Higher-order Confidence Intervals for Stochastic Programming using Bootstrapping

Cosmin G. Petra

Joint work with Mihai Anitescu

Mathematics and Computer Science DivisionArgonne National Laboratory

petra@mcs.anl.gov

INFORMS ANNUAL MEETING 2012

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Outline

Confidence intervals

Motivation– statistical inference for the stochastic optimization of power grid

Our statistical estimator for the optimal value

Bootstrapping

Second-order bootstrapped confidence intervals

Numerical example

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Confidence intervals (CIs) for a statistic

* ,( [ ])LP U

** Want an interval [L,U] where resides with high probability

Need the knowledge of the probability distribution Example: Confidence intervals for the mean of Gaussian (normal) random

variable

Normal distribution, also called Gaussian or "bell curve“ distribution. Image source: Wikipedia.

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Approximating CIs

* [ ( )]f X E 1

1ˆ ( )N

ii

f X f XN

In many cases the distribution function is not known.

Such intervals are approximated based on the central limit theorem (CLT)

Normal approximation for equal-tailed 95% CI

Notation

*

1/2

ˆ(0,1), as (by CLT)

ˆN

N

N

1/2 1/2/2 /2

ˆ ˆˆ ˆ, ][ ,N NU N z NL z

1( ) is the normal quantilez

x) is t( ) ( ( he 0,1 normal cdf)x P N

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Optimal value in stochastic programming

Monotonically shrinking negative bias: Consistency

Arbitrary slow convergence

Non-normal bias

* min ( ) : ( , )f x F x Ex X 1

1min ( ) , ) : (N

N N ii

f x F xN

x X

Sample average approximation (SAA)Stochastic programming (SP) problem

Properties*

1NN E E*

N

* 1/2( )N O N E1/2 *( ) inf ( )N x S

Y xN

D

Stochastic unit commitment with wind power

Wind Forecast – WRF(Weather Research and Forecasting) Model– Real-time grid-nested 24h simulation – 30 samples require 1h on 500 CPUs (Jazz@Argonne)

6

1min COST

s.t. , ,

, ,

ramping constr., min. up/down constr.

wind

wind

p u dsjk jk jk

s j ks

sjk kj

windsjk

j

wik ksj

ndsk

jjk

j

c c cN

p D s k

p D R s k

p

p

S N T

N

N

N

N

S T

S T

Slide courtesy of V. Zavala & E. Constantinescu

Wind farmThermal generator

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The specific of stochastic optimization of energy systems

SAA

discrete continuous

Sampling

Statistical inference

uncertainty

is expensive

Only a small number of samples are available.

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Standard methodology for stochastic programming – Linderoth, Shapiro, Wright (2004)

Lower bound CI CI for based on M batches of N samples

Upper bound CI CI for (obtained similarly)

Needs a relatively large number of samples (2MN)

First-order correct and therefore unreliable for small number of samples

1

1 ( , )N

ii

F xN

1/2 1/2/

1/

12 2

1 1,M M

m

m m m mN N N N

m

M z MM M

z

N

Correctness of a CI – order k if* /2[ , ])( )( k

N NL U O NP

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Our approach for SP with low-size samples

1. Novel estimator

2. Bootstrapping

Converges one order faster than– Excepting for a set whose measure converges exponentially to 0.

N

Allows the construction of reliable CIs in the low-size samples situation. Bootstrap CIs are second-order correct

M. Anitescu, C. Petra: “Higher-Order Confidence Intervals for Stochastic Programming using Bootstrapping”, submitted to Math. Prog.

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The estimator

12* ( , ) ( , ) ) ( , )1( )

2 0(

( 0) 0N

TN N N N N T N N

N

L x L x L xF x

J xJ x

E E EE[

L is the Lagrangian of SP and J is the Jacobian of the constraints is the solution of the SAA problem – obtained using N samples Intended for nonlinear recourse terms

Theorem 1: (Anitescu & P.) Under some regularity and smoothness conditions

Proof: based on the theory of large deviations.

