李宏毅机器学习课程笔记Lesson1-Regression - Case Study

machine learning 有三个步骤：
step 1 ：选择 a set of function, 即选择一个 model

step 2 ：评价goodness of function

step 3 ：选出 best function

采用梯度下降得到最佳答案
gradient descent 的步骤是：
先选择参数的初始值，再向损失函数对参数的负梯度方向迭代更新，learning rate控制步子大小、学习速度。梯度方向是损失函数等高线的法线方向。

基于以上步骤，测试y=w*x+b的代码实现

import np
import matplotlib.pyplot as plt  # plt

x_data = [1,2,3,4,5,6]
y_data = [3,5,7,9,11,13]

# yadata = b + w*xdata
b = -1 # intial b
w = 1 # intial w
lr = 0.000001 # learning rate
iteration = 100000000#iterate times

for i in range(iteration):

    b_grad = 0.0
    w_grad = 0.0
    for n in range(len(x_data)):
        b_grad = b_grad - 2.0*(y_data[n] - (b + w*x_data[n]))*1.0
        w_grad = w_grad - 2.0*(y_data[n] - (b + w*x_data[n]))*x_data[n]

    # update parameters
    b = b - lr*b_grad
    w = w - lr*w_grad

b = round(b,1)
w = round(w,1)
print(b,w)
plt.scatter(x_data, y_data, s=200, label = '$test1$', c = 'blue', marker='.', alpha = None, edgecolors= 'blue' )
x = np.arange(0, 20, 1)
plt.plot(x, b+w*x,color='red')  
plt.legend()
plt.show()

实验结果：

测试y=w1*x^2+w2*x+b的代码实现

import np
import matplotlib.pyplot as plt  # plt

x_data = []
y_data = []
for i in range(10):
    x_data.append(i)
    y_data.append(2*i*i+i+1)
# y= w1*x^2 + w2*x+b
w1 = 1#intial w1
w2 = 0.5#intial w2
b = 0.5#intial b
lr = 0.00000001# learning rate
iteration = 10000000#iterate times

for i in range(iteration):
    w1_grad = 0.0
    w2_grad = 0.0
    b_grad = 0.0
    for n in range(len(x_data)):
        t = 2*(y_data[n]-(w1*x_data[n]*x_data[n]+w2*x_data[n]+b))
        w1_grad = w1_grad + t*(-x_data[n]*x_data[n])
        w2_grad = w2_grad + t*(-x_data[n])
        b_grad = b_grad + t*(-1)
    w1 = w1 - lr*w1_grad
    w2 = w2 - lr*w2_grad
    b = b - lr*b_grad

w1 = round(w1,1)
w2 = round(w2,1)
b = round(b,1)
print(w1,w2,b)

x = np.arange(0, 20, 1)
v = 0.0
for n in range(len(x_data)):
    v = v +  (y_data[n] - (w1*x_data[n]*x_data[n]+w2*x_data[n]+b))**2
v = v/(len(x_data))
v = round(v,3)
plt.plot(x, w1*x*x+w2*x+1,color='red')  
l = 'w1 = %s,w2 = %s, b = %s ,variance=%s' % (w1,w2,b,v) 
plt.scatter(x_data, y_data, s=200, label = '$test2$', c = 'blue', marker='.', alpha = None, edgecolors= 'blue' )
plt.xlabel(l)
plt.legend()
plt.show()

实验结果：设置的w1=2,w2=1,b=1,实验结果有误差，可通过调整learning rate 和迭代次数、增加测试数据等调试