Convergence of asynchronous matrix iterations subject to diagonal dominance

Cover title.Includes bibliographical references.Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058by Paul Tseng

On some properties of contracting matrices

The concepts of paracontracting, pseudocontracting and nonexpanding operators have been shown to be useful in proving convergence of asynchronous or parallel iteration algorithms. The purpose of this paper is to give characterizations of these operators when they are linear and finite-dimensional. First we show that pseudocontractivity of stochastic matrices with respect to sup-norm is equivalent to the scrambling property, a concept first introduced in the study of inhomogeneous Markov chains. This unifies results obtained independently using different approaches. Secondly, we generalize the concept of pseudocontractivity to set-contractivity which is a useful generalization with respect to the Euclidean norm. In particular, we demonstrate non-Hermitian matrices that are set-contractive for ||.||_2, but not pseudocontractive for ||.||_2 or sup-norm. For constant row sum matrices we characterize set-contractivity using matrix norms and matrix graphs. Furthermore, we prove convergence results in compositions of set-contractive operators and illustrate the differences between set-contractivity in different norms. Finally, we give an application to the global synchronization in coupled map lattices.Comment: 17 page

Convergence of the D-iteration algorithm: convergence rate and asynchronous distributed scheme

In this paper, we define the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We show how this can be applied to prove and improve the convergence of a fixed point problem associated to the matrix iteration scheme, including for distributed computation framework. The approach can be understood as a decomposition of the matrix-vector product operation in elementary operations at the vector entry level.Comment: 9 page

Average Consensus in the Presence of Delays and Dynamically Changing Directed Graph Topologies

Classical approaches for asymptotic convergence to the global average in a distributed fashion typically assume timely and reliable exchange of information between neighboring components of a given multi-component system. These assumptions are not necessarily valid in practical settings due to varying delays that might affect transmissions at different times, as well as possible changes in the underlying interconnection topology (e.g., due to component mobility). In this work, we propose protocols to overcome these limitations. We first consider a fixed interconnection topology (captured by a - possibly directed - graph) and propose a discrete-time protocol that can reach asymptotic average consensus in a distributed fashion, despite the presence of arbitrary (but bounded) delays in the communication links. The protocol requires that each component has knowledge of the number of its outgoing links (i.e., the number of components to which it sends information). We subsequently extend the protocol to also handle changes in the underlying interconnection topology and describe a variety of rather loose conditions under which the modified protocol allows the components to reach asymptotic average consensus. The proposed algorithms are illustrated via examples.Comment: 37 page

Parallel and distributed iterative algorithms : a selective survey

Cover title.Includes bibliographical references.Supported by the NSF with matching funds from Bellcore, Inc. and IBM Inc. ECS-8519058 ECS-8552419 Supported by the ARO. DAAL03-86-K-0171Dimitri P. Bertsekas, John N. Tsitsiklis

A survey of some aspects of parallel and distributed iterative algorithms

Cover title.Includes bibliographical references (p. 29-33).Research supported by the NSF. ECS-8519058 ECS-8552419 Research supported by the ARO. DAAL03-86-K-0171 Research supported by Bellcore, Du Pont and IBM.Dimitri P. Bertsekas, John N. Tsitsiklis

Red Light Green Light Method for Solving Large Markov Chains

Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. Without loss of generality, and drawing our motivation from applications to large networks, we interpret this problem as one of computing the stationary distribution of a random walk on a graph. We propose a new controlled, easily distributed algorithm for this task, briefly summarized as follows: at the beginning, each node receives a fixed amount of cash (positive or negative), and at each iteration, some nodes receive `green light' to distribute their wealth or debt proportionally to the transition probabilities of the Markov chain; the stationary probability of a node is computed as a ratio of the cash distributed by this a node to the total cash distributed by all nodes together. Our method includes as special cases a wide range of known, very different, and previously disconnected methods including power iterations, Gauss-Southwell, and online distributed algorithms. We prove exponential convergence of our method, demonstrate its high efficiency, and derive scheduling strategies for the green-light, that achieve convergence rate faster than state-of-the-art algorithms