Modeling Latent Variable Uncertainty for Loss-based Learning

1. Modeling Latent Variable Uncertainty for Loss-based Learning M. Pawan Kumar ÉcoleCentrale Paris Écoledes PontsParisTech INRIA Saclay, Île-de-France Ben Packer Stanford University Daphne Koller Stanford University

2. Aim Accurate learning with weakly supervised data Train Input xi Output yi Input x Bison Deer Elephant Giraffe Output y = “Deer” Latent Variable h Llama Object Detection Rhino

3. Aim Accurate learning with weakly supervised data Input x Feature Ψ(x,y,h) (e.g. HOG) Function f : Ψ(x,y,h) (-∞, +∞) Output y = “Deer” Latent Variable h • (y(f),h(f)) = argmaxy,h f(Ψ(x,y,h)) Prediction

4. Aim Accurate learning with weakly supervised data Input x Feature Ψ(x,y,h) (e.g. HOG) Function f : Ψ(x,y,h) (-∞, +∞) Output y = “Deer” Latent Variable h • f* = argminfObjective(f) Learning

5. Aim Find a suitable objective function to learn f* Input x Feature Ψ(x,y,h) (e.g. HOG) Function f : Ψ(x,y,h) (-∞, +∞) Encourages accurate prediction Output y = “Deer” User-specified criterion for accuracy Latent Variable h • f* = argminfObjective(f) Learning

6. Outline • Previous Methods • Our Framework • Optimization • Results

7. Latent SVM Linear function parameterized by w Prediction (y(w), h(w)) = argmaxy,hwTΨ(x,y,h) Learning minwΣiΔ(yi,yi(w),hi(w)) ✔ Loss based learning ✖ Loss function has a restricted form ✖ Doesn’t model uncertainty in latent variables

8. Expectation Maximization exp(θTΨ(x,y,h)) Joint probability Pθ(y,h|x) = Z Prediction (y(θ), h(θ)) = argmaxy,hθTΨ(x,y,h) Learning maxθΣi Σhilog (Pθ(yi,hi|xi)) ✔ Models uncertainty in latent variables ✖ Doesn’t model accuracy of latent variable prediction ✖ No user-defined loss function

9. Problem Model Uncertainty in Latent Variables Model Accuracy of Latent Variable Predictions

10. Outline • Previous Methods • Our Framework • Optimization • Results

11. Solution Use two different distributions for the two different tasks Model Uncertainty in Latent Variables Model Accuracy of Latent Variable Predictions

12. Solution Use two different distributions for the two different tasks Pθ(hi|yi,xi) hi Model Accuracy of Latent Variable Predictions

13. Solution Use two different distributions for the two different tasks Pθ(hi|yi,xi) hi Pw(yi,hi|xi) (yi,hi) (yi(w),hi(w))

14. The Ideal Case No latent variable uncertainty, correct prediction Pθ(hi|yi,xi) hi hi(w) Pw(yi,hi|xi) (yi,hi(w)) (yi,hi)

15. In Practice Restrictions in the representation power of models Pθ(hi|yi,xi) hi Pw(yi,hi|xi) (yi,hi) (yi(w),hi(w))

16. Our Framework Minimize the dissimilarity between the two distributions Pθ(hi|yi,xi) hi User-defined dissimilarity measure Pw(yi,hi|xi) (yi,hi) (yi(w),hi(w))

17. Our Framework Minimize Rao’s Dissimilarity Coefficient Pθ(hi|yi,xi) hi ΣhΔ(yi,h,yi(w),hi(w))Pθ(h|yi,xi) Pw(yi,hi|xi) (yi,hi) (yi(w),hi(w))

18. Our Framework Minimize Rao’s Dissimilarity Coefficient Pθ(hi|yi,xi) hi Hi(w,θ) - β Σh,h’Δ(yi,h,yi,h’)Pθ(h|yi,xi)Pθ(h’|yi,xi) Pw(yi,hi|xi) (yi,hi) (yi(w),hi(w))

19. Our Framework Minimize Rao’s Dissimilarity Coefficient Pθ(hi|yi,xi) hi Hi(w,θ) - β Hi(θ,θ) - (1-β) Δ(yi(w),hi(w),yi(w),hi(w)) Pw(yi,hi|xi) (yi,hi) (yi(w),hi(w))

20. Our Framework Minimize Rao’s Dissimilarity Coefficient Pθ(hi|yi,xi) hi minw,θ Σi Hi(w,θ) - β Hi(θ,θ) Pw(yi,hi|xi) (yi,hi) (yi(w),hi(w))

21. Outline • Previous Methods • Our Framework • Optimization • Results

22. Optimization minw,θΣiHi(w,θ) - β Hi(θ,θ) Initialize the parameters to w0 and θ0 Repeat until convergence Fix w and optimize θ Fix θ and optimize w End

23. Optimization of θ minθΣiΣhΔ(yi,h,yi(w),hi(w))Pθ(h|yi,xi) - β Hi(θ,θ) Pθ(hi|yi,xi) hi hi(w) Case I: yi(w) = yi

24. Optimization of θ minθΣiΣhΔ(yi,h,yi(w),hi(w))Pθ(h|yi,xi) - β Hi(θ,θ) Pθ(hi|yi,xi) hi hi(w) Case I: yi(w) = yi

25. Optimization of θ minθΣiΣhΔ(yi,h,yi(w),hi(w))Pθ(h|yi,xi) - β Hi(θ,θ) Pθ(hi|yi,xi) hi Case II: yi(w) ≠ yi

26. Optimization of θ minθΣiΣhΔ(yi,h,yi(w),hi(w))Pθ(h|yi,xi) - β Hi(θ,θ) Stochastic subgradient descent Pθ(hi|yi,xi) hi Case II: yi(w) ≠ yi

27. Optimization of w minwΣiΣhΔ(yi,h,yi(w),hi(w))Pθ(h|yi,xi) Expected loss, models uncertainty Form of optimization similar to Latent SVM Concave-Convex Procedure (CCCP) Observation: When Δ is independent of true h, our framework is equivalent to Latent SVM

28. Outline • Previous Methods • Our Framework • Optimization • Results

29. Object Detection Train Input xi Output yi Input x Bison Deer Elephant Output y = “Deer” Latent Variable h Giraffe Mammals Dataset Llama 60/40 Train/Test Split 5 Folds Rhino

30. Results – 0/1 Loss Statistically Significant

31. Results – Overlap Loss

32. Action Detection Train Input xi Output yi Input x Jumping Phoning Playing Instrument Reading Riding Bike Output y = “Using Computer” Riding Horse Running Latent Variable h PASCAL VOC 2011 Taking Photo 60/40 Train/Test Split UsingComputer 5 Folds Walking

33. Results – 0/1 Loss Statistically Significant

34. Results – Overlap Loss Statistically Significant

35. Conclusions • Two separate distributions • Conditional probability of latent variables • Delta distribution for prediction • Generalizes latent SVM • Future work • Large-scale efficient optimization • Distribution over w • New applications

36. Code available at http://cvc.centrale-ponts.fr/personnel/pawan