We revisit Wald's celebrated Sequential Probability Ratio Test for sequential tests of two simple hypotheses, under privacy constraints. We propose DP-SPRT, a wrapper that can be calibrated to achieve desired error probabilities and privacy constraints, addressing a significant gap in previous work. DP-SPRT relies on a private mechanism that processes a sequence of queries and stops after privately determining when the query results fall outside a predefined interval. This OutsideInterval mechanism improves upon naive composition of existing techniques like AboveThreshold, achieving a factor-of-2 privacy improvement and thus potentially benefiting other continual monitoring procedures. We prove generic upper bounds on the error and sample complexity of DP-SPRT that can accommodate various noise distributions based on the practitioner's privacy needs. We exemplify them in two settings: Laplace noise (pure Differential Privacy) and Gaussian noise (Rényi differential privacy). In the former setting, by providing a lower bound on the sample complexity of any $\varepsilon$-DP test with prescribed type I and type II errors, we show that DP-SPRT is near optimal when both errors are small and the two hypotheses are close. Moreover, we conduct an experimental study revealing its good practical performance.

翻译：我们重新审视了Wald著名的序贯概率比检验，该检验用于两个简单假设的序贯检验，并在隐私约束条件下进行研究。我们提出了DP-SPRT，这是一个可以通过校准以满足指定错误概率和隐私约束的封装方法，从而填补了先前研究中的一个重要空白。DP-SPRT依赖于一种隐私机制，该机制处理一系列查询，并在通过隐私方式确定查询结果超出预定义区间后停止。这种OutsideInterval机制改进了现有技术（如AboveThreshold）的朴素组合方式，实现了隐私度量的两倍提升，因此可能惠及其他持续监控过程。我们证明了DP-SPRT在错误率和样本复杂度上的通用上界，该上界能够根据实践者的隐私需求适配多种噪声分布。我们在两种设置中进行了示例说明：拉普拉斯噪声（纯差分隐私）和高斯噪声（Rényi差分隐私）。在前一种设置中，通过给出任何具有指定第一类与第二类错误的$\varepsilon$-DP检验的样本复杂度下界，我们证明了当两类错误均较小且两个假设接近时，DP-SPRT是接近最优的。此外，我们通过实验研究揭示了其良好的实际性能。