Maximizing Monotone DR-submodular Continuous Functions by Derivative-free Optimization

TL;DR

This paper introduces a derivative-free algorithm, LDGM, for maximizing monotone DR-submodular continuous functions, achieving comparable approximation guarantees to gradient-based methods and demonstrating robustness and effectiveness in empirical tests.

Contribution

The paper presents the first derivative-free algorithm for monotone DR-submodular maximization with proven approximation guarantees under convex constraints.

Findings

LDGM achieves a (1-e^{-eta}-ε)-approximation after O(1/ε) iterations.

A variant of LDGM attains a ((α/2)(1-e^{-α})-ε)-approximation for submodular functions.

LDGM is more robust than gradient-based algorithms like Frank-Wolfe under noise.

Abstract

In this paper, we study the problem of monotone (weakly) DR-submodular continuous maximization. While previous methods require the gradient information of the objective function, we propose a derivative-free algorithm LDGM for the first time. We define $\beta$ and $\alpha$ to characterize how close a function is to continuous DR-submodulr and submodular, respectively. Under a convex polytope constraint, we prove that LDGM can achieve a $(1-e^{-\beta}-\epsilon)$-approximation guarantee after $O(1/\epsilon)$ iterations, which is the same as the best previous gradient-based algorithm. Moreover, in some special cases, a variant of LDGM can achieve a $((\alpha/2)(1-e^{-\alpha})-\epsilon)$-approximation guarantee for (weakly) submodular functions. We also compare LDGM with the gradient-based algorithm Frank-Wolfe under noise, and show that LDGM can be more robust. Empirical results on budget allocation verify the effectiveness of LDGM.