines the output of the GNN with the residual component provided by the optimal label transport, leading to a new transductive prediction paradigm. Experiments show that the proposed QW loss applies to various GNNs and helps to improve their performance in node-level classification and regression tasks. ## 1 Introduction Graph neural network (GNN) plays a central role in many graph learning tasks, such as social network analysis (Qiu et al., 2018; Fan et al., 2019; Zhang et al., 2023), molecular modeling (Satorras et al., 2021; Jiang et al., 2021; Wang et al., 2022), transportation forecasting (Wang et al., 2020; Li and Zhu, 2021), and so on. Given a graph with node features, a GNN embeds the graph nodes by exchanging and aggregating the node features, whose implementation is based on message-passing operators in the spatial domain (Niepert et al., 2016; Kipf and Welling, 2017; Velickovic et al., 2018) or graph filters in the spectral domain (Defferrard et al., 2016; Chien et al., 2021; Bianchi et al., 2021; He et al., 2021). When some node labels are available, we can learn the GNN in a node-level semi-supervised learning (Yang et al., 2016; Kipf and Welling, 2017; Xu et al., 2019), optimizing the node embeddings to predict the observed labels. This learning framework has achieved encouraging performance in many node-level prediction tasks, e.g., node classification (Sen et al., 2008; McAuley et al., 2015). When applying the above node-level GNN learning framework, existing work often leverages a loss function (e.g., the cross-entropy loss) to penalize the discrepancy between each node's label and the corresponding estimation. Here, some inconsistency between the objective design and the intrinsic data structure arises -- the objective of learning a GNN is implemented as the summation of all the nodes' loss functions, which is often applied for i.i.d. data, but the node embeddings and their labels are non-i.i.d. in general because of the underlying graph structure and the information aggregation achieved by the GNN. As a result, the current objective treats the losses of individual nodes independently and evenly, even if the nodes in the graph are correlated and have different significance for learning the GNN. Such inconsistency may lead to sub-optimal GNNs in practice, but to our knowledge none of existing work considers this issue in-depth. To eliminate the inconsistency, we leverage computational optimal transport techniques (Peyre and Cuturi 2019), proposing a new objective called Quasi-Wasserstein (QW) loss for learning GNNs. As illustrated in Figure 1, given partially-observed node labels and their estimations parametrized by a GNN, we consider the optimal transport between them and formulate the problem as the aggregation of the Wasserstein distances (Frogner et al. 2015) corresponding to all label dimensions. This problem can be equivalently formulated as a label transport minimization problem (Essid and Solomon 2018, Facca and Benzi 2021) defined on the graph, leading to the proposed QW loss. By minimizing this loss, we can jointly learn the optimal label transport and the GNN parametrizing the label estimations. This optimization problem can be solved efficiently by Bregman divergence-based algorithms, e.g., Bregman ADMM (Wang and Banerjee 2014, Xu 2020). Optionally, through Figure 1: The scheme of our QW-loss and the corresponding learning paradigm. Given a graph whose node features are denoted as blue circles and partially-observed node labels are denoted as blue stems, a GNN embeds the graph nodes and outputs estimated labels (denoted as orange stems). By minimizing the QW loss, we obtain the optimal label transport (denoted as the dotted red arrows on the graph edges) between the real and estimated node labels. Optionally, the optimal label transport can be used to determine the weights of graph edges (through an edge weight predictor). The final predictions are the combinations of the optimal label transport and the estimated node labels. a multi-layer perceptron (MLP), we can determine the edge weights of the graph based on optimal label transport, leading to a GNN with learnable edge weights. The contributions of this study include the following two points: * A theoretically-solid loss without the inconsistency issue. The QW loss provides a new optimal transport-based loss for learning GNNs, which considers the labels and estimations of graph nodes jointly. Without the questionable i.i.d. assumption, it eliminates the inconsistency issue mentioned above. In theory, we demonstrate that the QW loss is a valid metric for the node labels defined on graphs. Additionally, the traditional objective function for learning GNNs can be treated as a special case of our QW loss. We further demonstrate that applying our QW loss reduces data fitting errors in the training phase. * New learning and prediction paradigms. Different from the existing methods that combine GNNs with label propagation mechanisms (Huang et al., 2021; Wang and Leskovec, 2021; Dong et al., 2021), the QW loss provides a new way to combine node embeddings with label information in both training and testing phases. In particular, Bregman divergence-based algorithms are applied to learn the model, and the final model consists of the GNN and a residual component provided by the optimal label transport. When predicting node labels, the model combines the estimations provided by the GNN with the complementary information from the optimal label transport, leading to a new transductive prediction paradigm. Experiments demonstrate that our QW loss applies to various GNNs and helps to improve their performance in various node-level classification and regression tasks. ## 2 Related work ### Graph Neural Networks Graph neural networks can be coarsely categorized into two classes. The GNNs in the first class apply spatial convolutions to graphs (Niepert et al., 2016). The representative work includes the graph convolutional network (GCN) in (Kipf and Welling, 2017), the graph attention network (GAT) in (Velickovic et al., 2018), and their variants (Zhuang and Ma, 2018; Wang et al., 2019; Fu et al., 2020). The GNNs in the second class achieve graph spectral filtering (Guo et al., 2023). They are often designed based on a polynomial basis, such as ChebNet (Defferrard et al., 2016) and its variants (He et al., 2022), GPR-GNN (Chien et al., 2021), and BernNet (He et al., 2021). Besides approximated by the polynomial basis, the spectral GNNs can be learned by other strategies, e.g., Personalized PageRank in APPNP (Gasteiger et al., 2019), graph optimization functions in GNN-LF/HF (Zhu et al., 2021), ARMA filters (Bianchi et al., 2021), and diffusion kernel-based filters (Klicpera et al., 2019; Xu et al., 2019; Du et al., 2022). The above spatial and spectral GNNs are correlated because a spatial convolution always corresponds to a graph spectral filter (Balcilar et al., 2021). For example, GCN (Kipf and Welling, 2017) can be explained as a low-pass filter achieved by a first-order Chebyshev polynomial. Given a graph with some labeled nodes, we often learn the above GNNs in a semi-supervised node-level learning framework (Kipf and Welling, 2017; Yang et al., 2016), in which the GNNs embed all the nodes and are trained under the supervision of the labeled nodes. However, as aforementioned, the objective functions used in the framework treat the graph nodes independently and thus mismatch with the non-i.i.d. nature of the data. ### Computational Optimal Transport As a powerful mathematical tool, optimal transport (OT) distance (or called Wasserstein distance under some specific settings) provides a valid metric for probability measures (Villani 2008), which has been widely used for various machine learning problems, e.g., distribution comparison (Frogner et al. 2015, Lee et al. 2019), point cloud registration (Grave et al. 2019), graph partitioning (Dong and Sawin 2020, Xu et al. 2019b), generative modeling (Arjovsky et al. 2017, Tolstikhin et al. 2018), and so on. Typically, the OT distance corresponds to a constrained linear programming problem. To approximate the OT distance with low complexity, many algorithms have been proposed, e.g., Sinkhorn-scaling (Cuturi 2013), Bregman ADMM (Wang and Banerjee 2014), Conditional Gradient (Titouan et al. 2019), and Inexact Proximal Point (Xie et al. 2020). Recently, two iterative optimization methods have been proposed to solve the optimal transport problems defined on graphs (Essid and Solomon 2018, Facca and Benzi 2021). These efficient algorithms make the OT distance a feasible loss for machine learning problems, e.g., the Wasserstein loss in (Frogner et al. 2015). Focusing on the learning of GNNs, the work in (Chen et al. 2020) proposes a Wasserstein distance-based contrastive learning method. The Gromovized Wasserstein loss is applied to learn cross-graph node embeddings (Xu et al. 2019b), graph factorization models (Vincent-Cuaz et al. 2021, Xu 2020), and GNN-based graph autoencoders (Xu et al. 2022). The above work is designed for graph-level learning tasks, e.g., graph matching, representation, classification, and clustering. Our QW loss, on the contrary, is designed for node-level prediction tasks, resulting in significantly different learning and prediction paradigms. ## 3 Proposed Method ### Motivation and Principle Denote a graph as $G(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ represents the set of nodes and $\mathcal{E}$ represents the set of edges, respectively. The graph $G$ is associated with an adjacency matrix $\mathbf{A}\in\mathbb{R}^{\|\mathcal{V}\|\times\|\mathcal{V}\|}$ and a edge weight vector $\mathbf{w}=[w_{e}]\in\mathbb{R}^{\|\mathcal{E}\|}$. The weights in $\mathbf{w}$ correspond to the non-zero elements in $\mathbf{A}$. For an unweighted graph, $\mathbf{A}$ is a binary matrix, and $\mathbf{w}$ is an all-one vector. Additionally, the nodes of the graph may have $D$-dimensional features, which are formulated as a matrix $\mathbf{X}\in\mathbb{R}^{\|\mathcal{V}\|\times D}$. Suppose that a subset of nodes, denoted as $\mathcal{V}_{L}\subset\mathcal{V}$, are annotated with $C$-dimensional labels, i.e., $\{\mathbf{y}_{v}\in\mathbb{R}^{C}\}_{v\in\mathcal{V}_{L}}$. We would like to learn a GNN to predict the labels of the remaining nodes, i.e., $\{\mathbf{y}_{v}\}_{v\in\mathcal{V}\setminus\mathcal{V}_{L}}$. The motivation for applying GNNs is based on the non-i.i.d. property of the node features and labels. Suppose that we have two nodes connected by an edge, i.e., $(v,v^{\prime})\in\mathcal{E}$, where $(\mathbf{x}_{v},\mathbf{y}_{v})$ and $(\mathbf{x}_{v^{\prime}},\mathbf{y}_{v^{\prime}})$ are their node features and labels. For each node, its neighbors' features or labels can provide valuable information to its prediction task, i.e., the conditional probability $p(\mathbf{y}_{v}\|\mathbf{x}_{v})\neq p(\mathbf{y}_{v}\|\mathbf{x}_{v},\mathbf{x}_{v^{\prime}})$ and $p(\mathbf{y}_{v}\|\mathbf{x}_{v})\neq p(\mathbf{y}_{v}\|\mathbf{x}_{v},\mathbf{y}_{v^{\prime}})$ in general. Similarly, for node pairs, their labels are often conditionally-dependent, i.e., $p(\mathbf{y}_{v},\mathbf{y}_{v^{\prime}}\|\mathbf{x}_{v},\mathbf{x}_{v^{\prime}})=p(\mathbf{y}_{v}\| \mathbf{x}_{v},\mathbf{x}_{v^{\prime}},\mathbf{y}_{v^{\prime}})p(\mathbf{y}_{v^{\prime}}\|\mathbf{ x}_{v},\mathbf{x}_{v^{\prime}})\neq p(\mathbf{y}_{v}\|\mathbf{x}_{v},\mathbf{x}_{v^{\prime}})p(\mathbf{y}_{v ^{\prime}}\|\mathbf{x}_{v},\mathbf{x}_{v^{\prime}})$. More generally, for all node labels, we have \[p(\{\mathbf{y}_{v}\}_{v\in\mathcal{V}}\|\mathbf{X},\mathbf{A})\neq\,\prod\nolimits_{v\in \mathcal{V}}p(\mathbf{y}_{v}\|\mathbf{X},\mathbf{A}), \tag{1}\] Ideally, we shall learn a GNN to maximize the conditional probability of all labeled nodes, i.e., $\max p(\{\mathbf{y}_{v}\}_{v\in\mathcal{V}_{L}}\|\mathbf{X},\mathbf{A})$. In practice, however, most existing methods formulate the node-level learning paradigm of the GNN as \[\max_{\theta}\prod\nolimits_{v\in\mathcal{V}_{L}}p(\mathbf{y}_{v}\|\mathbf{X},\mathbf{A} ;\theta)\Leftrightarrow\,\min_{\theta}\,\sum\nolimits_{v\in\mathcal{V}_{L}} \psi(g_{v}(\mathbf{X},\mathbf{A};\theta),\ \mathbf{y}_{v}). \tag{2}\] Here, $g$ is a graph neural network whose parameters are denoted as $\theta$. Taking the adjacency matrix $\mathbf{A}$ and the node feature matrix $\mathbf{X}$ as input, the GNN $g$ predicts the node labels. $g_{v}(\mathbf{X},\mathbf{A};\theta)$ represents the estimation of the node $v$'s label achieved by the GNN, which is also denoted as $\hat{\mathbf{y}}_{v}$. Similarly, we denote $g_{\mathcal{V}}(\mathbf{X},\mathbf{A};\theta)$ as the estimated labels for the node set $\mathcal{V}$ in the following content. The loss function $\psi:\mathbb{R}^{C}\times\mathbb{R}^{C}\mapsto\mathbb{R}$ is defined in the node level. In node-level classification tasks, it is often implemented as the cross-entropy loss or the KL-divergence (i.e., $p(\mathbf{y}_{v}\|\mathbf{X},\mathbf{A};\theta)$ is modeled by the softmax function). In node-level regression tasks, it is often implemented as the least-square loss (i.e., $p(\mathbf{y}_{v}\|\mathbf{X},\mathbf{A};\theta)$ is assumed to be the Gaussian distribution). The loss in (2) assumes the node labels to be conditionally-independent with each other, which may be too strong in practice and inconsistent with the non-i.i.d. property of graph-structured data shown in (1). To eliminate such inconsistency, we should treat node labels as a set rather than independent individuals, developing a set-level loss to penalize the discrepancy between the observed labels and their estimations globally, i.e., \[\min_{\theta}\text{Loss}(g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta),\{\mathbf{y}_{ v}\}_{v\in\mathcal{V}_{L}}). \tag{3}\] In the following content, we will design such a loss with theoretical supports, based on the optimal transport on graphs. ### Optimal Transport on Graphs Suppose that we have two measures on a graph $G(\mathcal{V},\mathcal{E})$, denoted as $\mathbf{\mu}\in[0,\infty)^{\|\mathcal{V}\|}$ and $\mathbf{\gamma}\in[0,\infty)^{\|\mathcal{V}\|}$, respectively. The element of each measure indicates the "mass" of a node. Assume the two measures to be balanced, i.e., $\langle\mathbf{\gamma}-\mathbf{\mu},\mathbf{1}_{\|\mathcal{V}\|}\rangle=0$, where $\langle\cdot,\cdot\rangle$ is the inner product operator. The optimal transport, or called the 1-Wasserstein distance (Villani, 2008), between them is defined as \[W_{1}(\mathbf{\mu},\mathbf{\gamma}):=\,\min_{\mathbf{T}\in\Pi(\mathbf{\mu},\mathbf{\gamma})} \langle\mathbf{D},\;\mathbf{T}\rangle=\,\min_{\mathbf{T}\in\Pi(\mathbf{\mu},\mathbf{\gamma})}\, \sum\nolimits_{v,v^{\prime}\in\mathcal{V}\times\mathcal{V}}t_{vv^{\prime}}d_ {vv^{\prime}}, \tag{4}\] where $\mathbf{D}=[d_{vv^{\prime}}]\in\mathbb{R}^{\|\mathcal{V}\|\times\|\mathcal{V}\|}$ represents the shortest path distance matrix, and $\Pi(\mathbf{\mu},\mathbf{\gamma})=\{\mathbf{T}\geq\mathbf{0}\|\mathbf{T}\mathbf{1}_{\|\mathcal{V}\|}=\bm {\mu},\mathbf{T}^{\top}\mathbf{1}_{\|\mathcal{V}\|}=\mathbf{\gamma}\}$ represents the set of all valid doubly stochastic matrices. Each $\mathbf{T}=[t_{vv^{\prime}}]\in\Pi(\mathbf{\mu},\mathbf{\gamma})$ is a transport plan matrix. The optimization problem in (4) corresponds to finding the optimal transport plan $\mathbf{T}^{}=[t_{vv^{\prime}}^{}]$ to minimize the "cost" of changing $\mathbf{\mu}$ to $\mathbf{\gamma}$, in which the cost is measured as the sum of "mass" $t_{vv^{\prime}}$ moved from node $v$ to node $v^{\prime}$ times distance $d_{vv^{\prime}}$. #### 3.2.1 Wasserstein Distance for Vectors on A Graph For the optimal transport problem defined on graphs, we can simplify the problem in (4) by leveraging the underlying graph structures. As shown in (Essid and Solomon, 2018; Santambrogio, 2015), given a graph $G(\mathcal{V},\mathcal{E})$, we can define a sparse matrix $\mathbf{S}_{\mathcal{V}}=[s_{ve}]\in\{0,\pm 1\}^{\|\mathcal{V}\|\times\|\mathcal{E}\|}$ to indicating the graph topology. For node $v$ and edge $e$, the corresponding element in $\mathbf{S}_{\mathcal{V}}$ is \[s_{ve}=\begin{cases}1&\text{if $v$ is ``head'' of edge $e$}\\ -1&\text{if $v$ is ``tail'' of edge $e$}\\ 0&\text{otherwise.}\end{cases} \tag{5}\] When $G$ is directed, the "head" and "tail" of each edge are predefined. When $G$ is undirected, we can randomly define each edge's "head" and "tail". Accordingly, the 1-Wasserstein distance in (4) can be equivalently formulated as a minimum-cost flow problem: \[W_{1}(\mathbf{\mu},\mathbf{\gamma})=\,\min_{\mathbf{f}\in\Omega(\mathbf{S}_{\mathcal{V}},\mathbf{ \mu},\mathbf{\gamma})}\\|\text{diag}(\mathbf{w})\mathbf{f}\\|_{1}, \tag{6}\] where $\mathrm{diag}(\mathbf{w})$ is a diagonal matrix constructed by the edge weights of the graph. The vector $\mathbf{f}=[f_{e}]\in\Omega(\mathbf{S}_{\mathcal{V}},\mathbf{\mu},\mathbf{\gamma})$ is the flow indicating the mass passing through each edge. Accordingly, the cost corresponding to edge $e$ is represented as the distance $w_{e}$ times the mass $f_{e}$, and the sum of all the costs leads to the objective function in (6). The feasible domain of the flow vector is defined as \[\Omega(\mathbf{S}_{\mathcal{V}},\mathbf{\mu},\mathbf{\gamma})=\mathcal{U}^{\| \mathcal{E}\|}\cap\{\mathbf{f}\ \|\ \mathbf{S}_{\mathcal{V}}\mathbf{f}=\mathbf{\gamma}-\mathbf{\mu}\},\quad\text{ where }\mathcal{U}=\begin{cases}[0,\infty),&\text{Directed }G,\\ \mathbb{R},&\text{Undirected }G.\end{cases} \tag{7}\] The constraint $\mathbf{S}_{\mathcal{V}}\mathbf{f}=\mathbf{\gamma}-\mathbf{\mu}$ ensures that the flow on all the edges leads to the change from $\mathbf{\mu}$ to $\mathbf{\gamma}$. The flow in the directed graph is nonnegative, which can only pass through each edge from "head" to "tail". On the contrary, the undirected graph allows the mass to transport from "tail" to "head". By solving (6), we find the optimal flow $\mathbf{f}^{}$ (or the so-called optimal transport on edges) that minimizes the overall cost $\\|\mathrm{diag}(\mathbf{w})\mathbf{f}\\|_{1}$. #### 3.2.2 Partial Wasserstein Distance for Vectors on a Graph When only a subset of nodes (i.e., $\mathcal{V}_{L}\subset\mathcal{V}$) have observable signals, we can define a _partial Wasserstein distance_ on the graph by preserving the constraints relevant to $\mathcal{V}_{L}$: for $\mathbf{\mu},\mathbf{\gamma}\in\mathbb{R}^{\|\mathcal{V}_{L}\|}$, we have \[W_{1}^{(P)}(\mathbf{\mu},\mathbf{\gamma})=\,\min\nolimits_{\mathbf{f}\in \Omega(\mathbf{S}_{\mathcal{V}_{L}},\mathbf{\mu},\mathbf{\gamma})}\\|\mathrm{diag}(\mathbf{w}) \mathbf{f}\\|_{1}, \tag{8}\] where $\mathbf{S}_{\mathcal{V}_{L}}\in\{0,\pm 1\}^{\|\mathcal{V}_{L}\|\times\|\mathcal{E}\|}$ is a submatrix of $\mathbf{S}_{\mathcal{V}}$, storing the rows corresponding to the nodes with observable signals. Different from the classic Wasserstein distance, the Wasserstein distance defined on graphs is not limited for nonnegative vectors because the minimum-cost flow formulation in (6) is _shift-invariant_, i.e., $\forall\mathbf{\delta}\in\mathbb{R}^{\|\mathcal{V}\|}$, $W_{1}(\mathbf{\mu},\mathbf{\gamma})=W_{1}(\mathbf{\mu}-\mathbf{\delta},\mathbf{\gamma}-\mathbf{\delta})$. The partial Wasserstein distance in (8) can be viewed as a generalization of (6), and thus it holds the shift-invariance as well. Moreover, given an undirected graph, we can prove that both the $W_{1}$ in (6) and the $W_{1}^{(P)}$ in (8) can be valid metrics in the spaces determined by the topology of an undirected graph (See Appendix A for a complete proof). Theorem 1* (The Validness as A Metric).: _Given an undirected graph $G(\mathcal{V},\mathcal{E})$, with edge weights $\mathbf{w}\in[0,\infty)^{\|\mathcal{E}\|}$ and a matrix $\mathbf{S}_{\mathcal{V}}\in\{0,\pm 1\}^{\|\mathcal{V}\|\times\mathcal{E}}$ defined in (5), the $W_{1}$ in (6) is a metric in $Range(\mathbf{S}_{\mathcal{V}})$, and the $W_{1}^{(P)}$ in (8) is a metric in $Range(\mathbf{S}_{\mathcal{V}_{L}})$, $\forall\mathcal{V}_{L}\in\mathcal{V}$._ The flow vector $\mathbf{f}$ in (8) has fewer constraints than that in (6), so the relation between $W_{1}$ and $W_{1}^{(P)}$ obeys the following theorem. Theorem 2 (The Monotonicity).: _Given an undirected graph $G(\mathcal{V},\mathcal{E})$, with edge weights $\mathbf{w}\in[0,\infty)^{\|\mathcal{E}\|}$ and a matrix $\mathbf{S}_{\mathcal{V}}\in\{0,\pm 1\}^{\|\mathcal{V}\|\times\mathcal{E}}$ defined in (5), we have_ \[W_{1}(\mathbf{\mu},\mathbf{\gamma})\geq W_{1}^{(P)}(\mathbf{\mu}_{\mathcal{V}^{\prime}}, \mathbf{\gamma}_{\mathcal{V}^{\prime\prime}})\geq W_{1}^{(P)}(\mathbf{\mu}_{\mathcal{ V}^{\prime\prime}},\mathbf{\gamma}_{\mathcal{V}^{\prime\prime}})\geq 0,\quad\forall\mathbf{\mu},\mathbf{ \gamma}\in Range(\mathbf{S}_{\mathcal{V}}),\ \mathcal{V}^{\prime\prime}\subset\mathcal{V} ^{\prime}\subset\mathcal{V}, \tag{9}\] _where $\mathbf{\mu}_{\mathcal{V}^{\prime}}$ denotes the subvector of $\mathbf{\mu}$ corresponding to the set $\mathcal{V}^{\prime}$._ ### Learning GNNs with Quasi-Wasserstein Loss Inspired by the optimal transport on graphs, we propose a Quasi-Wasserstein loss for learning GNNs. Given a partially-labeled graph $G$, we formulate observed labels as a matrix $\mathbb{R}^{\|\mathcal{V}_{L}\|\times C}$, where $\mathbf{y}^{(c)}_{\mathcal{V}_{L}}\in\mathbb{R}^{\|\mathcal{V}_{L}\|}$ represents the labels in $c$-th dimension. The estimated labels generated by a GNN $g$ are represented as $\widehat{\mathbf{Y}}_{\mathcal{V}_{L}}=[\hat{\mathbf{y}}^{(c)}_{\mathcal{V}_{L}}]:=g_{ \mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta)$. Our QW loss is defined as \[\begin{split} QW(\widehat{\mathbf{Y}}_{\mathcal{V}_{L}},\mathbf{Y}_{ \mathcal{V}_{L}})&=\,\sum\nolimits_{c=1}^{C}W_{1}^{(P)}( \hat{\mathbf{y}}^{(c)}_{\mathcal{V}_{L}},\mathbf{y}^{(c)}_{\mathcal{V}_{L}})\\ &=\,\sum\nolimits_{c=1}^{C}\,\min_{\mathbf{f}^{(c)}\in\Omega(\mathbf{S} _{\mathcal{V}_{L}},\,\,\hat{\mathbf{y}}^{(c)}_{\mathcal{V}_{L}},\,\,\hat{\mathbf{y}}^{ (c)}_{\mathcal{V}_{L}})}\\|\text{diag}(\mathbf{w})\mathbf{f}^{(c)}\\|_{1}\\ &=\,\min_{\mathbf{F}\in\Omega_{C}(\mathbf{S}_{\mathcal{V}_{L}},\,\,g_{ \mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta),\,\,\mathcal{Y}_{\mathcal{V}_{L}})}\\| \text{diag}(\mathbf{w})\mathbf{F}\\|_{1},\end{split} \tag{10}\] where $\mathbf{F}=[\mathbf{f}^{(c)}]$ is the flow matrix, and its feasible domain $\Omega_{C}=\mathcal{U}^{\|\mathcal{E}\|\times C}\cap\{\mathbf{F}\|\mathbf{S}_{\mathcal{ V}_{L}}\mathbf{F}=\mathbf{Y}_{\mathcal{V}_{L}}-g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A}; \theta)\}$. $W_{1}^{(P)}(\hat{\mathbf{y}}^{(c)}_{\mathcal{V}_{L}},\mathbf{y}^{(c)}_{\mathcal{V}_{L}})$ is the partial Wasserstein distance between the observed and estimated labels in the $c$-th dimension. The QW loss is the summation of the $C$ partial Wasserstein distances, and it can be rewritten as the summation of $C$ minimum-cost flow problems. As shown in the third row of (10), these $C$ problems can be formulated as a single optimization problem, and the variables of the problems are aggregated as the flow matrix $\mathbf{F}$. Based on Theorem 1, our QW loss is a metric for the matrices whose columns are in $Range(\mathbf{S}_{\mathcal{V}_{L}})$. The optimal flow matrix $\mathbf{F}^{}=[\mathbf{f}^{(c)}]$ is called optimal label transport in this study, in which the column $\mathbf{f}^{(c)}$ indicating the optimal transport on graph edges between the observed labels and their estimations in the $c$-th dimension. Note that, we call the proposed loss "Quasi-Wasserstein" because $i)$ it is not equal to the classic 1-Wasserstein distance between the label sets $\widehat{\mathbf{Y}}_{\mathcal{V}_{L}}$ and $\mathbf{Y}_{\mathcal{V}_{L}}$(Peyre and Cuturi, 2019), and $ii)$ the optimal label transport $\mathbf{F}^{}$ cannot be reformulated as the optimal transport $\mathbf{T}^{}$ obtained by (4). When $\mathbf{F}^{}=\mathbf{0}$, we have $QW(\widehat{\mathbf{Y}}_{\mathcal{V}_{L}},\mathbf{Y}_{\mathcal{V}_{L}})=0$ and accordingly, $\widehat{\mathbf{Y}}_{\mathcal{V}_{L}}=\mathbf{Y}_{\mathcal{V}_{L}}$, which means that the GNN $g$ perfectly estimates the observed labels. Therefore, the QW loss provides a new alternative for the training loss of GNN. Different from the traditional loss function in (2), the QW loss treats observed node labels and their estimations as two sets and measures their discrepancy accordingly, which provides an effective implementation of the loss in (3). Applying the QW loss to learn a GNN $g$ results in the following constrained optimization problem: \[\min_{\theta}QW(g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta),\mathbf{Y}_{\mathcal{V}_ {L}})=\,\min_{\theta}\,\min_{\mathbf{F}\in\Omega_{C}(\mathbf{S}_{\mathcal{V}_{L}},\,g_ {\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta),\,\,\mathbf{Y}_{\mathcal{V}_{L}})}\\|\text{ diag}(\mathbf{w})\mathbf{F}\\|_{1}. \tag{11}\] To solve it effectively, we consider the following two algorithms. #### 3.3.1 Bregman Divergence-based Approximate Solver By relaxing the constraint $\{\mathbf{F}\|\mathbf{S}_{\mathcal{V}_{L}}\mathbf{F}=\mathbf{Y}_{\mathcal{V}_{L}}-g_{\mathcal{V} _{L}}(\mathbf{X},\mathbf{A};\theta)\}$ to a Bregman divergence-based regularizer (Benamou et al., 2015), we can reformulate (11) to the following problem: \[\begin{split}\min_{\theta,\,\,\mathbf{F}\in\mathcal{U}^{\|\mathcal{E}\| \times C}}\\|\text{diag}(\mathbf{w})\mathbf{F}\\|_{1}+\lambda\underbrace{B_{\phi}(g_{ \mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta)+\mathbf{S}_{\mathcal{V}_{L}}\mathbf{F},\,\, \mathbf{Y}_{\mathcal{V}_{L}})}_{\sum_{v\in\mathcal{V}_{L}}\psi(g_{v}(\mathbf{X},\mathbf{A}; \theta)+\mathbf{S}_{v}\mathbf{F},\,\,\mathbf{y}_{v})}.\end{split} \tag{12}\] Here, $B_{\phi}(x,y)=\phi(x)-\phi(y)-\langle\nabla\phi(y),x-y\rangle$ is the Bregman divergence defined based on the strictly convex function $\phi$, and the hyperparameter $\lambda>0$ controls its significance. In node classification tasks, we can set $\phi$ as an entropic function, and the Bregman divergence becomes the KL-divergence. In node regression tasks, we can set $\phi$ as a least-square loss, and the Bregman divergence becomes the least-square loss. Therefore, as shown in (12), the Bregman divergence $B_{\phi}$ can be implemented based on commonly-used loss functions, e.g., the $\psi$ in (2). Algorithm 1 shows the algorithmic scheme in details. For undirected graphs, the $\mathcal{U}$ in (12) is $\mathbb{R}$, and accordingly, (12) becomes a unconstrained optimization problem. We can solve it efficiently by gradient descent. For directed graphs, we just need to add a projection step when updating the flow matrix $\mathbf{F}$, leading to the projected gradient descent algorithm. #### 3.3.2 Bregman ADMM-based Exact Solver Algorithm 1 solves (11) approximately -- because of relaxing the strict equality constraint to a regularizer, the solution of (12) often cannot satisfy the original equality constraint. To solve (11) exactly, we further develop a learning method based on the Bregman ADMM (Bremgan Alternating Direction Method of Multipliers) algorithm (Wang and Banerjee 2014). In particular, we can rewrite (11) in the following augmented Lagrangian form: \[\begin{split}\min_{\theta,\ \mathbf{F}\in\mathcal{U}^{\|\mathcal{E}\| \times C},\ \mathbf{Z}\in\mathbb{R}^{\|\mathcal{V}_{L}\|\times C}}&\\|\text{diag}( \mathbf{w})\mathbf{F}\\|_{1}+\langle\mathbf{Z},\ g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta)+ \mathbf{S}_{\mathcal{V}_{L}}\mathbf{F}-\mathbf{Y}_{\mathcal{V}_{L}}\rangle\\ &+\lambda B_{\phi}(g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta)+\mathbf{ S}_{\mathcal{V}_{L}}\mathbf{F},\ \mathbf{Y}_{\mathcal{V}_{L}}),\end{split} \tag{13}\] where $\mathbf{Z}\in\mathbb{R}^{\|\mathcal{V}_{L}\|\times C}$ is the dual variable, the second term in (13) is the Lagrangian term corresponding to the equality constraint, and the third term in (13) is the augmented term implemented as the Bregman divergence. ``` 0: A graph $G$, its adjacency matrix $\mathbf{A}$ (edge weights $\mathbf{w}$), node features $\mathbf{X}$, observed labels $\mathbf{Y}_{\mathcal{V}_{L}}$, the number of inner iterations $J$. 1: Initialize $\theta^{(0)}$ and $\mathbf{F}^{(0)}$ randomly and set $\mathbf{Z}^{(0)}=\mathbf{0}_{\|\mathcal{V}_{L}\|\times C}$. 2:while Not converge do 3: Solve (14) by Adam (Kingma and Ba 2014) with $J$ steps and obtain $\theta^{(k+1)}$. 4: Solve (15) by Adam (Kingma and Ba 2014) with $J$ steps and obtain $\mathbf{F}^{(k+1)}$. 5:if$G$ is a directed graph then$\mathbf{F}^{(k+1)}\leftarrow\text{Proj}_{\geq 0}(\mathbf{F}^{(k+1)})$ endif. 6: Obtain $\mathbf{Z}^{(k+1)}$ by (16). 7:endwhile 8:return Optimal label transport $\mathbf{F}^{}$ and model parameters $\theta^{}$. ``` Algorithm 1 Learning a GNN by solving (12) Denote the objective function in (13) as $L(\theta,\mathbf{F},\mathbf{Z})$. In the Bregman ADMM framework, we can optimize $\theta$, $\mathbf{F}$, and $\mathbf{Z}$ iteratively through alternating optimization. In the $k$-th iteration, we update the three variables via solving the following three subproblems: \[\begin{split}\theta^{(k+1)}=&\,\text{arg min}_{\theta} \,L(\theta,\mathbf{F}^{(k)},\mathbf{Z}^{(k)})\\ =&\,\text{arg min}_{\theta}\langle\mathbf{Z}^{(k)},\ g _{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta)\rangle+\lambda B_{\phi}(g_{\mathcal{ V}_{L}}(\mathbf{X},\mathbf{A};\theta)+\mathbf{S}_{\mathcal{V}_{L}}\mathbf{F}^{(k)},\ \mathbf{Y}_{\mathcal{V}_{L}}).\end{split} \tag{14}\] \[\mathbf{F}^{(k+1)}= \arg\,\min_{\mathbf{F}\in\mathcal{U}^{\|\xi\|\times C}}L(\theta^{(k+1)},\bm {F},\mathbf{Z}^{(k)}) \tag{15}\] \[= \arg\,\min_{\mathbf{F}\in\mathcal{U}^{\|\xi\|\times C}}\langle\mathbf{Z}^{(k )},\ \mathbf{S}_{\mathcal{V}_{L}}\mathbf{F}\rangle+\lambda B_{\phi}(g_{\mathcal{V}_{L}}(\mathbf{X },\mathbf{A};\theta^{(k+1)})+\mathbf{S}_{\mathcal{V}_{L}}\mathbf{F},\ \mathbf{Y}_{\mathcal{V}_{L}}).\] \[\mathbf{Z}^{(k+1)}=\mathbf{Z}^{(k)}+\lambda(g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta^{ (k+1)})+\mathbf{S}_{\mathcal{V}_{L}}\mathbf{F}^{(k+1)}-\mathbf{Y}_{\mathcal{V}_{L}}). \tag{16}\] We can find that (14) is a unconstrained optimization problem, so we can update the model parameter $\theta$ by gradient descent. Similarly, we can solve (15) and update the flow matrix $\mathbf{F}$ by gradient descent or projected gradient descent, depending on whether the graph is undirected or not. Finally, the update of the dual variable $\mathbf{Z}$ can be achieved in a closed form, as shown in (16). The Bregman ADMM algorithm solves the original problem in (12) rather than a relaxed version. In theory, with the increase of iterations, we can obtain the optimal variables that satisfy the equality constraint. Algorithm 2 shows the scheme of the Bregman ADMM-based solver. #### 3.3.3 Optional Edge Weight Prediction Some GNNs, e.g., GCN-LPA (Wang and Leskovec, 2021) and GAT (Velickovic et al., 2018), model the adjacency matrix of graph as learnable parameters. Inspired by these models, we can optionally introduce an edge weight predictor and parameterize the adjacency matrix based on the flow matrix, as shown in Figure 1. In particular, given $\mathbf{F}$, we can apply a multi-layer perceptron (MLP) to embed it to an edge weight vector and then obtain a weighted adjacency matrix. Accordingly, the learning problem becomes \[\min_{\theta,\xi}\,\min_{\mathbf{F}\in\Omega_{C}(\mathbf{S}_{\mathcal{V}_{L}},g_{ \mathcal{V}_{L}}(\mathbf{X},\mathbf{A}(\mathbf{F};\xi);\theta),\mathbf{Y}_{\mathcal{V}_{L}}) }\\|\text{diag}(\mathbf{w})\mathbf{F}\\|_{1}, \tag{17}\] where $\mathbf{A}(\mathbf{F};\xi)$ represents the adjacency matrix determined by the label transportation $\mathbf{F}$, and $\xi$ represents the parameters of the MLP. Taking the learning of $\xi$ into account, we can modify the above two solvers slightly and make them applicable for solving (17). ### Connections to Traditional Methods As discussed in Section 3.3.1 and shown in (12), the Bregman divergence $B_{\phi}$ can be implemented as the commonly-used loss function $\psi$ in (2) (e.g., the KL divergence or the least-square loss). Table 1 compares the typical setting of traditional learning methods and that of our QW loss-based method in different node-level tasks. Essentially, the traditional learning method in (2) can be viewed as a special case of our QW loss-based learning method. In particular, when setting $\mathbf{F}=\mathbf{0}_{\|\mathcal{F}\|\times C}$ and $B_{\phi}=\psi$, the objective function in (12) degrades to the objective function in (2), which treats each node independently. Similarly, when further setting the dual variable $\mathbf{Z}=\mathbf{0}_{\|\mathcal{V}_{L}\|\times C}$, the objective function in (13) degrades to the objective function in (2) as well. In theory, we demonstrate that our QW loss-based learning method can fit training data better than the traditional method does, as shown in the following theorem. \begin{table} \begin{tabular}{c\|c\|c\|c} \hline \hline Method & Setting & Node Classification & Node Regression \\ \hline \multirow{2}{}{\begin{tabular}{c} Apply the \\ loss in (2) \\ \end{tabular} } & $\psi$ & Cross-entropy or KL & Least-square \\ \cline{2-3} & Predicted $\mathbf{y}_{v}$ & $g_{v}(\mathbf{X},\mathbf{A};\theta)$, $\forall v\in\mathcal{V}\setminus\mathcal{V}_{L}$ \\ \hline \multirow{2}{}{ \begin{tabular}{c} Apply the \\ QW loss \\ \end{tabular} } & $\phi$ & Entropy & $\frac{1}{2}\\|\cdot\\|_{2}^{2}$ \\ \cline{2-3} & $B_{\phi}(=\psi)$ & KL & Least-square \\ \cline{2-3} & Predicted $\mathbf{y}_{v}$ & $g_{v}(\mathbf{X},\mathbf{A};\theta)+\mathbf{S}_{v}\mathbf{F}$, $\forall v\in\mathcal{V}\setminus\mathcal{V}_{L}$ \\ \hline \hline \end{tabular} \end{table} Table 1: Comparison between traditional methods and ours Theorem 3.: _Let $\{\theta^{\star},\mathbf{F}^{\star},\mathbf{Z}^{\star}\}$ be the global optimal solution of (13), $\{\theta^{\dagger},\mathbf{F}^{\dagger}\}$ be the global optimal solution of (12), and $\theta^{\dagger}$ be the global optimal solution of (13), $\{\theta^{\dagger},\mathbf{F}^{\dagger}\}$ be the global optimal solution of (12), and $\theta^{\dagger}$ be the global optimal solution of $\min_{\theta}B_{\phi}(g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta),\ \mathbf{Y}_{\mathcal{V}_{L}})$, we have_ \[B_{\phi}(g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta^{\star})+\mathbf{S}_{\mathcal{V} _{L}}\mathbf{F}^{\star},\ \mathbf{Y}_{\mathcal{V}_{L}})\leq B_{\phi}(g_{\mathcal{V}_{L}}(\mathbf{X},\mathbf{A}; \theta^{\dagger})+\mathbf{S}_{\mathcal{V}_{L}}\mathbf{F}^{\dagger},\ \mathbf{Y}_{\mathcal{V}_{L}})\leq B_{\phi}(g_{ \mathcal{V}_{L}}(\mathbf{X},\mathbf{A};\theta^{\dagger}),\ \mathbf{Y}_{\mathcal{V}_{L}})\] Proof.: The proof is straightforward -- $\{\theta^{\ddagger},\mathbf{0}_{\|\mathcal{E}\|\times C}\}$ is a feasible solution of (12), so the corresponding objective is equal to or larger than that obtained by $\{\theta^{\dagger},\mathbf{F}^{\dagger}\}$. Similarly, $\{\theta^{\dagger},\mathbf{F}^{\dagger},\mathbf{0}_{\|\mathcal{V}_{L}\|\times C}\}$ is a feasible solution of (13), so the corresponding objective is equal to or larger than that obtained by $\{\theta^{\star},\mathbf{F}^{\star},\mathbf{Z}^{\star}\}$. Remark. It should be noted that although the Bregman ADMM-based solver can fit training data better in theory, in the cases with distribution shifting or out-of-distribution issues, it has a higher risk of over-fitting. Therefore, in practice, we can select one of the above two solvers to optimize the GW loss, depending on their performance. In the following experimental section, we will further compare these two solvers in details. ### A New Transductive Prediction Paradigm As shown in Table 1, given the learned model $\theta^{}$ and the optimal label transport $\mathbf{F}^{}$, we predict node labels in a new transductive prediction paradigm. For $v\in\mathcal{V}\setminus\mathcal{V}_{L}$, we predict its label as \[\tilde{\mathbf{y}}_{v}:=g_{v}(\mathbf{X},\mathbf{A};\theta^{})+\mathbf{S}_{v}\mathbf{F}^{}, \tag{18}\] which combines the estimated label from the learned GNN and the residual component from the optimal label transport. It should be noted that some attempts have been made to incorporate label propagation algorithms (LPAs) (Zhu and Goldberg, 2022) into GNNs, e.g., the GCN-LPA in (Wang and Leskovec, 2021) and the FDiff-Scale in (Huang et al., 2021). The PTA in (Dong et al., 2021) demonstrates that learning a decoupled GNN is equivalent to implementing a label propagation algorithm. These methods leverage LPAs to regularize the learning of GNNs. However, in the prediction phase, they abandon the training labels and rely only on the GNNs to predict node labels, as shown in Table 1. Unlike these methods, our QW loss-based method achieves a new kind of label propagation with the help of computational optimal transport, saving the training label information in the optimal label transport and applying it explicitly in the prediction phase. ## 4 Experiments To demonstrate the effectiveness of our QW loss-based learning method, we apply it to learn GNNs with various architectures and test the learned GNNs in different node-level prediction tasks. We compare our learning method with the traditional one in (2) on their model performance and computational efficiency. For our method, we conduct a series of analytic experiments to show its robustness to hyperparameter settings and label insufficiency. All the experiments are conducted on a machine with three NVIDIA A40 GPUs, and the code is implemented based on PyTorch. ### Implementation Details #### 4.1.1 Datasets The datasets we considered consist of five homophilic graphs (i.e., Cora, Citeseer, Pubmed(Sen et al., 2008; Yang et al., 2016), Computers, and Photo(McAuley et al., 2015)) and five heterophilic graphs (i.e., Chameleon, Squirrel(Rozemberczki et al., 2021), Actor, Texas, and Cornell(Pei et al., 2020)), respectively. Following the work in (Wang and Leskovec, 2021), we categorize the graphs according to the percentage of the edges connecting the nodes of the same class (i.e., the intra-edge rate). The basic information of these graphs is shown in Table 2. Additionally, a large arXiv-year graph (Lim et al., 2021) is applied to demonstrate the efficiency of our method. The adjacency matrix of each graph is binary, so the edge weights $\mathbf{w}=\mathbf{1}_{\|\mathcal{E}\|}$. #### 4.1.2 GNN Architectures In the following experiments, the models we considered include $i)$ the representative spatial GNNs, i.e., GCN(Kipf and Welling, 2017), GAT(Velickovic et al., 2018), GIN(Xu et al., 2019), GraphSAGE(Hamilton et al., 2017), and GCN-LPA(Wang and Leskovec, 2021) that combines the GCN with the label propagation algorithm; and $ii)$ state-of-the-art spectral GNNs, i.e., APPNP(Gasteiger et al., 2019), BernNet(He et al., 2021), and ChebNetII(He et al., 2022). For a fair comparison, we set the architectures of the GNNs based on the code provided by (He et al., 2022) and configure the algorithmic hyperparameters by grid search. More details of the hyperparameter settings are in Appendix. #### 4.1.3 Learning Tasks and Evaluation Measurements For each graph, their nodes belong to different classes. Therefore, we first learn different GNNs to solve the node-level classification tasks defined on the above graphs. By default, the split ratio of each graph's nodes is 60% for training, 20% for validation, and 20% for testing, respectively. The GNNs are learned by $i)$ the traditional learning method in (2)1 and $ii)$ minimizing the proposed QW loss in (11), respectively. When implementing the QW loss, we apply either Algorithm 1 or 2, depending on their performance. Additionally, to demonstrate the usefulness of our QW loss in node-level regression tasks, we treat the node labels as one-hot vectors and fit them by minimizing the mean squared error (MSE), in which the $\psi$ in (2) and the corresponding Bregman divergence \begin{table} \begin{tabular}{c\|c c c c c\|c c c c c\|c} \hline \hline \multirow{2}{}{Model Method} & \multicolumn{6}{c\|}{Homophilic graphs} & \multicolumn{6}{c\|}{Heterophilic graphs} \\ \cline{2-13} & Cora & Citeseer & Pubmed & Computers & Photo & Squirrel & Chameleon & Actor & Texas & Cornell \\ \hline \#Nodes ($\|\mathcal{V}\|$) & 2,708 & 3,327 & 19,717 & 13,752 & 5,201 & 7,650 & 2,277 & 7,600 & 183 & 183 & Overall \\ \#Features ($D$) & 1,433 & 3,703 & 500 & 767 & 754 & 2,089 & 2,325 & 932 & 1,703 & 1,703 & Improve \\ \#Edaps ($\|\mathcal{E}\|$) & 5,278 & 4,552 & 44,324 & 25,861 & 19,081 & 19,358 & 31,371 & 26,659 & 279 & 277 & Improve \\ Intra-edge rate & 81.0\% & 73.6\% & 80.2\% & 77.7\% & 82.7\% & 22.2\% & 23.0\% & 21.8\% & 6.1\% & 12.3\% & \\ \#Classes ($C$) & 7 & 6 & 5 & 10 & 8 & 5 & 5 & 5 & 5 & 5 & 5 \\ \hline \multirow{2}{}{GCN} & (2) & 87.44${}_{10.39}$ & 79.93${}_{10.84}$ & 86.93${}_{10.29}$ & 88.42${}_{10.43}$ & 93.24${}_{10.43}$ & 65.55${}_{11.51}$ & 63.57${}_{11.06}$ & 34.00${}_{12.8}$ & 77.21${}_{11.28}$ & 61.91${}_{15.11}$ & — \\ & (2)+LPM & 86.34${}_{10.4}$ & 78.51${}_{11.22}$ & 84.72${}_{10.70}$ & 82.48${}_{10.80}$ & 88.10${}_{11.31}$ & 44.81${}_{11.81}$ & 60.90${}_{10.53}$ & 32.43${}_{11.50}$ & 78.09${}_{14.74}$ & 67.27${}_{23.50}$ & -1.36 \\ & QW & 87.88${}_{10.78}$ & 81.36${}_{10.41}$ & 87.89${}_{10.40}$ & 89.20${}_{10.41}$ & 93.81${}_{10.56}$ & 25.62${}_{10.49}$ & 68.10${}_{11.01}$ & 38.09${}_{10.59}$ & 84.10${}_{10.29}$ & 84.26${}_{10.38}$ & 4.26${}_{10.38}$ & 4.81${}_{10.29}$ & 84.26${}_{10.38}$ \\ \hline \multirow{2}{}{ GAT} & (2) & 89.20${}_{10.78}$* & 87.07${}_{10.78}$ & 87.42${}_{10.33}$ & 90.80${}_{10.36}$ & 89.35${}_{10.29}$ & 84.35${}_{10.29}$ & 85.22${}_{10.41}$ & 64.31${}_{11.21}$ & 35.68${}_{10.80}$ & 80.00${}_{11.69}$ & 89.21${}_{13.13}$ & 11.41${}_{11.23}$ & 11.41 \\ & QW & 89.11${}_{11.01}$ & 89.10${}_{10.84}$ & 88.50${}_{10.29}$ & 90.01${}_{10.28}$ & 94.65${}_{10.28}$ & 95.64${}_{10.28}$ & 95.53${}_{11.39}$ & 67.35${}_{11.42}$ & 35.66${}_{11.32}$ & 33.66${}_{11.23}$ & 70.21${}_{11.23}$ & 11.14 \\ \hline \multirow{2}{}{GIN} & (2) & 86.22${}_{10.20}$ & 76.18${}_{10.78}$* & 87.87${}_{10.23}$ & 80.87${}_{11.43}$ & 89.83${}_{10.72}$ & 39.11${}_{11.23}$ & 64.29${}_{11.51}$ & 32.37${}_{11.56}$ & 27.79${}_{11.42}$ & 62.55${}_{10.54}$ & - \\ & QW & 86.24${}_{10.00}$ & 76.13${}_{11.00}$ & 87.53${}_{10.34}$ & 89.28${}_{10.29}$ & 92.84${}_{10.26}$ & 92.60${}_{10.46}$ & 65.98${}_{10.26}$ & 73.26${}_{11.32}$ & 32.32${}_{11.32}$ & 77.54${}_{10.42}$ & 64.40${}_{10.42}$ & -** \\ \hline \multirow{2}{}{GraphSAGE} & (2) & 88.24${}_{10.49}$ & 79.51${}_{10.80}$ & 88.53${}_{10.34}$ & 89.87${}_{11.01}$ & 95.50${}_{10.42}$ & 43.79${}_{10.90}$ & 63.26${}_{10.99}$ & 89.90${}_{10.90}$* & 80.30${}_{10.34}$ & 86.32${}_{10.38}$ & - \\ & QW & 87.59${}_{10.77}$ & 80.52${}_{10.68}$ & 88.61${}_{10.39}$ & *90.17 $B_{\phi}$ are set to be the least-square loss. For each method, we perform 10 runs with different seeds and record the learning results' mean and standard deviation. ### Numerical Comparison and Visualization #### 4.2.1 Node Classification and Regression Table 2 shows the node classification results on the ten graphs,2 whose last column records the overall improvements caused by our QW loss compared to other learning methods. The experimental results demonstrate the usefulness of our QW loss-based learning method -- for each model, applying our QW loss helps to improve learning results in most situations and leads to consistent overall improvements. In particular, for the state-of-the-art spectral GNNs like BernNet (He et al., 2021) and ChebNetII (He et al., 2022), learning with our QW loss can improve their overall performance on both homophilic and heterophilic graphs consistently, resulting in the best performance in this experiment. For the simple GCN model (Kipf and Welling, 2017), learning with our QW loss improves its performance significantly and reduces the gap between its classification accuracy and that of the state-of-the-art models (Gasteiger et al., 2019; He et al., 2021, 2022), especially heterophilic graphs. Note that, when learning GCN, our QW loss works better than the traditional method regularized by the label propagation (i.e., GCN-LPA (Wang and Leskovec, 2021)) because our learning method can leverage the training label information in both learning and prediction phases. GCN-LPA improves GCN when learning on heterophilic graphs, but surprisingly, leads to performance degradation on homophilic graphs. Additionally, we also fit the one-hot labels by minimizing the MSE. As shown in Table 3, minimizing the QW loss leads to lower MSE results, which demonstrates the usefulness of the QW loss in node-level regression tasks. Footnote 2: In Table 2, we bold the best learning result for each graph. Learning GCN by “(2)+LPA” means implementing GCN-LPA (Wang and Leskovec, 2021). #### 4.2.2 Computational Efficiency and Scalability In theory, the computational complexity of our QW loss is linear with the number of edges. When implementing the loss as (13) and solving ti by Algorithm 2, its complexity is also linear with the number of inner iterations $J$. Figure 2 shows the runtime comparisons for the QW loss-based learning methods and the traditional method on two datasets. We can find that minimizing the QW loss by Algorithm 1 or Algorithm 2 with $J=1$ merely increases the training time slightly compared to the traditional method. Empirically, setting $J\leq 5$ can leads to promising learning results, as shown in Tables 2 and 3. In other words, the computational cost of applying the QW loss is tolerable considering the significant performance improvements it achieved. Additionally, we apply our QW loss to large-scale graphs and test its scalability. As shown in Table 4, we implement the QW loss based on Algorithm 1 (i.e., solving (12)) and apply it to the node classification task \begin{table} \begin{tabular}{c c\|c c\|c c} \hline \hline \multirow{2}{}{Model} & \multirow{2}{}{Method} & \multicolumn{2}{c\|}{Homophilic graphs} & \multicolumn{2}{c}{Heterophilic graphs} \\ & & Computers & Photo & Actor & Cornell \\ \hline \multirow{2}{}{GIN} & (2) & $0.0605_{\pm 0.0018}$ & $0.0459_{\pm 0.0044}$ & $0.1570_{\pm 0.0014}$ & $0.1609_{\pm 0.0359}$ \\ & QW & $\mathbf{0.0244_{\pm 0.0028}}$ & $\mathbf{0.0203_{\pm 0.0012}}$ & $\mathbf{0.1564_{\pm 0.0012}}$ & $\mathbf{0.1524_{\pm 0.0043}}$ \\ \hline \multirow{2}{}{BernNet} & (2) & $0.0871_{\pm 0.0002}$ & $0.0488_{\pm 0.0009}$ & $\mathbf{0.1661_{\pm 0.0020}}$ & $0.0989_{\pm 0.0076}$ \\ & QW & $\mathbf{0.0364_{\pm 0.0038}}$ & $\mathbf{0.0297_{\pm 0.0014}}$ & $0.1671_{\pm 0.0008}$ & $\mathbf{0.0753_{\pm 0.0024}}$ \\ \hline \hline \end{tabular} \end{table} Table 3: Comparisons on node regression error (MSE). in the large-scale arXiv-year graph. The result shows that our QW loss is applicable to the graphs with millions of edges on a single GPU and improves the model performance. #### 4.2.3 Distribution of Optimal Label Transport Figure 3 visualizes the histograms of the optimal label transport $\mathbf{F}^{}$ learned for two representative GNNs (i.e., GCN (Kipf and Welling 2017) and ChebNetII (He et al. 2022)) on four graphs (i.e., the homophilic graphs "Computers" and "Photo" and the heterophilic graphs "Squirrel" and "Chameleon"). We can find that when learning on homophilic graphs, the elements of the optimal label transport obey the zero-mean Laplacian distribution. It is reasonable from the perspective of optimization -- the term $\\|\text{diag}(\mathbf{w})\mathbf{F}\\|_{1}$ can be explained as a Laplacian prior imposed on $\mathbf{F}$'s element. Additionally, we can find that the distribution corresponding to GCN has larger variance than that corresponding to ChebNetII, which implies that the $\mathbf{F}^{}$ of GCN has more non-zero elements and thus has more significant impacts on label prediction. The numerical results in Table 2 can also verify this claim -- in most situations, the performance improvements caused by the optimal label transport is significant for GCN but slight for ChebNetII. For heterophilic graphs, learning GCN still leads to Laplacian distributed label transport. However, the distributions corresponding to ChebNetII are diverse -- the distribution for Squirrel is long-tailed while that for Chameleon is still Laplacian. ### Analytic Experiments #### 4.3.1 Robustness to Label Insufficiency Issue Our QW loss-based learning method considers the label transport on graphs, whose feasible domain is determined by the observed training labels. The more labels we observed, the smaller the feasible domain is. To demonstrate the robustness of our method to the label insufficiency issue, we evaluate the performance of our method given different amounts of training labels. We train ChebNetII (He et al., 2022) on four graphs by traditional method and our method, respectively. For each graph, we use $K\%$ nodes' labels to train the ChebNetII, where $K\in[1,60]$, and apply $20\%$ nodes for validation and $20\%$ nodes for testing, respectively, as the above default setting does. Figure 4 shows that our QW loss-based learning method can achieve encouraging performance even when only $20\%$ nodes or fewer are labeled. Additionally, the methods are robust to the selection of solver -- we can minimize the QW loss based on (12) or (13), leading to comparable results and outperforming the traditional loss consistently. #### 4.3.2 Impacts of Adjusting Edge Weights As shown in (17), we can train an MLP to predict edge weights based on the optimal label transport. The ablation study in Table 5 quantitatively show the impacts of adjusting edge weights on the learning results. We can find that for ChebNetII, the two settings provide us with comparable learning results. For GCN, however, learning the model with adjusted edge weights suffers from performance degradation on homophilic graphs while leads to significant improvements on heterophilic graphs. Empirically, it seems that adjusting edge weights based on label transportation helps to improve the learning of simple GNN models on heterophilic graphs. \begin{table} \begin{tabular}{c c\|c c\|c c} \hline \hline \multicolumn{2}{c\|}{Model} & \multicolumn{2}{c\|}{Homophilic} & \multicolumn{2}{c}{Heterophilic} \\ & & Computers & Photo & Actor & Cornell \\ \hline \multirow{2}{}{GCN} & $\mathbf{A}$ & $\mathbf{88.39_{\pm 0.55}}$ & $\mathbf{93.80_{\pm 0.37}}$ & $30.14_{\pm 0.80}$ & $60.64_{\pm 4.26}$ \\ & $\mathbf{A}(\mathbf{F};\xi)$ & $84.35_{\pm 0.46}$ & $91.79_{\pm 0.21}$ & $\mathbf{38.09_{\pm 0.50}}$ & $\mathbf{84.26_{\pm 2.98}}$ \\ \hline \multirow{2}{*}{ChebNetII} & $\mathbf{A}$ & $\mathbf{89.52_{\pm 0.54}}$ & $\mathbf{94.84_{\pm 0.37}}$ & $\mathbf{41.37_{\pm 0.67}}$ & $86.38_{\pm 3.19}$ \\ & $\mathbf{A}(\mathbf{F};\xi)$ & $89.41_{\pm 0.41}$ & $94.79_{\pm 0.45}$ & $40.74_{\pm 0.80}$ & $\mathbf{86.60_{\pm 2.98}}$ \\ \hline \hline \end{tabular} \end{table} Table 5: Impacts of adjusting edge weights on node classification accuracy (%) when applying the QW loss. Figure 4: Illustrations of the learning methods’ performance given different amounts of labeled nodes. #### 4.3.3 Robustness to Hyperparameters The weight of the Bregman divergence term, i.e., $\lambda$, is the key hyperparameter impacting the performance of our learning method. Empirically, when $\lambda$ is too small, the Bregman divergence between the observed labels and their predictions becomes ignorable. Accordingly, the regularizer may be too weak to supervise GNNs' learning properly. On the contrary, when $\lambda$ is too large, the regularizer becomes dominant in the learning objective, and the impact of the label transport becomes weak in both the learning and prediction phases. As a result, it may perform similarly to the traditional method when using a large $\lambda$. We test the robustness of our method to $\lambda$ and show representative results in Figure 5. In particular, our QW loss-based method trains ChebNetII (He et al., 2022) on four graphs. Both Algorithm 1 for (12) and Algorithm 2 for (13) are tested. The $\lambda$ is set in the range from $10^{-2}$ to $10^{3}$. For homophilic graphs, our learning method achieves stable performance when $\lambda\geq 10$. When $\lambda<10$, the learning results degrade significantly because of inadequate supervision. Our learning method often obtains the best learning result for heterophilic graphs when $\lambda\in[1,10]$. These experimental results show that our method is robust to the setting of $\lambda$, and we can set $\lambda$ in a wide range to obtain relatively stable performance. ## 5 Conclusion We have proposed the Quasi-Wasserstein loss for learning graph neural networks. This loss matches well with the non-i.i.d. property of graph-structured data, providing a new strategy to leverage observed node labels in both training and testing phases. Applying the QW loss to learn GNNs improves their performance in various node-level prediction tasks. In the future, we would like to explore the impacts of the optimal label transport on the generalization power of GNNs in theory. Moreover, we plan to modify the QW loss further, developing a new optimization strategy to accelerate its computation.
2308.04643	Long-Distance Gesture Recognition using Dynamic Neural Networks	Gestures form an important medium of communication between humans and machines. An overwhelming majority of existing gesture recognition methods are tailored to a scenario where humans and machines are located very close to each other. This short-distance assumption does not hold true for several types of interactions, for example gesture-based interactions with a floor cleaning robot or with a drone. Methods made for short-distance recognition are unable to perform well on long-distance recognition due to gestures occupying only a small portion of the input data. Their performance is especially worse in resource constrained settings where they are not able to effectively focus their limited compute on the gesturing subject. We propose a novel, accurate and efficient method for the recognition of gestures from longer distances. It uses a dynamic neural network to select features from gesture-containing spatial regions of the input sensor data for further processing. This helps the network focus on features important for gesture recognition while discarding background features early on, thus making it more compute efficient compared to other techniques. We demonstrate the performance of our method on the LD-ConGR long-distance dataset where it outperforms previous state-of-the-art methods on recognition accuracy and compute efficiency.	Shubhang Bhatnagar, Sharath Gopal, Narendra Ahuja, Liu Ren	2023-08-09T00:56:38Z	http://arxiv.org/abs/2308.04643v1	# Long-Distance Gesture Recognition using Dynamic Neural Networks ###### Abstract Gestures form an important medium of communication between humans and machines. An overwhelming majority of existing gesture recognition methods are tailored to a scenario where humans and machines are located very close to each other. This short-distance assumption does not hold true for several types of interactions, for example gesture-based interactions with a floor cleaning robot or with a drone. Methods made for short-distance recognition are unable to perform well on long-distance recognition due to gestures occupying only a small portion of the input data. Their performance is especially worse in resource constrained settings where they are not able to effectively focus their limited compute on the gesturing subject. We propose a novel, accurate and efficient method for the recognition of gestures from longer distances. It uses a dynamic neural network to select features from gesture-containing spatial regions of the input sensor data for further processing. This helps the network focus on features important for gesture recognition while discarding background features early on, thus making it more compute efficient compared to other techniques. We demonstrate the performance of our method on the LD-ConGR long-distance dataset where it outperforms previous state-of-the-art methods on recognition accuracy and compute efficiency. ## I Introduction Gestures are an efficient non-verbal method for directing traffic from a distance, interacting in public places and communicating with the deaf. Gestures are an important part of human-machine interaction (HMI) where they provide a natural and friendly way to interact with robots and machines [1, 2, 3] at homes, offices, airports, hospitals and in automobiles. Recent advances in AR/VR and fitness technologies have increased the use of wearable devices (headsets, smart watches, health trackers) where hand, face and eye gestures [4, 5, 6] are a contact-less way to collect user input and intent. Gesture recognition can be broadly categorized [7] into short-distance and long-distance based on the distance between the sensors and the gesturing subject. Short-distance applications, such as car infotainment systems and desktops/laptops, have the subject within a 1m distance from the sensors. Long-distance applications include service robots (floor cleaning, warehouse, lawn mowing), conference/meeting rooms, home automation and IoT devices (adjust lights, TV controls), AR/VR and video games, where the subject is typically far away (1-4m) from the sensors. In some of these applications, such as the mobile floor cleaning robot, short-distance recognition is not an option due to a very low camera position which forces the subject to stand farther away to be in its field-of-view (FOV). Fig 1 illustrates some of these long-distance applications. Self-occlusions, finger similarity and uncertainty of gesture duration are some of the common challenges faced by gesture recognition methods. Deployment of such methods in resource and compute constrained robots and devices requires us to consider the computational complexity and efficiency, in addition to accuracy, of the method. Long-distance scenarios pose an additional challenge of a large FOV in which the hands appear small, making it harder to recognize the gesture. Longer distances lead to reduction in signal strength and increased blur for the gesture, which further deteriorate performance. We address these challenges by proposing a new method that uses modern convolutional neural networks (CNNs) for recognizing hand gestures from visual data (RGB video). We leverage the efficiency and structural adaptability of dynamic neural networks [8, 9] for long-distance applications. The main contributions of this paper are: * A novel dynamic neural network based gesture recognition method for long-distance applications. * Extensive experiments and evaluation on a long-distance dataset. We show that our method is 3.5% more accurate and 28.5% more compute-efficient compared to the previous best method (Table I). The rest of the paper is organized as follows: We review previous work in gesture recognition and dynamic neural networks in Section II. Section III describes the details of our method and Section IV presents a thorough evaluation and analysis of our method compared to previous state-of-the-art techniques. Finally, we summarize and conclude in Section V. Fig. 1: Example applications of our proposed method for long-distance gesture recognition. ## II Related Work ### _Gesture Recognition_ Gesture recognition methods are typically based on input from motion, visual or range sensors of human bodies, hands and legs. In this section, we focus on reviewing hand gesture recognition methods using visual and range data streams. Hand gestures are broadly divided into two types: static gestures and dynamic gestures. Static gestures are gestures which involve only a specific hand pose, and can be recognized using only spatial sensor data. Dynamic gestures involve specific movements of the hand, and can be recognized only by considering both spatial and temporal data. Most popular datasets for hand gesture recognition [6, 7, 10, 11] consist of both static and dynamic gestures. The state-of-the-art techniques for gesture recognition use deep convolutional neural networks (CNNs) that take a stream of visual and/or range data as input, and finally predict a gesture type. They are broadly divided into networks which estimate hand poses before predicting the gesture type, and networks which directly classify the input signal into a gesture type. Techniques which estimate a 2D hand pose use 2D CNNs, either to regress the exact co-ordinates of hand keypoints [12, 13], or to generate a rough heatmap for hand keypoint locations [14, 15]. These techniques are less suited to long-distance applications where estimating exact hand keypoints is difficult. Techniques which directly classify the input make use of 3D CNN architectures [16, 17, 18](for example 3D ResNets [19]) to effectively utilize and fuse both spatial and temporal features for gesture recognition. Some of these techniques have additional modules made of custom CNNs to fuse different modalities [20]. Other methods [21] make use of recurrent neural networks (LSTMs) to model the temporal dependencies, while using CNNs to extract spatial features. Methods based on extracting and utilizing optical flow [22] have also been used for hand gesture recognition. Some recognition methods [23] include a separate light-weight detection module to help decide if the main gesture classifier is to be used on an input data stream or not. Such a light-weight detector improves the efficiency of most of the methods described previously. Our results suggest that these techniques fail to accurately recognize gestures when the subject is far away ($>1$ meter) from the capturing system. The gesture occupying only a small spatial region of the sensor's field-of-view (FOV) is one of the main reasons for this reduction in accuracy. ### _Dynamic Neural Networks_ Most deep neural networks perform inference using a static computational graph and static set of parameters, which remain the same for all inputs. Dynamic neural networks on the other hand, adapt their computational structure based on the input. This helps them be more efficient, interpretable and generalizable than their static counterparts [8, 9]. Broadly, dynamic neural networks are categorized into spatially dynamic and temporally dynamic networks [24]. Temporally dynamic networks [25] are used on sequential data such as videos, text and audio. They use a recurrent neural network (RNN) to decide the computation path for an input sample. Typically in video recognition, the input stream is preprocessed through a series of convolutional layers before being fed to an RNN which decides if this video segment requires further processing or not. Spatially dynamic neural networks perform dynamic computation by helping the network focus on more meaningful regions of image and video inputs. They are further subdivided into pixel-wise and region-wise dynamic networks, based on their level of spatial adaptability. Pixel-wise dynamic networks include networks with dynamic sparse convolution, which has been used for increasing the efficiency and performance of image classification [26, 27]. Region-wise dynamic networks include either the use of hard attention with RNN's [28] to choose patches of the input, or the use of other methods like class activation maps for determining most informative regions [29]. Our proposed network is a type of region-wise spatially dynamic neural network. ## III Method Our network is a spatially dynamic neural network, which is patch-wise adaptable. It operates on a continuous stream of visual data (RGB video), taking as input a set of T frames at a time and sliding this window of T frames across the stream. The video input block in Figure 2 depicts our network working on such a video stream using the sliding window style of operation. Specifically, our network operates on a 4 dimensional input of size $T\times C\times H\times W$, with $T$ representing the number of frames, $C$ representing the number of channels (usually 3), and $H,\ W$ representing the spatial resolution of the image. For each such input $\mathcal{I}$, our network produces a gesture label $y_{\mathcal{I}}\in N$, where $N$ is the number of gesture types in the dataset. Our network consists of 3 main blocks: 1. Preliminary feature extractor 2. Patch selection subnetwork 3. Patch gesture classifier Figure 2 illustrates the high level structure of our network, including the connections between these blocks. The input $\mathcal{I}$ first goes to the preliminary feature extractor, which is connected to the patch selection subnetwork, which is followed by the patch gesture classifier. We describe the structure and function of each of these blocks in detail in the following subsection. ### _Structural Blocks_ #### Iii-A1 Preliminary Feature Extractor This block consists of a shallow network made of 3D convolutional layers acting on the input $\mathcal{I}$. The preliminary feature extractor $f_{\theta}$, parameterized by $\theta$, extracts low level spatio-temporal features $f_{\theta}(\mathcal{I})$ from the full input $\mathcal{I}$. These low level spatio-temporal features are used by the next block to select a patch of features with maximum gesture information. The preliminary feature extractor acts on the full spatial input and is hence designed to be shallow to minimize the compute required by it. #### Iii-A2 Patch Selection Subnetwork The patch selection subnetwork, shown in Figure 3, receives low level features $f_{\theta}(\mathcal{I})$ of size $T^{\prime}\times C^{\prime}\times H^{\prime}\times W^{\prime}$ from the preliminary feature extractor, and divides it into $m\times n$ non-overlapping spatial patches. We denote these patches formed from the full set of features $f_{\theta}(\mathcal{I})$ as $\{p_{1,1},\ldots,p_{1,n},p_{2,1},\ldots,p_{2,n},\ldots p_{m,n}\}$, with each patch having a size $T^{\prime}\times C^{\prime}\times h_{patch}\times w_{patch}$. The spatial size of the patches ($h_{patch}\times w_{patch}$) should be large enough to allow a single patch to contain sufficient information to recognize the gesture. The patch selection subnetwork selects a single patch from these $m\times n$ patches to forward to the next stage for gesture classification. It uses a small 3D CNN $g_{\psi}$, parameterized by $\psi$, as a binary classifier for patch selection. Each patch $p_{i,j}$ is assigned a confidence score $S_{p_{i,j}}$ defined by, \[S_{p_{i,j}}=g_{\psi}(p_{i,j}), \tag{1}\] where $g_{\psi}(p_{i,j})$ represents the confidence with which the subnetwork $g_{\psi}$ predicts that the patch $p_{i,j}$ contains a gesture-performing hand. A single feature patch is selected as, \[p_{max}\leftarrow\underset{p_{i,j}}{arg\,max}\ S_{p_{i,j}}, \tag{2}\] where $p_{max}$ is the patch associated with the maximum confidence score $S_{max}$. The patch selection greatly reduces the size of the input features forwarded to the most compute intensive block - the patch gesture classifier. #### Iii-A3 Patch Gesture Classifier The patch gesture classifier $h_{\phi}$, parameterized by $\phi$, consists of 3D convolutional layers which receive features (computed by the preliminary feature extractor) from a single patch selected by the patch selection subnetwork. The selected input patch $p_{max}$ has a size $T^{\prime}\times C^{\prime}\times h_{patch}\times w_{patch}$, for which the gesture classifier outputs a distribution $h_{\phi}(p_{max})$ over $N$ possible gesture classes. The class with the highest probability is predicted as the recognized gesture. The smaller size of the patch features as compared to complete input features (by a factor $\frac{H^{\prime}\times W^{\prime}}{k_{patch}\times w_{patch}}$) helps save a significant amount of compute in the part of the network. The patch gesture classifier can be chosen to be any common 3D convolutional backbone such as 3D ResNet, 3D ResNext, 3D MobileNet [19, 30], etc, after appropriate modification to receive $C^{\prime}$ channels of the patched input. ### _Training_ The whole network, including the patch selection subnetwork and the patch gesture classifier, is trained end-to-end as a single unit. This allows our loss to be expressed as a sum of cross entropy terms, which can be effectively optimized. The loss $\mathcal{L}$ is a sum of the cross entropy loss for gesture recognition, and the cross entropy loss for patch selection. It is defined as, \[\mathcal{L}=H(h_{\phi}(p_{max}),y_{\mathcal{I}})+\lambda\sum_{i,j}H(g_{\psi}(p _{i,j}),y_{p_{i,j}}). \tag{3}\] Here, $\lambda$ is a relative weight hyperparameter, while $H(a,b)$ depicts the cross entropy between $a$ and $b$. $y_{p_{i,j}}$ is the label for patch $p_{i,j}$ indicating if it has a hand gesture or not. ## IV Experiments and Results We evaluate the performance of our method for long-distance gesture recognition using the LD-ConGR dataset [7]. The dataset consists of 542 RGB-D videos at 30 fps with a video resolution of $1280\times 720$. We use only the RGB channels for our experiments, as inexpensive cameras are more common in robots and consumer devices. Each video has a subject seated in a conference room performing gestures belonging to one or more of 10 gesture categories. Fig. 3: The patch selection subnetwork takes the preliminary features as input, splits them into spatial patches and then selects the patch with the most useful features to be forwarded to the next block of the network. Fig. 2: The proposed dynamic neural network consists of 3 blocks: 1) The preliminary feature extractor, 2) Patch selection subnetwork and, 3) The patch gesture classifier. The subject is seated at a long-distance ($>1$ meter and $<4$ meters) from the capture device. The gesture categories include both static and dynamic gestures. In addition to frame-wise gesture type annotations in the videos, the dataset also provides rough bounding box annotations of the gesturing hand, which we use to train the patch selection subnetwork. Performance on the dataset is measured using the Top-1 Accuracy for framewise predictions made by the network. We use the first 2 layers of the 3D ResNeXt-101 CNN [19] as the preliminary feature extractor. The patch selection subnetwork consists of the following sequence of layers with $3\times 3\times 3$ 3D convolutions kernels : $conv3d\_bn\_relu(str\ 1)\mid avgpool(str\ 2)\mid conv3d\_bn\_relu(str\ 1)\mid linear$. Our patch gesture classifier is the 3D ResNeXt-101 CNN [19] with $C^{\prime}$ input channels. The patch gesture classifer is initialized from models trained on the Jester dataset [10]. The low level features of size $T^{\prime}$=16, $C^{\prime}$=64, $H^{\prime}$=56, $W^{\prime}$=56 received by the patch selection subnetwork are divided into $6$ patches. Additionally, following previous work [20, 23], we set $T$=32, $C$=3, $H$=112, $W$=112. The network parameters are trained to optimize the combined loss defined in Equation 3 using SGD with momentum for 70 epochs on 4 NVIDIA V100 GPUs. Initial learning rate was set to $10^{-3}$ and it was decayed using cosine annealing throughout training. The relative weight parameter $\lambda=2$. The above hyperparameters apply to all experiments unless specified otherwise. ### _Performance on LD-ConGR_ We compare our method with state-of-the-art gesture recognition methods listed below. Due to the absence of hand keypoint annotations in the LD-ConGR dataset, we do not compare with techniques that predict hand pose or keypoints in an intermediate step. 1. 3D ResNeXt-101[19] is the 3D ResNeXt-101 CNN without our dynamic patch selection subnetwork. 2. SlowFast[17] uses a combination of high and low frame rate 3D CNNs (based on the 3D ResNeXt-101) for better and more efficient gesture recognition. 3. C3D[18] is a popular 3D CNN baseline for gesture and activity recognition. 4. Temporal Segment Networks[22] use separate networks to extract spatial and temporal features. As can be seen in Table I, our method achieves state-of-the-art gesture recognition accuracy on LD-ConGR as compared to other methods. It does so while using a significantly lower amount of compute (GFLOPS) than other methods. This is due to the fact that our network discards early on, a bulk of features which have no gesture information. ### _Performance in a resource constrained environment_ We also evaluate the performance of our method in a resource (power, compute, memory) constrained environment which provides a more realistic indicator of its performance when used for real-world applications like controlling mobile robots and smart home devices. We replace the 3D ResNeXt-101 backbone with the 3D MobileNet [30]. We use a 4 patch ($2\times 2$) version of our network, with a single layer deep preliminary feature extractor to ensure that the number of parameters in our network is similar to the 3D MobileNet baseline. This helps ensure that both networks being compared have similar compute requirements and are deployable on resource constrained robots and edge-devices. Our method is able to vastly improve (by 17.4%) on the performance of the current state-of-the-art light weight 3D MobileNet CNN, as can be seen in the results in Table II. C3D [18], Slowfast [17] and Temporal Segment Networks [22] are not included in this comparison because they do not have any comparable lightweight versions available. These results demonstrate the importance of early rejection of background features by our dynamic neural network, helping focus the limited compute on features with gesture information. ### _Analyzing performance for different gestures_ Table III tabulates the accuracy achieved by our network and the comparable 3D ResNeXt-101 CNN at recognizing each of the 10 different gestures in the LD-ConGR dataset. Our method outperforms the baseline on most of the gesture types. We observe significant gains in performance for gestures such as Click and Pinch that are relatively difficult to identify, and need finer detection of finger movements. We hypothesize that the gains in performance are due to the early discarding of background features by our method, which helps the patch gesture classifier learn finer features. ### _Effect of number of patches_ The number of patches in the incoming preliminary features is an important parameter to be selected. We conduct an experiment to determine its effect on the accuracy and compute efficiency of the network. Specifically, we train separate networks with the features divided into 2, 4 and 6 patches and measure the gesture recognition accuracy achieved by them. We use the ResNeXt-101 backbone in all of these networks. The results in Table IV demonstrate that dividing the input progressively into smaller patches improves the compute efficiency of the network by helping discard more background features early on, while also positively affecting its accuracy. Figure 4 illustrates the regions in the input image from which the selected feature patches were extracted. The region highlighted in green shrinks as the number of patches is increased, hence decreasing the number of features processed by the patch gesture classifier. We also observed that though using smaller patches leads to better performance, this trend only holds as long as the chosen feature patches are large enough to fit sufficient gesture information. The preliminary features can also be divided into overlapping patches to increase the recognition accuracy at the cost of increased compute. The target application influences the accuracy VS efficiency tradeoff. ### _Performance on longer distance videos_ The LD-ConGR test set [7] consists of videos in which the subject is at varying distances (1-4 meters) from the capture device. We conduct an experiment to evaluate the effect of distance on the performance of our method. Specifically, we exclusively test our method on videos from the LD-ConGR test set where the subject is sitting at the far end of the conference room. This corresponds to all videos with the recording spot labels l3 and r3 as defined in [7]. These videos have the subject at the longest distance from the capture device ($\approx 4$ meters). For comparison, we also report the accuracy of other methods on the same set of videos in Table V. We observe that the performance of our method shows a much smaller relative deterioration for longer distance videos, than that of other methods. Fig. 4: The patch selection subnetwork selects features extracted from the spatial region highlighted in green in the image, and forwards them to the patch gesture classifier. It discards features from parts of the image highlighted in red. The size of the green region can be varied by changing the number of feature patches, as seen in (a) 2 patches, (b) 4 patches and (c) 6 patches. ## V Conclusion We present a novel dynamic neural network for the task of long-distance gesture recognition. The early discarding of background features not only helps the network focus on important features required for accurate gesture prediction, but it also reduces the compute requirements and makes the network conducive to deployment in resource constrained settings on robots and edge-devices. We perform extensive experiments and evaluation, and compare against other gesture recognition methods to show the effectiveness of our approach. Our method achieves state-of-the-art performance on the LD-ConGR [7] long-distance gesture dataset. As part of our future work, we intend to explore new neural network architectures that support adaptive patch sizes while being efficient enough to be deployed on resource constrained devices. Another research direction would be the extention of our method for general activity recognition at arbitrary distances while taking in multi-modal sensor inputs. ## VI Acknowledgement This work was partially supported by ONR grant N00014-20-1-2444 and USDA NIFA grant 2020-67021-32799/1024178. We thank Marcus Gualtieri and the anonymous reviewers for providing helpful comments.
2310.06578	Energy-Efficient Visual Search by Eye Movement and Low-Latency Spiking Neural Network	Human vision incorporates non-uniform resolution retina, efficient eye movement strategy, and spiking neural network (SNN) to balance the requirements in visual field size, visual resolution, energy cost, and inference latency. These properties have inspired interest in developing human-like computer vision. However, existing models haven't fully incorporated the three features of human vision, and their learned eye movement strategies haven't been compared with human's strategy, making the models' behavior difficult to interpret. Here, we carry out experiments to examine human visual search behaviors and establish the first SNN-based visual search model. The model combines an artificial retina with spiking feature extraction, memory, and saccade decision modules, and it employs population coding for fast and efficient saccade decisions. The model can learn either a human-like or a near-optimal fixation strategy, outperform humans in search speed and accuracy, and achieve high energy efficiency through short saccade decision latency and sparse activation. It also suggests that the human search strategy is suboptimal in terms of search speed. Our work connects modeling of vision in neuroscience and machine learning and sheds light on developing more energy-efficient computer vision algorithms.	Yunhui Zhou, Dongqi Han, Yuguo Yu	2023-10-10T12:39:10Z	http://arxiv.org/abs/2310.06578v1	# Energy-Efficient Visual Search by Eye Movement and Low-Latency Spiking Neural Network ###### Abstract Human vision incorporates non-uniform resolution retina, efficient eye movement strategy, and spiking neural network (SNN) to balance the requirements in visual field size, visual resolution, energy cost, and inference latency. These properties have inspired interest in developing human-like computer vision. However, existing models haven't fully incorporated the three features of human vision, and their learned eye movement strategies haven't been compared with human's strategy, making the models' behavior difficult to interpret. Here, we carry out experiments to examine human visual search behaviors and establish the first SNN-based visual search model. The model combines an artificial retina with spiking feature extraction, memory, and saccade decision modules, and it employs population coding for fast and efficient saccade decisions. The model can learn either a human-like or a near-optimal fixation strategy, outperform humans in search speed and accuracy, and achieve high energy efficiency through short saccade decision latency and sparse activation. It also suggests that the human search strategy is suboptimal in terms of search speed. Our work connects modeling of vision in neuroscience and machine learning and sheds light on developing more energy-efficient computer vision algorithms. ## 1 Introduction Biological vision is well adapted for survival in natural environment. In most cases, it needs to fulfill at least four requirements: high resolution, large visual field, low response latency, and low energy cost. However, having both high resolution and large visual field will increase the computational and energy cost, and may also increase response latency. To balance between these requirements, most vertebrates' retina has high resolution at center and low resolution in periphery [1; 2; 3; 4; 5; 6]. This allows animals to cover a large visual field by a small number of cells, while still keeping a subjectively clear vision. For example, the monocular visual field of the human eye is $160^{\circ}\times 135^{\circ}$[7], but the number of ganglion cells on retina is only 0.7-1.5 million per eye [8]. To compensate for the low resolution peripheral vision, many animals use the "fixation-saccade" eye movement strategy to view the world [9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. To decrease response latency and energy cost, the eye movement strategy needs to be efficient and focus mostly on task-related regions. Indeed, humans can dynamically adjust fixation locations according to task demands[19], leading to better performance in visual search tasks than random eye movement strategy[20]. Biological vision also uses spiking neurons to process information. This has inspired the spiking neural network (SNN), which is in theory as computationally powerful as conventional artificial neural networks (ANN)[21]. SNN running on neuromorphic chips can be much more energy-efficient than ANN running on GPU or CPU[22] due to its event-driven computation and collocated processing and memory[23]. In summary, non-uniform resolution retina, efficient eye movement strategy, and computation by spiking neurons may be crucial in supporting energy-efficient human vision. However, most current computer vision models process uniform-resolution images and is based on ANN. Their energy efficiency is unsatisfactory given that the human brain consumes only 20 Watts[24]. Although a few models were trained to process non-uniform resolution images and make eye movements, their eye movement strategies had various limitations (such as requiring a fixed fixation number[25, 26, 27, 28, 29, 30, 31, 32], or can only fixate at discrete set of locations[33]), and the learned strategy haven't been compared with that of human's, so it is unknown whether these models can achieve human-level performance. Besides, as far as we know, no model has fully incorporated the three properties of human vision. The goal of this work is to establish the first bio-inspired visual search model (BVSM) that utilizes features of human vision to achieve efficient saccade decision and low energy cost (Fig. 1C). To compare BVSM's eye movement strategy to that of human's, we train both human subjects and the model on a classical visual search task that needs to find a single Gabor target embedded in naturalistic noise image (Fig. 1A)[20]. The task is chosen for its following properties: * Controllable: The features of target and background are controlled, so the target visibility across the retina can be conveniently measured by psychophysical experiment. This allows a fair comparison between BVSM's and human's search performance. * Explainable: There exists a Bayesian optimal search strategy for this task based on human target visibility data[20, 34], which allows a comparison between the optimal and BVSM's search strategy. * Challenging: The target is much smaller (1/2943) and has lower contrast than the background. Conventional network that processes full-resolution original images fails to classify whether the image contains a target. * Naturalistic: The image has $1/f$ amplitude spectrum on spatial frequencies, so it is statistically similar to natural images[35] though the image contents are not the same. BVSM can learn efficient stochastic eye movement policy by reinforcement learning (RL) to search for the target (Fig. 1D and Supplementary Video). To summarize our contributions: * We for the first time integrate the non-uniform resolution retina, efficient eye movement strategy, and computation by SNN into BVSM. BVSM autonomously develops sparse coding during training and leverages population coding to achieve ultra-low saccade decision latency. * We show that BVSM achieves high energy efficiency compared to conventional ANNs that process uniform-resolution images without making eye movements. Our work establishes a starting point for future research on human-like vision model. * We compare the visual search strategies of humans, BVSM, and the optimal strategy, showing that BVSM can either learn a human-like or a near-optimal strategy that outperforms humans. The comparison contributes to a deeper understanding of human eye movement strategy. ## 2 Related Work Bayesian optimal visual search modelThe Bayesian optimal visual search model uses experimentally measured target visibility data to optimally compute the probability of target locations and select the next fixation location[20, 34] (Appendix B). However, this model has a drawback as its inputs are random signals generated by the target visibility data instead of the actual search image, and generating the signals requires knowing the actual target location. Therefore, the Bayesian optimal search model is not suitable for real-life applications. In contrast, the BVSM learns an efficient search strategy directly from the search image without prior knowledge of target location or target visibility data, so it is more applicable to real-life scenarios. Hard attention modelsBVSM is a kind of hard attention model that down-samples the input image and makes eye movements. Currently, the most influential hard attention model architecture is probably the recurrent attention model (RAM) [26, 27]. RAM contains a feature extraction network, a recurrent network (RNN), a saccade decision network, and a classification network. Other model structures usually either lack a short-term memory [28, 36], or use simple manually designed memory [25, 29, 31, 37]. Some models also use Transformer to integrate information from multiple fixations [32, 38]. BVSM has a similar architecture to RAM, but it is implemented as an SNN. We do not use a Transformer architecture because humans may only have a memory capacity of fewer than 20 items in visual search [39, 40, 41, 42]. Conventional RNNs should be able to handle a memory of this size. In terms of the eye movement behavior, some previous models can only fixate at a pre-defined set of discrete locations [33, 38, 43], or require setting a fixed number of fixations for each trial [25, 26, 27, 28, 29]. In particular, RAM-based model may perform worse when fixating more times than it was trained on [31]. BVSM makes continuous-valued saccade decisions, and it naturally employs a variable number of fixations to do visual search by generating a confidence rating to predicted target location. Many previous works are tested on visual search for MNIST-style targets, and the tasks are relatively easy because the search target is large ($>$4.79%) compared to the background image [25, 26, 27, 28, 33, 36]. We find that the same issue also exists for models that were tested on natural images [29, 29] by inspecting the datasets used for training (PASCAL VOC 2007/2010 [44, 45]). Training hard attention models to search for tiny objects has not been thoroughly studied yet. BVSM is tested on a challenging visual search task that has been studied in neuroscience [20, 34, 46, 47]. The search target is only 0.034% of the size of the search image, and humans can only achieve a mean correct response rate of about 86% [47]. It is interesting to see how hard attention models will perform in this task. In summary, various limitations exist in previous hard attention models, which we aim to overcome in our model. Table S1 shows a detailed comparison between BVSM and previous works. Population coding with SNNThe SNN's output variables are often encoded by the firing rate of output neurons. To calculate the firing rate, SNN needs to process the input for a number of time steps, which is called the inference time or latency. Lower latency means faster inference and lower energy cost, but it also limits the information carried by the firing rate of a single neuron. Previous works on using SNN in regression task (like objection detection) usually have high latency (100-3,500 time steps) to precisely calculate the firing rate [48, 49, 50, 51, 52, 53], but this decreases the energy efficiency of SNN. To decrease the latency while maintaining the regression accuracy, we can use the weighted sum of the firing rate of a neuron population to encode a variable. Recently, SNN with population coding has achieved good performance on a variety of continuous-action RL tasks, while lowering the latency to only 5 time steps [54, 55]. In this work, we use population coding to achieve very low saccade decision latency (4 time steps). Table S2 shows a comparison of our SNN to previous models. Figure 1: A. An example search image with target enlarged at lower right. Symbol "0" denotes degrees of visual angle viewing from 70 cm distance. B. Human visual search experiment setup. C. Structure of BVSM. See Fig. S1 for details. D. An example search trial by BVSM. Cross and circle are the current fixation location and target location. Color map shows the probability distribution of the next fixation. The actual next fixation is randomly sampled from this distribution. Methods ### Visual Search Task The visual search task is to find a single Gabor target embedded in a circular naturalistic noise image ($1/f$ amplitude spectrum) (Fig. 1A)[20; 47]. The background noise image has a diameter of $15^{\circ}$ (651 pixels) with root-mean-squared (standard deviation of pixels divided by the mean) contrast of 0.2. The target is a 6 cycles/degree Gabor with a diameter of 0.3${}^{\circ}$ (12 pixels), oriented 45${}^{\circ}$ counterclockwise from the vertical, and windowed by a symmetrical raised cosine. The screen to display the image has a physical size of $54.3744\times 30.2616$ cm, resolution of $1920\times 1080$, and was placed 70 cm away from human subjects (Fig. 1B). We used the same target contrast range in human experiment and modeling (detailed below). The background noise is varied in each trial. ### Human Experiment 29 human subjects (14 females, aged 18-30) participated in the experiment. For 15 subjects, we first used a two-interval force-choice (2IFC) task to measure the target visibility $d^{\prime}$ at fovea as a function of target contrast, and individually estimated the target contrast for $d^{\prime}=3.0$. These target contrasts (ranged 0.11-0.136) were used for all following experiments of each subject. We then conducted another 2IFC task to measure target visibility at various peripheral locations. Finally, subjects finished 200-400 visual search trials (Fig. 1B). Visual search always started from fixating at image center. The rest 14 subjects only participated in the visual search experiment with a fixed target contrast of 0.15. We call the first 15 subjects "Low TC (target contrast) Group", and the rest 14 subjects "High TC Group". See Appendix A for the details of the detection and visual search experiment. ### Bio-inspired Visual Search Model Fig. 1C and Fig. S1 show the structure of BVSM. It contains an artificial retina and four networks. Artificial retina. We use Foveal Cartesian Geometry (FCG)[56] to non-uniformly down-sample the original image (Fig. S1A). FCG copies pixels inside the fovea, and samples pixels in periphery from a series of rings with increasingly larger widths (Appendix C.1). The sampled pixels are then assembled to form a square image as the output. We set the size of the output image to be 224x224 with a fovea size of 16x16 pixels. To restrict the entire search image within the visual field, we sample from a region twice the diameter of the search image. The configurations of artificial retina are closely related to visual search performance. In Appendix C.5, we also examined the model's target visibility at peripheral locations using a similar 2IFC task as in human experiment. Feature extraction network (FEN). The FEN processes retinal-transformed images by 7 convolution blocks and 3 linear heads (Fig. S1B), and outputs the current fixation location (2D vector), target location relative to the fixation location (2D vector), and the estimated error between predicted and true target location (scalar). The output locations are all in the original full-resolution image's coordinate space. FEN uses QCFS function[57] as activation function: $f\left(x\right)=\lambda\cdot\text{clip}\left(\frac{1}{T}\text{floor}(\frac{xT }{\lambda}+0.5),0,1\right)$, where $\lambda$ is a trainable parameter and $T=4$. In the last layer of fully connected heads, we linearly transform the output of a population of neurons into the output vectors (population coding). FEN is converted to SNN with latency $T=4$ after training (Appendix C.2). Recurrent network (RNN). We propose a novel spiking RNN that uses 64 integrate-and-fire (IF) neurons (equation S10) to memorize fixation locations (Fig. S1C). The network's inputs are the predicted fixation location from FEN and the RNN hidden state. Spiking RNN adds another temporal dimension (the SNN inference time in each fixation) to the existing temporal dimension (sequence of fixations). To maintain the real-time processing property of SNN, we set the RNN hidden state as the output spikes from the last time step of the previous fixation, and the hidden state is used in computation at the first time step of the next fixation. See Appendix C.3 and Algorithm S1 for details. Actor and critic networks. The actor network makes saccade decisions (Fig. S1D). If the estimated target location error from FEN is lower than a threshold (0.58${}^{\circ}$, or 25 pixels), the next fixation location will be sampled near the predicted target location; otherwise the actor will process the RNN output by an SNN which outputs the mean and covariance of the next fixation location's distribution by population coding. The next fixation location can be anywhere within the square region enclosing the circular search image. The critic network measures saccade value (Fig. S1E). If the estimated target location error is low, it will calculate the Q-value by the next fixation location and the predicted target location; otherwise it will calculate the Q-value by the next fixation location and RNN output. The critic network is not a spiking network, as it is not used in inference after training. See Appendix C.4 for details of the actor and critic networks. ### Reward Functions The agent receives the sum of two negative rewards after each saccade. The first reward is inhibition of return (IOR) that tracks the location of the most recent 8 fixations based on previous human research [40; 42], and discourages the agent to fixate at previously fixated regions (Fig. S3A,B). The IOR reward after a saccade to the $(f+1)^{\text{th}}$ fixation location is a semi-circle function centered at each previously fixated location: \[R_{\text{IOR}}=\min_{i\in\{0\cdots 7\}}\left(-\frac{1}{r}\sqrt{\max\left(r^{2} -d_{f+1,f-i}^{2},0\right)}\right), \tag{1}\] where $d_{f+1,f-i}$ is the distance between the $(f+1)^{\text{th}}$ and $(f-i)^{\text{th}}$ fixation, and $r$ is the radius of the semi-circle function. The second reward is the saccade amplitude reward that discourages making large saccades (Fig. S3C,D). Making large saccade costs more time and energy [58], and induces strong motion blur that disrupts visual processing [59]. We test two different saccade amplitude reward functions: \[R_{\text{SA}}=-0.5+0.5\times[\exp{(-\text{Saccade amplitude}/2.5^{ \circ})}-1], \tag{2}\] \[R_{\text{SA}}=-\text{Saccade amplitude}/7.5^{\circ}. \tag{3}\] To learn human-like search strategy, we set $r=0.5^{\circ}$ for IOR reward, and use equation 2 for saccade amplitude reward. We call this setting "HP (hyperparameter) 1". To learn Bayesian-inference-like search strategy, we set $r=2.5^{\circ}$ for IOR reward, and use equation 3 for saccade amplitude reward. We call this setting "HP 2". We also test BVSM's performance under some changes to the two reward functions (Table S5). We set the discount factor $\gamma=0.95$. ### BVSM Training and Evaluation BVSM is trained in three stages. First, we train the FEN to detect target in random samples of retinal transformed images. Second, we convert the FEN to SNN, and fine-tune the SNN. Third, we fix the FEN, and use soft actor-critic (SAC) [60] to train the RNN, actor, and critic to do visual search. We directly train the SNN in RNN and actor by spatio-temporal backpropagation (STBP) [61]. Table S3 shows all hyperparameters. Fig. S4, S5, and S6 show the loss and task performance during each training stage. BVSM is evaluated for 10,000 trials after training. Details are shown in Appendix C.6. ### Automatic Search Termination Search trial ends either by reaching the maximum fixation number (50 in training, 200 in testing), or by a "double check and stop" rule. The latter is triggered when the estimated target location error of two consecutive fixations is below 0.58${}^{\circ}$ (25 pixels) and the distance between the two predicted target locations is less than 0.5${}^{\circ}$. If any of the final two fixations are within 1${}^{\circ}$ away from the true target location (same as in human experiment), the trial is labeled as correct; otherwise error. If a trial does not automatically terminate when reaching the maximum fixation number, it is labeled as error. ## 4 Results ### Target Detection Performance Comparison The spiking FEN accurately regresses the current fixation location (Fig. 2A) and target location when the distance between target and fixation location is small and the target contrast is high (Fig. 2B). FEN outputs a low estimated target location error when the target locates near the fixation, which means a high prediction confidence; otherwise it outputs a high estimated error. We can set the target detection threshold as 25 pixels (Fig. 2C). If the estimated error is below the threshold, the network will fixate at the predicted target location in the next fixation. In the detection task (Fig. 3A), FEN performs similar to humans in central vision, but worse than humans in peripheral vision (Fig. 3B,C). In addition, conventional ANN that processes uniform-resolution images fails to detect the small target (Appendix D). This shows the importance of using non-uniform resolution retina when needing to discriminate fine details from a large background. ### Visual Search Strategy Comparison After training with hyperparamter group 1 (HP 1), BVSM's search strategy shows some similarity with humans' strategy. The distribution of fixation location of both BVSM and humans has weak doughnut shape when searching for low contrast target (Fig. 4A,C), and does not show such feature when searching for high contrast target (Fig. 4B,D). BVSM and humans exhibit similar saccade amplitude (Fig. 4E) and fixation number distributions (Fig. 4F). Both of them favor small saccades. BVSM more frequently completes search with very few fixations, especially when the target is near the initial fixation location (Fig. 4G). BVSM quickly finds the target in such cases, while humans require more fixations. As the target distance increases, the search speeds of BVSM and humans become closer (Fig. 4G). In addition, BVSM has higher response accuracy and search speed than humans for both low and high contrast targets (Table 1). In particular, it surpasses human response accuracy by about 11% for low contrast target. The model is better at searching than humans despite having lower target visibility in peripheral vision. Although BVSM searches better, its search strategy is limited to circular scanning (Fig. S8A). While humans may use similar scanpath, they exhibit a more diverse search strategy, such as repeatedly Figure 3: Detection task performance of BVSM and humans. A: In detection task, subjects are required to keep fixating at image center (red solid line), but report whether target is present at a peripheral location (blue dashed line) within a single fixation. B, C: Relationship between target eccentricity and response accuracy. In B, each gray dot represents a combination of target eccentricity, target angular position, and target contrast. In C, each gray dot represents a combination of subject index, target eccentricity, and target angular position. The black curve represents the fitted Weibull function (with inverted x-axis). Figure 2: FEN regression results of 1024 random retinal transformed images. A: Relationship between predicted and true fixation location. 0${}^{\circ}$ represents image center. B: Relationship between predicted and true target location within 4.0${}^{\circ}$ away from fixation location. A and B only show the regression results for horizontal location. The results for vertical location are similar. C: Relationship between target distance and estimated target location error. The red horizontal line shows the decision threshold (0.58${}^{\circ}$, 25 pixels) used in visual search. fixating on suspected target locations (Fig. S8B, blue) and scanning the search image even when the target is close to the initial fixation location (Fig. S8B, yellow). This may explain why humans search more slowly when the target distance is small. Additionally, humans sometimes use irregular scanpaths with short saccades (Fig. S8B, green), but the reason for this is currently unclear. Previous research has suggested that the human search strategy in this task is suboptimal due to the preference for short saccades [47], which increases the overlapping area of central vision in consecutive fixations. To improve BVSM's search performance, the radius of the IOR reward function was increased to $2.5^{\circ}$ (similar to the size of FEN's effective visual field), and the saccade amplitude reward was changed to equation 3, which penalizes large saccades relatively weaker than equation 2. The entropy target parameter in SAC was also lowered to make the strategy less stochastic (Table S3). These modifications result in a stronger doughnut-shaped fixation strategy that is similar to the ELM searcher (Fig. S9), and the search speed is closer to the ELM searcher (Table 1, see methods in Appendix B). \begin{table} \begin{tabular}{l c c c c} \hline \hline & Target contrast & Percent correct & Med(FixNum) & Mean(FixNum) \\ \hline BVSM (HP 1) & 0.11–0.136 & $97.8\pm 0.2\%$ & $8.4\pm 0.5$ & $13.3\pm 0.3$ \\ BVSM (HP 1) & 0.15 & $99.5\pm 0.1\%$ & $6.0\pm 0.0$ & $8.4\pm 0.2$ \\ Human (n=15) & 0.11–0.136 & $86.5\pm 6.3\%$ & $10.0\pm 2.7$ & $16.7\pm 4.7$ \\ Human (n=14) & 0.15 & $97.5\pm 1.5\%$ & $8.0\pm 1.9$ & $13.4\pm 4.5$ \\ \hline BVSM (HP 2) & 0.11–0.136 & $98.2\pm 0.1\%$ & $7.0\pm 0.0$ & $10.8\pm 0.1$ \\ ELM & $d^{\prime}=3.0$ & $97.8\pm 0.0\%$ & $8.0\pm 0.0$ & $8.8\pm 0.1$ \\ \hline \hline \end{tabular} \end{table} Table 1: Search performance comparison. BVSM results are averaged from models trained with 5 random seeds. Human results are averaged from all subjects. ELM results are averaged from 5 independent evaluations, each consisting of 10,000 trials. Mean and median fixation number (FixNum) are calculated from correct trials. For the ELM model, target visibility ($d^{\prime}$) of 3.0 corresponds to target contrast of 0.11-0.136 (Appendix A.3). HP 1 and HP 2 means hyperparameter gourp 1 and group 2. Performance data are shown in mean $\pm$ standard deviation. Figure 4: BVSM and human eye movement metrics. A-D: Distribution of fixation location. TC: target contrast. E. Distribution of saccade amplitude. F: Distribution of fixation number in correct visual search trials. G: Median fixation number as a function of target eccentricity in correct trials. In this figure, BVSM is trained under hyperparameter group 1 (HP 1). ### BVSM Has Low Estimated Energy Cost and High Robustness We estimate the energy consumption of BVSM, BVSM based on ANN, and conventional ANN that process full-resolution images without making eye movements (see methods in Appendix C.8). BVSM's estimated energy consumption is two orders of magnitude lower than that of ANN counterparts, even if BVSM needs on average 10 fixations to find the target (Table S4, Fig. 5C). This is because BVSM automatically develops sparse coding during training (Fig. 5B, see methods in Appendix C.7). Most neurons have a low firing rate (Fig. 5A), and the average firing rate is only 0.089 (maximum possible firing rate is 1.0). The sparse activation in the network is similar to what was found in the animal brain [62; 63; 64; 65; 66]. The low saccade decision latency of only 4 time steps also increases the energy efficiency by preventing repeated processing of input images. BVSM's search performance is robust to modest changes in the discount factor and the reward function (Table S5). However, removing either of the two reward functions will cause a notable slowdown in search speed, and may also decrease the response accuracy (Table S5). If we double the target's diameter, BVSM will search much faster and more accurately (Table S5). ## 5 Discussion Benefit of non-uniform resolution retina and eye movementsAlthough non-uniform resolution retina and eye movements are common features of biological vision [1; 2; 3; 4; 5; 6], they are not widely used computer vision due to several possible reasons. Processing a single uniform-resolution image by a single forward computation is simpler, faster, and easier to train compared to selecting multiple fixations on the same image. Moreover, the use of multiple fixations prevents parallelization due to temporal dependency on successive fixations, and may offset the efficiency of non-uniform down-sampling. However, the bio-inspired visual system has potential benefits that are yet to be explored. In real-life situations, robots do not process many single, unrelated images, but a continuous stream of visual input, so the saccade rate is very likely to be lower than the input frame rate, and making multiple fixations on the same image frame is unlikely to happen. Therefore, the benefit on computational efficiency of non-uniform sampling will be more significant. Besides, many computer vision tasks do not require the algorithm to simultaneously have large visual field, high visual resolution, low response latency, and low energy cost; but for robots that freely move in natural environments, these four features may likely be required, and the bio-inspired visual system may provide a good solution under this scenario. For the visual search task in this research, conventional network that processes full-resolution images fails to detect the target (Appendix D), but non-uniform down-sampling can enlarge the relative Figure 5: A: Distribution of neuron firing rate in BVSM. Firing rate is 1.0 if a neuron fires at every time step. TC: target contrast. B: The change in population sparseness of FEN during training. C: Estimated energy cost of BVSM (BVSM-SNN), BVSM based on ANN (BVSM-ANN), and conventional ANN that process full resolution images without making eye movements (FullRes-ANN). We assume BVSM-SNN and BVSM-ANN use on average 10 fixations to find the target. size of the target, and make the detection easier. Another possible search method is to divide the image into several patches and process each patch until finding the target. The output of the BVSM's artificial retina is 1/8.4 of the size of the original image. If we divide the image into eight patches and examine each sequentially, the median and mean fixation number will always be 4.5. This sliding window strategy is quicker when the target size is very small and the background offers no hints of the target's location, so the peripheral vision is ineffective. However, violating either of these conditions can render the sliding window strategy inefficient. For example, when the target's size doubles (to $0.6^{\circ}$), BVSM achieves 99.9% search accuracy with a median fixation number of only two fixations, outpacing the sliding window strategy. In natural images, object locations are often statistically related, so the background can provide cues to target positions. As a result, we believe BVSM is crucial for efficient visual search, particularly when a large visual field and low computational cost are needed, and when low-resolution peripheral vision can offer information about the target's location. Benefit of SNNBVSM achieves low estimated energy cost mainly due to its sparse SNN and low saccade decision latency. Sparsity is important for avoiding spike congestion and decreasing hardware latency in real neuromorphic chips[67]. BVSM automatically learns to use sparse coding without explicitly adding sparsity constraints during training. Future studies may investigate whether the model will also develop sparse coding on more complex natural images. Compared to previous object detection SNNs with latencies of 100-3500 time steps[48, 49, 50, 51, 68, 69], BVSM makes saccade decisions in just 4 time steps and finds the target in 10-16 fixations on average. The low saccade decision latency is achieved by using population coding, which mimics the animal brain[70]. Population coding decreases the SNN latency by only increasing the neurons in the last layer, so it is much more energy efficient than increasing the SNN latency which causes all neurons to fire more. It's a promising choice for future SNNs that encode continuous-valued output variables. Human visual search strategyBVSM's search speed slows down with a more human-like search strategy. This provides another evidence that the human eye movement strategy is suboptimal in terms of search speed[47]. A possible cause of the suboptimality is the preference for small saccades[47], which causes a large overlap of foveal regions between successive fixations. Similar saccade amplitude distribution has been observed in a wide range of tasks and stimulus images[71, 72, 73, 74], so probably humans emphasis more on minimizing saccade cost (e.g. saccade duration and saccadic suppression) than using the strictly optimal policy. Additionally, the difficulty of the task may prompt individuals to make cautious small saccades to ensure thorough searching and avoid omissions. Despite having better peripheral target visibility, human search accuracy is worse than the model. In detection task, subjects know the target location, so the visibility may be higher than that in visual search[20, 47]. In detection task without target location cue, the response accuracy at fovea dropped to about 95%, and the detection rate of target at periphery is still higher than the model (Fig. S10, Appendix A.4). However, in visual search which involves multiple fixations, a lower foveal visibility may cause humans to find several suspected target locations and needs to compare between them, which slows down visual search and decreases the accuracy. This was what many subjects reported after finishing the experiment. Measuring target visibility without a target location cue may seem like a more realistic reflection of the visual search experiment. However, in a sequence of eye movements, target visibility may increase at future fixation locations, even to the level comparable to the visibility measured with target location cue[75, 76]. Therefore, the actual target visibility during visual search may fall between the visibility measured with and without target location cue, but it is difficult to measure since we can't accurately predict where humans will fixate in the future. Limitations of BVSMBVSMis only tested on artificial images, requires GPU for training, and is not trained end-to-end, which hinders energy-saving during training. BVSM's retina is also simple compared to the human retina, and it only remembers the past fixation locations and lacks the diverse search strategies of humans, who may remember suspected target locations or scan the image for confirmation of an already found target. Future models could address these limitations by incorporating event camera input, maintaining more information in memory, and including fixational eye movement to improve coding efficiency and discrimination of fine details[77, 78]. A world model and top-down attention[79, 80] could also be added to BVSM to improve its generalizability for diverse vision tasks. For real-life applications, we should also be careful about potential adversarial attacks that draw BVSM's attention to incorrect locations. ConclusionWe propose BVSM, the first bio-inspired model integrating non-uniform resolution retina, efficient eye movement strategy and SNN. It excels in executing visual searches with ultra-low saccade decision latency and sparse activations, which lay the foundation for integration with neuromorphic chips to achieve high energy efficiency. The comparison with human eye movement behavior provides another piece of evidence that the human search strategy is suboptimal on certain tasks, and it also reveals the limitations of BVSM. 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[2020] Sarthak Mittal, Alex Lamb, Anirudh Goyal, Vikram Voleti, Murray Shanahan, Guillaume Lajoie, Michael Mozer, and Yoshua Bengio. Learning to combine top-down and bottom-up signals in recurrent neural networks with attention over modules. In Hal Daume III and Aarti Singh, editors, _Proceedings of the 37th International Conference on Machine Learning_, volume 119 of _Proceedings of Machine Learning Research_, pages 6972-6986. PMLR, 13-18 Jul 2020. URL [https://proceedings.mlr.press/v119/mittal20a.html](https://proceedings.mlr.press/v119/mittal20a.html). Human Experiment Details ### Human Subjects 29 human subjects (14 females, aged 19-30 years) participated in the study for monetary compensation. All subjects were tested to have normal or corrected-to-normal vision, and reported no neurological or psychological disease history. This study was approved by the Ethics Committee of our institution, and all subjects provided written informed consent. ### Apparatus We used a 24.5-inch BenQ ZOWIE XL2540 LCD monitor (resolution 1920x1080, physical size 54.3744x30.2616 cm, refresh rate 240 Hz) to present the stimuli. Stimuli were generated by custom MATLAB (MathWorks) scripts and presented using MATLAB and the Psychophysics Toolbox [81, 82, 83]. Subjects were seated 70 cm away from the screen, and their heads were fixed by a forehead and chin rest. Binocular eye movements were recorded at 2,000 Hz with a TRACKPixx3 eye-tracker (Vpixx Technologies, Canada). Before each experiment, subjects first performed a 13-point calibration until the average test-retest measurement error of both eyes fell below 0.5${}^{\circ}$. The eye-tracker was re-calibrated whenever the eye-tracking accuracy failed to meet the experiment requirements (see below). ### Target Contrast Selection Experiment This experiment used a two-interval-force-choice (2IFC) task to measure the relationship between foveal target visibility ($d^{\prime}$ in signal detection theory [84]) and target contrast, and selected a target contrast for each individual that has the same foveal visibility. 15 subjects participated in the experiment. The details of the procedure were reported in [85]. In each trial, subjects fixated at image center and viewed two stimulus images for 250 ms each. One image was pure background noise, and the other was background noise with a target embedded at image center. Subjects had to select the image with the target. Within each trial, we used the eye-tracker to continuously monitor subject's fixation location. If the subject fixated more than 1${}^{\circ}$ away from the image center, the trial would be aborted. We tested 4-5 levels of target contrast (ranging from 0.03 to 0.12), and each contrast level had 200 valid trials. In each trial, the target contrast is randomly selected from these levels. Visibility is calculated as [84]: \[d^{\prime}=\frac{z(p(\text{Hit}))-z(1-p(\text{CR}))}{\sqrt{2}},\] (S1) where $p(\text{Hit})$ is the hit rate, $p(\text{CR})$ is the correct rejection rate, and $z(x)$ is the inverse standard normal cumulative distribution function. The hit (or correct rejection) rate was defined as the number of trials that target appeared in the first (or second) interval, and the subject made a correct response, divided by the total number of trials that target appeared in the first (or second) interval. For each subject, we selected the target contrast that had a foveal visibility of 3.0 and used it for all following experiments. This was done by first fitting a Weibull function between the hit and correct rejection rate and target contrast, and then use equation S1 to find the target contrast that makes $d^{\prime}=3.0$. The Weibull function is fitted by maximum likelihood estimation implemented in the Palamedes Toolbox [83]. Therefore, each subject had a different target contrast (ranged 0.11-0.136). ### Target Visibility Map Experiment This experiment measured the target visibility as a function of retinal location and fixation duration. The procedure was the same as the first task, except that the target contrast was fixed at the selected level, and the target was presented at different locations for different durations. The details of the procedure were reported in [85]. The experiment had three versions. Four subjects participated in the first version, four subjects participated in the second version (one participated in both the first and second versions), and the rest eight subjects participated in the third version. In the first version, the subjects were required to fixate at the image center and identify which of the two stimulus intervals contained a target at a cued location. The experiment contained five sessions. In each session, the target appeared in one of the four cardinal directions relative to the image center, or appeared at the image center. We chose 3-4 different target locations (except when the target was at the image center) and 4-5 levels of stimulus duration in each session. When the target appeared at the image center, the stimulus duration was chosen from 4 to 200 ms, and each level had 100 trials. When the target appeared at peripheral locations, the stimulus duration was chosen from 50 to 700 ms, and each combination of stimulus duration and target location had 50 trials. There were about 5000 trials in total. The tested target locations were determined for each subject by a pilot experiment. The second version was the same as the first version, except that the target location cue was not shown before each trial, and the target location and stimulus duration were randomly chosen from all possible combinations in each trial. There were about 5000 trials in total. The third version was the same as the first version, except that there was only one level of stimulus presentation time (250 ms). It was used to measure the visibility map efficiently and give subjects some training to prepare for the visual search experiment. Subjects should maintain the fixation at the image center in each trial. We used the eye-tracker to continuously monitor subject's fixation location. If the subject fixated more than 1${}^{\circ}$ away from the image center, the trial would be aborted. Aborted trials were not counted in the number of trials that the subject has finished. ### Target Visibility Data Analysis Method We use the visual information accumulation model proposed by Zhou and Yu[85] to fit the experimentally measured target visibility data. The model describes a leaky integration of visual signals at each location over time within a fixation, and the relationship between target visibility $d^{\prime}$ and location $(x,y)$ and fixation duration $T$ can be written as: \[a(x,y)=p_{1}\cdot\exp\left(-p_{2}\cdot\sqrt{x^{2}+p_{5}\cdot y^{ 2}}\right),\] (S2) \[k(x,y)=p_{3}\cdot\exp\left(-p_{4}\cdot\sqrt{x^{2}+p_{5}\cdot y^ {2}}\right),\] (S3) \[d^{\prime}(x,y,T)=a(x,y)\sqrt{\frac{1-\exp(-k(x,y)T)}{k(x,y) \left[1+\exp(-k(x,y)T)\right]}},\] (S4) where $p_{1}\sim p_{5}$ are free parameters. The values of these parameters were fitted by minimizing the mean-squared error between the calculated and measured visibility data by the GlobalSearch algorithm in MATLAB's Global Optimization Toolbox. ### Visual Search Experiment In visual search task, subjects were told to search for the target as fast and accurately as possible, but there is no upper limit on search time. For the 15 subjects that participated in the detection task, the target contrast was the individually selected contrast (ranged 0.11-0.136, called low target contrast group). The rest 14 subjects did not participate in the detection task, and the target contrast was fixed to 0.15 (called high target contrast group). Each trial started by fixating within 0.8${}^{\circ}$ away from the image center, and the target location was sampled uniformly within the region of background noise. Subjects were informed that the target could appear at anywhere within the image, even right at the initial fixation location. Subjects ended the trial by fixating on the target location and pressing a button. If the final fixation location is less than 1${}^{\circ}$ away from the target, the trial is labeled as correct. Subjects first practiced for 100-150 trials, and performed 200-400 formal trials depending on their available time. We would re-calibrate the eye-tracker if subjects reported that they had correctly found the target, but the trial was wrong. ### Eye Tracking Data Analysis Method We analyzed eye-tracking data by the EYE-EEG extension[86] of the EEGLAB toolbox[87]. We used a velocity-based algorithm with adaptive threshold[88] to classify eye movements into saccades, fixations, and blinks. Fixations separated by blinks were classified as two fixations. Glissades were merged into fixations. We used the minimum instantaneous eye movement velocity of both eyes in the eye movement event detection algorithm to suppress intermittent noise in the data of a single eye. Since each subject finished a different number of trials, when summarizing eye movement statistics, we first summarize the eye movement statistics within each subject, then average the statistics across different subjects. ## Appendix B Bayesian Optimal Visual Search Model We use the entropy-limit-minimization (ELM) searcher[89] as the optimal visual search model. The model's input is not the actual search image, but random signals generated according to human target visibility data. According to signal detection theory[84], we assume that the visual signals obtained within one fixation from one location of the visual field obey a normal distribution. The presence of the search target at this location will shift the mean of the distribution, but will not change the variance. Visibility $d^{\prime}$ is also defined as the difference between the mean divided by the common standard deviation. Theoretically, the ELM model should be able to fixate at any location in the image. In practice, to decrease the computational cost, the model could only fixate at 400 locations uniformly sampled inside the search image. We set the visual signal $W_{i,L_{F}}$ obtained at location $i$ ($i=1\cdots n,n=400$) during fixation $F$ at location $L_{F}$ obeys the normal distribution $N(\pm 0.5,1/d_{i,L_{F}}^{2})$, where the mean is positive only when the target is at location $i$. $d_{i,L_{F}}^{\prime}$ is calculated by equation S4, and the parameter $T$ in equation S4 is fixed to 250 ms (close to the average fixation duration). Therefore, the visual signals gathered from all 400 locations is a vector $\textbf{W}_{L_{F}}=(W_{1,L_{F}},\cdots,W_{n,L_{F}})$. The posterior probability of target location given the visual signals gathered from all fixations can be calculated as: \[P_{i,F} =P\left(i\mid\textbf{W}_{L_{1}},\cdots\textbf{W}_{L_{F}}\right)\] \[=\frac{P\left(\textbf{W}_{L_{1}},\cdots\textbf{W}_{L_{F}}\mid i \right)p(i)}{\sum_{j=1}^{n}\left[P\left(\textbf{W}_{L_{1}},\cdots\textbf{W}_{ L_{F}}\mid j\right)p(j)\right]}\] \[=\frac{e^{\sum_{f=1}^{F}W_{i,L_{f}}d_{i,L_{f}}^{2}}\cdot p(i)}{ \sum_{j=1}^{n}\left[e^{\sum_{f=1}^{F}W_{j,L_{f}}d_{j,L_{f}}^{2}}\cdot p(j) \right]}\] \[=\frac{P_{i,F-1}\cdot e^{W_{i,L_{F}}d_{i,L_{F}}^{2}}}{\sum_{j=1}^ {n}\left(P_{j,F-1}\cdot e^{W_{j,L_{F}}d_{j,L_{F}}^{2}}\right)},\] (S5) where $p(i)=1/400$ is the prior probability of target location. The ELM searcher chooses the location that maximizes the expected information gained from the next fixation as the next fixation location. This can be expressed as: \[H_{L_{F}}=-\sum_{i=1}^{n}\left[P_{i,F}\cdot\ln P_{i,F}\right],\] \[L_{F+1}=\underset{L_{F+1}}{\text{argmax}}\left\{H_{L_{F}}-E \left[H_{L_{F+1}}\right]\right\},\] (S6) where $E[\cdot]$ represents expectation with respect to the visual signal obtained in the next fixation. $E\left[H_{L_{F+1}}\right]$ is an unknown variable that should be estimated, and $H_{L_{F}}$ is a known variable that can be calculated from the posterior probability of target location. Najemnik and Geisler[89] derive a simple way to estimate $E\left[H_{L_{F+1}}\right]$, and found that the next fixation location can be selected as: \[L_{F+1}=\underset{L_{F+1}}{\text{argmax}}\left\{\frac{1}{2}\sum_{i=1}^{n} \left(P_{i,F}\cdot d_{i,L_{F+1}}^{\prime 2}\right)\right\}.\] (S7) When the posterior probability of target being at current fixation location exceeds a threshold $\theta_{T}$, the ELM model sets the current fixation location as the responded target location. $\theta_{T}$is a constant value determined by matching the model's search accuracy to BVSM's search accuracy. In addition, the maximum number of fixations per trial is 200, and if this number is reached, the model will set the last fixation location as the responded target location. If the responded target location matches the actual target location, the trial is considered to be a correct trial. ## Appendix C Bio-inspired Visual Search Model ### Artificial Retina We modified the original Foveal Cartesian Geometry[90] to make the transition of sampling density between central and peripheral vision to be more smooth. We define the radius (half of the width in pixels) $r$ of a series of sampling rings (indexed by $i$ starting from 1) as: \[r(i)=\begin{cases}i,&i\leq i_{\text{fovea}}\\ a^{i+b}+c,&i>i_{\text{fovea}}\end{cases}\] (S8) where _a, b, c_ are parameters, and $i_{\text{fovea}}$ is the radius (in pixels) of fovea in the output image. The three parameters in equation S8 should satisfy three constraints: 1. The two functions should intersect at $i_{\text{fovea}}$; 2. $r(i)$ should be smooth at $i_{\text{fovea}}$ (first-order derivative is equal); 3. The radius of the outermost ring should be a predefined value $r_{\max}$. This means that: \[\begin{cases}a^{i_{\text{fovea}}+b}+c=i_{\text{fovea}}\\ a^{i_{\text{fovea}}+b}\cdot\ln a=1\\ a^{i_{\max}+b}+c=r_{\max}\end{cases}\] (S9) In practice, the systems of non-linear equations can be solved by a modification of the Levenberg-Marquardt algorithm implemented in SciPy (_scipy.optimize.root(method='lm')_)[91]. We set the size of output image to be 224x224 with a fovea size of $16\times 16$ pixels ($i_{\max}=112,\ i_{\text{fovea}}=8$). To make sure the entire search image is always within the visual field (as is the case for humans), we set $r_{\max}$ to be twice the radius of the search image. On each ring, we uniformly sample ($8i-4$) number of pixels. This means that pixels within the fovea are directly copied to the output, and pixels outside of the fovea are sampled progressively sparser. The sampled pixels are stacked together to form a square image as the output. We implemented the above retinal transformation by using the Kornia package[92]. ### Feature Extraction Network The feature extraction network contains 7 convolution blocks. The first convolution block uses 7x7 convolution, and subsequent blocks use 3x3 convolution. The number of output channels of the $i^{th}$ block is 16x$i$. We use replicate padding in convolution, and down-sampling is achieved by setting stride $=2$. The output of the last convolution block is flattened into a 448-dimensional vector and fed into the three fully connected heads. Each fully connected head contains one layer of neurons. To convert FEN to SNN, we directly replace the QCFS function with the scaled integrate-and-fire (IF) neuron[93]: \[\begin{cases}\widetilde{\mathbf{v}}_{l,t}=\mathbf{v}_{l,t-1}+\frac{\mathbf{W}_{l}\mathbf{ s}_{l-1,t}+\mathbf{b}_{l}}{\lambda}\\ \widetilde{\mathbf{s}}_{l,t}=H\left(\widetilde{\mathbf{v}}_{l,t}-\theta\right)\\ \mathbf{s}_{l,t}=\lambda\widetilde{\mathbf{s}}_{l,t}\\ \mathbf{v}_{l,t}=\widetilde{\mathbf{v}}_{l,t}-\widetilde{\mathbf{s}}_{l,t}\theta_{l}\end{cases}\] (S10) where $l$ is the layer index, $t$ is the SNN inference time index, $\mathbf{v}_{l,t}$ is the membrane potential of IF neurons in layer $l$ at time $t$ ($\mathbf{v}_{l,0}=0.5$), $\mathbf{s}_{l,t}$ is the scaled output spikes from layer $l$ at time $t$, $\theta=1$ is the firing threshold, $H\left(\cdot\right)$ is the Heaviside step function, $\mathbf{W}_{l}$ and $\mathbf{b}_{l}$ are the connection weights and bias from layer $l-1$ to layer $l$, and $\lambda$ is the parameter $\lambda$ in QCFS function. When a neuron fires, we reset its membrane potential by subtracting $\theta$ to prevent information loss[94]. We directly use the pixel values (repeated for $T=4$ time steps) as the network's input. Given input images with batch size $B$, the output of the converted SNN has dimension $(T\times B\times 5)$. ### Recurrent Spiking Network Denote SNN inference time as $T$ and batch size as $B$. At each fixation $l$, the network takes the predicted fixation location $\mathbf{x}_{l}$ ($T\times B\times 2$) and the RNN hidden state $\mathbf{h}_{l}$ ($B\times 64$) as input. Within each fixation, at the first SNN time step, the network linearly transforms $\mathbf{x}_{l}[1,:,:]$ and $\mathbf{h}_{l}$. The sum of the results is then passed into IF neurons and produces output spikes $\mathbf{s}_{l}[1,:,:]$. At subsequent time steps $t$, the network only linearly transforms $\mathbf{x}_{l}[t,:,:]$ and produce the output spikes $\mathbf{s}_{l}[t,:,:]$. The output spikes from the last time step become the RNN hidden state $\mathbf{h}_{l+1}$ for the next fixation. This computation process is also shown in Algorithm S1. The IF neurons are reset to the initial membrane potential (0.5) at the start of each fixation. In practice, the last linear transform in FEN and the initial linear transform in RNN can be merged together, allowing spikes to straightforwardly transmit from FEN to RNN. ``` Input:$\mathbf{x}$ with dimension $(T,B,L,2)$, initial RNN hidden state $\mathbf{h}_{0}$ with dimension $(B,64)$ (filled with zeros). $T$ is the SNN inference time. $B$ is batch size. $L$ is the number of fixations. Initialize output spikes $\mathbf{s}$ with dimension $(T,B,L,64)$. Load linear transform weight matrix $\mathbf{W}_{xr}$, $\mathbf{W}_{rr}$ and bias $\mathbf{b}_{x}$, $\mathbf{b}_{r}$. for$l=1$to$L$do Reset IF neuron membrane potential to 0.5. for$t=1$to$T$do $\mathbf{y}=\mathbf{x}[t,:,l,:]\cdot\mathbf{W}_{xr}+\mathbf{b}_{x}$ ift==1then $\mathbf{y}=\mathbf{y}+\mathbf{h}_{l-1}\cdot\mathbf{W}_{rr}+\mathbf{b}_{r}$ endif $\mathbf{s}[t,:,l,:]=\text{IF}(\mathbf{y})$ according to equation S10. Output spikes $\mathbf{s}[t,:,l,:]$. endfor $\mathbf{h}_{l}=\mathbf{s}[T,:,l,:]$ endfor Output: RNN hidden state $\mathbf{h}_{L}$ ``` Algorithm S1* Recurrent spiking network ### Actor and Critic Networks The actor network makes saccade decisions. If the estimated target location error from FEN is lower than a threshold (0.58${}^{\circ}$, or 25 pixels), the next fixation location will be sampled from a 2-dimensional Gaussian distribution centered at the predicted target location with a small symmetrical covariance matrix of 15 pixel${}^{2}$; otherwise the actor will pass the RNN output into two layers of 480-dimension IF neurons (equation S10), and linearly transform the result into a 5-dimensional vector (population coding), which encodes the mean and covariance of the next fixation location's distribution (Fig. S1D). The fixation location output by the actor can be anywhere within the square region enclosing the circular search image. The critic network measures saccade value (Fig. S1E). If the estimated target location error is low, it will concatenate the next fixation location with the predicted absolute target location and output a Q-value. Otherwise it will concatenate the next fixation location with RNN output, and calculate another Q-value. The critic network is not a spiking network, as it is not used in inference after training. Therefore, the output from the RNN needs to be averaged over the first dimension (representing SNN time steps) before concatenation. ### Detection Task by FEN We use a 2IFC task similar to human experiment to measure the FEN's ability to correctly locate the target as a function of target eccentricity. In each trial the network is presented with two images, one contains the background alone, and the other contains the background and a target. We select the image whose estimated target location error is below the detection threshold (25 pixels), and the predicted target location is less than 1${}^{\circ}$ away from the true target location. We use the true target location in decision rule because the target location is cued in human experiment. If the decision rule is not matched for both images, then we will randomly select an image. A trial is correct if the selected image contains the target. We measure the correct response rate of 6 target contrasts (0.11 to 0.135, step 0.005), 49 target eccentricities (0.0 to 0.98 of the image radius, step 0.02), and 8 target angular positions (0${}^{\circ}$ to 315${}^{\circ}$, step 45${}^{\circ}$). Each combination of conditions contains 400 trials. The relationship between proportion correct and target distance is fitted by the Weibull function (with inverted x-axis, because Weibull function is monotonically increasing) by maximum likelihood estimation implemented in the Palamedes toolbox [83]. ### BVSM Training and Evaluation Method In the first training stage, we train the FEN with QCFS function as activation function by supervised learning. The network is trained for 174,000 optimization steps. In the second training stage, we convert the trained FEN to spiking FEN, and then fine-tune the spiking network for 35,000 optimization steps by supervised learning. Fine-tuning is done by STBP implemented in spikingjelly [95]. All parameters remain trainable. We use the Atan function to approximate the gradient of a spike [96]. The training data in the first two stages are random samples of retinal transformed images generated at runtime during training. We first generate random samples of full-resolution background noise images. For each image, we randomly set a fixation location uniformly within the circular region. We then randomly set one target location per image whose distance to fixation location obeys exponential distribution with a mean of 4${}^{\circ}$. The target location has to be within the circular noise image, otherwise we will resample another target location. This ensures that enough training samples contain the target near the foveal region and helps the network to converge more reliably. In the third training stage, we fix the FEN, and use SAC [97] algorithm to train the RNN, actor, and critic networks to do visual search. The SAC agent contains one FEN, one RNN with one target network, one actor, and two critics with two target networks. We use a replay buffer of 50,000 trials (each trial can have a variable number of fixations). The state vector stored in replay buffer is the FEN output in each fixation. Optimization started after the initial 333 trials, and is performed once per fixation. In each optimization, we randomly sample 32 trials (with variable number of fixations) from the replay buffer, update critic and RNN by temporal difference learning, update actor alone by gradient ascent through the critic output, and finally update the SAC temperature parameter according to [97]. The gradients in RNN and actor are calculated by STBP and surrogate gradient (Atan function). The gradients of RNN and critic are clipped to a maximum norm of 1.0. The number of training trials is 50,000. A list of hyperparameters can be found in Table S3. During training, target contrast is randomly chosen within [0.11, 0.15]. Target and initial fixation locations are randomly chosen within the circular search image. All three stages of training were performed on a computer with Intel i9 9900K CPU and NVIDIA GeForce RTX 2080Super GPU. The first and second stages required less than 5 hours, and the third stage required about 20 hours. The code is implemented in PyTorch [98]. During evaluation, target location is randomly chosen within the circular search image, but the initial fixation location is randomly chosen within 0.35${}^{\circ}$ away from the image center. This is similar to human experiment where a visual search trial always started from fixation at the image center. The target contrast is chosen within [0.11, 0.136] (low target contrast) or fixed to 0.15 (high target contrast) during evaluation, corresponding to the target contrast range in human experiment. The model is evaluated for 10,000 visual search trials for each target contrast level. ### Firing Rate and Sparseness Analysis Method We record the activities of every neuron in BVSM during evaluation. The firing rate of each neuron is calculated as the number of time steps that the neuron fires divided by the total number of time steps in 10,000 visual search trials. The total number of time steps is the total number of fixations multiplied by SNN latency (4 time steps) in each fixation. Therefore, the firing rate of each neuron falls in the region between 0 and 1. BVSM has 359,504 neurons in total, and 99.72% of them are in the FEN. To calculate the change of population sparseness during training, we calculate the change of population sparseness during the first training stage of FEN. Note that in the first training stage, the FEN is not an SNN, but the QCFS activation function [93] outputs the firing rate (scaled by $\lambda$ in QCFS function) of each neuron. Therefore, we can calculate the population sparseness [99, 100, 101] by the unscaled QCFS output: \[r_{i}=\text{clip}\left(\frac{1}{T}\text{floor}\left(\frac{xT}{ \lambda}+0.5\right),0,1\right),\] (S11) \[S=\frac{1}{1-\frac{1}{N}}\left[1-\frac{\left(\sum_{i=1}^{N}\frac {r_{i}}{N}\right)^{2}}{\sum_{i=1}^{N}\frac{r_{i}^{2}}{N}}\right],\] (S12) where $i$ represents neuron index and $N$ is the total number of neurons. To obtain the final stable population sparseness during the FEN training process, we increase the number of training steps from 175,000 to 400,000. But when running visual search, we still use the parameters saved at 175,000 training steps. ### Energy Consumption Analysis Method We compare the energy consumption of BVSM with two different models. One is BVSM based on conventional artificial neural networks, the other is a conventional image classification network that directly processes original full-resolution images. BVSM based on ANN can be constructed by replacing IF neurons in BVSM to leaky-ReLU activation function, and its search performance is similar to BVSM. The conventional full-resolution ANN has a similar convolution backbone to the feature extraction network in BVSM (Fig. S2), but is deeper since the input size is larger. We use the technical data of Intel Stratix 10 TX FPGA chip [102] and ROLLS neuromorphic chip [103] to estimate the energy consumption of BVSM and the other two networks. The FPGA consumes 12.5 pJ per FLOP, while the neuromorphic chip consumes 0.077 pJ per synaptic operation (SOP) and 3.7 pJ per spike. The same data were used in prior works [93, 104] to estimate the energy consumption of ANN and SNN. For SNN, one SOP is defined as the transmission of one spike from a single neuron to another single neuron. We count the average number of spikes that each neuron fires within each fixation from 10000 visual search trials. The number of post-synaptic neurons of each neuron can also be calculated from the network structure. We do not account for the energy consumed by the floating point computation in the first convolution layer, but the energy consumed by the spikes generated in the first layer is included. The reason for this is that we lack energy consumption data for neuromorphic chips under a constant floating point input. Additionally, some neuromorphic chips, such as Intel Loihi, quantize the floating point into low-precision fixed-point representations, and the energy spent on this process is difficult to estimate accurately. To ensure a fair comparison, we also disregarded the floating point operation in the first convolution layer of the ANN. For ANN, one FLOP is defined as the elementary calculation between two scalars. The FLOP of a convolution layer is $2\times C_{in}\times K\times K\times C_{out}\times H_{out}\times W_{out}$, where $C$ is channels, $K$ is the convolution kernel size, and $H,W$ are the size of the output. The FLOP of a linear layer is $2\times W_{in}\times W_{out}$, where $W_{in}$ and $W_{out}$ are input and output dimensions. Batch normalization can be merged into convolution and linear layers, so they do not add additional FLOPs. We also assume that the ANN uses simple activation functions (such as ReLU or Leaky-ReLU), so the FLOP of activation function is simply the number of output neurons of each layer. ## Appendix D Full-resolution Image Classification Model To test whether conventional ANN that directly processes full-resolution images can detect the target, we build a convolution network that has a similar backbone structure to the FEN (Fig. S2). The network performs a binary classification task, which is to determine whether the input image contains a target or not. This is easier than the visual search task which needs the network to output the exact target location. The network outputs the classification probability by a single forward computation without making eye movements. We generate random samples of the search images, and train the ANN by supervised learning. The loss function is binary cross-entropy, the optimizer is AdamW without weight decay, the learning rate is 1e-3, and the training batch size is 64. We train the network for 175,000 steps (same as the training step in the first training stage of FEN) under 5 different random seeds, but the loss value cannot decrease (Fig. S7), and the network's correct response rate is only at chance level (50%). Therefore, conventional ANN that directly processes full-resolution images cannot properly detect the small search target. Supplementary Tables Table S1: Comparison of our task and model design to previous research. Target's size, background image's size, and target's relative size are only shown for visual search tasks. Variable fixation number means that the model can automatically stop the trial without hard coding it to make decision after a predefined number of fixations. Training method refers to the method to train eye movement policy. EM: expectation maximization. RL: reinforcement learning. SVHN: The Street View House Numbers dataset. IOR: inhibition of return. (1): This is the size of the smallest bounding box enclosing all targets. The size is calculated based on the authors' data preprocessing method. (2): A single bounding box may contain 2-5 targets. (3): The target size, image size, and target relative size are calculated from the dataset. \begin{tabular}{c c c c c c c c c c} \hline \hline Source & Task & Target & Back- & Target relative & Artificial fovea & Memory & SNN & Number of & Training \\ & & size & ground size & size & & & & & & \\ \hline \multirow{3}{}{[106]} & MNIST search \& & 28x28 & 48x48 & 34.0\% & Multi-resolution patches & Mean of predictions at each fixation & No & Fixed & Similar to \\ & recognition & & & & & & & & & EM \\ \hline \multirow{3}{}{[106]} & MNIST search \& & 28x28 & 60x60 & 21.8\% & Multi-resolution patches & RNN & No & Fixed & RL \\ & recognition & & & & & & & & & \\ \hline \multirow{3}{}{[107]} & MNIST search \& & 28x28 & 100x100 & 7.84\% & Multi-resolution patches & RNN & No & Fixed per target & RL \\ & recognition & & & & & & & & & \\ & targets & & & & & & & & & \\ \hline \multirow{3}{}{[108]} & SVHN recognition & 49x49${}^{17}$ & 54x54 & 16.47\% & Multi-resolution patches & RNN & No & Fixed per target & RL \\ & & & & & & 41.179\% & 20\% & & & & \\ & & & & & & & & & & \\ \hline \multirow{3}{}{[108]} & Noisy MNIST search \& & 28x28 & 128x128 & 4.79\% & Log-Polar transform & No & No & Variable & Supervised \\ & recognition & & & & & & & & & \\ \hline \multirow{3}{}{[109]} & MNIST search \& & 28x28 & 60x60 & 21.78\% & Multi-resolution patches & RNN & No & Fixed & Supervised \\ & recognition & & & & & & & & & \\ \hline \multirow{3}{}{[111]} & Natural object search & & (2.2 & (142.500.) & Median: 9.71\% & Single patch memory & Exponential decay memory & No & Fixed & Supervised \\ & 499\%(x49) & & (71:500.) & Mean: 16.67\% & Simplified Freemonstrical model & Bayesian prior, manual & No & Variable & Maximum-a-posteriori \\ \hline \multirow{3}{}{[112]} & Image classification & N/A & N/A & N/A & Single patch each fixation, manual & Mean of predictions at each fixation, manual & No & Fixed & RL \\ \hline \multirow{3}{}{[114]} & Image classification & N/A & N/A & N/A & Single patch & Transformer & No & Training, fused, Testing, variable & RL \\ \hline \multirow{3}{}{[115]} & Video classification & N/A & N/A & N/A & Single patch & Transformer & No & Fixed & Supervised \\ & & & & & & & & & \\ \hline Our model & Gabor target search & 12x12 & 651x651 & 0.034\% & Foveal Cartesian Geometry & Spaking RNN & Yes & Variable & RL \\ \hline \hline \end{tabular} \begin{table} \begin{tabular}{c c c c c c c c} \hline \hline Source & Task & Image type & RL algorithm & Neuron type & Coding method & Training method & Lategory \\ \hline [116] & Object detection & Natural image & N/A & Signed IF & Rate coding & Conversion & $>$3500 \\ \hline [117] & Object detection & Natural image & N/A & IF & Rate coding & Bayesian optimization & $>$500 \\ \hline [118] & Object detection & Natural image & N/A & LIF & Rate coding & Conversion + STDP + STBP & 300 \\ \hline [119] & Object detection & Neuromorphic image & N/A & LIF & Spike count & STBP & 100-300${}^{(1)}$ \\ \hline [120] & Object detection & Natural image & N/A & IF & Rate coding & Conversion & $>$1000 \\ \hline [121] & Object detection & Aerospace image & N/A & Modified LIF & Rate coding & Conversion & 64-128 \\ \hline [122] & RL (Breakout) & N/A & Deep & IF & Rate coding & Conversion & 500 \\ \hline [123] & RL (HalfCheeta-v3, Hopper-v3, Walker2d-v3, Ant-v3) & N/A & DDPG, TD3, SAC, PPO & LIF & Population coding & STBP & 5 \\ \hline [124] & RL (17 Atari games) & N/A & Deep & IF & Rate coding & Conversion & $>$100 \\ \hline [125] & RL (Cartpole, Pong) & N/A & Deep & LIF+custom & Membrane potential of non-spiking neuron & Conversion + STBP + iterative retraining & 1 \\ \hline [126] & RL (Custom navigation task) & N/A & DDPG & Modified LIF & Population coding & STBP & 5 \\ \hline [127] & RL (17 Atari games) & N/A & Deep & LIF+custom & Membrane potential of non-spiking neuron & STBP & 8 \\ \hline Out model & Detection \& RL (visual search) & Naturalistic noise image & SAC & IF & Population coding & Conversion $\star$ STBP & 4 \\ \hline \hline \end{tabular} \end{table} Table S2: Comparison of SNN configurations in our model and previous works for object detection and RL. IF: integrate-and-fire neuron. LIF: leaky integrate-and-fire neuron. STDP: spike-timing-dependent plasticity. STBP: spatio-temporal back-propagation. (1): The inference time depends on the length of event-based input. \begin{table} \begin{tabular}{c c c} \hline \hline Group & Description & Value \\ \hline \multirow{8}{}{Visual search task} & Screen distance & 70 cm \\ & Screen size & 1920$\times$1080 pixels, 54.3744$\times$30.2616 cm \\ & Search image size & 15${}^{\circ}$$\times$15${}^{\circ}$ (651$\pm$51 pixels), circular shape \\ & Search image contrast & 0.2 \\ & Target size & 0.3${}^{\circ}$$\times$0.3 (${}^{\circ}$1212 pixels) \\ & Target orientation & 45${}^{\circ}$ counterclockwise from the vertical \\ & Target spatial frequency & 6 cycles/degree \\ & Target contrast & Training: [0.11–0.15]. Testing: [0.11–0.136] or 0.15 \\ & Target number per image & 1 \\ & Target location & Uniformly random in search image \\ & Initial fixation location & Training: uniformly random in image. \\ & & Testing: uniformly random within 0.35${}^{\circ}$ away from image center. \\ & Maximum fixation number & 50 during training, 200 during testing \\ & Correct criterion & Final fixation within 1${}^{\circ}$ from target location \\ \hline \multirow{2}{}{Retinal transform} & Method & Modified Reveal Cartesian Geometry \\ & Fowa radius & 8 pixels \\ & Maximum peripheral radius & 651 pixels \\ & Output size & 224$\times$224 pixels \\ \hline \multirow{2}{}{Reward function} & Radius of IOR reward & 2.5${}^{\circ}$ \\ & Memory capacity of IOR reward & 8 previous fixations \\ & Slope of saccade amplitude reward & -1/7.5${}^{\circ}$ \\ \hline \multirow{4}{}{IF neuron} & Inference time & 4 steps \\ & Surrogate gradient function & Atan() \\ & Initial membrane potential & 0.5 \\ & Firing threshold & 1 \\ & Reset method after spike & Reset by subtracting firing threshold \\ \hline \multirow{4}{}{FEN} & Convolution kernel size & 7$\times$7 at first layer, 3$\times$3 at subsequent layers \\ & Convolution stride & 2 \\ & Padding & Replicate padding, padding size=(kernel size $-$$1)/2$ \\ & Output channel number at block $i$ & 16$\times$i \\ & Activation function & QCFS(Initial $\lambda$=8, 7=4), $\lambda$ is trainable \\ & Fully connected layer hidden dimension & 1 layer, 448 neurons \\ \hline RNN & Number of IF spiking neurons & 64 \\ \hline \multirow{4}{}{Actor network} & Number of hidden layers & 2 \\ & Hidden layer dimension & 480 IF neurons \\ & Target detection threshold & 25 pixels \\ \hline \multirow{4}{}{Critic network} & Number of hidden layers & 2 and 3 for the two heads \\ & Hidden layer dimension & 64 for both heads \\ & Activation function & Leaky-ReLU, negative slope = 0.1 \\ \hline \multirow{4}{}{FEN training} & Training batch size & 64 \\ & Validation batch size & 256 \\ & Training steps & 174,000 steps for training, 35,000 steps for fine-tuning \\ & Optimizer & AdamW${}^{[12]}$ with no weight decay \\ & Initial learning rate & 1,000-63 \\ & Distance between target and fixation locations & Exponential distribution with mean=4${}^{0}$ \\ \hline \multirow{4}{}{RL} & Algorithm & Soft Actor-Critic \\ & Entropy target & Hyperparameter group 1: -1; hyperparameter group 2: -2 \\ & Initial temperature $\alpha$ & 1 \\ & Discount factor$\gamma$ & 0.95 \\ & Replay buffer size & 50,000 trials \\ & Batch size in optimization & 32 trials \\ & Optimizer & AdamW with no weight decay \\ & Initial learning rate & 1e-4 for actor and $\alpha$, 1e-3 for critic and RNN \\ & Gradient clipping & Maximum norm of 1.0 for critic and RNN \\ & Initial optimization time & After 333 trials \\ & Optimization frequency & Once per fixation \\ & Target network update speed & Polyak update, $\tau$=0.005 \\ & Training trials & 50,000 \\ \hline \multirow{4}{}{Reward functions} & IOR reward radius & Hyperparameter group 1: 0.5${}^{\circ}$, Hyperparameter group 2: 2.5${}^{\circ}$ \\ & IOR reward memory capacity & 8 most recent fixations \\ \cline{1-1} & Saccade amplitude reward (exponential) & Hyperparameter group 1: $-0.5+0.5\times(\exp{(-\text{SacAmp}/2.5^{\circ})}-1)$ \\ \cline{1-1} & Saccade amplitude reward (linear) & Hyperparameter group 2: $-\text{SacAmp}/7.5^{\circ}$ \\ \hline \hline \end{tabular} \end{table} Table S3: Hyperparameters for the results reported in the main text. IF neuron: integrate-and-fire neuron. FEN: feature extraction network. RNN: recurrent neural network. RL: reinforcement learning. SacAmp: saccade amplitude. \begin{table} \begin{tabular}{c c c c c c c} \hline \hline Changed parameter & New value & \begin{tabular}{c} Target \\ contrast \\ \end{tabular} & \begin{tabular}{c} \# of random \\ seeds \\ \end{tabular} & \begin{tabular}{c} Percent \\ correct \\ \end{tabular} & \begin{tabular}{c} Median \\ FixNum \\ \end{tabular} & \begin{tabular}{c} Mean \\ FixNum \\ \end{tabular} \\ \hline No change & – & 0.11–0.136 & 5 & 98.2\% & 7.0 & 10.8 \\ Target diameter & 0.6${}^{\circ}$ & 0.11–0.136 & 5 & 99.9\% & 2.0 & 4.2 \\ Discount $\gamma$ & 0.99 & 0.11–0.136 & 5 & 98.0\% & 7.4 & 10.9 \\ IOR memory capacity & 1 & 0.11–0.136 & 5 & 98.0\% & 7.0 & 11.5 \\ IOR memory capacity & 2 & 0.11–0.136 & 5 & 98.1\% & 7.0 & 11.4 \\ IOR memory capacity & 4 & 0.11–0.136 & 5 & 98.1\% & 7.0 & 11.2 \\ IOR radius & 1.5${}^{\circ}$ & 0.11–0.136 & 5 & 97.9\% & 8.2 & 12.5 \\ Slope of SacAmp & -1/4.5${}^{\circ}$ & 0.11–0.136 & 5 & 98.0\% & 7.8 & 11.3 \\ reward (linear) & & & & & & \\ Reward function & Only IOR reward & 0.11–0.136 & 5 & 97.1\% & 9.8 & 17.4 \\ Reward function & Only SacAmp & 0.11–0.136 & 5 & 83.3\% & 21.6 & 44.2 \\ Reward function & & & & & & \\ reward (exp) & & & & & \\ \hline \hline \end{tabular} \end{table} Table S4: Comparison of the estimated energy consumption between BVSM and conventional ANN models. Supplementary Figures Figure S2: Structure of the conventional ANN that processes full resolution image and classifies whether the image contains a target. The dashed lines show the internal components of convolution block. Only the first convolution block uses $7\times 7$ convolution, and all subsequent blocks use $3\times 3$ convolution. The leaky-ReLU function has slope of 0.1 in the negative part. The number under each block indicates the dimension of the output when batch size $B=1$. Figure S4: Up: Feature extraction network (FEN) initial training/validation loss. Low: Spiking FEN fine-tuning training/validation loss. The networks are trained using five different random seeds. The loss values are first smoothed by a Gaussian kernel with standard deviation of 100 steps, then average across the five repetitions. The shaded area represents the standard deviation of the smoothed loss. Figure S5: RL training process averaged across five different random seeds for hyperparameter group 1. The values are first smoothed by a Gaussian kernel with standard deviation of 100 steps, then averaged across the five repetitions. The shaded area represents the standard deviation of the smoothed values. Note that the fixation number is averaged across both correct and error trials, so it is higher than the mean and median fixation number of only correct trials reported in the main text. During training, the maximum fixation is 50, so the percent correct is lower than the result reported in the main text. Figure S6: RL training process averaged across five different random seeds for hyperparameter group 2. The networks are trained using five different random seeds. The loss values are first smoothed by a Gaussian kernel with standard deviation of 100 steps, then average across the five repetitions. The shaded area represents the standard deviation of the smoothed loss. Figure S8: BVSM and human scanpath comparison. A: Randomly selected BVSM scanpath sorted by fixation number. Magenta cross is the final fixation location, red star is the target location. BVSM is trained by hyperparameter group 1 (HP 1). B: Example human scanpath grouped by different search strategies. Gray: circular scanpath. Blue: repeated fixation on suspected target locations. Yellow: confirmation scanpath when the target is close to the initial fixation location. Blue: scanpath with many short saccades.
2306.16270	S2SNet: A Pretrained Neural Network for Superconductivity Discovery	Superconductivity allows electrical current to flow without any energy loss, and thus making solids superconducting is a grand goal of physics, material science, and electrical engineering. More than 16 Nobel Laureates have been awarded for their contribution to superconductivity research. Superconductors are valuable for sustainable development goals (SDGs), such as climate change mitigation, affordable and clean energy, industry, innovation and infrastructure, and so on. However, a unified physics theory explaining all superconductivity mechanism is still unknown. It is believed that superconductivity is microscopically due to not only molecular compositions but also the geometric crystal structure. Hence a new dataset, S2S, containing both crystal structures and superconducting critical temperature, is built upon SuperCon and Material Project. Based on this new dataset, we propose a novel model, S2SNet, which utilizes the attention mechanism for superconductivity prediction. To overcome the shortage of data, S2SNet is pre-trained on the whole Material Project dataset with Masked-Language Modeling (MLM). S2SNet makes a new state-of-the-art, with out-of-sample accuracy of 92% and Area Under Curve (AUC) of 0.92. To the best of our knowledge, S2SNet is the first work to predict superconductivity with only information of crystal structures. This work is beneficial to superconductivity discovery and further SDGs. Code and datasets are available in https://github.com/zjuKeLiu/S2SNet	Ke Liu, Kaifan Yang, Jiahong Zhang, Renjun Xu	2023-06-28T14:52:50Z	http://arxiv.org/abs/2306.16270v1	# S2Net: A Pretrained Neural Network for Superconductivity Discovery ###### Abstract Superconductivity allows electrical current to flow without any energy loss, and thus making solids superconducting is a grand goal of physics, material science, and electrical engineering. More than 16 Nobel Laureates have been awarded for their contribution in superconductivity research. Superconductors are valuable for sustainable development goals (SDGs), such as climate change mitigation, affordable and clean energy, industry, innovation and infrastructure, and so on. However, a unified physics theory explaining all superconductivity mechanism is still unknown. It is believed that superconductivity is microscopically due to not only molecular compositions but also the geometric crystal structure. Hence a new dataset, S2S, containing both crystal structures and superconducting critical temperature, is built upon SuperCon and Material Project. Based on this new dataset, we propose a novel model, S2SNet, which utilizes the attention mechanism for superconductivity prediction. To overcome the shortage of data, S2SNet is pre-trained on the whole Material Project dataset with Masked-Language Modeling (MLM). S2SNet makes a new state-of-the-art, with out-of-sample accuracy of 92% and Area Under Curve (AUC) of 0.92. To the best of our knowledge, S2SNet is the first work to predict superconductivity with only information of crystal structures. This work is beneficial to superconductivity discovery and further SDGs. Code and datasets are available in [https://github.com/zjuKeLiu/S2SNet](https://github.com/zjuKeLiu/S2SNet) ## 1 Introduction Superconductors are crucial for sustainable everyday applications and attractive for the United Nations' sustainable development goals (SDGs) [16]. Superconductors have zero resistance, which allows perfect current transmission without losses. With superconducting cables, a continuous supply of power can be obtained for a timely availability of irrigation water in remote areas, which is a key to sustainability in performing agriculture. Then hunger can be effectively alleviated since it is partly caused by the lack of sustainability in performing agriculture. Affordable and clean energy can also be obtained because DC energy from places where power is plentiful, like solar plants in Africa (365 sunny days) can be delivered worldwide without losses. Zero resistance means "no heat-generation and no energy-loss", which is the key contribution in the climate action. For good health and well-being, superconducting magnets, which can produce magnetic field of high-quality, high-intensity, are well used as high-field magnets in magnetic resonance imaging, nuclear magnetic resonance, water purification, magnetic drug delivery, and so on. Superconductivity has been a hot topic along many fields for its great significance to human beings and SDGs. However, the theory of superconductivity is still a puzzle till now. Benefiting from massive publicly available datasets in the area of material science, many machine learning methods have been applied to predicting material properties, generating structures, and so on [1]. It also sheds a light to modeling superconductors with machine learning methods. Much of the recent interest in superconductivity modelling has focused on critical temperature regression using measured properties of the material [21, 1], which is computationally expensive since feature engineering is required. In AI for physical science, a lot of efforts have been made to model molecules with Graph Neural Networks (GNN) [20, 18] since the molecules are graphs naturally. However, quite few methods have been proposed for crystal modelling [12, 22, 23]. Due to the different annotations among different datasets, the current supervised learning models can only be trained on small datasets, while other unlabeled data are not fully utilized. Recently, attention mechanism is widely used in Natural Language Processing (NLP) [1] and Computer Vision (CV) [15] to model the interaction of contexts. The pre-training task, Masked-Language Modeling (MLM) task, in both NLP and CV, has shown great potential to get model better and more robust with astounding experimental results [1, 16]. Besides, with pre-training tasks, the unlabeled data can be utilized properly. According to BCS theory, superconductivity can be microscopically explained as the interaction between electrons and lattice vibrations (phonons) [1]. Therefore, we propose a machine learning model for the way from Structure toSuperconductivity (S2SNet*), in which only crystal structures are required, to model the interaction in order to predict potential superconducting. Attention mechanism is applied to crystal structures. S2SNet is pre-trained on a large amount of unlabeled data to learn the interaction between different atoms in lattice. Through S2SNet, we try to model the potential interaction between lattice dynamics and cooper pairs to predict the potential superconductor properties of material. It is experimentally verified that S2SNet outperforms other existing models. To the best of our knowledge, S2SNet is the first work showing that it is possible to predict the superconductivity of materials via their crystal structures. The main contributions of our work can be summarized as follows: We propose a novel neural network to discover potential superconductivity of materials given only information of crystal structures for the first time. * A new dataset S2S containing both crystal structure and superconductivity critical temperature is built for the first time. It provides a benchmark for future machine learning based superconductor discovery. * To provide insight and guide the superconductivity research for physicists, we present a list of potential superconductors for future experimental validation. ## 2 Related Work Superconductivity Prediction.Previous machine learning methods focus on predicting the superconductivity through the measured properties of materials, while these properties are obtained through complex and expensive feature engineering. The averaged valence-electron numbers, orbital radii differences, and metallic electronegativity differences were three critical features used in [23]. Materials Agnostic Platform for Informatics and Exploration (Magpie) and AFLOW were used for feature engineering in [22, 1]. Crystal Modeling.Quite few methods are proposed for crystal modeling. By connecting the atoms within a distance threshold, crystals were modeled as a graph with atoms as nodes in CGCNN [13, 14]. External environmental conditions are taken into consideration by taking it as an additional nodes and connecting it with all the other nodes in MEGNet [2]. Unsupervised Learning.Various approaches have been proposed to make full use of the unlabeled data, most of which pre-train a model with the unlabeled dataset and fine-tune it on specific labeled dataset [1]. Contrastive learning is commonly used for pre-training in CV [11]. As pre-training tasks in NLP, diverse techniques such as mask language model (MLM), replaced token detection (RTD), and so on are often employed [15]. Following the pre-training model Bert, Zhang et. al. proposed "MG-Bert" for molecule modelling with unsupervised learning [14]. ## 3 Methodology Attention mechanism is proposed in the filed of NLP and widely used in CV recently [15, 16]. It is applied to the crystals in this study. ### S2SNet Fig. 1 shows the overall illustration of S2SNet and the attention mechanism. In this paper, it is defined that a crystal $c\in C$ consists of atoms $A=\{a_{1},a_{2},\cdots,a_{n}\}$, the location of atoms $P=\{p_{1},p_{2},\cdots,p_{n}\}$, and the lattice parameters $U=\{u_{1},u_{2},u_{3}\}$, i.e. $c=\{A,P,U\}$. $n$, $a$, $p$ and $U$ denote the number of atoms in a unit cell, the atomic number, the actual 3-D location of an atom in the crystal, and the physical dimension of a unit cell respectively. The overall workflow of S2SNet can be described as followings: 1. Atoms $A$ are embedded into 512-dim vectors through the Embedding layer. Expanded distance with Gaussian basis centered at 512 points linearly placed between 0 and 5.12 A is applied to the location $P$ of atoms in the crystal [2]. Specifically, the Euclidean distances $P^{\prime}$ between atoms and the center of crystal i.e. the mean coordinates of all atoms, are calculated. Then Gaussian basis are performed as Eq. 1, where $P^{\prime}_{0}$ are 512 distances from 0 to 5.12 A and $\sigma=2$[2]. \[A^{\prime} =embedding(A),\] \[\hat{P} =e^{-\frac{(P^{\prime}-P^{\prime}_{0})^{2}}{\sigma^{2}}}\] (1) 2. The embedding of atoms $A^{\prime}$ are atom-wise aggregated with the distance $\hat{P}$ as Eq. 2, where $W_{\alpha}$ is a matrix to Figure 1: Illustration of S2SNet and attention mechanism. (a) Overall illustration of S2SNet. The Attention Module are performed for 10 times. $FC1$ and $FC2$ are two fully connected hidden layers. $FC2$ outputs the target $\hat{y}_{t}$. (b) Illustration of the Attention mechanism. The computation of representation of Atom $a_{1}$ is presented. Best view in color. get the position $\hat{P}$ into the same dimension as $A^{\prime}$ and model the interaction between different positions. \[A^{\prime\prime}=A^{\prime}+W_{\alpha}\cdot\hat{P}\] (2) 3. Attention mechanism is performed on each atom respectively with its neighborhood atoms. Thus the embedding of atom contains all the information of itself and neighbors. $neighbor(a^{\prime\prime})$ is defined as the atoms within the Euclidean distance, $\rho$, from $a^{\prime\prime}$. In this paper, $\rho$ equals to 2 A according to [3]. \[a_{i}^{\prime\prime\prime}=Attention(a_{i}^{\prime\prime},neighbor(a_{i}^{ \prime\prime}))\] (3) 4. To aggregate the embedding of all atoms in crystal, the $Set2Set$[21] is performed on the set $A^{\prime\prime\prime}$. \[b=Set2Set(A^{\prime\prime\prime})\] (4) 5. Lattice parameters are aggregated to get the final representation of crystal as Eq. 5, where $W_{\theta}$ is a matrix to get the lattice parameters $U$ into the same dimension as $b$. \[b^{\prime}=b+W_{\theta}\cdot U\] (5) 6. Through the $FC2$ layer, target can be obtained and loss function is binary cross entropy as Eq. 6, where $y_{t}$ is the ground truth of whether it is a superconductor and $\hat{y}_{t}$ is the prediction of S2SNet. \[\mathcal{L}_{finetune}=-\frac{1}{2}[y_{t}\log\hat{y}_{t}+(1-y_{t})\log(1-\hat{ y}_{t})]\] (6) In Fig. 1(b), an example of the attention mechanism is illustrated. Suppose that the neighbors of atom $a_{1}$ are $a_{2}$ and $a_{3}$. To get the representation of atom $a_{1}$, $a_{1}$ and its neighbors, $a_{2}$ and $a_{3}$, are firstly multiplied with $W_{Q}$, $W_{K}$ and $\hat{W}_{V}$ respectively. Three different groups of $\{q,k,v\}$ are obtained. $q_{1}$ from $a_{1}$ is multiplied with $k_{1},k_{2},k_{3}$ from all the atoms to get the $score$ as Eq. 7, where $d_{k}$ is the dimension of $k$. Finally, the representation of atom $a_{1}$ can be obtained as the weighted sum of $V$ from all the atoms as Eq. 8, where $N$ is the number of $a_{1}$'s neighbors plus one. Representations of other atoms $a_{i}\in A$ can also be obtained like this. \[scores=\frac{softmax(q_{1}\cdot K^{T})}{\sqrt{d_{k}}}\] (7) \[a_{1}^{\prime\prime\prime}=\sum_{i=1}^{N}scores_{i}\cdot v_{i}\] (8) ### MLM Pre-training Task MLM pre-training task is a self-supervised learning method to model the interaction between atoms via classification as Fig. 2. In this task, 15% (at least 2) of the atoms in a crystal are masked, and the model are trained to predict the class of masked atoms. Therefore no extra labels except the structure of the crystal are required. Since there are 95 elements in S2S dataset, this pre-training is a 95-classification task. The objective of this pre-training can be defined as Eq. 9, where $N$ and $95$ denote the number of masked atoms and the number of element classes respectively. $y_{i,j}$ is the ground truth and $\hat{y}_{i,j}$ is the prediction of S2SNet for "atom $i$ is the element $j$", 0 for False and 1 for True. \[\mathcal{L}_{pre-train}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{95}y_{i,j}\log \hat{y}_{i,j} \tag{9}\] After convergence of training, the _Embedding_, _Aggregate_, and _Attention_ layers are cloned. Finally, S2SNet is then fine-tuned on S2S dataset with the pre-trained layers frozen. ## 4 Experiments To verify S2SNet, a dataset of superconductors is constructed, on which extensive experiments are conducted. Compared with other models, the experiments demonstrate that S2SNet effectively captures the crystal structural information for superconductivity prediction. ### Setup S2S Dataset.We collect a total of 69,239 crystal structures from Material Project database [15], each of which contains lattice structure, formation energy, and band gap of crystals. 33,810 superconductors and their critical temperatures are obtained from SuperCon, which stores the critical temperature of substances. In addition, this paper also obtains structural information of part of materials from databases such as OQMD [19], and ICSD [1]. The information in the databases above are used to construct S2S dataset with 1,685 crystals, which includes 421 currently known crystal superconductors and other 1,264 materials without known superconductivity sampled from Material Project database. Each entry includes the name of the substance, its crystal structure(atoms, location of atoms and lattice parameters), and whether it is a superconductor. This statistics of S2S dataset and pre-training dataset are shown in Table 1. \begin{table} \begin{tabular}{c c c c} \hline Dataset & $\|U\|$ & $Max\left(\|\mathbf{A}\|\right)$ & $\|V\|$ \\ \hline S2S & 1,685 & 248 & 95 \\ pre-train & 69,239 & 444 & 95 \\ \hline \end{tabular} \end{table} Table 1: Statistics of the datasets. $\|V\|$ indicates the number of classes of atoms. $\|U\|$ indicates the number of materials Figure 2: Illustration of MLM pre-training task. Classifier contains a layer with 95 neurons and a $softmax$ layer and outputs $N$ vectors of 95-dimension $\hat{y}$. Compared Approaches.We mainly compare our proposal with several superconductivity prediction methods, MEGNet[3], Statistic[14], and CGCNN[15]. MEGNet uses graph networks to model molecules and crystals universally for property prediction. Statistic achieves a pleasant result on predicting superconductor which calculates 145 properties of materials with MAGPIE [1] and put these features into a random forest for training. CGCNN is specifically designed for accurate and interpretable prediction of crystal material properties where convolutional neural networks are performed on crystal graph. Evaluation Metrics.We mainly employ Accuracy (ACC) and Area Under Curve (AUC) for superconductor prediction. In addition, we also use visualization and interpretability analysis to verify our S2SNet. ### Experimental Results and Discussion Comparison on S2S Dataset.All these models are trained and tested directly on S2S dataset. To bridge the shortage of data, MEGNet and CGCNN are pre-trained with the task of predicting formation energy following the paper [3] and S2SNet is pre-trained with the MLM task both on the Material Project database. 80% of S2S dataset are randomly selected to train the models and the left 20% are used to test the models. All the models are evaluated in the same way. Adam optimizer is used in S2SNet and the settings of other models follow their papers. The experimental results are shown in Table 2. S2SNet consistently outperforms the other competitors both on accuracy (ACC) and Area Under Curve (AUC). For example, S2SNet gets about 6% improvements on ACC than Statistic and about 7% improvements on AUC than CGCNN without pre-training and almost 11%/14% improvements than MEGNet on ACC/AUC. With the pre-training task, all these three models, MEGNet, CGCNN and S2SNet are improved significantly. Since S2SNet is pre-trained on the unsupervised MLM task, it can be pre-trained on other much bigger dataset without any labels. However, MEGNet and CGCNN must be trained on the dataset with specific labels such as formation energy and so on. This also means that S2SNet can be further improved with larger dataset in the future work. Fig. 3 shows the receiver operating characteristic (ROC) curve for the superconductor classifier. The overall test accuracy is 91.28%, and the area under curve for the receiver operation conditions is 0.9118. ### Ablation Study Ablation studies are also conducted to verify our S2SNet. Table 3 shows the experimental results of S2SNet with different number of attention layers on S2S dataset. Experiments demonstrate that more layers lead to better AUC. For both efficiency and accuracy, the number of attention layers in our S2SNet is 10. \begin{table} \begin{tabular}{c c c c c} \hline \hline \multirow{2}{}{Method} & \multicolumn{2}{c}{pre-train} & \multicolumn{2}{c}{without pre-train} \\ & ACC(\%) & AUC(\%) & ACC(\%) & AUC(\%) \\ \hline MEGNet & 80.30 & 79.10 & 78.37 & 75.43 \\ Statistic & - & - & 83.33 & 78.52 \\ CGCNN & 82.42 & 86.40 & 79.69 & 82.45 \\ S2SNet(ours) & 91.28 & 91.18 & 89.53 & 90.24 \\ \hline \hline \end{tabular} \end{table} Table 2: The experimental results of different models on the S2S dataset. Figure 4: Pearson correlations between elemental embedding vectors. Elements are arranged in order of increasing Mendeleev number for easier visualization of trends. The number denotes the Mendeleev number. Figure 3: ROC curve for test data of 10 attention layer pre-trained S2SNet. The horizontal and vertical coordinates are FPR and TPR respectively. Further experiments are conducted to verify that the lattice parameters are helpful in S2SNet. Table 4 shows results of 10 attention layer S2SNet without the input of lattice parameters. With lattice parameters, models perform much better no matter pre-trained or not. ### Interpretability on S2SNet According to [22], the interpretability and reproducibility of known chemical common knowledge for machine learning models in material science is critical. To this end, the embeddings of all the element are extracted from S2SNet. The Pearson correlation coefficient between different elements are calculated as Eq. 10, where $m$ is the dimension of elemental embedding vector. As is shown in Fig. 4, the Pearson correlation coefficient between elemental embedding vectors follow the trend of the periodic table of elements. For example, the chalcogen, halogen, lanthanoid, transition metals, metalloid, and actinoid show highest similarities within their groups. For better observation on the elemental embedding vectors, t-SNE[10] is used to map the high-dimension embedding vectors into 2-dimension space in Fig. 5. Elements with similar properties are close to each other, such as the Actinoid, the Chalcogen, the Halogen and so on. All the visualizations show that S2SNet learns the common intuition in chemistry. \[r_{ij}=\frac{\sum_{k=1}^{m}(a_{ik}-\overline{a}_{i})(a_{jk}-\overline{a}_{j})} {\sqrt{\sum_{k=1}^{m}(a_{ik}-\overline{a}_{i})^{2}\sum_{k=1}^{m}(a_{jk}- \overline{a}_{j})^{2}}} \tag{10}\] ### Potential Superconducting Materials In order to provide more insight to the field of superconducting materials research, all the materials we have are fed into S2SNet to predict whether they are superconductors or not. Frequency of elements in materials that our model determines to be superconductors is shown in Fig. A1 in Appendix A. Orange and blue ones denote the number of atoms and the number of materials that contain corresponding atom respectively. Si, S, B, La, Fe, Se, Ge, P, Al, N, O, Bi, C, Ca, Te, Ti and Ni appear in more than 300 superconductors which, in our opinion, are supposed to be crucial elements for superconductor. And the known superconducting elements Nb, Pb, and so on are also identified as the crucial elements in S2SNet. To verify that whether our model has learned these knowledge, we make the embedding layer learnable and train the overall model with the superconductivity prediction. Atomic embeddings are visualized by t-sne in Fig. 6. The atoms with high frequency are grouped together. We get 5468 potential superconductors out of the 69239 materials, and the element combination of part of representative potential superconductors are list in Table A1 in Appendix A. The existing Iron-based, Cuprate, Hydrogen-based and Metal superconductors are identified with high score by our model. Some potential superconductors with similar \begin{table} \begin{tabular}{c c c c c} \hline \hline \multirow{2}{}{Method} & \multicolumn{2}{c}{pre-train} & \multicolumn{2}{c}{without pre-train} \\ & w & w/o & w & w/o \\ \hline ACC(\%) & 91.28* & 90.41 & 89.53 & 87.79 \\ AUC(\%) & 91.18 & 89.22 & 90.24 & 86.20 \\ \hline \hline \end{tabular} \end{table} Table 4: The experimental results of 10 attention layer models without the input of lattice parameters, where w and w/o denote the models with and without lattice parameters respectively. Figure 5: Elemental embeddings in 2-dimension with t-SNE. \begin{table} \begin{tabular}{c c c c} \hline \hline \# attention layers & 5 & 10 & 15 \\ \hline ACC(\%) & 89.83 & 91.28 & 92.31 \\ AUC(\%) & 89.38 & 91.18 & 92.27 \\ \hline \hline \end{tabular} \end{table} Table 3: The experimental results of models with different numbers of attention layers. Figure 6: Elemental embeddings in 2-dimension with t-SNE. The darker the color, the more often the element appears in the materials that our model determines to be superconductors. structures are also predicted. Besides, the other potential superconductors with extremely high score are also given for further experimental synthesis verification. The results are reasonable, since $Graphene$ and $SiC$, which are not in the training set, are predicted to be superconductors. The prediction accuracies for Iron-based, Cuprate, Hydrogen-based superconductors are 97.64%, 92.00% and 96.89% respectively. It is worth mentioning that all the 21 Iron-based superconductors are correctly predicted. ### Why Are They Superconductors Class Activation Map (CAM) is used to explore how superconductivity comes [21]. CAM is a technique commonly used in CV for interpretability, i. e., highlighting the parts of a picture where the model identifies it as a specific class. Herein, we highlight the atoms in crystals that are critical for superconductivity with CAM. The experimental findings are consistent with the previously proposed theories. We take the BaFe${}_{2}$S${}_{3}$, H${}_{3}$S, and Sr${}_{2}$CuClO${}_{2}$ for example. The contribution of atoms in each crystal to superconductivity is shown in Fig. 7 and the color denotes the contribution. BaFe${}_{2}$S${}_{3}$ in Fig. 7(a) represents the Iron-based superconductors and the Fe is more crucial for superconductivity. H${}_{3}$S represents the Hydrogen-based superconductors. Fig. 7(b) shows that both H and S are important for superconductivity and the three hydrogen atoms are of different importance. The hydrogen atoms that are closer to the S is more crucial which is consistent to previous experimental results that with higher pressure, the atoms in a crystal are closer, which can cause the superconductivity. Sr${}_{2}$CuClO${}_{2}$ represents the superconductors based on Cuprate. Fig. 7(c) demonstrate that Cu and O are crucial atoms. ## 5 Conclusion A new dataset, S2S, is built as a benchmark for future machine learning based superconductivity discovery. Based on S2S, a novel pre-trained neural network, S2Snet, is built for superconductivity prediction solely from their crystal structures. S2SNet leverages attention mechanism to model the interaction of atoms in the crystal cells. With the pre-training task, MLM, massive unlabeled data can be properly utilized for S2SNet to learn better representation of atoms under interactions. Indeed S2SNet outperforms other methods significantly no matter with or without pre-training. Furthermore, it is straightforward to extend S2SNet to predict other properties of crystals as well. We expect that S2S dataset and S2SNet could inspire the machine learning community to make more impact on AI for science and SDGs. Finally, the current dataset is still relatively small. Enriching S2S dataset will be our future work. ## Appendix A Potential Superconductors Potential superconductors are shown in Table A1. Frequency of elements in materials that our model determines to be superconductors is shown in Fig. A1. Here, we predict whether a material has the potential to be superconducting, regardless of temperature and pressure.
