梯度下降算法几点说明
给定数据集即样本点
求出拟合的直线，给定模型f(x)=kx b,k,b为要求的参数
定义损失函数(Loss function)，回归问题里常用的是平方损失函数
初始化模型f(x)=x 1,即k,b都为1
步长即学习率alpha
代码如下：
代码语言：txt复制import numpy as np
import matplotlib.pyplot as plt
# Size of the points dataset.
m = 20
# Points x-coordinate and dummy value (x0, x1).
X0 = np.ones((m, 1))
X1 = np.arange(1, m 1).reshape(m, 1)
X = np.hstack((X0, X1))
# Points y-coordinate
y = np.array([
3, 4, 5, 5, 2, 4, 7, 8, 11, 8, 12,
11, 13, 13, 16, 17, 18, 17, 19, 21
]).reshape(m, 1)
# The Learning Rate alpha.
alpha = 0.01
def plot_graph(theta):
x = np.linspace(1, 20, 100)
fx = theta[1, 0] * x theta[0, 0]
plt.plot(x, fx)
def error_function(theta, X, y):
'''Error function'''
diff = np.dot(X, theta) - y
return (1./(2*m)) * np.dot(np.transpose(diff), diff)
def gradient_function(theta, X, y):
'''Gradient function'''
diff = np.dot(X, theta) - y
return (1.0/m)* np.dot(np.transpose(X), diff)
def gradient_descent(X, y, alpha):
'''Perform gradient descent.'''
theta = np.array([1, 1]).reshape(2, 1)
last_error = error_function(theta, X, y)[0, 0]
while True:
#plot_graph(theta)
gradient = gradient_function(theta, X, y)
theta = theta - alpha * gradient
new_error = error_function(theta, X, y)[0, 0]
if(np.absolute(last_error-new_error) <= 1e-5):
break
last_error = new_error
#print(gradient)
return theta
optimal = gradient_descent(X, y, alpha)
print('optimal:', optimal)
print('error function:', error_function(optimal, X, y)[0,0])
x=np.linspace(1,20,100)
fx=optimal[1,0]*x optimal[0,0]
plt.plot(x,fx)
plt.scatter(np.transpose(X1),np.transpose(y))
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph')
plt.show()拟合效果：
myplot.png
