Towards Q-learning the Whittle Index for Restless Bandits

We consider the multi-armed restless bandit problem (RMABP) with an infinite horizon average cost objective. Each arm of the RMABP is associated with a Markov process that operates in two modes: active and passive. At each time slot a controller needs to designate a subset of the arms to be active, of which the associated processes will evolve differently from the passive case. Treated as an optimal control problem, the optimal solution of the RMABP is known to be computationally intractable. In many cases, the Whittle index policy achieves near optimal performance and can be tractably found. Nevertheless, computation of the Whittle indices requires knowledge of the transition matrices of the underlying processes, which are sometimes hidden from decision makers. In this paper, we take first steps towards a tractable and efficient reinforcement learning algorithm for controlling such a system. We setup parallel Q-learning recursions, with each recursion mapping to individual possible values of the Whittle index. We then update these recursions as we control the system, learning an approximation of the Whittle index as time evolves. Tested on several examples, our control outperforms naive priority allocations and nears the performance of the fully-informed Whittle index policy.