L1-penalized logistic regression ($\hat{\beta }$)

L1-penalized logistic regression minimizes empirical logistic loss plus $\lambda \Vert \beta {\Vert }_1$. Its subgradient optimality conditions set each logistic score coordinate equal to $-\lambda z_j$, where $z_j=\mathrm{sgn}({\beta }_j)$ off zero and $z_j\in [-1,1]$ at zero.