ECE 6504: Advanced Topics in Machine Learning Probabilistic Graphical Models and Large-Scale Learning

Transcription

1 ECE 6504: Advanced Topics in Machine Learning Probabilistic Graphical Models and Large-Scale Learning Topics Summary of Class Advanced Topics Dhruv Batra Virginia Tech

2 HW1 Grades Mean: 28.5/38 ~= 74.9% #Students Score (C) Dhruv Batra 2

3 HW2 Grades Mean: 33.2/45 ~= 73.8% #Students Score (C) Dhruv Batra 3

4 Administrativia (Mini-)HW4 Out now Due: May 7, 11:55pm Implementation: Parameter Learning with Structured SVMs and Cutting-Plane Final Project Webpage Due: May 7, May 13 11:55pm Can use late days Can t use late days any more 1-3 paragraphs Goal Illustrative figure Approach Results (with figures or tables) Take Home Final Out: May 8 Due: May 13, 11:55pm No late days Open book, open notes, open internet. Cite your sources. No discussions! (C) Dhruv Batra 4

5 A look back: PGMs One of the most exciting advancements in statistical AI in the last years Marriage Graph Theory + Probability Compact representation for exponentially-large probability distributions Exploit conditional independencies Generalize naïve Bayes logistic regression Many more (C) Dhruv Batra 5

6 A look back: what you learnt Directed Graphical Models (Bayes Nets) Representation: Directed Acyclic Graphs (DAGs), Conditional Probability Tables (CPTs), d-separation, v-structures, Markov Blanket, I-Maps Parameter Learning: MLE, MAP, EM Structure Learning: Chow-Liu, Decomposable scores, hill climbing Inference: Marginals, MAP/MPE, Variable Elimination Undirected Graphical Models (MRFs/CRFs) Representation: Junction trees, Factor graphs, treewidth, Local Makov Assumptions, Moralization, Triangulation Inference: Belief Propagation, Message Passing, Linear Programming Relaxations, Dual-Decomposition, Variational Inference, Mean Field Parameter Learning: MLE, gradient descent Structured Prediction: Structured SVMs, Cutting-Plane training Large-Scale Learning Online learning: perceptrons, stochastic (sub-)gradients Distributed Learning: Dual Decomposition, Alternating Direction Method of Multipliers (ADMM) (C) Dhruv Batra 6

7 Main Issues in PGMs Representation How do we store P(X 1, X 2,, X n ) What does my model mean/imply/assume? (Semantics) Inference How do I answer questions/queries with my model? such as Marginal Estimation: P(X 5 X 1, X 4 ) Most Probable Explanation: argmax P(X 1, X 2,, X n ) Learning How do we learn parameters and structure of P(X 1, X 2,, X n ) from data? What model is the right for my data? (C) Dhruv Batra 7

8 What is this class about? Making global predictions from local observations (C) Dhruv Batra 8

9 A look forward Stuff we couldn t teach you A.K.A: Stuff that s not on the exam! What do people in this area work on? What is being published in PGMs / Structured Prediction? (C) Dhruv Batra 9

10 Error Decomposition Reality model class (C) Dhruv Batra 10

11 Error Decomposition Reality (C) Dhruv Batra 11

12 Error Decomposition Higher-Order Potentials model class Reality (C) Dhruv Batra 12

13 Error Decomposition Higher-Order Potentials model class Reality Focus of MAP Inference (C) Dhruv Batra 13

14 Continuous-Variable PGMs (C) Dhruv Batra 14

15 Multivariate Gaussian

16 Canonical form Standard form and canonical forms are related: Conditioning is easy in canonical form Marginalization easy in standard form

17 Sampling (C) Dhruv Batra 17

18 What you ve learned so far VE & Junction Trees Exact inference Exponential in tree-width Belief Propagation, Mean Field Approximate inference for marginals/conditionals Fast, but can get inaccurate estimates Sample-based Inference Approximate inference for marginals/conditionals With enough samples, will converge to the right answer (or a high accuracy estimate) (If you want to be cynical, replace enough with ridiculously many )

19 Goal Often we want expectations given samples x[1] x[m] from a distribution P. E P [f] 1 M P (X = x) 1 M M f(x[m]) m=1 M 1(x[m] = x) m=1 x[i] P (X) Discrete Random Variables: Number of samples from P(X): M X = {X 1,..., X n }

20 Forward Sampling Sample nodes in topological order Assignment to parents selects P(X Pa(X)) End result is one sample from P(X) Repeat to get more samples D I G S L x[m, D] (Easy : 0.6, Hard : 0.4) D = Easy x[m, I] (Low : 0.7, High : 0.3) I = High x[m, G D = d, I = i] ([80, 100] : 0.9, [50, 80) : 0.08, [0, 50) : 0.02) x[m, S I = i] (Bad : 0.2, Good : 0.8) S = Bad x[m, L G = g] (F ail : 0.1, P ass : 0.9) L = Pass G = [80,100]

