  A signal recovery problem is considered, where the same binary testing
problem is posed over multiple, independent data streams. The goal is to
identify all signals, i.e., streams where the alternative hypothesis is
correct, and noises, i.e., streams where the null hypothesis is correct,
subject to prescribed bounds on the classical or generalized familywise error
probabilities. It is not required that the exact number of signals be a priori
known, only upper bounds on the number of signals and noises are assumed
instead. A decentralized formulation is adopted, according to which the sample
size and the decision for each testing problem must be based only on
observations from the corresponding data stream. A novel multistage testing
procedure is proposed for this problem and is shown to enjoy a high-dimensional
asymptotic optimality property. Specifically, it achieves the optimal, average
over all streams, expected sample size, uniformly in the true number of
signals, as the maximum possible numbers of signals and noises go to infinity
at arbitrary rates, in the class of all sequential tests with the same global
error control. In contrast, existing multistage tests in the literature are
shown to achieve this high-dimensional asymptotic optimality property only
under additional sparsity or symmetry conditions. These results are based on an
asymptotic analysis for the fundamental binary testing problem as the two error
probabilities go to zero. For this problem, unlike existing multistage tests in
the literature, the proposed test achieves the optimal expected sample size
under both hypotheses, in the class of all sequential tests with the same error
control, as the two error probabilities go to zero at arbitrary rates. These
results are further supported by simulation studies and extended to problems
with non-iid data and composite hypotheses.
