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 $\gamma \to 0$.

翻译：平滑灵敏度是设计实用差分隐私机制最常用的技术之一。该方法通过计算给定查询$q$在输入$D$上的平滑灵敏度，并添加与该平滑灵敏度成比例的噪声来发布$q(D)$。然而，一个关键问题仍未解决：我们应当从何种分布中选取噪声？本文提出了一类适用于平滑灵敏度框架的新型噪声分布，命名为PolyPlace分布。该分布在标准差指标上相对于当前最优的学生t分布实现了任意倍数的改进，改进程度取决于平滑灵敏度框架中必须设置的“平滑参数”$\gamma$。此外，我们的分布定义了更广泛的$\gamma$参数范围，可带来显著的性能提升。我们进一步证明当$\gamma \rightarrow 0$时，PolyPlace分布及其方差均收敛于拉普拉斯分布，这意味着拉普拉斯机制是PolyPlace机制在极限情况下的特例。由此可推知，在$\gamma \to 0$的极限意义上，我们的机制具有某种最优性。