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Minimax Semiparametric Learning With Approximate Sparsity

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Document pages: 51 pages

Abstract: This paper is about the ability and means to root-n consistently andefficiently estimate linear, mean square continuous functionals of a highdimensional, approximately sparse regression. Such objects include a widevariety of interesting parameters such as the covariance between two regressionresiduals, a coefficient of a partially linear model, an average derivative,and the average treatment effect. We give lower bounds on the convergence rateof estimators of such objects and find that these bounds are substantiallylarger than in a low dimensional, semiparametric setting. We also giveautomatic debiased machine learners that are $1 sqrt{n}$ consistent andasymptotically efficient under minimal conditions. These estimators use nocross-fitting or a special kind of cross-fitting to attain efficiency withfaster than $n^{-1 4}$ convergence of the regression. This rate condition issubstantially weaker than the product of convergence rates of two functionsbeing faster than $1 sqrt{n},$ as required for many other debiased machinelearners.

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