Anytime-Valid Tests for Sparse Anomalies

We consider the problem of testing sequentially for the presence of sparse anomalies among a large number of data streams. To this end, we design and analyze Anytime-Valid (AV) tests, which retain type-I error control at arbitrary stopping times. Existing results address exclusively the nonsequential case, which exhibits a subtle phase transition between two regimes where tests are either powerless or powerful. In our sequential setting, we argue, two challenges arise: (1) the standard analysis of AV tests cannot be executed in the relevant sample-size regime; and (2) standard constructions of parameter-adaptive AV tests are either analytically intractable or computationally unfeasible. This work addresses these challenges. Borrowing insights from the nonsequential literature, we propose a framework to analyze AV tests and their shortest possible sample sizes. Under this framework, we show that, in the Gaussian location setting, the oracle AV test has a delicate threshold behavior that is related to -- but not implied by -- the phase transition observed in optimal nonsequential tests. Our main results include a computationally efficient, parameter-adaptive AV test; we show that it achieves the same threshold behavior as the oracle AV test. Numerical simulations illustrate these theoretical findings.