composite likelihood data augmentation for within-network ... · composite likelihood data...

Composite Likelihood Data Augmentation for Within-Network Statistical Relational Learning

Joseph J. Pfeiffer III1Jennifer Neville1Paul Bennett2

1Purdue University2Microsoft Research

ICDM 2014, Shenzhen

• Items: stock brokers

• Label: fraudulent

• Relationships: phone calls

• Attributes: other observable item traits

- social network users, papers, movies, products, etc.

- buy product, sales rank, etc.,

- friendships, emails, citations, co-purchases, etc.

Relational Machine Learning

2

Within-network relational machine learning

Within-Network Relational Machine Learning

3

Learn model parameters using the observed data

⇥Y

Infer (predict) remaining labels using the learned parameters and available data⇥Y

Will Show: Learning/inference approximations prevent

semi-supervised algorithms from converging

Will Develop/Demonstrate: Methods that model parameter uncertainty

overcome approximation errors

• Introduction • Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments• Conclusions

Outline

4

• Introduction• Background: Semi-supervised

Relational Machine Learning • Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments• Conclusions

Outline

5

Collective Inference

ˆ

⇥Y = argmax

⇥Y

X

vi2VL

logPY (yi|YMBL(vi),xi,⇥Y )

• Relational Machine Learning

- Approximate Learning:

- ApproximateInference:

• Many conditional forms to choose from(e.g., Generative, Logistic)

Background - Semi-Supervised Relational LearningLabels Edges

Attributes

yi

yjxi

MB(vi)

M-Step:

E-Step: (For all unlabeled instances)

Composite Likelihood (GO)

Pseudolikelihood (GL)ˆ

⇥Y = argmax

⇥Y

X

YU2YU

PY (YU )

X

vi2VL

logPY (yi| ˜YMB(vi),xi,⇥Y )

PY (yi|YMB(vi),xi,⇥Y )

P (Y|X,E,⇥Y )

yi

yjxi

MB(vi)

Semi-supervised Relational EM(Xiang & Neville, 2009)

How do these approximations affect semi-supervised relational learning?

Both learning and inference require approximations

• Introduction• Background: Semi-supervised

Relational Machine Learning • Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments• Conclusions

Outline

7

• Introduction• Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments• Conclusions

Outline

8

Collective Inference Approximation• (Relational EM:)

• Over-Propagation Error - Both neighboring yellow- Yellow neighbors are

not predictive • Over-Correction

Impact of Approximations in Semi-Supervised Relational EM

M-Step:

E-Step: (For all unlabeled instances)

Composite Likelihood Approximation

ˆ

⇥Y = argmax

⇥Y

X

YU2YU

PY (YU )

X

vi2VL

logPY (yi| ˜YMB(vi),xi,⇥Y )

PY (yi|YMB(vi),xi,⇥Y )

??

? P (Purple|Neighbors) ⇡ 0.61

P (Purple|Purple Neighbor) ⇡ 0.96

• Does over propagation during prediction happen in real world, sparsely labeled networks?

Impact of Approximations in Semi-Supervised Relational Learning

0 1000 2000 3000 4000 5000Total Positives

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2000

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lNeg

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0 1000 2000 3000 4000 5000Total Positives

0

1000

2000

3000

4000

5000

Tota

lNeg

ativ

es

Trial 1Trial 2Trial 3Trial 4Trial 5

Relational Naive Bayes Relational Logistic Regression

Yes

Class Prior Class Prior

• Does over correction happen during parameter estimation for semi-supervised relational learning for real world, sparsely labeled networks?

Impact of Approximations in Semi-Supervised Relational Learning

Relational Naive Bayes Relational Logistic Regression

Yes

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• Introduction• Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments• Conclusions

Outline

12

• Introduction• Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation • Experiments• Conclusions

Outline

13

Relational Stochastic EM andRelational Data Augmentation

14

ParametersPredictions

Fixed Point Stochastic

Fixed Point Relational EM —

Stochastic Relational Stochastic EM

Relational Data Augmentation

• Stochastic approximation to relational EM algorithm• Sample from the joint conditional

distribution of labels • Maximize the composite likelihood

• Contrasts with Relational EM, which utilizes expectations of unlabeled items

• Average over the parameters to reduce the variance (Celex et al., 2001)