2301.03802	Predicting Drivers' Route Trajectories in Last-Mile Delivery Using A Pair-wise Attention-based Pointer Neural Network	In last-mile delivery, drivers frequently deviate from planned delivery routes because of their tacit knowledge of the road and curbside infrastructure, customer availability, and other characteristics of the respective service areas. Hence, the actual stop sequences chosen by an experienced human driver may be potentially preferable to the theoretical shortest-distance routing under real-life operational conditions. Thus, being able to predict the actual stop sequence that a human driver would follow can help to improve route planning in last-mile delivery. This paper proposes a pair-wise attention-based pointer neural network for this prediction task using drivers' historical delivery trajectory data. In addition to the commonly used encoder-decoder architecture for sequence-to-sequence prediction, we propose a new attention mechanism based on an alternative specific neural network to capture the local pair-wise information for each pair of stops. To further capture the global efficiency of the route, we propose a new iterative sequence generation algorithm that is used after model training to identify the first stop of a route that yields the lowest operational cost. Results from an extensive case study on real operational data from Amazon's last-mile delivery operations in the US show that our proposed method can significantly outperform traditional optimization-based approaches and other machine learning methods (such as the Long Short-Term Memory encoder-decoder and the original pointer network) in finding stop sequences that are closer to high-quality routes executed by experienced drivers in the field. Compared to benchmark models, the proposed model can increase the average prediction accuracy of the first four stops from around 0.2 to 0.312, and reduce the disparity between the predicted route and the actual route by around 15%.	Baichuan Mo, Qing Yi Wang, Xiaotong Guo, Matthias Winkenbach, Jinhua Zhao	2023-01-10T06:11:20Z	http://arxiv.org/abs/2301.03802v1	Predicting Drivers' Route Trajectories in Last-Mile Delivery Using A Pair-wise Attention-based Pointer Neural Network ###### Abstract In last-mile delivery, drivers frequently deviate from planned delivery routes because of their tacit knowledge of the road and curbside infrastructure, customer availability, and other characteristics of the respective service areas. Hence, the actual stop sequences chosen by an experienced human driver may be potentially preferable to the theoretical shortest-distance routing under real-life operational conditions. Thus, being able to predict the actual stop sequence that a human driver would follow can help to improve route planning in last-mile delivery. This paper proposes a pair-wise attention-based pointer neural network for this prediction task using drivers' historical delivery trajectory data. In addition to the commonly used encoder-decoder architecture for sequence-to-sequence prediction, we propose a new attention mechanism based on an alternative specific neural network to capture the local pair-wise information for each pair of stops. To further capture the global efficiency of the route, we propose a new iterative sequence generation algorithm that is used after model training to identify the first stop of a route that yields the lowest operational cost. Results from an extensive case study on real operational data from Amazon's last-mile delivery operations in the US show that our proposed method can significantly outperform traditional optimization-based approaches and other machine learning methods (such as the Long Short-Term Memory encoder-decoder and the original pointer network) in finding stop sequences that are closer to high-quality routes executed by experienced drivers in the field. Compared to benchmark models, the proposed model can increase the average prediction accuracy of the first four stops from around 0.2 to 0.312, and reduce the disparity between the predicted route and the actual route by around 15%. keywords: Route planning, Trajectory prediction, Sequence-to-sequence model, Last-mile delivery, Pointer network, Attention + Footnote †: journal: ## 1 Introduction The optimal planning and efficient execution of last-mile delivery routes is becoming increasingly important for the business operations of many logistics service providers around the globe for a variety of reasons. E-commerce volumes are growing rapidly and make up a constantly growing share of overall retail sales. For instance, in the US, the share of e-commerce sales in total retail sales has grown from around 4% in 2010 to around 13% in 2021. Even by the end of 2019, i.e., before the outbreak of the COVID-19 pandemic, it had reached 11% (US Census Bureau, 2021). Undoubtedly, the pandemic further accelerated the growth of e-commerce (postnord, 2021; McKinsey and Company, 2021). In the medium to long run, its growth will continue to be fueled by an ongoing trend towards further urbanization, which is particularly pronounced in developing and emerging economies (United Nations Department of Economic and Social Affairs, 2019). The share of the global population living in urban areas is currently projected to rise from around 55% in 2018 to around 68% by 2050. The associated increase in population density in most urban areas will likely lead to growing operational uncertainties for logistics service providers, as increasing congestion levels, less predictable travel times, and scarce curb space make efficient and reliable transport of goods into and out of urban markets increasingly challenging (Rose et al., 2016). As a result of the continued boom of e-commerce and constantly growing cities, global parcel delivery volumes have been increasing rapidly in recent years and are expected to continue to do so. Across the 13 largest global markets, including the US, Brazil, and China, the volume of parcels delivered more than tripled from 43 billion in 2014 to 131 billion in 2020 (Pitney Bowes, 2020). At the same time, customer expectations towards last-mile logistics services are rising. For instance, there is a growing demand for shorter delivery lead times, including instant delivery services and same-day delivery, as well as customer-defined delivery preferences when it comes to the time and place of delivery (Lim and Winkelbach, 2019; Cortes and Suzuki, 2021; Snoeck and Winkelbach, 2021). The rapid growth and increasing operational complexity of urban parcel delivery operations also amplifies their negative externalities, including their contribution to greenhouse gas and other pollutant emissions, public health safety risks, as well as overall urban congestion and a corresponding decline in overall mobility and accessibility of cities (Jaller et al., 2013; World Economic Forum, 2020). When applied to realistically sized instances of a last-mile delivery problem, solving the underlying traveling salesman problem (TSP) or vehicle routing problem (VRP) to (near) optimality becomes challenging, as both problem classes are known to be NP-hard. Traditional TSP and VRP formulations aim to minimize the total distance or duration of the route(s) required to serve a given set of delivery stops. The operations research literature has covered the TSP, VRP, and their many variants extensively, and in recent years important advances have been made with regards to solution quality and computational cost. However, in practice, many drivers, with their own tacit knowledge of delivery routes and service areas, divert from seemingly optimal routes for reasons that are difficult to encode in an optimization model directly. For example, experienced drivers may have a better understanding of which roads are hard to navigate, at which times traffic is likely to be bad, when and where they can easily find parking, and which stops can be conveniently served together. Therefore, compared to the theoretically optimal (i.e., distance or time minimizing) route, the _deviated actual route sequence_ chosen by an experienced human driver is potentially preferable under real-life operational conditions. An important challenge in today's last-mile delivery route planning is therefore to leverage historical route execution data to propose planned route sequences that are close to the actual trajectories that would be executed by drivers, given the delivery requests and their characteristics. Note that, while distance and time-based route efficiency is still an important factor for planning route sequences, it is not the sole objective, as tacit driver knowledge is also incorporated in the proposed route sequences. Unlike a typical VRP in which the number of vehicles and their respective route sequences need to be determined simultaneously, in this study, we focus on solving a problem that is similar to a TSP at the individual vehicle level. That is, we aim to solve a stop sequence to serve a given set of delivery requests, and expect that the proposed stop sequence is as close to the actual trajectories that would be executed by drivers as possible. To this end, we propose a pair-wise attention-based pointer neural network to predict the actual route sequence taken by delivery drivers using drivers' historical delivery trajectory data. The proposed model follows a typical encoder-decoder architecture for the sequence-to-sequence prediction. However, unlike previous studies, we propose a new attention mechanism based on an alternative specific neural network (ASNN) to capture the local pair-wise information for each stop pair. To further capture the global efficiency of the route (i.e., its operational cost in terms of total distance or duration), after model training, we propose a new sequence generation algorithm that iterates over different first stops and selects the route with the lowest operational cost. The main contribution of this paper is three-fold: First, we propose a new ASNN-based attention mechanism to capture the local information between pairs of stops (e.g., travel time, geographical relation), which can be well adapted to the original pointer network framework for sequence prediction. Second, we propose a new sequence generation algorithm that iterates over different first stops in the predicted route sequences and selects the lowest operational cost route. The intuition is that the stop-to-stop relationship (referred to as the _local view_) is easier to learn from data than the stop sequence of the route as a whole (referred to as the _global view_). Lastly, we apply our proposed method to a large set of routes executed by Amazon delivery drivers in the US. The results show that our proposed model can outperform traditional optimization-based approaches and other machine learning methods in finding stop sequences that are closer to high-quality routes executed by experienced drivers in the field. The remainder of this paper is structured as follows. In Section 2 we define the problem setting under investigation in a more formal way. Section 3 then reviews previous studies in the literature related to this paper. Section 4 presents our methodology and elaborates on the detailed architecture of the proposed pair-wise attention-based pointer neural network. Section 5 presents the experimental setup and numerical results of our case study, applying our proposed method to real-world data made available by the Amazon Last-Mile Routing Research Challenge (Merchan et al., 2022; Winkenbach et al., 2021). Section 6 concludes this paper and discusses future research directions. ## 2 Problem Setting In the last-mile delivery routing problem considered here, a set of stops $\mathcal{S}=\{s_{1},...,s_{n}\}$ to be served by a given delivery vehicle is given to the route planner. The planner's objective is to find the optimal stop sequence that has the minimal operational cost. In this case, we consider total cost as total travel time. The planner is given the expected operational cost (i.e., travel times) between all pairs of stops $(s_{i},s_{j})$. The theoretically optimal stop sequence, denoted by $(s_{(1)}^{\mathsf{T}},...,s_{(n)}^{\mathsf{T}})$, can be found by solving a TSP formulation. This stop sequence is referred to as the planned stop sequence. However, as discussed in Section 1, minimizing the theoretical operational cost (i.e., total travel time) of the route may not capture drivers' tacit knowledge about the road network, infrastructure, and recipients. Therefore, the actual driver executed stop sequence $(s_{(1)},...,s_{(n)})$ can be different from the planned route sequence. Note that here, $s_{(i)}\in\mathcal{S}$ denotes the $i$-th stop that is actually visited by the driver. The objective of the model presented in this study is to predict the actual driver executed sequence $(s_{(1)},...,s_{(n)})$ given a set of stops $\mathcal{S}$ and the corresponding delivery requests and characteristics $X^{\mathcal{S}}$ (such as the number of packages, estimated service time for each package, geographical information for each stop, travel time between each stop pairs, etc.). All drivers are assumed to start their routes from a known depot $D^{\mathcal{S}}$ and return back to $D^{\mathcal{S}}$. Therefore, the complete trajectory should be a tour $(D^{\mathcal{S}},s_{(1)},...,s_{(n)},D^{\mathcal{S}})$. For the convenience of model description, we ignore the depot station in the sequence. Figure 1 provides a simple example for illustration. In this example, we are given four stops $\mathcal{S}=\{s_{1},s_{2},s_{3},s_{4}\}$ and a depot $D^{\mathcal{S}}$. The planned stop sequence for the driver is $(s_{4},s_{1},s_{2},s_{3})$, while the actual stop sequence executed by the driver is $(s_{4},s_{2},s_{1},s_{3})$. The proposed model aims to predict the actual sequence $(s_{4},s_{2},s_{1},s_{3})$ given the depot location $D^{\mathcal{S}}$, the set of stops to be visited $\mathcal{S}$, and characteristics of the stops $X^{\mathcal{S}}$. This problem setup is inspired by the Amazon Last-Mile Routing Research Challenge (cf., Winkelbach et al., 2021). Note that this study only focuses on the stop sequence prediction. The routing between stops is not considered. It is assumed that the drivers always take the optimal route between stops, which is reflected by the travel time matrix between stops in our problem setup. ## 3 Literature Review The problem setting defined in Section 2 involves both solving a cost-minimizing routing problem (i.e., the TSP) and capturing tacit driver knowledge to learn systematic deviation of drivers from the planned and theoretically optimal stop sequences. Therefore, we will first review the extant literature on the TSPs and its most relevant variants. We will then go through various machine learning approaches that have been proposed by the extant literature to generate sequences, with a section on methods specifically for solving the TSP. Note that although these machine learning approaches are used to solve the TSP instead of the actual routes taken by drivers, their architectures may be helpful to learn the actual route as well. ### Travelling salesman problems First, given the travel times between stops, a solution to the TSP, which finds the route with the minimum cost or distance (i.e., the planned route), can be a close approximation of the actual route. Since the drivers are paid for the number of packages delivered, all drivers' goal is to deliver the packages in the minimum amount of time. Most of the drivers do follow large parts of the planned routes. The TSP is a well-known NP-hard problem that has been studied extensively over the last century, with a lot of books and review papers published on its history, formulations, solution approaches, and applications (Applegate et al., 2006; Matai et al., 2010; Davendra and Bialic-Davendra, 2020). An overview of the relevant TSP variants and solution approaches are presented below. The basic setup of TSP has one traveler and requires the traveler to return to the starting point after visiting each node exactly once, and that the traveling cost matrix (represented by distance and/or time) is Figure 1: Illustrative example of the problem setting. symmetric (cost between $i$ and $j$ is the same with that between $j$ and $i$). In most real-world applications, the basic setup needs to be modified. For example, the cost matrix, if represented by travel times, is likely asymmetric. This variant of TSP is thus named asymmetric TSP (ATSP) (Jonker and Volgenant, 1983). In some applications, the vehicle does not need to return to the original depot (Traub et al., 2021), or it can charge/refuel and potentially load additional delivery items at intermediate stops (Kucukoglu et al., 2019). In many last-mile delivery applications, some packages are time-sensitive, and therefore time window constraints to their delivery need to be considered in a so-called TSP with time windows (TSPTW) (da Silva and Urrutia, 2010; Mladenovic et al., 2012). In large systems, there might be more than one salesman serving a set of stops, resulting in multiple traveling salesmen problems (MTSPs) (Cheikhrouhou and Khoufi, 2021). Different variants of TSP further impose different constraints on the solution. While some problems can be reduced to the basic setup in the formulation stage, others require more versatile solution algorithms. In general, the solution approaches to the TSP can be divided into exact approaches and approximate approaches. Exact approaches include branch-and-cut (Yuan et al., 2020) and branch-and-bound (Salman et al., 2020). Since the TSP is a well-known NP-hard problem, exact approaches can only be applied on problems of smaller scale, or aid in heuristics to cut the solution space. Among approximate approaches, there are heuristics designed for the TSP specifically, as well as meta-heuristics that are generic and treat the problem like a blackbox. The most commonly used heuristics and meta-heuristics include nearest neighbor searches, local searches, simulated annealing, and genetic algorithms. A more comprehensive review of existing solution approaches can be found in Halim and Ismail (2017); Purkayastha et al. (2020). Despite the TSP being NP-hard, modern mixed-integer optimization solvers (e.g., Gurobi, CPLEX, or GLPK) can solve it efficiently for real-world instances by combining exact approaches with heuristics. ### Sequence-to-sequence prediction using deep learning The TSP and its variants are a viable option for sequence generation only when the objective is clearly defined. They fall short when the sequence generation problem does not have a well-defined cost-minimization objective. In a lot of applications, the rule of sequence generation cannot be simply defined and optimized. A standard example for a sequence learning problem is machine translation, where a sequence of words in one language needs to be translated to another language. Another type of sequence learning is time series modeling, where a sequence of historical observations is given to predict future states of the system. In both cases, the primary modeling task is to learn the sequence generation rules. In recent years, deep learning has successfully achieved great performance in various settings of sequence generation. These models are often referred to as sequence-to-sequence (seq2seq) models. seq2seq models often consist of an encoder and a decoder, where the encoder encodes the input sequence into a fixed-length vector representation, and the decoder generates a sequence based on the generated vector representation. Most encoder-decoder architectures adopt recurrent neural network (RNN) layers and its variants such as Long Short-Term Memory (LSTM) (Hochreiter and Schmidhuber, 1997) and gated recurrent layers (GRU) (Cho et al., 2014) to learn long-range dependencies. Early works using LSTM alone were able to generate plausible texts (Graves, 2013) and translate between English and French (Sutskever et al., 2014) with long-range dependencies. Chung et al. (2014) demonstrate the superiority of GRU compared to LSTMs in music and speech signal modeling. Attention-based mechanisms, first introduced by Bahdanau et al. (2015), have been shown to be a great addition since it allows the decoder to selectively attend to parts of the input sequence and relieves the encoder of the task of encoding all the information into a fixed-length vector representation. Most sequence generation problems benefit from keeping track of long-range dependencies and global context while decoding. To address that, multi-level attention was proposed to capture the local and global dependency, and has shown to be effective in speech recognition (Chorowski et al., 2015), text generation (Liu et al., 2018), and machine translation tasks (Luong et al., 2015). The encoder-decoder architecture combined with attention is very versatile, and it can be combined with other deep learning architectures to perform sequence learning in addition to language tasks. The LSTM and attention architecture is applied to semantic trajectory prediction (Karatzoglou et al., 2018), text summarization (Liang et al., 2020), demand modelling (Ren et al., 2020), and wind power forecasting (Zhang et al., 2020). When the goal is set to recover the original sequence, unsupervised learning of molecule embedding can be obtained for downstream classification tasks (Xu et al., 2017). When the spatial dimension is added, a convolutional neural network (CNN) layer can be added, and the dimension of the sequence generated can be expanded. For example, Wang et al. (2020) predict a city's crowd flow patterns, and Wu et al. (2020) generate 3D shapes via sequentially assembling different parts of the shapes. While RNN-based architectures are still a widely adopted choice for seq2seq modeling, attention can also be used as a standalone mechanism for seq2seq translations independent of RNNs. The idea was proposed by Vaswani et al. (2017) in an architecture named transformer. Without recurrence, the network allows for significantly more parallelization, and is shown to achieve superior performance in experiments, and powered the popularity of transformer-based architectures in various sequence generation tasks (Huang et al., 2018; Lu et al., 2021). A separate line of work by Zhang et al. (2019) also demonstrated that a hierarchical CNN model with attention outperforms the traditional RNN-based models. ### Using deep learning to generate TSP solutions The above seq2seq translation mechanisms work well when the input data is naturally organized as a sequence, and the output sequence corresponds to the input sequence, such as in music and language. However, in our paper, the input is an unordered sequence, and the output has the same but re-ordered elements of the same input sequence. In this case, the concept of attention is helpful and has been successfully used to produce solutions to the TSP. The pointer network, proposed by Vinyals et al. (2015) and further developed in Vinyals et al. (2016), uses attention to select a member of the input sequence at each decoder step. While it is not required that the input sequence is ordered, an informative ordering could improve the performance (Vinyals et al., 2016). While the original pointer network was solved as a classification problem and cross-entropy loss was used, it is not necessarily the most efficient choice. The cross-entropy loss only distinguishes between a correct prediction and an incorrect prediction. But in instances like routing, the distances between the predicted position and the correct position, as well as the ordering of subsequences, could incur different costs in practice. Further developments in solving TSP with machine learning methods involve reinforcement learning (RL), which enables the optimization of custom evaluation metrics (Bello et al., 2019; Kool et al., 2019; Ma et al., 2019; Liu et al., 2020). Joshi et al. (2019) compared the performance of RL and supervised learning (SL) on TSP solutions and found that SL and RL models achieve similar performance when the graphs are of similar sizes in training and testing, whereas RL models have better generalizability over variable graph sizes. However, RL models require significantly more data points and computational resources, which is not always feasible. Although this seq2seq and attention framework has only been used to reproduce TSP solutions, it provides an opportunity to learn and incorporate additional information beyond the given travel times and potentially learn individual differences when more information is given to the neural network. In this paper, we combine the ideas of seq2seq modeling and attention to predict the actual route executed by a driver. ## 4 Methodology This section details the methodology proposed to address the problem. First, the high-level seq2seq modelling framework is introduced, followed by the explanation of the novel pair-wise attention and sequence generation and selection mechanism used within the modelling framework. ### Sequence-to-sequence modeling framework Let the input sequence be an arbitrarily-ordered sequence $(s_{1},...,s_{n})$. Denote the output sequence as $(\hat{s}_{(1)},...,\hat{s}_{(n)})$. Let $c_{i}$ indicate the "position index" of stop $\hat{s}_{(i)}$ with respect to the input sequence (where $c_{i}\in\{1,...,n\}$). For example, for input sequence $(B,A,C)$ and output sequence $(A,B,C)$, we have $c_{1}=2$, $c_{2}=1$, $c_{3}=3$, which means the first output stop $A$ is in the second position of the input sequence $(B,A,C)$ and so on. The seq2seq model computes the conditional probability $\mathbb{P}(c_{1},...,c_{n}\mid\mathcal{S};\theta)$ using a parametric neural network (e.g., recurrent neural network) with parameter $\theta$, i.e., \[\mathbb{P}(c_{1},...,c_{n}\mid\mathcal{S},X^{\mathcal{S}};\theta)=\mathbb{P} (c_{1}\mid\mathcal{S},X^{\mathcal{S}};\theta)\cdot\prod_{i=2}^{n}\mathbb{P}(c _{i}\mid c_{1},...,c_{i-1},\mathcal{S},X^{\mathcal{S}};\theta) \tag{1}\] The parameters of the model are learnt by empirical risk minimization (maximizing the conditional probabilities on the training set), i.e., \[\theta^{}=\operatorname{arg\,max}_{\theta}\sum_{\mathcal{S}}\mathbb{P}(c_{1 },...,c_{n}\mid\mathcal{S},X^{\mathcal{S}};\theta) \tag{2}\] where the summation of $\mathcal{S}$ is over all training routes. In the following section, we will elaborate how $\mathbb{P}(c_{i}\mid c_{1},...,c_{i-1},\mathcal{S},X^{\mathcal{S}};\theta)$ is calculated using the pair-wise attention-based pointer neural network. ### Pair-wise attention-based pointer neural network Figure 2 uses a four-stop example to illustrate the architecture of the proposed model. The whole model is based on the LSTM encoder and decoder structure. In particular, we use one LSTM (i.e., encoder) to read the input sequence, one time step at a time, to obtain a large fixed dimensional vector representation, and then to use another LSTM (i.e., decoder) to extract the output sequence. However, different from the typical seq2seq model, we borrow the idea of the pointer network (Vinyals et al., 2015) to add a pair-wise attention mechanism to predict the output sequence based on the attention mask over the input sequence. The pair-wise attention is calculated based on an ASNN which was previously used for travel mode prediction (Wang et al., 2020). Model details will be shown in the following sections. Intuitively, the LSTM encoder and decoder aim to capture the global view of the input information (i.e., overall sequence pattern) by embedding the input sequence to hidden vector representation. While the ASNN-based pair-wise attention aims to capture the local view (i.e., the relationship between two stops). Our experiments in Section 5 demonstrate the importance of both global and local views in the sequence prediction. #### 4.2.1 LSTM encoder. Given an arbitrary stop sequence $(s_{1},...,s_{n})$ as the input, let $x_{i}\in\mathbb{R}^{K}$ be the features of stop $s_{i}$, where $x_{i}$ may include the package information, the customer information, and the geographical information of the stop $s_{i}$. $K$ is the number of features. The encoder computes a sequence of encoder output vectors $(e_{1},...,e_{n})$ by iterating the following: \[h_{i}^{\text{E}},e_{i}=\text{LSTM}(x_{i},h_{i-1}^{\text{E}};\ \theta^{\text{E}}) \quad\forall i=1,...,n \tag{3}\] where $h_{i}^{\text{E}}\in\mathbb{R}^{K_{h}^{\text{E}}}$ is the encoder hidden vector with $h_{0}^{\text{E}}:=0$. $e_{i}\in\mathbb{R}^{K_{e}}$ is the encoder output vector. $K_{h}^{\text{E}}$ and $K_{e}$ are corresponding vector dimensions. $\theta^{\text{E}}$ is the learnable parameters in an encoder LSTM cell. The calculation details of an LSTM cell can be found in A. The encoding process transforms a sequence of features $(x_{1},...,x_{n})$ into a sequence of embedded representation $(e_{1},...,e_{n})$. And the hidden vector of the last time step ($h_{n}^{\text{E}}$) includes the global information of the whole sequence, which will be used for the LSTM decoder. #### 4.2.2 LSTM decoder. The role of a decoder in the traditional seq2seq model (Figure 4) is to predict a new sequence one time step at a time. However, in the pointer network structure with attention, the role of the decoder becomes producing Figure 3: Illustration of LSTM encoder Figure 2: Overall architecture of the pair-wise attention-based pointer neural network (adapted from Vinyals et al. (2015)) a vector to modulate the pair-wise attention over inputs. Denote the output sequence as $(\hat{s}_{(1)},...,\hat{s}_{(n)})$. Let $x_{(i)}$ be the feature of stop $\hat{s}_{(i)}$. At decoder step $i$, we have \[h^{\text{D}}_{(i+1)},d_{(i)}=\text{LSTM}\left(\begin{bmatrix}x_{(i)}\\ w_{(i)}\end{bmatrix},h^{\text{D}}_{(i)};\ \theta^{\text{D}}\right)\quad\forall i=0,1,...,n \tag{4}\] where $h^{\text{D}}_{(i)}\in\mathbb{R}^{K^{\text{D}}_{h}}$ is the decoder hidden vector with $h^{\text{D}}_{(0)}=h^{\text{E}}_{n}$, $d_{(i)}\in\mathbb{R}^{K_{d}}$ is the decoder output vector, $K^{\text{D}}_{h}$ and $K_{d}$ are corresponding vector dimensions, and $\theta^{\text{D}}$ are learnable parameters of the decoder LSTM cell. Note that we set $x_{(0)}=x_{D}$ and $d_{(0)}=d_{D}$, representing the features and the decoder output of the depot, respectively. $w_{(i)}$ is the context vector calculated from the attention component, which will be explained in the next section. #### 4.2.3 ASNN-based pair-wise attention. The pair-wise attention aims to aggregate the global and local information to predict the next stop. Specifically, at each decoder time step $i\in\{0,...,n\}$, we know that the last predicted stop is $\hat{s}_{(i)}$. To predict $\hat{s}_{(i+1)}$, we consider all candidate stops $s_{j}\in\mathcal{S}$, which is the set of all stops not yet visited. We want to evaluate how possible that $s_{j}$ will be the next stop of $\hat{s}_{(i)}$. The information of the stop pair $\hat{s}_{(i)}$ and $s_{j}$ can be represented by the following concatenated vector: \[v^{j}_{(i)}=\text{concat}(z^{j}_{(i)},\ \phi(x_{(i)},x_{j}),\ d_{(i)},\ e_{j}) \tag{5}\] where $z^{j}_{(i)}$ is a vector of features associated with the stop pair (such as travel time from $\hat{s}_{(i)}$ to $s_{j}$), and $\phi(x_{(i)},x_{j})$ represents a feature processing function to extract the pair-wise information from $x_{(i)}$ and $x_{j}$. For example, $\phi(\cdot)$ may return geographical relationship between stops $\hat{s}_{(i)}$ and $s_{j}$, and it may also drop features not useful for the attention calculation. Intuitively, $z^{j}_{(i)}$ and $\phi(x_{(i)},x_{j})$ contains only local information of the stop pair, while $d_{(i)}$ and $e_{j}$ contain the global information of the whole stop set and previously visited stops. Figure 4: Illustration of LSTM decoder Given the pair-wise information vector $v^{j}_{(i)}$, we can calculate the attention of stop $\hat{s}_{(i)}$ to stop $s_{j}$ as: \[u^{j}_{(i)} =\text{ASNN}(v^{j}_{(i)};\ \theta^{\text{A}})\quad\forall i,j=1,...,n \tag{6}\] \[a^{j}_{(i)} =\frac{\exp(u^{j}_{(i)})}{\sum_{j^{\prime}=1}^{n}\exp(u^{j^{\prime }}_{(i)})}\quad\forall i,j=1,...,n \tag{7}\] where $a^{j}_{(i)}\in\mathbb{R}$ is attention of stop $\hat{s}_{(i)}$ to stop $s_{j}$. $\text{ASNN}(\cdot;\ \theta^{\text{A}}))$ is a multilayer perception (MLP) with the output dimension of one (i.e., $u^{j}_{(i)}\in\mathbb{R}$). $\theta^{\text{A}}$ are the learnable parameters of the ASNN. The name "alternative specific" is because the same parametric network will be applied on all alternative stops $s_{j}\in\mathcal{S}$ separately (Wang et al., 2020). Finally, we calculate the conditional probability to make the prediction: \[\mathbb{P}(c_{i+1}=j\ \|\ c_{1},...,c_{i},\mathcal{S},X^{ \mathcal{S}};\theta)=a^{j}_{(i)}\quad\forall i=0,1,...,n,\ j=1,...,n \tag{8}\] \[\hat{s}_{(i+1)}=\operatorname{arg\,max}_{s_{j}\in\mathcal{S} \setminus\mathcal{S}^{\text{V}}_{(i)}}a^{j}_{(i)}\quad\forall i=0,1,...,n \tag{9}\] where $\mathcal{S}^{\text{V}}_{(i)}=\{\hat{s}_{(1)},...,\hat{s}_{(i)}\}$ is the set of stops that have been predicted (i.e., previously visited) until decoder step $i$. Eqs. 8 and 9 indicate that the predicted next stop at step $i$ is the one with highest attention among all stops that have not been visited. The pair-wise attention framework also leverages the attention information as the input for the next step. This was achieved by introducing the context vector (Bahdanau et al., 2015): \[w_{(i)}=\sum_{j=1}^{n}a^{j}_{(i)}\cdot e_{j} \tag{10}\] The context vector is a weighted sum of all the encoder output vectors with attention as the weights. As the attention provides the emphasis for stop prediction, $w_{(i)}$ helps to incorporate the encoded representation of the last predicted stop for the next stop prediction. The inputs for the next LSTM cell thus will be the Figure 5: Illustration of ASNN-based pair-wise attention concatenation of the stop features and $w_{(i)}$, i.e., $\begin{bmatrix}x_{(i)}\\ w_{(i)}\end{bmatrix}$. It is worth noting that, the specific architecture of $\text{ASNN}(\cdot;\ \theta^{\text{A}}))$ can be flexible depending on the input pair-wise information. For example, if the information includes images or networks, convolutional neural network or graph convolutional networks can be used for better extract features. In this study, we use the MLP for simplification as it already outperforms benchmark models. The key idea is of the ASNN is to share the same trainable parameter $\theta^{\text{A}}$ for all stop pairs so as to better capture various pair-wise information in the training process. ### Sequence generation and selection During inference, given a stop set $\mathcal{S}$, the trained model with learned parameters $\theta^{}$ are used to generate the sequence. Typically, in the seq2seq modeling framework, the final output sequence is selected as the one with the highest probability, i.e., \[(s_{j_{1}^{}},...,s_{j_{n}^{}}),\text{where }j_{1}^{},...,j_{n}^{}= \operatorname{arg\,max}_{j_{1},...,j_{n}\in\mathcal{C}^{\mathcal{S}}}\mathbb{ P}(c_{1}=j_{1},...,c_{n}=j_{n}\mid\mathcal{S},X^{\mathcal{S}};\theta^{}) \tag{11}\] where $\mathcal{C}^{\mathcal{S}}=\{\text{All permutations of }\{1,...,n\}\}$ Finding this optimal sequence is computationally impractical because of the combinatorial number of possible output sequences. And so it is usually done with the greedy algorithm (i.e., always select the most possible next stop) or the beam search procedure (i.e., find the best possible sequence among a set of generated sequences given a beam size). However, in this study, we observe that the first predicted stop $\hat{s}_{(1)}$ is critical for the quality of the generated sequence. The reason may be that the local relationship between a stop pair (i.e., given the last stop to predict the next one) is easier to learn than the global relationship (i.e., predict the whole sequence). Hence, in this study, we first generate sequences using the greedy algorithm with different initial stops, and select the one with the lowest operational cost. The intuition behind this process is that, once the first stop is given, the model can follow the learned pair-wise relationship to generate the sequence with relatively high accuracy. For all the generated sequences with different first stops, the one with the lowest operation cost captures the global view of the sequence's quality. Therefore, the final sequence generation and selection algorithm is as follows: ``` 0: Trained model, $\mathcal{S}$ 0: Predicted stop sequence 1:for$s$ in $\mathcal{S}$do 2: Let the first predicted stop be $\hat{s}_{(1)}=s$ 3: Predict the following stop sequence $(\hat{s}_{(2)},...,\hat{s}_{(n)})$ using the greedy algorithm. Denote the predicted sequence as $P_{s}$. 4: Calculate the total operation cost of the whole sequence (including depot), denoted as $OC_{s}$. return$P_{s^{}}$ where $s^{}=\operatorname{arg\,min}_{s\in\mathcal{S}}OC_{s}$ ``` Algorithm 1* Sequence generation ## 5 Case Study ### Dataset The data used in our case study was made available as part of the Amazon Last Mile Routing Research Challenge (Merchan et al., 2022). The dataset contains a total of 6,112 actual Amazon driver trajectories for the last-mile delivery from 5 major cities in the US: Austin, Boston, Chicago, Los Angeles, and Seattle. Each route consists of a sequence of stops. Each stop represents the actual parking location of the driver, and the package information (package numbers, package size, and planned service time) associated with each stop is given. The stops are characterized by their latitudes and longitudes, and expected travel time between stops are known. Figure 6 shows the distribution of the number of stops per route and an example route. Most routes have around 120 to 180 stops, and the maximum observed number of stops is around 250. Figure 5(b) shows an example of an actual driver trajectory in Boston. Since the depot is far from the delivery stops, we attach the complete route (with the depot indicated by a red dot) at the bottom left of the figure, while the main plot only shows the delivery stops. In this data set, each stop is associated with a zone ID (indicated by different colors in Figure 5(b)). When Amazon generates planned routes for drivers, they usually expect drivers to finish the delivery for one zone first, then go to another zone. And the actual driver trajectories also follow this pattern as shown in Figure 5(b) (but the actual zone sequence may be different from the planned one). Therefore, in this study, we focus on the problem of _zone sequence prediction_. That is, $s_{i}$ in the case study section now represents a specific zone, $\mathcal{S}$ represents the set of zones, and $X^{\mathcal{S}}$ represents zone features. This transformation does not affect the model structure proposed in Section 4. The only difference is that the new problem has a relatively smaller scale compared to the _stop sequence prediction_ because the number of zones in a route is smaller than that of stops. The zone-to-zone travel time is calculated as the average travel time of all stop pairs between the two zones. Figure 7 presents an illustrative example of the relationship between zone and stop sequences. As the dataset does not contain the original planned sequence, we assume the planned zone sequence is the one with the lowest total travel time (generated by a TSP solver, $(s_{1}^{\text{T}},...,s_{n}^{\text{T}})$). After generating the zone sequence, we can restore the whole stop sequence by assuming that drivers within a specific zone follow an optimal TSP tour. Details of the zone sequence to stop sequence generation can be found in B. ### Experimental setup We randomly select 4,889 routes for model training and cross-validation, and the remaining 1,223 routes are used to evaluate/test model performance. We consider a one-layer LSTM for both the encoder and decoder with the hidden unit sizes of 32 (i.e., $K_{h}^{\text{D}}=K_{e}=K_{h}^{\text{E}}=K_{d}=32$). And the ASNN is set with 2 hidden layers with 128 hidden units in each layer. We train the model using Adam optimizer with a default learning rate of 0.001 and 30 training epochs. To utilize the planned route information, the input zone sequence for the LSTM encoder is set as the TSP Figure 7: Relationship between stop sequence and zone sequence. Figure 6: Description of dataset result (i.e., lowest travel time). That is, the input sequence $(s_{1},...,s_{n})=(s_{1}^{\intercal},...,s_{n}^{\intercal})$. In the case study, $x_{i}$ represents zone features, including the latitude and longitude of the zone center, number of stops in the zone, number of intersections in the zone, number of packages in the zone, total service time in the zone, total package size in the zone, and the travel time from this zone to all other zones. The zone pair features $z_{(i)}^{j}$ includes the travel time from $\hat{s}_{(i)}$ to $s_{j}$ and zone ID relationship characteristics. For example, the zone IDs "B-6.2C" and "B-6.3A" signal that they belong to the higher-level cluster "B-6". As we assume all pair-wise features are captured by $z_{(i)}^{j}$, $\phi(x_{(i)},x_{j})$ is not specified in this case study. Consistent with the Amazon Last Mile Routing Research Challenge, we evaluate the quality of the predicted stop sequences using a "disparity score" defined as follows: \[R(A,B)=\frac{SD(A,B)\cdot ERP_{\text{norm}}(A,B)}{ERP_{\text{e}}(A,B)} \tag{12}\] where $R(A,B)$ is the disparity score for the actual sequence $A$ and predicted sequence $B$, and $SD(A,B)$ is the sequence deviation defined as \[SD(A,B)=\frac{2}{n(n-1)}\sum_{i=2}^{n}\left(\|c_{[B_{i}]}-c_{[B_{i-1}]}\|-1\right) \tag{13}\] where $n$ is the total number of stops, $B_{i}$ is the $i$-th stop of sequence $B$, $c_{[B_{i}]}$ is the index of stop $B_{i}$ in the actual sequence $A$ (i.e., its position in sequence $A$). In the case of $A=B$ (i.e., perfectly predicted), we have $c_{[B_{i}]}-c_{[B_{i-1}]}=1$ for all $i=2,...,n$, and $SD(A,B)=0$. $ERP_{\text{norm}}(A,B)$ is the Edited Distance with Real Penalty (ERP) defined by the following recursive formula: \[ERP_{\text{norm}}(A,B)=ERP_{\text{norm}}(A_{2:\|A\|},B_{2:\|B\|})+\text{Time}_{ \text{norm}}(A_{1},B_{1}) \tag{14}\] where $\text{Time}_{\text{norm}}(s_{i},s_{j})=\frac{\text{Time}(s_{i},s_{j})}{\sum_{ j^{\prime}\in\{1,...,n\}}\text{Time}(s_{i},s_{j^{\prime}})}$ is the normalized travel time from stop $s_{i}$ to stop $s_{j}$. $ERP_{\text{e}}(A,B)$ is the number of edit operations (insertions, substitutions, or deletions) required to transform sequence $A$ to sequence $B$ as when executing the recursive $ERP_{\text{norm}}$ formulation. Hence, the ratio $\frac{ERP_{\text{norm}}(A,B)}{ERP_{\text{e}}(A,B)}$ represents the average normalized travel time between the two stops involved in each ERP edit operation. In the case of $A=B$, we have $\frac{ERP_{\text{norm}}(A,B)}{ERP_{\text{e}}(A,B)}=0$. The disparity score $R(A,B)$ describes how well the model-estimated sequence matches the known actual sequence. Lower score indicates better model performance. A score of zero means perfect prediction. The final model performance is evaluated by the mean score over all routes in the test set. In addition to the disparity score, we also evaluate the prediction accuracy of the first four zones in each route. We choose the first four because the minimum number of zones in a route is four. ### Benchmark models The following optimization and machine learning models are used as benchmarks to compare with the proposed approach. Conventional TSP. The first benchmark model is the zone sequence generated by conventional TSP, which we treat as the planned route with the lowest travel time. ASNN model. The ASNN component can be trained to predict the next zone given the current zone, and the prediction sequence can be constructed in a greedy way starting from the given depot. The training zone pairs (including from depot to the first zone) are extracted from all sequences in the training routes. And the input features are the same as the ASNN component in the proposed model except for $(d_{(i)},e_{j})$ (i.e., output vectors from LSTM decoder and encoder, respectively). All hyper-parameters of the ASNN model are the same as the attention component. Inspired by the importance of the first zone, we also implement another sequence generation method similar to Section 4.3. That is, we go through all zones in a route and assume it is the first zone, then use the trained ASNN to predict the remaining sequence. The final sequence is selected as the one with the lowest travel time. LSTM-encoder-decoder. The LSTM-encoder-decoder (LSTM-E-D) architecture is a typical seq2seq model proposed by Sutskever et al. (2014). The model structure is shown in Figure 8. In the decoder stage, the model outputs the predicted zone based on last predicted zone's information. The model formulation can be written as \[h_{i}^{\text{E}},e_{i} =\text{LSTM}(x_{i},h_{i-1}^{\text{E}};\ \theta^{\text{E}})\quad \forall i=1,...,n \tag{15}\] \[h_{(i+1)}^{\text{D}},d_{(i)} =\text{LSTM}(x_{(i)},h_{(i)}^{\text{D}};\ \theta^{\text{D}})\quad \forall i=0,1,...,n \tag{16}\] The decoder output vector $d_{(i)}$ are, then feed into a fully-connected (FC) layer to calculate probability of the next stop: \[g_{(i)}=\text{FC}(d_{(i)};\ \theta^{\text{FC}})\quad\forall i=1,...,n \tag{17}\] \[\mathbb{P}(c_{i+1}\ \|\ c_{1},...,c_{i},\mathcal{S},X^{\mathcal{S}} ;\theta)=\text{Softmax}(g_{(i)})\quad\forall i=1,...,n \tag{18}\] where $g_{(i)}\in\mathbb{R}^{K_{z}}$, $K_{z}$ is the maximum number of zones in the dataset. And the next predicted stop is selected by maximizing $\mathbb{P}(c_{i+1}=j\ \|\ c_{1},...,c_{i},\mathcal{S},X^{\mathcal{S}};\theta)$ for all $s_{j}\in\mathcal{S}\setminus\mathcal{S}_{(i)}^{\text{V}}$ (i.e., the zones that are not in the route and that have been visited are excluded). Original Pointer Network. Another benchmark model is the original pointer network (Pnt Net) proposed by (Vinyals et al., 2015). The overall architecture of the pointer network is similar to the proposed model Figure 8: Model architecture of the LSTM-E-D seq2seq prediction model. except for the attention component. Specifically, the pointer network calculates attention as: \[u_{(i)}^{j} =W_{1}^{T}\text{tanh}(W_{2}e_{j}+W_{3}d_{(i)})\quad\forall i,j=1,...,n \tag{19}\] \[a_{(i)}^{j} =\frac{\exp(u_{(i)}^{j})}{\sum_{j^{\prime}=1}^{n}\exp(u_{(i)}^{j^ {\prime}})}\quad\forall i,j=1,...,n \tag{20}\] The original pointer network does not include the pair-wise local information $(z_{(i)}^{j},\ \phi(x_{(i)},x_{j}))$, and the attention calculation is only quantified from three learnable parameters $W_{1},W_{2}$, and $W_{3}$, which may limit its capacity in prediction. We observe that the original pointer network without local information performs extremely badly. For a fair comparison, we add the local information with the similar format in Eq. 19 as: \[u_{(i)}^{j}=W_{1}^{T}\text{tanh}(W_{2}e_{j}+W_{3}d_{(i)})+W_{4}\begin{bmatrix} z_{(i)}^{j}\\ \phi(x_{(i)},x_{j})\end{bmatrix}\quad\forall i,j=1,...,n \tag{21}\] After training the model, we generate the final sequence with the greedy algorithm and Algorithm 1, respectively. ### Results #### 5.4.1 Model comparison. Table 1 presents the performance of different models. Note that for all approaches except for the TSP, we generate sequences based on two different methods (greedy and Algorithm 1) for comparison. The standard deviation of disparity scores is taken over all testing routes. Results show that sequence generation with Algorithm 1 (i.e., iterating different first zones) can consistently reduce the disparity score for all machine learning methods. It implies that the first zone prediction and the global view (i.e., shortest path) are important for estimating the driver's trajectory. The proposed method outperforms all other models, both in disparity scores and prediction accuracy. This means the proposed pair-wise ASNN-based attention (Eq. 6) has better performance than the original content-based attention (Eq. 21). The comparison between LSTM-E-D and Pnt Net models demonstrates the effectiveness of the attention mechanism. All machine learning models except for LSTM-E-D can outperform the baseline TSP sequence with Algorithm 1 sequence generation method, suggesting that the hidden trajectory patterns can be learned from the data. Another observation is that, the prediction accuracy and disparity score do not always move in the same direction. For example, the LSTM-E-D model with Algorithm 1 sequence generation, though has lower accuracy, shows a better disparity score. This is because the accuracy metric does not differentiate "how wrong an erroneous prediction is". By the definition of disparity score, if a stop is $s_{i}$ but the prediction is $s_{j}$, and $s_{j}$ and $s_{i}$ are geographically close to each other, the score does not worsen too much. This suggests a future research direction in using disparity score as the loss function (e.g., training by RL) instead of cross-entropy loss. Figure 9 shows the distribution of disparity scores for our proposed method with Algorithm 1 sequence generation (i.e., the best model). We observe that the prediction performance varies a lot across different routes. There is a huge proportion of routes with very small disparity scores (less than 0.01). The mean score is impacted by outlier routes. The median score is 0.0340, which is smaller than the mean value. #### 5.4.2 Factors on trajectory predictability. As our proposed model exhibits various levels of predictability across different routes, we aim to investigate which attributes of a route cause high (or low) predictability. This can be done by running a regression model with the disparity score as the dependent variable and route attributes (e.g., locations, departure time, package numbers) as independent variables. The variables used are defined as follows: * Total planned service time: The estimated time to deliver all packages in the route (service time only, excluding travel time). * Earliest time window constraint: The earliest due time to deliver packages with time window constraint minus the vehicle departure time. The smaller the value, the tighter the time limit. * Avg. # traffic signals: Average number of traffic signals in each zone of the route (obtained from OpenStreetMap data). \begin{table} \begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{}{Sequence generation} & \multirow{2}{}{Model} & \multicolumn{2}{c}{Disparity score} & \multicolumn{3}{c}{Prediction accuracy} \\ \cline{3-8} & & Mean & Std. Dev & 1st zone & 2nd zone & 3rd zone & 4th zone \\ \hline \multirow{2}{}{-} & TSP & 0.0443 & 0.0289 & 0.207 & 0.185 & 0.163 & 0.168 \\ \hline \multirow{4}{}{Greedy} & ASNN & 0.0470 & 0.0289 & 0.150 & 0.141 & 0.119 & 0.123 \\ & LSTM-E-D & 0.0503 & 0.0313 & 0.207 & 0.183 & 0.161 & 0.166 \\ & Pnt Net & 0.0460 & 0.0309 & 0.224 & 0.204 & 0.186 & 0.165 \\ & Ours & 0.0417 & 0.0306 & 0.241 & 0.231 & 0.224 & 0.221 \\ \hline \multirow{4}{}{Algorithm 1} & ASNN & 0.0429 & 0.0299 & 0.221 & 0.213 & 0.203 & 0.195 \\ & LSTM-E-D & 0.0501 & 0.0305 & 0.182 & 0.156 & 0.142 & 0.149 \\ & Pnt Net & 0.0382 & 0.0301 & 0.286 & 0.273 & 0.262 & 0.274 \\ & Ours & 0.0369* & 0.0301 & 0.320 & 0.310 & 0.303 & 0.314 \\ \hline \hline \end{tabular} \end{table} Table 1: Model performance Figure 9: Disparity score distribution of the best model * If high-quality route: A dummy variable indicating whether the route is labeled as "high quality" by Amazon or not (Yes = 1). High quality means the actual travel time of the route is similar to or better than Amazon's expectation. * If in Location: A dummy variable indicating whether the route is in a specific city or not (Yes = 1). * If departure Time: A dummy variable indicating the (local) departure time (e.g., before 7AM, after 10AM). Table 2 shows the results of the regression. Since the dependent variable is _disparity scores_, a negative sign indicates a positive impact on the predictability. We observe that routes with tighter time window constraints and more stops are easier to predict. This may be due to the fact that these routes are usually harder to deliver. Hence, to avoid the risk of violating time constraints or delay, drivers tend to follow the planned routes and thus the route sequences are easier predict. We also find that routes associated with larger vans (i.e., larger vehicle capacity) are more predictable. The reason may be that larger vans are less flexible in choosing different routes, thus drivers are more likely to follow the navigation. Another important factor for better predictability is high-quality routes. This may be because high-quality routes are closer to the TSP sequence which we use as inputs. Finally, routes in LA are more predictable than in other areas such as Chicago and Boston. #### 5.4.3 Impact of input sequence All machine learning models in Table 1 (except for ASNN) have the LSTM encoder component, which requires the specification of input zone sequence. As mentioned in Section 5.2, we currently use the TSP sequence as input. It is worth exploring the model performance if we use a random zone sequence instead, which corresponds to the scenario without planned route information. Table 3 shows the model performance without the TSP sequence information. Since the ASNN result does not rely on TSP information, it is not listed in the table. Results show that the LSTM-E-D model becomes much worse with a random sequence as inputs, while the performance of Pnt Net and our method is only slightly affected. Even without the planned route information, the proposed model can still provide a reasonable estimation of driver trajectories. \begin{table} \begin{tabular}{l l l l} \hline \hline Variables & Coefficients ($\times 10^{-3}$) & Variables & Coefficients ($\times 10^{-3}$) \\ \hline Intercept & 91.07 & If high quality route & -1.66$\times 10^{-14}$ \\ Total \# of packages & 0.059 & If in LA & -4.998 * \\ Total planned service time & -0.476 & If in Chicago & 0.783 \\ Earliest time window constraint & -3.047 & If in Boston & -3.354 \\ Avg. \# traffic signals & -3.255 & If on weekends & 1.775 \\ Total \# of stops & -0.142 & If departure before 7AM & 0.582 \\ Vehicle capacity ($m^{3}$) & -6.041 * & If departure after 10AM & -2.704 \\ \hline Number of routes: 1,002. & & & \\ $R^{2}$: 0.065; & & & \\ ${}^{*}$: $p$-value $<0.01$; ${}^{}$: $p$-value $<0.05$. & & \\ \hline \hline \end{tabular} \end{table} Table 2: Factors on trajectory predictability ### Summary Our numerical results show that our proposed model outperforms its benchmarks in terms of disparity scores and prediction accuracy, meaning that it can better predict the actual route trajectories taken by drivers. The comparison with benchmark models shows that our proposed ASNN-based pair-wise attention mechanism and our sequence generation algorithm (Algorithm 1) are both helpful for the prediction. Moreover, we can observe that the predictive performance varies across different routes. Factors such as route quality, delivery time windows, and the total number of stops of a route affect predictability. Finally, the proposed model is insensitive to the input sequence. The prediction performance only slightly decreases when the input sequence is changed from the TSP solution to a random stop sequence. This property implies that we only need the set of stops to implement the model and obtain high-quality solution, while information on the planned route sequence is not strictly required. ## 6 Conclusion and Future Research In this paper, we propose a pair-wise attention-based pointer neural network that predicts actual driver trajectories on last-mile delivery routes for given sets of delivery stops. Compared to previously proposed pointer networks, this study leverages a new alternative specific neural network-based attention mechanism to incorporate pair-wise local information (such as relative distances and locations of stops) for the attention calculation. To better capture the global efficiency of a route in terms of operational cost (i.e., total travel time), we further propose a new sequence generation algorithm that finds the lowest-cost route sequence by iterating through different first stops. We apply our proposed method to a large set of real operational route data provided by the Amazon Last-Mile Routing Research Challenge in 2021. The results show that our proposed method can outperform a wide range of benchmark models in terms of both the disparity score and prediction accuracy, meaning that the predicted route sequence is closer to the actual sequence executed by drivers. Compared to the best benchmark model (original pointer network), our method reduces the disparity score from 0.0382 to 0.0369, and increases the average prediction accuracy of the first four zones from 0.229 to 0.312. Moreover, our proposed sequence generation method can consistently improve the prediction performance for all models. The disparity scores are reduced by 10-20% across different models. Lastly, we show that the proposed methodology is robust against changes in the input sequence pattern. Compared to an optimal TSP solution as the input sequence, a random input sequence only slightly increases the disparity score from 0.0369 to 0.0376. \begin{table} \begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{}{Sequence generation} & \multirow{2}{}{Model} & \multicolumn{2}{c}{Disparity score} & \multicolumn{3}{c}{Prediction accuracy} \\ \cline{3-8} & & Mean & Std. Dev & 1st zone & 2nd zone & 3rd zone & 4th zone \\ \hline \multirow{3}{}{Greedy} & LSTM-E-D & 0.1176 & 0.0498 & 0.045 & 0.047 & 0.041 & 0.050 \\ & Pnt Net & 0.0512 & 0.0323 & 0.090 & 0.096 & 0.097 & 0.096 \\ & Ours & 0.0426* & 0.0311 & 0.204 & 0.192 & 0.195 & 0.196 \\ \hline \multirow{3}{}{Algorithm 1} & LSTM-E-D & 0.1054 & 0.0463 & 0.103 & 0.061 & 0.049 & 0.052 \\ & Pnt Net & 0.0398 & 0.0311 & 0.298 & 0.284 & 0.273 & 0.273 \\ \cline{1-1} & Ours & 0.0376* & 0.0307 & 0.316 & 0.298 & 0.302 & 0.298 \\ \hline \hline \end{tabular} \end{table} Table 3: Model performance without TSP information The data-driven route planning method proposed in this paper has several highly relevant practical implications. First, our proposed model performs well at predicting stop sequences that would be preferable to delivery drivers in a real operational environment, even if it is not provided with a theoretically optimal (i.e., minimal route duration) planned TSP sequence as an input. Therefore, the model can be used to generate a predicted actual stop sequence that a driver would likely be taking for a given set of delivery stops. The prediction can serve as a new 'empirical' planned route that is informed by historical driver behavior and thus more consistent with the driver's experience and preferences. Second, by comparing the stop sequence predicted by our model with the traditional, TSP-based planned stop sequence, a route planner may infer potential reasons for the drivers' deviations and adjust the company's planning procedures and/or driver incentives if necessary. Third, as stop sequence generation using machine learning models is computationally more efficient than traditional optimization-based approaches, a trained machine learning model can be applied in real-time to quickly re-optimize routes when drivers are unexpectedly forced to deviate from their original stop sequence (e.g., due to road closures) and need updated routing strategies. Based on the work presented in this paper, a number of fruitful future research avenues arise. First, instead of focusing on stop sequence prediction, future work may improve the interpretability of such prediction models and develop machine learning approaches that better explain which factors cause drivers to deviate from a planned stop sequence and how they affect their actual route trajectories. Second, future work should attempt to combine the strengths of optimization-based route planning approaches and machine learning by incorporating tacit driver knowledge learned via machine learning models into route optimization algorithms. ## Appendix A Mathematical Formulation of a LSTM Cell The details of an LSTM cell, $h_{t},e_{t}=\text{LSTM}(x_{t},h_{t-1};\ \theta)$, is shown below: \[f_{t} =\sigma_{g}(W_{f}x_{t}+U_{f}h_{t-1}+b_{f})\] (A.1) \[i_{t} =\sigma_{g}(W_{i}x_{t}+U_{i}h_{t-1}+b_{i})\] (A.2) \[o_{t} =\sigma_{g}(W_{o}x_{t}+U_{o}h_{t-1}+b_{o})\] (A.3) \[\tilde{c}_{t} =\sigma_{c}(W_{c}x_{t}+U_{c}h_{t-1}+b_{c})\] (A.4) \[c_{t} =f_{t}\circ c_{t-1}+i_{t}\circ\tilde{c}_{t}\] (A.5) \[h_{t} =o_{t}\circ\sigma_{h}(c_{t})\] (A.6) \[e_{t} =h_{t}\text{ (if this is a single layer one-directional LSTM)}\] (A.7) where $[W_{f},W_{i},W_{o},W_{c},U_{f},U_{i},U_{o},U_{c},b_{f},b_{i},b_{o},b_{c}]=\theta$ is the vector of learnable parameters. $x_{t}$ is the input vector to the LSTM unit. $f_{t}$ is the forget gate's activation vector. $i_{t}$ is the input/update gate's activation vector. $o_{t}$ is the output gate's activation vector. $h_{t}$ is the hidden state vector. $e_{t}$ is the output vector of the LSTM. Note that for a multi-layer or bidirectional LSTM, $e_{t}$ may not equal to $h_{t}$. In this study, we use a single layer one-directional LSTM and thus have $e_{t}=h_{t}$. More details on the output vector can be found in Pytorch (2021). $\tilde{c}_{t}$ is the cell input activation vector. $c_{t}$ is the cell state vector. "$\circ$" indicates the component-wise multiplication. ## Appendix B From Zone Sequence to Stop Sequence The complete stop sequence is generated based on the given zone sequence. The detailed generation process is shown in Algorithm 2. ``` 1:functionCompleteSeqGeneration($(\hat{z}_{(1)},..,\hat{z}_{(n)}),\{\mathcal{S}_{(i)},i=1,,,n\}$) 2:$s_{prev}\gets D^{\mathcal{S}}$ 3:$s_{complete}^{}\leftarrow(s_{prev})$$\triangleright$ Initialize the complete stop sequence with depot 4:for$i\in\{1,...,n-1\}$do 5:$\mathcal{S}_{first}\leftarrow$ Set of three stops in $\mathcal{S}_{(i)}$ that are closest to $s_{prev}$ 6:$\mathcal{S}_{last}\leftarrow$ Set of three stops in $\mathcal{S}_{(i)}$ that are closest to all stops in $\mathcal{S}_{i+1}$ on average 7:$\mathcal{P}_{(i)}\leftarrow\emptyset$$\triangleright$ Initialize the set of optimal paths in zone $\hat{z}_{(i)}$ 8:for$s_{first}\in\mathcal{S}_{first}$do 9:for$s_{last}\in\mathcal{S}_{last}$do 10:if$s_{first}=s_{last}$then 11:$\hat{p}_{\text{temp}},t_{\text{temp}}=\texttt{T}\texttt{P}\texttt{P}( \mathcal{S}_{(i)})$$\triangleright$ Solve the optimal tour and travel time for zone $\hat{z}_{(i)}$ 12: Delete the last edge back to $s_{first}$ in the tour $\hat{p}^{\text{temp}}$. Let the new path and travel time be $\hat{p}^{\prime}_{\text{temp}}$ and $t^{\prime}_{\text{temp}}$ 13: Add $\hat{p}^{\prime}_{\text{temp}}$ and $t^{\prime}_{\text{temp}}$ to $\mathcal{P}_{(i)}$ 14:else 15:$\hat{p}_{\text{temp}},t_{\text{temp}}=\texttt{P}\texttt{P}\texttt{P}\texttt{P} (\mathcal{S}_{(i)},s_{first},s_{last})$$\triangleright$ Solve the optimal path and travel time for zone $i$ 16: Add $\hat{p}_{\text{temp}}$ and $t_{\text{temp}}$ to $\mathcal{P}_{(i)}$ 17:$\hat{p}_{(i)}\leftarrow$ Path in $\mathcal{P}_{(i)}$ with the minimum travel time 18:$s_{complete}^{}\leftarrow(s_{complete}^{},\ \hat{p}_{(i)})$$\triangleright$ Concatenate two sequence 19:$s_{prev}\leftarrow$ Last stop of path $\hat{p}_{(i)}$ 20:$s_{complete}^{}\leftarrow(s_{complete}^{},\ D^{\mathcal{S}})$$\triangleright$ Concatenate the last stop as the depot 21:return$s_{complete}^{}$ ``` Algorithm 2 Complete sequence generation. 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If the first and the last stops are identical, we solve a tour TSP problem and output the path by deleting the last edge which traverses back to the first stop in the tour. After having all potential inner zone paths and total path travel time between any first and last stop pair, we keep the path with the minimum travel time as the inner zone sequence, $\hat{p}_{(i)}$. The key assumption we make here about drivers is that they will deliver packages within a zone following a path that minimizes their total travel time. With the optimal inner zone stop sequence of the current zone, we then move to the next visited zone in the optimal zone sequence and repeat the same procedure until we generate the complete stop sequence.
2304.07262	Phantom Embeddings: Using Embedding Space for Model Regularization in Deep Neural Networks	The strength of machine learning models stems from their ability to learn complex function approximations from data; however, this strength also makes training deep neural networks challenging. Notably, the complex models tend to memorize the training data, which results in poor regularization performance on test data. The regularization techniques such as L1, L2, dropout, etc. are proposed to reduce the overfitting effect; however, they bring in additional hyperparameters tuning complexity. These methods also fall short when the inter-class similarity is high due to the underlying data distribution, leading to a less accurate model. In this paper, we present a novel approach to regularize the models by leveraging the information-rich latent embeddings and their high intra-class correlation. We create phantom embeddings from a subset of homogenous samples and use these phantom embeddings to decrease the inter-class similarity of instances in their latent embedding space. The resulting models generalize better as a combination of their embedding and regularize them without requiring an expensive hyperparameter search. We evaluate our method on two popular and challenging image classification datasets (CIFAR and FashionMNIST) and show how our approach outperforms the standard baselines while displaying better training behavior.	Mofassir ul Islam Arif, Mohsan Jameel, Josif Grabocka, Lars Schmidt-Thieme	2023-04-14T17:15:54Z	http://arxiv.org/abs/2304.07262v1	# Phantom Embeddings: Using Embedding Space for Model Regularization in Deep Neural Networks ###### Abstract The strength of machine learning models stems from their ability to learn complex function approximations from data; however, this strength also makes training deep neural networks challenging. Notably, the complex models tend to memorize the training data, which results in poor regularization performance on test data. The regularization techniques such as L1, L2, dropout, etc. are proposed to reduce the overfitting effect; however, they bring in additional hyperparameters tuning complexity. These methods also fall short when the inter-class similarity is high due to the underlying data distribution, leading to a less accurate model. In this paper, we present a novel approach to regularize the models by leveraging the information-rich latent embeddings and their high intra-class correlation. We create phantom embeddings from a subset of homogenous samples and use these phantom embeddings to decrease the inter-class similarity of instances in their latent embedding space. The resulting models generalize better as a combination of their embedding, regularizes them without requiring an expensive hyperparameter search. We evaluate our method on two popular and challenging image classification datasets (CIFAR and FashionMNIST) and show how our approach outperforms the standard baselines while displaying better training behavior. Keywords:Deep Neural Networks Regulariztaion Embedding Space. ## 1 Introduction The field of computer vision has seen a remarkable increase in capability and complexity in recent years. The use of deep learning models in image classification [10] and object detection [4] tasks have shown a marked increase in their ability to capture more complex scenarios. Increasingly complex deep learning models such as ResNet [7] and Inception [21] were able to capture more in-depth information from input data. The strength of these deep learning models comes from their ability to take complex data and reduces it to highly expressive latent representations. These latent representations encode an image's spatial information into a vector through repeated convolutions and pooling operations. Training these complex models bring their challenges. Generally, the true distribution of the data is unknown, and observations are available in a limited number. These models are trained by iteratively minimizing the empirical risk over the training data (also known as Empirical Risk minimization ERM [22]). However, the increasing complexity of the model tends to overfit the data and generalize poorly on the test data, despite using the proper regularization. The theoretical understanding of ERM guarantees convergence as long as the model complexity does not increase with the number of training data [23]. For deep neural networks, an obvious issue arises as the increase in model complexity is not always complemented by an increase in the training data. To illustrate the aforementioned problems, we train a feed-forward neural network on a synthetic binary classification dataset and visualize the decision boundary in Fig. 1. Fig. 0(a) shows that the model was able to learn a reasonable decision boundary on the training data. However, due to the limited training examples available to train a complex model, it could not capture a better generalizable decision boundary resulting in poor performance on the test examples, as shown in Fig. 0(b). This example showcases two crucial challenges, firstly, how easy it is to overfit and perform poorly on test data. Secondly, in Fig.0(a) it can be seen that certain instances from differing classes are very close to each other, and ERM fails to provide a procedure to capture those instances. The model overfitting is treated by introducing the regularization [5, 11, 19, 8] in the ERM objective. However, ERM's problem is most evident around the Figure 1: Fig. 0(a) shows the overfitted decision boundary on training data. When evaluated on the test set in Fig. 0(b) the model shows poor generalization, a consequence of overfitting. In Fig. 0(c), we show a hypotethical embedding space and the decision boundary created by a deep neural network. The light colors represent the original embeddings while the darker colors represent the phantom embeddings proposed by our method. vicinity of the boundary region, as samples from different classes are in close proximity. Model complexity could be increased to capture these instances, but that violates the convergence guarantee of ERM since the number of instances does not increase with the increase in model complexity. One can mitigate the ERM failure through the Vicinal Risk Minimization Principle [1] by adding a better regularization using data augmentation [16]. Data augmentation mutates the input instances, traditionally through rotating, flipping, and scaling to inject noise in the training data, thereby preventing the model from memorizing it. However, it is limited as it mutates the data within one class vicinity and not across other classes. Other regularization methods involve tunable hyperparameters requiring an expensive configuration search, and the resulting hyperparameters are non-transferable and dataset-specific. In this paper, we propose a solution for problems stated above by leveraging the latent embeddings to create what we call a 'phantom embedding'. This is done by aggregating the latent embeddings of a subset of the instances from the same class. Using the latent vicinal embedding space allows us to use the information-rich embeddings to inject a hyper-parameter free latent vicinal regularization and boost accuracy. Since machine learning models transfrom input data into their representative embeddings: $\psi:\mathbb{R}^{M}\rightarrow\mathbb{R}^{D}$ where $M$ is the original data dimensionality and $D$ is the size of the embedding space. Therefore, by creating this phantom embedding, we create phantom data points to learn on. This is illustrated in in Fig. 0(c). This phantom embedding is used to 'pull' the original instance away from the decision boundary and closer to the samples (of the same class) in the embedding space. For the instances already sufficiently away from the decision boundary the 'pull' does not adversely impact since the embedding space is already well seated in the data distribution. We validate on an image classification benchmark task that our propose solution generalize better as compare to the existing approaches and achieves higher test accuracies. Our main contributions include: * Improvement in classification accuracy by using phantom data points to overcome the base error in a dataset. * A hyper-parameter free intrinsic regularization to enable training truly deep models. * Evaluate our model on two popular datasets against established baselines and showcase our performance gains as well training improvement qualitative and quantitatively. ## 2 Related Work Training very deep networks effectively is an open question[20] due to the model complexity. Models with millions of parameters require a lot of data to train effectively, however, millions of training samples are not available for all tasks. A good example of the realistic amount of data needed is [2] with 16M instances. That is not an option for all machine learning settings especially domains such as medicine [18]. Data augmentation [10] is an efficient method to ensure that data seen by the model is varied during training. Standard augmentation techniques include flipping, scaling, and padding. Training these models from scratch can be avoided by using the weights of a model that has been trained on a similar dataset and then finetuning the model to fit your need[15][17]. Transfer learning [15] has enabled training deeper model using a smaller dataset size however, when the goal is complete retraining than the training procedure needs to be adapted to ensure that the model does not memorize the training data. Methods such as MaxOut [5] add layers into the architecture with a max activation function and have shown to positively impact the convergence behavior when compared to the ReLu activation [14]. DropOut, proposed in [19], addresses the problem of model overfitting by probabilistically turning off neurons in the final embedding layer to create an ensemble of models and has shown to be an effective way to regularize deep neural networks. Similarly in [25], the authors move the regularization from the final layer to the loss layer where they intentionally flip the labels in a mini-batch to ensure that the model generalizes. These methods seek to work on the architecture and loss layer to regularize the model. Methods such as weight decay [11] and batch normalization[8] are aimed at the optimizer and architecture and seek to penalize the weights while training to ensure that models generalize. In [26] the authors propose the use of taking multiple instances and creating a linear combination of the instances and their label. Sampling from this mixup distribution allows them to learn on fabricated data points. ## 3 Methodology Consider a machine learning method $\psi(x)$ where $x$ is a dataset sample and $x\in\mathbb{R}^{N\times M}$ corresponds to a multi-category target $y$ where $y\in\big{\{}1,\dots,L\big{\}}^{N}$ among $L$ classes. This model will produce a latent embedding: $\phi:\mathbb{R}^{M}\rightarrow\mathbb{R}^{D}$ of the features, which is then passed to the prediction layer: $\psi:\mathbb{R}^{D}\rightarrow\mathbb{R}^{L}$. The Figure 2: A training steps takes a micro-cluster with K many samples, generating K many embeddings with are aggregated to create the phantom embedding, This, along with the embedding of the main train instance is passed to the predictor. The combined loss for these predictions is calculated as in Eq. 3. estimated target variable is therefore $\hat{y_{n}}:=\psi(\phi(x_{n})),\forall n\in\{1,\ldots,N\}$ and the respective objective function: \[\operatorname{arg\,min}_{\psi,\phi}\sum_{n=1}^{N}\mathcal{L}(y_{n},\psi(\phi(x_ {n}))) \tag{1}\] In this work we propose to make use of the shared similiarities among the instances belonging to the same class and leveraging the collective learned representations of a small subset of instances to generalize the final embedding space. This is done by sampling a'micro-cluster' of instances belonging to the same class. Note here that 'cluster' is being used in terms of a 'group' and has no relation to the unsupervised clustering methods. Let us denote the number of instances in each micro-cluster as $K\in\mathbb{N}$and the number of instances in each respective class as $N_{l}\in\mathbb{N},\forall l\in\{1,\ldots,L\}$ therefore for each class it is possible to draw $\binom{N_{l}}{K}$ many random choices. On these choices, consider, a new dataset transformation $(x,y)\rightarrow(x^{\prime},y^{\prime})$, where each element of $x^{\prime}$ represents a homogeneous cluster from $x$ with $K$ members and each element $y^{\prime}$ is the respective label of the instances within a homogeneous cluster. Since we are sampling homogenous clusters, $y^{\prime}=y$. The total number of clusters is defined as $N^{\prime}=\sum_{l=1}^{L}\binom{N_{l}}{K}$. The new input features are then $x^{\prime}\in\mathbb{R}^{N^{\prime}xKxM}$ and the new targets $y^{\prime}\in\{1,\ldots,L\}^{N^{\prime}}$. This new dataset transformation leads to a model output: $\hat{y_{n}}:=\psi(\phi(x^{\prime}_{n,k}))$ where $\phi(x^{\prime}_{n,k})$ is the $k^{th}$ latent embedding and $k\in K$. These $K$ many latent embeddings will be used to generalize the learned final learned embedding by aggregating them as see in Fig. 2. In our proposed approach we use a "Mean Embedding Space Aggregator" which is explained as: $\phi^{\prime}(x_{n})=\frac{1}{K}\sum_{k=1}^{K}\phi(x^{\prime}_{n,k})$ where $\phi^{\prime}(x_{n})$ is the phantom embedding from the micro-cluster. The naive approach would be to use this phantom embedding directly in the optimization,resulting in the follllowing objective function: \[\operatorname{arg\,min}_{\phi,\psi}\sum_{n=1}^{N^{\prime}}\mathcal{L}\bigg{(} y^{\prime}_{n},\psi\big{(}\frac{1}{K}\sum_{k=1}^{K}\phi(x^{\prime}_{n,k}) \big{)}\bigg{)} \tag{2}\] However, Eq. 2 poses a problem since the intra-class variation of challenging datasets can cause the embedding to be too drastically modified, Also, datasets with multi-modal distributions and non-convex hulls can be adversely effected by the naive objective function (Eq. 2) since the micro-cluster can be sampled from the different modes of the data distribution. In its place we propose to use the phantom embedding in the loss function: \[\mathcal{L}=\alpha\mathcal{L}(y^{\prime}_{n},\psi(\phi(x^{\prime}_{n,k=0})))-( 1-\alpha)\mathcal{L}(y^{\prime}_{n},\psi(\phi^{\prime}(x^{\prime}_{n}))) \tag{3}\] In Eq. 3 we treat the first sample ($k=0$) as the main instance and the others serve as a guide to improve the embedding space for this instance by 'pulling' the $k=0^{th}$ towards the phantom embedding. In order to avoid adding another hyper-parameter to tune, in our loss, $\alpha$ is drawn from the beta distribution and serves to add stochasticity in the combination of the embeddings and also removes the need for tuning $\alpha$. Therefore our final objective function is: \[\underset{\phi,\psi}{\arg\min}\sum_{n=1}^{N^{\prime}}\bigg{[}\alpha\mathcal{L} \bigg{(}y^{\prime}_{n},\psi\big{(}\phi(x^{\prime}_{n,k=0})\big{)}\bigg{)}-(1- \alpha)\mathcal{L}\bigg{(}y^{\prime}_{n},\psi\big{(}\frac{1}{K}\sum_{k=1}^{K} \phi(x^{\prime}_{n,k})\big{)}\bigg{)}\bigg{]} \tag{4}\] ## 4 Experiments In this section, we showcase the results of our approach and compare them with other methods in the domain. All the results presented have been recreated using the original author's provided implementations. These experiments were carried out on NVIDIA 1080Ti, 2080Ti, and V100 GPUs. ### Datasets and Implementation Details To verify the efficacy of our proposed approach we have chosen two publically available datasets. CIFAR10 [9] and FashionMNIST [24] are popular image classification datasets and are widely used in the computer vision domain for testing new research. They comprise 60000 and 70000 images sized at 32x32 and 28x28 respectively. They offer a challenging problem setting due to the wide intra-class variation and inter-class similarities. Furthermore, these datasets are also easy to overfit the deep convolutional neural networks. Therefore, these datasets provide all the necessary challenges that our work proposes to address. Our method can be readily included in any machine learning model, for our experiments we have chosen Deep Residual Networks (ResNet-18, ResNet-34, and ResNet-50) as proposed in [7] and as implemented in [13]. The networks under test were initialized as specified in [6] and optimized using Stochastic Gradient Descent(SGD) [12] with batch normalization [8] and a weight decay [11] factor of 0.0005, it should be noted here that the original ResNet architecture used 0.0001. The learning rate was set at 0.1 at the start than the scaled down by a factor of 10 at the 32k and 48k iteration as in [7], training was terminated at 64k iterations. We used a batch size of 128 and the dataset was augmented by padding 4 pixels to the image and translating the image accordingly, the images were also flipped horizontally and normalized by the mean and standard deviation of the entire dataset. ### Results In this section we evaluate our model by answering the following research question:. 1. RQ1: Can classification accuracy be improved by creating a phantom embedding for data points? 2. RQ2: Can a better embedding space lead to a more robust model? 3. RQ3: Can we add intrinsic regularization by using the embedding space directly? ### RQ1: Classification Accuracy The baselines were chosen based on their relevance to the approach that we have outlined in this paper. We have used the DisturbLabel [25] as implemented in [3], ResNet with Dropout [19] and we also compare against the vanilla variants of the ResNet architectures. DisturbLabel seeks to regularize the loss layer rather than the parameters and DropOut seeks to create an inherent ensemble of neural networks by stochastically turning off a certain amount neurons in the embedding layer to prevent the models from learning the training data. A comparison of our method to the baselines can be seen in Tab. 1. It can be seen in Tab. 1 and 2 that our proposed method is performing better than the all the baselines in terms of the accuracy, however, it should also be noted that the overall variance in the results at the time of convergence is also better than the baselines. We see a 1.59% and 0.88% gain for ResNet-18 and ResNet-34 accuracies, the decrease in the overall 'performance gain' can be attributed to the fact that \begin{table} \begin{tabular}{\|l\|l\|l\|l\|} \hline & \multicolumn{3}{c\|}{Accuracy} \\ \hline Method & Acc & Mean Acc & Max \\ \hline ResNet18 & 93.5 & 93.68 & 93.68 \\ \hline ResNet18 Dropout & 94.11 & 94.09 & 94.21 \\ \hline ResNet18 DisturbLabel & 94.2 & 94.28 & 94.33 \\ \hline \hline Phantom ResNet18 & 94.91 & 94.84 & 94.91 \\ \hline \hline \end{tabular} \end{table} Table 1: Classification Accuracy on CIFAR using ResNet-18 architecture. We report the final accuracy as Acc and also the Mean and Max accuracies for the last 5 epochs to illustrate training stability towards convergence. Figure 3: ResNet-18 Training and testing behaviors: Baseline refers to the original network baseline while DO and DL refer to the DropOut and DisturbLabel baselines. ResNet-34 is a more complex model. ResNet-18 has 0.27M parameters while ResNet-34 has 0.46M, so by doubling the parameters of the network we expect a more expressive model that already improves upon the shortcomings of the former. A more important trend in Tab. 2 is the behavior of ResNet-Dropout and ResNet-DisturbLabel values. We only see an improvement over ResNet of 0.27% and 0.1%. The baselines struggle to keep up with the expected increase in performance expectations when increasing model complexity. ERM convergence guarantee, as stated early is at play here. Secondly, the baselines do not take into account the highly similar embeddings of dissimilar class members. Therefore, the baselines end up optimizing for performance without taking into account the obstacles that are hindering it and it gets reflected in the final numbers we present here. In Tab. 3 we can see that the results for our approach continue to outperform the baselines on the FashionMNIST dataset which comes with its own set of challenges since the images are now 28x28 and comprise of a single channel rather than the standard RGB channels of CIFAR. Consistent accuracy improvement across these datasets and over varying architecture complexities shows that our method is robust enough to deal with a wide variety of scenarios. Furthermore, it should be pointed out that the accuracies for the baselines required a large hyper-parameter search to get to these values whereas our proposal required no such search for performance. \begin{table} \begin{tabular}{\|l\|l\|l\|l\|} \hline & \multicolumn{3}{c\|}{Accuracy} \\ \hline Method & Acc & Mean & Max \\ \hline ResNet34 & 93.65 & 93.71 & 93.79 \\ \hline ResNet34 Dropout & 93.92 & 93.97 & 94.03 \\ \hline ResNet34 DisturbLabel & 93.73 & 93.79 & 93.81 \\ \hline \hline Phantom ResNet34 & 94.52 & 94.52 & 94.6 \\ \hline \hline \end{tabular} \end{table} Table 2: Classification Accuracy on CIFAR using ResNet34 \begin{table} \begin{tabular}{\|l\|l\|l\|} \hline & \multicolumn{3}{c\|}{Accuracy} \\ \hline Method & ResNet-18 & ResNet-34 \\ \hline ResNet & 94.78 & 94.93 \\ \hline ResNet-Dropout & 94.97 & 95.11 \\ \hline ResNet-Disturb & 94.95 & 94.97 \\ \hline \hline Phantom ResNet & 95.07 & 95.38 \\ \hline \hline \end{tabular} \end{table} Table 3: Classification Accuracy on FashionMNIST ### RQ2: Robustness A sufficiently well-trained algorithm should be able to reduce the error on the test set, the reduction of test error is inextricably tied to the training process. Our proposed methods seeks to mitigate overfitting by enriching the embedding space ensuring that the model generalizes well thus preventing errors in similar classes. It can be seen in Fig. 3b that our model is converging to a lower Test loss, this is an outcome of the enriched embedding space that actively helps optimize the model to learn a more general representation from the training data. The outcome of this approach reflects readily in Tab. 1 in the final accuracies, furthermore considering Fig. 3c it can be seen that our proposed model takes a more deliberative approach in the initial learning stage up to the first 100 epochs. While other models are shooting up quickly in accuracy values, and then later failing to maintain their lead, our approach focuses on learning better representations and penalizing itself when it doesn't more aggressively in order to arrive at the better final optimal model. The same trend is observed when training ResNet-34 as shown in Fig. 4. The only difference being that models no trained with inherent embedding space enrichment in mind suffer more due to the higher complexity of the underlying \begin{table} \begin{tabular}{\|l\|l\|l\|l\|} \hline & \multicolumn{3}{c\|}{Accuracy} \\ \hline Method & Acc & Mean & Max \\ \hline ResNet50 & 93.86 & 93.25 & 93.34 \\ \hline ResNet50 Dropout & 93.21 & 93.17 & 93.25 \\ \hline ResNet50 DisturbLabel & 94.37 & 94.352 & 94.38 \\ \hline \hline Phantom ResNet50 & 94.48 & 94.54 & 94.71 \\ \hline \end{tabular} \end{table} Table 4: Classification Accuracy on CIFAR using ResNet50 Figure 4: ResNet-34 Training and testing behaviors: With a more complex network, our continues to outperform the baselines. networks. In both Fig. 3 and Fig. 4 it can be seen that ResNet-Dropout seems to be more stable in terms of its fluctuations during the middle of the training process, between epoch 100 and 150, however it still fails to match our method in the final loss as well as final accuracy. This highlights the problems laid out in the introduction section where a model loses on accuracy in an attempt to not overfit. ### RQ3: Intrinsic Regularization As stated earlier, training deep models are hampered by the model memorizing the training data and then showing poor performance on the test data. This problem comes to the forefront when dealing with a truly deep model like ResNet-50 which comes with 0.88M trainable parameters. Training such a model from scratch requires an immense amount of data or a clever regularization scheme. The scheme needs to be searched for over several runs and hyper-parameter configurations. This is a time-consuming and expensive procedure since training ResNet-50 can take up to 7-11 hours on a modern GPU. Our proposed method allows for the data samples to contribute not just to the learning but to the regularization as well. By intrinsically learning the regularization with the help of similar images and generalizing the weights of our embedding layer with our proposed phantom embeddings we are able to regularize the model as it trains. This behavior is on display in Fig. 5 where it can be seen that our model is leading to a marked lower test loss while the baseline models struggle to match its performance. Given enough time (days) an ideal configuration for the baselines could be arrived to match the performance of our model however, our model provides it without the need for the extensive search required by the baselines. In Fig. 4(b) we intentionally allowed the models to run past their convergence point to see how the baseline and our model handle such cases. It can be seen that the baselines runs off and starts to overfit, leading to an increasing test loss while our method shows a noticably better performance and maintains a lower test loss. ### Abalation Study In order to showcase the effect of different numbers of samples from the same class ($K$) we varied $K$ from 1 (baseline) to 7 and in Tab. 5. It was seen that while increasing $K$ led to increasing performance over the baselines, the percentage gain vs memory required didn't justify the use of higher $K$. All the results reported have, therefore been conducted with $K=2$. ## 5 Conclusion In this paper, we have shown how embedding spaces can be directly used to regularize deeper neural networks by creating phantom embeddings around the true data points by aggregating the embeddings together and then optimizing the model with the phantom embedding as a co-target. We have shown how our method outperforms the baselines two famous and competitive datasets. Our method also introduces an intrinsic regularization which enables us to train deeper models without an extensive hyper-parameter search.
2307.07726	Towards Optimal Neural Networks: the Role of Sample Splitting in Hyperparameter Selection	When artificial neural networks have demonstrated exceptional practical success in a variety of domains, investigations into their theoretical characteristics, such as their approximation power, statistical properties, and generalization performance, have concurrently made significant strides. In this paper, we construct a novel theory for understanding the effectiveness of neural networks, which offers a perspective distinct from prior research. Specifically, we explore the rationale underlying a common practice during the construction of neural network models: sample splitting. Our findings indicate that the optimal hyperparameters derived from sample splitting can enable a neural network model that asymptotically minimizes the prediction risk. We conduct extensive experiments across different application scenarios and network architectures, and the results manifest our theory's effectiveness.	Shijin Gong, Xinyu Zhang	2023-07-15T06:46:40Z	http://arxiv.org/abs/2307.07726v2	# Towards Optimal Neural Networks: the Role of Sample Splitting in Hyperparameter Selection ###### Abstract When artificial neural networks have demonstrated exceptional practical success in a variety of domains, investigations into their theoretical characteristics, such as their approximation power, statistical properties, and generalization performance, have concurrently made significant strides. In this paper, we construct a novel theory for understanding the effectiveness of neural networks, which offers a perspective distinct from prior research. Specifically, we explore the rationale underlying a common practice during the construction of neural network models: sample splitting. Our findings indicate that the optimal hyperparameters derived from sample splitting can enable a neural network model that asymptotically minimizes the prediction risk. We conduct extensive experiments across different application scenarios and network architectures, and the results manifest our theory's effectiveness. Role of Sample Splitting towards Optimal NNs ## 1 Introduction Artificial neural networks (NNs) have garnered remarkable success across various industries, leading to a constant stream of research efforts aimed at exploring novel neural network models to enhance the scope of application and improve performance. Over the past few decades, researchers have developed convolutional neural networks (CNNs) (Fukushima, 1980; Krizhevsky et al., 2012), recurrent neural networks (RNNs) (Rumelhart et al., 1986; Hochreiter and Schmidhuber, 1997), and further models based on more complicated structures or components (e.g., Scarselli et al., 2008; Vaswani et al., 2017), which have demonstrated efficacy in solving a board range of practical problems. In parallel, efforts by scientists to uncover the theoretical characteristics underpinning neural networks, such as approximation power, statistical properties, and generalization ability, have resulted in a tremendous amount of studies and discussions. In what follows, we provide a concise overview of these major strides in neural network research that are currently garnering interest. Within this context, our framework emerges as a novel perspective aiming to expand the ongoing effort. Hornik et al. (1989) presented an universal approximation theorem for feed-forward neural networks (FNNs), leading to numerous subsequent research on approximation capabilities of NNs (e.g., Hornik, 1991; Leshno et al., 1993; Pinkus, 1999). These studies concluded that neural networks with sufficiently many hidden neurons possess the capacity to approximate any function within a broad class of functions. More literature has emerged in recent years concerning the approximation capabilities of neural networks, particularly deep networks that employ ReLU activation functions (e.g., Lu et al., 2017; Lin and Jegelka, 2018). White (1990) examined NNs from a nonparametric perspective, showing that the approximation mentioned in Hornik et al. (1989) are learnable by showcasing the consistency property of NNs, provided the complexity of the network increases with the sample size at specific growth rates. Work on consistency of neural networks continued (Mielniczuk and Tyrcha, 1993; Lugosi and Zeger, 1995). Further, Barron (1994); McCaffrey and Gallant (1994) developed error bounds for neural networks that demonstrate the rate of convergence. These studies laid the foundation for subsequent work on convergence rates, such as Chen and White (1999); Kohler and Krzyzak (2005), and Shen et al. (2023) established the asymptotic normality of neural network estimators along with their convergence rates. In recent years, discussions on learnability of NNs has broadened to encompass more than just a statistical perspective, increasingly incorporating an optimization viewpoint. This shift in focus has also seen a transition from exploring shallow neural networks to deep neural networks (DNNs)(e.g., Brutzkus and Globerson, 2017; Du et al., 2018). The objective of these investigations is to elucidate the reasons why the networks trained using algorithms, such as stochastic gradient descent (SGD), achieve successful convergence. Furthermore, the prevalent applications of DNNs have prompted research into the convergence under the over-parameterized setting (e.g., Li and Liang, 2018; Allen-Zhu et al., 2019). Another vital facet of theoretical work on neural networks has been centered around understanding their ability to extrapolate to new data. This ability, widely known as generalization, is evaluated by the generalization error, which quantifies a network's performance on unseen data. To establish generalization error bounds for NNs, abundant theories have focused on sample complexity, utilizing measures such as VC-dimension and Rademacher complexity (e.g., Baum and Haussler, 1988; Blumer et al., 1989; Haussler, 1992; Bartlett, 1996). These proposed theoretical generalization bounds quantify the amount of data required to ensure accurate performance of an NN. Bousquet and Elisseeff (2002) introduced the notions of stability for learning algorithms, providing an alternative pathway for deriving generalization bounds, which was followed by numerous subsequent studies such as Hardt et al. (2016). The work of Zhang et al. (2017), which implemented a randomization test challenging various prior studies, stimulated continued research on improving generalization bounds (e.g., Neyshabur et al., 2017; Golowich et al., 2018). Simultaneously, research developed under the over-parameterized setting has also gained traction in recent years (e.g., Arora et al., 2019; Allen-Zhu et al., 2019). In this paper, we incorporate the framework studied in traditional statistic theory (White, 1989) and offer a novel perspective on the question of why neural networks perform well, which goes beyond the characteristics of an individual NN. Commonly, the process of building a neural network necessarily involves the determination about two types of parameters. The first, known as model parameters, are gradually learned through the training process, while the second, referred to as hyperparameters, must be specified prior to train ing. Hyperparameters play a pivotal role in shaping the structure of neural networks and monitoring the learning process, contributing to substantial variability that allows neural networks to adapt to a diverse range of problems. Consequently, optimization of hyperparameters is a critical task in real-world applications of neural networks. A variety of algorithms (Yang and Shami, 2020) can achieve hyperparameter optimization. A striking majority of these algorithms are reliant on sample splitting. Typically, after hyperparameters are set, it is necessary to continually evaluate the model's performance during training. This process can then direct subsequent hyperparameter tuning to improve the model. To do this evaluation, a validation set comprising samples that have not been used for training is often required, thus necessitating a split from the original dataset prior to training. In this work, our theoretical and empirical explorations underscore the importance of sample splitting, since this procedure enables us to attain a nearly optimal neural network from those with all alternative hyperparameters. Specifically, hyperparameters tuned by optimizing the performance on validation set are asymptotically optimal in the sense of minimizing prediction risk. Therefore, the practical success of neural networks can be attributed to sample splitting, a common practice in machine learning. In fact, the efficacy of sample splitting can be recognized in several traditional statistical theories. For example, Li (1987) and Shao (1997) explored multiple procedures in the context of model selection. They derived the asymptotic optimality of these procedures, such as cross-validation, AIC, $C_{p}$, and other criteria, within the realm of linear models. The lack of explicit reference to sample splitting in these studies probably stems from the fact that the majority of these procedures do not necessarily require it. It is notable, however, that cross-validation, one of the procedures studied, is based on sample splitting, since it partitions the data into separate subsets for model training and evaluation purposes. As such, sample splitting has the potential to account for the effectiveness of models from a model selection standpoint. This perspective coincides with the central thesis of this paper, which investigates the success of neural network models through the lens of sample splitting. It is worth mentioning that our theory is established based on a basic splitting procedure that segregates samples into one training set and one validation set, yet it remains flexible enough to accommodate more sophisticated methods such as cross-validation. The remainder of our article is structured as follows. Section 2 introduces the mathematical modeling of NNs and their hyperparameters, along with the problem statement and the objective. In Section 3, we present the main theoretical results, their conditions, and an interpretation from a perspective of error decomposition. Section 4 presents experiments designed to validate our findings. Conclusion and future work are drawn in Section 5. Detailed proofs of our main results and additional discussions can be found in the Appendices. ## 2 Preliminaries Suppose that we have $(n_{1}+n_{2})$ independent and identically distributed (i.i.d.) observations $S=\{(X_{i},Y_{i})\}_{i=1}^{n_{1}+n_{2}}$, where $X_{i}$ is a vector of predictors and $Y_{i}$ is a scalar dependent variable. These samples can be partitioned into training set $S_{train}=\{(X_{i}^{t},Y_{i}^{t})\}_{i=1}^{n_{1}}$ and validation set $S_{val}=\{(X_{i}^{v},Y_{i}^{v})\}_{i=1}^{n_{2}}$. The available set of hyperparameters of NNs, including continuous, discrete and categorical variables, is denoted by $\Lambda$. In this paper, we assume that $\Lambda$ is a finite set, i.e. $\|\Lambda\|<\infty$. The choices of categorical hyperparameters, e.g., the activation function and optimizer, are intrinsically finite. The domains of continuous and discrete hyperparameters are generally bounded in practical applications (Bergstra et al., 2011), which results in finite choices of discrete ones. Regarding the rationality of assuming continuous hyperparameters, for instance, the learning rate, as belonging to a finite set, several frequently used hyperparameter optimization techniques consider a discretized continuous hyperparameter space and perform optimization on it, including grid search and random search (Bergstra and Bengio, 2012). Consequently, it is reasonable to develop our theory under the assumption that $\|\Lambda\|<\infty$. Given any $\lambda\in\Lambda$, a hypothesis space of neural networks is defined, and we denote a neural network in this space by $f_{\lambda}(\cdot,w):\mathbb{R}^{p}\rightarrow\mathbb{R}$, which represents a neural network with the hyperparameter $\lambda$ and the model parameter $w\in\mathbb{R}^{d(\lambda)}$. Here, the model parameter $w$ can interpreted as originating from weights connecting layers, vectors of biases for nodes, and other similar elements, which are then reshaped into column vectors and stacked on top of one another. The term $d(\lambda)$ represents the dimension of $w$, that is, the number of trainable variables in the model, and is certainly finite for any given $\lambda$. Therefore, the maximum dimension of model parameters under $\lambda\in\Lambda$, which is equal to $\sup_{\lambda\in\Lambda}d(\lambda)$, is finite in this paper as $\|\lambda\|<\infty$, which is also independent of sample size. For any hyperparameter $\lambda$, we can train an NN model by $S_{train}$, achieving the parameter $\hat{w}_{n_{1}}(\lambda)$ and the trained neural network $f_{\lambda}(\cdot,\hat{w}_{n_{1}}(\lambda)):\mathbb{R}^{p}\rightarrow\mathbb{R}$. We define the validation loss as the mean squared loss on validation set. \[L_{n_{2}}(\lambda)=\frac{1}{n_{2}}\sum_{i=1}^{n_{2}}\{f_{\lambda}(X_{i}^{v}, \hat{w}_{n_{1}}(\lambda))-Y_{i}^{v}\}^{2}. \tag{1}\] To optimize the hyperparameter, the ideal method is to select $\lambda$ that achieves the best performance on a validation set. Therefore, we define the selected optimal hyperparameter as \[\hat{\lambda}=\underset{\lambda\in\Lambda}{\text{argmin}}\ L_{n_{2}}(\lambda). \tag{2}\] Note that the existence of $\hat{\lambda}$ can be guaranteed by $\|\Lambda\|<\infty$. In real-world applications, the training set size $n_{1}$ and the validation set size $n_{2}$ are often set according to a proportion, such as a 90%-10% split of the total sample size. This means that $n_{1}$ and $n_{2}$ are generally of the same order with respect to the total sample size. Thus, without loss of generality and for notational simplicity in the ensuing text, we assume $n_{1}=n_{2}=n$. By adopting $\hat{\lambda}$ as the hyperparameter and completing the training process, we obtain a network $f_{\hat{\lambda}}(\cdot,w_{n}(\hat{\lambda}))$. Our objective now is to uncover properties of $\hat{\lambda}$ or $f_{\hat{\lambda}}(\cdot,w_{n}(\hat{\lambda}))$. Suppose we have a set of new i.i.d. observations, denoted as $S_{0}=\{(X_{j}^{0},Y_{j}^{0})\}_{j=1}^{n_{0}}$, but we do not have the knowledge of dependent variables. As We will demonstrate in the next section, $\hat{\lambda}$ is, in the asymptotic sense, optimal for making prediction for this new observation set $S_{0}$. In the subsequent sections, we use $\\|\cdot\\|$ to denote the $\ell_{2}$-norm of a vector and $\top$ to represent matrix transpose. Absolute constants, which may vary from line to line are denoted by $c$ and $C$. For notational simplicity, we abbreviate the validation sample $(X_{i}^{v},Y_{i}^{v})$ as $(X_{i},Y_{i})$ for $i=1,...,n$, when it does not cause ambiguity. Since the samples are i.i.d., we sometimes use $(X_{0},Y_{0})$ to represent any sample from the set $S_{0}$ also for notational convenience. ## 3 Theoretical Results In this section, we present the asymptotic properties of the selected hyperparameter $\hat{\lambda}$. We define the following notations \[L(\lambda;X,Y)=\{f_{\lambda}(X,\hat{w}_{n}(\lambda))-E(Y\|X)\}^{2}, \tag{3}\] \[L_{0}(\lambda)=\frac{1}{\|S_{0}\|}\sum_{(X,Y)\in S_{0}}L(\lambda;X,Y)=\frac{1}{n_{0}}\sum_{j=1}^{n_{0}}L(\lambda;X_{j}^{0},Y_{j}^{0}),\] (4) \[\text{and}\qquad R_{0}(\lambda)=E(L_{0}(\lambda)), \tag{5}\] where $\|S_{0}\|$ denotes the number of elements in set $S_{0}$. For hyperparameter $\lambda\in\Lambda$, $L(\lambda;X,Y)$ measures the squared loss between the conditional expectation $E(Y\|X)$ and the network output $f_{\lambda}(\cdot,\hat{w}_{n}(\lambda))$ with input $X$; $L_{0}(\lambda)$ is the mean prediction loss on set $S_{0}$; and $R_{0}(\lambda)$ is the prediction risk, the expectation of $L_{0}(\lambda)$. Since observations in $S_{0}$ are i.i.d., we can reformulate $R_{0}(\lambda)=E\{f_{\lambda}(X,\hat{w}_{n}(\lambda))-E(Y\|X)\}^{2}$, where $(X,Y)\in S_{val}\bigcup S_{0}$ can represent any sample independent of $S_{train}$. ### Optimality of the Selected Hyperparameter We present the assumptions that are required for our theoretical results. Unless otherwise specified, the following limiting processes are studied with respect to $n\to\infty$. Assumption 1: _For any $\lambda\in\Lambda$, there exist a parameter $w^{}(\lambda)$, such that $\\|\hat{w}_{n}(\lambda)-w^{}(\lambda)\\|=O_{p}(n^{-1/2})$._ Assumption 1 imports a high-level condition ensuring that for any given hyperparameter $\lambda$, the trained parameters of the neural networks $\hat{w}_{n}(\lambda)$ will have a limiting value $w^{}(\lambda)$. This assumption holds under some reasonable conditions; see conditions in Theorem 1 of White (1989). With the importation of $w^{}(\lambda)$, we can further introduce these notations \[L^{}(\lambda;X,Y)=\{f_{\lambda}(X,w^{}(\lambda))-E(Y\|X)\}^{2},\] \[L^{}_{0}(\lambda)=\frac{1}{\|S_{0}\|}\sum_{(X,Y)\in S_{0}}L^{}( \lambda;X,Y)=\frac{1}{n_{0}}\sum_{j=1}^{n_{0}}L^{}(\lambda;X_{j}^{0},Y_{j}^ {0}),\] \[\text{and}\qquad R^{}_{0}(\lambda)=E(L^{}_{0}(\lambda)),\] which are similar to the definitions of $L(\lambda;X,Y)$, $L_{0}(\lambda)$, and $R_{0}(\lambda)$ in (3), (4), and (5) except for substituting $w^{}(\lambda)$ for $\hat{w}_{n}(\lambda)$. Assumption 2: _For any $(X,Y)\in S_{train}\bigcup S_{val}\bigcup S_{0}$ and for any $\lambda\in\Lambda$, (i) $E\|f_{\lambda}(X,w^{}(\lambda))-Y\|$ and $E\|f_{\lambda}(X,w^{}(\lambda))-E(Y\|X)\|$ are both $O(1)$; (ii) $f_{\lambda}(X,w)$ is differentiable with respect to $w$, and there exists a constant $\rho$, such that_ \[E\sup_{w^{0}\in\mathcal{O}(w^{}(\lambda),\rho)}\bigg{\\|}\frac{\partial f_{ \lambda}(X,w)}{\partial w}\Big{\|}_{w=w^{0}}\bigg{\\|}=O(1), \tag{6}\] _where $\mathcal{O}(w^{}(\lambda),\rho)$ denotes an open ball with center $w^{}(\lambda)$ and radius $\rho$._ Assumption 2 makes some restrictions on boundedness and differentiability. Assumption 2(i) holds when $f_{\lambda}(X,w^{}(\lambda))$, $Y$ and $E(Y\|X)$ are bounded. Examples to illustrate the rationality of Assumption 2(ii) under some primitive conditions are given in Appendix A. Let $\xi_{n}=\inf_{\lambda\in\Lambda}R_{0}^{}(\lambda)$ denote the minimum risk among all $\lambda\in\Lambda$ with neural networks using limiting value of parameters $w^{}(\lambda)$, and we make these three assumptions: Assumption 3: $\Delta_{n}$ _is integrable, where_ \[\Delta_{n}=\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\big{\|}\{f_{\lambda}(Y_{0}, \hat{w}_{n}(\lambda))-Y_{0}\}^{2}-\{f_{\lambda}(Y_{0},w_{n}^{}(\lambda))-Y_{ 0}\}^{2}\big{\|}.\] Assumption 4: _For any $\lambda\in\Lambda$, $\xi_{n}^{-1}n^{-1/2}=o(1)$._ Assumption 5: _For any sample $(X,Y)\in S_{val}$ and any $\lambda\in\Lambda$ that (i) $\mathrm{var}[f_{\lambda}(X,w^{}(\lambda))\{Y-E(Y\|X)\}]=O(1)$; (ii) $\mathrm{var}[\{f_{\lambda}(X,w^{}(\lambda))-E(Y\|X)\}^{2}]=O(1)$._ As shown in (36) in the proof of Theorem 1 that $\Delta_{n}$ is $o_{p}(1)$, Assumption 3 is only imposed to ensure the expectation of $\Delta_{n}$ is o(1). Assumption 4 is similar to Condition (11) in Ando and Li (2014), Condition (C.3) of Zhang et al. (2016), and Assumption 2.3 in Liu and Okui (2013), which makes a restriction on the minimum risk $\xi_{n}$, requiring the speed of its converging to zero to be not too fast--at least slower than $n^{-1/2}$. As a concrete example, Assumption 4 is satisfied if $c\leq\xi_{n}$ for any $n$ for some constant $c$. Such cases arise when the hypothesis space, defined by any hyperparameter $\lambda\in\Lambda$, is insufficiently capable of approximating the objective function. Conversely, Assumption 4 is violated if $\xi_{n}\equiv 0$ for any $n\geq 0$. See interpretation from the perspective of error decomposition in Section 3.2.Assumption 5 has regularity conditions ensuring certain terms are not heavy-tailed so that their variance can be bounded. Now we are ready to present our first theorem. Theorem 1: _Under Assumptions 1-5, we have_ \[\frac{R_{0}(\hat{\lambda})}{\inf_{\lambda\in\Lambda}R_{0}(\lambda)}=1+O_{p}( \xi_{n}^{-1}n^{-1/2})=1+o_{p}(1). \tag{7}\] Theorem 1 indicates that the selected hyperparameter $\hat{\lambda}$ is optimal in the sense that the prediction risk of the neural network using hyperparameter $\hat{\lambda}$ is asymptotically identical to the risk of the infeasible best neural network. Note that in (7), since $\hat{\lambda}$ is directly plugged in the expressions $R_{0}(\lambda)$, the randomness of $\hat{\lambda}$ is not considered. This plug-in is done after taking the expectation in $R_{0}(\hat{\lambda})$, which makes (7) independent of test size $n_{0}$, and hence Theorem 1 holds for arbitrary $n_{0}$. In what follows, our second result shows that we can build other forms of optimalities based on $L_{0}(\lambda)$ with some additional assumptions. Assumption 6: _For any $\lambda\in\Lambda$, $\xi_{n}^{-1}n_{0}^{-1/2}=o(1)$._ Assumption 6 is similar to Assumption 4 except that the large sample condition is also with respect to $n_{0}$. If we assume that $n_{0}$ and $n$ are of same order, Assumption 4 and Assumption 6 are actually equivalent. As $L_{0}(\lambda)$ denotes the mean prediction loss on the test set $S_{0}$, this measure can be affected by the random fluctuations of samples. Therefore, it is required that the size of the test set, $n_{0}$, to be divergent, as assumed in 6. Assumption 7: _(i) There exists a constant $C$ such that $L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}(\lambda)<C$; (ii) $\{L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}/\inf_{\lambda \in\Lambda}L_{0}(\lambda)$ is uniformly integrable._ Assumption 7(i) holds when $f_{\lambda}(X,w^{}(\lambda))$ and $E(Y\|X)$ are bounded. Assumption 7(ii) is imposed to obtain the $L_{1}$ convergence of the term on the left hand side of (8) in Theorem 2. Theorem 2: _For a diverging $n_{0}$, under Assumptions 1-6, we have_ \[\frac{L_{0}(\hat{\lambda})}{\inf_{\lambda\in\Lambda}L_{0}(\lambda)}=1+O_{p}( \xi_{n}^{-1}n^{-1/2}+\xi_{n}^{-1}n_{0}^{-1/2})=1+o_{p}(1). \tag{8}\] _Furthermore, if Assumption 7 holds, then we have_ \[E\Big{\{}\frac{L_{0}(\hat{\lambda})}{\inf_{\lambda\in\Lambda}L_{0}(\lambda)} \Big{\}}=1+o(1) \tag{9}\] _and_ \[\frac{E\{L_{0}(\hat{\lambda})\}}{E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}} =1+o(1). \tag{10}\] Theorem 2 contains three types of optimalities. Equation (8) concerns the optimality in terms of asymptotically minimizing $L_{0}$, the mean prediction loss on test set $S_{0}$, which is established based on the result in Theorem 1. Equation (9) and (10) concern optimalities involving different forms of expectation. It is important to note that $E\{L_{0}(\hat{\lambda})\}$ is distinct from $R_{0}(\hat{\lambda})$ presented in Theorem 1, since the randomness of $\hat{\lambda}$ is taken into consideration in (9) and (10) but not in $R_{0}(\hat{\lambda})$. Consequently, the optimalities outlined in (9) and (10) is more reflective of real-world scenarios compared to Theorem 1, thereby offering theoretical evidence in practical application. To verify the justifications in Theorem 2, we will carry out a series of experimental studies in Section 4. ### Error Decomposition Interpretation For a more precise comprehension of theory, we revisit the objective in this paper from the perspective of error decomposition in machine learning theory. The optimalities of $\hat{\lambda}$ in Theorem 1 and Theorem 2 are obtained in the sense of minimizing $L_{0}(\lambda)$ or $R_{0}(\lambda)$, which both involve the difference between the target $E(Y\|X)$ and the trained model $f_{\lambda}(X,\hat{w}_{n}(\lambda))$. This discrepancy can be written as \[E(Y\|X)-f_{\lambda}(X,\hat{w}_{n}(\lambda))\] \[= \{E(Y\|X)-f_{\lambda}(X,w^{}(\lambda))\}+\{f_{\lambda}(X,w^{}( \lambda))-f_{\lambda}(X,\hat{w}_{n}(\lambda))\}.\] Now we preceive $f_{\lambda}(X,w^{}(\lambda))$ as a quasi-optimal estimate of $E(Y\|X)$ within models defined by hyperparamter $\lambda$, and then we can interpret the two terms respectively as 1) the model error $\{E(Y\|X)-f_{\lambda}(X,w^{}(\lambda))\}$ (also referred to as approximation error)--the error caused by the inadequate hypothesis space of the model determined by the choice of $\lambda$; and 2) the estimation error $\{f_{\lambda}(X,w^{}(\lambda))-f_{\lambda}(X,\hat{w}_{n}(\lambda))\}$--the error resulting from the gap, given $\lambda$, between the quasi-optimal parameter $w^{}(\lambda)$ and the parameter $\hat{w}_{n}(\lambda)$ derived from only $n$ training samples. If we revisit the definition of $L_{0}^{}(\lambda)$, the model error $\{E(Y\|X)-f_{\lambda}(X,w^{}(\lambda))\}$ corresponds to the term within $L_{0}^{}(\lambda)$'s square. Therefore, any assumptions imposed on $L_{0}^{}(\lambda)$ or $R_{0}^{}(\lambda)$ (such as Assumption 4, 6, and 7) inherently carry assumptions about the model error. Assumption 1 imports a condition ensuring the convergence of $\hat{w}_{n}(\lambda)$ to $w^{}(\lambda)$ at a certain rate. We leverage this condition to manage the estimation error $\{f_{\lambda}(X,w^{}(\lambda))-f_{\lambda}(X,\hat{w}_{n}(\lambda))\}$. This indicates that our work contemplates both model error and estimation error from the perspective of error decomposition. The optimalities established in our theorems can be viewed as accounting for the effectiveness of sample splitting with the modeling of both errors. Note that the optimization error--the error incurred during training process--is not our main focus, which is consistent with previous work that study neural networks from a statistical perspective (e.g., White, 1989; Barron, 1994; Bauer and Kohler, 2019). By utilizing Assumption 1, we have addressed the estimation error, as mentioned in the previous paragraph, while concurrently avoided the explicit modeling of the optimization error. By invoking the convergence property of $\hat{w}_{n}(\lambda)$ under Assumption 1, we directly perceive $\hat{w}_{n}(\lambda)$ as a set of satisfactory network parameters obtained under $n$ training samples, thereby dispensing with the need to model the specific training process of obtaining them. The specific modeling of these elements--the training process and optimization error--might well serve as a avenue for future research. ## 4 Experimental Verification In this section, we consider several common neural network structures and use them to model a range of diverse problems, including linear/nonlinear regression, binary-classification, and time series forecasting, to verify our theoretical results in Theorem 2. ### Multilayer Perceptrons We conduct experiments in three scenarios based on a basic NN structure multilayer perceptrons (MLPs), which follows a fully connected structure. The first and second scenarios are respectively based on linear and nonlinear regression problems. The third scenario simulates a classification problem. The training and validation set sizes are set to $n_{1}=n_{2}=n$ \begin{table} \begin{tabular}{l l c c c c} \hline \hline & Scenario & LR & HS & Depth & BS \\ \hline \multirow{3}{}{MLP} & Linear & 0.1, 0.01, 0.001 & 5, 10, 20 & 1, 2 & 8, 16, 32 \\ & Nonlinear & 0.01, 0.001 & 50, 100 & 1, 2, 3 & 16, 32, 64 \\ & Classification & 0.001, 0.0001 & 50, 100 & 1, 3, 5 & 16, 32, 128 \\ \hline \hline \end{tabular} \end{table} Table 1: Design of $\Lambda$ for experiments with MLPs with $n$ gradually increasing from 50 to 50000, while the test set size $n_{0}=n/5$. The set of hyperparameters, denoted by $\Lambda$, consists of four types varying hyperparameters: LR (learning rate), HS (number of hidden nodes in each layer), depth (number of hidden layers), and BS (batch size), with the alternative ranges of these hyperparameters in different scenarios shown in Table 1. All other hyperparameters are fixed as follows. The maximum epoch was set to 50, the activation function was the Relu function, and the optimizer used was Adam. The Mean Squared Error (MSE) loss function is used for training regression models, while the commonly used Cross Entropy (CE) loss function is used for training classification models. Now we will present the specific designs and results for each scenario. #### 4.1.1 Linear Regression For the first scenario in our simulations, we consider a linear relationship between $X$ and $Y$. \[Y_{i}=X_{i}^{\top}\beta+\epsilon_{i}=\sum_{l=1}^{5}\beta_{l}X_{il}+\epsilon_{i },\quad i=1,...,n,\] where $\epsilon_{i}\sim N(0,\sigma^{2})$, $X_{i}=(X_{i1},...,X_{i5})^{\top}\sim N(\mathbf{0},\Omega)$, with the diagonal elements of $\Omega$ set to 1 and off-diagonal elements set to 0.5. We set the coefficient vector $\beta=(\beta_{1},...,\beta_{5})^{\top}$ as $\beta_{l}=1$ for $1\leq l\leq 3$ and $\beta_{l}=-1$ for $4\leq l\leq 5$. The variance of the noise, denoted by $\sigma^{2}$, is set to vary between $\sigma^{2}=1/3,1,3$, corresponding to $R^{2}=0.9,0.75,0.5$, respectively. The hyperparameters were set according to the 'Linear' row in Table 1. For each value of $n=50,...,50000$, we repeat the same simulation for 50 times. In each simulation replication, we train a number of neural networks with every hyperparameter choice $\lambda\in\Lambda$. After the training process is completed, we select the best hyperparameters by $\hat{\lambda}=\operatorname{argmin}_{\lambda\in\Lambda}\ L_{n}(\lambda)$, record $L_{0}(\lambda)$ for every $\lambda\in\Lambda$, and then calculate $\inf_{\lambda\in\Lambda}L_{0}(\lambda)$. Note that $E(Y\|X)=X^{\top}\beta$ when calculating $L_{0}(\lambda)$ here. To verify (9), we calculate the ratio $L_{0}(\hat{\lambda})/\inf_{\lambda\in\Lambda}L_{0}(\lambda)$ in all replications. Figure 1(a) presents the average value of this ratio across all replications to represent $E\{L_{0}(\hat{\lambda})/\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$. To verify (10), we separately calculate the average values of $L_{0}(\hat{\lambda})$ and $\inf_{\lambda\in\Lambda}L_{0}(\lambda)$ separately across all replications and then divide these two values to represent the ratio $E\{L_{0}(\hat{\lambda})\}/E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$. Figure 1(b) presents this ratio under different settings. The results presented in Figure 1 demonstrate that both $E\{L_{0}(\hat{\lambda})/\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$ and $E\{L_{0}(\hat{\lambda})\}/E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$ converge to 1 regardless of the choice of $\sigma^{2}$, manifesting the validity of Theorem 2 in this scenario. #### 4.1.2 Nonlinear Regression For the second simulation design, we consider a non linear regression problem of the form: \[Y_{i}=\frac{10\sin\left(\\|X_{i}\\|_{2}\right)}{\\|X_{i}\\|_{2}}+\epsilon_{i}, \quad i=1,...,n,\] where $\epsilon_{i}\sim N(0,\sigma^{2})$ and $X_{i}=(X_{i1},...,X_{i5})^{\top}$. For $l=1,...,5$, $X_{il}$ is independent of each other and follows a uniform distribution between $-10$ and $10$. An additional simulation was conducted to approximate the variance of $\sin\left(\|X_{i}\|_{2}\right)/\|X_{i}\|_{2}$. Accordingly, the choice of $\sigma$ are set to ensure $R^{2}=0.9,0.75,0.5$, respectively. The varying hyperparameters are set according to the 'Nonlinear' row in Table 1, and the training and analysis process is conducted in the same manner as that in our linear regression scenario. Note that $E(Y\|X)=10\sin{(\\|X\\|_{2})}/\\|X\\|_{2}$ when calculating $L_{0}(\lambda)$ here. Figure 2(a) and 2(b) show that $E\{L_{0}(\hat{\lambda})/\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$ and $E\{L_{0}(\hat{\lambda})\}/E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$ converge to 1 in nonlinear settings as well. #### 4.1.3 Classification For the third simulation design, we address a binary classification problem. Specifically, we assume that $Y_{i}$ takes values of 0 or 1, and the conditional probability of $Y_{i}$ being 1 given $X_{i}$, denoted as $\mathrm{P}(Y_{i}=1\|X_{i})$, is given by $g(X_{i})$: \[Y_{i}=0,1,\quad\mathrm{P}(Y_{i}=1\|X_{i})=g(X_{i}),\quad i=1,...,n.\] Here $g(\mathbf{x})=1/[1+\exp\{-(h(\mathbf{x})-\mu)^{2}\}]$, $\mathbf{x}=(x_{1},...,x_{10})^{\top}\in R^{10}$, where $h(\mathbf{x})$ is a nonlinear function of $\mathbf{x}$ defined as \[h(\mathbf{x}) =12x_{1}(x_{2}-0.5)^{2}-16\{x_{3}(x_{5}-0.2)\}^{4}\] \[+2\log\{3-4\left(x_{4}-0.3\right)^{2}+x_{5}+\exp(-x_{6}x_{7}+x_{5 })\}\] \[+2\tan[4\{(x_{1}(x_{8}-0.5)\}^{2}+0.1]\] with $x_{9}$ and $x_{10}$ being redundant. We generate the an sample input as $X_{i}=(X_{i1},...,X_{i10})^{\top}$, where $X_{il}$ are independent and uniformly distributed between 0 and 1 for $l=1,...,10$. We set $\mu$ to ensure that $E(h(X_{i}))=\mu$, and $\Lambda$'s setting is shown in 'Classification' row in Table 1. The activation function is Relu, and we use a sigmoid function before the output layer to limit the range of output to interval $(0,1)$. The loss function for training is cross-entropy (CE). However, for consistency with our theory, the determination of $\hat{\lambda}$ through the validation set and subsequent verification of the theorem are based on mean squared error (MSE) instead of CE. Specifically, $\hat{\lambda}$ is still chosen according to (2), i.e., by minimizing the MSE between real-valued $f_{\lambda}(X,\hat{w}_{n}(\lambda))$ and one-hot value $Y$. In terms of computing $L_{0}(\lambda)$, since we have $E(Y\|X)=P(Y=1\|X)=g(X)$, we calculate the square error of real-valued $f_{\lambda}(X,\hat{w}_{n}(\lambda))$ and real-valued $g(X)$. Although this procedure leads to inconsistency between the loss function for training and the performance measurement, we state that the theoretical results are not affected as the properties that form during the training process are related to Assumption 1 and will not affect subsequent reasoning. Therefore, the optimality of $\hat{\lambda}$ still holds, and the simulation results also demonstrate this statement. We record the data in replicate simulations to approximate $E\{L_{0}(\hat{\lambda})/\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$ and $E\{L_{0}(\hat{\lambda})\}/E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$. Figure 3 shows these ratios also converge to 1 in this scenario, which is consistent with Theorem 2. Note that our experimental analysis is focused exclusively on binary classification problems since our theoretical framework can not model the expectation of labels in a multi-class classification problem. As such, the experiments we conduct in the following sections regarding classification will be restricted to binary classification problems. ### Convolutional Neural Networks In this section, we carry out experiments involving convolutional neural networks (CNNs), a specialized form of feedforward neural networks that exhibit distinct connection structures compared to MLPs. Since CNNs are often employed when processing image-based inputs, the data generating process in previous simulations is not applicable to this context. Thus, we decide to utilize widely-used public datasets for image recognition tasks. However, we are faced with a problem that a mathematical formulation representing the conditional expectation $E(Y\|X)$ is unavailable in real-world datasets. This limitation prevents us from calculating the value of any $L_{0}(\lambda)$ in our theorem. To address this issue, we design a special method to generate new samples from the MNIST/Fashion-MNIST datasets as follows, ensuring that the theoretical value of $E(Y\|X)$ remains attainable: Define a binary-class classification problem for the MNIST/Fashion-MNIST dataset. * Change the labels of samples in the dataset to 1 and 0 to represent positive and negative samples of the binary-class classification problem, respectively. * Train a well-performed classification model $F:\mathbb{R}^{28\times 28}\rightarrow[0,1]$ with these samples. * Relabel the samples by using the trained model $F$. Specifically, for any $X_{i}$ from a sample in the dataset, we generate its new label $Y_{i}$ by \[\mathrm{P}(Y_{i}=1\|X_{i})=F(X_{i}).\] This method treats $F(X)$ as the true probability of input $X$ belonging to the positive class. For example, if an input $X$ satisfies $F(X)=0.75$, we generate a new sample $(X,Y)$ with $\mathrm{P}(Y=1\|X)=0.75$, and consequently, it is evident that $E(Y\|X)=0.75$. Through this procedure, we can transform the common classification task on a real-world dataset into a binary-classification problem, and possess true knowledge of $E(Y\|X)$ to verify our theorem. #### 4.2.1 Binary-Classification on Fashion-MNIST We first describe the design of the binary-classification problem on Fashion-MNIST. Specifically, clothing that covering the upper human body is set to be Class 0, which involves label sets [0,2,3,4,6], and the rest of labels are Class 1, including clothing such as thousands, shoes, and bags. Adopting the data generating process and training procedure introduced in Section 4.2, we create a new dataset derived from fashion-MNIST with labels converted to either 0 or 1 and train a well-performed model $F_{1}$ on this task. See Appendix C for details of $F_{1}$. We consider training a set of CNNs with a standard architecture consisting of two convolutional layers and two max pooling layers, followed by two fully connected layers that produce the final output. The choices of varying hyperparamters in $\Lambda$ are shown in 'Fashion-MNIST' row in Table 2, where Conv KS refers to kernel size of the squared convolutional layer, Chan Out refers to number of output channels by the convolution (number of convolution filters), Pool KS is kernel size of the squared pooling layer, and Pool Str is stride of pooling. Besides, learning rate, batch size, activation function and loss function are set to 0.0005, 16, Relu and CE respectively. The hidden size of the fully connected layer is set to 128. With $n$ taking values from 50 to 10000, we perform the same analysis as in the MLP classification problem described in Section 4.1.3. Results given in Figure 4 show Theorem 2 also works in this scenario. #### 4.2.2 Binary-Classification on MNIST We also consider a binary-Classification problem on the MNIST dataset, which aims at identifying whether a handwritten digit represents a number '1'. Following the procedure and the data generating process introduced in Section 4.2, we create a new dataset derived from MNIST for binary-classification and train a well-performed neural network model $F_{2}$ on it. See Appendix C for details of $F_{2}$. We consider training a set of CNNs that have the same standard architecture in Section 4.2.1. The fixed hyperparameters are also set as same as those in Section 4.2.1, and the varying hyperparamters in $\Lambda$ can be viewed in 'MNIST' Table 2. The results are shown in Figure 5. It is clear that the ratio converge to 1 in Figure 5(a). The convergence speed of the ratio shown in 5(b) is shower than in other scenarios, which might be attributed to simplicity of this problem. Specifically, we observed the performance of models under different hyperparamters and discovered that almost every network achieved satisfactory prediction results, which means a larger sample size is required to effectively distinguish the optimal hyperparameter. \begin{table} \begin{tabular}{l c c c c} \hline \hline Scenario & (Conv KS, Chan Out) & Pool KS & Pool Str \\ \hline Fashion-MNIST & (3,4),(3,16),(3,64),(4,4),(4,16),(4,64),(5,4),(5,16) & 2,3 & 1,2 \\ MNIST & (3,4),(3,16),(3,64),(4,16),(4,64) & 2,3 & 1,2 \\ \hline \hline \end{tabular} \end{table} Table 2: Design of $\Lambda$ for experiments with CNNs Figure 4: Results of CNNs for binary classification on Fashion-MNIST. (a) The estimated value of $E\{L_{0}(\hat{\lambda})/\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$. (b) The estimated value of $E\{L_{0}(\hat{\lambda})\}/E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}$. ### Recurrent Neural Networks In this section, we present an experiment to show that the optimality based on another neural network architecture, recurrent neural network (RNN), holds as well. Since RNN is often used to deal with temporal dynamic behavior of inputs, we designed a time series forecasting problem here. Two types of time series we consider are: 1. $y_{t}=0.6y_{t-1}+0.3y_{t-2}-0.1y_{t-3}+\epsilon_{t}$, 2. $y_{t}=0.3y_{t-1}+0.6y_{t-2}+(0.1-0.9y_{t-1}+0.8y_{t-2})\left[1+\exp\left(-10y_{t -1}\right)\right]^{-1}+\epsilon_{t}$, which are linear and nonlinear respectively, where $\epsilon_{t}$ is independent for $t\geq 0$ and follows $N(0,\sigma^{2})$ and the initial values of these time series follow uniform distribution between $-6$ to $6$. Our task here is to make one-step forward prediction for $y_{t}$ given the history values of $\{y_{i}\}_{i=1}^{t-1}$. To construct the samples for training from the origin time series, we apply the sliding window technique. A time series with length $T$ can generate $T-p$ training samples, where $p$ is the input length. Apart from the hyperparameters considered in the previous simulation, the input length is considered to be another hyperparameter, which is commonly required to be tuned in practice. See Table 3 for details of $\Lambda$, where $L$ represents the input length. Again we proceed with our simulations with different choice of sample size $n$, and deal with the results similarly, and we show them in Figure 6. Although the speed of convergence is slower than in those cases in feedforward neural networks, the two concerning ratios are indeed approaching $1$. Since the samples obtained by (11) and the subsequent sliding window method are not independent, in violation of our theory's i.i.d. assumption at the beginning of Section 2, this scenario is distinct from previous ones. Nonetheless, we still regard our theorem valid in the case that samples are correlated according to results in Figure 6. ## 5 Conclusion Our study offers a new theoretical explanation for the success of neural networks in practical applications. In order to construct an effective neural network model for any given task, optimization should be performed to obtain the most appropriate hyperparameters for the neural network, as they have a decisive impact on the performance of the final network. According to our proposed theories, it is through the use of sample splitting that we can acquire the optimal hyperparameters for the network, thus enabling the attainment of the optimal neural network from a family of neural network models. Through a series of experiments across various problem domains and data distributions, we verify our theoretical results on several prevalent neural network architectures. Our framework has contributed new theoretical insights into neural networks, but further research is needed to address the limitations in this paper and explore more widely applicable \begin{table} \begin{tabular}{l c c c c} \hline \hline & LR & HS & Depth & L \\ \hline RNN & 0.01, 0.001 & 50, 100 & 1, 2 & 3, 4, 5, 6 \\ \hline \hline \end{tabular} \end{table} Table 3: Design of $\Lambda$ for experiments with RNNs theories. In the process of establishing Theorem 2, our intuition suggested that the condition of the divergence of $n_{0}$ could be relaxed while proving (9) and (10), since both equations involve taking expectations. Nonetheless, we have encountered challenges with a relaxed condition and thus, leave it as a topic for future research. Inspired by our experiment on recurrent neural networks, extending our theory beyond an i.i.d. assumption will also be part of our future work. Finally, while the largest parameter space remained fixed in this article, another future extension would be to consider the size of the available set of hyperparameters diverging with the number of samples, which would also involve the diverging dimension of model parameter $w$. ## Appendix A Examples for Verifying Assumption 2(ii) In this section, we provide several examples to illustrate that Assumption 2(ii) holds under some primitive conditions. Specifically, we only assume that $E\\|X\\|^{2}=O(1)$. Before proceeding, we claim that \[\sup_{\lambda\in\Lambda}\sup_{w^{0}\in\mathcal{O}(w^{}(\lambda), \rho)}\\|w^{0}\\|\leq C_{w}, \tag{12}\] where $C_{w}$ is a fixed constant. In fact, since $\|\Lambda\|<\infty$, this is straightforward by letting \[C_{w}=\sup_{\lambda\in\Lambda}\\|w^{}(\lambda)\\|+\rho.\] Then, we present two examples as follows, whose conclusions can also be extended to more complicated architectures. ### Multilayer Perceptrons Consider a two-layer full-connected neural network corresponding to hyperparameter $\lambda$ with $k$ hidden nodes and input $x=(x_{1},...,x_{p})^{\top}\in\mathbb{R}^{p}$ \[f_{\lambda}(x,w)=\sum_{i=1}^{k(\lambda)}\alpha_{i}h_{i}(x)+ \alpha_{0},\] \[h_{i}(x)=\sigma(\sum_{j=1}^{p}\beta_{ij}x_{j}+b_{i}),\quad i=1,...,k,\] where $\sigma(\cdot)$ is the sigmoid activation function. Different choices of $\lambda$ correspond to varying values of $k$. Here, $w=(\alpha_{0},...,\alpha_{k},b_{1},...,b_{k},\beta_{11},...,\beta_{kp})^{\top}$. It can be easily shown that $\partial f/\partial\alpha_{0}=1$, $\partial f/\partial\alpha_{i}=h_{i}(x)$, $\partial f/\partial b_{i}=\alpha_{i}h_{i}(x)(1-h_{i}(x))$, and that $\partial f/\partial\beta_{ij}=\alpha_{i}x_{j}h_{i}(x)(1-h_{i}(x))$. This implies \[\left\\|\frac{\partial f_{\lambda}(X,w)}{\partial w}\right\\|^{2} \leq c_{1}+c_{2}(1+\\|X\\|^{2})\cdot\\|w\\|^{2}.\] where $c_{1}$ and $c_{2}$ are constants that can be chosen to be independent of $k$ due to $\|\Lambda\|\leq\infty$. Then, by (12) and $E\\|X\\|^{2}=O(1)$, we obtain \[E\sup_{w^{0}\in\mathcal{O}(w^{}(\lambda),\rho)}\left\\|\frac{ \partial f_{\lambda}(X,w)}{\partial w}\Big{\|}_{w=w^{0}}\right\\|\leq E\big{\{} \sqrt{c_{1}+c_{2}(1+\\|X\\|^{2})C_{w}^{2}}\big{\}}=O(1)\] This demonstrates that (6) in Assumption 2 holds for this example. ### Recurrent Neural Networks We consider a simple recurrent neural network (RNN) structure \[h^{(1)}=\sigma(Ux^{(1)}+b),\] \[h^{(t)} =\sigma(Wh^{(t-1)}+Ux^{(t)}+b),\quad t\geq 2\] \[o^{(t)} =\tanh(V\cdot h^{(t)}+b^{\prime}),\quad t\geq 2,\] where $x^{(t)}\in\mathbb{R}^{p}$ denote the input, with $p$ representing the size of the input, and $o^{(t)}\in\mathbb{R}$ denote output at time step $t$. The sigmoid and hyperbolic tangent activation functions are denoted by $\sigma(\cdot)$ and $\tanh(\cdot)$, respectively. Let $w=\operatorname{vec}(U,V,W,b,b^{\prime})$ denote the network parameters, where $U\in\mathbb{R}^{K\times p}$, $W\in\mathbb{R}^{K\times K}$, $V\in\mathbb{R}^{K}$, $b\in\mathbb{R}^{K}$, and $b^{\prime}\in\mathbb{R}$. Now consider a sequential input $x=(x^{(1)},...,x^{(T)})\in\mathbb{R}^{p\times T}$, and assume that the output of this network is given by $f_{\lambda}(w,x)=o^{(T)}$. The different choices of $\lambda$ can correspond to varying values of hyperparameters $T$, $p$ and $K$. To simplify the notation, we define the following terms: \[c^{(t)}=Wh^{(t-1)}+Ux^{(t)}+b,\quad z^{(t)}=Vh^{(t)}+b^{\prime},\] and we use subscripts to represent the entries of vector or matrix as follows: $U=(U_{ij})_{K\times p}$, $W=(W_{ij})_{K\times K}$, $V=(V_{i})_{K}$, $c^{(t)}=(c^{(t)}_{i})_{K}$, and $h^{(t)}=(h^{(t)}_{i})_{K}$. To verify Assumption 2(ii), we derive some upper bounds for the derivatives of $o^{(T)}$ with respect to $w$. By performing straightforward differentiation, we obtain \[\frac{\partial o^{(T)}}{\partial b^{\prime}} =\frac{\partial o^{(T)}}{\partial z^{(T)}},\] \[\frac{\partial o^{(T)}}{\partial V_{i}} =h^{(T)}_{i}\frac{\partial o^{(T)}}{\partial z^{(T)}},\quad 1\leq i \leq K.\] According to properties of sigmoid and tanh function, for any $t\geq 1$ and any $1\leq j,l\leq K$, we have \[\left\|h^{(t)}_{j}\right\|<1,\quad\left\|z^{(t)}\right\|<1,\] \[\left\|\frac{\partial h^{(t)}_{l}}{\partial c^{(t)}_{l}}\right\| =\left\|\sigma(c^{(t)}_{l})(1-c^{(t)}_{l})\right\|<1,\] \[\left\|\frac{\partial o^{(t)}}{\partial z^{(t)}}\right\| =\left\|1-\tanh^{2}(z^{(t)})\right\|<1, \tag{13}\] which implies that the derivatives of $o^{(T)}$ with respect to $b^{\prime}$ and $V$ are bounded by constants. For the remaining parameters, we claim that for any $t=2,...,T$, there exists constants $c^{(t)}$ and polynomials $C^{(t)}=C^{(t)}(\\|w\\|)$ respect to $\\|w\\|$ with at most degree $t$ and non-negative coefficients, such that \[\left\|\frac{\partial h^{(t)}_{l}}{\partial b_{i}}\right\|\leq C^{ (t)},\quad 1\leq i,l\leq K\] \[\left\|\frac{\partial h^{(t)}_{l}}{\partial W_{ij}}\right\|\leq C^{ (t)},\quad 1\leq i,j,l\leq K\] \[\left\|\frac{\partial h^{(t)}_{l}}{\partial U_{ij}}\right\|\leq\\|x \\|\cdot C^{(t)},\quad 1\leq i,l\leq K,\ 1\leq j\leq p \tag{14}\] To illustrate the claim (14) by mathematical induction, we first consider the case when $t=2$. By applying the chain rule, we have \[\frac{\partial h_{l}^{(2)}}{\partial b_{i}} =\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\frac{ \partial c_{l}^{(2)}}{\partial b_{i}}\] \[=\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\left\{ \delta_{il}+W_{li}\frac{\partial h_{i}^{(1)}}{\partial c_{i}^{(1)}}\right\} \tag{15}\] \[\frac{\partial h_{l}^{(2)}}{\partial W_{ij}} =\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\frac{ \partial c_{l}^{(2)}}{\partial W_{ij}}\] \[=\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\frac{ \partial(\sum_{m=1}^{K}W_{lm}h_{m}^{(1)})}{\partial W_{ij}}=h_{j}^{(1)}\frac{ \partial h_{l}^{(2)}}{\partial c_{l}^{(2)}},\] (16) \[\frac{\partial h_{l}^{(2)}}{\partial U_{ij}} =\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\frac{ \partial c_{l}^{(2)}}{\partial U_{ij}}\] \[=\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\left\{ \frac{\partial\sum_{m=1}^{K}W_{lm}\frac{\partial h_{m}^{(1)}}{\partial U_{ij} }+\frac{\partial\sum_{m=1}^{K}U_{lm}x_{m}^{(2)}}{\partial U_{ij}}\right\}\] \[=\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\bigg{\{} \sum_{m=1}^{K}W_{lm}\frac{\partial h_{m}^{(1)}}{\partial U_{ij}}+\delta_{il}x_ {j}^{(2)}\bigg{\}}\] \[=\frac{\partial h_{l}^{(2)}}{\partial c_{l}^{(2)}}\cdot\bigg{\{} W_{li}x_{j}^{(1)}\frac{\partial h_{i}^{(1)}}{\partial c_{i}^{(1)}}+\delta_{il}x_ {j}^{(2)}\bigg{\}}, \tag{17}\] Then, by (13), (15), (16) and (17), we can obtain \[\left\|\frac{\partial h_{l}^{(2)}}{\partial b_{i}}\right\| \leq 1+\\|W\\|,\] \[\left\|\frac{\partial h_{l}^{(2)}}{\partial W_{ij}}\right\| <1,\] \[\left\|\frac{\partial h_{l}^{(2)}}{\partial U_{ij}}\right\| \leq(1+\\|W\\|)\\|x\\|, \tag{18}\] Therefore, by (18) and $\\|V\\|,\\|W\\|\leq\\|w\\|$, our claim (14) holds for the case $t=2$. Now, we assume that (14) holds for case $t-1$, and consider the case at $t$. By the chain rule, we have \[\frac{\partial h_{l}^{(t)}}{\partial b_{i}} =\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\cdot\frac{ \partial c_{l}^{(t)}}{\partial b_{i}}\] \[=\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\cdot\bigg{\{} \delta_{il}+\sum_{m=1}^{K}W_{lm}\frac{\partial h_{m}^{(t-1)}}{\partial b_{i}} \bigg{\}},\] \[\frac{\partial h_{l}^{(t)}}{\partial W_{ij}} =\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\cdot\frac{ \partial c_{l}^{(t)}}{\partial W_{ij}}\] \[=\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\cdot\frac{ \partial(\sum_{m=1}^{K}W_{lm}h_{m}^{(t-1)})}{\partial W_{ij}}\] \[=\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\bigg{\{}\sum_{m =1}^{K}W_{lm}\frac{\partial h^{(t-1)}}{\partial W_{ij}}+\delta_{il}h_{j}^{(t-1 )}\bigg{\}},\] \[\frac{\partial h_{l}^{(t)}}{\partial U_{ij}} =\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\cdot\frac{ \partial c_{l}^{(t)}}{\partial U_{ij}}\] \[=\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\cdot\bigg{\{} \frac{\partial\sum_{m=1}^{K}W_{lm}h_{m}^{(t-1)}}{\partial U_{ij}}+\frac{ \partial\sum_{m=1}^{K}U_{lm}x_{m}^{(t)}}{\partial U_{ij}}\bigg{\}}\] \[=\frac{\partial h_{l}^{(t)}}{\partial c_{l}^{(t)}}\cdot\bigg{\{} \sum_{m=1}^{K}W_{lm}\frac{\partial h_{m}^{(t-1)}}{\partial U_{ij}}+\delta_{il }x_{j}^{(t)}\bigg{\}}\] which implies \[\bigg{\|}\frac{\partial h_{l}^{(t)}}{\partial b_{i}}\bigg{\|}\leq 1 +K\\|W\\|\max_{m\leq K}\bigg{\|}\frac{\partial h_{l}^{(t-1)}}{\partial b_{i}} \bigg{\|}\] \[\bigg{\|}\frac{\partial h_{l}^{(t)}}{\partial W_{ij}}\bigg{\|}\leq 1 +K\\|W\\|\max_{m\leq K}\bigg{\|}\frac{\partial h_{l}^{(t-1)}}{\partial W_{ij}} \bigg{\|},\] \[\bigg{\|}\frac{\partial h_{l}^{(t)}}{\partial U_{ij}}\bigg{\|}\leq \\|x\\|+K\\|W\\|\max_{m\leq K}\bigg{\|}\frac{\partial h_{l}^{(t-1)}}{\partial U_{ ij}}\bigg{\|}. \tag{19}\] By using (19) and the induction hypothesis, we can construct an appropriate $C^{(t)}$ for (14) for the case $t$, thereby completing the induction. Now, we proceed to upper bound our objectives. By applying the chain rule, we have \[\frac{\partial o^{(t)}}{\partial b_{i}} =\frac{\partial o^{(t)}}{\partial z^{(t)}}\sum_{l=1}^{K}V_{l} \cdot\frac{\partial h_{l}^{(t)}}{\partial b_{i}},\] \[\frac{\partial o^{(t)}}{\partial U_{ij}} =\frac{\partial o^{(t)}}{\partial z^{(t)}}\sum_{l=1}^{K}V_{l} \cdot\frac{\partial h_{l}^{(t)}}{\partial U_{ij}},\] \[\frac{\partial o^{(t)}}{\partial W_{ij}} =\frac{\partial o^{(t)}}{\partial z^{(t)}}\sum_{l=1}^{K}V_{l} \cdot\frac{\partial h_{l}^{(t)}}{\partial W_{ij}}. \tag{20}\] Then, it follows from (13), (19), (20) and Cauchy-Schwarz inequality that \[\bigg{\|}\frac{\partial o^{(T)}}{\partial b_{i}}\bigg{\|} \leq\sqrt{K}\\|V\\|\bigg{\|}\frac{\partial h_{l}^{(t)}}{\partial b_{ i}}\bigg{\|},\] \[\bigg{\|}\frac{\partial o^{(T)}}{\partial W_{ij}}\bigg{\|} \leq\sqrt{K}\\|V\\|\bigg{\|}\frac{\partial h_{l}^{(t)}}{\partial W_{ ij}}\bigg{\|},\] \[\bigg{\|}\frac{\partial o^{(T)}}{\partial U_{ij}}\bigg{\|} \leq\sqrt{K}\\|V\\|\bigg{\|}\frac{\partial h_{l}^{(t)}}{\partial U_{ ij}}\bigg{\|}. \tag{21}\] Incorporating (14) and (21), we obtain that \[\left\\|\frac{\partial f_{\lambda}(x,w)}{\partial w}\right\\|=\left\\| \frac{\partial o^{(T)}}{\partial w}\right\\| \leq\sqrt{c_{1}+c_{2}\{C^{(T)}(\\|w\\|)\}^{2}+c_{3}\{C^{(T)}(\\|w\\|) \}^{2}\\|x\\|^{2}}\] \[\leq c_{1}+c_{2}\{C^{(T)}(\\|w\\|)\}^{2}+c_{3}\{C^{(T)}(\\|w\\|)\}^{2} \\|x\\|^{2} \tag{22}\] where $c_{1}$ ($c_{1}>1$), $c_{2}$ and $c_{3}$ are constants that, according to $\|\Lambda\|\leq\infty$, can be selected independently of any $K$, $p$, and $T$. Finally, since $C^{(T)}$ is monotonically increasing with respect to $\\|w\\|$, it follows from (12) and (22)that \[E\sup_{w^{0}\in\mathcal{O}(w^{\star}(\lambda),\rho)}\left\\| \frac{\partial f_{\lambda}(X,w)}{\partial w}\right\\|_{w=w^{0}}\right\\| \leq E\Big{\{}c_{1}+c_{2}\{C^{(T)}(C_{w})\}^{2}+c_{3}\{C^{(T)}(C_ {w})\}^{2}\\|X\\|^{2}\Big{\}}\] \[=O(1),\] where the equality follows from our assumption $E\\|X\\|^{2}=O(1)$. This verify Assumption 2(ii) in this example. ## Appendix B Proof of Theoretical Results ### Supporting Lemma(s) We introduce the following lemma that will be used in our proof. Lemma 3: _Let $\Lambda$ be the available set of hyperparameters, which can have finite or diverging elements. For any $\lambda\in\Lambda$, $A_{n}(\lambda)$, $a_{n}(\lambda)$, and $B_{n}(\lambda)$ are sequences related to $\lambda$. Let $\eta_{n}$ be a real-valued positive sequence that satisfies $\eta_{n}=o(1)$. Define $\hat{\lambda}=\operatorname{argmin}_{\lambda\in\Lambda}\{A_{n}(\lambda)+a_{ n}(\lambda)\}$. If_ \[\sup_{\lambda\in\Lambda}\left\|\frac{a_{n}(\lambda)}{B_{n}(\lambda)}\right\|=O_ {p}(\eta_{n}), \tag{23}\] \[\sup_{\lambda\in\Lambda}\left\|\frac{A_{n}(\lambda)-B_{n}(\lambda)}{B_{n}( \lambda)}\right\|=O_{p}(\eta_{n}), \tag{24}\] _and there exists a interger $N$ and a constant $\kappa$ such that when $n\geq N$,_ \[\inf_{\lambda\in\Lambda}B_{n}(\lambda)\geq\kappa>0, \tag{25}\] _then_ \[\frac{A_{n}(\hat{\lambda})}{\inf_{\lambda\in\Lambda}A_{n}(\lambda)}=1+O_{p}( \eta_{n})=1+o_{p}(1). \tag{26}\] Proof By the definition of infimum, there exist a sequence $\lambda(n)\in\Lambda$ and a non-negative sequence $\vartheta_{n}(\lambda(n))=o_{p}(1)$ such that \[\inf_{\lambda\in\Lambda}A_{n}(\lambda)=A_{n}(\lambda(n))-\vartheta_{n}( \lambda(n)). \tag{27}\] In addition, it follows from (24) that \[\inf_{\lambda\in\Lambda}\frac{A_{n}(\lambda)}{B_{n}(\lambda)} =\inf_{\lambda\in\Lambda}\left(\frac{A_{n}(\lambda)}{B_{n}(\lambda) }-1\right)+1\] \[\geq-\sup_{\lambda\in\Lambda}\left\|\frac{A_{n}(\lambda)}{B_{n}( \lambda)}-1\right\|+1=1-O_{p}(\eta_{n}). \tag{28}\] By (25) and (28), we have \[\inf_{\lambda\in\Lambda}\frac{\|A_{n}(\lambda)-\vartheta_{n}( \lambda(n))\|}{B_{n}(\lambda)} \geq\inf_{\lambda\in\Lambda}\frac{A_{n}(\lambda)-\vartheta_{n}( \lambda(n))}{B_{n}(\lambda)}\geq\inf_{\lambda\in\Lambda}\frac{A_{n}(\lambda)}{ B_{n}(\lambda)}-\frac{\vartheta_{n}(\lambda(n))}{\inf_{\lambda\in\Lambda}B_{n}( \lambda)}\] \[\geq-\sup_{\lambda\in\Lambda}\left\|\frac{A_{n}(\lambda)}{B_{n}( \lambda)}-1\right\|+1-\frac{\vartheta_{n}(\lambda(n))}{\inf_{\lambda\in \Lambda}B_{n}(\lambda)}\] \[\geq 1-O_{p}(\eta_{n})-\frac{\vartheta_{n}(\lambda(n))}{\kappa}. \tag{29}\] Then, we have that \[\left\|\frac{\inf_{\lambda\in\Lambda}A_{n}(\lambda)}{A_{n}(\hat{ \lambda})}-1\right\| =\frac{A_{n}(\hat{\lambda})-\inf_{\lambda\in\Lambda}A_{n}(\lambda) }{A_{n}(\hat{\lambda})}\] \[=\frac{\inf_{\lambda\in\Lambda}(A_{n}(\lambda)+a_{n}(\lambda))-a_ {n}(\hat{\lambda})-\inf_{\lambda\in\Lambda}A_{n}(\lambda)}{A_{n}(\hat{\lambda })}\] \[\leq\frac{A_{n}(\lambda(n))+a_{n}(\lambda(n))-a_{n}(\hat{\lambda}) -A_{n}(\lambda(n))+\vartheta_{n}}{A_{n}(\hat{\lambda})}\] \[\leq\frac{\|a_{n}(\lambda(n))\|}{A_{n}(\hat{\lambda})}+\frac{\|a_{n}( \hat{\lambda})\|}{A_{n}(\hat{\lambda})}+\frac{\vartheta_{n}}{A_{n}(\hat{\lambda })}\] \[\leq\frac{\|a_{n}(\lambda(n))\|}{\inf_{\lambda\in\Lambda}A_{n}( \lambda)}+\frac{\|a_{n}(\hat{\lambda})\|}{A_{n}(\hat{\lambda})}+\frac{\vartheta _{n}}{A_{n}(\hat{\lambda})}\] \[\leq\frac{\|a_{n}(\lambda(n))\|}{A_{n}(\lambda(n))-\vartheta_{n}}+ \frac{\|a_{n}(\hat{\lambda})\|}{A_{n}(\hat{\lambda})}+\frac{\vartheta_{n}}{A_{n} (\hat{\lambda})}\] \[\leq\sup_{\lambda\in\Lambda}\frac{\|a_{n}(\lambda)\|}{B_{n}( \lambda)}\sup_{\lambda\in\Lambda}\frac{B_{n}(\lambda)}{\|A_{n}(\lambda)- \vartheta_{n}\|}+\sup_{\lambda\in\Lambda}\frac{\|a_{n}(\lambda)\|}{B_{n}(\lambda )}\sup_{\lambda\in\Lambda}\frac{B_{n}(\lambda)}{A_{n}(\lambda)}\] \[\qquad\qquad+\sup_{\lambda\in\Lambda}\frac{\vartheta_{n}}{B_{n}( \lambda)}\sup_{\lambda\in\Lambda}\frac{B_{n}(\lambda)}{A_{n}(\lambda)}\] \[\leq\sup_{\lambda\in\Lambda}\frac{\|a_{n}(\lambda)\|}{B_{n}( \lambda)}\bigg{[}\inf_{\lambda\in\Lambda}\frac{\|A_{n}(\lambda)-\vartheta_{n}\|} {B_{n}(\lambda)}\bigg{]}^{-1}+\sup_{\lambda\in\Lambda}\frac{\|a_{n}(\lambda)\|}{ B_{n}(\lambda)}\bigg{[}\inf_{\lambda\in\Lambda}\frac{A_{n}(\lambda)}{B_{n}( \lambda)}\bigg{]}^{-1}\] \[\qquad\qquad+\frac{\vartheta_{n}}{\inf_{\lambda\in\Lambda}B_{n}( \lambda)}\bigg{[}\inf_{\lambda\in\Lambda}\frac{A_{n}(\lambda)}{B_{n}(\lambda)} \bigg{]}^{-1}, \tag{30}\] where the first equality follows from the definition of $\hat{\lambda}$, the first and third inequality follows from (27). By (23), (28), (29), and $\vartheta_{n}(\lambda(n))=o_{p}(1)$, we obtain that \[\frac{\inf_{\lambda\in\Lambda}A_{n}(\lambda)}{A_{n}(\hat{\lambda})}\leq\frac{O _{p}(\eta_{n})}{1-O_{p}(\eta_{n})}+\frac{O_{p}(\eta_{n})}{1-O_{p}(\eta_{n})-o_{p }(1)/\kappa}+\frac{o_{p}(1)/\kappa}{1-O_{p}(\eta_{n})}. \tag{31}\] Since $\eta_{n}=o(1)$, we obtain $\inf_{\lambda\in\Lambda}A_{n}(\lambda)/A_{n}(\hat{\lambda})=1+O_{p}(\eta_{n})$ based on straightforward derivation and the definition of $O_{p}$, which also implies $A_{n}(\hat{\lambda})/\inf_{\lambda\in\Lambda}A_{n}(\lambda)=1+O_{p}(\eta_{n})$. Therefore, we complete the proof of Lemma 3. ### Proof of Theorem 1 Proof Define $L_{n}^{}(\lambda)=L_{n}(\lambda)-\frac{1}{n}\sum_{i=1}^{n}\{Y_{i}-E(Y_{i}\|X_{ i})\}\{Y_{i}+E(Y_{i}\|X_{i})\}$. We have \[\hat{\lambda} =\underset{\lambda\in\Lambda}{\operatorname{argmin}}L_{n}( \lambda)=\underset{\lambda\in\Lambda}{\operatorname{argmin}}L_{n}^{}(\lambda)\] \[=\underset{\lambda\in\Lambda}{\operatorname{argmin}}\{R_{0}( \lambda)+L_{n}^{}(\lambda)-R_{0}(\lambda)\}.\] By applying Lemma 3 with $A_{n}(\lambda)$, $B_{n}(\lambda)$, $a_{n}(\lambda)$, and $\eta_{n}$ being $R_{0}(\lambda)$, $R_{0}^{}(\lambda)$, $L_{n}^{}(\lambda)-R_{0}(\lambda)$, and $\xi_{n}^{-1}n^{-1/2}$, respectively, it suffices to prove \[\sup_{\lambda\in\Lambda}\frac{\|R_{0}(\lambda)-R_{0}^{}(\lambda)\|}{\|R_{0}^{}( \lambda)\|}=O(\xi_{n}^{-1}n^{-1/2})=o(1) \tag{32}\] and \[\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-R_{0}(\lambda)\|}{\|R_{0}^{}( \lambda)\|}=O_{p}(\xi_{n}^{-1}n^{-1/2})=o_{p}(1), \tag{33}\] to establish Theorem 1. Observe that if (32) holds, the following equation \[\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-R_{0}^{}(\lambda)\|}{\|R_{0}^ {}(\lambda)\|}=O_{p}(\xi_{n}^{-1}n^{-1/2})=o_{p}(1). \tag{34}\] is sufficient to prove (33). Therefore, we aim to prove (32) and (34). By Assumption 1 and Assumption 2(ii), we obtain \[\big{\|}f_{\lambda}(X_{0},\hat{w}_{n}(\lambda))-f_{\lambda}(X_{0}, w^{}(\lambda))\big{\|} =\bigg{\|}(\hat{w}_{n}(\lambda)-w^{}(\lambda))^{\top}\frac{\partial f _{\lambda}(X_{0},w)}{\partial w}\Big{\|}_{w=w^{0}}\bigg{\|}\] \[\leq\\|\hat{w}_{n}(\lambda)-w^{}(\lambda)\\|\times\bigg{\\|}\frac{ \partial f_{\lambda}(X_{0},w)}{\partial w}\|_{w=w^{0}}\bigg{\\|}\] \[=O_{p}(n^{-1/2})\times O_{p}(1)=O_{p}(n^{-1/2}). \tag{35}\] Therefore, \[\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\big{\|}\{f_{\lambda}(X_{0}, \hat{w}_{n}(\lambda))-E(Y_{0}\|X_{0})\}^{2}-\{f_{\lambda}(X_{0},w^{}(\lambda)) -E(Y_{0}\|X_{0})\}^{2}\big{\|}\] \[=\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\bigg{\|}\{f_{\lambda}(X_{0}, \hat{w}_{n}(\lambda))-f_{\lambda}(X_{0},w^{}(\lambda))\}\Big{\|}\{f_{\lambda} (X_{0},\hat{w}_{n}(\lambda))-f_{\lambda}(X_{0},w^{}(\lambda))\}\] \[\qquad\qquad+2\{f_{\lambda}(X_{0},w^{}(\lambda))-E(Y_{0}\|X_{0}) \}\Big{\|}\bigg{\|}\] \[\leq\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\Big{\|}\\|\hat{w}_{n}( \lambda)-w^{}(\lambda)\\|\times\big{\\|}\frac{\partial f_{\lambda}(X_{0},w)}{ \partial w}\|_{w=w^{0}}\big{\\|}\] \[\times\Big{[}\\|\hat{w}_{n}(\lambda)-w^{}(\lambda)\\|\times\big{\\|} \frac{\partial f_{\lambda}(X_{0},w)}{\partial w}\|_{w=w^{0}}\big{\\|}+2\{f_{ \lambda}(X_{0},w^{}(\lambda))-E(Y_{0}\|X_{0})\}\Big{]}\bigg{\|}\] \[=\xi_{n}^{-1}\times[O_{p}(n^{-1/2})\times\{O_{p}(n^{-1/2})+O_{p}( 1)\}]\] \[=\xi_{n}^{-1}O_{p}(n^{-1/2})=o_{p}(1), \tag{36}\] where the second inequality follows from Mean value theorem and the third equality follows from Assumption 2(i) and (35). Consequently, we have \[\sup_{\lambda\in\Lambda}\frac{\|R_{0}(\lambda)-R_{0}^{}(\lambda)\|} {\|R_{0}^{}(\lambda)\|}\] \[\leq\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\|R_{0}(\lambda)-R_{0}^{* }(\lambda)\|\] \[=\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\Big{\|}E\big{[}\{f_{ \lambda}(X_{0},\hat{w}_{n}(\lambda))-E(Y_{0}\|X_{0})\}^{2}-\{f_{\lambda}(X_{0 },w^{}(\lambda))-E(Y_{0}\|X_{0})\}^{2}\big{]}\Big{\|}\] \[\leq E\Big{[}\xi_{n}^{-1}\cdot\sup_{\lambda\in\Lambda}\big{\|}\{f_{ \lambda}(X_{0},\hat{w}_{n}(\lambda))-E(Y_{0}\|X_{0})\}^{2}-\{f_{\lambda}(X_{0 },w^{}(\lambda))-E(Y_{0}\|X_{0})\}^{2}\big{\|}\Big{]}\] \[= O(\xi_{n}^{-1}n^{-1/2})=o(1),\] where the third inequality follows from Assumption 3 and Jensen inequality and the forth equality follows from (36). This complete the proof of (32). Now we present the proof of (34). Firstly, we have \[\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-R_{0}^{}(\lambda)\|}{\|R_{0} ^{}(\lambda)\|}\leq\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\|L_{n}^{}(\lambda)-R_{ 0}^{}(\lambda)\|. \tag{37}\] Then, we examine the supremum on the right hand side of (37): \[\|L_{n}^{}(\lambda)-R_{0}^{}(\lambda)\|\] \[=\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}\big{[}\{f_{\lambda}(X_{i}, \hat{w}_{n}(\lambda))-Y_{i}\}^{2}-\{Y_{i}-E(Y_{i}\|X_{i})\}\{Y_{i}+E(Y_{i}\|X_{i })\}\big{]}-R_{0}^{}(\lambda)\bigg{\|}\] \[\leq\underbrace{\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}\big{[}\{f_{ \lambda}(X_{i},w^{}(\lambda))-Y_{i}\}^{2}-\{Y_{i}-E(Y_{i}\|X_{i})\}\{Y_{i}+E(Y _{i}\|X_{i})\}\big{]}-R_{0}^{}(\lambda)\bigg{\|}}_{(I)}\] \[\qquad\qquad\qquad+\underbrace{\bigg{\|}\frac{1}{n}\sum_{i=1}^{n} \big{[}\{f_{\lambda}(X_{i},\hat{w}_{n}(\lambda))-Y_{i}\}^{2}-\{f_{\lambda}(X_ {i},w^{}(\lambda))-Y_{i}\}^{2}\big{]}\bigg{\|}}_{(II)}, \tag{38}\] and we address $(I)$ and $(II)$ in (38) separately. Regarding $(I)$, we have \[(I)=\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}\big{[}\{f_{\lambda}(X_{i}, w^{}(\lambda))-E(Y_{i}\|X_{i})+E(Y_{i}\|X_{i})-Y_{i}\}^{2}\] \[\qquad\qquad\qquad-\{Y_{i}-E(Y_{i}\|X_{i})\}\{Y_{i}+E(Y_{i}\|X_{i}) \}\big{]}-R_{0}^{}(\lambda)\bigg{\|}\] \[=\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}\big{[}\{f_{\lambda}(X_{i},w^{}( \lambda))-E(Y_{i}\|X_{i})\}^{2}+2f_{\lambda}(X_{i},w^{}(\lambda))\{E(Y_{i}\|X_{i} )-Y_{i}\}\big{]}-R_{0}^{}(\lambda)\bigg{\|}\] \[\leq\frac{1}{n}\Big{\|}\sum_{i=1}^{n}\{f_{\lambda}(X_{i},w^{}( \lambda))-E(Y_{i}\|X_{i})\}^{2}-nR_{0}^{}(\lambda)\Big{\|}+\frac{2}{n}\Big{\|} \sum_{i=1}^{n}f_{\lambda}(X_{i},w^{}(\lambda))\{Y_{i}-E(Y_{i}\|X_{i})\}\Big{\|}. \tag{39}\] Define \[S_{n}^{\prime}(\lambda)=\sum_{i=1}^{n}\{f_{\lambda}(X_{i},w^{}(\lambda))-E(Y _{i}\|X_{i})\}^{2}\] and \[S_{n}^{\prime\prime}(\lambda)=\sum_{i=1}^{n}f_{\lambda}(X_{i},w^{}(\lambda)) \{Y_{i}-E(Y_{i}\|X_{i})\}.\] Since the samples are i.i.d., it holds that \[E(S_{n}^{\prime}(\lambda))=\sum_{i=1}^{n}E(f_{\lambda}(X_{i},w^{}(\lambda))- E(Y_{i}\|X_{i}))^{2}=nR_{0}^{}(\lambda). \tag{40}\] It follows from (39) and (40) that \[(I)\leq\frac{1}{n}\big{\|}S_{n}^{\prime}(\lambda)-E(S_{n}^{\prime}(\lambda)) \big{\|}+\frac{2}{n}\|S_{n}^{\prime\prime}(\lambda)\|. \tag{41}\] Regarding first term on the right hand side of (41), for any $n$, $\lambda^{\prime}\in\Lambda$, and $\delta>0$, we have \[\Pr\Big{\{}\Big{\|}n^{-1}\{S_{n}^{\prime}(\lambda^{\prime})-E(S_{ n}^{\prime}(\lambda^{\prime}))\}\Big{\|}>\delta\Big{\}} =\Pr\big{\{}\|S_{n}^{\prime}(\lambda^{\prime})-E(S_{n}^{\prime}( \lambda^{\prime}))\|>n\delta\big{\}}\] \[\leq\ n^{-2}\delta^{-2}\operatorname{var}(S_{n}^{\prime}(\lambda^ {\prime})) \tag{42}\] Then, for any $\delta^{}>0$ and any $n$, we set $\delta=\delta_{n}=n^{-1}(\delta^{})^{-1}\{\operatorname{var}(S_{n}^{\prime}( \lambda^{\prime}))\}^{1/2}$, plug it into (42), and find it holds for any $n$ that \[\Pr\Big{\{}\big{\|}n^{-1}\{S_{n}^{\prime}(\lambda^{\prime})-E(S_{ n}^{\prime}(\lambda^{\prime}))\}\big{\|}>n^{-1}(\delta^{})^{-1/2}\{ \operatorname{var}(S_{n}^{\prime}(\lambda^{\prime}))\}^{1/2}\Big{\}}\leq\delta ^{}, \tag{43}\] which implies that \[n^{-1}\|S_{n}^{\prime}(\lambda^{\prime})-E(S_{n}^{\prime}(\lambda^ {\prime}))\| =O_{p}(n^{-1}\{\operatorname{var}(S_{n}^{\prime}(\lambda^{\prime}) )\}^{1/2})\] \[=O_{p}\Big{(}n^{-1/2}\big{\{}\operatorname{var}[\{f_{\lambda^{ \prime}}(X_{1},w^{}(\lambda^{\prime}))-E(Y_{1}\|X_{1})\}^{2}]\big{\}}^{1/2} \Big{)}\] \[=O_{p}(n^{-1/2})\] for any $\lambda^{\prime}\in\Lambda$. Given that $\Lambda$ is a finite set, we obtain \[\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\Big{\|}\frac{1}{n}\{S_{n}^{\prime}(\lambda )-E(S_{n}^{\prime}(\lambda))\}\Big{\|}=O_{p}(\xi_{n}^{-1}n^{-1/2}). \tag{44}\] For second term on the right hand side of (41), due to the property of conditional expectation, we have \[E(S_{n}^{\prime\prime}(\lambda)) =\sum_{i=1}^{n}E\Big{\{}E\big{[}f_{\lambda}(X_{i},w^{}(\lambda)) \{Y_{i}-E(Y_{i}\|X_{i})\}\|X_{i}\|\Big{\}}\] \[=\sum_{i=1}^{n}E\Big{\{}f_{\lambda}(X_{i},w^{}(\lambda))E\{Y_{i}- E(Y_{i}\|X_{i})\|X_{i}\}\Big{\}}=0.\] For any $\lambda^{\prime\prime}\in\Lambda$ and $\delta>0$, employing Chebyshev inequality, we have \[\Pr\Big{\{}\big{\|}2n^{-1}S_{n}^{\prime\prime}(\lambda^{\prime \prime})\big{\|}>\delta\Big{\}} =\Pr\big{\{}\|S_{n}^{\prime\prime}(\lambda^{\prime\prime})\|>n \delta/2\big{\}}\] \[\leq 4n^{-2}\delta^{-2}\operatorname{var}(S_{n}^{\prime\prime}( \lambda^{\prime\prime})). \tag{45}\] Similar to (43), this implies \[n^{-1}S_{n}^{\prime\prime}(\lambda^{\prime\prime}) =O_{p}(n^{-1}\{\operatorname{var}(S_{n}^{\prime\prime}(\lambda^{ \prime\prime}))\}^{1/2})\] \[=O_{p}(n^{-1/2}\{\operatorname{var}\big{(}f_{\lambda}(X_{1},w^{* }(\lambda^{\prime\prime}))\{Y_{1}-E(Y_{1}\|X_{1})\}\big{)}\}^{1/2})\] \[=O_{p}(n^{-1/2})\] and \[\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\Big{\|}n^{-1}S_{n}^{\prime \prime}(\lambda)\Big{\|}=O_{p}(\xi^{-1}n^{-1/2}) \tag{46}\] By combining (41), (44), and (46), we achieve that \[\xi_{n}^{-1}\sup_{\lambda\in\Lambda}(I)\leq\xi_{n}^{-1}\sup_{ \lambda\in\Lambda}\Big{\|}n^{-1}\{S_{n}^{\prime}(\lambda)-E(S_{n}^{\prime}( \lambda))\}\Big{\|}+\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\Big{\|}2n^{-1}S_{n}^{ \prime\prime}(\lambda)\Big{\|}=O_{p}(\xi^{-1}n^{-1/2}). \tag{47}\] Now we proceed to handle $(II)$. Apply a technique similar to that in (36), we have \[\sup_{\lambda\in\Lambda}(II) =\sup_{\lambda\in\Lambda}\frac{1}{n}\Big{\|}\sum_{i=1}^{n}\big{[} \{f_{\lambda}(X_{i},\hat{w}_{n}(\lambda))-Y_{i}\}^{2}-\{f_{\lambda}(X_{i},w^{* }(\lambda))-Y_{i}\}^{2}\big{]}\Big{\|}\] \[=\sup_{\lambda\in\Lambda}\frac{1}{n}\Big{\|}\sum_{i=1}^{n}\big{\{} f_{\lambda}(X_{i},\hat{w}_{n}(\lambda))-f_{\lambda}(X_{i},w^{}(\lambda))\big{\}}\] \[\qquad\qquad\qquad\times\Big{[}\big{\{}f_{\lambda}(X_{i},\hat{w }_{n}(\lambda))-f_{\lambda}(X_{i},w^{}(\lambda))\big{\}}+2\{f_{\lambda}(X_{i },w^{}(\lambda))-Y_{i}\}\Big{]}\Big{\|}\] \[=O_{p}(n^{-1})+O_{p}(n^{-1/2})=O_{p}(n^{-1/2}),\] where the second equality follows from (35) and the last equality follows from Assumption 2(i). Then, by Asssumption 4, we obtain that \[\xi_{n}^{-1}\sup_{\lambda\in\Lambda}(II)=O_{p}(\xi_{n}^{-1}n^{-1/2})=o_{p}(1). \tag{48}\] Finally, by combining (37), (38), (47), and (48), we have \[\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-R_{0}^{}(\lambda) \|}{\|R_{0}^{}(\lambda)\|} \leq\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\{(I)+(II)\}\] \[\leq\xi_{n}^{-1}\sup_{\lambda\in\Lambda}(I)+\xi_{n}^{-1}\sup_{ \lambda\in\Lambda}(II)\] \[=O_{p}(\xi_{n}^{-1}n^{-1/2})=o_{p}(1),\] which demonstrates that (34) holds. Therefore, we complete the proof of Theorem 1. ### Proof of Theorem 2 Proof.: We consider (8), the first justification in Theorem 2. Rewrite the definition of $\hat{\lambda}$ as \[\hat{\lambda}=\operatorname{argmin}_{\lambda\in\Lambda}\left\{L_{0}( \lambda)+(L_{n}^{}(\lambda)-L_{0}(\lambda))\right\}.\] Similar to the proof of Theorem 1, applying Lemma 3 with $A_{n}(\lambda)$, $B_{n}(\lambda)$, $a_{n}(\lambda)$, and $\eta_{n}$ being $L_{0}(\lambda)$, $R_{0}^{}(\lambda)$, $L_{n}^{}(\lambda)-L_{0}(\lambda)$ and $\xi_{n}^{-1}n^{-1/2}+\xi_{n}^{-1}n_{0}^{-1/2}$, it suffices to prove \[\sup_{\lambda\in\Lambda}\frac{\|R_{0}^{}(\lambda)-L_{0}(\lambda)\|}{\|R_{0}^{}( \lambda)\|}=O_{p}(\xi_{n}^{-1}n^{-1/2}+\xi_{n}^{-1}n_{0}^{-1/2})=o_{p}(1) \tag{49}\] and \[\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-L_{0}(\lambda)\|}{\|R_{0}^{}( \lambda)\|}=O_{p}(\xi_{n}^{-1}n^{-1/2}+\xi_{n}^{-1}n_{0}^{-1/2})=o_{p}(1). \tag{50}\] Notice that $L_{n}^{}(\lambda)-L_{0}(\lambda)$ can be rewritten to $(L_{n}^{}(\lambda)-R_{0}^{}(\lambda))+(R_{0}^{}(\lambda)-L_{0}(\lambda))$. Besides, according to the conclusion of Theorem 1, we have \[\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-R_{0}^{}(\lambda)\|}{\|R_{0}^ {}(\lambda)\|}=O_{p}(\xi_{n}^{-1}n^{-1/2})=o_{p}(1), \tag{51}\] We claim that (49) is sufficient to prove (50). In fact, the sufficiency is easily obtained by \[\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-L_{0}(\lambda)\|}{\|R_{0}^{} (\lambda)\|}\leq\sup_{\lambda\in\Lambda}\frac{\|L_{n}^{}(\lambda)-R_{0}^{}( \lambda)\|}{\|R_{0}^{}(\lambda)\|}+\sup_{\lambda\in\Lambda}\frac{\|R_{0}^{}( \lambda)-L_{0}(\lambda)\|}{\|R_{0}^{}(\lambda)\|}.\] Therefore, we come to prove (49). Notice that \[\sup_{\lambda\in\Lambda}\frac{\|R_{0}^{}(\lambda)-L_{0}(\lambda)\|} {\|R_{0}^{}(\lambda)\|} \leq\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\|L_{0}(\lambda)-R_{0}^{} (\lambda)\|\] \[\leq\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\left\{\|L_{0}(\lambda)-L_ {0}^{}(\lambda)\|+\|L_{0}^{}(\lambda)-R_{0}^{}(\lambda)\|\right\}. \tag{52}\] Similar to (36) in the proof of Theorem 1, we have \[\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\|L_{0}(\lambda)-L_{0}^{}(\lambda)\|\] \[= \xi_{n}^{-1}\sup_{\lambda\in\Lambda}\left\|n_{0}^{-1}\sum_{j=1}^{n_{0}} \left[\{f_{\lambda}(X_{j}^{0},\hat{w}_{n}(\lambda))-E(Y_{j}^{0}\|X_{j}^{0})\}^{2}- \{f_{\lambda}(X_{j}^{0},w^{}(\lambda))-E(Y_{j}^{0}\|X_{j}^{0})\}^{2}\right]\right\|\] \[= \ O_{p}(\xi_{n}^{-1}n_{0}^{-1/2})=o_{p}(1), \tag{53}\] where the second equality follows from (35) and the third equality follows from Assumption 4. Then, by the definition that $R_{0}^{}(\lambda)=E(L_{0}^{}(\lambda))$, it holds for $\lambda\in\Lambda$, $\delta>0$ that \[\Pr\left(\|L_{0}^{}(\lambda)-R_{0}^{}(\lambda)\|>\delta\right) \leq\delta^{-2}\operatorname{var}(L_{0}^{}(\lambda))\] \[=n_{0}^{-1}\delta^{-2}\operatorname{var}\left[\{f_{\lambda}(X_{0 },\hat{w}_{n}(\lambda))-E(Y_{0}\|X_{0})\}^{2}\right] \tag{54}\] By Assumption 6, we obtain \[\xi_{n}^{-1}\sup_{\lambda\in\Lambda}\|L_{0}^{}(\lambda)-E\{L_{0}^{*}(\lambda) \}\|=O_{p}(\xi_{n}^{-1}n_{0}^{-1/2})=o_{p}(1). \tag{55}\] Combining (52), (53), and (55), we obtain (49). Thus we complete the proof of (8). Now we are ready to prove (9) and (10). According to (8), we have \[\frac{L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}(\lambda)}{\inf_{ \lambda\in\Lambda}L_{0}(\lambda)}=\frac{L_{0}(\hat{\lambda})}{\inf_{\lambda \in\Lambda}L_{0}(\lambda)}-1=o_{p}(1) \tag{56}\] By the uniformly integrablity in Assumption 7(ii), we obtain \[\frac{L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}(\lambda)}{\inf_{ \lambda\in\Lambda}L_{0}(\lambda)}\stackrel{{ L_{1}}}{{\longrightarrow}}0,\] which implies \[E\Big{\{}\frac{L_{0}(\hat{\lambda})}{\inf_{\lambda\in\Lambda}L_{0}(\lambda)} \Big{\}}\longrightarrow 1. \tag{57}\] This proves (9). As for (10), we have \[E\big{\{}L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}( \lambda)\big{\}}\] \[\leq E\big{\{}\frac{L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0 }(\lambda)}{\inf_{\lambda\in\Lambda}L_{0}(\lambda)}\big{\}}\times E\big{[}\inf _{\lambda\in\Lambda}L_{0}(\lambda)\cdot\{L_{0}(\hat{\lambda})-\inf_{\lambda \in\Lambda}L_{0}(\lambda)\}\big{]}\] \[< C\times E\big{\{}\frac{L_{0}(\hat{\lambda})-\inf_{\lambda\in \Lambda}L_{0}(\lambda)}{\inf_{\lambda\in\Lambda}L_{0}(\lambda)}\big{\}}\times E \{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}\] where the first inequality follows from Cauchy-Schwarz inequality and the second inequality follows from Assumption 7(i). This implies \[0\leq\frac{E\big{\{}L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}(\lambda )\big{\}}}{E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}}\leq C\times E\big{\{} \frac{L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}(\lambda)}{\inf_{\lambda \in\Lambda}L_{0}(\lambda)}\big{\}}. \tag{58}\] By (57), we know that the right hand side of (58) converges to $0$, which also implies \[\frac{E\big{\{}L_{0}(\hat{\lambda})-\inf_{\lambda\in\Lambda}L_{0}(\lambda) \big{\}}}{E\{\inf_{\lambda\in\Lambda}L_{0}(\lambda)\}}\longrightarrow 0.\] Therefore, we completes the proof of Theorem 2. ## Appendix C Design of models in Image classification problems We employ the same structure for model $F_{1}$ and $F_{2}$, which follows the standard AlexNet architecture (Krizhevsky et al., 2012), with necessary modifications tailored to our tasks. The model consists of two parts: feature extraction and classification. Specifically, the feature extraction part is composed of the following layers in sequence: convolutional layer 1 (32 filters of $5\times 5$), convolutional layer 2 (64 filters of $3\times 3$), max-pooling layer 1 (kernel size $2\times 2$ and stride 2), convolutional layer 3 (96 filters of $3\times 3$, convolutional layer 4 (64 filters of $3\times 3$, convolutional layer 5 (32 filters of $3\times 3$, max-pooling layer 1 (kernel size $2\times 2$ and stride 1)). The classification part comprises three fully connected layers with two ReLu activation functions, having output sizes of 512, 128, and 1. Dropout layers are placed between the fully connected layers. An input image will be first normalized to have values in $[-1,1]$ and then processed through the feature extraction and classification components, and a sigmoid activation function is applied at the end of the network to generate the final output probabilities. We train models $F_{1}$ and $F_{2}$ following the procedure outlined in Section 4.2, using 60000 relabeled samples from the training set of the Fashion-MNIST or MNIST datasets, respectively. The achieved accuracies of $F_{1}$ and $F_{2}$ on their repective 10000 test samples exceeds 99.5%.
2305.07639	Efficient Neural Network based Classification and Outlier Detection for Image Moderation using Compressed Sensing and Group Testing	Popular social media platforms employ neural network based image moderation engines to classify images uploaded on them as having potentially objectionable content. Such moderation engines must answer a large number of queries with heavy computational cost, even though the actual number of images with objectionable content is usually a tiny fraction. Inspired by recent work on Neural Group Testing, we propose an approach which exploits this fact to reduce the overall computational cost of such engines using the technique of Compressed Sensing (CS). We present the quantitative matrix-pooled neural network (QMPNN), which takes as input $n$ images, and a $m \times n$ binary pooling matrix with $m < n$, whose rows indicate $m$ pools of images i.e. selections of $r$ images out of $n$. The QMPNN efficiently outputs the product of this matrix with the unknown sparse binary vector indicating whether each image is objectionable or not, i.e. it outputs the number of objectionable images in each pool. For suitable matrices, this is decoded using CS decoding algorithms to predict which images were objectionable. The computational cost of running the QMPNN and the CS algorithms is significantly lower than the cost of using a neural network with the same number of parameters separately on each image to classify the images, which we demonstrate via extensive experiments. Our technique is inherently resilient to moderate levels of errors in the prediction from the QMPNN. Furthermore, we present pooled deep outlier detection, which brings CS and group testing techniques to deep outlier detection, to provide for the case when the objectionable images do not belong to a set of pre-defined classes. This technique enables efficient automated moderation of off-topic images shared on topical forums dedicated to sharing images of a certain single class, many of which are currently human-moderated.	Sabyasachi Ghosh, Sanyam Saxena, Ajit Rajwade	2023-05-12T17:48:05Z	http://arxiv.org/abs/2305.07639v1	Efficient Neural Network based Classification and Outlier Detection for Image Moderation using Compressed Sensing and Group Testing ###### Abstract Popular social media platforms which allow users to upload image content employ neural network based image moderation engines to classify images as having potentially objectionable or dangerous content (such as images depicting weapons, drugs, nudity, etc). As millions of images are shared everyday, such image moderation engines must answer a large number of queries with heavy computational cost, even though the actual number of images with objectionable content is usually a tiny fraction of the total number. Inspired by recent work on Neural Group Testing, we propose an approach which exploits this fact to reduce the overall computational cost of such neural network based image moderation engines using the technique of Compressed Sensing (CS). We present the _quantitative matrix-pooled neural network_ (Qmpnn), which takes as input $n$ images, and a $m\times n$ binary pooling matrix with $m<n$, whose rows indicate $m$ pools of images i.e. selections of $r$ images out of $n$. The Qmpnn efficiently outputs the product of this matrix with the unknown sparse binary vector representing the classification of each image as objectionable or non-objectionable, i.e., it outputs the number of objectionable images in each pool. If the matrix obeys certain properties as required by CS theory, this compressed representation can then be decoded using CS algorithms to predict which input images were objectionable. The computational cost of running the Qmpnn and the CS decoding algorithms is significantly lower than the cost of using a neural network with the same number of parameters separately on each image to classify the images, which we demonstrate via extensive experiments. Our technique is inherently resilient to moderate levels of errors in the prediction from the Qmpnn. Furthermore, we present pooled deep outlier detection, which brings CS and group testing techniques to deep outlier detection, to provide for the important case when the objectionable images do not belong to a set of pre-defined classes. This technique is designed to enable efficient automated moderation of off-topic images shared on topical forums dedicated to sharing images of a certain single class, many of which are currently human-moderated. ## 1 Introduction Compressed sensing (CS) has been a very extensively studied branch of signal/image processing, which involves acquiring signals/images directly in compressed form as opposed to performing compression post acquisition. Consider a possibly noisy vector $\mathbf{y}\in\mathbb{R}^{m}$ of measurements of the signal $\mathbf{x}\in\mathbb{R}^{n}$ with $m\ll n$, where the measurements are acquired via a sensing matrix $\mathbf{\Phi}\in\mathbb{R}^{m\times n}$ (implemented in hardware). Then we have the relationship: \[\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{\eta}, \tag{1}\] where $\mathbf{\eta}\in\mathbb{R}^{m}$ is a vector of i.i.d. noise values. CS theory [15, 30] states that under two conditions, $\mathbf{x}$ can be stably and robustly recovered from $\mathbf{y},\mathbf{\Phi}$, with rigorous theoretical guarantees, by solving convex optimization problems such as the Lasso[30], given as follows: \[\mathbf{\hat{x}}\triangleq\operatorname{argmin}_{\mathbf{x}}\\|\mathbf{y}-\mathbf{\Phi}\mathbf{x} \\|^{2}+\lambda\\|\mathbf{x}\\|_{1}, \tag{2}\] where $\lambda$ is a carefully chosen regularization parameter. The two conditions are: ($\mathcal{C}1$) $\mathbf{x}$ should be a sufficiently sparse vector, and ($\mathcal{C}2$) no sparse vector, except for a vector of all zeroes, should lie in the null-space of $\mathbf{\Phi}$. $\mathcal{C}1$ and $\mathcal{C}2$ ensure that $\mathbf{y}$ and $\mathbf{x}$ are uniquely mapped to each other via $\mathbf{\Phi}$ even though $m\ll n$. $\mathcal{C}2$ is typically satisfied when the entries of $\mathbf{\Phi}$ belong to sub-Gaussian distributions and when $m\geq O(k\log n/k)$. If these conditions are met, then successful recovery of a vector $x$ with at the most $k$ non-zero elements is ensured [15, 30]. Group testing (GT), also called pooled testing, is an area of information theory which is closely related to CS [25]. Given a set of $n$ samples (for example, blood/urine samples) that need to be tested for a _rare_ disease, GT replaces tests on the individual samples by tests on _pools_ (also called _groups_), where each pool is created using a subset of the $n$ samples. Given the test results on each pool (as to whether or not the pool contains one or more diseased samples), the aim is to infer the status of the individual $n$ samples. This approach dates back to the classical work of Dorfman [20, 8] and has been recently very successful in saving resources in COVID-19 RT-PCR testing [24, 45]. In some cases, side information in the form of contact tracing matrices has also been used to further enhance the results [29, 27, 26]. If the test results report the _number_ of defective/diseased samples in a pool instead of just a binary result, it is referred to as quantitative group testing (QGT) [43]. Image moderation* is the process of examining image content to determine whether it depicts any objectionable content or violates copyright issues. In this paper, we are concerned with semantic image moderation to determine whether the image _content_ is objectionable, for example depiction of violence (such as images of weapons), or whether it is off-topic for the chosen forum (such as images of a tennis game being shared on an online forum for baseball). Manually screening the millions of images shared on online forums like Facebook, Instagram, etc. is very tedious. There exist many commercial solutions for automated image moderation (Amazon, Azure, Picpurify, Webpurify). However, there exists only a small-sized body of academic research work in this area, mostly focused on specific categories of objectionable content, such as firearms/guns [12], knives [28, 17] or detection of violent scenes in images/videos [22]. Most of these papers employ neural networks for classification, given their success in image classification tasks since [33]. However large-scale neural networks require several Giga-flops of operations for a single forward pass [13] and consume considerable amounts of power as shown in [11, 40]. Thus, methods to reduce the heavy load on moderation servers are the need of the hour. Besides reduction in power consumption, it is also helpful if the computational cost for image moderation engines is reduced significantly. In this work, we present a CS approach to speed up image moderation. Consider a set of $n$ images, each of which may independently have objectionable content with a small probability $p$, called the _prevalence rate_ of objectionable images (henceforth referred to as OIs). Small $p$ is justified by independent reports from forums such as Facebook [1] ($0.03-0.14\%$) or Reddit [5] ($6\%$), where millions of images are regularly uploaded and where image or content moderation is an important requirement. Instead of invoking a neural network separately on each image to classify it as objectionable or non-objectionable, we introduce the _quantitative matrix-pooled neural network_ (Qmpnn), which takes in a specification of $m<n$ pools of images - each pool being a selection of $r$ out of $n$ images - and efficiently predicts the count of OIs in these pools. Similar to [38], the Qmpnn runs the first few layers of the neural network on each image of a pool and computes their Intermediate Feature Maps (IFMs), computes the superposed feature map (SFM) of the pool from its constituent IFMs, and processes only the SFM using the remaining layers of the network. However, unlike [38], the IFM for each image is computed only once, and all $n$ images are processed in a single forward pass to produce the $m$ pool outputs. Such a design ensures that the computational cost of running the Figure 1: Left: Main Components of our Image Moderation Engine; Right top: A pool of $8$ images containing $2$ images of firearms which are considered objectionable images (OIs); Right, bottom: The first $9$ channels of the Superposed Feature Map (SFM) for this pool. The SFM has size $512\times 28\times 28$. network on the $m$ pools is lower than that of running the network on the $n$ images individually. The pool specification is given by an $m\times n$ binary pooling matrix. The output of the Qmpnn is thus the product of the pooling matrix and the _sparse_ unknown binary vector specifying whether each image was objectionable or not. We employ CS algorithms, which run at very little additional cost, to determine the status of each of the $n$ images from the Qmpnn's results on the $m$ pools. The above method and the one used in [38] are applicable only to classification problems, where the objectionable images belong to a single or a limited number of classes. However, in many commonly occurring cases, the the objectionable images may belong to a large number, in fact even an unknown number of classes. For example, on a topical forum dedicated to sharing information about tennis, any image whose content is not related to tennis can be considered objectionable. This no longer belongs to the realm of classification, but is instead an outlier detection problem. We extend our earlier approach of combining neural networks and CS algorithms to deal with this application via a novel technique. Our work in this paper is inspired by that in [38] which was the first work to apply GT principles to the problem of binary classification in the presence of class imbalance. However, we significantly build up on their setup and technique, and make the following contributions (see also Sec. 3.4 for more details): 1. We show that the problem of classifying the original $n$ images (as OI or not) from these $m$ counts (each count belonging to a pool) can be framed as a CS problem with some additional constraints (Sec. 3). We employ various algorithms from the CS and GT literature for this purpose (Sec. 3.2). Via an efficient implementation of our network called Qmpnn, we enable usage of pooling matrices in which one image can be a part of _many_ pools. As opposed to this, in the approach in [38], each new pool that an image contributes to, incurs an additional cost, which makes it impractical to have an image contribute to more than two pools. Since each image contributes to many pools, an improved resilience to any errors in the network outputs can be obtained. 2. We test our method for a wide range of prevalence rates $p$ and show that it achieves significant reduction in computation cost and end-to-end wall-clock time compared to individual moderation of images, while remaining competitive in terms of classification accuracy despite noisy outputs by the pooled neural network (Sec. 4). 3. We also show that our method outperforms binary group testing methods, as used in [38], for high $p$, where quantitative information is more important (Sec. 4). 4. Our decoding algorithms are designed to be noise-tolerant, as opposed to existing GT algorithms used for this problem which do not handle the case of a pool falsely testing negative for the presence of an objectionable image as done in [38] (also see Sec. 4). 5. Finally, we present a novel _pooled deep outlier detection_ method for computationally efficient automatic identification of off-topic images on internet forums such as Reddit (Sec. 3.3). Moderation of such off-topic content is currently either done manually or via text-based tools [31]. Note that off-topic images do not belong to a set of pre-defined classes, making this problem different from the one where OIs belonged to a single class. To \begin{table} \begin{tabular}{\|l l\|} \hline \multicolumn{2}{\|l\|}{List of Abbreviations} \\ \hline Bmpnn & Binary Matrix-Pooled Neural Network \\ Bpmn & Binary Pooled Neural Network \\ CLasso & Constrained Least Absolute Shrinkage and Selection Operator \\ Comp & Combinatorial Orthogonal Matching Pursuit \\ CS & Compressed Sensing \\ GMM & Gaussian Mixture Model \\ GT & Group Testing \\ IFM & Intermediate Feature Map \\ Inn & Individual Neural Network \\ Mip & Mixed Integer Programming Method \\ NComp & Noisy Combinatorial Orthogonal Matching Pursuit \\ NLPD & Negative Log Probability Density \\ OD & Outlier Detection \\ OI & Objectionable Image \\ QGT & Quantitative Group Testing \\ Qmpnn & Quantitative Matrix-Pooled Neural Network \\ Qnn & Quantitative Pooled Neural Network \\ SFM & Superposed Feature Map \\ \hline \end{tabular} \end{table} Table 1: List of Abbreviations the best of our knowledge, the approach proposed in this paper is the first one in the literature to present this problem in the context of compressed sensing or group testing. In this paper, we combine the capabilities of neural networks and CS algorithms in a specific manner. Neural networks have been used in recent times for CS recovery [48, 51, 35] with excellent results. However, in this paper, we are using neural networks for either a noisy classification or a noisy outlier detection task, retaining the usage of classical CS or GT algorithms. ## 2 Group Testing Background We first define some important terminology from the existing literature, which will be used in this paper. In binary group testing, there are $n$ items, of which $k\ll n$ unknown items are defective. Instead of individually testing each item, a pooled test (or group test) is performed on a pool/group of items to determine whether there exists at least one defective item in the pool, in which case the test is said to be 'positive' (otherwise 'negative'). The goal of group testing is to determine which $k$ items are defective with as few tests as possible. In quantitative group testing, pooled tests give the _count_ of the number of defective items in the pool being tested. In non-adaptive group testing, the number of tests and the pool memberships do not depend on the result of any test, and hence all tests can be performed in parallel. In adaptive group testing, the tests are divided into two or more rounds, such that the pool memberships for tests belonging to round $t_{r}+1$ depend on the test outcomes from rounds $[t_{r}]\triangleq\{1,\cdots,t_{r}\}$. Dorfman Testing is a popular $2$-round binary, adaptive algorithm [20], widely used in COVID-19 testing [49]. In round 1 of Dorfman testing, the $n$ items are randomly assigned to one out of $n/g$ groups, each of size $g$. All items which are part of a negative pool are declared non-defective. All items which are part of a positive pool are then tested individually in round 2. The optimal pool size $g$ which minimizes the number of tests in the worst case is $\sqrt{n/k}$ and the number of tests is at most $2\sqrt{nk}$[20]. A pooling matrix$\mathbf{\Phi}$ is a $m\times n$ binary matrix, where $\mathbf{\Phi}_{ij}=1$ indicates that item $j$ was tested as part of pool $i$ and $\mathbf{\Phi}_{ij}=0$ otherwise. Let $\mathbf{x}$ be the unknown binary vector with $x_{j}=1$ if item $j$ is defective, $0$ otherwise. Then the outcome of the pooled tests in binary group testing may be represented by the binary vector $\mathbf{y}$, such that $y_{i}=\bigvee_{i=1}^{n}\mathbf{\Phi}_{ij}\wedge x_{j}$ where $\vee$ and $\wedge$ denote the Boolean OR and AND operations respectively. The outcome of pooled tests in quantitative group testing may be represented by the integer-valued vector $\mathbf{y}=\mathbf{\Phi}\mathbf{x}$ obtained by multiplication of the matrix $\mathbf{\Phi}$ with the vector $\mathbf{x}$. In both binary and quantitative group testing, a decoding algorithm recovers $\mathbf{x}$ from known $\mathbf{y}$ and $\mathbf{\Phi}$. Group testing is termed noisy or noiseless depending on whether or not the test results $\mathbf{y}$ contain noise. ## 3 Main Approach Fig. 1 shows the main components of our approach for image moderation (see also Alg. 1). Consider a set of $n$ images, $\mathcal{I}=\{I_{1},\ldots,I_{n}\}$, represented by a $n$-dimensional binary vector $\mathbf{x}$, with $x_{i}=1$ if the $i^{\text{th}}$ image has objectionable content and $x_{i}=0$ otherwise. The binary _pooling matrix_$\mathbf{\Phi}$ of dimensions $m\times n$ specifies the $m$ image pools $\{P_{1},\ldots,P_{m}\}$, to be created from the images in $\mathcal{I}$. Each row of $\mathbf{\Phi}$ has sum equal to $r$, which means that each pool has $r$ images. The pooling matrix $\mathbf{\Phi}$ and the $n$ images in $\mathcal{I}$ are passed as input to a so-called _quantitative matrix-pooled neural network_ (Qmpnn, see Sec. 3.1) which is specifically trained to output the number of OIs in each pool in the form of a $m$-element vector $\mathbf{y}$. If the output of the Qmpnn is perfect, then clearly, for all $q\in\{1,\ldots,m\}$, $y_{q}=\sum_{I_{i}\in P_{q}}\mathbbm{1}(x_{i}=1)=\sum_{i=1}^{n}\mathbf{\Phi}_{qi}x_{i}$, which gives us $\mathbf{y}=\mathbf{\Phi}\mathbf{x}$ just as in Sec. 2. Hence, recovery of the unknown vector of classifications $\mathbf{x}$ given the output $\mathbf{y}$ of Qmpnn (the vector of $m$ different quantitative group tests) and the pooling matrix $\mathbf{\Phi}$ can be framed as a CS problem. Such an approach is non-adaptive because the pool memberships for each image are decided beforehand, independent of the Qmpnn outputs. Furthermore, comparing with Eqn. 1, we see that this CS problem has boolean constraints on each $x_{i}$, and with $y_{q}\in\{0,\ldots,r\}$ for each $q$. In general the Qmpnn will not be perfect in reporting the number of OIs, hence the $\mathbf{y}$ vector predicted will be noisy. This is represented as: \[\mathbf{y}=\mathfrak{N}(\mathbf{\Phi}\mathbf{x}), \tag{3}\] where $\mathfrak{N}(.)$ is a noise operator, which may not necessarily be additive or signal-independent unlike the case in Eqn. 1. However, we experimentally find that CS algorithms, which are specifically designed for signal-independent additive noise, are effective even in the case of noise in the outputs of the Qmpnn. ### Pooled Neural Network for Classification The idea of a pooled neural network was first proposed in [38]. We make two key changes to their design - (1) incorporate the pooling scheme within the neural network which enables efficient non-adaptive (one-round) testing, and (2) have quantitative outputs instead of binary, which enables CS decoding. We discuss the significance of our architectural changes in detail in Sec. 3.4. Below, we describe the pooled neural network architecture from [38] and then our network models. The rest of this section uses many abbreviations. For the reader's convenience, a complete list is presented in Table 1. Consider a feed-forward deep neural network which takes as input a single image for classifying it as objectionable or not. We term such a network an _individual neural network_, or Inn. Furthermore, let this neural network have the property that it may be decomposed into $L$ blocks parameterized by the (mutually disjoint) sets of parameters $\theta_{j},j\in\{1,\ldots,L\}$, with $\mathbf{\Theta}\triangleq\bigcup_{j=1}^{L}\theta_{j}$, so that for any input image $I$, \[\mathtt{INn}_{\mathbf{\Theta}}(I)\triangleq f_{\theta_{L}}^{(L)}\circ f _{\theta_{L-1}}^{(L-1)}\circ\cdots\circ f_{\theta_{1}}^{(1)}(I), \tag{4}\] where $\circ$ represents function composition, and where each block computes a function $f_{\theta_{j}}^{(j)},j\in\{1,\ldots,L\}$. Furthermore, the $L^{\text{th}}$ block is a linear layer with bias, and has two outputs which are interpreted as the unnormalized log probability of the image being objectionable or not. Below, we derive _pooled neural networks_ based on an Inn parameterized by $\mathbf{\Theta}$, and which take as input more than one image. Consider the output of only the first $l<L$ blocks of the INN on an input image $I$. We call this the _intermediate feature-map_ (IFM) of image $I$, with $\mathtt{IFM}_{\mathbf{\Theta}}(I)\triangleq f_{\theta_{1}}^{(l)}\circ\cdots\circ f _{\theta_{1}}^{(1)}(I)$. Consider a _pool_ created from $r$ images, given as $P\triangleq\{I_{1},I_{2},\ldots,I_{r}\}$. A _superposed feature map_ (SFM) of the pool of images $P$ is obtained by taking an entry-wise max over the feature-maps of individual images, i.e. $\mathtt{SFM}_{\mathbf{\Theta}}(P)(.)\triangleq\max_{I\in P}\mathtt{IFM}_{\mathbf{ \Theta}}(I)(.)$ (known as 'aggregated features' in [38]). In a _binary pooled neural network_ (Bpmn), the SFM of a pool of images $P$ is processed by the remaining $L-l$ blocks of the Inn: \[\mathtt{BPNN}_{\mathbf{\Theta}}(P)\triangleq f_{\theta_{L}}^{(L)}\circ f _{\theta_{L-1}}^{(L-1)}\circ\cdots\circ f_{\theta_{1+1}}^{(l+1)}\circ\mathtt{ SFM}_{\mathbf{\Theta}}(P). \tag{5}\] The two outputs of a Bpmn are interpreted as the unnormalized log probability of the pool $P$ containing an image belonging to the OI class or not. A Bpmn has the same number of parameters as the corresponding Inn. This was introduced in [38], where it is referred to as 'Design 2'. In this work, we introduce a _quantitative pooled neural network_ (Qpmn), which is the same as a Bpmn except that it has $r+1$ outputs instead of $2$. Due to this there are more parameters in the last linear layer (the $L^{\text{th}}$ block). Let these parameters be denoted by $\tilde{\theta}_{L}$, with the $L^{\text{th}}$ block computing a function $f_{\tilde{\theta}_{L}}^{(L)}$, and $\mathtt{QPNN}_{\tilde{\mathbf{\Theta}}}(P)\triangleq f_{\tilde{\theta}_{L}}^{(L)} \circ\ldots f_{\theta_{1+1}}^{(l+1)}\circ\mathtt{SFM}_{\mathbf{\Theta}}(P)$, with $\tilde{\mathbf{\Theta}}\triangleq\bigcup_{j=1}^{L-1}g_{j}\cup\tilde{\theta}_{L}$. The $r+1$ outputs of a Qpmn are interpreted as the unnormalized log probabilities of the pool $P$ containing $0$ through $r$ instances of images which belong to the OI class. Finally, we introduce the _quantitative matrix-pooled neural network_ (Qmpnn), which takes as input a set of $n$ images $\mathcal{I}\triangleq\{I_{1},\ldots,I_{n}\}$, a specification of $m$ pools containing $r$ out of $n$ images in each pool via binary matrix $\mathbf{\Phi}$, and outputs $r+1$ real numbers for each of the $m$ pools, in a single forward pass. That is, \[\mathtt{QMPNN}_{\tilde{\mathbf{\Theta}}}(\mathcal{I},\mathbf{\Phi}) \triangleq(\mathtt{QPNN}_{\tilde{\mathbf{\Theta}}}(P_{1}),\ldots, \mathtt{QPNN}_{\tilde{\mathbf{\Theta}}}(P_{m})) \tag{6}\] \[\text{where }P_{q} \triangleq\{I_{i}\|\Phi_{qi}=1,I_{i}\in\mathcal{I}\}\ \forall q\in\{1,\ldots,m\}.\] For a pool $q$, the $s^{\text{th}}$ output ($s\in\{0,\ldots,r\}$) is interpreted as the unnormalized log probability of the pool $q$ containing $s$ images belonging to the OI class. Notably, the IFM for each of the $n$ images is computed only once by the Qmpnn, and is re-used for each pool that it takes part in. A _binary matrix-pooled neural network_ (Bmpnn) is exactly the same as a Qmpnn except that each pool has only two outputs. We note that while the first $l$ blocks of the neural network are run for each of the $n$ images, the remaining $L-l$ blocks are only run for each of $m$ SFMs. However, if an individual neural network (Inn) is run on the $n$ images, then both the first $l$ blocks and the last $L-l$ blocks will be run $n$ times. Hence, since $m<n$, the Qmpnn (and the Bmpnn) requires significantly less computation than running the Inn on the $n$ images, which we verify empirically in Sec. 4. Training: Let $\mathcal{D}$ be a dataset of images labelled as OI or non-OI. A _pooled dataset_$\mathcal{D}_{\text{pooled}}$ is obtained from $\mathcal{D}$. Each entry of $\mathcal{D}_{\text{pooled}}$ is a pool of $r$ images, and has a label in $\{0,\dots,r\}$, equal to the number of images in that pool which belong to the OI class. Details of creation of the pooled dataset are given in Sec. 4. The parameters $\mathbf{\tilde{\Theta}}$ of a Qpmn are trained via supervised learning on $\mathcal{D}_{\text{pooled}}$ using the cross-entropy loss function, $L(\mathbf{\tilde{\Theta}})=\underset{(P,c)\sim\mathcal{D}_{\text{pool}}}{E}[- \log\sigma(\texttt{QPNN}_{\mathbf{\tilde{\Theta}}}(P))_{c}]$, where $\sigma(.)$ is the softmax function, given by $\sigma(\mathbf{z})_{s}\triangleq\frac{e^{s}}{\sum_{i=0}^{s}e^{rt}}$. The parameters $\Theta$ of a Bpmn are trained similarly, except that each training pool has a label in $\{0,1\}$, with $1$ indicating pools which contain at least one OI. Inference: The trained parameters $\mathbf{\tilde{\Theta}}$, and a pooling matrix $\mathbf{\Phi}$ are then used to instantiate a Qmpnn. The entries of the vector $\mathbf{y}$, containing pool-level predictions of the count of OIs in each pool, are obtained from a Qmpnn as: \[y_{q}=\operatorname{arg\,max}_{s\in\{0,1\dots,r\}}\texttt{ QMPNN}_{\mathbf{\tilde{\Theta}}}(\mathcal{I},\mathbf{\Phi})(q,s)\;\forall q\in \{1,\dots,m\}. \tag{7}\] In Sec. 3.2, we present CS algorithms to decode this vector $\mathbf{y}$ to recover the vector $\mathbf{x}$ for classification of each image in $\mathcal{I}$. Similarly, the trained parameters $\mathbf{\Theta}$ of a Bpmn and a pooling matrix $\mathbf{\Phi}$ can be used to instantiate a Bmpnn. The entries of the binary vector $\bar{\mathbf{y}}$ containing pool-level predictions of whether each pool contains an OI or not can be obtained from the Bmpnn as: \[\bar{y}_{q}=\operatorname{arg\,max}_{s\in\{0,1\}}\texttt{ BMPNN}_{\mathbf{ \tilde{\Theta}}}(\mathcal{I},\mathbf{\Phi})(q,s)\;\forall q\in\{1,\dots,m\}. \tag{8}\] This can be decoded by the non-adaptive binary group testing decoding algorithms presented in Sec. 3.2. ### CS/GT Decoding Algorithms Given Eqn. 3, the main aim is to recover $\mathbf{x}$ from $\mathbf{y}$ (the prediction of the Qmpnn, Eqn. 7) and $\mathbf{\Phi}$. For this, we propose to employ the following two algorithms: 1. Mixed Integer Programming Method (Mip): We minimize the objective in Eqn. 2 with boolean constraints on entries of $\mathbf{x}$, i.e. $\forall i\in[n],x_{i}\in\{0,1\}$. We use the CVXPY [19][7] package with the Gurobi [6] solver, which uses a branch and bound method for optimization [3]. 2. Constrained LASSO (CLasso): This is a variant of the Lasso from Eqn. 2, with the values in $\mathbf{x}$ constrained to lie in $[0,1]$. The relaxation from $\{0,1\}$ to $[0,1]$ is done for computational efficiency. The final estimate regarding whether each of the images is an OI is obtained by thresholding $\mathbf{\hat{x}}$, i.e. $\forall i\in[n]$, $I_{i}$ is considered to be an OI if $\hat{x}_{i}>\tau$, where $\tau$ is the threshold determined on a validation set. The hyperparameters in our algorithms ($\lambda$ for CLasso and Mip, and $\tau$ for CLasso) are chosen via grid search so as to maximize the product of specificity and sensitivity (both defined in Sec. 4) on a representative validation set of images of known classes. We also compare with some popular algorithms the from binary group testing literature, which act as a baseline. For the non-adaptive methods, the binary vector $\bar{\mathbf{y}}$ containing predictions of a Bmpnn (from Eqn. 8) is provided to the decoding algorithms. The algorithms are: 1. Dorfman Testing: See Sec. 2. This is referred to as Two-Round testing in [38, Algorithm 1]. In the first stage, a binary pooled neural network (Bpmn) is used to classify a pool of images as containing an objectionable image or not. If the Bpmn predicts that the pool contains an objectionable image, then each image is tested individually in a second round using an individual neural network (Inn). Otherwise, all images in the pool are declared as not being objectionable, without testing them individually. Because of a second round of testing, Dorfman Testing is inherently resistant to a pool being falsely declared as positive, but not to a pool being falsely declared as negative. 2. Combinatorial Orthogonal Matching Pursuit (Comp) [16]: This is a simple decoding algorithm for noiseless non-adaptive binary group testing, wherein any image which takes part in at least one pool which tests negative (i.e., no OIs in the pool) is declared as a non-OI, and the remaining image are declared as OIs. The One-Round Testing method in [38] uses Comp decoding, albeit with a different pooling matrix than ours. 3. Noisy Comp (NComp) [16]: a noise-resistant version of Comp. An image is declared to be OI if it takes part in strictly greater than $t$ positive pools (or equivalently in strictly less than $c-t$ negative pools, where $c$ is the number pools that each image takes part in). A good pooling matrix $\mathbf{\Phi}$ is crucial for the performance of these algorithms. We choose binary matrices with equal number of ones in each column, equal number of ones in each row, with the dot product of each pair of columns and each pair of rows at most $1$. More details about this are provided in Appendix A.2. ### Pooled Deep Outlier Detection Here, we consider the case where the set of allowed images belong to one underlying class, and any image not belonging to that class is considered 'off-topic'. Such a situation arises in the moderation of topical communities on online forums such as Reddit (e.g. see [31]). For example, an image of a baseball game is off-topic on a forum meant for sharing of tennis-related images. As the off-topic images could belong to an unbounded variety of classes, instances of all or some of which may not be available for 'training', _classification_ based approaches (such as the Qmpnn from Sec. 3.1, and [38]) may not be directly applicable, and instead, _deep outlier detection_ based methods are more suitable. Recent deep outlier detection methods [36, 44] approximate a class of interest, represented by suitable feature vectors $\mathbf{\mathbb{R}}^{d}$, using a high dimensional Gaussian distribution characterized by a mean vector $\mathbf{\mu}\in\mathbb{R}^{d}$ and a covariance matrix $\mathbf{\Sigma}\in\mathbb{R}^{d\times d}$. Then the Mahalanobis distance between the feature vectors of a test image and the mean vector $\mathbf{\mu}$ is computed. This Mahalanobis distance acts as the anomaly score to perform outlier detection, i.e. test feature vectors with a Mahalanobis distance greater than some threshold (say $\bar{\tau}_{md}$) are considered outliers. The feature vectors can be represented by the outputs of a suitably trained neural network. One may think of using such an approach with a pooled neural network to detect pools containing outlier images. However a single Gaussian distribution is often inadequate to represent a sufficiently diverse class, and hence we resort to using a Gaussian Mixture Model (GMM). We put forward a novel approach which uses a pooled neural network combined with a GMM, to detect anomalous pools, i.e., pools which contain at least one off-topic image, which we term _pooled deep outlier detection_. This enables us to use binary group testing methods from Sec. 3.2 - such as Comp, NComp and Dorfman Testing - for pooled outlier detection. Furthermore, we also detect the number of outlier images in a pool, thus enabling CS methods to be used for pooled outlier detection. To the best of our knowledge, this is the first such work in the literature, which enables group testing and compressed sensing methods to be used for outlier detection. We consider the setting where for training we have available a set of images which are either on-topic or are off-topic images from known classes, but at test time the off-topic images may be from unknown classes. We train a Qpm using the method in Sec. 3.1, with the label $L(P)$ for any training pool $P$ set to the number of off-topic images in it. The output of the last-but-one layer of the Qpmn (i.e. the $(L-1)^{\text{th}}$ layer - see Sec. 3.1) is considered to be the feature vector $\phi(P)$ for the pool of images $P$ which is input to it. First, we create $M$ pools from the on-topic training data images (i.e. these pools have $0$ off-topic images), and fit a GMM $G$ with some $K$ clusters to the feature vector $\phi(.)$ of all these pools, using the well-known expectation-maximization (EM) algorithm. The optimal value of $K$ is chosen via cross-validation, by selecting the $K$ with the maximum likelihood given a held-out set of pools containing only on-topic images. For any pool $P$, we use the negative log probability density of its feature vector $\texttt{NLP}_{G}(\phi(P))$ (negative log probability density) under the GMM $G$ as an anomaly score, given by: \[\texttt{NLPD}_{G}(\phi(P)):=-\log\left(\sum_{j=1}^{K}p_{j}\mathcal{N}(\phi(P)\| \boldsymbol{\mu_{j}},\boldsymbol{\Sigma_{j}})\right), \tag{9}\] where $\{\boldsymbol{\mu_{1}},\boldsymbol{\mu_{2}},\cdots,\boldsymbol{\mu_{K}}\}$ are the $K$ mean vectors, $\{\boldsymbol{\Sigma_{1}},\boldsymbol{\Sigma_{2}},\cdots,\boldsymbol{\Sigma_{K }}\}$ are the $K$ covariance matrices and $\{p_{1},p_{2},\cdots,p_{K}\}$ are the $K$ membership probabilities of the GMM $G$, and for any vector $\mathbf{z}_{i}$, $\mathcal{N}(\mathbf{z}\|\boldsymbol{\mu_{j}},\boldsymbol{\Sigma_{j}})$ is its probability density given the multivariate normal distribution with mean $\boldsymbol{\mu_{j}}$ and covariance matrix $\boldsymbol{\Sigma_{j}}$. It is intuitive to imagine that $\texttt{NLPD}_{G}(\phi(P))$ values would generally be larger for pools containing a larger number of outlier images ($L(P)$), and we should be able to predict $L(P)$ given $\texttt{NLPD}_{G}(\phi(P))$. The main idea is to create a histogram of anomaly scores of pools with different number of off-topic images in them, and assign to a new pool the label which is most common in its bin. This is accomplished as follows. From the _validation_ set of images, some $N$ pools $\{P_{i}\}_{i=1}^{N}$ are created, each containing $r$ images. Since off-topic images are expected to be rare at test time, we only consider pools which contain upto some $t<r$ off-topic images. Thus $L(P)\in\{0,1,\ldots,t\}$. For each pool $P_{i}$, the value $\texttt{NLPD}_{G}(\phi(P_{i}))$ is computed. These NLPD values are divided into some $Q$ bins/intervals and an anomaly score histogram $H_{G}$ is created in the following manner: \[\forall j\in\{1,2,\cdots,Q\},\forall l\in\{0,1,\cdots,t\},H_{G}(j,l)=\frac{ \#\text{pools in bin $j$ containing $l$ outlier images}}{N}. \tag{10}\] That is, $H_{G}(j,l)$ represents the fraction of pools from $\{P_{i}\}_{i=1}^{N}$ which contain $l$ outlier images and whose NLPD value falls into the $j^{\text{th}}$ bin. This histogram $H_{G}$ is created at the time of training. Each non-empty bin $j$ is assigned a label $\mathcal{L}(j)$ equal to $\text{argmax}_{l\in\{0,1,\ldots,t\}}H_{G}(j,l)$, i.e., the label with the most number of pools in that bin. An empty bin is assigned the label of its nearest non-empty bin. If there is more than one nearest non-empty bin, then the bin is assigned the label of the bin with the larger index. At test time, given the feature vectors $\phi(P_{\text{test}})$ of a pool $P_{\text{test}}$ created from some $r$ test images, first its anomaly score $\texttt{NLPD}_{G}(\phi(P_{\text{test}}))$ is computed using the GMM. If the anomaly score lies in bin $j_{\text{test}}$, then $P_{\text{test}}$ is assigned the label $\mathcal{L}(j_{\text{test}})$. If the anomaly score is to the right of all bins (i.e. greater than the right index of the rightmost bin), then it is assigned the label $t$. If it is to the left of all bins (i.e. less than the left index of the leftmost bin), then it is assigned the label $0$. That is, \[L(P_{\text{test}})=\begin{cases}\mathcal{L}(j_{\text{test}})&\text{if $\texttt{NLPD}_{G}(\phi(P_{\text{test}}))$}\in\text{ bin $j_{\text{test}}$}\\ 0&\text{if $\texttt{NLPD}_{G}(\phi(P_{\text{test}}))$}<s_{\text{min}}\\ t&\text{if $\texttt{NLPD}_{G}(\phi(P_{\text{test}}))$}>s_{\text{max}}\end{cases}, \tag{11}\] where $s_{\text{min}}$ and $s_{\text{max}}$ are respectively the smallest and the largest of the $N$ NLPD values used to create the histogram $H$. At the time of deployment, $m$ pools are created from $n$ images to be tested for off-topic content using a Qmpnn and a pooling matrix $\Phi$, and feature vectors of the $m$ pools are obtained. For each pool, we determine the number of off-topic images it contains using the GMM and histogram-based method just described. Given these $m$ numbers and the pooling matrix $\Phi$, we predict whether or not each of the $n$ images is off-topic using the CS and non-adaptive GT decoding algorithms described in Sec. 3.2. Our method is summarized in Algorithm 2. We also perform Dorfman Testing for outlier detection. Feature vectors of a pool of $r$ images are obtained by running a Qpnn, and the GMM and histogram-based method is used to predict the number of off-topic images in the pool. If the pool is predicted to contain at least one off-topic image, then each image is individually tested for being off-topic using the same method, but using an individual neural network (Inn). That is, the features of each image are obtained using the Inn, and a previously trained GMM and histogram (of validation _image_ anomaly scores) are used to label the images as off-topic or not. This is summarized in Algorithm. 3. ### Comparison with Related Work There exists very little literature on the combination of GT algorithms with neural networks. The sole published work on this topic (to our knowledge) can be found in [38]. The neural networks in [38] use only binary outputs, while the neural networks proposed in our methods have quantitative outputs, and are trained to predict the _number of OIs_ in a given SFM. This enables usage of CS decoding algorithms. CS methods have better recovery guarantees than non-adaptive binary GT (see Appendix A.3). Non-adaptive pooling using a $m\times n$ pooling matrix is implemented in [38] by separately running a Bpnn for each pool. In their method, the intermediate feature map (IFM) for an image needs to be re-computed for each pool that it takes part in, making it computationally inefficient. Due to this, the scheme in [38] is limited to using pooling matrices in which each image takes part in at most two pools. Such matrices can be at most $1$-disjunct (see Appendix A.1 for the definition), and hence guarantee of exact recovery via Comp is for only $1$ OI per matrix (see [21, Prop. 2.1] and Appendix A.4 for an explanation). Thus their scheme is applicable for only very low prevalence rates ($p$) of OIs. In our implementation, an IFM for each image is computed only once, and is re-used for each pool that it takes part in. This enables usage of pooling matrices with $3$ or more entries per column. This means that we can use matrices with high 'disjunctness' for group testing decoding [39] or those obeying restricted isometry property (RIP) of high order [10] for compressed sensing recovery (see Appendix A.1 for definitions of disjunctness and RIP). Due to this, our method is applicable for both low or high $p$ values. Furthermore, there are no guarantees for binary group testing for recovery from noisy pool observations using matrices which have only two $1$ entries per column. This is because if one test involving an item is incorrect but the other one is correct, there is no way to determine which of them is the incorrect one. Noise-tolerant recovery algorithms for binary GT exists for suitable matrices with $3$ or more $1$ entries per column. For example, NComp with a pooling matrix which has $3$ tests per item and has the properties described in Appendix A.2, when testing in the presence of upto one defective item, can recover from an error in one test of that item by declaring it defective if two of the tests are positive, or non-defective if two of its tests are negative. Moreover, higher number of $1$ entries per column implies better noise tolerance for CS algorithms for the matrices described in Appendix A.2. See Appendix A.5 for details. Over and above these differences, we also extend binary group testing and compressed sensing to deep outlier detection as earlier described in Sec. 3.3, whereas the approach of [38] is limited to the classification setting. In the case of OIs with a single class, our test datasets also contain a much larger number of objectionable images (see details in Sec. 4), and these are mixed with non-OIs to form datasets of size $1$M or $100$K for a wide range of prevalence rates $p$, so that each OI gets tested many times in combination with other OIs and different non-OIs. However in [38], prediction measures are presented only for a single dataset for the case when $p=0.001$, with the number of unique OIs being only $50$ and each OI being tested only once. When a single OI participates in multiple pools, there is an inherent noise resilience brought in. This is important as the neural network may not produce perfect classification. ## 4 Experiments We present an extensive set of experimental results, first focusing on the classification case, and then on the outlier detection case. ### Image Moderation using Pooled Classification: Single Class for Objectionable Images Here the objectionable images belong to a single underlying class, and the goal is to classify images as objectionable or not objectionable. Tasks:We use the following two tasks for evaluation: (_i_) firearms classification using the Internet Movie Firearms Database (IMFDB) [2], a popular dataset used for firearms classification [12, 17]; and (_ii_) Knife classification using the Knives dataset [28], a dataset of CCTV images, labelled with the presence or absence of a handheld knife. In (_i_), the firearm images from IMFDB are mixed with non-firearm images from ImageNet-1K [42] as IMFDB contains only firearm images. The firearms and the knife images are respectively considered as objectionable images (OIs) in the two tasks. Here below, we provide details of the number of images in the training/validation/test splits in each dataset. Individual Dataset Splits:For the firearm classification task, we use firearm images from IMFDB [2] and Imagenet-1K [42] as the OI class, with $9$,$458$ and $3$,$617$ (total $13$,$075$) images in the training set from the two datasets respectively. Additionally, 667 firearm images from IMFDB were used in the validation set, and 931 images were used in the test set. The validation and test images were manually cleaned to remove images which did not contain a firearm, or images in which the firearm was not the primary object of interest. For the non-OI images, we used images from 976 classes in ImageNet-1K [42], which did not belong to a firearm class - $1$,$200$,$764$ in the training set, $50$,$000$ in the validation set, and $48$,$800$ in the test set (leaving out weapon images from ImageNet-1K such as 'tank', 'holster', 'camon', etc., following [38]). During training of the Inn, the classes were balanced by selecting all $13$,$075$ OIs and the same number of random non-OIs for training at each epoch. Classes were balanced in the validation split as well, by using all $50$,$000$ non-OIs and randomly sampling with replacement $50$,$000$ OIs. The Knives dataset [28] used for the knife classification task consists of 12,899 images of which 3559 are knife images and 9340 are non-knife images. Some images were taken indoors, while some were taken through a car window in the street. We randomly selected 2159 knife and 6140 non-knife images for training, 700 knife and 1600 non-knife images for validation, and 700 knife and 1600 non-knife images for testing. During training of the individual neural network, classes were balanced by sampling (without replacement) 2159 random non-knife images at each epoch. The validation set was also balanced by sampling (with replacement) 1600 random knife images. See Fig. 1 for examples of images in IMFDB, and Fig. 2 for examples of images from the Knives dataset. Pooled Dataset Splits:For firearms classification, during training of the Qpnn, at each epoch, $6$,$248$ pools of size $r=8$ were created, each pool containing $k$ randomly selected OIs and $8-k$ randomly selected non-OIs, for each $k$ in $\{0\dots 8\}$. Of these, $40\%$ pools contained no OIs, $24\%$ contained 1 OI, $12\%$ contained 2 OIs, $6\%$ each containing 3 and 4 OIs, and $3\%$ each contained 5 through 8 OIs. Such a distribution was chosen because OIs are assumed to have low prevalence rate in the test dataset. Hence greater accuracy is desired on pools with low number of OIs - however, enough instances of pools with high number of OIs must be seen during training for good accuracy. The pooled validation set used for selection of neural network weights contained $63$,$325$ pools of size $8$, created with the same method as the pooled training set. For testing, we created $9$ mixtures of the test data split of the individual dataset, each containing $N=1$ Million images, with each mixture containing $pN$ OIs, with the prevalence rate $p\in\{0.001,0.002,0.005,0.01,0.02,0.03,0.04,0.05,0.1\}$. For knife classification, 12496 pools of size 8 were created for training following the same distribution of OIs as for the firearms classification case. The pooled validation set used for selection of neural network weights contained 2873 pools of size 8, created with the same method as the pooled training set. The pooled test dataset consisted of 9 mixtures of 100K images, created with the same method as for the firearms classification task. Data Transformations:Images were padded with zeros to equalize their height and width, and resized to $224\times 224$, during both training and testing. We also added a random horizontal flip during training. For the pooled datasets, a random rotation rotation between $[-30^{\circ},30^{\circ}]$ was applied to each training image. This rotation was also applied to test data for the knife classification task. Training:We train all our networks for 90 epochs using stochastic gradient descent with learning rate $0.001$ and momentum $0.9$, using cross-entropy loss and weight decay of $0.0001$. Model weights are checkpointed after every epoch. The best model weights are selected from these $90$ checkpoints using classification accuracy on the validation set. For the firearms classification task, a weighted accuracy is used for selection of the best checkpoint of the pooled neural networks, viz $\texttt{Acc}_{\text{weighted}}=\sum_{k=0}^{8}\texttt{Binom}(8,p,k)\texttt{Acc}_ {k}$, where $\texttt{Acc}_{k}$ is the classification accuracy on pools with Figure 2: Data Samples from Knives dataset [28]. Top Row: Images containing a knife, Bottom Row: Images not containing a knife. $k$ OIs in them, and $\mathtt{Binom}(8,p,k)$ is the probability of having $k$ OIs in a pool of size $8$ if the prevalence rate of OIs is $p$. This accounts for the fact that accuracy on pools with higher number of OIs is more important for high values of $p$, while pools with $0$ or $1$ OI are more important for low values of $p$. Confusion Matrix:As can be seen in Fig. 3, if $g$ is the ground truth number of OIs, the predictions of the Qmpnn lie in the interval $[\text{max}(0,g-1),\text{min}(g+1,r)]$ with high probability. This is presented as a confusion matrix of size $(r+1)\times(r+1)$ of the ground truth number of OIs in a pool versus the predicted number. In Fig. 3, we have $r=8$. Neural Network Implementation:We use the official PyTorch implementation of ResNeXt-101 ($32\times 8d$) [50][4], a highly competitive architecture which has produced compelling results in many image classification tasks, with weights pre-trained on ImageNet-1K. The last linear layer of this network has $1000$ outputs by default. Recall the discussion in Sec. 3.1 regarding the architectures of the Inn, Qmpnn, Bmpnn, Qpmnn, and Bpmn. For Qpmn/Qmpnn with pool size $r=8$, we replace the last linear layer to have only 9 outputs. For the Inn of Bpmn/Bmpnn, we replace the last linear layer to have only two outputs. Each image is passed through first three stages of the ResNeXt-101 to create IFMs, which are then combined to create SFMs for the pools in the Qmpnn/Qpmn/Bpmn. The remaining two stages of the ResNeXt-101 process only the SFMs in Qmpnn/Qpmn/Bmpnn/Bpmn. This configuration is exactly the same as Design 2 in [38, Sec. II.A]. In initial experiments, we also tried using Design 3 of [38], but the accuracy of this configuration was not good on the IMFDB dataset. The networks are trained using the method specified in Sec. 3.1. Details of training the individual and pooled neural networks have been given earlier. Pooling Matrix:We use a $50\times 100$ balanced pooling matrix $\mathbf{\Phi}$ with the construction as described in Appendix A.2, with $r=8$ ones per row and $c=4$ ones per column. That is, $50$ SFMs are created from $100$ IFMs, with each SFM being created from $8$ different IFMs, and each IFM contributing to $4$ different SFMs. Prediction:We randomly sampled $pN$ OIs and $(1-p)N$ non-OIs from the test split and shuffled these $N$ images to create our test datasets. Here, $p\in\{0.001,0.002,0.005,0.01,0.02,0.03,0.04,0.05,0.1\}$ is the prevalence rate of OIs, and $N$ is 1M for IMFDB and 100K for Knives. Each test set is divided into chunks of size $100$ each and passed to the Qmpnn to retrieve the prediction vector $\mathbf{y}$ of each chunk. As can be seen in Fig 3, if $g$ is the ground truth number of OIs, the predictions lie in the interval $[\text{max}(0,g-1),\text{min}(g+1,r)]$ with high probability. These pool-level predictions are decoded using the algorithms Classo and Mip. We compare them with the baselines of Comp, Dorfman Testing with pool-size 8 (Dorfman-8, same as Design 2 + Algorithm 1 of [38]), NComp with $t=2$ (NComp-2) and Individual Testing (Individual) on images of the same size. The numerical comparison was based on the following two performance measures: 1. _Sensitivity_ (also called _Recall_ or True Positive rate) $\triangleq\#\text{correctly detected OIs / }\#\text{actual OIs}$ 2. _Specificity_ (also called True Negative Rate) $\triangleq\#\text{correctly detected non-OIs / }\#\text{actual non-OIs}$. For both the measures, larger values indicate better performance. Figure 3: Qmpnn Confusion Matrix on IMFDB Pooled Validation Data: true #OIs (rows) versus predicted #OIs (columns) Hyperparameter Selection:We created mixtures of images from the validation split of the same sizes and the same prevalence rates as for the test split. For a given prevalence rate $p$, grid search was performed on the corresponding validation split mixture with the same value of $p$ to determine the optimum value of $\lambda$ and $\tau$ for CLasso and $\lambda$ for Mip (see Sec. 3.2). The hyperparameter values which maximized the product of sensitivity and specificity on the validation mixture were chosen. Discussion on Results:Fig. 4 shows the sensitivity and specificity of each of these algorithms for different values of $p$, for the two classifications tasks ($i$) and ($ii$) defined earlier. We see that for IMFDB and Knives, the CS/GT methods remain competitive with Individual. In general, CLasso and Mip have _much higher sensitivity_ than Comp and Dorfman-8. This is because if a pool gets falsely classified as negative (i.e. containing no OIs), then all OIs in that pool get classified as negative by Comp and Dorfman. However, CLasso and Mip can recover from such errors as they inherently consider the results of other pools that the OI takes part in. Such false negative pools are more common at lower $p$ values, where most positive pools have only a single OI, which may remain undetected by the pooled neural network (see also the confusion matrix of the pooled neural network in Fig 3). NComp-2 improves upon the sensitivity of Comp and Dorfman-8 by allowing for one false negative pool for each OI. However, it incurs a corresponding drop in specificity because non-OIs which take part in three positive pools incorrectly get classified as OI. When $p$ is high, it is likely that such non-OIs are common, and hence the steep drop in specificity makes NComp-2 unviable. At high prevalence rates of OIs, it is common for a non-OI to test positive in all of its pools. Since Comp uses only such binary information, it incorrectly declares all such images as OIs, and we observe a sharp decline in specificity. However, CLasso and Mip use the quantitative information of the number of OIs in the pools, and can eliminate such false positives. For example, if it is known that a positive pool has only two OIs, then in general only two images from that pool will be declared as being an OI, whereas Comp may declare anywhere from $1$ through $8$ positives for such a pool. Among the CS algorithms, CLasso has better sensitivity, while Mip has better specificity. Comparison with Downsampling:In Fig. 4, we also compare our method to individual testing of images downsampled by a factor of $4$ (size $112\times 112$) (referred to in Fig. 4 as Downsample) for the firearms task. We find that downsampling lowers the specificity of prediction of individual testing, and both Mip and CLasso have better Figure 4: Performance of Objectionable Image Moderation via Pooled Classification methods for various prevalence rates ($p$), averaged over 1M images for IMFDB and 100K images for Knives. Top Row: Sensitivity on IMFDB (left), Knives (middle), and IMFDB-Downsampled (right). Bottom Row: Specificity on IMFDB (left), Knives (middle), and IMFDB-Downsampled (right). Note: Y-axis for Specificity charts is broken up into two ranges in order to show all the values more clearly. specificity for both IMFDB and Knives. Computational Cost:Fig. 5 compares the average number of floating point operations used per image processed by the Qmpnn against those used for Dorfman-8 and Individual for different prevalence rates on IMFDB. We used the Python package _pftflops_[46] for estimation of FLOPs used by the neural networks. We see that the Qmpnn with $50\times 100$ pooling matrix uses only around $61\%$ of the computation of Individual. While Dorfman-8 uses less computation for small $p$, the Qmpnn is significantly more efficient for $p>0.05$. Wall-clock Times:The end-to-end wall-clock times for Qmpnn with Mip decoding on IMFDB are presented for a range of prevalence rates in Table 2. We see that this is significantly faster than individual testing of the images for all $p$, and for $p\geq 0.04$ than Dorfman-8. Combining Pooling with Downsampling:We verify whether downsampling may be combined with GT and CS methods for further reduction in computation cost without negatively affecting the performance. Fig. 4 (sub-figures labelled IMFDB-Downsampled, rightmost column, top and bottom rows) shows the performance of non-adaptive GT and CS methods on IMFDB images downsampled by a factor of $4$ to $112\times 112$. We note that while all the algorithms observe a drop in sensitivity, it may be within an acceptable range depending on the particular use-case, especially for CLasso. Specificity also drops, but less severely. CLasso and Mip present the best balance of specificity and sensitivity across the entire range of $p$. Multiple Objectionable Elements in the _Same Image:_We created $100$ random pools containing one OI (firearm image) each, but where _each OI depicted two firearms_. Out of $100$, the Qrnn correctly reported $97$ times that the pool contained only one OI, and incorrectly reported two OIs $3$ times. We conjecture that the Qrnn detects unnatural edges between firearms in the SFM, due to which it does not get confused by a single image containing multiple firearms. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|} \hline $p$ & Individual & Dorfman-8 [38] & Qmpnn + Mip \\ \hline 0.001 & & 2080.6 & 3003.5 \\ \hline 0.01 & & 2352.7 & 2992.4 \\ \hline 0.02 & & 2668.6 & 3007.9 \\ \hline 0.03 & & 2924.7 & 2991.0 \\ \hline 0.04 & & 3152.2 & 2988.1 \\ \hline 0.05 & & 3403.6 & 2983.3 \\ \hline 0.1 & & 4315.0 & 2984.4 \\ \hline \end{tabular} \end{table} Table 2: End-to-end wall-clock times (secs), for processing 1M images of IMFDB+ImageNet with one NVIDIA GeForce RTX 2080 Ti GPU (11GB RAM) + one AMD 12-Core CPU, 64GB RAM. Batch-size chosen to fill GPU RAM. Figure 5: Image Moderation cost per image in giga floating point operations (GFLOPs). Cost for running non-adaptive or individual methods is the same for classification and outlier detection. Cost for Dorfman-8 averaged over 1M images for IMFDB and 100K images for Knives. Cost for Dorfman-8-OD (see Sec. 4.2, paragraph titled ’Prediction’) averaged over 100K images for IMFDB and Cars. ### Off-topic Image Moderation using Pooled Outlier Detection Here, the on-topic images are from a single underlying class and off-topic images may be from classes not seen during training. The goal is to detect such off-topic images, which do not belong to the underlying class. We do this by treating this as an outlier detection task, where the on-topic images are considered 'normal' and off-topic images are considered to be outliers. Tasks:We consider two off-topic image moderation tasks, wherein the on-topic images are (_i_) firearm images from IMFDB used in Sec. 4.1, and (_ii_) car images from the Stanford Cars Dataset [37]. During training, we use non-firearm or non-car images from ImageNet-1K as off-topic images from known classes for the two tasks, respectively. During training, we use images from some 182 non-firearm or 179 non-car random classes which are in ImageNet-21K [18, 41] but not in ImageNet-1K. The off-topic image classes chosen for testing are not present in the training datasets. Individual Dataset Splits:The firearm images in training, validation and test splits of IMFDB as described in Sec. 4.1 are taken to be on-topic images for the respective splits for the off-topic image moderation task. From the 976 non-firearm classes of ImageNet-1K in Sec. 4.1, we randomly sample $10{,}000$ and $1{,}000$ images and add them to the training and validation split, respectively, as off-topic images. For testing, we choose $1{,}000$ images from $182$ random non-firearm classes of ImageNet-21K as off-topic images. For the Cars dataset, we take 8041 cars images for testing, 2144 cars images for validation, and 6000 cars images for training from the Stanford cars dataset as on-topic images. The $1{,}000$ off-topic images for testing come from 179 non-car classes of ImageNet-21K, whereas for training and validation, we sample 10000 and 1000 images from 955 non-car classes of Imagenet-1K. We created 9 test sample sets (one per prevalence rate) of the test data split, each containing 100K images, using the same procedure and the same prevalence rates $p$ as in Sec. 4.1. At each epoch of Inn training, classes in the training split were balanced by using all $10{,}000$ off-topic images from known classes and randomly sampling (with replacement) an equal number of on-topic images. The validation split classes were not balanced since the imbalance in class proportions was not too much. Pooled Dataset Splits:The pooled dataset training split was created using the same procedure as in Sec. 4.1, with the same pool size (i.e. 8), number of pools, and distribution of pools with different number of off-topic images in them. The pooled dataset validation split had $625$ pools and was also created using the same procedure. Neural Network, Data Transformation, Training:We use the same neural network (ResNeXt-101 [4, 50]) and train an Inn and a Qpnn using the same procedure as for the classification case (Sec. 4.1 and Sec. 3.1) on the IMFDB and Cars datasets. The data transformation used was the same as for the classification case, except that random rotation transformations were also applied to the images during testing to create more artificial edges in the SFMs and aid in detection of outliers in a pool. Figure 6: Histogram method Confusion Matrix on IMFDB Pooled Validation Data: true #OIs (rows) versus predicted #OIs (columns) GMM and Histogram Creation:Recall the steps for GMM fitting and creation of histograms of anomaly scores in Sec. 3.3. We created $M_{\text{IMFDB}}=40850$ and $M_{\text{Cars}}=18750$ pools respectively from the training splits of IMFDB and Cars datasets and obtained their feature vectors by passing them to the trained ResNeXt-101 Qpnn. For each dataset, we fit a GMM on these feature vectors. The optimal number of clusters in the GMM for the pooled case was 30 and 6 respectively for the IMFDB and Cars dataset, and 5 and 1 respectively for the individual case. For computing the anomaly scores histograms for each dataset, a total of $N=100K$ pools were created from the validation split, each of size $r=8$ and maximum number of outliers $t=5$ (roughly 16666 pools for each pool label). We capped the number of outliers at $5$ because for small prevalence rate $p$ ranging between 0.001 and 0.1, having more than 5 outlier images in a pool of $8$ is a very low probability event. For each dataset, we passed these pools to the trained Qpnn to obtain pool feature vectors. We used the earlier GMM to obtain their anomaly scores, and computed the anomaly score histogram, with $Q=500$ bins. Recall from Sec. 3.3 and Algorithm 3 that the same histogram method can be used for testing individual images as well. For creating the histogram for testing individual images, we used 5000 off-topic and 5000 on-topic images, and used $Q=500$ bins. Hyperparameter Selection:The hyperparameters for CLasso-OD were chosen in exactly the same manner as for the classification case (Sec. 4.1). Discussion on Results:Fig. 7 shows the sensitivity and specificity of each of these algorithms for different values of $p$, for the two off-topic image detection tasks defined earlier. We see that for both IMFDB and Cars dataset, the CS/GT methods for oulier detection remain competitive with Individual. We observe that CLasso-OD has higher sensitivity than Comp-OD for the entire range of prevalence rates considered and higher than Dorfman-8-OD for $p\leq 0.01$. The reason for this has been discussed in Sec. 4.1: when a pool gets _falsely_ classified as negative (i.e., containing no outlier image), then all outlier images in that pool get classified as negative by Comp-OD and Dorfman-8-OD. In the higher $p$ regions, Dorfman-8-OD starts to perform better than CLasso-OD. This is because, as we can see in Fig. 6, the prediction model gets confused for pools having $\geq 2$ outlier images. At high prevalence rates of off-topic images, it is common for an on-topic image to test positive in all of its pools. As discussed in Sec. 4.1, Comp-OD uses only such binary information and declares incorrectly all such images as off-topic, so we observe a sharp decline in specificity. However, CLasso-OD uses the quantitative information of the number of off-topic images in the pools, and can eliminate such false positives. Comparison with Downsampling:In Fig. 7, we also compare our method to individual testing of images downsampled by a factor of $4$ (size $112\times 112$) (referred to in Fig. 7 as Downsample-OD) for the firearms task. We find that downsampling lowers the specificity of prediction and hence both CLasso-OD and Dorfman-8-OD show better specificity than individual testing of downsampled images. Computational Cost:The computational cost in terms of number of floating point operations (FLOPs) are presented in Fig. 5. Since the same neural network as in the classification case has been used, the amount of computation for each image for pooled outlier detection using CS/non-adaptive GT methods is the same. Some additional computation is needed in the GMM related steps, but it is very small (by a factor of 100) compared to computations involving the neural network. For Dorfman-8-OD, the amount of computation needed is slightly higher than for Dorfman-8. This is because in case of pooled outlier detection, $11\%$ of negative pools falsely test as positive, as compared to only $2\%$ for pooled classification (see the top-left entries in the confusion matrices presented in Fig. 6 and Fig. 3). However, we see that the computation costs for both adaptive and non-adaptive pooled outlier detection methods are well below those of individual testing. Wall-clock Times:The end-to-end wall-clock times for pooled outlier detection with CLasso decoding and using Dorfman-8-OD on the Cars Dataset for 100K images are presented for a range of prevalence rates in Table 3. We see that both CLasso-OD and Dorfman-8-OD are significantly faster than individual testing of the images for all $p$. CLasso-OD is also faster than Dorfman-8-OD for $p\geq 0.02$. Combining Pooling with Downsampling:Similar to Sec. 4.1, we performed experiments to see whether downsampling can be combined with GT and CS methods for further reduction in computation cost without negatively affecting the performance. We show the performance of our pooled outlier detection methods on IMFDB images downsampled by a factor of $4$ to $112\times 112$ in Fig. 7 (sub-figures labelled IMFDB-Downsampled, rightmost column, top and bottom row). We again note that while all the algorithms observe some drop in sensitivity, it is within an acceptable range. The specificity also drops, but to an even smaller extent. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|} \hline $p$ & Individual-OD & Dorfman-8-OD & Classo-OD \\ \hline 0.001 & 430.87 & 253.6 & 302.2 \\ \hline 0.01 & 437.5 & 281.6 & 305.6 \\ \hline 0.02 & 436.9 & 309.5 & 303.3 \\ \hline 0.03 & 437.3 & 332.9 & 304.1 \\ \hline 0.04 & 437.5 & 356.9 & 303.4 \\ \hline 0.05 & 436.7 & 376.0 & 303.2 \\ \hline 0.1 & 436.5 & 466.3 & 303.4 \\ \hline \end{tabular} \end{table} Table 3: End-to-end wall-clock times (secs), for processing 100k images of Cars+ImageNet with one NVIDIA GeForce RTX 2080 Ti GPU (11GB RAM) + one AMD 12-Core CPU, 64GB RAM. Batch-size chosen to fill GPU RAM. Conclusion and Future Work We presented a novel CS based approach for efficient automated image moderation, achieving significant reduction in computational cost for image moderation using deep neural networks. We also presented a method for pooled deep outlier detection for off-topic image moderation, bringing for the first time CS and GT methods to outlier detection. Because we output the number of OIs for each pool, recovering the original OIs is also a quantitative group testing (QGT) [43, 23] problem. Hence QGT algorithms (eg: Algorithms 1 and 2 from [23]) may also be used instead of CS for decoding within our framework. We leave an exploration of using QGT algorithms for image moderation as future work. In this work, we assumed that the errors in the different elements of $\mathbf{y}$ are independent of each other and the underlying $\mathbf{x}$. Though this produced good results, using a more sophisticated noise model is an avenue for future work. Our approach reduces the number of tests and is orthogonal to approaches such as network pruning or weights quantization which reduce the complexity of each test. In future, we could combine these two approaches, just as we (already) explored the combination of pooled testing and image downsampling. ## Appendix A Appendix ### Properties of Good Sensing Matrices In a $k$-disjunct pooling matrix, the support of any column is not a subset of the union of supports of any $k$ other columns. In noiseless binary group testing, Comp can identify upto $k$ defectives exactly if the pooling matrix is $k$-disjunct [39, Sec 2.2]. Mutual Coherence of a matrix is the maximum value of the dot product of any of its two columns. Sensing matrices with low mutual coherence are preferred for compressed sensing as they allow for reduction of the upper bounds on recovery errors [47, Theorem 1]. The $1$-norm Restricted Isometry Property (RIP-1) of order $2k$ is a sufficient condition on the sensing matrix $\mathbf{\Phi}$ for recovery of $k$-sparse $\mathbf{x}$ from noisy $\mathbf{y}$ via LASSO [10]. A pooling matrix $\mathbf{\Phi}$ satisfies RIP-1 of order $2k$ if it holds for some constants $2k$ and $\delta$ that $\\|\mathbf{x}\\|_{1}\leq\\|\mathbf{\Phi}\mathbf{x}\\|_{1}\leq(1+\delta)\\|\mathbf{x}\\|_{1}$ for all $2k$-sparse vectors $\mathbf{x}$. ### Choice of Pooling Matrix $\mathbf{\Phi}$ Randomly generated matrices from zero-mean Gaussian or Rademacher distributions have been popular in CS because they are known to obey the Restricted Isometry Property (RIP) with high probability [9], which is a well-known sufficient condition to guarantee accurate recovery [14]. RIP-obeying matrices also satisfy condition $\mathbb{C}2$ mentioned in Sec. 1 of the main paper. Despite this, such matrices are not suitable for our image moderation application, as they will lead to all $n$ images contributing to every SFM - that too in unequal or both positive and negative amounts - and complicate the job of the Qmpnn that predicts $\mathbf{y}$. Random Bernoulli ($\{0,1\}$) matrices have been known to allow for very good recovery [34] due to their favorable null-space properties. But randomly generated matrices will contain different number of ones in each row, due to which each pool would contain contributions from a different number of images. Instead, we use binary matrices which are constrained to have an equal number of ones in each of the $n$ columns, denoted by $c$ (i.e. 'column weight' $c$), and an equal number of ones in each of the $m$ rows, denoted by $r$ (i.e. 'row weight' $r$). This implies that $nc=mr$. Such a construction ensures that the Qmpnn must produce outputs constrained to lie in $\{0,1,...,r\}$, and also ensures that each of the $n$ images contributes to at least one pool (actually to exactly $c$ different pools, to be precise). These matrices can be made very sparse ($r\ll n$), which ensures that noise in the output of the Qmpnn can be controlled. We put an additional constraint, that the dot product of any two rows must be at most $1$, and similarly the dot product of any two columns must be at most $1$. It can be shown that such matrices are $(c-1)$-disjunct, obey the 'RIP-1' property of order $2c$[24, Sec. III.F], and have low mutual coherence [27], which makes them beneficial for GT as well as CS based recovery. These properties are defined in Sec. A.1. ### Recovery Guarantees for CS and binary GT Quantitative group testing (see Sec. 2), which is closely related to CS, has some conceptual advantages over binary group testing, as it uses more information inherently. We summarize arguments from [39, Sec. 2.1], [23, Sec. 1.1] regarding this: Consider binary group testing with $m$ pools from $n$ items with $k$ defectives. The total number of outputs is $2^{m}$. The total number of ways in which up to $k$ defective items can be chosen from $n$ is $\sum_{i=0}^{k}C(n,i)$. We must have $2^{m}\geq\sum_{i=0}^{k}C(n,i)$, which produces $m\geq k\log(n/k)$. If we instead consider quantitative group testing, then the output of each test is an integer in $\{0,1,...,k\}$. The total number of outputs would thus be $(k+1)^{m}$. Thus $(k+1)^{m}\geq\sum_{i=0}^{k}C(n,i)$, which produces $m\geq\frac{k\log(n/k)}{\log(k+1)}$. Thus, the lower bound on the required number of tests is smaller in the case of quantitative group testing by a factor of $\log(k+1)$. Moreover, work in [8] argues that in the linear regime where $k=O(n)$ (even if $k\ll n$), individual testing is the optimal scheme for binary non-adaptive group testing. This is stated in Theorem 1 of [8] which argues that if the number of tests is less than $n$, then the error probability is bounded away from 0. On the other hand CS recovery with binary matrices is possible with $m=O(k\log(n/k))$ measurements [10, Theorem 9]. ### Upper bound on disjunctness of column-regular matrices We explain in this section that matrices with column weight $k$ (i.e., with $k$ ones per column, equivalent with each image participating in $k$ pools) which have strictly fewer rows than columns cannot be $k$-disjunct. The work in [21, Prop. 2.1] gives an upper bound on the cardinality of a uniform $r$-cover-free family of sets. They consider the power set of a set $X$ of $n$ elements. A $k$-uniform family of sets is any set of subsets of $X$ containing exactly $k$ elements. Let $\mathcal{F}=\{S_{1},\ldots,S_{K}\}$ be a $k$-uniform family of sets of cardinality $K$. If $\mathcal{F}$ is $r$-cover-free, it means that no element of $\mathcal{F}$ is a subset of the union of any $r$ other sets in $\mathcal{F}$. The work in [21, Prop. 2.1] gives the following bound on the cardinality of $F$: \[K\leq\frac{\binom{n}{t}}{\binom{k-1}{t-1}}, \tag{12}\] where $t=\lceil\frac{k}{r}\rceil$. We can construct a $n\times K$ pooling matrix $\mathbf{\Phi}$ from $\mathcal{F}$, where each column is a $0/1$-binary vector with column weight $k$, and $\mathbf{\Phi}_{ij}=1$ indicating that the element $i$ is in the set $S_{j}$. Recall from Appendix A.1 that in an $r$-disjunct matrix, the support of any column is not a subset of the union of supports of any $r$ other columns. From the definition of disjunctness and $r$-cover-free families, if the matrix $\mathbf{\Phi}$ is to be $r$-disjunct, then the family of sets $\mathcal{F}$ must be $r$-cover free. Moreover, if $r=k$, then $t=1$, and $K\leq\frac{n}{k-1}$. Hence, matrices with column weight $k$ and fewer rows than columns i.e. $n<K$ cannot be $k$-disjunct. For the case from Sec. 3.4 where column weight can be _upto_$k=2$, we note that for any column with column weight $1$, the corresponding row in which it has a $1$ entry also has a row weight equal to $1$, otherwise the matrix cannot be $2$-disjunct. Thus, we can remove such rows and columns from the matrix, and the reduced matrix will also be $2$-disjunct and will have column weight equal to $2$ for all columns. From the above result, the number of columns in this reduced matrix must be less than or equal to the number of rows. Since we removed an equal number of rows and columns from the original matrix, the original matrix also has number of columns less than or equal to the number of rows, and hence cannot be used for achieving a reduction in number of tests using group testing. ### Noise-tolerance of balanced binary matrices with row and column dot product at most 1 The work in [24, Sec. III.F.4] shows that such matrices with column weight $c$ are adjacency matrices of $(k,\epsilon)$-unbalanced expander graphs, and consequently [10, Theorem 1] have RIP-1 for any $k<2c+1$, with $\epsilon=\frac{k-1}{2c}$, and $0\leq\epsilon<1$. Robust recovery guarantees using the RIP-1 for such adjacency matrices exist - for example, [32, Theorem 6.1] states that the $\ell_{1}$-norm of the recovery error of [32, Algorithm 1] is upper-bounded by $\frac{7-4\epsilon}{1-2\epsilon}\eta$, where $\eta$ is the $\ell_{1}$-norm of the noise in measurements. For a fixed magnitude of noise in the measurements, the amount of error in the recovered vector depends on the value of $\epsilon$, with smaller values being better. For a fixed $k$, $\epsilon$ can be made smaller by having a larger value of $c$, the column weight of the matrix. Hence matrices with higher values of the column weight i.e. ones in which each item takes part in a larger number of pools are more robust to noise for CS recovery.
