Constructive approaches to concentration inequalities with independent random variables
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.
Cite
@article{arxiv.2408.16480,
title = {Constructive approaches to concentration inequalities with independent random variables},
author = {Celine Moucer and Adrien Taylor and Francis Bach},
journal= {arXiv preprint arXiv:2408.16480},
year = {2024}
}