Profile Reconstruction from Private Sketches

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