  In contemporary problems involving genetic or neuroimaging data, thousands of
hypotheses need to be tested. Due to their high power, and finite sample
guarantees on type-1 error under weak assumptions, Monte-Carlo permutation
tests are often considered as gold standard for these settings. However, the
enormous computational effort required for (thousands of) permutation tests is
a major burden. Recently, Fischer and Ramdas (2024) constructed a permutation
test for a single hypothesis in which the permutations are drawn sequentially
one-by-one and the testing process can be stopped at any point without
inflating the type I error. They showed that the number of permutations can be
substantially reduced (under null and alternative) while the power remains
similar. We show how their approach can be modified to make it suitable for a
broad class of multiple testing procedures. In particular, we discuss its use
with the Benjamini-Hochberg procedure and illustrate the application on a large
dataset.
