How do deep residual networks overcome the vanishing gradient problem?

Residual network is a popular deep learning model that solves the vanishing gradient problem by introducing residual blocks. This article starts from the essential cause of the vanishing gradient problem and explains in detail the solution to the residual network.

1. The essential reason for the vanishing gradient problem

In a deep neural network, the output of each layer is the combination of the input of the previous layer and It is obtained by multiplying the weight matrix and calculating it through the activation function. As the number of network layers increases, the output of each layer will be affected by the output of previous layers. This means that even small changes in the weight matrix and activation function will have an impact on the output of the entire network. In the backpropagation algorithm, gradients are used to update the weights of the network. The calculation of gradient requires passing the gradient of the next layer to the previous layer through the chain rule. Therefore, the gradients of previous layers will also affect the calculation of gradients. This effect is accumulated when weights are updated and propagated throughout the network during training. Therefore, each layer in a deep neural network is interconnected, and their outputs and gradients influence each other. This requires us to carefully consider the selection of weights and activation functions of each layer, as well as the calculation and transmission methods of gradients when designing and training the network, to ensure that the network can effectively learn and adapt to different tasks and data.

In deep neural networks, when there are many network layers, gradients often "disappear" or "explode". The reason why the gradient disappears is that when the derivative of the activation function is less than 1, the gradient will gradually shrink, causing the gradient of the further layers to become smaller and eventually become unable to be updated, causing the network to be unable to learn. The reason for gradient explosion is that when the derivative of the activation function is greater than 1, the gradient will gradually increase, causing the gradient of the further layers to become larger, eventually causing the network weight to overflow, and also causing the network to be unable to learn.

2. Solution of Residual Network

The residual network solves the problem of gradient disappearance by introducing residual blocks. Between each network layer, the residual block adds the input directly to the output, making it easier for the network to learn the identity mapping. This cross-layer connection design enables gradients to propagate better and effectively alleviates the phenomenon of gradient disappearance. Such a solution can improve the training efficiency and performance of the network.

Specifically, the structure x of the residual block represents the input, F(x) represents the mapping obtained by network learning, and H(x) represents the identity mapping. The output of the residual block is H(x) F(x), which is the input plus the learned mapping.

The advantage of this is that when the network learns an identity mapping, F(x) is 0, and the output of the residual block is equal to the input, that is, H( x) 0=H(x). This avoids the vanishing gradient problem because even if the gradient of F(x) is 0, the gradient of H(x) can still be passed to the previous layer through the cross-layer connection, thus achieving better gradient flow.

In addition, the residual network also uses technologies such as "batch normalization" and "pre-activation" to further enhance the performance and stability of the network. Among them, batch normalization is used to solve the problems of gradient disappearance and gradient explosion, while pre-activation can better introduce nonlinearity and improve the expressive ability of the network.

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