Online Submodular Maximization via Online Convex Optimization
This provides a theoretical framework for online combinatorial optimization, applicable to dynamic, bandit, and optimistic settings, but is incremental as it builds on existing OCO methods.
The paper tackles online monotone submodular maximization under matroid constraints by reducing it to online convex optimization for weighted threshold potential functions, achieving sublinear regret.
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.