quantum machine learning for election modeling - max henderson... · title: 2018-04-04 - max...
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DataAnalytics|QuantumComputing|SystemsEngineering
Established2014inWashingtonD.C./London/AdelaideTeamof~20softwareandsystemsengineers,datascientistsClients:
GlobalInvestmentBanksAssetManagementFirms
TechnologyCompaniesGovernmentEnergyPharmaceutical
QxBranchOverview
2
QxBranchdeliversrevolutionarydataanalyticssoftwareenabledbyclassicalandemergingquantum
computingcapabilitiesthatdrivebusinessvalue
ApplydataanalyticsexpertiseandsoftwarecapabilitiestomanagecomplexdataandprovideactionableinsightsacrossmultipleverticalsBusinessdomainexpertiseinfinance,aerospace,defence,andtechnologydomainsResearch&Developmentpartnershipswithclientsandacademiatoidentifybusinesschallengesthatcanbesolvedthroughcutting-edgeapplicationsofquantumcomputing(universalandadiabatic)andadvanceddataanalytics
QxBranch,Inc.2018
Election2016:Casestudyinthedifficultlyofsampling
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Wheredidthemodelsgowrong?
State-by-statecorrelations
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• Majorissue:failuretomodelcorrelations1-3betweenstates
• Mostmodelsassumedindependencebetweenresultsofeachstate
• Anaccuratecorrelationmatrixcancapturehigher-level,richerstructureinthedataandaccountforsystemicerrorsinpolls
1. http://www.independent.co.uk/news/world/americas/sam-wang-princeton-election-consortium-poll-hillary-clinton-donald-trump-victory-a7399671.html2. http://elections.huffingtonpost.com/2016/forecast/president3. http://money.cnn.com/2016/11/01/news/economy/hillary-clinton-win-forecast-moodys-analytics/index.html4. http://fivethirtyeight.com/
Difficultyofsamplingfromcorrelatedgraphs
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• Evenwithperfectdataoncorrelationsbetweenstates,usingthecorrelationmatrixisdifficultduetothecomputationalcostofsamplingfromfully-connectedgraphs
• Samplingfromfully-connectedgraphsisanalogoustosamplingfromaproperlytrainedBoltzmannmachine• TrainingcoefficientsofBoltzmannmachinesrequires
performingcalculationsonallpossiblestatesofthemodel• Asthisisintractableonlargeproblemsizes,heuristicsor
othermodelsaretypicallyimplementedinstead
Forecastingelectionsonaquantumcomputer
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• Quantumcomputing(QC)researchhasshownpotentialspeedupsintrainingdeepneuralnetworks1-3
• Coreidea:ByusingQC-trainedmodelstosimulateelectionresultswecanachieve:• Moreefficientsampling/training• Intrinsic,tuneablestatecorrelations• Inclusionofadditionalerrormodels
1. Adachi,StevenH.,andMaxwellP.Henderson."Applicationofquantumannealingtotrainingofdeepneuralnetworks." arXiv preprintarXiv:1510.06356 (2015).2. Benedetti,Marcello,etal."Estimationofeffectivetemperaturesinquantumannealers forsamplingapplications:Acasestudywithpossibleapplicationsindeep
learning." PhysicalReviewA 94.2(2016):022308.3. Benedetti,Marcello,etal."Quantum-assistedlearningofgraphicalmodelswitharbitrarypairwiseconnectivity." arXiv preprintarXiv:1609.02542 (2016).
WhatweAREdoingvs.whatweAREN’T
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SubjectMatterExpertiseModel(s) SimulationModel(s)
DatatoModel
1. Individualstatepredictions
2.StateCorrelations
SimulationResults
WhatweAREdoingvs.whatweAREN’T
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SubjectMatterExpertiseModel(s)
PreviousVotingresults
Currentstatepollingresults
Race
Gender
Urbanvsruralpopulationdistribution
Totalstatepopulation
Voterexcitability
Education
NumberofRussianbotsonTwitter
WhatweAREdoingvs.whatweAREN’T
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SubjectMatterExpertiseModel(s) SimulationModel(s)
DatatoModel
1. Individualstatepredictions
2.StateCorrelations
SimulationResults
Step1:MappinganelectiontoaBoltzmannmachine
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1. http://www.fivethirtyeight.com
Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
𝒔𝒊 𝑫
Availabledataislimited
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• Whatwewouldlike:• Detailedbreakdownsofdemographics• Meticulouslycuratedbiasesandcorrelations• Allofthedatathat538hasspentyearsandthousandsof
dollarscurating
• Whatwehave:• PubliclyavailableresultsofpreviousUSelections• Stateprobabilities,astoldbypolls• Publiclyaccessibledatafrom538
Calculatingthemissingsecondordermoments
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• Inlieuofbettercurateddataconcerningsecondordermoments,wecalculatedourowntermsfrompreviousUSelectionresults
• Ourmethodologyshouldnot“break”firstordermoments
• Assumptionsinthismodel:• Ineachpreviouselection,iftwostateshadthesameelectionresult,that
increasedtheircorrelation• Electionsthatweremorerecenthaveahigherweight
Step2:MappingaBoltzmannmachinetotheQC
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Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
Figure 2. (A) Example map of 538 state-by-state voting probabilities and the resulting national probability. (B) State
probabilities are formed from a time series averaging technique, and (C) the candidates lead translates into an overall probability.
