Daily Papers

How to reduce compute and memory requirements of neural networks (NNs) without sacrificing performance? Many recent works use sparse Mixtures of Experts (MoEs) to build resource-efficient large language models (LMs). Here we introduce several novel perspectives on MoEs, presenting a general framework that unifies various methods to approximate two-layer NNs (e.g., feedforward blocks of Transformers), including product-key memories (PKMs). Leveraging insights from this framework, we propose methods to improve both MoEs and PKMs. Unlike prior work that compares MoEs with dense baselines under the compute-equal condition, our evaluation condition is parameter-equal, which is crucial to properly evaluate LMs. We show that our MoEs are competitive with the dense Transformer-XL on both the WikiText-103 and enwiki8 datasets at two different scales, while being much more resource efficient. This demonstrates that MoEs are relevant not only to extremely large LMs but also to any-scale resource-efficient LMs. Our code is public.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Printed Electronics (PE) stands out as a promisingtechnology for widespread computing due to its distinct attributes, such as low costs and flexible manufacturing. Unlike traditional silicon-based technologies, PE enables stretchable, conformal,and non-toxic hardware. However, PE are constrained by larger feature sizes, making it challenging to implement complex circuits such as machine learning (ML) classifiers. Approximate computing has been proven to reduce the hardware cost of ML circuits such as Multilayer Perceptrons (MLPs). In this paper, we maximize the benefits of approximate computing by integrating hardware approximation into the MLP training process. Due to the discrete nature of hardware approximation, we propose and implement a genetic-based, approximate, hardware-aware training approach specifically designed for printed MLPs. For a 5% accuracy loss, our MLPs achieve over 5x area and power reduction compared to the baseline while outperforming state of-the-art approximate and stochastic printed MLPs.

Prompt tuning is a parameter-efficient way to deploy large-scale pre-trained models to downstream tasks by adding task-specific tokens. In terms of vision-language pre-trained (VLP) models, prompt tuning often requires a large number of learnable tokens to bridge the gap between the pre-training and downstream tasks, which greatly exacerbates the already high computational overhead. In this paper, we revisit the principle of prompt tuning for Transformer-based VLP models, and reveal that the impact of soft prompt tokens can be actually approximated via independent information diffusion steps, thereby avoiding the expensive global attention modeling and reducing the computational complexity to a large extent. Based on this finding, we propose a novel Approximated Prompt Tuning (APT) approach towards efficient VL transfer learning. To validate APT, we apply it to two representative VLP models, namely ViLT and METER, and conduct extensive experiments on a bunch of downstream tasks. Meanwhile, the generalization of APT is also validated on CLIP for image classification and StableDiffusion for text-to-image generation. The experimental results not only show the superior performance gains and computation efficiency of APT against the conventional prompt tuning methods, e.g., +7.01% accuracy and -82.30% additional computation overhead on METER, but also confirm its merits over other parameter-efficient transfer learning approaches.

Approximate inference in Gaussian process (GP) models with non-conjugate likelihoods gets entangled with the learning of the model hyperparameters. We improve hyperparameter learning in GP models and focus on the interplay between variational inference (VI) and the learning target. While VI's lower bound to the marginal likelihood is a suitable objective for inferring the approximate posterior, we show that a direct approximation of the marginal likelihood as in Expectation Propagation (EP) is a better learning objective for hyperparameter optimization. We design a hybrid training procedure to bring the best of both worlds: it leverages conjugate-computation VI for inference and uses an EP-like marginal likelihood approximation for hyperparameter learning. We compare VI, EP, Laplace approximation, and our proposed training procedure and empirically demonstrate the effectiveness of our proposal across a wide range of data sets.

It is common to reject undesired outputs of Large Language Models (LLMs); however, current methods to do so require an excessive amount of computation, or severely distort the distribution of outputs. We present a method to balance the distortion of the output distribution with computational efficiency, allowing for the generation of long sequences of text with difficult-to-satisfy constraints, with less amplification of low probability outputs compared to existing methods. We show through a series of experiments that the task-specific performance of our method is comparable to methods that do not distort the output distribution, while being much more computationally efficient.

Integrating a notion of symmetry into point cloud neural networks is a provably effective way to improve their generalization capability. Of particular interest are E(3) equivariant point cloud networks where Euclidean transformations applied to the inputs are preserved in the outputs. Recent efforts aim to extend networks that are E(3) equivariant, to accommodate inputs made of multiple parts, each of which exhibits local E(3) symmetry. In practical settings, however, the partitioning into individually transforming regions is unknown a priori. Errors in the partition prediction would unavoidably map to errors in respecting the true input symmetry. Past works have proposed different ways to predict the partition, which may exhibit uncontrolled errors in their ability to maintain equivariance to the actual partition. To this end, we introduce APEN: a general framework for constructing approximate piecewise-E(3) equivariant point networks. Our primary insight is that functions that are equivariant with respect to a finer partition will also maintain equivariance in relation to the true partition. Leveraging this observation, we propose a design where the equivariance approximation error at each layers can be bounded solely in terms of (i) uncertainty quantification of the partition prediction, and (ii) bounds on the probability of failing to suggest a proper subpartition of the ground truth one. We demonstrate the effectiveness of APEN using two data types exemplifying part-based symmetry: (i) real-world scans of room scenes containing multiple furniture-type objects; and, (ii) human motions, characterized by articulated parts exhibiting rigid movement. Our empirical results demonstrate the advantage of integrating piecewise E(3) symmetry into network design, showing a distinct improvement in generalization compared to prior works for both classification and segmentation tasks.

We define and investigate the problem of c-approximate window search: approximate nearest neighbor search where each point in the dataset has a numeric label, and the goal is to find nearest neighbors to queries within arbitrary label ranges. Many semantic search problems, such as image and document search with timestamp filters, or product search with cost filters, are natural examples of this problem. We propose and theoretically analyze a modular tree-based framework for transforming an index that solves the traditional c-approximate nearest neighbor problem into a data structure that solves window search. On standard nearest neighbor benchmark datasets equipped with random label values, adversarially constructed embeddings, and image search embeddings with real timestamps, we obtain up to a 75times speedup over existing solutions at the same level of recall.

Text-to-image generation using diffusion models has seen explosive popularity owing to their ability in producing high quality images adhering to text prompts. However, production-grade diffusion model serving is a resource intensive task that not only require high-end GPUs which are expensive but also incurs considerable latency. In this paper, we introduce a technique called approximate-caching that can reduce such iterative denoising steps for an image generation based on a prompt by reusing intermediate noise states created during a prior image generation for similar prompts. Based on this idea, we present an end to end text-to-image system, Nirvana, that uses the approximate-caching with a novel cache management-policy Least Computationally Beneficial and Frequently Used (LCBFU) to provide % GPU compute savings, 19.8% end-to-end latency reduction and 19% dollar savings, on average, on two real production workloads. We further present an extensive characterization of real production text-to-image prompts from the perspective of caching, popularity and reuse of intermediate states in a large production environment.

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

In this work, we conceptualize the learning process as information compression. We seek to equip generative pre-trained models with human-like learning capabilities that enable data compression during inference. We present a novel approach that utilizes the Generative Pre-trained Transformer (GPT) to approximate Kolmogorov complexity, with the aim of estimating the optimal Information Distance for few-shot learning. We first propose using GPT as a prior for lossless text compression, achieving a noteworthy compression ratio. Experiment with LLAMA2-7B backbone achieves a compression ratio of 15.5 on enwik9. We justify the pre-training objective of GPT models by demonstrating its equivalence to the compression length, and, consequently, its ability to approximate the information distance for texts. Leveraging the approximated information distance, our method allows the direct application of GPT models in quantitative text similarity measurements. Experiment results show that our method overall achieves superior performance compared to embedding and prompt baselines on challenging NLP tasks, including semantic similarity, zero and one-shot text classification, and zero-shot text ranking.

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick k centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of n points in a metric space (P,d) where each point belongs to one of the ell different demographics (i.e., P = P_1 uplus P_2 uplus cdots uplus P_ell) and a set of ell intervals [alpha_1, beta_1], cdots, [alpha_ell, beta_ell] on desired number of centers from each group, the goal is to pick a set of k centers C with minimum ell_p-clustering cost (i.e., (sum_{vin P} d(v,C)^p)^{1/p}) such that for each group iin ell, |Ccap P_i| in [alpha_i, beta_i]. In particular, the fair range ell_p-clustering captures fair range k-center, k-median and k-means as its special cases. In this work, we provide efficient constant factor approximation algorithms for fair range ell_p-clustering for all values of pin [1,infty).

Estimating truncated density models is difficult, as these models have intractable normalising constants and hard to satisfy boundary conditions. Score matching can be adapted to solve the truncated density estimation problem, but requires a continuous weighting function which takes zero at the boundary and is positive elsewhere. Evaluation of such a weighting function (and its gradient) often requires a closed-form expression of the truncation boundary and finding a solution to a complicated optimisation problem. In this paper, we propose approximate Stein classes, which in turn leads to a relaxed Stein identity for truncated density estimation. We develop a novel discrepancy measure, truncated kernelised Stein discrepancy (TKSD), which does not require fixing a weighting function in advance, and can be evaluated using only samples on the boundary. We estimate a truncated density model by minimising the Lagrangian dual of TKSD. Finally, experiments show the accuracy of our method to be an improvement over previous works even without the explicit functional form of the boundary.

