2Content Introduction Learning with Perfect Domain Theories: PROLOG-EBG Difference between Inductive and Analytical LearningLearning with Perfect Domain Theories: PROLOG-EBGRemarks on Explanation-Based LearningSummary
3IntroductionExplanation is used to distinguish the relevant features of the training examples from the irrelevant ones, so that the examples can be generalisedPrior knowledge and deductive reasoning is used to augment the information provided by the training examplesPrior knowledge is used to reduce the complexity of hypothesis spaceAssumption: learner's prior knowledge is correct and complete
4Introduction 2 Example: Learn to recognise important classes of games Goal: Recognise chessboard positions in which black will lose its queen within two movesInduction can be employed <=> Problem: thousands of training examples similar to this one are neededSuggested target hypothesis: board position in which the black king and queen are simultaneously attackedNot suggested: board position in which four white pawns are still in their original location
5Introduction 3Explanations of human beings provide the information needed to rationally generalise from detailsPrior knowledge: e.g. knowledge about the rules of chess: legal moves, how is the game won, ...Given just this prior knowledge it is possible in principle to calculate the optimal chess move for any board position <=> in practice it will be frustratingly complexGoal: learning algorithm that automatically constructs and learns a move from such explanations
6Difference between Inductive and Analytical Learning Analytical learning methods seek a hypothesis that fits the learner's prior knowledge and covers the training examplesExplanation based learning is a form of analytical learning in which the learner processes each new training example byExplaining the observed target value for this example in terms of the domain theoryAnalysing this explanation to determine the general conditions under which the explanation holdsRefining its hypothesis to incorporate these general conditions
7Difference between Inductive and Analytical Learning (2) Difference: They assume two different formulations of the learning problem:Inductive learning: input: hypothesis space H + set of training examples output: hypothesis h, that is consistent with these training examplesAnalytical learning: input: hypothesis space H + set of training examples domain theory B consisting of background knowledge (used to explain the training examples) output: hypothesis h, that is consistent with both the training examples D and the domain theory B
8Difference between Inductive and Analytical Learning (3) Illustration: is True if is a situation in which black will lose its queen within two moves and False otherwise H: set of Horn-clauses where predicates used by the rules refer to the position or relative position of specific pieces B: formalisation of the rules of chess
9New Example Given: Target concept: Instance space X: Each instance describes a pair of objects represented by the predicates Type, Color, Volume, Owner, Material, Density and On.Hypothesis space H: Each hypothesis is a set of Horn clauses. The head of each clause is a literal of the target predicate SafeToStack. The body of each Horn clause is a conjunction of literals based on the same predicates used to describe the instances + LessThan|Equal|GreaterThan + function: plus|minus|timesTarget concept:Training examples: On(Obj1, Obj2) Owner (Obj1, Fred) Type(Obj1, Box) Owner (Obj2, Louise) Type(Obj2, Endtable) Density(Obj1, 0.3) Color(Obj1, red) Material (Obj1, Carboard) Color(Obj2, Blue) Material (Obj1, Wood) Volume(Obj1, 2)
10New Example 2 Determine: Domain Theory B: A hypothesis from H consistent with the training examples and the domain theory
11Content Introduction Learning with Perfect Domain Theories: PROLOG-EBG An Illustrative TraceRemarks on Explanation-Based LearningExplanation-Based Learning of Search Control KnowledgeSummary
12Learning with Perfect Domain Theories: PROLOG-EBG A domain theory is said to be correct if each of its assertions is a truthful statement about the worldA domain theory is complete with respect to a given target concept and instance space, if the domain theory covers every positive example in the instance space.It is not required that the domain theory is able to prove that negative examples do not satisfy the target concept.
13Learning with Perfect Domain Theories: PROLOG-EBG (2) Question: The learner had a perfect domain theory, why would it need to learn?Answer:There are cases in which it is feasible to provide a perfect domain theoryIt is unreasonable to assure that a perfect domain theory is available. A realistic assumption is that plausible explanations based on imperfect domain theories must be used, rather than exact proofs based on perfect knowledge.
