$42.$ We construct a function that realizes the sparse group Lasso. Fill in the blanks, and execute the procedure in Excercise $3\mathrm{.}8$ to examine the validity.

$```$

def sparse$\_$gr$($x$,$ y$,$ lam, alpha$)$:

p $=$ X$.$shape $[1]$

nu $=1/$ np$.$max$($np$\mathrm{.}11$nalg.e$1$gvals$($x$.$T $($ x$))$

beta $=$ np$.$zeros$($p$)$

beta$\_$old $=$ np$.$zeros$($p$)$

eps $=1$

while eps $>0\mathrm{.}001$:

gamma $=$ beta $+$ nu $*$ x$.$T $($y $-$ x $0$ beta$)$

delta $=$ soft$\_$th$($lan $*$ alpha, gamma$)$

beta $=$ np$.$zeros$($p$)$

if np$.$l$1$nalg.norm$($delta$,2)>0$:

beta $-$ ## Blank$($l$)$ ##

eps $=$ np$.$max$($np$.$abs$($beta $-$ beta$\_$old$))$

beta$\_$old $=$ copy.copy$($beta$)$

return beta

$```$

$```$

def sparse$\_$group$\_$lasso$($z$,$ y$,1$am$-0,$ alpha$-0)$:

J $-$ len$($z$)$

theta $-[]$

for $1$ in range$($J$)$:

theta.append $($np$.$ zeros$($z$[1].$shape$[1]))$

for n in range$(10)$:

for J in range$($J$)$:

r $=$ copy $.$copy $($y$)$

for k in range$($J$)$:

if k $!-$ J:

r $=$ r $-$ z$[$k$]0$ theta$[$k$]$

theta$[1]-$ ## Blank$(2)$ ##

return theta

$```$