We consider the problem of designing scalable, robust protocols for computing statistics about sensitive data. Specifically, we look at how best to design differentially private protocols in a distributed setting, where each user holds a private datum. The literature has mostly considered two models: the "central" model, in which a trusted server collects users' data in the clear, which allows greater accuracy; and the "local" model, in which users individually randomize their data, and need not trust the server, but accuracy is limited. Attempts to achieve the accuracy of the central model without a trusted server have so far focused on variants of cryptographic secure function evaluation, which limits scalability.In this paper, we initiate the analytic study of a shuffled model for distributed differentially private algorithms, which lies between the local and central models. This simple-to-implement model, a special case of the ESA framework of Bittau et al. [5], augments the local model with an anonymous channel that randomly permutes a set of user-supplied messages. For sum queries, we show that this model provides the power of the central model while avoiding the need to trust a central server and the complexity of cryptographic secure function evaluation. More generally, we give evidence that the power of the shuffled model lies strictly between those of the central and local models: for a natural restriction of the model, we show that shuffled protocols for a widely studied selection problem require exponentially higher sample complexity than do central-model protocols.

We propose a hybrid model of differential privacy that considers a combination of regular and opt-in users who desire the differential privacy guarantees of the local privacy model and the trusted curator model, respectively. We demonstrate that within this model, it is possible to design a new type of blended algorithm that improves the utility of obtained data, while providing users with their desired privacy guarantees. We apply this algorithm to the task of privately computing the head of the search log and show that the blended approach provides significant improvements in the utility of the data compared to related work. Specifically, on two large search click data sets, comprising 1.75 and 16 GB, respectively, our approach attains NDCG values exceeding 95% across a range of privacy budget values.

An asymptotic model for extreme behavior of certain Markov chains is the "tail chain". Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and non-extreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

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