We study privately releasing column sums of a $d$-dimensional table with entries from a universe $\chi$ undergoing $T$ row updates, called histogram under continual release. Our mechanisms give better additive $\ell_\infty$-error than existing mechanisms for a large class of queries and input streams. Our first contribution is an output-sensitive mechanism in the insertions-only model ($\chi = \{0,1\}$) for maintaining (i) the histogram or (ii) queries that do not require maintaining the entire histogram, such as the maximum or minimum column sum, the median, or any quantiles. The mechanism has an additive error of $O(d\log^2 (dq^*)+\log T)$ whp, where $q^*$ is the maximum output value over all time steps on this dataset. The mechanism does not require $q^*$ as input. This breaks the $\Omega(d \log T)$ bound of prior work when $q^* \ll T$. Our second contribution is a mechanism for the turnstile model that admits negative entry updates ($\chi = \{-1, 0,1\}$). This mechanism has an additive error of $O(d \log^2 (dK) + \log T)$ whp, where $K$ is the number of times two consecutive data rows differ, and the mechanism does not require $K$ as input. This is useful when monitoring inputs that only vary under unusual circumstances. For $d=1$ this gives the first private mechanism with error $O(\log^2 K + \log T)$ for continual counting in the turnstile model, improving on the $O(\log^2 n + \log T)$ error bound by Dwork et al. [ASIACRYPT 2015], where $n$ is the number of ones in the stream, as well as allowing negative entries, while Dwork et al. [ASIACRYPT 2015] can only handle nonnegative entries ($\chi=\{0,1\}$).

翻译：本文研究在$T$次行更新的动态环境下，如何以差分隐私方式持续发布$d$维表格（取值域为$\chi$）的列和，即持续发布场景下的直方图问题。针对广泛类型的查询和输入流，我们的机制相比现有方案实现了更优的加性$\ell_\infty$误差界。第一项贡献是在仅插入模型（$\chi = \{0,1\}$）中提出输出敏感的机制，用于维护：（i）完整直方图，或（ii）无需维护完整直方图的查询，如最大/最小列和、中位数及任意分位数。该机制以高概率实现$O(d\log^2 (dq^*)+\log T)$的加性误差，其中$q^*$表示该数据集在所有时间步上的最大输出值。该机制无需预先获知$q^*$参数。当$q^* \ll T$时，此结果突破了现有工作中$\Omega(d \log T)$的误差下界。第二项贡献是针对允许负值更新（$\chi = \{-1, 0,1\}$）的旋转门模型设计机制，该机制以高概率实现$O(d \log^2 (dK) + \log T)$的加性误差，其中$K$表示连续两行数据发生变化的次数，且该机制无需预先获知$K$参数。该机制特别适用于监测仅在异常情况下发生变化的输入流。当$d=1$时，本工作首次为旋转门模型中的持续计数问题提供了误差为$O(\log^2 K + \log T)$的隐私机制，改进了Dwork等人[ASIACRYPT 2015]提出的$O(\log^2 n + \log T)$误差界（其中$n$为数据流中1的个数），同时支持负值输入，而Dwork等人[ASIACRYPT 2015]的方案仅能处理非负输入（$\chi=\{0,1\}$）。