Minimax Semiparametric Learning With Approximate Sparsity

This paper is about the feasibility and means of root-n consistently estimating linear, mean-square continuous functionals of a high dimensional, approximately sparse regression. Such objects include a wide variety of interesting parameters such as regression coefficients, average derivatives, and the average treatment effect. We give lower bounds on the convergence rate of estimators of a regression slope and an average derivative and find that these bounds are substantially larger than in a low dimensional, semiparametric setting. We also give debiased machine learners that are root-n consistent under either a minimal approximate sparsity condition or rate double robustness. These estimators improve on existing estimators in being root-n consistent under more general conditions that previously known.