nips nips2012 nips2012-246 knowledge-graph by maker-knowledge-mining

246 nips-2012-Nonparametric Max-Margin Matrix Factorization for Collaborative Prediction

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Author: Minjie Xu, Jun Zhu, Bo Zhang

Abstract: We present a probabilistic formulation of max-margin matrix factorization and build accordingly a nonparametric Bayesian model which automatically resolves the unknown number of latent factors. Our work demonstrates a successful example that integrates Bayesian nonparametrics and max-margin learning, which are conventionally two separate paradigms and enjoy complementary advantages. We develop an efﬁcient variational algorithm for posterior inference, and our extensive empirical studies on large-scale MovieLens and EachMovie data sets appear to justify the aforementioned dual advantages. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 cn Abstract We present a probabilistic formulation of max-margin matrix factorization and build accordingly a nonparametric Bayesian model which automatically resolves the unknown number of latent factors. [sent-5, score-0.72]

2 We develop an efﬁcient variational algorithm for posterior inference, and our extensive empirical studies on large-scale MovieLens and EachMovie data sets appear to justify the aforementioned dual advantages. [sent-7, score-0.118]

3 1 Introduction Collaborative prediction is a task of predicting users’ potential preferences on currently unrated items (e. [sent-8, score-0.08]

4 One typical setting formalizes it as a matrix completion problem, i. [sent-11, score-0.078]

5 Among other popular approaches, factor-based models have been used extensively in collaborative prediction. [sent-18, score-0.075]

6 The underlying idea behind such models is that there is only a small number of latent factors inﬂuencing the preferences. [sent-19, score-0.147]

7 In a linear factor model, a user’s rating of an item is modeled as a linear combination of these factors, with user-speciﬁc coefﬁcients and item-speciﬁc factor values. [sent-20, score-0.204]

8 Thus, given a N × M preference matrix for N users and M items, a K-factor model ﬁts it with a N × K coefﬁcient matrix U and a M × K factor matrix V as U V . [sent-21, score-0.467]

9 Various computational methods have been successfully developed to implement such an idea, including probabilistic matrix factorization (PMF) [13, 12] and deterministic reconstruction/approximation error minimization, e. [sent-22, score-0.386]

10 , max-margin matrix factorization (M3 F) with hinge loss [14, 11, 16]. [sent-24, score-0.397]

11 One common problem in latent factor models is how to determine the number of factors, which is unknown a priori. [sent-25, score-0.137]

12 On the other hand, probabilistic matrix factorization models have lend themselves naturally to leverage recent advances in Bayesian nonparametrics to bypass explicit model selection [17, 1]. [sent-29, score-0.473]

13 However, it remains largely unexplored how to borrow such advantages into deterministic max-margin matrix factorization models, particularly the very successful M3 F. [sent-30, score-0.354]

14 To address the above problem, this paper presents inﬁnite probabilistic max-margin matrix factorization (iPM3 F), a nonparametric Bayesian-style M3 F model that utilizes nonparametric Bayesian techniques to automatically resolve the unknown number of latent factors in M3 F models. [sent-31, score-0.834]

15 The ﬁrst key step towards iPM3 F is a general probabilistic formulation of the standard M3 F, which is based on the maximum entropy discrimination principle [4]. [sent-32, score-0.216]

16 We can then principally extend it to a non1 parametric model, which in theory has an unbounded number of latent factors. [sent-33, score-0.093]

17 To avoid overﬁtting we impose a sparsity-inducing Indian buffet process prior on the latent coefﬁcient matrix, selecting only an appropriate number of active factors. [sent-34, score-0.243]

18 We develop an efﬁcient variational method to infer posterior distributions and learn parameters (if ever exist) and our extensive empirical results on MovieLens and EachMovie demonstrate appealing performances. [sent-35, score-0.165]

19 2 Max-margin matrix factorization Given a preference matrix Y ∈ RN ×M , which is partially observed and usually sparse, we denote the observed entry indices by I. [sent-38, score-0.391]

20 The task of traditional matrix factorization is to ﬁnd a low-rank matrix X ∈ RN ×M to approximate Y under some loss measure, e. [sent-39, score-0.43]

21 , the commonly used squared error, and use Xij as the reconstruction of the missing entries Yij wherever ij ∈ I. [sent-41, score-0.09]

22 Max-margin / matrix factorization (M3 F) [14] extends the model by using a sparsity-inducing norm regularizer for a low-norm factorization and adopting hinge loss for the error measure, which is applicable to binary, discrete ordinal, or categorical data. [sent-42, score-0.664]

23 For the binary case where Yij ∈ {±1} and one predicts by Yij = sign(Xij ), the optimization problem of M3 F is deﬁned as min X X ∗ h (Yij Xij ) , +C (1) ij∈I where h(x) = max(0, 1 − x) is the hinge loss and X ∗ is the nuclear norm of X. [sent-43, score-0.16]

