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Profile Reconstruction from Private Sketches

Abstract

Given a multiset of nn items from D\mathcal{D}, the \emph{profile reconstruction} problem is to estimate, for t=0,1,,nt = 0, 1, \dots, n, the fraction f[t]\vec{f}[t] of items in D\mathcal{D} that appear exactly tt 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 f=(f[0],,f[n])\vec{f} = (\vec{f}[0], \dots, \vec{f}[n]). 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 O(d+nlogn)O(d + n \log n), where d=Dd = |\mathcal{D}|, and analyze the achievable error in the 1\ell_1, 2\ell_2 and \ell_\infty norms. In all cases the dependency of the error on dd is O(1/d)O( 1 / \sqrt{d}) -- we give an information-theoretic lower bound showing that this dependence on dd is asymptotically optimal among all private, updatable sketches for the profile reconstruction problem with a high-probability error guarantee.

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