Generalized differential privacy: Regions of priors that admit robust optimal mechanisms

Differential privacy is a notion of privacy that was initially designed for statistical databases, and has been recently extended to a more general class of domains. Both differential privacy and its generalized version can be achieved by adding random noise to the reported data. Thus, privacy is obtained at the cost of reducing the data's accuracy, and therefore their utility. In this paper we consider the problem of identifying optimal mechanisms for generalized differential privacy, i.e. mechanisms that maximize the utility for a given level of privacy. The utility usually depends on a prior distribution of the data, and naturally it would be desirable to design mechanisms that are universally optimal, i.e., optimal for all priors. However it is already known that such mechanisms do not exist in general. We then characterize maximal classes of priors for which a mechanism which is optimal for all the priors of the class does exist. We show that such classes can be defined as convex polytopes in the priors space. As an application, we consider the problem of privacy that arises when using, for instance, location-based services, and we show how to define mechanisms that maximize the quality of service while preserving the desired level of geo-indistinguishability.