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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
–
–
–
–
a1
a2
a3
a4
GUI present (yes/no)
Deletes files (yes/no)
Allocates memory (yes/no)
Creates new thread (yes/no)
• Distribution?
• Hypotheses?
Instance
a1
a2
a3
a4
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
True Error
• Definition: The true error (denoted errorD(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.
errorD (h)  Pr [c( x)  h( x)]
xD
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Error of h with respect to c
Instance space X
-
-
-
c
+
+
h
+
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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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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
errorD(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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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 PACLearnability
• 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 errorD(h)  , in time that is
polynomial in 1/, 1/, n, and size(c).
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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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Block Diagram of PAC
Learning Model
Control Parameters
, 
Training sample
{ xi , c( xi )}
n
i 1
Hypothesis
Learning
algorithm L
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h
Examples of second
requirement
• Consider executables problem where
instances are conjunctions of boolean
features:
a1=yes  a2=no  a3=yes  a4=no
• Concepts are conjunctions of a subset
of the features
a1=yes  a3=yes  a4=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 PAClearnable by L, L must learn from a
polynomial number of training examples.
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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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Recall definition of VS
• The version space, denoted VSH,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
VS H , D  {h  H | Consistent(h,D)}
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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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-exhausted
• Definition: Consider a hypothesis space H,
target concept c, instance distribution D, and
set of training examples D of c. The version
space VSH,D is said to be -exhausted with
respect to c and D, if every hypothesis h in
VH,D has error less than  with respect to c and
D.
(h  VH , D )errorD (h)  
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Exhausting the version
space
Hypothesis Space H
error = 0.2
error = 0.1
r=0.2
r=0
error = 0.1
error = 0.3
r=0.2
error = 0.3
VSH,D
r=0
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r=0.4
error = 0.2
r=0.3
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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Theorem 7.1
• Theorem 7.1 -exhausting the version
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 VSH,D is
not -exhausted (with respect to c) is less
than or equal to
|H|e-m
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Proof of theorem
• See text
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Number of Training
Examples (Eq. 7.2)
H e m  
1
H e m  0

ln
1

 ln H  m  0
m  ln H  ln
m
1

1

(ln H  ln
1

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)
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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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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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
m  2 (ln | H |  ln( 1 /  ))
2
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Concepts that are PACLearnable
• Proofs that a type of concept is PACLearnable 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/)(ln|H|+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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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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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
What is VC(H)?
What is |H|?
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6.7
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) = ?
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|H| = ?
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)?
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Shattering conjunctions of
literals
• Approach: construct a set of instances of size 3 that can be
shattered. Let instance i have positive literal li and all other
literals negative. Representation of instances that are
conjunctions of literals l1, l2 and l3 as bit strings:
Instance1: 100
Instance2: 010
Instance3: 001
• Construction of dichotomy: To exclude an instance, add
appropriate li 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)?)
m
1

4 log 2 (2 /  )  8VC ( H ) log 2 (13 /  )
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Comparing the Bounds
m
m
1
2
1

2
(ln H  ln( 1 ))
4 log

2
(2 /  )  8VC H log 2 (13 /  ) 
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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
VC (C )  1
1
max  log( 1 /  ),


32



• Then with probability at least , L outputs a
hypothesis h having errorD(h) > .
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