In this tutorial you will:

Run Gradient Descent on a data set with multiple features.
Explore the impact of the learning rate alpha on gradient descent.
Improve performance of gradient descent by feature scaling using z-score normalization.

import numpy as np
import matplotlib.pyplot as plt
from lab_utils_multi import  load_house_data, run_gradient_descent
from lab_utils_multi import  norm_plot, plt_equal_scale, plot_cost_i_w
from lab_utils_common import dlc
np.set_printoptions(precision=2)
plt.style.use('./deeplearning.mplstyle')

Notation:

Screenshot 2025-01-22 at 11.47.25 AM.png

Problem Statement:

We will use the motivating example of housing price prediction. The training data set contains many examples with 4 features (size, bedrooms, floors and age) shown in the table below. Note, in this lab, the Size feature is in sqft.

We would like to build a linear regression model using these values so we can then predict the price for other houses - say, a house with 1200 sqft, 3 bedrooms, 1 floor, 40 years old.

Dataset:

Size (`sqft`)	Number of Bedrooms	Number of floors	Age of Home	Price (1000s dollars)
952	2	1	65	271.5
1244	3	2	64	232
1947	3	2	17	509.8
...	...	...	...	...

# load the dataset
X_train, y_train = load_house_data()
X_features = ['size(sqft)','bedrooms','floors','age']

Let's view the dataset and its features by plotting each feature versus price.

fig,ax=plt.subplots(1, 4, figsize=(12, 3), sharey=True)
for i in range(len(ax)):
    ax[i].scatter(X_train[:,i],y_train)
    ax[i].set_xlabel(X_features[i])
ax[0].set_ylabel("Price (1000's)")
plt.show()

Plotting each feature vs. the target, price, provides some indication of which features have the strongest influence on price. Above, increasing size also increases price. Bedrooms and floors don't seem to have a strong impact on price. Newer houses have higher prices than older houses.

Gradient Descent With Multiple Variables:

Here are the equations you developed in the last lab on gradient descent for multiple variables.:

$$ \begin{align*} \text{repeat}&\text{ until convergence:} \; \lbrace \newline\; & w_j := w_j - \alpha \frac{\partial J(\mathbf{w},b)}{\partial w_j} \tag{1} \; & \text{for j = 0..n-1}\newline &b\ \ := b - \alpha \frac{\partial J(\mathbf{w},b)}{\partial b} \newline \rbrace \end{align*} $$