  We consider a multiobjective multiarmed bandit problem with lexicographically
ordered objectives. In this problem, the goal of the learner is to select arms
that are lexicographic optimal as much as possible without knowing the arm
reward distributions beforehand. We capture this goal by defining a
multidimensional form of regret that measures the loss of the learner due to
not selecting lexicographic optimal arms, and then, consider two settings where
the learner has prior information on the expected arm rewards. In the first
setting, the learner only knows for each objective the lexicographic optimal
expected reward. In the second setting, it only knows for each objective
near-lexicographic optimal expected rewards. For both settings we prove that
the learner achieves expected regret uniformly bounded in time. The algorithm
we propose for the second setting also attains bounded regret for the
multiarmed bandit with satisficing objectives. In addition, we also consider
the harder prior-free case, and show that the learner can still achieve
sublinear in time gap-free regret. Finally, we experimentally evaluate
performance of the proposed algorithms in a variety of multiobjective learning
problems.
