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📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) = , what is the projected variance of the dataset in the direction of the first principal component?
📗 Answer: .
📗 [3 points] Suppose the states are integers between \(1\) and \(x\). The initial state is \(1\), and the goal state is . The successors of a state \(i\) are \(2 i\) and \(2 i + 1\), if exist. What is the smallest value of \(x\) so that the worst case space complexity (number of states stored in the list (queue or stack)) of DFS (Depth First Search) is larger than or equal to BFS (Breadth First Search)?
📗 Note: the worst case space complexity for BFS is \(b^{d}\) and for DFS is \(\left(b - 1\right) D + 1\). "Worst case" means you can re-order the successors and search in the order that maximizes the space requirement.
📗 Answer: .
📗 [4 points] Given the list of states in the priority queue (frontier) and the current cost \(g\) and heuristic cost \(h\), what is the largest value of \(x\) so that state \(0\) will be removed (expanded) from the priority queue next in all three informed search strategies: UCS (Uniform Cost Search), (Best First) Greedy Search, and A Search? Break ties by expanding the state with the smallest index.
State
0
1
2
3
4
5
g
h
\(x\)
📗 Answer: .
📗 [4 points] Suppose the score (fitness) of a state \(\left(d_{1}, d_{2}, d_{3}, d_{4}\right)\) is \(d_{1} + d_{2} + d_{3} + d_{4}\), and only 1-point crossover with the cross-over point between \(d_{2}\) and \(d_{3}\) is used in a genetic algorithm (i.e. mutation probabilities are 0). Two states are chosen as parents at random according to the reproduction probabilities, what is the probability that one of their children is the optimal state (i.e. \(\left(1, 1, 1, 1\right)\)? Enter a number between 0 and 1.
📗 Note: the two parents are sampled with replacement, meaning the probability that two states are chosen as parents is the product of their reproduction probabilities.
Index
1
2
3
4
State
📗 Answer: .
📗 [4 points] For a zero-sum game in which moves first and if the action Left is chosen, then Chance (Chn) moves Left with probability \(p\) and Right with probability \(1 - p\), and if the action Right is chosen, then Chance moves Left with probability and Right with probability . Suppose the player who moves first uses a mixed strategy \(\dfrac{1}{2}\) Left and \(\dfrac{1}{2}\) Right in a solution, what is the value of \(p\)? If it's impossible, enter \(-1\).
📗 Note: in case the diagram is not clear, the values on the leafs (each sub-branch is a row): .
📗 Answer: .
📗 [3 points] There are \(n\) = cookies. The brother first proposes a division of these cookies into two piles (two integers adding up to \(n\)) and then the sister take one of the two piles. Both the brother and the sister want to maximize the number of cookies they take. What is the value of the game to the brother (measured by the number of cookies he gets)? Enter an integer.
📗 Answer: .
📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) is a diagonal matrix, what is the smallest value of \(K\) so that the Manhattan distance between the vector \(\begin{bmatrix} 1 \\ 1 \\ ... \\ 1 \end{bmatrix}\) with ones (\(1\)'s) and its reconstruction using the first \(K\) principal components is less than or equal to ?
📗 Answer: .
📗 [4 points] In a by grid, Tom is located at (, ) and Jerry is located at (, ). Tom uses to find Jerry and the successors of a state (one cell in the grid) are the four neighboring states on the grid (the cells above, below, to the left and to the right). What is the imum number of states that need to be expanded to find (and expand) the goal state? The order in which the successors are added can be arbitrary. Do not count repeated expansion of the same state. Include both the initial and the goal states.
📗 Answer: .
📗 [3 points] There are \(n\) = students in CS540, for simplicity, assume student \(0\) gets grade \(g = 0\), student \(1\) gets grade \(g = 1\), ..., student \(n - 1\) gets grade \(g = n - 1\). The payoff for each student who drop the course is \(0\), the payoff for the students who stay is if the student has the lowest grade among all students who decide to stay in the class, and the otherwise. If each student only uses actions that are rationalizable (i.e. survive the iterated elimination of strictly dominated actions), how many students will stay in the course? If there are multiple correct answers, enter one of them.
📗 Answer: .
📗 [4 points] What is the projected variance of and onto the principal component ? Use the MLE (Maximum Likelihood Estimate) formula for the variance: \(\sigma^{2} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}\) with \(\mu = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} x_{i}\).
📗 Answer: .
📗 [3 points] Given data and initial k-means cluster centers \(c_{1}\) = and \(c_{2}\) = , what is the initial total distortion (do not take the square root). Use Euclidean distance. Break ties by assigning points to the first cluster.
📗 Answer: .
📗 [3 points] You have a dataset with unique data points which you want to use k-means clustering on. You setup the experiment as follows: you apply k-means with different k's: \(k\) = . Which \(k\) value will minimize the total distortion? Enter -1 if the answer depends on the data points.
📗 Answer: .
📗 [4 points] Suppose the state space has \(n\) = states that form a tree with root state \(0\). What is the shape of the tree that makes iterative deepening realize that a goal does not exist as quickly as possible (i.e. one that minimizes the number of expanded nodes)? Enter the number of nodes searched in this case.
📗 Answer: .
📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmax}}_{t \in \left\{0, ..., T\right\}} x_{t} = 1\) with \(x_{0} = 1\) (i.e. the index of the last feature that is 1). What is the reproduction probability of the state with the highest reproduction probability?
📗 Answer: .
📗 [4 points] Run search algorithm on the following graph, starting from state 0 with the goal state being . Write down the expansion path (in the order of the states expanded). The heuristic function \(h\) is shown as subscripts. Break tie by expanding the state with a smaller index.
