ECE 5984: Introduction to Machine Learning
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1 ECE 5984: Introduction to Machine Learning Topics: (Finish) Expectation Maximization Principal Component Analysis (PCA) Readings: Barber Dhruv Batra Virginia Tech
2 Administrativia Poster Presentation: May 8 1:30-4:00pm 310 Kelly Hall: ICTAS Building Print poster (or bunch of slides) Format: Portrait Make 1 dimen = 36in Board size = 30x40 Less text, more pictures. Best Project Prize! (C) Dhruv Batra 2
3 Administrativia Final Exam When: May 11 7:45-9:45am Where: In class Format: Pen-and-paper. Open-book, open-notes, closed-internet. No sharing. What to expect: mix of Multiple Choice or True/False questions Prove this statement What would happen for this dataset? Material Everything! Focus on the recent stuff. Exponentially decaying weights? Optimal policy? (C) Dhruv Batra 3
4 Recap of Last Time (C) Dhruv Batra 4
5 K-means 1. Ask user how many clusters they d like. (e.g. k=5) 2. Randomly guess k cluster Center locations 3. Each datapoint finds out which Center it s closest to. 4. Each Center finds the centroid of the points it owns 5. and jumps there 6. Repeat until (C) Dhruv Batra terminated! Slide Credit: Carlos Guestrin 5
6 K-means as Co-ordinate Descent Optimize objective function: min µ 1,...,µ k min a 1,...,a N F (µ, a) = Fix µ, optimize a (or C) Assignment Step min µ 1,...,µ k min a 1,...,a N NX kx a ij x i µ j 2 j=1 Fix a (or C), optimize µ Recenter Step (C) Dhruv Batra Slide Credit: Carlos Guestrin 6
7 GMM (C) Dhruv Batra Figure Credit: Kevin Murphy 7
8 K-means vs GMM K-Means AppletKM.html GMM (C) Dhruv Batra 8
9 EM Expectation Maximization [Dempster 77] Often looks like soft K-means Extremely general Extremely useful algorithm Essentially THE goto algorithm for unsupervised learning Plan EM for learning GMM parameters EM for general unsupervised learning problems (C) Dhruv Batra 9
10 EM for Learning GMMs Simple Update Rules E-Step: estimate P(z i = j x i ) M-Step: maximize full likelihood weighted by posterior (C) Dhruv Batra 10
11 Gaussian Mixture Example: Start (C) Dhruv Batra Slide Credit: Carlos Guestrin 11
12 After 1st iteration (C) Dhruv Batra Slide Credit: Carlos Guestrin 12
13 After 2nd iteration (C) Dhruv Batra Slide Credit: Carlos Guestrin 13
14 After 3rd iteration (C) Dhruv Batra Slide Credit: Carlos Guestrin 14
15 After 4th iteration (C) Dhruv Batra Slide Credit: Carlos Guestrin 15
16 After 5th iteration (C) Dhruv Batra Slide Credit: Carlos Guestrin 16
17 After 6th iteration (C) Dhruv Batra Slide Credit: Carlos Guestrin 17
18 After 20th iteration (C) Dhruv Batra Slide Credit: Carlos Guestrin 18
19 General Mixture Models P(x z) P(z) Name Gaussian Categorial GMM Multinomial Categorical Mixture of Multinomials Categorical Dirichlet Latent Dirichlet Allocation (C) Dhruv Batra 19
20 Mixture of Bernoullis (C) Dhruv Batra 20
21 The general learning problem with missing data Marginal likelihood x is observed, z is missing: ll( NY : D) = log P (x i ) NX = log P (x i ) = NX log X z P (x i, z ) (C) Dhruv Batra 21
22 Applying Jensen s inequality Use: log z P(z) f(z) z P(z) log f(z) ll( : D) = NX log X z Q i (z) P (x i, z ) Q i (z) (C) Dhruv Batra 22
23 Convergence of EM Define potential function F(θ,Q): ll( : D) F (,Q i )= NX X z Q i (z) log P (x i, z ) Q i (z) EM corresponds to coordinate ascent on F Thus, maximizes lower bound on marginal log likelihood (C) Dhruv Batra 23
24 EM is coordinate ascent ll( : D) F (,Q i )= E-step: Fix θ (t), maximize F over Q: F (,Q i )= = = NX X Realigns F with likelihood: z NX X z NX X F ( (t),q (t) )=ll( (t) : D) z Q i (z) log P (x i, z (t) ) Q i (z) Q i (z) log P (z x i, )P (x i (t) ) Q i (z) NX X Q i (z) log P (x i z = ll( (t) : D) (t) )+ Q i (z) log P (x i, z ) Q i (z) NX X NX KL(Q i (z) P (z x i, Q (t) i (z) =P (z x i, z Q i (z) log P (z x i, (t) ) Q i (z) (t) ) (t) ) (C) Dhruv Batra 24
25 EM is coordinate ascent ll( : D) F (,Q i )= NX X M-step: Fix Q (t), maximize F over θ F (,Q i )= = = NX X Corresponds to weighted dataset: <x 1,z=1> with weight Q (t+1) (z=1 x 1 ) <x 1,z=2> with weight Q (t+1) (z=2 x 1 ) <x 1,z=3> with weight Q (t+1) (z=3 x 1 ) <x 2,z=1> with weight Q (t+1) (z=1 x 2 ) <x 2,z=2> with weight Q (t+1) (z=2 x 2 ) z NX X z NX X z Q (t) <x 2,z=3> with weight Q (t+1) (z=3 x 2 ) 25 (C) Dhruv Batra z i (z) log P (x i, z ) Q (t) (z) Q (t) i (z) log P (x i, z ) Q (t) i (z) log P (x i, z )+ i Q i (z) log P (x i, z ) Q i (z) NX X NX z Q (t) i H(Q (t) i ) {z } constant (z) log Q (t) i (z)
26 EM Intuition Q(θ,θ t ) Q(θ,θ t+1 ) l(θ) θ t θ t+1 θ t+2 (C) Dhruv Batra 26
27 What you should know K-means for clustering: algorithm converges because it s coordinate ascent EM for mixture of Gaussians: How to learn maximum likelihood parameters (locally max. like.) in the case of unlabeled data EM is coordinate ascent Remember, E.M. can get stuck in local minima, and empirically it DOES General case for EM (C) Dhruv Batra Slide Credit: Carlos Guestrin 27
28 Tasks x Classification y Discrete x Regression y Continuous x Clustering c Discrete ID x Dimensionality z Continuous Reduction (C) Dhruv Batra 28
29 New Topic: PCA
30 Synonyms Principal Component Analysis Karhunen Loève transform Eigen-Faces Eigen-<Insert-your-problem-domain> PCA is a Dimensionality Reduction Algorithm Other Dimensionality Reduction algorithms Linear Discriminant Analysis (LDA) Independent Component Analysis (ICA) Local Linear Embedding (LLE) (C) Dhruv Batra 30
31 Dimensionality reduction Input data may have thousands or millions of dimensions! e.g., images have 5M pixels Dimensionality reduction: represent data with fewer dimensions easier learning fewer parameters visualization hard to visualize more than 3D or 4D discover intrinsic dimensionality of data high dimensional data that is truly lower dimensional
32 Dimensionality Reduction Demo (C) Dhruv Batra 32
33 PCA / KL-Transform De-correlation view Make features uncorrelated No projection yet Max-variance view: Project data to lower dimensions Maximize variance in lower dimensions Synthesis / Min-error view: Project data to lower dimensions Minimize reconstruction error All views lead to same solution (C) Dhruv Batra 33
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