Stochastic approximation procedure with impulsive Markov perturbations
In this paper we discuss asymptotic behavior of the stochastic approximation procedure in case when the regression function is perturbed by the Markov impulsive process. Also we consider the stochastic approximation procedure stability conditions in the terms of existence of Lyapunov's function...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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| Цитувати: | Stochastic approximation procedure with impulsive Markov perturbations / Ya.M. Chabanyuk, S.A. Semenyuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2011. — Вип. 5. — С. 244-252. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-488102013-09-04T03:03:53Z Stochastic approximation procedure with impulsive Markov perturbations Chabanyuk, Ya.M. Semenyuk, S.A. In this paper we discuss asymptotic behavior of the stochastic approximation procedure in case when the regression function is perturbed by the Markov impulsive process. Also we consider the stochastic approximation procedure stability conditions in the terms of existence of Lyapunov's function for the averaged evolution system. Розглянуто асимптотичну поведінку процедури стохастичної апроксимації для випадку, коли функція регресії збурена марковським імпульсним процесом. Одержано достатні умови збіжності процедури в умовах існування функції Ляпунова для усередненої еволюційної системи. 2011 Article Stochastic approximation procedure with impulsive Markov perturbations / Ya.M. Chabanyuk, S.A. Semenyuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2011. — Вип. 5. — С. 244-252. — Бібліогр.: 11 назв. — англ. XXXX-0059 http://dspace.nbuv.gov.ua/handle/123456789/48810 519.21 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| description |
In this paper we discuss asymptotic behavior of the stochastic approximation procedure in case when the regression function is perturbed by the Markov impulsive process. Also we consider the stochastic approximation procedure stability conditions in the terms of existence of Lyapunov's function for the averaged evolution system. |
| format |
Article |
| author |
Chabanyuk, Ya.M. Semenyuk, S.A. |
| spellingShingle |
Chabanyuk, Ya.M. Semenyuk, S.A. Stochastic approximation procedure with impulsive Markov perturbations Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| author_facet |
Chabanyuk, Ya.M. Semenyuk, S.A. |
| author_sort |
Chabanyuk, Ya.M. |
| title |
Stochastic approximation procedure with impulsive Markov perturbations |
| title_short |
Stochastic approximation procedure with impulsive Markov perturbations |
| title_full |
Stochastic approximation procedure with impulsive Markov perturbations |
| title_fullStr |
Stochastic approximation procedure with impulsive Markov perturbations |
| title_full_unstemmed |
Stochastic approximation procedure with impulsive Markov perturbations |
| title_sort |
stochastic approximation procedure with impulsive markov perturbations |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/48810 |
| citation_txt |
Stochastic approximation procedure with impulsive Markov perturbations / Ya.M. Chabanyuk, S.A. Semenyuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2011. — Вип. 5. — С. 244-252. — Бібліогр.: 11 назв. — англ. |
| series |
Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| work_keys_str_mv |
AT chabanyukyam stochasticapproximationprocedurewithimpulsivemarkovperturbations AT semenyuksa stochasticapproximationprocedurewithimpulsivemarkovperturbations |
| first_indexed |
2025-07-04T09:33:17Z |
| last_indexed |
2025-07-04T09:33:17Z |
| _version_ |
1836708374300852224 |
| fulltext |
Математичне та комп’ютерне моделювання
244
2. Ленюк М. П. Обчислення невласних інтегралів методом гібридних інтег-
ральних перетворень (Фур’є, Бесселя, Лежандра) / М. П. Ленюк. — Чер-
нівці : Прут, 2005. — Т. 5. — 368 с.
3. Ленюк М. П. Гібридні інтегральні перетворення (Фур’є, Бесселя, Лежан-
дра) / М. П. Ленюк, М. І. Шинкарик. — Тернопіль : Економічна думка,
2004. — Ч. 1. — 368 с.
4. Степанов В. В. Курс дифференциальных уравнений / В. В. Степанов. —
М. : Физматгиз, 1959. — 468 с.
5. Шилов Г. Е. Математический анализ. Второй специальный курс /
Г. Е. Шилов. — М. : Наука, 1965. — 328 с.