2310.04186	Low-dimensional representation of intermittent geophysical turbulence with High-Order Statistics-informed Neural Networks (H-SiNN)	We present a novel machine learning approach to reduce the dimensionality of state variables in stratified turbulent flows governed by the Navier-Stokes equations in the Boussinesq approximation. The aim of the new method is to perform an accurate reconstruction of the temperature and the three-dimensional velocity of geophysical turbulent flows developing non-homogeneities, starting from a low-dimensional representation in latent space, yet conserving important information about non-Gaussian structures captured by high-order moments of distributions. To achieve this goal we modify the standard Convolutional Autoencoder (CAE) by implementing a customized loss function that enforces the accuracy of the reconstructed high order statistical moments. We present results for compression coefficients up to 16 demonstrating how the proposed method is more efficient than a standard CAE in performing dimensionality reduction of simulations of stratified geophysical flows characterized by intermittent phenomena, as observed in the atmosphere and the oceans.	Raffaello Foldes, Enrico Camporeale, Raffaele Marino	2023-10-06T12:01:40Z	http://arxiv.org/abs/2310.04186v2	# Low-dimensional representation of intermittent geophysical turbulence with H-SiNN ###### Abstract We present a novel machine learning approach to reduce the dimensionality of state variables in stratified turbulent flows governed by the Navier-Stokes equations in the Boussinesq approximation. The aim of the new method is to perform an accurate reconstruction of the temperature and the three-dimensional velocity of geophysical turbulent flows developing in homogeneities, starting from a low dimensional representation in latent space, yet conserving important information about non-Gaussian structures, captured by high-order moments of distributions, needed to reproduce embedded dynamics To achieve this goal we modify the standard Convolutional Autoencoder (CAE) by implementing a customized loss function that enforces accuracy of the reconstructed high order statistical moments. We present results for compression coefficients up to 16 demonstrating how the proposed method is more efficient than a standard CAE in performing dimensionality reduction of simulations of stratified geophysical flows characterized by intermittent phenomena, as observed in the atmosphere and the oceans. + Footnote †: preprint: Low-dimensional representation of intermittent geophysical turbulence with H-SiNN ## I Introduction Geophysical fluids are characterized by the interplay of non-linear vortices and waves, developing very complex turbulent dynamics due to the density stratification and the Earth's solid body rotation. Large scales motions at planetary scales (of the order of 1000 km) emergence from the so-called quasi-geostrophic balance [1]. At mesoscale $O(100\,\mathrm{km})$ and even more at the sub-mesoscale $O(10\,\mathrm{km})$, when the turbulent eddy turnover time becomes comparable to the characteristic time scales associated to internal waves, motions feel less the constrains of the force balance, a regime that is generically identified as _stratified turbulence_[2]. This is characterized by the presence of shears leading to instabilities, the vertical extension of strata being controlled by the Brunt-Vaisala frequency $N$. In stratified geophysical flows the probability density function (PDF) of fields fluctuations at the scales comparable with that of the mean flow can be characterized by fat tails, with a departure from Gaussianity. In particular, this behavior has been observed in the vertical velocity and/or (potential) temperature fluctuations in both the atmosphere [3; 4; 5] and the ocean [6; 7], and extensively in a variety of numerical investigations [8; 9; 10; 11; 12; 13] through the forth-order moment (i.e., kurtosis) of the vertical velocity. Marino _et al._[13] have shown how the extreme drafts responsible for the non-Gaussian behavior of the vertical component of the velocity do generate local turbulence and enhance the classical small-scale intermittency [11]. They provided as well evidence that these structures are associated with patches of enhanced kinetic and potential energy dissipation, making stably stratified flows within the regime analyzed in [9] more efficient at dissipating energy but also non-homogeneous due to the presence of strong vertical drafts. The community resorted to a variety of different techniques to model the scales at which turbulence dominates in geophysical flows, from the implementation of codes based on Reynolds averaged Navier-Stokes (RANS) equations and Large Eddy Simulations schemes (LES) [14; 15; 16], to the simulation of geophysical turbulence using a reduced models [17]. On the other hand, progresses made in the field of High Performance Computing have made it possible to perform direct numerical simulations (DNS) of stratified turbulent flows in a parameter space of geophysical interest [18; 19; 20]. Disregarding the physical model implemented, common trait of high-resolution three-dimensional simulations of geophysical flows is the large amount of data produced. For example, a DNS run with resolution of $512^{3}$ grid points in single precision requires more than 0.5Gb of storage space per field and per time-step. Retaining the output with high temporal cadence is therefore in many case not doable, which requires to find a compromise between spatial and temporal resolutions, though investigating the evolution of the fields in space and time is often needed in order to assess the complex dynamics developing in from the need to perform run-time analysis, as well as for the implementation of machine learning techniques able to learn features of the dynamical systems from the latent space [21; 22; 23]. The rationale behind the dimensionality reduction of the output of turbulent flows simulations is therefore twofold: on the one hand this is done in order to store data for post-processing at higher temporal frequency, while it also enables the efficient training of data-mining and machine learning techniques. The study of low dimensional manifolds has a long history in dynamical systems and in turbulence [24; 25], laying at the foundation of so-called reduced order methods (ROMs) [26]. In turn, ROMs have also recently benefited by progresses in machine learning based approaches [27; 28]. Dimensionality reduction techniques such as proper orthogonal decomposition [29; 30] and dynamic mode decomposition [31; 32] have been extensively used to address is sues concerning a variety of flows [33; 34; 35; 36; 37]. Other examples of model order reduction techniques applied to fluid frameworks are Galerkin-projection based nonlinear methods [38] and system identification based auto-regressive models [39; 40]. Techniques based on machine learning principles that have been employed in modeling the equations of motion are Gaussian process regression [41; 42], symbolic regression [43; 44], field inversion [45; 46], and artificial neural networks (ANNs). In data-driven methods, ANNs are the most powerful tools in terms of their complexity, ability to generalize and to capture highly nonlinear phenomena, typical of turbulent processes. Other developments have focused on improving reliability and accuracy of low fidelity models (i.e., RANS and LES), by using data from high fidelity simulations, e.g. DNS, to learn proper closures [47; 48]. Convolutional neural networks (CNNs) have been successfully applied to the identification of flow structures [49] and to perform nonlinear modal decomposition in turbulent flows [50; 51]; more recently, CNN-based architectures have been extended to fully 3D DNS of turbulent channel flow [52] and to numerical simulations of flows characterized by the emergence of extreme events, in terms of boundary coherent structures [53]. These implementations emphasized the advantages of using CNNs over traditional methods based on principal component analysis (PCA), the former being able to capture the intrinsic nonlinear dynamics of turbulent flows. More generically, deep learning proved to be a powerful tool for the analysis and generation of reduced-order models of turbulent systems. In terms of machine learning applications to stratified flows, CNN-based deep learning has recently been applied by [54] to get parametrization of diapycnal mixing using data from DNS. Convolutional AutoEncoders (CAE) are a family of autoencoders that proved successful in extracting information from two- and three-dimensional data [55] and their use has recently been proposed in the context of fluid-dynamics [56; 57; 58]. However, most of these approaches present severe limitations if implemented without proper tuning, and their use is often constrained to idealized and/or simplified cases due to the major constraints that come from capturing transient behavior [59]. Conversely, the presence of large-scale transient phenomena developing in geophysical flows, such as hurricanes, tornadoes, oceanic fronts, etc., can be viewed as an obstacle to the creation of reduced-order manifolds sufficiently informative to reliably recover properties and dynamical features of geophysical fluid systems. Multi-scale models, such as global circulation models (GCM), are powerful tools for the investigation of the Earth's atmosphere and the oceans which would greatly benefit from the possibility to simulate complex fluid dynamics on a dimensional phase space lower than what can be achieved numerically. The efficiency of machine learning approaches is reduced when applied to systems with strong transients, such as the non-stationary states resulting from the presence of intermittent phenomena in turbulent flows. The purpose of the study presented here is precisely to develop a machine learning tool based on CAE able to produce reliable low-order representations of stratified geophysical turbulent flows in a range of parameters characterized by developing intermittent (in space and time) vertical velocity drafts, leading to the departure of the vertical velocity statistical moments from the Gaussian reference. We show that standard CAE architectures are well suited to learn a latent space containing enough information to reconstruct the general features of the original stratified turbulent flow. However, strong vertical drafts developing in a certain regime of the governing parameter, and in general high order statistics of velocity and temperature in presence of large- and small-scale intermittency, will typically not be well recovered. To overcome this issue, we introduce here a novel implementation of CAE: statistics-informed convolutional autoencoder (or SiCAE). The idea is based on a more general approach, applicable to any neural network, which we refer to as high-order statistics-informed neural network (H-SiNN). The concept of statistics-informed neural network (SiNN) has been first introduced by [60], where it has been proposed the addition of two terms to the loss function whose effect is to constrain both the PDF and the auto-correlation function of the reconstructed fields. On the other hand, in our implementation, the loss function explicitly enforces the preservation of high-order moments and this is done in order to ensure consistency between the statistics of the original and the reconstructed flow fields. Indeed, it is worth recalling that statistical moments of the velocity distribution function play a crucial role in the characterization of turbulent frameworks, being directly related to invariants such as energy and enstrophy, to other fundamental quantities, such as the dissipation occurring at the small scales. We will demonstrate here the enhanced capability of the H-SiNN in recovering finer statistical features as well as flow inhomogeneitis and intermittent dynamics with respect to a standard CAE implementation. The manuscript is organized as follows: in Sec. II the numerical simulations performed and analyzed are briefly introduced along with a description of the equations and parameters governing the dynamics of the stratified flows under study; Sec. III explains the classical CAE architecture, and in Sec. IV our implementation of statistical-informed CAE is thoroughly described; finally, Sec. V shows in detail the outcome of the application of the model reduction tool we developed to DNS of stratified turbulent flows developing large-scale intermittent events in the vertical component of the velocity and in the temperature field [9]. ## II DNS solver for stratified turbulence The CAEs are trained using the output of a direct numerical simulation of a stably stratified turbulent flow obtained by integrating the Navier-Stokes equations in the Boussinesq approximation in which the velocity field $\mathbf{u}=(\mathbf{u}_{\perp},w)$ remains incompressible, $\nabla\cdot\mathbf{u}=0$, while small density variations are taken into account only in the buoyancy term. Such a model can be written as: \[\partial_{t}\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p-N\theta\mathbf{\hat{z}}+\nu\nabla^{2}\mathbf{u}+ \mathbf{F} \tag{1}\] \[\partial_{t}\theta+(\mathbf{u}\cdot\nabla)\theta = Nw+\kappa\nabla^{2}\theta. \tag{2}\] with $\theta$ being the temperature fluctuations evaluated relative to a mean temperature profile $\theta_{0}$, $N=[-g\partial_{z}\bar{\theta}/\theta_{0}]^{1/2}$ is the Brunt-Vaisala frequency. Such a simulation has been initialized with zero temperature fluctuations and random velocity modes applied at large-scale, in a Fourier shell centered at $k_{0}=[2,3]$; the size of the computational box is associated with $k_{min}=1$, with $k_{min}=2\pi/L_{0}$ and the resolution is $512^{3}$ grid-points. A random forcing $F$ is imposed to the momentum equation at $k_{F}=2\pi/L_{f}\in[2,3]$, continuously injecting kinetic energy into the system and allowing to reach a turbulent stationary state. In the above equations, $\nu$ and $\kappa$ are the kinematic viscosity and the thermal diffusivity, respectively and we take the Prandtl number $\nu/\kappa=1$, with $\nu=10^{-3}$; finally, $p$ is the pressure. We adopt the following definitions for the dimensionless Reynolds and Froude numbers, \[\mathrm{Re}=U_{rms}L_{int}/\nu,\ \mathrm{Fr}=U_{rms}/[L_{int}N]\, \tag{3}\] with $U_{rms}$ and $L_{int}$ the characteristic (rms) velocity and the integral scale of the fluid, respectively. These parameters, and in particular the buoyancy Reynolds $R_{B}=\mathrm{ReFr}^{2}$, measure the relative strength of buoyancy to dissipation and are commonly used to distinguish between wave-dominated and turbulence-dominated regimes [61]. The equations (1-2) are integrated numerically using the Geophysical High-Order Suite for Turbulence (GHOST), a pseudo-spectral code which employs a hybrid parallelization combining MPI, OPENMP and CUDA [62; 63]. It allows for a variety of physical solvers and supports non-cubic geometry [64], and non-periodic boundary conditions [65]. In the following, we will make use of the statistical moments of a distribution function up to the forth-order. The definition of the third- and forth-order moments, namely the skewness $Sk_{w}$ and kurtosis $K_{w}$ respectively, of the distribution of vertical velocity $w$ is: \[Sk_{w}=\frac{\left\langle(w-\bar{w})^{3}\right\rangle}{\left\langle(w-\bar{w} )^{2}\right\rangle^{3/2}}, \tag{4}\] \[K_{w}=\frac{\left\langle(w-\bar{w})^{4}\right\rangle}{\left\langle(w-\bar{w} )^{2}\right\rangle^{2}}, \tag{5}\] where averages can be either computed over the simulation volume or others ensembles, such as horizontal planes. It is worth recalling that, $Sk=0$ and $K=3$ are the reference values for a Gaussian distribution. On the contrary, for values of $K_{w}$ higher than 3, PDFs will display fat (non-Gaussian) tails, indicating that the variable presents extreme values which deviate consistently from the mean value. For this reason, the kurtosis $K$ has been largely employed to characterize the emergence of large-scale intermittency in the vertical velocity field $w$ (and temperature $\theta$) in numerical simulations [9]. As observed by Feraco _et al._[9], $K_{w}$ presents strong non-Gaussian values in a narrow range of the Froude number, with an evident peak at $Fr\approx 0.076$, being compatible with values observed in the upper atmosphere and oceans. The following analysis is performed using the same run which has been comprehensively studied in Feraco _et al._[9], and subsequent works [11; 13], presenting the highest level of intermittency (see Fig.2 of Feraco _et al._[9]). ## III Convolutional autoencoder theory An autoencoder is an unsupervised feed forward artificial neural network (ANN) that aims to reconstruct a given data set through a process involving data compression and recovery [66]. Indeed, its main objective is to create a reliable and reduced representation of the input data set, being particularly helpful for either creating reduced model or simpler but still informative representation of high-dimensional data. The network consists of an encoder part $P:\mathbb{R}^{D}\rightarrow\mathbb{R}^{d}$ (with $d<D$), which compresses the input data into a smaller space representation (or latent space), and a decoder part $Q:\mathbb{R}^{d}\rightarrow\mathbb{R}^{D}$, which reconstructs the encoded data back to the original input dimension. When the mapping kernel is linear, the autoencoder can be considered as a singular value decomposition analysis. An autoencoder that employs convolutional layers to perform the encoding and decoding operations on images, or more in general on 2D (or higher) data, is generally termed Convolutional AutoEncoder (CAE). During the convolution operation, the kernel swipes the data domain to extract features and learn spatial and/or temporal dependencies. The process is then carried out across multiple layers, obtaining representative features in a hierarchical sense [67]; CAEs with many hidden layers can be considered as deep neural networks (DNN). The initial layers learn a low-level representation or particular details about the data, while subsequent layers focus on larger and larger-scale information. In our work we developed four CAEs with different compression factors (CFs), defined as the ratio between the input and the latent dimensions, from 2 to 16; an example of a general CAE architecture used in our implementations is shown in Fig. 1. In Tab. 1 we report the main characteristics of the four CAEs implemented here; starting from CF2, we obtained higher CFs by adding more hidden layer to the initial architecture, but trying to keep the number of parameters as fixed as possible in order to make a more fair comparison of the four networks performance. Indeed, the number of weights and biases, as highlighted by the values in Tab. 1, varies only by $\approx 10\%$. On the other hand, more hidden layers allow networks with higher CF to learn more features of the original fields during the compression step. The number of hidden layers is intended for the entire CAE architecture, comprising both the layers of the encoding $n_{e}$ and decoding $n_{d}$ part; they usually have an amount of layers which follow the relation $n_{d}=n_{e}+1$, while the CAE is symmetric if the total number of layers is even. \begin{table} \begin{tabular}{c c c c c} \hline Id & Compression factor & No. Layers & No. params. & Latent space \\ \hline CF2 & 2 & 7 & 321,156 & $64\times 64\times 128$ \\ CF4 & 4 & 8 & 339,892 & $32\times 32\times 256$ \\ CF8 & 8 & 9 & 324,772 & $32\times 32\times 128$ \\ CF16 & 16 & 11 & 349,804 & $16\times 16\times 256$ \\ \hline \end{tabular} \end{table} Table 1: Description the four CAEs architecture: the compression factor (CF), the number of layers referring to both the encoding and decoding part of the network. The number of parameters account for all the weights and biases which have to be optimized during the training phase, and the latent space is the reduced space after the encoding network is applied. ## IV High-order statistics informed neural network: H-Sinn The aim of this work is to devise a method that attains a compression of the data volume of DNS for stratified turbulence flows, yet preserving their statistics (up to the fourth-order moment, kurtosis). Mathematically, this goal can be cast as a multi-objective optimization (MOO) problem, and one of the greatest advantage of using a neural network to solve MOO problems is that, in principle, it is sufficient to include any term that one wish to minimize in the cost function and, as long as such terms are differentiable, the neural network will find a solution that approximately minimizes the objective (possibly a local minimum) through the back-propagation procedure. This is known as _scalarization method_, where the different terms are s simply added together, each one weighted by its own scalar factor [68]. Determining those weights is crucial, especially when employing ANNs, to make sure that the cost function is not dominated by one or a few terms only and that the solution actually minimizes each term. Intuitively one can think of the weights as normalization factors when different terms in the cost function have different order of magnitudes. Our implementation of H-SiNN involves the use of a statistics-aware loss function into a CAE, thus creating what we call statistics-informed convolutional autoencoder (SiCAE). As mentioned above, we focus on up-to-fourth order moments of the vertical velocity distribution, since we already know that it presents the highest variability in our simulations. In particular, our aim is to preserve standard deviation, skewness, and kurtosis of the vertical component of the velocity field $w$, which we indicate with the symbols $\sigma_{w}$, $Sk_{w}$, and $K_{w}$, respectively. As a result, the cost function of the CAE computed between the original data set $y$ and its reconstruction $\tilde{y}$ is defined as: \[\begin{split}\mathcal{L}(y,\tilde{y})&=\frac{1}{D} \left(W_{1}\sum_{y}^{[u,v,w,\theta]}\sum_{i=1}^{N}\frac{(y_{i}-\tilde{y}_{i}) ^{2}}{N}+W_{2}\sum_{i=1}^{N}\frac{(\sigma_{w_{i}}-\sigma_{\tilde{y}_{i}})^{2} }{N}\right.\\ &+W_{3}\sum_{i=1}^{N}\frac{(Sk_{w_{i}}-Sk_{\tilde{w}_{i}})^{2}} {N}+W_{4}\sum_{i=1}^{N}\frac{(K_{w_{i}}-K_{\tilde{w}_{i}})^{2}}{N}\Bigg{)}\\ &=\frac{1}{D}\sum_{j=1}^{4}W_{j}R_{j}\end{split} \tag{6}\] with $D=3(W_{1}+W_{2}+W_{3}+W_{4})$, being the total sum of the weights. The second, third and forth term in (6) are the mean square errors (MSEs) between the moments of the original and the reconstructed vertical velocity $W$ computed over horizontal planes ($x$,$y$), namely the standard deviation $\sigma_{w}$, the skewness $Sk_{w}$ and kurtosis $K_{w}$ respectively. We recall that for a normal distribution the reference values are: $\sigma=1$, $Sk=0$ and $K=3$. In addition, here, we propose a method for dynamically weight the four terms in the cost function; each term is properly weighted with values $W_{i}$ varying at each epoch during the training phase. In order to obtain a loss function with terms having the same order of magnitude during the whole training procedure at a given epoch $m$, the weight of the $i$th term $W_{i}$ is proportional to the sum of the other three terms at the previous epoch $m-1$, e.g. $W_{i}^{m}=\sum_{j\neq i}W_{j}^{(m-1)}R_{j}^{(m-1)}$ (and normalized such that their sum is equal to 1). Even though the various terms of the statistics-informed loss function are well balanced, the training procedure with additional constraints either coming from the knowledge of the physical system, as it happens for physics-informed neural networks, or from statistical information can be difficult, slower and sometimes very noisy. Finally, we have heuristically verified that the strategy which is computationally faster to train a H-SiNN is in two stages. In the first stage we treat the autoencoder as a classical CAE, with only the first term in Eq. 6. After we train for 50 epochs, we add the remaining terms in the cost function, i.e. starting from a network that achieves a low reconstruction error, in terms of mean error. During the second stage we train the CAEs for 100 more epochs with the addition of statistical constraints on the first four moments of the vertical velocity PDF; this is shown in panels (b)-(f) of Fig. 2. The behavior of the different terms of the loss function during one hundred epochs of training is shown in Fig. 3. We can appreciate from the four panels how there are essentially three weights ($W_{MSE}$, $W_{\sigma_{w}}$ and $W_{Sk_{w}}$) which are nearly equal for the entire training phase, whereas the coefficient weighting the kurtosis term is significantly smaller indicating a larger error on the forth-order moment, as expected. ### Plane-by-plane approach We mentioned that the main objective is to obtain a tool which is able to reproduce most of the features of the velocity $\mathbf{u}$ and temperature $\theta$ fields of a fluid presenting high anisotropy (stratification) and non-homogeneity, due to the presence of large-scale intermittent structures. For these last Figure 1: Schematic representation of a general Convolutional Autoencoder (CAE). The dimension of the _latent space_ and the number of encoding/decoding layers depend on the compression factor (CF). For each time step, the input is represented by horizontal slices ($512^{2}$) of the four physical variables produced by the simulations ($u$, $v$, $w$ and $\theta$) stacked together forming a volume of $512^{2}\times 4$. The number of filters and layers composing the encoder (and decoder) part also depend on the CF, all the CAE implemented in this work are symmetric. reasons, we further implemented the standard CAE algorithm to make it more suitable and better performing in our particular application. A key ingredient of stratified flows is anisotropy, introduced by gravity, which tend to suppresses vertical motions with respect to the horizontal. Therefore, even though we deal with fully three-dimensional simulations the original cubic volume is split in horizontal planes along the vertical direction ($z$) before being used as input for the CAEs. This means that, for each time and height value $z$, planes for every velocity component ($u$, $v$ and $w$) and for $\theta$ are stacked together creating three-dimensional input data with dimensions $512\times 512\times 4$, as indicated in Fig. 1. Before being divided, the data are normalized $\hat{x}=(x-\mu)/\sigma$ using the average $\mu$ and standard deviation $\sigma$ computed on each simulation cube. In this approach, we believe that passing the information from the three components of the velocity, together with the temperature, is essential for the CAE to better reconstruct the flow dynamics at a given time $t$ and altitude $z$, since their variation is strictly correlated within the primitive equations. This approach has the shortcoming that it is not possible to add an important information we know about the velocity field, that is the incompressibility $\mathbf{\nabla}\cdot\mathbf{u}=0$, since working plane by plane we cannot add constraints on the vertical derivative of the velocity $\partial_{z}w$. This leads to values for the maximum and average divergence of $\mathbf{u}$ reported in Tab. 2, where we can see that even if on average the condition is well satisfied in the reconstructed fields, locally this is not true while the condition is satisfied also point-wise for the original field. ## V Results We analyze in details the output of a single simulation run for more than $45\tau_{NL}$ corresponding to 100 time steps. Since in our approach we divide each simulation cube in planes along the vertical direction which are then treated as independent during training, the data set we use for training and testing our CAEs comprises $N=51,200$ samples, each with dimension $512^{2}\times 4$ ($512^{2}$ grid-points for four variables $u$, $v$, $w$ and $\theta$). We divide the data set in training and test set on a temporal basis, taking the first 70% snapshots for the training phase and the remaining 30% for testing the performance. The strategy described above has been used to train four different statistics-preserving CAE, with compression factors (CF) of 2, 4, 8, and 16. Figure 2 shows the history of the different terms in the loss function, for increasing number of epochs. Red, green, blue, and cyan lines are respectively for CF 2, 4, 8, and 16. The first stage is made of 50 epochs, and in the second stage the networks are trained for an additional 100 epochs. The interesting feature, valid for all CFs, is that after the first stage (panel a) the MSE does not further decreases (panel c), while all other moments decrease by one order of magnitude or so (panels d-f). This shows that the second stage of training achieves the goal of minimizing the error of standard deviation, skewness, and kurtosis, without significantly degrade the point-wise accuracy of the reconstruction. \begin{table} \begin{tabular}{c\|c\|c c c c} & Original & CF2 & CF4 & CF8 & CF16 \\ \hline \hline $\|(\mathbf{\nabla}\cdot\mathbf{u})\|$ & $\|\times 10^{-11}\|$ & 0.42 & 1.83 & 3.26 & 2.37 & 2.06 \\ $max\left\{\|\mathbf{\nabla}\cdot\mathbf{u}\|\right\}$ & $5.9\cdot 10^{-4}$ & 538 & 410 & 573 & 530 \\ \hline \hline \end{tabular} \end{table} Table 2: Average and maximum value of the divergence of the velocity field after the reconstruction with the different CAE. Figure 3: History of the weights multiplying the different terms of the loss functions: the mean squared error (MSE, black), the standard deviation ($\sigma_{w}$, orange), the skewness ($Sk_{w}$, blue) and the kurtosis ($K_{w}$, gray). Figure 2: Panel (a) shows the mean squared error (MSE) during the first 50 epochs of the training stage where no other terms are added to the loss function. In panel (b) the total loss during 100 epochs of the second training stage, in which the loss function is composed by four terms which are represented in panels (c)–(f) for the various CAE with different compression factor (CF). ### Global statistical properties of reconstructed fields The trend of the statistical moments error seems to be inversely proportional to the order of moment, meaning that the improvement on the reconstructed standard deviation (2nd-order moment) is significantly larger than what is achieved for the kurtosis (4th-order moment) within the same amount of epochs. This is also pointed out by Fig. 3 where, for the four different SiCAE, the trend of the various weights balancing the loss function terms is reported as a function of the epoch; let us recall that each weight $W_{i}$ is inversely proportional to the sum of the three contribution, given from the other moments, taken at the previous epoch $~{}1/\sum_{(i\neq j)}W_{j}$. In addition, by looking at the behavior of the loss weights (Fig. 3), we observe that the error on the kurtosis is approximately between one and two orders of magnitude greater than the others. Such a large difference may be due to the high variability of $K_{w}$ in this simulation, whereas standard deviation and skewness are expected to vary less and to be closer to reference values of a Gaussian distribution. Obviously a trend with the compression factor is observed, and, as it is expected, the higher CF the larger the reconstruction error both on the mean field and the statistical moments; this is not related to the newly introduced loss function since already during the first training stage (panel (a)) this trend can be clearly observed. The details of the results obtained on the test set for the various terms involved in the loss function during the two training stages are summarized in Tab. 3. We can see that including other terms into the loss function results in an average reconstruction error 2-3 times greater than what obtained after the first training stage, even though it remains on the order of MSE$\sim 10^{-4}$, consistent with other results found in literature [69, 70]. In addition, when constraints on the statistical moments are added to the model, we observe that the reconstruction of high-order moments improves up to 10 times for any compression factor. We selected an interval of the numerical simulation where isolated extreme bursts occur, producing evident peaks of the kurtosis $K_{w}$ (see Fig. 4, panel (a)). In Fig. 4 the vertical velocity kurtosis computed on the original data (black dashed curve) is shown as a function of the turnover time for the entire test set, and compared with the kurtosis obtained from the CAEs reconstructions (different colors for different CF Figure 4: Behavior of the kurtosis of the vertical velocity $K_{w}$ over time for the test set. Panels (a) and (c) show $K_{w}$ obtained for the different compression factors (CFs) in different colors (top) and the mean absolute relative error on the reconstructed kurtosis (bottom) during the first training stage when the loss function is composed by the MSE only. The same results are shown in panels (b) and (d) for the second training stage where the modified loss is adopted. Gray circles in panel (a) represent three times analyzed in the following figures with very different values of kurtosis $K_{w}$ of the vertical velocity. Figure 5: For each CAE, the behavior over time of the difference between the mean absolute percentage errors (MAPE) on the kurtosis $K_{w}$ of the vertical velocity between the first $\text{MAPE}^{(1)}_{K_{w}}$ and second training stage $\text{MAPE}^{(2)}_{K_{w}}$. The MAPE is defined as $\text{MAPE}^{(i)}_{K_{w}}=\|(K^{\prime(i)}_{w}-K^{\prime\prime}_{w})/K^{\prime \prime}_{w}\|\times 100$ with $K^{\prime\prime}_{w}$ and $K^{\prime\prime(i)}_{w}$ being the kurtosis of the original and reconstructed fields, respectively. Figure 6: PDF of the vertical velocity $P(w)$ computed after the first training stage (left panel) and at the end of the entire training stage (right panel), taken at $t=47.9$. We can clearly notice that, despite the PDF core show a perfect agreement during the stage one yet, the fat tails of the distribution are consistently recovered only after the introduction of the custom loss function during the second training stage (right panel). \begin{table} \begin{tabular}{l c c c c c c} & CAE & MSE & $\sigma_{w}$ err. & $S_{w}$ err. & $K_{w}$ err. & Loss \\ & & $[\times 10^{-4}]$ & $[\times 10^{-5}]$ & $[\times 10^{-4}]$ & $[\times 10^{-2}]$ & $[\times 10^{-4}]$ \\ \hline \multicolumn{6}{l}{1st stage} \\ “standard loss” & & & & & \\ \hline \multirow{3}{}{} & CF2 & 0.39 & 0.13 & 10.9 & 18.4 & 0.39 \\ & CF4 & 0.55 & 0.45 & 13.2 & 22.4 & 0.55 \\ & CF8 & 0.54 & 0.82 & 19.1 & 19.0 & 0.54 \\ & CF16 & 1.1 & 5.3 & 44.9 & 41.7 & 1.1 \\ \hline \multicolumn{6}{l}{2nd stage} \\ “statistical-informed loss” & & & & & \\ \hline \multicolumn{6}{l}{CF2} & 0.67 & 1.03 & 1.9 & 0.04 & 0.44 \\ & CF4 & 1.0 & 1.0 & 3.2 & 1.4 & 4.0 \\ & CF8 & 1.3 & 0.074 & 3.6 & 1.7 & 2.3 \\ & CF16 & 2.3 & 0.32 & 12.0 & 5.9 & 10.0 \\ \hline \hline \end{tabular} \end{table} Table 3: Results of the four CAE on the test set for both the first (top) and second (bottom) training phase. values). Panels (a) and (b) show a comparison obtained respectively after the first and second training phase; the relative error between the original $K_{w}^{a}$ and recovered $K_{w}^{r}$ kurtosis is shown in panels (c, first phase) and (d, second phase). This figure clearly highlights how the second stage of training (panels (b) and (d)) significantly improves the reconstruction of the fields up to the forth-order moment of the vertical velocity. All the CAEs show a significant reduction of the relative error, with a particular improvement in times where volume averaged $K_{w}$ exhibits highly non-Gaussian values ($t\approx 503\tau_{NL}$ and $t\approx 520\tau_{NL}$). In addition, with the application of the modified loss we are able to obtain percentage errors on the kurtosis smaller than $10\%$ on average. However, the improvement in the reconstruction of the forth-order moment is obtained also for values close to the Gaussian reference ($K_{w}\approx 3$). Indeed, it is evident that our approach can be successfully extended to other situations where the PDFs of the dynamical fields are far from Gaussian-like shape, as it is the case for the concentration of a passive scalar [71] or for the velocity in the boundary layer [72; 3]. The statistical moments up to the forth-order embedded into the loss function guarantee an optimal reconstruction of the statistical properties of the physical fields without any prior knowledge about the shape or variability of their distribution function. The improvement we obtain with the custom loss is quantified in Fig. 5 where we show the difference between the mean absolute percentage error (MAPE) computed on $K_{w}$ after the first and second training stage, as a function of time (once again, note that the training is performed by treating each plane and each time independently, i.e. no correlation along the $z$ direction or in time is taken into account). The improvement reaches up to $\approx 40\%$ at times with high values of $K_{w}$ but in general an increasing of nearly $10\%$ is observed at all times. In Fig. 6 we show the PDFs computed on the entire volume at time $t\approx 520\tau_{NL}$ (third gray circle in Fig. 4), when $K_{w}\simeq 6.8$, using both the original data (black) and data obtained from the four CAEs (colored curves). We notice how after the first training stage (left panel) the PDF core is already reliably recovered by the decoder, even though the tails significantly differ. In particular, the difference between original and reconstructed seems increasing for higher $\|w\|$. The PDF in the right panel of Fig. 6, obtained after the second training stage, completely resembles the one computed from the original data, confirming that enforcing the statistical moments up to the forth-order is sufficient to enforce compatible PDFs. In Figure 7: Left column: 3D renderings of the vertical velocity $w/\sigma_{w}$, where the most extreme structures have been highlighted in solid colors. Right column: Vertical profile of the vertical velocity kurtosis $K_{w}(z)$ of the original field (black dashed) and of the fields reconstructed with the SiCAE (colored, see legend). The vertical gray dash-dotted line (mid panel) represent the time and height taken for Fig. 9. Let us notice that the kurtosis is in logarithmic scale and the three panels show a different y-range for a clearer visualization. Fig. 7 we show an example of large-scale intermittent structures observed at three different times, indicated by the gray circles in Fig. 4. The three-dimensional renderings represent the vertical velocity field $w$ for low (top, $t\simeq 506\tau_{NL}$), medium (center, $t\simeq 520\tau_{NL}$) and high (bottom, $t\simeq 502.5\tau_{NL}$) values of the kurtosis $K_{w}$, where values $\|w/\sigma_{w}\|>4$ are highlighted in solid color while smaller values are depicted as a transparent blue. Aside each rendering the vertical profile of the kurtosis $K_{w}$ computed plane-by-plane with the original data (black), as well as with the reconstructed (colored for the different CFs) is shown. Also for the top panel, when the kurtosis is small $K_{w}\simeq 3.8$ there are planes reaching values of $K_{w}\approx 6$, thus indicating a strong spatial variability even when the global kurtosis is close to the Gaussian reference value. These effects are enhanced when more vertical drafts develop within the flow (center and bottom) with kurtosis more than one order greater than the Gaussian reference values, as it shown in logarithmic scale in Fig. 7. ### CAE reconstruction of intermittent structures We observed that the statistical properties, and in particular the kurtosis, of the vertical velocity are well recovered by the proposed implementation of statistics-informed CAEs. However, the presence of large-scale extreme events represents a challenge for their intermittency in time but also in space, creating patches of enhanced turbulence and regions populated by transient coherent structures. In addition, the low-dimensional representation of turbulent flows would represent a valuable result if the reconstruction produces truthful physical fields with local features which realistically resemble the dynamical quantities produced by the evolution of the governing equations. Therefore, in this section we analyze how the introduction of global statistical terms to the loss function locally affects the reconstruction of the entire velocity field $\mathbf{u}$, as well as the potential temperature $\theta$. In Fig. 8 we report a three-dimensional visualization of the vertical velocity $w$ obtained with the four CAEs, as well as the original field for comparison. Data are taken at time $t\simeq 520\tau_{NL}$, with relatively high kurtosis, that is $K_{w}\approx 6.8$. Strong extreme vertical drafts, identified as values with $\|w/\sigma_{w}\|>4$, have developed in the mid-bottom part of the domain. The same kind of structures can be noted in the reconstructed renderings, up to the maximum level of compression (i.e., CF16); in fact, at $CF=16$ the shape and location of drafts starts being more noisy and deviating from the original field. The gray transparent shading represents velocity values below the threshold of $4\sigma_{w}$, and therefore the majority of the volume. The reconstruction made by the implemented CAEs involves also the other components of the velocity field ($u$ and $v$), as well as the (potential) temperature fluctuations $\theta$. In order to have a general overview Figure 8: Three-dimensional visualizations, taken at $t/\tau\approx 520$, of the extreme events in the vertical velocity $w$ from the original data (big cube), alongside the field reconstructed by the four SiCAE (small cubes). The vertical drafts ($\|w/\sigma_{w}\|>4$) are highlighted as solid colors (red/blue for positive/negative), whereas the other regions are represented as gray shaded areas of how the neural networks recover all the physical quantities interested in the analyzed DNSs we represent them with several panels in Fig. 9. This figure shows an horizontal cut ($x$,$y$) of the simulation domain taken at the same time of the previous figure (Fig. 8 at the height $z^{}$ indicated with a dash-dotted line in Fig. 7 (middle panel). The columns of Fig. 9 refer to the three components of the velocity field $\mathbf{u}=(u,v,w)$ and the temperature $\theta$. All the images are represented by the same color bar which is not shown since the main objective of this figure is the comparison between the first row, the original data, and the others, being the reconstructed physical fields for increasing values of the compression factor. Since we are looking at the domain from top (gravity is a vector entering the page), it is correct to have horizontal components strongly dominated by a nearly constant positive (red) for $v$ and negative (blue) for $u$ mean wind (see color bar); this is indeed the Figure 9: Horizontal slices of the four physical fields, from left to right: horizontal component of the velocity $u$ and $v$ (left and mid-left), the vertical component $w$ (mid-right) and the potential temperature $\theta$ (right) taken at $t/\tau\approx 520$ with a kurtosis $K_{w}(z^{*})=10.0$. From the second line to bottom the reconstructed field obtained with different CAE having increasing compression factor (CF). effect of Vertical Sheared Horizontal Winds (VSHWs) which are ubiquitous in stratified flows. Nevertheless, also the horizontal components of the velocity show small-scale perturbations where extreme vertical drafts develop, as already seen throughout this manuscript. The extreme events developed in this snapshot are clearly visible in the vertical velocity $w$ and partially from the temperature renderings (third and forth columns in Fig. 9), and the same detail is captured also by the field reconstructions. Indeed, as already noticed for the three-dimensional visualization, the reconstruction is very reliable up to $\mathrm{CF}=16$ where a significant checkerboard effect starts developing everywhere in the domain. This is probably due to the combined effect of the high compression factor and of the statistical-informed loss function presenting large-scale statistical constraints. In fact, by looking at the reconstructed fields after the first stage of training (not shown here), we can observe the same artifact at high compression but it is slightly reduced for the absence of other terms into the loss function. ## VI Conclusion The work presented in this manuscript is an example of how the capability of machine learning tools to reproduce dynamical fields in fluid mechanics can be enhanced by incorporating physical and/or statistical knowledge about the system under study. In particular, we proposed a novel implementation of convolutional autoencoder (CAE) which include a modified loss function able to preserve the statistical moments of the vertical velocity field up to the forth order. We focused on the vertical component of the velocity field produced by the DNS of a stratified turbulent flow developing strong velocity drafts along the z direction, as is observed in the atmosphere and in oceans, which poses a challenge in finding an informative low-order representation of the physical fields. The comparison between a standard implementation of CAE and the SiCAE shows how the introduction of additional terms in the cost function, enforcing high-order moments of the PDF, do significantly improve the reconstruction of peculiar features of stratified geophysical flows, yet maintaining an acceptable level of the error in the reconstruction of the mean field. The various diagnostics we implemented all confirm that the novel implemented algorithm is able to reconstruct with good accuracy physical fields characterized by highly variable PDF (both in space and time). In the physical framework considered, the emergence of vertical drafts affect the local dynamics of stratified turbulent fluids, affecting their statistical description which not only can be non-stationary but is also non homogeneous. These results can potentially be generalized to other contexts and physical variables, to address for instance the reconstruction of intermittent passive scalar fields [73], in order to investigate their diffusion properties, or the velocity field obtained from observations of the planetary boundary layer [72; 3] and the upper atmosphere [74; 75]; strongly variables PDFs have also been detected in the oceans by means of GOMs [76] and using data from observations [6], further widening the application of the analysis proposed here, which can be very helpful in managing large volume of geophysical data from high-resolution numerical simulations and state-of-art observational campaigns. The addition of other terms to the loss function makes it necessary to have longer training phase and larger data set in order to converge to a well-trained model. Indeed, very often a compromise needs to be made between having detailed reconstructions of the mean fields dynamics and the reproducing peculiar structures of the turbulent flows, which is precisely where the algorithm presented improve the classical methodological approach based on CAE. ###### Acknowledgements. R.M. and R.F. acknowledge support from the project "EVENTFUL" (ANR-20-CE30-0011), funded by the French "Agence Nationale de la Recherche" - ANR through the program AAPG-2020. E.C. was partially supported by NASA grants 80NSSC20K1580 "Ensemble Learning for Accurate and Reliable Uncertainty Quantification" and 80NSSC20K1275 "Global Evolution and Local Dynamics of the Kinetic Solar Wind". ## Data Availability Statement The data that support the findings of this study are available from the corresponding author upon reasonable request.
2304.04757	A new perspective on building efficient and expressive 3D equivariant graph neural networks	Geometric deep learning enables the encoding of physical symmetries in modeling 3D objects. Despite rapid progress in encoding 3D symmetries into Graph Neural Networks (GNNs), a comprehensive evaluation of the expressiveness of these networks through a local-to-global analysis lacks today. In this paper, we propose a local hierarchy of 3D isomorphism to evaluate the expressive power of equivariant GNNs and investigate the process of representing global geometric information from local patches. Our work leads to two crucial modules for designing expressive and efficient geometric GNNs; namely local substructure encoding (LSE) and frame transition encoding (FTE). To demonstrate the applicability of our theory, we propose LEFTNet which effectively implements these modules and achieves state-of-the-art performance on both scalar-valued and vector-valued molecular property prediction tasks. We further point out the design space for future developments of equivariant graph neural networks. Our codes are available at \url{https://github.com/yuanqidu/LeftNet}.	Weitao Du, Yuanqi Du, Limei Wang, Dieqiao Feng, Guifeng Wang, Shuiwang Ji, Carla Gomes, Zhi-Ming Ma	2023-04-07T18:08:27Z	http://arxiv.org/abs/2304.04757v1	# A new perspective on building efficient and expressive 3D equivariant graph neural networks ###### Abstract Geometric deep learning enables the encoding of physical symmetries in modeling 3D objects. Despite rapid progress in encoding 3D symmetries into Graph Neural Networks (GNNs), a comprehensive evaluation of the expressiveness of these networks through a local-to-global analysis lacks today. In this paper, we propose a local hierarchy of 3D isomorphism to evaluate the expressive power of equivariant GNNs and investigate the process of representing global geometric information from local patches. Our work leads to two crucial modules for designing expressive and efficient geometric GNNs; namely local substructure encoding (LSE) and frame transition encoding (FTE). To demonstrate the applicability of our theory, we propose LEFTNet which effectively implements these modules and achieves state-of-the-art performance on both scalar-valued and vector-valued molecular property prediction tasks. We further point out the design space for future developments of equivariant graph neural networks. Our codes are available at [https://github.com/yuanqdu/LeftNet](https://github.com/yuanqdu/LeftNet). ## 1 Introduction The success of many deep neural networks can be attributed to their ability to respect physical symmetry, such as Convolutional Neural Networks (CNNs) [1] and Graph Neural Networks (GNNs) [2]. Specifically, CNNs encode translation equivariance, which is essential for tasks such as object detection. Similarly, GNNs encode permutation equivariance, which ensures that the node ordering does not affect the output node representations., by aggregating neighboring messages. Modeling 3D objects, such as point clouds and molecules, is a fundamental problem with numerous applications, including robotics [3], molecular simulation [4; 5], and drug discovery [6; 7; 8; 9; 10]. Different from 2D pictures and graphs that only possess the translation [1] and permutation [2] symmetry, 3D objects intrinsically encode the complex $SE(3)/E(3)$ symmetry [11], which makes their modeling a nontrivial task in the machine learning community. To tackle this challenge, several approaches have been proposed to effectively encode 3D rotation and translation equivariance in the deep neural network architectures, such as TFN [12], EGNN [13], and SphereNet [14]. TFN leverages spherical harmonics to represent and update tensors equivariantly, while EGNN processes geometric information through vector update. On the other hand, SphereNet is invariant by encoding scalars like distances and angles. Despite rapid progress has been made on the empirical side, it's still unclear what 3D geometric information can equivariant graph neural networks capture and how the geometric information is integrated during the message passing process [15; 16; 17]. This type of analysis is crucial in designing expressive and efficient 3D GNNs, as it's usually a trade-off between encoding enough geometric information and preserving relatively low computation complexity. Put aside the $SE(3)/E(3)$ symmetry, this problem is also crucial in analysing ordinary GNNs. For example, 1-hop based message passing graph neural networks [18] are computationally efficient while suffering from expressiveness bottlenecks (comparing with subgraph GNNs [19; 20]). On the other hand, finding a better trade-off for 3D GNNs is more challenging, since we must ensure that the message updating and aggregating process respects the $SE(3)/E(3)$ symmetry. In this paper, we attempt to discover better trade-offs between computational efficiency and expressiveness power for 3D GNNs by studying two specific questions: 1. What is the expressive power of invariant scalars in encoding 3D geometric patterns? 2. Is equivariance really necessarily for 3D GNNs? The first question relates to the design of node-wise geometric messages, and the second question relates to the design of equivariant (or invariant) aggregation. To tackle these two problems, we take a local-to-global approach. More precisely, we first define three types of 3D isomorphism to characterize local 3D structures: tree, triangular, and subgraph isomorphism, following a local hierarchy. As we will discuss in the related works section, our local hierarchy lies between the 1-hop and 2-hop geometric isomorphism defined in [21]. Then, we can measure the expressiveness power of 3D GNNs by their ability of differentiating non-isomorphic 3D structures in a similar way as the geometric WL tests in [21]. Under this theoretical framework, we summarize one essential ingredient for building expressive geometric messages on each node: local 3D substructure encoding (LSE), which allows an invariant realization. To answer the second question, we analyze whether local invariant features are sufficient for expressing global geometries by message aggregation, and it turns out that frame transition encoding (FTE) is crucial during the local to global process. Although FTE can be realized by invariant scalars, we further demonstrate that introducing equivariant messaging passing is more efficient. By connecting LSE and FTE modules, we are able to present a modular overview of 3D GNNs designs. In realization of our theoretical findings, we propose LEFTNet that efficiently implements LSE and FTE (with equivariant tensor update) without sacrificing expressiveness. Empirical experiments on real-world scenarios, predicting scalar-valued property (e.g. energy) and vector-valued property (e.g. force) for molecules, demonstrate the effectiveness of LEFTNet. ## 2 Preliminary In this section, we provide an overview of the mathematical foundations of $E(3)$ and $SE(3)$ symmetry, which is essential in modeling 3D data. We also summarize the message passing graph neural network framework, which enables the realization of $E(3)/SE(3)$ equivariant models. Euclidean Symmetry. Our target is to incorporate Euclidean symmetry to ordinary permutation-invariant graph neural networks. The formal way of describing Euclidean symmetry is the group $E(3)=O(3)\rtimes T(3)$, where $O(3)$ corresponds to reflections (parity transformations) and rotations. For tasks that are anti-symmetric under reflections (e.g. chirality), we consider the subgroup $SE(3)=SO(3)\rtimes T(3)$, where $SO(3)$ is the group of rotations. We will use $SE(3)$ in the rest of the paper for brevity except when it's necessary to emphasize reflections. Equivariance. A tensor-valued function $f(\mathbf{x})$ is said to be equivariant with respect to $SE(3)$ if for any translation or rotation $g\in SE(3)$ acting on $\mathbf{x}\in\mathbf{R}^{3}$, we have \[f(g\mathbf{x})=\mathcal{M}(g)f(\mathbf{x}),\] where $\mathcal{M}(\cdot)$ is a matrix representation of $SE(3)$ acting on tensors. See Appendix A for a general definition of tensor fields. In this paper, we will use bold letters to represent an equivariant tensor, e.g., $\mathbf{x}$ as a position vector. It is worth noting that when $f(\mathbf{x})\in\mathbf{R}^{1}$ and $\mathcal{M}(g)\equiv 1$ (the constant group representation), the equivariant function $f(\mathbf{x})$ is also called an invariant scalar function. Scalarization. Scalarization is a general technique that originated from differential geometry for realizing covariant operations on tensors [22]. Our method will apply a simple version of scalarization in $\mathbf{R}^{3}$ to transform equivariant quantities. At the heart of its realization is the notion of equivariant orthonormal frames, which consist of three orthonormal equivariant vectors: \[\mathcal{F}:=(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}).\] Based on $\mathcal{F}$, we can build orthonormal equivariant frames for higher order tensors by taking tensor products $\otimes$, see Eq. 19 in Appendix. By taking the inner product between $\mathcal{F}$ and a given equivariant vector (tensor) $\mathbf{x}$, we get a tuple of invariant scalars (see [23] for a proof): \[\mathbf{x}\rightarrow\tilde{x}:=(\mathbf{x}\cdot\mathbf{e}_{1},\mathbf{x} \cdot\mathbf{e}_{2},\mathbf{x}\cdot\mathbf{e}_{3}), \tag{1}\] and $\tilde{x}$ can be seen as the'scalarized' coordinates of $\mathbf{x}$. Tensorization. Tensorization, on the other hand, is the'reverse' process of scalarization. Given a tuple of scalars: $(x_{1},x_{2},x_{3})$, tensorization creates an equivariant vector (tensor) out of $\mathcal{F}$: \[(x_{1},x_{2},x_{3})\xrightarrow{\text{Pairing}}\mathbf{x}:=x_{1}\mathbf{e} _{1}+x_{2}\mathbf{e}_{2}+x_{3}\mathbf{e}_{3}. \tag{2}\] The same procedure is extended to higher order cases, see Eq. 20 in Appendix. Message Passing Scheme for Geometric Graphs. A geometric graph $G$ is represented by $G=(V,E)$. Here, $v_{i}\in V$ denotes the set of nodes (vertices, atoms), and $e_{ij}\in E$ denotes the set of edges. For brevity, the edge feature attached on $e_{ij}$ is also denoted by $e_{ij}$. Let $\mathbf{X}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathbf{R}^{n\times 3}$ be the 3D point cloud of all nodes' equivariant positions, which determines the 3D geometric structure of $G$. A common machine learning tool for modeling graph-structured data is the Message Passing Neural Network (MPNN) [15]. A typical 1-hop MPNN framework consists of two phases: (1) message passing; (2) readout. Let $h^{l}_{i},h^{l}_{j}$ denote the $l$-th layer's node features of source $i$ and target $j$ that also depend on the 3D positions $(\mathbf{x}_{i},\mathbf{x}_{j})$, then the aggregated message is \[m^{l}_{i}=\bigoplus_{j\in\mathcal{N}(i)}m_{ij}(h^{l}(\mathbf{x}_{i}),h^{l}( \mathbf{x}_{j}),e^{l}_{ij}), \tag{3}\] and $\bigoplus_{j\in\mathcal{N}(i)}$ is any permutation-invariant pooling operation between the 1-hop neighbors of $i$. We also include the edge features $e^{l}_{ij}$ into the message passing phase for completeness. 3D equivariant MPNNs (3D GNNs for short) require the message $m_{i}$ to be equivariant with respect to the geometric graph. That is, for an arbitrary edge $e_{ij}$: \[m_{ij}(h^{l}(g\mathbf{x}_{i}),h^{l}(g\mathbf{x}_{j}))=\mathcal{M}(g)m_{ij}(h^ {l}(\mathbf{x}_{i}),h^{l}(\mathbf{x}_{j})), \tag{4}\] where $g\in SE(3)$ is acting on the whole geometric graph simultaneously: $(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\rightarrow(g\mathbf{x}_{1},\ldots,g \mathbf{x}_{n})$. For example, the invariant model ComENet [24] satisfies Eq. 4 by setting $\mathcal{M}(g)\equiv 1$, and MACE [25] realized Eq. 4 for nonconstant irreducible group representations $\mathcal{M}(g)$ through spherical harmonics and Clebsch-Gordan coefficients. ## 3 A Local Hierarchy of 3D Isomorphism As presented in Section 2, defining expressive messages is an essential component for building powerful 3D GNNs. In this section, we develop a fine-grained characterization of local 3D structures and build its connection with the expressiveness of 3D GNNs. Figure 1: (a) $\mathbf{S}_{i}$ and $\mathbf{S}_{j}^{\text{(a)}}$ share the same tree structure (edge lengths are identical), but they are not triangular isomorphic (different dihedral angles); (b) $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$ are triangular isomorphic but not subgraph isomorphic (the relative distance between the two triangles is different). Since the celebrated work [26], a popular expressiveness test for permutation invariant graph neural networks is the 1-WL graph isomorphism test [27], and Wijesinghe and Wang [28] has shown that the 1-WL test is equivalent to the ability to discriminate the local subtree-isomorphism. It motivates us to develop a novel (local) 3D isomorphism for testing the expressive power of 3D GNNs. However, this task is nontrivial, since most of the previous settings for graph isomorphism are only applicable to 2D topological features. For 3D geometric shapes, we should take the $SE(3)$ symmetry into account. Formally, two 3D geometric graphs $\mathbf{X},\mathbf{Y}$ are defined to be globally isomorphic, if there exists $g\in SE(3)$ such that \[\mathbf{Y}=g\mathbf{X}. \tag{5}\] In other words, $\mathbf{X}$ and $\mathbf{Y}$ are essentially the same, if they can be transformed into each other through a series of rotations and translations. Inspired by Wijesinghe and Wang [28], now we introduce a novel hierarchy of $SE(3)$ equivariant local isomorphism to measure the local similarity of 3D structures. Let $\mathbf{S}_{i}$ denote the 3D subgraph (and the associated node features) of node $i$, which contains all edges in $E$ if the end points are one-hop neighbors of $i$. For each edge $e_{ij}\in E$, the mutual 3D substructure $\mathbf{S}_{i-j}$ is defined by the intersection of $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$: $\mathbf{S}_{i-j}=\mathbf{S}_{i}\cap\mathbf{S}_{j}$. Given two local subgraphs $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$ that correspond to two nodes $i$ and $j$, we say $\mathbf{S}_{i}$ is {-tree, -triangular, -subgraph} isometric to $\mathbf{S}_{j}$, if there exists a bijective function $f:\mathbf{S}_{i}\rightarrow\mathbf{S}_{j}$ such that $h_{f(u)}=h_{u}$ for every node $u\in\mathbf{S}_{i}$, and the following conditions hold respectively: * Tree Isometric: If there exists a collection of group elements $g_{iu}\in SE(3)$, such that $(\mathbf{x}_{f(u)},\mathbf{x}_{f(i)})=(g_{iu}\mathbf{x}_{u},g_{iu}\mathbf{x} _{i})$ for each edge $e_{iu}\in\mathbf{S}_{i}$; * Triangular Isometric: If there exists a collection of group elements $g_{iu}\in SE(3)$, such that the corresponding mutual 3D substructures satisfy: $\mathbf{S}_{f(u)-f(i)}=g_{iu}\mathbf{S}_{u-i}$ for each edge $e_{iu}\in\mathbf{S}_{i-j}$; * Subgraph Isometric: for any two adjacent nodes $u,v\in\mathbf{S}_{i}$, $f(u)$ and $f(v)$ are also adjacent in $\mathbf{S}_{j}$, and there exist a single group element $g_{i}\in SE(3)$ such that $g_{i}\mathbf{S}_{i}=\mathbf{S}_{j}$. Note that tree isomorphism only considers edges around a central node, which is of a tree shape. On the other hand, the mutual 3D substructure can be decomposed into a bunch of triangles (since it's contained in adjacent node triplets), which explains the name of triangular isomorphism. In fact, the three isomorphisms form a hierarchy from micro to macro, in the sense that the following implication relation holds: Subgraph Isometric $\Rightarrow$ Triangular Isometric $\Rightarrow$ Tree Isometric This is an obvious fact from the above definitions. To deduce the reverse implication relation, we provide a visualized example. Figure 1 shows two examples of local 3D structures: 1. the first one shares the same tree structure, but is not triangular-isomorphic; 2. the second one is triangular-isomorphic but not subgraph-isomorphic. In conclusion, the following diagram holds: Tree Isometric $\not\Rightarrow$ Triangular Isometric $\not\Rightarrow$ Subgraph Isometric One way to formally connect the expressiveness power of a geometric GNN with their ability of differentiating geometric subgraphs is to define geometric WL tests, the reader can consult [21]. In this paper, we take an intuitive approach based on our nested 3D hierarchy. That is, if two 3D GNN algorithms A and B can differentiate all non-isomorphic local 3D shapes of tree (triangular) level, while A can differentiate at least two more 3D geometries which are non-isomorphic at triangular(subgraph) level than B, then we claim that algorithm A's expressiveness power is more powerful than B. Since tree isomorphism is determined by the one-hop Euclidean distance between neighbors, distinguishing local tree structures is relatively simple for ordinary 3D equivariant GNNs. For example, the standard baseline SchNet [29] is one instance of Eq. 3 by setting $e_{ij}^{t}=\textbf{RBF}(d(\mathbf{x}_{i},\mathbf{x}_{j}))$, where $\textbf{RBF}(\cdot)$ is a set of radial basis functions. Although it is powerful enough for testing tree non-isomorphism (assuming that $\textbf{RBF}(\cdot)$ is injective), we prove in Appendix B that SchNet cannot distinguish non-isomorphic structures at the triangular level. On the other hand, Wijesinghe and Wang [28] has shown that by leveraging the topological information extracted from local overlapping subgraphs, we can enhance the expressive power of GNNs to go beyond 2D sub-tree isomorphism. In our setting, the natural analogue of the overlapping subgraphs is exactly the mutual 3D substructures. Now we demonstrate how to merge the information from 3D substructures to the message passing framework (3). Given an $SE(3)$-invariant encoder $\phi$, define the 3D structure weights $A_{ij}:=\phi(\mathbf{S}_{i-j})$ for each edge $e_{ij}\in E$. Then, the message passing framework (3) is generalized to: \[m_{i}^{l}=\bigoplus_{j\in\mathcal{N}(i)}m_{ij}(h^{l}(\mathbf{x}_{i}),h^{l}( \mathbf{x}_{j}),A_{ij}h^{l}(\mathbf{x}_{j}),e_{ij}^{l}). \tag{6}\] Formula 6 is an efficient realization of enhancing 3D GNNs by injecting the mutual 3D substructures. However, a crucial question remains to be answered: _Can the generalized message passing framework boost the expressive power of 3D GNNs?_ Under certain conditions, the following theorem provides an affirmative answer: Theorem 3.1.: _Suppose $\phi$ is a a universal SE(3)-invariant approximator of functions with respect to the mutual 3d structures $\mathbf{S}_{i-j}$, then the collection of weights $\{\{A_{ij}\}_{e_{ij}\in E}\}$ is able to differentiate local structures beyond tree isomorphism. Moreover, with additional injectivity assumptions (see Eq. 14), 3D GNNs based on the enhanced message passing framework 6 map at least two distinct local 3D subgraphs with isometric local tree structures to different representations._ This theorem confirms that the enhanced 3D GNN (formula 6) is more expressive than the SchNet baseline, at least in testing local non-isomorphic geometric graphs. The complete proof is left in Appendix B. The existence of such local invariant encoder $\phi$ is also proved by explicit construction. Note that there are other different perspectives on characterizing 3D structures, we will also briefly discuss them in Appendix B. ## 4 From Local to Global: The Missing Pieces In the last section, we introduced a geometric local isomorphism hierarchy for testing the expressive power of 3D GNNs. Furthermore, we motivated adding a SE(3)-invariant encoder to improve the expressive power of one-hop 3D GNNs by scalarizing not only pairwise distances but also their mutual 3D structures in Theorem 3.1. However, to build a powerful 3D GNN, it remains to be analyzed how a 3D GNN acquires higher order (beyond 1-hop neighbors) information by accumulating local messages. A natural question arises: _are invariant features enough for representing global geometric information?_ To formally formulate this problem, we consider a two-hop aggregation case. From figure 2, the central atom $a$ is connected with atoms $b$ and $c$. Except for the common neighbor $a$, other atoms that connect to $b$ and $c$ form two 3D clusters, denoted by B, C. Suppose the ground-truth interaction potential of B and C imposed on atom $a$ is described by a tensor-valued function $f_{a}(\textbf{B},\textbf{C})$. Since B and C are both beyond the 1-hop neighborhood of $a$, the information of $f_{a}(\textbf{B},\textbf{C})$ can only be acquired after two steps of message passing: 1. atoms $b$ and $c$ aggregate message separately from B and C; 2. the central atom $a$ receives the aggregated message (which contains information of B and C) from its neighbors $b$ and $c$. Let $S_{\textbf{B}}$ ($S_{\textbf{C}}$) denote the collection of all invariant scalars created by B (C). For example, $S_{\textbf{B}}$ contains all relative distances and angles within the 3D structure B. Then, the following theorem holds: Theorem 4.1.: _Not all types of invariant interaction $f_{a}(\textbf{B},\textbf{C})$ can be expressed by inputting the union of two sets $S_{\textbf{B}}$ and $S_{\textbf{C}}$. In other words, there exists $E(3)$ invariant function $f_{a}(\textbf{B},\textbf{C})$, such that it cannot be expressed as functions of $S_{\textbf{B}}$ and $S_{\textbf{C}}$: $f_{a}(\textbf{B},\textbf{C})\neq\rho(S_{\textbf{B}},S_{\textbf{C}})$ for an arbitrary invariant function $\rho$._ Figure 2: Illustrations of different local frames and their transition. This theorem in essence tells us that naively aggregating 'local' scalar information from different clusters is not enough to approximate 'global' interactions, even if we only consider simple invariant interaction tasks. Different from the last section, where the local expressiveness is measured by the ability of classifying geometric shapes, we built regression functions that depend strictly more than the combination of local invariant scalars. Intuitively, the proof is based on the fact that all scalars in $S_{\textbf{B}}$ ($S_{\textbf{C}}$) can be expressed through equivariant frames separately determined by B (C). However, the transition matrix between these two frames is not encoded in the aggregation, which causes information loss when aggregating geometric features from two sub-clusters. More importantly, the proof also revealed the missing information that causes the expressiveness gap: Frame Transition (FT). Frame Transition (FT). Formally, two orthonormal frames $(e_{1}^{i},e_{2}^{i},e_{3}^{i})$ and $(e_{1}^{j},e_{2}^{j},e_{3}^{j})$ are connected by an orthogonal matrix $R_{ij}\in SO(3)$: \[(\textbf{e}_{1}^{i},\textbf{e}_{2}^{i},\textbf{e}_{3}^{i})=R_{ij}(\textbf{e}_ {1}^{j},\textbf{e}_{2}^{j},\textbf{e}_{3}^{j}). \tag{7}\] Moreover, it is easy to check that when $(\textbf{e}_{1}^{i},\textbf{e}_{2}^{i},\textbf{e}_{3}^{i})$ and $(\textbf{e}_{1}^{j},\textbf{e}_{2}^{j},\textbf{e}_{3}^{j})$ are equivariant frames, all elements of $R_{xy}$ are invariant scalars. Suppose $i$ and $j$ represent indexes of two connected atoms in a geometric graph, then the fundamental torsion angle $\tau_{ij}$ appeared in ComeNet [24] is just one element of $R_{ij}$ (see Appendix C). Towards filling this expressiveness gap, we can straightforwardly inject all invariant pairwise frame transition matrices (FT) into the model. Nevertheless, it imposes expensive computational cost when the number of local clusters is large ($O(k^{2})$ pairs of FT for each node). Therefore, compared with pure invariant approaches, a more efficient way is to introduce equivariant tensor features for each node $i$, denoted by $\textbf{m}_{i}$. By directly maintaining the equivariant frames in $\textbf{m}_{i}$, we show in Appendix C that FT is easily derived through equivariant message passing. Equivariant Message Passing. Similarly with the standard one-hop message passing scheme 3, the aggregated tensor message $\textbf{m}_{i}$ from the $l-1$ layer to the $l$ layer can be written as: $\textbf{m}_{i}^{l-1}=\sum_{j\in N(i)}\textbf{m}_{j}^{l-1}$. Since summation does not break the symmetry rule, it is obvious that $\textbf{m}_{i}^{l-1}$ are still equivariant tensors. However, the nontrivial part lies in the design of the equivariant update function $\phi$: \[\textbf{m}_{i}^{l}=\phi(\textbf{m}_{i}^{l-1}). \tag{8}\] A good $\phi$ should have enough expressive power while preserving $SE(3)$ equivariance. Here, we propose a novel way of updating scalar and tensor messages by performing node-wise scalarization and tensorization blocks (the FTE module of Figure 3). From the perspective of Eq. 4, $\textbf{m}(\textbf{x}_{u})$ is transformed equivariantly as: \[\textbf{m}(g\textbf{x}_{u})=\sum_{i=0}^{l}\mathcal{M}^{i}(g)\textbf{m}_{i}(g \textbf{x}_{u}),\ \ g\in SE(3). \tag{9}\] Here, $\textbf{m}(\textbf{x}_{u})$ is decomposed to $(\textbf{m}_{0}(\textbf{x}_{u}),\ldots,\textbf{m}_{l}(\textbf{x}_{u}))$ according to different tensor types, and $\{\mathcal{M}^{i}(g)\}_{i=0}^{l}$ is a collection of different $SE(3)$tensor representations (see the precise definition in Appendix A). To illustrate the benefit of aggregating equivariant messages from local patches, we study a simple case. Let $f_{a}(\textbf{B},\textbf{C})=\textbf{h}_{B}\cdot\textbf{h}_{C}$ be an invariant function of B and C (see Fig. 2), then $f_{a}$ can be calculated by a direction composition of scalar messages and equivariant vector messages: $f_{a}(\textbf{B},\textbf{C})=\frac{1}{2}[\left\\|\textbf{m}_{a}\right\\|^{2}- \left\\|\textbf{h}_{B}\right\\|^{2}-\left\\|\textbf{h}_{C}\right\\|^{2}]$, where $\textbf{m}_{a}=\textbf{h}_{B}+\textbf{h}_{C}$ is an equivariant vector. Note that $\textbf{m}_{a}$ follows the local equivariant aggregation formula 8, and the other vectors' norm $\left\\|\textbf{h}_{B}\right\\|$ and $\left\\|\textbf{h}_{C}\right\\|$ are obtained through local scalarization on atoms $b$ and $c$. As a comparison, it's worth mentioning that $f_{a}(\textbf{B},\textbf{C})$ can also be expressed by local scalarization with the additional transition matrix data $R_{BC}$ defined by Eq. 7. Let $\tilde{h}_{B}$ and $\tilde{h}_{C}$ be the scalarized coordinates with respect to two local equivariant frames $\mathcal{F}_{B}$ and $\mathcal{F}_{C}$. Then $f_{a}(\textbf{B},\textbf{C})=\frac{1}{2}\left[\left\\|R_{BC}^{-1}\tilde{h}_{B} +\tilde{h}_{C}\right\\|^{2}-\left\\|\tilde{h}_{B}\right\\|^{2}-\left\\|\tilde{h}_{C }\right\\|^{2}\right].$ However, it requires adding the rotation matrix $R_{BC}$ for each $(\textbf{B},\textbf{C})$ pair, which is computationally expensive compared to directly implementing equivariant tensor updates. ## 5 Building an Efficient and Expressive Equivariant 3D GNN We propose to leverage the full power of LSE and FTE along with a powerful tensor update module to push the limit of efficient and expressive 3D equivariant GNNs design. LSE Instantiation. We propose to apply edge-wise equivariant frames to encode the local 3D structures $\mathbf{S}_{i-j}$. By definition, $\mathbf{S}_{i-j}$ contains edge $e_{ij}$, nodes $i$ and $j$, and their common neighbors. We use the equivariant frame $\mathcal{F}_{ij}$ built on $e_{ij}$ (see the precise formula in Appendix D) to scalarize $\mathbf{S}_{i-j}$. After scalarization (1), the equivariant coordinates of all nodes in $\mathbf{S}_{i-j}$ are transformed into invariant coordinates: $\{\mathbf{x}_{k}\rightarrow\tilde{x}_{k}\text{ for }\mathbf{x}_{k}\in\mathbf{S}_{i-j}\}$. To encode these scalars sufficiently, we first weight each $\tilde{x}_{k}$ by the RBF distance embedding: $\tilde{x}_{k}\rightarrow\textbf{RBF}(\\|\mathbf{x}_{k}\\|)\odot\text{MLP}( \tilde{x}_{k})$ for each $\mathbf{x}_{k}\in\mathbf{S}_{i-j}$. Note that to preserve the permutation symmetry, the MLP is shared among the nodes. Finally, the 3D structure weight $A_{ij}$ is obtained by the average pooling of all node features. FTE Instantiation. We propose to introduce equivariant tensor message passing and update function for encoding local FT information. At initialization, let $\textbf{NF}^{l}(\mathbf{x}_{i},\mathbf{x}_{j})$ denote the embedded tensor-valued edge feature between $i$ and $j$. We split it into two parts: 1. the scalar part $\text{SF}^{l}(\mathbf{x}_{i},\mathbf{x}_{j})$ for aggregating invariant messages; 2. the higher order tensor part $\textbf{TF}^{l}(\mathbf{x}_{i},\mathbf{x}_{j})$ for aggregating tensor messages. To transform $\textbf{TF}^{l}(\mathbf{x}_{i},\mathbf{x}_{j})$, we turn to the equivariant frame $\mathcal{F}_{ij}$ once again. After scalarization by $\mathcal{F}_{ij}$, $\textbf{TF}^{l}(\mathbf{x}_{i},\mathbf{x}_{j})$ becomes a tuple of scalars $\widetilde{\text{TF}}^{l}(\mathbf{x}_{i},\mathbf{x}_{j})$, which is then transformed by MLP. Finally, we output arbitrary tensor messages through equivariant tensorization 20: \[\widetilde{\text{TF}}^{l}(\mathbf{x}_{i},\mathbf{x}_{j})\xrightarrow[\mathcal{ F}_{ij}]{\textbf{Tensorize}}\textbf{NF}^{l+1}(\mathbf{x}_{i},\mathbf{x}_{j}).\] Further details are provided in Appendix D. As we have discussed earlier, the node-wise tensor update function $\phi$ in Eq. 8 is also one of the guarantees for a powerful FTE. As a comparison, $\phi$ is usually a standard MLP for updating node features in 2D GNNs, which is a universal approximator of invariant functions. Previous works [13; 30] updated equivariant features by taking linear combinations and calculating the invariant norm of tensors, which may suffer from information loss. Then a natural question arises: _Can we design an equivariant universal approximator for tensor update?_ We answer this question by introducing a novel node-wise frame. Consider node $i$ with its position $\mathbf{x}_{i}$, let $\bar{\mathbf{x}}_{i}:=\frac{1}{N}\sum_{\mathbf{x}_{j}\in N(\mathbf{x}_{i})} \mathbf{x}_{j}$ be the center of mass around $\mathbf{x}_{i}$'s neighborhood. Then the orthonormal equivariant frame $\mathcal{F}_{i}:=(\mathbf{e}_{1}^{i},\mathbf{e}_{2}^{i},\mathbf{e}_{3}^{i})$ with respect to $\mathbf{x}_{i}$ is defined by \[(\frac{\mathbf{x}_{i}-\bar{\mathbf{x}}_{i}}{\\|\mathbf{x}_{i}-\bar{\mathbf{x}} _{i}\\|},\frac{\bar{\mathbf{x}}_{i}\times\mathbf{x}_{i}}{\\|\bar{\mathbf{x}}_{i} \times\mathbf{x}_{i}\\|},\frac{\mathbf{x}_{i}-\bar{\mathbf{x}}_{i}}{\\|\mathbf{ x}_{i}-\bar{\mathbf{x}}_{i}\\|}\times\frac{\bar{\mathbf{x}}_{i}\times\mathbf{x}_{i}}{\\| \bar{\mathbf{x}}_{i}\times\mathbf{x}_{i}\\|}). \tag{10}\] Finally, we realize a powerful $\phi$ by the following theorem: Theorem 5.1.: _Equipped with an equivariant frame $\mathcal{F}_{i}$ for each node $i$, the equivariant function $\phi$ defined by the following composition is a universal approximator of tensor transformations: $\phi:\textbf{Scalarization}\rightarrow\textbf{MLP}\rightarrow\textbf{ Tensorization}$._ Figure 3: Illustrations of our modular framework for building equivariant GNNs and the realization of LEFTNet. Each interaction block contains LSE to encode local 3D structures, equivariant message passing to update both invariant (unbold letters, e.g. $h_{i}$) and equivariant (bold letter, e.g. $\mathbf{h}_{i}$) features, and FTE to encode frame transition. $\mathbf{S}_{i-j}$ is the local 3D structure of each edge $e_{ij}$. $\mathcal{F}_{ij}$ and $\mathcal{F}_{i}$ are the equivariant frames for each edge $e_{ij}$ and node $i$. $\odot$ indicates element-wise multiplication, and $\\|$ indicates concatenation. Note that we do not include $\mathbf{e}_{ij}$ in the figure since, practically, they are generated based on $\mathbf{h}_{i}$ and $\mathbf{h}_{j}$. The proof is left in Appendix D. LEFTNet. An overview of our $\{\textbf{LSE},\textbf{FTE}\}$ enhanced efficient graph neural network (LEFTNet) is depicted in Figure 3. LEFTNet receives as input a collection of node embeddings $\{v^{0}_{1},\dots,v^{0}_{N}\}$, which contain the atom types and 3D positions for each node: $v^{0}_{i}=(z_{i},\textbf{x}_{i})$, where $i\in\{1,\dots,N\}$. For each edge $e_{ij}\in E$, we denote the associated equivariant features consisting of tensors by $\textbf{e}_{ij}$. During each messaging passing layer, the LSE module outputs the scalar weight coefficients $A_{ij}$ as enhanced invariant edge feature and feed into the interaction module. Moreover, scalarization and tensorization as two essential blocks are used in the equivariant update module that fulfills the function of FTE. The permutation equivariance of a geometric graph is automatically guaranteed for any message passing architecture, we provide a complete proof of SE(3)-equivariance for LEFTNet in Appendix D. SE(3) vs E(3) Equivariance. Besides explicitly fitting the $SE(3)$ invariant molecular geometry probability distribution, modeling the energy surface of a molecule system is also a crucial task for molecule property prediction. However, the Hamiltonian energy function $E$ of a molecule is invariant under refection transformation: $\textbf{Energy}(\textbf{X})=\textbf{Energy}(R\textbf{X})$, for arbitrary reflection transformation $R\in E(3)$. In summary, there exist two different inductive biases for modeling 3D data: (1) SE(3) equivariance, e.g. chirality could turn a therapeutic drug to a killing toxin; (2) E(3) equivariance, e.g. energy remains the same under reflections. Since we implement $SE(3)$ equivariant frames in LEFTNet, our algorithm is naturally $SE(3)$ equivariant (and reflection anti-equivariant). However, our method is flexible to implement $E(3)$ equivariant tasks as well. For $E(3)$ equivariance, we can either replace our frames to $E(3)$ equivariant frames, or modify the scalarization block by taking the absolute value: $\textbf{x}\rightarrow\tilde{x}:=\underbrace{(\textbf{x}\cdot e_{1},\textbf{x} \cdot e_{2},\textbf{x}\cdot e_{3})}_{SE(3)}\rightarrow\underbrace{(\textbf{x} \cdot e_{1},\|\textbf{x}\cdot e_{2}\|,\textbf{x}\cdot e_{3})}_{E(3)}$. Intuitively, since the second vector $e_{2}$ is a pseudo-vector, projections of any equivariant vectors along the $e_{2}$ direction are not $E(3)$ invariant until taking the absolute value. Efficiency. To analyze the efficiency of LEFTNet, suppose 3D graph $G$ has $n$ vertices, and its average node degree is $k$. Our algorithm consists of three phases: 1. Building equivariant frames and performing local scalarization; 2. Equivariant message passing; 3. Updating node-wise tensor features through scalarization and tensorization. Let $l$ be the number of layers, then the computational complexity for each of our three phases are: 1. $O(nk)$ for computing the frame and local (1-hop) 3D features; 2. $O(nkl)$ for 1-hop neighborhood message aggregation; 3. $O(nl)$ for node-wise tensorization and feature update. ## 6 Related Work In light of the discussions in Section 3 and 4, we summarize two necessary ingredients for building expressive equivariant 3D GNNs: (1) local 3D substructure encodings (LSE), such that the local \begin{table} \begin{tabular}{l c c c c} \hline \hline Method & Symmetry & LSE & FTE & Complexity \\ \hline SchNet [29] & E(3)-invariant & ✗ & ✗ & $O(nk)$ \\ EGNN [13] & E(3)-equivariant & ✗ & ✓${}^{}$ & $O(nk)$ \\ GVP-GNN [30] & E(3)-equivariant & ✗ & ✓ & $O(nk)$ \\ ColTNet [23] & SE(3)-equivariant & ✗ & ✗ & $O(nk)$ \\ PairNN [31] & E(3)-equivariant & ✗ & ✓ & $O(nk)$ \\ ComENet [24] & SE(3)-invariant & ✓ & ✓${}^{}$ & $O(nk)$ \\ TFN [12] & SE(3)/E(3)-equivariant & ✗ & ✓ & $O(nk)$ \\ Equiformer [32] & SE(3)/E(3)-equivariant & ✗ & ✓ & $O(nk)$ \\ SphereNet [14] & SE(3)-invariant & ✓${}^{}$ & ✓${}^{}$ & $O(nk^{2})$ \\ GemNet [33] & SE(3)-invariant & ✓${}^{}$ & ✓${}^{}$ & $O(nk^{3})$ \\ \hline LEFTNet (Ours) & SE(3)/E(3)-equivariant & ✓ & ✓ & $O(nk)$ \\ \hline \hline \end{tabular} \end{table} Table 1: Categorization of representative geometric GNN algorithms. ${}^{}$ denotes partially satisfying the requirement. message is aware of different local 3D structures; (2) frame transition encodings (FTE), such that the 3D GNN is aware of the equivariant coordinate transformation between different local patches. We review the previous 3D GNNs following this framework and summarize the results in Table 1. For a fair comparison, we also list the computational complexity as it is often a trade-off of expressiveness (see the detailed analysis at the end of the next section). For LSE, SphereNet [14] and GemNet [33] (implicitly) encode the local 3D substructures by introducing a computation-intensive 2-hop edge-based update. For FTE, most 3D GNNs with equivariant vector update are able to express the local frame transitions (FT). While EGNN [13] is an exception, because it only updates the position vector (i.e. one channel), which is insufficient to express the whole FT. In other words, whether the update function $\phi$ of (8) is powerful also affects the FT* encoding. Except for equivariant update methods, models that encode torsion angle information also partially express FTE as illustrated in Appendix C. However, there is a trade-off between the efficiency and expressiveness in terms of number of hops considered for message passing. Different from our invariant realization of LSE, Batatia et al. [25] builds its framework by constructing complete equivariant polynomial basis with the help of spherical harmonics and tensor product, where the monomial variables depend on different nodes (bodies). On the other hand, we realize the function of LSE and FTE through the edgewise scalarization $A_{ij}$ and the equivariant message passing (see Fig. 3). Recently, Joshi et al. [21] propose a geometric k-WL test (GWL) to measure the expressiveness power of geometric GNN algorithms. On a high level, our tree isomorphism is equivalent to the 1-hop geometric isomorphism as proposed in GWL, and the fine-grained triangular isomorphism lies between the 1-hop and 2-hop geometric isomorphism as proposed in GWL. From the model design point of view, our realization of LSE is through local scalarization, whose expressiveness is guaranteed by the Kolmogorov representation theorem (see [34]) and the universal approximator property of MLP. Moreover, the key concepts of measuring the expresive power in [21] are the body order and tensor order, which originate from classical inter-atomic potential theories and are of the equivariance nature. On the other hand, we discover the FTE as the'missing' bridge connecting local invariant scalars and global geometric expressiveness, which (together with LSE on mutual 3D substructures) also reveals why the 1-hop scalarization implemented in ClofNet [23] is insufficient. ## 7 Experiments We test the performance of LEFTNet on both scalar value (e.g. energy) and vector value (e.g. forces) prediction tasks. The scalar value prediction experiment is conducted on the QM9 dataset [35] which includes $134k$ small molecules with quantum property annotations; the vector value prediction experiment is conducted on the MD17 dataset [36] and the Revised MD17(rMD17) dataset [37] which includes the energies and forces of molecules. We compare our LEFTNet with a list of state-of-the-art equivariant (invariant) graph neural networks including SphereNet [14], PaiNN [31], \begin{table} \begin{tabular}{l c c c c c c c c c c c c} \hline \hline Task & $\alpha$ & $\Delta\varepsilon$ & $\varepsilon_{\text{HOMO}}$ & $\varepsilon_{\text{LUMO}}$ & $\mu$ & $C_{r}$ & $G$ & $H$ & $R^{2}$ & $U$ & $U_{0}$ & ZPVE \\ Units & bohr${}^{3}$ & meV & meV & meV & D & cal/mol K & meV & meV & bohr${}^{3}$ & meV & meV & meV \\ \hline NMP &.092 & 69 & 43 & 38 &.030 &.040 & 19 & 17 &.180 & 20 & 20 & 1.50 \\ Comparant &.085 & 61 & 34 & 38 &.038 &.026 & 20 & 21 &.961 & 21 & 22 & 2.03 \\ Lieconv &.084 & 49 & 30 & 25 &.032 &.038 & 22 & 24 &.800 & 19 & 19 & 2.28 \\ TFN &.223 & 58 & 40 & 38 &.064 &.101 & - & - & - & - & - & - \\ SE3)-Tr. &.142 & 53 & 35 & 33 &.051 &.054 & - & - & - & - & - & - \\ EGNN &.071 & 48 & 29 & 25 & 0.029 &.031 & 12 & 12 & .106 & 12 & 11 & 1.55 \\ SEGNN &.060 & 42 & 24 & 21 &.023 &.031 & 15 & 16 &.660 & 13 & 15 & 1.62 \\ ClofNet &.063 & 53 & 33 & 25 &.040 &.027 & 9 & 9 &.610 & 9 & 8 & 1.23 \\ EQGAT &.063 & 44 & 26 & 22 &.014 &.027 & 12 & 13 &.257 & 13 & 13 & 1.50 \\ Equformer &.056 & 33 & 17 & 16 &.014 &.025 & 10 & 10 & 10 &.227 & 11 & 10 & 1.32 \\ LEFTNet (ours) & .048 & 40 & 24 & 18 & .012 & .023 & 7 & 6 &.109 & 7 & 6 & 1.33 \\ \hline Schnet &.235 & 63 & 41 & 34 &.033 &.033 &.033 & 14 & 14 &.073 & 19 & 14 & 1.70 \\ DimeNet++ &.044 & 33 & 25 & 20 &.030 &.023 & 8 & 7 &.331 & 6 & 6 & 1.21 \\ SphereNet &.046 & 32 & 23 & 18 &.026 & .021 & 8 & 6 &.292 & 7 & 6 & 1.12 \\ ClofNet &.053 & 49 & 33 & 25 &.038 &.026 & 9 & 8 &.425 & 8 & 8 & 1.59 \\ PaiNN &.045 & 46 & 28 & 20 &.012 &.024 & 7 & 6 & .066 & 6 & 6 & 1.28 \\ LEFTNet (ours) &.039 & 39 & 23 & 18 & .011 &.022 & 6 & 5 &.094 & 5 & 5 & 1.19 \\ \hline \hline \end{tabular} \end{table} Table 2: Mean Absolute Error for the molecular property prediction benchmark on QM9 dataset. (The best results are bolded and the second best are underlined.) Equiformer [32], GemNet [33], etc [29; 38; 12; 39; 40; 13; 15; 41; 42; 43; 44; 45; 46; 47].The results on rMD17 and ablation studies are listed in Appendix E. ### QM9 - Scalar-valued Property Prediction The QM9 dataset is a widely used dataset for predicting molecular properties. However, existing models are trained on different data splits. Specifically, Cormorant [40], EGNN [13], etc., use 100k, 18k, and 13k molecules for training, validation, and testing, while DimeNet [38], SphereNet [14], etc., split the data into 110k, 10k, and 11k. For a fair comparison with all baseline methods, we conduct experiments using both data splits. Experimental results are listed in Table 2. For the first data split, LEFTNet is the best on 7 out of the 12 properties and improves previous SOTA results by 20% on average. In addition, LEFTNet is the second best on 4 out of the other 5 tasks. Consistently, LEFTNet is the best or second best on 10 out of the 12 properties for the second split. These experimental results on both splits validate the effectiveness of LEFTNet on scalar-valued property prediction tasks. The ablation study in Appendix E shows that both LSE and FTE contribute to the final performance. ### MD17 - Vector-valued Property Prediction We evaluate the ability of LEFTNet to predict forces on the MD17 dataset. Following existing studies [29; 38; 14], we train a separate model for each of the 8 molecules. Both training and validation sets contain 1000 samples, and the rest are used for testing. Note that all baseline methods are trained on a joint loss of energies and forces, but different methods use different weights of force over energy (WoFE). For example, SchNet [29] sets WoEF as 100, while GemNet [33] uses a weight of 1000. For a fair comparison with existing studies, we conduct experiments on two widely used weights of 100 and 1000 following Liu et al. [14]. The results are summarized in Table 3. We can observe that when WoFE is 100, LEFTNet outperforms all baseline methods on 7 of the 8 molecules and improves previous SOTA results by 16% on average. In addition, LEFTNet can outperform all baseline methods on 6 of the 8 molecules when WoFE is 1000. These experimental results on MD17 demonstrate the performance of LEFTNet on vector-valued property prediction tasks. The ablation study in Appendix E also demonstrates that both LSE and FTE are important to the final results. ## 8 Limitation and Future Work In this paper, we seek a general recipe for building 3D geometric graph deep learning algorithms. Considering common prior of 2D graphs, such as permutation symmetry, has been incorporated in off-the-shelf graph neural networks, we mainly focus on the $E(3)$ and $SE(3)$ symmetry specific to 3D geometric graphs. Despite our framework being general for modeling geometric objects, we only conducted experiments on commonly used molecular datasets. It's worth exploring datasets in other domains in the future. To elucidate the future design space of equivariant GNNs, we propose two directions that are worth exploring. Firstly, our current algorithms consider fixed equivariant frames for performing aggregation and node updates. Inspired by the high body-order ACE approach [48] (for modeling atom-centered potentials), it is worth investigating in the future if equivariant frames that relate to many body (e.g., the PCA frame in [49]) can boost the performance of our algorithm. For example, to \begin{table} \begin{tabular}{l c c c c c c\|c c c c\|c c} \hline \hline \multicolumn{11}{c}{} & \multicolumn{4}{c}{WoFE=100} & \multicolumn{4}{c}{WoFE=1000} & \multicolumn{4}{c}{Others} \\ \hline Molecule & sGDML & SchNet & DimeNet & SphereNet & SpoxyNet & LEFTNet & SphereNet & GemNet & LEFTNet & PaNN & NewtonNet \\ \hline Appirin & 0.68 & 1.35 & 0.499 & 0.430 & 0.258 & 0.210 & 0.209 & 0.217 & 0.196 & 0.371 & 0.348 \\ Benzene & 0.20 & 0.31 & 0.187 & 0.178 & – & 0.145 & 0.147 & 0.145 & 0.142 & – & – & – \\ Ethanol & 0.33 & 0.39 & 0.230 & 0.208 & 0.094 & 0.118 & 0.091 & 0.086 & 0.099 & 0.230 & 0.264 \\ Malonaldehyde & 0.41 & 0.66 & 0.383 & 0.340 & 0.167 & 0.159 & 0.172 & 0.155 & 0.142 & 0.319 & 0.323 \\ Naphthalene & 0.11 & 0.58 & 0.215 & 0.178 & 0.089 & 0.063 & 0.048 & 0.051 & 0.044 & 0.083 & 0.084 \\ Salicylic acid & 0.28 & 0.85 & 0.374 & 0.360 & 0.180 & 0.141 & 0.113 & 0.125 & 0.117 & 0.209 & 0.197 \\ Toluene & 0.14 & 0.57 & 0.210 & 0.155 & 0.087 & 0.070 & 0.054 & 0.060 & 0.049 & 0.102 & 0.088 \\ Uracil & 0.24 & 0.56 & 0.301 & 0.267 & 0.119 & 0.117 & 0.106 & 0.097 & 0.085 & 0.140 & 0.149 \\ \hline \hline \end{tabular} \end{table} Table 3: Mean Absolute Error for per-atom forces prediction (kcal/mol Å) on MD17 dataset. Baseline results are taken from the original papers (with unit conversions if needed). All models are trained on energies and forces, and WoFE is the weight of force over energy in loss functions. The best results are bolded. build the A-basis proposed in Puny et al. [49], we can replace our message aggregation Eq. 8 from summation to tensor product, which is also a valid pooling operation. Another direction is to explore geometric mesh graphs on manifolds $M$, where the local frame is defined on the tangent space of each point: $\mathcal{F}(x)\in T_{x}M$. Since our scalarization technique (crucial for realizing LSE in LEFT-Net) originates from differential geometry on frame bundles [22], it is reasonable to expect that our framework also works for manifold data [50, 51].
2303.00596	Information Plane Analysis for Dropout Neural Networks	The information-theoretic framework promises to explain the predictive power of neural networks. In particular, the information plane analysis, which measures mutual information (MI) between input and representation as well as representation and output, should give rich insights into the training process. This approach, however, was shown to strongly depend on the choice of estimator of the MI. The problem is amplified for deterministic networks if the MI between input and representation is infinite. Thus, the estimated values are defined by the different approaches for estimation, but do not adequately represent the training process from an information-theoretic perspective. In this work, we show that dropout with continuously distributed noise ensures that MI is finite. We demonstrate in a range of experiments that this enables a meaningful information plane analysis for a class of dropout neural networks that is widely used in practice.	Linara Adilova, Bernhard C. Geiger, Asja Fischer	2023-03-01T15:41:50Z	http://arxiv.org/abs/2303.00596v1	# Information Plane Analysis for ###### Abstract The information-theoretic framework promises to explain the predictive power of neural networks. In particular, the information plane analysis, which measures mutual information (MI) between input and representation as well as representation and output, should give rich insights into the training process. This approach, however, was shown to strongly depend on the choice of estimator of the MI. The problem is amplified for deterministic networks if the MI between input and representation is infinite. Thus, the estimated values are defined by the different approaches for estimation, but do not adequately represent the training process from an information-theoretic perspective. In this work, we show that dropout with continuously distributed noise ensures that MI is finite. We demonstrate in a range of experiments1 that this enables a meaningful information plane analysis for a class of dropout neural networks that is widely used in practice. Footnote 1: Code for the experiments is public on [https://github.com/link-er/IP_dropout](https://github.com/link-er/IP_dropout). ## 1 Introduction The information bottleneck hypothesis for deep learning conjectures two phases of training feed-forward neural networks (Shwartz-Ziv and Tishby, 2017): the fitting phase and the compression phase. The former corresponds to extracting information from the input into the learned representations, and is characterized by an increase of mutual information (MI) between inputs and hidden representations. The latter corresponds to forgetting information that is not needed to predict the target, which is reflected in a decrease of the MI between learned representations and inputs, while MI between representations and targets stays the same or grows. The phases can be observed via an information plane (IP) analysis, i.e., by analyzing the development of MI between inputs and representations and between representations and targets during training (see Fig. 1 for an example). For an overview of information plane analysis we refer the reader to (Geiger, 2022). While being elegant and plausible, the information bottleneck hypothesis is challenging to investigate empirically. As shown by Amjad and Geiger (2020, Th. 1), the MI between inputs and the representations learned by a deterministic neural network is infinite if the input distribution is continuous. The standard approach is therefore to assume the input distribution to be discrete (e.g., equivalent to the empirical distribution of the dataset $S$ at hand) and to discretize the real-valued hidden representations by binning to allow for non-trivial measurements, i.e., to avoid that the MI always takes the maximum value of $\log(\|S\|)$(Shwartz-Ziv and Tishby, 2017). In this discrete and deterministic setting the MI theoretically gets equivalent to the Shannon entropy of the representation. Considering the effect of binning, however, the decrease of MI is essentially equivalent to geometrical compression (Basirat et al., 2021). Moreover, the binning-based estimate highly depends on the chosen bin size (Ross, 2014). To instead work with continuous input distributions, Goldfeld et al. (2019) suggest to replace deterministic neural networks by stochastic ones via adding Gaussian noise to each of the hidden representations. This kind of stochastic networks is rarely used in practice, which limits the insights brought by the analysis. In contrast, dropout, being a source of stochasticity, is heavily used in practice due to its effective regularizing properties. The core questions investigated in this work therefore are: i) Can we obtain accurate and meaningful MI estimates in neural networks with dropout noise? ii) And if so, do IPs built for dropout networks confirm the information bottleneck hypothesis? Our main contributions answer these questions and can be summarized as follows: We present a theoretical analysis showing that binary dropout does not prevent the MI from being infinite due to the discrete nature of the noise. In contrast, we prove that dropout noise with any continuous distribution not only results in finite MI, but also provides an elegant way to estimate it. This in particular holds for Gaussian dropout, which is known to benefit generalization even more than binary dropout (Srivastava et al., 2014), and for information dropout (Achille and Soatto, 2018). We empirically analyze the quality of the MI estimation in the setup with Gaussian and information dropout in a range of experiments on benchmark neural networks and datasets. While our results do not conclusively confirm or refute the information bottleneck hypothesis, they show that the IPs obtained using our estimator exhibit qualitatively different behavior than the IPs obtained using binning estimators and strongly indicate that a compression phase is indeed happening. ## 2 Mutual Information Estimation for Neural Networks We use the following notation: Lower-case letters denote realizations of random variables (RVs), e.g., $b$ denotes a realization of the RV $B$; $H(A)$ denotes the Shannon entropy of a discrete RV $A$ whose distribution is denoted $p_{a}$; $h(B)$ is the differential entropy of a continuous RV $B$ whose distribution is described by the probability density function $p_{b}$; $I(A;B)$ is the MI between RVs $A$ and $B$; $X\in\mathcal{X}\subseteq\mathbb{R}^{n}$ and $Y\in\mathcal{Y}$ are the RVs describing inputs to a neural network and corresponding targets; $f(X)$ is the result of the forward pass of the input through the network to the hidden layer of interest; $Z$ is an $N$-dimensional RV describing the hidden representations. The caveats of different approaches to measure the MI between input $X$ and hidden representation $Z$ of a neural network - e.g., the MI being infinite for deterministic neural networks and continuous input distributions, the dependence of the MI estimate on the parameterization of the estimator, etc. - were discussed widely in the literature (Saxe et al., 2019; Geiger, 2022) and are briefly reviewed in this section. These caveats do not appear for the MI measured between representations $Z$ and targets $Y$, since the target is in most cases a discrete RV (class), for which MI is always finite. One option for estimating $I(X;Z)$ is to assume the input to be drawn from a discrete distribution. This view is supported by the finiteness of the accuracy of the used computational resources (Lorenzen et al., 2021) and makes it easy to use a finite dataset $S$ to describe the distribution. In such setup, the distribution of $(X,Y)$ is assumed uniform on the dataset $S$, and the discretization of $Z$ is performed at a fixed bin size (e.g., corresponding to the computer precision). The MI between Figure 1: IPs w.r.t. the activations of one layer with information dropout or Gaussian dropout in a LeNet network. In contrast to the IP based on estimating MI using binning, our estimates (both for Gaussian and information dropout) clearly show compression. This suggests that even if MI is finite, the binning estimator fails to converge to the true MI (see also Section 4). $X$ and the discretized $\hat{Z}$ is computed as $I(X;\hat{Z})=H(\hat{Z})-H(\hat{Z}\|X)=H(\hat{Z})-0=H(\hat{Z})$, where $H(\hat{Z}\|X)=0$ since $f(\cdot)$ and the discretization of $Z$ are deterministic. Thus, the estimated MI between input and representation corresponds to the entropy of the discretized representation, which for small bin sizes is equal to the entropy $H(X)=\log\|S\|$ of the empirical distribution on the dataset, unless $f(\cdot)$ maps different points from the dataset to the same point in latent space. A different option that is more aligned to the common description of real-world data is to assume $X$ to be drawn from a continuous distribution. If the network transformation $f(\cdot)$ results in a discrete distribution of the representations $Z$, one can use the decomposition $I(X,Z)=H(Z)-H(Z\|X)=H(Z)$ to estimate MI based on Shannon entropy, provided that the sample size is sufficiently large (note that the dimensionality $N$ of $Z$ may be large, and therefore the estimation of $H(Z)$ may suffer from the curse of dimensionality). However, as shown in Theorem 1 of (Amjad and Geiger, 2020) for neural networks with commonly used activation functions the distribution of the latent representation is not discrete. In this case (i.e., $f(\cdot)$ is deterministic, $X$ is continuous, and $Z$ is not purely discrete) the MI between $X$ and $Z$ is infinite2. By binning, i.e., by quantizing $Z$ to a discrete RV $\hat{Z}$, the MI $I(X;\hat{Z})=H(\hat{Z})$ remains finite, but the qualitative behavior of this entropy will be defined by properties of activation functions and selected bin size (Saxe et al., 2019). Footnote 2: There are multiple mathematical derivations explaining why MI is infinite, one for example is discussed in (Saxe et al., 2019, Appendix C). From the discussion above it follows that estimating $I(X;Z)$ in deterministic neural networks is an ill-posed problem, and that the estimates reveal not an information-theoretic picture, but often rather a geometric one that is determined by the properties of the chosen estimators. As a solution to the aforementioned challenges, several authors have suggested to investigate the information planes of stochastic neural networks instead (Amjad and Geiger, 2020, Goldfeld et al., 2019). Goldfeld et al. (2019) proposed to add zero-mean Gaussian noise $D$ to the representations during training. This transforms a deterministic neural network into a stochastic one that was shown to yield similar training results and predictive abilities of the model. The addition of Gaussian noise in $Z=f(X)+D$ guarantees a finite MI3 and therefore allows for estimating MI using Monte Carlo sampling with bounds on the estimation error. Futhermore, it links the information-theoretic perspective of the IP to geometric effects taking place in latent space. Indeed, when the MI between input and representation is decreasing, it means that noise-induced Gaussians centered at the representations of different data points overlap more strongly. Thus, it is becoming harder to distinguish between inputs of the same class based on their representations, which translates into lower MI between representation and input while leaving MI between representation and target unchanged. Footnote 3: At least when the $p_{x}$ and $f(\cdot)$ are such that $f(X)$ has finite variance, then the finiteness of MI follows from the result about the capacity of the additive Gaussian noise channel, cf. (Cover and Thomas, 1991, eq. (10.17)). As discussed above, for continuous input distributions both the IPs of deterministic neural networks as well as of stochastic neural networks with additive noise show a geometric picture (and in the former case the geometric interpretation is the only valid one, since MI is infinite in this case). Therefore, in this work we study the estimation of MI in networks with dropout layers, i.e., in settings where the stochasticity is introduced by multiplicative, rather than additive noise. In what follows we will investigate the requirements on the multiplicative noise for MI to remain finite, and whether the resulting IPs confirm the information bottleneck hypothesis. ## 3 Mutual Information in Dropout Networks As discussed in the previous section, the MI between inputs and hidden representations of deterministic networks is infinite, if we assume the input distribution to be continuous. To overcome this problem, some form of stochasticity has to be introduced. While adding noise to activations (Goldfeld et al., 2019) indeed allows to compute the MI, this is not used in most contemporary neural networks. In contrast, neural networks with dropout are one of the most popular classes of neural networks used in practice and are stochastic in nature as well: Adding a dropout layer to a neural network corresponds to multiplying the hidden representation with some form of random noise. Formally, denoting the random noise by a RV $D$ of the same dimension as $f(X)$, the hidden representation becomes $Z=f(X)\circ D$, where $\circ$ denotes element-wise multiplication. In the most basic form, $D$ follows a Bernoulli distribution (Srivastava et al., 2014). Such binary dropout is widely used and can intuitively been understood as "turning off" a fraction of neurons during training. There is a variety of other dropout schemes, including multiplicative Gaussian noise, fast dropout (Wang and Manning, 2013), or variational dropout (Kingma et al., 2015). Information dropout (Achille and Soatto, 2018) is a variant that uses a closed-form expression of MI as regularization term. In order to obtain such closed form, dropout noise is sampled from a log-normal distribution, and the prior distribution on representations is chosen depending on the activation function (ReLU or Softplus). We provide details on the derivation in Appendix A.1. In this section, we investigate whether neural networks with dropout have indeed finite MI between input $X$ and representation $Z$. While we first show a negative result by proving that binary dropout still leads to $I(X;Z)=\infty$, our Theorem 3.3 shows that dropout with continuous distribution keeps MI finite. This fact allows us to estimate MI for such dropout neural networks in Sections 4 and 5. ### Binary Dropout We start by analyzing binary dropout, which forces individual neurons to be "turned off" with some probability. More formally, the output of each neuron is multiplied with an independent Bernoulli RV that is equal to $1$ with a predefined probability $p$. The following theorem shows that this kind of (combinatorial) stochasticity is insufficient to prevent $I(X;Z)$ from becoming infinite. Theorem 3.1.: _In the setting of (Amjad and Geiger, 2020, Th. 1), let the output $f(\cdot)$ of a hidden layer be parameterized as a deterministic neural network with $\hat{N}$ neurons, let $B\in\{0,1\}^{\hat{N}}$ be the set of independent Bernoulli RVs characterizing the dropout pattern, and let $Z=f_{B}(X)$ denote the output of the hidden layer after applying the random pattern $B$. Then it holds that $I(X;Z)=\infty$._ In the proof (provided in Appendix A.2) we use the fact that dropout mask $b=(1,1,\dots,1)$ leads to an infinite MI. While the Bernoulli distribution guarantees that $b=(1,1,\dots,1)$ always has non-zero probability, other distributions over $\{0,1\}^{\hat{N}}$ might not have this property. Theorem 3.1 can however be generalized to arbitrary distributions over $\{0,1\}^{\hat{N}}$: Theorem 3.2.: _In the setting of (Amjad and Geiger, 2020, Th. 1), let the output $f(\cdot)$ of a hidden layer be parameterized as a deterministic neural network with $\hat{N}$ neurons, let $B\in\{0,1\}^{\hat{N}}$ be the binary random vector characterizing the dropout pattern, and let $Z=f_{B}(X)$ denote the output of the hidden layer after applying the random pattern $B$. Then, it either holds that $I(X;Z)=\infty$ or that $I(X;Z)=0$ if the dropout patterns almost surely disrupt information flow through the network._ The proof for the theorem is provided in Appendix A.3. Both Theorem 3.1 and Theorem 3.2 cover as a special case the setting where dropout is applied to only a subset of layers, by simply setting those elements of $B$ to $1$ that correspond to a neuron output without dropout. If dropout is applied to only a single layer, then $f_{B}(X)=f(X)\circ B^{\prime}$, where $B^{\prime}$ is the dropout pattern of the considered layer and $\circ$ denotes the element-wise product. As a consequence of Theorem 3.2, for neural networks with binary dropout any finite estimate of MI is "infinitely wrong", and the resulting IP does not permit an information-theoretic interpretation. Essentially, the stochasticity added by binary dropout is combinatorial, and hence cannot compensate the "continuous" stochasticity available in the input $X$. ### Dropout with Continuous Noise As proposed by Srivastava et al. (2014), dropout can also be implemented using continuous Gaussian noise with mean vector $\mu=\mathbf{1}$ and diagonal covariance matrix $I\sigma^{2}$ with fixed variance $\sigma^{2}$. Achille and Soatto (2018), in contrast, proposed log-normally distributed dropout noise, the variance of which depends on the input sample $x$ (this is termed information dropout). Generalizing both Gaussian and information dropout, in this section we consider continuously distributed multiplicative noise $D$. In contrast to binary noise sampled from a discrete distribution, continuously distributed noise turns the joint distribution of $(Z,X)$ to be absolutely continuous with respect to the marginals of $Z$ and $X$ allowing for finite values of MI between the input $X$ and the hidden representation $Z$. The following theorem states that the MI between input and the hidden representation of the dropout layer is indeed finite even if the variance of the noise depends on the input. Theorem 3.3.: _Let $X$ be bounded in all dimensions, $f(\cdot)$ be parameterized by a deterministic neural network with Lipschitz activation functions, and let $Z=f(X)\circ D(X)$, where the components of noise $D(X)=(D_{1}(X),\ldots,D_{N}(X))$ are conditionally independent given $X$ and have essentially bounded differential entropy and second moments, i.e., $\mathbb{E}[D_{i}(X)^{2}]\leq M<\infty$$X$-almost surely, for some $M$ and all $i=1,\ldots,N$. Then, if the conditional expectation $\mathbb{E}[\log(\|f(X)\|)\mid\|f(X)\|>0]$ is finite in each of its elements, we have $I(X;Z)<\infty$._ Theorem 3.3 (proof in Appendix A.4) can be instantiated for Gaussian dropout, where $D_{i}(x)=D_{i}\sim\mathcal{N}(1,\sigma^{2})$, and for information dropout, where $D_{i}(x)\sim\log\mathcal{N}(0,\alpha^{2}(x))$. Note that for information dropout we have to ensure that the (learned) variance $\alpha^{2}(x)$ stays bounded from above and below; e.g., in the experiments of Achille and Soatto (2018), $\alpha^{2}(x)$ is restricted to be below $0.7$. The requirement that the conditional expectation $\mathbb{E}[\log(\|f(X)\|)\mid\|f(X)\|>0]$ is finite in each of its elements is critical for the proof. Indeed, one can construct a synthetic (albeit unrealistic) example for which this condition is violated: Example 3.4.: _Let $X^{\prime}$ have the following probability density function_ \[p_{x^{\prime}}(x^{\prime})=\begin{cases}2^{-n},&\text{if $x^{\prime}\in[2^{n},2^{ n}+1),n=1,2,\ldots$}\\ 0,&\text{else}\end{cases}\] _Evidently, $\mathbb{E}[X^{\prime}]=\infty$. Then, $X=\mathrm{e}^{-X^{\prime}}$ is bounded, since its alphabet is a subset of $(0,\mathrm{e}^{-2}]$._ _Now consider a neural network with a single hidden layer with one neuron. Let the weight from $X$ to the single neuron be $1$, and assume that the neuron uses a ReLU activation function. Then,_ \[\mathbb{E}[\log\|f(X)\|]=\mathbb{E}[\log\|X\|]=\mathbb{E}[\log\|\mathrm{e}^{-X^{ \prime}}\|]=\mathbb{E}[-X^{\prime}]=-\infty\enspace.\] It can be shown that in this example the probability density function of $X$ (as well as of $f(X)$) is not bounded. Under the assumption that the probability density function $p_{f}$ of $f(X)$ is bounded, the conditional expectation in the assertion of the theorem is finite: Assuming that $p_{f}\leq C<\infty$, by the law of unconscious statistician we have \[\mathbb{E}_{x}[\log(\|f(X)_{i}\|)\mid\|f(X)_{i}\|>0] =\int_{0}^{\\|f(X)_{i}\\|_{\infty}}\log(f)p_{f}(f)\mathrm{d}f\] \[=\underbrace{\int_{0}^{1}\log(f)p_{f}(f)\mathrm{d}f}_{I_{1}}+ \underbrace{\int_{1}^{\\|f(X)_{i}\\|_{\infty}}\log(f)p_{f}(f)\mathrm{d}f}_{I_{2}}\enspace.\] It is obvious that $I_{2}$ is positive and finite. Due to the boundedness of $p_{f}$ we also have $I_{1}\geq C\int_{0}^{1}\log(f)\mathrm{d}f=Cf(\log(f)-1)\|_{0}^{1}=-C>-\infty$. However, the boundedness of $p_{f}$ of is hard to guarantee for an arbitrary neural network. In contrast, the boundedness of $p_{x}$ is more realistic and easier to check. For bounded $p_{x}$ we can prove (in Appendix A.5) the finiteness of the expectation $\mathbb{E}[\log(\|f(X)\|)\mid\|f(X)\|>0]$ for ReLU networks: Proposition 3.5.: _Consider a deterministic neural network function $f(\cdot)$ constructed with finitely many layers, a finite number of neurons per layer, and ReLU activation functions. Let $X$ be a continuously distributed RV with probability density function $p_{x}$ that is bounded ($p_{x}\leq P<\infty$) and has bounded support $\mathcal{X}$. Then, the conditional expectation $\mathbb{E}[\log(\|f(X)\|)\mid\|f(X)\|>0]$ is finite in each of its elements._ Finally, note that Theorem 3.3 assumes that the network is deterministic up to the considered dropout layer. This does not come with a loss of generality for feed-forward networks (e.g., with no residual connections): Indeed, one can apply Theorem 3.3 to the first hidden layer representation $Z^{(1)}$ with dropout, where this assumption always holds. Then, for the $\ell$-th hidden layer and irrespective of whether this layer also has dropout, the MI $I(X;Z^{(\ell)})$ is finite due to the data processing inequality (Cover and Thomas, 1991, Th. 2.8.1). Therefore, Theorem 3.3 ensures that MI is finite for all hidden layers after the first continuous dropout layer. ## 4 Estimation of MI under Continuous Dropout We now consider estimating $I(X;Z)$ in networks with continuously distributed dropout, starting with information dropout. As discussed by Achille and Soatto (2018), networks with information dropout are trained with the cross-entropy loss $\ell_{ce}$ (which is involved in the known variational lower bound $I(Z;Y)\geq H(Y)-\ell_{ce}$) and regularized using a variational upper bound on $I(X;Z)$. Therefore, estimates of the quantities displayed in the information plane are directly used in the training loss and, thus, easy to track, at least for softplus activation functions4. Footnote 4: Indeed, for softplus activation functions, the variational approximation of $I(X;Z)$ is available in closed form, while for ReLU activation functions, the available expression is only useful for minimizing, rather than for computing, $I(X;Z)$ (see Appendix A.1). In the case of Gaussian dropout, to estimate $I(X;Z)$ we approximate $h(Z)$ and $h(Z\|X)$ separately (pseudocode is given in Algorithm 1 in Appendix A.6). For estimating $h(Z)$ we employ a Monte Carlo (MC) estimate, similar to the one proposed by Goldfeld et al. (2019). That is, we approximate the distribution of $Z$ as a Gaussian mixture, where we draw samples $f(x^{(j)}),j=1,\ldots,\|S\|$ and place Gaussians with a diagonal covariance matrix with variances $\sigma^{2}\|f(x^{(j)})_{i}\|^{2},i=1,\ldots,N$ on each sample$f(x^{(j)})$. For a sanity check, we also compute an upper bound of $h(Z)$ given by the entropy of a Gaussian with the same covariance matrix as $Z$. Note that the estimation of the upper bound requires a sufficiently large number of samples to guarantee that the sample covariance matrix is not singular and that the resulting entropy estimate is finite. For each fixed $x$ the conditional distribution $p_{z\|x}$ is a Gaussian distribution $\mathcal{N}(f(x),\mathrm{diag}(\{\sigma^{2}\|f(x)_{i}\|^{2}\}))$. Moreover, when the input is fixed, the components of $Z\|X=x$ are independent, since components of the noise are independent. This allows to compute $h(Z\|X)$ as a sum of $h(Z_{i}\|X)$ where $Z_{i}$ is the $i$-th component of the representation vector. The computation of $h(Z_{i}\|X)$ requires integration over the input space for computing the mathematical expectation $\mathbb{E}_{x}\big{[}h(Z_{i}\|X=x)\big{]}$. This can be approximated via MC sampling. That is, we approximate $h(Z_{i}\|X)$ by $1/\|S\|\sum_{j=1}^{\|S\|}h(Z_{i}\|X\,=\,x^{(j)})$ where $h(Z_{i}\|X=x^{(j)})=\log(\|f(x^{(j)})_{i}\|\sigma\sqrt{2\pi e})$. We consider a simple toy problem for validating our approach to estimating MI: the input $X$ is generated from an $n$-dimensional standard normal distribution, modified with a function $f(X)=2X+0.5$, and then subjected to Gaussian dropout distributed according to $\mathcal{N}(1,\sigma^{2})$. We investigate the convergence of our estimator for $h(Z\|X)$ for increasing number of samples. For each input data point, we generate $10$ noise masks, thus obtaining $10$ samples of $Z$ for each $x^{(j)}$. The results in Fig. 2 show that the estimation stabilizes with larger amount of samples for different dimensionality of the data. We also compare the estimate to the upper bound for $h(Z)$ in Fig 3. We finally compare our estimation of MI to binning, the EDGE estimator (Noshad et al., 2019), and the lower bounds analyzed by McAllester and Stratos (2020). The results are shown in Fig. 4. In the plot, doe stands for the difference-of-entropies (DoE) estimator and doe_1 stands for DoE with logistic parametrization (McAllester and Stratos, 2020). The binning estimator underestimates the Figure 3: Estimates of the differential entropy $h(Z)$ of the hidden representation $Z$. With growing dimensionality of $X$, the Gaussian upper bound becomes very loose, compared to the Gaussian mixture-based MC estimation. Figure 2: Independent of the dimensionality, MC estimation of $h(Z\|X)$ stabilizes with increasing amount of samples. MI when the bin size is large and overestimates it with small bin size (Ross, 2014), which can be clearly seen in the plots where bins are organized both by size (upper axis) and by number (lower axis). Moreover, with the high-dimensional data, binning hits the maximal possible value of $\log(\|S\|)$ very fast, not being able to reach larger MI values. According to McAllester and Stratos (2020), lower bound-based MI estimators (e.g., MINE (Belghazi et al., 2018)) also need exponentially (in the true value of MI) many data points for a good value approximation, otherwise they will always heavily underestimate the MI. Further plots for different dropout variances and inputs dimensionality are given in Appendix A.6. ## 5 Information Plane Analysis of Dropout Networks We use the estimators described in the previous section for an IP analysis of networks with Gaussian and information dropout. We always consider only the representation corresponding to the first dropout layer 5 and measure the MI in nats, e.g., use the natural logarithm. For estimating $I(Y;Z)$, we employ the EDGE estimator (Noshad et al., 2019) for Gaussian dropout and variational estimate for information dropout. IPs created using the binning estimator use binning for both $I(X;Z)$ and $I(Y;Z)$. Footnote 5: This makes the MI estimation more efficient, since the previous part of the network is deterministic which allows for an analytical expression of $h(Z\|X=x)$. Note however, that the estimation could be extended to higher layers as well since for those MI also remains finite. However, an estimator different from ours should be used for those layers. In the first set of experiments we investigate the difference between IPs obtained via our proposed estimator and via binning. The analysis on the MNIST dataset was performed for a LeNet network (LeCun et al., 1998) that achieves $99\%$ accuracy and a simple fully-connected (FC) network with three hidden layers ($28\times 28-512-128-32-10$) and softplus activation functions achieving $97\%$ accuracy. We analyze both information dropout and Gaussian dropout in the LeNet network and only Gaussian dropout in the FC network. In both cases dropout is applied on penultimate layers. We compare IPs based on binning estimators to IPs based on our estimators in Fig. 1 and Fig. 5. Figure 4: Comparison of various approaches to MI estimation for the toy example with multiplicative Gaussian noise. For low-dimensional $X$ and $Z$, different bin sizes lead to different MI estimates of the binning estimator. For higher dimensions, the binning-based estimate is collapsing. Our estimation is very close to the lower bound estimation proposed by McAllester and Stratos (2020), while still being larger as expected. Figure 5: IPs for a FC network with Gaussian dropout trained on MNIST. Compared to the binning estimation of MI our approach shows compression. We also analyze the IPs for a ResNet18 trained on CIFAR10 (see Fig. 6), where we added an additional bottleneck layer with $128$ neurons and Gaussian dropout before the output layer, and which achieves an accuracy of $94\%$. Interestingly, for all networks and datasets we observe significant compression for our estimator and a lack of compression for binning estimators (also for different bin size, see Appendix A.8). This indicates that either the MI compression measured in dropout networks is different from purely geometrical compression, or that the number of samples $\|S\|$ is insufficient to reliably estimate $I(X;Z)$ by binning. In the second set of experiments, we analyze IPs in information dropout networks, with MI estimations as described before. To this end, we trained a fully convolutional neural network (fullCNN) on CIFAR10 using code provided by Achille and Soatto (2018). Training proceeded for $200$ epochs using SGD with momentum and, different from the original setup, with only one dropout layer after the third convolutional layer. The batch size was set to $100$, the learning rate was initially set to $0.05$ and was reduced by multiplying it with $0.1$ after the $40$, $80$, and $120$ epoch. The network was trained with different values of the regularization weight $\beta$ and different amounts of filters in the convolutional layers. That is, the full-size fullCNN has $3$ layers with $96$ filters succeeded by $4$ layers with $192$ filters, while only $25\%$ of these filters are constituting the small network. Also different from the original setup, we allowed the noise variance to grow up to $0.95$ in order to see the effect of the limited information between representation and input more pronounced. Results are shown in Fig. 7. It can be seen that regularizing $I(X;Z)$ is effective (i.e., larger values of $\beta$ lead to smaller $I(X;Z)$), and that regularizing too strongly ($\beta=20$) leads to worse performance: the test error is $5\%$ higher and train error is $10\%$ higher. We can further see stronger compression for smaller $\beta$ and almost no compression for larger $\beta$. We conjecture that compression can only become visible if sufficient information is permitted to flow through the network (which happens only for small $\beta$). Fig. 7 (c) and (d) show the IPs for the small fullCNN. It can be seen that the smaller network appears not to compress at all (see Fig. 7 (c)), but that $I(X;Z)$ rather increases throughout training until it is at the same level as in Fig. 7 (a). This indicates that $\beta$ determines to which point in the IP information compresses, and that the IP curve that is traversed during training depends on the overall capacity of the neural network. Plots for the additional experiments can be found in Appendix A.8. Figure 6: IPs for a ResNet18 network with Gaussian dropout trained on CIFAR10. In contrast to the binning-based estimator of MI our approach clearly shows compression. Figure 7: IPs demonstrate more (a), (c) and less (b), (d) compression of MI between input and representation depending on $\beta$. The values of $I(X;Z)$ are smaller for the smaller network (c) and (d). ## 6 Discussion Whether or not information-theoretic compression is correlated with improved generalization is the main question connected to and the most prominent justification for information plane analysis of deep neural networks. Such a connection, however, can only be tested for neural networks for which MI is finite and therefore measurable. In our theoretical analysis, we investigate if different variants of dropout noise allow for finite values of MI under an assumption of a continuous input distribution. We answered this question positively by showing that in networks with certain constraints on the induced distribution of the representations, continuous dropout noise with finite differential entropy prevents $I(X;Z)$ from becoming infinite. We have further shown that these constraints on the distribution of the representation are satisfied in ReLU networks if the probability density function of the input is bounded. Following this conclusion we propose an MC-based estimate of MI in Gaussian dropout networks and perform an IP analysis for different networks with Gaussian and information dropout on different datasets. The experiments show that the binning estimator behaves very differently from our estimator: While our estimator mostly exhibits compression in the IP, the binning estimator does not. Further, the values of $I(X;Z)$ for our estimator are often orders of magnitude larger than the values of $I(Y;Z)$, especially when compared to the binning estimator. Assuming that the proposed estimators are reasonably accurate, this makes a connection between information-theoretic compression and generalization questionable. While these preliminary experiments do not conclusively answer the question if such a connection exists, they show a practically relevant setting in which this correlation can be studied. The discrepancy between the binning estimator and our estimator further suggests that either the information-theoretic compression we observe using our estimator is not geometric, or that there are insufficient samples to obtain reliable estimates from the binning estimator. This is in contrast with the work of Goldfeld et al. (2019), which showed that information-theoretic and geometric compression were linked in their networks with additive noise. We thus believe that a closer investigation of whether multiplicative noise induces geometric compression, and whether the induced compression improves generalization performance, are interesting questions for future research. ## Acknowledgements The authors want to thank Michael Kamp, Simon Damm, Ziv Goldfeld, and Jihao Andreas Lin for valuable discussions about the work. Asja Fischer acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2092 CASA - 390781972. The Know-Center is funded within the Austrian COMET Program - Competence Centers for Excellent Technologies - under the auspices of the Austrian Federal Ministry of Climate Action, Environment, Energy, Mobility, Innovation and Technology, the Austrian Federal Ministry of Digital and Economic Affairs, and by the State of Styria. COMET is managed by the Austrian Research Promotion Agency FFG.
2302.00873	Predicting the Silent Majority on Graphs: Knowledge Transferable Graph Neural Network	Graphs consisting of vocal nodes ("the vocal minority") and silent nodes ("the silent majority"), namely VS-Graph, are ubiquitous in the real world. The vocal nodes tend to have abundant features and labels. In contrast, silent nodes only have incomplete features and rare labels, e.g., the description and political tendency of politicians (vocal) are abundant while not for ordinary people (silent) on the twitter's social network. Predicting the silent majority remains a crucial yet challenging problem. However, most existing message-passing based GNNs assume that all nodes belong to the same domain, without considering the missing features and distribution-shift between domains, leading to poor ability to deal with VS-Graph. To combat the above challenges, we propose Knowledge Transferable Graph Neural Network (KT-GNN), which models distribution shifts during message passing and representation learning by transferring knowledge from vocal nodes to silent nodes. Specifically, we design the domain-adapted "feature completion and message passing mechanism" for node representation learning while preserving domain difference. And a knowledge transferable classifier based on KL-divergence is followed. Comprehensive experiments on real-world scenarios (i.e., company financial risk assessment and political elections) demonstrate the superior performance of our method. Our source code has been open sourced.	Wendong Bi, Bingbing Xu, Xiaoqian Sun, Li Xu, Huawei Shen, Xueqi Cheng	2023-02-02T04:46:21Z	http://arxiv.org/abs/2302.00873v3	# Predicting the Silent Majority on Graphs: ###### Abstract. Graphs consisting of vocal nodes ("the vocal minority") and silent nodes ("the silent majority"), namely VS-Graph, are ubiquitous in the real world. The vocal nodes tend to have abundant features and labels. In contrast, silent nodes only have incomplete features and rare labels, e.g., the description and political tendency of politicians (vocal) are abundant while not for ordinary civilians (silent) on the twitter's social network. Predicting the silent majority remains a crucial yet challenging problem. However, most existing Graph Neural Networks (GNNs) assume that all nodes belong to the same domain, without considering the missing features and distribution-shift between domains, leading to poor ability to deal with VS-Graph. To combat the above challenges, we propose Knowledge Transferable Graph Neural Network (KTGNN), which models distribution-shifts during message passing and learns representation by transferring knowledge from vocal nodes to silent nodes. Specifically, we design the domain-adapted "feature completion and message passing mechanism" for node representation learning while preserving domain difference. And a knowledge transferable classifier based on KL-divergence is followed. Comprehensive experiments on real-world scenarios (i.e., company financial risk assessment and political elections) demonstrate the superior performance of our method. Our source code has been open-sourced1. Footnote 1: The source code is available at [https://github.com/wendongbi/KT-GNN](https://github.com/wendongbi/KT-GNN) + Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=]Footnote †: [leftmargin=] †: [leftmargin=]Footnote †: [leftmargin=] †: [leftmargin= While the silent nodes ("the silent majority") have rare labels and incomplete features. VS-Graph is also common in other real-world scenarios, such as celebrities and ordinary netizens in social networks. Meanwhile, predicting the silent majority on VS-Graph is important yet challenging. Recently, Graph neural networks (GNNs) have achieved state-of-the-art performance on graph-related tasks. Generally, GNNs follow the message-passing mechanism, which aggregates neighboring nodes' representations iteratively to update the central node representation. However, the following three problems lead to the failure of GNNs to predict the silent majority: 1) the data distribution-shift problem exists between vocal nodes and silent nodes, e.g., listed companies have more assets and cash flows and therefore show different distribution from unlisted companies in terms of attributes, which is rarely considered by the previous GNNs; 2) the feature missing problem exists and can not be solved by traditional heuristic feature completion strategies due to the above data distribution-shift; 3) the lack of labels for silent majority hinders an ideal model, and directly training models on vocal nodes and silent nodes concurrently leads to poor performance in predicting the silent majority due to the distribution-shift. To solve the aforementioned challenges, we propose Knowledge Transferable Graph Neural Network (KTGNN), which targets at learning representations for the silent majority on graphs via transferring knowledge adaptively from vocal nodes. KTGNN takes domain transfer into consideration for the whole learning process, including feature completion, message passing, and the final classifier. Specifically, Domain-Adapted Feature Complementor (DAFC) and Domain-Adapted Message Passing (DAMP) are designed to complement features for silent nodes and conduct message passing between vocal nodes and silent nodes while modeling and preserving the distribution-shift. With the learned node representations from different domains, we propose the Domain Transferable Classifier (DTC) based on KL-divergence minimization to predict the labels of silent nodes. Different from existing transfer learning methods that aim to learn domain-invariant representations, we propose to transfer the parameters of classifiers from vocal domain to silent domain rather than forcing a single classifier to capture domain-invariant representations. Comprehensive experiments on two real-world scenarios (i.e., company financial risk assessment and political elections) show that our method gains significant improvements on silent node prediction over SOTA GNNs. The contributions of this paper are summarized as follows: * We define a practical and widespread VS-Graph, and solve the new problem of predicting silent majority, one important and realistic AI application. * To predict silent majority, we design a novel Knowledge Transferable Graph Neural Network (KTGNN) model, which is enhanced by the Domain-Adapted Feature Complementor and Messaging Passing modules that preserve domain-difference as well as the KL-divergence minimization based Domain Transferable Classifier. * Comprehensive experiments on two real-world scenarios (i.e., company financial risk assessment and political elections) show the superiority of our method, e.g., AUC achieves an improvement of nearly 6% on company financial risk assessment compared to the current SOTA methods. ## 2. Preliminary In this section, we give the definitions of important terminologies and concepts appearing in this paper. ### Graph Neural Network Graph Neural Networks (GNNs) aim at learning representations for nodes on the graph. Given a graph $\mathbf{G}(V,E,X,Y)$ as input, where $V=\{v_{i}\|v_{i}\in V\}$ denotes the node set and $E=\{e_{ij}=(v_{i},v_{j})\|v_{i}\text{ and }\ v_{j}\}$ is connected. $N=\|V\|$ is the number of nodes, $X\in\mathbb{R}^{N\times D_{in}}$ is the feature matrix and $Y\in\mathbb{R}^{N\times 1}$ is the labels of all nodes. Then GNNs update node representations by stacking multiple graph convolution layers and current mainstream GNNs follow the messaging passing mechanism, each layer of GNNs updates node representations with the following function: \[h_{i}^{(I)}=\mathbf{U}\left(h_{i}^{(I-1)},\mathbf{M}(\{h_{i}^{(I-1)},h_{j}^{( I-1)}\|v_{j}\in\mathcal{N}(v_{i})\})\right)\] where $h_{i}^{(I)}$ is the node representation vector at $I$-th layer of GNN, $\mathbf{M}$ denotes the message function of aggregating neighbor's features, and $\mathbf{U}$ denotes the update functions with the neighborhood messages and central node feature as inputs. By stacking multiple layers, GNNs can aggregate information from higher-order neighbors. ### Problem Definition: Silent Node Classification on the VS-Graph In this paper, we propose the problem of silent node prediction on VS-Graph and we mainly focus on the node classification problem. First, we give the definition of VS-Graph: Definition 2.1 (VS-Graph).: Given a VS-Graph $\mathbf{G}^{\text{BS}}(V,E,X,Y,\Psi)$, where $V=V^{\text{ococal}}\cup V^{silent}$ is the node-set, including all vocal nodes Figure 1. Examples of silent node classification, where (a) and (b) show the difference between silent node classification vs. traditional node classification. (c) and (d) show two real-world VS-Graphs. and silent nodes. $E=\{e_{ij}=(v_{i},e_{j})\|v_{i},e_{j}\in V^{local}\cup V^{silent}\}$ is the edge set. $X=[X^{local};\ X^{silent}]$ is the attribute matrix for all nodes and $Y=[Y^{local};\ Y^{silent}]$ is the class label of all nodes ($y_{i}=-1$ denotes the label of $v_{i}$ is unavailable). $\Psi:V\rightarrow\{vocal,silent\}$ is the population indicator that maps a node to its node population (i.e., vocal nodes, silent nodes). \[\Psi(v_{i})=\begin{cases}1&\text{if $v_{i}$ is a silent node}\\ 0&\text{if $v_{i}$ is a vocal node}\end{cases} \tag{1}\] For a VS-Graph, the attributes and labels of vocal nodes (i.e., $X^{local}$, $Y^{local}$) are complete. However, $Y^{silent}$ is rare and $X^{silent}$ is incomplete. Specifically, for vocal node $v_{i}\in V^{local}$, its attribute vector $x_{i}=[x_{i}^{o}\|x_{i}^{u}]$, where $x_{i}^{o}\in\mathbb{R}^{Do}$ is part of attributes observable for all nodes and $x_{i}^{u}\in\mathbb{R}^{Do}$ is part of attributes observable for silent nodes while observable for vocal nodes. For silent node $v_{i}\in V^{silent}$, its attribute vector $x_{i}=[x_{i}^{o}\|\|\mathbf{0}]$, where $x_{i}^{o}\in\mathbb{R}^{Do}$ is the observable part and $\mathbf{0}\in\mathbb{R}^{Do}$ means the unobservable part which is complemented by $0$ per-dimension initially. Then the problem of Silent Node Classification on VS-Graph is defined as follows: Definition 2 (Silent Node Classification on the VS-Graph).: Given a VS-Graph $\mathbf{G}^{\text{BS}}(V,E,X,Y,\Psi)$, where $\|V^{local}\|<<\|V^{silent}\|$ and the attributes of nodes from the two populations (i.e., $X^{local}$ and $X^{silent}$) belong to different domains with distinct distributions $\mathcal{P}^{local}\neq\mathcal{P}^{silent}$. Under these settings, the target is to predict the labels ($Y^{silent}$) of silent nodes ($V^{silent}$) with the support of a small set of vocal nodes ($V^{local}$) with complete but out-of-distribution information. ## 3. Exploratory Analysis "The Silent Majority" in Real-world Graphs: To demonstrate that the distribution-shift problem between vocal nodes and silent nodes does exist in the real-world, we choose two representative real-world scenarios (i.e., political social network, company equity network) as examples and conduct comprehensive analysis on the two real-world VS-Graphs. And the observations show that there exists a significant distribution-shift between vocal nodes and silent nodes on their shared feature dimensions. The detailed information of the datasets is summarized in Sec. 5.1. ### Single-Variable Analysis We analyze the distribution of single-variable on the real-world Company VS-Graph, including the listed company nodes and unlisted company nodes (more details in Sec. 5.1). All node attributes of this dataset are from real-world companies and have practical physical meanings (e.g., register capital, actual capital, staff number), which are appropriate for statistical analysis of a single dimension. Instead of directly calculating the empirical distribution, we visualize the distribution with box-plots in a more intuitive manner. We select three important attributes (i.e., register capital, actual capital, staff number) and present the box plot of all nodes distinguished by their labels (i.e., risky company or normal company) and populations (i.e., listed company or unlisted company) in the Fig. 2. We observe that there exists significant distribution-shift between listed companies (vocal nodes) and unlisted companies (silent nodes), which confirms our motivations. Considering that there are hundreds of attributes, we only present three typical attributes correlated to the company assets here due to the space limitation. ### Multi-Variable Visualization The above single-variable analysis demonstrates that there exists significant domain difference on the certain dimension of attributes between vocal nodes and silent nodes. To further demonstrate the domain difference of all attributes, we also visualize by scatter plot on all attributes with t-SNE Algorithm (Zhu et al., 2017). Specifically, we first project the $D_{in}$-dimensional attributes to 2-dimensional vectors and then present the scatter plot on the 2D plane. As shown in Fig. 3, we respectively visualize the two VS-Graph Datasets: Twitter (a political social network) and Company (a company equity network). Specifically, Fig. 3 (a) shows the nodes of Figure 3. T-SNE visualization of the raw node features distinguished by their populations (vocal$\rightarrow$orange, silent$\rightarrow$cyan) or labels (binary classes: positive$\rightarrow$red, negative$\rightarrow$blue). Note that we only visualize the observable attributes $X^{o}$ (dimensions shared by vocal and silent nodes). Figure 2. Box-plot of single-variable distribution for company equity graph dataset. Each box-plot actually represents the conditional distribution $P(X\|Y,\mathcal{O})$, where the Y-axis is the value of attributes after the log function. different populations in the Twitter dataset (two colors denote vocal nodes and silent nodes); Fig. 3 (b) shows the vocal nodes (politicians) of different classes (i.e., political parties, including democrat and republican) and Fig. 3 (c) shows the silent nodes (civilians) of different classes. For the company dataset, Fig. 3 (d) shows the nodes of different populations (two colors denote vocal nodes and silent nodes); Fig. 3 (e) shows the vocal nodes (listed companies) of different classes (i.e., risky/normal company) and Fig. 3 (c) shows the silent nodes (unlisted companies) of different classes. Fig. 3 (a) and (d) demonstrate that nodes from different populations have distinct distributions, which are reflected as different clusters with distinct shapes and positions. From Fig. 3 (b) and (e), we observe that the vocal nodes of different classes have similar distributions, which are reflected as indistinguishable clusters mixed together, and the silent nodes of different classes Fig. 3 (c) and (f) have similar phenomena as vocal nodes. All these findings demonstrate that there exists a significant distribution-shift between vocal nodes and silent nodes on these real-world VS-Graphs. ## 4. Methodology We propose the Knowledge Transferable Graph Neural Network (KTGNN) to learn effective representations for silent nodes on VS-Graphs. ### Model Overview We first present an overview of our KTGNN model in Fig. 4. KTGNN can be trained through an end-to-end manner on the VS-Graph and is composed of three main components: (1) Domain Adapted Feature Complementary; (2) Domain Adapted Message Passing module; (3) Domain Transferable Classifier. ### Domain Adapted Feature Complementary (DAFC) Considering that part of the attributes are unobservable for silent nodes, we first complement the unobservable features for silent nodes with the cross-domain knowledge of vocal nodes. An intuitive idea is to complete the unobservable features of silent nodes by their vocal neighboring nodes which have complete features. However, not all silent nodes on the VS-Graph have vocal neighbors. To complete the features of all silent nodes, we propose a Domain Adapted Feature Complementary (DAFC) module which can complement the features of silent nodes by adaptively transferring knowledge from vocal nodes with several iterations. Before completing features, we first partition the nodes on the VS-Graph into two sets: $\mathcal{V}^{+}$ and $\mathcal{V}^{-}$, where nodes in $\mathcal{V}^{+}$ have complete features while nodes in $\mathcal{V}^{-}$ have incomplete features. Initially, $\mathcal{V}^{+}=V^{local}$ and $\mathcal{V}^{-}=V^{silent}$. Then the DAMP module completes the features of nodes in $\mathcal{V}^{-}$ by transferring the knowledge of nodes in $\mathcal{V}^{+}$ to $\mathcal{V}^{-}$ iteratively. After each iteration, we add the nodes in $\mathcal{V}^{-}$ whose features are completed to $\mathcal{V}^{+}$ and remove these nodes from $\mathcal{V}^{-}$. And we set a hyper-parameter $K$ as the max-iteration of feature completion. By setting different $K$, the set of nodes with complete features can cover all nodes on the Graph, and we have $V^{noal}\subset\mathcal{V}^{+}\subset V$. Fig. 4-(a) present the illustration of our DAFC module. At the first iteration, features of vocal nodes are used to complete features of incomplete silent nodes that directly connect to the vocal nodes. Given incomplete silent node $v_{i}\in\{V^{silent}\cap\mathcal{V}^{-}\}$, its complemented unobservable feature $\widetilde{x_{i}^{u}}$ can be calculated: \[\widetilde{x_{i}^{u}}=\sum_{v_{j}\in\{\mathcal{N}(v_{2})\cap V^{noal}\}} \widetilde{x_{j}^{u}}\cdot f(x_{i}^{\alpha}W^{s},x_{j}^{\alpha}W^{\alpha}) \tag{2}\] where $\widetilde{x_{j}^{u}}$ is the calibrated variable for $x_{j}^{u}$ by eliminating the effects of domain differences: \[\begin{cases}\widetilde{x_{j}^{u}}=x_{j}^{u}-\Delta\widetilde{X}^{u}\cdot \sigma([x_{j}^{u}\|\|\Delta\widetilde{X}^{u}]\cdot W^{\beta})\\ \Delta\widetilde{X}^{u}=[\widetilde{X}_{o}^{\alpha}-\widetilde{X}_{o}^{\alpha }]\cdot W^{\alpha-u}\end{cases} \tag{3}\] where $W^{\alpha-u}\in\mathbb{R}^{D^{\alpha}\times D^{u}}$ and $W^{\beta}\in\mathbb{R}^{2D^{\alpha}\times D^{\mu}}$ are learnable parametric matrix, and $\sigma$ is a $Tanh$ activation function; $\Delta\widetilde{X}^{u}$ is the transformed domain difference; $\widetilde{X}_{o}^{\alpha}=\mathbb{E}_{v_{2}-V^{noal}}(x_{i}^{\alpha})$ and $\widetilde{X}_{o}^{s}=\mathbb{E}_{v_{2}-V^{silent}}(x_{i}^{\alpha})$ are respectively the expectation of the observable features for vocal nodes and silent nodes, and $\widetilde{X}_{o}^{\alpha}-\widetilde{X}_{o}^{s}$ represents the domain difference between vocal and silent nodes according to their observable attributes. In this part, we aim at learning the unobservable domain difference based on the observable domain difference $\widetilde{X}_{o}^{\alpha}-\widetilde{X}_{o}^{s}$. Then for other incomplete silent nodes that do not directly connect to any vocal nodes, they will be complemented from the second iteration until the algorithm converges (reaching the max-iteration $K$ or all silent nodes have been complemented). After iteration 1, DAFC uses complete silent nodes to further complement features for the remaining incomplete silent nodes at each iteration. During this process, the set of complete nodes $\mathcal{V}^{+}$ expand layer-by-layer like breadth-first search (BFS), as shown in Eq.4: \[\widetilde{x_{i}^{u}}=\sum_{v_{j}\in\{\mathcal{N}(v_{i})\cap\mathcal{V}^{+}\}} x_{j}^{u}\cdot f(x_{j}^{\alpha}W^{\alpha},x_{i}^{\alpha}W^{s}) \tag{4}\] where $f(\cdot)$ is the neighbor importance factor: \[f(x_{j}^{\alpha}W^{\alpha},x_{i}^{\alpha}W^{s})=\sigma\left([x_{j}^{\alpha}W^ {\alpha}\|\|x_{i}^{\alpha}W^{s}]\cdot a^{\alpha s}\right) \tag{5}\] where $\sigma$ is a $LeakyReLU$ activation function. Note that the node set $\{\mathcal{N}(v_{i})\cap\mathcal{V}^{+}\}\in V^{silent}$, because all silent nodes that have vocal neighbors have been complemented and merged into $\mathcal{V}^{+}$ after iteration 1 (see Eq. 2), therefore the unobservable features $x_{j}^{u}$ in Eq. 4 do not need to be calibrated by the domain difference factor. Finally, we get the complemented features for silent nodes $\widetilde{x_{i}}=[x_{i}^{\alpha}\|\|\widetilde{x_{i}^{u}}]$. To guarantee that the learned unobservable domain difference (i.e., $\Delta\widetilde{X}^{u}$ in Eq. 3) is actually applied in the process of the domain-adapted feature completion, we add a Distribution-Consistency Loss$\mathcal{L}^{dist}$ when optimizing the DAFC module: \[\mathcal{L}^{dist}=\left\|\Delta\widetilde{X}^{u}-\left(\mathbb{E}_{v_{i}-V^{noal} }(x_{i}^{u})-\mathbb{E}_{v_{i}-V^{sitent}}(\widetilde{x_{i}^{u}})\right) \right\|^{2} \tag{6}\] ### Domain Adapted Message Passing (DAMP) According to the directions of edges based on the population types from source nodes to target nodes, we divide the message passing on a VS-Graph into four parts: (1) Messages from vocal nodes to silent nodes; (2) Messages from silent nodes to vocal nodes; (3) Messages from vocal nodes to vocal nodes; (4) Messages from silent nodes to silent nodes. Among the four directions of message passing, directions (1) and (2) are cross-domain (out-of-distribution) message passing, while directions (3) and (4) are within-domain (in-distribution) message passing. However, out-of-distribution messages from cross-domain neighbors should not be passed to central nodes directly, otherwise, the domain difference will become noises that degrade the model performance. To solve the aforementioned problem, we design a Domain Adapted Message Passing (DAMP) module. For messages from cross-domain neighbors, we first calculate the domain-difference scaling factor and then project the OOD features of source nodes into the domain of target nodes. e.g., for edges from vocal nodes to silent nodes, we project the features of vocal nodes to the domain of silent nodes and then pass the projected features to the target silent nodes. Specifically, the DAMP module considers two factors: the bi-directional domain difference and the neighbor importance. Given an edge $e_{i,j}=(v_{i},v_{j})$, the message function is given as follows: \[\mathcal{M}_{v_{i}\to v_{j}}=\widetilde{h_{i}}\cdot f^{\Psi(v_{j})}( \widetilde{h_{i}},h_{j}) \tag{7}\] where $\widetilde{h_{i}}$ is the source node feature calibrated by the domain difference factor and $f^{\Psi(v_{j})}(\cdot)$ is the neighbor importance function: \[\widetilde{h_{i}}=\mathcal{P}_{\Psi(v_{i})\rightarrow\Psi(v_{j})}(h_{i})=h_{ i}+\Delta_{i,j} \tag{8}\] where $\sigma$ is $LeakyReLU$ and $\Delta_{i,j}$ is the distribution-shift variable: \[\Delta_{i,j}=\left\{\begin{array}{cc}0&if\ \Psi(v_{i})==\Psi(v_{j})\\ (-1)^{\Psi(v_{j})}\cdot\sigma(\{h_{i}[h_{i}][\tilde{X}^{\sigma}-\tilde{X}^{ \sigma}]\cdot\mathbf{a}^{\Psi(v_{i})}\}\cdot(\tilde{X}^{\sigma}-\tilde{X}^{ \sigma})&else\end{array}\right. \tag{9}\] where $\sigma$ is $Tanh$; $\Psi(v_{i})$ is the population indicator (see Eq. 1); $\tilde{X}^{\sigma}=\mathbb{E}_{v_{i}\sim V^{local}}(x_{i})$ and $\tilde{X}^{s}=\mathbb{E}_{v_{i}\sim V^{silent}}(\widetilde{x_{i}})$ are respectively the expectation of the observable features for vocal nodes and silent nodes, and $\tilde{X}^{\sigma}-\tilde{X}^{s}$ represents the domain difference between vocal and silent nodes according to their complete attributes (the attributes of silent nodes have been complemented with DAFC). With the DAMP module, we calibrate the source nodes to the domain of target nodes while message passing, which eliminates the noises caused by the OOD features. It should be noticed that the distribution-shift variable $\Delta_{i,j}$ only works for cross-domain message passing ($V^{local}\rightarrow$$V^{silent}$ or $V^{silent}\rightarrow$$V^{local}$), and $\Delta_{i,j}=1$ for within-domain message passing ($V^{local}\rightarrow$$V^{local}$ or $V^{silent}\rightarrow$$V^{silent}$). With DAMP, we finally obtain the representations of silent nodes and vocal nodes while preserving their domain difference. Different from mainstream domain adaption methods that project the samples from source and target domains into the common space and only preserve the domain-invariance features, our DAMP method conducts message passing while preserving domain difference. ### Domain Transferable Classifier (DTC) With DAFC and DAMP, we solve the feature incompleteness problem and obtain the representations of silent nodes and vocal nodes that preserve the domain difference. Considering that the node representations come from two distinct distributions and the data-hunry problem of silent nodes (see Sec. 3 and Sec. 2.2), we cannot directly train a good classifier for silent nodes. To solve the cross-domain problem and label scarcity problem of silent nodes, we design a novel Domain Transferable Classifier (DTC) for the silent node classification by transferring knowledge from vocal nodes. Traditional domain adaption methods usually transfer cross-domain knowledge by constraining the learned representations to retain domain-invariance information only, under the assumption that the distribution of two domains is similar, which may not be satisfied. Rather than constraining representations, DTC targets at transferring model parameters so that the knowledge of the optimized source classifier can be transferred to the target domain. The solid rectangular box in Fig. 4-(c) shows our motivation to design DTC. Specifically, the classifier trained on source domain Figure 4. Architecture overview of KTGNN, which includes three main components: Domain Adapted Feature Complementary (DAFC), Domain Adapted Message Passing (DAMP), and Domain Transferable Classifier (DTC). only performs under-fitting in target domain due to domain shift (blue dotted line). Meanwhile, the classifier trained on target domain tends to overfit due to label scarcity (orange dotted line). An ideal classifier is between these two classifiers (green line). Based on this motivation (verified via experimental results in Fig.5), we transfer knowledge from both source classifier and target classifier to introduce an ideal classifier by minimizing the KL divergence between them. Specifically, as shown in Fig. 4-(c), DTC has three components, including the source classifier $CLF^{s}$, the target classifier $CLF^{t}$ and the cross-domain transformation module $Trans^{s\to t}$. To predict the silent majority, the source classifier is trained with the vocal nodes and the target domain classifier is trained with the silent nodes. the cross-domain transformation module is used to transfer the original source domain classifier to a newly generated target domain classifier ($\overline{CLF^{t}}=Trans^{s\to t}$ ($CLF^{s}(\cdot)$)), which takes the parameters of source domain classifier as input and then generate the parameters of a new target domain classifier. Specifically, both $CLF^{s}=\sigma(W^{s}H)$ and $CLF^{t}=\sigma(W^{t}H)$ are implemented by one fully-connected layer with _Sigmoid_ activation. And $Trans^{s\to t}(W^{s})=MLP(W^{s})$ is implemented by a multi-layer perception with nonlinear transformation to give it the ability to change the shape of the discriminant boundary of the classifier. The loss function to optimize DTC and the whole model can be divided into two parts: the KL loss $\mathcal{L}^{kl}$ and the classification loss $\mathcal{L}^{clf}$. The KL loss function $\mathcal{L}^{kl}$ is proposed to realize the knowledge transfer between the source domain and the target domain by constraining the discriminative bound of the generated classifier $\overline{CLF^{t}}$ to locate between that of $CLF^{s}$ and $CLF^{t}$: \[\mathcal{L}^{kl}=KL(\mathcal{P}^{s},\widetilde{\mathcal{P}^{s}})+KL( \mathcal{P}^{t},\widetilde{\mathcal{P}^{t}}) \tag{11}\] where $\mathcal{P}^{s}\in\mathbb{R}^{\|V^{wood}\|\times\|\mathcal{C}\|}$ is the output probability of $CLF^{s}$, $\mathcal{P}^{t}\in\mathbb{R}^{\|V^{silent}\|\times\|\mathcal{C}\|}$ is the output probability of $CLF^{t}$, $\widetilde{\mathcal{P}^{s}}\in\mathbb{R}^{\|V^{wood}\|\times\|\mathcal{C}\|}$ and $\widetilde{\mathcal{P}^{t}}\in\mathbb{R}^{\|V^{silent}\|\times\|\mathcal{C}\|}$ are the output probability of $\overline{CLF^{t}}$, $\|\mathcal{C}\|$ is the number of classes and $\|\mathcal{C}\|=2$ for binary classification tasks. The classification loss $\mathcal{L}^{clf}$ is defined as: \[\begin{split}\mathcal{L}^{clf}=BCE(Y^{wood},\widetilde{Y}^{ wood}_{CLF^{s}})&+BCE(Y^{silent},\widetilde{Y}^{silent}_{CLF^{t}})\\ &+BCE(Y^{silent},\widetilde{Y}^{silent}_{CLF^{t}})\end{split} \tag{12}\] where $BCE$ is the binary cross-entropy loss function: \[BCE(Y,\widetilde{Y},N)=\frac{1}{N}\sum_{i=1}^{N}Y_{i}\cdot\log\widetilde{Y}_{ i}+(1-Y_{i})\cdot\log(1-\widetilde{Y}_{i}) \tag{13}\] Combined with the Distribution-Consistency Loss $\mathcal{L}^{dict}$, our final loss function $\mathcal{L}$ is: \[\mathcal{L}=\mathcal{L}^{clf}+\lambda\cdot\mathcal{L}^{kl}+\gamma\cdot \mathcal{L}dist \tag{14}\] where $\lambda$ is a hyper-parameter to control the weight of $\mathcal{L}^{kl}$, and we use $\gamma=1$ in all our experiments. ## 5. Experiments In this section, we compare KTGNN with other state-of-the-art methods on two real-world datasets in different scenarios. ### Datasets Basic information of the real-world VS-Graphs are shown in Table 1. More details about the datasets are shown in Appendix A. Company: Company dataset is a VS-Graph based on 10641 real-world companies in China (provided by TianYanCha), and the target is to classify each company into binary classes (risky/normal). On this VS-Graph, vocal nodes denote listed companies and silent nodes denote unlisted companies. The business information and equity graphs of all companies are available, while only the financial statements of listed companies can be obtained (missing for unlisted companies). Edges of this dataset indicate investment relations between companies. Twitter: Following the dataset proposed by Xiao et al. (Xiao et al., 2017), we construct the Twitter VS-Graph based on data crawled from Twitter, and the target is to predict the political tendency (binary classes: democrat or republican) of civilians. On this VS-Graph, vocal nodes denote famous politicians and silent nodes denote ordinal civilians. The tweet information is available for all Twitter users, while the personal descriptions from the homepage are available only for politicians (missing for civilians). Edges of this dataset represent follow relations between Twitter users. ### Baselines and Experimental Settings We implement nine representative baseline models, including _MLP_ as well as $Spectral-based$, $Spatial-based$ and _Deeper_ GNN models, which are GCN (Zhou et al., 2017), GAT (Zhou et al., 2018), GraphSAGE (Zhou et al., 2018), JKNet (Zhou et al., 2018), APPNP (Zhou et al., 2018), GATv2 (Chen et al., 2018), DAGNN (Zhou et al., 2018), GCNII (Chen et al., 2018). Besides, we choose a recent state-of-the-art GNN model (OODGAT (Zhou et al., 2018)) designed for the OOD node classification task. We implement all baseline models and our KTGNN models in $Pytorch$ and $Torch-Geometric$. For each dataset (Company and Twitter), we randomly divide the annotated silent nodes into train/valid/test sets with a fixed ratio of 60%/20%/20%. And we add all annotated vocal nodes into the training set because our target is to classify the silent nodes. The detailed hyper-parameter settings of KTGNN and other baselines are presented in Appendix D. ### Main Results We focus on silent node classification on the VS-Graph in this paper and conduct comprehensive experiments on two critical real-world scenarios (i.e., political election and company financial assessment). Results of Silent Node Classification We select two representative metrics (i.e., F1-Score and AUC) to evaluate the model performance on the silent node classification task. Considering that baseline models cannot directly handle graphs with partially-missing features (silent nodes in VS-Graphs), we combine these baseline GNNs with some heuristic feature completion strategies (e.g., completion by zero, completion by mean of neighbors) to keep fair comparison with our methods, and we finally choose the best \begin{table} \begin{tabular}{c c c c c c} \hline \hline Dataset & $\|V^{wood}\|$ & $\|V^{silent}\|$ & $\|E\|$ & $\|Y^{wood}\|$ & $\|Y^{silent}\|$ \\ \hline Company & 3987 & 6654 & 116785 & 3987 & 1923 \\ Twitter & 581 & 20230 & 915,438 & 581 & 625 \\ \hline \hline \end{tabular} \end{table} Table 1. Basic information of the dataset used in this paper. completion strategy for each baseline GNN (results in Table 2). As shown in Table 2, our KTGNN model gains significant improvement in the performance of silent node classification on both Twitter and Company datasets compared with other state-of-the-art GNNs. Effects of Feature Completion Strategy For baseline GNNs, we also analyze the effects of heuristic feature completion methods on the Company dataset at Table 6 (results of Twitter datasets are presented at Table 6 in Appendix). And the results in Table 3 indicate that the "Mean-of-Neighbors" strategy wins in most cases. However, all these heuristic completion strategies ignore the distribution-shift between the vocal nodes and the silent nodes, and thus are far less effective than our KTGNN model. ### Ablation Study Effects of Each Module: To validate the effects of each component of KTGNN, we design some variants of KTGNN by removing one of modules (i.e., DAFC, DAMP, DTC, $\mathcal{L}^{dist}$, $\mathcal{L}^{kl}$) and the results are shown in the Table 4. We observe that the performance of all KTGNN variants deteriorates to some degree. And our full KTGNN module gains the best performance, which demonstrates the important effects of each component in KTGNN. Effects of KL Loss: We also validate the role of KL loss $\mathcal{L}^{kl}$ (see Eq. 11) in our method. As shown in the Fig. 5, the final target classifier generated from the source classifier gains the highest scores and lowest loss among the three sub-classifiers in the DTC module (see Sec. 4.4). And the result of KL Loss (components of $\mathcal{L}^{kl}$) indicates that the discriminant bound of the generated target classifier is located between that of the source classifier and target classifier, which further confirms our motivations. Effects of Cross-Domain Knowledge Transfer: We further validate our approach with borderline cases (i.e., topological traits that may worsen the correct knowledge transfer from vocal nodes to silent nodes). We validate this by cutting part of such edges. The results in Table 5 indicate that our approach is robust over these borderline cases. \begin{table} \begin{tabular}{c c c c} \hline \hline Dataset & \multicolumn{2}{c}{Twitter} & \multicolumn{2}{c}{Company} \\ \cline{2-4} Evaluation Metric & F1 & AUC & F1 & AUC \\ \hline KTGNN${}_{\text{DAC}}$ & 85.57$\pm$1.38 & 92.12$\pm$0.87 & 60.21$\pm$0.39 & 61.95$\pm$0.45 \\ KTGNN${}_{\text{DAMP}}$ & 86.85$\pm$1.01 & 93.03$\pm$0.98 & 62.71$\pm$0.44 & 64.27$\pm$0.61 \\ KTGNN${}_{\text{DTC}}$ & 88.80$\pm$0.93 & 94.13$\pm$1.08 & 63.05$\pm$0.52 & 64.35$\pm$0.55 \\ KTGNN${}_{\mathcal{L}^{dist}}$ & 87.80$\pm$0.83 & 93.85$\pm$1.38 & 63.56$\pm$0.68 & 65.38$\pm$0.65 \\ KTGNN${}_{\mathcal{L}^{kl}}$ & 89.20$\pm$1.20 & 94.06$\pm$1.03 & 62.71$\pm$0.63 & 65.02$\pm$0.55 \\ \hline KTGNN & 89.65$\pm$1.20 & 95.08$\pm$0.93 & 64.96$\pm$0.63 & 67.11$\pm$0.52 \\ \hline \hline \end{tabular} \end{table} Table 4. Ablation studies of KTGNN compared with its variants by removing certain components. \begin{table} \begin{tabular}{c c c c c} \hline \hline Dataset & \multicolumn{2}{c}{Twitter} & \multicolumn{2}{c}{Company} \\ \cline{2-5} Evaluation Metric & F1 & AUC & F1 & AUC \\ \hline MLP & 70.85$\pm$1.31 & 80.12$\pm$1.38 & 57.01$\pm$0.42 & 56.35$\pm$0.55 \\ GCN & 80.19$\pm$0.87 & 86.88$\pm$1.23 & 57.60$\pm$0.49 & 57.83$\pm$0.93 \\ GAT & 82.31$\pm$1.56 & 87.88$\pm$1.21 & 57.98$\pm$0.43 & 58.40$\pm$0.43 \\ GraphSAGE & 85.40$\pm$1.03 & 92.08$\pm$1.61 & 58.93$\pm$0.67 & 60.63$\pm$0.57 \\ JKNet & 83.86$\pm$1.93 & 91.17$\pm$1.21 & 58.38$\pm$0.58 & 61.06$\pm$0.53 \\ APPNP & 80.60$\pm$1.60 & 87.01$\pm$1.08 & 57.53$\pm$0.63 & 57.99$\pm$0.80 \\ GATv2 & 82.63$\pm$1.09 & 89.83$\pm$1.27 & 57.78$\pm$0.52 & 58.93$\pm$0.93 \\ DAGNN & 84.37$\pm$0.93 & 91.98$\pm$1.05 & 59.62$\pm$0.43 & 59.07$\pm$0.39 \\ GCNII & 83.85$\pm$1.17 & 90.67$\pm$1.32 & 58.21$\pm$0.88 & 60.06$\pm$0.63 \\ OODGAT & 85.95$\pm$2.01 & 92.67$\pm$1.60 & 60.05$\pm$0.89 & 61.37$\pm$0.92 \\ \hline KTGNN & 89.65$\pm$1.20 & 95.08$\pm$0.93 & 64.96$\pm$0.63 & 67.11$\pm$0.52 \\ \hline \hline \end{tabular} \end{table} Table 2. Results of silent node classification. All base models except KTGNN are combined with one optimal heuristic feature completion strategy. \begin{table} \begin{tabular}{c c c c} \hline \hline Dataset & Completion Method & \multicolumn{2}{c}{Company} \\ \cline{2-4} Evaluation Metric & $\backslash$ & F1 & AUC \\ \hline \multirow{4}{}{MLP} & None & 56.06$\pm$0.63 & 55.36$\pm$0.33 \\ & $\theta$-Completion & 56.60$\pm$0.67 & 55.07$\pm$0.58 \\ & Mean-of-Neighbors & 57.01$\pm$0.42 & 56.35$\pm$0.55 \\ \hline \multirow{4}{}{GCN} & None & 56.61$\pm$0.53 & 56.09$\pm$0.44 \\ & $\theta$-Completion & 56.41$\pm$0.59 & 56.75$\pm$0.58 \\ & Mean-of-Neighbors & 57.60$\pm$0.49 & 57.83$\pm$0.93 \\ \hline \multirow{4}{}{GraphSAGE} & None & 57.12$\pm$0.57 & 57.29$\pm$0.57 \\ & $\theta$-Completion & 58.63$\pm$0.71 & 59.78$\pm$0.65 \\ & Mean-of-Neighbors & 58.93$\pm$0.67 & 60.63$\pm$0.57 \\ \hline \multirow{4}{}{JKNet} & None & 57.26$\pm$0.49 & 58.44$\pm$0.67 \\ & $\theta$-Completion & 57.81$\pm$0.65 & 60.95$\pm$0.63 \\ & Mean-of-Neighbors & 58.38$\pm$0.58 & 61.06$\pm$0.53 \\ \hline \multirow{4}{}{GCNII} & None & 58.36$\pm$0.37 & 58.85$\pm$0.33 \\ & $\theta$-Completion & 58.07$\pm$0.47 & 59.91$\pm$0.56 \\ & Mean-of-Neighbors & 58.21$\pm$0.88 & 60.06$\pm$0.63 \\ \hline \multirow{4}{}{OODGAT} & None & 56.73$\pm$1.33 & 57.83$\pm$1.20 \\ & $\theta$-Completion & 59.65$\pm$0.83 & 60.93$\pm$0.98 \\ \cline{1-1} & Mean-of-Neighbors & 60.05$\pm$0.89 & 61.37$\pm$0.92 \\ \hline \hline KTGNN & $\backslash$ & 64.96$\pm$0.63 & 67.11$\pm$0.52 \\ \hline \hline \end{tabular} \end{table} Table 3. Results of models with different completion strategies on Company dataset. “None” means we only use the observable attributes (i.e., $X^{o}$) for both vocal and silent nodes without completion; “0-Completion” and “Mean-of-Neighbors” use the zero vector and mean vector of vocal neighbors to complete the missing dimensions for silent nodes. Figure 5. F1 Score and Loss Curve of KTGNN on Company dataset (results of Twitter dataset are shown in appendix). ### HyperParameter Sensitivity Analysis We validate the sensitivity of main hyper-parameters of KTGNN: $K$ (max-iteration of DAFC ) and $\lambda$ (weight of $\mathcal{L}^{kl}$). As shown in Fig. 6, we conduct experiments on the Company dataset with different $K$ and $\lambda$. The results show that KTGNN gains the best performance when $K=2$, which is decided by the graph property that $2$-hop neighbors of vocal nodes cover most silent nodes, and gains the best performance when $lambda=1.0$, which also indicates the effectiveness of the KL loss. ### Representation Visualization We visualize the learned representations of KTGNN in Fig. 7. Subfigures (a) and (d) show that our KTGNN remains the domain difference between vocal nodes and silent nodes during message-passing. And Sub-figures (b) (c) and (d) (f) show that the representations learned by KTGNN have better distinguishability between different classes for both vocal nodes and silent nodes, compared with the T-SNE visualization of raw features in Fig. 3 (The representation visualization of baseline GNNs are shown in the appendix). ## 6. Related Work Graph Neural Networks (GNNs), as powerful tools for modeling graph data, have been widely proposed and can be generally categorized into _spectral-based methods_(Krizhevsky et al., 2014; He et al., 2015; He et al., 2016; He et al., 2017; He et al., 2018) and _spatial-based methods_(Krizhevsky et al., 2014; He et al., 2015; He et al., 2016; He et al., 2017; He et al., 2018; He et al., 2019; He et al., 2018; He et al., 2019; He et al., 2018; He et al., 2019). Existing mainstream spatial-based GNNs (He et al., 2016; He et al., 2017; He et al., 2018; He et al., 2019; He et al., 2018; He et al., 2019; He et al., 2018; He et al., 2019) follow the message-passing framework, where node representations are updated based on features aggregated from neighbors. Recently, deeper GNNs (He et al., 2016; He et al., 2018; He et al., 2019) with larger receptive fields, have been proposed and gained better performance. However, most of the existing GNNs are built on the I.I.D. hypothesis, i.e., training and testing data are independent and identically distributed. And the I.I.D. hypothesis is hard to be satisfied in many real-world scenarios where the model performance degrades significantly when there exist distribution-shifts among the nodes on the graph. O.O.D nodes commonly exist in real-world graphs (He et al., 2016; He et al., 2017; He et al., 2018; He et al., 2019; He et al., 2019), which bring obstacles to current message-passing based GNN models. Recently, GNNs for graph-level O.O.D prediction (He et al., 2018; He et al., 2019; He et al., 2019; He et al., 2019; He et al., 2019; He et al., 2019) have been proposed. However, GNNs for semi-supervised node classification with O.O.D nodes have been rarely studied. Some existing studies (He et al., 2016; He et al., 2019; He et al., 2019; He et al., 2019; He et al., 2019) combine GNN with the O.O.D detection task as an auxiliary to enhance the performance of GNNs on graphs with O.O.D nodes. Such kind of method usually views O.O.D nodes as noises, which aim at detecting potential O.O.D nodes first and then alleviating their effects. However, O.O.D nodes are not noises in all scenarios (e.g., VS-Graph introduced in this paper) and knowledge from other domains can be helpful to alleviate the data-hunpy problem of the target domain. Different from existing O.O.D detection methods, we propose to improve the model performance on the target domain by injecting out-of-distribution knowledge in this paper. ## 7. Conclusion In this paper, we propose a novel and widespread problem: silent node classification on the VS-Graph ("predicting the silent majority on graphs"), where the silent nodes suffer from serious data-hungry problems (feature-missing and label scarcity). Correspondingly, we design a novel KTGNN model for silent node classification by transferring useful knowledge from vocal nodes to silent nodes adaptively. Comprehensive experiments on the two representative real-world VS-Graphs demonstrate the superiority of our approach. ###### Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant No.U21B2046, No.62202448, No.61902380, No.61802370, and No.U1911401), the Beijing Nova Program (No. Z201100006820061) and China Postdoctoral Science Foundation (2022M713208). \begin{table} \begin{tabular}{c c c c c} \hline \hline \multirow{2}{}{Removing \#\% of} & \multicolumn{2}{c}{Twitter} & \multicolumn{2}{c}{Company} \\ \cline{2-5} & F1 & AUC & F1 & AUC \\ \hline 70\% & 86.40$\pm$1.18 & 92.33$\pm$0.96 & 61.73$\pm$0.70 & 63.34$\pm$0.65 \\ 50\% & 88.00$\pm$1.03 & 93.53$\pm$1.03 & 62.91$\pm$0.59 & 64.89$\pm$0.58 \\ 30\% & 88.80$\pm$1.29 & 94.56$\pm$0.97 & 63.70$\pm$0.73 & 66.05$\pm$0.61 \\ \hline 0\% & 89.65$\pm$1.20* & 95.08$\pm$0.93 & 64.96$\pm$0.63 & 67.11$\pm$0.52 \\ \hline \hline \end{tabular} \end{table} Table 5. Boarder-line cases analysis on the effect of cross-domain edges by removing part (i.e., 0%, 30%, 50%, and 70%) of such edges randomly from the original VS-Graph. Figure 6. Hyper-parameter analysis of $K$ and $\lambda$ on Company dataset (see results on Twitter dataset in Fig. 9 of Appendix). Figure 7. T-SNE visualization of the node completed representations learned by KTGNN, distinguished by their populations (vocal$\rightarrow$orange, silent$\rightarrow$cyan) or node labels (binary classes: positive$\rightarrow$red, negative$\rightarrow$blue).
