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Sequential change-point detection when the distribution parameters areunknown is a fundamental problem in statistics and machine learning. When thepost-change parameters are unknown, we consider a set of detection proceduresbased on sequential likelihood ratios with non-anticipating estimatorsconstructed using online convex optimization algorithms such as online mirrordescent, which provides a more versatile approach to tackle complex situationswhere recursive maximum likelihood estimators cannot be found. When theunderlying distributions belong to a exponential family and the estimatorssatisfy the logarithm regret property, we show that this approach is nearlysecond-order asymptotically optimal. This means that the upper bound for thefalse alarm rate of the algorithm (measured by the average-run-length) meetsthe lower bound asymptotically up to a log-log factor when the threshold tendsto infinity. Our proof is achieved by making a connection between sequentialchange-point and online convex optimization and leveraging the logarithmicregret bound property of online mirror descent algorithm. Numerical and realdata examples validate our theory.

Nearly second-order asymptotic optimality of sequential change-point detection with one-sample updates