一元线性回归梯度下降代码

线性回归模型
$$\hat{y}=f_{w,b}(x)=wx+b$$

代价函数 cost function
平均平方误差函数 squared error function
$$J(w,b)=\frac{1}{2m}\sum _{i=1}^m({\hat{y}}^{(i)}-y^{(i)}{)}^2$$

计算偏导
$$\begin{gathered} \frac{\partial }{\partial w}J(w,b)=\frac{\partial }{\partial w}\frac{1}{2m}\sum _{i=1}^m(w{x}^{(i)}+b-y^{(i)}{)}^2 \\ =\frac{1}{2m}\sum _{i=1}^{m}2(w{x}^{(i)}+b-y^{(i)})x^{(i)}=\frac{1}{m}\sum _{i=1}^m{x}^{(i)}(w{x}^{(i)}+b-y^{(i)}) \end{gathered}$$
$$\begin{gathered} \frac{\partial }{\partial b}J(w,b)=\frac{\partial }{\partial b}\frac{1}{2m}\sum _{i=1}^m(w{x}^{(i)}+b-y^{(i)}{)}^2 \\ =\frac{1}{2m}\sum _{i=1}^{m}2(w{x}^{(i)}+b-y^{(i)})=\frac{1}{m}\sum _{i=1}^m(w{x}^{(i)}+b-y^{(i)}) \end{gathered}$$

梯度下降
$w=w-\alpha \frac{\partial J}{\partial w},b=b-\alpha \frac{\partial J}{\partial w}$

import nummpy as np

#代价函数
def compute_cost(x, y, w, b):
   
    m = x.shape[0] 
    cost = 0
    
    for i in range(m):
        f_wb = w * x[i] + b
        cost = cost + (f_wb - y[i])**2
    total_cost = 1 / (2 * m) * cost

    return total_cost

#计算梯度函数
def compute_gradient(x, y, w, b): 

    m = x.shape[0]    
    dj_dw = 0
    dj_db = 0
    
    for i in range(m):  
        f_wb = w * x[i] + b 
        dj_dw_i = (f_wb - y[i]) * x[i] 
        dj_db_i = f_wb - y[i] 
        dj_db += dj_db_i
        dj_dw += dj_dw_i 
    dj_dw = dj_dw / m 
    dj_db = dj_db / m 
        
    return dj_dw, dj_db

#梯度下降函数
def gradient_descent(x, y, w_in, b_in, alpha, num_iters, cost_function, gradient_function): 
   '''
     x:输入向量,numpy.ndarray
     y:输出向量，numpy.ndarray
     w_in:初始w
     b_in:初始b
     alpha:学习率
     num_iters:迭代次数
     cost_function:代价函数
     gradient_function:计算梯度函数
   '''
   
    J_history = [] #记录训练过程中的所有代价
    p_history = [] #记录训练过程中所有(w,b)
    b = b_in
    w = w_in
    
    for i in range(num_iters):
        # 计算偏导，更新参数w，b
        dj_dw, dj_db = gradient_function(x, y, w , b)     
        b = b - alpha * dj_db                            
        w = w - alpha * dj_dw                            

        # 保存当前代价J和参数(w,b)->可用于后续可视化
        J_history.append( cost_function(x, y, w , b))
        p_history.append([w,b])
        # 打印其中十次训练信息
        if i% math.ceil(num_iters/10) == 0:
            print(f"Iteration {i}: Cost {J_history[-1]} ",
                  f"dj_dw: {dj_dw}, dj_db: {dj_db}  ",
                  f"w: {w}, b:{b}")
    print{f'final w:{w},b:{b}'}
    
    #输出目标值与预测值对比
    y_hat=w*x+b 
    for i in range(x.shape[0])：
         print(f'target value:{y[i]},predicted value:{y_hat[i]},error:{y[i]-y_hat[i]}')
    
    return w, b, J_history, p_history