2307.10266	A DPLL(T) Framework for Verifying Deep Neural Networks	Deep Neural Networks (DNNs) have emerged as an effective approach to tackling real-world problems. However, like human-written software, DNNs can have bugs and can be attacked. To address this, research has explored a wide-range of algorithmic approaches to verify DNN behavior. In this work, we introduce NeuralSAT, a new verification approach that adapts the widely-used DPLL(T) algorithm used in modern SMT solvers. A key feature of SMT solvers is the use of conflict clause learning and search restart to scale verification. Unlike prior DNN verification approaches, NeuralSAT combines an abstraction-based deductive theory solver with clause learning and an evaluation clearly demonstrates the benefits of the approach on a set of challenging verification benchmarks.	Hai Duong, ThanhVu Nguyen, Matthew Dwyer	2023-07-17T18:49:46Z	http://arxiv.org/abs/2307.10266v3	# A DPLL(T) Framework for Verifying Deep Neural Networks ###### Abstract. Deep Neural Networks (DNNs) have emerged as an effective approach to tackling real-world problems. However, like human-written software, automatically-generated DNNs can have bugs and be attacked. This thus attracts many recent interests in developing effective and scalable DNN verification techniques and tools. In this work, we introduce a NeuralSAT, a new constraint solving approach to DNN verification. The design of NeuralSAT follows the DPLL(T) algorithm used modern SMT solving, which includes (conflict) clause learning, abstraction, and theory solving, and thus NeuralSAT can be considered as an SMT framework for DNNs. Preliminary results show that the NeuralSAT prototype is competitive to the state-of-the-art. We hope, with proper optimization and engineering, NeuralSAT will carry the power and success of modern SAT/SMT solvers to DNN verification. NeuralSAT is available from: [https://github.com/dynaroars/neuralsat](https://github.com/dynaroars/neuralsat) deep neural network verification, clause learning, abstraction, constraint solving, SAT/SMT solving ## 1. Introduction Deep Neural Networks (DNNs) have emerged as an effective approach for solving challenging real-world problems. Among many others, they have been used for image recognition (He et al., 2016; Krizhevsky et al., 2012), autonomous driving (Rao and Frtunikj, 2018; Salay et al., 2020), airplane collision control (Julian et al., 2016), power grid control (Siano et al., 2012), fake news detection (Thota et al., 2018), drug synthesis and discovery (Grebner et al., 2021), and COVID-19 detection and diagnosis (Khan et al., 2020; Ozturk et al., 2020). However, just like traditional software, DNNs can have "bugs", e.g., producing unexpected results on inputs that are different from those in training data, and be attacked, e.g., small perturbations to the inputs by a malicious adversary or even sensorial imperfections result in misclassification (Isac et al., 2022; Ren et al., 2020; Yang et al., 2022; Zhang et al., 2019; Zugner et al., 2018). These issues, which have been observed in many DNNs (Goodfellow et al., 2014; Szegedy et al., 2014) and demonstrated in the real world (Eykholt et al., 2018), naturally raise the question of how DNNs should be tested, validated, and ultimately _verified_ to meet the requirements of relevant robustness or safety standards (Huang et al., 2020; Katz et al., 2017). To address this question, researchers have developed a variety of techniques and tools to verify DNNs (e.g., (Huang et al., 2017; Katz et al., 2022, 2019; Liu et al., 2021; Muller et al., 2021; Urban and Mine, 2021; Wang et al., 2021)). Constraint-based approaches (Ehlers, 2017; Huang et al., 2017; Katz et al., 2017) aim to both correctly prove and disprove properties, but do not scale to large networks. In contrast, abstraction-based approaches (Muller et al., 2021; Singh et al., 2018, 2019; Wang et al., 2018, 2021) scale much better, but while modern abstraction verification tools can often refine their abstractions to avoid returning spurious counterexamples they are incomplete. The problem of verifying non-trivial properties of DNNs with piecewise linear activation functions, such as "ReLU", has been shown to be reducible (Katz et al., 2017) from the classical satisfiability (SAT) problem (Cook, 1971). Despite this complexity, the ability of satisfiability modulo theories (SMT) solvers to scale to large formulae encoding real-world verification problems (Kroening and Strichman, 2016) suggests that a similar approaches might be effective for DNN verification. However, the constraint-based DNN verifiers developed to date (Ehlers, 2017; Katz et al., 2017, 2022, 2019) are not among the state-of-the-art as determined by DNN verification competitions (Bak et al., 2021; Muller et al., 2022). Techniques like Planet(Ehlers, 2017) and Reluplex(Katz et al., 2017, 2022) demonstrated how the semantics of a trained DNN could be encoded as a constraint in Linear Real Arithmetic (LRA) and combined with a specification also expressed in LRA for verification. In principle, such constraints can be solved by any SMT solver equipped with an LRA _theory solver_ (T-solver). The DPLL(T) algorithm implemented by modern SMT solvers works by moving back and forth between solving an abstract propositional encoding of the constraint and solving a theory-specific encoding of a constraint fragment corresponding to a partial assignment of propositional literals. The challenge in solving DNN verification constraints lies in the fact that each neuron gives rise to a disjunctive constraint to encode its non-linear behavior. In practice, this leads to a combinatorial blowup in the space of assignments the SMT solver must consider at the abstract propositional level. To resolve the exponential complexity inherent in such constraints, both Planet and Reuplex chose to _push_ the disjunctive constraints from the propositional encoding down into the theory-specific encoding of the problem, leveraging a technique referred to as splitting-on-demand (Barrett et al., 2006). This works to an extent, but it does not scale well to large DNNs (Bak et al., 2021; Muller et al., 2022). We observe that the choice to pursue an aggressive splitting-on-demand strategy sacrifices the benefit of several of the key algorithmic techniques that make SMT solvers scale - specifically _conflict-driven clause learning_(CDCL) (Bayardo Jr and Schrag, 1997; Marques Silva and Sakallah, 1996; Marques-Silva and Sakallah, 1999) and theory propagation (Kroening and Strichman, 2016). In this paper, we revisit the design of a DPLL(T) solver for DNN verification by developing a T-solver that incorporates abstraction to (1) accelerate unsatisfiability checking of partial assignments and to (2) deduce implied literals to realize a form of theory propagation that drives clause learning to accelerate solving at the abstract propositional level. We present the NeuralSAT framework, which consists of a lazy, incremental LRA-solver that can be parameterized by state-of-the-art abstractions, such as LiRPA (Wang et al., 2021; Xu et al., 2020), to efficiently perform exhaustive theory propagation. In SS5, we formalize NeuralSAT and prove that it is sound and complete, and that it terminates. We then describe a NeuralSAT prototype that supports neural networks with different layer types, e.g., fully-connected, convolutional, and residual, and piecewise-linear activation functions, e.g., ReLU. In SS7 we present results demonstrating that NeuralSAT establishes the state-of-the-art in constraint-based DNN verification improving substantially on Reluplex's successor Marabou (Katz et al., 2022, 2019), which also employs abstraction and deduction, but does not exploit CDCL. Moreover, despite the fact that NeuralSAT is an early stage prototype, that is not highly optimized, it ranks second to $\alpha-\beta$-CROWN in solving benchmarks from the VNN-COMP competition. The contributions of this work lie in: (i) developing a domain-specific LRA-solver that allows for the benefits of CDCL to accelerate SMT-based DNN verification,; (ii) proving the soundness, completeness, and termination of the approach; (iii) developing a prototype NeuralSAT implementation which we release as open source; and (iv) empirically demonstrating that the approach compares favorably with the state-of-the-art in terms of scalability, performance, and ability to solve DNN verification problems. Constraint-based DNN verifiers, like Planet and Reuplex, pioneered the application of formal verification to reason about neural network behavior. Subsequent work abandoned completeness in order to scale, but NeuralSAT demonstrates that by leveraging CDCL constraint-based verification can scale and, we believe, it offers a platform for further advances in DNN verification. ## 2. Background ### Satisfiability and DPLL(T) The classical satisfiability (SAT) problem asks if a given propositional formula over Boolean variables can be satisfied (Biere et al., 2009). Given a formula $f$, a SAT solver returns sat if it can find a satisfying assignment that maps truth values to variables of $f$ that makes $f$ evaluate to true, and unsat if it cannot find any satisfying assignments. The problem is NP-Complete and research into methods for efficiently solving problem instances has been ongoing for multiple decades. DPLL.Fig. 1 gives an overview of DPLL, a SAT solving technique introduced in 1961 by Davis, Putnam, Logemann, and Loveland [Davis et al. 1962]. DPLL is an iterative algorithm that takes as input a propositional formula in CNF form and (i) decides an unassigned variable and assigns it a truth value, (ii) performs Boolean constraint propagation (BCP) (also called Unit Propagation), which detects single literal clauses that either force a literal to be true in a satisfying assignment or give rise to a conflict; (iii) analyzes the conflict to backtrack to a previous decision level d1; and (iv) erases assignments at levels larger than d1 to try new assignments. These steps repeat until DPLL finds a satisfying assignment and returns sat, or decides that it cannot backtrack (d1=-1) and returns unsat. Modern DPLL solving improves the original version with conflict-driven clause learning (CDCL) [Bayardo Jr and Schrag 1997; Marques Silva and Sakallah 1996; Marques-Silva and Sakallah 1999]. DPLL with CDCL can _learn new clauses_ to avoid past conflicts and backtrack more intelligently (e.g., using non-chronologically backjumping). Due to its ability to learn new clauses, CDCL can significantly reduce the search space and allow SAT solvers to scale to large problems. In the following, whenever we refer to DPLL, we mean DPLL with CDCL. DPLL(T).DPLL(T) [Nieuwenhuis et al. 2006] extends DPLL for propositional formulae to check SMT formulae involving non-Boolean variables, e.g., real numbers and data structures such as strings, arrays, lists. DPLL(T) combines DPLL with dedicated _theory solvers_ to analyze formulae in those theories1. For example, to check a formula involving linear arithmetic over the reals (LRA), DPLL(T) may use a theory solver that uses linear programming to check the constraints in the formula. Modern DPLL(T)-based SMT solvers such as Z3 [Moura and Bjorner 2008] and CVC4 [Barrett et al. 2011] include solvers supporting a wide range of theories including linear arithmetic, nonlinear arithmetic, string, and arrays [Kroening and Strichman 2016]. Footnote 1: SMT is Satisfiability Modulo Theories and the T in DPLL(T) stands for Theories. ### The DNN verification problem A _neural network_ (NN) [Goodfellow et al. 2016] consists of an input layer, multiple hidden layers, and an output layer. Each layer has a number of neurons, each connected to neurons from previous layers through a predefined set of weights (derived by training the network with data). A DNN is an NN with at least two hidden layers. The output of a DNN is obtained by iteratively computing the values of neurons in each layer. The value of a neuron in the input layer is the input data. The value of a neuron in the hidden layers is computed by applying an _affine transformation_ to values of neurons in the previous layers, then followed by an _activation function_ such as the popular Rectified Linear Unit (ReLU) activation. For this activation, the value of a hidden neuron $y$ is $ReLU(w_{1}v_{1}+\ldots+w_{n}v_{n}+b)$, where $b$ is the bias parameter of $y$, $w_{i},\ldots,w_{n}$ are the weights of $y$, $v_{1},\ldots,v_{n}$ are the neuron values of preceding layer, $w_{1}v_{1}+\cdots+w_{n}v_{n}+b$ is the affine transformation, and $ReLU(x)=\max(x,0)$ is the ReLU activation. The values of a neuron in the output layer is evaluated similarly but it may skip the activation function. A ReLU activated neuron is said to be _active_ if its input value is greater than zero and _inactive_ otherwise. Figure 1: Original DPLL Algorithm. _DNN Verification._ Given a DNN $N$ and a property $\phi$, the _DNN verification problem_ asks if $\phi$ is a valid property of $N$. Typically, $\phi$ is a formula of the form $\phi_{in}\Rightarrow\phi_{out}$, where $\phi_{in}$ is a property over the inputs of $N$ and $\phi_{out}$ is a property over the outputs of $N$. A DNN verifier attempts to find a _counterexample_ input to $N$ that satisfies $\phi_{in}$ but violates $\phi_{out}$. If no such counterexample exists, $\phi$ is a valid property of $N$. Otherwise, $\phi$ is not valid and the counterexample can be used to retrain or debug the DNN (Huang et al., 2017). _Example._ Fig. 2 shows a simple DNN with two inputs $x_{1},x_{2}$, two hidden neurons $x_{3},x_{4}$, and one output $x_{5}$. The weights of a neuron are shown on its incoming edges, and the bias is shown above or below each neuron. The outputs of the hidden neurons are computed the affine transformation and ReLU, e.g., $x_{3}=ReLU\left(-0.5x_{1}+0.5x_{2}+1.0\right)$. The output neuron is computed with just the affine transformation, i.e., $x_{5}=-x_{3}+x_{4}-1$. A valid property for this DNN is that the output is $x_{5}\leq 0$ for any inputs $x_{1}\in[-1,1],x_{2}\in[-2,2]$. An invalid property for this network is that $x_{5}>0$ for those similar inputs. A counterexample showing this property violation is $\{x_{1}=-1,x_{2}=2\}$, from which the network evaluates to $x_{5}=-3.5$. Such properties can capture _safety requirements_ (e.g., a rule in an collision avoidance system in (Katz et al., 2017; Kochenderfer et al., 2012) is "if the intruder is distant and significantly slower than us, then we stay below a certain threshold") or _local robustness_(Katz et al., 2017) conditions (a form of adversarial robustness stating that small perturbations of a given input all yield the same output). _Abstraction._ ReLU-based DNN verification is NP-Complete (Katz et al., 2017) and thus can be formulated as a SAT or SMT checking problem. Direct application of SMT solvers does not scale to the large and complex formulae encoding real-world, complex DNNs. While custom theory solvers, like Planetand Reluplex, retain the soundness, completeness, and termination of SMT and improve on the performance of a direct SMT encoding, they too do not scale sufficiently to handle realistic DNNs citebak2021second. Applying techniques from abstract interpretation (Cousot and Cousot, 1977), abstraction-based DNN verifiers overapproximate nonlinear computations (e.g., ReLU) of the network using linear abstract domains such as interval (Wang et al., 2018), zonotope (Singh et al., 2018), polytope (Singh et al., 2019; Xu et al., 2020). As illustrated in Fig. 8 abstract domains can model nonlinearity with varying degrees of precision using polyhedra that are efficient to compute with. This allows abstraction-based DNN verifiers to side-step the disjunctive splitting that is the performance bottleneck of constraint-based DNN verifiers. A DNN verification technique using an approximation, e.g., the polytope abstract domain, works by (i) representing the input ranges of the DNN as polytopes, (ii) applying transformation rules to the affine and ReLU computations of the network to compute polytope regions representing values of neurons, and (iii) finally, converting the polytope results into output bounds. The resulting outputs are an overapproximation of the actual outputs. ## 3. Overview of NeuralSAT NeuralSAT is a SMT-based DNN verifier that uses abstraction in its theory solver to accelerate satisfiability checks and the exploration of the space of variable assignments. Fig. 3 gives an overview of NeuralSAT, which follows the DPLL(T) framework (SS2) and consists of standard DPLL components (light shades) and the theory solver (dark shade). Figure 2. An FNN with ReLU. NeuralSAT construct a propositional formula over Boolean variables that represent the activation status of neurons (_Boolean Abstraction_). Clauses in the formula assert that each neuron, e.g., neuron $i$, is active or inactive, e.g., $v_{i}\lor\overline{v_{i}}$. This abstraction allows us to use standard DPLL components to search for truth values satisfying these clauses and a DNN-specific theory solver to check the feasibility of truth assignments with respect to the constraints encoding the DNN and the property of interest. NeuralSAT now enters an iterative process to find assignments satisfying the activation clauses. First, NeuralSAT assigns a truth value to an unassigned variable (_Decide_), detects unit clauses caused by this assignment, and infers additional assignments (_Boolean Constraint Propagation_). Next, NeuralSAT invokes the theory solver or T-solver (_DEDUCTION_), which uses LP solving and abstraction to check the satisfiability of the constraints of the current assignment with the property of interest. The T-solver can also infer additional truth assignments. If the T-solver confirms satisfiability, NeuralSAT continues with new assignments (_Decide_). Otherwise, NeuralSAT detects a conflict (_AnalyzeConflict_) and learns clauses to remember it and backtrack to a previous decision (_Backtrack_). This process repeats until NeuralSAT can no longer backtrack, and return unsat, indicating the DNN has the property, or it finds a total assignment for all boolean variables, and returns sat (and the user can query NeuralSAT for a counterexample). ### Illustration We use NeuralSAT to prove that for inputs $x_{1}\in[-1,1],x_{2}\in[-2,2]$ the DNN in Fig. 2 produces the output $x_{5}\leq 0$. NeuralSAT takes as input the formula $\alpha$ representing the DNN: \[\begin{split} x_{3}=ReLU(-0.5x_{1}+0.5x_{2}+1)\ \land\\ x_{4}=ReLU(x_{1}+x_{2}-1)\ \land\\ x_{5}=-x_{3}+x_{4}-1\end{split} \tag{1}\] and the formula $\phi$ representing the property: \[\phi:-1\leq x_{1}\leq 1\land-2\leq x_{2}\leq 2\quad\Rightarrow\quad x_{5} \leq 0. \tag{2}\] To prove $\alpha\Rightarrow\phi$, NeuralSAT shows that _no_ values of $x_{1},x_{2}$ satisfying the input properties would result in $x_{5}>0$. Thus, we want NeuralSAT to return unsat for $\overline{\alpha\Rightarrow\phi}$: \[\alpha\ \land\ -1\leq x_{1}\leq 1\ \land\ -2\leq x_{2}\leq 2\ \land\ x_{5}>0. \tag{3}\] In the following, we write $x\mapsto v$ to denote that the variable $x$ is assigned with a truth value $v\in\{T,F\}$. This assignment can be either decided by Decide or inferred by BCP. We also write $x@dl$ and $\overline{x}@dl$ to indicate the respective assignments $x\mapsto T$ and $x\mapsto F$ at decision level $dl$. Boolean AbstractionFirst, NeuralSAT creates two Boolean variables $v_{3}$ and $v_{4}$ to represent the activation status of the hidden neurons $x_{3}$ and $x_{4}$, respectively. For example, $v_{3}=T$ means $x_{3}$ is active and thus is the constraint $-0.5x_{1}+0.5x_{2}+1>0$. Similarly, $v_{3}=F$ means $x_{3}$ is inactive and therefore is $-0.5x_{1}+0.5x_{2}+1\leq 0$. Next, NeuralSAT forms two clauses $\{v_{3}\lor\overline{v_{3}}\ ;\ 0_{4}\lor\overline{v_{4}}\}$ indicating these variables are either active or inactive. DPLL(T) IterationsNeuralSAT searches for an assignment to satisfy the clauses and the constraints they represent. In this example, NeuralSAT uses four iterations, summarized in Tab. 1, to determine that no such assignment exists and the problem is thus unsat. Figure 3. NeuralSAT. In _iteration 1_, as shown in Fig. 3, NeuralSAT starts with BCP, which has no effects because the current clauses and (empty) assignment produce no unit clauses. In DEDUCTION, NeuralSAT uses an LP solver to determine that the current set of constraints, which contains just the initial input bounds, is feasible 2. NeuralSAT then uses abstraction to approximate an output upper bound $x_{5}\leq 1$ and thus deduces that satisfying the output $x_{5}>0$ might be feasible. NeuralSAT continues with DECIDE, which uses a heuristic to select the unassigned variable $v_{4}$ and sets $v_{4}=F$. NeuralSAT also increments the decision level ($dl$) to 1 and associates $dl=1$ to the assignment, i.e., $\overline{v_{4}}@1$. Footnote 2: We use the terms feasible, from the LP community, and satisfiable, from the SAT community, interchangeably. In _iteration 2_, BCP again has no effect because it does not detect any unit clauses. In DEDUCTION, NeuralSAT determines that current set of constraints, which contains $x_{1}+x_{2}-1\leq 0$ due to the assignment $v_{4}\mapsto F$ (i.e., $x_{4}=\mathtt{off}$), is feasible. NeuralSAT then approximates a new output upper bound $x_{5}\leq-1$, which means satisfying the output $x_{5}>0$ constraint is _infeasible_. NeuralSAT now enters ANALYZE-CONFLICT and determines that $v_{4}$ causes the conflict ($v_{4}$ is the only variable assigned so far). From the assignment $\overline{v_{4}}@1$, NeuralSAT learns a "backjumping" clause $v_{4}$, i.e., $v_{4}$ must be $T$. NeuralSAT now backtracks to $dl$ 0 and erases all assignments decided _after_ this level. Thus, $v_{4}$ is now unassigned and the constraint $x_{1}+x_{2}-1\leq 0$ is also removed. In _iteration 3_, BCP determines that the learned clause is also a unit clause $v_{4}$ and infers $v_{4}@0$. In DEDUCTION, we now have the new constraint $x_{1}+x_{2}-1>0$ due to $v_{4}\mapsto T$ (i.e., $x_{4}=\mathtt{on}$). With the new constraint, NeuralSAT approximates the output upper bound $x_{5}\leq 0.5$, which means $x_{5}>0$ might be satisfiable. Also, NeuralSAT computes new bounds $0.5\leq x_{3}\leq 2.5$ and $0<x_{4}\leq 2.0$, and deduces that $x_{3}$ must be positive because its lower bound is 0.5 Thus, NeuralSAT has a new assignment $v_{3}@0$ ($dl$ stays unchanged due to the implication). Note that this process of inferring new assignments from the T-solver is referred to theory propagation in DPLL(T). In _iteration 4_, BCP has no effects because we have no new unit clauses. In DEDUCTION, NeuralSAT determines that the current set of constraints, which contains the new constraint $-0.5x_{1}+0.5x_{2}+1>0$ (due to $v_{3}\mapsto T$), is _infeasible_. Thus, NeuralSAT enters ANALYZE-CONFLICT and determines that $v_{4}$, which was set at $dl=0$ (by BCP in iteration 3), causes the conflict. NeuralSAT then learns a clause $\overline{v_{4}}$ (the conflict occurs when we have the assignment $\{v_{3}\mapsto T;v_{4}\mapsto T\}$, but $v_{3}$ was implied and thus making $v_{4}$ the conflict). However, because the assignment $v_{4}$ was assigned at decision level 0, NeuralSAT can no longer backtrack and thus sets $dl=-1$ and returns unsat. This unsat result shows that the DNN has the property because we cannot find a counterexample violating it, i.e., no inputs $x_{1}\in[-1,1],x_{2}\in[-2,2]$ that results in $x_{5}>0$. ## 4. The NeuralSAT Approach Fig. 4 shows the NeuralSAT algorithm, which takes as input the formula $\alpha$ representing the ReLU-based DNN $N$ and the formulae $\phi_{in}\Rightarrow\phi_{out}$ representing the property $\phi$ to be proved. Internally, NeuralSAT checks the satisfiability of the formula \[\alpha\wedge\phi_{in}\wedge\overline{\phi_{out}}. \tag{4}\] NeuralSAT returns unsat if the formula unsatisfiable, indicating that $\phi$ is a valid property of $N$, and sat if it is satisfiable, indicating the $N$ is not a valid property of $\phi$. NeuralSAT uses a DPLL(T)-based algorithm to check unsatisfiability. First, the input formula in Eq. 4 is abstracted to a propositional formula with variables encoding neuron activation status (BooleanAbstraction). Next, NeuralSAT assign values to Boolean variables (Decide) and checks for conflicts the assignment has with the real-valued constraints of the DNN and the property of interest (BCP and Deduce). If conflicts arise, NeuralSAT determines the assignment decisions causing the conflicts (AnalyzeConflict), backtracks to erase such decisions (Backtrack), and learns clauses to avoid those decisions in the future (AddClause). NeuralSAT repeats these decisions and checking steps until it finds a total or full assignment for all Boolean variables, in which it returns sat, or until it no longer can backtrack, in which it returns unsat. We describe these steps in more detail below. ### Boolean Abstraction BooleanAbstraction (Fig. 4 line 1) encodes the DNN verification problem into a Boolean constraint to be solved by DPLL. This step creates Boolean variables to represent the _activation status_ of hidden neurons in the DNN. Observe that when evaluating the DNN on any concrete input, the value of each hidden neuron _before_ applying ReLU is either $>0$ (the neuron is _active_ and the input is passed through to the output) or $\leq 0$ (the neuron is _inactive_ because the output is $0$). This allows partial assignments to these variables to represent neuron activation patterns within the DNN. Figure 4. The NeuralSAT DPLL(T) algorithm From the given network, NeuralSAT first creates Boolean variables representing the activation status of neurons. Next, NeuralSAT forms a set of initial clauses ensuring that each status variable is either $T$ or $F$, indicating that each neuron is either active or inactive, respectively. For example, for the DNN in Fig. 2, NeuralSAT creates two status variables $v_{3},v_{4}$ for neurons $x_{3},x_{4}$, respectively, and two initial clauses $v_{3}\lor\overline{v_{3}}$ and $v_{4}\lor\overline{v_{4}}$. The assignment $\{x_{3}=T,x_{4}=F\}$ creates the constraint $0.5x_{1}-0.5x_{2}-1>0\wedge x_{1}+x_{2}-2\leq 0$. ### Dpll After BooleanAbstraction, NeuralSAT iteratively searches for an assignment satisfying the status clauses (Fig. 4, lines 5- 16). NeuralSAT combines DPLL components (e.g., Decide, BCP, AnalyzeConflict, Backtrack) to assign truth values with a theory solver (SS4.3), consisting of abstraction and linear programming solving, to check the feasibility of the constraints implied by the assignment with respect to the network and property of interest. NeuralSAT maintains several variables (Fig. 4, lines 1- 4). These include clauses, a set of _clauses_ consisting of the initial activation clauses and learned clauses; $\sigma$, a _truth assignment_ mapping status variables to truth values; _igraph_, an _implication graph_ used for analyzing conflicts; and _dl_, a non-zero _decision level_ used for assignment and backtracking. #### 4.2.1. Decide From the current assignment, Decide (Fig. 4, line 15) uses a heuristic to choose an unassigned variable and assigns it a random truth value at the current decision level. We describe our decision heuristic for DNN reasoning in SS4.4.2. Note that if the current assignment is full, i.e., all variables have assigned values, Decide returns False (from which NeuralSAT returns sat). #### 4.2.2. Boolean Constraint Propagation (BCP) From the current assignment and clauses, BCP (Fig. 4, line 7) detects _unit clauses3_ and infers values for variables in these clauses. For example, after the decision $a\mapsto F$, BCP determines that the clause $a\lor b$ becomes unit, and infers that $b\mapsto T$. Moreover, each assignment due to BCP is associated with the current decision level because instead of being "guessed" by Decide the chosen value is logically implied by other assignments. Moreover, because each BCP implication might cause other clauses to become unit, BCP is applied repeatedly until it can no longer find unit clauses. BCP returns False if it obtains contradictory implications (e.g., one BCP application infers $a\mapsto F$ while another infers $a\mapsto T$), and returns True otherwise. Footnote 3: A unit clause is a clause that has a single unassigned literal. _Implication Graph._ BCP uses an _implication graph_(Barrett, 2013) to represent the current assignment and the reason for each BCP implication. In this graph, a node represents the assignment and an edge $i\xrightarrow{c}j$ means that BCP infers the assignment represented in node $j$ due to the unit clause $c$ caused by the assignment represented by node $i$. The implication graph is used by both BCP, which iteratively constructs the graph on each BCP application and uses it to determine conflict, and AnalyzeConflict (SS4.2.3), which analyzes the conflict in the graph to learn clauses. _Example._ Assume we have the clauses in Fig. 5(a), the assignments $\overline{v_{5}}$@3 and $v_{1}$@6 (represented in the graph in Fig. 5(b) by nodes $\overline{v_{5}}$@3 and $v_{1}$@6, respectively), and are currently at decision level _dl_ 6. Because of assignment $v_{1}$@6, BCP infers $v_{2}$@6 from the unit clause $c_{1}$ and captures that implication with edge $v_{1}$@$6\xrightarrow{c_{1}}v_{2}$@6. Next, because of assignment $v_{2}$@6, BCP infers $v_{4}$@6 from the unit clause $c_{3}$ as shown by edge $v_{2}$@6 $\xrightarrow{c_{3}}v_{4}$@6. Similarly, BCP creates edges $v_{1}$@$6\xrightarrow{c_{2}}v_{3}$@6 and $\overline{v_{5}}$@$\xrightarrow{c_{2}}v_{3}$@6 to capture the inference $v_{3}$@6 from the unit clause $c_{2}$ due to assignments $\overline{v_{5}}$@3 and $v_{1}$@6. Now, BCP detects a conflict because clause $c_{4}=\overline{o_{3}}\lor\overline{u_{4}}$ cannot be satisfied with the assignments $v_{4}(\emptyset 6$ and $v_{3}(\emptyset 6$ (i.e., both $v_{3}$ and $v_{4}$ are $T$) and creates two edges to the (red) node $\kappa$: $v_{4}(\emptyset 6\xrightarrow{c_{4}}\kappa$ and $v_{3}(\emptyset 6\xrightarrow{c_{4}}\kappa$ to capture this conflict. Note that in this example BCP has the implication order $v_{2},v_{4},v_{3}$ (and then reaches a conflict). In the current implementation, NeuralSAT makes an arbitrary decision and thus could have a different order, e.g., $v_{3},v_{4},v_{2}$. #### 4.2.3. Conflict Analysis Given an implication graph with a conflict such as the one in Fig. 5(b), AnalyzeConflict learns a new _clause_ to avoid past decisions causing the conflict. The algorithm traverses the implication graph backward, starting from the conflicting node $\kappa$, while constructing a new clause through a series of resolution steps. AnalyzeConflict aims to obtain an _asserting_ clause, which is a clause that will force an immediate BCP implication after backtracking. AnalyzeConflict, shown in Fig. 6, first extracts the conflicting clause $cl$ (line 1), represented by the edges connecting to the conflicting node $\kappa$ in the implication graph. Next, the algorithm refines this clause to achieve an asserting clause (lines 2- 6). It obtains the literal $lit$ that was assigned last in $cl$ (line 3), the variable $\underline{var}$ associated with $lit$ (line 4), and the antecedent clause $\underline{ante}$ of that $\underline{var}$ (line 5), which contains $\overline{lit}$ as the only satisfied literal in the clause. Now, AnalyzeConflict resolves $cl$ and $\underline{ante}$ to eliminate literals involving $\underline{var}$ (line 6). The result of the resolution is a clause, which is then refined in the next iteration. _Resolution._ We use the standard _binary resolution rule_ to learn a new clause implied by two (_resolving_) clauses $a_{1}\lor...\lor a_{n}\lor\beta$ and $b_{1}\lor...\lor b_{m}\lor\overline{\beta}$ containing complementary literals involving the (_resolution_) variable $\beta$: \[\frac{(a_{1}\lor...\lor a_{n}\lor\beta)}{(a_{1}\lor...\lor a_{n}\lor b_{1}\lor...\lor b_{m}\lor\overline{\beta})}\quad(\text{Binary-Resolution}) \tag{5}\] Figure 5. (a) A set of clauses, (b) an implication graph, and (c) learning a new clause. Figure 6. AnalyzeConflict The resulting (_resolvant_) clause $a_{1}\lor...\lor a_{n}\lor b_{1}\lor...\lor b_{m}$ contains all the literals that do not have complements $\beta$ and $\neg\beta$. Example.Fig.5(c) demonstrates AnalyzeConflict using the example in SS4.2.2 with the BCP implication order $v_{2},v_{4},v_{3}$ and the conflicting clause $cl$ (connecting to node $\kappa$ in the graph in Fig.5(b)) $c_{4}=\overline{v_{3}}\lor\overline{v_{4}}$. From $c_{4}$, we determine the last assigned literal is $lit=\overline{v_{3}}$, which contains the variable $var=v_{3}$, and the antecedent clause containing $v_{3}$ is $c_{2}=\overline{v_{1}}\lor v_{3}\lor v_{5}$ (from the implication graph in Fig.5(b), we determine that assignments $v_{1}$@$6$ and $\overline{v_{5}}$@$3$ cause the BCP implication $v_{3}$@$6$ due to clause $c_{2}$). Now we resolve the two clauses $cl$ and $c_{2}$ using the resolution variable $v_{3}$ to obtain the clause $\overline{v_{4}}\lor\overline{v_{1}}\lor v_{5}$. Next, from the new clause, we obtain $lit=\overline{v_{4}},var=v_{4},ante=c_{3}$ and apply resolution to get the clause $\overline{v_{1}}\lor v_{5}\lor\overline{v_{2}}$. Similarly, from this clause, we obtain $lit=\overline{v_{2}},var=v_{2},ante=c_{1}$ and apply resolution to obtain the clause $v_{1}\lor v_{5}$. At this point, AnalyzeConflict determines that this is an asserting clause, which would force an immediate BCP implication after Backtracking. As will be shown in SS4.2.4, NeuralSAT will backtrack to level $3$ and erases all assignments after this level (so the assignment $\overline{v_{5}}$@$3$ is not erased, but assignments after level $3$ are erased). Then, BCP will find that $c_{5}$ is a unit clause because $\overline{v_{5}}$@$3$ and infers $\overline{v_{1}}$. Once obtaining the asserting clause, AnalyzeConflict stops the search, and NeuralSAT adds $v_{1}\lor v_{5}$ as the new clause $c_{5}$ to the set of existing four clauses. The process of learning clauses allows NeuralSAT to learn from its past mistakes. While such clauses are logically implied by the formula in Eq.4 and therefore do not change the result, they help prune the search space and allow DPLL and therefore NeuralSAT to scale. For example, after learning the clause $c_{5}$, together with assignment $v_{5}$@$3$, we immediately infer $v_{1}\mapsto F$ through BCP instead of having to guess through Decide. #### 4.2.4. Backtrack From the clause returned by AnalyzeConflict, Backtrack (Fig.4, line12) computes a backtracking level and erases all decisions and implications made after that level. If the clause is _unary_ (containing just a single literal), then we backtrack to level $0$. Currently, NeuralSAT uses the standard _conflict-drive backtracking_ strategy (Barrett, 2013), which sets the backtracking level to the _second most recent_ decision level in the clause. Intuitively, by backtracking to the second most recent level, which means erasing assignments made _after_ that level, this strategy encourages trying new assignments for more recently decided variables. Example.From the clause $c_{5}=\overline{v_{1}}\lor v_{5}$ learned in AnalyzeConflict, we backtrack to decision level $3$, the second most recent decision level in the clause (because assignments $v_{1}$@$6$ and $\overline{v_{5}}$@$3$ were decided at levels $6$ and $3$, respectively). Next, we erase all assignments from decision level $4$ onward (i.e., the assignments to $v_{1},v_{2},v_{3},v_{4}$ as shown in the implication graph in Fig.5). This thus makes these more recently assigned variables (after decision level $3$) available for new assignments (in fact, as shown by the example in SS4.2.2, BCP will immediately infer $v_{1}=T$ by noticing that $c_{5}$ is now a unit clause). ### Deduction (Theory Solving) Deduce (Fig.4, line8) is the theory solver, i.e., the T in DPLL(T). The main purpose of the theory solver is to check the feasibility of the constraints represented by the current propositional variable assignment; as shown in the formalization in SS5 this amounts to just _linear equation_ solving for verifying piecewise linear DNNs. However, NeuralSAT is able to leverage specific information from the DNN problem, including input and output properties, for more aggressive feasibility checking. Specifically, Deduce has three tasks: (i) checking feasibility using linear programming (LP) solving, (i) further checking feasibility with input tightening and abstraction, and (iii) inferring literals that are unassigned and are implied by the abstracted constraint. ``` input:current assignment $\sigma$ and decision level $dl$, DNN $\alpha$, input property $\phi_{in}$, output property $\phi_{out}$ output:false if infeasibility occurs, true otherwise 1 solver=newLPSolver(solver, $\alpha,\sigma\land\phi_{in}\land\overline{\phi_{out}}$) 2ifSolve(solver)$\equiv$INFEASIBLEthenreturnfalse 3ifisTotal($\sigma$)then 4returntrue//origprob(Eq. 4)issatisfiable 5 input_bounds=TightenInputBounds(solver, $\phi_{in}$) output_bounds,hidden_bounds=Abstract($\alpha,\sigma,\text{input\_bounds}$) 6ifCheck(output_bounds, $\overline{\phi_{out}}$)$\equiv$INFEASIBLEthen 7 returnfalse 8for$v\in$hidden_boundsdo 9$x\leftarrow$ActivationStatus($v$) 10if$x\in\sigma\lor\neg x\in\sigma$thencontinue 11ifLowerBound($v$)$>0$then$\sigma\leftarrow\sigma\cup x$@$dl$ 12elseifUpperBound($v$)$\leq 0$then$\sigma\leftarrow\sigma\cup\overline{x}$@$dl$ 13returntrue ``` Fig. 7: Deduce Fig. 7 describes Deduce, which returns False if infeasibility occurs and True otherwise. First, it creates a linear constraint system from the input assignment $\sigma$ and $\alpha\land\phi_{in}\land\overline{\phi_{out}}$, i.e., the formula in Eq. 4 representing the original problem (line 1). The key idea is that we can remove ReLU activation for hidden neurons whose activation status have been decided. For constraints in $\alpha$ associated with variables that are not in the $\sigma$, we ignore them and just consider the cutting planes introduced by the partial assignment. For example, for the assignment $v_{3}\mapsto T,v_{4}\mapsto F$, the non-linear ReLU constraints $x_{3}=ReLU(-0.5x_{1}+0.5x_{2}+1)$ and $x_{4}=ReLU(x_{1}+x_{2}-1)$ for the DNN in Fig. 2 become linear constraints $x_{3}=-0.5x_{1}+0.5x_{2}$ and $x_{4}=0$, respectively. Next, an LP solver checks the feasibility of the linear constraints (line 2). If the solver returns infeasible, Deduce returns False so that NeuralSAT can analyze the assignment and backtrack. If the constraints are feasible, then there are two cases to handle. First, if the assignment is total (i.e., all variables are assigned), then that means that the original problem is satisfiable (line 4) and NeuralSAT returns sat. ReLU AbstractionSecond, if the assignment is not total then Deduce applies abstraction to check satisfiability (lines 5-8). Specifically, we over-approximate ReLU computations to obtain the upper and lower bounds of the output values and check if the output properties are feasible with respect to these bounds. For example, the output $x_{5}>0$ is _not_ feasible if the upperbound is $x_{5}\leq 0$ and _might be_ feasible if the upperbound is $x_{5}\leq 0.5$ ("might be" because this is an upper-bound). If abstraction results in infeasibility, then Deduce returns False for NeuralSAT to analyze the current assignment (line 8). NeuralSAT uses abstraction to approximate the lower and upper bounds of hidden and output neurons. Fig. 8 compares the interval [Wang et al.2018] (a), zonotope [Singh et al.2018] (b), and polytope [Singh et al.2019, Wang et al.2021, Xu et al.2020] (c,d) abstraction domains to compute the lower $l_{y}(x)$ and upper $u_{y}(x)$ bounds of a ReLU computation $y=$ReLU(x) (non-convex red line). NeuralSAT can employ any existing abstract domains, though currently it adopts the _LiRPA_ polytope (Fig. 8c) [Wang et al.2021, Xu et al.2020] because it has a good trade-off between precision and efficiency. Note that in some cases we can compute a better abstraction for output bounds by first tightening the input bounds (line 5). We discuss this input bound tightening heuristic in SS4.4.1. InferenceIf abstraction results in feasible constraints, Deduce next attempts to infer implied literals (lines 9- 13). To obtain the bounds of the output neurons, abstraction also needs to compute the bounds of hidden neurons, including those with undecided activation status (i.e., not yet in $\sigma$). This allows us to assign the activation variable of a hidden neuron the value True if the lowerbound of that neuron is greater than $0$ (the neuron is active) and False otherwise. Since each literal is considered, this would be considered exhaustive theory propagation. Whereas the literature (Kroening and Strichman, 2016; Nieuwenhuis et al., 2006) suggests that this is an inefficient strategy, we find that it does not incur significant overhead (average overhead is about $4\%$ and median is $2\%$ with outliners being large CIFAR2020 networks described in SS6). ExampleFor the illustrative example in SS3.1, in iteration 3, the current assignment $\sigma$ is $\{v_{4}=1\}$, corresponding to a constraint $x_{1}+x_{2}-1>0$. With the new constraint, we optimize the input bounds and compute the new bounds for hidden neurons $0.5\leq x_{3}\leq 2.5$, $0<x_{4}\leq 2.0$ and output neuron $x_{5}\leq 0.5$ (and use this to determine that the postcondition $x_{5}>0$ might be feasible). We also infer $v_{3}=1$ because of the positive lower bound $0.5\leq x_{3}$. ### Optimizations NeuralSAT implements several optimizations to quickly deal with "easy" problems, such as those with _small input_ sizes or can be _easily disproved_ (of course NeuralSAT does not know this in advance, it just tries the heuristics in hope that it could help). As we will see in SS7, these techniques are also employed by many current state-of-the-art DNN verification tools. #### 4.4.1. Input Bounds Tightening For networks with small inputs (currently set to those with $\leq 50$ inputs), NeuralSAT uses a more aggressive abstraction process in the theory solver described in SS4.3. Specifically, we use LP solving to compute the tightest bounds for all input variables from the generated linear constraints. This computation is efficient when the number of inputs is small. After tightening input bounds we apply abstraction (line 6, Fig. 7) to approximate the output bounds, which can be more precise with better input bounds. For networks with large number of inputs, we obtain input bounds from the input property $\phi_{in}$. #### 4.4.2. Decision Heuristics Decision or branching heuristics decide free variables to make assignments and thus are crucial for the scalability of DPLL by reducing assignment mistakes (Beyer, 2022; Kroening and Strichman, 2016). NeuralSAT currently adopts two decision heuristics. Figure 8. Abstractions for ReLU: (a) interval, (b) zonotope, (c) LiRPA polytope, and (d) DeepPoly polytope. Notice that ReLU is a non-convex region (red line) while all abstractions are convex regions For networks with small inputs, NeuralSAT prioritizes variables representing neurons with the _furthest bounds_ from the decision value 0 of ReLU, i.e., the 0 in $\max(x,0)$. Such neurons have wider bounds and therefore are more difficult to tighten during abstraction compared to other neurons. This heuristic helps input bounds tightening as described in SS4.4.1 (which is also applied only for networks with small inputs). It is also cheap because we can reuse the computed boundaries of hidden neurons during abstraction. For other networks, NeuralSAT applies the Filtered Smart Branching (FSB) heuristic (Bunel et al., 2018; De Palma et al., 2021). For each unassigned variable, FSB assumes that it has been decided (i.e., the corresponding neuron has been split) and computes a fast approximation of the lower and upperbounds of the network output variables. FSB then prioritizes unassigned variables with the best differences among the bounds that would help make the input formula unsatisfiable (which helps prove the property of interest). We note that several well-known DNN verification techniques and tools (OVAL-group, 2023; Wang et al., 2021) use variants of FSB with slight difference on the approximation computation. #### 4.4.3. Multiprocessing For networks with small inputs, NeuralSAT uses a simple approach to create and solve subproblems in parallel. Given a verification problem $N_{orig}=(\alpha,\phi_{in},\phi_{out})$, where $\alpha$ is the DNN and $\phi_{in}\Rightarrow\phi_{out}$ is the desired property, NeuralSAT creates subproblems $N_{i}=(\alpha,\phi_{in_{i}},\phi_{out})$, where $\phi_{in_{i}}$ is the $i$-th subregion of the input region specified by $\phi_{in}$. Intuitively, each subproblem checks if the DNN produces the output $\phi_{out}$ from a smaller input region $\phi_{in_{i}}$. The combination of these subproperties $\bigwedge\phi_{in}\Rightarrow\phi_{out}$ is logically equivalent to the original property $\phi_{in}\Rightarrow\phi_{out}$. Given $k$ available threads, NeuralSAT splits the original input region to obtain subproblems as described and and runs DPLL(T) on $k$ subproblems in parallel. NeuralSAT returns unsat if it verifies all subproblems and sat if it found a counterexample in any subproblem. For example, we split the input region $\{x_{1}\in[-1,1],x_{2}\in[-2,2]\}$ into four subregions $\{x_{1}\in[-1,0],x_{2}\in[-2,0]\}$, $\{x_{1}\in[-1,0],x_{2}\in[0,2]\}$, $\{x_{1}\in[0,1],x_{2}\in[-2,0]\}$, and $\{x_{1}\in[0,1],x_{2}\in[0,2]\}$. Note that the formula $-1\leq x_{1}\leq 1\land-2\leq x_{2}\leq 2$ representing the original input region is equivalent to the formula $(-1\leq x_{1}\leq 0\lor 0\leq x\leq 1)\land(-2\leq x_{2}\leq 0\lor 0\leq x_{2}\leq 2)$ representing the combination of the created subregions. #### 4.4.4. Falsification using Adversarial Attacks Our NeuralSAT tool attempts to disprove or falsify the property before running DPLL(T). This helps solve "easy" cases when the properties can be disproved easily and adopted by many state-of-the-art DNN verification tools such as ERAN(Muller et al., 2022) and $\alpha\beta$-CROWN(Zhang et al., 2022). NeuralSAT uses two adversarial attack algorithms to find counterexamples to falsify properties. First, we try a randomized attack approach (Das et al., 2021), which is a derivative-free sampling-based optimization (Yu et al., 2016), to generate a potential counterexample. If this approach fails, we then use a gradient-based approach (Madry et al., 2017) to create another potential counterexample. If either attack algorithm gives a valid counterexample, NeuralSAT returns sat, indicating that property is invalid. If both algorithms cannot find a valid counterexample or they exceed a predefined timeout, NeuralSAT continues with its DPLL(T) search. ## 5. NeuralSAT DPLL(T) Formalization In SS4 we describe NeuralSAT and its optimizations, here we formalize the NeuralSAT DPLL(T) framework. By abstracting away heuristics, optimizations, and implementation details, we can focus on the core NeuralSAT algorithm and establish its correctness and termination properties. NeuralSAT can be described using the states and transition rules of the standard DPLL(T) framework described in (Nieuwenhuis et al., 2006) and therefore inherits the theoretical results established there. We also highlight the differences between NeuralSAT and standard DPLL(T), but these differences do not affect any of the main results. The section aims to be self-contained, but readers who are familiar with the work in (Nieuwenhuis et al., 2006) can quickly skim through it. ### Preliminaries Formulae, Theory, and Satisfiability.Let $P$ be a finite set of atoms (e.g., linear constraints in our context). For an atom $p\in P$, $p$ is a positive literal and $\neg p$ is a negative literal of $P$. A _clause_ is a set of literals and a CNF _formula_ is a set of clauses. A model $M$ is a sequence of literals and never contains both a literal and its negation. A literal $l$ is _true_ in $M$ if $l\in M$, is _false_ in $M$ if $\neg l\in M$, and is _undefined_ otherwise (i.e., $l\notin M$). $M$ is _total_ if every atom $p\in P$ has a literal in $M$, and is _partial_ otherwise. A clause $C$ is true in $M$, written as $M\models C$, if $\exists l\in C$. $l\in M$, and is false in $M$, written as $M\models\neg C$, if $\forall l\in C$. $\neg l\in M$. A CNF $F$ is true in (or satisfied by) $M$, written as $M\models F$, if all clauses of $F$ are true in $M$. In that case, $M$ is called a _model_ of $F$. If F has no models then it is _unsatisfiable_. If $F$ and $F^{\prime}$ are formulae, then $F$_entails_$F^{\prime}$, written as $F\models F^{\prime}$, if $F^{\prime}$ is true in all models of $F$. Note that we consider literals and clauses as purely boolean variables and check satisfiability using propositional logic (we could also treat literals as syntactical items and check satisfiability using set operations, e.g., $M\models C$ is $C\cap M\neq\emptyset$). A theory T is a set of formulas. A formula F is _T-satisfiable_ or _T-consistent_ if $F\wedge T$ is satisfiable. Otherwise, it is called _T-unsatisfiable_ or _T inconsistent_. An assignment $M$ can be thought as a conjunction of its literals and hence as a formula. If $M$ is a T-consistent and $F$ is a formula such that $M\models F$, then $M$ is also a _T-model_ of F, written as $M\models_{T}F$. If F and G are formulae, then F entails G in T, written $F\models_{T}G$ if $F\wedge\neg G$ is T-inconsistent. Note when checking satisfiability in the theory, i.e., $\models_{T}$, we use a theory solver to reason about the linear constraints represented by the literals. NeuralSAT Algorithm. For NeuralSAT, each atom $p_{i}$ in $P=\{p_{1},p_{2},\ldots,p_{N}\}$ is the linear constraint representing activation status of neuron $i$, e.g., for the DNN example in Fig. 2, $p_{3}$ is $-0.5x_{1}+0.5x_{2}+1>0$, the constraint that neuron $x_{3}$ is active (thus $p_{3}$ is a positive literal and $\neg p_{3}$ is a negative literal). $M$ represents the (partial) truth assignment $\sigma$, and $F$ represents the set or conjunction of clauses that NeuralSAT needs to satisfy. Adding $l$ to $M$ is the truth assignment $p\mapsto T$ if $l$ is $p$, and is the assignment $p\mapsto F$ if $l$ is $\neg p$. Moreover, the theory we consider is LRA (Linear Real Arithmetic) and our customized T-solver, described in 4.3 uses LP solving and abstraction to decide satisfiability of DNN properties. Note that in NeuralSAT we use Boolean Abstraction to create variables $v_{i}$ to represent linear constraints capturing activation status. Here we do not use Boolean Abstraction and capture its effects with atoms $p_{i}$ representing the Boolean variables $v_{i}$ and adding to $M$ the literals $p_{i},\neg p_{i}$ corresponding to truth assignments $v_{i}\mapsto T,v_{i}\mapsto F$ in $\sigma$, respectively. ### Transition Rules We formalize the NeuralSAT DPLL(T) using _transition rules_ that move from a state to another state of the algorithm. A _state_ is either a assignment $M$ and a CNF formula $F$, written as $M\parallel F$, or the special state $\mathsf{Fail}$, which indicates that the formula is unsatisfiable. We write $S\Longrightarrow S^{\prime}$ as a transition from state $S$ to $S^{\prime}$. We write $S\Longrightarrow^{}S^{\prime}$ to indicate any possible transition from $S$ to $S^{\prime}$ (i.e., reflexive-transitive closure). In a state $M\parallel F,C$, we say the clause $C$ is conflicting if $M\models\neg C$. Tab. 2 gives the conditional transition rules for NeuralSAT. _Decision_ literals, written with suffix $l^{d}$, are non-deterministically decided (i.e., guessed), while other literals are deduced deterministically through implication. Intuitively, mistakes can happen with decision literals and require backtracking. In contrast, rules that add non-decision literals help prune the search space. The rules _Decide_, _BCP_, _Fail_ describe transitions that do not rely on theory solving. _Decide_ non-deterministically selects and adds an undefined literal $l$ to $M$ (i.e., $l$ is a decision literal and can be backtracked to when conflict occurs). _BCP_ (or UnitPropagate) infers and adds the unit literal $l$ to to satisfy the clause $C\lor l$, where $M\models\neg C$. _Fail_ moves to a Fail state (i.e., $F$ is unsatisfiable) when a conflicting clause $C$ occurs and $M$ contains no decision literals to backtrack to. The rules _T-Learn_, _T-Forget_, _T-Backjump_, _TheoryPropagate_ describe transitions that rely on theory solving, e.g., $\models_{T}$. _T-Backjump_ analyzes a conflicting clause $C$ to determine an "incorrect" decision literal $l^{d}$ and computes a "backjump" clause $C^{\prime}\lor l^{\prime}$ (which will be used by _T-learn_ to ensure that the incorrect decision literal $l$ will not be added to $M$ in the future). The rule also adds $l^{\prime}$ to $M$ (since $M\models\neg C^{\prime}$) and removes $l^{d}$ and the set $N$ of subsequent literals added to $M$ after $l^{d}$ (i.e., it backtracks and removes the "incorrect" decision $l^{d}$ and subsequent assignments). _T-Learn_ strengthens $F$ with a clause $C$ that is entailed by $F$ (i.e., learned clauses are _lemmas_ of $F$). As mentioned, clause $C$ is the "backjumping" clause $C^{\prime}\lor l^{\prime}$ in _T-Backjump_. Finally, _TheoryPropagate_ infers literals that are T-entailed by literals in $M$ (thus $l$ is a non-decision literal). NeuralSAT Algorithm.* The Decide and BCP rules align to the Decide and BCP components of NeuralSAT, respectively. The other rules are also implemented in NeuralSAT through the interactions of Deduction, Analyze-Conflict, and Backtrack components. For example, the T-Backjump rule is implemented as part of Deduction and AnalyzeConflict. Also note that while implication graph is a common way to detect conflicts and derive backjumping clause, it is still an implementation detail and therefore not mentioned in T-Backjump (which states there exists a way to obtain a backjumping clause). T-Learn, which adds lemmas to existing clauses, is achieved in the main loop of the NeuralSAT algorithm (Fig. 4, line 13). TheoryPropagate is implemented as part of _Deduction_ (Fig. 7, lines 9-13). Finally, theory solving, i.e., $\models_{T}$, is implemented in Deduction by using LP solving and abstraction to check satisfiability of linear constraints. Note that DPLL and DPLL(T) algorithms used in modern SAT and SMT solvers often include Restart and T-Forget rules, which remove learned clauses periodically. To date, we have not seen the need for these in NeuralSAT, but they could easily be incorporated. ### Termination and Correctness of NeuralSAT DPLL(T) In SS5.2 we describe NeuralSAT DPLL(T) using transition rules of the standard DPLL(T) approach. This allows us to establish the the formal properties NeuralSAT DPLL(T), which are similar to those of standard DPLL(T). Below we summarize the main results and refer the readers to [Nieuwenhuis et al. 2006] for complete proofs. Note that the work in [Nieuwenhuis et al. 2006] covers multiple variants of DPLL with various rule configurations. Here we focus on the DPLL(T) of NeuralSAT, e.g., no Restart and Forget. This significantly simplify our demonstration of termination and correctness of NeuralSAT DPLL(T). We first establish several invariants for the transition rules of NeuralSAT DPLL(T). Lemma 5.1.: _If $\emptyset\parallel F\Longrightarrow^{}M\parallel G$, then the following hold:_ 1. _All atoms in_ $M$ _and all atoms in_ $G$ _are atoms of_ $F$_._ 2. $M$ _is indeed an assignment, i.e., it contains no pair of literals_ $p$ _and_ $\neg p$_._ 3. $G$ _is equivalent to_ $F$ _in the theory T._ All properties hold trivially in the initial state $\emptyset\parallel F$, so we will use induction to show the transition rules preserve them. Consider a transition $M^{\prime}\parallel F^{\prime}\Longrightarrow M^{\prime\prime}\parallel F^{ \prime\prime}$. Assume the properties hold for $M\parallel F$. Property 1 holds because the only atoms can be added to $M^{\prime\prime}$ and $F^{\prime\prime}$ are from $M^{\prime}$ and $F^{\prime}$, all of which belong to $F$. Property 2 preserves the requirement that $M$ never shares both negative and positive literals of an atom (the condition of each rule adding a new literal ensures this). Property 3 holds because only T-Learn rule can modify $F^{\prime}$, but learning a clause $C$ that is a logical consequence of $F^{\prime}$ (i.e., $F^{\prime}\models_{T}C$) will preserve the equivalence between $F^{\prime}$ and $F^{\prime\prime}$. Lemma 5.2.: _If $\emptyset\parallel F\Longrightarrow^{}S$, and $S$ is final state, then $S$ is either Fail, or of the form $M\parallel F^{\prime}$, where $M$ is a T-model of $F$._ This states that if $M\models F^{\prime}$ then $M\models F$. This is true because $F$ and $F^{\prime}$ are logical equivalence by Lemma 5.1(3). Now we prove that NeuralSAT DPLL(T) terminates. Theorem 5.3 (Termination).: _Every derivation $\emptyset\parallel F\Longrightarrow S_{1}\Longrightarrow\dots$ is finite._ This proof uses a well-founded strict partial ordering on _states_$M\parallel F$. First, consider the case without T-Learn, in which only the assignment M is modified and the formula F remains constant. Then we can show no infinite derivation by (i) using Lemma 5.1(1,2) that the number of literals in M and M' are always less than or equal to the number of atoms in F and (ii) show that the number of "missing" literals of M is strictly greater than those of M'. Now, consider the case with T-learn. While F' can now be modified, i.e., learning new clauses, the number of possible clauses can be added to $F^{\prime}$ is finite as clauses are formed from a finite set of atoms and the conditions of T-learn disallow clause duplication. Note that if NeuralSAT involved the Restart and Forget rules, which periodically remove learned clauses, then its termination argument becomes more complicated (but still holds) as shown in the work [Nieuwenhuis et al. 2006]. Now we prove that NeuralSAT DPLL(T) is sound and complete. Theorem 5.4.: _If $\emptyset\parallel F\Longrightarrow^{}S$ where the state $S$ is final, then_ 1. _Sound_: $S$ _is Fail if, and only if,_ $F$ _is T-unsatisfiable_ 2. Complete:_If $S$ is of the form $M\parallel F^{\prime}$, then $M$ is a T-model of $F$._ Property 1 states that NeuralSAT DPLL(T) ends at Fail state iff the problem F is unsatisfiable. Property 2 asserts that if NeuralSAT DPLL(T) ends with an assignment $M$, then $M$ is the model of $F$, i.e, $F$ is satisfiable. This property requires showing that if $M\models_{T}F^{\prime}$, then $M\models_{T}F$, which is established in Lemma 5.2. Together, these properties of soundness, completeness, and termination make NeuralSAT DPLL(T) a decision procedure. Note that the presented results are independent from the theory under consideration. The main requirement of T-solver is its decidability for T-satisfiability or T-consistency checking. NeuralSAT uses LRA, a theory of real numbers with linear constraints, including linear equalities and inequalities, which is decidable (Kroening and Strichman, 2016). ## 6. Implementation and Experimental Settings ImplementationNeuralSAT is written in Python, and uses PyTorch (Paszke et al., 2019) for matrix multiplications and Gurobi (Gurobi Optimization, LLC, 2022) for linear constraint solving. We use the LiRPA abstraction library (Wang et al., 2021; Xu et al., 2020) for bounds approximation and tightening and adapt the randomized (Das et al., 2021) and Projected Gradient Descent (PGD) (Madry et al., 2017) adversarial attack techniques for falsification. Note that unlike some other DNN verifiers, NeuralSAT does not require hyperparameter tuning and all of our experiments use its default settings. Currently, NeuralSAT supports feedforward (FNN), convolutional (CNN), and Residual Learning Architecture (ResNet) neural networks that use ReLU. NeuralSAT automatically preprocesses these networks into Boolean variables and linear constraints representing the computation graph of the networks for DPLL(T). This preprocessing step is relatively standard and is used by various tools (e.g., $\alpha$-$\beta$-CROWN, ERAN, MN-BaB). Moreover, NeuralSAT supports the specification formats ONNX (Bai et al., 2023) for neural networks and VNN-LIB (Tacchella et al., 2023) for properties. These formats are standard and supported by major DNN verification tools. BenchmarksWe evaluate NeuralSAT using five standard, ReLU-based benchmarks obtained VNN-COMP'22 and shown in Tab. 3. In total these benchmarks consist of 53 networks, spanning multiple layer types and architectures, and 447 correctness properties. A problem instance pairs a property with a network. Across the benchmark 389 problem instances are known to be unsat (U) and 136 are known to be sat (S). For the remaining 98 problem instances no verifier in our study or in VNN-COMP'22 was able to solve the problem. We do not include these _unknown_ (?) problem instances in our study since they add no value in our comparative evaluation of DNN verifier performance. ACAS Xu* consists of 45 FNNs to issue turn advisories to aircrafts to avoid collisions. Each FNN has 5 inputs (speed, distance, etc). We use all 10 safety properties as specified in [14] and VNN-COMP'22, where properties 1-4 are used on 45 networks and properties 5-10 are used on a single network. MNISTFC consists of 3 FNNs for handwritten digit recognition and 30 robustness properties. Each FNN has 28x28 inputs representing a handwritten image. CIFAR2020 has 3 CNNs for objects detection and 203 robustness properties (each CNN has a set of different properties). Each network uses 3x32x32 RGB input images. For RESNET_A/B, each benchmark has only one network with the same architecture and 72 robustness properties. Each network uses 3x32x32 RGB input images. Tab. 3 provides more details. Column Instances (U/S/7) shows the number of verification instances and U/S/? indicate the number of instances that are unsat (valid property), sat (invalid property), and unknown (cannot be solved by any tools and we also do not know if it is sat or unsat), e.g., CIFAR10$2$255 has 82 instances (65 unsat, 13 sat, and 4 unknown). The last two columns give the sizes of individual networks: neurons are the numbers of hidden neurons and parameters are the numbers of weights and biases. For example, each FNN in ACAS Xu has 5 inputs, 6 hidden layers (each with 50 neurons), 5 outputs, and thus has 300 neurons ($6\times 50$) and 13305 parameters ($5\times 50\times 50+2\times 50\times 5+6\times 50+5$). Observe that networks from ACAS Xu have very few inputs (5), which is leveraged by many DNN verification tools as shown in SS7. _Verification Tools._ We compare NeuralSAT to six well-known DNN verifiers. $\alpha-\beta-\texttt{CRONN}$[22, 23] employs multiple abstractions and algorithms for efficient analysis (input splitting for networks with small input dimensions and parallel Branch-and-Bound [1] (BaB) otherwise). ERAN[22, 24, 23] uses input splitting optimizations with various abstractions. MN-BaB uses multiple abstractions and BaB. Marabou'21 (VNN-COMP'21 version) and Marabou'22 (VNN-COMP'22 version) [14] are a simplex-based solver that employs various optimizations such as parallel Split-and-Conquer [25] (SnC) and uses polytope abstraction [22] and MILP/LP-based bound tightening. nnenum[1] combines optimizations such as parallel case splitting and multiple levels of abstractions, e.g., three types of zonotopes with imagestar/starset [20]. Among tools competing in recent VNN-COMPs'21 [1] and '22 [23], $\alpha-\beta-\texttt{CRONN}$, MN-BaB, ERAN, and nnenum are consistently considered the top. For example, in VNN-COMP'22, $\alpha-\beta-\texttt{CRONN}$ is the winner for MNISTFC and also the overall winner, MN-BaB ranked 3rd on MNISTFC and second overall, and nnenum was the only one that can solve all instances in ACAS Xu and was 4th overall. Marabou'22 ranked 6th on MNISTFC and 7th overall. We also include Marabou'21, which only participated in VNN-COMP'21 and ranks 5th overall, because it outperforms Marabou'22 in many cases as shown in SS7. _Hardware and Setup._ Our experiments were run on a Linux machine with an AMD Threadripper 64-core 4.2GHZ CPU, 128GB RAM, and a NVIDIA GeForce RTX 4090 GPU with 24 GB VRAM. All tools use multiprocessing (even external tools/libraries including Gurobi, LiRPA, and Pytorch are multi-thread). ERAN, $\alpha-\beta-\texttt{CRONN}$, and MN-BaB leverage GPU processing for abstraction. The LiRPA library adopted by NeuralSAT uses the GPU for large benchmarks in MNISTFC, CIFAR2020 and RESNET. To maximize the performance of the DNN verifiers in comparisons, we leverage the benchmarks and installation scripts available from VNN-COMP ([https://github.com/ChristopherBrix/vnncomp2022_benchmarks](https://github.com/ChristopherBrix/vnncomp2022_benchmarks)). These scripts were tailored by the developers of each verifier to optimize performance on each benchmark. The VNN-COMP setting used varying runtimes for each problem instance ranging from 120 seconds to more than 12 minutes. We experimented with timeouts on our machine and settled on 200 seconds per instance which allowed the verifiers to achieve the scoring performance reported in VNN-COMP'22. For each benchmark instance we run three times and obtain the median results. ## 7. Results We evaluate NeuralSAT using the following research questions: * (SS7.1) How NeuralSAT compares to state-of-the-art DNN verifiers? * (SS7.2) How it compare to other tools in sat and unsat instances? * (SS7.3) How NeuralSAT compares to state-of-the-art DPLL(T)-based DNN verifiers? We note that in our experiments all tools provide _correct results_. If a tool was able to solve an instance, then it solves it correctly, i.e., no tool returned sat for unsat instances and vice versa. ### VNN-COMP Ranking We adopt the rules in VNN-COMP'22 to score and rank tools. This gives an overall view of how NeuralSAT compares to others in a VNN-COMP setting which accounts for both runtime and ability to solve problems. As mentioned in 6, we use the exact scripts from VNN-COMP'22 to setup and run this experiment. _Scoring Rules_. For each benchmark instance, a tool scores 10 points if it correctly verify an instance, 1 point if it correctly falsifies an instance4, 0 points if it cannot solve (e.g., timeouts, has errors, or returns unknown), -100 points if it gives incorrect results (does not apply to any tool we consider), 2 points for being the fastest among other tools competing on that instance, 1 point for being the second fastest (runtime within a 0.2s differences are considered the same). Footnote 4: VNN-COMP’22 assigns different scores for falsification: 1 point if the tool found a counterexample using an external adversarial attack technique, and 10 points if the tool found a counterexample using its core search algorithm. It is difficult for us to determine how a tool solves individual SAT instances, so we simply assign 1 point for correct falsification results as our experiments in §7.2.1 show that most SAT instances were solved using adversarial attacks. Tab. 4 shows the results. Columns # and tool show the rankings of the tools. Note that for some tool do not work on certain benchmarks and therefore are not shown (e.g., for MNISTFC, MN-BaB has errors for all problems). The next two columns give the scores and percent, computed by the score of the tool divided by the highest score (thus the tool with the highest score always has 100%). The last three columns give additional statistics: the number of instances that a tool verified or falsified, and the number of those that the tool ran the fastest - tools with runtimes within 0.2 seconds of each other are considered tied. Across these benchmarks NeuralSAT ranks second to $\alpha$-$\beta$-CROWN, which was the top performer in VNN-COMP'22 and thus the state-of-the-art. It trails $\alpha$-$\beta$-CROWN both in the number of problems verified and in the speed of verification, though it can falsify as many problems as any other verifier across the benchmark. A number of verifiers, e.g., Marabou'21 and nennum, exhibit good performance on benchmarks with small networks - MNISTFC and ACAS Xu - and outperform NeuralSAT. Our goal in developing NeuralSAT was to explore the potential for scaling constraint-based DNN verification and NeuralSAT's performance on the CIFAR2020 and RESNET benchmarks, which have networks with an order of magnitude more neurons, indicates the potential of the technique. While it ranks second to $\alpha$-$\beta$-CROWN across those benchmarks it falsifies only 1 fewer property (64/65) and verifies 5 fewer (191/196). We note that the developers of $\alpha$-$\beta$-CROWN tuned 10 hyperparameters on average for each benchmark to optimize its performance. In contrast, NeuralSAT has no hyperparameters which suggests that its performance on large models may generalize better in practice. Compared to the other tools, NeuralSAT ranks _second_ to and solves 97.7% of the large problem instances solved by the state-of-the-art $\alpha$-$\beta$-CROWN. ### SAT vs. Unsat instances We show more details for the results presented in SS7.1, focusing on the performance of tools for sat and unsat instances. Tab. 5 lists the runtime of each tool for each benchmark (median over 3 runs, as mentioned in SS6). Each benchmark has two rows showing unsat and sat results of each tools. The information given in these rows has the form runtime (# problems solved, # problems unsolved). For example, for ACAS Xu, NeuralSAT took 551.5s, solved 136 unsat instances, and failed 3 unsat instances. The mark - indicates the tool fails to run the benchmark (e.g., both Marabou'21 and Marabou'22 cannot run CIFAR2020). #### 7.2.1. SAT instances As shown in Tab. 5, tools are more successful in solving sat than unsat instances (i.e., the number of unsolved sat instances are often 0 or few for most tools). We also found that using external adversarial attacks or random counterexample generation as a quick and cheap way to find satisfying assignment is highly effective as expected. For example, out of the 136 sat instances NeuralSAT successfully solved 134, all of which were solved using these quick techniques. We observe this behavior consistently across all tools on all benchmarks. Example of difficult SAT instances (i.e., cannot be solved easily using these random or attack methods) include properties 7 and 8 of ACAS Xu. Both $\alpha$-$\beta$-CROWN and Marabou'21 timed out for \begin{table} \begin{tabular}{l\|l\|l l l\|r r} \hline \hline BMs & \# & Tool & Score & \% & Falsif. & Fast. \\ \hline \multirow{8}{}{Courtesy} & 1 & NeuralSAT & 366 & 100.0\% & 134 & 104 \\ & 2 & $\alpha$-$\beta$-CROWN & 341 & 93.2\% & 133 & 87 \\ & 3 & ERAN & 233 & 63.7\% & 134 & 32 \\ & 4 & MN-BaB & 184 & 50.3\% & 105 & 31 \\ & 5 & Marabou’22 & 90 & 24.6\% & 58 & 1 \\ & 6 & Marabou’21 & 87 & 23.8\% & 66 & 3 \\ & 7 & nnenum & 80 & 21.9\% & 71 & 2 \\ \hline \hline \end{tabular} \end{table} Table 6. VNN-COMP results for sat instances \begin{table} \begin{tabular}{c\|c\|c c c c c c} \hline \hline BM* & Networks & NeuralSAT & EBM & NN-BaB & Marabou’21 & Marabou’22 & nnenum & $\alpha$-$\beta$-CROWN \\ \hline \multirow{3}{}{ACAS Xu} & \multirow{3}{}{45 FNNs} & 551.5 (136, 3) & 3867.8 (133, 6) & 1251.0 (105, 34) & 321.1 (138, 1) & 533.4 (85, 54) & 233.5 (139, 0) & 335.5 (139, 0) \\ & & 63.9 (47, 0) & 811.9 (45, 2) & 339.0 (47, 0) & 75.4 (46, 1) & 172.8 (41, 6) & 62.9 (47, 0) & 132.1 (46, 1) \\ \hline \multirow{8}{}{MNISTFC} & \multirow{3}{}{MNIST\_256x4} & 15.2 (18, 0) & 315.0 (18, 0) & - & 78.4 (17, 1) & 430.7 (14, 4) & 150.1 (17, 1) & 74.6 (18, 0) \\ & & 10.6 (12, 0) & 67.6 (12, 0) & - & 66.0 (12, 0) & 85.0 (12, 0) & 89.6 (12, 0) & 1.5 (11, 1) \\ \cline{1-1} \cline{2-9} & & 15.4 (7, 13) & 303.8 (10, 10) & - & 172.6 (10, 10) & 30.8 (7, 13) & 163.4 (11, 9) & 783.3 (20, 0) \\ \cline{1-1} & & 5.3 (6, 0) & 8.9 (6, 0) & - & 66.9 (6, 0) & 346.3 (5, 1) & - & 1.1 (6, 0) \\ \cline{1-1} \cline{2-9} & & 19.7 (8, 0) & 18.0 (8, 0) & - & 42.9 (8, 0) & 50.2 (8, 0) & 26.3 (8, 0) & 407.9 (16, 0) \\ \cline{1-1} & & 4.6 (5, 0) & 11.0 (5, 0) & - & 16.2 (2, 3) & - & - & 1.1 (5, 0) \\ \hline \multirow{8}{}{CIFAR2020} & \multirow{3}{}{CIFAR10\_255} & 726.1 (65, 0) & 784.9 (51, 14) & 820.9 (65, 0) & - & - & - & 2252 (65, 0) \\ & & 15.9 (13, 0) & 48.8 (13, 0) & 274.3 (11, 2) & - & - & - & 173.3 (10, 0) \\ \cline{1-1} \cline{2-9} & & 331.9 (30, 0) & 1170.4 (21, 9) & 380.7 (28, 2) & - & - & 1238.4 (16, 14) & 253.7 (30, 0) \\ \cline{1-1} & & 24.1 (20, 2) & 90.9 (22, 0) & 87.1 (17, 5) & - & - & 1730.7 (12, 10) & 26.9 (21, 1) \\ \cline{1-1} \cline{2-9} & & 940.7 (53, 0) & 813.4 (42, 11) & 668.7 (49, 4) & - & - & - & 349.6 (53, 0) \\ \cline{1-1} & & 9.9 (8, 0) & 33.3 (8, 0) & 97.6 (7, 1) & - & - & - & 9.9 (8, 0) \\ \hline \multirow{2}{}{RESNET\_A} & \multirow{2}{}{3B2\_ADV} & 383.7 (17, 3) & 132.0 (8, 12) & 177.6 (13, 7) & - & - & - & 438.4 (20, 0) \\ & & 14.6 (12, 0) & 47.8 (12, 0) & 14.0 (12, 0) & - & - & - & 27.6 (12, 0) \\ \hline \multirow{2}{}{RESNET\_B} & \multirow{2}{}{3B2\_SSADV} & 678.4 (26, 2) & 405.2 (16, 12) & 302.1 (21, 7) & - & - & - & 335.0 (28, 0) \\ & & 13.5 (11, 0) & 31.4 (11, 0) & 12.5 (11, 0) & - & - & - & 25.6 (11, 0) \\ \hline \hline \end{tabular} \end{table} Table 5. Results for sat and unsat instances. Time are wall-clock and in seconds. Each benchmark has two rows listing runtime (solved, unsolved) results for unsat and sat instances. - means the tool cannot run the benchmark. property 7 (but NeuralSAT solved it). NeuralSAT took around 34s over multiple DPLL(T) iterations to find a counterexample for property 8. In general, the effectiveness of falsification is mainly due to external techniques, and not part of the main algorithm of DNN verification tool (this explains why most modern tools, including NeuralSAT, runs these external techniques initially). Tab. 6 shows the VNN-COMP rankings over sat instances. Here, NeuralSAT ranked first, $\alpha-\beta-$CROWN second, and ERAN third. While all three tools was able to solve similar sat instances (133 and 134), NeuralSAT appears to solve them quicker (104 instances) and therefore had the most points. This is rather surprising as we, just like other tools, mainly rely on external adversarial techniques to generate counterexamples as described in SS4. #### 7.2.2. Unsat instances Unlike sat instances that can be effectively handled by external techniques, unsat instances truly exercise the power of DNN verification algorithms. As shown in Tab. 5, solving unsat instances often take more time and result in more unsolved results. Both nnenum and Marabou'21 were heavily optimized for networks with small input dimensions. These tools performed well for ACAS Xu networks, e.g., nnenum solved all 139 unsat instances in ACAS Xu and while Marabou'21 solved 1 fewer unsat instances, it has the most fastest solved instances for ACAS Xu. However, these tools failed to run on the larger benchmarks CIFAR2020 or RESNET_A/_B. Note that $\alpha-\beta-$CROWN, NeuralSAT, ERAN also perform well in ACAS Xu because they all have heuristics in some way to leverage low dimension inputs. MNISTFC is difficult and has 10 unsat instances that no tool can solve. $\alpha-\beta-$CROWN used a MILP solver in a preprocessing step to tighten the bounds of all hidden neurons. This helps $\alpha-\beta-$CROWN to solve 20 difficult unsat instances on larger MNIST_256x4 and MNIST_256x6 networks that other tools failed. For other benchmarks CIFAR2020 and RESNET_A/B, $\alpha-\beta-$CROWN ranked first and NeuralSAT followed closely (solved 2 fewer unsat CIFAR2020 instances and 5 fewer RESNET). It is worth recalling that $\alpha-\beta-$CROWN has custom run scripts that on average tuned 10 hyperparmeters for each benchmark. Tab. 7 shows the VNN-COMP rankings over unsat instances. Here, $\alpha-\beta-$CROWN ranked first, NeuralSAT second, and ERAN third. Marabou'21 and Marabou'22 ranked last because they fail to run many benchmarks. In summary, for unsat, the overall ranking is consistent with the overall VNN-COMP ranking given in Tab. 4. ### Comparison with DPLL(T)-based DNN verification The state-of-the-art in DPLL(T)-based DNN verification is Marabou. It improves on Reluplex, incorporates abstraction and deduction techniques, and has been entered in VNN-COMP'22 in recent years. This makes it a reasonable point of comparison for NeuralSAT especially in understanding the benefit of the addition of CDCL on the scalability of DNN verification. Overall both versions of Marabou ranked poorly. While Marabou'21 is the best performer for ACAS Xu, which consists of networks with low input dimensions, it does not scale for larger benchmarks and often either cannot handle such networks (CIFAR2020, RESNET_A/B) or ranked last compared to other tools (MNIST). Below we consider the case of smaller and larger networks separately. _Small networks._ ACAS Xu networks are small in two ways. First, they have very few neurons (300). Second, they only have 5 input dimensions. Marabou'21 employs multiple optimizations to target the scale of these networks. For example, a variant of the Split and Conquer algorithm (Wu et al., 2020) subdivides the input space to generate separate verification problems. Partitioning a 5 dimensional input space is one thing, but the number of partitions grows exponentially with input dimension and this approach is not cost effective for the other networks in the benchmarks. Tab. 5 shows how for MNIST networks with larger input dimension (784) the performance of Marabou degrades with increasing number of neurons. As the network sizes increase the number of problems solved by Marabou'21 decreases - from 29, to 16, and then to 10. There is a similar reduction in problems solved, from 30 to 13, for NeuralSAT going from the 512 neuron MNIST network to the 1024 neuron network, but then there is no reduction in problems solved going from the 1024 to the 1536 neuron networks. This is due to the fact that the parameters of the 1536 neuron network define a set of theory constraints that can be efficiently over-approximated. In fact, NeuralSAT only takes a single iteration to prove this property and, consequently, other algorithmic features of NeuralSAT, such as CDCL, do not have a chance to show their benefit. _Larger Networks._ Marabou could not scale to any of the larger CIFAR2020 or RESNET problems, so a direct comparison with NeuralSAT is not possible. Instead, we observe that NeuralSAT performed well on these problems - ranking better than it did on the smaller problems. We conjecture that this is because problems of this scale give ample time for clause learning and CDCL to significantly prune the search performed by DPLL(T). Evidence for this can be observed in data on the learned clauses recorded during runs of NeuralSAT on unsat problems. Since NeuralSAT's propositional encodings have a number of variables proportional to the number of neurons ($n$) in the network the effect of a learned clause of size $c$ is that it has the potential to block a space of assignments of size $2^{n-c}$. In other words, as problems grow the reduction through CDCL grows combinatorially. In the largest problem in the benchmarks, with $n=62464$ we see clauses on average of size $c=16$ which allows BCP to prune an enormous space of assignments - of size $2^{62448}$. The ability of NeuralSAT to scale well beyond other DPLL(T) approaches to DNN verification suggests that other constraint-based DNN verification approaches may want to incorporate CDCL. ## 8. Related Work The literature on DNN verification is rich and is steadily growing (cf. (Liu et al., 2021; Urban and Mine, 2021)). Here we summarize well-known techniques with tool implementations. Constraint-based approaches such as DLV (Huang et al., 2017), Planet (Ehlers, 2017), and Reluplex (Katz et al., 2017) and its successor Marabou (Katz et al., 2022, 2019) transform DNN verification into a constraint problem, solvable using an SMT (Planet, DLV) or DPLL-based search with a customized simplex and MILP solver (Reluplex, Marabou) solvers. Abstraction-based techniques and tools such as AI${}^{2}$(Gehr et al., 2018), ERAN (Muller et al., 2021; Singh et al., 2018, 2019) (DeepZ (Singh et al., 2018), RefineZono (Singh et al., 2018), DeepPoly (Singh et al., 2019), K-ReLU (Singh et al., 2019)), MN-BaB (Ferrari et al., 2022)), Reluval (Wang et al., 2018), Neurify (Wang et al., 2018), VeriNet (Henriksen and Lomuscio, 2020), NNV (Tran et al., 2021), nnenum (Bak, 2021; Bak et al., 2020), CROWN (Zhang et al., 2018), $\alpha$-$\beta$-CROWN (Wang et al., 2021), use abstract domains such as interval (Reluval/Neurify), zonotope (DeepZ, nnenum), polytope (DeepPoly), starset/imagestar (NNV, nnenum) to scale verification. OVAL (OVAL-group, 2023) and DNNV (Shriver et al., 2021) are frameworks employing various existing DNN verification tools. Our NeuralSAT, which is most related to Marabou, is a DPLL(T) approach that integrates clause learning and abstraction in theory solving. Well-known abstract domains for DNN verification include interval, zonotope, polytope, and starset/imagestar. Several top verifiers such as ERAN, MN-BaB and nnenum use multiple abstract domains (e.g., ERAN uses zonotope and polytope, nnenum adopts deeply, zonotope and imagestar. The work in (Goubault et al., 2021) uses the general max-plus abstraction (Heidergott et al., 2006) to represent the non-convex behavior of ReLU. NeuralSAT currently uses polytope in its theory solver though it can also use other abstract domains. Modern SAT solvers benefit from effective heuristics, e.g., VSIDS and DLIS strategies for decision (branching), random restart (Moskewicz et al., 2001) and shortening (Chinneck and Dravnieks, 1991) or deleting clauses (Moskewicz et al., 2001) for memory efficiency and avoiding local maxima caused by greedy strategies. Similarly, modern DNN verifiers such as nnenum, ERAN, $\alpha-\beta-$CROWN, and Marabou include many optimizations to improve performance, e.g., Branch-and-Bound (Bunel et al., 2020) and Split-and-Conquer (Katz et al., 2022, 2019; Wu et al., 2020) for parallelization, and various optimizations for abstraction refinement (Bak, 2021; Singh et al., 2018)) and bound tightening (Bak, 2021; Katz et al., 2019; Wang et al., 2021). NeuralSAT has many opportunities for improvements such as new decision heuristics and parallel DPLL(T) search algorithms (SS9). ## 9. Conclusion and Future Work We introduce NeuralSAT, a DPLL(T) approach and prototype for DNN verification. NeuralSAT includes the standard DPLL components such as clause learning and backtracking and a theory solver for DNN reasoning. We formalize NeuralSAT DPLL(T) using conditional transition rules and evaluate the NeuralSAT prototype with standard FNNs, CNNs, and Resnets. Our evaluation shows that NeuralSAT is competitive to the state-of-the-art DNN verification tools. We have many opportunities to improve the performance of NeuralSAT. For example, we plan to develop a native parallel DPLL(T) approach to exploit multiprocessing and exploring new decision heuristics and backtracking strategy for NeuralSAT (e.g., the first UIP strategy used in the CHAFF solver (Moskewicz et al., 2001) has been shown to create fewer clauses and have more effective backtracking in real-world scenarios(Marques-Silva et al., 2021) By using DPLL with implication graphs, NeuralSAT inherits a native mechanism to verify its own results, e.g., using these graphs and conflicting clauses to obtain resolution graphs/proofs and unsat cores as proofs of unsatisfiability (Asin et al., 2008; Kroening and Strichman, 2016; Zhang and Malik, 2003).
2303.02395	Quasi-normal modes of near-extremal black holes in dRGT massive gravity using Physics-Informed Neural Networks (PINNs)	In this study, we demonstrate the use of physics-informed neural networks (PINNs) for computing the quasinormal modes (QNMs) of black holes in de Rham-Gabadadze-Tolley (dRGT) massive gravity. These modes describe the oscillation frequencies of perturbed black holes and are important in understanding the behavior of these objects. We show that by carefully selecting the hyperparameters of the PINN, including the network architecture and the training data, it is possible to achieve good agreement between the computed QNMs and the approximate analytical formula in the near-extremal limit for the smallest mode number. Our results demonstrate the effectiveness of PINNs for solving inverse problems in the context of QNMs and highlight the potential of these algorithms for providing valuable insights into the behavior of black holes.	Marwan Ait Haddou	2023-03-04T12:05:46Z	http://arxiv.org/abs/2303.02395v2	Quasi-normal modes of near-extremal black holes in dRGT massive gravity using Physics-Informed Neural Networks (PINNs) ###### Abstract In this study, we demonstrate the use of physics-informed neural networks (PINNs) for computing the quasinormal modes (QNMs) of black holes in de Rham-Gabadadze-Tolley (dRGT) massive gravity. These modes describe the oscillation frequencies of perturbed black holes and are important in understanding the behavior of these objects. We show that by carefully selecting the hyperparameters of the PINN, including the network architecture and the training data, it is possible to achieve good agreement between the computed QNMs and the approximate analytical formula in the near-extremal limit for the smallest mode number. Our results demonstrate the effectiveness of PINNs for solving inverse problems in the context of QNMs and highlight the potential of these algorithms for providing valuable insights into the behavior of black holes. ## 1 Introduction Quasinormal modes (QNMs) are complex numbers that describe the oscillation modes of a black hole that has been perturbed. They play a crucial role in our understanding of the behavior of black holes. The real part of QNMs corresponds to the oscillation frequency of the black hole, while the imaginary part corresponds to the damping rate of the oscillation. The calculation of QNMs is a difficult task, as it involves solving a differential equation with appropriate boundary conditions in a highly complex spacetime. In recent years, Physics-informed neural networks (PINNs) have emerged as a machine learning algorithm that excels at tackling complex physics problems. These algorithms are especially useful for solving inverse problems, where the goal is to determine the unknown parameters of a system based on observed data. PINNs stand out due to their unique capability of incorporating physical constraints and laws directly into the neural network architecture. By doing so, they learn the solution to a problem by accounting for the underlying physics, rather than just fitting to a set of data points. PINNs have been increasingly utilized to compute Quasinormal modes (QNMs) of black holes in recent times. When the black hole is near-extremal, the effective potential of the radial wave equation for QNMs can be expressed in terms of the well-known Poschl-Teller potential, for which an exact solution exists. As a result, the QNMs problem can be seen as an inverse problem that can be solved using PINNs. By utilizing the known solution of the Poschl-Teller potential to train a PINN and approximate the QNMs [2], highly precise approximations of the QNMs can be obtained for a broad range of black hole spacetimes. Selecting the right hyperparameters can be a major challenge when utilizing PINNs to calculate QNMs. These parameters, such as the network architecture, activation functions, and training data, can have a significant impact on the performance of the algorithm. Therefore, it is crucial to carefully choose these parameters in order to achieve favorable outcomes. In conclusion, PINNs have emerged as a promising method for calculating QNMs of black holes and can offer valuable insights into the behavior of these entities. With the progress of the machine learning field, PINNs are expected to play an increasingly vital role in tackling complex problems in astrophysics and other areas of physics. ## 2 Computing QNMs with Physics-Informed Neural Networks Deep learning is a powerful machine learning technique that uses input data to generate predictions. This method identifies patterns in the data and utilizes them to minimize the difference between the predicted and actual output, while also enabling generalization to new inputs. One of the most popular types of deep learning algorithms is neural networks, which analyze the correlation between input and output data. A neural network is composed of various layers, including an input layer, hidden layers, and an output layer. The input layer collects data, the hidden layers adjust weights and parameters to recognize patterns within the data, and the output layer generates predictions based on these patterns. Physics-informed neural networks (PINNs) are a specific type of deep learning algorithm that incorporates the physical principles of the problem, which are often represented by nonlinear partial differential equations. In 2017, Raissi et al. were the first to suggest using PINNs to solve such equations. To compute quasinormal frequencies (QNFs), we can use the DeepXDE library, which is designed for PINN applications. DeepXDE simplifies the process by providing built-in modules for the computational domain, PDE equations, boundary and initial conditions, constraints, training data, neural network architecture, and training hyperparameters. The steps to solve differential equations using DeepXDE are as follows: ``` 1:The physical equations that drive the problem are all set and passed to deepxde.data...(...) as parameters. 2:Construct a neural network using the deepxde.nn...(...) module. 3:The PINN model is a combination of the neural network and physical constraints defined as parameters to deepxde.Model(...). 4:The Model.Compile(...) function is used to define training parameters such as the specific choice for the optimiser. 5:The PINN model can then be run using Model.train(...) for a defined number of training epochs. 6:Call Model.predict(...) to predict the solution. ``` Algorithm 1 The process for building PINN models in DeepXDE ## 3 Quasi-normal modes of scalar perturbations in a pure de Sitter space ### System Setup The static metric of a four-dimensional de Sitter (dS) space is given by: \[ds^{2}=-f(r)dt^{2}+f^{-1}(r)^{2}+r^{2}d\Omega^{2} \tag{1}\] Here, $f(r)=1-r^{2}/z^{2}$, where $z$ is the minimal radius of the dS space, and $r^{2}d\Omega^{2}$ represents the metric on the two-dimensional sphere $S^{2}$ of radius $r$. For a massive scalar field $\Psi$ that satisfies the Klein-Gordon equation: \[\Psi^{:\nu};\nu=m\Psi \tag{2}\] $\Psi$ can be separated as: \[\Psi=\frac{u}{r}e^{-\omega t}Yl(\Omega)\] where $Y$ is the spherical harmonic function. By using the tortoise coordinate $dx=dr/f(r)=ztanh^{-1}(r/z)$, we can express the radial part in a Schrodinger-like equation: \[\frac{d^{2}u}{dx^{2}}+(\omega^{2}-V(x))u=0 \tag{3}\] where the effective potential is given by: \[V(x)=-\frac{2-m^{2}z^{2}}{z^{2}cosh^{2}(x/z)}+\frac{l(l+1)}{z^{2}sinh^{2}(x/z)} \tag{4}\] By introducing a new variable $a=1/cosh^{2}(x/l)$, equation (3) can be written as: \[a(1-a)u^{{}^{\prime\prime}}+(1-\frac{3}{2}a)u^{{}^{\prime}}+\frac{1}{4}(\frac {\omega^{2}z}{a}-\frac{l(l+1)}{1-a}+2-m^{2}z^{2})u=0 \tag{5}\] The exact solution of this equation is given by: \[u(a)\propto a\frac{-i\omega l}{2}(1-a)\frac{l+1}{2}{}_{2}F_{1}(b-c+1,d-c+1,2-c,a) \tag{6}\] Figure 1: The PINNs algorithm in DeepXDE [1] where: \[b =\alpha+\beta+\frac{1}{4}(1+\sqrt{1+4(2-m^{2}z^{2})}),\] \[c =2\alpha+1,\] \[b =\alpha+\beta+\frac{1}{4}(1+\sqrt{1-4(2-m^{2}z^{2})}),\] \[\alpha =-j\omega z/2\] The corresponding quasi-normal frequencies are: \[\omega=-j\frac{1}{2}(2n+l+\frac{3}{2}) \tag{7}\] The results were generated using the following hyperparameters: two fully connected neural networks ($PFNN$) with a depth of 4 (i.e., 3 hidden layers) and the structure $[1,[20,20],[20,20],[20,20],2]$ were used. The non-linear activation function $tanh$ was employed, along with $Adam$ optimisers with a learning rate of 0.0001 and 160,000 training epochs. The training data consisted of 250 domain points, and the dataset used for training contained 250 actual values of the Quasi-normal Modes (QNMs) ($\Psi(y)$) uniformly distributed in the domain [1.1, 2.5] for x. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|c\|} \hline (n, l) & Ref [4] & FNN [5] & PINN & PE (FNN) [5] & PE (PINN) \\ \hline (0, 0) & 0 - 3000j & 0 - 3017j & 0.0 - 2997.10j & 0.56 & 0.0965 \\ \hline (0, 1) & 0 - 4000j & 0 - 3994j & 0.0 - 3998.61j & 0.15 & 0.0346 \\ \hline (1, 1) & 0 - 6000j & 0 - 6012j & 0.0 - 5999.67j & 0.2 & 0.0054 \\ \hline (0, 2) & 0 - 5000j & 0 - 5024j & 0.0 - 5000.22j & 0.48 & 0.0045 \\ \hline (1, 2) & 0 - 7000j & 0 - 7019j & 0.0 - 6999.14j & 0.27 & 0.0122 \\ \hline (2, 2) & 0 - 9000j & 0 - 9037j & 0.0 - 8999.58j & 0.41 & 0.0046 \\ \hline (0, 3) & 0 - 6000j & 0 - 6072j & 0.0 - 5999.45j & 1.2 & 0.0090 \\ \hline (1, 3) & 0 - 8000j & 0 - 8056j & 0.0 - 8000.19j & 0.7 & 0.0024 \\ \hline (2, 3) & 0 - 10000j & 0 - 10094j & 0.0 - 9998.26j & 0.94 & 0.0173 \\ \hline (3, 3) & 0 - 12000j & 0 - 12103j & 0.0 - 11995.40j & 0.85 & 0.0382 \\ \hline \end{tabular} \end{table} Table 1: Comparison of PINN QNMs with FNN QNMs, for the AdS spacetime for m = 0, z = 0.001. PE represents percentual error. Quasi-normal modes of near-extremal black holes in dRGT massive gravity ### Perturbation equations and Quasi-normal modes The study of black hole perturbations is crucial in understanding their behavior and properties. These perturbations are small deviations from the background metric and can be described by linearized equations. For the dRGT black hole, the perturbations can be decomposed into scalar, vector and tensor modes. Here, we will focus on the scalar perturbations. The perturbed metric function $h(r,t)$ can be written as: \[ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})+ r^{2}h(r,t)dt^{2}. \tag{8}\] The perturbations can be further decomposed into time and spatial parts as $h(r,t)=H(r)e^{-i\omega t}$, where $\omega$ is the frequency of the perturbation. Substituting this into the linearized Einstein equations and solving for $H(r)$ gives the following differential equation: \[\frac{d^{2}H}{dr_{}^{2}}+(\omega^{2}-V(r))H=0, \tag{9}\] where $r_{}$ is the tortoise coordinate defined as : \[dr_{}=\frac{dr}{f(r)}\] And the effective potential $V(r)$ is given by: \[V(r)=f(r)\left[\frac{l(l+1)}{r^{2}}+\frac{f^{\prime}(r)}{r}\right], \tag{10}\] where $l$ is the angular momentum quantum number. The quasi-normal modes (QNMs) of black holes are the solutions to the differential equation 9 with appropriate boundary conditions. These QNMs are complex frequencies, with the real part representing the oscillation frequency of the perturbation, while the imaginary part represents the damping rate. In our study, we use the numerical shooting method to find the QNMs of the dRGT black hole. We solve the differential equation 9 numerically from the horizon towards infinity, and match the solution with an outgoing wave at infinity. The QNMs are then obtained by finding the complex frequencies that satisfy the boundary conditions at both the horizon and infinity. The metric $f(r)$ of a near-extremal black hole with $r_{h}\sim r_{c}$ can be written in the tortoise coordinate $r_{}$ as follows [3]: \[f(r_{})\sim\frac{r_{h}k_{h}}{(1+\zeta+\gamma r_{h}-2Q^{2}/r_{h}^{2})\cosh^{2} \left(k_{h}r_{}\right)}, \tag{11}\] where $k_{h}$ is the surface gravity, $\zeta$ and $\gamma$ are dimensionless parameters, and $Q$ is the electric charge of the black hole. The wave equation for a neutral scalar field in a near-extremal black hole is given by the Klein-Gordon equation: \[\Box\Phi=0. \tag{12}\] The scalar field $\Phi$ can be expressed as: \[\Phi=\sum_{l}^{\infty}\sum_{m=0}^{l}e^{-i\omega t}\frac{\psi(r)}{r}Y_{lm}( \theta,\phi),\] where $Y_{lm}(\theta,\phi)$ are the spherical harmonics, $l$ is the azimuthal quantum number, and $\omega$ is the frequency. The radial wave equation for $\psi(r)$ is then given by: \[\frac{d^{2}\psi}{dr_{}^{2}}+[\omega^{2}-V(r)]\psi=0, \tag{13}\] where $V(r)$ is the potential defined as: \[V(r)=f(r)(m_{s}^{2}+\frac{l(l+1)}{r^{2}}+\frac{f^{\prime}(r)}{r}), \tag{14}\] with $m_{s}$ being the mass of the scalar field. In the near-extremal limit, the Klein-Gordon radial equation (13) becomes: \[\frac{d^{2}\psi}{dr^{2}}+[\omega^{2}-\frac{V_{0}}{cosh^{2}(\kappa_{h}r)}]\psi=0, \tag{15}\] where $V_{0}=\frac{\kappa_{h}^{2}}{1+\zeta+\gamma r_{h}-2Q^{2}/r^{2}}(m_{s}^{2}r^{2}+ l(l+1))$, and $\kappa_{h}$ is the surface gravity. The potential $V(r)$ is the well-known Poschl-Teller potential [6]. Applying the boundary condition for quasi-normal modes, the following quasi-normal frequencies can be obtained [2]: \[\omega_{n}=\sqrt{V_{0}-\frac{\kappa_{h}^{2}}{4}}-i\kappa_{h}(n+\frac{1}{2}), \tag{16}\] where $n$ is a non-negative integer. The associated quasi-normal modes are given by [2]: \[\psi=(D(D-1))^{-i\omega/2k_{h}}._{2}F_{1}(1/2+B-\frac{i\omega}{k_{h}},1/2-B- \frac{i\omega}{k_{h}};1-\frac{i\omega}{k_{h}};D) \tag{17}\] where : $D=\frac{1}{1+e^{-2k_{h}r_{}}}$ and $B=\sqrt{\frac{1}{4}-\frac{V_{0}}{k_{h}^{2}}}$ ### Results To address the infinite problem domain, we introduced a new coordinate $y=tanh(k_{h}r_{})$, which enabled us to create a finite domain of $(-1,1)$ for easier implementation in the code. Using $y$ as a function, we expressed the perturbation equations (15) for the near-extremal dRGT black holes in terms of $y$ as follows: Figure 2: The metric function $f(r)$ profil for $M=1$, $\Lambda=5.0001$, $\gamma=0.05$, $\zeta=2.51465$, $Q=0$. Event horizon $r_{h}=0.8460$, cosmological horizon $r_{c}=0.8509$ \[\kappa_{h}^{2}\left(1-y^{2}\right)^{2}\cdot\frac{d^{2}\Psi(y)}{dy^{2}}-2\kappa_{h}^ {2}y\left(1-y^{2}\right)\cdot\frac{d\Psi(y)}{dy}+\left[\omega^{2}-V_{0}\left(1- y^{2}\right)\right]\Psi(y)=0 \tag{18}\] To facilitate implementation in the DeepXDE package, we separated equations 18 into real and imaginary parts. We generated results using two fully connected neural networks ($PFNN$) of depth 4 (i.e., 3 hidden layers) with the following structure $[1,[36,36],[20,20],[20,20],2]$; $swish$ as a non-linear activation function; $L-BFGS-B$ and $Adam$ optimisers with learning rate = 0.0001 and loss weights = $[0.01,0.01,0.001,0.001,100,100,100,100]$; 150,000 training epochs; training data of 100 domain points, and a dataset of 100 uniformly distributed actual values of the QNMs ($\Psi(y)$) in the domain [-0,7,0,7]. Table 2 and Table 3 present the PINN approximations of the QNFs for massless scalar perturbations of dRGT neutral black holes, with $l=1$ and $n=0$, respectively. Based on the table, it appears that for $n=0$, the PINN approximation and the QNFs formula result in the same value of $0.0-0.0j$. For $n=1$ to $n=7$, the relative errors are less than 1%, indicating a good approximation. However, for $n=8$, the relative error is quite large, at 3.1830 %, indicating a poorer approximation. Overall, the PINN approximation seems to provide reasonable results for the QNFs of dRGT neutral black holes with massless scalar perturbations, for $l=1$ and $n$ values up to 7. Table 3 presents the results of PINN approximations for massless scalar perturbations of dRGT neutral black holes, specifically for $n=0$ and different values of $l$. The table lists the PINN approximations, the corresponding results obtained using a specific formula (Formula (16)), and the relative error between the two. The parameters used for the calculations are also listed. The results show that for $l=0$ and $l=1$, the PINN approximations are in agreement with the formula. For higher values of $l$, the relative errors are very small, indicating the high accuracy of the PINN method. The table also shows that the relative error decreases as $l$ increases, and the last entry for $l=50$ has a relative error of only 0.0018230%. Overall, the results demonstrate the effectiveness of the PINN method for approximating QNFs of black holes. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|} \hline \multicolumn{4}{\|c\|}{$l=1$} \\ \hline $n$ & PINN & Formula (16) & relative error \\ \hline 0 & 0.000031 - 0.000003j & 0.0 - 0.0j &... \\ \hline 1 & 0.0 - 0.012175j & 0.0 - 0.012175j & 0.0002072\% \\ \hline 2 & 0.0 - 0.024352j & 0.0 - 0.024352j & 0.001769\% \\ \hline 3 & 0.0 - 0.036527j & 0.0 - 0.036527j & 0.004005\% \\ \hline 4 & 0.0 - 0.048703j & 0.0 - 0.048703j & 0.00032755 \% \\ \hline 5 & 0.0 - 0.060879j & 0.0 - 0.060879j & 0.0001240\% \\ \hline 6 & 0.0 - 0.073055j & 0.0 - 0.073055j & 0.0004933\% \\ \hline 7 & 0.0 - 0.085233j & 0.0 - 0.085231j & 0.002220\% \\ \hline 8 & 0.0 - 0.097407j & 0.0 - 0.097407j & 0.0\% \\ \hline \end{tabular} \end{table} Table 2: The PINN approximations results of the QNFs for massless scalar perturbations of dRGT neutral black holes, for $l=1$ and different value of $n$, $M=1$, $\Lambda=5.0001$, $\gamma=0.05$, $\zeta=2.51465$, $Q=0$, $m_{s}=0$, $r_{h}=0.8460$, $r_{c}=0.8509$, $k_{h}=0.012175$ ## 5 Discussion In this study, we have employed Physics-Informed Neural Networks (PINNs) to compute the Quasinormal modes (QNMs) of scalar perturbations for the neutral dRGT black holes. In the process, we have introduced a new coordinate $y=tanh(k_{h}r_{})$ that allowed us to convert the infinite domain of $r_{}$ to a finite domain of $y\in(-1,1)$. The use of PINNs allowed us to avoid the need for the boundary conditions, which are not available in the near-extremal black hole cases. The results presented in Tables 2 and 3 indicate that the PINN approximations of the QNFs are in excellent agreement with the known analytical solutions. The relative errors between the PINN approximations and the analytical solutions were found to be less than $0.01\%$. It is interesting to note that the PINN approximations improved with an increase in the overtone number ($n$), with the highest relative error being less than $0.002\%$ for $n=8$. These results confirm the accuracy and efficiency of the PINN method in computing the QNMs of dRGT black holes. It is worth mentioning that the PINN method does not require any prior knowledge of the analytical solutions, and it can be used to compute QNMs for black holes with different parameters or geometries. This makes the method particularly useful in situations where analytical solutions are not available or difficult to obtain. The method can also be used to compute QNMs for other types of perturbations, such as electromagnetic and gravitational waves, and for black holes with charges and/or angular momenta. In conclusion, we have demonstrated that the PINN method is an accurate and efficient tool for computing the QNMs of scalar perturbations for neutral dRGT black holes. The method can be extended to other types of perturbations and geometries, making it a valuable tool for studying the properties of black holes in general relativity and beyond. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|} \hline $n$ & $l$ & PINN Approximation & Formula (16) \\ \hline 0 & 0 & $0.000031-0.000003j$ & $0.0-0.0j$ \\ \hline & 1 & $0.000031-0.000003j$ & $0.0-0.0j$ \\ \hline & 2 & $0.006804-0.006088j$ & $0.006804-0.006088j$ \\ \hline & 3 & $0.014593-0.006088j$ & $0.014599-0.006088j$ \\ \hline & 4 & $0.021519-0.006088j$ & $0.021519-0.006088j$ \\ \hline & 5 & $0.028222-0.006088j$ & $0.028223-0.006088j$ \\ \hline & 6 & $0.034832-0.006088j$ & $0.034833-0.006088j$ \\ \hline & 7 & $0.041394-0.006088j$ & $0.041394-0.006088j$ \\ \hline & \multicolumn{3}{c\|}{...} \\ \hline 0 & 30 & $0.190327-0.006088j$ & $0.190327-0.006088j$ \\ \cline{2-4} & 50 & $0.319495-0.006082j$ & $0.319498-0.006088j$ \\ \hline \end{tabular} \end{table} Table 3: The PINN approximations results of the QNFs for massless scalar perturbations of dRGT neutral black holes, for $n=0$ and different values of $l$, $M=1$, $\Lambda=5.0001$, $\gamma=0.05$, $\zeta=2.51465$, $Q=0$, $m_{s}=0$, $r_{h}=0.8460$, and $r_{c}=0.8509$. The relative error is also provided. Conclusion In conclusion, we have presented a deep learning approach for computing the quasinormal frequencies of massless scalar perturbations in the background of the dRGT black holes. By expressing the perturbation equations in terms of a new coordinate, we were able to transform the infinite problem domain into a finite one, making it more amenable to implementation in the code. Using two fully connected neural networks, we obtained highly accurate approximations of the QNFs with relative errors ranging from 0.0% to 0.001769%. Our results demonstrate that deep learning techniques can be effectively used to calculate the quasinormal frequencies of black holes, providing a promising avenue for future research in this field. The proposed method can be extended to other types of black holes and other perturbation fields, potentially offering a more efficient way to compute QNFs in astrophysical and gravitational wave studies. Important links* DeepXDE : [https://github.com/lululxvi/deepxde](https://github.com/lululxvi/deepxde)
2308.11925	Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System	In this work, we investigate a neural network based solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order optimality system of the optimal control problem, and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme, and provide $L^2(\Omega)$ error bounds on the state, control and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.	Yongcheng Dai, Bangti Jin, Ramesh Sau, Zhi Zhou	2023-08-23T05:18:19Z	http://arxiv.org/abs/2308.11925v2	# Solving Elliptic Optimal Control Problems using Physics Informed Neural Networks+ ###### Abstract In this work, we present and analyze a numerical solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. The approach is based on a coupled system derived from the first-order optimality system of the optimal control problem, and applies physics informed neural networks (PINNs) to solve the coupled system. We present an error analysis of the numerical scheme, and provide $L^{2}(\Omega)$ error bounds on the state, control and adjoint state in terms of deep neural network parameters (e.g., depth, width, and parameter bounds) and the number of sampling points in the domain and on the boundary. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the approach and compare it with three existing approaches. Key words: optimal control, physics informed neural network, error estimate, elliptic problem ## 1 Introduction Optimal control problems subject to partial differential equations (PDE) constraints represent a very important class of problems in practice and have found various important applications in science and engineering, e.g., fluid flow, heat conduction, structural optimization, microelectronics, crystal growth, vascular surgery, and cardiac medicine, to name just a few. The mathematical theory of distributed / boundary optimal control problems subject to linear / nonlinear elliptic / parabolic PDEs is well understood (see, e.g., the monographs [27, 39, 30]). Numerically, one of the most common approaches to solve such problems utilizes the first-order optimality system and introduces either adjoint state or Lagrangian multiplier, and then iteratively updates the control using established optimization algorithms, e.g., (accelerated / stochastic) gradient descent method, semismooth Newton method, and alternating direction method of multipliers. In practical implementation, the resulting continuous formulation is then often discretized by finite element method (FEM) and spectral methods. In the past few years, deep neural networks (DNNs) have demonstrated remarkable performance across a wide range of applications, e.g., computer vision, imaging, and natural language processing. These successes have permeated into several scientific disciplines, and DNNs have also become very popular tools to approximate solutions to various PDEs. The ease of implementation, and the potential to overcome the notorious curse of dimensionality have sparked intensive interest in revisiting classical computational problems from a deep-learning perspective. In the last few years, a large number of neural solvers based on DNNs have been proposed for solving PDEs, e.g., physics informed neural network (PINN) [34], deep Ritz method [14], deep Galerkin method [36], weak adversarial networks [41], and deep least-squares method [9]. These methods employ DNNs as ansatz functions to approximate the PDE solution directly. Among these neural solvers, PINN is undoubtedly the most prominent one, which is based on PDE residual minimization, penalty method for enforcing the boundary condition and computing the derivatives with automatic differentiation. The approach has been successfully applied to a wide range of scientific and engineering problems (see the reviews [23, 8]). See Section 2.2 for a more detailed description of PINN. These impressive developments on neural PDE solvers have also sparked interest in solving optimal control problems governed by PDE constraint with deep learning techniques, and several different methods have been proposed. One first class of methods is based on the classical penalty methods, including standard penalty method (soft constraint) [32], and augmented Lagrangian method [28]. The idea of the penalty based methods is to transform the PDE-constrained optimization problem into an unconstrained one by introducing an augmented loss function that incorporates the PDE residual with a suitable penalty parameter. The idea is simple yet quite effective. The main drawback of penalty-based methods is that the optimization problem becomes increasingly ill-conditioned (and hence more challenging to optimize) as the penalty weights become large [5, 19, 28], but imposing the PDE constraint accurately does require a large penalty. Moreover, in the presence of multiple constraints, one has to tune multiple penalty weights, which is crucial to yield reasonable results yet the tuning can be highly nontrivial, time-consuming and in fact very tedious. The augmented Lagrangian method partly alleviates the necessity of using a large penalty weight [19]. The second class of methods employs the adjoint technique [10], which solves the adjoint state equation and computes the derivative of the reduced objective with respect to the control variable. One method in the class is the direct adjoint looping (DAL) [27, 20], which iteratively updates the control using a gradient descent scheme computed via the adjoint system. Using the idea of DAL, very recently, Yin et al [40] proposed the adjoint-oriented neural network (AONN). It can be viewed as a neural network realization of the adjoint-based gradient descent scheme. It involves solving the state and adjoint and then updates the control sequentially, and thus consists of inner and outer loops. The authors reported excellent empirical performance on a number of challenging settings, e.g., (parametric) Navier-Stokes problem. See Section 3.4 for further descriptions of these methods for elliptic optimal control problems. Demo et al [12] proposed an extended PINN, which augments DNN inputs with parameters so that the Karush-Kuhn-Tucker (KKT) system and neural networks are combined for parametric optimal control problems; see also [2]. Recently, Song et al [38] develop a PINN and ADMM based approach for efficiently solving nonsmooth PDE constrained optimization. In this work, we investigate a class of numerical solvers based on DNNs for solving elliptic optimal control problems using PINN, which is termed coupled Physics Informed Neural Network (C-PINN in short). The solver is derived as follows. First, we derive a coupled system (i.e., KKT system) for the continuous optimal control problem, which consists of a state equation, an adjoint state equation, and an equality / variational inequality for the unconstrained and constrained cases, respectively. The variational equality allows recovering the optimal control from the optimal adjoint, and the variational inequality allows realizing the optimal control as a min-max projection of the optimal adjoint onto the constrained set. Therefore, the KKT system is equivalent to a (reduced) coupled system for the state and adjoint, which consists of a state equation with a min-max projection operator in the source, and an adjoint state equation. Second, we construct a (empirical) PINN loss for the (reduced) coupled system, which consists of the PDE residuals and boundary penalties. The method is illustrated for the unconstrained case, constrained case, and semilinear case. Third, we give an error analysis of the approach for linear elliptic optimal control problems, and derive an $L^{2}(\Omega)$ error estimate for the control, state, and adjoint state approximations. The error bounds can be decomposed into two parts: approximation and statistical errors. The approximation error arises from approximating the solution of the control problem using neural networks and is bounded using approximation theory of DNNs with general activation functions [15], and the statistical error arises from using Monte Carlo methods to approximate the (potentially) high-dimensional integrals and is bounded using offset Rademacher complexity [26, 13] and certain technical estimates on the DNN functions. Fourth and last, we present a set of numerical experiments in two- and multi-dimensions, and give a comparative study with existing neural solvers for optimal control problems. The derivation of the new computational technique based on the coupled system and PINN, its error analysis and the numerical evaluation represent the main contributions of the work. To the best of our knowledge, the coupled PINN represents the first DNN based solver with convergence guarantees for elliptic optimal control problems. The rest of the paper is organized as follows. In Section 2, we describe some preliminaries on neural networks and PINN. In section 3, we describe elliptic optimal control problems via PINN based on the coupled system reformulation of the optimal control problems. In Section 4, we present an error analysis for both constrained and unconstrained cases, which includes a detailed analysis of the approximation error and statistical errors, which in turn relies on suitable technical estimates on DNNs in Section 5. Finally, in Section 6, we present numerical examples to validate the theoretical results. Throughout the notation $c$ denotes a generic constant which may change at each occurrence, but it is always independent of the neural network parameters. ## 2 Preliminaries In this section we review preliminaries on deep neural networks and physics informed neural networks. ### Deep neural networks Due to their extraordinary expressivity for approximating functions, deep neural networks (DNNs) have been widely used, and play a vital role in the great success of deep learning. In this work, we consider fully connected feedforward neural networks. Fix $L\in\mathbb{N}$, and integers $\{n_{\ell}\}_{\ell=0}^{L}\subset\mathbb{N}$, with $n_{0}=d$ and $n_{L}=1$. A neural network $f_{\theta}:\mathbb{R}^{d}\to\mathbb{R}$, with $\theta\in\mathbb{R}^{N_{\theta}}$ being the DNN parameters, is defined recursively by \[f^{(0)}(x) =x,\] \[f^{(\ell)}(x) =\rho(A^{(\ell)}f^{(\ell-1)}+b^{(\ell)}),\quad\ell=1,\dots,L-1,\] \[f_{\theta}:=f^{(L)}(x) =A^{(L)}f^{(\ell-1)}+b^{(L)},\] where $A^{(\ell)}=[a_{ij}^{(\ell)}]\in\mathbb{R}^{n_{\ell}\times n_{\ell-1}}$, and $b^{(\ell)}=(b_{i}^{(\ell)})\in\mathbb{R}^{n_{\ell}}$ are known as the weight matrix and bias vector at the $\ell$th layer, respectively. $L$ is called the depth of the neural network, and $W:=\max(n_{\ell},\ell=1,\dots,L)$ is called the width of the neural network. We denote by $\boldsymbol{n}_{i}$, $i=1,\dots,L$, as the number of nonzero weights on the first $i$ layers. Then $\boldsymbol{n}_{L}$ is the total number of nonzero weights. We use the notation $\mathcal{N}_{\rho}(L,\boldsymbol{n}_{L},R)$ to refer to the collection of DNN functions generated by $\rho$ of depth $L$, total number $\boldsymbol{n}_{L}$ of nonzero weights and each weight bounded by $R$. Throughout, we focus on two activation functions $\rho$: hyperbolic tangent $\rho(t)=\frac{e^{t}-e^{-t}}{e^{t}+e^{-T}}$ and sigmoid $\rho(t)=\frac{1}{1+e^{-t}}$, which are popular in the literature on PINN. Lemma 2.1.: _The following estimates hold._ 1. _If_ $\rho(t)=\tanh(t)$_, then_ $\\|\rho^{(i)}\\|_{L^{\infty}(\mathbb{R})}\leq 1$_,_ $i=0,1,2$_, and_ $\\|\rho^{\prime\prime\prime}\\|_{L^{\infty}(\mathbb{R})}\leq 2$_._ 2. _If_ $\rho(t)=\frac{1}{1+e^{-t}}$_, then_ $\\|\rho^{(i)}\\|_{L^{\infty}(\Omega)}\leq 1$_,_ $i=0,1,2,3$_._ Proof.: The lemma follows from direct computation. Indeed, for $\rho(t)=\tanh(t)$, we have $\\|\rho\\|_{L^{\infty}(\mathbb{R})}\leq 1$ and $\rho^{\prime}(t)=1-\rho(t)^{2}$, $\rho^{\prime\prime}(t)=-2\rho(t)(1-\rho(t)^{2})$, $\rho^{\prime\prime\prime}(t)=(6\rho(t)^{2}-2)(1-\rho(t)^{2})$. Similarly, for $\rho(t)=\frac{1}{1+e^{-t}}$, clearly $\rho(t)\in(0,1)$, and $\rho^{\prime}(t)=\frac{e^{-t}}{(1+e^{-t})^{2}}=\rho(t)(1-\rho(t))\in(0,\frac{ 1}{2})$, $\rho^{\prime\prime}(t)=\rho(t)(1-\rho(t))(1-2\rho(t))$, $\rho^{\prime\prime\prime}(t)=\rho(t)(1-\rho(t))[1-6\rho(t)+6\rho(t)^{2}]$. Then the desired assertions follow immediately. ### Physics informed neural networks Now we describe physics informed neural networks (PINNs) of Rassi et al [34], for solving the elliptic boundary value problem \[\begin{cases}-\Delta y=f,&\text{in }\Omega,\\ \quad y=0,&\text{on }\partial\Omega,\end{cases} \tag{2.1}\] where $\Omega\subset\mathbb{R}^{d}$ is a smooth domain with a boundary $\partial\Omega$, and $f\in L^{2}(\Omega)$ is the source. PINN is based on the principle of PDE residual minimization. For problem (2.1), the continuous loss $\mathcal{L}(u)$ is given by \[\mathcal{L}(y)=\frac{1}{2}\\|\Delta y+f\\|_{L^{2}(\Omega)}^{2}+\frac{\alpha}{2} \\|y\\|_{L^{2}(\partial\Omega)}^{2}, \tag{2.2}\] where the penalty term $\\|y\\|_{L^{2}(\partial\Omega)}^{2}$ (with weight $\alpha>0$) is to approximately enforce the zero Dirichlet boundary condition. We seek an approximation $y_{\theta}\in\mathcal{Y}=\mathcal{N}_{\rho}(L,\mathbf{n}_{L},R)$, and then discretize relevant integrals using Monte Carlo method. Let $U(D)$ be the uniform distribution over a set $D$, and $\|D\|$ the Lebesgue measure of $D$, and $\mathbb{E}_{\nu}$ taking expectation with respect to a probability distribution $\nu$. Then the loss $\mathcal{L}(y_{\theta})$ is given by \[\mathcal{L}(y_{\theta})=\frac{1}{2}\|\Omega\|\mathbb{E}_{U(\Omega)}[(\Delta y_{ \theta}(X)+f(X))^{2}]+\frac{\alpha}{2}\|\partial\Omega\|\mathbb{E}_{U(\partial \Omega)}[(y_{\theta}(Y))^{2}],\] and hence also known as population loss. Let the sampling points $\{X_{i}\}_{i=1}^{n_{d}}$ and $\{Y_{j}\}_{j=1}^{n_{b}}$ be identically and independently distributed (i.i.d.), uniformly on the domain $\Omega$ and boundary $\partial\Omega$, respectively, i.e., $\mathbb{X}=\{X_{i}\}_{i=1}^{n_{d}}\sim U(\Omega)$ and $\mathbb{Y}=\{Y_{j}\}_{j=1}^{n_{b}}\sim U(\partial\Omega)$. Then the empirical loss $\widehat{\mathcal{L}}(y_{\theta})$ is given by \[\widehat{\mathcal{L}}(y_{\theta})=\frac{\|\Omega\|}{2n_{d}}\sum_{i=1}^{n_{d}}( \Delta y_{\theta}(X_{i})+f(X_{i}))^{2}+\frac{\alpha\|\partial\Omega\|}{2n_{b}} \sum_{j=1}^{n_{b}}(y_{\theta}(Y_{j}))^{2}.\] Note that the resulting optimization problem is well-posed due to the box constraint on the DNN parameters $\theta$, i.e., $\|\theta\|_{\ell^{\infty}}\leq R$ for suitable $R$, which induces a compact set in $\mathbb{R}^{N_{\theta}}$. Meanwhile the empirical loss $\widehat{\mathcal{L}}(y_{\theta})$ is continuous in $\theta$, when the activation function $\rho$ is smooth. In the absence of the box constraint, the optimization problem might not have a finite minimizer, due to the lack of weak closedness of the DNN function class [33]. The theoretical analysis of the PINNs has been carried out in [21, 35, 31, 29]. ## 3 Elliptic optimal control via PINN In this section, we develop the PINN for solving elliptic optimal control problems via the coupled optimality system, and term the resulting method as C-PINN. We also survey three existing neural solvers in the context of elliptic optimal control problems. ### Optimal control without control constraint Let $\Omega\subset(-1,1)^{d}\subset\mathbb{R}^{d}$ ($d\geq 1$) be a smooth domain with a boundary $\partial\Omega$. Consider the following distributed optimal control problem \[\min\;J(y,u):=\frac{1}{2}\\|y-y_{d}\\|_{L^{2}(\Omega)}^{2}+\frac{\lambda}{2}\\|u \\|_{L^{2}(\Omega)}^{2}, \tag{3.1}\] subject to the following elliptic PDE constraint \[\begin{cases}-\Delta y=f+u,&\text{in }\Omega,\\ y=0,&\text{on }\partial\Omega,\end{cases} \tag{3.2}\] where $u\in L^{2}(\Omega)$ is the control, $f\in L^{2}(\Omega)$ and $y_{d}\in L^{2}(\Omega)$ are the given source and target function, respectively, and the scalar $\lambda>0$ balances the two terms in the objective $J(y,u)$. It is known that the optimal control problem (3.1)-(3.2) has a unique solution $(\bar{y},\bar{u})$[39, Theorem 2.17]. The main goal is to approximate the solution $(\bar{y},\bar{u})$ by a DNN pair $(\bar{y}_{\theta},\bar{u}_{\kappa})$ via PINN. It is known that problem (3.1) is equivalent to the following Karush-Kuhn-Tucker (KTT) system for the optimal tuple $(\bar{y},\bar{p},\bar{u})$ (see, e.g., [27, Chapter 2, Theorem 1.4]): \[\begin{cases}-\Delta y=f+u,&\text{in }\Omega,\\ y=0,&\text{on }\partial\Omega,\\ -\Delta p=y-y_{d},&\text{in }\Omega,\\ p=0,&\text{on }\partial\Omega,\\ u=-\lambda^{-1}p,&\text{in }\Omega.\end{cases} \tag{3.3}\] The last line is the first-order necessary optimality condition of problem (3.1)-(3.2). The starting point of the coupled approach is system (3.3). Upon eliminating the first-order necessary condition $u=\lambda^{-1}p$, problem (3.1)-(3.2) amounts to solving the following reduced system in $y$ and $p$: \[\begin{cases}-\Delta y=f-\lambda^{-1}p,&\text{in }\Omega,\\ y=0&\text{on }\partial\Omega,\\ -\Delta p=y-y_{d}&\text{in }\Omega,\\ p=0&\text{on }\partial\Omega.\end{cases} \tag{3.4}\] In view of the (reduced) coupled system (3.4), we define the following continuous loss \[\mathcal{L}(y,p)=\left\\|\Delta y+f-\lambda^{-1}p\right\\|_{L^{2}(\Omega)}^{2}+ \alpha_{i}\left\\|\Delta p+y-y_{d}\right\\|_{L^{2}(\Omega)}^{2}+\alpha_{b}^{y} \left\\|y\right\\|_{L^{2}(\partial\Omega)}^{2}+\alpha_{b}^{p}\left\\|p\right\\|_ {L^{2}(\partial\Omega)}^{2}, \tag{3.5}\] where the positive scalars $\alpha_{i},\alpha_{b}^{y}$ and $\alpha_{b}^{p}$ are penalty parameters to enforce the PDE residual (of $p$) and zero Dirichlet boundary conditions of $y$ and $p$. Below we denote by $\mathbf{\alpha}=(\alpha_{i},\alpha_{b}^{y},\alpha_{b}^{p})$. Let $U(\Omega)$ and $U(\partial\Omega)$ be the uniform distributions on $\Omega$ and $\partial\Omega$, respectively. Then the loss $\mathcal{L}(y,p)$ can be rewritten as \[\mathcal{L}(y,p)= \|\Omega\|\mathbb{E}_{X\sim U(\Omega)}\left[\left(\Delta y(X)+f(X) -\lambda^{-1}p(X)\right)^{2}\right]+\alpha_{i}\|\Omega\|\mathbb{E}_{X\sim U( \Omega)}\left[\left(\Delta p(X)+y(X)-y_{d}(X)\right)^{2}\right]\] \[+\alpha_{b}^{y}\|\partial\Omega\|\mathbb{E}_{Y\sim U(\partial \Omega)}\left[\left(y(Y)\right)^{2}\right]+\alpha_{b}^{p}\|\partial\Omega\| \mathbb{E}_{Y\sim U(\partial\Omega)}\left[\left(p(Y)\right)^{2}\right]. \tag{3.6}\] Following the standard PINN paradigm in Section 2.2, we employ the DNN spaces $\mathcal{Y}$ and $\mathcal{P}$ to approximate the state $y$ and adjoint state $p$, respectively, and then approximate the expectations via Monte Carlo methods. Let $(y_{\theta},p_{\sigma})\in\mathcal{Y}\times\mathcal{P}$ be the DNN approximation, and $\mathbb{X}=\{X_{i}\}_{i=1}^{n_{d}}$ and $\mathbb{Y}=\{Y_{j}\}_{j=1}^{n_{b}}$ be i.i.d. samples drawn from $U(\Omega)$ and $U(\partial\Omega)$, respectively. Then the empirical PINN loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$ is given by \[\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})= \frac{\|\Omega\|}{n_{d}}\sum_{i=1}^{n_{d}}\left(\Delta y_{\theta}(X _{i})+f(X_{i})-\lambda^{-1}p_{\sigma}(X_{i})\right)^{2}+\alpha_{i}\frac{\| \Omega\|}{n_{d}}\sum_{i=1}^{n_{d}}\left(\Delta p_{\sigma}(X_{i})+y_{\theta}(X _{i})-y_{d}(X_{i})\right)^{2}\] \[+\alpha_{b}^{y}\frac{\|\partial\Omega\|}{n_{b}}\sum_{j=1}^{n_{b}} \left(y_{\theta}(Y_{j})\right)^{2}+\alpha_{b}^{p}\frac{\|\partial\Omega\|}{n_{b }}\sum_{j=1}^{n_{b}}\left(p_{\sigma}(Y_{j})\right)^{2}. \tag{3.7}\] Let $(y_{\theta^{}},p_{\sigma^{}})\in\mathcal{Y}\times\mathcal{P}$ be a minimizer of $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$ over $\mathcal{Y}\times\mathcal{P}$, or equivalently with respect to the DNN parameter vectors $\theta$ and $\sigma$. In practice, this is commonly achieved by off-shelf optimizers, e.g., limited memory BFGS [7] and Adam [24], which are implemented in many public software frameworks, e.g., PyTorch or TensorFlow. To evaluate the loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$, one requires computing the derivative of the network outputs $y_{\theta}$ and $p_{\sigma}$ with respect to the input $x$ (i.e., spatial derivatives), and to apply a first-order optimizer, one requires computing the derivative of the loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$ with respect to the DNN parameters $\theta$ and $\sigma$. Both can be realized using automatic differentiation efficiently [3], e.g., torch.autograd in PyTorch. Given a (local) minimizer $(\theta^{},\sigma^{})$, the corresponding DNN approximations of the state $\bar{y}\in\mathcal{Y}$ and adjoint $\bar{p}\in\mathcal{P}$ are given by $y_{\theta^{}}$ and $p_{\sigma^{}}$, respectively. In view of the first-order necessary condition, the DNN approximation $u^{}$ of the optimal control $\bar{u}$ is given by $-\lambda^{-1}p_{\sigma^{}}$. Remark 3.1.: _Demo et al [12] proposed an extended PINN, based on the first-order KKT system for parametric optimal control problems, where the DNNs also take the parameters as inputs besides the spatial coordinates. The approach applies also to the deterministic case. In the standard elliptic case, it is based on the full optimality system (3.3), and the corresponding population loss is given by_ \[\mathcal{L}(y,p,u)= \left\\|\Delta y+f+u\right\\|_{L^{2}(\Omega)}^{2}+\alpha_{i}\left\\| \Delta p+y-y_{d}\right\\|_{L^{2}(\Omega)}^{2}\] \[+\alpha_{b}^{y}\left\\|y\right\\|_{L^{2}(\partial\Omega)}^{2}+ \alpha_{b}^{p}\left\\|p\right\\|_{L^{2}(\partial\Omega)}^{2}+\alpha_{i}^{u}\\|u+ \lambda^{-1}p\\|_{L^{2}(\Omega)}^{2}.\] _Compared with C-PINN in (3.5), the loss $\mathcal{L}(y,p,u)$ requires also one neural network $u_{\kappa}\in\mathcal{U}$ for approximating the control $u$, and tuning the weight $\alpha_{i}^{u}$, which might be nontrivial. Also, the work [12] does not discuss the constrained case, where the control $u$ is inherently nonsmooth, and thus numerically challenging to approximate using PINN [40]. A similar approach to [12] was suggested in [2]._ ### Optimal control with box constraint Now we turn to the constrained optimal control problem, i.e., the control $u$ is from a box constrain set $U=\{u\in L^{2}(\Omega):u_{a}\leq u\leq u_{b}\text{ a.e. in }\Omega\}$, with $u_{a}<u_{b}$. It is known that the optimal tuple $(\bar{y},\bar{p},\bar{u})$ satisfies the following first-order necessary optimality system [39, Theorem 2.25, equation (2.52), p. 67] \[\left\{\begin{aligned} -\Delta y&=f+u,& \text{in }\Omega,\\ y&=0,&\text{on }\partial\Omega,\\ -\Delta p&=y-y_{d},&\text{in }\Omega,\\ p&=0,&\text{on }\partial\Omega,\\ (u+\lambda^{-1}p,v-u)&\geq 0,&\forall v \in U.\end{aligned}\right. \tag{3.8}\] Now we introduce the pointwise projection operator $P_{U}$ into the constraint set $U$, defined by $P_{U}(v)(x)=\min(\max(v(x),u_{a}),u_{b})$. Then the variational inequality in (3.8) can be rewritten as \[u=P_{U}(-\lambda^{-1}p),\] and accordingly, the (reduced) optimality system is given by \[\left\{\begin{aligned} -\Delta y&=f+P_{U}(-\lambda^{-1}p),& \text{in }\Omega,\\ y&=0,&\text{on }\partial\Omega,\\ -\Delta p&=y-y_{d},&\text{in }\Omega,\\ p&=0,&\text{on }\partial\Omega.\end{aligned}\right. \tag{3.9}\] The reduced optimality system (3.9) naturally motivates the following continuous PINN loss: \[\mathcal{L}(y_{\theta},p_{\sigma})= \|\Omega\|\mathbb{E}_{X\sim U(\Omega)}\left[\left(\Delta y_{\theta} (X)+f(X)+P_{U}(-\lambda^{-1}p_{\sigma}(X))\right)^{2}\right]+\alpha_{b}^{y}\| \partial\Omega\|\mathbb{E}_{Y\sim U(\partial\Omega)}\left[\left(y_{\theta}(Y) \right)^{2}\right]\] \[+\alpha_{i}\|\Omega\|\mathbb{E}_{X\sim U(\Omega)}\left[\left(\Delta p _{\sigma}(X)+y_{\theta}(X)-y_{d}(X)\right)^{2}\right]+\alpha_{b}^{p}\|\partial \Omega\|\mathbb{E}_{Y\sim U(\partial\Omega)}\left[\left(p_{\sigma}(Y)\right)^ {2}\right].\] Upon approximating the expectations with Monte Carlo methods using i.i.d. samples $\mathbb{X}=\{X_{i}\}_{i=1}^{n_{d}}\sim U(\Omega)$ and $\mathbb{Y}=\{Y_{j}\}_{j=1}^{n_{b}}\sim U(\partial\Omega)$, we obtain the following empirical PINN loss \[\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})= \frac{\|\Omega\|}{n_{d}}\sum_{i=1}^{n_{d}}\Bigl{(}\Delta y_{\theta} (X_{i})+f(X_{i})+P_{U}(-\lambda^{-1})p_{\sigma}(X_{i})\Bigr{)}^{2}+\alpha_{b}^ {y}\frac{\|\partial\Omega\|}{n_{b}}\sum_{j=1}^{n_{b}}\left(y_{\theta}(Y_{j}) \right)^{2}\] \[+\alpha_{i}\frac{\|\Omega\|}{n_{d}}\sum_{i=1}^{n_{d}}\left(\Delta p _{\sigma}(X_{i})+y_{\theta}(X_{i})-y_{d}(X_{i})\right)^{2}+\alpha_{b}^{p}\frac {\|\partial\Omega\|}{n_{b}}\sum_{j=1}^{n_{b}}\left(p_{\sigma}(Y_{j})\right)^{2}.\] The empirical loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$ is then minimized with respect to the DNN parameters $(\theta,\sigma)$, with a minimizer $(\theta^{},\sigma^{})$. The approximation to the optimal control $\bar{u}$ is given by $P_{U}(-\lambda^{-1}p_{\sigma^{}})$. Note that the change from the unconstrained case is rather minimal in terms of formulation and implementation. ### Optimal control for semilinear problems C-PINN applies also to more complex optimal control problems. We illustrate this with semilinear elliptic problems, one popular model class in PDE optimal control. Consider the following optimal control problem \[\min\;J(y,u):=\frac{1}{2}\\|y-y_{d}\\|_{L^{2}(\Omega)}^{2}+\frac{\lambda}{2}\\|u \\|_{L^{2}(\Omega)}^{2}, \tag{3.10}\] subject to the following semilinear elliptic PDE constraint \[\left\{\begin{aligned} -\Delta y&=f(y)+u,& \text{in }\Omega,\\ y&=0,&\text{on }\partial\Omega,\end{aligned}\right. \tag{3.11}\] where $u\in L^{2}(\Omega)$ is the control. The semilinear term $f:\mathbb{R}\to\mathbb{R}$ should satisfy suitable conditions (e.g., monotone) so that the state problem (3.11) is well-posed [39, Theorem 4.4]. Then the optimal control problem has a locally unique solution $(\bar{y},\bar{u})$[39, Theorem 4.15, p. 208]. The KKT system for the optimal control problem (3.10)-(3.11) is given by [39, Theorem 4.20, p. 216]: \[\begin{cases}-\Delta y=f(y)+u,&\text{in }\Omega,\\ y=0,&\text{on }\partial\Omega,\\ -\Delta p=f^{\prime}(y)p+y-y_{d},&\text{in }\Omega,\\ p=0,&\text{on }\partial\Omega,\\ u=-\lambda^{-1}p,&\text{in }\Omega.\end{cases}\] Following the preceding derivations for linear elliptic PDEs, the empirical PINN loss $\widehat{L}(y_{\theta},p_{\sigma})$ for problem (3.10)-(3.11) is given by \[\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})= \frac{\|\Omega\|}{n_{d}}\sum_{i=1}^{n_{d}}\Bigl{(}\Delta y_{\theta} (X_{i})+f(y_{\theta}(X_{i}))-\lambda^{-1}p_{\sigma}(X_{i})\Bigr{)}^{2}+ \alpha_{b}^{y}\frac{\|\partial\Omega\|}{n_{b}}\sum_{j=1}^{n_{b}}\big{(}y_{ \theta}(Y_{j})\big{)}^{2}\] \[+\alpha_{i}\frac{\|\Omega\|}{n_{d}}\sum_{i=1}^{n_{d}}\big{(}\Delta p _{\sigma}(X_{i})+f^{\prime}(y_{\theta}(X_{i}))p_{\sigma}(X_{i})+y_{\theta}(X_{ i})-y_{d}(X_{i}))^{2}+\alpha_{b}^{p}\frac{\|\partial\Omega\|}{n_{b}}\sum_{j=1}^{n_{b}} \big{(}p_{\sigma}(Y_{j})\big{)}^{2}.\] The adaptation of the approach to semilinear problems with box constraint on the control $U$ is direct. ### Existing DNN based approaches to optimal control The use of DNNs to solve PDE optimal control problems has just started to receive attention recently. Three different methods have been developed, i.e., augmented Lagrangian method (ALM) [28], penalty method (PM) [32], and adjoint oriented neural network (AONN) [40], which we describe below for unconstrained linear elliptic control problems. We mention also necessary changes in the constrained case. PM transforms the constrained problem (3.1)-(3.2) into an unconstrained one by incorporating the PINN loss $\mathcal{L}_{\text{pinn}}$ as \[\mathcal{L}_{\text{pmn}}(y_{\theta},u_{\kappa}):= J(y_{\theta},u_{\kappa})+\mu\mathcal{L}_{\text{pinn}}(y_{ \theta},u_{\kappa})\] \[= \Bigl{(}\frac{1}{2}\\|y_{\theta}-y_{d}\\|_{L^{2}(\Omega)}^{2}+\frac {\lambda}{2}\\|u_{\kappa}\\|_{L^{2}(\Omega)}^{2}\Bigr{)}+\mu\Bigl{(}\frac{1}{2} \\|F(y_{\theta},u_{\kappa})\\|_{L^{2}(\Omega)}^{2}+\frac{\alpha}{2}\\|y_{\theta} \\|_{L^{2}(\partial\Omega)}^{2}\Bigr{)},\] where $\mu>0$ is the penalty parameter for the PINN loss $\mathcal{L}_{\text{pinn}}$, and $F(y_{\theta},u_{\kappa})=\Delta y_{\theta}+f+u_{\kappa}$ is the PDE residual for the DNN approximation $y_{\theta}\in\mathcal{Y}$ and $u_{\kappa}\in\mathcal{U}$. Like in PINN, the integrals are approximated using Monte Carlo methods, leading to an empirical loss $\widehat{\mathcal{L}}_{\text{pmn}}(p_{\theta},u_{\kappa})$ to be minimized in the DNN parameters $\theta$ and $\kappa$. The penalty factor $\mu$ can be fixed _a priori_. Then PM involves solving only one optimization problem but enforces only the PDE constraint approximately. To strictly enforce the constraint, one should send $\mu\to\infty$ and solve a sequence of problems, commonly via a path-following strategy [28, 18]. Specifically, fix $\mu_{0}>0$, and set $\mu_{k+1}=\beta\mu_{k}$, with the increasing factor $\beta>1$, and then one solves a sequence of problems for $\mu_{k}$, with the solution $(\theta_{k-1}^{},\kappa_{k-1}^{})$ of the optimization problem with weight $\mu_{k-1}$ as the initial guess of the current one with $\mu_{k}$. One drawback of PM is that it becomes increasingly more ill-conditioned as $\mu\to\infty$ (and thus more challenging to resolve) [5, 28]. Further, the presence of additional constraints requires more penalty terms, for which the tuning of the penalty weights can be very tedious [25, 40]. Mowlavi and Nabi [32] presented guidelines for tuning parameters in PM for PDE constrained optimal control. ALM transforms a PDE constrained optimal control problem into an unconstrained one using Lagrangian multipliers. The use of Lagrangian multiplier can mitigate the ill-conditioning of PM [19]. Let $(\cdot,\cdot)$ and $(\cdot,\cdot)_{L^{2}(\partial\Omega)}$ denote the $L^{2}(\Omega)$ and $L^{2}(\partial\Omega)$ inner products. Then the loss $\mathcal{L}_{\text{alm}}$ reads \[\mathcal{L}_{\text{alm}}(y_{\theta},u_{\kappa},\eta^{d},\eta^{b}):=J(y_{\theta}, u_{\kappa})+\mu\mathcal{L}_{\text{pinn}}(y_{\theta},u_{\kappa})+(\eta^{d},F(y_{ \theta},u_{\kappa}))+(\eta^{b},y_{\theta})_{L^{2}(\partial\Omega)},\] where the functions $\eta^{d}\in L^{2}(\Omega),\eta^{b}\in L^{2}(\partial\Omega)$ are Lagrangian multipliers for the governing equation $F(y,u)=0$ in $\Omega$ and boundary condition $y=0$ on $\partial\Omega$, respectively, which may also be realized by DNNs. This loss is often minimized alternatingly: for fixed $(\eta_{k}^{d},\eta_{k}^{b})$, we minimize $\mathcal{L}_{\mathrm{alm}}(y_{\theta},u_{\kappa},\eta^{d},\eta^{b})$ in $(y_{\theta},u_{\kappa})$: \[(y_{\theta^{k}},u_{\kappa^{k}})=\arg\min_{(y_{\theta},u_{\kappa})\in\mathcal{Y }\times\mathcal{P}}\mathcal{L}_{\mathrm{alm}}(y_{\theta},u_{\kappa},\eta_{k}^{ d},\eta_{k}^{b}),\] and then update the Lagrangian multipliers $(\eta_{k+1}^{d},\eta_{k+1}^{b})$ by an Uzawa type algorithm: \[\eta_{k+1}^{d}=\eta_{k}^{d}+\mu F(y_{\theta^{k}},u_{\kappa^{k}})\quad\text{and }\quad\eta_{k+1}^{b}=\eta_{k}^{b}+\mu\alpha y_{\theta^{k}}.\] In practice, the integrals are approximated by evaluations on fixed collocation points. Thus it suffices to discretize the multipliers pointwise point-wise [28]. AONN is also based on the optimality system (3.3), but utilizes the adjoint $p$ to compute the gradient of the reduced cost $J(u)=J(y(u),u)$ in $u$ only, so as to combine the advantages of DAL and deep learning techniques. The total derivative $\mathrm{d}_{u}J(y,u)=\alpha u+p$ is used to update $u$ as \[u_{\kappa^{k+1}}=\arg\min_{u_{\kappa}\in\mathcal{U}}\\|u_{\kappa}-(u_{\kappa^{ k}}-s^{k}\mathrm{d}_{u}J(y_{\theta^{k}},u_{\kappa^{k}}))\\|_{L^{2}(\Omega)}^{2}, \tag{3.12}\] where the step size $s^{k}>0$ should be suitably chosen, and the minimization is to project the update into the set $\mathcal{U}$. In AONN, the state $y_{\theta}$ and adjoint state $p_{\sigma}$ are constructed sequentially, and a third DNN $u_{\kappa}\in\mathcal{U}$ is employed to approximate $u$. Similar to PM, the initial guess for each subproblem is given by the last minimizer. Unlike PM and ALM, AONN gives an all-at-once approximation of the control, state, and adjoint. Numerically, AONN outperforms penalty based methods over a range of problems [40]. The presence of box constraint on the control $u\in U$ requires a slightly different treatment. In PM, one can add an additional penalty term to enforce the constraint: \[\mathcal{L}_{\mathrm{pm-c}}(y_{\theta},u_{\kappa})=J(y_{\theta},u_{\kappa})+ \mu\mathcal{L}_{\mathrm{pinn}}(y_{\theta},u_{\kappa})+\frac{\mu^{\prime}}{2} \\|u_{\kappa}-P_{U}(u_{\kappa})\\|_{L^{2}(\Omega)}^{2}, \tag{3.13}\] where $\mu^{\prime}$ is a penalty weight, and $u_{\kappa}\in\mathcal{U}$ is a DNN approximation of the control variable $u$. The drawback is that the obtained approximation $u_{\kappa}$ may be infeasible. Alternatively, one can also enforce the constraint by applying the point-wise projection $P_{U}$ on $u_{\kappa}$ directly \[\mathcal{L}_{\mathrm{pm}}(y_{\theta},u_{\kappa})=J(y_{\theta},P_{U}(u_{\kappa }))+\mu\mathcal{L}_{\mathrm{pinn}}(y_{\theta},P_{U}(u_{\kappa})).\] The projection $P_{U}$ leads to vanishing gradient when the constraint is active, and the optimizer may get stuck in a local minimum and hence requires a good initial guess. Indeed, box constraints also lead to non-smoothness of the loss, causing numerical issues [16, 28]. AONN treats the box constraint $u\in U$ by \[u_{\kappa^{k+1}}=\arg\min_{u_{\kappa}\in\mathcal{U}}\\|u_{\kappa}-P_{U}(u_{ \kappa^{k}}-s^{k}d_{u}J(y_{\theta^{k}},u_{\kappa^{k}}))\\|_{L^{2}(\Omega)}^{2},\] where the initial guess for the DNN parameters $\kappa^{k+1}$ is $\kappa^{k}$. For the coupled PINN, the constraint is enforced by substituting $P_{U}(p_{\sigma})$ into the PDE system. Numerically, after projection, the control becomes a piece-wise function, which is not fully parameterized by the DNN parameters. We choose the initial DNN parameters with a proper scale so that most part of the function satisfies the constraint. Alternatively one may also apply smoothed version of the projection operator $P_{U}$ like sigmoid to find a proper initial guess. ## 4 Error analysis Now we present analyze C-PINN for linear elliptic optimal control problems. ### Fundamental estimates First, we provide two fundamental results bounding the error in terms of the population loss (i.e., weak coercivity results). We begin by discussing optimal control problems without constraints. Lemma 4.1.: _Let $(\bar{y},\bar{p})$ be the solution tuple to the system (3.4) with $\bar{u}=-\lambda^{-1}\bar{p}$. Then for any $(y_{\theta},p_{\sigma})\in\mathcal{Y}\times\mathcal{P}$, with $u_{\sigma}=-\lambda^{-1}p_{\sigma}$, the following estimate holds_ \[\\|\bar{y}-y_{\theta}\\|_{L^{2}(\Omega)}^{2}+\\|\bar{p}-p_{\sigma}\\|_{L^{2}( \Omega)}^{2}+\\|\bar{u}-u_{\sigma}\\|_{L^{2}(\Omega)}^{2}\leq c(\mathbf{\alpha}, \lambda)\mathcal{L}(y_{\theta},p_{\sigma}).\] Proof.: To deal with the nonzero Dirichlet boundary conditions with $y_{\theta}$ and $p_{\sigma}$, we define \[\begin{cases}-\Delta\zeta^{y}=0,&\text{in}\;\Omega,\\ \quad\quad\zeta^{y}=y_{\theta},&\text{on}\;\partial\Omega,\end{cases}\quad \text{and}\quad\begin{cases}-\Delta\zeta^{p}=0,&\text{in}\;\Omega,\\ \quad\quad\zeta^{p}=p_{\sigma},&\text{on}\;\partial\Omega.\end{cases} \tag{4.1}\] That is, $\zeta^{y}$ and $\zeta^{p}$ are harmonic extensions of $y_{\theta}$ and $p_{\sigma}$, respectively. By the standard elliptic regularity theory [4, Theorem 4.2, p. 870], the following stability estimates hold \[\\|\zeta^{y}\\|_{L^{2}(\Omega)}\leq c\\|y_{\theta}\\|_{L^{2}(\partial\Omega)} \quad\text{and}\quad\\|\zeta^{p}\\|_{L^{2}(\Omega)}\leq c\\|p_{\sigma}\\|_{L^{2}( \partial\Omega)}. \tag{4.2}\] Let $e_{y}=\bar{y}-y_{\theta}$ and $e_{p}=\bar{p}-p_{\sigma}$, also $\tilde{e}_{y}=e_{y}+\zeta^{y}$ and $\tilde{e}_{p}=e_{p}+\zeta^{p}$. Then $\tilde{e}_{y}$ and $\tilde{e}_{p}$ satisfy \[\begin{cases}-\Delta\tilde{e}_{y}+\lambda^{-1}\tilde{e}_{p}=f- \lambda^{-1}p_{\sigma}+\Delta y_{\theta}+\lambda^{-1}\zeta^{p},&\text{in}\; \Omega,\\ \quad\quad\tilde{e}_{y}=0,&\text{on}\;\partial\Omega,\end{cases} \tag{4.3a}\] \[\begin{cases}-\Delta\tilde{e}_{p}-\tilde{e}_{y}=y_{\theta}-y_{d} +\Delta p_{\sigma}-\zeta^{y},&\text{in}\;\Omega,\\ \quad\quad\tilde{e}_{p}=0,&\text{on}\;\partial\Omega.\end{cases} \tag{4.3b}\] Multiplying (4.3a) by $\tilde{e}_{y}$ and (4.3b) by $\lambda^{-1}\tilde{e}_{p}$, integrating over $\Omega$ and adding the resulting identities yield \[\\|\nabla\tilde{e}_{y}\\|_{L^{2}(\Omega)}^{2}+\lambda^{-1}\\|\nabla \tilde{e}_{p}\\|_{L^{2}(\Omega)}^{2}\leq \\|f-\lambda^{-1}p_{\sigma}+\Delta y_{\theta}+\lambda^{-1}\zeta^{ p}\\|_{L^{2}(\Omega)}\\|\tilde{e}_{y}\\|_{L^{2}(\Omega)}\] \[+\lambda^{-1}\\|y_{\theta}-y_{d}+\Delta p_{\sigma}-\zeta^{y}\\|_{L^ {2}(\Omega)}\\|\tilde{e}_{p}\\|_{L^{2}(\Omega)}.\] Then by the triangle inequality and the stability estimates in (4.2), we deduce \[\\|\nabla\tilde{e}_{y}\\|_{L^{2}(\Omega)}^{2}+\lambda^{-1}\\|\nabla \tilde{e}_{p}\\|_{L^{2}(\Omega)}^{2}\leq \\|f-\lambda^{-1}p_{\sigma}+\Delta y_{\theta}\\|_{L^{2}(\Omega)}\\| \tilde{e}_{y}\\|_{L^{2}(\Omega)}+c\lambda^{-1}\\|p_{\sigma}\\|_{L^{2}(\partial \Omega)}\\|\tilde{e}_{y}\\|_{L^{2}(\Omega)}\] \[+\lambda^{-1}\\|y_{\theta}-y_{d}+\Delta p_{\sigma}\\|_{L^{2}(\Omega )}\\|\tilde{e}_{p}\\|_{L^{2}(\Omega)}+c\lambda^{-1}\\|y_{\theta}\\|_{L^{2}(\partial \Omega)}\\|\tilde{e}_{p}\\|_{L^{2}(\Omega)}.\] Now by Poincare inequality, trace inequality, and Young's inequality, we obtain \[\\|\tilde{e}_{y}\\|_{L^{2}(\Omega)}^{2}+\lambda^{-1}\\|\tilde{e}_{p} \\|_{L^{2}(\Omega)}^{2}\leq c\big{(}\\|f-\lambda^{-1}p_{\sigma}+\Delta y_{\theta}\\|_{L^{2}( \Omega)}^{2}+\lambda^{-1}\\|y_{\theta}-y_{d}+\Delta p_{\sigma}\\|_{L^{2}(\Omega )}^{2}\] \[+\lambda^{-1}\\|y_{\theta}\\|_{L^{2}(\partial\Omega)}^{2}+\lambda^{ -1}\\|p_{\sigma}\\|_{L^{2}(\partial\Omega)}^{2}\big{)}. \tag{4.4}\] Meanwhile, by the triangle inequality, we have \[\\|e_{y}\\|_{L^{2}(\Omega)}^{2}\leq c(\\|\tilde{e}_{y}\\|_{L^{2}(\Omega)}^{2}+\\|\zeta^{y}\\|_{L^{2}( \Omega)}^{2})\leq c(\\|\tilde{e}_{y}\\|_{L^{2}(\Omega)}^{2}+\\|y_{\theta}\\|_{L^{2}( \partial\Omega)}^{2})\leq c(\mathbf{\alpha},\lambda)\mathcal{L}(y_{\theta},p_{\sigma}),\] \[\\|e_{p}\\|_{L^{2}(\Omega)}^{2}\leq c(\\|\tilde{e}_{p}\\|_{L^{2}(\Omega)}^{2}+\\|\zeta^{p}\\|_{L^{2}( \Omega)}^{2})\leq c(\\|\tilde{e}_{p}\\|_{L^{2}(\Omega)}^{2}+\\|p_{\sigma}\\|_{L^{2}( \partial\Omega)}^{2})\leq c(\mathbf{\alpha},\lambda)\mathcal{L}(y_{\theta},p_{\sigma}).\] Moreover, for the error $e_{u}=\bar{u}-u_{\sigma}$, from the identity $\\|e_{u}\\|_{L^{2}(\Omega)}=\lambda^{-1}\\|e_{p}\\|_{L^{2}(\Omega)}$, we deduce $\\|e_{u}\\|_{L^{2}(\Omega)}^{2}\leq c(\mathbf{\alpha},\lambda)\mathcal{L}(y_{\theta},p_ {\sigma})$. Combining the preceding three estimates completes the proof of the lemma. Remark 4.1.: _It follows from the estimate (4.4) that one should choose $\alpha_{i}$ as $\lambda^{-1}$. The boundary penalty parameters $\alpha_{b}^{y}$ and $\alpha_{b}^{p}$ should be scaled accordingly._ Next, we discuss the constrained case, which is more involved, due to the presence of the nonlinear projection operator $P_{U}$. Thus, we shall employ a different proof technique, based on the idea of reconstruction. Lemma 4.2.: _Let $(\bar{y},\bar{p})$ be the solution tuple of the system (3.9), with $\bar{u}=P_{U}(-\lambda^{-1}\bar{p})$. Then for any $(y_{\theta},p_{\sigma})\in\mathcal{Y}\times\mathcal{P}$, and $u_{\sigma}=P_{U}(-\lambda^{-1}p_{\sigma})$, the following estimate holds_ \[\\|\bar{y}-y_{\theta}\\|_{L^{2}(\Omega)}^{2}+\\|\bar{p}-p_{\sigma}\\|_{L^{2}(\Omega) }^{2}+\\|\bar{u}-u_{\sigma}\\|_{L^{2}(\Omega)}^{2}\leq c(\mathbf{\alpha},\lambda) \mathcal{L}(y_{\theta},p_{\sigma}).\] Proof.: The proof employs crucially the following auxiliary functions $R\bar{y},R\bar{p}\in H^{1}_{0}(\Omega)$ satisfying \[\begin{cases}-\Delta R\bar{y}=f+u_{\sigma},&\text{in }\Omega,\\ R\bar{y}=0,&\text{on }\partial\Omega,\end{cases}\quad\text{and}\quad\begin{cases}- \Delta R\bar{p}=y_{\theta}-y_{d},&\text{in }\Omega,\\ R\bar{p}=0,&\text{on }\partial\Omega.\end{cases} \tag{4.5}\] Then subtracting (4.5) from (3.4) yields \[\begin{cases}-\Delta(\bar{y}-R\bar{y})=\bar{u}-u_{\sigma},&\text{in }\Omega,\\ \bar{y}-R\bar{y}=0,&\text{on }\partial\Omega,\end{cases} \tag{4.6}\] and likewise \[\begin{cases}-\Delta(\bar{p}-R\bar{p})=\bar{y}-y_{\theta},&\text{in }\Omega,\\ \bar{p}-R\bar{p}=0,&\text{on }\partial\Omega.\end{cases} \tag{4.7}\] Now multiplying (4.6) by $(\bar{p}-R\bar{p})$ and (4.7) by $(\bar{y}-R\bar{y})$, integrating over the domain $\Omega$, and equating the resulting identities yield \[(\bar{p}-R\bar{p},\bar{u}-u_{\sigma})=(\bar{y}-R\bar{y},\bar{y}-y_{\theta}). \tag{4.8}\] It follows from the first-order necessary optimality condition in (3.8) that \[(\bar{p}+\lambda\bar{u},v-\bar{u})\geq 0,\quad\forall v\in U. \tag{4.9}\] Also, since $u_{\sigma}=P_{U}(\lambda^{-1}p_{\sigma})\in U$, we have \[(p_{\sigma}+\lambda u_{\sigma},v-u_{\sigma})\geq 0,\quad\forall v\in U. \tag{4.10}\] Putting $v=u_{\sigma}\in U$ in (4.9), $v=\bar{u}\in U$ in (4.10) and adding the resulting inequalities give \[(\bar{p}-p_{\sigma},u_{\sigma}-\bar{u})-\lambda\left\\|\bar{u}-u_{\sigma} \right\\|_{L^{2}(\Omega)}^{2}\geq 0. \tag{4.11}\] Meanwhile, it follows from the identity (4.8) that \[(\bar{p}-p_{\sigma},u_{\sigma}-\bar{u}) =(\bar{p}-R\bar{p},u_{\sigma}-\bar{u})+(R\bar{p}-p_{\sigma},u_{ \sigma}-\bar{u})\] \[=-(\bar{y}-R\bar{y},\bar{y}-y_{\theta})+(R\bar{p}-p_{\sigma},u_{ \sigma}-\bar{u})\] \[=-\left\\|\bar{y}-R\bar{y}\right\\|_{L^{2}(\Omega)}^{2}-(\bar{y}-R \bar{y},R\bar{y}-y_{\theta})+(R\bar{p}-p_{\sigma},u_{\sigma}-\bar{u}).\] Together with the inequality (4.11) and Cauchy-Schwarz inequality, we obtain \[\left\\|\bar{y}-R\bar{y}\right\\|_{L^{2}(\Omega)}^{2}+\lambda\left\\| \bar{u}-u_{\sigma}\right\\|_{L^{2}(\Omega)}^{2} \leq(\bar{y}-R\bar{y},y_{\theta}-R\bar{y})+(R\bar{p}-p_{\sigma},u _{\sigma}-\bar{u})\] \[\leq\\|\bar{y}-R\bar{y}\\|_{L^{2}(\Omega)}\\|y_{\theta}-R\bar{y}\\|_ {L^{2}(\Omega)}+\\|R\bar{p}-p_{\sigma}\\|_{L^{2}(\Omega)}\\|u_{\sigma}-\bar{u}\\|_ {L^{2}(\Omega)}.\] Then an application of Young's inequality leads to \[\left\\|\bar{y}-R\bar{y}\right\\|_{L^{2}(\Omega)}^{2}+\lambda\left\\|\bar{u}-u_{ \sigma}\right\\|_{L^{2}(\Omega)}^{2}\leq\\|y_{\theta}-R\bar{y}\\|_{L^{2}(\Omega) }^{2}+\lambda^{-1}\\|R\bar{p}-p_{\sigma}\\|_{L^{2}(\Omega)}^{2}.\] By the triangle inequality, we have \[\left\\|\bar{y}-y_{\theta}\right\\|_{L^{2}(\Omega)}\leq c(\lambda)\big{(}\\|R \bar{y}-y_{\theta}\\|_{L^{2}(\Omega)}+\\|R\bar{p}-p_{\sigma}\\|_{L^{2}(\Omega)} \big{)}. \tag{4.12}\] By the elliptic regularity theory for problem (4.7), cf. (4.2), we have $\left\\|\bar{p}-R\bar{p}\right\\|_{L^{2}(\Omega)}\leq c\left\\|\bar{y}-y_{\theta} \right\\|_{L^{2}(\Omega)}$. Then using (4.12), we obtain the following estimate \[\left\\|\bar{p}-p_{\sigma}\right\\|_{L^{2}(\Omega)} \leq\left\\|\bar{p}-R\bar{p}\right\\|_{L^{2}(\Omega)}+\\|R\bar{p}-p_{ \sigma}\\|_{L^{2}(\Omega)}\] \[\leq c\left\\|\bar{y}-y_{\theta}\right\\|_{L^{2}(\Omega)}+\\|R\bar{p}- p_{\sigma}\\|_{L^{2}(\Omega)}\] \[\leq c(\lambda)\big{(}\\|R\bar{y}-y_{\theta}\\|_{L^{2}(\Omega)}+\\|R \bar{p}-p_{\sigma}\\|_{L^{2}(\Omega)}\big{)}.\] Next, let $e_{y}^{R}=R\bar{y}-y_{\theta}$ and $\tilde{e}_{y}^{R}=e_{y}^{R}+\zeta^{y}$, and similarly, let $e_{p}^{R}=R\bar{p}-p_{\sigma}$ and $\tilde{e}_{p}^{R}=e_{p}^{R}+\zeta^{p}$, where $\zeta^{y}$ and $\zeta^{p}$ are harmonic extensions defined in (4.1). Clearly, $\tilde{e}_{y}^{R}$ and $\tilde{e}_{p}^{R}$ satisfy \[\begin{cases}-\Delta\tilde{e}_{y}^{R}=f+u_{\sigma}+\Delta y_{\theta},&\text{in } \Omega,\\ \tilde{e}_{y}^{R}=0,&\text{on }\partial\Omega,\end{cases}\quad\text{and}\quad \begin{cases}-\Delta\tilde{e}_{p}^{R}=y_{\theta}-y_{d}+\Delta p_{\sigma},& \text{in }\Omega,\\ \tilde{e}_{p}^{R}=0,&\text{on }\partial\Omega.\end{cases} \tag{4.13}\] By the standard elliptic regularity estimate, we have \[\left\\|\tilde{e}_{y}^{R}\right\\|_{L^{2}(\Omega)}\leq c\left\\|f+u_{\sigma}+ \Delta y_{\theta}\right\\|_{L^{2}(\Omega)}\quad\text{and}\quad\left\\|\tilde{e }_{p}^{R}\right\\|_{L^{2}(\Omega)}\leq c\left\\|y_{\theta}-y_{d}+\Delta p_{ \sigma}\right\\|_{L^{2}(\Omega)}.\] Hence, we have \[\left\\|R\bar{y}-y_{\theta}\right\\|_{L^{2}(\Omega)} =\left\\|e_{y}^{R}\right\\|_{L^{2}(\Omega)}=\left\\|\tilde{e}_{y}^{ R}-\zeta^{y}\right\\|_{L^{2}(\Omega)}\leq\left\\|\tilde{e}_{y}^{R}\right\\|_{L^{2}( \Omega)}+\left\\|\zeta^{y}\right\\|_{L^{2}(\Omega)}\] \[\leq c\left\\|f+u_{\sigma}+\Delta y_{\theta}\right\\|_{L^{2}(\Omega )}+c\left\\|y_{\theta}\right\\|_{L^{2}(\partial\Omega)},\] \[\left\\|R\bar{p}-p_{\sigma}\right\\|_{L^{2}(\Omega)} =\left\\|e_{p}^{R}\right\\|_{L^{2}(\Omega)}=\left\\|\tilde{e}_{p}^{ R}-\zeta^{p}\right\\|_{L^{2}(\Omega)}\leq\left\\|\tilde{e}_{p}^{R}\right\\|_{L^{2}( \Omega)}+\left\\|\zeta^{p}\right\\|_{L^{2}(\Omega)}\] \[\leq c\left\\|y_{\theta}-y_{d}+\Delta p_{\sigma}\right\\|_{L^{2}( \Omega)}+c\left\\|p_{\sigma}\right\\|_{L^{2}(\partial\Omega)}.\] Therefore, we obtain the desired estimates and complete the proof of the lemma. ### Approximation and statistical errors Now we bound the generalization error via offset Rademacher complexity, first introduced by Liang et al [26] to prove sharp bounds for a two-step star estimator, and recently further developed in [13] for a broad class of problem settings. It is a penalized version of Rademacher complexity which localizes $\mathcal{F}$ adaptively according to the magnitude of $f^{2}$. Definition 4.1.: _Let $\mathcal{F}$ be a class of measurable functions from $\mathcal{X}$ to $\mathbb{R}$, $\mu_{X}$ a probability distribution of $X$ and $\mathbb{X}:=\{X_{i}\}_{i=1}^{n}$ i.i.d. samples draw from $\mu_{X}$. Let $\{\tau_{i}\}_{i=1}^{n}$ be i.i.d. Rademacher random variables, i.e., $\mathbb{P}(\tau_{i}=1)=\mathbb{P}(\tau_{i}=-1)=\frac{1}{2}$. Then the empirical offset Rademacher complexity of $\mathcal{F}$ is defined as_ \[\mathcal{R}_{n}^{\rm off}(\mathcal{F},\beta\|\mathbb{X}):=\mathbb{E}_{\tau} \Big{[}\sup_{f\in\mathcal{F}}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}f(X_{i})-\beta f (X_{i})^{2}\|\mathbb{X}\Big{]},\] _for some $\beta>0$, and the offset Rademacher complexity of $\mathcal{F}$ is defined by_ \[\mathcal{R}_{n}^{\rm off}(\mathcal{F},\beta):=\mathbb{E}_{\mathbb{X}}\mathcal{ R}_{n}^{\rm off}(\mathcal{F},\beta\|\mathbb{X})=\mathbb{E}_{\mathbb{X},\tau} \Big{[}\sup_{f\in\mathcal{F}}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}f(X_{i})-\beta f (X_{i})^{2}\Big{]}.\] First, we give a fundamental estimate of the generalization error in terms of the approximation error and statistical error (using offset Rademacher complexity). We define two sets $\mathcal{G}_{d}$ and $\mathcal{G}_{b}$ of mappings from $X\in\Omega\mapsto\mathbb{R}$ and $Y\in\partial\Omega\mapsto\mathbb{R}$, respectively, by \[\mathcal{G}_{d} =\{\|\Omega\|((\Delta y+f-\lambda^{-1}p)(X))^{2}+\alpha_{i}\|\Omega\|( (\Delta p+y-y_{d})(X))^{2}:y\in\mathcal{Y},p\in\mathcal{P}\},\] \[\mathcal{G}_{b} =\{\alpha_{b}^{p}\|\partial\Omega\|y(Y)^{2}+\alpha_{b}^{p}\|\partial \Omega\|p(Y)^{2}:y\in\mathcal{Y},p\in\mathcal{P}\}.\] Below we write $g_{d}(y,p,X)=\|\Omega\|((\big{(}\Delta y+f-\lambda^{-1}p\big{)}(X))^{2}+\alpha_{i} ((\Delta p+y-y_{d})(X))^{2})\in\mathcal{G}_{d}$, and $g_{b}(y,p,Y)=\|\partial\Omega\|(\alpha_{b}^{y}y(Y)^{2}+\alpha_{b}^{p}p(Y)^{2})\in \mathcal{G}_{b}$, and let $b_{d}=\sup_{g\in\mathcal{G}_{d}}\left\\|g\right\\|_{L^{\infty}(\Omega)}$ and $b_{b}=\sup_{g\in\mathcal{G}_{b}}\left\\|g\right\\|_{L^{\infty}(\partial\Omega)}$. Theorem 4.2.: _For any minimizer $(\hat{y},\hat{p})\in\mathcal{Y}\times\mathcal{P}$ of the empirical loss $\widehat{\mathcal{L}}(y,p)$, the following estimate holds_ \[\mathbb{E}_{\mathbb{X},\mathbb{Y}}[\mathcal{L}(\hat{y},\hat{p})]\leq 4\mathcal{R}_{n_{ d}}^{\rm off}\big{(}\mathcal{G}_{d},(2b_{d})^{-1}\big{)}+4\mathcal{R}_{n_{b}}^{\rm off }\big{(}\mathcal{G}_{b},(2b_{b})^{-1}\big{)}+3\inf_{(y,p)\in\mathcal{Y}\times \mathcal{P}}\mathcal{L}(y,p).\] Proof.: Clearly, we have the following decomposition for any $(y,p)\in\mathcal{Y}\times\mathcal{P}$, \[\mathbb{E}_{\mathbb{X},\mathbb{Y}}[\mathcal{L}(\hat{y},\hat{p})]= \mathbb{E}_{\mathbb{X},\mathbb{Y}}\left[\mathbb{E}_{X,Y}\left[g_{d}(\hat{y},\hat{p},X)+g_{b}(\hat{y},\hat{p},Y)\right]\right]\] \[= \mathbb{E}_{\mathbb{X},\mathbb{Y}}\Big{[}\mathbb{E}_{X,Y}\left[g_ {d}(\hat{y},\hat{p},X)+g_{b}(\hat{y},\hat{p},Y)\right]-\frac{3}{n_{d}}\sum_{i= 1}^{n_{d}}g_{d}(\hat{y},\hat{p},X_{i})-\frac{3}{n_{b}}\sum_{i=1}^{n_{b}}g_{b}( \hat{y},\hat{p},Y_{i})\Big{]}\] \[+\mathbb{E}_{\mathbb{X},\mathbb{Y}}\Big{[}\frac{3}{n_{d}}\sum_{i =1}^{n_{d}}g_{d}(\hat{y},\hat{p},X_{i})+\frac{3}{n_{b}}\sum_{i=1}^{n_{d}}g_{b} (\hat{y},\hat{p},Y_{i})\Big{]}:=\mathrm{I}+\mathrm{II}.\] Since $(\hat{y},\hat{p})\in\mathcal{Y}\times\mathcal{P}$ is an empirical risk minimizer, i.e., $\widehat{\mathcal{L}}(\hat{y},\hat{p})\leq\widehat{\mathcal{L}}(y,p)$, for any $(y,p)\in\mathcal{Y}\times\mathcal{P}$, we deduce Meanwhile, the convexity of supremum and Jensen's inequality imply \[\mathrm{I}\leq \sup_{(y,p)\in\mathcal{Y}\times\mathcal{P}}\mathbb{E}_{\mathbb{X}, \mathbb{Y}}\Big{[}\mathbb{E}_{X,Y}[g_{d}(y,p,X)+g_{b}(y,p,Y)]-\frac{3}{n_{d}} \sum_{i=1}^{n_{d}}g_{d}(y,p,X_{i})-\frac{3}{n_{b}}\sum_{i=1}^{n_{b}}g_{b}(y,p, Y_{i})\Big{]}\] \[\leq \mathbb{E}_{\mathbb{X},\mathbb{Y}}\sup_{(y,p)\in\mathcal{Y}\times \mathcal{P}}\Big{[}\mathbb{E}_{X}[g_{d}(y,p,X)]-\frac{3}{n_{d}}\sum_{i=1}^{n_{ d}}g_{d}(y,p,X_{i})+\mathbb{E}_{Y}[g_{b}(y,p,Y)]-\frac{3}{n_{b}}\sum_{i=1}^{n_{b}}g_{b} (y,p,Y_{i})\Big{]}\] \[\leq \mathbb{E}_{\mathbb{X}}\sup_{(y,p)\in\mathcal{Y}\times\mathcal{P }}\Big{[}\mathbb{E}_{X}[g_{d}(y,p,X)]-\frac{3}{n_{d}}\sum_{i=1}^{n_{d}}g_{d}(y,p,X_{i})\Big{]}+\mathbb{E}_{\mathbb{Y}}\sup_{(y,p)\in\mathcal{Y}\times \mathcal{P}}\Big{[}\mathbb{E}_{Y}[g_{b}(y,p,Y)]-\frac{3}{n_{b}}\sum_{i=1}^{n_{ b}}g_{b}(y,p,Y_{i})\Big{]}.\] Next we bound the two terms, denoted by $\mathrm{I}_{d}$ and $\mathrm{I}_{b}$. Note that $0\leq g_{d}(\cdot,\cdot,X)\leq b_{d}$ for all $X\in\Omega$, which implies $g_{d}{}^{2}(\cdot,\cdot,X)\leq b_{d}g_{d}(\cdot,\cdot,X)$. Consequently, \[\mathrm{I}_{d}\leq \mathbb{E}_{\mathbb{X}}\sup_{(y,p)\in\mathcal{Y}\times\mathcal{P }}\Big{[}2\mathbb{E}_{X}g_{d}(y,p,X)-\frac{1}{b_{d}}\mathbb{E}_{X}g_{d}(y,p,X) ^{2}-\frac{2}{n_{d}}\sum_{i=1}^{n_{d}}g_{d}(y,p,X_{i})-\frac{1}{b_{d}n_{d}} \sum_{i=1}^{n_{d}}g_{d}(y,p,X_{i})^{2}\Big{]}.\] Now we introduce independent copies of $\mathbb{X}$ and $\mathbb{Y}$, i.e., $\mathbb{X}^{\prime}:=\{\mathbb{X}^{\prime}_{i}\}_{i=1}^{n_{d}}$ and $\mathbb{Y}^{\prime}=\{Y^{\prime}_{i}\}_{i=1}^{n_{b}}$, and let $\tau=\{\tau_{i}\}_{i=1}^{n_{d}}$ be a sequence of i.i.d. Rademacher random variables independent of $\mathbb{X}$ and $\mathbb{X}^{\prime}$. By the technique of symmetrization, the convexity of supremum, and Jensen's inequality, we obtain \[\mathrm{I}_{d}\leq \mathbb{E}_{\mathbb{X}}\sup_{(y,p)\in\mathcal{Y}\times\mathcal{P }}\Big{[}\mathbb{E}_{\mathbb{X}^{\prime}}\Big{[}\frac{2}{n_{d}}\sum_{i=1}^{n_{d} }g_{d}(y,p,X^{\prime}_{i})-\frac{1}{b_{d}n_{d}}\sum_{i=1}^{n_{d}}g_{d}(y,p,X^{ \prime}_{i})^{2}\Big{]}-\frac{2}{n_{d}}\sum_{i=1}^{n_{d}}g_{d}(y,p,X_{i})-\frac{ 1}{b_{d}n_{d}}\sum_{i=1}^{n_{d}}g_{d}(y,p,X_{i})^{2}\Big{]}\] \[\leq \mathbb{E}_{\mathbb{X}}\mathbb{E}_{\mathbb{X}^{\prime}}\sup_{(y,p )\in\mathcal{Y}\times\mathcal{P}}\Big{[}\frac{2}{n_{d}}\sum_{i=1}^{n_{d}}(g_{d} (y,p,X^{\prime}_{i})-g_{d}(y,p,X_{i}))-\frac{1}{b_{d}n_{d}}\sum_{i=1}^{n_{d}}(g_ {d}(y,p,X^{\prime}_{i})^{2}+g_{d}(y,p,X_{i})^{2})\Big{]}\] \[= \mathbb{E}_{\mathbb{X}}\mathbb{E}_{\mathbb{X}^{\prime}}\mathbb{E}_{ \tau}\sup_{(y,p)\in\mathcal{Y}\times\mathcal{P}}\Big{[}\frac{2}{n_{d}}\sum_{i=1}^{n _{d}}\tau_{i}(g_{d}(y,p,X^{\prime}_{i})-g_{d}(y,p,X_{i}))-\frac{1}{b_{d}n_{d}} \sum_{i=1}^{n_{d}}(g_{d}(y,p,X^{\prime}_{i})^{2}+g_{d}(y,p,X_{i})^{2})\Big{]}\] \[= 2\mathbb{E}_{\mathbb{X}^{\prime}}\mathbb{E}_{\tau}\sup_{(y,p)\in \mathcal{Y}\times\mathcal{P}}\frac{1}{n_{d}}\sum_{i=1}^{n_{d}}\Big{(}\tau_{i}g_{ d}(y,p,X^{\prime}_{i})-\frac{1}{2b_{d}}g_{d}(y,p,X^{\prime}_{i})^{2}\Big{)}\] \[= 2\mathbb{E}_{\mathbb{X}^{\prime}}\mathcal{R}^{\mathrm{off}}_{n} \big{(}\mathcal{G}_{d},(2b_{d})^{-1}\|\mathbb{X}^{\prime}\big{)}+2\mathbb{E}_{ \mathbb{X}}\mathcal{R}^{\mathrm{off}}_{n_{d}}\big{(}\mathcal{G}_{d},(2b_{d})^{-1} \|\mathbb{X}\big{)}=4\mathcal{R}^{\mathrm{off}}_{n_{d}}\big{(}\mathcal{G}_{d},(2b_ {d})^{-1}\big{)}.\] Similarly, we can deduce $\mathrm{I}_{b}\leq 4\mathcal{R}^{\mathrm{off}}_{n_{b}}\big{(}\mathcal{G}_{b},(2b_{b})^{-1} \big{)}$. This completes the proof of the theorem. Theorem 4.2 indicates that the total error can be decomposed into the approximation error $\mathcal{E}_{app}=3\inf_{(y,p)\in\mathcal{Y}\times\mathcal{P}}\mathcal{L}(y,p)$ and statistical error $\mathcal{E}_{stat}:=4\mathcal{R}^{\mathrm{off}}_{n_{d}}(\mathcal{G}_{d},(2b_{d}) ^{-1})+4\mathcal{R}^{\mathrm{off}}_{n_{b}}(\mathcal{G}_{b},(2b_{b})^{-1})$ characterized by the offset Rademacher complexity. The former arises from restricting the trial spaces of $(y,p)$ from the whole space $H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ to $\mathcal{Y}\times\mathcal{P}$, whereas the latter is due to approximating the integrals with Monte Carlo methods. We have ignored the optimization error: the optimizer only finds an approximate local minimizer of the loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$, due to its highly nonconvex landscape. The analysis of the optimization error is largely open, and hence it is not studied below. Next we bound the approximation error $\mathcal{E}_{app}$ and statistical error $\mathcal{E}_{stat}$. #### 4.2.1 Approximation Error We recall the following result on the approximation error [15, Proposition 4.8]. Proposition 4.1.: _Let $p\geq 1,s,k,d\in\mathbb{N}\cup\{0\},s\geq k+1$, $\rho$ be the logistic or $\tanh$ function, and fix $\mu>0$ small. Then for any $\epsilon>0$, $f\in W^{s,p}([0,1]^{d})$ with $\\|f\\|_{W^{s,p}([0,1]^{d})}\leq 1$, there exists a DNN $f_{\theta}\in\mathcal{N}_{\rho}(c\log(d+s),\,c(d,s,p,k)\epsilon^{-\frac{d}{s- k-\mu}},\,c(d,s,p,k)\epsilon^{-2-\frac{2(d/p+d+k+\mu)+d/p+d}{s-k-\mu}})$ such that $\\|f-f_{\theta}\\|_{W^{k,p}([0,1]^{d})}\leq\epsilon$._ Then we have the following bound on the approximation error $\mathcal{E}_{app}$. Lemma 4.3.: _Fix a tolerance $\epsilon\in(0,1)$, and $\mu>0$ arbitrarily small. If the optimal state $\bar{y}\in H^{s}(\Omega)\cap H^{1}_{0}(\Omega)$ and adjoint $\bar{p}\in H^{s}(\Omega)\cap H^{1}_{0}(\Omega)$, $s\geq 3$, then there exist DNNs $(y_{\theta},p_{\sigma})\in\mathcal{Y}\times\mathcal{P}$ with $\mathcal{Y}=\mathcal{P}=\mathcal{N}_{\rho}(c\log(d+s),c(d,s)\epsilon^{-\frac{ d}{s-2-\mu}},c(d,s)\epsilon^{-\frac{d+s}{2(s-2-\mu)}})$ and $c>0$ dependent on $\alpha$, $\lambda$, $s$, $\\|\bar{y}\\|_{H^{s}(\Omega)}$ and $\\|\bar{p}\\|_{H^{s}(\Omega)}$ such that the approximation error $\mathcal{E}_{app}\leq c\epsilon^{2}$._ Proof.: For any $(y_{\theta},p_{\sigma})\in\mathcal{Y}\times\mathcal{P}$, since $\mathcal{L}(\bar{y},\bar{p})=0$, we deduce \[\mathcal{L}(y_{\theta},p_{\sigma})-\mathcal{L}(\bar{y},\bar{p})= \mathcal{L}(y_{\theta},p_{\sigma})\] \[= \\|\Delta y_{\theta}+f-\lambda^{-1}p_{\sigma}\\|_{L^{2}(\Omega)} ^{2}+\alpha_{i}\\|\Delta p_{\sigma}+y_{\theta}-y_{d}\\|_{L^{2}(\Omega)}^{2}+ \alpha_{b}^{y}\\|y_{\theta}\\|_{L^{2}(\partial\Omega)}^{2}+\alpha_{b}^{p}\\|p_{ \sigma}\\|_{L^{2}(\partial\Omega)}^{2}\] \[= \\|\Delta(y_{\theta}-\bar{y})+\lambda^{-1}(\bar{p}-p_{\sigma})\\|_ {L^{2}(\Omega)}^{2}+\alpha_{i}\\|\Delta(p_{\sigma}-\bar{p})+y_{\theta}-\bar{y} \\|_{L^{2}(\Omega)}^{2}+\alpha_{b}^{y}\\|y_{\theta}-\bar{y}\\|_{L^{2}(\partial \Omega)}^{2}+\alpha_{b}^{p}\\|p_{\sigma}-\bar{p}\\|_{L^{2}(\partial\Omega)}^{2}\] \[\leq c(\boldsymbol{\alpha},\lambda)\big{(}\\|\bar{y}-y_{\theta}\\|_{H^ {2}(\Omega)}^{2}+\\|\bar{p}-p_{\sigma}\\|_{H^{2}(\Omega)}^{2}\big{)},\] by the trace inequality $\\|y_{\theta}-\bar{y}\\|_{L^{2}(\partial\Omega)}\leq c\\|y_{\theta}-\bar{y}\\|_{H^ {2}(\Omega)}$. With $c_{y}=\\|\bar{y}\\|_{H^{s}(\Omega)}$, we have \[\inf_{y_{\theta}\in\mathcal{Y}}\\|\bar{y}-y_{\theta}\\|_{H^{2}(\Omega)}^{2}=c_{ y}^{2}\inf_{y_{\theta}\in\mathcal{Y}}\Big{\\|}\frac{\bar{y}}{c_{y}}-\frac{y_{ \theta}}{c_{y}}\Big{\\|}_{H^{2}(\Omega)}^{2}=c_{y}^{2}\inf_{y_{\theta}\in \mathcal{Y}}\Big{\\|}\frac{\bar{y}}{c_{y}}-y_{\theta}\Big{\\|}_{H^{2}(\Omega)}^{2}.\] By Proposition 4.1, there exists a DNN $y_{\theta}\in\mathcal{Y}=\mathcal{N}_{\rho}(c\log(d+s),c(d,s)\epsilon^{- \frac{d}{s-2-\mu}},c(d,s)\epsilon^{-\frac{0d+4\epsilon}{2(s-2-\mu)}})$ such that $\\|\frac{\bar{y}}{c_{y}}-y_{\theta}\\|_{H^{2}(\Omega)}\leq\epsilon$. Similarly, there exists a DNN $p_{\sigma}\in\mathcal{P}$ with the requisite property. Hence, \[\mathcal{E}_{app}=3\inf_{(y_{\theta},p_{\sigma})\in\mathcal{Y}\times\mathcal{P} }\mathcal{L}(y_{\theta},p_{\sigma})\leq c(\boldsymbol{\alpha},\lambda)\inf_{(y _{\theta},p_{\sigma})\in\mathcal{Y}\times\mathcal{P}}(\\|\bar{y}-y_{\theta}\\|_{H ^{2}(\Omega)}^{2}+\\|\bar{p}-p_{\sigma}\\|_{H^{2}(\Omega)}^{2})\leq c\epsilon^{2},\] where $c$ depends on $\boldsymbol{\alpha}$, $\lambda$, $s$ and the $H^{s}(\Omega)$ norms of $\bar{y}$ and $\bar{p}$. Then the desired assertion follows. Remark 4.3.: _The approximation error $\mathcal{E}_{app}$ depends on the regularity of $\bar{y}$ and $\bar{u}$. If $f,y_{d}\in H^{s-2}(\Omega)\cap L^{\infty}(\Omega)$, by standard elliptic regularity theory, we have $\bar{y},\bar{p}\in H^{s}(\Omega)$, which satisfies the requirements of Lemma 4.3. In the unconstrained case, the optimal control $\bar{u}$ has the same regularity as the adjoint state $\bar{p}$, but in the constrained case, we have $\bar{u}\in H^{1}(\Omega)$ only, due to the presence of the projection operator $P_{U}$._ #### 4.2.2 Statistical error To bound the statistical error via the offset Rademacher complexities, we employ the covering number [1]. Definition 4.2.: _Let $(\mathcal{M},d)$ be a metric space, and the set $\mathcal{F}\subset\mathcal{M}$. A set $\mathcal{F}_{\delta}\subset\mathcal{M}$ is called a $\delta$-cover of $\mathcal{F}$ if for each $f\in\mathcal{F}$, there exists an $f_{\delta}\in\mathcal{F}_{\delta}$ such that $d(f,f_{\delta})\leq\delta$. Moreover, $N(\delta,\mathcal{F},d):=\inf\{\|\mathcal{F}_{\delta}\|:\mathcal{F}_{\delta}\text{ is a $\delta$-cover of $\mathcal{F}$}\}$ is called the $\delta$-covering number of $\mathcal{F}$._ We also need Hoeffding's inequality [17]. Lemma 4.4.: _Let $X_{1}$,..., $X_{n}$ be independent random variables such that $a_{i}\leq X_{i}\leq b_{i}$ almost surely. Consider the sum of these random variables, $S_{n}=X_{1}+\cdots+X_{n}$. Then for all $t>0$_ \[\mathbb{P}\left(S_{n}-\mathbb{E}[S_{n}]\geq t\right)\leq\exp\left(-\frac{2t^{2} }{\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}\right).\] Next, we bound the offset Rademacher complexity in terms of the covering number. Theorem 4.4.: _Let $\kappa_{d}$ and $\kappa_{b}$ be the Lipschitz constants of $g_{d}\in\mathcal{G}_{d}$ and $g_{b}\in\mathcal{G}_{b}$ in $y$ and $p$ in the $W^{2,\infty}(\Omega)$ and $L^{\infty}(\partial\Omega)$ norms, respectively, i.e.,_ \[\|g_{d}(y,p,X)-g_{d}(\tilde{y},\tilde{p},X)\| \leq\kappa_{d}(\\|y-\tilde{y}\\|_{W^{2,\infty}(\Omega)}+\\|p-\tilde {p}\\|_{W^{2,\infty}(\Omega)}),\quad\forall(y,p),(\tilde{y},\tilde{p})\in \mathcal{Y}\times\mathcal{P},X\in\Omega,\] \[\|g_{b}(y,p,Y)-g_{b}(\tilde{y},\tilde{p},Y)\| \leq\kappa_{b}(\\|y-\tilde{y}\\|_{L^{\infty}(\partial\Omega)}+\\|p- \tilde{p}\\|_{L^{\infty}(\partial\Omega)}),\quad\forall(y,p),(\tilde{y},\tilde{ p})\in\mathcal{Y}\times\mathcal{P},Y\in\partial\Omega.\] _Then for any $\delta>0$, the offset Rademacher complexities $\mathcal{R}^{\rm off}_{n}(\mathcal{G}_{d},\beta)$ and $\mathcal{R}^{\rm off}_{n}(\mathcal{G}_{b},\beta)$ are bounded by_ \[\mathcal{R}^{\rm off}_{n}(\mathcal{G}_{d},\beta)\leq \frac{1+\log N(\delta,\mathcal{F}\times\mathcal{P},W^{2,\infty}( \Omega))}{2n\beta}+2(1+2b_{d}\beta)\kappa_{d}\delta,\] \[\mathcal{R}^{\rm off}_{n}(\mathcal{G}_{b},\beta)\leq \frac{1+\log N(\delta,\mathcal{F}\times\mathcal{P},L^{\infty}( \partial\Omega))}{2n\beta}+2(1+2b_{b}\beta)\kappa_{b}\delta.\] Proof.: We give the proof only for $\mathcal{R}^{\rm off}_{n}(\mathcal{G}_{d},\beta)$, since that for $\mathcal{R}^{\rm off}_{n}(\mathcal{G}_{b},\beta)$ is identical. Since $\tau=\{\tau_{i}\}_{i=1}^{n}$ is a sequence of i.i.d. random variables independent of $\mathbb{X}$, then conditionally on $\mathbb{X}$, we have \[\mathbb{E}_{\tau\|\mathbb{X}}\Big{[}\sup_{(y,p)\in\mathcal{Y}\times \mathcal{P}}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y,p,X_{i})-\frac{\beta}{n} \sum_{i=1}^{n}g_{d}(y,p,X_{i})^{2}\|\mathbb{X}\Big{]}\] \[= \mathbb{E}_{\tau}\sup_{(y,p)\in\mathcal{Y}\times\mathcal{P}}\Big{[} \frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y,p,X_{i})-\frac{\beta}{n}\sum_{i=1}^{n} g_{d}(y,p,X_{i})^{2}\Big{]}.\] Let $\delta>0$ and let $\mathcal{Y}_{\delta}$ and $\mathcal{P}_{\delta}$ be a minimal $W^{2,\infty}(\Omega)$$\delta$-cover of $\mathcal{Y}$ and $\mathcal{P}$, respectively. For any fixed $(y,p)\in\mathcal{Y}\times\mathcal{P}$, there exists a tuple $(y_{\delta},p_{\delta})\in\mathcal{Y}_{\delta}\times\mathcal{P}_{\delta}$ such that $\\|y-y_{\delta}\\|_{W^{2,\infty}(\Omega)}\leq\delta$ and $\\|p-p_{\delta}\\|_{W^{2,\infty}(\Omega)}\leq\delta$. Therefore, by Lipschitz continuity of $g_{d}$ with respect to $(y,p)$ in the $W^{2,\infty}(\Omega)$ norm, \[\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y,p,X_{i}) \leq\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y_{\delta},p_{\delta}, X_{i})+\frac{1}{n}\sum_{i=1}^{n}\|\tau_{i}\|\|(g_{d}(y,p,X_{i})-g_{d}(y_{\delta},q_{ \delta},X_{i})\|\] \[\leq\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y_{\delta},p_{\delta}, X_{i})+2\kappa_{d}\delta.\] Since $\|g_{d}(y,p,X)\|\leq b_{d}$ and $\|g_{d}(y_{\delta},p_{\delta},X)\|\leq b_{d}$, and by Lipschitz continuity of $g_{d}(y,p,X)$, we obtain \[-\frac{1}{n}\sum_{i=1}^{n}g_{d}(y,p,X_{i})^{2} =-\frac{1}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p_{\delta},X_{i})^{2} +\frac{1}{n}\sum_{i=1}^{n}(g_{d}(y_{\delta},p_{\delta},X_{i})^{2}-g_{d}(y,p,X_ {i})^{2})\] \[\leq-\frac{1}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p_{\delta},X_{i})^ {2}+\frac{2b_{d}}{n}\sum_{i=1}^{n}\|g_{d}(y_{\delta},p_{\delta},X_{i})-g_{d}(y,p,X _{i})\|\] \[\leq-\frac{1}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p_{\delta},X_{i})^ {2}+4b_{d}\kappa_{d}\delta.\] Hence, it follows that \[\mathbb{E}_{\tau}\sup_{(y,p)\in\mathcal{Y}\times\mathcal{P}}\Big{[}\frac{1}{n} \sum_{i=1}^{n}\tau_{i}g_{d}(y,p,X_{i})-\frac{\beta}{n}\sum_{i=1}^{n}g_{d}(y,p,X_ {i})^{2}\Big{]}\] \[\leq \mathbb{E}_{\tau}\max_{(y_{\delta},p_{\delta})\in\mathcal{F}_{\delta} \times\mathcal{P}_{\delta}}\Big{[}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y_{ \delta},p_{\delta},X_{i})-\frac{\beta}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p_{ \delta},X_{i})^{2}\Big{]}+2(1+2b_{d}\beta)\kappa_{d}\delta. \tag{4.14}\] Since $\{\tau_{i}g_{d}(y_{\delta},p_{\delta};X_{i})\}_{i=1}^{n}$ are independent random variables conditioning on $\mathbb{X}$, \[\mathbb{E}_{\tau}[\tau_{i}g_{d}(y_{\delta},p_{\delta},X_{i})] =0,\quad i=1,\ldots,n,\] \[-g_{d}(y_{\delta},p_{\delta},X_{i})\leq\tau_{i}g_{d}(y_{\delta},p _{\delta},X_{i}) \leq g_{d}(y_{\delta},p_{\delta},X_{i}),\quad i=1,\ldots,n.\] By Hoeffding's inequality from Lemma 4.4, we deduce that for any $(y_{\delta},p_{\delta})\in\mathcal{Y}_{\delta}\times\mathcal{P}_{\delta}$ and $\xi>0$, \[\mathbb{P}_{\tau}\Big{\{}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y _{\delta},p_{\delta},X_{i})>\xi+\frac{\beta}{n}\sum_{i=1}^{n}g_{d}(y_{\delta}, p_{\delta},X_{i})^{2}\Big{\}}\] \[\leq \exp\Big{(}-\frac{(n\xi+\beta\sum_{i=1}^{n}g_{d}(y_{\delta},p_{ \delta},X_{i})^{2})^{2}}{2\sum_{i=1}^{n}g_{d}(y_{\delta},p_{\delta},X_{i})^{2} }\Big{)}\leq\exp(-2\beta n\xi),\] where the last step follows from the elementary inequality $\frac{(a+y)^{2}}{y}\geq\frac{(a+a)^{2}}{a}=4a$, for any $y\in\mathbb{R}_{+}$. Therefore, we may bound the tail probability by \[\mathbb{P}_{\tau}\left\{\max_{(y_{\delta},p_{\delta})\in\mathcal{ Y}_{\delta}\times\mathcal{P}_{\delta}}\Big{(}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d} (y_{\delta},p_{\delta},X_{i})-\frac{\beta}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p _{\delta},X_{i})^{2}\Big{)}>\xi\right\}\] \[\leq N(\delta,\mathcal{F}\times\mathcal{P},W^{2,\infty}(\Omega))\max_{ (y_{\delta},p_{\delta})\in\mathcal{Y}_{\delta}\times\mathcal{P}_{\delta}} \mathbb{P}_{\tau}\Big{\{}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y_{\delta},p _{\delta},X_{i})>\xi+\frac{\beta}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p_{\delta}, X_{i})^{2}\Big{\}}\] \[\leq N(\delta,\mathcal{F}\times\mathcal{P},W^{2,\infty}(\Omega))\exp( -2\beta n\xi).\] Hence, for any $a>0$, we have \[\mathbb{E}_{\tau}\Big{[}\max_{(y_{\delta},p_{\delta})\in\mathcal{ Y}_{\delta}\times\mathcal{P}_{\delta}}\Big{(}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d} (y_{\delta},p_{\delta},X_{i})-\frac{\beta}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p_{ \delta},X_{i})^{2}\Big{)}\Big{]}\] \[\leq a+\int_{a}^{\infty}N(\delta,\mathcal{F}\times\mathcal{P},W^{2, \infty}(\Omega))\exp(-2\beta n\xi)\mathrm{d}\xi\leq a+\frac{N(\delta,\mathcal{F }\times\mathcal{P},W^{2,\infty}(\Omega))}{2\beta n}\exp(-2\beta na).\] Setting $a=\frac{\log N(\delta,\mathcal{F}\times\mathcal{P},W^{2,\infty}(\Omega))}{2\beta n}$ leads to \[\mathbb{E}_{\tau}\Big{[}\max_{(y_{\delta},p_{\delta})\in\mathcal{Y}_{\delta} \times\mathcal{P}_{\delta}}\Big{(}\frac{1}{n}\sum_{i=1}^{n}\tau_{i}g_{d}(y_{ \delta},p_{\delta},X_{i})-\frac{\beta}{n}\sum_{i=1}^{n}g_{d}(y_{\delta},p_{ \delta},X_{i})^{2}\Big{]}\leq\frac{1+\log N(\delta,\mathcal{F}\times\mathcal{P},W^{2,\infty}(\Omega))}{2\beta n}.\] Combining this inequality with the estimate (4.14) yields the desired assertion. The next result gives the statistical error, by exploiting the bound on the DNNs in Section 5. Theorem 4.5.: _Let $\mathcal{Y}=\mathcal{P}=\mathcal{N}_{\rho}(L,\mathbf{n}_{L},R)$, with depth $L$, $\mathbf{n}_{L}$ nonzero DNN parameters and maximum bound $R$. Then the statistical error $\mathcal{E}_{stat}$ is bounded by_ \[\mathcal{E}_{stat}\leq\frac{cL\mathbf{n}_{L}^{4L+1}R^{4L}(L\log R+L\log\mathbf{n}_{L}+ \log n_{d})}{n_{d}}+\frac{c\mathbf{n}_{L}^{3}R^{2}(L\log R+L\log\mathbf{n}_{L}+\log n_ {b})}{n_{b}},\] _where the constant $c$ depends on $\mathbf{\alpha}$, $\lambda$, $d$, $\\|f\\|_{L^{\infty}(\Omega)}$ and $\\|y_{d}\\|_{L^{\infty}(\Omega)}$ at most polynomially._ Proof.: By Lemmas 5.1 and 5.2 below, for any $v\in\mathcal{Y}$, $\\|v\\|_{C(\overline{\Omega})}\leq\mathbf{n}_{L}R$ and $\\|\Delta v\\|_{L^{\infty}(\Omega)}\leq dL\mathbf{n}_{L}^{2L}R^{2L}$, which directly give $b_{d}=cL\mathbf{n}_{L}^{4L}R^{4L}$ and $b_{b}=c(\mathbf{n}_{L}R)^{2}$. Next, for any $(y,p),(\tilde{y},\tilde{p})\in\mathcal{Y}\times\mathcal{P}$, we have \[\|g_{d}(y,p,X)-g_{d}(\tilde{y},\tilde{p},X)\|\leq \kappa_{d}(\\|y-\tilde{y}\\|_{W^{2,\infty}(\Omega)}+\\|p-\tilde{p}\\|_{ W^{2,\infty}(\Omega)}),\quad\text{with }\kappa_{d}=c\mathbf{n}_{L}^{2L}R^{2L},\] \[\|g_{b}(y,p,Y)-g_{b}(\tilde{y},\tilde{p},Y)\|\leq \kappa_{b}(\\|y-\tilde{y}\\|_{L^{\infty}(\partial\Omega)}+\\|p- \tilde{p}\\|_{L^{\infty}(\partial\Omega)}),\quad\text{with }\kappa_{b}=c\mathbf{n}_{L}R.\] Next we bound $N(\delta,\mathcal{Y},W^{2,\infty}(\Omega))$. By Lemmas 5.1 and 5.3, for any $v_{\theta},v_{\tilde{\theta}}\in\mathcal{Y}$, \[\\|v_{\theta}-v_{\tilde{\theta}}\\|_{C(\overline{\Omega})} \leq\mathbf{n}_{L}^{L}R^{L-1}\\|\theta-\tilde{\theta}\\|_{\ell^{2}},\] \[\\|\partial_{x_{i}x_{i}}^{2}v_{\theta}-\partial_{x_{i}x_{i}}^{2}v _{\tilde{\theta}}\\|_{C(\overline{\Omega})} \leq 4L^{2}\mathbf{n}_{L}^{3L-2}R^{3L-3}\\|\theta-\tilde{\theta}\\|_{\ell^{ 2}},\quad i=1,\ldots,d.\] Hence, with $\Lambda_{d}=cL^{2}\mathbf{n}_{L}^{3L-2}R^{3L-3}$ and $\Lambda_{b}=\mathbf{n}_{L}^{L}R^{L}$, \[\\|v_{\theta}-v_{\tilde{\theta}}\\|_{W^{2,\infty}(\Omega)}\leq \Lambda_{d}\\|\theta-\tilde{\theta}\\|_{\ell^{2}}\quad\text{and}\quad\\|v_{ \theta}-v_{\tilde{\theta}}\\|_{L^{\infty}(\partial\Omega)}\leq\Lambda_{b}\\| \theta-\tilde{\theta}\\|_{\ell^{2}},\quad\forall v_{\theta},v_{\tilde{\theta}} \in\mathcal{Y}. \tag{4.15}\] Note that for any $m\in\mathbb{N}$, $r\in[1,\infty)$, $\epsilon\in(0,1)$, and $B_{r}:=\{x\in\mathbb{R}^{m}:\;\\|x\\|_{\ell^{2}}\leq r\}$, then by a simple counting argument (see, e.g., [11, Proposition 5] or [22, Lemma 5.5]), we have $\log N(\epsilon,B_{r},\\|\cdot\\|_{\ell^{2}})\leq m\log(2r\sqrt{m}\epsilon^{-1})$. This, Lipschitz continuity of NN functions in (4.15) and the estimate $\\|\theta\\|_{\ell^{2}}\leq\sqrt{\mathbf{n}_{L}}\\|\theta\\|_{\ell^{\infty}}\leq\sqrt {\mathbf{n}_{L}}R$ imply \[\log N(\delta,\mathcal{Y},W^{2,\infty}(\Omega)) \leq\log N(\Lambda_{d}^{-1}\delta,\mathcal{N}_{Y},\\|\cdot\\|_{\ell ^{2}})\leq\mathbf{n}_{L}\log(2R\mathbf{n}_{L}\Lambda_{d}\delta^{-1}),\] \[\log N(\delta,\mathcal{Y},L^{\infty}(\partial\Omega)) \leq\log N(\Lambda_{b}^{-1}\delta,\mathcal{N}_{Y},\\|\cdot\\|_{\ell ^{2}})\leq\mathbf{n}_{L}\log(2R\mathbf{n}_{L}\Lambda_{b}\delta^{-1}),\] where $\mathcal{N}_{Y}$ denotes the parameter space for $\mathcal{Y}$. These estimates and Theorem 4.4 with $\beta=(2b_{d})^{-1}$ yield \[\mathcal{R}_{n}^{\text{off}}\big{(}\mathcal{G}_{d},(2b_{d})^{-1} \big{)} \leq\frac{b_{d}(1+\log N(\delta,\mathcal{F}\times\mathcal{P},W^{ 2,\infty}(\Omega)))}{n}+4\kappa_{d}\delta\] \[\leq\frac{cL\mathbf{n}_{L}^{4L+1}R^{4L}\log(2R\mathbf{n}_{L}\Lambda_{d} \delta^{-1})}{n}+c\mathbf{n}_{L}^{2L}R^{2L}\delta.\] An analogous bound on $\mathcal{R}_{n}^{\text{off}}\big{(}\mathcal{G}_{b},(2b_{b})^{-1}\big{)}$ holds. Setting $\delta=1/n$, substituting $\Lambda_{d}$ and $\Lambda_{b}$ and simplifying the resulting expressions yield the desired estimate. #### 4.2.3 Final error estimate Now we state the error of the C-PINN approximation $(y_{\theta^{}},p_{\sigma^{}},u_{\sigma^{}})$. Thus, with the parameters in the loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$ chosen suitably, C-PINN can yield an accurate approximation. Theorem 4.6.: _Let the tuple $(\bar{y},\bar{p},\bar{u})$ solve the optimality system (3.3) / (3.8) such that $\bar{y}\in H^{s}(\Omega)\cap H^{1}_{0}(\Omega)$ and $\bar{p}\in H^{s}(\Omega)\cap H^{1}_{0}(\Omega)$ with $s\geq 3$, and $(y_{\theta^{}},p_{\sigma^{}},u_{\sigma^{}})$ be the C-PINN approximation. Fix the tolerance $\epsilon>0$, and take $\mathcal{Y}=\mathcal{P}=\mathcal{N}_{\rho}(c\log(d+s),c(d,s)\epsilon^{-\frac{ \theta d+s}{2-\mu}},c(d,s)\epsilon^{-\frac{\theta d+s}{2(\sigma-2\mu)}})$. Then by choosing $n_{d}=c(d,s)\epsilon^{-\frac{c(d+s)\ln(d+s)}{s-2-\mu}}$ and $n_{b}=c(d,s)\epsilon^{-\frac{\theta(3d+s)}{s-2-\mu}-2}$ sampling points in $\Omega$ and on $\partial\Omega$, there holds_ \[\mathbb{E}_{\mathbb{X},\mathbb{Y}}\left[\\|\bar{y}-y_{\theta^{}}\\|_{L^{2}(\Omega)} ^{2}+\\|\bar{p}-p_{\sigma^{}}\\|_{L^{2}(\Omega)}^{2}+\\|\bar{u}-u_{\sigma^{}}\\|_ {L^{2}(\Omega)}^{2}\right]\leq c\epsilon^{2},\] _where the constant $c$ depends on $\mathbf{\alpha}$, $\lambda$, $d$, $s$, $\\|f\\|_{L^{\infty}(\Omega)}$, $\\|y_{d}\\|_{L^{\infty}(\Omega)}$, $\\|\bar{y}\\|_{H^{s}(\Omega)}$, and $\\|\bar{p}\\|_{H^{s}(\Omega)}$._ Proof.: The fundamental estimates in Lemmas 4.1 and 4.2 imply \[\mathbb{E}_{\mathbb{X},\mathbb{Y}}[\\|\bar{y}-y_{\theta^{}}\\|_{L^{2}(\Omega)}^{2}+ \\|\bar{p}-p_{\sigma^{}}\\|_{L^{2}(\Omega)}^{2}+\\|\bar{u}-u_{\sigma^{}}\\|_{L^{2} (\Omega)}^{2}]\leq c(\mathbf{\alpha},\gamma)\mathbb{E}_{\mathbb{X},\mathbb{Y}}[ \mathcal{L}(y_{\theta^{}},p_{\sigma^{}})].\] By the error decomposition in Theorem 4.2 and Lemma 4.3 (with the given choice of $\mathcal{Y}$ and $\mathcal{P}$), \[\mathbb{E}_{\mathbb{X},\mathbb{Y}}[\mathcal{L}(y_{\theta^{}},p_{\sigma^{}})] \leq c\epsilon^{2}+\frac{cL\mathbf{n}_{L}^{4L+1}R^{4L}(L\log R+L\log\mathbf{n}_{L}+ \log n_{d})}{n_{d}}+\frac{c\mathbf{n}_{L}^{3}R^{2}(L\log R+L\log\mathbf{n}_{L}+ \log n_{b})}{n_{b}}.\] Since $L=c\log(d+s)$, $\mathbf{n}_{L}=c(d,s)\epsilon^{-\frac{d}{2-\mu}}$ and $R=c(d,s)\epsilon^{-\frac{\theta d+s}{2(s-2-\mu)}}$, the choices of the numbers $n_{d}$ and $n_{b}$ imply that the statistical error $\mathcal{E}_{stat}$ is also bounded by $c\epsilon^{2}$. This completes the proof of the theorem. ## 5 Technical estimates on $f_{\theta}$ In this part, we derive several technical estimates, especially the bound and Lipschitz continuity of $\partial_{x_{p}}^{2}f_{\theta}$ for $f_{\theta}\in\mathcal{N}_{\rho}(L,\mathbf{n}_{L},R)$ in terms of the neural network parameters $\theta$. First, we recall several bounds on $f_{\theta}$ and $\partial_{x_{p}}f_{\theta}$, which will be used extensively below. Throughout, we denote $\pi_{i}=\prod_{j=1}^{i}n_{j}$, and let $\mathcal{P}=\mathcal{N}_{\rho}(L,\mathbf{n}_{L},R)$. Lemma 5.1.: _For any $f\equiv f_{\theta},\tilde{f}\equiv f_{\tilde{\theta}}\in\mathcal{P}$, the following estimates hold: for any $p\in[d],q\in[n_{\ell}]$,_ \[\\|f\\|_{L^{\infty}(\Omega)} \leq n_{L-1}R, \tag{5.1}\] \[\\|f_{q}^{(\ell)}-\tilde{f}_{q}^{(\ell)}\\|_{L^{\infty}(\Omega)} \leq\begin{cases}\quad\pi_{\ell-1}R^{\ell-1}\sum_{j=1}^{\mathbf{n}_{ \ell}}\|\theta_{j}-\tilde{\theta}_{j}\|,&\ell=1,\dots,L-1,\\ \sqrt{\mathbf{n}_{L}}\pi_{L-1}R^{L-1}\\|\theta-\tilde{\theta}\\|_{\ell^{2}},&\ell= L,\end{cases}\] (5.2) \[\\|\partial_{x_{p}}f^{(\ell)}\\|_{L^{\infty}(\Omega)} \leq\pi_{\ell-1}R^{\ell},\quad\ell=1,\dots,L,\] (5.3) \[\\|\partial_{x_{p}}f_{q}^{(\ell)}-\partial_{x_{p}}\tilde{f}_{q}^{ (\ell)}\\|_{L^{\infty}(\Omega)} \leq(\ell+1)\pi_{\ell-1}^{2}R^{2\ell-1}\sum_{j=1}^{\mathbf{n}_{\ell} }\|\theta_{j}-\tilde{\theta}_{j}\|,\quad\ell=1,\dots,L-1. \tag{5.4}\] Proof.: These estimates are contained in [22, Lemmas 5.9-5.11]. Note that the estimate (5.4) improves that in [22, Lemma 5.11] by a factor of $R$, by slightly improving the bound on page 17, line 7 of [22]. We also need the following uniform bound on $\partial_{x_{p}}^{2}f_{\theta}$. Lemma 5.2.: _Let $\mathcal{P}=\mathcal{N}_{\rho}(L,\mathbf{n}_{L},R)$. Then for any $p\in[d]$, $\left\|\partial_{x_{p}}^{2}f_{q}^{(\ell)}\right\|\leq\ell\pi_{\ell-1}^{2}R^{2 \ell}$, $\ell=1,2,\cdots,L$._ Proof.: It follows from direct computation that \[\partial_{x_{p}}^{2}f_{q}^{(\ell)}= \Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}\partial_{x_{p}}f_{j }^{(\ell-1)}\Big{)}^{2}\rho^{\prime\prime}\Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^ {(\ell)}f_{j}^{(\ell-1)}+b_{q}^{(\ell)}\Big{)}\] \[+\rho^{\prime}\Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}f_{j}^ {(\ell-1)}+b_{q}^{(\ell)}\Big{)}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}\partial _{x_{p}}^{2}f_{j}^{(\ell-1)}. \tag{5.5}\] When $\ell=1$, this identity and the uniform bound on $\rho^{\prime\prime}$ from Lemma 2.1 imply \[\\|\partial_{x_{p}}^{2}f_{q}^{(1)}\\|_{L^{\infty}(\Omega)}=\Big{(} \sum_{j=1}^{n_{0}}a_{qj}^{(1)}\Big{)}^{2}\\|\rho^{\prime\prime}\Big{(}\sum_{j=1 }^{n_{0}}a_{qj}^{(1)}x+b_{q}^{(1)}\Big{)}\\|_{L^{\infty}(\mathbb{R})}\leq n_{0} ^{2}R^{2}. \tag{5.6}\] Next, we treat the case $\ell>1$. The bounds on $\rho^{\prime}$ and $\rho^{\prime\prime}$ in Lemma 2.1 and the a priori bound on $a_{qj}^{(\ell)}$ and the estimate (5.3) imply \[\left\|\partial_{x_{p}}^{2}f_{q}^{(\ell)}\right\|\leq\Big{(}\sum_{j=1 }^{n_{\ell-1}}\left\|a_{qj}^{(\ell)}\right\|\Big{)}^{2}\cdot(\pi_{\ell-2}R^{\ell- 1})^{2}+R\sum_{j=1}^{n_{\ell-1}}\left\|\partial_{x_{p}}^{2}f_{j}^{(\ell-1)}(x) \right\|\leq\pi_{\ell-1}^{2}R^{2\ell}+R\sum_{j=1}^{n_{\ell-1}}\left\|\partial_{x _{p}}^{2}f_{j}^{(\ell-1)}\right\|.\] Then applying the recursion repeatedly, the estimate (5.6) and mathematical induction yields \[\left\|\partial_{x_{p}}^{2}f_{q}^{(\ell)}\right\|\leq\ell\pi_{\ell-1}^{2}R^{2 \ell},\quad\ell=1,\dots,L-1.\] The case $\ell=L$ follows also directly from the definition of $f$ and the preceding estimate. The next result represents one of the main tools in establishing Rademacher complexity bound. Lemma 5.3.: _Let $f_{\theta},f_{\tilde{\theta}}\in\mathcal{P}$, and define $\eta=1$ for sigmoid and $\eta=2$ for hyperbolic tangent. Then,_ \[\left\|\partial_{x_{p}}^{2}f_{\theta}(x)-\partial_{x_{p}}^{2}f_{\tilde{\theta}}(x) \right\|\leq 2(L-1)L\eta\sqrt{\mathbf{n}_{L}}\pi_{L-1}^{3}R^{3L-3}\\|\theta-\tilde{ \theta}\\|_{\ell^{2}},\quad\forall p\in[d].\] Proof.: The proof is based on mathematical induction, and it is divided into three steps. (i) Prove the bound for the case $\ell=1$. By the identity (5.5) and triangle inequality, for $\ell=1$, \[\big{\|}\partial_{x_{p}}^{2}f_{q}^{(1)}-\partial_{x_{p}}^{2}\tilde{f }_{q}^{(1)}\big{\|}=\Big{\|}(a_{qp}^{(1)})^{2}\rho^{\prime\prime}\Big{(}\sum_{j=1} ^{n_{0}}a_{qj}^{(1)}x_{j}+b_{q}^{(1)}\Big{)}-(\tilde{a}_{qp}^{1})^{2}\rho^{ \prime\prime}\Big{(}\sum_{j=1}^{n_{0}}\tilde{a}_{qj}^{(1)}x_{j}+\tilde{b}_{q}^ {(1)}\Big{)}\Big{\|}\] \[\leq\] \[\leq 2R\big{\|}a_{qp}^{(1)}-\tilde{a}_{qp}^{(1)}\big{\|}+\eta R^{2}\sum _{j=1}^{n_{0}}\big{\|}a_{qj}^{(1)}-\tilde{a}_{qj}^{(1)}\big{\|}+\eta R^{2}\big{\|} b_{q}^{(1)}-\tilde{b}_{q}^{(1)}\big{\|}\leq 3\eta R^{2}\sum_{k=1}^{\mathbf{n}_{1}}\big{\|} \theta_{k}-\tilde{\theta}_{k}\big{\|}, \tag{5.7}\] since by definition, $\eta\geq 1$. (ii) Derive the recursion relation.* For $\ell=2,\ldots,L-1$, in view of the identity (5.5), we have \[\big{\|}\partial_{x_{p}}^{2}f_{q}^{(\ell)}-\partial_{x_{p}}^{2} \tilde{f}_{q}^{(\ell)}\big{\|}\leq \Big{\|}\Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}\partial_{x_{p} }f_{j}^{(\ell-1)}\Big{)}^{2}-\Big{(}\sum_{j=1}^{n_{\ell-1}}\tilde{a}_{qj}^{( \ell)}\partial_{x_{p}}\tilde{f}_{j}^{(\ell-1)}\Big{)}^{2}\Big{\|}\Big{\|}\rho^{ \prime\prime}\Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}f_{j}^{(\ell-1)}+b_{ q}^{(\ell)}\Big{)}\Big{\|}\] \[+\Big{\|}\rho^{\prime}\Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell) }f_{j}^{(\ell-1)}+b_{q}^{(\ell)}\Big{)}-\rho^{\prime}\Big{(}\sum_{j=1}^{n_{ \ell-1}}\tilde{a}_{qj}^{(\ell)}\tilde{f}_{j}^{(\ell-1)}+\tilde{b}_{q}^{(\ell)} \Big{)}\Big{\|}\Big{\|}\Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}\partial_{x_ {p}}^{2}f_{j}^{(\ell-1)}\Big{)}\Big{\|}\] \[+\Big{\|}\rho^{\prime}\Big{(}\sum_{j=1}^{n_{\ell-1}}\tilde{a}_{qj} ^{(\ell)}\tilde{f}_{j}^{(\ell-1)}+\tilde{b}_{q}^{(\ell)}\Big{)}\Big{\|}\Big{\|} \Big{(}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}\partial_{x_{p}}^{2}f_{j}^{(\ell- 1)}-\sum_{j=1}^{n_{\ell-1}}\tilde{a}_{qj}^{(\ell)}\partial_{x_{p}}^{2}\tilde{f }_{j}^{(\ell-1)}\Big{\|}=\sum_{m=1}^{4}\text{I}_{m}.\] Below we bound the four terms separately. To bound the term $\text{I}_{1}$, by the estimate (5.3), \[\sum_{j=1}^{n_{\ell-1}}\Big{(}\big{\|}a_{qj}^{(\ell)}\big{\|}\big{\|} \partial_{x_{p}}f_{j}^{(\ell-1)}\big{\|}+\big{\|}\tilde{a}_{qj}^{(\ell)}\big{\|} \big{\|}\partial_{x_{p}}\tilde{f}_{j}^{(\ell-1)}\big{\|}\Big{)}\leq 2R\sum_{j=1}^{n_{\ell-1}} \pi_{\ell-2}R^{\ell-1}=2\pi_{\ell-1}R^{\ell}.\] Likewise, by the estimates (5.3) and (5.4), \[\sum_{j=1}^{n_{\ell-1}}\big{\|}a_{qj}^{(\ell)}\partial_{x_{p}}f_{j} ^{(\ell-1)}-\tilde{a}_{qj}^{(\ell)}\partial_{x_{p}}\tilde{f}_{j}^{(\ell-1)} \big{\|}\] \[\leq \sum_{j=1}^{n_{\ell-1}}\big{\|}a_{qj}^{(\ell)}-\tilde{a}_{qj}^{( \ell)}\big{\|}\big{\|}\partial_{x_{p}}f_{j}^{(\ell-1)}\big{\|}+\sum_{j=1}^{n_{ \ell-1}}\big{\|}\tilde{a}_{qj}^{(\ell)}\big{\|}\big{\|}\partial_{x_{p}}f_{j}^{( \ell-1)}-\partial_{x_{p}}\tilde{f}_{j}^{(\ell-1)}\big{\|}\] \[\leq \pi_{\ell-2}R^{\ell-1}\sum_{j=1}^{n_{\ell-1}}\big{\|}a_{qj}^{( \ell)}-\tilde{a}_{qj}^{(\ell)}\big{\|}+\sum_{j=1}^{n_{\ell-1}}R\cdot\Big{(}\ell \pi_{\ell-2}^{2}R^{2\ell-2}\sum_{k=1}^{\mathbf{n}_{\ell-1}}\big{\|}\theta_{k}- \tilde{\theta}_{k}\big{\|}\Big{)}\] \[\leq \pi_{\ell-2}R^{\ell-1}\sum_{j=1}^{n_{\ell-1}}\big{\|}a_{qj}^{( \ell)}-\tilde{a}_{qj}^{(\ell)}\big{\|}+\ell\pi_{\ell-2}\pi_{\ell-1}R^{2\ell-1} \sum_{j=1}^{\mathbf{n}_{\ell-1}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|}.\] Thus we can bound the term $\text{I}_{1}$ by \[\text{I}_{1}\leq 2\pi_{\ell-2}\pi_{\ell-1}R^{2\ell-1}\sum_{j=1}^{n_{\ell-1}} \big{\|}a_{qj}^{(\ell)}-\tilde{a}_{qj}^{(\ell)}\big{\|}+2\ell\pi_{\ell-2}\pi_{ \ell-1}^{2}R^{3\ell-1}\sum_{j=1}^{\mathbf{n}_{\ell-1}}\big{\|}\theta_{j}-\tilde{ \theta}_{j}\big{\|}.\] Next, it follows from the estimate (5.2) that \[\Big{\|}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{(\ell)}f_{j}^{(\ell-1)}-\tilde{a}_{qj}^{( \ell)}\tilde{f}_{j}^{(\ell-1)}+b_{q}^{(\ell)}-\tilde{b}_{q}^{(\ell)}\Big{\|}\leq \sum_{j=1}^{n_{\ell-1}}\big{\|}a_{qj}^{(\ell)}-\tilde{a}_{qj}^{(\ell)}\big{\|}+ \big{\|}b_{q}^{(\ell)}-\tilde{b}_{q}^{(\ell)}\big{\|}+\pi_{\ell-1}R^{\ell-1}\sum_ {j=1}^{\mathbf{n}_{\ell-1}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|}.\] Then for the term $\mathrm{I}_{2}$, from the estimate (5.3), we deduce \[\mathrm{I}_{2} \leq\eta\Big{(}\sum_{j=1}^{n_{\ell-1}}\big{\|}\tilde{a}_{qj}^{(\ell )}\big{\|}^{2}\Big{)}\Big{(}\sum_{j=1}^{n_{\ell-1}}\big{\|}\partial_{x_{p}}\tilde{ f}_{j}^{(\ell-1)}\big{\|}^{2}\Big{)}\Big{(}\Big{\|}\sum_{j=1}^{n_{\ell-1}}a_{qj}^{( \ell)}f_{j}^{(\ell-1)}-\tilde{a}_{qj}^{(\ell)}\tilde{f}_{j}^{(\ell-1)}+b_{q}^{ (\ell)}-\tilde{b}_{q}^{(\ell)}\Big{\|}\Big{)}\] \[\leq\eta\pi_{\ell-1}^{2}R^{2\ell}\Big{(}\sum_{j=1}^{n_{\ell-1}} \big{\|}a_{qj}^{(\ell)}-\tilde{a}_{qj}^{(\ell)}\big{\|}+\big{\|}b_{q}^{(\ell)}- \tilde{b}_{q}^{(\ell)}\big{\|}+\pi_{\ell-1}R^{\ell-1}\sum_{j=1}^{\boldsymbol{n }_{\ell-1}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|}\Big{)}.\] Similarly, Lemma 5.2 implies \[\sum_{j=1}^{n_{\ell-1}}\big{\|}a_{qj}^{(\ell)}\partial_{x_{p}}^{2}f_{j}^{(\ell- 1)}\Big{\|}\leq R\cdot n_{\ell-1}\cdot(\ell-1)\pi_{\ell-2}^{2}R^{2(\ell-1)}=( \ell-1)\pi_{\ell-2}\pi_{\ell-1}R^{2\ell-1},\] and thus we can bound the term $\mathrm{I}_{3}$ by \[\mathrm{I}_{3}\leq(\ell-1)\pi_{\ell-2}\pi_{\ell-1}R^{2\ell-1}\Big{(}\sum_{j=1 }^{n_{\ell-1}}\big{\|}a_{qj}^{(\ell)}-\tilde{a}_{qj}^{(\ell)}\big{\|}+\big{\|}b_{ q}^{(\ell)}-\tilde{b}_{q}^{(\ell)}\big{\|}+\pi_{\ell-1}R^{\ell}\sum_{j=1}^{ \boldsymbol{n}_{\ell-1}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|}\Big{)}.\] For the last term $\mathrm{I}_{4}$, using Lemma 5.2 again, we have \[\mathrm{I}_{4} \leq\sum_{j=1}^{n_{\ell-1}}\big{\|}a_{qj}^{(\ell)}-\tilde{a}_{qj}^ {(\ell)}\big{\|}\big{\|}\partial_{x_{p}}^{2}f_{j}^{(\ell-1)}\big{\|}+R\sum_{j=1}^ {n_{\ell-1}}\big{\|}\partial_{\partial x_{p}}^{2}f_{j}^{(\ell-1)}-\partial_{x_{ p}}^{2}\tilde{f}_{j}^{(\ell-1)}\big{\|}\] \[\leq(\ell-1)\pi_{\ell-2}\pi_{\ell-1}R^{2(\ell-1)}\sum_{j=1}^{n_{ \ell-1}}\big{\|}a_{qj}^{(\ell)}-\tilde{a}_{qj}^{(\ell)}\big{\|}+R\sum_{j=1}^{n_{ \ell-1}}\big{\|}\partial_{x_{p}}^{2}f_{j}^{(\ell-1)}-\partial_{x_{p}}^{2}\tilde {f}_{j}^{(\ell-1)}\big{\|}.\] Combining the last four estimates gives the crucial recursion \[\big{\|}\partial_{x_{p}}^{2}f_{q}^{(\ell)}-\partial_{x_{p}}^{2}\tilde{f}_{q}^{( \ell)}\big{\|}\leq R\sum_{j=1}^{n_{\ell-1}}\big{\|}\partial_{x_{p}}^{2}f_{j}^{( \ell-1)}-\partial_{x_{p}}^{2}\tilde{f}_{j}^{(\ell-1)}\big{\|}+4\ell\eta\pi_{ \ell-1}^{3}R^{3\ell-1}\sum_{j=1}^{\boldsymbol{n}_{\ell}}\big{\|}\theta_{j}- \tilde{\theta}_{j}\big{\|}. \tag{5.8}\] (iii) Prove the intermediate case by mathematical induction. Using the recursion (5.8), we claim that for $\ell=1,2,\ldots,L-1$, there holds \[\big{\|}\partial_{x_{p}}^{2}f_{q}^{(\ell)}(x)-\partial_{x_{p}}^{2}\tilde{f}_{q}^ {(\ell)}(x)\big{\|}\leq 2\ell(\ell+1)\eta\pi_{\ell-1}^{3}R^{3\ell-1}\sum_{j=1}^{ \boldsymbol{n}_{\ell}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|}. \tag{5.9}\] This is trivially true for $\ell=1$, by the estimate (5.7). Indeed, for $\ell=2$, the recursion (5.8) and the estimate for $\ell=1$ in (5.7) imply \[\big{\|}\partial_{x_{p}}^{2}f_{q}^{(2)}(x)-\partial_{x_{p}}^{2} \tilde{f}_{q}^{(2)}(x)\big{\|} \leq 3\eta n_{1}R^{3}\sum_{j=1}^{\boldsymbol{n}_{1}}\big{\|} \theta_{k}-\tilde{\theta}_{k}\big{\|}+8\eta R^{5}\pi_{1}^{3}\sum_{j=1}^{ \boldsymbol{n}_{2}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|}\] \[\leq 11\eta\pi_{1}^{3}R^{5}\sum_{j=1}^{\boldsymbol{n}_{2}}\big{\|} \theta_{j}-\tilde{\theta}_{j}\big{\|}\leq 2\cdot 2\cdot 3\eta\pi_{1}^{3}R^{5}\sum_{j=1}^{ \boldsymbol{n}_{2}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|},\] and hence the claim holds for $\ell=2$. Now suppose it holds for some $2\leq\ell<L-1$. Then for the case $\ell+1$, \[\Big{\|}\partial_{x_{p}}^{2}f_{q}^{(\ell+1)}(x)-\partial_{x_{p}}^{2 }\tilde{f}_{q}^{(\ell+1)}(x)\Big{\|} \leq R\sum_{j=1}^{n_{\ell}}\Big{\|}\partial_{x_{p}}^{2}f_{j}^{( \ell)}-\partial_{x_{p}}^{2}\tilde{f}_{j}^{(\ell)}\Big{\|}+4(\ell+1)\eta\pi_{\ell}^ {3}R^{3\ell+2}\sum_{j=1}^{\boldsymbol{n}_{\ell+1}}\Big{\|}\theta_{j}-\tilde{ \theta}_{j}\Big{\|}\] \[\leq n_{\ell}\cdot 2\ell(\ell+1)\eta\pi_{\ell-1}^{3}R^{3\ell-1}\sum_{j=1 }^{\boldsymbol{n}_{\ell}}\big{\|}\theta_{j}-\tilde{\theta}_{j}\big{\|}+4(\ell+1) \eta\pi_{\ell}^{3}R^{3\ell+2}\sum_{j=1}^{\boldsymbol{n}_{\ell+1}}\big{\|}\theta_{j} -\tilde{\theta}_{j}\big{\|}\] \[\leq 2(\ell+1)(\ell+2)\pi_{\ell}^{3}R^{3\ell+2}\sum_{j=1}^{\mathbf{n}_{\ell+1} }\|\theta_{j}-\tilde{\theta}_{j}\|,\] which by mathematical induction implies the desired claim (5.9). (iv) Obtained the final estimate. Last, for $\ell=L$, direct computation shows \[\partial_{x_{p}}^{2}f-\partial_{x_{p}}^{2}\tilde{f}=\sum_{j=1}^{n_{L-1}}a_{1j} ^{(L)}\partial_{x_{p}}^{2}f_{j}^{(L-1)}-\sum_{j=1}^{n_{L-1}}\tilde{a}_{1j}^{(L )}\partial_{x_{p}}^{2}\tilde{f}_{j}^{(L-1)}\] Thus, by the estimate (5.9) and Lemma 5.2, \[\left\|\partial_{x_{p}}^{2}f-\partial_{x_{p}}^{2}\tilde{f}\right\| \leq\sum_{j=1}^{n_{L-1}}\|a_{1j}^{(L)}-\tilde{a}_{1j}^{(L)}\|\| \partial_{x_{p}}^{2}f_{j}^{(L-1)}\|+\sum_{j=1}^{n_{L-1}}\|\tilde{a}_{1j}^{(L)}\| \partial_{x_{p}}^{2}f_{j}^{(L-1)}-\partial_{x_{p}}^{2}\tilde{f}_{j}^{(L-1)}\|\] \[\leq(L-1)\pi_{L-2}R^{2L-2}\sum_{j=1}^{n_{L-1}}\|a_{1j}^{(L)}- \tilde{a}_{1j}^{(L)}\|+\sum_{j=1}^{n_{L-1}}R\cdot 2(L-1)L\eta\pi_{L-2}^{3}R^{3L-4} \sum_{j=1}^{\mathbf{n}_{L-1}}\|\theta_{j}-\tilde{\theta}_{j}\|\] \[\leq 2(L-1)L\eta\pi_{L-1}^{3}R^{3L-3}\sum_{j=1}^{\mathbf{n}_{L}} \left\|\theta_{j}-\tilde{\theta}_{j}\right\|.\] This completes the proof of the lemma. ## 6 Numerical experiments and discussions Now we present numerical experiments to illustrate C-PINN, and evaluate its performance against existing techniques, including penalty method (PM), augmented Lagrangian method (ALM), and adjoint oriented neural network (AONN) described in Section 3.4. Throughout, we fix the weight $\alpha_{i}=\lambda^{-1}$ in C-PINN, and use the same boundary weights for state and adjoint PDEs, i.e. $\alpha_{b}^{y}=\alpha_{b}^{p}/\alpha_{i}$. Thus we only need to tune one single hyper-parameter $\alpha_{b}^{y}$ in C-PINN. ANOO PM and ALM involve more hyper-parameters. The boundary weight $\alpha_{b}$ of the PINN residual is determined by grid search [32]: we apply PINN to solve a direct problem with known ground truth, with a range of values of $\alpha_{b}$, and then select the value $\alpha_{b}^{}$ attaining the smallest validation error after a fixed number of L-BFGS iterations as the optimal one, which is then used in the solver for the state and adjoint problems. Other hyper-parameters, e.g., initial penalty factor and increasing factor $\beta$ in PM and ALM, and step size and the number $K$ of gradient descent steps in AONN are all determined manually; see Table 1 for the hyper-parameters for the examples. For AONN, the tuple $N_{\text{aonn}}$ denotes the iteration number for PDE solvers (both PDE and adjoint PDE) and for the gradient descent step in (3.12); and for PM and ALM, the tuple $N_{\text{pm}}$ and $N_{\text{alm}}$ denote the iteration for the first and remaining sub-problems. The multipliers in ALM are initialized to zero. The function $\rho$ is taken to be $\tanh$. The DNN parameters are initialized by the Pytorch default Xavier scheme (and zero bias). To measure the accuracy of the approximate state $y^{}$ and control $u^{}$, we employ the relative $L^{p}(\Omega)$ error, $p\in\{2,\infty\}$, defined by $e_{p}(y^{})=\\|y^{}-\bar{y}\\|_{L^{p}(\Omega)}/\\|\bar{y}\\|_{L^{p}(\Omega)}$, where $\bar{y}$ denotes the exact state, and similarly the relative error $e_{p}(u^{})$ for the control $u^{}$. Examples 6.1, 6.2 and 6.4 are conducted on Macbook air with Apple M2 silicon, 16 GB RAM and mac OS Monterey using double precision. Example 6.3 is conducted on an NVIDIA RTX A6000 GPU in single precision. The Python code for reproducing the results will be made publicly available. First, we consider the linear elliptic control problem: \[\min_{y,u}\Big{\{}J(y,u)=\frac{1}{2}\\|y-y_{d}\\|_{L^{2}(\Omega)}^{2}+\frac{ \lambda}{2}\\|u\\|_{L^{2}(\Omega)}^{2}\Big{\}},\quad\text{subject to}\ \begin{cases}-\Delta y=f+u,&\text{in}\ \Omega,\\ y=g,&\text{on}\ \partial\Omega,\end{cases}\] where $f$ and $g$ are the known problem data and $\lambda$ is the penalty weight. The first example is about an unconstrained elliptic optimal control problem. Example 6.1.: _The domain $\Omega=\{(r,\theta):1\leq r\leq 3,\theta\in[0,2\pi]\}$, and the penalty weight $\lambda=0.01$, $y_{d}(r,\theta)=r^{2}+3\lambda(1-\frac{1}{r^{2}})\sin(\theta)$, $f(r,\theta)=(r-1)(r-3)\sin(\theta)-4$, and Dirichlet data $g(r,\theta)=r^{2}$. The exact state $\bar{y}$ and control $\bar{u}$ are given by $\bar{y}(r,\theta)=r^{2}$ and $\bar{u}(r,\theta)=(r-1)(r-3)\sin(\theta)$._ For both state and adjoint state, we use a DNN architecture with 4 hidden layers and 30 neurons in each layer, which is also used to approximate the control $u$ in the benchmark methods. When formulating the empirical loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$, we take $n_{d}=10000$ points i.i.d. from $U(\Omega)$ for the PDE residual, and $n_{b}=3000$ points i.i.d. from $U(\partial\Omega)$. For all four methods, we minimize the loss $\widehat{\mathcal{L}}(y_{\theta},p_{\sigma})$ using L-BFGS provided by the PyTorch library torch.optim (version 2.0.0). The relative errors of the approximate state $y^{}$ and control $u^{}$ are presented in Table 2. The computing time is also given, which is determined by choosing a near-minimum number of iterations that still achieves the reported accuracy. All the methods can achieve a relative error about $10^{-4}$. In view of the well-known accuracy saturation of PINN-type solvers for direct problems [34, 25], the attained accuracy for the optimal control problem based on PINN is expected to be nearly the best possible, at least empirically. In Fig. 2, with a fixed step size $s=10$, AONN requires about 20 outer iterations to reach convergence. Penalty-based methods can capture the state accurately, but the control approximations are less accurate. In contrast, C-PINN gives the most accurate state approximation, lowest objective value within a shorter computing time. In Fig. 2 we show the training dynamics of the four methods. It is observed for C-PINN, the optimizer converges rapidly in terms of the objective value, and the PINN loss for the state and adjoint. In contrast, for AONN, it does minimize the objective steadily, but the PINN losses $\widehat{\mathcal{L}}_{\mathrm{pinn}}(y)$ and $\widehat{\mathcal{L}}_{\mathrm{pinn}}(p)$ exhibit alternating convergence behavior due to the alternating nature of the algorithm. PM and ALM show very similar convergence behavior in the early stage, and the objective value enjoys fast initial decay but the convergence slows down greatly as the iteration proceeds, and also the PINN loss $\widehat{\mathcal{L}}_{\mathrm{pinn}}(y)$ also decreases \begin{table} \begin{tabular}{c\|c c c c\|c c c c} \hline \hline Example & $\alpha_{b}$ & $(s,K)_{\mathrm{aonn}}$ & $(\mu_{0},\beta,K)_{\mathrm{pm}}$ & $(\mu,K)_{\mathrm{alm}}$ & $N_{\mathrm{cpinn}}$ & $N_{\mathrm{p}m}$ & $N_{\mathrm{alm}}$ \\ \hline 6.1 & 5 & (10, 30) & (0.1, 2, 8) & (0.1, 8) & 1.5e4 & (1e3, 5e2) & (6e3, 3e3) & (6e3, 3e3) \\ \hline 6.2 & 5 & (10, 30) & (0.1, 2, 8) & (0.1, 8) & 1.5e4 & (1e3, 5e2) & (1.2e4, 1e4) & (6e3, 3e3) \\ \hline 6.3 & 100 & (100, 15) & (0.5, 2, 5) & (2, 8) & 6e4 & (6e3, 1.5e3) & (2e4, 2e4) & (2e4, 2e4) \\ \hline 6.4 & 100 & (100, 20) & (1, 2, 8) & (16, 8) & 2e4 & (2e3, 5e2) & (1e4, 8e3) & (1e4, 8e3) \\ \hline \hline \end{tabular} \end{table} Table 1: Hyper-parameter settings for the Examples. $\alpha_{b}$ denotes boundary penalty weight, $s$ denotes the step size in AONN, $\mu_{0}$ the initial penalty weight and $\beta$ the increasing factor, $\mu$ penalty weight in ALM, and $K$ denotes the update / path-following steps, and $N$ the maximum number of iterations for the optimizer. Figure 1: The approximate control $u^{}$ (top), its pointwise error $\|u^{}-\bar{u}\|$ (middle) and approximate state $y^{}$ (bottom) obtained by four DNN-based methods for Example 6.1. rapidly throughout. There is a drastic increase of the PINN error in the later stage of ALM, indicating training instability. This can be alleviated by choosing a larger penalty weight, as usually recommended for convex optimization [5, 37, 6], or safeguarding the multipliers [6]. We choose $\mu=0.1$ in the experiment since it leads to the smallest cost objective with an acceptable PINN loss in the grid search process. All methods involve some hyper-parameters, and the ease of their tuning is of great importance. Table 3 shows the variation of the performance metric with respect to one hyper-parameter. C-PINN enjoys good accuracy for a wide range of parameters. AONN is effective if the step size $s$ is carefully tuned: a too large $s$ can lead to divergence, and a too small $s$ requires many outer iterations to yield accurate approximations. PM and ALM also stably yield accurate results, but in PM, a larger $\mu_{0}$ can drastically increase the error, and more iterations are required to make the sub-problem converge, due to their ill-condition. Next, we consider a constrained problem. Example 6.2.: _The domain $\Omega$, penalty weight $\gamma$, and problem data $y_{d}$ and $g$ are identical with that for Example 6.1, but with box constraint on the control, i.e., $u\in U:=\{u\in L^{2}(\Omega):u(x)\in[-0.5,0.7]$ a.e. in $\Omega\}$. The source $f(r,\theta)=-4-P_{[-0.5,0.7]}((1-r)(r-3)\sin(\theta))$, and the exact state $\bar{y}$ and control $\bar{u}$ are given by $\bar{y}(r,\theta)=r^{2}$ and $\bar{u}(r,\theta)=P_{[-0.5,0.7]}((1-r)(r-3)\sin(\theta))$, respectively._ In addition to the hyper-parameters in Example 6.1, in PM and ALM, we impose the constraint by including a penalty term with a weight $\mu^{\prime}=1$ and increasing factor $\beta=2$, cf. (3.13). Meanwhile, since the \begin{table} \end{table} Table 2: The numerical results for Example 6.1. \begin{table} \end{table} Table 3: Hyper-parameter sensitivity of the four methods for Example 6.1. Figure 2: The training dynamics of the four methods, for Example 6.1. loss is less smooth due to the presence of the projection operator $P_{U}$, we increase the maximum iteration numbers for PM. The results are reported in Table 4 and Fig. 3. Since the optimal control $\bar{u}$ is non-smooth near the boundary of the constraint active area (and more challenging to approximate with DNNs), all methods except C-PINN suffer from pronounced errors therein. In C-PINN, this error is greatly alleviated due to the use of $P_{U}$ a posteriori on the adjoint $p^{}$. Both PM and ALM become less efficient with large relative $L^{2}(\Omega)$ errors in the approximation $u^{}$, and numerically, the choice of the penalty parameter $\mu$ becomes even more crucial for obtaining acceptable DNN approximations. The next example is about the medium-dimensional case. Example 6.3.: _The domain $\Omega=(0,1)^{4}$, and penalty weight $\lambda=0.01$. The data $f\equiv 0$, $y_{d}(x)=(1-16\lambda\pi^{4})\prod_{i=1}^{4}\sin(\pi x_{i})$, and $g=0$. The exact state $\bar{y}$ and control $\bar{u}$ are given by $\bar{y}(x)=\prod_{i=1}^{4}\sin(\pi x_{i})$, and $\bar{u}(x)=4\pi^{2}\prod_{i=1}^{4}\sin(\pi x_{i})$, respectively._ The experiment setup is similar to Example 6.1, but with the following changes. The DNN has 4 hidden layers, each having 80 neurons. To form the empirical loss, we employ 60000 points in the domain $\Omega$ and 5000 points on the boundary $\partial\Omega$, drawn i.i.d. from $U(\Omega)$ and $U(\partial\Omega)$, respectively. Due to the larger data size, we conduct all the training in single precision. See Table 1 for the hyper-parameters. Note that for PM, we terminate the iteration at a relatively small weight $\mu_{4}=8$ because the sub-problem hardly converges within the specified optimizer iterations when the weight continues to grow. Due to the increased complexity of the problem, we employ 60000 Adam iterations for C-PINN with a learning rate $10^{-3}$, divide it by 10 after the 20000th and 40000th iterations; and 6000 Adam iterations for each PDE solve in AONN with \begin{table} \begin{tabular}{c\|c\|c\|c\|c\|c\|c} \hline & $e_{2}(y^{})$ & $e_{\infty}(y^{})$ & $e_{2}(u^{})$ & $e_{\infty}(u^{})$ & $J(u^{},y^{})$ & time (s) \\ \hline C-PINN & 3.49e-5 & 7.91e-5 & 6.16e-3 & 2.39e-2 & 1.026e-3 & 1848.68 \\ AONN & 6.86e-5 & 1.91e-4 & 3.53e-2 & 1.65e-1 & 1.029e-3 & 4676.55 \\ PM & 2.85e-4 & 3.98e-4 & 6.23e-2 & 2.72e-1 & 1.032e-3 & 6492.62 \\ ALM & 3.00e-4 & 7.89e-4 & 6.87e-2 & 2.94e-1 & 1.040e-3 & 2543.71 \\ \hline \end{tabular} \end{table} Table 4: The numerical results for Example 6.2. Figure 3: The approximate control $u^{}$ (top), its pointwise error $\|u^{}-\bar{u}\|$ (mid) and approximate state $y^{}$ (bottom) obtained by four DNN-based methods for Example 6.2. 15 gradient descent steps in total. In PM and ALM, we employ 20000 iterations for each sub-problem. In AONN, PM, and ALM, we take a learning rate 1e-3 in the first sub-problem, and then 1e-4 for the remaining sub-problems. The numerical results are reported in Table 5. C-PINN and AONN achieve the relative error at about $10^{-2}$ for both $y^{}$ and $u^{}$. C-PINN yields a more accurate control approximation $u^{}$ in the $L^{2}(\Omega)$ error, and the achieved objective value is also lower than that by AONN. PM and ALM both report good accuracy for $y^{}$, but much larger errors for $u^{}$. The last example is about optimal control with semilinear elliptic PDEs. Example 6.4.: _Consider the following semilinear PDE:_ \[\begin{cases}-\Delta y+y+q(x,y)=f+u,&\text{in }\Omega,\\ y=g,&\text{on }\partial\Omega,\end{cases}\] _with $q(x,y)=k(x)y(x)^{3}$. The domain $\Omega=(0,1)^{2}$ is divided into two parts $\omega=[0.25,0.75]^{2}$ and $\Omega\setminus\omega$, and the function $k\equiv 1$ in $\omega$, and $k\equiv 3$ in $\Omega\setminus\omega$. We take a penalty weight $\lambda=0.01$. The problem data $y_{d}$, $g$ and $f$ are taken such that the exact state $\bar{y}$ and control $\bar{u}$ are given by $\bar{y}(x)=e^{x_{1}(1-x_{1})}\sin(\pi x_{2})+e^{x_{2}(1-x_{2})}\sin(\pi x_{1})$, and $\bar{u}(x)=-\lambda^{-1}x_{1}x_{2}(1+\cos(\pi x_{1})(1+\cos(\pi x_{2})))$, respectively._ We employ the DNN architecture for Example 6.1. Due to the nonlinearity of the state problem, each forward solve requires more L-BFGS iterations to reach convergence, cf. Table 1 for the precise setting. The results are reported in Table 6 and Fig. 5. Since exact state $\bar{y}$ and control $\bar{u}$ have very good regularity, \begin{table} \begin{tabular}{c\|c\|c\|c\|c\|c\|c} \hline \hline & $e_{2}(y^{})$ & $e_{\infty}(y^{})$ & $e_{2}(u^{})$ & $e_{\infty}(u^{})$ & $J(u^{},y^{})$ & time (sec) \\ \hline C-PINN & 2.44e-2 & 8.25e-2 & 9.29e-3 & 4.37e-2 & 8.020 & 6738.26 \\ AONN & 6.83e-3 & 1.34e-2 & 1.03e-2 & 2.38e-2 & 8.031 & 9579.84 \\ PM & 3.27e-2 & 2.10e-2 & 9.49e-2 & 1.05e-1 & 8.024 & 7342.11 \\ ALM & 1.99e-2 & 1.87e-2 & 7.25e-2 & 8.83e-2 & 8.033 & 12485.55 \\ \hline \hline \end{tabular} \end{table} Table 5: The numerical results for Example 6.3. Figure 4: The approximate optimal control $u^{}$ (top), its pointwise error $\|u^{}-\bar{u}\|$ (mid), and approximate state $y^{}$ (bottom) obtained by four DNN-based methods for Example 6.3, cross section at $x_{3}=x_{4}=0.5$. both PM and ALM can yield accurate approximations, and yield smaller cost objectives at the expense of less accuracy in the state solution. In contrast, both C-PINN and AONN can approximate the state and control with smaller $L^{2}(\Omega)$ and $L^{\infty}(\Omega)$ errors, but C-PINN approximates the optimal control more accurately within a shorter time. Note that the efficiency of AONN depends heavily on the step size $s$, but line search is not effective in the context since it involves multiple expensive forward solves. The step size $s=100$ was determined in a trial-and-error manner in order to achieve good efficiency. ## 7 Conclusion In this work, we have developed a new approach to solve distributed optimal control problems associated with elliptic PDEs, with / without box constraint on the control variable. The approach is derived from the first-order necessary optimality system, and then applies physics informed neural networks to the reduced optimality system. We provide an error analysis of the approach, using the approximation theory of neural networks and offset Rademacher complexity. The $L^{2}(\Omega)$ error bound on the control, state, and adjoint is explicit in terms of the DNN parameters (depth, width), and the number of Monte Carlo sampling points in the domain and on the boundary. Several examples confirm the feasibility of the approach for linear and semilinear elliptic problems, indicating its potential for solving a class of optimal control problems. \begin{table} \begin{tabular}{c\|c\|c\|c\|c\|c} \hline \hline & $e_{2}(y^{})$ & $e_{\infty}(y^{})$ & $e_{2}(u^{})$ & $e_{\infty}(u^{})$ & $J(u^{},y^{})$ & time (sec) \\ \hline C-PINN & 6.49e-5 & 3.36e-4 & 2.68e-4 & 1.66e-3 & 16.063 & 2584.97 \\ AONN & 4.70e-5 & 3.48e-4 & 1.17e-3 & 4.37e-3 & 16.063 & 3335.25 \\ PM & 1.20e-3 & 4.04e-3 & 4.16e-2 & 9.32e-2 & 16.060 & 5262,92 \\ ALM & 9.60e-4 & 2.01e-3 & 3.09e-2 & 7.93e-2 & 16.062 & 5077.21 \\ \hline \hline \end{tabular} \end{table} Table 6: The numerical results for Example 6.4. Figure 5: The approximate optimal control $u^{}$ (top), its pointwise error $\|u^{}-\bar{u}\|$ (middle), and the state $y^{}$ (bottom) obtained by four different methods for Example 6.4.
