  Large-margin classifiers are popular methods for classification. We derive
the asymptotic expression for the generalization error of a family of
large-margin classifiers in the limit of both sample size $n$ and dimension $p$
going to $\infty$ with fixed ratio $\alpha=n/p$. This family covers a broad
range of commonly used classifiers including support vector machine, distance
weighted discrimination, and penalized logistic regression. Our result can be
used to establish the phase transition boundary for the separability of two
classes. We assume that the data are generated from a single multivariate
Gaussian distribution with arbitrary covariance structure. We explore two
special choices for the covariance matrix: spiked population model and two
layer neural networks with random first layer weights. The method we used for
deriving the closed-form expression is from statistical physics known as the
replica method. Our asymptotic results match simulations already when $n,p$ are
of the order of a few hundreds. For two layer neural networks, we reproduce the
recently developed `double descent' phenomenology for several classification
models. We also discuss some statistical insights that can be drawn from these
analysis.
