Concentration Inequalities in Locally Dependent Spaces

TL;DR

This paper investigates how local dependence affects concentration inequalities for functions of random variables, introducing hypergraph dependence to establish new concentration results under certain conditions.

Contribution

It introduces hypergraph dependence as a specific form of local dependence and demonstrates its implications for concentration inequalities in various metrics.

Findings

Hypergraph dependence implies concentration when the maximal neighborhood size is small.

Concentration results are established for Hamming and Talagrand distances.

Self-bounding functions also satisfy concentration under hypergraph dependence.

Abstract

This paper studies concentration inequalities for functions of locally dependent random variables. We show that the usual definition of local dependence does not imply concentration for general Hamming Lipschitz functions. We define hypergraph dependence, which is a special case of local dependence, and show that it implies concentration if the maximal neighborhood size is small. We prove concentration in Hamming distance, Talagrand distance, and for self-bounding functions of a particular type under this dependence structure.