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Advanced Artificial Intelligence Lecture 3: Learning. Bob McKay School of Computer Science and Engineering College of Engineering Seoul National University. Outline. Defining Learning Kinds of Learning Generalisation and Specialisation Some Simple Learning Algorithms. References.

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Advanced artificial intelligence lecture 3 learning l.jpg

Advanced Artificial IntelligenceLecture 3: Learning

Bob McKay

School of Computer Science and Engineering

College of Engineering

Seoul National University


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Outline

  • Defining Learning

  • Kinds of Learning

  • Generalisation and Specialisation

  • Some Simple Learning Algorithms


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References

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


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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”


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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


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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


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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


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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


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Structure of a Learning System

  • Four modules

    • The Performance System

    • The Critic

    • The Generaliser (or sometimes Specialiser)

    • The Experiment Generator


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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)


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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


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Generaliser/Specialiser Module

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


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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


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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 V’s.

  • 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


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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’


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Search Bias

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


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The Downside:No Free Lunches

  • Wolpert and MacReady’s 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


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No Free Lunch:Does it Matter?

  • Alternative view

    • We aren’t 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


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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


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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


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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


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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


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    Stochastic vs Deterministic;Symbolic vs Subsymbolic

    • Classifying handwritten symbols (stochastic, subsymbolic)

  • vs

    • Predicting plant distributions (stochastic, symbolic)

  • vs

    • Cloud classification (deterministic, symbolic)

  • vs

    • ? (deterministic, subsymbolic)


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    Clean vs Noisy Data

    • Learning to diagnose errors in programs

  • vs

    • Greater gliders in the Coolangubra


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    Discrete vs Continuous Variables

    • Quinlan's chess end games

  • vs

    • Pearson's clouds (eg cloud heights)


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    Attibute vs Relational Learning

    • Predicting plant distributions

  • vs

    • Predicting animal distributions

      • (because plants can’t move, they don’t care - much - about spatial relationships)


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    The importance of Background Knowledge

    • Learning about faults in a satellite power supply

      • general electric circuit theory

      • knowledge about the particular circuit


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    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.


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    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


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    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).


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    A Common Misunderstanding

    • Suppose we have two rules,

      • 1) A ∧ Β → G

      • 2) A ∧ Β ∧ 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 ((A^B^C) → (A∧Β)) 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


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    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 <G G' as shorthand for 'S is less general than G'.

    • Axioms:

      • Transitivity: If A <G B and B <G C then also A <G C

      • Antisymmetry: If A <G B then it's not true that B <G A

      • Top: there is a unique element, ⊥, for which it is always true that A <G⊥.

      • Bottom: there is a unique element, T, for which it is always true that T <GA.


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    Picturing Generalisaion

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


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    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


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    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'


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    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.


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    Completeness and Generalisaion

    • Many approaches to axiomatising generalisation add an extra axiom:

      • Completeness: For any set Σ of members of the generalisation hierarchy, there is a unique 'least general generalisation' L, which satisfies two properties:

        • 1) for every S in Σ, S <GL

        • 2) if any other L' satisfies 1), then L <GL'

      • 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


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    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


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    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.


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    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


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    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 ‘φ’ 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.


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    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


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    Running Find-S

    • Initial Hypothesis

      • The most specific hypothesis (water sports are never enjoyed):

      • h ← (φ,φ,φ,φ,φ,φ)

    • 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)


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    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,?,?)


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    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 it’s 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


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    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


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    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


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    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


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    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.


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    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:


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    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


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    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


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    Summary

    • Defining Learning

    • Kinds of Learning

    • Generalisation and Specialisation

    • Some Simple Learning Algorithms

      • Find-S

      • Version Spaces

        • List-then-Eliminate

        • Candidate Elimination



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