Binary Principal Component Analysis in the Netflix Collaborative Filtering Task

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Binary Principal Component Analysis in the Netflix Collaborative Filtering Task László Kozma, Alexander Ilin, Tapani Raiko first.last@tkk.fi Helsinki University of Technology Adaptive Informatics Research Center Saarbrücken, 19th October 2009

Introduction Recommender systems Predict user preference in order to offer more relevant items (Amazon.com, Netflix, itunes Genius, Pandora) Content-based approach: use similarity between items (e.g. book text, genre, title, author, actors in a movie, etc.) Collaborative filtering: predict preferences from previous user-item relationships

Collaborative filtering: Netflix prize Netflix data set Training set: 100,480,507 ratings from n = 480,189 users on m = 17,770 movies Probe (test) set: 1,408,395 ratings are given for validation Quiz set: 2,817,131 user/movie pairs with ratings withheld Train, Probe, Quiz sets contain ratings from the same users and movies Ratings from 1 to 5, time of voting provided

Collaborative filtering: Netflix prize Netflix prize competition Score: RMSE on quiz set ratings, goal: 10% better RMSE than Cinematch (Netflix own algorithm trained on same data: 0.9514) Leading solution: A blend of many methods (>100): k-nn, factorization (SVD), restricted Boltzmann machines and many other http://www.netflixprize.com/ Robert M. Bell, Yehuda Koren and Chris Volinsky: The BellKor solution to the Netflix Prize

Collaborative filtering..can be formulated as missing value reconstruction: 5 2 2? 5 X n m = 3 1?? 3 5? n = 480,189 users, m = 17,770 movies, 100,480,507 ratings (over 98% missing)

Singular Value Decomposition Rank-c approximation of ratings-matrix X: X n m A n c S c m, min A,S X AS 2 F Since X is incomplete, the cost function is: (x ij a i s j ) 2 + λ( A 2 F + S 2 F ) (i,j) O The last term is added to avoid overfitting O is the set of observed ratings Unknown ratings are predicted as x ij = a i s j, (i, j) / O Code: http://www.cis.hut.fi/alexilin/software

Logistic PCA Similar to SVD, but we use non-linearity: Method P(y ij = 1) = σ(a i s j ) (1) Sigmoid function: σ(x) = 1 1 + e x (2) To avoid having a bias term, we fix the last row of S to all ones. Then, the elements in the last column of A play the role of the bias term.

Method Binarizing Data Each rating value (1-5) encoded on 4 bits: 1 0000 2 0001 3 0011 4 0111 5 1111 We create 4 binary matrices from the original matrix of ratings Each element tells whether a rating is greater or smaller than a threshold Binarization scheme could be used also for continuous data

Method

Method

Method Logistic PCA P(y ij = 1) = σ(a i s j ) (3) Both A and S are unknown and have to be estimated to fit the available ratings. We express the likelihood as a product of Bernoulli distributions based on Y L = σ(a T i s j ) y ij (1 σ(a i s j )) (1 y ij) ij O O are the observed ratings in Y (4)

Method Regularization Maximum likelihood estimate prone to overfitting We use Gaussian priors for A and S P(s kj ) = N (0, 1) P(a ik ) = N (0, v k ) i = 1,...,m; k = 1,...,c; j = 1,...,n

Method Learning We use MAP-estimation for the model parameters: F = y ij log σ(a T i s j ) ij O + (1 y ij )log(1 σ(a T i s j )) ij O m i=1 k=1 c k=1 j=1 c [ 1 2 log 2πv k + 1 ] a 2 ik 2v k n [ 1 2 log 2π + 1 2 kj] s2 (5)

Learning The derivatives of the log-posterior: F = a i j ij O F = s j i ij O Method s T j (y ij σ(a T i s j )) diag(1/v k )a i (6) a T i (y ij σ(a T i s j )) s j, (7) Gradient-ascent simultaneous update rules: a i a i + α F (8) a i m F s k s k + α (9) n s j

