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Vinterbo","submitted_at":"2020-12-18T21:35:26Z","abstract_excerpt":"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). 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