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arXiv 2202.11205cs.DScs.LG
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常数重要:差分隐私持续观测的细粒度复杂度

Constant matters: Fine-grained Complexity of Differentially Private Continual Observation

Google Research(谷歌研究院);University of Vienna(维也纳大学);Stanford University(斯坦福大学);Rutgers University(罗格斯大学)

Hendrik Fichtenberger, Monika Henzinger, Jalaj Upadhyay

AI总结:

本文针对持续观测下的差分隐私计数问题,通过显式分解计数矩阵并分析其cb-范数,给出了细粒度的误差上界,并首次为多种问题提供了具体误差界,改进了已有28年的最佳界。

摘要:

我们研究了持续观测下计数问题的差分隐私算法的细粒度误差界。我们的主要见解是,使用下三角矩阵的矩阵机制可以用于持续观测模型。更具体地说,我们给出了计数矩阵 $M_\mathsf{count}$ 的显式分解,并显式地给出了误差的上界。我们还进行了细粒度分析,指定了上界中的精确常数。我们的分析基于 $M_\mathsf{count}$ 的完全有界范数(cb-范数)的上下界。在此过程中,我们改进了 Mathias(SIAM Journal on Matrix Analysis and Applications, 1993)关于 $M_\mathsf{count}$ 的 cb-范数的最佳已知界(该界已存在28年),适用于 $M_\mathsf{count}$ 维数的大范围。此外,我们是第一个为持续观测下的各种问题给出具体误差界的人,这些问题包括二进制计数、维护直方图、发布近似保持切割的合成图、许多基于图的统计量,以及子串和情节计数。最后,我们注意到我们的结果可用于获得非交互式局部学习的细粒度误差界,以及 $(\epsilon,\delta)$-差分隐私计数在持续观测下加性误差的首个下界。在本工作之后,Henzinger 等人(SODA2023)表明我们的分解也实现了细粒度的均方误差。

英文摘要:

We study fine-grained error bounds for differentially private algorithms for counting under continual observation. Our main insight is that the matrix mechanism when using lower-triangular matrices can be used in the continual observation model. More specifically, we give an explicit factorization for the counting matrix $M_\mathsf{count}$ and upper bound the error explicitly. We also give a fine-grained analysis, specifying the exact constant in the upper bound. Our analysis is based on upper and lower bounds of the {\em completely bounded norm} (cb-norm) of $M_\mathsf{count}$. Along the way, we improve the best-known bound of 28 years by Mathias (SIAM Journal on Matrix Analysis and Applications, 1993) on the cb-norm of $M_\mathsf{count}$ for a large range of the dimension of $M_\mathsf{count}$. Furthermore, we are the first to give concrete error bounds for various problems under continual observation such as binary counting, maintaining a histogram, releasing an approximately cut-preserving synthetic graph, many graph-based statistics, and substring and episode counting. Finally, we note that our result can be used to get a fine-grained error bound for non-interactive local learning {and the first lower bounds on the additive error for $(ε,δ)$-differentially-private counting under continual observation.} Subsequent to this work, Henzinger et al. (SODA2023) showed that our factorization also achieves fine-grained mean-squared error.

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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