Asynchronous Approximation of a Single Component of the Solution to a Linear System

We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations <inline-formula><tex-math notation="LaTeX">Ax = b</tex-math></inline-formula>, where <inline-formula><tex-math notation=&quo...

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Veröffentlicht in:IEEE transactions on network science and engineering Jg. 7; H. 3; S. 975 - 986
Hauptverfasser: Ozdaglar, Asuman, Shah, Devavrat, Yu, Christina Lee
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Piscataway IEEE 01.07.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Approximation
Approximation algorithms
asynchronous randomized algorithms
Computation
Computational modeling
Convergence
distributed algorithms
Jacobian matrices
Linear equations
Linear system of equations
local computation
Mathematical analysis
Mathematical model
Matrix methods
Operators (mathematics)
Probabilistic analysis
Random walk
Symmetric matrices
Task analysis
Time dependence
ISSN:2327-4697, 2334-329X
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Abstract We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations <inline-formula><tex-math notation="LaTeX">Ax = b</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">A</tex-math></inline-formula> is a positive definite real matrix and <inline-formula><tex-math notation="LaTeX">b \in \mathbb {R}^n</tex-math></inline-formula>. This can equivalently be formulated as solving for <inline-formula><tex-math notation="LaTeX">x_i</tex-math></inline-formula> in <inline-formula><tex-math notation="LaTeX">x = Gx + z</tex-math></inline-formula> for some <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">z</tex-math></inline-formula> such that the spectral radius of <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> is less than 1. Our algorithm relies on the Neumann series characterization of the component <inline-formula><tex-math notation="LaTeX">x_i</tex-math></inline-formula>, and is based on residual updates. We analyze our algorithm within the context of a cloud computation model motivated by frameworks, such as Apache Spark, in which the computation is split into small update tasks performed by small processors with shared access to a distributed file system. We prove a robust asymptotic convergence result when the spectral radius <inline-formula><tex-math notation="LaTeX">\rho (|G|) < 1</tex-math></inline-formula>, regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds that depend on the order of update tasks performed, analyzing both deterministic update rules via counting weighted random walks, as well as probabilistic update rules via concentration bounds. The probabilistic analysis requires analyzing the product of random matrices that are drawn from distributions that are time and path dependent. We specifically consider the setting where <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> is large, yet <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> is sparse, e.g., each row has at most <inline-formula><tex-math notation="LaTeX">d</tex-math></inline-formula> nonzero entries. This is motivated by applications in which <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> is derived from the edge structure of an underlying graph. Our results prove that if the local neighborhood of the graph does not grow too quickly as a function of <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>, our algorithm can provide significant reduction in computation cost as opposed to any algorithm that computes the global solution vector <inline-formula><tex-math notation="LaTeX">x</tex-math></inline-formula>. Our algorithm obtains an <inline-formula><tex-math notation="LaTeX">\epsilon \Vert x\Vert _2</tex-math></inline-formula> additive approximation for <inline-formula><tex-math notation="LaTeX">x_i</tex-math></inline-formula> in constant time with respect to the size of the matrix when the maximum row sparsity <inline-formula><tex-math notation="LaTeX">d = O(1)</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">1/(1-\Vert G\Vert _2) = O(1)</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">\Vert G\Vert _2</tex-math></inline-formula> is the induced matrix operator 2-norm.
