Sharp Asymptotics and Optimal Performance for Inference in Binary Models

Understand

We study convex empirical risk minimization for high-dimensional inference in binary models.

• Our first result sharply predicts the statistical performance of such estimators in the linear asymptotic regime under isotropic Gaussian features.
• Importantly, the predictions hold for a wide class of convex loss functions, which we exploit in order to prove a bound on the best achievable performance among them.
• Notably, we show that the proposed bound is tight for popular binary models (such as Signed, Logistic or Probit), by constructing appropriate loss functions that achieve it.