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

The standard closed form lower bound on $\sigma$ for providing $(\epsilon, \delta)$-differentialprivacybyaddingzeromeanGaussiannoisewithvariance$\sigma^2$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 \geq \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 \leq 1/2$and$(a,s)=(\pi/4,-1)$otherwise\mathrm{.}Ourboundisvalidforall$\epsilon > 0$ and is always lower (better). We also present a sufficient condition for $(ϵ,\delta )$-differential privacy when adding noise distributed according to even and log-concave densities supported everywhere.