CSE 471/598 Intro to AI (Kambhampati)

Homework 5 Solutions

 

 

 

 

Question IV [Short answer questions:

ProblemVI. [30pt] George Costanza wants to throw a party and he wants it to be a success. Over the years he has thrown a fair number of parties, and some have been successful and some have been unsuccessful. He typically calls some subset of Seinfeld, Newman and Kramer. Here is the data about the past parties--described in terms of who turned up at the party and whether the party was a success.

 

 

Seinfeld invited

Newman invited

Kramer invited

Good Party

Party1

N

N

N

N

Party2

N

N

Y

N

Party3

N

Y

N

Y

Party4

N

Y

Y

Y

Party5

Y

N

N

Y

Party6

Y

N

Y

Y

Party7

Y

Y

N

N

Party8

Y

Y

Y

N

Part A.[3pt] We want to help George out by using decision trees to learn the pattern behind successful parties, so that George can then use the learned decision tree to predict whether or not a new party he is trying to throw will be a success. Since this is a small-enough example, YOU should be able to see the pattern in the data. Can you express the pattern in terms of a compact decision tree (If you are completely lost on this part, you can ask me for a hint at the expense of some points; so you can continue with the rest of the sub-parts)

 

Clearly, as any true-blue seinfeld aficianado will have no trouble recognizing, the party will be bad if both seinfeld and Newman are invited (they are sworn enemies, afterall), and will be good otherwise.

So, the concept is ~(Seinfeld .xor. Newman)

Part B.[6pt] Suppose you want to go with the automated decision-tree construction algorithm (which uses the information value of splitting on each attribute). Show the information values of splitting on each attribute, and show which attribute will be selected at the first level. Assume that if two attributes have the same information value, you will split the tie using the alphabetical ordering (this should be a reasonable assumption since decision trees don't care how you choose among equally valued attributes).

There are 4 parties with Seinfeld yes and 4 with seinfeld no.

In each set, 2 are good parties and 2 are bad

So, I(seinfeld) = 4/8 I(2/4,2/4) + 4/8 I(2/4,2/4)

= .5 * 1 + .5 * 1 ( as I(1/2,1/2) is 1)

= 1

Similar reasoning lets us realize that I(Kramer) = I(Newman) = 1

Since original set has 4 good and 4 bad parties, the information in the initial set is 1 too.

So, information gain associated with each of the attributes is 0.

Since all have same information gain, we break tie in alphabetical order, picking Kramer first

 

Part C.[3] Do you think the tree that you will ultimately get by continuing to refine the partial tree in Part B will be better or worse than the tree that you wrote in Part A If it is worse, is it theoretically consistent that you could do better than the automated information-theoretic method for constructing decision trees (You don't need to show any explicit information value calculations here).

 

Since we picked cramer first, we will wind up making a tree which will refer to all attributes (since we know that the good or bad party depends on seinfeld and newman and that cramer is a basically irrelevant question).

Clearly, the resulting tree is worse than the smallest tree we can get by just realizing that it is an XOR concept on Seinfeld and Newman.

Since we are using a greedy approach we are not guaranteed to get smallest tree.

Part D.[4] Consider the super-attribute "Seinfeld-Newman", which in essence is the result of merging the seinfeld and newman attributes. This super attribute takes four values: YY, YN, NY, NN. Compute the information value of this super-attribute, and compare it to the attribute information values in Part B. If you had a chance of considering this super attribute along with the other attributes, which would you have chosen to split on at the first level

For this case, the super attribute splits the examples as follows

YY 0+ 2-

YN 2+ 0-

NY 2+ 0-

NN 0+ 2-

2/8 I(0,1) + 2/8 I(1,0) + 2/8 I(1,0) + 2/8 I(0,1)

= 0

So the information gain is 1-0=1

Thus if you have this super attribute, clearly it has the best gainócompared to others which are all 0.

If we choose that super attribute, then we will get a single layer tree in terms of that attribute, which can be translated into a two layer tree interms of the single attributes.

 

Part E.[4] Does the answer to Part D suggest you a way of always getting optimal decision trees If so, what is the computational cost of this method relative to the standard method (that you used in Part B)

 

Yes, the idea is to consider not just single attributes but also super attributes corresponding to subsets of all attributes. Unfortunately however, there are 2N super attributes and single attributes, given the N single attributes. This will make the node ordering an exponential cost operation!

 

Part F.[4] Now, George is gettin' upset at all this dilly-dallying with the decision trees, and would like to switch over to neural networks for finding pattern in his successful parties. What is the minimum number of layers of threshold units that he would need in his neural network so he can converge and learn the pattern behind his successful parties Explain how you reached your answer.

Since we know that the underlying concept is an XOR on two of the three attributes, we know that we cannot get by with asingle layer perceptron. We need at least 2 layers. (Since XOR is not linearly separable).

Part G.[6] Show an actual neural network (with weights and thresholds) that represents the concept. Assume that the inputs are going to be 1 (for Y) and 0 (for N). We want the output to be 1 (for successful party) and 0 (for unsuccessful party). I am not asking you to learn a network using the gradient descent method. I am asking you to show one directly. (hint 1: Use your answer to part A). (Hint 2: You might want to consult figure 19.6 in page 570 of the textbook in doing this problem.)