We present a simple variant of the Gaussian mechanism for answering differentially private queries when the sensitivity space has a certain common structure. Our motivating problem is the fundamental task of answering $d$ counting queries under the add/remove neighboring relation. The standard Gaussian mechanism solves this task by adding noise distributed as a Gaussian with variance scaled by $d$ independently to each count. We show that adding a random variable distributed as a Gaussian with variance scaled by $(\sqrt{d} + 1)/4$ to all counts allows us to reduce the variance of the independent Gaussian noise samples to scale only with $(d + \sqrt{d})/4$. The total noise added to each counting query follows a Gaussian distribution with standard deviation scaled by $(\sqrt{d} + 1)/2$ rather than $\sqrt{d}$. The central idea of our mechanism is simple and the technique is flexible. We show that applying our technique to another problem gives similar improvements over the standard Gaussian mechanism.

翻译：针对敏感度空间具有特定常见结构时回答差分隐私查询的问题，我们提出了一种简单的高斯机制变体。我们的核心动机是解决在添加/删除相邻关系下回答$d$个计数查询这一基础任务。标准高斯机制通过向每个计数独立添加方差按$d$缩放的高斯分布噪声来解决此问题。我们证明，若向所有计数添加方差按$(\sqrt{d} + 1)/4$缩放的高斯分布随机变量，则可将独立高斯噪声样本的方差降低至仅按$(d + \sqrt{d})/4$缩放。每个计数查询的总噪声服从标准差按$(\sqrt{d} + 1)/2$而非$\sqrt{d}$缩放的高斯分布。本机制的核心思想简洁，且技术方法具有灵活性。我们通过将该技术应用于另一问题，展示了其相对于标准高斯机制的类似改进效果。