Encyclopaedia of Mathematics: Stochastic Approximation — Zygmund Class of Functions

542 Pages·1993·25.57 MB·English

by  M. Hazewinkel (auth.)

ENCYCLOPAEDIA OF MATHEMATICS
Volume 9
ENCYCLOPAEDIA OF MATHEMATICS
Managing Editor
M. Hazewinkel
Scientific Board
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genkamp, G. Rozenberg, W. Rudin, S. N. M. Ruysenaars, A. Salam, A. Salomaa, J. P. M.
Schalkwijk, C. L. Scheffer, R. Schneider, J. A. Schouten, F. Schurer, J. J. Seidel, A. Shenitzer,
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Veldkamp, P. M. B. Vitanyi, N. J. Vlaar, H. A. van der Vorst, J. de Vries, F. Waldhausen, B.
Wegner, J. J. O. O. Wiegerinck, J. C. Willems, J. M. Wills, B. de Wit, S. A. Wouthuysen, S.
Yuzvinskil, L. Zalcman
ENCYCLOPAEDIA
OF
MATHEMATICS
Volume 9
Stochastic Approximation - Zygmund Class of Functions
An updated and annotated translation of the Soviet
'Mathematical Encyclopaedia'
KLUWER ACADEMIC PUBLISHERS
Dordrecht / Boston / London
Library of Congress Cataloging-in-Publication Data
Matematicheskaia entsiklopediia. English.
Encyclopaedia of mathematics.
1. Mathematics--Dictionaries. I. Hazewinkel, Michie!. II. Title.
QA5.M3713 1987 510'.3'21 87-26437
ISBN 978-90-481-8238-1 ISBN 978-94-015-1233-6 (eBook)
DOl 10.1007/978-94-015-1233-6
Published by Kluwer Academic Publishers,
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E. G. Poznyak, Yu. V. Prokhorov (Deputy Editor-in-Chief), A.I. Shirshov, A. G. Sveshnikov,
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PREFACE
This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe
matics. It is a translation with updates and editorial comments of the Soviet Mathematical
Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in
1977-1985. The annotated translation consists of ten volumes including a special index
volume.
There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type
articles dealing with the various main directions in mathematics (where a rather fme subdivi
sion has been used). The main requirement for these articles has been that they should give a
reasonably complete up-to-date account of the current state of affairs in these areas and that
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areas and, depending on the specific subject, to specialists in other domains of science, en
gineers and teachers of mathematics. These articles treat their material at a fairly general level
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in question. They also contain background and motivation rather than precise statements of
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constructions.
The second kind of article, of medium length, contains more detailed concrete problems,
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means of which more details and more material on the topic can be found. Most articles were
specially written for the encyclopaedia and in such cases the names of the original Soviet
authors are mentioned. Some articles have another origin such as the Great Soviet Ency
clopaedia (Bol'shaya Sovetskaya Entsiklopediya or BSE).
Communication between mathematicians in various parts of the world has certainly greatly
improved in the last decennia. However, this does not mean that there are so-to-speak 'one-to
one onto' translations of the terminology, concepts and tools used by one mathematical school
to those of another. There also are varying traditions of which questions are important and
which not, and what is considered a central problem in one tradition may well be besides the
point from the point of view of another. Even for well-established areas of mathematical
inquiry, terminology varies across languages and even within a given language domain. Fur
ther, a concept, theorem, algorithm, ... , which is associated with one proper name within one
tradition may well have another one in another, especially if the result or idea in question was
indeed discovered independently and more-or-less simultaneously. Finally, mathematics is a
very dynamic science and much has happened since the original articles were finalized (mostly
around 1977). This made updates desirable (when needed). All this, as well as providing
vii
PREFACE
additional references to Western literature when needed, meant an enormous amount of work
for the board of experts as a whole; some indeed have done a truly impressive amount of work.
I must stress though that I am totally responsible for what is finally included and what is not of
all the material provided by the members of the board of experts.
Many articles are thus provided with an editorial comment section in a different and some
what smaller typeface. In particular, these annotations contain additional material, amplifica
tions, alternative names, additional references, . . . . Modifications, updates and other extra
material provided by the original Soviet authors (not a rare occurrence) have been incorporated
in the articles themselves.
The final (10-th) volume of the ENCYCLOPAEDIA OF MATHEMATICS will be an index volume.
This index will contain all the titles of the articles (some 6600) and in addition the names of all
the definitions, named theorems, algorithms, lemmas, scholia, constructions, ... , which occur
in the various articles. This includes, but is by no means limited to, all items which are printed
in bold or italic. Bold words or phrases, by the way, always refer to another article with
(precisely) that title.
