Consider a training set of data points and labels $\{(x^{(n)},y^{(n)}){\}}_{n-1}^4$ where xinR and

yin$\{0,1\}$ :

$x^{(1)}=-1,y^{(1)}=0$

$x^{(2)}=-2,y^{(2)}=0$

$x^{(3)}=2,y^{(3)}=1$

$x^{(4)}=1,y^{(4)}=1$

Now consider a linear classifier which consists of a linear model $f(x)=wx+b$ and a

threshold function that outputs a class prediction hat$(y)$ :

hat$(y)=\{\begin{array}{ccc}1& if& f(x)0\mathrm{.}5\\ 0& if& f(x)<mn>0</mn><mn>.</mn><mn>5</mn>\end{array}$

To learn the parameters of the linear model $(w,b)$ we can minimise the mean squared

error loss $($MSE$)$ across our training data:

$L_{\text{M}\text{S}\text{E}}=\frac{1}{4}{\displaystyle \underset{n=1}{\overset{4}{}}}(f(x^{(n)})-y^{(n)}{)}^2$

$($a$)$ Plot the training data. Annotate this plot with the classifier's decision boundary

for initial parameters $w_{t-0}=1$ and $b_{t-0}=1.$

$($b$)$ Determine which side of the decision boundary is allocated to which class, thence

classify a test point at $x^{(t)}=-0\mathrm{.}6.$

$($c$)$ Derive an expression for $\frac{del{L}_{MSBx}}{\text{d}\text{e}\text{l}\text{w}}$ and $\frac{del{L}_{\text{M}\text{S}\text{E}\text{x}}}{\text{d}\text{e}\text{l}\text{o}}$ and explain how these may be used

in gradient descent to update the classifier's parameters.

$($d$)$ Perform a single iteration of gradient descent using your expressions from $($c$)$ to

compute $w_{l-1}$ and $b_{l-1}.$ Use a learning rate $=0\mathrm{.}1.$

$($c$)$ Determine where the decision boundary is after this update, and reclassify the

test point at $x^{(t)}=0\mathrm{.}6.$