LambdaMF: Learning Nonsmooth Ranking Functions in Matrix Factorization Using Lambda

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1 2015 IEEE International Conference on Data Mining LambdaMF: Learning Nonsmooth Ranking Fnctions in Matrix Factorization Using Lambda Gang-He Lee Department of Compter Science and Information Engineering National Taiwan University Sho-De Lin Department of Compter Science and Information Engineering National Taiwan University Abstract This paper emphasizes optimizing ranking measres in a recommendation problem. Since ranking measres are nondifferentiable, previos works have been proposed to deal with this problem via approximations or lower/pper bonding of the loss. However, sch mismatch between ranking measres and approximations/bonds can lead to non-optimal ranking reslts. To solve this problem, we propose to model the gradient of non-differentiable ranking measre based on the idea of virtal gradient, which is called lambda in learning to rank. In addition, noticing the difference between learning to rank and recommendation models, we prove that nder certain circmstance the existence of poplar items can lead to nlimited norm growing of the latent factors in a matrix factorization model. We frther create a novel reglarization term to remedy sch concern. Finally, we demonstrate that or model, LambdaMF, otperforms several state-of-the-art methods. We frther show in experiments that in all cases or model achieves global optimm of normalized discont cmlative gain dring training. Detailed implementation and spplementary material can be fond at ( b /). I. INTRODUCTION With the prosperos emergence of e-commerce, recommendation system has played a big role in online stores. In general, sers might prefer a recommendation system that selects a sbset of candidate items for them to choose. In this sense the top-n recommended list is commonly adopted to evalate a ranking-oriented recommendation system. Sch evalation consists of an Information Retrieval (IR) measre, i.e. ranking measre, and a ct-off vale N. For example, Precision@5 specifies the precision vale given top 5 recommended items. However, the commonly sed ranking measres are either indiscriminate or discontinos over certain model space. Fig. 1 illstrates this behavior: adding a small Δs to the score of docment c, s(c) cannot alter the ranking of c. Ths a derivative eqal to 0 appears in the measring objective fnction. However, adding a larger vale Δs to the score of c can case the ranking of docment c to immediately jmp from 3 rd to 2 nd, which leads to discontinity on the ranking measre de to the swapping of two docments. The property of zero or discontinity gradient discorages the sage of common gradient descent optimization techniqes, which sally leads to non-optimal soltions. To handle sch drawback, researchers have designed novel loss fnction for optimization. They are sally derived from a Figre 1. Ranking based on different model score smooth approximation [1] or a bond [2] of the original ranking measres. However, there is a concern that the mismatch between the loss fnction and the ranking measre wold reslt in a gap between the optimal and learned models [3]. In the field of learning to rank, lambda-based methods [4] have been proposed to bypass converting challenging ranking measres into loss fnctions, by sing a proxy of the gradient for ranking measres. The reqired ranking and sorting in ranking measres are then incorporated into the model by calclating the virtal gradient, a.k.a lambda, after sorting. Inheriting the idea of lambda, we design a new recommendation model named Lambda Matrix Factorization (LambdaMF) to combine the idea of lambda with the highly-sccessfl matrix factorization model in collaborative filtering. To or knowledge, LambdaMF is the first ranking matrix factorization model to achieve global training optimm of a ranking measre: the Normalized Discont Cmlative Gain (NDCG). Or experiment also shows that LambdaMF directly optimize NDCG 98.4% of time dring the optimization process. Moreover, we have observed an interesting effect while learning LambdaMF. Sometimes the norm of some latent parameters can grow nlimitedly de to the lack of sppression power to constrain the effect of poplar items dring learning. In this paper we prove that sch phenomenon can always happen with the existence of some poplar items and some ranking-oriented matrix factorization models that satisfy certain criteria. We frther prpose a novel reglarization term to combat norm growing and empirically show the nmerical stability of or soltion /15 $ IEEE DOI /ICDM

