DTIC ADA443633: Sparse Forward-Backward for Fast Training of Conditional Random Fields

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Sparse Forward-Backward for Fast Training of
Conditional Random Fields
CharlesSutton,ChrisPalandAndrewMcCallum
UniversityofMassachusettsAmherst
Dept.ComputerScience
Amherst,MA01003
{casutton,pal,mccallum}@cs.umass.edu
Abstract
Complextasksinspeechandlanguageprocessingoftenincluderandom
variableswith large state spaces, both in speech tasks that involvepre-
dicting words and phonemes, and in joint processing of pipelined sys-
tems,inwhichthestatespacecanbethelabelingofanentiresequence.
In large state spaces, however, discriminative training can be expen-
sive, because it often requires many calls to forward-backward. Beam
search is a standard heuristic for controlling complexity during Viterbi
decoding, but during forward-backward, standard beam heuristics can
bedangerous,astheycanmaketrainingunstable. We introducesparse
forward-backward,avariationalperspectiveonbeammethodsthatuses
anapproximatingmixtureofKroneckerdeltafunctions. Thismotivates
anovelminimum-divergencebeamcriterionbasedonminimizingKLdi-
vergencebetweentherespectivemarginaldistributions.Ourbeamselec-
tion approach is not only more efficient for Viterbi decoding, but also
more stable within sparse forward-backward training. For a standard
text-to-speech problem, we reduce CRF training time fourfold—from
overadaytosixhours—withnolossinaccuracy.
1 Introduction
Complex tasks in speech and language processing often include random variables with
largestatespaces. Trainingsuchmodelscanbeexpensive,evenforlinearchains,because
standard estimation techniques, such as expectation maximization and conditionalmaxi-
mum likelihood, often requirerepeatedlyrunningfoward-backwardoverthe trainingset,
whichrequiresquadratictimeinthenumberofstates. DuringViterbidecoding,astandard
techniquetoaddressthisproblemisbeamsearch,thatis,ignoringvariableconfigurations
whose estimated max-marginal is sufficiently low. For sum-product inference methods
such as forward-backward, beam methods can be dangerous, however, because standard
beam selection criteria can inappropriatelydiscard probabilitymass in a way that makes
trainingunstable.
Inthispaper,weintroduceaperspectiveonbeamsearchthatmotivatesitsusewithinsum-
productinference.Inparticular,wecastbeamsearchasavariationalprocedurethatapprox-
imatesadistributionwithalargestatespacebyamixtureofmanyfewerKroneckerdelta
functions.Thismotivatessparseforward-backward,anovelmessage-passingalgorithmin
whichaftereachmessagepass,approximatemarginalpotentialsarecompressedaftereach
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Sparse Forward-Backward for Fast Training of Conditional Random
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Standard Form 298 (Rev. 8-98)
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pass. Essentially, this extends beam search from max-productinference to sum-product.
Our perspectivealso motivatesthe minimum-divergencebeam, a new beamcriterionthat
selectsacompressedmarginaldistributionwithafixedKullback-Leibler(KL)divergence
ofthetruemarginal. Notonlydoesthiscriterionperformbetterthanstandardbeamcrite-
riaforViterbidecoding,ititeractsmorestablywithtraining. Ononereal-worldtask, the
NetTalktext-to-speechdataset[5],wecannowtrainaconditionalrandomfield(CRF)in
about6hoursforwhichtrainingpreviouslyrequiredoveraday,withnolossinaccuracy.
2 Sparse Forward-Backward
Standard beam search can be viewed as maintaining sparse local marginal distributions
such that together they are as close as possible to a large distribution. In this section,
we formalize this intuition using a variationalargument, which motivates our new beam
criterionforsparseforward-backward.
Consideradiscretedistributionp(y),whereyisassumedtohaveverymanypossiblecon-
figurations. We approximatepbya sparsedistributionq, whichwe writeasa mixtureof
Kroneckerdeltafunctions:
q(y)= q δ (y), (1)
i i
Xi∈I
whereI ={i1,...,ik}isthesetofindicesisuchthatq(y =i)isnon-zero,andδi(y)=1
ify =i. WerefertothesetI asthebeam.
