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

Understand

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.