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Discriminative, Unsupervised, Convex Learning Dale Schuurmans Department of Computing Science University of Alberta MITACS Workshop, August 26, 2005

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2 Current Research Group PhD Tao Wang reinforcement learning PhD Ali Ghodsi dimensionality reduction PhD Dana Wilkinson action-based embedding PhD Yuhong Guo ensemble learning PhD Feng Jiao bioinformatics PhD Jiayuan Huang transduction on graphs PhD Qin Wang statistical natural language PhD Adam Milstein robotics, particle filtering PhD Dan Lizotte optimization, everything PhD Linli Xu unsupervised SVMs PDF Li Cheng computer vision

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3 Current Research Group PhD Tao Wang reinforcement learning PhD Dana Wilkinson action-based embedding PhD Feng Jiao bioinformatics PhD Qin Wang statistical natural language PhD Dan Lizotte optimization, everything PDF Li Cheng computer vision

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4 Today I will talk about: One Current Research Direction Learning Sequence Classifiers (HMMs) Discriminative Unsupervised Convex EM?

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5 Outline Unsupervised SVMs Discriminative, unsupervised, convex HMMs Tao, Dana, Feng, Qin, Dan, Li

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Unsupervised Support Vector Machines Joint work with Linli Xu

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8 Main Idea Unsupervised SVMs (and semi-supervised SVMs) Harder computational problem than SVMs Convex relaxation – Semidefinite program (Polynomial time)

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9 Background: Two-class SVM Supervised classification learning Labeled data linear discriminant Classification rule: Some better than others? +

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10 Maximum Margin Linear Discriminant Choose a linear discriminant to maximize

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11 Unsupervised Learning Given unlabeled data, how to infer classifications? Organize objects into groups — clustering

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12 Idea: Maximum Margin Clustering Given unlabeled data, find maximum margin separating hyperplane Clusters the data Constraint: class balance: bound difference in sizes between classes

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13 Challenge Find label assignment that results in a large margin Hard Convex relaxation – based on semidefinite programming

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14 How to Derive Unsupervised SVM? Two-class case: 1.Start with Supervised Algorithm Given vector of assignments, y, solve Inv. sq. margin

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15 How to Derive Unsupervised SVM? 2.Think of as a function of y If given y, would then solve Goal: Choose y to minimize inverse squared margin Problem: not a convex function of y Inv. sq. margin

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16 How to Derive Unsupervised SVM? 3.Re-express problem with indicators comparing y labels If given y, would then solve New variables: An equivalence relation matrix Inv. sq. margin

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17 How to Derive Unsupervised SVM? 3.Re-express problem with indicators comparing y labels If given M, would then solve New variables: An equivalence relation matrix Maximum of linear functions is convex Inv. sq. margin Note: convex function of M

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18 How to Derive Unsupervised SVM? 4.Get constrained optimization problem Solve for M encodes an equivalence relation iff Not convex! Class balance

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19 How to Derive Unsupervised SVM? 4.Get constrained optimization problem Solve for M encodes an equivalence relation iff

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20 How to Derive Unsupervised SVM? 5.Relax indicator variables to obtain a convex optimization problem Solve for M

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21 How to Derive Unsupervised SVM? 5.Relax indicator variables to obtain a convex optimization problem Solve for M Semidefinite program

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22 Multi-class Unsupervised SVM? 1.Start with Supervised Algorithm Given vector of assignments, y, solve (Crammer & Singer 01) Margin loss

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23 Multi-class Unsupervised SVM? 2.Think of as a function of y If given y, would then solve (Crammer & Singer 01) Margin loss Goal: Choose y to minimize margin loss Problem: not a convex function of y

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24 Multi-class Unsupervised SVM? 3.Re-express problem with indicators comparing y labels If given y, would then solve (Crammer & Singer 01) Margin loss New variables: M & D

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25 Multi-class Unsupervised SVM? 3.Re-express problem with indicators comparing y labels If given M and D, would then solve New variables: M & D Margin loss convex function of M & D

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26 Multi-class Unsupervised SVM? 4.Get constrained optimization problem Solve for M and D Class balance

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27 Multi-class Unsupervised SVM? 5.Relax indicator variables to obtain a convex optimization problem Solve for M and D

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28 Multi-class Unsupervised SVM? 5.Relax indicator variables to obtain a convex optimization problem Solve for M and D Semidefinite program

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29 Experimental Results SemiDef SpectralClustering Kmeans

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30 Experimental Results

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31 Percentage of misclassification errors Experimental Results Digit dataset

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32 Extension to Semi-Supervised Algorithm Matrix M :

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33 Experimental Results Percentage of misclassification errors Face dataset

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34 Experimental Results

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Discriminative, Unsupervised, Convex HMMs Joint work with Linli Xu With help from Li Cheng and Tao Wang

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37 Hidden Markov Model Joint probability model Viterbi classifier “hidden” state observations Must coordinate local classifiers

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38 HMM Training: Supervised Given Maximum likelihood Conditional likelihood Models input distribution Discriminative (CRFs)

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39 HMM Training: Unsupervised Given only Now what? EM! Marginal likelihood Exactly the part we don’t care about

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40 HMM Training: Unsupervised Given only The problem with EM: Not convex Wrong objective Too popular Doesn’t work

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41 HMM Training: Unsupervised Given only The dream: Convex training Discriminative training When will someone invent unsupervised CRFs?

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42 HMM Training: Unsupervised Given only The question: How to learn effectively without seeing any y’s?

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43 HMM Training: Unsupervised Given only The question: How to learn effectively without seeing any y’s? The answer: That’s what we already did! Unsupervised SVMs

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44 HMM Training: Unsupervised Given only The plan: supervised unsupervised single sequence SVMM3N unsup SVM?

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45 M3N: Max Margin Markov Nets Relational SVMs Supervised training: Given Solve factored QP

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46 Unsupervised M3Ns Strategy Start with supervised M3N QP y-labels re-express in local M,D equivalence relations Impose class-balance Relax non-convex constraints Then solve a really big SDP But still polynomial size

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47 Unsupervised M3Ns SDP

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48 Some Initial Results Synthetic HMM Protein Secondary Structure pred.

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50 Current Research Group PhD Tao Wang reinforcement learning PhD Dana Wilkinson action-based embedding PhD Feng Jiao bioinformatics PhD Qin Wang statistical natural language PhD Dan Lizotte optimization, everything PDF Li Cheng computer vision

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51 Brief Research Background Sequential PAC Learning Linear Classifiers: Boosting, SVMs Metric-Based Model Selection Greedy Importance Sampling Adversarial Optimization & Search Large Markov Decision Processes

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