Smooth Sensitivity Revisited: Towards Optimality

TL;DR

This paper introduces the PolyPlace distribution, a new noise distribution for smooth sensitivity in differential privacy, which outperforms existing distributions and generalizes the Laplace mechanism as a special case.

Contribution

The authors propose the PolyPlace distribution, improving noise addition in smooth sensitivity, and establish its convergence to the Laplace distribution, indicating near-optimality.

Findings

PolyPlace distribution reduces noise standard deviation compared to Student's T.

The distribution is applicable over a wider range of the smoothness parameter γ.

PolyPlace converges to Laplace as γ approaches zero, demonstrating its optimality in that limit.

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 $\gamma \to 0$.