Analytical Learning Introduction Lehrstuhl Explanation is used to distinguish the relevant features of the training examples from the irrelevant ones, so that the examples can be generalised Introduction 3 Prior knowledge and deductive reasoning is used to augment the information provided by the training examples Prior knowledge is used to reduce the complexity of hypothesis space Assumption: learner's prior knowledge is correct and complete Difference between Inductive and Analytical Learning Introduction 2 für Informatik 2 Example: Learn to recognise important classes of games Difference Analytical Learning between Inductive and Goal: Recognise chessboard positions in which black will lose its queen within two moves Induction can be employed <=> Problem: thousands of training examples similar to this one are needed Suggested target hypothesis: board position in which the black king and queen are simultaneously attacked Not suggested: board position in which four white pawns are still in their original location Explanations of human beings provide the information needed to rationally generalise from details Prior 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 complex Goal: learning algorithm that automatically constructs and learns a move from such explanations Analytical learning methods seek a hypothesis that fits the learner's prior knowledge and covers the training examples Explanation based learning is a form of analytical learning in which the learner processes each new training example by Explaining the observed target value for this example in terms of the domain theory Analysing this explanation to determine the general conditions under which the explanation holds Refining its hypothesis to incorporate these general conditions 1
Difference Analytical between Learning Inductive (2) and New Example Difference: They assume two different formulations of the learning problem: Inductive learning: input: hypothesis space H + set of training examples D= { x,f 1 ( x 1),..., x n,f ( xn ) } output: hypothesis h, that is consistent with these training examples Analytical learning: input: hypothesis space H + set of training examples D= x,f 1 x 1,..., x n,f xn + 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 { ( ) ( ) } Given: 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 times Target concept: SafeToStack ( x,y) Volume ( x,vx ) Volume ( y,vy ) LessThan ( vx,vy ) 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) Difference Analytical between Learning Inductive (3) and New Example 2 Learning PROLOG-EBG with Perfect Domain Theories: Illustration: f ( x i ) is True if x i 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 Domain Theory B: SafeToStack ( x, y ) Fragile ( y ) SafeToStack ( x,y ) Lighter ( x, y) Lighter ( x, y) Weight ( x,wx ) Weight ( y,wy ) LessThan ( wx, wy) Weight ( x, w ) Volume( x, v) Density ( x, d ) Equal( w, ( v,d )) W eight ( x,5) Type ( x, Endtable) Fragile ( x ) M aterial( x, Glass ) Determine: A hypothesis from H consistent with the training examples and the domain theory Learning with Perfect Domain Theories: PROLOG-EBG An Illustrative Trace Explanation-Based Learning of Search Control Knowledge A domain theory is said to be correct if each of its assertions is a truthful statement about the world A 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. 2
Learning PROLOG-EBG with (2) Perfect Domain Theories: Learning PROLOG-EBG with (4) Perfect Domain Theories: An Illustrative Trace (3) 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 theory It 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. Repeatedly: Domain theory is correct and complete this explanation constitutes a proof that the training examples satisfy the target concept PROLOG-EBG(TargetConcept, TrainingExamples, Domain Theory) LearnedRules { } Pos the positive examples from TrainingExamples for each PositiveExample in Pos that is not covered by LearnedRules do Explain Explanation an explanation (proof) in terms of the DomainTheory that PositiveExample satisfies the TargetConcept Analyse Sufficient Condition the most general set of features of PositiveExample sufficient to satisfy the TargetConcept according to the Explanation Refine LearnedRules LearnedRules + NewHornClause, where NewHornClause is of the form TargetConcept SufficientConditions In 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. General rule justified by the domain theory: SafeToStack ( x,y) Volume ( x,2) Density ( x,0.3 ) Type ( y,endtable ) leaf node in the proof tree expects Equal(0.6,times(2,03) and LessThan(0.6,5) Learning PROLOG-EBG with (3) Perfect Domain Theories: An Illustrative Trace (2) An Illustrative Trace (4) PROLOG-EBG (Kedar-Cabelli and McCarthy 1987) Sequential covering algorithm When 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 examples Output: set of logically sufficient conditions for the target concept, according the domain theory 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 Explanation of the training example forms the proof for the correctness of this rule PROLOG-EBG computes the most general rule that can be justified by the explanation, by computing the weakest preimage of the explanation Definition: 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: SafeToStack ( x,y) Volume ( x,vx ) Density ( x,dx ) Equal ( wx,times ( vx,dx )) LessThan ( wx,5) Type( y,endtable) PROLOG_EBG computes the weakest preimage of the target concept with respect to the explanation, using a general procedure called regression Regression: go iteratively backward through the explanation, first computing the weakest preimage of the target concept with respect to the final proof step in the explanation Computing the weakest preimage of the resulting expressions with respect to the proceeding step and so on 3
An Illustrative Trace (5) Remarks on Explanation-Based Learning Remarks Explanation-Based Learning 2 Discovering new features An Illustrative Trace 5 Remarks on Explanation-Based Learning Remarks Explanation-Based Learning 3 Key properties: PROLOG-EBG produces justified general hypotheses by using prior knowledge to analyse individual examples The explanation about the way how an example satisfies the target concept determines which example attributes are relevant: the ones mentioned by the explanation Regressing the target concept to determine its weakest preimage with respect to the explanation allows deriving more general constraints on the values of relevant features Each learned Horn clause corresponds to a sufficient condition for satisfying the target concept The 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 considered Implicitly assumes that the domain theory is correct and complete 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 learning EBL as example guided reformulation of theories: Method for reformulating the domain theory into more operational form: Creating rules that: Deductively follow the domain theory Classify the observed training examples in a single inference step 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 great Situation: 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. 4
Remarks Explanation-Based Learning 4 Knowledge Compilation: EBL involves reformulating the domain theory to produce general rules that classify examples in a single inference step Summary Discovering new features are needed to describe the general rule underlying the training examples Summary Interesting capability: Ability to formulate new features that are not explicitly in the description of the training examples but that This feature is similarly represented by the hidden units of neural networks Like the BACKPROPAGATION algorithm, PROLOG_EBG automatically formulates such features in its attempt to fit the training data BUT: In neural networks it's developed in a statistical process PROLOG-EBG it's derived in an analytical process Example: derives the feature Volume Density >5 PROLOG-EBG Uses first order Horn clauses in its domain theory and in its learned hypotheses The explanation is a PROLOG proof The hypothesis extracted from the explanation is the weakest preimage of this proof Analytical 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 solving Disadvantage: purely deductive implementations such as PROLOG- EBG produce a correct output if the domain theory is also correct 5