A stochastic approximation algorithm with multiplicative step size modification
Autor(a) principal: | |
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Data de Publicação: | 2009 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/6176 |
Resumo: | An algorithm of searching a zero of an unknown function $\vphi : \, \R \to \R$ is considered: $\, x_{t} = x_{t-1} - \gamma_{t-1} y_t$,\, $t=1,\ 2,\ldots$, where $y_t = \varphi(x_{t-1}) + \xi_t$ is the value of $\vphi$ measured at $x_{t-1}$ and $\xi_t$ is the measurement error. The step sizes $\gam_t > 0$ are modified in the course of the algorithm according to the rule: $\, \gamma_t = \min\{u\, \gamma_{t-1},\, \mstep\}$ if $y_{t-1} y_t > 0$, and $\gamma_t = d\, \gamma_{t-1}$, otherwise, where $0 < d < 1 < u$,\, $\mstep > 0$. That is, at each iteration $\gam_t$ is multiplied either by $u$ or by $d$, provided that the resulting value does not exceed the predetermined value $\mstep$. The function $\vphi$ may have one or several zeros; the random values $\xi_t$ are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on $\vphi$, $\xi_t$, and $\mstep$, the conditions on $u$ and $d$ guaranteeing a.s. convergence of the sequence $\{ x_t \}$, as well as a.s. divergence, are determined. In particular, if $\P (\xi_1 > 0) = \P (\xi_1 < 0) = 1/2$ and $\P (\xi_1 = x) = 0$ for any $x \in \R$, one has convergence for $ud < 1$ and divergence for $ud > 1$. Due to the multiplicative updating rule for $\gam_t$, the sequence $\{ x_t \}$ converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximates one of the zeros of $\vphi$. By adjusting the parameters $u$ and $d$, one can reach arbitrarily high precision of the approximation; higher precision is obtained at the expense of lower convergence rate. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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A stochastic approximation algorithm with multiplicative step size modificationStochastic approximationAccelerated convergenceStep size adaptationAn algorithm of searching a zero of an unknown function $\vphi : \, \R \to \R$ is considered: $\, x_{t} = x_{t-1} - \gamma_{t-1} y_t$,\, $t=1,\ 2,\ldots$, where $y_t = \varphi(x_{t-1}) + \xi_t$ is the value of $\vphi$ measured at $x_{t-1}$ and $\xi_t$ is the measurement error. The step sizes $\gam_t > 0$ are modified in the course of the algorithm according to the rule: $\, \gamma_t = \min\{u\, \gamma_{t-1},\, \mstep\}$ if $y_{t-1} y_t > 0$, and $\gamma_t = d\, \gamma_{t-1}$, otherwise, where $0 < d < 1 < u$,\, $\mstep > 0$. That is, at each iteration $\gam_t$ is multiplied either by $u$ or by $d$, provided that the resulting value does not exceed the predetermined value $\mstep$. The function $\vphi$ may have one or several zeros; the random values $\xi_t$ are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on $\vphi$, $\xi_t$, and $\mstep$, the conditions on $u$ and $d$ guaranteeing a.s. convergence of the sequence $\{ x_t \}$, as well as a.s. divergence, are determined. In particular, if $\P (\xi_1 > 0) = \P (\xi_1 < 0) = 1/2$ and $\P (\xi_1 = x) = 0$ for any $x \in \R$, one has convergence for $ud < 1$ and divergence for $ud > 1$. Due to the multiplicative updating rule for $\gam_t$, the sequence $\{ x_t \}$ converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximates one of the zeros of $\vphi$. By adjusting the parameters $u$ and $d$, one can reach arbitrarily high precision of the approximation; higher precision is obtained at the expense of lower convergence rate.Springer Verlag2012-02-10T12:03:19Z2009-01-01T00:00:00Z2009info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/6176eng1066-530710.3103/S1066530709020057Plakhov, AlexanderCruz, João Pedro Antunes Ferreira dainfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T11:09:19Zoai:ria.ua.pt:10773/6176Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:43:54.221670Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
A stochastic approximation algorithm with multiplicative step size modification |
title |
A stochastic approximation algorithm with multiplicative step size modification |
spellingShingle |
A stochastic approximation algorithm with multiplicative step size modification Plakhov, Alexander Stochastic approximation Accelerated convergence Step size adaptation |
title_short |
A stochastic approximation algorithm with multiplicative step size modification |
title_full |
A stochastic approximation algorithm with multiplicative step size modification |
title_fullStr |
A stochastic approximation algorithm with multiplicative step size modification |
title_full_unstemmed |
A stochastic approximation algorithm with multiplicative step size modification |
title_sort |
A stochastic approximation algorithm with multiplicative step size modification |
author |
Plakhov, Alexander |
author_facet |
Plakhov, Alexander Cruz, João Pedro Antunes Ferreira da |
author_role |
author |
author2 |
Cruz, João Pedro Antunes Ferreira da |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Plakhov, Alexander Cruz, João Pedro Antunes Ferreira da |
dc.subject.por.fl_str_mv |
Stochastic approximation Accelerated convergence Step size adaptation |
topic |
Stochastic approximation Accelerated convergence Step size adaptation |
description |
An algorithm of searching a zero of an unknown function $\vphi : \, \R \to \R$ is considered: $\, x_{t} = x_{t-1} - \gamma_{t-1} y_t$,\, $t=1,\ 2,\ldots$, where $y_t = \varphi(x_{t-1}) + \xi_t$ is the value of $\vphi$ measured at $x_{t-1}$ and $\xi_t$ is the measurement error. The step sizes $\gam_t > 0$ are modified in the course of the algorithm according to the rule: $\, \gamma_t = \min\{u\, \gamma_{t-1},\, \mstep\}$ if $y_{t-1} y_t > 0$, and $\gamma_t = d\, \gamma_{t-1}$, otherwise, where $0 < d < 1 < u$,\, $\mstep > 0$. That is, at each iteration $\gam_t$ is multiplied either by $u$ or by $d$, provided that the resulting value does not exceed the predetermined value $\mstep$. The function $\vphi$ may have one or several zeros; the random values $\xi_t$ are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on $\vphi$, $\xi_t$, and $\mstep$, the conditions on $u$ and $d$ guaranteeing a.s. convergence of the sequence $\{ x_t \}$, as well as a.s. divergence, are determined. In particular, if $\P (\xi_1 > 0) = \P (\xi_1 < 0) = 1/2$ and $\P (\xi_1 = x) = 0$ for any $x \in \R$, one has convergence for $ud < 1$ and divergence for $ud > 1$. Due to the multiplicative updating rule for $\gam_t$, the sequence $\{ x_t \}$ converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximates one of the zeros of $\vphi$. By adjusting the parameters $u$ and $d$, one can reach arbitrarily high precision of the approximation; higher precision is obtained at the expense of lower convergence rate. |
publishDate |
2009 |
dc.date.none.fl_str_mv |
2009-01-01T00:00:00Z 2009 2012-02-10T12:03:19Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/6176 |
url |
http://hdl.handle.net/10773/6176 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1066-5307 10.3103/S1066530709020057 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Springer Verlag |
publisher.none.fl_str_mv |
Springer Verlag |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
|
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1799137488934010880 |