Many objects of interest can be expressed as a linear, mean square continuous functional of a least squares projection (regression). Often the regression may be high dimensional, depending on many variables. This paper gives minimal conditions for root-n consistent and efficient estimation of such objects when the regression and the Riesz representer of the functional are approximately sparse and the sum of the absolute value of the coefficients is bounded. The approximately sparse functions we consider are those where an approximation by some t regressors has root mean square error less than or equal to Ct^{-\xi for C, \xi>0. We show that a necessary condition for efficient estimation is that the sparse approximation rate 1 for the regression and the rate 2 for the Riesz representer satisfy \max\{1 ,2\}>1/2. This condition is stronger than the corresponding condition 1+2>1/2 for Holder classes of functions}  . We also show that Lasso based, cross-fit, debiased machine learning estimators are asymptotically efficient under these conditions. In addition we show efficiency of an estimator without cross-fitting when the functional depends on the regressors and the regressionsparse approximation rate satisfies \xi_{1 1/ 2.
This paper is about the ability and means to root-n consistently and efficiently estimate linear, mean square continuous functionals of a high dimensional, for C, \xi>0. We show that a necessary condition for efficient estimation is that the sparse approximation rate 1 for the regression and the rate 2 for the Riesz representer satisfy \max\{1 ,2\}>1/2. This condition is stronger than the corresponding condition 1+2>1/2 for Holder classes of functions} 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/\sqrt{n 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 1/ \sqrt{n
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Many objects of interest can be expressed as a
This paper is about the ability and means to root-n consistently and efficiently estimate
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functional of a least squares projection (regression). Often the regression may be
functionals of a
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depending on many variables. This paper gives minimal conditions for root-n consistent and efficient estimation of such objects when the regression and the Riesz representer of the functional are approximately sparse and the sum of the absolute value of the coefficients is bounded. The approximately sparse functions we consider are those where an approximation by some t regressors has root mean square error less than or equal to Ct^{-\xi
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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
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show that Lasso based, cross-fit, debiased machine learning estimators are
give automatic debiased machine learners that are 1/\sqrt{n
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these conditions. In addition we show efficiency of an estimator without
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when the functional depends on the regressors and the regressionsparse approximation rate satisfies \xi_{1
or a special kind of cross-fitting to attain efficiency with faster than n^{-1/4
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