Online Learning for Non-monotone Submodular Maximization: From Full Information to Bandit Feedback
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In this paper, we revisit the online non-monotone continuous DR-submodular maximization problem over a down-closed convex set, which finds wide real-world applications in the domain of machine learning, economics, and operations research. At first, we present the Meta-MFW algorithm achieving a 1/e-regret of O(√(T)) at the cost of T^3/2 stochastic gradient evaluations per round. As far as we know, Meta-MFW is the first algorithm to obtain 1/e-regret of O(√(T)) for the online non-monotone continuous DR-submodular maximization problem over a down-closed convex set. Furthermore, in sharp contrast with ODC algorithm <cit.>, Meta-MFW relies on the simple online linear oracle without discretization, lifting, or rounding operations. Considering the practical restrictions, we then propose the Mono-MFW algorithm, which reduces the per-function stochastic gradient evaluations from T^3/2 to 1 and achieves a 1/e-regret bound of O(T^4/5). Next, we extend Mono-MFW to the bandit setting and propose the Bandit-MFW algorithm which attains a 1/e-regret bound of O(T^8/9). To the best of our knowledge, Mono-MFW and Bandit-MFW are the first sublinear-regret algorithms to explore the one-shot and bandit setting for online non-monotone continuous DR-submodular maximization problem over a down-closed convex set, respectively. Finally, we conduct numerical experiments on both synthetic and real-world datasets to verify the effectiveness of our methods.
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