（1）导入数据

问题引入：假设你是老板，要考虑在不同的城市开一家新店；
x_train 是不同城市的人口数量
y_train 是那个城市一家餐馆的利润

import math
import copy
import numpy as npx_train = np.array([6.1101, 5.5277, 8.5186, 7.0032, 5.8598, 8.3829, 7.4764, 8.5781, 6.4862, 5.0546, 5.7107, 14.164, 5.734, 8.4084, 5.6407, 5.3794, 6.3654, 5.1301, 6.4296, 7.0708, 6.1891, 20.27, 5.4901, 6.3261, 5.5649, 18.945, 12.828, 10.957, 13.176, 22.203, 5.2524, 6.5894, 9.2482, 5.8918, 8.2111, 7.9334, 8.0959, 5.6063, 12.836, 6.3534, 5.4069, 6.8825, 11.708, 5.7737, 7.8247, 7.0931, 5.0702, 5.8014, 11.7, 5.5416, 7.5402, 5.3077, 7.4239, 7.6031, 6.3328, 6.3589, 6.2742, 5.6397, 9.3102, 9.4536, 8.8254, 5.1793, 21.279, 14.908, 18.959, 7.2182, 8.2951, 10.236, 5.4994, 20.341, 10.136, 7.3345, 6.0062, 7.2259, 5.0269, 6.5479, 7.5386, 5.0365, 10.274, 5.1077, 5.7292, 5.1884, 6.3557, 9.7687, 6.5159, 8.5172, 9.1802, 6.002, 5.5204, 5.0594, 5.7077, 7.6366, 5.8707, 5.3054, 8.2934, 13.394, 5.4369])
y_train = np.array([17.592, 9.1302, 13.662, 11.854, 6.8233, 11.886, 4.3483, 12., 6.5987, 3.8166, 3.2522, 15.505, 3.1551, 7.2258, 0.71618, 3.5129, 5.3048, 0.56077, 3.6518, 5.3893, 3.1386, 21.767, 4.263, 5.1875, 3.0825, 22.638, 13.501, 7.0467, 14.692, 24.147, -1.22, 5.9966, 12.134, 1.8495, 6.5426, 4.5623, 4.1164, 3.3928, 10.117, 5.4974, 0.55657, 3.9115, 5.3854, 2.4406, 6.7318, 1.0463, 5.1337, 1.844, 8.0043, 1.0179, 6.7504, 1.8396, 4.2885, 4.9981, 1.4233, -1.4211, 2.4756, 4.6042, 3.9624, 5.4141, 5.1694, -0.74279, 17.929, 12.054, 17.054, 4.8852, 5.7442, 7.7754, 1.0173, 20.992, 6.6799, 4.0259, 1.2784, 3.3411, -2.6807, 0.29678, 3.8845, 5.7014, 6.7526, 2.0576, 0.47953, 0.20421, 0.67861, 7.5435, 5.3436, 4.2415, 6.7981, 0.92695, 0.152, 2.8214, 1.8451, 4.2959, 7.2029, 1.9869, 0.14454, 9.0551, 0.61705])

（2）画出城市人口与利润图

通过 python 包 matplotlib.pyplot 画图

import matplotlib.pyplot as pltplt.scatter(x_train, y_train, marker='x', c='g') plt.title("利润*10000/人口数量*10000")
plt.ylabel('利润*10000')
plt.xlabel('人口数量*10000')plt.show()

（3）计算损失值

已知模型为：
$$f_{\vec{w},b}({\vec{x}}^{(i)})=\vec{w}\cdot {\vec{x}}^{(i)}+b$$

损失函数为：
$$cos{t}^{(i)}=(f_{\vec{w},b}({\vec{x}}^{(i)})-y^{(i)}{)}^2$$

总损失函数为：
$$J(\vec{w}.b)=\frac{1}{2m}\sum _{i=0}^{m-1}cos{t}^{(i)}$$

程序实现如下：

def compute_cost(x,y,w,b):m = x.shape[0]total_cost = 0.for i in range(m):f_wb = np.dot(w,x[i]) + bcost = (f_wb - y[i]) ** 2total_cost += costtotal_cost = total_cost / (2 * m)return total_cost

