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In this paper, we propose a listwise approach for constructing user-specific rankings in recommendation systems in a collaborative fashion. We contrast the listwise approach to previous pointwise and pairwise approaches, which are based on treating either each rating or each pairwise comparison as an independent instance respectively. By extending the work of (Cao et al. 2007), we cast listwise collaborative ranking as maximum likelihood under a permutation model which applies probability mass to permutations based on a low rank latent score matrix. We present a novel algorithm called SQL-Rank, which can accommodate ties and missing data and can run in linear time. We develop a theoretical framework for analyzing listwise ranking methods based on a novel representation theory for the permutation model. Applying this framework to collaborative ranking, we derive asymptotic statistical rates as the number of users and items grow together. We conclude by demonstrating that our SQL-Rank method often outperforms current state-of-the-art algorithms for implicit feedback such as Weighted-MF and BPR and achieve favorable results when compared to explicit feedback algorithms such as matrix factorization and collaborative ranking. 

In this paper, we consider the Collaborative Ranking (CR) problem for recommendation systems. Given a set of pairwise preferences between items for each user, collaborative ranking can be used to rank un-rated items for each user, and this ranking can be naturally used for recommendation. It is observed that collaborative ranking algorithms usually achieve better performance since they directly minimize the ranking loss; however, they are rarely used in practice due to the poor scalability. All the existing CR algorithms have time complexity at least O(|Ω|r) per iteration, where r is the target rank and |Ω| is number of pairs which grows quadratically with number of ratings per user. For example, the Netflix data contains totally 20 billion rating pairs, and at this scale all the current algorithms have to work with significant subsampling, resulting in poor prediction on testing data. In this paper, we propose a new collaborative ranking algorithm called Primal-CR that reduces the time complexity toO(|Ω|+d1d2r), where d1 is number of users and d2 is the averaged number of items rated by a user. Note that d1, d2 is strictly smaller and open much smaller than |Ω|. Furthermore, by exploiting the fact that most data is in the form of numerical ratings instead of pairwise comparisons, we propose Primal-CR++ with O(d1d2(r + log d2)) time complexity. Both algorithms have better theoretical time complexity than existing approaches and also outperform existing approaches in terms of NDCG and pairwise error on real data sets. To the best of our knowledge, this is the first collaborative ranking algorithm capable of working on the full Netflix dataset using all the 20 billion rating pairs, and this leads to a model with much better recommendation compared with previous models trained on subsamples. Finally, compared with classical matrix factorization algorithm which also requires O(d1 d2r) time, our algorithm has almost the same efficiency while making much better recommendations since we consider the ranking loss.