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Inductive Learning of Rules

Mushroom Edible?Spores Spots Color

Y N Brown NY Y Grey YN Y Black YN N Brown NY N White NY Y Brown Y

Y N BrownN N Red

Don’t try this at home...

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Types of Learning

What is learning? Improved performance over time/experience Increased knowledge

Speedup learning No change to set of theoretically inferable facts Change to speed with which agent can infer them

Inductive learning More facts can be inferred

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Mature Technology

Many Applications Detect fraudulent credit card transactions Information filtering systems that learn user

preferences Autonomous vehicles that drive public highways

(ALVINN) Decision trees for diagnosing heart attacks Speech synthesis (correct pronunciation) (NETtalk)

Data mining: huge datasets, scaling issues

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Defining a Learning Problem

Experience:Task:Performance Measure:

A program is said to learn from experience E with respect to task T and performance measure P, if it’s performance at tasks in T, as measured by P, improves with experience E.

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Example: Checkers

Task T: Playing checkers

Performance Measure P: Percent of games won against

opponentsExperience E:

Playing practice games against itself

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Example: Handwriting RecognitionTask T:

Performance Measure P:

Experience E:

Recognizing and classifying handwritten words within images

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Example: Robot Driving

Task T:

Performance Measure P:

Experience E:

Driving on a public four-lane highway using vision sensors

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Example: Speech Recognition

Task T:

Performance Measure P:

Experience E:

Identification of a word sequence from audio recorded from arbitrary speakers ... noise

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Issues

What feedback (experience) is available?What kind of knowledge is being

increased? How is that knowledge represented?What prior information is available? What is the right learning algorithm?How avoid overfitting?

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Choosing the Training ExperienceCredit assignment problem:

Direct training examples: E.g. individual checker boards + correct move for

each Indirect training examples :

E.g. complete sequence of moves and final resultWhich examples:

Random, teacher chooses, learner chooses

Supervised learningReinforcement learningUnsupervised learning

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Choosing the Target Function

What type of knowledge will be learned?How will the knowledge be used by the

performance program?E.g. checkers program

Assume it knows legal moves Needs to choose best move So learn function: F: Boards -> Moves

hard to learn

Alternative: F: Boards -> R

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The Ideal Evaluation Function

V(b) = 100 if b is a final, won board V(b) = -100 if b is a final, lost boardV(b) = 0 if b is a final, drawn boardOtherwise, if b is not final

V(b) = V(s) where s is best, reachable final board

Nonoperational…Want operational approximation of V: V

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How Represent Target Function

x1 = number of black pieces on the board

x2 = number of red pieces on the board

x3 = number of black kings on the board

x4 = number of red kings on the board

x5 = number of black pieces threatened by red

x6 = number of red pieces threatened by black

V(b) = a + bx1 + cx2 + dx3 + ex4 + fx5 + gx6

Now just need to learn 7 numbers!

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Target Function

Profound Formulation: Can express any type of inductive learning as

approximating a functionE.g., Checkers

V: boards -> evaluation E.g., Handwriting recognition

V: image -> wordE.g., Mushrooms

V: mushroom-attributes -> {E, P}Inductive bias

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Theory of Inductive Learning

Dx

x)Pr(

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Theory of Inductive Learning

Suppose our examples are drawn with a probability distribution Pr(x), and that we learned a hypothesis f to describe a concept C.

We can define Error(f) to be:

where D are the set of all examples on which f and C disagree.

Dx

x)Pr(

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PAC LearningWe’re not perfect (in more than one way). So why

should our programs be perfect?What we want is:

Error(f) < for some chosen But sometimes, we’re completely clueless: (hopefully,

with low probability). What we really want is: Prob ( Error(f) < .

As the number of examples grows, and should decrease.

We call this Probably approximately correct.

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Definition of PAC Learnability

Let C be a class of concepts.We say that C is PAC learnable by a hypothesis space H

if: there is a polynomial-time algorithm A, a polynomial function p, such that for every C in C, every probability distribution Pr,

and and , if A is given at least p(1/, 1/) examples, then A returns with probability 1- a hypothesis whose error

is less than .k-DNF, and k-CNF are PAC learnable.

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Version Spaces: A Learning Alg.

Key idea: Maintain most specific and most general hypotheses at

every point. Update them as examples come in.We describe objects in the space by attributes:

faculty, staff, student 20’s, 30’s, 40’s. male, female

Concepts: boolean combination of attribute-values: faculty, 30’s, male, female, 20’s.

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Generalization and Specializ...

A concept C1 is more general than C2 if it describes a superset of the objects: C1={20’s, faculty} is more general than C2={20’s,

faculty, female}. C2 is a specialization of C1.

Immediate specializations (generalizations).The version space algorithm maintains the

most specific and most general boundaries at every point of the learning.

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ExampleT

male female faculty student 20’s 30’s

male, fac male,stud female,fac female,stud fac,20’s fac, 30’s

male,fac,20 male,fac,30 fem,fac,20 male,stud,30