Recurrent Submodular Welfare and Matroid Blocking Bandits

Abstract: A recent line of research focuses on the study of the stochastic multi-armed bandits problem (MAB), in the case where temporal correlations of specific structure are imposed between the player's actions and the reward distributions of the arms (Kleinberg and Immorlica [FOCS18], Basu et al. [NeurIPS19]). As opposed to the standard MAB setting, where the optimal solution in hindsight can be trivially characterized, these correlations lead to (sub-)optimal solutions that exhibit interesting dynamical patterns -- a phenomenon that yields new challenges both from an algorithmic as well as a learning perspective. In this work, we extend the above direction to a combinatorial bandit setting and study a variant of stochastic MAB, where arms are subject to matroid constraints and each arm becomes unavailable (blocked) for a fixed number of rounds after each play. A natural common generalization of the state-of-the-art for blocking bandits, and that for matroid bandits, yields a $(1-\frac{1}{e})$-approximation for partition matroids, yet it only guarantees a $\frac{1}{2}$-approximation for general matroids. In this paper we develop new algorithmic ideas that allow us to obtain a polynomial-time $(1 - \frac{1}{e})$-approximation algorithm (asymptotically and in expectation) for any matroid, and thus to control the $(1-\frac{1}{e})$-approximate regret. A key ingredient is the technique of correlated (interleaved) scheduling. Along the way, we discover an interesting connection to a variant of Submodular Welfare Maximization, for which we provide (asymptotically) matching upper and lower approximability bounds.