Chapter 11: Analytical Learning

1. Chapter 11: Analytical Learning Inductive learning training examples Analytical learning prior knowledge + deductive reasoning Explanation based learning - prior knowledge : analyze, explain how each training example satisfies the target concept - distinguish relevant features = generalization based on logical reasoning - applied to learning search control rules

2. Introduction • Inductive learning poor performance when insufficient data • Explanation based learning (1) accept explicit prior knowledge (2) generalize more accurately than inductive system (3) prior knowledge = reduce complexity of hypotheses space reduce sample complexity improve generalization accuracy

3. Task of learning play chess - target concept chessboard positions in which black will lose its queen within two moves - human explain/analyze training examples by prior knowledge - knowledge = legal rules of chess

4. Chapter summary - Learning algorithm that automatically construct/learn from explanation - Analytical learning problem definition - PROLOG-EBG algorithm - General properties, relationship to inductive learning algorithm - Application to improving performance at large state-space search problems

5. Inductive Generalization Problem • Given: Instances Hypotheses Target Concept Training examples of target concept • Determine: Hypotheses consistent with the training examples

6. Analytical Generalization Problem • Given: Instances Hypotheses Target Concept Training Examples Domain Theory • Determine: Hypotheses consistent with training examples and domain theory

7. Example of an analytical learning problem • Instance space : describe a pair of objects Color, Volume, Owner, Material, Density, On • Hypothesis space H SafeToStack(x,y) Volume(x,vx) ^ Volume(y,vy) ^ LessThan(vx,vy) • Target concept : SafeToStack(x,y) “pairs of physical objects, such that one can be stacked safely on the other”

8. 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,Cardboard) • Color(Obj2,Blue) Material(Obj2,Wood) • Volume(Obj1,2) • 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,times(v,d)) Weight(x,5) Type(x,Endtable) Fragile(x) Material(x,Glass) .......

9. Domain Theory B - explain why certain pairs of objects can be safely stacked on one another - described by a collection of Horn clause : enable system to incorporate any learned hypotheses into subsequent domain theories

10. Learning with Perfect Domain Theories : PROLOG-EBG • Correct = assertions are truthful statements • Complete = covers every positive examples • Perfect domain theory is available? Yes • Why does it need to learn when perfect domain theory is given?

11. PROLOG-EBG • Operation (1) Leaning a single Horn clause rule (2) Removing positive examples covered (3) Iterating this process • Given a complete/correct domain theory --> output a hypothesis (correct, cover observed positive training examples)

12. PROLOG-EBG Algorithm PROLOG-EBG(Target Concept,Training Examples,Domain Theory) • Learned Rules {} • Pos the positive examples from Training Examples • for each Positive Examples in Pos that is not covered by Learned Rules, do 1. Explain: - Explanation explanation in terms of Domain Theory that Positive Examples satisfies the Target Concept 2. Analyze: - Sufficient Conditions the most general set of features of Positive Examples sufficient to satisfy the Target Concept according to the Explanation 3. Refine: - Learned RulesLearned Rules + NewHornClause Target ConceptSufficient Conditions • Return Learned Rules

14. Weakest Preimage 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 the most general rules SafeToStack(x,y) Volume(x,vx) ^ Density(x,dx) ^ Equal(wx,times(vx,dx)) ^ LessThan(wx,5) ^ Type(y,Endtable)

17. Remarks on Explanation-Based Learning • Properties (1) justified general hypotheses by using prior knowledge (2) explanation determines relevant attributes (3) regressing the target concept allows deriving more general constraints (4) learned Horn clause = sufficient condition to satisfy target concept (5) implicitly assume the complete/correct domain theory (6) generality of the Horn clause depends on the formulation of the domain theory

18. Perspectives on example based learning (1) EBL as theory-guided generalization (2) EBL as example-guided reformation of theories (3) EBL as “just” restating what the learner already “knows” • Knowledge compilation - reformulate the domain theory to produce general rules that classify examples in a single inference step - transformation = efficiency improving task without altering correctness of system’s knowledge

19. Characteristics • Discovering New Features - learned feature : feature by hidden units of neural networks • Deductive Learning - background knowledge of ILP : enlarge the set of hypotheses - domain theory : reduce the set of acceptable hypotheses • Inductive Bias - inductive bias of PROLOG-EBG = domain theory B - Approximate inductive bias of PROLOG-EBG = domain theory B + preference for small sets of maximally general Horn clauses

20. LEMMA-ENUMERATOR algorithm - enumerate all proof trees - for each proof tree, calculate the weakest preimage and construct a Horn clause - ignore the training data - output a superset of Horn clauses output by PROLOG-EBG • Role of training data focus algorithm on generating rules that cover the distribution of instances that occur in practice • Observed positive example allow generalizing deductively to other unseen instances IF ((PlayTennis = YES) (Humidity=x)) THEN ((PlayTennis = YES) (Humidity <= x)

21. Knowledge-level learning - the learned hypothesis entails predictions that go beyond those entailed by the domain theory - deductive closure : set of all predictions entailed by a set of assertions • Determinations - some attribute of the instance is fully determined by certain other attributes, without specifying the exact nature of the dependency - example target concept : “people who speak Portuguese” domain theory : “ the language spoken by a person is determined by their nationality” training example : “Joe, 23-year-old Brazilian, speaks Portuguese” conclusion : “all Brazilians speak Portuguese”

22. Explanation-based Learning of Search Control Knowledge • Speed up complex search programs • Complete/Correct domain theory for learning search control knowledge = definition of legal search operator + definition of the search objective • Problem find a sequence of operators that will transform an arbitrary initial state S to some final state F that satisfies the goal predicate G

23. PRODIGY • Domain-independent planning system • find a sequence of operators that leads from S to O • means-ends planner decompose problems into subgoals solve them combine their solution into a solution for the full problem

24. SORA System • Support a broad variety of problem-solving strategies • Learned by explaining situations in which its current strategy leads to inefficiencies

25. Practical Problems applying EBL to learning search control • The number of control rules that must be learned is very large (1) efficient algorithms for matching rules (2) utility analysis : estimating the computational cost and benefit of each rule (3) identify types of rules that will be costly to match re-expressing such rules in more efficient forms optimizing rule-matching algorithm

26. Constructing the explanations for the desired target concept is intractable (1) example - states for which operator A leads toward the optimal solution (2) “lazy” or “incremental” explanation - heuristics are used to produce partial/approximate and tractable explanation - learned rules may be imperfect - monitoring performance of the rule on subsequent cases - when error, original explanation is elaborated to cover new case, - more refined rules is extracted