Collaborative Ordinal Regression

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Collaborative Ordinal Regression

• Shipeng Yu
• Joint work with Kai Yu, Volker Tresp
• and Hans-Peter Kriegel
• University of Munich, Germany
• Siemens Corporate Technology
• shipeng.yu_at_gmail.com

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Motivations
Features
Ratings
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• Superman
• The Pianist
• Star Wars
• The Matrix
• The Godfather
• American Beauty

Genre
Actors
Very Dislike
Very Like
Directors
Descriptions



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Ordinal Regression
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Motivations (Cont.)
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• Superman
• The Pianist
• Star Wars
• The Matrix
• The Godfather
• American Beauty

Features
Ratings



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Collaborative Ordinal Regression
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Outline

• Motivations
• Ranking Problem
• Bayesian Framework for Ordinal Regression
• Collaborative Ordinal Regression
• Learning and Inference
• Experiments
• Conclusion and Extensions

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Ranking Problem

• Goal Assign ranks to objects
• Different from classification/regression problem
• Binary classification Has only 2 labels
• Multi-class classification Ignores ordering
  property
• Regression Only deals with real outputs

Ordinal Regression
Preference Learning
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Ordinal Regression

• Goal Assign ordered labels to objects
• Applications
• User preference prediction
• Web ranking for search engines

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One-task vs Multi-task

• Common in real world problems
• Collaborative filtering preference learning for
  multiple users
• Web ranking ranking of web pages for different
  queries
• Question How to learn related ranking tasks
  jointly?

Different ranking functions are correlated
Each function only ranked part of data
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Outline

• Motivations
• Ranking Problem
• Bayesian Framework for Ordinal Regression
• Collaborative Ordinal Regression
• Learning and Inference
• Experiments
• Conclusion and Extensions

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Bayesian Ordinal Regression

• Conditional model on ranking outputs
• Ranking likelihood Conditional on the latent
  function
• Prior Gaussian Process prior for latent function
• Marginal ranking likelihood Integrate out latent
  function values
• Ordinal regression likelihood

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Bayesian Ordinal Regression (1)

• Need to define ranking likelihood
• Example Model (1)
• GP Regression (GPR)
• Assume a Gaussian form
• Regression on the ranking label directly

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Bayesian Ordinal Regression (2)

• Need to define ranking likelihood
• Example Model (2)
• GP Ordinal Regression (GPOR) (Chu Ghahramani,
  2005)
• A probit ranking likelihood
• Assign labels based on the surrounding area

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Outline

• Motivations
• Ranking Problem
• Bayesian Framework for Ordinal Regression
• Collaborative Ordinal Regression
• Learning and Inference
• Experiments
• Conclusion and Extensions

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Multi-task Setting?

• Naïve approach 1 Learn a GP model for each task
• No share of information between tasks
• Naïve approach 2 Fit one parametric kernel
  jointly
• The parametric kernel is too restrictive to fit
  all tasks
• The collaborative effects
• Common preferences
• Functions share similar regression labels on
  some items
• Similar variabilities
• Functions tend to have same predictability on
  similar items

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Collaborative Ordinal Regression

• Hierarchical GP model for multi-task ordinal
  regression
• mean function
• model common preferences
• covariance matrix
• model similar variabilities
• Both mean function and
• (non-stationary) covariance
• matrix are learned from data

GP Prior
Ordinal Regression Likelihood
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COR The Model

• Hierarchical Bayes model on functions
• All the latent functions are sampled from the
  same GP prior
• Allow different parameter settings for different
  tasks
• We may only observe part of rank labels for each
  function

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COR The Key Points

• The GP prior connects all ordinal regression
  tasks
• Model the first and second sufficient statistics
• The lower level features are incorporated
  naturally
• More general than pure collaborative filtering
• We dont fix a parametric form for the kernel
• Instead we assign the conjugate prior
• We can make predictions for new input data and
  new tasks

