Sum-max Submodular Bandits

Many online decision-making problems correspond to maximizing a sequence of submodular functions. In this work, we introduce sum-max functions, a subclass of monotone submodular functions capturing several interesting problems, including best-of-K-bandits, combinatorial bandits, and the bandit versions on facility location, M-medians, and hitting sets. We show that all functions in this class satisfy a key property that we call pseudo-concavity. This allows us to prove big(1 - frac1ebig)-regret bounds for bandit feedback in the nonstochastic setting of the order of √MKT (ignoring log factors), where T is the time horizon and M is a cardinality constraint. This bound, attained by a simple and efficient algorithm, significantly improves on the widetildeObig(T2/3big) regret bound for online monotone submodular maximization with bandit feedback.