Minimax Semiparametric Learning With Approximate Sparsity

There is a close correspondence between the minimax rate and the behavior of remainder terms in an asymptotic expansion of a doubly robust estimator around the average of the efficient influence function. A dominating remainder term is the product of the mean square norms of estimation errors for the regression and Riesz representer. Other remainder terms will be smaller order than this term. By virtue of the sum of the absolute values of the regression and Riesz representer coefficients being bounded, the estimation errors for both the regression and Riesz representer converge nearly at root-mean-square rate {ln(p)/n}1/4, as known for Lasso regression from Chatterjee and Jafarov (2015) and for the Riesz representer by Chernozhukov et al. (2018) and Chernozhukov, Newey, and Singh (2018). The minimax rate for the object of interest is ln(p)/n when max{ξ1,ξ2} 1/2, which is nearly the product of the two rates, i.e. the size of the dominating remainder.