Optimistic Whittle Index Policy: Online Learning for Restless Bandits

Restless multi-armed bandits (RMABs) extend multi-armed bandits to allow for stateful arms, where the state of each arm evolves restlessly with different transitions depending on whether that arm is pulled. However, solving RMABs requires information on transition dynamics, which is often not available upfront. To plan in RMAB settings with unknown transitions, we propose the first online learning algorithm based on the Whittle index policy, using an upper confidence bound (UCB) approach to learn transition dynamics. Specifically, we formulate a bilinear program to compute the optimistic Whittle index from the confidence bounds in transition dynamics. Our algorithm, UCWhittle, achieves sublinear $O(\sqrt{T \log T})$ frequentist regret to solve RMABs with unknown transitions. Empirically, we demonstrate that UCWhittle leverages the structure of RMABs and the Whittle index policy solution to achieve better performance than existing online learning baselines across three domains, including on real-world maternal and childcare data aimed at reducing maternal mortality.