$6.$ Neural networks and backpropagation $(10$ points$).$ Consider a simple two$-$layer network in the lecture slides. Given

n

training data

$($x

i

$,$y

i

$)$

$,$

i$=1,...,$n

$,$ the cost function used to training the neural networks $\backslash [\backslash$ell$($w$,\backslash$alpha$,\backslash$beta$)=\backslash$sum$\_\{$i$=1\}^\{$n$\}\backslash$left$($y$^\{$i$\}-\backslash$sigma$\backslash$left$($w$^\{$T$\}$ z$^\{$i$\}\backslash$right$)\backslash$right$)^\{2\}\backslash ]$ where

$\backslash$sigma $($x$)=1/(1+$e

$$x

$)$

is the sigmoid function,

z

i

is a two$-$dimensional vector such that

z

$1$

i

$$

$=\backslash$sigma $(\backslash$alpha

T

x

i

$)$

$,$ and

z

$2$

i

$$

$=\backslash$sigma $(\backslash$beta

T

x

i

$)$

$.2($a$)(5$ points$)$ Show that the gradient is given by $\backslash [\backslash$frac$\{\backslash$partial $\backslash$ell$($w$,\backslash$alpha$,\backslash$beta$)\}\{\backslash$partial w$\}=-\backslash$sum$\_\{$i$=1\}^\{$n$\}2\backslash$left$($y$^\{$i$\}-\backslash$sigma$\backslash$left$($u$^\{$i$\}\backslash$right$)\backslash$right$)\backslash$sigma$\backslash$left$($u$^\{$i$\}\backslash$right$)\backslash$left$(1-\backslash$sigma$\backslash$left$($u$^\{$i$\}\backslash$right$)\backslash$right$)$ z$^\{$i$\},\backslash ]$ where

u

i

$=$w

T

z

i

$.$ This is also known as backpropagation. $($b$)(5$ points$)$ Also show the gradient of

$($w$,\backslash$alpha $,\backslash$beta $)$

with respect to

$\backslash$alpha

and

$\backslash$beta

and write down their expression.

Advanced Math $-$ Calculus