Advanced Artificial Intelligence Lecture 3: Learning

1
Advanced Artificial IntelligenceLecture 3
Learning

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

2
Outline

• Defining Learning
• Kinds of Learning
• Generalisation and Specialisation
• Some Simple Learning Algorithms

3
References

• Mitchell, Tom M Machine Learning, McGraw-Hill,
  1997, ISBN 0 07 115467 1

4
Defining a Learning System (Mitchell)

• A program is said to learn from experience E
  with respect to some class of tasks T and
  performance measure P, if its performance at
  tasks in T, as measured by P, improves with
  experience E

5
Specifying a Learning System

• Specifying the task T, the performance P and the
  experience E defines the learning problem.
  Specifying the learning system requires us to
  define
• Exactly what knowledge is to be learnt
• How this knowledge is to be represented
• How this knowledge is to be learnt

6
Specifying What is to be Learnt

• Usually, the desired knowledge can be represented
  as a target valuation function V I ? D
• It takes in information about the problem and
  gives back a desired decision
• Often, it is unrealistic to expect to learn the
  ideal function V
• All that is required is a good enough
  approximation, V I ? D

7
Specifying How Knowledge is to be Represented

• The function V must be represented symbolically,
  in some language L
• The language may be a well-known language
• Boolean expressions
• Arithmetic functions
• .
• Or for some systems, the language may be defined
  by a grammar

8
Specifying How the Knowledge is to be Learnt

• If the learning system is to be implemented, we
  must specify an algorithm A, which defines the
  way in which the system is to search the language
  L for an acceptable V
• That is, we must specify a search algorithm

9
Structure of a Learning System

• Four modules
• The Performance System
• The Critic
• The Generaliser (or sometimes Specialiser)
• The Experiment Generator

10
Performance Module

• This is the system which actually uses the
  function V as we learn it
• Learning Task
• Learning to play checkers
• Performance module
• System for playing checkers
• (I.e. makes the checkers moves)

11
Critic Module

• The critic module evaluates the performance of
  the current V
• It produces a set of data from which the system
  can learn further

12
Generaliser/Specialiser Module

• Takes a set of data and produces a new V for the
  system to run again

13
Experiment Generator

• Takes the new V
• Maybe also uses the previous history of the
  system
• Produces a new experiment for the performance
  system to undertake

14
The Importance of Bias

• Important theoretical results from learning
  theory (PAC learning) tell us that learning
  without some presuppositions is infeasible.
• Practical experience, of both machine and human
  learning, confirms this.
• To learn effectively, we must limit the class of
  Vs.
• Two approaches are used in machine learning
• Language bias
• Search Bias
• Combined Bias
• Language and search bias are not mutually
  exclusive most learning systems feature both

15
Language Bias

• The language L is restricted so that it cannot
  represent all possible target functions V
• This is usually on the basis of some knowledge we
  have about the likely form of V
• It introduces risk
• Our system will fail if L does not contain an
  acceptable V

16
Search Bias

• The order in which the system searches L is
  controlled, so that promising areas for V are
  searched first

17
The DownsideNo Free Lunches

• Wolpert and MacReadys No Free Lunch Theorem
  states, in effect, that averaged over all
  problems, all biases are equally good (or bad).
• Conventional view
• The choice of a learning system cannot be
  universal
• It must be matched to the problem being solved
• In most systems, the bias is not explicit
• The ability to identify the language and search
  biases of a particular system is an important
  aspect of machine learning
• Some more recent systems permit the explicit and
  flexible specification of both language and
  search biases

18
No Free LunchDoes it Matter?

• Alternative view
• We arent interested in all problems
• We are only interested in prolems which have
  solutions of less than some bounded complexity
• (so that we can understand the solutions)
• The No Free Lunch Theorem may not apply in this
  case

19
Some Dimensions of Learning

• Induction vs Discovery
• Guided learning vs learning from raw data
• Learning How vs Learning That (vs Learning a
  Better That)
• Stochastic vs Deterministic Symbolic vs
  Subsymbolic
• Clean vs Noisy Data
• Discrete vs continuous variables
• Attribute vs Relational Learning
• The Importance of Background Knowledge

