Multi-Armed Sequential Hypothesis Testing by Betting

2026-03-18

AI Summary

This work addresses the problem of sequential multi-armed hypothesis testing, where the goal is to efficiently reject a global composite null hypothesis asserting that all arms satisfy the null, without prior knowledge of which data source (arm) provides the strongest evidence against it. The authors integrate sequential testing with betting mechanisms to introduce log-optimality and expected rejection time optimality criteria tailored to the multi-armed setting, and establish matching upper and lower bounds. By combining e-processes, Kelly’s theory of wealth growth, non-asymptotic concentration inequalities, and UCB-type strategies, the proposed algorithm nearly attains oracle performance when multiple non-null arms exist. Notably, the paper derives the first non-asymptotic concentration bound on the optimal wealth growth rate in this context.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Sequential Decision Making
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

We consider a variant of sequential testing by betting where, at each time step, the statistician is presented with multiple data sources (arms) and obtains data by choosing one of the arms. We consider the composite global null hypothesis $\mathscr{P}$ that all arms are null in a certain sense (e.g. all dosages of a treatment are ineffective) and we are interested in rejecting $\mathscr{P}$ in favor of a composite alternative $\mathscr{Q}$ where at least one arm is non-null (e.g. there exists an effective treatment dosage). We posit an optimality desideratum that we describe informally as follows: even if several arms are non-null, we seek $e$-processes and sequential tests whose performance are as strong as the ones that have oracle knowledge about which arm generates the most evidence against $\mathscr{P}$. Formally, we generalize notions of log-optimality and expected rejection time optimality to more than one arm, obtaining matching lower and upper bounds for both. A key technical device in this optimality analysis is a modified upper-confidence-bound-like algorithm for unobservable but sufficiently "estimable" rewards. In the design of this algorithm, we derive nonasymptotic concentration inequalities for optimal wealth growth rates in the sense of Kelly [1956]. These may be of independent interest.

Problem

• Multi-Armed Sequential Testing
• Composite Hypothesis Testing
• e-Processes
• Optimality
• Sequential Analysis

Innovation

• sequential hypothesis testing
• multi-armed bandits
• e-processes
• log-optimality
• nonasymptotic concentration inequalities