In this paper, we explore the asymptotically optimal tuning parameter choice in ridge regression for estimating nuisance functions of a statistical functional that has recently gained prominence in conditional independence testing and causal inference. Given a sample of size n, we study estimators of the Expected Conditional Covariance (ECC) between variables Y and Agiven a high-dimensional covariate X Rp. Under linear regression models for Y and A on X and the proportional asymptotic regime p/n c (0,), we evaluate three existing ECC estimators and two sample splitting strategies for estimating the required nuisance functions. Since no consistent estimator of the nuisance functions exists in the proportional asymptotic regime without imposing further structure on the problem, we first derive debiased versions of the ECC estimators that utilize the ridge regression nuisance function estimators. We show that our bias correction strategy yields n-consistent estimators of the ECC across different sample splitting strategies and estimator choices. We then derive the asymptotic variances of these debiased estimators to illustrate the nuanced interplay between the sample splitting strategy, estimator choice, and tuning parameters of the nuisance function estimators for optimally estimating the ECC. Our analysis reveals that prediction-optimal tuning parameters (i.e., those that optimally estimate the nuisance functions) may not lead to the lowest asymptotic variance of the ECC estimator - thereby demonstrating the need to be careful in selecting tuning parameters based on the final goal of inference. Finally, we verify our theoretical results through extensive numerical experiments.

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 spiked covariance models, by introducing and analyzing a broad class of estimators, namely spectral shrinkage estimators. We establish that for spiked covariance matrices with $s$ spikes, $s$-step self-distillation achieves optimal performance 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 the special subclass of isotropic covariances, we show that optimally tuned Ridge regression performs best among spectral shrinkage estimators. 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 elucidate why self-distillation improves predictive performance and provide a broader statistical framework connecting it with classical shrinkage-based methods.