Maximizing Monotone DR-submodular Continuous Functions by Derivative-free Optimization
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In this paper, we study the problem of monotone (weakly) DR-submodular continuous maximization. While previous methods require the gradient information of the objective function, we propose a derivative-free algorithm LDGM for the first time. We define β and α to characterize how close a function is to continuous DR-submodulr and submodular, respectively. Under a convex polytope constraint, we prove that LDGM can achieve a (1-e^-β-ϵ)-approximation guarantee after O(1/ϵ) iterations, which is the same as the best previous gradient-based algorithm. Moreover, in some special cases, a variant of LDGM can achieve a ((α/2)(1-e^-α)-ϵ)-approximation guarantee for (weakly) submodular functions. We also compare LDGM with the gradient-based algorithm Frank-Wolfe under noise, and show that LDGM can be more robust. Empirical results on budget allocation verify the effectiveness of LDGM.
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