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：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，一类广泛的子模函数（即加权阈值势函数）的在线优化可以归约为在线凸优化（OCO）。根本原因在于此类函数存在凹松弛形式；因此，结合适当的舍入策略，OCO方法能够在组合优化场景中实现次线性遗憾。我们进一步证明，该归约方法可推广至在线学习问题的多种变体，包括动态遗憾、多臂老虎机以及乐观学习等设定。