Convergence rate of moments in stochastic approximation with simultaneous perturbation gradient

12 Pages·1999·0.36 MB·English

894 IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.44,NO.5,MAY1999
Convergence Rate of Moments in Stochastic
Approximation with Simultaneous Perturbation
Gradient Approximation and Resetting
La´szlo´ Gerencse´r
Abstract—The sequence of recursive estimators for function order SPSA methods are developed and their convergence
minimization generated by Spall’s simultaneous perturbation properties are established.
stochasticapproximation(SPSA)method,presentedin[25],com-
The boundedness of the estimator sequence is a priori
bined with a suitable restarting mechanism is considered. It is
assumed also in [17], and a similar but weaker boundedness
provedthatthissequenceconvergesundercertainconditionswith
rate O(n(cid:0)(cid:12)=2) for some (cid:12)>0, the best value being (cid:12) = 2=3, condition is assumed in [19]. Namely, it is assumed that the
where the rate is measured by the L -norm of the estimation estimatorsequencevisitsacertainfixeddomaininfinitelyoften
q
error for any 1(cid:20)q<1. The authors also present higher order (cf., Theorem 2.3.5).
SPSA methods. It is shown that the error exponent (cid:12)/2 can be Theratethatwegetis—notsurprisingly—identicaltowhat
arbitrarilycloseto1/2iftheHessianmatrixofthecostfunctionat
appears in the CLT (cf., [25, Proposition 2]). It is expected
theminimizingpointhasallitseigenvaluestotherightof1/2,the
that the present rate of convergence result will play a role in
cost functionis sufficiently smooth, and a sufficiently high-order
approximation of the derivative is used. solving practical problems, such as the analysis of the effect
of parametric uncertainty on performance. The present paper
IndexTerms—L-mixingprocesses,limittheoremformoments,
alsoextendsresultsgivenin[11]inwhichacompleteanalysis
linearstochasticsystems,maximalinequalities,recursiveestima-
tion. fortheconvergenceofLjung’sscheme(cf.,[5],[4],[21],[1])
with resetting is given.
Higher order Kiefer–Wolfowitz methods were first con-
I. INTRODUCTION
sidered in [7], where the rate of mean-squared error for
THE aim of this paper is to prove a rate of convergence globally defined procedures had been established. The results
theoremforaclassofstochasticapproximationprocesses of Section IV complement these results: we consider SPSA
for function minimization developed by Spall in [25]. The methods rather than Kiefer–Wolfowitz methods, the proce-
mainfeatureofSpall’smethodisanewwayofestimatingthe dures are localized using a resetting mechanism, and the rate
gradient using only two measurements at properly selected of higher order moments of the error process is established.
random parameter values. One of the main application areas The analysis given in [25] is based on the early work of
ofsimultaneousperturbationstochasticapproximation(SPSA) Fabian in connection with the Kiefer–Wolfowitz method (cf.,
is direct stochastic adaptive control (cf., [25]). [6]). It is hoped that the ideas that had emerged since then in
A Kiefer–Wolfowitz method using randomized differences the theory of recursive identification yield a more transparent
was first proposed in [17, Sec. III-B]. However, the number proof that can be adapted to future needs.
of measurements required for this method is the same as for The present paper also complements recent results of
the standard Kiefer–Wolfowitz method, i.e., twice the number [2], where an almost sure convergence rate has been given
of dimension. The argument behind the use of randomized [2, Th. 3] for a modified version of the SPSA algorithm.
differences was that by their use, sensitivity with respect to
biasisreduced.ArandomdirectionKiefer–Wolfowitzmethod
II. THE PROBLEM FORMULATION
with just two measurements has been proposed in [19, Sec.
The -dimensionalEuclideanspace willbedenoted by
II-C.5], using random unit vectors as perturbing vectors; see
TheEuclideannormofavector willbedenotedby The
also [20].
operator norm of a matrix will be denoted be i.e.,
The main advances of this paper are that a crucial bound-
Finally a convention: in the various
edness hypothesis, given as [25, Assumptions A3 and A5, p.
estimations below we shall frequently have constants which
335] isremovedby forcingthe estimatortostay ina bounded
depend only on the constants that appear in the conditions
domain, and we get the rate of convergence of higher order
below. These constants will be called system constants.
momentsoftheestimationerror.Finally,inSection IV,higher
We consider the following problem: minimize the function
defined for where is an open domain,
Manuscript received November 1, 1996; revised September 2, 1997 and
for which only noise corrupted measurements are available,
July9,1998.RecommendedbyAssociateEditor,G.G.Yin.
