The biobjective multiarmed bandit: learning approximate lexicographic optimal allocations

We consider a biobjective sequential decision-making problem where an allocation (arm) is called epsilon lexicographic optimal if its expected reward in the first objective is at most epsilon smaller than the highest expected reward, and its expected reward in the second objective is at least the expected reward of a lexicographic optimal arm. The goal of the learner is to select arms that are epsilon lexicographic optimal as much as possible without knowing the arm reward distributions beforehand. For this problem, we first show that the learner's goal is equivalent to minimizing the epsilon lexicographic regret, and then, propose a learning algorithm whose epsilon lexicographic gap-dependent regret is bounded and gap-independent regret is sublinear in the number of rounds with high probability. Then, we apply the proposed model and algorithm for dynamic rate and channel selection in a cognitive radio network with imperfect channel sensing. Our results show that the proposed algorithm is able to learn the approximate lexicographic optimal rate-channel pair that simultaneously minimizes the primary user interference and maximizes the secondary user throughput.