Stochastic Conditional Gradient Methods: From Convex Minimization to Submodular Maximization
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This paper considers stochastic optimization problems for a large class of objective functions, including convex and continuous submodular. Stochastic proximal gradient methods have been widely used to solve such problems; however, their applicability remains limited when the problem dimension is large and the projection onto a convex set is costly. Instead, stochastic conditional gradient methods are proposed as an alternative solution relying on (i) Approximating gradients via a simple averaging technique requiring a single stochastic gradient evaluation per iteration; (ii) Solving a linear program to compute the descent/ascent direction. The averaging technique reduces the noise of gradient approximations as time progresses, and replacing projection step in proximal methods by a linear program lowers the computational complexity of each iteration. We show that under convexity and smoothness assumptions, our proposed method converges to the optimal objective function value at a sublinear rate of O(1/t^1/3). Further, for a monotone and continuous DR-submodular function and subject to a general convex body constraint, we prove that our proposed method achieves a ((1-1/e)OPT-) guarantee with O(1/^3) stochastic gradient computations. This guarantee matches the known hardness results and closes the gap between deterministic and stochastic continuous submodular maximization. Additionally, we obtain ((1/e)OPT -) guarantee after using O(1/^3) stochastic gradients for the case that the objective function is continuous DR-submodular but non-monotone and the constraint set is down-closed. By using stochastic continuous optimization as an interface, we provide the first (1-1/e) tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a matroid constraint and (1/e) approximation guarantee for the non-monotone case.
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