Smooth Sensitivity Revisited: Towards Optimality

Abstract

Smooth sensitivity is one of the most commonly used techniques for designing practical differentially private mechanisms. In this approach, one computes the smooth sensitivity of a given query $q$ on the given input $D$ and releases $q(D)$ with noise added proportional to this smooth sensitivity. One question remains: what distribution should we pick the noise from? In this paper, we give a new class of distributions suitable for the use with smooth sensitivity, which we name the PolyPlace distribution. This distribution improves upon the state-of-the-art Student's T distribution in terms of standard deviation by arbitrarily large factors, depending on a "smoothness parameter" $\gamma$, which one has to set in the smooth sensitivity framework. Moreover, our distribution is defined for a wider range of parameter $\gamma$, which can lead to significantly better performance. Moreover, we prove that the PolyPlace distribution converges for $\gamma \rightarrow 0$ to the Laplace distribution and so does its variance. This means that the Laplace mechanism is a limit special case of the PolyPlace mechanism. This implies that out mechanism is in a certain sense optimal for $\gamma \to 0$.