# Exploring the Limit of Outcome Reward for Learning Mathematical Reasoning

arXiv: https://arxiv.org/abs/2502.06781 · HTML: https://mlsanity.com/paper/2502.06781/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 39 of its 62 references that are in the map.
- First posted on arXiv 10 Feb 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 30 of 39)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [DeepSeek-V3 Technical Report](https://mlsanity.com/paper/2412.19437/) (2024)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [Scaling Relationship on Learning Mathematical Reasoning with Large Language Models](https://mlsanity.com/paper/2308.01825/) (2023)
- [Beyond Human Data: Scaling Self-Training for Problem-Solving with Language Models](https://mlsanity.com/paper/2312.06585/) (2023)
- [rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking](https://mlsanity.com/paper/2501.04519/) (2025)
- [Towards Large Reasoning Models: A Survey of Reinforced Reasoning with Large Language Models](https://mlsanity.com/paper/2501.09686/) (2025)
- [From r to Q^*: Your Language Model is Secretly a Q-Function](https://mlsanity.com/paper/2404.12358/) (2024)
- [Reward Model Ensembles Help Mitigate Overoptimization](https://mlsanity.com/paper/2310.02743/) (2023)
- [VinePPO: Refining Credit Assignment in RL Training of LLMs](https://mlsanity.com/paper/2410.01679/) (2024)
- [Imitate, Explore, and Self-Improve: A Reproduction Report on Slow-thinking Reasoning Systems](https://mlsanity.com/paper/2412.09413/) (2024)
- [Training Language Models with Language Feedback at Scale](https://mlsanity.com/paper/2303.16755/) (2023)
- [Evaluation of OpenAI o1: Opportunities and Challenges of AGI](https://mlsanity.com/paper/2409.18486/) (2024)
- [BoNBoN Alignment for Large Language Models and the Sweetness of Best-of-n Sampling](https://mlsanity.com/paper/2406.00832/) (2024)
- [InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning](https://mlsanity.com/paper/2402.06332/) (2024)
- [Back to Basics: Revisiting REINFORCE Style Optimization for Learning from Human Feedback in LLMs](https://mlsanity.com/paper/2402.14740/) (2024)
- [O1 Replication Journey -- Part 2: Surpassing O1-preview through Simple Distillation, Big Progress or Bitter Lesson?](https://mlsanity.com/paper/2411.16489/) (2024)

