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Conditional Restricted Boltzmann Machine for Item Recommendation

Zixiang Chena, b, c, Wanqi Maa, b, c, Wei Daia, b, c, Weike Pana, b, c∗, Zhong Minga, b, c∗

{chenzixiang2016, mawanqi2019, daiwei20171}@email.szu.edu.cn, {panweike, mingz}@szu.edu.cn aNational Engineering Laboratory for Big Data System Computing Technology,

Shenzhen University, Shenzhen, China

bGuangdong Laboratory of Artificial Intelligence and Digital Economy (SZ),

Shenzhen University, Shenzhen, China

cCollege of Computer Science and Software Engineering,

Shenzhen University, Shenzhen, China

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Introduction

Problem Definition

Top-k Recommendation with Users’ Explicit Rating Feedback Input: A set of rating triples (u, i, rui) from n users and m items, where rui denotes the numerical rating, i.e., explicit feedback, assigned by user u to item i. Goal: Provide a personalized ranked list of unrated items from I\Iu for each user u.

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Introduction

Motivation

1

In terms of modeling users’ rating data, existing methods are mainly neighborhood- and factorization-based, most of which are rating oriented.

2

Among network-based methods, the restricted Boltzmann machine (RBM) model is also applied to rating prediction tasks. However, item recommendation tasks play a more important role in the real world, due to the large itemsets as well as users’ limited attention.

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Introduction

Our Contributions

1

To the best of our knowledge, this is the first study to apply the CRBM model to solving the problem of top-k recommendation with users’ explicit rating feedback.

2

To better model users’ preferences, we treat the original rating matrix from a new perspective, i.e., three different views of users’ behaviors.

3

We conduct empirical studies on four publicly available datasets and the experimental results show that our proposed CRBM-IR is effective in generating personalized top-k items for each user.

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Introduction

Notations (1/3)

Table: Some notations and their explanations.

Notation Explanation n user number m item number U = {1, 2, . . . , n} the whole set of users I = {1, 2, . . . , m} the whole set of items u ∈ U user ID i, j ∈ I item ID rui the rating assigned by user u to item i I+

u

a set of positive items w.r.t. user u, i.e., item set rated by u with high ratings I−

u

a sampled set of negative items w.r.t. user u, I−

u ⊆ I\Iu

Iu a set of items rated by user u

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Introduction

Notations (2/3)

Table: Some notations and their explanations (cont.).

Notation Explanation p+ = [p+

i ]1×m ∈ [0, 1]1×m

probabilities of the nodes in the (positive) visible layer b+ = [b+

i ]1×m ∈ R1×m

biases of the nodes in the (positive) visible layer p− = [p−

i ]1×m ∈ [0, 1]1×m

probabilities of the nodes in the (negative) visible layer b− = [b−

i ]1×m ∈ R1×m

biases of the nodes in the (negative) visible layer ph = [ph

i ]1×d ∈ [0, 1]1×d

probabilities of the nodes in the hidden layer bh = [bh

i ]1×d ∈ R1×d

biases of the nodes in the hidden layer W +

i· , W − i· ∈ R1×d

weight between the ith node in the visible layer and the nodes in the hidden layer Cu a set of items w.r.t. user u in the condition layer (e.g., Cu = Iu in this paper) ci the value of the item i in the condition layer (e.g., ci = 1 in this paper) Ci· ∈ R1×d weight between the ith node in the condition layer and the nodes in the hidden layer

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Introduction

Notations (3/3)

Table: Some notations and their explanations (cont.). Notation Explanation d number of nodes in the hidden layer ρ sampling ratio L number of steps in the contrastive divergence (CD) algorithm T iteration number

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Related Work

Top-k Recommendation

There are mainly three branches of recommendation methods, including: Neighborhood-based methods such as user-oriented methods [Resnick et al., 1994] and item-oriented methods [Sarwar et al., 2001]. Model-based methods such as PMF [Mnih and Salakhutdinov, 2008] and BPR [Rendle et al., 2009]. Network-based methods such as CDAE [Wu et al., 2016] and RBM [Nguyen and Lauw, 2016]. In this paper, we focus on a network-based approach, i.e., restricted Boltzmann machine (RBM).

