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1. 正则化 regularization

   

   虚线是真实函数g(x) 圆点是加入的噪声：

   

   不应用正则化的实现:

   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)#加入噪声
   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)/sigmatrain_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,]).TX = 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)#不应用正则化的实现#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_errorerror = 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()
   

   过拟合的曲线：

   

   应用正则化的实现:

   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)#加入噪声
   train_x = np.linspace(-2,2,8)
   train_y = g(train_x) + np.random.randn(train_x.size)*0.05x = np.linspace(-2,2,100)#标准化
   mu = train_x.mean()
   sigma = train_x.std()
   def standardize(x):return (x-mu)/sigmatrain_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,]).TX = 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)#应用正则化的实现
   #正则化常量
   LAMBDA = 1#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:#正则化项 偏置项不适合用正则化 所以为0reg_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_errorerror = 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()