21 Multinomial Sampling Given an assignment to its parents, X i is a multinomial random variable. x[m, G D = d, I = i] (v 1 : 0.9, v 2 : 0.08, v 3 : 0.02) U ~ Unif[0,1] v 1 v 2 v

22 Sample-based probability estimates Have a set of M samples from P(X) Can estimate any probability by counting records: Marginals: ˆP (D = Easy, S = Bad) = 1 M M m=1 1(x[m, D] = Easy, x[m, S] = Bad) Conditionals: ˆP (D = Easy S = Bad) = M m=1 1(x[m, D] = Easy, x[m, S] = Bad) M m=1 1(x[m, S] = Bad) Rejection sampling: once the sample and evidence disagree, throw away the sample. Rare events: If the evidence is unlikely, i.e., P(E = e) small, then the sample size for P(X E=e) is low

23 (C) Dhruv Batra 23

24 Simulated Annealing (C) Dhruv Batra Image Credit: [BVZ, PAMI01] Alpha-Expansion 24

25 Simulated Annealing (C) Dhruv Batra Image Credit: [BVZ, PAMI01] Alpha-Expansion 25

26 Effect of Exact MAP a (C) Dhruv Batra Image Credit: [BVZ, PAMI01] 26

27 Sontag NIPS10 (C) Dhruv Batra 27

28 Soft-Margin Structured SVM Minimize subject to 1 2 w2 + C! N! j j w T!(x j, y j )! w T!(x j, y)+ "(y j, y)#! j Too many constraints! (C) Dhruv Batra Slide Credit: Thorsten Joachims 28

29 Cutting-Plane Method 1 2 w2 + C! N! j j w T!(x j, y j )! w T!(x j, y)+ "(y j, y)#! j Key insight of NIPS10 paper What if we replace exponentially many constraints with a smaller set (without using Cutting-Plane)? Key contribution There exist a rich set of distributions where this approximation results in optimal parameter in the infinite data setting (C) Dhruv Batra 29

30 Meshi ICML10 (C) Dhruv Batra 30

31 Tarlow UAI10 (C) Dhruv Batra 31

32 Ladicky IJCV12 (C) Dhruv Batra 32

33 Lempitsky ICCV09 (C) Dhruv Batra 33

34 Kohli & Rother (C) Dhruv Batra 34

35 Perturb and MAP S(y) = i V θ i (y i )+ θ ij (y i,y j ) (i,j) E Approach Perturb: θ = θ +, p() MAP: argmax y S θ(y) (C) Dhruv Batra 35

36 Perturb and MAP Perturb: θ = θ +, p() Theorem: If IID Gumbel, then EXACT samples. [Papandreou & Yuille, ICCV11] [Hazan & Jaakkola, ICML12] (C) Dhruv Batra 36

37 Perturb and MAP Full Order Gumbel is hard! (C) Dhruv Batra 37

38 Perturb and MAP Reduced Order Gumbel Approach Perturb: S(y) = θ i (y i )+ i V θ = θ +, p() (i,j) E θ ij (y i,y j ) θ i = θ i +, p() MAP: argmax y S θ(y) (C) Dhruv Batra 38

39 My Research: Multiple Predictions P (y) x x x x y y MAP Sampling Porway & Zhu, 2011" TU & Zhu, 2002" Rich History" (C) Dhruv Batra 39

40 My Research: Multiple Predictions P (y) y y MAP Sampling Porway & Zhu, 2011" TU & Zhu, 2002" Rich History" M-Best MAP Flerova et al., 2011" Fromer et al., 2009" Yanover et al., 2003" " Ideally: M-Best Modes (C) Dhruv Batra 40

41 My Research: Multiple Predictions P (y) y y MAP Our work: Diverse M-Best in MRFs [ECCV 12] " Sampling M-Best MAP - Don t hope for diversity. Explicitly encode it. Ideally: Porway & Zhu, 2011" Flerova et al., 2011" TU & Zhu, 2002" - Not guaranteed Fromer et to al., be 2009" modes. M-Best Modes Rich History" Yanover et al., 2003" (C) Dhruv Batra 41

42 What next? Seminars: CV-ML Reading Group Conferences: Neural Information Processing Systems (NIPS) International Conference in Machine Learning (ICML) Uncertainty in Artificial Intelligence (UAI) Artificial Intelligence & Statistics (AISTATS) Classes (at some point in the near future) ECE6504: Fundamental Ideas in Machine Learning Paper reading class showing lineage of ideas Story from 89 first backprop papers to CNNs today ECE6504: Machine Learning for Big Data Large-Scale Distributed Machine Learning Use frameworks such as Graphlab Implement things in CloudCV (C) Dhruv Batra 42

43 Feedback Student Perception of Teaching (SPOT) Tell us how we re doing What would you like to see more What would you like to see less ENDS MAY 8 (C) Dhruv Batra 43