Daily Papers

Generative AI and large language models hold great promise in enhancing programming education by automatically generating individualized feedback for students. We investigate the role of generative AI models in providing human tutor-style programming hints to help students resolve errors in their buggy programs. Recent works have benchmarked state-of-the-art models for various feedback generation scenarios; however, their overall quality is still inferior to human tutors and not yet ready for real-world deployment. In this paper, we seek to push the limits of generative AI models toward providing high-quality programming hints and develop a novel technique, GPT4Hints-GPT3.5Val. As a first step, our technique leverages GPT-4 as a ``tutor'' model to generate hints -- it boosts the generative quality by using symbolic information of failing test cases and fixes in prompts. As a next step, our technique leverages GPT-3.5, a weaker model, as a ``student'' model to further validate the hint quality -- it performs an automatic quality validation by simulating the potential utility of providing this feedback. We show the efficacy of our technique via extensive evaluation using three real-world datasets of Python programs covering a variety of concepts ranging from basic algorithms to regular expressions and data analysis using pandas library.

Machine learning research has advanced in multiple aspects, including model structures and learning methods. The effort to automate such research, known as AutoML, has also made significant progress. However, this progress has largely focused on the architecture of neural networks, where it has relied on sophisticated expert-designed layers as building blocks---or similarly restrictive search spaces. Our goal is to show that AutoML can go further: it is possible today to automatically discover complete machine learning algorithms just using basic mathematical operations as building blocks. We demonstrate this by introducing a novel framework that significantly reduces human bias through a generic search space. Despite the vastness of this space, evolutionary search can still discover two-layer neural networks trained by backpropagation. These simple neural networks can then be surpassed by evolving directly on tasks of interest, e.g. CIFAR-10 variants, where modern techniques emerge in the top algorithms, such as bilinear interactions, normalized gradients, and weight averaging. Moreover, evolution adapts algorithms to different task types: e.g., dropout-like techniques appear when little data is available. We believe these preliminary successes in discovering machine learning algorithms from scratch indicate a promising new direction for the field.

This document aims to be a self-contained, mathematically precise overview of transformer architectures and algorithms (*not* results). It covers what transformers are, how they are trained, what they are used for, their key architectural components, and a preview of the most prominent models. The reader is assumed to be familiar with basic ML terminology and simpler neural network architectures such as MLPs.

In this case study, the renowned Travelling Salesmen problem has been studied. Travelling Salesman problem is a most demanding computational problem in Computer Science. The Travelling Salesmen problem has been solved by two different ways using Hopfield Network. The main theory of the problem is to find distance and connectedness between nodes in a graph having edges between the nodes. The basic algorithm used for this problem is Djikstra's Algorithm. But till now , a number of such algorithms have evolved. Among them(some other algorithms) , are distinct and have been proved to solve the travelling salesmen problem by graph theory.

We initiate a formal investigation into the design and analysis of LLM-based algorithms, i.e. algorithms that contain one or multiple calls of large language models (LLMs) as sub-routines and critically rely on the capabilities of LLMs. While LLM-based algorithms, ranging from basic LLM calls with prompt engineering to complicated LLM-powered agent systems and compound AI systems, have achieved remarkable empirical success, the design and optimization of them have mostly relied on heuristics and trial-and-errors, which is largely due to a lack of formal and analytical study for these algorithms. To fill this gap, we start by identifying the computational-graph representation of LLM-based algorithms, the design principle of task decomposition, and some key abstractions, which then facilitate our formal analysis for the accuracy and efficiency of LLM-based algorithms, despite the black-box nature of LLMs. Through extensive analytical and empirical investigation in a series of case studies, we demonstrate that the proposed framework is broadly applicable to a wide range of scenarios and diverse patterns of LLM-based algorithms, such as parallel, hierarchical and recursive task decomposition. Our proposed framework holds promise for advancing LLM-based algorithms, by revealing the reasons behind curious empirical phenomena, guiding the choices of hyperparameters, predicting the empirical performance of algorithms, and inspiring new algorithm design. To promote further study of LLM-based algorithms, we release our source code at https://github.com/modelscope/agentscope/tree/main/examples/paper_llm_based_algorithm.

In Islam, readers must apply a set of pronunciation rules called Tajweed rules to recite the Quran in the same way that the angel Jibrael taught the Prophet, Muhammad. The traditional process of learning the correct application of these rules requires a human who must have a license and great experience to detect mispronunciation. Due to the increasing number of Muslims around the world, the number of Tajweed teachers is not enough nowadays for daily recitation practice for every Muslim. Therefore, lots of work has been done for automatic Tajweed rules' mispronunciation detection to help readers recite Quran correctly in an easier way and shorter time than traditional learning ways. All previous works have three common problems. First, most of them focused on machine learning algorithms only. Second, they used private datasets with no benchmark to compare with. Third, they did not take into consideration the sequence of input data optimally, although the speech signal is time series. To overcome these problems, we proposed a solution that consists of Mel-Frequency Cepstral Coefficient (MFCC) features with Long Short-Term Memory (LSTM) neural networks which use the time series, to detect mispronunciation in Tajweed rules. In addition, our experiments were performed on a public dataset, the QDAT dataset, which contains more than 1500 voices of the correct and incorrect recitation of three Tajweed rules (Separate stretching , Tight Noon , and Hide ). To the best of our knowledge, the QDAT dataset has not been used by any research paper yet. We compared the performance of the proposed LSTM model with traditional machine learning algorithms used in SoTA. The LSTM model with time series showed clear superiority over traditional machine learning. The accuracy achieved by LSTM on the QDAT dataset was 96%, 95%, and 96% for the three rules (Separate stretching, Tight Noon, and Hide), respectively.

Do large language models (LLMs) solve reasoning tasks by learning robust generalizable algorithms, or do they memorize training data? To investigate this question, we use arithmetic reasoning as a representative task. Using causal analysis, we identify a subset of the model (a circuit) that explains most of the model's behavior for basic arithmetic logic and examine its functionality. By zooming in on the level of individual circuit neurons, we discover a sparse set of important neurons that implement simple heuristics. Each heuristic identifies a numerical input pattern and outputs corresponding answers. We hypothesize that the combination of these heuristic neurons is the mechanism used to produce correct arithmetic answers. To test this, we categorize each neuron into several heuristic types-such as neurons that activate when an operand falls within a certain range-and find that the unordered combination of these heuristic types is the mechanism that explains most of the model's accuracy on arithmetic prompts. Finally, we demonstrate that this mechanism appears as the main source of arithmetic accuracy early in training. Overall, our experimental results across several LLMs show that LLMs perform arithmetic using neither robust algorithms nor memorization; rather, they rely on a "bag of heuristics".

Artificial intelligence (AI) has achieved astonishing successes in many domains, especially with the recent breakthroughs in the development of foundational large models. These large models, leveraging their extensive training data, provide versatile solutions for a wide range of downstream tasks. However, as modern datasets become increasingly diverse and complex, the development of large AI models faces two major challenges: (1) the enormous consumption of computational resources and deployment difficulties, and (2) the difficulty in fitting heterogeneous and complex data, which limits the usability of the models. Mixture of Experts (MoE) models has recently attracted much attention in addressing these challenges, by dynamically selecting and activating the most relevant sub-models to process input data. It has been shown that MoEs can significantly improve model performance and efficiency with fewer resources, particularly excelling in handling large-scale, multimodal data. Given the tremendous potential MoE has demonstrated across various domains, it is urgent to provide a comprehensive summary of recent advancements of MoEs in many important fields. Existing surveys on MoE have their limitations, e.g., being outdated or lacking discussion on certain key areas, and we aim to address these gaps. In this paper, we first introduce the basic design of MoE, including gating functions, expert networks, routing mechanisms, training strategies, and system design. We then explore the algorithm design of MoE in important machine learning paradigms such as continual learning, meta-learning, multi-task learning, and reinforcement learning. Additionally, we summarize theoretical studies aimed at understanding MoE and review its applications in computer vision and natural language processing. Finally, we discuss promising future research directions.

Training algorithms, broadly construed, are an essential part of every deep learning pipeline. Training algorithm improvements that speed up training across a wide variety of workloads (e.g., better update rules, tuning protocols, learning rate schedules, or data selection schemes) could save time, save computational resources, and lead to better, more accurate, models. Unfortunately, as a community, we are currently unable to reliably identify training algorithm improvements, or even determine the state-of-the-art training algorithm. In this work, using concrete experiments, we argue that real progress in speeding up training requires new benchmarks that resolve three basic challenges faced by empirical comparisons of training algorithms: (1) how to decide when training is complete and precisely measure training time, (2) how to handle the sensitivity of measurements to exact workload details, and (3) how to fairly compare algorithms that require hyperparameter tuning. In order to address these challenges, we introduce a new, competitive, time-to-result benchmark using multiple workloads running on fixed hardware, the AlgoPerf: Training Algorithms benchmark. Our benchmark includes a set of workload variants that make it possible to detect benchmark submissions that are more robust to workload changes than current widely-used methods. Finally, we evaluate baseline submissions constructed using various optimizers that represent current practice, as well as other optimizers that have recently received attention in the literature. These baseline results collectively demonstrate the feasibility of our benchmark, show that non-trivial gaps between methods exist, and set a provisional state-of-the-art for future benchmark submissions to try and surpass.

Large language models (LLMs) have achieved remarkable advancements in natural language processing, showcasing exceptional performance across various tasks. However, the expensive memory and computational requirements present significant challenges for their practical deployment. Low-bit quantization has emerged as a critical approach to mitigate these challenges by reducing the bit-width of model parameters, activations, and gradients, thus decreasing memory usage and computational demands. This paper presents a comprehensive survey of low-bit quantization methods tailored for LLMs, covering the fundamental principles, system implementations, and algorithmic strategies. An overview of basic concepts and new data formats specific to low-bit LLMs is first introduced, followed by a review of frameworks and systems that facilitate low-bit LLMs across various hardware platforms. Then, we categorize and analyze techniques and toolkits for efficient low-bit training and inference of LLMs. Finally, we conclude with a discussion of future trends and potential advancements of low-bit LLMs. Our systematic overview from basic, system, and algorithm perspectives can offer valuable insights and guidelines for future works to enhance the efficiency and applicability of LLMs through low-bit quantization.

Sampling is a basic operation in many inference-time algorithms of large language models (LLMs). To scale up inference efficiently with a limited compute, it is crucial to find an optimal allocation for sample compute budgets: Which sampling configurations (model, temperature, language, etc.) do we use? How many samples do we generate in each configuration? We formulate these choices as a learning problem and propose OSCA, an algorithm that Optimizes Sample Compute Allocation by finding an optimal mix of different inference configurations. Our experiments show that with our learned mixed allocation, we can achieve accuracy better than the best single configuration with 128x less compute on code generation and 25x less compute on 4 reasoning tasks. OSCA is also shown to be effective in agentic workflows beyond single-turn tasks, achieving a better accuracy on SWE-Bench with 3x less compute than the default configuration. Our code and generations are released at https://github.com/LeiLiLab/OSCA.

Sparse expert models are a thirty-year old concept re-emerging as a popular architecture in deep learning. This class of architecture encompasses Mixture-of-Experts, Switch Transformers, Routing Networks, BASE layers, and others, all with the unifying idea that each example is acted on by a subset of the parameters. By doing so, the degree of sparsity decouples the parameter count from the compute per example allowing for extremely large, but efficient models. The resulting models have demonstrated significant improvements across diverse domains such as natural language processing, computer vision, and speech recognition. We review the concept of sparse expert models, provide a basic description of the common algorithms, contextualize the advances in the deep learning era, and conclude by highlighting areas for future work.

The complexity of laboratory environments requires solutions that simplify instrument interaction and enhance measurement automation. Traditional tools often require configuration, software, and programming skills, creating barriers to productivity. Previous approaches, including dedicated software suites and custom scripts, frequently fall short in providing user-friendly solutions that align with programming practices. We present LABIIUM, an AI-enhanced, zero-configuration measurement automation system designed to streamline experimental workflows and improve user productivity. LABIIUM integrates an AI assistant powered by Large Language Models (LLMs) to generate code. LABIIUM's Lab-Automation-Measurement Bridges (LAMBs) enable seamless instrument connectivity using standard tools such as VSCode and Python, eliminating setup overhead. To demonstrate its capabilities, we conducted experiments involving the measurement of the parametric transfer curve of a simple two-transistor inverting amplifier with a current source load. The AI assistant was evaluated using different prompt scenarios and compared with multiple models, including Claude Sonnet 3.5, Gemini Pro 1.5, and GPT-4o. An expert solution implementing the Gradient-Weighted Adaptive Stochastic Sampling (GWASS) method was used as a baseline. The solutions generated by the AI assistant were compared with the expert solution and a uniform linear sweep baseline with 10,000 points. The graph results show that the LLMs were able to successfully complete the most basic uniform sweep, but LLMs were unable to develop adaptive sweeping algorithms to compete with GWASS. The evaluation underscores LABIIUM's ability to enhance laboratory productivity and support digital transformation in research and industry, and emphasizes the future work required to improve LLM performance in Electronic Measurement Science Tasks.

We present a framework for using transformer networks as universal computers by programming them with specific weights and placing them in a loop. Our input sequence acts as a punchcard, consisting of instructions and memory for data read/writes. We demonstrate that a constant number of encoder layers can emulate basic computing blocks, including embedding edit operations, non-linear functions, function calls, program counters, and conditional branches. Using these building blocks, we emulate a small instruction-set computer. This allows us to map iterative algorithms to programs that can be executed by a looped, 13-layer transformer. We show how this transformer, instructed by its input, can emulate a basic calculator, a basic linear algebra library, and in-context learning algorithms that employ backpropagation. Our work highlights the versatility of the attention mechanism, and demonstrates that even shallow transformers can execute full-fledged, general-purpose programs.

Transformer based large language models have achieved tremendous success. However, the significant memory and computational costs incurred during the inference process make it challenging to deploy large models on resource-constrained devices. In this paper, we investigate compression and efficient inference methods for large language models from an algorithmic perspective. Regarding taxonomy, similar to smaller models, compression and acceleration algorithms for large language models can still be categorized into quantization, pruning, distillation, compact architecture design, dynamic networks. However, Large language models have two prominent characteristics compared to smaller models: (1) Most of compression algorithms require finetuning or even retraining the model after compression. The most notable aspect of large models is the very high cost associated with model finetuning or training. Therefore, many algorithms for large models, such as quantization and pruning, start to explore tuning-free algorithms. (2) Large models emphasize versatility and generalization rather than performance on a single task. Hence, many algorithms, such as knowledge distillation, focus on how to preserving their versatility and generalization after compression. Since these two characteristics were not very pronounced in early large models, we further distinguish large language models into medium models and ``real'' large models. Additionally, we also provide an introduction to some mature frameworks for efficient inference of large models, which can support basic compression or acceleration algorithms, greatly facilitating model deployment for users.

The aligned Large Language Models (LLMs) are powerful language understanding and decision-making tools that are created through extensive alignment with human feedback. However, these large models remain susceptible to jailbreak attacks, where adversaries manipulate prompts to elicit malicious outputs that should not be given by aligned LLMs. Investigating jailbreak prompts can lead us to delve into the limitations of LLMs and further guide us to secure them. Unfortunately, existing jailbreak techniques suffer from either (1) scalability issues, where attacks heavily rely on manual crafting of prompts, or (2) stealthiness problems, as attacks depend on token-based algorithms to generate prompts that are often semantically meaningless, making them susceptible to detection through basic perplexity testing. In light of these challenges, we intend to answer this question: Can we develop an approach that can automatically generate stealthy jailbreak prompts? In this paper, we introduce AutoDAN, a novel jailbreak attack against aligned LLMs. AutoDAN can automatically generate stealthy jailbreak prompts by the carefully designed hierarchical genetic algorithm. Extensive evaluations demonstrate that AutoDAN not only automates the process while preserving semantic meaningfulness, but also demonstrates superior attack strength in cross-model transferability, and cross-sample universality compared with the baseline. Moreover, we also compare AutoDAN with perplexity-based defense methods and show that AutoDAN can bypass them effectively.

A basic algorithmic task in automated video surveillance is to separate background and foreground objects. Camera tampering, noisy videos, low frame rate, etc., pose difficulties in solving the problem. A general approach that classifies the tampered frames, and performs subsequent analysis on the remaining frames after discarding the tampered ones, results in loss of information. Several robust methods based on robust principal component analysis (PCA) have been introduced to solve this problem. To date, considerable effort has been expended to develop robust PCA via Principal Component Pursuit (PCP) methods with reduced computational cost and visually appealing foreground detection. However, the convex optimizations used in these algorithms do not scale well to real-world large datasets due to large matrix inversion steps. Also, an integral component of these foreground detection algorithms is singular value decomposition which is nonrobust. In this paper, we present a new video surveillance background modelling algorithm based on a new robust singular value decomposition technique rSVDdpd which takes care of both these issues. We also demonstrate the superiority of our proposed algorithm on a benchmark dataset and a new real-life video surveillance dataset in the presence of camera tampering. Software codes and additional illustrations are made available at the accompanying website rSVDdpd Homepage (https://subroy13.github.io/rsvddpd-home/)

This presentation focuses on the importance of web crawling and page ranking algorithms in dealing with the massive amount of data present on the World Wide Web. As the web continues to grow exponentially, efficient search and retrieval methods become crucial. Web crawling is a process that converts unstructured data into structured data, enabling effective information retrieval. Additionally, page ranking algorithms play a significant role in assessing the quality and popularity of web pages. The presentation explores the background of these algorithms and evaluates five different crawling algorithms: Shark Search, Priority-Based Queue, Naive Bayes, Breadth-First, and Depth-First. The goal is to identify the most effective algorithm for crawling web pages. By understanding these algorithms, we can enhance our ability to navigate the web and extract valuable information efficiently.

Many fields use search algorithms, which automatically explore a search space to find high-performing solutions: chemists search through the space of molecules to discover new drugs; engineers search for stronger, cheaper, safer designs, scientists search for models that best explain data, etc. The goal of search algorithms has traditionally been to return the single highest-performing solution in a search space. Here we describe a new, fundamentally different type of algorithm that is more useful because it provides a holistic view of how high-performing solutions are distributed throughout a search space. It creates a map of high-performing solutions at each point in a space defined by dimensions of variation that a user gets to choose. This Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) algorithm illuminates search spaces, allowing researchers to understand how interesting attributes of solutions combine to affect performance, either positively or, equally of interest, negatively. For example, a drug company may wish to understand how performance changes as the size of molecules and their cost-to-produce vary. MAP-Elites produces a large diversity of high-performing, yet qualitatively different solutions, which can be more helpful than a single, high-performing solution. Interestingly, because MAP-Elites explores more of the search space, it also tends to find a better overall solution than state-of-the-art search algorithms. We demonstrate the benefits of this new algorithm in three different problem domains ranging from producing modular neural networks to designing simulated and real soft robots. Because MAP- Elites (1) illuminates the relationship between performance and dimensions of interest in solutions, (2) returns a set of high-performing, yet diverse solutions, and (3) improves finding a single, best solution, it will advance science and engineering.

Among the most important properties of algorithms investigated in computer science are soundness, completeness, and complexity. These properties, however, are rarely analyzed for the vast collection of recently proposed methods for planning with large language models. In this work, we alleviate this gap. We analyse these properties of using LLMs for planning and highlight that recent trends abandon both soundness and completeness for the sake of inefficiency. We propose a significantly more efficient approach that can, at the same time, maintain both soundness and completeness. We exemplify on four representative search problems, comparing to the LLM-based solutions from the literature that attempt to solve these problems. We show that by using LLMs to produce the code for the search components we can solve the entire datasets with 100\% accuracy with only a few calls to the LLM. We argue for a responsible use of compute resources; urging research community to investigate sound and complete LLM-based approaches that uphold efficiency.

This paper discusses about a sorting algorithm which uses the concept of buckets where each bucket represents a certain number of digits. A two dimensional data structure is used where one dimension represents buckets i. e; number of digits and each bucket's corresponding dimensions represents the input numbers that belong to that bucket. Each bucket is then individually sorted. Since every preceding bucket elements will always be smaller than the succeeding buckets no comparison between them is required. By doing this we can significantly reduced the time complexity of any sorting algorithm used to sort the given set of inputs.

We discuss an algorithm which produces the meaning of a sentence given meanings of its words, and its resemblance to quantum teleportation. In fact, this protocol was the main source of inspiration for this algorithm which has many applications in the area of Natural Language Processing.

We introduce a new type of programming challenge called programming puzzles, as an objective and comprehensive evaluation of program synthesis, and release an open-source dataset of Python Programming Puzzles (P3). Each puzzle is defined by a short Python program f, and the goal is to find an input which makes f return True. The puzzles are objective in that each one is specified entirely by the source code of its verifier f, so evaluating f is all that is needed to test a candidate solution. They do not require an answer key or input/output examples, nor do they depend on natural language understanding. The dataset is comprehensive in that it spans problems of a range of difficulties and domains, ranging from trivial string manipulation problems, to classic programming puzzles (e.g., Tower of Hanoi), to interview/competitive-programming problems (e.g., dynamic programming), to longstanding open problems in algorithms and mathematics (e.g., factoring). We develop baseline enumerative program synthesis, GPT-3 and Codex solvers that are capable of solving puzzles -- even without access to any reference solutions -- by learning from their own past solutions. Codex performs best, solving up to 18% of 397 test problems with a single try and 80% of the problems with 1,000 tries per problem. In a small user study, we find a positive correlation between puzzle-solving performance and coding experience, and between the puzzle difficulty for humans and AI solvers. Therefore, further improvements on P3 could have a significant impact on many program synthesis areas.

