梯度下降法

梯度下降法是一种常用迭代方法，其目的是让输入向量找到一个合适的迭代方向，使得输出值能达到局部最小值。在拟合线性回归方程时，我们把损失函数视为以参数向量为输入的函数，找到其梯度下降的方向并进行迭代，就能找到最优的参数值。

1.计算对于给定的线性模型 (y = wx + b) 的均方误差（MSE）。它接受截距 (b)、斜率 (w) 和点集 (points)，然后遍历所有点，计算每个点的预测值，与真实值之差的平方和，最后返回平均误差。

2.更新w和b

3.多次迭代后，得到最优的w和b，也就是y=wx+b这个模型对于给定数据集的最优

这里给定数据集：100个（x,y）

import torch
import numpy as np

#计算给定点集的线性回归的误差  y = wx + b
def compute_error_for_line_given_points(b,w,points):
    total_error = 0
    for i in range(len(points)):
        x = points[i,0]
        y = points[i,1]
        total_error += (y - (w*x + b))**2
    return total_error/float(len(points))

#梯度下降法求解线性回归  w = w - learning_rate * w_gradient, b = b - learning_rate * b_gradient
def step_gradient(b_current,w_current,points,learning_rate):
    b_gradient = 0
    w_gradient = 0
    n = float(len(points))
    for i in range(len(points)):
        x = points[i,0]
        y = points[i,1]
        b_gradient += -(2/n) * (y - ((w_current*x) + b_current))
        w_gradient += -(2/n) * x * (y - ((w_current*x) + b_current))
    new_b = b_current - (learning_rate * b_gradient)
    new_w = w_current - (learning_rate * w_gradient)
    return [new_b,new_w]

#迭代梯度下降法求解线性回归
def gradient_descent_runner(points,starting_b,starting_w,learning_rate,num_iterations):
    b = starting_b
    w = starting_w
    for i in range(num_iterations):
        b,w = step_gradient(b,w,points,learning_rate)
    return [b,w]

def run():
    points = np.genfromtxt('data.csv', delimiter=',')
    learning_rate = 0.0001
    initial_b = 0
    initial_w = 0
    num_iterations = 1000
    print("Starting gradient descent at b = {0}, w = {1}, error = {2}".format(initial_b,initial_w,compute_error_for_line_given_points(initial_b,initial_w,points)))
    print("Running...")
    [b,w] = gradient_descent_runner(points,initial_b,initial_w,learning_rate,num_iterations)
    print("After {0} iterations b = {1}, w = {2}, error = {3}".format(num_iterations,b,w,compute_error_for_line_given_points(b,w,points)))

if __name__ == '__main__':
    run()

执行结果：