A Personalized Preference Learning Framework for Caching in Mobile Networks

04/15/2019 ∙ by Adeel Malik, et al. ∙ Yonsei University 0

This paper comprehensively studies a content-centric mobile network based on a preference learning framework, where each mobile user is equipped with finite-size cache. We consider a practical scenario where each user requests a content file according to its own preferences, which is motivated by the existence of heterogeneity in file preferences among different users. Under our model, we consider a single-hop-based device-to-device (D2D) content delivery protocol and characterize the average hit ratio for the following two file preference cases: the personalized file preferences and the common file preferences. By assuming that the model parameters such as user activity levels, user file preferences, and file popularity are unknown and thus need to be inferred, we present a collaborative filtering (CF)-based approach to learning these parameters. Then, we reformulate the hit ratio maximization problems into a submodular function maximization and propose several greedy algorithms to efficiently solve the cache allocation problems. We analyze the computational complexity of each algorithm along with the corresponding level of the approximation that it can achieve compared to the optimal solution. Using a real-world dataset, we demonstrate that the proposed framework employing the personalized file preferences brings substantial gains over its counterpart for various system parameters.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

The growing trend in mobile data traffic drives a need of a new wireless communication technology paradigm in which radios are capable of learning and decision making in order to autonomously determine the optimal system configurations. In this context, equipping the communication functionality with machine learning-based or data-driven algorithms has received a considerable attention both in academia as well as in industrial communities.

1.1 Prior Work

Cache-enabled (or content-centric) wireless systems are equipped with finite storage capacity, which restricts us from storing the entire content library at the local cache. In order to bring content objects closer to requesting users, deciding which content needs to be cached at which user’s or helper’s cache plays a crucial role on the overall performance of content-centric wirelss networks. This cache placement problem has recently attracted a wide attention. In general, the prior work on caching can be divided into two categories: caching at small-cell base stations (or helper nodes) [1, 2, 3, 4, 5] and caching at users (or devices) [6, 7, 8, 9]. Since the optimal caching problems in [1, 2, 3, 4, 6, 7] cannot be solvable in polynomial time, approximate solutions with performance guarantees were presented. It was shown in [1, 6] that the optimal cache placement problem falls in the category of monotone submodular maximization over a matroid constraint [10, 11], and a proposed time-efficient greedy algorithm achieves an approximation within the factor of of the optimum. In [6], a dynamic programming algorithm whose complexity increases exponentially with the number of users was also proposed to obtain the optimal solution. Similarly, a joint routing and caching problem [2], a multicast-aware caching problem [3], and a mobility-aware caching problem [4] were studied, while proving the NP-hardness of their caching problems and presenting polynomial-time greedy algorithms to obtain their approximate solutions.

On the one hand, studies on integrating machine learning-based or data-driven algorithms into caching frameworks in wireless networks have recently received attention [12, 13, 14, 15, 16]. In [12, 13, 14]

, the optimal caching problems were studied when the file popularity profile is unknown—each problem was modeled as a combinatorial multi-armed bandit problem to learn the content popularity distribution for caching at one base station. Moreover, a transfer learning-based technique exploiting the rich contextual information (e.g., users’ content viewing history, social ties, etc.) was proposed in

[15]

to estimate the content popularity distribution. The training time in a transfer learning-based approach to obtaining a good estimate of the content popularity distribution for caching was also investigated in

[16].

On the other hand, the importance of personalized file preferences in content-centric networks was studied in [7, 17, 18, 19]. In [7], a low-complexity semigradient-based cooperative caching scheme was designed in mobile social networks by incorporating probabilistic modeling of user mobility and heterogeneous interest patterns. It was assumed in [7] that all users have the same activity level (i.e., the same number of requests generated by each user) and the user file preferences are known, which may be hardly realistic. Later, the optimal caching policy via a greedy algorithm for cache-enabled device-to-device (D2D) communications was presented in [17] by modeling the behavior of user requests resorting to probabilistic latent semantic analysis. In [18], when there is some coordination between recommender systems and caching decisions in a wireless network with small cells, possible benefits that can arise for the end users were investigated. In [19], a recommender system-aided caching placement scheme was also developed in wireless social networks. Recently, the benefits of user preference learning over the common file preferences were studied in wireless edge-caching scenarios, where the network edge such as fog access points [20], small-cell base stations [21], and macro base stations [22] is taken into account so that users fetch the content of interest from edge caches. However, different from existing literature in [20, 21, 22], our focus is on D2D caching at mobile users by inferring different user activity levels and user file preferences for each user.

As another caching framework, coded caching [23, 24, 25, 8] and maximum distance separable (MDS)-coded caching [26, 1, 9] have received a lot of attention in content-centric wireless networks. The content placement is optimized so that several different demands can be supported simultaneously with a single coded multicast transmission, while MDS-coded subpackets of content objects are stored in local caches and the requested content objects are retrieved using unicast transmission.

1.2 Main Contribution

In this paper, we present a parameter learning framework for a content-centric mobile caching network in which mobile users equipped with finite-size cache are served by one access point (AP) having access to the whole file library. Each user requests a content file from the library independently at random according to its own predicted personalized file preferences. Unlike most of the prior work (e.g., [27, 28, 29, 30, 9] and references therein) in which a user requests according to the common file popularity distribution, our model is motivated by the existence of heterogeneity in file preferences among different mobile users, e.g., applications involving on-demand videos requested by mobile users. In our mobile network model, we consider single-hop D2D content delivery and characterize the average hit ratio. Then, we aim to develop caching strategies in the sense of maximizing the hit ratio (also known as the offloading ratio) for two file preference cases: the personalized file preferences and the common file preferences. The most distinct feature in our framework is to incorporate the notion of collaborative filtering (CF), which was originally introduced in the design of recommender systems [31, 32, 33, 34, 35], into caching problems in order to infer/predict the unknown model parameters such as user activity levels, user file preferences, and file popularity. Despite the existence of several techniques to predict the file popularity, CF is the most widely used method to predict users’ interests in a particular content object (e.g., the content ratings in recommender systems), since it can be implemented easily without requiring in-depth domain knowledge, producing satisfactory prediction accuracy. The main idea of CF is to exploit users’ past behaviors and infer the hidden user preferences. We note that the accessing frequency of a file in [17] may not be a suitable metric for predicting unknown model parameters. This is because content files such as season episodes or news talk shows do not need to be accessed more than once as users tend to watch them only once. In our study, we thus exploit the user rating history as a metric for learning the above model parameters instead of using the number of times that each user accesses a file. The main technical contributions of this paper are summarized as follows:

  • We first introduce a file preference learning framework in cache-enabled mobile network, which intelligently combines caching and CF techniques.

