We provide  adaptive inference methods, based on l1 regularizationmethods , for regular (semi-parametric) and non-regular (nonparametric) linear functionals of the conditional expectation function. Examples of regular functionals include average treatment effects, policy effects from covariate distribution shifts and stochastic transformations, and average derivatives. Examples of non-regular functionals include the local linear functionals defined as local averages that approximate perfectly localized quantities: average treatment , average policy effects, and average derivatives , conditional on a covariate subvector fixed at a point. Our construction relies on building Neyman orthogonal equations for the target parameter that are approximately invariant to small perturbations of the nuisance parameters. To achieve this property  we include the linear Riesz representer for the functionals in the equations as the additional nuisance parameter. We use l1-regularized methods to learn approximations to the regression function and the linear representer, in settings where dimension of (possibly overcomplete) dictionary of basis functions P is much larger than N. We then estimate the linear functional by the solution to the empirical analog of the orthogonal equations. Our key result is that under weak assumptions the estimator of the functional concentrates in a L/ root(n) neighborhood of the target with deviations controlled by the Gaussian law , provided L/root(n)%DIFDELCMD < \to %%% 0; L is the operator norm  of the functional , measuring the degree of its non-regularity, with L diverging for local functionals (or under weak identification of the global functionals) .
We provide novel adaptive inference methods, based on \ell_1 regularization , for regular (semi-parametric) and non-regular (nonparametric) linear functionals of the conditional expectation function. Examples of regular functionals include average treatment effects, policy effects , and derivatives. Examples of non-regular functionals include localized quantities: average treatment effects, policy effects, and derivatives conditional on a covariate subvector fixed at a point. We construct a Neyman orthogonal equation for the target parameter that is approximately invariant to small perturbations of the nuisance parameters. To achieve this property , we include the linear Riesz representer for the functional as an additional nuisance parameter. Under weak assumptions, the estimator of the functional concentrates in a L/ \sqrt{n neighborhood of the target with deviations controlled by the Gaussian law %DIFDELCMD < \to %%% . We require that the operator norm L of the functional is small compared to \sqrt{n .
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Our construction relies on building Neyman orthogonal equations
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We use l1-regularized methods to learn approximations to the regression function and the linear representer, in settings where dimension of (possibly overcomplete) dictionary of basis functions P is much larger than N. We then estimate the linear functional by the solution to the empirical analog of the orthogonal equations. Our key result is that under weak assumptions
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