  Regularized regression has become very popular nowadays, particularly on
high-dimensional problems where the addition of a penalty term to the
log-likelihood allows inference where traditional methods fail. A number of
penalties have been proposed in the literature, such as lasso, SCAD, ridge and
elastic net to name a few. Despite their advantages and remarkable performance
in rather extreme settings, where $p \gg n$, all these penalties, with the
exception of ridge, are non-differentiable at zero. This can be a limitation in
certain cases, such as computational efficiency of parameter estimation in
non-linear models or derivation of estimators of the degrees of freedom for
model selection criteria. With this paper, we provide the scientific community
with a differentiable penalty, which can be used in any situation, but
particularly where differentiability plays a key role. We show some desirable
features of this function and prove theoretical properties of the resulting
estimators within a regularized regression context. A simulation study and the
analysis of a real dataset show overall a good performance under different
scenarios. The method is implemented in the R package DLASSO freely available
from CRAN.
