Need help doing this by HAND only. No code should be added. Please do each update weight, and show the calculations done.

Need help to update the weights for each point specifically. Please be specific.

Forward

$-$

Calculation

Let's start with the following weights:

w

$1$

$1$

$=$

$1$

w

$1$

$2$

$=$

$1$

w

$1$

$0$

$=$

$0$

w

$2$

$1$

$=$

$1$

w

$2$

$2$

$=$

$1$

w

$2$

$0$

$=$

$0$

w

$3$

$1$

$=$

$1$

w

$3$

$2$

$=$

$1$

w

$3$

$0$

$=$

$0$

$($

All of the connections are

$1$

$$and all of the biases are zero

$)$

$.$

We'll work on training the logical

$-$

AND function that we previously used for the single layer perceptron. Here is its truth table:

x

$\_$

$1$

$$ x

$\_$

$2$

$|$

$$target label

$($

t

$)$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$-$

$$

$0$

$0$

$0$

$$

$0$

$1$

$0$

$$

$1$

$0$

$0$

$$

$1$

$1$

$1$

First, use the network as configured to attempt to classify each point. Perform a forward

$-$

pass through the network by hand for each of the four points. What results do you obtain?

for each hidden layer node j:

$$ y

$\_$

j

$=$

$$the weighted sum of inputs to node j

$$ o

$\_$

j

$=$

$$sigmoid

$($

y

$\_$

j

$)$

for the output layer node k:

$$ y

$\_$

k

$=$

$$the weighted sum of hidden layer o

$\_$

j values to node k

$$ o

$\_$

k

$=$

$$sigmoid

$($

y

$\_$

k

$)$

if o

$\_$

k

$<$

$=$

$.$

$5$

:

$$ assign class

$0$

else:

$$ assign class

$1$

Train

We'll now train the network by hand using the backpropagation update rules. I

$\text{'}$

m not repeating the update rules here, because they're hard to type, but you have them in your notes.

Iterate one point at a time. Use a learning rate of

$.$

$1$

$.$

Classify the point using the forward pass. In doing so

$,$

$$you$\text{'}$ll obtain values for each yj and oj at the hidden

$-$

layer nodes, and yk and ok at the output layer.

Update the weights

$($

and the bias

$)$

$$at the output node using its update rule.

Update the weights and bias at each of the two hidden

$-$

layer nodes using their update rules.

The updates take place even if the assigned class matches the target class. Use the value of the output sigmoid function $ok$ to calculate the error

Do one epoch, training over all four points in the logical

$-$

AND function, updating the weights after each point.

After you update the weights, verify that they have moved in a direction that makes the classification more correct, even if the output has not crossed the threshold of

$.$

$5$

$$to actually change its predicted class.