Wu Enda 471 Machine Learning Introduction Course 1 Week 1 - Gradient Descent

import math, copy
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('./deeplearning.mplstyle')
from lab_utils_uni import plt_house_x, plt_contour_wgrad, plt_divergence, plt_gradients

1 Load the dataset

# train data
x = np.array([1.0,2.0])
y = np.array([300.0,500.0])

2 Calculate COST(mean square difference, 1/2m(y_pre - y)²)

#通过w*x+b得到y_pre,
def costFun(x,y,w,b):
    m = x.shape[0] #数据长度,一维数组的长度
    cost = 0 #初始化
    for i in range(m):
        f_wb = w*x[i]+b
        cost += (f_wb-y[i])**2
    return 1/(2*m)*cost

3 Calculate the gradient

$$\begin{array}{rl}\frac{\partial J(w,b)}{\partial w}& =\frac{1}{m}\sum _{i=0}^{m-1}(f_{w,b}(x^{(i)})-y^{(i)})x^{(i)}\\ \frac{\partial J(w,b)}{\partial b}& =\frac{1}{m}\sum _{i=0}^{m-1}(f_{w,b}(x^{\left(i\right)})-y^{(i)})\end{array}\text{\hspace{1em}(4)(5)}$$

def gradientFun(x,y,w,b):
    m = x.shape[0]
    dj_dw = 0
    dj_db = 0
    
    for i in range(m):
        f_wb = w*x[i]+b
#         dj_dw += (f_wb - y[i])*x[i]
#         dj_db += f_wb - y[i]
#     dj_dw /= m
#     dj_dw /= m
        dj_dw_i = (f_wb - y[i]) * x[i] 
        dj_db_i = f_wb - y[i] 
        dj_db += dj_db_i
        dj_dw += dj_dw_i 
    dj_dw = dj_dw / m 
    dj_db = dj_db / m 
    return dj_dw,dj_db

4 Draw the cost curve

plt_gradients(x,y,costFun,gradientFun)
plt.show()

5 gradient descent

def gradient_descent(x,y,w_in,b_in,a,num_item,costFun,gradientFun):
    w = copy.deepcopy(w_in) #避免改变原来的w
    # 用数组存储用于在每次迭代时存储成本 J 和 w，主要用于以后绘制图形
    J_history = []
    p_history = []
    b = b_in
    w = w_in
    
    for i in range(num_item):
        #计算梯度
        dj_dw,dj_db = gradientFun(x,y,w,b)
        # 更新w,b
        b = b - a * dj_db                            
        w = w - a * dj_dw 
        # 保存每一次成本J
        if i < 100000:#避免资源浪费
            J_history.append(costFun(x,y,w,b))
            p_history.append([w,b])
        #输出10次结果，打印出来  
        #math.ceil(a),取a的最小整数 4.2取5
        if i % math.ceil(num_item/10) == 0:
            '''
            Iteration {i:4}:：表示当前循环迭代次数，用 4 位数字的方式呈现，比如第一次迭代就是 "Iteration 1:"。
            Cost {J_history[-1]:0.2e}：表示模型当前的代价函数值，用科学计数法表示，保留两位小数。
            dj_dw: {dj_dw: 0.3e}, dj_db: {dj_db: 0.3e}：表示代价函数对权值和偏置的梯度值，用科学计数法表示，保留三位小数。
            w: {w: 0.3e}, b:{b: 0.5e}：表示当前的权值 w 和偏置 b 的值，用科学计数法表示，分别保留三位和五位小数。
            '''
            print(f"Iteration {
      
      i:4}: Cost {
      
      J_history[-1]:0.2e} ",
                  f"dj_dw: {
      
      dj_dw: 0.3e}, dj_db: {
      
      dj_db: 0.3e}  ",
                  f"w: {
      
      w: 0.3e}, b:{
      
      b: 0.5e}")
    return w,b,J_history,p_history

# 初始化
w_init = 0
b_init = 0
iterations = 10000
tmp_alpha = 1.0e-2
# 
w_final, b_final, J_hist, p_hist = gradient_descent(x ,y, w_init, b_init, tmp_alpha, iterations, costFun, gradientFun)
print(f"(w,b) found by gradient descent: ({
      
      w_final:8.4f},{
      
      b_final:8.4f})")

Iteration    0: Cost 7.93e+04  dj_dw: -6.500e+02, dj_db: -4.000e+02   w:  6.500e+00, b: 4.00000e+00
Iteration 1000: Cost 3.41e+00  dj_dw: -3.712e-01, dj_db:  6.007e-01   w:  1.949e+02, b: 1.08228e+02
Iteration 2000: Cost 7.93e-01  dj_dw: -1.789e-01, dj_db:  2.895e-01   w:  1.975e+02, b: 1.03966e+02
Iteration 3000: Cost 1.84e-01  dj_dw: -8.625e-02, dj_db:  1.396e-01   w:  1.988e+02, b: 1.01912e+02
Iteration 4000: Cost 4.28e-02  dj_dw: -4.158e-02, dj_db:  6.727e-02   w:  1.994e+02, b: 1.00922e+02
Iteration 5000: Cost 9.95e-03  dj_dw: -2.004e-02, dj_db:  3.243e-02   w:  1.997e+02, b: 1.00444e+02
Iteration 6000: Cost 2.31e-03  dj_dw: -9.660e-03, dj_db:  1.563e-02   w:  1.999e+02, b: 1.00214e+02
Iteration 7000: Cost 5.37e-04  dj_dw: -4.657e-03, dj_db:  7.535e-03   w:  1.999e+02, b: 1.00103e+02
Iteration 8000: Cost 1.25e-04  dj_dw: -2.245e-03, dj_db:  3.632e-03   w:  2.000e+02, b: 1.00050e+02
Iteration 9000: Cost 2.90e-05  dj_dw: -1.082e-03, dj_db:  1.751e-03   w:  2.000e+02, b: 1.00024e+02
(w,b) found by gradient descent: (199.9929,100.0116)

w_final

199.99285075131766

b_final

100.011567727362

 np.arange(len(J_hist[1000:]))

array([   0,    1,    2, ..., 8997, 8998, 8999])

J_hist[1000:]

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fig, (ax1, ax2) = plt.subplots(1, 2, constrained_layout=True, figsize=(12,4))
ax1.plot(J_hist[:100])
ax2.plot(1000 + np.arange(len(J_hist[1000:])), J_hist[1000:])
ax1.set_title("Cost vs. iteration(start)");  ax2.set_title("Cost vs. iteration (end)")
ax1.set_ylabel('Cost')            ;  ax2.set_ylabel('Cost') 
ax1.set_xlabel('iteration step')  ;  ax2.set_xlabel('iteration step') 
plt.show()

​

​

print(f"1000 sqft house prediction {
      
      w_final*1.0 + b_final:0.1f} Thousand dollars")
print(f"1200 sqft house prediction {
      
      w_final*1.2 + b_final:0.1f} Thousand dollars")
print(f"2000 sqft house prediction {
      
      w_final*2.0 + b_final:0.1f} Thousand dollars")

1000 sqft house prediction 300.0 Thousand dollars
1200 sqft house prediction 340.0 Thousand dollars
2000 sqft house prediction 500.0 Thousand dollars