Unifying Laplace Mechanism with Instance Optimality in Differential Privacy

TL;DR

This paper introduces a piecewise Laplace mechanism for differential privacy that adapts noise to local dataset sensitivity, achieving near instance optimality and outperforming previous methods like inverse sensitivity.

Contribution

It proposes a novel piecewise Laplace mechanism that defies traditional assumptions, providing a more optimal privacy-preserving noise addition method based on local sensitivity.

Findings

The mechanism strictly dominates the inverse sensitivity approach.

It reduces to the standard Laplace mechanism in worst-case scenarios.

An approximate variant extends applicability to higher dimensions.

Abstract

We adapt the canonical Laplace mechanism, widely used in differentially private data analysis, to achieve near instance optimality with respect to the hardness of the underlying dataset. In particular, we construct a piecewise Laplace distribution whereby we defy traditional assumptions and show that Laplace noise can in fact be drawn proportional to the local sensitivity when done in a piecewise manner. While it may initially seem counterintuitive that this satisfies (pure) differential privacy and can be sampled, we provide both through a simple connection to the exponential mechanism and inverse sensitivity along with the fact that the Laplace distribution is a two-sided exponential distribution. As a result, we prove that in the continuous setting our \textit{piecewise Laplace mechanism} strictly dominates the inverse sensitivity mechanism, which was previously shown to both be nearly instance optimal and uniformly outperform the smooth sensitivity framework. Furthermore, in the worst-case where all local sensitivities equal the global sensitivity, our method simply reduces to a Laplace mechanism. We also complement this with an approximate local sensitivity variant to potentially ease the computational cost, which can also extend to higher dimensions.