L1 regularization

L1 regularisation (lasso) adds $\lambda \Vert \mathbf{w}{\Vert }_1=\lambda {\sum }_j|w_j|$. Its gradient has the same size for every non-zero weight, no matter how small, so weak features are pushed all the way to exactly zero.

The result is a sparse model that uses only a subset of features — built-in feature selection, which makes the model easier to interpret and cheaper to run.

L1 is not differentiable at zero, so it is optimised with coordinate descent, sub-gradients or proximal methods (soft-thresholding).