The author is with the Computer and Automation Institute, Hungarian given in the form
AcademyofSciences,H-1111Budapest,Kende13-17,Hungary.
PublisherItemIdentifierS0018-9286(99)03941-0.
0018–9286/99$10.00 ª 1999 IEEE
GERENCSE´R:CONVERGENCERATEOFMOMENTS 895
where is a random variable oversome probability To minimize we need an estimator of its gradient,
space It is assumed that the measured values of denoted by
can be obtained for each via a physical
(4)
experiment, and if necessary the experiment can be repeated.
Note that the measurement-noise does not depend on
The conventional finite difference approximation of partial
thus we have what is called a state-independent noise. The
derivativesthatrequiresalargenumberoffunctionevaluations
extension of the results of this paper to state-dependent noise
is replaced by an approximation using simultaneous random
is possible, but a number of additional technical details have
perturbationsofthecomponentsof Let denotetheiteration
to be clarified. This is the subject of a forthcoming paper.
time for the stochastic gradient algorithm to be developed. At
Condition2.1: The function is three times continu-
time we take a random vector over some probability space
ously differentiable with respect to for Let
denoteanupperboundoftheoperatornormsofthederivatives
up to order three. It is assumed that the minimizing value of
is unique in and will be denoted by
Condition2.2: The measurement noise process Condition2.3: is a double sequence of
is assumed to be a zero-mean -mixing process with i.i.d., symmetrically distributed, bounded random variables
respect to a pair of families of -fields such that for any (cf., [25, Sec. III]).
For the definition of -mixing, cf., Definition 5.2 of the Remark: Note that the processes and
Appendix and [9]. Actually we need less than -mixing, are defined on different probability spaces.
namely it is sufficient that an “improved Ho¨lder inequality” From now on we consider the product space
given below in (1), (cf., [9, Lemma 2.3] for continuous and write and
time),issatisfied.Analogousinequalitiesforuniformlymixing Thus the processes and are independent
stationary sequences are given in [16] and for strong mixing over Mathematicalexpectationwill
stationarysequencesin[3].Thequotedresultisthefollowing: be always meant to be taken over the mentioned probability
let be a zero-mean -mixing process with respect space unless otherwise stated.
to and let be an -measurable random variable A standard perturbation that will be used in the rest of the
for some such that its moments, which appear in paper is the double sequence
the inequality below, are finite. Then
(1)
Let
for every such that
The class of -mixing processes has been first system-
and
atically studied in [9] and later in [13]. The usefulness of
(5)
-mixing processes in stochastic systems theory has been
demonstratedinanumberofpapers,asurveyofwhichisgiven
Since is an i.i.d. sequence of bounded random variables
in [10] and [12]. A basic example of -mixing processes is
it follows that is -mixing with respect to
obtained as follows: let be an -bounded (cf.,
Now let be a fixed sequence of positive
Definition5.1oftheAppendix)independentsequenceofreal-
numbers and let be a compact convex domain specified
or vector-valued random variables and define a vector-valued
in Condition 2.4 below. For each we take two
process by
measurements that are denoted by
and defined as
(2)
with stableand Thenitiseasytoseethattheprocess
is -mixing with respect to defined as Then the estimator of the gradient at time for is
defined as
(3)
An important property of -mixing processes is that if
is a vector-valued -mixing process and is a function
such that the function itself and the function (6)
are polynomially increasing,
then the process is also -mixing. Thus the sum or A convenient representation is obtained if we define the
product of -mixing processes is -mixing. Furthermore, if random vector
is -mixing and is generated by (2), then is
also -mixing.
