Version space - Concept Learning

Concept Learning
• We incrementally learn general concepts from
specific training examples.
• Each concept can be viewed as describing some
subset of objects or events defined over a larger
set.
• Concept : A boolean-valued function defined over a
larger set.
– E.g.: A function defined over all engineering course,
whose value is true for computer science and false for
other courses.

Concept Learning
• Concept learning is the task of automatically
inferring the general definition of some
concept, given examples labelled as members
or non members of the concept.
– Inferring a boolean-valued function from training
examples of its input and output.

Concept Learning - Task
• Example: Task of learning the target concept “Days on which
my friend suzen enjoys his favorite water sport"
Yes and No training examples for the target concept “EnjoySport”
• The attribute EnjoySport indicates whether or not suzen
enjoys his favorite water sport on this day.
• The task is to learn to predict the value of EnjoySport for an
arbitrary day, based on the values of its other attributes.

Hypothesis Representation
• Hypothesis : h, a conjunction of constraints on the
instance attributes.
• Let each hypothesis be a vector of six constraints,
specifying the values of the six attributes (Sky, AirTemp,
Humidity, Wind, Water, and Forecast).
• Each constraint can be
– A specific value ( e.g., Water = Warm)
– Don’t care ( e.g., Water = ? )
– No value allowed ( e.g., Water = "ɸ”)
• Example: Sky, AirTemp, Humidity, Wind, Water, Forecast
< Sunny , ? , High, ? , ? , ? >

• If some instance x satisfies all the constraints of
hypothesis h, then h classifies x as a positive example
(h(x) = 1).
• Example: The hypothesis that Suzen enjoys his
favourite sport only on warm temperature with
Strong wind (independent of the values of the other
attributes) is represented by the expression
< ?, Warm,?, Strong, ?, ? >

• The most general hypothesis-that every day is
a positive example - is represented by
<?, ?, ?, ?, ?, ?>
• The most specific possible hypothesis-that no
day is a positive example-is represented by
< ɸ, ɸ, ɸ, ɸ, ɸ, ɸ >

• In general, any concept learning task can be
described by
– the set of instances over which the target function
is defined
– the target function
– the set of candidate hypotheses considered by
the learner, and
– the set of available training examples

Version space - Concept Learning - Machine learning

Concept Learning Task- EnjoySport Prototype

• The set of items over which the concept is defined is called the
set of instances, which we denote by X.
• Example: X is the set of all possible days, each represented by
the attributes Sky, AirTemp, Humidity, Wind, Water, and Forecast.
• The concept or function to be learned is called the target
concept, which we denote by c.
– In general, c can be any boolean-valued function defined over the
instances X; that is, c : X -> {0, 1}
• Example: The target concept corresponds to the value of the
attribute EnjoySport
(i.e., c(x) = 1 if EnjoySport = Yes, and c(x) = 0 if EnjoySport = No)

• When learning the target concept, the learner is presented a
set of training examples, each consisting of an instance x
from X, along with its target concept value c(x)
• Instances for which c(x) = 1 are called positive examples, or
members of the target concept.
• Instances for which c(x) = 0 are called negative examples,
or non members of the target concept.
• Ordered pair <x, c(x)> describes the training example
consisting of the instance x and its target concept value c(x).
• D denotes the set of available training examples.

• Given a set of training examples of the target concept c,
the problem faced by the learner is to hypothesize, or
estimate, c.
• H denotes the set of all possible hypotheses that the
learner may consider regarding the identity of the target
concept.
• In general, each hypothesis h in H represents a boolean-
valued function defined over X; that is, h : X  {0, 1}.
• The goal of the learner is to find a hypothesis h such
that h(x) = c(x) for all x in X.

The Inductive Learning Hypothesis
• Learning Task :To determine a hypothesis h identical to the
target concept c over the entire set of instances X
– The only information available about c is its value over the
training examples.
• Inductive learning algorithms guarantee that the output
hypothesis fits the target concept over the training data.
• Assumption : The best hypothesis regarding unseen instances
is the hypothesis that best fits the observed training data.
• The 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.

