Minimax Semiparametric Learning With Approximate Sparsity
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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^-ξ for C,ξ>0. We show that a necessary condition for efficient estimation is that the sparse approximation rate ξ_1 for the regression and the rate ξ_2 for the Riesz representer satisfy max{ξ_1 ,ξ_2}>1/2. This condition is stronger than the corresponding condition ξ_1+ξ_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 regression sparse approximation rate satisfies ξ_1>1/2.
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