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

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_mathsfcount 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_mathsfcount. 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_mathsfcount for a large range of the dimension of M_mathsfcount. 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.