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CSCE 580
Artificial Intelligence
Ch.18: Learning from Observations
Fall 2008
Marco Valtorta
[email protected]
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Acknowledgment
• The slides are based on the textbook [AIMA] and other
sources, including other fine textbooks and the accompanying
slide sets
• The other textbooks I considered are:
– David Poole, Alan Mackworth, and Randy Goebel.
Computational Intelligence: A Logical Approach. Oxford,
1998
• A second edition (by Poole and Mackworth) is under development.
Dr. Poole allowed us to use a draft of it in this course
– Ivan Bratko. Prolog Programming for Artificial Intelligence,
Third Edition. Addison-Wesley, 2001
• The fourth edition is under development
– George F. Luger. Artificial Intelligence: Structures and
Strategies for Complex Problem Solving, Sixth Edition.
Addison-Welsey, 2009
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Outline
• Learning agents
• Inductive learning
• Decision tree learning
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Department of Computer Science and Engineering
Learning
• Learning is essential for unknown environments,
– i.e., when designer lacks omniscience
• Learning is useful as a system construction method,
– i.e., expose the agent to reality rather than trying to
write it down
• Learning modifies the agent's decision mechanisms to
improve performance
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Learning agents
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Department of Computer Science and Engineering
Learning element
• Design of a learning element is affected by
– Which components of the performance element are to
be learned
– What feedback is available to learn these components
– What representation is used for the components
• Type of feedback:
– Supervised learning: correct answers for each example
– Unsupervised learning: correct answers not given
– Reinforcement learning: occasional rewards
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Inductive learning
• Simplest form: learn a function from examples
f is the target function
An example is a pair (x, f(x))
Problem: find a hypothesis h
such that h ≈ f
given a training set of examples
This is a highly simplified model of real learning:
– Ignores prior knowledge
– Assumes examples are given
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Inductive learning method
• Construct/adjust h to agree with f on training set
• (h is consistent if it agrees with f on all examples)
• E.g., curve fitting:
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Inductive learning method
• Construct/adjust h to agree with f on training set
• (h is consistent if it agrees with f on all examples)
• E.g., curve fitting:
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Inductive learning method
• Construct/adjust h to agree with f on training set
• h is consistent if it agrees with f on all examples
• E.g., curve fitting:
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Inductive learning method
• Construct/adjust h to agree with f on training set
• (h is consistent if it agrees with f on all examples)
• E.g., curve fitting:
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Inductive learning method
• Construct/adjust h to agree with f on training set
• (h is consistent if it agrees with f on all examples)
• E.g., curve fitting:
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Inductive learning method
• Construct/adjust h to agree with f on training set
• (h is consistent if it agrees with f on all examples)
• E.g., curve fitting:
• Ockham’s razor: prefer the simplest hypothesis consistent with data
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Curve Fitting and Occam’s Razor
• Data collected by Galileo
in1608 – ball rolling down an
inclined plane, then continuing in
free-fall
• Occam's razor ( suggests the
simpler model is better; it has a
higher prior probability
• The simpler model may have a
greater posterior probability
(the plausibility of the model):
Occam’s razor is not only a
good heuristic, but it can be
shown to follow from more
fundmental principles
• Jefferys, W.H. and Berger, J.O.
1992. Ockham's razor and
Bayesian analysis. American
Scientist 80:64-72
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Learning decision trees
Problem: decide whether to wait for a table at a restaurant, based on the
following attributes:
1. Alternate: is there an alternative restaurant nearby?
2. Bar: is there a comfortable bar area to wait in?
3. Fri/Sat: is today Friday or Saturday?
4. Hungry: are we hungry?
5. Patrons: number of people in the restaurant (None, Some, Full)
6. Price: price range ($, $$, $$$)
7. Raining: is it raining outside?
8. Reservation: have we made a reservation?
9. Type: kind of restaurant (French, Italian, Thai, Burger)
10. WaitEstimate: estimated waiting time (0-10, 10-30, 30-60, >60)
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Attribute-based representations
•
•
Examples described by attribute values (Boolean, discrete, continuous)
E.g., situations where I will/won't wait for a table:
•
•
Classification of examples is positive (T) or negative (F)
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Department of Computer Science and Engineering
Decision trees
• One possible representation for hypotheses
• E.g., here is the “true” tree for deciding whether to wait:
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Department of Computer Science and Engineering
Expressiveness
•
•
Decision trees can express any function of the input attributes
E.g., for Boolean functions, truth table row → path to leaf
•
Trivially, there is a consistent decision tree for any training set with one path to leaf for
each example (unless f nondeterministic in x) but it probably won't generalize to new
examples
•
Prefer to find more compact decision trees
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Hypothesis spaces
How many distinct decision trees with n Boolean attributes?