  • We characterize the average hit ratio under our content-centric mobile network according to both personalized file preferences and common file preferences.

  • We present a CF-based approach to infer the model parameters such as user activity levels, user file preferences, and file popularity by utilizing the user rating history.

  • We formulate hit ratio maximization problems for our two caching strategies and analyze the computational complexity of the optimal solutions.

  • We reformulate the hit ratio maximization problems into a submodular function maximization subject to matroid constraints in order to obtain approximate solutions within provable gaps by showing computationally efficient greedy algorithms.

  • We present several greedy algorithms and analyze corresponding computational complexities along with the corresponding level of the approximation that they can achieve compared to the optimal solution.

  • Using a real-world dataset adopted for evaluating recommender systems, we perform intensive numerical evaluation and demonstrate that 1) the proposed framework employing the personalized file preferences leads to substantial gains over the baseline employing the common file preferences with respect to various system parameters and 2) the performance of our greedy approaches is quite comparable to the optimal one while they manifest significant complexity reduction.

To the extent of our knowledge, this paper is the first attempt to present personalized parameter learning in cache-enabled mobile networks.

1.3 Organization

The rest of this paper is organized as follows. In Section 2, the network and caching models are described. In Section 3, problem definitions for both proposed and baseline caching strategies are presented. In Section 4, a CF-based parameter learning method is introduced. The optimal and greedy caching algorithms are presented with complexity analysis in Section 5. In Section 6, numerical results are shown to analyze the impact of personalized file preferences in our network. Finally, Section 7 summarizes the paper with some concluding remarks.

1.4 Notations

Throughout this paper,

is the probability,

is the empty set, and denotes the cardinality of a set . We use the asymptotic notation means that there exist constants and such that for all .

2 System Overview

2.1 Network Model

Let us consider a wireless network illustrated in Fig. 1, having an AP serving mobile users denoted by the set . In our content-centric mobile network, we assume a practical scenario where each mobile user is equipped with a local cache of the same finite storage capacity (i.e., the cache can store up to content files) and acts as a helper to share the cached files via D2D single-hop communications. The AP is able to have access to the content library server (e.g., YouTube) via infinite-speed backhaul and initiates the caching process. We consider a practical scenario that the number of content files in the library is dynamic and keeps updated due to the continuous addition of new user-generated content (UGC). It has been observed that the traffic toward the content library server is dominated by the small portion of popular content files [36]. To adopt this behavior, we assume that users are interested in requesting the content file from a set of popular content files that dominates the request traffic, where the size of each file is the same.

Fig. 1: Our content-centric mobile network model.

In our network model, the time is divided into independent slots and each user generates requests for content files during its allocated time slot. Note that each user can generate the request of content files that are not in the set , but for our content-centric network, we are interested in caching only the popular content files that dominate the request traffic. Due to the continuous addition of new UGC to the content library, the popularity of the files changes over time. Thus, we assume that the popularity of content files is fixed for a certain period of time , and after every time period , the AP updates the set of popular content files and initiates the content placement phase to fill the cache of each user.

In addition, we would like to illuminate two important real-world observations that were largely overlooked in most studies on cache-enabled wireless networks [27, 28, 29, 30, 9]: the file popularity is not the same as the preference of each individual user; and only a small portion of users are active in creating the data traffic. On the other hand, the main focus of this study is on comprehensively studying the impact and benefits of personalized file preferences in our mobile network by assuming that each user has a different activity level and a different personalized file preference . To this end, we formally define three model parameters as follows:

  • Activity level of user is defined as the probability that user sends a content request, where and . In our work, indicates the ratio of the number of files that user has experienced (e.g., the number of rated items in movie recommender systems) to the total number of files that all users has experienced.

  • Personalized file preference of user is defined as the conditional probability that user requests file given that the user sends a request, where and . More specifically, indicates the ratio of the preference that user has given to file to the total amount of preferences that the same user has given to all its belonging files.

  • File popularity is defined as the probability that a file is requested, where and . This probability represents how popular a content file is in a network. Note that this is the case where all users have the common file preferences (i.e., ) as in [27, 28, 29, 30, 9].

It will be described in Section 4 how to learn these three parameters based on the CF-based inference approach.

2.2 Caching Model

In content-centric wireless networks, a caching mechanism can be divided into the following two stages: the content placement phase and the content delivery phase. We first describe the content placement phase which determines the strategy for caching the content objects in the cache of mobile users. During the content placement process, each user

stores the content files according to a caching vector

(will be optimized later on the basis of the personalized file preferences), where is given by

(1)

In order to have a feasible cache allocation strategy , should satisfy the following constraints:

(2)
(3)

Note that the constraint in (2) is the individual user’s storage capacity constraint. The popularity of content files is assumed to be fixed only for a period of time . We also assume that the caching strategy should be determined by the AP after every time period during the off-peak time, representing the time period when network resources are in abundance (i.e., the data traffic is very low). One example includes such a midnight time that the content request activity from the network is very low as most of mobile users are not active during this time.

We now move on to the delivery phase which allows the requested content objects to be delivered from the source to the requesting user over wireless channels. In our network model, the users are assumed to prefetch a part of (popular) content objects in their local cache from the content library server or the AP when they are indoors. For example, during off-peak times, the AP can initiate the content placement phase and fill the cache of each user. On the other hand, for the case when the actual requests take place, we confine our attention to an outdoor environment where users are moving. For such cases, since file reception from the server may not be cost-effective, only D2D communications are taken into account for content delivery, i.e., the server or the AP does not participate in the delivery phase. In our content-centric mobile network, we focus on the peak time in which each mobile user may not have a reliable connection with the AP and thus retrieves the requested content either from his/her own cache or via single-hop D2D communications from one of the mobile users storing the requested content file in their caches within the collaboration distance of the requested user (refer to Fig. 1).

3 Problem Definition

In this section, we first define the average hit ratio as a performance metric used throughout this paper. Then, we introduce our problem formulation in terms of maximizing the average hit ratio in our content-centric mobile network employing the personalized file preferences. For comparison, we present a baseline strategy that employs the common file preferences. Finally, we show a way to reformulate the hit ratio maximization problems into submodular function maximization subject to matroid constraints.

3.1 Performance Metric

The goal of our cache-enabled mobile network is to reduce the traffic caused by the infrastructure mode in which users retrieve their requested content through the AP, which will be enabled either during off-peak times or when the content is not available via D2D communications. To this end, the performance metric of interest is the average hit ratio at any time slot and is defined as follows [17, 7].