Despite the great success of Transformer networks in various applications such as natural language processing and computer vision, their theoretical aspects are not well understood. In this paper, we study the approximation and estimation ability of Transformers as sequence-to-sequence functions with infinite dimensional inputs. Although inputs and outputs are both infinite dimensional, we show that when the target function has anisotropic smoothness, Transformers can avoid the curse of dimensionality due to their feature extraction ability and parameter sharing property. In addition, we show that even if the smoothness changes depending on each input, Transformers can estimate the importance of features for each input and extract important features dynamically. Then, we proved that Transformers achieve similar convergence rate as in the case of the fixed smoothness. Our theoretical results support the practical success of Transformers for high dimensional data.

The Shapley value is arguably the most popular approach for assigning a meaningful contribution value to players in a cooperative game, which has recently been used intensively in explainable artificial intelligence. The meaningfulness is due to axiomatic properties that only the Shapley value satisfies, which, however, comes at the expense of an exact computation growing exponentially with the number of agents. Accordingly, a number of works are devoted to the efficient approximation of the Shapley values, most of them revolve around the notion of an agent's marginal contribution. In this paper, we propose with SVARM and Stratified SVARM two parameter-free and domain-independent approximation algorithms based on a representation of the Shapley value detached from the notion of marginal contributions. We prove unmatched theoretical guarantees regarding their approximation quality and provide empirical results including synthetic games as well as common explainability use cases comparing ourselves with state-of-the-art methods.

Advanced video analytic systems, including scene classification and object detection, have seen widespread success in various domains such as smart cities and autonomous transportation. With an ever-growing number of powerful client devices, there is incentive to move these heavy video analytics workloads from the cloud to mobile devices to achieve low latency and real-time processing and to preserve user privacy. However, most video analytic systems are heavyweight and are trained offline with some pre-defined latency or accuracy requirements. This makes them unable to adapt at runtime in the face of three types of dynamism -- the input video characteristics change, the amount of compute resources available on the node changes due to co-located applications, and the user's latency-accuracy requirements change. In this paper we introduce ApproxDet, an adaptive video object detection framework for mobile devices to meet accuracy-latency requirements in the face of changing content and resource contention scenarios. To achieve this, we introduce a multi-branch object detection kernel (layered on Faster R-CNN), which incorporates a data-driven modeling approach on the performance metrics, and a latency SLA-driven scheduler to pick the best execution branch at runtime. We couple this kernel with approximable video object tracking algorithms to create an end-to-end video object detection system. We evaluate ApproxDet on a large benchmark video dataset and compare quantitatively to AdaScale and YOLOv3. We find that ApproxDet is able to adapt to a wide variety of contention and content characteristics and outshines all baselines, e.g., it achieves 52% lower latency and 11.1% higher accuracy over YOLOv3.

Conducting text retrieval in a dense learned representation space has many intriguing advantages over sparse retrieval. Yet the effectiveness of dense retrieval (DR) often requires combination with sparse retrieval. In this paper, we identify that the main bottleneck is in the training mechanisms, where the negative instances used in training are not representative of the irrelevant documents in testing. This paper presents Approximate nearest neighbor Negative Contrastive Estimation (ANCE), a training mechanism that constructs negatives from an Approximate Nearest Neighbor (ANN) index of the corpus, which is parallelly updated with the learning process to select more realistic negative training instances. This fundamentally resolves the discrepancy between the data distribution used in the training and testing of DR. In our experiments, ANCE boosts the BERT-Siamese DR model to outperform all competitive dense and sparse retrieval baselines. It nearly matches the accuracy of sparse-retrieval-and-BERT-reranking using dot-product in the ANCE-learned representation space and provides almost 100x speed-up.

Videos take a lot of time to transport over the network, hence running analytics on the live video on embedded or mobile devices has become an important system driver. Considering that such devices, e.g., surveillance cameras or AR/VR gadgets, are resource constrained, creating lightweight deep neural networks (DNNs) for embedded devices is crucial. None of the current approximation techniques for object classification DNNs can adapt to changing runtime conditions, e.g., changes in resource availability on the device, the content characteristics, or requirements from the user. In this paper, we introduce ApproxNet, a video object classification system for embedded or mobile clients. It enables novel dynamic approximation techniques to achieve desired inference latency and accuracy trade-off under changing runtime conditions. It achieves this by enabling two approximation knobs within a single DNN model, rather than creating and maintaining an ensemble of models (e.g., MCDNN [MobiSys-16]. We show that ApproxNet can adapt seamlessly at runtime to these changes, provides low and stable latency for the image and video frame classification problems, and show the improvement in accuracy and latency over ResNet [CVPR-16], MCDNN [MobiSys-16], MobileNets [Google-17], NestDNN [MobiCom-18], and MSDNet [ICLR-18].

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Many poker systems, whether created with heuristics or machine learning, rely on the probability of winning as a key input. However calculating the precise probability using combinatorics is an intractable problem, so instead we approximate it. Monte Carlo simulation is an effective technique that can be used to approximate the probability that a player will win and/or tie a hand. However, without the use of a memory-intensive lookup table or a supercomputer, it becomes infeasible to run millions of times when training an agent with self-play. To combat the space-time tradeoff, we use deep learning to approximate the probabilities obtained from the Monte Carlo simulation with high accuracy. The learned model proves to be a lightweight alternative to Monte Carlo simulation, which ultimately allows us to use the probabilities as inputs during self-play efficiently. The source code and optimized neural network can be found at https://github.com/brandinho/Poker-Probability-Approximation

Zipf's law appears in many application areas but does not have a closed form expression, which may make its use cumbersome. Since it coincides with the truncated version of the Zeta distribution, in this paper we propose three approximate closed form expressions for the truncated Zeta distribution, which may be employed for Zipf's law as well. The three approximations are based on the replacement of the sum occurring in Zipf's law with an integral, and are named respectively the integral approximation, the average integral approximation, and the trapezoidal approximation. While the first one is shown to be of little use, the trapezoidal approximation exhibits an error which is typically lower than 1\%, but is as low as 0.1\% for the range of values of the Zipf parameter below 1.

We seek to learn an effective policy for a Markov Decision Process (MDP) with continuous states via Q-Learning. Given a set of basis functions over state action pairs we search for a corresponding set of linear weights that minimizes the mean Bellman residual. Our algorithm uses a Kalman filter model to estimate those weights and we have developed a simpler approximate Kalman filter model that outperforms the current state of the art projected TD-Learning methods on several standard benchmark problems.

Large Language Models (LLMs) have showcased remarkable capabilities across various domains. Accompanying the evolving capabilities and expanding deployment scenarios of LLMs, their deployment challenges escalate due to their sheer scale and the advanced yet complex activation designs prevalent in notable model series, such as Llama, Gemma, and Mistral. These challenges have become particularly pronounced in resource-constrained deployment scenarios, where mitigating inference efficiency bottlenecks is imperative. Among various recent efforts, activation approximation has emerged as a promising avenue for pursuing inference efficiency, sometimes considered indispensable in applications such as private inference. Despite achieving substantial speedups with minimal impact on utility, even appearing sound and practical for real-world deployment, the safety implications of activation approximations remain unclear. In this work, we fill this critical gap in LLM safety by conducting the first systematic safety evaluation of activation approximations. Our safety vetting spans seven sota techniques across three popular categories, revealing consistent safety degradation across ten safety-aligned LLMs.

Printed Electronics (PE) feature distinct and remarkable characteristics that make them a prominent technology for achieving true ubiquitous computing. This is particularly relevant in application domains that require conformal and ultra-low cost solutions, which have experienced limited penetration of computing until now. Unlike silicon-based technologies, PE offer unparalleled features such as non-recurring engineering costs, ultra-low manufacturing cost, and on-demand fabrication of conformal, flexible, non-toxic, and stretchable hardware. However, PE face certain limitations due to their large feature sizes, that impede the realization of complex circuits, such as machine learning classifiers. In this work, we address these limitations by leveraging the principles of Approximate Computing and Bespoke (fully-customized) design. We propose an automated framework for designing ultra-low power Multilayer Perceptron (MLP) classifiers which employs, for the first time, a holistic approach to approximate all functions of the MLP's neurons: multiplication, accumulation, and activation. Through comprehensive evaluation across various MLPs of varying size, our framework demonstrates the ability to enable battery-powered operation of even the most intricate MLP architecture examined, significantly surpassing the current state of the art.

