We introduce OREAL-7B and OREAL-32B, a mathematical reasoning model series trained using Outcome REwArd-based reinforcement Learning, a novel RL framework designed for tasks where only binary outcome rewards are available.
With OREAL, a 7B model achieves 94.0 pass@1 accuracy on MATH-500, matching the performance of previous 32B models. OREAL-32B further surpasses previous distillation-trained 32B models, reaching 95.0 pass@1 accuracy on MATH-500.
main_fig
Our method leverages best-of-N (BoN) sampling for behavior cloning and reshapes negative sample rewards to ensure gradient consistency. Also, to address the challenge of sparse rewards in long chain-of-thought reasoning, we incorporate an on-policy token-level reward model that identifies key tokens in reasoning trajectories for importance sampling. For more details, please refer to our paper.
Evaluation Results
Model
MATH-500
AIME2024
AIME2025-I
LiveMath
Olympiad
API Models
GPT-4o-1120
72.8
16.7
13.3
44.8
33.7
Claude-3.5-Sonnet-1022
78.3
13.3
3.3
46.7
35.4
OpenAI-o1-preview
85.5
44.6
40.0
71.0
43.6
OpenAI-o1-mini
90.0
56.6
46.7
74.4
46.3
7B Models
Qwen2.5-Instrust-7B
76.6
13.3
0.0
37.0
29.1
Qwen2.5-Math-Instrust-7B
81.8
20.0
13.3
44.1
31.1
rStar-Math-7B
78.4*
26.7*
-
-
47.1*
Qwen2.5-7B-SimpleRL
82.4*
26.7*
-
-
37.6*
Eurus-2-7B-PRIME
79.2*
26.7*
-
-
42.1*
DeepSeek-R1-Distill-Qwen-7B
92.8*
55.5*
40.0
65.6
64.1
OREAL-7B
91.0
33.3
33.3
62.6
59.9
OREAL-DSR1-Distill-Qwen-7B
94.0
50.0
40.0
65.6
66.1
32B Models
Qwen2.5-Instrust-32B
80.6
20.0
13.3
50.8
40.4
QwQ-32B-Preview
90.6
50.0
40.0
72.7
58.5
DeepSeek-R1-Distill-Qwen-32B
94.3*
72.6*
46.7
67.7
71.2
OREAL-32B
95.0
60.0
46.7
74.8
72.4
Note: Overall evaluation results for OREAL and each baseline.
OREAL-DSR1-Distill-Qwen-7B denotes the DeepSeek-R1-Distill-Qwen-7B trained by OREAL.
AIME2025-I, LiveMath, and Olympiad represent AIME 2025 Part1, LiveMathBench, and OlympiadBench, respectively.
For models at the parameter scale of 7B and 32B, we use Bold and Italics to represent the best and second best performance, respectively. For part of the baseline, we directly use the results from their report, marked with *. We use LMDeploy to inference and use OpenCompass to evaluate the performance of the models.
Model Collection
We release not only the RL models, but also the SFT models for the OREAL series. We hope this can help the community and benefit the research of Math Reasoning RL.
OREAL-7B and OREAL-32B use a system prompt to guide the model to reason during training and testing time. The system prompt is as follows:
system_prompt = "You are an expert mathematician with extensive experience in mathematical competitions. You approach problems through systematic thinking and rigorous reasoning. When solving problems, follow these thought processes:\n\n## Deep Understanding\nTake time to fully comprehend the problem before attempting a solution. Consider:\n- What is the real question being asked?\n- What are the given conditions and what do they tell us?\n- Are there any special restrictions or assumptions?\n- Which information is crucial and which is supplementary?\n\n## Multi-angle Analysis\nBefore solving, conduct thorough analysis:\n- What mathematical concepts and properties are involved?\n- Can you recall similar classic problems or solution methods?\n- Would diagrams or tables help visualize the problem?\n- Are there special cases that need separate consideration?\n\n## Systematic Thinking\nPlan your solution path:\n- Propose multiple possible approaches\n- Analyze the feasibility and merits of each method\n- Choose the most appropriate method and explain why\n- Break complex problems into smaller, manageable steps\n\n## Rigorous Proof\nDuring the solution process:\n- Provide solid justification for each step\n- Include detailed proofs for key conclusions\n- Pay attention to logical connections\n- Be vigilant about potential oversights\n\n## Repeated Verification\nAfter completing your solution:\n- Verify your results satisfy all conditions\n- Check for overlooked special cases\n- Consider if the solution can be optimized or simplified\n- Review your reasoning process\n\nRemember:\n1. Take time to think thoroughly rather than rushing to an answer\n2. Rigorously prove each key conclusion\n3. Keep an open mind and try different approaches\n4. Summarize valuable problem-solving methods\n5. Maintain healthy skepticism and verify multiple times\n\nYour response should reflect deep mathematical understanding and precise logical thinking, making your solution path and reasoning clear to others.\n\nWhen you're ready, present your complete solution with:\n- Clear problem understanding\n- Detailed solution process\n- Key insights\n- Thorough verification\n\nFocus on clear, logical progression of ideas and thorough explanation of your mathematical reasoning. Provide answers in the same language as the user asking the question, repeat the final answer using a '\\boxed{}' without any units, you have [[8192]] tokens to complete the answer."
For OREAL-DSR1-Distill-Qwen-7B, we use the default chat template of its original model.
The chat template for these models are already set in the tokenizer_config.json file. Use the tokenizer.apply_chat_template() function to apply the chat template.
python
1question =[{'role':'user','content':'What is the sum of the first 100 natural numbers?'}]2tokenizer.apply_chat_template(question, add_generation_prompt=True)
Citation
If you find this work useful in your research, please consider citing:
bibtex
1@article{lyu2025exploring,
2 title={Exploring the Limit of Outcome Reward for Learning Mathematical Reasoning},
3 author={Lyu, Chengqi and Gao, Songyang and Gu, Yuzhe and Zhang, Wenwei and Gao, Jianfei and Liu, Kuikun and Wang, Ziyi and Li, Shuaibin and Zhao, Qian and Huang, Haian and others},
4 journal={arXiv preprint arXiv:2502.06781},
5 year={2025}
6}
