The Annals of Applied Probability

Woodroofe’s one-armed bandit problem revisited

Alexander Goldenshluger and Assaf Zeevi

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We consider the one-armed bandit problem of Woodroofe [J. Amer. Statist. Assoc. 74 (1979) 799–806], which involves sequential sampling from two populations: one whose characteristics are known, and one which depends on an unknown parameter and incorporates a covariate. The goal is to maximize cumulative expected reward. We study this problem in a minimax setting, and develop rate-optimal polices that involve suitable modifications of the myopic rule. It is shown that the regret, as well as the rate of sampling from the inferior population, can be finite or grow at various rates with the time horizon of the problem, depending on “local” properties of the covariate distribution. Proofs rely on martingale methods and information theoretic arguments.

Article information

Ann. Appl. Probab., Volume 19, Number 4 (2009), 1603-1633.

First available in Project Euclid: 27 July 2009

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Zentralblatt MATH identifier

Primary: 62L05: Sequential design
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62C20: Minimax procedures

Sequential allocation online learning estimation bandit problems regret inferior sampling rate minimax rate-optimal policy


Goldenshluger, Alexander; Zeevi, Assaf. Woodroofe’s one-armed bandit problem revisited. Ann. Appl. Probab. 19 (2009), no. 4, 1603--1633. doi:10.1214/08-AAP589.

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