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

Abstract

On-policy distillation (OPD) is increasingly adopted in the post-training of large language models. However, a stronger teacher may fail to improve the student. Prior work attributes this to a lack of new knowledge, but what that knowledge is remains unclear. We show that OPD learns not only how to branch, i.e., which candidate tokens to consider, but also where to branch, i.e., where to keep reasoning paths open. The existing token-overlap metric captures only the former. We introduce the entropy shape, the sequence of next-token entropies, to characterize the latter, and propose the Entropy Shape Correlation (ESC) to measure student-teacher agreement in entropy shape. Tracking ESC throughout training and intervening on a teacher's entropy while preserving its token preferences, we empirically show that where to branch causally drives OPD gains, independently of how to branch. Finally, we formulate OPD outcome prediction as a teacher selection task and establish an evaluation protocol for it, under which pre-OPD ESC selects the best teacher for both students and outperforms existing predictors. Code is available at https://github.com/maybenotime/opd-esc.