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Loss-based Learning with Weak Supervision

Loss-based Learning with Weak Supervision. M. Pawan Kumar. About the Talk. Methods that use latent structured SVM A little math-y Initial stages. Outline. Latent SSVM Ranking Brain Activation Delays in M/EEG Probabilistic Segmentation of MRI.

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Loss-based Learning with Weak Supervision

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  1. Loss-based Learning with Weak Supervision M. Pawan Kumar

  2. About the Talk • Methods that use latent structured SVM • A little math-y • Initial stages

  3. Outline • Latent SSVM • Ranking • Brain Activation Delays in M/EEG • Probabilistic Segmentation of MRI Andrews et al., NIPS 2001; Smola et al., AISTATS 2005; Felzenszwalb et al., CVPR 2008; Yu and Joachims, ICML 2009

  4. Weakly Supervised Data x Input x h Output y  {-1,+1} Hidden h y = +1

  5. Weakly Supervised Classification x Feature Φ(x,h) h Joint Feature Vector Ψ(x,y,h) y = +1

  6. Weakly Supervised Classification x Feature Φ(x,h) h Joint Feature Vector Φ(x,h) Ψ(x,+1,h) = y = +1 0

  7. Weakly Supervised Classification x Feature Φ(x,h) h Joint Feature Vector 0 Ψ(x,-1,h) = y = +1 Φ(x,h)

  8. Weakly Supervised Classification x Feature Φ(x,h) h Joint Feature Vector Ψ(x,y,h) y = +1 Score f : Ψ(x,y,h)  (-∞, +∞) Optimize score over all possible y and h

  9. Latent SSVM Scoring function wTΨ(x,y,h) Prediction y(w),h(w) = argmaxy,hwTΨ(x,y,h)

  10. Learning Latent SSVM Training data {(xi,yi), i= 1,2,…,n} w* = argminwΣiΔ(yi,yi(w)) Minimize empirical risk specified by loss function Highly non-convex in w Cannot regularize w to prevent overfitting

  11. Learning Latent SSVM Training data {(xi,yi), i= 1,2,…,n} wTΨ(x,yi(w),hi(w)) + Δ(yi,yi(w)) - wTΨ(x,yi(w),hi(w)) ≤ wTΨ(x,yi(w),hi(w)) + Δ(yi,yi(w)) - maxhiwTΨ(x,yi,hi) ≤ maxy,h{wTΨ(x,y,h) + Δ(yi,y)} - maxhiwTΨ(x,yi,hi)

  12. Learning Latent SSVM Training data {(xi,yi), i= 1,2,…,n} minw ||w||2 + C Σiξi wTΨ(xi,y,h) + Δ(yi,y) - maxhiwTΨ(xi,yi,hi)≤ ξi Difference-of-convex program in w Local minimum or saddle point solution (CCCP)

  13. CCCP Start with an initial estimate of w Impute hidden variables Loss independent hi*= argmaxhwTΨ(xi,yi,h) Update w Loss dependent minw ||w||2 + C Σiξi wTΨ(xi,y,h) + Δ(yi,y) - wTΨ(xi,yi,hi*)≤ ξi Repeat until convergence

  14. Recap Scoring function wTΨ(x,y,h) Prediction y(w),h(w) = argmaxy,hwTΨ(x,y,h) Learning minw ||w||2 + C Σiξi wTΨ(xi,y,h) + Δ(yi,y) - maxhiwTΨ(xi,yi,hi)≤ ξi

  15. Outline • Latent SSVM • Ranking • Brain Activation Delays in M/EEG • Probabilistic Segmentation of MRI Joint Work with AseemBehl and C. V. Jawahar

  16. Ranking Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Average Precision = 1

  17. Ranking Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Average Precision = 1 Accuracy = 1 Average Precision = 0.92 Average Precision = 0.81 Accuracy = 0.67

  18. Ranking During testing, AP is frequently used During training, a surrogate loss is used Contradictory to loss-based learning Optimize AP directly

  19. Outline • Latent SSVM • Ranking • Supervised Learning • Weakly Supervised Learning • Latent AP-SVM • Experiments • Brain Activation Delays in M/EEG • Probabilistic Segmentation of MRI Yue, Finley, Radlinski and Joachims, 2007

  20. Supervised Learning - Input P N = {HP,HN} Training images X Bounding boxes H

  21. Supervised Learning - Output Ranking matrix Y +1 if i is better ranked than k Yik = -1 if k is better ranked than i 0 if i and k are ranked equally Optimal ranking Y*

  22. SSVM Formulation Joint feature vector ΣiPΣkNYik (Φ(xi,hi)-Φ(xk,hk)) Ψ(X,Y,{HP,HN}) = |P||N| Scoring function wTΨ(X,Y,{HP,HN})

