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

2026-06-07

AI Summary

This work addresses the long-standing trade-off between privacy and utility in additive noise mechanisms for high-dimensional real-valued vector queries under differential privacy, where no universally optimal solution has previously existed. The study establishes, for the first time, the asymptotic optimality of the Gaussian mechanism in high dimensions and introduces a novel family of spherical generalized gamma mechanisms that significantly outperform existing approaches in low-dimensional settings. This mechanism family unifies the Gaussian and ℓ₂ mechanisms within a single framework and resolves the open problem of tight composition bounds for the ℓ₂ mechanism. Consequently, the authors derive tight privacy composition bounds applicable across both low- and high-dimensional regimes.

Abstract

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 $T\to\infty$, no additive-noise mechanism can asymptotically improve on the Gaussian mechanism's privacy--utility tradeoff for the strong privacy settings typically used.We also develop a new family of \emph{Spherical Generalized Gamma} DP mechanisms, which contains both the Gaussian mechanism and the recently studied $\ell_2$ mechanism (Joseph \emph{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 \emph{et al.}~regarding the $\ell_2$ mechanism.

Problem

• differential privacy
• additive noise mechanism
• privacy-utility tradeoff
• Gaussian mechanism
• high-dimensional

Innovation

• Gaussian mechanism
• Spherical Generalized Gamma
• differential privacy
• asymptotic optimality
• tight composition