Local Glivenko-Cantelli

TL;DR

This paper investigates dimension-free bounds on the maximum deviation of empirical means from true means over the Boolean cube, establishing necessary conditions and sharp decay rates for various measures, including product measures.

Contribution

It introduces the concept of local Glivenko-Cantelli bounds, providing necessary and sufficient conditions for convergence and deriving sharp rates, along with a novel sub-gamma maximal inequality.

Findings

Established conditions for $ riangle_n o 0$ for various measures.

Derived sharp decay rates for the maximal deviation.

Discovered a new sub-gamma maximal inequality for shifted Bernoullis.

Abstract

If $\mu$ is a distribution over the $d$-dimensional Boolean cube $\{0,1\}^d$, our goal is to estimate its mean $p\in[0,1]^d$ based on $n$ iid draws from $\mu$. Specifically, we consider the empirical mean estimator $\hat p_n$ and study the expected maximal deviation $\Delta_n=\mathbb{E}\max_{j\in[d]}|\hat p_n(j)-p(j)|$. In the classical Universal Glivenko-Cantelli setting, one seeks distribution-free (i.e., independent of $\mu$) bounds on $\Delta_n$. This regime is well-understood: for all $\mu$, we have $\Delta_n\lesssim\sqrt{\log(d)/n}$ up to universal constants, and the bound is tight. Our present work seeks to establish dimension-free (i.e., without an explicit dependence on $d$) estimates on $\Delta_n$, including those that hold for $d=\infty$. As such bounds must necessarily depend on $\mu$, we refer to this regime as {\em local} Glivenko-Cantelli (also known as $\mu$-GC), and are aware of very few previous bounds of this type -- which are either ``abstract'' or quite sub-optimal. Already the special case of product measures $\mu$ is rather non-trivial. We give necessary and sufficient conditions on $\mu$ for $\Delta_n\to0$, and calculate sharp rates for this decay. Along the way, we discover a novel sub-gamma-type maximal inequality for shifted Bernoullis, of independent interest.