Understanding the Forgetting of (Replay-based) Continual Learning via Feature Learning: Angle Matters

Continual learning (CL) is crucial for advancing human-level intelligence, but its theoretical understanding, especially regarding factors influencing forgetting, is still relatively limited. This work aims to build a unified theoretical framework for understanding CL using feature learning theory. Different from most existing studies that analyze forgetting under linear regression model or lazy training, we focus on a more practical two-layer convolutional neural network (CNN) with polynomial ReLU activation for sequential tasks within a signal-noise data model. Specifically, we theoretically reveal how the angle between task signal vectors influences forgetting that:acute or small obtuse angles lead to benign forgetting, whereas larger obtuse angles result in harmful forgetting. Furthermore, we demonstrate that the replay method alleviates forgetting by expanding the range of angles corresponding to benign forgetting. Our theoretical results suggest that mid-angle sampling, which selects examples with moderate angles to the prototype, can enhance the replay method's ability to mitigate forgetting. Experiments on synthetic and real-world datasets confirm our theoretical results and highlight the effectiveness of our mid-angle sampling strategy.