Final 10-701 Exam, Spring 2007
1. Personal info: Name: Andrew account: E-mail address: 2. There should be 16 numbered pages in this exam (including this cover sheet). 3. You can use any material you brought: any book, class notes, your print outs of class materials that are on the class website, including my annotated slides and relevant readings, and Andrew Moore's tutorials. You cannot use materials brought by other students. Calculators are allowed, but no laptops, PDAs, phones or Internet access. 4. If you need more room to work out your answer to a question, use the back of the page and clearly mark on the front of the page if we are to look at what's on the back. 5. Work efficiently. Some questions are easier, some more difficult. Be sure to give yourself time to answer all of the easy ones, and avoid getting bogged down in the more difficult ones before you have answered the easier ones. 6. Note there are extra-credit sub-questions. The grade curve will be made without considering students' extra credit points. The extra credit will then be used to try to bump your grade up without affecting anyone else's grade. 7. You have 180 minutes. 8. Good luck!
Question 1 2 3 4 5 6 7
Topic Short questions SVM and slacks GNB Feature Selection Irrelevant Features Neural Nets Learning theory 1
Max. score 21 + 0.911 extra 16 8 10 14 + 3 extra 16 + 5 extra 15
Score
1
[ Points] Short Questions
The following short questions should be answered with at most two sentences, and/or a picture. For the (true/false) questions, answer true or false. If you answer true, provide a short justification, if false explain why or provide a small counterexample. 1. [ points] Your billionaire friend needs your help. She needs to classify job applications into good/bad categories, and also to detect job applicants who lie in their applications using density estimation to detect outliers. To meet these needs, do you recommend using a discriminative or generative classifier? Why?
2. [ points] Your billionaire friend also wants to classify software applications to detect bug-prone applications using features of the source code. This pilot project only has a few applications to be used as training data, though. To create the most accurate classifier, do you recommend using a discriminative or generative classifier? Why?
3. [ points] Finally, your billionaire friend also wants to classify companies to decide which one to acquire. This project has lots of training data based on several decades of research. To create the most accurate classifier, do you recommend using a discriminative or generative classifier? Why?
4. [ points] Assume that we are using some classifier of fixed complexity. Draw a graph showing two curves: test error vs. the number of training examples and cross-validation
2
error vs. the number of training examples.
5. [ points] Assume that we are using an SVM classifier with a Gaussian kernel. Draw a graph showing two curves: training error vs. kernel bandwidth and test error vs. kernel bandwidth
6. [ points] Assume that we are modeling a number of random variables using a Bayesian Network with n edges. Draw a graph showing two curves: Bias of the estimate of the joint probability vs. n and variance of the estimate of the joint probability vs. n.
7. [ points] (a) Both PCA and linear regression can be thought of as algorithms for minimizing a sum of squared errors. Explain which error is being minimized in each algorithm.
8. [ points] A long time ago there was a village amidst hundreds of lakes. Two types of fish lived in the region, but only one type in each lake. These types of fish both looked exactly the same, smelled exactly the same when cooked, and had the exact same delicious taste - except one was poisonous and would kill any villager who ate it. The only other difference between the fish was their effect on the pH (acidity) of the lake they occupy. The pH for lakes occupied by the non-poisonous type of fish was 2 distributed according to a Gaussian with unknown mean (saf e ) and variance (saf e ) 3
and the pH for lakes occupied by the poisonous type was distributed according to a 2 different Gaussian with unknown mean (deadly ) and variance (deadly ). (Poisonous fish tended to cause slightly more acidic conditions). Naturally, the villagers turned to machine learning for help. However, there was much debate about the right way to apply EM to their problem. For each of the following procedures, indicate whether it is an accurate implementation of ExpectationMaximization and will provide a reasonable estimate for parameters and 2 for each class. (a) Guess initial values of and 2 for each class. (1) For each lake, find the most likely class of fish for the lake. (2) Update the and 2 values using their maximum likelihood estimates based on these predictions. Iterate (1) and (2) until convergence.
(b) For each lake, guess an initial probability that it is safe. (1) Using these probabilities, find the maximum likelihood estimates for the and values for each class. (2) Use these estimates of and to reestimate lake safety probabilities. Iterate (1) and (2) until convergence.
(c) Compute the mean and variance of the pH levels across all lakes. Use these values for the and 2 value of each class of fish. (1) Use the and 2 values of each class to compute the belief that each lake contains poisonous fish. (2) Find the maximum likelihood values for and 2 . Iterate (1) and (2) until convergence.
4
2
[ points] Reinforcement Learning
Consider the following Markov Decision Process:
r=1
r=1
r=1
r = 10
S1
r=1
S2
r=1
S3
r=1
S4
r=1
S5
We have states S1 , S2 , S3 , S4 , and S5 . We have actions Left and Right, and the chosen action happens with probability 1. In S1 the only option is to go back to S2 , and similarly in S5 we can only go back to S4 . The reward for taking any action is r = 1, except for taking action Right from state S4 , which has a reward r = 10. For all parts of this problem, assume that = 0.8. 1. What is the optimal policy for this MDP?
