Efficient Concentration with Gaussian Approximation

TL;DR

This paper introduces new concentration inequalities that combine the advantages of classical bounds and the CLT, providing accurate, finite-sample confidence intervals for bounded variables without prior variance knowledge.

Contribution

We develop computable, asymptotically optimal concentration inequalities using Stein's method, improving confidence interval accuracy for any sample size and eliminating the need for prior variance knowledge.

Findings

New inequalities outperform classical bounds in finite samples

Efficient confidence intervals with correct coverage for all sample sizes

No prior knowledge of population variance required

Abstract

Concentration inequalities for the sample mean, like those due to Bernstein, Hoeffding, and Bentkus, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem (CLT) provides asymptotic confidence intervals with optimal width, but these are invalid for all sample sizes. To resolve this tension, we develop new computable concentration inequalities for bounded variables with asymptotically optimal size, finite-sample validity, and sub-Gaussian decay. These bounds enable the construction of efficient confidence intervals with correct coverage for any sample size and efficient empirical Berry-Esseen bounds that require no prior knowledge of the population variance. We derive our inequalities by tightly bounding non-uniform Kolmogorov and Wasserstein distances to a Gaussian using zero-bias couplings and Stein's method of exchangeable pairs and demonstrate practical improvements over the Bernstein, Hoeffding, Bentkus, Berry-Esseen, Feller-Cram\'er, Romano-Wolf, empirical Bernstein, empirical Bentkus, and coin-betting inequalities.