Definition 1 (Average hit ratio ).

Let denote the hit probability that a user requesting a content file at any time slot retrieves the requested content either from his/her own cache or from the cache of a user which falls in a given time slot within collaboration distance between the requesting user and another user holding the requested content file. In this case, the average hit ratio that the content file is successfully retrieved via D2D communications is defined as

(4)

3.2 Problem Formulation

In this subsection, we present our problem formulation for the optimal cache allocation strategy that maximizes the average hit ratio in (4) for the content-centric mobile network employing the personalized file preferences in Section 2.1.

Let denote the contact probability that that users are within the communication distance

at a time slot. Then, given three probability distributions of the activity level

of each user, the personalized file preference of each user, and the contact probability , the average hit ratio depends solely on the cache allocation strategy . Among all the cache allocation strategies, the optimal one will be the one that leads to the maximum . Now, from the caching constraints in (2) and (3), the optimal cache allocation strategy for the content-centric mobile network employing the personalized file preferences can thus be the solution to the following optimization problem:

(5a)
subject to
(5b)
(5c)

where is defined in (1). The term in  (5a) indicates the hit probability in (4) to retrieve the content file via D2D single-hop communications, which thus corresponds to the probability that the requesting user comes in contact with another user storing the desired content. This implies that is the probability of the event that there is no user storing the content that user has requested within the distance . If user is holding the content requested by the user that is within the communication distance (i.e., ), then the D2D communication can be initiated.

As described in Section 2.2, the model parameters such as user activity levels and user file preferences are assumed to be fixed only for a period of time so that the caching strategy should be determined by the AP with updated parameters after every time period during the off-peak time. Fig. 2 is an illustration of the proposed content placement cycle, which basically consists of two tasks. First, the AP learn the model parameters such as user activity levels and user file preferences based on the users’ content request history. Afterwards, by using the predicted parameters, the AP determines the caching strategy .

Fig. 2: Content placement cycle.

3.3 Baseline Strategy

In this subsection, we describe a baseline caching strategy for comparison. While the focus of our study is on analyzing the impact of personalized file preferences in a content-centric mobile network, for our baseline, we use the case adopted in many previous studies [27, 28, 29, 30, 9] where all users have the common file preferences (i.e., ), where is the probability that a file is requested (i.e., the file popularity).

Now, from the caching constraints in (2) and (3), the optimal cache allocation strategy for the content-centric mobile network employing the common file preferences can thus be the solution to the following optimization problem:

(6a)
subject to
(6b)
(6c)

where is the contact probability that two users are within the communication distance and is defined in (1).

In practice, since users have different content file preferences, incorporating the knowledge of personalized file preferences into the cache allocation problem is expected to bring a substantial performance gain. This implies that content placement according to the caching strategy via the solution to the optimization problem in (6) will not perform better than the one obtained by solving the optimization problem in (5), which will be empirically demonstrated in Section 6.

3.4 Problem Reformulation Using Submodular Properties

In this subsection, we show that each of the optimization problems in (5) and (6) can be formulated as optimization (i.e., maximization) of a submodular function subject to matroid constraints [10]. Since the problems in (5) and (6) are NP-hard [7] and thus solving them for large system parameters such as and may not be tractable, a common way to solve these problems is the use of computationally efficient greedy algorithms. The computational efficiency comes at the cost of performance degradation. In this context, the problem reformulation enables us to obtain an approximate solution within provable gaps from greedy algorithms. Implementation details of greedy algorithms and the corresponding level of approximation that they can achieve compared to the optimal solution will be specified in Section 5.

First, we begin with formulating an equivalent optimization problem in (5) similarly as in [7]. Let denote a ground set, where represents the configuration that user caches file and the ground set contains all possible configurations of file placement. Then, we aim to find the subset of that maximizes the average hit ratio under the constraint (2). We define a discrete set function on subsets of the ground set as follows:

(7)

where indicates the possible placement configuration for content file and is a discrete set function, which gives us the average hit ratio for given subsets (i.e., the caching strategy) of the ground set , when personalized file preferences of users are available . Then, from (7), we are capable of establishing the following equivalent optimization problem to the original one in (5):

(8a)
subject to
(8b)

where is a family of feasible sets that satisfies the cache size constraint (8b) and indicates the possible cache placement configuration for user ’s cache. Note that each user’s cache size constraint in (8b) is a matroid constraint corresponding to (2). We are now ready to show the following submodular property.

Lemma 1.

The function in (7) is a monotone non-decreasing submodular function as adding one more configuration to given configuration subsets (i.e., ) leads to an increment of .

Proof.

The proof essentially follows the same steps as those in [7, Lemma 1] by just replacing (i.e., the term assuming the same activity level for all users) by the probability with which a user generates a request (i.e., heterogeneous activity levels of users). ∎

Next, we move on to the baseline strategy where all users have the common file preferences (i.e., . According to the above problem reformulation argument, the following equivalent optimization problem to the original one in (6) can be established:

(9a)
subject to
(9b)

where for the baseline strategy is given by

(10)
Lemma 2.

The function in (10) is a monotone non-decreasing submodular function as adding one more configuration to given configuration subsets (i.e., ) leads to an increment of .

We shall design caching algorithms based on greedy approaches in terms of maximizing a submodular function subject to the matroid constraints in Section 5.

4 Parameter Learning

Thus far, we have characterized the caching optimization problems in Section 3 under the assumption that the key model parameters such as activity levels , file preferences , and file popularity are available. In this section, we focus on the parameter learning phase of the content placement cycle illustrated in Fig. 2 in order to completely solve the caching optimization problems in Section 3. We describe a method to learn these three parameters based on the user rating behaviors, i.e., the rating matrix containing the ratings of users to content files (items). We first present a CF-based inference approach to predicting missing ratings for acquiring a complete rating matrix, denoted by , from the fact that users tend to give ratings to only few items. We then show how to compute three parameters using the complete rating matrix.

4.1 Predicting Missing Ratings

Collaborative filtering (CF) [31] is one of prevalent techniques widely used in recommender systems mostly to predict ratings of unrated items [37] or to suggest the most favorable top- items [38] for a user by analyzing users’ past behaviors on items. In CF, user behaviors are presented along with ratings (explicit) or clicks (implicit) on content files. In our study, we focus on predicting the missing ratings using the user–item rating matrix based on the explicit feedback and each rating reflects a user preference

for a certain item. The CF-based inference is built based upon the basic assumptions that (i) two users tend to have similar behavior patterns in the future when the two users share similar behavior patterns in the past; and (ii) the user behavior patterns are consistent over time. The CF-based techniques are generally classified into memory-based approaches including user-based CF 

[32] and item-based CF [33]

and model-based approaches such as singular value decomposition (SVD)-based CF 

[34]

and autoencoder-based CF 

[35]. In CF, the missing ratings are predicted to obtain the complete rating matrix containing the ratings of users to files (items).

