Anomaly Search With Multiple Plays Under Delay and Switching Costs

Abstract

The problem of searching for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> anomalous processes among <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> processes is considered. At each time, the decision maker can observe a subset of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> processes (i.e., multiple plays). The measurement drawn when observing a process follows one of two different distributions, depending on whether the process is normal or abnormal. The goal is to design a policy that minimizes the Bayes risk which balances between the sample complexity, detection errors, and the switching cost associated with switching across processes. We develop a policy, dubbed consecutive controlled sensing (CCS), to achieve this goal. On the one hand, by contrast to existing studies on controlled sensing, the CCS policy senses processes consecutively to reduce the switching cost. On the other hand, the policy controls the sensing operation in a closed-loop manner to switch between processes when necessary to guarantee reliable inference. We prove theoretically that CCS is asymptotically optimal in terms of minimizing the Bayes risk as the detection error approaches zero (i.e., the sample complexity increases). Simulation results demonstrate strong performance of CCS in the finite regime as well. Index Terms - Anomaly detection, controlled sensing, active hypothesis testing, sequential design of experiments.