Bracketing numbers of convex and $m$-monotone functions on polytopes

TL;DR

This paper establishes upper bounds for bracketing numbers of convex and m-monotone functions on polytopes, advancing understanding of statistical estimation under shape constraints without Lipschitz restrictions.

Contribution

It provides new bracketing number bounds for convex functions without Lipschitz constraints and introduces the multivariate m-monotone functions, extending univariate concepts.

Findings

Derived bracketing number bounds for convex functions on polytopes.

Extended bracketing bounds to multivariate m-monotone functions.

Applied proof techniques to new classes of multivariate functions.

Abstract

We study bracketing covering numbers for spaces of bounded convex functions in the $L_p$ norms. Bracketing numbers are crucial quantities for understanding asymptotic behavior for many statistical nonparametric estimators. Bracketing number upper bounds in the supremum distance are known for bounded classes that also have a fixed Lipschitz constraint. However, in most settings of interest, the classes that arise do not include Lipschitz constraints, and so standard techniques based on known bracketing numbers cannot be used. In this paper, we find upper bounds for bracketing numbers of classes of convex functions without Lipschitz constraints on arbitrary polytopes. Our results are of particular interest in many multidimensional estimation problems based on convexity shape constraints. Additionally, we show other applications of our proof methods; in particular we define a new class of multivariate functions, the so-called $m$-monotone functions. Such functions have been considered mathematically and statistically in the univariate case but never in the multivariate case. We show how our proof for convex bracketing upper bounds also applies to the $m$-monotone case.