14Learning with Perfect Domain Theories: PROLOG-EBG (3) PROLOG-EBG (Kedar-Cabelli and McCarthy 1987)Sequential covering algorithmWhen given a complete and correct domain theory, the method is guaranteed to output a hypothesis (set of rules) that is correct and that covers the observed positive training examplesOutput: set of logically sufficient conditions for the target concept, according the domain theory
15Learning with Perfect Domain Theories: PROLOG-EBG (4) Repeatedly: Domain theory is correct and complete this explanation constitutes a proof that the training examples satisfy the target conceptPROLOG-EBG(TargetConcept, TrainingExamples, Domain Theory)LearnedRulesPos the positive examples from TrainingExamplesfor each PositiveExample in Pos that is not covered by LearnedRules doExplain Explanation an explanation (proof) in terms of the DomainTheory that PositiveExample satisfies the TargetConceptAnalyse Sufficient Condition the most general set of features of PositiveExample sufficient to satisfy the TargetConcept according to the ExplanationRefine LearnedRules LearnedRules + NewHornClause, where NewHornClause is of the form TargetConcept SufficientConditionsIn general there may be multiple possible explanations Any or all of the explanations may be used. Explanation is generated using backward chaining search as performed by PROLOG.
16An Illustrative Trace (2) The imprtant question in the generalising-process: Of the many features that happen to be true of the current training example, which ones are generally relevant to the target concept?Explanation constructs the answer: Precisely the features mentioned in the explanation
17An Illustrative Trace (3) General rule justified by the domain theory: leaf node in the proof tree expects Equal(0.6,times(2,03) and LessThan(0.6,5)
18An Illustrative Trace (4) Explanation of the training example forms the proof for the correctness of this rulePROLOG-EBG computes the most general rule that can be justified by the explanation, by computing the weakest preimage of the explanationDefinition: The weakest preimage of a conclusion C with respect to a proof P is the most general set of initial assertions A, such that A entails C according to P.Example:PROLOG_EBG computes the weakest preimage of the target concept with respect to the explanation, using a general procedure called regressionRegression: go iteratively backward through the explanation,first computing the weakest preimage of the target concept with respect to the final proof step in the explanationComputing the weakest preimage of the resulting expressions with respect to the proceeding step and so on
21Content Introduction Learning with Perfect Domain Theories: PROLOG-EBG Remarks on Explanation-Based LearningDiscovering new featuresSummary
22Remarks on Explanation-Based Learning Key properties:PROLOG-EBG produces justified general hypotheses by using prior knowledge to analyse individual examplesThe explanation about the way how an example satisfies the target concept determines which example attributes are relevant: the ones mentioned by the explanationRegressing the target concept to determine its weakest preimage with respect to the explanation allows deriving more general constraints on the values of relevant featuresEach learned Horn clause corresponds to a sufficient condition for satisfying the target conceptThe generality of the learned Horn clauses will depend on the formulation of the domain theory and on the sequence in which the training examples are consideredImplicitly assumes that the domain theory is correct and complete
23Remarks Explanation-Based Learning 2 Related perspectives to help to understand its capabilities and limitations:EBL as theory guided generalisation of examples: Rational generalisation from examples allows to avoid the bounds on sample complexity that occured in pure inductive learningEBL as example guided reformulation of theories: Method for reformulating the domain theory into more operational form: Creating rules that:Deductively follow the domain theoryClassify the observed training examples in a single inference step
24Remarks Explanation-Based Learning 3 Related perspectives to help to understand its capabilities and limitations:EBL is „just“ restating what the learner already „knows“: In what sense does this quality help to learn then?Knowledge reformulation: In many tasks the difference between what one knows in principle and what one can efficiently compute in practice may be greatSituation: Complete perfect domain theory is already known to the (human) learner, and further learning is „simple“! So it's a matter of reformulating this knowledge into a form in which it can be used more effectively to select appropriate moves.
25Remarks Explanation-Based Learning 4 Knowledge Compilation: EBL involves reformulating the domain theory to produce general rules that classify examples in a single inference step
26Discovering new features Interesting capability: Ability to formulate new features that are not explicitly in the description of the training examples but that are needed to describe the general rule underlying the training examplesThis „feature“ is similarly represented by the hidden units of neural networksLike the BACKPROPAGATION algorithm, PROLOG_EBG automatically formulates such features in its attempt to fit the training dataBUT:In neural networks it's developed in a statistical processPROLOG-EBG it's derived in an analytical processExample: derives the feature
27Content Introduction Learning with Perfect Domain Theories: PROLOG-EBG Remarks on Explanation-Based LearningSummary
28SummaryPROLOG-EBGUses first order Horn clauses in its domain theory and in its learned hypothesesThe explanation is a PROLOG proofThe hypothesis extracted from the explanation is the weakest preimage of this proofAnalytical learning methods construct useful intermediate features as a side effect of analysing individual training examples.Other deductive learning procedures can extend the deductive closure of their domain.PRODIGY and SOAR have demonstrated the utility of explanation based learning methods for automatically acquiring effective search control knowledge that speeds up problem solvingDisadvantage: purely deductive implementations such as PROLOG-EBG produce a correct output if the domain theory is also correct