Bo\c{t}-Nguyen Acceleration, Weighted Mean Ergodic Iteration, and the Beta-Binomial Distribution

TL;DR

This paper analyzes a specific linear case of Boț-Nguyen accelerated algorithms, revealing connections to weighted mean ergodic iterations and the beta-binomial distribution, with convergence properties depending on parameters.

Contribution

It demonstrates that the linear case of Boț-Nguyen acceleration aligns with weighted mean ergodic iterations and links to the beta-binomial distribution, providing convergence insights.

Findings

Weak limit identified as projection onto fixed point set

Weights relate to beta-binomial distribution

Strong convergence achieved when parameter equals 4

Abstract

In 2023, Bo\c{t} and Nguyen introduced a new class of accelerated algorithms for finding a fixed point of a nonexpansive operator as the weak limit of a sequence. In this paper, we analyze a particular instance of their algorithm when the nonexpansive operator is assumed to be linear. Surprisingly, the Bo\c{t}-Nguyen acceleration then fits naturally into the framework of weighted mean ergodic iterations. This allows us to identify the weak limit as the projection of the starting point onto the fixed point set. Moreover, the weights involved are closely related to the beta-binomial distribution. Finally, when the parameter is equal to 4, then we obtain strong convergence of the iterates.