Title: Robust Multi-Hypothesis Testing with Moment-Constrained Uncertainty Sets
Abstract: Hypothesis testing is a fundamental problem in statistical decision-making, in which the goal is to decide between given hypotheses based on observed data. Detection problems have a wide range of applications in economics, communications, signal processing and life sciences. In general, the distributions under the hypotheses may be unknown, and may need to be estimated from historical data. However, deviations of the estimates from the true underlying distributions can result in significant performance degradation of the likelihood ratio test constructed using the estimated distributions. The robust hypothesis testing framework was proposed to address this problem. The problem of robust multi-hypothesis testing in the Bayesian setting is studied in this work. Under the m ≥ 2 hypotheses, the data-generating distributions are assumed to belong to uncertainty sets constructed through some moment functions; i.e., the sets contain distributions whose moments are centered around the empirical moments obtained from some training data sequences. The goal is to design a test that performs well under all distributions in the uncertainty sets, i.e., a test that minimizes the worst-case probability of error over the uncertainty sets. In the special case of binary hypothesis testing (m = 2), insights on the need for optimization-based approaches to solve the robust testing problem with moment constrained uncertainty sets are provided. Using these insights, the optimal test is obtained in the finite-alphabet case for multi-hypothesis testing. In the infinite-alphabet case, a tractable finite-dimension approximation is derived that converges to the optimal value of the original problem. A robust test is constructed for the entire alphabet from the solution of the approximation problem, and guarantees on the worst-case error of the proposed robust test over all distributions in the uncertainty sets are provided. To the best of our knowledge, ours is the first work to address the robust multi-hypothesis (m ≥ 2) testing setting. Numerical results demonstrate the performance of the proposed robust tests.
Bio: Akshayaa Magesh is a final year PhD student at the Department of Electrical and Computer Engineering at the University of Illinois Urbana-Champaign (UIUC), advised by Prof. Venu Veeravalli. Her research interests include statistical inference, robust machine learning and reinforcement learning, and information theory. She is currently working on problems in uncertainty quantification, in particular on out-of-distribution detection, and on developing principled algorithms for inference robust to distribution shifts. Prior to this, she received her Masters degree from the University of Illinois Urbana-Champaign in 2020, and her Bachelors from the Indian Institute of Technology, Madras in 2018.