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We study an extension of the classic stochastic multi-armed bandit problemwhich involves multiple plays and Markovian rewards in the rested banditssetting. In order to tackle this problem we consider an adaptive allocationrule which at each stage combines the information from the sample means of allthe arms, with the Kullback-Leibler upper confidence bound of a single armwhich is selected in round-robin way. For rewards generated from aone-parameter exponential family of Markov chains, we provide a finite-timeupper bound for the regret incurred from this adaptive allocation rule, whichreveals the logarithmic dependence of the regret on the time horizon, and whichis asymptotically optimal. For our analysis we devise several concentrationresults 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 wealso establish asymptotically optimal, finite-time guarantees for the case ofmultiple plays, and i.i.d. rewards drawn from a one-parameter exponentialfamily of probability densities. Additionally, we provide simulation resultsthat illustrate that calculating Kullback-Leibler upper confidence bounds in around-robin way, is significantly more efficient than calculating them forevery arm at each round, and that the expected regrets of those two approachesbehave similarly.

Finite-Time Analysis of Round-Robin Kullback-Leibler Upper Confidence  Bounds for Optimal Adaptive Allocation with Multiple Plays and Markovian  Rewards