Explanation-Based Learning

1
Explanation-Based Learning

• (borrowed from mooney et al)

2
Explanation-Based Learning (EBL)

• One definition
• Learning general problem-solving techniques by
  observing and analyzing solutions to specific
  problems.

3
SBL (vs. EBL)lots of data (examples)

• Similarity-based learning (SBL) are inductive
• generalizes from training data
• empirically identifies patterns that distinguish
  between positive and negative examples of a
  target concept.
• Inductive results are justified empirically
  (e.g., by statistical arguments such as those
  used in establishing theoretical results in PAC
  learning).
• Generally requires significant numbers of
  training examples in order to produce
  statistically justified conclusions.
• Generally does not require or exploit background
  knowledge.

4
EBL (vs. SBL)lots of knowledge

• Explanation-based learning (EBL) is (usually)
  deductive
• uses prior knowledge to explain each training
  example
• Explanation identifies what properties are
  relevant to the target function and which are
  irrelevant.
• Prior knowledge is used to reduce the hypothesis
  space and focus the learner on hypotheses that
  are consistent with prior knowledge about the
  target concept.
• Accurate learning is possible from very few (0)
  training examples (typically 1 example per
  learned rule).

5
The EBL Hypothesis

• By understanding why an example is a member of a
  target concept, one can learn the essential
  properties of the concept
• Trade-off
• the need to collect many examples
• for
• the ability to explain single examples (via a
  domain theory)
• This assumes the domain theory is competent
• Correct does not entail that any negative
  example is positive
• Complete each positive example can be
  explained
• Tractable an explanation can be found for each
  positive example.

6
SBL vs. EBLentailment constraints

• SBL
• Hypothesis Descriptions Classifications
• Hypothesis is selected from restricted
  hypothesis space.
• EBL
• Hypothesis Descriptions Classifications
• Background Hypothesis

7
EBL Task

• In addition to a set of training examples, EBL
  also takes as input a domain theory, background
  knowledge about the target concept that is
  usually specified as a set of logical rules (Horn
  clauses) and operationality criteria.
• The goal is to find an efficient or operational
  definition of the target concept that is
  consistent with both the domain theory and the
  training examples.

8
EBL Task operationalityobservable vs.
unobservable

• Operationality is often imposed by restricting
  the hypothesis space to using only certain
  predicates (e.g., those that are directly used to
  describe the examples).
• Observable predicates used to describe examples
• Unobservable the target concept
• In classical EBL the learned definition is
• logically entailed by the domain theory
• a more efficient definition of the target concept
  
• requires only look-up (pattern matching) using
  observable predicates rather than search (logical
  inference) mapping observables to unobservables.

9
EBL Task

• Given
• Goal concept
• Training example
• Domain Theory
• Operationality Criteria
• Find a generalization of the training example
  that is a sufficient criteria for the target
  concept and satisfies the operationality criteria

10
EBL Example

• Goal concept SafeToStack(x,y)
• Training Examples One example
• SafeToStack (Obj1,Obj2)
• On(Obj1,Obj2)
  Owner(Obj1,Molly)
• Type(Obj1,Box) Owner(Obj2,
  Muffet)
• Type(Obj2,Endtable) Fragile(Obj2)
• Color(Obj1,Red)
  Material(Obj1,Cardboard)
• Color(Obj2,Blue)
  Material(Obj2,Wood)
• Volume(Obj1, 0.1) Density(Obj1,0.1)

11
EBL Example

• Domain Theory
• SafeToStack(x,y) - not(Fragile(y)).
• SafeToStack(x,y) - Lighter(x,y).
• Lighter(x,y) - Weight(x,wx), Weight(y,wy), wx lt
  wy.
• Weight(x,w) - Volume(x,v), Density(x,d),
  wvd.
• Weight(x,5) - Type(x,Endtable).
• Fragile(x) - Material(x,Glass).
• Opertional predicates Type, Color, Volume,
  Owner, Fragile, Material, Density, On, lt, gt, .

