Georgia Tech 2017 March Talk

Deterministic, Randomized and BayesianPerspectives on Simulating the Mean

Fred J. HickernellDepartment of Applied Mathematics, Illinois Institute of Technology

[email protected] mypages.iit.edu/~hickernell

Thanks to Lan Jiang, Tony Jiménez Rugama, Jagadees Rathinavel,and the rest of the the Guaranteed Automatic Integration Library (GAIL) team

Supported by NSF-DMS-1522687

Thanks for your kind invitation

mailto:[email protected]

[email protected]

http://mypages.iit.edu/~hickernell

mypages.iit.edu/~hickernell

Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature General Error References

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2/16

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Monte Carlo Simulation in the August 12, 2016 News

Sampling with a computer can be fastHow big is our error?

3/16


Problemµ = E[f (X)] =

ż

Rdf (x)ν(dx) =?, µ̂n =

nÿ

i=1wif (xi)

Given εa, choose n to guarantee

|µ´ µ̂n| ď εa (with high probability)

adaptively and automatically

If µ P [pµn ´ errn, pµn + errn] (with high probability), then the optimal andsuccessful choice is

{answer =ans´max(εa, εr |ans+|) + ans+ max(εa, εr |ans´|)

max(εa, εr |ans+|) + max(εa, εr |ans´|)in general ‰ answer(pµn)

where ans˘ :=

"

supinf

*

µ P [pµn ´ errn, pµn + errn]answer(µ)

providedans+ ´ ans´

max(εa, εr |ans+|) + max(εa, εr |ans´|)ď 1 (H. et al., 2017+)

4/16



ż


nÿ

i=1wif (xi)




E.g.,

option price =

ż

Rdpayoff(x)

e´xTΣ´1x/2

(2π)d/2 |Σ|1/2 dx, Σ =

(min(i, j)T/d

)d

i,j=1

Gaussian probability =

ż

[a,b]

e´xTΣ´1x/2

(2π)d/2 |Σ|1/2 dx

Sobol’ indexj =

ş

[0,1]2d

[output(x)´ output(xj : x

1´j)]output(x 1)dxdx 1

ş

[0,1]d output(x)2 dx´[ş

[0,1]d output(x)dx]2

Bayesian estimatej =

ş

Rd βj prob(data|β) probprior(β)dβş

Rd prob(data|β) probprior(β)dβ




where ans˘ :=

"

supinf

*




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ż


nÿ

i=1wif (xi)




Need errn based on data t(xi, f (xi))uni=1 such that

|µ´ µ̂n| ď errn (with high probability)

then increase n untilerrn ď εa




where ans˘ :=

"

supinf

*




4/16


Central Limit Theorem Stopping Rule for IID Monte Carlo

µ =

ż

Rdf (x)ν(dx)

µ̂n =1n

nÿ

i=1f (xi), xi

IID„ ν

P[|µ´ µ̂n| ď errn] « 99%for Φ

(´?

n errn /(1.2σ̂))= 0.005

by the Central Limit Theorem (CLT)

where σ̂2 is the sample variance.

5/16


Central Limit Theorem Stopping Rule for IID Monte Carlo

µ =

ż

Rdf (x)ν(dx)

µ̂n =1n

nÿ

i=1f (xi), xi

IID„ ν

P[|µ´ µ̂n| ď errn] « 99%for Φ

(´?

n errn /(1.2σ̂))= 0.005

by the Central Limit Theorem (CLT)

where σ̂2 is the sample variance. But the CLT is only an asymptotic result, and1.2σ̂ may be an overly optimistic upper bound on σ.

5/16


Berry-Esseen Stopping Rule for IID Monte Carlo

µ =

ż

Rdf (x)ν(dx)

µ̂n =1n

nÿ

i=1f (xi), xi

IID„ ν

P[|µ´ µ̂n| ď errn] ě 99%for Φ

(´?

n errn /(1.2σ̂nσ))

+∆n(´?

n errn /(1.2σ̂nσ), κmax) = 0.0025

by the Berry-Esseen Inequality

where σ̂2nσ is the sample variance using an independent sample from that used to

simulate the mean, and provided that kurt(f (X)) ď κmax(nσ) (H. et al., 2013;Jiang, 2016)

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Adaptive Low Discrepancy Sampling Cubature

µ =

ż

[0,1]df (x)dx

µ̂n =1n

nÿ

i=1f (xi), xi Sobol’ or lattice

|µ´ µ̂n| ď

∞ÿ

λ=1

∣∣∣pfλn

∣∣∣where pfκ are the coefficients of the FourierWalsh or complex exponential expansion of f

