---
title: "WizardMath: Empowering Mathematical Reasoning for — Luo 2023"
description: "Open-source LLMs still lag behind proprietary systems on multi-step math word problems."
authors: ["Haipeng Luo", "Qingfeng Sun", "Can Xu", "Pu Zhao", "Jianguang Lou", "Chongyang Tao", "Xiubo Geng", "Qingwei Lin", "Shifeng Chen", "Yansong Tang", "Dongmei Zhang"]
year: 2023
venue: "arXiv:2308.09583"
source: "https://arxiv.org/abs/2308.09583"
doi: "10.48550/arxiv.2308.09583"
arxiv: "2308.09583"
licence: "arXiv — non-exclusive licence to distribute"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/luo-2023-wizardmath-empowering-mathematical-reasoning-large/"
---
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# WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct

**Authors:** Haipeng Luo, [Qingfeng Sun](https://md.research.elamur.ai/free/en/authors/qingfeng-sun/), [Can Xu](https://md.research.elamur.ai/free/en/authors/can-xu/), [Pu Zhao](https://md.research.elamur.ai/free/en/authors/pu-zhao/), Jianguang Lou, [Chongyang Tao](https://md.research.elamur.ai/free/en/authors/chongyang-tao/), [Xiubo Geng](https://md.research.elamur.ai/free/en/authors/xiubo-geng/), [Qingwei Lin](https://md.research.elamur.ai/free/en/authors/qingwei-lin/), Shifeng Chen, Yansong Tang, Dongmei Zhang · **Year:** 2023 · **Venue:** arXiv:2308.09583 · **Licence:** [arXiv — non-exclusive licence to distribute](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/2308.09583) · [PDF](https://arxiv.org/pdf/2308.09583) · [DOI 10.48550/arxiv.2308.09583](https://doi.org/10.48550/arxiv.2308.09583) · [arXiv 2308.09583](https://arxiv.org/abs/2308.09583) · [OpenAlex](https://openalex.org/W4386044080)

**Citations:** 28 citations (openalex-snapshot · 2026-10-14) · [338](#cited-by) in our library, of the 140673 papers whose bibliographies we read

## Summary
Open-source LLMs still lag behind proprietary systems on multi-step math word problems. This paper introduces WizardMath, trained with Reinforcement Learning from Evol-Instruct Feedback (RLEIF) to improve chain-of-thought (CoT) solving without external Python tools. The pipeline has three stages:

1. **Math Evol-Instruct and SFT.** Seed problems from GSM8k and MATH are rewritten in two directions: simplified ("downward evolution") or made harder ("upward evolution"). The resulting data is used for supervised fine-tuning.
2. **Reward models.** GPT-4 ranks the evolved instructions and checks each solution step. Two models are learned from these labels: one that scores instructions (IRM) and one that supervises the solution process (PRM).
3. **RL training.** Step-level PPO uses a reward that multiplies the IRM and PRM scores.

All training data is generated by AI, with no human annotation, which makes the approach scalable. WizardMath beats open-source models of the same size on GSM8k and MATH. The 70B version also outperforms several proprietary systems, including GPT-3.5-Turbo, Claude 2, Gemini Pro and an early GPT-4. The work shows that instruction evolution combined with automated process supervision can substantially improve math reasoning.

## Key points
- **70B results:** WizardMath-70B scores 92.8 on GSM8k and 58.6 on MATH, compared with 82.3 and 26.6 for MetaMath-70B.
- **7B results:** WizardMath-Mistral-7B reaches 90.7 on GSM8k and 55.4 on MATH, compared with 77.9 and 28.6 for MetaMath-Mistral-7B.
- **Downward evolution:** Two rounds raised GSM8k from 59.7 to 74.5 and MATH from 15.1 to 34.7. PPO with IRM and PRM added roughly 5–8 points over the SFT models.
- **Cheaper labeling:** Using Llama-3.1-405B-Instruct instead of GPT-4 to label PRM data still gave 85.8 on GSM8k and 51.5 on MATH. This is only about 1.2–1.4 points below the GPT-4-labeled version.
- **Out-of-domain tests:** On 7 tasks from MWPBench, WizardMath models beat DART-Math, Xwin-Math and MathScale. The Mistral-7B RL model averages 45.7.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

Fields: [Mathematics & statistics](https://md.research.elamur.ai/free/en/shelves/mathematics/)

## Cited by

Cited by

338 papers of our library cite it (of the 140673 whose bibliographies we read); 27 of them have a page here, 311 not yet.

- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://md.research.elamur.ai/free/en/papers/ahn-2024-large-language-models-mathematical-reasoning/)
- [Quiet-STaR: Language models can teach themselves to think before speaking](https://md.research.elamur.ai/free/en/papers/al-2024-quiet-star-language-models-teach/)
- [Llemma: An Open Language Model For Mathematics](https://md.research.elamur.ai/free/en/papers/azerbayev-2023-llemma-open-language-model-mathematics/)
- [Qwen Technical Report](https://md.research.elamur.ai/free/en/papers/bai-2023-qwen-technical-report/)
- [ReConcile: Round-Table Conference Improves Reasoning via Consensus among Diverse LLMs](https://md.research.elamur.ai/free/en/papers/chen-2024-reconcile-round-table-conference-improves/)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://md.research.elamur.ai/free/en/papers/chen-2025-towards-reasoning-era-survey-long/)
- [ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving](https://md.research.elamur.ai/free/en/papers/gou-2023-tora-tool-integrated-reasoning-agent/)
- [rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking](https://md.research.elamur.ai/free/en/papers/guan-2025-rstar-math-small-llms-can/)
- [GSM-Plus: A Comprehensive Benchmark for Evaluating the Robustness of LLMs as Mathematical Problem Solvers](https://md.research.elamur.ai/free/en/papers/li-2024-gsm-plus-comprehensive-benchmark-evaluating/)
- [From System 1 to System 2: A Survey of Reasoning Large Language Models](https://md.research.elamur.ai/free/en/papers/li-2025-system-1-system-2-survey/)
- [On LLMs-Driven Synthetic Data Generation, Curation, and Evaluation: A Survey](https://md.research.elamur.ai/free/en/papers/long-2024-llms-driven-synthetic-data-generation/)
- [Orca-Math: Unlocking the potential of SLMs in Grade School Math](https://md.research.elamur.ai/free/en/papers/mitra-2024-orca-math-unlocking-potential-slms/)
- [s1: Simple test-time scaling](https://md.research.elamur.ai/free/en/papers/muennighoff-2025-s1-simple-test-time-scaling/)
- [Iterative Reasoning Preference Optimization](https://md.research.elamur.ai/free/en/papers/pang-2024-iterative-reasoning-preference-optimization/)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://md.research.elamur.ai/free/en/papers/shao-2024-deepseekmath-pushing-limits-mathematical-reasoning/)
- [Large Language Models for Data Annotation and Synthesis: A Survey](https://md.research.elamur.ai/free/en/papers/tan-2024-large-language-models-data-annotation/)
- [ReFT: Reasoning with Reinforced Fine-Tuning](https://md.research.elamur.ai/free/en/papers/trung-2024-reft-reasoning-reinforced-fine-tuning/)
- [Aya Model: An Instruction Finetuned Open-Access Multilingual Language Model](https://md.research.elamur.ai/free/en/papers/ustun-2024-aya-model-instruction-finetuned-open/)
- [MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning](https://md.research.elamur.ai/free/en/papers/wang-2023-mathcoder-seamless-code-integration-llms/)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://md.research.elamur.ai/free/en/papers/wang-2024-math-shepherd-verify-reinforce-llms/)
- [Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch](https://md.research.elamur.ai/free/en/papers/yu-2023-language-models-super-mario-absorbing/)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://md.research.elamur.ai/free/en/papers/yu-2023-metamath-bootstrap-your-own-mathematical/)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://md.research.elamur.ai/free/en/papers/yue-2023-mammoth-building-math-generalist-models/)
- [MAmmoTH2: Scaling Instructions from the Web](https://md.research.elamur.ai/free/en/papers/yue-2024-mammoth2-scaling-instructions-web/)
- [MathVerse: Does Your Multi-modal LLM Truly See the Diagrams in Visual Math Problems?](https://md.research.elamur.ai/free/en/papers/zhang-2024-mathverse-multi-modal-llm-truly/)
- [A Survey of Large Language Models](https://md.research.elamur.ai/free/en/papers/zhao-2023-survey-large-language-models/)
- [JudgeLM: Fine-tuned Large Language Models are Scalable Judges](https://md.research.elamur.ai/free/en/papers/zhu-2023-judgelm-fine-tuned-large-language/)

## Cite

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{luo2023wizardmath,
  title = {{WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct}},
  author = {Luo, Haipeng and Sun, Qingfeng and Xu, Can and Zhao, Pu and Lou, Jianguang and Tao, Chongyang and Geng, Xiubo and Lin, Qingwei and Chen, Shifeng and Tang, Yansong and Zhang, Dongmei},
  year = {2023},
  doi = {10.48550/arxiv.2308.09583},
  eprint = {2308.09583},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2308.09583}
}
```

- [← previous WizardLM: Empowering large pre-trained language models to follow complex instructions](https://md.research.elamur.ai/free/en/papers/xu-2023-wizardlm-empowering-large-pre-trained/)
- [next → xCodeEval: A Large Scale Multilingual Multitask Benchmark for Code Understanding, Generation, Translation and Retrieval](https://md.research.elamur.ai/free/en/papers/khan-2023-xcodeeval-large-scale-multilingual-multitask/)
- [↑ back to papers of 2023](https://md.research.elamur.ai/free/en/papers/2023/page/8/)

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