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Lower Bounds on the Error of Query Sets Under the Differentially-Private Matrix Mechanism

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Abstract

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 β€œeasy” or β€œhard” to answer. We consider this issue in the particular case of answering sets of linear counting queries using the matrix mechanism (Li et al. 2010), 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 when using the matrix mechanism. 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. Under (πœ–, Ξ΄)-differential privacy, we prove that this bound is tight for many common workloads such as the set of all predicate queries and the set of all k-way marginals. Our empirical study also indicates this bound is close-to-tight on workloads consisting of random interval queries or random marginals.

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Notes

  1. This can also be done in an interactive (or online) setting, in which the workload queries are not known in advance, but our focus is the non-interactive (or batch) setting.

  2. Although there is no benefit in sensitivity when the workload W is used as the strategy, there are in fact some workloads for which this special instance of the matrix mechanism has lower error than the Gaussian mechanism. This happens because the matrix mechanism is able to combine related query answers from the strategy to form more accurate answers to the workload queries.

  3. The approaches in [20,30] were originally proposed in the context of πœ–-differential privacy, but their behavior is similar under (πœ–,Ξ΄)-differential privacy.

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Acknowledgments

We appreciate the thoughtful comments of the anonymous reviewers. Li was supported by NSF CNS-1012748. Miklau was partially supported by NSF CNS-1012748, NSF CNS-0964094, NSF CNS-1409143, and the European Research Council under the Webdam grant, No. 226513.

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  1. University of Massachusetts, Amherst, MA, USA

    Chao LiΒ &Β Gerome Miklau

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  1. Chao Li
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  2. Gerome Miklau
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Correspondence to Chao Li.

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Li, C., Miklau, G. Lower Bounds on the Error of Query Sets Under the Differentially-Private Matrix Mechanism. Theory Comput Syst 57, 1159–1201 (2015). https://doi.org/10.1007/s00224-015-9610-z

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