final2002-solution - 15-781 Final Exam, Fall 2002 . Write...

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Unformatted text preview: 15-781 Final Exam, Fall 2002 . Write your name and your and rew email address below. Name: A AA FQNU MOOFQ. Andrew ID: . There should be 17 pages in this exam (excluding this cover sheet). . 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. . You should attempt to answer all of the questions. . You may use any and all notes7 as well as the class textbook. ., All questions are worth an equal amount. They are not all equally difficult. . You have 3 hours. . Good luck! 1 Computational Learning Theory 1.1 PAC learning for Decison Lists A decision list is a list of if—then rules where each condition is a literal (a variable or its negation). It can be thought of as a decision tree with just one path, For example, say that I like to go for a walk if it’s warm or if it’s snowing and I have a jacket, as long as it’s not raining. We could describe this as the following decision list: (a) (d) if rainy then no else if warm then yes else if not(have-jacket) then no else if snowy then yes else no. Describe an algorithm to learn DLs given a data set, for example Your algorithm should have the characteristic that it should always classify exam— ples that it has already seen correctly (ie, it should be consistent with the data).If it’s not possible to continue to produce a decision list that’s consistent with the data, your algorithm should terminate and announce that it has failed. Find the size of the hypothesis space, IHI, for decisions lists of k attributes Find an expression for the number of examples needed to learn a decision list of k attributes with error at most .10 with probability 90%,. What if the learner is trying to learn a decision list, but the representation that it is using is a conjunction of k literals? Find the expression for the number of examples needed to learn the decision list with error at most 1.10 with 90% probability. (b) L322 00 00 M 000 5‘” o 00X 00 ’9 / ’4' o OO K-means and Gaussian Mixture Models What is the effect on the means found by k—means (as opposed to the true means) of overlapping clusters? M3 are. PWSLQOX {MW-tr quark“ HM“ HA2 W Mew/LS wasme lye. Run k—means manually for the following dataset‘. Circles are data points and squares are the initial cluster centers. Draw the cluster centers and the decision boundaries that define each cluster, Use as many pictures as you need until convergence. Note: Execute the algorithm such that if a mean has no points assigned to it, it stays Where it is for that iteration, (0) NOW draw (approximately) What a Gaussian mixture model of three gaussians with the same initial centers as for the k—means problem would converge to. Assume that the model puts no restrictions on the form of the covariance matrices and that EM updates both the means and covariance matrices” (d) Is the classification given by the mixture model the same as the, classification given by k—means? Why or why not? ‘ 3 HMMS Andrew lives a simple life. Some days he’s Angry and some days he’s Happy. But he hides his emotional state, and so all you can observe is whether he smiles, frowns, laughs, or yells, We start on day 1 in the Happy state, and there’s one transition per day. Aneg Happy p(smile)=0 ,5 p(frown)=0 1 p(laugh)=0,2 p(yell)=0,2 p(smile)=0 ,1 p(frown)=0 5 p(laugh)=0 2 p(yell)=0 2 Definitions: ‘ qt : state on day t,’ 0,, = observation on day t., (a) What is P(q2 = Happy)? 0, g (b) WhatisP(0:frown)? ,8. .L L .L 2 _8_, Jo _ i 2 onxo‘tgo"; Iw+To‘-5‘ro.85 (c) What is P(Q2 = Happleg = frown)? 01-: 92"“ l apt: H) P 1L: H) 4X3. : )0 lo _ 4/ (d) What is P(O100 = yell)? ( ll; —— ? F(Ouw=U£\l) 2‘- P(O.