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.

翻译：给定定义域 $\mathcal{D}$ 上的一个包含 $n$ 个元素的多重集合，\emph{分布轮廓重构}问题旨在估计对于 $t = 0, 1, \dots, n$，定义域 $\mathcal{D}$ 中恰好出现 $t$ 次的元素所占的比例 $\vec{f}[t]$。我们研究在分布式、空间受限场景下的差分隐私轮廓估计问题，目标是维护一个可更新的、私有的多重集合草图，该草图允许我们计算 $\vec{f} = (\vec{f}[0], \dots, \vec{f}[n])$ 的近似值。通过使用经离散拉普拉斯噪声隐私化的直方图，我们展示了如何利用 Dwork 等人（ITCS '10）的方法来“逆转”噪声。我们将其基于线性规划的技术从多项式时间加速至 $O(d + n \log n)$，其中 $d = |\mathcal{D}|$，并分析了在 $\ell_1$、$\ell_2$ 和 $\ell_\infty$ 范数下可达到的误差。在所有情况下，误差对 $d$ 的依赖为 $O( 1 / \sqrt{d})$——我们给出了一个信息论下界，证明对于具有高概率误差保证的分布轮廓重构问题，在所有私有、可更新的草图中，这种对 $d$ 的依赖是渐近最优的。