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

Sequential change-point detection when the distribution parameters are unknown is a fundamental problem in statistics and machine learning. When the underlying distributions belong to the exponential family, we show that detection procedures based on sequential likelihood ratios with simple one-sample update estimates such as online mirror descent are nearly second-order asymptotic optimal, under some mild conditions for the expected Kullback-Leibler divergence between the estimators and the true parameters. This means that the upper bound for the false alarm rate of the algorithm (measured by the average-run-length) meets the lower bound asymptotically up to a log-log factor when the threshold tends to infinity. This is a blessing since although the generalized likelihood ratio (GLR) statistics are asymptotically optimal in theory, they cannot be computed recursively and thus the exact computation can be time-consuming. We prove the nearly second-order asymptotic optimality by making a connection between sequential change-point and online convex optimization and leveraging the logarithmic regret bound property of online mirror descent algorithm. Numerical and real data examples validate our theory.