Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

TL;DR

This paper develops new concentration inequalities for functions with bounded higher-order derivatives under Sobolev-type inequalities, extending previous results for Gaussian polynomials to broader classes of functions and measures.

Contribution

It introduces a general concentration inequality framework for non-Lipschitz functions with bounded derivatives, applicable under Sobolev inequalities, and extends results to polynomial functions and sub-Gaussian vectors.

Findings

Provides concentration bounds expressed via tensor-product norms of derivatives.

Establishes reverse inequalities for polynomial functions under Gaussian measures.

Applies inequalities to random matrix eigenvalues and Erdős-Rényi graph cycle counts.

Abstract

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Lata{\l}a we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\|\nabla g\|_p.$ Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or the Poincar\'e inequality (then $C(p) \le Cp$). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of $f$. When the underlying measure is Gaussian and $f$ is a polynomial (non-necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erd\H{o}s-R{\'e}nyi random graphs, obtaining new estimates, optimal in a certain range of parameters.