Preprint Article Version 1 This version is not peer-reviewed

# On Near Optimality of One-sample Update for Joint Detection and Estimation

Version 1 : Received: 1 September 2017 / Approved: 1 September 2017 / Online: 1 September 2017 (05:24:10 CEST)

A peer-reviewed article of this Preprint also exists.

Cao, Y.; Xie, L.; Xie, Y.; Xu, H. Sequential Change-Point Detection via Online Convex Optimization. Entropy 2018, 20, 108. Cao, Y.; Xie, L.; Xie, Y.; Xu, H. Sequential Change-Point Detection via Online Convex Optimization. Entropy 2018, 20, 108.

Journal reference: Entropy 2018, 20, 108
DOI: 10.3390/e20020108

## Abstract

Sequential hypothesis test and change-point detection when the distribution parameters are unknown is a fundamental problem in statistics and machine learning. We show that for such problems, detection procedures based on sequential likelihood ratios with simple one-sample update estimates such as online mirror descent are nearly second-order optimal. This means that the upper bound for the algorithm performance meets the lower bound asymptotically up to a log-log factor in the false-alarm rate when it tends to zero. This is a blessing, since although the generalized likelihood ratio (GLR) statistics are optimal theoretically, but they cannot be computed recursively, and their exact computation usually requires infinite memory of historical data. We prove the nearly second-order optimality by making a connection between sequential analysis and online convex optimization and leveraging the logarithmic regret bound property of online mirror descent algorithm. Numerical and real data examples validate our theory.

## Subject Areas

sequential methods; change-point detection; online algorithms

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