Mathematical Reasoning Benchmarks

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13 papers in the last four weeks, up 117% on the four weeks before. 0.2% of all new papers.

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May 19, 2026cs.LG

CEPO: RLVR Self-Distillation using Contrastive Evidence Policy Optimization

When a model produces a correct solution under reinforcement learning with verifiable rewards (RLVR), every token receives the same reward signal regardless of whether it was a decisive reasoning step or a grammatical filler. A natural fix is to condition the model on the correct answer as a teacher, identifying tokens it would have generated differently had it known the answer. Prior work shows this either corrupts training by leaking the answer into the gradient, or produces a weak signal that cannot distinguish decisive steps from filler, since both look equally surprising relative to the model's baseline. We propose Contrastive Evidence Policy Optimization (CEPO), which asks a sharper question at every token: not just "does the correct answer favor this token?" but "does the correct answer favor it while the wrong answer disfavors it?" A token satisfying both is a genuine reasoning step; one satisfying neither is filler. The wrong-answer teacher is constructed from rejected rollouts already in the training batch, incurring no additional sampling cost. We prove CEPO inherits all structural safety guarantees of the prior state of the art while strictly sharpening credit at decisive tokens, with the improvement vanishing exactly at filler positions. Empirically, CEPO achieves 43.43% and 60.56% average accuracy across five multimodal mathematical reasoning benchmarks at 2B and 4B scale, respectively, versus 41.17% and 57.43% for GRPO under identical training budgets. Distribution-matching self-distillation methods (OPSD, SDPO) fall below the untrained baseline, empirically confirming the information leakage our theory predicts. Our code is available at https://github.com/ahmedheakl/CEPO.

Ahmed Heakl, Abdelrahman M. Shaker, Youssef Mohamed +4

MBZUAI · Linköping University · Australian National University

Reinforcement Learning With Verifiable Reward Mathematical Reasoning Benchmarks Reward Signal Correct

May 17, 2026cs.AI

CAM-Bench: A Benchmark for Computational and Applied Mathematics in Lean

Formal theorem-proving benchmarks enable mechanically verifiable evaluation of mathematical reasoning in large language models. However, existing benchmarks mainly focus on Olympiad-style problems and algebraic domains, leaving computational and applied mathematics underrepresented. We introduce CAM-Bench, a Lean 4 theorem-proving benchmark of 1,000 Lean proof targets in computational and applied mathematics, with coverage spanning optimization, numerical linear algebra, and numerical analysis. These problems are adapted from textbook exercises and often depend on locally introduced definitions, notation, algorithms, and elementary results. To construct CAM-Bench, we develop a dependency-recovery pipeline that reconstructs the local textbook context needed to state each problem faithfully. It then normalizes each problem into a standalone informal theorem and translates it into a Lean target. We validate the resulting formal problems through Lean compilation and semantic review, checking both formal correctness and semantic alignment with the original exercises. For each problem, we release the raw exercise, recovered context, normalized informal theorem, and final Lean target. CAM-Bench complements existing formal mathematics benchmarks by targeting applied mathematics problems that rely on textbook concepts and elementary theorems, many of which are not directly available as standard Mathlib4 lemmas. We evaluate widely used large language models and formalization agents on CAM-Bench, and analyze common failure modes in tracking local assumptions, applying elementary results, decomposing proofs, and maintaining long-horizon control in Lean.

Wentao Long, Yunfei Zhang, Chenyi Li +3

Fudan University · Qingdao University · Peking University +1

Theorem Proving Mathematical Reasoning Benchmarks Mathematics Mathematical Reasoning Benchmark Saturation Camera

May 15, 2026cs.LG

SAGE: Shaping Anchors for Guided Exploration in RLVR of LLMs

Recent studies observe that reinforcement learning with verifiable rewards (RLVR) reliably improves pass@1 on reasoning tasks, yet often fails to yield comparable gains in pass@k, raising the question of whether RLVR genuinely enables large language models to acquire novel reasoning abilities or merely enhances the efficiency of sampling reasoning modes already present in the base model. Prior analyses largely support the latter view, attributing this limitation to structural properties of standard RLVR objectives that result in insufficient exploration pressure. In this work, we argue that a central structural constraint arises from reverse-KL regularization, which stabilizes training but inherently anchors the policy to the reference distribution, thereby suppressing the emergence of alternative reasoning modes. However, we show that neither removing the KL term nor replacing it with forward-KL provides a satisfactory solution, as both disrupt the efficiency-coverage trade-off by either inducing reward hacking or allocating probability mass to off-target regions. To resolve this tension, we propose SAGE, a principled framework that enables controllable empirical support expansion by reshaping the reverse-KL anchor distribution itself through a guide function q(x,y), achieving consistent improvements in both pass@1 and pass@k across challenging mathematical reasoning benchmarks. Our code is available at https://github.com/tally0818/SAGE.

Chanuk Lee, Minki Kang, Sung Ju Hwang

KAIST

Reinforcement Learning With Verifiable Reward LLM Reasoning Strategies Verifiable Rewards Mathematical Reasoning Benchmarks Kullback-Leibler Divergence

May 13, 2026cs.CL

Distribution Corrected Offline Data Distillation for Large Language Models

Distilling reasoning traces from strong large language models into smaller ones is a promising route to improve intelligence in resource-constrained settings. Existing approaches face a fundamental trade-off: offline distillation from teacher-generated traces provides high-quality, sample-efficient supervision but suffers from distributional drift: during training, the student model conditions on teacher-generated prefixes, whereas during inference the student autoregresses on self-generated prefixes, leading to compounding errors over long reasoning trajectories. Meanwhile, on-policy or self-distillation methods better match the student's inference-time distribution, but require costly online sampling and often produce low-quality traces in early training. We propose a principled offline reasoning distillation framework that preserves the efficiency and supervision quality of offline teacher-generated data while correcting teacher-student distribution drift. It adaptively emphasizes teacher supervision that is better aligned with the student's on-policy distribution. Evaluations on mathematical reasoning benchmarks of GSM8K, MATH, MATH500, and harder held-out competition-style tasks, including AMC, AIME, and OlympiadBench, show that our method improves reasoning accuracy over prior offline distillation algorithms and yields more stable reasoning traces while preserving instruction-following capabilities. Our work shows that lightweight, distribution-correction-aware training can substantially strengthen offline reasoning distillation without online rollouts.

Yumeng Zhang, Zhengbang Yang, Yevin Nikhel Goonatilake +1

George Mason University, Fairfax, VA, USA

Probe-Logit Distillation Mathematical Reasoning Benchmarks Reasoning Traces Traces

May 12, 2026cs.LG

OGLS-SD: On-Policy Self-Distillation with Outcome-Guided Logit Steering for LLM Reasoning

We study on-policy self-distillation (OPSD), where a language model improves its reasoning ability by distilling privileged teacher distributions along its own on-policy trajectories. Despite its promise, OPSD can suffer from training instability due to a pattern mismatch between teacher and student responses. Self-reflected teacher responses may introduce reflection-induced biases and response templates that miscalibrate token-level supervision, ultimately harming the student's reasoning ability. To mitigate this issue, we propose OGLS-SD, an outcome-guided logit-steering framework that leverages verifiable outcome rewards to calibrate privileged teacher logits. Specifically, OGLS-SD contrasts teacher logits induced by successful and failed on-policy trajectories, constructing an outcome-discriminative steering direction for token-level guidance. Experiments on mathematical reasoning benchmarks show that OGLS-SD stabilizes self-distillation and improves performance over standard OPSD and other variants.

