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Online Max-Margin Weight Learning with Markov Logic Networks Tuyen N. Huynh and Raymond J. Mooney Machine Learning Group Department of Computer Science The University of Texas at Austin Star AI 2010, July 12, 2010

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Outline 2 Motivation Background Markov Logic Networks Primal-dual framework New online learning algorithm for structured prediction Experiments Citation segmentation Search query disambiguation Conclusion

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Motivation Most of the existing weight learning for MLNs are in the batch setting. Need to run inference over all the training examples in each iteration Usually take a few hundred iterations to converge Cannot fit all the training examples in the memory Conventional solution: online learning 3

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Background 4

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An MLN is a weighted set of first-order formulas Larger weight indicates stronger belief that the clause should hold Probability of a possible world (a truth assignment to all ground atoms) x: Markov Logic Networks (MLNs) Weight of formula iNo. of true groundings of formula i in x [Richardson & Domingos, 2006] 2.5 Center(i,c) => InField(Ftitle,i,c) 1.2 InField(f,i,c) ^ Next(j,i) ^ ¬HasPunc(c,i)=> InField(f,j,c) 5

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Existing discriminative weight learning methods for MLNs maximize the Conditional Log Likelihood (CLL) [Singla & Domingos, 2005], [Lowd & Domingos, 2007], [Huynh & Mooney, 2008] maximize the margin, the log ratio between the probability of the correct label and the closest incorrect one [Huynh & Mooney, 2009] 6

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Online learning 7

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A general and latest framework for deriving low- regret online algorithms Rewriting the regret bound as an optimization problem (called the primal problem), then considering the dual problem of the primal one A condition that guarantees the increase in the dual objective in each step Incremental-Dual-Ascent (IDA) algorithms. For example: subgradient methods Primal-dual framework [Shalev-Shwartz et al., 2006] 8

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Primal-dual framework (cont.) 9 Proposed a new class of IDA algorithms called Coordinate-Dual-Ascent (CDA) algorithm: The CDA update rule only optimizes the dual w.r.t the last dual variable A closed-form solution of CDA update rule CDA algorithms have the same cost as subgradient methods but increase the dual objective more in each step converging to the optimal value faster

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Primal-dual framework (cont.) 10

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CDA algorithms for max-margin structured prediction 11

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Max-margin structured prediction 12

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Steps for deriving new CDA algorithms 13 1. Define the regularization and loss functions 2. Find the conjugate functions 3. Derive a closed-form solution for the CDA update rule

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1. Define the regularization and loss functions 14 Label loss function

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1. Define the regularization and loss functions (cont.) 15

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2. Find the conjugate functions 16

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2. Find the conjugate functions (cont.) 17

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18 Optimization problem: Solution: 3. Closed-form solution for the CDA update rule

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CDA algorithms for max-margin structured prediction 19

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Experiments 20

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Citation segmentation 21 Citeseer dataset [Lawrence et.al., 1999] [ Poon and Domingos, 2007 ] 1,563 citations, divided into 4 research topics Each citation is segmented into 3 fields: Author, Title, Venue Used the simplest MLN in [ Poon and Domingos, 2007] Similar to a linear chain CRF: Next(j,i) ^ !HasPunc(c,i) ^ InField(c,+f,i) => InField(c,+f,j)

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Experimental setup Systems compared: MM: the max-margin weight learner for MLNs in batch setting [Huynh & Mooney, 2009] 1-best MIRA [Crammer et al., 2005] Subgradient [Ratliff et al., 2007] CDA1/PA1 CDA2 22

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Experimental setup (cont.) 4-fold cross-validation Metric: CiteSeer: micro-average F 1 at the token level Used exact MPE inference (Integer Linear Programming) for all online algorithms and approximate MPE inference (LP-relaxation) for the batch one. Used Hamming loss as the label loss function 23

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Average F1 24

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Average training time in minutes 25

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Microsoft web search query dataset 26 Used the clean-up dataset created by Mihalkova & Mooney [2009] Has thousands of search sessions where an ambiguous queries was asked Goal: disambiguate search query based on previous related search sessions Used 3 MLNs proposed in [Mihalkova & Mooney, 2009]

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Experimental setup Systems compared: Contrastive Divergence (CD) [Hinton 2002]: used in [Mihalkova & Mooney, 2009] 1-best MIRA Subgradient CDA1/PA1 CDA2 Metric: Mean Average Precision (MAP): how close the relevant results are to the top of the rankings 27

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MAP scores 28

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Conclusion 29 Derived CDA algorithms for max-margin structured prediction Have same computational cost as existing online algorithms but increase the dual objective more Experimental results on two real-world problems show that the new algorithms generally achieve better accuracy and also have more consistent performance.

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Thank you! 30 Questions?

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1 Copyright © 2010, Elsevier Inc. All rights Reserved Fig 2.1 Chapter 2.

1 Copyright © 2010, Elsevier Inc. All rights Reserved Fig 2.1 Chapter 2.

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