BordaRank: A Ranking Aggregation Based Approach to Collaborative Filtering


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1 BordaRank: A Ranking Aggregation Based Approach to Collaborative Filtering Yeming TANG Department of Computer Science and Technology Tsinghua University Beijing, China Qiuli TONG Information Technology Center Tsinghua, University Beijing, China Abstract Recommender systems are widely used in today s online applications. Traditional ratingoriented methods predict user ratings on items, but they fail to capture user preference among different items. This paper regards recommendation problem as a ranking task and proposes a new rankingoriented collaborative filtering framework based on ranking aggregation methods. In this framework, recommendation lists are generated according to item rankings given by users who are similar to the target user. Then, a twostep method called BordaRank is proposed to further explain the framework. The method first uses item collaborative filtering to predict unknown ratings and then uses Borda count method to aggregate item rankings of neighbors. Finally, BordaRank is modified as a pure rankingoriented method, which could be directly applied on the sparse rating matrix without rating prediction as an intermediate step. The methods are evaluated on real world movie rating data. Experimental results show that BordaRank improves the precision and recall of original ratingoriented methods and modified BordaRank also outperforms traditional methods. Index Terms Recommender System, Ranking Aggregation, Collaborative Filtering, Information Retrieval I. INTRODUCTION Recommender systems helps people filter out the information of interest from the mass quickly and effectively. Broadly speaking, recommendation problem is defined as a rating prediction problem. Since Netflix Prize competition was held in 2006, many rating prediction algorithms were proposed to solve recommendation problems. Most of these algorithms are based on collaborative filtering techniques. One category of these algorithms uses neighborhoodbased collaborative filtering methods to generate item recommendations. Such as Grouplens [1], one of the early generation of collaborative filtering algorithms, uses user rating data to calculate user similarity or item similarity. According to the calculated similarity, Grouplens predicts how well users will like new items based on similar users or items. Another category of these algorithms uses modelbased collaborative filtering methods to generate item recommendations. These algorithms use user rating data to train models to predict user ratings by machine learning or data mining algorithms [2]. For example, Paterek [3] improves the set of predictors used in Netflix Cinematch by adding biases to Singular Value Decomposition (SVD) models and performs a lower error rate than the original models. Liu et al. [4] extend Restricted Boltzmann Machine (RBM) and propose Content Based Restricted Boltzmann Machine (CBRBM); the model is applied on both rows and columns of rating matrix to predict a better result. Apart from ratingoriented recommendation algorithms, some researchers tried to combine information retrieval techniques into recommender systems, such as Learning to Rank (L2R), Normalized Discounted Cumulative Gain (NDCG) evaluation metric, etc. For example, Liu and Yang [5] propose EigenRank which improves memory based collaborative filtering methods by pairwise methods in learning to rank. In this paper, we propose an novel algorithm of rankingoriented collaborative filtering which uses ranking aggregation method to generate item recommendations. We first describe the framework of ranking aggregation based collaborative filtering algorithms. In this framework, we introduce an algorithm which uses Borda count method to aggregate item rankings derived from the predicted useritem rating matrix. We further modified our algorithm to a pure rankingoriented collaborative filtering algorithm which could be applied directly on the original sparse rating matrix. Finally, we perform a 5fold cross validation on MovieLens data set [6] to validate the effectiveness of our algorithms. We also discussed the impact of different neighborhood sizes and different scoring functions in section IV. II. RELATED WORKS There are mainly three parts of related works: neighborhoodbased collaborative filtering, rankingoriented collaborative filtering and ranking aggregation methods. A. Neighborhoodbased collaborative filtering Neighborhoodbased collaborative filtering (also called memorybased collaborative filtering) uses neighborhood to estimate the target user s ratings. The ratings are represented as a useritem rating matrix, which is highly sparse due to the huge number of users and items. The goal of neighborhoodbased collaborative filtering is to fill the useritem rating matrix. Neighborhoodbased methods fall into two categories: User Collaborative Filtering (UserCF) and Item Collaborative Filtering (ItemCF) /16/$31.00 copyright 2016 IEEE ICIS 2016, June 2629, 2016, Okayama, Japan
