Consider the following MLP network.

After some training, the units in the neural network have the following weight vectors:

$v_1=([-2],[2],[2]),v_2=([1],[1],[-1]),w_1=(\frac{1}{-3\mathrm{.}5}),w_2=(\frac{0\mathrm{.}5}{-1\mathrm{.}2}),$ and $w_3=(\frac{0\mathrm{.}3}{0\mathrm{.}6})$

Assume that all units have sigmoid activation functions given by

$f(x)=\frac{1}{1+e^{-x}}$

and that each unit has a bias $=0($zero$).$

The inputs of the network are given by $x=([2,3,1]{)}^T$ and the corresponding desired outputs

are given by $d=([1,0,1]{)}^T.$

a$)$ What is the error $($local gradient$)$ for each of the output units?

b$)$ Update the output weights.

c$)$ What is the error for each of the hidden units?

d$)$ Update the input weights.