CS 9633
Machine Learning
Computational Learning Theory
Adapted from notes by Tom Mitchell
http://www

2.cs.cmu.edu/~tom/mlbook

chapter

slides.html
Theoretical Characterization
of Learning Problems
•
Under what conditions is successful
learning possible and impossible?
•
Under what conditions is a particular
learning algorithm assured of learning
successfully?
Two Frameworks
•
PAC (Probably Approximately Correct)
Learning Framework: Identify classes
of hypotheses that can and cannot be
learned from a polynomial number of
training examples
–
Define a natural measure of complexity for
hypothesis spaces that allows bounding
the number of training examples needed
•
Mistake Bound Framework
Theoretical Questions of
Interest
•
Is it possible to identify classes of learning problems
that are inherently difficult or easy, independent of
the learning algorithm?
•
Can one characterize the number of training
examples necessary or sufficient to assure
successful learning?
•
How is the number of examples affected
–
If observing a random sample of training data?
–
if the learner is allowed to pose queries to the trainer?
•
Can one characterize the number of mistakes that a
learner will make before learning the target function?
•
Can one characterize the inherent computational
complexity of a class of learning algorithms?
Computational Learning
Theory
•
Relatively recent field
•
Area of intense research
•
Partial answers to some questions on
previous page is yes.
•
Will generally focus on certain types of
learning problems.
Inductive Learning of Target
Function
•
What we are given
–
Hypothesis space
–
Training examples
•
What we want to know
–
How many training examples are sufficient
to successfully learn the target function?
–
How many mistakes will the learner make
before succeeding?
Questions for Broad Classes of
Learning Algorithms
•
Sample complexity
How many training examples do we need to
converge to a successful hypothesis with a high
probability?
•
Computational complexity
How much computational effort is needed to
converge to a successful hypothesis with a high
probability?
•
Mistake Bound
How many training examples will the learner
misclassify before converging to a successful
hypothesis?
PAC Learning
•
Probably Approximately Correct
Learning Model
•
Will restrict discussion to learning
boolean

valued concepts in noise

free
data.
Problem Setting:
Instances and Concepts
•
X
is set of all possible instances over which
target function may be defined
•
C
is set of target concepts learner is to learn
–
Each target concept c in C is a subset of X
–
Each target concept c in C is a boolean function
c: X
{0,1}
c(x) = 1 if x is positive example of concept
c(x) = 0 otherwise
Problem Setting: Distribution
•
Instances generated at random using some
probability distribution
D
–
D
may be any distribution
–
D
is generally not known to the learner
–
D
is required to be stationary (does not change
over time)
•
Training examples
x
are drawn at random
from
X
according to
D
and presented with
target value
c(x)
to the learner.
Problem Setting:
Hypotheses
•
Learner
L
considers set of hypotheses
H
•
After observing a sequence of training
examples of the target concept
c
,
L
must output some hypothesis
h
from
H
which is its estimate of
c
Example Problem
(Classifying Executables)
•
Three Classes (Malicious, Boring, Funny)
•
Features
–
a
1
GUI present (yes/no)
–
a
2
Deletes files (yes/no)
–
a
3
Allocates memory (yes/no)
–
a
4
Creates new thread (yes/no)
•
Distribution?
•
Hypotheses?
Instance
a
1
a
2
a
3
a
4
Class
1
Yes
No
No
Yes
B
2
Yes
No
No
No
B
3
No
Yes
Yes
No
F
4
No
No
Yes
Yes
M
5
Yes
No
No
Yes
B
6
Yes
No
No
No
F
7
Yes
Yes
Yes
No
M
8
Yes
Yes
No
Yes
M
9
No
No
No
Yes
B
10
No
No
Yes
No
M
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CS 9633 Machine Learning
True Error
•
Definition: The
true error
(denoted
error
D
(h))
of hypothesis
h
with respect to target concept
c
and distribution
D
, is the probability that h will
misclassify an instance drawn at random
according to
D
.
)]
(
)
(
[
Pr
)
(
x
h
x
c
h
error
D
x
D
Computer Science Department
CS 9633 Machine Learning
Error of h with respect to c
Instance space
X
+
+
+
c
h