Assuming that the pfκ do not decay erratically, the discrete transform, trfn,κun´1κ=0,

may be used to bound the error reliably (H. and Jiménez Rugama, 2016; JiménezRugama and H., 2016; H. et al., 2017+):

|µ´ µ̂n| ď errn := C(n, `)2`´1ÿ

κ=2`´1

∣∣∣rfn,κ∣∣∣

6/16


Adaptive Low Discrepancy Sampling Cubature

µ =

ż

[0,1]df (x)dx

µ̂n =1n

nÿ

i=1f (xi), xi Sobol’ or lattice

|µ´ µ̂n| ď

∞ÿ

λ=1

∣∣∣pfλn

∣∣∣where pfκ are the coefficients of the FourierWalsh or complex exponential expansion of f

Assuming that the pfκ do not decay erratically, the discrete transform, trfn,κun´1κ=0,

may be used to bound the error reliably (H. and Jiménez Rugama, 2016; JiménezRugama and H., 2016; H. et al., 2017+):

|µ´ µ̂n| ď errn := C(n, `)2`´1ÿ

κ=2`´1

∣∣∣rfn,κ∣∣∣6/16


Asian Option Pricing

fair price =

ż

Rde´rT max

1d

dÿ

j=1Sj ´ K, 0

e´xTΣ´1x/2

(2π)d/2 |Σ|1/2 dx « $13.12

Sj = S0e(r´σ2/2)jT/d+σxj = stock price at time jT/d,

Σ =(

min(i, j)T/d)d

i,j=1

Worst 10% Worst 10%εa = 1E´4 Method % Accuracy n Time (s)

Sobol’ Sampling 100% 2.1E6 4.3Sobol’ Sampling w/ control variates 97% 1.0E6 2.1

The coefficient of the control variate for low discrepancy sampling is different thanfor IID Monte Carlo (H. et al., 2005; H. et al., 2017+)

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Bayesian Cubature—f Is Random

µ =

ż

Rdf (x)ν(dx) « µn =

nÿ

i=1wi f (xi)

Assume f „ GP(0, s2Cθ) (Diaconis, 1988;O’Hagan, 1991; Ritter, 2000; Rasmussen andGhahramani, 2003)

c0 =

ż

RdˆRdCθ(x, t)ν(dx)ν(dt)

c =

(ż

RdCθ(xi, t)ν(dt)

)n

i=1, C = (Cθ(xi, xj))

ni,j=1

Choosingw = (wi)ni=1 = C´1c is optimal

P[|µ´ µ̂n| ď errn] = 99% for errn = 2.58c

(c0 ´ cTC´1c)yTC´1y

n

where y = (f (xi))ni=1.

But, θ needs to be inferred (by MLE), and C´1 typicallyrequires O(n3) operations

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Bayesian Cubature—f Is Random

µ =

ż

Rdf (x)ν(dx) « µn =

nÿ

i=1wi f (xi)

Assume f „ GP(0, s2Cθ) (Diaconis, 1988;O’Hagan, 1991; Ritter, 2000; Rasmussen andGhahramani, 2003)

c0 =

ż

RdˆRdCθ(x, t)ν(dx)ν(dt)

c =

(ż

RdCθ(xi, t)ν(dt)

)n

i=1, C =

(Cθ(xi, xj)

)ni,j=1

Choosingw =(wi)n

i=1 = C´1c is optimal

P[|µ´ µ̂n| ď errn] = 99% for errn = 2.58c(

c0 ´ cTC´1c) yTC´1y

n

where y =(f (xi)

)ni=1. But, θ needs to be inferred (by MLE), and C´1 typically

requires O(n3) operations8/16


Maximum Likelihood Estimation of the Covariance Kernelf „ GP(0, s2Cθ), Cθ =

(Cθ(xi, xj)

)ni,j=1

y =(f (xi)

)ni=1, µ̂n = cT

θ̂C´1θ̂y

θ̂ = argminθ

yTC´1θ y

[det(C´1θ )]1/n

P[|µ´ µ̂n| ď errn] = 99% for errn =2.58?

n

b(c0,θ̂ ´ c

Tθ̂C´1θ̂cθ̂)yTC´1

θ̂y

There is a de-randomized interpretation of Bayesian cubature (H., 2017+)

f P Hilbert space w/ reproducing kernel Cθ and with best interpolant rfy

θ̂ = argminθ

yTC´1θ y

[det(C´1θ )]1/n

= argminθ

vol(

z P Rn :∥∥rfz‖θ ď ‖rfy‖θ

(

)|µ´ µ̂n| ď

2.58?

