xn = x n 1 α f(xn 1 + β n) f(xn 1 β n)

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Hubert Maxwell
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1 Proceedigs of the 005 Witer Simulatio Coferece M E Kuhl, N M Steiger, F B Armstrog, ad J A Joies, eds BALANCING BIAS AND VARIANCE IN THE OPTIMIZATION OF SIMULATION MODELS Christie SM Currie School of Mathematics Uiversity of Southampto Southampto, SO7 BJ, UK Russell CH Cheg School of Mathematics Uiversity of Southampto Southampto, SO7 BJ, UK ABSTRACT We cosider the problem of idetifyig the optimal poit of a objective i simulatio experimets where the objective is measured with error The best stochastic approximatio algorithms exhibit a covergece rate of /6 which is somewhat differet from the / rate more usually ecoutered i statistical estimatio We describe some simple simulatio experimetal desigs that emphasize the statistical aspects of the process Whe the objective ca be represeted by a Taylor series ear the optimum, we show that the best rate of covergece of the mea square error is whe the variace ad bias compoets balace each other More specifically, whe the objective ca be approximated by a quadratic with a cubic bias, the the fastest declie i the mea square error achievable is /3 Some elmetary theory as well as umerical examples will be preseted INTRODUCTION We cosider the problem of idetifyig the optimal poit of a o-liear objective fuctio i simulatio experimets where the objective is measured with error This problem may arise i a umber of settigs, i particular whe determiig the best set up for a stochastic system, such as that described i a previous paper Cheg ad Currie 004 I order to demostrate the priciples ad ivestigate some of the theoretical issues, we cosider oly a simple oe-dimesioal example, yx = ηx + ε, where ηx is the uderlyig objective fuctio, ad ε N0,σ is a radom error term, which we assume to be ormally distributed We assume that it is possible to expad the objective fuctio as a Taylor series ear the optimum ad so cosider the covergece of the mea square error whe the objective ca be estimated as a quadratic with a higher-powered bias We further assume that i the rage uder cosideratio, the objective fuctio is strogly covex has oly oe miimum, ie dη/dx < 0 for x<x ad dη/dx > 0 for x>x Two experimetal desigs are cosidered, both ivolvig makig observatios either side of the optimum, at a distace that decreases with the order of the observatio The differece betwee the two desigs is i the estimate of the optimal poit I the first desig, we cosider the distributio of poits aroud a kow optimum Although urealistic, this serves to demostrate some of the statistical properties of these desigs I the secod desig, we use our curret maximum likelihood estimate of the optimum to set the desig poit for the ext iteratio The bias ad variace both deped o the umber of observatios made, with the depedece beig determied by the rate at which the desig poits coverge o the optimum We show that, uder the optimal settigs for the first desig, the cotributio of the bias ad the variace to the mea square error are balaced Although cocered with experimetal desig, the methodology we propose for choosig desig poits has close liks with the techique of stochastic approximatio Robbis ad Moro 95 were the first to give a formal mathematical treatmet of stochastic approximatio, applyig it to fidig the solutio to the equatio yθ = M, where the output of the process, yθ, is a oisy fuctio of its iputs Kiefer ad Wolfowitz 95 adapted their work, ad used the techiques of stochastic approximatio to fid the maximum or miimum of a oisy fuctio Stochastic approximatio is a sequetial method i which the poit chose for the ext experimet is depedet o the poit of the previous experimet ad the most recet observatios For example, the approach used by Kiefer ad Wolfowitz is based o makig a fiite-differece approximatio at each iteratio, such that the estimate of the miimum after the th iteratio is x = x α fx + β fx β, β 485

2 Currie ad Cheg ad x is also the th desig poit The covergece rate of the methodology will deped o the properties of the objective fuctio fx We assume that fxis cotiuous ad show that the covergece rate of the algorithm depeds o the order of its differetiability I most situatios of iterest, the cubic term will domiate the Taylor series ear the optimum, ad the fuctio ca be regarded as beig thrice differetiable I this case we fid that the desig poits should coverge o the optimum at a rate of /6, with the mea square error decayig as /3 This covergece rate matches the optimum covergece rate for stochastic approximatio algorithms for a fuctio which is thrice differetiable, give by Dupač 957 Covergece rates for stochastic approximatio are also discussed i Wasa s book Wasa 969 We apply the two desigs to a umerical example of a oisy quadratic fuctio with a cubic perturbatio i Sectio 3, demostratig the performace of the algorithms at a umber of differet settigs for the covergece of the desig poits The umerical results from the secod, more practical, desig suggest that this has a similar covergece rate to the first desig The theoretical covergece properties of the first desig are also cosidered i Sectio METHODOLOGY We cosider perturbatios to the fuctio y = x of the form ax q, where q is a iteger, ie the fuctio ηx = x + ax q This has a local miimum at x = 0 We assume that observatios are subject to a additive oise term, ε N0,σ, such that a observatio yx = x + ax q + ε I the methodology that we propose, the i th desig poit will be at h i = i K i p We show iitially that this is equivalet to havig / observatios at each of h ad h The mea value of the positive observatios is / i p, i= ad the mea value of the egative observatios is / i= i p 3 To evaluate these sums, we make use of the Maclauri- Cauchy formula, lim { i= } i p x p dx L, where 0 L The sum i ca therefore be rewritte as / i= i p = p / i= i p, ad so, i the limit that, the mea value of the positive observatios, / i= i p p L + = p p p L p / p + p p As is large, this behaves as if all of the positive observatios were take at h = p p K p for p< Similarly, rewritig the sum i 3, / i= i p = i= / i p j= j p, ad i the limit of, we ca write the mea value of the egative observatios as / i= i p [ [ L + p ] + p p p L = p p + p p ] p p L p As with the positive observatios, this shows that, i the limit that, this methodology is equivalet to all of the egative observatios beig take at h = p p K p 486

