A Privileged Teacher That Only Scores Cannot Search: Critique-Conditioned On-Policy Self-Distillation for Reasoning

On-policy self-distillation aims to improve upon reinforcement learning from verifiable rewards (RLVR) by providing token-level scores derived from privileged information, such as reference solutions or critic feedback. These scores are treated as estimates of token-level action values, yet they answer a fundamentally different question: how the model's prediction changes when its input context is enriched, rather than how the expected outcome changes when a token is changed. We examine this gap along three dimensions: (i) whether the token-level score tracks task success; (ii) whether feedback generated from the same rollout causes the score to reflect agreement with its own description, and whether using feedback from other rollouts in the group mitigates this self-referential effect; and (iii) what behavior the resulting training objective actually reinforces. In experiments on AIME 2025, the implemented score distinguishes correct from incorrect rollouts at approximately chance level (AUC=0.505); using feedback from a different rollout does not consistently improve this discrimination; and all training configurations achieve only 24.2-33.9% Avg@4, compared with 64.2% for outcome-only GRPO. Moreover, the highest-entropy token decile accounts for 57-71% of the total absolute token-advantage mass, despite the score being least informative about reasoning quality in this regime. By contrast, similar experiments on SciKnowEval Biology improves held-out Avg@8 by 28.0%, while its corresponding trajectory scores achieve AUCs of 0.81-0.92. Together, these results suggest that dense credit assignment through distillation can be effective when its likelihood-based scores are empirically validated as meaningful proxies for outcome-relevant credit. When this alignment does not hold, however, the resulting supervision can fail to generalize and may substantially underperform outcome-based RL.

On-policy distillation (OPD) trains a student model on its own rollouts with token-level supervision from a teacher, assuming that the teacher can reach a correct answer from any prefix the student writes. We call this property the recoverability of the teacher and show it fails on incorrect rollouts: the teacher follows student errors, although it recognizes them. We propose CriticOPD, using error recognition to restore recoverability. The teacher critiques an incorrect rollout, allowing the student to rewrite its solution following the error. The resulting output, including the critique as privileged information, is scored by the teacher while the student uses the revision without the critique for training. On two Qwen3 students and six mathematical reasoning benchmarks, CriticOPD improves OPD, with the largest gains on competition problems.

Post-training of reasoning language models is commonly driven by supervised distillation and reinforcement learning with verifiable rewards. Distillation often relies on chain-of-thought annotations that are expensive to obtain and may themselves be noisy, incomplete, or partially incorrect; even when the final solution is correct, an imperfect rationale can interfere with learning. Reinforcement learning with verified rewards, on the other hand, typically compresses evaluative feedback into a scalar signal, obscuring which aspects of a response should be improved. We propose Rubric-Conditioned Self-Distillation, a framework that incorporates rubrics as structured, fine-grained feedback for on-policy self-distillation. Our method conditions the teacher model on criterion-level rubrics and uses it to provide token-level guidance on the student's own sampled trajectories. This design avoids treating a single reference rationale as the sole supervision target. Instead, rubrics specify what a strong response should satisfy, enabling more fine-grained credit assignment over the reasoning process than scalar reward optimization. We instantiate this framework with a two-stage pipeline that first learns to generate task-specific rubrics and then trains a rubric-guided reasoner. We evaluate on a diverse suite of science reasoning benchmarks and results show that rubric-conditioned self-distillation effectively converts rubric-level criteria into token-level guidance over the reasoning process, surpassing GRPO by 1.0 points and OPSD by 0.9 points on average.