ONLINE STATISTICAL INFERENCE FOR LOW-RANK REINFORCEMENT LEARNING

We propose a fully online procedure to conduct statistical inference with adaptively collected data. The low-rank structure of the model parameter and the adaptivity nature of the data collection process make this task challenging: standard low-rank estimators are biased and cannot be obtained in a sequential manner while existing inference approaches in sequential decision-making algorithms fail to account for the low-rankness and are also biased. To tackle the challenges previously outlined, we first develop an online low-rank estimation process employing Stochastic Gradient Descent with noisy observations. Subsequently, to facilitate statistical inference using the online low-rank estimator, we introduced a novel online debiasing technique designed to address both sources of bias simultaneously. This method yields an unbiased estimator suitable for parameter inference. Finally, we developed an inferential framework capable of establishing an online estimator for performing inference on the optimal policy value. In theory, we establish the asymptotic normality of the proposed online debiased estimators and prove the validity of the constructed confidence intervals for both inference tasks. Our inference results are built upon a newly developed low-rank stochastic gradient descent estimator and its non-asymptotic convergence result, which is also of independent interest.