Generalized fluctuation bounds for stochastic algorithms in the presence of compactness

TL;DR

This paper develops a quantitative convergence analysis for stochastic sequences in metric spaces under compactness assumptions, introducing a finitary martingale theory and applying it to stochastic fixed point algorithms.

Contribution

It provides the first explicit metastable convergence rates for stochastic quasi-Fejér sequences in metric spaces using a finitary martingale approach.

Findings

Derived a metastable rate of pointwise convergence for stochastic sequences.

Developed a finitary Robbins-Siegmund theorem for martingales.

Applied results to stochastic fixed point algorithms in Hadamard spaces.

Abstract

We provide a convergence result for sequences of random variables taking values in a metric space that satisfy a stochastic quasi-Fej\'er monotonicity condition, in the context of a (local) compactness assumption. Our result is quantitative in that we derive an explicit and effective construction which, in terms of only a few moduli representing quantitative witnesses to key properties of the sequence of random variables and the underlying metric space involved, provides a metastable rate of pointwise convergence, a type of generalized fluctuation bound. That quantitative result in particular relies on the development of a finitary theory of martingales, culminating in a fully finitary Robbins-Siegmund theorem. We outline how this result particularises to the circumstances of the seminal work of Combettes and Pesquet on stochastic quasi-Fej\'er monotone sequences in separable Hilbert spaces, and we provide an initial application by illustrating how these results can be used to provide a metastable rate of pointwise convergence for a stochastic Krasnoselskii-Mann scheme solving a stochastic common fixed point problem for nonexpansive maps over proper Hadamard spaces. This work is set in the context of recent applications of the logic-based methodology of proof mining to probability theory, and represents its most sophisticated case study to date.