Dual Decomposition for Parsing with Non-Projective Head ...people.csail.mit.edu/srush/talk2.pdf ·...

Dual Decomposition for Parsingwith Non-Projective Head Automata

Terry Koo, Alexander M. Rush, Michael Collins,David Sontag, and Tommi Jaakkola

The Cost of Model Complexity

We are always looking for better ways to model natural language.

Tradeoff: Richer models ⇒ Harder decoding

Added complexity is both computational and implementational.

Tasks with challenging decoding problems:

I Speech Recognition

I Sequence Modeling (e.g. extensions to HMM/CRF)

I Parsing

I Machine Translation

y∗ = arg maxy

f (y) Decoding

I Parsing

y∗ = arg maxy

f (y) Decoding

I Parsing

y∗ = arg maxy

f (y) Decoding

Non-Projective Dependency Parsing

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

Important problem in many languages.

Problem is NP-Hard for all but the simplest models.

Dual Decomposition

A classical technique for constructing decoding algorithms.

Solve complicated models

y∗ = arg maxy

by decomposing into smaller problems.

Upshot: Can utilize a toolbox of combinatorial algorithms.

I Dynamic programming

I Minimum spanning tree

I Shortest path

I Min-Cut

A Dual Decomposition Algorithmfor Non-Projective Dependency Parsing

Simple - Uses basic combinatorial algorithms

Efficient - Faster than previously proposed algorithms

Strong Guarantees - Gives a certificate of optimality when exact

Solves 98% of examples exactly, even though the problem isNP-Hard

Widely Applicable - Similar techniques extend to other problems

Roadmap

Algorithm

Experiments

Derivation

Non-Projective Dependency Parsing

I Starts at the root symbol *

I Each word has a exactly one parent word

I Produces a tree structure (no cycles)

I Dependencies can cross

Algorithm Outline

Arc-Factored Model

Dual Decomposition

Sibling Model

Algorithm Outline

Arc-Factored Model

Dual Decomposition

Sibling Model

Arc-Factored

f (y) =

score(head =∗0,mod =saw2) +score(saw2, John1)

+score(saw2,movie4) +score(saw2, today5)

+score(movie4, a3) + ...

e.g. score(∗0, saw2) = log p(saw2|∗0) (generative model)

or score(∗0, saw2) = w · φ(saw2, ∗0) (CRF/perceptron model)

y∗ = arg maxy

f (y) ⇐ Minimum Spanning Tree Algorithm

Arc-Factored

f (y) = score(head =∗0,mod =saw2)

+score(saw2, John1)

y∗ = arg maxy

Arc-Factored

f (y) = score(head =∗0,mod =saw2) +score(saw2, John1)

y∗ = arg maxy

Arc-Factored

+score(saw2,movie4)

+score(saw2, today5)

y∗ = arg maxy

Arc-Factored

y∗ = arg maxy

Arc-Factored

y∗ = arg maxy

Arc-Factored

y∗ = arg maxy

Arc-Factored

y∗ = arg maxy

Arc-Factored

y∗ = arg maxy

Sibling Models

f (y) =

score(head = ∗0, prev = NULL,mod = saw2)

+score(saw2,NULL, John1) +score(saw2,NULL,movie4)

+score(saw2,movie4, today5) + ...

e.g. score(saw2,movie4, today5) = log p(today5|saw2,movie4)

or score(saw2,movie4, today5) = w · φ(saw2,movie4, today5)

y∗ = arg maxy

f (y) ⇐ NP-Hard

Sibling Models

f (y) = score(head = ∗0, prev = NULL,mod = saw2)

y∗ = arg maxy

f (y) ⇐ NP-Hard

Sibling Models

+score(saw2,NULL, John1)

+score(saw2,NULL,movie4)

y∗ = arg maxy

f (y) ⇐ NP-Hard

Sibling Models

y∗ = arg maxy

f (y) ⇐ NP-Hard

Sibling Models

y∗ = arg maxy

f (y) ⇐ NP-Hard

Sibling Models

y∗ = arg maxy

f (y) ⇐ NP-Hard

Sibling Models

y∗ = arg maxy

f (y) ⇐ NP-Hard

Sibling Models

y∗ = arg maxy

f (y) ⇐ NP-Hard

Thought Experiment: Individual Decoding

score(saw2,NULL, John1) + score(saw2,NULL,movie4)+score(saw2,movie4, today5)

score(saw2,NULL, John1) + score(saw2,NULL, that6)

score(saw2,NULL, a3) + score(saw2, a3,he7)

2n−1

possibilities

Under Sibling Model, can solve for each word with Viterbi decoding.

2n−1

possibilities

2n−1

possibilities

2n−1

possibilities

2n−1

possibilities

2n−1

possibilities

Thought Experiment Continued

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

Idea: Do individual decoding for each head word using dynamicprogramming.

If we’re lucky, we’ll end up with a valid final tree.

But we might violate some constraints.

