Privacy and Utility Tradeoff in Approximate Differential Privacy

We characterize the minimum noise amplitude and power for noise-adding mechanisms in $(\epsilon, \delta)$-differential privacy for single real-valued query function. We derive new lower bounds using the duality of linear programming, and new upper bounds by proposing a new class of $(\epsilon,\delta)$-differentially private mechanisms, the \emph{truncated Laplacian} mechanisms. We show that the multiplicative gap of the lower bounds and upper bounds goes to zero in various high privacy regimes, proving the tightness of the lower and upper bounds and thus establishing the optimality of the truncated Laplacian mechanism. In particular, our results close the previous constant multiplicative gap in the discrete setting. Numeric experiments show the improvement of the truncated Laplacian mechanism over the optimal Gaussian mechanism in all privacy regimes.