Logarithmic Regret for Unconstrained Submodular Maximization Stochastic Bandit

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 $O(d{T}^{2/3}\log (dT{)}^{1/3})$ 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.

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