Daily Papers

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 study the optimisation problem associated with Gaussian process regression using squared loss. The most common approach to this problem is to apply an exact solver, such as conjugate gradient descent, either directly, or to a reduced-order version of the problem. Recently, driven by successes in deep learning, stochastic gradient descent has gained traction as an alternative. In this paper, we show that when done rightx2014by which we mean using specific insights from the optimisation and kernel communitiesx2014this approach is highly effective. We thus introduce a particular stochastic dual gradient descent algorithm, that may be implemented with a few lines of code using any deep learning framework. We explain our design decisions by illustrating their advantage against alternatives with ablation studies and show that the new method is highly competitive. Our evaluations on standard regression benchmarks and a Bayesian optimisation task set our approach apart from preconditioned conjugate gradients, variational Gaussian process approximations, and a previous version of stochastic gradient descent for Gaussian processes. On a molecular binding affinity prediction task, our method places Gaussian process regression on par in terms of performance with state-of-the-art graph neural networks.

This study proposes a hybrid deep-learning-metaheuristic framework with a bi-level architecture for road network design problems (NDPs). We train a graph neural network (GNN) to approximate the solution of the user equilibrium (UE) traffic assignment problem and use inferences made by the trained model to calculate fitness function evaluations of a genetic algorithm (GA) to approximate solutions for NDPs. Using three test networks, two NDP variants and an exact solver as benchmark, we show that on average, our proposed framework can provide solutions within 1.5% gap of the best results in less than 0.5% of the time used by the exact solution procedure. Our framework can be utilized within an expert system for infrastructure planning to determine the best infrastructure planning and management decisions under different scenarios. Given the flexibility of the framework, it can easily be adapted to many other decision problems that can be modeled as bi-level problems on graphs. Moreover, we foreseen interesting future research directions, thus we also put forward a brief research agenda for this topic. The key observation from our research that can shape future research is that the fitness function evaluation time using the inferences made by the GNN model was in the order of milliseconds, which points to an opportunity and a need for novel heuristics that 1) can cope well with noisy fitness function values provided by deep learning models, and 2) can use the significantly enlarged efficiency of the evaluation step to explore the search space effectively (rather than efficiently). This opens a new avenue for a modern class of metaheuristics that are crafted for use with AI-powered predictors.

Diffusion probabilistic models (DPMs) are emerging powerful generative models. Despite their high-quality generation performance, DPMs still suffer from their slow sampling as they generally need hundreds or thousands of sequential function evaluations (steps) of large neural networks to draw a sample. Sampling from DPMs can be viewed alternatively as solving the corresponding diffusion ordinary differential equations (ODEs). In this work, we propose an exact formulation of the solution of diffusion ODEs. The formulation analytically computes the linear part of the solution, rather than leaving all terms to black-box ODE solvers as adopted in previous works. By applying change-of-variable, the solution can be equivalently simplified to an exponentially weighted integral of the neural network. Based on our formulation, we propose DPM-Solver, a fast dedicated high-order solver for diffusion ODEs with the convergence order guarantee. DPM-Solver is suitable for both discrete-time and continuous-time DPMs without any further training. Experimental results show that DPM-Solver can generate high-quality samples in only 10 to 20 function evaluations on various datasets. We achieve 4.70 FID in 10 function evaluations and 2.87 FID in 20 function evaluations on the CIFAR10 dataset, and a 4sim 16times speedup compared with previous state-of-the-art training-free samplers on various datasets.

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

Recently, various methods have been proposed to address the inconsistency issue of DDIM inversion to enable image editing, such as EDICT [36] and Null-text inversion [22]. However, the above methods introduce considerable computational overhead. In this paper, we propose a new technique, named bi-directional integration approximation (BDIA), to perform exact diffusion inversion with neglible computational overhead. Suppose we would like to estimate the next diffusion state z_{i-1} at timestep t_i with the historical information (i,z_i) and (i+1,z_{i+1}). We first obtain the estimated Gaussian noise boldsymbol{epsilon}(z_i,i), and then apply the DDIM update procedure twice for approximating the ODE integration over the next time-slot [t_i, t_{i-1}] in the forward manner and the previous time-slot [t_i, t_{t+1}] in the backward manner. The DDIM step for the previous time-slot is used to refine the integration approximation made earlier when computing z_i. A nice property of BDIA-DDIM is that the update expression for z_{i-1} is a linear combination of (z_{i+1}, z_i, boldsymbol{epsilon}(z_i,i)). This allows for exact backward computation of z_{i+1} given (z_i, z_{i-1}), thus leading to exact diffusion inversion. It is demonstrated with experiments that (round-trip) BDIA-DDIM is particularly effective for image editing. Our experiments further show that BDIA-DDIM produces markedly better image sampling qualities than DDIM for text-to-image generation. BDIA can also be applied to improve the performance of other ODE solvers in addition to DDIM. In our work, it is found that applying BDIA to the EDM sampling procedure produces consistently better performance over four pre-trained models.

Large linear systems play an important role in high-energy theory, appearing in amplitude bootstraps and during integral reduction. This paper introduces FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear systems over the rationals. The solver interfaces directly with Mathematica, is straightforward to install, and seamlessly replaces Mathematica's native solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude faster than Mathematica and uses an order of magnitude less memory. The package also compares favorably against other public solvers in FiniteFieldSolve's intended use cases. As the name of the package suggests, solutions are obtained via well-known finite field methods. These methods suffer from introducing an inordinate number of modulo (or integer division) operations with respect to different primes. By automatically recompiling itself for each prime, FiniteFieldSolve converts the division operations into much faster combinations of instructions, dramatically improving performance. The technique of compiling the prime can be applied to any finite field solver, where the time savings will be solver dependent. The operation of the package is illustrated through a detailed example of an amplitude bootstrap.

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.

Automated math word problem solvers based on neural networks have successfully managed to obtain 70-80\% accuracy in solving arithmetic word problems. However, it has been shown that these solvers may rely on superficial patterns to obtain their equations. In order to determine what information math word problem solvers use to generate solutions, we remove parts of the input and measure the model's performance on the perturbed dataset. Our results show that the model is not sensitive to the removal of many words from the input and can still manage to find a correct answer when given a nonsense question. This indicates that automatic solvers do not follow the semantic logic of math word problems, and may be overfitting to the presence of specific words.

Large language models (LLMs) have exhibited great potential in mathematical reasoning. However, there remains a performance gap in this area between existing open-source models and closed-source models such as GPT-4. In this paper, we introduce MathGenie, a novel method for generating diverse and reliable math problems from a small-scale problem-solution dataset (denoted as seed data). We augment the ground-truth solutions of our seed data and train a back-translation model to translate the augmented solutions back into new questions. Subsequently, we generate code-integrated solutions for the new questions. To ensure the correctness of the code-integrated solutions, we employ rationale-based strategy for solution verification. Various pretrained models, ranging from 7B to 70B, are trained on the newly curated data to test the effectiveness of the proposed augmentation technique, resulting in a family of models known as MathGenieLM. These models consistently outperform previous open-source models across five representative mathematical reasoning datasets, achieving state-of-the-art performance. In particular, MathGenieLM-InternLM2 achieves an accuracy of 87.7% on GSM8K and 55.7% on MATH, securing the best overall score among open-source language models.

The recently released GPT-4 Code Interpreter has demonstrated remarkable proficiency in solving challenging math problems, primarily attributed to its ability to seamlessly reason with natural language, generate code, execute code, and continue reasoning based on the execution output. In this paper, we present a method to fine-tune open-source language models, enabling them to use code for modeling and deriving math equations and, consequently, enhancing their mathematical reasoning abilities. We propose a method of generating novel and high-quality datasets with math problems and their code-based solutions, referred to as MathCodeInstruct. Each solution interleaves natural language, code, and execution results. We also introduce a customized supervised fine-tuning and inference approach. This approach yields the MathCoder models, a family of models capable of generating code-based solutions for solving challenging math problems. Impressively, the MathCoder models achieve state-of-the-art scores among open-source LLMs on the MATH (45.2%) and GSM8K (83.9%) datasets, substantially outperforming other open-source alternatives. Notably, the MathCoder model not only surpasses ChatGPT-3.5 and PaLM-2 on GSM8K and MATH but also outperforms GPT-4 on the competition-level MATH dataset. The dataset and models will be released at https://github.com/mathllm/MathCoder.

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.

While Large Language Models (LLMs) demonstrate impressive performance in mathematics, existing math benchmarks come with significant limitations. Many focus on problems with fixed ground-truth answers, and are often saturated due to problem simplicity or the viability of guessing or memorization. Crucially, they capture only a narrow subset of relevant math problems. To address this research gap, we introduce \mc, a new benchmark of 126 challenging problems sourced from various math competitions, which targets constructive proofs, a widely encountered problem type requiring the construction of mathematical objects with specific properties. These proofs are particularly suitable for LLM evaluation, as solution correctness can be easily verified. Our automated verifiers also enable MathConstruct to generate problem variations, used to evaluate robustness. State-of-the-art LLMs solve only 54% of MathConstruct problems, highlighting its complexity and importance for LLM evaluation.

Large Language Models (LLMs) have demonstrated remarkable performance in solving math problems, a hallmark of human intelligence. Despite high success rates on current benchmarks; however, these often feature simple problems with only one or two unknowns, which do not sufficiently challenge their reasoning capacities. This paper introduces a novel benchmark, BeyondX, designed to address these limitations by incorporating problems with multiple unknowns. Recognizing the challenges in proposing multi-unknown problems from scratch, we developed BeyondX using an innovative automated pipeline that progressively increases complexity by expanding the number of unknowns in simpler problems. Empirical study on BeyondX reveals that the performance of existing LLMs, even those fine-tuned specifically on math tasks, significantly decreases as the number of unknowns increases - with a performance drop of up to 70\% observed in GPT-4. To tackle these challenges, we propose the Formulate-and-Solve strategy, a generalized prompting approach that effectively handles problems with an arbitrary number of unknowns. Our findings reveal that this strategy not only enhances LLM performance on the BeyondX benchmark but also provides deeper insights into the computational limits of LLMs when faced with more complex mathematical challenges.

Diffusion models have demonstrated remarkable generation quality but at the cost of numerous function evaluations. Recently, advanced ODE-based solvers have been developed to mitigate the substantial computational demands of reverse-diffusion solving under limited sampling steps. However, these solvers, heavily inspired by Adams-like multistep methods, rely solely on t-related Lagrange interpolation. We show that t-related Lagrange interpolation is suboptimal for diffusion model and reveal a compact search space comprised of time steps and solver coefficients. Building on our analysis, we propose a novel differentiable solver search algorithm to identify more optimal solver. Equipped with the searched solver, rectified-flow models, e.g., SiT-XL/2 and FlowDCN-XL/2, achieve FID scores of 2.40 and 2.35, respectively, on ImageNet256 with only 10 steps. Meanwhile, DDPM model, DiT-XL/2, reaches a FID score of 2.33 with only 10 steps. Notably, our searched solver outperforms traditional solvers by a significant margin. Moreover, our searched solver demonstrates generality across various model architectures, resolutions, and model sizes.

EasyMath is a compact benchmark for practical math reasoning in small language models. It covers thirteen categories, from basic arithmetic and order of operations to word problems, algebraic expressions, edge cases, and omits specialist topics. We tested 23 models (14M to 4B parameters) using exact, numerical, and symbolic checks on free-form answers in a zero-shot setting. Accuracy rises with size and training, chain-of-thought adds modest gains, and consistency improves at scale.

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/

Recent math benchmarks for large language models (LLMs) such as MathArena indicate that state-of-the-art reasoning models achieve impressive performance on mathematical competitions like AIME, with the leading model, o3-mini, achieving scores comparable to top human competitors. However, these benchmarks evaluate models solely based on final numerical answers, neglecting rigorous reasoning and proof generation which are essential for real-world mathematical tasks. To address this, we introduce the first comprehensive evaluation of full-solution reasoning for challenging mathematical problems. Using expert human annotators, we evaluated several state-of-the-art reasoning models on the six problems from the 2025 USAMO within hours of their release. Our results reveal that all tested models struggled significantly, achieving less than 5% on average. Through detailed analysis of reasoning traces, we identify the most common failure modes and find several unwanted artifacts arising from the optimization strategies employed during model training. Overall, our results suggest that current LLMs are inadequate for rigorous mathematical reasoning tasks, highlighting the need for substantial improvements in reasoning and proof generation capabilities.

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.

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.

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.

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.

The ability of large language models to solve complex mathematical problems has progressed significantly, particularly for tasks requiring advanced reasoning. However, the scarcity of sufficiently challenging problems, particularly at the Olympiad level, hinders further advancements. In this work, we introduce PromptCoT, a novel approach for automatically generating high-quality Olympiad-level math problems. The proposed method synthesizes complex problems based on mathematical concepts and the rationale behind problem construction, emulating the thought processes of experienced problem designers. We provide a theoretical analysis demonstrating that an optimal rationale should maximize both the likelihood of rationale generation given the associated concepts and the likelihood of problem generation conditioned on both the rationale and the concepts. Our method is evaluated on standard benchmarks including GSM8K, MATH-500, and AIME2024, where it consistently outperforms existing problem generation methods. Furthermore, we demonstrate that PromptCoT exhibits superior data scalability, consistently maintaining high performance as the dataset size increases, outperforming the baselines. The implementation is available at https://github.com/zhaoxlpku/PromptCoT.

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.

Recently, deep learning models have made great progress in MWP solving on answer accuracy. However, they are uninterpretable since they mainly rely on shallow heuristics to achieve high performance without understanding and reasoning the grounded math logic. To address this issue and make a step towards interpretable MWP solving, we first construct a high-quality MWP dataset named InterMWP which consists of 11,495 MWPs and annotates interpretable logical formulas based on algebraic knowledge as the grounded linguistic logic of each solution equation. Different from existing MWP datasets, our InterMWP benchmark asks for a solver to not only output the solution expressions but also predict the corresponding logical formulas. We further propose a novel approach with logical prompt and interpretation generation, called LogicSolver. For each MWP, our LogicSolver first retrieves some highly-correlated algebraic knowledge and then passes them to the backbone model as prompts to improve the semantic representations of MWPs. With these improved semantic representations, our LogicSolver generates corresponding solution expressions and interpretable knowledge formulas in accord with the generated solution expressions, simultaneously. Experimental results show that our LogicSolver has stronger logical formula-based interpretability than baselines while achieving higher answer accuracy with the help of logical prompts, simultaneously. The source code and dataset is available at https://github.com/yangzhch6/InterMWP.

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a O(1/k^2) and O(1/k) rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves O(1/k^2) and mathcal{O}(1/k^2) rates. We then provide a matching complexity lower bound to establish that Theta(1/k^2) is indeed the optimal accelerated rate.

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.

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.