2 Finally, we condct experiments with MovieLens and Netflix datasets. Empirical comparison shows that or method otperforms several state-of-the-art models. In a ntshell, or main contribtion in this paper can be listed as follows: We create a new matrix factorization algorithm modeling the gradient of ranking measre. To or best knowledge, LambdaMF is the first model to show that it empirically achieves global optimm of training data. We prove that some ranking-oriented matrix factorization model wold sffer from nlimited norm growing de to poplar item effect. We create a novel reglarization term to deal with the poplar item effect on recommendation systems. We condct experiments to show that the model is scalable to large recommendation datasets and otperforms several state-of-the-art models. The remaining sections are organized as follows. Section II presents several related works and comparative discssion. In Section III, the proposed LambdaMF and some related discssion are presented. Afterwards, experiment reslts and conclsion are in Section IV and Section V. II. RELATED WORK The designed model is strongly related to researches on both collaborative filtering and learning to rank. We discss them briefly below. A. Learning to Rank In the context of Learning to Rank (LTR), the problem is sally formlated with qeries Q, docments D and relevance levels R for docments in respect to each qery. A featre vector is often given for each docment-qery pair as the focs is on optimizing the ranking measre for each qery. The essential difference from a recommendation model is ths revealed: there is no qeries or featres in a collaborative filtering task. Nevertheless, people sally evalate the reslts of Top-N recommendation by correlating sers to qeries and items to docments. To be general, we wold se the same ser to qery and item to docment analogy to describe the LTR task in the remaining paragraphs. There are several different LTR measres that can be simply divided into explicit feedback and implicit relevance settings. For explicit feedback, for each observed item, sers explicitly provide a specific score for it. In this scenario, the performance is often evalated sing normalized disconted cmlative gain (NDCG) with a ct off vale K. Given relevance level R i and predicted ranking l i for every recommended item i in the sorted list based on sorted model score, the NDCG@K is generated as follows: NDCG@K = DCG@K K max(dcg@k),dcg@k = 2 Ri 1 log 2 l i +1 l i=1 B. Collaborative Filtering Classically, the recommendation task is eqivalent to the task of rating prediction. Well-known examples can be fond in competitions sch as the Netflix Prize. Common methods can be divided into memory based and model based methods. The major distinction between memory based and model based methods lies in whether the process of learning a model is condcted. Both methods can be regarded as collaborative filtering methods, as they meet the basic assmption that similar sers tend to share similar interests. However, recently the ranking-oriented CF methods [1], [5] [8] attract more attention becase ranking prediction can better captre the activity of personal preference ordering of items. Example can be fond in CLiMF [1], optimizing an approximation of mean reciprocal rank (MRR), and CoFi Rank [2], optimizing a bond of NDCG. However, in CLiMF, the 1 ranking of an item is approximated as the sigmoid fnction of its model score. It only considers the actal vale of the model score and rather than the ordinal ranking of the model score. In contrast, LambdaMF directly deals with actal ranking de to is incorporation of sorting. For CoFi Rank, the loss fnction is derived from a bond of NDCG. However, there is no garantee abot the distance between the bond and actal real ranking fnction. Conversely, LambdaMF directly formlate the ranking fnction into its gradient and therefore there is no need to worry abot the similar extent of approximation or bond. Comparing to the above models, LambdaMF is more general; it presents a template model which can be fine-tned for different types of ranking fnctions given sparse data. III. LAMBDA MATRIX FACTORIZATION This section first introdces the backgrond of lambda, then discsses how lambda gradient can be incorporated into a matrix factorization model. To reflect the difference between learning to rank and recommendation system, a novel reglarization term is created to address the concern of overflow from sing lambda. Finally, a time complexity analysis is presented to demonstrate the efficiency of the model A. RankNet and LambdaRank In a series of researches on learning to rank [4], RankNet [9] was first proposed to handle the problem sing gradient. In this model, given a qery q, for every pair of docments i and j with different relevance level R q,i >R q,j, RankNet aims to promote the relative predicted probability of these pair in terms of their model score s i, s j. Then the relative probability, docment i ranked higher than docment j, generated from this model is defined as (1) P q i,j = 1 (1) 1+exp( σ(s i s j )) σ is simply a parameter controlling the shape of sigmoid fnction. Finally, the loss fnction is defined as the cross entropy between the pair-wise preference ordering of the training data and the relative probability of the model. Given the same assmption with relevance level R q,i >R q,j, the 824

