A correct teacher solution becomes useful supervision when the receiving student can continue its reasoning. We measure this compatibility with prefix recovery: after revealing 25%, 50%, or 75% of a verified solution, we test whether the student completes it correctly. We connect recovery to the cosine conflict between cross-entropy and reverse-KL gradients over the full vocabulary. Across adjacent Qwen3 teacher-student pairs from 0.6B to 8B parameters, reverse-KL distillation delivers its most consistent mathematical and code improvements for the two students below 2B parameters. On a fixed cohort of 1,000 trajectories, average prefix recovery rises from 71.0% to 91.9% as student size increases from 0.6B to 4B, and the robust-fragile recovery gap contracts from 46.0 to 14.4 percentage points. With the teacher fixed at 8B, conflict separation falls from 0.993 to 0.233. An independent objective intervention finds the largest reverse-KL rescue on fragile trajectories. The three measurements locate the same capacity-dependent transfer regime: distribution matching has the greatest headroom when correct traces remain unevenly recoverable. Prefix recovery provides a practical diagnostic for selecting costly distribution-level distillation.
Language model distillation transfers a larger model’s capabilities to smaller or specialized models using outputs, probabilities, or internal representations.
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.
Kaixiang Zhao, Tianrun Yu, Shangzhe Li +4
Brigham Young University · University of North Carolina at Chapel Hill
On-policy distillation transfers reasoning capabilities by training a student model on its own generated trajectories using token-level feedback from a teacher. However, we identify a critical bottleneck, Supervision Fidelity Decay (SFD): as student-generated prefixes lengthen, the teacher's next-token distribution becomes less confident and less discriminative. Consequently, the teacher-dependent corrective signal in reverse-KL distillation weakens, causing student drift to compound across long reasoning chains. To mitigate SFD, we introduce Lookahead Group Reward (\ours{}). Building on the insight that next-step teacher confidence reflects the discriminative strength of future reverse-KL supervision, \ours{} evaluates the student's top-K candidate tokens by the teacher confidence they induce at the subsequent step and assigns a group-normalized reward. To maintain computational efficiency, we further design an entropy-triggered tree-attention mechanism. Across six math and code benchmarks, \ours{} improves mean@8 by 2.57 points over OPD for a 7B student, with gains increasing in longer-generation and reaching +4.92 points on AIME-26 at 39k tokens.
Yanjiang Liu, Jie Lou, Xinyan Guan +7
University of Chinese Academy of Sciences · 2Chinese Information Processing Laboratory Institute of Software, Chinese Academy of Sciences 2University of Chinese Academy of Sciences, Beijing, China · 3Xiaohongshu
On-policy distillation (OPD) improves the reasoning capabilities of small language models through token-level teacher supervision on student-generated trajectories. Yet can teachers that excel at solving problems independently also guide student reasoning effectively? Prior work shows that when student prefixes follow reasoning paths that differ from the teacher's own or contain errors, teachers can be less accurate when continuing from these prefixes than when solving problems independently. To this end, we propose Prep-OPD, which uses reinforcement learning (RL) before distillation to train the teacher to adapt to the student's existing reasoning state and correct course when errors arise. Training optimizes teacher continuations from fixed student prefixes using final-answer correctness as the reward. The prepared teacher then trains the student through trajectory guidance and token-level supervision. We evaluate Prep-OPD on eight mathematical reasoning benchmarks, using Qwen3-4B-Instruct-2507 as the teacher and Qwen3-0.6B and Qwen3-1.7B as students. With the 4B teacher and 1.7B student, Prep-OPD improves average accuracy over standard OPD and the strongest baseline, Relay-OPD, by 8.28 and 2.30 percentage points, respectively. Controlled experiments further show that teacher RL conditioned on student-generated prefixes yields higher student accuracy than problem-start teacher RL with and without handoff on Qwen3-1.7B. Reusing the same prepared teacher also improves Qwen3-0.6B.
Xiaoyu Ma, Haoyue Liu, Zhichao Wang +2
School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen · Shenzhen Future Network of Intelligence Institute (FNiI-Shenzhen) · Guangdong Provincial Key Laboratory of Future Networks of Intelligence, CUHK(SZ)