Constructive approaches to concentration inequalities with independent random variables

TL;DR

This paper introduces a convex optimization framework to derive and improve concentration inequalities for independent random variables, extending classical methods with new variational and polynomial techniques.

Contribution

It develops a systematic approach combining variational and sum-of-squares methods to extend and refine classical concentration inequalities for independent variables.

Findings

Refined Hoeffding's, Bennett's, and Bernstein's inequalities.

Extended the generalized problem of moments to independent variables.

Provided improved worst-case guarantees for concentration bounds.

Abstract

Concentration inequalities, a major tool in probability theory, quantify how much a random variable deviates from a certain quantity. This paper proposes a systematic convex optimization approach to studying and generating concentration inequalities with independent random variables. Specifically, we extend the generalized problem of moments to independent random variables. We first introduce a variational approach that extends classical moment-generating functions, focusing particularly on first-order moment conditions. Second, we develop a polynomial approach, based on a hierarchy of sum-of-square approximations, to extend these techniques to higher-moment conditions. Building on these advancements, we refine Hoeffding's, Bennett's and Bernstein's inequalities, providing improved worst-case guarantees compared to existing results.