nips nips2012 nips2012-74 knowledge-graph by maker-knowledge-mining

74 nips-2012-Collaborative Gaussian Processes for Preference Learning

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Author: Neil Houlsby, Ferenc Huszar, Zoubin Ghahramani, Jose M. Hernández-lobato

Abstract: We present a new model based on Gaussian processes (GPs) for learning pairwise preferences expressed by multiple users. Inference is simpliﬁed by using a preference kernel for GPs which allows us to combine supervised GP learning of user preferences with unsupervised dimensionality reduction for multi-user systems. The model not only exploits collaborative information from the shared structure in user behavior, but may also incorporate user features if they are available. Approximate inference is implemented using a combination of expectation propagation and variational Bayes. Finally, we present an efﬁcient active learning strategy for querying preferences. The proposed technique performs favorably on real-world data against state-of-the-art multi-user preference learning algorithms. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Inference is simpliﬁed by using a preference kernel for GPs which allows us to combine supervised GP learning of user preferences with unsupervised dimensionality reduction for multi-user systems. [sent-2, score-0.858]

2 The model not only exploits collaborative information from the shared structure in user behavior, but may also incorporate user features if they are available. [sent-3, score-0.708]

3 Finally, we present an efﬁcient active learning strategy for querying preferences. [sent-5, score-0.139]

4 The proposed technique performs favorably on real-world data against state-of-the-art multi-user preference learning algorithms. [sent-6, score-0.41]

5 1 Introduction Preference learning is concerned with making inference from data consisting of pairs of items and corresponding binary labels indicating user preferences. [sent-7, score-0.465]

6 A popular modeling approach assumes the existence of a utility function f such that f (x) gives the utility of an item with feature vector x; f (xi ) > f (xj ) indicates that item i is preferred to item j. [sent-9, score-0.518]

7 Bayesian methods can be used to learn f , for example, by modeling f independently for each user as a draw from a Gaussian process (GP) prior [4]. [sent-10, score-0.318]

8 However, when data from many users is available, such methods do not leverage similarities in preferences across users. [sent-11, score-0.332]

9 Current multi-user approaches require that features are available for each user and assume that users with similar features have similar preferences [2], or perform single-user learning, ignoring user features, but tie information across users with a hierachical prior [1]. [sent-12, score-1.182]

10 These methods are not ﬂexible and can only address one of two possible scenarios: a) user features are available and they are useful for prediction and b) when this is not the case. [sent-13, score-0.351]

11 Our approach, by contrast, can address both a) and b) by combining informative user features with collaborative information. [sent-16, score-0.416]

12 The proposed method is based on a connection between preference learning and GP binary classiﬁcation. [sent-22, score-0.435]

13 We show that both problems are equivalent when a covariance function called the preference kernel is used. [sent-23, score-0.489]

14 Finally, in real scenarios, querying users for preference may be costly and intrusive, so it is desirable to learn preferences using the least data possible. [sent-25, score-0.717]

15 With this objective, we present BALD (Bayesian active learning by disagreement), an efﬁcient active learning strategy for binary classiﬁcation problems with GP priors. [sent-26, score-0.25]

16 2 Pairwise preference learning as special case of binary classiﬁcation The problem of pairwise preference learning can be recast as a special case of binary classiﬁcation. [sent-27, score-0.934]

17 Let us consider two items i and j with corresponding feature vectors xi , xj ∈ X . [sent-28, score-0.269]

18 In the pairwise preference learning problem, we are given pairs of feature vectors xi and xj and corresponding class labels y ∈ {−1, 1} such that y = 1 if the user prefers item i to item j and y = −1 otherwise. [sent-29, score-1.292]

19 This problem can be addressed by introducing a latent preference function f : X → R such that f (xi ) > f (xj ) whenever the user prefers item i to item j and f (xi ) < f (xj ) otherwise [4]. [sent-31, score-1.096]

20 The preference learning problem can be solved by combining a GP prior on f with the likelihood function in (1) [4]. [sent-33, score-0.425]

21 The posterior for f can then be used to make predictions on the user preferences for new pairs of items. [sent-34, score-0.523]

22 Let g : X 2 → R be the latent function g(xi , xj ) = f (xi ) − f (xj ). [sent-36, score-0.183]

23 When the evaluation of g is contaminated with standard Gaussian noise, the likelihood for g given xi , xj and y is P(y|xi , xj , g) = Φ[g(xi , xj )y] . [sent-38, score-0.425]

24 The covariance function kpref of the GP prior on g can be computed from the covariance function k of the GP on f as kpref ((xi , xj ), (xk , xl )) = k(xi , xk )+k(xj , xl )−k(xi , xl )−k(xj , xk ). [sent-40, score-0.523]

25 However, to our knowledge, the preference kernel has not been used previously for GP-based models. [sent-44, score-0.439]

26 The combination of (2) with a GP prior based on the preference kernel allows us to transform the pairwise preference learning problem into binary classiﬁcation with GPs. [sent-45, score-0.937]

27 This means that state-ofthe-art methods for GP binary classiﬁcation, such as expectation propagation [14], can be applied directly to preference learning. [sent-46, score-0.469]

28 3 Multi-user preference learning Consider I items with feature vectors xi ∈ X for i = 1, . [sent-48, score-0.544]

29 Our approach to the multiuser problem is to assume common structure in the user latent functions. [sent-53, score-0.386]

30 In particular, we assume a set of D shared latent functions, hd : X 2 → R for d = 1, . [sent-54, score-0.268]

31 , D, such that the user latent functions are generated by a linear combination of these functions, namely D gu (xj , xk ) = wu,d hd (xj , xk ) , d=1 (3) here wu,d ∈ R is the weight given to function hd for user u. [sent-57, score-1.056]

32 We place a GP prior over the shared latent functions h1 , . [sent-58, score-0.145]

33 , hD using the preference kernel described in the previous section. [sent-61, score-0.439]

34 This model allows the preferences of the different users to share some common structure represented by the latent functions h1 , . [sent-62, score-0.374]

35 2 We may extend this model further to the case in which, for each user u, there is a feature vector uu ∈ U containing information that might be useful for prediction. [sent-67, score-0.378]

36 The user features are incorporated now by placing a separate GP prior over the users weights. [sent-72, score-0.527]

37 These weight functions describe the contribution of shared latent function hd to the user latent function gu as a function of the user feature vector uu . [sent-74, score-1.039]

38 The data consists of L, the sets of feature vectors for the users U (if available), the item features X = {x1 , . [sent-79, score-0.401]

39 , xI }, and U sets of preference judgements, one for each user, D = {{zu,i , yu,i }Mu }U , where zu,i indexes the i=1 u=1 i-th pair evaluated by user u, yi,u = 1 if this user prefers the ﬁrst item in the pair to the second and yi,u = −1 otherwise. [sent-82, score-1.188]

40 Mu is the number of preference judgements made by the u-th user. [sent-83, score-0.42]

41 1 Probabilistic description To address the task of predicting preference on unseen item pairs we cast the model into a probabilistic framework. [sent-85, score-0.593]

42 Let G be an U ×P ‘user-function’ matrix, where each row corresponds to a particular user’s latent function, that is, the entry in the u-th column and i-th row is gu,i = gu (xα(i) , xβ(i) ) and α(i) and β(i) denote respectively the ﬁrst and second item in the i-th pair from L. [sent-86, score-0.381]

43 Let H be a D × P ‘shared-function’ matrix, where each row represents the shared latent functions, that is, the entry in the d-th row and i-th column is hd,i = hd (xα(i) , xβ(i) ). [sent-87, score-0.352]

44 Finally, we introduce the U × D weight matrix W such that each row contains a user’s weights, that is, the entry in the u-th row and d-th column of this matrix is wd (uu ). [sent-88, score-0.123]

45 If user features are unavailable, Kusers becomes the identity matrix. [sent-104, score-0.324]

46 , hD is sampled a priori from a GP with zero mean and covariance function given by a preference kernel. [sent-108, score-0.438]

47 Let Kitems be the P × P preference covariance matrix for the item pairs in L. [sent-109, score-0.643]

48 BALD samples from the region of greatest uncertainty in the latent function. [sent-116, score-0.146]

49 k is the prior variance of hd (xα(P +1) , xβ(P +1) ) and k is a P -dimensional vector that contains the prior covariances between hd (xα(P +1) , xβ(P +1) ) and hd (xα(1) , xβ(1) ), . [sent-117, score-0.554]

50 We want to learn user preferences with the proposed model from the least amount of data possible. [sent-124, score-0.419]

