Title: Continuous DR-submodular Maximization: Structure and Algorithms,
Abstract: DR-submodular continuous functions are important objectives with wide
real-world applications spanning MAP inference in determinantal point processes
(DPPs), and mean-field inference for probabilistic submodular models, amongst
others. DR-submodularity captures a subclass of non-convex functions that
enables both exact minimization and approximate maximization in polynomial
time.
In this work we study the problem of maximizing non-monotone DR-submodular
continuous functions under general down-closed convex constraints. We start by
investigating geometric properties that underlie such objectives, e.g., a
strong relation between (approximately) stationary points and global optimum is
proved. These properties are then used to devise two optimization algorithms
with provable guarantees. Concretely, we first devise a "two-phase" algorithm
with $1/4$ approximation guarantee. This algorithm allows the use of existing
methods for finding (approximately) stationary points as a subroutine, thus,
harnessing recent progress in non-convex optimization. Then we present a
non-monotone Frank-Wolfe variant with $1/e$ approximation guarantee and
sublinear convergence rate. Finally, we extend our approach to a broader class
of generalized DR-submodular continuous functions, which captures a wider
spectrum of applications. Our theoretical findings are validated on synthetic
and real-world problem instances.