Online Continuous Submodular Maximization: From Full-Information to Bandit Feedback

TL;DR

This paper introduces three novel online algorithms for continuous and discrete submodular maximization, reducing gradient evaluations and achieving new regret bounds in full-information and bandit feedback settings.

Contribution

The paper presents the first bandit algorithm for continuous DR-submodular maximization and extends it to discrete cases, improving efficiency and regret bounds.

Findings

Mono-Frank-Wolfe reduces gradient evaluations to 1 per function.

Bandit-Frank-Wolfe is the first bandit algorithm for continuous DR-submodular maximization.

Responsive-Frank-Wolfe achieves regret bounds in the discrete bandit setting.

Abstract

In this paper, we propose three online algorithms for submodular maximisation. The first one, Mono-Frank-Wolfe, reduces the number of per-function gradient evaluations from $T^{1/2}$ [Chen2018Online] and $T^{3/2}$ [chen2018projection] to 1, and achieves a $(1-1/e)$-regret bound of $O(T^{4/5})$. The second one, Bandit-Frank-Wolfe, is the first bandit algorithm for continuous DR-submodular maximization, which achieves a $(1-1/e)$-regret bound of $O(T^{8/9})$. Finally, we extend Bandit-Frank-Wolfe to a bandit algorithm for discrete submodular maximization, Responsive-Frank-Wolfe, which attains a $(1-1/e)$-regret bound of $O(T^{8/9})$ in the responsive bandit setting.