Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses...

InterpolatingBetweenStochasticandWorst-caseOptimization

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Risk– calculablegamble

Handlinginputrisk:Averagecaseanalysis

Uncertainty– incalculableunknown

'Symbol of Uncertainty', John Gilbert

Handlinginputuncertainty:Worst-casecompetitiveanalysis

Commoncomplaints

• Competitiveanalysisistoopessimistic

• Stochasticanalysisneedtoodetailedinformation

PossibleResponses• Relaxpessimism:Addrestrictionstoworst-caseinputscenariostoreduceimpact

• Temperoptimism:Makestochasticanalysisrobusttochangesininputdistribution

• “Haveitall”:Buildalgorithmsthatachievethe‘bestofbothworlds’andworkforbothkindsofinputs

• InterpolateModelANDPerformance:Buildhybridinputmodelsinterpolatingstochasticandadversarialinputstoderivenewalgorithmsthatdeterioratesmoothlyinperformance

Outline

• Motivation• Relatedwork*(incompleteandrepresentative)• AttemptatSpecificModel

RelaxPessimism:Restrictclassesofinputs• Betteralgorithmsforboundedtree-widthinputs

• Betterapproximationalgorithmsforboundedgenusgraphs

RelaxPessimism:SmoothedAnalysis• [Spielman-Teng 2001]Usesdistributionaldisturbanceovergivenworst-caseinput,tosmoothout(expected)performance

• Resultingrunningtimesarepolynomialininputandperturbationsize

• Goal:Explain“unreasonableeffectiveness”ofpopularalgorithms(likeSimplexforLP)

Temperoptimism:StochasticProgrammingVariants

Stochastic Combinatorial Optimization with controllable risk aversion levelSo, Zhang & Ye, APPROX ‘06, Math of OR

TemperOptimism:DistributionallyRobustOptimizationAddambiguitytorisk• Newsvendorproblemwithonlymeanandvariance,notdistribution[Scarf1958]

• DistributionallyRobustStochasticOptimization[Zackova,Dupacova,Bertsimas+,Sim+,Ye+,…]

TemperOptimism:CorrelationRobustness

Correlation gap: Loss in performance by ignoring correlations and assuming independence

Haveitall:UniversalApproximationsOnesinglesolutionwhoseinducedsolutionisgoodforanysubsetinput [Bartholdi-Platzman,Jia+]

Haveitall:Robust/IncrementalSolutions• Matchings[Hassin-Rubinstein,SIDMA2002]Givenaweightedgraphfindasinglematchingandanorderingofitsedgessuchthatforeveryk,theprefixofkedgesisnearoptimalmaximumweightmatchingofsizek

• Metrick-median[Mettu-Plaxton,SICOMP2003]Findanorderingoffacilitiessuchthatforeveryk,theprefixofkfacilitiesisanearoptimalk-median

Haveitall/Bestofboth:Trade-offtwoguaranteesTrytofindthebestpossibleratioswithrespecttothe“pessimistic”and“optimistic”extremes

Online Algorithms with Uncertain InformationMahdian, Nazerzadeh & Saberi, EC ’07, TALG 2012

Noteperformanceisw.r.t.thatofthegivenalgorithmsbutnotthe“optimal”solutions

Bestofboth:OnlineResourceAllocation

Bestofboth:AdWords

Maintain(1– 1/e)-worstcaseguaranteeintheworstcaseanddoquantifiablybetterforrandomarrivalmodel“Inthispaperwedesignalgorithmsthatachieveacompetitiveratiobetterthan1−1/eonaverage,whilepreservinganearlyoptimalworstcasecompetitiveratio.Ideally,wewanttoachievethebestofbothworlds,i.e,todesignanalgorithmwiththeoptimalcompetitiveratioinboththeadversarialandrandomarrivalmodels.Weachievethisforunweightedgraphs,butshowthatitisnotpossibleforweightedgraphs.”

Simultaneous Approximations for Adversarial and Stochastic Online Budgeted AllocationMirrokni, Oveis-Gharan & Zadimoghaddam, SODA ‘12

Bestofboth:Balancedguaranteesforbandits

AudienceParticipation:OtherRelatedWork?

RecallPossibleResponses• Relaxpessimism:Addrestrictionstoworst-caseinputscenariostoreduceimpact

• Temperoptimism:Makestochasticanalysisrobusttoslightchangesininputdistribution

• “Haveitall”:Buildalgorithmsthatachievethe‘bestofbothworlds’andworkforbothkindsofinputs

• InterpolateModelANDPerformance:Buildhybridinputmodelsinterpolatingstochasticandadversarialinputstoderivenewalgorithmsthatdeterioratesmoothlyinperformance

Proposal:InterpolateModelsANDPerformance• Modelinterpolation:Inputmodelshouldallowsmoothinterpolationbetweenstochasticoptimismandworst-casepessimism

• Performanceinterpolation:Algorithmshouldhaveperformanceratiothatinterpolatessmoothlybetweenthebetterguaranteeforstochasticinputsandtheworseguaranteefortheworst-case

Restofthetalk[JointwithGuyBlelloch,Kedar Dhamdhere &Suporn Pongnumkul,Summer2004]