In this report we review memory-based meta-learning as a tool for building sample-efficient strategies that learn from past experience to adapt to any task within a target class. Our goal is to equip the reader with the conceptual foundations of this tool for building new, scalable agents that operate on broad domains. To do so, we present basic algorithmic templates for building near-optimal predictors and reinforcement learners which behave as if they had a probabilistic model that allowed them to efficiently exploit task structure. Furthermore, we recast memory-based meta-learning within a Bayesian framework, showing that the meta-learned strategies are near-optimal because they amortize Bayes-filtered data, where the adaptation is implemented in the memory dynamics as a state-machine of sufficient statistics. Essentially, memory-based meta-learning translates the hard problem of probabilistic sequential inference into a regression problem.

When trying to solve a computational problem, we are often faced with a choice between algorithms that are guaranteed to return the right answer but differ in their runtime distributions (e.g., SAT solvers, sorting algorithms). This paper aims to lay theoretical foundations for such choices by formalizing preferences over runtime distributions. It might seem that we should simply prefer the algorithm that minimizes expected runtime. However, such preferences would be driven by exactly how slow our algorithm is on bad inputs, whereas in practice we are typically willing to cut off occasional, sufficiently long runs before they finish. We propose a principled alternative, taking a utility-theoretic approach to characterize the scoring functions that describe preferences over algorithms. These functions depend on the way our value for solving our problem decreases with time and on the distribution from which captimes are drawn. We describe examples of realistic utility functions and show how to leverage a maximum-entropy approach for modeling underspecified captime distributions. Finally, we show how to efficiently estimate an algorithm's expected utility from runtime samples.

We study the algorithm configuration (AC) problem, in which one seeks to find an optimal parameter configuration of a given target algorithm in an automated way. Recently, there has been significant progress in designing AC approaches that satisfy strong theoretical guarantees. However, a significant gap still remains between the practical performance of these approaches and state-of-the-art heuristic methods. To this end, we introduce AC-Band, a general approach for the AC problem based on multi-armed bandits that provides theoretical guarantees while exhibiting strong practical performance. We show that AC-Band requires significantly less computation time than other AC approaches providing theoretical guarantees while still yielding high-quality configurations.

We revisit the noisy binary search model of Karp and Kleinberg, in which we have n coins with unknown probabilities p_i that we can flip. The coins are sorted by increasing p_i, and we would like to find where the probability crosses (to within varepsilon) of a target value tau. This generalized the fixed-noise model of Burnashev and Zigangirov , in which p_i = 1{2} pm varepsilon, to a setting where coins near the target may be indistinguishable from it. Karp and Kleinberg showed that Theta(1{varepsilon^2} log n) samples are necessary and sufficient for this task. We produce a practical algorithm by solving two theoretical challenges: high-probability behavior and sharp constants. We give an algorithm that succeeds with probability 1-delta from \[ 1{C_{\tau, \varepsilon}} \cdot \left(\lg n + O(\log^{2/3} n \log^{1/3} 1{\delta} + \log 1{\delta})\right) \] samples, where C_{tau, varepsilon} is the optimal such constant achievable. For delta > n^{-o(1)} this is within 1 + o(1) of optimal, and for delta ll 1 it is the first bound within constant factors of optimal.

We study the ranking of individuals, teams, or objects, based on pairwise comparisons between them, using the Bradley-Terry model. Estimates of rankings within this model are commonly made using a simple iterative algorithm first introduced by Zermelo almost a century ago. Here we describe an alternative and similarly simple iteration that provably returns identical results but does so much faster -- over a hundred times faster in some cases. We demonstrate this algorithm with applications to a range of example data sets and derive a number of results regarding its convergence.

We evaluate ChatGPT's ability to solve algorithm problems from the CLRS benchmark suite that is designed for GNNs. The benchmark requires the use of a specified classical algorithm to solve a given problem. We find that ChatGPT outperforms specialist GNN models, using Python to successfully solve these problems. This raises new points in the discussion about learning algorithms with neural networks and how we think about what out of distribution testing looks like with web scale training data.

Algorithmic reasoning refers to the ability to understand the complex patterns behind the problem and decompose them into a sequence of reasoning steps towards the solution. Such nature of algorithmic reasoning makes it a challenge for large language models (LLMs), even though they have demonstrated promising performance in other reasoning tasks. Within this context, some recent studies use programming languages (e.g., Python) to express the necessary logic for solving a given instance/question (e.g., Program-of-Thought) as inspired by their strict and precise syntaxes. However, it is non-trivial to write an executable code that expresses the correct logic on the fly within a single inference call. Also, the code generated specifically for an instance cannot be reused for others, even if they are from the same task and might require identical logic to solve. This paper presents Think-and-Execute, a novel framework that decomposes the reasoning process of language models into two steps. (1) In Think, we discover a task-level logic that is shared across all instances for solving a given task and then express the logic with pseudocode; (2) In Execute, we further tailor the generated pseudocode to each instance and simulate the execution of the code. With extensive experiments on seven algorithmic reasoning tasks, we demonstrate the effectiveness of Think-and-Execute. Our approach better improves LMs' reasoning compared to several strong baselines performing instance-specific reasoning (e.g., CoT and PoT), suggesting the helpfulness of discovering task-level logic. Also, we show that compared to natural language, pseudocode can better guide the reasoning of LMs, even though they are trained to follow natural language instructions.

A major technique in learning-augmented online algorithms is combining multiple algorithms or predictors. Since the performance of each predictor may vary over time, it is desirable to use not the single best predictor as a benchmark, but rather a dynamic combination which follows different predictors at different times. We design algorithms that combine predictions and are competitive against such dynamic combinations for a wide class of online problems, namely, metrical task systems. Against the best (in hindsight) unconstrained combination of ell predictors, we obtain a competitive ratio of O(ell^2), and show that this is best possible. However, for a benchmark with slightly constrained number of switches between different predictors, we can get a (1+epsilon)-competitive algorithm. Moreover, our algorithms can be adapted to access predictors in a bandit-like fashion, querying only one predictor at a time. An unexpected implication of one of our lower bounds is a new structural insight about covering formulations for the k-server problem.

Code generation with large language models has shown significant promise, especially when employing retrieval-augmented generation (RAG) with few-shot examples. However, selecting effective examples that enhance generation quality remains a challenging task, particularly when the target programming language (PL) is underrepresented. In this study, we present two key findings: (1) retrieving examples whose presented algorithmic plans can be referenced for generating the desired behavior significantly improves generation accuracy, and (2) converting code into pseudocode effectively captures such algorithmic plans, enhancing retrieval quality even when the source and the target PLs are different. Based on these findings, we propose Plan-as-query Example Retrieval for few-shot prompting in Code generation (PERC), a novel framework that utilizes algorithmic plans to identify and retrieve effective examples. We validate the effectiveness of PERC through extensive experiments on the CodeContests, HumanEval and MultiPL-E benchmarks: PERC consistently outperforms the state-of-the-art RAG methods in code generation, both when the source and target programming languages match or differ, highlighting its adaptability and robustness in diverse coding environments.

We propose a general two-stage algorithm that enjoys a provable scaling law for the test-time compute of large language models (LLMs). Given an input problem, the proposed algorithm first generates N candidate solutions, and then chooses the best one via a multiple-round knockout tournament where each pair of candidates are compared for K times and only the winners move on to the next round. In a minimalistic implementation, both stages can be executed with a black-box LLM alone and nothing else (e.g., no external verifier or reward model), and a total of N times (K + 1) highly parallelizable LLM calls are needed for solving an input problem. Assuming that a generated candidate solution is correct with probability p_{gen} > 0 and a comparison between a pair of correct and incorrect solutions identifies the right winner with probability p_{comp} > 0.5 (i.e., better than a random guess), we prove theoretically that the failure probability of the proposed algorithm decays to zero exponentially with respect to N and K: $P(final output is incorrect) le (1 - p_{gen})^N + lceil log_2 N rceil e^{-2 K (p_{comp} - 0.5)^2}.$ Our empirical results with the challenging MMLU-Pro benchmark validate the technical assumptions, as well as the efficacy of the proposed algorithm and the gains from scaling up its test-time compute.

Byte-Pair Encoding (BPE) is a popular algorithm used for tokenizing data in NLP, despite being devised initially as a compression method. BPE appears to be a greedy algorithm at face value, but the underlying optimization problem that BPE seeks to solve has not yet been laid down. We formalize BPE as a combinatorial optimization problem. Via submodular functions, we prove that the iterative greedy version is a 1{{sigma(mu^star)}}(1-e^{-{sigma(mu^star)}})-approximation of an optimal merge sequence, where {sigma(mu^star)} is the total backward curvature with respect to the optimal merge sequence mu^star. Empirically the lower bound of the approximation is approx 0.37. We provide a faster implementation of BPE which improves the runtime complexity from Oleft(N Mright) to Oleft(N log Mright), where N is the sequence length and M is the merge count. Finally, we optimize the brute-force algorithm for optimal BPE using memoization.

Vectors are universal mathematical objects that can represent text, images, speech, or a mix of these data modalities. That happens regardless of whether data is represented by hand-crafted features or learnt embeddings. Collect a large enough quantity of such vectors and the question of retrieval becomes urgently relevant: Finding vectors that are more similar to a query vector. This monograph is concerned with the question above and covers fundamental concepts along with advanced data structures and algorithms for vector retrieval. In doing so, it recaps this fascinating topic and lowers barriers of entry into this rich area of research.

In this paper, we revisit one of the simplest problems in data structures: the task of inserting elements into an open-addressed hash table so that elements can later be retrieved with as few probes as possible. We show that, even without reordering elements over time, it is possible to construct a hash table that achieves far better expected search complexities (both amortized and worst-case) than were previously thought possible. Along the way, we disprove the central conjecture left by Yao in his seminal paper ``Uniform Hashing is Optimal''. All of our results come with matching lower bounds.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for e, e^2, tan(1), and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.

The performance of an algorithm often critically depends on its parameter configuration. While a variety of automated algorithm configuration methods have been proposed to relieve users from the tedious and error-prone task of manually tuning parameters, there is still a lot of untapped potential as the learned configuration is static, i.e., parameter settings remain fixed throughout the run. However, it has been shown that some algorithm parameters are best adjusted dynamically during execution, e.g., to adapt to the current part of the optimization landscape. Thus far, this is most commonly achieved through hand-crafted heuristics. A promising recent alternative is to automatically learn such dynamic parameter adaptation policies from data. In this article, we give the first comprehensive account of this new field of automated dynamic algorithm configuration (DAC), present a series of recent advances, and provide a solid foundation for future research in this field. Specifically, we (i) situate DAC in the broader historical context of AI research; (ii) formalize DAC as a computational problem; (iii) identify the methods used in prior-art to tackle this problem; (iv) conduct empirical case studies for using DAC in evolutionary optimization, AI planning, and machine learning.

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern P of length m and a text T of length n, both over a polynomial-size alphabet, compute the Hamming distance between P and T[i, ., . , i+m-1] for every shift i, under the standard Word-RAM model with Theta(log n)-bit words. - We provide an O(nm) time Las Vegas randomized algorithm for this problem, beating the decades-old O(n m log m) running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher O(nm(log mloglog m)^{1/4}) running time. Our randomized algorithm extends to the k-bounded setting, with running time Obig(n+nk{m}big), removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uzna\'{n}ski, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020]. - For the (1+epsilon)-approximate version of Text-to-Pattern Hamming Distances, we give an O(epsilon^{-0.93}n) time Monte Carlo randomized algorithm, beating the previous O(epsilon^{-1}n) running time [Kopelowitz and Porat, FOCS 2015; Kopelowitz and Porat, SOSA 2018]. Our approximation algorithm exploits a connection with 3SUM, and uses a combination of Fredman's trick, equality matrix product, and random sampling; in particular, we obtain new results on approximate counting versions of 3SUM and Exact Triangle, which may be of independent interest. Our exact algorithms use a novel combination of hashing, bit-packed FFT, and recursion; in particular, we obtain a faster algorithm for computing the sumset of two integer sets, in the regime when the universe size is close to quadratic in the number of elements. We also prove a fine-grained equivalence between the exact Text-to-Pattern Hamming Distances problem and a range-restricted, counting version of 3SUM.

This research proposes a new recommender system algorithm for online grocery shopping. The algorithm is based on the perspective that, since the grocery items are usually bought in bulk, a grocery recommender system should be capable of recommending the items in bulk. The algorithm figures out the possible dishes a user may cook based on the items added to the basket and recommends the ingredients accordingly. Our algorithm does not depend on the user ratings. Customers usually do not have the patience to rate the groceries they purchase. Therefore, algorithms that are not dependent on user ratings need to be designed. Instead of using a brute force search, this algorithm limits the search space to a set of only a few probably food categories. Each food category consists of several food subcategories. For example, "fried rice" and "biryani" are food subcategories that belong to the food category "rice". For each food category, items are ranked according to how well they can differentiate a food subcategory. To each food subcategory in the activated search space, this algorithm attaches a score. The score is calculated based on the rank of the items added to the basket. Once the score exceeds a threshold value, its corresponding subcategory gets activated. The algorithm then uses a basket-to-recipe similarity measure to identify the best recipe matches within the activated subcategories only. This reduces the search space to a great extent. We may argue that this algorithm is similar to the content-based recommender system in some sense, but it does not suffer from the limitations like limited content, over-specialization, or the new user problem.

Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction to some of the ideas in quantum computing. The paper begins by motivating the central ideas of quantum mechanics and quantum computation with simple toy models. From there we move on to a formal presentation of the small fraction of (finite dimensional) quantum mechanics that we will need for basic quantum computation. Central notions of quantum architecture (qubits and quantum gates) are described. The paper ends with a presentation of one of the simplest quantum algorithms: Deutsch's algorithm. Our presentation demands neither advanced mathematics nor advanced physics.

We study the problem of (learning) algorithm comparison, where the goal is to find differences between models trained with two different learning algorithms. We begin by formalizing this goal as one of finding distinguishing feature transformations, i.e., input transformations that change the predictions of models trained with one learning algorithm but not the other. We then present ModelDiff, a method that leverages the datamodels framework (Ilyas et al., 2022) to compare learning algorithms based on how they use their training data. We demonstrate ModelDiff through three case studies, comparing models trained with/without data augmentation, with/without pre-training, and with different SGD hyperparameters. Our code is available at https://github.com/MadryLab/modeldiff .

We show that feedforward neural networks with ReLU activation generalize on low complexity data, suitably defined. Given i.i.d. data generated from a simple programming language, the minimum description length (MDL) feedforward neural network which interpolates the data generalizes with high probability. We define this simple programming language, along with a notion of description length of such networks. We provide several examples on basic computational tasks, such as checking primality of a natural number, and more. For primality testing, our theorem shows the following. Suppose that we draw an i.i.d. sample of Theta(N^{delta}ln N) numbers uniformly at random from 1 to N, where deltain (0,1). For each number x_i, let y_i = 1 if x_i is a prime and 0 if it is not. Then with high probability, the MDL network fitted to this data accurately answers whether a newly drawn number between 1 and N is a prime or not, with test error leq O(N^{-delta}). Note that the network is not designed to detect primes; minimum description learning discovers a network which does so.

Algorithm configuration (AC) is concerned with the automated search of the most suitable parameter configuration of a parametrized algorithm. There is currently a wide variety of AC problem variants and methods proposed in the literature. Existing reviews do not take into account all derivatives of the AC problem, nor do they offer a complete classification scheme. To this end, we introduce taxonomies to describe the AC problem and features of configuration methods, respectively. We review existing AC literature within the lens of our taxonomies, outline relevant design choices of configuration approaches, contrast methods and problem variants against each other, and describe the state of AC in industry. Finally, our review provides researchers and practitioners with a look at future research directions in the field of AC.

ML-augmented algorithms utilize predictions to achieve performance beyond their worst-case bounds. Producing these predictions might be a costly operation -- this motivated Im et al. '22 to introduce the study of algorithms which use predictions parsimoniously. We design parsimonious algorithms for caching and MTS with action predictions, proposed by Antoniadis et al. '20, focusing on the parameters of consistency (performance with perfect predictions) and smoothness (dependence of their performance on the prediction error). Our algorithm for caching is 1-consistent, robust, and its smoothness deteriorates with the decreasing number of available predictions. We propose an algorithm for general MTS whose consistency and smoothness both scale linearly with the decreasing number of predictions. Without the restriction on the number of available predictions, both algorithms match the earlier guarantees achieved by Antoniadis et al. '20.

We design a suite of minimal algorithmic tasks that are a loose abstraction of open-ended real-world tasks. This allows us to cleanly and controllably quantify the creative limits of the present-day language model. Much like real-world tasks that require a creative, far-sighted leap of thought, our tasks require an implicit, open-ended stochastic planning step that either (a) discovers new connections in an abstract knowledge graph (like in wordplay, drawing analogies, or research) or (b) constructs new patterns (like in designing math problems or new proteins). In these tasks, we empirically and conceptually argue how next-token learning is myopic and memorizes excessively; comparatively, multi-token approaches, namely teacherless training and diffusion models, excel in producing diverse and original output. Secondly, in our tasks, we find that to elicit randomness from the Transformer without hurting coherence, it is better to inject noise right at the input layer (via a method we dub hash-conditioning) rather than defer to temperature sampling from the output layer. Thus, our work offers a principled, minimal test-bed for analyzing open-ended creative skills, and offers new arguments for going beyond next-token learning and softmax-based sampling. We make part of the code available under https://github.com/chenwu98/algorithmic-creativity

Choosing automatically the right algorithm using problem descriptors is a classical component of combinatorial optimization. It is also a good tool for making evolutionary algorithms fast, robust and versatile. We present Shiwa, an algorithm good at both discrete and continuous, noisy and noise-free, sequential and parallel, black-box optimization. Our algorithm is experimentally compared to competitors on YABBOB, a BBOB comparable testbed, and on some variants of it, and then validated on several real world testbeds.

We propose prioritized unit propagation with periodic resetting, which is a simple but surprisingly effective algorithm for solving random SAT instances that are meant to be hard. In particular, an evaluation on the Random Track of the 2017 and 2018 SAT competitions shows that a basic prototype of this simple idea already ranks at second place in both years. We share this observation in the hope that it helps the SAT community better understand the hardness of random instances used in competitions and inspire other interesting ideas on SAT solving.

This paper presents a detailed analysis of the scalability and parallelization of local search algorithms for the Satisfiability problem. We propose a framework to estimate the parallel performance of a given algorithm by analyzing the runtime behavior of its sequential version. Indeed, by approximating the runtime distribution of the sequential process with statistical methods, the runtime behavior of the parallel process can be predicted by a model based on order statistics. We apply this approach to study the parallel performance of two SAT local search solvers, namely Sparrow and CCASAT, and compare the predicted performances to the results of an actual experimentation on parallel hardware up to 384 cores. We show that the model is accurate and predicts performance close to the empirical data. Moreover, as we study different types of instances (random and crafted), we observe that the local search solvers exhibit different behaviors and that their runtime distributions can be approximated by two types of distributions: exponential (shifted and non-shifted) and lognormal.

We introduce fast algorithms for correlation clustering with respect to the Min Max objective that provide constant factor approximations on complete graphs. Our algorithms are the first purely combinatorial approximation algorithms for this problem. We construct a novel semi-metric on the set of vertices, which we call the correlation metric, that indicates to our clustering algorithms whether pairs of nodes should be in the same cluster. The paper demonstrates empirically that, compared to prior work, our algorithms sacrifice little in the objective quality to obtain significantly better run-time. Moreover, our algorithms scale to larger networks that are effectively intractable for known algorithms.

Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify these using category theory in order to ease correctness proofs and guide the design of new algorithms. In this paper, we extend CALF to cover learning of algebraic structures that may not have a coalgebraic presentation. Furthermore, we provide a detailed algorithmic account of an abstract version of the popular L* algorithm, which was missing from CALF. We instantiate the abstract theory to a large class of Set functors, by which we recover for the first time practical tree automata learning algorithms from an abstract framework and at the same time obtain new algorithms to learn algebras of quotiented polynomial functors.

The field of Automatic Machine Learning (AutoML) has recently attained impressive results, including the discovery of state-of-the-art machine learning solutions, such as neural image classifiers. This is often done by applying an evolutionary search method, which samples multiple candidate solutions from a large space and evaluates the quality of each candidate through a long training process. As a result, the search tends to be slow. In this paper, we show that large efficiency gains can be obtained by employing a fast unified functional hash, especially through the functional equivalence caching technique, which we also present. The central idea is to detect by hashing when the search method produces equivalent candidates, which occurs very frequently, and this way avoid their costly re-evaluation. Our hash is "functional" in that it identifies equivalent candidates even if they were represented or coded differently, and it is "unified" in that the same algorithm can hash arbitrary representations; e.g. compute graphs, imperative code, or lambda functions. As evidence, we show dramatic improvements on multiple AutoML domains, including neural architecture search and algorithm discovery. Finally, we consider the effect of hash collisions, evaluation noise, and search distribution through empirical analysis. Altogether, we hope this paper may serve as a guide to hashing techniques in AutoML.

