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In this paper, we propose three online algorithms for submodularmaximisation. The first one, Mono-Frank-Wolfe, reduces the number ofper-function gradient evaluations from $T^{1/2}$ [Chen2018Online] and $T^{3/2}$[chen2018projection] to 1, and achieves a $(1-1/e)$-regret bound of$O(T^{4/5})$. The second one, Bandit-Frank-Wolfe, is the first bandit algorithmfor continuous DR-submodular maximization, which achieves a $(1-1/e)$-regretbound of $O(T^{8/9})$. Finally, we extend Bandit-Frank-Wolfe to a banditalgorithm for discrete submodular maximization, Responsive-Frank-Wolfe, whichattains a $(1-1/e)$-regret bound of $O(T^{8/9})$ in the responsive banditsetting.

Online Continuous Submodular Maximization: From Full-Information to Bandit Feedback