数学推理

Large Reasoning Models (LRMs) excel at complex reasoning but suffer from inefficient reasoning, like overthinking and overshoot. These issues stem from excessive or misdirected reasoning triggered by the model's "doubt", manifested as self-validation and exploratory extension, increasing computational cost and degrading performance. Existing efficient reasoning methods seek to regulate reasoning via internal signals or static schedules, lacking specialization in the "doubt" characteristics of LRMs. To address this, we propose a Conjunction-Guided Intervention (CGI) reasoning framework that intervenes when the model shows signs of doubt. Our key insight is that overthinking and overshoot in LRMs arise from conjunction-triggered extensions where LRMs exhibit "doubt" through transitional conjunctions, extending redundant self-validation and exploration without timely state-based correction. Building on this insight, CGI pauses reasoning at conjunction markers of doubt for external state-based feedback, adaptively extending or terminating reasoning to reduce redundancy while preserving accuracy. The feedback is generated via criteria evaluation (rationality and completeness) and comes from either human or LLM proxies. We train the target model by Group Relative Policy Optimization (GRPO) to adapt to the interactive mode. Experiments show that our framework achieves a superior balance between accuracy and reasoning length.

Asynchronous reinforcement learning (RL) has shown notable success in accelerating the post-training of large language models (LLMs). However, its decoupled data generation and training paradigm introduces a fundamental distributional mismatch between data generated by stale behavior policies and current policy, leading to unstable training and degraded performance. To address this challenge, we propose D-ARL, a **D**istribution-matched **A**synchronous **R**einforcement **L**earning framework that selects high-quality asynchronous samples whose distributions are well aligned with the current policy for policy optimization. Specifically, D-ARL maintains a replay buffer that collects samples from the most recent $K$ behavior policies and proposes a variance-guided metric to select distribution-matched data. During training, D-ARL introduces a multi-behavior policy optimization algorithm to leverage the multi-source nature of the selected samples for policy update. Experiments on six widely used reasoning benchmarks show that D-ARL outperforms state-of-the-art asynchronous methods, achieving an average improvement of 6.4\% in reasoning performance and 34.7\% in sample efficiency.

Large language models can exhibit emergent reasoning behaviors, often manifested as recurring lexical patterns (e.g., “wait,” indicating verification). However, complex reasoning trajectories remain sparse in unconstrained sampling, and standard RL often fails to guarantee the acquisition of diverse reasoning behaviors. We propose a systematic discovery and reinforcement of diverse reasoning patterns through structured reasoning, a paradigm that requires targeted exploration of specific reasoning patterns during the RL process. To this end, we propose Ctrl-R, a framework for learning structured reasoning via tractable trajectory control that actively guides the rollout process, incentivizing the exploration of diverse reasoning patterns that are critical for complex problem-solving. The resulting behavior policy enables accurate importance-sampling estimation, supporting unbiased on-policy optimization. We further introduce a power-scaling factor on the importance-sampling weights, allowing the policy to selectively learn from exploratory, out-of-distribution trajectories while maintaining stable optimization. Experiments demonstrate that Ctrl-R enables effective exploration and internalization of previously unattainable reasoning patterns, yielding consistent improvements across language and vision–language models on mathematical reasoning tasks.

What does it mean to understand the world? Is it simply to predict future video frames? Developmental cognitive science suggests that understanding the world is fundamentally the process of constructing internal theories of how it works rather than mere prediction, even before language is acquired. However, in machine learning, it remains unclear how to endow AI systems with such theory-building capability from raw, non-textual observation alone. In this paper, we introduce Learning-to-Theorize (L2T), a learning paradigm in which an AI system acquires the ability to construct theories represented as executable programs directly from observation alone. To instantiate this paradigm, we propose the Neural Language-of-Thought Programmer, a neural model that induces and executes latent programs as explanations rather than task-specific predictors or policies. In experiments, we show that this formulation enables explanation-driven generalization, allowing observations to be understood in terms of the programs that generate them.

