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Bracketing numbers of convex and m-monotone functions on polytopes

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Abstract

We study bracketing covering numbers for spaces of bounded convex functions in the Lp 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.

Section snippets

Introduction and motivation

To quantify the size of an infinite dimensional set, the pioneering work of [34] studied the so-called metric entropy of the set, which is the logarithm of the metric covering number of the set. In this paper, we are interested in a related quantity, the bracketing entropy for a class of functions, which serves a similar purpose as metric entropy. Metric or bracketing entropies quantify the amount of information it takes to approximate any element of a set with a given accuracy ϵ>0. This

Bracketing with lipschitz constraints

If we have sets DiRd, i=1,,M, for MN, and Di=1MDi then for ϵi>0, 0<p<, and any class of functions F, N[]((i=1Mϵip)1p,F,Lp)i=1MN[]ϵi,F|Di,Lp,where, for a set G, we let F|G denote the class f|G:fF where f|G is the restriction of f to the set G. We will apply (5) to a cover of D by sets G with the property that CD,1|GCG,1,Γ for some bounded vector Γ, so that we can apply bracketing results for classes of convex functions with Lipschitz bounds. Thus, in this section, we develop the

Bracketing without lipschitz constraints

In the previous section we bounded bracketing entropy for classes of functions with Lipschitz constraints. In this section we remove those Lipschitz constraints. With Lipschitz constraints we could consider arbitrary domains D, but without the Lipschitz constraints we need more restrictions: now we will take D to be a simple polytope (defined below). We now define notation and assumptions we will use for the remainder of the document.

Properties of Gi,j

In this section we show how to embed the domains Gi,j, which partition D, into hyperrectangles. We used this in the proof of Theorem 3.5 so we could apply Theorem 2.1. Theorem 2.1 says that the bracketing entropy of convex functions on domain D with Lipschitz constraints along directions e1,,ek depends on w(D,ei) (since that gives the maximum “rise” in “rise over run”). In our proof of Theorem 3.5 we partitioned D into sets related to parallelotopes. Thus we will study these parallelotopes. We

Further applications

We now consider further entropy bounds that rely on the above ideas, results, or their proofs. In Section 5.1 we consider so-called univariate and multivariate m-monotone functions. In Section 5.2 we briefly consider estimation of level sets of convex functions and the question of adaptation to polytopal level sets. Further discussion is given at the beginning of the two subsections.

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    Supported by National Science Foundation grant DMS-1712664..

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