Q$3)$ Gradient Descent. Please use the file $$assign$2$ q$3.$py$$ provided along with this assignment. $(3$ marks$)$

The python source file implements the gradient descent algorithm. You are

expected to fill in the missing lines of code to get the algorithm to work as

expected. The script takes no additional parameters.

There are four places where you need to add your lines of code. Your lines

of code should go in$-$between the lines with the comment:

$$##### To be Updated #####$$

and

$$#########################$$

NOTE: You should NOT modify any other lines of the script.

Deliverables: A$)$ A screenshot of the $3$D plot obtained when you execute the

python script after adding your lines of code. B$)$ A screenshot of the console

output obtained obtained when you execute the python script after adding

your lines of code. C$)$ The equations of the Loss function and gradient of

loss function on which the gradient descent was performed. D$)$ A snapshot

of the lines of code that was added by you in the designated places. Do not

write the entire code. $(3$ marks$)$

For your benefit, we have given below the sample output plot and console

output for a fictitious student id $023$xx$12345.$

import numpy as np

import matplotlib.pyplot as plt

import random

##### To be Updated #####

# e$.$g$.,$if your BITS email id is $023$ab$12345$@wilp.bits$-$pilani.com

# update the below line as student$\_$id $="023$xx$12345"$

student$\_$id $="2021$da$04122"$

#########################

student$\_$id $=\text{'}\text{'}.$join$([$i for i in student$\_$id if i$.$isdigit$()])$

random.seed$($student$\_$id$)$

# set the number of iterations and learning rate

iters $=$ random.randint$(100,300)$

learning$\_$rate $=0\mathrm{.}01$

# Evaluate the function at x

def C$($x$)$:

##### To be Updated #####

# NOTE: return value of this function will

# $**$ALSO$**$ change for Q$4$ of the assignment

#########################

return $($x$.$T@np$.$array$([[2,1],[1,20]])$@x$)-($np$.$array$([5,3]).$reshape$(2,1).$T@x$)$

# Evaluate the gradient of function at x

def dC$($x$)$:

##### To be Updated #####

# $1.$ Compute and return the gradient

return

#########################

def plot$\_$grad$\_$change$($X$,$Y$,$Z$,$ c$,$ grad$\_$xs$0,$ grad$\_$xs$1,$ grad$\_$ys$)$:

fig $=$ plt$.$figure$()$

title$\_$str $=$ "Gradient Descent:"$+"$lr$="+$str$($learning$\_$rate$)$

plt$.$title$($title$\_$str$)$

ax $=$ fig.add$\_$subplot$($projection$=\text{'}3$d$\text{'})$

ax$.$plot$\_$surface$($X$,$ Y$,$ Z$,$ cmap$=$plt$.$cm$.$YlGnBu$\_$r$,$alpha$=0\mathrm{.}7)$

for i in range$($len$($grad$\_$xs$0))$:

ax$.$plot$([$grad$\_$xs$0[$i$]],[$grad$\_$xs$1[$i$]],$ grad$\_$ys$[$i$][0],$ markerfacecolor$=\text{'}$r$\text{'},$ markeredgecolor$=\text{'}$r$\text{'},$ marker$=\text{'}$o$\text{'},$ markersize$=7)$

ax$.$text$($grad$\_$xs$0[-1],$grad$\_$xs$1[-1],$grad$\_$ys$[-1][0][0],$

$"("+$str$($round$($grad$\_$xs$0[-1],2))+","+$

str$($round$($grad$\_$xs$1[-1],2))+"),"+$

str$($round$($grad$\_$ys$[-1][0][0],2)))$

plt$.$show$()$

def GD$($start$,$x$,$y$,$z$,$ c$,$ dc$,$ iters, eta$)$:

px $=$ start.astype$($float$)$

py $=$ c$($px$).$astype$($float$)$

print$("$GD Start Point:",px$,$py$)$

print$("$Num steps:",iters$)$

grad$\_$xs$0,$ grad$\_$xs$1,$ grad$\_$ys $=[$px$[0][0]],[$px$[1][0]],[$py$]$

for iter in range$($iters$)$:

##### To be Updated #####

# $2.$ Update px using gradient descent

px $=$

# $3.$ Update py

py $=$

#########################

grad$\_$xs$0.$append$($px$[0][0])$

grad$\_$xs$1.$append$($px$[1][0])$

grad$\_$ys$.$append$($py$)$

print$("$Converged Point:",px$,$py$)$

plot$\_$grad$\_$change$($x$,$y$,$z$,$ c$,$ grad$\_$xs$0,$grad$\_$xs$1,$ grad$\_$ys$)$

lo $=-10$

hi $=10$

x$1=$ round$($random$.$uniform$($lo$,0),4)$

x$2=$ round$($random$.$uniform$($lo$,0),4)$

x $=$ np$.$linspace$($lo$,1,$ hi$)$

y $=$ np$.$linspace$($lo$,1,$ hi$)$

X$,$ Y $=$ np$.$meshgrid$($x$,$ y$)$

Z $=$ np$.$zeros$\_$like$($X$)$

for i in range$($X$.$shape$[0])$:

for j in range$($X$.$shape$[1])$:

Z$[$i$][$j$]=$ C$($np$.$array$([$X$[$i$][$j$],$Y$[$i$][$j$]]).$reshape$(2,1))$

# start Gradient Descent

GD$($np$.$array$([$x$1,$x$2]).$reshape$(2,1),$X$,$Y$,$Z$,$ C$,$ dC$,$ iters, learning$\_$rate$)$

narddarryassign$2-$q$3.$pyDownload assign$2-$q$3.$py $(2\mathrm{.}54$ KB$)$import numpy as npAmport matplotlib. pyplot as pltimport random###$*$## To be Updated ##$*$### e$.$g$.,$ if your BITS email id is $023$ab$123450$wilp. bits $-$pilani. comstudent$\_$ id $=$