Entropy, combinatorial dimensions and random averages

Abstract

In this article we introduce a new combinatorial parameter which generalizes the VC dimension and the fat-shattering dimension, and extends beyond the function-class setup. Using this parameter we establish entropy bounds for subsets of the n-dimensional unit cube, and in particular, we present new bounds on the empirical covering numbers and gaussian averages associated with classes of functions in terms of the fat-shattering dimension.