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    Learning from Observations

    Chapter 18

    Section 1 3, 6

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    Learning

    Learning is essential for unknownenvironments, i.e., when designer lacks omniscience

    Learning is useful as a system constructionmethod, i.e., expose the agent to reality rather than trying

    to write it down

    Learning modifies the agent's decisionmechanisms to improve performance

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    Types of Learning

    Supervised learning: correct answer for eachexample. Answer can be a numeric variable,categorical variable etc.

    Unsupervised learning: correct answers not given just examples (e.g. the same figures as above ,without the labels)

    Reinforcement learning: occasional rewards

    M M MF F F

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    Inductive learning

    Simplest form: learn a function from examples

    fis the target function

    An example is a pair (x, f(x))

    Problem: find a hypothesis hsuch that hf

    given a trainingset of examples

    (This is a highly simplified model of real learning: Ignores prior knowledge Assumes examples are given)

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    Inductive learning method

    Construct/adjust hto agree with fon training set

    (his consistent if it agrees with fon all examples)

    E.g., curve fitting:

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    Inductive learning method

    Construct/adjust hto agree with fon training set

    (his consistent if it agrees with fon all examples)

    E.g., curve fitting:

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    Inductive learning method

    Construct/adjust hto agree with fon training set

    (his consistent if it agrees with fon all examples)

    E.g., curve fitting:

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    Inductive learning method

    Construct/adjust hto agree with fon training set

    (his consistent if it agrees with fon all examples)

    E.g., curve fitting:

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    Inductive learning method

    Construct/adjust hto agree with fon training set

    (his consistent if it agrees with fon all examples)

    E.g., curve fitting:

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    Inductive learning method

    Construct/adjust hto agree with fon training set (his consistent if it agrees with fon all examples) E.g., curve fitting:

    Ockhams razor: prefer the simplest hypothesisconsistent with data

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    Learning decision trees

    Problem: decide whether to wait for a table at arestaurant, based on the following attributes:1. Alternate: is there an alternative restaurant nearby?2. Bar: is there a comfortable bar area to wait in?

    3. Fri/Sat: is today Friday or Saturday?4. Hungry: are we hungry?5. Patrons: number of people in the restaurant (None, Some,

    Full)6. Price: price range ($, $$, $$$)

    7. Raining: is it raining outside?8. Reservation: have we made a reservation?9. Type: kind of restaurant (French, Italian, Thai, Burger)10. WaitEstimate: estimated waiting time (0-10, 10-30, 30-60,

    >60)

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    Attribute-basedrepresentations

    Examples described by attribute values (Boolean, discrete, continuous) E.g., situations where I will/won't wait for a table:

    Classification of examples is positive (T) or negative (F)

    The set of examples used for learning is called training set.

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    Decision trees

    One possible representation for hypotheses

    E.g., here is the true tree for deciding whether towait:

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    Expressiveness

    Decision trees can express any function of the input attributes. E.g., for Boolean functions, truth table row path to leaf:

    Trivially, there is a consistent decision tree for any training set withone path to leaf for each example (unless fnondeterministic in x) but itprobably won't generalize to new examples

    Prefer to find more compact decision trees

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    Finding compact decision trees

    Motivated by Ockhams razor.

    However, finding the smallest decisiontree is an unsolved problem (NPc).

    There are heuristics that findreasonable decision trees in mostpractical cases.

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    Hypothesis spaces

    How many distinct decision trees with nBoolean attributes?

    = number of Boolean functions

    = number of distinct truth tables with 2n rows = 22n

    E.g., with 6 Boolean attributes, there are18,446,744,073,709,551,616 trees

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    Decision tree learning

    Aim: find a small tree consistent with the training examples

    Idea: (recursively) choose "most significant" attribute as rootof (sub)tree

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    Choosing an attribute

    Idea: a good attribute splits the examples intosubsets that are (ideally) "all positive" or "allnegative"

    Patrons?is a better choice

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    Example contd.

    Decision tree learned from the 12 examples:

    Substantially simpler than true tree---a morecomplex hypothesis isnt justified by small amount ofdata

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    Using information theory

    To implement Choose-Attribute in the DTLalgorithm

    Information Content (Entropy):

    I(P(v1), , P(vn)) = i=1 -P(vi) log2 P(vi) For a training set containingppositive

    examples and nnegative examples:

    npn

    npn

    npp

    npp

    npn

    nppI

    22 loglog),(

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    Information gain

    A chosen attribute Adivides the training set Eintosubsets E1, , Evaccording to their values for A,where Ahas vdistinct values.

    Information Gain (IG) or reduction in entropy fromthe attribute test:

    Choose the attribute with the largest IG

    v

    i ii

    i

    ii

    iii

    np

    n

    np

    p

    Inp

    np

    Aremainder1

    ),()(

    )(),()( Aremaindernp

    n

    np

    pIAIG

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    Information gain

    For the training set,p = n = 6, I(6/12, 6/12) = 1bit

    Consider the attributes Patronsand Type(and others too):

    Patronshas the highest IG of all attributes and so is chosen by the

    DTL algorithm as the root

    Given Patronsas root node, the next attribute chosen is Hungry?,with IG(Hungry?) = I(1/3, 2/3) ( 2/3*1 + 1/3*0) = 0.252

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    Next step

    Given Patronsas root node, the next attribute chosenis Hungry?, with IG(Hungry?) = I(1/3, 2/3) ( 2/3*1 +1/3*0) = 0.252

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    Computational Learning Theory why learning works

    PAC learning ) Probably Approximately Correct) This has been a breakthrough in the theory of

    machine learning. Basic idea: A really bad hypothesis will be easy to

    identify, With high probability it will err on one ofthe training examples.

    A consistent hypothesis will be probablyapproximately correct.

    Notice that if there are more training example, thenthe probability of approximately correct becomeshigher!

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    Braodening the applicability of

    Decision Trees Missing data

    Multivalued attributes

    Continuous or integer values attributes Continuous output attributes

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    Computational Learning Theory how many examples are needed

    Let:

    X the set of all possible examples

    D the distribution from which samples are

    drawn H the set of possible hypothesis

    m the number of examples in the training set

    And assume that f, the true function is in H.

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    error

    Error(h) = P( h(x) f(x) | x drawn from D) An approximately correct hypothesis h, is a hypothesis that

    satisfies Error(h) <

    m 1/ (ln|H| - ln ) , where is the probability that Hbadcontains a hypothesis consistent with all examples

    H

    Hbadf

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    Examples

    Learning a Boolean function;

    Learning a conjunction of n literals

    Learning decision lists

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    Boolean function

    A general boolean

    function on n

    attributes can be

    represented by itstruth table

    Size of truth table 2n

    A B C F(A,B,C)

    T T T F

    T T F T

    T F T T

    T F F F

    F T T F

    F T F F

    F F T F

    F F F T

    Arbitrary boolean function

    on 3 attributes

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