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With the fast development of big data, it has been easier than before tolearn the optimal decision rule by updating the decision rule recursively andmaking online decisions. We study the online statistical inference of modelparameters in a contextual bandit framework of sequential decision-making. Wepropose a general framework for online and adaptive data collection environmentthat can update decision rules via weighted stochastic gradient descent. Weallow different weighting schemes of the stochastic gradient and establish theasymptotic normality of the parameter estimator. Our proposed estimatorsignificantly improves the asymptotic efficiency over the previous averaged SGDapproach via inverse probability weights. We also conduct an optimalityanalysis on the weights in a linear regression setting. We provide a Bahadurrepresentation of the proposed estimator and show that the remainder term inthe Bahadur representation entails a slower convergence rate compared toclassical SGD due to the adaptive data collection.

Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent