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### Structured Prediction:A Large Margin Approach

Outline

Ben Taskar

University of Pennsylvania

Joint work with:

V. Chatalbashev, M. Collins, C. Guestrin, M. Jordan, D. Klein, D. Koller, S. Lacoste-Julien, C. Manning

Bilingual Word Alignment

En

vertu

de

les

nouvelles

propositions

,

quel

est

le

coût

prévu

de

perception

de

les

droits

?

x

y

What

is

the

anticipated

cost

of

collecting

fees

under

the

new

proposal

?

What is the anticipated cost of collecting fees under the new proposal?

En vertu des nouvelles propositions, quel est le coût prévu de perception des droits?

Combinatorial structure

Protein Structure and Disulfide Bridges

AVITGACERDLQCG

KGTCCAVSLWIKSV

RVCTPVGTSGEDCH

PASHKIPFSGQRMH

HTCPCAPNLACVQT

SPKKFKCLSK

Protein: 1IMT

Outline

- Structured prediction models
- Sequences (CRFs)
- Trees (CFGs)
- Associative Markov networks (Special MRFs)
- Matchings
- Structured large margin estimation
- Margins and structure
- Min-max formulation
- Linear programming inference
- Certificate formulation

Associative Markov Nets

Edge features

Point features

spin-images, point height

length of edge, edge orientation

“associative”

restriction

j

yj

jk

yk

Bilingual Word Alignment

- position
- orthography
- association

En

vertu

de

les

nouvelles

propositions

,

quel

est

le

coût

prévu

de

perception

de

le

droits

?

What

is

the

anticipated

cost

of

collecting

fees

under

the

new

proposal

?

k

j

Disulfide Bonds: Non-bipartite Matching

RSCCPCYWGGCPWGQNCYPEGCSGPKV

1 2 3 4 5 6

2

3

1

5

1

4

2

6

4

6

5

3

Fariselli & Casadio `01, Baldi et al. ‘04

3

1

4

6

5

Scoring FunctionRSCCPCYWGGCPWGQNCYPEGCSGPKV

1 2 3 4 5 6

RSCCPCYWGGCPWGQNCYPEGCSGPKV

1 2 3 4 5 6

- amino acid identities
- phys/chem properties

Structured Models

Mild assumptions:

linear combination

sum of part scores

scoring function

space of feasible outputs

Supervised Structured Prediction

Model:

Prediction

Learning

Data

Estimatew

Example:

Weighted matching

Generally:

Combinatorialoptimization

Local

(ignores

structure)

Margin

Likelihood

(intractable)

Outline

- Structured prediction models
- Sequences (CRFs)
- Trees (CFGs)
- Associative Markov networks (Special MRFs)
- Matchings
- Structured large margin estimation
- Margins and structure
- Min-max formulation
- Linear programming inference
- Certificate formulation

S

A

E

B

F

G

C

H

D

S

S

S

S

A

A

A

A

B

B

B

B

S

C

C

C

C

D

D

D

D

A

B

D

F

Parsing Example- We want:
- Equivalently:

‘It was red’

‘It was red’

‘It was red’

‘It was red’

‘It was red’

a lot!

…

‘It was red’

‘It was red’

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

Alignment Example- We want:
- Equivalently:

‘What is the’

‘Quel est le’

1

2

3

1

2

3

‘What is the’

‘Quel est le’

‘What is the’

‘Quel est le’

‘What is the’

‘Quel est le’

‘What is the’

‘Quel est le’

a lot!

…

‘What is the’

‘Quel est le’

‘What is the’

‘Quel est le’

S

B

B

D

E

A

A

C

C

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

S

A

B

S

C

D

A

E

C

D

1

2

3

1

2

3

Structured Lossb c a r e

2

b r o r e

2

b r o c e

1

b r a c e

0

0 1 2 2

0 1 2 3

‘What is the’

‘Quel est le’

‘It was red’

Large margin estimation

- Given training examples , we want:

- Maximize margin
- Mistake weighted margin:

# of mistakes in y

*Collins 02, Altun et al 03, Taskar 03

Large margin estimation

- Eliminate
- Add slacks for inseparable case

Large margin estimation

- Brute force enumeration
- Min-max formulation
- ‘Plug-in’ linear program for inference

Min-max formulation

Structured loss (Hamming):

Inference

LP Inference

Key step:

discrete optim.

continuous optim.

