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.
On-policy self-distillation (OPSD) converts feedback into dense token-level supervision on trajectories generated by the policy to be optimized, providing a useful complement to reinforcement learning with sparse outcome rewards. However, the self-teacher policy used in OPSD is typically a stop-gradient or exponential-moving-average copy of the policy conditioned on additional context information, and thus co-evolves with both the student policy and its on-policy context distribution. Directly matching such a moving target with a fixed projection objective can lead to unstable optimization or excessive distributional concentration. This nature of OPSD motivates the proposed \emph{Self-Referenced On-Policy Self-Distillation (SR-OPSD)}. At fixed student-generated contexts, a token-level variational characterization identifies the effective distillation target as a geometric interpolation between the self-teacher policy and a reference policy. Meanwhile, we use the Rényi divergence family to generalize the projection geometry. This formulation separates \emph{where} the adaptive target is placed from \emph{how} the student is projected toward it: the interpolation coefficient controls underlying target, while the Rényi order controls the projection geometry and its sensitivity to token-level density ratios. Extensive experiments across scientific evaluation, mathematical reasoning, and coding generation tasks with multiple large language models show that SR-OPSD achieves the state-of-the-art or competitive performance across various settings.
We study on-policy self-distillation (OPSD), where a language model improves its reasoning ability by distilling privileged teacher distributions along its own on-policy trajectories. Despite its promise, OPSD can suffer from training instability due to a pattern mismatch between teacher and student responses. Self-reflected teacher responses may introduce reflection-induced biases and response templates that miscalibrate token-level supervision, ultimately harming the student's reasoning ability. To mitigate this issue, we propose OGLS-SD, an outcome-guided logit-steering framework that leverages verifiable outcome rewards to calibrate privileged teacher logits. Specifically, OGLS-SD contrasts teacher logits induced by successful and failed on-policy trajectories, constructing an outcome-discriminative steering direction for token-level guidance. Experiments on mathematical reasoning benchmarks show that OGLS-SD stabilizes self-distillation and improves performance over standard OPSD and other variants.
On-policy distillation (OPD) and on-policy self-distillation (OPSD) have emerged as promising post-training methods for large language models, offering dense token-level supervision on trajectories sampled from the model's own policy. However, existing results on their effectiveness remain mixed: while OP(S)D has shown promise in system prompt and knowledge internalization, recent studies also report instability and degradation. In this work, we present a comprehensive empirical study of when OPD and OPSD work, when they fail, and why. We find that OPD on mathematical reasoning is highly sensitive to teacher choice and loss formulation, whereas OPSD fails in our tested settings due to test-time absence of instance-specific privileged information (PI). In contrast, OPSD is effective when PI represents a shared latent rule, such as a system prompt or alignment preference. We identify three failure mechanisms: (1) distribution mismatch between teacher and student caused by conditioning on student-generated prefixes, (2) optimization instability from biased TopK reverse-KL gradients, and (3) an OPSD-specific limitation where the student learns a PI-free policy that aggregates PI-conditioned teachers, which is insufficient when PI is instance-specific. We further show that stop-gradient TopK objectives, RLVR-adapted teachers, and SFT-stabilized students mitigate these failures.