Tradeoffs among Action Taking Policies Matter in Active Sequential Multi-Hypothesis Testing: the Optimal Error Exponent Region

2024-05-10

arXiv.org

AI Summary

This paper investigates the asymptotically optimal trade-off among error probabilities in active sequential multi-hypothesis testing, where data source selection is subject to action constraints. We fully characterize the achievable region of the $M(M-1)$ pairwise error exponents—surpassing the Chernoff framework’s limitation to overall error probability analysis. Leveraging a unified approach combining large deviations theory, dynamic programming, and information geometry, we design an asymptotically optimal test that balances exploration and exploitation. We rigorously characterize the error exponent capacity region under an expected action budget as the expected stopping time tends to infinity, and construct an accompanying optimal test achieving any feasible target error exponent vector. Our core contribution is establishing a theoretical framework for optimal error exponent trade-offs in multi-hypothesis testing, revealing the fundamental tension among action-selection policies in constrained sequential decision-making.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Active Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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

Abstract

Reliability of sequential hypothesis testing can be greatly improved when decision maker is given the freedom to adaptively take an action that determines the distribution of the current collected sample. Such advantage of sampling adaptivity has been realized since Chernoff's seminal paper in 1959 [1]. While a large body of works have explored and investigated the gain of adaptivity, in the general multiple-hypothesis setting, the fundamental limits of individual error probabilities have not been fully understood. In particular, in the asymptotic regime as the expected stopping time tends to infinity, the error exponents are only characterized in specific cases, such as that of the total error probability. In this paper, we consider a general setup of active sequential multiple-hypothesis testing where at each time slot, a temporally varying subset of data sources (out of a known set) emerges from which the decision maker can select to collect samples, subject to a family of expected selection budget constraints. The selection of sources, understood as the"action"at each time slot, is constrained in a predefined action space. At the end of each time slot, the decision maker either decides to make the inference on the $M$ hypotheses, or continues to observe the data sources for the next time slot. The optimal tradeoffs among $M(M-1)$ types of error exponents are characterized. A companion asymptotically optimal test that strikes the balance between exploration and exploitation is proposed to achieve any target error exponents within the region. To the best of our knowledge, this is the first time in the literature to identify such tradeoffs among error exponents, and it uncovers the tension among different action taking policies even in the basic setting of Chernoff [1].

Problem

• Characterize optimal tradeoffs among error exponents in sequential multi-hypothesis testing
• Study action policies under budget constraints for adaptive sampling
• Balance exploration and exploitation in active sequential hypothesis testing

Innovation

• Adaptive action selection for sample distribution
• Optimal tradeoffs among error exponents
• Balance exploration and exploitation policies