### 基础公式 1、线性回归方程为：$y=ax+b$ 2、损失函数 $loss=\sum_i{(w*x_i+b-y_i)^2}$ ### 求解回归问题 1、先随机初始化一个$w$和$b$,然后求解出$w'$和$b'$，然后根据求导的方向再更新w和b。 2、已知有1000个数据点[(x0,y0),(x1,y1)....(x1000,y1000)] 3、把1000个数据点带入$loss$公式，求得loss 具体代码如下 ```py def compute_error_for_line_given_points(b, w, points): totalError = 0 for i in range(0, len(points)): x = points[i, 0] y = points[i, 1] # computer mean-squared-error totalError += (y - (w * x + b)) ** 2 # average loss for each point return totalError / float(len(points)) ``` 4、求$loss$对$w$导数 (gradient) $\frac{\partial loss}{\partial w}=2\sum_i(wx_i+b-y_i)x_i$ 5、求$loss$对$b$导数 (gradient) $\frac{\partial loss}{\partial b}=2\sum_i(wx_i+b-y_i)$ 6、求解新的w和b(learningRate 为前进的步长，gradient可以求一个平均值) $new_w=w-learningRate*\frac{\partial loss}{\partial w}$ $new_b=b-learningRate*\frac{\partial loss}{\partial b}$ 具体代码如下： ```py 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] # grad_b = 2(wx+b-y) b_gradient += (2/N) * ((w_current * x + b_current) - y) # grad_w = 2(wx+b-y)*x w_gradient += (2/N) * x * ((w_current * x + b_current) - y) # update w' new_b = b_current - (learningRate * b_gradient) new_w = w_current - (learningRate * w_gradient) return [new_b, new_w] ``` 7、最后反复用求得的w和b计算loss，求得最好的loss，从而得到最好的w和b。