  Parametric high-dimensional regression analysis requires the usage of
regularization terms to get interpretable models. The respective estimators can
be regarded as regularized M-functionals which are naturally highly nonlinear.
We study under which conditions these M-functionals are compactly
differentiable, so that the corresponding estimators admit an asymptotically
linear expansion. In a one-step construction, for a suitably consistent
starting estimator, this linearization replaces solving optimization problems
by evaluating the corresponding influence curves at the given data points. We
show under which conditions the asymptotic linear expansion is valid and
provide concrete examples of machine learning algorithms that fit into this
framework.
