Mitigating Distribution Sharpening in Math RLVR via Distribution-Aligned Hint Synthesis and Backward Hint Annealing

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.

Reinforcement learning for vision-language math reasoning starves under sparse reward: on a pool of 20,830 visual-math problems where Qwen2-VL-2B answers 3.6% of rollouts correctly, 85-97% of GRPO rollout groups are entirely wrong and contribute zero gradient. We train eleven methods under identical conditions in this regime, each injecting a different prior: text (reference-solution hints), distribution (on-policy distillation from a 7B teacher), and value (a value-pretrained critic with an MSE or HL-Gauss categorical loss). A prior helps exactly when it is delivered: the six arms whose prior effectively reaches the policy separate with no overlap from the remaining five -- the no-prior baseline and four arms whose prior is teacher-capped, gated away, or lost to a mis-parameterized critic -- both on the pooled in-domain metric and on cross-domain transfer (DynaMath). The central finding, however, concerns evaluation: one slice of the in-domain pool -- long used as this project's general-distribution check -- anti-correlates with genuine cross-domain transfer (Spearman rho = -0.74, n = 11 arms, permutation p = 0.011), while the hardest in-domain slice predicts it closely (rho = +0.89, p < 0.001). We attribute the inversion to a near-chance multiple-choice subset that rewards models for not having changed; read through it, the best cross-domain method looked mediocre and the worst looked like the champion. Among the methods, hint-guided exploration -- not UFT's auxiliary loss -- drives hint gains, and replacing the critic's MSE loss with HL-Gauss cross-entropy is worth +14.4 points in-domain. All accuracies are blind-judged, with paired exact tests.

Recently, Reinforcement Learning from Verifiable Rewards (RLVR) has been established as a highly effective technique for augmenting the math reasoning skills of Large Language Models (LLMs) based on a single instance. Current state-of-the-art 1-shot RLVR models adopt heuristics for selecting instances, mostly based on historical variance in rewards, which we find to be inherently misleading as a measure of transferability value. In this paper, we propose a Selector-Guided Autonomous Curriculum (SGAC) approach, which employs a learnable selector model on a multi-dimensional feature space consisting of success probability, reward variance, output disagreement (entropy), and semantic difficulty level, instead of the static reward variance heuristic. In our empirical evaluation on pools of candidate problems, we observed that output disagreement, rather than reward variance, is the strongest predictor of reasoning gains in subsequent iterations. Leveraging this finding, we develop an autonomous curriculum algorithm for dynamically siphoning candidate problems from a large pool, ranking them by the learned selector, and running micro-bursts of 1-shot GRPO. Our framework is evaluated using the Hendrycks MATH benchmark, with the Qwen2.5-Math-1.5B model serving as the baseline. Our framework obtains an accuracy of 68.0% on the hold-out dataset, which is better than the accuracy obtained from the state-of-the-art model, 64.0%, as well as the 1-shot RLVR checkpoint proposed by Wang et al., which achieved an accuracy of 66.0%. The results confirm that entropy-based intelligent data curation leads to strict reasoning improvement over static training methods, particularly in severely limited data conditions.