On the Effective Throughput of Coded Caching: A Game Theoretic Perspective

11/29/2019 ∙ by Yawei Lu, et al. ∙ Princeton University Tsinghua University 0

Recently coded caching has emerged as a promising means to handle continuously increasing wireless traffic. However, coded caching requires users to cooperate in order to minimize the overall transmission rate. How users with heterogeneous preferences cooperate in coded caching and how to calculate the resulting caching gains are still open problems. In this paper, a two-phase cache-aided network is investigated, in which users with heterogeneous preferences are served by a base station through a shared link. Effective throughput is considered as a performance metric. It is proved that the achievable domain of effective throughputs is a convex set and can be characterized by its boundary. A special type of caching schemes, named uncoded placement absolutely-fair (UPAF) caching, is studied. For the two-user case, games are formulated to allocate effective throughput gains for the two users. For the general multiuser case, a UPAF policy is proposed to organize user cooperation. It is shown that users with more concentrated preferences can obtain higher effective throughputs.

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I Introduction

Thanks to the popularization of mobile devices and the rise of multimedia applications, wireless traffic has undergone dramatic growth in recent years. In the era of 5G, various wireless communication technologies have been developed to improve network capacity and handle the ever-increasing traffic, including polar codes, massive multi-input and multi-output, millimeter wave, etc. Notice that a significant feature of current wireless traffic is that content items are generated much earlier than they are requested. Caching has emerged as an inexpensive and powerful solution to cope with increasing wireless traffic by pre-storing popular content items in user terminals before users request them [1].

By shifting traffic at peak hours to idle hours, caching can exploit idle communication resources. Wireless networks benefit from caching in many aspects. Since content items can be provided for users from closer nodes rather than the core network, caching contributes to lower latency communications [2]. From an information theoretic perspective, the fundamental latency-cache tradeoff was revealed in [3] by introducing a performance metric termed normalized delivery time. Considerable attention has also been paid to improving energy efficiency through caching in wireless networks [4]-[6]. In [4], an in-memory storage method was proposed to reduce energy consumption in edge caching. The mismatch between the randomly arrived energy and content requests was eliminated by caching in energy harvesting networks [5]. The energy efficiency gains from caching in interference channels were revealed in [6]. In millimeter wave communications, caching helps to enhance mobility support and reduce handover failures [7]. A cost-optimal caching scheme was proposed for device-to-device (D2D) networks with user mobility in [8]. The effective throughput resulting from caching, which indicates the reduction in the real-time traffic, was investigated in [9]-[12].

Recently, a caching method termed coded caching has attracted considerable attention [13]-[19], because it can provide global caching gains. The seminal works [13] and [14] studied a two-phase (placement phase and delivery phase) system and first revealed the fundamental memory-rate tradeoff for coded caching. Thereafter, the exact memory-rate tradeoff for coded caching with uncoded prefetching was characterized in [15]. Coded caching was further extended to D2D communications in [16]. It was shown that coded caching can create multicasting opportunities for overloaded multi-input single-output channels [17]. The work [18] provided a coded caching scheme for multiple-server networks. An order-optimal coded caching scheme was proposed to meet users’ privacy requirement in [19].

In present multimedia applications, a few popular content items typically account for the majority of network traffic [20]. User preferences on these popular content items have been shown to play a critical role in designing caching schemes [21]. Various caching schemes were developed to improve network capability by taking advantage of content popularity [22]-[26]. The work [22] proposed femtocaching to reduce downloading delay according to network topology and popularity distributions. A file grouping scheme was developed to handle the uneven preferences in cache-aided networks [23]. For content items with discrete popularities, [24] proposed a near-optimal coded caching scheme to reduce network load. The memory-rate tradeoff for coded caching under uneven popularities was revealed in [25] and [26]. Game theoretic techniques were also adopted to design caching and pricing strategies [27]-[28].

In previous studies, the focus has been on improving network performance for the sake of servers. For example, [13]-[14] devoted to reduce the load that the server sustains and [29] tried to lower the overall energy consumption. Less attention has been paid to the revenue that an individual user can obtain from caching. Thanks to the progress of big data technologies, user preferences can be analyzed based on private browsing history and social relationships [30]-[31]

. Existing studies have mostly ignored users’ personal differences by assuming that all the users have the same probability of requesting a content item

[22]-[28]. However, user preferences have a significant impact on caching gains. If a user is interested in only a few content items, i.e., the user requests are deterministic or near-deterministic, this user’s buffer can be filled with these content items. This user can find the desired content items directly when it issues requests. As a result, this user obtains a high effective throughput from caching. If a user is interested in many content items, the above mechanism does not work due to the buffer size constraint. It can cooperate with other users by coded caching to split the transmission cost. Few works have focused on how to calculate each user’s caching gain in this case.

