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Stochastic Conditional Gradient++

In this paper, we consider the general non-oblivious 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++ ($\text{SFW}{++} $), an efficient variant of the conditional gradient method for minimizing a smooth non-convex function subject to a convex body constraint. We show that $\text{SFW}{++} $ converges to an $ε$-first order stationary point by using $O(1/ε^3)$ stochastic gradients. Once further structures are present, $\text{SFW}{++}$'s theoretical guarantees, in terms of the convergence rate and quality of its solution, improve. In particular, for minimizing a convex function, $\text{SFW}{++} $ achieves an $ε$-approximate optimum while using $O(1/ε^2)$ 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 $\text{SFW}{++} $, called Stochastic Continuous Greedy++ ($\text{SCG}{++} $), achieves a tight $[(1-1/e)\text{OPT} -ε]$ solution while using $O(1/ε^2)$ stochastic gradients. Through an information theoretic argument, we also prove that $\text{SCG}{++} $'s convergence rate is optimal. Finally, for maximizing a non-monotone continuous DR-submodular function, we can achieve a $[(1/e)\text{OPT} -ε]$ solution by using $O(1/ε^2)$ 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-)covex and continuous submodular settings.