Online Non-Monotone DR-Submodular Maximization

In this paper, we study fundamental problems of maximizing DR-submodular continuous functions that have real-world applications in the domain of machine learning, economics, operations research and communication systems. It captures a subclass of non-convex optimization that provides both theoretical and practical guarantees. Here, we focus on minimizing regret for online arriving non-monotone DR-submodular functions over down-closed and general convex sets. First, we present an online algorithm that achieves a 1/eapproximation ratio with the regret of O(T3/4 ) for maximizing DR-submodular functions over any down-closed convex set. Note that, the approximation ratio of 1/e matches the bestknown guarantee for the offline version of the problem. Next, we give an online algorithm that achieves an approximation guarantee (depending on the search space) for the problem of maximizing non-monotone continuous DR-submodular functions over a general convex set (not necessarily down-closed). To best of our knowledge, no prior algorithm with approximation guarantee was known for non-monotone DR-submodular maximization in the online setting. Finally we run experiments to verify the performance of our algorithms on problems arising in machine learning domain with the real-world datasets.

Authors

• Nguyễn Kim Thắng
• Abhinav Srivastav

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