51 Therefore we desire to query users actively about their preferences on the most informative pairs of items [3]. [sent-125, score-0.444]

52 This method exploits the preference kernel and so may be trivially generalized to GP binary classiﬁcation problems also. [sent-127, score-0.486]

53 4 Bayesian active learning by disagreement The goal of active learning is to choose item pairs such that we learn the preference functions for the users using minimal data. [sent-128, score-0.973]

54 Information theoretic approaches to active learning are popular because they do not require prior knowledge of loss functions or test domains. [sent-129, score-0.126]

55 For preference learning (see Section 2), this implies choosing the new item features xi and xj that maximize H[P(g|D)] − EP(y|xi ,xj ,D) [H[P(g|y, xi , xj , D)]] , (9) where D are the user preferences observed so far and H[p(x)] = − p(x) log p(x) dx represents the Shannon entropy. [sent-131, score-1.351]

56 Using this we can rearrange this equation to compute entropies in y space: H[P(y|xi , xj , D)] − EP(g|D) [H [P(y|xi , xj , g)]] . [sent-139, score-0.269]

57 Expression (10) also provides us with an intuition about the objective; we seek the xi and xj for which a) the model is marginally uncertain about y (high H[P(y|xi , xj , D)]) and b) conditioned on a particular value of g the model is conﬁdent about y (low EP(g|D) [H[P(y|xi , xj , g])]). [sent-144, score-0.395]

58 This can be interpreted as seeking the pair xi and xj for which the latent functions g, under the posterior, ‘disagree’ with each other the most about the outcome, that is, the preference judgement. [sent-145, score-0.649]

59 In the following section we show how (10) can be applied to binary classiﬁcation with GPs, and hence via the preference kernel also to any preference learning problem. [sent-148, score-0.874]

60 1 BALD in binary classiﬁcation with GPs Most approximate inference methods for the problem of binary classiﬁcation with GPs produce a Gaussian approximation to the posterior distribution of f , the latent function of interest. [sent-150, score-0.275]

61 In 4 the binary GP classiﬁer, the entropy of y given the corresponding value of f can be expressed in terms of the binary entropy function, h[f ] = −f log f − (1 − f ) log(1 − f ). [sent-151, score-0.182]

62 When a Gaussian is used to approximate the posterior of f , we have 2 that for each x, fx = f (x) will follow a Gaussian distribution with mean µx and variance σx . [sent-153, score-0.188]

63 The ﬁrst term in (10), that is, H[p(y|x, D)], can be handled analytically in this case: H[p(y|x, D)] ≈ 2 2 h Φ(fx )N (fx |µx , σx )dfx = h Φ µx (σx + 1)−1/2 , where ≈ represents here the Gaussian approximation to the posterior of fx . [sent-154, score-0.163]

64 This result is obtained by using the Gaussian approximation to the posterior of fx 2 and then approximating h[Φ(fx )] by the squared exponential curve exp(−fx /π log 2) (details can be found in Section 3 of the supplementary material). [sent-156, score-0.202]

65 To summarize, the BALD algorithm for active binary GP classiﬁcation / preference learning ﬁrst 2 applies any approximate inference method to obtain the posterior mean µx and variance σx of f at each point of interest x. [sent-157, score-0.635]

66 x (11) BALD assigns a high value to the feature vector x when the model is both uncertain about the label 2 (µx close to 0) and there is high uncertainty about fx (σx is large). [sent-159, score-0.155]

67 By contrast, BALD samples data from the region of greatest uncertainty in the latent function. [sent-163, score-0.146]

68 5 Expectation propagation and variational Bayes Approximate inference in our model is implemented using a combination of expectation propagation (EP) [13] and variational Bayes (VB) [7]. [sent-164, score-0.152]

69 , f4 by minimizing the Kullback-Leibler ˆ (KL) divergence between the product of Q\a and fa and the product of Q\a and fa , where Q\a is ˆ . [sent-189, score-0.12]

70 However, this does not perform well for reﬁning f2 ; details on this ˆ the ratio between Q and fa problem can be found in Section 4 of the supplementary material and in [19]. [sent-190, score-0.127]

71 In our case, reﬁning f3 has cost O(DU 3 ), where U is the number of users, and D the number of shared latent functions. [sent-202, score-0.13]

