1. CoFi Rank : Maximum Margin Matrix Factorization for Collaborative Ranking Markus Weimer, Alexandros Karatzoglou, Quoc Viet Le and Alex Smola NIPS’07

2. Idea Maximum Margin Matrix Factorization Structured Estimation for Ranking Bundle Method Solver

3. Collaborative Filtering Based on partial observed matrix to predict unobserved entries

4. Matrix Factorization Low Rank Approximation SVD for fully observed Y Non-convex

5. Maximum Margin Matrix Factorization Trace norm+Hinge loss: Convex Semi-Definite Programming

6. Regularized Matrix Factorization Formulation Probabilistic Matrix Factorization (PMF) CoFi Rank Linear Convex Upper Bound Non-Convex Solved by linear programming Alternating optimizing

7. How to Compute Loss? Linear Convex Upper Bound Solved by Linear Programming Can this explain in simple way?

8. Useful Links CoFi Rank http://www.cofirank.org MMMF http://ttic.uchicago.edu/~nati/mmmf/ MF http://helikoid.si/mf/index.html

9. Famous Researchers in Optimization Yurii Nesterov – “Introductory Lectures on Convex Optimization: A Basic Course” http://www.core.ucl.ac.be/~nesterov/ Arkadi Nemirovski – “Efficient methods in convex programming” http://www2.isye.gatech.edu/~nemirovs/ Stephen P. Boyd – “Convex Optimization” http://www.stanford.edu/~boyd/ Stephen J. Wright – “Numerical Optimization” http://pages.cs.wisc.edu/~swright/ Dimitri Bertsekas – “Nonlinear Programming”Nonlinear Programming http://web.mit.edu/dimitrib/www/home.html

10. Questions?

11. Normalized Discounted Cumulative Gain (NDCG)

12. How to set c ? c i is set decreasing,  is maximized with respect to π for argsort(f) c i =(i+1) -0.25

13. Convex Upper Bound Linear Convex Upper Bound

14. Bundle Method General convex optimization solver with tight convergence bound O(1/  )

15. Bundle MethodBundle Method for CoFi Rank