Privately Learning High-Dimensional Distributions
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We design nearly optimal differentially private algorithms for learning two fundamental families of high-dimensional distributions in total variation distance: multivariate Gaussians in R^d and product distributions on the hypercube. The sample complexity of both our algorithms approaches the sample complexity of non-private learners up to a small multiplicative factor and an additional additive term that is lower order for a wide range of parameters, showing that privacy comes essentially for free for these problems. Our algorithms use a novel technical approach to reducing the sensitivity of the estimation procedure that we call recursive private preconditioning and may find additional applications.
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