Abstract
We consider sequential decision problems in which we adaptively choose one of finitely many alternatives and observe a stochastic reward. We offer a new perspective on interpreting Bayesian ranking and selection problems as adaptive stochastic multiset maximization problems and derive the first finite-time bound of the knowledge-gradient policy for adaptive submodular objective functions. In addition, we introduce the concept of prior-optimality and provide another insight into the performance of the knowledge-gradient policy based on the submodular assumption on the value of information. We demonstrate submodularity for the two-alternative case and provide other conditions for more general problems, bringing out the issue and importance of submodularity in learning problems. Empirical experiments are conducted to further illustrate the finite-time behavior of the knowledge-gradient policy.
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