HW$4$A$.$ Training Perceptron $($paper and pencil ONLY$)$

Due By Oct. $10$ midnight

$20$ Points

Please put your name in the file$\_$name.doc

Scan your work or type in word doc, then upload it to the Brightspace

No programming for this HW$4$A$.$

A logic gate AND problem can be seen as a linear separable problem $($see the fig. below$).$ The input X and output Y are given in the following table. In the table, x$0$ is constant. The x$1$ and x$2$ values are the real input data. The Y values are the outputs which can be seen as binary class problem $(0$ or $1).$ A single perceptron can be used to solve the linear separable problem.

x$0$ x$1$ x$2$ y

$1000$

$1100$

$1010$

$1111$

Logic AND gate vs linear separable problem

Given the conditions above, you are required:

Train the perceptron to find the optimum weights W$.$

Find the Decision Boundary Function x$2=$ f$($x$1)$

Draw the decision boundary with given data X

You need to write down your calculations step by step

For Your Reference: Training step

Given initial weights W$\_($i$=0)=((1$@$1$@$1)),$ X$=((1$&$0$&$0$@$1$&$0$&$1$@$1$&$1$&$0$@$1$&$1$&$1)),$ and Y$=((0$@$0$@$0$@$1))$ above, where the i is the number of the epoch. For i $=0,$ it means the first epoch.

Training can be done as follows, starting from i $=0$

Find the sigma $($or the weighted sum$)$ using Wi:

$\backslash$Sigma i $=$ X Wi

Find the prediction $($Y$\_$i $)$ using the Step function as an activation function,

For each sample j: $($y$\_$j $)=1$ If any of $\backslash$Sigma ij $>=0$

$($y$\_$j $)=0$ otherwise

Find the Errors:

E$\_$i$=$Y$-($Y$\_$i $)$

Update the Wi to Wi$+1$ if Ei $!=0,$ otherwise the training is finished.

Wi$+1=$ Wi $+$ alpha$*$XT$*$Ei where alpha $=0\mathrm{.}2$

Go to Step $1.$