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📗 [3 points] Given the variance matrix of a data set \(V\) = , a principal component \(u\) = , what is the projected variance of the data set in the direction \(u\)?
📗 Answer: .
📗 [3 points] Suppose the UCB1 (Upper Confidence Bound) Algorithm is used to select arms in a multi-armed bandit problem, and in round \(t\) = , the arms pulls and empirical means \(\hat{\mu}\) for the arms are summarized in the following table, and in period \(t + 1\), an arm is pulled according to the UCB1 Algorithm and the reward is . Compute the updated empirical means of the arms after period \(t + 1\), i.e. updated \(\hat{\mu}_{1}, \hat{\mu}_{2}, ...\). Use \(c\) = .
Arms
arm pulls (\(n_{k}\))
empirical means \(\hat{\mu}_{k}\)
upper confidence bounds \(\hat{\mu}_{k} + c \sqrt{2 \dfrac{\log t}{n_{k}}}\)
\(k = 1\)
\(k = 2\)
\(k = 3\)
📗 Answer (comma separated vector): .
📗 [3 points] In an infinite horizon MDP (Markov Decision Process), there are \(n\) = states: initial state \(s_{0}\), and absorbing states \(s_{1}, s_{2}, ..., s_{n-1}\). In state \(s_{0}\), the agent can stay or move to any other state, but in all other absorbing states the agent can only choose to stay. The reward from staying in those states are summarized in the following table. Compute the Q value (under the optimal policy, not from Q learning) \(Q\left(s_{0}, \text{stay}\right)\). Use the discount factor \(\gamma\) = .
State
\(s_{0}\)
\(s_{1}\)
\(s_{2}\)
\(s_{3}\)
\(s_{4}\)
Reward from stay
Reward from move
-
-
-
-
📗 Answer: .
📗 [3 points] Consider a vector \(x\) = , if the principal component is = , what is the reconstruction of \(x\) using only the first principal components?
📗 Answer (comma separated vector): .
📗 [2 points] Given attention weight from \(q\) to \(k_{1}\), \(w_{1}\) = , from \(q\) to \(k_{2}\), \(w_{2}\) = , given values \(v_{1}\) = , \(v_{2}\) = , calculate the output vector.
📗 Answer (comma separated vector): .
📗 [3 points] Given the pairwise distance matrix \(d\), what is the linkage distance between the cluster {} and {}? The columns and rows are indexed \(1, 2, 3, ...\), i.e. row \(i\) column \(j\) is the distance between point \(i\) and point \(j\).
d =
📗 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 first state: ?
📗 Answer: .
📗 [4 points] In simulated annealing we move from \(s\) to an inferior neighbor \(t\) with probability \(\exp\left(\dfrac{- \left| f\left(s\right) - f\left(t\right) \right|}{T}\right)\), where \(T\) is the temperature parameter. Suppose \(f\left(s\right)\) = and \(f\left(t\right)\) = and \(T\) = . What is the probability we move to \(t\)?
📗 Note: we are minimizing the score.
📗 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: .
📗 [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: .
📗 [3 points] Let \(h_{1}\) be an admissible heuristic from a state to the optimal goal, A* search with which ones of the following \(h\) will be admissible? 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] Consider the following zero-sum game tree. player moves first. Draw a new game tree by re-ordering the children of each internal node (including the root), such that the new game is equivalent to the tree above, but alpha-beta pruning will prune as many nodes as possible. (You do not have to submit the drawing.) Enter the number of nodes pruned.
📗 Note: in case the diagram is not clear, the values on the leafs (each sub-branch is a row): .
📗 Answer: .
📗 [3 points] There are two players and \(k\) coins on the table. Players move sequentially with player 1 moving first. Each player chooses to take either one or two coins from the table. The player who takes the last coin wins. For which of the following values of \(k\) = {} does the first player has a winning strategy? Enter the values of \(k\), not the indices.
📗 Answer (comma separated vector): .
📗 [4 points] Given the following game payoff table, suppose the row player uses a pure strategy, and column player uses a mixed strategy playing L with probability \(q\). What is the smallest and largest value of \(q\) 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): .
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📗 Answer: .
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