Korbin

交叉熵损失的反向传播

对于一个单标签多分类问题,假设网络的输出层的输入为Zin=[z1,⋯ ,zi,⋯ ,zn]Z_{in}=[z_1,\cdots,z_i,\cdots,z_n], 输出为$\hat Y=[\hat y_1,\cdots,\hat y_i,\cdots,\hat y_n]$, 真实类标签为$Y = [y_1,\cdots,y_i,\cdots,y_n]$,$n$为类别数(输出层神经元数),通常有:

Y^=Softmax(Zin)(1)\hat Y = Softmax(Z_{in})\tag{1}

y^i=ezi∑j=0nezj(2)\hat y_i = \frac {e^{z_i}}{\sum_{j=0}^n e^{z_j}}\tag{2}

其中$Softmax$为:

Softmax(Zin)=[⋯ ,ezi∑j=1nezj,⋯ ](3)Softmax(Z_{in}) = [\cdots,\frac {e^{z_i}}{\sum_{j=1}^n e^{z_j}},\cdots]\tag{3}

交叉熵损失

Loss(Y,Y^)=−∑i=1nyi∗ln⁡(y^i)(4)Loss(Y,\hat Y) = -\sum_{i=1}^ny_i*\ln(\hat y_i)\tag{4}

损失对神经网络输出的偏导(标量对向量求偏导)为:

∂Loss(Y,Y^)∂Y^=[−y1y^1,⋯ ,−yiy^i,⋯ ,−yny^n]\label5(5)\frac {\partial Loss(Y,\hat Y)}{\partial \hat Y} = [-\frac {y_1}{\hat y_1},\cdots,-\frac {y_i}{\hat y_i},\cdots,-\frac {y_n}{\hat y_n}]\label{5}\tag{5}

后向传播推导中遇到的所有量都是变量,最终的目的是找到损失关于某变量的偏导,程序中也只用这个公式求得对应输入点的梯度

Softmax的偏导

求$\hat y_i​$对$z_i​$的偏导,根据{2}可得:

求$\hat Y$对$Z$的偏导(向量对向量求导)

这里把$y$的坐标写作$k$

$k=i$时:

∂y^i∂zi=ezi∗1∑j=1nezj+ezi∗(−1(∑j=1nezj)2)∗ezi=ezi∑j=1nezj−(ezi∑j=1nezj)2=y^i−y^i2(6)\begin{split}
\frac {\partial \hat y_i}{\partial z_i}
&= e^{z_i} * \frac {1}{\sum_{j=1}^ne^{z_j}} +
   e^{z_i} * (-\frac {1}{(\sum_{j=1}^n e^{z_j})^2}) * e^{z_i} \cr
&= \frac {e^{z_i}}{\sum_{j=1}^n e^{z_j}} - (\frac {e^{z_i}}{\sum_{j=1}^n e^{z_j}})^2 \cr
&= \hat y_i - \hat y_i^2
\end{split}
\tag{6}

$k \neq i$时:

∂y^k∂zi=ezk∗(−1(∑j=1nezj)2)∗ezi=−ezk∗ezi(∑j=1nezj)2=−y^ky^i(7)\begin{split}
\frac {\partial \hat y_k}{\partial z_i}
&= e^{z_k} * (-\frac {1}{(\sum_{j=1}^n e^{z_j})^2}) * e^{z_i} \cr
&= -\frac {e^{z_k}*e^{z_i}}{(\sum_{j=1}^n e^{z_j})^2} \cr
&= - \hat y_k\hat y_i
\end{split}
\tag{7}

写成矩阵:

∂Y^∂Zin=[y^1−y^12−y^2y^1⋯−y^ny^1−y^1y^2y^2−y^22⋯−y^ny^2⋮⋮⋱⋮−y^1y^n−y^2y^n⋯y^n−y^n2](8)\frac {\partial \hat Y}{\partial Z_{in}} = 
\begin{bmatrix}
\hat y_1-\hat y_1^2 & -\hat y_2\hat y_1 & \cdots & -\hat y_n\hat y_1\cr
-\hat y_1\hat y_2 & \hat y_2-\hat y_2^2 & \cdots & -\hat y_n\hat y_2\cr
\vdots & \vdots & \ddots & \vdots\cr
-\hat y_1\hat y_n & -\hat y_2\hat y_n & \cdots & \hat y_n-\hat y_n^2\cr
\end{bmatrix}\tag{8}

这是一个对称矩阵,在链式求导时加不加转置都一样

根据(5)和(8),损失$L$对输入$Z$的偏导(标量对向量求偏导):

∂L(Y,Y^)∂Z=∂L(Y,Y^)∂Y^(∂Y^∂Z)T=[−y1y^1,⋯ ,−yiy^i,⋯ ,−yny^n][y^1−y^12−y^2y^1⋯−y^ny^1−y^1y^2y^2−y^22⋯−y^ny^2⋮⋮⋱⋮−y^1y^n−y^2y^n⋯y^n−y^n2]T=[(y^1−1)y1+y^1y2+⋯+y^1yn,y^2y1+(y^2−1)y2+⋯+y^1yn,⋯ ]=[y^1∑i=1nyi−y1,⋯ ,y^j∑i=1nyi−yj,⋯ ,y^n∑i=1nyi−yn]=[y^1−y1,⋯ ,y^j−yj,⋯ ,y^n−yn]    (∑i=1nyi=1)=Y^−Y\begin{align}
\frac {\partial L(Y,\hat Y)}{\partial Z}
&= \frac {\partial L(Y,\hat Y)}{\partial \hat Y} (\frac {\partial \hat Y}{\partial Z})^T\cr
&= [-\frac {y_1}{\hat y_1},\cdots,-\frac {y_i}{\hat y_i},\cdots,-\frac {y_n}{\hat y_n}]
\begin{bmatrix}
\hat y_1-\hat y_1^2 & -\hat y_2\hat y_1 & \cdots & -\hat y_n\hat y_1\cr
-\hat y_1\hat y_2 & \hat y_2-\hat y_2^2 & \cdots & -\hat y_n\hat y_2\cr
\vdots & \vdots & \ddots & \vdots\cr
-\hat y_1\hat y_n & -\hat y_2\hat y_n & \cdots & \hat y_n-\hat y_n^2\cr
\end{bmatrix}^T\cr
&= [(\hat y_1-1)y_1+\hat y_1y_2+\cdots+\hat y_1y_n,\hat y_2y_1+(\hat y_2-1)y_2+\cdots+\hat y_1y_n,\cdots]\cr
&= [\hat y_1\sum_{i=1}^ny_i-y_1,\cdots,\hat y_j\sum_{i=1}^ny_i-y_j,\cdots,\hat y_n\sum_{i=1}^ny_i-y_n]\cr
&= [\hat y_1-y_1,\cdots,\hat y_j-y_j,\cdots,\hat y_n-y_n]\ \ \ \ (\sum_{i=1}^ny_i=1)\cr
&= \hat Y-Y
\end{align}