【梯度下降算法】{3} ——实现线性回归中的梯度下降法

此处准备实现线性回归中的梯度下降法。

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一、构造数据：

import numpy as np
import matplotlib.pyplot as plt

x = 2 * np.random.random(size=100) # 在[0,1)范围内返回随机生成的实数
y = x * 3. + 4. + np.random.normal(size=100) # 添加噪声

X = x.reshape(-1, 1) # 变成只有一列

plt.scatter(x, y)
plt.show()

Output：

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二、使用梯度下降法训练：

回顾一下梯度下降法：

import numpy as np

def J(theta, X_b, y):
    try:
        return np.sum((y - X_b.dot(theta))**2) / len(X_b)
    except:
        return float('inf')

def dJ(theta, X_b, y):
    res = np.empty(len(theta))
    res[0] = np.sum(X_b.dot(theta) - y)
    for i in range(1, len(theta)):
        res[i] = (X_b.dot(theta) - y).dot(X_b[:,i])
    return res * 2 / len(X_b)

def gradient_descent(X_b, y, initial_theta, eta, n_iters = 1e4, epsilon = 1e-8):
    
    theta = initial_theta
    i_iter = 0
    
    while i_iter < n_iters:
        gradient = dJ(theta, X_b, y)
        last_theta = theta
        theta = theta - eta * gradient
        
        if(abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
            break
            
        i_iter += 1
        
    return theta
    
X_b = np.hstack([np.ones((len(x), 1)), x.reshape(-1, 1)])
initial_theta = np.zeros(X_b.shape[1])
eta = 0.01

theta = gradient_descent(X_b, y, initial_theta, eta)
print(theta)

Output：

最初的theta从0开始，theta0是截距、theta1是斜率。

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三、封装梯度下降法：

在playML文件夹中写入LinearRegression模块如下：

import numpy as np
from .metrics import r2_score

class LinearRegression:

    def __init__(self):
        """初始化Linear Regression模型"""
        self.coef_ = None
        self.intercept_ = None
        self._theta = None

    def fit_gd(self, X_train, y_train, eta=0.01, n_iters=1e4):
    	"""根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型"""
        def J(theta, X_b, y):
            try:
                return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
            except:
                return float('inf')

        def dJ(theta, X_b, y):
            res = np.empty(len(theta))
            res[0] = np.sum(X_b.dot(theta) - y)
            for i in range(1, len(theta)):
                res[i] = (X_b.dot(theta) - y).dot(X_b[:, i])
            return res * 2 / len(X_b)
            # return X_b.T.dot(X_b.dot(theta) - y) * 2. / len(X_b)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y) # 计算当前点的梯度
                last_theta = theta # 记录移动前的位置
                theta = theta - eta * gradient # 往代价函数小的方向移动
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break

                cur_iter += 1

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

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四、使用封装之后的梯度下降法：

from playML.LinearRegression import LinearRegression

lin_reg = LinearRegression()
lin_reg.fit_gd(X, y)

print(lin_reg.coef_)
print(lin_reg.intercept_)

感觉代码没问题，但是输出的斜率和截距和封装前的就是不一样……

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参考资料：bobo老师机器学习教程