梯度下降
y = x 2 ∗ s i n ( x ) y ′ = 2 x s i n x + x 2 c o s x x 1 = x − Δ y ′ ( x ) 其中 Δ 表示学习率, y ′ ( x ) 代表 y 在 x 点处关于 x 的梯度。 y = x^2 * sin(x) \\ y' = 2xsinx+x^2cosx \\ x_1 = x - \Delta y'(x) \\ 其中 \Delta 表示学习率, y'(x)代表y在x点处关于x的梯度。 y=x2∗sin(x)y′=2xsinx+x2cosxx1=x−Δy′(x)其中Δ表示学习率,y′(x)代表y在x点处关于x的梯度。
注意:
在单变量中,梯度就等于导数。
在多变量中,梯度方向表示函数值增加最快的方向。
1、学习率过大会使算法难以收敛,且波动很大
2、学习率过低可能导致算法收敛过慢
具体代码:
csharp
import numpy as np
# y = wx+b
# loss = (wx+b-y)^2
# 计算损失
def compute_loss(w,b,points):
totalError = 0
for i in range(0, len(points)):
x = points[i, 0]
y = points[i, 1]
totalError += (y - (w * x + b)) ** 2
return totalError / float(len(points))
# 梯度下降
# w' = w - lr * dloss/dw
# dl/dw = 2(wx+b-y)x
# dl/db = 2(wx+b-y)
def step_gradient(b_current, w_current, points, learningRate):
b_gradient = 0
w_gradient = 0
N = float(len(points))
for i in range(0, len(points)):
x = points[i, 0]
y = points[i, 1]
b_gradient += (2/N) * (((w_current * x + b_current )) - y) * x
w_gradient += (2/N) * (w_current * x + b_current - y)
new_b = b_current - (learningRate * b_gradient)
new_w = w_current - (learningRate * w_gradient)
return [new_b, new_w]
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, np.array(points), learning_rate)
return [b, w]
def generate_data(num_points):
np.random.seed(42)
x = 2 * np.random.rand(num_points,1)
y = 4 + 3 * x + np.random.randn(num_points, 1)
points = np.hstack((x,y))
return points
def run():
# points = np.genfromtxt("data.csv", delimiter=',')
points = generate_data(100)
learning_rate = 0.0001
initial_b = 0
initial_w = 0
num_iterations = 100
print("Starting gradient descent at b = {0}, m = {1}, error = {2}".format(initial_b,initial_w,compute_loss(initial_b,initial_w,points)))
print("Runing....")
[b , w] = gradient_descent_runner(points, initial_b, initial_w, learning_rate, num_iterations)
print("After {0} iterations b = {1}, m = {2}, loss = {3}".format(num_iterations, b, w, compute_loss(b,w,points)))
if __name__ == '__main__':
run()