All articles have been provided with one or more AMS classification numbers according to
the 1980 classification scheme (not, for various reasons, the 1985 revision), as have all items
occurring in the index. A phrase or word from an article which is included in the index always
inherits all the classification numbers of the article in question. In addition, it may have been
provided with its own classification numbers. In the index volume these numbers will be listed
with the phrase in question. Thus e.g. the Quillen - Suslin theorem of algebraic K-theory will
have its own main classification numbers (these are printed in bold; in this case that number is
18F25) as well as a number of others, often from totally different fields, pointing e.g. to parts
of mathematics where the theorem is applied, or where there occurs a problem related to it (in
this case e.g. 93DI5). The index volume will also contain the inversion of this list which will,
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of the 'content' of that classification number (as far as this ENCYCLOPAEDIA is concerned). For
more details on the index volume, its structure and organisation, and what kind of things can
be done with it, cf. the (future) special preface to that volume.
Classifying articles is a subjective matter. Opinions vary greatly as to what belongs where
and thus this attempt will certainly reflect the tastes and opinions of those who did the clas
sification work. One feature of the present classification attempt is that the general basic
concepts and definitions of an area like e.g. 55N (Homology and Cohomology theories) or 60J
(Markov processes) have been assigned classification numbers like 55NXX and 60JXX if there
was no finer classification number different from ... 99 to which it clearly completely belongs.
Different parts of mathematics tend to have differences in notation. As a rule, in this
ENCYCLOPAEDIA in a given article a notation is used which is traditional in the corresponding
field. Thus for example the (repeated index) summation convention is used in articles about
topics in fields where that is traditional (such as in certain parts of differential geometry (tensor
geometry» and it is not used in other articles (e.g. on summation of series). This pertains
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For proper names in Cyrillic the British Standards Institute transcription system has been
used (cf. Mathematical Reviews). This makes well known names like S. N. Bernstein come out
as BernshteYn.
In such cases, especially in names of theorems and article titles, the traditional spelling has
been retained and the standard transcription version is given between brackets.
Ideally an encyclopaedia should be complete up to a certain more-or-Iess well defined level
Vlll
PREFACE
of detail. In the present case I would like to aim at the completeness level whereby every
theorem, concept, definition, lemma, construction which has a more-or-less constant and
accepted name by which it is referred to by a recognizable group of mathematicians occurs
somewhere, and can be found via the index. It is unlikely that this completeness ideal will be
reached with this present ENCYCLOPAEDIA OF MATHEMATICS, but it certainly takes substantial
steps in this direction. Everyone who uses this ENCYCLOPAEDIA and fmds items which are not
covered, which, he feels, should have been included, is invited to inform me about it. When
enough material has come in this way supplementary volumes will be put together.
The ENCYCLOPAEDIA is alphabetical. Many titles consist of several words. Thus the problem
arises how to order them. There are several systematic ways of doing this of course, for in
stance using the fIrst noun. All are unsatisfactory in one way or another. Here an attempt has
been made to order things according to words or natural groups of words as they are daily used
in practice. Some sample titles may serve to illustrate this: Statistical mechanics, mathemati
cal problems in; Lie algebra; Free algebra; Associative algebra; Absolute continuity;
Abstract algebraic geometry; Boolean functions, normal forms of. Here again taste plays a
role (and usages vary). The index will contain all permutations. Meanwhile it will be advisable
for the reader to try out an occasional transposition himself. Titles like K-theory are to be
found under K, more precisely its lexicographic place is identical with 'K theory', i.e. '-' =
'space' and comes before all other symbols. Greek letters come before the corresponding Latin
ones, using the standard transcriptions. Thus X2-distribution (chi-squared distribution) is at the
beginning of the letter C. A * as in C*-algebra and *-regular ring is ignored lexicographically.
Some titles involve Greek letters spelled out in Latin. These are of course ordered just like any
other 'ordinary' title.
This volume has been computer typeset using the (Unix-based) system of the CW!, Amster
dam. The technical (mark-up-language) keyboarding was done by Rosemary Daniels,
Chahrzade van 't Hoff and Joke Pesch. To meet the data-base and typesetting requirements of
this ENCYCLOPAEDIA substantial amounts of additional programming had to be done. This was
done by Johan Wolleswinkel. Checking the translations against the original texts, and a lot of
desk editing and daily coordination was in the hands of Rob Hoksbergen. All these persons, the
members of the board of experts, and numerous others who provided information, remarks and
material for the editorial comments, I thank most cordially for their past and continuing efforts.