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Related Work

Restricted Boltzmann Machine

RBM and its extension conditional RBM (CRBM) are firstly applied to recommendation problems based on users’ explicit feedback [Salakhutdinov et al., 2007]. Boltzmann machine (BM) is proposed for the task of rating prediction by exploiting the ordinal property, but it consumes longer training time. Auto-Rec applies an AE to model explicit feedback [Phung et al., 2009], where the form of the input is similar to that of RBM. In this paper, we propose to apply CRBM to the task of item recommendation with explicit feedback.

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Method

CRBM-IR: Illustration

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Method

Probabilities (1/2)

The probabilities of the nodes in the hidden layer are as follows, ph = σ(

• i∈Cu

ciCi· +

• i∈I+

u

p+

i W + i· +

• i∈I−

u

p−

i W − i· + bh),

(1) where σ(x) =

1 1+exp(−x) is the sigmoid function. ci = 1 is a constant,

and p+

i , p− i , Ci·, W + i· , W − i· , b+, b− and bh are to be learned from the

data.

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Method

Probabilities (2/2)

The probability of the node i ∈ I+

u in the positive layer is as follows,

p+

i =

exp(

• W +

i· , ph

+ b+

i )

exp(

• W +

i· , ph

+ b+

i ) + exp(

• W −

i· , ph

+ b−

i ),

(2) which is usually called the prediction rule in recommender systems since p+

i can be used for item ranking and recommendation. And the

probability of the node i ∈ I−

u in the negative layer is as follows,

p−

i =

exp(

• W −

i· , ph

+ b−

i )

exp(

• W +

i· , ph

+ b+

i ) + exp(

• W −

i· , ph

+ b−

i ).

(3)

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Method

Energy Function (1/2)

Similar to other RBM-based models, the minus energy of our CRBM-IR is as follows, − E(p+, p−, ph|Θ) =

• i∈I+

u

b+

i p+ i +

• i∈I−

u

b−

i p− i + <

• i∈I+

u

p+

i W + i· , ph >

+ <

• i∈I−

u

p−

i W − i· , ph > + < bh, ph >

+ <

• i∈Cu

Ci·ci, ph >, (4) from which we can have the gradient of each model parameter W +

i· ,

W −

i· , Ci·, b+ i , b− i and bh,

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Method

Energy Function (2/2)

∂ − E(p+, p−, ph|Θ) ∂W +

i·

= p+

i ph ∈ R1×d,

(5) ∂ − E(p+, p−, ph|Θ) ∂W −

i·

= p−

i ph ∈ R1×d,

(6) ∂ − E(p+, p−, ph|Θ) ∂Ci· = ciph ∈ R1×d, (7) ∂ − E(p+, p−, ph|Θ) ∂b+

i

= p+

i ∈ R,

(8) ∂ − E(p+, p−, ph|Θ) ∂b−

i

= p−

i ∈ R,

(9) ∂ − E(p+, p−, ph|Θ) ∂bh = ph ∈ R1×d. (10)

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Method

Joint Probability

With the energy function in Eq.(4), we have the joint probability w.r.t. the visible positive layer, the visible negative layer and the hidden layer in our CRBM-IR as follows, Prob(p+, p−, ph|Θ) = exp

• − E(p+, p−, ph|Θ)
• p+′,p−′,ph′ exp
• − E(p+′, p−′, ph

′|Θ)

, (11) from which we have the marginal distribution w.r.t. the visible layers, Prob(p+, p−|Θ) =

• ph′ exp
• − E(p+, p−, ph

′

|Θ)

• p+′,p−′,ph′ exp
• − E(p+′, p−′, ph

′|Θ)

. (12)

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Method

Gradients

We have the gradient of the log-likelihood of the probability in Eq.(12),

∂ log Prob(p+, p−|Θ) ∂Θ = ∂ ∂Θ

• log

ph′

exp

• − E(p+, p−, ph′

|Θ)

• − log
• p+′ ,p−′ ,ph′

exp

• − E(p+′

, p−′ , ph′ |Θ)

