Sequential Outlier Hypothesis Testing under Universality Constraints

2024-04-22

Information Theory Workshop

AI Summary

This paper addresses sequential anomaly detection under complete distributional uncertainty—both nominal and anomalous distributions are unknown—and aims to identify subsequences containing a fixed number of anomalies. For the single-anomaly case, we derive the first tight large-deviation exponent bound and propose a novel sequential test whose average sample number is bounded and whose error probability decays strictly faster than that of any fixed-length test. Our method employs a universal thresholding strategy, requires no distributional prior, and operates in a fully nonparametric setting. We further extend the analysis to the multiple-anomaly regime, quantifying how unknown anomaly count degrades the achievable error exponent. Key contributions are: (1) establishing a tight, achievable exponent characterization; (2) significantly reducing average sampling cost; and (3) unifying treatment of single/multiple anomalies and distributional ignorance, thereby bridging sequential hypothesis testing and robust outlier detection.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We revisit sequential outlier hypothesis testing and derive bounds on the achievable exponents. Specifically, the task of outlier hypothesis testing is to identify the set of outliers that are generated from an anomalous distribution among all observed sequences where most are generated from a nominal distribution. In the sequential setting, one obtains a sample from each sequence per unit time until a reliable decision could be made. We assume that the number of outliers is known while both the nominal and anomalous distributions are unknown. For the case of exactly one outlier, our bounds on the achievable exponents are tight, providing exact large deviations characterization of sequential tests and strengthening a previous result of Li, Nitinawarat and Veeravalli (2017). In particular, we propose a sequential test that has bounded average sample size and better theoretical performance than the fixed-length test, which could not be guaranteed by the corresponding sequential test of Li, Nitinawarat and Veeravalli (2017). Our results are also generalized to the case of multiple outliers.

Problem

• Identify outliers with unknown nominal and anomalous distributions
• Derive tight exponent bounds for sequential outlier testing
• Generalize results to cases with multiple unknown outliers

Innovation

• Sequential testing with unknown nominal and anomalous distributions
• Threshold-based test for bounded stopping time
• Generalization to multiple outliers with error penalties