DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learning

Theorem proving serves as a major testbed for evaluating complex reasoning abilities in large language models (LLMs). However, traditional automated theorem proving (ATP) approaches rely heavily on formal proof systems that poorly align with LLMs' strength derived from informal, natural language knowledge acquired during pre-training. In this work, we propose DeepTheorem, a comprehensive informal theorem-proving framework exploiting natural language to enhance LLM mathematical reasoning. DeepTheorem includes a large-scale benchmark dataset consisting of 121K high-quality IMO-level informal theorems and proofs spanning diverse mathematical domains, rigorously annotated for correctness, difficulty, and topic categories, accompanied by systematically constructed verifiable theorem variants. We devise a novel reinforcement learning strategy (RL-Zero) explicitly tailored to informal theorem proving, leveraging the verified theorem variants to incentivize robust mathematical inference. Additionally, we propose comprehensive outcome and process evaluation metrics examining proof correctness and the quality of reasoning steps. Extensive experimental analyses demonstrate DeepTheorem significantly improves LLM theorem-proving performance compared to existing datasets and supervised fine-tuning protocols, achieving state-of-the-art accuracy and reasoning quality. Our findings highlight DeepTheorem's potential to fundamentally advance automated informal theorem proving and mathematical exploration.

Two Experts Are All You Need for Steering Thinking: Reinforcing Cognitive Effort in MoE Reasoning Models Without Additional Training

Mixture-of-Experts (MoE) architectures within Large Reasoning Models (LRMs) have achieved impressive reasoning capabilities by selectively activating experts to facilitate structured cognitive processes. Despite notable advances, existing reasoning models often suffer from cognitive inefficiencies like overthinking and underthinking. To address these limitations, we introduce a novel inference-time steering methodology called Reinforcing Cognitive Experts (RICE), designed to improve reasoning performance without additional training or complex heuristics. Leveraging normalized Pointwise Mutual Information (nPMI), we systematically identify specialized experts, termed ''cognitive experts'' that orchestrate meta-level reasoning operations characterized by tokens like ''<think>''. Empirical evaluations with leading MoE-based LRMs (DeepSeek-R1 and Qwen3-235B) on rigorous quantitative and scientific reasoning benchmarks demonstrate noticeable and consistent improvements in reasoning accuracy, cognitive efficiency, and cross-domain generalization. Crucially, our lightweight approach substantially outperforms prevalent reasoning-steering techniques, such as prompt design and decoding constraints, while preserving the model's general instruction-following skills. These results highlight reinforcing cognitive experts as a promising, practical, and interpretable direction to enhance cognitive efficiency within advanced reasoning models.

Trust, But Verify: A Self-Verification Approach to Reinforcement Learning with Verifiable Rewards

Large Language Models (LLMs) show great promise in complex reasoning, with Reinforcement Learning with Verifiable Rewards (RLVR) being a key enhancement strategy. However, a prevalent issue is ``superficial self-reflection'', where models fail to robustly verify their own outputs. We introduce RISE (Reinforcing Reasoning with Self-Verification), a novel online RL framework designed to tackle this. RISE explicitly and simultaneously trains an LLM to improve both its problem-solving and self-verification abilities within a single, integrated RL process. The core mechanism involves leveraging verifiable rewards from an outcome verifier to provide on-the-fly feedback for both solution generation and self-verification tasks. In each iteration, the model generates solutions, then critiques its own on-policy generated solutions, with both trajectories contributing to the policy update. Extensive experiments on diverse mathematical reasoning benchmarks show that RISE consistently improves model's problem-solving accuracy while concurrently fostering strong self-verification skills. Our analyses highlight the advantages of online verification and the benefits of increased verification compute. Additionally, RISE models exhibit more frequent and accurate self-verification behaviors during reasoning. These advantages reinforce RISE as a flexible and effective path towards developing more robust and self-aware reasoners.

DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning

The capacity for complex mathematical reasoning is a key benchmark for artificial intelligence. While reinforcement learning (RL) applied to LLMs shows promise, progress is significantly hindered by the lack of large-scale training data that is sufficiently challenging, possesses verifiable answer formats suitable for RL, and is free from contamination with evaluation benchmarks. To address these limitations, we introduce DeepMath-103K, a new, large-scale dataset comprising approximately 103K mathematical problems, specifically designed to train advanced reasoning models via RL. DeepMath-103K is curated through a rigorous pipeline involving source analysis, stringent decontamination against numerous benchmarks, and filtering for high difficulty (primarily Levels 5-9), significantly exceeding existing open resources in challenge. Each problem includes a verifiable final answer, enabling rule-based RL, and three distinct R1-generated solutions suitable for diverse training paradigms like supervised fine-tuning or distillation. Spanning a wide range of mathematical topics, DeepMath-103K promotes the development of generalizable reasoning. We demonstrate that models trained on DeepMath-103K achieve significant improvements on challenging mathematical benchmarks, validating its effectiveness. We release DeepMath-103K publicly to facilitate community progress in building more capable AI reasoning systems: https://github.com/zwhe99/DeepMath.

