  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.
