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Algorithmic Stability in Infinite Dimensions: Characterizing Unconditional Convergence in Banach Spaces

Przemysław Spyra

TL;DR

This work characterizes unconditional convergence in Banach spaces, revealing seven equivalent conditions that connect permutation invariance, net convergence, subseries and sign stability, bounded multipliers, and weak uniform convergence. By bridging topological, algebraic, and functional-analytic viewpoints, it clarifies why unconditional convergence is strictly stronger than plain convergence but weaker than absolute convergence in infinite dimensions, as highlighted by the Dvoretzky-Rogers gap. The main result yields rigorous proofs and then translates the theory into algorithmic guidance for stochastic gradient accumulation and frame-based coefficient processing, demonstrating order-independence and numerical robustness under appropriate conditions. The practical impact is a rigorous foundation for stable, order-insensitive summation in high- or infinite-dimensional settings, informing both distributed optimization and signal-processing pipelines.

Abstract

The distinction between conditional, unconditional, and absolute convergence in infinite-dimensional spaces has fundamental implications for computational algorithms. While these concepts coincide in finite dimensions, the Dvoretzky-Rogers theorem establishes their strict separation in general Banach spaces. We present a comprehensive characterization theorem unifying seven equivalent conditions for unconditional convergence: permutation invariance, net convergence, subseries tests, sign stability, bounded multiplier properties, and weak uniform convergence. These theoretical results directly inform algorithmic stability analysis, governing permutation invariance in gradient accumulation for Stochastic Gradient Descent and justifying coefficient thresholding in frame-based signal processing. Our work bridges classical functional analysis with contemporary computational practice, providing rigorous foundations for order-independent and numerically robust summation processes.

Algorithmic Stability in Infinite Dimensions: Characterizing Unconditional Convergence in Banach Spaces

TL;DR

This work characterizes unconditional convergence in Banach spaces, revealing seven equivalent conditions that connect permutation invariance, net convergence, subseries and sign stability, bounded multipliers, and weak uniform convergence. By bridging topological, algebraic, and functional-analytic viewpoints, it clarifies why unconditional convergence is strictly stronger than plain convergence but weaker than absolute convergence in infinite dimensions, as highlighted by the Dvoretzky-Rogers gap. The main result yields rigorous proofs and then translates the theory into algorithmic guidance for stochastic gradient accumulation and frame-based coefficient processing, demonstrating order-independence and numerical robustness under appropriate conditions. The practical impact is a rigorous foundation for stable, order-insensitive summation in high- or infinite-dimensional settings, informing both distributed optimization and signal-processing pipelines.

Abstract

The distinction between conditional, unconditional, and absolute convergence in infinite-dimensional spaces has fundamental implications for computational algorithms. While these concepts coincide in finite dimensions, the Dvoretzky-Rogers theorem establishes their strict separation in general Banach spaces. We present a comprehensive characterization theorem unifying seven equivalent conditions for unconditional convergence: permutation invariance, net convergence, subseries tests, sign stability, bounded multiplier properties, and weak uniform convergence. These theoretical results directly inform algorithmic stability analysis, governing permutation invariance in gradient accumulation for Stochastic Gradient Descent and justifying coefficient thresholding in frame-based signal processing. Our work bridges classical functional analysis with contemporary computational practice, providing rigorous foundations for order-independent and numerically robust summation processes.
Paper Structure (18 sections, 7 theorems, 31 equations)

This paper contains 18 sections, 7 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Motivation and Context
  3. The Infinite-Dimensional Challenge
  4. Contributions
  5. Related Work
  6. Organization
  7. Preliminaries
  8. Finite-Dimensional Equivalence
  9. The Infinite-Dimensional Gap
  10. The Characterization Theorem
  11. Applications
  12. Stochastic Gradient Descent and Data Ordering
  13. Signal Processing: Frame Expansions and Thresholding
  14. The Fragility of Conditional Convergence
  15. Conclusion and Open Problems
  16. ...and 3 more sections

Key Result

Lemma 3.1

Let $\{c_n\}_{n=1}^\infty \subset \mathbb{C}$. Then $\sum c_n$ converges absolutely if and only if it converges unconditionally.

Theorems & Definitions (23)

  • Definition 2.1: Banach and Hilbert Spaces
  • Definition 2.2: Series Convergence
  • Remark 2.3
  • Definition 2.4: Directed Sets and Nets
  • Definition 2.5: Convergence via Nets
  • Remark 2.6
  • Lemma 3.1: Complex Scalars
  • proof
  • Proposition 3.2: Finite-Dimensional Spaces
  • proof
  • ...and 13 more