PLEASE DO NOT USE AI$!($I$\text{'}$ll know if you do$)$

Consider the following loss function on $win{R}^4$ :

$L(w)=w_1^2+w_2^2+w_3^2+2w_1{w}_2+w_4^2+4w_1+4w_2+4$

$($a$)$ What is gradL$(w)?$ What is the minimum value of $L(w)?$

$($b$)$ Suppose we run gradient descent on $L$ :

$w_t=w_{t-1}-gradL(w_{t-1})$ for $alt1$

If the step size is $>0$ and $w_0=(0,0,1,1),$ what is $w_t$ after $t$ step?

$($c$)$ Suppose we use gradient descent on $L$ with ridge regularization:

$L_{}(w)=L(w)+||w|{|}^2$

If $>0,0,$ and $w_0=(0,0,1,1),$ what is $w_t$ after $t$ step?

$($d$)$ Implement the gradient descent on $L_{}$ for the following nine combinations $=1,0\mathrm{.}1,0\mathrm{.}01$

and $=0,1,10.$ Include your code and plot $L(w_t)$ where the $x-$axis is the number of step

and $w_0=(0,0,1,1).$