We introduce FrontierMath, a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians. The questions cover most major branches of modern mathematics -- from computationally intensive problems in number theory and real analysis to abstract questions in algebraic geometry and category theory. Solving a typical problem requires multiple hours of effort from a researcher in the relevant branch of mathematics, and for the upper end questions, multiple days. FrontierMath uses new, unpublished problems and automated verification to reliably evaluate models while minimizing risk of data contamination. Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community. As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.

We introduce Goedel-Prover, an open-source large language model (LLM) that achieves the state-of-the-art (SOTA) performance in automated formal proof generation for mathematical problems. The key challenge in this field is the scarcity of formalized math statements and proofs, which we tackle in the following ways. We train statement formalizers to translate the natural language math problems from Numina into formal language (Lean 4), creating a dataset of 1.64 million formal statements. LLMs are used to check that the formal statements accurately preserve the content of the original natural language problems. We then iteratively build a large dataset of formal proofs by training a series of provers. Each prover succeeds in proving many statements that the previous ones could not, and these new proofs are added to the training set for the next prover. The final prover outperforms all existing open-source models in whole-proof generation. On the miniF2F benchmark, it achieves a 57.6% success rate (Pass@32), exceeding the previous best open-source model by 7.6%. On PutnamBench, Goedel-Prover successfully solves 7 problems (Pass@512), ranking first on the leaderboard. Furthermore, it generates 29.7K formal proofs for Lean Workbook problems, nearly doubling the 15.7K produced by earlier works.

In this study, we introduce an innovative deep learning framework that employs a transformer model to address the challenges of mixed-integer programs, specifically focusing on the Capacitated Lot Sizing Problem (CLSP). Our approach, to our knowledge, is the first to utilize transformers to predict the binary variables of a mixed-integer programming (MIP) problem. Specifically, our approach harnesses the encoder decoder transformer's ability to process sequential data, making it well-suited for predicting binary variables indicating production setup decisions in each period of the CLSP. This problem is inherently dynamic, and we need to handle sequential decision making under constraints. We present an efficient algorithm in which CLSP solutions are learned through a transformer neural network. The proposed post-processed transformer algorithm surpasses the state-of-the-art solver, CPLEX and Long Short-Term Memory (LSTM) in solution time, optimal gap, and percent infeasibility over 240K benchmark CLSP instances tested. After the ML model is trained, conducting inference on the model, reduces the MIP into a linear program (LP). This transforms the ML-based algorithm, combined with an LP solver, into a polynomial-time approximation algorithm to solve a well-known NP-Hard problem, with almost perfect solution quality.

Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely high-dimensional combinatorial and nonlinear hypothesis spaces. Traditional methods of equation discovery largely focus on extracting equations from data alone, often neglecting the rich domain-specific prior knowledge that scientists typically depend on. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data in an efficient manner. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeletons, drawing from its physical understanding, which are then optimized against data to estimate skeleton parameters. We demonstrate LLM-SR's effectiveness across three diverse scientific domains, where it discovers physically accurate equations that provide significantly better fits to in-domain and out-of-domain data compared to the well-established equation discovery baselines

As a seemingly self-explanatory task, problem-solving has been a significant component of science and engineering. However, a general yet concrete formulation of problem-solving itself is missing. With the recent development of AI-based problem-solving agents, the demand for process-level verifiability is rapidly increasing yet underexplored. To fill these gaps, we present a principled formulation of problem-solving as a deterministic Markov decision process; a novel framework, FPS (Formal Problem-Solving), which utilizes existing FTP (formal theorem proving) environments to perform process-verified problem-solving; and D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment. The expressiveness, soundness and completeness of the frameworks are proven. We construct three benchmarks on problem-solving: FormalMath500, a formalization of a subset of the MATH500 benchmark; MiniF2F-Solving and PutnamBench-Solving, adaptations of FTP benchmarks MiniF2F and PutnamBench. For faithful, interpretable, and human-aligned evaluation, we propose RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification. We evaluate four prevalent FTP models and two prompting methods as baselines, solving at most 23.77% of FormalMath500, 27.47% of MiniF2F-Solving, and 0.31% of PutnamBench-Solving.

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

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular low displacement rank) for integrating tensor fields defined on weighted trees. Several applications of the resulting fast tree-field integrators (FTFIs) are presented, including (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) Topological Transformers (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as three extra learnable parameters per Transformer layer, leading to 1.0-1.5%+ accuracy gains. Importantly, most of FTFIs are exact methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide 5.7-13x speedups. We also provide an extensive theoretical analysis of our methods.

Solving algebraic word problems requires executing a series of arithmetic operations---a program---to obtain a final answer. However, since programs can be arbitrarily complicated, inducing them directly from question-answer pairs is a formidable challenge. To make this task more feasible, we solve these problems by generating answer rationales, sequences of natural language and human-readable mathematical expressions that derive the final answer through a series of small steps. Although rationales do not explicitly specify programs, they provide a scaffolding for their structure via intermediate milestones. To evaluate our approach, we have created a new 100,000-sample dataset of questions, answers and rationales. Experimental results show that indirect supervision of program learning via answer rationales is a promising strategy for inducing arithmetic programs.

Large Language Models (LLMs) have limited performance when solving arithmetic reasoning tasks and often provide incorrect answers. Unlike natural language understanding, math problems typically have a single correct answer, making the task of generating accurate solutions more challenging for LLMs. To the best of our knowledge, we are not aware of any LLMs that indicate their level of confidence in their responses which fuels a trust deficit in these models impeding their adoption. To address this deficiency, we propose `MathPrompter', a technique that improves performance of LLMs on arithmetic problems along with increased reliance in the predictions. MathPrompter uses the Zero-shot chain-of-thought prompting technique to generate multiple Algebraic expressions or Python functions to solve the same math problem in different ways and thereby raise the confidence level in the output results. This is in contrast to other prompt based CoT methods, where there is no check on the validity of the intermediate steps followed. Our technique improves over state-of-the-art on the MultiArith dataset (78.7%rightarrow92.5%) evaluated using 175B parameter GPT-based LLM.

Geometry problem solving has attracted much attention in the NLP community recently. The task is challenging as it requires abstract problem understanding and symbolic reasoning with axiomatic knowledge. However, current datasets are either small in scale or not publicly available. Thus, we construct a new large-scale benchmark, Geometry3K, consisting of 3,002 geometry problems with dense annotation in formal language. We further propose a novel geometry solving approach with formal language and symbolic reasoning, called Interpretable Geometry Problem Solver (Inter-GPS). Inter-GPS first parses the problem text and diagram into formal language automatically via rule-based text parsing and neural object detecting, respectively. Unlike implicit learning in existing methods, Inter-GPS incorporates theorem knowledge as conditional rules and performs symbolic reasoning step by step. Also, a theorem predictor is designed to infer the theorem application sequence fed to the symbolic solver for the more efficient and reasonable searching path. Extensive experiments on the Geometry3K and GEOS datasets demonstrate that Inter-GPS achieves significant improvements over existing methods. The project with code and data is available at https://lupantech.github.io/inter-gps.

This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We advocate representing the PDE in the form of a computational graph, facilitating the seamless integration of both symbolic and numerical information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed to generate mesh-free predicted solutions. Following pretraining on data exhibiting a certain level of diversity, our model achieves zero-shot accuracies on benchmark datasets that surpass those of adequately trained expert models. Additionally, PDEformer demonstrates promising results in the inverse problem of PDE coefficient recovery.

Advances in Large Language Models (LLMs) have sparked interest in their ability to solve Olympiad-level math problems. However, the training and evaluation of these models are constrained by the limited size and quality of available datasets, as creating large-scale data for such advanced problems requires extensive effort from human experts. In addition, current benchmarks are prone to contamination, leading to unreliable evaluations. In this paper, we present an automated pipeline that leverages the rich resources of the Art of Problem Solving (AoPS) forum, which predominantly features Olympiad-level problems and community-driven solutions. Using open-source LLMs, we develop a method to extract question-answer pairs from the forum, resulting in AoPS-Instruct, a dataset of more than 600,000 high-quality QA pairs. Our experiments demonstrate that fine-tuning LLMs on AoPS-Instruct improves their reasoning abilities across various benchmarks. Moreover, we build an automatic pipeline that introduces LiveAoPSBench, an evolving evaluation set with timestamps, derived from the latest forum data, providing a contamination-resistant benchmark for assessing LLM performance. Notably, we observe a significant decline in LLM performance over time, suggesting their success on older examples may stem from pre-training exposure rather than true reasoning ability. Our work presents a scalable approach to creating and maintaining large-scale, high-quality datasets for advanced math reasoning, offering valuable insights into the capabilities and limitations of LLMs in this domain. Our benchmark and code is available at https://github.com/DSL-Lab/aops

We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically outperforms proof search only. We also observe that when applied to a collection of formal statements of sufficiently varied difficulty, expert iteration is capable of finding and solving a curriculum of increasingly difficult problems, without the need for associated ground-truth proofs. Finally, by applying this expert iteration to a manually curated set of problem statements, we achieve state-of-the-art on the miniF2F benchmark, automatically solving multiple challenging problems drawn from high school olympiads.

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.

Automatic math problem solving has recently attracted increasing attention as a long-standing AI benchmark. In this paper, we focus on solving geometric problems, which requires a comprehensive understanding of textual descriptions, visual diagrams, and theorem knowledge. However, the existing methods were highly dependent on handcraft rules and were merely evaluated on small-scale datasets. Therefore, we propose a Geometric Question Answering dataset GeoQA, containing 4,998 geometric problems with corresponding annotated programs, which illustrate the solving process of the given problems. Compared with another publicly available dataset GeoS, GeoQA is 25 times larger, in which the program annotations can provide a practical testbed for future research on explicit and explainable numerical reasoning. Moreover, we introduce a Neural Geometric Solver (NGS) to address geometric problems by comprehensively parsing multimodal information and generating interpretable programs. We further add multiple self-supervised auxiliary tasks on NGS to enhance cross-modal semantic representation. Extensive experiments on GeoQA validate the effectiveness of our proposed NGS and auxiliary tasks. However, the results are still significantly lower than human performance, which leaves large room for future research. Our benchmark and code are released at https://github.com/chen-judge/GeoQA .

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.

Math word problems are critical K-8 educational tools, but writing them is time-consuming and requires domain expertise. We suggest that language models can support K-8 math education by automatically generating problems at scale. To be educational, generated problems must be 1) solvable, 2) accurate, and 3) appropriate. Existing datasets are unlabeled for these criteria, making them ill-suited for training problem generators. We introduce MATHWELL, a Llama-2 (70B) model iteratively finetuned to generate K-8 math word problems using data from expert annotation. Using MATHWELL, we generate the largest English word problem dataset to date, containing 20,490 problems. 3,484 are scored by domain experts who find MATHWELL has a 40% higher share of problems that have executable solutions and meet all criteria than alternatives, with 74% of its problems with executable solutions being solvable, accurate, and appropriate.

The spatiotemporal resolution of Partial Differential Equations (PDEs) plays important roles in the mathematical description of the world's physical phenomena. In general, scientists and engineers solve PDEs numerically by the use of computationally demanding solvers. Recently, deep learning algorithms have emerged as a viable alternative for obtaining fast solutions for PDEs. Models are usually trained on synthetic data generated by solvers, stored on disk and read back for training. This paper advocates that relying on a traditional static dataset to train these models does not allow the full benefit of the solver to be used as a data generator. It proposes an open source online training framework for deep surrogate models. The framework implements several levels of parallelism focused on simultaneously generating numerical simulations and training deep neural networks. This approach suppresses the I/O and storage bottleneck associated with disk-loaded datasets, and opens the way to training on significantly larger datasets. Experiments compare the offline and online training of four surrogate models, including state-of-the-art architectures. Results indicate that exposing deep surrogate models to more dataset diversity, up to hundreds of GB, can increase model generalization capabilities. Fully connected neural networks, Fourier Neural Operator (FNO), and Message Passing PDE Solver prediction accuracy is improved by 68%, 16% and 7%, respectively.

Advanced applied mathematics problems are underrepresented in existing Large Language Model (LLM) benchmark datasets. To address this, we introduce HARDMath, a dataset inspired by a graduate course on asymptotic methods, featuring challenging applied mathematics problems that require analytical approximation techniques. These problems demand a combination of mathematical reasoning, computational tools, and subjective judgment, making them difficult for LLMs. Our framework auto-generates a large number of problems with solutions validated against numerical ground truths. We evaluate both open- and closed-source LLMs on HARDMath-mini, a sub-sampled test set of 366 problems, as well as on 40 word problems formulated in applied science contexts. Even leading closed-source models like GPT-4 achieve only 43.8% overall accuracy with few-shot Chain-of-Thought prompting, and all models demonstrate significantly lower performance compared to results on existing mathematics benchmark datasets. We additionally conduct a detailed error analysis to gain insights into the failure cases of LLMs. These results demonstrate limitations of current LLM performance on advanced graduate-level applied math problems and underscore the importance of datasets like HARDMath to advance mathematical abilities of LLMs.

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.

Constraint Programming (CP) is a declarative programming paradigm that allows for modeling and solving combinatorial optimization problems, such as the Job-Shop Scheduling Problem (JSSP). While CP solvers manage to find optimal or near-optimal solutions for small instances, they do not scale well to large ones, i.e., they require long computation times or yield low-quality solutions. Therefore, real-world scheduling applications often resort to fast, handcrafted, priority-based dispatching heuristics to find a good initial solution and then refine it using optimization methods. This paper proposes a novel end-to-end approach to solving scheduling problems by means of CP and Reinforcement Learning (RL). In contrast to previous RL methods, tailored for a given problem by including procedural simulation algorithms, complex feature engineering, or handcrafted reward functions, our neural-network architecture and training algorithm merely require a generic CP encoding of some scheduling problem along with a set of small instances. Our approach leverages existing CP solvers to train an agent learning a Priority Dispatching Rule (PDR) that generalizes well to large instances, even from separate datasets. We evaluate our method on seven JSSP datasets from the literature, showing its ability to find higher-quality solutions for very large instances than obtained by static PDRs and by a CP solver within the same time limit.

We present a novel approach to formalise and solve search-based problems using large language models, which significantly improves upon previous state-of-the-art results. We demonstrate the efficacy of this approach on the logic puzzles benchmark ZebraLogicBench. Instead of letting the LLM attempt to directly solve the puzzles, our method prompts the model to formalise the problem in a logic-focused domain-specific language (DSL) called Logic.py. This formalised representation is then solved using a constraint solver, leveraging the strengths of both the language model and the solver. Our approach achieves a remarkable 65% absolute improvement over the baseline performance of Llama 3.1 70B on ZebraLogicBench, setting a new state-of-the-art with an accuracy of over 90%. This significant advancement demonstrates the potential of combining language models with domain-specific languages and auxiliary tools on traditionally challenging tasks for LLMs.