n

b

c0,θ̂ ´ cTθ̂C´1θ̂cθ̂

loooooooooomoooooooooon

‖error representer‖θ̂

b

yTC´1θ̂y

looooomooooon

‖rfy‖θ̂

if∥∥f ´rfy

∥∥θ̂ď

2.58∥∥rf∥∥

θ̂?n

9/16



(Cθ(xi, xj)

)ni,j=1

y =(f (xi)

)ni=1, µ̂n = cT

θ̂C´1θ̂y

θ̂ = argminθ

yTC´1θ y

[det(C´1θ )]1/n


n

b(c0,θ̂ ´ c


θ̂y



θ̂ = argminθ

yTC´1θ y

[det(C´1θ )]1/n

= argminθ

vol(


(

)

|µ´ µ̂n| ď2.58?

n

b




b

yTC´1θ̂y

looooomooooon

‖rfy‖θ̂

if∥∥f ´rfy

∥∥θ̂ď

2.58∥∥rf∥∥

θ̂?n

9/16



(Cθ(xi, xj)

)ni,j=1

y =(f (xi)

)ni=1, µ̂n = cT

θ̂C´1θ̂y

θ̂ = argminθ

yTC´1θ y

[det(C´1θ )]1/n


n

b(c0,θ̂ ´ c


θ̂y



θ̂ = argminθ

yTC´1θ y

[det(C´1θ )]1/n

= argminθ

vol(


(

)|µ´ µ̂n| ď

2.58?

n

b




b

yTC´1θ̂y

looooomooooon

‖rfy‖θ̂

if∥∥f ´rfy

∥∥θ̂ď

2.58∥∥rf∥∥

θ̂?n

9/16



ż


nÿ

i=1wif (xi)




Need errn based on data

(xi, f (xi))(n

i=1 such that

|µ´ µ̂n| ď errn (with high probability)

then increase n untilerrn ď εa




where ans˘ :=

"

supinf

*




10/16


More General Problemanswer(µ) =?, µ = E[f(X)] =

ż

Rdf(x)ν(dx) P Rp, pµn =

nÿ

i=1wif(xi)

Given εa, εr, choose n and {answer dependent on pµn to guarantee∣∣answer(µ)´ {answer∣∣ ď max

(εa, εr |answer(µ)|

)(with high probability)





where ans˘ :=

"

supinf

*




10/16


Problemanswer(µ) =?, µ = E[f(X)] =

ż

Rdf(x)ν(dx) P Rp, pµn =

nÿ

i=1wif(xi)

Given εa, εr, choose n and {answer dependent on pµn to guarantee∣∣answer(µ)´ {answer∣∣ ď max

(εa, εr |answer(µ)|

)(with high probability)





where ans˘ :=

"

supinf

*




10/16


Gaussian Probability

µ =

ż

[a,b]

exp(´ 1

2tTΣ´1t

)a

(2π)d det(Σ)dt Genz (1993)

=

ż

[0,1]d´1f (x)dx

For some typical choice of a, b, Σ, d = 3, εa = 0; µ « 0.6763Worst 10% Worst 10%

εr Method % Accuracy n Time (s)IID Monte Carlo 100% 8.1E4 1.8E´2

1E´2 Sobol’ Sampling 100% 1.0E3 5.1E´3Bayesian Lattice 100% 1.0E3 2.8E´3

IID Monte Carlo 100% 2.0E6 3.8E´11E´3 Sobol’ Sampling 100% 2.0E3 7.7E´3

Bayesian Lattice 100% 1.0E3 2.8E´3

1E´4 Sobol’ Sampling 100% 1.6E4 1.8E´2Bayesian Lattice 100% 8.2E3 1.4E´2

Bayesian lattice cubature uses covariance kernel C for which C is circulant,and operations on C require only O(n log(n)) operations 11/16


Sobol’ IndicesY = output(X), where X „ U[0, 1]d; Sobol’ Indexj(µ) describes how muchcoordinate j of input X influences output Y (Sobol’, 1990; 2001):

Sobol’ Indexj(µ) :=µ1

µ2 ´ µ23, j = 1, . . . , d

µ1 :=

ż

[0,1)2d[output(x)´ output(xj : x

1´j)]output(x 1)dxdx 1

µ2 :=

ż

[0,1)doutput(x)2 dx, µ3 :=

ż

[0,1)doutput(x)dx.