Learning Gradient-ascent update rules: Method a i a i + α F (10) a i m F s k s k + α (11) n s j Scaling Update of the learning rate α: decrease by half if step unsuccessful, increase by 20% if successful Variance-parameters point-estimated to maximize F : v k = 1 d a 2 ik (12) d i=1

Method Prediction We split σ(a T i s j) into X1, X2, X3, X4. The probability of each rating value is then: P x=1 = (1 x 1 )(1 x 2 )(1 x 3 )(1 x 4 ) (13) P x=2 = (1 x 1 )(1 x 2 )(1 x 3 )x 4 (14) P x=3 = (1 x 1 )(1 x 2 )x 3 x 4 (15) P x=4 = (1 x 1 )x 2 x 3 x 4 (16) P x=5 = x 1 x 2 x 3 x 4 (17) We estimate the rating as expectation: ˆx = 1P x=1 + 2P x=2 + 3P x=3 + 4P x=4 + 5P x=5 P x=1 + P x=2 + P x=3 + P x=4 + P x=5. (18)

Experiments: MovieLens # components c Train rms Test rms binary PCA, movies users 10 0.8956 0.9248 binary PCA, users movies 10 0.8449 0.9028 20 0.8413 0.9053 30 0.8577 0.9146 VBPCA, users movies 10 0.7674 0.8905 20 0.7706 0.8892 30 0.7696 0.8880 Table: Performance obtained for MovieLens 100K data

Experiments: MovieLens Predictions on the test set with PCA (x-axis) and logistic PCA (y-axis):

Experiments: Netflix Method Probe rms Quiz subset VBPCA, 50 components 0.9055 0.9070 BinPCA, 20 components 0.9381 0.9392 Blend 0.9046 0.9062 Table: Performance obtained for Netflix data Training and test error: 0.5 0.45 0.4 0.35 PCAbin(20) PCAbin(30) PCAbin(45) 0.3 0 1 2 4 8 16 32 64 128 0.5 0.45 0.4 0.35 0.3 0 1 2 4 8 16 32 64 128

Experiments: Netflix

Conclusions We introduced an algorithm for binary logistic PCA that scales well to very high dimensional and very sparse data We experimented on a large scale collaborative filtering task The method captures different aspects of the data than traditional PCA We list some possible improvements in the paper Questions?

Experiments: Netflix Predicted 5 True X1 X2 X3 X4 4 3 2 1

Experiments: Netflix 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Experiments: Netflix A1 A2 10 10 5 0 0 10 20 10 5 0 10 20 30 10 10 0 10 A3 10 10 5 5 0 0 5 5 10 5 0 20 30 A4 5 10 10 5 0 5 10

Experiments: Netflix 3 3 2 2 1 1 0 2 0 2 0 2 0 2

Experiments: Netflix 1.01 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.345 0.35 0.355 0.36 0.365 0.37 0.375 0.38

Blending Linear regression using least squares error on known ratings from probe set [ ] predictions of method 1 predictions of method 2 [ α1 α 2 ] = [ probe set ratings ]

A real recommender system Low RMSE Accuracy on top few picks Easy to interpret Speed of response Update state (learn online) Discover novel items Collaborative filtering

Implementation Implementation in Matlab Extension of SVD code from http://www.cis.hut.fi/alexilin/software Most critical parts (sigmoid(), A S) implemented in C++, linked with mex library Use only sparse matrices, make sure they are never internally transformed into full matrices Example: Y = spfun(@(x) 1./x, Y); Three different values: missing 0, 0 ǫ, 1 1 Minibatch: no need to keep whole Y in memory at all times, either load from disk in every iteration or recompute

We need to compute L 1 = log(σ(x)) L 2 = log(1 σ(x)) where σ(x) = 1 1+e x For numerical stability we compute if(x >= 0) L 1 = log(1 + e x ) L 2 = x + L 1 if(x < 0) L 2 = log(1 + e x ) L 1 = x + L 2 Implementation

Collaborative filtering Other ideas Trouble with data: spam accounts, more people with same account Use information about quiz data set: data is not missing (entirely) at random Use external data: identify movies/users Rating depending on time: influenced by previously seen movies If we both hate Titanic which everyone liked, that tells more about our similar tastes than if we both liked it etc.

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