Bandit Multi-linear DR-Submodular Maximization and Its Applications on Adversarial Submodular Bandits

TL;DR

This paper introduces a new online bandit learning algorithm for monotone multi-linear DR-submodular functions, achieving improved regret bounds and addressing submodular bandit problems with partition matroid constraints.

Contribution

The authors develop the first sublinear regret algorithm for submodular bandits with partition matroid constraints, improving upon previous regret bounds for related problems.

Findings

Achieves $O(T^{2/3}\log T)$ regret for monotone multi-linear DR-submodular functions.

Reduces submodular bandit with partition matroid constraint to online learning, attaining similar regret bounds.

First to provide a sublinear regret algorithm for submodular bandit with partition matroid constraints.

Abstract

We investigate the online bandit learning of the monotone multi-linear DR-submodular functions, designing the algorithm $\mathtt{BanditMLSM}$ that attains $O(T^{2/3}\log T)$ of $(1-1/e)$-regret. Then we reduce submodular bandit with partition matroid constraint and bandit sequential monotone maximization to the online bandit learning of the monotone multi-linear DR-submodular functions, attaining $O(T^{2/3}\log T)$ of $(1-1/e)$-regret in both problems, which improve the existing results. To the best of our knowledge, we are the first to give a sublinear regret algorithm for the submodular bandit with partition matroid constraint. A special case of this problem is studied by Streeter et al.(2009). They prove a $O(T^{4/5})$ $(1-1/e)$-regret upper bound. For the bandit sequential submodular maximization, the existing work proves an $O(T^{2/3})$ regret with a suboptimal $1/2$ approximation ratio (Niazadeh et al. 2021).