The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime

TL;DR

This paper analyzes the generalization error of max-margin linear classifiers in high-dimensional settings, revealing conditions for benign overfitting and providing exact error formulas, with implications for neural network features.

Contribution

It derives exact asymptotic expressions for generalization error in high-dimensional max-margin classification and identifies conditions for benign overfitting, extending understanding beyond linear models.

Findings

Exact formulas for generalization error in high-dimensional regimes

Conditions for benign overfitting in max-margin classifiers

Application to neural network feature representations

Abstract

Modern machine learning classifiers often exhibit vanishing classification error on the training set. They achieve this by learning nonlinear representations of the inputs that maps the data into linearly separable classes. Motivated by these phenomena, we revisit high-dimensional maximum margin classification for linearly separable data. We consider a stylized setting in which data $(y_i,{\boldsymbol x}_i)$, $i\le n$ are i.i.d. with ${\boldsymbol x}_i\sim\mathsf{N}({\boldsymbol 0},{\boldsymbol \Sigma})$ a $p$-dimensional Gaussian feature vector, and $y_i \in\{+1,-1\}$ a label whose distribution depends on a linear combination of the covariates $\langle {\boldsymbol \theta}_*,{\boldsymbol x}_i \rangle$. While the Gaussian model might appear extremely simplistic, universality arguments can be used to show that the results derived in this setting also apply to the output of certain nonlinear featurization maps. We consider the proportional asymptotics $n,p\to\infty$ with $p/n\to \psi$, and derive exact expressions for the limiting generalization error. We use this theory to derive two results of independent interest: $(i)$ Sufficient conditions on $({\boldsymbol \Sigma},{\boldsymbol \theta}_*)$ for `benign overfitting' that parallel previously derived conditions in the case of linear regression; $(ii)$ An asymptotically exact expression for the generalization error when max-margin classification is used in conjunction with feature vectors produced by random one-layer neural networks.