On Dimension-dependent concentration for convex Lipschitz functions in product spaces

TL;DR

This paper establishes sharp, dimension-dependent concentration inequalities for convex Lipschitz functions of subgaussian vectors, revealing how concentration behavior varies with dimension and contrasting with bounded-norm cases.

Contribution

It provides optimal deviation bounds for convex Lipschitz functions of subgaussian vectors, highlighting the dimension-dependent nature of concentration in product spaces.

Findings

Derived explicit concentration inequalities with logarithmic dimension dependence.

Proved the bounds are optimal through matching lower bounds.

Contrasted subgaussian case with bounded norm variables showing different concentration behaviors.

Abstract

Let $n\geq 1$, $K>0$, and let $X=(X_1,X_2,\dots,X_n)$ be a random vector in $\mathbb{R}^n$ with independent $K$--subgaussian components. We show that for every $1$--Lipschitz convex function $f$ in $\mathbb{R}^n$ (the Lipschitzness with respect to the Euclidean metric), $$ \max\big(\mathbb{P}\big\{f(X)-{\rm Med}\,f(X)\geq t\big\},\mathbb{P}\big\{f(X)-{\rm Med}\,f(X)\leq -t\big\}\big)\leq \exp\bigg( -\frac{c\,t^2}{K^2\log\big(2+\frac{ n}{t^2/K^2}\big)}\bigg),\quad t>0, $$ where $c>0$ is a universal constant. The estimates are optimal in the sense that for every $n\geq \tilde C$ and $t>0$ there exist a product probability distribution $X$ in $\mathbb{R}^n$ with $K$--subgaussian components, and a $1$--Lipschitz convex function $f$, with $$ \mathbb{P}\big\{\big|f(X)-{\rm Med}\,f(X)\big|\geq t\big\}\geq \tilde c\,\exp\bigg( -\frac{\tilde C\,t^2}{K^2\log\big(2+\frac{n}{t^2/K^2}\big)}\bigg). $$ The obtained deviation estimates for subgaussian variables are in sharp contrast with the case of variables with bounded $\|X_i\|_{\psi_p}$--norms for $p\in[1,2)$.