for Markov Logic Networks Tuyen N Huynh and Raymond J Mooney Machine Learning Group Department of Computer Science The University of Texas at Austin SDM 2011 April 29 2011 Motivation 2 D McDermott and J Doyle ID: 536533
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Slide1
Online Max-Margin Weight Learning for Markov Logic Networks
Tuyen N. Huynh and Raymond J. Mooney
Machine Learning GroupDepartment of Computer ScienceThe University of Texas at Austin
SDM 2011, April 29, 2011Slide2
Motivation2
D. McDermott and J. Doyle.
Non-monotonic Reasoning I. Artificial Intelligence, 13: 41-72, 1980.
D. McDermott and J. Doyle.
Non-monotonic Reasoning I.
Artificial Intelligence, 13: 41-72, 1980.
D. McDermott and J. Doyle. Non-monotonic Reasoning I. Artificial Intelligence, 13: 41-72, 1980.
D. McDermott and J. Doyle. Non-monotonic Reasoning I. Artificial Intelligence, 13: 41-72, 1980.
D. McDermott and J. Doyle. Non-monotonic Reasoning I. Artificial Intelligence, 13: 41-72, 1980.
D. McDermott and J. Doyle. Non-monotonic Reasoning I. Artificial Intelligence, 13: 41-72, 1980.
D. McDermott and J. Doyle. Non-monotonic Reasoning I. Artificial Intelligence, 13: 41-72, 1980.
[A0 He] [AM-MOD would] [AM-NEG n’t] [V accept] [A1 anything of value] from [A2 those he was writing about]
[A0 He] [AM-MOD would] [AM-NEG n’t] [V accept] [A1 anything of value] from [A2 those he was writing about]
[A0 He] [AM-MOD would] [AM-NEG n’t] [V accept] [A1 anything of value] from [A2 those he was writing about]
[A0 He] [AM-MOD would] [AM-NEG n’t] [V accept] [A1 anything of value] from [A2 those he was writing about]
[A0 He] [AM-MOD would] [AM-NEG n’t] [V accept] [A1 anything of value] from [A2 those he was writing about]
[A0 He] [AM-MOD would] [AM-NEG n’t] [V accept] [A1 anything of value] from [A2 those he was writing about]
[A0 He] [AM-MOD would] [AM-NEG n’t] [V accept] [A1 anything of value] from [A2 those he was writing about]
Citation segmentation
Semantic role labelingSlide3
Motivation (cont.)3
Markov Logic Networks (MLNs) [Richardson & Domingos
, 2006] is an elegant and powerful formalism for handling those complex structured dataExisting weight learning methods 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
May not
fit all the training examples in
main memory
do not scale to
problems having a large number of examplesPrevious work applied an existing online algorithm to learn weights for MLNs but did not compare to other algorithms Introduce a new online weight learning algorithm and extensively compare to other existing methodsSlide4
Outline4
MotivationBackgroundMarkov Logic NetworksPrimal-dual framework for online learningNew online learning algorithm for max-margin structured prediction
Experiment EvaluationSummarySlide5
5
Markov Logic Networks [Richardson &
Domingos, 2006]Set of weighted first-order formulas
Larger weight indicates stronger belief that the formula should hold.
The formulas are called the
structure
of the MLN.MLNs are templates for constructing Markov networks for a given set of constants
MLN Example: Friends & Smokers
*Slide from
[
Domingos, 2007]Slide6
Example: Friends & Smokers
Two constants:
Anna
(A) and
Bob
(B)
6
*Slide from
[
Domingos, 2007]Slide7
Example: Friends & Smokers
Cancer(A)
Smokes(A)
Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants:
Anna
(A) and
Bob (B)
7*Slide from [Domingos, 2007]Slide8
Example: Friends & Smokers
Cancer(A)
Smokes(A)
Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants:
Anna
(A) and
Bob (B)8
*Slide from [Domingos, 2007]Slide9
Example: Friends & Smokers
Cancer(A)
Smokes(A)
Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants:
Anna
(A) and
Bob
(B)9
*Slide from [Domingos, 2007]Slide10
Weight of formula
i
No. of true groundings of formula
i
in
x
10
Probability of a possible world
A possible world becomes exponentially less likely as the total weight of all the grounded clauses it violates increases.
a possible worldSlide11
Max-margin weight learning for MLNs[Huynh & Mooney
, 2009]maximize the separation margin
: log of the ratio of the probability of the correct label and the probability of the closest incorrect oneFormulate as 1-slack Structural SVM [Joachims
et al., 2009]
Use cutting plane method
[
Tsochantaridis
et.al., 2004] with an approximate inference algorithm based on Linear Programming11Slide12
Online learning12
For i=1 to T:Receive an example
The learner choose a vector and uses it to predict a label
Receive the correct label
Suffer a loss:
Goal: minimize the regret
The accumulative loss of the online learner
The accumulative loss of the
best batch
learnerSlide13
A general and latest framework for deriving low-regret online algorithmsRewrite the regret bound as an optimization problem (called the primal problem), then considering the dual problem of the primal oneDerive a condition that guarantees the increase in the dual objective in each step
Incremental-Dual-Ascent (IDA) algorithms. For example: subgradient methods [
Zinkevich, 2003]Primal-dual framework for online learning[
Shalev-Shwartz
et al., 2006]
13Slide14
Primal-dual framework for online learning (cont.)
