## 3 Simulation Results  
We demonstrate the classification task for two types of teacher data (linear classification boundary and non-linear classification boundary) using the proposed method. The learning of the optimal \((\alpha,\beta,\epsilon)\) parameters was performed by minimizing the cost function. Since the \((\alpha,\beta,\epsilon)\) parameters are associated with discrete ranking, we chose gradient-free optimization instead of gradient-based optimization. We implemented the Nelder-Mead method and particle swarm optimization (PSO) algorithms. We observed that PSO generally gives better results; hence, we focused on the results obtained by PSO. The hyperparameter \(N\) is the number of particles for PSO. The learned support vectors, classification boundary and learning curves for \(N=10,50,100\) are depicted in Figure 6. The learned parameters are shown in Table 2. In all the cases, the total number of support vectors was fixed at \(n_{sv}=10\), where five of them belonged to \(+1\) and the other five belonged to \(-1\). The \(\alpha\) and the \(\epsilon\) parameters were fixed to \(\alpha=3\) and \(\epsilon=0.4\). The optimization was performed by minimizing the loss function Equation (3) for the optimal values of \(\beta\). We observed that the classification could be performed in all experiments. We also observed that, for larger values of \(N\), the optimized loss function had a lower value, but the  
Figure 5: (**a**) Fully-connected mean field Ising model. (**b**) A relaxed model where there are only one-to-one couplings between support vectors and test data. test accuracy was also lower. This suggests possible overfitting for large values of \(N\) in PSO optimization. We now compare the proposed method with scikit-learn SVM. The verification results are shown in Figure 7. We used 28 teacher data (small dots) for each verification. The detail numbers of data used are listed in Table 3. The large sphere symbols are the support vectors. In the classification using the proposed algorithm, we set the radius of the scan circle to \(\epsilon=0.5\) (linear data) and \(\epsilon=0.6\) (non-linear data). The power of the reciprocal of the distance among data was \(\beta=1\) and the criterion for determining the support vectors from the ranking was \(\alpha=3\). For the traditional SVM, we used scikit-learn.svm.SVC with default values. From Figure 7, we found that both the linear classification and the non-linear classification performed by the proposed method could give a classification \begin{table}
\begin{tabular}{c c c} \hline \hline \(N\) & \(\beta\) & **Test Accuracy** \\ \hline
10 & 1.128 & 0.9675 \\
50 & 0.6356 & 0.9610 \\
100 & 0.6157 & 0.9335 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Particle swarm optimization results. \(N\) is the number of particles and \(\beta\) is the hyperparameter being optimized. The optimization is conducted by minimizing the cross-entropy loss function Equation (3). The optimized results are visualized in Figure 6.  
Figure 6: Support vector distribution, prediction probability and classification boundary from particle swarm optimization (PSO) results. The experimental parameters and optimal values are listed in Table 2. (**a**) \(N=10\). (**b**) \(N=50\). (**c**) \(N=100\). The small dots are the training data. The symbols bordered by white circles are the test data that were classified correctly. The crossed circles are the test data that were classified incorrectly. The large sphere symbols are the support vectors. (**d**) The learning curves for four experiments. The orange line is the Nelder–Mead experiment which does not converge. The other three lines are PSO results. The black line is \(N=10\), the red line is \(N=50\) and the blue line is \(N=100\), respectively. (**e**) The training and testing accuracy suggest overfitting for larger values of \(N\). boundary similar to that of scikit-learn. However, if we changed only the positions of teacher data without changing the number of teacher data, there were cases where it could not be classified well.