Faster First-Order Algorithms for Monotone Strongly DR-Submodular Maximization

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 $1-\frac{c}{e}$ approximation ratio after only $\lceil\frac{L}{\mu}\rceil$ iterations, where $c\in[0,1]$ and $\mu\geq 0$ 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 $\frac{1}{1+c}$ approximation ratio (compared to $\frac{1}{2}$ in prior works) and a linear convergence rate. Experimental results illustrate and validate the efficiency and effectiveness of our proposed algorithms.