  $\ell_1$ regularization has been used for logistic regression to circumvent
the overfitting and use the estimated sparse coefficient for feature selection.
However, the challenge of such a regularization is that the $\ell_1$ norm is
not differentiable, making the standard algorithms for convex optimization not
applicable to this problem. This paper presents a simple projection neural
network for $\ell_1$-regularized logistics regression. In contrast to many
available solvers in the literature, the proposed neural network does not
require any extra auxiliary variable nor any smooth approximation, and its
complexity is almost identical to that of the gradient descent for logistic
regression without $\ell_1$ regularization, thanks to the projection operator.
We also investigate the convergence of the proposed neural network by using the
Lyapunov theory and show that it converges to a solution of the problem with
any arbitrary initial value. The proposed neural solution significantly
outperforms state-of-the-art methods with respect to the execution time and is
competitive in terms of accuracy and AUROC.