• Inference is still performed using a single, fixed point set of parameters

Relational Stochastic EM

15

Parameters

Predictions

Fixed Point Stochastic

Fixed Point Relational EM —

Stochastic Relational Stochastic EM

Relational Data Augmentation

Alternate Between:

Gibbs sample of labels

Maximizing Parameters

Aggregate Parameters:

Final Inference:

YtU ⇠ P t

Y (YU |YL,X,E, ⇥t�1Y )

˜

⇥

tY = arg max

⇥Y

X

vi2VL

logPY (yi| ˜YtMB(vi)

,xi,⇥Y )

⇥Y =1

T

X

t

⇥tY

PY (YU |YL,X,E, ⇥Y )

• Data Augmentation is a Bayesian viewpoint of EM- Parameters are random variables.- Computes posterior predictive

distribution (Tanner&Wong,1987)• Developed a version for relational semi-

supervised learning• Final inference is over a distribution

of parameter values• Requires prior distributions over the

parameters and corresponding sampling methods- RNB: Beta (conjunctive prior)- RLR: Normal

(Metropolis-Hastings sampler)

Relational Data Augmentation

16

Parameters

Predictions

Fixed Point Stochastic

Fixed Point Relational EM —

Stochastic Relational Stochastic EM

Relational Data Augmentation

Alternate Between:

Gibbs sample of labels

Sample Parameters

Final Parameters:

Final Inference:

YtU ⇠ P t

Y (YU |YL,X,E, ⇥t�1Y )

⇥Y =1

T

X

t

⇥tY

YtU =

1

T

X

t

YtU

⇥tY ⇠ P t(⇥Y |Yt

U ,YL,X,E)

• Introduction• Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation • Experiments• Conclusions

Outline

17

• Introduction• Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments • Conclusions

Outline

18

• Compare on four real world networks• Competing Models

- Independent- Relational (No CI and CI)- Relational EM- Relational SEM - Relational DA

• Conditional Models- Relational Naive Bayes- Relational Logistic Regression

• Error measures- Zero-One Loss (0-1 Loss)- Mean absolute error (MAE)

Experiments - Setup

19

Dataset Vertices Edges Attributes

Facebook 5,906 73,374 2

IMDB 11,280 426,167 28

DVD 16,118 75,596 28

Music 56,891 272,544 26

Experiments - DVD

20

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Percentage of Graph Labeled

0.340.360.380.400.420.440.460.480.50

0-1

Loss

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Percentage of Graph Labeled

0.300.320.340.360.380.400.420.440.460.48

0-1

Loss

Naive Bayes Logistic Regression

Relational EM Relational EM

Relational DA Relational DA

Relational EM’s instability in sparsely labeled domains causes poor performance

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Percentage of Graph Labeled

0.40

0.45

0.50

0.55

0.60

MA

E

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Percentage of Graph Labeled

0.300.320.340.360.380.400.420.440.46

MA

EExperiments - Facebook

21

Naive Bayes Logistic Regression

Relational EM Relational EM

Relational DA Relational DA

Relational SEM

Relational Data Augmentation can outperform Relational Stochastic EM

(In Paper): Cast collective inference as a nonlinear dynamical system to

analyze the convergence of Relational SEM

(Finding): Similar to Relational EM, Relational SEM can sometimes learn

parameters that result in an unstable system

• Introduction• Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments • Conclusions

Outline

22

• Introduction• Background: Semi-supervised

Relational Machine Learning• Analysis of Relational EM convergence

in real world networks • Relational Stochastic EM and

Relational Data Augmentation• Experiments• Conclusions

Outline

23

• Findings - Fixed point approximation error led Relational EM to not converge- Overpropagation and Overcorrection - (In Paper) Fixed point EM methods can result in unstable systems during

inference

• Developed - Relational Stochastic EM method has lower variance in parameter estimates- Relational Data Augmentation for computing a posterior predictive

distribution for the unlabeled instances- Models the uncertainty over the parameter estimates, meaning it can

outperform the Relational Stochastic EM approach- Both work well in conjunction with Composite Likelihood approximation- Both significantly outperformed a variety of competitors under a number of

testing scenarios

Conclusions

24

Thank you!jpfeiffer@purdue.edu neville@cs.purdue.edu paul.n.bennett@microsoft.com

JenniferNeville

PaulBennett