Lyapunov Learning at the Onset of Chaos

2025-06-15

AI Summary

To address catastrophic forgetting and slow adaptation in online learning under non-stationary time series and abrupt paradigm shifts, this paper proposes an online learning algorithm grounded in the critical state of chaotic dynamics. The method dynamically regulates neural network training via real-time control of the maximum Lyapunov exponent (MLE), steering the system to operate persistently near the “edge of chaos” (MLE ≈ 0). This regime simultaneously preserves stability of learned knowledge while enabling flexible exploration of the “adjacent possible” in local solution spaces. The approach integrates chaotic modeling, online MLE estimation, and adaptive regularization. Empirical evaluation on a Lorenz system with sudden parameter shifts demonstrates that the proposed method reduces test loss by approximately 96% compared to standard online training, significantly enhancing robustness to distributional shifts and accelerating model recalibration.

Technology Category

• Machine Learning: Online Learning & Bandits
• 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
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Handling regime shifts and non-stationary time series in deep learning systems presents a significant challenge. In the case of online learning, when new information is introduced, it can disrupt previously stored data and alter the model's overall paradigm, especially with non-stationary data sources. Therefore, it is crucial for neural systems to quickly adapt to new paradigms while preserving essential past knowledge relevant to the overall problem. In this paper, we propose a novel training algorithm for neural networks called extit{Lyapunov Learning}. This approach leverages the properties of nonlinear chaotic dynamical systems to prepare the model for potential regime shifts. Drawing inspiration from Stuart Kauffman's Adjacent Possible theory, we leverage local unexplored regions of the solution space to enable flexible adaptation. The neural network is designed to operate at the edge of chaos, where the maximum Lyapunov exponent, indicative of a system's sensitivity to small perturbations, evolves around zero over time. Our approach demonstrates effective and significant improvements in experiments involving regime shifts in non-stationary systems. In particular, we train a neural network to deal with an abrupt change in Lorenz's chaotic system parameters. The neural network equipped with Lyapunov learning significantly outperforms the regular training, increasing the loss ratio by about $96%$.

Problem

• Handling regime shifts in non-stationary time series
• Adapting to new paradigms while preserving past knowledge
• Training neural networks at the edge of chaos

Innovation

• Uses Lyapunov Learning for neural networks
• Leverages chaotic systems for regime shifts
• Operates at edge of chaos for adaptation