Localization of VC Classes: Beyond Local Rademacher Complexities

Abstract

In statistical learning the excess risk of empirical risk minimization (ERM) is controlled by \(\left( \frac{\text COMP_n(\mathcal F)}n\right) ^{\alpha }\), where n is a size of a learning sample, \(\text COMP_n(\mathcal F)\) is a complexity term associated with a given class \(\mathcal F\) and \(\alpha \in [\frac{1}2, 1]\) 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 \(\text COMP_n(\mathcal F)\) 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.