In this paper, we investigate effective throughput from caching for users with heterogeneous preferences. More specifically, multiple users are served by a base station through a shared link and user preferences are characterized by a probability measure. Similar to the model considered in [13] and [14], the network works in two phases, a placement phase and a delivery phase. In the placement phase, network load is light and user buffers are filled through idle spectra. In the delivery phase, user requests are revealed and the base station transmits messages to help the users recover the desired content items. Effective throughput is used as a performance metric, which indicates the reduction in the transmission cost in the delivery phase. Each user’s effective throughput is calculated individually. To characterize the whole achievable domain of effective throughputs, we need to investigate all the feasible placement and delivery policies, which is however of prohibitive complexity. Upon that, we prove the convexity of the achievable domain and focus on a special type of policies, termed uncoded placement absolutely-fair (UPAF) policies. The achievable domain under UPAF policies is shown to be a polygon in the two-user case.

The higher the effective throughput a user obtains, the lower the real-time transmission cost this user affords. If the users are selfish and each user wishes to maximize its own effective throughput, the users form a game relationship. Based on the analysis on achievable domain in the two-user case, a noncooperative game is formulated to investigate the effective throughput equilibrium between the two users. In addition, a cooperative game is studied to allocate the revenue of cooperation, which helps in designing pricing policies. Suffering from the hardness in finding a Nash equilibrium in noncooperative games, a low-complexity numerical algorithm is proposed to give a reasonable revenue allocation for the two users. An algorithm is also presented to organize user cooperation in the general multiuser case.

The rest of the paper is organized as follows. Section II presents our system model and the formal definition of placement and delivery policies. Section III proves the convexity of the achievable domain of effective throughputs and investigates the achievable domain for the two-user case in detail. Games among two users are studied in Section IV. Section V presents a cooperation scheme for the general multiuser case. Simulation results are given in Section VI. Finally, Section VII concludes this paper and suggests some directions for future research.

Ii Problem Setting

We introduce the system model in Subsection II-A, provide the definition of effective throughput in Subsection II-B, and then present an example to illustrate the research motivation in Subsection II-C.

Ii-a System Model

Consider a Base Station (BS) connected with users through a shared error-free link. The BS has access to a database of content items, denoted by . Assume that all the content items have an identical size of bits. Let denote the collection of bits of the content items.111For a positive integer , denotes the set User is equipped with a buffer of bits, or equivalently, content items. We refer to

as buffer size vector. Let

indicate whether user asks for content item , i.e., if user asks for and otherwise. We refer to

as the demand matrix, which is a random matrix with support set

. Then user preferences can be characterized by a probability measure on , denoted as . In practice, user preferences can be analyzed based on private browsing history and social relationships [30]-[31]. The probability that user requests is given by . The matrix is referred to as the user preference matrix. It should be noted that we do not assume that each user requests only one content item. In other words, the summations and may not be 1.

This cache-aided network operates in two phases, namely a placement phase and a delivery phase. In the placement phase, user requests are not specific. The users prefetch data from the BS and cache them in their buffers with the knowledge of the probability measure . In the delivery phase, the users issue requests for the content items. The demand matrix reduces to a deterministic matrix . Local buffers provide useful information in recovering the requested content items. The users probably also need to turn to the BS in order to recover all the requested content items. The network described above is referred to as -Caching.

Ii-B Formal Problem Statement

We provide a formal description of placement and delivery policies for -Caching.

Definition 1.

A placement and delivery policy consists of the following three types of functions.
i) Content prefetching function : For each user , determines the bits prefetched from the BS in the placement phase and thus gives this user’s buffer state in the beginning of the delivery phase. Specifically, we have

(1)

Due to the buffer size constraint, should satisfy .222For a set , denotes its cardinality.
ii) Message generation function : In the delivery phase, user requests are revealed and hence is known. For each subset of , generates a message according to and , i.e.,

(2)

The message is generated to help users in recover the requested content items.
iii) Content recovering function : After receiving the transmitted messages, each user attempts to recover the requested content items by , i.e.,

(3)

where represents the set of content items requested by user .333 traverses all the subsets of that contain .

stands for the estimated

.

