💬 NLP

GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization

The paper introduces GeometryZero, an affordable family of geometry reasoning models that leverages a novel Group Contrastive Policy Optimization (GCPO) framework to effectively train smaller models on auxiliary construction and geometric reasoning, outperforming existing reinforcement learning baselines on standard benchmarks.

Original authors: Yikun Wang, Yibin Wang, Dianyi Wang, Zimian Peng, Qipeng Guo, Dacheng Tao, Jiaqi Wang

Published 2026-04-21

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Yikun Wang, Yibin Wang, Dianyi Wang, Zimian Peng, Qipeng Guo, Dacheng Tao, Jiaqi Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

🎨 The Big Picture: Teaching a Robot to Draw

Imagine you are trying to teach a robot to solve geometry problems. Some problems are easy: "If a triangle has two 45-degree angles, what is the third?" The robot can just think and answer.

But other problems are like a tangled knot. To untie them, you need to draw a new line (an "auxiliary line") that isn't in the original picture. This new line acts like a secret shortcut that reveals the answer.

The problem is, most AI models are either:

Too dumb: They can't draw the lines or reason well.
Too expensive: They rely on massive, super-powerful (and costly) computers like GPT-4o to figure it out.
Too reckless: They try to draw a new line on every single problem, even when it's not needed. This is like trying to fix a flat tire by replacing the whole car engine—it wastes time and often makes things worse.

GeometryZero is a new, affordable, and smart robot that learns exactly when to draw a new line and when to just think.

🧠 The Core Problem: The "Indiscriminate Drawer"

The researchers noticed that previous methods (using a technique called GRPO) were like a student who was told: "You get a gold star every time you draw a new line!"

So, the student started drawing lines everywhere.

Problem A: Needed a line. The student drew one. ✅ Gold star.
Problem B: Didn't need a line. The student drew one anyway. ❌ Gold star (because the rule was just "draw").
Result: The student got confused, wasted energy, and sometimes drew the wrong line, leading to the wrong answer.

The AI was being rewarded for doing the action, not for solving the problem.

💡 The Solution: "Group Contrastive Policy Optimization" (GCPO)

The authors invented a new training method called GCPO. Let's call it the "Smart Coach" method.

Instead of just saying "Draw a line!", the Smart Coach runs a mental simulation before giving a reward. Here is how it works:

1. The "What If" Game (Group Contrastive Masking)

Imagine the AI is a detective. For every mystery (geometry problem), the Coach asks the AI to try two different strategies:

Group A (The Drawer): "Try to solve this by drawing a new line."
Group B (The Thinker): "Try to solve this without drawing a new line."

The Coach then looks at the results:

Scenario 1: The Drawer solved it, but the Thinker failed.
- Coach's Verdict: "Great job drawing that line! It was necessary. Reward!"
Scenario 2: The Thinker solved it easily, but the Drawer got confused and failed.
- Coach's Verdict: "You wasted time drawing a line that wasn't needed. In fact, it hurt you. Penalty!"
Scenario 3: Both solved it (or both failed).
- Coach's Verdict: "It didn't matter. No reward, no penalty."

This teaches the AI to be discriminating. It learns that drawing a line is a tool, not a habit. It only uses the tool when it actually helps.

2. The "Deep Thinker" Bonus (Length Reward)

Solving hard geometry problems often requires a long chain of thoughts. If the AI rushes to the answer, it might miss a step.
The Coach adds a small bonus for thinking longer and deeper (as long as the answer is correct). This encourages the AI to build a solid logical bridge rather than guessing.

🏆 The Result: GeometryZero

By using this "Smart Coach" method, the researchers created GeometryZero.

It's Small: It runs on models as small as 1.5 billion or 7 billion parameters (much cheaper than the giant models).
It's Smart: It knows when to stop and think, and when to grab a pencil and draw a new line.
It Wins: In tests (like Geometry3K and MathVista), GeometryZero beat other AI models that used older, "reckless" training methods.

🚀 The Takeaway

Think of GeometryZero as a student who has learned the difference between studying hard and studying smart.

Old AI: "I will draw a line on every page!" (Wasteful, confusing).
GeometryZero: "Hmm, this page looks tricky. I'll draw a line here. But this next page is simple; I'll just read it." (Efficient, accurate).

This paper proves that you don't need a super-computer to solve complex geometry; you just need a smarter way to teach the computer when to use its tools.

1. Problem Statement

Geometric problem solving remains a significant challenge for Large Language Models (LLMs), particularly when problems require auxiliary constructions (e.g., adding lines, points, or shapes to a diagram to reveal hidden relationships).

Current Limitations: Existing methods either rely on massive supervised datasets (which lack self-correction capabilities) or depend on extremely large models (e.g., GPT-4o, Gemini) that are computationally expensive.
The Reinforcement Learning (RL) Gap: While Reinforcement Learning with Verifiable Rewards (RLVR), specifically GRPO (Group Relative Policy Optimization), has shown promise in training smaller models for reasoning, naively applying it to geometry fails. Standard GRPO uses unconditional rewards, encouraging models to indiscriminately apply auxiliary constructions even when they are unnecessary or harmful. This leads to "over-thinking" or incorrect reasoning paths.
Core Challenge: How to train moderate-sized models to autonomously decide when to use auxiliary constructions and when to rely on direct reasoning, without relying on massive parameter counts or expensive human annotations.

