We consider a neural network with the diagram

The neural network has two inputs $1$ and $2$ and two outputs $1$ and $2.$ The

equations that represent this neural network are:

$1=(1$

$11+2$

$12)$

$1$

$2=(1$

$21)$

$2$

$2=(2$

$21)$

$1=1$

$2$

$2=2$

$2$

The activation functions for both layers are sigmoid $()=1$

$1+,$ whose

derivative is $()=$

$(1+)2.$

Let us consider the weights are $1$

$1=1,2$

$1=0\mathrm{.}2,1$

$2=0\mathrm{.}2,$ and $2$

$2=0\mathrm{.}1$

a$)$ For each neuron calculate its induced local field and output for the following

input value $1=0\mathrm{.}5,2=1.6$

b$)$ If the desired output to the previous input is $=[0\mathrm{.}5,0\mathrm{.}5]$T$,$ use the

backpropagation algorithm to compute the local gradients $1,1$

$2,2$

$2,$ one for

each neuron.

$9$

c$)$ Calculate the gradients of the error with respect to the weights. $8$

d$)$ If the learning rate is $0\mathrm{.}5,$ calculate the updated weights. $4$

e$)$ Calculate the output to the input value $1=0\mathrm{.}5,2=1$ with the updated

weights. Discuss the result