神经网络学习笔记 - 损失函数的定义和微分证明

神经网络学习笔记 - 损失函数的定义和微分证明

损失函数 Loss function (cross entropy loss)

损失函数，反向传播和梯度计算构成了循环神经网络的训练过程。

激活函数softmax和损失函数会一起使用。
激活函数会根据输入的参数（一个矢量，表示每个分类的可能性），计算每个分类的概率(0, 1)。
损失函数根据softmax的计算结果\(\hat{y}\)和期望结果\(y\)，根据交叉熵方法(cross entropy loss) 可得到损失\(L\)。

cross entropy loss函数

\[L_t(y_t, \hat{y_t}) = - y_t \log \hat{y_t} \\ L(y, \hat{y}) = - \sum_{t} y_t \log \hat{y_t} \\ \frac{ \partial L_t } { \partial z_t } = \hat{y_t} - y_t \\ \text{where} \\ z_t = s_tV \\ \hat{y_t} = softmax(z_t) \\ y_t \text{ : for training data x, the expected result y at time t. which are from training data} \]

证明

\[\begin{align} \frac{ \partial L_t } { \partial z_t } & = \frac{ \partial \left ( - \sum_{k} y_k \log \hat{y_k} \right ) } { \partial z_t } \\ & = - \sum_{k} y_k \frac{ \partial \log \hat{y_k} } { \partial z_t } \\ & = - \sum_{k} y_k \frac {1} {\hat{y_k}} \cdot \frac{ \partial \hat{y_k} } { \partial z_t } \\ & = - \left ( y_t \frac {1} {\hat{y_t}} \cdot \frac{ \partial \hat{y_t} } { \partial z_t } \right ) - \left ( \sum_{k \ne t} y_k \frac {1} {\hat{y_k}} \cdot \frac{ \partial \hat{y_k} } { \partial z_t } \right ) \\ & \because \text{softmax differentiation formula } \\ & = - \left ( y_t \frac {1} {\hat{y_t}} \cdot ( 1 - \hat{y_t} ) \hat{y_t} \right ) - \left ( \sum_{k \ne t} y_k \frac {1} {\hat{y_k}} \cdot (-\hat{y_t} \hat{y_k}) \right ) \\ & = - \left ( y_t \cdot ( 1 - \hat{y_t} ) \right ) - \left ( \sum_{k \ne t} y_k \cdot (-\hat{y_t}) \right ) \\ & = - y_t + y_t \hat{y_t} + \left ( \sum_{k \ne t} y_k \hat{y_t} \right ) \\ & = - y_t + \hat{y_t} \left ( \sum_{k} y_k \right ) \\ & \because \sum_{k} y_k = 1 \\ & = \hat{y_t} - y_t \end{align} \]

参照

• 机器学习 - 激活函数的作用、定义和微分证明