Consider the problem of finding the distribution q(y) of smallest weight such that
KL(qkp) ≤ ǫ. First, suppose the set I = {i1,...,ik} is fixed in advance, and we wish
to choose the probabilities q to minimize KL(qkp). Then the optimal choice is simply
i
q =p / p ,aresultwhichcanbeverifiedusingLagrangemultipliersonthenormal-
i i i∈I i
izationconstraintofq.
P
Second,supposewewishtodeterminethesetofindicesIofafixedsizekwhichminimize
KL(qkp). Then the optimal choice is when I = {i1,...,ik} consists of the indices of
thelargestk valuesofthediscretedistributionp. First, defineZ(I) = p , thenthe
i∈I i
optimalapproximatingdistributionis:
P
q
i
argminKL(qkp)=argmin argmin q log (2)
i
q I (cid:26) {qi}Xi∈I pi(cid:27)
p p /Z(I)
i i
=argmin log (3)
I (cid:26)Xi∈I Z(I) pi (cid:27)
=argmax logZ(I) (4)
I
(cid:8) (cid:9)
That is, the optimalchoice of indicesis the one that retains mostprobabilitymass. This
meansthatitis straightforwardto findthe discretedistributionq of minimalweightsuch
that KL(q||p) ≤ ǫ. We can sort the elements of the probability vector p, truncate after
logZ(I)exceeds−ǫ,andrenormalizetoobtainq.
To apply these ideas to forward-backward, essentially we compress the marginal beliefs
aftereverymessagepass. Wecallthismethodsparseforward-backward,whichwedefine
asfollows. Consideralinear-chainprobabilitydistributionp(y,x) ∝ tΨt(yt,yt−1,x),
such as an hidden Markov model (HMM) or conditional random fielQd (CRF). Let αt(i)
denotetheforwardmessages,β (i)thebackwardmessages,andγ (i)=α (i)β (i)bethe
t t t t
computedmarginals.Thenthesparseforwardrecursionis:
1. Passthemessageinthestandardway:
αt(j)← Ψt(i,j,x)αt−1(i) (5)
Xi
2. Computethenewdensebeliefγ as
t
γ (j)∝α (j)β (j) (6)
t t t
3. Compressintoasparsebeliefγ′(j),maintainingKL(γ′kγ)≤ ǫ. Thatis,sortthe
elementsofγandtruncateafterlogZ(I)exceeds−ǫ. CalltheresultingbeamI .
t
4. Compressα (j)torespectthenewbeamI .
t t
The backward recursion is defined similarly. Note that in every compression operation,
the beam I is recomputed from scratch; therefore, during the backward pass, variable
t
configurationscan both leave and enter the beam on the basis of backward information.
Just as in standardforward-backward,it can be shown by recursion thatthe sum of final
alphasyieldsthemassofthebeam.Thatis,ifIisthesetofallstatesequencesinthebeam,
then jaT(j)= y∈I tΨt(yt,yt−1,x). Therefore,becausebackwardrevisionstothe
beamPdonotdecrePasetheQlocalsumofbetas,theydonotdamagethequalityoftheglobal
beamoversequences.
The criterion in step 3 for selecting the beam is novel, and we call it the miniumum-
divergence criterion. Alternatively, we could take the top N states, or all states within
athreshold.Inthenextsectionwewillcomparetothesealternatecriteria.
Finally, we discuss a few practical considerations. We have found improved results by
addingaminimumbeliefsizeconstraintK,whichpreventsabeliefstateγ′(j)frombeing
t
compressedbelowK non-zeroentries. Also,wehavefoundthattheminimum-divergence
criterionusuallyfindsagoodbeamafterasingleforwardpass. Minimizingthenumberof
passesis desirable,becauseif findinga goodbeamrequiresmanyforwardandbackward
passes,onemayaswelldoexactforward-backward.
