$\alpha$-Wasserstein Mechanism for R\'{e}nyi Pufferfish Privacy

This paper introduces the $\alpha$-Wasserstein mechanism for achieving R\'{e}nyi Pufferfish Privacy using Laplace and Gaussian noise. By leveraging H\"{o}lder's inequality, we demonstrate that the scale parameter of the Laplace mechanism can be calibrated via an upper bound on the $W_\alpha$ metric to satisfy $(\alpha, \epsilon)$-R\'{e}nyi Pufferfish Privacy for $\alpha \in (1, \infty]$. We show that at the limit $\alpha = \infty$, this framework recovers the established $W_\infty$ mechanism for $\epsilon$-pufferfish privacy. This result is subsequently extended to the exponential mechanism. Furthermore, we propose a $W_\alpha$ mechanism for Gaussian noise for $\alpha \in (1, \infty)$, demonstrating that it generalizes existing results within the R\'enyi Differential Privacy framework. Experimental evaluations reveal that our $\alpha$-Wasserstein mechanism significantly reduces noise power compared to the conventional $W_\infty$-based approach, with the Gaussian mechanism providing superior utility over the Laplace mechanism. Notably, the mechanisms derived in this work achieve exact $(\alpha, \epsilon)$-R\'{e}nyi Pufferfish Privacy without requiring additional relaxations, such as $\delta$-approximations.