Learning IILearning II

Introduction to Artificial Intelligence

CS440/ECE448

Lecture 21

Last lectureLast lecture

• The (in)efficiency of exact inference with Bayes nets(in)efficiency of exact inference with Bayes nets• The learning problemThe learning problem• Decision treesDecision trees

This lectureThis lecture• Identification treesIdentification trees• Neural networks: PerceptronsNeural networks: Perceptrons

ReadingReading• Chapters 18 and 20

Inductive learning methodInductive learning method

• Construct/adjust h to agree with f on training set.

(h is consistent if it agrees with f on all examples)• E.g., curve fitting:

Inductive learning methodInductive learning method

• Construct/adjust h to agree with f on training set.

(h is consistent if it agrees with f on all examples)• E.g., curve fitting:

Inductive learning methodInductive learning method

• Construct/adjust h to agree with f on training set.

(h is consistent if it agrees with f on all examples)• E.g., curve fitting:

Inductive learning methodInductive learning method

Ockham’s razor: prefer the simplest consistent hypothesis.

• Construct/adjust h to agree with f on training set.

(h is consistent if it agrees with f on all examples)• E.g., curve fitting:

Inductive LearningInductive Learning

• Given examples of some concepts and a description (features) for these concepts as training data, learn how to classify subsequent descriptions into one of the concepts.– Concepts (classes)– Features– Training set– Test set

• Here, the function has discrete outputs (the classes).

Decision TreesDecision Trees“Should I play tennis today?”

Note: A decision tree can be expressed as a disjunction of conjunctions

(Outlook = sunny) (Humidity = normal)

(Outlook = overcast) (Wind=Weak)

Outlook

HumidityWind

No Yes No Yes

No

SunnyRain

Overcast

High Low Strong Weak

Learning Decision TreesLearning Decision Trees

• Inductive Learning.

• Given a set of positive and negative training examples of a concept, can we learn a decision tree that can be used to appropriately classify other examples?

• Identification Trees: ID3 [ Quinlan, 1979 ].

What on Earth causes people to get sunburns?

I don’t know, so let’s go to the beach and collect some data.

Sunburn dataSunburn dataName Hair Height

Swim Suit Color

Lotion Result

Sarah Blond Average Yellow No Sunburned

Dana Blond Tall Red Yes Fine

Alex Brown Short Red Yes Fine

Annie Blond Short Red No Sunburned

Emily Red Average Blue No Sunburned

Pete Brown Tall Blue No Fine

John Brown Average Blue No Fine

Katie Blond Short Yellow Yes Fine

There are 3 x 3 x 3 x 2 = 54 possible feature vectors

Exact Matching MethodExact Matching Method• Construct a table recording observed cases.

• Use table lookup to classify new data.

• Problem: For realistic problems, exact matching can’t be used.

8 people and 54 possible feature vectors:

15% chance of finding an exact match.

Another example:

106 Examples

12 features

5 values per feature

106/512 0.4%

How can we do the classification?How can we do the classification?

• Nearest-neighbor method (but only if we can establish a distance between feature vectors).

• Use identification trees:

An identification tree is a decision tree in which each set of possible conclusions is implicitly established by a list of samples of known class.

An ID tree consistent with the dataAn ID tree consistent with the data

Hair Color

Lotion Used

Sarah

Annie

Dana

Katie

EmilyAlexPeteJohn

Blond Red Brown

No Yes

Sunburned

Not Sunburned

Sunburned

Not Sunburned

Another consistent ID treeAnother consistent ID tree

Katie

Hair Color

Alex

Blond RedBrown

Sunburned

Not Sunburned

Hair Color

Emily

Red

Suit Color

Height

Annie

Short

Suit Color

AverageDanaPete

Tall

Sarah

John

Yellow RedBlue

Yellow BlueRed

Brown

An ideaAn ideaSelect tests that divide as well as possible Select tests that divide as well as possible people into sets with homogeneous labelspeople into sets with homogeneous labels

Hair Color

SarahAnnieDanaKatie

Emily AlexPeteJohn

Blond Red Brown

Lotion used

SarahAnnieEmilyPeteJohn

DanaAlexKatie

NoYes

Height Suit Color

Then among blonds...Then among blonds...

Lotion used

SarahAnnie

DanaKatie

NoYes

Height

KatieAnnie Sarah

Dana

Short Av Tall

Suit Color

SarahKatie

DanaAnnie

Yellow Red Blue This is perfectly

homogeneous...