3 Results and Analysis
Inthissectionweevaluatesparseforward-backwardforbothmax-productandsum-product
inferenceinHMMsandCRFsandthewellknownNetTalktext-to-speechdataset[5]which
contains20,008Englishwords. Thetaskistoproducetheproperphonesgivenastringof
lettersasinput.
3.1 DecodingExperiments
Inthissectionwecomparetheourminimum-divergencecriteriontotraditionalbeamsearch
criteriaduringViterbidecoding. We generatesyntheticdata fromanHMM oflength75.
Transition matrix entries are sampled from a Dirichlet with every α = .1. Emission
j
matricesaregeneratedfromamixtureoftwodistributions: (a)alowentropy,sparsecon-
ditionaldistributionwith10non-zeroelementsand(b)ahighentropyDirichletwithevery
α = 104, with mixture weights of .75 and .25 respectively. The goal is to simulate a
j
regimewheremoststatesarehighlyinformativeabouttheirdestination,butafewareless
informative.Wecomparedthreebeamcriteria:(1)afixedbeamsize,(2)anadaptivebeam
wheremessageentriesareretainediftheirlogscoreiswithinafixedthresholdofthebest
sofar,and(3)ourminimum-divergencecriterionwithKL≤0.001andanadditionalmin-
imumbeamsize constraintofK ≥ 4. Ourminimum-divergencecriterionfindstheexact
Viterbipathanaverageonly9.6statespervariable. Ontheotherhand,thefixedbeamre-
quiresbetween20and25statestoreadsthesameaccuracy,andthesimplethresholdbeam
requires30.4statespervariable. WehavesimilarresultsontheNetTalkdata(omitteddue
tospace).
3.2 TrainingExperiments
Inthissection,wepresentresultsshowingthatsparseforward-backwardcanbeembedded
withinCRFtraining,yieldingsignificantspeedupsintrainingtimewithnolossintesting
performance.
Computation Time vs. Likelihood Computation Time vs. Likelihood
−28
−1
−2 −30
−3 −32
3×Log Likelihood 10 −−−−−87654 FKFETiihLxxxar eee≤cdd stε hBB F&oeeol daar|Iw mmj|B a≥ erSo daNfi zmBAeav Ngck. wKaLr sdize 3×Log Likelihood 10−−−−43330864 FKEiLxxa e≤cd tε B F&eo ar|Iwmj| a≥ ro dNf BAavgck. wKaLr sdize
−9
−42
−10
−44
200 400 600 800 1000 1200 1400 1 2 3 4 5 6 7 8
Computation Time (seconds) Computation Time (seconds) x 104
(a) (b)
Figure1:Comparisonofsparseforward-backwardmethodsforCRFtrainingonbothsyntheticdata
(left) and on the NetTalk data set (right). Both graphs plot log likelihood on the training data as
afunctionoftrainingtime. Inbothcases,sparseforward-backwardperformsequivalentlytoexact
trainingonbothtrainingandtestaccuracyusingonlyaquarterofthetrainingtime.
First, we train CRFs using synthetic data generated from a 100 state HMM in the same
mannerasintheprevioussection. Weuse50sequencesfortrainingand50sequencesfor
testing.InallcasesweuseexactViterbidecodingtocomputetestingaccuracy.Wecompare
fivedifferentmethodsfordiscardingprobabilitymass: (1)theminimum-divergencebeam
with KL ≤ 0.5 and minimum beam size K ≥ 30 (2) a fixed beam of size K = 30,
(3)afixedbeamwhosesizewastheaveragesizeusedbytheminimum-divergencebeam,
(4) a threshold based beam which explores on average the same number of states as the
minimum-divergencebeam,and(5)exactforwardbackward. Learningcurvesareshown
inFigure1(a).
Comparedtoexacttraining,sparseforward-backwardusesone-fourthofthetimeofexact
trainingwithnolossinaccuracy.Also,wefinditisimportantforthebeamtobeadaptive,
by comparing to the fixed beam whose size is the average number of states used by our
minimum-divergencecriterion. Although minimum divergence and the fixed beam con-
vergetothesamesolution,minimumdivergencefinishesfaster,indicatingthattheadaptive
beam does help training time. Most of the benefit occurs later in training, as the model
becomesfartherfromuniform.
