ML

Cost Function损失函数

Squared error function/Mean squared function均方误差: 解析失败 (语法错误): {\displaystyle J(θ)=\frac{1}{2m}\sum_{i=1}^m(h_θ(x_i)-y_i)^2}
Cross entropy交叉熵: 解析失败 (语法错误): {\displaystyle 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

解析失败 (语法错误): {\displaystyle 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：
解析失败 (语法错误): {\displaystyle α\frac{∂}{∂θ_j}J(θ)= \frac{∂}{∂θ_j}(\frac{1}{2m}\sum_{i=1}^m(h_θ(x_i)-y_i)^2)}

解析失败 (语法错误): {\displaystyle = \frac{1}{2m}\frac{∂}{∂θ_j}(\sum_{i=1}^m(h_θ(x_i)-y_i)^2)}

解析失败 (语法错误): {\displaystyle = \frac{1}{2m}\sum_{i=1}^m( \frac{∂}{∂θ_j}(h_θ(x_i)-y_i)^2 )}

解析失败 (语法错误): {\displaystyle = \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}

解析失败 (语法错误): {\displaystyle = \frac{1}{m}\sum_{i=1}^m( (h_θ(x_i)-y_i) \frac{∂}{∂θ_j}x_iθ ) }

解析失败 (语法错误): {\displaystyle = \frac{1}{2m}\frac{∂}{∂θ_j} \sum_{i=1}^m(x_iθ-y_i)^2 }

解析失败 (语法错误): {\displaystyle = \frac{1}{m}\frac{∂}{∂θ_j} \sum_{i=1}^mx_{ij}θ_j //链式求导法式}