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A Logic Based Classification Technique

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### A Logic Based Classification Technique

General-to-Specific Ordering

Logic Based

Like Decision Tree

Tree questions

Sky? Sunny, ok, Wind? Strong, ok yes enjoy sport

Logic Based Classification

Candidate Elimination

With candidate elimination object is to predict class through the use of expressions

?’s are like wild cards

Expressions represent conjunctions

Expression

<Sunny,?,?,Strong,?,?>

Means will enjoy sport only when sky is sunny andwind is strong, don’t care about other attributes

Logic Based Classification

First Approach

Finding a maximally specific hypothesis

Start with most restrictive (specific) one can get and relax to satisfy each positive training sample

Ø’s mean nothing will match it

Most general (all dimensions can be any value)

<?,?,?,?,?,?>

Most restrictive (no dimension can be anything

<Ø, Ø, Ø, Ø, Ø,Ø>

Logic Based Classification

That pesky Ø

What if a relation has a single Ø? (remember, the expression is a conjunction)

Ø

Logic Based Classification

Find-S Algorithm

Initialize h to most specific hypothesis in H (<Ø, Ø, Ø, Ø, Ø, Ø>)

For each positive training instance x

For each attribute constraint ai in h

If the constraint ai is satisfied by x then do nothing

Else replace ai in h by the next more general constraint that is satisfied by x

Return h

Order of generality

? is more general than a specific attribute value which is more specific than Ø

Logic Based Classification

First positive (x)

<Sunny,Warm,Normal,Strong,Warm,Same>

Which attributes of x are satisfied by h? None?

Replace each ai with a relaxed form from x

<Sunny,Warm,Normal,Strong,Warm,Same>

ExampleLogic Based Classification

<Sunny,Warm,Normal,Strong,Warm,Same>

Next positive

<Sunny,Warm,High,Strong,Warm,Same>

Which attributes of x are satisfied by h? Not humidity

Replace h with

<Sunny,Warm,?,Strong,Warm,Same>

ExampleLogic Based Classification

<Sunny,Warm,?,Strong,Warm,Same>

Next positive

<Sunny,Warm,High,Strong,Cool,Change>

Which attributes of x are satisfied by h? Not water or forcast

Replace h with

<Sunny,Warm,?,Strong,?,?>

ExampleReturn <Sunny,Warm,?,Strong,?,?>

Can one use this to “test” a new instance?

Logic Based Classification

Next: Version Space

What if want all hypotheses that are consistent with a training set (called a version space)

A hypothesis is consistent with a set of training examples if and only if h(x)=c(x) for each training example

<?, Warm, ?, Strong, ?, ?>

<Sunny, ?, ?, Strong, ?, ?>

<?,Warm,?,?,?,?>

<Sunny,Warm,?,Strong,?,? >

<Sunny,?,?,?,?,?>

<Sunny, Warm, ?, ?, ?, ?>

<?,?,?,?,?,Same>

Logic Based Classification

List-Then-Eliminate

Algorithm

a list containing every hypothesis in

For each training example

Remove from any hypothesis for which

Output the list of hypotheses in

- Number of hypotheses 5,120 that can be represented (5*4*4*4*4*4)
- But a single Ø represents an empty set
- So semantically distinct hypotheses 973

Exhaustive

Logic Based Classification

Next: Candidate Elimination

More compact representation

Just those hypotheses at the extreme ends

Those that are the most general and those that are the most specific

All else between would necessarily be in the

Process of Elimination

Logic Based Classification

Definitions

And now for something totally formal:

The general boundary G, with respect to hypothesis space consistent with , is the set of maximally general members of consistent with .

G is identical to the set of all g that are members of H such that g is consistent with D and there does not exist a g’ in H such that it is more general than g and it (g’) is consistent with the training data

Logic Based Classification

Definitions

The specific boundary S, with respect to hypothesis space consistent with , is the set of minimally general members of consistent with .

S is identical to the set of all s that are members of H such that s is consistent with D and there does not exist a s’ in H such that it is more specific than s and it (s’) is consistent with the training data

Logic Based Classification

Example

All yes’s are sunny, warm, and strong

But “strong” isn’t enough to identify a yes

S:{<Sunny, Warm, ?, Strong, ?, ?>}

<Sunny, ?, ?, Strong, ?, ?> <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?>

G: {<Sunny, ?, ?, ?, ?, ?>, <?, Warm, ?, ?, ?, ?> }

3 ?’s

4 ?’s

5 ?’s

Logic Based Classification

Approach

Start with two extremes

Most general (all dimensions can be any value)

<?,?,?,?,?,?>

Most restrictive (no dimension can be anything

<Ø, Ø, Ø, Ø, Ø, Ø>

Slowly work inward

Specific General

Logic Based Classification

Algorithm

Initialize G to the set of maximally general hypotheses in H

Initialize S to the set of maximally specific hypotheses in H

For each training example d, do

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 generalizations 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

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 specializations 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

