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 $d_1 d_2 \ldots$, where the $m \star$-th class contains the optimal policy, and we design an algorithm that achieves $\tilde{O}l(T {2/3}d {1/3}_{m \star})$ regret with no prior knowledge of the optimal dimension $d_{m \star}$. This is the first model selection result for contextual bandits with non-vacuous regret for all values of $d_{m \star}$, 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.