Differentially private binary- and matrix-valued data query: an XOR mechanism

Differential privacy has been widely adopted to release continuous- and scalar-valued information on a database without compromising the privacy of individual data records in it. The problem of querying binary- and matrix-valued information on a database in a differentially private manner has rarely been studied. However, binary- and matrix-valued data are ubiquitous in real-world applications, whose privacy concerns may arise under a variety of circumstances. In this paper, we devise an exclusive or (XOR) mechanism that perturbs binary- and matrix-valued query result by conducting an XOR operation on the query result with calibrated noises attributed to a matrix-valued Bernoulli distribution. We first rigorously analyze the privacy and utility guarantee of the proposed XOR mechanism. Then, to generate the parameters in the matrix-valued Bernoulli distribution, we develop a heuristic approach to minimize the expected square query error rate under ϵ-differential privacy constraint. Additionally, to address the intractability of calculating the probability density function (PDF) of this distribution and efficiently generate samples from it, we adapt an Exact Hamiltonian Monte Carlo based sampling scheme. Finally, we experimentally demonstrate the efficacy of the XOR mechanism by considering binary data classification and social network analysis, all in a differentially private manner. Experiment results show that the XOR mechanism notably outperforms other state-of-the-art differentially private methods in terms of utility (such as classification accuracy and F1 score), and even achieves comparable utility to the non-private mechanisms.