  We consider inference about a scalar parameter under a non-parametric model
based on a one-step estimator computed as a plug in estimator plus the
empirical mean of an estimator of the parameter's influence function. We focus
on a class of parameters that have influence function which depends on two
infinite dimensional nuisance functions and such that the bias of the one-step
estimator of the parameter of interest is the expectation of the product of the
estimation errors of the two nuisance functions. Our class includes many
important treatment effect contrasts of interest in causal inference and
econometrics, such as ATE, ATT, an integrated causal contrast with a continuous
treatment, and the mean of an outcome missing not at random. We propose
estimators of the target parameter that entertain approximately sparse
regression models for the nuisance functions allowing for the number of
potential confounders to be even larger than the sample size. By employing
sample splitting, cross-fitting and $\ell_1$-regularized regression estimators
of the nuisance functions based on objective functions whose directional
derivatives agree with those of the parameter's influence function, we obtain
estimators of the target parameter with two desirable robustness properties:
(1) they are rate doubly-robust in that they are root-n consistent and
asymptotically normal when both nuisance functions follow approximately sparse
models, even if one function has a very non-sparse regression coefficient, so
long as the other has a sufficiently sparse regression coefficient, and (2)
they are model doubly-robust in that they are root-n consistent and
asymptotically normal even if one of the nuisance functions does not follow an
approximately sparse model so long as the other nuisance function follows an
approximately sparse model with a sufficiently sparse regression coefficient.
