Title: Continuous dr-submodular maximization: Structure and algorithms

Authors: A. Bian, K. Levy, A. Krause, and J. M. Buhmann

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 continuous DR- submodular 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 approxi- mation 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 continu- ous functions, which captures a wider spectrum of applications. Our theoretical findings are validated on synthetic and real-world problem instances.

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