  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. Solving RMABs requires
information on transition dynamics, which are often unknown 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 estimate
confidence bounds of the transition probabilities and formulate a bilinear
program to compute optimistic Whittle indices using these estimates. Our
algorithm, UCWhittle, achieves sublinear $O(H \sqrt{T \log T})$ frequentist
regret to solve RMABs with unknown transitions in $T$ episodes with a constant
horizon $H$. 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 one
constructed from a real-world maternal and childcare dataset.