Test-time compute scaling allocates inference computation uniformly, uses fixed sampling strategies, and applies verification only for reranking. In contrast, we propose a verifier-guided adaptive framework treating reasoning as iterative trajectory generation and selection. For each problem, the agent runs multiple inference iterations. In each iteration, it optionally produces a high-level plan, selects a set of reasoning tools and a compute strategy together with an exploration parameter, and then generates a candidate reasoning trajectory. A process reward model (PRM) serves as a unified control signal: within each iteration, step-level PRM scores are aggregated to guide pruning and expansion during generation, and across iterations, aggregated trajectory rewards are used to select the final response. Across datasets, our dynamic, PRM-guided approach consistently outperforms direct test-time scaling, yielding large gains on MATH-500 and several-fold improvements on harder benchmarks such as AIME24 and AMO-Bench. We characterize efficiency using theoretical FLOPs and a compute intensity metric penalizing wasted generation and tool overhead, demonstrating that verification-guided allocation concentrates computation on high-utility reasoning paths.

Informal mathematics has been central to modern large language model (LLM) reasoning, offering flexibility and efficient construction of arguments. However, purely informal reasoning is prone to logical gaps and subtle errors that are difficult to detect and correct. In contrast, formal theorem proving provides rigorous, verifiable mathematical reasoning, where each inference step is checked by a trusted compiler, but lacks the exploratory freedom of informal problem-solving. This mismatch leaves current LLM-based math agents without a principled way to combine the strengths of both paradigms. In this work, we introduce Hermes, the first tool-assisted agent that explicitly interleaves informal reasoning with formally verified proofs in Lean. The framework performs intermediate formal checking to prevent reasoning drift and a memory module for proof continuity across multi-step reasoning chains, enabling both exploration and verification. We evaluate Hermes on four challenging mathematical reasoning benchmarks using LLMs of varying parameter scales, from small models to state-of-the-art systems. Across all settings, Hermes reliably improves the reasoning accuracy of base models while substantially reducing reasoning token usage and computational cost compared to reward-based approaches. On difficult datasets such as AIME and HARDMath2, Hermes@1 achieves up to a 40% accuracy improvement while using 80% fewer total inference FLOPs. When scaled at test time, Hermes@5 boosts accuracy further by 20%.

It is widely recognized that reinforcement learning (RL) fine-tuning of large language models often leads to \textit{diversity collapse}, where outputs lack variety. Prior work has proposed a range of heuristics to counteract this effect, but these methods are ad hoc: they frequently trade off correctness for diversity, their effectiveness varies across tasks, and in some cases they even contradict one another. In this work, we place these observations on a rigorous foundation. We first provide a formal proof of why RL fine-tuning exhibits diversity collapse via a selection and reinforcement bias. Next, we make a key observation that any reward modification to address diversity collapse only needs to be applied on the correct trajectories. Building directly on this analysis, we introduce a principled method---\textit{differential smoothing}---that provably improves both correctness and diversity, outperforming vanilla RL as well as widely used entropy-based heuristics. Our theory precisely characterizes when existing heuristics help and why they fail, while showing that differential smoothing is universally superior. Extensive experiments with models from 1B to 7B parameters, across domains including CountDown and real-world mathematical reasoning, demonstrate consistent gains. Differential smoothing improves both Pass@1 and Pass@k, with up to 6.7\% improvements on AIME24 dataset.

Grading in large undergraduate STEM courses often yields minimal feedback due to heavy instructional workloads. We present a large-scale empirical study of AI grading on real, handwritten single-variable calculus work from the University of California, Irvine. Using OCR-conditioned large language models with structured, rubric-guided prompting, our system produces scores and formative feedback for thousands of handwritten, in-person proctored free-response quiz submissions from nearly 800 students included in the paper’s empirical analysis. In a setting with no single ground-truth label, we evaluate performance against official teaching-assistant grades, student surveys, and independent human review, finding strong alignment with TA scoring and a large majority of AI-generated feedback rated correct or acceptable across quizzes. Beyond calculus, this setting highlights core challenges in OCR-conditioned mathematical reasoning and partial-credit assessment. We analyze key failure modes, propose practical rubric- and prompt-design principles, and introduce a multi-perspective evaluation protocol for reliable real-course deployment. Building on the dataset and evaluation framework developed here, we outline a path toward a future standardized benchmark for AI grading of handwritten mathematics to support reproducible evaluation, transparent comparison, reliable deployment, and future research.

