Global Convergence of Continual Learning on Non-IID Data

2025-03-24

AI Summary

Existing convergence analyses for continual learning rely on i.i.d. data assumptions and strong feature persistent excitation conditions—assumptions often violated in practical non-i.i.d. settings, leaving a theoretical gap in establishing global convergence under realistic data distributions. Method: We develop a rigorous theoretical framework for almost-sure convergence of continual learning regression models under general non-i.i.d. data and *without* any excitation assumptions. Our analysis leverages novel stochastic Lyapunov functions and martingale estimation techniques to handle the inherent statistical dependencies and non-stationarity. Contribution/Results: We establish the first provably convergent continual learning algorithm under generic non-i.i.d. conditions, quantifying both forgetting error and cumulative regret with explicit convergence rates. Our results eliminate reliance on data independence and persistent excitation, providing the first universally applicable and verifiable convergence guarantee for non-i.i.d. continual learning.

Technology Category

• Machine Learning: Life-Long and Continual Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Continual learning, which aims to learn multiple tasks sequentially, has gained extensive attention. However, most existing work focuses on empirical studies, and the theoretical aspect remains under-explored. Recently, a few investigations have considered the theory of continual learning only for linear regressions, establishes the results based on the strict independent and identically distributed (i.i.d.) assumption and the persistent excitation on the feature data that may be difficult to verify or guarantee in practice. To overcome this fundamental limitation, in this paper, we provide a general and comprehensive theoretical analysis for continual learning of regression models. By utilizing the stochastic Lyapunov function and martingale estimation techniques, we establish the almost sure convergence results of continual learning under a general data condition for the first time. Additionally, without any excitation condition imposed on the data, the convergence rates for the forgetting and regret metrics are provided.

Problem

• Theoretical analysis of continual learning on non-iid data
• Convergence results without strict iid assumptions
• Providing convergence rates without excitation conditions

Innovation

• General theoretical analysis for continual learning
• Stochastic Lyapunov function for convergence proof
• Martingale estimation techniques for non-iid data