In online algorithm selection (OAS), instances of an algorithmic problem class are presented to an agent one after another, and the agent has to quickly select a presumably best algorithm from a fixed set of candidate algorithms. For decision problems such as satisfiability (SAT), quality typically refers to the algorithm's runtime. As the latter is known to exhibit a heavy-tail distribution, an algorithm is normally stopped when exceeding a predefined upper time limit. As a consequence, machine learning methods used to optimize an algorithm selection strategy in a data-driven manner need to deal with right-censored samples, a problem that has received little attention in the literature so far. In this work, we revisit multi-armed bandit algorithms for OAS and discuss their capability of dealing with the problem. Moreover, we adapt them towards runtime-oriented losses, allowing for partially censored data while keeping a space- and time-complexity independent of the time horizon. In an extensive experimental evaluation on an adapted version of the ASlib benchmark, we demonstrate that theoretically well-founded methods based on Thompson sampling perform specifically strong and improve in comparison to existing methods.

The algorithm of Shor for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number theory. In this contribution, we present the relevant results and proofs from the theory of continued fractions in detail (even in more detail than in text books) filling the gap to allow a complete comprehension of the algorithm of Shor. Similarly, we provide a detailed computation of the estimation of the probability that convergents will provide the period required for determining a prime factor.

Program synthesis is challenging largely because of the difficulty of search in a large space of programs. Human programmers routinely tackle the task of writing complex programs by writing sub-programs and then analyzing their intermediate results to compose them in appropriate ways. Motivated by this intuition, we present a new synthesis approach that leverages learning to guide a bottom-up search over programs. In particular, we train a model to prioritize compositions of intermediate values during search conditioned on a given set of input-output examples. This is a powerful combination because of several emergent properties. First, in bottom-up search, intermediate programs can be executed, providing semantic information to the neural network. Second, given the concrete values from those executions, we can exploit rich features based on recent work on property signatures. Finally, bottom-up search allows the system substantial flexibility in what order to generate the solution, allowing the synthesizer to build up a program from multiple smaller sub-programs. Overall, our empirical evaluation finds that the combination of learning and bottom-up search is remarkably effective, even with simple supervised learning approaches. We demonstrate the effectiveness of our technique on two datasets, one from the SyGuS competition and one of our own creation.

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of zeta(3). Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

This paper explores the limits of the current generation of large language models for program synthesis in general purpose programming languages. We evaluate a collection of such models (with between 244M and 137B parameters) on two new benchmarks, MBPP and MathQA-Python, in both the few-shot and fine-tuning regimes. Our benchmarks are designed to measure the ability of these models to synthesize short Python programs from natural language descriptions. The Mostly Basic Programming Problems (MBPP) dataset contains 974 programming tasks, designed to be solvable by entry-level programmers. The MathQA-Python dataset, a Python version of the MathQA benchmark, contains 23914 problems that evaluate the ability of the models to synthesize code from more complex text. On both datasets, we find that synthesis performance scales log-linearly with model size. Our largest models, even without finetuning on a code dataset, can synthesize solutions to 59.6 percent of the problems from MBPP using few-shot learning with a well-designed prompt. Fine-tuning on a held-out portion of the dataset improves performance by about 10 percentage points across most model sizes. On the MathQA-Python dataset, the largest fine-tuned model achieves 83.8 percent accuracy. Going further, we study the model's ability to engage in dialog about code, incorporating human feedback to improve its solutions. We find that natural language feedback from a human halves the error rate compared to the model's initial prediction. Additionally, we conduct an error analysis to shed light on where these models fall short and what types of programs are most difficult to generate. Finally, we explore the semantic grounding of these models by fine-tuning them to predict the results of program execution. We find that even our best models are generally unable to predict the output of a program given a specific input.

We present two different methods for estimating the cost of solving SAT problems. The methods focus on the online behaviour of the backtracking solver, as well as the structure of the problem. Modern SAT solvers present several challenges to estimate search cost including coping with nonchronological backtracking, learning and restarts. Our first method adapt an existing algorithm for estimating the size of a search tree to deal with these challenges. We then suggest a second method that uses a linear model trained on data gathered online at the start of search. We compare the effectiveness of these two methods using random and structured problems. We also demonstrate that predictions made in early restarts can be used to improve later predictions. We conclude by showing that the cost of solving a set of problems can be reduced by selecting a solver from a portfolio based on such cost estimations.

We introduce BM25S, an efficient Python-based implementation of BM25 that only depends on Numpy and Scipy. BM25S achieves up to a 500x speedup compared to the most popular Python-based framework by eagerly computing BM25 scores during indexing and storing them into sparse matrices. It also achieves considerable speedups compared to highly optimized Java-based implementations, which are used by popular commercial products. Finally, BM25S reproduces the exact implementation of five BM25 variants based on Kamphuis et al. (2020) by extending eager scoring to non-sparse variants using a novel score shifting method. The code can be found at https://github.com/xhluca/bm25s

LLM test-time compute (or LLM inference) via search has emerged as a promising research area with rapid developments. However, current frameworks often adopt distinct perspectives on three key aspects (task definition, LLM profiling, and search procedures), making direct comparisons challenging. Moreover, the search algorithms employed often diverge from standard implementations, and their specific characteristics are not thoroughly specified. In this survey, we provide a comprehensive technical review that unifies task definitions and provides modular definitions of LLM profiling and search procedures. The definitions enable precise comparisons of various LLM inference frameworks while highlighting their departures from conventional search algorithms. We also discuss the applicability, performance, and efficiency of these methods. For further details and ongoing updates, please refer to our GitHub repository: https://github.com/xinzhel/LLM-Agent-Survey/blob/main/search.md

A supervised learning algorithm searches over a set of functions A to B parametrised by a space P to find the best approximation to some ideal function fcolon A to B. It does this by taking examples (a,f(a)) in Atimes B, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.

This work proposes a procedure for designing algorithms for specific adaptive data collection tasks like active learning and pure-exploration multi-armed bandits. Unlike the design of traditional adaptive algorithms that rely on concentration of measure and careful analysis to justify the correctness and sample complexity of the procedure, our adaptive algorithm is learned via adversarial training over equivalence classes of problems derived from information theoretic lower bounds. In particular, a single adaptive learning algorithm is learned that competes with the best adaptive algorithm learned for each equivalence class. Our procedure takes as input just the available queries, set of hypotheses, loss function, and total query budget. This is in contrast to existing meta-learning work that learns an adaptive algorithm relative to an explicit, user-defined subset or prior distribution over problems which can be challenging to define and be mismatched to the instance encountered at test time. This work is particularly focused on the regime when the total query budget is very small, such as a few dozen, which is much smaller than those budgets typically considered by theoretically derived algorithms. We perform synthetic experiments to justify the stability and effectiveness of the training procedure, and then evaluate the method on tasks derived from real data including a noisy 20 Questions game and a joke recommendation task.

In the classical transformer attention scheme, we are given three n times d size matrices Q, K, V (the query, key, and value tokens), and the goal is to compute a new n times d size matrix D^{-1} exp(QK^top) V where D = diag( exp(QK^top) {bf 1}_n ). In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in n. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations: bullet On the positive side, if all entries of the input matrices are bounded above by o(sqrt[3]{log n}) then we show how to approximate the ``tensor-type'' attention matrix in n^{1+o(1)} time. bullet On the negative side, we show that if the entries of the input matrices may be as large as Omega(sqrt[3]{log n}), then there is no algorithm that runs faster than n^{3-o(1)} (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory). We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.

Recent automatic curriculum learning algorithms, and in particular Teacher-Student algorithms, rely on the notion of learning progress, making the assumption that the good next tasks are the ones on which the learner is making the fastest progress or digress. In this work, we first propose a simpler and improved version of these algorithms. We then argue that the notion of learning progress itself has several shortcomings that lead to a low sample efficiency for the learner. We finally propose a new algorithm, based on the notion of mastering rate, that significantly outperforms learning progress-based algorithms.

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erdos, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

We present a method to formulate algorithm discovery as program search, and apply it to discover optimization algorithms for deep neural network training. We leverage efficient search techniques to explore an infinite and sparse program space. To bridge the large generalization gap between proxy and target tasks, we also introduce program selection and simplification strategies. Our method discovers a simple and effective optimization algorithm, Lion (Evo\textbf{Lved Sign Momentum}). It is more memory-efficient than Adam as it only keeps track of the momentum. Different from adaptive optimizers, its update has the same magnitude for each parameter calculated through the sign operation. We compare Lion with widely used optimizers, such as Adam and Adafactor, for training a variety of models on different tasks. On image classification, Lion boosts the accuracy of ViT by up to 2% on ImageNet and saves up to 5x the pre-training compute on JFT. On vision-language contrastive learning, we achieve 88.3% zero-shot and 91.1% fine-tuning accuracy on ImageNet, surpassing the previous best results by 2% and 0.1%, respectively. On diffusion models, Lion outperforms Adam by achieving a better FID score and reducing the training compute by up to 2.3x. For autoregressive, masked language modeling, and fine-tuning, Lion exhibits a similar or better performance compared to Adam. Our analysis of Lion reveals that its performance gain grows with the training batch size. It also requires a smaller learning rate than Adam due to the larger norm of the update produced by the sign function. Additionally, we examine the limitations of Lion and identify scenarios where its improvements are small or not statistically significant. The implementation of Lion is publicly available.

The class of tree-adjoining languages can be characterized by various two-level formalisms, consisting of a context-free grammar (CFG) or pushdown automaton (PDA) controlling another CFG or PDA. These four formalisms are equivalent to tree-adjoining grammars (TAG), linear indexed grammars (LIG), pushdown-adjoining automata (PAA), and embedded pushdown automata (EPDA). We define semiring-weighted versions of the above two-level formalisms, and we design new algorithms for computing their stringsums (the weight of all derivations of a string) and allsums (the weight of all derivations). From these, we also immediately obtain stringsum and allsum algorithms for TAG, LIG, PAA, and EPDA. For LIG, our algorithm is more time-efficient by a factor of O(n|N|) (where n is the string length and |N| is the size of the nonterminal set) and more space-efficient by a factor of O(|Gamma|) (where |Gamma| is the size of the stack alphabet) than the algorithm of Vijay-Shanker and Weir (1989). For EPDA, our algorithm is both more space-efficient and time-efficient than the algorithm of Alonso et al. (2001) by factors of O(|Gamma|^2) and O(|Gamma|^3), respectively. Finally, we give the first PAA stringsum and allsum algorithms.

This paper presents our winning submission to the AI Mathematical Olympiad - Progress Prize 2 (AIMO-2) competition. Our recipe for building state-of-the-art mathematical reasoning models relies on three key pillars. First, we create a large-scale dataset comprising 540K unique high-quality math problems, including olympiad-level problems, and their 3.2M long-reasoning solutions. Second, we develop a novel method to integrate code execution with long reasoning models through iterative training, generation, and quality filtering, resulting in 1.7M high-quality Tool-Integrated Reasoning solutions. Third, we create a pipeline to train models to select the most promising solution from many candidates. We show that such generative solution selection (GenSelect) can significantly improve upon majority voting baseline. Combining these ideas, we train a series of models that achieve state-of-the-art results on mathematical reasoning benchmarks. To facilitate further research, we release our code, models, and the complete OpenMathReasoning dataset under a commercially permissive license.

The problem of designing NLP solvers for math word problems (MWP) has seen sustained research activity and steady gains in the test accuracy. Since existing solvers achieve high performance on the benchmark datasets for elementary level MWPs containing one-unknown arithmetic word problems, such problems are often considered "solved" with the bulk of research attention moving to more complex MWPs. In this paper, we restrict our attention to English MWPs taught in grades four and lower. We provide strong evidence that the existing MWP solvers rely on shallow heuristics to achieve high performance on the benchmark datasets. To this end, we show that MWP solvers that do not have access to the question asked in the MWP can still solve a large fraction of MWPs. Similarly, models that treat MWPs as bag-of-words can also achieve surprisingly high accuracy. Further, we introduce a challenge dataset, SVAMP, created by applying carefully chosen variations over examples sampled from existing datasets. The best accuracy achieved by state-of-the-art models is substantially lower on SVAMP, thus showing that much remains to be done even for the simplest of the MWPs.

Code search is an important task that has seen many developments in recent years. However, previous attempts have mostly considered the problem of searching for code by a text query. We argue that using a code snippet (and possibly an associated traceback) as a query and looking for answers with bugfixing instructions and code samples is a natural use case that is not covered by existing approaches. Moreover, existing datasets use comments extracted from code rather than full-text descriptions as text, making them unsuitable for this use case. We present a new SearchBySnippet dataset implementing the search-by-code use case based on StackOverflow data; it turns out that in this setting, existing architectures fall short of the simplest BM25 baseline even after fine-tuning. We present a new single encoder model SnippeR that outperforms several strong baselines on the SearchBySnippet dataset with a result of 0.451 Recall@10; we propose the SearchBySnippet dataset and SnippeR as a new important benchmark for code search evaluation.

We consider the task of mapping pseudocode to long programs that are functionally correct. Given test cases as a mechanism to validate programs, we search over the space of possible translations of the pseudocode to find a program that passes the validation. However, without proper credit assignment to localize the sources of program failures, it is difficult to guide search toward more promising programs. We propose to perform credit assignment based on signals from compilation errors, which constitute 88.7% of program failures. Concretely, we treat the translation of each pseudocode line as a discrete portion of the program, and whenever a synthesized program fails to compile, an error localization method tries to identify the portion of the program responsible for the failure. We then focus search over alternative translations of the pseudocode for those portions. For evaluation, we collected the SPoC dataset (Search-based Pseudocode to Code) containing 18,356 programs with human-authored pseudocode and test cases. Under a budget of 100 program compilations, performing search improves the synthesis success rate over using the top-one translation of the pseudocode from 25.6% to 44.7%.

Paging is a prototypical problem in the area of online algorithms. It has also played a central role in the development of learning-augmented algorithms -- a recent line of research that aims to ameliorate the shortcomings of classical worst-case analysis by giving algorithms access to predictions. Such predictions can typically be generated using a machine learning approach, but they are inherently imperfect. Previous work on learning-augmented paging has investigated predictions on (i) when the current page will be requested again (reoccurrence predictions), (ii) the current state of the cache in an optimal algorithm (state predictions), (iii) all requests until the current page gets requested again, and (iv) the relative order in which pages are requested. We study learning-augmented paging from the new perspective of requiring the least possible amount of predicted information. More specifically, the predictions obtained alongside each page request are limited to one bit only. We consider two natural such setups: (i) discard predictions, in which the predicted bit denotes whether or not it is ``safe'' to evict this page, and (ii) phase predictions, where the bit denotes whether the current page will be requested in the next phase (for an appropriate partitioning of the input into phases). We develop algorithms for each of the two setups that satisfy all three desirable properties of learning-augmented algorithms -- that is, they are consistent, robust and smooth -- despite being limited to a one-bit prediction per request. We also present lower bounds establishing that our algorithms are essentially best possible.

We present an accessible first course on diffusion models and flow matching for machine learning, aimed at a technical audience with no diffusion experience. We try to simplify the mathematical details as much as possible (sometimes heuristically), while retaining enough precision to derive correct algorithms.

Programming is a powerful and ubiquitous problem-solving tool. Developing systems that can assist programmers or even generate programs independently could make programming more productive and accessible, yet so far incorporating innovations in AI has proven challenging. Recent large-scale language models have demonstrated an impressive ability to generate code, and are now able to complete simple programming tasks. However, these models still perform poorly when evaluated on more complex, unseen problems that require problem-solving skills beyond simply translating instructions into code. For example, competitive programming problems which require an understanding of algorithms and complex natural language remain extremely challenging. To address this gap, we introduce AlphaCode, a system for code generation that can create novel solutions to these problems that require deeper reasoning. In simulated evaluations on recent programming competitions on the Codeforces platform, AlphaCode achieved on average a ranking of top 54.3% in competitions with more than 5,000 participants. We found that three key components were critical to achieve good and reliable performance: (1) an extensive and clean competitive programming dataset for training and evaluation, (2) large and efficient-to-sample transformer-based architectures, and (3) large-scale model sampling to explore the search space, followed by filtering based on program behavior to a small set of submissions.

This textbook covers principles behind main modern deep reinforcement learning algorithms that achieved breakthrough results in many domains from game AI to robotics. All required theory is explained with proofs using unified notation and emphasize on the differences between different types of algorithms and the reasons why they are constructed the way they are.

We introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets.

Discovering efficient algorithms for solving complex problems has been an outstanding challenge in mathematics and computer science, requiring substantial human expertise over the years. Recent advancements in evolutionary search with large language models (LLMs) have shown promise in accelerating the discovery of algorithms across various domains, particularly in mathematics and optimization. However, existing approaches treat the LLM as a static generator, missing the opportunity to update the model with the signal obtained from evolutionary exploration. In this work, we propose to augment LLM-based evolutionary search by continuously refining the search operator - the LLM - through reinforcement learning (RL) fine-tuning. Our method leverages evolutionary search as an exploration strategy to discover improved algorithms, while RL optimizes the LLM policy based on these discoveries. Our experiments on three combinatorial optimization tasks - bin packing, traveling salesman, and the flatpack problem - show that combining RL and evolutionary search improves discovery efficiency of improved algorithms, showcasing the potential of RL-enhanced evolutionary strategies to assist computer scientists and mathematicians for more efficient algorithm design.

Large Language Models (LLMs) have exhibited exceptional performance across a broad range of tasks and domains. However, they still encounter difficulties in solving mathematical problems due to the rigorous and logical nature of mathematics. Previous studies have employed techniques such as supervised fine-tuning (SFT), prompt engineering, and search-based methods to improve the mathematical problem-solving abilities of LLMs. Despite these efforts, their performance remains suboptimal and demands substantial computational resources. To address this issue, we propose a novel approach, BEATS, to enhance mathematical problem-solving abilities. Our method leverages newly designed prompts that guide the model to iteratively rewrite, advance by one step, and generate answers based on previous steps. Additionally, we introduce a new back-verification technique that uses LLMs to validate the correctness of the generated answers. Furthermore, we employ a pruning tree search to optimize search time while achieving strong performance. Notably, our method improves Qwen2-7b-Instruct's score from 36.94 to 61.52, outperforming GPT4's 42.5 on the MATH benchmark.

This paper explores the relationship between matrix factorizations and linear matrix equations. It shows that every matrix factorization defines two hidden projectors, one for the column space and one for the row space of a matrix, and how to calculate them. The projectors can be applied to solve linear matrix equations, generate low-rank approximations, or design randomized matrix algorithms. But also, as demonstrated, they can be applied in cryptography to encrypt and decrypt messages. The paper discusses some of the security implications of this application and leaves some questions open for further investigation. The basic concepts are illustrated with source code listings. Finally, this work shares some personal reflections on the meaning and importance of understanding in the time of the artificial intelligence revolution.

We introduce a large-scale dataset of math word problems and an interpretable neural math problem solver that learns to map problems to operation programs. Due to annotation challenges, current datasets in this domain have been either relatively small in scale or did not offer precise operational annotations over diverse problem types. We introduce a new representation language to model precise operation programs corresponding to each math problem that aim to improve both the performance and the interpretability of the learned models. Using this representation language, our new dataset, MathQA, significantly enhances the AQuA dataset with fully-specified operational programs. We additionally introduce a neural sequence-to-program model enhanced with automatic problem categorization. Our experiments show improvements over competitive baselines in our MathQA as well as the AQuA dataset. The results are still significantly lower than human performance indicating that the dataset poses new challenges for future research. Our dataset is available at: https://math-qa.github.io/math-QA/

Quantum algorithms based on variational approaches are one of the most promising methods to construct quantum solutions and have found a myriad of applications in the last few years. Despite the adaptability and simplicity, their scalability and the selection of suitable ans\"atzs remain key challenges. In this work, we report an algorithmic framework based on nested Monte-Carlo Tree Search (MCTS) coupled with the combinatorial multi-armed bandit (CMAB) model for the automated design of quantum circuits. Through numerical experiments, we demonstrated our algorithm applied to various kinds of problems, including the ground energy problem in quantum chemistry, quantum optimisation on a graph, solving systems of linear equations, and finding encoding circuit for quantum error detection codes. Compared to the existing approaches, the results indicate that our circuit design algorithm can explore larger search spaces and optimise quantum circuits for larger systems, showing both versatility and scalability.