Implantable wireless bioelectronic devices enable communication and/or power transfer through RF wireless connections with external nodes. These devices encounter notable design challenges due to the lossy nature of the host body, which significantly diminishes the radiation efficiency of the implanted antenna and tightens the wireless link budget. Prior research has yielded closed-form approximate expressions for estimating losses occurring within the lossy host body, known as the in-body path loss. To assess the total path loss between the implanted transmitter and external receiver, this paper focuses on the free-space path loss of the implanted antenna, from the body-air interface to the external node. This is not trivial, as in addition to the inherent radial spreading of spherical electromagnetic waves common to all antennas, implanted antennas confront additional losses arising from electromagnetic scattering at the interface between the host body and air. Employing analytical modeling, we propose closed-form approximate expressions for estimating this free-space path loss. The approximation is formulated as a function of the free-space distance, the curvature radius of the body-air interface, and the permittivity of the lossy medium. This proposed method undergoes thorough validation through numerical calculations, simulations, and measurements for different implanted antenna scenarios. This study contributes to a comprehensive understanding of the path loss in implanted antennas and provides a reliable analytical framework for their efficient design and performance evaluation.

Fine-tuning a pre-trained language model (PLM) emerges as the predominant strategy in many natural language processing applications. However, even fine-tuning the PLMs and doing inference are expensive, especially on edge devices with low computing power. Some general approaches (e.g. quantization and distillation) have been widely studied to reduce the compute/memory of PLM fine-tuning, while very few one-shot compression techniques are explored. In this paper, we investigate the neural tangent kernel (NTK)--which reveals the gradient descent dynamics of neural networks--of the multilayer perceptrons (MLP) modules in a PLM and propose to coin a lightweight PLM through NTK-approximating MLP fusion. To achieve this, we reconsider the MLP as a bundle of sub-MLPs, and cluster them into a given number of centroids, which can then be restored as a compressed MLP and surprisingly shown to well approximate the NTK of the original PLM. Extensive experiments of PLM fine-tuning on both natural language understanding (NLU) and generation (NLG) tasks are provided to verify the effectiveness of the proposed method MLP fusion. Our code is available at https://github.com/weitianxin/MLP_Fusion.

We prove an inverse approximation theorem for the approximation of nonlinear sequence-to-sequence relationships using recurrent neural networks (RNNs). This is a so-called Bernstein-type result in approximation theory, which deduces properties of a target function under the assumption that it can be effectively approximated by a hypothesis space. In particular, we show that nonlinear sequence relationships that can be stably approximated by nonlinear RNNs must have an exponential decaying memory structure - a notion that can be made precise. This extends the previously identified curse of memory in linear RNNs into the general nonlinear setting, and quantifies the essential limitations of the RNN architecture for learning sequential relationships with long-term memory. Based on the analysis, we propose a principled reparameterization method to overcome the limitations. Our theoretical results are confirmed by numerical experiments. The code has been released in https://github.com/radarFudan/Curse-of-memory

Speech produced by human vocal apparatus conveys substantial non-semantic information including the gender of the speaker, voice quality, affective state, abnormalities in the vocal apparatus etc. Such information is attributed to the properties of the voice source signal, which is usually estimated from the speech signal. However, most of the source estimation techniques depend heavily on the goodness of the model assumptions and are prone to noise. A popular alternative is to indirectly obtain the source information through the Electroglottographic (EGG) signal that measures the electrical admittance around the vocal folds using dedicated hardware. In this paper, we address the problem of estimating the EGG signal directly from the speech signal, devoid of any hardware. Sampling from the intractable conditional distribution of the EGG signal given the speech signal is accomplished through optimization of an evidence lower bound. This is constructed via minimization of the KL-divergence between the true and the approximated posteriors of a latent variable learned using a deep neural auto-encoder that serves an informative prior. We demonstrate the efficacy of the method at generating the EGG signal by conducting several experiments on datasets comprising multiple speakers, voice qualities, noise settings and speech pathologies. The proposed method is evaluated on many benchmark metrics and is found to agree with the gold standard while proving better than the state-of-the-art algorithms on a few tasks such as epoch extraction.

This study reveals an unexpected parallel between instructible vision-language models (VLMs) and human cognitive disorders, specifically constructive apraxia. We tested 25 state-of-the-art VLMs, including GPT-4 Vision, DALL-E 3, and Midjourney v5, on their ability to generate images of the Ponzo illusion, a task that requires basic spatial reasoning and is often used in clinical assessments of constructive apraxia. Remarkably, 24 out of 25 models failed to correctly render two horizontal lines against a perspective background, mirroring the deficits seen in patients with parietal lobe damage. The models consistently misinterpreted spatial instructions, producing tilted or misaligned lines that followed the perspective of the background rather than remaining horizontal. This behavior is strikingly similar to how apraxia patients struggle to copy or construct simple figures despite intact visual perception and motor skills. Our findings suggest that current VLMs, despite their advanced capabilities in other domains, lack fundamental spatial reasoning abilities akin to those impaired in constructive apraxia. This limitation in AI systems provides a novel computational model for studying spatial cognition deficits and highlights a critical area for improvement in VLM architecture and training methodologies.

Cross-validation (CV) is one of the most popular tools for assessing and selecting predictive models. However, standard CV suffers from high computational cost when the number of folds is large. Recently, under the empirical risk minimization (ERM) framework, a line of works proposed efficient methods to approximate CV based on the solution of the ERM problem trained on the full dataset. However, in large-scale problems, it can be hard to obtain the exact solution of the ERM problem, either due to limited computational resources or due to early stopping as a way of preventing overfitting. In this paper, we propose a new paradigm to efficiently approximate CV when the ERM problem is solved via an iterative first-order algorithm, without running until convergence. Our new method extends existing guarantees for CV approximation to hold along the whole trajectory of the algorithm, including at convergence, thus generalizing existing CV approximation methods. Finally, we illustrate the accuracy and computational efficiency of our method through a range of empirical studies.

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.

Exploratory data analytics (EDA) is a sequential decision making process where analysts choose subsequent queries that might lead to some interesting insights based on the previous queries and corresponding results. Data processing systems often execute the queries on samples to produce results with low latency. Different downsampling strategy preserves different statistics of the data and have different magnitude of latency reductions. The optimum choice of sampling strategy often depends on the particular context of the analysis flow and the hidden intent of the analyst. In this paper, we are the first to consider the impact of sampling in interactive data exploration settings as they introduce approximation errors. We propose a Deep Reinforcement Learning (DRL) based framework which can optimize the sample selection in order to keep the analysis and insight generation flow intact. Evaluations with 3 real datasets show that our technique can preserve the original insight generation flow while improving the interaction latency, compared to baseline methods.

We show that deep belief networks with binary hidden units can approximate any multivariate probability density under very mild integrability requirements on the parental density of the visible nodes. The approximation is measured in the L^q-norm for qin[1,infty] (q=infty corresponding to the supremum norm) and in Kullback-Leibler divergence. Furthermore, we establish sharp quantitative bounds on the approximation error in terms of the number of hidden units.

While Attention has come to be an important mechanism in deep learning, there remains limited intuition for why it works so well. Here, we show that Transformer Attention can be closely related under certain data conditions to Kanerva's Sparse Distributed Memory (SDM), a biologically plausible associative memory model. We confirm that these conditions are satisfied in pre-trained GPT2 Transformer models. We discuss the implications of the Attention-SDM map and provide new computational and biological interpretations of Attention.

Efficiently approximating local curvature information of the loss function is a key tool for optimization and compression of deep neural networks. Yet, most existing methods to approximate second-order information have high computational or storage costs, which can limit their practicality. In this work, we investigate matrix-free, linear-time approaches for estimating Inverse-Hessian Vector Products (IHVPs) for the case when the Hessian can be approximated as a sum of rank-one matrices, as in the classic approximation of the Hessian by the empirical Fisher matrix. We propose two new algorithms as part of a framework called M-FAC: the first algorithm is tailored towards network compression and can compute the IHVP for dimension d, if the Hessian is given as a sum of m rank-one matrices, using O(dm^2) precomputation, O(dm) cost for computing the IHVP, and query cost O(m) for any single element of the inverse Hessian. The second algorithm targets an optimization setting, where we wish to compute the product between the inverse Hessian, estimated over a sliding window of optimization steps, and a given gradient direction, as required for preconditioned SGD. We give an algorithm with cost O(dm + m^2) for computing the IHVP and O(dm + m^3) for adding or removing any gradient from the sliding window. These two algorithms yield state-of-the-art results for network pruning and optimization with lower computational overhead relative to existing second-order methods. Implementations are available at [9] and [17].

A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in mathbb R^d with continuous limit measures.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

Learning compatible representations enables the interchangeable use of semantic features as models are updated over time. This is particularly relevant in search and retrieval systems where it is crucial to avoid reprocessing of the gallery images with the updated model. While recent research has shown promising empirical evidence, there is still a lack of comprehensive theoretical understanding about learning compatible representations. In this paper, we demonstrate that the stationary representations learned by the d-Simplex fixed classifier optimally approximate compatibility representation according to the two inequality constraints of its formal definition. This not only establishes a solid foundation for future works in this line of research but also presents implications that can be exploited in practical learning scenarios. An exemplary application is the now-standard practice of downloading and fine-tuning new pre-trained models. Specifically, we show the strengths and critical issues of stationary representations in the case in which a model undergoing sequential fine-tuning is asynchronously replaced by downloading a better-performing model pre-trained elsewhere. Such a representation enables seamless delivery of retrieval service (i.e., no reprocessing of gallery images) and offers improved performance without operational disruptions during model replacement. Code available at: https://github.com/miccunifi/iamcl2r.