By the method of comparison of decisions, built on the arctic landmark
with the two point of interface for the separate system of differential
equalizations of Lezhandra, Bessel and Euler by the method of functions
Koshi and by the method of the proper hybrid integral transformation,
polyparametric family of known integrals is calculated.
Key words: Not own integrals, functions Koshi, the main decisions,
hybrid integrated transformation, the basic identity, condition of unequivo-
cal resolvabitili, the logic scheme.
Отримано: 05.06.2011
УДК 519.21
Я. М. Чабанюк, д-р фіз.-мат. наук,
С. А. Семенюк, аспірант
Національний університет “Львівська політехніка”, м. Львів
STOCHASTIC APPROXIMATION PROCEDURE WITH
IMPULSIVE MARKOV PERTURBATIONS
In this paper we discuss asymptotic behavior of the stochastic
approximation procedure in case when the regression function is
perturbed by the Markov impulsive process. Also we consider the
stochastic approximation procedure stability conditions in the terms of
existence of Lyapunov's function for the averaged evolution system.
Key words: stochastic approximation procedure, Markov
process, impulsive perturbation.
Introduction. The goal of the Robbins-Monro Stochastic Approxi-
mation Procedure (SAP) [1] is to find the solution of the equation
( ) = 0C u in the case when the measurements of regression function ( )C u
are made with some errors. It is widely used in the mathematical statistics
[2], control theory, image, signal and voice recognition theory [3], etc.
Let us consider the situation when estimated function errors are de-
fined by impulsive process. Then the continuous SAP is defined by the
differential equation
© Я. М. Чабанюк, С. А. Семенюк, 2011
Серія: Фізико-математичні науки. Випуск 5
245
4( ) = ( ) ( ), / ( ) ,du t a t C u t x t dt d t
(1)
with 2( , ) ( )nC u C R .
Where a Markov process ( ), 0x t t in the standard phase space
,X is defined by the generator:
Q ( ) = ( ) ( , ) ( ) ( ) , ( ),
X
x q x P x dy y x B X
here ( )B X is the Banach space of real bounded functions with supremum-
norm || ||= | ( ) |max
x X
x
.
A uniformly ergodic embedded Markov chain = ( )n nx x , 0n
with stationary distribution ( ),B B is defined by the stochastic ker-
nel , , ,P x B x X B . A stationary distribution ( ),B B of the
Markov process ( ), 0x t t is defined by the representation:
( ) ( ) = ( ), = ( ) ( ).
X
dx q x q dx q dx q x
Lets denote by 0R potential operator of the generator Q :
1
0 = QR
, where ( ) = ( ) ( )
X
x dy y is the projector on the
zeros subspace = : Q = 0QN of the generator Q .
Impulsive Perturbation Process. An impulsive perturbation process
(IPP) ( ), 0t t is defined by the representation [4]:
4
0
( ) = ; / ;
t
t ds x s (2)
where the family of independent increment processes ( , ), 0,t x t x X
is defined by the generators:
4 2( ) ( ) = ( ) ; , .
R
x w w v w dv x x X (3)
Generator (3) can be rewritten in the asymptotic form
2 2
1 2( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ),x w x w x w x w
where
1 1 1( ) ( ) = ( ) ( ); ( ) = ;
R
x w b x w b x v dv x (4)
2
2 2 2
1
( ) ( ) = ( ) ( ); ( ) = ; ,
2 R
x w b x w b x v dv x (5)
Математичне та комп’ютерне моделювання
246
and the remaining term satisfies the condition 2 ( ) ( ) 0x w while
0 .
Let also the following balance conditions take place:
1 1 1( ) = ( ) ( ) = 0; ( ) = ; .
X R
b x dx b x b x v dv x (6)
Stochastic Approximation Procedure behaviour. Let us consider
the continuous SAP (1) convergence under the exponential stability condi-
tions of the averaged evolution system:
( )
= ( ) ,
du t
C u t
dt
(7)
where
( ) = ( ) ( , ).
X
C u dx C u x
Balance condition must be satisfied for the averaged evolution sys-
tem equilibrium point 0u existence:
0 0, = ( ) , = 0.
X
C u x dx C u x (8)
Without limiting the generality, further assume that 0 = 0u .
