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

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 results show that if an effective rank condition holds, then PCR performs comparably to the oracle method obtained by replacing empirical principal components by their population counterparts. On the other hand, if this 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 dimensions.