Logic Based Classification

Example

First record

- S1: {<Sunny,Warm,Normal,Strong,Warm,Same>}
- G0 G1:{<?,?,?,?,?,?>}

Logic Based Classification

Example

Second

Modify previous S minimally to keep consistent with d

- S2: {<Sunny,Warm, ? ,Strong,Warm,Same>}
- G0G1G2: {<?,?,?,?,?,?>}

Logic Based Classification

Example

Third

Replace {<?,?,?,?,?,?>} with all one member expressions (minimally specialized)

- S2S3: {<Sunny,Warm, ? ,Strong,Warm,Same>}
- G3:{<Sunny,?,?,?,?,?>, <?,Warm,?,?,?,?>, <?,?,?,?,?,Same>}

Logic Based Classification

Example

Fourth

Back to positive, replace warm and same with “?” and remove “Same” from General

- S4: {<Sunny,Warm, ? ,Strong, ? , ? >}
- G3G4: {<Sunny,?,?,?,?,?>, <?,Warm,?,?,?,?>, <?,?,?,?,?,Same>}

Then can calculate the interior expressions

<Sunny, ?, ?, Strong, ?, ?> <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?>

Logic Based Classification

What if

Have two identical records but different classes?

If positive shows up first it, first step in evaluating a negative states “Remove from S any hypothesis that is not consistent with d” (S is now empty)

For each hypothesis g in G that is not consistent with d

Remove g from G (all ?’s is inconsistent with No, G is empty)

Add to G all minimal specializations h of g such that h is consistent with d, and some member of S is more specific than h

No matter what add to G it will violate either d or S (remains empty)

Both are empty, broken. Known as converging to an empty version space

Established by first positive

- S1: {<Sunny,Warm,Normal,Strong,Warm,Same>}
- G0 G1:{<?,?,?,?,?,?>}

Logic Based Classification

What if

Have two identical records but different classes?

If negative shows up first it, first step in evaluating a positive states “Remove from G any hypothesis that is not consistent with d”

This is all of them, leaving an empty set

For each hypothesis s in S that is not consistent with d

Remove s from S

Add to S all minimal generalizations h of s such that h is consistent with d and some member of G is more general than h

No minimal generalization exists except <?,?,?,?,?,?>

- S0: <Ø, Ø, Ø, Ø, Ø, Ø>
- G0G1:{<Rainy,?,?,?,?,?>, <Cloudy,?,?,?,?,?>,
- <?,Cold,?,?,?,?>,<?,?,High,?,?,?>,<?,?,?,Light,?,?>,
- <?,?,?,?,Cool,?>,<?,?,?,?,?,Change>}

Established by first negative

Logic Based Classification

Brittle

Bad with noisy data

Similar effect with false positives or negatives

Logic Based Classification

Will it converge?

Yes provided

There are no errors in the training examples

There is some hypothesis in H that correctly describes the target concept

For example: if the target concept is a disjunction () of feature attributes and the hypothesis space supports only conjunctions

Logic Based Classification

Classifying

Never before seen data

Vote

All training samples were strong wind

No

- S4: {<Sunny,Warm, ? ,Strong, ? , ? >}
- G3G4: {<Sunny,?,?,?,?,?>, <?,Warm,?,?,?,?>, <?,?,?,Strong,?,?>}

Proportion can be a confidence metric

<Sunny, ?, ?, Strong, ?, ?> <Sunny, Warm, ?, ?, ?, ?> <?, Warm, ?, Strong, ?, ?>

No

No

Yes

No

Yes

Yes

Logic Based Classification

A UnanimousVote

Same confidence as if already converged to the single correct target concept

Regardless of which hypothesis in the version space is eventually found to be correct, it will be positive for at least some of the hypotheses in the current set, and the test case is unanimously positive

100% as good as most specific match

Logic Based Classification

Now for…

Have seen 4 classifiers

Naïve Bayesian

KNN

Decision Tree

Candidate Elimination

Now for some theory

Logic Based Classification

Have already…

Curse of dimensionality

Overfitting

Lazy/Eager

Radial basis

Normalization

Gradient descent

Entropy/Information gain

Occam’s razor

Logic Based Classification

Biased Hypothesis Space

Another way of measuring whether a hypothesis captures the learning concept

Candidate Elimination

Conjunction of constraints on the attributes

Logic Based Classification

Biased toward linear solutions

Naïve Bayes

Biased to a given distribution or bin selection

KNN

Biased toward solutions that assume cohabitation of similarly classed instances

Decision Tree

Short trees

Biased Hypothesis SpaceLogic Based Classification

Unbiased learner?