In our study, for ease of explanation with intuition, we adopt the memory-based CF approaches that perform predictions based on the similarity among users or items  [32, 33].111

Since our parameter learning framework is CF-agnostic, model-based approaches and deep-learning approaches can also be adopted with a slight modification.

There exist two memory-based CF techniques: user-based CF [32] and item-based CF [33].

In user-based CF, the missing rating of user for content file is predicted using the ratings given by the users similar to the target user , and the similarity between the users is measured by referring to their ratings to other items. Let be the set of users who have rated content file and has similar rating behaviors to the target user . Then, the predicted rating of user to content file is given by

(11)

where denotes the rating given by user to file , denotes the biased rating value for to file , and denotes the similarity weight between users .

In item-based CF, the missing rating of user for the target content file is predicted using the ratings to the items similar to the target item , and the similarity between the items is measured by referring to the ratings given by users. Let be the set of items that have been rated by user and have similar rating behaviors to the target file . Then, the predicted rating of user to content file is given by

(12)

where is the similarity weight between the content files .

For the user-based CF approach, the steps to predict each missing rating to a content file by user are summarized below:

  1. First, we measure the similarity between users (files in case of item-based CF) using the available rating values. The popular similarity measures include the Pearson correlation, cosine similarity, and Euclidean distance. As an example, the similarity

    between two users using the Pearson correlation is given by

    (13)

    where is the set of items that have been rated by both users and is the mean value of the ratings given by user .

  2. Next, we use the similarity values to identify the most similar users (i.e., ) to the target user (the most similar files (i.e., ) to the target file in case of item-based CF), where either the top users (files in case of item-based CF) or only the users (files in case of item-based CF) whose similarities are larger than a pre-defined threshold are taken into account.

  3. Finally, we calculate the missing rating of user for the content file using (11) ((12) in case of item-based CF).

To further improve the prediction accuracy, it is also possible to apply both matrix factorization techniques such as SVD-based CF [34] to learn the hidden user patterns based on the observed ratings and recently emerging deep learning techniques such as autoencoder-based CF [35] into our prediction framework. In the next subsection, we explain how to calculate the model parameters based on the complete rating matrix .

4.2 Parameter Calculation

In this section, we calculate user activity levels , user file preferences , and file popularity using the rating matrix . Let denote the rating given by user to file and also denote the total number of ratings given (not predicted) by user . Then, similarly as in [17], the user activity level is given by

(14)

In addition, the file preferences of each user and the file popularity are given by

(15)

and

(16)

respectively.

Fig. 3: Block diagram of parameter learning with an example of .

Fig. 3 illustrates the parameter learning procedure. Consider an example of and as shown in Fig. 3. We start with the observed rating matrix containing the missing ratings and let the matrix pass through one of CF algorithms for predicting the missing ratings. The output of the CF algorithm is a complete rating matrix . Then, user activity levels can be calculated from (14). For example, the probability that user generates a request is as the total number of ratings given (not predicted) by user is and the total number of ratings given (not predicted) by all users is . Next, user file preferences can be calculated from (15). For example, the probability that user prefers file

at the moment of requesting the content is

as the rating given by user to file is and the sum of all the ratings of user to all files is . Lastly, the file popularity can be calculated from (16). For example, the probability that file is preferred at the moment of requesting the content is as the sum of ratings given by all users to file is and the sum of all the ratings given by all users to all four files is .

Remark 1.

In [17], user file preferences and file popularity are given by and , respectively, where is the cumulative number of requests from to content file . On the other hand, in our study, we utilize the user ratings as a metric to learn the above model parameters instead of using the number of times that each user accesses a file. Although the frequency of accessing a file may reflect user file preferences, it is true only such types of content files that tend to be accessed repeatedly such as music videos. This assumption may not hold for other types of content files which do not need to be accessed more than once as a user tends to watch them only once. In this context, the user rating history would be proper for both types of content objects stated above and can reflect a better understanding of each user behavior.

5 Caching Algorithms

In this section, we present the implementation details of several caching algorithms to solve the problems in Section 3. We also analyze the computational complexity for each algorithm along with the corresponding level of the approximation that it can achieve compared to the optimal solution. We first show an approach via exhaustive search to obtaining the optimal solution to the problems in (8) and (9). Moreover, we present two computationally efficient greedy algorithms at the cost of slight performance degradation compared to the optimal approach.

5.1 Optimal Approach

This subsection deals with an exhaustive search method over all feasible caching strategies to obtain the optimal caching strategy by solving the problems in (8) and (9), which is summarized in the Optimal Algorithm. For given input parameters such as the activity levels , the user file preferences (the file popularity for the baseline strategy), and the contact probability , the algorithm starts with an empty set , representing the optimal caching strategy, and zero cost (i.e., hit ratio). The algorithm runs for iterations, where is a family of feasible subsets (i.e, caching strategies) that satisfies the storage capacity constraint (8b) and . At each iteration, the hit ratio of a feasible subset is computed according to (7) ((10) for the baseline strategy) and the subset with the maximum hit ratio is selected as an optimal caching strategy .

Optimal Algorithm
Input: , , , and ,
Output:
Initialization: ;
01: for each do
02:    Calculate
03:    if then
04:      
05:      
06:    end if
07: end for

Next, we turn to analyzing the computational complexity of the Optimal Algorithm.

Remark 2.

The time complexity of the Optimal Algorithm is as the algorithm runs for iterations (i.e., the total number of subsets representing all the possible configurations that user caches file for and is ). In each iteration, the evaluation of takes times according to (7) ((10) for the baseline strategy). This finally gives us the time complexity of .

The high complexity of this algorithm makes it impossible to find the optimal solution within polynomial time even for small system parameters such as , , and . To overcome this issue, we would like to introduce computationally efficient greedy algorithms in the next subsection.

5.2 Greedy Approaches

In this subsection, we present two types of greedy algorithms, which guarantee the provable gaps between an approximate solution obtained from greedy algorithms and the optimal solution.

