Multi-objective Contextual Multi-armed Bandit Problem with a Dominant Objective

In this paper, we propose a new multi-objective contextual multi-armed bandit (MAB) problem with two objectives, where one of the objectives dominates the other objective. Unlike single-objective MAB problems in which the learner obtains a random scalar reward for each arm it selects, in the proposed problem, the learner obtains a random reward vector, where each component of the reward vector corresponds to one of the objectives and the distribution of the reward depends on the context that is provided to the learner at the beginning of each round. We call this problem contextual multi-armed bandit with a dominant objective (CMAB-DO). In CMAB-DO, the goal of the learner is to maximize its total reward in the non-dominant objective while ensuring that it maximizes its total reward in the dominant objective. In this case, the optimal arm given a context is the one that maximizes the expected reward in the non-dominant objective among all arms that maximize the expected reward in the dominant objective. First, we show that the optimal arm lies in the Pareto front. Then, we propose the multi-objective contextual multi-armed bandit algorithm (MOCMAB), and define two performance measures: the 2-dimensional (2D) regret and the Pareto regret. We show that both the 2D regret and the Pareto regret of MOC-MAB are sublinear in the number of rounds. We also compare the performance of the proposed algorithm with other state-of-the-art methods in synthetic and real-world datasets. The proposed model and the algorithm have a wide range of real-world applications that involve multiple and possibly conflicting objectives ranging from wireless communication to medical diagnosis and recommender systems.