896 IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.44,NO.5,MAY1999
Then the gradient estimator at can be written as This is obtained by a simple change of time-scale
ItiseasytoseethatinCondition2.4 canbetakentobe
(7)
the smallest eigenvalue of the Hessian-matrix of at
The common numerator of these differences can be written TheSPSAMethod: Let be a fixed sequence of positive
as where is the numbers with denoting the stepsize at time Then, in
compound measurement error defined by the original form of Spall’s method, we start with an initial
estimate and then a sequence of estimated parameters,
(8)
denoted by is generated
Define Thenitiseasytoseethat recursively by
is -mixing with respect to (14)
Thus we can write as
A standard choice is with and
The central limit theorem (CLT) given as
(9) [25, Proposition 2] indicates that for any fixed choice
of the best rate of convergence is obtained if we choose
A standard choice for is
Thus, in the sequel we shall assume that
with some (10) There seems to be no technical difficulty to extend the
analysis presented in this paper to the case
Notethatin[25]theconditionimposedonthemeasurement
The almost sure convergence of the estimator process for
noise is expressed in terms of the compound measurement
the casewhen the noiseis amartingaledifferenceprocesshas
noise which is assumed to be a martingale difference
been established in [25] using results of [22]. In the same
process (cf., the condition preceding [25, (2.2), p. 333]). The
paper asymptotic normality of a properly scaled estimation
condition of the present paper given as Condition 2.2 is a
error process is established by a nontrivial application of [6].
possible alternative to Spall’s condition.
The scaling is nonstandard compared to classical statistical
The ordinary differential equation (ODE)
theory: assuming and a normal limit
(11) distribution of exists for
A main advance of the present paper is that the “bound-
will be called the associated differential equation. edness conditions” [25, Conditions A3 and A5] repeatedly
is defined in and it has continuous partial derivatives up criticized in the literature on recursive identification (cf., [1,
to second order. Under the condition above (11) has a unique remarks, pp. 46 and 47]) are removed by the application of
solution in which we denote by It is well a resetting or truncation mechanism as described below. A
known that is a continuously differentiable function further advance is that by the application of the methods of
of [11] weget an upperboundfor the rate ofconvergenceofthe
Condition2.4: Let denote a compact convex moments of the error process, which is likely to be tight (cf.,
domain such that and the closure of the the remarks following the theorem).
neighborhoodof ofradius denotedas isinside Resetting: Assumethattheinitialestimate isin De-
For every is defined and fine,following(14)atentativevalue
we have with some
and then set
(12) when
and
Here denotestheoperatornormofamatrix.Furthermore,
when (15)
we assume that the initial condition
where is a compact domain which is invariant for (11)
Remark: The above resetting mechanism seems to lose
and that for any
information obtained up to time However, if the noise
sequence is “untypical” so that it drives the estimator out
of the domain then we can not expect to extract much
and for the solution trajectory converges information. An alternative resetting mechanism would be to
to chose However, may be at a position from
Inequality (12) is equivalent to the condition that the dif- which the solution of the ODE does not converge to at all,
ferential equation or hits the boundary of the truncation domain, and this would
force the estimator to be reset infinitely many times.
(13)
The choice of the initial estimate and of the set
requires some a priori knowledge of the problem. However,
is exponentiallyasymptotically stable withexponent i.e.,if
if the associated ODE is globally asymptotically stable in
the solution of (13) is denoted by then we have
the domain of definition of then for any initial condition
a “sufficiently large” is a suitable truncation
domain. This approach is practical if the parameterization of
GERENCSE´R:CONVERGENCERATEOFMOMENTS 897
the domain is simple and it is easy to describe what is The validity of the conclusion of Theorem 2.1 under such
a large domain As an example consider the problem of weakconditionimposedonthenoiseisasurprisingresult.This
system identification. Let the parameter denote the system’s remarkable feature of SPSA has been first observed in [2] in
parametersofastable,linearsingle-input/single-output(SISO) the context of almost sure convergence. Note that no similar
system. If we use the balanced parameterization of Ober result is known for the standard Kiefer–Wolfowitz method.
(cf., [24]), then the parameter-space is very simple, and the HoweverrandomizedKiefer–Wolfowitzmethods(cf.,[17]and
proposed procedure is feasible. [19,Sec.II-C.5])doexhibitsimilarrobustnesswithrespectto
In the theorem below the notation means that the noise.
