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Risk– calculablegamble
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Handlinginputrisk:Averagecaseanalysis
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Uncertainty– incalculableunknown
'Symbol of Uncertainty', John Gilbert
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Handlinginputuncertainty:Worst-casecompetitiveanalysis
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Commoncomplaints
• Competitiveanalysisistoopessimistic
• Stochasticanalysisneedtoodetailedinformation
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PossibleResponses• Relaxpessimism:Addrestrictionstoworst-caseinputscenariostoreduceimpact
• Temperoptimism:Makestochasticanalysisrobusttochangesininputdistribution
• “Haveitall”:Buildalgorithmsthatachievethe‘bestofbothworlds’andworkforbothkindsofinputs
• InterpolateModelANDPerformance:Buildhybridinputmodelsinterpolatingstochasticandadversarialinputstoderivenewalgorithmsthatdeterioratesmoothlyinperformance
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Outline
• Motivation• Relatedwork*(incompleteandrepresentative)• AttemptatSpecificModel
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RelaxPessimism:Restrictclassesofinputs• Betteralgorithmsforboundedtree-widthinputs
• Betterapproximationalgorithmsforboundedgenusgraphs
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RelaxPessimism:SmoothedAnalysis• [Spielman-Teng 2001]Usesdistributionaldisturbanceovergivenworst-caseinput,tosmoothout(expected)performance
• Resultingrunningtimesarepolynomialininputandperturbationsize
• Goal:Explain“unreasonableeffectiveness”ofpopularalgorithms(likeSimplexforLP)
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Temperoptimism:StochasticProgrammingVariants
Stochastic Combinatorial Optimization with controllable risk aversion levelSo, Zhang & Ye, APPROX ‘06, Math of OR
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TemperOptimism:DistributionallyRobustOptimizationAddambiguitytorisk• Newsvendorproblemwithonlymeanandvariance,notdistribution[Scarf1958]
• DistributionallyRobustStochasticOptimization[Zackova,Dupacova,Bertsimas+,Sim+,Ye+,…]
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TemperOptimism:CorrelationRobustness
Correlation gap: Loss in performance by ignoring correlations and assuming independence
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Haveitall:UniversalApproximationsOnesinglesolutionwhoseinducedsolutionisgoodforanysubsetinput [Bartholdi-Platzman,Jia+]
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Haveitall:Robust/IncrementalSolutions• Matchings[Hassin-Rubinstein,SIDMA2002]Givenaweightedgraphfindasinglematchingandanorderingofitsedgessuchthatforeveryk,theprefixofkedgesisnearoptimalmaximumweightmatchingofsizek
• Metrick-median[Mettu-Plaxton,SICOMP2003]Findanorderingoffacilitiessuchthatforeveryk,theprefixofkfacilitiesisanearoptimalk-median
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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
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Bestofboth:OnlineResourceAllocation
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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
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Bestofboth:Balancedguaranteesforbandits
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AudienceParticipation:OtherRelatedWork?