Solving mathematical problems requires advanced reasoning abilities and presents notable challenges for large language models. Previous works usually synthesize data from proprietary models to augment existing datasets, followed by instruction tuning to achieve top-tier results. However, our analysis of these datasets reveals severe biases towards easy queries, with frequent failures to generate any correct response for the most challenging queries. Hypothesizing that difficult queries are crucial to learn complex reasoning, we propose Difficulty-Aware Rejection Tuning (DART), a method that allocates difficult queries more trials during the synthesis phase, enabling more extensive training on difficult samples. Utilizing DART, we have created new datasets for mathematical problem-solving that focus more on difficult queries and are substantially smaller than previous ones. Remarkably, our synthesis process solely relies on a 7B-sized open-weight model, without reliance on the commonly used proprietary GPT-4. We fine-tune various base models on our datasets ranging from 7B to 70B in size, resulting in a series of strong models called DART-MATH. In comprehensive in-domain and out-of-domain evaluation on 6 mathematical benchmarks, DART-MATH outperforms vanilla rejection tuning significantly, being superior or comparable to previous arts, despite using much smaller datasets and no proprietary models. Furthermore, our results position our synthetic datasets as the most effective and cost-efficient publicly available resources for advancing mathematical problem-solving.

Despite their success in many natural language tasks, solving math problems remains a significant challenge for large language models (LLMs). A large gap exists between LLMs' pass-at-one and pass-at-N performance in solving math problems, suggesting LLMs might be close to finding correct solutions, motivating our exploration of fine-tuning methods to unlock LLMs' performance. Using the challenging MATH dataset, we investigate three fine-tuning strategies: (1) solution fine-tuning, where we fine-tune to generate a detailed solution for a given math problem; (2) solution-cluster re-ranking, where the LLM is fine-tuned as a solution verifier/evaluator to choose among generated candidate solution clusters; (3) multi-task sequential fine-tuning, which integrates both solution generation and evaluation tasks together efficiently to enhance the LLM performance. With these methods, we present a thorough empirical study on a series of PaLM 2 models and find: (1) The quality and style of the step-by-step solutions used for fine-tuning can make a significant impact on the model performance; (2) While solution re-ranking and majority voting are both effective for improving the model performance when used separately, they can also be used together for an even greater performance boost; (3) Multi-task fine-tuning that sequentially separates the solution generation and evaluation tasks can offer improved performance compared with the solution fine-tuning baseline. Guided by these insights, we design a fine-tuning recipe that yields approximately 58.8% accuracy on the MATH dataset with fine-tuned PaLM 2-L models, an 11.2% accuracy improvement over the few-shot performance of pre-trained PaLM 2-L model with majority voting.

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.

In this work, we introduce Boolformer, the first Transformer architecture trained to perform end-to-end symbolic regression of Boolean functions. First, we show that it can predict compact formulas for complex functions which were not seen during training, when provided a clean truth table. Then, we demonstrate its ability to find approximate expressions when provided incomplete and noisy observations. We evaluate the Boolformer on a broad set of real-world binary classification datasets, demonstrating its potential as an interpretable alternative to classic machine learning methods. Finally, we apply it to the widespread task of modelling the dynamics of gene regulatory networks. Using a recent benchmark, we show that Boolformer is competitive with state-of-the art genetic algorithms with a speedup of several orders of magnitude. Our code and models are available publicly.

Symbolic regression plays a crucial role in modern scientific research thanks to its capability of discovering concise and interpretable mathematical expressions from data. A grand challenge lies in the arduous search for parsimonious and generalizable mathematical formulas, in an infinite search space, while intending to fit the training data. Existing algorithms have faced a critical bottleneck of accuracy and efficiency over a decade when handling problems of complexity, which essentially hinders the pace of applying symbolic regression for scientific exploration across interdisciplinary domains. To this end, we introduce a parallelized tree search (PTS) model to efficiently distill generic mathematical expressions from limited data. Through a series of extensive experiments, we demonstrate the superior accuracy and efficiency of PTS for equation discovery, which greatly outperforms the state-of-the-art baseline models on over 80 synthetic and experimental datasets (e.g., lifting its performance by up to 99% accuracy improvement and one-order of magnitude speed up). PTS represents a key advance in accurate and efficient data-driven discovery of symbolic, interpretable models (e.g., underlying physical laws) and marks a pivotal transition towards scalable symbolic learning.

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 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.

The correctness of computations remains a significant challenge in computer science, with traditional approaches relying on automated testing or formal verification. Self-testing/correcting programs introduce an alternative paradigm, allowing a program to verify and correct its own outputs via randomized reductions, a concept that previously required manual derivation. In this paper, we present Bitween, a method and tool for automated learning of randomized (self)-reductions and program properties in numerical programs. Bitween combines symbolic analysis and machine learning, with a surprising finding: polynomial-time linear regression, a basic optimization method, is not only sufficient but also highly effective for deriving complex randomized self-reductions and program invariants, often outperforming sophisticated mixed-integer linear programming solvers. We establish a theoretical framework for learning these reductions and introduce RSR-Bench, a benchmark suite for evaluating Bitween's capabilities on scientific and machine learning functions. Our empirical results show that Bitween surpasses state-of-the-art tools in scalability, stability, and sample efficiency when evaluated on nonlinear invariant benchmarks like NLA-DigBench. Bitween is open-source as a Python package and accessible via a web interface that supports C language programs.

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

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.

Mathematical reasoning represents a critical frontier in advancing large language models (LLMs). While step-by-step approaches have emerged as the dominant paradigm for mathematical problem-solving in LLMs, the quality of reasoning steps in training data fundamentally constrains the performance of the models. Recent studies has demonstrated that more detailed intermediate steps can enhance model performance, yet existing methods for step expansion either require more powerful external models or incur substantial computational costs. In this paper, we introduce MathFimer, a novel framework for mathematical reasoning step expansion inspired by the "Fill-in-the-middle" task from code completion. By decomposing solution chains into prefix-suffix pairs and training models to reconstruct missing intermediate steps, we develop a specialized model, MathFimer-7B, on our carefully curated NuminaMath-FIM dataset. We then apply these models to enhance existing mathematical reasoning datasets by inserting detailed intermediate steps into their solution chains, creating MathFimer-expanded versions. Through comprehensive experiments on multiple mathematical reasoning datasets, including MathInstruct, MetaMathQA and etc., we demonstrate that models trained on MathFimer-expanded data consistently outperform their counterparts trained on original data across various benchmarks such as GSM8K and MATH. Our approach offers a practical, scalable solution for enhancing mathematical reasoning capabilities in LLMs without relying on powerful external models or expensive inference procedures.

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

In this article, we describe an algorithm for solving Quadratic Unconstrained Binary Optimization problems on the Intel Loihi 2 neuromorphic processor. The solver is based on a hardware-aware fine-grained parallel simulated annealing algorithm developed for Intel's neuromorphic research chip Loihi 2. Preliminary results show that our approach can generate feasible solutions in as little as 1 ms and up to 37x more energy efficient compared to two baseline solvers running on a CPU. These advantages could be especially relevant for size-, weight-, and power-constrained edge computing applications.

Existing math datasets evaluate the reasoning abilities of large language models (LLMs) by either using the final answer or the intermediate reasoning steps derived from static examples. However, the former approach fails to surface model's uses of shortcuts and wrong reasoning while the later poses challenges in accommodating alternative solutions. In this work, we seek to use symbolic programs as a means for automated evaluation if a model can consistently produce correct final answers across various inputs to the program. We begin by extracting programs for popular math datasets (GSM8K and MATH) using GPT4-o. For those executable programs verified using the original input-output pairs, they are found to encapsulate the proper reasoning required to solve the original text questions. We then prompt GPT4-o to generate new questions using alternative input-output pairs based the extracted program. We apply the resulting datasets to evaluate a collection of LLMs. In our experiments, we observe significant accuracy drops using our proposed evaluation compared with original static examples, suggesting the fragility of math reasoning in state-of-the-art LLMs.

Large language models (LLMs) have achieved impressive success on many benchmarks for mathematical reasoning. However, there is growing concern that some of this performance actually reflects dataset contamination, where data closely resembling benchmark questions leaks into the training data, instead of true reasoning ability. To investigate this claim rigorously, we commission Grade School Math 1000 (GSM1k). GSM1k is designed to mirror the style and complexity of the established GSM8k benchmark, the gold standard for measuring elementary mathematical reasoning. We ensure that the two benchmarks are comparable across important metrics such as human solve rates, number of steps in solution, answer magnitude, and more. When evaluating leading open- and closed-source LLMs on GSM1k, we observe accuracy drops of up to 13%, with several families of models (e.g., Phi and Mistral) showing evidence of systematic overfitting across almost all model sizes. At the same time, many models, especially those on the frontier, (e.g., Gemini/GPT/Claude) show minimal signs of overfitting. Further analysis suggests a positive relationship (Spearman's r^2=0.32) between a model's probability of generating an example from GSM8k and its performance gap between GSM8k and GSM1k, suggesting that many models may have partially memorized GSM8k.

Math reasoning is becoming an ever increasing area of focus as we scale large language models. However, even the previously-toughest evals like MATH are now close to saturated by frontier models (90.0% for o1-mini and 86.5% for Gemini 1.5 Pro). We introduce HARP, Human Annotated Reasoning Problems (for Math), consisting of 5,409 problems from the US national math competitions (A(J)HSME, AMC, AIME, USA(J)MO). Of these, 4,780 have answers that are automatically check-able (with libraries such as SymPy). These problems range six difficulty levels, with frontier models performing relatively poorly on the hardest bracket of 197 problems (average accuracy 41.1% for o1-mini, and 9.6% for Gemini 1.5 Pro). Our dataset also features multiple choices (for 4,110 problems) and an average of two human-written, ground-truth solutions per problem, offering new avenues of research that we explore briefly. We report evaluations for many frontier models and share some interesting analyses, such as demonstrating that frontier models across families intrinsically scale their inference-time compute for more difficult problems. Finally, we open source all code used for dataset construction (including scraping) and all code for evaluation (including answer checking) to enable future research at: https://github.com/aadityasingh/HARP.

Recent theoretical work has identified surprisingly simple reasoning problems, such as checking if two nodes in a graph are connected or simulating finite-state machines, that are provably unsolvable by standard transformers that answer immediately after reading their input. However, in practice, transformers' reasoning can be improved by allowing them to use a "chain of thought" or "scratchpad", i.e., generate and condition on a sequence of intermediate tokens before answering. Motivated by this, we ask: Does such intermediate generation fundamentally extend the computational power of a decoder-only transformer? We show that the answer is yes, but the amount of increase depends crucially on the amount of intermediate generation. For instance, we find that transformer decoders with a logarithmic number of decoding steps (w.r.t. the input length) push the limits of standard transformers only slightly, while a linear number of decoding steps, assuming a slight generalization to standard pre-norm, adds a clear new ability (under standard complexity conjectures): recognizing all regular languages. Our results also imply that linear steps keep transformer decoders within context-sensitive languages, and polynomial steps with generalized pre-norm make them recognize exactly the class of polynomial-time solvable problems -- the first exact characterization of a type of transformers in terms of standard complexity classes. Together, our results provide a nuanced framework for understanding how the length of a transformer's chain of thought or scratchpad impacts its reasoning power.

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given precision level. One challenge in PCG solvers is the selection of preconditioners, as different problem-dependent systems can benefit from different preconditioners. We present a new method to introduce inductive bias in preconditioning conjugate gradient algorithm. Given a system matrix and a set of solution vectors arise from an underlying distribution, we train a graph neural network to obtain an approximate decomposition to the system matrix to be used as a preconditioner in the context of PCG solvers. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach in solving various 2D and 3D linear second-order PDEs.

Recent advancements in Chain-of-Thoughts (CoT) and Program-of-Thoughts (PoT) methods have greatly enhanced language models' mathematical reasoning capabilities, facilitating their integration into instruction tuning datasets with LLMs. However, existing methods for large-scale dataset creation require substantial seed data and high computational costs for data synthesis, posing significant challenges for scalability. We introduce InfinityMATH, a scalable instruction tuning dataset for programmatic mathematical reasoning. The construction pipeline emphasizes decoupling numbers from mathematical problems to synthesize number-independent programs, enabling efficient and flexible scaling while minimizing dependency on specific numerical values. Fine-tuning experiments with open-source language and code models, such as Llama2 and CodeLlama, demonstrate the practical benefits of InfinityMATH. These fine-tuned models, showed significant relative improvements on both in-domain and out-of-domain benchmarks, ranging from 184.7% to 514.3% on average. Additionally, these models exhibited high robustness on the GSM8K+ and MATH+ benchmarks, which are enhanced version of test sets with simply the number variations. InfinityMATH ensures that models are more versatile and effective across a broader range of mathematical problems. The data is available at https://huggingface.co/datasets/flagopen/InfinityMATH.

A fundamental challenge in formal theorem proving by LLMs is the lack of high-quality training data. Although reinforcement learning or expert iteration partially mitigates this issue by alternating between LLM generating proofs and finetuning them on correctly generated ones, performance quickly plateaus due to the scarcity of correct proofs (sparse rewards). To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The conjecturer is trained iteratively on previously generated conjectures that are barely provable by the current prover, which incentivizes it to generate increasingly challenging conjectures over time. The prover attempts to prove the conjectures with standard expert iteration. We evaluate STP with both Lean and Isabelle formal versifiers. With 19.8 billion tokens generated during the training in Lean, STP proves 26.3% of the statements in the LeanWorkbook dataset, doubling the previous best result of 13.2% achieved through expert iteration. The final model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (61.7%, pass@3200), Proofnet-test (23.1%, pass@3200) and PutnamBench (8/644, pass@3200).

Mathematical word problem-solving has long been recognized as a complex task for small language models (SLMs). A recent study hypothesized that the smallest model size, needed to achieve over 80% accuracy on the GSM8K benchmark, is 34 billion parameters. To reach this level of performance with smaller models, researcher often train SLMs to generate Python code or use tools to help avoid calculation errors. Additionally, they employ ensembling, where outputs of up to 100 model runs are combined to arrive at a more accurate result. Result selection is done using consensus, majority vote or a separate a verifier model used in conjunction with the SLM. Ensembling provides a substantial boost in accuracy but at a significant cost increase with multiple calls to the model (e.g., Phi-GSM uses top-48 to boost the performance from 68.2 to 81.5). In this work, we present Orca-Math, a 7-billion-parameter SLM based on the Mistral-7B, which achieves 86.81% on GSM8k without the need for multiple model calls or the use of verifiers, code execution or any other external tools. Our approach has the following key elements: (1) A high quality synthetic dataset of 200K math problems created using a multi-agent setup where agents collaborate to create the data, (2) An iterative learning techniques that enables the SLM to practice solving problems, receive feedback on its solutions and learn from preference pairs incorporating the SLM solutions and the feedback. When trained with Supervised Fine-Tuning alone, Orca-Math achieves 81.50% on GSM8k pass@1 metric. With iterative preference learning, Orca-Math achieves 86.81% pass@1. Orca-Math surpasses the performance of significantly larger models such as LLAMA-2-70B, WizardMath-70B, Gemini-Pro, ChatGPT-3.5. It also significantly outperforms other smaller models while using much smaller data (hundreds of thousands vs. millions of problems).

