From Order to Distribution: A Spectral Characterization of Forgetting in Continual Learning
Summary: arXiv:2604.13460v1 Announce Type: cross
Abstract
A central challenge in continual learning is forgetting, the loss of performance on previously learned tasks induced by sequential adaptation to new ones. While forgetting has been extensively studied empirically, rigorous theoretical characterizations remain limited. A notable step in this direction is the work by Evron et al. (2022), which analyzes forgetting under random orderings of a fixed task collection in overparameterized linear regression.
Introduction
In recent years, continual learning has gained significant attention due to its applicability in real-world scenarios where models must adapt to new information without losing knowledge of previously learned tasks. However, one of the most daunting challenges arising in this field is the phenomenon of forgetting. This article presents a novel perspective on the issue of forgetting by shifting the focus from the order of tasks to the underlying distribution from which tasks are sampled.
Key Contributions
We explore the implications of sampling tasks independently and identically distributed (i.i.d.) from a task distribution, referred to as $\Pi$. Our main contributions are as follows:
- Derivation of an exact operator identity for the forgetting quantity, revealing a recursive spectral structure.
- Establishment of an unconditional upper bound on the forgetting rate.
- Identification of the leading asymptotic term in the forgetting process.
- Characterization of the convergence rate in generic nondegenerate cases, providing insight into the dynamics of forgetting.
- Clarification of the relationship between the convergence rate and geometric properties of the task distribution.
Theoretical Framework
Our framework builds on the existing literature by incorporating a more nuanced understanding of how the generating distribution influences forgetting. By employing spectral analysis, we uncover a recursive structure that governs the forgetting dynamics in this exact-fit linear regime.
Results
Through our analysis, we establish that the rate of forgetting is not merely a function of the order in which tasks are presented but is fundamentally linked to the geometric characteristics of the task distribution. This insight allows us to differentiate between scenarios that result in slow versus fast forgetting, thereby providing a pathway for developing strategies to mitigate forgetting in continual learning systems.
Conclusion
In summary, our research contributes a significant advancement in understanding the theoretical underpinnings of forgetting in continual learning. By shifting the focus from order to distribution, we provide a comprehensive framework for analyzing and addressing the challenges associated with forgetting, paving the way for more effective continual learning strategies in the future.
Future Work
Future research should aim to explore the implications of our findings in practical applications and investigate methods to leverage the identified properties of task distributions to enhance the performance of continual learning models.