  The sequential hypothesis testing problem is a class of statistical analyses
where the sample size is not fixed in advance. Instead, the decision-process
takes in new observations sequentially to make real-time decisions for testing
an alternative hypothesis against a null hypothesis until some stopping
criterion is satisfied. In many common applications of sequential hypothesis
testing, the data can be highly sensitive and may require privacy protection;
for example, sequential hypothesis testing is used in clinical trials, where
doctors sequentially collect data from patients and must determine when to stop
recruiting patients and whether the treatment is effective. The field of
differential privacy has been developed to offer data analysis tools with
strong privacy guarantees, and has been commonly applied to machine learning
and statistical tasks.
  In this work, we study the sequential hypothesis testing problem under a
slight variant of differential privacy, known as Renyi differential privacy. We
present a new private algorithm based on Wald's Sequential Probability Ratio
Test (SPRT) that also gives strong theoretical privacy guarantees. We provide
theoretical analysis on statistical performance measured by Type I and Type II
error as well as the expected sample size. We also empirically validate our
theoretical results on several synthetic databases, showing that our algorithms
also perform well in practice. Unlike previous work in private hypothesis
testing that focused only on the classical fixed sample setting, our results in
the sequential setting allow a conclusion to be reached much earlier, and thus
saving the cost of collecting additional samples.
