Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

In this paper we study the fundamental problems of maximizing a continuous non monotone submodular function over a hypercube, with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first 1/2 approximation algorithm for continuous submodular function maximization; this approximation factor of is the best possible for algorithms that use only polynomially many queries. For the special case of DR-submodular maximization, we provide a faster 1/2-approximation algorithm that runs in (almost) linear time. Both of these results improve upon prior work [Bian et al., 2017, Soma and Yoshida, 2017, Buchbinder et al., 2012].