ML ML 0

Cost Function损失函数

Squared error function/Mean squared function均方误差: $$J(θ)=\frac{1}{2m}\sum_{i=1}^m(h_θ(x_i)-y_i)^2$$
Cross entropy交叉熵: $$J(θ)=-\frac{1}{m}\sum_{i=1}^m[y^{(i)}*logh_θ(x^{(i)})+(1-y^{(i)})*log(1-h_θ(x^{(i)}))]$$

Gradient Descent梯度下降

$$θ_j:=θ_j+α\frac{∂}{∂θ_j}J(θ)$$
对于线性模型，其损失函数为均方误差，故有（这里输入训练数据x为m*n矩阵, 线性参数 $$θ$$ 为n*1）：
$$\frac{∂}{∂θ_j}J(θ)= \frac{∂}{∂θ_j}(\frac{1}{2m}\sum_{i=1}^m(h_θ(x_i)-y_i)^2)$$

$$= \frac{1}{2m}\frac{∂}{∂θ_j}(\sum_{i=1}^m(h_θ(x_i)-y_i)^2)$$

$$= \frac{1}{2m}\sum_{i=1}^m( \frac{∂}{∂θ_j}(h_θ(x_i)-y_i)^2 )$$

$$= \frac{1}{m}\sum_{i=1}^m( (h_θ(x_i)-y_i) \frac{∂}{∂θ_j}h_θ(x_i) ) //链式求导法式$$

$$= \frac{1}{m}\sum_{i=1}^m( (h_θ(x_i)-y_i) \frac{∂}{∂θ_j}x_iθ )$$

$$= \frac{1}{m}\sum_{i=1}^m( (h_θ(x_i)-y_i) \frac{∂}{∂θ_j}\sum_{k=0}^{n-1}x_{ik}θ_k )$$

对于j>=1:

$$= \frac{1}{m}\sum_{i=1}^m( (h_θ(x_i)-y_i) x_{ij} )$$

$$= \frac{1}{m} (h_θ(x)-y) x_{j}$$