Online Two-Stage Submodular Maximization

TL;DR

This paper introduces an online version of the two-stage submodular maximization problem, providing algorithms with sublinear regret bounds for important subclasses of submodular functions, and validating their effectiveness through experiments.

Contribution

It formulates the online 2SSM problem for weighted threshold functions and develops algorithms with provable regret bounds under matroid constraints.

Findings

Achieves sublinear regret bounds for online 2SSM.

Provides a state-of-the-art offline bound for 2SSM.

Empirically validates the online algorithm on real datasets.

Abstract

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.