ML_Lecture_2 well posed algorithm find s.ppt

1
Machine Learning: Lecture 2
Concept Learning
and
Version Spaces
(Based on Chapter 2 of Mitchell T..,
Machine Learning, 1997)

2
What is a Concept?
 A Concept is a a subset of objects or events defined over
a larger set [Example: The concept of a bird is the subset of all
objects (i.e., the set of all things or all animals) that belong to the
category of bird.]
Alternatively, a concept is a boolean-valued function
defined over this larger set [Example: a function defined over
all animals whose value is true for birds and false for every other
animal].
Things
Animals
Birds
Cars

3
What is Concept-Learning?
Given a set of examples labeled as members or
non-members of a concept, concept-learning
consists of automatically inferring the general
definition of this concept.
In other words, concept-learning consists of
approximating a boolean-valued function from
training examples of its input and output.

4
Example of a Concept Learning task
 Concept: Good Days for Water Sports (values: Yes, No)
 Attributes/Features:

Sky (values: Sunny, Cloudy, Rainy)

AirTemp (values: Warm, Cold)

Humidity (values: Normal, High)

Wind (values: Strong, Weak)

Water (Warm, Cool)

Forecast (values: Same, Change)
 Example of a Training Point:
<Sunny, Warm, High, Strong, Warm, Same, Yes>
class

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Day Sky AirTemp Humidity Wind Water Forecast WaterSport
1 Sunny Warm Normal Strong Warm Same Yes
2 Sunny Warm High Strong Warm Same Yes
3 Rainy Cold High Strong Warm Change No
4 Sunny Warm High Strong Cool Change Yes
Database:
Chosen Hypothesis Representation:
Conjunction of constraints on each attribute where:
• “?” means “any value is acceptable”
• “0” means “no value is acceptable”
Example of a hypothesis: <?,Cold,High,?,?,?>
(If the air temperature is cold and the humidity high then
it is a good day for water sports)
class

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Goal: To infer the “best” concept-description from the set
of all possible hypotheses (“best” means “which best
generalizes to all (known or unknown) elements of the
instance space”. . concept-learning is an ill-
defined task)
Most General Hypothesis: Everyday is a good day for
water sports <?,?,?,?,?,?>
Most Specific Hypothesis: No day is a good day for water
sports <0,0,0,0,0,0>

7
Terminology and Notation
 The set of items over which the concept is defined is called the set of instances
(denoted by X)
 The concept to be learned is called the Target Concept (denoted by c: X--> {0,1})
 The set of Training Examples is a set of instances, x, along with their target
concept value c(x).
 Members of the concept (instances for which c(x)=1) are called positive examples.
 Nonmembers of the concept (instances for which c(x)=0) are called negative
examples.
 H represents the set of all possible hypotheses. H is determined by the human
designer’s choice of a hypothesis representation.
 The goal of concept-learning is to find a hypothesis h:X --> {0,1} such that
h(x)=c(x) for all x in X.

8
Concept Learning as Search
 Concept Learning can be viewed as the task of
searching through a large space of hypotheses
implicitly defined by the hypothesis representation.
 Selecting a Hypothesis Representation is an
important step since it restricts (or biases) the
space that can be searched. [For example, the
hypothesis “If the air temperature is cold or the humidity
high then it is a good day for water sports” cannot be
expressed in our chosen representation.]

9
General to Specific Ordering of
Hypotheses
 Definition: Let hj and hk be boolean-valued functions defined over X.
Then hj is more-general-than-or-equal-to hk iff For all x in X, [(hk(x)
= 1) --> (hj(x)=1)]
 Example:

h1 = <Sunny,?,?,Strong,?,?>

h2 = <Sunny,?,?,?,?,?>
Every instance that are classified as positive by h1 will also be classified
as positive by h2 in our example data set. Therefore h2 is more general
than h1.
 We also use the ideas of “strictly”-more-general-than, and more-
specific-than (illustration [Mitchell, p. 25])

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Find-S, a Maximally Specific
Hypothesis Learning Algorithm
 Initialize h to the most specific hypothesis in H
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
Output hypothesis h

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Shortcomings of Find-S
 Although Find-S finds a hypothesis consistent
with the training data, it does not indicate
whether that is the only one available
 Is it a good strategy to prefer the most specific
hypothesis?
 What if the training set is inconsistent (noisy)?
 What if there are several maximally specific
consistent hypotheses? Find-S cannot backtrack!

