  In online multiple testing the hypotheses arrive one-by-one over time and at
each time it must be decided on the current hypothesis solely based on the data
and hypotheses observed so far. In this paper we relax this setup by allowing
initially accepted hypotheses to be rejected due to information gathered at
later steps. We propose online e-BH, an online ARC (online with
acceptance-to-rejection changes) version of the e-BH procedure. Online e-BH is
the first nontrivial online procedure which provably controls the FDR at
data-adaptive stopping times and under arbitrary dependence between the test
statistics. Online e-BH uniformly improves e-LOND, the existing method for
e-value based online FDR control. In addition, we introduce new boosting
techniques for online e-BH to increase the power in case of locally dependent
e-values. Furthermore, based on the same proof technique as used for online
e-BH, we show that all existing online procedures with valid FDR control under
arbitrary dependence also control the FDR at data-adaptive stopping times.
