# Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities

## TL;DR

This paper establishes multilevel concentration inequalities for bounded functionals of random variables, extending to dependent cases and providing applications in empirical processes, chaos, U-statistics, and random graphs.

## Contribution

It introduces new concentration inequalities based on generalized log-Sobolev inequalities, applicable to both independent and dependent variables, with explicit constants involving higher order differences.

## Key findings

- Derived tail bounds for empirical processes and chaos.
- Provided concentration inequalities for U-statistics with bounded kernels.
- Extended results to dependent variables and random graph models.

## Abstract

In this paper we prove multilevel concentration inequalities for bounded functionals $f = f(X_1, \ldots, X_n)$ of random variables $X_1, \ldots, X_n$ that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of $k$-tensors of higher order differences of $f$. We provide applications in both dependent and independent random variables. This includes deviation inequalities for empirical processes $f(X) = \sup_{g \in \mathcal{F}} \lvert g(X) \rvert$ and suprema of homogeneous chaos in bounded random variables in the Banach space case given by $f(X) = \sup_{t} \lVert \sum_{i_1 \neq \ldots \neq i_d} t_{i_1 \ldots i_d} X_{i_1} \cdots X_{i_d}\rVert_{\mathcal{B}}$. The latter application is comparable to earlier results of Boucheron-Bousquet-Lugosi-Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for $U$-statistics with bounded kernels $h$ and for the number of triangles in an exponential random graph model.