## Built on by (most cited first, 30 of 52)

- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [A Survey of Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2509.08827/) (2025)
- [Pass@k Training for Adaptively Balancing Exploration and Exploitation of Large Reasoning Models](https://mlsanity.com/paper/2508.10751/) (2025)
- [Self-Rewarding Vision-Language Model via Reasoning Decomposition](https://mlsanity.com/paper/2508.19652/) (2025)
- [A Sober Look at Progress in Language Model Reasoning: Pitfalls and Paths to Reproducibility](https://mlsanity.com/paper/2504.07086/) (2025)
- [Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2504.12501/) (2025)
- [Intern-S1: A Scientific Multimodal Foundation Model](https://mlsanity.com/paper/2508.15763/) (2025)
- [Consistent Paths Lead to Truth: Self-Rewarding Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.08745/) (2025)
- [ReasonFlux-PRM: Trajectory-Aware PRMs for Long Chain-of-Thought Reasoning in LLMs](https://mlsanity.com/paper/2506.18896/) (2025)
- [RL Tango: Reinforcing Generator and Verifier Together for Language Reasoning](https://mlsanity.com/paper/2505.15034/) (2025)
- [Ultra-FineWeb: Efficient Data Filtering and Verification for High-Quality LLM Training Data](https://mlsanity.com/paper/2505.05427/) (2025)
- [Open Vision Reasoner: Transferring Linguistic Cognitive Behavior for Visual Reasoning](https://mlsanity.com/paper/2507.05255/) (2025)
- [100 Days After DeepSeek-R1: A Survey on Replication Studies and More Directions for Reasoning Language Models](https://mlsanity.com/paper/2505.00551/) (2025)
- [Tina: Tiny Reasoning Models via LoRA](https://mlsanity.com/paper/2504.15777/) (2025)
- [A Technical Survey of Reinforcement Learning Techniques for Large Language Models](https://mlsanity.com/paper/2507.04136/) (2025)
- [RFTF: Reinforcement Fine-tuning for Embodied Agents with Temporal Feedback](https://mlsanity.com/paper/2505.19767/) (2025)
- [Hybrid Reinforcement: When Reward Is Sparse, It's Better to Be Dense](https://mlsanity.com/paper/2510.07242/) (2025)
- [ToolRM: Outcome Reward Models for Tool-Calling Large Language Models](https://mlsanity.com/paper/2509.11963/) (2025)
- [DRPO: Efficient Reasoning via Decoupled Reward Policy Optimization](https://mlsanity.com/paper/2510.04474/) (2025)
- [Alignment and Safety in Large Language Models: Safety Mechanisms, Training Paradigms, and Emerging Challenges](https://mlsanity.com/paper/2507.19672/) (2025)
- [Sailing by the Stars: A Survey on Reward Models and Learning Strategies for Learning from Rewards](https://mlsanity.com/paper/2505.02686/) (2025)
- [Training LLMs for EHR-Based Reasoning Tasks via Reinforcement Learning](https://mlsanity.com/paper/2505.24105/) (2025)
- [Aligning Multimodal LLM with Human Preference: A Survey](https://mlsanity.com/paper/2503.14504/) (2025)
- [The Synergy Dilemma of Long-CoT SFT and RL: Investigating Post-Training Techniques for Reasoning VLMs](https://mlsanity.com/paper/2507.07562/) (2025)
- [Don't Waste Mistakes: Leveraging Negative RL-Groups via Confidence Reweighting](https://mlsanity.com/paper/2510.08696/) (2025)
- [DeepSearch: Overcome the Bottleneck of Reinforcement Learning with Verifiable Rewards via Monte Carlo Tree Search](https://mlsanity.com/paper/2509.25454/) (2025)
- [Unleashing Embodied Task Planning Ability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2506.23127/) (2025)
- [Reinforcement Learning Fine-Tuning Enhances Activation Intensity and Diversity in the Internal Circuitry of LLMs](https://mlsanity.com/paper/2509.21044/) (2025)
- [SELAUR: Self Evolving LLM Agent via Uncertainty-aware Rewards](https://mlsanity.com/paper/2602.21158/) (2026)

## Abstract (arXiv)

Reasoning abilities, especially those for solving complex math problems, are crucial components of general intelligence. Recent advances by proprietary companies, such as o-series models of OpenAI, have made remarkable progress on reasoning tasks. However, the complete technical details remain unrevealed, and the techniques that are believed certainly to be adopted are only reinforcement learning (RL) and the long chain of thoughts. This paper proposes a new RL framework, termed OREAL, to pursue the performance limit that can be achieved through Outcome REwArd-based reinforcement Learning for mathematical reasoning tasks, where only binary outcome rewards are easily accessible. We theoretically prove that behavior cloning on positive trajectories from best-of-N (BoN) sampling is sufficient to learn the KL-regularized optimal policy in binary feedback environments. This formulation further implies that the rewards of negative samples should be reshaped to ensure the gradient consistency between positive and negative samples. To alleviate the long-existing difficulties brought by sparse rewards in RL, which are even exacerbated by the partial correctness of the long chain of thought for reasoning tasks, we further apply a token-level reward model to sample important tokens in reasoning trajectories for learning. With OREAL, for the first time, a 7B model can obtain 94.0 pass@1 accuracy on MATH-500 through RL, being on par with 32B models. OREAL-32B also surpasses previous 32B models trained by distillation with 95.0 pass@1 accuracy on MATH-500. Our investigation also indicates the importance of initial policy models and training queries for RL. Code, models, and data will be released to benefit future researchfootnotehttps://github.com/InternLM/OREAL.