5.2.1 Greedy Algorithm 1 (G1 Algorithm)

The first greedy approach is summarized in the G1 Algorithm. For given input parameters , , , and , the algorithm starts by initially setting , , and to , , , respectively, where denotes the set of content files not cached by user , denotes the set of users with residual cache space. Given the system parameters , , and , the algorithm runs for iterations. In each iteration, one feasible element that satisfies the storage capacity constraint (8b) is added and the highest marginal increase in the hit ratio is offered if added to the set .

Greedy Algorithm 1 (G1 Algorithm)
Input: , , , and ,
Output:
Initialization: ; , ;
01: for from 1 to do
02:   
03:   
04:   
05:   if then
06:     
07:   end if
08: end for

Now, we analyze the performance gap between the G1 Algorithm and the Optimal Algorithm in Section 5.1 as follows.

Proposition 1.

The caching strategy obtained from the G1 Algorithm achieves the hit ratio within the factor of of the optimum.

Proof.

The proof technique basically lies in the similar steps to those established in [11] since the G1 Algorithm is design based on the reformulated problem using submodular properties. ∎

Next, we turn to analyzing the computational complexity of the G1 Algorithm.

Remark 3.

The time complexity of G1 Algorithm is as the algorithm runs for iterations (refer to lines 1–8). Each iteration involves evaluating the marginal value (i.e., in line 2) of at most elements, that is, the number of unassigned elements in each iteration is given by , and each evaluation of takes times. This finally gives us the time complexity of .

Remark 4.

Note that there exist algorithms that can perform better than the G1 Algorithm at the cost of higher time complexity. In [39], a greedy algorithm was designed in such a way that maximizes a general monotone submodular function subject to matroid constraints, while guaranteeing an approximate solution within the factor of of the optimum. The approximate solution is briefly described according to the following two steps. First, the combinatorial integer programming problem is replaced with a continuous problem by relaxing the integer value constraint, and then the solution to the continuous problem is found. Second, the pipage rounding method is applied to the solution to the continuous problem to obtain a feasible approximate solution. Although this algorithm guarantees a better approximate solution within the factor of compared to the G1 Algorithm, this comes at the cost of high computational complexity as the time complexity of this algorithm is . The high time complexity of this algorithm makes it impractical for large-scale mobile networks.

Since the complexity of the G1 Algorithm may be still high for large system parameters , , and , we shall seek a possibility to further reduce the complexity with acceptable performance degradation. To this end, we design another greedy algorithm that has almost comparable performance to that of the G1 Algorithm in the next subsection.

5.2.2 Greedy Algorithm 2 (G2 Algorithm)

The second greedy approach is summarized in the G2 Algorithm. For given input parameters , , , , and the number of iterations, denoted by , the algorithm starts by initially setting , , and to , , and , respectively, where is the set of most preferred content files for user , denotes the set of content files not cached by user , and denotes the set of users with residual cache space. Given the system parameters , , and , the algorithm runs for iterations, each of which is divided into two parts. The first part runs times, where at each time, an element is removed from the set that offers the lowest marginal decrease in the hit ratio if removed from the set . Then, the second part runs times, and at each time, one feasible element that satisfies the storage capacity constraint (8b) is added and the highest marginal increase in the hit ratio is offered if added to the set .

Greedy Algorithm 2 (G2 Algorithm)
Input: , , , , and ,
Output:
Initialization: ; ;
                   ,
01: for from 1 to do
02:   for from 1 to do
03:     
04:     
05:     
06:     if and then
07:       
08:     end if
09:   end for
10:   for from 1 to do
11:     
12:     
13:     
14:     if then
15:       
16:     end if
17:   end for
18: end for

Next, let us analyze the computational complexity of the G2 Algorithm.

Remark 5.

The complexity of the G2 Algorithm is as the algorithm runs for iterations (refer to lines 1–8). The first part of each iteration involves evaluating the marginal value (i.e., in line 3) of elements times. The second part of each iteration involves evaluating the marginal value (i.e., in line 11) of at most elements, that is, , times. Since each evaluation of takes times, this finally gives us the time complexity of . The G2 Algorithm is computationally faster than the G1 Algorithm as is just a constant that does not scale with other system parameters.

6 Experimental Evaluation

In this section, we perform data-intensive simulations with finite system parameters , , , and for obtaining numerical solutions to the optimization problems in (8) and (9). We compare the results of the proposed caching strategies in Section 3.2 with the baseline strategies in Section 3.3 to analyze the impact of personalized file preferences in our content-centric mobile network. We first elaborate on our experimental setup including the real-world dataset and network settings. Then, we present the numerical results according to various system parameters.

6.1 Experimental Setup

6.1.1 Dataset and Pre-Filtering

We use MovieLens 1M222http://grouplens.org/datasets/movielens. for our experiments, which is one of publicly available real-world datasets for evaluating movie recommender systems[31, 33, 32, 34, 40, 41]. The dataset includes 3,952 items (movies), 6,040 users, and 1,000,209 ratings, where the ratings take integer values ranging from 1 (i.e., the worst) to 5 (i.e., the best).

Note that the sparsity (i.e., the ratio of the number of missing cells in a matrix to the total number of cells) of the rating matrix obtained from the original dataset is 95.8%, which is very high and often causes performance degradation. To resolve the sparsity issue, there have been a variety of studies (see [40, 41, 42, 43, 44] and references therein). One popular solution to the data sparsity problem is the use of data imputation [40, 43, 44]. The zero injection method was proposed in [40] by defining the pre-use preference as judgment of each user on items before using or purchasing them. Due to the belief that uninteresting items corresponding to the ones having low ranks of inferred pre-use preference scores would not satisfy the belonging users even if recommended, zeros are given to all of the uninteresting items in a rating matrix. Meanwhile, the pureSVD method [43] and the allRank method [44]

that assign zeros and twos, respectively, to all missing cells in a rating matrix were introduced. Even if such data imputation techniques are known to significantly improve the prediction accuracy, we do not employ them in our experiments since solving the data sparsity problem is not our primary focus.

As an alternative, we use a naïve pre-filtering method to obtain a less sparse dataset before applying the CF-based rating prediction approach since our main focus is not on improving the performance of recommender systems. Our pre-filtering method is given below:

  1. First, among 6,040 users, we select users who have rated at least 400 items (i.e., movies) out of 3,952 items. This gives us a reduced rating matrix.

  2. From the reduced dataset, we select only the items that have been rated by at least 180 users. This finally gives us a rating matrix with sparsity of 45.5%.