20
Induction vs Discovery

• Has the target concept been previously
  identified?
• Pearson cloud classifications from satellite
  data
• vs
• Autoclass and H - R diagrams
• AM and prime numbers
• BACON and Boyle's Law

21
Guided Learning vs Learning from Raw Data

• Does the learning system require carefully
  selected examples and counterexamples, as in a
  teacher student situation?
• (allows fast learning)
• CIGOL learning sort/merge
• vs
• Garvan institute's thyroid data

22
Learning How vs Learning That vs Learning a
Better That

• Classifying handwritten symbols
• Distinguishing vowel sounds (Sejnowski
  Rosenberg)
• Learning to fly a (simulated!) plane
• vs
• Michalski learning diagnosis of soy diseases
• vs
• Mitchell learning about chess forks

23
Stochastic vs DeterministicSymbolic vs
Subsymbolic

• Classifying handwritten symbols (stochastic,
  subsymbolic)
• vs
• Predicting plant distributions (stochastic,
  symbolic)
• vs
• Cloud classification (deterministic, symbolic)
• vs
• ? (deterministic, subsymbolic)

24
Clean vs Noisy Data

• Learning to diagnose errors in programs
• vs
• Greater gliders in the Coolangubra

25
Discrete vs Continuous Variables

• Quinlan's chess end games
• vs
• Pearson's clouds (eg cloud heights)

26
Attibute vs Relational Learning

• Predicting plant distributions
• vs
• Predicting animal distributions
• (because plants cant move, they dont care -
  much - about spatial relationships)

27
The importance of Background Knowledge

• Learning about faults in a satellite power supply
• general electric circuit theory
• knowledge about the particular circuit

28
Generalisation and Learning

• What do we mean when we say of two propositions,
  S and G, that G is a generalisation of S?
• Suppose skippy is a grey kangaroo.
• We would regard Kangaroos are grey as a
  generalisation of Skippy is grey.
• In any world in which kangaroos are grey is
  true, Skippy is grey will also be true.
• In other words, if G is a generalisation of
  specialisation S, then G is 'at least as true' as
  S,
• That is, S is true in all states of the world in
  which G is, and perhaps in other states as well.

29
Generalisation and Inference

• In logic, we assume that if S is true in all
  worlds in which G is, then
• G ? S
• That is, G is a generalisation of S exactly when
  G implies S
• So we can think of learning from S as a search
  for a suitable G for which G ? S
• In propositional learning, this is often used as
  a definition
• G is more general than S if and only if G ? S

30
Issues

• Equating generalisation and logical implication
  is only useful if the validity of an implication
  can be readily computed
• In the propositional calculus, validity is an
  exponential problem
• in the predicate calculus, validity is an
  undecidable problem
• so the definition is not universally useful
• (although for some parts of logic - eg learning
  rules - it is perfectly adequate).

31
A Common Misunderstanding

• Suppose we have two rules,
• 1) A ? ? ? G
• 2) A ? ? ? C ? G
• Clearly, we would want 1 to be a generalisation
  of 2
• This is OK with our definition, because
• ((A B ? G) ? (A B C ? G))
• is valid
• But the confusing thing is that ((ABC) ? (A??))
  is valid
• Iif you only look at the hypotheses of the rule,
  rather than the whole rule, the implication is
  the wrong way around
• Note that some textbooks are themselves confused
  about this

32
Defining Generalisaion

• We could try to define the properties that
  generalisation must satisfy,
• So let's write down some axioms. We need some
  notation.
• We will write 'S ltG G' as shorthand for 'S is
  less general than G'.
• Axioms
• Transitivity If A ltG B and B ltG C then also A
  ltG C
• Antisymmetry If A ltG B then it's not true that B
  ltG A
• Top there is a unique element, ?, for which it
  is always true that A ltG ?.
• Bottom there is a unique element, T, for which
  it is always true that T ltG A.

