Finite-time Analysis of Kullback-Leibler Upper Confidence Bounds for Optimal Adaptive Allocation with Multiple Plays and Markovian Rewards
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We study an extension of the classic stochastic multi-armed bandit problem which involves Markovian rewards and multiple plays. In order to tackle this problem we consider an index based adaptive allocation rule which at each stage combines calculations of sample means, and of upper confidence bounds, using the Kullback-Leibler divergence rate, for the stationary expected reward of Markovian arms. For rewards generated from a one-parameter exponential family of Markov chains, we provide a finite-time upper bound for the regret incurred from this adaptive allocation rule, which reveals the logarithmic dependence of the regret on the time horizon, and which is asymptotically optimal. For our analysis we devise several concentration results for Markov chains, including a maximal inequality for Markov chains, that may be of interest in their own right. As a byproduct of our analysis we also establish, asymptotically optimal, finite-time guarantees for the case of multiple plays, and IID rewards drawn from a one-parameter exponential family of probability densities.
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