The statistics for the original and pre-filtered datasets are summarized in Table I. We then predict all the missing ratings in the pre-filtered rating matrix via the CF-based approach. In our experiments, we randomly select users and items out of 333 users and 261 items, respectively, in the pre-filtered rating matrix. We use the Pearson correlation in (13) to evaluate the similarity.

Original dataset
(MovieLens 1M)
Pre-filtered
dataset
Number of users 6,040 333
Number of items 3,952 261
Total number of cells 23,870,080 86,913
Number of missing cells 22,869,871 39,541
Sparsity 95.8% 45.5%
TABLE I: Statistics of original and pre-filtered datasets

6.1.2 Network Setup

Now, let us turn to describing our network setup for experiments. We assume that each user moves independently according to the random walk mobility model [27, 9] in a square torus network of unit area. In the mobility model, the position of a user at time slot is updated by , where

is a sequence of independent and identically distributed (i.i.d.) random variables that represent a user’s flight vector. In order to calculate the contact probability

in our experiments, we create a synthetic mobility trajectory database of users for time slots according to the following steps:

  • At , users are distributed independently and uniformly on a square network, which is divided into smaller square cells of equal size.

  • Then at , each user moves independently in any direction from its current location with a flight length uniformly and randomly selected from 0 to the size of each square cell, .

As addressed in Section 2.2, a requesting user can successfully retrieve its desired content from his/her own cache or by communicating directly to other mobile users within the collaboration distance . For given , using the mobility trajectory database, it is possible to calculate the average contact probability at each time slot as the ratio of the time slots when two users and are within the distance to the total number of time slots, .

6.2 Numerical Results

In this section, we perform numerical evaluation via intensive computer simulations using the pre-filtered MovieLens dataset. We evaluate the hit ratio for the following five algorithms: the optimal algorithm via exhaustive search (Opt-P1), the G1 algorithm for solving (8) (G1-P3), the G2 algorithm for solving (8) (G2-P3), the G1 algorithm for solving (9) (G1-P4), and the G2 algorithm for solving (9) (G2-P4). In our experiments, the user-based CF technique is adopted as one of CF-based prediction approaches, but other CF techniques such as item-based CF and model-based approaches (e.g., SVD and autoencoder-based CF methods) can also be applied for predicting the ratings. The number of square cells is set to for simplicity.

6.2.1 Comparison Between the Optimal and Greedy Approaches

In Fig. 4, we first show how the average hit ratio behaves according to different values of the cache size to see how close performance of our two greedy algorithms G1-P3 and G2-P3 is to that of the optimal one Opt-P1, where and . In this experiment, small values for system parameters and are chosen due to the extremely high complexity of Opt-P1 as searches over the cache placement are necessary even with . In Fig. 4, it is obvious that the hit ratio grows monotonically with increasing since the contact probability becomes higher with increasing . More interestingly, since the performance of G1-P3 and G2-P3 is quite comparable to that of Opt-P1, using our scalable greedy strategies can be a good alternative in large-scale mobile networks. In the subsequent experiments, we evaluate the performance of our four greedy algorithms in Section 5 and perform a comparative analysis among them.

Fig. 4: The average hit ratio versus the cache size , where algorithms Opt-P1, G1-P3, and G2-P3 are compared.
Fig. 5: The average hit ratio versus the cache size , where algorithms G1-P3, G2-P3, G1-P4, and G2-P4 are compared.
Fig. 6: The average hit ratio versus the number of users, , where algorithms G1-P3, G2-P3, G1-P4, and G2-P4 are compared.
Fig. 7: The average hit ratio versus the library size , where algorithms G1-P3, G2-P3, G1-P4, and G2-P4 are compared.
Fig. 8: The average hit ratio versus the collaboration distance , where algorithms G1-P3, G2-P3, G1-P4, and G2-P4 are compared.
Fig. 9: The average hit ratio versus the masking percentage (%), where four cases G1-P3 (No masking), G1-P3 (Masking), G1-P4 (No masking), and G1-P4 (Masking) are compared.

6.2.2 Comparison Between the Proposed and Baseline Approaches

In Fig. 5, the average hit ratio versus the cache size is illustrated to see how much caching strategies employing the personalized file preferences are beneficial over the case with the common file preferences, where , , and . As expected, is enhanced with increasing . We can observe that knowledge on the personalized file preferences brings a significant gain compared to the baseline strategy in which all users have the common file preferences (i.e., ). The performance difference between two caching strategies becomes significant for a higher value of as depicted in the figure.

In Fig. 6, we illustrate the average hit ratio according to different values of the number of users, , where , , and . It is observed that the proposed strategy (G1-P3 and G2-P3) consistently outperforms its counterpart scheme employing the common file preferences (G1-P4 and G2-P4). Another interesting observation is that the performance gap between the two caching strategies tends to be reduced with increasing as depicted in the figure. This is because an increment of not only increases the overall storage capacity of the network but also the chance of finding a source user within increases by virtue of an increased user density within the .

In Fig. 7, the average hit ratio versus the content library size is illustrated, where , , and . As expected, decreases with increasing since the storage capacity of the network does not increase with . We also observe that the performance difference between the two caching strategies employing the personalized file preferences and the common file preferences is significant for almost all values of .

In Fig. 8, the average hit ratio versus the collaboration distance is illustrated, where , , and . It is seen that the performance difference between the two caching strategies is noticeable especially for small values of as depicted in the figure. This is due to the fact that increasing leads to an increment of the user density within , which can raise the chance such that a requesting user can interact with more users to retrieve their content, eventually leading to the case where the performance of both caching strategies approaches the upper bound .

From Figs. 5–8, we conclude that the caching strategies employing the personalized file preferences performs significantly better than the baseline caching strategies. We also observe that the performance of the G2 algorithm is quite comparable to or slightly lower than that of the G1 algorithm, whereas the complexity of the G2 algorithm can further be reduced (see Remarks 3 and 5).

6.2.3 The Effect of Predicted Ratings

Furthermore, it is worthwhile to examine the effect of predicted ratings on the hit ratio . To this end, we generate another rating matrix based on the pre-filtered rating matrix (namely, the non-masked rating matrix) as follows. Another pre-filtered rating matrix (the masked rating matrix) is obtained by randomly masking of the total given ratings in the original pre-filtered rating matrix. For example, when , the sparsity of the masked rating matrix becomes 50.9%. We then predict all the missing ratings in both non-masked and masked pre-filtered rating matrices via CF, where the case using the non-masked rating matrix provides an upper bound on the hit ratio performance. The rest of environmental settings is the same as those in Section 6.1.1.

