Distilling the Shape of Uncertainty: What On-Policy Distillation Learns and How to Predict It

The paper investigates on‑policy distillation (OPD) for large language models, showing that OPD learns not only which tokens to consider but also where to keep reasoning paths open. It introduces the entropy shape—a sequence of next‑token entropies—to capture this "where to branch" aspect and proposes the Entropy Shape Correlation (ESC) metric to measure student‑teacher agreement. By tracking ESC during training and adjusting a teacher’s entropy while preserving token preferences, the authors demonstrate that controlling where to branch causally drives OPD gains, and they establish a teacher‑selection protocol where pre‑OPD ESC predicts the best teacher more accurately than existing methods.

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