This paper investigates the relation between three different notions of privacy: identifiability, differential privacy and mutual-information privacy. Under a unified privacy-distortion framework, where the distortion is defined to be the Hamming distance of the input and output databases, we establish some fundamental connections between these three privacy notions. Given a distortion level @math , define @math to be the smallest (best) identifiability level, and @math to be the smallest differential privacy level. We characterize @math and @math , and prove that @math for @math in some range, where @math is a constant depending on the distribution of the original database @math , and diminishes to zero when the distribution of @math is uniform. Furthermore, we show that identifiability and mutual-information privacy are consistent in the sense that given distortion level @math , the mechanism that optimizes the mutual-information privacy also minimizes the identifiability level.
Differential privacy, as an analytical foundation for privacy-preserving data analysis, was developed by a line of work (see, e.g., @cite_16 @cite_27 @cite_1 ). @cite_16 proposed the Laplace mechanism which adds Laplace noise to each query result, with noise amplitude proportional to the global sensitivity of the query function. @cite_4 later generalize the mechanism using the concept of local sensitivity. The notion of @math -differential privacy @cite_1 has also been proposed as a relaxation of @math -differential privacy.
@cite_27 @cite_16 @cite_1 @cite_4
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Differential privacy mechanism design has traditionally been tailored for a scalar-valued query function. Although many mechanisms such as the Laplace and Gaussian mechanisms can be extended to a matrix-valued query function by adding i.i.d. noise to each element of the matrix, this method is often suboptimal as it forfeits an opportunity to exploit the structural characteristics typically associated with matrix analysis. To address this challenge, we propose a novel differential privacy mechanism called the Matrix-Variate Gaussian (MVG) mechanism, which adds a matrix-valued noise drawn from a matrix-variate Gaussian distribution, and we rigorously prove that the MVG mechanism preserves (e,δ)-differential privacy. Furthermore, we introduce the concept of directional noise made possible by the design of the MVG mechanism. Directional noise allows the impact of the noise on the utility of the matrix-valued query function to be moderated. Finally, we experimentally demonstrate the performance of our mechanism using three matrix-valued queries on three privacy-sensitive datasets. We find that the MVG mechanism can notably outperforms four previous state-of-the-art approaches, and provides comparable utility to the non-private baseline. Differential privacy is a robust privacy standard that has been successfully applied to a range of data analysis tasks. But despite much recent work, optimal strategies for answering a collection of related queries are not known. We propose the matrix mechanism, a new algorithm for answering a workload of predicate counting queries. Given a workload, the mechanism requests answers to a different set of queries, called a query strategy, which are answered using the standard Laplace 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 this problem can be formulated as a rank-constrained semidefinite program. Finally, we analyze two seemingly distinct techniques, whose similar behavior is explained by viewing them as instances of the matrix mechanism. e-Differential privacy is a property that seeks to characterize privacy in data sets. It is formulated as a query-response method, and computationally achieved by output perturbation. Several noise-addition methods to implement such output perturbation have been proposed in the literature. We focus on data-independent noise, that is, noise whose distribution is constant across data sets. Our goal is to find the optimal data-independent noise distribution to achieve e-differential privacy. We propose a general optimality criterion based on the concentration of the probability mass of the noise distribution around zero, and we show that any noise optimal under this criterion must be optimal under any other sensible criterion. We also show that the Laplace distribution, commonly used for noise in e-differential privacy, is not optimal, and we build the optimal data-independent noise distribution. We compare the Laplace and the optimal data-independent noise distributions. For univariate query functions, both introduce a similar level of distortion; for multivariate query functions, optimal data-independent noise offers responses with substantially better data quality. A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism M* -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user u, no matter what its side information and preferences, derives as much utility from M* as from interacting with a differentially private mechanism Mu that is optimally tailored to u. More precisely, for every user u there is an optimal mechanism Mu for it that factors into a user-independent part (the geometric mechanism M*) followed by user-specific post-processing that can be delegated to the user itself. The first part of our proof of this result characterizes the optimal differentially private mechanism for a fixed but arbitrary user in terms of a certain basic feasible solution to a linear program with constraints that encode differential privacy. The second part shows that all of the relevant vertices of this polytope (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range.