Better Gaussian Mechanism using Correlated Noise

TL;DR

This paper introduces a modified Gaussian mechanism for differential privacy that uses correlated noise to reduce total noise variance, improving accuracy for counting queries with shared sensitivity structures.

Contribution

It proposes a simple, flexible variant of the Gaussian mechanism that adds correlated Gaussian noise, reducing total noise variance in differential privacy applications.

Findings

Reduces noise variance from $d$ to $(rac{ ext{sqrt}(d)+1}{4})$ per query.

Achieves lower total noise standard deviation scaled by $( ext{sqrt}(d)+1)/2$.

Demonstrates applicability to multiple problems beyond counting queries.

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.