深度学习: BP (反向传播) 计算 & 链式法则

BP

每个epoch： \qquad 每个batch： \qquad\qquad 每个level (n = N, … to 1，即从顶层往底层)： \qquad\qquad\qquad 分别计算出该层误差（对该层参数、该层输入数据）的导数： \qquad\qquad\qquad\quad 1. ∂L∂ωn=∂L∂xn+1∂xn+1∂ωn∂L∂ωn=∂L∂xn+1∂xn+1∂ωn\frac{\partial L}{\partial \omega^{n}} = \frac{\partial L}{\partial x^{n+1}} \frac{\partial x^{n+1}}{\partial \omega^{n}} (更新本level的ωnωn\omega^{n}时即用) \qquad\qquad\qquad\quad 2. ∂L∂xn=∂L∂xn+1∂xn+1∂xn∂L∂xn=∂L∂xn+1∂xn+1∂xn\frac{\partial L}{\partial x^{n}} = \frac{\partial L}{\partial x^{n+1}} \frac{\partial x^{n+1}}{\partial x^{n}} (留给底一层的level用) \qquad\qquad\qquad 更新参数： \qquad\qquad\qquad\quad 1. ωn←ωn−η∂L∂ωnωn←ωn−η∂L∂ωn\omega^{n} \leftarrow \omega^{n} - \eta \frac{\partial L}{\partial \omega^{n}} \qquad\qquad\qquad\quad 2. bn←bn−η∂L∂bnbn←bn−η∂L∂bnb^{n} \leftarrow b^{n} - \eta \frac{\partial L}{\partial b^{n}}

Arg:

• ωω\omega：欧米茄；
• ηη\eta：艾塔。

Note：

• BP中的 ∂L∂ωn∂L∂ωn\frac{\partial L}{\partial \omega^{n}} 和 ∂L∂xn∂L∂xn\frac{\partial L}{\partial x^{n}} 的计算结果 来源于 对 前馈计算时 的 L=f(wnxn)L=f(wnxn)L = f(w^{n}x^{n}) 的求导 。

链式法则

∂L∂ωn=∂L∂xn+1∂xn+1∂ωn=(∂L∂xn+1)∂xn+1∂ωn=(∂L∂xn+2∂xn+2∂xn+1)∂xn+1∂ωn=(∂L∂xn+3∂xn+3∂xn+2∂xn+2∂xn+1)∂xn+1∂ωn∂L∂ωn=∂L∂xn+1∂xn+1∂ωn=(∂L∂xn+1)∂xn+1∂ωn=(∂L∂xn+2∂xn+2∂xn+1)∂xn+1∂ωn=(∂L∂xn+3∂xn+3∂xn+2∂xn+2∂xn+1)∂xn+1∂ωn\frac{\partial L}{\partial \omega^{n}} = \frac{\partial L}{\partial x^{n+1}} \frac{\partial x^{n+1}}{\partial \omega^{n}} = (\frac{\partial L}{\partial x^{n+1}}) \frac{\partial x^{n+1}}{\partial \omega^{n}} = (\frac{\partial L}{\partial x^{n+2}} \frac{\partial x^{n+2}}{\partial x^{n+1}}) \frac{\partial x^{n+1}}{\partial \omega^{n}} = (\frac{\partial L}{\partial x^{n+3}} \frac{\partial x^{n+3}}{\partial x^{n+2}} \frac{\partial x^{n+2}}{\partial x^{n+1}}) \frac{\partial x^{n+1}}{\partial \omega^{n}}