Towards optimal noise distribution for privacy preserving in data aggregation

In aggregation applications, individual privacy is a crucial factor to determine the effectiveness, for which the noise-addition method (i.e., a random noise value is added to the true value) is a simple yet powerful approach. However, improper additive noise could result in bias for the aggregate result. It demands an optimal noise distribution to reduce the deviation. In this paper, we develop a mathematical framework to derive the optimal noise distribution that provides privacy protection under the constraint of a limited value deviation. Specifically, we first derive a generic system dynamic function that the optimal noise distribution must satisfy, and further investigate the special case that the original values obey Gaussian distribution. Then we detailedly investigate the general cases that the original values obey arbitrary continuous distribution, which can be expressed by Gaussian Mixture Model (GMM). Our theoretical analysis suggests that for the Gaussian input Gaussian distribution is the optimal solution, and for the general input, the optimal solution is composed of infinite number of Gaussian components. We further find the general term formula of the components, which reduces the number of unknowns from infinite to three, i.e. the parameters of the first component (variance, expectation, weight). Based on it, we investigate the properties and propose an algorithm in order to calculate the asymptotically optimal solution composed of finite Gaussian components. The numerical evaluation shows that the results has little deviation to the optimal solutions.