Continual learning (CL) has emerged as a dominant paradigm for acquiring knowledge from sequential tasks while avoiding catastrophic forgetting. Although many CL methods have been proposed to show impressive empirical performance, the theoretical understanding of their generalization behavior remains limited, particularly for replay-based approaches. This paper establishes a unified theoretical framework for replay-based CL, deriving a series of information-theoretic generalization bounds that explicitly elucidate the impact of the memory buffer alongside the current task on generalization performance. Specifically, our hypothesis-based bounds capture the trade-off between the number of selected exemplars and the information dependency between the hypothesis and the memory buffer. Our prediction-based bounds yield tighter and computationally tractable upper bounds on the generalization error by leveraging low-dimensional variables. Theoretical analysis is general and broadly applicable to a wide range of learning algorithms, exemplified by stochastic gradient Langevin dynamics (SGLD) as a representative method. Comprehensive experimental evaluations demonstrate the effectiveness of our derived bounds in capturing the generalization dynamics in replay-based CL settings.

翻译：持续学习（CL）已成为从顺序任务中获取知识同时避免灾难性遗忘的主导范式。尽管已提出许多CL方法并展现出令人印象深刻的实证性能，但对其泛化行为的理论理解仍然有限，特别是对于基于回放的方法。本文为基于回放的CL建立了一个统一的理论框架，推导出一系列信息论泛化边界，这些边界明确阐明了记忆缓冲区与当前任务对泛化性能的影响。具体而言，我们基于假设的边界捕捉了所选样本数量与假设和记忆缓冲区之间信息依赖性之间的权衡。我们基于预测的边界通过利用低维变量，得到了泛化误差更紧且计算上易处理的上界。理论分析具有一般性，广泛适用于多种学习算法，以随机梯度朗之万动力学（SGLD）作为代表性方法进行了示例说明。全面的实验评估证明了我们推导的边界在捕捉基于回放的CL设置中泛化动态的有效性。