Stochastic optimization is one of the central problems in Machine Learning and Theoretical Computer Science. In the standard model, the algorithm is given a fixed distribution known in advance. In practice though, one may acquire at a cost extra information to make better decisions. In this paper, we study how to buy information for stochastic optimization and formulate this question as an online learning problem. Assuming the learner has an oracle for the original optimization problem, we design a 2-competitive deterministic algorithm and a e/(e-1)-competitive randomized algorithm for buying information. We show that this ratio is tight as the problem is equivalent to a robust generalization of the ski-rental problem, which we call super-martingale stopping. We also consider an adaptive setting where the learner can choose to buy information after taking some actions for the underlying optimization problem. We focus on the classic optimization problem, Min-Sum Set Cover, where the goal is to quickly find an action that covers a given request drawn from a known distribution. We provide an 8-competitive algorithm running in polynomial time that chooses actions and decides when to buy information about the underlying request.

k-Clustering in R^d (e.g., k-median and k-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality n, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for k-clustering in R^d with O(nkd^{3/2}) query complexity. Our coreset reduces the input size from n to poly(kepsilon^{-1}d), so that existing alpha-approximation algorithms for clustering can run on top of it and yield (1 + epsilon)alpha-approximation. This eventually yields a quadratic speedup for various k-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make Omega(nk) queries in order to achieve even O(1)-approximation for k-clustering.

We present AI-SARAH, a practical variant of SARAH. As a variant of SARAH, this algorithm employs the stochastic recursive gradient yet adjusts step-size based on local geometry. AI-SARAH implicitly computes step-size and efficiently estimates local Lipschitz smoothness of stochastic functions. It is fully adaptive, tune-free, straightforward to implement, and computationally efficient. We provide technical insight and intuitive illustrations on its design and convergence. We conduct extensive empirical analysis and demonstrate its strong performance compared with its classical counterparts and other state-of-the-art first-order methods in solving convex machine learning problems.

We consider non-clairvoyant scheduling with online precedence constraints, where an algorithm is oblivious to any job dependencies and learns about a job only if all of its predecessors have been completed. Given strong impossibility results in classical competitive analysis, we investigate the problem in a learning-augmented setting, where an algorithm has access to predictions without any quality guarantee. We discuss different prediction models: novel problem-specific models as well as general ones, which have been proposed in previous works. We present lower bounds and algorithmic upper bounds for different precedence topologies, and thereby give a structured overview on which and how additional (possibly erroneous) information helps for designing better algorithms. Along the way, we also improve bounds on traditional competitive ratios for existing algorithms.

Besides natural language processing, transformers exhibit extraordinary performance in solving broader applications, including scientific computing and computer vision. Previous works try to explain this from the expressive power and capability perspectives that standard transformers are capable of performing some algorithms. To empower transformers with algorithmic capabilities and motivated by the recently proposed looped transformer (Yang et al., 2024; Giannou et al., 2023), we design a novel transformer block, dubbed Algorithm Transformer (abbreviated as AlgoFormer). Compared with the standard transformer and vanilla looped transformer, the proposed AlgoFormer can achieve significantly higher expressiveness in algorithm representation when using the same number of parameters. In particular, inspired by the structure of human-designed learning algorithms, our transformer block consists of a pre-transformer that is responsible for task pre-processing, a looped transformer for iterative optimization algorithms, and a post-transformer for producing the desired results after post-processing. We provide theoretical evidence of the expressive power of the AlgoFormer in solving some challenging problems, mirroring human-designed algorithms. Furthermore, some theoretical and empirical results are presented to show that the designed transformer has the potential to be smarter than human-designed algorithms. Experimental results demonstrate the empirical superiority of the proposed transformer in that it outperforms the standard transformer and vanilla looped transformer in some challenging tasks.

We propose a novel approach to scaling LLM inference for code generation. We frame code generation as a black box optimization problem within the code space, and employ optimization-inspired techniques to enhance exploration. Specifically, we introduce Scattered Forest Search to enhance solution diversity while searching for solutions. Our theoretical analysis illustrates how these methods avoid local optima during optimization. Extensive experiments on HumanEval, MBPP, APPS, CodeContests, and Leetcode reveal significant performance improvements. For instance, our method achieves a pass@1 rate of 67.1% on HumanEval+ and 87.2% on HumanEval with GPT-3.5, marking improvements of 8.6% and 4.3% over the state-of-the-art, while also halving the iterations needed to find the correct solution. Furthermore, our method scales more efficiently than existing search techniques, including tree search, line search, and repeated sampling.

We assessed the performance of commercial Large Language Models (LLMs) GPT-3.5-Turbo and GPT-4 on tasks from the 2023 BioASQ challenge. In Task 11b Phase B, which is focused on answer generation, both models demonstrated competitive abilities with leading systems. Remarkably, they achieved this with simple zero-shot learning, grounded with relevant snippets. Even without relevant snippets, their performance was decent, though not on par with the best systems. Interestingly, the older and cheaper GPT-3.5-Turbo system was able to compete with GPT-4 in the grounded Q&A setting on factoid and list answers. In Task 11b Phase A, focusing on retrieval, query expansion through zero-shot learning improved performance, but the models fell short compared to other systems. The code needed to rerun these experiments is available through GitHub.

The execution of graph algorithms using neural networks has recently attracted significant interest due to promising empirical progress. This motivates further understanding of how neural networks can replicate reasoning steps with relational data. In this work, we study the ability of transformer networks to simulate algorithms on graphs from a theoretical perspective. The architecture that we utilize is a looped transformer with extra attention heads that interact with the graph. We prove by construction that this architecture can simulate algorithms such as Dijkstra's shortest path algorithm, Breadth- and Depth-First Search, and Kosaraju's strongly connected components algorithm. The width of the network does not increase with the size of the input graph, which implies that the network can simulate the above algorithms for any graph. Despite this property, we show that there is a limit to simulation in our solution due to finite precision. Finally, we show a Turing Completeness result with constant width when the extra attention heads are utilized.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

Gradient descent optimization algorithms, while increasingly popular, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by. This article aims to provide the reader with intuitions with regard to the behaviour of different algorithms that will allow her to put them to use. In the course of this overview, we look at different variants of gradient descent, summarize challenges, introduce the most common optimization algorithms, review architectures in a parallel and distributed setting, and investigate additional strategies for optimizing gradient descent.

ProSper is a python library containing probabilistic algorithms to learn dictionaries. Given a set of data points, the implemented algorithms seek to learn the elementary components that have generated the data. The library widens the scope of dictionary learning approaches beyond implementations of standard approaches such as ICA, NMF or standard L1 sparse coding. The implemented algorithms are especially well-suited in cases when data consist of components that combine non-linearly and/or for data requiring flexible prior distributions. Furthermore, the implemented algorithms go beyond standard approaches by inferring prior and noise parameters of the data, and they provide rich a-posteriori approximations for inference. The library is designed to be extendable and it currently includes: Binary Sparse Coding (BSC), Ternary Sparse Coding (TSC), Discrete Sparse Coding (DSC), Maximal Causes Analysis (MCA), Maximum Magnitude Causes Analysis (MMCA), and Gaussian Sparse Coding (GSC, a recent spike-and-slab sparse coding approach). The algorithms are scalable due to a combination of variational approximations and parallelization. Implementations of all algorithms allow for parallel execution on multiple CPUs and multiple machines for medium to large-scale applications. Typical large-scale runs of the algorithms can use hundreds of CPUs to learn hundreds of dictionary elements from data with tens of millions of floating-point numbers such that models with several hundred thousand parameters can be optimized. The library is designed to have minimal dependencies and to be easy to use. It targets users of dictionary learning algorithms and Machine Learning researchers.

Although deep learning has made great progress in recent years, the exploding economic and environmental costs of training neural networks are becoming unsustainable. To address this problem, there has been a great deal of research on *algorithmically-efficient deep learning*, which seeks to reduce training costs not at the hardware or implementation level, but through changes in the semantics of the training program. In this paper, we present a structured and comprehensive overview of the research in this field. First, we formalize the *algorithmic speedup* problem, then we use fundamental building blocks of algorithmically efficient training to develop a taxonomy. Our taxonomy highlights commonalities of seemingly disparate methods and reveals current research gaps. Next, we present evaluation best practices to enable comprehensive, fair, and reliable comparisons of speedup techniques. To further aid research and applications, we discuss common bottlenecks in the training pipeline (illustrated via experiments) and offer taxonomic mitigation strategies for them. Finally, we highlight some unsolved research challenges and present promising future directions.

This paper uses the concept of algorithmic efficiency to present a unified theory of intelligence. Intelligence is defined informally, formally, and computationally. We introduce the concept of Dimensional complexity in algorithmic efficiency and deduce that an optimally efficient algorithm has zero Time complexity, zero Space complexity, and an infinite Dimensional complexity. This algorithm is used to generate the number line.

Large Language Models (LLMs) have demonstrated strong capabilities across various domains, with recent advancements in challenging reasoning tasks such as mathematics and programming. However, solving reasoning tasks often requires long decoding chains (of thoughts), which incur O(N) time and memory consumption, where N is the chain length. To mitigate O(N) time and memory consumption, existing sparsity-based algorithms propose retaining only the most critical token's intermediate data (i.e., key-value cache) and discarding the rest. However, these existing algorithms struggle with the ``impossible trinity'' of accuracy, time, and memory. For example, the state-of-the-art algorithm, Quest, achieves high accuracy with O(L) time but O(N) memory (L is the cache budget, L ll N). To address this issue, in this paper, we identify a new attention pattern during the decode stage of reasoning tasks, where milestone tokens (analogous to lemmas in mathematical proofs) emerge, are utilized, and then become unimportant afterward. Based on this pattern, we propose a new algorithm named RaaS that identifies and retains milestone tokens only until they are no longer needed, achieving high accuracy with O(L) time and O(L) memory complexity.

The goal of this chapter is to present a survey of homomorphic encryption techniques and their applications. After a detailed discussion on the introduction and motivation of the chapter, we present some basic concepts of cryptography. The fundamental theories of homomorphic encryption are then discussed with suitable examples. The chapter then provides a survey of some of the classical homomorphic encryption schemes existing in the current literature. Various applications and salient properties of homomorphic encryption schemes are then discussed in detail. The chapter then introduces the most important and recent research direction in the filed - fully homomorphic encryption. A significant number of propositions on fully homomorphic encryption is then discussed. Finally, the chapter concludes by outlining some emerging research trends in this exicting field of cryptography.

Transformer models serve as the backbone of many state-ofthe-art language models, and most use the scaled dot-product attention (SDPA) mechanism to capture relationships between tokens. However, the straightforward implementation of SDPA has quadratic compute and memory complexity with respect to the sequence length. On processor architectures such as GPUs and TPUs, there is a robust body of prior work. However, little work has been performed on non-processor architectures.In this work, we show how the architecture and execution model of Streaming Dataflow Accelerators can help tackle this challenge. We first define abstract hardware that adopts a streaming execution model, and we implement a cycle-accurate simulator of the abstract hardware using the Dataflow Abstract Machine simulation framework. Second, we implement the naive SDPA algorithm on this abstract hardware and show it requires linear (O(N)) intermediate memory. Third, we then modify the naive algorithm, taking inspiration from prior processor-oriented works, by reordering the multiplication and division operations. Finally, we map the modified algorithm to abstract hardware, and confirm that the implementation computes SDPA at full throughput while only using a constant amount (O(1)) of intermediate memory.

We present an online method for estimating the cost of solving SAT problems. Modern SAT solvers present several challenges to estimate search cost including non-chronological backtracking, learning and restarts. Our method uses a linear model trained on data gathered at the start of search. We show the effectiveness of this method using random and structured problems. We demonstrate that predictions made in early restarts can be used to improve later predictions. We also show that we can use such cost estimations to select a solver from a portfolio.

Despite recent success in large language model (LLM) reasoning, LLMs struggle with hierarchical multi-step reasoning tasks like generating complex programs. For these tasks, humans often start with a high-level algorithmic design and implement each part gradually. We introduce Parsel, a framework enabling automatic implementation and validation of complex algorithms with code LLMs. With Parsel, we automatically decompose algorithmic tasks into hierarchical natural language function descriptions and then search over combinations of possible function implementations using tests. We show that Parsel can be used across domains requiring hierarchical reasoning, including program synthesis and robotic planning. We find that, using Parsel, LLMs solve more competition-level problems in the APPS dataset, resulting in pass rates over 75\% higher than prior results from directly sampling AlphaCode and Codex, while often using a smaller sample budget. Moreover, with automatically generated tests, we find that Parsel can improve the state-of-the-art pass@1 performance on HumanEval from 67\% to 85\%. We also find that LLM-generated robotic plans using Parsel are more than twice as likely to be considered accurate than directly generated plans. Lastly, we explore how Parsel addresses LLM limitations and discuss how Parsel may be useful for human programmers. We release our code at https://github.com/ezelikman/parsel

Programming-by-Examples (PBE) aims to generate an algorithm from input-output examples. Such systems are practically and theoretically important: from an end-user perspective, they are deployed to millions of people, and from an AI perspective, PBE corresponds to a very general form of few-shot inductive inference. Given the success of Large Language Models (LLMs) in code-generation tasks, we investigate here the extent to which LLMs can be said to have `solved' PBE. We experiment on classic domains such as lists and strings, and an uncommon graphics programming domain not well represented in typical pretraining data. We find that pretrained models are not effective at PBE, but that they can be fine-tuned for much higher performance, provided the test problems are in-distribution. We analyze empirically what causes these models to succeed and fail, and take steps toward understanding how to achieve better out-of-distribution generalization. Collectively these results suggest that LLMs make strong progress toward solving the typical suite of PBE tasks, potentially increasing the flexibility and applicability of PBE systems, while also identifying ways in which LLMs still fall short.

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.

Recently similarity graphs became the leading paradigm for efficient nearest neighbor search, outperforming traditional tree-based and LSH-based methods. Similarity graphs perform the search via greedy routing: a query traverses the graph and in each vertex moves to the adjacent vertex that is the closest to this query. In practice, similarity graphs are often susceptible to local minima, when queries do not reach its nearest neighbors, getting stuck in suboptimal vertices. In this paper we propose to learn the routing function that overcomes local minima via incorporating information about the graph global structure. In particular, we augment the vertices of a given graph with additional representations that are learned to provide the optimal routing from the start vertex to the query nearest neighbor. By thorough experiments, we demonstrate that the proposed learnable routing successfully diminishes the local minima problem and significantly improves the overall search performance.

Maximizing a monotone submodular function under cardinality constraint k is a core problem in machine learning and database with many basic applications, including video and data summarization, recommendation systems, feature extraction, exemplar clustering, and coverage problems. We study this classic problem in the fully dynamic model where a stream of insertions and deletions of elements of an underlying ground set is given and the goal is to maintain an approximate solution using a fast update time. A recent paper at NeurIPS'20 by Lattanzi, Mitrovic, Norouzi{-}Fard, Tarnawski, Zadimoghaddam claims to obtain a dynamic algorithm for this problem with a 1{2} -epsilon approximation ratio and a query complexity bounded by poly(log(n),log(k),epsilon^{-1}). However, as we explain in this paper, the analysis has some important gaps. Having a dynamic algorithm for the problem with polylogarithmic update time is even more important in light of a recent result by Chen and Peng at STOC'22 who show a matching lower bound for the problem -- any randomized algorithm with a 1{2}+epsilon approximation ratio must have an amortized query complexity that is polynomial in n. In this paper, we develop a simpler algorithm for the problem that maintains a (1{2}-epsilon)-approximate solution for submodular maximization under cardinality constraint k using a polylogarithmic amortized update time.

Data pruning algorithms are commonly used to reduce the memory and computational cost of the optimization process. Recent empirical results reveal that random data pruning remains a strong baseline and outperforms most existing data pruning methods in the high compression regime, i.e., where a fraction of 30% or less of the data is kept. This regime has recently attracted a lot of interest as a result of the role of data pruning in improving the so-called neural scaling laws; in [Sorscher et al.], the authors showed the need for high-quality data pruning algorithms in order to beat the sample power law. In this work, we focus on score-based data pruning algorithms and show theoretically and empirically why such algorithms fail in the high compression regime. We demonstrate ``No Free Lunch" theorems for data pruning and present calibration protocols that enhance the performance of existing pruning algorithms in this high compression regime using randomization.

We study single-machine scheduling of jobs, each belonging to a job type that determines its duration distribution. We start by analyzing the scenario where the type characteristics are known and then move to two learning scenarios where the types are unknown: non-preemptive problems, where each started job must be completed before moving to another job; and preemptive problems, where job execution can be paused in the favor of moving to a different job. In both cases, we design algorithms that achieve sublinear excess cost, compared to the performance with known types, and prove lower bounds for the non-preemptive case. Notably, we demonstrate, both theoretically and through simulations, how preemptive algorithms can greatly outperform non-preemptive ones when the durations of different job types are far from one another, a phenomenon that does not occur when the type durations are known.

This note describes the development of an exact solver for Minimal Directed Feedback Vertex Set as part of the PACE 2022 competition. The solver is powered largely by aggressively trying to reduce the DFVS problem to a Minimal Cover problem, and applying reduction rules adapted from Vertex Cover literature. The resulting problem is solved as an Integer Linear Program (ILP) using SCIP. The resulting solver performed the second-best in the competition, although a bug at submission time disqualified it. As an additional note, we describe a new vertex cover reduction generalizing the Desk reduction rule.

We begin the study of list-decodable linear regression using batches. In this setting only an alpha in (0,1] fraction of the batches are genuine. Each genuine batch contains ge n i.i.d. samples from a common unknown distribution and the remaining batches may contain arbitrary or even adversarial samples. We derive a polynomial time algorithm that for any nge tilde Omega(1/alpha) returns a list of size mathcal O(1/alpha^2) such that one of the items in the list is close to the true regression parameter. The algorithm requires only mathcal{O}(d/alpha^2) genuine batches and works under fairly general assumptions on the distribution. The results demonstrate the utility of batch structure, which allows for the first polynomial time algorithm for list-decodable regression, which may be impossible for the non-batch setting, as suggested by a recent SQ lower bound diakonikolas2021statistical for the non-batch setting.

Computationally intensive decoding procedures--including search, reranking, and self-critique--can improve the quality of language model (LM) outputs in problems spanning code generation, numerical reasoning, and dialog. Existing work typically applies the same decoding procedure for every input to an LM. But not all inputs require the same amount of computation to process. Can we allocate decoding computation adaptively, using more resources to answer questions whose answers will be harder to compute? We present an approach that predicts the distribution of rewards given an input and computation budget, then allocates additional computation to inputs for which it is predicted to be most useful. We apply this approach in two decoding procedures: first, an adaptive best-of-k procedure that dynamically selects the number of samples to generate as input to a reranker; second, a routing procedure that dynamically responds to a query using a decoding procedure that is expensive but accurate, or one that is cheaper but less capable. Across a suite of programming, mathematics, and dialog tasks, we show that accurate computation-allocation procedures can be learned, and reduce computation by up to 50% at no cost to response quality, or improve quality by up to 10% at a fixed computational budget.

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an O(k^2) amortized update time (in the number of additions and deletions) and yields a 4-approximate solution, where k is the rank of the matroid.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

Recent advances in neural algorithmic reasoning with graph neural networks (GNNs) are propped up by the notion of algorithmic alignment. Broadly, a neural network will be better at learning to execute a reasoning task (in terms of sample complexity) if its individual components align well with the target algorithm. Specifically, GNNs are claimed to align with dynamic programming (DP), a general problem-solving strategy which expresses many polynomial-time algorithms. However, has this alignment truly been demonstrated and theoretically quantified? Here we show, using methods from category theory and abstract algebra, that there exists an intricate connection between GNNs and DP, going well beyond the initial observations over individual algorithms such as Bellman-Ford. Exposing this connection, we easily verify several prior findings in the literature, produce better-grounded GNN architectures for edge-centric tasks, and demonstrate empirical results on the CLRS algorithmic reasoning benchmark. We hope our exposition will serve as a foundation for building stronger algorithmically aligned GNNs.

Despite the rapid progress in the capabilities of Large Language Models (LLMs), they continue to have difficulty following relatively simple, unambiguous instructions, especially when compositions are involved. In this paper, we take inspiration from recent work that suggests that models may follow instructions better when they are expressed in pseudo-code. However, writing pseudo-code programs can be tedious and using few-shot demonstrations to craft code representations for use in inference can be unnatural for non-expert users of LLMs. To overcome these limitations, we propose fine-tuning LLMs with instruction-tuning data that additionally includes instructions re-expressed in pseudo-code along with the final response. We evaluate models trained using our method on 11 publicly available benchmarks comprising of tasks related to instruction-following, mathematics, and common-sense reasoning. We conduct rigorous experiments with 5 different models and find that not only do models follow instructions better when trained with pseudo-code, they also retain their capabilities on the other tasks related to mathematical and common sense reasoning. Specifically, we observe a relative gain of 3--19% on instruction-following benchmark, and an average gain of upto 14% across all tasks.

This paper reports on the second "Throughput" phase of the Tracking Machine Learning (TrackML) challenge on the Codalab platform. As in the first "Accuracy" phase, the participants had to solve a difficult experimental problem linked to tracking accurately the trajectory of particles as e.g. created at the Large Hadron Collider (LHC): given O(10^5) points, the participants had to connect them into O(10^4) individual groups that represent the particle trajectories which are approximated helical. While in the first phase only the accuracy mattered, the goal of this second phase was a compromise between the accuracy and the speed of inference. Both were measured on the Codalab platform where the participants had to upload their software. The best three participants had solutions with good accuracy and speed an order of magnitude faster than the state of the art when the challenge was designed. Although the core algorithms were less diverse than in the first phase, a diversity of techniques have been used and are described in this paper. The performance of the algorithms are analysed in depth and lessons derived.

