## 3 Our Proposed FNNS
### Improved FNNRW Learning Algorithm  
In large-scale data, as shown in Figure 2, the sample is randomly divided into \(m\) parts, \(s=\{s_{1},s_{2},\cdots,s_{m}\}\). For each subset \(s_{i}\), derive the corresponding submodel and initialize the same input weights and biases. Calculate the hidden layer output matrix \(H_{si}\), and \(H_{si}\) is a positive definite matrix. The whole problem can be described as:  
\[f_{s}(\beta)=\frac{1}{m}\sum_{i=1}^{m}f_{si}(\beta), \tag{7}\]  
\[f_{si}(\beta)=\frac{1}{2}\|H_{si}\beta-T_{si}\|_{2}^{2},i=1,2,\ldots,m, \tag{8}\]  
The \(i\)th local model's hidden output matrix and target output, respectively, are denoted by the symbols \(H_{si}\) and \(T_{si}\). Figure 1: Mathematical model of FNNRWs. The convergence of the output weights of the unified learning model is provided by the FNNRW, which divides the whole training sample into many tiny subsets and utilizes each subset to create a local learning model to integrate the unified classifier. In this algorithm, the most common sigmoid function is used for the activation function. For convenience, the activation function can be denoted as:  
\[g(x)=\frac{1}{1+e^{-(\omega x+b)}}, \tag{9}\]  
The parameters \(\omega\) and \(b\) are used to govern the movement of the \(g(x)\) picture on the \(x\) axis. The derivative of the activation function is shown in Equation (10). When \(\omega>0\), the derivative is also greater than zero, so the slope of \(g(x)\) is greater than zero. Similarly, when \(\omega<0\), the slope of the activation function \(g(x)\) is less than zero, so \(\omega\) can be used as the slope parameter of \(g(x)\). According to Figure 3, when \(x=0\), \(g(x)=0.5\); when \(x=1\), \(g(x)=r\). \[g^{\prime}(x)=\frac{a\cdot e^{-(a\cdot x+b)}}{\left(1+e^{-(a\cdot x+b)}\right) ^{2}}. \tag{10}\]  
When \(x=1\), \(g(x)=r\), and \(b=0\), we get:  
Figure 3: The flatter fragments of the activation function. The figure corresponds to the range when the activation function’s value range is [0, 1]. Figure 2: General organization. The link between the local model \(f_{si}(\beta)\) and the global model \(f_{s}(\beta)\) is depicted in the picture. A big data collection may be broken up into smaller subsets, each of which is made up of \(s=\{s_{1},s_{2}\cdots s_{m}\}\), and \(m\) submodels that are derived from each subset. \[\frac{1}{1+e^{-(\omega\cdot 1+0)}}=r, \tag{11}\]  
After transformations we obtain:  
\[\omega=-\ln(\frac{1-r}{r})=\omega_{{}_{1}}, \tag{12}\]  
Assuming that the fitting curve of the activation function is not as flat as that of the sigmoid function, then:  
\[\omega\leq-|\omega_{1}|\quad\text{or}\quad\omega\geq|\omega_{1}|, \tag{13}\]  
In order to determine the range of parameters more accurately, let \(\omega_{2}=s\cdot\omega_{1}\), where \(s>1\) is used to define the maximum input weight and the steepest part of the activation function. The ranges may be used to calculate the slope parameter of the \(i\)-th activation function:  
\[f_{si}(\beta)=\frac{1}{2}\|H_{si}\beta-T_{si}\|_{2}^{2},i=1,2,\ldots,m, \tag{8}\]0  
After substituting Equation (12) into Equation (14), we obtain:  
\[f_{si}(\beta)=\frac{1}{2}\|H_{si}\beta-T_{si}\|_{2}^{2},i=1,2,\ldots,m, \tag{8}\]1  
Parameter \(s\) determines the steepest part of the activation function, whose specific value is determined by the target function. As for the determination of parameter \(b\), when \(x\in[0,1]\), according to Figure 3 we can get:  
\[f_{si}(\beta)=\frac{1}{2}\|H_{si}\beta-T_{si}\|_{2}^{2},i=1,2,\ldots,m, \tag{8}\]2  
Following transformations, we get:  
\[f_{si}(\beta)=\frac{1}{2}\|H_{si}\beta-T_{si}\|_{2}^{2},i=1,2,\ldots,m, \tag{8}\]3  
For \(x=0\) we get the first boundary of \(b\):\(b_{1}=0\); for \(x=1\), we get the second boundary of \(b\):\(b_{2}=-\omega\). It can be seen that bias \(b\) depends on the input weight \(\omega\), so the range of bias \(b\) can be obtained:  
\[b_{i}\in\bigg{\{} \begin{array}{l}[0,\omega_{i}],\omega_{i}\leq 0\\ [-\omega_{i},0],\omega_{i}>0\end{array}