# An Integer-Fractional Gradient Algorithm for Back Propagation Neural Networks
## 4 Numerical Experiments
### Training for Different Training Sets' Sizes  
A larger training set can provide more sample data for the model to learn from, which can help to improve the accuracy and generalizability of the model. Typically, the size of the training set can have an impact on the training results. Here, we investigate the performance of the three neural networks by adjusting various training set sizes, and the experimental results are displayed in Table 4, which shows that a larger training set can aid the model in better-capturing patterns and features from the data. Table 4 shows that, in terms of training accuracy and the number of layers to stop training, the MIFBPNN performs best on training datasets of all sizes, followed by the FBPNN. It is worth noting that change in the MIFBPNN's accuracy from the training dataset of 10,000 to 60,000 was the smallest, followed by that of the FBPNN. This can also be viewed as evidence that the MIFBPNN is more general and has a stronger capacity for convergence and learning while maintaining high accuracy. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Training Dataset Size**} & \multicolumn{2}{c}{**BP**} & \multicolumn{2}{c}{**FBP**} & \multicolumn{2}{c}{**MIFBP**} \\  & **Accuracy** & **Training Epoch** & **Accuracy** & **Training Epoch** & **Accuracy** & **Training Epoch** \\ \hline
10,000 & 91.69\% & 124 & 92.99\% & 86 & 94.55\% & 79 \\
20,000 & 93.71\% & 176 & 93.29\% & 76 & 96.06\% & 64 \\
30,000 & 93.81\% & 140 & 93.85\% & 68 & 96.94\% & 69 \\
40,000 & 94.20\% & 109 & 94.23\% & 58 & 97.17\% & 55 \\
50,000 & 95.23\% & 157 & 95.69\% & 71 & 97.28\% & 61 \\
60,000 & 95.65\% & 163 & 96.95\% & 68 & 97.64\% & 71 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of different neural networks under optimal hyperparameters on various sizes of training sets. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\eta=0.7\)} & \multirow{2}{*}{\(\alpha=0.1\)} & \multirow{2}{*}{\(\alpha=0.2\)} & \multirow{2}{*}{\(\alpha=0.3\)} & \multirow{2}{*}{\(\alpha=0.4\)} & \multirow{2}{*}{\(\alpha=0.5\)} & \multirow{2}{*}{\(\alpha=0.6\)} & \multirow{2}{*}{\(\alpha=0.7\)} & \multirow{2}{*}{\(\alpha=0.8\)} & \multirow{2}{*}{\(\alpha=0.9\)} \\  & & & & & & & & & \\ \hline Accuracy & 96.28\% & 95.86\% & 95.87\% & 95.67\% & 96.01\% & 96.65\% & 96.13\% & 95.93\% & 95.70\% \\ Training epoch & 79 & 75 & 55 & 63 & 70 & 68 & 67 & 73 & 57 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Test set accuracy of the MIFBPNN with \(\eta=0.7\), \(\alpha=0.6\) and various parameter values of \(\tau\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\eta=0.7\)} & \multirow{2}{*}{\(\tau=0.1\)} & \multirow{2}{*}{\(\tau=0.2\)} & \multirow{2}{*}{\(\tau=0.3\)} & \multirow{2}{*}{\(\tau=0.4\)} & \multirow{2}{*}{\(\tau=0.5\)} & \multirow{2}{*}{\(\tau=0.6\)} & \multirow{2}{*}{\(\tau=0.7\)} & \multirow{2}{*}{\(\tau=0.8\)} & \multirow{2}{*}{\(\tau=0.9\)} \\  & & & & & & & & & \\ \hline Accuracy & 97.52\% & 97.64\% & 97.34\% & 97.30\% & 97.27\% & 97.41\% & 97.46\% & 97.49\% & 97.36\% \\ Training epoch & 63 & 71 & 54 & 55 & 50 & 51 & 47 & 56 & 55 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Test set accuracy of the MIFBPNN with \(\eta=0.7\), \(\alpha=0.6\) and various parameter values of \(\tau\). The MIFBPNN has the highest accuracy, followed by the FBPNN, as can be seen in Figure 3. In terms of convergence speed, the accuracy of the MIFBPNN and FBPNN reaches a relatively stable state in about 50 and 80 epochs, while the BP neural network experiences this process in about 120 epochs. The MIFBPNN has the smallest loss and can reach the minimum loss fastest, both in the training set loss and the test set loss, as shown by Figures 3 and 4. Based on the above experimental results, it can be concluded that MIFBPNN exhibits high accuracy and fast convergence speed in various training set sizes under different experimental conditions.