实现代码：

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import numpy as np
import matplotlib.pyplot as plt
 
# 真正的函数
def g(x):
    return 0.1 * (x ** 3 + x ** 2 + x)
 
# 随意准备一些向真正的函数加入了一点噪声的训练数据
train_x = np.linspace (-2,2,8)
train_y =g(train_x)+ np.random.randn(train_x.size) * 0.05
 
# 绘图确认
x = np.linspace(-2,2,100)
plt.plot(train_x, train_y, 'o')
plt.plot(x, g(x), linestyle = 'dashed')
plt.ylim(-1, 2)
plt.show()
 
##########
 
# 标准化
mu = train_x.mean ()
sigma = train_x.std()
def standardize (x):
    return (x - mu)/ sigma
 
train_z = standardize(train_x)
 
# 创建训练数据的矩阵
def to_matrix(x):
    return np.vstack([ 
        np.ones(x.size),
        x,    
        x ** 2,
        x ** 3,
        x ** 4,
        x ** 5,
        x ** 6,
        x ** 7,                                           
        x ** 8,
        x ** 9, 
        x ** 10,
        ]).T 
 
X = to_matrix(train_z)
 
#参数初始化
theta= np.random.randn(X.shape[1])
 
# 预测函数
def f (x):
    return np.dot (x,theta)
 
##########
 
# 目标函数
def E(x, y):
     return 0.5 * np.sum((y - f(x)) ** 2)
 
# 学习率
ETA = 1e-4
 
# 误差
diff = 1
# 重复学习
error = E(X,train_y)
while diff > 1e-6:
    theta= theta - ETA * np.dot(f(X) - train_y,X)
    current_error = E(X,train_y)
    diff = error - current_error
    error = current_error
    
# 对结果绘图
z = standardize(x)
plt.plot(train_z, train_y, 'o')
plt.plot(z, f(to_matrix(z)))
plt.show()
 
##########
 
# 保存未正则化的参数，然后再次参数初始化
theta1 = theta
theta = np.random.randn(X.shape[1])
 
# 正则化常量
LAMBDA = 1
 
#误差
diff = 1
 
#重复学习（包含正则化项）
error = E(X, train_y)
while diff > 1e-6:
    # 正则化项。偏置项不适用正则化，所以为 0
    reg_term = LAMBDA * np.hstack([0, theta[1:]])
    # 应用正则化项，更新参数
    theta = theta - ETA *(np.dot(f(X) - train_y,X)+ reg_term) 
    current_error = E(X, train_y)
    diff = error - current_error
    error = current_error
 
# 对结果绘图
plt.plot(train_z,train_y,'o')
plt.plot(z,f(to_matrix(z)))
plt.show()
 
##########
 
# 保存应用了正则化的参数
theta2 = theta
 
plt.plot(train_z, train_y, 'o')
 
# 画出未应用正则化的结果
theta = theta1
plt.plot(z, f(to_matrix(z)), linestyle = 'dashed')
# 画出应用了正则化的结果
theta = theta2
plt.plot(z, f(to_matrix(z)))
 
plt.show()