The role of regularization in classification of high-dimensional noisy Gaussian mixture

TL;DR

This paper rigorously analyzes how regularized convex classifiers perform in high-dimensional noisy Gaussian mixture models, revealing effects like reaching Bayes-optimal performance and the interpolation peak.

Contribution

It provides a theoretical analysis of regularized classifiers' generalization error in high-dimensional Gaussian mixtures, highlighting surprising effects of regularization.

Findings

Regularization can enable Bayes-optimal performance in high-dimensional noisy settings.

Interpolation peak occurs at low regularization levels.

The size imbalance of clusters affects classifier performance.

Abstract

We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number $n$ of samples and their dimension $d$ go to infinity while their ratio is fixed to $\alpha= n/d$. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances. We also illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.