Covering Numbers for Convex Functions

TL;DR

This paper establishes optimal bounds for the covering numbers of convex functions in multiple dimensions, providing insights into their complexity and implications for statistical estimation.

Contribution

It derives the first optimal bounds for covering numbers of convex functions in multiple dimensions and offers new proofs and summaries of existing results.

Findings

Established optimal upper and lower bounds for covering numbers

Provided alternative proofs for known results

Implications for convergence rates in convex function estimation

Abstract

In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the $\epsilon$-covering number of $\C([a, b]^d, B)$, in the $L_p$-metric, $1 \le p < \infty$, in terms of the relevant constants, where $d \geq 1$, $a < b \in \mathbb{R}$, $B>0$, and $\C([a,b]^d, B)$ denotes the set of all convex functions on $[a, b]^d$ that are uniformly bounded by $B$. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.