$$ Q$2.$ Gradient Descent $(2$pt$)$

Given N training data points $\{(x_k,y_k)\},k=1,2,$dots,$N,x_k$ in $R^d,$ and labels as $y_k$ in $\{-1,1\}($either $-1$ or

$,$ we seek a linear discriminant function $f(x_k)=w*x_k={\displaystyle \underset{j=1}{\overset{d}{}}}{w}_j{x}_{k,j}($where $x_{k,j}$ is the feature value

of attribute $j$ of a data point $x_k)$ optimizing a special loss function $L(z)=e^{-z},$ where $z=yf(x).$

Let $>0$ be the learning rate, please derive the gradient update $w_k$ for a randomly selected data point

$k$ in the stochastic gradient descent $($SGD$)$ method.

Hint: Note that SGD randomly pick one data sample $k$ for gradient update per iteration. We can write

$z_k=y_{k}f(x_k)=y_k({\displaystyle \underset{j=1}{\overset{d}{}}}{w}_j{x}_{k,j})$ where $x_{k,j}$ is the feature value of attribute $j$ of a data point $x_k.$

You need to first write $w_j($partial derivative with respect to attribute $j)$ and then get $w_k($the vector

consisting of partial derivatives with respect to the $d$ attributes$).$