ML

Week1

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（这里输入训练数据x为m*n矩阵, 线性参数解析失败 (语法错误): {\displaystyle θ} 为n*1，$x_i$代表训练矩阵中的第i行，$x_{ik}$代表第i行第k列）：
解析失败 (语法错误): {\displaystyle \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}{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:

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

解析失败 (语法错误): {\displaystyle = \frac{1}{m} (h_θ(x)-y) x_{j} }

Week2

multivariate linear regression

解析失败 (语法错误): {\displaystyle h_θ(x) = θ_0x_0 + θ_1x_1 + θ_2x_2 + ... + θ_nx_n = θ^Tx}
其中，
解析失败 (语法错误): {\displaystyle x=\begin{vmatrix} x_0 \\ x_1 \\ x_2 \\ ... \\ x_m \end{vmatrix} = \begin{vmatrix} x_0^{(0)} & x_0^{(1)} & x_0^{(2)} & ... & x_0^{(n)} \\ x_1^{(0)} & x_1^{(1)} & x_1^{(2)} & ... & x_1^{(n)} \\ ... & ... & ... & ... & ...\\ x_m^{(0)} & x_m^{(1)} & x_m^{(2)} & ... & x_m^{(n)} \\ \end{vmatrix} , θ=\begin{vmatrix} θ_0 \\ θ_1\\ θ_2\\ ...\\ θ_n \end{vmatrix}, m为训练数据组数，n为特征个数。 }