（4）计算梯度下降

梯度下降公式为：
repeat until convergence:{0000b:=b−α∂J(w,b)∂b0000w:=w−α∂J(w,b)∂w}\begin{align*}& \text{repeat until convergence:} \; \lbrace \newline \; & \phantom {0000} b := b - \alpha \frac{\partial J(w,b)}{\partial b} \newline \; & \phantom {0000} w := w - \alpha \frac{\partial J(w,b)}{\partial w} \; & \newline & \rbrace\end{align*}​repeat until convergence:{0000b:=b−α∂b∂J(w,b)​0000w:=w−α∂w∂J(w,b)​}​​

详解 gradient 部分为：
$$\begin{gathered} \frac{\partial J(w,b)}{\partial b}=\frac{1}{m}\sum _{i=0}^{m-1}(f_{w,b}(x^{(i)})-y^{(i)}) \\ \frac{\partial J(w,b)}{\partial w}=\frac{1}{m}\sum _{i=0}^{m-1}(f_{w,b}(x^{(i)})-y^{(i)})x^{(i)} \end{gathered}$$

代码实现 gradient 部分：

def compute_gradient(x,y,w,b):m = x.shape[0]dj_dw = 0dj_db = 0for i in range(m):f_wb = np.dot(w,x[i]) + bdj_dw += (f_wb - y[i]) * x[i]dj_db +=  f_wb - y[i]dj_dw = dj_dw / mdj_db = dj_db / mreturn dj_dw, dj_db

代码实现梯度下降 gradient descent：

def gradient_descent(x,y,w_in,b_in,cost_function,gradient_function,alpha,num_iters):J_history = []w_history = []w = copy.deepcopy(w_in)b = b_infor i in range(num_iters):dj_dw, dj_db = gradient_function(x,y,w,b)w = w - alpha * dj_dwb = b - alpha * dj_dbif i<100000:cost = cost_function(x,y,w,b)J_history.append(cost)if i% math.ceil(num_iters/10) == 0:		# math.ceil: 将传入的参数向上取整为最接近的整数w_history.append(w)print(f"Iteration {i:4}: Cost {float(J_history[-1]):8.2f}   ")return w, b, J_history, w_history

（5）开始训练

initial_w = 0.
initial_b = 0.
iterations = 1500
alpha = 0.01w,b,_,_ = gradient_descent(x_train ,y_train, initial_w, initial_b, compute_cost, compute_gradient, alpha, iterations)
print("w,b found by gradient descent:", w, b)

（6）画出训练好的模型

因为我们已经训练完成模型，所以直接用参数 向量w 与 参数b 的值进行绘图：

m = x_train.shape[0]
predicted = np.zeros(m)for i in range(m):predicted[i] = w * x_train[i] + bplt.plot(x_train, predicted, c = "b")
plt.scatter(x_train, y_train, marker='x', c='g') plt.title("Profits*10000/Population*10000")
plt.ylabel('Profits*10000')
plt.xlabel('Population*10000')

（7）做出预测

predict1 = 3.5 * w + b
print('For population = 35,000, we predict a profit of $%.2f' % (predict1*10000))predict2 = 7.0 * w + b
print('For population = 70,000, we predict a profit of $%.2f' % (predict2*10000))