General full-wave electromagnetic solvers, such as those utilizing the finite-difference time-domain (FDTD) method, are computationally demanding for simulating practical GPR problems. We explore the performance of a near-real-time, forward modeling approach for GPR that is based on a machine learning (ML) architecture. To ease the process, we have developed a framework that is capable of generating these ML-based forward solvers automatically. The framework uses an innovative training method that combines a predictive dimensionality reduction technique and a large data set of modeled GPR responses from our FDTD simulation software, gprMax. The forward solver is parameterized for a specific GPR application, but the framework can be extended in a straightforward manner to different electromagnetic problems.

Large Language Models (LLMs) have shown excellent performance in language understanding, text generation, code synthesis, and many other tasks, while they still struggle in complex multi-step reasoning problems, such as mathematical reasoning. In this paper, through a newly proposed arithmetical puzzle problem, we show that the model can perform well on multi-step reasoning tasks via fine-tuning on high-quality synthetic data. Experimental results with the open-llama-3B model on three different test datasets show that not only the model can reach a zero-shot pass@1 at 0.44 on the in-domain dataset, it also demonstrates certain generalization capabilities on the out-of-domain datasets. Specifically, this paper has designed two out-of-domain datasets in the form of extending the numerical range and the composing components of the arithmetical puzzle problem separately. The fine-tuned models have shown encouraging performance on these two far more difficult tasks with the zero-shot pass@1 at 0.33 and 0.35, respectively.

Scientific equation discovery is a fundamental task in the history of scientific progress, enabling the derivation of laws governing natural phenomena. Recently, Large Language Models (LLMs) have gained interest for this task due to their potential to leverage embedded scientific knowledge for hypothesis generation. However, evaluating the true discovery capabilities of these methods remains challenging, as existing benchmarks often rely on common equations that are susceptible to memorization by LLMs, leading to inflated performance metrics that do not reflect discovery. In this paper, we introduce LLM-SRBench, a comprehensive benchmark with 239 challenging problems across four scientific domains specifically designed to evaluate LLM-based scientific equation discovery methods while preventing trivial memorization. Our benchmark comprises two main categories: LSR-Transform, which transforms common physical models into less common mathematical representations to test reasoning beyond memorized forms, and LSR-Synth, which introduces synthetic, discovery-driven problems requiring data-driven reasoning. Through extensive evaluation of several state-of-the-art methods, using both open and closed LLMs, we find that the best-performing system so far achieves only 31.5% symbolic accuracy. These findings highlight the challenges of scientific equation discovery, positioning LLM-SRBench as a valuable resource for future research.

This paper introduces the novel task of multimodal puzzle solving, framed within the context of visual question-answering. We present a new dataset, AlgoPuzzleVQA designed to challenge and evaluate the capabilities of multimodal language models in solving algorithmic puzzles that necessitate both visual understanding, language understanding, and complex algorithmic reasoning. We create the puzzles to encompass a diverse array of mathematical and algorithmic topics such as boolean logic, combinatorics, graph theory, optimization, search, etc., aiming to evaluate the gap between visual data interpretation and algorithmic problem-solving skills. The dataset is generated automatically from code authored by humans. All our puzzles have exact solutions that can be found from the algorithm without tedious human calculations. It ensures that our dataset can be scaled up arbitrarily in terms of reasoning complexity and dataset size. Our investigation reveals that large language models (LLMs) such as GPT4V and Gemini exhibit limited performance in puzzle-solving tasks. We find that their performance is near random in a multi-choice question-answering setup for a significant number of puzzles. The findings emphasize the challenges of integrating visual, language, and algorithmic knowledge for solving complex reasoning problems.

Large Language Models (LLMs) have achieved human-level proficiency across diverse tasks, but their ability to perform rigorous mathematical problem solving remains an open challenge. In this work, we investigate a fundamental yet computationally intractable problem: determining whether a given multivariate polynomial is nonnegative. This problem, closely related to Hilbert's Seventeenth Problem, plays a crucial role in global polynomial optimization and has applications in various fields. First, we introduce SoS-1K, a meticulously curated dataset of approximately 1,000 polynomials, along with expert-designed reasoning instructions based on five progressively challenging criteria. Evaluating multiple state-of-the-art LLMs, we find that without structured guidance, all models perform only slightly above the random guess baseline 50%. However, high-quality reasoning instructions significantly improve accuracy, boosting performance up to 81%. Furthermore, our 7B model, SoS-7B, fine-tuned on SoS-1K for just 4 hours, outperforms the 671B DeepSeek-V3 and GPT-4o-mini in accuracy while only requiring 1.8% and 5% of the computation time needed for letters, respectively. Our findings highlight the potential of LLMs to push the boundaries of mathematical reasoning and tackle NP-hard problems.

Geometry problem solving is a well-recognized testbed for evaluating the high-level multi-modal reasoning capability of deep models. In most existing works, two main geometry problems: calculation and proving, are usually treated as two specific tasks, hindering a deep model to unify its reasoning capability on multiple math tasks. However, in essence, these two tasks have similar problem representations and overlapped math knowledge which can improve the understanding and reasoning ability of a deep model on both two tasks. Therefore, we construct a large-scale Unified Geometry problem benchmark, UniGeo, which contains 4,998 calculation problems and 9,543 proving problems. Each proving problem is annotated with a multi-step proof with reasons and mathematical expressions. The proof can be easily reformulated as a proving sequence that shares the same formats with the annotated program sequence for calculation problems. Naturally, we also present a unified multi-task Geometric Transformer framework, Geoformer, to tackle calculation and proving problems simultaneously in the form of sequence generation, which finally shows the reasoning ability can be improved on both two tasks by unifying formulation. Furthermore, we propose a Mathematical Expression Pretraining (MEP) method that aims to predict the mathematical expressions in the problem solution, thus improving the Geoformer model. Experiments on the UniGeo demonstrate that our proposed Geoformer obtains state-of-the-art performance by outperforming task-specific model NGS with over 5.6% and 3.2% accuracies on calculation and proving problems, respectively.

Large Language Models (LLMs) have achieved impressive results across a broad array of tasks, yet their capacity for complex, domain-specific mathematical reasoning-particularly in wireless communications-remains underexplored. In this work, we introduce WirelessMathBench, a novel benchmark specifically designed to evaluate LLMs on mathematical modeling challenges to wireless communications engineering. Our benchmark consists of 587 meticulously curated questions sourced from 40 state-of-the-art research papers, encompassing a diverse spectrum of tasks ranging from basic multiple-choice questions to complex equation completion tasks, including both partial and full completions, all of which rigorously adhere to physical and dimensional constraints. Through extensive experimentation with leading LLMs, we observe that while many models excel in basic recall tasks, their performance degrades significantly when reconstructing partially or fully obscured equations, exposing fundamental limitations in current LLMs. Even DeepSeek-R1, the best performer on our benchmark, achieves an average accuracy of only 38.05%, with a mere 7.83% success rate in full equation completion. By publicly releasing WirelessMathBench along with the evaluation toolkit, we aim to advance the development of more robust, domain-aware LLMs for wireless system analysis and broader engineering applications.

Expert problem-solving is driven by powerful languages for thinking about problems and their solutions. Acquiring expertise means learning these languages -- systems of concepts, alongside the skills to use them. We present DreamCoder, a system that learns to solve problems by writing programs. It builds expertise by creating programming languages for expressing domain concepts, together with neural networks to guide the search for programs within these languages. A ``wake-sleep'' learning algorithm alternately extends the language with new symbolic abstractions and trains the neural network on imagined and replayed problems. DreamCoder solves both classic inductive programming tasks and creative tasks such as drawing pictures and building scenes. It rediscovers the basics of modern functional programming, vector algebra and classical physics, including Newton's and Coulomb's laws. Concepts are built compositionally from those learned earlier, yielding multi-layered symbolic representations that are interpretable and transferrable to new tasks, while still growing scalably and flexibly with experience.

As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently. In this work, we design a general solution operator for two different time-independent PDEs using graph neural networks (GNNs) and spectral graph convolutions. We train the networks on simulated data from a finite elements solver on a variety of shapes and inhomogeneities. In contrast to previous works, we focus on the ability of the trained operator to generalize to previously unseen scenarios. Specifically, we test generalization to meshes with different shapes and superposition of solutions for a different number of inhomogeneities. We find that training on a diverse dataset with lots of variation in the finite element meshes is a key ingredient for achieving good generalization results in all cases. With this, we believe that GNNs can be used to learn solution operators that generalize over a range of properties and produce solutions much faster than a generic solver. Our dataset, which we make publicly available, can be used and extended to verify the robustness of these models under varying conditions.

We introduce SATBench, a benchmark for evaluating the logical reasoning capabilities of large language models (LLMs) through logical puzzles derived from Boolean satisfiability (SAT) problems. Unlike prior work that focuses on inference rule-based reasoning, which often involves deducing conclusions from a set of premises, our approach leverages the search-based nature of SAT problems, where the objective is to find a solution that fulfills a specified set of logical constraints. Each instance in SATBench is generated from a SAT formula, then translated into a story context and conditions using LLMs. The generation process is fully automated and allows for adjustable difficulty by varying the number of clauses. All 2100 puzzles are validated through both LLM-assisted and solver-based consistency checks, with human validation on a subset. Experimental results show that even the strongest model, o4-mini, achieves only 65.0% accuracy on hard UNSAT problems, close to the random baseline of 50%. SATBench exposes fundamental limitations in the search-based logical reasoning abilities of current LLMs and provides a scalable testbed for future research in logical reasoning.

A fundamental problem in combinatorial optimization is identifying equivalent formulations, which can lead to more efficient solution strategies and deeper insights into a problem's computational complexity. The need to automatically identify equivalence between problem formulations has grown as optimization copilots--systems that generate problem formulations from natural language descriptions--have proliferated. However, existing approaches to checking formulation equivalence lack grounding, relying on simple heuristics which are insufficient for rigorous validation. Inspired by Karp reductions, in this work we introduce quasi-Karp equivalence, a formal criterion for determining when two optimization formulations are equivalent based on the existence of a mapping between their decision variables. We propose EquivaMap, a framework that leverages large language models to automatically discover such mappings, enabling scalable and reliable equivalence verification. To evaluate our approach, we construct the first open-source dataset of equivalent optimization formulations, generated by applying transformations such as adding slack variables or valid inequalities to existing formulations. Empirically, EquivaMap significantly outperforms existing methods, achieving substantial improvements in correctly identifying formulation equivalence.

Despite the rapid development of large language models (LLMs), a fundamental challenge persists: the lack of high-quality optimization modeling datasets hampers LLMs' robust modeling of practical optimization problems from natural language descriptions (NL). This data scarcity also contributes to the generalization difficulties experienced by learning-based methods. To address these challenges, we propose a scalable framework for synthesizing a high-quality dataset, named OptMATH. Starting from curated seed data with mathematical formulations (MF), this framework automatically generates problem data (PD) with controllable complexity. Then, a back-translation step is employed to obtain NL. To verify the correspondence between the NL and the PD, a forward modeling step followed by rejection sampling is used. The accepted pairs constitute the training part of OptMATH. Then a collection of rejected pairs is identified and further filtered. This collection serves as a new benchmark for optimization modeling, containing difficult instances whose lengths are much longer than these of NL4OPT and MAMO. Through extensive experiments, we demonstrate that models of various sizes (0.5B-32B parameters) trained on OptMATH achieve superior results on multiple modeling benchmarks, thereby validating the effectiveness and scalability of our approach. Our dataset is publicly available at https://github.com/AuroraLHL/OptMATH.

The art of mathematical reasoning stands as a fundamental pillar of intellectual progress and is a central catalyst in cultivating human ingenuity. Researchers have recently published a plethora of works centered around the task of solving Math Word Problems (MWP) - a crucial stride towards general AI. These existing models are susceptible to dependency on shallow heuristics and spurious correlations to derive the solution expressions. In order to ameliorate this issue, in this paper, we propose a framework for MWP solvers based on the generation of linguistic variants of the problem text. The approach involves solving each of the variant problems and electing the predicted expression with the majority of the votes. We use DeBERTa (Decoding-enhanced BERT with disentangled attention) as the encoder to leverage its rich textual representations and enhanced mask decoder to construct the solution expressions. Furthermore, we introduce a challenging dataset, Psmall{ARAMAWPS}, consisting of paraphrased, adversarial, and inverse variants of selectively sampled MWPs from the benchmark Msmall{AWPS} dataset. We extensively experiment on this dataset along with other benchmark datasets using some baseline MWP solver models. We show that training on linguistic variants of problem statements and voting on candidate predictions improve the mathematical reasoning and robustness of the model. We make our code and data publicly available.

Large Language Model (LLM) reasoning for complex tasks inherently involves a trade-off between solution accuracy and computational efficiency. The subsequent step of verification, while intended to improve performance, further complicates this landscape by introducing its own challenging trade-off: sophisticated Generative Reward Models (GenRMs) can be computationally prohibitive if naively integrated with LLMs at test-time, while simpler, faster methods may lack reliability. To overcome these challenges, we introduce FlexiVe, a novel generative verifier that flexibly balances computational resources between rapid, reliable fast thinking and meticulous slow thinking using a Flexible Allocation of Verification Budget strategy. We further propose the Solve-Detect-Verify pipeline, an efficient inference-time scaling framework that intelligently integrates FlexiVe, proactively identifying solution completion points to trigger targeted verification and provide focused solver feedback. Experiments show FlexiVe achieves superior accuracy in pinpointing errors within reasoning traces on ProcessBench. Furthermore, on challenging mathematical reasoning benchmarks (AIME 2024, AIME 2025, and CNMO), our full approach outperforms baselines like self-consistency in reasoning accuracy and inference efficiency. Our system offers a scalable and effective solution to enhance LLM reasoning at test time.

Large language models (LLMs) have exhibited their problem-solving abilities in mathematical reasoning. Solving realistic optimization (OPT) problems in application scenarios requires advanced and applied mathematics ability. However, current OPT benchmarks that merely solve linear programming are far from complex realistic situations. In this work, we propose OptiBench, a benchmark for End-to-end optimization problem-solving with human-readable inputs and outputs. OptiBench contains rich optimization problems, including linear and nonlinear programming with or without tabular data, which can comprehensively evaluate LLMs' solving ability. In our benchmark, LLMs are required to call a code solver to provide precise numerical answers. Furthermore, to alleviate the data scarcity for optimization problems, and to bridge the gap between open-source LLMs on a small scale (e.g., Llama-3-8b) and closed-source LLMs (e.g., GPT-4), we further propose a data synthesis method namely ReSocratic. Unlike general data synthesis methods that proceed from questions to answers, \ReSocratic first incrementally synthesizes formatted optimization demonstration with mathematical formulations step by step and then back-translates the generated demonstrations into questions. Based on this, we synthesize the ReSocratic-29k dataset. We further conduct supervised fine-tuning with ReSocratic-29k on multiple open-source models. Experimental results show that ReSocratic-29k significantly improves the performance of open-source models.

