Mixed Similarity Learning for Recommendation with Implicit Feedback Mengsi Liu, Weike Pan # , Miao Liu, Yaofeng Chen, Xiaogang Peng ∗ and Zhong Ming ∗ College of Computer Science and Software Engineering Shenzhen University Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 1 / 28
Introduction Problem Definition and Illustration Recommendation with implicit feedback Input: Implicit feedback in the form of (user, item) pairs Output: A personalized ranked list of unexamined items for each user Figure: Illustration of mixed similarity learning. Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 2 / 28
Introduction Notations Table: Some notations. user set, u ∈ U , |U| = n U item set, i , i ′ , j ∈ I , |I| = m I users that examined i U i items examined by u I u nearest neighbors of item i N i R = { ( u , i ) } examination records s ( p ) predefined similarity between i and i ′ ii ′ s ( ℓ ) learned similarity between i and i ′ ii ′ s ( m ) mixed similarity between i and i ′ ii ′ V i · , W i ′ · ∈ R 1 × d item-specific latent feature vector b i , b j item bias r ( p ) r ( m ) r ( ℓ ) ˆ ui , ˆ ui , ˆ predicted preference ui T iteration number λ s , α tradeoff parameter Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 3 / 28
Introduction Overall of Our Solution ICF [Deshpande and Karypis, TOIS 2004]: item-oriented collaborative filtering with predefined similarity BPR [Rendle et al., UAI 2009]: recommendation with pairwise preference learning FISMauc [Kabbur, Ning and Karypis, KDD 2013]: recommendation with learned similarity We combine predefined similarity, learned similarity and pairwise preference learning in a single framework P-FMSM (pairwise factored mixed similarity model) P-FISM (pairwise factored item similarity model) is a special case of P-FMSM with learned similarity only. Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 4 / 28
Method Prediction Rule of FISMauc and P-FISM The predicted rating of user u on item i ( i ∈ I u ), 1 r ui = b i + s ( ℓ ) � ˆ (1) ii ′ |I u \{ i }| � i ′ ∈I u \{ i } ii ′ = V i · W T where s ( ℓ ) i ′ · is the learned similarity between item i and item i ′ . The predicted rating of user u on item j ( j ∈ I\I u ), 1 r uj = b j + s ( ℓ ) � ˆ (2) ji ′ � |I u | i ′ ∈I u ji ′ = V j · W T where s ( ℓ ) i ′ · is the learned similarity between item j and item i ′ . Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 5 / 28
Method Prediction Rule of P-FMSM The predicted rating of user u on item i ( i ∈ I u ), 1 s ( m ) r ui = b i + � ˆ (3) ii ′ |I u \{ i }| � i ′ ∈I u \{ i } where s ( m ) ii ′ + λ s s ( p ) = ( 1 − λ s ) s ( ℓ ) ii ′ s ( ℓ ) ii ′ is the mixed similarity . ii ′ The predicted rating of user u on item j ( j ∈ I\I u ), 1 s ( m ) r uj = b j + � ˆ (4) ji ′ � |I u | i ′ ∈I u where s ( m ) ji ′ + λ s s ( p ) = ( 1 − λ s ) s ( ℓ ) ji ′ s ( ℓ ) ji ′ is the mixed similarity . ii ′ Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 6 / 28
Method Objective Function The objective function of BPR, P-FISM and P-FMSM, � � � f uij min (5) Θ u ∈U i ∈I u j ∈I\I u where f uij = − ln σ (ˆ r uij )+ α 2 � V i · � 2 + α 2 � V j · � 2 + α i ′ ∈I u || W i ′ · || 2 2 � b i � 2 + α 2 � b j � 2 , F + α � 2 r uij = ˆ r ui − ˆ r uj , and Θ = { W i · , V i · , b i , i = 1 , 2 , . . . , m } denotes the set of ˆ parameters to be learned. Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 7 / 28
Method Gradients of P-FISM For a triple ( u , i , j ) , we have the gradients, ∂ f uij ∇ b j r uij )( − 1 ) + α b j , = − σ ( − ˆ = (6) ∂ b j ∂ f uij ∇ V j · r uij )( − ¯ U u · ) + α V j · , = − σ ( − ˆ = (7) ∂ V j · ∂ f uij ∇ b i r uij ) + α b i , = = − σ ( − ˆ (8) ∂ b i ∂ f uij U − i ∇ V i · r uij )¯ u · + α V i · , = − σ ( − ˆ = (9) ∂ V i · ∂ f uij V i · V j · ∇ W i ′ · r uij )( = − σ ( − ˆ = |I u \{ i }| α − |I u | α ) ∂ W i ′ · α W i ′ · , i ′ ∈ I u \{ i } + (10) ∂ f uij r uij ) − V j · ∇ W i · |I u | α + α W i · = − σ ( − ˆ = (11) ∂ W i · U − i where ¯ U u · = i ′ ∈I u W i ′ · and ¯ i ′ ∈I u \{ i } W i ′ · . 1 1 √ � u · = √ � |I u \{ i }| |I u | Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 8 / 28