2. What is V (S5 )? It is acceptable to state it in terms of , but not in terms of state values.
3. Consider executing Q-learning on this MDP. Assume that the Q values for all (state,action) pairs are initialized to 0, that = 0.5, and that Q-learning uses a greedy exploration policy, meaning that it always chooses the action with maximum Q value. The algorithm breaks ties by choosing Left. What are the first 10 (state, action) pairs if our
5
robot learns using Q-learning and starts in state S3 (e.g. (S3 , Lef t), (S2 , Right), (S3, Right), . . . )?
4. Now consider executing Rmax on this MDP. Assume that we trust an observed P (x |x, a) transition probability after a single observation, that the value of Rmax = 100, and that we update our policy each time we observe a transition. Also, assume that Rmax breaks ties by choosing a policy of Left. What are the first 10 (state, action) pairs if our robot learns using Rmax and starts in state S3 (e.g. (S3 , Lef t), (S2 , Right), (S3 , Right), . . . )?
6
3
[ Points] Bayes Net Structure Learning
Finding the most likely Bayes Net structure given data is generally intractable. However, if certain restrictions are imposed on the structure, the most likely one can be found efficiently. One such restriction imposes a fixed ordering on the variables of the Bayes Net. This ordering restricts all edges to be directed forward in the ordering. For example, an edge X Y can only exist if X comes before Y in the ordering. 1. We'll now explore the effect that the ordering on has the number of parameters and independence assumptions of Bayes Nets. In each box you are given a Bayes Net that obeys a fixed ordering ABCD (1A and 1B). Draw a Bayes Net (part 2A) for the fixed ordering DCBA that can model the same distribution as the Bayes Net of part 1A. It should have no additional independence assumptions that are not present in part 1A, but also no unnecessary edges. Repeat for 1B and 2B. Count the number of parameters in each Bayes Net. Each variable is binary - it can take on 2 values. Identify an independence assumption of Bayes Net 1A that doesn't exist in Bayes Net 2A, if such an independence assumption exists. Repeat for Bayes Nets 1B and 2B. Hint: Pay close attention to V-structures - both existing ones and ones you create!!!
Number of parameters for Bayes Net 1A
Number of parameters for Bayes Net 1B
Number of parameters for Bayes Net 2A
Number of parameters for Bayes Net 2B
List an independence assumption of 1A not present in 2A (if there is one)
List an independence assumption of 1B not present in 2B (if there is one)
7
2. Given a fixed ordering over variables: X1 , X2 , X3 , ..., Xn , show that the choice of parents n is independent of the choice of other parents 1 , ...n-1 . In other words, show that: max1 ,...,n log P (X1 , ..., Xn |1 , ..., n ) = maxn f (X1 , ..., Xn , n ) + max1 ,...,n-1 g(X1, ..., Xn , 1 , ..., n-1 ) for some functions f and g.
8
3. For fixed orderings with a limit of k on the number of parents for each node, the best structure can be obtained by combinatoric search. For each variable, all subsets of variables from earlier in the ordering of size k or less are considered. For each set, log P (child|parents) is computed. We saw in part 1 of this question that the ordering can change the number of parameters required to model the joint probability. In this question we'll consider the efficiency of modifying the ordering. This approach can be used to greedily search for a good ordering of variables.
Consider the scenario where you are given a fixed ordering and the most likely Bayes Net structure for that fixed ordering. We would like to find the most likely structure after we switch two adjacent variables in the ordering. How many calculations of log P (child|parent) would this require in the worst case? Explain.
Local swapping of variables is prone to getting stuck in local maxima. Instead, let's consider changing the fixed ordering so that the two variables we swap have j variables in between. How many log P (child|parent) calculations are required to find the most likely structure for this new ordering in the worst case? Explain.
9
4
[ Points] Decision Trees
In class, we discussed greedy algorithms for learning decision trees from training data. These algorithms partition the feature space into labeled regions by greedily optimizing some metric (information gain) in hope of producing simple trees that partition the feature space into regions that perfectly classify the training data. As with most greedy approaches, if we consider finding a good tree with a limited depth, this approach is not guaranteed to produce the set of regions that best maximize this metric. We can always be less greedy. Instead of greedily making one decision and then greedily making the next decision, we can consider the outcome of all possible pairs of those two decisions and choose the best of those. We'll now explore the benefits and costs of being less greedy. In a standard decision tree, each level of the recursion will find one decision boundary (e.g., X=3) that partitions the feature space into two regions (e.g., X > 3, X 3) so to maximize the metric. Each region is then partitioned recursively using the same procedure. In a point-based look-ahead decision tree, the feature space is partitioned into four regions by a single point (e.g., X, Y = (3, 4) gives regions [X > 3, Y > 4], [X > 3, Y 4], [X 3, Y > 4], and [X 3, Y 4]. In a boundary-based look-ahead decision tree, three decision boundaries are considered in each level of the recursive decision-tree construction. The first decision boundary splits the feature space into two regions, and the two additional decision boundaries split those two regions for a total of 4 regions (e.g., X = 3, Y = 4 for X < 3, Y = 2 for X > 3) which yields regions [X > 3, Y > 4], [X > 3, Y 4], [X 3, Y > 2], and [X 3, Y 2]. 1. Draw a dataset on the following 2 plots so that a standard decision tree with two levels (4 regions) will poorly classify the data, but a...