The document discusses the concept of PAC (Probably Approximately Correct) learning. It begins by describing a learning scenario where a hidden hypothesis is chosen by nature, and a learner tries to approximate this hypothesis based on randomly generated training data. It then defines what it means for a learned hypothesis to be "bad" or have high test error, and shows that by choosing a large enough random training set, the probability of learning a bad hypothesis can be bounded. Finally, it provides the formula for calculating the minimum size of the random training set needed to guarantee this probability bound.

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PAC-learning
Andrew W. Moore
Associate Professor
School of Computer Science
Carnegie Mellon University
www.cs.cmu.edu/~awm
awm@cs.cmu.edu
412-268-7599
Copyright © 2001, Andrew W. Moore Nov 30th, 2001
Probably Approximately Correct
(PAC) Learning
• Imagine we’re doing classification with categorical
inputs.
• All inputs and outputs are binary.
• Data is noiseless.
• There’s a machine f(x,h) which has H possible
settings (a.k.a. hypotheses), called h1, h 2 .. hH.
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 2
1

2. Example of a machine
• f(x,h) consists of all logical sentences about X1, X2
.. Xm that contain only logical ands.
• Example hypotheses:
• X1 ^ X3 ^ X19
• X3 ^ X18
• X7
• X1 ^ X2 ^ X2 ^ x4 … ^ Xm
• Question: if there are 3 attributes, what is the
complete set of hypotheses in f?
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 3
Example of a machine
• f(x,h) consists of all logical sentences about X1, X2
.. Xm that contain only logical ands.
• Example hypotheses:
• X1 ^ X3 ^ X19
• X3 ^ X18
• X7
• X1 ^ X2 ^ X2 ^ x4 … ^ Xm
• Question: if there are 3 attributes, what is the
complete set of hypotheses in f? (H = 8)
True X2 X3 X2 ^ X3
X1 X1 ^ X2 X1 ^ X3 X1 ^ X2 ^ X3
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 4
2

3. And-Positive-Literals Machine
• f(x,h) consists of all logical sentences about X1, X2
.. Xm that contain only logical ands.
• Example hypotheses:
• X1 ^ X3 ^ X19
• X3 ^ X18
• X7
• X1 ^ X2 ^ X2 ^ x4 … ^ Xm
• Question: if there are m attributes, how many
hypotheses in f?
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 5
And-Positive-Literals Machine
• f(x,h) consists of all logical sentences about X1, X2
.. Xm that contain only logical ands.
• Example hypotheses:
• X1 ^ X3 ^ X19
• X3 ^ X18
• X7
• X1 ^ X2 ^ X2 ^ x4 … ^ Xm
• Question: if there are m attributes, how many
hypotheses in f? (H = 2m)
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 6
3

4. And-Literals Machine
• f(x,h) consists of all logical
sentences about X1, X2 ..
Xm or their negations that
contain only logical ands.
• Example hypotheses:
• X1 ^ ~X3 ^ X19
• X3 ^ ~X18
• ~X7
• X1 ^ X2 ^ ~X3 ^ … ^ Xm
• Question: if there are 2
attributes, what is the
complete set of hypotheses
in f?
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 7
And-Literals Machine
• f(x,h) consists of all logical
sentences about X1, X2 ..
Xm or their negations that
True True
contain only logical ands.
• Example hypotheses: True X2
• X1 ^ ~X3 ^ X19 True ~X2
• X3 ^ ~X18 X1 True
• ~X7 X1 ^ X2
• X1 ^ X2 ^ ~X3 ^ … ^ Xm X1 ^ ~X2
• Question: if there are 2 ~X1 True
attributes, what is the
~X1 ^ X2
complete set of hypotheses
~X1 ^ ~X2
in f? (H = 9)
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 8
4

