一、准备工作
1. 测试函数:
向这个g(x)函数加入一点噪声的训练数据,然后将它们画成图
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()
虚线就是正确的g(x)的图形,圆点就是加入了一点噪声的训练数据。先准备了8个数据。
2. 创建训练矩阵,10次多项式
# 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)
二、不应用正则化的实现
代码如下:
# 目标函数
def E(x, y):
return 0.5 * np.sum((y - f(x)) ** 2)
# 正则化常量
LAMBDA = 0.5
# 学习率
ETA = 1e-4
# 误差
diff = 1
# 重复学习(不应用正则化)
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
# 绘图
z = standardize(np.linspace(-2, 2, 100))
theta = theta1 # 未应用正则化
plt.plot(z, f(to_matrix(z)), linestyle='dashed')
plt.show()
图像如下:
三、应用了正则化的实现
代码如下:
theta1 = theta
# 重复学习(应用正则化)
theta = np.random.randn(X.shape[1])
diff = 1
error = E(X, train_y)
while diff > 1e-6:
reg_term = LAMBDA * np.hstack([0, theta[1:]])
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
theta = theta - ETA * (np.dot(f(X) - train_y, X) + reg_term)
plt.plot(train_z, train_y, 'o')
plt.plot(z, f(to_matrix(z)))
plt.show()
图像如下:
为了便于比较,把未应用和应用了正则化这两种情况展示在一 张图上,完整代码如下:
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
# 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)
# 正则化常量
LAMBDA = 0.5
# 学习率
ETA = 1e-4
# 误差
diff = 1
# 重复学习(不应用正则化)
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
theta1 = theta
# 重复学习(应用正则化)
theta = np.random.randn(X.shape[1])
diff = 1
error = E(X, train_y)
while diff > 1e-6:
reg_term = LAMBDA * np.hstack([0, theta[1:]])
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
theta2 = theta
# 绘图确认
plt.plot(train_z, train_y, 'o')
z = standardize(np.linspace(-2, 2, 100))
theta = theta1 # 未应用正则化
plt.plot(z, f(to_matrix(z)), linestyle='dashed')
theta = theta2 # 应用正则化
plt.plot(z, f(to_matrix(z)))
plt.show()
效果如图:
正则化的实际效果,更拟合数据点!