Differential privacy is often studied under two different models of neighboring datasets: the add-remove model and the swap model. While the swap model is frequently used in the academic literature to simplify analysis, many practical applications rely on the more conservative add-remove model, where obtaining tight results can be difficult. Here, we study the problem of one-dimensional mean estimation under the add-remove model. We propose a new algorithm and show that it is min-max optimal, achieving the best possible constant in the leading term of the mean squared error for all $\epsilon$, and that this constant is the same as the optimal algorithm under the swap model. These results show that the add-remove and swap models give nearly identical errors for mean estimation, even though the add-remove model cannot treat the size of the dataset as public information. We also demonstrate empirically that our proposed algorithm yields at least a factor of two improvement in mean squared error over algorithms frequently used in practice. One of our main technical contributions is a new hour-glass mechanism, which might be of independent interest in other scenarios.

翻译：差分隐私通常在两种相邻数据集模型下进行研究：增删模型和交换模型。尽管交换模型在学术文献中常被用于简化分析，但许多实际应用依赖于更为保守的增删模型，而在此模型下获取紧致结果往往较为困难。本文研究增删模型下的一维均值估计问题。我们提出了一种新算法，并证明该算法是极小极大最优的，能够在所有$\epsilon$下达到均方误差主导项的最优常数，且该常数与交换模型下的最优算法相同。这些结果表明：即使增删模型无法将数据集规模视为公开信息，两种模型在均值估计中的误差仍近乎一致。我们还通过实验证明，相较于实际中常用的算法，所提算法在均方误差上至少实现了两倍的改进。我们的主要技术贡献之一是新型沙漏机制，该机制在其他场景中可能具有独立的研究价值。