  We propose a novel nonparametric sequential test for composite hypotheses for
means of multiple data streams. Our proposed method, \emph{peeking with
expectation-based averaged capital} (PEAK), builds upon the testing-by-betting
framework and provides a non-asymptotic $\alpha$-level test across any stopping
time. Our contributions are two-fold: (1) we propose a novel betting scheme and
provide theoretical guarantees on type-I error control, power, and asymptotic
growth rate/$e$-power in the setting of a single data stream; (2) we introduce
PEAK, a generalization of this betting scheme to multiple streams, that (i)
avoids using wasteful union bounds via averaging, (ii) is a test of power one
under mild regularity conditions on the sampling scheme of the streams, and
(iii) reduces computational overhead when applying the testing-as-betting
approaches for pure-exploration bandit problems. We illustrate the practical
benefits of PEAK using both synthetic and real-world HeartSteps datasets. Our
experiments show that PEAK provides up to an 85\% reduction in the number of
samples before stopping compared to existing stopping rules for
pure-exploration bandit problems, and matches the performance of
state-of-the-art sequential tests while improving upon computational
complexity.