2 User collaborative filtering uses user neighborhood to predict the target user s ratings. The user neighborhood contains users whose ratings are similar to the target user s existing ratings. User collaborative filtering estimates the target user s unknown ratings by taking the similarity weighted average ratings of users in his or her neighborhood. The recommendation list could be generated by sorting the target user s unrated items in descending order and selecting topn recommendations. On the contrast, item collaborative filtering uses item neighborhood to predict the target user s ratings. The item neighborhood contains items which are similar to the target user s previous rated items. Item collaborative filtering estimates the item rating by taking the similarity weighted average ratings of the item s neighborhoods. Similarity measurement is the key concept in neighborhoodbased collaborative filtering. Because of the sparsity of the useritem rating matrix, one of the shortcomings of neighborhoodbased collaborative filtering is that it is unlikely to find highly similar users or items. The performance of neighborhoodbased collaborative filtering decreases when the rating matrix is sparse. To alleviate this problem, dimension reduction methods such as Singular Value Decomposition (SVD), Principle Component Analysis (PCA) etc. are applied to represent user vector in a reduced space. Many hybrid methods, such as contentboosted approaches [4], are also proposed. B. Rankingoriented collaborative filtering Traditional ratingoriented recommendation algorithms is aimed at minimizing the predicted rating errors. However, ratingoriented methods are deficient in capturing the item preference of the target user, due to user ratings could be biased depending on different users. Some users prefer to use higher ratings, while others prefer to use lower ratings. Though methods that normalize user ratings when aggregate the ratings of neighborhoods are proposed, Instead of estimating useritem rating matrix, rankingoriented collaborative filtering generates recommendation lists directly. Rankingoriented collaborative filtering methods combine thoughts from learning to rank, especially pairwise or listwise algorithms. For pairwise algorithm, the input is item pairs which indicate the target user s preference between rated items; the output is predicted preference on unrated items. One of the state of the art pairwise collaborative filtering algorithms is EigenRank [5]. It defines a preference function on item pairs, uses Kendall Rank Correlation Coefficient (KRCC) to measure similarity between users, and generate item recommendations by maximizing the loss function based on user preference. Listwise algorithm models user preference on item lists and directly learns on the lists to optimize model parameters. Unlike most of ratingoriented approaches use RootMean Square Error (RMSE) or Mean Average Error (MAE) as their loss functions, listwise approaches define the loss functions on whole lists. For example, Weimer et al. [7] propose CofiRank which uses Maximum Margin Matrix Factorization (MMMF) optimizes NDCG. C. Ranking aggregation methods Ranking aggregation methods have been used in recommender systems for aggregating multiple results from different recommendation algorithms [8]. However, rankingoriented collaborative filtering based on ranking aggregation is rarely researched due to most existing rankingoriented collaborative filtering methods are based on ranking generation. The most representative ranking aggregation methods falls into two categories: unsupervised methods and supervised methods. For example, for unsupervised methods, Borda count method [9] is voting mechanism which considers voter s preference on all candidates; for supervised methods, Cranking which proposed in [10] uses Markov model on permutations to aggregate multiple rankings. III. RANKING AGGREGATION BASED COLLABORATIVE FILTERING In this section, we presented a new rankingoriented collaborative filtering approach based on ranking aggregation methods. We first introduce a twostep recommendation method based on both ratingoriented and rankingoriented method. To alleviate the rating data sparsity problem, we use ratingoriented collaborative filtering to fill the rating matrix. We applied original Borda count method described in section IIIC to aggregate items rankings of the target user s neighbors. At last, we propose a pure ranking aggregation based collaborative filtering method, which could be directly applied on the original sparse rating matrix. A. Feature extraction and problem definition In general, a recommender system will collect users demographic information and items metadata. These are raw data to be extracted and transformed to feature representations. For categorical data (gender, occupation etc.), because they are hard to compare their similarities, categorical data are represented as vectors using oneofk coding. If a data field can take k different discrete values, the field should be represented as a kdimension binary vector, which corresponds with k possible values. Oneofk coding representation can also be used for multivalued data fields. For numerical data (age, date etc.), such transformation is not needed. In feature selection and extraction, we extract various features from raw data, for example, rate time in a day, rate date interval, ratings of different category items etc. For all numerical features, we further extract the standard deviation, skewness, mean, median, minimum, maximum etc. to describe the distribution attributes of the values. In order to adjusting features which are measured on different scales, all features are normalized using x x x = (1) x max x min where x is the feature vector and x is the normalized feature vector.
