  One important partition of algorithms for controlling the false discovery
rate (FDR) in multiple testing is into offline and online algorithms. The first
generally achieve significantly higher power of discovery, while the latter
allow making decisions sequentially as well as adaptively formulating
hypotheses based on past observations. Using existing methodology, it is
unclear how one could trade off the benefits of these two broad families of
algorithms, all the while preserving their formal FDR guarantees. To this end,
we introduce $\text{Batch}_{\text{BH}}$ and $\text{Batch}_{\text{St-BH}}$,
algorithms for controlling the FDR when a possibly infinite sequence of batches
of hypotheses is tested by repeated application of one of the most widely used
offline algorithms, the Benjamini-Hochberg (BH) method or Storey's improvement
of the BH method. We show that our algorithms interpolate between existing
online and offline methodology, thus trading off the best of both worlds.