Low-rank approximation is a fundamental technique in modern data analysis, widely utilized across various fields such as signal processing, machine learning, and natural language processing. Despite its ubiquity, the mechanics of low-rank approximation and its application in adaptation can sometimes be obscure, leaving practitioners and researchers with questions about its true capabilities and limitations. This paper seeks to clarify low-rank approximation and adaptation by offering a comprehensive guide that reveals their inner workings and explains their utility in a clear and accessible way. Our focus here is to develop a solid intuition for how low-rank approximation and adaptation operate, and why they are so effective. We begin with basic concepts and gradually build up to the mathematical underpinnings, ensuring that readers of all backgrounds can gain a deeper understanding of low-rank approximation and adaptation. We strive to strike a balance between informal explanations and rigorous mathematics, ensuring that both newcomers and experienced experts can benefit from this survey. Additionally, we introduce new low-rank decomposition and adaptation algorithms that have not yet been explored in the field, hoping that future researchers will investigate their potential applicability.

Global channel pruning (GCP) aims to remove a subset of channels (filters) across different layers from a deep model without hurting the performance. Previous works focus on either single task model pruning or simply adapting it to multitask scenario, and still face the following problems when handling multitask pruning: 1) Due to the task mismatch, a well-pruned backbone for classification task focuses on preserving filters that can extract category-sensitive information, causing filters that may be useful for other tasks to be pruned during the backbone pruning stage; 2) For multitask predictions, different filters within or between layers are more closely related and interacted than that for single task prediction, making multitask pruning more difficult. Therefore, aiming at multitask model compression, we propose a Performance-Aware Global Channel Pruning (PAGCP) framework. We first theoretically present the objective for achieving superior GCP, by considering the joint saliency of filters from intra- and inter-layers. Then a sequentially greedy pruning strategy is proposed to optimize the objective, where a performance-aware oracle criterion is developed to evaluate sensitivity of filters to each task and preserve the globally most task-related filters. Experiments on several multitask datasets show that the proposed PAGCP can reduce the FLOPs and parameters by over 60% with minor performance drop, and achieves 1.2xsim3.3x acceleration on both cloud and mobile platforms.

To achieve scalable and accurate inference for latent Gaussian processes, we propose a variational approximation based on a family of Gaussian distributions whose covariance matrices have sparse inverse Cholesky (SIC) factors. We combine this variational approximation of the posterior with a similar and efficient SIC-restricted Kullback-Leibler-optimal approximation of the prior. We then focus on a particular SIC ordering and nearest-neighbor-based sparsity pattern resulting in highly accurate prior and posterior approximations. For this setting, our variational approximation can be computed via stochastic gradient descent in polylogarithmic time per iteration. We provide numerical comparisons showing that the proposed double-Kullback-Leibler-optimal Gaussian-process approximation (DKLGP) can sometimes be vastly more accurate for stationary kernels than alternative approaches such as inducing-point and mean-field approximations at similar computational complexity.

In pursuit of faster computation, Efficient Transformers demonstrate an impressive variety of approaches -- models attaining sub-quadratic attention complexity can utilize a notion of sparsity or a low-rank approximation of inputs to reduce the number of attended keys; other ways to reduce complexity include locality-sensitive hashing, key pooling, additional memory to store information in compacted or hybridization with other architectures, such as CNN. Often based on a strong mathematical basis, kernelized approaches allow for the approximation of attention with linear complexity while retaining high accuracy. Therefore, in the present paper, we aim to expand the idea of trainable kernel methods to approximate the self-attention mechanism of the Transformer architecture.

Transformers have greatly advanced the state-of-the-art in Natural Language Processing (NLP) in recent years, but present very large computation and storage requirements. We observe that the design process of Transformers (pre-train a foundation model on a large dataset in a self-supervised manner, and subsequently fine-tune it for different downstream tasks) leads to task-specific models that are highly over-parameterized, adversely impacting both accuracy and inference efficiency. We propose AxFormer, a systematic framework that applies accuracy-driven approximations to create optimized transformer models for a given downstream task. AxFormer combines two key optimizations -- accuracy-driven pruning and selective hard attention. Accuracy-driven pruning identifies and removes parts of the fine-tuned transformer that hinder performance on the given downstream task. Sparse hard-attention optimizes attention blocks in selected layers by eliminating irrelevant word aggregations, thereby helping the model focus only on the relevant parts of the input. In effect, AxFormer leads to models that are more accurate, while also being faster and smaller. Our experiments on GLUE and SQUAD tasks show that AxFormer models are up to 4.5% more accurate, while also being up to 2.5X faster and up to 3.2X smaller than conventional fine-tuned models. In addition, we demonstrate that AxFormer can be combined with previous efforts such as distillation or quantization to achieve further efficiency gains.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

We present a new approach for the approximate K-nearest neighbor search based on navigable small world graphs with controllable hierarchy (Hierarchical NSW, HNSW). The proposed solution is fully graph-based, without any need for additional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. Hierarchical NSW incrementally builds a multi-layer structure consisting from hierarchical set of proximity graphs (layers) for nested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an exponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Small World (NSW) structures while additionally having the links separated by their characteristic distance scales. Starting search from the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a logarithmic complexity scaling. Additional employment of a heuristic for selecting proximity graph neighbors significantly increases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the proposed general metric space search index is able to strongly outperform previous opensource state-of-the-art vector-only approaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation.

While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer -- a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset -- thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times-from minutes to milliseconds-while achieving log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.

Learned sparse text embeddings have gained popularity due to their effectiveness in top-k retrieval and inherent interpretability. Their distributional idiosyncrasies, however, have long hindered their use in real-world retrieval systems. That changed with the recent development of approximate algorithms that leverage the distributional properties of sparse embeddings to speed up retrieval. Nonetheless, in much of the existing literature, evaluation has been limited to datasets with only a few million documents such as MSMARCO. It remains unclear how these systems behave on much larger datasets and what challenges lurk in larger scales. To bridge that gap, we investigate the behavior of state-of-the-art retrieval algorithms on massive datasets. We compare and contrast the recently-proposed Seismic and graph-based solutions adapted from dense retrieval. We extensively evaluate Splade embeddings of 138M passages from MsMarco-v2 and report indexing time and other efficiency and effectiveness metrics.

Low-rank adapters have become standard for efficiently fine-tuning large language models (LLMs), but they often fall short of achieving the performance of full fine-tuning. We propose a method, LoRA Silver Bullet or LoRA-SB, that approximates full fine-tuning within low-rank subspaces using a carefully designed initialization strategy. We theoretically demonstrate that the architecture of LoRA-XS, which inserts a learnable (r x r) matrix between B and A while keeping other matrices fixed, provides the precise conditions needed for this approximation. We leverage its constrained update space to achieve optimal scaling for high-rank gradient updates while removing the need for hyperparameter tuning. We prove that our initialization offers an optimal low-rank approximation of the initial gradient and preserves update directions throughout training. Extensive experiments across mathematical reasoning, commonsense reasoning, and language understanding tasks demonstrate that our approach exceeds the performance of standard LoRA while using 27-90 times fewer learnable parameters, and comprehensively outperforms LoRA-XS. Our findings establish that it is possible to simulate full fine-tuning in low-rank subspaces, and achieve significant efficiency gains without sacrificing performance. Our code is publicly available at https://github.com/RaghavSinghal10/lora-sb.

Training data compositions for Large Language Models (LLMs) can significantly affect their downstream performance. However, a thorough data ablation study exploring large sets of candidate data mixtures is typically prohibitively expensive since the full effect is seen only after training the models; this can lead practitioners to settle for sub-optimal data mixtures. We propose an efficient method for approximating data ablations which trains individual models on subsets of a training corpus and reuses them across evaluations of combinations of subsets. In continued pre-training experiments, we find that, given an arbitrary evaluation set, the perplexity score of a single model trained on a candidate set of data is strongly correlated with perplexity scores of parameter averages of models trained on distinct partitions of that data. From this finding, we posit that researchers and practitioners can conduct inexpensive simulations of data ablations by maintaining a pool of models that were each trained on partitions of a large training corpus, and assessing candidate data mixtures by evaluating parameter averages of combinations of these models. This approach allows for substantial improvements in amortized training efficiency -- scaling only linearly with respect to new data -- by enabling reuse of previous training computation, opening new avenues for improving model performance through rigorous, incremental data assessment and mixing.

The widespread adoption of machine learning algorithms necessitates hardware acceleration to ensure efficient performance. This acceleration relies on custom matrix engines that operate on full or reduced-precision floating-point arithmetic. However, conventional floating-point implementations can be power hungry. This paper proposes a method to improve the energy efficiency of the matrix engines used in machine learning algorithm acceleration. Our approach leverages approximate normalization within the floating-point multiply-add units as a means to reduce their hardware complexity, without sacrificing overall machine-learning model accuracy. Hardware synthesis results show that this technique reduces area and power consumption roughly by 16% and 13% on average for Bfloat16 format. Also, the error introduced in transformer model accuracy is 1% on average, for the most efficient configuration of the proposed approach.

