A closed form scale bound for the (ε, δ)-differentially private Gaussian Mechanism valid for all privacy regimes
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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√(2)(ϵ^-1) √(log( 5/4δ^-1)) for ϵ∈ (0,1). We present a similar closed form bound σ≥Δ (ϵ√(2))^-1(√(az+ϵ) + s√(az)) for z=-log(4δ(1-δ)) and (a,s)=(1,1) if δ≤ 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.
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