Sharp Guarantees and Optimal Performance for Inference in Binary and Gaussian-Mixture Models †

We study convex empirical risk minimization for high-dimensional inference in binary linear classification under both discriminative binary linear models, as well as generative Gaussian-mixture models. Our first result sharply predicts the statistical performance of such estimators in the proportional asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit to prove bounds on the best achievable performance. Notably, we show that the proposed bounds are tight for popular binary models (such as signed and logistic) and for the Gaussian-mixture model by constructing appropriate loss functions that achieve it. Our numerical simulations suggest that the theory is accurate even for relatively small problem dimensions and that it enjoys a certain universality property.