Model selection for contextual bandits

We introduce the problem of model selection for contextual bandits, where a learner must adapt to the complexity of the optimal policy while balancing exploration and exploitation. Our main result is a new model selection guarantee for linear contextual bandits. We work in the stochastic realizable setting with a sequence of nested linear policy classes of dimension d1 < d2 < ldots, where the m^star-th class contains the optimal policy, and we design an algorithm that achieves tildeO(T2/3d1/3m^star) regret with no prior knowledge of the optimal dimension dm^star. The algorithm also achieves regret tildeO(T3/4 + √Tdm^star), which is optimal for d_mstar≥√T. This is the first model selection result for contextual bandits with non-vacuous regret for all values of dm^star, and to the best of our knowledge is the first positive result of this type for any online learning setting with partial information. The core of the algorithm is a new estimator for the gap in the best loss achievable by two linear policy classes, which we show admits a convergence rate faster than the rate required to learn the parameters for either class.