Recall that the $\backslash$emph$\{$hinge loss$\}$ is

$\backslash [$

L$\_\backslash$text$\{$hinge$\}(\backslash$vec w$,\backslash$vec x$,$ y$)=$

$\backslash$max $\backslash \{$

$0,1-$ y$\backslash ,\backslash$vec w $\backslash$cdot $\backslash$operatorname$\{$Aug$\}(\backslash$vec x$)$

$\backslash \}$

$\backslash ]$

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

R$(\backslash$vec w$)=\backslash$frac$\{$C$\}\{$n$\}\backslash$sum$\_\{$i$=1\}^\{$n$\}$ L$\_\backslash$text$\{$hinge$\}(\backslash$vec w$,\backslash$vec x$^\{($i$)\},$ y$\_$i$)+\backslash |\backslash$vec w$\backslash |^2$

$\backslash ]$

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

Hint: you will probably $\backslash$emph$\{$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.