Frequency estimation, a.k.a. histograms, is a workhorse of data analysis, and as such has been thoroughly studied under differentially privacy. In particular, computing histograms in the \emph{local} model of privacy has been the focus of a fruitful recent line of work, and various algorithms have been proposed, achieving the order-optimal $\ell_\infty$ error in the high-privacy (small $\varepsilon$) regime while balancing other considerations such as time- and communication-efficiency. However, to the best of our knowledge, the picture is much less clear when it comes to the medium- or low-privacy regime (large $\varepsilon$), despite its increased relevance in practice. In this paper, we investigate locally private histograms, and the very related distribution learning task, in this medium-to-low privacy regime, and establish near-tight (and somewhat unexpected) bounds on the $\ell_\infty$ error achievable. As a direct corollary of our results, we obtain a protocol for histograms in the \emph{shuffle} model of differential privacy, with accuracy matching previous algorithms but significantly better message and communication complexity.
Our theoretical findings emerge from a novel analysis, which appears to improve bounds across the board for the locally private histogram problem. We back our theoretical findings by an empirical comparison of existing algorithms in all privacy regimes, to assess their typical performance and behaviour beyond the worst-case setting.
Local, Private, Efficient Protocols for Succinct Histograms
The Pennsylvania State University（宾夕法尼亚州立大学）；Boston University（波士顿大学）；Harvard University（哈佛大学）
我们给出了有效的协议和匹配精度下界的频率估计的局部模型的差分隐私。在这个模型中，个人用户自己随机化他们的数据，将不同的私人报告发送到一个不受信任的服务器，该服务器将这些报告聚集在一起。我们研究的协议，产生一个简洁的直方图表示的数据。一个简洁的直方图是数据中最频繁的项目（通常称为“重打击者”）及其频率估计值的列表;所有其他项目的频率隐式估计为0。如果有$n$用户的项目来自一个大小为$d$的宇宙，我们的协议在$n$和$\log(d)$中以时间多项式运行。以高概率，他们估计每个项目的准确性，直到错误$O\left(\sqrt{\log(d)/(ε^2n)}\right)$，其中$ε$是隐私参数。此外，我们表明，这么多的错误是必要的，无论计算效率，甚至对于简单的设置，只有一个项目出现在数据集中的显着频率。先前用于该任务的方案（Mishra和Sandler，2006; Hsu，Khanna和Roth，2012）要么在$Ω(d)$中运行，要么具有更严重的错误（约为$\sqrt[6]{\log(d)/(ε^2n)}$），并且唯一已知的错误下限是$Ω(1/\sqrt{n})$。我们还将McGregor et al（2010）的结果适用于当地环境。在一个使用公共硬币的模型中，我们展示了每个用户只需要向服务器发送1比特。对于所有已知的本地协议（包括我们的），转换保持计算效率。
