Thompson Sampling in Partially Observable Contextual Bandits

Abstract

Contextual bandits constitute a classical framework for decision-making under uncertainty. In this setting, the goal is to learn the arms of highest reward subject to contextual information, while the unknown reward parameters of each arm need to be learned by experimenting that specific arm. Accordingly, a fundamental problem is that of balancing exploration (i.e., pulling different arms to learn their parameters), versus exploitation (i.e., pulling the best arms to gain reward). To study this problem, the existing literature mostly considers perfectly observed contexts. However, the setting of partial context observations remains unexplored to date, despite being theoretically more general and practically more versatile. We study bandit policies for learning to select optimal arms based on the data of observations, which are noisy linear functions of the unobserved context vectors. Our theoretical analysis shows that the Thompson sampling policy successfully balances exploration and exploitation. Specifically, we establish the followings: (i) regret bounds that grow poly-logarithmically with time, (ii) square-root consistency of parameter estimation, and (iii) scaling of the regret with other quantities including dimensions and number of arms. Extensive numerical experiments with both real and synthetic data are presented as well, corroborating the efficacy of Thompson sampling. To establish the results, we introduce novel martingale techniques and concentration inequalities to address partially observed dependent random variables generated from unspecified distributions, and also leverage problem-dependent information to sharpen probabilistic bounds for time-varying suboptimality gaps. These techniques pave the road towards studying other decision-making problems with contextual information as well as partial observations.