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

TL;DR

This paper demonstrates that sequential change-point detection procedures using online convex optimization estimators are nearly second-order asymptotically optimal in exponential family distributions, with theoretical validation and practical examples.

Contribution

It introduces a new approach combining online convex optimization with change-point detection, achieving near optimality under broad distributional assumptions.

Findings

Approach is nearly second-order asymptotically optimal.

False alarm rate bounds match lower bounds up to a log-log factor.

Numerical and real data validate the theoretical results.

Abstract

Sequential change-point detection when the distribution parameters are unknown is a fundamental problem in statistics and machine learning. When the post-change parameters are unknown, we consider a set of detection procedures based on sequential likelihood ratios with non-anticipating estimators constructed using online convex optimization algorithms such as online mirror descent, which provides a more versatile approach to tackle complex situations where recursive maximum likelihood estimators cannot be found. When the underlying distributions belong to a exponential family and the estimators satisfy the logarithm regret property, we show that this approach is nearly second-order asymptotically optimal. 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. Our proof is achieved 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.