${\hat{I}}^2$-OPSD: Deriving with Policy Optimization, Training with Self-Distillation

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 ${\hat{I}}^2=1$ member of a broader policy-optimization family, where ${\hat{I}}^2$ weights the KL penalty anchoring the student to a reference policy. This equivalence turns ${\hat{I}}^2$ 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 ${\hat{I}}^2$-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 ${\hat{I}}^2$ 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 ${\hat{I}}^2$-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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