CS G120 Artificial Intelligence

1
CS G120 Artificial Intelligence

• Prof. C. Hafner
• Class Notes March 26, 2009

2
Outline

• Learning agents
• Inductive learning
• Decision tree learning
• Bayesian learning

3
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

4
How do people learn ?

• By experience
• By being told (in person, reading, TV, etc.)
• Inductive learning framework
• Agent gets positive and negative examples of some
  concept
• A major problem is overfitting
• Compare with learning a new model or theory such
  as the balance of power among the 3 branches of
  US government
• (Skill acquisition not considered)

5
Inductive Learning Elements

• 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

6
Learning frequently applied to

• Classification problems
• Finite number of classes, pre-defined features
• Score Ci S wi Fi
• Apply supervised learning to select the weights
• Also applied to finding a good heuristic function
  for searching
• H S wi Fi
• Apply supervised or reinforcement learning to
  select the weights

7
Inductive Learning agents
8
Learning decision trees

• Problem decide whether to wait for a table at a
  restaurant, based on the following attributes
• Alternate is there an alternative restaurant
  nearby?
• Bar is there a comfortable bar area to wait in?
• Fri/Sat is today Friday or Saturday?
• Hungry are we hungry?
• Patrons number of people in the restaurant
  (None, Some, Full)
• Price price range (, , )
• Raining is it raining outside?
• Reservation have we made a reservation?
• Type kind of restaurant (French, Italian, Thai,
  Burger)
• WaitEstimate estimated waiting time (0-10,
  10-30, 30-60, gt60)

9
Decision trees

• One possible representation for hypotheses
• E.g., here is the true tree for deciding
  whether to wait

10
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)
  

11
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

12
Example contd.

• Decision tree learned from the 12 examples
• Substantially simpler than true tree---a more
  complex hypothesis isnt justified by small
  amount of data

13
Hypothesis spaces

• How 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

14
Decision tree learning

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

15
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

16
Using information theory

• To implement Choose-Attribute in the DTL
  algorithm
• Information Content (Entropy)
• I(P(v1), , P(vn)) Si1 -P(vi) log2 P(vi)
• For a training set containing p positive examples
  and n negative examples

17
Information gain

• A chosen attribute A divides a collection C 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

18
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

19
Information Theory and Entropy

• We measure quantity of information by the
  resources needed to represent/store/transmit the
  information
• Messages are sequences of 0s and 1s
  (dots/dashes) which we call bits (for binary
  digits)
• You need to send a message containing the
  identity of a spy
• It is known to be Mr. Brown or Mr. Smith
• You can send the message with 1 bit, therefore
  the event the spy is Smith has 1 bit of
  information

20
Calculating quantity of information

• Def A uniform distribution of a set of possible
  outcomes (X1 . . . Xn) means the outcomes are
  equally probable that is, they each have
  probability 1/n.
• Suppose there are 8 people who can be the spy.
  Then the message requires 3 bits. If there are
  64 possible spies the message requires 6 bits,
  etc. (assuming a uniform distribution)
• Def The information quantity of a message where
  the (uniform) probability of each value is p
• I -log p bits

21
Intuition and Examples

• Intuitively, the more surprising a message is,
  the more information it contains. If there are
  64 equally-probable spies we are more surprised
  by the identity of the spy than if there are only
  two equally probable spies.
• There are 26 letters in the alphabet. Assuming
  they are equally probable, how much information
  is in each letter I -log (1/26) log 26
  4.7 bits
• Assuming the digits from 0 to 9 are equally
  probable. Will the information in each digit be
  more or less than the information in each letter?

22
Sequences of messages

• Things get interesting when we looks beyond a
  single message to a long sequence of messages.
• Consider a 4-sided die, with symbols A, B, C, D
• Let 00 A, 01B, 10C, 11D
• Each message is 2 bits. If you throw the die 800
  times, you get a message 1600 bits long
• Thats the best you can do if A,B,C,D equally
  probable

23
Non-uniform distributions (cont.)

• Consider a 4-sided die, with symbols A, B, C, D
• But assume P(A) 7/8 and P(B)P(C)P(D) 1/24
• We can take advantage of that with a different
  code
• 0 A, 10 B, 110 C, 111 D
• If we throw the die 800 times, what is the
  expected length of the message? What is the
  entropy?
• ENTROPY is the average information (in bits) of
  events in a long repeated sequence

24
Entropy

• Formula for entropy with outcomes x1 . . . xn
• - S P(xi) log P(xi) bits
• For a uniform distribution this is the same as
  log P(x1) since all the P(xi) are the same.
• What does it mean? Consider 6-sided die,
  outcomes equally probable
• -log 1/6 2.58 tells us a long sequence of die
  throws can be transmitted using 2.58 bits per
  throw on the average and this is the
  theoretical best

25
Review/Explain Entropy

• Entropy is sometimes called disorder it
  represents the lack of predictability as to the
  outcome for any element of a sequence (or set)
• If a set has just one outcome, entropy 1
  -log(1) 0
• If there are 2 outcomes, then 50/50 probability
  gives the maximum entropy complete
  unpredictability. This generalizes to any
  uniform distribution for n outcomes.
• - (0.5 log(.5) 0.5 log(.5)) 1 bit
• Note log(1/2) -log(2) -1

26
Calculating Entropy

• Consider a biased coin P(heads) ¾ P(tails)
  ¼
• What is the entropy of a coin toss outcome?
• H ¼ -log(1/4) ¾ -log(3/4) 0.811 bits
• Using the Information Theory Log Table
• H 0.25 2.0 0.75 0.415 0.5 0.311
  .811
• A fair coin toss has more information
• The more unbalanced the probabilities, the more
  predictable the outcome, the less you learn from
  each message.

