Empirical measures: regularity is a counter-curse to dimensionality

TL;DR

This paper introduces a decomposition method demonstrating that regularity in observables significantly accelerates the convergence rate of empirical measures, especially in high-dimensional settings, with applications to Markov chains.

Contribution

The paper presents a novel decomposition approach showing that regularity assumptions lead to faster convergence rates for empirical measures in high-dimensional spaces.

Findings

Convergence rate of 1/√n for sufficiently regular observables

Applicability to Markov chains with geometric contraction

Enhanced convergence speed with regularity assumptions

Abstract

We propose a "decomposition method" to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables, the convergence is much faster than for, say, merely Lipschitz observables. Actually, assuming $s$ derivatives with $s < d/2$ ($d$ the dimension) ensures an optimal rate of convergence of $1/\sqrt{n}$ ($n$ the number of samples). The method is flexible enough to apply to Markov chains which satisfy a geometric contraction hypothesis, assuming neither stationarity nor reversibility, with the same convergence speed up to a power of logarithm factor. Our results are stated as controls of the expected distance between the empirical measure and its limit, but we explain briefly how the classical method of bounded difference can be used to deduce concentration estimates.