Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)
Dylan J. Foster, Akshay Krishnamurthy, Haipeng Luo
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<…, where the m⋆-th class contains the optimal policy, and we design an algorithm that achieves ˜Ol(T2/3d1/3m⋆) regret with no prior knowledge of the optimal dimension dm⋆. The algorithm also achieves regret ˜O(T3/4+√Tdm⋆), which is optimal for dm⋆≥√T. This is the first model selection result for contextual bandits with non-vacuous regret for all values of dm⋆, 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.