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The linear submodular bandit problem was proposed to simultaneously addressdiversified retrieval and online learning in a recommender system. If there isno uncertainty, this problem is equivalent to a submodular maximization problemunder a cardinality constraint. However, in some situations, recommendationlists should satisfy additional constraints such as budget constraints, otherthan a cardinality constraint. Thus, motivated by diversified retrievalconsidering budget constraints, we introduce a submodular bandit problem underthe intersection of $l$ knapsacks and a $k$-system constraint. Here $k$-systemconstraints form a very general class of constraints including cardinalityconstraints and the intersection of $k$ matroid constraints. To solve thisproblem, we propose a non-greedy algorithm that adaptively focuses on astandard or modified upper-confidence bound. We provide a high-probabilityupper bound of an approximation regret, where the approximation ratio matchesthat of a fast offline algorithm. Moreover, we perform experiments undervarious combinations of constraints using a synthetic and two real-worlddatasets and demonstrate that our proposed methods outperform the existingbaselines.

Submodular Bandit Problem Under Multiple Constraints