1. Matrix Factorization Bamshad Mobasher DePaul University Bamshad Mobasher DePaul University

2. The $1 Million Question 2

3. Ratings Data 134 355 455 3 3 222 5 211 3 3 1 17,700 movies 480,000 users

4. Training Data  100 million ratings (matrix is 99% sparse)  Rating = [user, movie-id, time-stamp, rating value]  Generated by users between Oct 1998 and Dec 2005  Users randomly chosen among set with at least 20 ratings  Small perturbations to help with anonymity 4

5. Ratings Data 134 355 455 3 3 2?? ? 21? 3 ? 1 Test Data Set (most recent ratings) 480,000 users 17,700 movies

6. Scoring  Minimize root mean square error  Does not necessarily correlate well with user satisfaction  But is a widely-used well-understood quantitative measure  RMSE Baseline Scores on Test Data  1.054 - just predict the mean user rating for each movie  0.953 - Netflix’s own system (Cinematch) as of 2006  0.941 - nearest-neighbor method using correlation  0.857 - required 10% reduction to win $1 million 6 Mean square error = 1/|R|  (u,i) e R ( r ui - r ui ) 2 ^

7. Matrix Factorization of Ratings Data  Based on the idea of Latent Factor Analysis  Identify latent (unobserved) factors that “explain” observations in the data  In this case, observations are user ratings of movies  The factors may represent combinations of features or characteristics of movies and users that result in the ratings 7 R Q P m users n movies m users n movies f f ~ ~ x r ui q T i p u ~ ~

8. Matrix Factorization of Ratings Data 8 Figure from Koren, Bell, Volinksy, IEEE Computer, 2009

9. Matrix Factorization of Ratings Data 9 Credit: Alex Lin, Intelligent Mining

10. Predictions as Filling Missing Data Credit: Alex Lin, Intelligent Mining

11. Learning Factor Matrices  Need to learn the feature vectors from training data  User feature vector: (a, b, c)  Item feature vector (x, y, z)  Approach: Minimize the errors on known ratings Credit: Alex Lin, Intelligent Mining

12. Learning Factor Matrices min q,p     u,i)  R ( r ui - q t i p u ) 2 r ui q t i p u ~ ~ min q,p     u,i)  R ( r ui - q t i p u ) 2 + ( |q i | 2 + |p u | 2 ) Add regularization 12

13. Stochastic Gradient Descent (SGD)  ui = r ui - q t i p u q i  q i +   ui p u - q i ) p u  p u +   ui q i - p u ) min q,p     u,i)  R ( r ui - q t i p u ) 2 + ( |q i | 2 + |p u | 2 ) regularization goodness of fit Online (“stochastic”) gradient update equations: 13

14. Components of a Rating Predictor user-movie interactionmovie biasuser bias User-movie interaction  Characterizes the matching between users and movies  Attracts most research in the field  Benefits from algorithmic and mathematical innovations Baseline predictor Separates users and movies Often overlooked Benefits from insights into users’ behavior Among the main practical contributions of the competition 14 Credit: Yehuda Koren, Google, Inc.

15. Modeling Systematic Biases r ui  + b u + b i + user-movie interactions ~ ~ overall mean rating mean rating for user u mean rating for movie i Example: Mean rating  = 3.7 You are a critical reviewer: your ratings are 1 lower than the mean -> b u = -1 Star Wars gets a mean rating of 0.5 higher than average movie: b i = + 0.5 Predicted rating for you on Star Wars = 3.7 - 1 + 0.5 = 3.2 q t i p u 15 Credit: Padhraic Smyth, University of California, Irvine

16. Objective Function min q,p     u,i)  R ( r ui - (  + b u + b i + q t i p u ) ) 2 + ( |q i | 2 + |p u | 2 + |b u | 2 + |b i | 2 ) } regularization goodness of fit Typically selected via grid-search on a validation set 16 Credit: Padhraic Smyth, University of California, Irvine

17. 5% 8% 17 Figure from Koren, Bell, Volinksy, IEEE Computer, 2009

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19. Explanation for increase? 19

20. Adding Time Effects r ui  + b u + b i + user-movie interactions ~ ~ ~ ~ r ui  + b u (t) + b i (t) + user-movie interactions Add time dependence to biases Time-dependence parametrized by linear trends, binning, and other methods For details see Y. Koren, Collaborative filtering with temporal dynamics, ACM SIGKDD Conference 2009 20 Credit: Padhraic Smyth, University of California, Irvine

21. Adding Time Effects r ui  + b u (t) + b i (t) + q t i p u (t) ~ ~ Add time dependence to user “factor weights” Models the fact that user’s interests over “genres” (the q’s) may change over time 21

22. Figure from Koren, Bell, Volinksy, IEEE Computer, 2009 5% 8% 22

23. The Kitchen Sink Approach….  Many options for modeling  Variants of the ideas we have seen so far  Different numbers of factors  Different ways to model time  Different ways to handle implicit information  ….  Other models (not described here)  Nearest-neighbor models  Restricted Boltzmann machines  Model averaging was useful….  Linear model combining  Neural network combining  Gradient boosted decision tree combining  Note: combining weights learned on validation set (“stacking”) 23 Credit: Padhraic Smyth, University of California, Irvine

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25. Other Aspects of Model Building  Automated parameter tuning  Using a validation set, and grid search, various parameters such as learning rates, regularization parameters, etc., can be optimized  Memory requirements  Memory: can fit within roughly 1 Gbyte of RAM  Training time  Order of days: but achievable on commodity hardware rather than a supercomputer  Some parallelization used 25 Credit: Padhraic Smyth, University of California, Irvine

26. Progress Prize 2008 Sept 2 nd  Only 3 teams qualify for 1% improvement over previous year Oct 2 nd  Leading team has 9.4% overall improvement Progress prize ($50,000) awarded to BellKor team of 3 AT&T researchers (same as before) plus 2 Austrian graduate students, Andreas Toscher and Martin Jahrer Key winning strategy: clever “blending” of predictions from models used by both teams Speculation that 10% would be attained by mid-2009 26

27. The Leading Team for the Final Prize  BellKorPragmaticChaos  BellKor:  Yehuda Koren (now Yahoo!), Bob Bell, Chris Volinsky, AT&T  BigChaos:  Michael Jahrer, Andreas Toscher, 2 grad students from Austria  Pragmatic Theory  Martin Chabert, Martin Piotte, 2 engineers from Montreal (Quebec) 27

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29. June 26 th 2009: after 1000 days & nights… 29

30. Million Dollars Awarded Sept 21 st 2009 30