Scale Analysis and Shape Selection for the Generalized Gaussian Mechanism under Approximate Differential Privacy
Differential privacy provides a rigorous framework for protecting private information, typically achieved by adding random noise to query results. The generalized Gaussian family is a flexible class of additive noise distributions indexed by the shape parameter $p$ and includes the Laplace and Gaussian distributions as special cases $p=1$ and $p=2$, respectively. This paper studies the privacy-feasible scale estimation and the shape parameter selection of the generalized Gaussian mechanism (GGM) under $(\varepsilon,δ)$-differential privacy. For a given sensitivity vector $Δ$ and $p\in[1,\infty]$, let $b(p)$ denote the smallest value of the scale parameter for which the mechanism satisfies this privacy requirement. In the one-dimensional case, $b(p)$ can be implicitly characterized by a system of equations. For vector-valued queries, we construct a computable upper approximation of $b(p)$ that preserves the privacy guarantee. Shapes are compared under a scale-homogeneous utility criterion, with the $m$-th absolute moment as the main example. We develop an interval-wise shape search algorithm with an approximation guarantee that can be made arbitrarily precise. We also establish the invariance of the optimal shape under rescaling of the sensitivity vector and characterize its limiting behaviour under high privacy limits. Computational experiments show that optimizing shape parameters can improve utility by reducing the variance of each coordinate by 5% to 20% across a variety of cases, with some cases showing even greater reductions, while maintaining the same level of privacy protection. Task-specific experiments further show that shape optimization can improve task-level utility, reduce attacker success, or achieve both.