= number of Boolean functions
n
= number of distinct truth tables with 2n rows = 22 (for each of the 2n rows of the
decision table, the function may return 0 or 1)
•
E.g., with 6 Boolean attributes, there are 18,446,744,073,709,551,616 (more
than 18 quintillion) trees
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Hypothesis spaces
How many distinct decision trees with n Boolean attributes?
= number of Boolean functions
n
= number of distinct truth tables with 2n rows = 22
•
E.g., with 6 Boolean attributes, there are 18,446,744,073,709,551,616 trees
How many purely conjunctive hypotheses (e.g., Hungry  Rain)?
• Each attribute can be in (positive), in (negative), or out
 3n distinct conjunctive hypotheses
• More expressive hypothesis space
– increases chance that target function can be expressed
– increases number of hypotheses consistent with training set
 may get worse predictions
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Decision tree learning
•
•
Aim: find a small tree consistent with the training examples
Idea: (recursively) choose "most significant" attribute as root of (sub)tree
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Choosing an attribute
• Idea: a good attribute splits the examples into subsets that are (ideally)
"all positive" or "all negative"
• Patrons? is a better choice
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Department of Computer Science and Engineering
Using information theory
• To implement Choose-Attribute in the DTL algorithm
• Information Content (Entropy):
I(P(v1), … , P(vn)) = Σi=1 -P(vi) log2 P(vi)
• For a training set containing p positive examples and n
negative examples:
I(
p
n
p
p
n
n
,
)
log 2

log 2
pn pn
pn
pn pn
pn
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Department of Computer Science and Engineering
Information gain
• A chosen attribute A divides the training set E into subsets
E1, … , Ev according to their values for A, where A has v
distinct values
• Information Gain (IG) or reduction in entropy from the
attribute test:
v
remainder ( A)  
i 1
p i ni
pi
ni
I(
,
)
p  n pi  ni pi  ni
• Choose the attribute with the largest IG
p
n
IG ( A)  I (
,
)  remainder ( A)
pn pn
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Information gain
• For the training set, p = n = 6, I(6/12, 6/12) = 1 bit
• Consider the attributes Patrons and Type (and others too):
2
4
6 2 4
IG ( Patrons)  1  [ I (0,1)  I (1,0)  I ( , )]  .0541 bits
12
12
12 6 6
2 1 1
2 1 1
4 2 2
4 2 2
IG (Type)  1  [ I ( , )  I ( , )  I ( , )  I ( , )]  0 bits
12 2 2 12 2 2 12 4 4 12 4 4
• Patrons has the highest IG of all attributes and so is chosen
by the DTL algorithm as the root
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Example contd.
• Decision tree learned from the 12 examples:
• Substantially simpler than “true” tree---a more complex
hypothesis isn’t justified by small amount of data
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Performance measurement
•
How do we know that h ≈ f ?
1.
Use theorems of computational/statistical learning theory
2.
Try h on a new test set of examples
(use same distribution over example space as training set)
Learning curve = % correct on test set as a function of training set size
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Summary (so far)
• Learning needed for unknown environments, lazy designers
• Learning agent = performance element + learning element
• For supervised learning, the aim is to find a simple
hypothesis approximately consistent with training examples
• Decision tree learning using information gain
• Learning performance = prediction accuracy measured on
test set
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Outline for Ensemble Learning and
Boosting
• Ensemble Learning
– Bagging
– Boosting
• Reading: [AIMA-2] Sec. 18.4
• This set of slides is based on
http://www.cs.uwaterloo.ca/~ppoupart/teaching/
cs486-spring05/slides/Lecture21notes.pdf
• In turn, those slides follow [AIMA-2]
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Department of Computer Science and Engineering
Ensemble Learning
• Sometimes each learning techniqueyields a
different hypothesis
• But no perfect hypothesis…
• Could we combine several imperfect hypotheses
into a better hypothesis?
