Online Weakly DR-Submodular Optimization with Stochastic Long-Term Constraints

The online optimization has been extensively studied under a variety of different settings. In this paper, we consider the online maximization problems with stochastic linear cumulative constraints, where the objective functions are the sum of $$\rho $$ -weakly DR-submodular functions and concave functions. Inspired by the penalty function strategy, we propose an algorithm of primal-dual type to solve this class of problems. Under mild conditions, we show that the algorithm achieves sublinear regret bounds and cumulative budget violation bounds with high probability.