For a placement and delivery policy , the error probability is defined as

(4)

where denotes the conditional probability. Given a -Caching, there are numerous placement and delivery policies that can satisfy the user requests. A traditional one is just to ignore user buffers and transmit the requested content items in the delivery phase directly. Caching and multicasting enable us to satisfy user requests in a more effective manner. For a policy , we define the effective throughput of user as

(5)

where denotes the mathematical expectation with respect to the random matrix . The summation stands for the number of bits transmitted to user in the delivery phase without caching and multicasting. Because is transmitted to users, each user incurs cost of the transmission. Then, indicates the total transmission cost afforded by user . In addition, we normalize the transmission cost by . Thus, the effective throughput defined in Eq. (5) represents the reduction in the transmission cost of user .

The summation of all users’ effective throughputs has

(6)

Note that and represent the expected number of content items requested by the users and the total number of bits transmitted by the BS, respectively. The right-hand side of Eq. (6) indicates the reduction in the number of the bits transmitted by the BS due to caching and multicasting (normalized by the content size ). Thus, indicates the revenue of user from caching. As caching schemes studied in previous literatures like [13]-[16], the proposed policies may incur a high signaling overhead. This overhead can be partly eliminated by large content items.

Users’ effective throughputs vary with policies. All the possible values of effective throughputs form an achievable domain.

Definition 2.

The vector is achievable if for every and every sufficiently large there exists a policy that achieves with error probability lower than . For a -Caching, the achievable domain of effective throughputs is defined as

(7)

If no confusion arises, we simply denote by . The achievable domain depends on the buffer size vector, the number of content items, as well as user preferences.

Ii-C A Motivating Example

In this subsection, we present a demo to illustrate the research motivation and the impact of user preferences on effective throughputs. As shown in Fig. 1, two users are interested in two content items, denoted as and . Each user at most caches one content item. User 1 requests the two content items with probability 99% and 1%, respectively. User 2 requests the two content items with an identical probability, 50%. In this demo, we assume each user requests only one content item. The coded caching scheme suggests to divide each content item into two portions and then the user requests can always be satisfied by transmitting a coded packet of size 0.5 [13]. Then both the two users achieve an effective throughput 0.75. The coded caching scheme requires the two users cooperate in both the two phases. Let us consider a noncooperation scheme that each user caches the most popular content items. The two users split the transmission costs if the delivered data are useful for both the two users. It is seen that user 1 achieves a much higher effective throughput in the noncooperation scheme than it does in coded caching. As a result, user preferences have a significant impact on caching gains and should be taken into account in cache-aided networks.

Fig. 1: Effective throughputs under different caching schemes. The pair (A,A) in the tables stands for the realization when both the two users request content and the other pairs are similar. ET is short for effective throughput.

Iii Property of the Achievable Domain

In this section, we prove the convexity of the achievable domain of effective throughputs and then focus on a special type of placement and delivery policies, termed UPAF policies. It will be shown that -Caching with two users has an achievable domain as a polygon under UPAF policies.

Iii-a Convexity of the Achievable Domain

The following theorem presents the convexity of the achievable domain of effective throughputs .

Theorem 1.

For any -Caching, the achievable domain is a convex set and for any point ,

(8)
Fig. 2: Convex set in the positive orthant. This set can be completely characterized by its boundary in the positive orthant.
Proof.

Let and be two points in , as illustrated in Fig. 2. To prove the convexity, we only need to show that also belongs to for any . Given and large enough, suppose and achieve and with error probability lower than . We divide each content item into two portions, containing bits and bits respectively. By further dividing each buffer into two portions of size bits and bits, the original -Caching can be viewed as two -Caching with different content sizes. The policies and can be respectively applied in the two -Caching. The error probability is bounded by . The effective throughput of user is given by

where and are messages generated by and respectively. For any , can be achieved by combining and for sufficiently large . Hence, is a convex set.

Eq. (8) implies that if a point is in , any point such that also belongs to . We prove that by constructing a policy achieving . Suppose achieves with error probability lower than . In the delivery phase, transmits to user user exclusively. Let us consider a new policy in which the BS additionally transmits a random message of bits to user . For this new policy, the error probability remains unchanged while the effective throughputs reduce to . Thus, is achievable. ∎

According to Theorem 1, we only need to pay attention to the boundary in the positive orthant in order to characterize , as illustrated in Fig. 2. In addition, we can have more insights on the achievable domain of -Caching. The values of different users’ effective throughputs are interchangeable. When a policy brings one user a high effective throughput, the other users may only obtain low effective throughputs.

It is intractable to investigate all the feasible policies and characterize the whole achievable domain of effective throughputs. In the paper, we focus on absolutely fair (AF) policies.

Definition 3.

A policy is absolutely fair if each user in can obtain the same amount of useful information from the message , i,e,

(9)

for .444 represents the conditional mutual information.

