Localization of VC classes: Beyond local Rademacher complexities

Abstract

In statistical learning the excess risk of empirical risk minimization (ERM) is controlled by (Formula Presented), where n is a size of a learning sample, COMPn(F) is a complexity term associated with a given class F and α ∈ [1/2,1] interpolates between slow and fast learning rates. In this paper we introduce an alternative localization approach for binary classification that leads to a novel complexity measure: fixed points of the local empirical entropy. We show that this complexity measure gives a tight control over COMPn(F) in the upper bounds under bounded noise. Our results are accompanied by a novel minimax lower bound that involves the same quantity. In particular, we practically answer the question of optimality of ERM under bounded noise for general VC classes.