®:UaulézmzH>P(’1/,w= H) 1- P{Om:\aeM\ 7,“:A>P(1J£A> : 2/10" ‘1’,“3V‘) +P{$,W:A3) : afoxl : 27/10 (e) Assume that 01 = frown, 02 = frown, 3 = frown, 04 = frown, and O5 2 frown, What is the most likely sequence of states? HAAAA 4 Bayesian Inference (a) Consider a dataset over 3 boolean attributes, X, Y, and Z. Of these sets of information, which are sufiicient to specify the joint distribution? Circle all that apply. , yes buaukt HM A. P(~ X|Z) P(~ X|~Z) P(~ Y|X/\Z) P(~ YlXANZ) P(~ Y! ~ X /\ Z) P(~ Y| N X/\ ~ Z) P(Z) ®a®—>® 85“ v Naifafiiize B, P(~ X] N Z) P(X] N Z) P(Y[X /\ Z) PmXA ~ Z) No ‘ R can P(Yl~X/\Z)P(Y|~X/\~Z)P(Z) MW £3042) on, P(X|Z) P(X| N Z) P(Y|X /\ Z) P(Y|X/\ N Z) 7:25 Ame A (NA cm P(Y| NXAZ) P(~ Y| ~X/\~Z) P(~ Z) 8"“ (39%) 9"“ “2) D), P(Xlz) P(X\ ~Z) P(YIXAZ) P(Y|XA~Z)j No, 0ch 3w P(~ Yl N X/\ N Z) P(Y| N X/\ N Z) P(Z) Given this dataset of 16 records: (b) Write down the probabilities needed to make a joint density bayes classifier P AAB\C> PUMB‘NC) at PEAANBl c.) (Dams) NC) WC) :1 47;: WNW). -.- f.(“A33LT'C)-‘—"§; Mia/r Mpg C) PQA'ATBLNElfé—‘AJM‘MJ mm 'or u— (C) Write down—the probaloilities needed to make a naive bayes classifier, P(Al c) = %%%; 9(a) :72 P(Al~c): “*9 ?(Blc)= 542 Hangs/g (d) Write the classification that the joint density bayes classifier would make for C given A=0,B=1‘. F<C [Ii/AME} 2: P(~AABIC>P(C) (’(«AAM C)? (c)+P (NA/*8 PC) PFC) 2: iii .2 J2- : j é$i+igxli 42.1% (5/8) (e) Write the classification that the naive bayes classifier would make for C given A=0,B=l. WWW) = Whisk)?“ \°(»A»£>lc)P(c)+P(~A~£I~C)P(~Q) ngAlc; gauging ] +- 10 4 — 1 5/» mu, ha» "LS 2315. renaming /8 2+832’1-35‘r3 3e 5 Support Vector Machines This picture shows a dataset with two real—valued inputs (x1 and :02) and one categorical output class. The positive points are shown as solid dots and the negative points are small circles. (a) e (b) Suppose you are using a linear SVM with no provision for noise (ie, a Linear SVM that is trying to maximize its margin while ensuring all datapoints are on their correct Lsides of the margin) ., Draw three lines on the above diagram, showing’the classification boundary and the two sides of the margin' Circle the supportivector(sy)_ Using the familiar LSVM classifier notation of class 2 sign(w.x 47gb), calculate the values of W and b learned for part (a) , ’ ‘ x ; Assume you are using a noise-tolerant LSVM which tries to minimize. . : ' 5 ' g ; .' .- ‘ / _; i 1 .R r f—ww + 0 2 6k 1‘ 2 k=1 1 (1) using the notation of yourpnotes and the Burges pap‘er.’ Question: is it possible to invent a datas‘i‘etgand a positive" value of C in which (a) the dataset is linearly separable but (b) the LSVM would nevertheless misclassify at least one training point? If it is possible to invent such an example, please sketch the ' ’6 example and suggest a value for C. If it is not possible, explain why not, -- l 6 Instance-based learning This picture shows a dataset with one real—valued input a: and one real-valued output yr. There are seven training points. 0 1 2 3 4 5 6 X—> Suppose you are training using kernel regression using some unspecified kernel function. The only thing you know about the kernel function is that it is a monotonically decreasing function of distance that decays to zero at a distance of 3 units (and is strictly greater than zero at a distance of less than 3 units) i. ‘ l (a) isthe predicted value of y when :r/2‘l? I I” I \ ‘ GEJB ’ I I (“SN/nah ‘0 W‘M : Matte/52 (b) What is the predicted value of y when x = 3? 5;"‘2- 5M £5W¢Qv V4“ 3"“ Amb‘fmug/ (\ 993+ 0351-6 0M (lav—35‘ wdfiweek vallj IA «#2ij wqmfi‘JL—r—‘J (c) What is the predicted value of y when x = 5? : M :. C C3 NSTQS q. (d) What is the predicted value of y when :3 = 6? 