NeurIPS 2025 Papers

Large Language Models (LLMs) have demonstrated remarkable performance in many applications, including challenging reasoning problems via chain-of-thought (CoT) techniques that generate ``thinking tokens'' before answering the questions. While existing theoretical works demonstrate that CoT with discrete tokens boosts the capability of LLMs, recent work on continuous CoT lacks a theoretical understanding of why it outperforms discrete counterparts in various reasoning tasks, such as directed graph reachability, a fundamental graph reasoning problem that includes many practical domain applications as special cases. In this paper, we prove that a two-layer transformer with $D$ steps of continuous CoT can solve the directed graph reachability problem, where $D$ is the diameter of the graph, while the best known result of constant-depth transformers with discrete CoT requires $O(n^2)$ decoding steps where $n$ is the number of vertices ($D<n$). In our construction, each continuous thought vector is a superposition state that encodes multiple search frontiers simultaneously (i.e., parallel breadth-first search (BFS)), while discrete CoT must choose a single path sampled from the superposition state, which leads to a sequential search that requires many more steps and may be trapped in local solutions. We also performed extensive experiments to verify that our theoretical construction aligns well with the empirical solution obtained via training dynamics. Notably, encoding of multiple search frontiers as a superposition state automatically emerges in training continuous CoT, without explicit supervision to guide the model to explore multiple paths simultaneously.

The recent DeepSeek-R1 release has demonstrated the immense potential of reinforcement learning (RL) in enhancing the general reasoning capabilities of large language models (LLMs). While DeepSeek-R1 and other follow-up work primarily focus on applying RL to competitive coding and math problems, this paper introduces SWE-RL, the first approach to scale RL-based LLM reasoning for real-world software engineering. Leveraging a lightweight rule-based reward (e.g., the similarity score between ground-truth and LLM-generated solutions), SWE-RL enables LLMs to autonomously recover a developer's reasoning processes and solutions by learning from extensive open-source software evolution data -- the record of a software's entire lifecycle, including its code snapshots, code changes, and events such as issues and pull requests. Trained on top of Llama 3, our resulting reasoning model, Llama3-SWE-RL-70B, achieves a 41.0% solve rate on SWE-bench Verified -- a human-verified collection of real-world GitHub issues. To our knowledge, this is the best performance reported for medium-sized (<100B) LLMs to date, even comparable to leading proprietary LLMs like GPT-4o. Surprisingly, despite performing RL solely on software evolution data, Llama3-SWE-RL has even emerged with generalized reasoning skills. For example, it shows improved results on five out-of-domain tasks, namely, function coding, library use, code reasoning, mathematics, and general language understanding, whereas a supervised-finetuning baseline even leads to performance degradation on average. Overall, SWE-RL opens up a new direction to improve the reasoning capabilities of LLMs through reinforcement learning on massive software engineering data.

Reinforcement learning (RL) has emerged as a powerful tool for fine-tuning large language models (LLMs) to improve complex reasoning abilities. However, state-of-the-art policy optimization methods often suffer from high computational overhead and memory consumption, primarily due to the need for multiple generations per prompt and the reliance on critic networks or advantage estimates of the current policy. In this paper, we propose $A^\star$-PO, a novel two-stage policy optimization framework that directly approximates the optimal advantage function and enables efficient training of LLMs for reasoning tasks. In the first stage, we leverage offline sampling from a reference policy to estimate the optimal value function $V^\star$, eliminating the need for costly online value estimation. In the second stage, we perform on-policy updates using a simple least-squares regression loss with only a single generation per prompt. Theoretically, we establish performance guarantees and prove that the KL-regularized RL objective can be optimized without requiring complex exploration strategies. Empirically, $A^\star$-PO achieves competitive performance across a wide range of mathematical reasoning benchmarks, while reducing training time by up to 2$\times$ and peak memory usage by over 30\% compared to PPO, GRPO, and REBEL. Implementation of $A^\star$-PO can be found at https://github.com/ZhaolinGao/A-PO.

