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

arXiv: https://arxiv.org/abs/2601.07665 · HTML: https://mlsanity.com/paper/2601.07665/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 34 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [A Three-Operator Splitting Scheme and its Optimization Applications](https://mlsanity.com/paper/1504.01032/) (2015)
- [Learning to Warm-Start Fixed-Point Optimization Algorithms](https://mlsanity.com/paper/2309.07835/) (2023)
- [Accelerating Quadratic Optimization with Reinforcement Learning](https://mlsanity.com/paper/2107.10847/) (2021)
- [Safeguarded Learned Convex Optimization](https://mlsanity.com/paper/2003.01880/) (2020)
- [Transformer-based Model Predictive Control: Trajectory Optimization via Sequence Modeling](https://mlsanity.com/paper/2410.23916/) (2024)
- [Accelerated Forward-Backward Optimization using Deep Learning](https://mlsanity.com/paper/2105.05210/) (2021)
- [Learning to Optimize in Model Predictive Control](https://mlsanity.com/paper/2212.02603/) (2022)
- [Compact Optimization Learning for AC Optimal Power Flow](https://mlsanity.com/paper/2301.08840/) (2023)
- [Convergence Rates with Inexact Non-expansive Operators](https://mlsanity.com/paper/1404.4837/) (2014)
- [Learning to optimize with convergence guarantees using nonlinear system theory](https://mlsanity.com/paper/2403.09389/) (2024)
- [Inertial Krasnoselskii-Mann Iterations](https://mlsanity.com/paper/2210.03791/) (2022)
- [Data-Driven Performance Guarantees for Classical and Learned Optimizers](https://mlsanity.com/paper/2404.13831/) (2024)
- [Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces](https://mlsanity.com/paper/1301.4506/) (2013)
- [SuperMann: a superlinearly convergent algorithm for finding fixed points of nonexpansive operators](https://mlsanity.com/paper/1609.06955/) (2016)
- [Line Search for Averaged Operator Iteration](https://mlsanity.com/paper/1603.06772/) (2016)
- [Learning to optimize with guarantees: a complete characterization of linearly convergent algorithms](https://mlsanity.com/paper/2508.00775/) (2025)
- [Linear and strong convergence of algorithms involving averaged nonexpansive operators](https://mlsanity.com/paper/1402.5460/) (2014)
- [Incorporating History and Deviations in Forward--Backward Splitting](https://mlsanity.com/paper/2208.05498/) (2022)

## Built on by (most cited first, 2 of 2)

- [Learning Over-Relaxation Policies for ADMM with Convergence Guarantees](https://mlsanity.com/paper/2604.26932/) (2026)
- [Reinforcement Learning to Accelerate Primal-Dual Hybrid Gradient for Linear Programming](https://mlsanity.com/paper/2610.01546/) (2026)

## Abstract (arXiv)

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.