Bounding the expectation of the supremum of an empirical process over a (weak) vc-major class

Abstract

Given a bounded class of functions G and independent random variables X1, . . . , Xn, we provide an upper bound for the expectation of the supremum of the empirical process over elements of G having a small variance. Our bound applies in the cases where G is a VC-subgraph or a VC-major class and it is of smaller order than those one could get by using a universal entropy bound over the whole class G . It also involves explicit constants and does not require the knowledge of the entropy of G