Exponentially Consistent Low Complexity Tests for Outlier Hypothesis Testing

Abstract

We revisit outlier hypothesis testing, propose exponentially consistent low complexity fixed-length and sequential tests and show that our tests achieve better tradeoff between detection performance and computational complexity than existing tests that use exhaustive search. Specifically, in outlier hypothesis testing, one is given a list of observed sequences, most of which are generated i.i.d. from a nominal distribution while the rest sequences named outliers are generated i.i.d. from another anomalous distribution. The task is to identify all outliers when both the nominal and anomalous distributions are unknown. There are two basic settings: fixed-length and sequential. In the fixed-length setting, the sample size of each observed sequence is fixed a priori while in the sequential setting, the sample size is a random number that can be determined by the test designer to ensure reliable decisions. For the fixed-length setting, we strengthen the results of Bu \emph{et. al} (TSP 2019) by i) allowing for scoring functions beyond KL divergence and further simplifying the test design when the number of outliers is known and ii) proposing a new test, explicitly bounding the detection performance of the test and characterizing the tradeoff among exponential decay rates of three error probabilities when the number of outliers is unknown. For the sequential setting, our tests for both cases are novel and enable us to reveal the benefit of sequentiality. Finally, for both fixed-length and sequential settings, we demonstrate the penalty of not knowing the number of outliers on the detection performance.