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We characterize the minimum noise amplitude and power for noise-addingmechanisms in $(\epsilon, \delta)$-differential privacy for single real-valuedquery function. We derive new lower bounds using the duality of linearprogramming, and new upper bounds by proposing a new class of$(\epsilon,\delta)$-differentially private mechanisms, the \emph{truncatedLaplacian} mechanisms. We show that the multiplicative gap of the lower boundsand upper bounds goes to zero in various high privacy regimes, proving thetightness of the lower and upper bounds and thus establishing the optimality ofthe truncated Laplacian mechanism. In particular, our results close theprevious constant multiplicative gap in the discrete setting. Numericexperiments show the improvement of the truncated Laplacian mechanism over theoptimal Gaussian mechanism in all privacy regimes.

Privacy and Utility Tradeoff in Approximate Differential Privacy