Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918284116361216 |
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| author | Martin, Andrea Belgioioso, Giuseppe |
| author_facet | Martin, Andrea Belgioioso, Giuseppe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07665 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees Martin, Andrea Belgioioso, Giuseppe Systems and Control Machine Learning Optimization and Control 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. |
| title | Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees |
| topic | Systems and Control Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2601.07665 |