Unifying Laplace Mechanism with Instance Optimality in Differential Privacy

TL;DR

The paper introduces the piecewise Laplace mechanism (PLM), a DP mechanism that adapts noise scale to local sensitivity by partitioning the outcome space into intervals bounded by data-dependent extrema. By framing the construction through the exponential mechanism with a carefully designed quality score and implementing a two-step sampling procedure, PLM achieves pure $\varepsilon$-DP and matches an inverse-sensitivity-like intuition while improving accuracy in the continuous setting. It is shown to strictly dominate the inverse sensitivity mechanism (in terms of utility) and to reduce to the standard Laplace mechanism in the worst case where local and global sensitivities coincide. The work also provides approximate extensions to ease computation, and discusses extensions to higher dimensions and connections to smoothing variants, offering a practical path toward instance-optimal private data analysis using local sensitivity.

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.

Figures (1)

• Figure 1: Two-step sampling procedure for piecewise Laplace distribution

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 1
• Corollary 3.1
• Corollary 3.2
• Remark
• Definition 3.1
• Theorem 1
• Corollary 3.3