2. Methodology: Group Contrastive Policy Optimization (GCPO)

The authors propose GCPO, a novel RL framework designed to teach models the conditional utility of auxiliary tools. GCPO modifies the standard GRPO algorithm with two key components:

A. Group Contrastive Masking

Instead of treating all auxiliary construction attempts as positive, GCPO introduces a dynamic masking mechanism based on the contextual utility of the construction.

Mechanism: During training, for a given problem, the model generates three groups of rollouts (responses):
1. Standard Group ( $O$ ): Normal reasoning.
2. With Construction Group ( $O_w$ ): Forced to include auxiliary construction attempts.
3. Without Construction Group ( $O_{wo}$ ): Forced to abstain from auxiliary construction.
Reward Logic: The system compares the average accuracy rewards of $O_w$ $O_{w}$ and $O_{wo}$ $O_{w o}$ .
- If $E(R_{acc}(O_w)) > E(R_{acc}(O_{wo})) + \epsilon$ : The auxiliary construction is deemed beneficial. The model receives a positive reward for constructing.
- If $E(R_{acc}(O_w)) + \epsilon < E(R_{acc}(O_{wo}))$ : The construction is detrimental. The model receives a negative reward (penalty) for constructing.
- Otherwise: The reward is masked to zero.
Goal: This teaches the model to distinguish between scenarios where "doing more" (adding lines) helps versus scenarios where it hurts.

B. Length Reward

Inspired by Length Controlled Policy Optimization (LCPO), GCPO introduces a reward component that encourages longer reasoning chains.

Rationale: Complex geometric reasoning, especially involving auxiliary constructions, requires deeper, multi-dimensional analysis.
Implementation: A reward is added based on the token length of the response (up to a maximum limit), encouraging the model to explore more thorough reasoning paths rather than rushing to a conclusion.

C. The GeometryZero Model Family

Built upon GCPO, the authors trained GeometryZero, a family of lightweight models (1.5B, 3B, and 7B parameters) based on the Qwen2.5 architecture. These models are trained to:

Convert visual diagrams into formal languages (TikZ code or logic forms).
Decide autonomously whether to generate auxiliary construction code.
Perform chain-of-thought reasoning to solve the problem.

3. Key Contributions

Conditional Tool Usage: The paper demonstrates that LLMs can be trained to selectively use auxiliary constructions rather than applying them indiscriminately. This is achieved through the novel Group Contrastive Masking mechanism.
GCPO Framework: A new RL algorithm that integrates verifiable rewards with conditional masking and length rewards, specifically tailored for geometric reasoning.
GeometryZero Models: A series of affordable, open-source models (1.5B–7B) that achieve state-of-the-art performance on geometry benchmarks, outperforming much larger models and other RL baselines.
Empirical Validation: Extensive experiments showing that GCPO enables models to develop "aha moments" (self-correction and reflection) and scale effectively across different model sizes.

4. Experimental Results

The models were evaluated on four benchmarks: Geomverse, Geometry3K (in-domain), MathVista, and OlympiadBench (out-of-domain).

Performance vs. Baselines:
- GeometryZero-7B achieved an average score of 57.47, outperforming the GRPO baseline (55.63) and the ToRL baseline (54.72).
- It significantly outperformed the base Qwen2.5-7B-Instruct (48.30) and SFT-only variants.
- On the OlympiadBench, GeometryZero-7B scored 45.69, surpassing the GRPO baseline (40.32).
Comparison with Large Models:
- GeometryZero-7B outperformed GPT-4o and Gemini on specific geometry tasks in some contexts, and significantly outperformed the 14B Qwen model without specialized training.
- In geometric proof tasks (UniGeo), GeometryZero-14B achieved 72.2%, closing the gap with AlphaGeometry (94.4%) and significantly beating GPT-4o (74.1%) and the base 14B model (64.8%).
Ablation Studies:
- Removing Group Contrastive Masking (reverting to unconditional rewards like ToRL) caused performance drops, confirming that when to construct is as important as how.
- Removing the Length Reward reduced performance, highlighting the need for extended reasoning chains.
Vision-Language Generalization: When applied to Qwen2.5-VL-7B (which processes images directly), the method still improved performance, though text-only reasoning with formal language representation proved slightly more effective than direct visual processing for these specific tasks.

5. Significance

Cost-Effectiveness: GeometryZero proves that high-level geometric reasoning does not require massive (e.g., 70B+) models or expensive proprietary APIs. Small, open-source models can achieve competitive results with the right training paradigm.
Strategic Reasoning: The work shifts the paradigm from "more tools = better" to "strategic tool use." By teaching models to recognize when auxiliary lines are counterproductive, it reduces hallucination and logical errors.
Scalability: The method shows consistent scaling laws; as model size increases from 1.5B to 7B, the benefits of GCPO become more pronounced, suggesting a viable path for future geometric reasoning systems.
Open Science: The code and models are open-sourced, providing a strong baseline for future research in multimodal reasoning and tool-augmented LLMs.

In summary, GeometryZero addresses the critical bottleneck of geometric reasoning by introducing a reinforcement learning framework that learns the timing and necessity of auxiliary constructions, enabling small models to solve complex geometry problems with the precision of much larger systems.