24 M3 F can be equivalently reformulated as a semi-deﬁnite programming (SDP) and thus learned using standard SDP solvers, but it is unfortunately very slow and can only scale up to thousands of users and items. [sent-44, score-0.189]

25 The fast M3 F model can scale up to millions of users and items. [sent-48, score-0.216]

26 But one unaddressed resulting problem is that it needs to specify the unknown number of latent factors, K, a priori. [sent-49, score-0.093]

27 Below we present a nonparametric Bayesian approach, which effectively bypasses the model selection problem and produces very robust prediction. [sent-50, score-0.105]

28 We also design a blockwise coordinate descent algorithm that directly solves problem (3) rather than working on a smoothing relaxation [11], and it turns out to be as efﬁcient and accurate. [sent-51, score-0.151]

29 3 Nonparametric Bayesian max-margin matrix factorization Now we present the nonparametric Bayesian max-margin matrix factorization models. [sent-53, score-0.731]

30 1 Maximum entropy discrimination We consider the binary classiﬁcation setting since it sufﬁces for our model. [sent-56, score-0.158]

31 Accordingly, MED takes expectation over the original discriminant function with respect to p(η) and has the new prediction rule y = sign (Ep [F (x; η)]) . [sent-58, score-0.133]

32 In fact, MED subsumes SVM as a special case and has been extended to incorporate latent variables [5, 18] and perform structured output prediction [21]. [sent-61, score-0.162]

33 Recent work has further extended MED to unite Bayesian nonparametrics and max-margin learning [20, 19], which have been largely treated as isolated topics, for learning better classiﬁcation models. [sent-62, score-0.093]

34 The present work contributes by introducing a novel generalization of MED to handle the challenging matrix factorization problems. [sent-63, score-0.313]

35 2 Probabilistic max-margin matrix factorization Like PMF [12], we treat U and V as random variables, whose joint prior distribution is denoted by p0 (U, V ). [sent-65, score-0.372]

36 Then, our goal is to infer their posterior distribution p(U, V )2 after a set of observations have been provided. [sent-66, score-0.099]

37 (6) Furthermore, since both U and V are random variables, we need to resolve the uncertainty in order to derive a prediction rule. [sent-69, score-0.074]

38 Hence, substituting the discriminant function (6) into (4), we have the prediction rule Yij = sign Ep [Ui Vj ] . [sent-71, score-0.133]

39 (7) Then following the principle of MED learning, we deﬁne probabilistic max-margin matrix factorization (PM3 F) as solving the following optimization problem min p(U,V ) KL(p(U, V ) p0 (U, V )) + C h Yij Ep [Ui Vj ] . [sent-72, score-0.419]

40 (8) ij∈I Note that our probabilistic formulation is strictly more general than the original M3 F model, which is in fact a special case of PM3 F under a standard Gaussian prior and a mean-ﬁeld assumption on p(U, V ). [sent-73, score-0.16]

41 Speciﬁcally, if we assume p0 (U, V ) = i N (Ui |0, I) j N (Vj |0, I) and p(U, V ) = p(U )p(V ), then one can prove p(U ) = i N (Ui |Φi , I), p(V ) = j N (Vj |Ψj , I) and PM3 F reduces accordingly to a M3 F problem (3), namely min Φ,Ψ 1 ( Φ 2 2 F + Ψ 2 F) +C h Yij Φi Ψj . [sent-74, score-0.11]

42 (9) ij∈I Ratings: For ordinal ratings Yij ∈ {1, 2, . [sent-75, score-0.278]

43 Speciﬁcally, we introduce thresholds θ0 ≤ θ1 ≤ · · · ≤ θL , where θ0 = −∞ and θL = +∞, to discretize R into L intervals. [sent-79, score-0.099]

44 The prediction rule is changed accordingly to Yij = max r|Ep [Ui Vj ] ≥ θr + 1. [sent-80, score-0.151]

45 The loss thus deﬁned is an upper bound to the sum of absolute −1 for r < Yij r where Tij = differences between the predicted ratings and the true ratings, a loss measure closely related to Normalized Mean Absolute Error (NMAE) [7, 14]. [sent-85, score-0.261]

46 Furthermore, we can learn a more ﬂexible model to capture users’ diverse rating criteria by replacing user-common thresholds θr in the prediction rule (10) and the loss (12) with user-speciﬁc ones θir . [sent-86, score-0.28]

47 Finally, we may as well treat the additionally introduced thresholds θir as random variables and infer their posterior distribution, hereby giving the full PM3 F model as solving L−1 min p(U,V,θ) 3. [sent-87, score-0.231]

48 (13) Inﬁnite PM3 F (iPM3 F) As we have stated, one common problem with ﬁnite factor-based models, including PM3 F, is that we need to explicitly select the number of latent factors, i. [sent-89, score-0.093]