3 The problem is defined as follows. Given a set of m users U = {u 1,u 2,...,u m }, a set of n items I = {i 1,i 2,...,i n }, and a useritem rating matrix R m n, for each user u find a permutation ˆπ u, which is a bijection from I to {1, 2,...,n}, to fit the users tastes best. The rating that the user u gives to item i is denoted by r u,i, while r u,i =0if the user u has not rated item i. The set of items rated by user u is denoted by I u and the set of users who have rated item i is denoted by U i. B. Similarity measurement For neighborhoodbased collaborative filtering, similarity measurement is a major step. We use Euclidean distance based similarity to measure the user and item similarity. Euclidean distance based similarity is defined as 1 s x,y = (2) 1+ x y 2 where x and y are user or item feature vectors. We select k nearest neighbors as the neighborhood of user and item. The neighborhood of user u is denoted by N u and the neighborhood of item i is denoted by N i. Although our algorithm uses Euclidean distance based similarity measurement, it can be easily replaced by other similarity measurement, such as cosine similarity [11] or Pearson correlation coefficient [1] [12]. C. Borda count based collaborative filtering In this section, we introduce the basic ranking aggregation based collaborative filtering framework. A twostep method called BordaRank is introduced to further explain the proposed framework. In the first step, BordaRank uses item collaborative filtering to fill the sparse rating matrix. This step allows Borda count method to be applied for aggregating neighbors rankings in our method. Then, BordaRank derives rankings from the predicted rating matrix and generates the item recommendation to the target user. In our ranking aggregation based collaborative filtering framework, the ranking of items given by user u is denoted by π u. A recommendation algorithm should estimate a permutation ˆπ u : I {1, 2,...,n} for each user u as the recommendation list. For rating based collaborative filtering, the recommendation list ˆπ u of the target user u is generated by ranking all items according his or her estimated ratings in descending order: ˆπ u = π u where π u is the item permutation in descending order of the target user u s predicted ratings. However, for ranking aggregation based collaborative filtering algorithm, the estimated permutation ˆπ u of the target user u are generated by the items permutations of his or her neighbors: ˆπ u = f(π v1,π v2,...,π vk ) v N u where k = N u, π v is the permutation which ranks all items according previous estimated ratings of user v in descending order, function f is the ranking aggregation algorithm, for example, Borda count method in our algorithm which will be discussed later. Borda count [9] is originally an election method. In the election, each voter ranks all candidates in order of preference. Borda count gives each candidate a score according to his or her rank in each ballot. The candidate who gains the highest score wins the election. In the following part of this section, we will give a formal description of how Borda count works in our algorithm. Noting that Borda count method requires each voter to rank all the candidates, the sparse rating matrix has to be filled. The unknown ratings are predicted by item collaborative filtering: j N ˆr u,i = i I u s i,j r u,j (3) j N i I u s i,j where ˆr u,i is the estimated rating which user u will give to item i, r u,j is the actual rating which user u gives to item j, s i,j is the similarity between item i and item j, N i is the neighborhood of item i and I u is the item set rated by user i. Though useritem rating matrix is filled using item collaborative filtering, there could still be unpredicted ratings in the rating matrix, because the rated item set of the target user s neighborhood may not cover all the items: I v I v N u We adopt a simple strategy to deal with these unpredicted ratings by filling them by 0 values. The next step is aggregating neighbors permutations which derived from rating matrix and generate the estimated permutation of the target user. Inspired by Borda count method, we apply a scoring function on each item according to its ranking in the permutation. Then, for each item, we adds up the all the similarity weighted scores which the item gains among all neighbors permutations: Γ u (i) = v N u s u,v score u (i) (4) where Γ u (i) is the similarity weighted score sum of item i for the target user u, s u,v is the similarity between user u and user v. The most common formula of the scoring function [13] is linear function score u (i) =n π u (i)+1 (5) where n is the total number of items, π v (i) is the ranking of item i in permutation π v. Our algorithm uses this scoring function as well. The estimated permutation ˆπ u for user u is generated by sorting the similarity weighted score sum Γ u (i) of each item i in descending order. Other variants of scoring function formula could also be applied. For example, the General Election in Nauru held on June 19, 2010 used reciprocal function formula; the election of the most valuable NBA player uses piecewise function formula. We will further compare and discuss the selection of scoring function formula in section IV.
