Off-policy estimation with adaptively collected data: the power of online learning

We consider estimation of a linear functional of the treatment effect using adaptively collected data. This task finds a variety of applications including the off-policy evaluation (\textsf{OPE}) in contextual bandits, and estimation of the average treatment effect (\textsf{ATE}) in causal inference. While a certain class of augmented inverse propensity weighting (\textsf{AIPW}) estimators enjoys desirable asymptotic properties including the semi-parametric efficiency, much less is known about their non-asymptotic theory with adaptively collected data. To fill in the gap, we first establish generic upper bounds on the mean-squared error of the class of AIPW estimators that crucially depends on a sequentially weighted error between the treatment effect and its estimates. Motivated by this, we also propose a general reduction scheme that allows one to produce a sequence of estimates for the treatment effect via online learning to minimize the sequentially weighted estimation error. To illustrate this, we provide three concrete instantiations in (\romannumeral 1) the tabular case; (\romannumeral 2) the case of linear function approximation; and (\romannumeral 3) the case of general function approximation for the outcome model. We then provide a local minimax lower bound to show the instance-dependent optimality of the \textsf{AIPW} estimator using no-regret online learning algorithms.