CSCE 580 Artificial Intelligence Ch.18: Learning from Observations

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

CSCE 580Artificial IntelligenceCh.18: Learning from

ObservationsFall 2008

Marco [email protected]


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


Outline• Learning agents• Inductive learning• Decision tree learning


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


Learning agents


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


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 hsuch that h ≈ fgiven a training set of examples

This is a highly simplified model of real learning:– Ignores prior knowledge– Assumes examples are given


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:


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


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


Learning decision treesProblem: 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)


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)•


Decision trees• One possible representation for hypotheses• E.g., here is the “true” tree for deciding whether to wait:


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


Hypothesis spacesHow many distinct decision trees with n Boolean attributes?= number of Boolean functions= number of distinct truth tables with 2n rows = 22n (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


Hypothesis spacesHow many distinct decision trees with n Boolean attributes?= number of Boolean functions= number of distinct truth tables with 2n rows = 22n

• 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


Decision tree learning• Aim: find a small tree consistent with the training examples• Idea: (recursively) choose "most significant" attribute as root

of (sub)tree


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


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:npn

npn

npp

npp

npn

nppI

22 loglog),(


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:

• Choose the attribute with the largest IG

v

i ii

i

ii

iii

npn

nppI

npnpAremainder

1

),()(

)(),()( Aremaindernpn

nppIAIG


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):

• Patrons has the highest IG of all attributes and so is chosen by the DTL algorithm as the root

bits 0)]42,

42(

124)

42,

42(

124)

21,

21(

122)

21,

21(

122[1)(

bits 0541.)]64,

62(

126)0,1(

124)1,0(

122[1)(

IIIITypeIG

IIIPatronsIG


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


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


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


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]


Ensemble Learning• Sometimes each learning

techniqueyields a different hypothesis• But no perfect hypothesis…• Could we combine several imperfect

hypotheses into a better hypothesis?


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


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


Bagging


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


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


Boosting• Most popular ensemble technique• Computes a weighted majority• Can “boost” a “weak learner”• Operates on a weighted training set


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


Boosting Framework

Read the figure left to right: the algorithm builds a hypothesis on a weighted set of four examples, one hypothesis per column


AdaBoost (Adaptive Boosting)

There are N examples.There are M “columns” (hypotheses), each of which has weight zm


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


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


Boosting Paradigm• When we already have a bunch of

hypotheses, boosting provides a principled approach to combine them

• Useful for– Sensor fusion– Combining experts


Boosting Applications• Any supervised learning task

– Spam filtering– Speech recognition/natural language

processing– Data mining– Etc.


Computational Learning Theory

The slides on COLT are from ftp://ftp.cs.bham.ac.uk/pub/authors/M.Kerber/Teaching/SEM2A4/l4.ps.gzand http://www.cs.bham.ac.uk/~mmk/teaching/SEM2A4/, which also has slides on version spaces


How many examples are needed?

This is the probability that Hεbad

contains a consistent hypothesis


How many examples?


Complexity and hypothesis language


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]

CSCE 580 Artificial Intelligence Ch.18: Learning from Observations

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