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3.7. ROBUSTNESS
341
...asecretweaponisavailabletodesignrobustdecisionrules
Thesecretweaponismax-mincontroltheory
Avalue-maximizingdecisionmakerenliststheaidofan(imaginary)value-minimizingmodel
choosertoconstructboundsonthevalueattainedbyagivendecisionruleunderdifferentmodels
ofthetransitiondynamics
Theoriginaldecisionmakerusesthoseboundstoconstructadecisionrulewithanassuredper-
formancelevel,nomatterwhichmodelactuallygovernsoutcomes
Note: Inreadingthislecture,pleasedon’tthinkthatourdecisionmakerisparanoidwhenhe
conductsaworst-caseanalysis. Bydesigningarulethatworkswellagainstaworst-case, , his
intentionistoconstructarulethatwillworkwellacrossasetofmodels.
SetsofModelsImplySetsOfValues Our“robust”decisionmakerwantstoknowhowwella
givenrulewillworkwhenhedoesnotknowasingletransitionlaw...
...hewantstoknowsetsofvaluesthatwillbeattainedbyagivendecisionruleFunderasetof
transitionlaws
Ultimately,hewantstodesignadecisionruleFthatshapesthesesetsofvaluesinwaysthathe
prefers
Withthisinmind,considerthefollowinggraph,whichrelatestoaparticulardecisionproblemto
beexplainedbelow
Thefigureshowsavalue-entropycorrespondenceforaparticulardecisionruleF
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
342
Theshadedsetisthegraphofthecorrespondence,whichmapsentropytoasetofvaluesassoci-
atedwithasetofmodelsthatsurroundthedecisionmaker’sapproximatingmodel
Here
• ValuereferstoasumofdiscountedrewardsobtainedbyapplyingthedecisionruleFwhen
thestatestartsatsomefixedinitialstatex
0
• Entropyisanonnegativenumberthatmeasuresthesizeofasetofmodelssurroundingthe
decisionmaker’sapproximatingmodel
– Entropyiszerowhenthesetincludesonlytheapproximatingmodel,indicatingthat
thedecisionmakercompletelytruststheapproximatingmodel
– Entropyisbigger,andthesetofsurroundingmodelsisbigger,thelessthedecision
makertruststheapproximatingmodel
Theshadedregionindicatesthatforallmodelshavingentropylessthanorequaltothenumber
onthehorizontalaxis,thevalueobtainedwillbesomewherewithintheindicatedsetofvalues
Nowlet’scomparesetsofvaluesassociatedwithtwodifferentdecisionrules,F
r
andF
b
Inthenextfigure,
• Theredsetshowsthevalue-entropycorrespondencefordecisionruleF
r
• Thebluesetshowsthevalue-entropycorrespondencefordecisionruleF
b
Thebluecorrespondenceisskinnierthantheredcorrespondence
ThisconveysthesenseinwhichthedecisionruleF
b
ismorerobustthanthedecisionruleF
r
• morerobustmeansthatthesetofvaluesislesssensitivetoincreasingmisspecificationasmea-
suredbyentropy
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
343
NoticethatthelessrobustruleF
r
promiseshighervaluesforsmallmisspecifications(smallen-
tropy)
(Butitismorefragileinthesensethatitismoresensitivetoperturbationsoftheapproximating
model)
Belowwe’llexplainindetailhowtoconstructthesesetsofvaluesforagivenF,butfornow...
Hereisahintaboutthesecretweaponswe’llusetoconstructthesesets
• We’llusesomeminproblemstoconstructthelowerbounds
• We’llusesomemaxproblemstoconstructtheupperbounds
WewillalsodescribehowtochooseFtoshapethesetsofvalues
Thiswillinvolvecraftingaskinniersetatthecostofalowerlevel(atleastforlowvaluesofentropy)
InspiringVideo Ifyouwanttounderstandmoreaboutwhyoneseriousquantitativeresearcher
isinterestedinthisapproach,werecommendLarsPeterHansen’sNobellecture
OtherReferences Ourdiscussioninthislectureisbasedon
• [HS00]
• [HS08]
TheModel
Forsimplicity,wepresentideasinthecontextofaclassofproblemswithlineartransitionlaws
andquadraticobjectivefunctions
TofitinwithourearlierlectureonLQcontrol,wewilltreatlossminimizationratherthanvalue
maximization
Tobegin,recalltheinfinitehorizonLQproblem,whereanagentchoosesasequenceofcontrolsfu
t
g
tominimize
¥
å
t=0
b
t
x
0
t
Rx
t
+u
0
t
Qu
t
 
