Scaling Neighbourhood Methods

Transcription

1 Quick Recap

2 Scaling Neighbourhood Methods

3 Collaborative Filtering m = #items n = #users Complexity : m * m * n

4 Comparative Scale of Signals ~50 M users ~25 M items Explicit Ratings ~ O(1M) (1 per billion) Purchase ~ O(100M) (100 per billion) Browse ~ O(10B) (10000 per billion)

5 Implicit Signals Used Bought History Browse History Compare History

6 Category-partitioned v/s Category independent

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9 Similarity Metric for boolean matrix Cosine Similarity A. Pair Count (p) - P1 and P2 B. Individual Count (n_i) - P1, P2 individually

10 Employing Map Reduce Calculate 'p' by forming pairs and counting Calculate 'n1' by making P1 as the key Calculate 'n2' by making P2 as the key Took 2 hours on 5 years of data Scales Horizontally

11 Map Reduce 1

12 Map Reduce 2

13 Map Reduce 3

14 Latent Variable Models and Factorization Models Akash Khandelwal, Avijit Saha, Mohit Kumar, Vivek Mehta 1 / 34

15 Outline Introduction Recommender Systems Recap Latent Variable Models Factorization Models Matrix Factorization Singular Value Decomposition BPMF Factorization Machine Conclusion 2 / 34

16 Recommender Systems (RSs) Collaborative Filtering (CF) Neighborhood Based - KNN Model Based - Cluster-based CF and Bayesian classifiers. - Latent variable models such as, LDA, plsa, and matrix factorization (MF). Content Based Knowledge Based Hybrid 3 / 34

17 Latent Variable Models Supplementing a set of observed variables with additional latent, or hidden, variables. Latent variable models are widely used in several domains such as machine learning, statistics, data mining. Reveals hidden structure which explains the data. Latent variable models consider a joint distribution over the hidden and observed variables. Hidden structure is found by calculating the posterior. LDA is an example of latent variable models. 4 / 34

18 Factorization Models One of the widely used latent variable models in the RSs community. preferences of a user are determined by a small number of unobserved latent factors. Matrix Factorization : Each user and item are mapped to a latent factor vector: u i R K v j R K Tensor Factorization : Mapping of each variable of each category type to a K dimensional latent factor vector. Many problem specific factorization models. 5 / 34

19 Focus Factorization Models Matrix Factorization (MF) Factorization Machine (FM) 6 / 34

20 Singular Value Decomposition Singular value decomposition (SVD) is a factorization of a matrix. Formally, the SVD of an R R I J is: R = UΣV, (1) where, U = I I unitary matrix Σ =I J rectangular diagonal matrix V = J J unitary matrix σ i,i of Σ are known as the singular values of R I columns of U and the J columns of V are called the left-singular vectors and right-singular vectors of R, respectively. 7 / 34

21 Dimensionality Reduction Using SVD Let, R R I J Estimate : Apply SVD: R = UΣV, (2) σ 1,1 σ 2,2... ˆR = U V. σk,k 0... (3) 8 / 34

22 Example of SVD (a) (b) (c) (d) 9 / 34

23 MF for RSs Figure 1: Matrix Factorization 10 / 34

24 MF for RSs Consider a user-movie matrix R R I J where the r ij cell represents the rating provided to the j th movie by the i th user. MF decomposes the matrix R into two low-rank matrices U = [u 1, u 2,..., u I ] T R I K and V = [v 1, v 2,..., v J ] T R J K : R UV T. (4) ( ) 2 r ij u T i v j (5) (i,j) Ω 11 / 34

25 MF for RSs SVD with K singular value would been the solution if R is fully observed. However, R is partially observed. Solution: Stochastic gradient descent to rank-1 update: Then iterate this for each rank K. e ij = r ij u T i v j u ik = u ik + ν(e ij v jk λu ik ) (6) v jk = v jk + ν(e ij u ik λv jk ) (7) 12 / 34

26 Problem with SVD approach Learning rate and regularization parameters needs to be tuned manually. Overfitting. Solution: Bayesian Probabilistic Matrix Factorization (BPMF) [1]. 13 / 34

27 BPMF Model The likelihood term of BPMF is as follows: p(r Θ) = N (r ij u T i v j, τ 1 ), (8) (i,j) Ω where u i is the latent factor vector for the i th user, v j is the latent factor vector for the j th item, τ is the model precision, Ω is the set of all observations, and Θ is the set of all the model parameters. 14 / 34

28 Priors Independent priors are placed on all the model parameters in Θ as: I p(u) = N (u i µ u, Λ 1 u ), (9) p(v ) = i=1 J N (v j µ v, Λ 1 v ). (10) j=1 15 / 34

29 Hyperpriors Further place Normal-Wishart priors are placed on all the hyperparameters Θ H = {{µ u, Λ u }, {µ v, Λ v }} as: where p(µ u, Λ u ) = N W (µ u, Λ u µ 0, β 0, W 0, ν 0 ), (11) = N (µ u µ 0, (β 0 Λ u ) 1 )W(Λ u W 0, ν 0 ) p(µ v, Λ v ) = N W (µ v, Λ v µ 0, β 0, W 0, ν 0 ). (12) W(Λ W 0, ν 0 ) = 1 C Λ ν 0 D 1 exp( 1 1 Tr(W 0 2 Λ)) 16 / 34