  23. Prediction using SSVM Y(w) = argmaxYwTΨ(X,Y, {HP,HN}) Sort by value of sample score wTΦ(xi,hi) Same as standard binary SVM

  24. Learning SSVM minw Δ(Y*,Y(w)) Loss = 1 – AP of prediction

  25. Learning SSVM wTΨ(X,Y(w),{HP,HN}) + Δ(Y*,Y(w)) - wTΨ(X,Y(w),{HP,HN})

  26. Learning SSVM wTΨ(X,Y(w),{HP,HN}) + Δ(Y*,Y(w)) - wTΨ(X,Y*,{HP,HN})

  27. Learning SSVM minw ||w||2+ C ξ wTΨ(X,Y,{HP,HN}) + maxY Δ(Y*,Y) - wTΨ(X,Y*,{HP,HN}) ≤ ξ

  28. Learning SSVM minw ||w||2+ C ξ wTΨ(X,Y,{HP,HN}) + maxY Δ(Y*,Y) - wTΨ(X,Y*,{HP,HN}) ≤ ξ Loss Augmented Inference

  29. Loss Augmented Inference Rank 1 Rank 2 Rank 3 Rank positives according to sample scores

  30. Loss Augmented Inference Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Rank negatives according to sample scores

  31. Loss Augmented Inference Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Slide best negative to a higher rank Terminate after considering last negative Continue until score stops increasing Slide next negative to a higher rank Continue until score stops increasing Optimal loss augmented inference

  32. Recap Scoring function wTΨ(X,Y,{HP,HN}) Prediction Y(w) = argmaxYwTΨ(X,Y, {HP,HN}) Learning Using optimal loss augmented inference

  33. Outline • Latent SSVM • Ranking • Supervised Learning • Weakly Supervised Learning • Latent AP-SVM • Experiments • Brain Activation Delays in M/EEG • Probabilistic Segmentation of MRI

  34. Weakly Supervised Learning - Input Training images X

  35. Weakly Supervised Learning - Latent Bounding boxes HP Training images X All bounding boxes in negative images are negative

  36. Intuitive Prediction Procedure Select the best bounding boxes in all images

  37. Intuitive Prediction Procedure Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Rank them according to their sample scores

  38. Weakly Supervised Learning - Output Ranking matrix Y +1 if i is better ranked than k Yik = -1 if k is better ranked than i 0 if i and k are ranked equally Optimal ranking Y*

  39. Latent SSVM Formulation Joint feature vector ΣiPΣkNYik (Φ(xi,hi)-Φ(xk,hk)) Ψ(X,Y,{HP,HN}) = |P||N| Scoring function wTΨ(X,Y,{HP,HN})

  40. Prediction using Latent SSVM maxY,HwTΨ(X,Y, {HP,HN})

  41. Prediction using Latent SSVM maxY,HwTΣiPΣkNYik (Φ(xi,hi)-Φ(xk,hk)) Choose best bounding box for positives Choose worst bounding box for negatives Not what we wanted

  42. Learning Latent SSVM minw Δ(Y*,Y(w)) Loss = 1 – AP of prediction

  43. Learning Latent SSVM wTΨ(X,Y(w),{HP(w),HN(w)}) + Δ(Y*,Y(w)) - wTΨ(X,Y(w),{HP(w),HN(w)})

  44. Learning Latent SSVM wTΨ(X,Y(w),{HP(w),HN(w)}) + Δ(Y*,Y(w)) - wTΨ(X,Y*,{HP,HN}) maxH

  45. Learning Latent SSVM minw ||w||2+ C ξ wTΨ(X,Y,{HP,HN}) + maxY,H Δ(Y*,Y) - wTΨ(X,Y*,{HP,HN}) ≤ ξ maxH

  46. Learning Latent SSVM minw ||w||2+ C ξ wTΨ(X,Y,{HP,HN}) + maxY,H Δ(Y*,Y) - wTΨ(X,Y*,{HP,HN}) ≤ ξ maxH Loss Augmented Inference Cannot be solved optimally

  47. Recap Unintuitive prediction Unintuitive objective function Non-optimal loss augmented inference Can we do better?

  48. Outline • Latent SSVM • Ranking • Supervised Learning • Weakly Supervised Learning • Latent AP-SVM • Experiments • Brain Activation Delays in M/EEG • Probabilistic Segmentation of MRI

  49. Latent AP-SVM Formulation Joint feature vector ΣiPΣkNYik (Φ(xi,hi)-Φ(xk,hk)) Ψ(X,Y,{HP,HN}) = |P||N| Scoring function wTΨ(X,Y,{HP,HN})

  50. Prediction using Latent AP-SSVM Choose best bounding box for all samples hi(w) = argmaxhwTΦ(xi,h) Optimize over the ranking Y(w) = argmaxYwTΨ(X,Y, {HP(w),HN(w)}) Sort by sample scores

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