  The linear submodular bandit problem was proposed to simultaneously address
diversified retrieval and online learning in a recommender system. If there is
no uncertainty, this problem is equivalent to a submodular maximization problem
under a cardinality constraint. However, in some situations, recommendation
lists should satisfy additional 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 form a very general class of constraints including cardinality
constraints and the intersection of $k$ matroid constraints. To solve this
problem, we propose a non-greedy algorithm that adaptively focuses on a
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 experiments under
various combinations of constraints using a synthetic and two real-world
datasets and demonstrate that our proposed methods outperform the existing
baselines.
