Off-policy estimation of linear functionals: Non-asymptotic theory for semi-parametric efficiency

TL;DR

This paper develops non-asymptotic bounds for estimating linear functionals in observational studies, highlighting the importance of weighted norm error minimization and proposing an optimal two-stage regression procedure.

Contribution

It introduces a non-asymptotic analysis framework for off-policy linear functional estimation and proposes an instance-dependent optimal two-stage method based on weighted norm regression.

Findings

Non-asymptotic upper bounds on mean-squared error are derived.

Optimal procedures depend on weighted norm error minimization.

The proposed method achieves finite-sample optimality via matching lower bounds.

Abstract

The problem of estimating a linear functional based on observational data is canonical in both the causal inference and bandit literatures. We analyze a broad class of two-stage procedures that first estimate the treatment effect function, and then use this quantity to estimate the linear functional. We prove non-asymptotic upper bounds on the mean-squared error of such procedures: these bounds reveal that in order to obtain non-asymptotically optimal procedures, the error in estimating the treatment effect should be minimized in a certain weighted $L^2$-norm. We analyze a two-stage procedure based on constrained regression in this weighted norm, and establish its instance-dependent optimality in finite samples via matching non-asymptotic local minimax lower bounds. These results show that the optimal non-asymptotic risk, in addition to depending on the asymptotically efficient variance, depends on the weighted norm distance between the true outcome function and its approximation by the richest function class supported by the sample size.