A B
C
jxix
The update equations for training the model:
∆𝑤&' = −1𝜂 𝑠&𝑠' - − 𝑠&𝑠' . ∆𝜃& = −
1𝜂 𝑠& - − 𝑠& .
Potential quantum advantage
Graphembedding– Qubitchains
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Exampleofembeddingaproblem(left)intoafixedgraphstructure(right)1
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1. Choi,Vicky."Minor-embeddinginadiabaticquantumcomputation:II.Minor-universalgraphdesign." arXiv preprinthttps://arxiv.org/pdf/1001.3116v2.pdf(2010).
Effectofembedding:Shortqubitchains
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• Tovalidatetheapproach,werandomlychosefirstandsecondordertermsforahypothetical5-statenation
• Usingthesmallestembeddingchains,thisnetworkwasunabletoproperlytrain• “Hopfield”likeresults;
optimalsolutionsratherthanprobabilisticresults
• Leadstohugechangesinweights/biases,causingnetworkinstability
Diagonal= 𝒔𝒊 𝑴Offdiagonal= 𝒔𝒊𝒔𝒋 𝑴
Effectofembedding:Longqubitchains
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• Forlargerproblemsizes,theembeddingwillnecessarilyhavelongerqubitchains
• Tosimulatethisforoursmallnetwork,weartificiallyincreasedthequbitchains
• Withthisapproach,arbitraryfirstandsecondordermomentswerelearnedbythenetworks
Diagonal= 𝒔𝒊 𝑴Offdiagonal= 𝒔𝒊𝒔𝒋 𝑴
Primaryexperiment
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• Goal:UsinghistoricaldataandtheQC-trainingmethodologypresentedhere,reproduceelectionforecastsovertime
• Somecaveats:• Multiplemodelsneededformodeling nationalerror;25were
usedhere• LimitedtimewindowsofD-Waveaccess,soresultswere
generatedeverytwoweeksinsteadofdaily• Limitedhardwaresizemadeusomit1stateandprovince
(sorryMarylandandDC…youalwaysvoteDanyway)• Forsimplification,MaineandNebraskawereconsidered
winner-take-all
Results– Trainingerrors
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Examplestestingextremesofcorrelations:negative,random,&positive
Redlines= 𝒔𝒊 𝑫Bluelines= 𝒔𝒊 𝑴
Results– Trainingerrors
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Largeerrorsemergewhenpollsareupdatedandlargechangesoccur
Results– Trainingerrors
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QC=QuantumtrainedTB=NationalTrumpbiasCB=NationalClintonbias
Themost“impactful”states
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• Pearsoncorrelationcoefficientsforthe10statesmost(top)andleast(bottom)correlatedwiththeelectionforecastingresults
Ourmodels 538
Stateerrors
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• Individualstateserrordistributionswashighlydependentonifthestatewasahardred,blue,orpurplestate
• Differentwaysofdealingwitherrorsofthisform:• Shimming• Multiplegauges
Summary
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• TheQC-trainednetworkswereabletolearnstructureinpollingdatatomakeelectionforecastsinlinewiththemodelsof538
• Trumpwasgivenahigherlikelihoodofvictory(comparedtootherpollsters),eventhoughthefirstordermomentsremainedunchanged• Ideallyinthefuture,wecouldrerunthismethodusing
correlationsknownwithmoredetailin-housefrom538• Eachiterationofthetrainingmodelquicklyproduced25,000
simulations(oneforeachnationalerrormodel),whicheclipsesthe20,000simulations538performseachtimetheyreruntheirmodels