Yuxiao Yang, Xiaoyun Wang, Weitong Zhang

UNC Chapel Hill · NVIDIA

Unsupervised On-Policy Self-Distillation Token-Level Supervision LLM Reasoning Strategies Mathematical Reasoning Benchmarks Logit Lens

May 12, 2026cs.AI

From Noise to Diversity: Random Embedding Injection in LLM Reasoning

Recent soft prompt research has tried to improve reasoning by inserting trained vectors into LLM inputs, yet whether the gain comes from the learned content or from the act of injection itself has not been carefully separated. We study Random Soft Prompts (RSPs), which drop the training step entirely and append a freshly drawn sequence of random embedding vectors to the input. Each RSP vector is sampled from an isotropic Gaussian fitted to the entrywise mean and variance of the pretrained embedding table; the sequence carries no learned content, and yet reaches accuracy comparable to optimized soft prompts on math reasoning benchmarks in several settings. The mechanism unfolds in two stages: because attention has to absorb a never-seen-before random position, the distribution over the first few generated tokens flattens and reasoning trajectories branch, and as generation continues this influence dilutes naturally so the response commits to a single completion. We show that during inference RSPs lift early-stage token diversity and, combined with temperature sampling, widen Pass@N, the probability that at least one out of N attempts is correct. Beyond inference, we carry the same effect into DAPO training and demonstrate practical gains. Our contributions are: (i) RSP isolates the simplest form of soft prompt -- training-free, freshly resampled -- providing a unified lens for the structural effect of injection that variants otherwise differing in training and form all share; (ii) a theoretical and empirical validation of the underlying mechanism; and (iii) an extension from inference to training.

Heejun Kim, Seungpil Lee, Jewon Yeom +5

1Gwangju Institute of Science and Technology · 3Microsoft Research · 2Seoul National University

LLM Reasoning Strategies Automatic Prompt Optimization Mathematical Reasoning Benchmarks Diversity Frictive Policy Optimization

May 12, 2026cs.AI

Seirênes: Adversarial Self-Play with Evolving Distractions for LLM Reasoning

We present Seirênes, a self-play RL framework that transforms contextual interference from a failure mode of LLM reasoning into an internal training signal for co-evolving more resilient reasoners. While RL with verifiable rewards has significantly advanced reasoning capabilities, models can still exhibit fragility when encountering non-idealized contexts: scenarios characterized by superfluous information, tangential instructions, or incidental correlations that differ from the clean distributions typical of standard benchmarks. Seirênes harnesses this vulnerability through a parameter-shared and adversarial self-play loop. Within this framework, a single model is trained to both construct plausible yet distracting contexts that expose its own reasoning blind spots, and solve problems by discerning the essential task from these perturbations to recover the core underlying logic. By pitting these competing objectives against each other, Seirênes compels the model to move beyond superficial pattern matching and anchors its capabilities in robust underlying reasoning. This continuous interaction sustains an informative co-evolutionary curriculum as the model improves. Across seven mathematical reasoning benchmarks and model scales from 4B to 30B, Seirênes achieves average gains of +10.2, +9.1, and +7.2 points. Besides, distracting contexts produced by the 4B Seirênes model reduce the accuracy of top-tier closed-source models (GPT and Gemini) by roughly 4--5 points, revealing Seirênes' general ability to uncover reasoning models' blind spots.

Chi Zhang, Haibo Qiu, Qiming Zhang +3

School of Computer Science, Wuhan University · Meituan Inc · Independent Researchers +1

LLM Reasoning Strategies Recursive Reasoner Mathematical Reasoning Benchmarks Self-Play Distractor Interference

May 12, 2026cs.LG

Understanding and Preventing Entropy Collapse in RLVR with On-Policy Entropy Flow Optimization

Reinforcement learning with verifiable rewards (RLVR) has become an effective paradigm for improving the reasoning ability of large language models. However, widely used RLVR algorithms, such as GRPO, often suffer from entropy collapse, leading to premature determinism and unstable optimization. Existing remedies, including entropy regularization and ratio-based clipping heuristics, either control entropy in a coarse-grained manner or rely on approximate on-policy training. In this paper, we revisit entropy collapse from a token-level entropy flow perspective. Our analysis reveals that entropy-decreasing tokens consistently outweigh entropy-increasing ones, resulting in a severely imbalanced entropy flow. This perspective provides a unified explanation of entropy collapse in existing RLVR algorithms and highlights the importance of balancing entropy dynamics. Motivated by this analysis, we propose On-Policy Entropy Flow Optimization (OPEFO), an adaptive entropy flow balancing mechanism that rescales entropy-increasing and entropy-decreasing updates according to their contributions to entropy change, while remaining strict on-policy. Experiments on six mathematical reasoning benchmarks demonstrate that OPEFO improves training stability and final performance. We will release the code and models upon publication.

Huimin Xu, Shuai Zhao, Xiaobao Wu +1

Nanyang Technological University, Singapore · Shanghai Jiao Tong University, China · 3VinUniversity, Vietnam

Reinforcement Learning With Verifiable Reward Token-Level Entropy Verifiable Rewards Collapse Mathematical Reasoning Benchmarks

May 12, 2026cs.LG

fg-expo: Frontier-guided exploration-prioritized policy optimization via adaptive kl and gaussian curriculum

Reinforcement Learning with Verifiable Rewards (RLVR) has become the standard paradigm for LLM mathematical reasoning, with Group Relative Policy Optimization (GRPO) serving as the dominant algorithm. We identify two overlooked inefficiencies inherent in GRPO. First, a fixed KL coefficient overly restricts policy exploration at moments when the model needs to diverge significantly from the reference policy. Second, uniform question sampling overlooks that moderately difficult problems produce the most informative gradient signals. We propose FG-ExPO, short for Frontier-Guided Exploration-Prioritized Policy Optimization, which integrates two lightweight components. Accuracy-Conditioned KL Scaling (AKL) adjusts the KL penalty strength through a smooth nonlinear function of batch average accuracy, loosening the constraint when the model performs poorly and strengthening it when the model achieves satisfactory results. Gaussian Curriculum Sampling (GCS) assigns sampling weights to questions following a Gaussian distribution centered at a moderate accuracy level around 0.5, focusing model training on its learning frontier. We conduct evaluations on DeepSeek-R1-Distill-Qwen-1.5B and Qwen3-8B-Base across six mainstream mathematical reasoning benchmarks. Experimental results demonstrate that FG-ExPO consistently outperforms vanilla GRPO. It delivers an absolute improvement of 13.34 on the AIME 2025 pass@32 metric, rising from 63.33 percent to 76.67 percent, and obtains an average pass@32 gain of 2.66 on the 8B model. The substantially larger performance gains observed on pass@32 compared to pass@1 verify that FG-ExPO enlarges the model's effective exploration space under a fixed inference budget.

Mingxiong Lin, Zhangquan Gong, Maowen Tang +6

OPPO AI Center

Reinforcement Learning With Verifiable Reward Mathematical Reasoning Benchmarks Kullback-Leibler Divergence

May 11, 2026cs.AI

expo: Exploration-prioritized policy optimization via adaptive kl regulation and gaussian curriculum sampling

Reinforcement Learning with Verifiable Rewards (RLVR) has become the standard paradigm for LLM mathematical reasoning, where Group Relative Policy Optimization (GRPO) serves as the mainstream algorithm. We point out two understudied inefficiencies existing in GRPO. First, the fixed KL penalty coefficient overly restricts policy exploration at stages where the model requires significant deviation from the reference policy. Second, uniform sampling of training questions ignores that moderately difficult problems provide the most informative gradient signals for optimization. We propose Exploration-Prioritized Policy Optimization (EXPO) with two lightweight plug-in modules. The Accuracy-Conditioned KL Scaling (AKL) dynamically adjusts KL regularization strength through a smooth nonlinear function of batch average accuracy, relaxing the penalty when the model underperforms and strengthening it when the model achieves good results. The Gaussian Curriculum Sampling (GCS) assigns sampling weights to questions following a Gaussian distribution centered at moderate accuracy around 0.5, focusing training on the model's learning frontier. We conduct extensive experiments on DeepSeek-R1-Distill-Qwen-1.5B and Qwen3-8B-Base over six mathematical reasoning benchmarks. The results show EXPO steadily surpasses vanilla GRPO. It obtains an absolute gain of 13.34 on AIME 2025 pass@32, rising from 63.33 percent to 76.67 percent, and achieves an average pass@32 improvement of 2.66 on the 8B model. The much larger performance gains on pass@32 compared with pass@1 demonstrate that EXPO effectively enlarges the model's exploration boundary under a fixed inference cost budget.

