🤖 AI Summary
This paper addresses the $d$-dimensional counting query problem under differential privacy with add/remove neighborhood relations. Conventional independent Gaussian mechanisms incur a per-query standard deviation of $sqrt{d}$, constrained by the fundamental variance lower bound. We propose a structure-aware Gaussian mechanism that jointly designs globally correlated and independent Gaussian noise, explicitly modeling the covariance matrix to capture the intrinsic geometric structure of the sensitivity space. This design reduces the per-query standard deviation to $(sqrt{d}+1)/2$, breaking the theoretical limitation of independent-noise mechanisms. Theoretical analysis establishes the mechanism’s generality, showing direct applicability to other multidimensional query tasks sharing similar sensitivity structures. Extensive experiments demonstrate significant improvements in total noise standard deviation over state-of-the-art baselines, achieving both rigorous $(varepsilon,delta)$-differential privacy guarantees and substantially enhanced statistical utility.
📝 Abstract
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.