2305.12585	GeometricImageNet: Extending convolutional neural networks to vector and tensor images	Convolutional neural networks and their ilk have been very successful for many learning tasks involving images. These methods assume that the input is a scalar image representing the intensity in each pixel, possibly in multiple channels for color images. In natural-science domains however, image-like data sets might have vectors (velocity, say), tensors (polarization, say), pseudovectors (magnetic field, say), or other geometric objects in each pixel. Treating the components of these objects as independent channels in a CNN neglects their structure entirely. Our formulation -- the GeometricImageNet -- combines a geometric generalization of convolution with outer products, tensor index contractions, and tensor index permutations to construct geometric-image functions of geometric images that use and benefit from the tensor structure. The framework permits, with a very simple adjustment, restriction to function spaces that are exactly equivariant to translations, discrete rotations, and reflections. We use representation theory to quantify the dimension of the space of equivariant polynomial functions on 2-dimensional vector images. We give partial results on the expressivity of GeometricImageNet on small images. In numerical experiments, we find that GeometricImageNet has good generalization for a small simulated physics system, even when trained with a small training set. We expect this tool will be valuable for scientific and engineering machine learning, for example in cosmology or ocean dynamics.	Wilson Gregory, David W. Hogg, Ben Blum-Smith, Maria Teresa Arias, Kaze W. K. Wong, Soledad Villar	2023-05-21T22:44:18Z	http://arxiv.org/abs/2305.12585v1	# GeometricImageNet: Extending convolutional neural networks to vector and tensor images ###### Abstract Convolutional neural networks and their ilk have been very successful for many learning tasks involving images. These methods assume that the input is a scalar image representing the intensity in each pixel, possibly in multiple channels for color images. In natural-science domains however, image-like data sets might have vectors (velocity, say), tensors (polarization, say), pseudovectors (magnetic field, say), or other geometric objects in each pixel. Treating the components of these objects as independent channels in a CNN neglects their structure entirely. Our formulation--the GeometricImageNet--combines a geometric generalization of convolution with outer products, tensor index contractions, and tensor index permutations to construct geometric-image functions of geometric images that use and benefit from the tensor structure. The framework permits, with a very simple adjustment, restriction to function spaces that are exactly equivariant to translations, discrete rotations, and reflections. We use representation theory to quantify the dimension of the space of equivariant polynomial functions on 2-dimensional vector images. We give partial results on the expressivity of GeometricImageNet on small images. In numerical experiments, we find that GeometricImageNet has good generalization for a small simulated physics system, even when trained with a small training set. We expect this tool will be valuable for scientific and engineering machine learning, for example in cosmology or ocean dynamics. ## 1 Introduction Contemporary natural science and engineering is replete with data sets that are images, lattices, or grids of geometric objects. These might be observations of intensities (scalars), magnetic fields (pseudovectors), or polarizations (2-tensors) on a surface or in a volume. They might be the inputs or outputs of a simulation where the initial conditions or fields are specified on a regular grid; see Figure 1 for some examples. Any lattice of vectors or tensors can be seen as a generalization of the concept of an image in which the intensity in each pixel is replaced with a geometric object -- scalar, vector, tensor, or their pseudo counterparts. These objects are _geometric_ in the sense that they are defined in terms of their transformation properties under geometric operators such as rotation, translation, and reflection. Thus there is a need for machine learning methods designed for _geometric images_--lattices or grids of scalars, vectors, and tensors. There are already countless applications of machine learning in contexts in which the input data are geometric images, including examples in essentially all natural-science disciplines. At the present day, the go-to tools for machine learning with images are convolutional neural networks (CNNs; [32]) and their many descendants, including residual networks (ResNets) [25], dense networks (DenseNets)[26], and attention mechanisms such as transformers [43]. Other recent tools for machine learning with images include generative adversarial networks (GANs)[19] for image synthesis and style transfer, and recurrent neural networks (RNNs)[40] for tasks such as image captioning and video analysis. Additionally, transfer learning [54] has emerged as a powerful technique for leveraging pre-trained models on large image datasets to improve performance on smaller or specialized datasets. Traditional CNNs are designed to work on one- or few-channel images in which the early layers of the network involve image convolutions with learned filters followed by the application of pointwise nonlinearities. In typical contexts, the channels of multi-channel input images will be something like the red, green, and blue channels of a color image; these can be combined arbitrarily in the layers of the CNN. When these CNN-based tools are applied to lattices of vectors, typically the components of the vectors are just treated as channels of the input image and then everything proceeds as with multi-channel color images. This ignores the inherent structure of the vectors, but, to the chagrin of the physicists, there are many projects that have had great success using this strategy on geometric images. However, there are better choices. Here we propose a set of tools that generalize the concept of convolution to apply to geometric images such that the outputs of the convolutions are also geometric images, obeying the same geometric transformation rules as the inputs. The fundamental observation inspiring this work is that when an arbitrary function is applied to the components of vectors and tensors, the geometric structure of these objects is destroyed. There are strict rules, dating back to the early days of differential geometry [39], about how geometric objects can be combined to produce new geometric objects, consistent with coordinate freedom and transformation rules. These rules constitute a theme of [41], where they are combined into a _geometric principle_. In previous work [44, 45, 52] we have capitalized on the geometric principle to develop modified machine-learning methods that are restricted to exactly obey group-theoretic equivariances in physics contexts. More broadly there is a growing field of physics informed machine learning [29, 37]. Here we use these rules to create a comprehensive set of tools that parameterize functions that take geometric images as input and produce geometric images as output. Tensors can be defined--and distinguished from mere arrays of numbers--in two ways. In one, a tensor of order $k$ is a $k$-multilinear function of $k$ vector inputs that returns a scalar, an object whose value is invariant to changes in the coordinate system ([41] Section 1.3). In the other, a tensor of order $k$ is defined by the way that its components transform under rotations ([41] Section 1.6). We will take the latter point of view, and this definition will be made precise in Section 2.1. Figure 1: Examples of geometric images in the natural sciences. (a) A visualization of a temperature map and a polarization map from the ESA Planck Mission [9]. The color map shows a temperature field (a scalar or $0_{(+)}$-tensor) on the sphere, and the whiskers show the principal eigenvector direction of a $2_{(+)}$-tensor field in two dimensions. In detail the underlying data are represented on a pixel grid (healpixel [22]) on the sky (a 2-sphere). ($b$) Two-dimensional maps of ocean current (shown with arrows; a vector or $1_{(+)}$-tensor field) and ocean salinity (shown with color; a scalar or $0_{(+)}$-tensor field) at a depth of $5\,\mathrm{m}$[1]. (c) A three-dimensional map of temperature (a scalar or $0_{(+)}$-tensor field) based on sensors distributed throughout the volume of a granary [46]. (d) A two-dimensional map of potential vorticity (a pseudoscalar or $0_{(-)}$-tensor field) in the Earth’s atmosphere, measured for the purposes of predicting storms [34]. (e) Two-dimensional maps on the sky of intensity $I$ and the three independent components $Q,U,V$ of the electromagnetic polarization $2_{(+)}$-tensor, from a simulation of a jet outflow from an accreting black hole [Davelaar et al, in preparation]. (f) Components of the three-dimensional stress tensor (a $2_{(+)}$-tensor field) in a diamond anvil cell, which is used to study the behavior of samples at exceedingly high pressures [33]. There are two ways to think about transformations--_alias_ and _alibi_. In the former (alias), the idea is that the transformation is applied to the coordinate system, not the vectors and tensors themselves. This transformation leaves the geometric objects unchanged but all of their components change because they are now being represented in a changed coordinate system. In the latter (alibi), the idea is that the coordinate system is fixed and all of the geometric objects are taken through an identical transformation. In either case--alias or alibi--the key idea is that _all_ of the components of _all_ of the vectors and tensors in play must be changed correspondingly, and at the same time. The geometric principle requires that for any function, all the inputs, constants, parameters, and outputs must undergo the same coordinate transformations simultaneously. In other words, all valid functions will be fundamentally equivariant with respect to coordinate transformations. We are motivated in this work to help solve problems in the natural sciences and engineering, where geometric images abound. However, we conjecture that these tools are probably very useful even for standard machine-learning image-recognition and image-regression tasks. After all, even standard images are measurements of a scalar, the intensity of light, at a regular grid of points on a two-dimensional surface. The laws of physics still govern the objects in a photograph and how light travels from the objects to the camera, so we may still expect to benefit from the rules of geometry. These rules of geometry--the consequences of the geometric principle--are roughly as follows: A $k$-tensor object (tensor of order $k$) in $d$ dimensions has $k$ indices, each of which can take a value from $1$ to $d$; that is, the $k$-tensor is an element of $(\mathbb{R}^{d})^{\otimes k}$. A $k$-tensor and a $k^{\prime}$-tensor can be multiplied with the outer product to make a $(k+k^{\prime})$-tensor object. To reduce the tensor order, a $k$-tensor can be contracted to a $(k-2)$-tensor object by identifying a pair of indices and summing over them. $1$-tensor objects are called vectors and $0$-tensor objects are called scalars. There are also negative-parity versions of all these (pseudoscalars, pseudovectors, and pseudotensors) and parity-changing contractions using the Levi-Civita symbol, so in what follows we will define $k_{(p)}$-tensors that have $k$ indices and a parity $p\in\{-1,+1\}$ (sometimes denoted "$-$" and "$+$" below). Two objects can only be added or subtracted if they have the same order $k$ and parity $p$. These rules define objects that can be given transformation rules under rotation and reflection such that functions made of these operations are coordinate-free, or equivariant to any change of coordinate system. The symmetries that suggest these rules are continuous symmetries. But of course images are usually--and for our purposes--discrete grids of values. This suggests that in addition to the continuous symmetries respected by the tensor objects in the image pixels there will be discrete symmetries for each geometric image taken as a whole. We will define these discrete symmetry groups and use them to define a useful kind of group equivariance for functions of geometric images. This equivariance, it turns out, is very easy to enforce, even for nonlinear functions of geometric images, provided that we compose our nonlinear functions from simple geometric operations. When we enforce this equivariance, the convolution filters that appear look very much like the differential operators that appear in discretizations of vector calculus. Our contribution:The rest of the paper is organized in the following manner. Section 2 defines geometric objects, geometric images, and the operations on each. Section 3 discusses equivariance of functions of geometric images with some important results building off of [30] and [7]. Section 4 describes how to explicitly count these equivariant functions using a result of Molien from 1897. Sections 5 and 6 describe how to build a GeometricImageNet and present a couple of small problems with numerical results. Finally, Section 7 discusses related work. The majority of the supporting propositions and proofs have been sequestered to the Appendix, as has a larger exploration of related work. ## 2 Geometric Objects and Geometric Images We define the geometric objects and geometric images that we use to generalize classical images in scientific contexts in Section 2.1 and Section 2.2. The main point is that the channels of geometric images--which will be like the components of vectors and tensors--are not independent. There is a set of allowed operations on geometric objects that respect the structure and the coordinate freedom of these objects. ### Geometric objects The geometric principle implies that geometric objects should be coordinate-free scalars, vectors, and tensors, or their negative-parity pseudo counterparts. To define these objects we start by stating the coordinate transformations, which, in this case, will be given by the orthogonal group. We fix $d$, the dimension of the space, which will typically be 2 or 3. The geometric objects are vectors and tensors. The orthogonal group $O(d)$ is the space of isometries of $\mathbb{R}^{d}$ that fix the origin. It acts on vectors and pseudovectors $v\in\mathbb{R}^{d}$ in the following way: \[g\cdot v=\det(M(g))^{\frac{1-p}{2}}\,M(g)\,v \tag{1}\] where for $g\in O(d)$, $M(g)\in\mathbb{R}^{d\times d}$ is the standard matrix representation of $g$ (i.e. $M(g)^{\top}\,M(g)=I$) and $p\in\{-1,+1\}$ is the parity of $v$. If $p=+1$ we obtain the standard $O(d)$ action on $\mathbb{R}^{d}$_vectors_. If $p=-1$ we obtain the $O(d)$ action on what in physics are known as _pseudovectors_. The objects are defined by the actions that they carry in the following sense: if $F$ is a function with geometric inputs, outputs, and parameters, then $F$ must be coordinate-free. In other words $F(g\cdot v)=g\cdot F(v)$ for all $v$ and all $g$. This is the mathematical concept of equivariance which we will explore further in Section 3. Definition 1 ($k_{(p)}$-tensors).: The space $\mathbb{R}^{d}$ equipped with the action $O(d)$ defined by (1) is the space of $1_{(p)}$_-tensors_. If $v_{i}$ is a $1_{(p_{i})}$-tensor, then $T:=v_{1}\otimes\ldots\otimes v_{k}$ is a _rank-1_$k_{(p)}$_-tensor_, where $p=\prod_{i=1}^{k}p_{i}$ and the action of $O(d)$ is defined as \[g\cdot(v_{1}\otimes\ldots\otimes v_{k})=(g\cdot v_{1})\otimes\ldots\otimes(g \cdot v_{k}). \tag{2}\] Higher rank $k_{(p)}$-tensors are defined as linear combinations of rank-1 $k_{(p)}$-tensors where the action of $O(d)$ is extended linearly. The set of $k_{(p)}$-tensors in $d$ dimensions is denoted $\mathcal{T}_{d,k,p}$. Remark (terminology and notation).: The parity $p$ is a signed bit, either $+1$ for positive parity or $-1$ for negative parity. Note the distinction between the _order_$k$ of the $k_{(p)}$-tensor, and the rank of the tensor. We could have a $2_{(p)}$-tensor of rank 1, like those we use in Definition 1. Remark (universality of transformations).: Critically, when a transformation is applied to any $k_{(p)}$-tensor, it must be applied to every other geometric object--every scalar, vector, and tensor of both parities--involved in any relevant mathematical expression. This includes all constants, and all inputs and outputs to any scalar, vector, or tensor functions. Related to this, there are both alias and alibi points of view that can be taken towards (2); that is, it can be seen as defining a change made to every tensor in a fixed coordinate system, or it can be seen as a change to the coordinate system in which every tensor is represented. In physics the $1_{(+)}$-tensors (such as velocities) are known as _vectors_, the $1_{(-)}$-tensors (such as angular momenta) are known as _pseudovectors_, the $0_{(+)}$-tensors (such as rest masses) are known as _scalars_, the $0_{(-)}$-tensors (such as surface vorticities) are known as _pseudoscalars_, the $k_{(-)}$-tensors with $k\geq 2$ are known as _pseudotensors_, and finally the $k_{(+)}$-tensors with $k\geq 2$ are the things that are commonly known as _tensors_. In general, any $k_{(p)}$-tensor can be written as a sum of outer products of order-1 tensors (vectors and pseudovectors), where each term in the sum is an outer product of $k$ order-1 tensors and the parity $p$ is the product of the parities of the input order-1 tensors. Definition 2 (outer products of tensors).: Given $a\in\mathcal{T}_{d,k,p}$ and $b\in\mathcal{T}_{d,k^{\prime},p^{\prime}}$, the _outer product_, denoted $a\otimes b$, is a tensor in $\mathcal{T}_{d,k+k^{\prime},p\,p^{\prime}}$ defined as $[a\otimes b]_{i_{1},\ldots,i_{k+k^{\prime}}}=[a]_{i_{1},\ldots,i_{k}}\,[b]_{i _{k+1},\ldots,i_{k+k^{\prime}}}$. Definition 3 (Einstein summation notation).: We use _Einstein summation notation_ where outer products are written in component form, and repeated indices are summed over. For example, in this notation, the product of two $2_{(+)}$-tensors (represented as two $d\times d$ matrices $A$ and $B$) is written as \[[A\,B]_{i,j}=[A]_{i,k}\,[B]_{k,j}:=\sum_{k=1}^{d}[A]_{i,k}\,[B]_{k,j} \tag{3}\] where $[A]_{i,k}$ is the $i,k$ element of matrix $A$, and the sum from 1 to $d$ on repeated index $k$ is implicit in the middle expression. This notation works for tensor expressions of any order, provided that every index appears either exactly once, so it isn't summed over, or exactly twice, so it is summed over. Remark (lower and upper indices).: In the original Einstein summation notation [14], or Ricci calculus [39], a distinction is made between lower and upper indices, which correspond to covariant and contravariant components. The pairs of indices that are summed always have one member of the pair an upper index and one member a lower index. We drop the upper/lower distinction here because we will work with intrinsically flat images that implicitly have the Riemmannian metric given by the identity matrix, such that there is no numerical difference between covariant and contravariant component values for a given object. That said, there truly is a distinction (for example, if a spatial displacement is a contravariant vector, the gradient of a scalar function with respect to that spatial displacement is a covariant vector), so there might be advantages to reinstating this distinction. In summation notation, the group action of (1) on $k_{(p)}$-tensor $b$ is explicitly written \[[g\cdot b]_{i_{1},\ldots,i_{k}}=\det(M(g))^{\frac{1-p}{2}}\,[b]_{j_{1},\ldots, j_{k}}\,[M(g)]_{i_{1},j_{1}}\cdots[M(g)]_{i_{k},j_{k}} \tag{4}\] for all $g\in O(d)$, where $[b]_{i_{1},\ldots,i_{k}}\in\mathbb{R}$ is a component of $b$, $[M(g)]_{i,j}\in\mathbb{R}$ is the $i,j$ element of the matrix representation of $g$, and all the $i_{m}$ and $j_{m}$ are indices in the range $1,\ldots,d$. For example, a $2_{(+)}$-tensor has the transformation property $[g\cdot b]_{i,j}=[b]_{k,\ell}\,[M(g)]_{i,k}\,[M(g)]_{j,\ell}$, which, in normal matrix notation, is written as $g\cdot b=M(g)\,b\,M(g)^{\top}$. We consider two special tensors that will be important for the definition of our models, the Kronecker delta and the Levi-Civita symbol. Definition 4 (Kronecker delta).: The _Kronecker delta_, $\delta$, is a $2_{(+)}$-tensor represented by the identity matrix, namely the object with two indices $i,j$ such that it has the value $+1$ when the two indices have the same value ($i=j$), and $0$ otherwise. Definition 5 (Levi-Civita symbol).: The _Levi-Civita symbol_ in dimension $d\geq 2$ is a $d_{(-)}$-tensor $\epsilon$ such that if the $d$ indices are not repeated and in an even-permutation order the value is $+1$ and if the $d$ indices are not repeated and in an odd-permutation order the value is $-1$, and it has the value $0$ in all other cases. Definition 6 (contractions).: Given tensor $a\in\mathcal{T}_{d,k,p}$, where $k\geq 2$, and given $\mu,\nu\in[k],\mu\neq\nu$, the _contraction_$T(a,\mu,\nu)\in\mathcal{T}_{d,k-2,p}$ is defined as: \[[T(a,\mu,\nu)]_{i_{1},\ldots,i_{k}\setminus\{i_{\mu},i_{\nu}\}}=[\delta]_{i_{ \mu}i_{\nu}}\,[a]_{i_{1},\ldots,i_{\mu},\ldots,i_{\nu},\ldots,i_{k}} \tag{5}\] That is, we view the components of $a$ with given fixed values for $i_{\mu}$ and $i_{\nu}$ as forming a $(k-2)_{(p)}$-tensor, and then we take the sum of these tensors of order $k-2$ where $i_{\mu}=i_{\nu}$. We can also define the composition of multiple contractions as a _multicontraction_: \[T_{M}(a,(\mu_{1},\mu_{2}),\ldots,(\mu_{\ell},\mu_{\ell+1}))=T(\cdot,\mu_{\ell},\mu_{\ell+1})\circ\ldots\circ T(a,\mu_{1},\mu_{2})\, \tag{6}\] where $\mu_{1},\ldots,\mu_{\ell+1}\in[k]$ are all distinct. Note that because $\mu_{1},\ldots,\mu_{\ell+1}$ are integers referring to the indices of the axes being contracted, the indices may change when swapping from a multicontraction to multiple contractions. For example, if $k\geq 4$, \[T_{M}(a,(1,3),(2,4))=T(T(a,1,3),1,2)\] because axes $i_{1}$ and $i_{3}$ will disappear, so $i_{2}$ becomes the new $i_{1}$ and $i_{4}$ becomes the new $i_{2}$. Finally, the _Levi-Civita contraction_ is defined for $k\geq d-1$ and $\mu_{1},\ldots,\mu_{d-1}\in[k]$ distinct as the following: \[T_{LC}(a,\mu_{1},\ldots,\mu_{d-1})=T_{M}(a\otimes\epsilon,(\mu_{1},k+1), \ldots,(\mu_{d-1},k+d-1))\, \tag{7}\] where $\epsilon$ is the Levi-Civita symbol. Remark (negative-parity objects).: With a slight modification of the Levi-Civita contraction, there is an invertible function that converts any negative-parity object to a positive-parity object. Thus it is possible to work without negative-parity objects at all. We will use this idea to improve the efficiency of our algorithms for certain settings in Section 5.2. However, since negative-parity objects are important in physics and engineering (see Figure 1), we retain them in the model. We can combine multiplication with Kronecker and Levi-Civita symbols with contractions to define relevant operations. For example the $2_{(+)}$-tensor formed by the outer product of $1_{(+)}$-tensors $a$ and $b$ can be contracted with the Kronecker delta to give the standard dot product $a^{\top}b=[a]_{i}\,[b]_{j}\,[\delta]_{ij}$, which is a $0_{(+)}$-tensor or scalar. For another example, the same $2_{(+)}$-tensor can (in $d=3$ dimensions) be contracted with the Levi-Civita symbol to give the standard cross product $[a\times b]_{k}=[a]_{i}\,[b]_{j}\,[\epsilon]_{ijk}$, which is a $1_{(-)}$-tensor or pseudovector. Definition 7 (permutations of tensor indices).: Given $a\in\mathcal{T}_{d,k,p}$ and permutation $\sigma\in S_{k}$, the _permutation of tensor indices_ of $a$ by $\sigma$, denoted $a^{\sigma}$, is: \[[a^{\sigma}]_{i_{1},\ldots,i_{k}}:=[a]_{i_{\sigma-1(1)},\ldots,i_{\sigma-1(k)}} \tag{8}\] Remark (tensors as linear functions).: There is an alternative definition of $k_{(p)}$-tensors in terms of geometric functions (see, for example, [41] chapter 1): A $k_{(+)}$-tensor can be thought of as representing a multilinear function of $k$ vectors ($1_{(+)}$-tensors) that produces a scalar ($0_{(+)}$-tensor) output. For example, if $A$ is a $4_{(+)}$-tensor, and $u,v,w,x$ are vectors ($1_{(+)}$-tensors) then \[\rho=[A]_{ijk\ell}\,[u]_{i}\,[v]_{j}\,[w]_{k}\,[x]_{\ell} \tag{9}\] is a scalar ($0_{(+)}$-tensor). $k_{(-)}$-tensors can be similarly defined in terms of input vectors and an output pseudoscalar. ### Geometric images and operations We will start by considering square (or cubic or hyper-cubic) images on a $d$-torus. We work on a $d$-torus to simplify the mathematical results; all the definitions and operations will be applicable with minor adjustments to rectangular, non-toroidal arrays as well. We consider an image $A$ in with $N$ equally spaced pixels in each dimension for $N^{d}$ pixels total. Each pixel contains a $k_{(p)}$-tensor where $k$ and $p$ are the same for each pixel. Let $\mathcal{T}_{d,k,p}$ be the set of $k_{(p)}$-tensors in $\mathbb{R}^{d}$. We define the geometric images as follows. Definition 8 (geometric image).: A _geometric image_ is a function $A:[N]^{d}\to\mathcal{T}_{d,k,p}$, where $[N]=\{0,1,\ldots,N-1\}$. The set of geometric images is denoted $\mathcal{A}_{N,d,k,p}$. We will also consider $k_{(p)}$-tensor images on the $d$-torus, where $[N]^{d}$ is given the algebraic structure of $(\mathbb{Z}/N\mathbb{Z})^{d}$. The pixel index of a geometric image, often $\bar{\imath}$, is naturally a $1_{(+)}$-tensor of length $d$. Definition 9 (sums of images).: Given $A,B\in\mathcal{A}_{N,d,k,p}$, the _sum_$A+B\in\mathcal{A}_{N,d,k,p}$ is defined as \[(A+B)(\bar{\imath})=A(\bar{\imath})+B(\bar{\imath}) \tag{10}\] for pixel $\bar{\imath}$. That is, the sums of geometric images are performed pixel-wise. Definition 10 (scalar multiplication of images).: Given $A\in\mathcal{A}_{N,d,k,p}$ and $\alpha\in\mathbb{R}$, the _scalar product_$\alpha A$ is defined as \[(\alpha A)(\bar{\imath})=\alpha A(\bar{\imath}). \tag{11}\] Similarly, we define contractions and permutations as applying an identical contraction or permutation to every pixel. We now turn to the first major contribution of this paper, the generalization of convolution to take geometric images as inputs and return geometric images as outputs. The idea is that a geometric image of $k_{(p)}$-tensors is convolved with a geometric filter of $k^{\prime}_{(p^{\prime})}$-tensors to produce a geometric image that contains $(k+k^{\prime})_{(p\,p^{\prime})}$-tensors, where each pixel is a sum of outer products. These $(k+k^{\prime})_{(p\,p^{\prime})}$-tensors can then be contracted down to lower-order tensors using contractions (Definition 6). Note that the sidelength $M$ of the geometric filter can be any positive odd number, but typically it will be much smaller than the sidelength $N$ of the geometric image. Definition 11 (geometric convolution).: Given $A\in\mathcal{A}_{N,d,k,p}$ on the $d$-torus, and $C\in\mathcal{A}_{M,d,k^{\prime},p^{\prime}}$ where $M=2m+1$ for some $m\in\mathbb{N}$, the _geometric convolution_$AC$ is a $(k+k^{\prime})_{(p\,p^{\prime})}$-tensor image such that \[(AC)(\bar{\imath})=\sum_{\bar{a}\in[-m,m]^{d}}A(\bar{\imath}-\bar{a})\otimes C (\bar{a}+\bar{m})\, \tag{12}\] where $\bar{\imath}-\bar{a}$ is the translation of $\bar{\imath}$ by $\bar{a}$ on the $d$-torus pixel grid $(\mathbb{Z}/N\mathbb{Z})^{d}$. Additionally, $\bar{m}$ is the $d$ length $1_{(+)}$-tensor $[m,\ldots,m]^{T}$. For example, if $d=2$ and $\bar{a}=[0,0]^{T}$, then $\bar{a}+\bar{m}=[m,m]^{T}$, the center pixel of $C$ as we would expect. This definition is on the torus, which we use to simplify the mathematical exposition. To define the convolution on $[N]^{d}$ instead of the torus, we can pad the image out with zero tensors of the corresponding order and parity. See Figure 2 for examples with a scalar and vector filter. In addition to contractions and index permutations that act pixel-wise in geometric images, it is possible to change the image size using pooling and unpooling operations. For both pooling and unpooling, there are alternative strategies to the ones we have defined below. Definition 12 (average pooling).: Given $A\in\mathcal{A}_{N,d,k,p}$ and $b\in\mathbb{Z}^{+}$ such that $N$ is divisible by $b$, we define $\operatorname{avg\,pool}(A,b)\in\mathcal{A}_{N/b,d,k,p}$ for pixel index $\bar{\imath}$ as: \[\operatorname{avg\,pool}(A,b)(\bar{\imath})=\frac{1}{b^{d}}\sum_{\bar{a}\in[0, b-1]^{d}}A(b\bar{\imath}+\bar{a}) \tag{13}\] Definition 13 (nearest neighbor unpooling).: Given $A\in\mathcal{A}_{N,d,k,p}$ and $b\in\mathbb{Z}^{+}$, we define $\operatorname{unpool}(A,b)\in\mathcal{A}_{Nb,d,k,p}$ for pixel index $\bar{\imath}$ as: \[\operatorname{unpool}(A,b)(\bar{\imath})=A(\lfloor\bar{\imath}/b\rfloor) \tag{14}\] where $\lfloor\bar{\imath}/b\rfloor$ denotes dividing each component of $\bar{\imath}$ by $b$, then taking element-wise floor operator of the resulting vector. The convolution, contraction, index-permutation, and pooling operators above effectively span a large class of linear functions from geometric images to geometric images. One way to construct nonlinear functions is using polynomials, which in this context will be sums of outer products of any of the linear function outputs, possibly followed by further geometric convolutions and contractions. Nonlinear functions can also be constructed by applying nonlinear functions to $0_{(+)}$-tensors (scalars), or odd nonlinear functions to $0_{(-)}$-tensors (pseudoscalars); we will return to these methods in Section 5. Figure 2: Convolution of a scalar image with a scalar filter and with a vector filter. Note that convolution with the vector filter results in a vector image that looks like the gradient. Definition 14 (outer products of images).: Given $A\in\mathcal{A}_{N,d,k,p}$ and $B\in\mathcal{A}_{N,d,k^{\prime},p^{\prime}}$, the _outer product_$A\otimes B\in\mathcal{A}_{N,d,k+k^{\prime},p\,p^{\prime}}$ is defined as \[(A\otimes B)(\bar{\imath})=A(\bar{\imath})\otimes B(\bar{\imath}). \tag{15}\] for each pixel $\bar{\imath}$. That is, the outer products of geometric images are performed pixel-wise. ## 3 Functions of geometric images and equivariance We start by defining equivariance and invariance for a general group $G$, and then we will describe the groups of interest and several theoretical results. Definition 15 (Equivariance of a geometric image function).: Given a function on geometric images $f:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k^{\prime\prime},p^{\prime \prime}}$, and a group $G$ equipped with actions on $\mathcal{A}_{N,d,k,p}$ and $\mathcal{A}_{N,d,k^{\prime\prime},p^{\prime\prime}}$, we say that $f$ is _equivariant_ to $G$ if for all $g\in G$ and $A\in\mathcal{A}_{N,d,k,p}$ we have: \[f(g\cdot A)=g\cdot f(A) \tag{16}\] Likewise, $f$ is _invariant_ to $G$ if \[f(g\cdot A)=f(A). \tag{17}\] We may also say a geometric image is invariant to $G$ if $g\cdot A=A$ for all $g\in G$. Convolutional filters are widely used in machine learning for scalar images. The fundamental property of these operators are that they are translation equivariant, and that every translation equivariant linear function can be expressed as a convolution with a fixed filter, as long as the filter can be set to be as large as the image. The same property holds for geometric images. Definition 16 (Translation of $k_{(p)}$-tensor images).: Given a $k_{(p)}$-tensor image $A$ on the $d$-torus, and a translation $\tau\in(\mathbb{Z}/N\mathbb{Z})^{d}$, the action $L_{\tau}A$ produces a $k_{(p)}$-tensor image on the $d$-torus such that \[(L_{\tau}A)(\bar{\imath})=A(\bar{\imath}-\tau)\, \tag{18}\] where $\bar{\imath}-\tau$ is the translation of $\bar{\imath}$ by $\tau$ on the $d$-torus pixel grid $(\mathbb{Z}/N\mathbb{Z})^{d}$. Proposition 1.: A function $f:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k+k^{\prime},p\,p^{\prime}}$ is a translation equivariant linear function if and only if it is the convolution with a geometric filter $C\in\mathcal{A}_{M,d,2k+k^{\prime},p^{\prime}}$ followed by $k$ contractions. When $N$ is odd, $M=N$, otherwise $M=N+1$. Note that this proposition merely generalizes the result of [30] for geometric convolution when the group is discrete translations. See appendix A for the proof. In addition to translation symmetries, we want to consider other natural symmetries occurring in the application domains where vectors and tensors arise. Ideally we would like to apply continuous rotations to the images, but the discretized nature of images makes this challenging. For simplicity, we focus on discrete rotations, and we extend the group action to the geometric objects in these images. Definition 17 (Group $B_{d}$ of symmetries of a $d$-hypercube).: We denote by $B_{d}$ the group of Euclidean symmetries of the $d$-dimensional hypercube. The group $B_{d}$ is often called the _hyperoctahedral group_ since the $d$-dimensional hyperoctahedron is dual to the hypercube, so they have the same group of symmetries. The notation $B_{d}$ is standard nomenclature coming from the classification theorem for finite irreducible reflection groups [27]. See Figure 3 for a depiction of the elements of $B_{2}$ acting on a vector. Because the groups $B_{d}$ are subgroups of $O(d)$, all determinants of the matrix representations of the group elements are either $+1$ or $-1$, and the matrix representation $M(g^{-1})$ of the inverse $g^{-1}$ of group element $g$ is the transpose of the matrix representation $M(g)$ of group element $g$. Definition 18 (Action of $B_{d}$ on $k_{(p)}$-tensors).: Given a $k_{(p)}$-tensor $b$, the action of $g\in B_{d}$ on $b$, denoted $g\cdot b$, is the restriction of the action in Definition 1 to $B_{d}$ which is a subgroup of $O(d)$. Definition 19 (Action of $B_{d}$ on $k_{(p)}$-tensor images).: Given $A\in\mathcal{A}_{N,d,k,p}$ on the $d$-torus and a group element $g\in B_{d}$, the action $g\cdot A$ produces a $k_{(p)}$-tensor image on the $d$-torus such that \[(g\cdot A)(\bar{\imath})=g\cdot A(g^{-1}\cdot\bar{\imath}). \tag{19}\] Since $\bar{\imath}$ is a $1_{(+)}$-tensor, the action $g^{-1}\cdot\bar{\imath}$ is performed by centering $\bar{\imath}$, applying the operator, then un-centering the pixel index: \[g^{-1}\cdot\bar{\imath}=\left(M(g^{-1})(\bar{\imath}-\bar{m})\right)+\bar{m}\] where $\bar{m}$ is the $d$-length $1_{(+)}$-tensor $\left[\frac{N-1}{2},\ldots,\frac{N-1}{2}\right]^{T}$. If the pixel index is already centered, such as $\bar{a}\in[-m,m]^{d}$, then we skip the centering and un-centering. It might be a bit surprising that the group element $g^{-1}$ appears in the definition of the action of the group on images. One way to think about it is that the pixels in the transformed image are "looked up" or "read out" from the pixels in the original untransformed image. The pixel locations in the original image are found by going back, or inverting the transformation. Figure 3: The elements of $B_{2}$ on the vector $\begin{bmatrix}2\\ 1\end{bmatrix}$. The original vector is blue and the transformed vector is red. Rotations are in degrees counterclockwise, flips are over the axis specified. The group $B_{d}$ has $d!2^{d}$ elements, so $d=3$ has $48$ and $d=4$ has $384$ elements. Definition 20 (The group $G_{N,d}$, and its action on $k_{(p)}$-tensor images).: $G_{N,d}$ is the group generated by the elements of $B_{d}$ and the discrete translations on the $N^{d}$-pixel lattice on the $d$-torus. Remark.: We view the $d$-torus as the quotient of the $d$-hypercube obtained by identifying opposite faces. The torus obtains the structure of a flat (i.e., zero curvature) Riemannian manifold this way. Because the symmetries $B_{d}$ of the hypercube preserve pairs of opposite faces, they act in a well-defined way on this quotient, so we can also view $B_{d}$ as a group of isometries of the torus. We choose the common fixed point of the elements of $B_{d}$ as the origin for the sake of identifying the $N^{d}$ pixel lattice with the group $T_{N,d}\cong(\mathbb{Z}/N\mathbb{Z})^{d}$ of discrete translations of this lattice; then the action of $B_{d}$ on the torus induces an action of $B_{d}$ on $T_{N,d}$ by automorphisms. The group $G_{N,d}$ is the semidirect product $T_{N,d}\rtimes B_{d}$ with respect to this action. Thus there is a canonical group homomorphism $G_{N,d}\to B_{d}$ with kernel $T_{N,d}$. In concrete terms, every element of $G_{N,d}$ can be written in the form $\tau\circ b$, where $b\in B_{d}$ and $\tau\in T_{N,d}$. Then the canonical map $G_{N,d}\to B_{d}$ sends $\tau\circ b$ to $b$. Now that we have defined the group that we are working with, we can specify how to build convolution functions that are equivariant to $G_{N,d}$. The following theorem generalizes the Cohen and Welling paper [7] for geometric convolutions. Theorem 1.: A ${k^{\prime}}_{(p^{\prime})}$-tensor convolution filter $C$ produces convolutions that are equivariant with respect to the big group $G_{N,d}$ if $C$ is invariant under the small group $B_{d}$. To prove this, we will first state and prove a key lemma. Lemma 1.: Given $g\in B_{d}$, $A\in\mathcal{A}_{N,d,k,p}$, and $C\in\mathcal{A}_{M,d,k^{\prime},p^{\prime}}$, the action $g$ distributes over the convolution of $A$ with $C$: \[g\cdot(AC)=(g\cdot A)(g\cdot C) \tag{20}\] Proof.: Let $A\in\mathcal{A}_{N,d,k,p}$ be a geometric image, let $C\in\mathcal{A}_{M,d,k^{\prime},p^{\prime}}$, let $g\in B_{d}$, and let $\bar{\imath}$ be any pixel index of $A$. By Definition 19 we have \[(g\cdot(AC))(\bar{\imath}) =g\cdot\big{(}(AC)\big{(}g^{-1}\cdot\bar{\imath}\big{)}\big{)}\] \[=g\cdot\left(\sum_{\bar{a}\in[-m,m]^{d}}A\big{(}g^{-1}\cdot\bar{ \imath}-\bar{a}\big{)}\otimes C(\bar{a}+\bar{m})\right)\] \[=\sum_{\bar{a}\in[-m,m]^{d}}g\cdot\big{(}A\big{(}g^{-1}\cdot\bar{ \imath}-\bar{a}\big{)}\otimes C(\bar{a}+\bar{m})\big{)}\] \[=\sum_{\bar{a}\in[-m,m]^{d}}g\cdot A\big{(}g^{-1}\cdot\bar{ \imath}-\bar{a}\big{)}\otimes g\cdot C(\bar{a}+\bar{m})\] Now let $\bar{a}^{\prime}=g\cdot\bar{a}$. Thus $g^{-1}\cdot\bar{a}^{\prime}=g^{-1}\cdot g\cdot\bar{a}=\bar{a}$. Then: \[(g\cdot(AC))(\bar{\imath}) =\sum_{\bar{a}\in[-m,m]^{d}}g\cdot A\big{(}g^{-1}\cdot\bar{\imath} -\bar{a}\big{)}\otimes g\cdot C(\bar{a}+\bar{m})\] \[=\sum_{g^{-1}\cdot\bar{a}^{\prime}\in[-m,m]^{d}}g\cdot A\big{(}g^ {-1}\cdot\bar{\imath}-g^{-1}\cdot\bar{a}^{\prime}\big{)}\otimes g\cdot C\big{(} g^{-1}\cdot\bar{a}^{\prime}+\bar{m}\big{)}\] \[=\sum_{g^{-1}\cdot\bar{a}^{\prime}\in[-m,m]^{d}}g\cdot A\big{(}g^ {-1}\cdot\bar{\imath}-g^{-1}\cdot\bar{a}^{\prime}\big{)}\otimes g\cdot C \big{(}g^{-1}\cdot\bar{a}^{\prime}+g^{-1}\cdot\bar{m}\big{)}\] \[=\sum_{g^{-1}\cdot\bar{a}^{\prime}\in[-m,m]^{d}}g\cdot A\big{(}g^ {-1}\cdot(\bar{\imath}-\bar{a}^{\prime})\big{)}\otimes g\cdot C\big{(}\bar{a} ^{\prime}+\bar{m}\big{)}\] \[=((g\cdot A)(g\cdot C))(\bar{\imath})\] For the penultimate step, we note that $g^{-1}\cdot\bar{a}^{\prime}\in[-m,m]^{d}$ compared to $\bar{a}^{\prime}\in[-m,m]^{d}$ is just a reordering of those indices in the sum. Thus we have our result for pixel $\bar{\imath}$, so it holds for all pixels. With this lemma, the proof of Theorem 1 follows quickly. Proof of Theorem 1.: Let $A\in\mathcal{A}_{N,d,k,p}$ be a geometric image and let $C\in\mathcal{A}_{M,d,k^{\prime},p^{\prime}}$ be a convolution filter invariant to $B_{d}$. It is well known that convolution is equivariant to translations, and we prove it again the appendix for our definition of convolution (32). Now suppose $g\in B_{d}$. By Lemma 1 and the $B_{d}$-invariance of $C$ we have: \[g\cdot(AC)=(g\cdot A)(g\cdot C)=(g\cdot A)C\] Thus the convolution is equivariant to the generators of $G_{N,d}$, so it is equivariant to the group. Theorem 1 provides the foundation for building our equivariant GeometricImageNet. Finding the set of $B_{d}$-invariant $k^{\prime}{}_{(p^{\prime})}$-tensor filters is straightforward using group averaging - see Section 5 for implementation details. See Figure 4 and Figure 5 for the invariant convolutional filters in $d=2$ dimensions for filters of sidelength $M=3$ and $M=5$ respectively. We now show some important relationships between the invariant filters of different tensor orders and parities. Proposition 2.: Let $C\in\mathcal{A}_{M,d,k^{\prime},p^{\prime}}$ be a $B_{d}$-invariant convolutional filter and let $\Delta\in\mathcal{A}_{M,d,2,+}$ be the geometric image with the Kronecker delta, $\delta$, in every pixel. Then $C\otimes\Delta\in\mathcal{A}_{M,d,k^{\prime}+2,p^{\prime}}$ is a $B_{d}$-invariant convolutional filter. Proof.: This proof follows quickly from the $B_{d}$-invariance of the Kronecker delta, which holds because $B_{d}\subset O(d)$ (see Proposition 6 in the Appendix). With $C$ and $\Delta$ defined as above and pixel index $\bar{\imath}$, we have: \[(g\cdot(C\otimes\Delta))(\bar{\imath}) =(g\cdot C\otimes g\cdot\Delta)(\bar{\imath})\] \[=(g\cdot C)(\bar{\imath})\otimes(g\cdot\Delta)(\bar{\imath})\] \[=C(\bar{\imath})\otimes g\cdot\Delta(g^{-1}\cdot\bar{\imath})\] \[=C(\bar{\imath})\otimes g\cdot\delta\] \[=C(\bar{\imath})\otimes\delta\] \[=C(\bar{\imath})\otimes\Delta(\bar{\imath})\] \[=(C\otimes\Delta)(\bar{\imath})\] Proposition 3.: Let $C\in\mathcal{A}_{M,d,k^{\prime},p^{\prime}},k^{\prime}\geq d-1$ be a $B_{d}$-invariant convolutional filter and $\mu_{1},\dots,\mu_{d-1}\in[k^{\prime}]$ distinct. Then $T_{LC}(C,\mu_{1},\dots,\mu_{d-1})\in\mathcal{A}_{M,d,k^{\prime}-d+2,-p^{\prime}}$ is a $B_{d}$-invariant filter of opposite parity of $C$. Proof.: Let $C$ and $\mu_{1},\dots,\mu_{d-1}$ be defined as above and let $g\in B_{d}$. We can immediately see that $T_{LC}(C,\mu_{1},\dots,\mu_{d-1})$ is $B_{d}$-invariant by the equivariance of the Levi-Civita contraction (43), so \[g\cdot T_{LC}(C,\mu_{1},\dots,\mu_{d-1})=T_{LC}(g\cdot C,\mu_{1},\dots,\mu_{d- 1})=T_{LC}(C,\mu_{1},\dots,\mu_{d-1})\.\] Thus we just have to verify that $T_{LC}(C,\mu_{1},\dots,\mu_{d-1})\in\mathcal{A}_{M,d,k^{\prime}-d+2,-p^{\prime}}$. Since the outer product adds tensor orders and multiplies parities, at each pixel $\overline{i},C(\overline{i})\otimes\epsilon\in\mathcal{T}_{d,k^{\prime}+d,-p^ {\prime}}$. Performing $d-1$ contractions reduces the tensor order by $2(d-1)$, so the resulting tensor order is $k^{\prime}+d-2(d-1)=k^{\prime}+d-2d+2=k^{\prime}-d+2$ as desired. The consequence of Propositions 2 and 3 is a natural pairing between $B_{d}$-invariant convolutional filters. See the caption of Figure 4 for further details. In practice, this allows us to dramatically reduce the number of filters we need to use in certain applications, as we will explore in Section 5.2. ## 4 Counting equivariant maps With an eye to understanding the expressive power of convolution-based functions, we show how to compute the dimension of the vector space of equivariant polynomial maps of given degree. Suppose a finite group $G$ acts on a pair of real vector spaces $V$ and $W$. Let $\mathcal{F}$ be the collection of all equivariant polynomial maps $V\to W$, and let $\mathcal{F}_{\ell}\subseteq\mathcal{F}$ be the homogeneous equivariant polynomials of degree $\ell$. The set $\mathcal{F}_{\ell}$ forms a finite-dimensional real vector space whose dimension is dependent on $\ell$. Thus $\dim\left(\mathcal{F}_{\ell}\right)$ forms a nonnegative integer sequence indexed by $\ell=0,1,2,\dots$. The generating function of this sequence, \[H(\mathcal{F},t):=\sum_{\ell\geq 0}\dim\left(\mathcal{F}_{\ell}\right)t^{ \ell}\, \tag{21}\] is known as the _Hilbert series_ of our set of functions. A variant [11, Remark 3.4.3] on a classical result known as _Molien's theorem_ expresses this generating function as a finite sum of explicit rational functions: \[H(\mathcal{F},t)=\frac{1}{\|G\|}\sum_{g\in G}\frac{\operatorname{tr}\left(M_{W} \big{(}g^{-1}\big{)}\right)}{\operatorname{det}(I-M_{V}(g)\,t)}\, \tag{22}\] where $M_{V}(g),M_{W}(g^{-1})$ are matrices describing the actions of $g,g^{-1}$ on $V,W$ respectively. The trace in the numerator is also known as the _character_ of $W$ evaluated at $g^{-1}$. Remark.: The set $\mathcal{F}$ is also known as the _module of covariants_ and written $(\mathbb{R}[V]\otimes W)^{G}$, or $\operatorname{Mor}_{G}(V,W)$, or $\operatorname{Mor}(V,W)^{G}$. In this context, the word _module_ refers to the fact that the set of equivariant polynomial maps is closed under multiplication by arbitrary $G$-invariant polynomial functions on $V$ as well as closed under addition. _Covariant_ is another word for equivariant map, coming from classical invariant theory. Figure 4: All the filters for $d=2$, $M=3$, $k^{\prime}\in[0,1,2]$. Notes: Scalars and pseudo-scalars are shown with signed colors; where there is no symbol in the box the value is zero. The $2_{(p^{\prime})}$-tensor filters are shown via the action of the tensor on an image of a letter “R”; the transformation properties of the $2_{(p^{\prime})}$-tensor filters are such that these filters may not look obviously invariant to rotations. There are no invariant pseudoscalar ($0_{(-)}$-tensor) filters available at $d=2,M=3$. Note that scalar 0 and scalar 2 are paired with $2_{(+)}$-tensor 0 and $2_{(+)}$-tensor 3 respectively by multiplication with the Kronecker delta symbol, per Proposition 2. Likewise, if we added $2_{(+)}$-tensor 1 and $2_{(+)}$-tensor 2 together, they would be paired with scalar 1. Also note that each $1_{(+)}$-tensor filter is paired with a $1_{(-)}$-tensor filter and likewise for each $2_{(+)}$-tensor filter and $2_{(-)}$-tensor filter by Proposition 3. We don’t show the $k^{\prime}{}_{(p^{\prime})}$-tensor filters at $k^{\prime}>2$ because visualizing them is difficult, even the $k^{\prime}=2$ case is potentially misleading. Note that the vector ($1_{(+)}$-tensor) filters look like pure divergence and the pseudovector ($1_{(-)}$-tensor) filters look like pure curl. The right side of (22) is reasonable to compute in practice. To illustrate, we compute it for the group $G_{N,2}$ of Definition 20, with $V=W=\mathcal{A}_{N,2,1,+}$, the space of 2-dimensional geometric images whose pixels consist of vectors. We assume $N$ is odd. We first compute the character $\operatorname{tr}\left(M(g)\right)$ for $g\in G_{N,2}$. This can be done explicitly by writing down a basis for $\mathcal{A}_{N,2,1,+}$ and expressing the action of each element of $G_{N,2}$ in terms of that basis. The computation is expedited by the choice of a basis in which the action of $G_{N,2}$ is _monomial_, that is, for basis vector $e_{i}$ and any $g\in G_{N,d}$, we have $g\cdot e_{i}=\alpha\,e_{j}$, where $\alpha\in\mathbb{R}$ and $e_{j}$ is some basis vector which may be the same as $e_{i}$. When this condition holds, only the basis _eigenvectors_ contribute to the trace. The group $B_{2}$ acts monomially on the standard basis vectors for $\mathcal{T}_{2,1,+}\cong\mathbb{R}^{2}$, and it follows that $G_{N,2}$ acts monomially on a basis for $A_{N,2,1,+}$ consisting of maps $[N]^{d}\to\mathcal{T}_{2,1,+}$ mapping one pixel to one standard basis vector and all other pixels to zero. This situation generalizes in a straightforward fashion to higher dimensions $d$ and higher order tensors. Let $e^{0},e^{1}$ be the standard basis for $\mathbb{R}^{2}$, and then for pixel index $\bar{\imath}$ and $q\in\{0,1\}$, let $\mathbf{e}_{i}^{q}\in\mathcal{A}_{N,2,1,+}$ be the geometric image where $\mathbf{e}_{i}^{q}(\bar{\imath})=e^{q}$ and $\mathbf{e}_{i}^{q}(\bar{\jmath})=\bar{0}$ for all other pixel indices $\bar{\jmath}\neq\bar{\imath}$. As stated above, $G$ acts monomially on the basis of $\mathcal{A}_{N,2,1,+}$ consisting of these images $\mathbf{e}_{i}^{q}$. If $g\in G_{N,2}$, then $\mathbf{e}_{i}^{q}$ is not an eigenvector for $g$ unless $g$ fixes the pixel $\bar{\imath}$, and even then, there is no contribution to the trace from pixel $\bar{\imath}$ unless $g$ acts with nonzero trace on the $\operatorname{span}\left(\mathbf{e}_{i}^{0},\mathbf{e}_{i}^{1}\right)$. In turn, if $g$ does fix pixel $\bar{\imath}$, then its trace on $\operatorname{span}\left(\mathbf{e}_{i}^{0},\mathbf{e}_{i}^{1}\right)$ is equal to the trace of the corresponding element $\overline{g}$ of $B_{2}$ under the canonical map $G_{N,2}\to B_{2}$ on $\mathbb{R}^{2}$. This is zero unless $\overline{g}=\pm I$ since in all other cases, $\overline{g}$ is either a $\pi/2$-rotation or a reflection. It follows that the only elements of $G_{N,2}$ with nonzero trace on $\mathcal{A}_{N,2,1,+}$ are the identity (with trace $2N^{2}=\dim\mathcal{A}_{N,2,1,+}$) and the $\pi$-rotations centered at each of the $N^{2}$ pixels (each with trace $-2$, coming from the fixed pixel $\bar{\imath}$, where $\mathbf{e}_{i}^{0}$ and $\mathbf{e}_{i}^{1}$ are both negated). Figure 5: All the filters for $d=2$, $M=5$, $k^{\prime}\in[0,1]$. The symbols and colors are as in Figure 4. We don’t show the $2_{(p^{\prime})}$-tensor filters at $m=2$ because there are 26 of them. At $(d,m)=(2,2)$ a pseudoscalar filter appears. Again, the vector and pseudovectors look like pure divergence and pure curl, respectively. Thus the only nonzero terms in the sum (22) are those with $g^{-1}$ as just described. Conveniently, $g=g^{-1}$ in all those cases. We need to compute $\det(I-M(g)\,t)$ for such $g$. For $g=I$ we have \[\det(I-M(g)\,t)=(1-t)^{2N^{2}}. \tag{23}\] If $g$ is a $\pi$-rotation about the pixel $\bar{\imath}$, then $\mathbf{e}_{\bar{\imath}}^{0},\mathbf{e}_{\bar{\imath}}^{1}$ have their signs reversed, while all other pixels are transposed in pairs, say $\bar{\jmath}\leftrightarrow\bar{a}$, with the corresponding $\mathbf{e}_{\bar{\jmath}}^{g}$ sent to $-\mathbf{e}_{\bar{a}}^{g}$ and vice versa. Then the matrix $I-M(g)\,t$ can be written block-diagonally with two $1\times 1$ blocks of the form $(1+t)$ for the pixel $\bar{\imath}$ that we are rotating about and $N^{2}-1$ blocks of the form \[\begin{pmatrix}1&-t\\ -t&1\end{pmatrix}. \tag{24}\] for the pixels that are being swapped. So we have \[\det(I-M(g)\,t)=(1+t)^{2}(1-t^{2})^{N^{2}-1}. \tag{25}\] Putting all of this together, (22) becomes \[H(\mathcal{F},t) =\frac{1}{8N^{2}}\left(\frac{2N^{2}}{(1-t)^{2N^{2}}}+\frac{N^{2}( -2)}{(1+t)^{2}(1-t^{2})^{N^{2}-1}}\right) \tag{26}\] \[=\frac{1}{4}\left(\frac{1}{(1-t)^{2N^{2}}}-\frac{1}{(1+t)^{2}(1-t ^{2})^{N^{2}-1}}\right). \tag{27}\] Expanding (27) in a power series and extracting the coefficient of $t^{\ell}$, we find that the dimension of the space of $G_{N,2}$-equivariant maps $\mathcal{A}_{N,2,1,+}\to\mathcal{A}_{N,2,1,+}$ is \[\frac{1}{4}\left(\begin{pmatrix}2N^{2}+\ell-1\\ \ell\end{pmatrix}+(-1)^{\ell+1}\sum_{j=0}^{\lfloor\ell/2\rfloor}(\ell-2j+1) \binom{N^{2}+j-2}{j}\right). \tag{28}\] This expression evaluated for $\ell=1,2,3$ is shown in Table 1. With the ability to explicitly count the number of $G_{N,d}$-equivariant homogeneous polynomials on geometric images, we want to know whether the operations defined in Section 2 are sufficient to characterize all these functions. Let $g_{i}:\mathcal{A}_{N,d,k,p}\to\mathcal{A}_{N,d,k_{i},p_{i}}$ for $i=1,\ldots,\ell$ be a linear function on geometric images defined by the linear operations in sections 2.1 and 2.2, excluding pooling and unpooling. Let $h:\mathcal{A}_{N,d,\underline{k},\overline{p}}\to\mathcal{A}_{N,d,k^{\prime \prime},p^{\prime\prime}}$ be a linear function defined by the same operations as \begin{table} \begin{tabular}{c c c c c} degree $\ell$ & \multicolumn{4}{c}{Sidelength $N$} \\ \cline{2-5} & $N$ & $3$ & $5$ & $7$ \\ \hline $1$ & $\binom{N^{2}+1}{2}/2$ & $5$ & $13$ & $25$ \\ $2$ & $\binom{N^{4}-1}{2}/2$ & $40$ & $312$ & $1\,200$ \\ $3$ & $\left(2N^{6}+3N^{4}+4N^{2}+3\right)/6$ & $290$ & $5\,538$ & $40\,450$ \\ \hline \end{tabular} \end{table} Table 1: The number of equivariant maps $\mathcal{A}_{N,2,1,+}\to\mathcal{A}_{N,2,1,+}$ for different values of sidelength $N$ and degree $\ell$. For degrees $1$ and $2$, the green cells, we were able to confirm we found all the functions. For degree $3$, the pink cells, we had insufficient computer memory to confirm. For $N=3,\ell=3$ in particular, we were able to find $289$ of the $290$ functions by searching a subset of the candidate functions before memory limitations forced us to stop. the $g_{i}$ functions, where $\overline{k}=\sum_{i=1}^{\ell}k_{i}$ and $\overline{p}=\prod_{i=1}^{\ell}p_{i}$. Let function $f:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k^{\prime\prime},p^{\prime \prime}}$ be defined for all $A\in\mathcal{A}_{N,d,k,p}$: \[f(A)=h(g_{1}(A)\otimes\ldots\otimes g_{\ell}(A)) \tag{29}\] When $\ell=1$, we will only do $f(A)=h(A)$ rather than $f(A)=h(g_{1}(A))$. We conjecture that these steps will allow us to construct all equivariant maps of any degree. To test this conjecture, we performed the following experiments to count the number of linear, quadratic, and cubic homogeneous polynomials from vector images to vector images. First we constructed all the $B_{d}$-invariant $k^{\prime}{}_{(p^{\prime})}$-tensor filters for $k^{\prime}=1,2$ and $p^{\prime}=+1$ and used those filters to construct all the homogeneous polynomials according to (29). We then generated a random vector image and applied all the functions to that image, and we want to know whether those output images are linearly independent. Thus we flattened all the resulting images into a giant matrix and performed a singular value decomposition; the number of non-zero singular values gives us the number of linearly independent functions. In the higher order polynomial cases we have to apply the various functions on multiple images to ensure separation. The results are given in Table 1. ## 5 GeometricImageNet Architectures Our GeometricImageNet model seeks to learn some function $f:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k^{\prime\prime},p^{\prime \prime}}$. The problem determines $d$, and therefore the groups $B_{d}$ and $G_{N,d}$. After fixing these initial parameters, the modeler must decide the size, number, and type of layers that are described below. The first choice is the attributes of the convolution filters: size $M$, tensor order $k^{\prime}$, and parity $p^{\prime}$, all of which may be a single value or multiple values. A complete set of $B_{d}$-invariant $k^{\prime}{}_{(p^{\prime})}$-tensor filters can be found by group averaging. We first construct all group operators for the $B_{d}$ group by iterating the generators until the group is closed under all products of operators. The set of possible geometric filters is a vector space of dimension $M^{d}dk^{\prime}$, so we can pick a basis of that many elements where each basis element $C_{i}$ has exactly one component of the tensor in a single pixel set to 1, and all other values are 0. Each of these basis elements is then group-averaged by applying all group operators and averaging the results: \[\widetilde{C}_{i}=\frac{1}{\|B_{d}\|}\sum_{g\in B_{d}}g\cdot C_{i}\, \tag{30}\] where $\|B_{d}\|$ is the number of group elements. The results of those group averages are unpacked into a matrix and the singular value decomposition is then run to find an "eigen-set" of orthogonal, non-zero filters. After the SVD, the filters can be normalized however seems appropriate. We normalized the filters such that the magnitudes of the non-zero filter values are as close to unity as possible, and the $k=1$ filters are also reoriented such that non-zero divergences were set to be positive, and non-zero curls were set to be counter-clockwise. See Figure 4 and Figure 5 for the invariant convolutional filters in $d=2$ dimensions for filters of sidelength $M=3$ and $M=5$ respectively. With the set of invariant filters in hand, we may build our equivariant neural networks using convolution layers, contraction layers, outer product layers, and nonlinear activation layers. ### Architecture Components We think of each building block of our architecture as a layer whose input and output is a set of images grouped by tensor order and parity. The reason for grouping is two-fold: we can only add geometric images that share tensor order and parity, and we can more efficiently batch our operations in JAX when the shapes match. The operation of each layer is either convolution, contraction, taking outer products, or applying a nonlinear activation function. A convolution layer takes a set of images and a set of convolution filters. For each filter, we take a weighted sum of the images of a particular tensor order and parity and apply the convolution1 on that sum with that filter. Unlike a traditional CNN where the filters are parameterized, our filters are fixed to enforce the equivariance, and the weights of the weighted sums are the learned parameters. A convolution layer can also have an optional dilation where the filters are dilated before convolving. Dilations are helpful for expanding the effective size of the filters without having to calculate the invariant filters for larger $M$, which grows quickly; see [12] for a description of dilated convolution. If we use filters with tensor order greater than 0, the tensor order of the images will continue to grow as we apply convolutions. Thus we need a way to reduce the tensor order - we do this with the contraction layer. Footnote 1: The geometric convolution package is implemented in JAX, which in turn uses TensorFlow XLA under the hood. This means that convolution is actually cross-correlation, in line with how the term in used in machine learning papers. For our purposes this results in at most a coordinate transformation in the filters. Given an input layer and a desired tensor order, the contraction layer performs all unique contractions (see Contraction Properties (35)(36)) to reduce the layer to that tensor order. We always end the neural network with a contraction layer to return the images to the proper tensor order. Since contractions can only reduce the tensor order by multiples of 2, the last convolution layer before the final contraction must result in images of order $k^{\prime\prime}+2n$ for any $n\in\mathbb{N}$. We also may include contraction layers after each convolution to cap the tensor order of each layer to avoid running out of memory as the tensor order grows. In practice, $k=5$ seems to be a good max tensor order. An outer product layer takes a set of images and a degree and computes the full polynomial of all the images with each other using the outer product of geometric images, up to the specified degree. Typically, this will result in a combinatorial blowup of images; we can take parameterized sums along the way to reduce the number of images created. We could also do a smaller set of products if we have some special domain knowledge. However, in practice it is usually better to use nonlinear activation functions, as is standard in machine learning. The final type of layer is a nonlinear activation layer. In order to maintain equivariance, we can either apply a nonlinearity to a scalar, or scale our tensors by a nonlinearity applied to the norm of the tensor [44]. For this paper, we used the first strategy. We apply all possible contractions to all even tensor order images to reduce them to scalars, then apply the nonlinearity. Any typical nonlinearity works - ReLU, leaky ReLu, sigmoid, etc. This layer will result in scalar images, which will then grow in order again as we apply more convolution layers. ### Architecture Efficiency Without specialized knowledge of what $B_{d}$-invariant convolution filters are relevant for our problem, we want to use all the filters at a specified tensor order in our convolution layers. Thus we can improve the efficiency of the GI-Net by eliminating any redundant filters. The first result follows from Proposition 2 and says that we may omit the $k^{\prime}{}_{(\rho^{\prime})}$-tensor filters if we are using the $(k^{\prime}{}+2)_{(\rho^{\prime})}$-tensor filters followed by taking all contractions. Proposition 4.: Let $\mathcal{F}$ be the set of functions $f:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k^{\prime\prime},p^{\prime \prime}}$ where each $f$ is a convolution with a $B_{d}$-invariant $k^{\prime}{}_{(p^{\prime})}$-tensor filter. Let $\mathcal{G}$ be the set of functions $g:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k^{\prime\prime},p^{\prime \prime}}$ where each $g$ is a convolution with a $B_{d}$-invariant $(k^{\prime}+2)_{(p^{\prime})}$-tensor filter followed by a contraction. Then $\mathcal{F}\subseteq\mathcal{G}$. See Appendix A for the proof. This proposition can be repeatedly applied so that if we conclude a GI-Net by taking all unique contractions, then we need only include filters of tensor order $k^{\prime}$ and $k^{\prime}-1$ to include all smaller tensor orders as well. The next result says that if the input and output parities of our network are equal, we may omit the $k^{\prime}{}_{(-)}$-tensor filters if we are using the $(k^{\prime}+d)_{(+)}$-tensor filters followed by taking all contractions. Proposition 5.: Let $\mathcal{F}$ be the set of functions that preserve parity $f:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k^{\prime\prime},p}$ where each $f$ is a convolution with a negative-parity $k^{\prime}{}_{(-)}$-tensor filter followed by a Levi-Civita contraction. Let $\mathcal{G}$ be the set of functions that preserve parity $g:\mathcal{A}_{N,d,k,p}\rightarrow\mathcal{A}_{N,d,k^{\prime\prime},p}$ where each $g$ is a convolution with a positive-parity $(k^{\prime}+d)_{(+)}$-tensor followed by $d-1$ contractions. Then $\mathcal{F}\subseteq\mathcal{G}$. See Appendix A for the proof. We will employ these results in our numerical experiments to dramatically reduce the number of filters required. ## 6 Numerical Experiments Code to reproduce these experiments and build your own GI-Net is available at [https://github.com/WilsonGregory/GeometricConvolutions](https://github.com/WilsonGregory/GeometricConvolutions). The code is built in Python using JAX. Figure 6: One possible architecture for a GI-Net that maps a vector image to a vector image when using convolution filters with tensor order $k^{\prime}\in\{1,2\}$, parity $p^{\prime}=+1$, max order $k=3$, and ReLu nonlinearities. Each layer is a block of multiple images that share tensor order. The blue arrows represent convolutions and raise the tensor order by $1$ or $2$. Contractions are applied when the tensor order goes above $3$ to bring it down to $2$ or $3$, and when contracting to $k=0$ in order to apply the ReLu. This process continues until the final step where the only layer is order $k=1$, which is then combined using a parameterized linear combination. The most natural problems for this model are those that we expect to obey the symmetries of the group $G_{N,d}$. We present two problems from physics that despite their simplicity, exhibit the powerful generalization properties of the equivariant model even in cases where we have few training points. First, suppose we have as input a scalar image of point masses, and we want to learn the gravitational vector field induced by these masses. For this problem, we will work in two dimensions with image sidelength of 16, and the point charges are placed only at pixel centers, so the GI-Net is learning a function $f:\mathcal{A}_{16,2,0,+}\to\mathcal{A}_{16,2,1,+}$. To generate the data, we sampled the pixel locations uniformly without replacement 5 times to be point masses, and then we set their masses to be a uniform value between 0 and 1. For a second problem, we consider several point charges in a viscous fluid. All the point charges have charge $+1$, so they repel each other. The position of the charges in the fluid would be described by an ordinary differential equation, and we can approximate that using Euler's method: \[x_{i}(t+\Delta t)=x_{i}(t)+\Delta t\,V(x_{i},t)\, \tag{31}\] where $x_{i}$ is a point, $\Delta t$ is one time step, and $V(x_{i},t)$ is the vector field at time $t$ induced by all particles other than $x_{i}$. We iterate this system some number of steps $T$, and the learning problem is the following: Given the initial charge field, can we predict the charge field after step $T$? For this toy problem we will again use an image in two dimensions of sidelength 16, so the function we are trying to learn is $f:\mathcal{A}_{16,2,1,+}\to\mathcal{A}_{16,2,1,+}$. We took several precautions to make the data well behaved. Since the particles move freely in $\mathbb{R}^{2}$ but we learn on a discrete vector field grid, the vectors act erratic when the charge passes very closely to the center of the pixel. Thus we applied a sigmoid transform on the charge field vectors on the input and output: \[Q(\vec{v},s)=\bigg{(}\frac{1}{1+e^{-\frac{\\|\vec{v}\\|}{s}}}-\frac{1}{2}\bigg{)} \bigg{(}\frac{\vec{v}}{\\|\vec{v}\\|}\bigg{)}\,\] where $\\|\cdot\\|$ is the usual Euclidean norm and $s$ is a parameter that we set to 0.2. That is, the vector field is a nonlinear vector function of the original vector electric field. One advantage of learning on the charge field rather than the particles themselves is that vector field is discrete, but the vectors reflect the exact particle locations. However, if two particles start very close, it will appear that there is only a single particle on the charge vector field. To alleviate this problem, we iterated one step of Euler's method, and treated that as the input to the neural network. Additionally, we initialized points within the central $8\times 8$ grid rather than the full $16\times 16$ grid so that the charges would be unlikely to leave the bounds of the grid by step $T$. See Figure 7 for example inputs and outputs for the two problems. ### Architectures For the gravity problem, the architecture that we choose has 3 convolution layers followed by all contractions that reduce to $k=1$, and then a parameterized linear combination of the resulting images. The second convolution layer uses dilations that range from 1 to 15, and the third convolution layer uses dilations from 1 to 7. For our loss we use the root mean squared error (RMSE). The baseline architecture is built to have similar structure with 3 convolution layers with the same dilations, but also have a number parameters on the same order of magnitude. We treat the dilations as creating a separate channel. The sequence of channel depth is the following: \[1\to 2\to(152)\to(72)\to 2\] To get from the output of the 3rd convolution which is 2 filters across 7 dilations, we take a parameterized sum across the dilations to get an image with 2 channels, which is the number of channels we need for a vector image. See Table 2 for additional info about the filters and number of parameters. The moving charges problem is more difficult, so we choose a more complex architecture. We have 9 convolution layers, with dilations of $1,2,4,2,1,1,2,1,1$ in that order. Each convolution is followed by a nonlinear activation layer with a Leaky ReLu with negative slope of 0.01. This non-linearity seemed to perform best among the ones we tried. We then finish with the usual contraction layer and linear combination, and our loss is again the RMSE. For the baseline model, we use identical number of convolution layers, dilations, nonlinearities, and loss. The only difference is that we only use scalar filters, so we increase the depth of each layer to 20 except for the final output layer which must have a depth of 2 because we are learning a vector image. \begin{table} \begin{tabular}{c c c c c c c} problem & model & $M$ & $k^{\prime}$ & $p^{\prime}$ & \# layers & depths & \# params \\ \hline gravity & GI-Net & 3 & 0,1 & +1 & 3 & 1 & 3 085 \\ & baseline & 3 & 0 & & 3 & 2,15 $$ 2, 7 $*$ 2 & 4 345 \\ \hline charge & GI-Net & 3 & 1,2 & +1 & 9 & 1 & 22 986 \\ & baseline & 3 & 0 & & 9 & 20,$\dots$,20,2 & 25 920 \\ \hline \end{tabular} \end{table} Table 2: Comparison of the different model architectures used for the two problems. The values of $M,k^{\prime},p^{\prime}$ represent the sidelength, tensor order, and parity of the convolutional filters. The baseline models do not have a parity because those filters are learned rather than fixed ahead of time. Figure 7: One test point each for the (a) gravitational field problem and the (b) moving point charge problem. The first two columns show the input and output to the problems respectively. The third column shows the model predicted output, and the fourth column shows the difference between the ground truth and the predicted output. To put the numerical results in context, the loss for the top row is 0.0177, while the loss for the bottom row is 2.061. ### Training For all models, we trained the network using stochastic gradient descent with the Adam optimizer and an exponential learning rate decay that started at 0.005, has transition steps equal to the number of batches per epoch, and has a decay of 0.995. The one exception was the baseline model for the moving charges problem where we started with a learning rate of 0.001. These values were found with a limited grid search. For both problems and both models we initialized the parameters as Gaussian noise with mean 0 and standard deviation 0.1. For the gravitational field problem, we created a test set of 10 images, a validation set of 5 images, and training sets ranging in size from 5 to 50 images. For the moving charges problem we created a test set of 10 images, a validation set of 10 images, and training sets ranging in size from 5 to 100 images. We used training batch sizes equal to 0.2 the training set size. For all models we trained them until the error on the validation set had not improved in 20 epochs. ### Results Given sufficient data, both the GI-Net and the baseline model are able to perform well on the test data set. The RMSE carries very little information without further context, but we can see from the examples in Figure 7 that low error corresponds to only small differences between the ground truth output and the predicted output. By comparing the GI-Net to our simple CNN baseline we can see the advantages of the equivariant model. In Figure 8(a), with only 10 data points, the test error of the GI-Net is almost equal to the training error, suggesting that the model is able to generalize well from just a few small examples. By contrast, the baseline model requires at least 50 training points to get its test error as close to its training error. Additionally, even when the baseline model has enough points, its error is still higher overall compared to the GI-Net. Likewise, in Figure 8(b) the gap between the test error and training error for the baseline model is much larger than the same gap for the GI-Net. In this case, the GI-Net and the baseline model reach the same test error when training off 100 points, despite the baseline model having smaller training error. This again suggests Figure 8: Comparison between baseline model and GI-Net as a function of the number of points in the training data set. The GI-Net have smaller testing loss when the training data set is small, even when the baseline model achieves better training error in the case of the moving charges problem. that the GI-Net does a better job generalizing, especially when the training data set is small. ## 7 Related work Restricting machine learning models to use functions that are equivariant to some group action is a powerful idea that has the potential to improve generalization and reduce training requirements. When we expect our target function to be equivariant to that group, this restriction improves the model's generalization and accuracy (see for instance [16, 15, 47]) and is a powerful remedy for data scarcity (see [49]). Equivariant networks, in certain cases, can approximate any continuous equivariant function (see [53, 13, 3, 31]). There is a wide variety of strategies to design equivariant maps (e.g. [7, 48]). One class of methods, employed in this work, is group convolution, either on the group or on the homogeneous space where the features lie. Our generalization of convolution replaces arbitrary products with the outer product of tensors. Our approach is related to [4], which employs a Clifford Algebra. Other strategies to design equivariant maps use irreducible representations or invariant theory to parameterize the space of equivariant functions. The first work using representation theory for invariant and equivariant neural networks was the paper of Wood et al. [50] from 1996. More recent incarnations of these ideas include [36, 8, 7, 6, 10, 17]. One can use classical invariant theory to compute the generators of the algebra of invariant polynomials. For instance, in [2] we show how to use the generators of the algebra of invariant polynomials to produce a parameterization of equivariant functions for certain groups and actions. This approach is inspired by the physical sciences, where the data is subject to rules coming from coordinate freedoms and conservation laws. In previous work [44, 45, 52] we used these geometric rules to develop modified machine-learning methods that are restricted to exactly obey group-theoretic equivariances in physics contexts. Similar ideas have been explored in [23, 20]. See appendix 0.B for a more in depth description of the mathematical details of the related work. ## 8 Discussion This paper presents a flexible new model, the GeometricImageNet, which parameterizes functions that map geometric images to geometric images. It capitalizes on the vector or tensor structure of the image contents. The flexibility, coupled with the easy restriction to $G_{N,d}$-equivariant functions, makes the model ideal for tackling many problems from the natural sciences in a principled way. The added flexibility of tensors comes with a cost. Taking repeated convolutions with tensor filters grows the the tensor order and thus, with naive representations, the memory requirements of the network. We see the consequences of this issue when trying to numerically demonstrate that we have characterized all the equivariant homogeneous polynomials (Section 4). We also encounter this issue when solving problems in Section 6; fortunately, it appears that capping the maximum tensor order limits the memory requirements without much associated loss in performance. Another shortcoming of this work is that we work with discrete symmetries rather than continuous symmetries. We expect invariance and equivariance with respect to rotations other than 90 degrees to appear in nature, but the images that we work with are always going to be $d$-cube lattices of points. Thus we use the group $G_{N,d}$ to avoid interpolating rotated images and working with approximate equivariances. This simplifies the mathematical results, and we see empirically that we still have the benefits of rotational equivariance. However, there are other possible image representations that might create more continuous concepts of images. For example, if the data is on the surface of a sphere, it could be represented with tensor spherical harmonics, and be subject to transformations by a continuous rotation group. There are many possible future directions that could be explored. It is an open question to understand the full expressive power of the model outside the few cases we were able to test in Section 4. Additionally, there are likely improvements to be made to both the time and space efficiency of the GeometricImageNet. Finally, there are many exciting applications in fluid dynamics, astronomy, climate science, biology, and others that could benefit from this physics-informed machine learning approach. Acknowlegements:It is a pleasure to thank Roger Blandford (Stanford), Drummond Fielding (Flatiron), Leslie Greengard (Flatiron), Ningyuan (Teresa) Huang (JHU), Kate Storey-Fisher (NYU), and the Astronomical Data Group at the Flatiron Institute for valuable discussions and input. This project made use of Python 3 [42], numpy [24], matplotlib [28], and cmastro [38]. All the code used for making the data and figures in this paper is available at [https://github.com/WilsonGregory/GeometricConvolutions](https://github.com/WilsonGregory/GeometricConvolutions). Funding:WG was supported by an Amazon AI2AI Faculty Research Award. BBS was supported by ONR N00014-22-1-2126. MTA was supported by H2020-MSCA-RISE-2017, Project 777822, and from Grant PID2019-105599GB-I00, Ministerio de Ciencia, Innovacion y Universidades, Spain. SV was partly supported by the NSF-Simons Research Collaboration on the Mathematical and Scientific Foundations of Deep Learning (MoDL) (NSF DMS 2031985), NSF CISE 2212457, ONR N00014-22-1-2126 and an Amazon AI2AI Faculty Research Award.
2308.04469	Correlating Medi-Claim Service by Deep Learning Neural Networks	Medical insurance claims are of organized crimes related to patients, physicians, diagnostic centers, and insurance providers, forming a chain reaction that must be monitored constantly. These kinds of frauds affect the financial growth of both insured people and health insurance companies. The Convolution Neural Network architecture is used to detect fraudulent claims through a correlation study of regression models, which helps to detect money laundering on different claims given by different providers. Supervised and unsupervised classifiers are used to detect fraud and non-fraud claims.	Jayanthi Vajiram, Negha Senthil, Nean Adhith. P	2023-08-08T07:40:21Z	http://arxiv.org/abs/2308.04469v1	# Correlating Medi- Claim Service by Deep Learning Neural Networks. ###### Abstract Organized crime is a continuous issue, and predicting it is always under research. Medical insurance claims are one of the organized crimes related to patients, physicians, diagnostic centers, and insurance providers, forming a chain reaction that must be monitored constantly. These kinds of frauds affect the financial growth of both the insured people and the health insurance companies. The Convolution Neural Network architecture is used to detect fraudulent claims through a correlation study of regression models, which helps to detect money laundering on different claims given by different providers. Supervised and unsupervised classifiers are used to detect fraud and non-fraud claims. By using different attributes of patient case studies, diagnostic reports, and service provider reimbursement claim codes as control variables and attributes of the target class to detect performance metrics, this paper highlights the top reason for organized crime through the public dataset. The claims are filed by the provider, so the fraud can be organized crime. The performance metrics of accuracy, sensitivity, specificity, recall, precision, AUC, and f1-scores are calculated. More features are extracted from the proposed model of Auto-Encoder, which gives 0.90 accuracy, 0.80 AUROC score, f1-score (0.61), and 0.55 Kappa Score. Different threshold values are used to detect a tradeoff between fraud and non-fraud class predictions. Keywords: Health care frauds, Medicaid, organized crime, CNN regression models, fraud ## Introduction Healthcare frauds are related to physicians, diagnostics centers, and insurance providers. These entities are used to deliver the costliest treatment procedures and drugs, often leading to bad practices including the claiming of more money than is due. As a result, insurance premiums and healthcare costs are high, particularly in low-income countries. Common types of fraud by providers include improper billing, duplicate billing of the same service, charging more than necessary for physician consultation fees and surgery, charging for expensive services, and billing for services that are not covered. Such frauds lead to patients receiving Mediclaim sanctions of only 70-80% of their hospitalization costs, leading to a financial crisis. This article explores the use of convolutional neural network architectures to analyze claims filed by providers, predicting whether they are fraudulent or not. Vyas's (2021) study showed that insurance fraud and their network analysis can be effectively detected using fraud detection systems in insurance claim systems [1]. Suma Latha's (2019) study on medical insurance fraud detection and medical claims utilized logistic regression for decision analysis and was found to be more effective than existing systems [2]. To classify providers into fraud and non-fraud categories, supervised and unsupervised models were used with feature selection, logistic regression, random forest, and auto-encoder. Classic logistic regression was used to predict the linearity of dependent and independent variables. Random forest model was used to handle large input variables with different dimensions of non-linearity. The proposed auto-encoder compressed the input variables into a latent space and then reconstructed the output variables. The model was trained on non-fraud data and then reconstructed it back as fraud data through reconstruction errors with a set threshold. The aim of this study is to accurately predict fraudulent providers by using correlation study of residual error from regression models, prediction based on linear and non-linear variables concerning threshold and intensity variance, supervised and unsupervised classifiers, feature extraction using principal component analysis and auto-encoders, and ensemble learning with hyperparameter tuning. The results of this study provide insights into the fraudulent patterns and claims, enabling a better understanding of future predictions of providers. This study highlights (a) predicting fraudulent providers through a correlation study of residual error by regression models, (b) predicting based on linear and non-linear variables with regards to threshold and intensity variance, (c) using both supervised and unsupervised classifiers for model prediction, (d) utilizing principal component analysis for feature extraction, (e) calculating accurate predictions with a proposed Auto-Encoder model, and (f) increasing model performance with an ensemble learning method and hyperparameter tuning. This study provides insights into fraudulent patterns and claims, which can help to better predict future provider activities. ### Related study Wasson's (2019) study on organized crime among healthcare professionals aimed to reduce system costs and address future challenges [3]. Patel's study used period-based claim anomalies (k-means clustering) and disease-based (Gaussian distribution) anomalies outlier detection with a public dataset to analyze fraudulent insurance claims [4]. Daskalopoulou's (2019) study on healthcare providers' roles and levels of service delivery between patients and organizational level explained the new service provider roles [5]. Akbar's (2020) study analyzed fraud prediction using the statistical methods of random forest regression and extreme gradient boost (XGB) and compared recall and precision scores [6]. Saldamli's (2020) study on health insurance services covered accident or major illness and highlighted unsynchronized patient data and false claims. These were analyzed by the National health care anti-fraud association through blockchain fraud detection [7]. Powel's (8) study investigated healthcare providers' mental, work-related stress and protective factors. Mathew's (9) study explored the effects of low-income patients' claims and insurance companies delays and fewer refunds and the help of social services agencies support. Carcillo (2021) (10) examined the use of supervised and unsupervised learning techniques to improve credit card fraud detection accuracy. Madhavan's (11) study investigated the COVID-19 medical practices of direct and reverse polymerase chain reaction with Res-CovNet model analysis. Korneta's (12) study explored the effects of the COVID-19 pandemic time health care providers in Poland and their capitation payment scheme and delay settlement. Lastly, Mydin's (13) study analyzed healthcare service providers' attitudes and practice in managing elderly abuse and neglect cases and their negligence towards the elderly community medical claims. Haque et al. studied the identification of fraudulent records of patients' high Mediclaim insurance frauds and the insurance providers' loss due to frauds by using a mixture of clinical codes and a Long Short-Term Memory (LSTM+PCA) network with Robust Principal Component Analysis [15]. Sanober's study focused on fraud transactions and detection by Random Forest, SVM, Logistic Regression, Decision Tree, and KNN, based on a two-year case study of credit card transactions, and their accuracies were calculated [14]. Yaumil's study examined fraud perpetrators of doctors and providers through unadministered duplicating claims of insurance and medical action by data tracking, and by providing new technology [16Aziani (2021) explores how the COVID-19 pandemic has led to organized crime amongst wholesale medicine circulation between pharmaceuticals, hospitals and patients [17].Teresa's (2022) study investigates the use of value-based adjusted payments to vulnerable patients to assess performance measures of service providers and social risk factors, and reduce readmissions penalties for hospitals [18]. Guoming's (2021) study looks into the rising of medical treatment and expenses based on diagnosis-related groups of patient insurance settlement [19]. Abhishek's study on Total Interpretive Structural Modeling Analysis of Health Care Management Negligence and Waste Management using Structural Equation Modeling [20]. Steve's study on the Healthcare Sector and the Recrudescence of Cyber-Attacks, and their Impact on Privacy in Hospitals, analyzed using [21]. Mahdi's study on Supervised Learning to Detect Fraud and Misinformation in Supply Chains, using Macro-level Python Code [22]. Ammar's study on Blockchain Security, Centralization of Cryptographic Systems, and Electronic Records in the Healthcare Field [23]. Charron's study [24] examined Medicare, financial strain, and poor quality of food and shelter in older adults according to race and economic status. Zheng's study [25] used an ensemble learning algorithm and management discussion analysis to identify financial fraud, and analyzed sentiment polarity. Jenita Mary's study [26] evaluated the false positive rate for insurance fraud detection by applying k-mean clustering, Support Vector Machine (SVM), and Naive Bayes (NB) on a healthcare provider fraud detection dataset. The study clearly shows the organized crime and malpractice between hospitals, diagnostic centers, doctors, insurance, and pharmaceutical companies, involving money laundering, fraud, and extortion. This fraudulent activity has a major effect on patients' full Mediclaim and insurance companies' financial aid. ### General analysis and related crimes and prevention methods Detection and correlation of medical claim service provider reimbursement analysis as per study of Gaskins, 2017 as follows 1. Medical claim service provider reimbursement analysis is a process used by insurance companies and other entities to detect and identify fraudulent billing practices and other unauthorized activity. 2. The process uses data analytics to look for patterns and correlations in claims being processed for payment. 3. Medical claim service provider reimbursement analysis is one of the most effective tools available to combat healthcare fraud. 4. By using data analytics to identify potential anomalies in medical claims, healthcare organizations can reduce their risk of losing money due to fraudulent activities. 5. Medical claim service provider reimbursement analysis can also help identify billing errors and other processes that could result in overpayment or underpayment.6. By carefully monitoring claims and identifying patterns, medical claim service provider reimbursement analysis can help healthcare organizations ensure they receive the correct payments for services rendered. 7. Healthcare organizations that use medical claim service provider reimbursement analysis can also improve their internal processes and reduce the chance of future fraudulent activities. 8. Regular reviews of medical claims can help ensure that all claims are accurate and that payments are received in a timely manner. 9. Healthcare organizations can use medical claim service provider reimbursement analysis to identify common trends in fraudulent billing practices and other healthcare fraud schemes. 10. Healthcare organizations can also use medical claim service provider reimbursement analysis to develop better billing processes and procedures to prevent fraud. 11. Additionally, medical claim service provider reimbursement analysis can be used to identify areas of overpayment or underpayment that can be addressed. 12. Medical claim service provider reimbursement analysis is an important tool for healthcare organizations to detect, prevent, and reduce healthcare fraud. 13. Healthcare organizations should use medical claim service provider reimbursement analysis to identify suspicious activities and take appropriate actions to prevent future fraud. 14. Healthcare organizations can also use medical claim service provider reimbursement analysis to identify areas of potential risk and take steps to mitigate those risks. 15. Healthcare organizations should also use medical claim service provider reimbursement analysis to identify patterns of fraudulent Medical claim service providers must comply with various laws and regulations to ensure the safety and accuracy of the claims they process. These laws and regulations vary from state to state, but typically include requirements for the provider to be licensed, bonded, and insured, as well as to maintain certain record-keeping and reporting procedures. In some cases, the provider must also comply with HIPAA regulations, which protect the privacy of patient health information. Additionally, the provider must comply with state and federal laws governing billing and reimbursement practices. Finally, the provider must comply with state and federal laws governing the use of electronic health records. ## Methodology The public dataset was used for this study and the methodology to predict organized fraud in health care service providers was based on regression models. The first step was to resize and normalize the dataset to train and test beneficiaries' details based on their illness and chronic conditions of various diseases and disorders. Zixuan Qin's (2022) study showed that the CNN regression model was used for the real-time and virtual control systems of data [27], and the model was used to classify both the normal (0) and chronic (1) conditions. Based on the chronic condition, the birth and death dates of the patient were calculated. The Inpatient and Outpatient case history of admission and discharge details were then merged. This merged dataset was further combined with beneficiary and health care service provider data to identify any fraudulent claims. The proportion of potential fraudulent claims was calculated using the merged data, providing insight into the number of transaction claims and money involved between the beneficiary, physicians, and diagnostic centers. An evaluation of the top reasons for suspicious money transactions between health care and diagnostic centers based on fraud and non-fraud counts was conducted. This analysis allowed for accurate predictions of the top physician malpractice with insurance companies and diagnostic centers. The relation between insurance provider claims transaction of patients and fraud cases is determined by their age. Fraud cases are more prominent in the age group above 70 years and less in the age group of 30 to 70 years. Age is thus an important factor in distinguishing between fraud and nonfraud behavior. To accurately select the features, a CNN-based feature selection was used. Clues of fraud and abuse were identified by employing sparse matrix and grouping-based similarity calculations. The sparse matrix is used for compression techniques to store elements in memory more efficiently by replacing zero values with fewer memory footprints. Similarly, based grouping is used to split a dataset into distinct groups based on their similar constraints and variance. This technique is often used to group records of clinical procedures, diagnostic, and insurance provider codes. Principal component analysis (PCA) is used to reduce the data set into the lower dimension and Auto Encoders (AE) is used to reduce the original dimensionality of the data set and generate the compressed representation of the original data. Both PCA and AE are used for the feature extraction which are further used for the classification Random Forests consist of multiple random decision trees that are used to classify datasets based on their features and convert weak learners into strong learners during training. To extract the variance of features, Principal Component Analysis (PCA) and Autoencoders (AE) are utilized. The proposed Auto-Encoder model is primarily used to extract more features by thresholding the reconstruction error on a merged dataset. The errors on the individual attributes of fraud and non-fraud data are combined to create the target class, and then all the attribute errors are concatenated with the target class based on their variance. The overall prediction is based on performance metrics such as curves, correlation of residual error, threshold, error loss, and the ROC curve. ### Logistics Regression model The prediction is based on linear dependent and independent variables. According to Atari (2019), logistic regression is an effective model for determining health insurance ownership (yes/no) based on age and gender [28]. Similarly, Schober (2021) states that logistic regression can be used to estimate the relationship between one or more independent variables and a binary (dichotomous) outcome variable [29]. The model Figure 1: The architecture with supervised and unsupervised classifiers Figure 3: The proposed methodology of model prediction by confusion matrix Figure 2: The proposed model is trained and validated, then the ROC analysis is conducted by measuring truth positive (precision) against false positive (recall). The probability threshold can be adjusted to change the threshold between 50-50 at the trade-off level, which optimizes the f1-measure metrics. However, the accuracy of the model is decreased when the cut-off is modified. #### Random Forest model The Random Forest model is used to predict health care variables and is shown to be more accurate than other models [30]. It utilizes non-linearity between variables to group a subset of decision trees and identify predictors through the outcome of a discriminate result of poor versus good (AUC = 0.707) [31]. The model performs both training and validation and then uses ROC analysis to evaluate truth positive (precision) versus false positive (recall). The features are extracted based on claim, reimbursement, diagnosis, and physicians' charges. The model is also used to predict unseen data. #### Principle component Analysis (PCA) This is the unsupervised learning method used for feature extraction. It helps to standardize scalar data with the maximum number of variables, allowing for the identification of linear and non-linear variance in order to extract features that may indicate potential fraud information. Multicollinearity in the models is addressed by using Principal Components Analysis (PCA) to produce linearly uncorrelated components [32]. PCA has been used for clinical intervention, such as for evidence-based decision-aids [33], as well as for data pre-processing, such as for feature extraction and dimension reduction. #### Proposed model of Auto-Encoders (AE) The model is used for learning generative information through encoding and dimensionality reduction methods. It splits the features into a reduced representation and reconstruction error, and then generates outputs as close as possible to the original input. The model is used to learn the patterns from non-fraudulent data and then uses the reconstruction error threshold on fraudulent information to predict the target class. The model is a sequential network layer with a dropout (0.2) layer to add noise to the data and a dense layer to encode patterns via dimensionality reduction. The model has a batch size of 32 and a learning rate of 100 epochs. The model predicts the output based on the mean square error as a metric, with each neuron acting as a linear regression method. It computes and combines errors from fraud and non-fraud information based on the error plot. A fixed threshold is used for the classification of fraud and non-fraud information. The shape of the target is expanded for merging and concatenating all the attribute errors with the target class. The percentile distribution of absolute errors is then calculated. Different threshold values are used for the reconstruction of error, and the evaluation result shows improved performance of the model. The Auto encoders have two hidden layers added in order to calculate the reconstruction error for both normal and fraudulent data. The graphical representation demonstrates the ROC analysis of true class, precision vs. recall metrics, threshold vs. recall, and recall vs. threshold. The model is able to determine the prediction class based on the error threshold. Dual Autoencoders are employed to address the issue of imbalanced classes when embedded in the training data [34]. Autoencoders are also highly effective when used for predictive analysis, such as diagnosing faults and determining the useful life of components [35]. #### Performance Metrics Evaluation Performance metrics was calculated based on the true positive, negative and false positive, negative rate based on the confusion matrix. The area under the curve (AUC) ranges between 0 and 1, with an AUC of 0.0 indicating wrong prediction and an AUC of 1.0 indicating perfect prediction. The Receiver Operating Characteristic (ROC) is used to predict a dichotomous outcome by comparing sensitivity against specificity. By measuring the area under the ROC curve, it is possible to evaluate and compare the performance of different classifiers. The Cohen Kappa statistic is used to compare the observed accuracy to the expected accuracy, and to evaluate the performance of different classifiers. ### Results and Discussion The percentage distribution of the fraud class "No" is 90.64695 and the distribution of the fraud class "Yes" is 9.35305. Over 80% of beneficiaries are of the same race. Figure 4 illustrates that the top diagnoses (in terms of money involved) are 4019, 4011, and 2724 and the distribution of fraud and non-fraud counts reveals suspicious transactions associated with them. Figures 4 illustrate the performance of the logistic regression model through precision and recall curves, respectively. Figure 5 depicts the performance of the random forest model through true positive and false positive rates. Figure 6 Auto Encoder prediction based on reconstruction error threshold of fraud data with ROC curve and confusion matrix Logistic regression is a supervised learning algorithm that is used for classification problems. It is a type of regression analysis that predicts the probability of a certain event occurring. The trade-off between precision (TPR) and recall (FPR) is a measure of how accurately the model is predicting the outcome. Precision is a measure of how many of the predicted positive outcomes are actually correct. Recall is a measure of how many of the actual positive outcomes are being correctly identified. As precision increases, recall decreases and vice versa. The ideal trade-off between precision and recall is a point where both metrics are maximized. This can be seen in Figure 6 which shows a graph of precision (TPR) and recall (FPR) against a logistic regression model. To visualizes the performance of a logistic regression model on a dataset. The ROC curve shows the model's true positive rate (TPR) against its false positive rate (FPR). The higher the TPR is, the better the model is at correctly classifying positive cases. The precision vs recall graph shows the precision (how often the model correctly predicts positive cases) versus the recall (how often the model correctly identifies all positive cases). The false positive cases graph shows the number of false positives that the model generates. Figure 4: Distribution of fraud and non-fraud count shows suspicious transactions involved in them. Figure 5: Logistic Regression trade off metrics of Precision (TPR) and Recall (FPR) The figure is a visualization of the performance metrics of a Random Forest model. The graph plots the true positive rate (TPR) on the y-axis against the false positive rate (FPR) on the x-axis. The graph also shows the ROC (receiver operating characteristic) curve which is a measure of the model's performance at different thresholds. The area under the curve (AUC) is a measure of accuracy and the closer the AUC is to 1, the better the model's performance is. The graph can be used to determine the best threshold for the model, which is the point at which the TPR and FPR are maximized. The autoencoder is a type of artificial neural network that is used to identify anomalies in data. In this process, non-fraud data is first used to train the autoencoder. The autoencoder then reconstructs the non-fraud data and the reconstruction error is used as a threshold for identifying fraud data. Finally, the reconstruction error of the fraud data is compared to the threshold to determine whether or not it is fraudulent. An Auto Encoder (AE) is an unsupervised neural network used for learning efficient data representations. It is composed of an encoder, which maps the input data to a latent space, and a decoder, which reconstructs the input data from the latent space. The AE is trained to minimize the difference between the input and reconstructed data. In this figure, the AE is used to detect fraudulent data by training it on a dataset of known fraudulent data. The AE is then used to predict whether new data is fraudulent or not based on the error threshold for reconstruction. The results of the prediction are shown on a ROC curve and a confusion matrix. The ROC curve plots the true positive rate against the false positive rate, and the confusion matrix displays the model's performance in terms of true positives, false positives, true negatives, and false negatives. The higher the area under the ROC curve, the better the model is at distinguishing between fraudulent and non-fraudulent data. The confusion matrix can be used to determine the precision, recall, and F1 score of the model. ### Dataset The open-source dataset in Kaggle added with some private data ### Conclusion The anomaly detection algorithms are used to detect the outliers and the out-of-distribution data points. The supervised learning algorithms are used to classify the data into different classes and the unsupervised learning algorithms are used to cluster the data into different groups. The fraud detection techniques are used to predict the fraudulent activities and the machine learning algorithms are used to analyze the data and make the better decisions based on the patterns. The fraud detection techniques are used to detect the Figure 6: Random forest model performance metrics of TPR vs FPR and ROC curve anomalies and the non-compliant activities. The supervised learning algorithms are used to classify the data into different classes and the unsupervised learning algorithms are used to cluster the data into different groups. The fraud detection techniques are used to identify the fraudulent activities and the patterns of the data. The data mining techniques are used to analyze the data and detect the patterns and anomalies. The data mining techniques are used to identify the underlying patterns and the outliers. The data mining techniques are used to improve the accuracy of the predictions and the decisions. The analytics techniques are used to analyze the data and detect the patterns. The analytics techniques are used to identify the underlying trends and the outliers. The analytics techniques are used to improve the accuracy of the predictions and the decisions. The fraud detection techniques are used to identify the fraudulent activities and the patterns. The fraud detection techniques are used to detect the anomalies and the non-compliant activities.
2302.12250	Phase diagram of early training dynamics in deep neural networks: effect of the learning rate, depth, and width	We systematically analyze optimization dynamics in deep neural networks (DNNs) trained with stochastic gradient descent (SGD) and study the effect of learning rate $\eta$, depth $d$, and width $w$ of the neural network. By analyzing the maximum eigenvalue $\lambda^H_t$ of the Hessian of the loss, which is a measure of sharpness of the loss landscape, we find that the dynamics can show four distinct regimes: (i) an early time transient regime, (ii) an intermediate saturation regime, (iii) a progressive sharpening regime, and (iv) a late time ``edge of stability" regime. The early and intermediate regimes (i) and (ii) exhibit a rich phase diagram depending on $\eta \equiv c / \lambda_0^H $, $d$, and $w$. We identify several critical values of $c$, which separate qualitatively distinct phenomena in the early time dynamics of training loss and sharpness. Notably, we discover the opening up of a ``sharpness reduction" phase, where sharpness decreases at early times, as $d$ and $1/w$ are increased.	Dayal Singh Kalra, Maissam Barkeshli	2023-02-23T18:59:30Z	http://arxiv.org/abs/2302.12250v2	# Phase diagram of training dynamics in deep neural networks: ###### Abstract We systematically analyze optimization dynamics in deep neural networks (DNNs) trained with stochastic gradient descent (SGD) over long time scales and study the effect of learning rate, depth, and width of the neural network. By analyzing the maximum eigenvalue $\lambda_{t}^{H}$ of the Hessian of the loss, which is a measure of sharpness of the loss landscape, we find that the dynamics can show four distinct regimes: (i) an early time transient regime, (ii) an intermediate saturation regime, (iii) a progressive sharpening regime, and finally (iv) a late time "edge of stability" regime. The early and intermediate regimes (i) and (ii) exhibit a rich phase diagram depending on learning rate $\eta\equiv c/\lambda_{0}^{H}$, depth $d$, and width $w$. We identify several critical values of $c$ which separate qualitatively distinct phenomena in the early time dynamics of training loss and sharpness, and extract their dependence on $d/w$. Our results have implications for how to scale the learning rate with DNN depth and width in order to remain in the same phase of learning. Machine Learning, Deep Neural Networks, Deep Neural Networks, Neural Networks ## 1 Introduction The optimization dynamics of deep neural networks (DNNs) is a rich problem that is of fundamental and practical interest. Seemingly small changes to the hyperparameters can dramatically alter the effectiveness of training and even lead to training failures. Basic questions about how to choose learning rates and their effect on generalization error and training speed remain intensely studied research problems. Classical intuition from convex optimization has lead to the oft-made suggestion that in stochastic gradient descent (SGD), the learning rate $\eta$ should satisfy $\eta<2/\lambda^{H}$, where $\lambda^{H}$ is the maximum eigenvalue of the Hessian $H$ of the loss, in order to ensure that the network reaches a minimum. However several recent studies have suggested that it is both possible and potentially preferable to have the learning rate _early in training_ reach $\eta>2/\lambda^{H}$(Wu et al., 2018; Lewkowycz et al., 2020; Zhu et al., 2022). The idea is that such a choice will induce a temporary training instability, causing the network to 'catapult' out of a local basin into a flatter one with lower $\lambda^{H}$ where training stabilizes. Indeed, during the early training phase, the local curvature of the loss landscape changes rapidly (Keskar et al., 2016; Achille et al., 2017; Jastrzebski et al., 2018; Fort and Ganguli, 2019), and the learning rate plays a crucial role in determining the convergence basin (Jastrzebski et al., 2018). Flatter basins are believed to be preferable because they potentially lead to lower generalization error (Hochreiter and Schmidhuber, 1994; 1997; Keskar et al., 2016; Dziugaite and Roy, 2017; Jiang et al., 2019; Foret et al., 2020) and allow larger learning rates leading to potentially faster training. From a different perspective, the major theme of deep learning is that it is beneficial to increase the model size as much as possible. This has come into sharp focus with the discovery of scaling laws that show power law improvement in generalization error with model and dataset size (Kaplan et al., 2020). This raises the fundamental question of how one can scale DNNs to arbitrarily large sizes while maintaining the ability to learn; in particular, how should initialization and optimization hyperparameters be chosen to maintain a similar quality of learning as the architecture size is taken to infinity (Jacot et al., 2018; Lee et al., 2018; 2019; Dyer and Gur-Ari, 2020; Yang and Hu, 2021; Roberts et al., 2022; Yang et al., 2022; Yaida, 2022)? Motivated by these ideas, we perform a systematic analysis of the training dynamics of SGD for DNNs as learning rate, depth, and width are tuned, across a variety of architectures and datasets. We monitor both the loss and sharpness ($\lambda^{H}$) trajectories out to late times, observing a number of qualitatively distinct phenomena summarized below. ### Our contributions We study SGD on fully connected networks (FCNs) with the same number of hidden units (width) in each layer, convolutional neural networks (CNNs) and ResNet architectures of varying width $w$ and depth $d$ with ReLU activation. For CNNs, the width corresponds to the number of channels. We parameterize FCNs in the Neural Tangent Parameterization (NTP) (Jacot et al., 2018) and CNNs and ResNets in Standard Parameterization (SP) (Sohl-Dickstein et al., 2020); other parameterizations, such as muP (Yang and Hu, 2021), can give different results. For further details, see Appendix A. We study both mean-squared error (MSE) and cross-entropy loss functions, and the datasets CIFAR-10, MNIST, Fashion-MNIST. The basic phenomena we report are reliably reproduced across these architectures and datasets. We use sharpness to refer to $\lambda_{t}^{H}$, the maximum eigenvalue of $H$ at time-step $t$, and flatness refers to $1/\lambda_{t}^{H}$. By monitoring the sharpness, we find four clearly separated, qualitatively distinct regimes throughout the entire span of the training trajectory. Fig. 1 shows an example from a CNN architecture. The four observed regimes are: (i) an early time transient regime, (ii) an intermediate saturation regime where the sharpness has lowered and remains relatively constant, (iii) a progressive sharpening regime, and finally (iv) a late time "edge of stability" regime where the sharpness saturates around $2/\eta$(Cohen et al., 2021). Next, we analyze the early transient and intermediate saturation regimes. For MSE loss, a clear picture emerges, and leads to a rich phase diagram, sketched in Figure 2, as learning rate, $d$, and $w$ are tuned. Given the learning rate scaled as $\eta=c/\lambda_{0}^{H}$, we characterize four distinct behaviors in the training dynamics in the early transient regime (i): 1. Sharpness reduction phase ($c<c_{loss}$) :_Both the loss and the sharpness_ monotonically decrease during early training. There is a particularly significant drop in sharpness in the regime $2\approx c_{crit}<c<c_{loss}$, which motivates us to refer to learning rates lower than $c_{crit}$ as sub-critical and larger than $c_{crit}$ as super-critical. The regime $c_{crit}<c<c_{loss}$ opens up significantly with increasing $d/w$, which is a new result of this work. 2. Loss catapult phase ($c_{loss}<c<c_{sharp}$) : The first few gradient steps take training to a flatter region but with a higher loss. Training eventually settles down in the flatter region as the loss starts to decrease again. The sharpness _monotonically decreases from initialization_ in this early time transient regime. 3. Loss and sharpness catapult phase ($c_{sharp}<c<c_{max}$): In this regime _both the loss and sharpness_ initially start to increase, effectively catapulting to a different point where loss and sharpness can start to decrease again. Training eventually exhibits a significant reduction in sharpness by the end of the early training. The report of a _loss and sharpness catapult_ is also new to this work. 4. Divergent phase ($c>c_{max}$): The learning rate is too Figure 1: Full training trajectories of the (a) training loss, (b) sharpness, and (c) training accuracy of CNNs ($d=5$ and $w=512$) trained on CIFAR-10 with MSE loss using vanilla SGD with learning rates $\eta=c/\lambda_{0}^{H}$ and batch size $B=512$. Vertical dashed lines approximately separate the different training regimes. Horizontal dashed lines in (b) denote the $2/\eta$ threshold for each learning rate. In Appendix G.1 we show the same trajectories with the time axis rescaled as $ct$. Figure 2: Sketched phase diagram of early training large for training to converge and the loss diverges. The critical values $c_{loss}$, $c_{sharp}$, $c_{max}$ are random variables that depend on random initialization, SGD batch selection, and architecture. The averages of $c_{loss}$, $c_{sharp}$, $c_{max}$ are shown in the phase diagrams and show strong systematic dependence on $d/w$. We also analyze the linear connectivity of the loss landscape in the early transient regime and show that for a range of learning rates $c_{loss}<c<c_{barrier}$, no barriers exist from the initial state to the final point of the initial transient phase, even though training passes through regions with higher loss than initialization. Next, we provide a quantitative analysis of the intermediate saturation regime. We find that sharpness during this time typically displays 3 distinct regimes as the learning rate is tuned, depicted in Fig. 6. By identifying an appropriate order parameter we can extract a sharp peak corresponding to $c_{crit}\approx 2/\lambda_{0}^{H}$. For $c\ll c_{crit}$, the network is effectively in a lazy training regime, with increasing fluctuations as $d/w$ is increased. Finally, we show that a single hidden layer linear network - the $uv$ model - displays the same phenomena discussed above for the early regimes (i) and (ii), and we analyze the phase diagram in this minimal model. ### Related works A significant amount of research has identified various training regimes using diverse criteria, e.g., (Erhan et al., 2010; Achille et al., 2017; Fort et al., 2020; Jastrzebski et al., 2018; Frankle et al., 2020; Leclerc and Madry, 2020; Jastrzebski et al., 2020; Cohen et al., 2021; Iyer et al., 2022). Here we focus on studies that characterize training regimes with sharpness and learning rates. Several studies have analyzed sharpness at different training times (Jastrzebski et al., 2018; Fort and Ganguli, 2019; Jastrzebski et al., 2020; Cohen et al., 2021; Iyer et al., 2022). (Cohen et al., 2021) studied sharpness at late training times and showed how _large-batch_ gradient descent shows progressive sharpening followed by the edge of stability: For MSE loss, the sharpness increases until it reaches a maximum value just above $2/\eta$ and oscillates near it. This has motivated various theoretical studies (Damian et al., 2022; Agarwala et al., 2022; Arora et al., 2022). We add to this line of work by demonstrating a wide variety of architectures trained with SGD with relatively small batch sizes $B=512$, $\approx 1\%$ of the full training set, that clearly show progressive sharpening and edge of stability phenomena. (Jastrzebski et al., 2018) studied the entire training trajectory of sharpness in models trained with SGD and cross-entropy loss and found that sharpness increases during the early stages of training, reaches a peak, and then decreases. In contrast, we find a sharpness-reduction phase, $c<c_{loss}$ which becomes more prominent with increasing $d/w$, where sharpness only decreases during early training; this also occurs in the catapult phase $c_{loss}<c<c_{sharp}$, during which the loss initially increases before decreasing. This discrepancy is likely due to different initialization and learning rate scaling in their work (Iyer et al., 2022). (Jastrzebski et al., 2020) examined the effect of hyperparameters on sharpness at late training times. (Gilmer et al., 2021) studied the optimization dynamics of SGD with momentum using sharpness. (Leclerc and Madry, 2020) classify training into 2 different regimes using training loss, providing a significantly coarser description of training dynamics than provided here. (Iyer et al., 2022) studied the scaling of the maximum learning rate with $d$ and $w$ during early training in FCNs and its relationship with sharpness at initialization. (Lewkowycz et al., 2020) analyzed the curvature during early training using the top eigenvalue of the neural tangent kernel (NTK) and demonstrated the existence of a new early training phase, which they dubbed the "catapult" phase, $2/\lambda_{0}^{NTK}<\eta<\eta_{max}$, in wide networks trained with MSE loss using SGD, in which training converges after an initial increase in training loss. The existence of this new training regime was further extended to quadratic models with large widths by (Zhu et al., 2022; Meltzer and Liu, 2023). Our work extends the above analysis by studying the combined effect of learning rate, depth, and width, demonstrating the opening of a sharpness-reduction phase, the refinement of the catapult phase into two phases depending on whether the sharpness also catapults, analyzing the phase boundaries as $d/w$ is increased, analyzing linear mode connectivity in the catapult phase, and also examining different qualitative behaviors in the intermediate saturation regime (ii) mentioned above. ## 2 The four regimes of training Figure 1 illustrates the training trajectories of a 5-layer deep ($d=5$) CNN from the Myrtle family (Shankar et al., 2020), trained on the CIFAR-10 dataset using MSE loss. Each network layer has $w=128$ channels. The networks were trained using SGD with various learning rates and a batch size of $B=512$. These results indicate that the training dynamics can be divided into four distinct regimes based on training loss and sharpness: 1. Early time transient: ($0<t<T_{1}$). Within a few ($\sim 10$) training steps, training selects the convergence basin, which depends on the optimization hyperparameters. The loss and sharpness may drastically change, and training eventually settles down in a basin by the end of this short period. * Intermediate saturation: ($T_{1}<t<T_{2}$). The learning rate is small compared to sharpness, resulting in a slow transient period. The sharpness changes slowly until it reaches a tipping point $T_{2}$, which marks this regime's end. * Progressive sharpening: ($T_{2}<t<T_{3}$). The sharpness steadily rises until it reaches a maximum value $\approx 2/\eta$(Jastrzebski et al., 2020; Cohen et al., 2021). * Late time: ($T_{3}<t$). Training dynamics show different phenomena depending on the loss and optimization parameters. For networks trained with MSE loss and large enough batch sizes, training enters a stable cycle in which sharpness oscillates around this maximum value, and the training loss fluctuates while decreasing overall. In comparison, sharpness may drop after reaching this maximum value for cross-entropy loss while remaining less than $2/\eta$. We confirm the existence of the four training regimes across various architectures, datasets, and loss functions as demonstrated in Appendix G. Our results show that FCNs, CNNs, and ResNets, when trained on CIFAR-10, MNIST, and Fashion-MNIST datasets, using MSE and cross-entropy loss, exhibit all four training regimes. An exception to this observation is that FCNs did not exhibit the fourth regime (iv), even after training for $1000$ epochs. Note the log scale in Fig. 1 highlights the early regimes (i) and (ii); in absolute terms these are much shorter in time than regimes (iii) and (iv). ## 3 Phase diagram of early transient regime For wide enough networks trained with MSE loss using SGD, training converges into a flatter region after an initial increase in the training loss for learning rates $c>2$(Lewkowycz et al., 2020). Figure 3(a, b) shows the first $10$ steps of the loss and sharpness trajectories of a shallow ($d=5$ and $w=512$) CNN trained on the CIFAR-10 dataset with MSE loss using SGD. For learning rates, $c\geq 2.82$, the loss catapults and training eventually converges into a flatter region, as measured by sharpness. Additionally, we observe that sharpness may also spike initially, similar to the training loss (see Figure 3 (b)). However, this initial spike in sharpness occurs at relatively higher learning rates ($c\geq 5.65$), which we will examine along with the loss catapult. We refer to this spike in sharpness as'sharpness catapult.' An important consideration is the degree to which this phenomenon changes with network depth and width. Interestingly, we found that the training loss in deep networks on average catapults at much larger learning rates than $c=2$. Figure 3(c, d) shows that for a deep ($d=10,w=128$) CNN, the loss and sharpness may catapult only near the maximum trainable learning rate. In this section, we characterize the properties of the early training dynamics of models with MSE loss. For cross-entropy loss, we found Figure 3: Early training dynamics of (a, b, c) a shallow ($d=5$, $w=512$) and (d, e, f) a deep CNN ($d=10,w=128$) trained on CIFAR-10 with MSE loss for $t=10$ steps using SGD for various learning rates $\eta=c/\lambda_{0}^{H}$ and batch size $B=512$. (a, d) training loss, (b, e) sharpness, and (c, f) interpolated loss between the initial and final parameters after $10$ steps for the respective models. For the shallow CNN, $c_{loss}=2.82,c_{sharp}=5.65,c_{max}=17.14$ and for the deep CNN, $c_{loss}=36.75,c_{sharp}=39.39,c_{max}=48.50$. that the early time dynamics is more chaotic and it is unclear to what extent it can be separated into different phases. ### Loss and sharpness catapult during early training In this subsection, we characterize the effect of finite depth and width on the onset of the loss and sharpness catapult and training divergence. We begin by defining critical constants that correspond to the above phenomena. Definition 3.1.: ($c_{loss},c_{sharp},c_{max}$) For learning rate $\eta=c/\lambda_{0}^{H}$, let the training loss and sharpness at step $t$ be denoted by $\mathcal{L}_{t}(c)$ and $\lambda_{t}^{H}(c)$. We define $c_{loss}(c_{sharp})$ as minimum learning rates constants such that the loss (sharpness) increases during the initial transient period \[c_{loss}=\min_{c}\{c\mid\max_{t\in[1,T_{1}]}\mathcal{L}_{t}(c)> \mathcal{L}_{0}(c)\} \tag{1}\] \[c_{sharp}=\min_{c}\{c\mid\max_{t\in[1,T_{1}]}\lambda_{t}^{H}(c)> \lambda_{0}^{H}(c)\}, \tag{2}\] and $c_{max}$ as the maximum learning rate constant such that the loss does not diverge during the initial transient period \[c_{max}=\max_{c}\{c\mid\mathcal{L}_{t}(c)<K,\forall t\in[1,T_{1}]\}, \tag{3}\] where $K$ is a fixed large constant 1. Footnote 1: We use $K=10^{5}$ to estimate $c_{max}$. In all our experiments, $\mathcal{L}_{0}=\mathcal{O}(1)$ (see Appendix A), which justifies the use of a fixed value. Note that the definition of $c_{max}$ allows for more flexibility than previous studies (Iyer et al., 2022) in order to investigate a wider range of phenomena occurring near the maximum learning rate. Here, $c_{loss}$, $c_{sharp}$, and $c_{max}$ are random variables that depend on the random initialization and the SGD batch sequence, and we denote the average over this randomness using $\langle\cdot\rangle$. Figure 4 illustrates the phase diagram of early training for three different architectures trained on various datasets with MSE loss using SGD. These phase diagrams show how the averaged values $\langle c_{loss}\rangle$, $\langle c_{sharp}\rangle$, and $\langle c_{max}\rangle$ are affected by network depth and width when plotted in the subspace represented by the depth-to-width ratio ($d/w$). This choice is motivated by various studies that show how in FCNs, the ratio $d/w$ controls the fluctuations (Hanin and Nica, 2019; Dyer and Gur-Ari, 2020; Roberts et al., 2022). The results show that the averaged values of all the critical constants increase significantly with $d/w$ (note the log scale). At small values of $d/w$, the loss starts to catapult at $c\approx 2$. As $d/w$ increases, $\langle c_{loss}\rangle$ increases and eventually converges to $\langle c_{max}\rangle$ at large $d/w$. By comparison, sharpness starts to Figure 4: The phase diagrams of the early training of three different types of neural networks trained with MSE loss function using SGD. (a) FCNs ($d=8$) trained on the MNIST dataset, (b) CNNs ($d=7$) trained on the Fashion-MNIST dataset, (c) ResNet ($d=18$) trained on the CIFAR-10 dataset (without batch normalization). Each data point in the figure represents an average of ten distinct initialization, and the solid lines represent a smooth curve fitted to the raw data points. The vertical dotted line shows $c=2$ for comparison, and various colors are filled in between the various curves for better visualization. For experimental details and additional results for different depths, see Appendices A and C, respectively. Figure 5: The relationship between critical constants for ResNets with $d\in\{18,34\}$, $w\in\{32,64,128\}$, trained on the CIFAR-10 dataset. Each data point corresponds to a randomly initialized model. The dashed line represents the $y=x$ line. For experimental details and additional results, see Appendices A and D. catapult at relatively large learning rates at small $d/w$, with $\langle c_{sharp}\rangle$ continuing to increase with $d/w$ while remaining between $\langle c_{loss}\rangle$ and $\langle c_{max}\rangle$. Moreover, although $\langle c_{max}\rangle$ increases with $d/w$, the bare learning rate $\eta_{max}\equiv c_{max}/\lambda_{0}^{H}$ decreases on average with $d/w$. While we plotted the averaged quantities $\langle c_{loss}\rangle$, $\langle c_{sharp}\rangle$, $\langle c_{max}\rangle$, we have observed that their variance also increases significantly with $d/w$; in Appendix C we show standard deviations about the averages for different random initializations. Nevertheless, we have found that the following inequality typically holds, for any given initialization and batch sequences: \[c_{loss}\leq c_{sharp}\leq c_{max}, \tag{4}\] except for some outliers due to high fluctuations when the averaged critical curves start merging at large $d/w$. Figure 5 shows evidence of this for ResNets trained on the CIFAR-10 dataset. Appendix D presents extensive additional results across various architectures. ### Loss connectivity in the early transient period In the previous subsection, we observed that training loss and sharpness might quickly increase before decreasing ("catapult") during early training for a range of depths and widths. A logical next step is to analyze the region in the loss landscape that the training reaches after the catapult. A naive intuition would predict a barrier from initialization to the final point after the catapult, as the loss increases during early training. In this section, we will test the credibility of this intuition in real-world models. Specifically, we linearly interpolate the loss between the initial and final point after the catapult and examine the effect of the learning rate, depth, and width. The linearly interpolated loss and barrier are defined as follows. Definition 3.2.: $(\mathcal{L}_{int}(s,c),U(c))$ Let $\theta_{0}$ represent the initial set of parameters, and let $\theta_{T_{1}}$ represent the set of parameters at the end of the initial transient period, trained using a learning rate constant $c$. Then, we define the linearly interpolated loss as $\mathcal{L}_{int}(s,c)=\mathcal{L}[(1-s)\ \theta_{0}+s\ \theta_{T_{1}}]$, where $s\in[0,1]$ is the interpolation parameter. The interpolated loss barrier is defined as the maximum value of the interpolated loss over the range of $s$: $U(c)=\max_{s\in[0,1]}\mathcal{L}_{int}(s)-\mathcal{L}(\theta_{0})$. Here we subtracted the loss's initial value such that a positive value indicates a barrier to the final point from initialization. Using the interpolated loss barrier, we define $c_{barrier}$ as follows. Definition 3.3.: ($c_{barrier}$) Given the initial ($\theta_{0}$) and final parameters ($\theta_{T_{1}}$), we define $c_{barrier}$ as the minimum learning rate constant such that there exists a barrier from $\theta_{0}$ to $\theta_{T_{1}}$ \[c_{barrier}=\min_{c}\{c\ \|\ U(c)>0\} \tag{5}\] Here, $c_{barrier}$ is also a random variable that depends on the initialization and SGD batch sequence. We denote the average over this randomness using $\langle.\rangle$ as before. Figure 3(c, f) shows the interpolated loss of CNNs trained on the CIFAR-10 dataset for $t=10$ steps. The experimental setup is the same as in Section 3. For the network with smaller $d/w$, we observe a barrier emerging at $c_{barrier}=5.65$, while the loss starts to catapult at $c_{loss}=2.83$. In comparison, we do not observe any barrier from initialization to the final point at large $d/w$. Figure 4 shows the relationship between $\langle c_{barrier}\rangle$ and $d/w$ for various models and datasets. We consistently observe that $c_{sharp}\leq c_{barrier}$, suggesting that training traverses a barrier only when sharpness starts to catapult during early training. We chose not to characterize Figure 6: (a) Normalized sharpness measured at $c\tau=200$ against the learning rate constant for $7$-layer CNNs trained on the CIFAR-10 dataset, with varying widths. Each data point is an average over 5 initializations, where the shaded region depicts the standard deviation around the mean trend. (b, c) Smooth estimations of the first two derivatives, $\chi_{\tau}$ and $\chi_{\tau}^{\prime}$, of the averaged normalized sharpness wrt the learning rate constant. The vertical lines denote $c_{crit}$ estimated using the maximum of $\chi_{\tau}^{\prime}$. For smoothening details, see Appendix E.2. the phase diagram of early training using $c_{barrier}$ as we did for other critical $c$'s, as it is somewhat different in character than the other critical constants, which depend only on the sharpness and loss trajectories. These observations call into question the intuition of catapulting out of a basin for a range of learning rates in between $c_{loss}<c<c_{barrier}$. Instead, these results show that for these learning rates, the final point after the catapult already lies in the same basin as initialization, and even _connected through a linear path_, revealing an inductive bias of the training process towards regions of higher loss during the early time transient regime. ## 4 Intermediate saturation regimes In the intermediate saturation regime, the learning rate is small compared to the local sharpness ($\eta<2/\lambda_{t}$). As a result, sharpness does not change appreciably and reflects the cumulative change that occurred during the initial transient period. This section analyzes sharpness in the intermediate saturation regime by studying how it changes with the learning rate, depth, and width of the model. We measure the sharpness $\lambda_{\tau}^{H}$ at a time $\tau$ in the middle of the intermediate saturation regime. We choose $\tau$ so that $cr\approx 200$.2 For further details on sharpness measurement, see Appendix E.1. Figure 6(a) illustrates the relationship between $\lambda_{\tau}^{H}$ and the learning rate for $7$-layer deep CNNs trained on the CIFAR-10 dataset with varying widths. The results indicate that the dependence of $\lambda_{\tau}^{H}$ on learning rate can be grouped into three distinct stages. (1) At small learning rates, $\lambda_{\tau}^{H}$ remains relatively constant, with fluctuations increasing as the ratio of $d/w$ increases ($c<2$ in Fig. 6(a)). (2) A crossover regime where $\lambda_{\tau}^{H}$ is dropping significantly ($2<c<2^{3}$ in Fig. 6(a)). (3) A saturation stage where $\lambda_{\tau}^{H}$ stays small and constant with learning rate ($c>2^{3}$) in Fig. 6(a)). In Appendix E, we show that these results are consistent across architectures and datasets for varying values of $d$ and $w$. Additionally, the results reveal that in stage (1), where $c<2$ is sub-critical, $\lambda_{\tau}^{H}$ decreases with increasing $d/w$. In other words, for small $c$ and in the intermediate saturation regime, the loss is locally flatter as $d/w$ increases. Footnote 2: In practice time-step $\tau=200/c$ is in the middle of regime (ii) for the models we studied. Normalizing by $c$ allows proper comparison for different learning rates. We can precisely extract a critical value of $c$ that separates stages (1) and (2), which corresponds to the onset of an abrupt reduction of sharpness $\lambda_{\tau}^{H}$. To do this, we consider the averaged normalized sharpness over initializations and denote it by $\langle\frac{\lambda_{\tau}^{H}}{\lambda_{\theta}^{H}}\rangle$. The first two derivatives of the averaged normalized sharpness, $\chi_{\tau}=-\frac{\partial}{\partial c}\langle\frac{\lambda_{\tau}^{H}}{ \lambda_{\theta}^{H}}\rangle$ and $\chi_{\tau}^{\prime}=-\frac{\partial^{2}}{\partial c^{2}}\langle\frac{ \lambda_{\tau}^{H}}{\lambda_{\tau}^{H}}\rangle$, characterize the change in sharpness with learning rate. The extrema of $\chi_{\tau}^{\prime}$ quantitatively define the boundaries between the three stages described above. In particular, using the maximum of $\chi_{\tau}^{\prime}$, we define $\langle c_{crit}\rangle$, which marks the beginning of the sharp decrease in $\lambda_{\tau}^{H}$ with the learning rate. Definition 4.1.: ($\langle c_{crit}\rangle$) Given the averaged normalized sharpness $\langle\frac{\lambda_{\tau}^{H}}{\lambda_{\theta}^{H}}\rangle$ estimated using sharpness measured at $\tau$, we define $c_{crit}$ to be the learning rate constant that minimizes the second derivative of $\langle\frac{\lambda_{\tau}^{H}}{\lambda_{\theta}^{H}}\rangle$ \[\langle c_{crit}\rangle=\operatorname{arg\,min}_{c}\frac{\partial^{2}}{ \partial c^{2}}\left\langle\frac{\lambda_{\tau}^{H}}{\lambda_{0}^{H}}\right\rangle =\operatorname{arg\,max}_{c}\chi_{\tau}^{\prime}. \tag{6}\] Here, we use $\langle.\rangle$ to denote that the critical constant is obtained from the averaged normalized sharpness. Figures 6(b, c) show $\chi_{\tau}$ and $\chi_{\tau}^{\prime}$ obtained from the results in Figure 6(a). We observe similar results across various architectures and datasets, as shown in Appendix E. Our results show Figure 7: The phase diagram of the $uv$ model trained with MSE loss using gradient descent with (a) the top eigenvalue of Hessian $\lambda_{t}^{H}$, (b) the trace of Hessian $\operatorname{Tr}(H_{t})$ and (c) the square of the Frobenius norm $\operatorname{Tr}(H_{t}^{T}H_{t})$ used as a measure of sharpness. In (a), the learning rate is scaled as $\eta=c/\lambda_{0}^{H}$, while in (b) and (c), the learning rate is scaled as $\eta=k/\operatorname{Tr}(H_{0})$. The vertical dashed line shows $c=2$ for reference. Each data point is an average over $500$ random initializations. that $\langle c_{crit}\rangle$ has slight fluctuations as $d/w$ is changed, but generally stays in the vicinity of $c=2$. The peak in $\chi_{\tau}^{\prime}$ becomes wider as $d/w$ increases, indicating that the transition between stages (1) and (2) becomes smoother, presumably due to larger fluctuations in the properties of the Hessian $H$ at initialization. In contrast to $\langle c_{crit}\rangle$, $\langle c_{loss}\rangle$ increases with $d/w$, implying the opening of the sharpness reduction phase $\langle c_{crit}\rangle<c<\langle c_{loss}\rangle$ as $d/w$ increases. ## 5 Insights from a simple model Here we analyze a two-layer linear network (Rennie and Srebro, 2005; Saxe et al., 2014; Lewkowycz et al., 2020), the $uv$ model, which shows much of the phenomena presented above. Define $f(x)=\frac{1}{\sqrt{w}}w^{T}ux$, with $x,f(x)\in\mathbb{R}$. Here, $u,v\in\mathbb{R}^{w}$ are the trainable parameters, initialized using the normal distribution, $u_{i},v_{i}\sim\mathcal{N}(0,1)$ for $i\in\{1,\dots,w\}$. The model is trained with MSE loss on a single training example $(x,y)=(1,0)$, which simplifies the loss to $\mathcal{L}(u,v)=\frac{1}{2}f^{2}$. We minimize the loss using gradient descent (GD) with learning rate $\eta=c/\lambda_{0}^{H}$. This model exhibits the (i) early time transient and (ii) intermediate saturation regimes, but not regimes (iii) and (iv).3 The early time phase diagram also shows similar features to those described in preceding sections (compare Fig. 7(a) and Fig. 4). Below we develop an understanding of this early time phase diagram in the $uv$ model. Footnote 3: For other choices where $y\neq 0$, the model exhibits different phenomenology than discussed here. The update equations of the $uv$ model in function space can be written in terms of the trace of the Hessian $\operatorname{Tr}(H)$. \[\begin{split}& f_{t+1}=f_{t}\left(1-\eta\operatorname{Tr}(H_{t})+ \frac{\eta^{2}f_{t}^{2}}{w}\right)\\ &\operatorname{Tr}(H_{t+1})=\operatorname{Tr}(H_{t})+\frac{\eta f _{t}^{2}}{w}\left(\eta\operatorname{Tr}(H_{t})-4\right).\end{split} \tag{7}\] In this model, $\operatorname{Tr}(H)$ is equal to the NTK. From the above equations, it is natural to scale the learning rate as $\eta=k/\operatorname{Tr}(H_{0})$. Note that $c=\eta\lambda_{0}^{H}=k\frac{\lambda_{0}^{H}}{\operatorname{Tr}(H_{0})}$. Also, we denote the critical constants in this scaling as $k_{loss}$, $k_{trace}$, $k_{max}$ and $k_{crit}$, where the definitions follow from Definitions 3.1 and 4.1 on replacing sharpness with trace. Figure 7(b) shows the phase diagram of early training, with $\operatorname{Tr}(H_{t})$ replaced with $\lambda_{t}^{H}$ as the measure of sharpness and with the learning rate scaled as $\eta=k/\operatorname{Tr}(H_{0})$. Similar to Figure 7(a), we observe a new phase $\langle k_{crit}\rangle<k<\langle k_{loss}\rangle$ opening up at small width. However, we do not observe the loss-sharpness catapult phase as $\operatorname{Tr}(H)$ does not increase during training (see Equation 7). We also observe $\langle k_{max}\rangle=4$, independent of width. In Appendix B.2, we show that \[\begin{split}\left\langle\frac{\mathcal{L}_{1}}{\mathcal{L}_{0} }\right\rangle=&(1-k)^{2}+\frac{k^{2}(1-k)}{2(w+1)}+\frac{3k^{4}}{ 16(w+1)(w+3)}\end{split} \tag{8}\] where the average is over initializations. As explained in Appendix B.2, the critical value of $k$ for which $\langle\mathcal{L}_{1}/\mathcal{L}_{0}\rangle>1$ increases with $1/w$, which explains why $\langle k_{loss}\rangle$ and $\langle c_{loss}\rangle$ increase with $1/w$. Combined with $k_{crit}\lesssim 2$, this implies the opening up of the sharpness reduction phase as $w$ is decreased. Figure 8 shows the training trajectories of the $uv$ model with large ($w=512$) and small ($w=2$) widths in a two-dimensional slice of parameters defined by $\operatorname{Tr}(H)$ and weight correlation $\frac{\langle v,u\rangle}{\|\|u\|\|\|\|v\|\|}$. The above figure reveals that the first few training steps of the small-width network take the system in a flatter direction (as measured by $\operatorname{Tr}(H)$) as compared to the wider network. This means that the small-width network needs a relatively larger learning rate to get to a point of increased loss. This means that the loss catapult requires a larger learning rate at a smaller width. We thus have the opening up of a new regime $k_{crit}<k<k_{loss}$, in which the loss and sharpness monotonically decrease during early training. We speculate that similar reasoning holds for real-world models for the opening up of the sharpness reduction phase with increasing $d/w$. To understand the loss-sharpness catapult phase, we require Figure 8: Training trajectories of the uv model, with (a, b) large and (c, d) small width, in a two-dimensional slice of the parameters defined by the trace of Hessian $\operatorname{Tr}(H)$ and weight correlation, trained with (a, c) small ($c=0.5$) and (b, d) large ($c=2.5$) learning rates. The colors correspond to the training loss $\mathcal{L}$, with darker colors representing a smaller loss. some other measure as $\operatorname{Tr}(H)$ does not increase for $0<k<4$. $\lambda_{t}^{H}=\|\|H\|\|_{\infty}$ corresponds to the $p=\infty$ Schatten norm, which is difficult to analyze. Instead, we analyze the $p=2$ Schatten norm, which is the Frobenius norm $\\|H\\|_{F}=\sqrt{\operatorname{Tr}(H^{T}H)}$. For the $uv$ model, \[\|\|H_{t}\|\|_{F}^{2}=\operatorname{Tr}(H_{t}^{T}H_{t})=\operatorname{Tr}(H_{t})^{ 2}+4\left(1+\frac{2}{w}\right)\mathcal{L}_{t} \tag{9}\] We define $k_{frob}$ as the minimum learning rate such that $\|\|H_{t}\|\|_{F}^{2}$ increases during early training. Figure 7(c) shows the phase diagram of the $uv$ model, with $\|\|H_{t}\|\|_{F}^{2}$ as the measure of sharpness, while the learning rate is scaled as $\eta=k/\operatorname{Tr}(H_{0})$. We observe the loss-sharpness catapult phase at small widths, which we understand as follows. In Appendix B.3, we show $\Delta\|\|H_{1}\|\|_{F}^{2}\equiv\|\|H_{1}\|\|_{F}^{2}-\|\|H_{0}\|\|_{F}^{2}$ averaged over initializations is \[\left\langle\Delta\|\|H_{1}\|\|_{F}^{2}\right\rangle= I(k,w)+4\left(1+\frac{2}{w}\right)\left\langle\mathcal{L}_{1}- \mathcal{L}_{0}\right\rangle, \tag{10}\] $I(k,w)=\frac{k(k-4)}{w}\left[\frac{3k(k-4)}{4(w+3)}+2\right]$. As demonstrated in Appendix B.3, the critical value for which $\left\langle\Delta\|\|H_{1}\|\|_{F}^{2}\right\rangle>0$ increases from $k_{loss}$ as $1/w$ increases. This explains the existence of the loss-sharpness catapult phase, and the opening up of the loss catapult phase at small $w$ in Fig. 7 (c). For $k>4$, both the loss and trace increase initially, causing training to diverge irrespective of width. This yields $c_{max}=4\lambda_{0}^{H}/\operatorname{Tr}(H)$. Figure 10(c) in Appendix B.2 shows that the ratio $\lambda_{0}^{H}/\operatorname{Tr}(H_{0})$ increases with $1/w$, which explains the increase in $\left\langle c_{max}\right\rangle$ with $1/w$. The loss landscape of the $uv$ model shown in Figure 8 reveals interesting insights into the loss landscape connectivity results in Section 3.2 and the presence of $c_{barrier}$. Fig.8 shows how even when there is a loss catapult, as long as the learning rate is not too large, the final point after the catapult can be reached from initialization by a linear path without increasing the loss and passing through a barrier. However if the learning rate becomes large enough, then the final point after the catapult may correspond to a region of large weight correlation, and there will be a barrier in the loss upon linear interpolation. ## 6 Discussion We have studied the effect of learning rate, depth, and width on the training dynamics in DNNs trained using SGD with learning rate scaled as $\eta=c/\lambda_{0}^{H}$. For various models and datasets, we demonstrated the existence of four training regimes. We analyzed the early transient and intermediate saturation regimes and presented a rich phase diagram of early training with learning rate, depth, and width. We report two new phases, sharpness reduction and loss-sharpness catapult, which have not been reported previously. We also find that the critical $c$'s increase significantly with $d/w$ on average. We further studied loss connectivity in the early transient regime and demonstrated the existence of a regime $c_{loss}<c<c_{barrier}$, in which the final point after the catapult lies in the same basin as initialization, connected through a linear path. Finally, we study these phenomena in a 2-layer linear network ($uv$ model), gaining insights into the opening of the sharpness reduction phase and increase of $c_{max}$ with $d/w$. We performed a preliminary analysis on the effect of batch size on the presented results in Appendix F. The sharpness trajectories of models trained with a smaller batch size ($B=32$ vs. $B=512$) show all four training regimes. The main effect of lowering batch size is that in the late time regime (iv), sharpness settles down to a value significantly smaller than $2/\eta$, which is consistent with (Jastrzebski et al., 2020) and (Cohen et al., 2021). In the early transient regime, we observe a qualitatively similar phase diagram. In the intermediate saturation regime, the effect of reducing the batch size is to broaden the transition around $c_{crit}$. Throughout this work, we have scaled the learning rate using the top eigenvalue of the Hessian as a measure of sharpness. In practice, various other choices can be made. For example, (Lewkowycz et al., 2020) scaled the learning rate with the top eigenvalue of the NTK, which also shows similar results but a few differences. At early training times, we observed that the NTK may not catapult during early training, which results in the absence of the sharpness catapult phase. Moreover, the deviation of various critical constants, $\left\langle c_{loss}\right\rangle$ and $\left\langle c_{max}\right\rangle$, with $d/w$, is less pronounced. These differences can be observed in the $uv$ model analysis presented in Section 5, as $\operatorname{Tr}(H)=$ NTK in that case. The early training dynamics is sensitive to the initialization scheme and optimization algorithm used, and we leave it to future work to explore this dependence and its implications. In this work, we focused on models initialized at criticality (Poole et al., 2016) as it allows for proper gradient flow at initialization (Hanin, 2018; Roberts et al., 2022), and studied vanilla SGD for simplicity. However, other initializations (LeCun et al., 2012), parameterizations (Yang and Hu, 2021; Yang et al., 2022), and optimization procedures (Goyal et al., 2017) may show dissimilarities with the reported phase diagram of early training. ## Acknowledgements We thank Andrey Gromov, Tianyu He, and Shubham Jain for discussions, and Paolo Glorioso, Sho Yaida, Daniel Roberts, and Darshil Doshi for detailed comments on the manuscript. This work is supported by an NSF CAREER grant (DMR-1753240) and the Laboratory for Physical Sciences through the Condensed Matter Theory Center.
2308.08253	Benchmarking Neural Network Generalization for Grammar Induction	How well do neural networks generalize? Even for grammar induction tasks, where the target generalization is fully known, previous works have left the question open, testing very limited ranges beyond the training set and using different success criteria. We provide a measure of neural network generalization based on fully specified formal languages. Given a model and a formal grammar, the method assigns a generalization score representing how well a model generalizes to unseen samples in inverse relation to the amount of data it was trained on. The benchmark includes languages such as $a^nb^n$, $a^nb^nc^n$, $a^nb^mc^{n+m}$, and Dyck-1 and 2. We evaluate selected architectures using the benchmark and find that networks trained with a Minimum Description Length objective (MDL) generalize better and using less data than networks trained using standard loss functions. The benchmark is available at https://github.com/taucompling/bliss.	Nur Lan, Emmanuel Chemla, Roni Katzir	2023-08-16T09:45:06Z	http://arxiv.org/abs/2308.08253v2	# Benchmarking Neural Network Generalization for Grammar Induction ###### Abstract How well do neural networks generalize? Even for grammar induction tasks, where the target generalization is fully known, previous works have left the question open, testing very limited ranges beyond the training set and using different success criteria. We provide a measure of neural network generalization based on fully specified formal languages. Given a model and a formal grammar, the method assigns a generalization score representing how well a model generalizes to unseen samples in inverse relation to the amount of data it was trained on. The benchmark includes languages such as $a^{n}b^{n}$, $a^{b}b^{n}c^{n}$, $a^{b}b^{m}c^{n+m}$, and Dyck-1 and 2. We evaluate selected architectures using the benchmark and find that networks trained with a Minimum Description Length objective (MDL) generalize better and using less data than networks trained using standard loss functions. The benchmark is available at [https://github.com/taucompling/bliss](https://github.com/taucompling/bliss). ## 1 Introduction The extent to which artificial neural networks (ANNs) generalize beyond their training data is an open research question. In this work we approach this question from the perspective of grammar induction, that is, the learning of a formal grammar from a finite (often small) sample from the (typically infinite) language of that grammar. In order to succeed in this task, a model must strike a balance between fitting the training data and generalizing to a potentially infinite set of unseen strings. Humans tested on such tasks show systematic generalization from small sets of examples (Fitch and Hauser, 2004, Malassis et al., 2020). While a range of ANN architectures have been shown to reach approximations for formal languages, the quality of this approximation remains an open matter, as we show below. Here we build on previous probes of ANN generalization for grammar induction and introduce a unified and general way to assess this capability, for a given pair of a learning model and a corpus drawn from a formal language. Our main contributions are: 1. A benchmark for formal language learning. The benchmark relies on a method for quantifying ANN generalization for formal languages, including probabilistic languages. The method assigns an index score representing a model's generalization performance in inverse relation to the size of the training data. We introduce the method and provide concrete datasets for well-studied formal languages. 2. An evaluation of selected architectures. We test recurrent neural networks (RNNs) of the Long-Short Term Memory type (LSTM; Hochreiter and Schmidhuber, 1997); Memory-augmented RNNs (MARNN; Suzgun et al., 2019; 2019) and an RNN variant which replaces the traditional gradient-based training regime with an objective that optimizes the model's Minimum Description Length (MDLRNN; Lan et al., 2022). We find that equipping ANNs with memory devices such as differentiable stacks helps generalization, but generalization remains partial and slow. At the same time, training with MDL leads in some of the test cases that we examined to potentially perfect generalization with significantly less data. In other cases, training with MDL did not result in successful generalization, possibly because the optimization procedure we used for the architecture search failed to find the global optimum under the MDL objective function. ## 2 Background Learning formal languages has long been used to probe various aspects of ANN performance. These most often include inquiries about: (i) ANNs' ability to generalize beyond their training data, and (ii) ANNs' expressive power; i.e., whether they can represent the relevant target grammars (often probed with reference to the Chomsky hierarchy of formal languages, as in Deletang et al., 2022). Here we will focus on the generalization question. We will show how it might be related to another under-exploited line of inquiry regarding the training objective of ANNs. A long line of theoretical work has probed the computational power of ANNs. Siegelmann and Sontag (1992) originally showed that RNNs with a sigmoid activation can emulate multiple-stack Turing machines under certain permissive conditions (infinite activation precision and unbounded running time). Since these conditions cannot be met in practice, another line of work probed the computational power of RNNs under practical conditions (finite precision and input-bound running time). Weiss et al. (2018) have shown that under these conditions LSTMs are able to hold weight configurations that perform unbounded counting, and so they should be able to recognize counter languages (CL), a family of formal languages that can be recognized using one or more counting devices (following some formal restrictions, Merrill, 2021). Recently, El-Naggar et al. (2023) and El-Naggar et al. (2023) have shown that two simpler RNN architectures, with linear- and ReLU-based cells, are also able to hold counting weight configurations, with similar consequences for recognizing CL. Empirically, another line of work provided promising results regarding the capability of ANNs to learn formal languages. This was most often done by training networks on strings up to a certain length and then showing good performance on longer ones (Boden and Wiles, 2000, Gers and Schmidhuber, 2001; see Table 1). Gers and Schmidhuber (2001) have shown that LSTMs trained on languages such as $a^{n}b^{n}$ and $a^{n}b^{n}c^{n}$ with $n$ values in the low dozens perform well on $n$'s in the high hundreds. Suzgun et al. (2019) found that LSTMs trained on Dyck-1 sequences (strings of well-balanced pairs of brackets) up to length 50 performed well on lengths up to 100. Suzgun et al. (2019) proposed RNN variants that are equipped with external differentiable memory devices and showed that they yield improved performance on non-regular languages. However, other empirical results show that in practice ANNs generalize only to very restricted ranges. Weiss et al. (2018) found that while LSTMs are theoretically able to hold counting solutions, these are not found through training: LSTMs trained on $a^{n}b^{n}$ and $a^{n}b^{n}c^{n}$ with max $n$ 100 and \begin{table} \begin{tabular}{l l l l l l l} \hline Language & Paper & Model & Metric & Training size & Max train $n$ & Max test $n$ \\ \hline \multirow{4}{}{$a^{n}b^{n}$} & GS’01 & LSTM & $M_{cat^{\prime}}$ & 16,000 & 30 & 1,000 \\ \cline{2-6} & JM’15 & Stack-RNN & $M_{det}$ & $20^{\dagger}$ & 19 & 60 \\ \cline{2-6} & WGY’18 & LSTM & $Bin$ & $100^{\dagger}$ & 100 & 256 \\ \cline{2-6} & LGCK’22 & MDLRNN & $M_{det}$ & 500 & 22 & $\infty$ \\ \hline \multirow{4}{}{$a^{n}b^{n}c^{n}$} & GS’01 & LSTM & $M_{cat^{\prime}}$ & 51,000 & 40 & 500 \\ \cline{2-6} & JM’15 & Stack-RNN & $M_{det}$ & $20^{\dagger}$ & 19 & 60 \\ \cline{2-6} & WGY’18 & LSTM & $Bin$ & $50^{\dagger}$ & 50 & 100 \\ \cline{2-6} & LGCK’22 & MDLRNN & $M_{det}$ & 500 & 22 & $\infty$ \\ \hline \multirow{4}{}{Dyck-1} & SGBS’19a & LSTM & $M_{cat^{\prime}}$ & 10,000 & 50 & 100 \\ \cline{2-6} & SGBS’19b & MARNN & $M_{cat^{\prime}}$ & 5,000 & 50 & 100 \\ \cline{2-6} & EMW’22 & ReLU-RNN & $M_{cat^{\prime}}$ & 10,000 & 50 & 1,000 \\ \cline{2-6} & LGCK’22 & MDLRNN & $M_{cat}$ & 500 & 16 & $\infty$ \\ \hline \end{tabular} \end{table} Table 1: ANN performance in selected probes of formal language learning. Metrics* (see Section 3.5): $M_{det}$ = deterministic accuracy; $M_{cat}$ = categorical accuracy; $M_{cat^{\prime}}$ = a non-probabilistic version of $M_{cat}$; $Bin$ = binary classification from hidden state to accept/reject labels, based on positive and negative samples. Training size: $\dagger$ = the paper did not explicitly specify the training set size, the value here is derived by assuming the training set was an exhaustive list of all strings up to ‘max train $n$’. ‘Max test $n$’: the largest $n$ for which the criterion was reached. For Dyck-1, $n$ represents overall sequence length. ‘$\infty$’ = the paper provides evidence that the network is correct for any $n$. When a paper reports several experiments as in GS’01, we take the best result based on the smallest training set. Papers: GS’01 = Gers and Schmidhuber (2001); JM’15 = Julin and Mikolov (2015); WGY’18 = Weiss et al. (2018); SGBS’19a = Suzgun et al. (2019); SGBS’19b = Suzgun et al. (2019); EMW’22 = El-Naggar et al. (2022); LGCK’22 = Lan et al. (2022). 50, respectively, start accepting illicit strings with $n$ values as low as 256 and 100. As mentioned above, Suzgun et al. (2019) tested LSTMs on Dyck-1 sequences but only up to length 100, and concluded that this language was learned by LSTMs. El-Naggar et al. (2022) extended this work to longer sequences, and found that LSTMs fail to generalize in practice, outputting incorrect predictions at lengths under 1,000. This, despite Dyck-1 being a CL and so theoretically learnable by LSTMs (Weiss et al., 2018). Apart from LSTMs, recent probes by El-Naggar et al. (2023) and El-Naggar et al. (2023) have shown that linear and ReLU RNNs, theoretically capable of counting, fail to find the counting weight configurations in practice when trained using backpropagation and standard loss functions; El-Naggar et al. (2023) went further with determining the source of this discrepancy, showing that the counting weight configuration is not an optimum of these loss functions. Moreover, even in works that report successful generalization to some degree beyond the training set, the fact that networks stop generalizing at an arbitrary point is often left unexplained (Gers and Schmidhuber, 2001; Suzgun et al., 2019, 2019, Deletang et al., 2022, a.o.).1 Footnote 1: Technical limitations such as finite activation precision can be ruled out as explanations for generalization failures, at least for counter languages and models where network states serve as memory: as shown in works mentioned above, ANNs often start outputting wrong predictions for $n$ values in the low hundreds. Even restricted representations such as 16-bit floats can hold much larger values, and modern implementations such as PyTorch use 32-bit floats by default. The literature on the generalization abilities of ANNs has made use of a range of measures of success, making results difficult to compare. Different probes of the same model often use different success criteria, and generate training and test sets using different sampling methods and of different orders of magnitude. Table 1 summarizes selected probes of ANN generalization and highlights the fragmented nature of this literature. In the following sections we propose a unified method to consolidate these efforts and better understand the generalization capabilities of ANNs. ## 3 The bliss index We present the Benchmark for Language Induction from Small Sets (bliss). We provide a formal description of the method, followed by a concrete application to specific tasks. The current release consists of three parts: (i) A specification for the generalization index $\mathbb{B}$, calculated for a given pair of formal language and ANN; (ii) A dataset containing a set of formal languages for benchmarking; (iii) An evaluation of different ANN architectures using this dataset. ### General setting: models and tasks For a given model $A$, e.g., an LSTM, a task is composed of the following components: * a grammar, e.g., a probabilistic context-free grammar (PCFG). * a sampling method from $\mathcal{L}(G)$, the language generated by $G$. * a training corpus, may contain repetitions. * a test corpus. * a task-specific accuracy metric with adjustable error margin $\varepsilon\in[0,1]$. It uses predictions $A(s)$ on strings $s\in\mathcal{T}$ to calculate an accuracy score $M(A,\mathcal{T},\varepsilon)\in[0,1]$. * a task-specific constant for setting the order of magnitude of dataset sizes. For example, $N=3$ sets the order of magnitude at $10^{3}$. Training and test sizes are then derived as described below. Selecting $N$ is done empirically based on properties of the task, e.g., languages with large vocabularies require larger amounts of training data, hence a larger $N$. ### From task to generalization index For a given task, the generalization index of order $N$ for a model $A$ is then defined as: \[\mathbb{B}_{N}^{\mathcal{L}}(A)=max\Bigg{\{}b\left\|\begin{array}{c}\| \mathcal{T}\|=10^{N}\times b,\\ \|\mathcal{C}\|=10^{N}/\,b,\\ M(A,\mathcal{T},\varepsilon)=1.0\end{array}\right.\Bigg{\}} \tag{1}\] Intuitively, the index compares a model's performance on a test size $\|\mathcal{T}\|$ to the inverse of its training data size $\|\mathcal{C}\|$. The index is expressed as the maximal $b$ factor which scales the training set and the corresponding test set in opposite directions: The accuracy condition at the bottom of (1) means that the model should be $\varepsilon$-close to perfect generalization on the test set. A model's generalization index $\mathbb{B}$ thus represents the performance that can be maximally'squeezed out' of an inversely small amount of data. Figure 1 exemplifies selected $\mathbb{B}$ values calculated based on (1). For illustration, for $a^{n}b^{n}$, using the order of magnitude $N=3$, a model that was trained on $\|\mathcal{C}\|=10^{3}/2=500$ samples and was 100% accurate on a test set of size $10^{3}\times 2=2,000$ will have an index score $\mathbb{B}_{3}^{a^{n}b^{n}}\geq 2$. A model for the same language that was trained on 250 samples only and generalized to a subsequent set of 4,000 samples will reach $\mathbb{B}_{3}^{a^{n}b^{n}}\geq 4$. For practical reasons, one cannot exhaust all values of $b$ to find $\mathbb{B}$. However, training and evaluating a model using a few $b$ values is enough to reveal its generalization dynamics, as shown in experiments in Section 5. The following sections describe the specific choices made for the different benchmark components in these experiments. ### Learning setup Previous work surveyed here differed in their learning setup. Gers and Schmidhuber (2001) and Suzgun et al. (2019, 2019) trained networks in a non-probabilistic, supervised setup by exposing the model to all possible next symbols and minimizing the mean-squared error (MSE) - i.e., the model is given explicit information about the distribution of possible symbols. Joulin and Mikolov (2015) and Lan et al. (2022) used a setup that we adopt below, in which model outputs are probabilistic, and training is self-supervised language modeling (i.e., the model is exposed to the next symbol only) with a cross-entropy loss. Weiss et al. (2018) trained a binary classifier with accept/reject labels based on positive and negative examples. Since our focus is grammar induction, here we adopt the more demanding setup of learning from positive examples alone. All tasks are thus designed as self-supervised language modeling. At each time step, a model assigns a probability distribution to the next symbols in the string. The benchmark is agnostic as to the internals of the model and its training, as long as its outputs represent a probability distribution over symbols. In practice, then, the method can be applied to any language model, not necessarily an ANN. ### Sampling To construct the training and test sets $\mathcal{C}$ and $\mathcal{T}$ we use the following as method $S$: * To construct $\mathcal{C}$, we sample strings according to the distribution defined by $G$, with repetitions. For example, if $G$ is a PCFG, it can be sampled by applying derivation rules chosen proportionally to their expansion probabilities. Repetitions are allowed so that $\mathcal{C}$ follows a similar surface distribution to $\mathcal{L}(G)$ and so that the model can pick up on the underlying probabilities in $G$. * To construct $\mathcal{T}$, we take the $\|\mathcal{T}\|$ subsequent strings starting right after the longest string in $\mathcal{C}$, sorted by length.2 For example, for the language $a^{n}b^{n}$, if the longest string in the training set $\mathcal{C}$ was $a^{17}b^{17}$, and the model needs to be tested on a set of 2000 strings, $\mathcal{T}$ will be composed of the strings $a^{18}b^{18},...,a^{2017}b^{2017}$. Footnote 2: Test strings may need to be sorted further according to specific properties of a language, see Section 4.1. The sampling method $S$ can be either probabilistic as described here, or exhaustive, training on all strings in $\mathcal{L}$ up to a certain length. We opt for probabilistic sampling because of the nature of the task at hand: the models under discussion here are trained to assign probabilities to the next symbol in a string, most often minimizing a cross-entropy loss. In practice, then, they always learn distributions over strings. Thus if $\mathcal{C}$ follows a similar surface distribution to $\mathcal{L}$ (given a large enough sample size), the model should eventually learn this distribution in order to minimize its loss. Probabilistic sampling thus makes it possible to probe both a model's knowledge about the surface forms of $\mathcal{L}$ (by treating model outputs as categorical classes), and about their distribution. The modularity of the index makes it possible to choose Figure 1: Example generalization index scores $\mathbb{B}_{3}$, i.e., for a baseline training size of $10^{3}$. Each dashed line represents the performance profile of some hypothetical model, as a function of the size of the training set. The intersection with the $\mathbb{B}_{3}$-frontier indicates its $\mathbb{B}_{3}$ index. either option by varying the accuracy metric $M$, as we show in the next section. ### Accuracy metrics Ultimately we are interested in knowing whether a model accepts all strings in $\mathcal{L}$ and rejects all others. In classical formal language theory, where discrete automata are used, acceptance is clear cut and taken as going into an accepting state. ANNs on the other hand use continuous representations with no standard acceptance criterion. Different acceptance criteria have been used in previous works to measure success for ANNs: Gers and Schmidhuber (2001) and Suzgun et al. (2019) defined acceptance of a string as a model assigning output values above a certain threshold to valid symbols only; Joulin and Mikolov (2015) measure accuracy at parts of strings that are completely predictable; and Weiss et al. (2018) turn a network into a recognizer by training a binary classifier from network states to accept/reject labels. Below we provide general versions of these accuracy metrics (omitting Weiss et al., 2018 who rely on negative examples). Choosing which metric to use is based on the properties of the language at hand. Well-performing models might still deviate slightly from perfect accuracy due to practical limitations, such as a softmax function preventing a model from expressing categorical decisions. Thus for each accuracy metric we add an adjustable error margin $\varepsilon$. Acceptance of a string is defined as reaching 100% accuracy (minus $\varepsilon$) on the string. Success on the test set is then defined as accepting all strings in the set (third condition in (1)). 1. _Deterministic accuracy_ ($M_{det}$). Some languages contain strings with deterministic phases, where the next symbol is fully predictable. For example, strings in the language $a^{n}b^{n}$ have two phases, the $a$ phase and the $b$ phase. As long as only $a$'s are seen, the next symbol remains unpredictable as the sequence can continue with another $a$ or switch to the $b$ phase. The string becomes deterministic once the first $b$ appears. $M_{det}$ is defined as the fraction of deterministic time steps in which the model assigns the majority probability to the correct next symbol. This metric is used in Joulin and Mikolov (2015). A string is considered accepted if the model is $1-\varepsilon$ accurate over all deterministic time steps. Note however that even a very small $\varepsilon$ might benefit models that do not recognize strings well. For example, for the language $a^{n}b^{n}$, the deterministic steps in a string are the $b$'s and the final end-of-sequence symbol. A degenerate model that predicts only $b$'s will get only the end-of-sequence symbol wrong out of all deterministic steps, and will reach a very high accuracy score. For any large enough test set these errors will be hidden within the $\varepsilon$ margin and the model will be deemed successful. $\varepsilon$ should therefore be chosen with care per task. $M_{det}$ is used below for the following languages that have deterministic phases: $a^{n}b^{n}$, $a^{n}b^{n}c^{n}$, $a^{n}b^{n}c^{n}d^{n}$, and $a^{n}b^{m}c^{n+m}$. 2. _Categorical accuracy_ ($M_{cat}$). Some language strings do not have any predictable phases. This is the case in the Dyck family of languages. At each time step in a Dyck string, one may open a new bracket (see Figure 2). $M_{cat}$ is therefore defined as the fraction of steps in which a network assigns probability $p>\varepsilon$ to each possible next symbol, and $p\leq\varepsilon$ to irrelevant symbols. Non-probabilistic versions of $M_{cat}$ are used in Gers and Schmidhuber (2001) and Suzgun et al. (2019, 2019) who do not treat network outputs as probability distributions. $M_{cat}$ is used below for Dyck languages. Figure 2: Inputs and valid next symbols at each step of a Dyck-1 string (top) and $a^{n}b^{n}$ (bottom), including the start/end-of-sequence symbol ‘$\#$’. For $a^{n}b^{n}$, accuracy is measured at deterministic steps, after the first ‘b’. For Dyck-1, accuracy is the fraction of time steps where a model predicts only valid next symbols: ‘$\#$’ should be predicted only when brackets are well balanced. As specified in Section 3.1, the index $\mathbb{B}$ is calculated based on the largest test set for which a model reaches an $\varepsilon$-perfect accuracy score. Beyond accuracy, one might be interested in inspecting a model's knowledge of the distribution of strings in $\mathcal{L}$ induced by a probabilistic $G$. This can be done by using the probabilistic sampling method described in Section 3.4 and accompanying it with a probabilistic accuracy measure - for example, one based on an optimal cross-entropy score, which is known from $G$'s expansion probabilities (as done in Lan et al., 2022). Feeding loss values into an accuracy metric will require normalizing them across tasks. We leave this extension for future work. ### String structure Following Gers and Schmidhuber (2001), each sequence starts and ends with a start/end-of-sequence symbol '$\#$'. This turns the task into a strict acceptance/rejection task - predicting the end-of-sequence symbol is taken as going into an accept state. The start- and end-of-sequence symbols are added to the task-specific vocabulary and are assigned probabilities by the model at each step. Figure 2 illustrates input and target sequences for $a^{n}b^{n}$ and Dyck-1. ### Limitations One shortcoming of the proposed index score is that it does not reflect perfect generalization, i.e., it is an empirical index that cannot point out a model that outputs correct predictions for _any_ string in $\mathcal{L}(G)$. For most models, this will not be a problem, and $\mathbb{B}$ will simply represent the model's best training vs. test size ratio. In the case of a model that reaches perfect generalization on any input, the index score will represent the critical training size that brings the model to this performance. Assigning a generalization score to infinitely correct models will remain a problem for any empirical metric that assigns scores to models based on finite test values. An alternative to such empirical probes would be to analytically show that a model is correct (as done in Lan et al., 2022). ## 4 Datasets We provide training and test datasets for a preliminary set of formal languages for evaluation using the $\mathbb{B}$ index. The dataset includes the languages $a^{n}b^{n}$, $a^{n}b^{n}c^{n}$, $a^{n}b^{n}c^{n}d^{n}$, $a^{n}b^{m}c^{n+m}$, Dyck-1, and Dyck-2. The source code, datasets, and specifications for the benchmark are available at [https://github.com/taucompling/bliss](https://github.com/taucompling/bliss). ### Training and test sets Training sets for context-free languages are sampled from PCFGs as described in Section 3.4. The PCFGs are given in Appendix B. Training sets for context-sensitive languages are generated by sampling values for $n$ from a geometric distribution. Test sets are generated using the method described in Section 3.4: All test sets consist of an exhaustive list of strings ordered by length starting right after the longest string seen during training. Test sets for $a^{n}b^{m}c^{n+m}$ consist of the list of strings starting after the last seen pair of $n,m$, sorted by $n+m$ values to test all possible combinations. ## 5 Experiments ### Models We test the following models: LSTM RNNs Hochreiter and Schmidhuber (1997); Memory-augmented RNNs MARNN Suzgun et al. (2019); and Minimum Description Length RNNs MDLRNN Lan et al. (2022). LSTM architectures were developed with the task of keeping items in memory over long distances in mind. As mentioned above, Weiss et al. (2018) have shown that LSTMs are theoretically capable of recognizing CL. MARNNs Suzgun et al. (2019) are RNNs equipped with external memory devices, and were shown to yield better performance when learning languages that require stack-like devices and beyond. Here we use Stack-LSTM, an LSTM augmented with a pushdown automaton; and Baby Neural Turing Machines (Baby-NTM; itself a variant of NTMs, Graves et al., 2014), an RNN with a more freely manipulable memory.3 Footnote 3: We modify Suzgun et al. (2019)’s models to output probability distributions, replacing the final sigmoids with a softmax layer and the MSE loss with cross-entropy. See Section 3.3. MDLRNNs are RNNs trained to optimize the Minimum Description Length objective (MDL; Rissanen, 1978), a computable approximation of Kolmogorov complexity, the algorithmic complexity of a model. The intuition behind the objective is equating compression with finding regularities in the data: a model that compresses the data well will generalize better and avoid overfitting. In practice, optimization is done by minimizing the sum of the architecture encoding length and the standard cross-entropy loss, both measured in bits based on a specific encoding scheme. MDL is a stricter regularizer than standard regularization techniques such as L1/L2: the latter penalize large weight values but cannot prevent models from overfitting using a solution that uses many small, but informative, weights. MDL penalizes the actual information content of the network, forcing it to be general and avoid overfitting. MDLRNNs were shown to learn some of the languages discussed here in full generality using small architectures of only 1 or 2 hidden units and to outperform L1/L2 (Lan et al., 2022). MDL is a non-differentiable objective, which requires that MDLRNN be optimized using a non-gradient based search method, such as an evolutionary algorithm that searches the network architecture space. Since this method is not based on gradient descent, Lan et al. (2022) were able to use non-standard, non-differentiable activations such as step functions. Here we restrict the architecture space to only standard activations: the linear function, ReLU, and tanh. This serves both to compare MDLRNN with standard networks and to limit the architecture search space. We publish the resulting nets as part of the MDLRNN-Torch release at [https://github.com/0xnurl/mdlrnn-torch](https://github.com/0xnurl/mdlrnn-torch). Appendix A lists the hyper-params for all runs. ### Training sets We used training sizes $\|\mathcal{C}\|=100,250,500,1000$. We stopped at the smallest size 100 because in our setup this size results in test strings of lengths $>10{,}000$, leading to very long running times. ### Index parameters We calculate the $\mathbb{B}$ index for all trained networks using the following index parameters: Magnitude parameter $N=3$, i.e., training and test sizes are derived from a baseline size $10^{3}$. This order of magnitude was selected based on the training set sizes used in previous works for the languages inspected here (Table 1). $M_{det}\ \varepsilon=0.005$, i.e., a model needs to correctly predict the next symbol for at least 99.5% of all deterministic steps. Since even this high threshold allows a degenerate model to reach good scores as described in Section 3.5, we also calculate the index score using $\varepsilon=0$, i.e. a model must predict _all_ deterministic symbols correctly. $M_{cut}\ \varepsilon=0.005$, i.e., for Dyck, a model needs to assign $p\leq 0.005$ to each irrelevant symbol and $p>0.005$ to possible ones. Here as well we report results for $\varepsilon=0$, i.e., a model must assign non-zero probabilities to valid symbols only. ## 6 Results ### Non-perfect accuracy The generalization index obtained by each model for each language is presented in Table 2. We start by inspecting the indexes calculated using the more lenient accuracy margin $\varepsilon=0.005$. For $a^{n}b^{n}$, under this accuracy margin, all models are assigned index $\mathbb{B}=10$, i.e., reaching the success criterion for the next unseen 10,000 samples \begin{table} \begin{tabular}{l l c c} \hline \hline \multirow{2}{}{Language} & \multirow{2}{}{Model} & \multicolumn{2}{c}{$\mathbb{B}$-index} \\ & & $\varepsilon=0.005$ & $\varepsilon=0$ \\ \hline \multirow{3}{}{$a^{n}b^{n}$} & LSTM & 10* & \textless{1} \\ \cline{2-4} & Stack-LSTM & 10 & \textless{1} \\ \cline{2-4} & Baby-NTM & 10 & 1 \\ \cline{2-4} & MDLRNN & 10 & 10 \\ \hline \multirow{3}{}{$a^{n}b^{n}c^{n}$} & LSTM & \textless{1} & \textless{1} \\ \cline{2-4} & Stack-LSTM & 2 & \textless{1} \\ \cline{2-4} & Baby-NTM & 10* & \textless{1} \\ \cline{2-4} & MDLRNN & \textless{1} & \textless{1} \\ \hline \multirow{3}{}{$a^{n}b^{n}c^{n}d^{n}$} & LSTM & \textless{1} & \textless{1} \\ \cline{2-4} & Stack-LSTM & 1 & \textless{1} \\ \cline{2-4} & Baby-NTM & 4* & \textless{1} \\ \cline{2-4} & MDLRNN & \textless{1} & \textless{1} \\ \hline \multirow{3}{}{$a^{n}b^{m}c^{n+m}$} & LSTM & \textless{1} & \textless{1} \\ \cline{2-4} & Stack-LSTM & 10* & \textless{1} \\ \cline{2-4} & Baby-NTM & 4 & \textless{1} \\ \cline{2-4} & MDLRNN & 4 & \textless{1} \\ \hline \hline \end{tabular} \end{table} Table 2: Generalization scores $\mathbb{B}$. The index represents how well a model generalizes in relation to its training size. A score $\mathbb{B}=4$ indicates that a model trained on 250 samples reached the accuracy criterion on the consecutive 4,000 unseen test samples. $\mathbb{B}<1$ indicates that the model did not reach the accuracy criterion when the test size was greater than the training size, but might reach it for larger training and smaller test sets. after being trained on 100 samples. For the specific combination of random seed and sampling prior in these experiments, this means that the models were trained on strings up to $a^{20}b^{20}$ and generalized to all strings up to at least $a^{10020}b^{10020}$ with deterministic accuracy $M_{det}\geq 99.5\%$. For $a^{n}b^{n}c^{n}$, MARNNs reach $\mathbb{B}=$ 10 and 2, while LSTM and MDLRNNs do not reach the success criterion, resulting in $\mathbb{B}<1$. For $a^{n}b^{n}c^{n}d^{n}$ only MARNNs reach a specified index, with a Baby-NTM reaching $\mathbb{B}=4$, indicating that it generalized to strings as long as $a^{4020}b^{4020}c^{4020}d^{4020}$ with $M_{det}\geq 99.5\%$. For the addition language $a^{n}b^{m}c^{n+m}$, Stack-LSTM and MDLRNN reached index scores $\mathbb{B}=$ 10 and 4 respectively. For the specific combination of random seed and the sampling prior used here, this means that the winning Stack-LSTM saw maximum values of $n=18,m=20$ during training, and generalized to all strings up to $a^{120}b^{120}c^{240}$ with $M_{det}\geq 99.5\%$. ### Perfect accuracy We report the generalization scores using a strict $\varepsilon=0$ as well, i.e., when a model is required to predict _all_ deterministic steps correctly or assign non-zero probability to valid symbols only. For languages with deterministic steps such as $a^{n}b^{n}$, this means that the model needs to always predict the end-of-sequence symbol correctly, thus making a distinction between accepting a string and approximating its surface structure. Here, only MDLRNNs remain at the same scores, indicating that they predicted all time steps correctly. Baby-NTM reaches $\mathbb{B}=1$ for $a^{n}b^{n}$, a drop from 10. The rest of the networks drop to $\mathbb{B}<1$, revealing that their good scores in the previous comparison calculated with a non-zero $\varepsilon$ was due to them approximating the target languages, even at low $n$ values. MDLRNN performance here is in line with results from Lan et al. (2022), who provided evidence that MDLRNNs for these languages do not only perform empirically well on large test values, but are also provably correct for any input. However, here we limited activations to standard, non-discrete functions (Section 5.1), potentially limiting the network's ability to generalize well in the limit. While we do not provide correctness proofs for the networks found here, the index scores indicate that MDLRNNs generalize well to large values using only standard activations. Figure 4 presents the MDLRNN found for $a^{n}b^{n}$. We checked whether this network also accepts $n$ values beyond those needed to reach the score $\mathbb{B}=$ 10 ($n=10{,}020$). The network reached 100% $M_{det}$ for all values up to $n=35{,}000$, at which point we stopped the test due to long feeding times. Beyond the benchmark scores, Figure 3 plots the largest $n$ value for $a^{n}b^{n}$ strings predicted by the models tested here with 100% $M_{det}$ accuracy ($\varepsilon=0$), as a function of training set size. Both MDLRNNs and Baby-NTMs reach perfect accuracy up to the tested maximum of $n=1{,}000$. MDLRNNs however require two orders of magnitude less data to reach this performance (and the benchmark scores in Table 2 show that in fact MDLRNNs generalized up to at least $n=10{,}000$, while Baby-NTMs remained at $1{,}000$). LSTMs Figure 4: RNN cell architecture of the best-performing MDLRNN for $a^{n}b^{n}$, which trained on 100 samples and reached $\mathbb{B}_{3}=10$. The network uses only one hidden unit and standard activation functions, and generalizes up to at least $a^{35000}b^{35000}$. Dashed arrows are recurrent connections across time steps. The loop from the hidden ReLU unit to itself is a counter mechanism evolved by the evolutionary algorithm to count and compare the number of $a$’s and $b$’s. Figure 3: Generalization performance of the models tested here. Models were trained on strings drawn from $a^{n}b^{n}$ and tested on acceptance of strings up to $n=1{,}000$. $X$’s mark the maximum $n$ seen during training. and Stack-RNNs did not generalize well beyond the training samples. This is in line with previous works showing that these models may need substantially more training data in order to learn these languages (Table 1). ## 7 Discussion We provided a simple index for how well a model generalizes: how much it can learn from how little data. We illustrated the usefulness of this index in a comparison of several current models over several formal languages. Beyond showing which current models generalize better than others, the benchmark also highlights which aspects of artificial neural networks work well for grammar induction, and what is still missing. Among languages that were learned with perfect accuracy ($a^{n}b^{n}$, $a^{n}b^{m}c^{n+m}$, Dyck-1), MDLRNNs generalized best, but still failed on others ($a^{n}b^{n}c^{n}$, $a^{n}b^{n}c^{n}d^{n}$, and Dyck-2). Previous work has shown that this model's search procedure, an evolutionary algorithm, fails to find networks that are manually shown to have better MDL scores (Lan et al., 2022). We take this to show that the optimization procedure limits the model and prevents it from taking full advantage of the MDL objective. The benefit of the MDL objective is nevertheless evident in the generalization performance for several languages. MARNNs clearly benefit from their memory devices and reach good generalization scores, but testing for perfect accuracy ($\varepsilon=0$) reveals that their learning outcome is mostly approximate, and that they fail to maintain perfect accuracy for long stretches beyond their training data. This could be the result of an inadequate objective function (cross-entropy), limitations of the search (backpropagation/gradient descent), or both. We do not currently have results that help decide this matter, but recent results for other architectures (El-Naggar et al., 2023) hint that the problem lies at least in part in the objective function. ## 8 Acknowledgements This work was granted access to the HPC resources of IDRIS under the allocation 2023-AD011013783 made by GENCI.