AbstractList We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations <inline-formula><tex-math notation="LaTeX">Ax = b</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">A</tex-math></inline-formula> is a positive definite real matrix and <inline-formula><tex-math notation="LaTeX">b \in \mathbb {R}^n</tex-math></inline-formula>. This can equivalently be formulated as solving for <inline-formula><tex-math notation="LaTeX">x_i</tex-math></inline-formula> in <inline-formula><tex-math notation="LaTeX">x = Gx + z</tex-math></inline-formula> for some <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">z</tex-math></inline-formula> such that the spectral radius of <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> is less than 1. Our algorithm relies on the Neumann series characterization of the component <inline-formula><tex-math notation="LaTeX">x_i</tex-math></inline-formula>, and is based on residual updates. We analyze our algorithm within the context of a cloud computation model motivated by frameworks, such as Apache Spark, in which the computation is split into small update tasks performed by small processors with shared access to a distributed file system. We prove a robust asymptotic convergence result when the spectral radius <inline-formula><tex-math notation="LaTeX">\rho (|G|) < 1</tex-math></inline-formula>, regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds that depend on the order of update tasks performed, analyzing both deterministic update rules via counting weighted random walks, as well as probabilistic update rules via concentration bounds. The probabilistic analysis requires analyzing the product of random matrices that are drawn from distributions that are time and path dependent. We specifically consider the setting where <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> is large, yet <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> is sparse, e.g., each row has at most <inline-formula><tex-math notation="LaTeX">d</tex-math></inline-formula> nonzero entries. This is motivated by applications in which <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula> is derived from the edge structure of an underlying graph. Our results prove that if the local neighborhood of the graph does not grow too quickly as a function of <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>, our algorithm can provide significant reduction in computation cost as opposed to any algorithm that computes the global solution vector <inline-formula><tex-math notation="LaTeX">x</tex-math></inline-formula>. Our algorithm obtains an <inline-formula><tex-math notation="LaTeX">\epsilon \Vert x\Vert _2</tex-math></inline-formula> additive approximation for <inline-formula><tex-math notation="LaTeX">x_i</tex-math></inline-formula> in constant time with respect to the size of the matrix when the maximum row sparsity <inline-formula><tex-math notation="LaTeX">d = O(1)</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">1/(1-\Vert G\Vert _2) = O(1)</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">\Vert G\Vert _2</tex-math></inline-formula> is the induced matrix operator 2-norm.
We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations [Formula Omitted], where [Formula Omitted] is a positive definite real matrix and [Formula Omitted]. This can equivalently be formulated as solving for [Formula Omitted] in [Formula Omitted] for some [Formula Omitted] and [Formula Omitted] such that the spectral radius of [Formula Omitted] is less than 1. Our algorithm relies on the Neumann series characterization of the component [Formula Omitted], and is based on residual updates. We analyze our algorithm within the context of a cloud computation model motivated by frameworks, such as Apache Spark, in which the computation is split into small update tasks performed by small processors with shared access to a distributed file system. We prove a robust asymptotic convergence result when the spectral radius [Formula Omitted], regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds that depend on the order of update tasks performed, analyzing both deterministic update rules via counting weighted random walks, as well as probabilistic update rules via concentration bounds. The probabilistic analysis requires analyzing the product of random matrices that are drawn from distributions that are time and path dependent. We specifically consider the setting where [Formula Omitted] is large, yet [Formula Omitted] is sparse, e.g., each row has at most [Formula Omitted] nonzero entries. This is motivated by applications in which [Formula Omitted] is derived from the edge structure of an underlying graph. Our results prove that if the local neighborhood of the graph does not grow too quickly as a function of [Formula Omitted], our algorithm can provide significant reduction in computation cost as opposed to any algorithm that computes the global solution vector [Formula Omitted]. Our algorithm obtains an [Formula Omitted] additive approximation for [Formula Omitted] in constant time with respect to the size of the matrix when the maximum row sparsity [Formula Omitted] and [Formula Omitted], where [Formula Omitted] is the induced matrix operator 2-norm.
Author Ozdaglar, Asuman
Yu, Christina Lee
Shah, Devavrat
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Snippet We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations <inline-formula><tex-math...
We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations [Formula Omitted], where...
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SubjectTerms Algorithms
Approximation
Approximation algorithms
asynchronous randomized algorithms
Computation
Computational modeling
Convergence
distributed algorithms
Jacobian matrices
Linear equations
Linear system of equations
local computation
Mathematical analysis
Mathematical model
Matrix methods
Operators (mathematics)
Probabilistic analysis
Random walk
Symmetric matrices
Task analysis
Time dependence
Title Asynchronous Approximation of a Single Component of the Solution to a Linear System
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