Ex am ple Qu es tio ns EEEE4119 The University of Nottingham DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING A LEVEL 4 MODULE, SPRING 2020-21 ARTIFICIAL INTELLIGENCE AND INTELLIGENT SYSTEMS Time allowed: TWO hours Candidates may complete the front cover of their answer book and sign their desk card but must NOT write any thing else until the start of the examination period is announced Answer ALL questions. Only a calculator from approved list A (or one functionally equivalent) may be used in this examination. Dictionaries are not allowed with one exception. Those whose first language is not English may use a standard translation dictionary to translate between that language and English provided that neither language is the subject of this examination. Subject specific translation dictionaries are not permitted. No electronic devices capable of storing and retrieving text, including electronic dictionaries, may be used. DO NOT turn examination paper over until instructed to do so ADDITIONAL MATERIAL: [none] INFORMATION FOR INVIGILATORS: Question papers should be collected in at the end of the exam – do not allow candidates to take copies from the exam room. EEEE4119 Turn over Ex am ple Qu es tio ns 1 EEEE4119 1. (a) Consider a dichotomisation problem defined as C1 = {a1, a2 ∈ R : a1 ≤ 2 ∧ a2 ≤ 3} ∪ {a1, a2 ∈ R : a1 ≥ 2 ∧ a2 ≥ 3} C0 = {a1, a2 ∈ R : a1, a2 /∈ C1} (i) Draw and annotate the dichotomisation problem in the input space. Then, provide a description of the given dichotomisation problem. [3] (ii) Using the Lippmann’s multilayer perceptron rule, describe and draw with detailed annotation the dichotomies process tree. [7] (iii) Based on the dichotomies process in (b), draw a neural network architecture and find suitable values for the weights in order to perform the required classification [9] (b) Now, consider the following case. An engineer was given a classification prob- lem to be solved using neural network system. The engineer proposed a single operational layer neural network and then attempted to design such network using the Perceptron Learning Law. The engineer found that the Perceptron Learning Law failed to converge. (i) Describe what is the meaning of this finding? [3] (ii) Give a recommendation and its rationale how to solve the classification prob- lem! [3] EEEE4119 Turn over Ex am ple Qu es tio ns 2 EEEE4119 2. A Support Vector Machine (SVM) classifier is to be built for a two class problem. There are a total of m, d-dimensional, training samples x1 to xm with associated labels y1 to ym where yi ∈ {−1, 1}. Note that our notation assumes a final constant term, i.e. x = [x1 · · ·xd1]⊤. (a) The decision boundary for linear SVMs has the form w⊤x = 0 How would you expand this model into higher dimensions without using a kernel approach? Give an example of how you might add 1 extra dimension. [2] Why do we need to use kernel functions to go to higher dimensions with SVMs? [2] (b) A polynomial kernel can be written in the following forms κ(xk,xt) = ( x⊤k xt )c (i) How does the parameter c impact the form of the decision boundary? [2] (ii) Discuss how you select an appropriate value for c that provides maximum accuracy while preventing overfitting. [4] (iii) Consider the kernel κ(xk,xt) = (xk⊤xt − 1)3 with dimension d = 2. Show that this equivalent to training an SVM on the following 4-dimensional trans- formed space: Φ ( x1 x2 ) = [ x31 √ 3x21x2 √ 3x1x 2 2 x 3 2 ]⊤ [6] (iv) If the kernel is modified to κ(xk,xt) = (xk⊤xt)3, how many dimensions are in this new transformed space? [4] (v) For a quadratic kernel, κ(xk,xt) = (xk⊤xt)2, show that the effective dimen- sionality of the space (including the constant term) in terms of d, the number of dimensions of the data points, is given by: d3 + 3d 2 + 1 [5] EEEE4119 Turn over Ex am ple Qu es tio ns 3 EEEE4119 3. Consider the following Markov Decision Process (MDP): Figure 1: Markov decision process 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 γ = 0.8. (a) What is the optimal policy for this MDP? [5] (b) What is the optimal value function of state S5, i.e. V ∗(S5)? It is acceptable to state it in terms of γ, but not in terms of state values. [6] (c) Consider executing Q-learning on this MDP. Assume that the Q values for all (state, action) pairs are initialized to 0, that α = 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 robot learns using Q-learning and starts in state S3? For example, a (not necessarily correct) sequence might read: {S3, left}, {S2, right}, {S3, right}, . . .). [7] (d) How would you modify the Q−learning approach above to improve performance in a realistic setting? [2] (e) If this MDP represents a robot navigating in an unknown environment, would it be better to use Q-learning, TD-learning or Monte-Carlo learning? Dis- cuss. [5] EEEE4119 Turn over Ex am ple Qu es tio ns 4 EEEE4119 4. Consider the following Bayesian network developed for graduates from Nottingham: Figure 2: Bayesian network (a) What is the probability that an EEEE4119 student (W = true) had quality in- struction (I = true) and became successful in life (S = true), but did not have raw talent (T = false) yet was hard-working (H = true) and confident (C = true). Leave your answer unsimplified in terms of constants from the probability ta- bles. [6] (b) What is probability of success in life (S = true) given that a student has high quality instruction (I = true)? Express your final answer in terms of expressions of probabilities that could be read off the Bayes Net. You do not need to simplify down to constants defined in the Bayes Net tables. You may use summations s necessary. [6] (c) What is the probability a student is hardworking (H = true), given that she was an EEEE4119 student (W = true)? Express your final answer in terms of expressions of probabilities that could be read off the Bayes Net. You do not need to simplify down to constants defined in the Bayes Net tables. You may use summations as necessary. [6] (d) A biased coin has a probability p of landing on heads. We flip the coin 10 times and find that 4 times it lands on heads. EEEE4119 Turn over Ex am ple Qu es tio ns 5 EEEE4119 (i) Write down Bayes rule, relating the posterior distribution of p, P (p|D) where D is the observed set of flips, to P (D|p). The prior distribution is P (p). [1] (ii) Using the binomial distribution write an expression for P (D|p) (whereD is the data set i.e. 4 heads in a set of 10). Hint: If you have a binary variable with probability p of success and you run n trials with k successes, the binomial distribution is P = ( n k ) pk(1− p)n−k [2] (iii) Calculate the ratio P (p=0.4|D)P (p=0.5|D) assuming a uniform distribution for the P (p). Comment on whether or not you think the coin is truly biased. [4] EEEE4119 END
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