3 probability of pairwise real preference ordering between i, j is defined as P q i,j =1. Finally, the loss fnction and gradient for the qery q, docment i, j is defined as (2), (3) C q i,j = P q i,j log P q i,j (1 P q i,j )log(1 P q i,j ) (2) C q i,j = log(1 + exp( σ(s i s j ))) σ = 1+exp(σ(s i s j )) = Cq i,j (3) Based on the idea of RankNet, formlating pair-wise ranking problem into gradient descent for a pair of docments, LambdaRank [3] is proposed to formlate gradient of a listwise ranking measre, which is called λ. Several possible λs have been tested in [3], while the athors report the best λ sed for optimizing NDCG as (4) with ΔNDCG defined as the absolte NDCG gain from swapping docments i and j C q i,j s i = λ q i,j = σ ΔNDCG 1+exp(σ(s i s j )) = Cq i,j s j (4) C q i,j w = Cq i,j w + Cq i,j w = λq i,j w λq i,j w B. Lambda in Matrix Factorization To introdce λ into a collaborative filtering model, the first thing to notice is that we do not assme the existences of ser or item featres de to privacy concern. Hence, we adopt MF as or latent factor model representing featres for sers and items. Given M sers and N items, there is a sparse rating matrix R M,N, in which each row represents a ser while each colmn represents an item. The vales in the matrix are only available when the corresponding ratings exist in the training dataset, and the task of MF is to gess the remaining ratings. Similar to Singlar Vale Decomposition, the idea behind MF is to factorize the rating matrix R M,N into two matrices U M,d and V N,d with pre-specified latent factor dimension d. Given a [1,M] and b [1,N], U a represents the latent factor of the a th ser while V b represents the latent factor of the b th item. Finally, the rating prediction can be formlated as the matrix prodct below. The inner-prodct of U,V i represents the predicted score for each (, i) pair: R = UV T Using MF as the basis of or latent model, now we are ready to introdce LambdaMF. Sppose the goal is to optimize a ranking measre fnction f, we first constrct sers and items latent factors of dimension d as or model basis. Then we assme that there is a virtal cost fnction C optimizing f, and the gradient of C in respect to model score is defined as λ, C i,j = λ i,j (5) However, to obtain the gradient in respect to model parameters, we need to first discss the differences between LambdaRank and LambdaMF. First, given a ser, when a pair of items (i, j) is chosen, nlike in LTR which only model weight vector is reqired to be pdated, in a recommendation task we have item latent factors of i and j and ser latent factor of to pdate. Second, in the original LambdaRank model, the score of the pair (, i) in the model is generated from the prediction otpts of a neral network, while in MF sch score is generated from the inner-prodct of U and V i. Hence, we have si = V i and si = U. Here we are ready to describe how to apply SGD to learn model parameters. Given the first difference described above, it is apparent that there are 2 item latent factors and 1 ser latent factor to be considered when generating the stochastic gradient. Moreover, given the second difference, we can exploit the chain rle dring the derivation of gradient. Finally, given a ser, a pair of item i and j, with R,i >R,j, the gradient can be compted as (6): Ci,j V i Ci,j V j Ci,j U = C i,j = C i,j = λ V i,ju (6) i = λ V i,ju j = C i,j + C i,j = λ U i,j(v i V j ) The specific definition of λ i,j is not given in this section. In contrast, we want to convey that the design of λ i,j can be simple and generic. In [10], to deal with LTR problems, several λ have been proposed for optimizing corresponding metrics. In or opinion, any fnction with positive range can be viewed as an legitimate lambda as long as it can promote the preferred item and penalize the less relevant item. Here we emphasize the positive vales of lambda as positive gradient which can always enlarge the gap between each pair of items with different relevance levels and leads to faster optimization. C. Poplar Item Effect As wold be shown later, we fond the existence of poplar items can case a serios problems while trying to apply the concept of Lambda in MF. Moreover, we will show that the phenomena is niversal for MF models with positive gradient C i,j > 0 for all j with R,i >R,j sch as BPRMF [5]. Sppose that there exists a poplar item î liked by all sers who have rated it, then for each ser among them, the rating for item