49 In this section, we present an inﬁnite PM3 F model which, through Bayesian nonparametric techniques, automatically adapts and selects the number of latent factors during learning. [sent-92, score-0.283]

50 Without loss of generality, we consider learning a binary3 coefﬁcient matrix Z ∈ {0, 1}N ×∞ . [sent-93, score-0.117]

51 For ﬁnite-sized binary matrices, we may deﬁne their prior as given by a Beta-Bernoulli process [8]. [sent-94, score-0.129]

52 Similar to the nonparametric matrix factorization model [17], we adopt IBP prior over unbounded binary matrices as previously established in [3] and furthermore, we focus on its stick-breaking construction [15], which facilitates the development of efﬁcient inference algorithms. [sent-96, score-0.552]

53 Then the IBP prior can be described as given by the following generative process i. [sent-98, score-0.086]

54 Speciﬁcally, given a ﬁnite data set (N < +∞), the probability of seeing the kth factor decreases exponentially with k and the number of active factors K+ follows a Poisson(αHN ), where HN is the N th harmonic number. [sent-112, score-0.098]

55 Alternatively, we can use a Beta process prior over Z as in [9]. [sent-113, score-0.086]

56 As for the counterpart, we place an isotropic Gaussian prior over the item factor matrix V . [sent-114, score-0.286]

57 Prior speciﬁed, we may follow the above probabilistic framework to perform max-margin training, with U replaced by Z. [sent-115, score-0.099]

58 For ordinal ratings, we augment the iPM3 F problem from (13) likewise and, apart from adopting the same prior assumptions for ν, Z and V , assume p0 (θ) = p0 (θ|ν, Z, V ) with θir ∼ N (ρr , ς 2 ) i. [sent-138, score-0.186]

59 3 Learning real-valued coefﬁcients can be easily done as in [3] by deﬁning U = Z ◦ W , where W is a real-valued matrix and ◦ denotes the Hadamard product or element-wise product. [sent-148, score-0.078]

60 This approach naturally ﬁts in our MEDbased model thanks to the adoption of the expectation operator when deﬁning prediction rule (7) and (10). [sent-151, score-0.074]

61 And note that exactly the same process applies as well to the full model for ordinal ratings. [sent-162, score-0.122]

62 4 Learning and inference under truncated mean-ﬁeld assumptions Now, we brieﬂy discuss how to perform learning and inference in iPM3 F. [sent-163, score-0.122]

63 Speciﬁcally, we introduce a simple variational inference method to approximate the optimal posterior, which turns out to perform well in practice. [sent-166, score-0.097]

64 We make the following truncated mean-ﬁeld assumption K p(ν, Z, V ) = p(ν)p(Z)p(V ) = k=1 where K is the truncation level and N p(νk ) · K i=1 k=1 p(Zik ) · p(V ), i. [sent-167, score-0.087]

65 For ordinal ratings, we make an additional mean-ﬁeld assumption (21) p(ν, Z, V, θ) = p(ν, Z, V )p(θ), where p(ν, Z, V ) is treated exactly the same as for binary data and p(θ) is left in free forms. [sent-188, score-0.138]

66 One noteworthy point is that given p(Z), we may calculate the expectation of the posterior effective dimensionality of the latent factor space as K N Ep [K+ ] = k=1 1− (1 − ψik ) . [sent-189, score-0.224]

67 ): Infer p(V ): The linear discriminant function and the isotropic Gaussian prior on V leads to an M isotropic Gaussian posterior p(V ) = j=1 N (Vj |Λj , σ 2 I) while the M mean vectors Λj can be obtained via solving M independent binary SVMs min Λj 1 Λj 2σ 2 2 +C h i|ij∈I 5 Yij Λj ψi . [sent-191, score-0.36]

68 We update ψi via coordinate descent, with each conditional optimal ψik sought by binary search. [sent-195, score-0.087]

69 Note that as ς → +∞, the Gaussian distribution regresses to a uniform distribution and problem (25) reduces accordingly to the corresponding conditional subproblem for θ in the original M3 F (Appendix B. [sent-200, score-0.117]

70 5 Experiments and discussions We conduct experiments on the MovieLens 1M and EachMovie data sets, and compare our results with fast M3 F [11] and two probabilistic matrix factorization methods, PMF [13] and BPMF [12]. [sent-202, score-0.386]

71 Data sets: The MovieLens data set contains 1,000,209 anonymous ratings (ranging from 1 to 5) of 3,952 movies made by 6,040 users, among which 3,706 movies are actually rated and every user has at least 20 ratings. [sent-203, score-0.472]

72 The EachMovie data set contains 2,811,983 ratings of 1,628 movies made by 72,916 users, among which 1,623 movies are actually rated and 36,656 users has at least 20 ratings. [sent-204, score-0.609]