Reinforcement learning with verifiable rewards can improve LLM reasoning, but learning remains sample-inefficient when terminal rewards are sparse. This has motivated a growing line of work on RL with textual feedback, where a critic model generates natural language feedback to guide a reasoning model (the actor), augmenting scalar rewards with richer learning signals. However, existing methods typically treat feedback as fixed or auxiliary, which misses a key property: feedback should not merely be correct, but should improve the policy (actor model) when provided in context. This motivates a paradigm of \textit{learnable textual feedback for RL}. Yet the learnability and usefulness of feedback depend on the policy's ability to learn from it, making RL with learnable feedback an inherently bilevel problem. We formalize this coupling as a Stackelberg bilevel program and derive \emph{Bilevel Natural Language Actor-Critic} (Bi-NAC), which jointly trains a critic to generate reward-improving feedback and an actor to exploit it. Across MATH-500, MBPP, and GPQA, Bi-NAC improves sample and parameter efficiency over RL and fixed-critic baselines: our 2B model outperforms the 3B GRPO baseline, achieving 46.6% versus 41.4% on MATH-500, while our 6B model surpasses the 7B GRPO baseline, achieving 49.3% versus 43.6% on GPQA. Our code is available at [GitHub Repository](https://github.com/sid-in-the-loop/Bi-Natural-Language-Actor-Critic/).

Chain-of-thought (CoT) reasoning and its variants have substantially improved the performance of language models on complex reasoning tasks, yet the precise mechanisms by which different strategies facilitate generalization remain poorly understood. While current explanations often point to increased test-time computation or structural guidance, establishing a consistent, quantifiable link between these factors and generalization remains challenging. In this work, we identify *intrinsic dimensionality* as a quantitative measure for characterizing the effectiveness of reasoning chains. Intrinsic dimensionality quantifies the minimum number of model dimensions needed to reach a given accuracy threshold on a given task. By keeping the model architecture fixed and varying the task formulation through different reasoning strategies, we demonstrate that effective reasoning strategies consistently reduce the intrinsic dimensionality of the task. Validating this on GSM8K with Gemma-3 1B and 4B, we observe a strong inverse correlation between the intrinsic dimensionality of a reasoning strategy and its generalization performance on both in-distribution and out-of-distribution data. Our findings suggest that effective reasoning chains facilitate learning by better compressing the task using fewer parameters, offering a new quantitative metric for analyzing reasoning processes.

A widely adopted strategy for model enhancement is to use synthetic data generated by a stronger model for supervised fine-tuning (SFT). However, for emerging reasoning models like Qwen3-8B, this approach often fails to improve reasoning capabilities and can even lead to a substantial drop in performance. In this work, we identify substantial stylistic divergence between teacher generated data and the distribution of student as a major factor impacting SFT. To bridge this gap, we propose a Teacher–Student Cooperation Data Synthesis framework (TESSY), which interleaves teacher and student models to alternately generate style and non-style tokens. Consequently, TESSY produces synthetic sequences that inherit the advanced reasoning capabilities of the teacher while maintaining stylistic consistency with the distribution of the student. In experiments on code generation using GPT-OSS-120B as the teacher, fine-tuning Qwen3-8B on teacher-generated data leads to performance drops of 3.25% on LiveCodeBench-Pro and 10.02% on OJBench, whereas TESSY achieves improvements of 11.25% and 6.68%.

Large Language Models have achieved strong performance on reasoning tasks with objective answers by generating step-by-step solutions, but diagnosing where a multi-step reasoning trace might fail remains difficult. Confidence estimation offers a natural diagnostic signal, yet existing methods are restricted to final answers or require internal model access. We introduce Stepwise Confidence Attribution (SCA), a framework for closed-source LLMs that assigns step-level confidence based only on generated reasoning traces. SCA applies the Information Bottleneck principle: steps aligning with consensus structures across correct solutions receive high confidence, while deviations are flagged as potentially erroneous. We propose two complementary methods: (1) NIBS, a non-parametric IB approach measuring consistency without graph structures, and (2) GIBS, a graph-based IB model that learns subgraphs through a differentiable mask to capture logical variability. Extensive experiments on mathematical reasoning and multi-hop question answering show that SCA reliably identifies low-confidence steps strongly correlated with reasoning errors. Moreover, using step-level confidence to guide self-correction improves the correction success rate by up to 13.5\% over answer-level feedback.

