Online Submodular Maximization via Online Convex Optimization

TL;DR

This paper introduces a method to handle online monotone submodular maximization problems under matroid constraints by reducing them to online convex optimization, enabling effective regret minimization across various settings.

Contribution

It establishes a reduction from online submodular maximization to online convex optimization for a broad class of functions, facilitating new algorithms with sublinear regret.

Findings

Reduction applies to weighted threshold potential functions

Extends to dynamic regret, bandit, and optimistic-learning settings

Achieves sublinear regret in online combinatorial optimization

Abstract

We study monotone submodular maximization under general matroid constraints in the online setting. We prove that online optimization of a large class of submodular functions, namely, weighted threshold potential functions, reduces to online convex optimization (OCO). This is precisely because functions in this class admit a concave relaxation; as a result, OCO policies, coupled with an appropriate rounding scheme, can be used to achieve sublinear regret in the combinatorial setting. We show that our reduction extends to many different versions of the online learning problem, including the dynamic regret, bandit, and optimistic-learning settings.