We study fine-grained error bounds for differentially private algorithms for averaging and counting under continual observation. Our main insight is that the factorization mechanism when using lower-triangular matrices, can be used in the continual observation model. We give explicit factorizations for two fundamental matrices, namely the counting matrix $M_{\mathsf{count}}$ and the averaging matrix $M_{\mathsf{average}}$ and show fine-grained bounds for the additive error of the resulting mechanism using the {\em completely bounded norm} (cb-norm) or {\em factorization norm}. Our bound on the cb-norm for $M_{\mathsf{count}}$ is tight up an additive error of 1 and the bound for $M_{\mathsf{average}}$ is tight up to $\approx 0.64$. This allows us to give the first algorithm for averaging whose additive error has $o(\log^{3/2} T)$ dependence. 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 present a fine-grained error bound for non-interactive local learning.
翻译:我们研究细微的误差, 用于在连续观测中平均和计算。 我们的主要洞察力是, 使用低三角矩阵时的系数化机制可以用于连续观测模式。 我们给两个基本矩阵给出明确的系数化, 即 $M ⁇ mathsf{count} $和平均矩阵$M ⁇ mathsf{平均=%1 $ 和平均矩阵$M ⁇ mathsf{ = 0. 64$。 这让我们能够给出使用 {em 完全封闭的规范} (c- norm) 或 {em 乘数规范} 来计算结果机制的添加错误的细度。 此外, 我们首先给持续观察下的 cb- norm 设置了具体错误, 例如 $M ⁇ mathsf{ 计数 $1 和 $M ⁇ mathsf{ 平均 { $Mgrofs} 和 平均矩阵 $\\\ max 0. 0. 64美元。 这让我们给出第一个算法 中, 其累积错误有 $o(log 3/3/2} T) 基础依赖 。 此外, 我们首先为在持续观察中给出了各种不精确的精确的错误, 问题中,,, 解算。