Guaranteed Non-convex Optimization: Submodular Maximization over Continuous Domains

TL;DR

This paper introduces new theoretical frameworks and algorithms for efficiently maximizing submodular continuous functions with approximation guarantees, applicable to diverse real-world problems.

Contribution

It characterizes submodular continuous functions via the weak DR property and proposes novel algorithms with provable guarantees for both monotone and non-monotone cases.

Findings

The proposed algorithms outperform baselines in experiments.

The weak DR property unifies submodularity across different domains.

Algorithms achieve near-optimal approximation guarantees.

Abstract

Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation guarantees. Specifically, i) We introduce the weak DR property that gives a unified characterization of submodularity for all set, integer-lattice and continuous functions; ii) for maximizing monotone DR-submodular continuous functions under general down-closed convex constraints, we propose a Frank-Wolfe variant with $(1-1/e)$ approximation guarantee, and sub-linear convergence rate; iii) for maximizing general non-monotone submodular continuous functions subject to box constraints, we propose a DoubleGreedy algorithm with $1/3$ approximation guarantee. Submodular continuous functions naturally find applications in various real-world settings, including influence and revenue maximization with continuous assignments, sensor energy management, multi-resolution data summarization, facility location, etc. Experimental results show that the proposed algorithms efficiently generate superior solutions compared to baseline algorithms.