Theorem. Let there exists Lyapunov function 3( ) ( )V u C R that
provides the exponential stability of the averaged system (7):
1: ( ) ( ), > 0.C CV u cV u c
(9)
Also let the additional conditions hold true:
2 2 2
3 3
1 0 1 4 4
1 0 5 5
0 1 6 6
0 7 7
2 0 1
2 : Γ( ) ( ) 1 ( ) , > 0,
3 : ( ) ( ) 1 ( ) , > 0,
4 : Γ( ) Γ( ) ( ) 1 ( ) , > 0,
5 : Γ( ) C( ) ( ) 1 ( ) , > 0,
6 : C( ) Γ( ) ( ) 1 ( ) , > 0,
7 : C( ) C( ) ( ) 1 ( ) , > 0,
8 : Γ( ) Γ
C x V u c V u c
C x V u c V u c
C x R x V u c V u c
C x R x V u c V u c
C x R x V u c V u c
C x R x V u c V u c
C x R
8 8
2 0 9 9
0 1 10 10
0 11 11
( ) ( ) 1 ( ) , > 0
9 : Γ( ) C( ) ( ) 1 ( ) , > 0,
10 : ( ) Γ( ) ( ) 1 ( ) , > 0,
11: ( ) C( ) ( ) 1 ( ) , > 0,
x V u c V u c
C x R x V u c V u c
C x R x V u c V u c
C x R x V u c V u c
(10)
Серія: Фізико-математичні науки. Випуск 5
247
with
C( ) = C( ) L.x x
In addition let the function ( , )C u x has the third derivative on u R ,
and is uniformly bounded on x X . Furthermore, let the balance condi-
tion (6) takes place. Let also pick control function ( ) > 0a t in such way
that it satisfies the conditions [2]:
2
0 0
( ) = , ( ) < .a t dt a t dt
Then the solution of the evolution equation (1) converges weakly (with
probability 1) to the equilibrium point 0 = 0u of the averaged system (7) for
all initial values (0) =u u and all 0< , where 0 is small enough, i.e.:
( ) = 0 = 1.lim
t
P u t
The proof of the theorem is made with the help of the singular per-
turbation problem solution and martingale characterization of the two-
component Markov process.
Let's consider Markov process
4( ), : / , 0.tu t x x t t (11)
This process is heterogeneous in time because the SAP ( )u t (1) de-
pends on the control function ( )a t .
Lemma 1. A generator of the two-component Markov process ( )u t ,
4/x t , 0t has the form
ε 4
tL ( ) ( , ) = Q ( , ) ( ) ( ) ( , ) ( )C( ) ( , ),x u x u x a t x u x a t x u x (12)
where C( ) ( , , ) = ( , ) ( , )ux u w x C u x u x ,
Proof. Let us denote ( ) = tu t u , 4/ = tx t x . Then according to the
Markov process generator definition [5] we obtain:
ε 4
t
0
4 4
0
1
L ( ) ( , ) = , /lim
( ), / | ( ) = , / = =
1
= , ( , ) =lim t t
x u x E u t x t
u t x t u t u x t x
E u x u x
0 0
1 1
= , , [ , ,lim limt t t tE u x u x E u x u x
Математичне та комп’ютерне моделювання
248
According to (1): = ( ) , ( ) ( ) ( )u t u a t C u x a t t o ,
therefore
0
0
1
, , =lim
1
( ) ( , ) ( ) ( ) ( ), ,lim
t t t
t t
E u x u x
E u a t C u x a t d t o x u x
Let us add and substitute ( ) ( , ) ( ), tu a t C u x o x in the last
limit. Thus we obtain:
0
0
0
1
, , =lim
1
( ) ( , ) ( ) ( ) ( ),lim
( ) ( , ) ( ),
1
( ) ( , ) ( ), ,lim
t t t
t
t
t t
E u x u x
E u a t C u x a t d t o x
u a t C u x o x
E u a t C u x o x u x
According to the generator ( )x definition:
0
1
( ) ( ) = ( ) ( )limx u E u t u
.
Let's also denote = ( ) ( , ) ( )v u a t C u x o (in this case v u
while 0 ). As a result we obtain:
0
1
( ) ( ), , = ( ) ( ) ( , )lim t tE v a t t x v x a t x u x
Then we can transform the second limit to the form
0
0
1
( ) ( , ) , , =lim
1
, ( ) ( , ) ( ) = ( ) ( , ) ( , ).lim
t t
u t u
E u a t C u x x u x
E u x a t C u x o a t C u x u x
So, using the process 4/x t generator definition:
4
0
1
, ( , ) = Q ( , ),lim tE u x u x u x
we obtain (12).
