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。