Machine learning question, use jupyter lab, no AI plz:

Import this:

import sys

from packaging import version

import sklearn

import matplotlib.pyplot as plt

import numpy as np

from sklearn.preprocessing import add$\_$dummy$\_$feature

import numpy as np

from sklearn import metrics #Import scikit$-$learn metrics module for accuracy calculation

from sklearn.model$\_$selection import train$\_$test$\_$split

from sklearn.metrics import confusion$\_$matrix, ConfusionMatrixDisplay

from sklearn.datasets import make$\_$blobs

print$("$Sklearn package",sys$.$version$\_$info$)$

print$("$Sklearn package",sklearn.$\_\_$version$\_\_)$

assert sys$.$version$\_$info $>=(3,10)$

#assert version.parse$($sklearn$.\_\_$version$\_\_)>=$ version.parse$("1\mathrm{.}2\mathrm{.}1")$

plt$.$rc$(\text{'}$font$\text{'},$ size$=12)$

plt$.$rc$(\text{'}$axes$\text{'},$ labelsize$=14,$ titlesize$=14)$

plt$.$rc$(\text{'}$legend$\text{'},$ fontsize$=14)$

plt$.$rc$(\text{'}$xtick$\text{'},$ labelsize$=10)$

plt$.$rc$(\text{'}$ytick$\text{'},$ labelsize$=10)$

$-----$

Questions:

$4.$ Q$2$ Early Stopping

$4\mathrm{.}1.$ Modify the gradient descent code for linear regression so that the gradient descent stops updating the weights when the changes in the loss is too small.

$4\mathrm{.}2.$ You can stop updating when abs$($new$\_$loss $-$ old$\_$loss$)$ is less than a tolerance threshold of $0\mathrm{.}00000001,$ tolerance $=1$e$-8.$

def f$($x$,$w$)$:

return$(($x$.$T@w$).$item$())$

def J$\_$vectorized$($X$,$y$,$w$)$:

m$=$ X$.$shape$[0]$

V $=$ X@w$-$y

sum$\_$squared$\_$error $=$ V$.$T @ V

return $(($sum$\_$squared$\_$error$/(2*$m$)).$item$())$

def J$\_$delta$\_$vectorized$($X$,$y$,$w$)$:

m$=$ X$.$shape$[0]$

sum$\_$w $=$ X$.$T @ $($X@w$-$y$)$

#return $(($sum$*2/$m$))$

return $(($sum$\_$w$/$m$))$

def simple$\_$gradient$\_$vectorized$($X$,$y$,$theta, n$\_$epochs, eta$)$:

theta$\_$path $=[$theta$]$

for epoch in range$($n$\_$epochs$)$:

gradients $=$ J$\_$delta$\_$vectorized$($X$,$y$,$theta$)$

theta $=$ theta $-$ eta $*$ gradients

theta$\_$path.append$($theta$)$

return$($theta$\_$path$)$

np$.$random.seed$(42)$ # to make this code example reproducible

m $=500$ # number of instances

X $=2*$ np$.$random.rand$($m$,1)$ # column vector

y $=4+3*$ X $+$ np$.$random.randn$($m$,1)$ # column vector

X$\_$orig $=$ X

X $=$ add$\_$dummy$\_$feature$($X$)$ # add x$0=1$ to each instance

print$("$Before adding a $\text{'}1\text{'}$ X$[0]=",$ X$\_$orig$[0])$

print$("$After adding a $\text{'}1\text{'}$ X$[0]=",$ X$[0])$

plt$.$plot$($X$\_$orig, y$,"$b$.")$

plt$.$grid$()$

plt$.$xlim$(0,2)$

plt$.$ylim$(2,12)$

##Without early stopping

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

n$\_$epochs $=5000$

m $=$ len$($X$)$ # number of instances

np$.$random.seed$(42)$

theta $=$ np$.$random.randn$(2,1)$ # randomly initialized model parameters

theta$\_$path $=$ simple$\_$gradient$\_$vectorized$($X$,$y$,$theta,n$\_$epochs, eta$)$

print$("$Loss at final theta",J$\_$vectorized$($X$,$y$,$theta$\_$path$[-1]))$

print$("$Number of updates made",len$($theta$\_$path$)-1)$

def simple$\_$gradient$\_$vectorized$\_$early$\_$stopping$($X$,$y$,$theta, n$\_$epochs, eta,tolerance$)$:

theta$\_$path $=[$theta$]$

current$\_$loss $=$ J$\_$vectorized$($X$,$y$,$theta$)$ #initial loss

#iterate over n$\_$epochs, update the gradient, calculate the new loss, break from the loop if

#the loss does not improve compared to the loss in the previous iteration.

#if$($np$.$abs$($current$\_$loss$-$prev$\_$loss$)<$ tolerance$)$

return$($theta$\_$path$)$

$4\mathrm{.}3.$ The code should stop after around $2712$ iterations

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

n$\_$epochs $=5000$

m $=$ len$($X$)$ # number of instances

np$.$random.seed$(42)$

theta $=$ np$.$random.randn$(2,1)$ # randomly initialized model parameters

theta$\_$path $=$ simple$\_$gradient$\_$vectorized$\_$early$\_$stopping$($X$,$y$,$theta,n$\_$epochs, eta,$1$e$-8)$

print$("$Loss at final theta",J$\_$vectorized$($X$,$y$,$theta$\_$path$[-1]))$

print$("$Number of updates made",len$($theta$\_$path$)-1)$