RNN反向传播推导

参考：

RNN正向传播推导

RNN原理图：

变量说明
- $W,U,V$为权重矩阵，在整个RNN网络中是共享的。
- $h^t$代表在序列$t$时模型的隐藏状态。$h^t$由$x^t$和$h^{t-1}$共同决定。
- $o^{t}$代表在序列$t$时模型的输出。$o^t$只由模型当前的隐藏状态$h^{t}$决定。
- $x^t$代表在序列$t$时训练样本的输入。
- $L_t$代表在序列$t$时模型的损失函数。
- $y^t$代表在序列$t$时训练样本序列的真实输出。
前向公式

$\begin{aligned} &h^t=\sigma(Wh^{t-1}+Ux^{t}+B) \\ &o^t=Vh^t+C\\ &\hat{y}^t=\sigma_{softmax}(o^t) \end{aligned}$

RNN反向传播推导

$L=\sum_TL_t$

$L$对$C$的梯度：
$\frac{\partial{L}}{\partial{C}} =\sum_T\frac{\partial{L_t}}{\partial{o^t}} \cdot \frac{\partial{o^t}}{\partial{C}}=\sum_T(\hat{y}^t-y^t)$
$L$ 对V的梯度：

$\frac{\partial{L}}{\partial{V}}=\sum_T\frac{\partial{L_t}}{\partial o^t}\cdot\frac{\partial o^t}{\partial V}=\sum_T(\hat{y}^t-y^t)(h^t)^T$

$L$ 对$h^t$的梯度：

$\begin{aligned} \delta^{t}&=\frac{\partial L}{\partial h^{t}} \\ &=\frac{\partial L}{\partial o^{t}} \frac{\partial o^{t}}{\partial h^{t}} + \frac{\partial L}{\partial h^{t+1}}\frac{\partial h^{t+1}}{\partial h^{t}} \\ &=V^T(\hat{y}^{(t)} - y^{(t)}) + W^T\delta^{(t+1)}diag(1-(h^{(t+1)})^2) \end{aligned}$

其中：

(1)

$\frac{\partial L}{\partial o^{t}} \frac{\partial o^{t}}{\partial h^{t}} = V^T(\hat{y}^{t} - y^{t})$

(2)

$\begin{aligned} \frac{\partial L}{\partial h^{t+1}}\frac{\partial h^{t+1}}{\partial h^{t}}&=\delta^{t+1} \frac{\partial h^{t+1}}{\partial h^{t}}\\ &=W^T\delta^{t+1}diag(1-(h^{t+1})^2) \end{aligned}$

详细推导：

$L$ 对$W$的梯度：
$\frac{\partial L}{\partial W} = \sum\limits_{t=1}^{\tau}\frac{\partial L}{\partial h^{t}} \frac{\partial h^{t}}{\partial W} = \sum\limits_{t=1}^{\tau}diag(1-(h^{t})^2)\delta^{t}(h^{t-1})^T$
$L$ 对$B$的梯度：
$\frac{\partial L}{\partial B}= \sum\limits_{t=1}^{\tau}\frac{\partial L}{\partial h^{t}} \frac{\partial h^{t}}{\partial b} = \sum\limits_{t=1}^{\tau}diag(1-(h^{(t)})^2)\delta^{t}$
$L$ 对$U$的梯度：
$\frac{\partial L}{\partial U} = \sum\limits_{t=1}^{\tau}\frac{\partial L}{\partial h^{t}} \frac{\partial h^{t}}{\partial U} = \sum\limits_{t=1}^{\tau}diag(1-(h^{t})^2)\delta^{t}(x^{t})^T$