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arXiv · 1810.06833

Maximizing Monotone DR-submodular Continuous Functions by Derivative-free Optimization

Yibo Zhang·Chao Qian·Ke Tang
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Abstract

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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Yibo Zhang, Chao Qian, Ke Tang. 2019-02-23. Maximizing Monotone DR-submodular Continuous Functions by Derivative-free Optimization. https://arxiv.org/abs/1810.06833

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