A closed form scale bound for the $(\epsilon ,\delta )$-differentially private Gaussian Mechanism valid for all privacy regimes

Abstract

The standard closed form lower bound on $\sigma$ for providing $(\epsilon ,\delta )$-differential privacy by adding zero mean Gaussian noise with variance ${\sigma }^2$ is $\sigma >\Delta \sqrt{2}({\epsilon }^{-1})\sqrt{\log \left(5/4{\delta }^{-1}\right)}$ for $\epsilon \in (0,1)$. We present a similar closed form bound $\sigma \ge \Delta (\epsilon \sqrt{2}{)}^{-1}\left(\sqrt{az+\epsilon }+s\sqrt{az}\right)$ for $z=-\log (4\delta (1-\delta ))$ and $(a,s)=(1,1)$ if $\delta \le 1/2$ and $(a,s)=(\pi /4,-1)$ otherwise. Our bound is valid for all $\epsilon >0$ and is always lower (better). We also present a sufficient condition for $(\epsilon ,\delta )$-differential privacy when adding noise distributed according to even and log-concave densities supported everywhere.

11 pages. Version 2 improves on the bound