Conditional Gradient Method for Stochastic Submodular Maximization: Closing the Gap

In this paper, we study the problem of constrained and stochastic continuous submodular maximization. Even though the objective function is not concave (nor convex) and is defined in terms of an expectation, we develop a variant of the conditional gradient method, called Stochastic Continuous Greedy, which achieves a tight approximation guarantee. More precisely, for a monotone and continuous DR-submodular function and subject to a general convex body constraint, we prove that Stochastic Continuous Greedy achieves a [(1 − 1/e)OPT − ] guarantee (in expectation) with O(1/ ) stochastic gradient computations. This guarantee matches the known hardness results and closes the gap between deterministic and stochastic continuous submodular maximization. By using stochastic continuous optimization as an interface, we also provide the first (1−1/e) tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a general matroid constraint.