Linear bandits with limited adaptivity and learning distributional optimal design

ABSTRACTMotivated by practical needs such as large-scale learning, we study the impact of adaptivity constraints to linear contextual bandits, a central problem in online learning and decision making. We consider two popular limited adaptivity models in literature: batch learning and rare policy switches. We show that, when the context vectors are adversarially chosen in d-dimensional linear contextual bandits, the learner needs O(d logd logT) policy switches to achieve the minimax-optimal regret, and this is optimal up to poly(logd, loglogT) factors; for stochastic context vectors, even in the more restricted batch learning model, only O(loglogT) batches are needed to achieve the optimal regret. Together with the known results in literature, our results present a complete picture about the adaptivity constraints in linear contextual bandits. Along the way, we propose the distributional optimal design, a natural extension of the optimal experiment design, and provide a both statistically and computationally efficient learning algorithm for the problem, which may be of independent interest.