Download Hypothesis

ArtificialIntelligence
Roman Barták
Department of Theoretical Computer Science and Mathematical Logic
Knowledgeinlearning
Sofarwelearntafunctioninput→ output.
Weonlyassumedtoknowtheformofthefunction(suchas
adecisiontree)definedbythehypothesisspace.
Canwetakeadvantageofpriorknowledgeaboutthe
world?
Inmostcasesthepriorknowledgeisrepresentedasgeneral
first-orderlogicaltheories.
Somemethods:
– current-best-hypothesis search
– versionspacelearning
– inductivelogicprogramming
Logicalformulationoflearning
Hypotheses, exampledescriptions,andclassificationwillberepresented
usinglogicalsentences.
Examples
– attributes becomeunarypredicates
Alternate(X1) ∧ ¬Bar(X1) ∧ ¬Fri/Sat(X1)∧Hungry(X1)∧…
– classification isgivenbyliteralusingthegoalpredicate
WillWait(X1)or¬WillWait(X1)
Hypothesiswillhavetheform
∀xGoal(x)⇔ Cj(x)
Cj iscalledtheextensionofthepredicate
∀r WillWait(r) ⇔ Patrons(r,Some)
∨ (Patrons(r,Full)∧ Hungry(r)∧ Type(r,French))
∨ (Patrons(r,Full)∧ Hungry(r)∧ Type(r,Thai) ∧ Fri/Sat(r))
∨ (Patrons(r,Full)∧ Hungry(r)∧ Type(r,Burger))
Hypothesisspace
Hypothesisspaceisthesetofallhypothesis.
Thelearningalgorithmbelievesthatonehypothesisis
correct,thatis,itbelievesthesentence:
h1 ∨ h2 ∨ h3 ∨…∨ hn
Hypothesesthatarenotconsistentwiththeexamplescan
berulesout.
Therearetwopossiblewaystobeinconsistent withan
example(thenotionsoriginatedinmedicinetodescribe
erroneousresultsfromlabtests)
– falsenegative– hypothesissaystheexampleshouldbe
negativebutinfactitispositive
– falsepositive– hypothesissaystheexampleshouldbe
positivebutinfactitisnegative
Current-best-hypothesissearch
Theideaistomaintainasinglehypothesis,and
toadjustitasnewexamplesarriveinorderto
maintainconsistency
iftheexampleisconsistent withthehypothesis
thendonotchangeit
iffalsenegative
thengeneralize thehypothesis
iffalsepositive
thenspecialize thehypothesis
Thecurrent-best-hypothesislearningalgorithm
Specializationandgeneralization
Howtoimplementspecializationandgeneralizationofthe
hypothesis?
•
Ifhypothesish1 isageneralization
ofhypothesish2,thenwemusthave
∀xC2(x)⇒ C1(x)
•
Ci istypicallyaconjunctionofpredicates
– generalizationcanberealizedby
droppingconditionsorbyaddingdisjuncts
– specializationcanberealizedbyadding
extraconditionsorbyremovingdisjuncts
Arestaurantexample:
– thefirstexampleispositive,attributeAlternate(X1)istrue,solettheinitialhypothesisbe
h1:∀xWillWait(x)⇔ Alternate(x)
– thesecondexampleisnegative,hypothesispredictsittobepositive,soitisafalsepositive; we
needtospecializebyaddingextracondition
h2:∀xWillWait(x)⇔ Alternate(x)∧ Patrons(x,Some)
– thethirstexampleispositive,thehypothesispredictsittobenegative,soitisafalsenegative;we
needtogeneralizebydroppingtheconditionAlternate
h3:∀xWillWait(x)⇔ Patrons(x,Some)
– Thefourthexampleispositive,thehypothesispredictsittobenegative,soitisafalsepositive;
weneedtogeneralizebyaddingadisjunct (wecannotdropthePatronscondition)
h3:∀xWillWait(x)⇔ Patrons(x,Some)∨ (Patrons(x,Full) ∧ Fri/Sat(x))
Current-best-hypothesis:properties
Aftereachmodificationofthehypothesisweneed
tocheckallthepreviousexamples.
Thereareseveralpossiblegeneralizationsand
specializationsandwemayneedtobacktrack
wherenosimplemodificationofthehypothesisis
consistentwithallthedata.
Thesourceofproblems– strongcommitment
– Thealgorithmhastochooseaparticularhypothesisas
itsbestguesseventhoughitdoesnothaveenough
datayettobesureofthechoice.
Asolutioncouldbeleast-commitmentsearch.
Versionspacelearning
Thehypothesisspacecanbeviewedasadisjunctivesentence
h1 ∨ h2 ∨ h3 ∨…∨ hn
Hypothesisinconsistentwithanewexampleisremovedfromthedisjunction.
Assumingtheoriginalhypothesisspacedoesinfactcontaintherightanswer,
thereduceddisjunctionmuststillcontaintherightanswer.
Thesetofhypothesisremainingiscalledtheversionspace.
Theversionspacelearningalgorithm(alsothecandidateelimination
algorithm).
Thisapproachisincremental:onenever hastogobackandreexaminetheold
examples
Representationofversionspace
Hypothesis spaceisenormous, sohowcanwepossiblywritedownthis
enormousdisjunction?
Wehaveanorderingofhypothesisspace(generalization/specialization)
sowecanspecifyboundaries,whereeachboundarywillbeasetof
hypothesis(aboundaryset).
G=amostgeneralboundary
• consistentwithallobservationssofar
• therearenoconsistenthypotheses
thataremoregeneral
• initiallyTrue
S=amostspecificboundary
• consistentwithallobservationssofar
• therearenoconsistenthypotheses
thataremorespecific
• initiallyFalse
Everythinginbetween G-setandS-setisguaranteedtobeconsistentwith
theexamplesandnothingelseisconsistent.
