On Augmented Stochastic Submodular Optimization: Adaptivity, Multi-Rounds, Budgeted, and Robustness

In this work we consider the problem of Stochastic Submodular Maximization, in which we would like to maximize the value of a monotone and submodular objective function, subject to the fact that the values of this function depend on the realization of stochastic events. This problem has applications in several areas, and in particular it well models basic problems such as influence maximization and stochastic probing. In this work, we advocate the necessity to extend the study of this problem in order to include several different features such as a budget constraint on the number of observations, the chance of adaptively choosing what we observe or the presence of multiple rounds. We here speculate on the possible directions that this line of research can take. In particular, we will discuss about interesting open problems mainly in the settings of robust optimization and online learning.