-norm of the left-hand side Aninterestingspecialcaseiswhenthereisnomeasurement
decreases with the rate given on the right hand side for any noise.Thenwehaveastandardoptimizationproblemwhichis
solved by a randomization method. The use of SPSA is justi-
Theorem2.1: Let Assume that fiedfor large scale problemswith lowprecision requirements.
the smallest eigenvalue of the Hessian matrix of at Theorem2.3: Let for all Choose and
denoted by satisfies Then under the conditions assume that the smallest eigenvalue of the Hessian matrix of
above, i.e., Conditions 2.1–2.4 we have at denoted by satisfies Then under
Conditions 2.1, 2.3, and 2.4 we have
(16)
(17)
For we have Finally for
we have for any III. THE ANALYSIS
The value of is maximized for
Step1—Continuous-TimeEmbedding: Some of the calcu-
from which we get and
lations to follow are easier to carry out in continuous time,
The best rate is then
hence we embed our discrete-time data and procedure into a
Remark: The proof of the theorem yields the following
continuous-time data and procedure. Consider the piecewise
stronger result: let and define
linear curve defined for as
where is the tentative value
of the estimator computed by (14). If then the
straightline willliecompletelyin since isconvex,
then, for and then we set On the other hand if
Remark: Therole of the relation between and can be
then let denote the moment when first
roughly explained as follows: it will be proved in Lemma 3.2
hits the boundary of Let us then reset to i.e., we
that the estimator sequence locally tracks the solution of the
define for and In the period
ODE given by (11) with an error irrespective of
we keep constant:
the value of But the solution of the ODE converges to
Furthermore, let the piecewise constant continuous-
with a rate If this rate is better than the rate of the
time extension of denoted as
local error, then the latter will dominate. Otherwise, it is the
be defined as for
other way around.
We define the continuous-time extensions of
Remark: The theorem does not claim that the given con-
the sequences similarly.
vergence rate is sharp. But the CLT given as [25, Proposition
Set for
2] indicates that in the case the convergence rate
Then it is easy to see, using, e.g., inequality (74) of the
given in the theorem is sharp, indeed.
Appendix, that the process is -mixing with respect to
Remark: Notethatwedonotneedtohavethelowerbound
Similarly, defining
as in [25, Proposition 2], this is only needed for
for the process is -mixing with respect
the asymptotic normality result of [25]. The reason for this
to
is that for the contribution of the third-order term of
Lemma3.1: In the period the straight line
the Taylor-series expansion (called in (27) below)
will be the solution of a differential equation of the form
dominates the error process [cf., (30) below], hence the
stochastic effect is suppressed and the existence of a limiting
(18)
distribution is not ensured.
For the next theorem we consider an alternative noise
where If the measurement noise is
condition, in which no dependence structure is imposed on
zero, then we have for any
This theorem is based on an observation in [2].
Proof: The correction term is defined by the re-
Condition2.5: The measurement noise process
quirement that the right-hand side yields the constant velocity
is assumed to be a bounded sequence of random
i.e., we require
variables.
From here we get
Theorem2.2: LettheconditionsofTheorem2.1besatisfied
so that Condition 2.2 is replaced by Condition 2.5. Then the
conclusions of Theorem 2.1 remain valid.
898 IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.44,NO.5,MAY1999
The time-dependent random field is easily seen to Step2—AnalysisonFiniteIntervals: Let and
be Lipschitz-continuous with respect to First note that for We consider the trajectory on the interval
we have Let denote the first moment after at which hits the
boundary of If does not hit the boundary of at all,
then we set Further, let denote the solution of
(11) with initial condition , i.e.,
A main technical tool used in the proof is the development of
where Asimilar
a locally uniform upper bound for the increments
expression can be obtained for
Note that the validity of the lemma is independent of the
Since the measurement noise is independent of we get,
smallest eigenvalue of the Hessian-matrix of at Let
subtracting the latter expression from the former one
Lemma3.2: Let and let
Usingasecond-orderTaylor-seriesexpansionfor
Then we have If
around it is easy to see that the
the measurement noise is zero, then taking we get
Euclidean norm of the integrand is bounded by and thus
the Lipschitz continuity of is established with a Lipschitz
Proof: First fix Since we have for
constant
To get an upper bound for we subtract and add
to get
(19)
For the first term we have
(21)
where is the cumulative error defined as
(22)
Thiserrorprocesswillbedecomposedaccordingtothesource
Now consider the expression for the random field
of the error. First write
given in (9). We have
(23)
To analyze substitute the free variable for Define
with some deterministic constant uniformly in for
On the other hand
Then Now we continue to decom-
pose taking into account the representation of
thus we get altogether that for the standard choice
given in (7): we write
We conclude that the contribution of the first term in (19) is
Forthesecondtermin(19)weget (24)
and the norm of the latter is majorated by Since
The first term, will be further decomposed using a
we conclude that the second term
third-order Taylor-series expansion. Write
in (19) is also of the order of magnitude and
thus finally we get that
(20)
which completes the proof.