33
Picturing Generalisaion

• We can draw a 'picture' of a generalisation
  hierarchy satisfying these axioms

34
Specifying Generalisaion

• In a particular domain, the generalisation
  hierarchy may be defined in either of two ways
• By giving a general definition of what
  generalisation means in that domain
• Example our earlier definition in terms of
  implication
• By directly specifying the specialisation and
  generalisation operators that may be used to
  climb up and down the links in the generalisation
  hierarchy

35
Learning and Generalisaion

• How does learning relate to generalisation?
• We can view most learning as an attempt to find
  an appropriate generalisation that generalises
  the examples.
• In noise free domains, we usually want the
  generalisation to cover all the examples.
• Once we introduce noise, we want the
  generalisation to cover 'enough' examples, and
  the interesting bit is in defining what 'enough'
  is.
• In our picture of a generalisation hierarchy,
  most learning algorithms can be viewed as methods
  for searching the hierarchy.
• The examples can be pictured as locations low
  down in the hierarchy, and the learning algorithm
  attempts to find a location that is above all (or
  'enough') of them in the hierarchy, but usually,
  no higher 'than it needs to be'

36
Searching the Generalisaion Hierarchy

• The commonest approaches are
• generalising search
• the search is upward from the original examples,
  towards the more general hypotheses
• specialising search
• the search is downward from the most general
  hypothesis, towards the more special examples
• Some algorithms use different approaches.
  Mitchell's version space approach, for example,
  tries to 'home in' on the right generalisation
  from both directions at once.

37
Completeness and Generalisaion

• Many approaches to axiomatising generalisation
  add an extra axiom
• Completeness For any set S of members of the
  generalisation hierarchy, there is a unique
  'least general generalisation' L, which satisfies
  two properties
• 1) for every S in S, S ltG L
• 2) if any other L' satisfies 1), then L ltG L'
• If this definition is hard to understand, compare
  it with the definition of 'Least Upper Bound' in
  set theory, or of 'Least Common Multiple' in
  arithmetic

38
Restricting Generalisation

• Let's go back to our original definition of
  generalisation
• G generalises S iff G ? S
• In the general predicate calculus case, this
  relation is uncomputable, so it's not very useful
• One approach to avoiding the problem is to limit
  the implications allowed

39
Generalisation and Substitution

• Very commonly, the generalisations we want to
  make involve turning a constant into a variable.
• So we see a particular black crow, fred, so we
  notice
• crow(fred) ? black(fred)
• and we may wish to generalise this to
• ?X(crow(X) ? black(X))
• Notice that the original proposition can be
  recovered from the generalisation by substituting
  'fred' for the variable 'X'
• The original is a substitution instance of the
  generalisation
• So we could define a new, restricted
  generalisation
• G subsumes S if S is a substitution instance of G
• An example of our earlier definition, because a
  substitution instance is always implied by the
  original proposition.

40
Learning Algorithms

• For the rest of this lecture, we will work with a
  specific learning dataset (due to Mitchell)
• Item Sky AirT Hum Wnd Wtr Fcst Enjy
• 1 Sun Wrm Nml Str Wrm Sam Yes
• 2 Sun Wrm High Str Wrm Sam Yes
• 3 Rain Cold High Str Wrm Chng No
• 4 Sun Wrm High Str Cool Chng Yes
• First, we look at a really simple algorithm,
  Maximally Specific Learning

41
Maximally Specific Learning

• The learning language consists of sets of tuples,
  representing the values of these attributes
• A ? represents that any value is acceptable for
  this attribute
• A particular value represents that only that
  value is acceptable for this attribute
• A f represents that no value is acceptable for
  this attribute
• Thus (?, Cold, High, ?, ?, ?) represents the
  hypothesis that water sport is enjoyed only on
  cold, moist days.
• Note that our language is already heavily biased
  only conjunctive hypotheses (hypotheses built
  with ) are allowed.