4»: (Laws “‘56) The final two parts of this question concern 1-nearest neighbor used as a classifier. The following dataset has two real valued inputs and one binary categorical output. The class is denoted by the color of the datapointi (e) Does there exist a choice of Euclidian distance metric for which 1--nearest—neighbor would achieve zero training set error on the above dataset? V yrs, 63. memo, L) = «(m—mole ugh—1°.sz * V. ,. ,(Lz’m‘ nor-g; 1y i: Ma 9%, ii“: at Hidiflv-u‘vivw% ANY Mm‘flflfl 0 W3 Now let’s consider a different dataset: «6» <0» o o go ‘o Elliot‘o o o ‘0'.” (f) Does there exist a choice of Euclidian distance metric for which l-nearest—neighbor would achieve zero training set error on the above dataset? 765, m3 NRC. 7 Nearest Neighbor and Cross-Validation Recipe for making training set of 10,000 dat— Recipe for making test set of 10,000 datapoints apoints with two real—valued inputs and one with two real-valued inputs and one binary binary output class: output class: No points in No points in gap between rectangles gap between rectangles 5000 points \1/ 5000 points 5000 points \L 5000 points with positions with positions with positions with positions chosen chosen chosen chosen randomly randomly randomly randomly uniformly uniformly uniformly uniformly in. this in this in this in this rectangle . rectangle » rectangle i. rectangle . 2595 have +ve 7596 have +ve none have +ve 1009s have +ve class. class" class» class. 7595 have -ve 2596 have —ve 100% have —ve none have -ve class" class» c1ass.. class. Using the above recipes for making training and test sets you will see that the training set is noisy: in either region, 25% of the data comes from the minority class“ The test set is noise—free. In each of the following questions, circle the answer that most closely defines the expected error rate, expressed as a fraction. (a) What is the expected training set error using one—nearest—neighbor? (8) 1/8 1/4 3/8 1/3 1/2 5/8 2/3 3/4 7/8 1 (b) What is the expected leave-one-out cross-validation error on the training set using _ - _ ‘ I; J... 3/ 4. 3/ aPL. - ,3 one nearest neighbor. if )1 1+. 9— L..— /g 0 1/8 1/4 3/8 1/3 1/2 5/8 2/3 3/4 7/8 1 (c) What is the expected test set error if we train on the training set, test on the test set, and use one—nearest—neighbor? O 1/8 1/4 3/8 1/3 1/2 5/8 2/3 3/4 7/8 1 (d) What is the expected training set error using 21—nearest—neighbor? 0 1/8 3/8 1/3 1/2 5/8 2/3 3/4 7/8 1 (e) What is the expected leave—one—out cross—validation error on the training set using 21—nearest—neighbor? O 1/8 1/4 3/8 1/3 1/2 5/8 2/3 3/4 7/8 1 (f) What is the expected test set error if we train on the training set, test on the test set, and use 21—nearest-neighbor? O 1/8 1/4 3/8 1/3 1/2 5/8 2/3 3/4 7/8 1 8 Learning Bayes Net Structure For each of the following training sets, draw the structure and CPTs that a Bayes Net Structure learner should learn, assuming that it tries to account for all the dependencies in the data as well as possible While minimizing the number of unnecessary links. In each case7 your Bayes Net will have three nodes, called A B and Cl, Some or all of these questions have multiple correct answersmyou need only supply one answer to each question. 9 Markov Decision Processes Consider the following MDP, assuming a discount factor of ’y = 0.5. Note that the action “Party” carries an immediate reward of +10. The action “Study” unfortunately carries no immediate reward, except during the senior year, when a reward of +100 is provided upon transition to the terminal state “Employed”. (a) What is the probability that a freshman will fail to graduate to the “Employed” state within four years, even if they study at every opportunity? (b) Draw the diagram for the Markov Process (not the MDP, the MP) that corresponds to the policy “study whenever possible.” (c) What is the value associated with the state “Junior” under the “study whenever pos— sible” policy? ' (d) Exactly how rewarding would parties have to be during junior year in order to make it advisable for a junior to party rather than study (assuming, of course, that they wish to optimize their cumulative discounted reward)? (e) Answer the following true or false. If true, give a one~sentence argument . If false, give a counterexample. 