Covering Numbers for Convex Functions

Abstract—In this paper, we study the covering numbers of the space of convex and uniformly bounded functions in multidimen-sion. We find optimal upper and lower bounds for the-covering number of, in the-metric, , in terms of the relevant constants, where, , , and denotes the set of all convex functions on that are uniformly bounded by.We summarize previously known re-sults 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 min-imization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems. Index Terms—Convexity constrained function estimation, empirical risk minimization, Hausdorff distance, Kolmogorov entropy,-metric, metric entropy, packing numbers