To remove redundant components of large language models (LLMs) without incurring significant computational costs, this work focuses on single-shot pruning without a retraining phase. We simplify the pruning process for Transformer-based LLMs by identifying a depth-2 pruning structure that functions independently. Additionally, we propose two inference-aware pruning criteria derived from the optimization perspective of output approximation, which outperforms traditional training-aware metrics such as gradient and Hessian. We also introduce a two-step reconstruction technique to mitigate pruning errors without model retraining. Experimental results demonstrate that our approach significantly reduces computational costs and hardware requirements while maintaining superior performance across various datasets and models.

Fine-tuning pretrained large models to downstream tasks is an important problem, which however suffers from huge memory overhead due to large-scale parameters. This work strives to reduce memory overhead in fine-tuning from perspectives of activation function and layer normalization. To this end, we propose the Approximate Backpropagation (Approx-BP) theory, which provides the theoretical feasibility of decoupling the forward and backward passes. We apply our Approx-BP theory to backpropagation training and derive memory-efficient alternatives of GELU and SiLU activation functions, which use derivative functions of ReLUs in the backward pass while keeping their forward pass unchanged. In addition, we introduce a Memory-Sharing Backpropagation strategy, which enables the activation memory to be shared by two adjacent layers, thereby removing activation memory usage redundancy. Our method neither induces extra computation nor reduces training efficiency. We conduct extensive experiments with pretrained vision and language models, and the results demonstrate that our proposal can reduce up to sim30% of the peak memory usage. Our code is released at https://github.com/yyyyychen/LowMemoryBP.

Zero-shot and in-context learning enable solving tasks without model fine-tuning, making them essential for developing generative model solutions. Therefore, it is crucial to understand whether a pretrained model can be prompted to approximate any function, i.e., whether it is a universal in-context approximator. While it was recently shown that transformer models do possess this property, these results rely on their attention mechanism. Hence, these findings do not apply to fully recurrent architectures like RNNs, LSTMs, and the increasingly popular SSMs. We demonstrate that RNNs, LSTMs, GRUs, Linear RNNs, and linear gated architectures such as Mamba and Hawk/Griffin can also serve as universal in-context approximators. To streamline our argument, we introduce a programming language called LSRL that compiles to these fully recurrent architectures. LSRL may be of independent interest for further studies of fully recurrent models, such as constructing interpretability benchmarks. We also study the role of multiplicative gating and observe that architectures incorporating such gating (e.g., LSTMs, GRUs, Hawk/Griffin) can implement certain operations more stably, making them more viable candidates for practical in-context universal approximation.

We introduce for continual learning Autodiff Quadratic Consolidation (AQC), which approximates the previous loss function with a quadratic function, and Neural Consolidation (NC), which approximates the previous loss function with a neural network. Although they are not scalable to large neural networks, they can be used with a fixed pre-trained feature extractor. We empirically study these methods in class-incremental learning, for which regularization-based methods produce unsatisfactory results, unless combined with replay. We find that for small datasets, quadratic approximation of the previous loss function leads to poor results, even with full Hessian computation, and NC could significantly improve the predictive performance, while for large datasets, when used with a fixed pre-trained feature extractor, AQC provides superior predictive performance. We also find that using tanh-output features can improve the predictive performance of AQC. In particular, in class-incremental Split MNIST, when a Convolutional Neural Network (CNN) with tanh-output features is pre-trained on EMNIST Letters and used as a fixed pre-trained feature extractor, AQC can achieve predictive performance comparable to joint training.

Many training data attribution (TDA) methods aim to estimate how a model's behavior would change if one or more data points were removed from the training set. Methods based on implicit differentiation, such as influence functions, can be made computationally efficient, but fail to account for underspecification, the implicit bias of the optimization algorithm, or multi-stage training pipelines. By contrast, methods based on unrolling address these issues but face scalability challenges. In this work, we connect the implicit-differentiation-based and unrolling-based approaches and combine their benefits by introducing Source, an approximate unrolling-based TDA method that is computed using an influence-function-like formula. While being computationally efficient compared to unrolling-based approaches, Source is suitable in cases where implicit-differentiation-based approaches struggle, such as in non-converged models and multi-stage training pipelines. Empirically, Source outperforms existing TDA techniques in counterfactual prediction, especially in settings where implicit-differentiation-based approaches fall short.

A neural architecture with randomly initialized weights, in the infinite width limit, is equivalent to a Gaussian Random Field whose covariance function is the so-called Neural Network Gaussian Process kernel (NNGP). We prove that a reproducing kernel Hilbert space (RKHS) defined by the NNGP contains only functions that can be approximated by the architecture. To achieve a certain approximation error the required number of neurons in each layer is defined by the RKHS norm of the target function. Moreover, the approximation can be constructed from a supervised dataset by a random multi-layer representation of an input vector, together with training of the last layer's weights. For a 2-layer NN and a domain equal to an n-1-dimensional sphere in {mathbb R}^n, we compare the number of neurons required by Barron's theorem and by the multi-layer features construction. We show that if eigenvalues of the integral operator of the NNGP decay slower than k^{-n-2{3}} where k is an order of an eigenvalue, then our theorem guarantees a more succinct neural network approximation than Barron's theorem. We also make some computational experiments to verify our theoretical findings. Our experiments show that realistic neural networks easily learn target functions even when both theorems do not give any guarantees.

Diffusion models have significantly advanced the state of the art in image, audio, and video generation tasks. However, their applications in practical scenarios are hindered by slow inference speed. Drawing inspiration from the approximation strategies utilized in consistency models, we propose the Sub-path Linear Approximation Model (SLAM), which accelerates diffusion models while maintaining high-quality image generation. SLAM treats the PF-ODE trajectory as a series of PF-ODE sub-paths divided by sampled points, and harnesses sub-path linear (SL) ODEs to form a progressive and continuous error estimation along each individual PF-ODE sub-path. The optimization on such SL-ODEs allows SLAM to construct denoising mappings with smaller cumulative approximated errors. An efficient distillation method is also developed to facilitate the incorporation of more advanced diffusion models, such as latent diffusion models. Our extensive experimental results demonstrate that SLAM achieves an efficient training regimen, requiring only 6 A100 GPU days to produce a high-quality generative model capable of 2 to 4-step generation with high performance. Comprehensive evaluations on LAION, MS COCO 2014, and MS COCO 2017 datasets also illustrate that SLAM surpasses existing acceleration methods in few-step generation tasks, achieving state-of-the-art performance both on FID and the quality of the generated images.

This work considers a rather general and broad class of Markov chains, Ito chains that look like Euler-Maryama discretization of some Stochastic Differential Equation. The chain we study is a unified framework for theoretical analysis. It comes with almost arbitrary isotropic and state-dependent noise instead of normal and state-independent one, as in most related papers. Moreover, our chain's drift and diffusion coefficient can be inexact to cover a wide range of applications such as Stochastic Gradient Langevin Dynamics, sampling, Stochastic Gradient Descent, or Stochastic Gradient Boosting. We prove an upper bound for W_{2}-distance between laws of the Ito chain and the corresponding Stochastic Differential Equation. These results improve or cover most of the known estimates. Moreover, for some particular cases, our analysis is the first.

Large language models (LLMs) are trained on massive internet corpora that often contain copyrighted content. This poses legal and ethical challenges for the developers and users of these models, as well as the original authors and publishers. In this paper, we propose a novel technique for unlearning a subset of the training data from a LLM, without having to retrain it from scratch. We evaluate our technique on the task of unlearning the Harry Potter books from the Llama2-7b model (a generative language model recently open-sourced by Meta). While the model took over 184K GPU-hours to pretrain, we show that in about 1 GPU hour of finetuning, we effectively erase the model's ability to generate or recall Harry Potter-related content, while its performance on common benchmarks (such as Winogrande, Hellaswag, arc, boolq and piqa) remains almost unaffected. We make our fine-tuned model publicly available on HuggingFace for community evaluation. To the best of our knowledge, this is the first paper to present an effective technique for unlearning in generative language models. Our technique consists of three main components: First, we use a reinforced model that is further trained on the target data to identify the tokens that are most related to the unlearning target, by comparing its logits with those of a baseline model. Second, we replace idiosyncratic expressions in the target data with generic counterparts, and leverage the model's own predictions to generate alternative labels for every token. These labels aim to approximate the next-token predictions of a model that has not been trained on the target data. Third, we finetune the model on these alternative labels, which effectively erases the original text from the model's memory whenever it is prompted with its context.

We discuss the numerical solution of initial value problems for varepsilon^2,varphi''+a(x),varphi=0 in the highly oscillatory regime, i.e., with a(x)>0 and 0<varepsilonll 1. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude O(varepsilon^{N}) where N refers to the truncation order of the underlying asymptotic series. When the optimal truncation order N_{opt} is chosen, the error behaves like O(varepsilon^{-2}exp(-cvarepsilon^{-1})) with some c>0.

