Stochastic Conditional Gradient++: (Non)Convex Minimization and Continuous Submodular Maximization

Abstract

In this paper, we consider the general nonoblivious stochastic optimization where the underlying stochasticity may change during the optimization procedure and depends on the point at which the function is evaluated. We develop Stochastic Frank--Wolfe++ (SFW++), an efficient variant of the conditional gradient method for minimizing a smooth nonconvex function subject to a convex body constraint. We show that SFW++ converges to an $\epsilon$-first order stationary point by using $O(1/\epsilon^3)$ stochastic gradients. Once further structures are present, SFW++'s theoretical guarantees, in terms of the convergence rate and quality of its solution, improve. In particular, for minimizing a convex function, SFW++ achieves an $\epsilon$-approximate optimum while using $O(1/\epsilon^2)$ stochastic gradients. It is known that this rate is optimal in terms of stochastic gradient evaluations. Similarly, for maximizing a monotone continuous DR-submodular function, a slightly different form of SFW++, called Stochastic Continuous Greedy++ (SCG++), achieves a tight $[(1-1/e){\text{OPT}} -\epsilon]$ solution while using $O(1/\epsilon^2)$ stochastic gradients. Through an information theoretic argument, we also prove that SCG++'s convergence rate is optimal. Finally, for maximizing a nonmonotone continuous DR-submodular function, we can achieve a $[(1/e){\text{OPT}} -\epsilon]$ solution by using $O(1/\epsilon^2)$ stochastic gradients. We should highlight that our results and our novel variance reduction technique trivially extend to the standard and easier oblivious stochastic optimization settings for (non)convex and continuous submodular settings.