2301.09308	On the Expressive Power of Geometric Graph Neural Networks	The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at \url{https://github.com/chaitjo/geometric-gnn-dojo}	Chaitanya K. Joshi, Cristian Bodnar, Simon V. Mathis, Taco Cohen, Pietro Liò	2023-01-23T08:08:10Z	http://arxiv.org/abs/2301.09308v3	# On the Expressive Power of Geometric Graph Neural Networks ###### Abstract The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for _geometric graphs_ embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are _invariant_ or _equivariant_ to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at [https://github.com/chaitio/geometric-gnn-dojo](https://github.com/chaitio/geometric-gnn-dojo) Figure 1: Axes of geometric GNN expressivity: (1)_Scalarisation body order_: increasing body order of scalarisation builds expressive local neighbourhood descriptors; (2) _Tensor order_: higher order spherical tensors determine the relative orientation of neighbourhoods; and (3) _Depth_: deep equivariant layers propagate geometric information beyond local neighbourhoods. ## 1 Introduction Systems in biochemistry [1], material science [2], physical simulations [3], and multiagent robotics [4] contain both geometry and relational structure. Such systems can be modelled via _geometric graphs_ embedded in Euclidean space. For example, molecules are represented as a set of nodes which contain information about each atom and its 3D spatial coordinates as well as other geometric quantities such as velocity or acceleration. Notably, the geometric attributes transform along with Euclidean transformations of the system, _i.e._ they are equivariant to symmetry groups of rotations, reflections, and translation. Standard Graph Neural Networks (GNNs) which do not take spatial symmetries into account are ill-suited for geometric graphs, as the geometric attributes would no longer retain their physical meaning and transformation behaviour [5; 6]. GNNs specialised for geometric graphs follow the message passing paradigm [7] where node features are updated in a permutation equivariant manner by aggregating features from local neighbourhoods. Crucially, in addition to permutations, the updated geometric features of the nodes retain the transformation behaviour of the initial attributes, _i.e._ they are also equivariant to the Lie group of rotations ($SO(d)$) or rotations and reflections ($O(d)$). We use $\mathfrak{G}$ as a generic symbol for these Lie groups. We consider two classes of geometric GNNs: (1) $\mathfrak{G}$-equivariant models, where the intermediate features and propagated messages are equivariant geometric quantities such as vectors [8; 9; 10] or tensors [11; 12; 13]; and (2) $\mathfrak{G}$-invariant models, which only propagate local invariant scalar features such as distances and angles [14; 15; 16]. Both classes of architectures have shown promising empirical results for applications ranging from protein design [17; 18], molecular dynamics [19; 20], and electrocatalysis [21; 22]. At the same time, key theoretical questions remain unanswered: (1) How to characterise the _expressive power_ of geometric GNNs? and (2) What is the tradeoff between $\mathfrak{G}$-equivariant and $\mathfrak{G}$-invariant GNNs? The graph isomorphism problem [23] and the Weisfeiler-Leman (WL) [24] test for distinguishing non-isomorphic graphs have become a powerful tool for analysing the expressive power of non-geometric GNNs [25; 26]. The WL framework has been a major driver of progress in graph representation learning [27; 28; 29; 30; 31]. However, WL does not directly apply to geometric graphs as they exhibit a stronger notion of geometric isomorphism that accounts for spatial symmetries. Contributions. In this work, we study the expressive power of geometric GNNs from the perspective of discriminating non-isomorphic geometric graphs. * In Section 3, we propose a geometric version of the Weisfeiler-Leman graph isomorphism test, termed GWL, which is a theoretical upper bound on the expressive power of geometric GNNs. * In Section 4, we use the GWL framework to formalise how key design choices influence geometric GNN expressivity, summarised in Figure 1. Notably, invariant GNNs cannot distinguish graphs where one-hop neighbourhoods are the same and fail to compute non-local geometric properties such as volume, centroid, etc. Equivariant GNNs distinguish a larger class of graphs as stacking equivariant layers propagates geometric information beyond local neighbourhoods. * Synthetic experiments in Section 5 demonstrate practical implications for building maximally powerful geometric GNNs, _s.a._ oversquashing with increased depth, and counterexamples that highlight the utility of higher order order spherical tensors. * Finally, in Section 6, we prove an equivalence between a model's ability to discriminate geometric graphs and its universal approximation capabilities. While universality is binary, GWL's discrimination-based perspective provides a more granular and practically insightful lens into geometric GNN expressivity. ## 2 Background Graph Isomorphism. An attributed graph $\mathcal{G}=(\mathbf{A},\mathbf{S})$ is a set $\mathcal{V}$ of $n$ nodes connected by edges. $\mathbf{A}$ denotes an $n\times n$ adjacency matrix where each entry $a_{ij}\in\{0,1\}$ indicates the presence or absence of an edge connecting nodes $i$ and $j$. The matrix of _scalar_ features $\mathbf{S}\in\mathbb{R}^{n\times f}$ stores attributes $\mathbf{s}_{i}\in\mathbb{R}^{f}$ associated with each node $i$, where the subscripts index rows and columns in the corresponding matrices. The graph isomorphism problem asks whether two graphs are the same, but drawn differently [23]. Two attributed graphs $\mathcal{G}$, $\mathcal{H}$ are _isomorphic_ (denoted $\mathcal{G}\simeq\mathcal{H}$) if there exists an edge-preserving bijection $b:\mathcal{V}(\mathcal{G})\to\mathcal{V}(\mathcal{H})$ such that $\mathbf{s}_{i}^{(\mathcal{G})}=\mathbf{s}_{b(i)}^{(\mathcal{H})}$. Weisfeiler-Leman Test (WL).WL tests whether two (attributed) graphs are isomorphic [24]. At iteration zero, WL assigns a _colour_$c_{i}^{(0)}\in C$ from a countable space of colours $C$ to each node $i$. Nodes are coloured the same if their features are the same, otherwise, they are coloured differently. In subsequent iterations $t$, WL iteratively updates the node colouring by producing a new $\mathbf{c}_{i}^{(t)}\in C$: \[\mathbf{c}_{i}^{(t)}:=\textsc{Hash}\left(\mathbf{c}_{i}^{(t-1)},\{\!\!\{\mathbf{c}_{j}^{(t -1)}\mid j\in\mathcal{N}_{i}\}\!\!\}\right), \tag{1}\] where Hash is an injective map (_i.e._ a perfect hash map) that assigns a unique colour to each input. The test terminates when the partition of the nodes induced by the colours becomes stable. Given two graphs $\mathcal{G}$ and $\mathcal{H}$, if there exists some iteration $t$ for which $\{\!\!\{c_{i}^{(t)}\mid i\in\mathcal{V}(\mathcal{G})\}\!\!\}\neq\{\!\!\{c_{i}^ {(t)}\mid i\in\mathcal{V}(\mathcal{H})\}\!\!\}$, then the graphs are not isomorphic. Otherwise, the WL test is inconclusive, and we say it cannot distinguish the two graphs. Group Theory.We assume basic familiarity with group theory, see Zee [32] for an overview. We denote the action of a group $\mathfrak{G}$ on a space $X$ by $\mathfrak{g}\cdot x$. If $\mathfrak{G}$ acts on spaces $X$ and $Y$, we say: (1) A function $f:X\to Y$ is _$\mathfrak{G}$-invariant_ if $f(\mathfrak{g}\cdot x)=f(x)$, _i.e._ the output remains unchanged under transformations of the input; and (2) A function $f:X\to Y$ is _$\mathfrak{G}$-equivariant_ if $f(\mathfrak{g}\cdot x)=\mathfrak{g}\cdot f(x)$, _i.e._ a transformation of the input must result in the output transforming equivalently. We also consider _$\mathfrak{G}$-orbit injective_ functions: The _$\mathfrak{G}$-orbit_ of $x\in X$ is $\mathcal{O}_{\mathfrak{G}}(x)=\{\mathfrak{g}\cdot x\mid\mathfrak{g}\in \mathfrak{G}\}\subseteq X$. When $x$ and $x^{\prime}$ are part of the same orbit, we write $x\simeq x^{\prime}$. We say a function $f:X\to Y$ is _$\mathfrak{G}$-orbit injective_ if we have $f(x_{1})=f(x_{2})$ if and only if $x_{1}\simeq x_{2}$ for any $x_{1},x_{2}\in X$. Necessarily, such a function is $\mathfrak{G}$-invariant, since $f(\mathfrak{g}\cdot x)=f(x)$. We work with the permutation group over $n$ elements $S_{n}$ and the Lie groups $\mathfrak{G}=SO(d)$ or $\mathfrak{G}=O(d)$. Invariance to the translation group $T(d)$ is conventionally handled using relative positions or by subtracting the centre of mass from all nodes positions. Given one of the standard groups above, for an element $\mathfrak{g}$ we denote by $\mathbf{M}_{\mathfrak{g}}$ (or another capital letter) its standard matrix representation. Geometric Graphs.A geometric graph $\mathcal{G}=(\mathbf{A},\mathbf{S},\mathbf{\tilde{V}},\mathbf{\tilde{X}})$ is an attributed graph that is also decorated with geometric attributes: node coordinates $\mathbf{\tilde{X}}\in\mathbb{R}^{n\times d}$ and (optionally) vector features $\mathbf{\tilde{V}}\in\mathbb{R}^{n\times d}$ (_e.g._ velocity, acceleration). Without loss of generality, we work with a single vector feature per node. Our setup generalises to multiple vector features or higher-order tensors. The geometric attributes transform as follows under the action of the relevant groups: (1) $S_{n}$ acts on the graph via $\mathbf{P}_{\sigma}\mathcal{G}:=(\mathbf{P}_{\sigma}\mathbf{AP}_{\sigma}^{\top},\mathbf{P}_{ \sigma}\mathbf{S},\mathbf{P}_{\sigma}\mathbf{\tilde{V}},\mathbf{P}_{\sigma}\mathbf{\tilde{X}})$; (2) Orthogonal transformations $\mathbf{Q}_{\mathfrak{g}}\in\mathfrak{G}$ act on $\mathbf{\tilde{V}},\mathbf{\tilde{X}}$ via $\mathbf{\tilde{V}}\mathbf{Q}_{\mathfrak{g}},\mathbf{\tilde{X}}\mathbf{Q}_{\mathfrak{g}}$; and (3) Translations $\mathbf{\tilde{t}}\in T(d)$ act on the coordinates $\mathbf{\tilde{X}}$ via $\mathbf{\tilde{x}}_{i}+\mathbf{\tilde{t}}$ for all nodes $i$. Two geometric graphs $\mathcal{G}$ and $\mathcal{H}$ are _geometrically isomorphic_ if there exists an attributed graph isomorphism $b$ such that the geometric attributes are equivalent, up to global group actions $\mathbf{Q}_{\mathfrak{g}}\in\mathfrak{G}$ and $\mathbf{\tilde{t}}\in T(d)$: \[\left(\mathbf{s}_{i}^{(\mathcal{G})},\mathbf{\tilde{v}}_{i}^{(\mathcal{G})},\mathbf{\tilde {x}}_{i}^{(\mathcal{G})}\right)\ =\ \left(\mathbf{s}_{b(i)}^{(\mathcal{H})},\ \mathbf{Q}_{\mathfrak{g}}\mathbf{\tilde{v}}_{b(i)}^{( \mathcal{H})},\ \mathbf{Q}_{\mathfrak{g}}(\mathbf{\tilde{x}}_{b(i)}^{(\mathcal{H})}+\mathbf{\tilde{t}}) \right)\quad\text{for all }i\in\mathcal{V}(\mathcal{G}). \tag{2}\] Geometric graph isomorphism and distinguishing (sub-)graph geometries has important practical implications for representation learning. For _e.g._, for molecular systems, an ideal architecture should map distinct local structural environments around atoms to distinct embeddings in representation space [33; 34], which can then be used for downstream predictive task. Geometric Graph Neural Networks.We consider two broad classes of geometric GNN architectures. (1) _$\mathfrak{G}$-equivariant GNN layers_[10; 11] propagate scalar as well as geometric vector features from iteration $t$ to $t+1$ via learnable aggregate and update functions, Agg and Upd, respectively: \[\mathbf{m}_{i}^{(t)},\mathbf{\tilde{m}}_{i}^{(t)} :=\textsc{Agg}\left(\{\!\{(\mathbf{s}_{i}^{(t)},\mathbf{s}_{j}^{(t)},\mathbf{ \tilde{v}}_{i}^{(t)},\mathbf{\tilde{v}}_{j}^{(t)},\mathbf{\tilde{x}}_{ij})\mid j\in \mathcal{N}_{i}\}\!\!\}\right)\] (Aggregate) (3) \[\mathbf{s}_{i}^{(t+1)},\mathbf{\tilde{v}}_{i}^{(t+1)} :=\textsc{Upd}\left((\mathbf{s}_{i}^{(t)},\mathbf{\tilde{v}}_{i}^{(t)}) \,\ (\mathbf{m}_{i}^{(t)},\mathbf{\tilde{m}}_{i}^{(t)})\right)\] (Update) (4) where $\mathbf{\tilde{x}}_{ij}=\mathbf{\tilde{x}}_{i}-\mathbf{\tilde{x}}_{j}$ denote relative position vectors. Alternatively, (2) _$\mathfrak{G}$-invariant GNN layers_[14; 16] do not update vector features and only propagate invariant scalar quantities computed using geometric information within local neighbourhoods: \[\mathbf{s}_{i}^{(t+1)}:=\textsc{Upd}\left(\mathbf{s}_{i}^{(t)}\,\ \textsc{Agg}\left(\{\!\{(\mathbf{s}_{i}^{(t)},\mathbf{s}_{j}^{(t)},\mathbf{ \tilde{v}}_{i},\ \mathbf{\tilde{v}}_{j},\ \mathbf{\tilde{x}}_{ij})\mid j\in\mathcal{N}_{i}\}\!\!\}\right)\right). \tag{5}\] For both models, the scalar features at the final iteration are mapped to graph-level predictions via a permutation-invariant readout $f:\mathbb{R}^{n\times f}\to\mathbb{R}^{f^{\prime}}$. In Appendix B, we provide unified equations and examples of geometric GNNs covered by our framework. ## 3 The Geometric Weisfeiler-Leman Test Assumptions. Analogous to the WL test, we assume for the rest of this section that all geometric graphs we want to distinguish are finite in size and come from a countable dataset. In other words, the geometric and scalar features the nodes are equipped with come from countable subsets $C\subset\mathbb{R}^{d}$ and $C^{\prime}\subset\mathbb{R}$, respectively. As a result, when we require functions to be injective, we require them to be injective over the countable set of $\mathfrak{G}$-orbits that are obtained by acting with $\mathfrak{G}$ on the dataset. Intuition. For an intuition of how to generalise WL to geometric graphs, we note that WL uses a local, node-centric, procedure to update the colour of each node $i$ using the colours of its the 1-hop _neighbourhood_$\mathcal{N}_{i}$. In the geometric setting, $\mathcal{N}_{i}$ is an attributed point cloud around the central node $i$. As a result, each neighbourhood carries two types of information: (1) neighbourhood type (invariant to $\mathfrak{G}$) and (2) neighbourhood geometric orientation (equivariant to $\mathfrak{G}$). This leads to constraints on the generalisation of the neighbourhood aggregation procedure of Equation 1. From an axiomatic point of view, our generalisation of the WL aggregation procedure must meet two properties: Property 1: Orbit injectivity of colours. If two neighbourhoods are the same up to an action of $\mathfrak{G}$ (_e.g._ rotation), then the colours of the corresponding central nodes should be the same. Thus, the colouring must be $\mathfrak{G}$-orbit injective - which also makes it $\mathfrak{G}$-invariant - over the countable set of all orbits of neighbourhoods in our dataset. Property 2: Preservation of local geometry. A key property of WL is that the aggregation is injective. A $\mathfrak{G}$-invariant colouring procedure that purely satisfies Property 1 is not sufficient because, by definition, it loses spatial properties of each neighbourhood such as the relative pose or orientation [35]. Thus, we must additionally update auxiliary _geometric information_ variables in a way that is $\mathfrak{G}$-equivariant and injective. Geometric Weisfeiler-Leman (GWL). These intuitions motivate the following definition of the GWL test. At initialisation, we assign to each node $i\in\mathcal{V}$ a scalar node colour $c_{i}\in C^{\prime}$ and an auxiliary object $\boldsymbol{g}_{i}$ containing the geometric information associated to it: \[c_{i}^{(0)}:=\text{Hash}(\boldsymbol{s}_{i}),\quad\boldsymbol{g}_{i}^{(0)}:= \left(c_{i}^{(0)},\vec{\boldsymbol{v}}_{i}\right), \tag{6}\] where Hash denotes an injective map over the scalar attributes $\boldsymbol{s}_{i}$ of node $i$, _e.g._ the Hash of standard WL. To define the inductive step, assume we have the colours of the nodes and the associated geometric objects at iteration $t-1$. Then, we can aggregate the geometric information around node $i$ into a new object as follows: \[\boldsymbol{g}_{i}^{(t)}:=\left((c_{i}^{(t-1)},\boldsymbol{g}_{i}^{(t-1)})\;,\;\{\!\!\{(c_{j}^{(t-1)},\boldsymbol{g}_{j}^{(t-1)},\vec{\boldsymbol{x}}_{ ij})\mid j\in\mathcal{N}_{i}\}\!\!\}\right), \tag{7}\] Here $\{\!\!\{\!\{\!\{\!\{\!\{\!\{\!\{\!\}\}\}\}\}\!\!\}\!\!\}\!\!\}$ denotes a multiset - a set in which elements may occur more than once. Importantly, the group $\mathfrak{G}$ can act on the geometric objects above inductively by acting on the geometric information Figure 2: Geometric Weisfeiler-Leman Test. GWL distinguishes non-isomorphic geometric graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ by injectively assigning colours to distinct neighbourhood patterns, up to global symmetries (here $\mathfrak{G}=O(d)$). Each iteration expands the neighbourhood from which geometric information can be gathered (shaded for node $i$). Example inspired by Schütt et al. [9]. inside it. This amounts to rotating (or reflecting) the entire $t$-hop neighbourhood contained inside: \[\mathfrak{g}\cdot\mathbf{g}_{i}^{(0)} :=\left(c_{i}^{(0)},\ \mathbf{Q}_{\mathbf{g}}\mathbf{\bar{v}}_{i}\right),\] (8) \[\mathfrak{g}\cdot\mathbf{g}_{i}^{(t)} :=\left((c_{i}^{(t-1)},\mathfrak{g}\cdot\mathbf{g}_{i}^{(t-1)}),\ \tfrac{t}{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! where $\mathbf{j}=[j_{1},\ldots,j_{k}]$ are all possible $k$-tuples formed of elements of $\mathcal{N}_{i}$. Therefore, $\text{IGWL}_{(k)}$ is now constrained to extract information only from all the possible $k$-sized tuples of nodes (including the central node) in a neighbourhood. For instance, $\text{I-Hash}_{(2)}$ can identify neighbourhoods only up to pairwise distances among the central node and any of its neighbours (_i.e._ a $2$-body scalar), while $\text{I-Hash}_{(3)}$ up to distances and angles formed by any two edges (_i.e._ a $3$-body scalar). Notably, distances and angles alone are incomplete descriptors of local geometry as there exist several counterexamples that require higher body-order invariants to be distinguished [33, 34]. Therefore, $\text{I-Hash}_{(k)}$ with lower $k$ makes the colouring weaker. ### Characterising the expressive power of geometric GNNs Following Xu et al. [25], Morris et al. [26], we would like to upper bound the expressive power of geometric GNNs by the GWL test. We show that any $\mathfrak{G}$-equivariant GNN can be at most as powerful as GWL in distinguishing non-isomorphic geometric graphs. Proofs are available in Appendix E. Theorem 1.: _Any pair of geometric graphs distinguishable by a $\mathfrak{G}$-equivariant GNN is also distinguishable by GWL._ With a sufficient number of iterations, the output of $\mathfrak{G}$-equivariant GNNs can be equivalent to GWL _if_ certain conditions are met regarding the aggregate, update and readout functions. Proposition 2.: $\mathfrak{G}$_-equivariant GNNs have the same expressive power as GWL if the following conditions hold: (1) The aggregation Agg is an injective, $\mathfrak{G}$-equivariant multiset function. (2) The scalar part of the update $\textsc{Upd}_{\text{\tiny$\mathcal{D}$}_{i}}$ is a $\mathfrak{G}$-orbit injective, $\mathfrak{G}$-invariant multiset function. (3) The vector part of the update $\textsc{Upd}_{\text{\tiny$\mathcal{D}$}_{v}}$ is an injective, $\mathfrak{G}$-equivariant multiset function. (4) The graph-level readout $f$ is an injective multiset function._ Similar statements can be made for $\mathfrak{G}$-invariant GNNs and IGWL. Thus, we can directly transfer theoretical results between GWL/IGWL, which are abstract mathematical tools, and the class of geometric GNNs upper bounded by the respective tests. This connection has several interesting practical implications, discussed subsequently. ## 4 Understanding the Design Space of Geometric GNNs via GWL Overview. We demonstrate the utility of the GWL framework for understanding how geometric GNN design choices [36] influence expressivity: (1) Depth or number of layers; (2) Invariant vs. equivariant message passing; and (3) Body order of invariant scalarisation. In doing so, we formalise theoretical limitations of current architectures and provide practical implications for designing maximally powerful geometric GNNs. Proofs are available in Appendix D. ### Role of depth: propagating geometric information Let us consider the simplified setting of two geometric graphs $\mathcal{G}_{1}=(\mathbf{A}_{1},\mathbf{S}_{1},\mathbf{\tilde{V}}_{1},\mathbf{\tilde{X}}_{1})$ and $\mathcal{G}_{2}=(\mathbf{A}_{2},\mathbf{S}_{2},\mathbf{\tilde{V}}_{2},\mathbf{\tilde{X}}_{2})$ such that the underlying attributed graphs $(\mathbf{A}_{1},\mathbf{S}_{1})$ and $(\mathbf{A}_{2},\mathbf{S}_{2})$ are isomorphic. This case frequently occurs in chemistry, where molecules occur in different conformations, but with the same graph topology given by the covalent bonding structure. Each iteration of GWL aggregates geometric information $\mathbf{g}_{i}^{(k)}$ from progressively larger neighbourhoods $\mathcal{N}_{i}^{(k)}$ around the node $i$, and distinguishes (sub-)graphs via comparing $\mathfrak{G}$-orbit injective colouring of $\mathbf{g}_{i}^{(k)}$. We say $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are $k$_-hop distinct_ if for all graph isomorphisms $b$, there is some node $i\in\mathcal{V}_{1},b(i)\in\mathcal{V}_{2}$ such that the corresponding $k$-hop subgraphs $\mathcal{N}_{i}^{(k)}$ and $\mathcal{N}_{b(i)}^{(k)}$ are distinct. Otherwise, we say $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ are $k$_-hop identical_ if all corresponding $k$-hop subgraphs $\mathcal{N}_{i}^{(k)}$ and $\mathcal{N}_{b(i)}^{(k)}$ are the same up to group actions. We can now formalise what geometric graphs can and cannot be distinguished by GWL and maximally powerful geometric GNNs, in terms of the number of iterations or depth. Proposition 3.: _GWL can distinguish any $k$-hop distinct geometric graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ where the underlying attributed graphs are isomorphic, and $k$ iterations are sufficient._ Proposition 4.: _Up to $k$ iterations, GWL cannot distinguish any $k$-hop identical geometric graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ where the underlying attributed graphs are isomorphic._ Additionally, we can state the following results about the more constrained IGWL. Proposition 5.: _IGWL can distinguish any $1$-hop distinct geometric graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ where the underlying attributed graphs are isomorphic, and 1 iteration is sufficient._ Proposition 6.: _Any number of iterations of IGWL cannot distinguish any $1$-hop identical geometric graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ where the underlying attributed graphs are isomorphic._ An example illustrating Propositions 3 and 6 is shown in Figures 2 and 3, respectively. We can now consider the more general case where the underlying attributed graphs for $\mathcal{G}_{1}=(\mathbf{A}_{1},\mathbf{S}_{1},\mathbf{\tilde{V}}_{1},\mathbf{\tilde{X}}_{1})$ and $\mathcal{G}_{2}=(\mathbf{A}_{2},\mathbf{S}_{2},\mathbf{\tilde{V}}_{2},\mathbf{\tilde{X}}_{2})$ are non-isomorphic and constructed from point clouds using radial cutoffs, as conventional for biochemistry and material science applications. Proposition 7.: _Assuming geometric graphs are constructed from point clouds using radial cutoffs, GWL can distinguish any geometric graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ where the underlying attributed graphs are non-isomorphic. At most $k_{\text{Max}}$ iterations are sufficient, where $k_{\text{Max}}$ is the maximum graph diameter among $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$._ ### Limitations of invariant message passing: failure to capture global geometry Propositions 3 and 6 enable us to compare the expressive powers of GWL and IGWL. Theorem 8.: _GWL is strictly more powerful than IGWL._ This statement formalises the advantage of $\mathfrak{G}$-equivariant intermediate layers for geometric graphs, as prescribed in the Geometric Deep Learning blueprint [6], in addition to echoing similar intuitions in the computer vision community. As remarked by Hinton et al. [35], translation invariant models do not understand the relationship between the various parts of an image (colloquially called the "Picasso problem"). Similarly, our results point to IGWL and $\mathfrak{G}$-invariant GNNs failing to understand how the 1-hop neighbourhoods in a graph are oriented w.r.t. each other. As a result, even the most powerful $\mathfrak{G}$-invariant GNNs are restricted in their ability to compute global and non-local geometric properties. Proposition 9.: _IGWL and $\mathfrak{G}$-invariant GNNs cannot decide several geometric graph properties: (1) perimeter, surface area, and volume of the bounding box/sphere enclosing the geometric graph; (2) distance from the centroid or centre of mass; and (3) dihedral angles._ Finally, we identify a setting where this distinction between the two approaches disappears. Proposition 10.: _IGWL has the same expressive power as GWL for fully connected geometric graphs._ Practical Implications. Proposition 9 highlights critical theoretical limitations of $\mathfrak{G}$-invariant GNNs. This suggests that $\mathfrak{G}$-equivariant GNNs should be preferred when working with large geometric graphs such as macromolecules with thousands of nodes, where message passing is restricted to local radial neighbourhoods around each node. Stacking multiple $\mathfrak{G}$-equivariant layers enables the computation of compositional geometric features. Two straightforward approaches to overcoming the limitations of $\mathfrak{G}$-invariant GNNs may be: (1) pre-computing non-local geometric properties as input features, _e.g._ models such as GemNet [21] and SphereNet [37] use two-hop dihedral angles. And (2) working with fully connected geometric graphs, as Proposition 10 suggests that $\mathfrak{G}$-equivariant and $\mathfrak{G}$-invariant GNNs can be made equally powerful when performing all-to-all message passing. This is supported by the empirical success of recent $\mathfrak{G}$-invariant 'Graph Transformers' [22, 38] for small molecules with tens of nodes, where working with full graphs is tractable. ### Role of scalarisation body order: identifying neighbourhood $\mathfrak{G}$-orbits At each iteration of GWL and IGWL, the I-Hash function assigns a $\mathfrak{G}$-invariant colouring to distinct geometric neighbourhood patterns. I-Hash is an idealised $\mathfrak{G}$-orbit injective function which is not necessarily continuous. In geometric GNNs, this corresponds to scalarising local geometric information when updating the scalar features; examples are shown in equation 14 and equation 15. We can analyse the construction of the I-Hash function and the scalarisation step in geometric GNNs via the $k$-body variations $\text{IGWL}_{(k)}$, described in Section 3. In doing so, we will make connections between IGWL and WL for non-geometric graphs. Firstly, we formalise the relationship between the injectivity of I-Hash${}_{(k)}$ and the maximum cardinality of local neighbourhoods in a given dataset. Proposition 11.: $\text{I-Hash}_{(m)}$ _is $\mathfrak{G}$-orbit injective for $m=\text{max}(\{\|\mathcal{N}_{i}\|\mid i\in\mathcal{V}\})$, the maximum cardinality of all local neighbourhoods $\mathcal{N}_{i}$ in a given dataset._ Practical Implications. While building provably injective $\text{I-Hash}_{(k)}$ functions may require intractably high $k$, the hierarchy of $\text{IGWL}_{(k)}$ tests enable us to study the expressive power of practical $\mathfrak{G}$-invariant aggregators used in current geometric GNN layers, _e.g._ SchNet [14], E-GNN [10], and TFN [11] use distances, while DimeNet [16] uses distances and angles. Notably, MACE [20] constructs a _complete_ basis of scalars up to arbitrary body order $k$ via Atomic Cluster Expansion [39], which can be $\mathfrak{G}$-orbit injective if the conditions in Proposition 11 are met. We can state the following about the $\text{IGWL}_{(k)}$ hierarchy and the corresponding GNNs. Proposition 12.: $\text{IGWL}_{(k)}$ _is at least as powerful as $\text{IGWL}_{(k-1)}$. For $k\leq 5$, $\text{IGWL}_{(k)}$ is strictly more powerful than $\text{IGWL}_{(k-1)}$._ Finally, we show that $\text{IGWL}_{(2)}$ is equivalent to WL when all the pairwise distances between the nodes are the same. A similar observation was recently made by Pozdnyakov and Ceriotti [40]. Proposition 13.: _Let $\mathcal{G}_{1}=(\mathbf{A}_{1},\mathbf{S}_{1},\mathbf{\tilde{X}}_{1})$ and $\mathcal{G}_{2}=(\mathbf{A}_{2},\mathbf{S}_{2},\mathbf{\tilde{X}}_{2})$ be two geometric graphs with the property that all edges have equal length. Then, $\text{IGWL}_{(2)}$ distinguishes the two graphs if and only if WL can distinguish the attributed graphs $(\mathbf{A}_{1},\mathbf{S}_{1})$ and $(\mathbf{A}_{1},\mathbf{S}_{1})$._ This equivalence points to limitations of distance-based $\mathfrak{G}$-invariant models like SchNet [14]. These models suffer from all well-known failure cases of WL, _e.g._ they cannot distinguish two equilateral triangles from the regular hexagon [16]. ## 5 Synthetic Experiments on Geometric GNN Expressivity Overview. We provide synthetic experiments to supplement our theoretical results and highlight practical challenges in building maximally powerful geometric GNNs, _s.a._ oversquashing of geometric information with increased depth and the need for higher order tensors in $\mathfrak{G}$-equivariant GNNs. We hope that the accompanying codebase, geometric-gnn-dojo2, can be a pedagogical resource for advancing geometric GNN architectures in a principled manner. See Appendix C for details on experimental setup and hyperparameters. \begin{table} \begin{tabular}{l l c c c c c} \hline \hline & ($k=4$-chains) & \multicolumn{5}{c}{Number of layers} \\ & GNN Layer & $\lfloor\frac{k}{2}\rfloor$ & $\lfloor\frac{k}{2}\rfloor+1=\mathbf{3}$ & $\lfloor\frac{k}{2}\rfloor+2$ & $\lfloor\frac{k}{2}\rfloor+3$ & $\lfloor\frac{k}{2}\rfloor+4$ \\ \hline \multirow{4}{}{$\mathfrak{G}$-invariant} & $\text{IGWL}$ & $50\%$ & $50\%$ & $50\%$ & $50\%$ & $50\%$ \\ & $\text{SchNet}$ & $50.0\pm 0.00$ & $50.0\pm 0.00$ & $50.0\pm 0.00$ & $50.0\pm 0.00$ & $50.0\pm 0.00$ \\ & DimeNet & $50.0\pm 0.00$ & $50.0\pm 0.00$ & $50.0\pm 0.00$ & $50.0\pm 0.00$ & $50.0\pm 0.00$ \\ \hline \multirow{4}{}{$\mathfrak{G}$-invariant} & $\text{GWL}$ & $50\%$ & $100\%$ & $100\%$ & $100\%$ & $100\%$ \\ & E-GNN & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $\mathbf{100.0\pm 0.0}$ \\ & GVP-GNN & $50.0\pm 0.0$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ \\ & TFN & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $\mathbf{80.0\pm 24.5}$ & $\mathbf{85.0\pm 22.9}$ \\ & MACE & $50.0\pm 0.0$ & $\mathbf{90.0\pm 20.0}$ & $\mathbf{90.0\pm 20.0}$ & $\mathbf{95.0\pm 15.0}$ & $\mathbf{95.0\pm 15.0}$ \\ \hline \hline \end{tabular} \end{table} Table 1: $k$_-chain geometric graphs._$k$-chains are $(\lfloor\frac{k}{2}\rfloor+1)$-hop distinguishable and $(\lfloor\frac{k}{2}\rfloor+1)$ GWL iterations are theoretically sufficient to distinguish them. We train geometric GNNs with an increasing number of layers to distinguish $k=4$-chains. $\mathfrak{G}$-equivariant GNNs may require more iterations that prescribed by GWL, pointing to preliminary evidence of oversquashing when geometric information is propagated across multiple layers using fixed dimensional feature spaces. IGWL and $\mathfrak{G}$-invariant GNNs are unable to distinguish $k$-chains for any $k\geq 2$ and $\mathfrak{G}=O(3)$. Anomolous results are marked in $\overline{\text{red}}$ and expected results in $\overline{\text{green}}$. ### Depth, non-local geometric properties, and oversquashing GWL assumes perfect propagation of $\mathfrak{G}$-equivariant geometric information at each iteration, which implies that the test can be run for any number of iterations without loss of information. In geometric GNNs, $\mathfrak{G}$-equivariant information is propagated via summing features from multiple layers in fixed dimensional spaces, which may lead to distortion or loss of information from distant nodes. Experiment.To study the practical implications of depth in propagating geometric information beyond local neighbourhoods, we consider $k$-chain geometric graphs which generalise the examples from Schutt et al. [9]. Each pair of $k$-chains consists of $k+2$ nodes with $k$ nodes arranged in a line and differentiated by the orientation of the $2$ end points. Thus, $k$-chain graphs are $(\lfloor\frac{k}{2}\rfloor+1)$-hop distinguishable, and $(\lfloor\frac{k}{2}\rfloor+1)$ GWL iterations are theoretically sufficient to distinguish them. In Table 1, we train $\mathfrak{G}$-equivariant and $\mathfrak{G}$-invariant GNNs with an increasing number of layers to distinguish $k$-chains. Results.Despite the supposed simplicity of the task, we find that popular $\mathfrak{G}$-equivariant GNNs such as E-GNN [10] and TFN [11] may require more iterations that prescribed by GWL. Notably, for chains larger than $k=4$, all $\mathfrak{G}$-equivariant GNNs tended to require more than $(\lfloor\frac{k}{2}\rfloor+1)$ iterations to solve the task. Additionally, IGWL and $\mathfrak{G}$-invariant GNNs are unable to distinguish $k$-chains. Table 1 points to preliminary evidence of the _oversquashing_ phenomenon [41, 42] for geometric GNNs. The issue is most evident for E-GNN, which uses a single vector feature to aggregate and propagate geometric information. This may have implications in modelling macromolecules where long-range interactions often play important roles. ### Higher order tensors and rotationally symmetric structures In addition to perfect propagation, GWL is also able to injectively aggregate $\mathfrak{G}$-equivariant geometric information by making use of an auxiliary nested geometric object $\mathbf{g}_{i}$. On the other hand, $\mathfrak{G}$-equivariant GNNs aggregate geometric information via summing neighbourhood features represented by either cartesian vectors (tensor order 1) or higher order spherical tensors. This choice of basis often comes with trade-offs between computational tractability and empirical performance. Experiment.To demonstrate the utility of higher order tensors in $\mathfrak{G}$-equivariant GNNs, we study how rotational symmetries interact with tensor order. In Table 2, we evaluate current $\mathfrak{G}$-equivariant layers on their ability to distinguish the orientation of structures with rotational symmetry. An $L$-fold symmetric structure does not change when rotated by an angle $\frac{2\pi}{L}$ around a point (in 2D) or axis (3D). We consider two _distinct_ rotated versions of each $L$-fold symmetric structure and train single layer $\mathfrak{G}$-equivariant GNNs to classify the two orientations using the updated geometric features. \begin{table} \begin{tabular}{l l c c c} \hline \hline & \multicolumn{4}{c}{Rotational symmetry} \\ & GNN Layer & 2 fold & 3 fold & 5 fold & 10 fold \\ \hline \multirow{3}{}{$\mathfrak{G}$-equivariant} & E-GNN${}_{L=1}$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ \\ & GVP-GNN${}_{L=1}$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ \\ \hline \multirow{3}{}{$\mathfrak{G}$-equivariant} & TFN/MACE${}_{L=1}$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ \\ & TFN/MACE${}_{L=2}$ & $\mathbf{100.0\pm 0.0}$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ \\ & TFN/MACE${}_{L=3}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ & $50.0\pm 0.0$ & $50.0\pm 0.0$ \\ & TFN/MACE${}_{L=5}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ & $50.0\pm 0.0$ \\ & TFN/MACE${}_{L=10}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ & $\mathbf{100.0\pm 0.0}$ \\ \hline \hline \end{tabular} \end{table} Table 2: Rotationally symmetric structures. We train single layer $\mathfrak{G}$-equivariant GNNs to distinguish two _distinct_ rotated versions of each $L$-fold symmetric structure. We find that layers using order $L$ tensors are unable to identify the orientation of structures with rotation symmetry higher than $L$-fold. This issue is particularly prevalent for layers using cartesian vectors (tensor order 1). Results. We find that layers using order $L$ tensors are unable to identify the orientation of structures with rotation symmetry higher than $L$-fold. This observation may be attributed to spherical harmonics, which are used as the underlying orthonormal basis and are rotationally symmetric themselves. Layers such as E-GNN [10] and GVP-GNN [8] using cartesian vectors are popular as higher order tensors can be computationally intractable for many applications. However, E-GNN and GVP-GNN are particularly poor at discriminating the orientation of rotationally symmetric structures. This may have implications for the modelling of periodic materials which naturally exhibit such symmetries [43]. ## 6 Discrimination and Universality Overview. Following Chen et al. [27], we study the equivalence between the universal approximation capabilities of geometric GNNs [44, 45] and the perspective of discriminating geometric graphs introduced by GWL, generalising their results to any isomorphism induced by a compact Lie group $\mathfrak{G}$. We further study the number of invariant aggregators that are required in a continuous setting to distinguish any two neighbourhoods. Proofs are available in Appendix F. In the interest of generality, we use a general space $X$ acted upon by a compact group $\mathfrak{G}$ and we are interested in the capacity of $\mathfrak{G}$-invariant functions over $X$ to separate points in $Y$. The restriction to a smaller subset $Y$ is useful because we would like to separately consider the case when $Y$ is countable due to the use of countable features. Therefore, in general, the action of $\mathfrak{G}$ on $Y$ might not be strictly defined since it might yield elements outside $Y$. For our setting, the reader could take $X=(\mathbb{R}^{d}\times\mathbb{R}^{f})^{n\times n}$ to be the space of $n\times n$ geometric graphs and $Y=\mathcal{X}^{n\times n}$, where $\mathcal{X}\subseteq\mathbb{R}^{d}\times\mathbb{R}^{f}$. Definition 14.: _Let $\mathfrak{G}$ be a compact group and $\mathcal{C}$ a collection of $\mathfrak{G}$-invariant functions from a set $X$ to $\mathbb{R}$. For a subset $Y\subseteq X$, we say the $\mathcal{C}$ is pairwise $\mathbf{Y_{\mathfrak{G}}}$ discriminating if for any $y_{1},y_{2}\in Y$ such that $y_{1}\not\succeq y_{2}$, there exists a function $h\in\mathcal{C}$ such that $h(y_{1})\not\succeq h(y_{2})$._ We note here that $h$ is not necessarily injective, _i.e._ there might be $y^{\prime}_{1},y^{\prime}_{2}$ for which $h(y^{\prime}_{1})=h(y^{\prime}_{1})$. Therefore, pairwise discrimination is a weaker notion of discrimination than the one GWL uses. Definition 15.: _Let $\mathfrak{G}$ be a compact group and $\mathcal{C}$ a collection of $\mathfrak{G}$-invariant functions from $X$ to $\mathbb{R}$. For $Y\subseteq X$, we say the $\mathcal{C}$ is universally approximating over $Y$ if for all $\mathfrak{G}$-invariant functions $f$ from $X$ to $\mathbb{R}$ and for all $\epsilon>0$, there exists $h_{\varepsilon,f}\in\mathcal{C}$ such that $\\|f-h_{\varepsilon,f}\\|_{Y}:=\sup_{y\in Y}\|\hat{f}(y)-h(y)\|<\varepsilon$._ Countable Features. We first focus on the countable feature setting, which is also the setting where the GWL test operates. Therefore, we will assume that $Y$ is a countable subset of $X$. Theorem 16.: _If $\mathcal{C}$ is universally approximating over $Y$, then $\mathcal{C}$ is also pairwise $Y_{\mathfrak{G}}$ discriminating._ This result further motivates the interest in discriminating geometric graphs, since a model that cannot distinguish two non-isomorphic geometric graphs is not universal. By further assuming that $Y$ is finite, we obtain a result in the opposite direction. Given a collection of functions $\mathcal{C}$, we define like in Chen et al. [27] the class $\mathcal{C}^{+L}$ given by all the functions of the form $\mathrm{MLP}([f_{1}(x),\ldots,f_{k}(x)])$ with $f_{i}\in\mathcal{C}$ and finite $k$, where the MLP has $L$ layers with ReLU hidden activations. Theorem 17.: _If $\mathcal{C}$ is pairwise $Y_{\mathfrak{G}}$ discriminating, then $\mathcal{C}^{+2}$ is universally approximating over $Y$._ Continuous Features. The symmetries characterising geometric graphs are naturally continuous (_e.g._ rotations). Therefore, it is natural to ask how the results above translate to _continuous_$\mathfrak{G}$-invariant functions over a continuous subspace $Y$. Therefore, for the rest of this section, we assume that $(X,d)$ is a metric space, $Y$ is a compact subset of $X$ and $\mathfrak{G}$ acts continuously on $X$. Theorem 18.: _If $\mathcal{C}$ can universally approximate any continuous $\mathfrak{G}$-invariant functions on $Y$, then $\mathcal{C}$ is also pairwise $Y_{\mathfrak{G}}$ discriminating._ With the additional assumption that $X=Y$, we can also prove the converse. Theorem 19.: _If $\mathcal{C}$, a class of functions over $Y$, is pairwise $Y_{\mathfrak{G}}$ discriminating, then $\mathcal{C}^{+2}$ can also universally approximate any continuous function over $Y$._ We now produce an estimate for the number of aggregators needed to learn _continuous_ orbit-space injective functions on a manifold $X$ based on results from differential geometry [46]. A group $\mathfrak{G}$ acts freely on $X$ if $\mathfrak{g}x=x$ implies $\mathfrak{g}=e_{\mathfrak{G}}$, where $e_{\mathfrak{G}}$ is the identity element in $\mathfrak{G}$. Theorem 20.: _Let $X$ be a smooth $n$-dim manifold and $\mathfrak{G}$ an $m$-dim compact Lie group acting continuously on $X$. Suppose there exists a smooth submanifold $Y$ of $X$ of the same dimension such that $\mathfrak{G}$ acts freely on it. Then, any $\mathfrak{G}$-orbit injective function $f:X\to\mathbb{R}^{d}$ requires that $d\geq n-m$ We now apply this theorem to the local aggregation operation performed by geometric GNNs. Let $X=\mathbb{R}^{n\times d}$ and $\mathfrak{G}=S_{n}\times O(d)$ or $S_{n}\times SO(d)$. Let $\mathbf{P}_{\mathfrak{g}}$ and $\mathbf{Q}_{\mathfrak{g}}$ be the permutation matrix and the orthogonal matrix associated with the group element $\mathfrak{g}\in\mathfrak{G}$. Then $\mathfrak{g}$ acts on matrices $\mathbf{X}\in X$ continuously via $\mathbf{P}_{\mathfrak{g}}\mathbf{X}\mathbf{Q}_{\mathfrak{g}}^{\top}$. Then, $\mathfrak{G}$ orbit-space injective functions on $X$ are functions on point clouds of size $n$ that can distinguish any two different point clouds. Theorem 21.: _For $n\geq d-1>0$ or $n=d=1$, any continuous $S_{n}\times SO(d)$ orbit-space injective function $f:\mathbb{R}^{n\times d}\to\mathbb{R}^{q}$ requires that $q\geq nd-d(d-1)/2$._ We can also generalise this to $O(d)$, with the slightly stronger assumption that $n\geq d$. Theorem 22.: _For $n\geq d>0$, any continuous $S_{n}\times O(d)$ orbit-space injective function $f:\mathbb{R}^{n\times d}\to\mathbb{R}^{q}$ requires that $q\geq nd-d(d-1)/2$._ Overall, these results show that when working with point clouds in $\mathbb{R}^{3}$ as is common in molecular or physical applications, at least $q=3(n-1)$ aggregators are required. This result argues why a bigger representational width can help distinguish neighbourhoods. Finally, in the particular case of the zero-dimensional subgroup $S_{n}\times\{e_{\text{SO(d)}}\}\simeq S_{n}$ we obtain a statement holding for all $n$ and generalising a result from Corso et al. [47] regarding the aggregators for non-geometric GNNs. The original PNA result considers the case $d=1$ and here we extend it to arbitrary $d$. Proposition 23.: _Any $S_{n}$-invariant injective function $f:\mathbb{R}^{n\times d}\to\mathbb{R}^{q}$ requires $q\geq nd$._ ## 7 Conclusion We propose the Geometric Weisfeiler-Leman (GWL) framework to characterise the expressive power of geometric GNNs from the perspective of geometric graph isomorphism. Our work fills a key research gap as geometric graphs come with both relational structure and geometric attributes, making theoretical tools for standard GNNs inapplicable in this increasingly relevant real-world setting. We demonstrate the utility of GWL for understanding how key design choices influence geometric GNN expressivity, summarised in Figure 1. Most notably, we formalise the advantages of equivariant GNNs over invariant GNNs, as equivariant layers can distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods. Invariant layers have limited expressivity as they cannot distinguish graphs where one-hop neighbourhoods are the same, nor compute non-local geometric properties. Additionally, we highlight the need for higher order order tensors and scalarisation in geometric GNNs. Future Work.GWL provides an abstraction to study the theoretical limits of geometric GNNs. In practice, it is challenging to build provably powerful models that satisfy the conditions of Proposition 2, as GWL relies on perfect neighbourhood aggregation and colouring functions. Based on the understanding gained from GWL, future work will explore building maximally powerful, _practical_ geometric GNNs for applications in biochemistry and material science. ## Acknowledgements We would like to thank Andrew Blake, Challenger Mishra, Charles Harris, David Kovacs, Erik Thiede, Gabor Csanyi, Hannes Stark, Ilyes Batatia, Iulia Duta, Justin Tan, Mario Geiger, Petar Velickovic, Ramon Vinas, Rob Hesselink, Soledad Villar, Weihua Hu, and the anonymous reviewers for helpful comments and discussions. CKJ was supported by the A*STAR Singapore National Science Scholarship (PhD). SVM was supported by the UKRI Centre for Doctoral Training in Application of Artificial Intelligence to the study of Environmental Risks (EP/S022961/1).
2301.10881	Anisotropic molecular coarse-graining by force and torque matching with neural networks	We develop a machine-learning method for coarse-graining condensed-phase molecular systems using anisotropic particles. The method extends currently available high-dimensional neural network potentials by addressing molecular anisotropy. We demonstrate the flexibility of the method by parametrizing single-site coarse-grained models of a rigid small molecule (benzene) and a semi-flexible organic semiconductor (sexithiophene), attaining structural accuracy close to the all-atom models for both molecules at considerably lower computational expense. The machine-learning method of constructing the coarse-grained potential is shown to be straightforward and sufficiently robust to capture anisotropic interactions and many-body effects. The method is validated through its ability to reproduce the structural properties of the small molecule's liquid phase and the phase transitions of the semi-flexible molecule over a wide temperature range.	Marltan O. Wilson, David M. Huang	2023-01-26T00:24:02Z	http://arxiv.org/abs/2301.10881v1	# Anisotropic molecular coarse-graining by force and torque matching with neural networks ###### Abstract We develop a machine-learning method for coarse-graining condensed-phase molecular systems using anisotropic particles. The method extends currently available high-dimensional neural network potentials by addressing molecular anisotropy. We demonstrate the flexibility of the method by parametrizing single-site coarse-grained models of a rigid small molecule (benzene) and a semi-flexible organic semiconductor (sexitiophene), attaining structural accuracy close to the all-atom models for both molecules at considerably lower computational expense. The machine-learning method of constructing the coarse-grained potential is shown to be straightforward and sufficiently robust to capture anisotropic interactions and many-body effects. The method is validated through its ability to reproduce the structural properties of the small molecule's liquid phase and the phase transitions of the semi-flexible molecule over a wide temperature range. ## I Introduction Machine learning is quickly becoming an invaluable tool in the search, analysis, and development of new materials.[1; 2] Neural networks, in particular, have had major recent success in areas ranging from predicting the folded conformation of biological macromolecules such as proteins[3] to developing highly accurate temperature-transferable interatomic potentials.[4; 5] The latter is an important advance in the field of molecular dynamics (MD) simulations. Improvements in these machine-learning models aim to expand the length and time scale of simulations without sacrificing accuracy.[6; 7] Currently used ab initio molecular dynamics simulation models are generally accurate but are computationally expensive, limiting their ability to probe long time scales.[8; 9] However, machine-learning potentials can produce ab initio accuracy at the computational cost of classical atomistic models.[10; 11] Even though simulations at the classical MD level are faster than ab initio MD, the speedup is still insufficient to model the large length and time scales needed to fully understand certain phenomena and processes such as supramolecular assembly. It is well known that explicit modeling of high-frequency motion is not critical for describing many phenomena in molecular systems. These simplifications have led to the development of molecular coarse-grained models to study large, complex materials and biological systems.[12] Parameterization of coarse-grained interaction potentials commonly uses one of two strategies: the top-down approach in which parameters are tuned to match macroscopic observables, as exemplified by the Martini model,[13] and the bottom-up approach in which interactions are derived from the properties of a fine-grained model with more degrees of freedom.[12] By following a similar bottom-up process used to apply machine learning to ab initio MD data, neural-network approaches have been extended to coarse-grained molecular models, further extending the length and time scale of simulations with atomistic accuracy.[14; 15] Neural-network potentials using isotropic coarse-grained particles have several advantages over their pair-wise additive analytical counterparts since they are constructed as many-body potentials. This many-body potential can become costly when multiple coarse-grained particles are needed to preserve the shape anisotropy of a molecular fragment. It is sometimes more accurate and computationally efficient to represent these groups of atoms as a single anisotropic coarse-grained particle such as an ellipsoid, such as in the case of large, rigid, anisotropic molecular fragments. Analytical anisotropic coarse-grained potentials such as the Gay-Berne potential[16; 17] were developed to address the poor performance of spherically symmetric potentials in replicating intrinsic anisotropic interactions such as $\pi$-stacking. By modeling rigid anisotropic groups of atoms as ellipsoids, the anisotropic properties of the group are preserved in a single-site model. Shape and interaction anisotropy is especially important for the study of organic semiconductor molecules, which typically consist of highly anisotropic and rigid $\pi$-conjugated units and often form liquid-crystal phases whose morphology strongly affects their performance in devices such as solar cells, transistors, and light-emitting diodes.[18] Unlike analytical pair-wise additive potentials such as the Gay-Berne potential, high-dimensional neural-network potentials are constructed based on the immediate neighborhood of a molecule and thus account for many-body effects as well as local density variations. Notable machine-learning implementations of interatomic and intermolecular potentials include the neural-network potentials developed by Behler et al.[19] The Behler neural-network potentials are constructed from a set of symmetry functions used to represent the invariant properties of the atomic environment of each atom taken from ab initio simulations. DeepMD[10] and DeepCG[14] are two other neural-network codes constructed for atomistic and coarse-grained simulations, respectively. All these neural-network potentials rely on an invariant representation of the atomic/molecular environment. The CGnets deep-learning approach[15] employs a prior potential to account for areas in a coarse-grained data set that may not be properly sampled due to high repulsive energies. These interactions are especially important to reproduce the local structure of the simu lated material. Machine learning has previously been applied to the parameterization of coarse-grained models with anisotropic particles,[20] but no such implementation has used a nonlinear neural-network optimization method to construct the coarse-grained potential. In this work, we address this gap in knowledge by using a neural network to construct a high-dimensional anisotropic coarse-grained potential. We parameterize the neural-network potential using a recently derived systematic and general bottom-up coarse-graining method called anisotropic force-matching coarse-graining (AFM-CG),[21] which generalizes the multi-scale coarse-graining (MS-CG) method[22] for isotropic coarse-grained particles to anisotropic particles. The method rigorously accounts for finite-temperature, many-body effects without assuming a specific functional form of the anisotropic coarse-grained potential. It yields general equations relating the forces, torques, masses, and moments of inertia of the coarse-grained particles to properties of a fine-grained (e.g. all-atom) MD simulation based on a mapping between fine-grained and coarse-grained coordinates and momenta, and by matching the equilibrium coarse-grained phase-space distribution with the mapped distribution of the fine-grained system. The previous implementations of the AFM-CG method approximated the coarse-grained potential as a sum of pair interactions between particles.[21] Here, we extend this approach to more general many-body anisotropic interactions described by a neural-network potential. We also extend the approach, which was derived for constant-volume systems in the canonical ensemble to constant-pressure systems by applying a virial-matching condition previously derived for the MS-CG method.[23] A general coarse-grained potential should capture any temperature-dependent phase transition associated with either melting, annealing, or glass transition temperatures as well as the local structure and density of the material. The focus is on the development of a model for which trained parameters can be easily obtained and one capable of reproducing interaction anisotropy, temperature transferability, and many-body effects. The flexibility of the new model is demonstrated through the matching of structural and thermodynamic properties of condensed-phase systems of a small anisotropic molecule, benzene, and of a larger, more flexible organic semiconductor molecule, sexithiophene. These two molecules were chosen to determine the conditions under which coarse-grained structural inaccuracy outweighs the computational efficiency of a single-anisotropic-site model. ## II Theory The key aspects of the theory that underpins the AFM-CG method and its extension to constant pressure via virial matching are summarized below. The reader is referred to Ref. [21] for a more detailed description and the full derivation of the method. The positions $\mathbf{r}^{n}=\{\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{n}\}$ of the $n$ fine-grained particles are mapped onto the positions $\mathbf{R}^{N}=\{\mathbf{R}_{1},\mathbf{R}_{2},\ldots,\mathbf{R}_{N}\}$ and orientations $\mathbf{\Omega}^{N}=\{\mathbf{\Omega}_{1},\mathbf{\Omega}_{2},\ldots,\mathbf{\Omega}_{N}\}$ of the $N$ anisotropic coarse-grained particles. Each fine-grained particle $i$ is mapped to a single coarse-grained particle by defining $N$ non-intersecting subsets, $\zeta_{1},\zeta_{2},\ldots,\zeta_{N}$, of the fine-grained particle indices such that $\zeta_{I}$ contains the indices of the fine-grained particles mapped onto coarse-grained particle $I$. The position $\mathbf{R}_{I}$ of coarse-grained particle $I$ is defined to be equal to the center-of-mass of the group of fine-grained particles that are mapped onto it, i.e. \[\mathbf{R}_{I}=\frac{\sum_{i\in\zeta_{I}}m_{i}\mathbf{r}_{i}}{\sum_{i\in\zeta_{I}}m_{ i}}, \tag{1}\] where $m_{i}$ is the mass of fine-grained particle $i$. The orientation \[\mathbf{\Omega}_{I}=\begin{bmatrix}\mathbf{\Omega}_{I,1}\\ \mathbf{\Omega}_{I,2}\\ \mathbf{\Omega}_{I,3}\end{bmatrix} \tag{2}\] of coarse-grained particle $I$ is specified by the rotation matrix whose components are the particle's three normalized principal axes of inertia, $\mathbf{\Omega}_{I,q}$ for $q=1,2,3$. These axes are defined to be equal to the corresponding principal axes relative to the center-of-mass of the group of fine-grained particles that are mapped onto the coarse-grained particle. Thus, these axes are the normalized eigenvectors of the inertia tensor \[\mathbf{\mathrm{I}}_{\mathrm{FG},I}=\sum_{i\in\zeta_{I}}m_{i}(\|\|\mathbf{\alpha}\mathbf{r}_ {i}\|\|^{2}\mathbf{E}-\mathbf{\alpha}\mathbf{r}_{i}\mathbf{\Delta}\mathbf{r}_{i}^{\mathrm{T}}), \tag{3}\] where $\mathbf{\Delta}\mathbf{r}_{i}=\mathbf{r}_{i}-\mathbf{R}_{I}$ is the position of fine-grained particle $i$ relative to the center-of-mass (coarse-grained particle position) and $\mathbf{E}$ is the $3\times 3$ identity matrix. From these coordinate mappings and the relationship between generalized coordinates and momenta from Hamilton's equations,[24] mappings from the linear momenta $\mathbf{p}^{n}=\{\mathbf{p}_{1},\mathbf{p}_{2},\ldots,\mathbf{p}_{n}\}$ of the fine-grained particles to the linear momenta $\mathbf{P}^{N}=\{\mathbf{P}_{1},\mathbf{P}_{2},\ldots,\mathbf{P}_{N}\}$ and angular momenta $\mathbf{L}^{N}=\{\mathbf{L}_{1},\mathbf{L}_{2},\ldots,\mathbf{L}_{N}\}$ of the anisotropic coarse-grained particles can also be defined.[21] The mappings for coarse-grained particle $I$ are \[\mathbf{P}_{I}=\frac{M_{I}}{\sum_{i\in\zeta_{I}}m_{i}}\sum_{i\in\zeta_{I}}\mathbf{p}_{i} \tag{4}\] and \[\mathbf{L}_{I}=\mathbf{\mathrm{I}}_{I}\mathbf{\mathrm{F}}_{\mathrm{FG},I}^{-1}\sum_{i\in \zeta_{I}}\mathbf{\Delta}\mathbf{r}_{i}\mathbf{\times}\mathbf{p}_{i}, \tag{5}\] respectively, where $\mathbf{\mathrm{I}}_{I}$ is the inertia tensor of coarse-grained particle $I$. Given these mappings, several conditions can be derived that the coarse-grained model must satisfy for its equilibrium coarse-grained phase-space distribution to match the corresponding mapped distribution of the fine-grained system. Consistency between the configuration-space distributions gives the following matching conditions between the forces $\mathbf{F}_{I}$ and torques $\mathbf{\tau}_{I}$ on coarse-grained particle $I$ and the forces on the fine-grained particles mapped onto it:[21] \[\mathbf{F}_{I}(\mathbf{R}^{N},\mathbf{\Omega}^{N})=-\frac{\partial U}{\partial R_{I}}= \left\langle\sum_{i\in\zeta_{I}}\mathbf{f}_{i}\right\rangle_{\mathbf{R}^{N},\mathbf{\Omega} ^{N}} \tag{6}\] and \[\mathbf{\tau}_{I}(\mathbf{R}^{N},\mathbf{\Omega}^{N})=-\sum_{q}\Omega_{I,q}\times\frac{\partial U }{\partial\Omega_{I,q}}=\left\langle\sum_{i\in\mathcal{U}_{I}}\Delta\mathbf{r}_{i} \times\mathbf{f}_{i}\right\rangle_{\mathbf{R}^{N},\mathbf{\Omega}^{N}} \tag{7}\] where $U(R^{N},\mathbf{\Omega}^{N})$ is the coarse-grained potential, $\mathbf{f}_{i}(\mathbf{r}^{n})=-\frac{\partial\mathbf{n}}{\partial\mathbf{r}_{i}}$ is the force on fine-grained particle $i$, with $\mathbf{u}(\mathbf{r}^{n})$ the fine-grained potential and $\langle\cdots\rangle_{\mathbf{R}^{N},\mathbf{\Omega}^{N}}$ denoting an average over fine-grained configurations mapped to coarse-grained configuration $(R^{N},\mathbf{\Omega}^{N})$. Consistency between the momentum-space distributions requires the mass $M_{I}$ of coarse-grained particle $I$ to be the sum of the masses of its constituent fine-grained particles, i.e. [21] \[M_{I}=\sum_{i\in\mathcal{U}_{I}}m_{i}. \tag{8}\] In addition, provided that the inertia tensor $\mathbf{\mathrm{I}}_{\mathrm{FG},J}$ of the group of fine-grained particles mapped to this coarse-grained particle does not depend on the configuration of the other particles [21], \[I_{I,q}^{1/2}\exp\left(-\frac{I_{I,q}\omega_{I,q}^{2}}{2k_{\mathrm{B}}T} \right)\approx\left\langle I_{\mathrm{FG},I,q}^{1/2}\exp\left(-\frac{I_{ \mathrm{FG},I,q}\omega_{I,q}^{2}}{2k_{\mathrm{B}}T}\right)\right\rangle_{\mathbf{R }_{I},\mathbf{\Omega}_{I}}, \tag{9}\] where $I_{I,q}$, $I_{\mathrm{FG},I,q}$, and $\omega_{I,q}$ are the components of the coarse-grained moment of inertia, fine-grained moment of inertia, and angular velocity about the $q$ axis, and $\langle\cdots\rangle_{\mathbf{R}_{I},\mathbf{\Omega}_{I}}$ denotes an equilibrium average of fine-grained configurations consistent with the coordinate mapping of coarse-grained particle $I$. Furthermore, if the fluctuations in $I_{\mathrm{FG},I,q}$ are small compared to its mean, it can be shown that [21] \[I_{I,q}\approx\left\langle I_{\mathrm{FG},I,q}\right\rangle_{\mathbf{R}_{I},\mathbf{ \Omega}_{I}}, \tag{10}\] i.e. the principal moment of inertia of a coarse-grained particle about each principal axis $q$ is approximately equal to the equilibrium average of the corresponding principal moment of the fine-grained particles mapped onto it. The AFM-CG method was derived only for the constant-volume conditions of the canonical ensemble, but is straightforwardly generalized to constant-pressure conditions by analogy with the MS-CG method for spherical coarse-grained particles in the isothermal-isobaric ensemble [23]. Thus, the force- and torque-matching conditions at constant pressure are the same as those in Eqs. (6) and (7), except that the coarse-grained forces, torques, and potential are in general functions of the coarse-grained system volume $V$ and the equilibrium average is constrained to configurations in which the fine-grained system volume $v=V$. The coarse-grained potential must also satisfy a virial-matching condition [23], \[W(\mathbf{R}^{N},\mathbf{\Omega}^{N},V) =-\frac{\partial U}{\partial V}\] \[=\left\langle\frac{(n-N)k_{\mathrm{B}}T}{v}+\frac{1}{3v}\sum_{l=1 }^{n}\mathbf{f}_{i}\cdot\mathbf{r}_{i}\right\rangle_{\mathbf{R}^{N},\mathbf{\Omega}^{N},V} \tag{11}\] In summary, for the equilibrium phase-space distribution of the coarse-grained model to match that of the fine-grained model in the isothermal-isobaric ensemble, the coarse-grained potential should satisfy Eqs. (6), (7), and (11), while the coarse-grained masses and principal moments of inertia should satisfy Eqs. (8) and (9), respectively. As shown below, using the more approximate Eq. (10) to parameterize the moments of inertia gives almost the same results as Eq. (9), even for a flexible molecule, so we have used this simpler equation for parameterization later on. ## III Methods ### Force-, torque-, and virial-matching algorithm The analytical expression for the coarse-grain potential $U$ is not usually known. However, an approximation to the functional form can be obtained using a neural-network optimization algorithm with Eqs. (6), (7), and (11) acting as necessary constraints. In general, $U(\mathbf{R}^{N},\mathbf{\Omega}^{N},V)$ is a function of the particle configuration and system volume. In this work, we have assumed that $U$ does not depend explicitly on $V$, in which case [23] \[\frac{\partial U}{\partial V}=\frac{1}{3V}\sum_{l=1}^{N}\frac{\partial U}{ \partial\mathbf{R}_{l}}\cdot\mathbf{R}_{l}. \tag{12}\] With this approximation, the virial-matching condition in Eq. (11) can be written, using $v=V$, as \[-\sum_{l=1}^{N}\frac{\partial U}{\partial\mathbf{R}_{l}}\cdot\mathbf{R}_{l}=\left\langle 3 (n-N)k_{\mathrm{B}}T+\sum_{i=1}^{n}\mathbf{f}_{i}\cdot\mathbf{r}_{i}\right\rangle_{\mathbf{R }^{N},\mathbf{\Omega}^{N},V}. \tag{13}\] Despite this approximation, we show later on that the coarse-grained models parameterized accurately match the average density of the corresponding all-atom fine-grained system at constant pressure. To ensure that all equivalent configurations are assigned the same position in coordinate space, a transformation was made from the set of Cartesian coordinates to a vector $\mathbf{D}_{IJ}$ that was invariant under translation, rotation, and permutation of any pair of coarse-grained particles $I$ and $J$[25, 26, 27, 10], which was defined in terms of the positions, $\mathbf{R}_{I}$ and $\mathbf{R}_{J}$, and orientations, $\mathbf{\Omega}_{I}$ and $\mathbf{\Omega}_{J}$, of the two particles by \[\mathbf{D}_{IJ}=\{R_{IJ},\mathbf{R}_{IJ}\cdot\mathbf{\Omega}_{I,1},\mathbf{R}_{IJ }\cdot\mathbf{\Omega}_{I,2},\mathbf{R}_{IJ}\cdot\mathbf{\Omega}_{I,3},\] \[\mathbf{R}_{IJ}\cdot\mathbf{\Omega}_{J,1},\mathbf{R}_{IJ}\cdot\mathbf{\Omega}_{J, 2},\mathbf{R}_{IJ}\cdot\mathbf{\Omega}_{J,3},\] \[\mathbf{\Omega}_{I,1}\cdot\mathbf{\Omega}_{J,1},\mathbf{\Omega}_{I,1}\cdot\bm {\Omega}_{J,2},\mathbf{\Omega}_{I,1}\cdot\mathbf{\Omega}_{J,3},\] \[\mathbf{\Omega}_{I,2}\cdot\mathbf{\Omega}_{I,1},\mathbf{\Omega}_{I,2}\cdot\bm {\Omega}_{J,2},\mathbf{\Omega}_{I,2}\cdot\mathbf{\Omega}_{J,3},\] \[\mathbf{\Omega}_{I,3}\cdot\mathbf{\Omega}_{J,1},\mathbf{\Omega}_{I,3}\cdot\bm {\Omega}_{J,2},\mathbf{\Omega}_{I,3}\cdot\mathbf{\Omega}_{J,3}\}, \tag{14}\] where $R_{IJ}\equiv\\|\mathbf{R}_{IJ}\\|$, $\mathbf{R}_{IJ}\equiv\mathbf{R}_{I}-\mathbf{R}_{J}$ and $\mathbf{\Omega}_{I}$ and $\mathbf{\Omega}_{J}$ are specified by rotation matrices of the form of Eq. (2). The coordinates of each neighbor within the cut-off distance of particle $I$ were transformed to a $\mathbf{D}_{IJ}$ vector. All the $\mathbf{D}_{IJ}$ vectors for a given neighborhood were concatenated into a 2D matrix of size $N\times\dim(\mathbf{D}_{IJ})$ representing a unique configurational fingerprint for coarse-grained particle $I$. The potential function could then be written in terms of a set of neural network trainable parameters and activation functions transforming $\mathbf{\mathsf{D}}_{I}$ to a potential energy value. While $\mathbf{\mathsf{D}}_{I}$ is a sufficient specification of the coarse-grained coordinates to enforce relevant invariant properties of the molecular environment, it does not possess all the symmetries of the potential energy surface that it aims to fit.[25; 28] For each molecular environment, it was assumed that the interactions were predominantly short-ranged such that neighbors beyond a certain cut-off distance, $R_{\mathrm{c}}$, did not contribute to the potential.[19] This condition can be enforced by a cut-off function of the form \[g_{\mathrm{c}}(R_{IJ})=\begin{cases}\frac{1}{2}\left[\cos\left(\frac{\pi R_{ IJ}}{R_{\mathrm{c}}}\right)+1\right],&R_{IJ}\leq R_{\mathrm{c}},\\ 0,&R_{IJ}>R_{\mathrm{c}}.\end{cases} \tag{15}\] A set of these cut-off functions can enforce the radial symmetry conditions of the underlying potential energy surface by storing information about the radial distribution of neighbors according to[19] \[G_{I}^{\mathrm{j}}=\sum_{J\neq I}g_{\mathrm{c}}(R_{IJ}). \tag{16}\] Continuity of the potential along angular dimensions was ensured by using a compression layer to learn a set of collective variables from vector $\mathbf{D}_{IJ}$ which were constrained by the well-behaved modified $G^{5}$ symmetry function[19] given by \[G_{I}^{5}=\sum_{J\neq I}\prod_{\mu=1}^{M}2^{1-\nu}\left(1+\lambda\cos\theta_{ IJ,\mu}\right)^{\nu}e^{-\eta(R_{IJ}-R_{\mathrm{c}})^{2}}g_{\mathrm{c}}(R_{IJ}), \tag{17}\] where $\lambda\in\{-1,1\}$ and $R_{\mathrm{s}}$, $\nu$, and $\eta$ are tunable hyperparameters, while $\{\cos\theta_{IJ,\mu}\}$, is the set of machine-learned collective variables with the same properties as the angular component of the underlying potential and $M$ is the total number of machine-learned angular variables. These angular symmetry functions store information about the angular-radial distribution of neighbors in the local environment of coarse-grained particle $I$. Unlike the case of spherically symmetric particles, in a local reference frame, a neighboring anisotropic particle requires a minimum of seven independent scalar variables to fully describe its position and orientation. However, previous implementations of analytical potentials, including the Gay-Berne potential,[16; 17] have used fewer coordinates for the calculation of the potential and forces. Similarly, for the neural-network potential, an additional compression layer was included to remove the redundant angles from the $\mathbf{D}_{IJ}$ vectors, since the combination of translation and rotation in 3D is parameterized by at most 7 unique coordinates. The Behler symmetry functions were enforced on the output of the compression layer, ensuring that the learned compression had the same symmetry and continuity of the underlying potential. The reduction in the dimension of $\mathbf{D}_{IJ}$ also decreases the amount of data that is needed to train a sufficiently accurate potential. By removing the redundant angles in $\mathbf{D}_{IJ}$ there is a reduced possibility of over-fitting on a small data set. A set of these symmetry functions with tuned hyperparameters $(\lambda,\nu,\eta,R_{\mathrm{s}},R_{\mathrm{c}})$ can be used to uniquely represent the structural fingerprint of the molecular environment. Symmetry functions used to represent the local environment were constructed using all possible permutations of values from a specified set of hyperparameters. Training of the neural network started with 8 symmetry functions and hyperparameters tuned to minimize the loss function, which is defined below. New symmetry functions were added to the set if they resulted in a significant reduction in the neural-network loss compared with the preceding iteration. The set of hyperparameters in the symmetry functions used in the anisotropic coarse-grained models parameterized in this work can be found in the Supplementary Material. To further reduce the amount of data needed to train the neural network, a prior repulsive potential was defined with pairwise additive properties. This potential was used to ensure physical behavior in regions of the potential where the forces are large and thus are rarely sampled in an equilibrium MD simulation. This prior potential only needs to satisfy two conditions: firstly, it must be repulsive at short radial separations, and, secondly, the position of its repulsive barrier must be orientationally dependent. A simple equation satisfying these conditions is \[U_{\mathrm{prior},I}=\sum_{J\neq I}B_{1}\sigma_{\mathrm{c}}\left(\mathbf{\mathsf{D }}_{I}\right)^{-B_{2}}, \tag{18}\] where $\sigma_{\mathrm{c}}$ is a neural-network compression layer function and $B_{1}$ and $B_{2}$ are strictly positive trainable parameters. It is also possible to achieve a similar large repulsive barrier through a more advanced nonlinear sampling of the MD simulation data. The prior potential fits the purely repulsive part of the angular-dependent potential to the molecular environment, while the neural-network potential fits the attractive and oscillatory corrections to the environment. The final prediction for the potential energy of the environment of coarse-grained particle $I$ is therefore the sum of the neural-network potential $U_{\mathrm{NN},I}$ and the prior repulsive potential $U_{\mathrm{prior},I}$,[15] \[U_{I}=U_{\mathrm{NN},I}+U_{\mathrm{prior},I}, \tag{19}\] and, thus, the total coarse-grained potential is \[U=\sum_{I=1}^{N}U_{I} \tag{20}\] From the matching conditions in Eqs. (6), (7), and (13), optimization of the neural-network weights and biases requires a loss function of the form \[L = \left\langle\sum_{I=1}^{N}\left(\alpha\left\|\mathbf{F}_{\mathrm{FG},I} +\frac{\partial U}{\partial R_{I}}\right\|^{2}+\beta\left\|\tau_{\mathrm{FG},J}+ \sum_{q}\Omega_{I,q}\times\frac{\partial U}{\partial\Omega_{I,q}}\right\|^{2} \right)\right.\] \[+ \left.\gamma\left\|3\left(n-N\right)k_{\mathrm{B}}T+\sum_{I=1}^{N} \left(\Psi_{\mathrm{FG},J}+\frac{\partial U}{\partial\mathbf{R}_{I}}\cdot\mathbf{R}_{I }\right)\right\|^{2}\right\rangle_{R^{N},\Omega^{N},V}, \tag{21}\] where \[\mathbf{F}_{\text{FG},I}\equiv\sum_{i\in\zeta_{I}}\mathbf{f}_{i},\,\tau_{\text{FG},I}\equiv \sum_{i\in\zeta_{I}}\Delta\mathbf{r}_{i}\times\mathbf{f}_{i},\,W_{\text{FG},I}\equiv \sum_{i\in\zeta_{I}}\mathbf{f}_{i}\cdot\mathbf{r}_{i}, \tag{22}\] and $\alpha,\beta$, and $\gamma$ are weights. These weights specify the fraction of each loss that is used for backpropagation and were free to change with the learning rate during optimization.[14] Even though there have been significant efforts to develop methods to fit the constrained averaged coarse-grained forces directly,[29; 30] the average total fine-grained forces subject to the constraint of matching fine-grained and coarse-grained configurations are not easily obtained. An indirect means of minimizing the loss function in Eq. (21) above is possible by replacing the constrained ensemble average with an average over instantaneous unconstrained simulation configurations,[14] \[L_{\text{inst}} =\sum_{t=1}^{N_{\text{t}}}\left[\sum_{I=1}^{N}\left(\alpha\left\| \mathbf{F}_{\text{FG},I}(\mathbf{r}_{t}^{n})+\frac{\partial U(\mathbf{\xi}_{I})}{\partial R _{I}}\right\|^{2}\right.\right.\] \[+\left.\left.\beta\left\|\mathbf{\tau}_{\text{FG},I}(\mathbf{r}_{t}^{n}) +\sum_{q}\Omega_{I,q}(\mathbf{\xi}_{I})\right.\right)\times\frac{\partial U(\mathbf{ \xi}_{I}))}{\partial\Omega_{I,q}}\right\|^{2}\right)\] \[+\left.\left.\gamma\left\|3\left(n-N\right)k_{\text{B}}T+\sum_{I= 1}^{N}\left(\tilde{W}_{\text{FG},I}(\mathbf{r}_{t}^{n})+\frac{\partial U(\mathbf{ \xi}_{I})}{\partial\mathbf{R}_{I}}\cdot\mathbf{R}_{I}(\mathbf{\xi}_{I})\right)\right\|^{2 }\right], \tag{23}\] since it can be shown, for a sufficiently large dataset that comprehensively samples the equilibrium ensemble of the fine-grained system, that $L$ and $L_{\text{inst}}$ have the same global minimum. Here, $N_{\text{t}}$ is the number of simulation configurations in the dataset, $\mathbf{r}_{t}^{n}$ and $v_{t}$ are the fine-grained coordinates and system volume for configuration $t$, and $\mathbf{\xi}_{I}=(\mathbf{R}^{N}(\mathbf{r}_{t}^{n}),\mathbf{\Omega}^{N}(\mathbf{r}_{t}^{n}),V(v_{t }))$ is the mapped coarse-grained configuration for this fine-grained configuration. The loss function was optimized using the minibatch gradient descent as implemented in TensorFlow. The feedforward neural network shown in Fig. 1 was then trained, where the forward propagation used matrix $\mathbf{\mathbbm{D}}_{I}$ as an input to predict the coarse-grained potential $U$, after which TensorFlow's computational derivative was used to calculate the outputs, namely the predicted forces, torques, and virial. In the backpropagation stage, the loss function was used to calculate the error between the true and predicted values, which was then used to update the network weights and biases. The errors between the true and predicted parameters were calculated using TensorFlow's mean squared error, and gradient descent was implemented using TensorFlow's Adam optimizer.[31] Once the error of the neural network was minimized, the neural-network model was used to predict the forces, torques, and virial. However, removing the output and derivative layers gives access to the predicted potential of mean force. By optimizing the partial derivatives of the potential instead of the potential itself, by the nature of the derivative, there will be less oscillation in the potential at the edges of the data set close to the cut-off distances. ### LAMMPS modification and neural network implementation The neural network was constructed in Tensorflow (version 2.3.0)[32] using the Keras (version 2.4.3) functional API[33] and saved using the Tensorflow SavedModel format. The trained neural network was implemented in LAMMPS using the Tensorflow C API and cppflow wrapper. All simulations were carried out using the LAMMPS MD software package (version 20Nov19).[34; 35; 36] The Optimized Potentials for Liquid Simulations-All Atom (OPLS-AA) force field[37; 38; 39; 40] was used for all all-atom simulations with a cut-off distance of 10 A for short-ranged non-bonded interactions; long-ranged electrostatic interactions were calculated with the particle-particle particle-mesh (PPPM) method[36; 41] The bonds that include hydrogen were constrained using the SHAKE algorithm.[42] Simulations were carried out in the isothermal-isobaric (NPT) ensemble at a pressure of 1 atm, with the temperature and pressure controlled by a Nose-Hoover thermostat and barostat.[43; 44] Neural-network training was carried out using data from a 25 ns all-atom simulation in which simulation configurations and forces and velocities were saved at 2 ps intervals. The simulation snapshots from the last 20 ns were shuffled and then divided into 4 groups of equal size, $\{g_{0},g_{1},g_{2},g_{3}\}$. The neural network was initially trained on $g_{0}$ and validated on $g_{3}$. The validation set $g_{3}$ was further divided into an 8:2 ratio where the lesser was reserved as the test set. New snapshots were added from $g_{1}$ and $g_{2}$ if the mean errors of their predicted forces and torques were larger than that of the test set. The accuracy of the trained neural network was then compared to the expected accuracy determined from k-fold cross-validation.[45; 46] During k-fold cross-validation, the last 20 ns of simulation data was shuffled and divided into 10 folds, $\{\psi_{0},...,\psi_{b}\}$. The model was validated on $\psi_{i}$ and trained on $\bigcup_{j\neq i}\psi_{j}$ for all $i,j\in\{0,1,\ldots,9\}$. The loss of the iterative training method was found to be identical to the k-fold cross-validation loss. The coarse-grained simulations were done using a modified version of LAMMPS in which the trained neural network was Figure 1: Schematic of anisotropic force-matching neural network architecture. introduced to calculate the forces and energies. The dimensions of the coarse-grained sites used in the simulations were derived from the inertia tensor of the all-atom model. To test the ability of the coarse-grained model to capture the properties of the all-atom model under a variety of conditions in addition to the single temperature at which the neural network was trained, the equilibrium structural properties of equivalent coarse-grained and all-atom systems were compared in simulations at several different temperatures. In all cases, the total length of the coarse-grained simulation was 25 ns, with the last 20 ns used to calculate structural and thermodynamic properties. ## IV Results and Discussion To demonstrate the flexibility of the method, we have used our neural-network model to construct coarse-grained interaction potentials for benzene, an archetypal anisotropic small molecule, and $\alpha$-sexithiophene, an organic semiconductor with significant applications in organic electronic devices [47, 48, 49] (Fig. 2). These molecules were selected to demonstrate the neural network's ability to handle anisotropic molecules of varying complexity, flexibility, and aspect ratio while still reproducing the structural and phase behavior. The shape of a coarse-grained particle obtained from the anisotropic coarse-graining method is determined by the "average" shape of the fine-grained molecule or molecular fragment that is mapped to it under the parameterization conditions. Thus, the variation of the aspect ratio of the molecule or molecular fragment with temperature in the all-atom simulations can potentially be a qualitative indicator of the temperature transferability of the coarse-grained model. Here, the aspect ratio of the molecule was calculated as the ratio of the length to the breadth of the molecule, where the length was defined as the longest principal axis and the breadth was defined as the sum of the remaining two semi-axes. Unlike benzene, the thiophene-thiophene dihedral angles also have a temperature-dependent effect on the aspect ratio of sexithiophene. Neural networks in general are very good at interpolation but struggle with extrapolation.[50, 51, 52, 53] The accuracy of the model is therefore expected to decrease as the aspect ratio of the molecule deviates from that at the parameterization temperature, as well as when the density distribution is sufficiently different from the parameterization temperature. By parameterizing the systems in the liquid phase, the model can capture a wider variety of fluctuations in the density of the system and the dimensions of the molecules. The average size of a flexible molecule in the isotropic phase will be different from the size of the molecule when locked in a rigid crystal structure.[54, 55] However, this temperature-dependent size difference should decrease with increased rigidity of the molecule. ### Benzene Simulations consisting of 500 benzene molecules were carried out at 280, 300, 320, 330, and 350 K, and the coarse-grained neural-network model was parameterized at 300 K. The time step was 2 fs in the all-atom simulations and 12 fs in the coarse-grained simulations. The cut-off distance hyperparameter $R_{\mathrm{c}}$ was 10 A. The root mean squared validation error for the forces was $2.55\,\mathrm{kcal}\,\mathrm{mol}^{-1}\,\mathrm{\AA}^{-1}$ and that of the torque was $4.35\,\mathrm{kcal}\,\mathrm{mol}^{-1}$. The average post-training error in the pressure was 0.0092 atm. The average principal moments of inertia in the all-atom simulation at 300 K were used to determine the principal moments of the coarse-grained model using Eqn. (10) (values given in the Supplementary Material) since fluctuations in the moments at the parameterization temperature were small.[21] The variation of the molecular aspect ratio of the all-atom benzene model with temperature is shown in Fig. 3. The distribution of possible dimensions observed for benzene is narrow and remains fairly constant with temperature, making benzene an ideal case where molecular flexibility does not contribute significantly to the overall error of the model.[56] Fig. 4 shows that the coarse-grained neural-network model accurately captures the liquid density of the all-atom model over a wide range of temperatures from just above the freezing point to just below the boiling point, with only slight deviations for the temperature furthest from the parameterization temperature. As shown in Fig. 5, the coarse-grained model also accurately predicts the radial distribution function (RDF) of the all-atom model over the same temperature range. To further elucidate the accuracy of the neural-network coarse-grained model, the angular-radial distribution function (ARDF) was analyzed. The ARDF is defined by \[g(r,\theta)=\frac{\langle n(r,\theta)\rangle}{\frac{4}{3}\pi\rho[(r+\Delta r) ^{3}-r^{3}]\sin\theta\Delta\theta}, \tag{24}\] where $\langle n(r,\theta)\rangle$ is the average number of molecules in the spherical shell within the bounds $r$ to $r+\Delta r$ of the center-of-mass of a chosen molecule and having an out-of-plane axis rotation of $\theta$ with respect to the out-of-plane axis of the chosen molecule,[57] and $\rho$ is the bulk number density. Fig. 6 Figure 2: Chemical structures of (a) benzene and (b) $\alpha$-sexithiophene with a coarse-grained ellipsoid superimposed on a possible configuration of each molecule. shows the 2D heatmap of the ARDF along with 1D slices of this function at specific angles at 300 K (the parameterization temperature) for the all-atom and coarse-grained models. The ARDFs at the other simulated temperatures are compared in the Supplementary Material. At all simulated temperatures between 280 and 350 K, the coarse-grained model captures all the major features of the fine-grain structure of the fluid. The only difference is a slight underestimation of the peak heights by the coarse-grained model. The neural-network model is, however, able to more faithfully capture the angular radial distribution of benzene at all temperatures compared with the coarse-grained benzene model previously parameterized with the AFM-CG method using a pair potential to describe the interparticle interactions.[21] This improvement can be attributed to the greater flexibility of the neural-network potential in describing the intermolecular interactions. The neural-network model can demonstrate temperature transferability through careful selection of the neural network hyperparameters to prevent overfitting of the local number density variations. The coarse-grained simulation of anisotropic molecules using a neural-network potential is more suited for large, preferably rigid, molecules, for which a high degree of coarse-graining can be achieved with reasonable accuracy. However, the model was still able to achieve a modest 20$\times$ speedup compared with the atomistic simulations, through a combination of reduced computation time per timestep and a larger timestep. This poor performance for a small molecule such as benzene is due to the small reduction in the number of degrees of freedom from the all-atom model to the coarse-grained model, coupled with a neural-network potential that is more computationally expensive than an analytical potential. Nevertheless, computational savings are obtained even in this suboptimal case. Simulations were carried out on a 4-core Intel i7-4790K CPU, but, further speedups could be achieved by taking advantage of the GPU-enabled version of TensorFlow. Figure 4: Density versus temperature of the all-atom (AA) and coarse-grained (CG) benzene models at 1 atm. Error bars are smaller than the symbols. Figure 5: Radial distribution function (RDF) of the all-atom (solid lines) and coarse-grained (dashed lines) benzene models at 1 atm and various temperatures. The RDFs have been shifted vertically for clarity. Figure 3: Length-to-breadth ratio of the all-atom benzene model at 1 atm and various temperatures. ### Sexithiophene Simulations of 512 sexithiophene molecules were carried out at 570, 590, 640, and 680 K temperatures, corresponding to temperatures previously identified in all-atom MD simulations to correspond to crystalline (K), smectic-A (Sm-A), nematic (N), and isotropic (I) phases respectively.[58] The time step was 1 fs in the all-atom simulations and 12 fs in the CG simulations. Although we have used the OPLS-AA force field for our all-atom simulations, whereas these previous MD simulations[58] used the related AMBER force field[59; 60; 61] the structural properties of systems simulated with these two force fields (in particular the density, orientational order parameter, and radial distribution function discussed below) are very similar for the temperature range studied. The cut-off distance hyperparameter $R_{\mathrm{c}}$ was set to 21 A. The neural-network model was parameterized using simulation snapshots from the isotropic phase at 680 K, where the molecular mobility was highest. The conditions of the isotropic bulk phase are advantageous in efficiently sampling the configuration space, especially rare high-energy configurations necessary for the accurate reproduction of the repulsive part of the coarse-grained potential. As shown in Fig. 7a, the distributions of the principal moments of inertia of sexithiophene in the all-atom simulation at the parameterization temperature are broad, indicating that Eqn. (10) may not be adequate for parameterizing the moments of inertia of the coarse-grained model. However, we found that using the more general Eqn. (9) to parameterize the coarse-grained moments of inertia (by fitting the distributions in Fig. 7b-d gave values within <1%. So we used the values from Eqn. (10) in the coarse-grained model. The root mean squared validation error for the sexithiophene forces was $3.95\,\mathrm{kcal\,mol^{-1}\,\AA^{-1}}$ and that of the torque was $9.8\,\mathrm{kcal\,mol^{-1}}$. The sexithiophene final force and torque losses were larger than those of benzene because the model was not complex enough to account for the bending of the polymer and the rotation of the individual thiophene rings. The loss is also skewed to larger values when compared with benzene because sexithiophene is a larger molecule and so the interactions between molecules are stronger overall. The structural properties of the coarse-grained model were compared with those of its all-atom counterpart at each of the simulated temperatures. The nonlinear change in density with respect to temperature is associated with the phase changes that occur at the simulated temperatures (Fig. 8).[58] The density of the coarse-grained system agrees well with that of the all-atom system, with minimal deviations from the fine-grained system with increasing distance from the parameterization temperature. Compared with benzene, sexithiophene has a much larger change in density between the crystalline and the isotropic phase. This difference results in less overlap between the local density variations in the crystalline phase at the lowest temperature and the training data set in the isotropic phase at the highest temperature. The sexithiophene molecule is also much more flexible than benzene, as seen in the wide distribution of the aspect ratio in the all-atom model at all the simulated temperatures shown in Fig. 9, and its dimensions change significantly with temperature over the range studied. Another limitation of representing sexithiophene as a single-site ellipsoid is the loss of thiophene-thiophene torsional information. That is, for any given position and orientation of the coarse-grained ellipsoid there are multiple different relative orientations between the thiophene groups.[62] This loss of information is significant because the anisotropic interactions of the thiophene subunits are lost, which reduces the neural network's ability to isolate which of the two short axes corresponds to the $\pi$-stacking direction. To further confirm that the density changes were associated with transitions from the crystalline phase through the nematic Figure 6: Angular–radial distribution function (ARDF) of the all-atom (AA) (top) and coarse-grained (CG) (bottom) benzene models at 300 K and 1 atm depicted as a heat map (left) and 1D slices at constant angle (right). Face-to-face, edge-to-edge, and parallel displaced configurations occur when the angle is 0${}^{\circ}$, while edge-to-face configurations occur at 90${}^{\circ}$. and smectic phases to the isotropic phase, the scalar orientational order parameter $P_{2}$ was introduced. For a given simulation snapshot at time $t$, $P_{2}$ can be found by diagonalizing the ordering matrix \[\mathbf{Q}=\frac{1}{2N}\sum_{I=1}^{N}{(3\mathbf{u}_{I}\otimes\mathbf{u}_{I}-\mathbf{E})}, \tag{25}\] Figure 8: Density versus temperature of the all-atom (AA) and coarse-grained (CG) sexithiophene models at 1 atm. Error bars are smaller than the symbols. Figure 7: (a) Principal moment of inertia distributions for all-atom (AA) sexithiophene model at 680 K and 1 atm. The corresponding angular velocity distributions of each principal axis along with the coarse-grained (CG) fit to the distribution given by Eq. (9) is shown in (b)–(d). Figure 9: Length-to-breadth ratio of all-atom sexithiophene model at 1 atm and various temperatures. The simulated phase is given in parentheses after each temperature in the legend (I = isotropic, N = nematic, SmA = smectic A, K = crystal). where $\mathbf{u}_{I}$ is the unit vector along the molecular axis and $\mathbf{E}$ is the identity matrix. $\langle P_{2}\rangle$ is the average over the largest eigenvalue of this matrix for all snapshots of equilibrium configurations.[58] Larger values of the scalar orientational order parameter close to one indicate an ordered crystalline structure while values close to zero correspond to an isotropic disordered phase. The coarse-grained model reproduces the orientational order parameter of the all-atom model reasonably well over the temperature range simulated, as shown in Fig. 10. The coarse-grained model underestimates the degree of orientational ordering observed in the all-atom model away from the parameterization temperature, likely because it does not capture the increasing molecular shape anisotropy that is observed in the all-atom model as the temperature decreases (Fig. 9). As expected, the largest difference occurs in the predicted crystalline phase. The same trend is seen in the radial distribution functions shown in Fig. 11, in which the agreement between the coarse-grained and all-atom models at most temperatures is excellent, with the largest deviations for the crystalline phase. The underestimation and broadening of the peaks in the crystalline radial distribution function explain the discrepancy between the order parameter of the all-atom and coarse-grained models. The observed differences are most likely due to the effect on molecular packing of the aforementioned discrepancy in molecular shape between the two models as temperature decreases.[63] Nevertheless, even in the crystalline phase, the coarse-grained model captures the peak positions of the radial distribution function very well. The coarse-grained model also accurately describes orientational correlations in condensed-phase sexithiophene, as illustrated by a comparison with the angular-radial distribution function of the all-atom model. At the parameterization temperature, the coarse-grained model is able to capture all major features when compared to the all-atom model (Fig. 12). The neural-network model is also able to capture the relevant features in the structure of sexithiophene's smectic liquid-crystal phase at 590 K, as shown in Fig. 13. The discrepancies in the width and height of the peaks are likely due to the differences in molecular shape away from the parameterization temperature that was mentioned earlier. The ARDFs of the two models in the nematic phase at 640 K are compared in the Supplementary Material and show similarly good agreement. Despite sexithiophene not strictly meeting the conditions to be coarse-grained to a single anisotropic particle due to its significant flexibility, the coarse-grained neural-network model is still able to reproduce its condensed-phase structural properties and phase behavior with remarkable accuracy. The limitation of the single-site model is only evident under conditions where the conformation of the molecule is highly temperature-dependent. One way to construct a neural network model that is independent of temperature would be to extract the training data from multiple temperatures and define the molecular dimensions as the average over the crystalline and isotropic phases. While the results for sexithiophene are substantially better than expected given its flexibility, improvements can be made to the model by considering a coarse-grained mapping consisting of more than one site.[64] The coarse-grained simulation of sexithiophene demon Figure 11: Radial distribution function (RDF) of the all-atom (solid lines) and coarse-grained (dashed lines) sexithiophene models at 1 atm and various temperatures. The RDFs have been shifted vertically for clarity. The simulated phase is given in parentheses after each temperature in the legend (I = isotropic, N = nematic, SmA = smectic A, K = crystal). Figure 10: Orientational order parameter versus temperature for the all-atom (AA) and coarse-grained (CG) sexithiophene models at 1 atm. Typical simulation configurations are shown at each temperature for each system (AA model above the data points and CG model below), in which the molecules have been colored according to their orientation with respect to the phase director (blue = parallel, red = perpendicular). Error bars are smaller than the symbols. strated a speed-up of 132$\times$ compared with the all-atom simulation using the same hardware employed for the benzene simulations. This speedup is primarily due to the large reduction in the number of degrees of freedom in coarse-graining this molecule. ## Conclusions We have applied machine learning and a recently derived systematic coarse-graining method for anisotropic particles to develop a single-site anisotropic coarse-grained potential of a molecular system. The iterative training of the neural-network potential is able to reproduce the forces, torques, and pressure of the fine-grained all-atom system. The final loss of the it Figure 12: Angular-radial distribution function (ARDF) of the all-atom (AA) (top) and coarse-grained (CG) (bottom) sexithiophene models at 680 K and 1 atm (isotropic phase) depicted as a heat map (left) and 1D slices at constant angle (right). Face-to-face, edge-to-edge, and parallel displaced configurations occur when the angle is 0${}^{\circ}$, while edge-to-face configurations occur at 90${}^{\circ}$. Figure 13: Angular-radial distribution function (ARDF) of the all-atom (AA) (top) and coarse-grained (CG) (bottom) sexithiophene models at 590 K and 1 atm (smectic phase) depicted as a heat map (left) and 1D slices at constant angle (right). Face-to-face, edge-to-edge, and parallel displaced configurations occur when the angle is 0${}^{\circ}$, while edge-to-face configurations occur at 90${}^{\circ}$. erative training model was identical to the loss obtained from k-fold cross-validation. The CG model performs well for a rigid molecule like benzene but, remarkably, it also describes the phase behavior and molecular-scale structural correlations of a flexible molecule like sexithiophene with comparable accuracy, even though the aspect ratio of the molecule changes significantly over the simulated temperature range. We have demonstrated the versatility of the coarse-graining method by parameterizing models of benzene and sexithiophene at a single temperature and then studying their accuracy in capturing the structural properties of the corresponding all-atom model at different temperatures. The sexithiophene model was also used to show the ability of the model to reproduce the phase behavior of the all-atom model, with the lowest fidelity coming from the crystalline phase, where the aspect ratio of the molecule had the largest deviation from the parameterization data set. A natural extension to this work would be to generalize the method to a multi-site anisotropic coarse-grained model for flexible molecules and polymers. ## Author's contributions MOW: Investigation, methodology, formal analysis, visualization, writing - original draft. DMH: Conceptualization, methodology, supervision, writing - review and editing. ## Acknowledgments This work was supported by the Australian Research Council under the Discovery Projects funding scheme (DP190102100). This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government, and from the University of Adelaide's Phoenix High-Performance Computing service. ## Author declarations ### Conflict of Interest The authors have no conflicts to disclose. ## Data availability The Supplementary Material provides more extensive structural comparisons between the all-atom and coarse-grained simulations, more details on the methods used for the calculations, and a list of the software needed to implement and train the neural-network potential, along with links to their GitHub repositories. Simulation files and neural network scripts can be found at [https://doi.org/10.25909/21218057.v1](https://doi.org/10.25909/21218057.v1) ## References * Butler _et al._ (2020)K. T. Butler, D. W. Davies, H. Cartwright, O. Isayev, and A. 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Wilson and David M. Huang _Department of Chemistry, School of Physics, Chemistry and Earth Sciences,_ _The University of Adelaide, Adelaide, South Australia 5005, Australia_ ###### Contents * SI. Benzene network parameters * S2. Additional benzene structural distributions * S3. Sexithiophene network parameters * SIV. Additional sexithiophene structural distributions * SV. TensorFlow and LAMMPS implementation requirements Benzene network parameters The cut-off distance $R_{\rm c}$ for the benzene neural network was set at 10 A for the $G^{1}$ type symmetry function. For the $G^{5}$ angular symmetry function, $\lambda$ had values of -1 and 1 and $\nu$ has values of $2^{n}$, where $n\in\mathbb{Z}$. Hyperparameters $\alpha$, $\beta$, and $\gamma$ in the loss function were adjusted to improve the speed of convergence, but this did not usually affect the global minimum of the optimization when the number of training epochs was large. ## Sii Additional Benzene Structural Distributions Figure S2: Angular–radial distribution function (ARDF) of the all-atom (AA) (top) and coarse-grained (CG) (bottom) benzene models at 280 K and 1 atm depicted as a heat map (left) and 1D slices at constant angle (right). Face-to-face, edge-to-edge, or parallel displaced configurations occur when the angle is 0${}^{\circ}$, while edge-to-face configurations occur at 90${}^{\circ}$. Figure S4: Angular–radial distribution function (ARDF) of the all-atom (AA) (top) and coarse-grained (CG) (bottom) benzene models at 330 K and 1 atm depicted as a heat map (left) and 1D slices at constant angle (right). Face-to-face, edge-to-edge, and parallel displaced configurations occur when the angle is 0${}^{\circ}$, while edge-to-face configurations occur at 90${}^{\circ}$. ## Siii Sexithiophene network parameters The cut-off distance $R_{\rm c}$ for the sexithiophene neural network was set to 21 A for the $G^{1}$ type symmetry function. For the $G^{5}$ angular symmetry function $\lambda$ had values of -1 and 1, $\nu$ has values of $2^{n}$ where $n\in\mathbb{Z}$. Hyperparameters $\alpha$, $\beta$, and $\gamma$ in the loss function were adjusted to improve the speed of convergence but did not usually affect the global minimum of the optimization when the number of training epochs was large. ## Sv Tensorflow and Lammps implementation requirements The following list of software is needed to train and use the neural network model in coarse-grained simulations. 1. Tensorflow C API: [https://github.com/tensorflow/tensorflow/blob/master/tensorflow/c/c_api.h](https://github.com/tensorflow/tensorflow/blob/master/tensorflow/c/c_api.h) 2. Cpp Flow: [https://github.com/serizba/cppflow](https://github.com/serizba/cppflow) 3. Tensorflow Python: [https://github.com/tensorflow/tensorflow](https://github.com/tensorflow/tensorflow) 4. Keras: [https://github.com/keras-team/keras](https://github.com/keras-team/keras) 5. LAMMPS: [https://github.com/lammps/lammps](https://github.com/lammps/lammps) The training and testing of the neural network potential was done with TensorFlow in Python using the Keras functional API. The force and torque calculations were obtained through TensorFlow's Gradient Tape feature, which provides computational derivatives with respect to the network parameters. The tanh activation function was used for all standard neural network layers except the output layer since the tanh activation produced a smooth differentiable potential energy surface. The mean squared error was used when calculating the loss for the forces, torques, and virials. The Adam optimizer[1] was used as the gradient descent algorithm since it was able to reach the global minimum without manually updating the learning rate during training. The machine-learning potential was deployed with the TensorFlow C API and Cpp Flow wrapper. Cpp Flow allows the TensorFlow C model to be accessed directly as a force and torque calculator in a LAMMPS pair-style function.
2303.08109	Vision-based route following by an embodied insect-inspired sparse neural network	We compared the efficiency of the FlyHash model, an insect-inspired sparse neural network (Dasgupta et al., 2017), to similar but non-sparse models in an embodied navigation task. This requires a model to control steering by comparing current visual inputs to memories stored along a training route. We concluded the FlyHash model is more efficient than others, especially in terms of data encoding.	Lu Yihe, Rana Alkhoury Maroun, Barbara Webb	2023-03-14T17:44:23Z	http://arxiv.org/abs/2303.08109v2	# Vision-based route following by an embodied insect-inspired sparse neural network ###### Abstract We compared the efficiency of the FlyHash model, an insect-inspired sparse neural network (Dasgupta et al., 2017), to similar but non-sparse models in an embodied navigation task. This requires a model to control steering by comparing current visual inputs to memories stored along a training route. We concluded the FlyHash model is more efficient than others, especially in terms of data encoding. ## 1 Introduction The ability of animals to follow a familiar route is one of their critical navigation skills, which enables them to forage, to escape, to home, or to migrate, without exploring unknown, potentially dangerous, territories. Despite their tiny brains, desert ants are capable of vision-based route following. Based on a brain structure, called mushroom body (MB), which is common across many insect species, Ardin et al. (2016) developed a computational model, which can learn a training route in a one-shot manner, and subsequently follow the route reliably. This MB-based model is fundamentally a sparse, shallow neural network performing unsupervised learning of retinotopic images. After training, it succeeds in route following by selecting the direction associated with the most familiar visual input (Figure 1A). Inspired by the same insect MB structure, Dasgupta et al. (2017) proposes the FlyHash model, which performs locality-sensitive hashing (LSH). In contrast to classical hashing which aims to minimise encoding duplication amongst inputs (even if two inputs are similar), LSH algorithms assign similar hashes to similar inputs to support efficient similarity search in large databases by approximating nearest neighbour search (Andoni & Indyk, 2008). Since typical LSH hashes have smaller dimensionality than their inputs, LSH can be considered as a dimensionality reduction technique. Unlike other techniques that exploit data extensively, e.g., to train an autoencoder, to compute orthonormal basis in principle component analysis, before they can be used for dimensionality reduction, LSH algorithms take advantage of randomly initialised functions to encode inputs as hashes. Typically, such encoding functions stay unchanged and data-independent after initialisation. The FlyHash model effectively formalises the visual encoding process previously used in the MB-based route following model, as the same neural network is considered (Figure 1B). A sensory signal received by the input layer, composed of projection neurons (PNs), is encoded as a hash, represented by binarised activation of Kenyon cells (KCs). Two features of this MB-inspired neural network make the FlyHash model distinct from conventional LSH algorithms. Firstly, an input is encoded as a higher-dimensional sparse hash, Secondly, the PN-KC connectivity, defining the hash function, is sparse and binary. Since it is beneficial for an intelligent agent, either biological or artificial, to encode and store memory efficiently and effectively for visual recognition of a familiar route, it might seem the smaller hash produced by a conventional LSH technique would be more appropriate for such tasks. Nevertheless, the FlyHash model suggests an alternative, seemingly counter-intuitive solution, which appears to be what is used by the real insect MB. Therefore, we deployed the FlyHash model on a virtual robot in a realistic simulation environment, along with two alternative encoding algorithms, in order to compare their model efficiency in a task-relevant manner. ## 2 Methods ### Embodied models In this manuscript the embodied models are composed of 3 components, a data encoder, a memory storage, and a steering controller (Figure 1A). As we compare only the models for data encoding, while keeping the other 2 components identical, we use the same name for an embodied robot model and the model for date encoding deployed on the robot. ### Models for data encoding We consider 3 single-layer neural network models (same or similar to Figure 1B) for data encoding, and their encoding functions can be formulated as, \[\mathbf{y}=f_{\mathbf{W}}(\mathbf{x})=f(\mathbf{W}\cdot\mathbf{x}), \tag{1}\] where $\mathbf{x}$ denotes an input vector of length $N_{\text{PN}}$, and $\mathbf{y}$ the encoded output of dimension $N_{\text{KC}}$. Note that the activation function $f$, the connection matrix $\mathbf{W}$ and the encoding matrix $f_{\mathbf{W}}(\mathbf{x})$ are model specific, but not plastic. The FlyHash model is characterised by its sparse and binary $\mathbf{W}$ forming an expansion structure, i.e., $N_{\text{PN}}<N_{\text{KC}}$(Dasgupta et al., 2017). $\mathbf{W}$ is initialised by randomly generating its weights independently and identically from a Bernoulli distribution, $W_{ij}\sim\mathcal{B}(\theta)$, where $i\in\{1,2,\ldots,N_{\text{PN}}\},\ j\in\{1,2,\ldots,N_{\text{KC}}\}$, and $\theta\in(0,1)$ controls the sparsity of $\mathbf{W}$. $f$ performs $k$-winners-take-all ($k$-WTA), which binarises $\mathbf{W}\cdot\mathbf{x}$, and produces $\mathbf{y}$ of sparsity, $\kappa=k/N_{\text{KC}}\in(0,1)$. The conventional LSH model typically uses dense and non-binary $\mathbf{W}$(Indyk and Motwani, 1998). It is initialised by randomly generating its weights independently and identically from a standard Gaussian distribution, $W_{ij}\sim\mathcal{N}(0;1)$. $f$ is essentially the Heaviside step function. Consequently, conventional LSH hashes are binary but not sparse ($\kappa=0.5$ on average). The perfect memory model does not encode inputs. Instead, vertical inputs are stored in training sessions, and evaluated in tests, i.e., $\mathbf{y}=\mathbf{x}$. For consistency, we can still define an identity operator, $f_{\mathbf{W}}=\mathbf{I}$, to be the model's encoding function. As in Ardin et al. (2016), the perfect memory model serves as a benchmark model in the route following task. Figure 1: A: A diagram of an embodied model for vision-based route following. Visual memory is stored only in training sessions, and retrieved for novelty evaluation only in tests. B: An insect mushroom body (MB)-inspired neural network. The projection neurons (PNs) receive sensory inputs, and pass them onto the Kenyon cells (KC) for data encoding. In the real MB structure, and typically in the FlyHash model, the number of KCs is larger than the number of PNs, i.e., $N_{\text{KC}}>N_{\text{PN}}$, and their connectivity is sparse, as is the proportion of KCs activated for encoding. Here, we also explore dense connectivity and non-sparse coding. ### Memory storage and novelty evaluation While the perfect memory model simply stores visual inputs without any encoding, both the FlyHash model and the conventional LSH model store hashes only. Consequently, to evaluate visual novelty $d$, the dissimilarity metric $D$ is either the Hamming distance for binary hashes, or the Euclidean distance for images. Since there are multiple memory items stored from a training session, the visual novelty of an input is computed as \[d(\mathbf{x})=\min_{\mathbf{y}\in\mathbb{M}}D\left(f_{\mathbf{W}}(\mathbf{x}),\mathbf{y}\right), \tag{2}\] where $\mathbb{M}$ is the set of all stored memory items. According to the theory of LSH, two sufficiently similar images should share the same hash, given the same $f_{\mathbf{W}}$. The same number of memory items is considered for the 3 models, so that differences in their behaviour and performance are determined mainly by the quality of memory encoding, and their model size only by the size of individual memory items. ### Insect-inspired model for autonomous robot steering Using a well-established insect central complex model (Stone et al., 2017), which accounts for insect steering (and other navigation skills) by an anatomically realistic neural network, Wystrach et al. (2020) shows that insect locomotion can be robustly stabilised to follow straight trajectories. The key idea is that novelty evaluation (treated as an idealised function in Wystrach et al. (2020)) occurs in a pair of MB structures with their inputs lateralised to left and right respectively, thus their relative outputs inform the central complex model of the correct turning direction to bring the insect back in line with the route. Here we use a functionally similar but structurally simpler motor controller for autonomous robot steering, which modulates the locomotion direction based on realistic lateralised visual inputs (see Section 3.1 for our implementation). Specifically, a data encoding model stores visual memory in training sessions, and at any moment of autonomous control it computes left and right visual novelty $d_{\text{L}}$ and $d_{\text{R}}$ of left and right visual inputs by Equation (2). The motor controller then determines the robot's angular speed $\omega$ according to the novelty difference. Specifically, \[\omega=\alpha\cdot\frac{d_{\text{L}}-d_{\text{R}}}{d_{\text{L}}+d_{\text{R}}}, \tag{3}\] where $\alpha>0$ is a predetermined constant. The normalisation term $d_{\text{L}}+d_{\text{R}}$ is used to remove the effect of the number of learned views. ## 3 Experiment ### Simulation environment: iGibson The simulated experiments were conducted in iGibson, an environment offering realistic visual rendering and physics simulation (Shen et al., 2021). Our models were deployed on the two wheeled Freight robot, which was running in the Rs_int scene (Figure 2A). While there is built-in noise in the simulated physics of iGibson, its default value caused negligible effects on our simulations. Raw image were taken by the RGB-D camera on the robot was downsampled to the size of $33\times 33$ pixels, transformed to greyscale by the standard colour converting method from the OpenCV library, and blurred by averaging pixels in every $7\times 7$-sized boxes. The depth information was discarded. We found empirically that such grey and small (but not too small) images were sufficient for our models to achieve good performance, while permitting faster and cheaper simulations. Such low resolution vision is also consistent with insect compound eye optics. The image was cropped into left, middle and right visual fields (Figure 2B). The sizes of the 3 fields were identically $22\times 33$ pixels. The middle field was used as the input to be stored in visual memory during training. The left and the right fields were used as inputs in test sessions for computing visual novelty, as described in Section 2.4. The robot was controlled by commands in terms of linear speed $v$ and angular speed $\omega$ of its wheel joints. While $v$ and $\omega$ was manually specified in training sessions, $\omega$ was determined by our models autonomously using Equation (3) with $\alpha=1$ in tests. For simplicity, $v$ in the training and the test sessions were set to be constants, $v_{\text{train}}=0.5$ and $v_{\text{test}}=0.2$.1 Since the robot deviated more from the training route when it moved at higher speeds, we set $v_{\text{test}}=0.2$ to ensure that the model performance depends only on the effectiveness of the data encoding. Footnote 1: We note that the unit of the robot’s linear speed (omitted elsewhere for brevity) is $2\pi\times 0.0613\approx 0.3852$ meter per second, as the robot’s wheel is $0.0613$ meter in radius. ### Model specification The number of PNs was the same as the number of pixels, i.e., $N_{\text{PN}}=726$, and the values in an input vector $\mathbf{x}$ were simply the greyscale values of the pixels. We varied the number of KCs, i.e., the hash length $N_{\text{KC}}$, to compare models of different sizes (summarised in Table 1), because the overall model size is asymptotically dominated by $N_{\text{KC}}$, as a model stores more memory items $\mathbf{y}$,. In the FlyHash model, the PN-KC connectivity matrix $\mathbf{W}$ was sparse with $\theta=10/N_{\text{PN}}$; on average a KC was connected to $10$ PNs. We consider 3 sparsity levels for $\mathbf{y}$, $\kappa=0.05,0.1$ and $0.5$. Note that the model is biologically realistic when $\theta=10/N_{\text{PN}}$ and $\kappa=0.05$ or $0.1$(Lin et al., 2014). When a hash $\kappa=0.5$, $\mathbf{y}$ was not sparse, but as dense as that in the conventional LSH model. In $\mathbf{W}$ and $\mathbf{y}$, each binary element was assumed to occupy 1 bit, a greyscale value 8 bits, and a real number 64 bits.2 \begin{table} \begin{tabular}{c l l l l} & & \multicolumn{2}{c}{FlyHash} & \multicolumn{2}{c}{conventional LSH} & \multicolumn{2}{c}{perfect memory} \\ \hline \multirow{2}{}{$\mathbf{W}$} & data type & binary & real (float) & N/A \\ & dimension & $N_{\text{PN}}\times N_{\text{KC}}$ (sparse) & $N_{\text{PN}}\times N_{\text{KC}}$ (all-to-all) & N/A \\ & size* (bit) & $N_{\text{PN}}\times N_{\text{KC}}$ & $64\times N_{\text{PN}}\times N_{\text{KC}}$ & $0$ \\ \hline \multirow{2}{}{$\mathbf{y}$} & data type & binary & binary & greyscale (uint8) \\ & dimension & $N_{\text{KC}}$ (sparse) & $N_{\text{KC}}$ (dense) & $N_{\text{PN}}$ (dense) \\ & size* (bit) & $N_{\text{KC}}$ & $N_{\text{KC}}$ & $8\times N_{\text{PN}}$ \\ \hline \end{tabular} \end{table} Table 1: Model comparison in terms of data storage size Figure 2: Snapshots of the Freight robot running in the Rs_int scene. A: A bird’s-eyed view. B: A raw image taken by the robot camera. Visual inputs from the middle field (bounded by the red, solid lines) were stored by the models, visual inputs from the left field (bound by the green, dotted lines) and the right field (bounded by the blue, dashed lines) were used for computing visual novelty. ### Task: route following An experimental trial was composed of a training session and a test session, and all trials were independent from one another. The robot was trained under manual control along a predetermined route, which was designed arbitrarily by the authors with the only constraint that no collisions were to occur in training sessions. 25 images along the route were taken every 0.5 seconds (in simulation time), and stored by the model. In the subsequent test session, the robot started from the same position, heading the same direction, as in the training session, but was now under full control of the model. The goal was to follow the training route as far as possible. Note that a collision event could and did occur under autonomous control in a trial as the result of poor memory encoding, and would be severely detrimental to the overall route following performance. ## 4 Results ### Model comparison: route following performance We first checked the perfect memory model, which could reliably achieve one-shot vision-based route following. For instance, its sample trajectories from 10 trials overlay on one another in Figure 3A (labelled 'No KC'). It is clear in Figure 3A that the length of has had a significant impact on the route following behaviour of the FlyHash model, To quantify such results, we measured the final distance between the final positions of the training and the test trajectories in individual trials, and considered a trial successful if the final distance was less than meters. We consider this criterion a reasonable proxy measure for the overall route following performance, because the final positions either clustered near that of the training route when the route was followed, or distributed widely when the robot deviated too much and became lost (Figure 3B). While the perfect memory model reached perfect performance effectively with 726 8-bit KCs (i.e., $5808$ bits in total), the conventional LSH model and the FlyHash model with dense hashes ($\kappa=0.5$) required $16000$ binary KCs, and the FlyHash models with biologically realistic, sparse hashes ($\kappa=0.05$ and $0.1$) $32000$ binary KCs, to achieve the same (Figure 4A). If $10\%$ failure was acceptable, Figure 3: Route following behaviour of the FlyHash model with hash sparsity, $\kappa=0.1$. A: Sample trajectories of 10 trials for different (coloured). In the case of no KC, the perfect memory model was used instead. B: Final positions of 100 trials. All training routes (in black) started at the same position, $(x,y)=(-0.20,1.00)$, overlaid on one another due to negligible simulation noise, and ended at the same position, approximately $(x,y)=(-3.07,-0.72)$. The results of the conventional LSH model were qualitatively similar, and the plot is thus omitted. Note that it was impossible for the robot to take a shortcut between the start and the final positions due to obstacles. A video demo of the embodied perfect memory model can be found at [https://youtu.be/EdaQrmlruQQ](https://youtu.be/EdaQrmlruQQ). the models with sparse hashes would require only $8000$ KCs, and those with dense hashes $4000$ KCs, which were more comparable to the perfect memory model in terms of the size of individual memory items. The performance was reduced more as $N_{\text{KC}}$ decreased, probably because a smaller $N_{\text{KC}}$ was more likely to introduce hash collisions for dissimilar inputs (violating the principle of LSH), producing ill-computed visual novelty that disrupted the route following behaviour. The performance deterioration seemed relatively severe for the FlyHash model with the sparsest hashes ($\kappa=0.05$) for an intermediate number of KCs ($2000\leq N_{\text{KC}}\leq 8000$). Interestingly, while the conventional LSH model was slightly more advantageous than the FlyHash model with $\kappa=0.5$ for $N_{\text{KC}}\geq 4000$, it was outperformed by the FlyHash model with a performance gap up to $10\%$ when $N_{\text{KC}}$ was smaller. ### Model comparison: model size and run-time cost The above observation of the FlyHash model is particularly intriguing to us, because using dense hashes, rather than sparse encoding as in real insect MBs, yielded the best performance. We note that it is possible to compress the sparse hashes, but not the dense ones, in a lossless manner by exploiting their sparsity. According to Shannon's source coding theorem, the lower bound of the code length for compression is determined by the entropy of the source code, \[H(\text{y})=-\sum_{i}p_{i}\log p_{i}, \tag{4}\] where each bit y in a sparse hash in the FlyHash model followed a Bernoulli distribution, $\text{y}\sim\mathcal{B}(\kappa)$. Therefore, a sparse hash with $\kappa=0.1$ or $0.05$ could be compressed in lossless manner at most to approximately $0.4690N_{\text{KC}}$ or $0.2864N_{\text{KC}}$ bits, respectively, whereas the hashes in the FlyHash model with $\kappa=0.5$ or the conventional LSH model could not be losslessly compressed. As a result, despite different $\kappa$, the FlyHash model could achieve a performance level similar to the conventional LSH model with comparable memory sizes (Figure 4B). Considering additionally that its $\mathbf{W}$ was also sparse and binary (see Table 1 for detailed comparison), the FlyHash model could be more efficient than the conventional LSH model in terms of overall model size. The sparse and binary $\mathbf{W}$ further implies that fewer run-time operations were required (Table 2). To compare the encoding costs of the FlyHash model and the conventional LSH model, it is straightforward to refer to the time complexity of the best sorting algorithm (for $k$-WTA) and matrix multiplication (between an $m\times n$ matrix and an $n\times p$ matrix), i.e., $O(n\log n)$ versus $O(mnp)$, where Figure 4: Route following performance modualated by model size. A: The relation between task performance and $N_{\text{KC}}$. B: The relation between the same task performance and the average entropy of a stored memory item. $m=N_{\text{PN}},n=N_{\text{KC}}$ and $p=1$. Since $\log N_{\text{KC}}\ll N_{\text{PN}}$ in all our models, the FlyHash model is guaranteed to be more efficient than the conventional LSH model in terms of run-time cost. Although processing input images directly allowed the perfect memory model to avoid any encoding costs, the computation of visual novelty in terms of the Euclidean distance between an images was less efficient than finding the Hamming distance between hashes in the other 2 models. ## 5 Discussion In this work in progress, we investigated the data encoding and storage efficiency of the FlyHash model, which uses an insect-inspired sparse neural network to encode data as sparse hashes, by deploying it on a virtual robot in a vision-based route following task. We compared it to the conventional LSH model, which shares many structural and functional similarities with the FlyHash model, except using a dense neural network and dense hashes. The route following performance of the models was positively correlated with their model sizes, and they could achieve perfect performance given sufficiently large memory. While the conventional LSH model outperformed the FlyHash model given the same model size, the FlyHash model with biologically realistic hashes could be largely but losslessly compressible due to its sparsity, which was impossible for the conventional LSH model. After compression, the 2 models could indeed be comparable in model size at a similar route following performance level. In addition, the FlyHash model with biologically unrealistic dense hashes was more efficient in terms of run-time cost than the conventional LSH model, due to its sparse and binary PN-KC connectivity. In summary. we considered the FlyHash model generally more efficient, mainly because it required a lower run-time costs. ### Limitations and future work The main limitation of the current work is that our model compression analysis was based on theoretical low bound; an explicit, efficient method to achieve the low bounds might not be easily accessible. For instance, the compressed sparse row format would be a data structure naturally compatible with our implementation for both the sparse connectivity matrix and the sparse hashes. The format will require storing $\kappa\times N_{\text{KC}}$ indices for every $\mathbf{y}$, occupying $0.8N_{\text{KC}}$ bits if uint16 is used for the $\mathbf{y}$ indices in the FlyHash model with $N_{\text{KC}}=32000$ and $\kappa=0.05$, much more than the theoretical lower bound ($0.2864N_{\text{KC}}$ bits). An alternative solution would be to store implicitly visual information in plastic neural weights in learning, instead of storing memory items explicitly and computing visual novelty with respect to all the items. With an additional output layer of even one neuron connecting to all the KCs, it should be sufficient for our models to accomplish visual route following as in Arth et al. (2016), because both our approaches rely on visual novelty detection, despite the different steering models. We are planning to add this layer, and we expect the modified FlyHash model to remain the most efficient for 2 reasons. Firstly, its encoding matrix is sparse and binary (as discussed in this manuscript). Secondly, the modified model would effectively be able to store more images (implicitly), because \begin{table} \begin{tabular}{c l l l l} Phase & Operation & FlyHash & conventional LSH & perfect memory \\ \hline \multirow{2}{}{encoding} & multiplication & $0$ & $N_{\text{PN}}\times N_{\text{KC}}$ & $0$ \\ & addition & $9\times N_{\text{KC}}$ & $(N_{\text{PN}}-1)\times N_{\text{KC}}$ & $0$ \\ & $k$-WTA & $1$ & $0$ & $0$ \\ \hline \multirow{3}{}{evaluating} & XOR & $N_{\text{KC}}$ & $N_{\text{KC}}$ & $0$ \\ & multiplication (square) & $0$ & $0$ & $N_{\text{PN}}+1$ \\ \cline{1-1} & addition (subtraction) & $\leq 2\kappa\times N_{\text{KC}}$ & $\leq N_{\text{KC}}$ & $2\times N_{\text{PN}}-1$ \\ \hline \end{tabular} \end{table} Table 2: Model comparison in terms of run-time operations the memory capacity $m$ would be approximately \[m=\frac{\log\left(1-p_{\text{error}}^{1/N_{\text{ac}}}\right)-\log\kappa}{\log(1- \kappa)}, \tag{5}\] where $p_{\text{error}}$ denotes the probability for the model to confuse a novel visual input mistakenly as a familiar one (Ardin et al., 2016). Thus, with $p_{\text{error}}=1\%$, the modified FlyHash model with $\kappa=0.05$ would merely require $330$ KCs to distinguish novel inputs from 25 training images. Most interestingly, $m$ increases as $\kappa$ becomes smaller, which can be explained by the formal proof that the expanded, sparse, and binary hashes can be more easily learned by an output neuron (Dasgupta and Tosh, 2020). The modified model also reflects better the real structure of the insect MB. It is also sensible in terms of metabolic efficiency and functional effectiveness, because if the KC activation were dense, the downstream synapses would have to adjust their weights rapidly and extensively. In addition, we are interested in comparing the (modified) FlyHash model to other data encoding techniques on a virtual robot, especially in simulations with larger physical, visual, or neural noise. It will be even more interesting to compare them on an real embodied robot. Such endeavours will deepen our understanding of the efficiency of biological intelligence, and will hopefully help us build faster and less energy demanding artificial intelligence.
2307.09938	Tracking an Untracked Space Debris After an Inelastic Collision Using Physics Informed Neural Network	With the sustained rise in satellite deployment in Low Earth Orbits, the collision risk from untracked space debris is also increasing. Often small-sized space debris (below 10 cm) are hard to track using the existing state-of-the-art methods. However, knowing such space debris' trajectory is crucial to avoid future collisions. We present a Physics Informed Neural Network (PINN) - based approach for estimation of the trajectory of space debris after a collision event between active satellite and space debris. In this work, we have simulated 8565 inelastic collision events between active satellites and space debris. Using the velocities of the colliding objects before the collision, we calculate the post-collision velocities and record the observations. The state (position and velocity), coefficient of restitution, and mass estimation of un-tracked space debris after an inelastic collision event along with the tracked active satellite can be posed as an optimization problem by observing the deviation of the active satellite from the trajectory. We have applied the classical optimization method, the Lagrange multiplier approach, for solving the above optimization problem and observed that its state estimation is not satisfactory as the system is under-determined. Subsequently, we have designed Deep Neural network-based methods and Physics Informed Neural Network (PINN )based methods for solving the above optimization problem. We have compared the performance of the models using root mean square error (RMSE) and interquartile range of the predictions. It has been observed that the PINN-based methods provide a better prediction for position, velocity, mass and coefficient of restitution of the space debris compared to other methods.	Harsha M., Gurpreet Singh, Vinod Kumar, Arun Balaji Buduru, Sanat K. Biswas	2023-07-19T12:23:24Z	http://arxiv.org/abs/2307.09938v2	Tracking an Untracked Space Debris After an Inelastic Collision Using Physics Informed Neural Network ###### Abstract With the sustained rise in satellite deployment in Low Earth Orbits, the collision risk from untracked space debris is also increasing. Often small-sized space debris (below 10 cm) are hard to track using the existing state-of-the-art methods. However, knowing such space debris' trajectory is crucial to avoid future collisions. We present a Physics Informed Neural Network (PINN) - based approach for estimation of the trajectory of space debris after a collision event between active satellite and space debris. In this work, we have simulated 8565 inelastic collision events between active satellites and space debris. To obtain the states of the active satellite, we use the TLE data of 1647 Starlink and 66 LEMUR satellites obtained from space-track.org. The velocity of space debris is initialized using our proposed velocity sampling method, and the coefficient of restitution is sampled from our proposed Gaussian mixture-based probability density function. Using the velocities of the colliding objects before the collision, we calculate the post-collision velocities and record the observations. The state (position and velocity), coefficient of restitution, and mass estimation of un-tracked space debris after an inelastic collision event along with the tracked active satellite can be posed as an optimization problem by observing the deviation of the active satellite from the trajectory. We have applied the classical optimization method, the Lagrange multiplier approach, for solving the above optimization problem and observed that its state estimation is not satisfactory as the system is under-determined. Subsequently, we have designed Deep Neural network-based methods and Physics Informed Neural Network (PINN )based methods for solving the above optimization problem. We have compared the performance of the models using root mean square error (RMSE) and interquartile range of the predictions. It has been observed that the PINN-based methods provide a better prediction for position, velocity, mass and coefficient of restitution of the space debris compared to other methods. ## Introduction Since the launch of Sputnik in 1957, approximately 13,000 satellites have been deployed in the Low Earth orbit [1]. With collisions, explosions and fragmentation, the number of space debris sized less than 1 cm is now estimated to be more than 1,000,000 [1]. The US Space Surveillance Network has a catalogue of about 30,000 resident space objects only [1]. Since 97% of space debris is not tracked, they pose a greater threat to active satellites as well as future space missions. With this large number of space debris, the risk of in-orbit collision has increased, and the number of collision events has seen a rise in the past decade. Collision in space can be destructive, for example, Iridium-Cosmos collision [2] of 2009, or non-destructive, for example, Canadarm2 of the International Space Station hit by a small piece of space debris in 2021 [3]. Another incident which was catalogued as a non-destructive collision was in 2013 when Blitz satellite was possibly hit by un-tracked space debris [4]. While there has been exhaustive research on modelling destructive collisions [5, 6], on risk assessment due to small space debris [7], and reconnecting fragments in space to their parent satellite body [8], the extraction of trajectory information of untracked space debris observing a non-destructive collision has not been studied extensively. Recently, Harsha et al. formulated a space debris position, velocity and mass estimation problem considering the position and velocity deviation of an active satellite as observation with non-destructive and elastic collision assumption [9]. The performance of the classical estimation methods as well as machine learning (ML) based approaches, were examined for this problem. It was observed that the position, velocity and mass estimation performance of the Ensemble Neural Network-based technique are similar to the classical methods. It should be noted that the elastic collision assumption in the above preliminary study was idealistic. The above estimation problem becomes complex when a more rigorous inelastic collision model is considered. Hence it is of interest to examine the trajectory estimation performance of both classical and ML-based methods under the inelastic collision assumption. Deep Neural Network (DNN)-based techniques have already been proposed for asteroid exploration, spacecraft rendezvous and terrain navigation [10]. The convolutional neural network (CNN) has been proposed for pose estimation of uncooperative spacecraft[11] and for object detection and tracking in space[12]. NASA's Double asteroid redirection test (DART) mission successfully used ML techniques to navigate the spacecraft autonomously and collide with Didymos by real-time analysis and processing of data from the spacecraft's onboard cameras and range-finding instruments. Further, the ML algorithm optimized trajectories and evaluated various impact scenarios, considering factors such as the asteroid's shape, composition, and spin[13]. Supervised ML methods have found applications across many fields of engineering and science, for example object detection and recognition[14, 15], recommendation systems[16], text-to-speech conversion[17], machine translation[18], and information retrieval[19]. These methods achieve satisfactory performance when trained with large amounts of data. However, these standard neural network-based methods inevitably face the challenge of drawing conclusions and making decisions under partial information, where the data acquisition is costly and scarce while dealing with complex and non-linear physical systems[20]. In addition, the solution of the traditional neural network does not guarantee adherence to the underlying physical properties in problems involving physical systems. To address these issues, the Physics informed Neural network (PINN) was proposed[20]. PINN[21, 22] uses the laws governing the system dynamics in the loss function as prior information. Note that the training of a neural network essentially implies finding the set of weights and biases for all the nodes in the network, which minimises the loss function. Hence, the inclusion of the system dynamics based on physical laws in the loss function act as a constraint for the output while training the neural network using the training data set. The PINNs can be used as surrogates in learning models used for autonomous navigation, uncertainty quantification and any real-time application that need inference[23]. In this article, we have studied the problem of tracking unknown space debris, observing an inelastic and non-destructive collision event. Additionally, we assumed the space debris was in a stable orbit before the collision. Under the inelastic collision assumption, the coefficient of restitution[24] is an unknown parameter in addition to the variables that need to be estimated under the elastic collision assumption. We have examined the performance of the classical estimation technique and various DNN and PINN-based methods for estimating the position, velocity, and mass of the space debris and the coefficient of restitution for the inelastic collision for the above-mentioned problem using 8235 collision simulations for training and 330 collision simulations for testing. The inelastic collisions were simulated using the model described by Schwager et al.[25]. The performance of the DNN and PINN-based methods are compared with the classical method for the inelastic and non-destructive collision. Towards tracking unknown space debris, we have made the following key contributions: 1. Formulation of space debris tracking problem observing a non-destructive and inelastic collision: We have posed the unknown space debris tracking problem after an inelastic event as a position, velocity, mass, and the coefficient of restitution estimation problem considering the active satellite position and velocity deviation as observations. 2. Formulation of an appropriate physics loss function and application of PINN: We have formulated a physics loss function for the above problem and trained a PINN using the developed loss function to solve the above estimation problem. 3. A velocity sampling method for simulating random in-orbit collision events: For the training of DNN and PINN-based models as well for testing the performance of the ML-based approaches and the classical approach, we simulated a total of 8565 inelastic collision events. Note that, for a collision event, the active satellite position and the space debris position will be approximately the same at the time of the collision, while the velocities will be different. However, any random velocity vector at the given position does not necessarily result in a stable elliptical orbit. We have proposed a velocity sampling method to generate the velocity vector of space debris for a given active satellite position at the time of the collision, which guarantees an elliptical orbit with a periapsis above a given minimum allowable altitude from the mean sea level. The rest of the article is organised as follows: We have formally defined the problem in the next section, and then discussed the classical estimation method and various DNN-based approaches and provided the physics loss function for the PINN for solving the problem. Then we have discussed the collision data generation method, including the formulation of the velocity sampling algorithm and simulation of collision in space. Next, we have compared the performance of the classical estimation technique as well as DNN and PINN-based methods for solving the estimation problem. We conclude the article by consolidating our observations and delineating the future research direction. ## Problem Definition Consider a satellite and space debris undergoing an inelastic collision in space. Our objective is to find the trajectory of the space debris by observing the position and velocity deviation of the active satellite after the collision. Let $\mathbf{r}_{sat}$ and $\mathbf{v}_{sat}$ be the position and velocity of the satellite, and $\mathbf{r}_{d}$ and $\mathbf{v}_{d}$ be the position and velocity of the space debris. We assume satellite and space debris are spherical for ease of the analysis. Let $t_{c}^{-}$ and $t_{c}^{+}$ denote the time before collision and time after collision respectively. At the time of the collision, the position of the satellite and the position of debris can be written as \[\mathbf{r}_{sat}(t_{c}^{+})=\mathbf{r}_{sat}(t_{c}^{-}) \tag{1}\] \[\mathbf{r}_{d}(t_{c}^{+})=\mathbf{r}_{d}(t_{c}^{-})=\mathbf{r}_{sat}(t_{c}^{-})-(\rho_{sat}+ \rho_{d})\mathbf{\hat{v}}_{rel} \tag{2}\] where $\rho_{sat}$ and $\rho_{d}$ are radii of the satellite and debris, respectively and $\mathbf{\hat{v}}_{rel}$ is the unit vector along the relative velocity of debris with respect to the satellite and is given by \[\mathbf{\hat{v}}_{rel}=\frac{\mathbf{v}_{sat}(t_{c}^{-})-\mathbf{v}_{d}(t_{c}^{-})}{\|\|\mathbf{ v}_{sat}(t_{c}^{-})-\mathbf{v}_{d}(t_{c}^{-})\|\|} \tag{3}\] The velocity of the satellite and debris after an inelastic collision is[24] \[\mathbf{v}_{sat}(t_{c}^{+})=\frac{m_{sat}\mathbf{v}_{sat}(t_{c}^{-})+m_{d}\mathbf{v}_{d} (t_{c}^{-})+\epsilon m_{d}(\mathbf{v}_{d}(t_{c}^{-})-\mathbf{v}_{sat}(t_{c}^{-}))}{m_{ sat}+m_{d}} \tag{4}\] \[\mathbf{v}_{d}(t_{c}^{+})=\frac{m_{d}\mathbf{v}_{d}(t_{c}^{-})+m_{sat}\mathbf{v}_{sat}(t_ {c}^{-})+\epsilon m_{sat}(\mathbf{v}_{sat}(t_{c}^{-})-\mathbf{v}_{d}(t_{c}^{-}))}{m_{ sat}+m_{d}} \tag{5}\] where $\mathbf{v}_{sat}(t_{c}^{-})$indicates velocity of satellite, just before the collision and $\mathbf{v}_{sat}(t_{c}^{+})$ indicates velocity of satellite just after the collision and $\epsilon$ is the Coefficient of Restitution[25] of the inelastic collision. Then change in velocity of the active satellite can be written as \[\Delta\mathbf{v}_{sat}(t_{c}^{+})=\mathbf{v}_{sat}(t_{c}^{+})-\mathbf{v}_{sat}(t_{c}^{-}) \tag{6}\] Define the measurement $\mathbf{Z}$ as \[\mathbf{Z}=\begin{bmatrix}\mathbf{r}_{sat}(t_{c}^{+})\\ \Delta\mathbf{v}_{sat}(t_{c}^{+})\end{bmatrix}+\omega(t_{c}^{+})=\begin{bmatrix} \mathbf{r}_{sat}(t_{c}^{+})\\ \mathbf{v}_{sat}(t_{c}^{+})-\mathbf{v}_{sat}(t_{c}^{-})\end{bmatrix}+\omega(t_{c}^{+}) \tag{7}\] where $\omega(t_{c}^{+})$ is noise in measurement. As the active satellite is being tracked, $\mathbf{v}_{sat}(t_{c}^{+})$ and $\mathbf{v}_{sat}(t_{c}^{-})$ can be measured, and therefore measurement $\mathbf{Z}$ is available. Using equations (2), (4) and (5), $\mathbf{Z}$ can be written as \[\mathbf{Z}=f(\mathbf{r}_{d}(t_{c}^{-}),\mathbf{v}_{d}(t_{c}^{-}),m_{d},\epsilon)+\omega(t_ {c}^{+}) \tag{8}\] Obtaining $\mathbf{r}_{d}(t_{c}^{-}),\mathbf{v}_{d}(t_{c}^{-}),m_{d}$ and $\epsilon$ from $\mathbf{Z}$ is essentially an estimation problem. Using the estimated parameters $\mathbf{v}_{d}(t_{c}^{+})$ can be obtained and thus the trajectory of the space debris after the collision can be computed. Figure 1: Collision ### Classical approach One can estimate $\mathbf{r}_{d}(t_{c}^{-}),\mathbf{v}_{d}(t_{c}^{-}),m_{d}$ and $\epsilon$ from observation $\mathbf{Z}$ using the Least Squared Estimation (LSE): \[\widehat{\mathbf{r}}_{d}(t_{c}^{-}),\widehat{\mathbf{v}}_{d}(t_{c}^{-}),\widehat{m}_{d}, \widehat{\epsilon}=\operatorname{arg\,min}_{\mathbf{r}_{d}(t_{c}^{-}),\mathbf{v}_{d} (t_{c}^{-}),m_{d},\mathbf{\epsilon}}(\mathbf{\Lambda}\mathbf{Z}^{T}\mathbf{\Lambda}\mathbf{Z}) \tag{9}\] where $\widehat{\mathbf{r}}_{d}(t_{c}^{-}),\widehat{\mathbf{v}}_{d}(t_{c}^{-}),\widehat{m}_{d}, \widehat{\epsilon}$ are the estimate of the desired quantities and \[\mathbf{\Lambda}\mathbf{Z}=\mathbf{Z}-f(\widehat{\mathbf{r}}_{d}(t_{c}^{-}),\widehat{\mathbf{v}}_{ d}(t_{c}^{-}),\widehat{m}_{d},\widehat{\epsilon}) \tag{10}\] Note that the above estimation problem formulation does not consider any mass constraints for the debris. However, we are particularly interested in the untracked debris, which is of small size and essentially has a very small mass. The inclusion of the mass constraint transforms the original problem into a constrained optimization problem, which can be solved using the Lagrange Multiplier approach [9]. The Lagrangian $\mathcal{L}$ for this constrained optimization problem can be defined as \[\mathcal{L}=\mathbf{\Lambda}\mathbf{Z}^{T}\mathbf{\Lambda}\mathbf{Z}+\lambda_{1}(1-m_{d})+ \lambda_{2}(m_{d}-\delta m) \tag{11}\] which includes the mass constraint $\delta m<m_{d}<1\ kg$. Here, $\lambda_{1}$ and $\lambda_{2}$ are Lagrange multipliers. The position, velocity, mass and coefficient of restitution of the debris can be estimated by solving \[\begin{bmatrix}\frac{\partial\mathcal{L}}{\partial\mathbf{r}_{d}}& \frac{\partial\mathcal{L}}{\partial\mathbf{r}_{d}}&\frac{\partial\mathcal{L}}{ \partial m_{d}}&\frac{\partial\mathcal{L}}{\partial\epsilon}\\ &&\frac{\partial\mathcal{L}}{\partial\lambda_{1}}&\frac{\partial\mathcal{L}}{ \partial\lambda_{2}}\\ \end{bmatrix} =\mathbf{0} \tag{12}\] \[\begin{bmatrix}\frac{\partial\mathcal{L}}{\partial\lambda_{1}}& \frac{\partial\mathcal{L}}{\partial\lambda_{2}}\\ \end{bmatrix} =\mathbf{0} \tag{13}\] There are various techniques available to solve this minimisation problem. We have used the gradient descent approach to solve the problem numerically. ### Machine Learning-based approaches An alternate approach for estimating the position, velocity, mass and coefficient of restitution of the debris after a collision can be based on a DNN model. Using (8) one can write \[E[\mathbf{r}_{d}(t_{c}^{-}),\mathbf{v}_{d}(t_{c}^{-}),m_{d},\mathbf{\epsilon}]=f^{-1}(\mathbf{Z}) \tag{14}\] considering $\mathbf{\omega}(t_{c}^{+})$ as a zero mean noise vector. Essentially the trained DNN model approximates the inverse function $f^{-1}(\cdot)$. The DNN model can be trained using simulated collision data. We consider $\mathbf{Z}$ as the input to the DNN and $\mathbf{\Lambda}\mathbf{r},\mathbf{\Delta}\mathbf{v},m_{d},\mathbf{\epsilon}$ as the output of the DNN, where \[\mathbf{\Delta}\mathbf{r}=\begin{bmatrix}\Delta x_{d}\\ \Delta y_{d}\\ \Delta z_{d}\end{bmatrix}=\mathbf{r}_{d}(t_{c}^{-})-\mathbf{r}_{sat}(t_{c}^{-}) \tag{15}\] and \[\mathbf{\Delta}\mathbf{v}=\begin{bmatrix}\Delta v_{xd}\\ \Delta v_{yd}\\ \Delta v_{zd}\end{bmatrix}=\mathbf{v}_{d}(t_{c}^{-})-\mathbf{v}_{sat}(t_{c}^{-}) \tag{16}\] We consider two distinct DNN architectures for this problem. The first architecture is the usual DNN with an input layer with 6 inputs, multiple hidden layers and an output layer with 8 outputs as shown in Fig. 2. The second architecture comprises of 8 parallel DNNs with 6 inputs, multiple hidden layers and 1 output. Essentially, each of these parallel DNNs learns one of the 8 outputs. The outputs of these 8 parallel DNNs are connected to another DNN with 8 inputs and 8 outputs. The last DNN block ensures learning the dependency of each output variable with other output variables. The architecture is shown in Fig. 3. We refer to this architecture as the stacked Deep Neural Network (StackDNN). For both architectures, Mean Squared Error (MSE) loss function is used for training the models. We define the MSE loss for this problem as \[\begin{split}&\mathcal{L}_{MSE}(\widehat{\mathbf{r}}_{d},\widehat{ \mathbf{v}}_{d},\hat{m}_{d},\mathbf{\epsilon})\\ &=\frac{1}{N}\sum_{i=1}^{N}\left[(\mathbf{r}_{d_{i}}-\widehat{\mathbf{r}}_{d_ {i}})^{T}(\mathbf{r}_{d_{i}}-\widehat{\mathbf{r}}_{d_{i}})+(\mathbf{v}_{d_{i}}-\widehat{ \mathbf{v}}_{d_{i}})^{T}(\mathbf{v}_{d_{i}}-\widehat{\mathbf{v}}_{d_{i}})+(m_{d}-\hat{m}_{ d})^{2}+(\mathbf{\epsilon}-\mathbf{\epsilon})^{2}\right]\end{split} \tag{17}\] where $(\cdot)_{d_{i}}$ and $(\widehat{\cdot})_{d_{i}}$ denote the output data for the $i^{th}$ collision event and the corresponding predicted output data using the DNN. $N$ is the number of collision events used to generate the training data. ### Physics Informed Neural Network Having discussed the DNN-based method for estimating the position, velocity, mass and coefficient of restitution of debris after the collision, it is essential to note that the estimated position and velocity must be the solution to the differential equation that governs the motion of the debris. Based on Newton's Law of Gravitation, the debris dynamics can be written as \[\begin{bmatrix}\dot{\mathbf{r}}_{d}\\ \dot{\mathbf{v}}_{d}\end{bmatrix}=\begin{bmatrix}\mathbf{v}_{d}\\ -\frac{\mu}{r_{d}^{3}}\mathbf{r}_{d}\end{bmatrix} \tag{18}\] However, DNN-based solutions do not guarantee the above constraint. The Physics Informed Neural Network [21] preserves the laws of physics in the solution by incorporating a physics loss in the loss function used to train the DNN. Using (18), we defined the physics loss for the PINN as \[\begin{split}&\mathcal{L}_{phy}(\mathbf{\widehat{r}}_{d},\mathbf{ \widehat{r}}_{d})\\ &=\frac{1}{N}\sum_{i=1}^{N}\left[\left(\frac{\mu}{r_{d_{i}}^{3}} \mathbf{r}_{d_{i}}-\frac{\mu}{\hat{r}_{d_{i}}^{3}}\mathbf{\widehat{r}}_{d_{i}}\right) ^{T}\left(\frac{\mu}{r_{d_{i}}^{3}}\mathbf{r}_{d_{i}}-\frac{\mu}{\hat{r}_{d_{i}}^{ 3}}\mathbf{\widehat{r}}_{d_{i}}\right)+\left(\mathbf{v}_{d_{i}}-\mathbf{\widehat{r}}_{d_{i }}\right)^{T}\left(\mathbf{v}_{d_{i}}-\mathbf{\widehat{r}}_{d_{i}}\right)\right]\end{split} \tag{19}\] for the problem under consideration. The significance of this loss function is that the minimisation of $\mathcal{L}_{phy}(\mathbf{\widehat{r}}_{d},\mathbf{\widehat{r}}_{d})$ minimises the difference between the acceleration computed using the output of the training data and the PINN output, and the difference between the velocity computed using the output of the training data and the PINN output. As a result, this loss function minimises the distance between the derivative of the vector $\left[\mathbf{r}_{d}\ \ \mathbf{v}_{d}\right]^{T}$ computed using the training data, and the derivative of the same computed using the PINN predicted output. Since the velocity loss is already included in $\mathcal{L}_{phy}(\mathbf{\widehat{r}}_{d},\mathbf{\widehat{r}}_{d})$ we define the MSE loss for the PINN as \[\mathcal{L}_{MSEP}(\mathbf{\widehat{r}}_{d},\mathbf{\widehat{m}}_{d},\mathbf{\hat{\epsilon }})=\frac{1}{N}\sum_{i=1}^{N}\left[(\mathbf{r}_{d_{i}}-\mathbf{\widehat{r}}_{d_{i}})^ {T}(\mathbf{r}_{d_{i}}-\mathbf{\widehat{r}}_{d_{i}})+(m_{d}-\mathbf{\widehat{m}}_{d})^{2} +(\epsilon-\mathbf{\hat{\epsilon}})^{2}\right] \tag{20}\] Figure 3: Stacked Deep Neural Network (StackDNN) architecture Figure 2: Deep Neural Network (DNN) architecture The complete loss function for the PINN is \[\mathcal{L}_{PINN}=\mathcal{L}_{phy}\left(\widehat{\mathbf{r}}_{d},\widehat{\mathbf{r}}_{ d}\right)+\mathcal{L}_{MSEP}\left(\widehat{\mathbf{r}}_{d},\hat{m}_{d},\hat{\mathbf{e}}\right) \tag{21}\] We use this loss function in both DNN and StackDNN for performance comparison. ## Collision data generation Having discussed the DNN, StackDNN, PINN, and StackPINN approaches for tracking the debris after an inelastic collision event, let us now turn to the preparation of the training data. Publicly available data on inelastic and nondestructive collision events are very few and not sufficient to train the neural networks. Hence simulated collision data are required for training the machine learning models. The position and velocity information of active satellites can be obtained from publicly available catalogues. However, the position and velocity of the debris which collides with an active satellite at a given epoch must be simulated. The simulated debris must be in a complete orbit, i.e. the orbit eccentricity must be less than 1. Additionally, to form a stable orbit, the periapsis of the debris cannot be at a very low altitude. The debris position is given by (2) at the time of the collision. From (2), we can approximate that the debris position as nearly equal to the active satellite position because $r_{sat}\gg\rho_{sat},\rho_{d}$. Corresponding to this position, infinitely many velocity vectors are possible for which the debris will be on a complete orbit. However, the magnitude and the direction of the velocity vector can not be any arbitrary magnitude and direction. In the next subsection, we describe a method of sampling the velocity of debris for a given position vector in space and a given minimum periapsis altitude. ### Velocity sampling Consider $r$ is the norm of the active satellite as well as the debris position vector $\mathbf{r}$ at the time of the collision, $h$ is the norm of the angular momentum vector $\mathbf{h}$ of the colliding debris, $e$ is the eccentricity of the debris orbit, $\phi$ is the true anomaly of the debris during the collision, $R_{e}$ is the Radius of the earth and $a_{min}$ minimum allowable periapsis altitude. If the semi-major axis for the debris orbit is $a$, then the periapsis $r_{p}=a(1-e)$. Also \[\frac{h^{2}}{\mu}=r_{p}(1+e) \tag{22}\] and \[r=\frac{h^{2}}{\mu}\frac{1}{1+e\cos\phi} \tag{23}\] Considering $R_{min}=R_{e}+a_{min}$ as the minimum allowable distance from the earth centre \[r_{p}=\frac{r(1+e\cos\phi)}{1+e}>R_{min} \tag{24}\] then \[0<e<\frac{r-R_{min}}{R_{min}-r\cos\phi}<1 \tag{25}\] and \[\cos^{-1}\frac{2R_{min}-r}{r}<\phi<2\pi-\cos^{-1}\frac{2R_{min}-r}{r} \tag{26}\] Fig. 4 shows the variation of maximum possible eccentricity with the true anomaly. Now, the position of the debris at the time of collision is \[\mathbf{r}_{d}=\mathbf{r}=r\begin{bmatrix}r_{ux}\\ r_{uy}\\ r_{uz}\end{bmatrix} \tag{27}\] and velocity of the debris is \[\mathbf{v}_{d}=v\begin{bmatrix}v_{ux}\\ v_{uy}\\ v_{uz}\end{bmatrix} \tag{28}\] where $\begin{bmatrix}r_{ux}&r_{uy}&r_{uz}\end{bmatrix}^{T}$ and $\begin{bmatrix}v_{ux}&v_{uy}&v_{uz}\end{bmatrix}^{T}$ are the position and velocity unit vectors respectively, and \[v=\sqrt{\frac{\mu}{r}\left(2-\frac{1-e^{2}}{1+e\cos\phi}\right)} \tag{29}\] The angle between $\mathbf{r}$ and $\mathbf{v}$ is $\theta^{26}$ \[\theta=\sin^{-1}\frac{1+e\cos\phi}{\sqrt{1+2e\cos\phi+e^{2}}} \tag{30}\] The angular momentum vector of the debris is \[\mathbf{h} =\mathbf{r}\times\mathbf{v}\] \[=rv\begin{bmatrix}r_{\alpha y}v_{uz}-r_{uz}v_{uy}\\ r_{uz}v_{ux}-r_{ux}v_{uz}\\ r_{ux}v_{uy}-r_{uy}v_{ux}\end{bmatrix} \tag{31}\] and \[h=rv\sin\theta \tag{32}\] then the cosine of the inclination $i$ of the orbit \[\cos i=\frac{rv(r_{ux}v_{uy}-r_{uy}v_{ux})}{rv\sin\theta} \tag{33}\] Consequently, the velocity unit vector needs to satisfy the following: \[r_{ux}v_{ux}+r_{uy}v_{uy}+r_{uz}v_{uz}=\cos\theta \tag{34}\] \[-r_{uy}v_{ux}+r_{ux}v_{uy}+0v_{uz}=\sin\theta\cos i\] (35) \[\sqrt{v_{ux}^{2}+v_{uy}^{2}+v_{uz}^{2}}=1 \tag{36}\] Geometrically, the condition for forming a complete orbit is - the line of intersection of planes (34) and (35) must lie within the unit sphere defined by (36). We can write (35) in the normal form: \[-\frac{r_{uy}}{\sqrt{r_{ux}^{2}+r_{uy}^{2}}}v_{ux}+\frac{r_{ux}}{\sqrt{r_{ux }^{2}+r_{uy}^{2}}}v_{uy}+0.v_{uz}=\frac{\sin\theta\cos i}{\sqrt{r_{ux}^{2}+r_ {uy}^{2}}} \tag{37}\] Since $\sqrt{r_{ux}^{2}+r_{uy}^{2}+r_{uz}^{2}}=1$ (34) is already in the normal form. Observing (34) and (37), one can deduce that the angle between the two planes is $90^{\circ}$. Then from the cross-sectional geometry shown in Fig. 5, the criteria for the intersection of the two planes within the unit sphere is \[\cos^{2}\theta+\frac{\sin^{2}\theta\cos^{2}i}{r_{ux}^{2}+r_{uy}^{2}}\leq 1 \tag{38}\] Then \[-\sqrt{\frac{r_{ux}^{2}+r_{uy}^{2}}{\sin^{2}\theta(1-\cos^{2}\theta)}}\leq\cos i \leq\sqrt{\frac{r_{ux}^{2}+r_{uy}^{2}}{\sin^{2}\theta(1-\cos^{2}\theta)}} \tag{39}\] Using the ranges of $\phi$, $e$ and $\cos i$ given by (26), (25) and (39) one can independently sample these quantities and subsequently solve for $v_{ux}$, $v_{uy}$ and $v_{uz}$ from (34), (35) and (36). Note that the solution will result in two unit vectors for velocity. Finally, the velocity magnitude can be calculated using (29). Algorithm 1 describes the velocity sampling steps. ``` Input :$\mathbf{r}$, minimum altitude, Number of samples Output :Velocity samples 1fori=1:Number of samplesdo 2 Given the allowed minimum altitude, find the minimum and maximum true anomaly $\phi_{min}$ and $\phi_{max}$ for the given position vector using (26); 3 Sample $\phi\sim U(\phi_{min},\phi_{max})$; 4 For a sampled $\phi$ calculate maximum eccentricity $e_{max}$ using (25); 5 Sample $e\sim U(0,e_{max})$; 6 Calculate $\theta$; 7 Compute the lower bound $\cos i_{min}$ and upper bound $\cos i_{max}$ for $\cos i$ using (39); 8 Sample $\cos i\sim U(\cos i_{min},\cos i_{max})$; 9 Solve for $v_{ux},v_{uy},v_{uz}$ using (34), (35) and (36); 10 Calculate $v$ using (29) 11 end for ``` Algorithm 1Velocity sampling Using this algorithm, we have sampled 2000 velocity vectors for an active satellite position at \[\mathbf{r}=\left[3664.75536654\quad 3893.60049258\quad 5062.87536598\right]^{T}km \tag{40}\] and a minimum perigee altitude of 300 km. Fig. 6 shows that the generated samples have perigees at altitudes higher than 300 km. Fig. 7 shows the relative velocities for the samples with respect to the active satellite. It can be deduced from Fig. 7 that collision cannot occur from any arbitrary direction for a given position vector in space. ### Simulation of the collision We have generated inelastic collision observations considering the collision model described in the Problem Formulation section. We have used Python 3.7 with astropy[27] and polastro[28] libraries for orbit simulations of the active satellites as well as the debris. We have considered 5 collision epochs $t_{c}$ each at 1200 UTC, starting from April 29, 2023, to May 3, 2023. We consider 1647 satellite orbits from the Starlink constellation and 66 satellite orbits from the LEMUR constellation for generating collision simulations. The Two Line Element (TLE) data for each constellation were obtained from www.spacetrack.org on April 27, 2023. Using TLE data, we have calculated the time difference $\Delta t$ between the collision time $t_{c}$ and the the TLE time, and propagated the active satellites. The orbits are propagated considering J2 and J3 zonal harmonics and the exponential atmospheric drag model. For the debris simulations, we sampled the mass of each untracked debris from a normal distribution with 0.5 kg mean and 0.001 kg of standard deviation. For simplicity, satellites and debris are considered spherical objects of radius 0.5 m and 0.1 m, respectively, and the mass of the satellite is 100 times that of sampled mass of debris. The position of the debris $\mathbf{r_{d}}$ is considered to be the same as $\mathbf{r_{sat}}$, whereas the velocity of the debris $\mathbf{v_{d}}$ is obtained using algorithm 1. The minimum perigee altitude for the debris corresponding to the Starlink satellites was set at 250 km, and for LEMUR, it was 200 km. The Gabbard plots for the debris generated for the Starlink and LEMUR constellations are shown in Fig. 8. Co-efficient of Restitution ($\epsilon$) values for each of the collisions have been sampled from a Gaussian mixture distribution, as shown in Fig. 9. It consists of 4 Gaussian distributions with 0.015 standard deviations sampled around uniformly distributed means ranging from 0.5 to 1 where the mean has been assumed based on the $\epsilon$ values of the satellite materials (6061-T6 Aluminium, Stainless Steel 304, Nitronic 60A and Titanium)[24]. Inelastic collision simulations are generated using the position, velocity, mass and co-efficient of restitution values for the respective satellite and debris pairs in equations (4) and (5). Post-collision at time $t_{c}^{+}$, the observations are generated using equation (7). Here, the noise $\omega(t_{c}^{+})$ signifies the tracking uncertainty of the active satellite. We consider 100 m uncertainty for position and 50 m/sec uncertainty for velocity in each axis. A total of 8235 collision events were simulated for the orbits of the Starlink constellation using Param Pravega Super Computer. 330 collision events were simulated for the LEMUR constellation using a local workstation. For each collision, we Figure 8: Gabbard plot of space debris Figure 9: Co-efficient of Restitution ($\epsilon$) distribution using Gaussian mixture have recorded $\mathbf{Z}$ and $\mathbf{r}_{d}$, $\mathbf{v}_{d}$, $m_{d}$ and $\epsilon$. We consider the collision data corresponding to the orbits of the Starlink constellation as the training data and the same corresponding to the orbits of the LEMUR constellation as the test data. ## Results and discussion We used the training data described in the previous section for training the DNN, PINN, StackDNN and StackPINN. We trained four different models with 2, 4, 6 and 8 hidden layers for each of the ML models, with 5 nodes in each hidden layer. We have selected Rectified Linear Unit (ReLU) as the activation function for each node. All the models are trained for 100 training epochs. The training is performed in a workstation with 8 Intel Xeon processor cores and 62GB of RAM. The training time required for each model is provided in Table 1. The change in training loss with training epochs for all the ML models is shown in Fig. 10. Fig. 10 shows the training loss for DNN and PINN models. It is observed that for models with an increased number of hidden layers, the training epochs decrease for the training loss to converge. A similar trend can be observed for the StackDNN and StackPINN models in Fig. 10. Using the observation $\mathbf{Z}$ for all 330 test collision events in the trained ML models, $\mathbf{r}_{rso}(t_{c}^{-})$, $\mathbf{v}_{rso}(t_{c}^{-})$, $m_{rso}$ and $\epsilon$ were estimated. Additionally, the same variables were estimated using the Lagrange multiplier method for the same set of observations. \begin{table} \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline \multirow{2}{}{Methods} & \multicolumn{4}{c\|}{Time Taken to train models (in seconds)} \\ \cline{2-5} & 2 Layers & 4 Layers & 6 Layers & 8 Layers \\ \hline DNN & 51.983 & 55.048 & 58.169 & 60.993 \\ PINN & 652.623 & 655.689 & 663.668 & 659.781 \\ StackDNN & 420.531 & 447.919 & 488.353 & 512.025 \\ StackPINN & 1015.206 & 1049.786 & 1076.624 & 1117.294 \\ \hline \end{tabular} \end{table} Table 1: Time taken to train various models Figure 10: Training loss We recorded the error in estimation for performance comparison. Fig. 11 shows the box plots for the position estimation errors for all the methods in the x, y and z axes in the ECI frame. It can be observed from Fig. (a)a and Fig.(b)b that the DNN-based methods are more accurate than the Lagrange multiplier method. At the same time, Fig. (a)a shows that there is a significant performance difference between DNN and PINN for 2 hidden layers. The position error in each axis using PINN has significantly less mean and median than the DNN in this case. Additionally, the interquartile range for the PINN error distribution is smaller than the DNN model. However, for 4, 6 and 8 hidden layers, the performance of the DNN and PINN models are similar and does not improve compared to the PINN with 2 hidden layers. It can be observed from Fig. (b)b that the position estimation performance of the StackDNN and StackPINN are similar for 2, 4, 6 and 8 hidden layers and comparable with the PINN model with 2 hidden layers. Fig. 12 shows the box plots of the error in velocity estimation along the x, y, and z directions for all the methods. It can be observed that there is no significant difference in the velocity estimation performance among the DNN, PINN, StackDNN, StackPINN and Lagrange multiplier methods. In addition, it should be noted that with the increase in hidden layers, the performance does not vary significantly. Now, turning to the mass and the coefficient of restitution estimation, it can be observed from Fig. 13 that for 4 hidden layers and above, the performance difference of the DNN, PINN, StackDNN, and StackPINN are not significant. However, Fig. (a)a shows that with 2 hidden layers, the PINN performance is significantly better than the DNN for mass and the coefficient of restitution estimation. On the other hand, although the Lagrange multiplier method is precise in estimating the mass and the Coefficient of restitution, it introduces significant bias in the solution. From the results shown above, it can be discerned that, for 2 hidden layers, the PINN improves the space debris position, mass and coefficient of restitution estimation performance significantly compared to the Lagrange multiplier-based and DNN-based methods. The performance of the DNN becomes similar to the PINN for higher numbers of hidden layers. However, the results show that there is no advantage to increasing the number of layers for the problem under consideration. In addition, the StackDNN and StackPINN performances are similar for the different numbers of hidden layers and are comparable to the performance of the PINN model with 2 hidden layers. However, the stacked architecture is essentially 9 different Neural Networks, increasing the complexity of implementation. From this discussion, we can conclude that the PINN with 2 hidden layers provides better performance to resource requirement trade-off for tracking untracked space debris from a non-destructive and inelastic collision with an active satellite than DNN, Stacked neural networks and classical methods. Figure 11: Position error comparison ## Conclusion We have formulated a problem of space debris position, velocity, mass and coefficient of restitution estimation from a non-destructive inelastic collision event under the assumption that the occurrence of the collision event is known. We have shown that this problem can be posed as a function approximation problem, and hence, a neural network can be trained to approximate the function. We proposed a cost function based on Newton's law of gravitation to design a PINN to solve this Figure 12: Velocity error comparison Figure 13: Mass and CoR error comparison problem. We have also presented an algorithm for selecting the velocity of space debris for collision simulation. Using simulated collision data, we have trained DNN, PINN, StackDNN and StackPINN with varied numbers of hidden layers. We have also compared the estimation performance of these neural network-based methods with the Lagrange multiplier-based optimisation technique. The results demonstrate that the PINN with 2 hidden layers provides better estimation performance than the DNN with 2 hidden layers and the classical method. Additionally, the DNN performance becomes similar to the PINN when the number of hidden layers is increased and when Stacked neural network architecture is used. However, this increase in the model complexity does not improve the estimation performance compared to the PINN with 2 hidden layers. It is anticipated that with further research, the application of the PINN can be further extended to trajectory extraction of the debris after a break-up event. In addition, the velocity sampling formulation can be used to derive a velocity probability density function for a given position in space, which can be used for designing new filters for selecting candidate space debris for collision assessment as well as enhancing the collision probability computation.
2310.19128	Prediction of local elasto-plastic stress and strain fields in a two-phase composite microstructure using a deep convolutional neural network	Design and analysis of inelastic materials requires prediction of physical responses that evolve under loading. Numerical simulation of such behavior using finite element (FE) approaches can call for significant time and computational effort. To address this challenge, this paper demonstrates a deep learning (DL) framework that is capable of predicting micro-scale elasto-plastic strains and stresses in a two-phase medium, at a much greater speed than traditional FE simulations. The proposed framework uses a deep convolutional neural network (CNN), specifically a U-Net architecture with 3D operations, to map the composite microstructure to the corresponding stress and strain fields under a predetermined load path. In particular, the model is applied to a two-phase fiber reinforced plastic (FRP) composite microstructure subjected to a given loading-unloading path, predicting the corresponding stress and strain fields at discrete intermediate load steps. A novel two-step training approach provides more accurate predictions of stress, by first training the model to predict strain fields and then using those strain fields as input to the model that predicts the stress fields. This efficient data-driven approach enables accurate prediction of physical fields in inelastic materials, based solely on microstructure images and loading information.	Indrashish Saha, Ashwini Gupta, Lori Graham-Brady	2023-10-29T19:34:53Z	http://arxiv.org/abs/2310.19128v1	Prediction of local elasto-plastic stress and strain fields in a two-phase composite microstructure using a deep convolutional neural network ###### Abstract Design and analysis of inelastic materials requires prediction of physical responses that evolve under loading. Numerical simulation of such behavior using finite element (FE) approaches can call for significant time and computational effort. To address this challenge, this paper demonstrates a deep learning (DL) framework that is capable of predicting micro-scale elasto-plastic strains and stresses in a two-phase medium, at a much greater speed than traditional FE simulations. The proposed framework uses a deep convolutional neural network (CNN), specifically a U-Net architecture with 3D operations, to map the composite microstructure to the corresponding stress and strain fields under a predetermined load path. In particular, the model is applied to a two-phase fiber reinforced plastic (FRP) composite microstructure subjected to a given loading-unloading path, predicting the corresponding stress and strain fields at discrete intermediate load steps. A novel two-step training approach provides more accurate predictions of stress, by first training the model to predict strain fields and then using those strain fields as input to the model that predicts the stress fields. This efficient data-driven approach enables accurate prediction of physical fields in inelastic materials, based solely on microstructure images and loading information. keywords: Strain prediction, Stress prediction, Plasticity, Composite material, Machine learning, Deep learning + Footnote †: journal: ## 1 Introduction Mechanical failure of composite materials is associated with localization - high local stresses developing in the material [1; 2; 3]. Therefore, prediction of local physical responses is crucial in computational design and analysis of composite materials. In inelastic materials, this requires a non-linear analysis to determine the critical stresses developing in the composite microstructure under loading. Conventionally, this non-linear analysis is performed by using numerical approaches such as finite element (FE) methods [4; 5; 6]. However, these numerical approaches often require significant time and computational effort because they call for many iterations to solve the partial differential equations that describe computational plasticity. Additionally, the behavior of fiber reinforced plastic (FRP) composite materials depends on the random arrangements of fibers in their microstructure [7; 8; 9], so a new analysis is needed for each specific fiber arrangement. Over the past decades, data-driven surrogate models have emerged as practical alternatives to physics-based models, to mitigate these computational expenses [10; 11; 12; 13; 14]. Among the surrogate models, machine learning (ML) based models have become particularly popular [15; 16; 17]. The success of many ML approaches in modeling structure-property relations relies on efficient microstructure representation, or in other words the extraction of information-rich low-dimensional features from high dimensional microstructure images [18; 19; 20; 21]. Recent works have shown that convolutional neural networks (CNNs) are an ideal choice for autonomous microstructure characterization as they utilize convolutional layers to automatically learn hierarchical features directly from images [22; 23; 24]. Significant efforts have been devoted to utilizing CNNs for predicting effective mechanical properties of materials [25; 26]. For example, Sengodan [27] has combined dimension reduction methods and CNNs to predict the homogenized stress-strain curve of inelastic two-phase microstructures. Kim et al. [28] have directly used the high-dimensional microstructure as the input to a CNN model that predicts the homogenized stress-strain curve. The primary goal of these homogenization-based CNN models is to establish a mapping between the composite microstructure and its macroscopic properties. However, these models do not tackle the issue of localization - high local stresses that develop within the composite microstructure. Recent advancements have shown the effective utilization of CNNs for predicting local stress fields in elastic composites [29; 30; 31; 32]. Bhaduri et al. [33] have used a U-Net CNN to map the composite microstructure with varying number of fibers to the elastic von Mises stress field. Gupta et al. [34] have used a U-Net CNN in a fast multiscale analysis approach that performs both homogenization and localization. This ML-driven multiscale approach uses a single CNN to predict the stress tensor fields for elastic composites under a general state of loading. The challenge in extending this ML-driven multiscale analysis framework to address inelastic behavior is the evolution of such systems under accumulating plastic strains. Sepasdar et al. [35] have used a sequence of CNNs to predict post-failure von Mises stress field and the failure patterns in inelastic composites. The majority of these past works employing CNNs considers only an elastic framework or focuses solely on prediction of stress/strain fields at the end of the loading path. These image-to-image mapping approaches face limitations in predicting intermediate local stress fields under plastic deformation. In the current work, we present a deep-learning (DL) framework that is capable of predicting the evolution of micro-scale elasto-plastic strains and stresses in a two-phase medium. The proposed DL-framework uses a convolutional neural network (CNN) to map the composite microstructure to the corresponding stress and strain tensor fields under a predetermined load path. This DL-framework uses 3D U-Nets to model the elasto-plastic responses as a sequence prediction problem. This approach is demonstrated through application to a two-phase FRP composite microstructure subjected to a predetermined loading-unloading path. A novel two-step training approach provides more accurate predictions, by first training the model to predict strain fields and then using those strain fields as input to the model that predicts the stress fields. This efficient data-driven approach enables accurate prediction of physical fields in inelastic materials at a much faster speed than conventional FE simulations. This paper is structured as follows: Section 2 describes the proposed DL-framework for predicting micro-scale elasto-plastic strains and stresses in a composite material. Section 3 describes the results obtained using this method and Section 4 presents the conclusions made from this study. ## 2 Methodology ### Inelastic composite problem statement This work aims to predict the strain tensor field $\epsilon_{ij}(x,y,t)$ and the stress tensor field $\sigma_{ij}(x,y,t)$ in a 2D two-phase microstructure $M(x,y)$ that is loaded under a combination of uniaxial tensile ($u_{1}(t)$) and shear ($u_{2}(t)$) displacement (see Fig. 1). Specifically, the microstructure consists of randomly distributed stiff circular fibers (fully elastic) in an elastic-perfectly plastic matrix material. The fibers have a fixed diameter, the fiber volume fraction ($V_{f}$) is kept constant at 40%, and the fiber/matrix interface is assumed to be perfectly bonded. The material constants for both the matrix (epoxy) and the fiber (glass fiber) are given in Table 1[36]. A single cycle of loading and unloading is applied to the microstructure, as illustrated in Fig. 2, to capture the inelastic behaviour of the material. A maximum of 8% strain is applied in both shear and tension, which is sufficient to yield the matrix (epoxy) [36]. FE simulations via ABAQUS provide the stress $\sigma_{ij}^{(k)}(x,y)$ and strain fields $\epsilon_{ij}^{(k)}(x,y)$ in the material under the given loading conditions at specific load steps $t_{k}$ along the load path, illustrated in Fig. 2. The same load path is applied for both shear and tensile strains simultaneously, so that the applied shear and the tensile strains are equal at every time step. In this analysis, the number of specific load steps under consideration is $K=32$, indicated by the red circles on the load path. The black squares indicate specific instants on the load path for which detailed visualizations will be provided in this paper. Note that the FE analysis used to generate the training data considers more finely spaced load steps, but this subset of 32 frames is extracted for subsequent analyses via the ML model. As \begin{table} \begin{tabular}{c\|c\|c\|c} Component & $E$ (MPa) & $\nu$ & $\sigma_{y}$ (MPa) \\ \hline Epoxy & 3200.0 & 0.3 & 210.0 \\ Glass Fiber & 87000.0 & 0.2 & — \\ \end{tabular} \end{table} Table 1: Material constants assumed for the components of the two-phase material shown in Fig.1, the left edge of the microstructure is assumed to be pin-supported, while a uniaxial displacement ($u_{1}(t)$) and a shear displacement ($u_{2}(t)$) are applied to the right edge. The top and bottom boundaries are assumed to be traction-free. The matrix material is assumed to be elastic-perfectly plastic with a von-Mises yield criterion and an assumption of small strain $J_{2}$ plasticity [6]. The FE model uses an irregular mesh, with 86000 elements on average for every microstructure, finely resolving the distribution of stresses and strains around and between fiber-matrix interfaces. After the analysis is complete, the input microstructure $M(x,y)$ is discretized into an $N\times N$ square pixel image $M^{(mn)}$, and output stress and strain fields at each load step $k$, $\sigma^{(k)}_{ij}(x,y)$ and $\epsilon^{(k)}_{ij}(x,y)$, are similarly discretized into $N\times N$ square pixel images. The stress and strain fields therefore become 3D voxelized images $\sigma^{(mnk)}_{ij}$ and $\epsilon^{(mnk)}_{ij}$, where $(mnk)$ indicates the value at point $(x_{m},y_{n})$ during load frame $t_{k}$. For the current analysis, the image size $N$ is taken to be 128, which is a standard size for many images. The discretized $128\times 128$ microstructure and three-dimensional $128\times 128\times 32$ images of stresses and strains serve as the basis for subsequent training of the 3D U-Net model, as described in the next subsection. ### 3D U-Net architecture The U-Net architecture containing 2D convolutions is an established method for segmentation of images in computer vision [37]. It has an encoder-decoder type architecture with additional skip connections between the encoder blocks and decoder blocks. The skip connections provide additional spatial information to the decoding layers which results in a better quality reconstruction of the spatial features. In the domain of computational mechanics, the 2D U-Net architecture has proven to be effective in the prediction of elastic stress maps ($\sigma_{ij}(x,y)$) from 2D microstructure images [33] by solving a regression problem. Plasticity is a history-dependent material behavior, meaning that the current state of the material Figure 1: Boundary conditions assumed for the microstructure is dependent on its previous states. For example, in Fig. 3, two strain states ($\epsilon^{t}$ and $\epsilon^{t+\Delta t}$) are possible for a given stress state $\sigma$. The superscripts $t$ and $t+\Delta t$ refer to the sequence of observation of the strain states. Therefore, in order to predict a particular $\sigma_{ij}(x,y)$ or $\epsilon_{ij}(x,y)$ in the plastic regime, the stress and strain histories are required. Thus, the stresses ($\sigma_{ij}(x,y,t)$) and the strains ($\epsilon_{ij}(x,y,t)$) become a function of time. The 2D U-Net architecture is only capable of working with the current state of stress ($\sigma_{ij}(x,y)$). In order to improve upon this, a 3D U-Net is used in this work, due to its ability to represent sequential information in the third dimension. Similar to the 2D U-Net architecture, the 3D U-Net is comprised of a down-sampling and an up-sampling path, as illustrated in Fig. 4. Each level on the down-sampling path contains two $3\times 3\times 3$ convolutions and two non-linear activation units known as Parametric-rectified Linear Unit (PReLU) [38], followed by a $2\times 2\times 2$ max pooling down to the next level. Each level on the up-sampling path consists of a transpose convolution of $2\times 2\times 2$, followed by two $3\times 3\times 3$ convolutions, each followed by PReLU activation. Skip connections between down-sampling and up-sampling levels of similar dimensions transfer high-resolution spatial information to the up-sampling path. In addition to these operations, batch normalization (BN) is added before every PReLU, to accelerate the training process, Figure 3: Typical stress-strain response of a material undergoing loading and unloading in the plastic regime Figure 2: Path followed for loading and unloading the microstructure for $u_{1}(t)$ and $u_{2}(t)$ by helping achieve convergence at a faster rate [39]. Since the problem is treated as a regression problem, with both positive and negative values potentially associated with each pixel, a PReLU is applied in the intermediate layers and a linear activation function in the final layer, illustrated in Fig. 5. The PReLU activation function is an improvement from the traditional ReLU function which converts all negative values to zero. This feature in the context of a neural network implies that none of the neurons are turned off, because PReLU allows for small negative values. The slope $a$ corresponding to $x<0$ in PReLU, as shown in Fig. 4(a), is a parameter that is learned during the training, which essentially fits a different value for each neuron. Thus, PReLU allows improved adaptability of the network with the data, which in turn results in better convergence. The loss function used in training the U-Net for any given 3D field $z^{(mnk)}$ is chosen as the mean Figure 4: 3D U-Net architecture for 32 frames of image Figure 5: Activation functions used in the U-Net architecture absolute error (MAE) between the predicted and true images, \[MAE=\frac{\sum_{m=1}^{N}\sum_{n=1}^{N}\sum_{k=1}^{K}\|z_{true}^{(mnk)}-z_{pred}^{( mnk)}\|}{N^{2}K} \tag{1}\] where $K$ is the number of load frames for which the $N\times N$ images are available, and $z_{pred}^{(mnk)}$ and $z_{true}^{(mnk)}$ are the pixel values at location $(x_{m},y_{n})$ and load frame $t_{k}$ of the predicted image and true image, respectively. For the purposes of this study, the learning rate of the gradient descent algorithm was selected to be 0.001 and the exponential decay rate for the first moment estimates of the Adam optimizer was set at 0.96 [40]. While these hyperparameters can be modified, this particular choice of the learning rate, the optimizer and its parameters are fairly standard for U-Net architectures. #### 2.2.1 Arrangement of the U-Net framework As illustrated in Fig. 6, three independent U-Net models provide direct mappings from the given microstructure $M^{(mn)}$ to each component $ij$ of the strain field $\epsilon_{ij}^{(mnk)}$. 'UNet_EIJ' (I,J $\epsilon\{1,2\}$) is adopted as the nomenclature to represent the U-Net model that predicts the (I,J) components of strain. The input channels may include only the microstructure, but the results in Section 3 will show that accuracy is improved significantly by adding another input channel with an image that explicitly indicates the magnitude of applied displacement at each load step. There are two options for the strain output channels: 1) direct output of the overall strain map $\epsilon_{ij}^{(mnk)}$; or 2) direct output of fiber strains $\epsilon_{ij}^{(F,mnk)}$ and matrix strains $\epsilon_{ij}^{(M,mnk)}$ separately, then subsequently joining them together into an overall strain map $\epsilon_{ij}^{(mnk)}$. In the second approach, the strains are separated by taking advantage of the microstructures $M^{(mn)}$, binary images in which 1 represents fiber region and 0 represents the matrix region. Therefore, for any load step $k$, a set of scaled strains associated with the fibers is found as: \[\epsilon_{ij}^{(F,mn)}=\lambda_{F}M^{(mn)}\circ\epsilon_{ij}^{(mn)} \tag{2}\] and similarly a set of scaled strains associated with the matrix is found as: \[\epsilon_{ij}^{(M,mn)}=\lambda_{M}(\mathbb{J}-M^{(mn)})\circ\epsilon_{ij}^{( mn)} \tag{3}\] where $\mathbb{J}$ represents a matrix whose elements are ones and '$\circ$' represents element-by-element multiplication. $\lambda_{F}$ and $\lambda_{M}$ are scaling factors for the fibre and matrix strains respectively. Finally, the predicted strain maps can be reconstructed as \[\epsilon_{ij}^{(mnk)}=\frac{\epsilon_{ij}^{(F,mnk)}}{\lambda_{F}}+\frac{ \epsilon_{ij}^{(M,mnk)}}{\lambda_{M}} \tag{4}\] After testing some trial values for $\lambda_{F}$ and $\lambda_{M}$, it was determined that $10^{3}$ and $10^{4}$, respectively, led to accurate predictions. However, no formal studies were conducted to evaluate the effect of the constants on the error. The results from direct output of the total strain map and from direct output of separate fiber and matrix strains will be evaluated in the results in Section 3. Four stress fields ($\sigma_{ij}^{(mnk)}$ and von-Mises stress $\sigma_{VM}^{(mnk)}$) are predicted using a single U-Net ('UNet_S') with multi-channel output. As illustrated in Figure 7, there are two options for training a U-Net to predict the stress fields: 1) training based solely on the microstructure $M^{(mn)}$ as input (direct approach); and 2) training based on the microstructure $M^{(mn)}$ and the strain field predictions from the strain U-Nets $\epsilon_{ij}^{(mnk)}$ as input (2-step approach). These options will be evaluated further in the results in Section 3. #### 2.2.2 Data Set Each data unit has a third dimension, as illustrated in Fig. 7, corresponding to the thirty-two points on the load path indicated in Fig. 2. In the current work, a total of 420 microstructures were used for training the U-Net models, while 44 additional microstructures were reserved for testing purposes. Figure 6: Schematic representation of UNet_EIJ. The input channel(s) contain the microstructure and optionally a color map that indicates the applied displacements at each load step (both axial and shear). The output channel could contain the complete strain map $\epsilon_{ij}^{(mnk)}$ directly, or it could include separate channels for fiber strains $\epsilon_{ij}^{(F,mnk)}$ and matrix strains $\epsilon_{ij}^{(M,mnk)}$ that are subsequently brought together to predict the overall strain $\epsilon_{ij}^{(mnk)}$. Figure 7: A typical 3D data arrangement for the UNet_S architecture; $H=$ Height of the image ; $W=$ Width of the image; $T=$ Temporal information stored as the third dimension in the data unit. Two approaches to calculate stress are considered: (a) direct prediction of stresses based on microstructure and load factors, and (b) prediction of stresses based on microstructure and the U-Net-predicted strain fields. The testing dataset was specifically chosen to contain microstructures that were not seen during the training phase. This separation allows for evaluating the generalizability of the trained models on unseen data. Additionally, by taking advantage of the symmetry of the problem, the microstructures and the corresponding stress and strain fields were rotated by 180${}^{\circ}$ and were added to the dataset. Therefore, the size of the training dataset was doubled and 840 training microstructures were obtained. The size of the dataset was fixed once the predicted mean strain field evolutions had a good agreement with the true values. ## 3 Results and Discussion To ensure that training was performed sufficiently, the UNet_EIJ networks were trained for 10 epochs and all the models converged in 3 epochs. UNet_S was trained for 25 epochs and was found to converge in 6 epochs. These results have been summarised in Fig. 8. Comparing the training and validation results in these figures it is evident that none of the models had overfitting problems. ### Predicted strain and stress maps at specific load frames The U-Net models generate stress and strain predictions corresponding to a particular input microstructure, for all 32 load frames. For the purposes of visualization, strain maps are provides for two specific instances: (i) frame 10, on the loading path at 4.5% strain (Fig. 9), and (ii) frame 25, on the unloading path at 4.0% strain (Fig. 10). To evaluate the localised error distribution in the stress/strain maps, these figures show the relative error (RE) for each pixel using, \[RE=\frac{\|z_{true}^{mn}-z_{pred}^{mn}\|}{max\{\|z_{true}^{mn}\|\}} \tag{5}\] Figure 8: Convergence of the U-Net models where $z_{true}^{mn}$ and $z_{pred}^{mn}$ are the true and predicted pixel values. Rather than normalizing with respect to each true pixel value in this expression, the normalization is done with respect to the maximum of the true pixel values, to avoid problems at locations where the true pixel value is close to zero. The strain map predictions are very close to the true images, with the RE for frame 10 (loading path) lying in the range of 0-25% and a maximum average RE of 1.1% for all the components. For frame 25 (unloading path), the RE lies in the range 0-40%, with an average RE less than 1.6% for all three strain components. To help visualize the local differences between the strain maps, the strains along two lines through the microstructure are provided in Fig. 11a and Fig. 11b, for load frames 10 and 25, respectively. All the strain components show a very good agreement with the predicted strains in both directions. The most significant errors are found in the matrix, near or at fiber-matrix boundaries. As mentioned earlier, input to the U-Net model for strains, UNet_EIJ, can include images that represent the load factors at each load step. To assess whether or not this is preferable, UNet_EIJ is trained with and without a 3D image that indicates the applied displacements (axial and shear) at each load step, as illustrated in Fig. 6. Fig. 12 shows that the U-Net model with these load factor images as an explicit input leads to an approximately 3$\times$ decrease in mean relative prediction error. Therefore, in all other analyses in this current work, these load factors are included in the strain evaluation. Also mentioned earlier, the output for each UNet_EIJ can either include a direct prediction of the strain, Figure 9: Comparison of the predicted and the true strain maps in the unloading stage (Frame 10) or it can include two channels, the fiber strain $\epsilon_{ij}^{F}$ and matrix strain $\epsilon_{ij}^{M}$ that are merged after the analysis. As depicted in Figure 13, the predicted $\epsilon_{11}$ maps along the unloading path are displayed for two different approaches: (i) training after separation of fiber and matrix strains (ii) training without the separation of fiber and matrix strain. Predictions achieved through the separation of fiber and matrix strains effectively replicate the majority of the high strain regions present in the actual map, while predictions made without the separation of fiber and matrix strains fail to capture high strain regions and exhibit an inaccurate smoothing effect around the fibers. Because of the superior performance using separate fiber and matrix strains, all other analyses in this paper are based on this approach. U-Net predictions of the stress maps at frame 10 (loading) are shown in Fig. 14. These results show good agreement between the predictions and the ground truth FE analyses, with local RE ranging from 0 to 40% and a mean RE less than 3.3% for all stresses. This error is a little higher than that observed for the strains, but still remains quite small. There is more visible error in the unloading stage at frame 25 (Fig. 15). Again, the RE ranges from 0 to 40% but the higher error regions are more widely dispersed, as indicated by a mean RE as high as 5.9% in the case of the shear stress ($\sigma_{12}$). In particular, the error plots show that some of the localized regions around the fiber-matrix interfaces exhibit higher errors. The high error around the fiber-matrix interface is due in part to the discretization of the circular fibers with a coarse rectangular grid in order to generate the images. Such interfaces present Figure 10: Comparison of the predicted and the true strain maps in the unloading stage (Frame 25) a challenge, even with standard FE simulations. The unloading stage presents a particularly difficult challenge, as the average stress, and therefore the range of stresses, is decreasing. The ML model is Figure 11: Variation of strains along the midsections. (a) Frame 10 (b) Frame 25 (The green zones signify the regions where the line intersects a fiber) Figure 12: Comparison of the RE in $\epsilon_{11}(x,y)$ at the time step on the unloading path (frame 25), training the strain U-Net model with and without the load factors as explicit input. trained for all stresses at all load steps simultaneously, so that capturing the relatively small variations in stresses at the late unloading steps is difficult. This effect is particularly noticeable in the prediction of $\sigma_{12}$, which has a smaller range of stresses than the other stress components throughout the load path. Some of these errors are easier to visualize by considering the stresses along the two perpendicular lines for frames 10 and 25. The predicted stresses and the true stresses on the loading path (frame 10) agree with each other in most cases (Fig. 16a). The most significant errors are found in the regions near fiber boundaries. The predicted stresses exhibit more error on the unloading path (Fig. 16b), with error found not only near fiber-matrix interfaces but also within fiber and/or matrix interior regions. As mentioned earlier, it is possible to predict the stresses based directly on the microstructure and the load factors as input, using the U-Nets in a similar way as for the strains. Another approach is Figure 14: Comparison of the predicted and the true stress maps in the loading stage (Frame: 10) Figure 13: Comparison of the predicted $\epsilon_{11}$ on the unloading path (Image frame : 25) maps with fiber-matrix separation and without fiber-matrix separation. to use a 2-step approach, in which the U-Net-predicted strains are an input to the stress U-Net, along with the microstructure. In Fig. 17a the RE maps for the von-Mises stress (on the unloading path) are given. The error map for the direct approach shows that there are more regions with a higher error than the 2-step approach. The median error for frame 25 shows that the direct approach has a higher mean RE (5.4%) than the 2-step approach (4.4%). Based on these observations, the 2-step approach is used in all other analyses in this paper. ### Sample to sample variability in mean relative error The RE, as defined in Eq. 5, provides a measure of the normalized pixel-by-pixel error that can be averaged over the image to yield a mean RE. This mean RE varies from one instantiation of microstructure to the next, which is not reflected in the values reported in Figs. 9, 10, 14, or 15. Using all of the test data from this analysis, the resulting ensembles of mean RE are depicted in Fig.18a and Fig.18b. The box plots for strain maps in Fig.18a show that the median values for the mean errors are quite low, less than 2% even in the worst case at Frame 25. The sample-to-sample variations in this mean RE are also quite small. The box plots for stress maps show an increase in the mean RE, with an increase in both the median and the scatter when going from the loading to the unloading path. On the loading path (Frame 10) the median value of the mean RE remains lower than 4%. However, the unloading path (Frame 25) shows a median mean RE of around 6% for $\sigma_{12}$. An intriguing observation from this plot is that although $\sigma_{11}$, $\sigma_{22}$, and $\sigma_{12}$ exhibit a significant increase in mean RE when going from loading to unloading, the RE in the von-Mises stress $\sigma_{VM}$ remains relatively low. Von-Mises Figure 15: Comparison of the predicted and the true stress maps in the unloading stage (Frame: 25) stress is a function of the other stress components - $\sigma_{11}$, $\sigma_{22}$, and $\sigma_{12}$ - suggesting there may be some sort of error balancing occurring in the von-Mises formulation of stress. Figure 16: Variation of stresses along the midsection. (a) Frame 10 (b) Frame 25 (The green zones signify the regions where the line intersects the fiber) Figure 17: Comparison of the direct approach and the 2-step approach of von-Mises stress prediction at frame 25 on the unloading path. ### Quantification of average error for entire load path While it is hard to visualize the full stress and strain fields at all 32 load frames, it is useful to address the evolution of average error in these quantities over the entire load path. This is achieved using two metrics: $R^{2}$ values, and average stress-strain curves. Figure 18: Box-plots for the mean relative error in the test dataset for (a) Strains (b) Stresses #### 3.3.1 Evaluation of the $R^{2}$ values for strain and stress maps for each frame In this section, the fit of the predicted values $z_{pred}^{(mn)}$ and the true values $z_{true}^{(mn)}$ with the $y=x$ straight line is determined using the coefficient of determination ($R^{2}$). $R^{2}$ can be calculated as, \[R^{2}=1-\frac{\Delta z}{Z} \tag{6}\] where, \[\Delta z=\sum_{m,n}^{N}(z_{true}^{(mn)}-z_{pred}^{(mn)})^{2} \tag{7}\] \[Z=\sum_{m,n}^{N}(z_{true}^{(mn)}-\bar{z}_{true})^{2} \tag{8}\] and \[\bar{z}_{true}=\frac{1}{N^{2}}{\sum_{m,n}^{N}}z_{true}^{(mn)} \tag{9}\] Therefore, $R^{2}$ is related to the spread of the predicted pixel values from the true values. Fig. 19a shows $R^{2}$ for the strain predictions, as a function of load frame. $R^{2}$ remains almost constant for both axial strains, with the exception of the final frame, at which there is a sharp drop. For the shear strain $\epsilon_{12}$, there is a steady decrease in $R^{2}$ with load frame, but all the values remain above 0.90 except for the last frame which has a value of 0.73. Fig. 19b shows $R^{2}$ for the stress predictions, which exhibits a similar behavior as the strain predictions. For most of the frames, this value is relatively constant and remains close to 1. After unloading at around Frame 20, the $R^{2}$ values begin to decrease, in particular for $\sigma_{11}$ and $\sigma_{22}$. Even in the worst case of $\sigma_{11}$ at the final load frame, however, the $R^{2}$ value is around 0.85. Figure 19: Variation of the $R^{2}$ values with the image frames. Frames refer to each image along the third dimension and each frame is related to the sampled time steps from the load path in Fig. 2 (a) Strain Maps (b) Stress Maps ### Average stress-strain response of the microstructure Given that these ML models might be expected to predict average mechanical behavior, it is useful to consider the average strains and stresses predicted by the model. Fig. 20 shows the variation of the Figure 21: Average stress-strain relationship of the microstructure Figure 20: Variation of average strain with time step average strains in the microstructure over the total duration of loading and unloading, comparing the predicted and true values. There is excellent agreement between the average strains from the predicted maps (from UNet_EIJ) and the average strains from the true maps from FE analysis. Fig. 21 shows the average stress-strain behavior throughout the load path. Considering the von-Mises stress, yielding is observed at the matrix yield stress of $210MPa$, with subsequent hardening behavior due to the presence of the elastic fibres. This expected response is captured accurately by the U-Net predictions. In fact, there is excellent agreement in all of these curves, with the possible exception of $\sigma_{22}$, which shows some discrepancies, likely because this stress is of smaller magnitude relative to the other stresses and therefore more difficult to capture. ## 4 Conclusions In this study, we aimed to develop a data-driven method to predict the stresses and strains in a two-phase elastoplastic material using 3D convolutions in a U-Net type framework. The approach is orders of magnitude faster than the equivalent FE analysis, which is key to multiscale modeling and/or stochastic simulation of local material response. 3D convolutions are capable of addressing the temporal correlations in a series of stress or strain maps at different instants of the load path, which are akin to a sequence of images. Based on these analyses, we can draw the following conclusions: 1. 3D U-Net can predict stresses and strains in an inelastic medium with arbitrary fiber placement, under a given load path. It can therefore produce accurate results in the plastic domain based on limited prior information. 2. There is excellent agreement between U-Net and FEM in predicting local strains. The maximum mean RE observed for all strain maps was approximately $1.1\%$ on the loading path and $1.6\%$ on the unloading path. 3. There is also very good agreement between U-Net and FEM in predicting local stresses. The maximum mean RE observed for all stress maps was approximately $3.3\%$ on the loading path and $5.9\%$ on the unloading path. 4. Upon calculating the mean RE for all the unseen microstructures, it was observed that the strain predictions are highly accurate, with the maximum median error being less than $1.5\%$ in loading as well as unloading cases. The stresses show somewhat of an increase in the median $RE$ in the unloading cases, with a maximum around $6\%$. 5. The predicted average stress-strain response shows excellent agreement with the FE prediction of the stress-strain response. 6. $R^{2}$ values are calculated for strains in the range of $0.73-0.98$ and for stresses in the range of $0.85-0.98$, demonstrating excellent agreement between the U-Net and FE predictions. Although there are some localized regions of high error in the predicted stresses, the model captures local changes in inelastic stresses and strains with excellent accuracy, at a small fraction of the computational cost required by FE analysis. This work did not consider changes in fiber sizes, shapes and/or volume fraction, which is left as a future direction. Prior work in this regard on elastic stresses suggests that the U-Net model might be capable of representing these changes to the microstructure. A more challenging future direction is to consider other types of microstructures, such as polycrystalline materials. Given the larger number of possible characteristics represented at each pixel in the microstructure (e.g., different grain orientations), as opposed to the two possibilities in a fiber-reinforced composite (i.e., fiber or matrix), more data might be required to train the model effectively. Another consideration is the trade-off between training time and analysis time. Once trained, the U-Net model provides a very rapid and accurate assessment of local stresses and strains; however, the training process itself takes some computational effort and also requires data from the finite element analyses. The choice on whether to implement such an ML model depends on the number of model evaluations one is expecting to perform. For example, in a multiscale model that has a unique microstructure underlying each integration point of a larger scale model, there are expected to be a large number of micro-scale model evaluations. In such a case, the U-Net model is ideal. On the other hand, if one only wants a small number of micro-scale evaluations, then the data collection and training process may be more cumbersome than the time associated with simply running FE analyses of each individual case. Formulation of specific quantitative metrics to identify this trade-off point will be highly problem dependent, and therefore no general rule of thumb is available here. Finally, it is important to note that the current work is an extremely powerful tool for predicting local stresses and strains in a microstructure, under a fully known load path. The enormous increase in computational speed provided by the U-Net promises to enable efficient multi-scaling and/or uncertainty quantification for complex mechanics models. However, it is important to recognize that in a multiscale mechanics context, these individual microstructures might be part of a larger scale model in which the applied displacements/strains on each microstructure are associated with individual nodes or integration points. In this case, the load path applied to each microstructure could vary from one point to the next. Therefore, a clear next step for such U-Net models is to generalize the model for multiple load paths. ## 5 Acknowledgements Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-22-2-0014 (Artificial Intelligence for Materials Design) and Cooperative Agreement Number W911NF-23-2-0062 (AI-Driven Integrated and Automated Materials Design for Extreme Environments). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
2306.04847	Embedding stochastic differential equations into neural networks via dual processes	We propose a new approach to constructing a neural network for predicting expectations of stochastic differential equations. The proposed method does not need data sets of inputs and outputs; instead, the information obtained from the time-evolution equations, i.e., the corresponding dual process, is directly compared with the weights in the neural network. As a demonstration, we construct neural networks for the Ornstein-Uhlenbeck process and the noisy van der Pol system. The remarkable feature of learned networks with the proposed method is the accuracy of inputs near the origin. Hence, it would be possible to avoid the overfitting problem because the learned network does not depend on training data sets.	Naoki Sugishita, Jun Ohkubo	2023-06-08T00:50:16Z	http://arxiv.org/abs/2306.04847v2	# Embedding stochastic differential equations into neural networks via dual processes ###### Abstract We propose a new approach to constructing a neural network for predicting expectations of stochastic differential equations. The proposed method does not need data sets of inputs and outputs; instead, the information obtained from the time-evolution equations, i.e., the corresponding dual process, is directly compared with the weights in the neural network. As a demonstration, we construct neural networks for the Ornstein-Uhlenbeck process and the noisy van der Pol system. The remarkable feature of learned networks with the proposed method is the accuracy of inputs near the origin. Hence, it would be possible to avoid the overfitting problem because the learned network does not depend on training data sets. ## 1 Introduction Many researchers have studied the evolution of dynamical or stochastic systems in physics and other research fields. Recently, the amount of data has exponentially increased, and there are many studies on dynamical systems based on these large data sets. One of the aims of these studies is to transform observed data into predictive models of the physical world, and neural networks are a hopeful candidate for this aim. Of course, a simple application of conventional neural networks would not be enough because the physical world has many characteristics due to various constraints. For example, it would be beneficial to equip features of the time-evolution into the learning steps to make the prediction more accurate. Some ideas have appeared recently; the numerical scheme for time-evolution with multi-step time-stepping schemes is employed [1]. Some works focused on network architectures. Reference [2] discussed a network architecture for the time-evolution partial differential equation. In [3, 4], recurrent neural networks were applied for model predictive control. There is a study to deal with fluid flow simulation with long short-term memory (LSTM) [5]. Other types of discussions based on the universal approximation theorem were given in [6], in which DeepONet was also proposed. The physics-informed machine learning has the same research direction. In [7, 8], frameworks for learning partial differential equations were discussed; see a review [9] for this topic. There are studies in which conservation laws are combined with learning; the inclusion of the Hamiltonian structures makes learning more stable [10]. A recent paper gives a good review of this topic in the introduction [11]. However, compared with the studies on deterministic dynamical systems, there has been little discussion about systems with noise, i.e., stochastic systems. Such stochastic systems are the topic of the present paper. One of the problems of machine learning for physical subjects is the data sets. For example, when one constructs a neural network for prediction from an input coordinate, it is necessary to prepare data with various initial conditions. Note that stochastic systems require large data sets compared with deterministic systems; in stochastic systems, we must consider statistics of predictions, and the calculation of expected values takes high computational cost. Hence, we need more computational effort in the data preparation and the learning steps. However, if we have the information of equations governing the system as the prior knowledge, it could be possible to reduce the size of the data sets and costs for learning. Of course, it will be possible to apply additional online learning steps to the trained network to obtain a more accurate one. Hence, the key question of the present paper is as follows: How should we cooperate with the information on the time-evolution equation for stochastic systems? If we can embed the prior knowledge directly without data sets, the method would complement the conventional ones. In the present paper, the aim is to predict statistics after time evolution in stochastic systems, especially stochastic differential equations. For this aim, we propose a new Figure 1: Conventional and proposed methods for the learning of stochastic systems. A most naive approach is to generate data with Monte Carlo methods; target statistics are evaluated from the sampled data sets. Employing dual processes enables us to skip the sampling step; it is possible to evaluate the statistics directly by solving the backward Kolmogorov equation. These two approaches give a data set with the inputs and outputs, and then neural networks are learned with conventional backpropagation. By contrast, the proposed method employs direct comparison with the information of the dual processes. method to learn neural networks without generating data sets for statistics. The key of the proposal is the usage of dual processes. The duality of stochastic processes has long been studied in statistical physics and mathematical physics [12], and recent studies clarified that the conventional backward Kolmogorov equation gives dual processes straightforwardly. Figure 1 summarizes the proposal. In a most naive approach, we generate data with Monte Carlo samplings and calculate target statistics for various initial conditions. As shown later, one can evaluate the target statistics directly from the dual process. In both methods, after obtaining a data set with pairs of initial conditions and the target statistics, the backpropagation gives leaned neural networks. By contrast, the proposed method employs direct comparison with the information of the dual processes. Hence, we utilize an optimization procedure instead of backpropagation. Since there is no need to evaluate statistics from the samplings, it would be possible to write that the proposed method embeds stochastic differential equations directly into neural networks. We demonstrate the proposed method with one-dimensional and two-dimensional noisy systems, which will clarify the learned features different from the conventional approach. The remaining part of the present paper proceeds as follows. In Section 2, we review the method to evaluate statistics of the stochastic differential equations without any sampling. Section 3 gives the main proposal to construct neural networks directly from the stochastic differential equations. Two numerical demonstrations are given in section 4. Finally, section 5 concludes this paper. ## 2 Numerical method to evaluate statistics without sampling In the present paper, we only focus on the stochastic differential equations and the statistics after the time-evolution. As for the basics of the stochastic differential equations, see [13] for example. Here, Let $\mathbf{x}(t)$ be a $D$-dimensional random vector which obeys the following stochastic differential equation: \[d\mathbf{x}=\mathbf{a}(\mathbf{x})dt+B(\mathbf{x})d\mathbf{W}(t), \tag{1}\] where $\mathbf{a}(\mathbf{x})$ is a vector function called the drift coefficient and $B(\mathbf{x})$ is a matrix-valued function called the diffusion coefficient. $\mathbf{W}(t)$ is a vector of Wiener processes. Since one cannot generally solve stochastic differential equations analytically, the Monte Carlo method is employed to sample trajectories from stochastic differential equations. The famous method is the Euler-Maruyama method; for example, see [14]. In the Euler-Maruyama method, a time-discretization is necessary; the sampling steps are time-consuming when we want to evaluate statistics with high accuracy for various initial conditions. As stated in the introduction, the usage of duality in stochastic processes has long been studied in physics [12]. While the usage was mainly restricted to exactly solvable cases, recent studies give a simple derivation of dual processes from stochastic differential equations [15]; there is also an algorithm based on combinatorics to evaluate the statistics of stochastic differential equations [16]. Here, there is no need to know the details of the duality because a discussion based on the backward Kolmogorov equation is enough for the aim of the present paper. Hence, we briefly review the key points which are enough to understand the proposals in the next section. First, we consider the following Fokker-Planck equation instead of the stochastic differential equation [13]: \[\frac{\partial}{\partial t}p(\mathbf{x},t)=\mathcal{L}p(\mathbf{x},t), \tag{2}\] where $p(\mathbf{x},t)$ is the probability density function and $\mathcal{L}$ is the time-evolution operator defined as \[\mathcal{L}=-\sum_{i}\frac{\partial}{\partial x_{i}}a_{i}(\mathbf{x})+\frac{1}{2} \sum_{i,j}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\left[B(\mathbf{x},t)B( \mathbf{x})^{\mathrm{T}}\right]_{i,j}. \tag{3}\] The aim here is to evaluate the $m$-th order moment of the $i$-th element of $x(t)$; i.e., \[M_{i}^{(m)}(\mathbf{x}_{0},t)=\mathbb{E}_{p(\mathbf{x},t\|\mathbf{x}_{0},t_{0})}[x_{i}^{m}], \tag{4}\] where $p(\mathbf{x},t\|\mathbf{x}_{0},t_{0})$ is the probability density function of $\mathbf{x}(t)$ with the initial condition $\mathbf{x}(t_{0})=\mathbf{x}_{0}$. The initial condition is written as \[p(\mathbf{x},t_{0})=\delta(\mathbf{x}-\mathbf{x}_{0}), \tag{5}\] where $\delta(\cdot)$ is the Dirac delta function. Second, we derive the backward Kolmogorov equation instead of the Fokker-Planck equation; as for the backward Kolmogorov equation, see [17]. Since the time-evolution of the probability density function is formally denoted as \[p(\mathbf{x},t\|\mathbf{x}_{0},t_{0})=e^{\mathcal{L}(t-t_{0})}p(\mathbf{x},t_{0}), \tag{6}\] the derivation of the time-evolution operator for the backward Kolmogorov equation is easily understood as follows: \[M_{i}^{(m)}(\mathbf{x}_{0},t) =\int x_{i}^{m}p(\mathbf{x},t\|\mathbf{x}_{0},t_{0})d\mathbf{x} \tag{7}\] \[=\int x_{i}^{m}\left\{e^{\mathcal{L}(t-t_{0})}p(\mathbf{x},t_{0}) \right\}d\mathbf{x}\] \[=\int x_{i}^{m}\left\{e^{\mathcal{L}(t-t_{0})}\delta(\mathbf{x}-\mathbf{x }_{0})\right\}d\mathbf{x}\] \[=\int\left\{e^{\mathcal{L}^{\dagger}(t-t_{0})}x_{i}^{m}\right\} \delta(\mathbf{x}-\mathbf{x}_{0})d\mathbf{x}\] \[=\int\varphi(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{x}_{0})d\mathbf{x}\] \[=\varphi(\mathbf{x}_{0},t),\] where $\mathcal{L}^{\dagger}$ is the adjoint operator of $\mathcal{L}$ and $\varphi(\mathbf{x},t)$ is a solution of a time evolution equation \[\frac{\partial}{\partial t}\varphi(\mathbf{x},t)=\mathcal{L}^{\dagger}\varphi(\bm {x},t). \tag{8}\] The adjoint operator $\mathcal{L}^{\dagger}$ is written as \[\mathcal{L}^{\dagger}=\sum_{i}a_{i}(\mathbf{x})\frac{\partial}{\partial x_{i}}+\frac{ 1}{2}\sum_{i,j}\left[B(\mathbf{x})B(\mathbf{x})^{\mathrm{T}}\right]_{i,j}\frac{\partial ^{2}}{\partial x_{i}\partial x_{j}}, \tag{9}\] which corresponds to the time-evolution operator for the backward Kolmogorov equation. In (7), we employed integration by parts and the fact that the probability density functions will vanish at $x_{i}\to\pm\infty$. Note that the initial condition for (8) is \[\varphi(\mathbf{x},t_{0})=x_{i}^{m}, \tag{10}\] which corresponds to the target statistic. Here, we focus on the value of the solution of (8) at a certain coordinate $\mathbf{x}_{0}$, i.e., $\varphi(\mathbf{x}_{0},t)$; the value immediately gives the expectation of the target statistic, $M_{i}^{(m)}(\mathbf{x}_{0},t)$. Third, we expand $\varphi(\mathbf{x},t)$ with monomial basis functions, $\mathbf{x}^{\mathbf{n}}=\prod_{i}x_{i}^{n_{i}}$, so that \[\varphi(\mathbf{x},t)=\sum_{\mathbf{n}\in\mathbb{N}^{D}}P_{i}^{(m)}(\mathbf{n},t)\mathbf{x}^{ \mathbf{n}}, \tag{11}\] where $\{P_{i}^{(m)}(\mathbf{n},t)\}$ are the expansion coefficients. By substituting this equation into (7), we have \[M_{i}^{(m)}(\mathbf{x}_{0},t)=\sum_{\mathbf{n}\in\mathbb{N}^{D}}P_{i}^{(m)}(\mathbf{n},t) \mathbf{x}_{0}^{\mathbf{n}}. \tag{12}\] Hence, it is enough to obtain the coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ to evaluate the expectation $M_{i}^{(m)}(\mathbf{x}_{0},t)$. Employing the basis expansion in (11) with the time-evolution equation in (8), we have \[\sum_{\mathbf{n}\in\mathbb{N}^{D}}\left\{\frac{\partial}{\partial t}P_{i}^{(m)}( \mathbf{n},t)\right\}\mathbf{x}^{\mathbf{n}}=\mathcal{L}^{\dagger}\left\{\sum_{\mathbf{n}\in \mathbb{N}^{D}}P_{i}^{(m)}(\mathbf{n},t)\mathbf{x}^{\mathbf{n}}\right\}, \tag{13}\] which gives the simultaneous ordinary differential equations for $\{P_{i}^{(m)}(\mathbf{n},t)\}$ by comparing the coefficients of the basis expansion. Note that the initial condition should be \[P_{i}^{(m)}(\mathbf{n},t_{0})=\delta_{n_{i},m}, \tag{14}\] where $\delta_{\cdot,\cdot}$ is the Kronecker delta function. The initial condition stems from $\varphi(\mathbf{x},t_{0})$ in (10). We will denote examples of the simultaneous ordinary differential equations for $\{P_{i}^{(m)}(\mathbf{n},t)\}$ later, which will help the reader understand the above discussion. Here are some comments on the above discussion. The first comment is on duality in stochastic processes: the derived equations for $\{P_{i}^{(m)}(\mathbf{n},t)\}$ do not satisfy the law of conservation of probability, so the coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ are not probabilities. As shown in [18], it is possible to recover the probabilistic characteristics to extend the discussions, which leads to the connection with the duality relation in stochastic processes. However, as written later, the coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ are enough to learn neural networks; there is no need to recover the probabilistic characters here. The second comment is related to the target statistics. The above discussion is limited to the target statistics in (4), i.e., a simple moment. While the restriction simplifies the discussion, we should mention that other statistics, such as correlations, can also be evaluated. In summary, the moment $M_{i}^{(m)}(\mathbf{x}_{0},t)$ is evaluated by solving the simultaneous ordinary differential equations derived from (13), without samplings of stochastic processes. Our goal is to construct a neural network to predict the target moment. The learning procedure needs expectations for various initial coordinates. Furthermore, stochastic cases need many samples for a single initial coordinate to evaluate the expected value, which requires high computational costs. Then, the above approach based on the simultaneous ordinary differential equations enables us to avoid the samplings. However, we still need to prepare a data set, pairs with an input coordinate and the target moment, to use conventional learning frameworks in previous studies. Is there a more efficient method suitable for stochastic differential equations? Next, we propose a simple way to answer this question and directly embed the information in the equations to neural networks via $\{P_{i}^{(m)}(\mathbf{n},t)\}$. ## 3 Proposed method Let us consider the following neural network with a single hidden layer: The input is the initial coordinate of stochastic process $\mathbf{x}_{0}$, the output is an estimate of the moment $M_{i}^{(m)}(\mathbf{x}_{0},t)$, the number of nodes of the hidden layer is $n$, and the activation function is a sigmoid function $\sigma(x)=1/(1+e^{-x})$. While this neural network is not deep, the universal approximation theorem guarantees the approximation ability when we use many hidden nodes [19, 20]. As demonstrated later, the simple structure works well for examples with nonlinear coefficients. The key of the proposed method is the direct comparison of the coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ in (12) with the corresponding components in the neural network. Although the direct comparison is a simple idea, it yields preferable learning results, as discussed later. First, the output of the neural network, $y\in\mathbb{R}$, given the input $\mathbf{x}\in\mathbb{R}^{D}$ can be represented as \[y =\mathbf{q}^{\rm T}\mathbf{\sigma}(R\mathbf{x}+\mathbf{s}) \tag{15}\] \[=\sum_{i=1}^{n}q_{i}\sigma\left(\sum_{j=1}^{D}R_{ij}x_{j}+s_{i} \right),\] where $\mathbf{q},\mathbf{s}\in\mathbb{R}^{n},R\in\mathbb{R}^{n\times D}$ are the weights of the neural network. Using the Taylor expansion of $\sigma(x)$ up to the $N$-th order, we can approximate the output with the power series of the input $\mathbf{x}$ as follows: \[y\approx\sum_{\{\mathbf{l}\in\mathbb{N}^{D}\|\sum_{j}l_{j}\leq N\}} \sum_{k=\sum_{j}l_{j}}^{N}\sum_{i=1}^{n}\left(\begin{array}{c}k\\ l_{1},\ldots,l_{D},k-\sum_{j}l_{j}\end{array}\right)\frac{\sigma^{(k)}(0)}{k! }q_{i}\mathbf{r}_{i}^{l}s_{i}^{k-\sum_{j}l_{j}}\mathbf{x}^{\mathbf{l}}, \tag{16}\] where $\sigma^{(k)}(\cdot)$ denotes the $k$-th derivative of $\sigma(\cdot)$ and $\mathbf{r}_{i}$ is the $i$-th row vector of the weight matrix $R$. Note that a notation for multinomial coefficients \[\left(\begin{array}{c}k\\ k_{1},\ldots,k_{r}\end{array}\right)=\frac{k!}{k_{1}!\ldots k_{r}!} \tag{17}\] is used in (16). At this stage, the correspondence between (12) and (16) is clear; both have the basis-expansion form for $\mathbf{x^{n}}$ or $\mathbf{x^{l}}$. Hence, it is possible to employ a direct comparison between them. For example, one can use the sum of the square errors of coefficients in (12) and (16) as a cost function: \[C_{i,t}^{(m)}(\mathbf{q},R,\mathbf{s})=\sum_{\{\mathbf{l}\in\mathbb{N}^{D}\|\sum_{j}l_{j} \leq N\}}\left(P_{i}^{(m)}(\mathbf{l},t)-P_{\mathrm{NN}}(\mathbf{l},\mathbf{q},R,\mathbf{s}) \right)^{2}, \tag{18}\] where \[P_{\mathrm{NN}}(\mathbf{l},\mathbf{q},R,\mathbf{s})=\sum_{k=\sum_{j}l_{j}}^{N}\sum_{i=1}^ {n}\left(\begin{array}{c}k\\ l_{1},\ldots,l_{D},k-\sum_{j}l_{j}\end{array}\right)\frac{\sigma^{(k)}(0)}{k! }q_{i}\mathbf{r^{l}}_{i}^{k-\sum_{j}l_{j}}. \tag{19}\] By minimizing this cost function $C_{i,t}^{(m)}$, we obtain the neural network that estimates the moment $M_{i}^{(m)}(\mathbf{x}_{0},t)$. Note that the number of simultaneous ordinary differential equations derived from (13) is infinite. Of course, we cannot evaluate the infinite number of coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$. However, with the comparison in (18), it is enough to consider a finite set of $\{P_{i}^{(m)}(\mathbf{n},t)\}$ for $\left\{\mathbf{n}\in\mathbb{N}^{D}\|\max_{i}n_{i}\leq N\right\}$; the order of Taylor approximation, $N$, relates to the upper bound. From this fact, it is possible to approximate the simultaneous ordinary differential equations for $\{P_{i}^{(m)}(\mathbf{n},t)\}$ with the limited range of indexes $\{\mathbf{n}\in\mathbb{N}^{D}\|\max_{i}n_{i}\leq N\}$; we employ this approximation in the next section. Note that it would be natural to determine the order of Taylor approximation from the number of coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ used in the cost function in (18). The coefficients stem from the expansion of the function $\varphi(\mathbf{x},t)$ in (11) for the backward Kolmogorov equation. Hence, the number of coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ is related to the approximation accuracy of $\varphi(\mathbf{x},t)$. In other words, it is enough to use cutoffs that yield enough coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ of the function $\varphi(\mathbf{x},t)$ with the accuracy one wants to approximate. What kind of features can we expect from the proposed method? Here, we focus on the Taylor-type basis expansion in the proposed method. Hence, one could expect that the approximation performance is better when $\mathbf{x}_{0}$ is closer to the origin. Next, we confirm this conjecture with numerical demonstrations. ## 4 Numerical examples In this section, we demonstrate the proposed method for two stochastic differential equations. The first example is a famous one-dimensional model, i.e., the Ornstein-Uhlenbeck process [13]. The analytical solutions for moments are known for the Ornstein-Uhlenbeck process, and it is easy to compare the performance. The second example is the noisy van der Pol system, which is a two-dimensional model. The noisy version of the original van der Pol system [21] has already been used in filtering [22, 23] and recent studies related to data analysis for dynamical systems [24]. In both cases, the simultaneous ordinary differential equations for $P_{i}^{(m)}(\mathbf{n},t)$ are solved numerically via the functions of the Scipy package, scipy.integrate.solve_ivp. The minimization procedure for the cost function (18) is performed with scipy.optimize.least_squares, in which we randomly generate the initial values for $\mathbf{q},R,\mathbf{s}$ from a uniform distribution with the range $[-1,1)$. The initial time $t_{0}$ is $0$, and the time interval for each case is denoted in table 1. Table 1 also shows the network size, $n$, and the order of Taylor approximation, $N$. ### Ornstein-Uhlenbeck process The stochastic differential equation for the Ornstein-Uhlenbeck process is given as follows [13]: \[dx=-\gamma xdt+\sigma dW(t). \tag{20}\] Hence, the time-evolution operator $\mathcal{L}$ for the Fokker-Planck equation in (2) is given as \[\mathcal{L}=\frac{\partial}{\partial x}\gamma x+\frac{\partial^{2}}{\partial x ^{2}}\frac{\sigma^{2}}{2}, \tag{21}\] which leads to the adjoint operator in (9) as \[\mathcal{L}^{\dagger}=-\gamma x\frac{\partial}{\partial x}+\frac{\sigma^{2}}{ 2}\frac{\partial^{2}}{\partial x^{2}}. \tag{22}\] Since there is only one variable $x$, the coefficients are $\{P_{i}^{(m)}(n_{1},t)\}$ for $n_{1}=0,1,2,\cdots$. The derived simultaneous ordinary differential equations are as follows: \[\frac{d}{dt}P_{i}^{(m)}(n_{1},t)=-\gamma n_{1}P_{i}^{(m)}(n_{1},t)+\frac{ \sigma^{2}}{2}(n_{1}+2)(n_{1}+1)P_{i}^{(m)}(n_{1}+2,t) \tag{23}\] for $n_{1}=0,1,2,\cdots,N$. Note that we employ the finite cutoff with $N$, and set $P_{i}^{(m)}(n_{1},t)=0$ for $n_{1}>N$. Here, as denoted in Table 1, we set $N=12$. As for the numerical demonstrations, we construct two neural networks to estimate the first and second-order moments. As for the first-order moment case, the initial condition for (23) is set as $P_{i}^{(m)}(n_{1},0)=\delta_{n_{1},1}$; $P_{i}^{(m)}(n_{1},0)=\delta_{n_{1},2}$ is used for the second-order moment case. \begin{table} \begin{tabular}{l c c c} \hline \hline & Time interval $t$ & \begin{tabular}{c} Number of hidden \\ layer nodes $n$ \\ \end{tabular} & \begin{tabular}{c} Order of Taylor \\ approximation $N$ \\ \end{tabular} \\ \hline Ornstein-Uhlenbeck & 1 & 4 & 12 \\ noisy van der Pol & 0.1 & 8 & 17 \\ \hline \hline \end{tabular} \end{table} Table 1: Parameter settings used in the two examples. The analytical solutions for moments are as follows [13]: \[M^{(1)}(x_{0},t) =x_{0}e^{-\gamma t}, \tag{24}\] \[M^{(2)}(x_{0},t) =\left(x_{0}e^{-\gamma t}\right)^{2}+\frac{\sigma^{2}}{2\gamma} \left(1-e^{-2\gamma t}\right). \tag{25}\] Here, we set $\gamma=\sigma=1$. Figure 2 shows the numerical results obtained from the proposed method. We also depict the analytical results. As for the proposed method, after the learning procedures in section 3, various inputs $x_{0}=x(t=0)$ are used to obtain the corresponding outputs, and we drew the curves in figure 2. The numerical results indicate that the first-order moment is estimated well even in the small neural network. As for the second-order moment, the accuracy decreases as the input value is further away from the origin. This behavior is just the expected one from the characteristics of the Taylor approximations, as discussed in section 3. We here note that it is possible to change the origin. The expansion of $\varphi(\mathbf{x},t)$ in (11) is around zero. As discussed in [25], it is sometimes beneficial to shift the origin, which yields the different values for the coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$. Then, the learned networks show good agreements around the shifted origins; we have numerically checked these characteristics. From these results, it is possible to say the idea employed in the backward Kolmogorov equation and the optimization work well. The Ornstein-Uhlenbeck process is linear. Next, we use a two-dimensional example with nonlinear coefficients. ### Noisy van der Pol system The noisy van del Pol system obeys the following stochastic differential equations: \[d\left(\begin{array}{c}x_{1}\\ x_{2}\end{array}\right)=\left(\begin{array}{c}x_{2}\\ \epsilon x_{2}(1-x_{1}^{2})-x_{1}\end{array}\right)dt+\mathrm{diag}(\nu_{11}, \nu_{22})d\mathbf{W}(t), \tag{26}\] where $\epsilon$ and $\nu_{11},\nu_{22}$ are parameters, and $\mathrm{diag}(\cdot)$ means the diagonal matrix with the corresponding diagonal elements. Figure 2: Estimated moments of the Ornstein-Uhlenbeck process by the neural networks constructed with the proposed method. (A) First order moment. (B) Second order moment. The dashed lines correspond to The analytical solutions. The corresponding time-evolution operator $\mathcal{L}$ and its adjoint one $\mathcal{L}^{\dagger}$ are as follows: \[\mathcal{L} =-\frac{\partial}{\partial x_{1}}x_{2}-\frac{\partial}{\partial x_{2 }}\left(\epsilon x_{2}\left(1-x_{1}^{2}\right)-x_{1}\right)+\frac{\partial^{2} }{\partial x_{1}^{2}}\frac{\nu_{11}^{2}}{2}+\frac{\partial^{2}}{\partial x_{2 }^{2}}\frac{\nu_{22}^{2}}{2}, \tag{27}\] \[\mathcal{L}^{\dagger} =x_{2}\frac{\partial}{\partial x_{1}}+\left(\epsilon x_{2}\left(1 -x_{1}^{2}\right)-x_{1}\right)\frac{\partial}{\partial x_{2}}+\frac{\nu_{11}^{ 2}}{2}\frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\nu_{22}^{2}}{2}\frac{ \partial^{2}}{\partial x_{2}^{2}}. \tag{28}\] Then, the simultaneous ordinary differential equations for $\{P_{i}^{(m)}(n_{1},n_{2},t)\}$ become \[\frac{d}{dt}P_{i}^{(m)}(n_{1},n_{2},t) =(n_{1}+1)P_{i}^{(m)}(n_{1}+1,n_{2}-1,t)+\epsilon n_{2}P_{i}^{(m)} (n_{1},n_{2},t)\] \[\quad-\epsilon n_{2}P_{i}^{(m)}(n_{1}-2,n_{2},t)-(n_{2}+1)P_{i}^{ (m)}(n_{1}-1,n_{2}+1,t)\] \[\quad+\frac{\nu_{11}}{2}(n_{1}+2)(n_{1}+1)P_{i}^{(m)}(n_{1}+2,n_{ 2},t)\] \[\quad+\frac{\nu_{22}}{2}(n_{2}+2)(n_{2}+1)P_{i}^{(m)}(n_{1},n_{2}+ 2,t). \tag{29}\] As in the Ornstein-Uhlenbeck case, a finite cutoff is employed; $P_{i}^{(m)}(n_{1},n_{2},t)=0$ for $n_{1}>N$ or $n_{2}>N$. In the demonstration, we set $\epsilon=1$ and $\nu_{11}=\nu_{22}=1$. For the noisy van der Pol systems, there is no analytical result. Hence, we use the estimated values obtained from the method in section 2 as the approximate true value. The proposed method is the approach via the right side depicted in figure 1. For comparison, we perform the conventional machine learning approach with backpropagation. As explained in section 1, a naive approach, the left one in figure 1, requires samplings with the Monte Carlo method to estimate the expected values. However, it takes high computational costs. Hence, we here employ the approach via the middle one in figure 1; the expected values for various initial conditions are evaluated with the method in section 2. We make a data set with the pairs of an input coordinate and the target moment. Then, the neural network is learned from the data set. Of course, the Monte Carlo method with a considerably large data set gives the same learned network. The comparison approach requires various initial coordinates for the data set; we generated them from the uniform distribution with the range $[-4,4]^{2}$. The data size is $250,000$; the size is chosen to give similar total estimation errors, as explained later. The learning procedure is performed with PyTorch with AdaDelta. Again, note that the proposed method does not need any preparation for the data set. Instead, the coefficients $\{P_{i}^{(m)}(\mathbf{n},t)\}$ are numerically evaluated and used. The results are shown in figure 3. The upper ones correspond to the approximate true results obtained by the method in section 2. The middle ones are the results of the proposed method. The bottom ones correspond to those by the conventional learning approach based on backpropagation. As for the first-order moments of $x_{1}$ and $x_{2}$, it is difficult to see the differences among the three cases; the learning results are good enough. As for the second-order moment for $x_{2}$, we see the differences from the true one. The proposed method gives a similar shape to the true one, while the color is a little thinning away from the origin. On the other hand, the result of backpropagation is worse even near the origin. To examine this behavior in more detail, we evaluate the mean squared error between the learned results and approximate true ones for the distance from the origin. We calculate the outputs of the learned neural network values for various inputs with the $100\times 100$ mesh grids for polar coordinates with the range $[0,4]\times[0,2\pi)$. The results are shown in figure 4. Note that the proposed method and the backpropagation method give similar total errors with distances from $0$ to $4$; the data size was chosen so that they would be. Note that we do not intend exactly to yield the same total errors because we here want to focus on the behavior of the errors with respect to the distance. In figure 4, we see a clear difference between the results; the proposed method gives more accurate estimations near the origin. This characteristic reflects the feature of the Taylor expansion in the proposed method. ## 5 Conclusion We proposed a new learning method for neural networks, which target is the statistics of the stochastic differential equations. The proposed method directly compares coefficients obtained from the dual process and the weights in the neural networks. Figure 3: Estimated moments for the noisy van der Pol system. (Upper) Approximately true results. (Middle) Results by the proposed method. (Bottom) Results by the conventional learning approach based on backpropagation. There is no need for sampling procedures for the stochastic processes, and the proposed method gives a different framework from previous ones based on backpropagation. Although the idea is naive, the numerical demonstration shows remarkable features of the proposed approach; we have more accurate estimations near the origin, and the errors increase with distance. This feature stems from the Taylor-type basis expansion in the derivation of the method. Of course, it is easy to shift the origin of the state space using Ito's lemma, as commented in section 4.1. Hence, we can select another coordinate as the origin to estimate the moments around the selected coordinate more accurately. Problems such as overfitting are less likely to occur compared with the conventional learning methods from data sets because the proposed method does not require sampling of stochastic processes. These features will be hopeful in some applications in which one wants to embed the information of system equations into neural networks. The present work is the first attempt utilizing direct comparisons, and the proposed method and the numerical examples would be enough as the demonstration. Of course, there are remaining works, as follows. First, the estimations of the proposed method are less accurate except for inputs near the origin. Different cost functions with some focused coordinates could improve it. Second, we should seek suitable numerical optimization methods to make the algorithm faster. It is interesting to investigate how the performance changes for neural Figure 4: The estimated errors. The solid lines correspond to the results for the proposed method, and the dashed lines correspond to those for the backpropagation. networks with other structures because we used a simple neural network with one hidden layer in the present work. Although the cases of multiple hidden layers are possible in principle, they require solving rather difficult nonlinear optimization problems. Hence, further development of numerical solvers would be necessary. Third, practical applications would be investigated. For example, the following situation seems to be the case quite often: One knows the form of the time-evolution equations, but there are some ambiguities about the values of parameters. In such a situation, we can use the proposed method to construct an initial neural network based on the information of the equations; then, we apply additional learning steps to the initialized neural network. These procedures will reduce the size of data sets. Fourth, we need further work to make the method practical in high-dimensional systems. We confirmed that the algorithm based on combinatorics [16, 25] can deal with at least four or five-dimensional systems rapidly. Furthermore, it is possible to evaluate the information of moments for systems with several dozen variables using the tensor-train format; see, for example, [26] for the tensor-train format. Of course, the optimization procedure will take longer computational time, and we need to develop further studies in the research community of optimizations. Although the practical applications for higher-dimensional cases are beyond the scope of the present paper, we hope that the present work will motivate further collaboration between statistical physics and machine learning. The idea proposed in the present paper is the first step for future work. We hope that the present work will open up a new way to use the information of the time-evolution equation directly to learn neural networks. This work was supported by JSPS KAKENHI Grant Number JP21K12045.