Computer Science Department
CS 9633 Machine Learning
Key Points
•
True error defined over entire instance space,
not just training data
•
Error depends strongly on the unknown
probability distribution
D
•
The error of
h
with respect to
c
is not directly
observable to the learner L
—
can only
observe performance with respect to training
data (training error)
•
Question: How probable is it that the
observed training error for h gives a
misleading estimate of the true error?
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CS 9633 Machine Learning
PAC Learnability
•
Goal: characterize classes of target concepts that
can be reliably learned
–
from a reasonable number of randomly drawn training
examples and
–
using a reasonable amount of computation
•
Unreasonable to expect perfect learning where
error
D
(h) = 0
–
Would need to provide training examples corresponding to
every possible instance
–
With random sample of training examples, there is always a
non

zero probability that the training examples will be
misleading
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CS 9633 Machine Learning
Weaken Demand on
Learner
•
Hypothesis error (Approximately)
–
Will not require a zero error hypothesis
–
Require that error is bounded by some constant
,
that can be made arbitrarily small
–
is the error parameter
•
Error on training data (Probably)
–
Will not require that the learner succeed on every
sequence of randomly drawn training examples
–
Require that its probability of failure is bounded by
a constant,
, that can be made arbitrarily small
–
is the confidence parameter
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Definition of PAC

Learnability
•
Definition: Consider a concept class
C
defined over a set of instances
X
of length
n
and a learner
L
using hypothesis space
H
.
C
is PAC

learnable by
L
using
H
if all
c
C
,
distributions
D
over
X
,
such that
0 <
< ½
,
and
such that
0 <
< ½
, learner
L
will with
probability at least
(1

)
output a hypothesis
h
H
such that
error
D
(h)
, in time that is
polynomial in
1/
,
1/
,
n
, and
size(c)
.
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CS 9633 Machine Learning
Requirements of Definition
•
L must with arbitrarily high probability (1

), out put a hypothesis having arbitrarily
low error (
).
•
L’s learning must be efficient
—
grows
polynomially in terms of
–
Strengths of output hypothesis (1/
, 1/
)
–
Inherent complexity of instance space (n)
and concept class C (size(c)).
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CS 9633 Machine Learning
Block Diagram of PAC
Learning Model
Learning
algorithm L
Training sample
n
i
i
i
x
c
x
1
)}
(
,
{
Control Parameters
,
Hypothesis
h
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CS 9633 Machine Learning
Examples of second
requirement
•
Consider executables problem where
instances are conjunctions of boolean
features:
a
1
=yes
a
2
=no
a
3
=yes
a
4
=no
•
Concepts are conjunctions of a subset
of the features
a
1
=yes
a
3
=yes
a
4
=yes
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Using the Concept of PAC
Learning in Practice
•
We often want to know how many training
instances we need in order to achieve a
certain level of accuracy with a specified
probability.
•
If L requires some minimum processing time
per training example, then for C to be PAC

learnable by L, L must learn from a
polynomial number of training examples.
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CS 9633 Machine Learning
Sample Complexity
•
Sample complexity of a learning
problem
is
the growth in the required training examples
with problem size.
•
Will determine the sample complexity for
consistent learners.
–
A learner is consistent if it outputs hypotheses
which perfectly fit the training data whenever
possible.
–
All algorithms in Chapter 2 are consistent learners.
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CS 9633 Machine Learning
Recall definition of VS
•
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
}

{
,
(h,D)
Consistent
H
h
VS
D
H
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CS 9633 Machine Learning
VS and PAC learning by
consistent learners
•
Every consistent learner outputs a hypothesis
belonging to the version space, regardless of
the instance space X, hypothesis space H, or
training data D.
•
To bound the number of examples needed by
any consistent learner, we need only to
bound the number of examples needed to
assure that the version space contains no
unacceptable hypotheses.
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CS 9633 Machine Learning

exhausted
•
Definition: Consider a hypothesis space H,
target concept c, instance distribution
D
, and
set of training examples D of c. The version
space VS
H,D
is said to be

exhausted with
respect to c and
D
, if every hypothesis h in
V
H,D
has error less than
with respect to c and
D.
)
(
)
(
,
h
error
V
h
D
D
H
Computer Science Department
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Exhausting the version
space
VS
H,D
error = 0.1
r=0.2
error = 0.3
r=0.2
error = 0.2
r=0
error = 0.1
r=0
error = 0.3
r=0.4
error = 0.2
r=0.3
Hypothesis Space H
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Exhausting the Version
Space
•
Only an observer who knows the identify of
the target concept can determine with
certainty whether the version space is

exhausted.
•
But, we can bound the probability that the
version space will be

exhausted after a
given number of training examples
–
Without knowing the identity of the target concept
–
Without knowing the distribution from which
training examples were drawn
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CS 9633 Machine Learning
Theorem 7.1
•
Theorem 7.1

