$$ You must hand in both your Python code and your output to receive credit for

all questions in this assignment

Questions

$1.$ Complete the program in the file $$grad descent shell.py$$ in order to implement the gradient

descent algorithm for the linear neural network. To do this, in order to compute the gradient

of the error function, you may use either

$$E $=$

N$$

k$=1$

$($W Ak $$ Yk$)$AT

k $(1)$

$($where N is the number of training pairs, Yk is the kth output training vector and Ak represents

a column in the matrix A discussed in class$)$ or the matrix form

$$E $=($W A $$ Y $)$AT $.(2)$

Complete the program by filling in Python code at the specified points within the file

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 for both pinv and linearnn along with

their sum squared difference: w sum abs diff.

$"""$

Complete the progran below by filling in Python

code at the specified points.

To be handed into Canvas assignments:

A$.$ Upload your Python script file

B$.$ Upload an image of your MSE plot in $.$png or $.$jpg format

C$.$ Make sure to show your weight matrices

for both pinv and linearnn along with sum squred difference: w$\_$sum$\_$abs$\_$diff

$"""$

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'

$4.$ Given the variable 'gradientofmsenorm', update the

network weight matrix, w using the gradient descent formula

given in class

$"""$

gradientofmsenorm$=$np$.$sqrt$($np$.$sum$($gradientofmse$*$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$)$

$"""$

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

#plt$.$show$()$

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