Asymptotic Optimality of the High-Dimensional Gaussian Mechanism and Improved Low-Dimensional Mechanisms for Differential Privacy

The additive noise mechanism is a foundational tool for differential privacy (DP) of $T$-dimensional real-valued vector queries. The Gaussian mechanism, utilizing Gaussian noise, is the mostly widely used such mechanism, due to its simplicity and strong privacy guarantees. In this work, we provide justification for this choice, showing that as the dimension $T\to\infty$, the Gaussian mechanism has the lowest error among all additive noise mechanisms for all meaningful privacy regimes. We also develop a new family of *Spherical Generalized Gamma* DP mechanisms, which contains both the Gaussian mechanism and the recently studied $\ell_2$ mechanism (Joseph *et al.*, ICML 2025). We identify members of this family that outperform both the Gaussian and $\ell_2$ mechanisms in certain low-dimensional settings, and show tight composition of all mechanisms in this family, answering an open question of Joseph *et al.* regarding the $\ell_2$ mechanism.