Online Two-Stage Submodular Maximization

Given a collection of monotone submodular functions, the goal of Two-Stage Submodular Maximization (2SSM) (Balkanski et al. 2016) is to restrict the ground set so an objective selected u.a.r. from the collection attains a high maximal value, on average, when optimized over the restricted ground set. We introduce the Online Two-Stage Submodular Maximization (O2SSM) problem, in which the submodular objectives are revealed in an online fashion. We study this problem for weighted threshold potential functions, a large and important subclass of monotone submodular functions that includes influence maximization, data summarization, and facility location, to name a few. We design an algorithm that achieves sublinear $(1 - 1/e)^2$-regret under general matroid constraints and $(1 - 1/e)(1-e^{-k}k^k/k!)$-regret in the case of uniform matroids of rank $k$; the latter also yields a state-of-the-art bound for the (offline) 2SSM problem. We empirically validate the performance of our online algorithm with experiments on real datasets.