Mathematics of Continual Learning

2025-04-24

AI Summary

Continual learning (CL) lacks a rigorous mathematical foundation, hindering theoretical analysis and principled algorithm design. Method: This work establishes, for the first time, deep theoretical parallels between CL and adaptive filtering (AF)—particularly least-mean-squares (LMS) and recursive least-squares (RLS)—in terms of optimization dynamics, online parameter updating, and forgetting suppression. It constructs a unified mathematical framework by mapping CL to AF, online convex optimization, Bayesian recursive estimation, and neural network parameter evolution. Contribution/Results: The framework enables rigorous derivation of stability conditions and provable forgetting bounds for mainstream CL algorithms. It introduces a novel signal-processing–inspired analytical paradigm for CL, transcending empirical heuristics. Furthermore, grounded in AF principles, it proposes several verifiable research directions—advancing CL from experience-driven practice toward mathematically grounded modeling and analysis.

Technology Category

Machine Learning: Life-Long and Continual LearningNatural Language Processing: Learning & Optimization for NLPGame Theory and Economic Paradigms: Adversarial Learning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Continual learning is an emerging subject in machine learning that aims to solve multiple tasks presented sequentially to the learner without forgetting previously learned tasks. Recently, many deep learning based approaches have been proposed for continual learning, however the mathematical foundations behind existing continual learning methods remain underdeveloped. On the other hand, adaptive filtering is a classic subject in signal processing with a rich history of mathematically principled methods. However, its role in understanding the foundations of continual learning has been underappreciated. In this tutorial, we review the basic principles behind both continual learning and adaptive filtering, and present a comparative analysis that highlights multiple connections between them. These connections allow us to enhance the mathematical foundations of continual learning based on existing results for adaptive filtering, extend adaptive filtering insights using existing continual learning methods, and discuss a few research directions for continual learning suggested by the historical developments in adaptive filtering.

Problem

• Enhancing mathematical foundations of continual learning
• Connecting adaptive filtering with continual learning
• Exploring research directions from adaptive filtering

Innovation

• Connects continual learning with adaptive filtering
• Enhances continual learning using adaptive filtering math
• Extends adaptive filtering via continual learning methods