Data Analytics with Differential Privacy

Differential privacy is the state-of-the-art definition for privacy, guaranteeing that any analysis performed on a sensitive dataset leaks no information about the individuals whose data are contained therein. In this thesis, we develop differentially private algorithms to analyze distributed and streaming data. In the distributed model, we consider the particular problem of learning -- in a distributed fashion -- a global model of the data, that can subsequently be used for arbitrary analyses. We build upon PrivBayes, a differentially private method that approximates the high-dimensional distribution of a centralized dataset as a product of low-order distributions, utilizing a Bayesian Network model. We examine three novel approaches to learning a global Bayesian Network from distributed data, while offering the differential privacy guarantee to all local datasets. Our work includes a detailed theoretical analysis of the distributed, differentially private entropy estimator which we use in one of our algorithms, as well as a detailed experimental evaluation, using both synthetic and real-world data. In the streaming model, we focus on the problem of estimating the density of a stream of users, which expresses the fraction of all users that actually appear in the stream. We offer one of the strongest privacy guarantees for the streaming model, user-level pan-privacy, which ensures that the privacy of any user is protected, even against an adversary that observes the internal state of the algorithm. We provide a detailed analysis of an existing, sampling-based algorithm for the problem and propose two novel modifications that significantly improve it, both theoretically and experimentally, by optimally using all the allocated "privacy budget."