Large reasoning models achieve strong performance through test-time scaling, but this incurs substantial computational overhead due to long decoding from short prompts. While sparse attention can reduce latency and memory usage, existing methods often degrade reasoning accuracy because selection errors accumulate over long generation horizons, or require costly retraining. We introduce LessIsMore, a training-free sparse attention mechanism for long-horizon reasoning. Our key insight is that token importance in reasoning is global and stable: critical tokens are largely shared across attention heads and remain stable over decoding steps. Guided by this structure, LessIsMore enforces cross-head unified token selection and preserves recent context via a stable recency window, yielding a globally consistent token set that can be reused across layers. Across multiple model families and challenging reasoning benchmarks, LessIsMore matches or improves accuracy while attending to substantially fewer tokens. With kernel-level optimizations, LessIsMore achieves up to $1.6\times$ end-to-end decoding speedup and up to $1.72\times$ faster sparse attention computation, with additional long-context results demonstrating the generality of our approach.

The veracious semantic alignment in autoformalization is significant for formal mathematical reasoning. However, existing evaluations provide only opaque binary verdicts or scalar scores, offering no interpretable insight into where or why translations fail. This opacity severely limits both human understanding and automated system improvement. To bridge this gap, we introduce FormalRx, a comprehensive diagnostic evaluation framework that transforms autoformalization assessment from black-box judgments into actionable feedback. At its core is SCI Error Taxonomy, a hierarchical classification scheme decomposing autoformalization errors into 28 distinct categories with strict priority ordering. Building on this taxonomy, FormalRx provides four critical diagnostic capabilities: alignment verdicts, error categorization, error localization, and correction. We instantiate the framework with a diagnostic model FormalRx-8B, trained on 56,287 synthetically generated samples with fine-grained diagnostic annotations, and release FormalRx-Test as the first fine-grained diagnostic benchmark. FormalRx-8B achieves F1-scores of 0.88 (verdict) and 0.71 (categorization), along with accuracies of 0.75 (localization) and 0.73 (correction), substantially outperforming both general-purpose LLMs and specialized baselines. By connecting evaluation with actionable insights, FormalRx enables systematic diagnosis and improvement of autoformalization systems.

Large Reasoning Models (LRMs) excel at solving complex problems by explicitly generating a reasoning trace before deriving the final answer. However, these extended generations incur substantial memory footprint and computational overhead, bottlenecking LRMs' efficiency. This work uses attention maps to analyze the influence of reasoning traces and uncover an interesting phenomenon: *only some decision-critical tokens in a reasoning trace steer the model toward the final answer, while the remaining tokens contribute negligibly.* Building on this observation, we propose **Dyn**amic **T**hinking-Token **S**election (**DynTS**). This method identifies decision-critical tokens and retains only their associated Key-Value (KV) cache states during inference, evicting the remaining redundant entries to optimize efficiency. Across six benchmarks, **DynTS** surpasses the state-of-the-art KV cache compression methods, improving Pass@1 by $2.6\\%$ under the same budget. Compared to vanilla Transformers, it reduces inference latency by $1.84–2.62\times$ and peak KV-cache memory footprint by $3.32–5.73\times$ without compromising LRMs' reasoning performance.

Self-consistency (SC) is a widely-used test-time inference technique for improving performance in chain-of-thought reasoning. It consists of generating multiple responses, or "samples," from a large language model (LLM) and selecting the most frequent answer. This procedure can naturally be viewed as a majority vote or empirical mode estimation. Despite its effectiveness, self-consistency is prohibitively expensive at scale when naively applied to datasets, and it lacks a unified theoretical treatment of sample efficiency and scaling behavior. In this paper, we provide the first comprehensive analysis of SC's scaling behavior and its variants, drawing on mode estimation and voting theory. We derive and empirically validate power law scaling for self-consistency across datasets, and analyze the sample efficiency for fixed-allocation and dynamic-allocation sampling schemes. From these insights, we introduce Blend-ASC, a novel variant of self-consistency that dynamically allocates samples to questions during inference, achieving state-of-the-art sample efficiency. Our approach uses $4.8\times$ fewer samples than vanilla SC on average, outperforming both fixed- and dynamic-allocation SC baselines, thereby demonstrating the superiority of our approach in terms of efficiency. In contrast to existing variants, we note that Blend-ASC is hyperparameter-free and can fit any budget of samples, ensuring it can be easily applied to any self-consistency application.

