# Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces

arXiv: https://arxiv.org/abs/1705.09340 · HTML: https://mlsanity.com/paper/1705.09340/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2017.

## Built on by (most cited first, 5 of 5)

- [Asymptotic and Finite Sample Analysis of Nonexpansive Stochastic Approximations with Markovian Noise](https://mlsanity.com/paper/2409.19546/) (2024)
- [Almost Sure Convergence of Linear Temporal Difference Learning with Arbitrary Features](https://mlsanity.com/paper/2409.12135/) (2024)
- [Non-Expansive Mappings in Two-Time-Scale Stochastic Approximation: Finite-Time Analysis](https://mlsanity.com/paper/2501.10806/) (2025)
- [A Class of Accelerated Fixed-Point-Based Methods with Delayed Inexact Oracles and Its Applications](https://mlsanity.com/paper/2512.13547/) (2025)
- [Quotient-Categorical Representations for Bellman-Compatible Average-Reward Distributional Reinforcement Learning](https://mlsanity.com/paper/2605.11289/) (2026)

## Abstract (arXiv)

We study the convergence of an inexact version of the classical Krasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kii-Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.