Outline

- Structured prediction models
- Sequences (CRFs)
- Trees (CFGs)
- Associative Markov networks (Special MRFs)
- Matchings
- Structured large margin estimation
- Margins and structure
- Min-max formulation
- Linear programming inference
- Certificate formulation

Markov Net Inference LP

normalization

agreement

Has integral solutions z for chains, trees

Can be fractional for untriangulated networks

Associative MN Inference LP

“associative”

restriction

- For K=2, solutions are always integral (optimal)
- For K>2, within factor of 2 of optimal

CFG Chart

- CNF tree = set of two types of parts:
- Constituents (A, s, e)
- CF-rules (A B C, s, m, e)

Matching Inference LP

En

vertu

de

les

nouvelles

propositions

,

quel

est

le

coût

prévu

de

perception

de

le

droits

?

k

What

is

the

anticipated

cost

of

collecting

fees

under

the

new

proposal

?

degree

j

Has integral solutions z

LP Duality

- Linear programming duality
- Variables constraints
- Constraints variables
- Optimal values are the same
- When both feasible regions are bounded

Min-max Formulation

LP duality

Min-max formulation summary

- Formulation produces concise QP for
- Low-treewidth Markov networks
- Associative MNs (K=2)
- Context free grammars
- Bipartite matchings
- Approximate for untriangulated MNs, AMNs with K>2

*Taskar et al 04

Factored Primal/Dual

By QP duality

Dual inherits structure from problem-specific inference LP

Variables correspond to a decomposition of variables of the flat case

Duals and Kernels

- Kernel trick works:
- Factored dual
- Local functions (log-potentials) can use kernels

Alternatives: Perceptron

- Simple iterative method
- Unstable for structured output: fewer instances, big updates
- May not converge if non-separable
- Noisy
- Voted / averaged perceptron [Freund & Schapire 99, Collins 02]
- Regularize / reduce variance by aggregating over iterations

Alternatives: Constraint Generation

- Add most violated constraint
- Handles more general loss functions
- Only polynomial # of constraints needed
- Need to re-solve QP many times
- Worst case # of constraints larger than factored

[Collins 02; Altun et al, 03; Tsochantaridis et al, 04]

pixels

quadratic

kernel

cubic

kernel

Handwriting RecognitionLength: ~8 chars

Letter: 16x8 pixels

10-fold Train/Test

5000/50000 letters

600/6000 words

Models:

Multiclass-SVMs*

CRFs

M3 nets

30

better

25

20

Test error (average per-character)

15

10

45% error reduction over linear CRFs

33% error reduction over multiclass SVMs

5

0

MC–SVMs

M^3 nets

CRFs

*Crammer & Singer 01

Hypertext Classification

- WebKB dataset
- Four CS department websites: 1300 pages/3500 links
- Classify each page: faculty, course, student, project, other
- Train on three universities/test on fourth

better

relaxed

dual

53% errorreduction over SVMs

38% error reduction over RMNs

loopy belief propagation

*Taskar et al 02

3D Mapping

Data provided by: Michael Montemerlo & Sebastian Thrun

Laser Range Finder

GPS

IMU

Label: ground, building, tree, shrub

Training: 30 thousand points Testing: 3 million points

Segmentation results

Hand labeled 180K test points

Word Alignment Results

Data: [Hansards – Canadian Parliament]

Features induced on 1 mil unsupervised sentences

Trained on 100 sentences (10,000 edges)

Tested on 350 sentences (35,000 edges)

[Taskar+al 05]

*Error: weighted combination of precision/recall

[Lacoste-Julien+Taskar+al 06]

- Structured prediction models
- Sequences (CRFs)
- Trees (CFGs)
- Associative Markov networks (Special MRFs)
- Matchings
- Structured large margin estimation
- Margins and structure
- Min-max formulation
- Linear programming inference
- Certificate formulation

3

1

4

6

5

Certificate formulation- Non-bipartite matchings:
- O(n3) combinatorial algorithm
- No polynomial-size LP known
- Spanning trees
- No polynomial-size LP known
- Simple certificate of optimality
- Intuition:
- Verifying optimality easier than optimizing
- Compact optimality condition of wrt.