î wold keep climbing dring the optimization process as C i,j is positive: it pshes the score of U Vi T pwards. Moreover, if the latent factors among all sers observing the item are similar, the increasing of predicted score of î for every ser observing î wold not case the decrease of predicted score of î for the other sers who observe î. Hence, the latent factor of item î wold keep growing and soon case overflow. Formally speaking, let s first denote the set of sers observing î in the training set as Rel(î), the rating of î for ser k as R, the latent factors in the iteration t as U t k,î /V t. Then the above discssion yields the Theorem 1, whose proof is shown in spplementary materials. Theorem 1 (Poplar Item Theorem): If there exists an item î, sch that for all sers k Rel(î), R R k,î k,j for all other observed item j of ser k. Frthermore, if after certain iteration τ, latent factors of all sers k Rel(î) converge to certain extent. That is, there exists a vector Ū t sch that for all 825

4 k Rel(î) in all iteration t>τ, inner-prodct(uk t, U τ ) > 0. Then the norm of Vî will eventally grow to infinity for any MF model satisfying the constraint that Cṷ i,j > 0 for all j sî with R >R,î,j, as shown below: lim n V n î 2 = Note that the existence of item î can be easily satisfied since the set size of Rel(î) can be arbitrary (even with only one ser). The other condition in Theorem 1, namely the convergence of latent factors of sers observing î, can also happen freqently in CF since we generally believe that sers with similar transaction history, i.e. bying the same item î, shall have similar latent factors. Frthermore, since the theorem only reqires Cṷ i,j > 0 for all j with sî R >R,î,j, it is not restricted to LambdaMF bt applies to all other matrix factorization models satisfying sch constraint. Empirical evidences will also be shown in the experiment section. D. Reglarization in LambdaMF To address the above concern, an intitive soltion is to add a reglarization term to the original cost C (we call this new cost Ċ). For example, a commonly sed L2 reglarization term, i.e. weight decay, can confine the norm of the matrix U and V and ths restrict the complexity of model. Then the new cost fnction Ċ can be formlated as Ċ = Ci,j α 2 ( U V 2 2 ) (7) all(,i,j) data However, since the norm of Vî can grow very fast, the L2 reglarization term may not effectively stop the norm of Vî from going to infinity when the effect of weight decay is not significant. On the other extreme, if α is large, the capability of the model is strongly restricted even when the norm of all parameters are small. Hence, we arge that the tilization of norm restriction based method is blind; it cannot adapt its reglarization intensity for different magnitdes of parameter norms. Here, we propose another reglarization to confine the inner-prodct of U Vi T (which is the prediction otcome of the model) to be as close to the actal rating R,i for every observed rating, i as possible, as defined in (8) Ĉ = C ˆ i,j ˆ C i,j V j C ˆ i,j all(,i,j) data C i,j α 2 all(,i) data (R i,j U V T i ) 2 (8) = λ i,ju i + α(r,i U V T i )U (9) = λ i,ju i + α(r,j U V T j )U = λ i,j(v i V j )+α(r,i U Vi T )V i + α(r,j U V T j )V j We choose the sqare error between R,i and U Vi T becase it is differentiable and ths can be optimized easily. This eqation is known as mean-sqare-error (MSE), which is often adopted in a rating prediction task as an objective fnction [11]. There are two advantages in adopting the MSE reglarization term. 1) Intensity-adaptive reglarization term: It can be derived from (8) that the gradient of C with respect to parameters is (9). Sch reglarization can adapt its intensity to the difference between the predicted score and the real ). When the predicted score varies, the reglarization power also adjst linearly. In conclsion, we wold like to point ot that the adaptation power of MSE reglarization is atomatic with no maniplation of vale α needed. 2) Indctive transfer with MSE: Indctive transfer [12] rating sing (R,i U V T i is a techniqe to se some sorce task T S to help the learning of target task T T. In or scenario, T S corresponds to MSE while T T is corresponds to the ranking measre f. Moreover, MSE presents a way to model learning to rank with point-wise learning. An optimal MSE=0 also sggests an optimal NDCG=1. Hence, the proposed reglarization can be viewed as adopting an indctive transfer strategy to enhance the performance of LambdaMF. E. 3.5 Algorithm and Complexity Analysis Once the lambda gradient and reglarization term are derived, we are ready to define the algorithm and analyze its time complexity. In the previos sections we have provided a general definition of λ, bt how to choose λ remains nsolved. In [9], the λ is defined as (4); however, we arge that the mltiplication