Covering numbers of $L_p$-balls of convex sets and functions

TL;DR

This paper establishes bounds on the covering numbers of convex functions and sets in Euclidean space under weaker integral constraints, extending previous results that required uniform boundedness.

Contribution

It introduces new bounds for covering numbers of convex classes under integral constraints, broadening the applicability of existing theories.

Findings

Bounds for covering numbers under weaker conditions

Extension of previous uniform boundedness results

Applicable to broader classes of convex functions and sets

Abstract

We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it with weaker integral constraints. Existing results can be recovered as special cases of our results.