CS 9633 Machine Learning Explanation Based Learning

1
CS 9633 Machine LearningExplanation Based
Learning
2
Analytical Learning

• Inductive learning
• Given a large set of examples generalize to find
  features that distinguish positive and negative
  examples
• Examples include NNs, GAs, Decision trees,
  support vector machines, etc.
• Problem is that they perform poorly with very
  small training sets
• Analytical learning combines examples and
  domain model

3
Learning by People

• People can often learn a concept from a single
  example.
• They appear to do this by analyzing the example
  in terms of previous knowledge to determine the
  most relevant features.
• Some inductive algorithms use domain knowledge to
  increase the hypothesis space
• Explanation based learning uses domain knowledge
  to decrease the size of the hypothesis space.

4
Example
Positive example of Chess positions in which
black will lose its queen within two moves
5
Inductive versus Analytical Learning

• Inductive Learning Given a hypothesis space H,
  set of training examples D, desired output is
  hypothesis consistent with training examples.

• Analytical Learning Given hypothesis space H,
  set of training examples D, and a domain theory
  B, the desired output is hypothesis consistent
  with B and D.

6
SafeToStack Problem Instances

• Instance Space Each instance describes a pair
  of objects represented by the predicates
• Type (Ex. Box, Endtable, )
• Color
• Volume
• Owner
• Material
• Density
• On

7
SafeToStack Hypothesis Space

• Hypothesis space H is a set of Horn clause rules.
• The head of each rule is a literal containing the
  target predicate SafeToStack
• The body of each rule is a conjunction of
  literals based on
• The predicates used to describe the instances
• Additional general purpose predicates like
• LessThan
• Equal
• Greater
• Additional general purpose functions like
• Plus
• Minus
• Times
• SafeToStack(x,y) ?Volume(x,vx) ? Volume(y,vy) ?
  LessThan(vx,vy)

8
SafeToStack Target Concept

• SafeToStack(x,y)

9
SafeToStack Training Examples

• SafeToStack(Obj1, Obj2)
• On(Obj1, Obj2)
• Type(Obj1,Box)
• Type(Obj2, Endtable)
• Color(Obj1, Red)
• Color(Obj2, Blue)
• Volume(Obj1, 2)
• Owner(Obj1, Fred)
• Owner(Obj2, Louise)
• Density(Obj1, 0.3)
• Material(Obj1, Cardboard)
• Material(Obj2, Wood)

10
SafeToStack 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)

11
Analytical Learning Problem

• We must provide a domain theory sufficient to
  explain why observed positive examples satisfy
  the target concept.
• The domain theory is a set of Horn clauses.

12
Learning with Perfect Domain Theories

• Prolog EBG is an example system.
• Domain theory must be
• Correct
• Complete with respect to target concept and
  instance space

13
Reasonableness of Perfect Domain Theories

• In some cases it is feasible to develop a perfect
  domain theory (chess is an example). Can help
  improve the performance of search intensive
  planning and optimization problems.
• It is often not feasible to develop a perfect
  domain theory. Must be able to generate
  plausible explanations

14
Prolog-EBL (see Table 11.2 for details)

• For each new positive training example not yet
  covered by a learned Horn clause, form a new Horn
  clause by
• Explaining the new positive training example by
  proving its truth
• Analyzing this explanation to determine an
  appropriate generalization
• Refine the current hypothesis by adding a new
  Horn clause to cover this positive example as
  well as other similar instances.

15
1. Explaining the Training Example

• Provide a proof that the training example
  satisfies the target concept.
• If the domain theory is correct and complete, use
  a proof procedure like resolution.
• If the domain theory is not correct and complete,
  must extend proof procedure to allow plausible
  approximate arguments.

