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

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 (yi,boldsymbol xi), i≤ n are i.i.d. with boldsymbol xi~mathsfN(boldsymbol 0,boldsymbol Sigma) a p-dimensional Gaussian feature vector, and yi ∈+1,-1 a label whose distribution depends on a linear combination of the covariates langle boldsymbol θ_*,boldsymbol xi 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→∞ with p/n→ 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 θ_*) 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.