Dobrushin Coefficients of Private Mechanisms Beyond Local Differential Privacy

TL;DR

This paper studies the Dobrushin coefficients of discrete Markov kernels with bounded pointwise maximal leakage, providing bounds and mechanisms that extend local differential privacy to broader settings.

Contribution

It introduces bounds on contraction for kernels with bounded PML and constructs mechanisms achieving these bounds, extending results beyond LDP.

Findings

Derived bounds on contraction using PML guarantees.

Constructed mechanisms that achieve the bounds.

Extended analysis to general f-divergences.

Abstract

We investigate Dobrushin coefficients of discrete Markov kernels that have bounded pointwise maximal leakage (PML) with respect to all distributions with a minimum probability mass bounded away from zero by a constant $c>0$. This definition recovers local differential privacy (LDP) for $c\to 0$. We derive achievable bounds on contraction in terms of a kernels PML guarantees, and provide mechanism constructions that achieve the presented bounds. Further, we extend the results to general $f$-divergences by an application of Binette's inequality. Our analysis yields tighter bounds for mechanisms satisfying LDP and extends beyond the LDP regime to any discrete kernel.