# Other Materials
📗 Videos Going through Past Exam Questions
X3Q9-10 (PCAs):
Link
X3Q11-12 (Hierarchical):
Link
X3Q13-15 (K-Means):
Link
X4Q1-3 (Search):
Link
X4Q4-5 (Informed):
Link
X4Q6-7 (Local Search):
Link
X4Q8-9 (Genetic):
Link
X4Q10-13 (Alpha Beta):
Link
X4Q14-15 (Mixed NE):
Link
X5Q1-4 (Pure NE):
Link
📗 Pre-recorded Videos from 2020
Lecture 23 (Repeated Games): Interactive Lecture (see Canvas Zoom recording)
Lecture 24 (Mechanism Design): Interactive Lecture (see Canvas Zoom recording)
📗 Relevant websites
2022 Online Exams:
F1A-C Permutations:
Link
F2A-C Permutations:
Link
FB-C Permutations:
Link
FA-E Permutations:
Link
FB-E Permutations:
Link
2021 Online Exams:
F1A-C Permutations:
Link
F1B-C Permutations:
Link
F2A-C Permutations:
Link
F2B-C Permutations:
Link
2020 Online Exams:
F1A-C Permutations:
Link
F1B-C Permutations:
Link
F2A-C Permutations:
Link
F2B-C Permutations:
Link
F1A-E Permutations:
Link
F1B-E Permutations:
Link
F2A-E Permutations:
Link
F2B-E Permutations:
Link
2019 In-person Exams:
Final Version A:
File
Version A Answers: CECBC DBBBA BEEDD BCACB CBEED DDCDC ACBCC ECABC
Final Version B:
File
Version B Answers: EEAEE AEACE BBDED BDAAA DCEEA CDACA AEAAA CCABB
Sample final:
Link
Video going through sample final very quickly:
Link
Past exam other professors made:
Professor Zhu:
Link
Professor Dyer:
Link
Relevant questions:
Midterms: F18Q1,2,3,4,5,6,7,8,9,10,11,12,13,14; F17Q1,2,3,4,5,6,7,8,9,10,11,12,13; F16Q1,2,3,4,5,6,7,8,9,10; F14Q1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20; F11Q1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20; F10Q1,2,3,4; F09Q1,3,4,5,6; F08Q1,2,3,5; F06Q1,2,3,4,5,6,7,8,9,10,11,12; F05Q1,2,3,4,5,6,7,8,9,10,11,14,19,20; F19Q1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32; S18Q1,2,3,4,5,6,7,8,9; S17Q1,2,3,4,5,6,7,8
Final Exams: F17Q1,2,3,4,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25; F16Q1,2,3,4,5,6,7,8,9,10,11,13,14,15,17,18; F14Q1,2,3,4,5,9,10,13,14,15,16,17,19,20; F13Q1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20; F12Q1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20; F10Q1,2,3,4,5,6,10,11,12,13,14,15,16,17,18,19,20; F09Q1,2,3,4,5,6,7,8,10,11,12,13,17,19,20; F08Q1,2,3,4,5,6,7; F06Q1,2,3,4,5,6,10,11,13,14,15,16,17,18,19,20; F05Q1,2,3,4,5,6,10,11,13,14,15,16,17,18,19,20; F19Q6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32; S18Q3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33; S17Q2,3,4,5,6,7,8,9,10
📗 YouTube videos from 2019 and 2020
L17Q1:
Link
L17Q2:
Link
M8Q6:
Link
M8Q8:
Link
M9Q2:
Link
M9Q7:
Link
M10Q1:
Link
M10Q4:
Link
M10Q5Q6:
Link
M12Q1:
Link
M12Q5:
Link
M12Q7:
Link
From Lectures
L16Q1 (UCS):
Link
L17Q1 (Hill-climbing SAT):
Link
L17Q2 (Genetic Algorithm):
Link
Other Final Exam Questions
Q4 (Shape of Quickest IDS):
Link
Q5 (Pure against Mixed):
Link
Q6 (Switch Lights):
Link
Q7 (K Means Cluster Assignment):
Link
Q8 (Vaccination Game):
Link
# Keywords and Notations
📗 Clustering
📗 Single Linkage: \(d\left(C_{k}, C_{k'}\right) = \displaystyle\min\left\{d\left(x_{i}, x_{i'}\right) : x_{i} \in C_{k}, x_{i'} \in C_{k'}\right\}\), where \(C_{k}, C_{k'}\) are two clusters (set of points), \(d\) is the distance function.
📗 Complete Linkage: \(d\left(C_{k}, C_{k'}\right) = \displaystyle\max\left\{d\left(x_{i}, x_{i'}\right) : x_{i} \in C_{k}, x_{i'} \in C_{k'}\right\}\).
📗 Average Linkage: \(d\left(C_{k}, C_{k'}\right) = \dfrac{1}{\left| C_{k} \right| \left| C_{k'} \right|} \displaystyle\sum_{x_{i} \in C_{k}, x_{i'} \in C_{k'}} d\left(x_{i}, x_{i'}\right)\), where \(\left| C_{k} \right|, \left| C_{k'} \right|\) are the number of the points in the clusters.