（8）完整代码

import copy
import math
import numpy as npx_train = np.array([6.1101, 5.5277, 8.5186, 7.0032, 5.8598, 8.3829, 7.4764, 8.5781, 6.4862, 5.0546, 5.7107, 14.164, 5.734, 8.4084, 5.6407, 5.3794, 6.3654, 5.1301, 6.4296, 7.0708, 6.1891, 20.27, 5.4901, 6.3261, 5.5649, 18.945, 12.828, 10.957, 13.176, 22.203, 5.2524, 6.5894, 9.2482, 5.8918, 8.2111, 7.9334, 8.0959, 5.6063, 12.836, 6.3534, 5.4069, 6.8825, 11.708, 5.7737, 7.8247, 7.0931, 5.0702, 5.8014, 11.7, 5.5416, 7.5402, 5.3077, 7.4239, 7.6031, 6.3328, 6.3589, 6.2742, 5.6397, 9.3102, 9.4536, 8.8254, 5.1793, 21.279, 14.908, 18.959, 7.2182, 8.2951, 10.236, 5.4994, 20.341, 10.136, 7.3345, 6.0062, 7.2259, 5.0269, 6.5479, 7.5386, 5.0365, 10.274, 5.1077, 5.7292, 5.1884, 6.3557, 9.7687, 6.5159, 8.5172, 9.1802, 6.002, 5.5204, 5.0594, 5.7077, 7.6366, 5.8707, 5.3054, 8.2934, 13.394, 5.4369])
y_train = np.array([17.592, 9.1302, 13.662, 11.854, 6.8233, 11.886, 4.3483, 12., 6.5987, 3.8166, 3.2522, 15.505, 3.1551, 7.2258, 0.71618, 3.5129, 5.3048, 0.56077, 3.6518, 5.3893, 3.1386, 21.767, 4.263, 5.1875, 3.0825, 22.638, 13.501, 7.0467, 14.692, 24.147, -1.22, 5.9966, 12.134, 1.8495, 6.5426, 4.5623, 4.1164, 3.3928, 10.117, 5.4974, 0.55657, 3.9115, 5.3854, 2.4406, 6.7318, 1.0463, 5.1337, 1.844, 8.0043, 1.0179, 6.7504, 1.8396, 4.2885, 4.9981, 1.4233, -1.4211, 2.4756, 4.6042, 3.9624, 5.4141, 5.1694, -0.74279, 17.929, 12.054, 17.054, 4.8852, 5.7442, 7.7754, 1.0173, 20.992, 6.6799, 4.0259, 1.2784, 3.3411, -2.6807, 0.29678, 3.8845, 5.7014, 6.7526, 2.0576, 0.47953, 0.20421, 0.67861, 7.5435, 5.3436, 4.2415, 6.7981, 0.92695, 0.152, 2.8214, 1.8451, 4.2959, 7.2029, 1.9869, 0.14454, 9.0551, 0.61705])import matplotlib.pyplot as pltplt.scatter(x_train, y_train, marker='x', c='g')plt.title("profits*10000/num(human in a city)*10000")
plt.ylabel('profits*10000')
plt.xlabel('num(human in a city)*10000')plt.show()def compute_cost(x, y, w, b):m = x.shape[0]total_cost = 0.for i in range(m):f_wb = np.dot(w, x[i]) + bcost = (f_wb - y[i]) ** 2total_cost += costtotal_cost = total_cost / (2 * m)return total_costdef compute_gradient(x,y,w,b):m = x.shape[0]dj_dw = 0dj_db = 0for i in range(m):f_wb = np.dot(w,x[i]) + bdj_dw += (f_wb - y[i]) * x[i]dj_db +=  f_wb - y[i]dj_dw = dj_dw / mdj_db = dj_db / mreturn dj_dw, dj_dbdef gradient_descent(x, y, w_in, b_in, cost_function, gradient_function, alpha, num_iters):J_history = []w_history = []w = copy.deepcopy(w_in)b = b_infor i in range(num_iters):dj_dw, dj_db = gradient_function(x, y, w, b)w = w - alpha * dj_dwb = b - alpha * dj_dbif i < 100000:cost = cost_function(x, y, w, b)J_history.append(cost)if i % math.ceil(num_iters / 10) == 0:  # math.ceil: 将传入的参数向上取整为最接近的整数w_history.append(w)print(f"Iteration {i:4}: Cost {float(J_history[-1]):8.2f}   ")return w, b, J_history, w_historyinitial_w = 0.
initial_b = 0.
iterations = 1500
alpha = 0.01w,b,_,_ = gradient_descent(x_train ,y_train, initial_w, initial_b, compute_cost, compute_gradient, alpha, iterations)
print("w,b found by gradient descent:", w, b)m = x_train.shape[0]
predicted = np.zeros(m)for i in range(m):predicted[i] = w * x_train[i] + bplt.plot(x_train, predicted, c = "b")
plt.scatter(x_train, y_train, marker='x', c='g')plt.title("Profits*10000/Population*10000")
plt.ylabel('Profits*10000')
plt.xlabel('Population*10000')plt.show()predict1 = 3.5 * w + b
print('For population = 35,000, we predict a profit of $%.2f' % (predict1*10000))predict2 = 7.0 * w + b
print('For population = 70,000, we predict a profit of $%.2f' % (predict2*10000))