3 Currie ad Cheg From the above aalysis, we ca therefore assume that we make / observatios at each of x = h ad h, such that the total umber of observatios made is The observed sample meas at h ad h will be y = h ah q + ε y = h + ah q + ε, where the ε i will deped o We cosider fittig the fuctio gx = θ f x + θ f x 4 to ηx, where the f i x are orthoormal basis fuctios These must be orthogoal, ie j= f k x j f l x j = 0 ad be ormalized, ie j= f k x j f k x j = for k =, ad l =, We assume that the basis fuctios are of the form f x = bx + c, f x = dx, where b, c, d are costats to be determied Differet results are obtaied for q odd ad q eve The case where q is assumed to be odd is more iterestig, ad we cosider that first Substitutig the f i ito the orthoormality coditios, we fid that b = h, c = 0 ad d = h, so that f x = f x = We ca therefore rewrite gx as x h x gx = ˆθ h + x ˆθ h x h 5 Usig the orthoormal properties of the basis fuctios, ad referrig back to 4, the least squares estimates of the θ i are ˆθ i = f i x j y j, j= ad for the set of observatios made, We wish to determie x, the value of x at the miimum Differetiatig, ad is zero at x, where x = h ˆθ ˆθ If ε +ε 3 ε h = ahq dg dx = ˆθ h + ˆθ x h, + ε ε ah q + ε + ε h is eglible as, the third term i the equatio will ted to oe Assumig that ε i = O p / ad rememberig that h = O p, this meas that + p <0, ad so p< 4 Assumig this is true, the i the limit of large, x = ahq + ε ε ah q We defie the optimal experimetal desig as oe that miimizes the mea square error MSE, which is defied to be MSE = bias + variace The bias i x is B = ahq ad the variace is V = σ h If h = K p the V = σ p ad K B = akq pq The MSE ca the be writte as where MSE = a K q pq + 3σ p 4 K = α pq + β p, α = a K q 4 β = σ K, which are both idepedet of ad p ˆθ = ˆθ = ah q + ε ε h + ε + ε 487

4 Currie ad Cheg We wish to fid the p that miimizes the MSE ad so differetiate with respect to p, dmse dp = α q pq l +β p l = l β p αq pq At the miimum, this is equal to zero ad p, the optimal value of p, obeys β p β αq Takig logs of both sides, = αq p q = p q 6 p = l q β αq l 7 Therefore, as, p /q With p equal to /q, the variace declies as V q q ad the square of the bias decays at the same rate, B q q The bias has a egative depedece o p ad the variace a positive depedece Therefore, with p = /q, we have a balace betwee the two With q eve, the bias term i the mea square error is zero ad the desig is chose simply to reduce the variace The variace decreases with decreasig p, ad so the best desig with a eve powered deviatio from the quadratic q eve is to choose poits further away from the optimum, as icreases If we are sufficietly close to the miimum, the domiat term i the Taylor series would be the cubic term, ie q = 3, suggestig a optimal value for p of /6, ad a mea squared error that decays as 3 This reproduces the results obtaied for the optimal covergece of stochastic approximatio algorithms, as put forward by Dupač NUMERICAL EXAMPLES I this sectio, we describe the implemetatio of two desigs for fidig the positio of the miimum of, where q = 3, a = 0 ad σ = 0 The simple algorithm discussed i Sectio is cosidered iitially, where we observe the objective fuctio at poits aroud its kow miimum value of zero Table 3 gives estimates of the optimum after 000 iteratios, averaged over 0 rus of the algorithm This shows that covergece to the optimum is fastest with p = /6 However, the results are very Table : Estimates of the Miimum Value with Differet Values of the Power p for the First Method Estimates are the Average of 0 Rus, each of 000 Iteratios p Estimate of x mi /8 003 /6 00 / variable, ad could ot be used as proof that the theory holds i practice If we ow istead assume that the miimum is ukow to us before the start of the experimet, which is a more realistic case, we ca choose desig poits x i such that x i = x i + i i p, where xi is our best estimate of the miimum after i iteratios of the algorithm The estimate of the miimum is obtaied by fittig a quadratic model to the data usig maximum likelihood methods The positio of the miimum ca the be easily deduced from the parameter values of the quadratic fuctio Fidig ew estimates of the maximum likelihood parameters at each step does ot ivolve a complete refit as we ca take advatage of the updatig routies described i Kedall ad Stewart 99 ad Bartlett 95 We write the desig matrix after iteratios as x x x x X =, x x ad write the parameters of the quadratic model that we are fittig as ˆθ = ˆθ ˆθ, such that the our estimate of the objective fucio after i iteratios is ŷ i = ˆθ i + ˆθ i x i + ˆθ 3i x i Usig this otatio, the maximum likelihood estimate of θ after iteratios is ˆθ = X X X y = A X y We ow make a additioal observatio y + at x Writig x = xx, we ca see that the ew desig matrix ca X be writte as x We ca therefore make use of the 488