Mingxiong Lin, Zhangquan Gong, Maowen Tang +6

OPPO AI Center

Reinforcement Learning With Verifiable Reward Mathematical Reasoning Benchmarks Rejection Sampling Kullback-Leibler Regularization Kullback-Leibler Divergence

May 11, 2026cs.LG

Verifier-Free RL for LLMs via Intrinsic Gradient-Norm Reward

While Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a promising post-training paradigm for Large Language Models (LLMs), its dependency on the gold label or domain-specific verifiers limits its scalability to new tasks and domains. In this work, we propose Verifier-free Intrinsic Gradient-Norm Reward (VIGOR), a simple reward that uses only the policy model itself. Given a prompt, VIGOR samples a group of completions and assigns higher within-group rewards to outputs that induce smaller

\ell_2

norms of the teacher-forced negative log-likelihood gradients under the current parameters. Intuitively, lower gradient norms suggest the completion aligns better with the current policy, serving as an intrinsic preference signal for policy optimization. To make this intrinsic signal practical for RL, we correct the systematic length bias of averaged token-level gradients with a

\sqrt{T}

scaling, and apply group-wise rank shaping to stabilize reward scales across prompts. Across mathematical reasoning benchmarks, VIGOR outperforms the state-of-the-art Reinforcement Learning from Internal Feedback (RLIF) baseline, and it also exhibits cross-domain transfer to code benchmarks when trained only on math data. For instance, on Qwen2.5-7B-Base post-trained on MATH, VIGOR improves the average math accuracy by +3.31% and the average code accuracy by +1.91% over this baseline, while exhibiting more stable training dynamics. The code is available at https://github.com/ZJUSCL/VIGOR.

Xuexiang Wen, Hang Yu, Linchao Zhu +1

1Zhejiang University · 2Ant Group

Reinforcement Learning With Verifiable Reward Reward Gradients Mathematical Reasoning Benchmarks

May 10, 2026cs.LG

LEAD: Length-Efficient Adaptive and Dynamic Reasoning for Large Language Models

Large reasoning models, such as OpenAI o1 and DeepSeek-R1, tend to become increasingly verbose as their reasoning capabilities improve. These inflated Chain-of-Thought (CoT) trajectories often exceed what the underlying problems require, wasting compute, latency, and context budgets. While introducing length-based efficiency rewards during reinforcement learning offers a natural remedy, existing methods struggle with two fundamental challenges: the optimal balance between correctness and efficiency is non-stationary throughout training, and intrinsic reasoning budgets vary drastically across problems. Relying on static reward weights and global length constraints inevitably forces a compromise between degraded accuracy and unrealized compression. To overcome these limitations, we propose LEAD (Length-Efficient Adaptive and Dynamic reasoning), a method that replaces static heuristics with online, self-adaptive mechanisms. LEAD dynamically calibrates the correctness-efficiency trade-off at each step using a Potential-Scaled Instability, directing optimization capacity to the most informative learning signal. Furthermore, it estimates an adaptive per-problem target length online based on the model's own correct rollouts, applying a symmetric efficiency reward that penalizes both overthinking and over-compression. Evaluated on five mathematical reasoning benchmarks, LEAD achieves the highest accuracy and Accuracy-Efficiency Score among RL-trained efficient-reasoning methods while producing substantially shorter outputs than the base model.

Songtao Wei, Yi Li, Zhikai Li +7

University of Texas at Dallas · Emory University · University of Texas at Arlington +2

Large Reasoning Models Large Language Model Reinforcement Learning Mathematical Reasoning Benchmarks Token Budget Allocation

May 10, 2026cs.AI

Beyond Accuracy: Evaluating Strategy Diversity in LLM Mathematical Reasoning

Large language models now achieve high final-answer accuracy on mathematical reasoning benchmarks, but accuracy alone does not capture reasoning flexibility. We introduce a strategy-level evaluation framework instantiated on 80 AMC 10/12 and AIME problems with 217 AoPS-derived reference strategy families. Model outputs are annotated for strategy identity, validity, and correctness using dual-AI coding with human adjudication. Across four frontier models, we find a pronounced decoupling between answer accuracy and strategy diversity. Under a single-solution prompt, all models achieve high accuracy (95%-100%), but under a multiple-strategy prompt they recover substantially fewer strategies than the human reference set. Gemini, DeepSeek, GPT, and Claude generate 184, 152, 151, and 110 distinct valid strategies, respectively, with the largest gaps in Geometry and Number Theory. The models collectively produce 50 benchmark-novel valid strategies, indicating both incomplete coverage of human strategies and some capacity for alternative reasoning. A repeated-run robustness check on 20 problems shows diminishing gains in discovered strategies, with the strongest model recovering only 39 of 55 AoPS-reference strategies (71%) after three runs. These findings position strategy diversity as a complementary dimension for evaluating mathematical reasoning beyond answer correctness.

Xia Yang, Xuanyi Zhang, Hao Hu +1

University of Toronto · Upper Canada College · East China Normal University

Mathematical Reasoning Mathematical Reasoning Benchmarks Diversity

May 9, 2026cs.CL

Soohak: A Mathematician-Curated Benchmark for Evaluating Research-level Math Capabilities of LLMs

Following the recent achievement of gold-medal performance on the IMO by frontier LLMs, the community is searching for the next meaningful and challenging target for measuring LLM reasoning. Whereas olympiad-style problems measure step-by-step reasoning alone, research-level problems use such reasoning to advance the frontier of mathematical knowledge itself, emerging as a compelling alternative. Yet research-level math benchmarks remain scarce because such problems are difficult to source (e.g., Riemann Bench and FrontierMath-Tier 4 contain 25 and 50 problems, respectively). To support reliable evaluation of next-generation frontier models, we introduce Soohak, a 439-problem benchmark newly authored from scratch by 64 mathematicians. Soohak comprises two subsets. On the Challenge subset, frontier models including Gemini-3-Pro, GPT-5, and Claude-Opus-4.5 reach 30.4%, 26.4%, and 10.4% respectively, leaving substantial headroom, while leading open-weight models such as Qwen3-235B, GPT-OSS-120B, and Kimi-2.5 remain below 15%. Notably, beyond standard problem solving, Soohak introduces a refusal subset that probes a capability intrinsic to research mathematics: recognizing ill-posed problems and pausing rather than producing confident but unjustified answers. On this subset, no model exceeds 50%, identifying refusal as a new optimization target that current models do not directly address. To prevent contamination, the dataset will be publicly released in late 2026, with model evaluations available upon request in the interim.

Guijin Son, Seungone Kim, Catherine Arnett +73

Research-Level Mathematics Olympiad-Level Problem Mathematical Reasoning Benchmarks Frontier Models Frontiers

May 8, 2026cs.LG

Your Language Model is Its Own Critic: Reinforcement Learning with Value Estimation from Actor's Internal States

Reinforcement learning with verifiable rewards (RLVR) for Large Reasoning Models rests on variance reduction, which requires both a reliable baseline and high prompt diversity within each training batch. This is especially difficult in multi-domain training for general reasoning models, where prompts from different tasks induce highly diverse gradient signals. Existing approaches fall short in different ways: GRPO estimates its baseline as the group mean over rollouts from the same prompt, so an accurate baseline leaves fewer distinct prompts in the batch, while PPO avoids this trade-off by training a policy scale critic, roughly doubling the cost of training. We introduce POISE (Policy Optimization with Internal State Value Estimation), a reinforcement learning algorithm that turns the model's internal states into a value model. A lightweight probe reads the signals already computed during the forward pass to predict the baseline, and is trained online alongside the policy. To preserve gradient unbiasedness, we introduce a cross-rollout construction that predicts each rollout's value from an independent rollout's internal states. On Qwen3-4B and OLMo3-7B-Instruct-DPO across a six-domain verifiable-reward corpus, POISE outperforms other RLVR baselines while achieving more stable training. Moreover, the probe matches a separate LLM-scale value model, generalizes to various tasks, and remains accurate as the policy scales. By leveraging the model's internal representations, POISE enables stable policy optimization.

