Structured Prediction, Dual Extragradient

1
Structured Prediction, Dual Extragradient
Bregman Projections

• Ben Taskar
• Simon Lacoste-Julien
• Michael Jordan
• UC Berkeley

2
Handwriting Recognition
x
y
brace
Sequential structure
3
Natural Language Parsing
x
y
The screen was a sea of red
Recursive structure
4
Object Segmentation
x
y
Spatial structure
5
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
6
Linear Structured Models
scoring function
space of feasible outputs

• Assumption
• linear combination of features

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Chain Markov Net (aka CRF)
y
x
Lafferty et al. 01
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Chain Markov Net (aka CRF)
y
x
Lafferty et al. 01
9
CFG Parsing
(NP ? DT NN) (PP ? IN NP) (NN ? sea)
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Bilingual Word Alignment
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 ?

• association
• position
• orthography

k
j
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Ising Models Min Cuts
Point features spin image
1
0
Edge features length, angle

• Find max y via min-cut if edge scores are
  non-negative
• Restrict edge features and
  weights

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Linear Structured Models
scoring function
space of feasible outputs

• Assumptions
• linear combination of features
• sum of part scores

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Learning w

• Training examples
• Probabilistic approach
• Computing Zw(x) can be P-complete
• Tractable models but intractable estimation
• Large margin approach
• Exact and efficient when prediction is tractable

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Alignment Example Loss

• We want
• Structured Loss
• Precision, Recall, F1, Hamming

What is the Quel est le
0 1 2 2
What is the Quel est le
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Alignment Example Constraints

• 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
Exponential number of constraints
What is the Quel est le
What is the Quel est le

What is the Quel est le
What is the Quel est le
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Large Margin Estimation

• Approximation constraint generation/sampling
  Collins02Altun03Tsochantaridis04Joachims0
  5
• Alternative approach
• Hinge loss
• Min-max formulation

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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
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Min-max Formulation
Structured loss (Hamming)
Inference
LP Inference
Key step
discrete optim.
continuous optim.
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y ? z Map for Markov Nets
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Markov Net Inference LP
normalization
agreement
Has integral solutions z for chains, trees Can be
fractional for untriangulated networks
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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
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Saddle-point Problem
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First Try Projected Gradient
Euclidean projection
Can oscillate!
no convergence guarantee
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Dual Extragradient for Structured Prediction
State -- cumulative gradient
Start
Prediction
Correction
Cumulative gradient update
Output
O(1/?) convergence rate
Nesterov03
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for Bipartite Matchings Min Cost Flow
t
s

• All capacities 1
• Min-cost quadratic flow computes projection
• O(N3) complexity for fixed precision (Nnum
  nodes)
• Well-studied problem, free code (Guerreiro
  Tseng02)
• See paper for flow-reduction for min-cuts

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Non-Euclidean Dual Extragradient
d( , ) Bregman divergence
Prediction
Correction
Cumulative dual gradient update
Output
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Bregman Divergence Updates

• Squared distance ? Euclidean projections
• KL-distance ? Multiplicative update
  normalization
• In case of sequences trees, can be computed via
  forward-backward and inside-outside

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Memory-efficient Version

• Required memory
• -- proportional to number of parameters
• -- proportional to number/size of
  examples
• Luckily, we dont need to maintain explicit
• Sufficient to maintain with memory proportional
  to number of parameters
• Similar trick works in Exponentiated Gradient
  Bartlett04 which requires decomposable models

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Experiments

• Word Alignment
• Training data
• 5000 sentences
• 555K edges
• Object Segmentation
• Training data
• 5 scenes
• 37K nodes
• 88K edges
• Compare to averaged perceptron

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(Averaged) Perceptron

• Perceptron for structured output Collins 2002
• For each example ,
• Predict
• Update
• Output averaged parameters

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Matchings
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Min-cuts
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Conclusion

• General technique for structured large-margin
  estimation
• Exact, compact, convex formulations
• Allow efficient use of kernels
• Tractable when other estimation methods are not
• Memory efficient learning algorithms
• See http//www.cs.berkeley.edu/taskar for paper