We study efficient differentially private algorithms for estimating monotone statistics, i.e., statistics that are monotone under the addition of new observations. The starting point for our investigation is subsample-and-aggregate: a classical paradigm that partitions the dataset into blocks, estimates the statistic on each block, and then privately aggregates the estimates. While practical and generically applicable, this approach is quite data-hungry. We improve upon this framework for the class of monotone statistics -- compared to subsample-and-aggregate, our algorithms save a factor of $t$ in sample complexity and pay a factor of $e^t$ in running time, where $t>0$ is a tunable parameter. We complement our results with a query-complexity lower bound, showing that our algorithms are essentially optimal for this task. As an application, we obtain improved results for private eigenvalue estimation, private loss estimation, and privately estimating a single parameter of a high-dimensional model, e.g., in linear regression.

翻译：我们研究用于估计单调统计量的高效差分隐私算法，这类统计量在添加新观测数据时具有单调性。研究的起点是子采样-聚合范式：这种经典方法将数据集划分为多个数据块，分别估计各块上的统计量，再通过隐私保护方式聚合这些估计值。虽然该方法实用且具有通用性，但对数据需求量较大。针对单调统计量类别，我们改进了该框架——相较于子采样-聚合方法，我们的算法在样本复杂度上节省了$t$倍因子，但运行时间增加了$e^t$倍因子（其中$t>0$为可调参数）。我们通过查询复杂度下界证明了该算法在此任务中本质最优。应用方面，我们获得了私有特征值估计、私有损失估计及高维模型（如线性回归）单参数私有估计的改进结果。