CONCEPT LEARNING AS SEARCH
• Concept Learning : Viewed as the task of searching through a large space
of hypotheses implicitly defined by the hypothesis representation.
• Goal of search: Find the hypothesis that best fits the training examples.
– By selecting a hypothesis representation, the designer of the learning algorithm
implicitly defines the space of all hypotheses that the program can ever
represent and therefore can ever learn.
• Example: Consider the instance X and hypotheses H in the EnjoySport
learning task.
– Given that the attribute Sky has three possible values, and that AirTemp,
Humidity, Wind, Water, and Forecast each have two possible values, the
instance space X contains exactly 3 .2. 2 .2 .2 .2 = 96 distinct instances.
– Further, there are 5.4.4.4.4.4 = 5120 syntactically distinct hypotheses within
H.

CONCEPT LEARNING AS SEARCH
• Every hypothesis containing one or more ɸ
symbols represents the empty set of instances
(classified as negative)
• Therefore, the number of semantically distinct
hypotheses is only 1+ (4.3.3.3.3.3)
• EnjoySport example is a very simple learning task,
with a relatively small, finite hypothesis space.
• Most practical learning tasks involve much larger,
sometimes infinite hypothesis spaces.

FIND-S: FINDING A MAXIMALLY specific HYPOTHESIS
• Use the more_general_than partial ordering to organize the search for a
hypothesis consistent with the observed training examples
– Begin with the most specific possible hypothesis in H, then generalize this
hypothesis each time it fails to cover an observed positive training example.
– A hypothesis "covers" a positive example if it correctly classifies the example
as positive
• FIND-S algorithm
1. Initialize h to the most specific hypothesis in H
2. For each positive training instance x
For each attribute constraint a, in h
If the constraint a, is satisfied by x Then do nothing
Else replace a, in h by the next more general constraint that is
satisfied by x
3. Output hypothesis h

FIND-S: Example
• Illustration: EnjoySport task
– The first step of FIND-S is to initialize h to the most specific
hypothesis in H
h < ɸ, ɸ, ɸ, ɸ, ɸ, ɸ >
– Observing the first training example in EnjoySport , which is a
positive example, it is clear that the hypothesis is too specific.
– None of the " ɸ " constraints in h are satisfied by this
example, so each is replaced by the next more general
constraint that fits the example; namely, the attribute values
for this training example.

FIND-S: Example
Hence, h  (Sunny, Warm, Normal, Strong, Warm, Same)
• This h is still very specific; it asserts that all instances are negative
except for the single positive training example observed.
• Next, the second training example (positive example) forces the
algorithm to further generalize h, this time substituting a "?' in place
of any attribute value in h that is not satisfied by the new example.
• The refined hypothesis in this case is
h  (Sunny, Warm, ?, Strong, Warm, Same)
• The third training example (a negative example) the algorithm makes
no change to h.
• FIND-S algorithm simply ignores every negative example, and hence
no revision in h, is needed.

FIND-S: Example
• The fourth (positive) example leads to a
further generalization of h.
h  ( Sunny, Warm, ?, Strong, ?, ? )
• The search moves from hypothesis to
hypothesis, searching from the most specific
to progressively more general hypotheses.

FIND-S: Example
Search in terms of the instance and hypothesis spaces:
• At each step, the hypothesis is generalized only as far as necessary to cover the new positive example.
• At each stage the hypothesis is the most specific hypothesis consistent with the training examples
observed up to this point (hence the name FIND-S).

FIND-S: Example
• The key property of the FIND-S algorithm is that for
hypothesis spaces described by conjunctions of attribute
constraints (such as H for the EnjoySport task),
– FIND-S is guaranteed to output the maximally specific
hypothesis within H that is consistent with the positive
training examples.
• Its final hypothesis will also be consistent with the
negative examples provided the correct target concept
is contained in H, and provided the training examples
are correct.

Version space - Concept Learning - Machine learning

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