Q$1)$ Gradient Descent.

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 lines of code. 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 lines of code.

B$)$ Console output obtained obtained when you execute the python script after adding lines of code.

C$)$ The equations of the Loss function and gradient of loss function on which the gradient descent was performed.

D$)$ State whether the algorithm converged to the global minima, a local minima or if it failed to converge in our case

Python Code:$$import numpy as np

import matplotlib.pyplot as plt

import random

student$\_$id $="023$xx$12345"$

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$2$ 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$)$