It has been shown that deep neural networks of a large enough width are universal approximators but they are not if the width is too small. There were several attempts to characterize the minimum width w_{min} enabling the universal approximation property; however, only a few of them found the exact values. In this work, we show that the minimum width for L^p approximation of L^p functions from [0,1]^{d_x} to mathbb R^{d_y} is exactly max{d_x,d_y,2} if an activation function is ReLU-Like (e.g., ReLU, GELU, Softplus). Compared to the known result for ReLU networks, w_{min}=max{d_x+1,d_y} when the domain is mathbb R^{d_x}, our result first shows that approximation on a compact domain requires smaller width than on mathbb R^{d_x}. We next prove a lower bound on w_{min} for uniform approximation using general activation functions including ReLU: w_{min}ge d_y+1 if d_x<d_yle2d_x. Together with our first result, this shows a dichotomy between L^p and uniform approximations for general activation functions and input/output dimensions.

Sublinear time complexity is required by the massively parallel computation (MPC) model. Breaking dynamic programs into a set of sparse dynamic programs that can be divided, solved, and merged in sublinear time. The rectangle escape problem (REP) is defined as follows: For n axis-aligned rectangles inside an axis-aligned bounding box B, extend each rectangle in only one of the four directions: up, down, left, or right until it reaches B and the density k is minimized, where k is the maximum number of extensions of rectangles to the boundary that pass through a point inside bounding box B. REP is NP-hard for k>1. If the rectangles are points of a grid (or unit squares of a grid), the problem is called the square escape problem (SEP) and it is still NP-hard. We give a 2-approximation algorithm for SEP with kgeq2 with time complexity O(n^{3/2}k^2). This improves the time complexity of existing algorithms which are at least quadratic. Also, the approximation ratio of our algorithm for kgeq 3 is 3/2 which is tight. We also give a 8-approximation algorithm for REP with time complexity O(nlog n+nk) and give a MPC version of this algorithm for k=O(1) which is the first parallel algorithm for this problem.

We introduce a new class of algorithms, Stochastic Generalized Method of Moments (SGMM), for estimation and inference on (overidentified) moment restriction models. Our SGMM is a novel stochastic approximation alternative to the popular Hansen (1982) (offline) GMM, and offers fast and scalable implementation with the ability to handle streaming datasets in real time. We establish the almost sure convergence, and the (functional) central limit theorem for the inefficient online 2SLS and the efficient SGMM. Moreover, we propose online versions of the Durbin-Wu-Hausman and Sargan-Hansen tests that can be seamlessly integrated within the SGMM framework. Extensive Monte Carlo simulations show that as the sample size increases, the SGMM matches the standard (offline) GMM in terms of estimation accuracy and gains over computational efficiency, indicating its practical value for both large-scale and online datasets. We demonstrate the efficacy of our approach by a proof of concept using two well known empirical examples with large sample sizes.

General function approximation is a powerful tool to handle large state and action spaces in a broad range of reinforcement learning (RL) scenarios. However, theoretical understanding of non-stationary MDPs with general function approximation is still limited. In this paper, we make the first such an attempt. We first propose a new complexity metric called dynamic Bellman Eluder (DBE) dimension for non-stationary MDPs, which subsumes majority of existing tractable RL problems in static MDPs as well as non-stationary MDPs. Based on the proposed complexity metric, we propose a novel confidence-set based model-free algorithm called SW-OPEA, which features a sliding window mechanism and a new confidence set design for non-stationary MDPs. We then establish an upper bound on the dynamic regret for the proposed algorithm, and show that SW-OPEA is provably efficient as long as the variation budget is not significantly large. We further demonstrate via examples of non-stationary linear and tabular MDPs that our algorithm performs better in small variation budget scenario than the existing UCB-type algorithms. To the best of our knowledge, this is the first dynamic regret analysis in non-stationary MDPs with general function approximation.

The study of universal approximation properties (UAP) for neural networks (NN) has a long history. When the network width is unlimited, only a single hidden layer is sufficient for UAP. In contrast, when the depth is unlimited, the width for UAP needs to be not less than the critical width w^*_{min}=max(d_x,d_y), where d_x and d_y are the dimensions of the input and output, respectively. Recently, cai2022achieve shows that a leaky-ReLU NN with this critical width can achieve UAP for L^p functions on a compact domain K, i.e., the UAP for L^p(K,R^{d_y}). This paper examines a uniform UAP for the function class C(K,R^{d_y}) and gives the exact minimum width of the leaky-ReLU NN as w_{min}=max(d_x+1,d_y)+1_{d_y=d_x+1}, which involves the effects of the output dimensions. To obtain this result, we propose a novel lift-flow-discretization approach that shows that the uniform UAP has a deep connection with topological theory.

Neural Controlled Differential Equations (NCDEs) are a state-of-the-art tool for supervised learning with irregularly sampled time series (Kidger, 2020). However, no theoretical analysis of their performance has been provided yet, and it remains unclear in particular how the irregularity of the time series affects their predictions. By merging the rich theory of controlled differential equations (CDE) and Lipschitz-based measures of the complexity of deep neural nets, we take a first step towards the theoretical understanding of NCDE. Our first result is a generalization bound for this class of predictors that depends on the regularity of the time series data. In a second time, we leverage the continuity of the flow of CDEs to provide a detailed analysis of both the sampling-induced bias and the approximation bias. Regarding this last result, we show how classical approximation results on neural nets may transfer to NCDEs. Our theoretical results are validated through a series of experiments.

We study multi-agent reinforcement learning in the setting of episodic Markov decision processes, where multiple agents cooperate via communication through a central server. We propose a provably efficient algorithm based on value iteration that enable asynchronous communication while ensuring the advantage of cooperation with low communication overhead. With linear function approximation, we prove that our algorithm enjoys an mathcal{O}(d^{3/2}H^2K) regret with mathcal{O}(dHM^2) communication complexity, where d is the feature dimension, H is the horizon length, M is the total number of agents, and K is the total number of episodes. We also provide a lower bound showing that a minimal Omega(dM) communication complexity is required to improve the performance through collaboration.

In this paper, we develop an approximation scheme for solving bilevel programs with equilibrium constraints, which are generally difficult to solve. Among other things, calculating the first-order derivative in such a problem requires differentiation across the hierarchy, which is computationally intensive, if not prohibitive. To bypass the hierarchy, we propose to bound such bilevel programs, equivalent to multiple-followers Stackelberg games, with two new hierarchy-free problems: a T-step Cournot game and a T-step monopoly model. Since they are standard equilibrium or optimization problems, both can be efficiently solved via first-order methods. Importantly, we show that the bounds provided by these problems -- the upper bound by the T-step Cournot game and the lower bound by the T-step monopoly model -- can be made arbitrarily tight by increasing the step parameter T for a wide range of problems. We prove that a small T usually suffices under appropriate conditions to reach an approximation acceptable for most practical purposes. Eventually, the analytical insights are highlighted through numerical examples.

Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.

It is often useful to compactly summarize important properties of model parameters and training data so that they can be used later without storing and/or iterating over the entire dataset. As a specific case, we consider estimating the Function Space Distance (FSD) over a training set, i.e. the average discrepancy between the outputs of two neural networks. We propose a Linearized Activation Function TRick (LAFTR) and derive an efficient approximation to FSD for ReLU neural networks. The key idea is to approximate the architecture as a linear network with stochastic gating. Despite requiring only one parameter per unit of the network, our approach outcompetes other parametric approximations with larger memory requirements. Applied to continual learning, our parametric approximation is competitive with state-of-the-art nonparametric approximations, which require storing many training examples. Furthermore, we show its efficacy in estimating influence functions accurately and detecting mislabeled examples without expensive iterations over the entire dataset.

Recently, remarkable progress has been made by approximating Nash equilibrium (NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE) through function approximation that trains a neural network to predict equilibria from game representations. Furthermore, equivariant architectures are widely adopted in designing such equilibrium approximators in normal-form games. In this paper, we theoretically characterize benefits and limitations of equivariant equilibrium approximators. For the benefits, we show that they enjoy better generalizability than general ones and can achieve better approximations when the payoff distribution is permutation-invariant. For the limitations, we discuss their drawbacks in terms of equilibrium selection and social welfare. Together, our results help to understand the role of equivariance in equilibrium approximators.

We study the problem of combining neural networks with symbolic reasoning. Recently introduced frameworks for Probabilistic Neurosymbolic Learning (PNL), such as DeepProbLog, perform exponential-time exact inference, limiting the scalability of PNL solutions. We introduce Approximate Neurosymbolic Inference (A-NeSI): a new framework for PNL that uses neural networks for scalable approximate inference. A-NeSI 1) performs approximate inference in polynomial time without changing the semantics of probabilistic logics; 2) is trained using data generated by the background knowledge; 3) can generate symbolic explanations of predictions; and 4) can guarantee the satisfaction of logical constraints at test time, which is vital in safety-critical applications. Our experiments show that A-NeSI is the first end-to-end method to solve three neurosymbolic tasks with exponential combinatorial scaling. Finally, our experiments show that A-NeSI achieves explainability and safety without a penalty in performance.