(3.54)
subjecttothelinearlawofmotion
x
t+1
=Ax
t
+Bu
t
+Cw
t+1
,
t=0,1,2,...
(3.55)
Asbefore,
• x
t
isn1,Aisnn
• u
t
isk1,Bisnk
• w
t
isj1,Cisnj
• RisnnandQiskk
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
344
Herex
t
isthestate,u
t
isthecontrol,andw
t
isashockvector.
Fornowwetakefw
t
g:=fw
t
g
¥
t=1
tobedeterministic—asinglefixedsequence
Wealsoallowformodeluncertaintyonthepartoftheagentsolvingthisoptimizationproblem
Inparticular,theagenttakesw
t
=0forallt0asabenchmarkmodel,butadmitsthepossibility
thatthismodelmightbewrong
Asaconsequence,shealsoconsidersasetofalternativemodelsexpressedintermsofsequences
fw
t
gthatare“close”tothezerosequence
Sheseeksapolicythatwilldowellenoughforasetofalternativemodelswhosemembersare
pinneddownbysequencesfw
t
g
Soonwe’llquantifythequalityofamodelspecificationintermsofthemaximalsizeoftheexpres-
sion
å
¥
t=0
b
t+1
w
0
t+1
w
t+1
ConstructingMoreRobustPolicies
Ifouragenttakesfw
t
gasagivendeterministicsequence,then,drawingonintuitionfromearlier
lecturesondynamicprogramming,wecananticipateBellmanequationssuchas
J
1
(x)=min
u
fx
0
Rx+u
0
Qu+bJ
t
(Ax+Bu+Cw
t
)g
(HereJdependsontbecausethesequencefw
t
gisnotrecursive)
Ourtoolforstudyingrobustnessistoconstructarulethatworkswellevenifanadversesequence
fw
t
goccurs
Inourframework,“adverse”means“lossincreasing”
Aswe’llsee,thiswilleventuallyleadustoconstructtheBellmanequation
J(x)=min
u
max
w
fx
0
Rx+u
0
Qu+b[J(Ax+Bu+Cwqw
0
w]g
(3.56)
Noticethatwe’veaddedthepenaltyterm qw
0
w
Sincew
0
w=kwk
2
,thistermbecomesinfluentialwhenwmovesawayfromtheorigin
Thepenaltyparameterqcontrolshowmuchwepenalizethemaximizingagentfor“harming”the
minmizingagent
Byraisingqmoreandmore,wemoreandmorelimittheabilityofmaximizingagenttodistort
outcomesrelativetotheapproximatingmodel
Sobiggerqisimplicitlyassociatedwithsmallerdistortionsequencesfw
t
g
AnalyzingtheBellmanequation SowhatdoesJin(3.56)looklike?
AswiththeordinaryLQcontrolmodel,JtakestheformJ(x)=x
0
Pxforsomesymmetricpositive
definitematrixP
OneofourmaintaskswillbetoanalyzeandcomputethematrixP
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
345
Relatedtaskswillbetostudyassociatedfeedbackrulesforu
t
andw
t+1
First,usingmatrixcalculus,youwillbeabletoverifythat
max
w
f(Ax+Bu+Cw)
0
P(Ax+Bu+Cwqw
0
wg
=(Ax+Bu)
0
D(P)(Ax+Bu(3.57)
where
D(P):=P+PC(qI C
0
PC)
1
C
0
P
(3.58)
andIisajjidentitymatrix.Substitutingthisexpressionforthemaximuminto(3.56)yields
x
0
Px=min
u
fx
0
Rx+u
0
Qu+b(Ax+Bu)
0
D(P)(Ax+Bu)g
(3.59)
Usingsimilarmathematics,thesolutiontothisminimizationproblemisFxwhere:=
(Q+bB0D(P)B1bB0D(P)A
Substitutingthis minimizer r back k into (3.59) and workingthrough the algebra gives x
