Q$4)$ Gradient Descent.

Implement the Gradient descent algorithm using the source code given for the loss function L $=3$ x$2+2$ y$2+20$ cos$($x$)$ cos$($y$)$

Modify the source code given only in$-$between the lines marked with com$-$ ment as described in Q$3.$ Do not modify any other line 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. E$)$ State whether the algorithm converged to the global minima, a local minima or if it failed to converge in your case.

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 $=$

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

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$)$