Covering Numbers for Convex Functions
This work addresses a foundational problem in mathematical statistics and machine learning, with direct implications for convergence rates in empirical minimization and convexity-constrained function estimation, though it is incremental as it builds on and refines known results.
The paper tackles the problem of determining optimal covering numbers for convex, uniformly bounded functions in multi-dimensional spaces under L_p metrics, establishing both upper and lower bounds in terms of relevant constants.
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 $ε$-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.