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 Tdimensional 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 → ∞, no additive-noise mechanism can asymptotically improve on the Gaussian mechanism's privacyutility tradeoff for the strong privacy settings typically used.We also develop a new family of Spherical Generalized Gamma DP mechanisms, which contains both the Gaussian mechanism and the recently studied ℓ 2 mechanism (Joseph et al., ICML 2025). We identify members of this family that outperform both the Gaussian and ℓ 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 ℓ 2 mechanism.