Optimality of the Laplace Mechanism in Differential Privacy

TL;DR

This paper introduces Lipschitz privacy, a refined privacy framework, and proves the Laplace mechanism's optimality in minimizing mean-squared error for identity queries under differential privacy, especially for high-dimensional data.

Contribution

It establishes the optimality of the Laplace mechanism within Lipschitz privacy and derives the best mechanism for high-dimensional data scenarios.

Findings

Laplace mechanism is optimal for identity queries under Lipschitz privacy.

A variation of the Laplace mechanism achieves minimal mean-squared error.

Derived the optimal mechanism for high-dimensional sensitive data submissions.

Abstract

In the highly interconnected realm of Internet of Things, exchange of sensitive information raises severe privacy concerns. The Laplace mechanism -- adding Laplace-distributed artificial noise to sensitive data -- is one of the widely used methods of providing privacy guarantees within the framework of differential privacy. In this work, we present Lipschitz privacy, a slightly tighter version of differential privacy. We prove that the Laplace mechanism is optimal in the sense that it minimizes the mean-squared error for identity queries which provide privacy with respect to the $\ell_{1}$-norm. In addition to the $\ell_{1}$-norm which respects individuals' participation, we focus on the use of the $\ell_{2}$-norm which provides privacy of high-dimensional data. A variation of the Laplace mechanism is proven to have the optimal mean-squared error from the identity query. Finally, the optimal mechanism for the scenario in which individuals submit their high-dimensional sensitive data is derived.