Generalized Gaussian Mechanism for Differential Privacy

The goal of this paper is multi-fold. First, we define a family of generalized Gaussian mechanism (GGM) based on the $l_p$ global sensitivity (GS) of queries, which includes the Laplace and Gaussian mechanisms as special cases. We present theoretical results on the requirement to reach $\epsilon$-differential privacy (DP) and $(\epsilon, \delta)$-probabilistic DP, respectively, for the GGM. Second, we present the Gaussian mechanism as an example of the GG mechanism and compare the utility of the sanitized results from the Gaussian mechanism of $(\epsilon, \delta)$-probabilistic DP and the Laplace mechanism in independent sanitization. We derive a new lower bound on the scale parameter for the Gaussian mechanism of $(\epsilon,\delta)$-probabilistic DP. The new bound is tighter than the existing one in the literature. Lastly, we investigate the connections and differences between the GGM and the generalized Gaussian distribution-based Exponential mechanism, and establish the relationship between the $l_p$ GS of queries and the GS of the utility function in the Exponential mechanism.