Logarithmic Regret for Unconstrained Submodular Maximization Stochastic Bandit

TL;DR

This paper introduces a new algorithm for online unconstrained submodular maximization with stochastic bandit feedback, achieving improved regret bounds and characterizing the problem's hardness transition.

Contribution

It proposes the DG-ETC algorithm, combining Double-Greedy with explore-then-commit, and provides new regret bounds along with a hardness measure for the problem.

Findings

Achieves $O(d ext{log}(dT))$ problem-dependent regret bound.

Achieves $O(dT^{2/3} ext{log}(dT)^{1/3})$ problem-free regret bound.

Introduces a hardness measure for the transition between regret regimes.

Abstract

We address the online unconstrained submodular maximization problem (Online USM), in a setting with stochastic bandit feedback. In this framework, a decision-maker receives noisy rewards from a non monotone submodular function taking values in a known bounded interval. This paper proposes Double-Greedy - Explore-then-Commit (DG-ETC), adapting the Double-Greedy approach from the offline and online full-information settings. DG-ETC satisfies a $O(d\log(dT))$ problem-dependent upper bound for the $1/2$-approximate pseudo-regret, as well as a $O(dT^{2/3}\log(dT)^{1/3})$ problem-free one at the same time, outperforming existing approaches. In particular, we introduce a problem-dependent notion of hardness characterizing the transition between logarithmic and polynomial regime for the upper bounds.