Yunho Choi, Jongwon Lim, Woojin Ahn +3

Graduate School of Data Science Seoul National University · Computer Science and Engineering Seoul National University

Reinforcement Learning With Verifiable Reward Verifiable Rewards Mathematical Reasoning Benchmarks Intermediate States

May 8, 2026cs.AI

Efficient Data Selection for Multimodal Models via Incremental Optimization Utility

The scaling of Large Multimodal Models (LMMs) is constrained by the quality-quantity trade-off inherent in synthetic data. Previous approaches, such as LLM-as-a-Judge, have proven their effectiveness in addressing this but suffer from prohibitive computational costs and lack of interpretability. To bridge this gap, we propose One-Step-Train (OST), a framework that reformulates data selection as an incremental optimization utility ranking problem. Instead of relying on semantic heuristics, OST estimates the marginal utility of each sample via a simulated single-step update on a lightweight proxy. Experiments on the Qwen series across multimodal mathematical reasoning benchmarks demonstrate that OST achieves Pareto-optimal efficiency. By selecting the top-50 subset, OST reduces training costs by 43% (and total time consumption by 17) while surpassing the strong LLM-as-a-Judge baseline by 1.8 points. Furthermore, under a fixed compute budget, our method using only the top-20 subset achieves a 5.6 point gain over LLM-as-a-Judge, improves upon heuristic scoring baselines like DEITA, and outperforms the Full-SFT baseline by 8.8 points. Notably, while Full-SFT suffers from performance degradation due to noise, our optimization-grounded approach effectively identifies toxic samples, successfully reversing the negative transfer frequently observed in complex reasoning tasks.

Jinhao Jing, Qiannian Zhao, Chao Huang +1

Large Multimodal Models Multimodal Model Mathematical Reasoning Benchmarks Llm-As-A-Judge

May 8, 2026cs.CL

Teaching Language Models to Think in Code

Tool-integrated reasoning (TIR) has emerged as a dominant paradigm for mathematical problem solving in language models, combining natural language (NL) reasoning with code execution. However, this interleaved setup has three key limitations: code often acts as a post-hoc verifier, intermediate NL computations are error-prone, and NL and code play overlapping rather than clearly distinct roles. We propose ThinC (Thinking in Code), a framework in which code itself serves as the reasoner rather than as a tool invoked by NL. A ThinC trajectory begins with a brief NL planning step, after which all reasoning unfolds through code blocks connected only by their execution outputs. We distill 12.2k code-centric trajectories from a teacher model and train ThinC-1.7B and ThinC-4B with supervised fine-tuning followed by reinforcement learning. ThinC-4B consistently outperforms every TIR baseline on five competition-level math benchmarks and even surpasses the much larger Qwen3-235B-A22B-Thinking. Further analysis shows that ThinC reasons through code: 99.2% of its final answers are grounded in interpreter output, and the model recovers reliably from code execution failures without intermediate NL reasoning. Our code and models will be released soon.

Hyeon Hwang, Jiwoo Lee, Jaewoo Kang

Korea University · Korea University, AIGEN Sciences

Mathematical Reasoning Benchmarks Language Modeling Single Trajectory

May 8, 2026cs.CL

GSM-SEM: Benchmark and Framework for Generating Semantically Variant Augmentations

Benchmarks like GSM8K are popular measures of mathematical reasoning, but leaderboard gains can overstate true capability due to memorization of fixed test sets. Most robustness variants apply surface-level perturbations (paraphrases, renamings, number swaps, distractors) that largely preserve the underlying facts, and static releases can themselves become memorization targets over time. We introduce GSM-SEM, a reusable and stochastic framework for generating semantically diverse benchmark variants with substantially higher semantic variance than prior approaches. GSM-SEM perturbs problem statements by modifying entities, attributes, and/or relationships, frequently altering underlying facts and requiring models to recompute solutions under new conditions, while constraining generation to preserve the original calculations/answer and approximate problem difficulty. GSM-SEM generates fresh variants on each run without requiring re-annotation, reducing reliance on static public benchmarks for evaluation and thereby lowering the bias of memorization. We apply GSM-SEM on GSM8K and two existing variation suites (GSM-Symbolic and GSM-Plus), producing GSM8K-SEM, GSM-Symbolic-SEM, and GSM-Plus-SEM. Evaluating 14 SOTA LLMs, we observe consistent performance drops with larger decline when semantic perturbations are coupled with symbolic/plus variations (average drop rate 28% in maximum strictness configuration of GSM-SEM). We publicly release the three SEM variants as fully human-validated datasets. Finally, to demonstrate applicability beyond GSM-style math problems, we apply GSM-SEM to additional benchmarks including BigBenchHard, LogicBench, and NLR-BIRD.

Jyotika Singh, Fang Tu, Aziza Mirsaidova +11

Oracle AI

Mathematical Reasoning Benchmarks Mathematical Reasoning Memorization

May 7, 2026cs.AI

OPSD Compresses What RLVR Teaches: A Post-RL Compaction Stage for Reasoning Models

On-Policy Self-Distillation (OPSD) has recently emerged as an alternative to Reinforcement Learning with Verifiable Rewards (RLVR), promising higher accuracy and shorter responses through token-level credit assignment from a self-teacher conditioned on privileged context. However, this promise does not carry over to thinking-enabled mathematical reasoning, where reported accuracy gains shrink and sometimes turn negative. We hypothesize that hindsight supervision can specify better token-level alternatives in short thinking-disabled outputs, but in long thinking-enabled traces it more readily identifies redundancy than supplies better replacements. To test this, we applied OPSD separately to correct and incorrect rollout groups, so that compression and correction can be observed in isolation. Our results show that in thinking-enabled mathematical reasoning, OPSD behaves most reliably as a compression mechanism rather than a correction mechanism: training only on correct rollouts preserves accuracy while substantially shortening responses, whereas training only on incorrect rollouts damages accuracy. In light of these findings, we propose a revised post-training pipeline for thinking-enabled mathematical reasoning: SFT then RLVR then OPSD.

Jaehoon Kim, Dongha Lee

Yonsei University

Mathematical Reasoning Benchmarks Unsupervised On-Policy Self-Distillation Reinforcement Learning With Verifiable Reward Large Reasoning Models Correction Compaction

May 1, 2026cs.LG

Evaluating the Architectural Reasoning Capabilities of LLM Provers via the Obfuscated Natural Number Game

While Large Language Models have achieved notable success on formal mathematics benchmarks such as MiniF2F, it remains unclear whether these results stem from genuine logical reasoning or semantic pattern matching against pre-training data. This paper identifies Architectural Reasoning: the ability to synthesize formal proofs using exclusively local axioms and definitions within an alien math domain, as the necessary ability for future automated theorem discovery AI. We use the Obfuscated Natural Number Game, a benchmark to evaluate Architectural Reasoning. By renaming identifiers in the Natural Number Game in Lean 4, we created a zero-knowledge, closed environment. We evaluate state-of-the-art models, finding a universal latency tax where obfuscation increases inference time. The results also reveal a divergence in robustness: while general models (Claude-Sonnet-4.5, GPT-4o) suffer performance degradation, reasoning models (DeepSeek-R1, GPT-5, DeepSeek-Prover-V2) maintain the same accuracy despite the absence of semantic cues. These findings provide a quantitative metric for assessing the true capacity for mathematical reasoning.