5 Currie ad Cheg Table : Estimates of the Miimum Value with Differet Values of the Power p for the Secod Method Estimates are the Average of 0 Rus, each of 000 Iteratios p Estimate of x mi /8 005 / /4 07 matrix theorem stated i Bartlett 95 to update A, A + = A A xx A + x A x, ad use this to update the maximum likelihood estimate of θ give the additioal observatio, such that ˆθ + = ˆθ + A x y + x ˆθ + x A x Updatig A ad ˆθ ivolves o ew matrix iversio, therefore usig these formulae it is oly ecessary to perform oe matrix iversio at the start of the procedure, speedig up the process cosiderably Table 3 gives estimates of the optimum after 000 iteratios, averaged over 0 rus of the algorithm These suggest that the best estimate of the optimum after 000 rus is obtaied for p = /8 These experimets suggest that the theoretical results which we have proved for the iitial desig, where the miimum is assumed to be kow, hold i practice, but that p = /6 will ot ecessarily be the optimal covergece rate for the secod, more practical desig Further work is eeded to ivestigate what the optimal covergece rate is for the secod desig 4 CONCLUSION We have demostrated the use of a simple experimetal desig for fidig the optimum of a stochastic objective fuctio The theoretical treatmet of the problem showed that, for the optimal experimetal desig, the cotributios of the variace ad the bias to the mea square error are balaced Uder the optimal desig we have show that the optimal covergece rate for the desig poits is p = /6 for a quadratic objective fuctio perturbed by a cubic term, ad that at this rate the mea square error declies as /3, matchig the covergece results recorded by Dupač 95 for stochastic approximatio algorithms The umerical results agree with this result, but suggest that the optimal covergece rate for the secod desig may ot be p = /6 Ivestigatig the differet covergece properties of the two desigs will be the focus of further work o this problem This experimetal desig has may similarities with stochastic approximatio The treatmet of the problem i this paper will hopefully highlight some of the iterestig statistical properties of the problem of maximisig a stochastic objective fuctio REFERENCES Bartlett, MS 95 A iverse matrix adjustmet arisig i discrimiat aalysis The Aals of Mathematical Statistics : 07 Cheg, RCH ad CSM Currie 004 Optimizatio by simulatio metamodellig methods I Proceedigs of the 004 Witer Simulatio Coferece, ed RG Igalls, MD Rossetti, JS Smith, ad BA Peters, Piscataway, New Jersey: Istitute of Electrical ad Electroics Egieers Dupač, V 95 O the Kiefer-Wolfowitz approximatio method Časopis Pro Pěstovai Matematiky 8 : Kedall, MG, A Stuart, ad JK Ord 99 Kedall s advaced theory of statistics Fifth ed New York: Oxford Uiversity Press Kiefer, J, ad J Wolfowitz 95 Stochastic estimatio of the maximum of a regressio fuctio The Aals of Mathematical Statistics 3 3: Robbis, H ad S Moro 95 A stochastic approximatio method The Aals of Mathematical Statistics 3: Wasa, MT 969 Stochastic Approximatio Cambridge: Cambridge Uiversity Press AUTHOR BIOGRAPHIES CHRISTINE SM CURRIE is a lecturer of operatioal research i the School of Mathematics i the Uiversity of Southampto, where she also obtaied her PhD I additio, she has a MPhys from Oxford Uiversity ad a MSc i Operatioal Research from the Uiversity of Southampto Her research iterests iclude mathematical modelig of epidemics, Bayesia statistics, variace reductio methods ad optimizatio of simulatio models Her address is <christiecurrie@sotoacuk> ad her web page is <wwwmathssotoacuk/staff/currie> RUSSELL CH CHENG is Professor, Head of Operatioal Research, ad Deputy Dea of the Faculty of Mathematical Studies at the Uiversity of Southampto He has a MA ad the Diploma i Mathematical Statistics from 489

6 Cambridge Uiversity, Eglad He obtaied his PhD from Bath Uiversity He is a former Chairma of the UK Simulatio Society, a Fellow of the Royal Statistical Society, Member of the Operatioal Research Society His research iterests iclude: variace reductio methods ad parametric estimatio methods He was a Joit Editor of the IMA Joural of Maagemet Mathematics His address is <rchcheg@mathssotoacuk>, ad his web page is <wwwmathssotoacuk/staff/cheg> Currie ad Cheg 490

Chapter 9: Numerical Differentiation

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