## 3 Proposed Method
### Network Training and Optimization  
The forward propagation algorithm and the back propagation algorithm are two processes for convolution neural network training. The process of forward propagation is the process of getting the corresponding output based on the input. The backpropagation algorithm reversely transmits the error value according to the error between the output value and the true value. Backpropagation algorithm corrects the connection weight between the neurons to obtain the network model with the smallest error.  
For a data set containing m samples \(\{(x^{(1)},y^{(1)}),\ldots,\langle x^{(m)},y^{(m)}\rangle\}\), \(O_{W,b}(x)\) represents the output value of the sample \((x,y)\) after classified by the network. To perform iterative optimization of the network, define the following cost function:  
\[J(W,b;x,y)=\frac{1}{2}\|o_{W,b}(x)-y\|^{2}. \tag{7}\]  
Considering m samples in the data set, an expression for their overall cost function can be obtained:  
\[J(W,b)=\left[\frac{1}{m}\sum_{i=1}^{m}J(W,b;x^{(i)},y^{(i)})\right]+\frac{ \lambda}{2}\sum_{l=1}^{n_{l}-1}\sum_{i=1}^{n_{l}}\sum_{j=1}^{n_{l+1}}\left(W_{ ji}^{(l)}\right)^{2}. \tag{8}\]  
In Formula (8), the first term represents the mean square error term, reflecting the deviation between the predicted value and the true value. The second term is a regularization term that prevents overfitting, expressed in terms of the complexity of the hyperparameters.  
The backpropagation algorithm uses gradient descent and chain derivation rules to calculate the parameters. It continuously corrects the values of the hyperparameters \(W\) and b in the iteration to gradually reduce the error. The backpropagation training method has the following steps: First, the forward propagation algorithm is used to obtain the output values of the network, the activation values and weights of each layer of the network. Then, the error of each output neuron can be calculated. The error calculation formula is as follows:  
\[\delta_{i}^{(n_{l})}=\frac{\partial}{\partial z_{i}^{(n_{l})}}\frac{1}{2}\|y- o_{W,b}(x)\|^{2}=-(y_{i}-a_{i}^{(n_{l})})\cdot f^{\prime}(z_{i}^{(n_{l})}).\] (Finally, the following formulas are used to update the network weights:  
\[W^{(l)}=W^{(l)}-\alpha\bigg{[}\bigg{(}\frac{1}{m}\Delta W^{(l)}\bigg{)}+\lambda W ^{(l)}\bigg{]}, \tag{12}\]  
\[b^{(l)}=b^{(l)}-\alpha\bigg{[}\frac{1}{m}\Delta b^{(l)}\bigg{]}. \tag{13}\]  
During the training process, the network weights are iteratively optimized until the termination condition is met. Common methods for judging convergence include setting the maximum number of iterations and determining whether the difference between adjacent errors meets the specified error interval. At the end of the training, parameters \(W\) and \(b\) are obtained which can correctly classify the samples. In the calculation of the pooling layer, the size transformation of the feature map is performed, and then the propagation and parameters updating are performed.  
A larger learning rate is used at the beginning of the network training, which can make the optimization step of the network larger. It can skip the local extreme point in the high-dimensional space. This way can obviously accelerate the convergence speed of the network. After a certain number of iterations, the parameters of the model can be closer to the true values, and the network converges at the optimal values.