High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification
This work provides theoretical insights for statisticians and machine learning researchers, though it is incremental as it builds on existing random matrix theory.
The authors tackled the problem of predicting the risk of ridge regression and regularized discriminant analysis in high-dimensional settings, deriving explicit formulas for limiting predictive risk that depend on covariance spectrum, signal strength, and aspect ratio, with results showing nuanced dependence on eigenvalue distributions.
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 $p/n \to γ\in (0, \, \infty)$, 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 $γ$. 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.