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

Abstract

Let $n\ge 1$, $K>0$, 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), $\forall t>0$,

$$max\left(\mathbb{P}\right\{f(X)-\mathrm{Med}f(X)\ge t\},\mathbb{P}\{f(X)-\mathrm{Med}f(X)\le -t\left\}\right)\le exp(-\frac{c{t}^2}{K^{2}log(2+\frac{K^{2}n}{t^2})}),$$

where $c>0$ is a universal constant. The estimates are optimal in the sense that for every $n\ge \tilde{C}$ 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

$$\mathbb{P}\left\{\right|f(X)-\mathrm{Med}f(X)|\ge t\}\ge \tilde{c}exp(-\frac{\tilde{C}{t}^2}{K^{2}log(2+\frac{K^{2}n}{t^2})}).$$

The obtained deviation estimates for subgaussian variables are in sharp contrast with the case of variables with bounded $\Vert X_i{\Vert }_{{\mathit{\psi }}_p}$–quasi-norms for $p\in (0,2)$.