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

TL;DR

This paper derives an explicit upper bound for the expected supremum of an empirical process over a bounded VC-major class, especially for functions with small variance, improving upon universal entropy bounds.

Contribution

It introduces a new bound applicable to VC-subgraph and VC-major classes that does not depend on the entropy of the entire class, with explicit constants.

Findings

Provides a tighter bound than universal entropy bounds for small-variance functions.

Applicable to VC-subgraph and VC-major classes.

Does not require knowledge of the class entropy.

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