  This paper offers a new approach for study the frequentist properties of the
penalized MLE for general nonlinear regression models. The idea of the approach
is to relax the nonlinear structural equation by introducing an auxiliary
parameter for the regression response and replacing the structural equation
with a penalty. This leads to a general semiparametric problem which is studied
using the SLS approach from \cite{Sp2022}. We state sharp bounds on
concentration and on the accuracy of the penalized MLE, Fisher and Wilks
expansions, evaluate the risk of estimation over smoothness classes, and a
number of further results. All the bounds are given in terms of effective
dimension and do not involve the ambient dimension of the parameter space.
