Equivalence of concentration inequalities for linear and non-linear functions

TL;DR

This paper demonstrates that, under certain conditions, concentration inequalities for linear and non-linear functions of a random variable in a topological vector space are equivalent, linking classical concentration results across different settings.

Contribution

It establishes the equivalence of concentration inequalities for linear and non-linear functions using a normal distance, extending classical results to infinite-dimensional spaces.

Findings

Concentration inequalities are equivalent for linear and non-linear functions under a normal distance.

Results are asymptotically optimal in high-dimensional limits.

Applicable to classical concentration scenarios like Gaussian and cube concentration.

Abstract

We consider a random variable $X$ that takes values in a (possibly infinite-dimensional) topological vector space $\mathcal{X}$. We show that, with respect to an appropriate "normal distance" on $\mathcal{X}$, concentration inequalities for linear and non-linear functions of $X$ are equivalent. This normal distance corresponds naturally to the concentration rate in classical concentration results such as Gaussian concentration and concentration on the Euclidean and Hamming cubes. Under suitable assumptions on the roundness of the sets of interest, the concentration inequalities so obtained are asymptotically optimal in the high-dimensional limit.