Python——对模型实现正则化

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！