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.

An AI system can create and maintain knowledge only to the extent that it can verify that knowledge itself. Recent work on long Chain-of-Thought reasoning has demonstrated great potential of LLMs on solving competitive problems, but their verification ability remains to be weak and not sufficiently investigated. In this paper, we propose Heimdall, the long CoT verification LLM that can accurately judge the correctness of solutions. With pure reinforcement learning, we boost the verification accuracy from 62.5% to 94.5% on competitive math problems. By scaling with repeated sampling, the accuracy further increases to 97.5%. Through human evaluation, Heimdall demonstrates impressive generalization capabilities, successfully detecting most issues in challenging math proofs, the type of which is not included during training. Furthermore, we propose Pessimistic Verification to extend the functionality of Heimdall to scaling up the problem solving. It calls Heimdall to judge the solutions from a solver model and based on the pessimistic principle, selects the most likely correct solution with the least uncertainty. Taking DeepSeek-R1-Distill-Qwen-32B as the solver model, Pessimistic Verification improves the solution accuracy on AIME2025 from 54.2% to 70.0% with 16x compute budget and to 83.3% with more compute budget. With the stronger solver Gemini 2.5 Pro, the score reaches 93.0%. Finally, we prototype an automatic knowledge discovery system, a ternary system where one poses questions, another provides solutions, and the third verifies the solutions. Using the data synthesis work NuminaMath for the first two components, Heimdall effectively identifies problematic records within the dataset and reveals that nearly half of the data is flawed, which interestingly aligns with the recent ablation studies from NuminaMath.

Solving math word problems requires deductive reasoning over the quantities in the text. Various recent research efforts mostly relied on sequence-to-sequence or sequence-to-tree models to generate mathematical expressions without explicitly performing relational reasoning between quantities in the given context. While empirically effective, such approaches typically do not provide explanations for the generated expressions. In this work, we view the task as a complex relation extraction problem, proposing a novel approach that presents explainable deductive reasoning steps to iteratively construct target expressions, where each step involves a primitive operation over two quantities defining their relation. Through extensive experiments on four benchmark datasets, we show that the proposed model significantly outperforms existing strong baselines. We further demonstrate that the deductive procedure not only presents more explainable steps but also enables us to make more accurate predictions on questions that require more complex reasoning.

We propose utilizing background operators for mathematical reasoning in large language models (LLMs). To achieve this, we define a set of fundamental mathematical predicates as the basic building blocks. For each mathematical problem, we develop a Prolog solution that includes problem-specific predicates and intermediate predicates derived from these background operators, ensuring that each solution adheres to the defined operator set. We introduce the MATH-Prolog corpus, which is derived from the counting and probability categories of the MATH corpus. For efficient data augmentation, we apply K-fold cross-validated self-training. This method incrementally generates new Prolog solutions for each fold, incorporating those verified as correct into the training set throughout the model training process. Our experimental results demonstrate that 5-fold crossvalidated self-training effectively identifies new, accurate Prolog solutions, achieving an accuracy of 84.6% on the cross-validated set, and 84.8% on the test set during fine-tuning the Meta-Llama-3.1-8B-Instruct model. This approach successfully uncovers new solutions with fully computable inference steps for previously unseen problems. Additionally, incorporating the background mathematical predicates into the prompt enhances solution coverage.

Despite their spectacular progress, language models still struggle on complex reasoning tasks, such as advanced mathematics. We consider a long-standing open problem in mathematics: discovering a Lyapunov function that ensures the global stability of a dynamical system. This problem has no known general solution, and algorithmic solvers only exist for some small polynomial systems. We propose a new method for generating synthetic training samples from random solutions, and show that sequence-to-sequence transformers trained on such datasets perform better than algorithmic solvers and humans on polynomial systems, and can discover new Lyapunov functions for non-polynomial systems.

Large language models (LLMs) now achieve near-human performance on standard math word problem benchmarks (e.g., GSM8K), yet their true reasoning ability remains disputed. A key concern is that models often produce confident, yet unfounded, answers to unanswerable problems. We introduce TreeCut, a synthetic dataset that systematically generates infinite unanswerable math word problems and their answerable counterparts, by representing each question as a tree and removing chosen necessary conditions. Experiments show TreeCut effectively induce hallucinations in large language models, including GPT-4o and o3-mini, with rates of 64% and 44% in their respective worst-case scenarios under zero-shot setting. Further analysis highlights that deeper or more complex trees, composite item names, and removing necessary condition near the middle of a path all increase the likelihood of hallucinations, underscoring the persistent challenges LLMs face in identifying unanswerable math problems. The dataset generation code and sample data are available at https://github.com/j-bagel/treecut-math.

We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra. We solve questions from a MATH dataset (on Prealgebra, Algebra, Counting and Probability, Intermediate Algebra, Number Theory, and Precalculus), the latest benchmark of advanced mathematics problems designed to assess mathematical reasoning. We randomly sample questions and generate solutions with multiple modalities, including numbers, equations, and plots. The latest GPT-3 language model pre-trained on text automatically solves only 18.8% of these university questions using zero-shot learning and 30.8% using few-shot learning and the most recent chain of thought prompting. In contrast, program synthesis with few-shot learning using Codex fine-tuned on code generates programs that automatically solve 81% of these questions. Our approach improves the previous state-of-the-art automatic solution accuracy on the benchmark topics from 8.8% to 81.1%. We perform a survey to evaluate the quality and difficulty of generated questions. This work is the first to automatically solve university-level mathematics course questions at a human level and the first work to explain and generate university-level mathematics course questions at scale, a milestone for higher education.

The research in AI-based formal mathematical reasoning has shown an unstoppable growth trend. These studies have excelled in mathematical competitions like IMO, showing significant progress. However, these studies intertwined multiple skills simultaneously, i.e., problem-solving, reasoning, and writing formal specifications, making it hard to precisely identify the LLMs' strengths and weaknesses in each task. This paper focuses on formal verification, an immediate application scenario of formal reasoning, and decomposes it into six sub-tasks. We constructed 18k high-quality instruction-response pairs across five mainstream formal specification languages (Coq, Lean4, Dafny, ACSL, and TLA+) in six formal-verification-related tasks by distilling GPT-4o. They are split into a 14k+ fine-tuning dataset FM-alpaca and a 4k benchmark FM-Bench. We found that LLMs are good at writing proof segments when given either the code, or the detailed description of proof steps. Also, the fine-tuning brought about a nearly threefold improvement at most. Interestingly, we observed that fine-tuning with formal data also enhances mathematics, reasoning, and coding abilities. We hope our findings inspire further research. Fine-tuned models are released to facilitate subsequent studies

In this paper, we introduce AceMath, a suite of frontier math models that excel in solving complex math problems, along with highly effective reward models capable of evaluating generated solutions and reliably identifying the correct ones. To develop the instruction-tuned math models, we propose a supervised fine-tuning (SFT) process that first achieves competitive performance across general domains, followed by targeted fine-tuning for the math domain using a carefully curated set of prompts and synthetically generated responses. The resulting model, AceMath-72B-Instruct greatly outperforms Qwen2.5-Math-72B-Instruct, GPT-4o and Claude-3.5 Sonnet. To develop math-specialized reward model, we first construct AceMath-RewardBench, a comprehensive and robust benchmark for evaluating math reward models across diverse problems and difficulty levels. After that, we present a systematic approach to build our math reward models. The resulting model, AceMath-72B-RM, consistently outperforms state-of-the-art reward models. Furthermore, when combining AceMath-72B-Instruct with AceMath-72B-RM, we achieve the highest average rm@8 score across the math reasoning benchmarks. We will release model weights, training data, and evaluation benchmarks at: https://research.nvidia.com/labs/adlr/acemath

The one-dimensional J_1-J_2 q-state Potts model is solved exactly for arbitrary q, based on using OpenAI's latest reasoning model o3-mini-high to exactly solve the q=3 case. The exact results provide insights to outstanding physical problems such as the stacking of atomic or electronic orders in layered materials and the formation of a T_c-dome-shaped phase often seen in unconventional superconductors. The work is anticipated to fuel both the research in one-dimensional frustrated magnets for recently discovered finite-temperature application potentials and the fast moving topic area of AI for sciences.

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

Modern Question Answering (QA) and Reasoning approaches based on Large Language Models (LLMs) commonly use prompting techniques, such as Chain-of-Thought (CoT), assuming the resulting generation will have a more granular exploration and reasoning over the question space and scope. However, such methods struggle with generating outputs that are faithful to the intermediate chain of reasoning produced by the model. On the other end of the spectrum, neuro-symbolic methods such as Faithful CoT (F-CoT) propose to combine LLMs with external symbolic solvers. While such approaches boast a high degree of faithfulness, they usually require a model trained for code generation and struggle with tasks that are ambiguous or hard to formalise strictly. We introduce Faithful Logic-Aided Reasoning and Exploration (\ours), a novel interpretable approach for traversing the problem space using task decompositions. We use the LLM to plan a solution, soft-formalise the query into facts and predicates using a logic programming code and simulate that code execution using an exhaustive multi-hop search over the defined space. Our method allows us to compute the faithfulness of the reasoning process w.r.t. the generated code and analyse the steps of the multi-hop search without relying on external solvers. Our methods achieve SOTA results on 7 out of 9 diverse reasoning benchmarks. We also show that model faithfulness positively correlates with overall performance and further demonstrate that {\ours} allows pinpointing the decisive factors sufficient for and leading to the correct answer with optimal reasoning during the multi-hop search.

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

Large language models (LLMs) have shown increasing in-context learning capabilities through scaling up model and data size. Despite this progress, LLMs are still unable to solve algorithmic reasoning problems. While providing a rationale with the final answer has led to further improvements in multi-step reasoning problems, Anil et al. 2022 showed that even simple algorithmic reasoning tasks such as parity are far from solved. In this work, we identify and study four key stages for successfully teaching algorithmic reasoning to LLMs: (1) formulating algorithms as skills, (2) teaching multiple skills simultaneously (skill accumulation), (3) teaching how to combine skills (skill composition) and (4) teaching how to use skills as tools. We show that it is possible to teach algorithmic reasoning to LLMs via in-context learning, which we refer to as algorithmic prompting. We evaluate our approach on a variety of arithmetic and quantitative reasoning tasks, and demonstrate significant boosts in performance over existing prompting techniques. In particular, for long parity, addition, multiplication and subtraction, we achieve an error reduction of approximately 10x, 9x, 5x and 2x respectively compared to the best available baselines.

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.

In recent years, the rapid development of large reasoning models has resulted in the saturation of existing benchmarks for evaluating mathematical reasoning, highlighting the urgent need for more challenging and rigorous evaluation frameworks. To address this gap, we introduce OlymMATH, a novel Olympiad-level mathematical benchmark, designed to rigorously test the complex reasoning capabilities of LLMs. OlymMATH features 200 meticulously curated problems, each manually verified and available in parallel English and Chinese versions. The problems are systematically organized into two distinct difficulty tiers: (1) AIME-level problems (easy) that establish a baseline for mathematical reasoning assessment, and (2) significantly more challenging problems (hard) designed to push the boundaries of current state-of-the-art models. In our benchmark, these problems span four core mathematical fields, each including a verifiable numerical solution to enable objective, rule-based evaluation. Empirical results underscore the significant challenge presented by OlymMATH, with state-of-the-art models including DeepSeek-R1 and OpenAI's o3-mini demonstrating notably limited accuracy on the hard subset. Furthermore, the benchmark facilitates comprehensive bilingual assessment of mathematical reasoning abilities-a critical dimension that remains largely unaddressed in mainstream mathematical reasoning benchmarks. We release the OlymMATH benchmark at the STILL project: https://github.com/RUCAIBox/Slow_Thinking_with_LLMs.

Recent LLMs have demonstrated remarkable performance in solving exam-like math word problems. However, the degree to which these numerical reasoning skills are effective in real-world scenarios, particularly in expert domains, is still largely unexplored. This paper introduces DocMath-Eval, a comprehensive benchmark specifically designed to evaluate the numerical reasoning and problem-solving capabilities of LLMs in the context of understanding and analyzing financial documents containing both text and tables. We evaluate a wide spectrum of 19 LLMs, including those specialized in coding and finance. We also incorporate different prompting strategies (i.e., Chain-of-Thoughts and Program-of-Thoughts) to comprehensively assess the capabilities and limitations of existing LLMs in DocMath-Eval. We found that, although the current best-performing system (i.e., GPT-4), can perform well on simple problems such as calculating the rate of increase in a financial metric within a short document context, it significantly lags behind human experts in more complex problems grounded in longer contexts. We believe DocMath-Eval can be used as a valuable benchmark to evaluate LLMs' capabilities to solve challenging numerical reasoning problems in expert domains. We will release the benchmark and code at https://github.com/yale-nlp/DocMath-Eval.

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.

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.

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.

Recent advancements in large language models (LLMs) have led to significant breakthroughs in mathematical reasoning capabilities. However, existing benchmarks like GSM8K or MATH are now being solved with high accuracy (e.g., OpenAI o1 achieves 94.8% on MATH dataset), indicating their inadequacy for truly challenging these models. To bridge this gap, we propose a comprehensive and challenging benchmark specifically designed to assess LLMs' mathematical reasoning at the Olympiad level. Unlike existing Olympiad-related benchmarks, our dataset focuses exclusively on mathematics and comprises a vast collection of 4428 competition-level problems with rigorous human annotation. These problems are meticulously categorized into over 33 sub-domains and span more than 10 distinct difficulty levels, enabling a holistic assessment of model performance in Olympiad-mathematical reasoning. Furthermore, we conducted an in-depth analysis based on this benchmark. Our experimental results show that even the most advanced models, OpenAI o1-mini and OpenAI o1-preview, struggle with highly challenging Olympiad-level problems, with 60.54% and 52.55% accuracy, highlighting significant challenges in Olympiad-level mathematical reasoning.

The rapid advancement of Large Language Models (LLMs) has demonstrated remarkable progress in complex reasoning tasks. However, a significant discrepancy persists between benchmark performances and real-world applications. We identify this gap as primarily stemming from current evaluation protocols and metrics, which inadequately capture the full spectrum of LLM capabilities, particularly in complex reasoning tasks where both accuracy and consistency are crucial. This work makes two key contributions. First, we introduce G-Pass@k, a novel evaluation metric that provides a continuous assessment of model performance across multiple sampling attempts, quantifying both the model's peak performance potential and its stability. Second, we present LiveMathBench, a dynamic benchmark comprising challenging, contemporary mathematical problems designed to minimize data leakage risks during evaluation. Through extensive experiments using G-Pass@k on state-of-the-art LLMs with LiveMathBench, we provide comprehensive insights into both their maximum capabilities and operational consistency. Our findings reveal substantial room for improvement in LLMs' "realistic" reasoning capabilities, highlighting the need for more robust evaluation methods. The benchmark and detailed results are available at: https://github.com/open-compass/GPassK.