4 D. Modified Borda count based collaborative filtering Although Borda count based collaborative filtering performs a good result, it still needs a rating prediction step to alleviate data sparsity problem. In this section, we describe a pure rankingoriented collaborative filtering algorithm which applies modified Borda count method directly on the original sparse rating matrix without any rating prediction. Modified Borda count method does not require users to rank all items. It allows that each user could only rank his or her topk favorite items, where 1 k I. In this section, π u is represented as the partial permutation of user u on items: π u : I {1, 2,...,k u } where k u is the total number of items which user u have actually ranked. The scoring function needs to be modified to score u (i) =k u π u (i) (6) where π u (i) is the rank position of item i in the partial permutation π u. Substitute the scoring function into equation (4), we can obtain the similarity weighted score sum as the following equation: Γ u (i) = s u,v (k u π u (i)) (7) v N u Our rankingoriented collaborative filtering algorithm generates the estimated permutation ˆπ u of the target user u by sorting the score sum Γ u (i) in descending order. IV. EXPERIMENTS We use MovieLens data set [6] to evaluate the proposed method. This data set consists of 100,000 ratings by 943 users on 1682 movies. The rating ranges from 1 to 5, where 1 means the worst and 5 means the best. Each user has rated at least 20 movies. The data set also contains basic information of users and items, such as user gender, movie release date. A. Comparison with other algorithms We perform a 5fold cross validation to evaluate BordaRank and modified BordaRank algorithm. The data is equally split into five disjoint sets, from u1 to u5. Each set is used for test data while others are used for training data. We also implement user collaborative filtering and item collaborative filtering as baseline algorithms. In the evaluation, a user gives an item rating larger than 3, i.e. the user rates 4 or 5, is regarded as the user likes the item. F 1 score and coverage are used to measure all the algorithms and the results are shown in table I and table II. According to the result, BordaRank has a higher F 1 score than item collaborative filtering, which indicates that ranking aggregation in BordaRank could have a better recommendation effort than simply sorting the item ratings of the target user. The modified BordaRank performs a better result than other rating prediction based algorithms, which indicates that rankingoriented collaborative filtering can capture user preference on items more precisely. However, the item coverages of BordaRank and modified Borda Rank are lower than the coverage of item collaborative filtering BordaRank modified BordaRank Fig. 1. F1 score of BordaRank and modified BordaRank 0.10 BordaRank modified BordaRank Fig. 2. Coverage of BordaRank and modified BordaRank B. Impact of neighborhood size The item coverage of a recommender system indicates how many items are able to be recommended by the system. In collaborative filtering, the item coverage is mainly affected by the neighborhood size. We compare the impact of different neighborhood size on BordaRank and modified BordaRank. The result is shown in figure 1 and figure 2. In figure 1, the F1 score of both BordaRank and modified BordaRank increases with neighborhood size increases. The F1 score tends to converge when neighborhood size is larger than 50. In figure 2, the coverage of BordaRank and modified BordaRank decreases with neighborhood size increases. The coverage decreases sharply while neighborhood size is smaller than 30.
5 TABLE I THE F1 SCORE OF EACH METHOD IN 5FOLD CROSS VALIDATION UserCF ItemCF BordaRank modified BordaRank TABLE II THE ITEM COVERAGE OF EACH METHOD IN 5FOLD CROSS VALIDATION UserCF ItemCF BordaRank modified BordaRank C. Impact of different scoring functions We designed four different scoring functions for Borda count based collaborative filtering: linear score u (i) =n π u (i)+1 reciprocal score u (i) = 1 π u (i) logarithmic score u (i) =log 2 (n π u (i)+1) polynomial score u (i) =(n +1) 2 2(n +1)π u (i)+π u (i) 2 The results of the methods using these scoring functions are shown in table III, table IV and table V. According to the result, the logarithmic scoring function performs better than linear and reciprocal scoring functions, though the latter are more common. We further implements and evaluates modified BordaRank using the logarithmic scoring function. In the pure rankingoriented collaborative filtering, the precision and recall of modified BordaRank using logarithmic scoring function is lower than using linear scoring function, while the