Lemma 2. A generator L ( )t x of the Markov process ( )u t ,
4/x t , 0t can be rewritten in the asymptotic form
Серія: Фізико-математичні науки. Випуск 5
249
ε 4 1
t 1L ( ) ( , ) = Q ( , ) ( ) ( ) ( , )
( )C( ) ( , ) ( ) ( , ),t
x u x u x a t x u x
a t x u x x u x
(13)
where
2
2( ) = ( ) ( ) ( ) ( , ),t x a t a t x u x
and generators 1 2( ), ( )x x are defined in (4) and (5) respectively.
Also the remaining term is such that ( )( ) ( , ) 0t x x u x while
0.
We can proof this lemma using the representation (3) of the generator
( )x and lemma 1.
Lemma 3. Under the conditions of the theorem, the singular pertur-
bation problem for the operator (13) on the perturbed Lyapunov function
3 4
1 0( , ) = ( ) ( ) ( , ) ( ) ( , )tV u x V u a t V u x a t V u x
has the solution in the form
L ( ) ( , ) = ( )L ( ) ( ) ( ),t t tx V u x a t V u x V u (14)
where the limit generator L is such that
L ( ) ( ) ( )V u ca t V u (15)
and the remaining term ( )t x satisfies the inequality:
* 2( ) ( ) ( ) 1 ( ) .t x V u c a t V u
Proof. Let us collect the similar terms with respect to in order to
prove equality (14):
ε 4 1
t 1
2
2
3 4
1 0
4 1
1 1
0 2
2
1
L ( ) ( , ) = Q ( ) ( ) ( )C( )
( ) ( ) ( ) ( )
( ) ( ) ( , ) ( ) ( , ) =
= Q ( ) ( ) Q ( , ) ( ) ( )
( ) Q ( , ) C( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
tx V u x a t x a t x
a t x a t x
V u a t V u x a t V u x
V u a t V u x x V u
a t V u x x V u a t x V u
a t a t x
1
3 2
1 0 1
4 2
0 2 1
5 6
2 0 1 0
( , ) ( ) ( )
( ) ( ) ( , ) C( ) ( , )
( ) C( ) ( , ) ( ) ( , )
( ) ( ) ( ) ( , ) ( ) ( , ) ( ) ( ) ( , ).
V u x x V u
a t x V u x x V u x
a t x V u x x V u x
a t a t x V u x x V u x a t x V u x
(16)
Since ( , )u w doesn't depend on x , then
Математичне та комп’ютерне моделювання
250
Q ( ) = 0, ( ) .QV u V u N
The following equation can be solved under the balance condition (6)
1 1Q ( , ) ( ) ( ) = 0.V u x x V u
That is why
1 0 1( , ) = ( ) ( ).V u x R x V u
We can obtain the limit process L using the solution condition of the
last equation:
0( )Q ( , ) ( )C( ) ( ) = ( )L ( ).a t V u x a t x V u a t V u
Thus
L ( ) = C( ) ( ) = ( , ) ( )V u x V u C u x V u
and then
1
0 0 0( , ) = ( ) ( )C( ) ( )L ( ) = C( ) ( ).V u x a t R a t x a t V u R x V u
We can obtain ( )t x from the rest of terms in (16):
2 1 1
2 2
1 0 1
3 2
0 2 1
4 5
2 0 1 0
( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( ) ( )
( ) ( ) ( , ) C( ) ( , )
( ) C( ) ( , ) ( ) ( , )
( ) ( ) ( ) ( , ) ( ) ( , ) ( ) ( ) ( , ),
t x V u a t x V u a t a t x V u x x V u
a t x V u x x V u x
a t x V u x x V u x
a t a t x V u x x V u x a t x V u x
and then, using the representation of the Lyapunov function perturbations
0 ( , )V u x and 1( , )V u x , we can obtain
2 1 0 1
2 2
1 0 0 1
3 2
0 2 0 1
4
2 0 0 1
( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) C( ) ( ) C( ) ( ) ( )
( ) C( ) C( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) C( ) ( ) ( ) ( ) ( )
t x V u a t x V u a t a t x R x V u x V u
a t x R x V u x R x V u
a t x R x V u x R x V u
a t a t x R x V u x R x V u
5
0( ) ( ) C( ) ( ).a t x R x V u
Let's denote by 0 5 corresponding expressions near and ( )a t
Since Lyapunov function ( )V u is smooth enough and under the con-
dition 2C , for 0 2= ( ) ( )x V u we obtain
0 2| | 1 ( ) .c V u