How to design reinforcement learning (RL) tasks that effectively unleash the reasoning capability of large language models (LLMs) remains an open question. Existing RL tasks (e.g., math, programming, and constructing reasoning tasks) suffer from three key limitations: (1) Scalability. They rely heavily on human annotation or expensive LLM synthesis to generate sufficient training data. (2) Verifiability. LLMs' outputs are hard to verify automatically and reliably. (3) Controllable Difficulty. Most tasks lack fine-grained difficulty control, making it hard to train LLMs to develop reasoning ability from easy to hard. To address these limitations, we propose Saturn, a SAT-based RL framework that uses Boolean Satisfiability (SAT) problems to train and evaluate LLMs reasoning. Saturn enables scalable task construction, rule-based verification, and precise difficulty control. Saturn designs a curriculum learning pipeline that continuously improves LLMs' reasoning capability by constructing SAT tasks of increasing difficulty and training LLMs from easy to hard. To ensure stable training, we design a principled mechanism to control difficulty transitions. We introduce Saturn-2.6k, a dataset of 2,660 SAT problems with varying difficulty. It supports the evaluation of how LLM reasoning changes with problem difficulty. We apply Saturn to DeepSeek-R1-Distill-Qwen and obtain Saturn-1.5B and Saturn-7B. We achieve several notable results: (1) On SAT problems, Saturn-1.5B and Saturn-7B achieve average pass@3 improvements of +14.0 and +28.1, respectively. (2) On math and programming tasks, Saturn-1.5B and Saturn-7B improve average scores by +4.9 and +1.8 on benchmarks (e.g., AIME, LiveCodeBench). (3) Compared to the state-of-the-art (SOTA) approach in constructing RL tasks, Saturn achieves further improvements of +8.8\%. We release the source code, data, and models to support future research.

This paper introduces Completion Pruning Policy Optimization (CPPO) to accelerate the training of reasoning models based on Group Relative Policy Optimization (GRPO). GRPO, while effective, incurs high training costs due to the need to sample multiple completions for each question. Our experiment and theoretical analysis reveal that the number of completions impacts model accuracy yet increases training time multiplicatively, and not all completions contribute equally to policy training---their contribution depends on their relative advantage. To address these issues, we propose CPPO, which prunes completions with low absolute advantages, significantly reducing the number needed for gradient calculation and updates. Additionally, we introduce a dynamic completion allocation strategy to maximize GPU utilization by incorporating additional questions, further enhancing training efficiency. Experiments show that CPPO achieves up to $7.98\times$ speedup on GSM8K and $3.48\times$ on Math while preserving or even enhancing the accuracy compared to the original GRPO. We release our code at https://github.com/lzhxmu/CPPO.

We study a novel language model architecture that is capable of scaling test-time computation by implicitly reasoning in latent space. Our model works by iterating a recurrent block, thereby unrolling to arbitrary depth at test-time. This stands in contrast to mainstream reasoning models that scale up compute by producing more tokens. Unlike approaches based on chain-of-thought, our approach does not require any specialized training data, can work with small context windows, and can capture types of reasoning that are not easily represented in words. We train a proof-of-concept model from scratch with 3.5 billion parameters and 800 billion tokens. We show that this model can effortlessly use varying levels of compute, significantly improving with additional compute especially on reasoning tasks, such as math and coding. Further, this architecture naturally reduces compute costs via zero-shot per-token adaptive compute, KV-cache sharing and speculative decoding.

Frontier large language models (LLMs) have shown great success in text modeling and generation tasks across domains. However, natural language exhibits inherent semantic hierarchies and nuanced geometric structure, which current LLMs do not capture completely owing to their reliance on Euclidean operations such as dot-products and norms. Furthermore, recent studies have shown that not respecting the underlying geometry of token embeddings leads to training instabilities and degradation of generative capabilities. These findings suggest that shifting to non-Euclidean geometries can better align language models with the underlying geometry of text. We thus propose to operate fully in $\textit{Hyperbolic space}$, known for its expansive, scale-free, and low-distortion properties. To this end, we introduce $\textbf{HELM}$, a family of $\textbf{H}$yp$\textbf{E}$rbolic Large $\textbf{L}$anguage $\textbf{M}$odels, offering a geometric rethinking of the Transformer-based LLM that addresses the representational inflexibility, missing set of necessary operations, and poor scalability of existing hyperbolic LMs. We additionally introduce a $\textbf{Mi}$xture-of-$\textbf{C}$urvature $\textbf{E}$xperts model, $\textbf{HELM-MiCE}$, where each expert operates in a distinct curvature space to encode more fine-grained geometric structure from text, as well as a dense model, $\textbf{HELM-D}$. For $\textbf{HELM-MiCE}$, we further develop hyperbolic Multi-Head Latent Attention ($\textbf{HMLA}$) for efficient, reduced-KV-cache training and inference. For both models, we further develop essential hyperbolic equivalents of rotary positional encodings and root mean square normalization. We are the first to train fully hyperbolic LLMs at billion-parameter scale, and evaluate them on well-known benchmarks such as MMLU and ARC, spanning STEM problem-solving, general knowledge, and commonsense reasoning. Our results show consistent gains from our $\textbf{HELM}$ architectures – up to 4\% – over popular Euclidean architectures used in LLaMA and DeepSeek with superior semantic hierarchy modeling capabilities, highlighting the efficacy and enhanced reasoning afforded by hyperbolic geometry in large-scale language model pretraining.