0 (True or False?) If partying during junior year an optimal action when it is assigned reward r, then it will also be an Optimal action for a freshman when assigned reward r., 0 (True or False?) If partying during junior year is an optimal action when it is assigned reward r, then it will also be an optimal action for a freshman when assigned reward r, 10 Q Learning Consider the robot grid world shown below, in which actions have deterministic outcomes, and for which the discount factor 7 z: 0.5. The robot receives zero immediate reward upon executing its actions, except for the few actions where an immediate reward has been written in on the diagram. Note the state in the upper corner allows an action in which the robot remains in that same state for one time tick. IMPORTANT: Notice the immediate reward for the state—action pair < 0, South > is —100, not +100. (a) Write in the Q value for each state-action pair, by writing it next to the corresponding arrow. (b) Write in the V* (5) value for each state, by writing its value inside the grid cell repre- senting that state. (0) Write down an equation that relates the V*(s) for an arbitrary state 8 to the Q(s, a) values associated with the same state. ((1) Describe one optimal policy, by circling only the actions recommended by this policy (8) Hand execute the deterministic Q learning algorithm, assuming the robot follows the trajectory shown below. Show the sequence of Q estimates (describe which entry in the Q table is being updated at each step): state action next-state immediate—reward updated-Q—estimates A East B 0 B East C 10 C Loop C O C South F -100 F West E O E North B O B East C 10 (f) Propose a change to the immediate reward function that results in a change to the Q function, but not to the V function. 11 Short Questions (a) Describe the difference between a maximum likelihood hypothesis and a maximum a posteriori hypothesis, MLE : MCKDLCMiLQ, Woiéx‘m‘j} gamers) b3 96W MAP 3 MMCkMiZQ. WU“ (5W3, M a.ch Er PM WU" FWS (b) Consider a learning problem defined over a set of instances X i. Assume the space of possible hypotheses, H , consists of all possible disjunctions over instances in X i. Le», the hypothesis 221 V $6 labels these two instances positive, and no others, What is the VC dimension of H 7 (0) Consider a naive Bayes classifier with 2 boolean input variables, X and Y, and one boolean output7 Z i, 0 Draw the equivalent Bayesian network. f9 on) o How many parameters must be estimated to train such a naive Bayes classifier? 5 Fri?) PC7! N?) PEKI~%) WHEY?) o How many parameters would have to be estimated if the naive Bayes assumption is not made, and we wish to learn the Bayes net for the joint distribution over X , Y, and Z? 7 a P<a> amok P<X=i,7=il2=k) True or False? If true, explain Why in at most two sentences" If false, explain Why or give a brief counterexample. 0 (True or False?) The error of a hypothesis measured over the training set provides a pessimistically biased eStimiiate of the true error of the hypothesis. 0 (True or False?) Boosting and the Weighted Majority algorithm are both methods for combining the votes of multiple classifiers, 0 (True or False?) Unlabeled data can be used to detect over'fitting, 0 (True or False?) Gradient descent has the problem of sometimes falling into local minima7 Whereas EM does not] 0 (True or False?) HMM’s are a special case of MDP’s. ...
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This note was uploaded on 07/10/2009 for the course INFORMATIC Inf taught by Professor Lanzi during the Spring '09 term at Politecnico di Milano.

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final2002-solution - 15-781 Final Exam, Fall 2002 . Write...

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