用学习方法加速不动点迭代，保证收敛且提升平均性能。
Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees
- 在标准迭代中注入可求和扰动，优化平均表现。
- 证明新方法局部线性收敛，偏差趋近于零。
- 适用于多种分裂算法，适合需要高效求解的场景。
我们提出一种针对一般非扩张映射的不动点问题的合理学习优化（L2O）框架。通过在标准Krasnosel'skii-Mann迭代中刻意引入可求和扰动，在保持收敛性保证的同时，提升特定问题分布下的平均性能。在度量次正则性假设下，我们证明所提参数化仅包含局部实现线性收敛（至可忽略偏差项）的迭代，并涵盖所有以足够快速率实现线性收敛的迭代。随后，我们展示了该框架如何用于增强多种常用分裂方法，以加速结构单调包含问题的求解，并通过使用L2O增强的Douglas-Rachford分裂算法在最佳逼近问题上验证了方法的有效性。
原文摘要 · Abstract (English)
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.
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