H$^2$R: Hierarchical Hint-Based Reward for Mathematical Reasoning via Reinforcement Learning

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Reinforcement Learning with Verifiable Rewards has shown great promise in enhancing mathematical reasoning in large language models. However, existing approaches provide limited intermediate supervision for complex, multi-step reasoning tasks. To bridge this gap, we propose \textbf{Hierarchical Hint-based Reward}, a framework that leverages the intrinsic sub-question structure of mathematical problems as structured intermediate conditioning signals during RLVR training. Unlike chain-of-thought prompting, these hierarchical hints serve as logically grounded intermediate milestones that effectively reduce the reasoning horizon toward the final answer. To validate this framework, we introduce \textbf{HiMath}, a benchmark of hierarchically structured mathematical problems. Across diverse LLMs, hierarchical hints consistently improve reasoning performance even without task-specific training. When integrated into RL training via H$^2$R, they boost Qwen2.5-Math-7B accuracy on HiMath from 30.4\% to 39.1\% using hierarchical hints, and up to 45.6\% with Oracle Hierarchical Hints. This pronounced improvement demonstrates that incorporating reliable intermediate checkpoints into reward design is essential for sustaining complex mathematical reasoning.