Method Gradients of P-FMSM For a triple ( u , i , j ) , we have the gradients, ∂ f uij ∇ b j r uij )( − 1 ) + α b j , = = − σ ( − ˆ (12) ∂ b j ∂ f uij ∇ V j · r uij )( − ¯ U u · ) + α V j · , = − σ ( − ˆ = (13) ∂ V j · ∂ f uij ∇ b i r uij ) + α b i , = − σ ( − ˆ = (14) ∂ b i ∂ f uij U − i ∇ V i · r uij )¯ u · + α V i · , = − σ ( − ˆ = (15) ∂ V i · ∂ f uij V i · V j · r uij )(( 1 − λ s ) + λ s s ( p ) ∇ W i ′ · = = − σ ( − ˆ ii ′ )( |I u \{ i }| α − |I u | α ) ∂ W i ′ · α W i ′ · , i ′ ∈ I u \{ i } + (16) r uij ) − (( 1 − λ s ) + λ s s ( p ) ii ′ ) V j · ∂ f uij ∇ W i · + α W i · = − σ ( − ˆ = (17) ∂ W i · |I u | α i ′ ∈I u (( 1 − λ s ) + λ s s ( p ) U u · = ii ′ ) W i ′ · and where ¯ 1 √ � |I u | i ′ ∈I u \{ i } (( 1 − λ s ) + λ s s ( p ) U − i ¯ ii ′ ) W i ′ · . 1 √ u · = � |I u \{ i }| Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 9 / 28
Method Update Rules of P-FISM and P-FMSM For a triple ( u , i , j ) , we have the gradients, b j b j − γ ∇ b j = (18) V j · V j · − γ ∇ V j · = (19) b i b i − γ ∇ b i = (20) V i · V i · − γ ∇ V i · = (21) W i ′ · − γ ∇ W i ′ · , i ′ ∈ I u \{ i } W i ′ · = (22) W i · W i · − γ ∇ W i · = (23) where γ is the learning rate. Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 10 / 28
Method Algorithm 1: Initialize the model parameters Θ 2: for t = 1 , . . . , T do for t 2 = 1 , . . . , |R| do 3: Randomly pick up a pair ( u , i ) ∈ R 4: Randomly pick up an item j from I\I u 5: Calculate the gradients via Eq.(12-17) 6: Update the model parameters via Eq.(18-23) 7: end for 8: 9: end for Figure: The SGD algorithm for P-FMSM. Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 11 / 28
Experiments Datasets For direct comparative empirical studies, we use four public datasets 1 . Table: Statistics of the data used in the experiments, including numbers of users ( |U| ), items ( |I| ), implicit feedback ( |R| ) of training records (Tr.) and test records (Te.), and densities ( |R| / |U| / |I| ) of training records. Data set |U| |I| |R| (Tr.) |R| (Te.) |R| / |U| / |I| (Tr.) MovieLens100K 943 1682 27688 27687 1 . 75 % MovieLens1M 6040 3952 287641 287640 1 . 21 % UserTag 3000 2000 123218 123218 2 . 05 % Netflix5K5K 5000 5000 77936 77936 0 . 31 % 1 https://sites.google.com/site/weikep/GBPRdata.zip Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 12 / 28
Experiments Evaluation Metrics We use Prec @ 5 and NDCG @ 5 in the experiments, 5 1 1 δ ( L u ( p ) ∈ I te Pre @ 5 � � , = u ) |U te | 5 u ∈U te p = 1 2 δ ( L u ( p ) ∈I u 5 te ) − 1 1 1 NDCG @ 5 � � = , |U te | log ( p + 1 ) � min ( 5 , | I te u | ) 1 p = 1 u ∈U te p = 1 log ( p + 1 ) where L u ( p ) is the p th item for user u in the recommendation list, and δ ( x ) is an indicator function with value of 1 if x is true and 0 otherwise. Note that U te and I te u denote the set of test users and the set of examined items by user u in the test data, respectively. Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 13 / 28
Experiments Baslines Ranking based on items’ popularity (PopRank) Item-oriented collaborative filtering with Cosine similarity (ICF) ICF with an amplifier on the similarity (ICF-A), which favors the items with higher similarity Factored item similarity model with AUC loss (FISMauc) and RMSE loss (FISMrmse) Hierarchical poisson factorization (HPF) Bayesian personalized ranking (BPR) A factored version of adaptive K nearest neighbors based recommendation KNN (FA-KNN) Group preference based BPR (GBPR) Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 14 / 28
Experiments Initialization of Model Parameters We use the statistics of training data to initialize the model parameters, n b i � y ui / n − µ = u = 1 n b j y uj / n − µ � = u = 1 V ik ( r − 0 . 5 ) × 0 . 01 , k = 1 , . . . , d = W i ′ k ( r − 0 . 5 ) × 0 . 01 , k = 1 , . . . , d = where r (0 ≤ r < 1) is a random variable, and µ = � n � m i = 1 y ui / n / m . u = 1 Liu et al. (CSSE, SZU) Mixed Similarity Learning KBS 2017 15 / 28
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