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home / papers / 2026 / GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization

GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization

Paper · 2026 · Yikun Wang, Yibin Wang, Dianyi Wang et al.

Summary

Geometry problems remain hard for LLMs, mainly because many of them require drawing extra lines or points. Earlier approaches either performed poorly or relied on very large proprietary systems such as GPT-4o or Gemini, which are costly to run. The authors show that reinforcement learning with verifiable rewards can teach smaller models to combine auxiliary construction with geometric reasoning.

Plain GRPO with tool-style rewards has a flaw: it rewards construction no matter what. Models then add lines even when this hurts the solution. To fix this, the paper introduces Group Contrastive Policy Optimization (GCPO). For each problem, it samples a set of rollouts that use auxiliary construction and a separate set that do not, then contrasts the two. From this comparison it decides whether construction should be rewarded or penalized. GCPO also adds a reward that favors longer, deeper reasoning.

The resulting GeometryZero models, ranging from 1.5B to 7B parameters, learn when to construct and when to reason directly. They beat GRPO and ToRL baselines on Geometry3K and MathVista. This makes capable geometry solvers far more affordable.

Key points

  • GCPO has two parts. Group Contrastive Masking gives conditional positive or negative construction rewards based on contextual utility. A Length Reward, inspired by LCPO, encourages extended reasoning.
  • The geometry diagrams are given to the model as formal language (Python drawing code). The model performs auxiliary construction by editing this code inside <aux> segments.
  • GeometryZero is built on Qwen2.5 models from 1.5B to 7B. It outperforms GRPO- and ToRL-trained counterparts on Geometry3K and MathVista and scales better with model size.
  • In ablations on the 3B model, removing the auxiliary reward, the group contrastive masking, or the length reward each reduced average benchmark scores.
  • Response length during training rises, falls, then rises again for GCPO models. The 1.5B GRPO model shows no final recovery. Code is available at github.com/ekonwang/GeometryZero.

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

 Institutions

🎓Fudan UniversityShanghai · China167 works · 20,987 citations🎓Nanyang Technological UniversitySingapore · Singapore156 works · 25,644 citations🎓Zhejiang UniversityHangzhou · China172 works · 12,988 citations🔬Shanghai Artificial Intelligence LaboratoryShanghai · China117 works · 11,281 citations🎓Shanghai Innovation InstituteShanghai · China24 works · 262 citations

 Fields, topics and concepts

📚 Shelves 7The field of science this work belongs to (OpenAlex fields).Computer Science · 16,667 worksArtificial Intelligence · 16,194 worksArts and Humanities · 16,039 worksLanguage and Linguistics · 15,891 worksMathematics · 177 worksComputational Theory and Mathematics · 252 worksApplied Mathematics · 46 works

Cite

Cite as Yikun Wang, Yibin Wang, Dianyi Wang, Zimian Peng, Qipeng Guo, Dacheng Tao and Jiaqi Wang. 2026. GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization. Findings of the Association for Computational Linguistics: ACL 2026.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.18653/v1/2026.findings-acl.1392

@inproceedings{wang2026geometryzero,
  title = {{GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization}},
  author = {Yikun Wang, Yibin Wang, Dianyi Wang, Zimian Peng, Qipeng Guo, Dacheng Tao and Jiaqi Wang},
  booktitle = {Findings of the Association for Computational Linguistics: ACL 2026},
  year = {2026},
  doi = {10.18653/v1/2026.findings-acl.1392},
  url = {https://doi.org/10.18653/v1/2026.findings-acl.1392}
}

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