Differential privacy provides a rigorous framework for privacy-preserving data analysis. This paper proposes the first differentially private procedure for controlling the false discovery rate (FDR) in multiple hypothesis testing. Inspired by the Benjamini-Hochberg procedure (BHq), our approach is to first repeatedly add noise to the logarithms of the $p$-values to ensure differential privacy and to select an approximately smallest $p$-value serving as a promising candidate at each iteration; the selected $p$-values are further supplied to the BHq and our private procedure releases only the rejected ones. Moreover, we develop a new technique that is based on a backward submartingale for proving FDR control of a broad class of multiple testing procedures, including our private procedure, and both the BHq step-up and step-down procedures. As a novel aspect, the proof works for arbitrary dependence between the true null and false null test statistics, while FDR control is maintained up to a small multiplicative factor.
翻译:不同的隐私为隐私保护数据分析提供了一个严格的框架。 本文件提出了在多个假设测试中控制虚假发现率(FDR)的第一个有差别的私人程序。在Benjamini-Hochberg程序(BHq)的启发下,我们的方法是首先在美元价值的对数中反复增加噪音,以确保有差异的隐私,并选择一个大约最小的美元价值,作为每次循环的有希望候选人;所选定的美元价值进一步提供给BHq,而我们的私人程序只释放了被拒绝的。此外,我们开发了一种新的技术,它基于一种落后的子边际技术,以证明FDR对包括我们的私人程序在内的广泛类别的多重测试程序的控制,以及BHq的升级和逐步下降程序。作为一个新颖的方面,证明真实的无效无效测试统计数据之间的任意依赖性是有效的,而FDR控制则保持在一个小的倍增系数。