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 $(\epsilon,\delta)$-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.

翻译：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。我们的主要洞见是，使用下三角矩阵的矩阵机制可应用于持续观测模型。具体而言，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上界。我们还提供了细粒度分析，明确了上界中的确切常数。该分析基于 $M_\mathsf{count}$ 的完全有界范数（cb-范数）的上界和下界。在此过程中，我们改进了 Mathias（SIAM Journal on Matrix Analysis and Applications, 1993）保持28年之久的 $M_\mathsf{count}$ 的 cb-范数最佳已知界，且适用于 $M_\mathsf{count}$ 维度的较大范围。此外，我们是首个为持续观测下的多种问题（如二元计数、维护直方图、发布近似割保持合成图、多种图统计量、子串与情节计数）给出具体误差界的工作。最后，我们指出，我们的结果可用于获得非交互式本地学习的细粒度误差界，并首次给出了持续观测下 $(\epsilon,\delta)$-差分隐私计数加性误差的下界。在本工作之后，Henzinger 等人（SODA2023）证明我们的分解方法还能实现细粒度的均方误差。