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 tackling 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.
Highlights
Sequential analysis is a classic topic in statistics concerning online inference from a sequence of observations
Inspired by the existing connection between sequential analysis and online convex optimization in [13,14], we prove the near optimality leveraging the logarithmic regret property of online mirror descent (OMD) and the lower bound established in statistical sequential change-point literature [4,15]
We focus on f θ being the exponential family for the following reasons: (i) exponential family [10] represents a very rich class of parametric and even many nonparametric statistical models [37]; (ii) the negative log-likelihood function for exponential family − log f θ ( x ) is convex, and this allows us to perform online convex optimization
Summary
Sequential analysis is a classic topic in statistics concerning online inference from a sequence of observations. An important sequential analysis problem commonly studied is sequential change-point detection [1]. It arises from various applications including online anomaly detection, statistical quality control, biosurveillance, financial arbitrage detection and network security monitoring (see, e.g., [2,3,4]). We are interested in the sequential change-point detection problem with known pre-change parameters but unknown post-change parameters. We further assume that the parameters before the change-point are known. This is reasonable since usually it is relatively easy to obtain the reference data for the normal state, so that the parameters in the normal state can be estimated with good accuracy. After the change-point, the values of the parameters switch to some unknown values, which represent anomalies or novelties that need to be discovered
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