Versionspaceupdate
ForeachnewexampleweupdatethesetsGandS:
– falsepositiveforSi
ÄthrowSi outoftheS-set
– falsenegativeforSi
ÄreplaceSi intheS-setbyallitsimmediategeneralizations
– falsepositiveforGi
ÄreplaceGi intheG-setbyallitsimmediatespecilaizations
– falsenegativeforGi
ÄthrowGi outoftheG-set
Thealgorithmcontinuesuntiloneofthreethingshappens:
– wehaveexactlyonehypothesisleftintheversionspace
– theversionspacecollapses(eitherSorGbecomesempty)
– werunoutofexamplesandhaveseveralhypothesisremaining
intheversionspace
• theversionspacerepresentsadisjunctionofhypotheses
• ifthehypothesisdisagreeinclassification,onepossibilityistotakethe
majorityvote
Propertiesofversionspacelearning
Ifthedomaincontainsnoiseorinsufficientattributesfor
exactclassification,theversionspacewillalwayscollapse.
– todate,nocompletely successfulsolutionhasbeenfound
Ifweallowunlimiteddisjunctioninthehypothesisspace,
– theS-setwillalwayscontainasinglemost-specifichypothesis
(thedisjunctionofthedescriptionsofpositiveexamples)
– theG-setwillcontainjustthenegationofthedisjunctionofthe
descriptionsofthenegativeexamples
– canbeaddressedbyallowingonlylimitedformsofdisjunction
byincludingageneralizationhierarchyofmoregeneral
predicates:
• insteadofWaitEstimate(x,30-60) ∨WaitEstimate(x,>60)wecanuse
LongWait(x)
Thepureversionspacealgorithmwasfirstappliedinthe
Meta-DENDRALsystem,whichwasdesignedtolearnrulesfor
predictinghowmoleculeswouldbreakintopiecesinmass
spectrometer.
Inductivelogicprogramming
Inductivelogicprogramming(ILP)combines
inductivemethodswiththepoweroffirst-order
representations(logicprograms).
ILPworkswellwithrelationships betweenobjects,
whichishardforattribute-onlyapproaches.
Inprinciplethegeneralknowledge-inductionproblem
istosolvetheentailmentconstraint:
Background∧ Hypothesis∧ Descriptions|=Classifications
TwoprincipalapproachestoILP:
– top-downinductivelearningmethods
(systemFOIL)
– inductivelearningwithinversededuction
(systemPROGOL)
ILPproblem
Background∧ Hypothesis∧ Descriptions|=Classifications
•
ExamplesaretypicallygivenasPrologfacts
Father(Philip,Charles), Father(Philip, Anne), …
Mother(Mum,Margaret), Mother(Mum, Elizabeth), …
Married(Diana, Charles), Married(Elizabeth, Philip), …
Male(Philip), Male(Charles), …
Female(Beatrice), Female(Margaret),…
•
Similarlyknownclassifications aregivenbyPrologfacts:
Grandparent(Mum,Charles), Gradparent(Elizabeth, Beatrice), …
¬Gradparent(Mum,Harry), ¬Grandparent(Spencer,Peter), …
•
Possiblehypothesis:
Grandparent(x,y) ⇔ [∃z
[∃z
[∃z
[∃z
•
Mother(x,z)
Mother(x,z)
Father(x,z)
Father(x,z)
∧
∧
∧
∧
Mother(z,y)] ∨
Father(z,y)] ∨
Mother(z,y)] ∨
Father(z,y)]
Wecanexploitbackgroundknowledge:
Parent(x,y) ⇔ Mother(x,y) ∨ Father(x,y)
•
Thenwecansimplifythehypothesis:
Grandparent(x,y) ⇔ [∃z Parent(x,z) ∧ Parent(z,y)]
Top-downlearning
• Startwithaclausewithanemptybody
Grandfather(x,y) ←
• Thisclauseclassifieseveryexampleaspositive,soitneeds
tobespecialized
– byaddingliteralsoneatatimetothebody
Grandfather(x,y) ← Father(x,y)
Grandfather(x,y) ← Parent(x,z)
Grandfather(x,y) ← Father(x,z)
…
• Wepreferthespecializationthatclassifiescorrectlymoreexamples
– specialize thisclausefurther
Grandfather(x,y) ← Father(x,z) ∧ Parent(z,y)
– ifbackgroundknowledge Parentisnotavailablewemayneedto
addmoreclauses
Grandfather(x,y) ← Father(x,z) ∧ Father(z,y)
Grandfather(x,y) ← Father(x,z) ∧ Mother(z,y)
• eachclausecoverssomepositiveexamplesandnonegativeexample
Top-downlearningalgorithm
Build new clauses covering
positive examples
Literals are chosen from known
predicates, equality/inequality
literals, and arithmetic comparisons:
• they have to include a variable
that is already in clause
• we can exploit types (number,
person,…)
• the choice of literal can be based
on information gain
SystemFOILsolvedalongsequence ofexercises onlist-processing functions
(forexampleappend,QuickSort).
Inverseresolution
Background∧ Hypothesis∧Descriptions|=Classifications
• ClassicalresolutiondeducesClassificationsfromBackground,
Hypothesis,Descriptions.
• Wecanruntheproofbackward,findHypothesissuchthat
theproofgoesthrough:
– forresolvent CproduceC1 andC2 (ifC2 isgiventhenproduceC1)
¬Parent(Elizabeth,Anne) ∨ Grandparent(George,Anne)
¬Parent(z,Anne) ∨ Grandparent(George,Anne)
¬Parent(z,y) ∨ Grandparent(George,y)
…
© 2016 Roman Barták
Department of Theoretical Computer Science and Mathematical Logic
[email protected]