kl

ij

3

1

4

6

5

Certificate for non-bipartite matchingAlternating cycle:

- Every other edge is in matching

Augmenting alternating cycle:

- Score of edges not in matching greater than edges in matching

Negate score of edges not in matching

- Augmenting alternating cycle = negative length alternating cycle

Matching is optimal no negative alternating cycles

Edmonds ‘65

3

1

4

6

5

Certificate for non-bipartite matchingPick any node r as root

= length of shortest alternating

path from r to j

Triangle inequality:

Theorem:

No negative length cycle distance function d exists

Can be expressed as linear constraints:

O(n) distance variables, O(n2) constraints

Certificate formulation

- Formulation produces compact QP for
- Spanning trees
- Non-bipartite matchings
- Any problem with compact optimality condition

*Taskar et al. ‘05

Disulfide Bonding Prediction

Data [Swiss Prot 39]

- 450 sequences (4-10 cysteines)
- Features:
- windows around C-C pair
- physical/chemical properties

C C CC C C C C C C

AVITGA ERDLQ GKGT AVSLWIKSVRV TPVGTSGED HPASHKIPFSGQRMHHT P APNLA VQTSPKKFK LSK

*Accuracy: % proteins with all correct bonds

[Taskar+al 05]

Formulation summary

- Brute force enumeration
- Min-max formulation
- ‘Plug-in’ convex program for inference
- Certificate formulation
- Directly guarantee optimality of

Omissions

- Kernels
- Non-parametric models
- Structured generalization bounds
- Bounds on hamming loss
- Scalable algorithms (no QP solver needed)
- Structured SMO (works for chains, trees)[Taskar 04]
- Structured ExpGrad (works for chains, trees)[Bartlett+al 04]
- Structured ExtraGrad (works for matchings, AMNs)[Taskar+al 06]

Open questions

- Statistical consistency
- Hinge loss not consistent for non-binary output
- [See Tewari & Bartlett 05, McAllester 07]
- Learning with approximate inference
- Does constant factor approximate inference guarantee anything about learning?
- No [See Kulesza & Pereira 07]
- Perhaps other assumptions needed
- Discriminative structure learning
- Using sparsifying priors

Conclusion

- Two general techniques for structured large-margin estimation
- Exact, compact, convex formulations
- Allow efficient use of kernels
- Tractable when other estimation methods are not
- Efficient learning algorithms
- Empirical success on many domains

References

Y. Altun, I. Tsochantaridis, and T. Hofmann. Hidden Markov support vector machines. ICML03.

M. Collins. Discriminative training methods for hidden Markov models: Theory and experiments with perceptron algorithms. EMNLP02

K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based vector machines. JMLR01

J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. ICML04

- More papers at http://www.cis.upenn.edu/~taskar

Modeling First Order Effects

- QAP NP-complete
- Sentences (30 words, 1k vars) few seconds (Mosek)
- Learning: use LP relaxation
- Testing: using LP, 83.5% sentences, 99.85% edges integral

Segmentation Model Min-Cut

Computing is hard in general, but

if edge potentials attractive min-cut algorithm

Multiway-cut for multiclass case use LP relaxation

Local evidence

0

1

Spatial smoothness

[Greig+al 89, Boykov+al 99, Kolmogorov & Zabih 02, Taskar+al 04]

Scalable Algorithms

- Batch and online
- Linear in the size of the data
- Iterate until convergence
- For each example in the training sample
- Run inference using current parameters (varies by method)
- Online: Update parameters using computed example values
- Batch: Update parameters using computed sample values

- Structured SMO (Taskar et al, 03; Taskar 04)
- Structured Exponentiated Gradient (Bartlett et al, 04)
- Structured Extragradient (Taskar et al, 05)

Experimental Setup

- Standard Penn treebank split (2-21/22/23)
- Generative baselines
- Klein & Manning 03 and Collins 99
- Discriminative
- Basic = max-margin version of K&M 03
- Lexical & Lexical + Aux
- Lexical features (on constituent parts only)

ts-1 [ts … te]te+1 predicted tags

xs-1 [xs … xe]xe+1

- Auxillary features
- Flat classifier using same features
- Prediction of K&M 03 on each span

Example

The Egyptian president said he would visit

Libya today to resume the talks.

Generative model: Libya todayis base NP

Lexical model: today is a one word constituent

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