Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

TL;DR

This paper introduces the first polynomial-query algorithms achieving a 1/2-approximation for maximizing continuous non-monotone submodular functions, with improved efficiency for DR-submodular cases, applicable in machine learning and economics.

Contribution

It presents the first 1/2-approximation algorithms for continuous non-monotone submodular maximization, including a quasilinear time method for DR-submodular functions, advancing prior work.

Findings

Achieved a 1/2-approximation for continuous non-monotone submodular maximization.

Developed a quasilinear time algorithm for DR-submodular maximization.

Validated algorithms through experiments in machine learning applications.

Abstract

In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first $\frac{1}{2}$-approximation algorithm for continuous submodular function maximization; this approximation factor of $\frac{1}{2}$ is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular functions is also coordinate wise concave along all coordinates, we provide a different $\frac{1}{2}$-approximation algorithm that runs in quasilinear time. Both of these results improve upon prior work [Bian et al, 2017, Soma and Yoshida, 2017]. Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications.