1. Curriculum Learning for Latent Structural SVM
(under submission)
M. Pawan Kumar
Benjamin Packer
Daphne Koller

2. Aim Input x Output y  Y Hidden Variable h  H
To learn accurate parameters for latent structural SVM
Input x
Output y  Y
Hidden Variable
h  H
“Deer”
Y = {“Bison”, “Deer”, ”Elephant”, “Giraffe”, “Llama”, “Rhino” }

3. Aim (y*,h*) = maxyY,hH wT(x,y,h) Feature (x,y,h) (HOG, BoW)
To learn accurate parameters for latent structural SVM
Feature (x,y,h)
(HOG, BoW)
Parameters w
(y*,h*) = maxyY,hH wT(x,y,h)

4. Motivation FAILURE … BAD LOCAL MINIMUM Real Numbers Imaginary Numbers
Math is for
losers !!
Real
Numbers
Imaginary
Numbers
eiπ+1 = 0
FAILURE … BAD LOCAL MINIMUM

5. Motivation SUCCESS … GOOD LOCAL MINIMUM Real Numbers Imaginary Numbers
Euler was
a Genius!!
Real
Numbers
Imaginary
Numbers
eiπ+1 = 0
SUCCESS … GOOD LOCAL MINIMUM
Curriculum Learning: Bengio et al, ICML 2009

6. Motivation Simultaneously estimate easiness and parameters
Start with “easy” examples, then consider “hard” ones
Simultaneously estimate easiness and parameters
Easiness is property of data sets, not single instances
Easy vs. Hard
Expensive
Easy for human
 Easy for machine

7. Outline Latent Structural SVM Concave-Convex Procedure
Curriculum Learning
Experiments

8. Latent Structural SVM Training samples xi Ground-truth label yi
Felzenszwalb et al, 2008, Yu and Joachims, 2009
Training samples xi
Ground-truth label yi
Loss Function
(yi, yi(w), hi(w))

9. (yi(w),hi(w)) = maxyY,hH wT(x,y,h)
Latent Structural SVM
(yi(w),hi(w)) = maxyY,hH wT(x,y,h)
min ||w||2 + C∑i(yi, yi(w), hi(w))
Non-convex Objective
Minimize an upper bound

10. Latent Structural SVM (yi(w),hi(w)) = maxyY,hH wT(x,y,h)
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

11. Outline Latent Structural SVM Concave-Convex Procedure
Curriculum Learning
Experiments

12. Concave-Convex Procedure
Start with an initial estimate w0
Update
hi = maxhH wtT(xi,yi,h)
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 12

13. Concave-Convex Procedure
Looks at all samples simultaneously
“Hard” samples will cause confusion
Start with “easy” samples, then consider “hard” ones 13

14. Outline Latent Structural SVM Concave-Convex Procedure
Curriculum Learning
Experiments

15. Curriculum Learning REMINDER
Simultaneously estimate easiness and parameters
Easiness is property of data sets, not single instances 15

16. wT(xi,yi,hi) - wT(xi,y,h)
Curriculum Learning
Start with an initial estimate w0
Update
hi = maxhH wtT(xi,yi,h)
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 16

17. wT(xi,yi,hi) - wT(xi,y,h)
Curriculum Learning
min ||w||2 + C∑i i
wT(xi,yi,hi) - wT(xi,y,h)
≥ (yi, y, h) - i 17

18. wT(xi,yi,hi) - wT(xi,y,h)
Curriculum Learning
vi  {0,1}
min ||w||2 + C∑i vii
wT(xi,yi,hi) - wT(xi,y,h)
≥ (yi, y, h) - i
Trivial Solution 18

19. Curriculum Learning min ||w||2 + C∑i vii - ∑ivi/K
wT(xi,yi,hi) - wT(xi,y,h)
≥ (yi, y, h) - i
Large K
Medium K
Small K 19

20. Curriculum Learning min ||w||2 + C∑i vii - ∑ivi/K
Biconvex
Problem
vi  [0,1]
min ||w||2 + C∑i vii - ∑ivi/K
wT(xi,yi,hi) - wT(xi,y,h)
≥ (yi, y, h) - i
Large K
Medium K
Small K 20

21. Curriculum Learning hi = maxhH wtT(xi,yi,h)
Start with an initial estimate w0
hi = maxhH wtT(xi,yi,h)
Update
Update wt+1 by solving a convex problem
min ||w||2 + C∑i vii - ∑i vi/K
wT(xi,yi,hi) - wT(xi,y,h)
≥ (yi, y, h) - i
Decrease K  K/ 21

22. Outline Latent Structural SVM Concave-Convex Procedure
Curriculum Learning
Experiments

23. Object Detection Input x - Image Output y  Y Latent h - Box
 - 0/1 Loss
Y = {“Bison”, “Deer”, ”Elephant”, “Giraffe”, “Llama”, “Rhino” }
Feature (x,y,h) - HOG

24. Object Detection Mammals Dataset 271 images, 6 classes
90/10 train/test split
5 folds

25. Object Detection
CCCP
Curriculum

26. Object Detection
CCCP
Curriculum

27. Object Detection
CCCP
Curriculum

28. Object Detection
CCCP
Curriculum

29. Object Detection
Objective value
Test error

30. Handwritten Digit Recognition
Input x - Image
Output y  Y
Latent h - Rotation
 - 0/1 Loss
MNIST Dataset
Y = {0, 1, … , 9}
Feature (x,y,h) - PCA + Projection

31. Handwritten Digit Recognition
- Significant Difference

32. Handwritten Digit Recognition
- Significant Difference

33. Handwritten Digit Recognition
- Significant Difference

34. Handwritten Digit Recognition
- Significant Difference

35. Feature (x,y,h) - Ng and Cardie, ACL 2002
Motif Finding
Input x - DNA Sequence
Output y  Y
Y = {0, 1}
Latent h - Motif Location
 - 0/1 Loss
Feature (x,y,h) - Ng and Cardie, ACL 2002

36. Motif Finding UniProbe Dataset 40,000 sequences 50/50 train/test split
5 folds

37. Motif Finding
Average Hamming Distance of Inferred Motifs

38. Motif Finding
Objective Value

39. Motif Finding
Test Error

40. Noun Phrase Coreference
Input x - Nouns
Output y - Clustering
Latent h - Spanning Forest over Nouns
Feature (x,y,h) - Yu and Joachims, ICML 2009

41. Noun Phrase Coreference
MUC6 Dataset
60 documents
50/50 train/test split
1 predefined fold

42. Noun Phrase Coreference
MITRE
Loss
Pairwise
Loss
- Significant Improvement
- Significant Decrement

43. Noun Phrase Coreference
MITRE
Loss
Pairwise
Loss

44. Noun Phrase Coreference
MITRE
Loss
Pairwise
Loss

45. 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