Woodroofe's one-armed bandit problem revisited

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Abstract

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.

Original languageEnglish
Pages (from-to)1603-1633
Number of pages31
JournalAnnals of Applied Probability
Volume19
Issue number4
DOIs
StatePublished - Aug 2009

Keywords

  • Bandit problems
  • Estimation
  • Inferior sampling rate
  • Minimax
  • Online learning
  • Rate-optimal policy
  • Regret
  • Sequential allocation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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