We have witnessed remarkable advances in LLM reasoning capabilities with the advent of DeepSeek-R1. However, much of this progress has been fueled by the abundance of internet question–answer (QA) pairs—a major bottleneck going forward, since such data is limited in scale and concentrated mainly in domains like mathematics. In contrast, other sciences such as physics lack large-scale QA datasets to effectively train reasoning-capable models. In this work, we show that physics simulators can serve as a powerful alternative source of supervision for training LLMs for physical reasoning. We generate random scenes in physics engines, create synthetic question–answer pairs from simulated interactions, and train LLMs using reinforcement learning on this synthetic data. Our models exhibit zero-shot sim-to-real transfer to real-world physics benchmarks: for example, training solely on synthetic simulated data improves performance on IPhO (International Physics Olympiad) problems by 5-10 percentage points across model sizes. These results demonstrate that physics simulators can act as scalable data generators, enabling LLMs to acquire deep physical reasoning skills beyond the limitations of internet-scale QA data. Code available at: https://sim2reason.github.io/.

Recent advances in vision-language models (VLMs) emphasize long chain-of-thought reasoning; yet, we find that their performance on visual tasks is primarily limited by a lack of visual perception as opposed to reasoning itself. In this work, we systematically study the interplay between perception and reasoning in VLM post-training by decomposing their capabilities into three separate training stages: visual perception, visual reasoning, and textual reasoning, incorporating specialized training data. We demonstrate that visual perception (a) requires targeted optimization with specialized data; (b) serves as a fundamental scaffold that should be solidified through staged training before refining visual reasoning; and (c) is more effectively learned via RL than caption-based SFT. Our experiments across multiple VLMs demonstrate that staged training consistently improves both visual perception and reasoning performance over merged training. Notably, models trained with our approach achieve 1.5% higher reasoning accuracy with 20.8% shorter reasoning traces, suggesting that superior perception reduces the need for excessive reasoning. Furthermore, we show that this capability-based staging represents a new curriculum dimension orthogonal to traditional difficulty-based curricula, and combining both yields further additive gains. Our staged-training models achieve superior performance among open-weight VLMs, establishing advanced results on several visual math and perception (e.g., +5.2% on WeMath and +3.7% on RealWorldQA) tasks compared with the base counterpart.

Knowledge Graph Question Answering (KGQA) offers grounded, interpretable reasoning, but existing methods often fail to provide reliable coverage guarantees over retrieved answers. While Conformal Prediction (CP) offers a principled framework for producing prediction sets with statistical guarantees, prior conformal KGQA methods suffer from two critical pitfalls: violated coverage guarantees due to invalid calibration, and weak score discriminability that yields excessively large prediction sets. We propose Conformal Path Reasoning (CPR), a novel trustworthy KGQA framework built on two key innovations. First, query-level conformal calibration over path-level scores preserves exchangeability to ensure valid coverage guarantees. Second, we introduce the Residual Conformal Value Network (RCVNet), a lightweight module trained via PUCT-guided exploration to learn discriminative path-level nonconformity scores. Extensive experiments show that CPR significantly improves the Empirical Coverage Rate by 45\% while reducing prediction set size by 52\% on average over conformal baselines across benchmark datasets, highlighting its effectiveness for reliable conformal reasoning over knowledge graphs.

Recent advances in Large Reasoning Models (LRMs) demonstrate remarkable performance improvements by iteratively reflecting, exploring, and executing complex tasks, yet suffer from inefficiencies due to redundant reasoning, known as ``overthinking''. Existing methods to mitigate this issue either rely on static difficulty estimates or require task-specific training, and thus fail to adapt to the dynamic complexity during reasoning. In this work, we empirically show that the problem difficulty evolves dynamically throughout the reasoning process and is linearly encoded in the LRM’s step-level embeddings. Building on this insight, we propose DyCon, a training-free framework that leverages latent step-level representations to explicitly model the evolving task difficulty, enabling the dynamic control of reasoning depth to mitigate the overthinking issue. Extensive experiments conducted on four models ranging from 4B to 32B, and across twelve benchmarks in math reasoning, general question answering, and coding tasks demonstrate that DyCon significantly enhances reasoning efficiency by reducing redundant steps without sacrificing accuracy or generalization. Project page and code are available at https://github.com/yu-lin-li/DyCon.