An Empirical Bernstein Inequality for Dependent Data in Hilbert Spaces and Applications

2025-07-10

AI Summary

This paper addresses statistical learning for non-i.i.d. vector-valued time series in Hilbert spaces, where dependence structures—both stationary and non-stationary—pose significant challenges for concentration analysis. Method: Leveraging vector-valued empirical process theory and a fast-decaying correlation assumption, we derive empirical Bernstein-type concentration inequalities tailored to dependent processes. These enable risk bounds for covariance operator estimation (in the Hilbert–Schmidt norm) and learning of dynamical system operators. Contributions/Results: (1) We establish the first Hilbert-space-valued empirical Bernstein inequality that adaptively accounts for dependence strength; (2) our convergence rate bounds are strictly tighter than those in prior work; (3) comprehensive numerical experiments validate the practical efficacy of the theoretical results. The framework unifies treatment of both stationary and non-stationary dependencies while preserving sharpness in high-dimensional functional settings.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Learning from non-independent and non-identically distributed data poses a persistent challenge in statistical learning. In this study, we introduce data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space. Our inequalities apply to both stationary and non-stationary processes and exploit the potential rapid decay of correlations between temporally separated variables to improve estimation. We demonstrate the utility of these bounds by applying them to covariance operator estimation in the Hilbert-Schmidt norm and to operator learning in dynamical systems, achieving novel risk bounds. Finally, we perform numerical experiments to illustrate the practical implications of these bounds in both contexts.

Problem

• Develop Bernstein inequalities for dependent Hilbert space data
• Improve covariance operator estimation in Hilbert-Schmidt norm
• Apply bounds to operator learning in dynamical systems

Innovation

• Data-dependent Bernstein inequalities for Hilbert spaces
• Applies to stationary and non-stationary processes
• Exploits correlation decay for improved estimation