# 3.4.2 The Lasso

The lasso is a shrinkage method, which is estimated by (3.51):
$$
\begin{equation}
\hat{\beta}^{lasso} = \underset{\beta}{argmin}\sum_{i=1}^N \left( y_i-\beta_0-\sum_{j=1}^p{x_{ij}\beta_j} \right)^2\\
\text{ subject to } \sum_{j=1}^p|\beta_j| \le t
\end{equation}
$$

we can re-parametrize the constant $\beta_0$ by standartizing the predictors; the solution for $\beta_0$ is $\overline{y}$.

We can also write the lasso in the equivalent *Lagragngian form* (3.52):
    
$$\hat{\beta}^{lasso} = \underset{\beta}{argmin} \left\{
\frac{1}{2}\sum_{i=1}^N \left( y_i-\beta_0-\sum_{j=1}^p{x_{ij}\beta_j} \right)^2
+\lambda\sum_{j=1}^p|\beta_j|
\right\}$$

The lasso solution is nonlinear in the $y_i$ and it is a quadratic programming problem.

c9cc34c04d38ff5301b22d308cc36967f42ecba8
ipynb
Jupyter Notebook
chapter-03/3.4.2-the-lasso.ipynb
leduran/ESL
fcb6c8268d6a64962c013006d9298c6f5a7104fe
MIT
2019-01-28T14:05:02.000Z
2022-03-27T00:11:21.000Z
chapter-03/3.4.2-the-lasso.ipynb
leduran/ESL
fcb6c8268d6a64962c013006d9298c6f5a7104fe
MIT
2020-07-06T16:51:40.000Z
2020-07-06T16:51:40.000Z
chapter-03/3.4.2-the-lasso.ipynb
leduran/ESL
fcb6c8268d6a64962c013006d9298c6f5a7104fe
MIT
2019-03-21T23:48:35.000Z
2022-03-31T13:05:10.000Z
Qwen/Qwen-72B
1. YES
2. YES