  We investigate the online bandit learning of the monotone multi-linear
DR-submodular functions, designing the algorithm $\mathtt{BanditMLSM}$ that
attains $O(T^{2/3}\log T)$ 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 $O(T^{2/3}\log T)$ 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 $O(T^{4/5})$
$(1-1/e)$-regret upper bound. For the bandit sequential submodular
maximization, the existing work proves an $O(T^{2/3})$ regret with a suboptimal
$1/2$ approximation ratio (Niazadeh et al. 2021).
