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Structured SVM Chen-Tse Tsai and Siddharth Gupta

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Outline Introduction to SVM Large Margin Methods for Structured and Interdependent Output Variables (Tsochantaridis et. al., 2005) Max-Margin Markov Networks (Taskar et. al., 2003) Learning Structural SVMs with Latent Variables (Yu and Joachims, 2009) 2

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SVM- The main idea

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Maximum margin Find w and b such that is maximized and for all (x i, y i ), i=1..n : y i (w T x i + b) ≥ 1 Find w and b such that Φ(w) = ||w|| 2 =w T w is minimized and for all (x i, y i ), i=1..n : y i (w T x i + b) ≥ 1 quadratic optimization problem r ρ

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Binary SVM Training examples: Primal form: Dual form:

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Multiclass SVM

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Structured Output Approach: view as multi-class classification task Every complex output is one class Problems: Exponentially many classes How to predict efficiently? How to learn efficiently? Potentially huge model Manageable number of features? The dog chased the cat x S VPNP DetNV NP DetN y2y2 S VP DetNV NP VN y1y1 S VP DetNV NP DetN ykyk … 7

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Multi-Class SVM (Crammer & Singer, 2001) Training Examples: Inference: Training: Find that solve 8

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Multi-Class SVM (Crammer & Singer, 2001) The dog chased the cat x S VPNP DetNV NP DetN y1y1 S VP DetNV NP VN y2y2 S VP NP y 58 S VPNP DetNV NP DetN y 12 S VPNP DetNV NP DetN y 34 S VPNP DetNV NP DetN y4y4 9

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Joint Feature Map Problem: exponential number of parameters Feature vector that describes match between x and y Learn single weight vector. Inference The dog chased the cat x S VPNP DetNV NP DetN y1y1 S VP DetNV NP VN y2y2 S VP DetNV NP DetN y 58 S VPNP DetNV NP DetN y 12 S VPNP DetNV NP DetN y 34 S VPNP DetNV NP DetN y4y4 10

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Joint Feature Map for Trees Weighted Context Free Grammar Each rule has a weight Score of a tree is the sum of its weight Find highest scoring tree Using CKY Parser The dog chased the cat S VPNP DetNV NP DetN Thecatthechaseddog x y 11

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Structured SVM Hard margin … 12

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Structured SVM Soft Margin SVM 1 SVM 2 13

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General Loss Function measures the difference between prediction y, and the true value y i. The y with high loss should be penalized more severely. Slack re-scaling Margin re-scaling 14

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A Cutting Plane Algorithm Only polynomial number of constraints are needed. 15

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A Cutting Plane Algorithm Cutting plane algorithm 16

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Computational problem Prediction: Get the most violated constraint: Approximate inference methods in MRF Training Structural SVMs when Exact Inference is Intractable. T. Finley, T. Joachims, ICML 2008 17

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Outline Large Margin Methods for Structured and Interdependent Output Variables (Tsochantaridis et. al., 2005) Max-Margin Markov Networks (Taskar et. al., 2003) Learning Structural SVMs with Latent Variables (Yu and Joachims, 2009) 18

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Max-Margin Markov Network Structured SVM entails a large number of constraints So far, handled by adding one constraint a time M 3 network A way to solve SVM 1 with margin re-scaling Use Markov network to encode dependency and generate features Reduce exponential to polynomial number of constraints. 19

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M 3 Network A way to generate features. Define features on the edges The k-th feature of this instance The loss function 20

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M 3 Network A way to solve SVM 1 with margin re-scaling Primal: Dual: Only need node and edge marginal probability to compute expectation 21

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Polynomial-Size Reformulation The key step 22 y0y0 y1y1 y2y2 Δt x (y)α x (y) All possible y 11110.1 1102 1010 1001 0112 01030.2 0011 00020.1 Gold y101 µ x (0)0.60.5 µ x (1)0.40.5

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Polynomial-Size Reformulation The key step Marginal dual variables New constraints Tree structure: 23

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Polynomial-Size Reformulation Factored dual QP #variables and #constraints: N2 M down to N(M 2 +M) N: number of instances, M: the length of y Problem If the structure is not simple, we may need exponential number of new constraints Enforce only local consistency of marginals, get an approximate result 24

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SMO Sequential minimal optimization In binary SVM, we have a linear constraint Working set selection: select the two variables to update M 3 net: 25

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Experimental Results Max-Margin Parsing (Taskar et. al, 2004) Apply M 3 Net to parsing Discussed how to extract features from a grammar 26

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Outline Large Margin Methods for Structured and Interdependent Output Variables (Tsochantaridis et. al., 2005) Max-Margin Markov Networks (Taskar et. al., 2003) Learning Structural SVMs with Latent Variables (Yu and Joachims, 2009) 27

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Latent Variable Models Widely used in machine learning and statistics Unobserved quantities/missing data in experiments Dimensionality Reduction Classical examples: Mixture models, PCA, LDA This paper: Latent variables in supervised prediction tasks

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Latent Variables in S-SVMs How can we extend structural SVM to handle latent variables?

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Structured SVM

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Latent S-SVM Formulation

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CCCP Algorithm

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aa

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Noun Phrase Co-reference

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Noun phrase co-reference results

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