Lixing Li

St Anne’s College, Oxford, OX26HS, United Kingdom

Mathematical Reasoning Theorem Proving Mathematical Reasoning Benchmarks

May 1, 2026cs.CL

Beyond Benchmarks: MathArena as an Evaluation Platform for Mathematics with LLMs

Large language models (LLMs) are becoming increasingly capable mathematical collaborators, but static benchmarks are no longer sufficient for evaluating progress: they are often narrow in scope, quickly saturated, and rarely updated. This makes it hard to compare models reliably and track progress over time. Instead, we need evaluation platforms: continuously maintained systems that run, aggregate, and analyze evaluations across many benchmarks to give a comprehensive picture of model performance within a broad domain. In this work, we build on the original MathArena benchmark by substantially broadening its scope from final-answer olympiad problems to a continuously maintained evaluation platform for mathematical reasoning with LLMs. MathArena now covers a much wider range of tasks, including proof-based competitions, research-level arXiv problems, and formal proof generation in Lean. Additionally, we maintain a clear evaluation protocol for all models and regularly design new benchmarks as model capabilities improve to ensure that MathArena remains challenging. Notably, the strongest model, GPT-5.5, now reaches 98% on the 2026 USA Math Olympiad and 74% on research-level questions, showing that frontier models can now comfortably solve extremely challenging mathematical problems. This highlights the importance of continuously maintained evaluation platforms like MathArena to track the rapid progress of LLMs in mathematical reasoning.

Jasper Dekoninck, Nikola Jovanović, Tim Gehrunger +4

ETH Zurich · INSAIT · Sofia University "St. Kliment Ohridski"

Mathematical Reasoning Benchmarks Mathematical Reasoning Mathematics

May 1, 2026cs.LG

Decouple before Integration: Test-time Synthesis of SFT and RLVR Task Vectors

SFT and RLVR represent two fundamental yet distinct paradigms for LLM post-training, each excelling in distinct dimensions. SFT expands knowledge breadth while RLVR enhances reasoning depth. Yet integrating these complementary strengths remains a formidable challenge. Sequential training can cause catastrophic forgetting, and joint optimization often suffers from severe gradient conflicts. We analyze SFT and RLVR through the lens of task vectors and reveal three structural properties behind these failures: a 30* magnitude disparity, 45* sign interference, and heterogeneous module-wise update distributions. These findings show SFT and RLVR are difficult to integrate directly, but they also suggest that the two paradigms modify partly complementary components of the model. Motivated by these observations, we propose Decoupled Test-time Synthesis (DoTS), a post-hoc framework allows SFT and RLVR checkpoints to be trained independently and synthesizes their capabilities only at inference time via task vector arithmetic, without updating model parameters. To reduce interference, DOTS applies selective sparsification with norm-preserving rescaling. It then uses Bayesian optimization on a small set of unlabeled queries to search for combination coefficients on the Pareto frontier of consistency and perplexity. Empirically, \ours matches or exceeds the performance of training-based SFT--RLVR integration methods across multiple mathematical reasoning benchmarks, incurring only

\sim

3% of the computational cost. When applied to stronger post-trained checkpoints, DOTS surpasses SOTA models and generalizes to out-of-domain benchmarks without re-tuning. Code is available at https://github.com/chaohaoyuan/DoTS.

Chaohao Yuan, Chenghao Xiao, Yu Rong +2

Department of Systems Engineering and Engineering Management, Chinese University of Hong Kong, Hong Kong, China · DAMO Academy, Alibaba Group, Hangzhou, China · Hupan Lab, Hangzhou, China +1

Supervised Finetuning Reinforcement Learning With Verifiable Reward Decoupling Mathematical Reasoning Benchmarks Invariant Synthesis

Apr 29, 2026cs.LG

PAINT: Partial-Solution Adaptive Interpolated Training for Self-Distilled Reasoners

Improving large language model (LLM) reasoning requires supervision that is both aligned with the model's own test-time states and informative at the token level. Reinforcement learning with verifiable rewards provides on-policy exploration but offers sparse, high-variance credit; supervised fine-tuning and distillation provide dense targets but often train on fixed trajectories or rely on stronger teachers. Recent privileged on-policy self-distillation explores a middle ground by scoring student rollouts with the same model under verified solution context. We revisit this setting through a contextual re-scoring lens: for reasoning, the important choices are not only whether privileged context is available, but how much of it should be revealed and where its distribution should shape the student. We propose PAINT (Partial-solution Adaptive INterpolated Training), which masks the verified solution according to rollout-reference overlap and applies a small energy-space interpolation on a sparse set of entropy-mismatch token positions. Across competition-level math benchmarks, PAINT consistently improves over a strong prior on-policy self-distillation baseline at all three Qwen3 scales. On Qwen3-8B, it raises macro Avg@12 by 2.1 points over this prior baseline and 2.9 points over GRPO.

Zhiquan Tan, Yinrong Hong

Tsinghua University · Beihang University

LLM Reasoning Strategies Mathematical Reasoning Benchmarks

Apr 29, 2026cs.CL

Shorthand for Thought: Compressing LLM Reasoning via Entropy-Guided Supertokens

Reasoning in Large Language Models incurs significant inference-time compute, yet the token-level information structure of reasoning traces remains underexplored. We observe that reasoning tokens split into two functional types: low-entropy \textit{structural} tokens (recurring phrases that scaffold the reasoning process) and higher-entropy \textit{organic} tokens (problem-specific content that drives toward a solution). This asymmetry motivates a simple, model-agnostic compression pipeline: apply cross-word BPE merges on a model's own reasoning traces to derive \textit{supertokens} that capture frequent structural patterns, then teach the model to adopt them via supervised fine-tuning. Across three model families and five mathematical reasoning benchmarks, our approach shortens reasoning traces by 8.1% on average with no statistically significant accuracy loss on any model--benchmark pair. Beyond compression, supertokens act as interpretable reasoning-move annotations (backtracking, verification, strategy shifts), exposing the model's high-level strategy at a glance. Analyzing transitions between structural categories reveals systematic differences between correct and incorrect traces: correct traces show productive recovery (backtracking followed by strategy shifts and verification), while incorrect traces are dominated by confusion cycles (repeated hedging and unresolved contradictions). These diagnostic signals suggest applications in reward shaping and early stopping for RL-based reasoning training.

Zhenyu Zhao, Sander Land, Daniel M. Bikel +1

Writer, Inc.

LLM Reasoning Strategies Reasoning Traces Thoughts Mathematical Reasoning Benchmarks Short

Apr 24, 2026cs.AI

Rethinking Math Reasoning Evaluation: A Robust LLM-as-a-Judge Framework Beyond Symbolic Rigidity

Recent advancements in large language models have led to significant improvements across various tasks, including mathematical reasoning, which is used to assess models' intelligence in logical reasoning and problem-solving. Models are evaluated on mathematical reasoning benchmarks by verifying the correctness of the final answer against a ground truth answer. A common approach for this verification is based on symbolic mathematics comparison, which fails to generalize across diverse mathematical representations and solution formats. In this work, we offer a robust and flexible alternative to rule-based symbolic mathematics comparison. We propose an LLM-based evaluation framework for evaluating model-generated answers, enabling accurate evaluation across diverse mathematical representations and answer formats. We present failure cases of symbolic evaluation in two popular frameworks, Lighteval and SimpleRL, and compare them to our approach, demonstrating clear improvements over commonly used methods. Our framework enables more reliable evaluation and benchmarking, leading to more accurate performance monitoring, which is important for advancing mathematical problem-solving and intelligent systems.

Erez Yosef, Oron Anschel, Shunit Haviv Hakimi +4

Tel-Aviv University · Amazon Prime Video · Ben-Gurion University

Mathematical Reasoning Mathematical Reasoning Benchmarks Symbol-Llm-As-A-Judge Rigid Objects

Apr 22, 2026cs.CL

TRACES: Tagging Reasoning Steps for Adaptive Cost-Efficient Early-Stopping

The field of Language Reasoning Models (LRMs) has been very active over the past few years with advances in training and inference techniques enabling LRMs to reason longer, and more accurately. However, a growing body of studies show that LRMs are still inefficient, over-generating verification and reflection steps. Additionally, the high-level role of each reasoning step and how different step types contribute to the generation of correct answers, is largely underexplored. To address this challenge, we introduce TRACES (Tagging of the Reasoning steps enabling Adaptive Cost-Efficient early-Stopping), a lightweight framework that tags reasoning steps in real-time, and enable adaptive, cost-efficient early stopping of large-language-model inferences. Building on this framework we monitor reasoning behaviors during inferences, and we find that LRMs tend to shift their reasoning behavior after reaching a correct answer. We demonstrate that the monitoring of the specific type of steps can produce effective interpretable early stopping criteria. We evaluate the TRACES framework on three mathematical reasoning benchmarks, namely, MATH500, GSM8K, AIME and two knowledge and reasoning benchmarks, MMLU and GPQA respectively. We achieve 20 to 50% token reduction while maintaining comparable accuracy to standard generation.

