Advanced Artificial Intelligence Lecture 5: Inductive Logic Programming

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Advanced Artificial IntelligenceLecture 5
Inductive Logic Programming

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University

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Outline

• Inductive Logic Programming
• FOIL

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What is Relational Learning?

• Relational Learning systems use a representation
  language which goes beyond the propositional
• Permitting learning about relationships between
  data items
• a definition of a sort or append function
• family relationships
• spatial or geometric relationships
• temporal relationships
• Relational systems
• Inductive Logic Programming
• Genetic Programming
• Recurrent neural networks can learn simple
  relationships
• Its often possible to turn a known relationship
  into a propositional learning problem
• Eg learning time series

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Relational Learning - status

• Still largely a research domain rather than
  practical applications
• Relational learning is difficult
• Computationally expensive
• Data expensive
• Finding good algorithms is difficult
• Hence achievements so far are limited
• An important research domain because
• Relational problems are widespread in practical
  applications
• There are often no alternative approaches to
  these problems

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Inductive Logic Programming

• Representation Language
• Prolog (ie first order predicate rules)
• Some systems slightly extend the representation
  language
• Learning Algorithms
• Usually Deterministic
• Some stochastic (evolutionary) approaches have
  been tried, but with limited success
• Generally gradient descent algorithms, often
  extended with special-purpose heuristic

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Some Terminology A Reminder

• An atom a is a formula of the form
• P(x1,, xn)
• (in general logic, x1,, xn may contain function
  symbols, but almost ILP systems are limited to
  the case where there are no function symbols)
• A literal is either an atom or the negation of an
  atom
• A (predicate calculus or relational) rule is a
  formula of the form
• L1 Ln ? L0
• Where all of the Li are literals
• And in particular, L0 is a positive literal (ie
  an atom)

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Learning Example

• For example, a relational learner might be asked
  to learn the member predicate from examples
• That is, given
• member(1,1).
• member(1,1,2).
• member(2,1,2).
• Etc.
• It should learn
• member(X,YYs) - X Y.
• member(X,YYs - member(X,Ys).

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Negative Examples

• Many relational learning systems require also
  negative examples for their learning, such as
• That is, given
• Not member(2,1).
• Not member(3,1,2).
• Not, member(4,1,2,3).
• Etc.
• Alternatively, such systems may rely on the
  Closed World Assumption
• I.e. anything not included in the positive
  examples is assumed to be a negative example

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FOIL (Quinlan)

• FOIL is probably the easiest of the well-known
  ILP systems to understand
• Outer loop of the algorithm
• Find a rule that covers some of the positive
  instances (and no negatives)
• Store the rule and remove the covered data from
  the dataset
• Re-run the algorithm on the reduced dataset

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FOIL Example

• In learning the member predicate, FOIL might
  first learn
• member(X,YYs) - X Y.
• Then FOIL would be left with the reduced dataset
• member(2,1,2).
• member(2,1,2,3).
• member(3,1,2,3).
• etc

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FOIL Inner Loop

• The inner loop generates individual rules
• A general-to-specific algorithm
• Learning a rule of the form Hyp ? Conc
• FOIL begins with Hyp empty
• ie with the rule that says the conclusion is
  always satisfied
• Adds literals one by one to Hyp
• Using a greedy (ie non-backtracking hillclimbing)
  search algorithm.

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FOIL Inner Loop Example

• In our example, FOIL would start with the clause
• member(X,YYs).
• Of course, this would have an unacceptable error
  rate.
• FOIL would try
• member(X,YYs) - P.
• with many different Ps, one being XY
• This would have an acceptable error rate, so the
  inner loop would terminate

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FOIL Heuristic

• How does FOIL decide what literal to add next?
• FOIL uses the information-gain heuristic
• (same as C4.5)
• The next literal added is that which produces the
  highest information gain relative to the data
• We will look at this in more detail later

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FOIL Heuristic

• However the information gain heuristic is biased
  by a couple of extra factors
• Introducing variables
• Its generally useful to introduce new variables
  into the developing rule
• So FOIL gives a small positive bonus to literals
  that introduce new variables
• Determinacy
• Next slide

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FOIL and Determinacy

• A literal is determinate relative to the literals
  already in the developing rule if
• For each combination of values of old variables
  already in the rule
• All new variables have only one corresponding
  value in the dataset
• Determinate literals are often useful in
  generating rules
• So FOIL has an if all else fails heuristic
• If no new literal produces sufficient
  information gain
• Where sufficient is a threshold set by the user
• Then FOIL introduces a new determinate literal
• To avoid possible infinite loops, FOIL sets a
  bound on the depth of determinate literals it can
  introduce
• Usually this heuristic introduces too many new
  literals
• However FOIL has a pruning stage that removes
  most of them again

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Entropy

• Denote the positive training instances by T, and
  the positive instances covered by a particular
  clause C as T?C
• Then we can estimate the probability that a
  particular instance is covered by C as pC T?C
  / T
• (there are some additional complications relating
  to the effects of new variables introduced by
  clause C, but these need not concern us here)
• Next, we can define the information entropy of
  the clause against the training data as
• Entropy(C) - pC log2 pC
• Entropy is a measure of the minimum number of
  bits required to encode the status of a
  randomly-drawn member of T

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Information Gain

• The change in entropy due to adding a new literal
  is known as the information gain
• The literal chosen by FOIL will that which leads
  to the greatest information gain
• (with the caveats noted before)

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FOIL Pruning Stage

• The information gain of each literal is
  calculated relative to the rest of the rule
• The literal with the least information gain is
  deleted
• Until no literal with information gain less than
  a predetermined threshold can be found

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