Information Contraction under $(\varepsilon,\delta)$-Differentially Private Mechanisms

TL;DR

This paper extends the understanding of information contraction under $(\, ext{ε,δ})$-differential privacy mechanisms, providing new inequalities that apply to a broader class of privacy settings and improve existing bounds.

Contribution

It derives new strong data-processing inequalities for hockey-stick and $f$-divergences valid for all $(\, ext{ε,δ})$-LDP mechanisms, generalizing previous results.

Findings

New inequalities for hockey-stick divergence under $(\, ext{ε,δ})$-LDP.

Generalization of contraction bounds to all $(\, ext{ε,δ})$-LDP mechanisms.

Improved bounds on information measure contraction for privacy analysis.

Abstract

The distinguishability quantified by information measures after being processed by a private mechanism has been a useful tool in studying various statistical and operational tasks while ensuring privacy. To this end, standard data-processing inequalities and strong data-processing inequalities (SDPI) are employed. Most of the previously known and even tight characterizations of contraction of information measures, including total variation distance, hockey-stick divergences, and $f$-divergences, are applicable for $(\varepsilon,0)$-local differential private (LDP) mechanisms. In this work, we derive both linear and non-linear strong data-processing inequalities for hockey-stick divergence and $f$-divergences that are valid for all $(\varepsilon,\delta)$-LDP mechanisms even when $\delta \neq 0$. Our results either generalize or improve the previously known bounds on the contraction of these distinguishability measures.