Second term 1 1 0 1= ( ) ( ) ( ) ( ) ( )x R x V u x V u under the condi-
tions 3C and 4C satisfies the inequality
Серія: Фізико-математичні науки. Випуск 5
251
1 3 4| | 1 ( ) .c c V u
Using the theorem conditions 5C and 6C , we can obtain the esti-
mate for the term 2 1 0 0 1= ( ) C( ) ( ) C( ) ( ) ( )x R x V u x R x V u :
2 5 6| | 1 ( ) .c c V u
Similarly for 3 0 2 0 1= C( ) C( ) ( ) ( ) ( ) ( )x R x V u x R x V u (according
to 7C and 8C ) we obtain:
3 7 8| | 1 ( ) ,c c V u
and for 4 2 0 0 1= ( ) C( ) ( ) ( ) ( ) ( )x R x V u x R x V u (according to 9C and
10C ) there is a similar estimate:
4 9 10| | 1 ( ) .c c V u
And, using the condition 11C , we can bound 5 0= ( ) C( ) ( )x R x V u
with:
5 11| | 1 ( ) .c V u
Thus, for the remainder term ( ) ( )t x V u cumulative estimate holds:
* 2( ) ( ) ( ) 1 ( ) .t x V u c a t V u (17)
Theorem proof. Using the condition 1C and inequality (17) we can
obtain:
ε * 2
tL ( ) ( , ) ( ) ( ) ( ) 1 ( ) .tx V u x ca t V u c a t V u (18)
Then the stochastic approximation procedure convergence follows
from the inequality (18) and the theorem 1 in [6].
Conclusions. Stochastic approximation procedure converges weakly
to the equilibrium point of the averaged system under the condition of the
exponential stability of that system and additional limitations on the re-
gression function smoothness. These results can be used to identify the
SAP asymptotic behaviour [7; 8; 9]. Such SAP can also be used as a
scratch for the stochastic optimization procedure (Kiefer-Wolfowitz
method [10; 11]) in impulsive perturbations case.
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Розглянуто асимптотичну поведінку процедури стохастичної апрок-
симації для випадку, коли функція регресії збурена марковським імпуль-
сним процесом. Одержано достатні умови збіжності процедури в умовах
існування функції Ляпунова для усередненої еволюційної системи.
Ключові слова: процедура стохастичної апроксимації, марковсь-
кий процес, імпульсне збурення.
Отримано: 23.04.2011
<<
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>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AllowImageBreaks true
/AllowTableBreaks true
/ExpandPage false
/HonorBaseURL true
/HonorRolloverEffect false
/IgnoreHTMLPageBreaks false
/IncludeHeaderFooter false
/MarginOffset [
0
0
0
0
]
/MetadataAuthor ()
/MetadataKeywords ()
/MetadataSubject ()
/MetadataTitle ()
/MetricPageSize [
0
0
]
/MetricUnit /inch
/MobileCompatible 0
/Namespace [
(Adobe)
(GoLive)
(8.0)
]
/OpenZoomToHTMLFontSize false
/PageOrientation /Portrait
/RemoveBackground false
/ShrinkContent true
/TreatColorsAs /MainMonitorColors
/UseEmbeddedProfiles false
/UseHTMLTitleAsMetadata true
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/BleedOffset [
0
0
0
0
]
/ConvertColors /ConvertToRGB
/DestinationProfileName (sRGB IEC61966-2.1)
/DestinationProfileSelector /UseName
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements true
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MarksOffset 6
/MarksWeight 0.250000
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PageMarksFile /RomanDefault
/PreserveEditing true
/UntaggedCMYKHandling /UseDocumentProfile
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [419.528 595.276]
>> setpagedevice
|