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arXiv · 2601.05073

Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward

Jianlong Chen·Daocheng Fu·Shengze Xu·Jiawei Chen·Yuan Feng·Yue Yang·Junchi Yan·Hongyuan Zha·Renqiu Xia
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Abstract

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.

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Jianlong Chen, Daocheng Fu, Shengze Xu, Jiawei Chen, Yuan Feng, Yue Yang, Junchi Yan, Hongyuan Zha, Renqiu Xia. 2026-01-08. Milestones over Outcome: Unlocking Geometric Reasoning with Sub-Goal Verifiable Reward. https://arxiv.org/abs/2601.05073

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