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  1. We consider the problem of estimating sparse discrete distributions under local differential privacy (LDP) and communication constraints. We characterize the sample complexity for sparse estimation under LDP constraints up to a constant factor, and the sample complexity under communication constraints up to a logarithmic factor. Our upper bounds under LDP are based on the Hadamard Response, a private coin scheme that requires only one bit of communication per user. Under communication constraints we propose public coin schemes based on random hashing functions. Our tight lower bounds are based on recently proposed method of chi squared contractions. 
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  2. We develop differentially private methods for estimating various distributional properties. Given a sample from a discrete distribution p, some functional f, and accuracy and privacy parameters alpha and epsilon, the goal is to estimate f(p) up to accuracy alpha, while maintaining epsilon-differential privacy of the sample. We prove almost-tight bounds on the sample size required for this problem for several functionals of interest, including support size, support coverage, and entropy. We show that the cost of privacy is negligible in a variety of settings, both theoretically and experimentally. Our methods are based on a sensitivity analysis of several state-of-the-art methods for estimating these properties with sublinear sample complexities. 
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  3. We study the fundamental problems of identity testing (goodness of fit), and closeness testing (two sample test) of distributions over k elements, under differential privacy. While the problems have a long history in statistics, finite sample bounds for these problems have only been established recently. In this work, we derive upper and lower bounds on the sample complexity of both the problems under (epsilon, delta)-differential privacy. We provide sample optimal algorithms for identity testing problem for all parameter ranges, and the first results for closeness testing. Our closeness testing bounds are optimal in the sparse regime where the number of samples is at most k. Our upper bounds are obtained by privatizing non-private estimators for these problems. The non-private estimators are chosen to have small sensitivity. We propose a general framework to establish lower bounds on the sample complexity of statistical tasks under differential privacy. We show a bound on di erentially private algorithms in terms of a coupling between the two hypothesis classes we aim to test. By carefully constructing chosen priors over the hypothesis classes, and using Le Cam’s two point theorem we provide a general mechanism for proving lower bounds. We believe that the framework can be used to obtain strong lower bounds for other statistical tasks under privacy. 
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