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

TL;DR

This paper introduces a new framework and benchmark for improving geometric reasoning in multimodal large language models by focusing on subgoal verification, leading to significant performance gains and better generalization.

Contribution

It presents the GeoGoal benchmark and the Sub-Goal Verifiable Reward (SGVR) framework, enabling more rigorous and effective geometric reasoning in large language models.

Findings

SGVR improves geometric reasoning accuracy by 9.7%.

Models trained with SGVR transfer gains to general math (+8.0%).

Enhanced reasoning generalizes to other domains (+2.8%).

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.