Polynomial convergence of iterations of certain random operators in Hilbert space

TL;DR

This paper investigates the convergence behavior of random iterative operators in Hilbert spaces, extending known results by broadening conditions for polynomial convergence and analyzing the impact of randomness.

Contribution

It introduces broader conditions for polynomial convergence of random operators in Hilbert spaces and characterizes the influence of randomness on convergence constants.

Findings

Established broader conditions for polynomial convergence

Proved almost sure convergence of the iterative sequence

Analyzed the role of randomness in convergence rates

Abstract

We study the convergence of a random iterative sequence of a family of operators on infinite dimensional Hilbert spaces, inspired by the Stochastic Gradient Descent (SGD) algorithm in the case of the noiseless regression, as studied in [1]. We identify conditions that are strictly broader than previously known for polynomial convergence rate in various norms, and characterize the roles the randomness plays in determining the best multiplicative constants. Additionally, we prove almost sure convergence of the sequence.