5. And-Literals Machine
• f(x,h) consists of all logical
sentences about X1, X2 ..
Xm or their negations that
True True
contain only logical ands.
• Example hypotheses: True X2
• X1 ^ ~X3 ^ X19 True ~X2
• X3 ^ ~X18 X1 True
• ~X7 X1 ^ X2
• X1 ^ X2 ^ ~X3 ^ … ^ Xm X1 ^ ~X2
• Question: if there are m ~X1 True
attributes, what is the size of
~X1 ^ X2
the complete set of
~X1 ^ ~X2
hypotheses in f?
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 9
And-Literals Machine
• f(x,h) consists of all logical
sentences about X1, X2 ..
Xm or their negations that
True True
contain only logical ands.
• Example hypotheses: True X2
• X1 ^ ~X3 ^ X19 True ~X2
• X3 ^ ~X18 X1 True
• ~X7 X1 ^ X2
• X1 ^ X2 ^ ~X3 ^ … ^ Xm X1 ^ ~X2
• Question: if there are m ~X1 True
attributes, what is the size of
~X1 ^ X2
the complete set of
~X1 ^ ~X2
hypotheses in f? (H = 3m)
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 10
5

6. Lookup Table Machine
• f(x,h) consists of all truth X1 X2 X3 X4 Y
tables mapping combinations 0 0 0 0 0
0 0 0 1 1
of input attributes to true 0 0 1 0 1
and false 0 0 1 1 0
0 1 0 0 1
• Example hypothesis: 0 1 0 1 0
0 1 1 0 0
0 1 1 1 1
• Question: if there are m 1 0 0 0 0
attributes, what is the size of 1 0 0 1 0
the complete set of 1 0 1 0 0
1 0 1 1 1
hypotheses in f? 1 1 0 0 0
1 1 0 1 0
1 1 1 0 0
1 1 1 1 0
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 11
Lookup Table Machine
• f(x,h) consists of all truth X1 X2 X3 X4 Y
tables mapping combinations 0 0 0 0 0
0 0 0 1 1
of input attributes to true 0 0 1 0 1
and false 0 0 1 1 0
0 1 0 0 1
• Example hypothesis: 0 1 0 1 0
0 1 1 0 0
0 1 1 1 1
• Question: if there are m 1 0 0 0 0
attributes, what is the size of 1 0 0 1 0
the complete set of 1 0 1 0 0
1 0 1 1 1
hypotheses in f? 1 1 0 0 0
1 1 0 1 0
H =2
m 1 1 1 0 0
2 1 1 1 1 0
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 12
6

7. A Game
• We specify f, the machine
• Nature choose hidden random hypothesis h*
• Nature randomly generates R datapoints
•How is a datapoint generated?
1.Vector of inputs x k = (x k1 ,x k2 , x km) is
drawn from a fixed unknown distrib: D
2.The corresponding output y k=f(x k , h*)
• We learn an approximation of h* by choosing
some hest for which the training set error is 0
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 13
Test Error Rate
• We specify f, the machine
• Nature choose hidden random hypothesis h*
• Nature randomly generates R datapoints
•How is a datapoint generated?
1.Vector of inputs x k = (x k1 ,x k2 , x km) is
drawn from a fixed unknown distrib: D
2.The corresponding output y k=f(x k , h*)
• We learn an approximation of h* by choosing
some hest for which the training set error is 0
• For each hypothesis h ,
• Say h is Correctly Classified (CCd) if h has
zero training set error
• Define TESTERR(h )
= Fraction of test points that h will classify
correctly
= P(h classifies a random test point
correctly)
• Say h is BAD if TESTERR(h) > ε
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 14
7