We take two key steps in automating the open-ended discovery of new mathematical theories, a grand challenge in artificial intelligence. First, we introduce Fermat, a reinforcement learning (RL) environment that models concept discovery and theorem-proving using a set of symbolic actions, opening up a range of RL problems relevant to theory discovery. Second, we explore a specific problem through Fermat: automatically scoring the interestingness of mathematical objects. We investigate evolutionary algorithms for synthesizing nontrivial interestingness measures. In particular, we introduce an LLM-based evolutionary algorithm that features function abstraction, leading to notable improvements in discovering elementary number theory and finite fields over hard-coded baselines. We open-source the \fermat environment at github.com/trishullab/Fermat.

Chest X-rays (CXRs) are the most frequently performed imaging examinations in clinical settings. Recent advancements in Medical Large Multimodal Models (MLMMs) have enabled automated CXR interpretation, improving diagnostic accuracy and efficiency. However, despite their strong visual understanding, current MLMMs still face two major challenges: (1) insufficient region-level understanding and interaction, and (2) limited accuracy and interpretability due to single-step prediction. In this paper, we address these challenges by empowering MLMMs with anatomy-centric reasoning capabilities to enhance their interactivity and explainability. Specifically, we propose an Anatomical Ontology-Guided Reasoning (AOR) framework that accommodates both textual and optional visual prompts, centered on region-level information to enable multimodal multi-step reasoning. We also develop AOR-Instruction, a large instruction dataset for MLMMs training, under the guidance of expert physicians. Our experiments demonstrate AOR's superior performance in both Visual Question Answering (VQA) and report generation tasks. Code and data are available at: https://github.com/Liqq1/AOR.

Existing benchmarks for evaluating mathematical reasoning in large language models (LLMs) rely primarily on competition problems, formal proofs, or artificially challenging questions---failing to capture the nature of mathematics encountered in actual research environments. We introduce \textsc{RealMath}, a novel benchmark derived directly from research papers and mathematical forums that assesses LLMs' abilities on authentic mathematical tasks. Our approach addresses three critical challenges: sourcing diverse research-level content, enabling reliable automated evaluation through verifiable statements, and designing a continually refreshable dataset to mitigate contamination risks. Experimental results across multiple LLMs reveal surprising capabilities in handling research mathematics compared to competition problems, suggesting current models may already serve as valuable assistants for working mathematicians despite limitations on highly challenging problems. The code and dataset for \textsc{RealMath} are publicly available.

In modern sequential decision-making systems, the construction of an optimal candidate action space is critical to efficient inference. However, existing approaches either rely on manually defined action spaces that lack scalability or utilize unstructured spaces that render exhaustive search computationally prohibitive. In this paper, we propose a novel framework named \textsc{DynaAct} for automatically constructing a compact action space to enhance sequential reasoning in complex problem-solving scenarios. Our method first estimates a proxy for the complete action space by extracting general sketches observed in a corpus covering diverse complex reasoning problems using large language models. We then formulate a submodular function that jointly evaluates candidate actions based on their utility to the current state and their diversity, and employ a greedy algorithm to select an optimal candidate set. Extensive experiments on six diverse standard benchmarks demonstrate that our approach significantly improves overall performance, while maintaining efficient inference without introducing substantial latency. The implementation is available at \url{https://github.com/zhaoxlpku/DynaAct}.