The study of hardest and easiest fitness landscapes is an active area of research. Recently, Kaufmann, Larcher, Lengler and Zou conjectured that for the self-adjusting (1,lambda)-EA, Adversarial Dynamic BinVal (ADBV) is the hardest dynamic monotone function to optimize. We introduce the function Switching Dynamic BinVal (SDBV) which coincides with ADBV whenever the number of remaining zeros in the search point is strictly less than n/2, where n denotes the dimension of the search space. We show, using a combinatorial argument, that for the (1+1)-EA with any mutation rate p in [0,1], SDBV is drift-minimizing among the class of dynamic monotone functions. Our construction provides the first explicit example of an instance of the partially-ordered evolutionary algorithm (PO-EA) model with parameterized pessimism introduced by Colin, Doerr and F\'erey, building on work of Jansen. We further show that the (1+1)-EA optimizes SDBV in Theta(n^{3/2}) generations. Our simulations demonstrate matching runtimes for both static and self-adjusting (1,lambda) and (1+lambda)-EA. We further show, using an example of fixed dimension, that drift-minimization does not equal maximal runtime.

The focus of recent meta-learning research has been on the development of learning algorithms that can quickly adapt to test time tasks with limited data and low computational cost. Few-shot learning is widely used as one of the standard benchmarks in meta-learning. In this work, we show that a simple baseline: learning a supervised or self-supervised representation on the meta-training set, followed by training a linear classifier on top of this representation, outperforms state-of-the-art few-shot learning methods. An additional boost can be achieved through the use of self-distillation. This demonstrates that using a good learned embedding model can be more effective than sophisticated meta-learning algorithms. We believe that our findings motivate a rethinking of few-shot image classification benchmarks and the associated role of meta-learning algorithms. Code is available at: http://github.com/WangYueFt/rfs/.

While programming is one of the most broadly applicable skills in modern society, modern machine learning models still cannot code solutions to basic problems. Despite its importance, there has been surprisingly little work on evaluating code generation, and it can be difficult to accurately assess code generation performance rigorously. To meet this challenge, we introduce APPS, a benchmark for code generation. Unlike prior work in more restricted settings, our benchmark measures the ability of models to take an arbitrary natural language specification and generate satisfactory Python code. Similar to how companies assess candidate software developers, we then evaluate models by checking their generated code on test cases. Our benchmark includes 10,000 problems, which range from having simple one-line solutions to being substantial algorithmic challenges. We fine-tune large language models on both GitHub and our training set, and we find that the prevalence of syntax errors is decreasing exponentially as models improve. Recent models such as GPT-Neo can pass approximately 20% of the test cases of introductory problems, so we find that machine learning models are now beginning to learn how to code. As the social significance of automatic code generation increases over the coming years, our benchmark can provide an important measure for tracking advancements.

These handouts are designed for people who is just starting involved with the topic artificial neural networks. We show how it works a single artificial neuron (McCulloch & Pitt model), mathematically and graphically. We do explain the delta rule, a learning algorithm to find the neuron weights. We also present some examples in MATLAB/Octave. There are examples for classification task for lineal and non-lineal problems. At the end, we present an artificial neural network, a feed-forward neural network along its learning algorithm backpropagation. ----- Estos apuntes est\'an dise\~nados para personas que por primera vez se introducen en el tema de las redes neuronales artificiales. Se muestra el funcionamiento b\'asico de una neurona, matem\'aticamente y gr\'aficamente. Se explica la Regla Delta, algoritmo deaprendizaje para encontrar los pesos de una neurona. Tambi\'en se muestran ejemplos en MATLAB/Octave. Hay ejemplos para problemas de clasificaci\'on, para problemas lineales y no-lineales. En la parte final se muestra la arquitectura de red neuronal artificial conocida como backpropagation.

We show that transformer-based large language models are computationally universal when augmented with an external memory. Any deterministic language model that conditions on strings of bounded length is equivalent to a finite automaton, hence computationally limited. However, augmenting such models with a read-write memory creates the possibility of processing arbitrarily large inputs and, potentially, simulating any algorithm. We establish that an existing large language model, Flan-U-PaLM 540B, can be combined with an associative read-write memory to exactly simulate the execution of a universal Turing machine, U_{15,2}. A key aspect of the finding is that it does not require any modification of the language model weights. Instead, the construction relies solely on designing a form of stored instruction computer that can subsequently be programmed with a specific set of prompts.

Competition-level code generation tasks pose significant challenges for current state-of-the-art large language models (LLMs). For example, on the LiveCodeBench-Hard dataset, models such as O1-Mini and O1-Preview achieve pass@1 rates of only 0.366 and 0.143, respectively. While tree search techniques have proven effective in domains like mathematics and general coding, their potential in competition-level code generation remains under-explored. In this work, we propose a novel token-level tree search method specifically designed for code generation. Leveraging Qwen2.5-Coder-32B-Instruct, our approach achieves a pass rate of 0.305 on LiveCodeBench-Hard, surpassing the pass@100 performance of GPT4o-0513 (0.245). Furthermore, by integrating Chain-of-Thought (CoT) prompting, we improve our method's performance to 0.351, approaching O1-Mini's pass@1 rate. To ensure reproducibility, we report the average number of generations required per problem by our tree search method on the test set. Our findings underscore the potential of tree search to significantly enhance performance on competition-level code generation tasks. This opens up new possibilities for large-scale synthesis of challenging code problems supervised fine-tuning (SFT) data, advancing competition-level code generation tasks.

In this article, we study the parameterized complexity of the Set Cover problem restricted to semi-ladder-free hypergraphs, a class defined by Fabianski et al. [Proceedings of STACS 2019]. We observe that two algorithms introduced by Langerman and Morin [Discrete & Computational Geometry 2005] in the context of geometric covering problems can be adapted to this setting, yielding simple FPT and kernelization algorithms for Set Cover in semi-ladder-free hypergraphs. We complement our algorithmic results with a compression lower bound for the problem, which proves the tightness of our kernelization under standard complexity-theoretic assumptions.

Solving a linear system Ax=b is a fundamental scientific computing primitive for which numerous solvers and preconditioners have been developed. These come with parameters whose optimal values depend on the system being solved and are often impossible or too expensive to identify; thus in practice sub-optimal heuristics are used. We consider the common setting in which many related linear systems need to be solved, e.g. during a single numerical simulation. In this scenario, can we sequentially choose parameters that attain a near-optimal overall number of iterations, without extra matrix computations? We answer in the affirmative for Successive Over-Relaxation (SOR), a standard solver whose parameter omega has a strong impact on its runtime. For this method, we prove that a bandit online learning algorithm--using only the number of iterations as feedback--can select parameters for a sequence of instances such that the overall cost approaches that of the best fixed omega as the sequence length increases. Furthermore, when given additional structural information, we show that a contextual bandit method asymptotically achieves the performance of the instance-optimal policy, which selects the best omega for each instance. Our work provides the first learning-theoretic treatment of high-precision linear system solvers and the first end-to-end guarantees for data-driven scientific computing, demonstrating theoretically the potential to speed up numerical methods using well-understood learning algorithms.

We present a very simple algorithm for attention that requires O(1) memory with respect to sequence length and an extension to self-attention that requires O(log n) memory. This is in contrast with the frequently stated belief that self-attention requires O(n^2) memory. While the time complexity is still O(n^2), device memory rather than compute capability is often the limiting factor on modern accelerators. Thus, reducing the memory requirements of attention allows processing of longer sequences than might otherwise be feasible. We provide a practical implementation for accelerators that requires O(n) memory, is numerically stable, and is within a few percent of the runtime of the standard implementation of attention. We also demonstrate how to differentiate the function while remaining memory-efficient. For sequence length 16384, the memory overhead of self-attention is reduced by 59X for inference and by 32X for differentiation.

We extend the capabilities of neural networks by coupling them to external memory resources, which they can interact with by attentional processes. The combined system is analogous to a Turing Machine or Von Neumann architecture but is differentiable end-to-end, allowing it to be efficiently trained with gradient descent. Preliminary results demonstrate that Neural Turing Machines can infer simple algorithms such as copying, sorting, and associative recall from input and output examples.

Weight sharing promises to make neural architecture search (NAS) tractable even on commodity hardware. Existing methods in this space rely on a diverse set of heuristics to design and train the shared-weight backbone network, a.k.a. the super-net. Since heuristics and hyperparameters substantially vary across different methods, a fair comparison between them can only be achieved by systematically analyzing the influence of these factors. In this paper, we therefore provide a systematic evaluation of the heuristics and hyperparameters that are frequently employed by weight-sharing NAS algorithms. Our analysis uncovers that some commonly-used heuristics for super-net training negatively impact the correlation between super-net and stand-alone performance, and evidences the strong influence of certain hyperparameters and architectural choices. Our code and experiments set a strong and reproducible baseline that future works can build on.

We present Llemma, a large language model for mathematics. We continue pretraining Code Llama on the Proof-Pile-2, a mixture of scientific papers, web data containing mathematics, and mathematical code, yielding Llemma. On the MATH benchmark Llemma outperforms all known open base models, as well as the unreleased Minerva model suite on an equi-parameter basis. Moreover, Llemma is capable of tool use and formal theorem proving without any further finetuning. We openly release all artifacts, including 7 billion and 34 billion parameter models, the Proof-Pile-2, and code to replicate our experiments.

In a compound AI system, components such as an LLM call, a retriever, a code interpreter, or tools are interconnected. The system's behavior is primarily driven by parameters such as instructions or tool definitions. Recent advancements enable end-to-end optimization of these parameters using an LLM. Notably, leveraging an LLM as an optimizer is particularly efficient because it avoids gradient computation and can generate complex code and instructions. This paper presents a survey of the principles and emerging trends in LLM-based optimization of compound AI systems. It covers archetypes of compound AI systems, approaches to LLM-based end-to-end optimization, and insights into future directions and broader impacts. Importantly, this survey uses concepts from program analysis to provide a unified view of how an LLM optimizer is prompted to optimize a compound AI system. The exhaustive list of paper is provided at https://github.com/linyuhongg/LLM-based-Optimization-of-Compound-AI-Systems.

We propose Step-by-Step Coding (SBSC): a multi-turn math reasoning framework that enables Large Language Models (LLMs) to generate sequence of programs for solving Olympiad level math problems. At each step/turn, by leveraging the code execution outputs and programs of previous steps, the model generates the next sub-task and the corresponding program to solve it. This way, SBSC, sequentially navigates to reach the final answer. SBSC allows more granular, flexible and precise approach to problem-solving compared to existing methods. Extensive experiments highlight the effectiveness of SBSC in tackling competition and Olympiad-level math problems. For Claude-3.5-Sonnet, we observe SBSC (greedy decoding) surpasses existing state-of-the-art (SOTA) program generation based reasoning strategies by absolute 10.7% on AMC12, 8% on AIME and 12.6% on MathOdyssey. Given SBSC is multi-turn in nature, we also benchmark SBSC's greedy decoding against self-consistency decoding results of existing SOTA math reasoning strategies and observe performance gain by absolute 6.2% on AMC, 6.7% on AIME and 7.4% on MathOdyssey.

Large Language Models (LLMs) have shown great success in code generation. LLMs take as the input a prompt and output the code. A key question is how to make prompts (i.e., Prompting Techniques). Existing prompting techniques are designed for natural language generation and have low accuracy in code generation. In this paper, we propose a new prompting technique named AceCoder. Our motivation is that code generation meets two unique challenges (i.e., requirement understanding and code implementation). AceCoder contains two novel mechanisms (i.e., guided code generation and example retrieval) to solve these challenges. (1) Guided code generation asks LLMs first to analyze requirements and output an intermediate preliminary (e.g., test cases). The preliminary is used to clarify requirements and tell LLMs "what to write". (2) Example retrieval selects similar programs as examples in prompts, which provide lots of relevant content (e.g., algorithms, APIs) and teach LLMs "how to write". We apply AceCoder to three LLMs (e.g., Codex) and evaluate it on three public benchmarks using the Pass@k. Results show that AceCoder can significantly improve the performance of LLMs on code generation. (1) In terms of Pass@1, AceCoder outperforms the state-of-the-art baseline by up to 56.4% in MBPP, 70.7% in MBJP, and 88.4% in MBJSP. (2) AceCoder is effective in LLMs with different sizes (i.e., 6B to 13B) and different languages (i.e., Python, Java, and JavaScript). (3) Human evaluation shows human developers prefer programs from AceCoder.

While scaling training compute has led to remarkable improvements in large language models (LLMs), scaling inference compute has not yet yielded analogous gains. We hypothesize that a core missing component is a lack of diverse LLM outputs, leading to inefficient search due to models repeatedly sampling highly similar, yet incorrect generations. We empirically demonstrate that this lack of diversity can be mitigated by searching over candidate plans for solving a problem in natural language. Based on this insight, we propose PLANSEARCH, a novel search algorithm which shows strong results across HumanEval+, MBPP+, and LiveCodeBench (a contamination-free benchmark for competitive coding). PLANSEARCH generates a diverse set of observations about the problem and then uses these observations to construct plans for solving the problem. By searching over plans in natural language rather than directly over code solutions, PLANSEARCH explores a significantly more diverse range of potential solutions compared to baseline search methods. Using PLANSEARCH on top of Claude 3.5 Sonnet achieves a state-of-the-art pass@200 of 77.0% on LiveCodeBench, outperforming both the best score achieved without search (pass@1 = 41.4%) and using standard repeated sampling (pass@200 = 60.6%). Finally, we show that, across all models, search algorithms, and benchmarks analyzed, we can accurately predict performance gains due to search as a direct function of the diversity over generated ideas.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for pi, ln(2), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness. To give an efficient yet principled implementation of the AD algorithm we define a sound and complete representation of hierarchical string diagrams as a class of hierarchical hypergraphs we call hypernets.

We propose a new learning paradigm called Deep Memory. It has the potential to completely revolutionize the Machine Learning field. Surprisingly, this paradigm has not been reinvented yet, unlike Deep Learning. At the core of this approach is the Learning By Heart principle, well studied in primary schools all over the world. Inspired by poem recitation, or by pi decimal memorization, we propose a concrete algorithm that mimics human behavior. We implement this paradigm on the task of generative modeling, and apply to images, natural language and even the pi decimals as long as one can print them as text. The proposed algorithm even generated this paper, in a one-shot learning setting. In carefully designed experiments, we show that the generated samples are indistinguishable from the training examples, as measured by any statistical tests or metrics.

In this paper, we compare the three most popular algorithms for hyperparameter optimization (Grid Search, Random Search, and Genetic Algorithm) and attempt to use them for neural architecture search (NAS). We use these algorithms for building a convolutional neural network (search architecture). Experimental results on CIFAR-10 dataset further demonstrate the performance difference between compared algorithms. The comparison results are based on the execution time of the above algorithms and accuracy of the proposed models.

Intermediate reasoning or acting steps have successfully improved large language models (LLMs) for handling various downstream natural language processing (NLP) tasks. When applying LLMs for code generation, recent works mainly focus on directing the models to articulate intermediate natural-language reasoning steps, as in chain-of-thought (CoT) prompting, and then output code with the natural language or other structured intermediate steps. However, such output is not suitable for code translation or generation tasks since the standard CoT has different logical structures and forms of expression with the code. In this work, we introduce the universal code (UniCode) as the intermediate representation. It is a description of algorithm steps using a mix of conventions of programming languages, such as assignment operator, conditional operator, and loop. Hence, we collect an instruction dataset UniCoder-Instruct to train our model UniCoder on multi-task learning objectives. UniCoder-Instruct comprises natural-language questions, code solutions, and the corresponding universal code. The alignment between the intermediate universal code representation and the final code solution significantly improves the quality of the generated code. The experimental results demonstrate that UniCoder with the universal code significantly outperforms the previous prompting methods by a large margin, showcasing the effectiveness of the structural clues in pseudo-code.

Computing olympiads contain some of the most challenging problems for humans, requiring complex algorithmic reasoning, puzzle solving, in addition to generating efficient code. However, it has been understudied as a domain to evaluate language models (LMs). In this paper, we introduce the USACO benchmark with 307 problems from the USA Computing Olympiad, along with high-quality unit tests, reference code, and official analyses for each problem. These resources enable us to construct and test a range of LM inference methods for competitive programming for the first time. We find GPT-4 only achieves a 8.7% pass@1 accuracy with zero-shot chain-of-thought prompting, and our best inference method improves it to 20.2% using a combination of self-reflection and retrieval over episodic knowledge. However, this is far from solving the benchmark. To better understand the remaining challenges, we design a novel human-in-the-loop study and surprisingly find that a small number of targeted hints enable GPT-4 to solve 13 out of 15 problems previously unsolvable by any model and method. Our benchmark, baseline methods, quantitative results, and qualitative analysis serve as an initial step toward LMs with grounded, creative, and algorithmic reasoning.

Neural networks -- especially those that use large, pre-trained language models -- have improved search engines in various ways. Most prominently, they can estimate the relevance of a passage or document to a user's query. In this work, we depart from this direction by exploring whether neural networks can effectively predict which of a document's passages are unlikely to be relevant to any query submitted to the search engine. We refer to this query-agnostic estimation of passage relevance as a passage's quality. We find that our novel methods for estimating passage quality allow passage corpora to be pruned considerably while maintaining statistically equivalent effectiveness; our best methods can consistently prune >25% of passages in a corpora, across various retrieval pipelines. Such substantial pruning reduces the operating costs of neural search engines in terms of computing resources, power usage, and carbon footprint -- both when processing queries (thanks to a smaller index size) and when indexing (lightweight models can prune low-quality passages prior to the costly dense or learned sparse encoding step). This work sets the stage for developing more advanced neural "learning-what-to-index" methods.

Clustering is a NP-hard problem. Thus, no optimal algorithm exists, heuristics are applied to cluster the data. Heuristics can be very resource-intensive, if not applied properly. For substantially large data sets computational efficiencies can be achieved by reducing the input space if a minimal loss of information can be achieved. Clustering algorithms, in general, face two common problems: 1) these converge to different settings with different initial conditions and; 2) the number of clusters has to be arbitrarily decided beforehand. This problem has become critical in the realm of big data. Recently, clustering algorithms have emerged which can speedup computations using parallel processing over the grid but face the aforementioned problems. Goals: Our goals are to find methods to cluster data which: 1) guarantee convergence to the same settings irrespective of the initial conditions; 2) eliminate the need to establish the number of clusters beforehand, and 3) can be applied to cluster large datasets. Methods: We introduce a method that combines probabilistic and combinatorial clustering methods to produce repeatable and compact clusters that are not sensitive to initial conditions. This method harnesses the power of k-means (a combinatorial clustering method) to cluster/partition very large dimensional datasets and uses the Gaussian Mixture Model (a probabilistic clustering method) to validate the k-means partitions. Results: We show that this method produces very compact clusters that are not sensitive to initial conditions. This method can be used to identify the most 'separable' set in a dataset which increases the 'clusterability' of a dataset. This method also eliminates the need to specify the number of clusters in advance.

Optimization can be found in many real-life applications. Designing an effective algorithm for a specific optimization problem typically requires a tedious amount of effort from human experts with domain knowledge and algorithm design skills. In this paper, we propose a novel approach called Algorithm Evolution using Large Language Model (AEL). It utilizes a large language model (LLM) to automatically generate optimization algorithms via an evolutionary framework. AEL does algorithm-level evolution without model training. Human effort and requirements for domain knowledge can be significantly reduced. We take constructive methods for the salesman traveling problem as a test example, we show that the constructive algorithm obtained by AEL outperforms simple hand-crafted and LLM-generated heuristics. Compared with other domain deep learning model-based algorithms, these methods exhibit excellent scalability across different problem sizes. AEL is also very different from previous attempts that utilize LLMs as search operators in algorithms.

Scientists often infer abstract procedures from specific instances of problems and use the abstractions to generate new, related instances. For example, programs encoding the formal rules and properties of a system have been useful in fields ranging from RL (procedural environments) to physics (simulation engines). These programs can be seen as functions which execute to different outputs based on their parameterizations (e.g., gridworld configuration or initial physical conditions). We introduce the term EFA (Executable Functional Abstraction) to denote such programs for math problems. EFA-like constructs have been shown to be useful for math reasoning as problem generators for stress-testing models. However, prior work has been limited to abstractions for grade-school math (whose simple rules are easy to encode in programs), while generating EFAs for advanced math has thus far required human engineering. We explore the automatic construction of EFAs for advanced math problems. We operationalize the task of automatically constructing EFAs as a program synthesis task, and develop EFAGen, which conditions an LLM on a seed math problem and its step-by-step solution to generate candidate EFA programs that are faithful to the generalized problem and solution class underlying the seed problem. Furthermore, we formalize properties any valid EFA must possess in terms of executable unit tests, and show how the tests can be used as verifiable rewards to train LLMs to become better writers of EFAs. We demonstrate that EFAs constructed by EFAGen behave rationally by remaining faithful to seed problems, produce learnable problem variations, and that EFAGen can infer EFAs across multiple diverse sources of competition-level math problems. Finally, we show downstream uses of model-written EFAs e.g. finding problem variations that are harder or easier for a learner to solve, as well as data generation.