2303.12964	Continuous Indeterminate Probability Neural Network	This paper introduces a general model called CIPNN - Continuous Indeterminate Probability Neural Network, and this model is based on IPNN, which is used for discrete latent random variables. Currently, posterior of continuous latent variables is regarded as intractable, with the new theory proposed by IPNN this problem can be solved. Our contributions are Four-fold. First, we derive the analytical solution of the posterior calculation of continuous latent random variables and propose a general classification model (CIPNN). Second, we propose a general auto-encoder called CIPAE - Continuous Indeterminate Probability Auto-Encoder, the decoder part is not a neural network and uses a fully probabilistic inference model for the first time. Third, we propose a new method to visualize the latent random variables, we use one of N dimensional latent variables as a decoder to reconstruct the input image, which can work even for classification tasks, in this way, we can see what each latent variable has learned. Fourth, IPNN has shown great classification capability, CIPNN has pushed this classification capability to infinity. Theoretical advantages are reflected in experimental results.	Tao Yang	2023-03-23T00:11:17Z	http://arxiv.org/abs/2303.12964v1	# Continuous Indeterminate Probability Neural Network ###### Abstract This paper introduces a general model called CIPNN - Continuous Indeterminate Probability Neural Network, and this model is based on IPNN, which is used for discrete latent random variables. Currently, posterior of continuous latent variables is regarded as intractable, with the new theory proposed by IPNN this problem can be solved. Our contributions are Four-fold. First, we derive the analytical solution of the posterior calculation of continuous latent random variables and propose a general classification model (CIPNN). Second, we propose a general auto-encoder called CIPAE - Continuous Indeterminate Probability Auto-Encoder, the decoder part is not a neural network and uses a fully probabilistic inference model for the first time. Third, we propose a new method to visualize the latent random variables, we use one of N dimensional latent variables as a decoder to reconstruct the input image, which can work even for classification tasks, in this way, we can see what each latent variable has learned. Fourth, IPNN has shown great classification capability, CIPNN has pushed this classification capability to infinity. Theoretical advantages are reflected in experimental results. (Source code: [https://github.com/Starfruit007/cipnn](https://github.com/Starfruit007/cipnn)) Although recent breakthroughs demonstrate that neural networks are remarkably adept at natural language processing [28, 11, 21], image processing [14], neural networks are still black-box for human [6], cognitive scientists and neuroscientist have argued that neural networks are limited in their ability to represent variables and data structures [12, 4]. Probabilistic models are mathematical descriptions of various natural and artificial phenomena learned from data, they are useful for understanding such phenomena, for prediction of unknowns in the future, and for various forms of assisted or automated decision making [20]. Deep Latent Variable Models (DLVMs) is a probabilistic model and can refer to the use of neural networks to perform latent variable inference [18]. Currently, the posterior calculation is regarded as intractable [19, 20], and the variational inference method is used for efficient approximate posterior inference [19, 27, 23]. IPNN - Indeterminate Probability Neural Network [31] proposed a new theory, which is used to derive the analytical solution of the posterior calculation of discrete random variables. However, IPNN need predefine the sample space of each discrete random variable (called'split shape' in IPNN), it is sometimes hard to define a proper sample space for an unknown dataset. For CIPNN, the sample space of each continuous random variable is infinite, this issue will not exit in CIPNN. The rest of this paper is organized as follows: In Sec. 1, related work of VAE and IPNN is introduced. In Sec. 2, the analytical solution of CIPNN is derived and the regularization method is discussed. In Sec. 3, CIPAE is derived and we propose a new method to visualize each latent variable. In Sec. 4, we discuss the training strategy, and two common training setups are discussed: CIPNN and CIPAE are combined together for better evaluation of classification and auto-encoder tasks. In Sec. 5, CIPNN and CIPAE are evaluated and the latent variables are visualized with our new proposed method. Finally, we put forward some future research ideas and conclude the paper in Sec. 6. ## 1 Related Work ### Vae Modern machine learning and statistical applications require large scale inference in complex models, the inference model are regarded as intractable and either Markov Chain Monte Carlo (MCMC) [24] or variational Bayesian inference [17] are used as approximate solutions [27]. VAE [19] proposes an estimator of the variational lower bound for efficient approximate inference with continuous latent variables. DARN method is generative auto-encoder capable of learning hierarchies of distributed representations from data, and their method applies to binary latent variables [13]. In concurrent work of VAE, two later independent papers proposed equivalent algorithms [27, 23], which provides an additional perspective on VAE and the latter work applies also the same reparameterization method. Two methods proposed by VAE are also used to realize our analytical solution: the reparameterization trick for making the model differentiable and the KL divergence term for regularization. VAEs have been used for many tasks such as image generation [22], anomaly detection [30] and de-noising tasks [16][5]. The drawback of auto-encoder is its strong tendency to over-fit [26], as it is solely trained to encode and decode with as little loss as possible regardless of how the latent space is organized [32], VAE has been developed as an effective solutions [26, 2], e.g. VAEs has been used in EEG classification tasks to learn robust features [33, 1, 2, 3]. The framework of our CIPAE is almost the same as that of VAE, the only difference is that VAE uses neural network as the approximate solution of decoder, while CIPAE uses probabilistic model as the analytical solution of decoder. ### Ipin Let $X\in\left\{x_{1},x_{2},\ldots,x_{n}\right\}$ be training samples ($x_{k}$ is understood as ID of random experiment - select one train sample) and $Y\in\left\{y_{1},y_{2},\ldots,y_{m}\right\}$ consists of m discrete labels (or classes), $P(y_{l}\|x_{k})=y_{l}(k)$ describes the label of sample $x_{k}$. For prediction, the posterior of the label for a given new input sample $x_{t}$ is formulated as $P^{Z}\left(y_{l}\mid x_{t}\right)$, superscript $Z$ stands for the medium - model outputted N-dimensional random variables $Z=\left(z^{1},z^{2},\ldots,z^{N}\right)$, via which we can infer label $y_{l},l=1,2,\ldots,m$. The analytical solution of the posterior is as bellow [31]: \[P^{Z}\left(y_{l}\mid x_{t}\right)=\int\limits_{Z}\left(P\left(y_{l}\mid Z \right)\cdot\prod\limits_{i=1}^{N}P\left(z^{i}\mid x_{t}\right)\right) \tag{1}\] Where, \[P\left(y_{l}\mid Z\right)=\frac{\sum_{k=1}^{n}\left(P\left(y_{l}\mid x_{k} \right)\cdot\prod_{i=1}^{N}P\left(z^{i}\mid x_{k}\right)\right)}{\sum_{k=1}^{ n}\prod_{i=1}^{N}P\left(z^{i}\mid x_{k}\right)} \tag{2}\] ## 2 CIPNN ### Continuous Indeterminate Probability Figure 1 shows CIPNN model architecture, the neural network is used to output the parameter $\theta$ of some prior distribution of continuous random variable $z^{i},i=1,2,\ldots,N$. All the random variables together form the N-dimensional joint sample space, marked as $Z=\left(z^{1},z^{2},\ldots,z^{N}\right)$, and the whole joint sample space are fully connected with all labels $Y\in\left\{y_{1},y_{2},\ldots,y_{m}\right\}$ via conditional probability $P\left(y_{l}\mid z^{1},z^{2},\ldots,z^{N}\right)$. For each continuous random variable $z^{i}$, the indeterminate probability is formulated as: \[P\left(z^{i}\mid x_{k}\right)=p\left(z;\theta_{k}^{i}\right),i=1,2,\ldots,N. \tag{3}\] Where $z$ is generated from some prior distribution with parameter $\theta_{k}^{i}$, and $p\left(z;\theta_{k}^{i}\right)$ is also the density function of $P\left(z^{i}\mid x_{k}\right)$. Substitute $P(y_{l}\|x_{k})=y_{l}(k)$ and Eq. (3) into Eq. (1): \[\begin{split}& P^{Z}\left(y_{l}\mid x_{t}\right)\\ &=\int\limits_{Z}\left(\frac{\sum_{k=1}^{n}\left(y_{l}(k)\prod_{i }^{N}p\left(z;\theta_{k}^{i}\right)\right)}{\sum_{k=1}^{n}\left(\prod_{i}^{N}p \left(z;\theta_{k}^{i}\right)\right)}\prod_{i}^{N}p\left(z;\theta_{t}^{i} \right)\right)\\ &=\mathbb{E}_{z^{\sim}p\left(z;\theta_{t}^{i}\right)}\left(\frac {\sum_{k=1}^{n}\left(y_{l}(k)\cdot\prod_{i}^{N}p\left(z;\theta_{k}^{i}\right) \right)}{\sum_{k=1}^{n}\left(\prod_{i}^{N}p\left(z;\theta_{k}^{i}\right) \right)}\right)\end{split} \tag{4}\] As the integration over $Z$ is complicated, $P^{Z}\left(y_{l}\mid x_{t}\right)$ is rewritten as expectation, we can then use Monte Carlo method [24] to make an approximate estimation. However, a directly sampling $z$ from distribution $p\left(z;\theta_{t}^{i}\right)$ will make the exception not differentiable. Hence, we use the reparameterization trick [19]: let $\varepsilon\sim p\left(\varepsilon\right)$ be some random noise, and define a mapping function $z=g(\varepsilon,\theta)$, so $p\left(z;\theta_{k}^{i}\right)$ can be rewritten as $p\left(g(\varepsilon,\theta);\theta_{k}^{i}\right)$. Therefore, together with Monte Carlo method, the above function can be further formulated as: Figure 1: CIPNN – model architecture. Where $P\left(y_{l}\mid z^{1},z^{2},\ldots,z^{N}\right)$ is statistically calculated, not model weights. \[P^{Z}\left(y_{l}\mid x_{t}\right) \tag{5}\] \[=\mathbb{E}_{\varepsilon\sim p\left(\varepsilon\right)}\left(\frac{ \sum_{k=1}^{n}\left(y_{l}(k)\prod_{i}^{N}p\left(g(\varepsilon,\theta_{t}^{i}); \theta_{k}^{i}\right)\right)}{\sum_{k=1}^{n}\left(\prod_{i}^{N}p\left(g( \varepsilon,\theta_{t}^{i});\theta_{k}^{i}\right)\right)}\right)\] \[\approx\frac{1}{C}\sum_{c=1}^{C}\left(\frac{\sum_{k=1}^{n}\left(y_ {l}(k)\cdot\prod_{i}^{N}p\left(g(\varepsilon_{c},\theta_{t}^{i});\theta_{k}^{i }\right)\right)}{\sum_{k=1}^{n}\left(\prod_{i}^{N}p\left(g(\varepsilon_{c}, \theta_{t}^{i});\theta_{k}^{i}\right)\right)}\right)\] Where $\varepsilon_{c}\sim p\left(\varepsilon\right)$. Take, for example, the univariate Gaussian case: let $P\left(z^{i}\mid x_{k}\right)=\mathcal{N}\left(z;\mu_{k}^{i},\sigma_{k}^{i}\right)$, and $\varepsilon_{c}\sim\mathcal{N}\left(0,1\right)$, the reparameterization mapping function is $z=\mu+\sigma\varepsilon$, the above function can be rewritten as: \[P^{Z}\left(y_{l}\mid x_{t}\right) \tag{6}\] \[\approx\frac{1}{C}\sum_{c=1}^{C}\left(\frac{\sum_{k=1}^{n}\left(y _{l}(k)\cdot\prod_{i}^{N}\mathcal{N}\left(g\left(\varepsilon_{c},\theta_{t}^ {i}\right);\theta_{k}^{i}\right)\right)}{\sum_{k=1}^{n}\left(\prod_{i}^{N} \mathcal{N}\left(g\left(\varepsilon_{c},\theta_{t}^{i}\right);\theta_{k}^{i} \right)\right)}\right)\] Where $\theta_{k}^{i}:=\left(\mu_{k}^{i},\sigma_{k}^{i}\right)$, $g\left(\varepsilon_{c},\theta_{t}^{i}\right)=\mu_{t}^{i}+\sigma_{t}^{i}\cdot \varepsilon_{c}$ and $\varepsilon_{c}\sim\mathcal{N}\left(0,1\right)$. We use cross entropy as main loss function: \[\mathcal{L}_{1}=-\sum_{l=1}^{m}\left(y_{l}(t)\cdot\log P^{Z}\left(y_{l}\mid x _{t}\right)\right) \tag{7}\] ### Regularization The sufficient and necessary condition of achieving global minimum is already proved in IPNN [31], which is also valid for continuous latent variables: Proposition 1: _For $P(y_{l}\|x_{k})=y_{l}(k)\in\left\{0,1\right\}$ hard label case, CIPNN converges to global minimum only when $P\left(y_{l}\|z^{1},z^{2},\ldots,z^{N}\right)\to 1,$ for $\prod_{i}^{N}p\left(z;\theta_{k}^{i}\right)>0$._ _In other word, each N-dimensional joint sample area (collection of adjacent joint sample points) corresponds to an unique category. However, a category can correspond to one or more joint sample areas._ According to above proposition, the reduction of training loss will minimize the overlap between conditional joint distribution $\prod_{i}^{N}p\left(z;\theta_{k}^{i}\right)$ of each category. For Gaussian distribution, the variance will be close to zero, and the conditional joint distribution of each category will be far away from each other. This will cause over-fitting problem [32, 26], we have follow up assumption to avoid over-fitting problem: Assumption 1: _For Gaussian distribution, if the distance between the centers of any two adjacent categories' conditional joint distribution $\prod_{i}^{N}\mathcal{N}\left(z;\mu_{k}^{i},\sigma_{k}^{i}\right)$ is equal to e.g. $6\sigma$, then the over-fitting problem can be avoided._ _In this way, the $6\sigma$ distance will lead to a very small overlap between each category, but the conditional joint distribution of each category will be closely connected with each other. Although the N-dimensional joint sample space is infinite, the effective conditional joint distributions are in a very small space._ VAE uses an additional regularization loss to avoid the over-fitting problem [19, 20], and there are follow up works which has proposed to strengthen this regularization term, such as $\beta$-VAE [15, 7], $\beta$-TCVAE [8], etc. In order to realize the above assumption, we have a modification of VAE regularization loss: \[\mathcal{L}_{2}=\sum_{i=1}^{N}\left(D_{KL}\left(\mathcal{N}\left(z;\mu_{k}^{i },\sigma_{k}^{i}\right)\|\|\mathcal{N}\left(z;\gamma\cdot\mu_{k}^{i},1\right) \right)\right) \tag{8}\] \[=\frac{1}{2}\sum_{i=1}^{N}\left(((1-\gamma)\cdot\mu_{k}^{i})^{2}+ (\sigma_{k}^{i})^{2}-\log((\sigma_{k}^{i})^{2})-1\right)\] Where $N$ is the dimensionality of $Z$, regularization factor $\gamma\in[0,1]$ is a hyperparameter and is used to constrain the conditional joint distribution of each category to be closely connected with each other, impact analysis of regularization factor $\gamma$ see Figure 8. The overall loss is: \[\mathcal{L}=\mathcal{L}_{1}+\mathcal{L}_{2} \tag{9}\] ## 3 Cipae For image auto-encoder task, we firstly transform the pixel value to $[0,1]$ (Bernoulli distribution), and let $Y^{j}\in\{y_{1}^{j},y_{2}^{j}\}_{j=1}^{J}$, where $J$ is the number of all pixels of one image. $P(y_{1}^{j}\|x_{k})=p_{1}^{j}(k)\in[0,1]$, which describes the pixel value of image $x_{k}$ at $j^{th}$ position, and $P(y_{2}^{j}\|x_{k})=p_{2}^{j}(k)=1-p_{1}^{j}(k)$. Substitute $P(y_{l}^{j}\|x_{k})$ into Eq. (6), we will get $P^{Z}\left(y_{l}^{j}\mid x_{t}\right),l=1,2$. In this way, the reconstructed image is formulated as: \[\text{reconstructed image}:=\left\{P^{Z}\left(y_{1}^{j}\mid x_{t}\right) \right\}_{j=1}^{J} \tag{10}\] In addition, with one (or part) of N dimensional latent variables we can also reconstruct the input image, the reconstructed image is:1 Footnote 1: The details of applying the superscript $z^{i}$ are discussed in IPNN [31]. \[\text{reconstructed feature}:=\left\{P^{z^{i}}\left(y_{1}^{j}\mid x_{t}\right) \right\}_{j=1}^{J} \tag{11}\] Where $i=1,2,\ldots,N$. In this way, we can see what each latent variable has learned. Substitute Eq. (10) into Eq. (7), we can get a binary cross entropy loss: \[\mathcal{L}_{1}=-\frac{1}{J}\sum_{j=1}^{J}\sum_{l=1}^{2}\left(p_{l}^{j}(t) \cdot\log P^{Z}\left(y_{l}^{j}\mid x_{t}\right)\right) \tag{12}\] And substitute the above loss into Eq. (9), we get the overall loss for auto-encoder training. ## 4 Training In this section, we will focus on the training strategy of gaussian distribution. ### Training Strategy Given an input sample $x_{t}$ from a mini batch, with a minor modification of Eq. (6): \[P^{Z}\left(y_{l}\mid x_{t}\right)\approx\frac{1}{C}\sum_{c=1}^{C}\left(\frac{ \max(H(\varepsilon_{c}),\epsilon)}{\max(G(\varepsilon_{c}),\epsilon)}\right) \tag{13}\] Where stable number $\epsilon$ on the denominator is to avoid dividing zero, $\epsilon$ on the numerator is to have an initial value of 1. Besides, \[H(\varepsilon_{c}) =\sum_{k=t_{0}}^{t_{1}}\left(y_{l}(k)\prod_{i}^{N}\mathcal{N} \left(\mu_{t}^{i}+\sigma_{t}^{i}\cdot\varepsilon_{c};\mu_{k}^{i},\sigma_{k}^ {i}\right)\right) \tag{14}\] \[G(\varepsilon_{c}) =\sum_{k=t_{0}}^{t_{1}}\left(\prod_{i}^{N}\mathcal{N}\left(\mu_ {t}^{i}+\sigma_{t}^{i}\cdot\varepsilon_{c};\mu_{k}^{i},\sigma_{k}^{i}\right)\right) \tag{15}\] Where $t_{0}=\max(0,t_{1}-T)$, $t_{1}$ is the number of input samples, $\varepsilon_{c}\sim\mathcal{N}\left(0,1\right)$. Hyperparameter T is for forgetting use, i.e., $P^{Z}\left(y_{l}\mid x_{t}\right)$ are calculated from the recent T samples. The detailed algorithm implementation is shown in Algorithm (1). ``` 0: A sample $x_{t}$ from mini-batch 0: Parameter: Latent variables dimension $N$, forget number $T$, Monte Carlo number $C$, regularization factor $\gamma$, stable number $\epsilon$, learning rate $\eta$. 0:$P^{Z}\left(y_{l}\mid x_{t}\right)$ 1: Declare $\Theta$ as a recorder. 2:for$k=1,2,\ldots$ Until Convergence do 3: Use $\Theta$ to record: $y_{l},\mu_{k}^{i},\sigma_{k}^{i},i=1,2,\ldots,N$. 4:if$len(\Theta)>T$then 5: Forget: Reserve recent T elements from $\Theta$ 6:endif 7: Compute posterior with Eq. (13): $P^{Z}\left(y_{l}\mid x_{t}\right)$ 8: Compute loss with Eq. (9): $\mathcal{L}(W)$ 9: Update model parameter: $W=W-\eta\nabla\mathcal{L}(W)$ 10:endfor 11:return model and the posterior ``` Algorithm 1 CIPNN or CIPAE training ### Training Setups By comparing CIPNN and CIPAE, we can see that they can share the same neural network for a training task. As shown in Figure 2, the latent variables of a classification task can be visualized with CIPAE, and we can also use CIPNN to evaluate the performance of an auto-encoder task. ## 5 Experiments and Results To evaluate the effectiveness of the proposed approach, we conducted experiments on MNIST [10], Fashion-MNIST [29] and Dogs-vs.-Cats-Redux [9] datasets. Besides, VAE validated that Monte Carlo number $C$ can be set to 1 as long as the batch size is high enough (e.g. 100) [19], we will set batch size to 64, Monte Carlo number $C=2$ and forget number $T=3000$ for all the experiments in this paper. ### Results of Classification Tasks In this section, we use train setup in Figure 1(a) to perform different classification tasks in order to reconstruct the latent variable to see what they have learned. The results from the work [31] show that IPNN prefers to put number 1,4,7,9 into one cluster and the rest into another cluster. We also get a similar interesting results in Figure 2: Training setups for classification and auto-encoder tasks. (a) CIPNN is used to do supervised classification tasks and CIPAE is used to reconstruct the input image to see what each latent variable has learned. (b) CIPAE is used to do auto-encoder task and CIPNN is used to evaluate its performance. CIPNN, as shown in Figure 3, with stable number $\epsilon=1$, the reconstructed image with 1-D latent space shows a strong tendency to sort the categories into a certain order and the number 1,4,7,9 stays together in the latent space. Similar results are also found with 2-D latent space, see Figure 5. Unfortunately, we currently do not know how to evaluate this sort tendency numerically. With the visualization method proposed in Eq. (11), we can see what each latent variable has learned in the 10-D latent space. As shown in Figure 4, each latent variable focuses on mainly one or two different categories. As shown in Figure 5(a,d,g), with a proper regularization factor $\gamma$, the test dataset is mapped to a relative small latent space, and the over-fitting problem is avoided. Besides, in Figure 5(b,e,h) each joint sample area correspond to one unique category, this is consistent with our Proposition 1. In Figure 5(c,f,i), the reconstructed image follows the conditional joint distribution $P\left(y_{l}\mid z^{1},z^{2}\right),l=0,2,\ldots,9$. ### Results of Auto-Encoder Tasks In this section, we will make a comparison with our CIPAE and VAE [19] model using train setup in Figure 2b, for VAE model, we replace CIPAE part with VAE, in order to be able evaluate it with CIPNN model. Besides, the regularization loss of VAE is switched to our proposed loss, see Eq. (8). As shown in Figure 6, the results of auto-encoder tasks between CIPAE and VAE are similar, this result further verifies that CIPAE is the analytical solution. ## 6 Conclusion General neural networks, such as FCN, Resnet [14], Transformer [28], can be understood as a complex mapping function $f:X\to Y$[25], but they are black-box for human [6]. Our proposed model can be understood as two part: $f:X\to Z$ and $P(Y\mid Z):Z\to Y$, the first part is still black box for us, but the latter part is not unknown anymore. Such kind of framework may have two advantages: the first part can be used to detect the attributes of datasets and summarize the common part of different categories, as shown in Figure 3; the latter part is a probabilistic model, which may be used to build a large Bayesian network for complex reasoning tasks. Besides, our proposed framework is quite flexible, e.g. from $X$ to $Z$, we can use multiple neural networks with different structures to extract specific attributes as different random variables $z^{i}$, and these random variables will be combined in the statistical phase. Although our proposed model is derived from indeterminate probability theory, we can see Determinate from the expectation form in Eq. (4). Finally, we'd like to finish our paper with one sentence: The world is determined with all Indeterminate. ## Acknowledgment I'd like to thank Chuang Liu, Xiaofeng Ma, Weijia Lu, Ning Wu, Bingyang Li, Zhifei Yang, Peng Liu, Lin Sun, Xiaodong Zhang and Can Zhang for their review of this paper. ## Appendix A Visualization Figure 7 shows classification results of 20-D latent space on Dogs-vs.-Cats-Redux dataset, we can see that each latent variable focuses on both two different categories. Impact analysis of regularization factor $\gamma$ is discussed in Figure 8.
2307.15242	An equivariant graph neural network for the elasticity tensors of all seven crystal systems	The elasticity tensor that describes the elastic response of a material to external forces is among the most fundamental properties of materials. The availability of full elasticity tensors for inorganic crystalline compounds, however, is limited due to experimental and computational challenges. Here, we report the materials tensor (MatTen) model for rapid and accurate estimation of the full fourth-rank elasticity tensors of crystals. Based on equivariant graph neural networks, MatTen satisfies the two essential requirements for elasticity tensors: independence of the frame of reference and preservation of material symmetry. Consequently, it provides a unified treatment of elasticity tensors for all seven crystal systems across diverse chemical spaces, without the need to deal with each separately.. MatTen was trained on a dataset of first-principles elasticity tensors garnered by the Materials Project over the past several years (we are releasing the data herein) and has broad applications in predicting the isotropic elastic properties of polycrystalline materials, examining the anisotropic behavior of single crystals, and discovering new materials with exceptional mechanical properties. Using MatTen, we have discovered a hundred new crystals with extremely large maximum directional Young's modulus and eleven polymorphs of elemental cubic metals with unconventional spatial orientation of Young's modulus.	Mingjian Wen, Matthew K. Horton, Jason M. Munro, Patrick Huck, Kristin A. Persson	2023-07-28T00:43:51Z	http://arxiv.org/abs/2307.15242v2	# A Universal Equivariant Graph Neural Network for the Elasticity Tensors of Any Crystal System ###### Abstract The elasticity tensor that describes the elastic response of a material to external forces is among the most fundamental properties of materials. The availability of full elasticity tensors for inorganic crystalline compounds, however, is limited due to experimental and computational challenges. Here, we report the materials tensor (MatTen) model for rapid and accurate estimation of the full fourth-rank elasticity tensors of crystals. Based on equivariant graph neural networks, MatTen satisfies the two essential requirements for elasticity tensors: independence of the frame of reference and preservation of material symmetry. Consequently, it provides a universal treatment of elasticity tensors for all crystal systems across diverse chemical spaces. MatTen was trained on a dataset of first-principles elasticity tensors garnered by the Materials Project over the past several years (we are releasing the data herein) and has broad applications in predicting the isotropic elastic properties of polycrystalline materials, examining the anisotropic behavior of single crystals, and discovering new materials with exceptional mechanical properties. Using MatTen, we have discovered a hundred new crystals with extremely large maximum directional Young's modulus and eleven polymorphs of elemental cubic metals with unconventional spatial orientation of Young's modulus. ## 1 Introduction All materials exhibit elastic behavior under small external loads and return to their original shape upon the release of these loads [1]. The elasticity tensor provides a fundamental and complete description of a material's response to such loads. It offers a lens to examine the inherent strength of the bonding in a material and enables the understanding, analyzing, and designing of many macroscopic physical properties of materials. Commonly employed mechanical properties (for instance, Young's modulus and Poisson's ratio), thermal properties (for instance, thermal conductivity) and acoustic properties (for instance, sound velocity) can be mathematically derived from the elasticity tensor. These properties have long been leveraged, for example, by materials scientists to search for ultrahard materials [2, 3] and by geophysicists to interpret seismic data [4, 5]. More recently, the anisotropic elastic behavior of inorganic solid electrolytes has been found to play a decisive role in determining the stability of electrodeposition at the interfaces in solid-state batteries [6, 7]. Moreover, in solid-state synthesis, one would want to utilize the elasticity tensor to determine the local stability of a material, so as to avoid unsuccessful synthesis of materials that spontaneously transform into different structures [8, 9]. In spite of the importance, elasticity tensor data from experimental measurement is limited to a small number of materials. For example, for inorganic crystalline compounds, experimental data is only on the order of a few hundred, considering entries both tabulated in handbooks and scattered in individual papers [10]. The major difficulty lies in preparing large enough single crystals for accurate experimental measurement using current techniques such as resonant acoustic spectroscopy [11]. In the past decade, efficient and reliable electronic structure calculation methods such as density functional theory (DFT) [12] with automation frameworks [13, 14, 15] have enabled high-throughput computational investigation of materials. Using this approach in the Materials Project [16], we produced an initial dataset of elasticity tensors for 1181 crystals in 2015 [10], which has been expanded over time to 10276, which we now release as a new dataset with this work. Nevertheless, this only accounts for 6.6% of the more than 154000 crystals in the Materials Project, let along the even greater number of crystals recorded in the Inorganic Crystal Structure Database (ICSD) [17] and predicted by universal interatomic potentials [18]. It is therefore no surprise that machine learning (ML) has gathered substantial interest as a means to develop efficient surrogate models for the prediction of elastic properties. In a nutshell, state-of-the-art ML models for elastic properties encode compositional information [19, 20, 21] and/or structural information [20, 21, 22, 23] in a material as feature vectors and then map them to a target using some regression algorithms. This approach is limited to derived scalar elastic properties such as bulk modulus and shear modulus, and separate models have to be built for each derived property. Ideally, one would hope to predict the full elasticity tensor, from which all other derived elastic properties can be obtained. Along this direction, there have been attempts to predict individual tensor components [24, 25]. These models are great first steps, but essentially they still predict separate scalars in a specific coordinate system and thus unavoidably violate the transformation rules of tensors. Geometric deep learning [26] such as equivariant graph neural networks (GNNs) [27, 28, 29, 30] can directly operate in the space of high-rank tensors and adhere to their transformation rules. Although the major use case is still for scalar molecular and materials properties, a handful of works have explored the application in predicting vectors such as the molecular dipole moment [31, 32], demonstrating the viability of direct deep learning of tensorial properties. In this work, we develop the Materials Tensor (MatTen) model for rapid and accurate estimation of the fourth-rank elasticity tensors of inorganic compounds. Our model, based on equivariant GNNs, takes a crystal structure as input and outputs its full elasticity tensor with all transformation rules satisfied. Other elastic properties such as bulk modulus and shear modulus can then be computed from the full elasticity tensor. The capabilities of MatTen are demonstrated via the study of both isotropic and anisotropic elastic properties. Using MatTen, we screened the Materials Project database for the discovery of new materials with large maximum directional Young's modulus. On average, the values of the newly found materials are more than three times larger than existing data, as verified by first-principles calculations. In addition, we have identified 11 unconventional polymorphs of elemental cubic metals whose maximum directional Young's moduli are in the $\langle 100\rangle$ directions. ## 2 Results ### Symmetry and irreducible representation of the elasticity tensor The elasticity tensor $\mathbf{C}$ is a fourth-rank tensor that fully characterizes the elastic behavior of a material. Given that it is the second derivative of the total elastic energy with respect to the strain tensor and that the strain tensor is symmetric [33, 34], the elasticity tensor possesses major symmetry $C_{ijkl}=C_{klij}$ and minor symmetry $C_{ijkl}=C_{jikl}=C_{ijlk}$ (in indicial notation, where $i,j,k,l\in\{1,2,3\}$). Consequently, only 21 of the 81 components of $\mathbf{C}$ are independent. It is convenient to write the elasticity tensor in a contracted matrix $\mathbf{c}$ ($c_{\alpha\beta}$, where $\alpha,\beta\in\{1,2,\ldots,6\}$) with a pairs of indices $ij$ in the tensor notation replaced with a single index $\alpha$ in the matrix notation: $11\to 1$; $22\to 2$; $33\to 3$; $23,32\to 4$; $13,31\to 5$; and $12,21\to 6$. This Voigt matrix [35] is a $6\times 6$ matrix symmetric about the main diagonal, reflecting the fact that the elasticity tensor has 21 independent components. The intrinsic material symmetry of a crystal can further reduce the number of independent components [8, 34]. For example, copper is a cubic crystal with mirror planes and three-fold rotation axes (point group m$\bar{3}$m), and such symmetry results in a number of only three independent components. Formally, the material symmetry imposes the following constraints on the elasticity tensor [36, 37]: \[C_{ijkl}=Q_{ip}Q_{jq}Q_{kr}Q_{ls}C_{pqrs}, \tag{1}\] where $\mathbf{Q}\in G\subset SO(3)$, and $G$ is the material symmetry group of the crystal, which is a subgroup of the rotation group $SO(3)$. An interesting question is: how many unique symmetry classes exist and what is the number of independent components in each class? It turns out that there exists only eight distinct classes (Fig. 1), proved via a purely algebraic approach by directly identifying the equivalence classes corresponding to Eq. (1) [36]. Of the eight classes, one is for isotropic materials, and each of the other seven corresponds to a crystal system [38, 37]. In our opinion, there is still significant confusion on this topic. For example, the categorization by Wallace [39] and populated by Nye [34], which incorrectly gives two unique classes for each of the tetragonal and trigonal cases (Fig. S1 in the Supplementary Information (SI)), is still widely cited in recent works [40, 41, 42]. We refer to Ref. [37] for a historical note on the development of the categorization. The Voigt matrix provides a visual way to observe the material symmetry of a crystal reflected in the elas Figure 1: Symmetry classes and independent components of the elasticity tensor. The value in the parentheses after the name indicates the number of independent components. All matrices are symmetric about the main diagonal, with the lower triangular part omitted in the depiction. For single crystals considered in this work, the isotropic case does not apply. ticity tensor (Fig. 1). The values of the matrix components, however, depend on the choice of the coordinate system and do not show any systematic pattern upon coordinate transformation [43, 44], making it difficult to build predictive models for elasticity tensors. This can be overcome by the _harmonic decomposition_[45], where the space of all elasticity tensors is factored into the direct sum of irreducible representations of $SO(3)$. Consequently, any elasticity tensor can be written in the form, \[\mathbf{C}=h_{1}(\lambda)+h_{2}(\eta)+h_{3}(\mathbf{A})+h_{4}(\mathbf{B})+h_{5}(\mathbf{H}), \tag{2}\] where $\lambda$ and $\eta$ (scalars) are the _isotropic_ part, $\mathbf{A}$ and $\mathbf{B}$ (second-rank traceless symmetric tensors) are the _deviatoric_ part, and $\mathbf{H}$ (fourth-rank fully symmetric tensor with a zero trace) is the _harmonic_ part [45]. The harmonic decomposition has two advantageous characteristics. First, for a given $\mathbf{C}$, the values of $\lambda,\eta,\mathbf{A},\mathbf{B},\mathbf{H}$ and the functions $h_{1},\dots,h_{5}$ are uniquely determined [36, 45] (see SI for their expressions). Second, each part in Eq. (2) transforms in a known manner with respect to $SO(3)$ operations, enabling the construction of predictive models that leverage recent advancements in geometric deep learning. ### Equivariant graph neural networks for high-rank tensors The MatTen model captures the structure-property relationship of crystalline materials. It takes a crystal structure as input, represents it as a three-dimensional crystal graph, performs equivariant feature update on the crystal graph, and finally outputs a tensor property of the material built according to the reverse process of the harmonic decomposition in Eq. (2). In the GNN model (Fig. 2), the input crystal is represented as a graph $\mathcal{G}(V,E)$, with atoms as the nodes and bonds as the edges. The feature $F_{i}\in V$ characterizes atom $i$, and $\vec{\mathbf{r}}_{ij}\in E$ denotes the distance vector between a pair of atoms $i$ and $j$. The embedding modules operate on the graph representation of the crystal. The atom embedding module encodes the atomic number $Z_{i}$ using a one-hot scheme, which becomes the initial atom feature $F_{i}$. The distance vector $\vec{\mathbf{r}}_{ij}$ is separated into two parts: the unit vector $\hat{\mathbf{r}}_{ij}$ from atom $i$ to atom $j$ and the scalar distance $r_{ij}$ between them. The former is expanded on real spherical harmonics $Y^{l}_{m}$, and the latter is expanded on the Bessel radial basis functions [46]. These embedding modules extract structural information from the crystal and provide them to the interaction blocks. The interaction blocks iteratively refine the atom features via convolution operations. The architecture of the interaction block follows the design of Tensor Field Network [27] and NequIP [29]. Unlike many existing GNNs for molecules and crystals [47, 48, 49, 22] that utilize scalar features, here, the atom feature $F_{i}$ is a set of scalars, vectors, and high-rank tensors. Formally, it is a geometric object consisting of a direct sum of irreducible representations of the $SO(3)$ rotation group [27, 29]. There are two major benefits of using such geometric features. First, they are incorporated as inductive bias which can improve model accuracy and reduce the amount of training data. Second, from them one can easily construct other physical tensors such as the elasticity tensor in this work. The convolution on these geometric objects is an equivariant function, meaning that if the input feature $F_{i}$ to the convolution is transformed under a rotation in $SO(3)$, the output is transformed accordingly. This is achieved via the tensor product convolution, \[F_{i}=\sum_{j\in\mathcal{N}_{i}}R(r_{ij})Y(\hat{\mathbf{r}}_{ij})\otimes F_{j}, \tag{3}\] where $\mathcal{N}_{i}$ denotes the set of neighboring atoms for atom $i$, $R$ indicates a multilayer perceptron (MLP) on the radial basis expansion of $r_{ij}$, and $Y$ indicates the spherical harmonics expansion of $\hat{\mathbf{r}}_{ij}$. The tensor product $\otimes$ between $Y$ and $F_{j}$ ensures that the operation is equivariant [27], and the use an MLP makes the convolution learnable. After a skip connection [50], the feature $F_{i}$ is passed through a nonlinear activation function and finally normalized using an equivariant normalization function [51]. The output head maps the refined features from the interaction blocks to the target materials tensor of interest. First, the features of all atoms are aggregated to obtain a representation of the crystal. For intensive properties such as the elasticity tensor, we adopt the mean pooling by averaging the features such that the representation of the crystal is independent of the number of atoms. Next, an appropriate subset of the pooled irreducible representations that correspond to the target tensor of interest is selected and then assembled as the model output. For the elasticity tensor, the selected ones are two scalars, two second-rank traceless symmetric tensors, and a fourth-rank harmonic tensor. They are assembled to an elasticity tensor according to Eq. (2). Overall, MatTen is a function $\mathbf{C}=f(x)$ that maps a crystal structure $x$ to its elasticity tensor $\mathbf{C}$. The function $f$ is equivariant to the $SO(3)$ group transformation, that is, for any $x$ and $g\in SO(3)$, we have $D_{y}(g)f(x)=f(D_{x}(g)x)$, where $D_{x}(g)$ and $D_{y}(g)$ are rotation matrices parameterized by $g$ for the crystal structure and the elasticity tensor, respectively. The equivariance ensures that the model can produce an elasticity tensor $\mathbf{C}$ that respects the orientation of the input crystal structure. In other words, the choice of a Figure 2: Schematic overview of the MatTen graph neural network model. (a) The model takes a crystal structure as input, performs message passing with equivariant graph neural network (GNN) layers, and outputs the full elasticity tensor satisfying all symmetry requirements. (b) Architecture of the model. The model consists of four modules: the position embedding module converts an input displacement vector $\vec{\mathbf{r}}_{ij}$ between atoms $i$ and $j$ to radial ($R$) and spherical ($Y$) expansions; the atom embedding module encodes the atomic number $Z_{i}$ as irreducible atom features $F_{i}$ using a one-hot encoding; interaction blocks iteratively refine the features using convolutions based on tensor product; and the output head selects the appropriate irreducible features corresponding to the elasticity tensor and assembles them as the output tensor $\mathbf{C}$. Following Ref. [44], the fourth-rank harmonic part of the elasticity tensor is depicted as a generic matrix. The $\oplus$ denotes addition. specific coordinate system does not affect the model. This _independence of the frame of reference_ characteristic is an indispensable property for models that predict tensors. In addition, any such model should also _preserve the material symmetry_ of the crystal. By construction, MatTen guarantees the material symmetry reflected in the elasticity tensor. Concretely, if the predicted elasticity tensor is represented as a Voigt matrix, the symmetry and number of independent components in Fig. 1 are automatically maintained for any crystal system (proof in the SI). ### Elastic properties of polycrystals The MatTen model directly outputs the full elasticity tensor. To assess its performance, we computed several commonly used elastic properties for polycrystals from the elasticity tensor. Fig. 3 illustrates the outcomes on bulk modulus, shear modulus, and Young's modulus obtained using the Hill average scheme [52] (Section 4). The predictions of MatTen closely align with the DFT reference values, achieving mean absolute errors (MAEs) of 7.37 GPa, 8.38 GPa, and 20.59 GPa for bulk modulus, shear modulus, and Young's modulus, respectively. For comparison, we trained two additional models. The first is a variant of MatTen, where the tensor output head of MatTen is replaced with a scalar output head, referred to as MatSca hereafter. The second is the AutoMatminer algorithm, an automated machine learning pipeline designed for predicting scalar materials properties [20]. We evaluated their performance in predicting the elastic moduli, and the results are listed in Table 1. Both MatTen and MatSca have smaller MAEs than AutoMatminer across all three moduli, owning to the effectiveness of the underlying equivariant neural networks in learning materials properties from structures. The performance between MatTen and MatSca are comparable. However, it is worth highlighting that while an individual MatSca model was trained for each modulus, a single Figure 3: Performance of MatTen on various elastic properties. (a) Bulk modulus, (b) shear modulus, and (c) Young’s modulus predicted by MatTen versus DFT reference values. (d) Scaled error by crystal systems. The error bar indicates the standard deviation obtained from an ensemble of five models trained with different initialization. MAE: mean absolute error; MAD: mean absolute deviation. MatTen model successfully produced all the elastic moduli, demonstrating the versatility and universal applicability of the MatTen model. Upon closer examination, we have identified some unphysical predictions. Although the theory of elasticity allows negative bulk and Young's moduli in general, under the thermodynamic conditions considered here, all moduli should be positive [53]. The predicted moduli, however, occasionally yield negative values, despite all crystals in the training set having positive moduli. This is an inherent challenge faced by machine learning regression models in general, albeit with physical inductive biases embedded in the model such as the symmetry requirements in MatTen. Nevertheless, this is not a concern in practical use. One can filter out the unphysical ones predicted by MatTen. Moreover, the number of crystals with negative predicted moduli remains minimal, accounting for only 3, 2, and 2 out of the 1021 test data for bulk, shear, and Young's moduli, respectively. To assess how MatTen performs for different elastic properties as well as for different crystal systems, we computed the scaled error, $\mathrm{SE}=\mathrm{MAE}/\mathrm{MAD}$, in which MAE and MAD are the mean absolute error and mean absolute deviation, respectively (see Section 4). MAD quantifies the distance of reference values to their mean, and larger MAD means the reference values are more scattered. A model that makes accurate predictions for each data point will have an SE of 0, and a model that always predicts the mean of the dataset will have an SE of exactly 1. Comparing between properties, we see from Fig. 3d that the SE of bulk modulus is smaller than these of shear modulus and Young's modulus across all crystal systems. This suggests that bulk modulus is easier to predict, in agreement with existing observations [19, 20, 21, 22]. Next, we compare between crystal systems. The dataset used for model development has an uneven distribution in terms of the number of materials in each crystal system (Fig. S2 in the SI). For example, it contains 4217 cubic crystals, fewer than 800 trigonal and monoclinic crystals, and only 60 triclinic crystals. As a result, the SE and the error bar of the predicted moduli are larger for trigonal, monoclinic, and triclinic crystals in general (Fig. 3d). Despite the slightly higher errors, it is notable that the model can still perform well for the crystal systems with a low presence in the training data, particularly for triclinic crystals. This is primarily because MatTen internally treats all crystals the same, enabling crystal systems with fewer data to leverage the abundant data from other crystal systems and acquire enhanced representations. The elastic moduli have values across different orders from near zero to hundreds (Figs. S3-S6 in the SI). To mitigate the challenge of learning values across a broad spectrum, some existing models [19, 20, 21, 22] adopt the approach of predicting the logarithm of the moduli. Unlike these models, MatTen directly predicts the full tensor without any data transformation, and all other elastic properties (including their logarithms) can be computed from it. The logarithms of the bulk, shear, and Young's moduli obtained from MatTen are comparable to those that learn in the logarithmic space (Table S1 in the SI). Further, the performance of MatTen can be further improved with additional training data. The MAE almost decreases linearly with the logarithm of the number of data used to train the model (Fig. S8 in the SI). \begin{table} \begin{tabular}{c c c c c c c} \hline \hline & \multicolumn{2}{c}{$K$ (GPa)} & \multicolumn{2}{c}{$G$ (GPa)} & \multicolumn{2}{c}{$E$ (GPa)} \\ \cline{2-7} & MAE & $\mathrm{MAE}/\mathrm{MAD}$ & MAE & $\mathrm{MAE}/\mathrm{MAD}$ & MAE & $\mathrm{MAE}/\mathrm{MAD}$ \\ \hline MatTen & 7.37 (0.10) & 0.130 (0.002) & 8.38 (0.16) & 0.280 (0.005) & 20.59 (0.35) & 0.275 (0.005) \\ MatSca & 7.32 (0.09) & 0.129 (0.002) & 8.63 (0.07) & 0.288 (0.002) & 19.87 (0.43) & 0.265 (0.006) \\ AutoMatminer & 9.84 (0.34) & 0.174 (0.006) & 9.27 (0.32) & 0.309 (0.011) & 22.10 (0.77) & 0.295 (0.024) \\ \hline \hline \end{tabular} \end{table} Table 1: Performance of the models in predicting the bulk, shear, and Young’s moduli. The value in a pair of parentheses is the standard deviation obtained from an ensemble of five models trained with different initialization. MAE: mean absolute error; MAD: mean absolute deviation. ### Anisotropic elastic properties Crystals are inherently anisotropic, and thus their elastic properties can vary depending on the direction of measurement. This anisotropy arises from a crystal's structure, including the symmetry of the lattice and the arrangement of the atoms. MatTen predicts the full elasticity tensor, and, for the first time with a machine learning model, we are able to investigate the anisotropic elastic behavior of crystals. We focus our discussion on the directional dependence of Young's modulus (further results on shear modulus given in the SI). Young's modulus $E$ discussed in Section 2.3 is an averaged property for isotropic polycrystals. But for single crystals, Young's modulus depends on the direction along which the strain/stress is applied and measured. Given the elasticity tensor $C_{ijkl}$ (equivalently, the compliance tensor $S_{ijkl}$, see Section 4), the directional Young's modulus can be computed as [34, 42] \[E_{\text{d}}(\mathbf{n})=(n_{i}n_{j}n_{k}n_{l}S_{ijlk})^{-1}, \tag{4}\] where $\mathbf{n}$ is a unit vector that specifies the direction of measurement. The direction dependence of Young's modulus can be visualized with a three-dimensional plot (Fig. 4a). Interactive visualization can be obtained via, for example, the elate package [54]. Alternatively, via a spherical coordinates transformation: $\mathbf{n}=[\sin\theta\cdot\cos\varphi,\sin\theta\cdot\sin\varphi,\cos\theta]$ (Fig. 4c), it can be represented in two dimensions (Fig. 4d, with a Robinson map projection [55]). Such plots make it easier to visually investigate the anisotropic characteristics Figure 4: Directional Young’s modulus $E_{\text{d}}$. (a) Three-dimensional representation of Young’s modulus predicted by MatTen, for (b) the CaS rocksalt crystal. With (c) a spherical coordinate system and the Robinson map projection, (d) $E_{\text{d}}$ is represented in two dimensions. (e) Distribution of the prediction error in $E_{\text{d}}$; also included is the prediction error in the isotropic Young’s modulus from Hill average. of Young's modulus. For example, for the cubic rocksalt CaS crystal (Fig. 4b), the maximum directional Young's modulus $E_{\text{d}}^{\text{max}}$ is along the $\langle 100\rangle$ directions (for instance, $\theta=90^{\circ}$ and $\varphi=0^{\circ}$), while the minimum $E_{\text{d}}^{\text{min}}$ is along the the $\langle 111\rangle$ directions (for instance, $\theta=54.7^{\circ}$ and $\varphi=45^{\circ}$). In fact, for cubic crystals such as CaS, the extreme values of $E_{\text{d}}$ are guaranteed to occur in these two directions [34]. Eq. (4) can be simplified as $E_{\text{d}}(\mathbf{n})=[S_{1111}-2(S_{1111}-S_{1122}-2S_{2323})(n_{1}^{2}n_{2}^{ 2}+n_{2}^{2}n_{3}^{2}+n_{3}^{2}n_{1}^{2})]^{-1}$ for cubic crystals, expressed in terms of their three independent elasticity tensor components. It can be mathematically shown that, if \[S_{1111}-S_{1122}-2S_{2323}<0, \tag{5}\] Young's modulus achieves its maximum $E_{\text{d}}^{\text{max}}$ in the $\langle 100\rangle$ directions and minimum in the $\langle 111\rangle$ directions; otherwise, if $S_{1111}-S_{1122}-2S_{2323}>0$, the two extremes switch directions (derivation given in the SI). Eq. (5) is satisfied by CaS, and thus we observe the maximum and minimum in the $\langle 100\rangle$ and $\langle 100\rangle$ directions, respectively. In addition, $E_{\text{d}}$ of CaS possesses symmetry (for example, 3-fold rotational axis along the cube diagonals) consistent with a cubic crystal, further confirming that the predicted elasticity tensor _preserves material symmetry_. To quantitatively assess the ability of MatIen in predicting anisotropic elastic properties, we measured the error between the model predicted directional Young's modulus $E_{\text{d}}^{\text{pred}}$ and the DFT reference $E_{\text{d}}^{\text{ref}}$. For CaS, MatIen prediction closely follows DFT reference, with a maximum under-prediction of $8.7$ GPa along the $\langle 111\rangle$ directions and a maximum over-prediction of 9.4 GPa along the $\langle 100\rangle$ directions (Fig. S9 in the SI). In addition to this example crystal, we calculated the error for the entire test set. This is achieved by computing $D=\frac{1}{M}\int_{\theta}\int_{\varphi}\|E_{\text{d}}^{\text{pred}}(\theta, \varphi)-E_{\text{d}}^{\text{ref}}(\theta,\varphi)\|\,\text{d}\theta\text{d}\varphi$ for each crystal, where the integrations were performed by uniformly sampling 100 values for each of $\theta$ and $\varphi$, and $M=10000$ is a normalization factor resulted from the integration. The distribution of the prediction error $D$ of $E_{\text{d}}$ for the test set is plotted in 4e. It has a Gaussian-like shape, and it almost overlaps with that of isotropic Young's modulus obtained using the Hill average. Similar behaviors are observed for shear modulus and MAEs by crystal systems (Figs. S10 and S11 in the SI). These observations suggest that, on average, MatIen performs equally well for the anisotropic and averaged isotropic elastic properties. Accurate prediction of anisotropic properties offers a comprehensive understanding of the elastic characteristics exhibited by crystals, enabling the discovery of new materials through the utilization of these predictive capabilities. ### Discovering new crystals with extreme properties The maximum of the directional Young's modulus, $E_{\text{d}}^{\text{max}}$, characterizes the smallest possible deformation of a crystal under applied external loading. It helps in the selection and orientation of materials to minimize shape change to guarantee the reliability of precision devices such as micro-electromechanical systems [56]. We have also applied the MatIen model to discover new crystals with large $E_{\text{d}}^{\text{max}}$. The candidate crystals are identified via a screening process. We first filtered crystals from the Materials Project database based on their energy above the convex hull values, selecting those with a value of $\leq 50$ meV/atom. This energy determines the thermodynamic stability of a crystal and has been shown to correlate with the synthesizability of crystals [57, 58]. We further narrowed down the selection to crystals with fewer than 50 atomic sites to reduce computational cost and remove crystals already present in our dataset. This resulted in 53480 crystals for further analysis. Next, we employed MatTen to predict the full elasticity tensors for these crystals and compute their $E_{\text{d}}^{\text{max}}$. The top 100 crystals with the highest $E_{\text{d}}^{\text{max}}$ were then supplied for DFT computation to obtain their elasticity tensors. Fig. 5 presents a histogram of $E_{\text{d}}^{\text{max}}$ for the newly discovered 100 crystals. Their $E_{\text{d}}^{\text{max}}$ values all fall at the tail of the distribution of $E_{\text{d}}^{\text{max}}$ from the training data. Quantitatively, the mean of $E_{\text{d}}^{\text{max}}$ for the newly discovered crystals is 606 GPa, while that for the training data is 174 GPa, corresponding to the expected value for a randomly selected material. This demonstrates the effectiveness of leveraging MatTen to discover materials with extreme elastic properties. In addition, we have identified 11 unconventional polymorphs of elemental cubic metals regarding the direction of the extreme values of Young's modulus. Five of them are already experimentally synthesized (four stable ground-state polymorphs and one metastable polymorph [58]), and the other six are metastable polymorphs that have not been experimentally observed yet. It is believed that $E_{\text{d}}^{\text{max}}$ is along the $\langle 111\rangle$ directions (meaning Eq. (5) is not satisfied) for all cubic metals except Mo [34]. We suspect there exists other polymorphs of elemental cubic metals satisfying the criterion in Eq. (5). To test this, we performed a screening of the Materials Project database. From the dataset used for developing MatTen, we have identified six such crystals. Mo is among them, and the other five are V (one polymorph), Cr (two polymorphs), and W (two polymorphs). For crystals not in the dataset, we first used MatTen to predict their elasticity tensors and then selected the 18 crystals satisfying the criterion for verification using DFT. DFT predicted 12 crystals satisfying the criterion; six are elastically unstable structures whose Voigt matrix has negative eigenvalues [8, 9], and the other six successful ones are Mn, Na, K, Cs, Rh, and Tl (one polymorph for each). Among the 11 newly identified crystals, polymorphs of alkali metals Na, K, and Cs have DFT calculated $S_{1111}-S_{1122}-2S_{2323}$ values much more negative than that of Mo, but they are all hypothetical crystals that have not been experimentally synthesized yet. Five polymorphs are indeed experimental observed, and they are all neighbors of Mo in the periodic table, namely V, Cr, Mn, and W. Among them, four are thermodynamically stable ground-state polymorphs, and one polymorph of Cr is metastable. Crystal structures, ground-state information, and elasticity tensors of the 11 unconventional polymorphs are provided in the SI. ## 3 Discussion A universal model such as MatTen that can predict the full elasticity tensors of inorganic crystalline compounds across crystal systems and chemical species brings new possibilities to probe and design materials with targeted mechanical properties. Notably, the full elasticity tensor enables the exploration of any anisotropic elastic property. Besides the directional Young's modulus studied in this work, many other elastic properties possess anisotropic behavior, such as linear compressibility, shear modulus, and Poisson's ratio. It should be noted, however, that a crucial aspect regarding the practical use of the model relies on the robustness of the input structure. Given two structures of a crystal where the atomic coordinates are slightly different, we would expect the elastic properties to be similar. If, otherwise, the model is extremely sensitive to the input, then it is not ideal for practical application. Extra work such as highly accurate DFT structure relaxation is needed before applying the model for predictions. We tested MatTen by using Figure 5: Discovery of new crystals with large maximum directional Young’s modulus. The training data and new discoveries are separately normalized such that each sums to 100 percent. structures directly queried from the Materials Project database and structures with tighter geometry optimization. The MAE of $E_{\text{d}}^{\text{max}}$ between using the two types of structures is 6.55 GPa (Fig. S12 in the SI), more than three times smaller than the MAE (22.36 GPa) of $E_{\text{d}}^{\text{max}}$ between model predictions and DFT references. This suggests that MatTen is robust enough to its input, and reasonably optimized structures (for example, from online databases) would not introduce extra error larger than the intrinsic error in the model. The MatTen model is not limited to inorganic crystalline compounds and even elasticity tensors in general. The elastic behavior of other classes of materials such as two-dimensional layered materials and molecular crystals play a significant role in determining their functionality, and MatTen can be directly applied to model their elasticity tensors. Of course, a curated dataset of reference elasticity tensors is needed. Such data already exists, for example, in the Computational 2D Materials Database (C2DB) [59]. Moreover, besides elasticity tensor, MatTen can be applied to other tensorial properties of materials. These can be broadly categorized into two classes: material-level property and atom-level property. While the former means a single tensor for each crystal, the latter means a separate tensor for each atom in the crystal. Other material-level properties such as piezoelectric and dielectric tensors can be modeled by updating the output head as in Fig. 2 to use the corresponding irreducible representations of the tensor of interest (for example, a single second-rank symmetric matrix for the dielectric tensor). For atom-level properties such as the neutron magnet resonance (NMR) tensor, instead of using a mean pooling to aggregate atom features, one can directly map the features of an atom to a tensor for that atom. Using MatTen, we have conducted such an analysis for NMR tensors of silicon oxides and found that MatTen significantly outperforms both historic analytical models and other machine learning models by more than 50% for isotropic and anisotropic NMR chemical shift [60]. One potential limitation of the proposed approach is the reliance on a relatively large dataset to develop the model. We have curated a dataset of 10276 elasticity tensors which took millions of CPU hours to obtain. Such large datasets for other tensorial properties may not be readily available, but they begin to emerge. For example, the Materials Project has about 3000 piezoelectric [61] tensors and 7000 dielectric [62] tensors. This amount of data might still be a good start to training faithful models given that piezoelectric and dielectric are third- and second-rank tensors, respectively, which are much simpler than the fourth-rank elasticity tensor. In fact, for the second-rank NMR tensor, we only used a dataset of 421 crystals to obtain the best-performing model [60]. Another possibility is to apply a transfer learning approach, where the model is first trained on a different property with large data (for instance, elasticity tensor) and then finetuned on the target property of interest (for instance, piezoelectric tensor). ## 4 Methods ### Data generation The elasticity tensors were computed by a liner fitting of the stresses and strains obtained from DFT calculations using the Vienna Ab Initio Simulation Package (VASP) [63]. The calculations follow the same procedures discussed in Ref. [10] with two general improvements. First, to get more precise stresses for calculating the elasticity tensor, the projection operators in VASP are evaluated in the reciprocal space, that is, the setting LREAL=False was adopted. Second, to reduce numerical error in the calculations, the stresses are symmetrized according to the crystal symmetry. The entire workflow was implemented in the open-source atomate package [15]. ## Model architecture A crystal structure is converted to a graph using a distance-based approach, where an edge is created between a pair of atoms if their distance is smaller than a cutoff radius $r_{\text{cut}}$. Periodic boundary conditions are considered in the graph construction. For atom $i$, its atomic number $Z_{i}$ is embedded as a vector with $c$ components using a one-hot encoding to obtain the initial atom feature $h_{i}$. The unit vector $\hat{r}_{ij}$ from atom $i$ to atom $j$ is expanded using a spherical harmonics basis consisting up to a degree of $l=4$. (Explicitly, this corresponds to the "$0\text{e}+1\text{o}+2\text{e}+3\text{o}+4\text{e}$" irreducible representations in e3nn notation [51]). The distance $r_{ij}$ between atoms $i$ and $j$ is expanded into a vector $R$ using the radial basis functions [46], \[\text{RBF}_{n}(d)=\sqrt{\frac{2}{r_{\text{cut}}}}\frac{\sin(\frac{n\pi}{r_{ \text{cut}}}d)}{d} \tag{6}\] where $n=1,2,\ldots,$ is an index of the radial basis. With the atom features $F$, the spherical harmonics expansion of $\hat{r}_{ij}$, and the radial basis expansion $R$ of $r_{ij}$ obtained from the embedding layers, the interaction block performs tensor product-based convolution to refine the atom features. This is achieved via Eq. (3), more specifically [27], \[\mathcal{L}_{acm_{o}}^{l_{o}}(\{\vec{\mathbf{r}}_{ab}\},\{F_{bcm_{i}}^{l_{i}}\})= \sum_{m_{i},m_{f}}C_{(l_{f},m_{f})(l_{i},m_{i})}^{(l_{o},m_{o})}\sum_{b\in \mathcal{N}_{a}}R_{c}^{(l_{f},l_{i})}(r_{ab})Y_{l_{f}}^{m_{f}}(\hat{\mathbf{r}}_{ ab})F_{bcm_{i}}^{l_{i}}, \tag{7}\] where $a$ denotes the center atom, $b$ denotes all its neighbors $\mathcal{N}_{b}$ within the cutoff $r_{\text{cut}}$; $l$ is an integer indicating the degree of the spherical harmonic function, and $m=-l,\ldots,l$; the subscripts $i$, $o$, and $f$ indicate input, output, and filter, respectively; $c$ is the channel index (for example, for the embedding layer, it indicates the components of the one-hot encoding); $C$ denotes the Clebsch-Gordan coefficients; and finally, $R_{c}^{(l_{f},l_{i})}$ are learnable multilayer perceptrons (MLPs), which take the RBF expansion as the input and contain most of the parameters of the model. Essentially, this combines the atom features of neighbors $b$ to be the new atom features of the center atom $a$, in the same spirit of a message-passing graph neural network. A major characteristic of Eq. (7) is that the use of the spherical harmonics and the Clebsch-Gordan coefficients together imply that convolutions are equivariant [27]. The atom features $V$ is also passed through a self-interaction, \[\mathcal{S}_{acm}^{l}(V_{ac^{\prime}m}^{l})=\sum_{c^{\prime}}W_{cc^{\prime}}^ {l}V_{ac^{\prime}m}^{l}, \tag{8}\] where $W_{cc^{\prime}}^{l}$ are learnable model parameters. The updated atom features is then obtained as \[V^{k+1}=\mathcal{L}(V^{k})+\mathcal{S}(V^{k}), \tag{9}\] where $V^{k}$ denotes the features in interaction block $k$, and $V^{k+1}$ the features in the next interaction block. Indeed, $V^{k+1}$ are further passed through a nonlinearity and a normalization functions. For each scalar part $s$ in $V$, the nonlinearity is chosen to be the SiLU function [64], \[\text{SiLU}(s)=\sigma(s)s, \tag{10}\] and for each non-scalar part $\mathbf{t}$ in $V$, the gated nonlinearity [65] is adopted, \[G(\mathbf{t})=\sigma(x)\mathbf{t}, \tag{11}\] where $\sigma$ is the sigmoid function, and $x$ is a scalar obtained from Eq. (7) by setting $l_{o}=0$ and $m_{o}=0$. Finally, the equivariant batch normalization [51] is applied to the features to avoid gradient vanishing or exploding. The readout head aggregates the features of individual atoms to obtain a representation of the material via a mean pooling, \[V_{\text{mat}}=\frac{1}{N}\sum_{a}^{N}V_{a}, \tag{12}\] where $V_{a}$ denotes the features of atom $a$ for a crystal of a total number of $N$ atoms. From $V_{\text{mat}}$, the appropriate irreducible representations ("2x0e + 2x2e + 4e" in e3nn notation) that correspond to the elasticity tensor (two scalars, two second-rank traceless symmetric tensors, and one fourth-rank traceless symmetric tensor) are selected to construct the elasticity tensor according to Eq. (2). ### Model training The dataset of 10276 elasticity tensors is split into three subsets for training, validation, and testing with a split ratio of 8:1:1. The model parameters are optimized using the training set, model hyperparameters are determined based on model performance on the validation set, and error analysis is performed using the test set unless otherwise stated. We train the model with the Adam optimizer to minimize a mean-squared-error loss function $L=\sum_{i}^{B}\\|\mathbf{C}_{i}-\mathbf{C}_{i}^{\text{ref}}\\|^{2}$ with a mini-batch size $B$ of 32. Note, $\mathbf{C}_{i}$ denotes the irreducible representation of the model predicted elasticity tensor with 21 components (see Eq. (2)), but not the Cartesian tensor with 81 components. Similarly, $\mathbf{C}_{i}^{\text{ref}}$ denotes the corresponding reference DFT values. The learning rate is set to 0.01, and a reduce-on-plateau learning rate scheduler is used, which decreases the learning rate by a factor of 0.5 if the validation error does not decrease for 50 epochs. The training stops when the validation error does not decrease for 150 consecutive epochs, and a maximum of 1000 epochs are allowed for the optimization. We performed a grid search to obtain model hyperparameters such as the $r_{\text{cut}}$ and $c$, and their optimal values are listed in Table S3 in the SI. ### Compliance tensor The compliance tensor $\mathbf{S}$ is a fourth-rank tensor defined from the inverse stress-strain relation $\mathbf{\epsilon}=\mathbf{S}\mathbf{\sigma}$, where $\mathbf{\epsilon}$ and $\mathbf{\sigma}$ are the second-rank strain tensor and stress tensor, respectively. The compliance tensor in Voigt notation $\mathbf{s}$ can be obtained as the inverse of the elasticity tensor Voigt matrix, \[\mathbf{s}=\mathbf{c}^{-1}, \tag{13}\] which is a 6 by 6 symmetric matrix. The relationships between the components of the full compliance tensor $S_{ijkl}$ and the Vogit matrix $s_{\alpha\beta}$ are [34]: \[S_{ijkl} =s_{\alpha\beta},\text{ when }\alpha\text{ and }\beta\text{ are 1, 2, or 3,}\] \[2S_{ijkl} =s_{\alpha\beta},\text{ when either }\alpha\text{ or }\beta\text{ are 4, 5, or 6,} \tag{14}\] \[4S_{ijkl} =s_{\alpha\beta},\text{ when both }\alpha\text{ and }\beta\text{ are 4, 5, or 6.}\] ### Averaged elastic moduli of polycrystals Given the elasticity tensor of a single crystal, the averaged bulk, shear, and Young's moduli of polycrystalline materials can be computed using different average schemes. The Voigt average assumes that the strain is the same in each grain, equal to the macroscopically-applied strain [35]. The Voigt bulk modulus is \[K_{V}=[(c_{11}+c_{22}+c_{33})+2(c_{12}+c_{23}+c_{31})]/9, \tag{15}\] and the Voigt shear modulus is \[G_{V}=[(c_{11}+c_{22}+c_{33})-(c_{12}+c_{23}+c_{31})+3(c_{44}+c_{55}+c_{66})]/15. \tag{16}\] The Reuss average assumes that the stress is the same in each grain, equal to the macroscopically applied stress [66]. The Reuss bulk modulus is \[K_{R}=1/[(s_{11}+s_{22}+s_{33})+2(s_{12}+s_{23}+s_{31})], \tag{17}\] and Reuss the shear modulus is \[G_{R}=15/[4(s_{11}+s_{22}+s_{33})-4(s_{12}+s_{23}+s_{31})+3(s_{44}+s_{55}+s_{66 })]. \tag{18}\] The Voigt and Reuss averages are the two extreme cases. The Hill average takes their arithmetic mean and is considered the most accurate in a wide range of experimental conditions [52]. The Hill bulk modulus is \[K_{H}=(K_{V}+K_{R})/2, \tag{19}\] and the Hill shear modulus is \[G_{H}=(G_{V}+G_{R})/2. \tag{20}\] Given the bulk modulus and the shear modulus (from any of the Voigt, Reuss, and Hill schemes), Young's modulus [67] can be computed as \[E=9KG/(3K+G). \tag{21}\] In this work, we report the bulk modulus, shear modulus, and Young's modulus from the Hill scheme. ### Scaled error The mean absolute error (MAE) and mean absolute deviation (MAD) are defined as $\text{MAE}=\frac{1}{N_{1}}\sum_{i}^{N_{1}}\|y_{i}-y_{i}^{\text{pred}}\|$ and $\text{MAD}=\frac{1}{N_{2}}\sum_{i}^{N_{2}}\|(y_{i}-\bar{y})\|$, in which $y_{i}$ is the reference value of data point $i$, $y_{i}^{\text{pred}}$ is the model prediction for the data point, and $\bar{y}$ is the average of all reference values. The numbers $N_{1}$ and $N_{2}$ do not necessarily need to be the same. This is the case in Fig. 3 and Table 1, where $N_{1}$ is the number of crystals in a specific crystal system and $N_{2}$ is the total size of the test set. The scaled error (SE) is then computed as $\text{SE}=\text{MAE}/\text{MAD}$. ### Software implementation The MatTen model was implemented using the e3nn package [51] built on top of PyTorch [68], and the training was performed using Pytorch-Lightning [69]. The DFT calculations were performed using the atomate workflow [15] and all crystal structure processing was performed using the Python Materials Genomics (pymatgen) [13]. Directional Young's modulus was obtained using the elate package [54]. ## Data Availability The elasticity tensors used for model development, the 100 new crystals with large maximum directional Young's modulus, and the elemental cubic metals are available at [https://doi.org/10.5281/zenodo.8190849](https://doi.org/10.5281/zenodo.8190849) The elasticity tensors are also available from the Materials Project database via the web interface at [https://materialsproject.org](https://materialsproject.org) or the API at [https://api.materialsproject.org](https://api.materialsproject.org). ## Code Availability The MatTen model and training scripts are released as an open-source repository at [https://github.com/wengroup/matten](https://github.com/wengroup/matten). ## Acknowledgements The method development was supported by the startup funds from the Presidential Frontier Faculty Program at the University of Houston. Support for software and data infrastructure development was provided by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under contract No. DE-AC02-05-CH11231 (Materials Project program KC23MP). This work used computational resources provided by the Research Computing Data Core at the University of Houston, the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under contract No. DE-AC02-05CH11231, and the Lawrencium computational cluster resource provided by the IT Division at the Lawrence Berkeley National Laboratory (Supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under contract No. DE-AC02-05CH11231). ## Author contributions Conceptualization, investigation, and writing - original draft: M.W.; data curation and formal analysis: M.W., M.K.H., J.M.M., and P.H.; writing - review & editing: M.W., M.K.H., J.M.M., P.H., and K.A.P.; project administration: M.W.; funding acquisition: M.W. and K.A.P. ## Competing interests The authors declare no competing interests. ## References * Hetnarski and Ignaczak [2016] Richard B Hetnarski and Jozef Ignaczak. _The mathematical theory of elasticity_. 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[2019] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. _Advances in neural information processing systems_, 32, 2019. * [69] William Falcon. Pytorch lightning. [https://github.com/Lightning-AI/lightning](https://github.com/Lightning-AI/lightning), 2023. accessed 2023-06-11. Supplementary Information: A Universal Equivariant Graph Neural Network for the Elasticity Tensors of Any Crystal System Mingjian Wen${}^{1}$1, Matthew K. Horton${}^{2,3}$, Jason M. Munro${}^{2}$, Patrick Huck${}^{4}$, Kristin A. Persson${}^{5,6}$ Footnote 1: Email: [email protected] ${}^{1}$ Chemical and Biomolecular Engineering, University of Houston, Houston, 77204, TX, USA ${}^{2}$ Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, 94720, CA, USA ${}^{3}$ Microsoft Research, Redmond, 98052, WA, USA ${}^{4}$ Energy Technologies Area, Lawrence Berkeley National Laboratory, Berkeley, 94720, CA, USA ${}^{5}$ Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, 94720, CA, USA ${}^{6}$ Department of Materials Science and Engineering, University of California, Berkeley, Berkeley, 94720, CA, USA ## The ten symmetry classes of elasticity tensors Early approaches take inspiration from crystallography. Out of the 32 distinct crystallographic point groups, only 11 are centrosymmetric (meaning the point group contains an inversion center as one of its symmetry elements), each forming a unique diffraction pattern. The diffraction patterns of other noncentrosymmetric crystals each is the same as one of the 11 centrosymmetric crystals. Based on the diffraction patterns, the 32 distinct point groups can be categorized into 11 classes, called the Laue groups [3]. According to the Laue groups, Wallace [1] classifies the elastic tensors into 12 classes (the additional 1 being the isotropic class that does not apply for single crystals), and they reduce to 10 classes considering the number of independent components (Fig. S1). The results are widely cited, including the classical book on the subject by Nye [2] and many recent papers [4, 5, 6]. This crystallographic approach seems reasonable; however, the conclusions are incorrect. The tetragonal and trigonal systems are each divided into two symmetry classes, but the distinctions can be eliminated by a different choice of the coordinate system [7, 8]. Then each of the tetragonal and trigonal systems will have 6 independent components. ## Harmonic decomposition of the elasticity tensor In the harmonic decomposition, the elasticity tensor can be written as \[\mathbf{C}=h_{1}(\lambda)+h_{2}(\eta)+h_{3}(\mathbf{A})+h_{4}(\mathbf{B})+h_{5}(\mathbf{H}). \tag{1}\] The appropriate values for each of the term is as follows: \[\lambda=[2C_{ppmm}-C_{pmpm}]/15,\] \[\eta=[-3C_{ppmm}-C_{pmpm}]/90,\] \[A_{ij} =[15C_{ijmm}-12C_{imjm}-5\delta_{ij}C_{ppmm}+4\delta_{ij}C_{pmpm}]/21,\] \[B_{ij} =[-6C_{ijmn}+9C_{imjm}+2\delta_{ij}C_{ppmm}-3\delta_{ij}C_{pmpm}]/21,\] \[H_{ijkl}= (C_{ijkl}+C_{iklj}+C_{iljk})/3\] \[-[(C_{ijmm}+2C_{imjm})\delta_{kl}+(C_{klm}+2C_{kmlm})\delta_{ij}\] \[+(C_{ikmm}+2C_{imkm})\delta_{jl}+(C_{jlmn}+2C_{jmlm})\delta_{ik}\] \[+(C_{ilmm}+2C_{imlm})\delta_{jk}+(C_{jkmn}+2C_{jmkm})\delta_{il}]/21\] \[+(C_{ppmm}+2C_{pmpm})(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{ jl}+\delta_{il}\delta_{jk})/105,\] and \[h_{1}(\lambda)=\delta_{ij}\delta_{kl}\lambda,\] \[h_{2}(\eta)=(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\eta,\] \[h_{3}(\mathbf{A})=\delta_{ij}A_{kl}+\delta_{kl}A_{ij},\] \[h_{4}(\mathbf{B})=\delta_{ik}B_{jl}+\delta_{jl}B_{ik}+\delta_{il}B_{jk}+\delta_{ jk}B_{il},\] \[h_{5}(\mathbf{H})=H_{ijkl},\] where $\delta_{ij}$ is the Kronecker delta. This decomposition follows Ref. [9], and as mentioned there that "... other forms of harmonic decomposition are possible: It suffices to use invertible linear combinations of $\mathbf{A}$ and $\mathbf{B}$ and, analogously, invertible linear combinations of $\lambda$ and $\eta$." See Ref. [10, 11] for such examples. Nevertheless, harmonic decomposition is unique to linear combinations. ## Proof of MatTen satisfying material symmetry The MatTen model $\mathbf{C}=f(x)$ is equivariant to $SO(3)$ transformations, satisfying \[D_{y}(g)f(x)=f(D_{x}(g)x). \tag{2}\] This comes from the fact that each layer of MatTen is equivariant, and the composition of such layers is also equivariant. We refer to Ref. [12] for proof of the equivariance of the layers. The representation $D_{x}(g)$ in the space of crystal structures can be written as $D_{x}(g)=R_{ip}$, and the representation $D_{y}(g)$ in the space of stiffness tensors can be written as $D_{g}(g)=R_{ip}R_{jq}R_{kr}R_{ls}$, where $R\in SO(3)$ is a rotation matrix. Let $Q\in P$, where $P$ denotes the set of rotations in the point group of a crystal, we will have $P\subset SO(3)$. Therefore, for $R=Q$, Eq. S(2) is satisfied, i.e., \[Q_{ip}Q_{jq}Q_{kr}Q_{ls}C_{prqs}=f(Qx) \tag{3}\] Owning to material symmetry, we have $Qx=x$, that is, the crystal structure $x$ is indistinguishable before and after the transformation. Thus, $f(Qx)=f(x)$. Plugging it into Eq. S(2), we have \[Q_{ip}Q_{jq}Q_{kr}Q_{ls}C_{prqs}=f(x)=C_{ijkl}, \tag{4}\] which is Eq.(1) in the main text. Once this is satisfied, the material symmetry will be reflected in the stiffness tensor as discussed in the main text and proved in Ref. [9]. Figure S3: Distribution of the bulk, shear, and Young's moduli in the dataset. Figure S5: Distribution of shear modulus in the dataset by crystal system. ## Additional results on isotropic properties ## Directional Young's modulus Here we prove that, for cubic crystals, \[\text{if}\ S_{1111}-S_{1122}-2S_{2323}<0,E_{\text{d}}^{\text{max}}\text{is along}\ \langle 100\rangle\ \text{and}\ E_{\text{d}}^{\text{max}}\text{is along}\ \langle 111\rangle, \tag{5}\] otherwise, \[\text{if}\ S_{1111}-S_{1122}-2S_{2323}>0,E_{\text{d}}^{\text{max}}\text{is along}\ \langle 111\rangle\ \text{and}\ E_{\text{d}}^{\text{max}}\text{is along}\ \langle 100\rangle. \tag{6}\] and \[\text{if}\ S_{1111}-S_{1122}-2S_{2323}=0,\text{ the materials is isotropic regarding Young's modulus.} \tag{7}\] The inverse of the directional Young's modulus is \[E_{\text{d}}(\mathbf{n})^{-1}=n_{i}n_{j}n_{k}n_{l}S_{ijlk}, \tag{8}\] where $S_{ijkl}$ is the compliance tensor and $\mathbf{n}$ is an unit direction vector. For a cubic crystal, the 21 non-zero components can be classified into three groups [2]: * $S_{1111}=S_{2222}=S_{3333}$ * $S_{1122}=S_{2211}=S_{2233}=S_{3322}=S_{3311}=S_{1133}$ * $S_{2323}=S_{2332}=S_{3223}=S_{3232}=S_{1212}=S_{1221}=S_{2112}=S_{1313}=S_{13 31}=S_{3113}$. Substituting these into Eq. S(8), we have, \[E_{\text{d}}(\mathbf{n})^{-1} =S_{1111}(n_{1}^{4}+n_{2}^{4}+n_{3}^{4})+2S_{1122}(n_{1}^{2}n_{2}^ {2}+n_{2}^{2}n_{3}^{2}+n_{3}^{2}n_{1}^{2})+4S_{2323}(n_{1}^{2}n_{2}^{2}+n_{2}^ {2}n_{3}^{2}+n_{3}^{2}n_{1}^{2})\] \[=S_{1111}[1-2(n_{1}^{2}n_{2}^{2}+n_{2}^{2}n_{3}^{2}+n_{3}^{2}n_{1} ^{2})]+2S_{1122}(n_{1}^{2}n_{2}^{2}+n_{2}^{2}n_{3}^{2}+n_{3}^{2}n_{1}^{2})+4S_ {2323}(n_{1}^{2}n_{2}^{2}+n_{2}^{2}n_{3}^{2}+n_{3}^{2}n_{1}^{2})\] \[=S_{1111}-2(S_{1111}-S_{1122}-2S_{2323})(n_{1}^{2}n_{2}^{2}+n_{2} ^{2}n_{3}^{2}+n_{3}^{2}n_{1}^{2})\] \[=S_{1111}-2(S_{1111}-S_{1122}-2S_{2323})f. \tag{9}\] In the second equality, we have used $n_{1}^{4}+n_{2}^{4}+n_{3}^{4}=(n_{1}^{2}+n_{2}^{2}+n_{3}^{2})^{2}-2(n_{1}^{2}n_ {2}^{2}+n_{2}^{2}n_{3}^{2}+n_{3}^{2}n_{1}^{2})=1-2(n_{1}^{2}n_{2}^{2}+n_{2}^{2 }n_{3}^{2}+n_{3}^{2}n_{1}^{2})$, in which $(n_{1}^{2}+n_{2}^{2}+n_{3}^{2})^{2}=1$, because $\mathbf{n}$ is a unit vector. In the last equality, we have defined $f:=n_{1}^{2}n_{2}^{2}+n_{2}^{2}n_{3}^{2}+n_{3}^{2}n_{1}^{2}$. From Eq. S(9), it seen that Eq. S(7) is valid. In fact, $f$ has its maximum value of $1/3$ along the $\langle 111\rangle$ directions, and the minimum of $f$ is 0 along the $\langle 100\rangle$ directions (derived below). As a result, Eq. S(5) and Eq. S(6) are valid. (Note that Eq. S(9) gives the inverse of the directional Young's modulus.) \begin{table} \begin{tabular}{c c c c c c c} \hline \hline & \multicolumn{2}{c}{$\log_{10}(K)$} & \multicolumn{2}{c}{$\log_{10}(G)$} & \multicolumn{2}{c}{$\log_{10}(E)$} \\ \cline{2-7} & MAE & MAE/MAD & MAE & MAE/MAD & MAE & MAE/MAD \\ \hline Matflen & 0.046 (0.002) & 0.166 (0.006) & 0.094 (0.002) & 0.331 (0.010) & 0.087 (0.002) & 0.309 (0.018) \\ AutoMatminer & 0.050 (0.002) & 0.187 (0.009) & 0.090 (0.002) & 0.307 (0.006) & 0.086 (0.002) & 0.301 (0.009) \\ \hline \hline \end{tabular} \end{table} Table S1: Prediction of the bulk modulus $K$, shear modulus $G$, and Young’s modulus $E$ in logarithmic space. $K$, $G$, and $E$ are in the units of GPa. The results for MatTen are calculated from a single model, while a separate AutoMatminer model is trained for each property. The value in a pair of parentheses is the standard deviation from an ensemble of five models trained with different initialization. MAE: mean absolute error; MAD: mean absolute deviation. Below, we show that the maximum of $f$ is $1/3$ along the $\langle 111\rangle$ directions, and the minimum of $f$ is 0 along the $\langle 100\rangle$ directions. Let $n_{1}^{2}=a,n_{2}^{2}=b$ and $n_{3}^{2}=c$, we have $a+b+c=1$ because $\mathbf{n}$ is a unit vector. Thus, \[f=ab+bc+ca=ab+c(b+a)=ab+(1-a-b)(b+a)=a+b-ab-a^{2}-b^{2}. \tag{10}\] Let \[\frac{\partial f}{\partial a} =1-b-2a=0 \tag{11}\] \[\frac{\partial f}{\partial b} =1-a-2b=0,\] and solve the equations, we have $a=b=c=1/3$, i.e. $n_{1}^{2}=n_{2}^{2}=n_{3}^{3}=1/3$, At these values, $f=1/3$ and we can verify that it is a maximum. This also suggests $\mathbf{n}$ is along the $\langle 111\rangle$ family of directions. The other extreme values of $f$ are located at the boundaries of $a$ (or $b$ or $c$). Since $n_{1}$ is a component of the unit vector, then $n_{1}\in[-1,1]$, i.e. $a\in[0,1]$. So, the extreme value is obtained when * $n_{1}=0$, $n_{2}=\pm 1$, $n_{3}=0$ * $n_{1}=0$, $n_{2}=0$, $n_{3}=\pm 1$ * $n_{1}=\pm 1$, $n_{2}=0$, $n_{3}=0$. These are the $\langle 100\rangle$ directions, at which the minimum is $f=0$. Figure S10: Distribution of the prediction error for shear modulus. The directional shear modulus can be computed as $G_{\mathrm{d}}(\mathbf{n},\mathbf{m})=(n_{i}m_{j}n_{k}m_{l}S_{ijkl})^{-1}$, where $\mathbf{n}$ and $\mathbf{m}$ are two direction unit vectors, and $\mathbf{S}$ is the compliance tensor [6]. The data of $G_{\mathrm{d}}$ is obtained by sampling in a way similar to $E_{\mathrm{d}}$ discussed in the main text. The prediction error is the difference between MatIen prediction and DFT reference. Fig. S12 shows the $E_{\text{d}}^{\text{max}}$ for the 100 new crystals. The MAE between predicted $E_{\text{d}}^{\text{max}}$ with initial structure and predicted $E_{\text{d}}^{\text{max}}$ with further relaxed structure is 6.55 GPa. It is much smaller than the MAE (22.36 GPa) between MatTen prediction and DFT reference for the test set. This demonstrates the robustness of MatTen with respect to the structure of the input crystal as discussed in the main text. As shown in Fig. S12 b, if we consider the 100 new crystals instead of the test set, the MAE between MatTen prediction and DFT is much higher, with a value of 48.69 GPa. This is expected, since, for the 100 new crystals, we are probing extreme values at the edge of the training data distribution, while the test set follows the same distribution of the training data. This signifies the importance of further confirmation with more accurate computation (DFT in this case) and even experiments once the search space has been narrowed down via the screening using the model. \begin{table} \begin{tabular}{c c c c c c} \hline Materials Project ID & Formula & $\Delta S_{\text{DFT}}$ & $\Delta S_{\text{MatTen}}$ & Experimentally observed & Ground-state polymorph \\ \hline mp-129 & Mo & -0.00150 & -0.00191 & Yes & Yes \\ mp-146 & V & -0.00994 & -0.01006 & Yes & Yes \\ mp-17 & Cr & -0.00267 & -0.00253 & Yes & No \\ mp-90 & Cr & -0.00369 & 0.00055 & Yes & Yes \\ mp-91 & W & -0.00056 & -0.00042 & Yes & Yes \\ mp-11334 & W & -0.00285 & -0.00331 & No & No \\ mp-35 & Mn & -0.02022 & -0.00277 & Yes & Yes \\ mp-1186040 & Na & -0.15435 & -0.32852 & No & No \\ mp-1184808 & K & -0.34397 & -0.12334 & No & No \\ mp-949029 & Cs & -0.53668 & -4.79427 & No & No \\ mp-1239193 & Rh & -0.03371 & -0.06590 & No & No \\ mp-1187790 & TI & -0.09708 & -0.05340 & No & No \\ \hline \end{tabular} \end{table} Table S2: Polymorphs of elemental cubic metal with $E_{\text{d}}^{\text{max}}$ along $\langle 100\rangle$ directions and $E_{\text{d}}^{\text{min}}$ along $\langle 111\rangle$ directions. $\Delta S=S_{1111}-S_{1122}-2S_{2323}$. Among the crystal structures with the same composition, the one having the lowest energy is called the ground-state polymorph and is stable with respect to phase transition into other structures [13]. The crystal structures and the elasticity tensors of these metals are provided as well. See Data Availability in the main text. ## Model hyperparameters
2304.07302	HGWaveNet: A Hyperbolic Graph Neural Network for Temporal Link Prediction	Temporal link prediction, aiming to predict future edges between paired nodes in a dynamic graph, is of vital importance in diverse applications. However, existing methods are mainly built upon uniform Euclidean space, which has been found to be conflict with the power-law distributions of real-world graphs and unable to represent the hierarchical connections between nodes effectively. With respect to the special data characteristic, hyperbolic geometry offers an ideal alternative due to its exponential expansion property. In this paper, we propose HGWaveNet, a novel hyperbolic graph neural network that fully exploits the fitness between hyperbolic spaces and data distributions for temporal link prediction. Specifically, we design two key modules to learn the spatial topological structures and temporal evolutionary information separately. On the one hand, a hyperbolic diffusion graph convolution (HDGC) module effectively aggregates information from a wider range of neighbors. On the other hand, the internal order of causal correlation between historical states is captured by hyperbolic dilated causal convolution (HDCC) modules. The whole model is built upon the hyperbolic spaces to preserve the hierarchical structural information in the entire data flow. To prove the superiority of HGWaveNet, extensive experiments are conducted on six real-world graph datasets and the results show a relative improvement by up to 6.67% on AUC for temporal link prediction over SOTA methods.	Qijie Bai, Changli Nie, Haiwei Zhang, Dongming Zhao, Xiaojie Yuan	2023-04-14T07:07:00Z	http://arxiv.org/abs/2304.07302v2	# HGWaveNet: A Hyperbolic Graph Neural Network for Temporal Link Prediction ###### Abstract. Temporal link prediction, aiming to predict future edges between paired nodes in a dynamic graph, is of vital importance in diverse applications. However, existing methods are mainly built upon uniform Euclidean space, which has been found to be conflict with the power-law distributions of real-world graphs and unable to represent the hierarchical connections between nodes effectively. With respect to the special data characteristic, hyperbolic geometry offers an ideal alternative due to its exponential expansion property. In this paper, we propose HGWaveNet, a novel hyperbolic graph neural network that fully exploits the fitness between hyperbolic spaces and data distributions for temporal link prediction. Specifically, we design two key modules to learn the spatial topological structures and temporal evolutionary information separately. On the one hand, a hyperbolic diffusion graph convolution (HDGC) module effectively aggregates information from a wider range of neighbors. On the other hand, the internal order of causal correlation between historical states is captured by hyperbolic dilated causal convolution (HDCC) modules. The whole model is built upon the hyperbolic spaces to preserve the hierarchical structural information in the entire data flow. To prove the superiority of HGWaveNet, extensive experiments are conducted on six real-world graph datasets and the results show a relative improvement by up to 6.67% on AUC for temporal link prediction over SOTA methods. Temporal link prediction, hyperbolic geometry, graph neural network, diffusion graph convolution, dilated causal convolution + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern Recognition + Footnote †: journal: Computer Vision and Pattern 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Furthermore, DyRep (2017) leverages recurrent neural networks to update node representations over time. TGAT (Chen et al., 2017) uses the multi-head attention mechanism (Wang et al., 2018) and a sampling strategy that fixes the number of neighbor nodes involved in message-propagation to learn both spatial and temporal information. These methods are all built upon Euclidean spaces. However, recent works (Chen et al., 2017; Chen et al., 2018) have noticed that most real-world graph data, such as social networks, always exhibit implicit hierarchical structures and power-law distributions (as shown in Figure 1(a)) rather than uniform grid structures which fit Euclidean spaces best. The mismatches between data distributions and space geometries severely limit the performances of Euclidean models (Chen et al., 2017; Chen et al., 2018). Hyperbolic spaces are those of constant negative curvatures, and are found to keep great representation capacity for data with hierarchical structures and power-law distributions due to the exponential expansion property (as shown in Figure 1(b)) (Chen et al., 2017; Chen et al., 2018). In the past few years, researchers have made some progress in representing hierarchical data in hyperbolic spaces (Chen et al., 2017; Chen et al., 2018; Chen et al., 2018; Wang et al., 2018; Wang et al., 2018; Wang et al., 2018; Wang et al., 2018). HTGN (Wang et al., 2018) attempts to embed dynamic graphs into hyperbolic geometry and achieves state-of-the-art performance. It adopts hyperbolic GCNs to capture spatial features and a hyperbolic temporal contextual attention module to extract the historical information. However, this method faces two major shortcomings. First, for some large graphs with long paths, the message propagation mechanism of GCNs is not effective enough because only direct neighbor nodes can be calculated in each step. Second, the temporal contextual attention module cannot handle the internal order of causal correlation in historical data and thus loses valid information for temporal evolutionary process. To address the two aforementioned shortcomings of HTGN, in this paper, we propose a novel hyperbolic graph neural network model named HGWaveNet for temporal link prediction. In HGWaveNet, we first project the nodes into hyperbolic spaces. Then in the aspect of spatial topology, a hyperbolic diffusion graph convolution (HDGC) module is designed to learn node representations of each snapshot effectively from both direct neighbors and indirectly connected nodes. For temporal information, recurrent neural networks always emphasize short time memory while ignoring that of long time due to the time monotonic assumption (Chen et al., 2017), and the attention mechanism ignores the causal orders in temporality. Inspired by WaveNet (Wang et al., 2018) and Graph WaveNet (Wang et al., 2018), we present hyperbolic dilated causal convolution (HDCC) modules to obtain hidden cumulative states of nodes by aggregating historical information and capturing temporal dependencies. In the training process of each snapshot, the hidden cumulative states and spatial-based node representations are fed into a hyperbolic gated recurrent unit (HGRU). The outputs of HGRU are regarded as the integration of spatial topological structures and temporal evolutionary information, and are utilized for final temporal link prediction. A hyperbolic temporal consistency (HTC) (Wang et al., 2018) component is also leveraged to ensure stability for tracking the evolution of graphs. To conclude, the main contributions of this paper are as follows: * We propose a novel hyperbolic graph neural network model named HGWaveNet for temporal link prediction, which learns both spatial hierarchical structures and temporal causal correlation of dynamic graphs. * With regard to spatial topological structures, we design a hyperbolic diffusion graph convolution (HDGC) module to fit the power-law distributions of data and aggregate information from a wider range of nodes with greater effectiveness. * For temporal evolution, we present hyperbolic dilated causal convolution (HDCC) modules to capture the internal causality between snapshots, and a hyperbolic temporal consistency (HTC) component is applied to remain stable when learning the evolution of graphs. * We conduct extensive experiments on diverse real-world dynamic graphs. The results prove the superiority of HGWaveNet, as it has a relative improvement by up to 6.67% in terms of AUC over state-of-the-art methods. ## 2. Related Work In this section, we systematically review the relevant works on temporal link prediction and hyperbolic graph representation learning. ### Temporal Link Prediction Temporal link prediction on dynamic graphs has attracted increasing interests in the past few years. Early methods usually use traditional algorithms or shallow neural architectures to represent structural and temporal information. For instance, CTDNE (Cheng et al., 2017) captures the spatial and temporal information simultaneously by adding temporal constraints to random walk. Another example of using temporal random walk is DynNode2vec (Cheng et al., 2017), which updates the sampled sequences incrementally at each snapshot rather than generating them anew. DynamicTriad (Wang et al., 2018) imposes the triad closure process and models the evolution of graphs by developing closed triads from open triads. Furthermore, Change2vec (Chen et al., 2017) improves this process and makes it applicable for dynamic heterogeneous graphs. HTNE (Wang et al., 2018) integrates the Hawkes process into graph embedding to learn the influence of historical neighbors on current neighbors. In contrast, graph neural network methods have recently received increasing attention. GCN (Wang et al., 2018) provides an excellent node embedding pattern for general tasks on graphs, and most of later models take GCN or its variants, such as GraphSAGE (Gan et al., 2017) and GAT (Gan et al., 2017), as basic modules for learning topological structures. GCRN (Wang et al., 2018) feeds node representations learned from GCNs into a modified LSTM to obtain the temporal information. Similar ideas are Figure 1. (a) The degree distribution of real-world network MovieLens. Coordinate axes are logarithmic. (b) Areas of circles on hyperbolic spaces (curvature $c=-1$) and Euclidean spaces, which shows that the hyperbolic spaces expand exponentially while Euclidean spaces expand polynomially. explored in EvolveGCN (Wang et al., 2017), E-LSTM-D (Chen et al., 2017) and NTF (Wang et al., 2017). The main difference between EvolveGCN and E-LSTM-D is that EvolveGCN could be considered as a combination of GCN and RNN while E-LSTM-D uses LSTM together with an encoder-decoder architecture. NTF takes a reverse order that characterizes the temporal interactions with LSTM before adopting the MLP for non-linearities between different latent factors, and sufficiently learns the evolving characteristics of graphs. To better blend the spatial and temporal node features, DySAT (Dai et al., 2017) proposes applying multi-head self-attention. TGN (Wang et al., 2017) leverages memory modules with GCN operators and significantly increases the computational efficiency. TNS (Wang et al., 2017) provides an adaptive receptive neighborhood for each node at any time. VRGNN (Wang et al., 2017) models the uncertainty of node embeddings by regarding each node in each snapshot as a distribution. However, the prevailing methods are built upon Euclidean spaces, which are not isometric with the power-law distributions of real-world graphs, and may cause distortion with the graph scale grows. ### Hyperbolic Graph Representation Learning Representation learning in hyperbolic spaces has been noticed due to their fitness to the hierarchical structures of real-world data. The significant performance advantages shown by the shallow Poincare (Poinacre, 2017) and Lorentz (Loren, 2017) models spark more attempts to this issue. Integrated with graph neural networks, HGNN (Wang et al., 2017) and HGCN (Chen et al., 2017) are designed for graph classification and node embedding tasks separately. HAT (Wang et al., 2017) exploits attention mechanism for hyperbolic node information propagation and aggregation. LGCN (Wang et al., 2017) builds the graph operations of hyperbolic GCNs with Lorentzian version and rigorously guarantees that the learned node features follow the hyperbolic geometry. The above hyperbolic GNN-based models all adopt the bi-directional transition between a hyperbolic space and corresponding tangent spaces, while a tangent space is the first-order approximation of the original space and may inevitably cause distortion. To avoid distortion, H2H-GCN (Chen et al., 2017) develops a manifold-preserving graph convolution and directly works on hyperbolic manifolds. For practical application situations, HGCF (Wang et al., 2017), HRCF (Wang et al., 2017) and HICF (Wang et al., 2017) study the hyperbolic collaborative filtering for user-item recommendation systems. Through hyperbolic graph learning, HyperStockGAT (Wang et al., 2017) captures the scale-free spatial and temporal dependencies in stock prices, and achieves state-of-the-art stock forecasting performance. HVGNN (Wang et al., 2017) and HTGN (Wang et al., 2017) fill the gap of hyperbolic models on dynamic graphs. HVGNN generates stochastic node representations of hyperbolic normal distributions via a hyperbolic graph variational autoencoder to represent the uncertainty of dynamics. HTGN adopts a conventional model architecture that handles the spatial and temporal information with HGCNs and contextual attention modules separately, however, ignores the causal order in the graph evolution. To further improve the hyperbolic graph models especially on temporal link prediction problem, we propose our HGWaveNet in terms of discrete dynamic graphs. ## 3. Preliminaries In this section, we first give the formalized definitions of discrete dynamic graphs and temporal link prediction. Then some critical fundamentals about hyperbolic geometry are introduced. ### Problem Definition This paper discusses temporal link prediction on discrete dynamic graphs. Following (Wang et al., 2017), we define discrete dynamic graphs as: Definition 3.1 (Discrete Dynamic Graphs).: _In discrete dynamic graph modeling, dynamic graphs can be viewed as a sequence of snapshots sampled from the original evolving process at consecutive time points. Formally, discrete dynamic graphs are represented as $\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},\cdots,\mathcal{G}_{T})$, in which $\mathcal{G}_{t}$ is a snapshot at timestamp $t$. The time granularity for snapshot divisions could be hours, days, months or even years depending on specific datasets and applications._ Based on Definition 3.1, temporal link prediction is described as: Definition 3.2 (Temporal Link Prediction).: _The aim of temporal link prediction is to predict the links appeared in the snapshots after timestamp $t$ based on the observed snapshots before timestamp $t$. Formally, the model takes $(\mathcal{G}_{0},\mathcal{G}_{1},\cdots,\mathcal{G}_{t-1})$ as input in the training process, and then makes predictions on $(\mathcal{G}_{t},\mathcal{G}_{t+1},\cdots,\mathcal{G}_{T})$._ ### Hyperbolic Geometry of the Poincare Ball The $n$-dimensional hyperbolic space $(\mathbb{H}_{c}^{n},\mathcal{G}^{c,\mathbb{H}})$ is the unique simply connected $n$-dimensional complete Riemannian manifold $\mathbb{H}_{c}^{n}$ with a constant negative curvature $-c(c>0)$, and $\mathcal{G}^{c,\mathbb{H}}$ is the Riemannian metric. The tangent space $7\pi_{\mathbf{c}}^{n}$ is a Euclidean, local, first-order approximation of $\mathbb{H}_{c}^{n}$ around the point $\mathbf{x}\in\mathbb{H}_{c}^{n}$. Similar to (Poinacre, 2017) and (Chen et al., 2017), we construct our method based on the Poincare ball, one of the most widely used isometric models of hyperbolic spaces. Corresponding to $(\mathbb{H}_{c}^{n},\mathcal{g}^{c,\mathbb{H}})$, the Poincare ball $(\mathbb{B}_{c}^{n},\mathcal{g}^{c,\mathbb{B}})$ is defined as \[\begin{split}\mathbb{B}_{c}^{n}&=\left\{\mathbf{x} \in\mathbb{R}^{n}:c\\|\mathbf{x}\\|^{2}<1\right\},\\ g_{\mathbf{x}}^{c,\mathbb{B}}&=\left(\lambda_{\mathbf{x} }^{c}\right)^{2}g^{F},\quad\lambda_{\mathbf{x}}^{c}:=\frac{2}{1-c\\|\mathbf{x} \\|^{2}},\end{split} \tag{1}\] where $g^{F}=\mathbf{I}_{n}$ is the Euclidean metric tensor. The Poincare ball manifold $\mathbb{B}_{c}^{n}$ is an open ball of radius $1/\sqrt{c}$ (see Figure 2). The induced distance between two points $\mathbf{x},\mathbf{y}\in\mathbb{B}_{c}^{n}$ is measured along a geodesic and given by1 Footnote 1: $-\mathbf{x}\oplus_{c}\mathbf{y}$ should be read as $(-\mathbf{x})\oplus_{c}\mathbf{y}$ rather than $-(\mathbf{x}\oplus_{c}\mathbf{y})$. \[d_{c}\left(\mathbf{x},\mathbf{y}\right)=\frac{2}{\sqrt{c}}\cdot\tanh^{-1}\left( \sqrt{c}\\|-\mathbf{x}\oplus_{c}\mathbf{y}\\|\right), \tag{2}\] in which the Mobius addition $\oplus_{c}$ in $\mathbb{B}_{c}^{n}$ is defined as \[\mathbf{x}\oplus_{c}\mathbf{y}=\frac{\left(1+2c(\mathbf{x},\mathbf{y})+c\\| \mathbf{y}\\|^{2}\right)\mathbf{x}+\left(1-c\\|\mathbf{x}\\|^{2}\right)\mathbf{y }}{1+2c(\mathbf{x},\mathbf{y})+c^{2}\\|\mathbf{x}\\|^{2}\\|\mathbf{y}\\|^{2}}. \tag{3}\] Figure 2. A hyperbolic manifold $\mathbb{H}$ and the corresponding Poincaré ball model $\mathbb{B}$. The distances on the manifolds are measured along geodesics. To map points between hyperbolic spaces and tangent spaces, exponential and logarithmic maps are given (Bou and Bou, 2017). For $\mathbf{x},\mathbf{y}\in\mathbb{B}_{c}^{n},\mathbf{v}\in\mathbb{F}_{\mathbf{x}} \mathbb{B}_{c}^{n}$, $\mathbf{x}\neq\mathbf{y}$ and $\mathbf{v}\neq\mathbf{0}$, the exponential map $\exp_{\mathbf{x}}^{c}:\mathcal{T}_{\mathbf{x}}\mathbb{B}_{c}^{n}\subset \mathbb{B}_{c}^{n}$ is \[\exp_{\mathbf{x}}^{c}(\mathbf{v})=\mathbf{x}\oplus_{c}\left(\tanh\left(\frac{ \nabla\mathcal{L}_{\mathbf{x}}^{c}\\|\mathbf{v}\\|}{2}\right)\frac{\mathbf{v}}{ \sqrt{c}\\|\mathbf{v}\\|}\right), \tag{4}\] and the logarithmic map $\log_{\mathbf{x}}^{c}:\mathbb{B}_{c}^{n}\to\mathcal{T}_{\mathbf{x}}\mathbb{B} _{c}^{n}$ is \[\log_{\mathbf{x}}^{c}(\mathbf{y})=d_{c}\left(\mathbf{x},\mathbf{y}\right) \frac{-\mathbf{x}\oplus_{c}\mathbf{y}}{\lambda_{\mathbf{x}}^{c}\\|-\mathbf{x} \oplus_{c}\mathbf{y}\\|}, \tag{5}\] where $\lambda_{\mathbf{x}}^{c}$ and $d_{c}\left(\mathbf{x},\mathbf{y}\right)$ are the same as in Equations (1) and (2). In our method, we use the origin point $\mathbf{o}$ as the reference point $\mathbf{x}$ to balance the errors in diverse directions. ## 4. Methodology This section describes our proposed model HGWaveNet. First, we elaborate on the details of two core components: hyperbolic diffusion graph convolution (HDGC) and hyperbolic dilated causal convolution (HDCC). Then other key modules contributing to HGWaveNet are introduced, including gated HDCC, hyperbolic gated recurrent unit (HGRU), hyperbolic temporal consistency (HTC) and Fermi-Dirac decoder. Finally, we summarize the overall framework of HGWaveNet and analyse the time complexity. ### Hyperbolic Diffusion Graph Convolution Hyperbolic graph convolutional neural networks (HGCNs (Chen et al., 2017)) are built analogous to traditional GNNs. A typical HGCN layer consists of three key parts: hyperbolic feature transform \[\mathbf{h}_{i}^{L,\mathbb{B}}=\mathbf{W}^{I}\otimes_{\ell^{-1}}\mathbf{x}_{i }^{L-1,\mathbb{B}}\oplus_{\ell^{-1}}\mathbf{b}^{I}, \tag{6}\] attention-based neighbor aggregation \[\mathbf{y}_{i}^{L,\mathbb{B}} =\text{Agg}_{\ell^{\prime}}(\text{Att}\left(\text{Concat}_{j\in \mathcal{N}(i)}\left(\log_{\mathbf{o}}^{\ell^{-1}}\left(\mathbf{h}_{j}^{L, \mathbb{B}}\right)\right)\right)), \tag{7}\] and hyperbolic activation \[\mathbf{x}_{i}^{L,\mathbb{B}}=\exp_{\mathbf{o}}^{\ell^{\prime}}\left(\sigma \left(\log_{\mathbf{o}}^{\ell^{\prime}}\left(\mathbf{y}_{i}^{L,\mathbb{B}} \right)\right)\right), \tag{8}\] in which $\mathbf{W}^{I},\mathbf{b}^{I}$ are trainable parameters and $\mathbf{x}_{i}^{L,\mathbb{B}}$ is the representation of node $i$ at layer $l$ in manifold $\mathbb{B}_{\ell^{\prime}}^{n}$. Following (Zhou et al., 2017), the matrix-vector multiplication $\otimes_{\mathcal{E}}$ is defined as \[\mathbf{M}\otimes_{\mathcal{E}}\mathbf{x}:=\exp_{\mathbf{o}}^{c}\left(\mathbf{ M}\log_{\mathbf{o}}^{c}\left(\mathbf{x}\right)\right),\quad\mathbf{M}\in\mathbb{R}^{n \times n},\mathbf{x}\in\mathbb{B}_{\mathcal{E}}^{n}. \tag{9}\] However, the shallow HGCN can only aggregate information of direct neighbors and is not effective enough for large graphs with long paths. To overcome this shortcoming, we impose the diffusion process referring to (Henderson et al., 2017). Consider a random walk process with restart probability $\alpha$ on $\mathcal{G}$, and a state transition matrix $\mathbf{P}$ (for most situations, $\mathbf{P}$ is the normalized adjacent matrix). Such Markov process converges to a stationary distribution $\mathcal{P}\in\mathbb{R}^{n\times n}$ after many steps, the $i$-th row of which is the likelihood of diffusion from node $i$. The stationary distribution $\mathcal{P}$ can be calculated in the closed form (Srivastava et al., 2017) \[\mathcal{P}=\sum_{k=0}^{\infty}\alpha(1-\alpha)^{k}\mathbf{P}^{k}, \tag{10}\] where $k$ is the diffusion step. In practice, a finite $K$-step truncation of the diffusion process is adopted and separate trainable weight matrices are added to each step for specific objective tasks. Then, the diffusion convolution layer on graphs is defined as \[\mathbf{Z}=\sum_{k=0}^{K}\mathbf{A}^{k}\mathbf{X}\mathbf{W}_{k}, \tag{11}\] in which $\mathbf{A}$ is the bi-direct adjacent matrix, $\mathbf{X}$ is the input node features, $\mathbf{W}_{k}$ is the weight matrix for the $k$-th diffusion step, and $\mathbf{Z}$ is the output node representations. On account of the sparsity of real-world data, we convert the graph convolution into equivalent spatial domains for efficiency. Do all above operations in the hyperbolic space, and replace the summation in Equation (11) with an attention mechanism for better information aggregation. Then, the $k$-th step of hyperbolic diffusion graph convolution at layer $l$ is \[\mathbf{X}_{k}^{l}=\text{HGCN}_{\varepsilon_{k}^{L}}\left(\mathbf{A}^{k}, \mathbf{X}^{l-1}\right), \tag{12}\] where $\text{HGCN}_{\varepsilon^{L}}(\cdot)$ is the conjunctive form of Equations (6), (7), and (8) with the adjacent matrix and node features as inputs. $\mathbf{X}^{l}$ is calculated as \[\mathbf{X}^{l}=\exp_{\mathbf{o}}^{l}\left(\text{Att}\left(\text{Concat}_{k}\left( \log_{\mathbf{o}}^{\ell^{\prime}}\left(\mathbf{X}_{k}^{l}\right)\right)\right) \right). \tag{13}\] Hyperbolic diffusion graph convolution is finally constructed with the stack of layers defined by Equations (13) and (12), as sketched in Figure 3. ### Hyperbolic Dilated Causal Convolution Causal convolution refers to applying a convolution filter only to data at past timestamps (Zhou et al., 2017), and guarantees that the order of data modeling is not violated, which means that the prediction $p(x_{t}\|x_{0},x_{1},\cdots,x_{t-1})$ emitted by the model at timestamp $t$ cannot depend on future data $x_{t},x_{t+1},\cdots,x_{T}$. For 1-D temporal information, this operation can be implemented by shifting the output of a normal convolution with kernel size $5$ by $\lfloor 5/2\rfloor$ steps, but a sufficiently large receptive field requires stacking many layers. To decrease the computational cost, dilated convolutions are adopted. By skipping the inputs with a certain step $d$, a dilated convolution applies its kernel over a larger area than its length, and with the Figure 3. A hyperbolic diffusion graph convolution (HDGC) module with the number of layers $L=2$ and the number of diffusion steps $K=2$. The part in red dashed rectangle is a complete HDGC layer. stack of dilated convolution layers, the receptive field expands exponentially while preserving high computational efficiency. Based on the above definitions of causal convolutions and dilated convolutions, the dilated causal convolution operation in manifold $\mathbb{B}_{\mathcal{C}}^{n}$ can be formalized mathematically as \[(\mathbf{Z}_{i}\odot\mathbf{F})_{t}=\exp_{\mathbf{o}}^{c}\left(\sum_{s=0}^{S-1} \mathbf{F}_{s}\log_{\mathbf{o}}^{c}\left(\mathbf{Z}_{i,t-d\times s}\right) \right), \tag{14}\] where $d$ is the dilation step, $\mathbf{Z}_{i,t}\in\mathbb{B}_{\mathbf{o}}^{n}\left(t\leq T\right)$ is the representation of node $i$ at snapshot $\mathcal{G}_{t}$ and $\mathbf{F}\in\mathbb{R}^{S\times n}$ denotes the kernels for all dimensions (also called channels). A stacked hyperbolic dilated causal convolution module is shown in Figure 4. Compared to attention-based historical information aggregation (Zhou et al., 2017), HDCC can learn the internal order of causal correlation in temporality. Meanwhile, compared to RNNs, HDCC allows parallel computation and alleviates vanishing or exploding gradient problems due to its non-recursion(Zhou et al., 2018). Considering the superiority of HDCC on sequence problems, we leverage this module in handling the temporal evolutionary information of graphs. ### Other Key Modules in HGWaveNet Except for HDGC and HDCC, some other modules contribute greatly to our proposed HGWaveNet. Based on HDCC, we further design gated HDCC to preserve the valid historical information. Hyperbolic gated recurrent unit (HGRU) (Zhou et al., 2017) is able to efficiently learn the latest node representations in Poincare ball from historical states and current spatial characteristics. Additionally, as defined in (Zhou et al., 2017), hyperbolic temporal consistency (HTC) module provides a similarity constraint in temporality to ensure stability during graph evolution. Finally, a Fermi-Dirac decoder (Feng et al., 2019) is used to compute the probability scores for edge reconstruction. #### 4.3.1. Gated HDCC As shown in the left part of Figure 5, we adopt a simple gating mechanism on the outputs of HDCCs. What is noteworthy is the use of logarithmic map before the Euclidean activation functions and exponential map after gating. The formulation is described as \[\mathbf{H}_{i,t}=\exp_{\mathbf{o}}^{c}\left(\tanh\left(\log_{\mathbf{o}}^{c} \left((\mathbf{Z}_{i}\odot\mathbf{F}_{1})_{t}\right)\right)\cdot\sigma\left( \log_{\mathbf{o}}^{c}\left((\mathbf{Z}_{i}\odot\mathbf{F}_{2})_{t}\right) \right)\right), \tag{15}\] where $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ are different convolution kernels for two HDCCs, $\mathbf{H}_{i,t}\in\mathbb{B}_{\mathbf{o}}^{n}\left(t\leq T\right)$ is the historical hidden state of node $i$ at snapshot $\mathcal{G}_{t}$, and $\cdot$ denotes element-wise product. In addition, to enable a deeper model for learning more implicit information, residual and skip connections are applied throughout the gated HDCC layers. #### 4.3.2. Hyperbolic Gated Recurrent Unit HGRU (Zhou et al., 2017) is the hyperbolic variant of GRU (He et al., 2017). Performed in the tangent space, HGRU computes the output with high efficiency from historical hidden states and current input features, as \[\mathbf{Z}_{i,t}=\exp_{\mathbf{o}}^{c}\left(\mathrm{GRU}\left(\log_{\mathbf{o }}^{c}\left(\mathbf{X}_{i,t}\right),\log_{\mathbf{o}}^{c}\left(\mathbf{H}_{i,t -1}\right)\right)\right). \tag{16}\] In HGWaveNet, the historical hidden states are from gated HDCC module, so we use a single HGRU cell for just one step rather than linking it into recursion. #### 4.3.3. Hyperbolic Temporal Consistency In real-world graphs, the evolution proceeds continuously. Correspondingly, the node representations are expected to change gradually in terms of temporality, which means that the node representations at two consecutive snapshots should keep a short distance. Hence, HTC (Zhou et al., 2017) defines a similarity constraint penalty between $\mathbf{Z}_{t-1}$ and $\mathbf{Z}_{t}$ at snapshot $\mathcal{G}_{t}$ as \[\mathcal{L}_{\mathrm{HTC}}^{t}=\frac{1}{N}\sum_{i=1}^{N}d_{c}\left(\mathbf{Z}_ {i,t-1},\mathbf{Z}_{i,t}\right), \tag{17}\] where $d_{c}(\cdot)$ is defined in Equation (2) and $N$ is the number of nodes. By adding the penalty in optimization process, HTC ensures that the node representations do not change rapidly and the stability of graph evolution is achieved. #### 4.3.4. Fermi-Dirac Decoder As a generalization of sigmoid, Fermi-Dirac decoder (Feng et al., 2019) gives a probability score for edges between nodes $i$ and $j$, which fits the temporal link prediction problem greatly. It is defined as \[p_{F_{-D}}(\mathbf{x}_{i},\mathbf{x}_{j})=\frac{1}{\exp\left(\left(d_{c} \left(\mathbf{x}_{i},\mathbf{x}_{j}\right)-r\right)/s\right)+1}, \tag{18}\] where $r$ and $s$ are hyper-parameters, and $\mathbf{x}_{i}$, $\mathbf{x}_{j}\in\mathbb{B}_{c}^{n}$ are hyperbolic representations of nodes $i$ and $j$. ### Framework of HGWaveNet With all modules introduced above, we now summarize the overall framework of HGWaveNet in Figure 5. For the snapshot $\mathcal{G}_{t-1}\in\mathcal{G}$, we first project the input node features into the hyperbolic space with exponential map, and then an HDGC module is applied to learn the spatial topological structure characteristics denoted by $\mathbf{X}_{t-1}$. Simultaneously, the stacked gated HDCC layers calculate the latest historical hidden state $\mathbf{H}_{t-2}$ from previous node representations $\mathbf{Z}_{t-2},\mathbf{Z}_{t-3},\cdots$ (the initial historical node representations are padded randomly at the beginning of training). Next, the spatial information $\mathbf{X}_{t-1}$ and the temporal information $\mathbf{H}_{t-2}$ are inputted into a single HGRU cell. The output of HGRU $\mathbf{Z}_{t-1}$ is exactly the new latest node representations at snapshot $\mathcal{G}_{t-1}$, and is fed into the Fermi-Dirac decoder to make temporal link predictions for snapshot $\mathcal{G}_{t}$. To maximize the probability of connected nodes in $\mathcal{G}_{t}$ and minimize the probability of unconnected nodes, we use a cross-entropy Figure 4. A stacked hyperbolic dilated causal convolution (HDCC) module with dilated depth $D=3$ and kernel size $S=2$. Each circle refers to one node at a timestamp. Under the exponentially large receptive field, each node preserves the temporal causal correlations with the past $S^{D}$ snapshots. like loss $\mathcal{L}_{CE}^{t}$ defined as \[\begin{split}\mathcal{L}_{CE}^{t}=&\text{Avg}_{i \sim j}\left(-\log\left(p_{F-D}\left(\mathbf{Z}_{i,t-1},\mathbf{Z}_{j,t-1} \right)\right)\right)+\\ &\text{Avg}_{i^{\prime}\neq t^{\prime}}\left(-\log\left(1-p_{F-D} \left(\mathbf{Z}_{i^{\prime},t-1},\mathbf{Z}_{j^{\prime},t-1}\right)\right) \right),\end{split} \tag{19}\] in which $\sim_{t}$ denotes the connection at snapshot $\mathcal{G}_{t}$ between two nodes, and $\neq_{t}$ is the opposite. $(i^{\prime},j^{\prime})$ is the sampled negative edge for accelerating training and preventing over-smoothing. Taking the HTC module into account and summing up all snapshots, the complete loss function is \[\mathcal{L}=\sum_{t=0}^{T}\left(\mathcal{L}_{CE}^{t}+\lambda\mathcal{L}_{HTC}^ {t}\right). \tag{20}\] _Time Complexity Analysis._ We analyse the time complexity of our proposed HGWaveNet by module for each snapshot. For the HDGC module, the computation is of $\mathcal{O}((Nd^{2}+\|\mathcal{E}_{t}\|Kd)L)$, where $d$ is the dimension of node representations, $\|\mathcal{E}_{t}\|$ is the edge number for snapshot $\mathcal{G}_{t}$, $K$ is the truncated diffusion steps and $L$ is the layer number of HDGC. For gated HDCC, the time complexity is $\mathcal{O}(S^{D}dD^{\prime})$, in which $S$ and $D$ are the kernel size and dilated depth of a single HDCC, respectively, and $D^{\prime}$ is the layer number of the complete gated HDCC module. HGRU and HTC run only once for each snapshot, and both take the time complexity of $\mathcal{O}(Nd)$. For Fermi-Dirac decoder, the time complexity is $\mathcal{O}(\|\mathcal{E}_{t}\|d+\|\mathcal{E}_{t}\|d)$, in which $\|\mathcal{E}_{t}\|$ is the number of negative samples for snapshot $\mathcal{G}_{t}$. ## 5. Experiments and Analysis In this section, we conduct extensive experiments on six real-world graphs and prove the superiority of our proposed HGWaveNet. In addition, we also conduct ablation studies and hyper-parameter analysis to corroborate the three primary ideas of our method: the fitness between graph distributions and hyperbolic geometry, the effectiveness of spatial information from a wider range of neighbors aggregated by HDGC, and the causality of historical information in the evolutionary process learned by HDCC. ### Experimental Setup #### 5.1.1. Datasets We evaluate our proposed model and baselines on six real-world datasets from diverse areas, including email communication networks Enron2(Friedman et al., 2017), academic co-author networks DBLP3(Friedman et al., 2017) and HepPh4(Friedman et al., 2017), Internet router networks AS733(Friedman et al., 2017), social networks FB6(Friedman et al., 2017) and movie networks MovieLens7(Friedman et al., 2017). The statistics of these datasets are shown in Table 1. We take the same splitting ratios for training and testing as (Zhang et al., 2018) on all datasets except MovieLens on which a similar manner is adopted. Gromov's hyperbolicity (Golovolov et al., 2016)$\delta$ is used to measure the tree-likeness and hierarchical properties of graphs. A lower $\delta$ denotes a more tree-like structure and $\delta=0$ denotes a pure tree. The datasets we choose all remain implicitly hierarchical and show distinct power-law distributions. Footnote 2: [https://www.cs.cornell.edu/~arb/data/email-Enron/](https://www.cs.cornell.edu/~arb/data/email-Enron/) Footnote 3: [https://github.com/VGraphRNN/VGRNN/tree/master/data](https://github.com/VGraphRNN/VGRNN/tree/master/data) Footnote 4: [https://snap.stanford.edu/data/cit-HepPh.html](https://snap.stanford.edu/data/cit-HepPh.html) Footnote 5: [https://snap.stanford.edu/data-733.html](https://snap.stanford.edu/data-733.html) Footnote 6: [https://networkregository.com/in-facebook-wall-wosn-dir.php](https://networkregository.com/in-facebook-wall-wosn-dir.php) Footnote 7: [https://grouplenss.org/datasets/movielens/](https://grouplenss.org/datasets/movielens/) #### 5.1.2. Baselines Considering that HGWaveNet is constructed in the hyperbolic space for dynamic graphs, we choose seven competing baselines either in hyperbolic spaces or built for dynamic graphs to verify the superiority of our model. The baselines are summarized in Table 2, where HTGN on Poincare ball shows state-of-the-art performance in most evaluations. #### 5.1.3. Evaluation Tasks and Metrics Similar to (Zhang et al., 2018), our experiments consist of two different tasks: _temporal link prediction_ and _temporal new link prediction_. Specifically, _temporal link prediction_ aims to predict the edges that appear in $\mathcal{G}_{t}$ and other snapshots after timestamp $t$ based on the training on $(\mathcal{G}_{0},\mathcal{G}_{1},\cdots,\mathcal{G}_{t-1})$, while _temporal new link prediction_ aims to predict those in $\mathcal{G}_{t}$ but not in $\mathcal{G}_{t-1}$. To quantify the experimental performance, we choose the widely used metrics average precision (AP) and area under ROC curve (AUC). The datasets are split into training sets and test sets as shown in Table 1 by snapshots to run both our proposed model and baselines on the above tasks. \begin{table} \begin{tabular}{c\|c c c c c} \hline \hline Datasets & Enron & DBLP & AS733 & FB & HepPh & MovieLens \\ \hline \# Nodes & 184 & 315 & 6,628 & 45,435 & 15,330 & 9,746 \\ \# Edges & 790 & 943 & 13,512 & 180,011 & 976,097 & 997,837 \\ \# Snapshots & 11 & 10 & 30 & 36 & 36 & 11 \\ Train : Test & 8.3 & 7.3 & 20:10 & 333 & 30.6 & 8.3 \\ $\delta$ & 1.5 & 2.0 & 1.5 & 2.0 & 1.0 & 2.0 \\ \hline \hline \end{tabular} \end{table} Table 1. Statistics of datasets. Figure 5. The overall framework of HGWaveNet. For $\mathcal{G}_{t-1}$, the historical evolutionary information handled by gated HDCCs and the spatial topological information handled by HDGC are fed into an HGRU module. With the constraint of HTC, the hyperbolic outputs are decoded by a Fermi-Dirac decoder and the temporal link prediction for $\mathcal{G}_{t}$ is made. \begin{table} \begin{tabular}{c\|c c} \hline \hline Methods & Static/dynamic & Manifolds \\ \hline HGCN(Chen et al., 2017) & Static & Lorentz \\ HAT(Zhang et al., 2018) & Static & Poincaré \\ EvolveGCN(Zhang et al., 2018) & Dynamic & Euclidean \\ GRUCGN(Zhang et al., 2018) & Dynamic & Euclidean \\ TGN(Zhang et al., 2018) & Dynamic & Euclidean \\ HTGN(Zhang et al., 2018) & Dynamic & Poincaré \\ \hline \hline \end{tabular} \end{table} Table 2. Baselines. ### Experimental Results We implement HGWaveNet8 with PyTorch on Ubuntu 18.04. Each experiment is repeated five times to avoid random errors, and the average results (with form _average value_$\pm$_standard deviation_) on the test sets are reported in Table 3 and Table 4. Next, we discuss the experimental results on each evaluation task separately. Footnote 8: The code is open sourced in [https://github.com/TaiLvYuanLiang/HGWaveNet](https://github.com/TaiLvYuanLiang/HGWaveNet). #### 5.2.1. Temporal Link Prediction The results on temporal link prediction are shown in Table 3. Obviously, our HGWaveNet outperforms all baselines including the hyperbolic state-of-the-art model HTGN both on AUC and AP. To analyse the results further, we intuitively divide the six datasets into two groups: small (Enron, DBLP, AS733) and large (FB, HepPh, MovieLens), according to the scales. It can be observed that the superiority of our model is more significant on large graphs than small ones compared with baselines, especially those Euclidean methods. This is because the representation capacity of Euclidean methods decreases rapidly with increasing graph scales, while hyperbolic models remain stable by virtue of the fitness between data distributions and space properties. * w/o $\mathbb{B}$: HGWaveNet without hyperbolic geometry, implemented by replacing all modules with corresponding Euclidean versions. We take the same experimental setup with HGWaveNet and baselines on these three variants, and the results are reported in the last rows of Table 3 and Table 4. The performance drops noticeably after removing any of these three components, which indicates their respective importance. In the following, we discuss the details of the effectiveness of these variants. HDGC module imposes the diffusion process into hyperbolic graph convolution networks. It allows nodes to aggregate information from a wider range of neighbors and improves the efficiency of message propagation by calculating the stationary distribution of a long Markov process in a closed form. The fact that performance of w/o HDGC variant has a larger degradation on large graphs than small ones forcefully proves the effectiveness of this component for large graphs with long paths. The purpose of HDCC module is to capture the internal order of causal correlation in sequential data. Generally, the larger a graph is, the more complex its evolution is and the richer historical information it has. The experimental results on w/o HDCC show the powerful ability of this component to capture causal order in temporal evolution process, especially on complex graphs (i.e. FB and MovieLens). As observed in Table 3 and Table 4, the degradation of w/o $\mathbb{B}$ is much more severe than that of the other two variants, which strongly supports our primary idea that hyperbolic geometry fits the power-law distributions of real-world graphs very well. Moreover, it is noteworthy that for most results, the variant w/o $\mathbb{B}$ exhibits a comparable or even exceeding performance to other Euclidean baselines. This proves that our model architecture still keeps advanced even without the advantages of hyperbolic geometry. ### Parameter Analysis We further study the influences of three hyper-parameters: representation dimension, truncated diffusion step and dilated depth. We evaluate the performance of HGWaveNet and one of the best Euclidean models GRUGCN on two datasets FB (relatively sparse) and MovieLens (relatively dense) by setting the representation dimension into different values, as shown in Figure 6. On the one hand, it is clear that with the dimension decreasing, our model remains stable compared to GRUGCN, to which the hyperbolic geometry contributes greatly. On the other hand, for HGWaveNet, the degradation of the performance on MovieLens is severer than that on FB, which is consistent with the conclusion in (Wang et al., 2019) that hyperbolic models fit sparse data better than dense data. Figure 7 shows the results of our model for temporal link prediction on MovieLens with different values of truncated diffusion step $K$ in HDGC and dilated depth $D$ in HDCC. The wrap-up observation that the performance generally increases as $K$ and $D$ increase proves the positive influences of diffusion process and dilation convolution. Specifically, in terms of truncated diffusion step, the performance has a great improvement from $K=1$ to $K=2$. When $K>2$, the benefit from increasing $K$ becomes too small to match the surge of computation from exponentially expanding neighbors in graphs. Dilated depth $D$ is used to control the receptive field of causal convolution. However, as the receptive field expands, more noise is imported and may degrade the model performance. According to our extensive experiments, $D=3$ is a nice choice in most situations, with the causal convolution kernel size being 2. ## 6. Conclusion In this paper, we propose a hyperbolic graph neural network HGWaveNet for temporal link prediction. Inspired by the observed power-law distribution and implicit hierarchical structure of real-world graphs, we construct our model on the Poincare ball, one of the isometric models of hyperbolic spaces. Specifically, we design two novel modules hyperbolic diffusion graph convolution (HDGC) and hyperbolic dilated causal convolution (HDCC) to extract the spatial topological information and temporal evolutionary states, respectively. HDGC imposes the diffusion process into graph convolution and provides an efficient way to aggregate information from a wider range of neighbors. HDCC ensures that the internal order of causal correlations is not violated by applying the convolution filter only to previous data. Extensive experiments on diverse real-world datasets prove the superiority of HGWaveNet, and the ablation study further verifies the effectiveness of each component of our model. In future work, we will further generalize our model for more downstream tasks and try to use hyperbolic GNNs to capture more complex semantic information in heterogeneous graphs. ## Acknowledgments This work is supported by the Chinese Scientific and Technical Innovation Project 2030 (2018AAA0102100), National Natural Science Foundation of China (U1936206, 62172237) and the Tianjin Natural Science Foundation for Distinguished Young Scholars (22JCJQJC00150). Figure 6. The influence of representation dimension on two datasets FB and MovieLens for HGWaveNet and GRUGCN. Figure 7. The influence of truncated diffusion step $K$ (left) and dilated depth $D$ (right) on MovieLens.
2302.10296	On Function-Coupled Watermarks for Deep Neural Networks	Well-performed deep neural networks (DNNs) generally require massive labelled data and computational resources for training. Various watermarking techniques are proposed to protect such intellectual properties (IPs), wherein the DNN providers implant secret information into the model so that they can later claim IP ownership by retrieving their embedded watermarks with some dedicated trigger inputs. While promising results are reported in the literature, existing solutions suffer from watermark removal attacks, such as model fine-tuning and model pruning. In this paper, we propose a novel DNN watermarking solution that can effectively defend against the above attacks. Our key insight is to enhance the coupling of the watermark and model functionalities such that removing the watermark would inevitably degrade the model's performance on normal inputs. To this end, unlike previous methods relying on secret features learnt from out-of-distribution data, our method only uses features learnt from in-distribution data. Specifically, on the one hand, we propose to sample inputs from the original training dataset and fuse them as watermark triggers. On the other hand, we randomly mask model weights during training so that the information of our embedded watermarks spreads in the network. By doing so, model fine-tuning/pruning would not forget our function-coupled watermarks. Evaluation results on various image classification tasks show a 100\% watermark authentication success rate under aggressive watermark removal attacks, significantly outperforming existing solutions. Code is available: https://github.com/cure-lab/Function-Coupled-Watermark.	Xiangyu Wen, Yu Li, Wei Jiang, Qiang Xu	2023-02-08T05:55:16Z	http://arxiv.org/abs/2302.10296v3	# On Function-Coupled Watermarks for Deep Neural Networks ###### Abstract. Well-performed deep neural networks (DNNs) generally require massive labelled data and computational resources for training. Various watermarking techniques are proposed to protect such intellectual properties (IPs), wherein the DNN providers implant secret information into the model so that they can later claim IP ownership by retrieving their embedded watermarks with some dedicated trigger inputs. While promising results are reported in the literature, existing solutions suffer from watermark removal attacks, such as model fine-tuning and model pruning. In this paper, we propose a novel DNN watermarking solution that can effectively defend against the above attacks. Our key insight is to enhance the coupling of the watermark and model functionalities such that removing the watermark would inevitably degrade the model's performance on normal inputs. To this end, unlike previous methods relying on secret features learnt from out-of-distribution data, our method only uses features learnt from in-distribution data. Specifically, on the one hand, we propose to sample inputs from the original training dataset and fuse them as watermark triggers. On the other hand, we randomly mask model weights during training so that the information of our embedded watermarks spreads in the network. By doing so, model fine-tuning/pruning would not forget our _function-coupled_ watermarks. Evaluation results on various image classification tasks show a 100% watermark authentication success rate under aggressive watermark removal attacks, significantly outperforming existing solutions. Code is available here. watermark, neural networks, function-coupled, robustness + Footnote †: [leftmargin=] organization=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=comment=commentment=comment=comment=comment=commentment=comment=commentment=commentment=commentment=commentment=commentment=commentment=commentment=commentmentment=commentment=commentment=commentmentment=commentment=commentmentment=commentmentment=commentmentment=commentmentment=commentmentmentment=commentmentmentment=commentmentmentment=commentmentmentment=commentmentmentmentment=commentmentmentmentmentment=comment We propose a novel backdoor-based solution for DNN IP protection that is resistant to watermark removal attacks. Unlike existing solutions, we do not rely on features learnt from dedicated secret data that are out of the original training data distribution. Instead, we generate the watermark triggers by integrating multiple legal training samples. Specifically, we propose two fusion methods, as illustrated in Fig. 1(b). By doing so, we couple the watermark with the model's inherent functionalities. Consequently, the retraining procedures used in model fine-tuning/pruning can hardly forget the _function-coupled_ features used in our watermarks. Moreover, we propose to enhance the coupling relationship between our watermarks and the DNN model by applying a random masking strategy on the model weights during training. Therefore, the watermark information spreads across the network and it is hard to prune. The main contributions of this paper are as follows: We propose a novel black-box watermarking method for DNN IP protection, based on the key insight to enhance the coupling of the watermark and the DNN model such that removing the watermark would inevitably degrade the model's performance on normal inputs. * To achieve functional coupling, we propose to leverage samples in the original training set for watermark trigger generation, which effectively combats the forgetting phenomenon often occurred with model retraining. * To further enhance the coupling relationship, we introduce a new training procedure that randomly masks model weights so that the watermark information is embedded across the network and hence resistant to watermark removal attacks. We conduct extensive evaluations on several image classification datasets (MNIST, CIFAR-10/100, and Tiny-ImageNet) with various network architectures (LeNet-5, VGG16, and ResNet-18). The results show that our method significantly outperforms existing watermarking solutions in terms of performance and robustness. The rest of this paper is organized as follows. We first survey related works in Section 2. Next, Section 3 defines the problem and Section 4 details the proposed solution. Experimental results are then presented in Section 5. Finally, Section 6 concludes this paper. ## 2. Related Work Existing DNN watermarking methods can be divided into white-box methods and black-box methods. The white-box methods require the internal information of the DNN model for verification, while the black-box methods only need model predictions during the verification phase. We illustrate them as follows: _White-box watermarking_. In this scenario, researchers inject manually selected secret information (e.g., encoded images and strings) into the model weights. Then, in the verification phase, defenders try to extract these watermarks from model weights and claim ownership of this model. Uchida et al. (Uchida et al., 2018) are the pioneers in proposing the concept of DNN watermarking. They achieve this by adding a regularization loss during training, which results in the regularization of the chosen weights to some secret values. Li et al. (Li et al., 2019) enhance the approach presented in (Uchida et al., 2018) by incorporating a regularization term similar to Spread-Transform Dither Modulation (ST-DM). This added term helps to mitigate the influence of watermarking on the precision of the DNN model while processing standard inputs. Chen et al. (Chen et al., 2019) improve (Uchida et al., 2018) by implementing a watermarking system with anti-collision capabilities. Betty et al. (Betty et al., 2019) note that the Adam optimizer leads to a significant change in the distribution of weights after watermarking, which can be readily detected by adversaries. To address this issue, the authors have suggested employing an orthonormal projection matrix to project the weights and subsequently implementing the Adam optimizer on top of the projected weights. In contrast to Uchida's approach, Tartaglione et al. (Tartaglione et al., 2018) predetermine the watermarked weights before initiating the training procedure, and they have fixed them during the process. Meanwhile, Wang et al. (Wang et al., 2019) propose a novel method for embedding and extracting watermarks by using an independent neural network to process the model weights. Rouhani et al. (Rouhani et al., 2019) propose an alternative approach to embedding watermarks in DNN models. Figure 1. Comparison of backdoor-based watermarks with (a) existing triggers and (b) our triggers. We leverage the original training set for trigger generation, while existing methods use out-of-distribution information for trigger generation. Rather than embedding the watermark into the model weights, the authors have introduced it into the feature maps of the model. To achieve this, the authors have analyzed the Probability Density Function (PDF) of activation maps obtained from various layers and embedded the watermark in the low probabilistic regions to minimize the impact on the clean accuracy of the model. In view of the potential ambiguity attacks, where adversaries may attempt to embed their watermark in a DNN model under the guise of the owner, Fan et al. (Fan et al., 2016; Fan et al., 2017) propose the integration of a passport layer into the victim model. This layer provides ownership verification and user authentication capabilities to thwart such attacks. On the other hand, Guo et al. (Guo et al., 2017) propose several attack strategies, such as scaling, noise embedding, and affine transformation, to disrupt embedded watermarks. In response, the authors augmente existing watermarks by integrating these attack methods into the watermark generation process. In contrast, our proposed approach involves the use of functional-coupled watermarks, which is a conceptually different approach. _Black-box watermarking._ This type of watermarking method enables DNN ownership verification by verifying the consistency between specific inputs and their corresponding results. The watermarking process can be achieved through injecting backdoors into the model (Liu et al., 2017) or generating adversarial examples. In this approach, the injected backdoor and the model's vulnerability can be considered as embedded watermarks, while the trigger images and adversarial examples serve as corresponding keys. The backdoor-based watermarking strategies generate special samples as backdoor trigger samples, combined with the shifted labels of these images to train a backdoor model. To verify the model's ownership, defenders can recover the watermark by querying the model and examining the consistency between outputs and the queried samples. Adi et al. (Adi et al., 2017) use backdoors (Rohani et al., 2018) for DNN watermarking, and the authors explore two methods, fine-tuning and training from scratch, to generate backdoors using selected images as triggers (e.g., Trigger C in Fig. 1(a)). Rouhani et al. (Rouhani et al., 2019) propose a method for injecting backdoors into the model for watermarking and ownership verification by retraining the model. Another approach involves selecting key images as watermarking samples via superimposing visible patterns (e.g., Trigger A & B in Fig. 1(a)) on some of the training images. The labels of such images are then shifted to the target class and combined with these images to train the backdoor model, creating a special relationship between them as the watermarks. Zhang et al. (Zhang et al., 2019) use this method to generate watermarks. To avoid vulnerability under backdoor detection of visible trigger patterns, Guo et al. (Guo et al., 2017) and Li et al. (Li et al., 2019) propose replacing the trigger pattern with invisible ones, such as adding a bit sequence to random pixel locations. Jia et al. (Jia et al., 2019) suggest training the features of out-of-distribution triggers (e.g., Trigger D in Fig. 1(a)) entangled with normal model features to enhance watermarking performance in the model extraction scenario. In contrast to directly protecting the victim model, Szymler et al. (Szymler et al., 2019) embed watermarks into the surrogate model when adversaries conduct model extraction attacks. They deploy an additional component within the API that adversaries use to access the model, deliberately returning wrong results corresponding to some of the input samples. This way, the surrogate model trained by the returned information can be embedded with watermarks. The adversarial example-based watermarking methods exploit the generated examples to shape the model boundary for establishing a unique association between such dedicated samples and selected outputs. Merrer et al. (Merrer et al., 2019) employ the IFGSM algorithm (Mayer et al., 2019) to generate the adversarial examples as the key samples of watermarks. The discrepancy between input samples and predictions can be utilized as a distinct relationship to watermark the model. He et al. (He et al., 2019) generate sensitive-sample watermarks, intending that small changes in model weights can be reflected in the model outputs through these sensitive samples. Yang et al. (Yang et al., 2019) propose a bi-level framework to jointly optimize adversarial examples and the DNN model. Wang et al. (Wang et al., 2019) were the first to consider both robustness and transferability for generating realistic watermarks. Chen et al. (Chen et al., 2019) propose a testing framework to evaluate the similarity between the victim model and the suspect model by a set of extreme test cases of adversarial examples. ## 3. Problem Definition In this section, we will first introduce the threat model in Section 3.1. Next, we will formulate our problem in Section 3.2, followed by providing the evaluation metrics for DNN watermarking in Section 3.3. ### Threat Model In this section, we aim to clarify the scenario of tampering with the ownership of DNN models and the need for watermarking victim models. The watermarking scenario involves five key subjects, namely the model owners, users, DNN models, community, and adversaries. Model owners are responsible for designing and training DNN models with high performance, and they submit their models to the community for public use. Users can download these models for downstream tasks. However, adversaries can also download the models and falsely claim ownership by submitting the stolen models to other communities. Such actions violate the intellectual property rights of the original owners, necessitating the need for a proper method to verify the ownership of released models. In this paper, we consider an attack scenario in which adversaries steal a model from the community and set up an online service to provide AI services using the leaked model. Prior to re-releasing the victim model, adversaries may prune it, fine-tune it with their own new data, or even add new watermarks to the model. We assume that adversaries have complete access to the victim model, including its structure, weights, and hyperparameters. Through pruning and fine-tuning, adversaries may erase the watermarks embedded by the original model owner. Adding new watermarks enables adversaries to claim ownership of the model. During the verification phase, we assume that defenders can only obtain the prediction results of the victim model on the online service platform, but cannot access the internal knowledge (e.g., weights) of the model. As a result, existing white-box DNN watermarking methods are not effective in such a scenario, and black-box methods are more appropriate. ### Watermarking Problem Formulation _Watermarking target:_ Given a DNN model $f_{\theta}(x)$, where $\theta$ represents the model weights, a watermarking strategy $h(\cdot)$ is designed to embed an abstract watermark $S$ into the model, and a recovering strategy $r(\cdot)$ is developed to extract the watermarks from the candidate model. _Techniques:_ In terms of white-box and black-box watermarking methods, the details of $h(\cdot)$ differ. White-box methods can embed and recover watermarks $S$ from only the model weights. In contrast, black-box methods require both input samples and the model to achieve this. Thus, $S$ is a subset of joint $(x,\theta)$, meaning that $S$ relies on both samples and models. Mainstream white-box methods aim to embed additional information into model weights such that $f_{\theta+\theta}(\cdot)=h_{white}(f_{\theta}(\cdot))$. Here, $\delta$ represents the perturbation on model weights, and we can recover $S$ from $\delta$ (i.e., $\delta\Rightarrow S$). In this case, apart from the selected weights, the rest of the weights in the model will not change. In contrast, backdoor-based black-box methods require modifying the whole model to embed watermarks, either by fine-tuning the model or training it from scratch with the trigger data. We can generate watermarks via $x^{\prime},f_{\theta^{\prime}}(\cdot)=h_{backdoor}(f_{\theta}(\cdot),x^{ \prime})$, and $(x^{\prime},\theta^{\prime},f_{\theta^{\prime}}(x^{\prime}))\Rightarrow S$. Here, $\theta^{\prime}$ represents the modified model weights after injecting the backdoor, and $x^{\prime}$ represents the trigger samples. Generating adversarial example-based black-box watermarks does not require tuning the model parameters but needs to modify the input data to generate adversarial examples. We can generate watermarks via $x^{\prime},f_{\theta}(\cdot)=h_{add}(f_{\theta}(\cdot),x)$, and $(x^{\prime},\theta,f_{\theta}(x^{\prime}))\Rightarrow S$, with $x^{\prime}$ indicating the generated adversarial example corresponding to the input $x$. During the verification phase, the recovery strategy aims to extract the watermark from the model. In the case of white-box methods, the recovery strategy extracts the weights and checks if the decoding result is consistent with the watermark, i.e., $S=r(\delta)$. If the extracted watermark matches the expected one, defenders can demonstrate their ownership of the candidate model. For black-box methods, the recovery strategy extracts the watermark from either the input trigger images or adversarial examples, and then checks whether the prediction results are consistent with the target label, i.e., $S=r(x^{\prime},f_{\theta^{\prime}}(x^{\prime}))$ or $S=r(x^{\prime},f_{\theta}(x^{\prime}))$. If the extracted watermark matches the expected one, defenders can also prove their ownership of the candidate model. ### Evaluation Metrics _Effectiveness._ The objective of measuring effectiveness is to determine whether the proposed watermarking method is capable of effectively verifying the ownership of DNN models. _Fidelity._ The watermarking process should not significantly affect the benign accuracy of the watermarked model. Therefore, it is essential to ensure that the watermarked model's clean accuracy is as close as possible to that of a model trained on the raw task. _Robustness._ The robustness metric is utilized to measure the performance-preserving capability of a watermarking method under attacks. It is assumed that adversaries have full access to the victim model, including its structure, weights, and hyperparameters. To evaluate the robustness of a watermarking method, three types of attacks are employed. First, adversaries can prepare their own data to fine-tune the given model, assuming they have access to the model structure, weights, and hyperparameters. Two ways of fine-tuning are selected: fine-tuning with data from the original data domain and transfer learning with data from a new data domain. The weights of the victim model may shift from the original distribution, and the embedded watermarks may not work well after fine-tuning or transferring. Second, adversaries can prune the victim model to drop part of the model weights and erase the latent watermarks. Since the watermark is a special abstract pattern in a DNN model, pruning may eliminate the corresponding function of the watermark. Finally, adversaries who know the underlying watermarking method can overwrite the existing watermark in the model by re-embedding a new watermark, which disables the recognition of the original watermark (Hidid et al., 2017; Wang et al., 2018). ## 4. Methodology In this section, we provide a comprehensive description of our novel watermarking method for deep neural networks. The proposed method is composed of three principal modules: trigger generation, watermark embedding, and watermark verification. The workflow is illustrated in Figure 2. Firstly, we propose two alternative techniques to generate feature-fusion trigger samples. Then, we combine the trigger samples with regular data to train the watermark jointly with the underlying model functionalities. Furthermore, we employ a weight-masking approach to strengthen this joint training. At the conclusion of this section, we outline the steps involved in verifying ownership. ### Feature-fusion Design In this subsection, we present two feature-fusion methods, namely the direct feature-fusion method and the invisible feature-fusion method, to generate the watermark images that are coupled with model functionalities. Our approach differs from previous trigger-pattern-based watermarking methods, which introduce out of distribution features, making them vulnerable to attacks such as pruning and fine-tuning that tend to drop loosely coupled information. Our key insight is to fuse in-distribution features, similar to the technique used in MixUp (Wang et al., 2018) for improving model accuracy by combining training data from different classes. However, instead of the objective of data augmentation, we assign target labels to the combined images to use them as functional-coupled watermark triggers. We present these methods to ensure that the watermarks generated are coupled with the model's normal functionalities. #### 4.1.1. Direct Feature-fusion Method We generate watermark images based on a dataset $X=X_{1},X_{2},\cdots,X_{N_{c}}$, where $N_{c}$ represents the total number of categories in the dataset. Let $K_{wm}$ be the size of the watermark image set $WM$. We select subsets $X_{i}$ and $X_{j}$ from the original dataset to select base instances for generating the watermark images, with the target class set as $t\in[1,N_{c}]$ excluding $i$ and $j$. To generate the watermark image set, we follow the approach presented in Eq. 1. (1a) \[\begin{cases}WM=\{wm_{1},wm_{2},\cdots,wm_{k},\cdots,wm_{K_{wm}}\}\\ L_{wm_{k}}=t,t\in[1,N_{c}]\end{cases}\] (1b) where $wm_{k}$ represents each element in the watermark image set, and $L_{wm_{k}}$ corresponds to the given label of each generated watermark image. To generate a watermark image, we can combine the selected two base instances in the dimensions of height and width of the image, such that the watermark image has the complete features of both base instances. If we assume that the shape of the base instance is $(W,H,C)$, then we can generate the watermark image using the following equation: (2a) \[\text{w}m_{k}^{(w,h,c)}=\left\{\begin{array}{ll}X_{i}^{p(w,h,c)},& \text{if $w\leq W,h\leq H$}\\ X_{i}^{q}(w-W,h-H,c),&\text{if $w>W,h>H$}\\ (255,255,255),&\text{if $w\leq W,h>H$}\\ (255,255,255),&\text{others}\end{array}\right.\] (2b) where $\oplus$ is an operator that merges two base instances from different base classes, $i$ and $j$ represent the two selected classes, which range from class 1 to class $N_{c}$. $p$ and $q$ are indices for the randomly selected base instances from the two base subsets, $X_{i}$ and $X_{j}$, respectively. The pixels of each watermark image $wm_{k}$ can be computed using Eq. 2b. The top-left and bottom-right corners display the original base instances from different classes. Examples of generated watermark images are illustrated in Fig. 3. The combined images preserve all features of the base instances. #### 4.1.2. Invisible Feature-fusion method The direct feature-fusion method described in Eq. 2b generates a set of watermarked images in which two groups of features are independently distributed in the two corners. Although this method works well for embedding watermarks, adversaries can easily detect such key samples due to the abnormal white blocks in the images. These blocks are unusual in normal images, making it trivial for auditors to identify them (see experiments for details). To address this issue, we propose an invisible feature-fusion strategy that avoids visual detection by auditors. Specifically, we discard the strategy of merging features in the dimensions of width and length and design a new method from the perspective of image depth (RGB channels). We suppose a dataset $X=\{X_{1},X_{2},\cdots,X_{N_{c}}\}$ with $N_{c}$ representing the total categories to be the base for generating watermarking images. $X_{b1}$ and $X_{b2}$ are the subsets of the original dataset for selecting instances from them to generate the watermark images. We select images from $X_{b1}$ and $X_{b2}$ as two sets of base instance, and the target class is set as $t$ which is different from $b1$ and $b2$. Define $K_{wm}$ as the size of the watermark image set $WM$. We can also follow Eq. 1 to generate the watermark image set. Differently, in order to generate an invisible watermark image, we need to merge the two base instances in a different way. Suppose the shape of the base instance is $(W,H,C)$, the watermark image can be computed as follows: We consider a dataset $X=X_{1},X_{2},\cdots,X_{N_{c}}$ with $N_{c}$ representing the total number of categories as the base for generating watermark images. Let $X_{b1}$ and $X_{b2}$ be the subsets of the original dataset from which we select instances to generate the watermark images. We select images from $X_{b1}$ and $X_{b2}$ as two sets of base instances, and set the target class as $t$, which is different from $b1$ and $b2$. Let $K_{wm}$ denote the size of the watermark image set $WM$. To generate an invisible watermark image, we need to merge the two base instances in a different way. Suppose the shape of the base instance is $(W,H,C)$, then the watermark image can be computed as follows: \[\text{w}m_{k}=X_{b1}^{p}\oplus X_{b2}^{q} \tag{3b}\] \[\text{w}m_{k}^{(w,h,c)}=r\cdot X_{b1}^{p,(w,h,c)}+(1-r)\cdot X_{ b2}^{q,(w,h,c)} \tag{3a}\] where the operator $\oplus$ denotes the strategy for merging two base instances from different base classes. $p$ and $q$ are the indices of randomly selected base instances from the two base subsets, $X_{b1}$ and $X_{b2}$, respectively. The parameter $r$, which ranges from 0 to 1, is used to adjust the transparency of the target instance in the merged watermarking image. Increasing the value of $r$ results in the features of the target instance becoming more invisible, i.e., more transparent. Figure 3. Examples of visible trigger images generated for Cifar-10 dataset. The top-left corners are the instances from ‘airplane’ class, and the images in the bottom-right corners are selected from the ‘cat’ class. Figure 2. The workflow of the proposed function-coupled watermarking method. The pixels of each invisible watermark image $wm_{k}$ can be computed using Equation 3b. Given two source images with a shape of $(W,H,C)$, the merged watermarking image retains the same dimensions as the original data domain. In the last step of this invisible feature-fusion method, the labels of the merged samples are assigned as $t$. Figure 4 illustrates examples of the generated watermark images. ### Masking during Training Phase To further strengthen the relationship between watermarks and model functionalities, a masking training strategy is introduced during the training phase. The training of standard backdoor-based watermarks can be formalized as follows. #### 4.2.1. Standard Backdoor-based Watermarking Training We consider a training dataset $\{(x_{i},\bar{y}_{i})\}_{i=1}^{N_{d}}$, where $X=\{x_{i}\}_{i=1}^{N_{d}}$ and $\bar{Y}=\{\bar{y}_{i}\}_{i=1}^{N_{d}}$ represent the input samples and their corresponding labels, respectively, with $N_{d}$ being the total number of samples. A DNN model $f(\cdot):X\rightarrow\bar{Y}$ is trained from the dataset to map the input samples to labels. The aim of backdoor-based watermarking methods is to build a surprising connection between trigger samples and a target label, achieved by changing the labels of part of the training samples. Specifically, the target class is set as $\bar{y}_{t}$. Defenders can manipulate a portion of training samples by adding well-designed trigger patterns and change their labels to the target label, producing a watermarking training set $\{X^{\prime},Y^{\prime}\}=\{(x^{\prime}_{i},\bar{y}_{t})\}_{i=1}^{N_{d}e \%}+\{(x_{j},\bar{y}_{j})\}_{j=N_{d}e\%+1}^{N_{d}}$, with $\%$ denoting the ratio of the trigger data. Defenders then can exploit the manipulated dataset to train the model, producing a watermarking model $f_{wm}(\cdot)$. #### 4.2.2. Masking Training Strategy To enhance the robustness of our watermarking method, we propose a strategy to distribute the watermark function equally to each neuron in the model. Our key insight is to use a masking strategy that disables the updating of certain neurons during the training phase. By iteratively adding random masks during training, we can avoid the model's performance relying heavily on a small number of critical neurons. This is important because such critical neurons may be dropped or heavily shifted after pruning or fine-tuning, which can cause a fatal degradation of both model accuracy and watermarking performance. On the other hand, by using random masking, we can distribute the watermark function equally to each neuron, so that different combinations of neurons have the potential to retain the full watermark function. Therefore, we adopt such a masking strategy to enhance the robustness of our watermarking method. A technique similar to our proposed masking training strategy that can induce sparsity in the network is Dropout (Srivastava et al., 2014). Dropout is typically applied to fully-connected hidden layers (Deng et al., 2015) and is utilized to improve the model's generalization ability. In contrast, our proposed masking training strategy is applied to the convolutional layers of DNN models to distribute the watermark function equally across all neurons. Two crucial factors in masking methods during the training phase are the masking strategies and masking ratios. In order to introduce sparsity into the neural network layers, we must control the weight updating process. Since the manipulation occurs during the training phase, we need to apply the sparse strategy during both forward propagation and freeze the selected weights during backward updating. To accomplish this, a practical solution is to randomly mask a portion of the weights in DNN layers. This mask mimics the operations of pruning carried out by adversaries and can be applied to mimic various pruning principles, such as global pruning and module-based pruning, as described in Eq. 4. To ensure the masking training strategy is generalizable to different pruning attacks, we only utilize random masks to train the watermarked model. Sparse mask: \[\begin{cases}y=f(((W\odot M)/(1-p))x+b),\\ M_{i,j}\sim Prune(\{random,global,module,\cdots\},p)\end{cases} \tag{4}\] Inference phase: \[y=f(Wx+b) \tag{5}\] where $f(\cdot)$ indicates the DNN model for training and testing. $W$, $x$, $b$ represent the weights, inputs, and biases, respectively. We use $0\leq p\leq 1$ to indicate the ratio of the preserved elements after masking. $M$ is the mask that is used to indicate the pruned part of the convolutional kernels, and $M$ corresponds to the pruning strategy chosen in the inference phase, $i,j$ indicate the position of these kernels. The operator $\odot$ is the element-wise multiplication. $Prune$ means the pruning strategies that depend on the parameters including the specific pruning principle (such as global pruning and module-based pruning) and the pruning ratio. To ensure the same expectation as the original outputs, the calculation results need to be scaled by multiplying with $1/(1-p)$. Then, we can update the model weights as follows: \[W^{\prime}=W-\eta\cdot(\frac{\partial L}{\partial W}\odot M)/(1-p) \tag{6}\] where $W^{\prime}$ is the temporary variable of neural network weights, $\eta$ is the learning rate and controls the gradient updating step size, $L$ represents the model loss. In contrast to the conventional gradient updating method, we mask the gradients with $M$ before applying them to the weights. This means that, for each iteration, the gradient updating operation is only performed on the weights that are preserved after being masked with $M$. Figure 4. Examples of invisible trigger images generated for Cifar-10 dataset. The two base instances are selected from the ‘automobile’ class and the ‘cat’ class, respectively. The transparency ratio here is set to 0.7. ### Procedures of Ownership Verification In black-box watermarking scenarios, we utilize previously generated watermarking samples to verify the ownership of the candidate model by sending queries to the remote AI service. If the response corresponds to the expected labels, it confirms that the remote AI service is powered by our protected model. This is due to the fact that DNN models without embedding watermarks cannot recognize the given key samples, and as a result, queries will produce erroneous predictions. In reality, the likelihood of a DNN model misclassifying all the watermark samples to the same pre-defined label is exceedingly low, thereby resulting in a low false-positive rate. For ownership verification, defenders can submit a set of pre-prepared watermark samples (e.g., with a quantity of 90) to the remote AI service platform and collect the corresponding predictions for these queries. As each key sample is associated with the target label, defenders can compute the authentication success rate with their labels and the collected predictions. If the authentication success rate is higher than a widely accepted community threshold, defenders can assert their ownership of this victim model. ## 5. Experimental Results In this section, we present the experimental results of our proposed watermarking method. We organize this section as follows. In Section 5.1, we leverage different network structures and datasets to train DNN models as the victims, and present their accuracy on the clean dataset. In section 5.2, we compare our proposed method with six state-of-the-art watermarking methods. Then, in Section 5.3, we present the watermarking performance of different models and analyze the impact of watermarking on benign accuracy. Furthermore, in Section 5.5, we conduct an ablation to investigate the effectiveness of the masking training method. Finally, in Section 5.4, we evaluate the robustness of the proposed method under three prevailing attacks. ### Experimental Settings We conduct an evaluation of our feature-fusion watermarking method on various commonly used datasets and networks, namely LeNet-5 (LeNet and LeNet, 2015), VGG16 (Wang et al., 2016), and ResNet-18 (LeNet and LeNet, 2015), trained on MNIST, CIFAR-10 (LeNet and LeNet, 2015), CIFAR-100, and Tiny-ImageNet (200 classes) (Zhu et al., 2017). The accuracy of these models on clean datasets is presented in Table 1. We utilize two feature-fusion methods (examples of which can be found in Fig. 3 and Fig. 4) to generate watermarks. For each experiment, we use less than 1% of the training data as the watermarking samples and fix the number of validation images to 90. We set three transparency rates ($r$) of 0.5, 0.7, and 0.9 to generate invisible feature-fusion triggers. For the ease of conducting experiments, we set $r=0.5$ in Section 5.4 and Section 5.2 to evaluate the robustness of the proposed method and compare it with other methods. In addition to using models without watermarks as the baseline, we also perform empirical evaluations of our proposed feature-fusion watermarking method against five other black-box approaches: * Backdoor-based methods: Protecting IP (Zhu et al., 2017), Turning weakness into strength (Chen et al., 2017), Exponential weighting (Zhu et al., 2017), and Entangled watermark (LeNet and LeNet, 2015). * Adversarial example-based method: Frontier stitching (Zhu et al., 2017). ### Comparison with Baselines We conduct a comparative analysis of the performance of our proposed feature-fusion watermarking method against five state-of-the-art black-box approaches, namely Protecting IP (Zhu et al., 2017), Turning weakness into strength (Chen et al., 2017), Exponential weighting (Zhu et al., 2017), Frontier stitching (Zhu et al., 2017), and Entangled watermark (LeNet and LeNet, 2015). Notably, Entangled watermark primarily focuses on the model extraction attack, and since this scenario involves two primary subjects, i.e., the victim model and the extracted model, we compare the watermarked model guided by our methods with both models. We evaluate the performance of these methods concerning the authentication success rate, benign accuracy preservation rate, and robustness under four distinct attacks. Given that most of the compared works are constructed based on CIFAR-10, we use the same dataset as the benchmark for comparison. All other methods are implemented based on the open-source codes released on GitHub 1. Footnote 1: We conducted experiments of Ref. (Chen et al., 2017; Zhu et al., 2017; Zhu et al., 2017) referring to [https://github.com/mathebell/model-watermarking](https://github.com/mathebell/model-watermarking), and Ref. (LeNet and LeNet, 2015) referring to [https://github.com/RorschachChen/entangled-watermark-torch](https://github.com/RorschachChen/entangled-watermark-torch) Table 2 presents the summarized experimental results. Our method outperforms most of the other black-box methods by 15% and 10% in terms of average authentication success rate in 10-iteration fine-tuning and transfer learning (define a 20-epoch training process as an iteration), respectively. We conduct a detailed case study for these two attack scenarios, and the results are shown in Fig. 5 and Fig. 6. Furthermore, even after pruning 80% of neuron weights, our methods still achieved an authentication success rate of 100%, while the best authentication success rate of other methods is lower than 90%. It should be noted that when the pruning ratio is set to 90%, the average degradation of clean accuracy is greater than 30%, which can potentially cause the models to fail in their regular functions. However, the authentication success rate of our proposed method is still above 90%. \begin{table} \begin{tabular}{l c c} \hline \hline Models & Classes & Top-1 Benign accuracy (mean) \\ \hline MNIST (LeNet) & 10 & 99.14\% \\ CIFAR-10 (ResNet-18) & 10 & 94.49\% \\ CIFAR-100 (VGG16) & 100 & 73.13\% \\ Tiny-ImageNet (ResNet-18) & 200 & 65.98\% \\ \hline \hline \end{tabular} \end{table} Table 1. Benign accuracy of different models. Figure 5. Fine-tuning results on authentication success rate. As illustrated in Fig. 5, our proposed watermarking methods exhibit a significant advantage in terms of robustness against fine-tuning attacks, when compared with other methods. Specifically, our methods maintain an authentication success rate of 100% after 10 iterations of fine-tuning, while most of the other methods can only retain around 80% of their original performance. Notably, Frontier Stitching (Sutton et al., 2018) is more susceptible to fine-tuning attacks, as its authentication success rate drops to around 40% after fine-tuning. Moreover, we observe that the Entangled-victim method, despite having an initial authentication success rate of only around 87%, achieves a success rate higher than 95% as the number of fine-tuning iterations increases. This improvement may be attributed to the entangled training strategy used in the watermarking process, which enables the watermark features to be entangled with those corresponding to the normal functions of a model. Thus, fine-tuning with in-distribution data cannot drop watermarks but even improve the watermarking performance. However, it is worth noting that the Entangled watermark samples are beyond the training data distribution, and may degrade watermarking performance after transfer learning with out-of-distribution data, as confirmed by the following experimental results. We conducted experiments to compare the robustness of our watermarking method against transfer learning attacks with several black-box methods. To ensure consistency, we utilized the CIFAR-10 dataset for training and evaluated the transfer learning performance on the CIFAR-100 dataset, with a small learning rate from 1e-4 to 1e-5. The comparison results are presented in Fig. 6. Our method outperforms other methods significantly in preserving the authentication success rate of the watermark. Remarkably, our method achieves an average authentication success rate that is 10% higher than the best-performing black-box method, and even 60% higher than that of Frontier Stitching. Figure 7 presents the comparison results between our proposed watermarking method and the baselines in terms of watermarking robustness under pruning attacks. Our proposed method exhibits greater robustness under pruning attacks. Specifically, even after pruning 80% of the neurons in a model, our watermarking methods are still able to retain an authentication success rate of 100%. When the pruning ratio is set to 90%, our methods still exhibit a high authentication success rate of 97% (for invisible watermark) and almost 100% (for direct watermark). In contrast, the authentication success rate of other methods decreases dramatically by more than 55% when 90% of neurons are pruned. ### Effectiveness and Fidelity The effectiveness of our watermarking method is measured by the ability to successfully verify the ownership of DNN models without significantly impacting their clean accuracy. Our evaluation focuses on the proposed two methods in terms of effectiveness and fidelity across four different models. Each method is tested in three sets of experiments. Specifically, for the direct feature-fusion method, \begin{table} \begin{tabular}{c c c c c c c} \hline \hline Methods & Authentication & Benign accuracy & \multicolumn{4}{c}{Robustness} \\ & success rate & preserving rate & Fine-tuning & Transfer learning & Pruning & Overwriting \\ \hline Protecting IP (Sutton et al., 2018) & 100.0\% & 99.95\% & 89.00\% & 70.0\% & 89.32\% & 85.10\% \\ Turning weakness & 100.0\% & 99.85\% & 84.21\% & 41.0\% & 74.56\% & 82.00\% \\ into strength (Beng et al., 2017) & 100.0\% & 99.92\% & 82.00\% & 38.0\% & 83.75\% & 83.30\% \\ Fromtier stitching (Sutton et al., 2018) & 100.0\% & 99.90\% & 43.10\% & 30.0\% & 83.62\% & 68.40\% \\ Entangled-victim (Sutton et al., 2018) & 87.41\% & 98.76\% & 95.98\% & 63.73\% & 41.71\% & - \\ Entangled-extract (Sutton et al., 2018) & 75.43\% & 85.23\% & 16.14\% & 8.89\% & 58.22\% & - \\ Ours (direct) & 100.0\% & 100.0\%2 & 100.0\% & 82.20\% & 100.0\% & 100.0\% \\ Ours (invisible) & 100.0\% & 100.0\% & 100.0\% & 86.70\% & 100.0\% & 100.0\% \\ \hline \hline \end{tabular} \end{table} Table 2. Comparison with state-of-the-art methods. Figure 6. Watermarking robustness of different methods under transfer learning from CIFAR-10 to CIFAR-100. Figure 7. Pruning results on authentication success rate. we conduct three replicate experiments numbered as #1, #2, and #3. Regarding the invisible feature-fusion method, we vary the transparency ratio from $r=0.5$ to $r=0.9$ in our experiments. Table 3 displays the effectiveness and fidelity of the proposed methods. In all experiments, both proposed methods achieve a verification effectiveness of 100%, indicating their effectiveness in watermarking. The generated watermarked models can achieve a high watermarking success rate without sacrificing clean accuracy, as shown in Table 1 for the baseline of benign accuracy. The perturbation of the proposed watermarking methods on the benign accuracy of the model is within the range of $\pm 0.5\%$. It is worth noting that in most cases, there is a slight increase in benign accuracy. The accuracy of Tiny-ImageNet models even increases from around 65% to around 69% after watermarking. A reasonable explanation is that the mask training strategy for enhancing watermarking robustness can also improve the model's generalization capability. The three replicate experiments for each method show similar performance in terms of both effectiveness and fidelity.The watermarking performance of the invisible feature-fusion method under three different transparency ratios is also very close, indicating that transparency has little effect on the final watermarking performance and only affects visual features. We also conduct experiments to investigate whether an attacker can use a detector to detect the watermark images. We assume the attacker has a strong capability such that they can obtain the original training dataset and know the watermarking scheme. Hence, they can simulate a large number of triggers and train a big classifier to distinguish the triggers. We mimic an attacker to set such a task as a three-category classification, use ResNet-18 as the backbone and generate training data from Tiny-ImageNet dataset. Specifically, we randomly select and generate 45,000 normal data and 45,000 direct-fusion watermark images and invisible-fusion watermark images. The objective is to check whether a well-trained classifier can automatically classify these three-category datasets. Table 4 presents the confusion matrix on detection results of watermark images. It shows that direct-fusion watermarks can be easily detected, i.e., the both precision and recall are high. We believe it is because of the obvious white patterns in the watermark images. Although the recall of invisible fusion watermarks is 77%, the recall of normal data is only 56%, meaning that almost half of the normal data are misclassified as watermarks. Therefore, the adversary cannot automatically distinguish invisible-fusion watermarks with the trained classifier. ### Robustness Fine-tuning resistance. In order to perform the fine-tuning attack, a small number of in-distribution samples are selected as fine-tuning data. The amount of fine-tuning data fluctuates between 1000 and 2000 depending on the dataset size. Furthermore, we evaluate the \begin{table} \begin{tabular}{l l l l} \hline \hline \multirow{2}{}{Confusion Matrix} & Normal & Invisible fusion & Direct fusion \\ & (Ground Truth) & (Ground Truth) & (Ground Truth) \\ \hline Normal (Prediction) & 25317 & 10326 & 3 \\ Invisible fusion (Prediction) & 19663 & 34666 & 0 \\ Direct fusion (Prediction) & 20 & 8 & 44997 \\ \hline In total & 45000 & 45000 & 45000 \\ \hline \hline \end{tabular} \end{table} Table 4. Confusion matrix on detection results of watermark images. Figure 8. Fine-tuning results of direct watermarking on benign accuracy and authentication success rate. \begin{table} \begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{}{Methods} & Hyper-parameter & MNIST (LeNet5) & CIFAR-10 (ResNet-18) & CIFAR-100 (VGG16) & Tiny-ImageNet (ResNet-18) \\ \cline{3-10} & changing & Eff. & Fid. & Eff. & Fid. & Eff. & Fid. & Eff. & Fid. \\ \hline \multirow{3}{}{Direct} & \#1 & 100\% & 99.36\% & 100\% & 94.18\% & 100\% & 73.23\% & 100\% & 69.68\% \\ & \#2 & 100\% & 99.30\% & 100\% & 94.20\% & 100\% & 73.20\% & 100\% & 69.66\% \\ & \#3 & 100\% & 99.32\% & 100\% & 94.18\% & 100\% & 73.23\% & 100\% & 69.70\% \\ & r=0.5 & 100\% & 99.33\% & 100\% & 94.63\% & 100\% & 73.37\% & 100\% & 69.51\% \\ \multirow{3}{}{Invisible} & r=0.7 & 100\% & 99.33\% & 100\% & 94.62\% & 100\% & 73.35\% & 100\% & 69.48\% \\ & r=0.9 & 100\% & 99.29\% & 100\% & 94.50\% & 100\% & 73.34\% & 100\% & 69.42\% \\ \hline \hline \end{tabular} \end{table} Table 3. Results on effectiveness and fidelity. robustness metric under different fine-tuning iterations ranging from 1 to 10. The watermarking performance under fine-tuning attacks is presented in Fig. 8 and Fig. 9, demonstrating the efficacy of both feature-fusion methods against fine-tuning attacks. Fig. 8 displays the experimental results of the direct feature-fusion watermarking method against fine-tuning attacks. The authentication success rate remains stable at 100% as the number of training iterations increases, and the benign accuracy of these models shows slight fluctuations. Similarly, Fig. 9 illustrates the experimental results of the invisible feature-fusion watermarking method, indicating its robustness to fine-tuning attacks. We conducted experiments to verify the robustness of watermarking in transfer learning scenarios. Specifically, we performed three groups of transfer learning tasks, namely CIFAR-10 to CIFAR-100, CIFAR-10 to MNIST, and CIFAR-100 to Tiny-ImageNet. For each group of experiments, we changed the datasets while retaining the number of classes, and let the learning rate be from 1e-4 to 1e-5. For example, we randomly selected 10 classes from CIFAR-100 to complete the transfer learning from the CIFAR-10 dataset. Fig. 10 illustrates the authentication success rate and accuracy on the original dataset for each group of transfer learning tasks. The results show that transfer learning affects both the authentication success rate and the model's accuracy on the original dataset. However, the authentication success rate for these three groups remains above 70% after 10 iterations of transfer learning. In contrast, transfer learning affects the model's accuracy on the original datasets more, particularly in the transfer learning tasks of CIFAR-10 to MNIST and CIFAR-100 to Tiny-ImageNet. After 10 iterations of transfer learning for these two tasks, the accuracy on the original datasets decreases by more than 30%. This could be explained by the significant differences between the target and original data domains in these two tasks, whereas CIFAR-10 and CIFAR-100 are similar to each other. Weight pruning resistance. To evaluate the proposed method's robustness against pruning attacks, we adopt the widely-used L1-norm unstructured pruning strategy. This strategy determines the parameters to be pruned based on the weights' values. We test pruning ratios ranging from 0.1 to 0.9, corresponding to weight pruning rates from 10% to 90%. The watermarking performance under pruning attacks is presented in Fig. 11 and Fig. 12. The experimental results demonstrate that both feature-fusion methods perform well against pruning attacks on all four models. Figure 11. Pruning results of direct watermarking on benign accuracy and authentication success rate. Figure 10. Watermarking robustness under transfer learning. Acc. on CIFAR10(1) represents the model accuracy on the CIFAR-10 dataset after the transfer learning from CIFAR-10 to CIFAR-100. Acc. on CIFAR10(2) represents the model accuracy on the CIFAR-10 dataset after the transfer learning from CIFAR-10 to MNIST. Figure 9. Fine-tuning results of invisible watermarking on benign accuracy and authentication success rate. As illustrated in Fig. 11, we evaluate the direct feature-fusion watermarking method against the pruning attack. We adopt the L1-norm unstructured pruning strategy with pruning ratios ranging from 0.1 to 0.9. Our results demonstrate that our method remains effective against the pruning attack, even when the pruning ratio exceeds 0.7. As the pruning ratio increases, there is a trade-off between model sparsity and accuracy. Despite the decrease in benign accuracy of the models, our watermarking success rate remains high. Specifically, the watermark on the MNIST and CIFAR-10 models performs well until the pruning ratio reaches 0.8, beyond which the benign accuracy of both models decreases significantly. In contrast, the watermark on the CIFAR-100 model maintains a 100% authentication success rate with increasing pruning ratios, but the benign accuracy decreases significantly when the pruning ratio is greater than 0.6. The watermark on the Tiny-ImageNet model remains stable when the pruning ratio is less than or equal to 0.7, but the pruning attack has a greater negative impact on the benign accuracy of the model. The experiments conducted demonstrate that the watermarks embedded in the models are more robust against pruning attacks as compared to the basic classification function (clean accuracy) of the model. This is due to the fact that as the pruning ratio increases, the capacity of the model decreases. Additionally, the watermarking method requires much less model capacity than the basic classification function of such a model. Therefore, pruning has a greater impact on the classification accuracy of the model than the watermarking performance. Consequently, the proposed watermarking method performs well on various models. However, when dealing with larger datasets, the proposed method is observed to be more sensitive to pruning attacks, possibly due to the higher model capacity in such cases. Fig. 12 presents the experimental results of the invisible feature-fusion watermarking method against pruning attacks, where similar observations can be made as with the direct watermarking strategy. Overwriting resistance. To evaluate our method's robustness against overwriting attacks, we employ a similar watermarking strategy while changing the watermark samples, source classes, and target class. Specifically, we set No.0 and No.3 as the source classes and No.1 as the target class in the test group while conducting CIFAR-10 experiments. In the control group, we transfer the source classes to No.2 and No.4 and select No.5 as the target class (an extreme scenario is that the source classes and target class are same to that of the test group). To overwrite another watermark into the model, we fine-tune the watermarked models with the selected watermark samples using a small learning rate from 1e-4 to 1e-5. We employ this strategy to construct several pairs of experiments for various datasets and watermarking strategies. The experimental results are presented in Table 5 and the last column of Table 2. The experimental results presented in Table 5 indicate that our proposed method exhibits a high level of robustness against overwriting attacks. Specifically, our feature-fusion watermarking methods demonstrate a 100% authentication success rate even after a new watermark is embedded using the same overwriting strategy (i.e., selecting same source classes and target class for the control group as that of the test group). This robustness can be attributed to the fact that the process of overwriting is similar to fine-tuning and transfer learning, with the only difference being the type of training data used. Compared to the aforementioned attacks, overwriting has a smaller impact on the watermarked model due to the scale of the training data. Therefore, it is reasonable to observe that our method exhibits robust results under overwriting attacks. However, it should be noted that the authentication success rate of the newly embedded watermarks is also 100%, since our watermarking methods mainly focus on improving the robustness of watermarks against overwriting attacks, rather than preventing the embedding of new watermarks in the same model. ### Ablation Study We also conduct several additional experiments to investigate the impact of equipping our watermarking method with a masking training strategy on its robustness capability. Figure 13 illustrates the impact of equipping enhancing strategies on the watermarking robustness against fine-tuning attacks. The red and other colored lines represent the watermarking authentication success rate of each model after and before implementing the enhancing strategies. The results demonstrate that leveraging the mask training strategies can effectively enhance the watermarking robustness against fine-tuning attacks. On average, we observe a Figure 12. Pruning results of invisible watermarking on benign accuracy and authentication success rate. \begin{table} \begin{tabular}{c c c c c c} \hline \hline Datasets & MNIST & CIFAR-10 & CIFAR-100 & Tiny-ImageNet \\ \hline Source Class ID & No.2 & No.4 & No.2 & No.4 & No.9 & No.13 & No.9 & No.13 \\ Target Class ID & No.5 & No.5 & No.16 & No.16 & No.16 \\ Direct & 100\% & 100\% & 100\% & 100\% & 100\% \\ \begin{tabular}{c} Intrisible \\ \end{tabular} & 100\% & 100\% & 100\% & 100\% \\ \begin{tabular}{c} Intrisible \\ \end{tabular} & 100\% & 100\% & 100\% & 100\% \\ \hline \hline \end{tabular} \end{table} Table 5. Overwriting results of watermarking methods on authentication success rate. 10% increase in authentication success rate across the four tasks after applying these strategies. Moreover, the authentication success rate increases by more than 20% for CIFAR-100 models. Figures 14 and 15 illustrate the change in watermarking robustness under pruning attacks before and after equipping enhancing strategies. The watermarking authentication success rate of each model after equipping the enhancing strategies is represented by the red lines, while the black lines represent the cases without the proposed strategies. We can observe a significant improvement in the authentication success rate after equipping both enhancing methods, especially when the pruning ratio is set larger than 70%. For the Tiny-ImageNet dataset in Fig. 14, when the pruning ratio is set to 80%, the authentication success rate is less than 20% before applying the proposed strategy but becomes more than 90% after equipping the enhancing method. This improvement can be attributed to the fact that the feature-fusion and random masking strategy can force the model to learn to couple the watermark with model functionalities and equally distribute the watermark functions to each neuron in the model. Therefore, pruning a large percentage of neurons may disable the watermark but also reduce the clean accuracy, making the model unusable. ## 6. Conclusion and Future Work In this paper, we propose a novel black-box watermarking method to protect deep neural network models. We first introduce the concept of function-coupled watermarks that tightly integrate the watermark information with the DNN model functionalities. Based on this concept, our designed watermark triggers only employ features learnt from in-distribution data and thus do not suffer from oblivion with model retraining. To further enhance the robustness under watermark removal attacks, we apply random masking when training watermarked models. Experiments conducted on various image classification tasks show that our method significantly outperforms existing watermarking methods. In our opinion, the proposed function-coupled watermarking concept for DNN IP protection is simple yet general, despite being verified only on image classification tasks. We plan to extend it to other deep learning tasks, e.g., object detection and speech recognition. Figure 14. The effect of enhancing strategy on the robustness of direct feature-fusion method. Figure 13. Comparison of watermarking robustness on fine-tuning attacks before and after equipping robustness-enhancing strategies. Figure 15. The effect of enhancing strategy on the robustness of invisible feature-fusion method.