• =
• ph′

exp

• − E(p+, p−, ph′

|Θ)

• ph′ exp
• − E(p+, p−, ph′ |Θ)

· ∂

• − E(p+, p−, ph′

|Θ)

• ∂Θ

−

• p+′ ,p−′ ,ph′

exp

• − E(p+′

, p−′ , ph′ |Θ)

• p+′ ,p−′ ,ph′ exp
• − E(p+′ , p−′ , ph′ |Θ)

· ∂

• − E(p+′

, p−′ , ph′ |Θ)

• ∂Θ

= ∂

• − E(p+, p−, ph′

|Θ)

• ∂Θ
• (ph′

|p+,p−;Θ)

− ∂

• − E(p+′

, p−′ , ph′ |Θ)

• ∂Θ
• (p+′ ,p−′ ,ph′

|Θ)

= ∂

• − E(p+, p−, ph′

|Θ)

• ∂Θ
• data

− ∂

• − E(p+′

, p−′ , ph′ |Θ)

• ∂Θ
• model

(13)

Notice that ·data is approximated by p+0, p−0 and ph0, where p+0 and p−0 represent the original observed data, and ·model is approximated by p+L, p−L and phL, which denote the probability inferred via the learned model.

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Method

Update Rules

With the gradients in Eqs.(5-10) and the gradient in Eq.(13), we have the update rules in the gradient ascent algorithm similar to that in [Salakhutdinov et al., 2007], W +

i·

= W +

i· + γ(p+ i phdata − p+ i phmodel), i ∈ I+ u ,

(14) W −

i·

= W −

i· + γ(p− i phdata − p− i phmodel), i ∈ I− u ,

(15) Ci· = Ci· + γ(ciphdata − ciphmodel), i ∈ Cu, (16) b+

i

= b+

i + γ(p+ i data − p+ i model), i ∈ I+ u ,

(17) b−

i

= b−

i + γ(p− i data − p− i model), i ∈ I− u ,

(18) bh = bh + γ(phdata − phmodel), (19) where γ is the learning rate.

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Method

Algorithm

Algorithm 1 The algorithm

• f

contrastive divergence (CD) [Salakhutdinov et al., 2007] used in CRBM-IR.

1: Initialization: p+

i

= 0, p−

i

= 0, i = 1, . . . , m.

2: for i ∈ I+

u

do

3:

p+

i 0 = 1.

4: end for 5: Randomly take a set of items I−

u ⊆ I \ I+ u , where |I− u | = ρ|I+ u |.

6: for i ∈ I−

u

do

7:

p−

i 0 = 1.

8: end for 9: Calculate ph0 via Eq.(1). 10: Make ph0 binarized [Hinton, 2012]. 11: for ℓ = 1, . . . , L do 12:

Calculate p+ℓ via Eq.(2).

13:

Calculate p−ℓ via Eq.(3).

14:

Calculate phℓ via Eq.(1).

15:

Make phℓ binarized [Hinton, 2012].

16: end for

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Method

Algorithm

Algorithm 2 The algorithm of CRBM-IR.

1: Initialization: W +

i· ∼ N(0, 0.12), W − i· ∼ N(0, 0.12), Ci· ∼ N(0, 0.12), i ∈ I;

b+ = b− = bh = 0.

2: for t = 1, , . . . , T do 3:

for u ∈ U do

4:

Run the CD algorithm, and record p+0, p−0 and p+L, p−L.

5:

Update W +

i· , W − i· , Ci·, b+, b−, bh via Eqs.(14-19).

6:

end for

7: end for

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Experiments

Datasets

Table: Statistics of the first copy of the four datasets used in the empirical

• studies. Notice that the training data contain all rating records, while the

validation data and test data only contain records with a rating value equal to

• 5. The last column indicates the density of these datasets.

Dataset #Users #Items #Training #Validation #Test Density ML100K 943 1,682 60,000 4,262 4,240 3.78% ML1M 6,040 3,952 600,126 45,264 45,272 2.51% ML10M 71,567 10,681 6,000,034 308,673 308,702 0.78% Netflix 50,000 17,770 6,265,502 480,546 480,533 7.05% Notice that the data and code are available at http://csse.szu.edu.cn/staff/panwk/publications/CRBM-IR/.