📗 Distortion (Euclidean distance): \(D_{K} = \displaystyle\sum_{i=1}^{n} d\left(x_{i}, c_{k^\star\left(x_{i}\right)}\left(x_{i}\right)\right)^{2}\), \(k^\star\left(x\right) = \mathop{\mathrm{argmin}}_{k = 1, 2, ..., K} d\left(x, c_{k}\right)\), where \(k^\star\left(x\right)\) is the cluster \(x\) belongs to.
📗 K-Means Gradient Descent Step: \(c_{k} = \dfrac{1}{\left| C_{k} \right|} \displaystyle\sum_{x \in C_{k}} x\).
📗 Projection: \(\text{proj} _{u_{k}} x_{i} = \left(\dfrac{u_{k^\top} x_{i}}{u_{k^\top} u_{k}}\right) u_{k}\) with length \(\left\|\text{proj} _{u_{k}} x_{i}\right\|_{2} = \left(\dfrac{u_{k^\top} x_{i}}{u_{k^\top} u_{k}}\right)\), where \(u_{k}\) is a principal direction.
📗 Projected Variance (Scalar form, MLE): \(V = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left(u_{k^\top} x_{i} - \mu_{k}\right)^{2}\) such that \(u_{k^\top} u_{k} = 1\), where \(\mu_{k} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} u_{k^\top} x_{i}\).
📗 Projected Variance (Matrix form, MLE): \(V = u_{k^\top} \hat{\Sigma} u_{k}\) such that \(u_{k^\top} u_{k} = 1\), where \(\hat{\Sigma}\) is the convariance matrix of the data: \(\hat{\Sigma} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left(x_{i} - \hat{\mu}\right)\left(x_{i} - \hat{\mu}\right)^\top\), \(\hat{\mu} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} x_{i}\).
📗 New Feature: \(\left(u_{1^\top} x_{i}, u_{2^\top} x_{i}, ..., u_{K^\top} x_{i}\right)^\top\).
📗 Reconstruction: \(x_{i} = \displaystyle\sum_{i=1}^{m} \left(u_{k^\top} x_{i}\right) u_{k} \approx \displaystyle\sum_{i=1}^{K} \left(u_{k^\top} x_{i}\right) u_{k}\) with \(u_{k^\top} u_{k} = 1\).
📗 Uninformed Search
📗 Breadth First Search (Time Complexity): \(T = 1 + b + b^{2} + ... + b^{d}\), where \(b\) is the branching factor (number of children per node) and \(d\) is the depth of the goal state.
📗 Breadth First Search (Space Complexity): \(S = b^{d}\).
📗 Depth First Search (Time Complexity): \(T = b^{D-d+1} + ... + b^{D-1} + b^{D}\), where \(D\) is the depth of the leafs.
📗 Depth First Search (Space Complexity): \(S = \left(b - 1\right) D + 1\).
📗 Iterative Deepening Search (Time Complexity): \(T = d + d b + \left(d - 1\right) b^{2} + ... + 3 b^{d-2} + 2 b^{d-1} + b^{d}\).
📗 Iterative Deepening Search (Space Complexity): \(S = \left(b - 1\right) d + 1\).
📗 Informed Search
📗 Admissible Heuristic: \(h : 0 \leq h\left(s\right) \leq h^\star\left(s\right)\), where \(h^\star\left(s\right)\) is the actual cost from state \(s\) to the goal state, and \(g\left(s\right)\) is the actual cost of the initial state to \(s\).
📗 Local Search
📗 Hill Climbing (Valley Finding), probability of moving from \(s\) to a state \(s'\) \(p = 0\) if \(f\left(s'\right) \geq f\left(s\right)\) and \(p = 1\) if \(f\left(s'\right) < f\left(s\right)\), where \(f\left(s\right)\) is the cost of the state \(s\).
📗 Simulated Annealing, probability of moving from \(s\) to a worse state \(s'\) = \(p = e^{- \dfrac{\left| f\left(s'\right) - f\left(s\right) \right|}{T\left(t\right)}}\) if \(f\left(s'\right) \geq f\left(s\right)\) and \(p = 1\) if \(f\left(s'\right) < f\left(s\right)\), where \(T\left(t\right)\) is the temperature as time \(t\).
📗 Genetic Algorithm, probability of get selected as a parent in cross-over: \(p_{i} = \dfrac{F\left(s_{i}\right)}{\displaystyle\sum_{j=1}^{n} F\left(s_{j}\right)}\), \(i = 1, 2, ..., N\), where \(F\left(s\right)\) is the fitness of state \(s\).
📗 Adversarial Search
📗 Sequential Game (Alpha Beta Pruning): prune the tree if \(\alpha \geq \beta\), where \(\alpha\) is the current value of the MAX player and \(\beta\) is the current value of the MIN player.
📗 Simultaneous Move Game (rationalizable): remove an action \(s_{i}\) of player \(i\) if it is strictly dominated \(F\left(s_{i}, s_{-i}\right) < F\left(s'_{i}, s_{-i}\right)\), for some \(s'_{i}\) of player \(i\) and for all \(s_{-i}\) of the other players.
📗 Simultaneous Move Game (Nash equilibrium): \(\left(s_{i}, s_{-i}\right)\) is a (pure strategy) Nash equilibrium if \(F\left(s_{i}, s_{-i}\right) \geq F\left(s'_{i}, s_{-i}\right)\) and \(F\left(s_{i}, s_{-i}\right) \geq F\left(s_{i}, s'_{-i}\right)\), for all \(s'_{i}, s'_{-i}\).