73 As in [7, 11], we discarded users with fewer than 20 ratings, leaving us with 2,579,985 ratings. [sent-205, score-0.189]

74 Protocol: As in [7, 11], we test our method in a pure collaborative prediction setting, neglecting any external information other than the user-item-rating triplets in the data sets. [sent-214, score-0.118]

75 We adopt as well the all-but-one protocol to partition the data set into training set and test set, that is to randomly withhold one of the observed ratings from each user into test set and use the rest as training set. [sent-215, score-0.345]

76 We partition the users accordingly as in [7, 11], namely 5,000 and 1,040 users for weak and strong respectively in MovieLens, and 30,000 and 6,565 in EachMovie. [sent-219, score-0.558]

77 Moreover, we ﬁnd that the fully Bayesian formulation of iPM3 F achieves comparable performances in most cases as iPM3 F and that our coordinate descent algorithm for M3 F (M3 F∗ ) performs quite similar to the original gradient descent algorithm for M3 F. [sent-288, score-0.172]

78 In summary, the effect of endowing M3 F models with a probabilistic formulation is intriguing in that not only the performance of the model is largely improved but with the help of Bayesian nonparametric techniques, the effort of selecting the number of latent factors is saved as well. [sent-289, score-0.394]

79 al almost all models perform worse on EachMovie Algorithm weak strong than on MovieLens. [sent-291, score-0.075]

80 0059 a user has rated an item as zero star, he might either BPMF [12] . [sent-301, score-0.155]

81 0067 we should treat such a declaration as less authoritative than a regular rating of zero star and hence omit it from the data set. [sent-318, score-0.098]

82 Again, the coordinate descent M3 F performs comparably with fast M3 F; iPM3 F performs better than all the other methods; And iBPM3 F performs comparably with iPM3 F. [sent-321, score-0.16]

83 5 partition #1 partition #2 partition #3 NMAE time 0. [sent-325, score-0.162]

84 (22), we may calculate the expectation of the effective dimensionality K+ of the latent factor space to roughly have a sense of how the iPM3 F model automatically chooses the latent dimensionality. [sent-344, score-0.296]

85 Since we take α = 3 in the IBP prior (15) and N ∼ 104 , the expected prior dimensionality αHN is about 30. [sent-345, score-0.153]

86 , 60 or 80, the expected posterior dimensionality very quickly saturates, 4 5 Note that M3 F models output discretized ordinal ratings while PMF models output real-valued ratings. [sent-348, score-0.365]

87 After discarding users with less than 20 normal ratings, we are left with 35,281 users and 2,315,060 ratings. [sent-349, score-0.378]

88 7 Table 3: Performance of iPM3 F with and without probabilistic treatment of θ Algorithm w/ prob. [sent-350, score-0.107]

89 (For each truncation level, we rerun our model and perform cross-validation to select the best regularization constant C. [sent-373, score-0.081]

90 Treating thresholds θ: When predicting ordinal ratings, the introduced thresholds θ are very important since they underpin the large-margin principle of max-margin matrix factorization models. [sent-378, score-0.606]

91 Nevertheless without a proper probabilistic treatment, the subproblems on thresholds (25) are not strictly convex, very often giving rise to a section of candidate thresholds that are “equally optimal” for the solution. [sent-379, score-0.271]

92 Under our probabilistic model however, we can easily get rid of this non-strict convexity by introducing for them a Gaussian prior as stated above in section 3. [sent-380, score-0.132]

93 We compare performances of iPM3 F both with and without the probabilistic treatment of θ and as shown in Table 3, the improvement is outstanding. [sent-382, score-0.107]

94 7m 25m 50 4 CPU and about 15h on EachMovie, which are fairBPMF [12] 19m 1h 50 ly acceptable for factorizing a matrix with millions M3 F∗ 4h 10h 50 3 of entries. [sent-387, score-0.105]

95 Furthermore, the blockwise coordinate descent algorithm can naturally be parallelized, since the sub-problems of learning different Ui (or Vj ) are not coupled. [sent-399, score-0.151]

96 6 Conclusions We’ve presented an inﬁnite probabilistic max-margin matrix factorization method, which utilizes the advantages of nonparametric Bayesian techniques to bypass the model selection problem of maxmargin matrix factorization methods. [sent-401, score-0.839]

97 We’ve also developed efﬁcient blockwise coordinate descent algorithms for variational inference and performed extensive evaluation on two large benchmark data sets. [sent-402, score-0.249]

98 Inﬁnite latent feature models and the Indian buffet process. [sent-427, score-0.157]

99 Bayesian matrix factorization with side information and Dirichlet process mixtures. [sent-468, score-0.34]

100 Bayesian probabilistic matrix factorization using Markov chain Monte Carlo. [sent-480, score-0.386]

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