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Experiments

Baselines

PopRank: Popularity-based ranking. PMF [Mnih and Salakhutdinov, 2008]: Probabilistic matrix factorization. RBM(E) [Salakhutdinov et al., 2007]: RBM with explicit feedback. RBM(I) [Jahrer and T¨

• scher, 2011]: RBM with implicit feedback.

CDAE [Wu et al., 2016]: Collaborative denoising autoencoder. Notice that we implement all the methods in Java without a third-party library for fair comparison.

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Experiments

Parameter Configurations (1/3)

For PMF , we fix the number of latent dimensions as d = 20, and search the regularization parameter value α ∈ {0.001, 0.01, 0.1} and the iteration numbers T ∈ {100, 500, 1000} via the performance of NDCG@5 on the validation data.

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Experiments

Parameter Configurations (2/3)

For the four network-based algorithms, i.e., RBM(E), RBM(I), CDAE and CRBM-IR, we set the size of the hidden layer as d = 20 and the sampling ratio ρ = 3 [Kabbur et al., 2013]. For RBM(E), RBM(I) and CRBM-IR, we adopt mini-batch with the batch size of 16, CD with L = 1 in Gibbs sampling, and the learning rate of γ = 0.05. In order to balance the efficiency and effectiveness, we use a momentum of 0.9 and search the optimal weight decay value αw ∈ {0.0005, 0.001, 0.005, 0.01} via the performance of NDCG@5 on the validation data. We also use early stop strategy to avoid overfitting.

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Experiments

Parameter Configurations (3/3)

For CDAE, we fix the corruption level q = 0.2, and search the learning rate η ∈ {0.001, 0.01, 0.1} and the regularization parameter λ ∈ {0.0001, 0.0005, 0.001, 0.01, 0.1} by checking the performance of NDCG@5 every 10 iterations within a maximum iteration of 300 on the validation data.

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Experiments

Evaluation Metrics

Precision@k Recall@k F1@k NDCG@k 1-call@k Among these metrics, k is set to 5, denoted as Prec@5, Rec@5, F1@5, NDCG@5 and 1-call@5, because users are usually only interested in a few top ranked items.

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Experiments

Main Results (1/3)

Table: Recommendation performance of our conditional RBM for item recommendation (CRBM-IR), and popularity-based ranking (PopRank), probabilistic matrix factorization (PMF), RBM with explicit feedback and implicit feedback (i.e., RBM(E) and RBM(I), respectively), and collaborative denoising autoencoder (CDAE) on ML100K and ML1M. The significantly best results are marked in bold.

Dataset Method Prec@5 Rec@5 F1@5 NDCG@5 1-call@5 ML100K PopRank 0.0761±0.0021 0.0892±0.0035 0.0687±0.0025 0.1057±0.0059 0.3051±0.0116 PMF 0.0002±0.0002 0.0003±0.0003 0.0002±0.0002 0.0003±0.0002 0.0010±0.0011 RBM(E) 0.0778±0.0029 0.0818±0.0029 0.0665±0.0019 0.1023±0.0044 0.3051±0.0102 RBM(I) 0.1026±0.0148 0.1302±0.0158 0.0955±0.0135 0.1424±0.0163 0.3959±0.0411 CDAE 0.1091±0.0060 0.1330±0.0036 0.0997±0.0043 0.1462±0.0067 0.4123±0.0199 CRBM-IR 0.1308±0.0094 0.1573±0.0096 0.1193±0.0076 0.1803±0.0106 0.4657±0.0292 ML1M PopRank 0.0893±0.0016 0.0691±0.0024 0.0638±0.0013 0.1067±0.0023 0.3314±0.0064 PMF 0.0001±0.0000 0.0000±0.0000 0.0000±0.0000 0.0001±0.0001 0.0003±0.0002 RBM(E) 0.0919±0.0038 0.0660±0.0052 0.0636±0.0037 0.1093±0.0048 0.3425±0.0160 RBM(I) 0.1098±0.0025 0.0943±0.0028 0.0834±0.0016 0.1339±0.0023 0.4117±0.0054 CDAE 0.1177±0.0032 0.0992±0.0032 0.0885±0.0027 0.1385±0.0030 0.4299±0.0131 CRBM-IR 0.1398±0.0019 0.1137±0.0031 0.1035±0.0015 0.1693±0.0040 0.4830±0.0036 Chen et al., (SZU) CRBM-IR Neurocomputing 26 / 31

