Optimal Unconstrained Self-Distillation in Ridge Regression: Strict Improvements, Precise Asymptotics, and One-Shot Tuning

Abstract

Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth labels and the teacher’s own predictions using the same architecture and training data. Although SD has been empirically shown to often improve generalization, its formal guarantees remain limited. We study SD for ridge regression with an unconstrained mixing weight $\xi \in \mathbb{R}$. Conditional on the training data and without any distributional assumptions, we prove that for any squared prediction risk $R$ (including out-of-distribution), the optimally mixed student strictly improves upon the ridge teacher at every regularization level $\lambda$ where the teacher risk is not stationary ($R'(\lambda) \neq 0$). We also characterize the optimal mixing weight $\xi^\star$ in terms of the risk derivative $R'$, showing that it can be negative, which is the case in over-regularized regimes. To quantify SD risk improvements, we derive exact risk asymptotics in the proportional asymptotics regime for general anisotropic covariance and deterministic signals. From a practical standpoint, we propose a consistent one-shot tuning method to estimate $\xi^\star$ without grid search, sample splitting, or refitting. Experiments on real-world datasets and pretrained neural network features support our theory and the one-shot tuning method.