output(x) = ´x1 + x1x2 ´ x1x2x3 + ¨ ¨ ¨+ x1x2x3x4x5x6 (Bratley et al., 1992)

εa = 1E´3, εr = 0 j 1 2 3 4 5 6n 65 536 32 768 16 384 16 384 2 048 2 048

Sobol’ Indexj 0.6529 0.1791 0.0370 0.0133 0.0015 0.0015{Sobol’ Indexj 0.6528 0.1792 0.0363 0.0126 0.0010 0.0012

Sobol’ Indexj(pµn) 0.6492 0.1758 0.0308 0.0083 0.0018 0.003912/16


Summary

Knowing when to stop a simulation of the mean is not trivialThe Berry-Esseen inequality can tell us when to stop an IID simulationFourier analysis can tell us when to stop a low discrepancy simulationBayesian cubature can tell us when to stop a simulation if you can afford thecomputational costAll methods can be fooled by nasty functions, fRelative error tolerances and problems involving functions of integrals canbe handled

13/16

Thank you

Slides available at www.slideshare.net/fjhickernell/georgia-tech-2017-march-talk

https://www.slideshare.net/fjhickernell/georgia-tech-2017-march-talk

https://www.slideshare.net/fjhickernell/georgia-tech-2017-march-talk


References I

Bratley, P., B. L. Fox, and H. Niederreiter. 1992. Implementation and tests of low-discrepancysequences, ACM Trans. Model. Comput. Simul. 2, 195–213.

Cools, R. and D. Nuyens (eds.) 2016. Monte Carlo and quasi-Monte Carlo methods: MCQMC,Leuven, Belgium, April 2014, Springer Proceedings in Mathematics and Statistics, vol. 163,Springer-Verlag, Berlin.

Diaconis, P. 1988. Bayesian numerical analysis, Statistical decision theory and related topics IV,Papers from the 4th Purdue symp., West Lafayette, Indiana 1986, pp. 163–175.

Genz, A. 1993. Comparison of methods for the computation of multivariate normal probabilities,Computing Science and Statistics 25, 400–405.

H., F. J. 2017+. Error analysis of quasi-Monte Carlo methods. submitted for publication,arXiv:1702.01487.

H., F. J., L. Jiang, Y. Liu, and A. B. Owen. 2013. Guaranteed conservative fixed width confidenceintervals via Monte Carlo sampling, Monte Carlo and quasi-Monte Carlo methods 2012, pp. 105–128.

H., F. J. and Ll. A. Jiménez Rugama. 2016. Reliable adaptive cubature using digital sequences,Monte Carlo and quasi-Monte Carlo methods: MCQMC, Leuven, Belgium, April 2014, pp. 367–383.arXiv:1410.8615 [math.NA].

H., F. J., Ll. A. Jiménez Rugama, and D. Li. 2017+. Adaptive quasi-Monte Carlo methods. submittedfor publication, arXiv:1702.01491 [math.NA].

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References II

H., F. J., C. Lemieux, and A. B. Owen. 2005. Control variates for quasi-Monte Carlo, Statist. Sci. 20,1–31.

Jiang, L. 2016. Guaranteed adaptive Monte Carlo methods for estimating means of randomvariables, Ph.D. Thesis.

Jiménez Rugama, Ll. A. and F. J. H. 2016. Adaptive multidimensional integration based on rank-1lattices, Monte Carlo and quasi-Monte Carlo methods: MCQMC, Leuven, Belgium, April 2014,pp. 407–422. arXiv:1411.1966.

O’Hagan, A. 1991. Bayes-Hermite quadrature, J. Statist. Plann. Inference 29, 245–260.

Rasmussen, C. E. and Z. Ghahramani. 2003. Bayesian Monte Carlo, Advances in Neural InformationProcessing Systems, pp. 489–496.

Ritter, K. 2000. Average-case analysis of numerical problems, Lecture Notes in Mathematics,vol. 1733, Springer-Verlag, Berlin.

Sobol’, I. M. 1990. On sensitivity estimation for nonlinear mathematical models, Matem. Mod. 2,no. 1, 112–118.

. 2001. Global sensitivity indices for nonlinear mathematical models and their monte carloestimates, Math. Comput. Simul. 55, no. 1-3, 271–280.

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Georgia Tech 2017 March Talk

Engineering

Transcript of Georgia Tech 2017 March Talk