We study differentially-private statistics in the fully dynamic continual observation model, where many updates can arrive at each time step and updates to a stream can involve both insertions and deletions of an item. Earlier work (e.g., Jain et al., NeurIPS 2023 for counting distinct elements; Raskhodnikova & Steiner, PODS 2025 for triangle counting with edge updates) reduced the respective cardinality estimation problem to continual counting on the difference stream associated with the true function values on the input stream. In such reductions, a change in the original stream can cause many changes in the difference stream, this poses a challenge for applying private continual counting algorithms to obtain optimal error bounds. We improve the accuracy of several such reductions by studying the associated $\ell_p$-sensitivity vectors of the resulting difference streams and isolating their properties. We demonstrate that our framework gives improved bounds for counting distinct elements, estimating degree histograms, and estimating triangle counts (under a slightly relaxed privacy model), thus offering a general approach to private continual cardinality estimation in streaming settings. Our improved accuracy stems from tight analysis of known factorization mechanisms for the counting matrix in this setting; the key technical challenge is arguing that one can use state-of-the-art factorizations for sensitivity vector sets with the properties we isolate. Empirically and analytically, we demonstrate that our improved error bounds offer a substantial improvement in accuracy for cardinality estimation problems over a large range of parameters.

翻译：我们研究完全动态连续观测模型下的差分隐私统计问题，在该模型中每个时间步可能到达多个更新，且数据流的更新可同时包含项目的插入与删除。先前工作（例如Jain等人于NeurIPS 2023提出的独立元素计数方法；Raskhodnikova与Steiner于PODS 2025提出的带边更新的三角形计数方法）将相应的基数估计问题转化为对输入流真实函数值所关联的差分流进行连续计数。在此类转化中，原始流的单次变化可能导致差分流的多次变化，这对应用私有连续计数算法获取最优误差界提出了挑战。我们通过研究所得差分流的相关$\ell_p$-敏感度向量并分离其特性，改进了多项此类转化的精度。我们证明所提框架为独立元素计数、度直方图估计及三角形计数估计（在稍宽松的隐私模型下）提供了更优的误差界，从而为流式场景下的私有连续基数估计提供了通用方法。我们的精度提升源于对该场景下计数矩阵已知分解机制的严密分析；关键技术挑战在于论证如何将最先进的分解方法应用于具有我们所分离特性的敏感度向量集。通过实证与理论分析，我们证明改进后的误差界在广泛参数范围内为基数估计问题提供了显著的精度提升。