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Structured Prediction: A Large Margin Approach

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Structured Prediction:A Large Margin Approach

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

“Don’t worry, Howard. The big questions are multiple choice.”

x

y

brace

Sequential structure

x

y

Spatial structure

x

y

The screen was

a sea of red

Recursive structure

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

AVITGACERDLQCG

KGTCCAVSLWIKSV

RVCTPVGTSGEDCH

PASHKIPFSGQRMH

HTCPCAPNLACVQT

SPKKFKCLSK

Protein: 1IMT

Classify using local information

Ignores correlations & constraints!

b

r

a

c

e

building

tree

shrub

ground

- Use local information
- Exploit correlations

b

r

a

c

e

building

tree

shrub

ground

- 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

Mild assumption:

linear combination

scoring function

space of feasible outputs

a-z

a-z

a-z

a-z

a-z

y

x

*Lafferty et al. 01

a-z

a-z

a-z

a-z

a-z

y

x

*Lafferty et al. 01

Edge features

Point features

spin-images, point height

length of edge, edge orientation

“associative”

restriction

j

yj

jk

yk

#(NP DT NN)

…

#(PP IN NP)

…

#(NN ‘sea’)

- 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

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

2

3

1

4

6

5

RSCCPCYWGGCPWGQNCYPEGCSGPKV

1 2 3 4 5 6

RSCCPCYWGGCPWGQNCYPEGCSGPKV

1 2 3 4 5 6

- amino acid identities
- phys/chem properties

Mild assumptions:

linear combination

sum of part scores

scoring function

space of feasible outputs

Model:

Prediction

Learning

Data

Estimatew

Example:

Weighted matching

Generally:

Combinatorialoptimization

Local

(ignores

structure)

Margin

Likelihood

(intractable)

- 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

- We want:
- Equivalently:

“brace”

“brace”

“aaaaa”

“brace”

“aaaab”

a lot!

…

“brace”

“zzzzz”

S

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

- 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’

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

1

2

3

- 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

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

b 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’

- Given training examples , we want:

- Maximize margin
- Mistake weighted margin:

# of mistakes in y

*Collins 02, Altun et al 03, Taskar 03

- Eliminate
- Add slacks for inseparable case

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

Structured loss (Hamming):

Inference

LP Inference

Key step:

discrete optim.

continuous optim.

- 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

normalization

agreement

Has integral solutions z for chains, trees

Can be fractional for untriangulated networks

“associative”

restriction

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

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

root

inside

outside

Has integral solutions z

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

- Linear programming duality
- Variables constraints
- Constraints variables

- Optimal values are the same
- When both feasible regions are bounded

LP duality

- 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

QP duality

Exponentially many constraints/variables

By QP duality

Dual inherits structure from problem-specific inference LP

Variables correspond to a decomposition of variables of the flat case

b c a r e

2

.2

b r o r e

2

.15

b r o c e

.25

1

b r a c e

.4

0

r

c

a

1

1

.65

.8

.6

e

b

c

r

o

.4

.35

.2

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

- 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

- 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]

raw

pixels

quadratic

kernel

cubic

kernel

Length: ~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

- 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

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

Hand labeled 180K test points

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

2

3

1

4

6

5

- 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

2

3

1

4

6

5

Alternating 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

2

3

1

4

6

5

Pick 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

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

*Taskar et al. ‘05

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]

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

- Certificate formulation
- Directly guarantee optimality of

- 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]

- 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

- 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

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

- 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

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]

- 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

- For each example in the training sample

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

- 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

*Trained only on sentences ≤20 words

*Taskar et al 04

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