Linearly parameterized bandits
We consider bandit problems involving a large (possibly infinite) collection of arms, in which the expected reward of each arm is a linear function of an r-dimensional random vector Z ∈ ℝ(superscript r), where r ≥ 2. The objective is to minimize the cumulative regret and Bayes risk. When the set of...
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| Format: | Article |
| Language: | English |
| Published: |
INFORMS,
2012-06-04T18:10:35Z.
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| Online Access: | Get fulltext |
| Summary: | We consider bandit problems involving a large (possibly infinite) collection of arms, in which the expected reward of each arm is a linear function of an r-dimensional random vector Z ∈ ℝ(superscript r), where r ≥ 2. The objective is to minimize the cumulative regret and Bayes risk. When the set of arms corresponds to the unit sphere, we prove that the regret and Bayes risk is of order Θ(r √T), by establishing a lower bound for an arbitrary policy, and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases. The phase-based policy is also shown to be effective if the set of arms satisfies a strong convexity condition. For the case of a general set of arms, we describe a near-optimal policy whose regret and Bayes risk admit upper bounds of the form O(r √T log(superscript 3/2)T). National Science Foundation (U.S.) (grant DMS-0732196) National Science Foundation (U.S.) (grant ECCS-0701623) National Science Foundation (U.S.) (grant CMMI-0856063) National Science Foundation (U.S.) (grant CMMI-0855928) |
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