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Machine Learning - Feature scaling and Learning Rate (Multi-variable)

Machine Learning - Feature scaling and Learning Rate (Multi-variable)#

Tools#

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')

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[1], line 3
      1 import numpy as np
      2 import matplotlib.pyplot as plt
----> 3 from lab_utils_multi import  load_house_data, run_gradient_descent 
      4 from lab_utils_multi import  norm_plot, plt_equal_scale, plot_cost_i_w
      5 from lab_utils_common import dlc

ModuleNotFoundError: No module named 'lab_utils_multi'

Problem Statements#

  • 주택 가격 예측을 위한 예제를 그대로 사용한다. 훈련 데이터셋에는 네가지 특징 (크기, 침실 수, 층 수, 햇수)을 포함한 세 가지 예제가 아래 표에 나와있다.

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']

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()

Gradient Descent With Multiple Variables#

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

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

where, n is the number of features, parameters \(w_j\), \(b\), are updated simultaneously and where

\[\begin{split} \begin{align} \frac{\partial J(\mathbf{w},b)}{\partial w_j} &= \frac{1}{m} \sum\limits_{i = 0}^{m-1} (f_{\mathbf{w},b}(\mathbf{x}^{(i)}) - y^{(i)})x_{j}^{(i)} \tag{2} \\ \frac{\partial J(\mathbf{w},b)}{\partial b} &= \frac{1}{m} \sum\limits_{i = 0}^{m-1} (f_{\mathbf{w},b}(\mathbf{x}^{(i)}) - y^{(i)}) \tag{3} \end{align} \end{split}\]

  • m is the number of training examples in the data set

  • \(f_{\mathbf{w},b}(\mathbf{x}^{(i)})\) is the model’s prediction, while \(y^{(i)}\) is the target value

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