  We consider a class of restless multi-armed bandit (RMAB) problems with
unknown arm dynamics. At each time, a player chooses an arm out of N arms to
play, referred to as an active arm, and receives a random reward from a finite
set of reward states. The reward state of the active arm transits according to
an unknown Markovian dynamics. The reward state of passive arms (which are not
chosen to play at time t) evolves according to an arbitrary unknown random
process. The objective is an arm-selection policy that minimizes the regret,
defined as the reward loss with respect to a player that always plays the most
rewarding arm. This class of RMAB problems has been studied recently in the
context of communication networks and financial investment applications. We
develop a strategy that selects arms to be played in a consecutive manner,
dubbed Adaptive Sequencing Rules (ASR) algorithm. The sequencing rules for
selecting arms under the ASR algorithm are adaptively updated and controlled by
the current sample reward means. By designing judiciously the adaptive
sequencing rules, we show that the ASR algorithm achieves a logarithmic regret
order with time, and a finite-sample bound on the regret is established.
Although existing methods have shown a logarithmic regret order with time in
this RMAB setting, the theoretical analysis shows a significant improvement in
the regret scaling with respect to the system parameters under ASR. Extensive
simulation results support the theoretical study and demonstrate strong
performance of the algorithm as compared to existing methods.