We introduce a large language model (LLM) based approach to answer complex questions requiring multi-hop numerical reasoning over financial reports. While LLMs have exhibited remarkable performance on various natural language and reasoning tasks, complex reasoning problems often rely on few-shot prompts that require carefully crafted examples. In contrast, our approach uses novel zero-shot prompts that guide the LLM to encode the required reasoning into a Python program or a domain specific language. The generated program is then executed by a program interpreter, thus mitigating the limitations of LLM in performing accurate arithmetic calculations. We evaluate the proposed approach on three financial datasets using some of the recently developed generative pretrained transformer (GPT) models and perform comparisons with various zero-shot baselines. The experimental results demonstrate that our approach significantly improves the accuracy for all the LLMs over their respective baselines. We provide a detailed analysis of the results, generating insights to support our findings. The success of our approach demonstrates the enormous potential to extract complex domain specific numerical reasoning by designing zero-shot prompts to effectively exploit the knowledge embedded in LLMs.

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.

Scientific problem-solving involves synthesizing information while applying expert knowledge. We introduce CURIE, a scientific long-Context Understanding,Reasoning and Information Extraction benchmark to measure the potential of Large Language Models (LLMs) in scientific problem-solving and assisting scientists in realistic workflows. This benchmark introduces ten challenging tasks with a total of 580 problems and solution pairs curated by experts in six disciplines - materials science, condensed matter physics, quantum computing, geospatial analysis, biodiversity, and proteins - covering both experimental and theoretical work-flows in science. We evaluate a range of closed and open LLMs on tasks in CURIE which requires domain expertise, comprehension of long in-context information,and multi-step reasoning. While Gemini Flash 2.0 and Claude-3 show consistent high comprehension across domains, the popular GPT-4o and command-R+ fail dramatically on protein sequencing tasks. With the best performance at 32% there is much room for improvement for all models. We hope that insights gained from CURIE can guide the future development of LLMs in sciences. Evaluation code and data are in https://github.com/google/curie

Large language models (LLMs) have demonstrated remarkable reasoning capability in solving mathematical problems. However, existing approaches primarily focus on improving the quality of correct training data, e.g., distilling high-quality correct solutions from advanced models, neglecting the value contained in error data, potentially hindering the model's reflective ability. Though some studies attempt to leverage error data, they often involve complex mechanisms, such as Monte Carlo Tree Search (MCTS) to explore error nodes. In this work, we propose to enhance LLMs' reasoning ability by Learning from Errors for Mathematical Advancement (LEMMA). LEMMA constructs data consisting of an incorrect solution with an erroneous step and a reflection connection to a correct solution for fine-tuning. Specifically, we systematically analyze the model-generated error types and introduce an error-type grounded mistake augmentation method to collect diverse and representative errors. Correct solutions are either from fixing the errors or generating a fresh start. Through a model-aware smooth reflection connection, the erroneous solution is transferred to the correct one. By fine-tuning on the constructed dataset, the model is able to self-correct errors autonomously within the generation process without relying on external critique models. Experimental results demonstrate that LEMMA achieves significant performance improvements over other strong baselines.

Automatic math word problem solving has attracted growing attention in recent years. The evaluation datasets used by previous works have serious limitations in terms of scale and diversity. In this paper, we release a new large-scale and template-rich math word problem dataset named Ape210K. It consists of 210K Chinese elementary school-level math problems, which is 9 times the size of the largest public dataset Math23K. Each problem contains both the gold answer and the equations needed to derive the answer. Ape210K is also of greater diversity with 56K templates, which is 25 times more than Math23K. Our analysis shows that solving Ape210K requires not only natural language understanding but also commonsense knowledge. We expect Ape210K to be a benchmark for math word problem solving systems. Experiments indicate that state-of-the-art models on the Math23K dataset perform poorly on Ape210K. We propose a copy-augmented and feature-enriched sequence to sequence (seq2seq) model, which outperforms existing models by 3.2% on the Math23K dataset and serves as a strong baseline of the Ape210K dataset. The gap is still significant between human and our baseline model, calling for further research efforts. We make Ape210K dataset publicly available at https://github.com/yuantiku/ape210k

Large language models (LLMs) have pushed the limits of natural language understanding and exhibited excellent problem-solving ability. Despite the great success, most existing open-source LLMs (\eg, LLaMA-2) are still far away from satisfactory for solving mathematical problem due to the complex reasoning procedures. To bridge this gap, we propose MetaMath, a fine-tuned language model that specializes in mathematical reasoning. Specifically, we start by bootstrapping mathematical questions by rewriting the question from multiple perspectives without extra knowledge, which results in a new dataset called {MetaMathQA}. Then we fine-tune the LLaMA-2 models on MetaMathQA. Experimental results on two popular benchmarks (\ie, GSM8K and MATH) for mathematical reasoning demonstrate that MetaMath outperforms a suite of open-source LLMs by a significant margin. Our MetaMath-7B model achieves 66.4% on GSM8K and 19.4% on MATH, exceeding the state-of-the-art models of the same size by 11.5% and 8.7%. Particularly, {MetaMath-70B} achieves an accuracy of 82.3% on {GSM8K}, slightly better than {GPT-3.5-Turbo}. We release the {MetaMathQA} dataset, the {MetaMath} models with different model sizes and the training code for public use.

Large language models have demonstrated impressive performance on challenging mathematical reasoning tasks, which has triggered the discussion of whether the performance is achieved by true reasoning capability or memorization. To investigate this question, prior work has constructed mathematical benchmarks when questions undergo simple perturbations -- modifications that still preserve the underlying reasoning patterns of the solutions. However, no work has explored hard perturbations, which fundamentally change the nature of the problem so that the original solution steps do not apply. To bridge the gap, we construct MATH-P-Simple and MATH-P-Hard via simple perturbation and hard perturbation, respectively. Each consists of 279 perturbed math problems derived from level-5 (hardest) problems in the MATH dataset (Hendrycksmath et. al., 2021). We observe significant performance drops on MATH-P-Hard across various models, including o1-mini (-16.49%) and gemini-2.0-flash-thinking (-12.9%). We also raise concerns about a novel form of memorization where models blindly apply learned problem-solving skills without assessing their applicability to modified contexts. This issue is amplified when using original problems for in-context learning. We call for research efforts to address this challenge, which is critical for developing more robust and reliable reasoning models.

We present rStar-Math to demonstrate that small language models (SLMs) can rival or even surpass the math reasoning capability of OpenAI o1, without distillation from superior models. rStar-Math achieves this by exercising "deep thinking" through Monte Carlo Tree Search (MCTS), where a math policy SLM performs test-time search guided by an SLM-based process reward model. rStar-Math introduces three innovations to tackle the challenges in training the two SLMs: (1) a novel code-augmented CoT data sythesis method, which performs extensive MCTS rollouts to generate step-by-step verified reasoning trajectories used to train the policy SLM; (2) a novel process reward model training method that avoids na\"ive step-level score annotation, yielding a more effective process preference model (PPM); (3) a self-evolution recipe in which the policy SLM and PPM are built from scratch and iteratively evolved to improve reasoning capabilities. Through 4 rounds of self-evolution with millions of synthesized solutions for 747k math problems, rStar-Math boosts SLMs' math reasoning to state-of-the-art levels. On the MATH benchmark, it improves Qwen2.5-Math-7B from 58.8% to 90.0% and Phi3-mini-3.8B from 41.4% to 86.4%, surpassing o1-preview by +4.5% and +0.9%. On the USA Math Olympiad (AIME), rStar-Math solves an average of 53.3% (8/15) of problems, ranking among the top 20% the brightest high school math students. Code and data will be available at https://github.com/microsoft/rStar.

Automated model selection is often proposed to users to choose which machine learning model (or method) to apply to a given regression task. In this paper, we show that combining different regression models can yield better results than selecting a single ('best') regression model, and outline an efficient method that obtains optimally weighted convex linear combination from a heterogeneous set of regression models. More specifically, in this paper, a heuristic weight optimization, used in a preceding conference paper, is replaced by an exact optimization algorithm using convex quadratic programming. We prove convexity of the quadratic programming formulation for the straightforward formulation and for a formulation with weighted data points. The novel weight optimization is not only (more) exact but also more efficient. The methods we develop in this paper are implemented and made available via github-open source. They can be executed on commonly available hardware and offer a transparent and easy to interpret interface. The results indicate that the approach outperforms model selection methods on a range of data sets, including data sets with mixed variable type from drug discovery applications.

We study the depth of grade-school math (GSM) problem-solving capabilities of LLMs. To this end, we evaluate their performance on pairs of existing math word problems together so that the answer to the second problem depends on correctly answering the first problem. Our findings reveal a significant reasoning gap in most LLMs, that is performance difference between solving the compositional pairs and solving each question independently. This gap is more pronounced in smaller, more cost-efficient, and math-specialized models. Moreover, instruction-tuning recipes and code generation have varying effects across LLM sizes, while finetuning on GSM can lead to task overfitting. Our analysis indicates that large reasoning gaps are not because of test-set leakage, but due to distraction from additional context and poor second-hop reasoning. Overall, LLMs exhibit systematic differences in their reasoning abilities, despite what their performance on standard benchmarks indicates.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the L^2 difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

The derivation of mathematical results in specialised fields using Large Language Models (LLMs) is an emerging research direction that can help identify models' limitations, and potentially support mathematical discovery. In this paper, we leverage a symbolic engine to generate derivations of equations at scale, and investigate the capabilities of LLMs when deriving goal equations from premises. Specifically, we employ in-context learning for GPT and fine-tune a range of T5 models to compare the robustness and generalisation of pre-training strategies to specialised models. Empirical results show that fine-tuned FLAN-T5-large (MathT5) outperforms GPT models on all static and out-of-distribution test sets in terms of absolute performance. However, an in-depth analysis reveals that the fine-tuned models are more sensitive to perturbations involving unseen symbols and (to a lesser extent) changes to equation structure. In addition, we analyse 1.7K equations and over 200 derivations to highlight common reasoning errors such as the inclusion of incorrect, irrelevant, and redundant equations, along with the tendency to skip derivation steps. Finally, we explore the suitability of existing metrics for evaluating mathematical derivations finding evidence that, while they capture general properties such as sensitivity to perturbations, they fail to highlight fine-grained reasoning errors and essential differences between models. Overall, this work demonstrates that training models on synthetic data can improve their mathematical capabilities beyond larger architectures.

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.

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

Advancing code reasoning in large language models (LLMs) is fundamentally limited by the scarcity of high-difficulty datasets, especially those with verifiable input-output test cases necessary for rigorous solution validation at scale. We introduce rStar-Coder, which significantly improves LLM code reasoning capabilities by constructing a large-scale, verified dataset of 418K competition-level code problems, 580K long-reasoning solutions along with rich test cases of varying difficulty. This is achieved through three core contributions: (1) we curate competitive programming code problems and oracle solutions to synthesize new, solvable problems; (2) we introduce a reliable input-output test case synthesis pipeline that decouples the generation into a three-step input generation method and a mutual verification mechanism for effective output labeling; (3) we augment problems with high-quality, test-case-verified long-reasoning solutions. Extensive experiments on Qwen models (1.5B-14B) across various code reasoning benchmarks demonstrate the superiority of rStar-Coder dataset, achieving leading performance comparable to frontier reasoning LLMs with much smaller model sizes. On LiveCodeBench, rStar-Coder improves Qwen2.5-7B from 17.4% to an impressive 57.3%, and Qwen2.5-14B from 23.3% to 62.5%, surpassing o3-mini (low) by3.1%. On the more challenging USA Computing Olympiad, our 7B model achieves an average pass@1 accuracy of 16.15%, outperforming the frontier-level QWQ-32B. Code and the dataset will be released at https://github.com/microsoft/rStar.

Instructing large language models (LLMs) to solve elementary school math problems has shown great success using Chain of Thought (CoT). However, the CoT approach relies on an LLM to generate a sequence of arithmetic calculations which can be prone to cascaded calculation errors. We hypothesize that an LLM should focus on extracting predicates and generating symbolic formulas from the math problem description so that the underlying calculation can be done via an external code interpreter. We investigate using LLM to generate Prolog programs to solve mathematical questions. Experimental results show that our Prolog-based arithmetic problem-solving outperforms CoT generation in the GSM8K benchmark across three distinct LLMs. In addition, given the insensitive ordering of predicates and symbolic formulas in Prolog, we propose to permute the ground truth predicates for more robust LLM training via data augmentation.

We introduce PHYSICS, a comprehensive benchmark for university-level physics problem solving. It contains 1297 expert-annotated problems covering six core areas: classical mechanics, quantum mechanics, thermodynamics and statistical mechanics, electromagnetism, atomic physics, and optics. Each problem requires advanced physics knowledge and mathematical reasoning. We develop a robust automated evaluation system for precise and reliable validation. Our evaluation of leading foundation models reveals substantial limitations. Even the most advanced model, o3-mini, achieves only 59.9% accuracy, highlighting significant challenges in solving high-level scientific problems. Through comprehensive error analysis, exploration of diverse prompting strategies, and Retrieval-Augmented Generation (RAG)-based knowledge augmentation, we identify key areas for improvement, laying the foundation for future advancements.

We introduce KnowledgeMath, a novel benchmark designed to evaluate LLMs' capabilities in applying financial knowledge to solve complex math word problems. Compared to prior works, this study features three core advancements. First, KnowledgeMath includes 1,259 problems with a hybrid of textual and tabular content and require college-level knowledge in the finance domain for effective resolution. Second, we provide expert-annotated, detailed solution references in Python program format, ensuring a high-quality benchmark for LLM assessment. Finally, we evaluate a wide spectrum of 14 LLMs with different prompting strategies like Chain-of-Thoughts and Program-of-Thoughts. The current best-performing system (i.e., GPT-4 with Program-of-Thoughts) achieves only 45.4% accuracy, leaving substantial room for improvement. While knowledge-augmented LLMs can improve the performance (e.g., from 23.9% to 32.0% for GPT-3.5), it is still significantly lower the estimated human expert performance of 94%. We believe that KnowledgeMath can facilitate future research on domain-specific knowledge retrieval and augmentation into the math word problem-solving process. We will release the benchmark and code at https://github.com/yale-nlp/KnowledgeMath.