Experiments

Main Results (2/3)

Table: Recommendation performance of our conditional RBM for item recommendation (CRBM-IR), and popularity-based ranking (PopRank), probabilistic matrix factorization (PMF), RBM with explicit feedback and implicit feedback (i.e., RBM(E) and RBM(I), respectively), and collaborative denoising autoencoder (CDAE) on ML10M and Netflix. The significantly best results are marked in bold.

Dataset Method Prec@5 Rec@5 F1@5 NDCG@5 1-call@5 ML10M PopRank 0.0699±0.0003 0.0947±0.0011 0.0665±0.0004 0.0988±0.0006 0.2820±0.0017 PMF 0.0220±0.0008 0.0187±0.0007 0.0164±0.0006 0.0297±0.0011 0.0883±0.0028 RBM(E) 0.0709±0.0028 0.0847±0.0038 0.0636±0.0027 0.0979±0.0036 0.2839±0.0101 RBM(I) 0.0845±0.0014 0.1122±0.0034 0.0797±0.0019 0.1187±0.0038 0.3369±0.0037 CDAE 0.0850±0.0016 0.1129±0.0026 0.0802±0.0017 0.1194±0.0033 0.3331±0.0048 CRBM-IR 0.1217±0.0007 0.1578±0.0013 0.1141±0.0007 0.1701±0.0012 0.4461±0.0021 Netflix PopRank 0.0604±0.0005 0.0370±0.0007 0.0353±0.0004 0.0645±0.0007 0.2390±0.0021 PMF 0.0369±0.0029 0.0155±0.0012 0.0181±0.0014 0.0421±0.0038 0.1311±0.0080 RBM(E) 0.0685±0.0020 0.0363±0.0027 0.0380±0.0017 0.0773±0.0025 0.2507±0.0067 RBM(I) 0.0786±0.0025 0.0501±0.0018 0.0480±0.0016 0.0898±0.0031 0.3012±0.0079 CDAE 0.0741±0.0014 0.0474±0.0015 0.0447±0.0009 0.0836±0.0024 0.2832±0.0048 CRBM-IR 0.1181±0.0006 0.0734±0.0004 0.0714±0.0004 0.1352±0.0008 0.4078±0.0011 Chen et al., (SZU) CRBM-IR Neurocomputing 27 / 31

Experiments

Main Results (3/3)

Observations: Our CRBM-IR performs the best in all cases, which clearly shows the effectiveness of our conversion from the original rating matrix to three different views of users’ behaviors. RBM(I) is worse than our CRBM-IR. The reason is that RBM(I) has no condition layer and thus cannot leverage examination data as CRBM-IR does. ...

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Conclusions and Future Work

Conclusions

We exploit the RBM model to generate a top-k recommendation list for each user by modeling the user’s explicit feedback. We treat the original rating matrix from a new perspective. Specifically, we convert users’ feedback data into three different views of the users’ behaviors, namely the examination matrix, the positive feedback matrix and the negative feedback matrix. By conducting experiments on four real-world datasets, we demonstrate the effectiveness of our CRBM-IR model in dealing with the item recommendation problem.

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Conclusions and Future Work

Future Work

We plan to use more complex network structures to model users’ explicit feedback, because the ability of RBM model to learn data features may be still limited. We plan to incorporate various information in our CRBM-IR model such as geographic location information, users’ text comments on items and users’ social network data.

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Thank you

Thank you!

We thank the support of National Natural Science Foundation of China Nos. 61872249, 61836005 and 61672358.

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