Ifthemeasurementnoiseiszero,then
and thus we get an error term for any
GERENCSE´R:CONVERGENCERATEOFMOMENTS 899
where and denotesappro- We further decompose in order to simplify the
priatetensorproducts.Asimilarexpansioncanbeobtainedfor randomness present in Let us write
Subtractingthelatterexpansionfromthe
onegivenabovethefirst-andthird-ordertermwillbedoubled
and the second-order terms will be cancelled, thus we get
Since is restricted to a compact domain in which the first
andsecondderivativesof arebounded,andthecomponents
of have absolute value equal to one, it follows from the
definition of [cf., (3.9)] that is Lipschitz-
continuous,say where
isadeterministicconstant,dependingonlyon and
(25)
Now write
Note that we have
The advantage of this approximation is that the randomness
of is purely due to the randomness of the initial condition
Define the modified error process
and thus the first term on the right-hand side of (25) can be
written as
(30)
(26)
Substituting into (21) and taking into account the inequality
In randomness is purely due to
wegetthat ismajorated
the random perturbation of the parameters. The conditions
by
imposed on imply that
Since is -mixing with respect to we get that
is also -mixing with respect to and we (31)
concludethat isazero-mean -mixingprocesswith
respectto Thesameholdstriviallyforitsfirstthree
Now we are in a position to apply the Bellman–Gronwall
derivatives with respect to
lemma with fixed. But first we need to get a tight upper
For the third-order term we introduce the notations
bound for the first term on the right-hand side.
Step3—EstimationoftheConstantTerm: Let us consider
the expressions
(27)
It is easy to see that when is restricted to we have
and hence
(28)
Thuswehavearrivedatthefollowingdecompositionofthe
error process
(29)
For the sake of further reference the relevant terms are
summarized as follows:
Note that since Condition 2.4 above implies that
is defined for all and hence the definition
of is correct. Defining the compound error term
(32)
we get from (31) for any with
(33)
900 IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.44,NO.5,MAY1999
Estimationof It is easy to see that for fixed we can write the above inequality as
and the process
(37)
Theadditionaldifficultyinestimating isinthehandling
andits firsttwopartialderivativeswithrespectto are ofthesupremumwithrespectto overasetofdilatinginter-
zero-mean -mixing processes over Using this vals. This will be done with the application of an appropriate
fact, [11, Lemma 3.1] implies that change of time scale. Let us set and define
(34) (38)
in the sense that the -norm of the left-hand
With this notation we have the following proposition.
side decreases with the rate given on the right-hand side for
Lemma3.3: The processes and
any Obviously, if we consider as a random
are -bounded.
variable overtheproductspace the
Proof: We have already shown that
same proposition holds.
