  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.
