Stochastic Systems

A linear response bandit problem

Alexander Goldenshluger and Assaf Zeevi

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We consider a two–armed bandit problem which involves sequential sampling from two non-homogeneous populations. The response in each is determined by a random covariate vector and a vector of parameters whose values are not known a priori. The goal is to maximize cumulative expected reward. We study this problem in a minimax setting, and develop rate-optimal polices that combine myopic action based on least squares estimates with a suitable “forced sampling” strategy. It is shown that the regret grows logarithmically in the time horizon $n$ and no policy can achieve a slower growth rate over all feasible problem instances. In this setting of linear response bandits, the identity of the sub-optimal action changes with the values of the covariate vector, and the optimal policy is subject to sampling from the inferior population at a rate that grows like $\sqrt{n}$.

Article information

Stoch. Syst., Volume 3, Number 1 (2013), 230-261.

First available in Project Euclid: 24 February 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L05: Sequential design
Secondary: 60G40, 62C20

Sequential allocation estimation bandit problems regret minimax rate–optimal policy


Goldenshluger, Alexander; Zeevi, Assaf. A linear response bandit problem. Stoch. Syst. 3 (2013), no. 1, 230--261. doi:10.1214/11-SSY032.

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