  We provide a unified analysis of the predictive risk of ridge regression and
regularized discriminant analysis in a dense random effects model. We work in a
high-dimensional asymptotic regime where $p, n \to \infty$ and $p/n \to \gamma
\in (0, \, \infty)$, and allow for arbitrary covariance among the features. For
both methods, we provide an explicit and efficiently computable expression for
the limiting predictive risk, which depends only on the spectrum of the
feature-covariance matrix, the signal strength, and the aspect ratio $\gamma$.
Especially in the case of regularized discriminant analysis, we find that
predictive accuracy has a nuanced dependence on the eigenvalue distribution of
the covariance matrix, suggesting that analyses based on the operator norm of
the covariance matrix may not be sharp. Our results also uncover several
qualitative insights about both methods: for example, with ridge regression,
there is an exact inverse relation between the limiting predictive risk and the
limiting estimation risk given a fixed signal strength. Our analysis builds on
recent advances in random matrix theory.