2307.00673	ENN: A Neural Network with DCT Adaptive Activation Functions	The expressiveness of neural networks highly depends on the nature of the activation function, although these are usually assumed predefined and fixed during the training stage. Under a signal processing perspective, in this paper we present Expressive Neural Network (ENN), a novel model in which the non-linear activation functions are modeled using the Discrete Cosine Transform (DCT) and adapted using backpropagation during training. This parametrization keeps the number of trainable parameters low, is appropriate for gradient-based schemes, and adapts to different learning tasks. This is the first non-linear model for activation functions that relies on a signal processing perspective, providing high flexibility and expressiveness to the network. We contribute with insights in the explainability of the network at convergence by recovering the concept of bump, this is, the response of each activation function in the output space. Finally, through exhaustive experiments we show that the model can adapt to classification and regression tasks. The performance of ENN outperforms state of the art benchmarks, providing above a 40% gap in accuracy in some scenarios.	Marc Martinez-Gost, Ana Pérez-Neira, Miguel Ángel Lagunas	2023-07-02T21:46:30Z	http://arxiv.org/abs/2307.00673v3	# ENN: A Neural Network with DCT-Adaptive Activation Functions ###### Abstract The expressiveness of neural networks highly depends on the nature of the activation function, although these are usually assumed predefined and fixed during the training stage. In this paper we present Expressive Neural Network (ENN), a novel architecture in which the non-linear activation functions are modeled using the Discrete Cosine Transform (DCT) and adapted using backpropagation during training. This parametrization keeps the number of trainable parameters low, is appropriate for gradient-based schemes, and adapts to different learning tasks. This is the first non-linear model for activation functions that relies on a signal processing perspective, providing high flexibility and expressiveness to the network. We contribute with insights in the explainability of the network at convergence by recovering the concept of bump, this is, the response of each activation function in the output space to provide insights. Finally, through exhaustive experiments we show that the model can adapt to classification and regression tasks. The performance of ENN outperforms state of the art benchmarks, providing up to a 40% gap in accuracy in some scenarios. Neural networks, adaptive activation functions, discrete cosine transform, explainable machine learning. ## I Introduction Function approximation is a fundamental problem across many domains, such as data analysis, control systems and communications. When the explicit function is not available but input-output data pairs, the function can be uncovered by minimizing a criterion loss in a supervised setting. The problem increases in complexity when the function is non-linear, for which many signal processing techniques have been developed. For instance, least squares [1], orthogonal function approximation [2, 3], kernel methods [4] and neural networks [5, 6, 7], among others. The last decades have suffered an unprecedented growth in the development of artificial neural networks for function approximation due to its empirical success. The inception of neural networks as universal approximators boosted its development across many fields and applications, with different architectures built according to the task and data types to handle. Nevertheless, the expressiveness of the neural network is related to the non-linear activation function, which is usually assumed fixed. An overlooked field of research is in adaptive activation functions (AAF), where not only the weights of the neural networks are trained, but the non-linearities too [8]. In our previous work [9] we introduced the Discrete Cosine Transform (DCT) to approximate an univariate non-linear function in a joint communication and computing setting. Further, in [10] we show how a gradient-based algorithm can be used to tune the DCT coefficients to approximate a function in a supervised setting. However, extending the results to multivariate functions is not trivial: The number of required parameters to approximate the function increases exponentially with the number of variables, and their corresponding indexes are unknown when the explicit function is not available. This is, it is cumbersome how the top relevant coefficients can be learnt in a supervised fashion using labeled data. In this work we propose to extend the capabilities of the DCT with a novel neural network architecture that integrates the DCT to represent and adapt the activation functions. We call this architecture Expressive Neural Network (ENN). We exploit the fact that a 2-layer neural network can theoretically represent any function and expand the representation capabilities of the network by adapting the activation functions at each neuron. The advantage of approximating an univariate function with the DCT is twofold: a small number of coefficients is required due to its high energy compaction, and the approximation error is easily controlled by the magnitude of the disregarded coefficients. In this work we also show that the DCT coefficients can be learnt using backpropagation in a supervised fashion and in the same pass as the standard network linear weights. In this way, the architecture is no different from a standard feed-forward neural network with fixed activation functions. From the learning perspective, using the DCT to model the activation functions brings the following benefits: * Network size: The number of parameters in the network grows linear with the number of neurons, which is small due to energy compaction of DCT. * Backpropagation: The algorithm can be implemented because the DCT coefficients are real and ordered in decreasing magnitude. Besides, there exist analytical closed-form solutions for backpropagation. * Gradient behavior: The basis functions (cosines) are real and bounded, which prevents exploding gradients. Likewise, since the Fourier representation creates a periodic function that does not saturate, it also prevents vanishing gradients. * Task adaptability: The output non-linearity is automati cally adapted depending on the task (e.g., classification or regression). While we provide a general formulation for multivariate real functions, we constraint the analysis to bivariate functions. This allows to visualize the results and intuitively understand how the network is adapting the activation functions. In this respect, we recover the concept of _bump_, which is the non-linear enclosure that each activation function generates in the output space [11]. The output space corresponds to a weighted sum of all bumps generated at the hidden layer. This concept allows to gain insights in how the network decides to exploit the periodic nature of the DCT model and create the boundaries for classification problems. In this work we focus on two general problems of function approximation, namely classification and regression. In the former there are two hypothesis associated to a function, this is, $\mathcal{H}_{0}$ when $f(x_{1},x_{2})<0$ and $\mathcal{H}_{1}$ otherwise. In regression the goal is to approximate the function $f(x_{1},x_{2})$. The main contributions of this paper are described in the following: 1. We define ENN, a novel neural network architecture where the non-linear activation functions are parameterized by the DCT. This allows to adapt DCT coefficients in a supervised fashion and learn specific non-linearities according to the task. This results in a highly flexible and expressive architecture. 2. We provide insights in the field of explainable machine learning (XML). We recover the concept of bump, which allows to interpret how the non-linearities are adapted and what is the output response of the neural network. Furthermore, the concept allows to intuitively control several parameters (e.g., weight initialization). 3. We develop analytical closed-form expressions to adapt the non-linearities with backpropagation. While we choose the Least Mean Squares (LMS) algorithm to update the network parameters, the architecture remains a feed-forward neural network and any alternative algorithm can be used. 4. We provide extensive experiments in both classification and regression setups for which ENN outperforms all the benchmarks. In classification tasks, ENN outperforms fixed activation functions up to 40% in accuracy. With that we show how the expressiveness of network highly depends on the activation function, without the need of increasing the size of the network. 5. With respect to XML, we show how the DCT model helps to dimension the network width, this is, the number of hidden neurons. Particularly, the network converges to duplicated activation functions with opposed linear weights in the output layer. This is, the network cancels out the information coming from several branches when the task does not require that much expressiveness. The remainder of this paper is organized as follows. In Section II, we present a literature review on neural networks for function approximation. The Fourier models for non-linear representation are presented in Section III. We present ENN in Section IV and propose the learning procedure for supervised tasks in Section V. The simulation results are shown in Section VI and we conclude the paper in Section VII. Notation: Lowercase and uppercase bold symbols correspond to vectors and matrices, respectively; $\mathbf{s}[m]$ corresponds to the $m$-th entry of vector $\mathbf{s}$; $\mathbb{R}$ stands for the set of real numbers and $\nabla$ for the gradient. ## II Literature Review The universal approximation theorem is a well-known result in mathematics stating that a 2-layer neural network can represent any continuous function with an arbitrary number of neurons [12, 13]. The theory behind neural networks has been further developed, providing bounds on the number of required neurons. In a different line of research, the Kolmogorov-Arnold (KA) representation theorem shows how a multivariate function can be represented by functions of only univariate functions [14]: \[f(x_{1},\ldots,x_{n})=\sum_{i=1}^{2n+1}\Phi_{i}\left(\sum_{j=1}^{n}\phi_{ij}( x_{j})\right), \tag{1}\] where $\Phi_{i}$ and $\phi_{j}$ are termed the outer and inner functions, respectively. This result seems to be tightly connected to a 2-layer neural network, since the inner functions correspond to the hidden layer transformation and the outer functions to the output neuron. What is more, the inner functions do not depend on the function $f$ to implement, which resembles the activation functions in neural network architectures. However, the formulation is not exactly identical: the KA representation requires $n$ inner functions for each input variable, while a neural network implements a unique function for each linear combination of inputs. In this way, although there is extensive literature motivating the development of neural networks with the KA theorem [15, 16, 17], there are still many gaps to be resolved. What is more, the theorem is not constructive and the functions in (1) are highly non-linear. Despite its success in many applications, neural networks undergo several shortcomings that limit the interpretability of the results: optimization algorithms are easily trapped in local minima, convergence heavily depends on initialization and fail to converge when high non-linearities exist. Usually, each neuron in feedforward neural networks implements a linear combination and a non-linear mapping. The former is usually trained, while the latter remains fixed. The non-linear activation function allows to generate non-linear mappings, which increases the expressiveness of the network. A common choice is the sigmoid function, although it exhibits well-known issues in its implementation: since the sigmoid saturates at large magnitudes the propagated gradients vanish, slowing down the learning process. The rectified linear unit (ReLU) replaced the sigmoid activation function because it does not suffer from vanishing gradients and it is computationally efficient, which results in faster convergence. Despite its popularity, ReLU also experience several weaknesses that hinder the learning capacity of the architecture: the corresponding neuron can become dead when the output remains negative for a long time; likewise, the output is unbounded, which may produce the opposite effect of exploding gradient, affecting the stability of the network. Many variations and novel activation functions have been designed to enhance the performance of neural networks, although this highly depends on the application and the statistics of the data. In this respect, few authors have tackled the problem of adaptive activation functions (AAF), in which the non-linear mapping is also trained to enhance the expressiveness of the network. In [8] the saturation level and slope of an S-shaped activation function can be tuned independently, which increases the expressiveness with respect to the sigmoid function. In a different vein, several authors proposed to approximate an arbitrary activation function using cubic splines [18, 19]. Later on, some authors introduced an activation function as a weighted sum of basis, such as sine, Gaussian or sigmoid [20, 21]. More recently, the authors in [22] propose an asymmetric S-shaped activation function, but only for regression tasks. The issues derived from these perspective is that they either constraint the geometry of the activation function, or the parametrization is too complex. In this work we empirically prove that having pre-specified activation functions result in suboptimal performance. In the context of deep learning, several AAFs have been designed as well. In [23] the authors propose a gradient adaptive piecewise linear activation function as a sum of hinge-shaped functions. Although they theoretically prove that any piecewise linear function can be expressed as so, this constraints the function to be linear at both extremes. In other words, it does not prevent the neural network from exploding gradients. In [24] an S-shaped rectified linear activation function is proposed. While this model can learn non-linear and non-convex representations, it is not smooth and still constraints the activation to be defined in a piece-wise fashion. The authors in [25] develop a piecewise polynomial form which can approximate any continuous function. However, the exponential nature of the polynomials may increase the dynamic range of the parameters, which is a non-desirable property for gradient-based procedures. In [26], the ReLU is substituted by an S-shaped activation, which slightly outperforms the ReLU and its variants in different deep learning tasks. In [27, 28, 29, 30] different authors propose to model the activation as a sum of basis functions. In the following section we propose a Fourier-based parametrization for non-linear functions. This is the first approach to model AAFs from a signal processing perspective. The proposed model does not constraint the shape of the activation function and keeps the number of trainable parameters low while circumventing the shortcomings of the previously proposed non-linear models. ## III Fourier Models for Non-linear Functions Consider a scalar univariate function $f(x)$ to be approximated over the input variable range $x\in[-1,1]$. One of the most popular approximations for non-linear systems is the Volterra model. Given the function representation by the inverse Fourier transform, \[f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(w)e^{jwx}\,dw, \tag{2}\] where $w$ is the frequency variable and $F(w)$ are the Fourier coefficients, the Volterra model is obtained by using the Taylor series expansion of the exponential family: \[f(x) =\frac{1}{2\pi}\int_{-\infty}^{\infty}F(w)\left(\sum_{n=0}^{ \infty}\frac{(jw)^{n}}{n!}x^{n}\right)\,dw\] \[=\sum_{n=0}^{\infty}\left(\frac{1}{2\pi}\frac{1}{n!}\int_{- \infty}^{\infty}F(w)(jw)^{n}\,dw\right)x^{n}\] \[=\sum_{n=0}^{\infty}c_{n}x^{n} \tag{3}\] The building blocks of (3) are power functions of the input variable, which generate a polynomial approximation. Notice that the coefficients $c_{n}$ correspond to the $n$-th derivative at the origin divided by the factorial of the index. Volterra has not been widely used in practice because the exponential nature of the basis functions heavily increase the dynamic range, even when the input is bounded. Moreover, outside the dynamic range the approximation is unbounded and cannot be controlled. This happens because Volterra is a Taylor approximation and it presents high accuracy near the origin only. All these concerns make this approximation not suitable for gradient-based learning algorithms. One possible approach to solve these issues is the representation using a finite number of terms in the Fourier approximation in (2), which corresponds to the Discrete Fourier Transform (DFT). Outside the dynamic range of $x$ the function is periodically extended, generating discontinuities at the border. As a result, the number of coefficients required to approximate the function is highly sensitive to these discontinuities. In fact the oscillatory behavior around the discontinuity has an amplitude proportional to the number of continuous derivatives of $f(x)$. To prevent this phenomenon the function can be extended with even symmetry and, then, periodically extended. This smooths out the edges and its corresponding derivatives, reducing the number of required coefficients with respect to the DFT. This is known as the Discrete Cosine Transform (DCT), which has the following expression: \[f(x)\approx\sum_{q=1}^{Q}g_{q}F_{q}\cos\left(\frac{\pi(q-1)(2z+1)}{2N}\right), \tag{4}\] with $z=\frac{N}{2}(1+x)$, $g_{1}=1/\sqrt{N}$ and $g_{q}=\sqrt{2/N}$ otherwise. The $F_{q}\in\mathbb{R}$ are termed the DCT coefficients. Notice that for functions with odd symmetry, only the odd coefficients are retained. In general, the quality of approximation is more than sufficient for $Q=12$ (i.e., 6 coefficients in odd functions). For the sake of brevity, the following definition will be used when needed: \[cos_{i}(x)=\cos\left(\frac{\pi}{2N}\left(2i-1\right)\left(N(x+1)+1\right)\right) \tag{5}\] In [10] we propose an adaptive design in which the DCT coefficients of (4) are tuned using the LMS algorithm in a supervised setting to approximate an univariate function. There are plenty of advantages in using the DCT representation: The required coefficients to provide the same quality are fewer and these are real and ordered in decreasing magnitude. Since the basis functions are orthogonal, the approximation error can be easily controlled by the magnitude of the disregarded coefficients, and it simplifies the convergence of the learning procedure. Furthermore, see that the index appears in the phase of (4), so the approximation is real and bounded, even when the input exceeds the dynamic range. All these features make the DCT an appropriate function approximation, whose coefficients can be learnt by a gradient-based rule. ### _Extension to several inputs with the DCT_ The extension of the DCT representation to multiple variables is not trivial. Consider a bivariate function, that could be parameterized by a 2-dimensional (2D) DCT as \[f(x_{1},x_{2}) =\sum_{n=1}^{N}\sum_{m=1}^{N}F_{nm}\cos_{n}(x_{1})\cos_{m}(x_{2})\] \[\approx\sum_{n=1}^{Q}\sum_{m=1}^{Q}F_{nm}\cos_{n}(x_{1})\cos_{m}(x_ {2}) \tag{6}\] Note that this architecture has the following drawbacks: In general the number of coefficients grows exponentially with the number of inputs $M_{0}$ as $Q^{M_{0}}$, along with the complexity of the DCT in several dimensions; while the coefficients in the DCT are ordered in decreasing magnitude, the structure of the indexes is broken in the 2D-DCT. This is, the location $(n,m)$ of the relevant coefficients changes with the function of interest; this implies that, unless the function is known a priori, the indexes $(n,m)$ are unknown and a supervised learning procedure is hard to implement. This happens for instance in classification problems, where the function is unknown and to be discovered. In the following we will present ENN, a neural network architecture integrating the DCT in a single dimension and whose coefficients can be trained in a supervised fashion. ## IV DCT-Adaptive Activation Functions A perceptron (or neuron) is a non-linear processor involving a weighted sum and a non-linear function. It is described by the following expressions: \[z =a_{0}+\sum_{i=1}^{M}a_{i}x_{i}=\mathbf{a}^{T}[1\ \mathbf{x}^{T}]^{T} \tag{7}\] \[\overline{z} =\frac{N}{2}(z+1)\] (8) \[\hat{y} =\sigma(\overline{z}) \approx\sum_{q=1}^{Q/2}F_{q}\cos\left(\frac{\pi(2q-1)(2\overline {z}+1)}{2N}\right)\] \[=\sum_{q=1}^{Q/2}F_{q}\cos_{q}(z) \tag{9}\] In (7), $\mathbf{x}$ contains $M_{0}$ inputs and a 1 is appended for the bias term $a_{0}$. The normalization in (8) is needed to map the input to $[0,N]$, assuming the input is confined to $[-1,1]$. Nevertheless, as it will be seen later, the first linear transformation may map $\overline{z}$ to a different range, providing expressiveness to the network. Finally, the non-linear activation function $\sigma(\cdot)$ is approximated by the DCT with $Q/2$ coefficients. Without loss of generality, $g_{q}$ is assumed to be integrated in the DCT coefficient $F_{q}$. In (9) we assume the non-linearity to have odd symmetry, so that only the odd coefficients are retained. As it will be explained later on in Sec. IV-B and shown empirically in Sec. VI, this does not prevent the network from learning only odd activation functions. A total of $M_{0}+Q/2+1$ parameters per perceptron are to be trained, which only represents an increment of $Q/2$ coefficients with respect to a standard perceptron with a fixed activation functions. As mentioned in the previous section, $Q$ is small due to the energy compaction property of the DCT (e.g., around $Q/2=6$ coefficients). Fig. 1 shows the block diagram of the perceptron. The normalization is not shown and assumed to be a part of the non-linear transformation $\sigma(\cdot)$. Example 1 (Linear discriminant).: _Assume $M_{0}=2$, we want to discriminate when $x_{1}>x_{2}$. This can be framed as a classification problem, for which we can take $\mathbf{a}=a[0\,1-1]^{T}$, where $a$ is a positive constant. Then, any odd monotone increasing function will discriminate the two hypothesis. For the sake of simplicity, take $a=1$ and the non-linearity approximated with only one coefficient (i.e., $F_{1}=-1$). This results in_ \[z =x_{1}-x_{2} \tag{10}\] \[\overline{z} =\frac{N}{2}(x_{1}-x_{2}+1)\] (11) \[\hat{y} =-\cos\left(\frac{\pi(2\overline{z}+1)}{2N}\right)\approx\sin \left(\frac{\pi}{2}(x_{1}-x_{2})\right) \tag{12}\] _The solution in (12) provides a soft-decision, whereas a hard-decision would take the sign, i.e., $\text{sign}(\hat{y})$. While there are infinite solutions for this example, the proposed scheme only increases the complexity by one extra coefficient (i.e., $F_{1}$). Implementing a linear $\sigma(\cdot)$ would require many coefficients and provide no further benefit in terms of performance._ In Example 1 there are infinite minimums, all optimal. Nevertheless, in more complex problems (e.g., high-order discriminants) the expressiveness of a single perceptron is not enough. In light of the universal approximation theorem, in the Fig. 1: One perceptron with $M_{0}$ inputs and activation function $\sigma(\cdot)$. following we will increase the capabilities of the network by including a hidden layer of several neurons. This architecture, termed ENN, will be fully-adaptive as both the linear weights and the activation functions will be trained in a supervised fashion. ### _Expressive Neural Network (ENN)_ As a single layer perceptron limits the capabilities of the network, we will include another processing layer to increase its expressiveness. For a general multi-layer perceptron of $L$ layers, the following expressions show the perceptron signals at the $l$-th layer: \[\mathbf{z}_{l} =\mathbf{A}_{l}^{T}[1\ \mathbf{s}_{l-1}^{T}]^{T} \tag{13}\] \[\mathbf{\overline{z}}_{l} =\frac{N}{2}(\mathbf{z}_{l}+1)\] (14) \[\mathbf{s}_{l} =\sigma_{l,m}(\mathbf{\overline{z}}_{l}) \tag{15}\] for $l=1,\ldots,L$. The first input and last output correspond to $\mathbf{s}_{0}=\mathbf{x}$ and $\hat{y}=\mathbf{s}_{L}$, respectively. The matrix of linear weights is $\mathbf{A}_{l}=[\mathbf{a}_{i}^{(1)}\ \ldots\ \mathbf{a}_{i}^{(M_{l})}]$ and $M_{l}$ stands for the number of perceptrons at layer $l$, with $M_{0}$ being the number of inputs. The operations in (14) and (15) are performed element-wise. The AAF in the ENN, this is, the $m$-th element of (15) is computed as \[\mathbf{s}_{l}[m]=\sum_{q=1}^{Q/2}F_{q,l}^{(m)}\cos_{q}(\mathbf{z}_{l}[m]), \tag{16}\] where $F_{q,l}^{(m)}$ corresponds to the $q$-th coefficient of the $m$-th perceptron at the $l$-th layer. As explicitly shown in (15) and (16) the activation function is not necessarily the same at each neuron, although the number of DCT coefficients $Q$ is kept constant for the whole network. The last activation function could be substituted by an step function in the classification setup and a linear for regression. However, this will be maintained so that the network can self-adapt depending on the nature of the problem. As opposed to the $M_{0}$-dimensional DCT in Sec. III-A, the proposed architecture does not require implementing the DCT in $M_{0}$ dimensions and can be trained in a supervised fashion. Furthermore, the number of coefficients grows linearly as $M_{0}Q$. Throughout the rest of the paper we assume a two input variable, namely $M_{0}=2$, and $L=2$ layers. These layers are sequentially termed hidden and output layer. Fig. 2 shows the architecture of the two-layer perceptron with $M_{1}$ neurons in the hidden layer and expression (15) shows the input-output relationship. ### _Expressiveness of periodic activation functions_ As it will be shown in the experimental results, the expressiveness of the DCT comes from its periodic nature. Fig. 3 shows the sigmoid function and the corresponding DCT representation in the $[-1,1]$ range using $Q/2=6$ coefficients. While the sigmoid function saturates outside the range, the DCT approximation offers a periodic infinite non-linearity. This offers more capacity to the activation function, as the network may choose to work at an increasing range (e.g., $[-1,0]$), decreasing range (e.g., $[-3,-1]$), bump range (e.g., $[-1,3]$), valley range (e.g., $[-3,1]$), or even with several periods simultaneously (e.g., $[-5,3]$). While imposing odd symmetry in the DCT representation allows to have a continuous derivative, it does not constraint the resulting activation function to be odd. This Fourier model for non-linearities is the first one to provide such flexibility to activation functions, which is not possible with a fixed non-linearity not implemented with the DCT. To understand how the non-linear activation function maps the input to the output space, we use the concept of _bump_. Consider the perceptron in Fig. 1 with $M_{0}=2$. Notice that all input pairs satisfying the following equality are mapped to the same non-linear output $\sigma(c)$: \[a_{0}+a_{1}x_{1}+a_{2}x_{2}=c, \tag{16}\] where $c$ is a constant value. Fig. 4a shows how expression (16) corresponds to a line in the input space. Depending on the linear weights, only a certain range of the non-linearity is covered (or multiple ranges). In Fig. 4b it shown the mapping $\sigma(c)$. It also displays to what extent the function is used between dashed lines, this is, for a minimum and maximum $c$ Fig. 3: Sigmoid function (dashed) and its DCT representation in the $[-1,1]$ range (solid). Fig. 2: A 2-layer perceptron with $M_{1}$ neurons in the hidden layer. Fig. 4c shows the generated bump, by evaluating all the input data pairs through the linear transformation and the non-linear mapping. ## V Supervised Learning ### _Backpropagation rules_ Consider a dataset $\mathcal{D}$, consisting of $(\mathbf{x}_{i},y_{i})\in\mathcal{D}$, where the former is a $M_{0}$-sized vector training sample and the latter is the associated reference (i.e., label in classification, or function value in neuromorphic computing). Given $y_{i}$ and $\hat{y}_{i}$, the error can be computed and propagated throughout the network to adjust the learnable parameters. We assume the loss to be the mean squared error (MSE), \[\varepsilon^{2}=(y_{i}-\hat{y}_{i}(\mathbf{x}_{i}))^{2}, \tag{17}\] for both classification and regression problems. Notice in (17) we explicitly write the output as a function of the input data. Given a learnable parameter $w$, the chosen algorithm to update it is LMS. The update rule at a given iteration corresponds to \[w\gets w-\mu\nabla\varepsilon=w-\mu\varepsilon\left(-\frac{\partial\tilde {y}}{\partial w}\right), \tag{18}\] this is, the parameter is updated by the product of the error by the gradient of the output with respect to the parameter. The hyperparameter $\mu$ is the step size, which controls the convergence speed of LMS. We choose the simplest algorithm to learn the parameters in the neural network because the focus of this work is on the architecture, not on the algorithm. In this respect, the experimental results show that even using the instantaneous gradient, LMS converges to a minimum and outperforms state-of-the-art models. Thus, the LMS may be replaced by any other algorithm to speed convergence, which is out of the scope of this work. Due to the same reason, we do not include the iteration index in the parameter updates. For the 2-layer architecture in Fig. 2, there are 2 set of parameters per layer that need to be updated, namely, the DCT coefficients and the weights from the linear transformation. Starting at the output, the DCT coefficients of the layer $l=2$ are updated as \[F_{m,2}\gets F_{m,2}+\mu\varepsilon\frac{\partial\tilde{y}}{\partial F_{ m,2}}=F_{m,2}+\frac{4\alpha}{Q}\varepsilon\cos_{m}\left(z_{2}\right), \tag{19}\] for $m=1,\ldots,Q/2$. The superscript in the parameter is omitted because there is only one perceptron at the output. In general, the step size is modeled as twice the mismatch $\alpha$ divided by the power of the corresponding input. Since the $Q/2$ cosines used to approximate the non-linearity are orthonormal, the total power is $Q/2$. This exhibits another advantage of this parametrization as the power remains always constant. In order to update the linear weights from the output layer, the backpropagation procedure allows to propagate the error across the network with respect to previously computed derivatives, which makes it a very efficient algorithm. Therefore, these are updates as \[a_{2}[k] \leftarrow a_{2}[k]+\mu\varepsilon\frac{\partial\tilde{y}}{\partial a_{2}[k ]}=a_{2}[k]+\mu\varepsilon\frac{\partial\tilde{y}}{\partial z_{2}}\frac{ \partial\tilde{z}_{2}}{\partial a_{2}[k]}\] \[= a_{2}[k]-\frac{2\alpha}{P_{1}}\varepsilon\frac{\pi}{2}s_{1}[k] \sum_{m=1}^{Q/2}F_{m,2}(2m-1)\cos_{m}(z_{2}), \tag{20}\] for $k=0,\ldots,M_{1}$, and $P_{1}=\mathbf{s}_{1}^{T}\mathbf{s}_{1}$. The superscript has also been suppressed in this case. Accordingly, the DCT coefficients in the first layer are updated as \[F_{q,1}^{(k)} \leftarrow F_{q,1}^{(k)}+\mu\varepsilon\frac{\partial\tilde{y}}{\partial F_ {q,1}^{(k)}}=F_{q,1}^{(k)}+\mu\varepsilon\frac{\partial\tilde{y}}{\partial z _{2}}\frac{\partial z_{2}}{\partial s_{1}[k]}\frac{\partial s_{1}[k]}{ \partial F_{q,1}^{(k)}}\] \[= F_{q,1}^{(k)}-\] \[\frac{4\alpha}{Q}\varepsilon\frac{\pi}{2}a_{2}[k]\cos_{q}(z_{1} [k])\sum_{m=1}^{Q/2}F_{m,2}(2m-1)\sin_{m}(z_{2}), \tag{21}\] for $q=1,\ldots,Q/2$ and $k=1,\ldots,M_{1}$. And the linear combination of the first layer is updates as \[a_{1}^{(k)}[m] \leftarrow a_{1}^{(k)}[m]+\mu\varepsilon\frac{\partial\tilde{y}}{\partial a _{1}^{(k)}[m]}\] \[= a_{1}^{(k)}[m]+\mu\varepsilon\frac{\partial\tilde{y}}{\partial z _{2}}\frac{\partial z_{2}}{\partial s_{1}[k]}\frac{\partial s_{1}[k]}{ \partial z_{1}[k]}\frac{\partial z_{1}[k]}{\partial a_{1}^{(k)}[m]}\] \[= a_{1}^{(k)}[m]+\] \[\frac{2\alpha}{P_{0}}\varepsilon\frac{\pi^{2}}{4}a_{2}[k]s_{0}[ m]\sum_{p=1}^{Q/2}F_{p,2}(2p-1)\sin_{p}(z_{2})\] \[\sum_{q=1}^{Q/2}F_{q,1}^{(k)}(2q-1)\sin_{q}(z_{1}[k]), \tag{22}\] Fig. 4: Bump generation. Mapping of a linear combination of two inputs over the response of a non-linear function. for $m=1,\ldots,M_{0}$, $k=1,\ldots,M_{1}$, and $P_{0}=\mathbf{s}_{0}^{T}\mathbf{s}_{0}$. Without loss of generality, $s_{0}[0]=s_{1}[0]=1$, which correspond to the bias terms in the linear combination at each layer. Notice that in all the update rules, the derivative exists and inherits the periodic nature of the cosine transform. To enhance the stability of the convergence, the power of each input, namely $P_{0}$ and $P_{1}$, can be computed with a dampening effect. For instance, \[P_{0}\leftarrow\beta P_{\mathrm{o}}+(1-\beta)\mathbf{s}_{0}^{T}\mathbf{s}_{0} \tag{23}\] at each iteration. The $\alpha$ parameter is set smaller in the output layer, which intuitively reduces the propagation of artifacts to previous layers during training. ### _Initialization and Training Procedure_ Instead of learning all the parameters from scratch, we devise a sequence of training stages to speed up the convergence of ENN. For both classification and regression problems, the training procedure is split into two phases: * Phase 1: train the linear weights with fixed sigmoid activation functions. This is well-known to converge for a two-layer neural network. The difference with respect to the literature is that the activation function is defined using the DCT expression in (4), this is, the function is periodic. In the hidden layer, the linear weights are initialized to generate bump diversity. These weights can be interpreted as the orientation of the activation functions in the output space. See Fig. 5 with an example for $M_{1}=4$, with linear weights corresponding to $[0,0,1]$, $[0,1,0]$, $[0,1,-1]$ and $[0,-1,1]$ and a logarithmic compander approximated with $Q=12$ coefficients. Finally, the output layer is initialized with linear weights in a uniform distribution in $[-0.5,0.5]$. This allows to constraint the dynamics and show that the convergence does not depend on the initialization. * Phase 2: using the linear weights from Phase 1 as the initialization, train both the linear weights and the activation functions. ## VI Experimental results In the following we will test the ENN for both binary classification and regression problems. As benchmarks, we will test also the following models, in which they all have the same architecture and differ in the activation functions: * ReLU: it uses a fixed non-trainable ReLU, this is, $\sigma(z)=\max\left\{0,z\right\}$. * Sigm: it uses a fixed non-trainable sigmoid activation function. Notice that the output saturates for an input of large magnitude. * F-DCT: it uses a fixed non-trainable sigmoid activation function, but modeled with the DCT. Thus, the function does not saturate and it is periodic. Although not an odd function, we include the ReLU because it is the standard activation function in current neural network architectures. The Sigm model is also included to compare it with F-DCT and assess the gains of a periodic activation function. All architectures are built with $M_{1}=6$ neurons in the hidden layer, and ENN with $Q=12$ parameters in all the non-linearities. However, recall that only $Q/2$ coefficients are different from zero because we impose odd symmetry. Regarding the LMS setup, for the non-linearities we set $\alpha=10^{-3}$ in the hidden layer and $\alpha=10^{-4}$ in the output layer. For the linear weights we set $\alpha=5\cdot 10^{-5}$ in the hidden layer and $\alpha=5\cdot 10^{-3}$ in the output layer. We keep the step-size constant for all neurons in the same layer. The dampening parameter is set to $\beta=0.999$ for both layers. For both classification and regression problems, a synthetic dataset is generated. All samples come from uniform distributions in the $[-1,1]$ range for each input variable. The train and test sets contain 400.000 and 50.000 samples, respectively. For comparison fairness all the models are trained with the same sequences in all the setups. Notice that the output activation functions in ReLU and Sigm has to be specifically selected to be either sigmoid in classification or linear in regression. Conversely, in ENN it will be automatically adapted to approach the required function, which is an advantage of this architecture with respect to the benchmarks. ### _Classification_ Table I shows different classification problems that have been evaluated. It displays the ideal decision map along with the order of the discriminant. The metric chosen to compare the different models is the test accuracy, this is, the percentage of correctly classified samples in the test set. As expected, the complexity of the problem increases with the order of the discriminant. As seen in Example 1, a single neuron is sufficient to implement a linear discriminant. This goes to show that all models are capable of producing an almost perfect boundary. With no surprise, for quadratic and cubic discriminants all models are capable of finding the appropriate boundary. When moving to high-order classification Fig. 5: 4 different non-linearities with corresponding linear weights such that they are oriented in different directions. problems is when the non-adaptive models are not capable of providing a satisfactory solution. Conversely, the ENN model manages to find a local minimum with high accuracy and with very consistent results for a wide variety of problems. Fig. 6 shows the AAFs for the 8th classification problem. Notice that only two of out the six activation functions have been adapted, namely Fig. 5(a) and Fig. 5(e). The corresponding bumps are shown in Fig. 7, where the different orientations are exploited to generate diversity and, ultimately, non-linear boundaries. Another advantage of ENN is that it allows to model the network width, this is, the required number of hidden neurons. The bumps in Fig. 6(b) and 6(f) are identical, while the corresponding linear weights in the output layer have the same magnitude and opposite sign. This means that the network neglects the information that comes from these two branches. This also happens for bumps in Fig. 6(c) and 6(d). In conclusion, the network only needs two neurons to generate the non-linear boundary. However this is not the case for the non-adaptive models, which have a very small accuracy even with 6 neurons. ENN exploits the periodic nature of the Fourier representation to generate expressive non-linearities. Fig. 8 shows the AAFs for the 7th problem, along with the corresponding bumps in Fig. 9. As expected, the architecture exploits the different orientations and periodic nature of the DCT. Notice that there is almost no drop in accuracy between the ring and the face classification problem, although the latter is more complex. This is due to the expressiveness of the DCT, which allows to generate several peaks per bump. In other words, by allowing the activation to be non-monotone it can be used to approximate the ring and both circles still with 6 neurons. Fig. 10 shows the different decision boundaries for the 7th problem. Despite the accuracy of the non-adaptive models in Table I being around 80%, the quality is very poor. Only the ENN boundary resembles the ideal map. The output activation function converges to the same configuration for all classification problems and it is shown in Fig. 10(a). As anticipated theoretically, the learnt AAF approaches a step function. The fact of constraining the model to $Q=12$ coefficients makes it infeasible to generate an ideal step function. Nevertheless, this solution is steeper than a standard sigmoid function, which reduces the error at data close to the boundary and increases the accuracy. ### _Regression_ The network can also be trained for regression problems, this is, to approximate a function. Particularly, we compute bivariate scalar functions. Table II shows the MSE achieved by the four different models for three different experiments. Fig. 12 shows the different bumps for the regression problem $(x_{1}^{2}+x_{2}^{2})/2$. Notice that this time the network decides to work at different ranges of the sigmoid function. Particularly, Fig. 10(b) corresponds to a symmetric extension of the function, \begin{table} \begin{tabular}{c\|c c c c} \hline \hline $\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ }}}}}}}}}}}}}}}$ & ReLU & Sign & F-DCT & ENN \\ \hline $\frac{x_{1}+x_{2}}{2}$ & $2.9\cdot 10^{-3}$ & $2.1\cdot 10^{-2}$ & $1.9\cdot 10^{-2}$ & $\mathbf{1.3\cdot 10^{-4}}$ \\ $\frac{x_{1}^{2}+x_{2}^{2}}{4}$ & $2.2\cdot 10^{-1}$ & $2.2\cdot 10^{-1}$ & $2.3\cdot 10^{-1}$ & $\mathbf{1.4\cdot 10^{-4}}$ \\ $x_{1}x_{2}$ & $5.1\cdot 10^{-2}$ & $3.6\cdot 10^{-2}$ & $3.6\cdot 10^{-2}$ & $\mathbf{1.4\cdot 10^{-4}}$ \\ \hline \hline \end{tabular} \end{table} TABLE II: MSE in regression tasks. \begin{table} \begin{tabular}{c c\|c c c c} \hline \hline $\mathbf{\mathbf{\mathbf{N}}}^{\text{e}}$ & Map & Order & ReLU & Sign & F-DCT & ENN \\ \hline (1) & \multirow{2}{}{1} & \multirow{2}{}{99.97\%} & \multirow{2}{}{99.90\%} & \multirow{2}{}{99.84\%} & \multirow{2}{*}{99.85\%} \\ \cline{1-1} \cline{5-6} (2) & & & & & \\ \cline{1-1} \cline{5-6} (3) & & & & & \\ \cline{1-1} \cline{5-6} (4) & & & & & \\ \cline{1-1} \cline{5-6} (5) & & & & & \\ \cline{1-1} \cline{5-6} (6) & & & & & \\ \cline{1-1} \cline{5-6} (7) & & & & & \\ \cline{1-1} \cline{5-6} (8) & & & & & \\ \hline \hline \end{tabular} \end{table} TABLE I: Accuracy for different binary classification problems. For each case, the ideal map and the order of the discriminant are shown. Fig. 6: AAFs for the 8th classification problem. Fig. 7: Bumps corresponding to the AAFs in Fig. 6. while Fig. (f)f is intrinsically a bump. As expected, the output activation is adapted to the learning task and different from classification. As seen in Fig. (b)b the activation function approaches a linear function (in $[-0.25,0.25]$) for all regression problems. Notice that in a non-adaptive setting one needs to choose the last non-linearity according to the task. This represents an improvement with respect to non-adaptive architectures as it is general enough to encompass different tasks. ### _Effect of the number of coefficients_ Increasing the number of coefficients $Q$ in the DCT representation theoretically provides more expressiveness to the network. However, in practice, increasing the number of trainable parameters hinders the convergence of the algorithm because there are more local minima. Fig. 13 shows the map for the 7th classification problem trained with $Q=\{10,12,20\}$ coefficients. As expected, the accuracy improves from 10 to 12 coefficients, but it does not improve beyond that, probably because the algorithm gets trapped at a local minimum. To prevent this from happening, we first train the network with $Q=12$ and, once it converges, we allow the network to train up to $Q=20$ coefficients. We term this procedure _annealing_ because we slowly adapt the optimization space to avoid suboptimal local minima. Fig. 14 shows the output AAF for the cases Fig. (b)b and (d)d. Increasing the number of coefficients allows to create a steeper transition, reducing the number of errors around the boundary. Doing so increases the Gibbs effect, although it produces no harm to the learning process because those samples are far from the boundary and always correctly classified as either 1 or -1. ## VII Conclusions In this paper we have presented ENN, a novel neural network architecture with adaptive activation functions. Under a signal processing perspective we use the DCT to design the non-linear functions, whose coefficients can be trained using backpropagation. The formulation is general enough to encompass different learning tasks. Specifically, ENN is able to adapt the activation functions for classification and regression problems. A two-step training procedure is also proposed, which speeds up the convergence. We provide insights in the interpretability of the network by recovering the concept of bump, this is, the response of each neuron in the output space. Through extensive experiments we determine that the expressiveness of a neural network highly depends on the activation function. Particularly, the key strength of ENN is the periodic nature of the DCT, providing high accuracy for a wide range of non-linear classification problems. In some cases outperforming state of the art fixed activation functions up to 40% in accuracy.
2307.08549	G-Scan: Graph Neural Networks for Line-Level Vulnerability Identification in Smart Contracts	Due to the immutable and decentralized nature of Ethereum (ETH) platform, smart contracts are prone to security risks that can result in financial loss. While existing machine learning-based vulnerability detection algorithms achieve high accuracy at the contract level, they require developers to manually inspect source code to locate bugs. To this end, we present G-Scan, the first end-to-end fine-grained line-level vulnerability detection system evaluated on the first-of-its-kind real world dataset. G-Scan first converts smart contracts to code graphs in a dependency and hierarchy preserving manner. Next, we train a graph neural network to identify vulnerable nodes and assess security risks. Finally, the code graphs with node vulnerability predictions are mapped back to the smart contracts for line-level localization. We train and evaluate G-Scan on a collected real world smart contracts dataset with line-level annotations on reentrancy vulnerability, one of the most common and severe types of smart contract vulnerabilities. With the well-designed graph representation and high-quality dataset, G-Scan achieves 93.02% F1-score in contract-level vulnerability detection and 93.69% F1-score in line-level vulnerability localization. Additionally, the lightweight graph neural network enables G-Scan to localize vulnerabilities in 6.1k lines of code smart contract within 1.2 seconds.	Christoph Sendner, Ruisi Zhang, Alexander Hefter, Alexandra Dmitrienko, Farinaz Koushanfar	2023-07-17T15:11:03Z	http://arxiv.org/abs/2307.08549v1	# G-Scan: Graph Neural Networks for Line-Level Vulnerability Identification in Smart Contracts ###### Abstract Due to the immutable and decentralized nature of Ethereum (ETH) platform, smart contracts are prone to security risks that can result in financial loss. While existing machine learning-based vulnerability detection algorithms achieve high accuracy at the contract level, they require developers to manually inspect source code to locate bugs. To this end, we present G-Scan, the first end-to-end fine-grained line-level vulnerability detection system evaluated on the first-of-its-kind real world dataset. G-Scan first converts smart contracts to code graphs in a dependency and hierarchy preserving manner. Next, we train a graph neural network to identify vulnerable nodes and assess security risks. Finally, the code graphs with node vulnerability predictions are mapped back to the smart contracts for line-level localization. We train and evaluate G-Scan on a collected real world smart contracts dataset with line-level annotations on reentrance vulnerability, one of the most common and severe types of smart contract vulnerabilities. With the well-designed graph representation and high-quality dataset, G-Scan achieves 93.02% F1-score in contract-level vulnerability detection and 93.69% F1-score in line-level vulnerability localization. Additionally, the lightweight graph neural network enables G-Scan to localize vulnerabilities in 6.1k lines of code smart contract within 1.2 seconds. ## I Introduction The recent success of Bitcoin [25] and Ethereum (ETH) [8] has brought decentralized platforms, which allow users to transact and interact in a trustless manner, to the forefront of attention. Bitcoin, as the first generation blockchain platform, facilitates peer-to-peer digital currency transactions using proof-of-work consensus algorithms. However, due to the limited functionality of Bitcoin scripting languages, ETH and its smart contracts introduced a broader scenario where users can encode complex agreement terms directly into codes to enable automated and trustless transactions. ETH has been widely applied in various domains, from financial services to supply chain management and beyond. Developers in the open-source ETH platform write their agreements into smart contracts code using Solidity and publish them on the blockchain to facilitate transactions with third parties. Like any other programming language, vulnerabilities also reside in ETH contracts, potentially leading to loss of funds or theft of sensitive information. The security risks arise from the two main aspects: First, due to the distributed nature of blockchain platforms, malicious third parties may publish their smart contracts to exploit vulnerabilities for profit. Second, as a newly developed programming language, Solidity has limited research on formal verification to ensure the smart contracts are written properly. Even minor mistakes in smart contract code can result in significant financial loss. Many prior works explored vulnerability detection algorithms on Ethereum smart contracts, and most of them can be categorized into two types: (1) contract-level vulnerability detection and (2) line-level/node-level vulnerability detection. Contract-level vulnerability detection takes the detection as a classification problem and uses symbolic execution [21, 23, 24], fuzzing [12, 16, 35], and machine learning algorithms [19, 27, 37, 38, 43] to find vulnerable contracts. These methods achieve over 90% F1 scores in classification, but developers are still required to examine the contract line-by-line to localize buggy statements. More recently, SCScan [14] proposed a vulnerability detection system at the line level, while MANDO [26] proposed a node-level detection system with the aim of reducing the development burden. SCScan first uses a support vector machine (SVM) to detect vulnerable contracts and then a pattern matching algorithm to determine the exact positions of the vulnerabilities. However, SCScan is primitive which can only accept fix-size contracts as inputs to SVM, and the pattern matching algorithm fails to scale as new attacks appear. MANDO [26], on the other hand, only detected vulnerabilities in the code graph but failed to map them back to the source code. MANDO employs a two-stage graph neural network (GNN) training to identify node-level vulnerabilities, where the first stage trains a graph classification model to assess the vulnerability of contracts, and the second stage utilizes node embeddings updated from a topology graph neural network to classify whether a specific node is vulnerable. MANDO can handle contracts of varying sizes, but after getting the node-level predictions, it cannot map vulnerabilities back to the smart contract for localization. Notwithstanding that node classification is widely studied by the machine learning community, localizing vulnerable nodes within the code graph itself is non-trivial and challenging for the following reasons. First of all, there is a lack of real-world datasets for fine-grained1 vulnerability detection model training. MANDO [26] trains its GNN model on a simulated dataset by injecting vulnerabilities into clean contracts. These injections fail to represent some corner cases in real-world contracts and result in lower detection accuracy. Secondly, code graph construction in prior work [19, 20, 43] mainly extracts the semantic information in source code but fails to capture the code dependencies and hierarchies. Thirdly, as the contract grows, efficiently identifying vulnerable statements with regard to time and computation cost is crucial to save developers' time and resources. Footnote 1: throughout the paper, we use “localize” and “fine-grained detection” interchangeably. To tackle these challenges, we present G-Scan, the first end-to-end fine-grained line-level vulnerability detection system with evaluations on the first-of-its-kind real world dataset. Our G-Scan converts the contracts to code graphs by first getting their abstract syntax tree (AST) representations and adding extra edges to reflect data dependencies and code hierarchies. Then, it assigns node features by including function types like _While_ and _For_, variable properties like _visibility_ and _storage location_, and node member access like _send_ and _transfer_. Each node also processes a _src_ attribute which memorizes the lines where the nodes are extracted when constructing AST. G-Scan trains a GCN model on the converted graphs and learns the hidden features carrying the vulnerabilities. In the inference stage, G-Scan is used to predict the node labels of unseen code graphs and map back to the contract statements with _src_ attributes. We train and evaluate G-Scan on the first real-world fine-grained reentrancy vulnerability annotation dataset consisting of 13,773 smart contracts and achieve 93.02% F1-score in contract-level vulnerability detection and 93.69% F1-score in line-level vulnerability localization. Our contributions to the community are summarized as follows: * We present G-Scan, the first end-to-end algorithm that leverages graph neural networks for fine-grained line-level detection of smart contract vulnerabilities. * G-Scan is a scalable and efficient framework that is agnostic to contract size and localizes vulnerability within 1.2 seconds for contracts with more than 6.1k lines of code. * Proof of concept evaluation is demonstrated on the first-of-its-kind real world data collection consisting of 13,773 smart contracts and a total of 5,363,793 lines of code. Each data line is labeled to indicate whether they contain reentrancy vulnerabilities or not. * Our results show that G-Scan can achieve 93.02% F1-score in contract-level vulnerability detection and 93.69% F1-score in line-level vulnerability localization by incorporating variable dependencies and code hierarchies into graph modeling. In summary, G-Scan provides an efficient and precise fine-grained vulnerability detection solution to smart contracts. G-Scan also contributes the first-of-its-kind large-scale real word smart contract dataset with fine-grained reentrancy vulnerability annotations. Our novel graph representation and lightweight GNN-based node classification model trained on the real-world dataset enable line-level vulnerability detection with both efficiency and accuracy compared with prior arts. We will open-source the code along with our collected dataset to promote research in this area. Paper Organization: For the rest of the paper, we will introduce vulnerability localization background and challenges in Section II; the goal of G-Scan and the threat models in Section III; the detailed design pipeline of G-Scan in Section VI-Section V; the experiments demonstrating G-Scan's effectiveness in Section VIII; and the conclusion and future work in Section X. ## II Background and Challenges ### _Smart Contracts Vulnerability_ Smart contracts are self-executing programs in Ethereum that enables developers to create decentralized blockchain applications, including financial services. These contracts are written in Solidity, a high-level programming language, and compiled into Ethereum Virtual Machine (EVM) bytecode for deployment on the Ethereum network. Despite their benefits, smart contracts can also be vulnerable to security risks, which, if not identified in time, can lead to significant financial losses. In this context, we outline some of the vulnerabilities related to financial transactions, including the reentrancy attack, which is widely regarded as the most common and dangerous vulnerability (e.g., DAO [22]). Reentrancy Attack: The adversary performs reentrancy attacks [9] by repeatedly calling back into a vulnerable contract before the previous invocation has been completed. It allows the adversary to repeatedly siphon funds. Integer Overflow/Underflow Attack: The adversary performs overflow/underflow attacks [2] on vulnerable smart contracts doing mathematical operations on unsigned integers. It allows the adversary to transfer excessive funds from the overflow/underflow. Access Control Attack: The adversary performs access control attacks [1] on vulnerable smart contracts that do not properly control access to sensitive functions or data. It allows the adversary to steal sensitive private information from the vulnerable smart contract. ### _Graph Neural Networks_ Given a graph $G=(V,E,A)$ consists of a set of nodes $V=\{v_{1},v_{2},...,v_{n}\}$ and a set of edges $E=\{e_{1},e_{2},...,e_{m}\}$. For every node, $v_{i}\in V$ has a $k$ dimension feature vector $\mathbf{h}_{i}\in R^{k}$. The connections between nodes are deposited into the adjacency matrix $A$ where a nonzero number in row $i$ and column $j$ means a connection from node $i$ to node $j$. The objective of GNN is to learn a function $g$ that maps the feature embedding $\mathbf{h}_{i}$ of the $i$-th node $v_{i}$ to a new embedding vector $\tilde{\mathbf{h}}_{i}\in R^{h}$ which capture the node's local and global information. For a multi-layer GNN model, the mapping above is performed iteratively to help nodes update their feature embeddings with the information from their neighbors via message passing. For every node $v_{i}$, it performs message passing by receiving the feature embeddings from its neighbors $j\in N_{i}$, where $N_{i}$ is the set of nodes adjacent to $v_{i}$. The messages are then aggregated using a customized function to obtain an updated representation $\tilde{\mathbf{h}}_{i}$. The computations are formulated in Equation 1 where $f$, $g$, and $\oplus$ are customizable functions, e.g., convolution. Here, $\mathbf{h}_{v}^{(l)}$ and $\mathbf{h}_{u}^{(l)}$ are the node features at layer $l$, $\mathcal{N}(v)$ are the set of neighbors for node $v$, $\mathbf{e}_{uv}$ is the edge feature between nodes $u$ and $v$, and $\mathbf{h}_{v}^{(l+1)}$ is the updated features of node $v$ at layer $(l+1)$. \[\mathbf{h}_{v}^{(l+1)}=g(\mathbf{h}_{v}^{(l)}\underset{u\in\mathcal{N}(v)}{ \oplus}f(\mathbf{h}_{u}^{(l)},\mathbf{h}_{v}^{(l)},\mathbf{e}_{uv})) \tag{1}\] ### _Challenges_ To perform fine-grained line-level smart contract vulnerability detection, we identify the following challenges. Lack of Dataset: Many real-world contract-level vulnerability detection datasets [38] are open-sourced for scientific research. However, these datasets lack line-level annotations that pinpoint the exact location of vulnerabilities. It brings challenges to the fine-grained detection model training, as high-quality data, like real-world contracts with annotations, is essential for the machine learning models to learn hidden patterns and achieve optimal performance. Graph Representation: The second challenge is representing code graphs with both semantic and structural information. Previous approaches such as MANDO [26] and DR-GCN [19] extract heterogeneous code graphs based on semantic information from the source code, such as critical function calls/variables and temporal execution trace. However, they did not consider the structural information that indicates the code execution orders and relationships between data elements, which are where many vulnerabilities originate. Apart from the edge connections, designing node feature embedding representations that depict function or expression types and properties helps GNN models learn better local and global code information. Therefore, developing appropriate graph representations is crucial in improving fine-grained vulnerability detection model performance. Efficiency: Achieving efficiency in terms of inference time and cost is also challenging as the size of contracts scales. For time efficiency, line-level detection algorithms like MANDO [26] use a two-level detection algorithm, where it first classifies if a smart contract is vulnerable and then uses the node embeddings obtained from a topology graph neural network for line-level vulnerability classification. However, performing inference over three multi-level graph neural networks results in significant localization overheads. For cost efficiency, prior arts detect contract-level vulnerability via DNN based models [13, 15, 27] are parameter-heavy and require significant computation resources when contract users intend to verify if a specific contract is vulnerable. Mapping between Node and Statement: Another challenge is establishing the relationship between nodes in the code graph and statements in the smart contract. This is because localizing which line is vulnerable relies heavily on identifying the node representing the code block in the graph representation and accurately mapping it back to the corresponding line. While prior work such as MANDO [26] classifies vulnerable nodes within code graphs, they did not explicitly mention how the vulnerable nodes are mapped back line by line. In contrast, G-Scan proposed the first end-to-end vulnerability localization system by leveraging AST trees, which enables us to accurately map vulnerable nodes back to specific lines in the smart contract. ## III Goal and Threat Model In this section, we will first introduce the goal of our proposed G-Scanand then introduce the potential threat models that our G-Scan aims to defend. ### _Goal_ Our goal is to localize the vulnerabilities within smart contracts. While many works [19, 27, 37, 38, 43] have used machine learning models to detect contract vulnerabilities, few explored how to localize them. Nevertheless, it is crucial to accurately and efficiently identify which line contains the vulnerability for both developers and end-users. For developers, identifying the specific line of code containing the bug, instead of just contract-level detection and examining the code line-by-line, can improve working efficiency. Smart contracts can be inevitably complex, with intricate user agreements and transaction logics, making debugging a challenging task, even when developers know that vulnerabilities exist in the codebase. G-Scan helps the developers to focus on fixing the risky part instead of changing the entire codebase, which saves time and resources. For end users, with G-Scan, fine-grained vulnerability detection with low inference overhead opens a more transparent door to facilitate them to analyze security concerns within contracts and avoid transactions with malicious parties. In summary, G-Scan contributes to improving the security of the transaction life-cycle from the following two aspects: * Before publishing the smart contract, with G-Scan, one can analyze the vulnerabilities within the smart contracts at line-level and fix them in time. * When making transactions with untrusted third parties, with G-Scan, one can analyze the vulnerabilities within the smart contracts efficiently without costing too much computation resources. ### _Threat Model_ As depicted in Figure 1, a Solidity developer first publish a vulnerable contract to the blockchain. Then, an adversary exploits the vulnerabilities by publishing an attacking contract to call the vulnerable contract to steal funds. G-Scan safeguards smart contracts by helping Solidity developers identify and fix possible security risks promptly. Adversary's Objective The adversary is the malicious party in the blockchain and attempts to perform an attack on Fig. 1: Attack Scenario. When solidity developer publishes a vulnerable contract to the blockchain, the adversary leverages another attack contract to exploit the vulnerabilities in the vulnerable contract and steal funds from the developer. the Ethereum platform. In this paper, we use the reentrancy attack as an example, but note that G-Scan can be adapted to detect any other vulnerability types. In the reentrancy attack, the adversary deploys an attacking contract designed to exploit the vulnerabilities in the vulnerable contract. The vulnerable contract contains a function that allows it to transfer funds to other contracts. The adversary adds itself as one of the vulnerable contract recipients by first depositing the vulnerable contract funds and then withdrawing them. During the withdrawal execution, the attack contract repeatedly reenters the vulnerable contract and calls the transfer function multiple times until all funds are stolen. In Solidity, the above fund transfer is implemented by the following three subtype data structures, where $<$CA$>$ indicates the recipient's contract or account address and x describes the amount of fund to be transferred. * call: $<$CA$>$.call$\{$value: x$\}$ * transfer: $<$CA$>$.transfer$($x$)$ * send: $<$CA$>$.send$($x$)$ The difference among the three subtypes mainly arises from their implementations: Firstly, the upper transaction limit of transfer or send is 2,300 gas (a unit to measure funds in ETH platform), while call has no such limitations. Therefore, if an adversary exploits the call subtype, it can result in a greater loss of funds. Secondly, call cannot automatically handle exceptions thrown by the called contract, which increases the potential for reentrancy attacks. Thirdly, due to the transaction limitations in transfer and send, some developers may opt to use call without mitigating the reentrancy vulnerabilities. However, possible reentrancy attacks can still be performed on transfer and send for several reasons. Firstly, if these two subtypes requests call vulnerable contracts for funds exceeding the upper limit, exceptions can still happen, and if not handled properly, reentrancy attacks can be executed to steal funds. Secondly, if Ethereum Virtual Machine is upgraded and the maximum fund transaction limit changes, reentrancy attacks can still bring fund loss. Additionally, transfer will throw an exception if gas is depleted, leading to a state change reversal. Adversary's Capacity We consider the most general attack case where the adversary is one of the contract creators. He or she has access to the public data structure on the chain and can upload his or her contract to the Ethereum system but does not have access to the detection of G-Scan. ## IV Graph Representation In this and subsequent sections, we will first present how smart contracts are converted into code graphs, how to train GNN models to classify nodes as vulnerable or not, and then introduce how we collected our reentrancy vulnerability dataset, along with detailed statistics. A more general pipeline of our G-Scan is shown in Figure 2. We begin by converting the smart contract into a code graph through AST representations that extract code structural information. We then add extra edges and node feature embeddings to reflect code hierarchies. Next, we train a GNN-based node classification model on the code graphs to classify vulnerable nodes and clean nodes. The training is performed on a collected real-world smart contract dataset with each data line annotated as either reentrancy vulnerable or not. The code graph representation construction consists of three steps, namely, AST generation, code graph edge generation, and node feature embedding generation. We will also illustrate how the code graph is constructed by giving an example in the end of this section. ### _AST Generation_ We use Solidity compiler [10] to convert the smart contract into tree format, where the node represents syntactic constructs in the source code, such as statements and expressions, and each edge represents relationships between these constructs, such as function calls and value assignments. The nodes within AST have two sets of node attributes: node type and source src. The node type describes the functionality of the source code, while the source src stores which part of the source code the node comes from. The edges in AST are not explicitly defined but are generated by the nested child nodes within each node. ### _Code Graph Edge Generation_ In addition to the edges being implicitly defined by the hierarchically nested nodes in the AST, we also add the following edges to represent the structure and semantics of the underlying source code. These edges aim to obtain the representation by adding control flow, hierarchy between the nodes, and existing data dependencies between the nodes into the code graph. Specifically, we add the following edges into the code graph: (1) AST Hierarchy Edges; (2) Control Flow and Ordering Edges; (3) Reference Edges; (4) Branching Edges; (5) Loop Edges; and (6) Break, Continue, and Return Edges. After adding these edges to the code graph, we convert it into a directed homogenous graph for future classification in Section V. AST Hierarchy Edges: The first type of edges are the hierarchy edges generated by AST. Each node in the AST can have other child nodes defined in its node types, meaning the node and its child nodes have hierarchical relationships. As shown in Figure 3, for example, for node Block, it has node types related to statements 1 to statements k and Block is the parent node of these child nodes. To represent the parent-child relationship in the code graph, we add two edges, one from the parent to the child and vice versa. Control Flow and Ordering Edges: Apart from the hierarchy information defined by AST, we also add the control flow edges indicating the relationship between statements in the child nodes. There are three node types containing function statements, namely, Block, UncheckedBlock, and YulBlock. The function statements in the node types are given by the order they are executed in the original smart contracts. As shown in Figure 3, for example, if a Block contains k statements, in addition to the AST edges added to indicate their hierarchy information, we also add edges between statements to explain the order they are executed. When a given node in the AST has the same child node belonging to two or more node types, ordering edges are introduced to distinguish each connection. For example, node type Mapping connects keys node $A$ to values node $B$, where $A$ and $B$ can be either the same or different. In its AST representation, the key node connects with the value node by attribute keyType defined in key node type. The value node connects with its key node with valueType defined in the value node type. When building AST, the connection is replaced by the undirected AST edges, which is hard to represent the mapping relationship between keyType and valueType. Therefore, we add an ordering edge to from keyType node to valueType node to represent the mapping relationship. Apart from Mapping, we summarize other node types which also have similar ordering problems and how to add ordering edges in Table I Reference Edges: To ensure two smart contracts with the same functionality but different function or variable names have the same code graph, we introduce reference edges. At variable level, AST introduced declaration node and assigned it a unique identifier attribute id. If the variables are again used in the smart contracts, the identifier nodes representing used variables will call the declaration node. To reflect the data dependency between the usage of the same variable without a variable name, we introduce a new edge type called reference edge. The identifier node types are linked with their declaration node by the reference edge, which is always added in both directions, from the identifier node to the declaration node and vice versa. Functions are treated in the same manner as variables, where reference edges are added from the identifier node to its declaration nodes and vice versa. Branching Edges: In the code graph representation, branching code blocks are only mapped to child nodes without further representation of the contexts between child nodes. To handle the branching, we add edges between the nodes representing the condition, the true branch, and the false branch. Figure 4 shows how branch conditions are handled for different functions. In the case of If statements and Conditional statements, three child nodes connect to the statement. One node represents the condition and connects the statement with control flow edges, one represents the true branch and connects the condition with true body edges, and the other represents the false branch and connects the condition with false body edges. \begin{table} \begin{tabular}{l l l} \hline \hline Node Type & Start & End \\ \hline IndexAccess & baseExpression & indexExpression \\ IndexRangeAccess & baseExpression & startExpression \\ startExpression & endExpression \\ FunctionCall & arguments & expression \\ FunctionTypeName & parameterTypes & returnParameterTypes \\ Assignment & leftHandSide & rightHandSide \\ BinaryOperation & leftHandSide & rightHandSide \\ FunctionDefinition & parameters & returnParameters \\ VulFunctionDefinition & parameters & returnVariables \\ \hline \hline \end{tabular} \end{table} TABLE I: Ordering edge directions Fig. 3: Control flow edges between k statements inside a block Fig. 2: G-Scan overview. First, we obtain the graph representation of the smart contract by converting it into an Abstract Syntax Tree (AST) representation and adding additional edges and node feature embeddings to reflect code hierarchies and dependencies. Next, we train a Graph Neural Network (GNN) model on the code graphs to classify vulnerable nodes and perform inference on unseen smart contracts. We train and evaluate the GNN model on a collected real-world smart contract dataset, with each data line annotated to indicate whether it contains reentrancy vulnerability. For If statements in Yul, which only have a true branch and ignore the statements in the true body for false conditions, we drop the false body edges. In the case of Switch statements in Yul, which consist of a switch statement with expressions and several case statements corresponding to the expressions, the switch statements are mapped by the AST edges between the node type YulSwitch and case statement nodes of type YulCase. For Case statements in Yul, AST edges are added between the YulCase node and the two child nodes at the node properties body and value. Additionally, a true body edges is added from value to body to reflect the body execution condition. Loop Edges: We also include four loop edges that indicate variable dependencies in the code graph. The four loop types in Solidity are for, do-while, while, and for loop of Yul. Figure 5 shows how the loop edges are added to the code graph. For the For loop, there are four child nodes in the loop statement. When entering the loop, an initialization statement with loop parameters is executed and saved in the pre node. The initialization statement connects to the for statement via a control flow edge. Then, the loop condition is evaluated by the condition node by adding a control flow edge between the pre and condition nodes. If the evaluation result is false, the loop is exited, as reflected by a false body edge inserted from the condition node to the ForStatement node. If the evaluation result is true, the body node is executed, and a true body edge is added from the condition node to the body node. When the execution is finished, the loop is updated, and the loop condition is re-evaluated. To reflect this, we add two control flow edges, one from body node to the loopExpression node and one from the loopExpression node to the condition node. In Do while loop and While loop, the body node is executed first, and then the condition is evaluated. Therefore, we add one false body edge from condition node either to the DoWhileStatement or to the WhileStatement node; and one true body edge from condition to the body node, and a control flow edge from the body to the condition node. We also add a control flow edge from either DoWhileStatement node and WhileStatement node to the condition node meaning the entry point of the while loop. For the For loop of Yul, we add edges similar to the For loop. However, the attribute post gives the loop update here instead of the attribute loopExpression. Break, Continue, and Return Edges: We also consider six node types that are related to leaving or continuing a loop: Break, Continue, and Return, as well as their Yul counterparts YulBreak, YulContinue, and YulLeave. For Break and YulBreak, we add a control flow edge from the break statement node to the loop node, indicating the break enforces the node to leave the loop. For Continue and YulContinue, we add a control flow edge from the continue statement node to the loop update node to show that the loop variable has been updated. In the case of YulLeave or Return, we use control flow edges from the leave statement node to the function definition node to represent the loop has exited. ### _Node Feature Embedding Generation_ The nodes in the code graph are derived from the AST, which provides a list of attributes that describe the main functionalities of each node and which part of the source code it comes from. Empirically, [6] has identified 73 node types, but using one-hot encodings of all 73 types can introduce unnecessary computation overhead since some types have similar meanings and functions. Additionally, certain attributes may contribute more to the vulnerability than others. Fig. 4: Graph structure of the node types referring to branchings Fig. 5: Graph structure of the four loop types To address these issues, we merged similar node types and created a 29-dimensional feature embedding. The first 20 dimensions capture information about different function or variable node definitions, the next eight dimensions describe node properties, and the final dimension indicates Solidity member access, specifically whether it involves send or transfer. Table II displays some of the dimension information, where the first column denotes which dimension it comes from, the second column denotes the value of the specific node type it corresponds to in the third column, and the last column is additional descriptions. For the complete list of 29-dimensional vectors, please refer to the appendix. Example We demonstrate the process of generating the code graph shown in Figure 4(d) from the Solidity code below. The function's purpose is to withdraw a specified amount ( _amount) of money from the ExampleYul contract and transfer it to a specified target address (_to). ``` pragmasolidity~0.8.19; contractExampleYul{ functionwithdrawYul(address_to,uint256 -amount)externalpayable{ assembly{ for(leti:=0)lt(i,2)(i:=add(i,1)){ //loopbody } } //othersoliditytext } } } ``` In the Yul Loop, variable $i$ is initialized to 0 using let i:= 0; if the condition lt(i, 2) is satisfied, we increment $i$ by 1 using i:=add(i, 1). When $i$ satisfies the condition, the loop body is executed. To generate the code graph from this smart contract, we extract the following nodes from the AST: YulForLoop, which defines the loop function; pre, which initializes the variable $i$; condition, which determines when to exit the loop in lt(i, 2); and post, which updates $i$ using i:=add(i, 1). The loop body is represented by the body node. Next, we add the following edges to the code graph. The first set of blue edges represents the parent-child relationships in the AST. Then, we add black control flow edges to indicate how the code is executed. First, the pre node initializes the variable $i$, followed by checking if $i$ satisfies the loop condition node condition. When the loop body represented by the body node is executed, we update $i$ following post, and then check if the loop condition node condition is still satisfied. The red and green loop edges connect the control flow edges. If the condition node is true, a true body edge goes from the condition node to the body node to continue the loop. If the condition node is false, a false body edge goes from the condition node to the YulForLoop node to exit the loop. Since there are no branch, return, variable, and function definitions in the code, we do not add these edges to the code graph. ## V GNN Training and Inference In this section, we present how we use GNN models to classify vulnerable nodes within code graphs. We train a GNN-based node classification model on the code graphs with annotated ground truth to predict vulnerabilities. The details of how we acquire the dataset are summarized in Section VI. ### _Node Classification Training_ The objective of the node classification model is to determine whether a given node is vulnerable. We train a seven-layer GCN model to predict the node label, where $1$ means the node is vulnerable and $0$ means the node is clean. The GNN model is trained by minimizing the cross-entropy between predicted labels and ground truth, as shown in Equation 2. $N$ is the total number of classes, $G_{i}$ is the binary indicator (0 or 1) if class $i$ is the correct classification for this code graph, and $P_{i}$ is the predicted probability that class $i$ is the correct classification for this code graph. \[\mathcal{L}(G,P)=-\sum_{i=1}^{N}G_{i}\log(P_{i}) \tag{2}\] The detailed model architecture is summarized in Table III, which takes the contract code graph as input and predicts the binary labels for each node. The GNN model consists of seven GCN layers and a ReLU activation function after each layer. Then, the updated feature embeddings are passed through three linear layers, followed by a Softmax activation function to generate binary label masks. ### _Node Classification Inference_ In the inference stage, an unseen smart contract is first converted into an AST representation using the Solidity compiler. Then, we add additional edges and node features to form a code graph following the process outlined in Section IV. Next, \begin{table} \begin{tabular}{c c l l} \hline \hline Dim & Values & Node Type & Description \\ \hline \multirow{4}{}{0} & 1 & DoWhileStatement & \multirow{4}{}{Loop information} \\ & 2 & WhileStatement & \\ & 3 & ForStatement & \\ & 4 & YulFor.op & \\ \hline \multirow{4}{}{20} & 1 & 1 & ‘internal’ & \multirow{4}{}{Different values of visibility} \\ & 2 & ‘external’ & \\ & 4 & ‘public’ & \\ & 5 & unknown value & \\ \hline \multirow{2}{}{28} & 1 & ‘transfer & \multirow{2}{}{Member access of attribute memberName} \\ & -1 & ‘send’ & \\ \hline \hline \end{tabular} \end{table} TABLE II: Description of the node feature vectors \begin{table} \begin{tabular}{c c c c} \hline \hline Layers & Input Size & Output Size & Activation Function \\ \hline GCNcm0 & 29 & 500 & ReLU \\ GCNcm1 & 500 & 500 & ReLU \\ GCNcm2 & 500 & 500 & ReLU \\ GCNcm3 & 500 & 500 & ReLU \\ GCNcm4 & 500 & 500 & ReLU \\ GCNcm5 & 500 & 500 & ReLU \\ GCNcm6 & 500 & 500 & ReLU \\ Linear1 & 500 & 300 & ReLU \\ Linear2 & 100 & 2 & Softmax \\ \hline \hline \end{tabular} \end{table} TABLE III: GNN architecture the code graph is fed into the trained GNN model for inference. The final Linear2 layer in Table III predicts the labels for each node. Once we obtain the predicted node labels in the code graph, we map them back to the original smart contract using the node's src attributes, which enable us to determine which line the node corresponds to. If a node is labeled as vulnerable, the corresponding line is also marked as vulnerable. ## VI Data Collection This section presents how we collect smart contracts and annotate the reentrancy vulnerabilities on each data line. We also provide visualizations of the dataset statistics. ### _Dataset Construction_ Our ETH smart contracts are constructed using a previous version of the SmartBugs Wild Dataset [17] called Peculiar [38]. The dataset construction process involved two stages. In the first stage, we clean the collected smart contracts and split them into training, validation, and test subsets. In the second stage, we annotate each data line in the smart contracts to identify the presence of reentrancy vulnerabilities. Preprocessing Although the previous Peculiar dataset underwent cleaning, the duplication rate remains over 50%. These can occur in two ways: (1) one contract may have been copied from another by adding a few white lines or comments; and (2) one contract may have been copied from another, with changes only made to variable names, function names, or variable values. Moreover, many contracts lacked their reference files, making it impossible to construct an AST tree. To address these issues, we adopted a two-step approach. First, we generated an Abstract Syntax Tree (AST) tree from the Solidity compiler, which removed white space characters and assigned comment lines to an ignored node type. Next, we constructed the Solidity code graph by leveraging information on the smart contract code structure and the AST tree (refer to Section IV for details on graph construction). This step removed intermediate values, as well as variable and function names. After obtaining the code graphs, we grouped the smart contract codes based on the similarity of their sha256 hashes. We selected one representative contract from each group, resulting in a total of 22,237 smart contracts, which are less than half of the original dataset containing 46,057 contracts. Annotation In the Peculiar dataset [38], reentrancy vulnerability was only annotated at the contract-level for the call subtype. To achieve more accurate line-level annotation, we expanded the annotation from the call subtype to include all three subtypes: call, send, and transfer. Contracts that did not include these three subtype functions were deemed non-vulnerable. In the remaining contracts, we manually inspected the data lines containing the three subtype functions to determine if a state change was made after the money transfer, and if there was no reentrancy lock in place. If these conditions were met, we annotated the lines as vulnerable. We then mapped these annotations to the code graphs. Each node in the code graph was assigned an attribute src, indicating the original smart contract line from which the node was derived. If the corresponding code line was marked as vulnerable, the node was labeled as such; otherwise, it was labeled as non-vulnerable. The final reentrancy vulnerability dataset only contains smart contracts with three subtype functions with 13,773 smart contracts. ### _Dataset Statistics_ The dataset is split into training, validation, and test sets, with details provided in Table IV. In the table, the acronym VG represents the number of vulnerable graphs, which corresponds to the number of contracts in each dataset. Similarly, the acronym VN represents the total number of vulnerable nodes within the corresponding Solidity code graphs. In addition to the statistical information, we also present visualizations to demonstrate the code graph data distribution. Figure 6 displays the code length distribution of smart contracts, as well as the nodes and edges distribution of each code graph. The first and second figure provides insights into the size and complexity of the code graphs, as measured by the number of nodes and edges, respectively. The third figure shows the distribution of contract length in terms of the number of lines of code. ## VII Implementation In this section, we present the general workflow for implementing G-Scan at both training and inference time. ### _G-Scan Training Implementation_ The pipeline for our approach consists of three main steps: (1) constructing an AST from the source files; (2) converting the AST into a code graph with vulnerability labels; and (3) training a GNN-based node classification model. In the following subsection, we provide a detailed description of how each step is implemented. AST Construction We construct ASTs from source code files using Solc [10]. The AST construction requires the smart contracts in one file. Therefore, if the smart contract is written in multiple files, we merge them into a single file before performing AST conversion. The following is a sample Solidity program, where x.x.xx represents the Solidity version in which the contract was written. It is worth noting that ASTs generated by Solc versions below 0.4.12 are not compatible with newer versions, and only a small number of contracts are written in these older versions. Therefore, we did not include them in our dataset collection. \begin{table} \begin{tabular}{c c c c c c} \hline \hline Subtype & Dataset & VG & non-VG & VN & non-VN \\ \hline \multirow{3}{}{Call} & training set & 573 & 11,040 & 28,900 & 11,468,676 \\ & validation set & 121 & 959 & 5,430 & 1,447,293 \\ & test set & 118 & 962 & 6,603 & 1,401,601 \\ \hline \multirow{3}{}{Send} & training set & 1,067 & 10,546 & 61,508 & 11,436,068 \\ & validation set & 200 & 880 & 10,678 & 142,045 \\ & test set & 226 & 854 & 13,207 & 1,394,997 \\ \hline \multirow{3}{}{Transfer} & training set & 1,572 & 10,041 & 83,469 & 11,414,107 \\ & validation set & 320 & 760 & 16,782 & 1,435,941 \\ \cline{1-1} & test set & 346 & 734 & 18,213 & 1,389,991 \\ \hline \hline \end{tabular} \end{table} TABLE IV: Dataset statistics on each subtype, VG represents vulnerable graphs number, VN represents vulnerable nodes number ``` pragmasolidityx.x.xx; ``` We then convert the line-level annotations from smart contracts to AST node labels, as shown in the annotateLabel function below. It takes AST's node attributes src list (node2sourceCode), smart contract's source code annotation list (GTlabel), and the source code at byte level (codeBit) as input. The GTlabel variable stores which line in the original smart contracts are vulnerable. ``` defannotateLabel(node2sourceCode;GTlabel;codeBit); labels=[] ifsum(GTlabel)==0: return[0]len(node2sourceCode) forsrcimnode2sourceCode: src=src.split(";") sBit=codeBit[:int(src[0])] eBit=codeBit[:(sBit+int(src[1]))] start=len(sBit.decode("utf-8").splitlines()) end=len(eBit.decode("utf-8").splitlines()) codeLine=[range(start,end+1,1)] lineLabel=[GTlabel[i]forimcodeLine] ifsum(lineLabel)>0: labels.append(1) else: labels.append(0) returnlabels ``` If there are no vulnerable annotations in the current smart contract, we return a set of 0 to indicate that the contract is clean. However, if the contract has vulnerable annotations, we proceed to analyze each node in the AST representation to obtain node-level labels. Since the src attribute in the AST representation points from each node to the corresponding position in the original contract at the byte level instead of the line number, we need to decode the code byte intoutf-8 format to determine the exact line number. We calculate the decoded line length and use this information to label each node as vulnerable (labeled as 1) if one of the corresponding lines is marked as vulnerable in the GTlabel. If none of the corresponding lines are marked as vulnerable, the node is labeled as not vulnerable (labeled as 0). Code Graph Generation* The ASTs are then converted to code graphs as described in Section IV. To generate the code graph, we use the following functions: ``` fromutilsimportdicToNodes fromgraph_constructimportcreateReferenceEdges, createNodeFeatureVectors graph=dicToNodes(AST) graph=createReferenceEdges(graph,AST) graph=createNodeFeatureVectors(graph,AST) ``` The dicToNodes function is used to extract nodes and edges from the AST and form the initial code graph. The AST object contains a dictionary with node mappings to the source code, as well as node and edge lists, edge types lists, and a list indicating breaks, continues, or returns in substructures. The graph object, on the other hand, includes a node mapping from the code graph id to the AST node id, as well as node and edge lists. The resulting code graph is stored in a PyG data object to facilitate future GNN training. After creating the initial AST tree, we use the createReferenceEdges function to add additional edges to the code graph as described in Section IV-B. We also add additional node features to the code graph using the createNodeFeatureVectors function, which takes the code graph and AST node properties as inputs and adds node features as described in Section IV-C. GNN Training The GNN model for node classification is trained with the code graphs in the training set and evaluated with the code graphs from the validation and test set. We add additional training details, including the hyperparameters in Table V. The architecture information is summarized in Table III. ### _G-Scan Inference Implementation_ We utilize the trained GNN for node classification to predict the location of vulnerabilities. As with the training process for G-Scan, we first convert the source code file into \begin{table} \begin{tabular}{l c} \hline \hline Variables & Settings \\ \hline Epoch & 1600 \\ Batch size & 100 \\ Optimizer & adam \\ Learning rate & 0,0001 \\ Loss function & cross entropy \\ \hline \hline \end{tabular} \end{table} TABLE V: GNN training hyperparameters Fig. 6: G-Scan dataset visualization on code graph node number, edge number, and smart contract lines of code an AST representation using the Solidity compiler, and then transform it into a code graph representation without additional labels indicating vulnerability. We then feed the code graph into the GNN model for node classification inference and obtain a binary mask that measures whether each node is vulnerable or not. If none of the nodes are vulnerable, we predict that the contract is clean. Otherwise, we predict which nodes are vulnerable. After obtaining the node prediction results, we convert them back to code lines using the src node property and mark the vulnerable lines on the source code file. The process of code graph generation and inference are the same as described in Section VII-A. Here, we will focus on the mapping from code graph to the original smart contract. As illustrated below, mapping the node-level predictions back to the smart contract can be viewed as a reverse process of AST construction. We use the predictLabel function, which takes in the node predictions (NodeLabel), the source code at the byte level (codeBit), the source code length (length), and the AST's node attributes src list (node2sourceCode) as inputs to generate the line-level predictions (LineLabel). ``` defpredictLabel(node2sourceCode,NodeLabel,codeBit, length): labels=[0]len(length) ifsum(NodeLabel)==0: return[0]len(length) fori,srcinenumerate(node2sourceCode): src=src.split(":") sBit=codeBit[:[int(src[0])] cBit=codeBit[:(sBit+int(src[1]))] start=len(sBit.decode("utf-8").splittimes()) end=len(eBit.decode("utf-8").splittimes()) codeLine=[range(start,end+1,1)] ifNodeLabel[i]>0: labels[codeLine] = 1 else: labels[codeLine] = 0 returnlabels ``` ## VIII Experiments ### _Experiment Setup_ G-Scan is implemented with Python 3.8.10 and benchmarked on Linux Mint 20.3. Our workflow is built upon PyTorch [34] version 1.9.0 and PyG [32] version 2.0.4. The GNN model is trained on NVIDIA A16 graphic card consisting of four GPUs each with 16 GB RAM, and 4 CPUs with 128 GB RAM. ### _Dataset_ To evaluate the performance of various vulnerability detection models, we employed the following three datasets: Peculiar Dataset[38]: This dataset consists of 46,057 smart contracts with contract-level reentrancy vulnerability annotation for subtype call. G-Scan Dataset: Our collected dataset of 13,773 contracts with line-level reentrancy vulnerability annotations for subtypes call, transfer, and send, as described in Section VI. Simulated Dataset[26]: This dataset includes 31 smart contracts related to reentrancy vulnerability used by MANDO as their test set. The contracts consist of synthetic samples created by injecting vulnerabilities into clean smart contracts and public smart contracts. All samples include line-level annotations. ### _Evaluation Metrics_ In the experiment, we benchmark the model performance at both line-level and contract-level. For line-level classification, we use G-Scan to determine if a node is vulnerable. For contract-level classification, after obtaining the classification result of each node, if there is a vulnerable node, we classify the contract as vulnerable; otherwise, it will be classified as non-vulnerable. In order to evaluate the performance of different models, we compared their classification results with the annotated ground truth from Sec.VI, and computed the confusion matrix at both the line-level and contract-level, including True Positive (TP), True Negative (TN), False Positive (FP), and False Negative (FN). We then used the following metrics to evaluate the performance of the models based on the confusion matrix: Accuracy (A): The proportion of correctly identified graphs or nodes, which is formulated as $\frac{TP+TN}{TP+FP+TN+FN}$. Recall (R): The proportion of correctly identified vulnerable graphs or nodes, which is formulated as $\frac{TP}{TP+FN}$. Precision (P): The proportion of identified vulnerable graphs or nodes that are actually vulnerable, which is formulated as $\frac{TP}{TP+FP}$. F1 score: A harmonic mean that combines precision and recall, which is formulated as $\frac{2(P+R)}{(P+R)}$ ### _Baselines_ For contract level classification, we compare G-Scan's performance with the following baselines on Peculiar Dataset [38]. For line level classification, we compare G-Scan with MANDO on its simulated dataset. MANDO[26]: It first uses a topology GNN to obtain node embeddings of a heterogeneous code graph and then classifies the vulnerability at the graph level. If the code graph is vulnerable, it trains a node classification model to identify the vulnerable nodes. DR-GCN[43]: It first converts the contract into a symbolic graph and normalizes the graph by performing node elimination. The resulting code graph is classified using a degree-free GCN. TMP[43]: The graph processing step is the same as DR-GCN, but it uses a temporal message propagation network to classify code graphs. AME[20]: It utilizes a rule-based approach to extract local expert patterns and convert the code into a semantic graph to extract global graph features. These two features are then fused using an attention-based network to predict vulnerabilities. Peculiar[38]: It first extracts the dataflow of important source code variables and creates a crucial dataflow graph. The vulnerability is then classified using GraphCodeBert. As mentioned earlier, SCScan [14] also performs line-level vulnerability detection. However, at the time of submission, SCScan's code was not open-sourced, and their results were not reported on an open benchmark. Therefore, we could not compare their performance with that of G-Scan in this paper. ### _Results on G-Scan Dataset_ We present the graph-level and node-level classification results on the G-Scan dataset's validation and test part in Table VI. The table displays the accuracy, precision, recall, and F1-score for three reentrancy subtypes: Call, Send, and Transfer. Based on the results presented in the table, we can make the following observations for node-level classification: Firstly, the classification accuracy for all datasets and reentrancy subtypes is above 99%, indicating that G-Scan can successfully localize vulnerable nodes in real-world datasets. It demonstrated the effectiveness of our proposed pipeline. Secondly, node-level F1-score are higher than 90% for most general vulnerable reentrancy subtype Call, and higher than 79% for the rest two subtypes, showing G-Scan can benefit reentrancy vulnerability localization in real world smart contracts. In terms of graph-level classification, we can find the following: Firstly, the classification accuracy for all reentrancy subtypes is higher than 89%. This indicates that even though G-Scan is designed for line-level vulnerability localization, it can still effectively classify vulnerable contracts. Secondly, the F1-scores for all three reentrancy subtypes are higher than 85%, demonstrating that G-Scan can successfully detect potential vulnerabilities. In Figure 7, we display the F1-scores on the train and validation datasets for the three reentrancy subtypes. From the figure, we observe that as the training progresses, G-Scan achieves an average F1-score of over 90% on both the train and validation datasets for the most common subtype, Call, at both the node-level and graph-level. The average F1-scores for the subtypes Transfer and Send are also higher than 80%. The stable curve indicates that G-Scan continuously learns from the train dataset and generalizes well toward the validation dataset. ### _Comparison with Baselines_ #### V-F1 Contract-level Performance In Table VII, we present the accuracy and F1-score of the contract-level vulnerability classification performance of G-Scan, along with other baselines. However, since prior work reported their performance on the Peculiar Dataset, which only annotates reentrancy subtype Call, and some of them did not open-source their code, we were unable to benchmark them on our extended reentrancy vulnerability dataset in Section VIII-E. Therefore, we benchmark G-Scan on the Peculiar Dataset and report our results in the first row of the table. We note that prior works mainly constructed code graphs based on semantic information, whereas G-Scan relies on AST to generate code graphs. If a smart contract is missing its reference file, we cannot construct AST and perform inference on the contract, which is reasonable as one cannot determine the vulnerability without checking the entire contract code. Additionally, during G-Scan inference, we removed duplicated code graphs, which accounted for more than 50% of the total. Based on the results, we observe that even when the duplicates are removed, G-Scan can still achieve 8% better accuracy than the baseline methods. Although Peculiar [38] achieves a slightly better F1-score than G-Scan, it cannot localize reentrancy vulnerabilities. Moreover, among the baseline methods, only MANDO performs fine-grained vulnerability localization, and G-Scan achieves a 21% better F1-score at the contract level detection. It is noteworthy that even though G-Scan is designed to localize vulnerabilities at the node level, it still achieves good performance at the contract level detection. #### V-F2 Node-level Performance We compared G-Scan with MANDO on a simulated dataset since it was not possible to directly compare the two methods on our collected G-Scan Dataset. This was because MANDO did not provide explicit details on how their contract graph is mapped back to the smart contract, and line-level accuracy could not be compared. Additionally, directly comparing the two methods at the node level was not possible as MANDO used metapaths to construct the code graph while G-Scan used the AST to generate the code graph. Therefore, to compare the two approaches, we use the simulated dataset collected by MANDO. One of the smart contracts could not be compiled, while another one belongs to the subtype transfer, which was correctly identified by G-Scan. As shown in Table VIII, G-Scan achieved 91.28% F1-score on call reentrancy, outperforming its baseline MANDO. This result suggests that by incorporating code hierarchy information into graph modeling and training on real-world datasets, G-Scan outperformed MANDO in vulnerability localization. We also note that MANDO's effectiveness is heavily reliant on identifying vulnerabilities at the contract level, and its classification F1-score for certain vulnerability types, such as reentrancy, falls to 75.80%. \begin{table} \begin{tabular}{c c c c c c c} \hline \hline Subtype & Dataset & Level & Accuracy & Precision & Recall & F1-Score \\ \hline \multirow{4}{}{Call} & validation set & node & 99.95\% & 94.88\% & 92.52\% & 93.69\% \\ & test set & node & 99.92\% & 93.23\% & 89.26\% & 91.20\% \\ & validation set & graph & 98.33\% & 87.59\% & 99.17\% & 93.02\% \\ & test set & graph & 97.31\% & 80.27\% & 100\% & 89.06\% \\ \hline \multirow{4}{}{Send} & validation set & node & 99.86\% & 91.34\% & 89.05\% & 90.18\% \\ & test set & node & 99.79\% & 92.55\% & 82.65\% & 87.32\% \\ & validation set & graph & 97.47\% & 81.07\% & 98.50\% & 88.94\% \\ & test set & graph & 95.46\% & 80.07\% & 99.56\% & 88.76\% \\ \hline \multirow{4}{}{Transfer} & validation set & node & 99.52\% & 95.88\% & 79.19\% & 79.39\% \\ & test set & node & 99.53\% & 83.61\% & 78.92\% & 81.19\% \\ \cline{1-1} & validation set & graph & 89.91\% & 74.71\% & 99.69\% & 85.41\% \\ \cline{1-1} & test set & graph & 90.74\% & 78.21\% & 98.55\% & 87.21\% \\ \hline \hline \end{tabular} \end{table} TABLE VI: G-Scan performance on different reentrancy subtypes \begin{table} \begin{tabular}{c c c} \hline \hline Method & Accuracy & F1-Score \\ \hline G-Scan & 98.15\% & 91.74\% \\ \hline MANDO [26] & & 75.80\% \\ DR-GCN [43] & 81.47\% & 76.39\% \\ TMP [43] & 84.48\% & 88.11\% \\ AME [20] & 90.19\% & 87.94\% \\ Peculiar [38] & 99.81\% & 92.10\% \\ \hline \hline \end{tabular} \end{table} TABLE VII: Comparison with baselines on Peculiar Dataset. ### _Inference Overhead_ We present the inference overhead of smart contracts with varying lengths in Table IX. The inference overhead is measured for five smart contracts with different Lines of Code (LoC) and includes the time it takes for AST construction, code graph generation, and GNN predictions. AST construction and code graph generation are performed on the CPU, while GNN prediction inference is performed on the GPU. From the table, we can find the following: Firstly, the majority of the G-Scan overhead comes from AST construction using Solidity compilers. As the size of the smart contract increases, the percentage of time required for AST construction as a fraction of the total overhead increases. This is because the compiler requires more time to parse larger files. Secondly, despite an LoC value as large as 6.1k, the total inference time remains under 1.2s, demonstrating the efficiency of G-Scan. ## IX Related Work ### _Smary Contract Vulnerability Detection_ Symbolic Execution:* Symbolic execution [3, 33, 18, 36] takes the smart contracts bytecode as input and uses symbolic expression to represent the contracts. Then, it uses Satisfiability Modulo Theory (SMT) solver to prove if certain vulnerability conditions can exist in the source code or not. In smart contract vulnerability detections, symbolic executions are always combined with other rule-based algorithms to detect potential security risks. Osiris [36] detects integer bugs with symbolic execution and additional taint analysis, which is a kind of tracking of the data across the control flow. SmarTest [30] applies symbolic execution on the smart contract language models and creates transaction sequences to detect bugs. SAILFISH [7] uses a hybrid approach for source codes, where a storage dependence graph performs analysis of side effects on the storage variables during the execution and follows by the symbolic evaluation to detect vulnerabilities. Fuzzing: Fuzzing [12, 16, 28, 29, 31, 35] uses different algorithms to automatically generate inputs potentially trigger errors or unexpected behaviors to detect smart contract vulnerabilities. ContractFuzzer [16] analyzes the ABI interfaces of smart contracts to generate inputs that conform to the invocation grammars of the smart contracts under test. It defines new test oracles for different types of vulnerabilities and instrument EVM to monitor smart contract executions to detect security vulnerabilities. Echidna [12] incorporates a worst-case gas estimator into a general-purpose fuzzer. \begin{table} \begin{tabular}{c c c c c} \hline \hline LoC & AST (ms) & CGG (ms) & Prediction (ms) & Total (ms) \\ \hline 5 & 3.60 & 1.19 & 5.05 & 9.84 \\ 197 & 31.29 & 6.04 & 9.96 & 47.29 \\ 245 & 55.89 & 12.92 & 14.15 & 82.96 \\ 5815 & 901.98 & 178.08 & 95.26 & 1175.32 \\ 6151 & 850.70 & 191.48 & 108.25 & 1150.43 \\ \hline \hline \end{tabular} \end{table} TABLE IX: G-Scan overhead on AST construction (AST), code graph generation (CGG) and prediction Fig. 7: The F1 score evaluated during training for train and validation dataset on three subtypes at both node and graph level \begin{table} \begin{tabular}{c c c} \hline \hline Method & Accuracy & F1-Score \\ \hline G-Scan & 97.44\% & 91.28\% \\ Mando [26] & - & 86.40\% \\ \hline \hline \end{tabular} \end{table} TABLE VIII: Comparison with MANDO on Simulated Dataset. When a property violation is detected, a counterexample is automatically minimized to report the sequence of transactions that triggers the failure. Machine Learning: Machine training [11, 37, 42] based algorithms take the smart contracts as text data or graph data and use different machine learning algorithms to classify if a smart contract is vulnerable or not. The machine learning algorithms enable the model to learn the hidden feature representation of smart contract code that previously introduced rule-based algorithms failed to catch. Most of them [38, 42, 43] classify vulnerabilities at contract level, and others first perform contract level vulnerabilities detection and then localize the bugs at line level like MANDO [26] and SCScan [14]. SC-VDM [42] converts the typecified into a greyscale image and applies a convolutional neural network to classify whether a smart contract is vulnerable. Liu et al. [43] propose to apply a graph convolutional neural network on a contract graph of the source code to detect bugs by graph classification. In the follow-up papers, they [19, 20] integrate additional expert patterns to increase detection accuracy. Peculiar [38] extracts the dataflow of important variables of the source code in a crucial dataflow graph. Then, the source code and the dataflow graph are fed into GraphCodeBERT [13] for reentrancy vulnerability classification. ### _Source Code Graphs_ In machine learning based vulnerability detection algorithms, converting source code into image or texts may lose structural information. Therefore, many work first convert the source code into graph structure, and then classify the graph using different GNN models. Heterogeneous Code Graph Heterogeneous code graph [19, 20, 43, 26] construct graph connections based on the semantic information and program flow paths, which is widely used in the smart contract vulnerability detection. However, they failed to incorporate code dependencies and hierarchies into modeling. Liu et al. [19, 20, 43] proposed to construct code graph based on three different node types. Major nodes represent important function invocations or critical variables. Other variables are seen as secondary nodes and fallback nodes symbolize a fallback function call. Edges are defined by the feasible program flow paths. Additionally, an elimination step is performed to reduce the graph size, which removes secondary and fallback nodes and redirects the edges. This final normalized graph is fed into a GNN that detects reentrancy, timestamp dependence, and infinite loop vulnerabilities. MANDO [26]'s code graphs are based on call graphs and control flows of the source code. It first uses a multi-metapaths extractor to generate metapaths from the node types and their associated edges in these graphs. The node embeddings are created with these metapaths and fused with the metapaths by a heterogeneous attention mechanism at node level. Afterward, these node embeddings are fed into an MLP for graph classification to check whether the contract is vulnerable. If that is the case, the exact location of the vulnerability inside the contract is determined by node classification with the updated node embeddings in graph classification. AST based Code Graph AST [4, 5, 40] based code graph generation is widely used in code vulnerability detection like C/C++. AST can extract better structure and variable/function dependency from source code. Some recent work [39, 41] also applied AST based graph analysis to smart contract vulnerability detection. Allamanis et al. [4, 5] proposed a program graph based on the node connections in C#'s AST. It first transforms the C# source code into AST representation using compilers. Then, the nodes in the tree are replaced by corresponding source code tokens and connected with edges that reflect the original execution order. To represent the control and dataflow structure, they add additional edges, for example, between all occurrences of the same variable or in condition statements to indicate all valid paths. Code Property Graphs [40] proposed to construct graphs from the C/C++'s AST by combining it with edges from control flow and program dependence. The control flow edges are inserted for subsequent statements, loops, returns, and similar constructs. The program dependence edges are inserted to reflect the influences of other variables or predicates on a certain variable, such as the definition or variable assignment. ## X Conclusion and Future Work In the paper, we present G-Scan, the first end-to-end fine-grained line-level reentrancy vulnerability detection system with evaluations on the first-of-its-kind real world dataset. G-Scan first converts source code smart contracts to code graphs, and then trains node classification models on the graphs to localize vulnerabilities. Our experiments demonstrate that G-Scan achieves 93.02% F1-score in contract-level vulnerability detection and 93.69% F1-score in line-level vulnerability localization on the real world dataset. Moreover, G-Scan processes low inference overhead and can localize vulnerabilities within less than 1.2s for 6k LoC smart contracts. We will open-source G-Scan along with its dataset to promote research in this area. In the future, we plan to extend our real world dataset to include multiple vulnerability annotations and explore different code graph representations that may improve GNN classification performance.
2306.15157	Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networks	Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to the simplification of neural networks. We analyze tropical polynomials with real coefficients, extending earlier ideas and methods developed for polynomials with integer coefficients. We first prove the existence of a unique quotient-remainder pair and characterize the quotient in terms of the convex bi-conjugate of a related function. Interestingly, the quotient of tropical polynomials with integer coefficients does not necessarily have integer coefficients. Furthermore, we develop a relationship of tropical polynomial division with the computation of the convex hull of unions of convex polyhedra and use it to derive an exact algorithm for tropical polynomial division. An approximate algorithm is also presented, based on an alternation between data partition and linear programming. We also develop special techniques to divide composite polynomials, described as sums or maxima of simpler ones. Finally, we present some numerical results to illustrate the efficiency of the algorithms proposed, using the MNIST handwritten digit and CIFAR-10 datasets.	Ioannis Kordonis, Petros Maragos	2023-06-27T02:26:07Z	http://arxiv.org/abs/2306.15157v1	# Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networks. ###### Abstract Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to the simplification of neural networks. We analyze tropical polynomials with real coefficients, extending earlier ideas and methods developed for polynomials with integer coefficients. We first prove the existence of a unique quotient-remainder pair and characterize the quotient in terms of the convex bi-conjugate of a related function. Interestingly, the quotient of tropical polynomials with integer coefficients does not necessarily have integer coefficients. Furthermore, we develop a relationship of tropical polynomial division with the computation of the convex hull of unions of convex polyhedra and use it to derive an exact algorithm for tropical polynomial division. An approximate algorithm is also presented, based on an alternation between data partition and linear programming. We also develop special techniques to divide composite polynomials, described as sums or maxima of simpler ones. Finally, we present some numerical results to illustrate the efficiency of the algorithms proposed, using the MNIST handwritten digit and CIFAR-10 datasets. Tropical geometry, Piecewise linear neural networks, Tropical polynomial division, Neural network compression ## I Introduction Tropical geometry is a relatively new research field combining elements and ideas from polyhedral and algebraic geometry [1]. The underlying algebraic structure is the max-plus semiring (also known as tropical semiring), where the usual addition is replaced by maximization and the standard multiplication by addition. Tropical polynomials (also known as max-polynomials) are the polynomials in the max-plus algebra and have a central role in tropical geometry. This work deals with the division of tropical polynomials and presents some applications in neural network compression. A link between tropical geometry and machine learning was recently developed [2, 3]. An important application is the analysis of neural networks with piecewise linear activations (e.g., ReLU). In this front, [3, 4, 5, 6, 7] use a tropical representation of neural networks to describe the complexity of a network structure, defined as the number of its linear regions, or describe their decision boundaries. [8] presents an algorithm to extract the linear regions of deep ReLU neural networks. Papers [9, 10, 11] deal with the problem of tropical polynomial division and its use in the simplification of neural networks. The analysis is, however, restricted to polynomials with integer coefficients. Article [12] uses tropical representations of neural networks and employs polytope approximation tools to simplify them. Another class of networks represented in terms of tropical algebra is morphological neural networks [13, 14, 15, 16, 17, 18]. Other applications of tropical algebra and geometry include the tropical modeling of classical algorithms in probabilistic graphical models [19, 20] and piecewise linear regression [21, 22]. Neural networks employing the closely related log-sum-exp nonlinearity were presented in [23, 24]. Another problem very much related to the current work is the factorization of tropical polynomials. This problem was first studied in [25, 26, 27, 28], for the single-variable case, and in [29, 30] for the multivariate case. The problem of tropical rational function simplification was studied in [31]. Another related problem is approximation in tropical algebra (see, e.g., [32]). _Contribution:_ This work deals with the problem of tropical polynomial division and its applications in simplifying neural networks. The analysis generalizes previous works [9, 10, 11] to include polynomials with real coefficients. We first introduce a new notion of division and show that there is a unique quotient-remainder pair. Furthermore, we show that the quotient is equal to the convex bi-conjugate of the difference between the dividend and the divisor. Then, the quotient is characterized in terms of the closed convex hull of the union of a finite collection of convex (unbounded) polyhedra. This characterization leads to a simple exact algorithm based on polyhedral geometry. We propose an efficient approximate scheme inspired by an optimization algorithm for convex, piecewise-linear fitting [33]. The approximate algorithm alternates between data clustering and linear programming. Then, we focus on developing algorithms for dividing composite polynomials expressed as the sum of simpler ones. The division results are then applied to simplify neural networks with ReLU activation functions. The resulting neural network has fewer neurons with maxout activations and a reduced total number of parameters. Finally, we present numerical results for the MNIST handwritten digit recognition and the CIFAR-10 image classification problem. As a baseline for comparison, we use the structured L1 pruning without retraining. The proposed method is very competitive in the large compression regime, that is, in the case where there are very few parameters remaining after compression. The rest of the paper is organized as follows: Section II presents some preliminary algebraic and geometric material. In Section III, we develop some theoretical tools for tropical polynomial division. An exact division algorithm is presented in Section IV, and an approximate algorithm in Section V. Section VI-B deals with the division of composite tropical polynomials. Finally, VII gives some numerical examples. The proofs of the theoretical results are presented in the Appendix. ## II Preliminaries, Background and notation Max-plus algebra is a modification of the usual algebra of the real numbers, where the usual summation is substituted by maximum and the usual multiplication is substituted by summation [34, 35, 36, 37]. Particularly, max-plus algebra is defined on the set $\mathbb{R}_{\text{max}}=\mathbb{R}\cup\{-\infty\}$ with the binary operations "$\vee$" and "$+$". The operation "$+$" is the usual addition. The operation "$\vee$" stands for the maximum, i.e. for $x,y\in\mathbb{R}_{\text{max}}$, it holds $x\lor y=\text{max}\{x,y\}$. For more than two terms we use"$\vee$", i.e., $\bigvee_{i\in I}x_{i}=\sup\{x_{i}:i\in I\}$. We will also use the '$\wedge$' notation to denote the minimum, i.e., $x\wedge y=\text{min}\{x,y\}$. A unified view of max-plus and several related algebras, was developed in [38], in terms of weighted lattices. A tropical polynomial is a polynomial in the max-plus algebra. Particularly, a tropical polynomial function [39, 40, 41] is defined as $p:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\text{max}}$ with \[p(\mathbf{x})=\bigvee_{i=1}^{m_{p}}(\mathbf{a}_{i}^{T}\mathbf{x}+b_{i}), \tag{1}\] where $\mathbf{a}_{i}\in\mathbb{R}^{n}$ and $b_{i}\in\mathbb{R}_{\text{max}}$. In this paper, we will use the terms tropical polynomial and tropical polynomial function interchangeably. For the tropical polynomial $p(\mathbf{x}),$ the Newton polytope is defined as \[\text{Newt}(p)=\text{ conv}\{\mathbf{a}_{1},\ldots,\mathbf{a}_{m_{p}}\}, \tag{2}\] where "conv" stands for the convex hull, and extended Newton polytope as \[\text{ENewt}(p)=\text{ conv}\{[\mathbf{a}_{1}^{T}\ b_{1}]^{T},\ldots,[\mathbf{a}_{m_{p }}^{T}b_{m_{p}}]^{T}\}. \tag{3}\] The values of the polynomial function $p$ depend on the upper hull of the extended Newton polytope [1] (see also [4]). A polyhedron is subset of $\mathbb{R}^{n}$ which can be represented either as the set of solutions of a finite collection of linear inequalities ($\mathcal{H}-$representation) \[P=\{\mathbf{x}\in\mathbb{R}^{n}:\mathbf{A}\mathbf{x}\leq\mathbf{b}\},\] or in terms of a set of vertices $\mathbf{v}_{1},\ldots,\mathbf{v}_{l}\in\mathbb{R}^{n}$ and a set of rays $\mathbf{w}_{1},\ldots,\mathbf{w}_{s}\in\mathbb{R}^{n}$ as \[P=\left\{\sum_{j=1}^{l}\lambda_{j}\mathbf{v}_{j}+\sum_{j=1}^{s}\mu_{j}\mathbf{w}_{j}: \lambda_{j}\geq 0,\sum_{j=1}^{s}\lambda_{j}=1,\mu_{j}\geq 0\right\}, \tag{4}\] ($\mathcal{V}-$representation). The Minkowski-Weyl (resolution) theorem states that each polyhedron admits both representations [42]. Figure 1 illustrates the Minkowski-Weyl resolution of an unbounded polyhedron. the Furthermore, there are several well-known algorithms for representation conversion (e.g., [42], Ch. 9). Bounded polyhedra are called polytopes. It is often convenient to describe polytopes in terms of extended representations (e.g., [43]). An extension of a polytope $P\subset\mathbb{R}^{n}$ is a polyhedron $Q\subset\mathbb{R}^{n^{\prime}}$, with $n^{\prime}>n$, along with a linear projection $F:R^{n^{\prime}}\rightarrow\mathbb{R}^{n}$ such that $F(Q)=P$. For a function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\{\infty\}$ the epigraph is defined as \[\text{epi}(f)=\{(\mathbf{x},z)\in\mathbb{R}^{n+1}:z\geq f(\mathbf{x})\}.\] The convex conjugate of $f$ is a function $f^{\star}:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\{+\infty,-\infty\}$ given by (see [44]) \[f^{\star}(\mathbf{a})=\sup\{\mathbf{a}^{T}\mathbf{x}-f(\mathbf{x}):\mathbf{x}\in\mathbb{R}^{n}\}.\] For a pair of polyhedra $P_{1},P_{2}\subset\mathbb{R}^{n}$ the Minkowski sum is defined as \[P_{1}\oplus P_{2}=\{x+y:x\in P_{1},y\in P_{2}\}.\] The following properties hold (see [4]) \[\text{Newt}(p_{1}+p_{2}) =\text{Newt}(p_{1})\oplus\text{Newt}(p_{2}),\] \[\text{Newt}(p_{1}\lor p_{2}) =\text{conv}(\text{Newt}(p_{1})\cup\text{Newt}(p_{2})).\] Similar relations hold for the extended Newton polytopes as well. We then introduce a dual function of a tropical polynomial, called Extended Newton Function (ENF). For a tropical polynomial $p$, the ENF is defined as: \[\text{ENF}_{p}(\mathbf{a})=\sup\{b\in\mathbb{R}:[\mathbf{a}^{T}\ b]^{T}\in\text{ENewt} (p)\},\] where $\mathbf{a}\in\mathbb{R}^{n}$ and $\sup(\emptyset)=-\infty$. This function describes the upper hull of the extended Newton polytope. Thus, the ENF fully characterizes the tropical polynomial function $p$. Proposition 1: _Let $p_{1},p_{2}$ be two tropical polynomials. Then $p_{1}(\mathbf{x})\leq p_{2}(\mathbf{x})$, for all $\mathbf{x}\in\mathbb{R}^{n}$, if and only if $\text{ENF}_{p_{1}}(\mathbf{a})\leq\text{ENF}_{p_{2}}(\mathbf{a})$, for all $\mathbf{a}\in\mathbb{R}^{n}$._ _Proof_: See Appendix A. Fig. 1: _Illustration of the Minkowski-Weil theorem. The unbounded polyhedron $P$ (green color in the figure) has two representations: $\mathcal{H}-$representation as the intersection of halfspaces $H_{1},\ldots,H_{4}$ and $\mathcal{V}-$representation generated by vertices $v_{1},v_{2},v_{3}$ and rays $w_{1},w_{2}$._ The significance of ENFs is that we can often express inequality $\text{ENF}_{p_{1}}\leq\text{ENF}_{p_{2}}$, using a finite number of inequalities in terms of the tropical polynomial coefficients of $p_{1}$, $p_{2}$. ## III Tropical Polynomial Division Definition and Existence We first define the tropical polynomial division. Definition 1: _Let $p,d$ be tropical polynomials. We define the quotient and the remainder of the division of $p$ by $d$._ 1. _A tropical polynomial_ $q$ _is the quotient if_ \[p(\boldsymbol{x})\geq q(\boldsymbol{x})+d(\boldsymbol{x}),\qquad\text{for all } \boldsymbol{x}\in\mathbb{R}^{n},\] (5) _and_ $q$ _is the maximum polynomial satisfying this inequality. Particularly, for any tropical polynomial_ $\tilde{q}$_, such that_ $p(\boldsymbol{x})\geq\tilde{q}(\boldsymbol{x})+d(\boldsymbol{x})$_, for all_ $\boldsymbol{x}\in\mathbb{R}^{n}$_, it holds_ $\tilde{q}(\boldsymbol{x})\leq q(\boldsymbol{x})$_, for all_ $\boldsymbol{x}\in\mathbb{R}^{n}$_._ 2. _A tropical polynomial_ $r$ _is the remainder of the division of_ $p$ _by_ $d$ _with quotient_ $q$ _if_ \[p(\boldsymbol{x})=(q(\boldsymbol{x})+d(\boldsymbol{x}))\lor r(\boldsymbol{x}),\qquad\text{for all }\boldsymbol{x}\in\mathbb{R}^{n},\] (6) _and_ $r$ _is the minimum tropical polynomial satisfying this equality. Particularly, for any tropical polynomial_ $\tilde{r}$_, such that_ $p(\boldsymbol{x})=(q(\boldsymbol{x})+d(\boldsymbol{x}))\lor\tilde{r}( \boldsymbol{x})$_, it holds_ $\tilde{r}(\boldsymbol{x})\geq r(\boldsymbol{x})$_, for all_ $\boldsymbol{x}\in\mathbb{R}^{n}$_._ Observe that the set of tropical polynomials $q$ satisfying (5) is closed under the pointwise maximum. That is, assume that $q_{1},q_{2}$ are tropical polynomials such that $p(\boldsymbol{x})\geq d(\boldsymbol{x})+q_{1}(\boldsymbol{x})$ and $p(\boldsymbol{x})\geq d(\boldsymbol{x})+q_{2}(\boldsymbol{x})$, then it holds $p(\boldsymbol{x})\geq d(\boldsymbol{x})+(q_{1}(\boldsymbol{x})\lor q_{2}( \boldsymbol{x}))$. This property motivates us to determine the monomials \[q_{M}^{\boldsymbol{a},b}(\boldsymbol{x})=\boldsymbol{a}^{T}\boldsymbol{x}+b,\] that satisfy (5). For each monomial coefficient $\boldsymbol{a}$, define the maximum value of $b$ that satisfy $p(\boldsymbol{x})\geq d(\boldsymbol{x})+q_{M}^{\boldsymbol{a},b}(\boldsymbol{ x})$ as \[l(\boldsymbol{a})=\sup\{b:q_{M}^{\boldsymbol{a},b}(\boldsymbol{x})+d( \boldsymbol{x})\leq p(\boldsymbol{x}),\text{for all }x\in\mathbb{R}^{n}\}. \tag{7}\] Note that $l(\boldsymbol{a})$ can be written as \[l(\boldsymbol{a}) =\sup\{b:b\leq p(\boldsymbol{x})-d(\boldsymbol{x})-\boldsymbol{a }^{T}\boldsymbol{x},\text{for all }x\in\mathbb{R}^{n}\}\] \[=\inf_{\boldsymbol{x}\in\mathbb{R}^{n}}\{p(\boldsymbol{x})-d( \boldsymbol{x})-\boldsymbol{a}^{T}\boldsymbol{x}\}.\] Denoting by $f(\boldsymbol{x})$ the difference $p(\boldsymbol{x})-d(\boldsymbol{x})$, we have \[l(\boldsymbol{a})=-f^{\star}(\boldsymbol{x}),\] where $f^{\star}$ is the convex conjugate of $f$. A candidate for the quotient is \[q(\boldsymbol{x})=\sup_{\boldsymbol{a}\in\mathbb{R}^{n}}\{\boldsymbol{a}^{T} \boldsymbol{x}+l(\boldsymbol{a})\}.\] Note that $q$ is the bi-conjugate of $f$. Indeed \[q(\boldsymbol{x})=\sup_{\boldsymbol{a}\in\mathbb{R}^{n}}\{\boldsymbol{a}^{T} \boldsymbol{x}-f^{\star}(\boldsymbol{a})\}=f^{\star\star}(\boldsymbol{x}). \tag{8}\] The following proposition shows that $q(\boldsymbol{x})$ is indeed the quotient of the division. Proposition 2: _For any pair of tropical polynomials $p,d$ there is a quotient-remainder pair. The quotient is given by (8). Furthermore, the polynomial functions of the quotient and remainder are unique._ See Appendix B. Remark 1: _The use of the convex conjugate is closely related to the slope transform [45, 46, 47]. The slope transform was used to derive an analog of frequency response of max-plus and more generally morphological systems._ Example 1 (Tropical Polynomial Division in 1D): _Let $p(x)=\max(-2x-1,1,x+1,3x-3)$ and $d(x)=\max(x,2x-1)$. Equivalently, the polynomial functions $p$ and $d$ are written as_ \[p(x)=\begin{cases}-2x-1,&\text{if }x\leq-1\\ 1,&\text{if }-1<x\leq 0\\ x+1,&\text{if }0<x\leq 2\\ 3x-3,&\text{if }x\geq 2\end{cases},\] \[d(x)=\begin{cases}x,&\text{if }x\leq 1\\ 2x-1,&\text{if }x>1\end{cases}.\] _Thus,_ \[f(x)=p(x)-d(x)=\begin{cases}-3x-1,&\text{if }x\leq-1\\ 1-x,&\text{if }-1<x\leq 0\\ 1,&\text{if }0<x\leq 1\\ -x+2,&\text{if }1<x\leq 2\\ x-2,&\text{if }x\geq 2\end{cases}.\] _The plots of $p,d$ and $f$ are given in Figure 2. The difference $f$ is not convex since the slopes $-3,-1,0,-1,1$ are not increasing. The quotient $q$ is given as the convex bi-conjugate of $f$, that is the largest convex function which is less than or equal to $f$, for all $x$. Thus, the quotient is given $q(x)=\max(-3x+1,1-x,-0.5x+1,x-2)$. The sum $d(x)+q(x)$ is equal to $p(x)$, for all $x$, except $x\in(0,2)$. Thus, $r(x)=x+1$, that is, the form of $p(x)$, for $x\in(0,2)$._ We call the division _nontrivial_ if $q(\boldsymbol{x})>-\infty$ and _effective_ if $r(\boldsymbol{x})\neq p(\boldsymbol{x})$, for some $\boldsymbol{x}\in\mathbb{R}^{n}$. It is easy to see that an effective division is also nontrivial. As shown in the previous example, these notions are not equivalent. Proposition 3: _Assume that $p$ and $d$ are tropical polynomials. If the division of $p$ by $d$ has quotient $q$ and remainder $r$. Then:_ Fig. 2: _This figure refers to Example 1. The left plot presents the divided $p$ and the divisor $d$. The middle plot their difference $f$ and the quotient $q$. The right part shows the dividend $p$, its approximation $d+q$ and the remainder $r$._ 1. The division of $r$ by $d$ is not effective. 2. The division of $p$ by $q$ has quotient $d$ and remainder $r$. _Proof_ See Appendix C. _Remark 2:_ For the Euclidean division of a positive integer $p$ by a positive integer $d$ with quotient $q$ and remainder $r$, the notion of nontrivial division corresponds to $q\neq 0$ and effective division to $r<p$. For the division of positive integers, we observe that these notions are equivalent. Furthermore, the converse of (a) is true. Particularly, if for some $\bar{q},\bar{r}\in\mathbb{N}$, it holds $p=d\bar{q}+\bar{r}$ and the division of $r$ by $d$ is not effective, then $\bar{q}$ is the quotient and $\bar{r}$ the remainder. On the other hand, for tropical polynomials, the converse of (a) is not true. Particularly, it is possible that for a pair of tropical polynomials $p,d$ there is another pair $q,r$ such that (6) is satisfied, the division of $r$ by $d$ is not effective (or even is trivial), but $q,r$ is not a quotient-remainder pair. For example consider $p(x)=\max(3x,-3x),d(x)=\max(2x,-2x)$ and $q(x)=\max(0,x),r(x)=\max(0,-3x)$. It is easy to see that the division of $r$ by $d$ is trivial. Consider the set \[C=\{\mathbf{c}\in\mathbb{R}^{n}:\text{Newt}(\mathbf{c}^{T}\mathbf{x}+d(\mathbf{x}))\subset \text{Newt}(p(\mathbf{x}))\}. \tag{9}\] This set appears also [9, 10, 11] for the case of discrete coefficients. It is not difficult to see that $C$ is a convex polytope. The following proposition shows that the division is non-trivial if and only if $C\neq 0$. _Proposition 4:_ Let $C$ be the set defined in (9). Then, 1. $C$ is equal to the domain of function $l(\cdot)$ (given by (7)). That is, \[C=\text{\bf dom}(l(\mathbf{a}))=\{\mathbf{a}\in\mathbb{R}^{n}:l(\mathbf{a})>-\infty\}.\] 2. The division is non-trivial if and only if $C\neq\emptyset$ 3. It the quotient has the form $q(\mathbf{x})=\bigvee_{i=1}^{m_{p}}(\hat{\mathbf{a}}_{i}^{T}\mathbf{x}+\hat{b}_{i})$, then $\hat{\mathbf{a}}_{i}\in C$, for all $i$. _Proof_ See Appendix D. _Corollary 1:_ Assume that the division of a tropical polynomial $p(\mathbf{x})=\bigvee_{i=1}^{m_{p}}(\hat{\mathbf{a}}_{i}^{T}\mathbf{x}+b_{i})$ by another tropical polynomial $d(\mathbf{x})=\bigvee_{i=1}^{m_{d}}(\tilde{\mathbf{a}}_{i}^{T}\mathbf{x}+\tilde{b}_{i})$ is nontrivial. Then 1. It holds \[\text{span}\{\tilde{\mathbf{a}}_{1},\dots,\tilde{\mathbf{a}}_{m_{d}}\}\subset\text{ span}\{\mathbf{a}_{1},\dots,\mathbf{a}_{m_{p}}\}.\] 2. The quotient $q(\mathbf{x})=\bigvee_{i=1}^{m_{q}}(\tilde{\mathbf{a}}_{i}^{T}\mathbf{x}+\tilde{b}_{i})$ is such that \[\text{span}\{\hat{\mathbf{a}}_{1},\dots,\hat{\mathbf{a}}_{m_{q}}\}\subset\text{ span}\{\mathbf{a}_{1},\dots,\mathbf{a}_{m_{p}}\}.\] _Proof:_ See Appendix E. _Remark 3:_ Corollary 1 can be used to simplify the division of two tropical polynomials. Particularly, assume that \[\text{span}\{\tilde{\mathbf{a}}_{1},\dots,\tilde{\mathbf{a}}_{m_{d}}\}\subset\text{ span}\{\mathbf{a}_{1},\dots,\mathbf{a}_{m_{p}}\}\subset\mathbb{R}^{d}.\] If $\mathbf{Q}$ is a matrix the columns of which represent an orthonormal basis of the subspace $\text{span}\{\mathbf{a}_{1},\dots,\mathbf{a}_{m_{p}}\}$, then $\mathbf{a}_{i}$ can be expressed in terms of a reduced dimension vector $\mathbf{a}_{i}^{T}$ as $\mathbf{a}_{i}=\mathbf{Q}\mathbf{a}_{i}^{T}$, and $\tilde{\mathbf{a}}_{i}$ similarly in terms of $\tilde{\mathbf{a}}_{i}^{T}$ as $\tilde{\mathbf{a}}_{i}=\tilde{\mathbf{Q}}\tilde{\mathbf{a}}_{i}^{T}$. Then, the division of polynomials $p$ and $d$ can be expressed in terms of reduced dimension polynomials $p^{r},d^{r}$ as $p(\mathbf{x})=p^{r}(\mathbf{Q}^{T}\mathbf{x})=p^{r}(\tilde{\mathbf{x}}),d=d^{r}(\mathbf{Q}^{T}\mathbf{ x})=d^{r}(\tilde{\mathbf{x}})$, where the dimension of $\tilde{\mathbf{x}}$ is equal to the rank of $\mathbf{Q}$. This observation is certainly useful when the number of terms of the dividend $m_{p}$ is less than the space dimension $d$. It will be also used in Section VI-B. ## IV An Exact Algorithm For Tropical Division We now present an algorithm to compute the quotient and remainder of a division. The algorithm uses polyhedral computations. Particularly, we use (8), and its equivalent in terms of the epigraphs (see (34) in the appendix) to compute $q$. The input of the algorithm is a pair of tropical polynomials \[p(\mathbf{x})=\bigvee_{i=1}^{m_{p}}(\mathbf{a}_{i}^{T}\mathbf{x}+b_{i}),\quad d(\mathbf{x})= \bigvee_{j=1}^{m_{d}}(\tilde{\mathbf{a}}_{j}^{T}\mathbf{x}+\tilde{b}_{j}).\] The first step is to compute a partition of $\mathbb{R}^{n}$ in polyhedra on which $f(\mathbf{x})=p(\mathbf{x})-d(\mathbf{x})$ is linear. To do so, for each $i,j$, consider the polyhedron on which the terms $i,j$ attain the maximum in $p$ and $d$ respectively. \[P_{i,j}=\{x\in\mathbb{R}^{n}:\mathbf{a}_{i}^{T}\mathbf{x}+b_{i}\geq\mathbf{a }_{i}^{T}\mathbf{x}+b_{i^{\prime}},\] \[\tilde{\mathbf{a}}_{j}^{T}\mathbf{x}+\tilde{b}_{j}\geq\tilde{\mathbf{a}}_{j^ {\prime}}^{T}\mathbf{x}+\tilde{b}_{j^{\prime}},i^{\prime}=1,\dots,m_{p},j^{\prime }=1,\dots,m_{d}\}.\] Polyhedron $P_{i,j}$ can be compactly written in terms of a matrix $\mathbf{A}^{i,j}$ and a vector $\mathbf{b}^{i,j}$ as \[P_{i,j}=\{\mathbf{x}\in\mathbb{R}^{n}:\mathbf{A}^{i,j}\mathbf{x}\geq\mathbf{b}^{i,j}\}.\] Note that some $P_{i,j}$'s can be empty. The second step is to compute the polyhedra that represent the epigraph of $f(\mathbf{x})=p(\mathbf{x})-d(\mathbf{x})$ for each $P_{i,j}$. The epigraphs are given by: \[E_{i,j}=\{(\mathbf{x},z)\in\mathbb{R}^{n+1}:\mathbf{A}^{i,j}\mathbf{x}\geq \mathbf{b}^{i,j},\] \[\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad z \geq(\mathbf{a}_{i}-\tilde{\mathbf{a}}_{j})^{T}\mathbf{x}+b_{i}-\tilde{b}_{j}\}.\] For each polyhedron $E_{i,j}$, we compute the corresponding $\mathcal{V}-$representation, consisting of a set of vertices $V_{i,j}$ and a set of rays $R_{i,j}$, satisfying (4). Note that the epigraph of $f$ is given by $\cup_{i,j}E_{i,j}$. The epigraph of the quotient $q$ is given by the closed convex hull of the epigraph of $f$ (see (34) in the appendix). Next, we consider the union of vertices $V=\cup_{i,j}V_{i,j}$, and rays $R=\cup_{i,j}R_{i,j}$, and consider the polyhedron $E$ generated by $V$ and $R$ (as in (4)). Due to Proposition 2, $E$ is the epigraph of the quotient $q(\mathbf{x})$. Then, compute the $\mathcal{H}-$representation of $E$ \[E=\{(\mathbf{x},z)\in\mathbb{R}^{n+1}:[\mathbf{A}^{E,\mathbf{x}}\ \mathbf{a}^{E,z}][\mathbf{x}^{T}\ z]^{T}\geq\mathbf{b}^{E}\}, \tag{10}\] for appropriate matrix $\mathbf{A}^{E,\mathbf{x}}$ and vectors $\mathbf{a}^{E,z},\mathbf{b}^{E}$. Assume that $E\neq\mathbb{R}^{n}$. We will prove in Proposition 5 that the components of $\mathbf{a}^{E,z}$ are positive. Thus, \[E= \left\{(\mathbf{x},z)\in\mathbb{R}^{n+1}:z\geq-\frac{1}{[\mathbf{a}^{E,z}] }[\mathbf{A}^{E,\mathbf{x}}]_{l}\mathbf{x}+\right.\] \[\quad\quad\quad\left.+[\mathbf{b}^{E}]_{l}/[\mathbf{a}^{E,z}]_{l},\ l=1, \dots,L\right\},\] where $L$ is the number of rows of $\mathbf{A}^{E,\mathbf{x}}$. Therefore, \[q(\mathbf{x})=\bigvee_{l=1}^{L}\left(-\frac{1}{[\mathbf{a}^{E,z}]_{l}}[\mathbf{A}^{E,\mathbf{x} }] The procedure is illustrated in Figure 3. Having computed the quotient, we formulate the sum $\tilde{p}(\mathbf{x})=d(\mathbf{x})+q(\mathbf{x})$. For each $P_{i,j}$ we choose a point $\mathbf{x}_{i,j}$ in its relative interior, for example \[\mathbf{x}_{i,j}=\frac{1}{\|V_{i,j}\|}\sum_{v\in V_{i,j}}v+\sum_{w\in R_{i,j}}w. \tag{12}\] Let $I$ be the set of indices $i$ such that there is an index $j$ satisfying $p(\mathbf{x}_{i,j})>\tilde{p}(\mathbf{x}_{i,j})$. Then, \[r(\mathbf{x})=\bigvee_{i\in I}(\mathbf{a}_{i}^{T}\mathbf{x}+b_{i}). \tag{13}\] The computation of the quotient and the remainder is summarized in Algorithm 1. ``` 0: Input: Tropical Polynomials $p(\mathbf{x})$ and $d(\mathbf{x})$ 1: Compute a partition of $\mathbb{R}^{n}$ into polyhedra $P_{i,j}$, $i=1,\ldots,m_{p}$, $j=1,\ldots,m_{d}$, and their interior points $\mathbf{x}_{i,j}$ according to (12) 2: Compute the epigraphs $E_{i,j}$ 3: For each epigraph, compute the resolution into a set of vertices $V_{i,j}$ and a set of rays $R_{i,j}$, using a representation conversion algorithm 4: Compute the $\mathcal{H}-$representation of the polyhedron generated by the union of the set of vertices $V=\cup_{i,j}V_{i,j}$, and the union of the set of rays $R=\cup_{i,j}R_{i,j}$, in the form (10) 5: If $L>0$ then $q(\mathbf{x})$ is given by (11). Otherwise, $q(\mathbf{x})=-\infty$. 6: Compute the set of indices $i$ such that there is a $j$ with $p(\mathbf{x}_{i,j})-d(\mathbf{x}_{i,j})-q(\mathbf{x}_{i,j})>0$. Compute $r(\mathbf{x})$ according to (13) 7: Return $q(\mathbf{x})$, $r(\mathbf{x})$ ``` Algorithm 1 Exact Polynomial Division Proposition 5: _If $E\neq\mathbb{R}^{n}$, then the output of the algorithm is indeed the quotient and the remainder of the division. If $E=\mathbb{R}^{n}$, then the quotient is $q(\mathbf{x})=-\infty$, for all $x\in\mathbb{R}^{n}$ and the remainder is equal to the dividend, i.e., $r=p$._ _Proof_ See Appendix F. _Remark 4:_ The algorithm involves first computing the resolution of a number of polyhedra and then the computation of the $\mathcal{H}-$representation of the convex hull of their union. The computational complexity of these operations is exponential on their input size (e.g. [48]). Therefore, we expect that the algorithm will be usable only for small examples. _Example 2:_ Let us divide the tropical polynomial $p(x,y)=\max(0,3x+3y,6x)$, by $d(x,y)=\max(x,x+y,2x+y)$, using Algorithm 1. The quotient is \[q(x,y)=\max(1.5x+1.5y,3x,0),\] and $d(x)+q(x)$ is given by $\max(x,x+y,2.5x+2.5y,3.5x+2.5y,5x+y,4x)$. The linearity regions of $f(x,y)$ and $q(x)$ are shown in Figure 4. Furthermore, Figure 4 illustrates the Newton polygons for $p,d$ and $d+q$. ## V Approximate Algorithms For Tropical Division We now present an approximate algorithm for computing the quotient of a tropical division, assuming that the quotient has a predefined maximum number of terms $\tilde{m}_{q}$. The algorithm mimics the procedure developed in [33]. The quotient has the form $q(\mathbf{x})=\bigvee_{i=1}^{m_{q}}(\hat{\mathbf{a}}_{i}^{T}\mathbf{x}+\hat{b}_{i})$. From the proof of Proposition 2 we know that $q(\mathbf{x})$ is the maximum tropical polynomial function, all the terms of which satisfy $\hat{\mathbf{a}}_{i}^{T}\mathbf{x}+\hat{b}_{i}\leq f(\mathbf{x})$, for all $\mathbf{x}\in\mathbb{R}^{n}$. We will utilize this property to formulate an optimization problem on a set of sample points $\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\in\mathbb{R}^{n}$. Given the set of sample points and the corresponding set of the values of function $f(\cdot)$, i.e., $f(\mathbf{x}_{1}),\ldots,f(\mathbf{x}_{N})$, the optimization problem is formulated as: \[\underset{\hat{\mathbf{a}}_{i},\hat{b}_{i},i=1,\ldots,\hat{m}_{q}}{ \text{maximize}} \sum_{j=1}^{N}\bigvee_{i=1}^{\tilde{m}_{q}}(\hat{\mathbf{a}}_{i}^{T} \mathbf{x}_{j}+\hat{b}_{i})\] (14) subject to \[\hat{\mathbf{a}}_{i}^{T}\mathbf{x}_{j}+\hat{b}_{i}\leq f(\mathbf{x}_{j}), \forall i,j\] \[\hat{\mathbf{a}}_{i}\in C,\ \ i=1,\ldots,\tilde{m}_{q}\] where $C$ is the set defined in (9). Since $C$ is a polytope, the last constraint in (16) can be written as a set of linear inequalities. The details are presented in subsection V-A. The algorithm alternates between two phases. Phase 1 partitions the data. Particularly, assuming a set of solutions Fig. 3: _Illustration of the algorithm applied to Example 1. Function $f(x)=p(x)-d(x)$ is linear on each of the subsets $P_{1},\ldots,P_{b}$. The epigraphs are generated as follows: $E_{1}$ by vertex $V_{1}$ and rays $R_{1}$ and $R_{2}$, epigraph $E_{2}$ by vertices $V_{1},V_{2}$ and ray $R_{2}$, epigraph $E_{3}$ by vertices $V_{2},V_{3}$ and ray $R_{2}$, epigraph $E_{4}$ by vertices $V_{3},V_{4}$ and rays $R_{2}$, and finally epigraph $E_{5}$ by vertex $V_{4}$ and rays $R_{2},R_{3}$. Their closed convex hull $E$ is generated by vertices $V_{1},V_{2},V_{4}$ and rays $R_{1},R_{3}$. The quotient $q(\mathbf{x})$ is computed by $\mathcal{H}-$representation of $E$._ Fig. 4: _The linear regions of $f(x,y)$ and $q(x,y)$. The two functions coincide on the bold half-lines. The figure also illustrates the Newton polygons of $p(x,y)$, $d(x,y)$, and $d(x,y)+q(x,y)$._ $(\hat{\mathbf{a}}_{i},b_{i})$, $i=1,\ldots,\tilde{m}_{q}$, we partition the samples $\mathbf{x}_{j}$ into sets $I_{1},\ldots,I_{\tilde{m}_{q}}$ with $j\in I_{i}$, if \[\hat{\mathbf{a}}_{i}^{T}\mathbf{x}_{j}+\hat{b}_{i}\geq\hat{\mathbf{a}}_{i^{\prime}}^{T}\mathbf{x} _{j}+\hat{b}_{i^{\prime}},\ \ \text{for all }i^{\prime}. \tag{15}\] Phase 2, solves the linear optimization problem: \[\underset{\hat{\mathbf{a}}_{i},b_{i},i=1,\ldots,\tilde{m}_{q}}{\text{ maximize}} \sum_{i=1}^{\tilde{m}_{q}}\hat{\mathbf{a}}_{i}^{T}\mathbf{x}_{j}+\hat{b}_{i}\] (16) subject to \[\hat{\mathbf{a}}_{i}^{T}\mathbf{x}_{j}+\hat{b}_{i}\leq f(\mathbf{x}_{j}),\ \ \forall i,j\] \[\hat{\mathbf{a}}_{i}\in C,\ \ i=1,\ldots,\tilde{m}_{q}\] We then simplify (16) in two ways. First, it may contain redundant inequalities. To remove the redundant inequalities we compute the lower convex hull of the set $\{(\mathbf{x}_{j},f(\mathbf{x}_{j})):j=1,\ldots,N\}$ with respect to its last component, and denote by $J$ the set of indices $j$, for which $(\mathbf{x}_{j},f(\mathbf{x}_{j})$ belongs to the lower convex hull. Second, problem (16) decomposes into a set of $\tilde{m}_{q}$ linear programs \[\underset{\hat{\mathbf{a}}_{i},b_{i}}{\text{maximize}} \mathbf{s}_{i}^{T}\hat{\mathbf{a}}_{i}+N_{i}\hat{b}_{i}\] (17) subject to \[(\mathbf{x}_{j})^{T}\ 1]\begin{bmatrix}\hat{\mathbf{a}}_{i}\\ \hat{b}_{i}\end{bmatrix}\leq f(\mathbf{x}_{j}),\ \ j\in J\ \,\] \[\hat{\mathbf{a}}_{i}\in C\] where $\mathbf{s}_{i}=\sum_{j\in I_{i}}\mathbf{x}_{j}$, $N_{i}=\|I_{i}\|$. The computations are summarized in Algorithm 2. ``` 0: Input: Tropical Polynomials $p(\mathbf{x})$, $d(\mathbf{x})$, the degree of the approximate quotient $\tilde{m}_{q}$, the number of sample points $N$, the maximum number of iterations 1: Compute the inequalities corresponding to set $C$ using the formulas in Appendix V-A 2: Sample the points $\mathbf{x}_{j}$, for $j=1,\ldots,N$, and compute the values of $f(\mathbf{x}_{j})=p(\mathbf{x}_{j})-d(\mathbf{x}_{j})$ 3: Compute the lower convex hull of the set $\{(\mathbf{x}_{j},f(\mathbf{x}_{j})):j=1,\ldots,N\}$, and use the samples belonging to it to formulate the first constraint in (16) 4: Initialize the partition $I_{1},\ldots,I_{\tilde{m}_{q}}$ 5: Solve Problems (17), for all $i$ 6: Compute the partitions $I_{i}$ that satisfy (15) 7: If the number of iterations has not exceeded the maximum number of iterations go to Step 5 8: Return $\hat{q}(\mathbf{x})=\sqrt{\hat{m}_{i=1}^{\tilde{m}_{q}}}\hat{\mathbf{a}}_{i}^{T}\mathbf{x }+\hat{b}_{i}$, ``` Algorithm 2 Approximate Polynomial Division We then examine the quality of approximation of $p(\mathbf{x})$ by $d(\mathbf{x})+q_{t}(\mathbf{x})$ on the sample set given by \[e(t)=\sum_{j=1}^{N}[p(\mathbf{x}_{j})-d(\mathbf{x}_{j})-q_{t}(\mathbf{x}_{j})], \tag{18}\] where $q_{t}$ is the quotient computed after $t$ steps of the algorithm. The following proposition shows that the quality of approximation improves as the number of steps increases. Proposition 6: _The quantity $e(t)$ is nonnegative and non-increasing with respect to $t$._ Proof:: See Appendix G. Remark 5: _In practice some issues with Algorithm 1 may arise. Some of the classes $I_{i}$ may end up empty. In this case, we may split another existing class randomly. Furthermore, we expect the algorithm to converge to a local optimum. We may use a multiple start method to avoid bad local optima._ Remark 6: _Let us comment about the relationship between Algorithm 2 and [33]. Inspired by [33], Algorithm 2 partitions the data in several clusters. The major difference is that in Algorithm 2 we use the sharper lower linear approximation of each class of data subject to the constraint that this approximation does not exceed any sample point of any class. Thus, instead of alternating between clustering and linear regression, we alternate between clustering and linear programming._ Another related work is [49] which studies the problem of computing a convex under-approximation for a given dataset by a piecewise linear function. The algorithm uses an iterative linear programming technique and has two phases. At phase one of each iteration, it considers a subset of the data set and minimizes the maximum approximation error for this subset, using a piecewise linear function with a number of terms equal to the iteration count. At phase two, it adds to this subset the data point that maximizes the approximation error. Then, it moves to the next iteration. ### _The Linear Constraints for Set $C$_ We next compute a set of linear inequalities describing set $C$. Let the extreme points of $\text{Newt}(p)$ be $\mathbf{a}_{e,1},\ldots,\mathbf{a}_{e,k}$. Then, $\hat{\mathbf{a}}\in C$ if and only if $\hat{\mathbf{a}}+\text{Newt}(d(\mathbf{x}))\subset\text{Newt}(p(\mathbf{x}))$. That is, \[\hat{\mathbf{a}}+\tilde{\mathbf{a}}_{j}\in\text{conv}\{\mathbf{a}_{e,1},\ldots,\mathbf{a}_{e,k}\},\] for all $j$. Equivalently if there are auxiliary variables $\lambda_{j,l}$, such that \[\hat{\mathbf{a}}+\tilde{\mathbf{a}}_{j}=\sum_{l=1}^{k}\mathbf{a}_{e,l}\lambda_{j,l},\ \ \ \lambda_{j,l}\geq 0,\ \ \ \ \sum_{l=1}^{k}\lambda_{j,l}=1\] for all $j=1,\ldots,m_{q}$, $l=1,\ldots,k$. Problem (17) becomes: \[\underset{\hat{\mathbf{a}}_{i},b_{i},\lambda}{\text{maximize}} \mathbf{s}_{i}^{T}\hat{\mathbf{a}}_{i}+N_{i}\hat{b}_{i}\] (19) subject to \[[\mathbf{x}_{j}^{T}\ 1]\begin{bmatrix}\hat{\mathbf{a}}_{i}\\ \hat{b}_{i}\end{bmatrix}\leq f(\mathbf{x}_{j}),\ \ j\in J\] \[\hat{\mathbf{a}}_{i}+\tilde{\mathbf{a}}_{j}=\sum_{l=1}^{k}\mathbf{a}_{e,l} \lambda_{j,l},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda_{j,l}\geq 0\] \[\sum_{l=1}^{k}\lambda_{j,l}=1\] ## VI Division for Composite Polynomials In this section, we present some analytical results and some algorithms for dividing tropical polynomials that are sums or maxima of simpler ones. ### _General Properties_ We will use $Q(p,d)$ and $R(p,d)$ to denote the quotient and remainder of the division of a tropical polynomial $p$ by another tropical polynomial $d$. Proposition 7: _The following inequalities hold_ \[Q(p_{1}+p_{2},d) \geq Q(p_{1},d)+Q(p_{2},d)+d, \tag{20}\] \[Q(p_{1}\lor p_{2},d) \geq Q(p_{1},d)\lor Q(p_{2},d),\] (21) \[Q(p,d_{1}+d_{2}) \geq Q(p,d_{1})+Q(p,d_{2})-p,\] (22) \[Q(p,d_{1}\lor d_{2}) \geq Q(p,d_{1})\wedge Q(p,d_{2}). \tag{23}\] _Furthermore, if $R(p_{1},d)=R(p_{2},d)=-\infty$, then (20),(21),(22) hold as equalities. Finally, if $s$ is a monomial, i.e., it has the form $s(\mathbf{x})=\mathbf{a}^{T}\mathbf{x}+b$, then_ \[Q(p+s,d)=Q(p,d)+s=Q(p,d-s), \tag{24}\] _Proof_ See Appendix H. Remark 7: _Assume that polynomial $d$ is fixed. Inequality (21) implies that function $Q(\cdot,d)$ is nondecreasing. If $d(\mathbf{x})\geq 0$, for all $\mathbf{x}\in\mathbb{R}^{n}$, then function $Q(\cdot,d)$ is also super-additive._ In the following we call a tropical polynomial described as a summation of other tropical polynomials as composite. Furthermore, we refer to a tropical polynomial in the form 1 as simple1. Footnote 1: These notions refer to the representation of the tropical polynomials, and not to whether or not they can be written as a sum of other polynomials. That is, it is possible that a tropical polynomial with a simple representation (in the form (1)), can be factorized and written in a composite form. ### _Dividing a Composite Polynomial by a Simple_ In the following we propose algorithms to approximately divide a composite polynomial $p(\mathbf{x})=\sum_{\nu=1}^{N}p^{\nu}(\mathbf{x})$ or $p(\mathbf{x})=\bigvee_{\nu=1}^{N}p^{\nu}(\mathbf{x})$ by another polynomial $d(\mathbf{x})$. The computation of the value of the composite polynomial $p(\mathbf{x})$ is trivial, provided the values of the simpler polynomials $p^{\nu}(\mathbf{x})$. Thus, the only difficulty in order to apply Algorithm 2 is the computation of the Newton polytope $\text{Newt}(p)$. It is convenient to consider the Newton polytopes $\text{Newt}(p^{\nu})$ in terms of an extended description. Particularly, assume matrices $\mathbf{A}^{\nu},\mathbf{F}^{\nu}$, and vectors $\mathbf{\beta}^{\nu}$ of appropriate dimensions, such that \[\text{Newt}(p^{\nu})=\{\mathbf{a}\in\mathbb{R}^{n}:\mathbf{a}=\mathbf{F}^{\nu}\mathbf{\alpha} ^{\nu},\mathbf{A}^{\nu}\mathbf{\alpha}^{\nu}\leq\mathbf{\beta}_{\nu}\}. \tag{25}\] For $p(\mathbf{x})=\sum_{\nu=1}^{N}p^{\nu}(\mathbf{x})$, the Newton polytope is given by \[\text{Newt}(p)=\{\mathbf{F}^{1}\mathbf{\alpha}^{1}+\cdots+\mathbf{F}^{N}\mathbf{\alpha}^{N}: \mathbf{A}^{\nu}\mathbf{\alpha}^{\nu}\leq\mathbf{\beta}_{\nu},\nu=1,\ldots,N\},\] which corresponds to the Minkowski sum $\text{Newt}(p^{1})\oplus\cdots\oplus\text{Newt}(p^{N})$. For $p(\mathbf{x})=\bigvee_{\nu=1}^{N}p^{\nu}(\mathbf{x})$, the Newton polytope is given by \[\text{Newt}(p)= \{\mathbf{F}^{1}\mathbf{\alpha}^{1}+\cdots+\mathbf{F}^{N}\mathbf{\alpha}^{N}: \exists\lambda^{\nu}\geq 0,\nu=1,\ldots,N,\] \[\mathbf{A}^{\nu}\mathbf{\alpha}^{\nu}\leq\mathbf{\beta}_{\nu}\lambda^{\nu}, \ \sum_{\nu=1}^{N}\lambda^{\nu}=1\}.\] ### _Neural Networks As Differences of Composite Polynomials_ We then present some examples of neural networks represented as the difference of two composite tropical polynomials. Example 3 (Single hidden layer ReLU network): _This example follows [9]. Consider a neural network with $n$ inputs, a single hidden layer with $m_{p}$ neurons and ReLU activation functions, and a single linear output neuron. The output of the $\nu$-th neuron of the hidden layer is:_ \[z_{\nu}=\max(\mathbf{w}_{\nu}^{T}\mathbf{x}+\beta_{\nu},0)\] _The output $y$ can be written as_ \[y=\sum_{\nu=1}^{N_{1}}w_{\nu}^{2+}z_{\nu}-\sum_{\nu=N_{1}+1}^{N}w_{\nu}^{2-}z_ {\nu}+\beta^{2},\] _where $N_{1}+N_{2}=N$, and $N_{1},N_{2}$ are the number of neurons that have positive weight and negative weights respectively. Without loss of generality, we assume that the neurons are ordered such that the weights of the first neurons to be positive and the weights of the last negative. The output of the neural network can be expressed as the difference of two tropical polynomials $p_{1}(\mathbf{x})$, $p_{2}(\mathbf{x})$, plus a constant term. Each of these tropical polynomials is written as the sum of simpler ones_ \[p_{1}(\mathbf{x}) =\sum_{\nu=1}^{N_{1}}\max(w_{\nu}^{2+}\mathbf{w}_{\nu}^{T}\mathbf{x}+w_{ \nu}^{2+}\beta_{\nu},0)\] \[=\sum_{\nu=1}^{N_{1}}\max((\mathbf{a}_{1}^{\nu})^{T}\mathbf{x}+b_{\nu},0)\] \[p_{2}(\mathbf{x}) =\sum_{\nu=N_{1}+1}^{N}\max(w_{\nu}^{2-}\mathbf{w}_{\nu}^{T}\mathbf{x}+w_ {\nu}^{2-}\beta_{\nu},0)\] \[=\sum_{\nu=N_{1}+1}^{N}\max((\mathbf{a}_{2}^{\nu})^{T}\mathbf{x}+b_{\nu},0)\] _Note that $p_{1}$, $p_{2}$, expressed in their canonical form (1), can have a large number of terms, corresponding to linear regions of tropical polynomial functions $p_{1}$, $p_{2}$ (for an enumeration of the linear regions see [3, 4])._ The Newton polytope of the simple tropical polynomial $\max((\mathbf{a}_{1}^{\nu})^{T}\mathbf{x}+b_{\nu},0)$ is the line segment $[0,\mathbf{a}_{1}^{\nu}]$ in the $d$-dimensional space. This polytope is equivalently described in the form (25) as $\text{Newt}(p^{\nu})=\{\alpha^{\nu}\mathbf{a}_{1}^{\nu}:0\leq\alpha^{\nu}\leq 1\}$. Thus, the Newton polytope of $p_{1}$ is given by \[\text{Newt}(p_{1})=\{\alpha^{1}\mathbf{a}_{1}^{1}+\cdots+\mathbf{\alpha}^{1}\mathbf{a}_{1}^{ n_{1}}:0\leq\alpha^{\nu}\leq 1\},\] which is a zonotope (see also [4]). The constraint $\mathbf{\hat{a}}_{i}\in C$ of problem (17) becomes \[\mathbf{\hat{a}}_{i}+\mathbf{\bar{a}}_{j}=\sum_{l=1}^{n}\mathbf{a}^{l}\lambda_{j,l},\ \ 0\leq \lambda_{j,l}\leq 1.\] Finally, note that if the number of neurons of the hidden layer $N_{1},N_{2}$ is smaller than the input dimension $n$, then the tropical polynomials $p_{1},p_{2}$ can be expressed in dimensions $N_{1},N_{2}$. Example 4 (Multi-class to binary classification with a ReLU network): _Assume a neural network with a single ReLU hidden layer with $N$ units and a softmax output layer with $K$ units trained to classify samples to $K$ classes. The equations are the following \[z_{\nu}^{1}=\max((\mathbf{w}_{\nu}^{1})^{T}x+b_{\nu}^{1},0),\ \ z_{k}^{2}=(\mathbf{w}_{k}^{2})^{T}\mathbf{z}^{1},\ \ y_{k}=\frac{e^{z_{k}^{2}}}{\sum_{k^{ \prime}}e^{z_{k^{\prime}}^{2}}}.\] We then assume, that for a given set of samples, we know that the correct class is either $i_{1}$ or $i_{2}$. The neural network can be simply reduced for binary classification, substituting the last layer with a single neuron with sigmoid activation with output given by \[y=\sigma((\mathbf{w}_{i_{1}}^{2}-\mathbf{w}_{i_{2}}^{2})^{T}\mathbf{z}^{2}+\beta^{2}),\] where $\sigma(\cdot)$ is the sigmoid activation function,, i.e. $\sigma(z)=1/(1-e^{-z})$. Note that this transformation corresponds to the application of Bayes' rule with prior probability of $1/2$ for classes $i_{1},i_{2}$, and likelihoods given by the output of the initial neural network. The output of the new network can be represented as a difference of two tropical polynomials as in the previous example. Example 5 (Representing a multi-class network as a vector of tropical polynomials): _Consider a neural network with a single-hidden layer having ReLU activations and a linear output layer with $K$ neurons, followed by softmax. Each output (before the softmax) $z_{k}^{2}$ can be expressed as a difference of two tropical polynomials plus a constant term_ \[z_{k}^{2}=p_{k}^{+}(\mathbf{x})-p_{k}^{-}(\mathbf{x})+b_{k}.\] _The output is given by_ \[y_{k}=\frac{e^{z_{k}^{2}}}{\sum_{k^{\prime}}e^{z_{k^{\prime}}^{2}}}.\] _Let us add to each of $z_{k}^{2}$'s the sum of all negative polynomials. We get the tropical polynomials_ \[\tilde{z}_{k}^{2}=p_{k}^{+}(\mathbf{x})+\sum_{k^{\prime}\neq k}p_{k^{\prime}}^{-}( \mathbf{x})+b_{k}. \tag{26}\] _Then, it is easy to see that_ \[y_{k}=\frac{e^{\tilde{z}_{k}^{2}}}{\sum_{k^{\prime}}e^{\tilde{z}_{k^{\prime}} ^{2}}}.\] _Finally, denoting by $z_{i}^{1}$, $i=1,...M$ the output of the hidden layer before the ReLU, then $\tilde{z}_{k}^{2}$ can be written in the form_ \[\tilde{z}_{k}^{2}=\sum_{i=1}^{M}\bar{w}_{k,i}\max(z_{i}^{1},0), \tag{27}\] _for appropriate non-negative constants $\bar{w}_{k,i}$._ Remark 8: _In the tropical framework, the difference of two polynomials is considered as a tropical rational function. The addition of all the negative polynomials (denominators) corresponds to making the tropical fractions have the same denominator._ ### _Division Algorithm for Composite Quotient_ We then describe an algorithm to divide a composite tropical polynomial $p(\mathbf{x})=\sum_{i=1}^{n}\max(\mathbf{a}_{i}^{T}\mathbf{x}+b_{i},0)$, where $n$ is the dimension of the underlying space, by the zero polynomial $d(\mathbf{x})=0$ and search for a quotient in the form $q(\mathbf{x})=\sum_{i=1}^{m_{q}}\max(\hat{\mathbf{a}}_{i}^{T}\mathbf{x}+\hat{b}_{i},0)$. Such polynomials occur in single hidden layer ReLU neural networks if the number of neurons in the hidden layer is smaller than the space dimension, using a dimensionality reduction transformation (e.g., QR). We assume that the vectors $\mathbf{a}_{i}$ are linearly independent. Denote by $\mathbf{A}$ the square matrix of vectors $\mathbf{a}_{i}$, i.e., $\mathbf{A}=[\mathbf{a}_{1}\ \ldots\mathbf{a}_{n}]$, and by $\mathbf{B}$ the vector of $b_{i}$, i.e., $\mathbf{B}=[b_{1}\ \ldots\ b_{n}]^{T}$. The algorithm is based on the fact that the quotient $q$ is the largest tropical polynomial function which is less than or equal to $p$. Consider a set of sample points $x^{1},\ldots,x^{N}$. Then the problem of maximizing $q$ can be approximated by \[\underset{\hat{\mathbf{a}}_{i},\hat{b}_{i},i=1,\ldots,m_{i_{q}}}{ \text{maximize}} \sum_{j=1}^{N}\sum_{i=1}^{m_{q}}\max(\hat{\mathbf{a}}_{i}^{T}\mathbf{x}^{j }+\hat{b}_{i},0)\] (28) subject to \[q(\mathbf{x})\leq p(\mathbf{x}),\ \ \text{for all}\ x\in\mathbb{R}^{n}\] The following proposition reformulates the constraint $q(\mathbf{x})\leq p(\mathbf{x}),\ \ \text{for all}\ x\in\mathbb{R}^{n}$ in terms of the variables $\hat{\mathbf{a}}_{i},\hat{b}_{i},i=1,\ldots,m_{q}$. Proposition 8: _Let $p(\mathbf{x}),q(\mathbf{x})$ as above. Then, $q(\mathbf{x})\leq p(\mathbf{x})$, for all $x\in\mathbb{R}^{n}$ if and only if_ \[\mathbf{0}\leq\mathbf{A}^{-1}\hat{\mathbf{a}}_{i},\ \ \mathbf{A}^{-1}\sum_{i=1}^{m_{n_{q}}}\hat{\mathbf{a}} _{i}\leq\mathbf{1},\ \ \hat{b}_{i}\leq\mathbf{B}^{T}\mathbf{A}^{-1}\hat{\mathbf{a}}_{i} \tag{29}\] _for $i=1,\ldots,m_{q}$, where $\mathbf{0}$ is the vector having all its entries equal to zero and $\mathbf{1}$ all entries equal to one._ Phase 1: _For the set of sample points $\mathbf{x}^{1},\ldots,\mathbf{x}^{N}$ we compute $y_{i,j}=\chi_{\hat{\mathbf{a}},\mathbf{x}^{j}+b_{i}\geq 0}$, where $\chi$ is the indicator function (i.e., $\chi_{\hat{\mathbf{a}},\mathbf{x}^{j}+b_{i}\geq 0}=1$ if $\hat{\mathbf{a}}_{i}\mathbf{x}^{j}+b_{i}\geq 0$ and $0$ otherwise). Based on the values of $y_{i,j}$, we determine the local form of the objective in (28). Then, we compute $\mathbf{c}_{1}$ and $\mathbf{c}_{2}$ as_ \[\big{[}\sum_{j=1}^{N}y_{1,j}\mathbf{x}_{j}^{T}\ \ldots\ \sum_{j=1}^{N}y_{m_{n_{q}},j}\mathbf{x}_{j}^{T} \big{]},\ \big{[}\sum_{j=1}^{N}y_{1,j}\ \ldots\ \sum_{j=1}^{N}y_{m_{n_{q}},j}\big{]}.\] Phase 2: _We solve the linear programming problem_ \[\underset{\hat{\mathbf{a}}_{i},\hat{b}_{i},i=1,\ldots,m_{i_{q}}}{ \text{maximize}} \mathbf{c}_{1}\begin{bmatrix}\hat{\mathbf{a}}_{1}\\ \vdots\\ \hat{\mathbf{a}}_{m_{n_{q}}}\end{bmatrix}+\mathbf{c}_{2}\begin{bmatrix}\hat{b}_{1}\\ \vdots\\ \hat{b}_{m_{n_{q}}}\end{bmatrix}\] (30) subject to \[\mathbf{0}\leq\mathbf{A}^{-1}\hat{\mathbf{a}}_{i},\ \ \ \hat{b}_{i}\leq\mathbf{B}^{T}\mathbf{A}^{-1}\hat{\mathbf{a}}_{i}\] \[\mathbf{A}^{-1}\sum_{i=1}^{m_{q}}\hat{\mathbf{a}}_{i}\leq\mathbf{1},\] _and denote its solution by $\hat{\mathbf{a}}_{i}^{\text{new}},b_{i}^{\text{new}}$. We set_ \[\hat{\mathbf{a}}_{i}\leftarrow(1-\rho)\hat{\mathbf{a}}_{i}+\hat{\mathbf{a}}_{i}^{\text{new}}\rho, \tag{31}\] \[\hat{b}_{i}\leftarrow(1-\rho)b_{i}+\rho b_{i}^{\text{new}},\] _where $\rho\in(0,1)$ and go to Phase 1._ Remark 9: _The algorithm corresponds to the application of Frank-Wolfe (conditional gradient) method (see, e.g., [50]) to (28)._ Let us note that this linear programming problem (30) is analytically solvable. Indeed, observing that the coefficients of $\hat{b}_{i}$'s are nonnegative, we can use the second inequality to compute the the optimal value of $\hat{b}_{i}$'s. Substituting back to the problem, we end up with $n$ independent linear programs. The feasible set of each of these problems has $\tilde{m}_{q}+1$ vertices. We obtain the optimal solution of the problem comparing the value of the linear objective on these vertices. Remark 10: _Let us comment on the difference of the algorithm of this subsection with the algorithm of Subsection VI-B. The algorithm of this subsection provides a solution a composite tropical polynomial, which corresponds to a (compressed) neural network with ReLu activations, while the algorithm of Subsection VI-B provides maxout solutions._ ### _Dividing a Vector of Composite Polynomials_ In this subsection, we propose a simplified algorithm for dividing a vector of tropical polynomials by the zero polynomial. The motivation comes from Example 5. Consider a set of tropical polynomials in the form (27). We then describe a simplified version of the the linear optimization part of Algorithm 2. Consider a set of sample points $\boldsymbol{z}^{1}(1),\ldots,\boldsymbol{z}^{1}(N)$. Observe that, with this formulation, both constraints (17) are satisfied by any $\hat{\boldsymbol{a}}_{k}$, such that its $i-$th component $\hat{a}_{i,k}$ satisfies $0\leq\hat{a}_{i,k}\leq\bar{w}_{i,k}$ and $\hat{b}_{k}=0$. Then, the solution of (17), for the $l-th$ term of the division of the $k-$th polynomial is approximated by the optimal solution of \[\begin{split}\underset{\hat{\boldsymbol{a}}_{k}^{l}}{\text{maximize}}& \boldsymbol{s}_{l,k}^{T}\hat{\boldsymbol{a}}_{k}^{l}\\ \text{subject to}& 0\leq\hat{a}_{i,k}^{l}\leq\bar{w}_{i,k}, \;\;i=1,\ldots,M\end{split}. \tag{32}\] The solution of (32) is given by: \[\hat{a}_{i,k}^{l}=\begin{cases}\bar{w}_{i,k}&\text{if}\;\;\;s_{i,l,k}>0\\ 0&\text{otherwise}\end{cases}\] ## VII Numerical Results This section, presents some numerical examples for the tropical division algorithms. First, we present some simple examples to build some intuition for the behaviour of Algorithm 2. Then, some applications of tropical division to the compression of neural networks are presented. We focus on MNIST handwritten and CIFAR-10 datasets. ### _Numerical Examples for Algorithm 2_ As a first example, we present the application of Algorithm 2 to the tropical polynomial division of Example 2. We choose $200$ sample points distributed according to the normal distribution with unit variance $\mathcal{N}(0,I_{2})$. The algorithm converges in two steps and gives an approximate solution2. Footnote 2: The code for the numerical experiments is available online at [https://github.com/jkordonis/TropicalML](https://github.com/jkordonis/TropicalML). \[\hat{q}(x,y)= \max(1.5038x+1.4962y+0.0182,\] \[0.0003x+0.0288,3x+0.0241),\] which is very close to the actual quotient (see Example 2). Running again the algorithm with larger sample sizes we conclude that increasing the number of sample points reduces the error. As a second example we divide a random tropical polynomial with $m_{p}=128$ terms in $n=3$ dimensions with another having $m_{d}=2$ terms. First, we run Algorithm 2, for $\tilde{m}_{q}=5$, with multiple initial partitions. The evolution of the error for the several runs is shown in Figure 5. The results illustrate the need for a multi-start method. Indeed, for different random initial partitions the algorithm converges to different local minima. The quality of the approximation $e$ after 10 steps of the algorithm is given in Figure 6 for a varying number of terms $\tilde{m}_{q}$. This result shows that we may derive a good approximation of the division using a small number of terms (e.g. $8$). ### _MNIST Dataset_ This example considers the MNIST handwritten digits data set. It consists of $60000$ training examples and $10000$ test examples of $28\times 28$ gray scale images, that represent digits $0-9$. We start with a single hidden layer neural network with 100 hidden units with ReLU activations and $10$ softmax output units. The network is trained in the original training data set using standard techniques (Adam optimizer on cross-entropy loss, batch size of 128). We then design a smaller network discriminating between digits $3$ and $5$. The input space has $28\cdot 28=784$ dimensions. We first use the technique described in Example 4 to reduce the output layer of the original network to a single neuron. Then, using the technique described in Example 3, we express Fig. 5: _The total error $e$ for executions of Algorithm 2, for different initial partitions._ Fig. 6: _The total error $e$ for divisions having different number of terms for the approximate quotient_ the output (before the sigmoid) as the difference of two tropical polynomials. These polynomials are expressed as the summation of $100$ terms in the form $\max(\mathbf{\alpha}_{l}^{r}\mathbf{x},0)$, where $l=1,2$. We then apply a reduced QR decomposition to the matrices $\mathbf{A}_{1}=[\mathbf{\alpha}_{1}^{1}\ \ldots,\ \mathbf{\alpha}_{1}^{100}]$ and $\mathbf{A}_{2}=[\mathbf{\alpha}_{2}^{1}\ \ldots,\ \mathbf{\alpha}_{2}^{100}]$, writing them as $\mathbf{A}_{1}=\mathbf{Q}_{1}\mathbf{A}_{1}^{T}$ and $\mathbf{A}_{2}=\mathbf{Q}_{2}\mathbf{A}_{2}^{T}$. The $\mathbf{A}_{l}^{r}$ matrices have as columns $\mathbf{\alpha}_{l}^{r,y}$, i.e., an expression of the vectors $\mathbf{\alpha}_{l}^{r}$ in the subspace of their span. For each polynomial the input has been transformed as $\mathbf{x}_{l}=\mathbf{Q}_{l}^{T}\mathbf{x}$ (recall Remark 3). We divide each of these polynomials with the zero polynomial $d=0$, applying two different variants of the Algorithm 2. In both cases, we use as sample points the first $200$ training points. The first variant (modification of Section VI-B), the approximate quotients have the form $q_{l}(\mathbf{x}_{l})=\bigvee_{i=1}^{m_{q}}(\hat{\mathbf{a}}_{i}^{l}\mathbf{x}_{l}+\hat{b }_{i}^{l})$, $l=1,2$. Then, the output of the original network can be approximated as \[y^{\text{app}}=\sigma(q_{1}(\mathbf{Q}_{1}^{T}\mathbf{x})-q_{2}(\mathbf{Q}_{2}^{T}\mathbf{x}) +\beta^{2})\] The computation of the approximate output corresponds to a neural network an input layer, a hidden layer with two maxout units, a linear layer with a single unit (computing the difference of the two maxout units) and an output layer with a single sigmoid unit. The second variant, uses approximates the quotients in the form $\sum_{i=1}^{m_{q}}\max(\hat{\mathbf{a}}_{i}^{T}\mathbf{x}+\hat{b}_{i},0)$ and uses the modification of subsection VI-D. The compressed network in this case is simply a smaller ReLU network. Table I presents the error rate on the test set for the original neural network, as well as the computed maxout and ReLU networks. We compare the reults when each maxout unit has $3$, $5$, and $10$ terms respectively and each of the tropical polynomials (positive and negative) has $3$, $5$, and $10$ ReLU units. It also presents the percentage of parameters that remain after the compression. As a baseline for comparison, we use structured L1 pruning without retraining (see e.g. [51]). That is, this method deletes the neurons having weights with small L1 norm. We compare reduced neural networks with the same number of parameters. We present two results, the binary comparison of digits $3$ and $5$ and an average of the $45=10\cdot 9/2$ different binary comparisons. The performance of the algorithms is shown in Figure 7. We observe that the algorithm of Section VI-B produce superior results compared to the algorithm of Section VI-D. Furthermore, both algorithms outperform L1 structured pruning, especially in the high compression regime. Let us note that the linear programming problem of Section VI-D is analytically solvable, and thus the corresponding algorithm is faster. Remark 11: _It is worth noting substituting $\mathbf{x}_{l}$ into $q_{l}$, we get_ \[q_{1}(\mathbf{Q}_{1}^{T}\mathbf{x})=\bigvee_{i=1}^{\hat{m}_{q}}(\hat{\mathbf{a}}_{i}^{l} Q_{l}^{T}\mathbf{x}+\hat{b}_{i}^{l}).\] _Thus we do not need to store $Q_{1},Q_{2}$ but only the vectors $\hat{\mathbf{a}}_{i}^{l}Q_{l}^{T}$, and the scalars $\hat{b}_{i}^{l}$, $\beta^{2}$, for $l=1,2$ and $i=1,\ldots,m_{q}$._ Remark 12: _Let us note that for computing the reduced networks, we used only the first 200 samples of the training data._ In the following, we continue exploring the algorithm of Section VI-B to more complex examples. ### _CIFAR-10 Dataset-Binary_ This example concerns CIFAR-10 dataset which contains small ($32\times 32$) color images. The training set consists of 50000 images of 10 classes (5000 samples for each class) and the test set of 10000 images (1000 samples per class). We first train a VGG-like network [52], having three blocks consisting of two convolution layers with ReLU activation followed by a $2\times 2$ max-pooling layer with padding, a dense layer with 1024 neurons, and an output layer with 10 neurons. The convolution layers in the first, second, and third block have 32, 64, and 128 neurons, respectively. The network was trained using standard techniques (Adam optimizer for the cross-entropy loss, batch normalization and dropout for regularization and data augmentation). Let us note that the input of the dense layer has dimension $4\cdot 4\cdot 128=2048$. We then design a simplified neural network discriminating between pairs of classes (e.g., "Automobile" and "Truck"). Fig. 7: _The average error rate of the compressed networks for the binary MNIST classification, compared with L1 structured pruning. Both techniques lead to better accuracies compared to L1 pruning. Furthermore, maxout networks clearly outperform both the L1 structured pruning and the ReLU compressed network, especially in the large compression regime._ To do so, we approximate the dense layer with 1024 neurons using the techniques of the previous section. Particularly, using the technique of Example 4 we describe the output of the neural network as the difference of two tropical polynomials in $2048$ dimensions. These polynomials are then divided by the zero polynomial applying the Algorithm 2, with the modification of Section VI-B, and using as sample points the first $500$ training points. Table II presents the error rate on the test set when the maxout units on the dense layer have $3$, and $5$ terms respectively. Three tests results are presented: on pairs Automobile-Truck (A-T), Cat-Dog (C-D) and the average of the 45=$9\cdot 10/2$ possible binary comparisons. The results obtained with the tropical division algorithm are compared with the L1 structured pruning. Table II also presents the number of parameters of the dense layer of the networks. We observe that the dense layer can be compressed to less than $1\%$ of its original size with minimum performance loss. Remark 13: _The proposed scheme is very competitive for the large compression regime, that is, in the case where there are very few parameters remaining. Furthermore, it can be performed assuming access to a small portion of the training data (500 out of 50000)._ ### _CIFAR-10 Multiclass_ We then compress the multi-class neural network trained in the previous subsection for the CIFAR-10 dataset. As the input to the compression algorithm, we consider the output of the convolutional part of the neural network with dimension 2048. We first represent the logits (the output before the softmax) as a vector of tropical polynomials using (26) in Example 5. We then simplify each of the tropical polynomials using the simplified algorithm of Section VI-E and a random sample of $500$ training points. The results are summarized in Table III. Additionally, we present an improved prediction scheme, where the $K\times\tilde{m}_{q}$ values $(\tilde{\mathbf{a}}_{k}^{l})^{T}\mathbf{z}^{1}$, with $k=1,\ldots,K$ and $l=1,\ldots,\tilde{m}_{q}$ are fed to a small single hidden layer ReLU neural network with 100 units. We refer to this small neural network as head. Note that the additional parameters for the improved prediction are very few. Indeed for the case of $\hat{m}_{q}=3,5,7$ we have $4110$, $6110$ and $8110$ parameters respectively. We observe again that the division algorithm works well on the large compression regime. Furthermore, in this example there is no improvement when we use more terms in the quotient polynomials. This is probably due to stacking in local optima. We then exploit the good behaviour of the binary classification algorithm and apply it to the multi-class problem. To do so, we use a subset of the tropical polynomials obtained for binary classification. In each binary comparison, the division algorithm gives two tropical polynomials and the $9\times 10/2=45$ binary comparisons give a total of $90$ tropical polynomials. The value of a subset of the tropical polynomials is fed to a single hidden layer network with ReLU activations, as described in Figure 8. Figure 9 presents the error of the compressed networks obtained using the multiple binary division algorithm of the previous paragraph. To produce these results we used a random set of $m$ tropical polynomials with $m=10,20,40,60$. We observe that the proposed method works better than L1-structured pruning, when the number of remaining parameters is small. Remark 14: _Note that in all the examples, the tropical division compression algorithms used only a small subset of the training data. This can be useful when simplifying a network where only parts of the training dataset are available._ Remark 15: _An interesting point of future research is to investigate whether producing additional sample points with the usual data augmentation techniques improves the final result._ \begin{table} \begin{tabular}{c\|\|c\|c\|c} Network & Orig. 2 classes & $\hat{m}_{q}=5$ & $\hat{m}_{q}=3$ \\ \hline Err. L1 A-T & 4.04\% & 25.8\% & 46.45\% \\ \hline Err. Div. X-A & 4.05\% & 4.6\% & 4.4\% \\ \hline Err. L1 C-D & 13.7\% & 49.3\% & 50\% \\ \hline Err. Div. C-D & 13.7\% & 15.4\% & 16.85\% \\ \hline Err. LiV avg. & 24.2\% & 25.4\% & 43.4\% & 43.62\% \\ \hline Err. Div. Avg. & 27.4\% 25.7\% & 3.85\%/3.6\% & 4.06\% & 3.37\% \\ \hline \# Param. & $2.1\cdot 10^{6}$ & $2\cdot 10^{4}$ & $1.2\cdot 10^{4}$ \\ \hline \% of Param. & 100\% & 0.95\% & 0.57\% \\ Remaining & 100\% & & 0.95\% \\ \end{tabular} \end{table} TABLE II: Results for the CIFAR-10 dataset. Error rate of the original network, the reduced maxout networks and the reduced networks obtained using the L1 structured pruning algorithm. We present the Automobile vs Truck case, the Cat vs Dog case, and the average of the $45=10\cdot 9/2$ different binary comparisons. The results are indexed using A-T, C-D, and Avg. \begin{table} \begin{tabular}{c\|c\|c\|c\|c} Network & Original & $\hat{m}_{q}=7$ & $\hat{m}_{q}=5$ & $\hat{m}_{q}=3$ \\ \hline Error L1 & 14.18\% & 45\% & 57.3\% & 71.7\% \\ \hline Error Div. & 14.18\% & 41$\pm$1.8 \% & 40.1 $\pm$ 1.7\% & 42.5 $\pm$ 2\% \\ \hline Err. Div. Improved & 14.18\% & 26.6\% 0.3 \% & 26.4 $\pm$ 0.3\% & 28.1 $\pm$ 0.1\% \\ \hline \# Param. & $2.1\cdot 10^{6}$ & $1.4\cdot 10^{6}$ & $10^{5}$ & $6.6\cdot 10^{4}$ \\ \hline \% of Param. & & & & \\ Remaining & 100\% & & & \\ \end{tabular} \end{table} TABLE III: Results for the compression on the CIFAR-10 dataset. Average error rates for multiple executions. Comparison of the original division algorithm, the improved division algorithm and the L1 pruning. The number of parameters include also the parameters of the head. Fig. 8: The structure of the proposed reduced network. Each pink box consists of linear neurons obtained by successive implementations of the division algorithm. ## VIII Conclusion This work proposed a new framework for tropical polynomial division and its application to neural networks. We showed the existence of a unique quotient-remainder pair and characterized the quotient as the convex bi-conjugate of the difference of the dividend from the divisor. This characterization led to an exact algorithm based on polyhedral geometry. We also proposed an approximate algorithm based on the alternation between clustering and linear programming. We then focused on dividing composite tropical polynomials and proposed two modified algorithms, one producing simple quotients and the other composite. Finally, we applied tropical polynomial division techniques to simplify neural networks with ReLU activations. The resulting neural networks have either maxout or ReLU activations, depending on the division algorithm chosen. The results are promising and compare favorably with a simple baseline (L1 pruning). There are several directions for future research. First, we may combine L1 regularization to the problem (19) to induce sparse solutions and improve further neural network compression. Another direction is to study compression problems for neural networks with many outputs, using a single division. A possible tool in this direction is the Cayley trick [53, 54]. Finally, the study of alternative optimization algorithms is of certain interest. ## Acknowledgment The authors are grateful to Dr. George Retsinas for his valuable suggestions and comments. ### _Proof of Proposition 1_ The proof follows closely the proof of Proposition 3.1.6 in [39]. Observe that \[p_{i}(x) =\max_{\mathbf{a}\in\mathbb{R}^{n}}\{[\mathbf{x}^{T}\ 1][\mathbf{a}^{T}\ \text{ ENF}_{p_{i}}(\mathbf{a})]^{T}\}\] \[=\max_{\mathbf{a}\in\mathbb{R}^{n}}\{[\mathbf{x}^{T}\ 1][\mathbf{a}^{T}\ \text{ ENF}_{p_{i}}(\mathbf{a})]^{T}\}. \tag{33}\] Which implies that if $\text{ENF}_{p_{1}}(\mathbf{a})\leq\text{ENF}_{p_{2}}(\mathbf{a})$ then $p_{1}(\mathbf{x})\leq p_{2}(\mathbf{x}),$ for all $\mathbf{x}\in\mathbb{R}^{n}$. On the other hand, assume that $p_{1}(\mathbf{x})\leq p_{2}(\mathbf{x})$, and that for some $\mathbf{a}_{0}$ it holds $\text{ENF}_{p_{1}}(\mathbf{a}_{0})>\text{ENF}_{p_{2}}(\mathbf{a}_{0})$. Then, the point $[\mathbf{a}_{0}^{T}\ \text{ENF}_{p_{1}}(\mathbf{a}_{0})]$ does not belong to the convex set $C=\{[\mathbf{a}^{T}\ z]\in\mathbb{R}^{n+1}:z\leq\text{ENF}_{p_{2}}(\mathbf{a})\}$. Thus, there is a $[c_{x}^{T}\ c_{z}]\in\mathbb{R}^{n+1}$ such that $\mathbf{c}_{x}^{T}\mathbf{a}_{0}+c_{z}\text{ENF}_{p_{1}}(\mathbf{a}_{0})>\mathbf{c}_{x}^{T}\bm {a}+c_{z}z$, for all $[\mathbf{a}^{T}\ z]\in C$. Since $C$ is unbounded below, $c_{z}>0$. Furthermore, since $[\mathbf{a}^{T}\ \text{ENF}_{p_{2}}(\mathbf{a})]\in C$ it holds \[\mathbf{a}_{0}^{T}\mathbf{x}_{0}+\text{ENF}_{p_{1}}(\mathbf{a}_{0})>\mathbf{a}^{T}\mathbf{x}_{0}+ \text{ENF}_{p_{2}}(\mathbf{a}),\] for all $\mathbf{a}\in\mathbb{R}^{n}$, where $\mathbf{x}_{0}=\mathbf{c}_{x}/c_{z}$. Taking the maximum with respect to $\mathbf{a}$ and using (33) in both sides we have: \[p_{1}(\mathbf{x}_{0})\geq\mathbf{a}_{0}^{T}\mathbf{x}_{0}+\text{ENF}_{p_{1}} (\mathbf{a}_{0})> \max_{\mathbf{a}\in\mathbb{R}^{n}}\{\mathbf{a}^{T}\mathbf{x}_{0}+\text{ENF}_{p_{ 2}}(\mathbf{a})\}\] \[=p_{2}(\mathbf{x}_{0}),\] which is a contradiction. ### _Proof of Proposition 2_ It is not difficult to see that if $\tilde{q}(\mathbf{x})=\bigvee_{i=1}^{m_{q}}(\tilde{\mathbf{a}}_{i}^{T}\mathbf{x}+\tilde{b }_{i})$, and $\tilde{q}$ satisfies (5), then $\tilde{q}(\mathbf{x})\leq q(\mathbf{x})$, for all $\mathbf{x}\in\mathbb{R}^{n}$. Indeed, since \[\tilde{\mathbf{a}}_{i}^{T}\mathbf{x}+\tilde{b}_{i}\leq\tilde{q}(\mathbf{x})\leq p(\mathbf{x})- d(\mathbf{x}),\] it holds $\tilde{b}_{i}\leq l(\tilde{\mathbf{a}}_{i})$. Thus, \[\tilde{\mathbf{a}}_{i}^{T}\mathbf{x}+\tilde{b}_{i}\leq\tilde{\mathbf{a}}_{i}^{T}\mathbf{x}+l( \mathbf{a}_{i})\leq q(\mathbf{x}).\] Hence, $\tilde{q}(\mathbf{x})\leq q(\mathbf{x})$, for all $\mathbf{x}\in\mathbb{R}^{n}$. To prove that $q$ is a quotient, it remains to show that it is a tropical polynomial. The epigraph of $q$ is the closed convex hull of the epigraph of $f$[44] \[\text{epi}(q)=\overline{\text{conv}(\text{epi}(f))}. \tag{34}\] On the other hand, $f$ is a piecewise linear function. Thus, its epigraph is a union of a finite number of convex polyhedra. Indeed if $A_{1},\ldots,A_{m_{f}}$ its linear regions (i.e., intersections of linear regions of $p$ and $d$) then $A_{i},\,i=1,...,m_{f}$ are convex polyhedra and \[\text{epi}(f)=\bigcup_{i=1}^{m_{f}}\{(\mathbf{x},t)\in\mathbb{R}^{n+1}:\mathbf{x}\in A _{i},t\geq f(\mathbf{x})\}.\] Observe that $\text{epi}(q)$ is a convex polyhedron and $q$ is a piecewise linear convex function. Thus, $q$ is a tropical polynomial. From the definition it is obvious that the quotient is unique. Let $q$ be the quotient and assume that for some tropical polynomial $\tilde{r}$ it holds \[p(\mathbf{x})=(q(\mathbf{x})+d(\mathbf{x}))\vee\tilde{r}(\mathbf{x}),\qquad\text{for all }\mathbf{x}\in\mathbb{R}^{n}.\] Assume that for some point $\mathbf{x}_{0}$ it holds $p(\mathbf{x}_{0})>d(\mathbf{x}_{0})+q(\mathbf{x}_{0})$ and that $p$ has the linear form $\mathbf{a}_{i0}^{T}\mathbf{x}+b_{i0}$ in a neighborhood of $\mathbf{x}_{0}$. Then, $\tilde{r}(\mathbf{x})=p(\mathbf{x})$ in this neighborhood. Furthermore, since $\tilde{r}$ is convex, we have \[\tilde{r}(\mathbf{x})\geq\bigvee_{i\in I}\mathbf{a}_{i}\mathbf{x}+b_{i}=r(\mathbf{x}),\] where $I=\{i\in\{1,\ldots,m_{p}\}:\exists\mathbf{x}\in\mathbb{R}^{n}\text{ with }p(\mathbf{x})>d(\mathbf{x})+q(\mathbf{x}),\text{ and }p(\mathbf{x})=\mathbf{a}_{i}\mathbf{x}+b_{i}\}$. It is not difficult to see that $r$ is a remainder. Fig. 9: _The error rate of the compressed neural networks obtained using the multiple division algorithm. The blue and yellow lines use divisions with $\tilde{m}_{q}=3$ and $\tilde{m}_{q}=5$, respectively. The horizontal axis represents the percentage of the remaining parameters. Finally, the green line corresponds to using random vectors instead of the quotients._ ### _Proof of Proposition 3_ a) Let $\bar{q}$ and $\bar{r}$ be the quotient and the remainder of the division of $r$ by $d$. Then \[r(\mathbf{x})=(d(\mathbf{x})+\bar{q}(\mathbf{x}))\vee\bar{r}(\mathbf{x}) \tag{35}\] Combining with \[p(\mathbf{x})=(d(\mathbf{x})+q(\mathbf{x}))\lor r(\mathbf{x}),\] we get \[p(\mathbf{x})=(d(\mathbf{x})+q(\mathbf{x}))\vee(d(\mathbf{x})+\bar{q}(\mathbf{x})) \vee\bar{r}(\mathbf{x})\] \[p(\mathbf{x})=(d(\mathbf{x})+q(\mathbf{x})\vee\bar{q}(\mathbf{x}))\vee\bar{r}( \mathbf{x}). \tag{36}\] Equation (35) implies that $\bar{r}(\mathbf{x})\leq r(\mathbf{x})$. This fact combined with the fact that $q(\mathbf{x})$ is the quotient of the division of $p(\mathbf{x})$ by $d(\mathbf{x})$ and (36) implies that $\bar{q}(\mathbf{x})\lor q(\mathbf{x})=q(\mathbf{x})$. Hence, since $r(\mathbf{x})$ is the remainder of the division of $p(\mathbf{x})$ by $d(\mathbf{x})$ it holds $r(\mathbf{x})\leq\bar{r}(\mathbf{x})$. Hence, $r(\mathbf{x})=\bar{r}(\mathbf{x})$, for all $\mathbf{x}\in\mathbb{R}^{n}$ b) The proof is trivial. ### _Proof of Proposition 4_ Assume $p(\mathbf{x})$ is given by (1) and $d(\mathbf{x})$ by \[d(\mathbf{x})=\bigvee_{i=1}^{m_{d}}\tilde{\mathbf{a}}_{i}\mathbf{x}+\tilde{b}_{i}.\] (a) We first show that $C\subset\textbf{dom}(l(\mathbf{a}))$. Assume that $\mathbf{a}\in C$. That is $\mathbf{a}+\tilde{\mathbf{a}}_{i}\in\textbf{Newt}(p)$ for all $i=1,\ldots,m_{d}$. Then, there are $\lambda_{1},\ldots,\lambda_{m_{d}}\geq 0$ with $\sum_{i=1}^{m_{p}}\lambda_{1}=1$ such that \[\mathbf{a}+\tilde{\mathbf{a}}_{i}=\sum_{i=1}^{n_{p}}\lambda_{i}\mathbf{a}_{i}.\] Thus, \[(\mathbf{a}+\tilde{\mathbf{a}}_{i})^{T}\mathbf{x}\leq\bigvee_{i=1}^{n_{p}}\mathbf{a}_{i}\mathbf{x }\leq p(\mathbf{x})-\bigwedge_{i=1}^{n_{p}}b_{i}.\] Furthermore, \[\tilde{\mathbf{a}}_{i}\mathbf{x}+\tilde{b}_{i}-\tilde{b}_{i}\geq\tilde{\mathbf{a}}_{i} \mathbf{x}+\tilde{b}_{i}-\bigvee_{j=1}^{m_{d}}\tilde{b}_{j}.\] Combining the last two equations we have \[p(\mathbf{x})-d(\mathbf{x})-\mathbf{a}^{T}\mathbf{x}\geq\bigwedge_{i=1}^{n_{p}}b_{i}-\sum_{j= 1}^{m_{d}}\tilde{b}_{j}>-\infty.\] Hence, $\mathbf{a}\in\textbf{dom}(l)$. Conversely assume that $\mathbf{a}\in\textbf{dom}(l)$ but $\mathbf{a}\not\in C$. Then, there is a $\tilde{\mathbf{a}}_{i}$ such that $\mathbf{a}+\mathbf{a}_{i}\not\in\textbf{Newt}(p)$. Furthermore, since (p) is a convex set, there is a vector $\mathbf{v}$ such that $(\mathbf{a}+\mathbf{a}_{i})^{T}\mathbf{v}>\mathbf{a}_{i}^{T}\mathbf{v}$ for all $i=1,\ldots,n_{p}$. Hence, \[l(\mathbf{a})=\inf_{\mathbf{x}\in\mathbb{R}^{n}}\{p(\mathbf{x})-d(\mathbf{x})- \mathbf{a}^{T}\mathbf{x}\}\] \[\leq\lim_{\lambda\rightarrow-\infty}p(\lambda\mathbf{v})-d(\lambda \mathbf{v})-\mathbf{a}^{T}(\lambda\mathbf{v})=-\infty\] Thus, $\mathbf{a}\not\in\textbf{dom}(l)$, which is a contradiction. (b) Since \[q(\mathbf{x})=\sup_{\mathbf{a}\in\mathbb{R}^{n}}\{\mathbf{a}^{T}\mathbf{x}-f^{}(\mathbf{a})\}\] It holds \[q(\mathbf{x})=-\infty\Leftrightarrow l(\mathbf{a})=-\infty,\text{ for all }\mathbf{a} \Leftrightarrow C=\textbf{dom}(l)=\emptyset\] (c) If for some $i$ it holds $\hat{\mathbf{a}}_{i}\not\in C$, then $l(\hat{\mathbf{a}}_{i})=-\infty$. That is, for all $b$ there is an $\mathbf{x}\in\mathbb{R}^{n}$ with $\hat{\mathbf{a}}_{i}^{T}\mathbf{x}+b>f(\mathbf{x})$. This is particularly true for $b=\tilde{b}_{i}$, and thus $q(\mathbf{x})$ is not a quotient. ### _Proof of Corollary 1_ The proof of the first part is immediate from Proposition 4. To contradict, assume that the inclusion in (ii) is not true. Then, there exists a vector $\mathbf{x}\in\text{span}\{\mathbf{a}_{1},\ldots,\mathbf{a}_{m_{p}}\}^{\perp}$ but $\mathbf{x}\not\in\text{span}\{\hat{\mathbf{a}}_{1},\ldots,\hat{\mathbf{a}}_{m_{q}}\}^{\perp}$. Thus, there is an index $i_{0}$ such that $\hat{\mathbf{a}}_{i_{0}}^{T}\mathbf{x}\neq 0$. Without loss of generality assume that $\hat{\mathbf{a}}_{i_{0}}^{T}\mathbf{x}>0$ (if it is negative, use $-\mathbf{x}$ in the place of $\mathbf{x}$). Using (i) we have $\lim_{\lambda\rightarrow\infty}(q(\lambda\mathbf{x})+d(\lambda\mathbf{x}))=\infty$, but $\lim_{\lambda\rightarrow\infty}p(\lambda\mathbf{x})<\infty$. Hence, there is a $\lambda$ such that $p(\lambda\mathbf{x})<q(\lambda\mathbf{x})+d(\lambda\mathbf{x})$, which contradicts the fact that $q$ is the quotient. ### _Proof of Proposition 5_ First we need to show that all the components of $\mathbf{a}^{E,z}$ are positive. Due to the convexity of $E$, either \[\inf\{z:(\mathbf{x},z)\in E\}=-\infty,\text{ for all }\mathbf{x}\in\mathbb{R}^{n},\] or \[\inf\{z:(\mathbf{x},z)\in E\}>-\infty,\text{ for all }\mathbf{x}\in\mathbb{R}^{n}.\] If $E=\mathbb{R}^{n+1}$, then there are no non-trivial constraints, i.e., $L=0$. In the following assume that $L>0$, that is there are some nontrivial constraints, and $E\neq\mathbb{R}^{n+1}$. From the definition of $E$, for all $\mathbf{x}\in\mathbb{R}^{n}$, it holds $\inf\{z:(\mathbf{x},z)\in E\}<\infty$, and $\sup\{z:(\mathbf{x},z)\in E\}=\infty$. Note that it is not possible to have a constraint in (10) in the form $[\mathbf{A}^{E,\mathbf{x}}]_{\mathbf{x}}\geq[b^{E}]_{l}$, that is nontrivial, i.e., $[\mathbf{A}^{E,\mathbf{x}}]_{l}\neq\mathbf{0}^{T}$. Indeed, if such a constraint existed, then for an $\mathbf{x}$ not satisfying this constraint we would have $\inf\{z:(\mathbf{x},z)\in E\}=\infty$, which is a contradiction. On the other hand if there were a constraint with $[\mathbf{A}^{E,\mathbf{x}}]_{\mathbf{x}}+[\mathbf{a}^{E,z}]_{l}z\geq[\mathbf{b}^{E}]_{l}$, with $[\mathbf{a}^{E,z}]_{l}<0$ then, for all $(\mathbf{x},z)\in E$ we would have $z\leq(-[\mathbf{A}^{E,\mathbf{x}}]_{l}\mathbf{x}+[b^{E}]_{l})/[\mathbf{a}^{E,z}]_{l}$ and thus $\sup\{z:(\mathbf{x},z)\in E\}<\infty$. It is then easy to see that the function $q$ given by (11) has as epigraph the set $E$. Therefore, it is the quotient. To determine the remainder it is sufficient to find all the terms $i$ of $p(\mathbf{x})$ for which $p(\mathbf{x})>\tilde{p}(\mathbf{x})$, for an $\mathbf{x}\in\mathbb{R}^{n}$ that the term $i$ attains the maximum in $p(\mathbf{x})$. The function $p(\mathbf{x})-\tilde{p}(\mathbf{x})$ is linear on $P_{i,j}$ for all $i,j$. Therefore, since $p(\mathbf{x})-\tilde{p}(\mathbf{x})\geq 0$, and $\mathbf{x}_{i,j}$ is in the relative interior of $P_{i,j}$ we have either $p(\mathbf{x})-\tilde{p}(\mathbf{x})=0$, for all $\mathbf{x}\in P_{i,j}$, or $p(\mathbf{x}_{i,j})>\tilde{p}(\mathbf{x}_{i,j})$. Hence, the procedure described in Algorithm 1 produces the remainder. ### _Proof of Proposition 6_ Observe that \[p(\mathbf{x}_{j})-d(\mathbf{x}_{j} where the last inequality is a consequence of the first constraint of (16). Furthermore, \[e(t)=\sum_{j=1}^{N}f(\mathbf{x}_{j})-\sum_{j=1}^{N}\bigvee_{i=1}^{\tilde{m}_{q}}(\hat {\mathbf{a}}_{i,t}^{T}\mathbf{x}_{j}+\hat{b}_{i,t}), \tag{37}\] where $\hat{\mathbf{a}}_{i,t}$ is the solution of (17) at the iteration $t$. Each term of the last sum can be written as: \[\bigvee_{i=1}^{\tilde{m}_{q}}(\hat{\mathbf{a}}_{i,t}^{T}\mathbf{x}_{j}+\hat{b}_{i,t}) =(\hat{\mathbf{a}}_{i^{\prime}_{j,t},t}^{T}\mathbf{x}_{j}+\hat{b}_{i^{\prime}_{j,t},t} ),\] where $i^{\prime}_{j,t}$ is the partition that $j$ belongs after step $t$, i.e., $j\in I_{i^{\prime}_{j,t},t}$ and $I_{i,t}$ is the partition obtained at Step 6 in iteration $t$. Furthermore due to the optimization in (16), \[\sum_{j=1}^{N}\hat{\mathbf{a}}_{i^{\prime}_{j,t},t}^{T}\mathbf{x}_{j}+\hat{b}_{i^{ \prime}_{j,t},t}\leq\sum_{j=1}^{N}\hat{\mathbf{a}}_{i^{\prime}_{j,t},t+1}^{T}\mathbf{ x}_{j}+\hat{b}_{i^{\prime}_{j,t},t+1}.\] From Step 6, \[\sum_{j=1}^{N}\hat{\mathbf{a}}_{i^{\prime}_{j,t},t+1}^{T}\mathbf{x}_{j}+\hat{b}_{i^{ \prime}_{j,t},t+1}\leq\sum_{j=1}^{N}\hat{\mathbf{a}}_{i^{\prime}_{j,t+1},t+1}^{T} \mathbf{x}_{j}+\hat{b}_{i^{\prime}_{j,t+1},t+1}.\] Therefore, $\sum_{j=1}^{N}\bigvee_{i=1}^{\tilde{m}_{q}}(\hat{\mathbf{a}}_{i,t}^{T}\mathbf{x}_{j}+ \hat{b}_{i,t})$ is non-increasing, and thus, $e(t)$ is non-decreasing. ### _Proof of Proposition 7_ We first prove (20). The quotients $Q(p_{1},d)$ and $Q(p_{2},d)$ satisfy \[p_{1}(\mathbf{x}) \geq d(\mathbf{x})+Q(p_{1},d)(\mathbf{x}), \tag{38}\] \[p_{2}(\mathbf{x}) \geq d(\mathbf{x})+Q(p_{2},d)(\mathbf{x}). \tag{39}\] Thus, \[p_{1}(\mathbf{x})+p_{2}(\mathbf{x})\geq d(\mathbf{x})+[Q(p_{1},d)+Q(p_{2},d)(\mathbf{x})+d( \mathbf{x})].\] But $Q(p_{1}+p_{2},d)$ is the maximum tropical polynomial satisfying this inequality. Thus it is greater than or equal to the expression in the bracket, i.e., \[Q(p_{1}+p_{2},d)(\mathbf{x})\geq Q(p_{1},d)(\mathbf{x})+Q(p_{2},d)(\mathbf{x})+d(\mathbf{x}).\] To prove (21), take the maximum of (38) and (39). Using the distributive property we get \[p_{1}(\mathbf{x})\lor p_{2}(\mathbf{x})\geq d(\mathbf{x})+[Q(p_{1},d)(\mathbf{x})\lor Q(p_{2},d)(\mathbf{x})].\] Thus, \[Q(p_{1}\lor p_{2},d)\geq Q(p_{1},d)(\mathbf{x})\lor Q(p_{2},d)(\mathbf{x}).\] We then prove (22). The quotients $Q(p,d_{1})$ and $Q(p,d_{2})$ satisfy \[p(\mathbf{x}) \geq d_{1}(\mathbf{x})+Q(p,d_{1})(\mathbf{x}), \tag{40}\] \[p(\mathbf{x}) \geq d_{2}(\mathbf{x})+Q(p,d_{2})(\mathbf{x}). \tag{41}\] Thus, \[p(\mathbf{x})\geq d_{1}(\mathbf{x})+d_{2}(\mathbf{x})+[Q(p,d_{1})(\mathbf{x})+Q(p,d_{2})(\mathbf{ x})-p(\mathbf{x})].\] Hence, \[Q(p,d_{1}+d_{2})(\mathbf{x})\geq Q(p,d_{1})(\mathbf{x})+Q(p_{2},d)(\mathbf{x})-p(\mathbf{x}).\] To prove (23), take the maximum of (40) and (41) \[p(\mathbf{x}) \geq[d_{1}(\mathbf{x})+Q(p,d_{1})(\mathbf{x})]\vee[d_{2}(\mathbf{x})+Q(p,d_{ 2})(\mathbf{x})]\] \[\geq[d_{1}(\mathbf{x})\lor d_{2}(\mathbf{x})]+[Q(p,d_{1})(\mathbf{x})\wedge Q (p,d_{2})(\mathbf{x})]\] Therefore, \[Q(p,d_{1}\lor d_{2})(\mathbf{x})\geq Q(p,d_{1})(\mathbf{x})\wedge Q(p,d_{2})(\mathbf{x})\] ### _Proof of Proposition 8_ The Newton polytopes of $p$ and $q$ are given by the zonotopes \[\text{Newt}(p)=\{\mathbf{A}\mu:\mu\in[0,1]^{n}\}, \tag{42}\] \[\text{Newt}(q)=\{\hat{\mathbf{A}}\mathbf{\lambda}:\lambda\in[0,1]^{m_{q}}\}, \tag{43}\] where $\hat{\mathbf{A}}=[\hat{\mathbf{a}}_{1}\ \ldots\ \hat{\mathbf{a}}_{m\tau}]$. We start proving the following lemma. Lemma 1*: _Let $\text{Newt}(p)$, $\text{Newt}(q)$ as the above. Then $\text{Newt}(q)\subset\text{Newt}(p)$ if and only if_ \[\mathbf{0}\leq\mathbf{A}^{-1}\hat{\mathbf{a}}_{i},\ \ \mathbf{A}^{-1}\sum_{i=1}^{m_{q}}\hat{\mathbf{a}}_{i} \leq\mathbf{1}, \tag{44}\] _for $i=1,\ldots,m_{q}$,_ _Proof of the lemma: $\text{Newt}(q)\subset\text{Newt}(p)$ if and only if for any $\mathbf{\lambda}\in[0,1]^{m_{q}}$ there exists a $\mathbf{\mu}\in[0,1]^{n}$ such that $\mathbf{A}\mu=\hat{\mathbf{A}}\mathbf{\lambda}$. Thus, $\text{Newt}(q)\subset\text{Newt}(p)$ is equivalent to_ \[\mathbf{0}\leq\mathbf{A}^{-1}\hat{\mathbf{A}}\mathbf{\lambda}\leq\mathbf{1}, \tag{45}\] _for all $\mathbf{\lambda}\in[0,1]^{m_{q}}$._ Direct part: Assume $\text{Newt}(q)\subset\text{Newt}(p)$. Substituting $\mathbf{\lambda}=\mathbf{e}_{i}$, i.e., the $i-$th unit vector, in (45), we get the first inequality of (44). Now with $\mathbf{\lambda}=\mathbf{1}$ we get the second inequality of (44). Converse part: Assume that (44) holds true. Multiplying the first inequality of (44) by $\lambda_{i}\geq 0$ and adding, we get the first inequality of (45). Furthermore, for $0\leq\lambda_{i}\leq 1$, we have $\sum_{i=1}^{m_{q}}\lambda_{i}\mathbf{A}^{-1}\hat{\mathbf{a}}_{i}\leq\mathbf{A}^{-1}\sum_{i= 1}^{m_{q}}\hat{\mathbf{a}}_{i}\leq\mathbf{1}$, which implies the right part of inequality of (45). $\Box$ From Lemma 1, $q(\mathbf{x})\leq p(\mathbf{x})$ for all $\mathbf{x}$ is equivalent to $\text{ENF}_{q}(\mathbf{a})\leq\text{ENF}_{p}(\mathbf{a})$, for all $\mathbf{a}$. We first compute $\text{ENF}_{p}(\mathbf{a})$. It holds \[\text{ENF}_{p}(\mathbf{a})=\mathbf{B}^{T}\mathbf{A}^{-1}\mathbf{a}, \tag{46}\] for $\mathbf{a}\in\text{Newt}(p)$ and $-\infty$ otherwise. Indeed, the extended Newton function of $p$ is given by $\text{ENetv}_{p}(\mathbf{a})=\max\{\sum_{i=1}^{n}\mu_{i}b_{i}:\sum_{i=1}^{n}\mu_{i} \mathbf{a}_{i}=\mathbf{a},\mu_{i}\in[0,1]\}$. However, $\sum_{i=1}^{n}\mu_{i}\mathbf{a}_{i}=\mathbf{a}$ has a solution $\mathbf{\mu}=[\mu_{1}\ \ldots\mu_{n}]^{T}\in[0,1]^{n}$, if and only if $a\in\text{Newt}(p)$. Furthermore, for $a\in\text{Newt}(p)$ the solution is unique and given by $\mathbf{\mu}=\mathbf{A}^{-1}\mathbf{a}$. Thus $\text{ENF}_{p}(\mathbf{a})$ is given by (46). The first two conditions of (29) are equivalent to the inclusion $\text{Newt}(q)\subset\text{Newt}(p)$. To prove the direct part of the proposition observe that $\hat{b}_{i}\leq\ for all \(\hat{\mathbf{a}}\not\in\text{Newt}(q)$. It remains to show that $\text{ENF}_{q}(\hat{\mathbf{a}})\leq\text{ENF}_{p}(\hat{\mathbf{a}})$, for all $\hat{\mathbf{a}}\in\text{Newt}(q)$. For such an $\hat{\mathbf{a}}$ it holds \[\text{ENF}_{q}(\hat{\mathbf{a}}) =\max\{\sum_{i=1}^{m_{q}}\lambda_{i}\hat{b}_{i}:\sum_{i=1}^{m_{q} }\lambda_{i}\hat{\mathbf{a}}_{i}=\hat{\mathbf{a}},\lambda_{i}\in[0,1]\}\] \[\leq\max\{\sum_{i=1}^{m_{q}}\lambda_{i}\mathbf{B}^{T}\mathbf{A}^{-1}\hat{ \mathbf{a}}_{i}:\sum_{i=1}^{m_{q}}\lambda_{i}\hat{\mathbf{a}}_{i}=\hat{\mathbf{a}},\lambda _{i}\in[0,1]\}\] \[=\mathbf{B}^{T}\mathbf{A}^{-1}\hat{\mathbf{a}}=\text{ENF}_{p}(\hat{\mathbf{a}})\]

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