30 Joint Distributions The joint distribution of the observations and the hidden variables can be written as: p(r, Θ, Θ H Θ 0 ) = p(r Θ)p(U)p(V )p(µ u, Λ u Θ 0 )p(µ v, Λ v Θ 0 ), (13) where Θ 0 = {µ 0, β 0, W 0, ν 0 } 17 / 34

31 Inference Evaluation of the joint distribution in Eq. (13) is intractable. However, all the model parameters are conditionally conjugate. So we Gibbs sampler has closed form updates. Replacing Eq. (8)-(12) in Eq. (13), the sampling distribution of u i can be written as follows: p(u i ) N ( u i µ, (Λ ) 1), (14) Λ = Λ u + τ v j v T j (15) j Ωi µ = (Λ ) 1 Λ u µ u + τ v j r ij (16) j Ωi 18 / 34

32 Results Figure 2: RMSE vs Iterations 19 / 34

33 Factorization Models Matrix factorization [2]. Tensor factorization [3]. Specific models like SVD++ [4], TimeSVD++ [5], FPMC [6], and BPTF [3], etc. have been developed. Several Learning technique like SGD, ALS, variational Bayes, MCMC Gibbs sampling have been developed. 20 / 34

34 Factorization Models Vs Feature Based Techniques Advantages of Factorization Model: Problem: Scalability and Performance. deriving inference techniques for each individual model is a time consuming task and requires considerable expertise. Advantages of Feature Based Techniques: Problem: Generic approach. Can be solved using standard tools like LIBSVM or SVMLight. Can not handle very sparse data. 21 / 34

35 Factorization Machine A Generic framework [7] proposed by Stephen Rendle. Combines advantages of both factorization models and feature based model. Can subsume many state of the art factorization model like SVD++, TimeSVD++, FPMC, PITF, etc. Performs well for sparse data where SVMs fails. 22 / 34

36 Feature Representation of Factorization Machine Example: (U1,M1,G1,2),(U1,M3,G2,5),(U2,M2,G3,4) and (U3,M1,G1,5) Feature x Target y User Movie Genre x y1 x y2 x y3 x y4 U1 U2 U3. M1 M2 M3. G1 G2 G3. 23 / 34

37 Factorization Machine Following is the equation for FM. D D y n = w 0 + w i x ni + D x ni x nj K i=1 i=1 j=i+1 k=1 Assumptions of FM are following: y n x n, θ N (ŷ(x n, θ), α 1 ) y n x n, θ Bernoulli(b(ŷ(x n, θ)) And L2 regularization on θ v ik v jk (17) 24 / 34

38 Mimic MF ŷ = w 0 + w u + w i + v T u v i (18) Feature representation of FM: D = U I x j = δ(j = u j = i) 25 / 34

39 Mimic SVD++ ŷ = w 0 + w u + w i + v T u v i + 1 N Feature representation of FM: D = U I L x j = 1 if j = u j = i = 1 N if j N u = 0 else l N u v T i v l (19) 26 / 34

40 Algorithms of Factorization Machine Stochastic gradient descent is the simplest algorithm to solve FM (SGD-FM). MCMC based Bayesian Factorization Machine gives state-of-the-art performance (MCMC-FM). Alternating least square and adaptive stochastic gradient descent. 27 / 34

41 Comparison of each Leaning Techniques SGD-FM Pros: SGD online algorithm and more scalable. Cons: Costly cross validation of parameters. MCMC-FM Pros: Performance. No cross validation required. 28 / 34

42 SGD Learning Following is the equation for FM. ŷ n = w 0 + Cost function is: D D D w i x ni + x ni x nj i=1 K i=1 j=i+1 k=1 v ik v jk (20) N (y n ŷ n ) 2 + L2 regularization (21) n=1 SGD update equations: v ik = v ik + ν 2 (y n ŷ n )x ni N l=1&l i x nl v nl 2λv ik (22) 29 / 34

43 MCMC-FM Likelihood: Prior: Hyperprior: y n N (y n ŷ n, τ) p(w 0 ) N (w 0 µ 0, σ 0 ) p(w i ) N (w i µ w, σ w ) p(v ik ) N (v ik µ k, σ k ) p(µ w ) N (µ w µ, σ w ν 0 ) p(σ w ) G(σ w α 0, β 0 ) p(µ k ) N (µ k µ, σ k ν 0 ) p(σ k ) G(σ k α 0, β 0 ) 30 / 34

44 Results 31 / 34

45 Datasets Yelp challenge Yelp datasets Users information Social information Location Time Ratings Reviews 32 / 34

46 References R. Salakhutdinov and A. Mnih, Bayesian probabilistic matrix factorization using markov chain monte carlo, in Proc. of ICML, pp , R. Salakhutdinov and A. Mnih, Probabilistic matrix factorization, in Proc. of NIPS, L. Xiong, X. Chen, T. K. Huang, J. Schneider, and J. G. Carbonell, Temporal collaborative filtering with bayesian probabilistic tensor factorization, in Proc. of SDM, Y. Koren, Factorization meets the neighborhood: A multifaceted collaborative filtering model, in Proc. of KDD, pp , Y. Koren, Collaborative filtering with temporal dynamics, in Proc. of KDD, pp , S. Rendle, C. Freudenthaler, and L. Schmidt-Thieme, Factorizing personalized markov chains for next-basket recommendation, in Proc. of WWW, pp , S. Rendle, Factorization machines, in Proc. of ICDM, / 34

47 THANK YOU 34 / 34