Higher order concentration in presence of Poincar\'e-type inequalities

TL;DR

This paper develops refined concentration inequalities for differentiable functions in Euclidean space, utilizing higher-order derivatives and Poincaré-type inequalities to achieve sharper bounds.

Contribution

It introduces higher-order concentration bounds based on derivatives of order d, extending classical results to more precise inequalities under Poincaré conditions.

Findings

Sharper concentration bounds using d-th order derivatives

Extension of measure concentration to higher orders

Applicability to functions satisfying Poincaré inequalities

Abstract

We show sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d \in \mathbb{N}$. Here we focus on differentiable functions on the Euclidean space in presence of a Poincar\'e-type inequality. The bounds are based on $d$-th order derivatives.