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Modern machine learning classifiers often exhibit vanishing classificationerror on the training set. They achieve this by learning nonlinearrepresentations of the inputs that maps the data into linearly separableclasses.  Motivated by these phenomena, we revisit high-dimensional maximum marginclassification for linearly separable data. We consider a stylized setting inwhich data $(y_i,{\boldsymbol x}_i)$, $i\le n$ are i.i.d. with ${\boldsymbolx}_i\sim\mathsf{N}({\boldsymbol 0},{\boldsymbol \Sigma})$ a $p$-dimensionalGaussian feature vector, and $y_i \in\{+1,-1\}$ a label whose distributiondepends on a linear combination of the covariates $\langle {\boldsymbol\theta}_*,{\boldsymbol x}_i \rangle$. While the Gaussian model might appearextremely simplistic, universality arguments can be used to show that theresults derived in this setting also apply to the output of certain nonlinearfeaturization 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 thistheory to derive two results of independent interest: $(i)$ Sufficientconditions on $({\boldsymbol \Sigma},{\boldsymbol \theta}_*)$ for `benignoverfitting' that parallel previously derived conditions in the case of linearregression; $(ii)$ An asymptotically exact expression for the generalizationerror when max-margin classification is used in conjunction with featurevectors produced by random one-layer neural networks.

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