Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward
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arXiv
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| Autori principali: | , , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918278248529920 |
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| author | Chen, Jianlong Fu, Daocheng Xu, Shengze Chen, Jiawei Feng, Yuan Yang, Yue Yan, Junchi Zha, Hongyuan Xia, Renqiu |
| author_facet | Chen, Jianlong Fu, Daocheng Xu, Shengze Chen, Jiawei Feng, Yuan Yang, Yue Yan, Junchi Zha, Hongyuan Xia, Renqiu |
| contents | Multimodal Large Language Models (MLLMs) struggle with complex geometric reasoning, largely because "black box" outcome-based supervision fails to distinguish between lucky guesses and rigorous deduction. To address this, we introduce a paradigm shift towards subgoal-level evaluation and learning. We first construct GeoGoal, a benchmark synthesized via a rigorous formal verification data engine, which converts abstract proofs into verifiable numeric subgoals. This structure reveals a critical divergence between reasoning quality and outcome accuracy. Leveraging this, we propose the Sub-Goal Verifiable Reward (SGVR) framework, which replaces sparse signals with dense rewards based on the Skeleton Rate. Experiments demonstrate that SGVR not only enhances geometric performance (+9.7%) but also exhibits strong generalization, transferring gains to general math (+8.0%) and other general reasoning tasks (+2.8%), demonstrating broad applicability across diverse domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05073 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward Chen, Jianlong Fu, Daocheng Xu, Shengze Chen, Jiawei Feng, Yuan Yang, Yue Yan, Junchi Zha, Hongyuan Xia, Renqiu Machine Learning Multimodal Large Language Models (MLLMs) struggle with complex geometric reasoning, largely because "black box" outcome-based supervision fails to distinguish between lucky guesses and rigorous deduction. To address this, we introduce a paradigm shift towards subgoal-level evaluation and learning. We first construct GeoGoal, a benchmark synthesized via a rigorous formal verification data engine, which converts abstract proofs into verifiable numeric subgoals. This structure reveals a critical divergence between reasoning quality and outcome accuracy. Leveraging this, we propose the Sub-Goal Verifiable Reward (SGVR) framework, which replaces sparse signals with dense rewards based on the Skeleton Rate. Experiments demonstrate that SGVR not only enhances geometric performance (+9.7%) but also exhibits strong generalization, transferring gains to general math (+8.0%) and other general reasoning tasks (+2.8%), demonstrating broad applicability across diverse domains. |
| title | Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2601.05073 |