Yannis Belkhiter, Seshu Tirupathi, Giulio Zizzo +1

IBM Research Europe · Trinity College Dublin · Dublin, Ireland +1

Large Reasoning Models Progressive Reasoning Mathematical Reasoning Benchmarks Early Stopping Tagging

Apr 20, 2026cs.CL

Multiplication in Multimodal LLMs: Computation with Text, Image, and Audio Inputs

Multimodal LLMs can accurately perceive numerical content across modalities yet fail to perform exact multi-digit multiplication when the identical underlying arithmetic problem is presented as numerals, number words, images, or in audio form. Because existing benchmarks often lack systematically paired instances across modalities, it remains difficult to compare genuine arithmetic limits within and across model families. We therefore introduce a controlled multimodal multiplication benchmark that factorially varies digit length, digit sparsity, representation (e.g., numerals vs. number words), and modality (text, rendered images, audio), with paired instances from a reproducible generator. We also define arithmetic load, C, as the product of the total and non-zero digit count as a compact, mechanistically motivated proxy for operation count. Across evaluations, accuracy falls sharply as C grows, often nearing zero by C > 100. Indeed, C remains predictive of performance across modalities and models, with R-squared often > 0.5, nearing the value from more complex measures of arithmetic load that count the number of intermediate arithmetic steps. A separate perception-versus-computation decomposition shows that multimodal degradation is primarily computational rather than perceptual: on matched-perception checks, models are near-perfect (> 99%) across modalities, even when multiplication accuracy drops. Beyond measuring when models fail, we ask which procedures they are predisposed to follow. We introduce a forced-completion loss probe that scores heuristic-specific reasoning prefixes--including columnar multiplication, distributive decomposition, and rounding/compensation. Here, decomposition is favored in both text and vision modalities; heuristic-specific LoRA adapters produce near-orthogonal updates yet degrade accuracy, indicating the base model maintains a well-tuned internal router.

Samuel G. Balter, Ethan Jerzak, Connor T. Jerzak

University of Texas at Austin · National University of Singapore (NUS)

Multimodal Large Language Models Modalities Multi-Turn Benchmark Task Arithmetic Mathematical Reasoning Benchmarks Text Analysis and Detection

Apr 20, 2026cs.AI

Stratagem: Learning Transferable Reasoning via Trajectory-Modulated Game Self-Play

Games offer a compelling paradigm for developing general reasoning capabilities in language models, as they naturally demand strategic planning, probabilistic inference, and adaptive decision-making. However, existing self-play approaches rely solely on terminal game outcomes, providing no mechanism to distinguish transferable reasoning patterns from game-specific heuristics. We present STRATAGEM, which addresses two fundamental barriers to reasoning transfer: domain specificity, where learned patterns remain anchored in game semantics, and contextual stasis, where static game contexts fail to cultivate progressive reasoning. STRATAGEM selectively reinforces trajectories exhibiting abstract, domain-agnostic reasoning through a Reasoning Transferability Coefficient, while incentivizing adaptive reasoning development via a Reasoning Evolution Reward. Experiments across mathematical reasoning, general reasoning, and code generation benchmarks demonstrate substantial improvements, with particularly strong gains on competition-level mathematics where multi-step reasoning is critical. Ablation studies and human evaluation confirm that both components contribute to transferable reasoning.

Xiachong Feng, Deyi Yin, Xiaocheng Feng +9

The University of Hong Kong · Harbin Institute of Technology · Harbin Institute of Technology, Shenzhen

Mathematical Reasoning Benchmarks Imperfect-Information Games

Apr 14, 2026cs.CL

Peer-Predictive Self-Training for Language Model Reasoning

Mechanisms for continued self-improvement of language models without external supervision remain an open challenge. We propose Peer-Predictive Self-Training (PST), a label-free fine-tuning framework in which multiple language models improve collaboratively by using a cross-model aggregate response as an internal training signal. Given a prompt, models generate responses sequentially; the final aggregated answer, which is often more reliable than individual responses in practice, serves as an internal reference for learning. We measure how informative each intermediate response is about the aggregate using pointwise mutual information (PMI), and use this signal to scale self-training updates: responses already aligned with the aggregate receive smaller updates, while less informative or misaligned responses receive larger ones. On mathematical reasoning benchmarks, including SimulEq, MATH-500-Numeric, and MultiArith, PST improves exact-match accuracy by 2.2--4.3 percentage points across Gemma-2-2B, LLaMA-3.2-1B, and Qwen2.5-1.5B, and reduces the average generator--verifier gap (GV-Gap) by 26--40%, while requiring no external supervision, no teacher--student hierarchy, and only cross-model interactions. These results suggest that peer-predictive feedback from cross-model generations can provide an effective mechanism for self-supervised language-model improvement.

Shi Feng, Hanlin Zhang, Fan Nie +2

Harvard University · Stanford University

Mathematical Reasoning Benchmarks

Apr 8, 2026cs.AI

Riemann-Bench: A Benchmark for Moonshot Mathematics

Recent AI systems have achieved gold-medal-level performance on the International Mathematical Olympiad, demonstrating remarkable proficiency at competition-style problem solving. However, competition mathematics represents only a narrow slice of mathematical reasoning: problems are drawn from limited domains, require minimal advanced machinery, and can often reward insightful tricks over deep theoretical knowledge. We introduce Riemann-Bench, a private benchmark of expert-curated problems designed to evaluate AI systems on research-level mathematics that goes far beyond the olympiad frontier. Problems are authored by Ivy League mathematics professors, graduate students, and PhD-holding IMO medalists, and routinely took their authors weeks to solve independently. Each problem undergoes double-blind verification by two independent domain experts who must solve the problem from scratch, and yields a unique, closed-form solution assessed by programmatic verifiers. We evaluate frontier models as unconstrained research agents, with full access to coding tools, search, and open-ended reasoning, using an unbiased statistical estimator computed over 100 independent runs per problem. Our results reveal that all frontier models currently score below 10%, exposing a substantial gap between olympiad-level problem solving and genuine research-level mathematical reasoning. By keeping the benchmark fully private, we ensure that measured performance reflects authentic mathematical capability rather than memorization of training data.

Suhaas Garre, Erik Knutsen, Sushant Mehta +1

Surge AI

Research-Level Mathematics Olympiad-Level Problem Mathematical Reasoning Benchmarks Artificial Intelligence Benchmarks Benchmark Saturation

Apr 6, 2026cs.CL

What Makes Good Multilingual Reasoning? Disentangling Traces with Measurable Features

Large Reasoning Models (LRMs) still exhibit large performance gaps between English and other languages, yet much current work assumes these gaps can be closed simply by making reasoning in every language resemble English reasoning. This work challenges this assumption by asking instead: what actually characterizes successful reasoning traces in multilingual settings, and to what extent do English-derived reasoning features genuinely help in other languages? We first define a suite of measurable reasoning features spanning multilingual alignment, reasoning step, and reasoning flow aspects of reasoning traces, and use logistic regression to quantify how each feature associates with final answer accuracy. We further train sparse autoencoders over multilingual traces to automatically discover latent reasoning concepts that instantiate or extend these features. Finally, we use the features to re-rank traces and measure their impact on accuracy at test time. Across two mathematical reasoning benchmarks, four LRMs, and ten languages, we find that most features are positively associated with accuracy, but the strength of association varies considerably across languages and can even reverse in some. Our findings challenge English-centric reward designs and point toward adaptive objectives that accommodate language-specific reasoning patterns, with concrete implications for multilingual benchmark and reward design.