therefore Let us now
Estimationof Using (28) and the assumption
take a small and estimate the moments of
we get
Write then we can write
The difference can obviously be majorated by
(35)
the sum of the following two terms:
with some
(39)
Estimationof We have [cf., (24)]
(40)
We estimate similarly to for any
By Condition 2.2 the process considered as a process
over is a zero-mean -mixing process with respect
to It is easy conclude that considered as a
processoverthe product space is a
zero-mean -mixing process with respect to
Similarly considered as a process over the
product space is a zero-mean -
mixing process with respect to Since where depends on and the processes but is
and are independent we conclude that is a independentof and whenthelatterisconfinedtoabounded
zero-mean -mixing process over interval.Substituting weget fortheright-handside
with respect to
Thusforfixed wecanestimatethemomentsoftheintegral
on the right-hand side using the maximal inequality given as
[9, Th. 5.1] and restated as Theorem 5.1 in the Appendix. We
get for any and fixed
where depends only on and Thus we get
(41)
where depends on and the processes but is Similar estimates can be obtained for thus we finally get
independentof and whenthelatterisconfinedtoabounded
interval.Substituting wegetfortheright-handside (42)
Now write
To proceed we need to develop an extension of [11, Lemma
3.1]. Defining
(36) Then, by (42) the first term on the right-hand side is
when is confined to a bounded interval. For the
GERENCSE´R:CONVERGENCERATEOFMOMENTS 901
second term we get by (37) and by the elementary inequality Step4—HittingProbabilities: Let denote the event that
that its order of a resetting takes place in the interval It has been
magnitude is Since for small we have shown in [11, Lemma 2.3] that Lemma 3.2 implies that
we finally get for all It follows that in the whole
interval we have
(43)
(51)
Thus Lemma 3.3 has been proved.
Now let be fixed and let Set
and Then applying the maximal inequality given where is the diameter of Note that the validity of this
as Theorem 5.2 of the Appendix to the congruent compact estimate is independent of
intervals with varying we get Step5—PastingTogethertheIntervalEstimates: Let us
now take a subdivision of by the points
(44)
with some Then
Observingthat [cf.,(38)],weim- (52)
mediately get that
Assume that Following the arguments on [11,
thus we conclude that
p. 1208] and using the stability condition imposed on the
(45) associated ODE, it follows that with denoting the solution
of the associated ODE starting from we have
If the measurement noise is zero, then obviously
for any
Estimationof Using (20) we get
Since the proposition
of the theorem follows. The case is handled anal-
ogously (cf., [11]), using a corrected version of [11, Lemma
7.4], given as Lemma 5.1 in the Appendix of this paper. The
(46)
case triviallyfollowsfromthepreviousresults.The
proof of Theorem 2.3 is completed analogously.
If the measurement noise is zero, then ProofofTheorem2.2: We have to reconsider only the
for any arguments given under the heading “Estimation of ”
Summarizing (34), (35), (45), (46) it is easy to see that Themainideaisthattheboundednessof impliesthat
for the dominant terms are arising from the for any fixed the process is a zero-mean
approximation error in the numerical differentiation scheme, -mixingprocessover withrespectto
and arising from the measurement error. Thus for the uniformly in (cf., [9]). Writing
compound error term defined under (32) we get
(53)
(47)
we get by the arguments given above that for all we
It follows that we can write
have
with (48)
(54)
Ifthemeasurementnoiseiszero,thensummarizing(34),(35),
(45), (46) we see that for the dominant term is where is independent of Integrating this inequality over
It follows that we have with respect to we get the desired inequality (45),
and the rest of the proof is the same.
(49)
Now from (33) we get using the Bellman–Gronwall lemma IV. HIGHER ORDER SCHEMES
with fixed
The idea behind SPSA can be generalized to get
(50) higher order approximations of the gradient. Higher order
Kiefer–Wolfowitzmethods werefirstconsideredin[7],where
with a special ad hoc approximation scheme is used. In contrast
to this we rely on approximation schemes the numerical
properties of which are known to be very good. The rate of
mean-squared error established in [7] is identical to the rate
Since the right-hand side of (50) is independent of we that we get for higher order moments of the error process.
can take supremum over on the left-hand side, and thus However, better numerical procedures may improve the
Lemma 3.2 is proved. asymptotic covariance of the estimation error.
902 IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.44,NO.5,MAY1999
Following [8, Ch. 2], let be a real-valued function estimator will be based on measurements taken for
of the real variable and let Then define the shift at A single measurement has
operator by the form where
is the measurement error. Note that for the positions
(55)
at which measurements are taken are halfway between and
and the central difference operator by the positions taken by standard SPSA. We assume that for
wehave Let denote
(56)
a generic measurement taken at , i.e.,
Then the gradient estimator at is defined as
Clearly we can write
(64)
(57)
On the other hand, for analytic, a Taylor-expansion around where the notation is self-explanatory. We
gives define the estimator sequence as in (14) replacing
by the expression given above in (64) and using the same
resetting rule.
