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.

翻译：加性噪声机制是保障$T$维实值向量查询差分隐私（DP）的基础工具。其中，采用高斯噪声的高斯机制因其简洁性与强隐私保证成为应用最广泛的此类机制。本研究为这一选择提供了理论依据：当维度$T\to\infty$时，在通常采用的强隐私设置下，任何加性噪声机制均无法在隐私-效用权衡上渐进优于高斯机制。同时，我们构建了一类新的\emph{球面广义伽马}DP机制簇，该簇同时包含高斯机制与近期研究的$\ell_2$机制（Joseph \emph{等}, ICML 2025）。我们确定了该机制簇中在特定低维场景下性能优于高斯机制与$\ell_2$机制的成员，并证明了该簇中所有机制具有紧致组合性，从而回答了Joseph \emph{等}关于$\ell_2$机制的开放性问题。