On the tightness of Gaussian concentration for convex functions

Abstract

The concentration of measure phenomenon in Gauss' space states that every L-Lipschitz map f on R n satisfies2L 2 , t > 0, where γn is the standard Gaussian measure on R n and M f is a median of f . In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when f is additionally assumed to be convex. In particular, we show that if the variance Var(f ) (with respect to γn) satisfies αL Var(f ) for some 0 < α 1, thenwhere c, C > 0 are constants depending only on α.

Improved one-sided deviation inequalities under regularity assumptions for product measures

“…as we have just seen on the basic example of the maximum of n independent standard Gaussian random variables. Ehrhard's inequality has also been used by Valettas in [22] where he proved that (1.1) is tight if the convex function f is not superconcentrated.…”

“…Indeed, as consequence of his inequality with Paouris, combined with transportation-type arguments, he obtained (cf. [22], section 2.1.3) concentration inequalities for nondecreasing, convex functions in a log-concave measures setting.…”

“…[6]). This kind of phenomenon occurs for different functionals of Gaussian random variables and has been studied in [4,19,20,18,22]. .…”

Remarks on Superconcentration and Gamma Calculus: Applications to Spin Glasses

“…As we have said in the introduction, this phenomenon has been studied in various manner : semigroup interpolation [14], Renyi's representation of order statistics [3], Optimal Transport [15], Ehrard's inequality [17],. .…”

Dichotomies, structure, and concentration in normed spaces

“…[LT91, Corollary 3.2], [LMS98, Statement 3.1] and [PVZ17, Proposition 2.10]). In the small deviation regime 0 < t < 1 there exist many important examples which show that the obtained bounds are suboptimal; see [PVZ17] and [Val17] for a detailed discussion. Ideally one would like to know what properties of the underlying function improve the concentration.…”