这篇论文把OPSD的隐式β变成可控参数，用蒸馏近似强化学习，数学推理上比原版更稳更强。
论文提出β-OPSD，将普通在线自蒸馏（OPSD）推广为β=1特例的策略优化家族。β作为正则化参数，控制学生模型相对参考策略与特权教师之间的权衡。最优策略被推导为参考策略与教师的几何插值，实现时混合二者的token级logits。在数学推理基准上，β-OPSD比vanilla OPSD更稳定、推理性能更好。
$β$-OPSD: Deriving with Policy Optimization, Training with Self-Distillation
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.