📗 In case the diagram is not clear: the weights are (with heuristic values on the diagonal entries): .
📗 Answer (comma separated vector): .
📗 [2 points] Consider a game board consisting of bits initially at . Each player can simultaneously flip any number of bits in a move, but needs to pay the other player one dollar for each bit flipped. The player who achieves wins and collects dollars from the other player. What is the game theoretic value (in dollars) of this game for the first player?
📗 Note: "game theoretic value" is what we called "value of the game" in the lectures.
📗 Answer: .
📗 [4 points] Given the following BoS (Battle of Sexes) game, what is the row (Romeo) player's (expected) value (i.e. payoff) in the mixed strategy Nash equilibrium?
Romeo \ Juliet
Bach
Stravinsky
Bach
Stravinsky
📗 Answer: .
📗 [4 points] In GoogSoft, software engineers A and B form a two-person team. Their year-end bonus depends on their relative performance. The bonus outcomes are summarized in the following table. The value of slacking to each person is \(s\) = . The total payoff to each person is the sum of the bonus and the value from slacking. What is the smallest value of \(x\) such that both players will work hard in a Nash equilibrium?
-
B works hard
B slacks
A works hard
\(x, x\)
A slacks
📗 Answer: .
📗 [4 points] You are given the distance table. Consider the next iteration of hierarchical clustering using linkage. What will the new values be in the resulting distance table corresponding to the new clusters? If you merge two columns (rows), put the new distances in the column (row) with the smaller index. For example, if you merge columns 2 and 4, the new column 2 should contain the new distances and column 4 should be removed, i.e. the columns and rows should be in the order (1), (2 and 4), (3).
\(d\) =
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [4 points] Suppose K-Means with \(K = 2\) is used to cluster the data set and initial cluster centers are \(c_{1}\) = and \(c_{2}\) = \(x\). What is the largest value of \(x\) if cluster 1 has \(n\) = points initially (before updating the cluster centers). Break ties by assigning the point to cluster 2.
📗 Answer: .
📗 [2 points] Consider \(n + 1\) = + \(1\) states. The initial state is \(1\), the goal state is \(n\). State \(0\) is a dead-end state with no successors. For each non-\(0\) state \(i\), it has two successors: \(i + 1\) and \(0\). We may expand the same states many times, because we do not keep track of which states are checked previously. How many states (including repeated ones) will be expanded by ? Break ties by expanding the state with the index first.
📗 Note: the tie-breaking rule may be different from the convention used during the lectures, please read the question carefully.
📗 Answer: .
📗 [4 points] Which order of goal check is possible with , without specifying the order of successors when putting them in the queue (i.e. you can rearrange the order of the branches)? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".
📗 Choices:
(1)
(2)
(3)
(4)
(5)
(6)
(7) None of the above
📗 Answer (comma separated vector):
📗 [4 points] Let the states be 3D integer points with integer coordinates \(\left(i, j, k\right)\) with boundary constrains and and . Each state \(\left(i, j, k\right)\) has six successors \(\left(i - 1, j, k\right), \left(i + 1, j, k\right), \left(i, j - 1, k\right), \left(i, j + 1, k\right), \left(i, j, k - 1\right), \left(i, j, k + 1\right)\) or a subset thereof subject to the boundary constraints. The score of state \(\left(i, j, k\right)\) is . Which local minimum will be reached if hill climbing is used starting from ? Enter the state, not the score.
📗 Answer (comma separated vector): .
📗 [3 points] If \(h_{1}\) and \(h_{2}\) are both admissible heuristic functions, which ones of following are also admissible heuristic functions? Enter the correct choices as a list, comma separated, without parentheses, for example, "1, 2, 4".
📗 Choices:
(1)
(2)
(3)
(4)
(5)
(6)
(7) None of the above
📗 Answer (comma separated vector): .
📗 [4 points] Perform iterated elimination of strictly dominated strategies (i.e. find rationalizable actions). Player A's strategies are the rows. The two numbers are (A, B)'s payoffs, respectively. Recall each player wants to maximize their own payoff. Enter the payoff pair that survives the process. If there are more than one rationalizable action, enter the pair that leads to the largest payoff for player A.
A \ B
I
II
III
IV
I
II
III
IV
📗 Answer (comma separated vector): .
📗 [4 points] Enter the largest integer value of \(A\) such that \(B\) will be alpha-beta pruned? Min player moves first. In the case alpha = beta, prune the node. Enter 100 if you think the answer is infinity.
📗 Answer: .
📗 [4 points] Given the following game payoff table, suppose the row player uses a mixed strategy playing U with probability \(p\), and column player uses a pure strategy. What is the smallest and largest value of \(p\) in a mixed strategy Nash equilibrium?
Row \ Col
L
R
U
D
Note: the following is a diagram of the best responses (make sure you understand what they are and how to draw them). The red curve is the best response for the column player and the blue curve is the best response for the row player.
📗 Answer (comma separated vector): .
📗 [3 points] Consider Depth First Search on a tree, where the nodes are denoted by numbers. Write down the sequence depth first search visited in the order they are expanded (i.e. expansion path). \(0\) is the initial state and is the goal state.
📗 Note: use the convention used in the lectures, push the rightmost successor into the stack first (i.e. expand the leftmost successor first).
📗 Answer (comma separated vector): .
📗 [4 points] You will receive 4 points for this question and you can choose to donate x points (a number between 0 and 4). Your final grade for this question is the points you keep plus twice the average donation (sum of the donations from everyone in your section divided by the number of people in your section, combining both versions). Enter the points you want to donate (an integer between 0 and 4).
📗 Answer: (The grade for this question will be updated later).
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📗 Answer: .
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