Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)
Amin Karbasi, Hamed Hassani, Aryan Mokhtari, Zebang Shen
In this paper, we develop \scg~(\text{SCG}{++}), the first efficient variant of a conditional gradient method for maximizing a continuous submodular function subject to a convex constraint. Concretely, for a monotone and continuous DR-submodular function, \SCGPP achieves a tight [(1−1/e)\OPT−ϵ] solution while using O(1/ϵ2) stochastic gradients and O(1/ϵ) calls to the linear optimization oracle. The best previously known algorithms either achieve a suboptimal [(1/2)\OPT−ϵ] solution with O(1/ϵ2) stochastic gradients or the tight [(1−1/e)\OPT−ϵ] solution with suboptimal O(1/ϵ3) stochastic gradients. We further provide an information-theoretic lower bound to showcase the necessity of \OM(1/ϵ2) stochastic oracle queries in order to achieve [(1−1/e)\OPT−ϵ] for monotone and DR-submodular functions. This result shows that our proposed \SCGPP enjoys optimality in terms of both approximation guarantee, i.e., (1−1/e) approximation factor, and stochastic gradient evaluations, i.e., O(1/ϵ2) calls to the stochastic oracle. By using stochastic continuous optimization as an interface, we also show that it is possible to obtain the [(1−1/e)\OPT−ϵ] tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a general matroid constraint after at most O(n2/ϵ2) calls to the stochastic function value, where n is the number of elements in the ground set.