42
Find-S

• Find-S is a simple algorithm its initial
  hypothesis is that water sport is never enjoyed
• It expands the hypothesis as positive data items
  are noted

43
Running Find-S

• Initial Hypothesis
• The most specific hypothesis (water sports are
  never enjoyed)
• h ? (f,f,f,f,f,f)
• After First Data Item
• Water sport is enjoyed only under the conditions
  of the first item
• h ? (Sun,Wrm,Nml,Str,Wrm,Sam)
• After Second Data Item
• Water sport is enjoyed only under the common
  conditions of the first two items
• h ? (Sun,Wrm,?,Str,Wrm,Sam)

44
Running Find-S

• After Third Data Item
• Since this item is negative, it has no effect on
  the learning hypothesis
• h ? (Sun,Wrm,?,Str,Wrm,Sam)
• After Final Data Item
• Further generalises the conditions encountered
• h ? (Sun,Wrm,?,Str,?,?)

45
Discussion

• We have found the most specific hypothesis
  corresponding to the dataset and the restricted
  (conjunctive) language
• It is not clear it is the best hypothesis
• If the best hypothesis is not conjunctive (eg if
  we enjoy swimming if its warm or sunny), it will
  not be found
• Find-S will not handle noise and inconsistencies
  well.
• In other languages (not using pure conjunction)
  there may be more than one maximally specific
  hypothesis Find-S will not work well here

46
Version Spaces

• One possible improvement on Find-S is to search
  many possible solutions in parallel
• Consistency
• A hypothesis h is consistent with a dataset D of
  training examples iff h gives the same answer on
  every element of the dataset as the dataset does
• Version Space
• The version space with respect to the language L
  and the dataset D is the set of hypotheses h in
  the language L which are consistent with D

47
List-then-Eliminate

• Obvious algorithm
• The list-then-eliminate algorithm aims to find
  the version space in L for the given dataset D
• It can thus return all hypotheses which could
  explain D
• It works by beginning with L as its set of
  hypotheses H
• As each item d of the dataset D is examined in
  turn, any hypotheses in H which are inconsistent
  with d are eliminated
• The language L is usually large, and often
  infinite, so this algorithm is computationally
  infeasible as it stands

48
Version Space Representation

• One of the problems with the previous algorithm
  is the representation of the search space
• We need to represent version spaces efficiently
• General Boundary
• The general boundary G with respect to language L
  and dataset D is the set of hypotheses h in L
  which are consistent with D, and for which there
  is no more general hypothesis in L which is
  consistent with D
• Specific Boundary
• The specific boundary S with respect to language
  L and dataset D is the set of hypotheses h in L
  which are consistent with D, and for which there
  is no more specific hypothesis in L which is
  consistent with D

49
Version Space Representation 2

• A version space may be represented by its general
  and specific boundary
• That is, given the general and specific
  boundaries, the whole version space may be
  recovered
• The Candidate Elimination Algorithm traces the
  general and specific boundaries of the version
  space as more examples and counter-examples of
  the concept are seen
• Positive examples are used to generalise the
  specific boundary
• Negative examples permit the general boundary to
  be specialised.

50
Candidate Elimination Algorithm

• Set G to the set of most general hypotheses in L
• Set S to the set of most specific hypotheses in L
• For each example d in D

51
Candidate Elimination Algorithm

• If d is a positive example
• Remove from G any hypothesis inconsistent with
  d
• For each hypothesis s in S that is not
  consistent with d
• Remove s from S
• Add to S all minimal generalisations h of s
  such that h is consistent with d, and some
  member of G is more general than h
• Remove from S any hypothesis that is more
  general than another hypothesis in S

52
Candidate Elimination Algorithm

• If d is a negative example
• Remove from S any hypothesis inconsistent with
  d
• For each hypothesis g in G that is not
  consistent with d
• Remove g from G
• Add to G all minimal specialisations h of g
  such that h is consistent with d, and some
  member of S is more specific than h
• Remove from G any hypothesis that is less
  general than another hypothesis in G

53
Summary

• Defining Learning
• Kinds of Learning
• Generalisation and Specialisation
• Some Simple Learning Algorithms
• Find-S
• Version Spaces
• List-then-Eliminate
• Candidate Elimination

54
?????