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BP算法

训练集 $\left\{\left(x^{(1)}, y^{(1)}\right), \ldots,\left(x^{(m)}, y^{(m)}\right)\right\}$

设 $\Delta_{i j}^{(l)}=0(\text { for all } l, i, j)$

$\begin{array}{l}{\text {For } i=1 \text { to } m}\end{array}$

$$\begin{array}{l}{\text { Set } a^{(1)}=x^{(i)}} \\ {\text { Perform forward propagation to compute } a^{(l)} \text { for } l=2,3, \ldots, L} \\ {\text { Using } y^{(i)}, \text { compute } \delta^{(L)}=a^{(L)}-y^{(i)}} \\ {\text { Compute } \delta^{(L-1)}, \delta^{(l+1)}, \ldots, \delta^{(2)}} \\ {\Delta_{i j}^{(l)} :=\Delta_{i j}^{(l)}+a_{j}^{(l)} \delta_{i}^{(l+1)}}\end{array}$$

$\begin{array}{l}{D_{i j}^{(l)} :=\frac{1}{m} \Delta_{i j}^{(l)}+\lambda \Theta_{i j}^{(l)}} & {\text { if } j \neq 0} \\ {D_{i j}^{(l)} :=\frac{1}{m} \Delta_{i j}^{(l)}} & {\text { if } j=0}\end{array}$

其中 $\frac{\partial}{\partial \Theta_{i j}^{(l)}} J(\Theta)=D_{i j}^{(l)}$

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