@leona1992 2022-08-03T15:38:10.000000Z 字数 2105 阅读 100

梯度下降法2

python 机器学习 梯度下降法

生成数据集

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(666)
x = 2 * np.random.random(size=100)
y = x * 3. + 4. + np.random.normal(size=100)
X = x.reshape(-1, 1)
plt.scatter(x,y,marker='.')
plt.title('database')
plt.show()

数据集

梯度下降法进行训练

目标：使得 $J(\theta)=\operatorname{MSE}(y, \hat{y})$ 尽可能小

$\frac{1}{m} \sum_{i=1}^{m}\left(y^{(i)}-\hat{y}^{(i)}\right)^{2}=\frac{1}{m} \sum_{i=1}^{m}\left(y^{(i)}-\theta_{0}-\theta_{1} X_{1}^{(i)}-\theta_{2} X_{2 k}^{(i)}-\ldots-\theta_{n} X_{n}^{(i)}\right)^{2}$
生成

$J(\theta)$ :

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

$\nabla J(\theta)=\left(\begin{array}{c} \partial J / \partial \theta_{0} \\ \partial J / \partial \theta_{1} \\ \partial J / \partial \theta_{2} \\ \ldots \\ \partial J / \partial \theta_{n} \end{array}\right)=\frac{2}{m} \cdot\left(\begin{array}{c} \sum_{i=1}^{m}\left(X_{b}^{(i)} \theta-y^{(i)}\right) \\ \sum_{i=1}^{m}\left(X_{b}^{(i)} \theta-y^{(i)}\right) \cdot X_{1}^{(i)} \\ \sum_{i=1}^{m}\left(X_{b}^{(i)} \theta-y^{(i)}\right) \cdot X_{2}^{(i)} \\ \ldots \\ \sum_{i=1}^{m}\left(X_{b}^{(i)} \theta-y^{(i)}\right) \cdot X_{n}^{(i)} \end{array}\right)$

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
    xb = (np.array(X_b)[:, 1])
    while i_iter < n_iters:
        plt.ion()
        gradient = dJ(theta, X_b, y)
        last_theta = theta
        theta = theta - eta * gradient
        #while (i_iter % 1000) == 1:
        while (i_iter < 150) & ((i_iter % 5) == 1):
            print(i_iter)
            xq = [0, 2]
            yq = theta[0] * np.array(xq) + theta[1] * np.ones(np.shape(theta))
            fig = plt.figure()
            fig = plt.scatter((np.array(X_b)[:, 1]), y)
            fig = plt.plot(xq, yq, linewidth=4, color='darkgoldenrod')
            fig = plt.legend(['data', 'simulate'], loc='upper left')
            fig = plt.title('gradient')
            fig = plt.xlabel(i_iter)
            plt.savefig(r"C:\\Users\\xiaoy\\Pictures\\s\\" + format(i_iter+100) +
                        '.png')
            plt.show()
            print(theta)
            plt.close()
            i_iter = i_iter + 1 
        if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
            break
        i_iter = i_iter + 1
    return theta
theta = gradient_descent(X_b, y, initial_theta, eta)
print("荣耀不朽")