• ListUpdateProblem:Competitive&AverageCaseAnalysis

• AHybridOnlineModel• SetupandResultsfromapreliminaryexperiment• Conjecture

ListUpdateProblemSelf-organizingsequentialsearch

• Unsortedlist

• Receivedasequenceofrequests

• Costofaccessingtheith elementofthelistisi.Afteraccess,canmoveitanywhereaheadinthelistforfree

• Cantransposeanypairofadjacentelementsatunitcost

y w z x v uL:

y w z x v uL:

ListUpdateExample

Action Cost Total Cost

y w z x v uL:

ListUpdateExample

Action Cost Total CostAccess x

y w z x v uL:

ListUpdateExample

Action Cost Total CostAccess x 4 4

y w z x v uL:

ListUpdateExample

Action Cost Total CostAccess x 4 4Access v

y w z x v uL:

ListUpdateExample

Action Cost Total CostAccess x 4 4Access v 5 9

y w z x v uL:

ListUpdateExample


Move v forward to before w

y v w z x uL:

ListUpdateExample


Move v forward to before w 0 9

y v w z x uL:

ListUpdateExample


Move v forward to before w 0 9Access y

y v w z x uL:

ListUpdateExample


Move v forward to before w 0 9Access y 1 10

y v w z x uL:

ListUpdateExample


Move v forward to before w 0 9Access y 1 10Access v

y v w z x uL:

ListUpdateExample


Move v forward to before w 0 9Access y 1 10Access v 2 12

y v z w x uL:

ListUpdateExample



Transpose w and z

y v w z x uL:

ListUpdateExample



Transpose w and z 1 13

AverageCaseAnalysis

• Assumeeachrequestcomesfromafixedprobabilitydistribution,independentofpreviousrequests.Supposetheith itemhasprobabilitypi.Designalgorithmstominimizetheexpectedcost.

• Optimalstrategyistokeepthelistsortedinnon-increasingorderofpi.

STAT=StaticList

• Listissortedinnon-increasingorderoftheprobabilities

• Nevermovesanything

• Goodforwhenwehaveagoodestimateoftheprobabilitydistribution.

FC:FrequencyCount

• Ifprobabilitydistributionisunknown,estimateitusingfrequencycounts

• Keeplistsortedaccordingtocounts

CompetitiveAnalysis

• Definition: Ananalysisinwhichtheperformanceofanonlinealgorithmiscomparedtothebestthatcouldhavebeenachievedifalltheinputshadbeenknowninadvance.

CompetitiveRatio

A:Our online algorithm

CA(s)

OPT:Optimal Offline

algorithm

COPT(s)

A is c-competitive if $ aCA(s) ≤ c COPT(s) + a

for all request sequences s

Move-to-Front(MTF)

[Sleator,Tarjan,CACM1985]Whenanelementisaccessed,moveittothefrontofthelist.

• Theorem:MTFhascompetitiveratio2againstoptimalofflinealgorithm.

PerformanceComparison

• E(FC)/E(STAT) =1[Rivest CACM1976]• E(MTF)/E(STAT)=𝜋/2 =1.58…[Chung,Hajela,SeymourSTOC85]

TStimestampalgorithm[AlbersSODA95,Albers&MitzenmacherAlgorithmica 1998]Ourworkismotivatedbythegoaltopresentauniversalalgorithmthatachievesagoodcompetitiveratiobutalsoperformsespeciallywellwhenrequestsaregeneratedbydistributions.(“Haveitall”)TS:Inserttherequesteditem,sayx,infrontofthefirstiteminthelistthathasbeenrequestedatmostoncesincethelastrequesttox.Ifxhasnotbeenrequestedsofar,leavethepositionofxunchanged.Theorem TSis2-competitive

E(TS)/E(STAT)=1.5whileforsomedistribution,E(MTF)/E(STAT)>1.57

NewHybridInterpolatingModel• Assumeafixedprobabilitydistribution

• Foreachrequest,withprobability,letadversarychangetherequest.

• ó AverageCaseAnalysis

• ó CompetitiveAnalysis

• ó Knownprobabilitydistributionwithuncertainty.

),...,,( 21 npppp =!

e0=e1=e10 << e e

Desiderata:InterpolatingAlgorithmforHybridModel

• Takesasinputestimatesofp ande

• Matchesbestaveragecaseperformancewhene islow,andmatchesbestcompetitiveratiowhene ishigh,andinterpolatesinbetween.

CandidateAlgorithm:Move-From-Back-Epsilon• Listinitiallysortedinnon-increasingorderofprobabilities.