In Fig. 9, the average hit ratio versus the masking percentage (%) is illustrated, where , , , and . From this figure, the following observations are found: the performance of G1-P3 (Masking) and G1-P4 (Masking) (i.e., caching based on the masked rating matrix) is degraded with increasing ; G1-P3 (Masking) is still quite superior to G1-P4 (No masking) (i.e., its counterpart scheme based on the non-masked rating matrix); and the performance gap between G1-P4 (No masking) and G1-P4 (Masking) (the case with the common file preferences) is negligible. The last observation comes from the fact that the file popularity in (16) is calculated by summing up the predicted ratings for a file over all users and thus prediction errors are averaged out.

As addressed before, we are capable of further improving the prediction accuracy by using not only more sophisticated CF but also data imputation techniques. Such approaches enable us to greatly enhance the hit ratio performance of the caching strategies based on the masked rating matrix (i.e., G1-P3 (Masking) and G1-P4 (Masking)).

7 Concluding Remarks

This paper investigated the impact and benefits of personalized file preferences in a content-centric mobile network by proposing a CF-aided learning framework that enables us to infer model parameters based on the user rating history. The hit ratio under our mobile network was characterized by adopting single-hop-based D2D content delivery according to the two caching strategies employing both personalized file preferences and common file preferences. The hit ratio maximization problems were then reformulated into a submodular function maximization, and several scalable greedy algorithms were presented to significantly reduce the computational complexity. The proposed algorithms were shown to achieve approximate solutions that are within provable gaps compared to the optimal solutions. In addition, data-intensive evaluation was performed using the MovieLens dataset, and it was demonstrated that the caching strategy employing the personalized file preferences has a significant performance gain over the case employing the common file preferences. More precisely, it was shown that the performance difference between the two caching strategies is significant under practical scenarios such that 1) the collaboration distance is small, 2) the library size is large, 3) the storage capacity is neither too high to cache almost entire library nor too low not to be even capable of caching few content files, and 4) the density of users within is not too high for users within to collectively store the entire content in the library.

An interesting direction for future research in this area includes the analysis of the performance by adopting mobility models that are inferred using real-life trajectories of human mobility. Potential avenues of another future research are to introduce other methods such as reinforcement learning and regression for predicting model parameters including the file preferences.

Acknowledgments

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1A09000835).

References

  • [1] K. Shanmugam, N. Golrezaei, A. G. Dimakis, A. F. Molisch, and G. Caire, “Femtocaching: Wireless content delivery through distributed caching helpers,” IEEE Trans. Inf. Theory, vol. 59, no. 12, pp. 8402–8413, Dec. 2013.
  • [2] K. Poularakis, G. Iosifidis, and L. Tassiulas, “Approximation algorithms for mobile data caching in small cell networks,” IEEE Trans. Commun., vol. 62, no. 10, pp. 3665–3677, Oct. 2014.
  • [3] K. Poularakis, G. Iosifidis, V. Sourlas, and L. Tassiulas, “Exploiting caching and multicast for 5G wireless networks,” IEEE Trans. Wireless Commun., vol. 15, no. 4, pp. 2995–3007, Apr. 2016.
  • [4] Y. Guan, Y. Xiao, H. Feng, C.-C. Shen, and L. Cimini, “MobiCacher: Mobility-aware content caching in small-cell networks,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), Austin, TX, Dec. 2014, pp. 4537–4542.
  • [5] Y. Cui, D. Jiang, and Y. Wu, “Analysis and optimization of caching and multicasting in large-scale cache-enabled wireless networks,” IEEE Trans. Wireless Commun., vol. 15, no. 7, pp. 5101–5112, Jul. 2016.
  • [6] R. Wang, J. Zhang, S. H. Song, and K. B. Letaief, “Mobility-aware caching in D2D networks,” IEEE Trans. Wireless Commun., vol. 16, no. 8, pp. 5001–5015, Aug. 2017.
  • [7] Y. Wu, S. Yao, Y. Yang, Z. Hu, and C.-X. Wang, “Semigradient-based cooperative caching algorithm for mobile social networks,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), Washington, DC, Dec. 2016, pp. 1–6.
  • [8] M. Ji, G. Caire, and A. F. Molisch, “Fundamental limits of caching in wireless D2D networks,” IEEE Trans. Inf. Theory, vol. 62, no. 2, pp. 849–869, Feb. 2016.
  • [9] A. Malik, S. H. Lim, and W.-Y. Shin, “On the effects of subpacketization in content-centric mobile networks,” IEEE J. Sel. Areas Commun., Jun. 2018.
  • [10] G. Nemhauser and L. A. Wolsey,