TheanalysisofhigherorderSPSAmethodsisanalogousto
which can be expressed in the form that of the second-order SPSA method given above. In view
where is the differentiation operator. But of the assumed independence of and we have
thus we can write formally
(58)
Furthermore by (63) the latter expression is equal to
Combining this with (57), we conclude that
(59)
from which we get for the representation
(60)
Theeffectofusinghigherorderapproximationschemesisthat
The first few terms of the Taylor-series expansion of the
theresidualterm definedunder(27)willbereplaced
right-hand side gives the approximation
by a higher order residual term for which we
have [cf., (28)]
(61)
(65)
Letthe th-orderformalTaylor-seriesexpansionoftheright-
Theestimationof willproceedasin(35):weget
hand side of (60) be denoted by It is stated (cf.,
[8, p. 22]) that for a function having continuous
derivatives, we have
(66)
(62)
whereinthelastterm isintherange
Thisapproximationofthederivativeisfavoredinnumerical with some Summarizing (34), (66), (45), (46) it
analysis because of its good convergence properties, namely is again easy to see that for the dominant terms
thecoefficientsoftheexpansionoftheright-handsideof(60) are and the latter arising from the
decay faster to zero, than for other expansions. measurement error.Thus for the compound error term [cf.,
Foramultivariablefunction weapproximatethepartial (32)] which is now defined as
derivatives analogously. For this we fix and define the
(67)
central difference operator in the direction by applying
theoperator tothefunction ofthescalar
we get the estimation
variable Thus if the function is continuously
differentiable, then we have (68)
(63) It follows that we can write
when is restricted to a compact domain. with (69)
Now take randomasinSection IIand let
be a fixed sequence of positive numbers. The gradient Thus we get in general the following result.
GERENCSE´R:CONVERGENCERATEOFMOMENTS 903
Theorem4.1: Let Assumethat Proof: Let and set Then
the smallest eigenvalue of the Hessian matrix of at we have after multiplying (71) by
denoted by satisfies Assume that the conditions
of Theorem 2.1, i.e., Conditions 2.1–2.4, are satisfied with
the following additions: the function is times
which can be solved explicitly for
continuously differentiable in and in Condition 2.4 the
neighborhood of of radius is inside Then
for the estimator sequence defined by (14), combined with
a resetting mechanism, with being defined under (64), we
have Using the triangle inequality for the norm and
the condition we get
(70)
For we have Finally for
we have for any from which the first proposition follows.
The value of is maximized for Ausefulreformulationoftheabovederivationisasfollows:
from which we get and write
The best rate that is
obtained is
Then we have
This rate can be arbitrarily close to if is sufficiently
large.
APPENDIX
(72)
In this section, we summarize some definitions and earlier
results that have been used in the paper. Let a probability
Thus it is sufficient to establish that for
space begivenandlet beanopendomain.
A parameter-dependent stochastic process or
equivalently a time-varying random field, is a sequence of
measurable mappings for from to
Here denotesthe -fieldofBorelsetsof
and this has been done above. The advantage of this refor-
Definition5.1: We say that the -valued parameter-
mulation is that the left-hand side is the convolution of the
dependent stochastic process is -bounded if for all
sequences and andthusitissymmetricin and
In the case when we use the same estimate for
buttheroleof and isinterchangedthusweget
If is -bounded we shall also write Sim-
ilarly, if is a positive sequence we write if Let be a monotone increasing family of -
Thedefinitiontriviallyextendstoparameter- algebras, and be a monotone decreasing family
independent processes. of -algebras. We assume that for all and are
The first part of the following result was stated in [11, independent. A standard example is
Lemma 7.4]. The second part of the quoted lemma was not
(73)
correctly stated and is therefore restated and proved here.
Lemma5.1: Let be an -bounded process where is an independent sequence of random
and define a process by variables.
Definition5.2: An -valuedstochasticprocess
(71)
is -mixing with respect to if it is -adapted,
where Then for any we have -bounded, and with being a nonnegative integer and
we have for any
On the other hand if then we have