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

TL;DR

This paper develops data-dependent Bernstein inequalities for dependent vector-valued data in Hilbert spaces, enabling improved estimation and learning in non-i.i.d. settings with applications to covariance and operator estimation.

Contribution

It introduces novel Bernstein inequalities for dependent Hilbert space data, applicable to stationary and non-stationary processes, with practical bounds for covariance and operator learning.

Findings

Improved risk bounds for covariance operator estimation.

Effective bounds for operator learning in dynamical systems.

Numerical experiments demonstrate practical benefits.

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.