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional E(u) = int_Omega L(x, u(x), nabla u(x)) - f(x) u(x)dx. We show that if composing a function with Barron norm b with partial derivatives of L produces a function of Barron norm at most B_L b^p, the solution to the PDE can be epsilon-approximated in the L^2 sense by a function with Barron norm Oleft(left(dB_Lright)^{max{p log(1/ epsilon), p^{log(1/epsilon)}}}right). By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating p, epsilon, B_L as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

Contrastive learning has recently achieved remarkable success in many domains including graphs. However contrastive loss, especially for graphs, requires a large number of negative samples which is unscalable and computationally prohibitive with a quadratic time complexity. Sub-sampling is not optimal and incorrect negative sampling leads to sampling bias. In this work, we propose a meta-node based approximation technique that can (a) proxy all negative combinations (b) in quadratic cluster size time complexity, (c) at graph level, not node level, and (d) exploit graph sparsity. By replacing node-pairs with additive cluster-pairs, we compute the negatives in cluster-time at graph level. The resulting Proxy approximated meta-node Contrastive (PamC) loss, based on simple optimized GPU operations, captures the full set of negatives, yet is efficient with a linear time complexity. By avoiding sampling, we effectively eliminate sample bias. We meet the criterion for larger number of samples, thus achieving block-contrastiveness, which is proven to outperform pair-wise losses. We use learnt soft cluster assignments for the meta-node constriction, and avoid possible heterophily and noise added during edge creation. Theoretically, we show that real world graphs easily satisfy conditions necessary for our approximation. Empirically, we show promising accuracy gains over state-of-the-art graph clustering on 6 benchmarks. Importantly, we gain substantially in efficiency; up to 3x in training time, 1.8x in inference time and over 5x in GPU memory reduction.

We study the approximability of an existing framework for clustering edge-colored hypergraphs, which is closely related to chromatic correlation clustering and is motivated by machine learning and data mining applications where the goal is to cluster a set of objects based on multiway interactions of different categories or types. We present improved approximation guarantees based on linear programming, and show they are tight by proving a matching integrality gap. Our results also include new approximation hardness results, a combinatorial 2-approximation whose runtime is linear in the hypergraph size, and several new connections to well-studied objectives such as vertex cover and hypergraph multiway cut.

Transformers have emerged as a preferred model for many tasks in natural langugage processing and vision. Recent efforts on training and deploying Transformers more efficiently have identified many strategies to approximate the self-attention matrix, a key module in a Transformer architecture. Effective ideas include various prespecified sparsity patterns, low-rank basis expansions and combinations thereof. In this paper, we revisit classical Multiresolution Analysis (MRA) concepts such as Wavelets, whose potential value in this setting remains underexplored thus far. We show that simple approximations based on empirical feedback and design choices informed by modern hardware and implementation challenges, eventually yield a MRA-based approach for self-attention with an excellent performance profile across most criteria of interest. We undertake an extensive set of experiments and demonstrate that this multi-resolution scheme outperforms most efficient self-attention proposals and is favorable for both short and long sequences. Code is available at https://github.com/mlpen/mra-attention.

Well-known activation functions like ReLU or Leaky ReLU are non-differentiable at the origin. Over the years, many smooth approximations of ReLU have been proposed using various smoothing techniques. We propose new smooth approximations of a non-differentiable activation function by convolving it with approximate identities. In particular, we present smooth approximations of Leaky ReLU and show that they outperform several well-known activation functions in various datasets and models. We call this function Smooth Activation Unit (SAU). Replacing ReLU by SAU, we get 5.12% improvement with ShuffleNet V2 (2.0x) model on CIFAR100 dataset.

Ensembles of independently trained neural networks are a state-of-the-art approach to estimate predictive uncertainty in Deep Learning, and can be interpreted as an approximation of the posterior distribution via a mixture of delta functions. The training of ensembles relies on non-convexity of the loss landscape and random initialization of their individual members, making the resulting posterior approximation uncontrolled. This paper proposes a novel and principled method to tackle this limitation, minimizing an f-divergence between the true posterior and a kernel density estimator (KDE) in a function space. We analyze this objective from a combinatorial point of view, and show that it is submodular with respect to mixture components for any f. Subsequently, we consider the problem of greedy ensemble construction. From the marginal gain on the negative f-divergence, which quantifies an improvement in posterior approximation yielded by adding a new component into the KDE, we derive a novel diversity term for ensemble methods. The performance of our approach is demonstrated on computer vision out-of-distribution detection benchmarks in a range of architectures trained on multiple datasets. The source code of our method is made publicly available at https://github.com/Oulu-IMEDS/greedy_ensembles_training.

Optical character recognition (OCR) is a widely used pattern recognition application in numerous domains. There are several feature-rich, general-purpose OCR solutions available for consumers, which can provide moderate to excellent accuracy levels. However, accuracy can diminish with difficult and uncommon document domains. Preprocessing of document images can be used to minimize the effect of domain shift. In this paper, a novel approach is presented for creating a customized preprocessor for a given OCR engine. Unlike the previous OCR agnostic preprocessing techniques, the proposed approach approximates the gradient of a particular OCR engine to train a preprocessor module. Experiments with two datasets and two OCR engines show that the presented preprocessor is able to improve the accuracy of the OCR up to 46% from the baseline by applying pixel-level manipulations to the document image. The implementation of the proposed method and the enhanced public datasets are available for download.

The state of the art in open-domain question answering (QA) relies on an efficient retriever that drastically reduces the search space for the expensive reader. A rather overlooked question in the community is the relationship between the retriever and the reader, and in particular, if the whole purpose of the retriever is just a fast approximation for the reader. Our empirical evidence indicates that the answer is no, and that the reader and the retriever are complementary to each other even in terms of accuracy only. We make a careful conjecture that the architectural constraint of the retriever, which has been originally intended for enabling approximate search, seems to also make the model more robust in large-scale search. We then propose to distill the reader into the retriever so that the retriever absorbs the strength of the reader while keeping its own benefit. Experimental results show that our method can enhance the document recall rate as well as the end-to-end QA accuracy of off-the-shelf retrievers in open-domain QA tasks.

In value-based reinforcement learning methods such as deep Q-learning, function approximation errors are known to lead to overestimated value estimates and suboptimal policies. We show that this problem persists in an actor-critic setting and propose novel mechanisms to minimize its effects on both the actor and the critic. Our algorithm builds on Double Q-learning, by taking the minimum value between a pair of critics to limit overestimation. We draw the connection between target networks and overestimation bias, and suggest delaying policy updates to reduce per-update error and further improve performance. We evaluate our method on the suite of OpenAI gym tasks, outperforming the state of the art in every environment tested.

The creation of practical deep learning data-products often requires parallelization across processors and computers to make deep learning feasible on large data sets, but bottlenecks in communication bandwidth make it difficult to attain good speedups through parallelism. Here we develop and test 8-bit approximation algorithms which make better use of the available bandwidth by compressing 32-bit gradients and nonlinear activations to 8-bit approximations. We show that these approximations do not decrease predictive performance on MNIST, CIFAR10, and ImageNet for both model and data parallelism and provide a data transfer speedup of 2x relative to 32-bit parallelism. We build a predictive model for speedups based on our experimental data, verify its validity on known speedup data, and show that we can obtain a speedup of 50x and more on a system of 96 GPUs compared to a speedup of 23x for 32-bit. We compare our data types with other methods and show that 8-bit approximations achieve state-of-the-art speedups for model parallelism. Thus 8-bit approximation is an efficient method to parallelize convolutional networks on very large systems of GPUs.

We consider a family of linear systems A_mu alpha=C with system matrix A_mu depending on a parameter mu and for simplicity parameter-independent right-hand side C. These linear systems typically result from the finite-dimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form A_muapproxsum_{m}beta_m(mu)A_{mu_m} for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices A_{mu_m}. As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.

The simulation of the metabolism in mammalian cells becomes a severe problem if spatial distributions must be taken into account. Especially the cytoplasm has a very complex geometric structure which cannot be handled by standard discretization techniques. In the present paper we propose a homogenization technique for computing effective diffusion constants. This is accomplished by using a two-step strategy. The first step consists of an analytic homogenization from the smallest to an intermediate scale. The homogenization error is estimated by comparing the analytic diffusion constant with a numerical estimate obtained by using real cell geometries. The second step consists of a random homogenization. Since no analytical solution is known to this homogenization problem, a numerical approximation algorithm is proposed. Although rather expensive this algorithm provides a reasonable estimate of the homogenized diffusion constant.

Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function "S". To demonstrate our proposed approach two examples are investigated in details.

Complex sensors such as LiDAR, RADAR, and event cameras have proliferated in autonomous robotics to enhance perception and understanding of the environment. Meanwhile, these sensors are also vulnerable to diverse failure mechanisms that can intricately interact with their operation environment. In parallel, the limited availability of training data on complex sensors also affects the reliability of their deep learning-based prediction flow, where their prediction models can fail to generalize to environments not adequately captured in the training set. To address these reliability concerns, this paper introduces STARNet, a Sensor Trustworthiness and Anomaly Recognition Network designed to detect untrustworthy sensor streams that may arise from sensor malfunctions and/or challenging environments. We specifically benchmark STARNet on LiDAR and camera data. STARNet employs the concept of approximated likelihood regret, a gradient-free framework tailored for low-complexity hardware, especially those with only fixed-point precision capabilities. Through extensive simulations, we demonstrate the efficacy of STARNet in detecting untrustworthy sensor streams in unimodal and multimodal settings. In particular, the network shows superior performance in addressing internal sensor failures, such as cross-sensor interference and crosstalk. In diverse test scenarios involving adverse weather and sensor malfunctions, we show that STARNet enhances prediction accuracy by approximately 10% by filtering out untrustworthy sensor streams. STARNet is publicly available at https://github.com/sinatayebati/STARNet.

