Provable Effects of Data Replay in Continual Learning: A Feature Learning Perspective

Continual learning (CL) aims to train models on a sequence of tasks while retaining performance on previously learned ones. A core challenge in this setting is \textit{catastrophic forgetting}, where new learning interferes with past knowledge. Among various mitigation strategies, data-replay methods—where past samples are periodically revisited—are considered simple yet effective, especially when memory constraints are relaxed. However, the theoretical effectiveness of full data replay, where all past data is accessible during training, remains largely unexplored. In this paper, we present the first theoretical framework for analyzing full data-replay training in continual learning from a feature learning perspective. Adopting a multi-view data model, we identify the signal-to-noise ratio (SNR) as a critical factor affecting forgetting. Focusing on task-incremental binary classification across $M$ tasks, our analysis verifies two key conclusions: (1) forgetting can still occur under full replay when the cumulative noise from later tasks dominates the signal from earlier ones; and (2) with sufficient signal accumulation, data replay can recover earlier tasks-even if their initial learning was poor. Notably, we uncover a novel insight into task ordering: prioritizing higher-signal tasks not only facilitates learning of lower-signal tasks but also helps prevent catastrophic forgetting—highlighting the importance of order-aware replay strategies. We validate our theoretical findings through synthetic experiments that visualize the interplay between signal learning and noise memorization across varying SNRs and task correlation regimes.