On the rate of convergence of Krasnosel’skiĭ-Mann iterations and their connection with sums of Bernoullis

Cominetti, Roberto; Soto, José A.; Vaisman, J.

doi:10.1007/s11856-013-0045-4

Cited by 52 publications

(72 citation statements)

References 26 publications

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“…Regarding the second case, notice that if T t is contractive then F t is contractive; however, the converse is not necessarily true. We start by outlining the following standard assumptions [1], [7].…”

Section: Convergencementioning

confidence: 99%

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On the Convergence of the Inexact Running Krasnosel’skiĭ–Mann Method

Dall’Anese

Simonetto

Bernstein

2019

IEEE Control Syst. Lett.

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This paper leverages a framework based on averaged operators to tackle the problem of tracking fixed points associated with maps that evolve over time. In particular, the paper considers the Krasnosel'skiȋ-Mann method in a settings where: (i) the underlying map may change at each step of the algorithm, thus leading to a "running" implementation of the Krasnosel'skiȋ-Mann method; and, (ii) an imperfect information of the map may be available. An imperfect knowledge of the maps can capture cases where processors feature a finite precision or quantization errors, or the case where (part of) the map is obtained from measurements. The analytical results are applicable to inexact running algorithms for solving optimization problems, whenever the algorithmic steps can be written in the form of (a composition of) averaged operators; examples are provided for inexact running gradient methods and the forward-backward splitting method. Convergence of the average fixed-point residual is investigated for the nonexpansive case; linear convergence to a unique fixed-point trajectory is showed in the case of inexact running algorithms emerging from contractive operators.

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Section: Convergencementioning

confidence: 99%

“…Funds for A. Bernstein were provided by ARPA-e NODES. constant value of λ k = λ as an example, one has that the average fixed-point residual of the map T after K iterations can be bounded as [1], [7]:…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

On the Convergence of the Inexact Running Krasnosel’skiĭ–Mann Method

Dall’Anese

Simonetto

Bernstein

2019

IEEE Control Syst. Lett.

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“…where ǫ k is the error of approximating T (x k ), and {α k } k∈N ∈ [0, 1] is a sequence of relaxation parameters. When ǫ k ≡ 0 for all k ∈ N, (7) reduces to the classical KM iteration [20], [22]. It has been shown that the (inexact) KM iteration converges Assumption 1 (Graph Connectivity and Weights Rule).…”

Section: Preliminaries and Problem Statementmentioning

confidence: 99%

“…Remark 1. It is worth pointing out that it is in general standard to leverage T (x) − x as a measure of the convergence speed for the centralized (inexact) KM iteration, since T (x) − x = 0 amounts to T (x) = x, see [20]- [25]. This is why F i (x i,k l ) − x i,k l is employed for measuring the convergence rate of the D-IKM iteration, as shown in (12).…”

Section: The D-ikm Iterationmentioning

confidence: 99%

“…Note that Picard iteration does not converge in general for a nonexpansive operator. The KM iteration is firstly proposed in [27], [28], which have so far received tremendous attention [20]- [26]. Moreover, the KM iteration provides a unified framework for analysis of various algorithms, such as Proximal point algorithms (PPA) [29], forward-backward splitting method (FBS) [30], Peaceman-Rachford splitting (PRS) [31], Douglas-Rachford splitting (DRS) [32], [33], alternating direction method of multipliers (ADMM) [34], and a threeoperator splitting [35].…”

Section: Introductionmentioning

confidence: 99%

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A distributed algorithm for computing a common fixed point of a family of strongly quasi-nonexpansive maps

Liu

Fullmer

Nedić

et al. 2017

2017 American Control Conference (ACC)

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This paper addresses the problem of seeking a common fixed point for a collection of nonexpansive operators over time-varying multi-agent networks in real Hilbert spaces, where each operator is only privately and approximately known to each individual agent, and all agents need to cooperate to solve this problem by propagating their own information to their neighbors through local communications over time-varying networks. To handle this problem, inspired by the centralized inexact Krasnosel'skiȋ-Mann (IKM) iteration, we propose a distributed algorithm, called distributed inexact Krasnosel'skiȋ-Mann (D-IKM) iteration. It is shown that the D-IKM iteration can converge weakly to a common fixed point of the family of nonexpansive operators. Moreover, under the assumption that all operators and their own fixed point sets are (boundedly) linearly regular, it is proved that the D-IKM iteration converges with a rate O(1/k ln(1/ξ) ) for some constant ξ ∈ (0, 1), where k is the iteration number. To reduce computational complexity and burden of storage and transmission, a scenario, where only a random part of coordinates for each agent is updated at each iteration, is further considered, and a corresponding algorithm, named distributed inexact block-coordinate Krasnosel'skiȋ-Mann (D-IBKM) iteration, is developed. The algorithm is proved to be weakly convergent to a common fixed point of the group of considered operators, and, with the extra assumption of (bounded) linear regularity, it is convergent with a rate O(1/k ln(1/ξ) ). Furthermore, it is shown that the convergence rate O(1/k ln(1/ξ) ) can still be guaranteed under a more relaxed (bounded) power regularity condition.Index Terms-Distributed algorithms, multi-agent networks, Krasnosel'skiȋ-Mann iteration, nonexpansive operators, fixed point, optimization.weakly to a fixed point of T under mild conditions [21], [25], [36], for example, when ∞ j=1 α j (1 − α j ) = ∞ for the KM iteration [36]. Now, let us introduce the graph theory for describing the communication pattern among all agents [11]. Specifically, the communication mode among N agents can be modeled by a digraph G = (V, E), where V = [N ] is the node or vertex set, and E ⊂ V × V is the edge set. An edge (i, j) ∈ E means that agent i is capable of transmitting its information to agent j, in which case agent i is called a neighbor of agent j. A directed path from i 1 to i l is a sequence of edges of the form (i 1 , i 2 ), (i 3 , i 4 ), . . . , (i l−1 , i l ). A graph is called strongly connected if there exists at least a directed path from any node to any other node in this graph. In this paper, the communication graph for all agents is assumed to be time-varying, that is, any two agents can have different communication status at different time steps. In this case, the graph is denoted asAt each time k ∈ N, there exists an adjacency matrix A k = (a ij,k ) ∈ R N ×N such that a ij,k > 0 if (j, i) ∈ E k , and a ij,k = 0 otherwise. Assume that a ii,k > 0 for all i ∈ [N ] and all k ∈ N. For communication graphs, we...

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