Residual Connections and Multilayer Perceptrons (MLP): Power Tools for Training Deep Networks, Solving Gradient Vanishing, Enhancing Model Performance

# 1. The Gradient Vanishing Problem in Deep Network Training

In deep neural network training, as the number of network layers increases, the gradient vanishing problem becomes increasingly pronounced. This is because during the backpropagation process, gradients tend to shrink with the increasing number of layers, leading to difficulties in updating weights in deep networks and thereby affecting the training effectiveness of the model.

The emergence of the gradient vanishing problem mainly has the following causes:

***Saturation of activation functions:** Commonly used activation functions (such as sigmoid, tanh) tend to saturate when the input is large or small, causing gradients to approach zero.
***Weight initialization:** Improper weight initialization, such as using uniform distribution or normal distribution, may lead to gradient vanishing.
***Excessive number of network layers:** As the number of network layers increases, gradients pass through more layers and are continuously reduced.

# 2. Theoretical Basis of Residual Connections

### 2.1 Structure and Principle of Residual Networks

Residual networks (ResNet) are a type of deep neural network that addresses the gradient vanishing problem by introducing residual connections. These connections directly link the input and output of the network, thus allowing gradients to propagate more efficiently.

The basic structure of a ResNet is as shown in the following diagram:

graph LR
subgraph Input
     A[Input]
end
subgraph Hidden Layer
     B[Hidden Layer 1]
     C[Hidden Layer 2]
     D[Hidden Layer 3]
end
subgraph Output
     E[Output]
end
A --> B
B --> C
C --> D
D --> E
A --> E

The residual connection is represented by the dotted arrow, directly linking input `A` to output `E`.

### 2.2 Mathematical Derivation of Residual Connections

Assuming the input of a residual block is `x`, and the output is `y`, the mathematical expression for the residual connection is:

y = x + F(x)

where `F(x)` represents the nonlinear transformation of the residual block, usually consisting of convolutional layers, activation functions, and normalization layers.

### 2.3 Advantages and Limitations of Residual Connections

**Advantages:**

***Solves the gradient vanishing problem:** Residual connections allow gradients to propagate more effectively through the network, alleviating the gradient vanishing problem.
***Improves training stability:** Residual connections provide additional pathways for the network, making the training process more stable.
***Enhances model performance:** Residual connections have been proven to significantly improve the performance of deep neural networks, especially in tasks such as image classification and object detection.

**Limitations:**

***Increases computational cost:** Residual connections require additional computation, which may increase the model's training and inference time.
***May introduce redundant information:** Residual connections may introduce redundant information, potentially reducing the model's generalization ability.

# 3. Introduction to Multilayer Perceptrons (MLP)

### 3.1 MLP Network Structure and Forward Propagation

Multilayer perceptrons (MLP) are feedforward neural networks composed of multiple fully connected layers. The MLP network structure is shown in the following diagram:

graph LR
subgraph Input Layer
     A[Input Layer]
end
subgraph Hidden Layers
     B[Hidden Layer 1]
     C[Hidden Layer 2]
     D[Hidden Layer 3]
end
subgraph Output Layer
     E[Output Layer]
end
A --> B
B --> C
C --> D
D --> E

The forward propagation process of the MLP is as follows:

1. The input layer receives the input