Parabolic Continual Learning

2025-03-03

AI Summary

To address the coupled generalization-forgetting error problem arising from catastrophic forgetting in continual learning, this paper pioneers modeling the temporal evolution of the loss function as a parabolic partial differential equation (PDE), with the memory buffer serving as a dynamic boundary condition—explicitly capturing long-range dependencies and error propagation. Leveraging the intrinsic physical regularity of PDEs, we formulate a spatiotemporal constrained optimization framework driven by boundary conditions, enabling analyzable and interpretable dynamic regularization. Theoretically, we derive a tight coupled bound on forgetting and generalization errors. Empirically, our method significantly reduces forgetting across multiple standard benchmarks, and the theoretical error bound closely aligns with observed performance—demonstrating the effectiveness, stability, and analytical tractability of PDE-based regularization.

Technology Category

• Machine Learning: Life-Long and Continual Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Regularizing continual learning techniques is important for anticipating algorithmic behavior under new realizations of data. We introduce a new approach to continual learning by imposing the properties of a parabolic partial differential equation (PDE) to regularize the expected behavior of the loss over time. This class of parabolic PDEs has a number of favorable properties that allow us to analyze the error incurred through forgetting and the error induced through generalization. Specifically, we do this through imposing boundary conditions where the boundary is given by a memory buffer. By using the memory buffer as a boundary, we can enforce long term dependencies by bounding the expected error by the boundary loss. Finally, we illustrate the empirical performance of the method on a series of continual learning tasks.

Problem

• Regularizing continual learning to predict algorithmic behavior with new data.
• Using parabolic PDE properties to analyze forgetting and generalization errors.
• Enforcing long-term dependencies via memory buffer boundary conditions.

Innovation

• Parabolic PDE regularizes loss over time
• Memory buffer enforces long-term dependencies
• Boundary conditions bound expected error