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RecallPossibleResponses• Relaxpessimism:Addrestrictionstoworst-caseinputscenariostoreduceimpact
• Temperoptimism:Makestochasticanalysisrobusttoslightchangesininputdistribution
• “Haveitall”:Buildalgorithmsthatachievethe‘bestofbothworlds’andworkforbothkindsofinputs
• InterpolateModelANDPerformance:Buildhybridinputmodelsinterpolatingstochasticandadversarialinputstoderivenewalgorithmsthatdeterioratesmoothlyinperformance
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Proposal:InterpolateModelsANDPerformance• Modelinterpolation:Inputmodelshouldallowsmoothinterpolationbetweenstochasticoptimismandworst-casepessimism
• Performanceinterpolation:Algorithmshouldhaveperformanceratiothatinterpolatessmoothlybetweenthebetterguaranteeforstochasticinputsandtheworseguaranteefortheworst-case
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Restofthetalk[JointwithGuyBlelloch,Kedar Dhamdhere &Suporn Pongnumkul,Summer2004]
• ListUpdateProblem:Competitive&AverageCaseAnalysis
• AHybridOnlineModel• SetupandResultsfromapreliminaryexperiment• Conjecture
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ListUpdateProblemSelf-organizingsequentialsearch
• Unsortedlist
• Receivedasequenceofrequests
• Costofaccessingtheith elementofthelistisi.Afteraccess,canmoveitanywhereaheadinthelistforfree
• Cantransposeanypairofadjacentelementsatunitcost
y w z x v uL:
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y w z x v uL:
ListUpdateExample
Action Cost Total Cost
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y w z x v uL:
ListUpdateExample
Action Cost Total CostAccess x
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y w z x v uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4
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y w z x v uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v
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y w z x v uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
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y w z x v uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w
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y v w z x uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w 0 9
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y v w z x uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w 0 9Access y
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y v w z x uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w 0 9Access y 1 10
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y v w z x uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w 0 9Access y 1 10Access v
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y v w z x uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w 0 9Access y 1 10Access v 2 12
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y v z w x uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w 0 9Access y 1 10Access v 2 12
Transpose w and z
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y v w z x uL:
ListUpdateExample
Action Cost Total CostAccess x 4 4Access v 5 9
Move v forward to before w 0 9Access y 1 10Access v 2 12
Transpose w and z 1 13
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AverageCaseAnalysis
• Assumeeachrequestcomesfromafixedprobabilitydistribution,independentofpreviousrequests.Supposetheith itemhasprobabilitypi.Designalgorithmstominimizetheexpectedcost.
• Optimalstrategyistokeepthelistsortedinnon-increasingorderofpi.
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STAT=StaticList
• Listissortedinnon-increasingorderoftheprobabilities
• Nevermovesanything
• Goodforwhenwehaveagoodestimateoftheprobabilitydistribution.
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FC:FrequencyCount
• Ifprobabilitydistributionisunknown,estimateitusingfrequencycounts
• Keeplistsortedaccordingtocounts
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CompetitiveAnalysis
• Definition: Ananalysisinwhichtheperformanceofanonlinealgorithmiscomparedtothebestthatcouldhavebeenachievedifalltheinputshadbeenknowninadvance.
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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
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Move-to-Front(MTF)
[Sleator,Tarjan,CACM1985]Whenanelementisaccessed,moveittothefrontofthelist.
• Theorem:MTFhascompetitiveratio2againstoptimalofflinealgorithm.
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PerformanceComparison
• E(FC)/E(STAT) =1[Rivest CACM1976]• E(MTF)/E(STAT)=𝜋/2 =1.58…[Chung,Hajela,SeymourSTOC85]
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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
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NewHybridInterpolatingModel• Assumeafixedprobabilitydistribution
• Foreachrequest,withprobability,letadversarychangetherequest.
• ó AverageCaseAnalysis
• ó CompetitiveAnalysis
• ó Knownprobabilitydistributionwithuncertainty.
),...,,( 21 npppp =!
e0=e1=e10 << e e
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Desiderata:InterpolatingAlgorithmforHybridModel
• Takesasinputestimatesofp ande
• Matchesbestaveragecaseperformancewhene islow,andmatchesbestcompetitiveratiowhene ishigh,andinterpolatesinbetween.
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CandidateAlgorithm:Move-From-Back-Epsilon• Listinitiallysortedinnon-increasingorderofprobabilities.
• Whenanelementx isaccessed,promoteitpastothersthathaveprobabilityuptopx +e.
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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
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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
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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
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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
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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
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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
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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
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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
![Page 57: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/57.jpg)
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
Access y
s: x v y vProb py = 0.5 pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02
![Page 58: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/58.jpg)
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
Access y 1 10
s: x v y vProb py = 0.5 pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02
![Page 59: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/59.jpg)
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
Access y 1 10
s: x v y vProb py = 0.5 pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02
![Page 60: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/60.jpg)
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
Access y 1 10Access v
s: x v y vProb py = 0.5 pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02
![Page 61: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/61.jpg)
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
Access y 1 10Access v 2 12
s: x v y vProb py = 0.5 pv = 0.03 px = 0.1 pw = 0.2 pz = 0.15 pu = 0.02
![Page 62: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/62.jpg)
DifficultiesinProvingPropertiesofMFBE• MustcomparewithOPTratherthanSTAT• OPTcanbecomputedbyDynamicprogramming
• Trivialway=O((n!)2m)• Improvement=O((2n)(n!)m)
[Reingold,Westbrook,1996]
• HardtoderiveusefulpropertiesofOPT
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Experiments(Pongnumkul ALADDINREU2004)• Motivation:Toseethebehaviorofalgorithmsinourhybridmodel.