8. Test Error Rate
P (h is CCd | h is bad ) =
P(∀k ∈ Training Set, f ( xk , h ) = yk | h is bad )
• We specify f, the machine
≤ (1 − ε ) R
• Nature choose hidden random hypothesis h*
• Nature randomly generates R datapoints
•How is a datapoint generated?
1.Vector of inputs x k = (x k1 ,x k2 , x km) is
drawn from a fixed unknown distrib: D
2.The corresponding output y k=f(x k , h*)
• We learn an approximation of h* by choosing
some hest for which the training set error is 0
• For each hypothesis h ,
• Say h is Correctly Classified (CCd) if h has
zero training set error
• Define TESTERR(h )
= Fraction of test points that i will classify
correctly
= P(h classifies a random test point
correctly)
• Say h is BAD if TESTERR(h) > ε
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 15
Test Error Rate
P (h is CCd | h is bad ) =
P(∀k ∈ Training Set, f ( xk , h ) = yk | h is bad )
• We specify f, the machine
≤ (1 − ε ) R
• Nature choose hidden random hypothesis h*
• Nature randomly generates R datapoints
P ( we learn a bad h ) ≤
•How is a datapoint generated?
1.Vector of inputs x k = (x k1 ,x k2 , x km) is
 the set of CCd h' s 
drawn from a fixed unknown distrib: D P
 containsa bad h  = 
 
2.The corresponding output y k=f(x k , h*)
• We learn an approximation of h* by choosing
P (∃h. h is CCd ^ h is bad)=
some hest for which the training set error is 0
• For each hypothesis h ,
 ( h1 is CCd ^ h1 is bad) ∨ 
• Say h is Correctly Classified (CCd) if h has  
zero training set error  ( h2 is CCd ^ h2 is bad) ∨ 
≤
P
• Define TESTERR(h )
:
 
= Fraction of test points that i will classify
 ( h is CCd ^ h is bad) 
H 
correctly H
= P(h classifies a random test point H
∑ P(h is CCd ^ hi is bad)≤
correctly) i
i =1
H
• Say h is BAD if TESTERR(h) > ε
∑ P(h is CCd | hi is bad) =
i
i =1
H × P (hi is CCd | hi is bad) ≤ H (1 − ε ) R
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 16
8

9. PAC Learning
• Chose R such that with probability less than δ we’ll
select a bad hest (i.e. an hest which makes mistakes
more than fraction ε of the time)
• Probably Approximately Correct
• As we just saw, this can be achieved by choosing
R such that
δ = P( we learn a bad h ) ≤ H (1 − ε ) R
• i.e. R such that
0.69  1
R≥  log 2 H + log 2 
ε δ
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 17
PAC in action
Machine Example H R required to PAC-
Hypothesis learn
0.69  1
And-positive- X3 ^ X7 ^ X8 2m  m + log 2 
ε δ
literals
And-literals X3 ^ ~X7 3m 0.69  1
 (log2 3)m + log2 
ε δ
Lookup X1 X2 X3 X4 Y
0.69  m 1
m
0 0 0 0 0
22  2 + log 2 
0 0 0 1 1
Table
0 0 1 0 1
ε δ
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
0 1 1 1 1
1 0 0 0 0
1 0 0 1 0
1 0 1 0 0
1 0 1 1 1
1 1 0 0 0
1 1 0 1 0
1 1 1 0 0
1 1 1 1 0
(X1 ^ X5) v
And-lits or 0. 69  1
(3m ) 2 = 3 2m  ( 2 log 2 3) m + log2 
ε δ
And-lits (X2 ^ ~X7 ^ X8)
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 18
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10. PAC for decision trees of depth k
• Assume m attributes
• Hk = Number of decision trees of depth k
• H0 =2
• Hk+1 = (#choices of root attribute) *
(# possible left subtrees) *
(# possible right subtrees)
= m * Hk * Hk
• Write Lk = log2 Hk
• L0 = 1
• Lk+1 = log2 m + 2Lk
• So Lk = (2 k -1)(1+log2 m) +1
• So to PAC-learn, need
0.69  k 1
R≥  ( 2 − 1)(1 + log 2 m) + 1 + log 2 
ε δ
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 19
What you should know
• Be able to understand every step in the math that
gets you to
δ = P( we learn a bad h ) ≤ H (1 − ε ) R
• Understand that you thus need this many records
to PAC-learn a machine with H hypotheses
0.69  1
R≥  log 2 H + log 2 
ε δ
• Understand examples of deducing H for various
machines
Copyright © 2001, Andrew W. Moore PAC-learning: Slide 20
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