Gradient Descent Algorithm in Hilbert Spaces under Stationary Markov Chains with $\phi$- and $\beta$-Mixing

TL;DR

This paper analyzes the convergence of a gradient descent algorithm in Hilbert spaces driven by stationary Markov chains, emphasizing the impact of mixing properties on convergence rates.

Contribution

It provides probabilistic convergence bounds for the algorithm under $ ext{phi}$- and $eta$-mixing conditions, extending analysis to general Hilbert spaces.

Findings

Convergence bounds depend on the decay rate of mixing coefficients.

Exponential and polynomial mixing decay influence convergence speed.

Results applicable to a broad class of Markov chain-driven algorithms.

Abstract

In this paper, we study a strictly stationary Markov chain gradient descent algorithm operating in general Hilbert spaces. Our analysis focuses on the mixing coefficients of the underlying process, specifically the $\phi$- and $\beta$-mixing coefficients. Under these assumptions, we derive probabilistic upper bounds on the convergence behavior of the algorithm based on the exponential as well as the polynomial decay of the mixing coefficients.