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8*0 John1 saw2 a3 movie4 today5 that6 he7 liked8

Dual Decomposition Idea

NoConstraints

TreeConstraints

Arc-Factored

MinimumSpanning Tree

SiblingModel

IndividualDecoding

Dual Decomposition Idea

NoConstraints

TreeConstraints

Arc-Factored

MinimumSpanning Tree

SiblingModel

IndividualDecoding

DualDecomposition

Dual Decomposition Structure

Goal y∗ = arg maxy∈Y

Rewrite as argmax

z∈ Z, y∈ Yf (z) + g(y)

such that z = y

Valid TreesAll Possible

Sibling Arc-Factored

Constraint

Rewrite as argmax

z∈ Z, y∈ Yf (z) + g(y)

such that z = y

Constraint

Rewrite as argmax

z∈ Z, y∈ Yf (z) + g(y)

such that z = y

Valid Trees

All Possible

Constraint

Rewrite as argmax

z∈ Z, y∈ Yf (z) + g(y)

such that z = y

Constraint

Rewrite as argmax

z∈ Z, y∈ Yf (z) + g(y)

such that z = y

Sibling

Arc-Factored

Constraint

Rewrite as argmax

z∈ Z, y∈ Yf (z) + g(y)

such that z = y

Constraint

Rewrite as argmax

z∈ Z, y∈ Yf (z) + g(y)

such that z = y

Constraint

Algorithm Sketch

Set penalty weights equal to 0 for all edges.

For k = 1 to K

z(k) ← Decode (f (z) + penalty) by Individual Decoding

y (k) ← Decode (g(y)− penalty) by Minimum Spanning Tree

If y (k)(i , j) = z(k)(i , j) for all i , j Return (y (k), z(k))

Algorithm Sketch

For k = 1 to K

Algorithm Sketch

For k = 1 to K

Algorithm Sketch

For k = 1 to K

Algorithm Sketch

For k = 1 to K

Else Update penalty weights based on y (k)(i , j)− z(k)(i , j)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

Minimum Spanning Tree

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Penaltiesu(i , j) = 0 for all i ,j

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

Convergedy∗ = arg max

y∈Yf (y) + g(y)

Keyf (z) ⇐ Sibling Model g(y) ⇐ Arc-Factored ModelZ ⇐ No Constraints Y ⇐ Tree Constraintsy(i , j) = 1 if y contains dependency i , j

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Individual Decoding

z∗ = arg maxz∈Z

(f (z) +∑i ,j

u(i , j)z(i , j))

y∗ = arg maxy∈Y

(g(y)−∑i ,j

u(i , j)y(i , j))

Iteration 1

u(8, 1) -1

u(4, 6) -1

u(2, 6) 1

u(8, 7) 1

Iteration 2

u(8, 1) -1

u(4, 6) -2

u(2, 6) 2

u(8, 7) 1

y∈Yf (y) + g(y)

Guarantees

TheoremIf at any iteration y (k) = z(k), then (y (k), z(k)) is the global

optimum.

In experiments, we find the global optimum on 98% of examples.

If we do not converge to a match, we can still return anapproximate solution (more in the paper).

Guarantees

TheoremIf at any iteration y (k) = z(k), then (y (k), z(k)) is the global

optimum.

In experiments, we find the global optimum on 98% of examples.

If we do not converge to a match, we can still return anapproximate solution (more in the paper).

Extensions

I Grandparent Models

f (y) =...+ score(gp =∗0, head = saw2, prev =movie4,mod =today5)

I Head Automata (Eisner, 2000)

Generalization of Sibling models

Allow arbitrary automata as local scoring function.

Roadmap

Algorithm

Experiments

Derivation

ExperimentsProperties:

I Exactness

I Parsing Speed

I Parsing Accuracy

I Comparison to Individual Decoding

I Comparison to LP/ILP

Training:I Averaged Perceptron (more details in paper)

Experiments on:

I CoNLL Datasets

I English Penn Treebank

I Czech Dependency Treebank

How often do we exactly solve the problem?

CzeEng

DanDut

PorSlo

SweTur

I Percentage of examples where the dual decomposition findsan exact solution.

Parsing Speed

CzeEng

DanDut

PorSlo

SweTur

Sibling model

CzeEng

DanDut

PorSlo

SweTur

Grandparent model

I Number of sentences parsed per second

I Comparable to dynamic programming for projective parsing

Accuracy

Arc-Factored Prev Best Grandparent

Dan 89.7 91.5 91.8Dut 82.3 85.6 85.8Por 90.7 92.1 93.0Slo 82.4 85.6 86.2Swe 88.9 90.6 91.4Tur 75.7 76.4 77.6Eng 90.1 — 92.5Cze 84.4 — 87.3

Prev Best - Best reported results for CoNLL-X data set, includes

I Approximate search (McDonald and Pereira, 2006)

I Loop belief propagation (Smith and Eisner, 2008)

I (Integer) Linear Programming (Martins et al., 2009)

Comparison to Subproblems

IndividualMSTDual

F1 for dependency accuracy

Comparison to LP/ILP

Martins et al.(2009): Proposes two representations ofnon-projective dependency parsing as a linear programmingrelaxation as well as an exact ILP.

I LP (1)I LP (2)I ILP

Use an LP/ILP Solver for decoding

We compare:

I AccuracyI ExactnessI Speed

Both LP and dual decomposition methods use the same model,features, and weights w .

