A note on the prediction error of principal component regression in high dimensions

TL;DR

The first main result shows that PCR performs comparably to the oracle method obtained by replacing empirical principal components by their population counterparts, provided that an effective rank condition holds.

Abstract

We analyze the prediction error of principal component regression (PCR) and prove high probability bounds for the corresponding squared risk conditional on the design. Our first main result shows that PCR performs comparably to the oracle method obtained by replacing empirical principal components by their population counterparts, provided that an effective rank condition holds. On the other hand, if the latter condition is violated, then empirical eigenvalues start to have a significant upward bias, resulting in a self-induced regularization of PCR. Our approach relies on the behavior of empirical eigenvalues, empirical eigenvectors and the excess risk of principal component analysis in high-dimensional regimes.

Theorems & Definitions (38)

• Lemma 1
• Proposition 1
• Remark 1
• Corollary 1
• Theorem 1
• Example 1: Exponential decay
• Example 2: Polynomial decay
• Theorem 2
• Example 3: The spiked covariance model
• Example 4: Benign overfitting phenomenon