We introduce DeepSeek-Prover-V2, an open-source large language model designed for formal theorem proving in Lean 4, with initialization data collected through a recursive theorem proving pipeline powered by DeepSeek-V3. The cold-start training procedure begins by prompting DeepSeek-V3 to decompose complex problems into a series of subgoals. The proofs of resolved subgoals are synthesized into a chain-of-thought process, combined with DeepSeek-V3's step-by-step reasoning, to create an initial cold start for reinforcement learning. This process enables us to integrate both informal and formal mathematical reasoning into a unified model. The resulting model, DeepSeek-Prover-V2-671B, achieves state-of-the-art performance in neural theorem proving, reaching 88.9% pass ratio on the MiniF2F-test and solving 49 out of 658 problems from PutnamBench. In addition to standard benchmarks, we introduce ProverBench, a collection of 325 formalized problems, to enrich our evaluation, including 15 selected problems from the recent AIME competitions (years 24-25). Further evaluation on these 15 AIME problems shows that the model successfully solves 6 of them. In comparison, DeepSeek-V3 solves 8 of these problems using majority voting, highlighting that the gap between formal and informal mathematical reasoning in large language models is substantially narrowing.

Optimization problems are pervasive across various sectors, from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers, as the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. We introduce OptiMUS, a Large Language Model (LLM)-based agent designed to formulate and solve MILP problems from their natural language descriptions. OptiMUS is capable of developing mathematical models, writing and debugging solver code, developing tests, and checking the validity of generated solutions. To benchmark our agent, we present NLP4LP, a novel dataset of linear programming (LP) and mixed integer linear programming (MILP) problems. Our experiments demonstrate that OptiMUS solves nearly twice as many problems as a basic LLM prompting strategy. OptiMUS code and NLP4LP dataset are available at https://github.com/teshnizi/OptiMUS{https://github.com/teshnizi/OptiMUS}

This study focuses on improving the performance of lightweight Large Language Models (LLMs) in mathematical reasoning tasks. We introduce a novel method for measuring mathematical logic similarity and design an automatic screening mechanism to construct a set of reference problems that integrate both semantic and logical similarity. By employing carefully crafted positive and negative example prompts, we guide the model towards adopting sound reasoning logic. To the best of our knowledge, this is the first attempt to utilize retrieval-enhanced generation for mathematical problem-solving. Experimental results demonstrate that our method achieves a 15.8% improvement over the Chain of Thought approach on the SVAMP dataset and a 21.5 % improvement on the GSM8K dataset. Further application of this method to a large-scale model with 175 billion parameters yields performance comparable to the best results on both aforementioned datasets. Finally, we conduct an analysis of errors during the reasoning process, providing valuable insights and directions for future research on reasoning tasks using large language models.

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.

This paper investigates the mathematical reasoning capabilities of large language models (LLMs) using 50 newly constructed high-school-level word problems. Unlike prior studies that focus solely on answer correctness, we rigorously analyze both final answers and solution steps to identify reasoning failures. Evaluating eight state-of-the-art models - including Mixtral, Llama, Gemini, GPT-4o, and OpenAI's o1 variants - we find that while newer models (e.g., o3-mini, deepseek-r1) achieve higher accuracy, all models exhibit errors in spatial reasoning, strategic planning, and arithmetic, sometimes producing correct answers through flawed logic. Common failure modes include unwarranted assumptions, over-reliance on numerical patterns, and difficulty translating physical intuition into mathematical steps. Manual analysis reveals that models struggle with problems requiring multi-step deduction or real-world knowledge, despite possessing broad mathematical knowledge. Our results underscore the importance of evaluating reasoning processes, not just answers, and caution against overestimating LLMs' problem-solving proficiency. The study highlights persistent gaps in LLMs' generalization abilities, emphasizing the need for targeted improvements in structured reasoning and constraint handling.

While reasoning models (e.g., DeepSeek R1) trained with reinforcement learning (RL), excel in textual reasoning, they struggle in scenarios requiring structured problem-solving, such as geometric reasoning, concise computation, or complex equation solving-areas where computational tools like code interpreters (CI) demonstrate distinct advantages. To bridge this gap, we propose ReTool, which enhances long-form reasoning with tool-integrated learning, including two key features: (1) dynamic interleaving of real-time code execution within natural language reasoning processes, and (2) an automated RL paradigm that allows policy rollouts with multi-turn real-time code execution and teaches the model in learning when and how to invoke tools based on outcome feedback. ReTool employs a systematic training framework, beginning with synthetic cold-start data generation to produce code-augmented long-form reasoning traces for fine-tuning base models. Subsequent RL training leverages task outcomes as rewards to iteratively refine the model's tool use strategy, enabling autonomous discovery of optimal tool invocation patterns without human priors. Experiments on the challenging MATH Olympiad benchmark AIME demonstrate ReTool's superiority: Our 32B model achieves 67% accuracy with 400 training steps, outperforming text-based RL baseline (40% accuracy, 1080 steps) in efficiency and performance. Remarkably, ReTool-32B attains 72.5% accuracy in extended settings, surpassing OpenAI's o1-preview by 27.9%. Further analysis reveals emergent behaviors such as code self-correction, signaling an ''aha moment'' in which the model autonomously masters adaptive tool use. These findings highlight the promise of outcome-driven tool integration for advancing complex mathematical reasoning and offer new insights into hybrid neuro-symbolic systems.

Despite recent progress made by large language models in code generation, they still struggle with programs that meet complex requirements. Recent work utilizes plan-and-solve decomposition to decrease the complexity and leverage self-tests to refine the generated program. Yet, planning deep-inside requirements in advance can be challenging, and the tests need to be accurate to accomplish self-improvement. To this end, we propose FunCoder, a code generation framework incorporating the divide-and-conquer strategy with functional consensus. Specifically, FunCoder recursively branches off sub-functions as smaller goals during code generation, represented by a tree hierarchy. These sub-functions are then composited to attain more complex objectives. Additionally, we designate functions via a consensus formed by identifying similarities in program behavior, mitigating error propagation. FunCoder outperforms state-of-the-art methods by +9.8% on average in HumanEval, MBPP, xCodeEval and MATH with GPT-3.5 and GPT-4. Moreover, our method demonstrates superiority on smaller models: With FunCoder, StableCode-3b surpasses GPT-3.5 by +18.6% and achieves 97.7% of GPT-4's performance on HumanEval. Further analysis reveals that our proposed dynamic function decomposition is capable of handling complex requirements, and the functional consensus prevails over self-testing in correctness evaluation.

Small-scale models offer various computational advantages, and yet to which extent size is critical for problem-solving abilities remains an open question. Specifically for solving grade school math, the smallest model size so far required to break the 80\% barrier on the GSM8K benchmark remains to be 34B. Our work studies how high-quality datasets may be the key for small language models to acquire mathematical reasoning. We introduce TinyGSM, a synthetic dataset of 12.3M grade school math problems paired with Python solutions, generated fully by GPT-3.5. After finetuning on TinyGSM, we find that a duo of a 1.3B generation model and a 1.3B verifier model can achieve 81.5\% accuracy, outperforming existing models that are orders of magnitude larger. This also rivals the performance of the GPT-3.5 ``teacher'' model (77.4\%), from which our model's training data is generated. Our approach is simple and has two key components: 1) the high-quality dataset TinyGSM, 2) the use of a verifier, which selects the final outputs from multiple candidate generations.

Many challenging reasoning tasks require not just rapid, intuitive responses, but a more deliberate, multi-step approach. Recent progress in large language models (LLMs) highlights an important shift from the "System 1" way of quick reactions to the "System 2" style of reflection-and-correction problem solving. However, current benchmarks heavily rely on the final-answer accuracy, leaving much of a model's intermediate reasoning steps unexamined. This fails to assess the model's ability to reflect and rectify mistakes within the reasoning process. To bridge this gap, we introduce FINEREASON, a logic-puzzle benchmark for fine-grained evaluation of LLMs' reasoning capabilities. Each puzzle can be decomposed into atomic steps, making it ideal for rigorous validation of intermediate correctness. Building on this, we introduce two tasks: state checking, and state transition, for a comprehensive evaluation of how models assess the current situation and plan the next move. To support broader research, we also provide a puzzle training set aimed at enhancing performance on general mathematical tasks. We show that models trained on our state checking and transition data demonstrate gains in math reasoning by up to 5.1% on GSM8K.

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.

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.

The CASH problem has been widely studied in the context of automated configurations of machine learning (ML) pipelines and various solvers and toolkits are available. However, CASH solvers do not directly handle black-box constraints such as fairness, robustness or other domain-specific custom constraints. We present our recent approach [Liu, et al., 2020] that leverages the ADMM optimization framework to decompose CASH into multiple small problems and demonstrate how ADMM facilitates incorporation of black-box constraints.

Though Rectified Flows (ReFlows) with distillation offers a promising way for fast sampling, its fast inversion transforms images back to structured noise for recovery and following editing remains unsolved. This paper introduces FireFlow, a simple yet effective zero-shot approach that inherits the startling capacity of ReFlow-based models (such as FLUX) in generation while extending its capabilities to accurate inversion and editing in 8 steps. We first demonstrate that a carefully designed numerical solver is pivotal for ReFlow inversion, enabling accurate inversion and reconstruction with the precision of a second-order solver while maintaining the practical efficiency of a first-order Euler method. This solver achieves a 3times runtime speedup compared to state-of-the-art ReFlow inversion and editing techniques, while delivering smaller reconstruction errors and superior editing results in a training-free mode. The code is available at https://github.com/HolmesShuan/FireFlow{this URL}.

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.

Large language models (LLMs) have shown promise in proving formal theorems using proof assistants such as Lean. However, existing methods are difficult to reproduce or build on, due to private code, data, and large compute requirements. This has created substantial barriers to research on machine learning methods for theorem proving. This paper removes these barriers by introducing LeanDojo: an open-source Lean playground consisting of toolkits, data, models, and benchmarks. LeanDojo extracts data from Lean and enables interaction with the proof environment programmatically. It contains fine-grained annotations of premises in proofs, providing valuable data for premise selection: a key bottleneck in theorem proving. Using this data, we develop ReProver (Retrieval-Augmented Prover): the first LLM-based prover that is augmented with retrieval for selecting premises from a vast math library. It is inexpensive and needs only one GPU week of training. Our retriever leverages LeanDojo's program analysis capability to identify accessible premises and hard negative examples, which makes retrieval much more effective. Furthermore, we construct a new benchmark consisting of 96,962 theorems and proofs extracted from Lean's math library. It features challenging data split requiring the prover to generalize to theorems relying on novel premises that are never used in training. We use this benchmark for training and evaluation, and experimental results demonstrate the effectiveness of ReProver over non-retrieval baselines and GPT-4. We thus provide the first set of open-source LLM-based theorem provers without any proprietary datasets and release it under a permissive MIT license to facilitate further research.

Software engineers mainly write code by editing existing programs. In contrast, large language models (LLMs) autoregressively synthesize programs in a single pass. One explanation for this is the scarcity of open-sourced edit data. While high-quality instruction data for code synthesis is already scarce, high-quality edit data is even scarcer. To fill this gap, we develop a synthetic data generation algorithm called LintSeq. This algorithm refactors existing code into a sequence of code edits by using a linter to procedurally sample across the error-free insertions that can be used to sequentially write programs. It outputs edit sequences as text strings consisting of consecutive program diffs. To test LintSeq, we use it to refactor a dataset of instruction + program pairs into instruction + program-diff-sequence tuples. Then, we instruction finetune a series of smaller LLMs ranging from 2.6B to 14B parameters on both the re-factored and original versions of this dataset, comparing zero-shot performance on code synthesis benchmarks. We show that during repeated sampling, edit sequence finetuned models produce more diverse programs than baselines. This results in better inference-time scaling for benchmark coverage as a function of samples, i.e. the fraction of problems "pass@k" solved by any attempt given "k" tries. For example, on HumanEval pass@50, small LLMs finetuned on synthetic edit sequences are competitive with GPT-4 and outperform models finetuned on the baseline dataset by +20% (+/-3%) in absolute score. Finally, we also pretrain our own tiny LMs for code understanding. We show that finetuning tiny models on synthetic code edits results in state-of-the-art code synthesis for the on-device model class. Our 150M parameter edit sequence LM matches or outperforms code models with twice as many parameters, both with and without repeated sampling, including Codex and AlphaCode.

We introduce MAmmoTH, a series of open-source large language models (LLMs) specifically tailored for general math problem-solving. The MAmmoTH models are trained on MathInstruct, our meticulously curated instruction tuning dataset. MathInstruct is compiled from 13 math datasets with intermediate rationales, six of which have rationales newly curated by us. It presents a unique hybrid of chain-of-thought (CoT) and program-of-thought (PoT) rationales, and also ensures extensive coverage of diverse fields in math. The hybrid of CoT and PoT not only unleashes the potential of tool use but also allows different thought processes for different math problems. As a result, the MAmmoTH series substantially outperform existing open-source models on nine mathematical reasoning datasets across all scales with an average accuracy gain between 13% and 29%. Remarkably, our MAmmoTH-7B model reaches 35% on MATH (a competition-level dataset), which exceeds the best open-source 7B model (WizardMath) by 25%, and the MAmmoTH-34B model achieves 46% accuracy on MATH, even surpassing GPT-4's CoT result. Our work underscores the importance of diverse problem coverage and the use of hybrid rationales in developing superior math generalist models.

Neural-symbolic learning, an intersection of neural networks and symbolic reasoning, aims to blend neural networks' learning capabilities with symbolic AI's interpretability and reasoning. This paper introduces an approach designed to improve the performance of neural models in learning reasoning tasks. It achieves this by integrating Answer Set Programming (ASP) solvers and domain-specific expertise, which is an approach that diverges from traditional complex neural-symbolic models. In this paper, a shallow artificial neural network (ANN) is specifically trained to solve Sudoku puzzles with minimal training data. The model has a unique loss function that integrates losses calculated using the ASP solver outputs, effectively enhancing its training efficiency. Most notably, the model shows a significant improvement in solving Sudoku puzzles using only 12 puzzles for training and testing without hyperparameter tuning. This advancement indicates that the model's enhanced reasoning capabilities have practical applications, extending well beyond Sudoku puzzles to potentially include a variety of other domains. The code can be found on GitHub: https://github.com/Fadi2200/ASPEN.

Code generation models have increasingly become integral to aiding software development, offering assistance in tasks such as code completion, debugging, and code translation. Although current research has thoroughly examined the correctness of code produced by code generation models, a vital aspect, i.e., the efficiency of the generated code, has often been neglected. This paper presents EffiBench, a benchmark with 1,000 efficiency-critical coding problems for assessing the efficiency of code generated by code generation models. EffiBench contains a diverse set of LeetCode coding problems. Each problem is paired with an executable human-written canonical solution. With EffiBench, we empirically examine the capability of 21 Large Language Models (13 open-sourced and 8 closed-sourced) in generating efficient code. The results demonstrate that GPT-4-turbo generates the most efficient code, significantly outperforming Palm-2-chat-bison, Claude-instant-1, Gemini-pro, GPT-4, and GPT-3.5. Nevertheless, its code efficiency is still worse than the efficiency of human-written canonical solutions. In particular, the average and worst execution time of GPT-4-turbo generated code is 1.69 and 45.49 times that of the canonical solutions.

