# Convergence Rates with Inexact Non-expansive Operators

arXiv: https://arxiv.org/abs/1404.4837 · HTML: https://mlsanity.com/paper/1404.4837/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Apr 2014.

## Built on by (most cited first, 10 of 10)

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- [Byzantine-Resilient Federated Learning via Distributed Optimization](https://mlsanity.com/paper/2503.10792/) (2025)
- [Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees](https://mlsanity.com/paper/2601.07665/) (2026)
- [A Class of Accelerated Fixed-Point-Based Methods with Delayed Inexact Oracles and Its Applications](https://mlsanity.com/paper/2512.13547/) (2025)
- [Last-Iterate Guarantees for Learning in Co-coercive Games](https://mlsanity.com/paper/2604.19065/) (2026)
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## Abstract (arXiv)

In this paper, we present a convergence rate analysis for the inexact Krasnosel'skii-Mann iteration built from nonexpansive operators. Our results include two main parts: we first establish global pointwise and ergodic iteration-complexity bounds, and then, under a metric subregularity assumption, we establish local linear convergence for the distance of the iterates to the set of fixed points. The obtained iteration-complexity result can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature, including the Forward-Backward, the Generalized Forward-Backward, Douglas-Rachford, alternating direction method of multipliers (ADMM) and Primal-Dual splitting methods. For these methods, we also develop easily verifiable termination criteria for finding an approximate solution, which can be seen as a generalization of the termination criterion for the classical gradient descent method. We finally develop a parallel analysis for the non-stationary Krasnosel'skii-Mann iteration. The usefulness of our results is illustrated by applying them to a large class of structured monotone inclusion and convex optimization problems. Experiments on some large scale inverse problems in signal and image processing problems are shown.