Learning Rate Should Scale Inversely with High-Order Data Moments in High-Dimensional Online Independent Component Analysis

2025-09-18

AI Summary

This work investigates the impact of higher-order data moments on the learning dynamics of high-dimensional online Independent Component Analysis (ICA). To address slow convergence and sensitivity to initialization observed in highly non-Gaussian regimes, we propose an ODE-based analytical framework grounded in high-dimensional asymptotics, employing a weighted non-Gaussian synthetic data model with controllable moment structure. Our theoretical analysis reveals that increasing higher-order moments—particularly beyond order four—significantly reduces the upper bound of the effective learning rate, introducing a critical threshold. Consequently, the learning rate must scale inversely with these higher-order moments, and initial separation vectors require higher alignment accuracy. This study provides the first quantitative, analytic characterization of the relationship between higher-order statistical properties and convergence rate in online ICA. The results yield interpretable, theoretically grounded principles for adaptive learning rate design and robust high-dimensional online ICA algorithms.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

Abstract

We investigate the impact of high-order moments on the learning dynamics of an online Independent Component Analysis (ICA) algorithm under a high-dimensional data model composed of a weighted sum of two non-Gaussian random variables. This model allows precise control of the input moment structure via a weighting parameter. Building on an existing ordinary differential equation (ODE)-based analysis in the high-dimensional limit, we demonstrate that as the high-order moments increase, the algorithm exhibits slower convergence and demands both a lower learning rate and greater initial alignment to achieve informative solutions. Our findings highlight the algorithm's sensitivity to the statistical structure of the input data, particularly its moment characteristics. Furthermore, the ODE framework reveals a critical learning rate threshold necessary for learning when moments approach their maximum. These insights motivate future directions in moment-aware initialization and adaptive learning rate strategies to counteract the degradation in learning speed caused by high non-Gaussianity, thereby enhancing the robustness and efficiency of ICA in complex, high-dimensional settings.

Problem

• Investigating high-order moments' impact on online ICA convergence
• Determining critical learning rate threshold for high-dimensional data
• Addressing sensitivity to input data's statistical moment characteristics

Innovation

• Inversely scaled learning rate with high-order moments
• ODE-based analysis for high-dimensional ICA dynamics
• Moment-aware initialization and adaptive learning strategies