Profile Reconstruction from Private Sketches

Given a multiset of items from , the \emph{profile reconstruction} problem is to estimate, for , the fraction of items in that appear exactly times. We consider differentially private profile estimation in a distributed, space-constrained setting where we wish to maintain an updatable, private sketch of the multiset that allows us to compute an approximation of . Using a histogram privatized using discrete Laplace noise, we show how to ``reverse'' the noise, using an approach of Dwork et al.~(ITCS '10). We show how to speed up their LP-based technique from polynomial time to , where , and analyze the achievable error in the , and norms. In all cases the dependency of the error on is -- we give an information-theoretic lower bound showing that this dependence on is asymptotically optimal among all private, updatable sketches for the profile reconstruction problem with a high-probability error guarantee.
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