數桡畳h楮朠瑨癥牳楯渠
space.
If the hypothesis space H is finite, D
is a sequence of m
1 independent
randomly drawn examples of some target
concept c, then for any 0
1, the
probability that the version space VS
H,D
is
not

exhausted (with respect to c) is less
than or equal to
He

m
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Proof of theorem
•
See text
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Number of Training
Examples (Eq. 7.2)
)
1
ln
(ln
1
1
ln
ln
0
ln
1
ln
0
1
H
m
H
m
m
H
e
H
e
H
m
m
Computer Science Department
CS 9633 Machine Learning
Summary of Result
•
Inequality on previous slide provides a general bound
on the number of trianing examples sufficient for any
consistent learner to successfully learn any target
concept in H, for any desired values of
and
.
•
This number m of training examples is sufficient to
assure that any consistent hypothesis will be
–
probably (with probability 1

)
–
approximately (within error
) correct.
•
The value of m grows
–
linearly with 1/
–
logarithmically with 1/
–
logarithmically with H
•
The bound can be a substantial overestimate.
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CS 9633 Machine Learning
Problem
•
Suppose we have the instance space described for
the EnjoySports problem:
–
Sky (Sunny, Cloudy, Rainy)
–
AirTemp (Warm, Cold)
–
Humidity (Normal, High)
–
Wind (Strong, Weak)
–
Water (Warm, Cold)
–
Forecast (Same, Change)
•
Hypotheses can be as before
(?, Warm, Normal, ?, ?, Same) (0, 0, 0, 0, 0, 0)
•
How many training examples do we need to have an
error rate of less than 10% with a probability of 95%?
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CS 9633 Machine Learning
Limits of Equation 7.2
•
Equation 7.2 tell us how many training
examples suffice to ensure (with probability
(1

) that every hypothesis having 0 training
error, will have a true error of at most
.
•
Problem: there may be no hypothesis that is
consistent with if the concept is not in H. In
this case, we want the minimum error
hypothesis.
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Agnostic Learning and
Inconsistent Hypotheses
•
An Agnostic Learner does not make the
assumption that the concept is contained in
the hypothesis space.
•
We may want to consider the hypothesis with
the minimum error
•
Can derive a bound similar to the previous
one:
))
/
1
ln(


(ln
2
1
2
H
m
Computer Science Department
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Concepts that are PAC

Learnable
•
Proofs that a type of concept is PAC

Learnable usually consist of two steps:
–
Show that each target concept in C can be
learned from a polynomial number of
training examples
–
Show that the processing time per training
example is also polynomially bounded
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PAC Learnability of Conjunctions
of Boolean Literals
•
Class C of target concepts described by
conjunctions of boolean literals:
GUI_Present
Opens_files
•
Is C PAC learnable? Yes.
•
Will prove by
–
Showing that a polynomial # of training examples
is needed to learn each concept
–
Demonstrate an algorithm that uses polynomial
time per training example
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Examples Needed to Learn
Each Concept
•
Consider a consistent learner that uses
hypothesis space H =C
•
Compute number m of random training
examples sufficient to ensure that L will, with
probability (1

), output a hypothesis with
maximum error
.
•
We will use m
(1/
)(lnH+ln(1/
))
•
What is the size of the hypothesis space?
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Complexity Per Example
•
We just need to show that for some algorithm, we can spend a
polynomial amount of time per training example.
•
One way to do this is to give an algorithm.
•
In this case, we can use Find

S as the learning algorithm.
•
Find

S incrementally computes the most specific hypothesis
consistent with each training example.
Old
Tired
+
Old
Happy +
Tired +
Old
Tired

Rich
Happy +
•
What is a bound on the time per example?
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Theorem 7.2
PAC

learnability of boolean conjunctions.
The class C of conjunctions of boolean
literals is PAC

learnable by the FIND

S
algorithm using H=C
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CS 9633 Machine Learning
Proof of Theorem 7.2
•
Equation 7.4 shows that the sample
complexity for this concept class id
polynomial in n, 1/
, and 1/
, and
independent of size(c). To incrematally
process each training example, the
FIND

