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arXiv 1810.02810cs.LGstat.ML
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局部差分隐私下的线性隐式估计

Linear Queries Estimation with Local Differential Privacy

The Ohio State University(俄亥俄州立大学)

Raef Bassily

摘要:

研究了在局部差分隐私约束下,基于$n$个体的敏感数据集,在域$\mathcal{J}=[J]$上估计关于未知分布$\mathbf{p}$的$d$线性查询集的问题.这个问题包含了广泛的估计任务,例如,分布估计和$d$维均值估计。我们为这个问题的离线(非自适应)和自适应版本提供了新的算法。在离线设置中,查询集在算法开始之前是固定的。在$n\lesssim d^2/\log(J)$的制度,我们的算法达到$L_2$估计误差是独立的$d$,是紧到一个因素的$\tilde{O}\left(\log^{1/4}(J)\right)$。对于分布估计的特殊情况,我们证明了将[Acharya et al. 2018]的算法的输出估计投影到概率单纯形上会产生一个$L_2$误差,该误差仅在$n\lesssim J^2/\log(J)$的区域中亚数学地依赖于$J$。这些结果表明,线性查询的准确估计的可能性,在高维设置下的$L_2$错误标准。在自适应设置中,查询是在$d$轮中生成的;每次生成一个查询。在每一轮中,可以基于先前查询和答案的所有历史来自适应地选择查询。我们给出了一个算法,这个问题的最佳$L_{\infty}$估计误差(最坏的错误估计值的查询w.r.t.数据分布)。我们的界限与该问题的离线版本的$L_{\infty}$错误的下限相匹配[Duchi et al. 2013]。

英文摘要:

We study the problem of estimating a set of $d$ linear queries with respect to some unknown distribution $\mathbf{p}$ over a domain $\mathcal{J}=[J]$ based on a sensitive data set of $n$ individuals under the constraint of local differential privacy. This problem subsumes a wide range of estimation tasks, e.g., distribution estimation and $d$-dimensional mean estimation. We provide new algorithms for both the offline (non-adaptive) and adaptive versions of this problem. In the offline setting, the set of queries are fixed before the algorithm starts. In the regime where $n\lesssim d^2/\log(J)$, our algorithms attain $L_2$ estimation error that is independent of $d$, and is tight up to a factor of $\tilde{O}\left(\log^{1/4}(J)\right)$. For the special case of distribution estimation, we show that projecting the output estimate of an algorithm due to [Acharya et al. 2018] on the probability simplex yields an $L_2$ error that depends only sub-logarithmically on $J$ in the regime where $n\lesssim J^2/\log(J)$. These results show the possibility of accurate estimation of linear queries in the high-dimensional settings under the $L_2$ error criterion. In the adaptive setting, the queries are generated over $d$ rounds; one query at a time. In each round, a query can be chosen adaptively based on all the history of previous queries and answers. We give an algorithm for this problem with optimal $L_{\infty}$ estimation error (worst error in the estimated values for the queries w.r.t. the data distribution). Our bound matches a lower bound on the $L_{\infty}$ error for the offline version of this problem [Duchi et al. 2013].

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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