By enforcing other users receive the bits needed only by a certain user, the transmission cost of this user can be simply reduced. In an AF policy, such enforcement is forbidden. In other words, all the users are fair and no one incurs transmission costs for other users. If an AF policy has an uncoded placement process, the policy is referred to as a UPAF policy. In the rest of the paper, our attention will be paid to UPAF policies. The significance of studying UPAF policies is twofold. Theoretically, UPAF policies provide an inner bound for the whole achievable domain. In addition, UPAF policies are practical, since all the users are fairly treated and the placement process is simpler compared with a coded one.

Iii-B The Achievable Domain for -Caching with Two Users

In this subsection, we investigate the achievable domain of -Caching with two users. A policy maps the content items and requests into the buffer states and the transmitted messages . Thus we can use and to represent the policy . To characterize for the two-user case, we only need to study all feasible and .

Fig. 3: Venn Diagram of and .

For a UPAF policy, we have . Let us define

(10)
(11)
(12)
(13)

Then, stands for the bits exclusively cached in the buffer of user for It is seen that and . In the delivery phase, the demand matrix is known. Then, user and user wish to recover the bits in and , respectively. We define

(14)
(15)
(16)

Then is the set of bits requested only by users in but not cached in the buffers of these users. As a result, users in want to recover from the transmitted messages. Fig. 3 illustrates the relations between and .

To satisfy user requests in the delivery phase, we can simply set and . Notice that user contains a part of bits that user requests and vice versa. Index coding can be applied to create more multicasting opportunities and improve the effective throughputs of both user and user [32]. In this case, the transmission costs of user and user are given by

(17)
(18)

Note that Eqs. (17) and (18) give the minimum transmission costs for given and , because all the possible multicasting opportunities have been exploited.

The transmission costs, i.e., Eqs. (17) and (18), can also be represented as functions of the two users’ buffer states. To this end, we denote . According to De Morgan’s laws, Eqs. (14)-(16) can be rewritten as

(19)
(20)
(21)

Defining and substituting Eqs. (19)-(21) into Eqs. (17)-(18) yield

(22)
(23)

The variable stands for how much accounts for .

For fixed uncoded placement and , the maximum effective throughputs achieved by UPAF policies are given by

(24)
(25)

From the proof of Theorem 1, one can see that Theorem 1 also holds for UPAF policies. Thus, the boundary of the achievable domain under UPAF policies can be given by solving the following optimization problem:

(26)

The first constraint is due to the fact that the sets are disjoint for . The second constraint forbids buffer overflows in the placement phase. By tuning from 0 to 1, we obtain the boundary of the achievable domain of effective throughputs under UPAF policies.

Problem (26

) is almost a linear programming (LP) problem except the minimizing operations. To simplify problem (

26), we introduce auxiliary variables and vectors

(27)
(28)
(29)
(30)

Then and represent the proportion of each content item cached in user 1 and user 2, respectively. The vector represents the proportion of each content item cached in both the two users’ buffers. The vector is a function of and . The last element of is a fixed constant , which is used to express the constant term in Eqs. (24)-(25) later. The constraints on are linear:

(31)
(32)

Then, problem (26) can be transformed into an equivalent LP problem555Let and be two -dimensional vectors. By , we mean for .

(33)

where gives the effective throughput of user and , as well as result from the constraints in problem (26) and Eqs. (31)-(32). It should be noted that the coefficients depend on and only on the user preferences. Based on problem (33), we have the following theorem.

Theorem 2.

For -Caching with two users, the achievable domain under UPAF policies is a polygon.

Proof.

To show the achievable domain under UPAF policies is a polygon, we only need to show its boundary is piecewise linear. The points in the boundary can be obtained by solving problem (33). Notice that the constraints of problem (33) are linear and are independent of . Therefore, the feasible domain of problem (33) remains unchanged with different values of .

According to the LP theory, the feasible domain of an LP problem is a convex polytope and the optimal solution is a vertex of the convex polytope [33].666If an LP problem has only one optimal solution, this solution must be a vertex of the convex polytope. If an LP problem has multiple optimal solutions, at least one of the optimal solutions is a vertex of the convex polytope. Since the number of vertices of a convex polytope is finite, problem (33) at most achieves finitely many different optimal solutions when goes from 0 to 1. As a result, the boundary can be characterized by finitely many points and therefore is piecewise linear. ∎

The achievable domain gathers all the possible values of effective throughputs that can be achieved by possible placement and delivery policies, no matter it is centralized or decentralized. The higher the effective throughput a user obtains, the lower the transmission cost this user affords. The boundary of the achievable domain represents the Pareto-efficient effective throughputs. If the users are selfish and each user only wants to maximize its own effective throughput, the users form a game relationship. In the next section, games are formulated to allocate caching gains for -Caching with two users.

