明月机器学习系列（五）：从零动手实现梯度下降

决定给这个系列起一个响亮的名字：明月机器学习系列

import numpy as np

# 使用梯度下降算法训练模型：y = ax + b
# 给定两个点(1,2)和(2,1)，过这两个点拟合一条曲线
points = [(1,2), (2,1)] # 样本数据
learning_rate = 0.1     # 学习率
epochs = 100            # 迭代次数
w = (0, 0)              # 初始权重

def cal_loss(a, b):
    """计算损失"""
    sum_loss = [(a*x+b-y)**2 for x, y in points]
    return sum(sum_loss) / len(points)

def gradient_a(a, b):
    """损失函数在a方向上的梯度：损失函数对a求偏导"""
    loss_grad = [2*(a*x+b-y)*x for x,y in points]
    return sum(loss_grad)

def gradient_b(a, b):
    """损失函数在b方向上的梯度：损失函数对b求偏导"""
    loss_grad = [2*(a*x+b-y) for x,y in points]
    return sum(loss_grad)

def update(a, b):
    """更新权重"""
    return (a - learning_rate * gradient_a(a, b), 
            b - learning_rate * gradient_b(a, b))
            
for i in range(epochs):
    w = update(w[0], w[1])
    loss = cal_loss(w[0], w[1])
    if i % 10 == 9:
        print("a=%.3f, b=%.3f, loss=%.3f" % (w[0], w[1], loss))

a=0.800, b=0.600, loss=1.800
a=0.440, b=0.480, loss=1.296
a=0.512, b=0.624, loss=1.166
a=0.426, b=0.667, loss=1.092
a=0.400, b=0.745, loss=1.028
。。。。。。
a=-0.916, b=2.865, loss=0.004

import numpy as np

# 使用梯度下降算法训练模型：Y = AX 
# 支持多个特征
# points = [(1,1), (2,2), (4,2)]        # 样本数据: y = ax + b
points = [(1,1,1), (1,2,2), (2,2,3)]  # 样本数据: y = ax1 + bx2 + c
learning_rate = 0.01     # 学习率，注意对于[(1,1), (2,2), (4,2)]，如果学习率为0.1，将不再收敛
epochs = 500             # 迭代次数

def parse_points(points):
    """生成特征集合目标集"""
    return np.array([np.append(row[:-1], 1) for row in points]), np.array([row[-1] for row in points])

def init_w(n):
    """初始化权重"""
    return np.zeros(n)

def cal_loss(w):
    """计算损失"""
    sum_loss = [(sum(w*x)-y)**2 for x, y in zip(train_X, train_Y)]
    return sum(sum_loss) / len(points)

def gradient(w, i):
    """计算损失函数在第i个权重上的梯度"""
    loss_grad = [2*(sum(w*x) - y)*x[i] for x, y in zip(train_X, train_Y)]
    grad = sum(loss_grad)
    return grad

def update(w):
    """更新权重"""
    new_w = [wi - learning_rate * gradient(w, i) for i, wi in zip(range(len(w)), w)]
    return np.array(new_w)

train_X, train_Y = parse_points(points)
print('特征字段：', train_X)
print('目标字段：', train_Y)
w = init_w(len(train_X[0]))
print('初始权重：', w)

for i in range(epochs):
    w = update(w)
    loss = cal_loss(w)
    if i % 10 == 9:
        print(w, loss)