12
Version Spaces and the
Candidate-Elimination Algorithm
 Definition: A hypothesis h is consistent with a set of
training examples D iff h(x) = c(x) for each example
<x,c(x)> in D.
 Definition: The version space, denoted VS_H,D, with
respect to hypothesis space H and training examples D, is
the subset of hypotheses from H consistent with the
training examples in D.
 NB: While a Version Space can be exhaustively
enumerated, a more compact representation is preferred.

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A Compact Representation for
Version Spaces
 Instead of enumerating all the hypotheses consistent with a
training set, we can represent its most specific and most general
boundaries. The hypotheses included in-between these two
boundaries can be generated as needed.
 Definition: The general boundary G, with respect to hypothesis
space H and training data D, is the set of maximally general
members of H consistent with D.
 Definition: The specific boundary S, with respect to hypothesis
space H and training data D, is the set of minimally general (i.e.,
maximally specific) members of H consistent with D.

14
Candidate-Elimination Learning
Algorithm
 The candidate-Elimination algorithm
computes the version space containing all
(and only those) hypotheses from H that are
consistent with an observed sequence of
training examples.
 See algorithm in [Mitchell, p.33].

15
Remarks on Version Spaces and
Candidate-Elimination
 The version space learned by the Candidate-Elimination Algorithm
will converge toward the hypothesis that correctly describes the
target concept provided: (1) There are no errors in the training
examples; (2) There is some hypothesis in H that correctly
describes the target concept.
 Convergence can be speeded up by presenting the data in a strategic
order. The best examples are those that satisfy exactly half of the
hypotheses in the current version space.
 Version-Spaces can be used to assign certainty scores to the
classification of new examples

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Inductive Bias I: A Biased
Hypothesis Space
Day Sky AirTemp Humidity Wind Water Forecast WaterSport
1 Sunny Warm Normal Strong Cool Change Yes
2 Cloudy Warm Normal Strong Cool Change Yes
3 Rainy Warm Normal Strong Cool Change No
Given our previous choice of the hypothesis space
representation, no hypothesis is consistent with the
above database: we have BIASED the learner to
consider only conjunctive hypotheses
Database:
class

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Inductive Bias II: An Unbiased
Learner
 In order to solve the problem caused by the bias of the
hypothesis space, we can remove this bias and allow the
hypotheses to represent every possible subset of instances.
The previous database could then be expressed as:
<Sunny, ?,?,?,?,?> v <Cloudy,?,?,?,?,?,?>
 However, such an unbiased learner is not able to generalize
beyond the observed examples!!!! All the non-observed
examples will be well-classified by half the hypotheses of the
version space and misclassified by the other half.

18
Inductive Bias III: The Futility of
Bias-Free Learning
 Fundamental Property of Inductive Learning A
learner that makes no a priori assumptions regarding
the identity of the target concept has no rational basis
for classifying any unseen instances.
 We constantly have recourse to inductive biases
Example: we all know that the sun will rise tomorrow.
Although we cannot deduce that it will do so based on
the fact that it rose today, yesterday, the day before,
etc., we do take this leap of faith or use this inductive
bias, naturally!

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Inductive Bias IV: A Definition
 Consider a concept-learning algorithm L for the set of
instances X. Let c be an arbitrary concept defined over
X, and let Dc = {<x,c(x)>} be an arbitrary set of training
examples of c. Let L(xi,Dc) denote the classification
assigned to the instance xi by L after training on the data
Dc. The inductive bias of L is any minimal set of
assertions B such that for any target concept c and
corresponding training examples Dc
(For all xi in X) [(B ^Dc^xi) |-- L(xi,Dc)]

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Ranking Inductive Learners
according to their Biases

Rote-Learner: This system simply memorizes
the training data and their classification--- No
generalization is involved.

Candidate-Elimination: New instances are
classified only if all the hypotheses in the
version space agree on the classification

Find-S: New instances are classified using the
most specific hypothesis consistent with the
training data
Bias
Strength
Weak
Strong

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