Combining these two together …Combining these two together …

Hair Color

Lotion Used

Sarah

Annie

Dana

Katie

EmilyAlexPeteJohn

Blond Red Brown

No Yes

Sunburned

Not Sunburned

Sunburned

Not Sunburned

Decision Tree Learning AlgorithmDecision Tree Learning Algorithm

• Problem:– For practical problems, it is unlikely that

any test will produce one completely homogeneous subset.

• Solution:– Minimize a measure of inhomogeneity or

disorder. – Available from information theory.

InformationInformation• Let’s say we have a question which has n possible

answers and call them vi.

• Let’s say that answer vi occurs with probability P(vi), then the information content (entropy) measured in bits of knowing the answer is:

• One bit of information is enough information to answer a yes or no question.

• E.g. consider flipping a fair coin, how much information do you have if you know which side comes up?

I(½, ½) = - (½ log2½ + ½ log2½) = 1bit

n

iiin vPvPvPvPI

121 )(log)())(,),((

Information at a nodeInformation at a node• In our decision tree for a given feature (e.g. hair color), we have

– b: number of branches (e.g. possible values for the feature)– Nb: number of samples in branch – Np: number of samples in all branches– Nbc: number of samples in class c in branch b.

• Using frequencies as an estimate of the probabilities, we have

• For a single branch, the information is simply

b

bc

c b

bc

b p

b

N

N

N

N

N

NnInformatio 2log

b

bc

c b

bc

N

N

N

NnInformatio 2log

ExampleExample• Consider a single branch (b=1) which only contains members of two

classes A and B.– If half of the points belong to A and half belong to B:

– What if all the points belong to A (or to B):

• We like the latter situation since the branches are homogeneous, so less information is needed to make a decision (maximize information

gain).

12

1

2

12

1log

2

1

2

1log

2

1

lognInformatio

22

2

b

bc

c b

bc

b p

b

N

N

N

N

N

N

000

0log01log1

lognInformatio

22

2

b

bc

c b

bc

b p

b

N

N

N

N

N

N

0log :Note lim0

xxx

What is the amount of information required for What is the amount of information required for classification after we have used the hair test?classification after we have used the hair test?

Hair Color

SarahAnnieDanaKatie

Emily AlexPeteJohn

Blond Red Brown

- 2/4 log22/4- 2/4 log22/4= 1

-1 log21-0 log20= 0

- 0 log20- 3/3 log23/3= 0

Information = 4/8*1 + 1/8*0 + 3/8*0 = 0.5

b

bc

c b

bc

b p

b

N

N

N

N

N

N

nInformatio

2log

b

bc

c b

bc

b p

b

N

N

N

N

N

N

nInformatio

2log

Selecting top level featureSelecting top level feature• Using the 8 samples we have so far, we get:

Test InformationHair 0.5Height 0.69Suit Color 0.94Lotion 0.61

• Hair wins, least additional information needed for rest of classification.

• This is used to build the first level of the identification tree:

Hair Color

SarahAnnieDanaKatie

Emily AlexPeteJohn

Blond Red Brown

Selecting second level featureSelecting second level feature

• Let’s consider the remaining features for the blond branch (4 samples)Test InformationHeight 0.5Suit Color 1Lotion 0

• Lotion wins, least additional information.

Hair Color

SarahAnnieDanaKatie

Emily AlexPeteJohn

Blond Red Brown

Thus we get to the tree we had arrived Thus we get to the tree we had arrived at earlierat earlier

Hair Color

Lotion Used

Sarah

Annie

Dana

Katie

EmilyAlexPeteJohn

Blond Red Brown

No Yes

Sunburned

Not Sunburned

Sunburned

Not Sunburned

Using the Identification tree as a Using the Identification tree as a classification procedureclassification procedure

Hair Color

Lotion Used

Sunburn OK

Sunburn OK

Blond Red Brown

No Yes

Rules:

•If Blond and uses lotion, then OK

•If Blond and does not use lotion, then gets burned

•If red-haired, then gets burned

•If brown hair, then OK

Rules:

•If Blond and uses lotion, then OK

•If Blond and does not use lotion, then gets burned

•If red-haired, then gets burned

•If brown hair, then OK

Performance measurementPerformance measurementHow do we know that h ≈ f ?

1. Use theorems of computational/statistical learning theory

2. Try h on a new test set of examples

(use same distribution over example space as training set)

Learning curve = % correct on test set as a function of training set size