Continuous Submodular Maximization: Beyond DR-Submodularity

In this paper, we propose the first continuous optimization algorithms that achieve a constant factor approximation guarantee for the problem of monotone continuous submodular maximization subject to a linear constraint. We first prove that a simple variant of the vanilla coordinate ascent, called Coordinate-Ascent+, achieves a (frace-12e-1-varepsilon)-approximation guarantee while performing O(n/varepsilon) iterations, where the computational complexity of each iteration is roughly O(n/√varepsilon+nlog n) (here, n denotes the dimension of the optimization problem). We then propose Coordinate-Ascent++, that achieves the tight (1-1/e-varepsilon)-approximation guarantee while performing the same number of iterations, but at a higher computational complexity of roughly O(n3/varepsilon2.5 + n3 log n / varepsilon2) per iteration. However, the computation of each round of Coordinate-Ascent++ can be easily parallelized so that the computational cost per machine scales as O(n/√varepsilon+nlog n).