with RankNet gradient is irrelevant to the target fnction. In addition, the comptation of RankNet gradient incldes exponential fnction, which is time consming. Therefore, given a ser, ranking measre f, and an item ranking list ξ, for every pair of items i and j with R,i >R,j, we adopt the absolte ranking measre difference between ξ and ξ, the same list as ξ except that the rankings of i and j are swapped. That is, λ i,j = f(ξ) f(ξ ) (10) The ranking measre can be applied to certain ranking measres. To be concrete, applying NDCG as the ranking measre and denoting the ranking of i as l i, (10) will be: f(ξ) f(ξ ) = (2Ri 2 Rj 1 )( log(1+l 1 i) log(1+l ) j) (11) max(dcg) Let s assme that the comptation of inner-prodct of latent factors takes O(1) time to simplify the following derivation. Given N observed items for a ser, the comptation of ξ takes O( N log N) time since it reqires sorting, so as the comptation of λ i,j. Hence, if SGD is applied, pdating a pair of items for a ser takes total O( N log N) time, which is expensive considering only a single pair of item is pdated. To overcome sch deficiency, we propose to condct the mini-batch gradient descent. After sorting the observed item list for a ser, we compte the gradient for all observed items. As a reslt, O( N log N) time is still needed to do 826

5 the sorting and max(dcg). Finally, the comptation of λ i,j takes constant time with O( N 2 ) pairs. Hence, the total comptation of pdating O( N 2 ) pairs takes O( N 2 ) time. Effectively, pdating each pair of items takes O(1) time as the sorting time is shadowed with mini-batch learning strctre. To sm p, the whole algorithm is depicted in Algorithm 1. Algorithm 1: Learning LambdaMF inpt : R M,N, learning rate η, α, latent factor dimension d, # of iterations n iter, and target ranking measre f Initialize U M,d,V N,d randomly; for t 1 to n iter do for 1 to M do C ˆ 0; for all observed i R do C ˆ 0; ξ = observed item list of sorted by its predicted scores for all observed pair i, j R,R,i >R,j do ξ = ξ with i, j swapped retrn U, V C ˆ i,j U ˆ Ci,j V ˆ i Ci,j V j C ˆ += as Eqation(9),(10) C ˆ += as Eqation(9),(10) C ˆ V j += as Eqation(9),(10) U += η C ˆ for all observed i R do V i += η C ˆ Since there are enormos amont of items in a recommender system, the O( N 2 ) time complexity seems to be nignorable. However, the sparsity of data does compensate this drawback. De to data sparsity, the nmber of observed items per ser N is negligible compared to the nmber of total items N. For example, with 99% sparsity, the comptation of all pairs takes 10 4 N 2 N 2. In [7], each λ is compted throgh all N items, leading to the failre of applying lambda. IV. EXPERIMENT In this section, we evalate the performance of LambdMF comparing to other state-of-the-art recommendation methods. Frthermore, we demonstrate the optimality of LambdaMF and the stability of reglarization. A. Comparison Methods In this work, we adopt NDCG as the target ranking fnction to demonstrate the performance of LambdaMF; applying LambdaMF to other ranking fnctions is also possible bt not inclded in this paper. We compare LambdaMF with several state-of-the-art LTR recommendation system methods. Table I STATISTICS OF DATASETS Netflix MovieLens # of sers 480, # of items 17,770 1,682 sparsity 98.82% 93.70% PopRec: Poplar Recommendation (PopRec) is a strong non-cf baseline. The idea of PopRec is to give sers nonpersonalized recommendation according to the poplarity of items in the training set. That is, the more times an item appears in training, the higher ranking the item gets. CoFi Rank: CoFi Rank is a state-of-the-art MF method minimizing a convex pper bond of (1-NDCG) [2]. CoFi Rank presents the choice of formlating an pper bond as the learning fnction, which differs from or choice of formlating gradient. In their work, several other loss fnctions are also evalated for NDCG. We compare with the NDCG loss fnction (denoted as CoFi NDCG) and the loss fnction with the best performance (denoted as CoFi Best) reported in the original paper. ListRank MF: ListRank MF is another state-of-the-art MF method optimizing cross-entropy of top-one probability [6], which is implemented sing softmax fnction. Thogh the top-one probability seems to be irrelevant to NDCG, the experiment shows the speriority over CoFi Rank in some experiments. B. Experiment Reslts We condct experiments on MovieLens 100K and Netflix dataset as in [2]. The statistics are illstrated in Table I. For each dataset, we apply weak generalization [2], [6] to prodce the experimental data. That is, we first remove sers who rated less than 20, 30, and 60 items (sch as movies). Then for the remaining sers, we sample N =10, 20, 50 ratings for each ser in the training dataset and pt the remaining ratings in testing dataset. As a reslt, a ser