论文标题
交互式与非相互作用的本地差异性私有估计：二次功能的两个肘部
Interactive versus non-interactive locally differentially private estimation: Two elbows for the quadratic functional
论文作者
论文摘要
当地的差异隐私最近已从统计社区受到越来越多的关注，作为保护单个数据所有者隐私而无需值得信赖的第三方的宝贵工具。与随机响应的经典概念类似，其想法是数据所有者在本地随机将其真实信息随机化，并且仅释放受扰动的数据。可以设计用于此类本地扰动过程的许多不同协议。然而，在迄今为止文献中研究的大多数估计问题中，纯粹的非交互协议和协议之间的最小值风险尚无显着差异，这些方案可以观察到可以观察到单个数据提供者之间的一定互动。在本文中，我们表明，为了估算密度的集成平方，就最小值估计速率而言，依次的交互过程超过了最佳的非交互过程。特别是，在非相互作用的情况下，我们确定了最小速率的肘部为$ s = \ frac34 $，而在依次交互式方案中，肘部为$ s = \ frac12 $。这与直接观察的情况明显不同，肘部众所周知，肘部为$ s = \ frac14 $，以及从将拉普拉斯噪声添加到原始数据中的情况下，其中肘部为$ s = \ frac94 $。我们还提供自适应估计器，以达到对数因素的最佳速率，我们与非参数拟合优度测试和更一般的积分功能的估计并进行了一系列数值实验。特定的本地私人但交互式机制的事实改善了简单的非交互式机制，对于当地差异性隐私的实际实现也非常重要。
Local differential privacy has recently received increasing attention from the statistics community as a valuable tool to protect the privacy of individual data owners without the need of a trusted third party. Similar to the classical notion of randomized response, the idea is that data owners randomize their true information locally and only release the perturbed data. Many different protocols for such local perturbation procedures can be designed. In most estimation problems studied in the literature so far, however, no significant difference in terms of minimax risk between purely non-interactive protocols and protocols that allow for some amount of interaction between individual data providers could be observed. In this paper we show that for estimating the integrated square of a density, sequentially interactive procedures improve substantially over the best possible non-interactive procedure in terms of minimax rate of estimation. In particular, in the non-interactive scenario we identify an elbow in the minimax rate at $s=\frac34$, whereas in the sequentially interactive scenario the elbow is at $s=\frac12$. This is markedly different from both, the case of direct observations, where the elbow is well known to be at $s=\frac14$, as well as from the case where Laplace noise is added to the original data, where an elbow at $s= \frac94$ is obtained. We also provide adaptive estimators that achieve the optimal rate up to log-factors, we draw connections to non-parametric goodness-of-fit testing and estimation of more general integral functionals and conduct a series of numerical experiments. The fact that a particular locally differentially private, but interactive, mechanism improves over the simple non-interactive one is also of great importance for practical implementations of local differential privacy.
