gradient descent python code implementations (multiple linear regression minimizing loss function)
1, the gradient descent method is mainly used to minimize the loss function, is a relatively common method of optimization, which specifically includes the following two different ways: batch gradient descent method (the fastest search along the direction of least gradient value) and the stochastic gradient descent method (mainly stochastic gradient descent, by iteration convergence to a minimum)
(1) batch gradient descent
(2) stochastic gradient descent method ( learning rate eta increases with reduced training times continuously, using the principle of simulated annealing, is no longer a fixed value)
2, the multiple linear regression mathematical quantized gradient descent calculation principle :
3, python principle function codes to achieve the following two methods:
(1) batch gradient descent method:
# multiple linear regression using a gradient descent method to obtain the minimum loss function
Import numpy AS NP
Import matplotlib.pyplot PLT AS
NP. random.seed (666)
X = np.random.random (size = 100)
Y = X + 3.0 * + np.random.normal. 4 (size = 100)
X-x.reshape = (-1,1)
Print (X- )
Print (x.shape)
Print (y.shape)
plt.scatter (X, Y)
plt.show ()
Print (X-)
Print (len (X-))
#1使用梯度下降法训练
def J1(theta,x_b,y):
return np.sum((y-x_b.dot(theta))**2)/len(x_b)
def DJ2(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]=np.sum((x_b.dot(theta)-y).dot(x_b[:,i]))
return res*2/len(x_b)
def DJ1(theta, x_b, y):
return x_b.T.dot(x_b.dot(theta)-y)*2/len(y)
def gradient_descent1(x_b,y,eta,theta_initial,erro=1e-8, n=1e4):
theta=theta_initial
i=0
while i<n:
gradient = DJ1(theta,x_b,y)
last_theta = theta
theta = theta - gradient * eta
if (abs(J1(theta,x_b,y) - J1(last_theta,x_b,y))) < erro:
break
i+=1
return theta
x_b=np.hstack([np.ones((len(X),1)),X])
print(x_b)
theta0=np.zeros(x_b.shape[1])
eta=0.1
theta1=gradient_descent1(x_b,y,eta,theta0)
print(theta1)
from sklearn.linear_model import LinearRegression
l=LinearRegression()
l.fit(X,y)
print(l.coef_)
print(l.intercept_)
#2随机梯度下降法的函数原理代码（多元线性回归为例）：
#1-1写出损失函数的表达式子
def J_SGD(theta, x_b, y):
return np.sum((y - x_b.dot(theta)) ** 2) / len(x_b)
#1-2写出梯度胡表达式
def DJ_SGD(theta, x_b_i, y_i):
return x_b_i.T.dot(x_b_i.dot(theta)-y_i)*2
#1-3写出SGD随机梯度的函数形式
def SGD(x_b, y, theta_initial, n):
t0=5
t1=50
def learning_rate(t):
return t0/(t+t1) #计算学习率eta的表达式，需要随着次数的增大而不断的减小
theta = theta_initial #定义初始化的点（列阵）
for i1 in range(n): #采用不断增加次数迭代计算的方式来进行相关的计算
rand_i=np.random.randint(len(x_b)) #生成随机的索引值，计算随机梯度
gradient = DJ_SGD(theta, x_b[rand_i], y[rand_i])
theta = theta - gradient *learning_rate(i1)
return theta
np.random.seed(666)
x=np.random.random(size=100)
y=x*3.0+4+np.random.normal(size=100)
X=x.reshape(-1,1)
print(X)
print(x.shape)
print(y.shape)
plt.scatter(x,y)
plt.show()
print(X)
print(len(X))
#1-4初始化数据x,y以及定义超参数theta0,迭代次数n
x_b=np.hstack([np.ones((len(X),1)),X])
print(x_b)
theta0=np.zeros(x_b.shape[1])
theta1=SGD(x_b,y,theta0,100000)
print(theta1)
