【python实现卷积神经网络】卷积层Conv2D反向传播过程

代码来源：https://github.com/eriklindernoren/ML-From-Scratch

卷积神经网络中卷积层Conv2D（带stride、padding）的具体实现：https://www.cnblogs.com/xiximayou/p/12706576.html

激活函数的实现（sigmoid、softmax、tanh、relu、leakyrelu、elu、selu、softplus）：https://www.cnblogs.com/xiximayou/p/12713081.html

损失函数定义（均方误差、交叉熵损失）：https://www.cnblogs.com/xiximayou/p/12713198.html

优化器的实现（SGD、Nesterov、Adagrad、Adadelta、RMSprop、Adam）：https://www.cnblogs.com/xiximayou/p/12713594.html

本节将根据代码继续学习卷积层的反向传播过程。

这里就只贴出Conv2D前向传播和反向传播的代码了：

def forward_pass(self, X, training=True):
        batch_size, channels, height, width = X.shape
        self.layer_input = X
        # Turn image shape into column shape
        # (enables dot product between input and weights)
        self.X_col = image_to_column(X, self.filter_shape, stride=self.stride, output_shape=self.padding)
        # Turn weights into column shape
        self.W_col = self.W.reshape((self.n_filters, -1))
        # Calculate output
        output = self.W_col.dot(self.X_col) + self.w0
        # Reshape into (n_filters, out_height, out_width, batch_size)
        output = output.reshape(self.output_shape() + (batch_size, ))
        # Redistribute axises so that batch size comes first
        return output.transpose(3,0,1,2)

    def backward_pass(self, accum_grad):
        # Reshape accumulated gradient into column shape
        accum_grad = accum_grad.transpose(1, 2, 3, 0).reshape(self.n_filters, -1)

        if self.trainable:
            # Take dot product between column shaped accum. gradient and column shape
            # layer input to determine the gradient at the layer with respect to layer weights
            grad_w = accum_grad.dot(self.X_col.T).reshape(self.W.shape)
            # The gradient with respect to bias terms is the sum similarly to in Dense layer
            grad_w0 = np.sum(accum_grad, axis=1, keepdims=True)

            # Update the layers weights
            self.W = self.W_opt.update(self.W, grad_w)
            self.w0 = self.w0_opt.update(self.w0, grad_w0)

        # Recalculate the gradient which will be propogated back to prev. layer
        accum_grad = self.W_col.T.dot(accum_grad)
        # Reshape from column shape to image shape
        accum_grad = column_to_image(accum_grad,
                                self.layer_input.shape,
                                self.filter_shape,
                                stride=self.stride,
                                output_shape=self.padding)

        return accum_grad

而在定义卷积神经网络中是在neural_network.py中　　

   def train_on_batch(self, X, y):
        """ Single gradient update over one batch of samples """
        y_pred = self._forward_pass(X)
        loss = np.mean(self.loss_function.loss(y, y_pred))
        acc = self.loss_function.acc(y, y_pred)
        # Calculate the gradient of the loss function wrt y_pred
        loss_grad = self.loss_function.gradient(y, y_pred)
        # Backpropagate. Update weights
        self._backward_pass(loss_grad=loss_grad)

        return loss, acc

还需要看一下self._forward_pas和self._backward_pass：

    def _forward_pass(self, X, training=True):
        """ Calculate the output of the NN """
        layer_output = X
        for layer in self.layers:
            layer_output = layer.forward_pass(layer_output, training)

        return layer_output

    def _backward_pass(self, loss_grad):
        """ Propagate the gradient 'backwards' and update the weights in each layer """
        for layer in reversed(self.layers):
            loss_grad = layer.backward_pass(loss_grad)

我们可以看到，在前向传播中会计算出self.layers中每一层的输出，把包括卷积、池化、激活和归一化等。然后在反向传播中从后往前更新每一层的梯度。这里我们以一个卷积层+全连接层+损失函数为例。网络前向传播完之后，最先获得的梯度是损失函数的梯度。然后将损失函数的梯度传入到全连接层，然后获得全连接层计算的梯度，传入到卷积层中，此时调用卷积层的backward_pass()方法。在卷积层中的backward_pass()方法中，如果设置了self.trainable，那么会计算出对权重W以及偏置项w0的梯度，然后使用优化器optmizer，也就是W_opt和w0_opt进行参数的更新，然后再计算对前一层的梯度。最后有一个colun_to_image()方法。

def column_to_image(cols, images_shape, filter_shape, stride, output_shape='same'):
    batch_size, channels, height, width = images_shape
    pad_h, pad_w = determine_padding(filter_shape, output_shape)
    height_padded = height + np.sum(pad_h)
    width_padded = width + np.sum(pad_w)
    images_padded = np.empty((batch_size, channels, height_padded, width_padded))

    # Calculate the indices where the dot products are applied between weights
    # and the image
    k, i, j = get_im2col_indices(images_shape, filter_shape, (pad_h, pad_w), stride)

    cols = cols.reshape(channels * np.prod(filter_shape), -1, batch_size)
    cols = cols.transpose(2, 0, 1)
    # Add column content to the images at the indices
    np.add.at(images_padded, (slice(None), k, i, j), cols)

    # Return image without padding
    return images_padded[:, :, pad_h[0]:height+pad_h[0], pad_w[0]:width+pad_w[0]]

该方法是将之间为了方便计算卷积进行的形状改变image_to_column()重新恢复成images_padded的格式。

像这种计算期间的各种的形状的变换就挺让人头疼的，还会碰到numpy中各式各样的函数，需要去查阅相关的资料。只要弄懂其中大致过程就可以了，加深相关知识的理解。