Regularized Online DR-Submodular Optimization

The utilization of online optimization techniques is prevalent in many fields of artificial intelligence, enabling systems to continuously learn and adjust to their surroundings. This paper outlines a regularized online optimization problem, where the regularizer is defined on the average of the actions taken. The objective is to maximize the sum of rewards and the regularizer value while adhering to resource constraints, where the reward function is assumed to be DR-submodular. Both concave and DR-submodular regularizers are analyzed. Concave functions are useful in describing the impartiality of decisions, while DR-submodular functions can be employed to represent the overall effect of decisions on all relevant parties. We have developed two algorithms for each of the concave and DR-submodular regularizers. These algorithms are easy to implement, efficient, and produce sublinear regret in both cases. The performance of the proposed algorithms and regularizers has been verified through numerical experiments in the context of online joke recommendation and internet advertising.