Problem $2.$

Recall that the hinge loss is

$L_{\text{h}\text{i}\text{n}\text{g}\text{e}}(vec(w),$vec$(x),y)=$max$\{0,1-$yvec$(w)*Aug(vec(x))\}$

The Soft$-$SVM problem aims to minimize the regularized empirical risk:

$R(vec(w))=\frac{C}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{L}_{\text{h}\text{i}\text{n}\text{g}\text{e}}((vec(w)),$vec$(x{)}^{(i)},y_i)+||vec(w)|{|}^2$

Show that $R(vec(w))$ is a convex function of vec$(w).$

Hint: you will probably not want to use the formal definition of convexity here. Instead, you'll want to show

that $R$ is composed of simpler functions which themselves are convex.