QuantumMachineLearningforElectionModeling

September28,2017MaxHenderson,Ph.D.

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).

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

!" #

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:

∆%&' = −1+ ,&,' - − ,&,' . ∆/& = −1+ ,& - − ,& .

Potential quantum advantage

Graphembedding– Qubitchains

11

Exampleofembeddingaproblem(left)intoafixedgraphstructure(right)

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Effectofembedding:Shortqubitchains

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• Tovalidatetheapproach,werandomlychosefirstandsecondordertermsforahypothetical5-statenation

• Usingthesmallestembeddingchains,thisnetworkwasunabletoproperlytrain• “Hopfield”likeresults;

optimalsolutionsratherthanprobabilisticresults

• Leadstohugechangesinweights/biases,causingnetworkinstability

Diagonal= !" 1Offdiagonal= !"!2 1

Effectofembedding:Longqubitchains

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• Forlargerproblemsizes,theembeddingwillnecessarilyhavelongerqubitchains

• Tosimulatethisforoursmallnetwork,weartificiallyincreasedthequbitchains

• Withthisapproach,arbitraryfirstandsecondordermomentswerelearnedbythenetworks

Diagonal= !" 1Offdiagonal= !"!2 1

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= !" #Bluelines= !" 1

Results– Trainingerrors

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Largeerrorsemergewhenpollsareupdatedandlargechangesoccur

Results– Trainingerrors

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QC=QuantumtrainedTB=NationalTrumpbiasCB=NationalClintonbias

Stateerrors

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• 538individualstateerrorsusedat-distributionwith10degreesoffreedom(df)

• ProbabilisticsamplingfromQCnaturallyledtostateerrorswithsimilardistributionandparameters

Summary

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• TheQC-trainednetworkswereabletolearnstructureinpollingdatatomakeelectionforecastsinlinewiththemodelsof538

• Trumpwasgivenahigherlikelihoodofvictory,eventhoughthefirstordermomentsremainedunchanged• Ideallyinthefuture,wecouldrerunthismethodusing

correlationsknownwithmoredetailin-housefrom538• Eachiterationofthetrainingmodelquicklyproduced25,000

simulations(oneforeachnationalerrormodel),whicheclipsesthe20,000simulations538performseachtimetheyreruntheirmodels

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