Comparison to LP/ILP: Accuracy

100LP(1)LP(2)

ILPDual

Dependency Accuracy

I All decoding methods have comparable accuracy

Comparison to LP/ILP: Exactness and Speed

100LP(1)LP(2)

ILPDual

Percentage with exact solution

14LP(1)LP(2)

ILPDual

Sentences per second

Roadmap

Algorithm

Experiments

Derivation

Deriving the Algorithm

Goal:y∗ = arg max

y∈Yf (y)

Rewrite:arg max

z∈Z,y∈Yf (z) + g(y)

s.t. z(i , j) = y(i , j) for all i , j

Lagrangian: L(u, y , z) = f (z) + g(y) +∑i,j

u(i , j) (z(i , j)− y(i , j))

The dual problem is to find minu

L(u) where

L(u) = maxy∈Y,z∈Z

L(u, y , z) = maxz∈Z

f (z) +∑i,j

u(i , j)z(i , j)

g(y)−∑i,j

u(i , j)y(i , j)

Dual is an upper bound: L(u) ≥ f (z∗) + g(y∗) for any u

Deriving the Algorithm

Goal:y∗ = arg max

y∈Yf (y)

Rewrite:arg max

z∈Z,y∈Yf (z) + g(y)

s.t. z(i , j) = y(i , j) for all i , j

Lagrangian: L(u, y , z) = f (z) + g(y) +∑i,j

u(i , j) (z(i , j)− y(i , j))

The dual problem is to find minu

L(u) where

L(u) = maxy∈Y,z∈Z

L(u, y , z) = maxz∈Z

f (z) +∑i,j

u(i , j)z(i , j)

g(y)−∑i,j

u(i , j)y(i , j)

Dual is an upper bound: L(u) ≥ f (z∗) + g(y∗) for any u

A Subgradient Algorithm for Minimizing L(u)

L(u) = maxz∈Z

f (z) +∑i,j

u(i , j)z(i , j)

+ maxy∈Y

g(y)−∑i,j

u(i , j)y(i , j)

L(u) is convex, but not differentiable. A subgradient of L(u) at uis a vector gu such that for all v ,

L(v) ≥ L(u) + gu · (v − u)

Subgradient methods use updates u′ = u − αgu

In fact, for our L(u), gu(i , j) = z∗(i , j)− y∗(i , j)

Related Work

I Methods that use general purpose linear programming orinteger linear programming solvers (Martins et al. 2009;Riedel and Clarke 2006; Roth and Yih 2005)

I Dual decomposition/Lagrangian relaxation in combinatorialoptimization (Dantzig and Wolfe, 1960; Held and Karp, 1970;Fisher 1981)

I Dual decomposition for inference in MRFs (Komodakis et al.,2007; Wainwright et al., 2005)

I Methods that incorporate combinatorial solvers within loopybelief propagation (Duchi et al. 2007; Smith and Eisner 2008)

Summary

y∗ = arg maxy

f (y) ⇐ NP-Hard

Arc-Factored Model

Dual Decomposition

Sibling Model

Summary

y∗ = arg maxy

f (y) ⇐ NP-Hard

Arc-Factored Model

Dual Decomposition

Sibling Model

Other Applications

I Dual decomposition can be applied to other decodingproblems.

I Rush et al. (2010) focuses on integrated dynamicprogramming algorithms.

I Integrated Parsing and Tagging

I Integrated Constituency and Dependency Parsing

Parsing and Tagging

y∗ = arg maxy

f (y) ⇐ Slow

S N V D N A D N V

HMM Model

Dual Decomposition

CFG Model

Parsing and Tagging

y∗ = arg maxy

f (y) ⇐ Slow

S N V D N A D N V

HMM Model

Dual Decomposition

CFG Model

Dependency and Constituency

y∗ = arg maxy

f (y) ⇐ Slow

Dependency Model

Dual Decomposition

Lexicalized CFG

Dependency and Constituency

y∗ = arg maxy

f (y) ⇐ Slow

Dependency Model

Dual Decomposition

Lexicalized CFG

Future Directions

There is much more to explore around dual decomposition in NLP.

I Known TechniquesI Generalization to more than two modelsI K-best decodingI Approximate subgradientI Heuristic for branch-and-bound type search

I Possible NLP ApplicationsI Machine TranslationI Speech RecognitionI “Loopy” Sequence Models

I Open QuestionsI Can we speed up subalgorithms when running repeatedly?I What are the trade-offs of different decompositions?I Are there better methods for optimizing the dual?

Appendix

Training the Model

f (y) = ... + score(saw2,movie4, today5) + ...

I score(saw2,movie4, today5) = w · φ(saw2,movie4, today5)

I Weight vector w trained using Averaged perceptron.

I (More details in the paper.)

Early Stopping

0 200 400 600 800 1000

Maximum Number of Dual Decomposition Iterations

% validation UAS% certificates

% match K=5000

Early Stopping

Caching

0 1000 2000 3000 4000 5000

Iterations of Dual Decomposition

% recomputed, g+s% recomputed, sib

Caching speed

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