Faster First-Order Algorithms for Monotone Strongly DR-Submodular Maximization
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Continuous DR-submodular functions are a class of generally non-convex/non-concave functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing work has studied monotone continuous DR-submodular maximization subject to a convex constraint and provided efficient algorithms with approximation guarantees. In many applications, such as computing the stability number of a graph, the monotone DR-submodular objective function has the additional property of being strongly concave along non-negative directions (i.e., strongly DR-submodular). In this paper, we consider a subclass of L-smooth monotone DR-submodular functions that are strongly DR-submodular and have a bounded curvature, and we show how to exploit such additional structure to obtain faster algorithms with stronger guarantees for the maximization problem. We propose a new algorithm that matches the provably optimal 1-c/e approximation ratio after only ⌈L/μ⌉ iterations, where c∈[0,1] and μ≥ 0 are the curvature and the strong DR-submodularity parameter. Furthermore, we study the Projected Gradient Ascent (PGA) method for this problem, and provide a refined analysis of the algorithm with an improved 1/1+c approximation ratio (compared to 1/2 in prior works) and a linear convergence rate. Experimental results illustrate and validate the efficiency and effectiveness of our proposed algorithms.
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