Exponentially Consistent Low Complexity Tests for Outlier Hypothesis Testing

TL;DR

The paper tackles outlier hypothesis testing with unknown nominal and anomalous distributions by proposing exponentially consistent, low-complexity fixed-length and sequential tests for both known and unknown numbers of outliers. By replacing exhaustive search with sorting-based and pairwise-score procedures, it achieves strong error-exponent guarantees while reducing computational burden, and shows clear performance advantages of sequential testing over fixed-length approaches. It extends the framework to unknown outlier counts, introducing detection-idetification two-phase schemes with bounded stopping times and explicit exponent trade-offs among misclassification, false-reject, and false-alarm events. The results provide a practical pathway for scalable OHT in large datasets, with quantitative insights into the cost of distribution-uncertainty and the gains from sequential operation. The work also highlights the versatility of generalized divergences, notably the generalized Jensen-Shannon divergence, in achieving favorable detection performance under non-parametric settings.

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.

Figures (8)

• Figure 1: Plot of the simulated misclassification probabilities as a function of running times of the fixed-length test in Algorithm \ref{['FA:known']} and the fixed-length test $\Phi_{\rm Li}$ in \ref{['test_Li']} when $M=10$ and $t=3$ and $(P_\mathrm{N},P_\mathrm{A})=\mathrm{Bern}(0.23,0.3)$. As observed, the low-complexity test in Algorithm \ref{['FA:known']} achieves the same misclassification probability with much less running time than the test $\Phi_{\rm Li}$.
• Figure 2: Numerical comparison of achievable misclassification exponents in Theorem \ref{['fixed:known']} for KL and GJS divergence scoring functions when $P_\mathrm{N}=\rm{Bern}(0.2)$ and $P_\mathrm{A}=\rm{Bern}(a)$ for different values of $a\in[0.01,0.55]$ such that $a\neq 0.2$. As observed, GJS divergence scoring function can yield larger misclassification exponent in certain cases.
• Figure 3: Plot of achievable misclassification exponents for the sequential test in Theorem \ref{['th:known']} and the fixed-length test in Theorem \ref{['fixed:known']} when the scoring function $f(\cdot)$ is the GJS divergence, $P_\mathrm{N}=\rm{Bern}(0.5)$, $P_\mathrm{A}=\rm{Bern}(a)$ for $a\in(0,1)$ such that $a\neq 0.5$, $\lambda_1=0.0005$ and $\lambda_2=f(P_\mathrm{A},P_\mathrm{N})-0.0001$ for each $a$. As observed, the achievable misclassification exponent for the sequential test is larger than that for the fixed-length test.
• Figure 4: Plot of the simulated misclassification probabilities as a function of expected stopping times for the sequential test in Algorithm \ref{['CA:known']} and fixed-length test in Algorithm \ref{['FA:known']} when $M=100$, $t=10$, the scoring function $f(\cdot)$ is the GJS divergence, $(P_\mathrm{N},P_\mathrm{A})=\mathrm{Bern}(0.32,0.25)$ and $(\lambda_1,\lambda_2)=(0.001,0.003)$. As observed, our sequential test achieves smaller misclassification probability than the fixed-length test.
• Figure 5: Numerical comparison of achievable misclassification exponents of our test in Theorem \ref{['th:known']} under KL and GJS divergence when $P_\mathrm{N}=\rm{Bern}(0.2)$ and $P_\mathrm{A}=\rm{Bern}(a)$ for different values of $a\in[0.01,0.99]$ and $a\neq 0.2$, with thresholds $\lambda_1=0.001$ and $\lambda_2=f(P_\mathrm{A},P_\mathrm{N})-0.0001$ for each $a$. As observed, GJS divergence can yield better performance in certain cases.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4