Modify your 'gradient descent assignment I$\text{'}$ code in order to train the single hidden layer

feedforward neural network $($FFNN$)$ discussed in class.

You must use the same code to build your training set from 'gradient descent assignment

I$\text{'}.$

Instead of initializing a single weight matrix to random values, you will now need to

initialize $W,$ and $$ to random values.

You must choose the number of neurons $m($a good starting point would be $m=2n).$

a$.$ Upload your working Python script file

b$.$ Make sure to output your training matrices $x$ and $Y$ or you will not receive credit for

this assignment.

c$.$ Upload an image of your mserror plot in $.$png or $.$jpg format

d$.$ Make sure to output your weight matrices $W,,.$

e$.$ Upload all your materials $($code$,$ output and plots$)$ in one single file in either pdf or docx

format.

import numpy as np

import matplotlib.pyplot as plt

def linearnn$($x$,$ y$,$ nits, eta$)$:

# initialize dimensions

xshape $=$ x$.$shape

yshape $=$ y$.$shape

n $=$ xshape$[0]$

N $=$ xshape$[1]$

k $=$ yshape$[0]$

# build the input data matrix

z $=$ np$.$ones$((1,$ N$))$

a $=$ np$.$concatenate$(($x$,$ z$),$ axis$=0)$

# initialize random weights

w $=$ np$.$random.normal$($size$=($k$,$ n $+1))$

# initialize mserror storage

mserror $=$ np$.$zeros$((1,$ nits$))$

# train the network using gradient descent

for itcount in range$($nits$)$:

$"""$

Section of code to update network weights

$...$

$3.$ Compute the gradient of the mean squared error and store it in a variable named 'gradientofmse'

$"""$

gradientofmse $=$ np$.$dot$(($np$.$dot$($w$,$ a$)-$ y$),$ a$.$T$)$

$"""$

$4.$ Given the variable 'gradientofmsenorm', update the network weight matrix, w using the gradient descent formula given in class

$"""$

w $=$ w $-$ eta $*$ gradientofmse

# calculate MSE

etemp $=0$

for trcount in range$($N$)$:

# get $($f$($Xk$)-$ Yk$)$ column vector

avec $=$ np$.$concatenate$(($x$[$:$,$ trcount$],[1]),$ axis$=0)$

vk $=$ np$.$dot$($w$,$ avec$)-$ y$[$:$,$ trcount$]$

# calculate the error

etemp $=$ etemp $+$ vk$.$transpose$()*$ vk

mserror$[0,$ itcount$]=(1/(2*$ N$))*$ etemp$[0]$

# test the output

netout $=$ np$.$dot$($w$,$ a$)$

return netout, mserror, w

# 'main' starts here

# set up the training set

n $=3$

k $=2$

N $=5$

x $=$ np$.$random.normal$($size$=($n$,$ N$))$

y $=$ np$.$random.normal$($size$=($k$,$ N$))$

# compute the closed form solution

z $=$ np$.$ones$((1,$ N$))$

a $=$ np$.$concatenate$(($x$,$ z$),$ axis$=0)$

wpinv $=$ np$.$dot$($y$,$ np$.$linalg.pinv$($a$))$

print$(\text{'}$Weight matrix computed using the pseudoinverse:$\text{'})$

print$($wpinv$)$

print$()$

$"""$

Section of code to call the gradient descent function to computer the weight matrix

$1.$ You must input a value for the number of iterations, nits

$2.$ You must input a value for the learning rate, eta $($You should run numerical tests with a few different values$)$

$"""$

nits $=1000$ # Set the number of iterations

eta $=0\mathrm{.}01$ # Set the learning rate

netout, mserror, w $=$ linearnn$($x$,$ y$,$ nits, eta$)$

print$(\text{'}$Weight matrix computed via gradient descent:$\text{'})$

print$($w$)$

print$()$

$"""$

Compute the sum of the absolute value of the difference between the weight matrices from pinv and gradient descent

$"""$

w$\_$sum$\_$abs$\_$diff $=$ np$.$sum$($np$.$absolute$($wpinv $-$ w$))$

print$(\text{'}$Sum of absolute value of the difference between weight matrices:$\text{'})$

print$($w$\_$sum$\_$abs$\_$diff$)$

# plot the mean squared error

plt$.$plot$($mserror$[0,$ :$])$

plt$.$title$(\text{'}$Mean Squared Error versus Iteration'$)$

plt$.$xlabel$(\text{'}$Iteration #$\text{'})$

plt$.$ylabel$(\text{'}$MSE$\text{'})$

plt$.$savefig$(\text{'}$plot$.$png$\text{'})$