## 3 Results  
First, 36 samples in the sample set were randomly selected from the overall 436 GGNs samples as a fixed test set. Four training subsets with different but increasing sample numbers were drawn from the remaining 400 samples as training sets; the proportion between positive (IA) and negative (non-IA) samples was 1:1. Therefore, the number of samples in the four group of training sets were 100, 200, 300, and 400, respectively. We repeated four sets of experiments with different training sets of four sample sizes and a fixed test set. With four sets of sample sizes, the 10-dimensional feature subsets obtained from the AlexNet were inputted into SVM for classifying IA and non-IA. The EXP error of GGNs in the test set and the EMP error in the training set under different sample sizes were calculated separately. We compared the actual value from the test set with the theoretical value computed by Equation (1) at 90% (i.e., \(\eta\) = 0.10) and 95% (i.e., \(\eta\) = 0.05) confidence levels. The experimental results are shown in Table 1.  
It can be seen from Table 1 that the actual values from the testing samples and the computed theoretical values of \(R_{exp}(w)\) in Equation (1) show a nearly consistent trend as the sample size increases. Their differences gradually decrease with increase in sample size \(n\), while not exceeding the upper limit of the theoretical value of \(R_{exp}(w)\). The value of confidence range \(\Phi(n/h)\) is inversely proportional to the sample size, given the same number of features used. Experimental results demonstrate that the generalization ability of the invasive classification model is proportional to the sample size. From the above conclusions, we can see that it is a necessary condition to minimize only EMP, but it is not a sufficient condition when the number of samples is limited. To increase the generalization and prediction capability of the model, the selected types of classifiers must have as small a VC dimension as is possible. Moreover, there is a requirement to minimize the confidence range as well. This process of selecting the classifier model and learning algorithm can be referred to as adjusting the confidence range. However, due to a lack of theoretical guidance, currently this process in most existing methods is based on prior knowledge and experience, resulting in an excessive dependence on researchers. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Number of** \\ **Samples** \\ \end{tabular} } & \multirow{2}{*}{\(R_{exp}(w)\)} & \multicolumn{2}{c}{**EXP (\(\eta\) = 0.05)**} & \multicolumn{2}{c}{**EXP (\(\eta\) = 0.10)**} & \multirow{2}{*}{\multirow{2}{*}{\(R_{exp}(w)\) in Test Set}} \\ \cline{3-3} \cline{5-6}  & & \(\Phi(w/h)\) **in** & & & & \\ \cline{3-3} \cline{5-6}  & & **Equation (2)** & & & & \\ \hline \(n=100\) & 0 & 0.6877 & 0.6877 & 0.6826 & 0.6826 & 0.361 \\ \(n=200\) & 0.005 & 0.524 & 0.529 & 0.5207 & 0.5257 & 0.222 \\ \(n=300\) & 0.003 & 0.4449 & 0.4479 & 0.4423 & 0.4453 & 0.194 \\ \(n=400\) & 0.002 & 0.3954 & 0.3974 & 0.3932 & 0.3952 & 0.139 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of theoretical and actual values of \(R_{exp}(w)\). Figure 2: Schematic diagram for classifying IA and non-IA based on SVM and GGNs. In practice, another critical challenge for building an effective classifier is the question of the number of samples necessary under a certain requirement of prediction accuracy. According to Equation (1), we continue to explore the lower bound on sample size under various generalization requirements. The experimental results are shown in Table 2.  
From Table 1, we see that the value of \(R_{emp}(w)\) is inversely proportional to the sample size when the VC dimension and confidence limits are unchanged, and the values are all lower than 1%. Therefore, we suppose \(R_{emp}(w)\) is 1% taking into consideration a degree of classification error, which means that the classification accuracy of the training set reaches 99%.