16
SafeToStack(Obj1,Obj2)
Weight(Obj1,0.6)
LessThan(0.6,5)
Weight(Obj2,5)
Type(Obj2,Endtable)
Volume(Obj1,2)
Density(Obj1,0.3)
Equal(0.6,20.3)
17
EndTable
Type
Obj2
On
Material
Wood
Owner
Obj1
Density
Volume
0.3
Color
Louise
2
Blue
Type
Material
Color
Owner
Box
Cardboard
Red
Fred
18
2. Generating a General Rule

• General rule from domain theory
• SafeToStack(x,y)?Volume(x,2)?Density(x,0.3)?Type(y
  , EndT)
• Note that we omitted the leaf nodes that are
  always satisfied independent of x and y
• Equal(0.6, times(2,0.3))
• LessThan(0.6, 5)
• However, we would like an even more general rule

19
Weakest Preimage

• Goal is to compute the most general rule that can
  be justified by the explanation.
• We do this by computing the weakest preimage
• Definition the weakest preimage of a conclusion
  C with respect to proof P is the most general set
  of assertions A, such that A entails C according
  to P.

20
Most General Rule

• The most general rule that can be justified by
  the explanation is
• SafeToStack(x,y)?Volume(x,vx)?Density(x,dx)?Equal(
  wx,times(vx,dx))?LessThan(wx,5) ?Type(y, EndT)
• Use general procedure called regression to
  generate this rule
• Start with the target concept with respect to the
  final step in the explanation
• Generate weakest preimage of the target concept
  with respect to the preceding step
• Terminate after iterating over all steps in the
  explanation.

21
SafeToStack(Obj1,Obj2) SafeToStack(x,y)
22
SafeToStack(Obj1,Obj2) SafeToStack(x,y)
Lighter(Obj1,Obj2) Lighter(x,y)
23
SafeToStack(Obj1,Obj2) SafeToStack(x,y)
Lighter(Obj1,Obj2) Lighter(x,y)
Weight(Obj1,0.6) Weight(x,wx)
LessThan(0.6,5) LessThan(wx,wy)
Weight(Obj2,5) Weight(y,wy)
24
SafeToStack(Obj1,Obj2) SafeToStack(x,y)
Lighter(Obj1,Obj2) Lighter(x,y)
Weight(Obj1,0.6) Weight(x,wx)
LessThan(0.6,5) LessThan(wx,wy)
Weight(Obj2,5) Weight(y,wy)
Volume(Obj1,2) Density(Obj1, 0.3)
Equal(0.6,20.3) Volume(x,xv)
Density(x,dx) Equal(wx,vxdx)
LessThan(wx,wv) Weight(y,wy)
25
SafeToStack(Obj1,Obj2) SafeToStack(x,y)
Lighter(Obj1,Obj2) Lighter(x,y)
Weight(Obj1,0.6) Weight(x,wx)
LessThan(0.6,5) LessThan(wx,wy)
Weight(Obj2,5) Weight(y,wy)
Volume(Obj1,2) Density(Obj1, 0.3)
Equal(0.6,20.3) Volume(x,xv)
Density(x,dx) Equal(wx,vxdx)
LessThan(wx,wy) Weight(y,wy)


Type((obj2, EndT) Volume(x,xv)
Density(x,dx) Equal(wx,vxdx)
LessThan(wx,5) Type(y,EndT)
26
3. Refine the Current Hypothesis

• The current hypothesis is the set of Horn clauses
  learned so far.
• At each stage, a new positive example is picked
  that is not yet covered by the current hypothesis
  and a new rule is developed to cover it.
• Only positive examples are covered by the rules.
• Instances not covered by the rules are classified
  as negative (negation-as-failure approach)

27
EBL Summary

• Individual examples are explained (proven) using
  prior knowledge
• Attributes included in the proof are considered
  relevant.
• Regression is used to generalize the rule.
• Generality of learned clauses depends on the
  formulation of the domain theory, the order in
  which examples are encountered, and other
  instances that share the same explanation.
• Assumes domain theory is complete and correct.

28
Different Perspectives on EBL

• EBL is a theory-guided generalization of
  examples.
• EBL is an example-guided reformulation of
  theories. Rules created that
• Follow deductively from the domain theory
• Classify the observed training examples in a
  single inference step
• EBL is just a restating of what the learner
  already knows (knowledge compilation)

29
Inductive Bias of EBL

• Domain theory
• Algorithm (sequential covering) used to choose
  among alternative Horn clauses.
• Generalization procedure favors small sets of
  Horn clauses.

30
EBL for Search Strategies

• Requirement for correct and complete domain
  theory is often difficult to meet, but can often
  be met in complex search tasks.
• This type of learning is called speedup learning.
• Can use EBL to learn efficient sequences of
  operators (evolve meta-operators)