S algorithm requires effort linear
in n and independent of 1/
, 1/
, and
size(c). Therefore, this concept class is
PAC

learnable by the FIND

S algorithm.
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Interesting Results
•
Unbiased learners are not PAC
learnable because they require an
exponential number of examples.
•
K

term Disjunctive Normal Form is not
PAC learnable
•
K

term Conjunctive Normal Form is a
superset of k

DNF, but it is PAC
learnable
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Sample Complexity with
Infinite Hypothesis Spaces
•
Two drawbacks to previous result
–
It often does not give a very tight bound on
the sample complexity
–
It only applies to finite hypothesis spaces
•
Vapnik

Chervonekis dimension of H
(VC dimension)
–
Will give tighter bounds
–
Applies to many infinite hypothesis spaces.
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Shattering a Set of
Instances
•
Consider a subset of instances S from the
instance space X.
•
Every hypothesis imposes dichotomies on S
{x
S  h(x) = 1}
{x
S  h(x) = 0}
•
Given some instance space S, there are 2
S
possible dichotomies.
•
The ability of H to shatter a set of concepts is
a measure of its capacity to represent target
concepts defined over these instances.
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Shattering a Hypothesis
Space
•
Definition: A set of instances S is
shattered by hypothesis space H if and
only if for every dichotomy of S there
exists some hypothesis in H consistent
with this dichotomy.
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Vapnik

Chervonenkis
Dimension
•
Ability to shatter a set of instances is
closely related to the inductive bias of
the hypothesis space.
•
An unbiased hypothesis space is one
that shatters the instance space X.
•
Sometimes H cannot be shattered, but
a large subset of it can.
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CS 9633 Machine Learning
Vapnik

Chervonenkis
Dimension
•
Definition: The Vapnik

Chervonenkis
dimension, VC(H) of hypothesis space
H defined over instance space X, is the
size of
the largest finite subset of X
shattered by H
. If arbitrarily large finite
sets of X can be shattered by H, then
VC(H) =
.
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Shattered Instance Space
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Example 1 of VC Dimension
•
Instance space X is the set of real
numbers X =
R
.
•
H is the set of intervals on the real
number line. Form of H is:
a < x < b
•
What is VC(H)?
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Shattering the real number
line

1.2
3.4

1.2
3.4
6.7
What is VC(H)?
What is H?
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Example 2 of VC Dimension
•
Set X of instances corresponding to
numbers on the x,y plane
•
H is the set of all linear decision
surfaces
•
What is VC(H)?
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Shattering the x

y plane
2 instances
3 instances
VC(H) = ?
H = ?
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Proving limits on VC
dimension
•
If we find any set of instances of size d
that can be shattered, then VC(H)
d.
•
To show that VC(H) < d, we must show
that no set of size d can be shattered.
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General result for r
dimensional space
The VC dimension of linear decision
surfaces in an r dimensional space is
r+1.
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Example 3 of VC dimension
•
Set X of instances are conjunctions of
exactly three boolean literals
young
happy
single
•
H is the set of hypothesis described by
a conjunction of up to 3 boolean literals.
•
What is VC(H)?
Computer Science Department
CS 9633 Machine Learning
Shattering conjunctions of
literals
•
Approach: construct a set of instances of size 3 that can be
shattered. Let instance
i
have positive literal
l
i
and all other
literals negative. Representation of instances that are
conjunctions of literals
l
1
,
l
2
and
l
3
as bit strings:
Instance
1
: 100
Instance
2
: 010
Instance
3
: 001
•
Construction of dichotomy: To exclude an instance, add
appropriate
l
i
to the hypothesis.
•
Extend the argument to n literals.
•
Can VC(H) be greater than n (number of literals)?
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Sample Complexity and the
VC dimension
•
Can derive a new bound for the number of
randomly drawn training examples that suffice
to probably approximately learn a target
concept (how many examples do we need to

exhaust the version space with probability (1

)?)
)
/
13
(
log
)
(
8
)
/
2
(
log
4
1
2
2
H
VC
m
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Comparing the Bounds
)
/
13
(
log
8
)
/
2
(
log
4
1
2
2
H
VC
m
))
1
ln(
(ln
2
1
2
H
m
Computer Science Department
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Lower Bound on Sample
Complexity
•
Theorem 7.3
Lower bound on sample
complexity.
Consider any concept class C
such that VC(C)
2, any learner L, and any 0 <
< 1/8, and 0 <
< 1/100. Then there exists a
distribution
D
and target concept in C such that
if L observes fewer examples than
•
Then with probability at least
, L outputs a
hypothesis h having error
D
(h) >
.
32
1
)
(
),
/
1
log(
1
max
C
VC
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