Model selection for contextual bandits
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We introduce the problem of model selection for contextual bandits, wherein 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 d_1 < d_2 < ..., where the m^-th class contains the optimal policy, and we design an algorithm that achieves Õ(T^2/3d^1/3_m^) regret with no prior knowledge of the optimal dimension d_m^. The algorithm also achieves regret Õ(T^3/4 + √(Td_m^)), which is optimal for d_m^≥√(T). This is the first contextual bandit model selection result with non-vacuous regret for all values of d_m^ and, to the best of our knowledge, is the first guarantee of its type in any contextual bandit setting. The core of the algorithm is a new estimator for the gap in best loss achievable by two linear policy classes, which we show admits a convergence rate faster than what is required to learn either class.
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