The matrix mechanism: optimizing linear counting queries under differential privacy

Differential privacy is a robust privacy standard that has been successfully applied to a range of data analysis tasks. We describe the matrix mechanism, an algorithm for answering a workload of linear counting queries that adapts the noise distribution to properties of the provided queries. Given a workload, the mechanism uses a different set of queries, called a query strategy, which are answered using a standard Laplace or Gaussian mechanism. Noisy answers to the workload queries are then derived from the noisy answers to the strategy queries. This two-stage process can result in a more complex, correlated noise distribution that preserves differential privacy but increases accuracy. We provide a formal analysis of the error of query answers produced by the mechanism and investigate the problem of computing the optimal query strategy in support of a given workload. We show that this problem can be formulated as a rank-constrained semidefinite program. We analyze two seemingly distinct techniques proposed in the literature, whose similar behavior is explained by viewing them as instances of the matrix mechanism. We also describe an extension of the mechanism in which nonnegativity constraints are included in the derivation process and provide experimental evidence of its efficacy.