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

• URL: http://arxiv.org/abs/2012.10523v2
• Date: Thu, 21 Jan 2021 14:01:55 GMT
• Title: A closed form scale bound for the $(\epsilon, \delta)$-differentially private Gaussian Mechanism valid for all privacy regimes
• Authors: Staal A. Vinterbo
• Abstract summary: We present a similar closed form bound $sigma geq Delta (epsilonsqrt2)-1 left(sqrtaz+epsilon + ssqrtazright)$ for $epsilon in (0,1)$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\sigma \geq \Delta (\epsilon\sqrt{2})^{-1} \left(\sqrt{az+\epsilon} + s\sqrt{az}\right)$ for $z=-\log(4\delta(1-\delta))$ and $(a,s)=(1,1)$ if $\delta \leq 1/2$ and $(a,s)=(\pi/4,-1)$ otherwise. 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.

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