Python 实现梯度下降算法总结

Python 实现梯度下降算法

最近重新回顾了一下机器学习的基础知识，关于梯度下降的知识，个人认为看懂原理和公式，仅仅是一方面，如果能从代码的角度重新实现或者走一遍可能会更加记忆深刻，下面两个参考链接讲的就非常好，大家一块来学习呀。

相关参考：

[1] 深入浅出–梯度下降法及其实现
[2] python写出梯度下降的代码

Python实现版本 [1]

import numpy as np

m = 20  # Size of the points dataset.

X0 = np.ones((m, 1))  # Points x-coordinate and dummy value (x0, x1).
X1 = np.arange(1, m + 1).reshape(m, 1)
X = np.hstack((X0, X1))

y = np.array([  # Points y-coordinate
    3, 4, 5, 5, 2, 4, 7, 8, 11, 8, 12,
    11, 13, 13, 16, 17, 18, 17, 19, 21
]).reshape(m, 1)

alpha = 0.01  # The Learning Rate alpha.


def error_function(theta, X, y):
    """Error function J definition."""
    diff = np.dot(X, theta) - y
    return (1. / 2 * m) * np.dot(np.transpose(diff), diff)


def gradient_function(theta, X, y):
    """Gradient of the function J definition."""
    diff = np.dot(X, theta) - y
    return (1. / m) * np.dot(np.transpose(X), diff)


def gradient_descent(X, y, alpha):
    """Perform gradient descent."""
    theta = np.array([1, 1]).reshape(2, 1)
    gradient = gradient_function(theta, X, y)
    while not np.all(np.absolute(gradient) <= 1e-5):
        theta = theta - alpha * gradient
        gradient = gradient_function(theta, X, y)
    return theta


optimal = gradient_descent(X, y, alpha)
error = error_function(optimal, X, y)[0, 0]
print('optimal:', optimal)
print('error function:', error)
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Python实现版本 [2]

import numpy as np

iterations = 1000  # 定义迭代次数和学习率
alpha = 0.1  # 学习率
m = 100  # 数据长度


def compute_error(X, y, theta):  # 定义损失函数
    predictions = np.dot(X, theta)
    sqrErrors = (predictions - y) ** 2
    J = 1 / (2 * m) * np.sum(sqrErrors)
    return J

def compute_gradient(theta, X, y):  # 梯度更新
    predictions = np.dot(X, theta)
    errors = predictions - y
    theta = theta - alpha / m * np.dot(X.T, errors)
    return theta

def gradient_descent(X, y, theta, num_iters):  # 梯度下降算法
    J_history = np.zeros((num_iters, 1))  # 记录损失函数
    for i in range(num_iters):
        theta = compute_gradient(theta, X, y)
        J_history[i] = compute_error(X, y, theta)
    return theta, J_history

# 测试代码
x = 2 * np.random.rand(m, 1)  # 随机生成一些数据
y = 100 + 3 * x + np.random.randn(m, 1)
X_b = np.c_[np.ones((m, 1)), x]  # 在数据中添加x0=1
theta = np.random.randn(2, 1)  # 初始化theta
theta, J_history = gradient_descent(X_b, y, theta, iterations)  # 运行梯度下降算法
print("最终参数值：", theta)  # 输出最终结果
print("损失函数值：", J_history[-1])
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声明： 总结学习，有问题或不当之处，可以批评指正哦，谢谢。