Data diversity is crucial for training a strong language model. Yet metrics of diversity often diverge from this goal, measuring variations in heuristic features—like n-grams or embeddings—that are detached from how the model actually performs on a target task. This motivates us to ask: *Can we redefine data diversity—beyond measuring variations in heuristic features—in a way that better predicts model generalization?* Through large-scale empirical analyses spanning over 300 training runs, carefully controlled for data scale and quality, we show that data diversity can be a strong predictor of generalization in LLM reasoning—as measured by average model performance on unseen out-of-distribution benchmarks. We introduce **G-Vendi**, a metric that quantifies diversity via the entropy of model-induced loss gradients. G-Vendi scales to million-sample datasets and yet consistently outperforms heuristic alternatives, achieving strong correlation ($\text{Spearman's } \rho \approx 0.9$) with out-of-distribution (OOD) performance across both natural language inference (NLI) and math reasoning tasks. Building on this insight, we present **Prismatic Synthesis**, a framework for generating diverse synthetic data by targeting underrepresented regions in gradient space. Experimental results show that Prismatic Synthesis consistently improves model performance as we scale synthetic data—not just on in-distribution test but across unseen, out-of-distribution benchmarks—significantly outperforming state-of-the-art models in both domains. For example, PrismMath-7B, our model distilled from a 32B LLM without human verification, outperforms R1-Distill-Qwen-7B—trained on proprietary data generated by 671B R1—on 6 out of 7 challenging math benchmarks.

Building robust and general reasoning ability is a central goal in the development of large language models (LLMs). Recent efforts increasingly turn to code as a rich training source, given its inherent logical structure and diverse reasoning paradigms—such as divide-and-conquer, topological ordering, and enumeration. However, reasoning in code is often expressed implicitly and entangled with syntactic or implementation noise, making direct training on raw code suboptimal. To address this, we introduce TraceMind, a large-scale corpus of 2.6 million samples that transforms code execution into explicit, step-by-step chain-of-thought style rationales, which we call Chain of Execution (CoE). The corpus spans domains including mathematics, classical algorithms and algorithmic competition, and is enriched with variable-tracing questions and code rewritings to enhance logical granularity and code diversity. We evaluate Tracepile using three training setups—continue-pretraining, instruction tuning after pretraining, and two-stage finetuning. Experiments across four base models (LLaMA 3, LLaMA 3.1, Qwen-2.5, and Qwen-2.5 Coder) and 20 benchmarks covering math, code, logic, and algorithms demonstrate consistent improvements. Notably, Tracepile boosts LLaMA3-8B by 9.2\% on average across nine math datasets and delivers clear gains on LiveCodeBench, CRUX, and Zebra Logic under two-stage finetuning.

Selecting an appropriate reasoning method for a given query remains a key challenge in language model generation. Existing approaches typically generate multiple candidate responses and use an aggregation strategy to select the output answer, often assuming that more candidate answers yield higher accuracy. We revisit this assumption through a rigorous theoretical analysis, deriving accuracy bounds for standard aggregation methods under fixed generation distributions and candidate sizes. Building on these insights, we introduce EPIC, an Ensemble Planning with Contrastive learning framework to learn a shared representation space that captures both model reasoning abilities and query-method compatibility. EPIC incorporates our probability bounds as a regularizer in a utility-driven optimization that balances accuracy and computational cost. Experiments on diverse mathematical reasoning tasks show that EPIC consistently selects optimal reasoning methods, improving accuracy while reducing computational overhead. Our code can be found at https://github.com/nguyenngocbaocmt02/EPIC.

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present $\textbf{HARDMath2}$, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We build the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.

For Large Language Models (LLMs) to be reliably deployed in both everyday and high-stakes domains, knowing when not to answer is equally critical as answering correctly. Real-world user queries, which can be underspecified, ill-posed, or fundamentally unanswerable, require LLMs to reason about uncertainty and selectively abstain---i.e., refuse to answer definitively. However, abstention remains understudied, without a systematic evaluation framework for modern LLMs. In this work, we introduce AbstentionBench: a large-scale benchmark for holistically evaluating abstention across 20 diverse datasets, including questions with unknown answers, underspecification, false premises, subjective interpretations, and outdated information. Evaluating 20 frontier LLMs reveals abstention is an unsolved problem, and one where scaling models is of little use. While recent reasoning LLMs have shown impressive results in complex problem solving, surprisingly, we find that reasoning fine-tuning degrades abstention (by 24\% on average), even for math and science domains on which reasoning models are explicitly trained. We find that while a carefully crafted system prompt can boost abstention in practice, it does not resolve models’ fundamental inability to reason about uncertainty. We release AbstentionBench to foster research into advancing LLM reliability.

