Contextual Blocking Bandits

We study a novel variant of the multi-armed bandit problem, where at each time step, the player observes an independently sampled context that determines the arms' mean rewards. However, playing an arm blocks it (across all contexts) for a fixed and known number of future time steps. The above contextual setting, which captures important scenarios such as recommendation systems or ad placement with diverse users, invalidates greedy solution techniques that are effective for its non-contextual counterpart (Basu et al., NeurIPS19). Assuming knowledge of the context distribution and the mean reward of each arm-context pair, we cast the problem as an online bipartite matching problem, where the right-vertices (contexts) arrive stochastically and the left-vertices (arms) are blocked for a finite number of rounds each time they are matched. This problem has been recently studied in the full-information case, where competitive ratio bounds have been derived. We focus on the bandit setting, where the reward distributions are initially unknown; we propose a UCB-based variant of the full-information algorithm that guarantees a $\mathcal{O}(\log T)$-regret w.r.t. an $\alpha$-optimal strategy in $T$ time steps, matching the $\Omega(\log(T))$ regret lower bound in this setting. Due to the time correlations caused by blocking, existing techniques for upper bounding regret fail. For proving our regret bounds, we introduce the novel concepts of delayed exploitation and opportunistic subsampling and combine them with ideas from combinatorial bandits and non-stationary Markov chains coupling.