Iv Games in -Caching with Two Users

In this section, a noncooperative game is formulated to investigate the equilibrium on effective throughputs for -Caching with two users. Based on the noncooperative game, a cooperative game is studied to allocate the caching gains. Furthermore, a low-complexity algorithm is presented to provide a reasonable effective throughput allocation.

Iv-a Noncooperative Game in -Caching with Two Users

In this subsection, we investigate a noncooperative game in -Caching with two users. More specifically, we assume that the two users fill their buffers individually in the placement phase. The BS satisfies the user requests in a manner that the number of transmitted bits is minimized. Each user wishes to maximize its own effective throughput from caching. It will be shown that the noncooperative game always has a mixed Nash equilibrium (NE). In addition, the noncooperative game has pure strategy Nash equilibria (PSNEs) when the user preferences are similar.

In the noncooperative game, the two users take the roles of players. The sets of bits cached in the user buffers in the placement phase, i.e., and , act as strategies. Throughout this subsection, we consider only UPAF policies. As a result, the strategy sets for this two users are given by and . The payoffs for this two users are the effective throughputs resulting from caching and thus are presented in Eqs. (24) and (25), respectively. We denote the above noncooperative game as .

Nash’s existence theorem guarantees that the noncooperative game in -Caching with two users at least has a mixed NE [34].

Theorem 3.

has a mixed NE.

Proof.

has finitely many players. In addition, the strategy sets are finite, i.e., for . Thus is a finite game. According to Nash’s existence theorem, has a mixed NE. ∎

A pure strategy is a bit-by-bit decision over the content items. However, a mixed NE needs not to divide a bit into smaller parts. Instead, a mixed NE chooses pure strategies according to a certain distribution. Having proved the existence of a mixed NE, we pay attention to PSNEs. However, it is computationally prohibitive to find a PSNE and corresponding payoffs for , due to the fact that the strategy sets are of exponential sizes. To overcome that, we construct an infinite game based on .

Note that the strategies and can be completely characterized by and . The payoff functions in can be rewritten as

(34)

Let us consider a two-player infinite game . The strategy sets are feasible domains of and , i.e., and . The payoff function is defined as

(35)

where

(36)

where the constraint is equivalent to the one in problem (33). One can see that problem (36) is also an LP problem. The basic idea to formulate is as follows. Each user decides the number of bits cached in its own buffer. Thus and act as strategies. When the values of and are selected, user 1 is granted the privilege to maximize its own effective throughput by adjusting the values of and and then yields the payoff function . Similarly, can be derived.

Lemma 1 presents the relationship between the existences of PSNEs of and .

Lemma 1.

If has a PSNE and there exist and satisfying , has a PSNE.

Proof.

Suppose is a PSNE of We have

(37)
(38)

The condition implies that problem (36) has the same optimal solution for both and . Thus, we have

(39)
(40)

Let us consider satisfying and . Eqs. (39) and (40) can be written as

(41)
(42)

Thus is a PSNE of . ∎

Lemma 1 reveals that has a PSNE if has a PSNE and the two users adopt the same pair values of and . By applying Debreu’s theorem [35], we can prove that has a PSNE.

Lemma 2.

has a PSNE.

Proof.

According to Debreu’s theorem, we only need to show that the strategy sets are nonempty convex compact subsets of an Euclidean space and the payoff functions are continuous and quais-concave.

The sets and are bounded and closed in and thus are also compact. The payoff functions are maximums of a series of linear functions:

(43)

It is seen that is continuous. To prove the quais-concavity of , we only need to show is quais-concave, according to the properties of quais-concavity. As a linear function, is quais-concave. ∎

Based on Lemmas 1 and 2, we have the following theorem.

Theorem 4.

Given any and , there exists a positive number such that if , has PSNEs and the PSNEs are not unique.

Proof.

According to Lemmas 1 and 2, we only need to show that problem (36) has the same optimal solution for and at the PSNE of , when is small enough. Notice that the feasible domain of problem (36) remains unchanged with different values of . The objective function of problem (36

) describes a group of hyperplanes with the same normal vector

. The optimal solution happens to be the intersection point of the feasible domain and a certain hyperplane [33]. If the normal vectors for and are close enough, problem (36) achieves its optimization value at the same vertex of its feasible domain. Thus has a PSNE when the difference between and is small.

Having proved the existence of PSNEs, we now show its non-uniqueness. Let us consider the dual problem of problem (36):

(44)

Problem (