Conv2d反向传播梯度的计算过程

我们用一个例子来说明：

令
$$x\ast w=y$$
并且
$$x=\left[\begin{array}{ccc}{x}_{11}& x_{12}& x_{13}\\ x_{21}& x_{22}& x_{23}\\ x_{31}& x_{32}& x_{33}\end{array}\right],w=\left[\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right],y=\left[\begin{array}{cc}{y}_{11}& y_{12}\\ y_{21}& y_{22}\end{array}\right]$$

则
$$\{\begin{array}{rl}& y_{11}=w_{11}{x}_{11}+w_{12}{x}_{12}+w_{21}{x}_{21}+w_{22}{x}_{22}\\ & y_{12}=w_{11}{x}_{12}+w_{12}{x}_{13}+w_{21}{x}_{22}+w_{22}{x}_{23}\\ & y_{21}=w_{11}{x}_{21}+w_{12}{x}_{22}+w_{21}{x}_{31}+w_{22}{x}_{32}\\ & y_{22}=w_{11}{x}_{22}+w_{12}{x}_{23}+w_{21}{x}_{32}+w_{22}{x}_{33}\end{array}$$

梯度
$$\frac{\partial L}{\partial x}=\frac{\partial L}{\partial y}\frac{\partial y}{\partial x}=\left[\begin{array}{ccc}{\delta }_{11}{w}_{11}& {\delta }_{11}{w}_{12}+{\delta }_{12}{w}_{11}& {\delta }_{12}{w}_{12}\\ {\delta }_{11}{w}_{21}+{\delta }_{21}{w}_{11}& {\delta }_{11}{w}_{22}+{\delta }_{12}{w}_{21}+{\delta }_{21}{w}_{12}+{\delta }_{22}{w}_{11}& {\delta }_{12}{w}_{22}+{\delta }_{22}{w}_{12}\\ {\delta }_{21}{w}_{21}& {\delta }_{21}{w}_{22}+{\delta }_{22}{w}_{21}& {\delta }_{22}{w}_{22}\end{array}\right]$$
其计算过程如下图所示（其中不同颜色块对应位置的值相加）：

另外，可以用卷积的形式计算$x$的梯度：
$$\frac{\partial L}{\partial x}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& {\delta }_{11}& {\delta }_{12}& 0\\ 0& {\delta }_{21}& {\delta }_{22}& 0\\ 0& 0& 0& 0\end{array}\right)\ast \left(\begin{array}{cc}{w}_{22}& w_{21}\\ w_{12}& w_{11}\end{array}\right)$$
此时卷积核$w$需要旋转180度，也即上下翻转一次，然后左右翻转一次。输入矩阵需要增加padding。

设正向卷积的padding为$p$，反向卷积的padding为$p^{\prime }$，则需要增加的padding为：
$$p^{\prime }=k-p-1$$

其中$k$是卷积核的大小，另外假设正向卷积的stride为1。