Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions

TL;DR

This paper establishes a bound on the subgaussian constant of restricted measures in metric probability spaces, leading to new concentration inequalities for non-Lipschitz functions.

Contribution

It introduces a universal bound on the subgaussian constant of restricted measures, enabling concentration results for non-Lipschitz functions.

Findings

Bound on subgaussian constant of restricted measures

Concentration inequalities for non-Lipschitz functions

Universal constant c in the bound

Abstract

We show that for any metric probability space $(M,d,\mu)$ with a subgaussian constant $\sigma^2(\mu)$ and any set $A \subset M$ we have $\sigma^2(\mu_A) \leq c \log\left(e/\mu(A)\right)\,\sigma^2(\mu)$, where $\mu_A$ is a restriction of $\mu$ to the set $A$ and $c$ is a universal constant. As a consequence we deduce concentration inequalities for non-Lipschitz functions.