0
Px =
x
0B(D(
P))xforallx,or,equivalently,
P=B(D(P))
whereDistheoperatordefinedin(3.58)and
B(P):=R b
2
A
0
PB(Q+bB
0
PB)
1
B
0
PA+bA
0
PA
TheoperatorBisthestandard(i.e.,non-robust)LQBellmanoperator,andP=B(P)isthestan-
dardmatrixRiccatiequationcomingfromtheBellmanequation—seethisdiscussion
Undersomeregularityconditions(see[HS08]),theoperatorBDhasauniquepositivedefinite
fixedpoint,whichwedenotebelowby
ˆ
P
Arobustpolicy,indexedbyq,isu=
ˆ
Fxwhere
ˆ
F:=(Q+bB
0
D(
ˆ
P)B)
1
bB
0
D(
ˆ
P)A
(3.60)
Wealsodefine
ˆ
K:=(qI C
ˆ
PC)
1
C
ˆ
P(A B
ˆ
F)
(3.61)
Theinterpretationof
ˆ
Kisthatw
t+1
=
ˆ
Kx
t
ontheworst-casepathoffx
t
g,inthesensethatthis
vectoristhemaximizerof(3.57)evaluatedatthefixedruleu=
ˆ
Fx
Notethat
ˆ
P,
ˆ
F,
ˆ
Karealldeterminedbytheprimitivesandq
Notealsothatifqisverylarge,thenDisapproximatelyequaltotheidentitymapping
Hence,whenqislarge,
ˆ
Pand
ˆ
FareapproximatelyequaltotheirstandardLQvalues
Furthermore,whenqislarge,
ˆ
Kisapproximatelyequaltozero
Conversely,smallerqisassociatedwithgreaterfearofmodelmisspecification,andgreaterconcern
forrobustness
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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3.7. ROBUSTNESS
346
RobustnessasOutcomeofaTwo-PersonZero-SumGame
Whatwehavedoneabovecanbeinterpretedintermsofatwo-personzero-sumgameinwhich
ˆ
F,
ˆ
KareNashequilibriumobjects
Agent1isouroriginalagent,whoseekstominimizelossintheLQprogramwhileadmittingthe
possibilityofmisspecification
Agent2isanimaginarymalevolentplayer
Agent2’smalevolencehelpstheoriginalagenttocomputeboundsonhisvaluefunctionacrossa
setofmodels
Webeginwithagent2’sproblem
Agent2’sProblem
Agent2
1. knowsafixedpolicyFspecifyingthebehaviorofagent1,inthesensethatu
t
Fx
t
forall
t
2. respondsbychoosingashocksequencefw
t
gfromasetofpathssufficientlyclosetothe
benchmarksequencef0,0,0,...g
Anaturalwaytosay“sufficientlyclosetothezerosequence”istorestrictthesummedinner
product
å
¥
t=1
w
0
t
w
t
tobesmall
However,toobtainatime-invariantrecusiveformulation,itturnsouttobeconvenienttorestrict
adiscountedinnerproduct
¥
å
t=1
b
t
w
0
t
w
t
h
(3.62)
NowletFbeafixedpolicy,andletJ
F
(x
0
,w)bethepresent-valuecostofthatpolicygivensequence
w:=fw
t
gandinitialconditionx
0
2R
n
Substituting Fx
t
foru
t
in(3.54),thisvaluecanbewrittenas
J
F
(x
0
,w):=
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
(3.63)
where
x
t+1
=(A BF)x
t
+Cw
t+1
(3.64)
andtheinitialconditionx
0
isasspecifiedintheleftsideof(3.63)
Agent2chooseswtomaximizeagent1’slossJ
F
(x
0
,w)subjectto(3.62)
UsingaLagrangianformulation,wecanexpressthisproblemas
max
w
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
bq(w
0
t+1
w
t+1
h)
 