Restart strategies are an important factor in the performance of conflict-driven Davis Putnam style SAT solvers. Selecting a good restart strategy for a problem instance can enhance the performance of a solver. Inspired by recent success applying machine learning techniques to predict the runtime of SAT solvers, we present a method which uses machine learning to boost solver performance through a smart selection of the restart strategy. Based on easy to compute features, we train both a satisfiability classifier and runtime models. We use these models to choose between restart strategies. We present experimental results comparing this technique with the most commonly used restart strategies. Our results demonstrate that machine learning is effective in improving solver performance.

In today's software world with its cornucopia of reusable software libraries, when a programmer is faced with a programming task that they suspect can be completed through the use of a library, they often look for code examples using a search engine and then manually adapt found examples to their specific context of use. We put forward a vision based on a new breed of developer tools that have the potential to largely automate this process. The key idea is to adapt code autocompletion tools such that they take into account not only the developer's already-written code but also the intent of the task the developer is trying to achieve next, formulated in plain natural language. We call this practice of enriching the code with natural language intent to facilitate its completion natural language-guided programming. To show that this idea is feasible we design, implement and benchmark a tool that solves this problem in the context of a specific domain (data science) and a specific programming language (Python). Central to the tool is the use of language models trained on a large corpus of documented code. Our initial experiments confirm the feasibility of the idea but also make it clear that we have only scratched the surface of what may become possible in the future. We end the paper with a comprehensive research agenda to stimulate additional research in the budding area of natural language-guided programming.

Recent work on neural algorithmic reasoning has investigated the reasoning capabilities of neural networks, effectively demonstrating they can learn to execute classical algorithms on unseen data coming from the train distribution. However, the performance of existing neural reasoners significantly degrades on out-of-distribution (OOD) test data, where inputs have larger sizes. In this work, we make an important observation: there are many different inputs for which an algorithm will perform certain intermediate computations identically. This insight allows us to develop data augmentation procedures that, given an algorithm's intermediate trajectory, produce inputs for which the target algorithm would have exactly the same next trajectory step. Then, we employ a causal framework to design a corresponding self-supervised objective, and we prove that it improves the OOD generalisation capabilities of the reasoner. We evaluate our method on the CLRS algorithmic reasoning benchmark, where we show up to 3times improvements on the OOD test data.

The emerging field of Diverse Intelligence seeks to identify, formalize, and understand commonalities in behavioral competencies across a wide range of implementations. Especially interesting are simple systems that provide unexpected examples of memory, decision-making, or problem-solving in substrates that at first glance do not appear to be complex enough to implement such capabilities. We seek to develop tools to help understand the minimal requirements for such capabilities, and to learn to recognize and predict basal forms of intelligence in unconventional substrates. Here, we apply novel analyses to the behavior of classical sorting algorithms, short pieces of code which have been studied for many decades. To study these sorting algorithms as a model of biological morphogenesis and its competencies, we break two formerly-ubiquitous assumptions: top-down control (instead, showing how each element within a array of numbers can exert minimal agency and implement sorting policies from the bottom up), and fully reliable hardware (instead, allowing some of the elements to be "damaged" and fail to execute the algorithm). We quantitatively characterize sorting activity as the traversal of a problem space, showing that arrays of autonomous elements sort themselves more reliably and robustly than traditional implementations in the presence of errors. Moreover, we find the ability to temporarily reduce progress in order to navigate around a defect, and unexpected clustering behavior among the elements in chimeric arrays whose elements follow one of two different algorithms. The discovery of emergent problem-solving capacities in simple, familiar algorithms contributes a new perspective to the field of Diverse Intelligence, showing how basal forms of intelligence can emerge in simple systems without being explicitly encoded in their underlying mechanics.

Eliciting reasoning capabilities from language models (LMs) is a critical direction on the path towards building intelligent systems. Most recent studies dedicated to reasoning focus on out-of-distribution performance on procedurally-generated synthetic benchmarks, bespoke-built to evaluate specific skills only. This trend makes results hard to transfer across publications, slowing down progress. Three years ago, a similar issue was identified and rectified in the field of neural algorithmic reasoning, with the advent of the CLRS benchmark. CLRS is a dataset generator comprising graph execution traces of classical algorithms from the Introduction to Algorithms textbook. Inspired by this, we propose CLRS-Text -- a textual version of these algorithmic traces. Out of the box, CLRS-Text is capable of procedurally generating trace data for thirty diverse, challenging algorithmic tasks across any desirable input distribution, while offering a standard pipeline in which any additional algorithmic tasks may be created in the benchmark. We fine-tune and evaluate various LMs as generalist executors on this benchmark, validating prior work and revealing a novel, interesting challenge for the LM reasoning community. Our code is available at https://github.com/google-deepmind/clrs/tree/master/clrs/_src/clrs_text.

As models become increasingly sophisticated, conventional algorithm benchmarks are increasingly saturated, underscoring the need for more challenging benchmarks to guide future improvements in algorithmic reasoning. This paper introduces OIBench, a high-quality, private, and challenging olympiad-level informatics dataset comprising 250 carefully curated original problems. We detail the construction methodology of the benchmark, ensuring a comprehensive assessment across various programming paradigms and complexities, and we demonstrate its contamination-resistant properties via experiments. We propose Time/Space Completion Curves for finer-grained efficiency analysis and enable direct human-model comparisons through high-level participant evaluations. Our experiments reveal that while open-source models lag behind closed-source counterparts, current SOTA models already outperform most human participants in both correctness and efficiency, while still being suboptimal compared to the canonical solutions. By releasing OIBench as a fully open-source resource (https://huggingface.co/datasets/AGI-Eval/OIBench), we hope this benchmark will contribute to advancing code reasoning capabilities for future LLMs.

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

The application of machine learning algorithms to source code has grown in the past years. Since these algorithms are quite sensitive to input data, it is not surprising that researchers experiment with input representations. Nowadays, a popular starting point to represent code is abstract syntax trees (ASTs). Abstract syntax trees have been used for a long time in various software engineering domains, and in particular in IDEs. The API of modern IDEs allows to manipulate and traverse ASTs, resolve references between code elements, etc. Such algorithms can enrich ASTs with new data and therefore may be useful in ML-based code analysis. In this work, we present PSIMiner - a tool for processing PSI trees from the IntelliJ Platform. PSI trees contain code syntax trees as well as functions to work with them, and therefore can be used to enrich code representation using static analysis algorithms of modern IDEs. To showcase this idea, we use our tool to infer types of identifiers in Java ASTs and extend the code2seq model for the method name prediction problem.

The Airbnb search system grapples with many unique challenges as it continues to evolve. We oversee a marketplace that is nuanced by geography, diversity of homes, and guests with a variety of preferences. Crafting an efficient search system that can accommodate diverse guest needs, while showcasing relevant homes lies at the heart of Airbnb's success. Airbnb search has many challenges that parallel other recommendation and search systems but it has a unique information retrieval problem, upstream of ranking, called location retrieval. It requires defining a topological map area that is relevant to the searched query for homes listing retrieval. The purpose of this paper is to demonstrate the methodology, challenges, and impact of building a machine learning based location retrieval product from the ground up. Despite the lack of suitable, prevalent machine learning based approaches, we tackle cold start, generalization, differentiation and algorithmic bias. We detail the efficacy of heuristics, statistics, machine learning, and reinforcement learning approaches to solve these challenges, particularly for systems that are often unexplored by current literature.

A subgraph is constructed by using a subset of vertices and edges of a given graph. There exist many graph properties that are hereditary for subgraphs. Hence, researchers from different communities have paid a great deal of attention in studying numerous subgraph problems, on top of the ordinary graph problems. Many algorithms are proposed in studying subgraph problems, where one common approach is by extracting the patterns and structures of a given graph. Due to the complex structures of certain types of graphs and to improve overall performances of the existing frameworks, machine learning techniques have recently been employed in dealing with various subgraph problems. In this article, we present a comprehensive review on five well known subgraph problems that have been tackled by using machine learning methods. They are subgraph isomorphism (both counting and matching), maximum common subgraph, community detection and community search problems. We provide an outline of each proposed method, and examine its designs and performances. We also explore non-learning-based algorithms for each problem and a brief discussion is given. We then suggest some promising research directions in this area, hoping that relevant subgraph problems can be tackled by using a similar strategy. Since there is a huge growth in employing machine learning techniques in recent years, we believe that this survey will serve as a good reference point to relevant research communities.

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

Since the emergence of large language models, prompt learning has become a popular method for optimizing and customizing these models. Special prompts, such as Chain-of-Thought, have even revealed previously unknown reasoning capabilities within these models. However, the progress of discovering effective prompts has been slow, driving a desire for general prompt optimization methods. Unfortunately, few existing prompt learning methods satisfy the criteria of being truly "general", i.e., automatic, discrete, black-box, gradient-free, and interpretable all at once. In this paper, we introduce metaheuristics, a branch of discrete non-convex optimization methods with over 100 options, as a promising approach to prompt learning. Within our paradigm, we test six typical methods: hill climbing, simulated annealing, genetic algorithms with/without crossover, tabu search, and harmony search, demonstrating their effectiveness in black-box prompt learning and Chain-of-Thought prompt tuning. Furthermore, we show that these methods can be used to discover more human-understandable prompts that were previously unknown, opening the door to a cornucopia of possibilities in prompt optimization. We release all the codes in https://github.com/research4pan/Plum.

We propose an online training procedure for a transformer-based automated theorem prover. Our approach leverages a new search algorithm, HyperTree Proof Search (HTPS), inspired by the recent success of AlphaZero. Our model learns from previous proof searches through online training, allowing it to generalize to domains far from the training distribution. We report detailed ablations of our pipeline's main components by studying performance on three environments of increasing complexity. In particular, we show that with HTPS alone, a model trained on annotated proofs manages to prove 65.4% of a held-out set of Metamath theorems, significantly outperforming the previous state of the art of 56.5% by GPT-f. Online training on these unproved theorems increases accuracy to 82.6%. With a similar computational budget, we improve the state of the art on the Lean-based miniF2F-curriculum dataset from 31% to 42% proving accuracy.

Given a set of dissimilarity measurements amongst data points, determining what metric representation is most "consistent" with the input measurements or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. Existing methods are restricted to specific kinds of metrics or small problem sizes because of the large number of metric constraints in such problems. In this paper, we provide an active set algorithm, Project and Forget, that uses Bregman projections, to solve metric constrained problems with many (possibly exponentially) inequality constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithm converges to the global optimal solution and that the L_2 distance of the current iterate to the optimal solution decays asymptotically at an exponential rate. We demonstrate that using our method we can solve large problem instances of three types of metric constrained problems: general weight correlation clustering, metric nearness, and metric learning; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes.

Active learning (AL) algorithms may achieve better performance with fewer data because the model guides the data selection process. While many algorithms have been proposed, there is little study on what the optimal AL algorithm looks like, which would help researchers understand where their models fall short and iterate on the design. In this paper, we present a simulated annealing algorithm to search for this optimal oracle and analyze it for several tasks. We present qualitative and quantitative insights into the behaviors of this oracle, comparing and contrasting them with those of various heuristics. Moreover, we are able to consistently improve the heuristics using one particular insight. We hope that our findings can better inform future active learning research. The code is available at https://github.com/YilunZhou/optimal-active-learning.

Neural networks are sensitive to hyper-parameter and architecture choices. Automated Machine Learning (AutoML) is a promising paradigm for automating these choices. Current ML software libraries, however, are quite limited in handling the dynamic interactions among the components of AutoML. For example, efficientNAS algorithms, such as ENAS and DARTS, typically require an implementation coupling between the search space and search algorithm, the two key components in AutoML. Furthermore, implementing a complex search flow, such as searching architectures within a loop of searching hardware configurations, is difficult. To summarize, changing the search space, search algorithm, or search flow in current ML libraries usually requires a significant change in the program logic. In this paper, we introduce a new way of programming AutoML based on symbolic programming. Under this paradigm, ML programs are mutable, thus can be manipulated easily by another program. As a result, AutoML can be reformulated as an automated process of symbolic manipulation. With this formulation, we decouple the triangle of the search algorithm, the search space and the child program. This decoupling makes it easy to change the search space and search algorithm (without and with weight sharing), as well as to add search capabilities to existing code and implement complex search flows. We then introduce PyGlove, a new Python library that implements this paradigm. Through case studies on ImageNet and NAS-Bench-101, we show that with PyGlove users can easily convert a static program into a search space, quickly iterate on the search spaces and search algorithms, and craft complex search flows to achieve better results.

The recently presented idea to learn heuristics for combinatorial optimization problems is promising as it can save costly development. However, to push this idea towards practical implementation, we need better models and better ways of training. We contribute in both directions: we propose a model based on attention layers with benefits over the Pointer Network and we show how to train this model using REINFORCE with a simple baseline based on a deterministic greedy rollout, which we find is more efficient than using a value function. We significantly improve over recent learned heuristics for the Travelling Salesman Problem (TSP), getting close to optimal results for problems up to 100 nodes. With the same hyperparameters, we learn strong heuristics for two variants of the Vehicle Routing Problem (VRP), the Orienteering Problem (OP) and (a stochastic variant of) the Prize Collecting TSP (PCTSP), outperforming a wide range of baselines and getting results close to highly optimized and specialized algorithms.

The paradigm of machine learning and artificial intelligence has pervaded our everyday life in such a way that it is no longer an area for esoteric academics and scientists putting their effort to solve a challenging research problem. The evolution is quite natural rather than accidental. With the exponential growth in processing speed and with the emergence of smarter algorithms for solving complex and challenging problems, organizations have found it possible to harness a humongous volume of data in realizing solutions that have far-reaching business values. This introductory chapter highlights some of the challenges and barriers that organizations in the financial services sector at the present encounter in adopting machine learning and artificial intelligence-based models and applications in their day-to-day operations.

In recent times, a plethora of Large Code Generation Models (LCGMs) have been proposed, showcasing significant potential in assisting developers with complex programming tasks. Benchmarking LCGMs necessitates the creation of a set of diverse programming problems, and each problem comprises the prompt (including the task description), canonical solution, and test inputs. The existing methods for constructing such a problem set can be categorized into two main types: manual methods and perturbation-based methods. However, manual methods demand high effort and lack scalability, while also risking data integrity due to LCGMs' potentially contaminated data collection, and perturbation-based approaches mainly generate semantically homogeneous problems with the same canonical solutions and introduce typos that can be easily auto-corrected by IDE, making them ineffective and unrealistic. In this work, we propose the idea of programming problem merging (PPM) and provide two implementation of this idea, we utilize our tool on two widely-used datasets and compare it against nine baseline methods using eight code generation models. The results demonstrate the effectiveness of our tool in generating more challenging, diverse, and natural programming problems, comparing to the baselines.

Recent Language Models (LMs) achieve breakthrough performance in code generation when trained on human-authored problems, even solving some competitive-programming problems. Self-play has proven useful in games such as Go, and thus it is natural to ask whether LMs can generate their own instructive programming problems to improve their performance. We show that it is possible for an LM to synthesize programming problems and solutions, which are filtered for correctness by a Python interpreter. The LM's performance is then seen to improve when it is fine-tuned on its own synthetic problems and verified solutions; thus the model 'improves itself' using the Python interpreter. Problems are specified formally as programming puzzles [Schuster et al., 2021], a code-based problem format where solutions can easily be verified for correctness by execution. In experiments on publicly-available LMs, test accuracy more than doubles. This work demonstrates the potential for code LMs, with an interpreter, to generate instructive problems and improve their own performance.

Language models typically need to be trained or finetuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this work, we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate kNN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. On benchmarks including code and mathematics, we find that the model is capable of making use of newly defined functions and theorems during test time.

Given a matrix Min R^{mtimes n}, the low rank matrix completion problem asks us to find a rank-k approximation of M as UV^top for Uin R^{mtimes k} and Vin R^{ntimes k} by only observing a few entries specified by a set of entries Omegasubseteq [m]times [n]. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli and Sanghavi~jns13 showed that if M has incoherent rows and columns, then alternating minimization provably recovers the matrix M by observing a nearly linear in n number of entries. While the sample complexity has been subsequently improved~glz17, alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time widetilde O(|Omega| k), which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require widetilde O(|Omega| k^2) time.

The correct use of model evaluation, model selection, and algorithm selection techniques is vital in academic machine learning research as well as in many industrial settings. This article reviews different techniques that can be used for each of these three subtasks and discusses the main advantages and disadvantages of each technique with references to theoretical and empirical studies. Further, recommendations are given to encourage best yet feasible practices in research and applications of machine learning. Common methods such as the holdout method for model evaluation and selection are covered, which are not recommended when working with small datasets. Different flavors of the bootstrap technique are introduced for estimating the uncertainty of performance estimates, as an alternative to confidence intervals via normal approximation if bootstrapping is computationally feasible. Common cross-validation techniques such as leave-one-out cross-validation and k-fold cross-validation are reviewed, the bias-variance trade-off for choosing k is discussed, and practical tips for the optimal choice of k are given based on empirical evidence. Different statistical tests for algorithm comparisons are presented, and strategies for dealing with multiple comparisons such as omnibus tests and multiple-comparison corrections are discussed. Finally, alternative methods for algorithm selection, such as the combined F-test 5x2 cross-validation and nested cross-validation, are recommended for comparing machine learning algorithms when datasets are small.

We show how to "compile" human-readable programs into standard decoder-only transformer models. Our compiler, Tracr, generates models with known structure. This structure can be used to design experiments. For example, we use it to study "superposition" in transformers that execute multi-step algorithms. Additionally, the known structure of Tracr-compiled models can serve as ground-truth for evaluating interpretability methods. Commonly, because the "programs" learned by transformers are unknown it is unclear whether an interpretation succeeded. We demonstrate our approach by implementing and examining programs including computing token frequencies, sorting, and parenthesis checking. We provide an open-source implementation of Tracr at https://github.com/google-deepmind/tracr.

In deep learning, performance is strongly affected by the choice of architecture and hyperparameters. While there has been extensive work on automatic hyperparameter optimization for simple spaces, complex spaces such as the space of deep architectures remain largely unexplored. As a result, the choice of architecture is done manually by the human expert through a slow trial and error process guided mainly by intuition. In this paper we describe a framework for automatically designing and training deep models. We propose an extensible and modular language that allows the human expert to compactly represent complex search spaces over architectures and their hyperparameters. The resulting search spaces are tree-structured and therefore easy to traverse. Models can be automatically compiled to computational graphs once values for all hyperparameters have been chosen. We can leverage the structure of the search space to introduce different model search algorithms, such as random search, Monte Carlo tree search (MCTS), and sequential model-based optimization (SMBO). We present experiments comparing the different algorithms on CIFAR-10 and show that MCTS and SMBO outperform random search. In addition, these experiments show that our framework can be used effectively for model discovery, as it is possible to describe expressive search spaces and discover competitive models without much effort from the human expert. Code for our framework and experiments has been made publicly available.

In this study, we propose a simple method for fault-tolerant Strassen-like matrix multiplications. The proposed method is based on using two distinct Strassen-like algorithms instead of replicating a given one. We have realized that using two different algorithms, new check relations arise resulting in more local computations. These local computations are found using computer aided search. To improve performance, special parity (extra) sub-matrix multiplications (PSMMs) are generated (two of them) at the expense of increasing communication/computation cost of the system. Our preliminary results demonstrate that the proposed method outperforms a Strassen-like algorithm with two copies and secures a very close performance to three copy version using only 2 PSMMs, reducing the total number of compute nodes by around 24\% i.e., from 21 to 16.

We present MIPS, a novel method for program synthesis based on automated mechanistic interpretability of neural networks trained to perform the desired task, auto-distilling the learned algorithm into Python code. We test MIPS on a benchmark of 62 algorithmic tasks that can be learned by an RNN and find it highly complementary to GPT-4: MIPS solves 32 of them, including 13 that are not solved by GPT-4 (which also solves 30). MIPS uses an integer autoencoder to convert the RNN into a finite state machine, then applies Boolean or integer symbolic regression to capture the learned algorithm. As opposed to large language models, this program synthesis technique makes no use of (and is therefore not limited by) human training data such as algorithms and code from GitHub. We discuss opportunities and challenges for scaling up this approach to make machine-learned models more interpretable and trustworthy.

The MineRL BASALT competition has served to catalyze advances in learning from human feedback through four hard-to-specify tasks in Minecraft, such as create and photograph a waterfall. Given the completion of two years of BASALT competitions, we offer to the community a formalized benchmark through the BASALT Evaluation and Demonstrations Dataset (BEDD), which serves as a resource for algorithm development and performance assessment. BEDD consists of a collection of 26 million image-action pairs from nearly 14,000 videos of human players completing the BASALT tasks in Minecraft. It also includes over 3,000 dense pairwise human evaluations of human and algorithmic agents. These comparisons serve as a fixed, preliminary leaderboard for evaluating newly-developed algorithms. To enable this comparison, we present a streamlined codebase for benchmarking new algorithms against the leaderboard. In addition to presenting these datasets, we conduct a detailed analysis of the data from both datasets to guide algorithm development and evaluation. The released code and data are available at https://github.com/minerllabs/basalt-benchmark .

