Unifying Laplace Mechanism with Instance Optimality in Differential Privacy

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.