Lasso regression

Lasso regression uses an L1 penalty: ${\min }_{\mathbf{w}}\Vert \mathbf{y}-X\mathbf{w}{\Vert }^2+\lambda \Vert \mathbf{w}{\Vert }_1$, where $\Vert \mathbf{w}{\Vert }_1={\sum }_j|w_j|$.

The L1 penalty's corners push many weights to exactly zero, so lasso performs automatic feature selection and produces sparse, interpretable models. There is no closed form; it is solved with coordinate descent or proximal gradient methods.

Geometric picture: the L1 constraint region is a diamond, and the elliptical loss contours usually first touch it at a corner, where some coordinates are zero. The L2 region is a circle with no corners, so weights shrink but stay non-zero.