We present lambeq, the first high-level Python library for Quantum Natural Language Processing (QNLP). The open-source toolkit offers a detailed hierarchy of modules and classes implementing all stages of a pipeline for converting sentences to string diagrams, tensor networks, and quantum circuits ready to be used on a quantum computer. lambeq supports syntactic parsing, rewriting and simplification of string diagrams, ansatz creation and manipulation, as well as a number of compositional models for preparing quantum-friendly representations of sentences, employing various degrees of syntax sensitivity. We present the generic architecture and describe the most important modules in detail, demonstrating the usage with illustrative examples. Further, we test the toolkit in practice by using it to perform a number of experiments on simple NLP tasks, implementing both classical and quantum pipelines.

When agents are acting together, they may need a simple mechanism to decide on joint actions. One possibility is to have the agents express their preferences in the form of a ballot and use a voting rule to decide the winning action(s). Unfortunately, agents may try to manipulate such an election by misreporting their preferences. Fortunately, it has been shown that it is NP-hard to compute how to manipulate a number of different voting rules. However, NP-hardness only bounds the worst-case complexity. Recent theoretical results suggest that manipulation may often be easy in practice. To address this issue, I suggest studying empirically if computational complexity is in practice a barrier to manipulation. The basic tool used in my investigations is the identification of computational "phase transitions". Such an approach has been fruitful in identifying hard instances of propositional satisfiability and other NP-hard problems. I show that phase transition behaviour gives insight into the hardness of manipulating voting rules, increasing concern that computational complexity is indeed any sort of barrier. Finally, I look at the problem of computing manipulation of other, related problems like stable marriage and tournament problems.

We present PutnamBench, a new multilingual benchmark for evaluating the ability of neural theorem-provers to solve competition mathematics problems. PutnamBench consists of 1697 hand-constructed formalizations of 640 theorems sourced from the William Lowell Putnam Mathematical Competition, the premier undergraduate-level mathematics competition in North America. All the theorems have formalizations in Lean 4 and Isabelle; a substantial subset also has Coq formalizations. Proving the theorems requires significant problem-solving ability and proficiency in a broad range of topics taught in undergraduate mathematics courses. We use PutnamBench to evaluate several established neural and symbolic theorem-provers. These approaches can only solve a handful of the PutnamBench problems, establishing the benchmark as a difficult open challenge for research on neural theorem-proving. PutnamBench is available at https://github.com/trishullab/PutnamBench.

Integer sequences are of central importance to the modeling of concepts admitting complete finitary descriptions. We introduce a novel view on the learning of such concepts and lay down a set of benchmarking tasks aimed at conceptual understanding by machine learning models. These tasks indirectly assess model ability to abstract, and challenge them to reason both interpolatively and extrapolatively from the knowledge gained by observing representative examples. To further aid research in knowledge representation and reasoning, we present FACT, the Finitary Abstraction Comprehension Toolkit. The toolkit surrounds a large dataset of integer sequences comprising both organic and synthetic entries, a library for data pre-processing and generation, a set of model performance evaluation tools, and a collection of baseline model implementations, enabling the making of the future advancements with ease.

In this paper, we present improved learning-augmented algorithms for the multi-option ski rental problem. Learning-augmented algorithms take ML predictions as an added part of the input and incorporates these predictions in solving the given problem. Due to their unique strength that combines the power of ML predictions with rigorous performance guarantees, they have been extensively studied in the context of online optimization problems. Even though ski rental problems are one of the canonical problems in the field of online optimization, only deterministic algorithms were previously known for multi-option ski rental, with or without learning augmentation. We present the first randomized learning-augmented algorithm for this problem, surpassing previous performance guarantees given by deterministic algorithms. Our learning-augmented algorithm is based on a new, provably best-possible randomized competitive algorithm for the problem. Our results are further complemented by lower bounds for deterministic and randomized algorithms, and computational experiments evaluating our algorithms' performance improvements.

There is growing excitement about the potential of Language Models (LMs) to accelerate scientific discovery. Falsifying hypotheses is key to scientific progress, as it allows claims to be iteratively refined over time. This process requires significant researcher effort, reasoning, and ingenuity. Yet current benchmarks for LMs predominantly assess their ability to generate solutions rather than challenge them. We advocate for developing benchmarks that evaluate this inverse capability - creating counterexamples for subtly incorrect solutions. To demonstrate this approach, we start with the domain of algorithmic problem solving, where counterexamples can be evaluated automatically using code execution. Specifically, we introduce REFUTE, a dynamically updating benchmark that includes recent problems and incorrect submissions from programming competitions, where human experts successfully identified counterexamples. Our analysis finds that the best reasoning agents, even OpenAI o3-mini (high) with code execution feedback, can create counterexamples for only <9% of incorrect solutions in REFUTE, even though ratings indicate its ability to solve up to 48% of these problems from scratch. We hope our work spurs progress in evaluating and enhancing LMs' ability to falsify incorrect solutions - a capability that is crucial for both accelerating research and making models self-improve through reliable reflective reasoning.

The paper focuses on deep learning semantic search algorithms applied in the HR domain. The aim of the article is developing a novel approach to training a Siamese network to link the skills mentioned in the job ad with the title. It has been shown that the title normalization process can be based either on classification or similarity comparison approaches. While classification algorithms strive to classify a sample into predefined set of categories, similarity search algorithms take a more flexible approach, since they are designed to find samples that are similar to a given query sample, without requiring pre-defined classes and labels. In this article semantic similarity search to find candidates for title normalization has been used. A pre-trained language model has been adapted while teaching it to match titles and skills based on co-occurrence information. For the purpose of this research fifty billion title-descriptions pairs had been collected for training the model and thirty three thousand title-description-normalized title triplets, where normalized job title was picked up manually by job ad creator for testing purposes. As baselines FastText, BERT, SentenceBert and JobBert have been used. As a metric of the accuracy of the designed algorithm is Recall in top one, five and ten model's suggestions. It has been shown that the novel training objective lets it achieve significant improvement in comparison to other generic and specific text encoders. Two settings with treating titles as standalone strings, and with included skills as additional features during inference have been used and the results have been compared in this article. Improvements by 10% and 21.5% have been achieved using VacancySBERT and VacancySBERT (with skills) respectively. The benchmark has been developed as open-source to foster further research in the area.

This paper presents EasyRAG, a simple, lightweight, and efficient retrieval-augmented generation framework for automated network operations. Our framework has three advantages. The first is accurate question answering. We designed a straightforward RAG scheme based on (1) a specific data processing workflow (2) dual-route sparse retrieval for coarse ranking (3) LLM Reranker for reranking (4) LLM answer generation and optimization. This approach achieved first place in the GLM4 track in the preliminary round and second place in the GLM4 track in the semifinals. The second is simple deployment. Our method primarily consists of BM25 retrieval and BGE-reranker reranking, requiring no fine-tuning of any models, occupying minimal VRAM, easy to deploy, and highly scalable; we provide a flexible code library with various search and generation strategies, facilitating custom process implementation. The last one is efficient inference. We designed an efficient inference acceleration scheme for the entire coarse ranking, reranking, and generation process that significantly reduces the inference latency of RAG while maintaining a good level of accuracy; each acceleration scheme can be plug-and-play into any component of the RAG process, consistently enhancing the efficiency of the RAG system. Our code and data are released at https://github.com/BUAADreamer/EasyRAG.

This paper presents the computational challenge on differential geometry and topology that happened within the ICLR 2021 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide creative contributions to the fields of computational geometry and topology through the open-source repositories Geomstats and Giotto-TDA. The challenge attracted 16 teams in its two month duration. This paper describes the design of the challenge and summarizes its main findings.

We study the accuracy of differentially private mechanisms in the continual release model. A continual release mechanism receives a sensitive dataset as a stream of T inputs and produces, after receiving each input, an accurate output on the obtained inputs. In contrast, a batch algorithm receives the data as one batch and produces a single output. We provide the first strong lower bounds on the error of continual release mechanisms. In particular, for two fundamental problems that are widely studied and used in the batch model, we show that the worst case error of every continual release algorithm is tilde Omega(T^{1/3}) times larger than that of the best batch algorithm. Previous work shows only a polylogarithimic (in T) gap between the worst case error achievable in these two models; further, for many problems, including the summation of binary attributes, the polylogarithmic gap is tight (Dwork et al., 2010; Chan et al., 2010). Our results show that problems closely related to summation -- specifically, those that require selecting the largest of a set of sums -- are fundamentally harder in the continual release model than in the batch model. Our lower bounds assume only that privacy holds for streams fixed in advance (the "nonadaptive" setting). However, we provide matching upper bounds that hold in a model where privacy is required even for adaptively selected streams. This model may be of independent interest.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter sigma > 0. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be O(sigma)-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as O(sigma)-approximation of the original BO, we propose first-order algorithms that find an epsilon-stationary solution by optimizing the penalty formulation with sigma = O(epsilon). When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an epsilon-stationary point of the penalty function, using in total O(epsilon^{-3}) and O(epsilon^{-7}) accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with O(1) samples per iteration, and achieves the improved oracle-complexity of O(epsilon^{-3}) and O(epsilon^{-5}), respectively.

There has been an increase of interest in code search using natural language. Assessing the performance of such code search models can be difficult without a readily available evaluation suite. In this paper, we present an evaluation dataset consisting of natural language query and code snippet pairs, with the hope that future work in this area can use this dataset as a common benchmark. We also provide the results of two code search models ([1] and [6]) from recent work. The evaluation dataset is available at https://github.com/facebookresearch/Neural-Code-Search-Evaluation-Dataset

We combine beam search with the probabilistic pruning technique of nucleus sampling to create two deterministic nucleus search algorithms for natural language generation. The first algorithm, p-exact search, locally prunes the next-token distribution and performs an exact search over the remaining space. The second algorithm, dynamic beam search, shrinks and expands the beam size according to the entropy of the candidate's probability distribution. Despite the probabilistic intuition behind nucleus search, experiments on machine translation and summarization benchmarks show that both algorithms reach the same performance levels as standard beam search.

Program synthesis with language models (LMs) has unlocked a large set of reasoning abilities; code-tuned LMs have proven adept at generating programs that solve a wide variety of algorithmic symbolic manipulation tasks (e.g. word concatenation). However, not all reasoning tasks are easily expressible as code, e.g. tasks involving commonsense reasoning, moral decision-making, and sarcasm understanding. Our goal is to extend an LM's program synthesis skills to such tasks and evaluate the results via pseudo-programs, namely Python programs where some leaf function calls are left undefined. To that end, we propose, Code Generation and Emulated EXecution (CoGEX). CoGEX works by (1) training LMs to generate their own pseudo-programs, (2) teaching them to emulate their generated program's execution, including those leaf functions, allowing the LM's knowledge to fill in the execution gaps; and (3) using them to search over many programs to find an optimal one. To adapt the CoGEX model to a new task, we introduce a method for performing program search to find a single program whose pseudo-execution yields optimal performance when applied to all the instances of a given dataset. We show that our approach yields large improvements compared to standard in-context learning approaches on a battery of tasks, both algorithmic and soft reasoning. This result thus demonstrates that code synthesis can be applied to a much broader class of problems than previously considered. Our released dataset, fine-tuned models, and implementation can be found at https://github.com/nweir127/CoGEX.

Large language models exhibit surprising emergent generalization properties, yet also struggle on many simple reasoning tasks such as arithmetic and parity. This raises the question of if and when Transformer models can learn the true algorithm for solving a task. We study the scope of Transformers' abilities in the specific setting of length generalization on algorithmic tasks. Here, we propose a unifying framework to understand when and how Transformers can exhibit strong length generalization on a given task. Specifically, we leverage RASP (Weiss et al., 2021) -- a programming language designed for the computational model of a Transformer -- and introduce the RASP-Generalization Conjecture: Transformers tend to length generalize on a task if the task can be solved by a short RASP program which works for all input lengths. This simple conjecture remarkably captures most known instances of length generalization on algorithmic tasks. Moreover, we leverage our insights to drastically improve generalization performance on traditionally hard tasks (such as parity and addition). On the theoretical side, we give a simple example where the "min-degree-interpolator" model of learning from Abbe et al. (2023) does not correctly predict Transformers' out-of-distribution behavior, but our conjecture does. Overall, our work provides a novel perspective on the mechanisms of compositional generalization and the algorithmic capabilities of Transformers.

Many graph algorithms can be viewed as sets of rules that are iteratively applied, with the number of iterations dependent on the size and complexity of the input graph. Existing machine learning architectures often struggle to represent these algorithmic decisions as discrete state transitions. Therefore, we propose a novel framework: GraphFSA (Graph Finite State Automaton). GraphFSA is designed to learn a finite state automaton that runs on each node of a given graph. We test GraphFSA on cellular automata problems, showcasing its abilities in a straightforward algorithmic setting. For a comprehensive empirical evaluation of our framework, we create a diverse range of synthetic problems. As our main application, we then focus on learning more elaborate graph algorithms. Our findings suggest that GraphFSA exhibits strong generalization and extrapolation abilities, presenting an alternative approach to represent these algorithms.

We present miniF2F, a dataset of formal Olympiad-level mathematics problems statements intended to provide a unified cross-system benchmark for neural theorem proving. The miniF2F benchmark currently targets Metamath, Lean, Isabelle (partially) and HOL Light (partially) and consists of 488 problem statements drawn from the AIME, AMC, and the International Mathematical Olympiad (IMO), as well as material from high-school and undergraduate mathematics courses. We report baseline results using GPT-f, a neural theorem prover based on GPT-3 and provide an analysis of its performance. We intend for miniF2F to be a community-driven effort and hope that our benchmark will help spur advances in neural theorem proving.

In the past decade, advances in deep learning have resulted in breakthroughs in a variety of areas, including computer vision, natural language understanding, speech recognition, and reinforcement learning. Specialized, high-performing neural architectures are crucial to the success of deep learning in these areas. Neural architecture search (NAS), the process of automating the design of neural architectures for a given task, is an inevitable next step in automating machine learning and has already outpaced the best human-designed architectures on many tasks. In the past few years, research in NAS has been progressing rapidly, with over 1000 papers released since 2020 (Deng and Lindauer, 2021). In this survey, we provide an organized and comprehensive guide to neural architecture search. We give a taxonomy of search spaces, algorithms, and speedup techniques, and we discuss resources such as benchmarks, best practices, other surveys, and open-source libraries.

The field of machine learning has taken a dramatic twist in recent times, with the rise of the Artificial Neural Network (ANN). These biologically inspired computational models are able to far exceed the performance of previous forms of artificial intelligence in common machine learning tasks. One of the most impressive forms of ANN architecture is that of the Convolutional Neural Network (CNN). CNNs are primarily used to solve difficult image-driven pattern recognition tasks and with their precise yet simple architecture, offers a simplified method of getting started with ANNs. This document provides a brief introduction to CNNs, discussing recently published papers and newly formed techniques in developing these brilliantly fantastic image recognition models. This introduction assumes you are familiar with the fundamentals of ANNs and machine learning.

The 2021 NeurIPS Machine Learning for Combinatorial Optimization (ML4CO) competition was designed with the goal of improving state-of-the-art combinatorial optimization solvers by replacing key heuristic components with machine learning models. The competition's main scientific question was the following: is machine learning a viable option for improving traditional combinatorial optimization solvers on specific problem distributions, when historical data is available? This was motivated by the fact that in many practical scenarios, the data changes only slightly between the repetitions of a combinatorial optimization problem, and this is an area where machine learning models are particularly powerful at. This paper summarizes the solution and lessons learned by the Huawei EI-OROAS team in the dual task of the competition. The submission of our team achieved the second place in the final ranking, with a very close distance to the first spot. In addition, our solution was ranked first consistently for several weekly leaderboard updates before the final evaluation. We provide insights gained from a large number of experiments, and argue that a simple Graph Convolutional Neural Network (GCNNs) can achieve state-of-the-art results if trained and tuned properly.

Recently, Offline Reinforcement Learning (RL) has achieved remarkable progress with the emergence of various algorithms and datasets. However, these methods usually focus on algorithmic advancements, ignoring that many low-level implementation choices considerably influence or even drive the final performance. As a result, it becomes hard to attribute the progress in Offline RL as these choices are not sufficiently discussed and aligned in the literature. In addition, papers focusing on a dataset (e.g., D4RL) often ignore algorithms proposed on another dataset (e.g., RL Unplugged), causing isolation among the algorithms, which might slow down the overall progress. Therefore, this work aims to bridge the gaps caused by low-level choices and datasets. To this end, we empirically investigate 20 implementation choices using three representative algorithms (i.e., CQL, CRR, and IQL) and present a guidebook for choosing implementations. Following the guidebook, we find two variants CRR+ and CQL+ , achieving new state-of-the-art on D4RL. Moreover, we benchmark eight popular offline RL algorithms across datasets under unified training and evaluation framework. The findings are inspiring: the success of a learning paradigm severely depends on the data distribution, and some previous conclusions are biased by the dataset used. Our code is available at https://github.com/sail-sg/offbench.

Recent reports claim that large language models (LLMs) now outperform elite humans in competitive programming. Drawing on knowledge from a group of medalists in international algorithmic contests, we revisit this claim, examining how LLMs differ from human experts and where limitations still remain. We introduce LiveCodeBench Pro, a benchmark composed of problems from Codeforces, ICPC, and IOI that are continuously updated to reduce the likelihood of data contamination. A team of Olympiad medalists annotates every problem for algorithmic categories and conducts a line-by-line analysis of failed model-generated submissions. Using this new data and benchmark, we find that frontier models still have significant limitations: without external tools, the best model achieves only 53% pass@1 on medium-difficulty problems and 0% on hard problems, domains where expert humans still excel. We also find that LLMs succeed at implementation-heavy problems but struggle with nuanced algorithmic reasoning and complex case analysis, often generating confidently incorrect justifications. High performance appears largely driven by implementation precision and tool augmentation, not superior reasoning. LiveCodeBench Pro thus highlights the significant gap to human grandmaster levels, while offering fine-grained diagnostics to steer future improvements in code-centric LLM reasoning.

Competitive programming benchmarks are widely used in scenarios such as programming contests and large language model assessments. However, the growing presence of duplicate or highly similar problems raises concerns not only about competition fairness, but also about the validity of competitive programming as a benchmark for model evaluation. In this paper, we propose a new problem -- similar question retrieval -- to address this issue. Due to the lack of both data and models, solving this problem is challenging. To this end, we introduce CPRet, a retrieval-oriented benchmark suite for competitive programming, covering four retrieval tasks: two code-centric (i.e., Text-to-Code and Code-to-Code) and two newly proposed problem-centric tasks (i.e., Problem-to-Duplicate and Simplified-to-Full), built from a combination of automatically crawled problem-solution data and manually curated annotations. Our contribution includes both high-quality training data and temporally separated test sets for reliable evaluation. In addition, we develop two task-specialized retrievers based on this dataset: CPRetriever-Code, trained with a novel Group-InfoNCE loss for problem-code alignment, and CPRetriever-Prob, fine-tuned for identifying problem-level similarity. Both models achieve strong results and are open-sourced for local use. Finally, we analyze LiveCodeBench and find that high-similarity problems inflate model pass rates and reduce differentiation, underscoring the need for similarity-aware evaluation in future benchmarks. Code and data are available at: https://github.com/coldchair/CPRet

This study evaluates large language models (LLMs) in generating code from algorithm descriptions from recent NLP papers. The task requires two key competencies: (1) algorithm comprehension: synthesizing information from papers and academic literature to understand implementation logic, and (2) coding expertise: identifying dependencies and correctly implementing necessary APIs. To facilitate rigorous evaluation, we introduce SciReplicate-Bench, a benchmark of 100 tasks from 36 NLP papers published in 2024, featuring detailed annotations and comprehensive test cases. Building on SciReplicate-Bench, we propose Sci-Reproducer, a multi-agent framework consisting of a Paper Agent that interprets algorithmic concepts from literature and a Code Agent that retrieves dependencies from repositories and implement solutions. To assess algorithm understanding, we introduce reasoning graph accuracy, which quantifies similarity between generated and reference reasoning graphs derived from code comments and structure. For evaluating implementation quality, we employ execution accuracy, CodeBLEU, and repository dependency/API recall metrics. In our experiments, we evaluate various powerful Non-Reasoning LLMs and Reasoning LLMs as foundational models. The best-performing LLM using Sci-Reproducer achieves only 39% execution accuracy, highlighting the benchmark's difficulty.Our analysis identifies missing or inconsistent algorithm descriptions as key barriers to successful reproduction. We will open-source our benchmark, and code at https://github.com/xyzCS/SciReplicate-Bench.

