Best of Both Worlds Model Selection

We study the problem of model selection in bandit scenarios in the presence of nested policy classes, with the goal of obtaining simultaneous adversarial and stochastic ("best of both worlds") high-probability regret guarantees. Our approach requires that each base learner comes with a candidate regret bound that may or may not hold, while our meta algorithm plays each base learner according to a schedule that keeps the base learner's candidate regret bounds balanced until they are detected to violate their guarantees. We develop careful mis-specification tests specifically designed to blend the above model selection criterion with the ability to leverage the (potentially benign) nature of the environment. We recover the model selection guarantees of the CORRAL algorithm for adversarial environments, but with the additional benefit of achieving high probability regret bounds, specifically in the case of nested adversarial linear bandits. More importantly, our model selection results also hold simultaneously in stochastic environments under gap assumptions. These are the first theoretical results that achieve best of both world (stochastic and adversarial) guarantees while performing model selection in (linear) bandit scenarios.