Contextual bandit has been widely used for sequential decision-making based on the current contextual information and historical feedback data. In modern applications, such context format can be rich and can often be formulated as a matrix. Moreover, while existing bandit algorithms mainly focused on reward-maximization, less attention has been paid to the statistical inference. To fill in these gaps, in this work we consider a matrix contextual bandit framework where the true model parameter is a low-rank matrix, and propose a fully online procedure to simultaneously make sequential decision-making and conduct statistical inference. The low-rank structure of the model parameter and the adaptivity nature of the data collection process makes this difficult: standard low-rank estimators are not fully online and are biased, while existing inference approaches in bandit algorithms fail to account for the low-rankness and are also biased. To address these, we introduce a new online doubly-debiasing inference procedure to simultaneously handle both sources of bias. In theory, we establish the asymptotic normality of the proposed online doubly-debiased estimator and prove the validity of the constructed confidence interval. 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.
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