There is a compelling necessity from enterprises for fine tuning LLMs (Large Language Models) o get them trained on proprietary domain knowledge. The challenge is to imbibe the LLMs with domain specific knowledge using the most optimial resource and cost and in the best possible time. Many enterprises rely on RAG (Retrieval Augmented Generation) which does not need LLMs to be ine-tuned but they are limited by the quality of vector databases and their retrieval capabilities rather than the intrinsic capabilities of the LLMs themselves. In our current work we focus on fine tuning LLaMA, an open source LLM using proprietary documents and code from an enterprise repository and use the fine tuned models to evaluate the quality of responses. As part of this work, we aim to guide beginners on how to start with fine tuning an LLM for documentation and code by making educated guesses on size of GPU required and options that are available for formatting the data. We also propose pre processing recipes for both documentation and code to prepare dataset in different formats. The proposed methods of data preparation for document datasets are forming paragraph chunks, forming question and answer pairs and forming keyword and paragraph chunk pairs. For code dataset we propose forming summary and function pairs. Further, we qualitatively evaluate the results of the models for domain specific queries. Finally, we also propose practical guidelines and recommendations for fine tuning LLMs.

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.

Heuristics are widely used for dealing with complex search and optimization problems. However, manual design of heuristics can be often very labour extensive and requires rich working experience and knowledge. This paper proposes Evolution of Heuristic (EoH), a novel evolutionary paradigm that leverages both Large Language Models (LLMs) and Evolutionary Computation (EC) methods for Automatic Heuristic Design (AHD). EoH represents the ideas of heuristics in natural language, termed thoughts. They are then translated into executable codes by LLMs. The evolution of both thoughts and codes in an evolutionary search framework makes it very effective and efficient for generating high-performance heuristics. Experiments on three widely studied combinatorial optimization benchmark problems demonstrate that EoH outperforms commonly used handcrafted heuristics and other recent AHD methods including FunSearch. Particularly, the heuristic produced by EoH with a low computational budget (in terms of the number of queries to LLMs) significantly outperforms widely-used human hand-crafted baseline algorithms for the online bin packing problem.

The problem of minimizing the maximum of N convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring O(Nepsilon^{-2/3} + epsilon^{-8/3}) queries to a first-order oracle to compute an epsilon-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of N convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of O(Nepsilon^{-5/3} + epsilon^{-8/3}). On the other hand, we prove that quantum algorithms must take Omega(Nepsilon^{-2/3}) queries to a first order quantum oracle, showing that our dependence on N is optimal up to poly-logarithmic factors.

This paper presents NT-Java-1.1B, an open-source specialized code language model built on StarCoderBase-1.1B, designed for coding tasks in Java programming. NT-Java-1.1B achieves state-of-the-art performance, surpassing its base model and majority of other models of similar size on MultiPL-E Java code benchmark. While there have been studies on extending large, generic pre-trained models to improve proficiency in specific programming languages like Python, similar investigations on small code models for other programming languages are lacking. Large code models require specialized hardware like GPUs for inference, highlighting the need for research into building small code models that can be deployed on developer desktops. This paper addresses this research gap by focusing on the development of a small Java code model, NT-Java-1.1B, and its quantized versions, which performs comparably to open models around 1.1B on MultiPL-E Java code benchmarks, making them ideal for desktop deployment. This paper establishes the foundation for specialized models across languages and sizes for a family of NT Models.

Given an unconditional diffusion model and a predictor for a target property of interest (e.g., a classifier), the goal of training-free guidance is to generate samples with desirable target properties without additional training. Existing methods, though effective in various individual applications, often lack theoretical grounding and rigorous testing on extensive benchmarks. As a result, they could even fail on simple tasks, and applying them to a new problem becomes unavoidably difficult. This paper introduces a novel algorithmic framework encompassing existing methods as special cases, unifying the study of training-free guidance into the analysis of an algorithm-agnostic design space. Via theoretical and empirical investigation, we propose an efficient and effective hyper-parameter searching strategy that can be readily applied to any downstream task. We systematically benchmark across 7 diffusion models on 16 tasks with 40 targets, and improve performance by 8.5% on average. Our framework and benchmark offer a solid foundation for conditional generation in a training-free manner.

Recent work has shown that leveraging learned predictions can improve the running time of algorithms for bipartite matching and similar combinatorial problems. In this work, we build on this idea to improve the performance of the widely used Ford-Fulkerson algorithm for computing maximum flows by seeding Ford-Fulkerson with predicted flows. Our proposed method offers strong theoretical performance in terms of the quality of the prediction. We then consider image segmentation, a common use-case of flows in computer vision, and complement our theoretical analysis with strong empirical results.

This paper presents the second ChatGPT4PCG competition at the 2024 IEEE Conference on Games. In this edition of the competition, we follow the first edition, but make several improvements and changes. We introduce a new evaluation metric along with allowing a more flexible format for participants' submissions and making several improvements to the evaluation pipeline. Continuing from the first edition, we aim to foster and explore the realm of prompt engineering (PE) for procedural content generation (PCG). While the first competition saw success, it was hindered by various limitations; we aim to mitigate these limitations in this edition. We introduce diversity as a new metric to discourage submissions aimed at producing repetitive structures. Furthermore, we allow submission of a Python program instead of a prompt text file for greater flexibility in implementing advanced PE approaches, which may require control flow, including conditions and iterations. We also make several improvements to the evaluation pipeline with a better classifier for similarity evaluation and better-performing function signatures. We thoroughly evaluate the effectiveness of the new metric and the improved classifier. Additionally, we perform an ablation study to select a function signature to instruct ChatGPT for level generation. Finally, we provide implementation examples of various PE techniques in Python and evaluate their preliminary performance. We hope this competition serves as a resource and platform for learning about PE and PCG in general.

This paper develops a new mathematical-statistical approach to analyze a class of Flajolet-Martin algorithms (FMa), and provides analytical confidence intervals for the number F0 of distinct elements in a stream, based on Chernoff bounds. The class of FMa has reached a significant popularity in bigdata stream learning, and the attention of the literature has mainly been based on algorithmic aspects, basically complexity optimality, while the statistical analysis of these class of algorithms has been often faced heuristically. The analysis provided here shows deep connections with mathematical special functions and with extreme value theory. The latter connection may help in explaining heuristic considerations, while the first opens many numerical issues, faced at the end of the present paper. Finally, the algorithms are tested on an anonymized real data stream and MonteCarlo simulations are provided to support our analytical choice in this context.

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.

Dynamic programming (DP) algorithms for combinatorial optimization problems work with taking maximization, minimization, and classical addition in their recursion algorithms. The associated value functions correspond to convex polyhedra in the max plus semiring. Existing Neural Algorithmic Reasoning models, however, rely on softmax-normalized dot-product attention where the smooth exponential weighting blurs these sharp polyhedral structures and collapses when evaluated on out-of-distribution (OOD) settings. We introduce Tropical attention, a novel attention function that operates natively in the max-plus semiring of tropical geometry. We prove that Tropical attention can approximate tropical circuits of DP-type combinatorial algorithms. We then propose that using Tropical transformers enhances empirical OOD performance in both length generalization and value generalization, on algorithmic reasoning tasks, surpassing softmax baselines while remaining stable under adversarial attacks. We also present adversarial-attack generalization as a third axis for Neural Algorithmic Reasoning benchmarking. Our results demonstrate that Tropical attention restores the sharp, scale-invariant reasoning absent from softmax.

We consider the problem of fairly allocating a sequence of indivisible items that arrive online in an arbitrary order to a group of n agents with additive normalized valuation functions. We consider both the allocation of goods and chores and propose algorithms for approximating maximin share (MMS) allocations. When agents have identical valuation functions the problem coincides with the semi-online machine covering problem (when items are goods) and load balancing problem (when items are chores), for both of which optimal competitive ratios have been achieved. In this paper, we consider the case when agents have general additive valuation functions. For the allocation of goods, we show that no competitive algorithm exists even when there are only three agents and propose an optimal 0.5-competitive algorithm for the case of two agents. For the allocation of chores, we propose a (2-1/n)-competitive algorithm for n>=3 agents and a square root of 2 (approximately 1.414)-competitive algorithm for two agents. Additionally, we show that no algorithm can do better than 15/11 (approximately 1.364)-competitive for two agents.

It is almost always easier to find an accurate-but-complex model than an accurate-yet-simple model. Finding optimal, sparse, accurate models of various forms (linear models with integer coefficients, decision sets, rule lists, decision trees) is generally NP-hard. We often do not know whether the search for a simpler model will be worthwhile, and thus we do not go to the trouble of searching for one. In this work, we ask an important practical question: can accurate-yet-simple models be proven to exist, or shown likely to exist, before explicitly searching for them? We hypothesize that there is an important reason that simple-yet-accurate models often do exist. This hypothesis is that the size of the Rashomon set is often large, where the Rashomon set is the set of almost-equally-accurate models from a function class. If the Rashomon set is large, it contains numerous accurate models, and perhaps at least one of them is the simple model we desire. In this work, we formally present the Rashomon ratio as a new gauge of simplicity for a learning problem, depending on a function class and a data set. The Rashomon ratio is the ratio of the volume of the set of accurate models to the volume of the hypothesis space, and it is different from standard complexity measures from statistical learning theory. Insight from studying the Rashomon ratio provides an easy way to check whether a simpler model might exist for a problem before finding it, namely whether several different machine learning methods achieve similar performance on the data. In that sense, the Rashomon ratio is a powerful tool for understanding why and when an accurate-yet-simple model might exist. If, as we hypothesize in this work, many real-world data sets admit large Rashomon sets, the implications are vast: it means that simple or interpretable models may often be used for high-stakes decisions without losing accuracy.

Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.

Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.

Recovering causal relationships from data is an important problem. Using observational data, one can typically only recover causal graphs up to a Markov equivalence class and additional assumptions or interventional data are needed for complete recovery. In this work, under some standard assumptions, we study causal graph discovery via adaptive interventions with node-dependent interventional costs. For this setting, we show that no algorithm can achieve an approximation guarantee that is asymptotically better than linear in the number of vertices with respect to the verification number; a well-established benchmark for adaptive search algorithms. Motivated by this negative result, we define a new benchmark that captures the worst-case interventional cost for any search algorithm. Furthermore, with respect to this new benchmark, we provide adaptive search algorithms that achieve logarithmic approximations under various settings: atomic, bounded size interventions and generalized cost objectives.

Context: Recent research indicates that Web queries written by software developers are not very successful in retrieving relevant results, performing measurably worse compared to general purpose Web queries. Most approaches up to this point have addressed this problem with software engineering-specific automated query reformulation techniques, which work without developer involvement but are limited by the content of the original query. In other words, these techniques automatically improve the existing query but can not contribute new, previously unmentioned, concepts. Objective: In this paper, we propose a technique to guide software developers in manually improving their own Web search queries. We examine a conversational approach that follows unsuccessful queries with a clarification question aimed at eliciting additional query terms, thus providing to the developer a clear dimension along which the query could be improved. Methods: We describe a set of clarification questions derived from a corpus of software developer queries and a neural approach to recommending them for a newly issued query. Results: Our evaluation indicates that the recommendation technique is accurate, predicting a valid clarification question 80% of the time and outperforms simple baselines, as well as, state-of-the-art Learning To Rank (LTR) baselines. Conclusion: As shown in the experimental results, the described approach is capable at recommending appropriate clarification questions to software developers and considered useful by a sample of developers ranging from novices to experienced professionals.

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.

The original Grover's algorithm suffers from the souffle problem, which means that the success probability of quantum search decreases dramatically if the iteration time is too small or too large from the right time. To overcome the souffle problem, the fixed-point quantum search with an optimal number of queries was proposed [Phys. Rev. Lett. 113, 210501 (2014)], which always finds a marked state with a high probability when a lower bound of the proportion of marked states is given. The fixed-point quantum search relies on a key lemma regarding the explicit formula of recursive quasi-Chebyshev polynomials, but its proof is not given explicitly. In this work, we give a detailed proof of this lemma, thus providing a sound foundation for the correctness of the fixed-point quantum search. This lemma may be of independent interest as well, since it expands the mathematical form of the recursive relation of Chebyshev polynomials of the first kind, and it also constitutes a key component in overcoming the souffle problem of quantum walk-based search algorithms, for example, robust quantum walk search on complete bipartite graphs [Phys. Rev. A 106, 052207 (2022)]. Hopefully, more applications of the lemma will be found in the future.

Recently, the retrieval models based on dense representations have been gradually applied in the first stage of the document retrieval tasks, showing better performance than traditional sparse vector space models. To obtain high efficiency, the basic structure of these models is Bi-encoder in most cases. However, this simple structure may cause serious information loss during the encoding of documents since the queries are agnostic. To address this problem, we design a method to mimic the queries on each of the documents by an iterative clustering process and represent the documents by multiple pseudo queries (i.e., the cluster centroids). To boost the retrieval process using approximate nearest neighbor search library, we also optimize the matching function with a two-step score calculation procedure. Experimental results on several popular ranking and QA datasets show that our model can achieve state-of-the-art results.

Which transformer scaling regimes are able to perfectly solve different classes of algorithmic problems? While tremendous empirical advances have been attained by transformer-based neural networks, a theoretical understanding of their algorithmic reasoning capabilities in realistic parameter regimes is lacking. We investigate this question in terms of the network's depth, width, and number of extra tokens for algorithm execution. Our novel representational hierarchy separates 9 algorithmic reasoning problems into classes solvable by transformers in different realistic parameter scaling regimes. We prove that logarithmic depth is necessary and sufficient for tasks like graph connectivity, while single-layer transformers with small embedding dimensions can solve contextual retrieval tasks. We also support our theoretical analysis with ample empirical evidence using the GraphQA benchmark. These results show that transformers excel at many graph reasoning tasks, even outperforming specialized graph neural networks.

How well do AI systems perform in algorithm engineering for hard optimization problems in domains such as package-delivery routing, crew scheduling, factory production planning, and power-grid balancing? We introduce ALE-Bench, a new benchmark for evaluating AI systems on score-based algorithmic programming contests. Drawing on real tasks from the AtCoder Heuristic Contests, ALE-Bench presents optimization problems that are computationally hard and admit no known exact solution. Unlike short-duration, pass/fail coding benchmarks, ALE-Bench encourages iterative solution refinement over long time horizons. Our software framework supports interactive agent architectures that leverage test-run feedback and visualizations. Our evaluation of frontier LLMs revealed that while they demonstrate high performance on specific problems, a notable gap remains compared to humans in terms of consistency across problems and long-horizon problem-solving capabilities. This highlights the need for this benchmark to foster future AI advancements.

More and more processes governing our lives use in some part an automatic decision step, where -- based on a feature vector derived from an applicant -- an algorithm has the decision power over the final outcome. Here we present a simple idea which gives some of the power back to the applicant by providing her with alternatives which would make the decision algorithm decide differently. It is based on a formalization reminiscent of methods used for evasion attacks, and consists in enumerating the subspaces where the classifiers decides the desired output. This has been implemented for the specific case of decision forests (ensemble methods based on decision trees), mapping the problem to an iterative version of enumerating k-cliques.

Neural Architecture Search (NAS) is an exciting new field which promises to be as much as a game-changer as Convolutional Neural Networks were in 2012. Despite many great works leading to substantial improvements on a variety of tasks, comparison between different methods is still very much an open issue. While most algorithms are tested on the same datasets, there is no shared experimental protocol followed by all. As such, and due to the under-use of ablation studies, there is a lack of clarity regarding why certain methods are more effective than others. Our first contribution is a benchmark of 8 NAS methods on 5 datasets. To overcome the hurdle of comparing methods with different search spaces, we propose using a method's relative improvement over the randomly sampled average architecture, which effectively removes advantages arising from expertly engineered search spaces or training protocols. Surprisingly, we find that many NAS techniques struggle to significantly beat the average architecture baseline. We perform further experiments with the commonly used DARTS search space in order to understand the contribution of each component in the NAS pipeline. These experiments highlight that: (i) the use of tricks in the evaluation protocol has a predominant impact on the reported performance of architectures; (ii) the cell-based search space has a very narrow accuracy range, such that the seed has a considerable impact on architecture rankings; (iii) the hand-designed macro-structure (cells) is more important than the searched micro-structure (operations); and (iv) the depth-gap is a real phenomenon, evidenced by the change in rankings between 8 and 20 cell architectures. To conclude, we suggest best practices, that we hope will prove useful for the community and help mitigate current NAS pitfalls. The code used is available at https://github.com/antoyang/NAS-Benchmark.

Premise selection is a crucial yet challenging step in mathematical formalization, especially for users with limited experience. Due to the lack of available formalization projects, existing approaches that leverage language models often suffer from data scarcity. In this work, we introduce an innovative method for training a premise retriever to support the formalization of mathematics. Our approach employs a BERT model to embed proof states and premises into a shared latent space. The retrieval model is trained within a contrastive learning framework and incorporates a domain-specific tokenizer along with a fine-grained similarity computation method. Experimental results show that our model is highly competitive compared to existing baselines, achieving strong performance while requiring fewer computational resources. Performance is further enhanced through the integration of a re-ranking module. To streamline the formalization process, we will release a search engine that enables users to query Mathlib theorems directly using proof states, significantly improving accessibility and efficiency. Codes are available at https://github.com/ruc-ai4math/Premise-Retrieval.

Code quality is and will be a crucial factor while developing new software code, requiring appropriate tools to ensure functional and reliable code. Machine learning techniques are still rarely used for software engineering tools, missing out the potential benefits of its application. Natural language processing has shown the potential to process text data regarding a variety of tasks. We argue, that such models can also show similar benefits for software code processing. In this paper, we investigate how models used for natural language processing can be trained upon software code. We introduce a data retrieval pipeline for software code and train a model upon Java software code. The resulting model, JavaBERT, shows a high accuracy on the masked language modeling task showing its potential for software engineering tools.

We extend conformal prediction to control the expected value of any monotone loss function. The algorithm generalizes split conformal prediction together with its coverage guarantee. Like conformal prediction, the conformal risk control procedure is tight up to an O(1/n) factor. We also introduce extensions of the idea to distribution shift, quantile risk control, multiple and adversarial risk control, and expectations of U-statistics. Worked examples from computer vision and natural language processing demonstrate the usage of our algorithm to bound the false negative rate, graph distance, and token-level F1-score.

Several recent advances in AI systems (e.g., Tree-of-Thoughts and Program-Aided Language Models) solve problems by providing a "scaffolding" program that structures multiple calls to language models to generate better outputs. A scaffolding program is written in a programming language such as Python. In this work, we use a language-model-infused scaffolding program to improve itself. We start with a seed "improver" that improves an input program according to a given utility function by querying a language model several times and returning the best solution. We then run this seed improver to improve itself. Across a small set of downstream tasks, the resulting improved improver generates programs with significantly better performance than its seed improver. Afterward, we analyze the variety of self-improvement strategies proposed by the language model, including beam search, genetic algorithms, and simulated annealing. Since the language models themselves are not altered, this is not full recursive self-improvement. Nonetheless, it demonstrates that a modern language model, GPT-4 in our proof-of-concept experiments, is capable of writing code that can call itself to improve itself. We critically consider concerns around the development of self-improving technologies and evaluate the frequency with which the generated code bypasses a sandbox.

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce well-structured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems.

Proof-oriented programs mix computational content with proofs of program correctness. However, the human effort involved in programming and proving is still substantial, despite the use of Satisfiability Modulo Theories (SMT) solvers to automate proofs in languages such as F*. Seeking to spur research on using AI to automate the construction of proof-oriented programs, we curate a dataset of 600K lines of open-source F* programs and proofs, including software used in production systems ranging from Windows and Linux, to Python and Firefox. Our dataset includes around 32K top-level F* definitions, each representing a type-directed program and proof synthesis problem -- producing a definition given a formal specification expressed as an F* type. We provide a program-fragment checker that queries F* to check the correctness of candidate solutions. We believe this is the largest corpus of SMT-assisted program proofs coupled with a reproducible program-fragment checker. Grounded in this dataset, we investigate the use of AI to synthesize programs and their proofs in F*, with promising results. Our main finding in that the performance of fine-tuned smaller language models (such as Phi-2 or StarCoder) compare favorably with large language models (such as GPT-4), at a much lower computational cost. We also identify various type-based retrieval augmentation techniques and find that they boost performance significantly. With detailed error analysis and case studies, we identify potential strengths and weaknesses of models and techniques and suggest directions for future improvements.

We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve similar OT problems between different measures by leveraging the knowledge and information present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and suboptimally re-solve each problem from scratch. We instantiate Meta OT models in discrete and continuous settings between grayscale images, spherical data, classification labels, and color palettes and use them to improve the computational time of standard OT solvers. Our source code is available at http://github.com/facebookresearch/meta-ot

Instruction tuning -- tuning large language models on instruction-output pairs -- is a promising technique for making models better adapted to the real world. Yet, the key factors driving the model's capability to understand and follow instructions not seen during training remain under-explored. Our investigation begins with a series of synthetic experiments within the theoretical framework of a Turing-complete algorithm called Markov algorithm, which allows fine-grained control over the instruction-tuning data. Generalization and robustness with respect to the training distribution emerge once a diverse enough set of tasks is provided, even though very few examples are provided for each task. We extend these initial results to a real-world application scenario of code generation and find that a more diverse instruction set, extending beyond code-related tasks, improves the performance of code generation. Our observations suggest that a more diverse semantic space for instruction-tuning sets greatly improves the model's ability to follow instructions and perform tasks.