Learning, page 1CSI 4106, Winter 2005

Symbolic learning

PointsDefinitionsRepresentation in logicWhat is an arch?Version spaces

• Candidate eliminationLearning decision treesExplanation-based learning

Learning, page 2CSI 4106, Winter 2005

Definitions

Learning is a change that helps improve future performance.(a paraphrase of Herbert Simon's definition)

Broad categories of machine learning methods and systems:

• Symbolic our focus in this brief presentation

• Statistical

• Neural / Connectionist

• Genetic / Evolutionary

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Definitions (2)

Various characterizations of symbolic learning

Learning can be• supervised—pre-classified data,• unsupervised (conceptual clustering)—raw data.

Data for supervised learning can be• many positive and negative examples,• a theory and one example.

The goal of learning can be• concept discovery, (section 10.6)

• generalization from examples (inductive learning),• a heuristic, a procedure, and so on.

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Representation in logic

The AI techniques in symbolic ML:

• search,

• graph matching,

• theorem proving.

Knowledge can be represented in logic:

• a list of elementary properties and facts about things,

• examples are conjunctive formulae with constants,

• generalized concepts are formulae with variables.

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Representation in logic (2)

Two instances of a concept:

size( i1, small ) colour( i1, red ) shape( i1, ball )

size( i2, large ) colour( i2, red ) shape( i2, brick )

A generalization of these instances:

size( X, Y ) colour( X, red ) shape( X, Z )

Another representation:

obj( small, red, ball )

obj( large, red, brick )

A generalization of these instances:

obj( X, red, Z )

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What is an arch?

Positive and negative examples

Learning the concept of an arch (section 10.1)

An arch:

A hypothesis: an arch has three bricks as parts.

Another example—not an arch:

Another hypothesis: two bricks support a third brick.

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What is an arch (2)

Another arch:

Two bricks support a pyramid. We can generalize both positive examples if we have a taxonomy of blocks: the supported object is a polygon.

Not an arch:

We have to specialize the hypothesis: two bricks that do not touch support a polygon.

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Version spaces

This is a method of learning a concept from positive and negative examples.

In the first stage, we have to select features that characterize the concepts, and their values. Our example:

size = {large, small}

colour = {red, white, blue}

shape = {ball, brick, cube}

We will represent a feature “bundle” like this:

obj( Size, Colour, Shape )

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Version spaces (2)

There are 18 specific objects and 30 classes of objects (expressed by variables instead of constants), variously generalized. They are all arranged into a graph called a version space. Here is part of this space for our example:

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Version spaces (3)

Generalization and specialization in this very simple representation are also simple.

To generalize, replace a constant with a variable:

obj( small, red, ball ) obj( small, X, ball )

More generalization requires introducing more (unique) variables.

obj( small, X, ball ) obj( small, X, Y )

To specialize, replace a variable with a constant (it must come from the set of allowed feature values):

obj( small, X, Y ) obj( small, blue, Y )

obj( small, blue, Y ) obj( small, blue, cube )

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Version spaces (4)

Other generalization operators (illustrated with a different representation) include the following.• Drop a conjunct.

size( i1, small ) colour( i1, red ) shape( i1, ball ) colour( i1, red ) shape( i1, ball )• Add a disjunct.

colour( i2, red ) shape( i2, ball ) (colour( i2, red ) colour( i2, blue )) shape( i2, ball )• Use a taxonomy (assuming that you have it!).

Suppose we have a hierarchy of colours where “red” is a subclass of primaryColour". We can generalize:

colour( i3, red ) colour( i3, primaryColour )

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Version spaces (5)

The candidate elimination algorithm

There are three variants: general-to-specific, specific-to-general and a combination of both directions.

All three methods work with sets of hypotheses — classes of concepts.

We consider, one by one, a series of examples, both positive and negative.

In the specific-to-general method, the set S is the (evolving) target concept, the set N stores negative examples.

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Version spaces (6)

Initialize the concept set S to the first positive example.

Initialize the concept set N to Ø. Then repeat:

For a positive example p

• Replace every s S that does not match p with the minimal (most specific) generalization that matches p.

• Remove any s S more general than another s’ S.

• Remove any s S that matches some n N.

For a negative example n

• Remove any s S that matches n.

• Add n to N for future use.

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Version spaces (7)

Example: obj(small, X, ball) minimally generalizesobj(small, white, ball) and obj(small, red, ball).

The conceptof a ball

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Version spaces (8)

Now, general-to-specific.

Initialize the concept set G to the most general concept.

Initialize the concept set P to Ø. Then repeat:

For a negative example n

• Replace every g G that matches n with the minimal (most general) specialization that does not match n.

• Remove any g G more specific than another g’ G.

• Remove any g G that does not match some p P.

For a positive example p

• Remove any g G that does not match p.

• Add p to P for future use.

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Version spaces (9)

Example: obj(large, Y, Z), obj(X, Y, ball), ...minimally specialize obj(X, Y, Z).

The conceptof a ball

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Version spaces (10)

Initialize G to the most general concept, S to the first positive example.

For a positive example p• Remove any g G that does not match p.• Replace every s S that does not match p with the most specific

generalization that matches p.• Remove any s S more general than another s’ S.• Remove any s S more general that some g G.

For a negative example n• Remove any s S that matches n.• Replace every g G that matches n with the most general

specialization that does not match n.• Remove any g G more specific than another g’ G.• Remove any g G more specific than some s S.

If G = S = {c}, the learning of c succeeds.

If G = S = Ø, learning fails.

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Version spaces (11)

The conceptof a red ball