Paper Detail

$β$-OPSD: Deriving with Policy Optimization, Training with Self-Distillation

Jiawei Xu, Minghui Liu, Juzheng Zhang, Tom Goldstein, Furong Huang

arxiv Score 11.3

Published 2026-07-30 · First seen 2026-07-31

General AI

Abstract

On-policy self-distillation (OPSD) is a promising approach to improve reasoning language models, but it remains brittle in practice: making it work reliably often requires substantial engineering effort. We identify a structural source of this difficulty: vanilla OPSD is precisely the $β=1$ member of a broader policy-optimization family, where $β$ weights the KL penalty anchoring the student to a reference policy. This equivalence turns $β$ from an implicit value fixed at one into a controllable regularization parameter, yielding a more general formulation that trades off proximity to a reference policy against privileged teacher guidance. We introduce $β$-OPSD and derive its optimal policy as a geometric interpolation between the reference policy and the privileged teacher. Directly optimizing this objective with reinforcement learning, however, would be costly and high-variance. Rather than optimize the RL objective directly, we turn its closed-form solution into a distillation target. Each value of $β$ selects a target along the reference-to-teacher path, which we implement efficiently by mixing their token-level logits. In this way, inexpensive distillation approximates the solution of expensive policy optimization. Return-to-go credit assignment further aligns token updates with the sequence-level objective while retaining the simplicity of OPSD. Experiments on mathematical reasoning benchmarks show that $β$-OPSD consistently outperforms vanilla OPSD, improving optimization stability and downstream reasoning performance. Our results provide a principled route from self-distillation to policy optimization and back without sacrificing the efficiency that makes OPSD practical.

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BibTeX

@article{xu2026opsd,
  title = {\$β\$-OPSD: Deriving with Policy Optimization, Training with Self-Distillation},
  author = {Jiawei Xu and Minghui Liu and Juzheng Zhang and Tom Goldstein and Furong Huang},
  year = {2026},
  abstract = {On-policy self-distillation (OPSD) is a promising approach to improve reasoning language models, but it remains brittle in practice: making it work reliably often requires substantial engineering effort. We identify a structural source of this difficulty: vanilla OPSD is precisely the \$β=1\$ member of a broader policy-optimization family, where \$β\$ weights the KL penalty anchoring the student to a reference policy. This equivalence turns \$β\$ from an implicit value fixed at one into a controllable},
  url = {https://arxiv.org/abs/2607.28582},
  keywords = {cs.LG, huggingface daily},
  eprint = {2607.28582},
  archiveprefix = {arXiv},
}

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