Tight on Budget? Tight Bounds for r-Fold Approximate Differential Privacy

Many applications, such as anonymous communication systems, privacy-enhancing database queries, or privacy-enhancing machine-learning methods, require robust guarantees under thousands and sometimes millions of observations. The notion of r-fold approximate differential privacy (ADP) offers a well-established framework with a precise characterization of the degree of privacy after r observations of an attacker. However, existing bounds for r-fold ADP are loose and, if used for estimating the required degree of noise for an application, can lead to over-cautious choices for perturbation randomness and thus to suboptimal utility or overly high costs. We present a numerical and widely applicable method for capturing the privacy loss of differentially private mechanisms under composition, which we call privacy buckets. With privacy buckets we compute provable upper and lower bounds for ADP for a given number of observations. We compare our bounds with state-of-the-art bounds for r-fold ADP, including Kairouz, Oh, and Viswanath\u27s composition theorem (KOV), concentrated differential privacy and the moments accountant. While KOV proved optimal bounds for heterogeneous adaptive k-fold composition, we show that for concrete sequences of mechanisms tighter bounds can be derived by taking the mechanisms\u27 structure into account. We compare previous bounds for the Laplace mechanism, the Gauss mechanism, for a timing leakage reduction mechanism, and for the stochastic gradient descent and we significantly improve over their results (except that we match the KOV bound for the Laplace mechanism, for which it seems tight). Our lower bounds almost meet our upper bounds, showing that no significantly tighter bounds are possible

Optimizing Batch Linear Queries under Exact and Approximate Differential Privacy

Differential privacy is a promising privacy-preserving paradigm for statistical query processing over sensitive data. It works by injecting random noise into each query result, such that it is provably hard for the adversary to infer the presence or absence of any individual record from the published noisy results. The main objective in differentially private query processing is to maximize the accuracy of the query results, while satisfying the privacy guarantees. Previous work, notably \cite{LHR+10}, has suggested that with an appropriate strategy, processing a batch of correlated queries as a whole achieves considerably higher accuracy than answering them individually. However, to our knowledge there is currently no practical solution to find such a strategy for an arbitrary query batch; existing methods either return strategies of poor quality (often worse than naive methods) or require prohibitively expensive computations for even moderately large domains. Motivated by this, we propose low-rank mechanism (LRM), the first practical differentially private technique for answering batch linear queries with high accuracy. LRM works for both exact (i.e., $\epsilon$-) and approximate (i.e., ($\epsilon$, $\delta$)-) differential privacy definitions. We derive the utility guarantees of LRM, and provide guidance on how to set the privacy parameters given the user's utility expectation. Extensive experiments using real data demonstrate that our proposed method consistently outperforms state-of-the-art query processing solutions under differential privacy, by large margins.Comment: ACM Transactions on Database Systems (ACM TODS). arXiv admin note: text overlap with arXiv:1212.230

Differentially Private Convex Optimization with Piecewise Affine Objectives

Differential privacy is a recently proposed notion of privacy that provides strong privacy guarantees without any assumptions on the adversary. The paper studies the problem of computing a differentially private solution to convex optimization problems whose objective function is piecewise affine. Such problem is motivated by applications in which the affine functions that define the objective function contain sensitive user information. We propose several privacy preserving mechanisms and provide analysis on the trade-offs between optimality and the level of privacy for these mechanisms. Numerical experiments are also presented to evaluate their performance in practice