• Whenanelementx isaccessed,promoteitpastothersthathaveprobabilityuptopx +e.

y w z x v uL:

MFBEExample(e =0.2)

Request Cost Total Cost

s: v y z yProb py = 0.5 pw = 0.2 pz = 0.15 px = 0.1 pv = 0.03 pu = 0.02

y w z x v uL:

MFBEExample(e =0.2)

Request Cost Total Cost

s: x v y vProb py = 0.5 pw = 0.2 pz = 0.15 px = 0.1 pv = 0.03 pu = 0.02

y w z x v uL:

MFBEExample(e =0.2)

Request Cost Total CostAccess x (and move to before w)

s: x v y vProb py = 0.5 pw = 0.2 pz = 0.15 px = 0.1 pv = 0.03 pu = 0.02

y x w z v uL:

MFBEExample(e =0.2)

Request Cost Total CostAccess x (and move to before w) 4 4

s: x v y vProb py = 0.5 px = 0.1 pw = 0.2 pz = 0.15 pv = 0.03 pu = 0.02

y x w z v uL:

MFBEExample(e =0.2)

Request Cost Total CostAccess x (and move to before w) 4 4Access v (and move to before x)

s: x v y vProb py = 0.5 px = 0.1 pw = 0.2 pz = 0.15 pv = 0.03 pu = 0.02

y v x w z uL:

MFBEExample(e =0.2)

Request Cost Total CostAccess x (and move to before w) 4 4Access v (and move to before x) 5 9

s: x v y vProb py = 0.5 pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02

y v x w z uL:

MFBEExample(e =0.2)


Access y


y v x w z uL:

MFBEExample(e =0.2)


Access y 1 10


y v x w z uL:

MFBEExample(e =0.2)


Access y 1 10Access v


y v x w z uL:

MFBEExample(e =0.2)


Access y 1 10Access v 2 12


DifficultiesinProvingPropertiesofMFBE• MustcomparewithOPTratherthanSTAT• OPTcanbecomputedbyDynamicprogramming

• Trivialway=O((n!)2m)• Improvement=O((2n)(n!)m)

[Reingold,Westbrook,1996]

• HardtoderiveusefulpropertiesofOPT

Experiments(Pongnumkul ALADDINREU2004)• Motivation:Toseethebehaviorofalgorithmsinourhybridmodel.

• Measurement:Wemeasuretheperformanceofanonlinealgorithmbytheaveragecompetitiveratio.

OurExperiment

• Variablesinourexperiment• TypeofListUpdateAlgorithm(MTF,STAT,MFBE)

• TypeofProbabilityDistribution• TypeofAdversary

• Epsilon:e

OurExperiment

• Wegeneratearequestsequenceoflength100,withachosenprobabilitydistribution

• Then,withprobabilitye changetherequestsequence adverserially

OurExperiment• Recordthecostincurredbytheonlinealgorithm=CostA(s)

• UseDynamicProgrammingtofindoptimumcostofthatrequestsequence=CostOPT(s).

• CompetitiveRatio=CostA(s)/CostOPT(s)

• Repeatthis100timestofindtheaveragecompetitiveratio.

Distribution

• GeometricDistribution:• P[i]/ 1/2i

• UniformDistribution:• P[i]=1/n,n=lengthofthelist

• ZipfianDistribution(Zipf(2)):• P[i]/ 1/i2

CruelAdversary

• Thisisanadaptiveadversary

• Looksatthecurrentlistandrequestthelastiteminthelist.

y w z x v uL:

CruelAdversary,GeometricDistribution,n=6

Average Competitive Ratio

1

1.5

2

2.5

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3.5

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4.5

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6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Epsilon

Aver

age

Com

petit

ive

Rat

io

STAT

MFBE

MTF

CruelAdversary,GeometricDistribution,n=6(zoomedin)


1

1.1

1.2

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1.9

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Epsilon

Aver

age

Com

petit

ive

Rat

io

STAT

MFBE

MTF

OtherAdversaries

• Geometricadversarychooseselementsrandomly,accordingtothegeometricdistributiononthereversedSTATorder

• Uniformadversaryrequestselementsfromthelistuniformlyatrandom

• ObliviousAdversarydoesn’tlookatthecurrentlist

UniformAdversary,Zipfian2Distribution,n=6


1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Epsilon

Aver

age

Com

petit

ive

Rat

io

STAT

MFBE

MTF

ReversedGeometricAdversary,GeometricDistribution,n=6


1

1.1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Epsilon

Aver

age

Com

petit

ive

Rat

io

STAT

MFBE

MTF

Observations

• Theperformanceofanyalgorithminthishybridmodeldependsheavilyonthetypeofadversary.

• MFBEseemsbetterthantheworseofSTATandMTF.

Conjecture

(Nottheworst)TheaveragecompetitiveratioofMFBEisdominatedbythemaximumoftheaveragecompetitiveratiosofSTATandMTF.

• (Bestofall)Whatwewantedbutprobablynottrue:Avg competitiveratioofMFBEisatmosttheminimumoftheaveragecompetitiveratiosofSTATandMTF.

InterpolateModelANDPerformance:Buildhybridinputmodelsinterpolatingstochasticandadversarialinputstoderivenewalgorithmsthatdeterioratesmoothlyinperformance• Prove/disprove“MFBEisnottheworst”• Findasettingwherethehybridmodelinterpolatesstochasticandworst-caseinputs,andthealgorithminterpolatestheperformanceofaveragecaseandworstcasealgorithms

• Aretheremoreeffectiveapproachestotrade-offstochasticoptimismandworst-casepessimism?

Summary

Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses...

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