Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods

TL;DR

This paper establishes a Bernstein-type inequality for weakly dependent Banach-valued sums and applies it to analyze error bounds in spectral regularization methods for kernel learning with dependent data.

Contribution

It introduces a new Bernstein inequality for Banach-valued sums under weak dependence and uses it to derive error bounds for spectral regularization in dependent data scenarios.

Findings

Derived a Bernstein-type inequality for Banach-valued sums with weak dependence.

Provided asymptotic error bounds for spectral regularization methods in dependent data.

Applied the inequality to kernel decision rules trained on $ au$-mixing processes.

Abstract

We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad family of spectral regularization methods for reproducing kernel decision rules, when trained on a sample coming from a $\tau-$mixing process.