Related papers: Minimax-optimal Halpern iterations for Lipschitz m…
We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving…
This paper investigates optimal error bounds and convergence rates for general Mann iterations for computing fixed-points of non-expansive maps. We look for iterations that achieve the smallest fixed-point residual after $n$ steps, by…
The Halpern algorithm is a powerful fixed point approximation method for finding the closest point in the fixed point set of a nonexpansive mapping to the initial point. However, in practice, it is not necessarily true that this algorithm…
The Banach-Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This paper extends the Banach-Picard iteration to distributed settings; specifically, we assume the map of which the fixed point is sought to…
We analyze the oracle complexity of the stochastic Halpern iteration with minibatch, where we aim to approximate fixed-points of nonexpansive and contractive operators in a normed finite-dimensional space. We show that if the underlying…
The Krasnosel'ski\u{\i}--Mann and Halpern iterations are classical schemes for approximating fixed points of nonexpansive mappings in Banach spaces, and have been widely studied in more general frameworks such as $CAT(\kappa)$ and, more…
This paper presents a tractable algorithm for estimating an unknown Lipschitz function from noisy observations and establishes an upper bound on its convergence rate. The approach extends max-affine methods from convex shape-restricted…
Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that…
In a recent paper~\cite{paper2}, we proposed the concept of optimal error bounds for an iterative process, which allows us to obtain the convergence result of the iterative sequence to the common fixed point of the nonexpansive mappings in…
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…
In this paper, we study a new approach related to the convergence analysis of Ishikawa-type iterative models to a common fixed point of two non-expansive mappings in Banach spaces. The main novelty of our contribution lies in the so-called…
Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting…
We propose an adaptive way to choose the anchoring parameters for the Halpern iteration to find a fixed point of a nonexpansive mapping in a real Hilbert space. We prove strong convergence of this adaptive Halpern iteration and obtain the…
In this paper, we introduce some new iterative optimisation algorithms on Riemannian manifolds and Hilbert spaces which have good global convergence guarantees to local minima. More precisely, these algorithms have the following properties:…
We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in…
We investigate the asymptotic behavior of Halpern-type iterations applied to quasi-nonexpansive operators arising in best approximation problems over the intersection of finitely many closed convex sets in $\mathbb{R}^n$. Assuming a local…
In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps $F$ between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames,…
In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, proof-theoretic techniques. We give effective rates of asymptotic regularity for…
In this study, we introduce a new iterative processes to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings…
We give a rate of metastability for Halpern's iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth Banach spaces, extracted from a proof due to Xu. In Hilbert space, the latter is…