CoFi Rank : Maximum Margin Matrix Factorization for Collaborative Ranking

1. CoFiRank : 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) • CoFiRank • Linear Convex Upper Bound Alternating optimizing Non-Convex Solved by linear programming

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

8. Useful Links • CoFiRank 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” http://web.mit.edu/dimitrib/www/home.html

10. Questions?

11. Normalized Discounted Cumulative Gain (NDCG)

12. How to set c? • ci is set decreasing,  is maximized with respect toπ for argsort(f) • ci =(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 Method for CoFiRank