14Propose 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 (the current example)A closed-form solution of CDA update rule
CDA algorithm has the same cost as subgradient methods but increase the dual objective more in each step better accuracySlide15
Steps for deriving a new CDA algorithm 15
Define the regularization and loss functionsFind the conjugate functions
Derive a closed-form solution for the CDA update ruleCDA algorithm for max-margin structured predictionSlide16
Max-margin structured prediction
16The output y belongs to some structure space YJoint feature function:
(x,y): X x Y
→
R
Learn a discriminant function f:
Prediction for a new input x:
Max-margin criterion:
MLNs: n(
x,y
)Slide17
1. Define the regularization and loss functions17
Regularization function:
Loss function:
Prediction based loss (PL):
the loss incurred by using the
predicted label
at each step
+
where
Label loss functionSlide18
1. Define the regularization and loss functions (cont.)
18Loss function:Maximal
loss (ML): the maximum loss an online learner could suffer at each step
where
Upper bound of the PL loss
more aggressive update better predictive accuracy on clean datasets
The
ML loss depends on the label loss function
can
only be used with some label loss functions
Slide19
2. Find the conjugate functions
19Conjugate function:
1-dimension:
is the negative of the y-intercept of the tangent line to the graph of f that has slope
Slide20
2. Find the conjugate functions (cont.)
20Conjugate function of the regularization function f(w):f(w)=(1/2)||w||
22 f*(µ) = (1/2)||µ||22Slide21
2. Find the conjugate functions (cont.)
21Conjugate function of the loss functions:
+
similar to Hinge loss
+
Conjugate function of Hinge loss:
[
Shalev-Shwartz
& Singer, 2007]
Conjugate functions of PL and
M
L loss:
Slide22
CDA’s update formula:
Compare with the update formula of the simple update,
subgradient
method
[Ratliff et al., 2007]
:
22
CDA’s learning rate combines the learning rate of the
subgradient
method with the loss incurred at each step
3. Closed-form solution for the CDA update ruleSlide23
Experiments
23Slide24
Experimental Evaluation24
Citation segmentation on CiteSeer datasetSearch query disambiguation on a dataset obtained from MicrosoftSemantic role labeling on noisy
CoNLL 2005 datasetSlide25
Citation segmentation25
Citeseer dataset [Lawrence et.al., 1999] [
Poon and Domingos, 2007]
1,563 citations, divided into 4 research topics
Task: segment each citation into 3 fields:
Author, Title, Venue
Used
the MLN for isolated segmentation model in [
Poon and Domingos, 2007]Slide26
Experimental setup4-fold cross-validationSystems compared:
MM: the max-margin weight learner for MLNs in batch setting [Huynh & Mooney, 2009]1-best MIRA [Crammer et al., 2005]
SubgradientCDACDA-PLCDA-MLMetric:
F
1
, harmonic mean of the precision and recall
26
Slide27
Average F1on CiteSeer
27Slide28
Average training time in minutes28Slide29
Search query disambiguation29
Used the dataset created by Mihalkova & Mooney [2009]
Thousands of search sessions where ambiguous queries were asked: 4,618 sessions for training, 11,234 sessions for testingGoal: disambiguate search query based on previous related search sessionsNoisy dataset since the true labels are based on which results were clicked by usersUsed the 3 MLNs proposed in [
Mihalkova
& Mooney, 2009]Slide30
Experimental setupSystems compared:Contrastive Divergence (CD)
[Hinton 2002] used in [Mihalkova & Mooney, 2009]
1-best MIRASubgradient CDACDA-PLCDA-MLMetric:Mean Average Precision (MAP): how close the relevant results are to the top of the rankings
30Slide31
MAP scores on Microsoft query search31Slide32
Semantic role labeling 32
CoNLL 2005 shared task dataset [Carreras & Marques, 2005]Task: For each target verb in a sentence, find and label all of its semantic components
90,750 training examples; 5,267 test examplesNoisy labeled experiment:Motivated by noisy labeled data obtained from crowdsourcing services such as Amazon Mechanical TurkSimple noise model:At p percent noise, there is p probability that an argument in a verb is swapped with another argument of that verb.Slide33
Experimental setupUsed the MLN developed in
[Riedel, 2007]Systems compared:1-best MIRASubgradient CDA-ML
Metric:F1 of the predicted arguments [Carreras & Marques, 2005]33Slide34
F1 scores on CoNLL 2005
34Slide35
Summary35
Derived CDA algorithms for max-margin structured predictionHave the same computational cost as existing online algorithms but increase the dual objective more Experimental results on several real-world problems show that the new algorithms generally achieve better accuracy and also have more consistent performance.Slide36
Thank you!
36
Questions?