wherefx
t
gsatisfied(3.64)andqisaLagrangemultiplieronconstraint(3.62)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.7. ROBUSTNESS
347
Forthemoment,let’stakeqasfixed,allowingustodroptheconstantbqhtermintheobjective
function,andhencewritetheproblemas
max
w
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
bqw
0
t+1
w
t+1
 
or,equivalently,
min
w
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
+bqw
0
t+1
w
t+1
 
(3.65)
subjectto(3.64)
What’sstrikingaboutthisoptimizationproblemisthatitisonceagainanLQdiscounteddynamic
programmingproblem,withw=fw
t
gasthesequenceofcontrols
TheexpressionfortheoptimalpolicycanbefoundbyapplyingtheusualLQformula(seehere)
WedenoteitbyK(F,q),withtheinterpretationw
t+1
=K(F,q)x
t
Theremainingstepforagent2’sproblemistosetqtoenforcetheconstraint(3.62),whichcanbe
donebychoosingq=q
h
suchthat
b
¥
å
t=0
b
t
x
0
t
K(F,q
h
)
0
K(F,q
h
)x
t
=h
(3.66)
Herex
t
isgivenby(3.64)—whichinthiscasebecomesx
t+1
=(A BF+CK(F,q))x
t
UsingAgent2’sProblemtoConstructBoundsontheValueSets
TheLowerBound Definetheminimizedobjectontherightsideofproblem(3.65)asR
q
(x
0
,F).
Because“minimizersminimize”wehave
R
q
(x
0
,F)
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
 
+bq
¥
å
t=0
b
t
w
0
t+1
w
t+1
,
wherex
t+1
=(A BF+CK(F,q))x
t
andx
0
isagiveninitialcondition.
Thisinequalityinturnimpliestheinequality
R
q
(x
0
,Fqent
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
 
(3.67)
where
ent:=b
¥
å
t=0
b
t
w
0
t+1
w
t+1
Theleftsideofinequality(3.67)isastraightlinewithslope q
Technically,itisa“separatinghyperplane”
Ataparticularvalueofentropy,thelineistangenttothelowerboundofvaluesasafunctionof
entropy
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.7. ROBUSTNESS
348
Inparticular,thelowerboundontheleftsideof(3.67)isattainedwhen
ent=b
¥
å
t=0
b
t
x
0
t
K(F,q)
0
K(F,q)x
t
(3.68)
Toconstructthelowerboundonthesetofvaluesassociatedwithallperturbationswsatisfyingthe
entropyconstraint(3.62)atagivenentropylevel,weproceedasfollows:
• Foragivenq,solvetheminimizationproblem(3.65)
• ComputetheminimizerR
q
(x
0
,F)andtheassociatedentropyusing(3.68)
• ComputethelowerboundonthevaluefunctionR
q
(x
0
,Fqentandplotitagainstent
• Repeattheprecedingthreestepsforarangeofvaluesofqtotraceoutthelowerbound
Note: Thisproceduresweepsoutasetofseparatinghyperplanesindexedbydifferentvaluesfor
theLagrangemultiplierq
TheUpperBound Toconstructanupperboundweuseaverysimilarprocedure
Wesimplyreplacetheminimizationproblem(3.65)withthemaximizationproblem
V
˜
q
(x
0
,F)=max
w
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
b
˜
qw
0
t+1
w
t+1
 
(3.69)
wherenow
˜
q>0penalizesthechoiceofwwithlargerentropy.
(Noticethat
˜
qqinproblem(3.65))
Because“maximizersmaximize”wehave
V
˜
q
(x
0
,F)
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
 
b
˜
q
¥
å
t=0
b
t
w
0
t+1
w
t+1
whichinturnimpliestheinequality
V
˜
q
(x
0
,F)+
˜
qent
¥
å
t=0
b
t
x
0
t
(R+F
0
QF)x
t
 
(3.70)
where
entb
¥
å
t=0
b
t
w
0
t+1
w
t+1
Theleftsideofinequality(3.70)isastraightlinewithslope
˜
q
Theupperboundontheleftsideof(3.70)isattainedwhen
ent=b
¥
å
t=0
b
t
x
0
t
K(F,
˜
q)
0
K(F,
˜
q)x
t
(3.71)
Toconstructtheupperboundonthesetofvaluesassociatedallperturbationswwithagivenen-
tropyweproceedmuchaswedidforthelowerbound
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.7. ROBUSTNESS
349
• Foragiven
˜
q,solvethemaximizationproblem(3.69)
• ComputethemaximizerV
˜
q
(x
0
,F)andtheassociatedentropyusing(3.71)
• ComputetheupperboundonthevaluefunctionV
˜
q
(x
0
,F)+
˜
qentandplotitagainstent
• Repeattheprecedingthreestepsforarangeofvaluesof
˜
qtotraceouttheupperbound
Reshapingthesetofvalues NowintheinterestofreshapingthesesetsofvaluesbychoosingF,
weturntoagent1’sproblem
Agent1’sProblem
Nowweturntoagent1,whosolves
min
fu
t
g
¥
å
t=0
b
t
x
0
t
Rx
t
+u
0
t
Qu
t
bqw
0
t+1
w
t+1
 