CIs constructed for are based on a second batch of N samples.

A total of 2N sample needed when using bootstrapping

Nx

2 ( )3/2 * *1 3| ) ( ) ( (( )| )ra cr N xP p r NN e òò ò ò

*

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Bootstrapping – a textbook example

1. 1930 population = 1920 population X mean of the ratios

2. needs the distribution of the ratios - not enough samples -> Bootstrapping– Sample the existing samples (with replacement)– For each sample compute the mean– Bootstrapping distribution is obtained– Build CIs based on the bootstrapping distribution

Histogram for the ratio of 1930 and 1920 populationsfor N=49 US cities

“Bootstrapped” distribution clearly not a GaussianBootstrap CIs outperform normal CIs.

US population known in 1920. 1930 population of 49 cities knownWant 1. estimation of the 1930 population2. CIs for the estimation

Solution

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The methodology of bootstrapping

BCa (bias corrected and accelerated) confidence intervals– second-order correct– the method of choice when an accurate estimate of the variance is not available

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What does bootstrapping do?

Edgeworth expansions for cdfs

Bootstrapping accounts also for the second term in the expansion

The quantiles are also second-order correct (Cornish-Fisher inverse expansions)

(Some) Bootstrapped CIs are second-order correct

1/2 ( 1)/21( ) ( ) (( ) ( ) ( ) ( )) k

kx N p x x p x x O NH x

1/2 1 1/21 1 1

ˆ ˆ ˆ( ) ( ) ( ) ( ), with( )) (pH x N p x x O N p Ox p N

Reference: Peter Hall, “The Bootstrap and Edgeworth Expansion”, 1994.

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Bootstrapping the estimator

* 1,

ˆ ) ( )( .aBCaP J O N

Theorem 2: (Anitescu & P.) Let be a second order bootstrapping confidence interval for . Then for any

,ˆBCaJ * 0,a

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Numerical order of correctness

Correctness order 0.32

Correctness order 0.82

Correctness order 1.14

Correctness order 2.11

Observed order of correctness*

*

4 4

1

1min ( ) : ( ) min ( ) : ( )

~ (0,

1

)

NN

u ii

f x E x u f x x uN

u U

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Coverage for small number of samples

*

1 2

1 2

2 21 2 1 2 1 2 , 1 2 1 2

2 21 2 1 2 , 1 2 1 2

1 1 1 2

2 2 1 2

5 5min ( , ) : 7.4 2.4 ( , ; , )2 2

1( , ; , ) min ( ) 2 22

s.t. 20 (2 6 )10 (3 3 )

u u

y y

f x x x x x x E Q x x u u

Q x x u u y y y y

y u x xy u x x

1

2

~ ( 10,10)~ ( 5, 5)

u Uu U

Concluding remarks and future work

Proposed and analyzed a novel statistical estimator for the optimal solution of nonlinear stochastic optimization

Almost second order correct confidence intervals using bootstrapping

Theoretical properties confirmed by numerical testing

Some assumptions are rather strict and can/should be relaxed

Parallelization of the CI computations for large problems needed

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Thank you for your attention!

Questions?

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Bootstrapping - theory

Edgeworth expansions for pdfs1/2 ( 1)/2

1( ) ( ) (( ) ( ) ( ) ( )) kkx N p x x p x x O NH x

1/2 ( 1)/211 1( ) ( ) ( )k

kz N p z p z O Ny

1/21 1ˆ ( )k kp p O N

Bootstrapping also accounts for the second term of in the expansion.

Cornish-Fisher expansion for quantiles (inverting Edgeworth expansion)

Bootstrapped quantiles possess similar expansion

But

(Some) Bootstrap CIs are second-order correct (Hall’s book is really detailed on this)

1/2 ( 1)/211 1ˆ ˆ ˆ( ) ( ) ( ),k

k py z N p z p z O N