72 The cost of ˆ reﬁning f4 is O(DP 3 ), where P is the number of observed item pairs. [sent-203, score-0.185]

73 These approximations allow us 2 2 ˆ ˆ to reﬁne f3 and f4 in O(DU0 U ) and O(DP0 P ) operations, where U0 and P0 are the number of pseudo-inputs for the users and for the item pairs, respectively. [sent-207, score-0.329]

74 6 Experiments and Discussion The performance of our collaborative preference model with the BALD active learning strategy is evaluated in a series of experiments with simulated and real-world data. [sent-210, score-0.567]

75 Two versions of the proposed collaborative preference (CP) model are used. [sent-215, score-0.453]

76 The ﬁrst version (CPU) takes into account the available user features, as described in Section 3. [sent-216, score-0.308]

77 This method does not use user features, and captures similarities between users with a hierarchical GP model. [sent-220, score-0.475]

78 In particular, a common GP prior is assumed for the preference function of each user; using this prior the model learns the full GP posterior for each user. [sent-221, score-0.524]

79 In this model there exists one high-dimensional function which depends on both the features of the two items to be compared and on the features of the user who makes the comparison. [sent-224, score-0.438]

80 Relationships between users’ behaviors are captured only via the user features. [sent-225, score-0.281]

81 We implement BO and BI using the preference kernel and EP for approximate inference1 . [sent-226, score-0.486]

82 BI does not include user features, but learns U GPs, so has complexity O(U P 3 ); the equivalent version of our model (CP) has cost O(N P + DP 3 ), which is lower because D << U . [sent-230, score-0.303]

83 Finally, we consider a single user approach (SU) which ﬁts a different GP classiﬁer independently to the data of each user. [sent-232, score-0.281]

84 1 Although this is not the same as the original implementations (sampling-based for BI, Laplace approximation for BO), the preference kernel and EP are likely to augment the performance of these algorithms, and provides the fairest comparison of the underlying models. [sent-233, score-0.439]

85 The available data were split randomly into training and test sets of item pairs, where the training sets contain 20 pairs per user in Sushi, MovieLens and Election, 15 pairs in Jura and 30 in Synthetic. [sent-334, score-0.555]

86 In our model, we can also estimate the kernel lengthscales by maximizing the EP approximation of the model evidence, as illustrated in Section 9 of the supplementary material. [sent-344, score-0.125]

87 Overall, CP and CPU outperform SU and BO, and breaks even with BI; the ﬁnal result is notable as BI learns the full mean and covariance structure across all users, ours uses only a few latent dimensions, which provides the key to scaling to many more users. [sent-353, score-0.12]

88 In the real-world datasets, users with similar features do not seem to have similar preferences and so correlating behavior of users with similar features is detrimental. [sent-355, score-0.556]

89 In this case, the unsupervised learning of similarities in user preferences is more useful for prediction than the user features. [sent-356, score-0.728]

90 We now use all the available users from each dataset, with a maximum of 1000 users. [sent-364, score-0.193]

91 For each user the available preference data are split randomly into training, pool and test sets with 5, 35 and 5 data points respectively in 7 Synthetic Sushi 0. [sent-365, score-0.724]

92 2 2 num samples 4 6 8 10 num samples Election 0. [sent-371, score-0.15]

93 1 0 2 4 6 8 10 num samples Figure 2: Average test error for CPU, CP and SU, using the strategies BALD (-B), entropy (-E) and random (-R) for active learning. [sent-397, score-0.208]

94 Average errors after 10 queries from the pool set of each user are summarized in Table 3. [sent-407, score-0.309]

95 For each model (CPU, CP and SU), the results of the best active learning strategy are highlighted in bold. [sent-408, score-0.157]

96 7 Conclusions We have proposed a multi-user model that combines collaborative ﬁltering methods with GP binary preference modeling. [sent-413, score-0.5]

97 We have shown that the task of learning user preferences can be recast as a particular case of binary classiﬁcation with GPs when a covariance function called the preference kernel is used. [sent-414, score-0.993]

98 We have also presented BALD, a novel active learning strategy for binary classiﬁcation models with GPs. [sent-415, score-0.161]

99 The proposed multi-user model with BALD performs favorably on simulated and real-world data against single-user methods and existing approaches for multi-user preference learning, whilst having signiﬁcantly lower computational times than competing multi-user methods. [sent-416, score-0.41]

100 Multi-task preference learning with an application to hearing aid personalization. [sent-423, score-0.388]

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