The original Soviet version had a printrun of 150,000 and is completely sold out. I hope that
this annotated and updated translation will turn out to be comparably useful.
Bussum, August 1987 MICHIEL HAZEWINKEL
ix
STOCHASTIC APPROXIMATION - A method for are fulfilled. The Kiefer-Wolfowitz procedure of
solving a class of problems of statistical estimation, in stochastic approximation also permits a multi
which the new value of the estimator is a modification dimensional generalization: instead of the right-hand
of an existing estimator, based on new information. side in (2), an approximate value of the gradient of the
The first procedure of stochastic approximation was function Yn(x) has to be substituted.
proposed in 1951 by H. Robbins and S. Monro. Procedures of stochastic approximation can naturally
Let every measurement Yn(Xn) of a function R (X), be generalized to a continuous observation process. For
x ERI, at a point Xn contain a random error with mean example, if an observation process is disturbed by a
zero. The Robbins-Monro procedure of stochastic Gaussian white noise, then the analogue of (1) takes the
approximation for finding a root of the equation form
dX(t) = a(t)dY(t),
R (x) = a takes the form
where
= =
Xn+1 Xn+an(Yn(Xn)-a). (1) dY(t) R(X(t))dt +a(X(t), t)dw(t)
If ~an = 00, ~a~ < 00, if R (x) is, for example, an is the differential of the process under observation and
increasing function, if R (x) increases no faster than w (t) is a Wiener process. The conditions of conver
1 1
a linear function, and if the random errors are indepen gence of continuous processes are analogous to those
dent, then Xn tends to a root Xo of the equation mentioned above for discrete time (see [2]). The basic
R (x) = a with probability 1 and in the quadratic mean instrument for proving the convergence of procedures
(see [l], [2]). It is clear from (1) that the process of of stochastic approximation is the theorem on the con
stochastic approximation is recursive, i.e. a new value vergence of non-negative supermartingales (see Mar
of the estimator can be obtained without recourse to tingale).
the old measurement Yn, and is convenient in cases The limit behaviour for an appropriate normalization
where the moment at which the estimator is to be of the difference Xn - Xo when n.-?oo has been studied.
represented is not known in advance. The estimator is In (1), let
formed continuously on the basis of observations relat = =
an an-I, Yn(Xn) R(Xn)+G(n, Xn,w)
ing to a given moment. These characteristics also per
and let Xn.-?XO almost certainly when n.-?oo. Given
tain to stochastic approximation with recursive filters,
certain restrictions, foremost among which are the
and explain the popularity of stochastic approximation
requirements
in theoretical and practical applications. The procedure
, ,1
(1) can be directly generalized to the multi-dimensional R (xo) < 0, aR (xo)+"2 < 0,
case.
Another procedure of stochastic approximation, used EG2(n, x, w) -') So
in finding a maximum point of a regression function where n.-?oo, X.-?Xo, the asymptotic normality of the
R(x), is attributed to J. Kiefer and J. Wolfowitz. Let variable Zn = Vn(Xn - xo) with parameters 0,
Yn(x) be an observation at the point x. The a 2 S / ( - 2aR' (x 0) - I) has been proved. The least vari
Kiefer-Wolfowitz procedure then takes the form ance of the limit distribution is obtained for
YiXn + en) - Yn(Xn - en) ao = -[R' (XO)]-I. This choice of a is impossible, since
Xn+I-Xn = 2e . (2)
the function R(x) and its derivative are unknown
n
I t has been proved that Xn converges to a maximum values to be observed. However, in a number of works,
point x max of the function R (x) if, for example, adaptive procedures have been constructed in which
R' (x )(x - x max)<O when x=¥=x max' if the regression a =a(n) depends on the observations and approximates
function and the variance of the random errors do not ao when n.-?oo. These procedures possess properties
increase too rapidly when 1x I.-?oo, and if the condi- that are asymptotically optimal in the sense of the
tions asymptotic variance.
an > 0, ~en = 00, en > 0, The results of asymptotic normality are also known
° in the multi-dimensional case. Let all roots of the
~anen < 00, en -') matrix
STOCHASTIC APPROXIMATION
A = a aaRx (x ) + 1. I Methods based on compactness and weak convergence
0 2 have also been introduced.
have negative real parts (1 is the identity matrix), let
References
EG(n, x, w)G*(n, x, w) = Sen, x) ----> So [A1] KUSHNER, H.J. and CLARK, D.S.: Stochastic approximation
methods for constrained and unconstrained systems,
when n~oo, let X~XO and Xn~XO almost certainly, Springer, 1978.