• Measurement:Wemeasuretheperformanceofanonlinealgorithmbytheaveragecompetitiveratio.
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OurExperiment
• Variablesinourexperiment• TypeofListUpdateAlgorithm(MTF,STAT,MFBE)
• TypeofProbabilityDistribution• TypeofAdversary
• Epsilon:e
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OurExperiment
• Wegeneratearequestsequenceoflength100,withachosenprobabilitydistribution
• Then,withprobabilitye changetherequestsequence adverserially
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OurExperiment• Recordthecostincurredbytheonlinealgorithm=CostA(s)
• UseDynamicProgrammingtofindoptimumcostofthatrequestsequence=CostOPT(s).
• CompetitiveRatio=CostA(s)/CostOPT(s)
• Repeatthis100timestofindtheaveragecompetitiveratio.
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Distribution
• GeometricDistribution:• P[i]/ 1/2i
• UniformDistribution:• P[i]=1/n,n=lengthofthelist
• ZipfianDistribution(Zipf(2)):• P[i]/ 1/i2
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CruelAdversary
• Thisisanadaptiveadversary
• Looksatthecurrentlistandrequestthelastiteminthelist.
y w z x v uL:
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CruelAdversary,GeometricDistribution,n=6
Average Competitive Ratio
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
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
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CruelAdversary,GeometricDistribution,n=6(zoomedin)
Average Competitive Ratio
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
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
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OtherAdversaries
• Geometricadversarychooseselementsrandomly,accordingtothegeometricdistributiononthereversedSTATorder
• Uniformadversaryrequestselementsfromthelistuniformlyatrandom
• ObliviousAdversarydoesn’tlookatthecurrentlist
![Page 72: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/72.jpg)
UniformAdversary,Zipfian2Distribution,n=6
Average Competitive Ratio
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
![Page 73: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/73.jpg)
ReversedGeometricAdversary,GeometricDistribution,n=6
Average Competitive Ratio
1
1.1
1.2
1.3
1.4
1.5
1.6
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
![Page 74: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/74.jpg)
Observations
• Theperformanceofanyalgorithminthishybridmodeldependsheavilyonthetypeofadversary.
• MFBEseemsbetterthantheworseofSTATandMTF.
![Page 75: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/75.jpg)
Conjecture
(Nottheworst)TheaveragecompetitiveratioofMFBEisdominatedbythemaximumoftheaveragecompetitiveratiosofSTATandMTF.
• (Bestofall)Whatwewantedbutprobablynottrue:Avg competitiveratioofMFBEisatmosttheminimumoftheaveragecompetitiveratiosofSTATandMTF.
![Page 76: Interpolating Between Stochastic and Worst-case Optimization · 2020. 1. 3. · Possible Responses • Relax pessimism: Add restrictions to worst-case input scenarios to reduce impact](https://reader036.vdocument.in/reader036/viewer/2022071508/6128c2ee7e19d535b44bab51/html5/thumbnails/76.jpg)
InterpolateModelANDPerformance:Buildhybridinputmodelsinterpolatingstochasticandadversarialinputstoderivenewalgorithmsthatdeterioratesmoothlyinperformance• Prove/disprove“MFBEisnottheworst”• Findasettingwherethehybridmodelinterpolatesstochasticandworst-caseinputs,andthealgorithminterpolatestheperformanceofaveragecaseandworstcasealgorithms
• Aretheremoreeffectiveapproachestotrade-offstochasticoptimismandworst-casepessimism?
Summary