Fast First-Order Methods for Monotone Strongly DR-Submodular Maximization

Continuous DR-submodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing works have studied monotone continuous DR-submodular maximization subject to a convex constraint and have proposed efficient algorithms with approximation guarantees. However, in many applications, e.g., computing the stability number of a graph and mean-field inference for probabilistic log-submodular models, the DR-submodular function has the additional property of being \emph{strongly} concave along non-negative directions that could be utilized for obtaining faster convergence rates. In this paper, we first introduce and characterize the class of \emph{strongly DR-submodular} functions and show how such a property implies strong concavity along non-negative directions. Then, we study $L$-smooth monotone strongly DR-submodular functions that have bounded curvature, and we show how to exploit such additional structure to obtain algorithms with improved approximation guarantees and faster convergence rates for the maximization problem. In particular, we propose the SDRFW 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 and a linear convergence rate. Given that both algorithms require knowledge of the smoothness parameter $L$, we provide a \emph{novel} characterization of $L$ for DR-submodular functions showing that in many cases, computing $L$ could be formulated as a convex problem, i.e., a geometric program, that could be solved efficiently.