12
EBL Method

• For each positive example not correctly covered
  by an
• operational rule do
• Explain Use the domain theory to construct a
  logical proof that the example is a member of the
  concept.
• Analyze Generalize the explanation to determine
  a rule that logically follows from the domain
  theory given the structure of the proof and is
  operational.
• Add the new rule to the concept definition.

13
EBL Example

• Training Example
• SafeToStack (Obj1,Obj2) Type(Obj2,Endtable)
• Volume(Obj1, 0.1)
  Density(Obj1,0.1)
• Domain Theory
• SafeToStack(x,y) - Lighter(x,y).
• Lighter(x,y) - Weight(x,wx), Weight(y,wy), wx lt
  wy.
• Weight(x,w) - Volume(x,v), Density(x,d),
  wvd.
• Weight(x,5) - Type(x,Endtable).

14
Example Explanation (Proof)
SafeToStack(Obj1,Obj2)
Lighter(Obj1,Obj2)
Weight(Obj1,0.6)
Weight(Obj2,5)
06.lt5
Volume(Obj1,2)
0.620.3
Type(Obj2.Endtable)
Density(Obj1,0.3)
15
Generalization

• Find the weakest preconditions A for a conclusion
  C such that A entails C using the given proof.
• The general target predicate is regressed through
  each rule used in the proof to produce
  generalized conditions at the leaves.
• To regress a set of literals P through a rule H
  - B1,...Bn (BB1,...Bn) using literal L
  element of P
• Let ? be the most general unifier of L and H
• apply the resulting substitution to all the
  literals in P and B
• and return P (P? - L?) U B?
• Also apply the substitution to update the
  conclusion CC?
• After regressing the general target concept
  through each rule used in the proof return C -
  P1,...Pn (PP1...Pn)

16
Generalization Example

• Regress SafeToStack(x,y) through
• SafeToStack(x1,y1) - Lighter(x1,y1).
• Unifier ? x/x1, y/y1
• Result Lighter(x,y)

17
Generalization Example

• Regress Lighter(x,y) through
• Lighter(x2,y2) - Weight(x2,wx2),
  Weight(y2,wy2), wx2 lt wy2.
• Unifier ? x/x2, y/y2
• ResultWeight(x,wx), Weight(y,wy), wx lt wy

18
Generalization Example

• Regress Weight(x,wx), Weight(y,wy), wx lt wy
  through
• Weight(x3,w) - Volume(x3,v), Density(x3,d),
  wvd.
• Unifeir ? x/x3, wx/w
• Result Volume(x,v), Density(x,d), wxvd,
• Weight(y,wy), wx lt wy

19
Generalization Example

• Regress Weight(y,wy) through
• Weight(x4,5) - Type(x4,Endtable).
• Unifier ? y/x4, 5/wy
• Result Volume(x,v), Density(x,d), wxvd,
• Type(y,Endtable), wx lt 5
• Learned Rule
• SafeToStack(x,y) - Volume(x,v),
  Density(x,d), wxvd,
• Type(y,Endtable), wx lt 5.

20
Re Generalization

• Simply substituting variables for constants in
  the proof will not work because
• Some constants (Endtable,5) may come from the
  domain theory and cannot be generalized and
  maintain soundness.
• Two instances of the same constant may or may not
  generalize to the same variable depending on
  structure of the proof (e.g. assume both the
  weight and density happened to be the same in the
  example, but they clearly dont have to be the
  same in general).
• Since generalization is basically performing a
  set of unifications and substitutions and these
  operations have linear time complexity,
  generalization is a quick, linear-time process.

21
Knowledge as Bias

• The hypotheses produced by EBL are obviously
  strongly biased by the domain theory it is given.
• Being able to alter the bias of a learning
  algorithm by supplying prior knowledge in
  declarative form (declarative bias) is very
  useful (e.g., by adding new rules and
  predicates).
• EBL assumes a complete and correct domain theory,
  but theory refinement and other methods can be
  biased by incomplete and incorrect domain
  theories.

