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					CS 5100 Artificial Intelligence

           Prof. C. Hafner
    Class Notes March 29, 2011
                Learning Paradigms

• Supervised Learning
• Reinforcement Learning (genetic algorithms)
• Unsupervised Learning (clustering algorithms)

We will consider supervised learning, the most widely
 used and the most successful paradigm (also the
 most expensive. Why ??)
            Supervised Learning Methods
•   Naïve Bayes
•   Decision Tree Learning (ID3 and C4.5)
•   Neural Nets
•   Support Vector Machines
• Naïve Bayes learning
   – Maximum likelihood estimation




   – Smoothing
      • Motivation
      • How to (simplest method)
                 Information Theory
• Information is about categories and classification
• We measure quantity of information by the
  resources needed to represent/store/transmit the
  information
• Messages are sequences of 0’s and 1’s (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
        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
              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?
               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, 01=B, 10=C, 11=D
   – Each message is 2 bits. If you throw the die 800 times,
     you get a message 1600 bits long

   That’s the best you can do if A,B,C,D equally probable
         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) = 3/24
   – 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
                       Entropy
Formula for entropy with outcomes x1 . . . xn :

      - Σ 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
                Review/Explain Entropy
• Let the possible outcomes be x1 . . . . xn
   – With probabilities p1 . . . pn that add up to 1
• Ex: an unfair coin where n = 2, x1 = H (3/4), x2 = T
  (1/4)
• In a long sequence of events E = e1 . . . ek, we
  assume that outcome xi will occur k * pi times, etc.
E = HHTHTHHHTTHHHHHTHHHHTTHTHTHHHH …….
If k = 10000, we can assume H occurs 7500 times, T
   2500.
Note: the concept TYPES vs. TOKENS. There are two
types and 10000 tokens in this scenario.
               Review/Explain Entropy

The entropy of E H(E) is the average information of the
 events in the sequence e1 . . . ek :
        k

 1/k * Σ I(ej) =   [now switch to summation over outcomes]
       j=1

       n                            n
 1/k * Σ I(xi) * (k*pi) = k/k * Σ I(xi) * pi
       i=1                          i=1

  n
 Σ -log(pi) * pi bits
 i=1
                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
                 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.
                         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
   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 attribute
   – Each link is labeled with one value of the parent’s
     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.
    A trivial example of a classification tree
         Record#        Color            Shape      Fruit
         1              red              round      apple
         2              yellow           round      lemon
         3              yellow           oblong     banana


                          Color
               red                       yellow


             apple                          Shape

                                 round                oblong


                                    lemon             banana


The goal is to create an “efficient” classification tree which always gives
the same answer as the table
 A well-known “toy” example: sunburn data

Name       Hair      Height     Weight     Lotion       Sunburned
Sarah    Blonde     Average      Light       No            Yes
Dana     Blonde       Tall     Average       Yes           No
AleX      Brown      Short     Average       Yes           No
Annie    Blonde      Short     Average       No            Yes
Emily      Red      Average     Heavy        No            Yes
Pete      Brown       Tall      Heavy        No            No
John      Brown     Average     Heavy        No            No
Katie    Blonde      Short       Light       Yes           No


   Predictor attributes: hair, height, weight, lotion
                                 Hair

                 Blonde                   Brown
                               Red



                                                     Not
            Lotion            Sunburned           Sunburned

                          N
     Y



   Not
Sunburned            Sunburned
                 Outline of the algorithm

1. Create the root, and make its COLLECTION the entire table
2. Select any non-singular leaf node N to SPLIT
   1. Choose the best attribute A for splitting N (use info theory)
   2. For each value of A (a1, a2, . .) create a child of N, Nai
   3. Label the links from N to its children: “A = ai”
   4. SPLIT the collection of N among its children according to
       their values of A
3. When no more non-singular leaf nodes exist, the tree is finished
4. Def: a singular node is one whose COLLECTION includes just one
   value for the result attribute (therefore its entropy = 0)
 Choosing the best attribute to SPLIT: the one
         that is MOST INFORMATIVE
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 child’s entropy is weighted by the “probability” of
   that child (estimated by the proportion of the parent’s
   collection that would be transferred to the child in the
   split)
                                 C(S1) = {S,D,X,A,E,P,J,K}(3,5)/____}
          S1: _______            Calculate entropy: - [3/8 log 3/8 + 5/8 log 5/8] =
                                  .53 + .424 = .954

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
Child 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
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

Child 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
                  Red                              Blond: C = {S,D,A,K}(2,2)/1
                                      Brown

       yes                       no                        S2:_______
                  S2:_________         C(S2) = {S,D,A,K}(2,2)/1


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
                Red                                  Blond: C = {S,D,A,K}(2,2)/1
                                      Brown

     yes                        no                            S2: Lotion

                                                    no                      yes

                                              yes                      no
Discuss assignment 5
  Perceptrons and Neural Networks:
Another Supervised Learning Approach
          Perceptron Learning (Supervised)

•   Assign random weights (or set all to 0)
•   Cycle through input data until change < target
•   Let α be the “learning coefficient”
•   For each input:
    – If perceptron gives correct answer, do nothing
    – If perceptron says yes when answer should be no,
      decrease the weights on all units that “fired” by α
    – If perceptron says no when answer should be yes,
      increase the weights on all units that “fired” by α

				
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posted:7/25/2011
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