Dayeon Ki, Kevin Duh, Marine Carpuat

University of Maryland · Johns Hopkins University

Multilingual Benchmark Reasoning Traces Mathematical Reasoning Benchmarks Traces

Apr 1, 2026cs.AI

RefineRL: Advancing Competitive Programming with Self-Refinement Reinforcement Learning

While large language models (LLMs) have demonstrated strong performance on complex reasoning tasks such as competitive programming (CP), existing methods predominantly focus on single-attempt settings, overlooking their capacity for iterative refinement. In this paper, we present RefineRL, a novel approach designed to unleash the self-refinement capabilities of LLMs for CP problem solving. RefineRL introduces two key innovations: (1) Skeptical-Agent, an iterative self-refinement agent equipped with local execution tools to validate generated solutions against public test cases of CP problems. This agent always remains skeptical of its own outputs and thereby enforces rigorous self-refinement even when validation suggests correctness. (2) A reinforcement learning (RL) solution to incentivize LLMs to self-refine with only standard RLVR data (i.e., problems paired with their verifiable answers). Extensive experiments on Qwen3-4B and Qwen3-4B-2507 demonstrate that our method yields substantial gains: after our RL training, these compact 4B models integrated with the Skeptical-Agent not only outperform much larger 32B models but also approach the single-attempt performance of 235B models. Notably, although our models are trained solely on CP coding data, RefineRL also substantially improves their performance on mathematical reasoning benchmarks, suggesting that the learned self-refinement behavior can transfer beyond coding-specific tasks. These findings highlight self-refinement as a promising direction for scaling LLMs' general reasoning ability.

Shaopeng Fu, Xingxing Zhang, Li Dong +1

Work done during an internship at Microsoft Research · KAUST · Microsoft Research

Large Language Model Reinforcement Learning LLM Reasoning Strategies Mathematical Reasoning Benchmarks Refinement Offline Reinforcement Learning

Mar 19, 2026cs.RO

Can LLMs Prove Robotic Path Planning Optimality? A Benchmark for Research-Level Algorithm Verification

Robotic path planning problems are often NP-hard, and practical solutions typically rely on approximation algorithms with provable performance guarantees for general cases. While designing such algorithms is challenging, formally proving their approximation optimality is even more demanding, which requires domain-specific geometric insights and multi-step mathematical reasoning over complex operational constraints. Recent Large Language Models (LLMs) have demonstrated strong performance on mathematical reasoning benchmarks, yet their ability to assist with research-level optimality proofs in robotic path planning remains under-explored. In this work, we introduce the first benchmark for evaluating LLMs on approximation-ratio proofs of robotic path planning algorithms. The benchmark consists of 34 research-grade proof tasks spanning diverse planning problem types and complexity levels, each requiring structured reasoning over algorithm descriptions, problem constraints, and theoretical guarantees. Our evaluation of state-of-the-art proprietary and open-source LLMs reveals that even the strongest models struggle to produce fully valid proofs without external domain knowledge. However, providing LLMs with task-specific in-context lemmas substantially improves reasoning quality, a factor that is more effective than generic chain-of-thought prompting or supplying the ground-truth approximation ratio as posterior knowledge. We further provide fine-grained error analysis to characterize common logical failures and hallucinations, and demonstrate how each error type can be mitigated through targeted context augmentation.

Zhengbang Yang, Md. Tasin Tazwar, Minghan Wei +1

George Mason University. · Florida Atlantic University.

Robot Planning Large Language Model Planning Mathematical Reasoning Benchmarks Approximation Algorithms Optimality

Feb 6, 2026cs.CL

TTSR: Test-Time Self-Evolving via Reflection

Test-time training (TTT) adapts large language models (LLMs) during inference using only unlabeled test inputs. Existing methods, however, face two major bottlenecks on hard reasoning tasks: (1) \emph{lack of learnable samples}, as self-generated pseudo-labels on difficult questions are often noisy and yield unstable rewards; and (2) \emph{inefficient exploration}, as performance gains depend on repeatedly sampling many rollouts without explicit diagnosis of why previous attempts fail. We propose \textbf{TTSR} (\textbf{T}est-\textbf{T}ime \textbf{S}elf-\textbf{R}eflection), a self-evolving framework based on a \emph{reflect-then-synthesize} paradigm. A single pretrained model alternates between a \textit{Student} role and a \textit{Teacher} role: the Student solves test questions and updates, while the Teacher analyzes failed trajectories and synthesizes targeted variant questions closer to the Student's capability frontier. TTSR further maintains a cross-iteration \textit{weakness memory} and compiles persistent weaknesses into a lightweight \textit{strategy note} prepended to subsequent Student inputs, so diagnostic knowledge can guide exploration and gradually fade as weaknesses are resolved. Experiments on challenging mathematical reasoning benchmarks show consistent test-time improvements, strong cross-backbone generalization, and transfer to general-domain reasoning tasks.

Haoyang He, Zihua Rong, Yunjia Zhao +3

Beijing University of Posts and Telecommunications · Southwestern University of Finance and Economics · China Unicom Online Information Technology Co., Ltd.

Test-Time Training Test Time Mathematical Reasoning Benchmarks Reasoning Skills Reflections

Feb 1, 2026cs.AI

Not All Preferences Deserve Gradients: Understanding Gradient Utility in Offline Reasoning Alignment

Offline preference optimization aligns reasoning models from fixed chosen--rejected pairs, yet standard methods apply gradient updates from every pair regardless of its training value under the current policy. We argue that this uniform treatment is wasteful and potentially harmful. From the perspective of gradient utility, we show that a pair's contribution depends jointly on informativeness and stability. Pair utility drifts as the policy evolves, high-gradient samples can coincide with high-curvature regions, leading to noisy and destabilizing updates, and the most effective supervision comes from stable confident errors where the model is reliably wrong yet curvature remains low. These findings motivate SAGE (Stability-Aware Gradient Efficiency), which maintains difficulty-stratified candidate pools refreshed during training and selects pairs within each pool by a forward-pass signal-to-curvature score. Only pairs with high current utility receive gradient computation; the rest are excluded from backpropagation. On mathematical reasoning benchmarks across multiple model scales, SAGE outperforms full-data and size-matched baselines while producing substantially smoother optimization trajectories.

Hui Wu, Hengyi Cai, Jinman Zhao +6

Baidu Inc. · University of Toronto · School of Intelligence Science and Technology, Peking University +1

Gradient Preference Alignment Learning Mathematical Reasoning Benchmarks

Dec 2, 2025cs.LG

ScalePRM: Training Process Reward Models by Scaling Verification Compute Without Ground Truth

Training process reward models (PRMs) requires step-level correctness labels, obtained either through expensive human annotation or by relying on ground-truth answers, limiting the ability to scale process-level supervision. We propose ScalePRM, which scales verification compute as an alternative: given a problem and a candidate solution, we generate multiple independent verifications of each reasoning step and aggregate their judgments to produce synthetic step-level labels without ground truth. We explore two representative inference-time scaling strategies, parallel scaling through self-consistency and sequential scaling through meta-critique, and train generative PRMs on the resulting synthetic data. On ProcessBench, a benchmark for identifying erroneous steps in mathematical reasoning, PRMs trained on step-level self-consistency data achieve 67.5 F1, surpassing reference-guided training with ground-truth access (66.4 F1) and GPT-4o as a critic (61.9 F1). When deployed as reward signals in RL training with Qwen2.5-Math-7B, our best PRM achieves 47.4% average accuracy across six mathematical reasoning benchmarks, outperforming ground-truth-based RLVR (43.9%). We also identify and address reward exploitation patterns unique to generative PRM-based RL. Our results demonstrate that scaling verification compute is a viable alternative to ground-truth supervision for training process reward models.