    Integer and Combinatorial Optimization

    .   New York, NY, USA: Wiley-Interscience, 1988.
  • [11] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of approximations for maximizing submodular set functions—I,” Math. Program., vol. 14, no. 1, pp. 265–294, Dec. 1978.
  • [12] P. Blasco and D. Gündüz, “Learning-based optimization of cache content in a small cell base station,” in Proc. IEEE Int. Conf. Commun. (ICC), Sydney, NSW, Jun. 2014, pp. 1897–1903.
  • [13] J. Song, M. Sheng, T. Q. S. Quek, C. Xu, and X. Wang, “Learning-based content caching and sharing for wireless networks,” IEEE Trans. Commun., vol. 65, no. 10, pp. 4309–4324, Oct. 2017.
  • [14] P. Blasco and D. Gündüz, “Multi-armed bandit optimization of cache content in wireless infostation networks,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Honolulu, HI, Jun./Jul. 2014, pp. 51–55.
  • [15] E. Bastuǧ, M. Bennis, and M. Debbah, “A transfer learning approach for cache-enabled wireless networks,” in Proc. Int. Symp. Modeling Opt. Mobile, Ad Hoc, Wireless (WiOpt), Mumbai, India, May 2015, pp. 161–166.
  • [16] B. N. Bharath, K. G. Nagananda, and H. V. Poor, “A learning-based approach to caching in heterogenous small cell networks,” IEEE Trans. Commun., vol. 64, no. 4, pp. 1674–1686, Apr. 2016.
  • [17] B. Chen and C. Yang, “Caching policy for cache-enabled D2D communications by learning user preference,” IEEE Trans. Commun., vol. 66, no. 12, pp. 6586–6601, Dec. 2018.
  • [18] L. E. Chatzieleftheriou, M. Karaliopoulos, and I. Koutsopoulos, “Caching-aware recommendations: Nudging user preferences towards better caching performance,” in Proc. IEEE Conf. Comput. Commun. (INFOCOM), Atlanta, GA, May. 2017, pp. 1–9.
  • [19] Y. Wang, M. Ding, Z. Chen, and L. Luo, “Caching placement with recommendation systems for cache-enabled mobile social networks,” IEEE Commun. Lett., vol. 21, no. 10, pp. 2266–2269, Oct. 2017.
  • [20] Y. Jiang, M. Ma, M. Bennis, F. Zheng, and X. You, “User preference learning-based edge caching for fog radio access network,” IEEE Trans. Commun., vol. 67, no. 2, pp. 1268–1283, Feb. 2019.
  • [21] P. Cheng, C. Ma, M. Ding, Y. Hu, Z. Lin, Y. Li, and B. Vucetic, “Localized small cell caching: A machine learning approach based on rating data,” IEEE Trans. Commun., vol. 67, no. 2, pp. 1663–1676, Feb. 2019.
  • [22] D. Liu and C. Yang, “Caching at base stations with heterogeneous user demands and spatial locality,” IEEE Trans. Commun., vol. 67, no. 2, pp. 1554–1569, Feb. 2019.
  • [23] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,” IEEE Trans. Inf. Theory, vol. 60, no. 5, pp. 2856–2867, May. 2014.
  • [24] S. H. Lim, C.-Y. Wang, and M. Gastpar, “Information-theoretic caching: The multi-user case,” IEEE Trans. Inf. Theory, vol. 63, no. 11, pp. 7018–7037, Nov. 2017.
  • [25] M. A. Maddah-Ali and U. Niesen, “Decentralized coded caching attains order-optimal memory-rate tradeoff,” IEEE/ACM Trans. Netw., vol. 23, no. 4, pp. 1029–1040, Aug. 2015.
  • [26] V. Bioglio, F. Gabry, and I. Land, “Optimizing MDS codes for caching at the edge,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), San Diego, CA, Dec. 2015, pp. 1–6.
  • [27] G. Alfano, M. Garetto, and E. Leonardi, “Content-centric wireless networks with limited buffers: When mobility hurts,” IEEE/ACM Trans. Netw., vol. 24, no. 1, pp. 299–311, Feb. 2016.
  • [28] T.-A. Do, S.-W. Jeon, and W.-Y. Shin, “How to cache in mobile hybrid IoT networks?” IEEE Access, vol. 7, no. 1, pp. 27 217–27 230, Mar. 2019.
  • [29] M. Mahdian and E. Yeh, “Throughput and delay scaling of content-centric ad hoc and heterogeneous wireless networks,” IEEE/ACM Trans. Netw., vol. 25, no. 5, pp. 3030–3043, Oct. 2017.
  • [30] C. Fricker, P. Robert, J. Roberts, and N. Sbihi, “Impact of traffic mix on caching performance in a content-centric network,” in Proc. IEEE Conf. Comput. Commun. (ICC) Workshops, Orlando, FL, Mar. 2012, pp. 310–315.
  • [31] X. Su and T. Khoshgoftaar, “A survey of collaborative filtering techniques,” Adv. Artif. Intell., vol. 2009, no. 4, pp. 1687–7470, Jan. 2009.
  • [32] J. Herlocker, J. Konstan, A. Borchers, and J. Riedl, “An algorithmic framework for performing collaborative filtering,” in Proc. Int. ACM SIGIR Conf. Research Development in Inf. Retrieval (SIGIR), Berkeley, CA, Aug. 1999, pp. 230–237.
  • [33] B. Sarwar, G. Karypis, J. Konstan, and J. Riedl, “Item-based collaborative filtering recommendation algorithms,” in Proc. ACM WWW Int. World Wide Web Conf. (WWW), Hong Kong, Hong Kong, May 2001, pp. 285–295.
  • [34] Y. Koren, R. Bell, and C. Volinsky, “Matrix factorization techniques for recommender systems,” IEEE Comput., vol. 42, no. 8, pp. 30–37, Aug. 2009.
  • [35] S. Sedhain, A. K. Menon, S. Sanner, and L. Xie, “AutoRec: Autoencoders meet collaborative filtering,” in Proc. ACM WWW Int. World Wide Web Conf. (WWW), New York, USA, May 2015, pp. 111–112.
  • [36] P. Gill, M. Arlitt, Z. Li, and A. Mahanti, “Youtube traffic characterization: a view from the edge,” in Proc. ACM SIGCOMM Conf. Internet Measurement (IMC), San Diego, CA, Oct. 2007, pp. 15–28.
  • [37] J. L. Herlocker, J. A. Konstan, L. G. Terveen, and J. T. Riedl, “Evaluating collaborative filtering recommender systems,” ACM Trans. Inf. Syst., vol. 22, no. 1, pp. 5–53, Jan. 2004.
  • [38] G. Karypis, “Evaluation of item-based top- recommendation algorithms,” in Proc. Int. Conf. on Inf. and Knowl. Management (CIKM), Atlanta, Georgia, USA, Oct. 2001, pp. 247–254.
  • [39] G. Calinescu, C. Chekuri, M. Pál, and J. Vondrák, “Maximizing a submodular set function subject to a matroid constraint,” in Proc. Integer Program. Combinatorial Optim. (IPCO), Ithaca, NY, Jun. 2007, pp. 182–196.
  • [40] W.-S. Hwang, J. Parc, S.-W. Kim, J. Lee, and D. Lee, “Told you I didn’t like it: Exploiting uninteresting items for effective collaborative filtering,” in Proc. IEEE Int. Conf. Data Eng. (ICDE), Helsinki, Finland, May 2016, pp. 349–360.
  • [41] H. Ma, I. King, and M. Lyu, “Effective missing data prediction for collaborative filtering,” in Proc. Int. ACM SIGIR Conf. Research Development in Inf. Retrieval (SIGIR), Amsterdam, Netherland, Jul. 2007, pp. 39–46.
  • [42] R. Pan and M. Scholz, “Mind the gaps: Weighting the unknown in large-scale one-class collaborative filtering,” in Proc. ACM SIGKDD Conf. Knowl. Discovery Data Mining (KDD), Paris, France, Jun./Jul. 2009, pp. 667–676.
  • [43] P. Cremonesi, Y. Koren, and R. Turrin, “Performance of recommender algorithms on top- recommendation tasks,” in Proc. ACM Conf. Recommender Syst. (RecSys), Barcelona, Spain, Sep. 2010, pp. 39–46.
  • [44] H. Steck, “Training and testing of recommender systems on data missing not at random,” in Proc. ACM SIGKDD Conf. Knowl. Discovery Data Mining (KDD), Washington, DC, USA, Jul. 2010, pp. 713–722.