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dc.contributor.authorRichtarik, Peter
dc.contributor.authorTakáč, Martin
dc.date.accessioned2017-12-28T07:32:16Z
dc.date.available2017-12-28T07:32:16Z
dc.date.issued2017-06-04
dc.identifier.urihttp://hdl.handle.net/10754/626554
dc.description.abstractWe develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several equivalent interpretations, allowing for researchers from various communities to leverage their domain specific insights. In particular, our reformulation can be equivalently seen as a stochastic optimization problem, stochastic linear system, stochastic fixed point problem and a probabilistic intersection problem. We prove sufficient, and necessary and sufficient conditions for the reformulation to be exact. Further, we propose and analyze three stochastic algorithms for solving the reformulated problem---basic, parallel and accelerated methods---with global linear convergence rates. The rates can be interpreted as condition numbers of a matrix which depends on the system matrix and on the reformulation parameters. This gives rise to a new phenomenon which we call stochastic preconditioning, and which refers to the problem of finding parameters (matrix and distribution) leading to a sufficiently small condition number. Our basic method can be equivalently interpreted as stochastic gradient descent, stochastic Newton method, stochastic proximal point method, stochastic fixed point method, and stochastic projection method, with fixed stepsize (relaxation parameter), applied to the reformulations.
dc.publisherarXiv
dc.relation.urlhttp://arxiv.org/abs/1706.01108v2
dc.relation.urlhttp://arxiv.org/pdf/1706.01108v2
dc.rightsArchived with thanks to arXiv
dc.titleStochastic Reformulations of Linear Systems: Algorithms and Convergence Theory
dc.typePreprint
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.departmentComputer Science Program
dc.eprint.versionPre-print
dc.contributor.institutionUniversity of Edinburgh
dc.contributor.institutionLehigh University
dc.identifier.arxividarXiv:1706.01108
kaust.personRichtarik, Peter
refterms.dateFOA2018-06-13T10:26:23Z


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