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Generalized fluctuation bounds for stochastic algorithms in the presence of compactness

Morenikeji Neri, Nicholas Pischke, Thomas Powell

TL;DR

This work derives 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.

Abstract

We provide a convergence result for sequences of random variables taking values in a metric space that satisfy a stochastic quasi-Fejér 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-Fejér 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.

Generalized fluctuation bounds for stochastic algorithms in the presence of compactness

TL;DR

This work derives 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.

Abstract

We provide a convergence result for sequences of random variables taking values in a metric space that satisfy a stochastic quasi-Fejér 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-Fejér 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.
Paper Structure (15 sections, 34 theorems, 294 equations)

This paper contains 15 sections, 34 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Overview of the main technical results
  4. Relations to logic and proof mining
  5. Summary and organization of the paper
  6. Preliminaries and notation
  7. Finitary martingales and quantitative martingale convergence
  8. A quantitative Robbins-Siegmund theorem
  9. Quantitative convergence of stochastic quasi-Fejér monotone sequences
  10. Stochastic $\liminf$-moduli
  11. Moduli of uniform stochastic quasi-Fejér monotonicity
  12. A quantitative almost sure convergence theorem under relative compactness
  13. Application: Common fixed point problems
  14. Common fixed points on separable Hadamard spaces
  15. A Krasnoselskii-Mann scheme for approximating common fixed points

Key Result

Lemma 1.1

Let $\varphi(n)\in\mathsf{F}$ be a measurable property for all $n\in\mathbb{N}$ and $p\in [0,1]$. If $\varphi$ is anti-monotone, i.e. $\varphi(n+1)\subseteq\varphi(n)$ for all $n\in\mathbb{N}$, then If $\varphi$ is monotone, i.e. $\varphi(n)\subseteq\varphi(n+1)$ for all $n\in\mathbb{N}$, then

Theorems & Definitions (93)

  • Lemma 1.1: Lemma 3.1 in NeriPowell2025a
  • Definition 2.1: Finitary supermartingale
  • Remark 2.2: For logicians
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5: For logicians
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 83 more