Must be able to accommodate every distinct subset as class definition

96 distinct instances (3*2*2*2*2*2)

Sky has three possible answers–rest two

Number of distinct subsets 296

Think binary: 1 indicates membership

Logic Based Classification

Number of hypotheses 5,120 that can be represented (5*4*4*4*4*4)

But a single Ø represents an empty set

So semantically distinct hypotheses 973

Each hypothesis represents a subset (due to wild cards)

1+(4*3*3*3*3*3)

Search Space- Candidate elimination can represent 973 different subsets
- But 296 is the number of distinct subsets
- Very biased

- S0: <Ø, Ø, Ø, Ø, Ø, Ø>
- G0: {<?,?,?,?,?,?>}

Logic Based Classification

I (5*4*4*4*4*4) think of bias as inflexibility in expressing hypotheses

Or, alternatively, what are the implicit assumptions of the approach

BiasInflexibility

Implicit Assumptions

Logic Based Classification

Next term: (5*4*4*4*4*4)inductive inference

The process by which a conclusion is inferred from multiple observations

What we’ve been doingTraining data

Classifier

Make prediction on New data

Logic Based Classification

The Hypothesis (5*4*4*4*4*4)

Inductive learning hypothesis

Any hypothesis found to approximate the target function well over a sufficiently large set of training examples will also approximate the target function well over other unobserved examples

Logic Based Classification

Next Term (5*4*4*4*4*4)

Concept learning

Automatically inferring the general definition of some concept, given examples labeled as members or nonmembers of the concept

Roughly equate “Concept” to “Class”

Logic Based Classification

is the set of all possible hypotheses that the learner may consider regarding the choice of hypothesis representation.

In general, each hypothesis in represents a boolean-valued function defined over ; that is, .

Note that this is for a two class system

The goal of the learner is to find a hypothesis such that for all in

is the target concept

HypothesesLogic Based Classification

Target Concept consider regarding the choice of hypothesis representation.

In regression

The various “y” values of the training instances

Function approximation

Naïve Bayes, KNN, and Decision Tree

Class

Logic Based Classification

Hypotheses consider regarding the choice of hypothesis representation.

In regression

Line; the coefficients (or other equation members such as exponents)

Naïve Bayes

Class of an instance is predicted by determining most probable class given the training data. That is, by finding the probability for each class for each dimension, multiplying these probabilities (across the dimensions for each class) and taking the class with the maximum probability as the predicted class

KNN

Class of an instance is predicted by examining an instance’s neighborhood

Decision Tree

Tree itself

Candidate Elimination

Conjunction of constraints on the attributes

Logic Based Classification

Something Else We’ve Been Doing consider regarding the choice of hypothesis representation.

Supervised Learning

Supervision from an oracle that knows the classes of the training data

Is there unsupervised learning?

Yes, covered in pattern rec

Seeks to determine how the data are organized

Clustering

PCA

Edge detection

Logic Based Classification

Definition of Machine Learning consider regarding the choice of hypothesis representation.

Machine learning addresses the question of how to build computer programs that improve their performance at some task through experience.

Finally

Logic Based Classification

Learning Checkers consider regarding the choice of hypothesis representation.

All about representation

Out representation

End game is to develop

function that returns the best next move

Logic Based Classification

chooseNextMove consider regarding the choice of hypothesis representation.

Look at every legal move

Determine goodness (score) of resultant board state

Return the highest score (argmax)

Logic Based Classification

How to Assess a Board State consider regarding the choice of hypothesis representation.

Score function, we will keep it simple

Work with a polynomial with just a few variables

X1: the number of black pieces on the board

X2: the number of red pieces on the board

X3: the number of black kings on the board

X4: the number of red kings on the board

X5: the number of black pieces threatened by red

X6: the number of red pieces threatened by black

Logic Based Classification

Score(b) consider regarding the choice of hypothesis representation.

Gotta learn them weights

But how?

X1: the number of black pieces on the board

X2: the number of red pieces on the board

X3: the number of black kings on the board

X4: the number of red kings on the board

X5: the number of black pieces threatened by red

X6: the number of red pieces threatened by black

Logic Based Classification

Training consider regarding the choice of hypothesis representation.

A bunch of board states (a series of games)

Use them to jiggle the weights

Must know the current real “score” vs. “predicted score” using polynomial

Train the scoring function

Logic Based Classification

A trick consider regarding the choice of hypothesis representation.

If my predictor is good then it will be self-consistent

That is, the score of my best move should lead to a good scoring board state

If it doesn’t maybe we should adjust our predictor

Precognition

Logic Based Classification

ScoreBasedUponSuccessor consider regarding the choice of hypothesis representation.

Successor returns the board state of the best move (returned by chooseNextMove(b))

It has been found to be surprisingly successful

Logic Based Classification

Learning consider regarding the choice of hypothesis representation.

For each training sample (board states from a series of games)

If win (zero opponent pieces on the board) could give some fixed score (100 if win, -100 if lose)

Look familiar?

LMS (least mean squares) weight update rule

Logic Based Classification

Is this a classifier? consider regarding the choice of hypothesis representation.

Is it Machine Learning?

Classifier?Logic Based Classification

Logic Based Classification consider regarding the choice of hypothesis representation.

Makes a big deal… consider regarding the choice of hypothesis representation.

At the beginning of candidate elimpg 29

Diff between satisfies and consistent with

Satisfies h when h(x)=1 regardless of whether x is a positive or negative example

Consistent with h depends on the target concept, whether h(x)=c(x)

Logic Based Classification

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