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.

Answering complex logical queries on incomplete knowledge graphs is a challenging task, and has been widely studied. Embedding-based methods require training on complex queries, and cannot generalize well to out-of-distribution query structures. Recent work frames this task as an end-to-end optimization problem, and it only requires a pretrained link predictor. However, due to the exponentially large combinatorial search space, the optimal solution can only be approximated, limiting the final accuracy. In this work, we propose QTO (Query Computation Tree Optimization) that can efficiently find the exact optimal solution. QTO finds the optimal solution by a forward-backward propagation on the tree-like computation graph, i.e., query computation tree. In particular, QTO utilizes the independence encoded in the query computation tree to reduce the search space, where only local computations are involved during the optimization procedure. Experiments on 3 datasets show that QTO obtains state-of-the-art performance on complex query answering, outperforming previous best results by an average of 22%. Moreover, QTO can interpret the intermediate solutions for each of the one-hop atoms in the query with over 90% accuracy. The code of our paper is at https://github.com/bys0318/QTO.

Designing solutions for complex engineering challenges is crucial in human production activities. However, previous research in the retrieval-augmented generation (RAG) field has not sufficiently addressed tasks related to the design of complex engineering solutions. To fill this gap, we introduce a new benchmark, SolutionBench, to evaluate a system's ability to generate complete and feasible solutions for engineering problems with multiple complex constraints. To further advance the design of complex engineering solutions, we propose a novel system, SolutionRAG, that leverages the tree-based exploration and bi-point thinking mechanism to generate reliable solutions. Extensive experimental results demonstrate that SolutionRAG achieves state-of-the-art (SOTA) performance on the SolutionBench, highlighting its potential to enhance the automation and reliability of complex engineering solution design in real-world applications.

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 introduce a framework for automatically defining and learning deep generative models with problem-specific structure. We tackle problem domains that are more traditionally solved by algorithms such as sorting, constraint satisfaction for Sudoku, and matrix factorization. Concretely, we train diffusion models with an architecture tailored to the problem specification. This problem specification should contain a graphical model describing relationships between variables, and often benefits from explicit representation of subcomputations. Permutation invariances can also be exploited. Across a diverse set of experiments we improve the scaling relationship between problem dimension and our model's performance, in terms of both training time and final accuracy. Our code can be found at https://github.com/plai-group/gsdm.

Code has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its precision and accuracy. Previous works involving continued mathematical pretraining often include code that utilizes math-related packages, which are primarily designed for fields such as engineering, machine learning, signal processing, or module testing, rather than being directly focused on mathematical reasoning. In this paper, we introduce a novel method for generating mathematical code accompanied with corresponding reasoning steps for continued pretraining. Our approach begins with the construction of a high-quality mathematical continued pretraining dataset by incorporating math-related web data, code using mathematical packages, math textbooks, and synthetic data. Next, we construct reasoning steps by extracting LaTeX expressions, the conditions needed for the expressions, and the results of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding code to accurately capture the mathematical reasoning process. Appending the generated code to each reasoning step results in data consisting of paired natural language reasoning steps and their corresponding code. Combining this data with the original dataset results in a 19.2B-token high-performing mathematical pretraining corpus, which we name MathCode-Pile. Training several popular base models with this corpus significantly improves their mathematical abilities, leading to the creation of the MathCoder2 family of models. All of our data processing and training code is open-sourced, ensuring full transparency and easy reproducibility of the entire data collection and training pipeline. The code is released at https://github.com/mathllm/MathCoder2 .

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.

In nature, the behaviors of many complex systems can be described by parsimonious math equations. Automatically distilling these equations from limited data is cast as a symbolic regression process which hitherto remains a grand challenge. Keen efforts in recent years have been placed on tackling this issue and demonstrated success in symbolic regression. However, there still exist bottlenecks that current methods struggle to break when the discrete search space tends toward infinity and especially when the underlying math formula is intricate. To this end, we propose a novel Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data. The RSRM model is composed of three key modules: (1) a Monte Carlo tree search (MCTS) agent that explores optimal math expression trees consisting of pre-defined math operators and variables, (2) a Double Q-learning block that helps reduce the feasible search space of MCTS via properly understanding the distribution of reward, and (3) a modulated sub-tree discovery block that heuristically learns and defines new math operators to improve representation ability of math expression trees. Biding of these modules yields the state-of-the-art performance of RSRM in symbolic regression as demonstrated by multiple sets of benchmark examples. The RSRM model shows clear superiority over several representative baseline models.

Recent advances in language models have demonstrated their capability to solve mathematical reasoning problems, achieving near-perfect accuracy on grade-school level math benchmarks like GSM8K. In this paper, we formally study how language models solve these problems. We design a series of controlled experiments to address several fundamental questions: (1) Can language models truly develop reasoning skills, or do they simply memorize templates? (2) What is the model's hidden (mental) reasoning process? (3) Do models solve math questions using skills similar to or different from humans? (4) Do models trained on GSM8K-like datasets develop reasoning skills beyond those necessary for solving GSM8K problems? (5) What mental process causes models to make reasoning mistakes? (6) How large or deep must a model be to effectively solve GSM8K-level math questions? Our study uncovers many hidden mechanisms by which language models solve mathematical questions, providing insights that extend beyond current understandings of LLMs.

Selecting the best code solution from multiple generated ones is an essential task in code generation, which can be achieved by using some reliable validators (e.g., developer-written test cases) for assistance. Since reliable test cases are not always available and can be expensive to build in practice, researchers propose to automatically generate test cases to assess code solutions. However, when both code solutions and test cases are plausible and not reliable, selecting the best solution becomes challenging. Although some heuristic strategies have been proposed to tackle this problem, they lack a strong theoretical guarantee and it is still an open question whether an optimal selection strategy exists. Our work contributes in two ways. First, we show that within a Bayesian framework, the optimal selection strategy can be defined based on the posterior probability of the observed passing states between solutions and tests. The problem of identifying the best solution is then framed as an integer programming problem. Second, we propose an efficient approach for approximating this optimal (yet uncomputable) strategy, where the approximation error is bounded by the correctness of prior knowledge. We then incorporate effective prior knowledge to tailor code generation tasks. Both theoretical and empirical studies confirm that existing heuristics are limited in selecting the best solutions with plausible test cases. Our proposed approximated optimal strategy B4 significantly surpasses existing heuristics in selecting code solutions generated by large language models (LLMs) with LLM-generated tests, achieving a relative performance improvement by up to 50% over the strongest heuristic and 246% over the random selection in the most challenging scenarios. Our code is publicly available at https://github.com/ZJU-CTAG/B4.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

This paper proposes a novel admission and routing scheme which takes into account arbitrarily assigned priorities for network flows. The presented approach leverages the centralized Software Defined Networking (SDN) capabilities in order to do so. Exact and heuristic approaches to the stated Priority Flow Admission and Routing (PFAR) problem are provided. The exact approach which provides an optimal solution is based on Integer Linear Programming (ILP). Given the potentially long running time required to find an exact and optimal solution, a heuristic approach is proposed; this approach is based on Genetic Algorithms (GAs). In order to effectively estimate the performance of the proposed approaches, a simulator that is capable of generating semi-random network topologies and flows has been developed. Experimental results for large problem instances (up 50 network nodes and thousands of network flows), show that: i) an optimal solution can be often found in few seconds (even milliseconds), and ii) the heuristic approach yields close-to-optimal solutions (approximately 95\% of the optimal) in a fixed amount of time; these experimental results demonstrate the pertinence of the proposed approaches.

We show that extensive LLM safety fine-tuning is easily subverted when an attacker has access to model weights. We evaluate three state-of-the-art fine-tuning methods-QLoRA, ReFT, and Ortho-and show how algorithmic advances enable constant jailbreaking performance with cuts in FLOPs and optimisation power. We strip safety fine-tuning from Llama 3 8B in one minute and Llama 3 70B in 30 minutes on a single GPU, and sketch ways to reduce this further.

The performance of large language models (LLMs) on existing reasoning benchmarks has significantly improved over the past years. In response, we present JEEBench, a considerably more challenging benchmark dataset for evaluating the problem solving abilities of LLMs. We curate 515 challenging pre-engineering mathematics, physics and chemistry problems from the highly competitive IIT JEE-Advanced exam. Long-horizon reasoning on top of deep in-domain knowledge is essential for solving problems in this benchmark. Our evaluation on various open-source and proprietary models reveals that the highest performance, even after using techniques like self-consistency, self-refinement and chain-of-thought prompting, is less than 40%. The typical failure modes of GPT-4, the best model, are errors in algebraic manipulation, difficulty in grounding abstract concepts into mathematical equations accurately and failure in retrieving relevant domain-specific concepts. We also observe that by mere prompting, GPT-4 is unable to assess risk introduced by negative marking for incorrect answers. For this, we develop a post-hoc confidence-thresholding method over self-consistency, which enables effective response selection. We hope that our challenging benchmark will guide future re-search in problem-solving using LLMs.

Automated Theorem Proving (ATP) in formal languages remains a formidable challenge in AI, demanding rigorous logical deduction and navigating vast search spaces. While large language models (LLMs) have shown promising performance, existing stepwise provers often suffer from biased search guidance, leading to inefficiencies and suboptimal proof strategies. This paper introduces the Multi-Perspective Search Prover (MPS-Prover), a novel stepwise ATP system designed to overcome these limitations. MPS-Prover incorporates two key innovations: a highly effective post-training data curation strategy that prunes approximately 40% of redundant training data without sacrificing performance, and a multi-perspective tree search mechanism. This search integrates a learned critic model with strategically designed heuristic rules to diversify tactic selection, prevent getting trapped in unproductive states, and enhance search robustness. Extensive evaluations demonstrate that MPS-Prover achieves state-of-the-art performance on multiple challenging benchmarks, including miniF2F and ProofNet, outperforming prior 7B parameter models. Furthermore, our analyses reveal that MPS-Prover generates significantly shorter and more diverse proofs compared to existing stepwise and whole-proof methods, highlighting its efficiency and efficacy. Our work advances the capabilities of LLM-based formal reasoning and offers a robust framework and a comprehensive analysis for developing more powerful theorem provers.

Large Language Models (LLMs) struggle to directly generate correct plans for complex multi-constraint planning problems, even with self-verification and self-critique. For example, a U.S. domestic travel planning benchmark TravelPlanner was proposed in Xie et al. (2024), where the best LLM OpenAI o1-preview can only find viable travel plans with a 10% success rate given all needed information. In this work, we tackle this by proposing an LLM-based planning framework that formalizes and solves complex multi-constraint planning problems as constrained satisfiability problems, which are further consumed by sound and complete satisfiability solvers. We start with TravelPlanner as the primary use case and show that our framework achieves a success rate of 93.9% and is effective with diverse paraphrased prompts. More importantly, our framework has strong zero-shot generalizability, successfully handling unseen constraints in our newly created unseen international travel dataset and generalizing well to new fundamentally different domains. Moreover, when user input queries are infeasible, our framework can identify the unsatisfiable core, provide failure reasons, and offers personalized modification suggestions. We show that our framework can modify and solve for an average of 81.6% and 91.7% unsatisfiable queries from two datasets and prove with ablations that all key components of our framework are effective and necessary. Project page: https://sites.google.com/view/llm-rwplanning.

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.

Large Language Models (LLMs) have demonstrated exceptional capabilities in various natural language tasks, often achieving performances that surpass those of humans. Despite these advancements, the domain of mathematics presents a distinctive challenge, primarily due to its specialized structure and the precision it demands. In this study, we adopted a two-step approach for investigating the proficiency of LLMs in answering mathematical questions. First, we employ the most effective LLMs, as identified by their performance on math question-answer benchmarks, to generate answers to 78 questions from the Math Stack Exchange (MSE). Second, a case analysis is conducted on the LLM that showed the highest performance, focusing on the quality and accuracy of its answers through manual evaluation. We found that GPT-4 performs best (nDCG of 0.48 and P@10 of 0.37) amongst existing LLMs fine-tuned for answering mathematics questions and outperforms the current best approach on ArqMATH3 Task1, considering P@10. Our Case analysis indicates that while the GPT-4 can generate relevant responses in certain instances, it does not consistently answer all questions accurately. This paper explores the current limitations of LLMs in navigating complex mathematical problem-solving. Through case analysis, we shed light on the gaps in LLM capabilities within mathematics, thereby setting the stage for future research and advancements in AI-driven mathematical reasoning. We make our code and findings publicly available for research: https://github.com/gipplab/LLM-Investig-MathStackExchange

Although Answer Set Programming (ASP) allows constraining neural-symbolic (NeSy) systems, its employment is hindered by the prohibitive costs of computing stable models and the CPU-bound nature of state-of-the-art solvers. To this end, we propose Answer Set Networks (ASN), a NeSy solver. Based on Graph Neural Networks (GNN), ASNs are a scalable approach to ASP-based Deep Probabilistic Logic Programming (DPPL). Specifically, we show how to translate ASPs into ASNs and demonstrate how ASNs can efficiently solve the encoded problem by leveraging GPU's batching and parallelization capabilities. Our experimental evaluations demonstrate that ASNs outperform state-of-the-art CPU-bound NeSy systems on multiple tasks. Simultaneously, we make the following two contributions based on the strengths of ASNs. Namely, we are the first to show the finetuning of Large Language Models (LLM) with DPPLs, employing ASNs to guide the training with logic. Further, we show the "constitutional navigation" of drones, i.e., encoding public aviation laws in an ASN for routing Unmanned Aerial Vehicles in uncertain environments.

In math reasoning with large language models (LLMs), fine-tuning data augmentation by query evolution and diverse reasoning paths is empirically verified effective, profoundly narrowing the gap between open-sourced LLMs and cutting-edge proprietary LLMs. In this paper, we conduct an investigation for such data augmentation in math reasoning and are intended to answer: (1) What strategies of data augmentation are more effective; (2) What is the scaling relationship between the amount of augmented data and model performance; and (3) Can data augmentation incentivize generalization to out-of-domain mathematical reasoning tasks? To this end, we create a new dataset, AugGSM8K, by complicating and diversifying the queries from GSM8K and sampling multiple reasoning paths. We obtained a series of LLMs called MuggleMath by fine-tuning on subsets of AugGSM8K. MuggleMath substantially achieves new state-of-the-art on GSM8K (from 54% to 68.4% at the scale of 7B, and from 63.9% to 74.0% at the scale of 13B). A log-linear relationship is presented between MuggleMath's performance and the amount of augmented data. We also find that MuggleMath is weak in out-of-domain math reasoning generalization to MATH. This is attributed to the differences in query distribution between AugGSM8K and MATH which suggest that augmentation on a single benchmark could not help with overall math reasoning performance. Codes and AugGSM8K will be uploaded to https://github.com/OFA-Sys/gsm8k-ScRel.