Introduction to Statistical Learning Eq. 6.12 and 6.13

Question

Can someone please explain me how the optimization of 6.12 leads to 6.14 and the optimization of 6.13 leads to 6.15?

Answer 1

For the first equation, it's the result of zero gradient; $$ \begin{aligned} S &= \sum_{j=1}^p (y_j-\beta_j)^2 +\lambda\sum_{j=1}^p\beta_j^2\\ \end{aligned} $$ at extrema, $$ \begin{aligned} \frac{\partial S}{\partial \beta_j} &=0\\ -2(y_j -\beta_j) +2\lambda\beta_j &= 0\\ \beta_j &= \frac{y_j}{1+\lambda}. \end{aligned} $$

I think you should be able to derive the other expression using the same technique shown above and use the fact that $$ \vert \beta_j \vert = \begin{cases} \beta_j \ \text{if} \ \beta_j > 0\\ -\beta_j \ \text{if} \ \beta_j < 0\end{cases}. $$