A closed form scale bound for the $(ε, δ)$-differentially private Gaussian Mechanism valid for all privacy regimes
This work offers an improved, more accurate bound for the Gaussian mechanism, which is a fundamental component for researchers and practitioners working with differential privacy.
This paper provides a new closed-form lower bound for the standard deviation (σ) of the Gaussian mechanism, which is used to achieve (ε, δ)-differential privacy. The new bound is always lower (better) than the existing standard bound and is valid for all ε > 0.
The standard closed form lower bound on $σ$ for providing $(ε, δ)$-differential privacy by adding zero mean Gaussian noise with variance $σ^2$ is $σ> Δ\sqrt {2}(ε^{-1}) \sqrt {\log \left( 5/4δ^{-1} \right)}$ for $ε\in (0,1)$. We present a similar closed form bound $σ\geq Δ(ε\sqrt{2})^{-1} \left(\sqrt{az+ε} + s\sqrt{az}\right)$ for $z=-\log(4δ(1-δ))$ and $(a,s)=(1,1)$ if $δ\leq 1/2$ and $(a,s)=(π/4,-1)$ otherwise. Our bound is valid for all $ε> 0$ and is always lower (better). We also present a sufficient condition for $(ε, δ)$-differential privacy when adding noise distributed according to even and log-concave densities supported everywhere.