Distributionally Robust Quickest Change Detection using Wasserstein Uncertainty Sets

Abstract

The problem of quickest detection of a change in the distribution of a sequence of independent observations is considered. It is assumed that the pre-change distribution is known (accurately estimated), while the only information about the post-change distribution is through a (small) set of labeled data. This post-change data is used in a data-driven minimax robust framework, where an uncertainty set for the post-change distribution is constructed using the Wasserstein distance from the empirical distribution of the data. The robust change detection problem is studied in an asymptotic setting where the mean time to false alarm goes to infinity, for which the least favorable post-change distribution within the uncertainty set is the one that minimizes the Kullback-Leibler divergence between the post- and the pre-change distributions. It is shown that the density corresponding to the least favorable distribution is an exponentially tilted version of the pre-change density and can be calculated efficiently. A Cumulative Sum (CuSum) test based on the least favorable distribution, which is referred to as the distributionally robust (DR) CuSum test, is then shown to be asymptotically robust. The results are extended to the case where the post-change uncertainty set is a finite union of multiple Wasserstein uncertainty sets, corresponding to multiple post-change scenarios, each with its own labeled data. The proposed method is validated using synthetic and real data examples