and let certain other not too-restrictive conditions be [A2] LJUNG, L.: 'Analysis of recursive stochastic algorithms', IEEE
Trans. Autom. Control AC-22 (1977), 551-575.
fulfilled. Then the vector Zn = Vn(Xn - xo) is asymp
totically normal with mean zero and with covariance AMS 1980 Subject Classification: 62L20
matrix
S = a2!exp(At)Soexp(A*t)dt. STOCHASTIC BASIS - A complete probability space
o (Q, :#, P) with an increasing family F=(,%t)/;;,o of
The above result for an asymptotically-optimal sub-a-algebras :#/ <;;;:#, which satisfies the (so-called
Robbins - Monro procedure can also be generalized to usual) conditions:
the multi-dimensional case. It has been proved that the 1) it must be continuous from the right, :#( =:#(
n
random process Zn converges to a Gaussian Markov (= '%s), t;:;:'O;
s>/
process in a logarithmic scale. Given certain conditions,
2) it must be complete, i.e. :#( contains all subsets
the convergence of the moments of the random variable
from ,% of P-measure zero.
Xn to the moments of the limit law has been proved. For stochastic bases, the notations (Q, ,%, F, P) or
Stochastic approximation-type procedures are con
(Q, :#, ('%"/)/;;'0, P) are also used.
venient in non-parametric situations, since they can be A.N. Shiryaev
used when a priori information on the regression func Editorial comments. An increasing family of (a-) alge
tion is scarce. However, they are also used in estimating bras is usually called a filtration.
the parameter 0 of the density f (x, 0) through indepen
AMS 1980 Subject Classification: 60G05, 60B99
dent observations XI, ... ,Xn with this density. Given
certain restrictions, the recursive procedure
STOCHASTIC BOUNDEDNESS, boundedness in pro
8n +] - 8n = 1.(1 -] )(8n) gradoln f (Xn ~ ], 8n) bability - The property of a stochastic process X(t),
n t E T, expressed by the condition: For an arbitrary (>0
(l (0) is the Fisher information matrix of the density f) there exists a C>O such that for all t E T,
is a consistent and asymptotically-efficient recursive
P{ I X(t) I >C} < {.
estimator of the parameter O. The same process is also A. V Prokhorov
possible in the case of continuous time.
AMS 1980 Subject Classification: 60G07
The behaviour of procedures of stochastic approxi
mation has been studied in the case where the regres
STOCHASTIC CONTINUITY - A property of the
sion function has several zeros (several extremal
sample functions of a stochastic process. A stochastic
points), and for different modifications and generaliza
process X (t) defined on a set T <;;; R I is called stochas
tions of procedures of stochastic approximation.
tically continuous on this set if for any (>0 and all
References
toET,
[1] WASAN, M.T.: Stochasric approximarioll, Cambridge Univ. lim P{p[X(t), X(to)]>{} = 0,
Press, 1969. /----7111
[2] NEVEL'SON. M.B. and KHAS'MINSKII. R.Z.: Sroe/wsric approxi where p is the distance between points III the
marion and reclirsil'C esrirl/alion, Amer. Math. Soc., 1976
corresponding space of values of X(t).
(translated from the Russian).
[3] TSYPKIN, Y A.Z.: Adaptioll and learnilw in automaric svstems. References
Acad. Press, 1973 (translated from the Russian).
[1] PROKHOROV, YU.V. and ROZAl\O\. YU.A.: Proba/Jilio' theOl)"
[4] TSYPKIN, Y A.Z.: Foundariol1s of" rhe rhco/T of' learning sl'Srems. Springer 1979 (translated from the Russian)A' V P kh
Acad. Press, 1973 (translated from the Russian). ' . . ro orov
R. Z. Khas'l1linskii
AMS 1980 Subject Classification: 60G07
Editorial comments. Convergence properties of stochas
tic approximation and other recursive algorithms have been
STOCHASTIC CONVERGENCE - The same as con
the subject of much research. One approach is the 'ordI
vergence in probability.
nary differential equations' method (rA1], [A2J), which is
based on interpreting sUitably rescaled versions of (1) and AMS 1980 Subject Classification: 60FXX, 60B10,
(2) as Euler approximations to the solution of an ordinary or 28A20
stochastic differential equation. AsymptotiC properties of the
algorithm can then be related to stability properties of the STOCHASTIC DEPENDENCE, prohahilistic depen
corresponding ordinary or stochastic differential equation. dence, statistical dependence - A dependence bet ween
2