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

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 $(\epsilon ,0)$-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 $(\epsilon ,\delta )$-LDP mechanisms even when $\delta \mathrm{\ne }0$. Our results either generalize or improve the previously known bounds on the contraction of these distinguishability measures.