22
Perspectives on EBL

• EBL as theory guided generalization of examples
• Explanations are used to distinguish relevant
  from irrelevant features.
• EBL as example guided reformulation of theories
• Examples are used to focus on which operational
  concept reformulations to learn are typical
• EBL as knowledge compilation Deductive
  consequences that are particularly useful (e.g.,
  for reasoning about the training examples) are
  compiled out to subsequently allow for more
  efficient reasoning.

23
Standard Approach to EBL
24
Knowledge-Level Learning (Newell, Dietterich)

• Knowledge closure
• all things that can be inferred from a collection
  of rules and facts
• Pure EBL only learns how to solve faster, not
  how to solve problems previously insoluble.
• Inductive learners make inductive leaps and hence
  can solve more after learning.
• EBL is often called Speed-up learning
• (not knowledge-level learning)
• What about considering resource-limits (e.g.,
  time) on problem solving?

25
Utility of Knowledge Compilation

• Deductive reasoning is difficult and frequently
  similar conclusions must be derived repeatedly.
• Some domains have complete and correct theories
  and learning involves deriving useful
  consequences that make reasoning more efficient,
  e.g. chess, mathematics, etc.

26
Utility of Knowledge Compilation

• Different types of knowledge compilation
• Static Not example-based, reformulate KB up
  front to make it more efficient for general
  inferences of a particular type.
• Dynamic Uses examples, perhaps, incrementally,
  to tune a system to improve efficiency on a
  particular distribution of problems.
• Dynamic systems like EBL make the inductive
  assumption that improving performance on a set of
  training cases will generalize to improved
  performance on subsequent test cases.

27
Utility Problem

• After learning many macro-operators, macro-rules,
  or search control rules, the time to match and
  search through this added knowledge may start to
  outweigh its benefits (Minton 1988)
• A learned rule must be useful in solving new
  problems frequently enough and save enough
  processing time in order to compensate for the
  time need to attempt to match it every time.
• Utility (AvgSavings x ApplicFreq) -
  AvgMatchCost
• EBL methods can frequently result in learning a
  set of rules with negative overall utility
  resulting in slowdown rather than the intended
  speedup.

28
Addressing the Utility Problem

• Improve Efficiency of Matching Preprocess
  learned rules to improve their match effiicency.
• Restrict Expressiveness Prevent learning of
  rules with combinatorial match costs.
• Selective Acquisition Only learn rules whose
  expected benefit outweighs their cost.
• Selective Retention Dynamically forget expensive
  rules that are rarely used.
• Selective Utilization Restrict the use of
  learned rules to avoid undue cost of application.

29
Imperfect Theories and EBL

• Incomplete Theory Problem
• Cannot build explanations of specific problems
  because of missing knowledge
• Intractable Theory Problem
• Have enough knowledge, but not enough computer
  time to build specific explanation
• Inconsistent Theory Problem
• Can derive inconsistent results from a theory
  (e.g., because of default rules)

30
Applications

• Planning (macro operators in STRIPS)
• Mathematics (search control in LEX)

31
Planning with Macro-Operators

• AI planning using Strips operators is search
  intensive.
• People seem to utilize canned plans to achieve
  everyday goals.
• Such pre-packaged planning sequences
  (macro-operators) can be learned by generalizing
  specific constructed or observed plans.
• Method is analogous to composing Horn-clause
  rules by generalizing proofs.
• A problem is solved by first trying to use
  learning macro-operators, falling back on general
  planning as a last resort.