(3.72)
wherefw
t+1
gsatisfiesw
t+1
=Kx
t
Inotherwords,agent1minimizes
¥
å
t=0
b
t
x
0
t
(R bqK
0
K)x
t
+u
0
t
Qu
t
 
(3.73)
subjectto
x
t+1
=(A+CK)x
t
+Bu
t
(3.74)
Onceagain,theexpressionfortheoptimalpolicycanbefoundhere—wedenoteitby
˜
F
NashEquilibrium
Clearlythe
˜
FwehaveobtaineddependsonK,which,inagent2’sproblem,
dependedonaninitialpolicyF
Holdingallotherparametersfixed,wecanrepresentthisrelationshipasamappingF,where
˜
F=F(K(F,q))
ThemapF7!F(K(F,q))correspondstoasituationinwhich
1. agent1usesanarbitraryinitialpolicyF
2. agent2bestrespondstoagent1bychoosingK(F,q)
3. agent1bestrespondstoagent2bychoosing
˜
F=F(K(F,q))
Asyoumayhavealreadyguessed,therobustpolicy
ˆ
Fdefinedin(3.60)isafixedpointofthe
mappingF
Inparticular,foranygivenq,
1. K(
ˆ
F,q)=
ˆ
K,where
ˆ
Kisasgivenin(3.61)
2. F(
ˆ
K)=
ˆ
F
Asketchoftheproofisgivenintheappendix
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
3.7. ROBUSTNESS
350
TheStochasticCase
Nowweturntothestochasticcase,wherethesequencefw
t
gistreatedasaniidsequenceof
randomvectors
Inthissetting,wesupposethatouragentisuncertainabouttheconditionalprobabilitydistribution
ofw
t+1
TheagenttakesthestandardnormaldistributionN(0,I)asthebaselineconditionaldistribution,
whileadmittingthepossibilitythatother“nearby”distributionsprevail
Thesealternativeconditionaldistributionsofw
t+1
mightdependnonlinearlyonthehistoryx
s
,s
t
Toimplementthisidea,weneedanotionofwhatitmeansforonedistributiontobenearanother
one
Hereweadoptaveryusefulmeasureofclosenessfordistributionsknownastherelativeentropy,
orKullback-Leiblerdivergence
Fordensitiesp,q,theKullback-Leiblerdivergenceofqfrompisdefinedas
D
KL
(p,q):=
Z
ln
p(x)
q(x)
p(x)dx
Usingthisnotation,wereplace(3.56)withthestochasticanalogue
J(x)=min
u
max
y2P
x
0
Rx+u
0
Qu+b
Z
J(Ax+Bu+Cw)y(dwqD
KL
(y,f)

(3.75)
HerePrepresentsthesetofalldensitiesonR
n
andfisthebenchmarkdistributionN(0,I)
Thedistributionfischosenastheleastdesirableconditionaldistributionintermsofnextperiod
outcomes,whiletakingintoaccountthepenaltytermqD
KL
(y,f)
Thispenaltytermplaysaroleanalogoustotheoneplayedbythedeterministicpenaltyqw
0
win
(3.56),sinceitdiscourageslargedeviationsfromthebenchmark
SolvingtheModel Themaximizationproblemin(3.75)appearshighlynontrivial—afterall,
wearemaximizingoveraninfinitedimensionalspaceconsistingoftheentiresetofdensities
However,itturnsoutthatthesolutionistractable,andinfactalsofallswithintheclassofnormal
distributions
First,wenotethatJhastheformJ(x)=x
0
Px+dforsomepositivedefinitematrixPandconstant
realnumberd
Moreover,itturnsoutthatif(q
1
C
0
PC)
1
isnonsingular,then
max
y2P
Z
(Ax+Bu+Cw)
0
P(Ax+Bu+Cw)y(dwqD
KL
(y,f)
=(Ax+Bu)
0
D(P)(Ax+Bu)+k(q,P(3.76)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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