Unbiased Top-$k$ Estimation for On-Policy Distillation

2026-09-28

AI Summary

This study addresses the biased gradient estimation and performance degradation in Top-k online distillation caused by truncating tail probabilities. To overcome this limitation, we propose TT-OPD, whose core innovation lies in recovering discarded probability mass using tokens sampled from student-generated rollouts. By integrating reverse KL divergence minimization with an expectation-based tail correction technique, our method achieves unbiased gradient estimation while maintaining low computational overhead. Experimental results demonstrate that TT-OPD significantly outperforms existing online distillation baselines, effectively enhancing both the reasoning capabilities and accuracy of student models in knowledge distillation for large language models.

Abstract

On-policy distillation (OPD) is becoming an important component of large language model (LLM) post-training for transferring the reasoning capability of a strong teacher LLM to a weaker student LLM. OPD trains the student by minimizing the reverse KL divergence between the teacher and the student via rollouts generated by the student's policy. However, estimating the gradient of the reverse KL divergence in OPD remains a challenge. Using only the sampled token from the student-generated rollout is computationally cheap but provides limited distributional supervision, which will degrade accuracy. In addition, using the full vocabulary provides complete distributional supervision but is computationally expensive. Therefore, recent works propose Top-$k$ OPD (TK-OPD) that use selected top-$k$ tokens, which provides richer distributional supervision than sampled-token estimation at substantially lower computational cost than full-vocabulary estimation. Unfortunately, using only the selected top-$k$ tokens induces bias, leading to accuracy degradation, as the probability mass outside the selected top-$k$ tokens is discarded. To address the bias of TK-OPD, we propose Tail-Corrected Top-$k$ On-Policy Distillation (TT-OPD). It preserves the advantages of TK-OPD, including rich distributional supervision and low computational cost, while providing an unbiased estimator of the gradient of the reverse KL divergence. The key insight of TT-OPD is to use not only the selected top-$k$ tokens, but also the sampled token from the student-generated rollout, thereby recovering the discarded probability mass in expectation, avoiding the bias. Experimental results demonstrate that TT-OPD significantly outperforms other tested OPD variants.

Problem

• On-Policy Distillation
• Top-k Estimation
• Reverse KL Divergence
• Estimation Bias
• Large Language Models

Innovation

• On-Policy Distillation
• Unbiased Top-k Estimation
• Reverse KL Divergence
• Tail Correction
• Large Language Models