  Motivated by clinical trials, we study bandits with observable
non-compliance. At each step, the learner chooses an arm, after, instead of
observing only the reward, it also observes the action that took place. We show
that such noncompliance can be helpful or hurtful to the learner in general.
Unfortunately, naively incorporating compliance information into bandit
algorithms loses guarantees on sublinear regret. We present hybrid algorithms
that maintain regret bounds up to a multiplicative factor and can incorporate
compliance information. Simulations based on real data from the International
Stoke Trial show the practical potential of these algorithms.
