Bandit Multi-linear DR-Submodular Maximization and Its Applications on Adversarial Submodular Bandits

We investigate the online bandit learning of the monotone multi-linear DR-submodular functions, designing the algorithm mathttBanditMLSM that attains O(T2/3log T) of (1-1/e)-regret. Then we reduce submodular bandit with partition matroid constraint and bandit sequential monotone maximization to the online bandit learning of the monotone multi-linear DR-submodular functions, attaining O(T2/3log T) of (1-1/e)-regret in both problems, which improve the existing results. To the best of our knowledge, we are the first to give a sublinear regret algorithm for the submodular bandit with partition matroid constraint. A special case of this problem is studied by Streeter et al.(2009). They prove a O(T4/5) (1-1/e)-regret upper bound. For the bandit sequential submodular maximization, the existing work proves an O(T2/3) regret with a suboptimal 1/2 approximation ratio (Niazadeh et al. 2021).