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

TL;DR

This paper proposes a learning-based framework to accelerate Krasnosel'skii-Mann fixed-point iterations, providing convergence guarantees and demonstrating improved performance on structured monotone inclusion problems.

Contribution

It introduces a novel L2O approach that injects perturbations into fixed-point iterations, ensuring local linear convergence with guarantees, and applies it to enhance operator splitting methods.

Findings

Accelerated convergence in fixed-point iterations.

Validated improvements on structured monotone inclusion problems.

Guarantees of local linear convergence under certain conditions.

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.