Curriculum Learning forLatent Structural SVM

M. Pawan Kumar

(under submission)

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Daphne Koller

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Benjamin Packer

AimTo learn accurate parameters for latent structural SVM

Input x

Output y Y

“Deer”

Hidden Variableh H

Y = {“Bison”, “Deer”, ”Elephant”, “Giraffe”, “Llama”, “Rhino” }

AimTo learn accurate parameters for latent structural SVM

Feature (x,y,h)(HOG, BoW)

(y*,h*) = maxyY,hH wT(x,y,h)

Parameters w

Motivation

Real Numbers

Imaginary Numbers

eiπ+1 = 0

Math is forlosers !!

FAILURE … BAD LOCAL MINIMUM

Motivation

Real Numbers

Imaginary Numbers

eiπ+1 = 0

Euler wasa Genius!!

SUCCESS … GOOD LOCAL MINIMUMCurriculum Learning: Bengio et al, ICML 2009

Motivation

Start with “easy” examples, then consider “hard” ones

Easy vs. Hard

Expensive

Easy for human Easy for machine

Simultaneously estimate easiness and parametersEasiness is property of data sets, not single instances

Outline

• Latent Structural SVM

• Concave-Convex Procedure

• Curriculum Learning

• Experiments

Latent Structural SVM

Training samples xi

Ground-truth label yi

Loss Function(yi, yi(w), hi(w))

Felzenszwalb et al, 2008, Yu and Joachims, 2009

Latent Structural SVM

(yi(w),hi(w)) = maxyY,hH wT(x,y,h)

min ||w||2 + C∑i(yi, yi(w), hi(w))

Non-convex Objective

Minimize an upper bound

Latent Structural SVM

min ||w||2 + C∑i i

maxhiwT(xi,yi,hi) - wT(xi,y,h)

≥ (yi, y, h) - i

Still non-convex Difference of convex

CCCP Algorithm - converges to a local minimum

(yi(w),hi(w)) = maxyY,hH wT(x,y,h)

Outline

• Latent Structural SVM

• Concave-Convex Procedure

• Curriculum Learning

• Experiments

Concave-Convex Procedure

Start with an initial estimate w0

Update

Update wt+1 by solving a convex problem

min ||w||2 + C∑i i

wT(xi,yi,hi) - wT(xi,y,h)≥ (yi, y, h) - i

hi = maxhH wtT(xi,yi,h)

Concave-Convex Procedure

Looks at all samples simultaneously

“Hard” samples will cause confusion

Start with “easy” samples, then consider “hard” ones

Outline

• Latent Structural SVM

• Concave-Convex Procedure

• Curriculum Learning

• Experiments

Curriculum Learning

REMINDER

Simultaneously estimate easiness and parametersEasiness is property of data sets, not single instances

Curriculum Learning

Start with an initial estimate w0

Update

Update wt+1 by solving a convex problem

min ||w||2 + C∑i i

wT(xi,yi,hi) - wT(xi,y,h)≥ (yi, y, h) - i

hi = maxhH wtT(xi,yi,h)

Curriculum Learning

min ||w||2 + C∑i i

wT(xi,yi,hi) - wT(xi,y,h)≥ (yi, y, h) - i

Curriculum Learning

min ||w||2 + C∑i vii

wT(xi,yi,hi) - wT(xi,y,h)≥ (yi, y, h) - i

vi {0,1}

Trivial Solution

Curriculum Learning

vi {0,1}

Large K Medium K Small K

min ||w||2 + C∑i vii - ∑ivi/K

wT(xi,yi,hi) - wT(xi,y,h)≥ (yi, y, h) - i

Curriculum Learning

vi [0,1]

min ||w||2 + C∑i vii - ∑ivi/K

wT(xi,yi,hi) - wT(xi,y,h)≥ (yi, y, h) - i

Large K Medium K Small K

BiconvexProblem

Curriculum LearningStart with an initial estimate w0

Update

Update wt+1 by solving a convex problem

min ||w||2 + C∑i vii - ∑i vi/K

wT(xi,yi,hi) - wT(xi,y,h)≥ (yi, y, h) - i

hi = maxhH wtT(xi,yi,h)

Decrease K K/

Outline

• Latent Structural SVM

• Concave-Convex Procedure

• Curriculum Learning

• Experiments

Object Detection

Feature (x,y,h) - HOG

Input x - Image

Output y Y

Latent h - Box

- 0/1 Loss

Y = {“Bison”, “Deer”, ”Elephant”, “Giraffe”, “Llama”, “Rhino” }

Object Detection

271 images, 6 classes

90/10 train/test split

5 folds

Mammals Dataset

Object DetectionCCCP Curriculum

Object DetectionCCCP Curriculum

Object DetectionCCCP Curriculum

Object DetectionCCCP Curriculum

44.14.24.34.44.54.64.74.84.9

Fold 1 Fold 2 Fold 3 Fold 4 Fold 5

Objective value

0

5

10

15

20

25

Fold 1 Fold 2 Fold 3 Fold 4 Fold 5CCCP Curriculum

Test error

Object Detection

Handwritten Digit Recognition

Feature (x,y,h) - PCA + Projection

Input x - Image

Output y Y

Y = {0, 1, … , 9}

Latent h - Rotation

MNIST Dataset

- 0/1 Loss

Handwritten Digit Recognition

- Significant Difference

C

C

C

Handwritten Digit Recognition

- Significant Difference

C

C

C

Handwritten Digit Recognition

- Significant Difference

C

C

C

Handwritten Digit Recognition

- Significant Difference

C

C

C

Motif Finding

Feature (x,y,h) - Ng and Cardie, ACL 2002

Input x - DNA Sequence

Output y Y

Y = {0, 1}

Latent h - Motif Location

- 0/1 Loss

Motif Finding

40,000 sequences

50/50 train/test split

5 folds

UniProbe Dataset

Motif FindingAverage Hamming Distance of Inferred Motifs

Motif Finding

020406080

100120140160

Fold 1 Fold 2 Fold 3 Fold 4 Fold 5

CCCPCurr

Objective Value

Motif Finding

0

10

20

30

40

50

Fold1

Fold2

Fold3

Fold4

Fold5

CCCPCurr

Test Error

Noun Phrase Coreference

Feature (x,y,h) - Yu and Joachims, ICML 2009

Input x - Nouns Output y - Clustering

Latent h - Spanning Forest over Nouns

Noun Phrase Coreference60 documents

50/50 train/test split 1 predefined fold

MUC6 Dataset

Noun Phrase Coreference

- Significant Improvement

- Significant Decrement

MITRELoss

PairwiseLoss

Noun Phrase Coreference

MITRELoss

PairwiseLoss

Noun Phrase Coreference

MITRELoss

PairwiseLoss

Summary

• Automatic Curriculum Learning

• Concave-Biconvex Procedure

• Generalization to other Latent models– Expectation-Maximization– E-step remains the same

– M-step includes indicator variables vi