Continuous-time Online Learning via Mean-Field Neural Networks: Regret Analysis in Diffusion Environments

2026-04-12

AI Summary

This work addresses online learning and cumulative regret control in continuous-time diffusion data streams with unknown coefficients by proposing a non-anticipative parameter update mechanism based on two-layer neural networks. The mean-field limit of this scheme corresponds to a stochastic Wasserstein gradient flow adapted to the data filtration. For the first time, mean-field neural networks are introduced into online learning under continuous-time diffusion settings, leveraging displacement convexity to establish a constant static regret bound and, in non-convex cases, an explicit linear dynamic regret bound that reveals the interplay among data evolution, entropy-driven exploration, and regularization. The theoretical analysis integrates mean-field limits, logarithmic Sobolev inequalities, and Malliavin calculus to rigorously bound the regret for both finite-particle systems and their mean-field counterparts. Numerical experiments confirm the method’s superiority and highlight the critical roles of network width and regularization parameters.

Technology Category

• Machine Learning: Online Learning & Bandits
• Natural Language Processing: Learning & Optimization for NLP
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• 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

Abstract

We study continuous-time online learning where data are generated by a diffusion process with unknown coefficients. The learner employs a two-layer neural network, continuously updating its parameters in a non-anticipative manner. The mean-field limit of the learning dynamics corresponds to a stochastic Wasserstein gradient flow adapted to the data filtration. We establish regret bounds for both the mean-field limit and finite-particle system. Our analysis leverages the logarithmic Sobolev inequality, Polyak-Lojasiewicz condition, Malliavin calculus, and uniform-in-time propagation of chaos. Under displacement convexity, we obtain a constant static regret bound. In the general non-convex setting, we derive explicit linear regret bounds characterizing the effects of data variation, entropic exploration, and quadratic regularization. Finally, our simulations demonstrate the outperformance of the online approach and the impact of network width and regularization parameters.

Problem

• online learning
• diffusion processes
• mean-field neural networks
• regret analysis
• continuous-time

Innovation

• mean-field neural networks
• continuous-time online learning
• stochastic Wasserstein gradient flow
• regret analysis
• propagation of chaos