will have at least 10 relevant items in testing set, which favors the sage of NDCG@10 as or evalation fnction. Similar to [2], [6] which fixed all model parameters in their experiment, we tne 5 set of parameters in MovieLens 100K N = 20 dataset and set η = 0.001,α = 0.5 and n iter = 250 for or experiments. For N = 10, 20, 50, we constrct different experimental dataset 10 times for each N to avoid bias. Finally, in each experimental dataset, we rn LambdaMF 10 times for MovieLens 100K and once for Netflix, following the same setting as [2]. Finally, we report average reslt of NDCG@10 across 10 experimental datasets for N =10, 20, 50. The experiment reslts for MovieLens 100K and Netflix are shown in Table II and Table III, respectively. Note that for LambdaMF, we report two experiment reslts of adopting different reglarization terms, L2 and MSE. For L2 reglarization, we apply the weight decay for each related latent factors once while pdating a single ser. For all experiment, 827

6 Table II (MEAN,STANDARD DEVIATION) OF IN MOVIELENS100K N=10 N=20 N=50 PopRec (0.5995,0.0000) (0.6202,0.0000) (0.6310,0.0000) CoFi NDCG (0.6400,0.0061) (0.6307,0.0062) (0.6076,0.0077) CoFi Best (0.6420,0.0252) (0.6686,0.0058) (0.7169,0.0059) ListRank MF (0.6943,0.0050) (0.6940,0.0036) (0.6881,0.0052) LambdaMF L2 (0.5518,0.0066) (0.5813,0.0074) (0.6471,0.0074) LambdaMF MSE (0.7119,0.0005) (0.7126,0.0008) (0.7172,0.0013) Table III (MEAN) OF IN NETFLIX; THE RESULTS OF COFI RANK FOR N=50 AND LISTRANK MF IS NOT AVAILABLE SINCE THE AUTHORS DO NOT REPORT ITS RESULTS AND DISCLOSE THE COMPLETE PARAMETERS N=10 N=20 N=50 PopRec (0.5175) (0.5163) (0.5293) CoFi NDCG (0.6081) (0.6204) (Not available) CoFi Best (0.6082) (0.6287) (Not available) LambdaMF L2 (0.7550) (0.7586) (0.7464) LambdaMF MSE (0.7171) (0.7163) (0.7218) LambdaMF MSE best (0.7558) (0.7586) (0.7664) we fond that LambdaMF with MSE reglarization achieves sperior performance comparing to all other competitors. In Netflix, as LambdaMF with L2 reglarization yields better reslt than MSE reglarization, it seems to be de to the tolerability of strong reglarization power in large Netflix dataset, bt MSE reglarization performs better in general for all experiments. Moreover, we fond that LambdaMF with MSE reglarization sing fewer iterations achieves the best reslt in all Netflix experimental settings sing the same parameters (LambdaMF MSE best). Finally, we condct npaired t test on the LambdaMF MSE with the other models. With two tailed p vale smaller than , we add to show extremely significant improvement. Besides, we also rn experiments for LambdaMF withot reglarization, and it lead to overflow after certain iterations in all cases. C. Optimality of LambdaMF Even with sperior performance, one may still arge that the learning of LambdaMF is only an approximation of optimizing ranking measre, bt not a real optimization of ranking measre. The qestion has once been answered in [10] sing Monte-Carlo hypothesis testing of the weighting vectors of the nderlined model. The reslt of testing shows that LambdaRank has achieved a local optimm with 99% confidence. However, in this work, we notice that LambdaMF with MSE reglarization is capable of achieving a global optimm in all or experimental datasets given long enogh iterations. That is, the training performance of NDCG@10 achieves 1.0. Frthermore, the NDCG@10 is nondecreasing for all iterations ntil convergence in Netflix dataset and nondecreasing for at least 98.4% of the iterations in MovieLens 100K dataset before reaching an optimm. V. CONCLUSION In this work, we prpose a novel framework for lambda in a recommendation scenario. After deriving a model incorporating MF and lambda, we frther prove that rankingbased MF model can be ineffective de to overflow in certain circmstances. To deal with this problem, an adjstment of or model sing reglarization is proposed. Note that sch reglarization is not emphasized in a typical LTR problem. In addition, the pdate step for each pair of items takes effectively O(1) time in LambdaMF. We empirically show that LambdaMF otperforms the state-of-the-arts in terms of NDCG@10. Extended experiments demonstrate that LambdaMF exhibits global optimality and directly optimizes target ranking fnction. The insensitivity of MSE reglarization in the experiments also demonstrates the stability of MSE reglarization. ACKNOWLEDGMENT This work is spported by Taiwan government MOST fnding nmber E MY2 and E MY2 REFERENCES [1] Y. Shi, A. Karatzoglo, L. Baltrnas, M. Larson, N. Oliver, and A. Hanjalic, Climf: learning to maximize reciprocal rank with collaborative less-is-more filtering, in Proceedings of the sixth ACM conference on Recommender systems. ACM, 2012, pp [2] M. Weimer, A. Karatzoglo, Q. V. Le, and A. 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