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Toy Problem (GPR Model)
New task prediction with base kernel (RBF)
Mean rank labels
Mean function
New task prediction with learned kernel
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Outline

• Motivations
• Ranking Problem
• Bayesian Framework for Ordinal Regression
• Collaborative Ordinal Regression
• Learning and Inference
• Experiments
• Conclusion and Extensions

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Learning

• Variational lower bound
• EM Learning
• E-step Approximate each posterior as a
  Gaussian
• Estimate the mean vector and covariance matrix
  using EP
• M-step Fix and maximize w.r.t.
  and

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E-step

• The true posterior distribution factorizes
• EP procedures
• Deletion Delete factor from the
  approximated Gaussian
• Moments matching Match moments by adding true
  likelihood
• Update Update the factor
• Can be done analytically for the example models
• For GPR model the EP step is exact

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M-step

• Update GP prior
• Does not depend on the form of ranking likelihood
• The conjugate prior corresponds to a smooth term
• Update likelihood parameter
• Do it separately for each task
• Have the same update equation as the single-task
  case

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Inference

• Ordinal Regression
• Non-stationary kernel on test data is unknown!
• Solution work in the dual space (Yu et al. 2005)
• Posterior
• By constraint , posterior
• For test data we have

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Outline

• Motivations
• Ranking Problem
• Bayesian Framework for Ordinal Regression
• Collaborative Ordinal Regression
• Learning and Inference
• Experiments
• Conclusion and Extensions

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Experiments

• Predict user ratings in movie data
• MovieLens 591 movies, 943 users
• 19 features from the Genre part of each movie
  (binary)
• EachMovie 1,075 movies, 72,916 users
• 23,753 features from online database (TF-IDF)
• Experimental Settings
• Pick up 100 users with the most ratings as
  tasks
• Randomly choose 10, 20, 50 ratings for each user
  for training
• Base kernel cosine similarity

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Comparison Metrics

• Ordinal Regression Evaluation
• Mean absolute error (MAE)
• Mean 0-1 error (MZOE)
• Use Macro Micro average over multiple tasks
• Ranking Evaluation
• Normalized Discounted Cumulative Gain (NDCG)
• NDCG_at_10 Only count the top 10 ranked items

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Results - MovieLens

• N Number of training items for each user
• MMMF Maximum Margin Matrix Factorization (Srebro
  et al 2005)
• State-of-the-art collaborative filtering model

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Results - EachMovie

• N Number of training items for each user
• MMMF Maximum Margin Matrix Factorization (Srebro
  et al 2005)
• State-of-the-art collaborative filtering model

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New Ranking Functions
Test on the rest users for MovieLens
Use different kernels
The more users we use for training, the better
kernel we obtain!
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Observations

• Collaborative models are always better than
  individual models
• We can learn a good non-stationary kernel from
  users
• GPR CGPR are fast in training and robust in
  testing
• Since there is no approximation
• GPOR CGPOR are slow and sometimes overfit
• Due to the numerical M-step
• We can use other ranking likelihood
• Then we may need to do numerical integration in
  EP step

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Outline

• Motivations
• Ranking Problem
• Bayesian Framework for Ordinal Regression
• Collaborative Ordinal Regression
• Learning and Inference
• Experiments
• Conclusion and Extensions

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Conclusion

• A Bayesian framework for multi-task ordinal
  regression
• An efficient EM-EP learning algorithm
• COR is better than individual OR algorithms
• COR is better than pure collaborative filtering
• Experiments show very encouraging results

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Extensions

• The framework is applicable to preference
  learning
• Collaborative version of GP preference learning
  (Chu Ghahramani, 2005)
• A probabilistic version of RankNet (Burges et al.
  2005)
• GP mixture model for multi-task learning
• Assign a Gaussian mixture model to each latent
  function
• Prediction uses a linear combination of learned
  kernels
• Connection to Dirichlet Processes

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Thanks!

• Questions?