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

Reinforcement Learning from Verifiable Rewards (RLVR) has become the dominant approach for improving mathematical reasoning in large language models, yet current methods reduce each correct rollout to a single reward bit, ignoring the geometric structure shared among their hidden states. Investigating this structure, we find that at the anchor token (the position immediately before the answer marker), correct rollouts converge naturally because they must produce the same answer (cosine similarity ~0.84), yet each retains residual variance from its unique reasoning path. Encouraging full alignment at this point pushes the model to extract a unified "correct decision" representation, reducing sensitivity to which reasoning path was taken. Based on this observation, we propose Hidden-Align, an auxiliary loss function that aligns the last-layer hidden states of correct rollouts at the anchor token during RL training, with zero overhead in both training and inference. On eight mathematical reasoning benchmarks, Hidden-Align improves average pass@1 over the DAPO baseline by 3.8, 6.2, and 5.4 percentage points on Qwen3-1.7B, 4B, and 14B respectively, with consistent pass@k gains across all three scales, supported by ablations on loss type, anchor position, layer depth, and loss weight.

Small language models (SLMs) often struggle with complex mathematical reasoning due to limited capacity to maintain long chains of intermediate steps and to recover from early errors. We address this challenge by introducing a hint-assisted reasoning framework that incrementally guides SLMs through multi-step mathematical problem solving. Our approach decomposes solutions into sequential reasoning steps and provides context-aware hints, where hints are generated by a separate SLM trained via distillation from a strong large language model. While the hint-generating SLM alone is not capable of solving the problems, its collaboration with a reasoning SLM enables effective guidance, forming a cooperative two-model system for reasoning. Each hint is generated conditionally on the problem statement and the accumulated reasoning history, providing stepwise, localized guidance without revealing full solutions. This reduces error propagation and allows the reasoning model to focus on manageable subproblems. Experiments across diverse mathematical benchmarks and models demonstrate that hint assistance consistently improves reasoning accuracy for SLMs, yielding substantial gains over standard prompting while preserving model efficiency. These results highlight that structured collaboration between SLMs-via hint generation and reasoning-offers an effective and lightweight mechanism for enhancing mathematical reasoning.

In reinforcement learning with verifiable rewards (RLVR), challenging math questions whose sampled groups remain uniformly incorrect yield no informative update, letting easier questions dominate actor updates. This imbalance can contribute to distribution sharpening: low-$k$ accuracy improves while problem coverage contracts. Hint scaffolding can restore informative updates by revealing a prefix of a verified teacher solution. However, existing hint-based approaches still face two issues: teacher--student distribution mismatch and the need to reduce hint exposure to match no-hint evaluation. We address these issues through two components. Distribution-Aligned Hint Synthesis (DAHS) constructs verified teacher hints conditioned on student-style responses. Backward Hint Annealing (BHA) anneals hint exposure across length buckets as bucket accuracy improves. BHA also uses per-question hint dropout to keep no-hint rollouts throughout RL training. Under the DAPO training framework, our method improves both pass@1 and pass@2048 over DAPO on AIME24/25/26 for Qwen3-1.7B-Base and Qwen3-4B-Base, and improves pass@1 on five secondary math benchmarks (Olympiad, MATH500, Minerva Math, AMC23, and GSM8K). On Llama-3.2-1B-Instruct, the gains concentrate in the high-$k$ regime without recovering the base model's coverage. Overall, when verified teacher prefixes are available, DAHS+BHA mitigates DAPO-induced contraction of high-$k$ problem coverage in the evaluated challenging-math RLVR settings.