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Ensemble Learning
• Analogies:
– Elections combine voters’ choices to pick a good
candidate
– Committees combine experts’ opinions to make better
decisions
• Intuitions:
– Individuals often make mistakes, but the “majority” is
less likely to make mistakes.
– Individuals often have partial knowledge, but a
committee can pool expertise to make better decisions
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Ensemble Learning
• Definition: method to select and
combine an ensemble of
hypotheses into a (hopefully)
better hypothesis
• Can enlarge hypothesis space
– Perceptron (a simple kind of
neural network)
• linear separator
– Ensemble of perceptrons
• polytope
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Department of Computer Science and Engineering
Bagging
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Department of Computer Science and Engineering
Bagging
• Assumptions:
– Each hi makes error with probability p
– The hypotheses are independent
• Majority voting of n hypotheses:
– k hypotheses make an error:
– Majority makes an error:
• – With n=5, p=0.1 error( majority ) < 0.01
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Weighted Majority
• In practice
– Hypotheses rarely independent
– Some hypotheses make fewer errors than others
• Let’s take a weighted majority
• Intuition:
– Decrease weight of correlated hypotheses
– Increase weight of good hypotheses
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Boosting
•
•
•
•
Most popular ensemble technique
Computes a weighted majority
Can “boost” a “weak learner”
Operates on a weighted training set
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Weighted Training Set
• Learning with a weighted training set
– Supervised learning -> minimize training error
– Bias algorithm to learn correctly instances with
high weights
• Idea: when an instance is misclassified by a
hypotheses, increase its weight so that the next
hypothesis is more likely to classify it correctly
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Boosting Framework
Read the figure left to right: the algorithm builds a hypothesis on a
weighted set of four examples, one hypothesis per column
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
AdaBoost (Adaptive Boosting)
There are N
examples.
There are M
“columns”
(hypotheses),
each of which
has weight zm
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
What can we boost?
• Weak learner: produces hypotheses at least as
good as random classifier.
• Examples:
– Rules of thumb
– Decision stumps (decision trees of one node)
– Perceptrons
– Naïve Bayes models
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Boosting Paradigm
• Advantages
– No need to learn a perfect hypothesis
– Can boost any weak learning algorithm
– Boosting is very simple to program
– Good generalization
• Paradigm shift
– Don’t try to learn a perfect hypothesis
– Just learn simple rules of thumbs and boost them
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering
Boosting Paradigm
• When we already have a bunch of hypotheses,
boosting provides a principled approach to
combine them
• Useful for
– Sensor fusion
– Combining experts
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Department of Computer Science and Engineering
Boosting Applications
• Any supervised learning task
– Spam filtering
– Speech recognition/natural language
processing
– Data mining
– Etc.
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Department of Computer Science and Engineering
Computational Learning Theory
The slides on COLT are from
ftp://ftp.cs.bham.ac.uk/pub/authors/M.Kerber/Teaching/SEM2A4/l4.ps.gz
and http://www.cs.bham.ac.uk/~mmk/teaching/SEM2A4/, which also has slides
on version spaces
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Department of Computer Science and Engineering
How many examples are needed?
This is the probability that Hεbad
contains a consistent hypothesis
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Department of Computer Science and Engineering
How many examples?
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Complexity and hypothesis language
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Department of Computer Science and Engineering
Learning Decision Lists
• A decision list consists of a series of tests, each of which is a
conjunction of literals. If the tests succeeds, the decision list
specifies the value to be returned. Otherwise, the processing
continues with the next test in the list
• Decision lists can represent any Boolean function hence are not
learnable (in polynomial time)
• A k-DL is a decision list where each test is restricted to at most k
literals
• K- Dl is learnable! [Rivest, 1987]
UNIVERSITY OF SOUTH CAROLINA
Department of Computer Science and Engineering