Valid Post-Contextual Bandit Inference

Abstract

We establish an asymptotic framework for the statistical analysis of the stochastic contextual multi-armed bandit problem (CMAB), which is widely employed in adaptively randomized experiments across various fields. While algorithms for maximizing rewards or, equivalently, minimizing regret have received considerable attention, our focus centers on statistical inference with adaptively collected data under the CMAB model. To this end we derive the limit experiment (in the Hajek-Le Cam sense). This limit experiment is highly nonstandard and, applying Girsanov's theorem, we obtain a structural representation in terms of stochastic differential equations. This structural representation, and a general weak convergence result we develop, allow us to obtain the asymptotic distribution of statistics for the CMAB problem. In particular, we obtain the asymptotic distributions for the classical t-test (non-Gaussian), Adaptively Weighted tests, and Inverse Propensity Weighted tests (non-Gaussian). We show that, when comparing both arms, validity of these tests requires the sampling scheme to be translation invariant in a way we make precise. We propose translation-invariant versions of Thompson, tempered greedy, and tempered Upper Confidence Bound sampling. Simulation results corroborate our asymptotic analysis.