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Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees

Andrea Martin, Giuseppe Belgioioso

TL;DR

This work tackles accelerating fixed-point computations for general nonexpansive mappings by learning perturbations to the Krasnosel'skii–Mann (KM) iteration that preserve convergence guarantees. It introduces a principled L2O framework where summable perturbations $\bm{v}$ are added to the KM update, yielding inexact KM iterations whose convergence is guaranteed and whose local linear behavior is captured under metric sub-regularity. The authors prove that, locally, the learned perturbations can represent all sufficiently fast locally linear-convergent iterations while maintaining a vanishing bias term, and they show how to apply the framework to operator-splitting methods such as the Davis–Yin (DY) and Douglas–Rachford (DR) splits, including an explicit bound ensuring summability of the perturbations. The approach is demonstrated on a best-approximation problem where an L2O-augmented DR splitting accelerates convergence, illustrating practical speedups with convergence guarantees and suggesting broad applicability to monotone inclusions and equilibrium problems.

Abstract

We introduce a principled learning to optimize (L2O) framework for solving fixed-point problems involving general nonexpansive mappings. Our idea is to deliberately inject summable perturbations into a standard Krasnosel'skii-Mann iteration to improve its average-case performance over a specific distribution of problems while retaining its convergence guarantees. Under a metric sub-regularity assumption, we prove that the proposed parametrization includes only iterations that locally achieve linear convergence-up to a vanishing bias term-and that it encompasses all iterations that do so at a sufficiently fast rate. We then demonstrate how our framework can be used to augment several widely-used operator splitting methods to accelerate the solution of structured monotone inclusion problems, and validate our approach on a best approximation problem using an L2O-augmented Douglas-Rachford splitting algorithm.

Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees

TL;DR

This work tackles accelerating fixed-point computations for general nonexpansive mappings by learning perturbations to the Krasnosel'skii–Mann (KM) iteration that preserve convergence guarantees. It introduces a principled L2O framework where summable perturbations are added to the KM update, yielding inexact KM iterations whose convergence is guaranteed and whose local linear behavior is captured under metric sub-regularity. The authors prove that, locally, the learned perturbations can represent all sufficiently fast locally linear-convergent iterations while maintaining a vanishing bias term, and they show how to apply the framework to operator-splitting methods such as the Davis–Yin (DY) and Douglas–Rachford (DR) splits, including an explicit bound ensuring summability of the perturbations. The approach is demonstrated on a best-approximation problem where an L2O-augmented DR splitting accelerates convergence, illustrating practical speedups with convergence guarantees and suggesting broad applicability to monotone inclusions and equilibrium problems.

Abstract

We introduce a principled learning to optimize (L2O) framework for solving fixed-point problems involving general nonexpansive mappings. Our idea is to deliberately inject summable perturbations into a standard Krasnosel'skii-Mann iteration to improve its average-case performance over a specific distribution of problems while retaining its convergence guarantees. Under a metric sub-regularity assumption, we prove that the proposed parametrization includes only iterations that locally achieve linear convergence-up to a vanishing bias term-and that it encompasses all iterations that do so at a sufficiently fast rate. We then demonstrate how our framework can be used to augment several widely-used operator splitting methods to accelerate the solution of structured monotone inclusion problems, and validate our approach on a best approximation problem using an L2O-augmented Douglas-Rachford splitting algorithm.
Paper Structure (8 sections, 28 equations, 2 figures)

This paper contains 8 sections, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Problem Formulation
  4. Main Results
  5. A characterization of convergent iterations
  6. Application to operator splitting methods
  7. Experiments
  8. Conclusion

Figures (2)

  • Figure 1: Comparison between the evolution of the first $20$ shadow iterates $x^t$ and $y^t$ generated by \ref{['eq:baDR']} and \ref{['eq:baDR-l2o']}. The orange point indicates the query point to be projected, and the shaded orange crosses denote the training samples.
  • Figure 2: Evolution of the distance between the solution of \ref{['eq:projecting-intersection-problem']} and the shadow iterates $x^t$ generated by \ref{['eq:baDR']} and \ref{['eq:baDR-l2o']}. Shaded areas and solid lines denote standard deviations and mean values, respectively.