High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification

TL;DR

This paper analyzes the predictive risk of ridge regression and regularized discriminant analysis in high-dimensional settings, providing explicit formulas that depend on the covariance spectrum and revealing nuanced effects of covariance structure.

Contribution

It offers a unified, explicit characterization of predictive risk for both methods in high dimensions, accounting for arbitrary covariance structures and connecting risk measures.

Findings

Predictive risk depends on the spectrum of the covariance matrix.

There is an inverse relation between predictive risk and estimation risk for ridge regression.

Eigenvalue distribution significantly influences classification accuracy.

Abstract

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.