Neural Information Processing Systems http://nips.cc/

We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent probability measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, C, are equivalent if the difference of their means lies in the Cameron-Martin space of C. In the case of releasing finite-dimensional summaries using elliptical perturbations, we show that the privacy parameter ɛ can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, t, Gaussian, and K-norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve ɛ-DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give ɛ-DP; only (ɛ, δ)-DP is possible.

Neural Information Processing Systems http://nips.cc/

We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent probability measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, C, are equivalent if the difference of their means lies in the Cameron-Martin space of C. In the case of releasing finite-dimensional summaries using elliptical perturbations, we show that the privacy parameter ɛ can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, t, Gaussian, and K-norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve ɛ-DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give ɛ-DP; only (ɛ, δ)-DP is possible.

We derive uniformly most powerful (UMP) tests for simple and one-sided hypotheses for a population proportion within the framework of Differential Privacy (DP), optimizing finite sample performance. We show that in general, DP hypothesis tests can be written in terms of linear constraints, and for exchangeable data can always be expressed as a function of the empirical distribution. Using this structure, we prove a'Neyman-Pearson lemma' for binomial data under DP, where the DP-UMP only depends on the sample sum. Our tests can also be stated as a post-processing of a random variable, whose distribution we coin "Truncated-Uniform-Laplace" (Tulap), a generalization of the Staircase and discrete Laplace distributions. Furthermore, we obtain exact p-values, which are easily computed in terms of the Tulap random variable. We show that our results also apply to distribution-free hypothesis tests for continuous data. Our simulation results demonstrate that our tests have exact type I error, and are more powerful than current techniques.