32
STRIPS

• Original planning system which used means-ends
  analysis and theorem proving in robot planning
• Sample actions
• GoThru(A,D,R1,R2)
• Preconditions In(A,R1), Connects(D,R1,R2)
• Effects In(A,R2), In(A,R1)
• PushThru(A,O,D,R1,R2)
• Preconditions In(A,R1), In(O,R1)
  Connects(D,R1,R2)
• Effects In(A,R2), In(O,R2),In(A,R1), In(O,R1)

33
STRIPS

• Sample Problem
• State
• In(r,room1), In(box,room2),
  Connects(d1,room1,room2),
• Connects(d2,room2,room3)
• Goal In(box,room1)
• Sample Solution
• GoThru(r,d1,room1,room2)
• PushThru(r,box,d1,room2,room1)

34
Learned Macro-Operator

• EBL generalizing this plan produces the following
  macro-operator
• GoThruPushThru(A,D1,R1,R2,O,D2,R3)
• Preconditions
• InRoom(A,R1), InRoom(O,R2), Connects(D1,R1,R2),
  Connects(D2,R2,R3), (AO R1R2)
• Effects
• InRoom(O,R3), InRoom(A,R3), InRoom(A,R2),
  InRoom(O,R2), (R3R1) ? InRoom(A,R1)
• Extra preconditions needed to prevent
  precondition clobbering during execution of
  generalized plan.
• Conditional effects come from possible deletions
  in the generalized plan.

35
Representing Plan MACROPS

• Strips actually used a triangle table to
  implicitly store macros for every subsequence of
  the actions in the plan.
• Plan State OP1 ? OP2 ? OP3 ? OP4 ? OP5 Goal
• Op1
  
• Op1 Op2
• Op1 Op2 Op3
• Op1 Op2 Op3 Op4
• Op1 Op2 Op3 Op4 Op5
• The triangle table supports treating any of the
  10 subsequence of the generalized plan as a
  macrop in future problems.

36
Experimental Results

• Planning time with and without learning (minsec)

trial 1 2 3 4 5
No learn 305 942 703 1409 --
learning 305 354 634 437 913
37
Learning Search Control

• Search control rules are used to select operators
  during search.
• IF the state is of the form ? r f(x) dx,
• THEN apply the operator MoveConstantOutsideIntegra
  l
• Such search control rules can be learned by
  explaining how the application of an operator in
  a sample problem led to a solution
• ? 3sin(x)dx ? 3 ? sin(x)dx ? 3
  cos(x)
• Positive examples of when to apply an operator
  are states in which applying that operator leads
  to a solution, negative examples are states in
  which applying the operator leads away from the
  solution (i.e. another operator leads to the
  solution).
• Induction and combinations of explanation and
  induction can also be used to learn search
  control rules.

38
EBL variations

• Generalizing to N handling recursive rules in
  proofs
• Knowledge Deepening explaining shallow rules
• Explanation-based induction and abductive
  generalization

39
Generalizing to N(Shavlik, BAGGER2)

• Handling recursive or iterative concepts
• (recursive rules in proofs).

goal
P
1
2
P
P
3
4
P
P
5
6
Learned rules Goal ? P gen-2 P
? gen-3 V gen-5 V gen-6 V recursive-gen-1
V recursive-gen-2

40
Knowledge Deepening

• When two proofs, A and B, exist for a
  proposition, and proof A involves a single
  (shallow) rule, P?Q, and the weakest
  preconditions of proof B is equivalent to P, then
  proof B explains rule P?Q.
• Shallow rule leaves are green
• Explanation leaves are green because they
  contain mesophylls, which contain chlorophyll,
  which is a green pigment.

41
Knowledge Deepening

• (leaf ?x)

(green ?x)
Part(?x ?y) (isa ?y Mesophyll)
(green ?y)
Part(?y ?z) (isa ?z Chrorophyll)
(green ?z)
The weakest preconditions of both proofs are the
same (leaf ?x) Use the more complicated proof to
explain the shallow rule.
42
Explanation-Based InductionTeleology function
suggests structure

• Identify a teleologic explanation
• Structural properties supporting physiological
  goal
• leaf dehydration is avoided by the cutilcle
  covering the leafs epidermis
• Identify the weakest preconditions of the
  explanation.
• Separate into
• Structural preconditions epidermis covered by
  cuticle
• Qualifying preconditions performs transpiration
• Find other organs satisfying the qualifying
  conditions stems, flowers, fruit.
• Hypothesize they also have the structural
  conditions
• are the epidermises of stems, flowers, and
  fruit also covered by a cuticle?