Self-distillation has emerged as a promising technique for improving model performance in modern machine learning systems. We develop the statistical foundations of self-distillation in high-dimensional linear regression by introducing and analyzing a broad class of estimators, namely spectral shrinkage estimators. We consider both spiked and general covariance matrices. For the former, we establish that when the covariance has $s$ distinct spikes, $s$-step self-distillation achieves optimal prediction among spectral shrinkage estimators, outperforming well-known estimators in statistics and machine learning. Moreover, we show that $s$ steps are necessary for optimality: any $(s-k)$-step distilled estimator is strictly suboptimal for $1 \leq k \leq s$. For general covariance matrices, we show that self-distillation approaches the optimal limiting prediction risk among continuous spectral shrinkage estimators as the number of distillation steps increases. We further develop a generalized cross-validation procedure for selecting the distillation parameters from data and establish an asymptotic oracle guarantee for the resulting estimator. We also study a federated approach where multiple data centers share spectral shrinkage estimators and a common server seeks to aggregate them to achieve optimal performance. In this case, we find that the best local rule again takes the form of self-distillation, though it differs from the optimal rule when data are hosted centrally on a single server. Together, our results explain why self-distillation improves predictive performance and provide a broader statistical framework that connects it to classical shrinkage-based methods.

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