Problem $5($Coding$)11$ pts:

For the following tasks use the following California House Price Data:

$[1]1$ from sklearn.datasets import fetch$\_$california$\_$housing

$2$ housing $=$ fetch$\_$california$\_$housing$()$

$[20]1$ data$\_$X$=$ pd$.$DataFrame$($housing$[\text{'}$data$\text{'}],$ columns $=$ housing.feature$\_$names$)$

$2$ data$\_$X$.$head$(2)$

$1$ data$\_$y $=$ housing.target

$2$ data$\_$y$|$

array$([4\mathrm{.}526,3\mathrm{.}585,3\mathrm{.}521,$dots,$0\mathrm{.}923,0\mathrm{.}847,0\mathrm{.}894])$

Split the dataset into training and test sets of $80$:$20$ ratio $($use random$\_$seed $2024)$ and test$\_$size $=$

$0\mathrm{.}20.$ You must train the linear regression model using the training data and compute MSE using the test

dataset.

Perform Multiple linear regression using gradient descent optimization technique. You must not use

any direct package for this. You should generate the parameters $($intercepts & coefficients$)$ initial values

from standard normal distribution. Number of iterations should be minimum of $50.$

a$.(2$ pts$)$ Mention: loss function, initial values of parameters $($should be generated from Standard

Normal Distribution: $xN(0,1)),$ hyperparameters, and partial derivatives of the loss function

with respect to intercept and coefficients

b$.(5$ pts$)$ Write a function that takes input $($as arguments$)$ of hyperparameters and provides

output: best set of optimal parameters $($intercept and coefficients$),$ and the loss $($cost$).$

c$.(1$ pt$)$ Plot learning curve where $x-$axis will be number of iteration and $y-$axis will be loss for each

iteration and explain the curve

d$.(1$ pts$)$ Form the regression equation using the optimal parameters and find $($ hat$(y))($predict for each

datapoints of $x_{?}$test$)-$ show in DataFrame including two columns: $y($y$\_$actual$)$ & hat$(y)($y$\_$predict$)$

e$.(1$ pts$)$ Finally, for the test dataset: Find Mean Squared Error $($MSE$)$

f$.(1$ pts$)$ Indicate whether your model is experiencing overfitting. Explain.

g$.$ Extra Points: The model that achieves the lowest MSE on the test data will be awarded $2$ bonus

points.