Measuring the achievable error of query sets under differential privacy

A common goal of privacy research is to release synthetic data that satisfies a formal privacy guarantee and can be used by an analyst in place of the original data. To achieve reasonable accuracy, a synthetic data set must be tuned to support a specified set of queries accurately, sacrificing fidelity for other queries. This work considers methods for producing synthetic data under differential privacy and investigates what makes a set of queries "easy" or "hard" to answer. We consider answering sets of linear counting queries using the matrix mechanism, a recent differentially-private mechanism that can reduce error by adding complex correlated noise adapted to a specified workload. Our main result is a novel lower bound on the minimum total error required to simultaneously release answers to a set of workload queries. The bound reveals that the hardness of a query workload is related to the spectral properties of the workload when it is represented in matrix form. The bound is tight and, because it satisfies important soundness criteria, it can serve as a reliable numerical measure of the "error complexity" of a workload.