# Low-Cost Active Anomaly Detection with Switching Latency
### Related Work  
The anomaly detection problem studied in this paper can be regarded as a variation of the Sequential Hypothesis Testing (SHT) problem [1] for a single process. Differing from the classic hypothesis testing problem with fixed sample size, SHT allows decision-maker to stop sampling from the process based on the gathered observations, which reduces the sample size without compromising the detection performance. The SHT problem was pioneered in [1] by [PERSON], who established the Sequential Probability Ratio Test (SPRT) for simple binary hypothesis testing and showed its optimality in terms of minimizing the expected sample size under error probability constraints. Various extensions for testing \(M\)-ary hypothesis or composite hypothesis for a single process were studied in [2; 3; 4], where asymptotically optimal performance can be obtained as the error constraints approach zero. The recent application and modification of the SPRT can be found in [5; 6; 7; 8], where the SPRT was modified based on the fuzzy hypotheses [5], or was used to identify and diagnose the latent leakage faults for the pneumatic units [6], track multiple targets [7], or to determine software reliability [8]. Since the decision-maker can choose which process to probe at each time based on the past observations, the anomaly detection problem has a connection with the sequential design of experiments problem studied by [PERSON] in 1959 [9] named as the Active Hypothesis Testing (AHT) problem. Compared with SPRT where the observation action under each hypothesis is fixed, active hypothesis testing allows the decision-maker to choose different experiments to conduct at each time and infer the state of the probed process. The test developed by [PERSON] (referred to as Chernoff test) is a randomized test that generates random actions based on historical observations. The actions, consequently the test statistics of log-likelihood ratio, thus become independent and identically distributed over time, allowing asymptotic analysis of the stopping time on the test statistic. [PERSON]'s work has been extended in various directions. It should be mentioned that [PERSON] [10] generated [PERSON]'s work to general multiple hypotheses testing. [PERSON] and [PERSON] in [11; 12; 13; 14; 15] studied this sequential problem from the perspective of minimizing Bayesian risk considering the detection error and detection delay. Specifically, in [11], they established the lower bounds for the Bayesian risk which characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability. They further investigated the roles of information utility [13], performance bounds [14], and sequentiality and adaptivity [15] in active hypothesis testing problem separately. Recently, [PERSON] and [PERSON] et al. [16; 17; 18; 19; 20] studied the AHT problem for anomaly detection of detecting anomalous processes among a fixed number of processes. In these studies, they were aimed at finding a sequential search strategy for the anomaly detection problem that minimizes the expected detection time subject to an error probability constraint. Specifically, in [16], in contrast to the random policies advocated in the Chernoff test, they introduced a simple deterministic model which offers the same asymptotic optimality yet with significant performance gain in the finite regime and considerable reduction in implementation complexity. In [17; 18], [PERSON] and [PERSON] considered the AHT model for the anomaly detection scenario where there are a few anomalous processes among a large number of processes, based on a binary-tree structured detection space. They further consider the anomaly detection problem in the scenario where the processes are heterogeneous [19]and the composite hypothesis case in which the observations follow a common distribution with an unknown parameter [20]. These above studies have focused on minimizing _the expected sample size_ while we focus on minimizing the expected cumulative cost in this paper.