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.

翻译：摘要：我们研究了在线环境下一般拟阵约束下的单调子模最大化问题。我们证明，一大类子模函数（即加权阈值势函数）的在线优化可简化为在线凸优化。这恰恰是因为该类函数存在凹松弛；因此，结合适当的舍入方案，在线凸优化策略可在此类组合环境中实现次线性遗憾。我们证明，该简化方法可推广至多种不同版本的在线学习问题，包括动态遗憾、赌博机以及乐观学习等场景。