$3-$ Consider a classification problem with training data $(x_i,y_i),i=1,$dots,$N$ where vector

$x_{i}in{R}^D$ and $y{i}_{i}in\{0,1\}.$ Also, $w$ is weight vector and linear activation as $a_i=w^T{}_i,$ and

activation functions of the form

$f_{\text{s}\text{i}\text{g}\text{m}\text{o}\text{i}\text{d}}(a_i)=\frac{1}{1+\text{e}\text{x}\text{p}(-a_i)},f_{\text{r}\text{e}\text{l}\text{u}}(a_i)=$max$(0,a_i).$

Comment $($prove with all the details$)$ on the convexity of squared loss as a function of activation

vector $a=[a_1$dotsaN$]$ for the following

a$)L_{\text{r}\text{e}\text{l}\text{u}}(a)={\displaystyle \underset{i=1}{\overset{N}{}}}(y_i-f_{\text{r}\text{e}\text{l}\text{u}}(a_i){)}^2$

b$)L_{\text{s}\text{i}\text{g}\text{m}\text{o}\text{i}\text{d}}(a)={\displaystyle \underset{i=1}{\overset{N}{}}}(y_i-f_{\text{s}\text{i}\text{g}\text{m}\text{o}\text{i}\text{d}}(a_i){)}^2$