coverage of modified BordaRank using logarithmic scoring function is better. V. CONCLUSIONS AND FUTURE WORK In this paper, we proposed a new rankingoriented collaborative filtering framework based on ranking aggregation methods. Differs from traditional ratingoriented collaborative filtering and ranking generation based collaborative filtering, our algorithm generates item recommendations from item rankings which are given by the neighbors of the target user. In the proposed framework, we use Borda count method as ranking aggregation method and provide BordaRank algorithm. BordaRank uses rating predict results from item collaborative filtering method and aggregates neighborhoods rankings to give item recommendations. Experimental results shows BordaRank improves precision and recall of item collaborative filtering method on a considerable scale. We further proposed modified BordaRank which is a pure rankingoriented collaborative filtering method. The modified BordaRank could be directly applied on the sparse rating matrix. Finally, we evaluated both methods on MovieLens data set and compared the impact of different neighborhood sizes and different selections of scoring function. Both BordaRank and modified BordaRank have higher precision and recall than other baseline methods. For future work, we would like to investigate different ranking aggregation methods for our ranking aggregation based collaborative filtering framework. We would also like to apply machine learning algorithms to optimize the parameters in the methods. REFERENCES [1] P. Resnick, N. Iacovou, M. Suchak, P. Bergstrom, and J. Riedl, GroupLens : An Open Architecture for Collaborative Filtering of Netnews, Proceedings of the 1994 ACM conference on Computer supported cooperative work, pp , [2] J. L. Herlocker, J. A. Konstan, L. G. Terveen, and J. T. Riedl, Evaluating collaborative filtering recommender systems, ACM Transactions on Information Systems (TOIS), vol. 22, no. 1, pp. 5 53, [Online]. Available: [3] A. Paterek, Improving regularized singular value decomposition for collaborative filtering, in Proceedings of KDD cup and workshop, 2007, pp [4] Y. Liu, Q. Tong, Z. Du, and L. Hu, ContentBoosted Restricted Boltzmann Machine for Recommendation, in Artificial Neural Networks and Machine Learning ICANN 2014, S. Wermter, C. Weber, W. Duch, T. Honkela, P. KoprinkovaHristova, S. Magg, G. Palm, and A. Villa, Eds. Springer International Publishing, 2014, pp [Online]. Available: 97 [5] N. N. Liu and Q. Yang, EigenRank, in Proceedings of the 31st annual international ACM SIGIR conference on Research and development in information retrieval  SIGIR 08, 2008, p. 83. [Online]. Available: s &partnerid=tzotx3y1 [6] F. M. Harper and J. A. Konstan, The MovieLens Datasets: History and Context, ACM Trans. Interact. Intell. Syst., vol. 5, no. 4, pp. 19:119:19, [Online]. Available: [7] M. Weimer, A. Karatzoglou, M. Bruch, Q. V. Le, and A. Smola, CofiRank Maximum Margin Matrix Factorization for Collaborative Ranking, Advances in neural information processing systems, pp. 1 3, [Online]. Available: [8] H. Wu, Y. Hua, B. Li, and Y. Pei, Personalized Recommendation via Rank Aggregation in Social Tagging Systems, in Proceedings of the 10th International Conference on Fuzzy Systems and Knowledge Discovery. Shenyang: IEEE, 2013, pp [9] M. Dummett, The Borda count and agenda manipulation, Social Choice and Welfare, vol. 15, no. 2, pp , 1998.
6 TABLE III THE PRECISION OF METHODS USING DIFFERENT SCORING FUNCTION linear reciprocal logarithmic polynomial modified linear modified logarithmic TABLE IV THE RECALL OF METHODS USING DIFFERENT SCORING FUNCTION linear reciprocal logarithmic polynomial modified linear modified logarithmic TABLE V THE COVERAGE OF METHODS USING DIFFERENT SCORING FUNCTION linear reciprocal logarithmic polynomial modified linear modified logarithmic [10] J. D. Lafferty, G. Lebanon, and J. D. Lafferty, Cranking: Combining Rankings Using Conditional Probability Models on Permutations, in Proceedings of the Nineteenth International Conference on Machine Learning, ser. ICML 02. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 2002, pp [Online]. Available: [11] S. Wang, J. Sun, B. J. Gao, and J. Ma, VSRank: A Novel Framework for RankingBased Collaborative Filtering, ACM Transactions on Intelligent Systems and Technology (TIST), vol. 5, no. 3, pp. 51:151:24, [Online]. Available: [12] J. Herlocker, J. Konstan, and J. Riedl, An Empirical Analysis of Design Choices in NeighborhoodBased Collaborative Filtering Algroithms, Information retrieval, pp , [Online]. Available: [13] P. Emerson, The original Borda count and partial voting, Social Choice and Welfare, vol. 40, no. 2, pp , 2013.
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