Large Reasoning Models (LRMs) demonstrate remarkable problem-solving capabilities through extended Chain-of-Thought (CoT) reasoning but often produce excessively verbose and redundant reasoning traces. This inefficiency incurs high inference costs and limits practical deployment. While existing fine-tuning methods aim to improve reasoning efficiency, assessing their efficiency gains remains challenging due to inconsistent evaluations. In this work, we introduce the ***reasoning efficiency frontiers***, empirical upper bounds derived from fine-tuning a base LRM (DeepSeek-R1-Distill-Qwen-1.5B/7B) across diverse approaches and training configurations. Based on these frontiers, we propose the ***Reasoning Efficiency Gap (REG)***, a unified metric quantifying deviations of any fine-tuned LRMs from these frontiers. Systematic evaluation on challenging mathematical benchmarks, AMC23, AIME24, and AIME25, reveals significant gaps in current methods: they either sacrifice accuracy for short length or use excessive tokens to achieve sub-optimal accuracies despite high overall accuracy. To reduce the efficiency gap, we propose ***REO-RL***, a Reinforcement Learning algorithm that optimizes reasoning efficiency by targeting a sparse set of token budgets. Leveraging numerical integration over strategically selected budgets, REO-RL approximates the full efficiency objective with low error using a small set of token budgets. Experiments show that, compared to vanilla RL with outcome reward, REO-RL reduces the reasoning efficiency gap by 74.5\% and 64.2\% in the 1.5B and 7B settings. The 7B LRM fine-tuned with REO-RL achieves reasoning conciseness surpassing frontier LRMs like Qwen3 and Claude Sonnet 3.7. Ablation studies confirm the efficacy of our token budget strategy and highlight REO-RL’s flexibility across design choices. This work establishes a systematic framework for evaluating and optimizing reasoning efficiency in LRMs. We will release the related code, data, and models to support future research on efficient reasoning in LRMs.

Small language models (SLMs) struggle to learn complex reasoning behaviors, especially when high-quality traces are scarce or difficult to learn from. A typical approach for training such models combines a supervised fine-tuning (SFT) stage, often to distill reasoning capabilities from a larger model, followed by a reinforcement learning (RL) stage such as Group Relative Policy Optimization (GRPO). In this paper, we investigate the fundamental limitations of this SFT + RL paradigm and propose methods to overcome them. Using a toy student-expert model over Markov chains, we demonstrate that the SFT + RL strategy can fail completely when (1) the expert's traces are too difficult for the small model to express, or (2) the small model's initialization achieves exponentially sparse rewards as task complexity grows. To address these, we introduce BREAD, a GRPO variant that bridges SFT and RL via partial expert guidance and branch rollouts. When self-generated traces fail, BREAD adaptively inserts short expert prefixes/hints, allowing the small model to complete the rest of the reasoning path, and ensuring that each update includes at least one successful trace. This mechanism both densifies the reward signal and induces a natural learning curriculum. BREAD requires fewer than 40\% of ground-truth traces, consistently outperforming standard GRPO while speeding up the training by about 3$\times$. Importantly, we find that BREAD helps the model solve problems that are otherwise unsolvable by the SFT + RL strategy, highlighting how branch rollouts and expert guidance can aid SLM reasoning.

Despite their many successes, transformer-based large language models (LLMs) continue to struggle with tasks that require complex reasoning over large parts of their input. We argue that these failures arise due to capacity limits on the accurate flow of information within LLMs. To formalize this issue, we introduce the bounded attention prefix oracle (BAPO) model, a new computational framework that models bandwidth constraints on attention heads, the mechanism for internal communication in LLMs. We show that several important reasoning problems like graph reachability require high communication bandwidth for BAPOs to solve; we call these problems BAPO-hard. Our experiments corroborate our theoretical predictions: GPT-4o, Claude, and Gemini succeed on BAPO-easy tasks and fail even on relatively small BAPO-hard tasks. BAPOs also reveal another benefit of chain of thought (CoT): we prove that breaking down a task using CoT can turn any BAPO-hard problem into a BAPO-easy one. Our results offer principled explanations for key LLM failures and suggest directions for architectures and inference methods that mitigate bandwidth limits.