## 3 Our Proposed FNNS
### Improved FNNRW Learning Algorithm
, \tag{18}\]  
The formula above illustrates the best decision range in the process of determining input weights and biases at random. In the process of determining the output weight, randomly initialize the output matrix as \(\beta(0)\) and, respectively, calculate the local and global gradients:  
\[\
abla f_{si}(\beta^{(t-1)})=H_{si}^{T}(H_{si}\beta^{(t-1)}-T_{si}), \tag{19}\]  
\[\
abla f_{s}(\beta^{(t-1)})=\frac{1}{m}\sum_{i=1}^{m}\
abla f_{si}(\beta^{t-1 })=\frac{1}{m}\sum_{i=1}^{m}H_{si}^{T}(H_{si}\beta^{(t-1)}-T_{si}), \tag{20}\]  
At the same time, calculate:  
\[\
abla^{2}f_{si}(\beta^{(t-1)})=H_{si}^{T}H_{si}, \tag{21}\]  
Each local model is locally optimized during each iteration:  
\[\beta^{(t)}_{i}=\arg\min_{\beta}\bigg{\{}f_{si}(\beta)-(\
abla f_{si}(\beta^{ (t-1)})-\eta\
abla f_{s}(\beta^{(t-1)}))^{T}\beta+\frac{\sigma}{2}\bigg{\|} \beta-\beta^{(t-1)}\bigg{\|}_{2}^{2}\bigg{\}}, \tag{22}\]The idea of Bregman divergence is presented in order to better comprehend this local model:  
\[B_{\psi}(\beta,\beta^{{}^{\prime}})=\psi(\beta)-\psi(\beta^{{}^{\prime}})-\
abla \psi(\beta^{{}^{\prime}})\cdot(\beta-\beta^{{}^{\prime}}) \tag{23}\]  
In this algorithm, for each \(f_{si}(\beta)\), there is  
\[F_{i}(\beta)=f_{si}(\beta)+\frac{\sigma}{2}\|\beta\|_{2}^{2}, \tag{24}\]  
The regularization parameter is \(\sigma>0\). Accordingly, the Bregman divergence is  
\[B_{i}(\beta,\beta^{{}^{\prime}})=B_{Fi}(\beta,\beta^{{}^{\prime}})=B_{fi}( \beta,\beta^{{}^{\prime}})+\frac{\sigma}{2}\left\|\beta-\beta^{{}^{\prime}} \right\|_{2}^{2}, \tag{25}\]  
According to the above equation, Equation (22) can be be changed to:  
\[\beta_{i}^{(t)}=\arg\min_{\beta}\left\{f_{s}(\beta^{(t-1)})+\
abla f_{s}(\beta ^{(t-1)})\cdot(\beta-\beta^{(t-1)})+\frac{1}{\eta}B_{i}(\beta,\beta^{(t-1)}) \right\}, \tag{26}\]  
[PERSON]'s extension, however, transforms the Bregman divergence into:  
\[B_{i}(\beta,\beta^{(t-1)})=\frac{1}{2}(\beta-\beta^{(t-1)})(\
abla^{2}f_{si}( \beta^{(t-1)})+\sigma I)(\beta-\beta^{(t-1)}), \tag{27}\]  
Thus, Equation (22) can be further transformed into:  
\[\beta_{i}^{(t)}=\beta^{(t-1)}-\eta(\
abla^{2}f_{si}(\beta^{(t-1)})+\sigma I)^ {-1}\
abla f_{s}(\beta^{(t-1)}), \tag{28}\]  
The ultimate output weight may be calculated using the preceding derivation as follows:  
\[\
abla f_{s}(\beta^{(t-1)})=\frac{1}{m}\sum_{i=1}^{m}\
abla f_{si}(\beta^{t-1 })=\frac{1}{m}\sum_{i=1}^{m}H_{si}^{T}(H_{si}\beta^{(t-1)}-T_{si}), \tag{20}\]0  
These are a few of the approaches for improvement this study employed. Algorithm 1 displays the particular FNNS algorithm, and Figure 4 displays the flowchart of algorithm development. **Step 1:** Divide the sample randomly into \(m\) parts, \(s=\{s_{1},s_{2},\cdots,s_{m}\}\). For each subset \(s_{i}\), derive the corresponding submodel. **Step 2:** Determine the range of input weights and biases that the activation function deems to be optimum,  
\[\
abla f_{s}(\beta^{(t-1)})=\frac{1}{m}\sum_{i=1}^{m}\
abla f_{si}(\beta^{t-1 })=\frac{1}{m}\sum_{i=1}^{m}H_{si}^{T}(H_{si}\beta^{(t-1)}-T_{si}), \tag{20}\]1  
\[b_{i}\in\left\{ \begin{array}{c}[0,\omega_{i}],\omega_{i}\leq 0\\ -\omega_{i},0],\omega_{i}>0\end{array} \right.. \tag{31}\]  
**Step 3:** Randomly initialize the same input weights \(\omega_{i}\) and biases \(b_{i}\) within the range of values. **Step 4:** Calculate the \(H_{si}\) hidden layer output matrix and initialize the \(\beta(0)\) output matrix at random. **Step 5:** Calculate the required components such as local gradients and global gradients. \[\
abla f_{s}(\beta^{(t-1)})=H_{si}^{T}(H_{si}\beta^{(t-1)}-T_{si}), \tag{32}\]  
\[\
abla f_{s}(\beta^{(t-1)})=\frac{1}{m}\sum_{i=1}^{m}{\
abla f_{si}(\beta^{(t -1)})}=\frac{1}{m}\sum_{i=1}^{m}H_{si}^{T}(H_{si}\beta^{(t-1)}-T_{si}), \tag{33}\]  
\[\
abla^{2}f_{si}(\beta^{(t-1)})=H_{si}^{T}H_{si}. \tag{34}\]  
**Step 6:** Calculate the output weight,  
\[\beta^{(t)}=\beta^{(t-1)}-\eta(\frac{1}{m}\sum_{i=1}^{m}{(\
abla^{2}f_{si}( \beta^{(t-1)})+\sigma I)