27
Maximum disorder
1
H (entropy in bits)
0 ½
1
Probability of x1
Entropy for a set containing 2 possible outcomes
(x1, x2) What if there are 3 possible outcomes?
for equal probability case H -log(1/3)
about 1.58
28
Define classification tree and ID3 algorithm

• Def Given a table with one result attribute and
  several designated predictor attributes, a
  classification tree for that table is a tree such
  that
• Each leaf node is labeled with a value of the
  result attribute
• Each non-leaf node is labeled with the name of a
  predictor
• Each link is labeled with one value of the
  parents predictor
• Def the ID3 algorithm takes a table as input and
  learns a classification tree that efficiently
  maps predictor value sets into their results from
  the table.

29
A trivial example of a classification tree
Color
yellow
red
Shape
apple
oblong
round
lemon
banana
The goal is to create an efficient
classification tree which always gives the same
answer as the table
30
A well-known toy example sunburn data
Predictor attributes hair, height, weight,
lotion
31
Blonde
Brown
Red
N
Y
32
Review the algorithm

• Create the root, and make its COLLECTION the
  entire table
• Select any non-singular leaf node N to SPLIT
• Choose the best attribute A for splitting N (use
  info theory)
• For each value of A (a1, a2, . .) create a child
  of N, Nai
• Label the links from N to its children A ai
• SPLIT the collection of N among its children
  according to their values of A
• When no more non-singular leaf nodes exist, the
  tree is finished
• Def a singular node is one whose COLLECTION
  includes just one value for the result attribute
• (therefore its entropy 0)

33
Choosing the best attribute to SPLIT the one
that is MOST INFORMATIVE (highest IG)that
reduces the entropy (DISORDER) the most

• Assume there are k attributes we can choose.
  For each one, we compute how much less entropy
  exists in the resulting children than we had in
  the parents
• H(N) weighted sum of H(children
  of N)
• Each childs entropy is weighted by the
  probability of that child (estimated by the
  proportion of the parents collection that would
  be transferred to the child in the split)

34
C S,D,X,A,E,P,J,K(3,5)/____
Calculate entropy - 3/8 log 3/8 5/8 log 5/8
.53 .424 .954
S1 _______CC
Find information gain (IG) for all 4 predictors
hair, height, weight, lotion Start with lotion
values (yes, no) Child 1 (yes) D,X,K(0,
3)/0 Child 2 (no) S,A,E,P,J(3,2)/ -3/5 log
3/5 2/5 log 2/5 .971 Child set entropy
3/8 0 5/8 .971 0.607 IG(Lotion) .954 -
.607 .347 Then try hair color values (blond,
brown, red) Child 1(blond) S,D,A,K(2,2)/1 Chil
d 2(brown) X,P,J(0,3)/0 Child 3(red)
E(1,0)/0 Child set entropy 4/8 1 3/8
0 1/8 0 0.5 IG(Hair color) .954 - 0.5
.454
35
Next try Height values (average, tall
short) Child1(average) S,E,J(2,1)/ -2/3 log
2/3 1/3 log 1/3 0.92 Child2(tall)
D,P(0,2)/0 Child3(short)X,A,K(1,2)/0.92 Chil
d set entropy 3/8 0.92 2/8 0 3/8 0.92
0.69 IG(Height) .954 - .69 0.26 Next try
Weight . . . IG(Weight) 0.954 0.94 0.014 So
Hair color wins Draw the first split and assign
the collections
N1 Hair Color
Blond C S,D,A,K(2,2)/1
Red
Brown
yes
no
S2_______
36
C S,D,A,K(2,2)/1
S2_________
Start with lotion values (yes, no) Child 1
(yes) D, K(0, 2)/0 Child 2 (no)
S,A(2,0)/ 0 Child set entropy 0 IG(Lotion)
1 0 1 No reason to go any farther

S1 Hair Color
Blond C S,D,A,K(2,2)/1
Red
Brown
yes
no
S2 Lotion
no
yes
yes
no
37
Limitations of DTL

• Inconsistency
• Can use majority, if enough data
• Missing data
• Overfitting (a problem with all inductive
  learning)
• Multivalued attributes
• Use gain ratio
• Numerical attributes
• Search for split points that maximize IG

38
Performance Evaluation of DTL

• Training set/test set division
• Addresses overfitting problem to some extent
• K-fold cross validation (5, 10, N)
• The problem of peeking (parameter setting and
  evaluation require separate test sets)

39
Performance measurement

• Learning curve correct on test set as a
  function of training set size