With the growth of model and data sizes, a broad effort has been made to design pruning techniques that reduce the resource demand of deep learning pipelines, while retaining model performance. In order to reduce both inference and training costs, a prominent line of work uses low-rank matrix factorizations to represent the network weights. Although able to retain accuracy, we observe that low-rank methods tend to compromise model robustness against adversarial perturbations. By modeling robustness in terms of the condition number of the neural network, we argue that this loss of robustness is due to the exploding singular values of the low-rank weight matrices. Thus, we introduce a robust low-rank training algorithm that maintains the network's weights on the low-rank matrix manifold while simultaneously enforcing approximate orthonormal constraints. The resulting model reduces both training and inference costs while ensuring well-conditioning and thus better adversarial robustness, without compromising model accuracy. This is shown by extensive numerical evidence and by our main approximation theorem that shows the computed robust low-rank network well-approximates the ideal full model, provided a highly performing low-rank sub-network exists.

Recent advances in efficient Transformers have exploited either the sparsity or low-rank properties of attention matrices to reduce the computational and memory bottlenecks of modeling long sequences. However, it is still challenging to balance the trade-off between model quality and efficiency to perform a one-size-fits-all approximation for different tasks. To better understand this trade-off, we observe that sparse and low-rank approximations excel in different regimes, determined by the softmax temperature in attention, and sparse + low-rank can outperform each individually. Inspired by the classical robust-PCA algorithm for sparse and low-rank decomposition, we propose Scatterbrain, a novel way to unify sparse (via locality sensitive hashing) and low-rank (via kernel feature map) attention for accurate and efficient approximation. The estimation is unbiased with provably low error. We empirically show that Scatterbrain can achieve 2.1x lower error than baselines when serving as a drop-in replacement in BigGAN image generation and pre-trained T2T-ViT. On a pre-trained T2T Vision transformer, even without fine-tuning, Scatterbrain can reduce 98% of attention memory at the cost of only 1% drop in accuracy. We demonstrate Scatterbrain for end-to-end training with up to 4 points better perplexity and 5 points better average accuracy than sparse or low-rank efficient transformers on language modeling and long-range-arena tasks.

Convolutional neural networks (CNN) are built upon the classical McCulloch-Pitts neuron model, which is essentially a linear model, where the nonlinearity is provided by a separate activation function. Several researchers have proposed enhanced neuron models, including quadratic neurons, generalized operational neurons, generative neurons, and super neurons, with stronger nonlinearity than that provided by the pointwise activation function. There has also been a proposal to use Pade approximation as a generalized activation function. In this paper, we introduce a brand new neuron model called Pade neurons (Paons), inspired by the Pade approximants, which is the best mathematical approximation of a transcendental function as a ratio of polynomials with different orders. We show that Paons are a super set of all other proposed neuron models. Hence, the basic neuron in any known CNN model can be replaced by Paons. In this paper, we extend the well-known ResNet to PadeNet (built by Paons) to demonstrate the concept. Our experiments on the single-image super-resolution task show that PadeNets can obtain better results than competing architectures.

Image denoising is essential for removing noise in images caused by electric device malfunctions or other factors during image acquisition. It helps preserve image quality and interpretation. Many convolutional autoencoder algorithms have proven effective in image denoising. Owing to their promising efficiency, quantum computers have gained popularity. This study introduces a quantum convolutional autoencoder (QCAE) method for improved image denoising. This method was developed by substituting the representative latent space of the autoencoder with a quantum circuit. To enhance efficiency, we leveraged the advantages of the quantum approximate optimization algorithm (QAOA)-incorporated parameter-shift rule to identify an optimized cost function, facilitating effective learning from data and gradient computation on an actual quantum computer. The proposed QCAE method outperformed its classical counterpart as it exhibited lower training loss and a higher structural similarity index (SSIM) value. QCAE also outperformed its classical counterpart in denoising the MNIST dataset by up to 40% in terms of SSIM value, confirming its enhanced capabilities in real-world applications. Evaluation of QAOA performance across different circuit configurations and layer variations showed that our technique outperformed other circuit designs by 25% on average.

In-context learning (ICL) is an astonishing emergent ability of large language models (LLMs). By presenting a prompt that includes multiple input-output pairs as examples and introducing a new query input, models can generate the corresponding output. However, the performance of models heavily relies on the quality of the input prompt when implementing in-context learning. Biased or imbalanced input prompts can significantly degrade the performance of language models. To address this issue, we introduce a reweighted algorithm called RICL (Reweighted In-context Learning). This algorithm fine-tunes language models using an unbiased validation set to determine the optimal weight for each input-output example to approximate unbiased in-context learning. Furthermore, we also introduce a low-cost reweighted algorithm, a linear optimal weight approximation algorithm called LARICL (Linear Approximation of Reweighted In-context Learning). This algorithm requires minimal training cost while providing effective results. We prove the convergence of our algorithm and validate its performance through experiments conducted on a numerical dataset. The experimental findings reveal a substantial improvement in comparison to benchmarks including the performance of casual prompt-based in-context learning and the performance of a classic fine-tuning method.

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the L^p variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e.g., linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond O(T) and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL.

We study reinforcement learning with linear function approximation and adversarially changing cost functions, a setup that has mostly been considered under simplifying assumptions such as full information feedback or exploratory conditions.We present a computationally efficient policy optimization algorithm for the challenging general setting of unknown dynamics and bandit feedback, featuring a combination of mirror-descent and least squares policy evaluation in an auxiliary MDP used to compute exploration bonuses.Our algorithm obtains an widetilde O(K^{6/7}) regret bound, improving significantly over previous state-of-the-art of widetilde O (K^{14/15}) in this setting. In addition, we present a version of the same algorithm under the assumption a simulator of the environment is available to the learner (but otherwise no exploratory assumptions are made), and prove it obtains state-of-the-art regret of widetilde O (K^{2/3}).

We consider the problem of global optimization with noisy zeroth order oracles - a well-motivated problem useful for various applications ranging from hyper-parameter tuning for deep learning to new material design. Existing work relies on Gaussian processes or other non-parametric family, which suffers from the curse of dimensionality. In this paper, we propose a new algorithm GO-UCB that leverages a parametric family of functions (e.g., neural networks) instead. Under a realizable assumption and a few other mild geometric conditions, we show that GO-UCB achieves a cumulative regret of O(T) where T is the time horizon. At the core of GO-UCB is a carefully designed uncertainty set over parameters based on gradients that allows optimistic exploration. Synthetic and real-world experiments illustrate GO-UCB works better than Bayesian optimization approaches in high dimensional cases, even if the model is misspecified.

Empirical neural tangent kernels (eNTKs) can provide a good understanding of a given network's representation: they are often far less expensive to compute and applicable more broadly than infinite width NTKs. For networks with O output units (e.g. an O-class classifier), however, the eNTK on N inputs is of size NO times NO, taking O((NO)^2) memory and up to O((NO)^3) computation. Most existing applications have therefore used one of a handful of approximations yielding N times N kernel matrices, saving orders of magnitude of computation, but with limited to no justification. We prove that one such approximation, which we call "sum of logits", converges to the true eNTK at initialization for any network with a wide final "readout" layer. Our experiments demonstrate the quality of this approximation for various uses across a range of settings.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

Transformers have emerged as a powerful tool for a broad range of natural language processing tasks. A key component that drives the impressive performance of Transformers is the self-attention mechanism that encodes the influence or dependence of other tokens on each specific token. While beneficial, the quadratic complexity of self-attention on the input sequence length has limited its application to longer sequences -- a topic being actively studied in the community. To address this limitation, we propose Nystr\"{o}mformer -- a model that exhibits favorable scalability as a function of sequence length. Our idea is based on adapting the Nystr\"{o}m method to approximate standard self-attention with O(n) complexity. The scalability of Nystr\"{o}mformer enables application to longer sequences with thousands of tokens. We perform evaluations on multiple downstream tasks on the GLUE benchmark and IMDB reviews with standard sequence length, and find that our Nystr\"{o}mformer performs comparably, or in a few cases, even slightly better, than standard self-attention. On longer sequence tasks in the Long Range Arena (LRA) benchmark, Nystr\"{o}mformer performs favorably relative to other efficient self-attention methods. Our code is available at https://github.com/mlpen/Nystromformer.

Since long, research on machine translation has been ongoing. Still, we do not get good translations from MT engines so developed. Manual ranking of these outputs tends to be very time consuming and expensive. Identifying which one is better or worse than the others is a very taxing task. In this paper, we show an approach which can provide automatic ranks to MT outputs (translations) taken from different MT Engines and which is based on N-gram approximations. We provide a solution where no human intervention is required for ranking systems. Further we also show the evaluations of our results which show equivalent results as that of human ranking.