This paper is about the ability and means to root-n consistently and eﬃciently estimate linear, mean-square continuous functionals of a high dimensional, approximately sparse regression. Such objects include a wide variety of interesting parameters such as the covariance between two regression residuals, a coefficient of a partially linear model, an average derivative, and the average treatment effect. We give lower bounds on the convergence rate of estimators of such objects and ﬁnd that these bounds are substantially larger than in a low dimensional, semiparametric setting. We also give automatic debiased machine learners that are 1/√ n consistent and asymptotically efficient under minimal conditions. These estimators use no cross-ﬁtting or a special kind of cross-ﬁtting to attain efficiency with faster than n−1/4 convergence of the regression. This rate condition is substantially weaker than the product of convergence rates of two functions being faster than 1/√ n, as required for many other debiased machine learners.
Minimax semiparametric learning with approximate sparsity
29 July 2021
Working Paper (CWP32/21)