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Convex Optimization in Machine Learning

This paper explores advanced convex optimization techniques in machine learning, including SDP, SOCP, QCQP, QP, and LP, and their application in solving problems such as MPM and robust MPM. It also discusses the use of kernelization in nonlinear classifiers and compares the results with existing literature, including SVMs.

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Convex Optimization in Machine Learning

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  1. Convex Optimization in Machine Learning MURI MeetingJuly 2002 Gert Lanckriet (gert@eecs.berkeley.edu) L. El Ghaoui, M. Jordan, C. Bhattacharrya, N. Cristianini, P. Bartlett U.C. Berkeley

  2. Convex Optimization in Machine Learning

  3. Advanced Convex Optimization in Machine Learning SDP SOCP QCQP QP LP

  4. Advanced Convex Optimization in Machine Learning

  5. Linear Programming (LP)

  6. Second Order Cone Programming (SOCP)

  7. Semi-Definite Programming

  8. Advanced Convex Optimization in Machine Learning

  9. MPM: Problem Sketch (1) aT z = b : decision hyperplane

  10. MPM: Problem Sketch (2)

  11. Probability of misclassification… … should be minimized ! … for worst-case class-conditional density… MPM: Problem Sketch (3)

  12. MPM: Main Result (5)

  13. MPM: Geometric Interpretation

  14. Robustness to Estimation Errors: Robust MPM (R-MPM)

  15. Robust MPM (R-MPM)

  16. Robust MPM (R-MPM)

  17. MPM: Convex Optimization to solve the problem Lemma Linear Classifier Convex Optimization: Second Order Cone Program (SOCP) Kernelizing Nonlinear Classifier ) competitive with Quadratic Program (QP) SVMs

  18. MPM: Empirical results a=1–b and TSA (test-set accuracy) of the MPM, compared to BPB (best performance in Breiman's report (Arcing classifiers, 1996)) and SVMs. (averages for 50 random partitions into 90% training and 10% test sets) • Comparable with existing literature, SVMs • a=1-b is indeed smaller than the test-set accuracy in all cases (consistent with b as worst-case bound on probability of misclassification) • Kernelizing leads to more powerfull decision boundaries (alinear decision boundary < anonlinear decision boundary (Gaussian kernel))

  19. Advanced Convex Optimization in Machine Learning

  20. Machine learning Kernel-based machine learning The idea (1)

  21. The idea (2)

  22. training set (labelled) test set (unlabelled) The idea (4)

  23. Hard margin SVM classifiers (3)

  24. Hard margin SVM classifiers (4)

  25. Hard margin SVM classifiers (5) SDP !

  26. Hard margin SVM classifiers (7) training set (labelled) test set (unlabelled) Learning the kernel matrix !

  27. Hard margin SVM classifiers (8) ?

  28. Hard margin SVM classifiers (11) Learning Kernel Matrix with SDP !

  29. Empirical results hard margin SVMs

  30. See also

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