Abstract: 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 $M$-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 $\big(1 - \frac{1}{e}\big)$-regret bounds for bandit feedback in the nonstochastic setting of the order of $\sqrt{MKT}$ (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 $\widetilde{\mathcal{O}}\big(T^{2/3}\big)$ regret bound for online monotone submodular maximization with bandit feedback. We also extend our results to a bandit version of the facility location problem.