use python jupyter lab:

$5.$Q$3$Parameters Tuning

$5\mathrm{.}1.$For the dataset in X$,$find good parameters than can achieve a very small loss.

$5\mathrm{.}2.$For example, you could get a loss $0\mathrm{.}5037178103$

$5\mathrm{.}3.$Use the original gradient descent without early stopping for this question

$5\mathrm{.}3\mathrm{.}1.$Run the gradient descent function for various settings,

$5\mathrm{.}3\mathrm{.}2.$Use eta from this set $[0\mathrm{.}00001,0\mathrm{.}0001,0\mathrm{.}001,0\mathrm{.}01,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3]$

$5\mathrm{.}3\mathrm{.}3.$Use n$\_$epochs from this set $[100,500,1000,5000,10000]$

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$=10)$

##

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$)$

##

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

n$\_$epochs $=100$

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

print$("$X$.$shape", X$.$shape$)$

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

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

print$("$Loss at initial theta",J$\_$vectorized$($X$,$y$,$theta$))$

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

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

##

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

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

best$\_$theta $=$ theta

best$\_$eta $=0\mathrm{.}1$

best$\_$n$\_$epochs $=100$

best$\_$loss $=$ J$\_$vectorized$($X$,$y$,$theta$)$

##Your code here..

## Iterate over the eta and n$\_$epochs and update the best values

## when the loss improves $($get smaller loss than best$\_$loss$)$

print$("$Best Loss", best$\_$loss$)$

print$("$Best theta", best$\_$theta.ravel$())$

print$("$Best eta", best$\_$eta$)$

print$("$Best n$\_$epochs", best$\_$n$\_$epochs$)$

##

#using sklearn package

#using sklearn package

#This code is provided to give you an idea of how small the loss can be$.$

#You should not aim to get the exact number, but you should get a close one.

from sklearn.linear$\_$model import LinearRegression

lin$\_$reg $=$ LinearRegression$($fit$\_$intercept$=$False$)$

lin$\_$reg.fit$($X$,$ y$)$

best$\_$theta $=$ lin$\_$reg.coef$\_.$reshape$(2,1)$

print$($J$\_$vectorized$($X$,$y$,$best$\_$theta$))$