## 5 Discussion
### Experiments on AC Loss Function  
For the AC loss proposed in the paper, we conduct experiments to verify the effectiveness of the AC loss function on restricting angle during network training and the improvement of CIL performance. Firstly, Figure 15a shows the trends of the total loss and our AC loss in 80 epochs. For better visual clarity, the values of both losses were normalized. As shown in Figure 15a, both the total loss and AC loss decrease smoothly without obvious fluctuations, showing that our model achieves effective and stable fitting, and adding our AC loss will not disrupt the convergence process or compromise training stability. Then, an ablation comparison about AC loss is applied, including \"only backpropagate CE loss\" and \"backpropagate CE loss and AC loss\". The training process of both methods adopts the same hyperparameters such as the learning rate and the optimizer. Figure 15b shows the \(\cos\theta\) trends of the above two methods, where \(\theta\) represents the angle between the new feature and the former classifier weight as described in Section 3.3. Lower cosine values correspond to angles that are closer to being orthogonal. From Figure 15b, the model trained with AC loss exhibits a faster decrease in the cosine value (\(\cos\theta\)), indicating a quicker progression toward orthogonality between the new class features and the old-class classifier weights. More importantly, as training proceeds, the model without AC loss shows a gradual increase in \(\cos\theta\), suggesting that the orthogonality is being weakened, and the new samples are increasingly interfering with the classification performance of old classes. In contrast, the model with AC loss maintains a relatively stable \(\cos\theta\) after convergence, demonstrating the effectiveness of the proposed constraint. This phenomenon explains why our method achieves better performance on old-class classification compared to other approaches. Figure 14: Ablation experiments on “Removing dropout” and “Late-stop”. Considering that the value of the loss is difficult to control in the learning process, we compare the cosine values of the angles between new class features and old classifier weights when network training reaches different accuracy rates, which is called sampling accuracies in this paper, including 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95. The experimental results are shown in Table 4. It should be noted that since the network training progress is difficult to intervene manually, it is verified in the test phase after each stage of training whether the accuracy exceeds the sampling accuracy for the first time, so as to acquire our experimental results. As shown in Figure 15b, regardless of whether our proposed AC loss is applied or not, the \(\cos\theta\) values exhibit unstable oscillations after 30 epochs. This instability contributes to the deviation observed at the 0.7 sampling point. We have taken measures to obtain experimental data that best meets the requirements. However, for the sampling accuracy of 0.7, the accuracies of the two networks are quite different which are 0.704 and 0.749, respectively. From Table 4, it can be concluded that when the network trained with AC loss reaches a certain accuracy, the angle between the new feature vector and old classifier weight is smaller than the network trained without AC loss, indicating that it is closer to orthogonality. This proves that our AC loss has a certain constraint effect on the angle during the network optimization process. Since our AC loss applies angular constraints only between the new feature vectors and the former classifier weights, we also conducted an experiment to analyze the effect of our AC loss on the angle between the former classifier weight vector and the former feature vector. The angular changes in training process are output as shown in Figure 16: (a) angular changes between old-class sample features and their corresponding classifier weight vectors during training.