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 • Ongoing and Future Work

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)) User-defined loss ✔ Loss based learning ✖ Loss independent of true (unknown) latent variable ✖ 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,hPθ(y,h|x)

9. 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 log (Pθ(yi|xi))

10. 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

11. Outline • Previous Methods • Our Framework • Optimization • Results • Ongoing and Future Work

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

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

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

15. 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))

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

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

18. 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)

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

20. 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))

21. 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))

22. 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))

23. 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))

24. 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))

25. Outline • Previous Methods • Our Framework • Optimization • Results • Ongoing and Future Work

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. Outline • Previous Methods • Our Framework • Optimization • Results • Ongoing and Future Work

33. 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

34. Results – 0/1 Loss Statistically Significant

35. Results – Overlap Loss

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

37. Results – 0/1 Loss Statistically Significant

38. Results – Overlap Loss Statistically Significant

39. Outline • Previous Methods • Our Framework • Optimization • Results • Ongoing and Future Work

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