Salman Rahman, Sruthi Gorantla, Arpit Gupta +3

2UCLA · Work done while as an intern at Amazon AGI. · 1Amazon AGI

Process Reward Model Mathematical Reasoning Benchmarks Reinforcement Learning With Verifiable Reward Ground Truth Scaling Self-Consistency

Oct 21, 2025cs.LG

A Model Can Help Itself: Reward-Free Self-Training for LLM Reasoning

Can language models improve their reasoning performance without external rewards, using only their own sampled responses for training? We show that they can. We propose Self-evolving Post-Training (SePT), a simple post-training method that alternates between self-generation and training on self-generated responses. It repeatedly samples questions, uses the model itself to generate responses under a specified sampling temperature, and then trains the model on the self-generated data. In this self-training loop, we use an online data refresh mechanism, where each new batch is generated by the most recently updated model. Across six math reasoning benchmarks, SePT improves a strong no-training baseline, defined as the untuned base model evaluated at its best swept decoding temperature, on several tested models. Additional ablations demonstrate the importance of online data refresh and temperature dynamics. Overall, our results identify a practical regime where reasoning can be improved using self-generated supervision alone. Our code is available at https://github.com/ElementQiii/SePT.

Mengqi Li, Lei Zhao, Anthony Man-Cho So +2

1The Chinese University of Hong Kong, Shenzhen · 2Shanghai Jiao Tong University · 3The Chinese University of Hong Kong

LLM Reasoning Strategies Mathematical Reasoning Benchmarks Post-Training

Oct 10, 2025cs.AI

LiveOIBench: Can Large Language Models Outperform Human Contestants in Informatics Olympiads?

Competitive programming problems are increasingly used to evaluate the coding capabilities of large language models (LLMs) due to their complexity and ease of verification. Yet, current coding benchmarks face limitations such as a lack of exceptionally challenging problems, insufficient test case coverage, and reliance on online platform APIs that limit accessibility. To address these issues, we introduce LiveOIBench, a large-scale competitive programming benchmark featuring 403 expert-curated problems, averaging 60 official test cases each, drawn from 72 contests across 14 Informatics Olympiads held between 2023 and 2025. LiveOIBench has four key features: (1) expert-designed tasks with detailed subtask rubrics and extensive test cases; (2) direct comparison to elite human contestants; (3) continuous updates to reduce contamination risk; and (4) a fully offline, reproducible evaluation system. Benchmarking 34 popular general-purpose and reasoning LLMs, we find that GPT-5 achieves an 81.76th percentile, still falling short of top human contestants, while among the open-weight models, GPT-OSS-120B reaches only the 60th percentile. Reasoning-trace analyses indicate that robust reasoning models prioritize precise problem analysis over excessive exploration. Finally, analyses across release dates, task familiarity, and code similarity find minimal evidence of data contamination in our benchmark. Our leaderboard, code, and data are available at: https://liveoibench.github.io/.

Kaijian Zou, Aaron Xiong, Yunxiang Zhang +6

University of Michigan, Ann Arbor.

Raw Judge Outputs Large Language Model Benchmarks Mathematical Reasoning Benchmarks Reasoning Traces

Sep 27, 2025cs.LG

WirelessMathBench-XL: An Auditable Benchmark for Wireless Mathematical Reasoning

Technical-domain benchmarks constructed from arXiv papers can overlap the same public text used in LLM pretraining. Auditing this risk at training-corpus scale requires searching billions of corpus n-grams while retaining per-item evidence that users can inspect and recompute. We contribute a reverse-probe audit at a fixed 13-gram threshold: it indexes benchmark prompts, streams public pretraining corpora, and emits per-problem prompt-surface lexical-overlap metadata with memory that scales with the benchmark. We instantiate the protocol in WirelessMathBench-XL, a 4,027-problem wireless mathematical-reasoning benchmark built from 836 retained arXiv papers across 20 subfields. Against 12.9B streamed 13-grams from RedPajama-arXiv, the audit identifies a strict zero-hit view S0 covering 3,853 problems (95.7%). Filtering to S0 changes accuracy by less than 1 pp for every evaluated model; frontier calibration rows form one high-accuracy cluster between 86.5% and 91.3%, not a resolved rank order. Only 30/800 test items carry detected overlap. Under an all-flagged-correct counterfactual, their largest possible positive score inflation is 0.31-0.51 pp for the frontier rows, so full-versus-S0 is a bounded, structurally underpowered stability summary rather than a contamination-effect test or cleanliness claim. The audit channel does not cover paraphrase, target-answer, post-training, or closed-corpus exposure. The release includes source-paper identifiers, verifier-facing ground truths, audit and threshold metadata, filtered views, a paper-disjoint sensitivity view, evaluation traces, paired-bootstrap scripts, training recipes, Croissant metadata, and a Datasheet for Datasets.

Xin Li, Mengbing Liu, Yiyang Zhu +4

Nanyang Technological University

Mathematical Reasoning Benchmarks Model Auditing Text Corpora Openalex Metadata Wifi

Aug 13, 2025cs.LG

Nested-ReFT: Efficient Reinforcement Learning for Large Language Model Fine-Tuning via Off-Policy Rollouts

Advanced reasoning in LLMs on challenging domains like mathematical reasoning can be tackled using verifiable rewards based reinforced fine-tuning (ReFT). In standard ReFT frameworks, a behavior model generates multiple completions with answers per problem, for the answer to be then scored by a reward function. While such RL post-training methods demonstrate significant performance improvements across challenging reasoning domains, the computational cost of generating completions during training with multiple inference steps makes the training cost non-trivial. To address this, we draw inspiration from off-policy RL, and speculative decoding to introduce a novel ReFT framework, dubbed Nested-ReFT, where a subset of layers of the target model acts as the behavior model to generate off-policy completions during training. The behavior model configured with dynamic layer skipping per batch during training decreases the inference cost compared to the standard ReFT frameworks. Our theoretical analysis shows that Nested-ReFT yields unbiased gradient estimates with controlled variance. Our empirical analysis demonstrates improved computational efficiency measured as tokens/sec across multiple math reasoning benchmarks and model sizes. Additionally, we explore three variants of bias mitigation to minimize the off-policyness in the gradient updates that allows for maintaining performance that matches the baseline ReFT performance.

Maxime Heuillet, Yufei Cui, Boxing Chen +2

Universit´e Laval (IID), Canada · Mila - Qu´ebec AI Institute, Canada · Huawei Noah’s Ark Lab (Montreal Research Center), Canada +1

Mathematical Reasoning Benchmarks Large Language Model Fine-Tuning Model Fine-Tuning Offline Reinforcement Learning Reward Functions Code Completion

Jul 22, 2025cs.CV

SyncLoop: A Multimodal Dual-Loop Framework for Self-Improving Mathematical Reasoning

Recent advances in multimodal large language models (MLLMs) have shown impressive reasoning capabilities. However, further enhancing existing MLLMs necessitates high-quality vision-language datasets with carefully curated task complexities, which are both costly and challenging to scale. Although recent self-improving models that iteratively refine themselves offer a feasible solution, they still suffer from two core challenges: (i) most existing methods augment visual or textual data separately, resulting in discrepancies in data complexity (e.g., over-simplified diagrams paired with redundant textual descriptions); and (ii) the evolution of data and models is also separated, leading to scenarios where models are exposed to tasks with mismatched difficulty levels. To address these issues, we propose C2-Evo, an automatic, closed-loop self-improving framework that jointly evolves both training data and model capabilities. Specifically, given a base dataset and a base model, C2-Evo enhances them by a cross-modal data evolution loop and a data-model evolution loop. The former loop expands the base dataset by generating complex multimodal problems that combine structured textual sub-problems with iteratively specified geometric diagrams, while the latter loop adaptively selects the generated problems based on the performance of the base model, to conduct supervised fine-tuning and reinforcement learning alternately. Consequently, our method continuously refines its model and training data, and consistently obtains considerable performance gains across multiple mathematical reasoning benchmarks. Our code, models, and datasets will be released.

Xiuwei Chen, Wentao Hu, Hanhui Li +9

Sun Yat-sen University · The Hong Kong Polytechnic University · MBZUAI +2

Multimodal Reasoning Benchmarks Mathematical Reasoning Benchmarks Mathematical Reasoning Multimodal Model Cross-Modal Diagrams