In the case of the smaller, fixed beam of size N, our L-BFGS optimizerterminatedwith
an erroras a resultof thenoisy gradientcomputation. Inthe case ofthe thresholdbeam,
the likelihood gradients were erratic, but L-BFGS did terminate normally. However the
recognitionaccuracyofthefinalmodelwaslow,at67.1%.
Finally, we present results from training on the real-world NetTalk data set. In Figure
1(b)wepresentruntime,modellikelihoodandaccuracyresultsfora52stateCRFforthe
NetTalkproblemthatwasoptimizedusing19075examplesandtestedusing934examples.
For the minimum divergence beam, we set the divergence threshold ǫ = .005 and the
minimum beam size K ≥ 10. We initialize the CRF parameters using a subset of 12%
ofthedata,beforetrainingonthefulldatauntilconvergence. Weusedthebeammethods
duringthecompletetrainingrunandduringthisinitializationperiod.
Duringthecompletetrainingrun,thethresholdbeamgradientestimatesweresonoisythat
ourL-BFGSoptimizerwasunabletotakeacompletestep. Exactforwardbackwardtrain-
ingproducedatestsetaccuracyof91.6%. Trainingusingthelargerfixedbeam(N =20)
terminated normallybut very noisy intermediategradientswere foundin the terminating
iteration.Theresultwasamuchloweraccuracyof85.7%.Incontrast,theminimumdiver-
gencebeamachievedanaccuracyof91.7%inlessthan25%ofthetimeittooktoexactly
traintheCRFusingforward-backward.
4 Related Work
Relatedtoourworkiszero-compressioninjunctiontrees[3],describedin[2],whichcon-
siderseverypotentialinacliquetree,andsetsthesmallestpotentialvaluestozero,withthe
constraintthatthetotalmassofthepotentialdoesnotfallbelowafixedvalueδ. Incontrast
toourwork,theyprunethemodel’spotentialsoncebeforeperforminginference,whereas
wedynamicallyprunethebeliefsduringinference,andindeedthebeamcanchangeduring
inference as new information arrives from other parts of the model. Also, Jordan et al.
[4], intheirworkonhiddenMarkovdecisiontrees, introduceavariationalalgorithmthat
usesadeltaonasinglebeststatesequence,buttheyprovidenoexperimentalevaluationof
thistechnique. Incomputervision, CoughlanandFerreira[1]haveuseda beliefpruning
methodwithinbeliefpropagationforloopymodelswhichisverysimilartoourthreshold
beambaseline.
5 Conclusions
Wehavepresentedaprincipledmethodforsignificantlyspeedingupdecodingandlearning
tasksinHMMsandCRFs. We alsohavepresentedexperimentalworkdemonstratingthe
utility of our approach. As future work, we believe a promising avenue of exploration
would be to explore adaptive strategies involving interaction of our L-BFGS optimizer,
detectingexcessivelynoisygradientsandautomaticallysettingǫvalues.Whileresultshere
were only with linear-chain models, we believe this approach should be more generally
applicable. For example, in pipelines of NLP tasks, it is often better to pass lattices of
predictionsratherthansingle-bestpredictions,inordertopreserveuncertaintybetweenthe
tasks. For such systems, the current work has implications for how to select the lattice
size, and how to pass information backwards through the pipeline, so that higher-level
informationfromlatertaskscanimproveperformanceonearliertasks.
ACKNOWLEDGMENTSThisworkwassupportedinpartbytheCenterforIntelligentInforma-
tionRetrieval,inpartbyTheCentralIntelligenceAgency,theNationalSecurityAgencyandNational
ScienceFoundationunderNSFgrants#IIS-0326249and#IIS-0427594, andinpartbytheDefense
Advanced ResearchProjectsAgency(DARPA),throughtheDepartment oftheInterior, NBC,Ac-
quisition Services Division, under contract number NBCHD030010. Any opinions, findings and
conclusionsorrecommendationsexpressedinthismaterialaretheauthor(s)anddonotnecessarily
reflectthoseofthesponsor.
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