A Predictive Law for On-Policy Self-Distillation From World Feedback
Authors: Tommy He, Jerome Sieber, Matteo Saponati
Organizations: Tufa Labs, Zürich, Switzerland
Abstract
Moving beyond simple scalar rewards toward richer world feedback is a natural path to more scalable RL post-training. On-policy self-distillation (OPSD) is a promising recent approach that uses arbitrary feedback as learning signal, yet its reliability compared to established methods, such as GRPO, remains unclear. We identify a strikingly consistent linear correlation between the initial student-self-teacher performance gap and the final performance improvement in OPSD. This relationship holds across context types and model families, providing a powerful predictive law for anticipating the outcome of an OPSD configuration without running the full training procedure. Interestingly, we show that this linear predictability holds with model scale, suggesting a potential basis for new empirical scaling laws on larger models with stronger in-context learning capabilities. In essence, our findings show that OPSD performance can be predicted and tuned before training, offering a principled way to incorporate world feedback as a first-class component of the post-training pipeline.
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.
On-policy self-distillation (OPSD) scores student-generated prefixes with a solution-conditioned self-teacher, yet transfers supervision only through next-token probabilities. We ask whether the aligned final-layer discrepancy offers a useful second channel, and how to test that channel without confusing its geometry with auxiliary strength. SCOPE-OPSD projects the privileged teacher-student residual onto a frozen rank-64 factor estimated from residual covariance and language-model-head Fisher sensitivity. It reuses the forwards already required by OPSD and adds neither rollouts nor inference-time modules. A matched Random control preserves the structured factor's rank and nonzero spectrum and uses per-arm gradient-RMS calibration, isolating the effect of the data-dependent orientation. Across the complete 25/50/75/100-step trajectories for Qwen3-1.7B, 4B, and 8B, Structured is never below Pure OPSD, with strict gains in 11 of the 12 model-checkpoint combinations and an exact tie at 4B step 25. Structured also exceeds matched Random in 10 of the 12 combinations. At step 75 on Qwen3-1.7B, Structured exceeds matched Random by 1.39 Macro Avg@12 points in each of two independent training reruns. A cross-fitted diagnostic also shows 4.40 times greater held-out privileged-gap capture than the matched random orientation. The results support a compact, Fisher-conditioned privileged subspace for short-budget OPSD.
Yunmeng Chen (Chongqing Ant Consumer Finance Co., Ltd), Kunyu Wang (Alibaba Cloud Computing Co. +18
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.