Dancing with Critiques: Enhancing LLM Reasoning with Stepwise Natural Language Self-Critique

Enhancing the reasoning capabilities of large language models (LLMs), particularly for complex tasks requiring multi-step logical deductions, remains a significant challenge. Traditional inference time scaling methods utilize scalar reward signals from process reward models to evaluate candidate reasoning steps, but these scalar rewards lack the nuanced qualitative information essential for understanding and justifying each step. In this paper, we propose a novel inference-time scaling approach -- stepwise natural language self-critique (PANEL), which employs self-generated natural language critiques as feedback to guide the step-level search process. By generating rich, human-readable critiques for each candidate reasoning step, PANEL retains essential qualitative information, facilitating better-informed decision-making during inference. This approach bypasses the need for task-specific verifiers and the associated training overhead, making it broadly applicable across diverse tasks. Experimental results on challenging reasoning benchmarks, including AIME and GPQA, demonstrate that PANEL significantly enhances reasoning performance, outperforming traditional scalar reward-based methods. Our code is available at https://github.com/puddingyeah/PANEL to support and encourage future research in this promising field.

RaSA: Rank-Sharing Low-Rank Adaptation

Low-rank adaptation (LoRA) has been prominently employed for parameter-efficient fine-tuning of large language models (LLMs). However, the limited expressive capacity of LoRA, stemming from the low-rank constraint, has been recognized as a bottleneck, particularly in rigorous tasks like code generation and mathematical reasoning. To address this limitation, we introduce Rank-Sharing Low-Rank Adaptation (RaSA), an innovative extension that enhances the expressive capacity of LoRA by leveraging partial rank sharing across layers. By forming a shared rank pool and applying layer-specific weighting, RaSA effectively increases the number of ranks without augmenting parameter overhead. Our theoretically grounded and empirically validated approach demonstrates that RaSA not only maintains the core advantages of LoRA but also significantly boosts performance in challenging code and math tasks. Code, data and scripts are available at: https://github.com/zwhe99/RaSA.

The First Few Tokens Are All You Need: An Efficient and Effective Unsupervised Prefix Fine-Tuning Method for Reasoning Models

Improving the reasoning capabilities of large language models (LLMs) typically requires supervised fine-tuning with labeled data or computationally expensive sampling. We introduce Unsupervised Prefix Fine-Tuning (UPFT), which leverages the observation of Prefix Self-Consistency -- the shared initial reasoning steps across diverse solution trajectories -- to enhance LLM reasoning efficiency. By training exclusively on the initial prefix substrings (as few as 8 tokens), UPFT removes the need for labeled data or exhaustive sampling. Experiments on reasoning benchmarks show that UPFT matches the performance of supervised methods such as Rejection Sampling Fine-Tuning, while reducing training time by 75% and sampling cost by 99%. Further analysis reveals that errors tend to appear in later stages of the reasoning process and that prefix-based training preserves the model's structural knowledge. This work demonstrates how minimal unsupervised fine-tuning can unlock substantial reasoning gains in LLMs, offering a scalable and resource-efficient alternative to conventional approaches.

Thoughts Are All Over the Place: On the Underthinking of o1-Like LLMs

Large language models (LLMs) such as OpenAI's o1 have demonstrated remarkable abilities in complex reasoning tasks by scaling test-time compute and exhibiting human-like deep thinking. However, we identify a phenomenon we term underthinking, where o1-like LLMs frequently switch between different reasoning thoughts without sufficiently exploring promising paths to reach a correct solution. This behavior leads to inadequate depth of reasoning and decreased performance, particularly on challenging mathematical problems. To systematically analyze this issue, we conduct experiments on three challenging test sets and two representative open-source o1-like models, revealing that frequent thought switching correlates with incorrect responses. We introduce a novel metric to quantify underthinking by measuring token efficiency in incorrect answers. To address underthinking, we propose a decoding strategy with thought switching penalty TIP that discourages premature transitions between thoughts, encouraging deeper exploration of each reasoning path. Experimental results demonstrate that our approach improves accuracy across challenging datasets without requiring model fine-tuning. Our findings contribute to understanding reasoning inefficiencies in o1-like LLMs and offer a practical solution to enhance their problem-solving capabilities.

Do NOT Think That Much for 2+3=? On the Overthinking of o1-Like LLMs

The remarkable performance of models like the OpenAI o1 can be attributed to their ability to emulate human-like long-time thinking during inference. These models employ extended chain-of-thought (CoT) processes, exploring multiple strategies to enhance problem-solving capabilities. However, a critical question remains: How to intelligently and efficiently scale computational resources during testing. This paper presents the first comprehensive study on the prevalent issue of overthinking in these models, where excessive computational resources are allocated for simple problems with minimal benefit. We introduce novel efficiency metrics from both outcome and process perspectives to evaluate the rational use of computational resources by o1-like models. Using a self-training paradigm, we propose strategies to mitigate overthinking, streamlining reasoning processes without compromising accuracy. Experimental results show that our approach successfully reduces computational overhead while preserving model performance across a range of testsets with varying difficulty levels, such as GSM8K, MATH500, GPQA, and AIME.

Draft Model Knows When to Stop: A Self-Verification Length Policy for Speculative Decoding

Speculative Decoding (SD) has become an important technique in accelerating the inference speed of large language models. Conventional SD methods employ a fixed draft length, which ignores the token generation difficulty across tasks. Consequently, in this paper, we address such an issue and introduce SVIP - a difficulty-aware dynamic draft length policy for speculative decoding systems. Based on a theoretical lower bound of draft token acceptance rate and its inference-time approximation, SVIP adaptively determines the lengths of draft sequences based on the entropy of each draft token distribution. Experimental results on mainstream SD benchmarks and frameworks demonstrate the superior performance of SVIP, achieving up to 20\% walltime speedup on SpecBench over baseline SD methods and 60\% speedup on MT-Bench for long-form generation of up to 8K tokens. Moreover, SVIP is totally training-free and compatible with any existing SD methods that generate draft tokens autoregressively. Experimental results also show that SVIP yields consistent walltime improvement on top of GliDe & CaPE and EAGLE-2.