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# Submodular Bandit Problem Under Multiple Constraints

*Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI)*, PMLR 124:191-200, 2020.

#### Abstract

The linear submodular bandit problemwas proposedto simultaneously address diversified retrieval and online learning in a recommender system.If there is no uncertainty, this problem is equivalent toa submodular maximization problem under a cardinality constraint.However, in some situations, recommendation lists should satisfyadditional constraints such as budget constraints,other than a cardinality constraint.Thus, motivated by diversified retrieval considering budget constraints,we introduce a submodular bandit problem under the intersection of$l$ knapsacks and a $k$-system constraint.Here $k$-system constraints forma very general class of constraints includingcardinality constraints and the intersection of $k$ matroid constraints.To solve this problem, we propose a non-greedy algorithm that adaptively focuses ona standard or modified upper-confidence bound.We provide a high-probability upper bound of an approximation regret,where the approximation ratio matches that of a fast offline algorithm.Moreover, we perform experimentsunder various combinations of constraints using asynthetic and two real-world datasetsand demonstrate that our proposed method outperforms the existing baselines.