1 UA V-Assisted Multi-Cluster Over-the-Air Computation

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UAV-Assisted Multi-Cluster Over-the-Air

Computation

Min Fu, Student Member, IEEE, Yong Zhou, Member, IEEE,

Yuanming Shi, Senior Member, IEEE, Chunxiao Jiang, Senior Member, IEEE,

and Wei Zhang, Fellow, IEEE

Abstract

In this paper, we study unmanned aerial vehicles (UAVs) assisted wireless data aggregation (WDA)

in multi-cluster networks, where multiple UAVs simultaneously perform different WDA tasks via over-

the-air computation (AirComp) without terrestrial base stations. This work focuses on maximizing the

minimum amount of WDA tasks performed among all clusters by optimizing the UAV’s trajectory

and transceiver design as well as cluster scheduling and association, while considering the WDA

accuracy requirement. Such a joint design is critical for interference management in multi-cluster

AirComp networks, via enhancing the signal quality between each UAV and its associated cluster for

signal alignment and meanwhile reducing the inter-cluster interference between each UAV and its non-

associated clusters. Although it is generally challenging to optimally solve the formulated non-convex

mixed-integer nonlinear programming, an efﬁcient iterative algorithm as a compromise approach is

developed by exploiting bisection and block coordinate descent methods, yielding an optimal transceiver

solution in each iteration. The optimal binary variables and a suboptimal trajectory are obtained by using

the dual method and successive convex approximation, respectively. Simulations show the considerable

performance gains of the proposed design over benchmarks and the superiority of deploying multiple

UAVs in increasing the number of performed tasks while reducing access delays.

Index Terms

M. Fu, Y. Zhou, and Y. Shi are with School of Information Science and Technology, ShanghaiTech University, Shanghai

201210, China (e-mail: {fumin, zhouyong, shiym}@shanghaitech.edu.cn).

C. Jiang is with the Tsinghua Space Center and the Beijing National Research Center for Information Science and Technology,

Tsinghua University, Beijing 100084, China (e-mail: jchx@tsinghua.edu.cn).

W. Zhang is with the School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney,

NSW 2052, Australia (e-mail: w.zhang@unsw.edu.au).

arXiv:2210.10963v1 [cs.IT] 20 Oct 2022

Over-the-air computation, UAV communications, wireless data aggregation, multi-cluster coopera-

tion, interference management.

I. INTRODUCTION

Machine-type communication (MTC) is one of the disruptive technologies promised by 5G

and beyond wireless networks [1]. Therein, it is crucial to collect and leverage Big Data effec-

tively for decision-making to automate various intelligent applications. However, collecting data

generated by an enormous number of devices in the future Internet of Things (IoT) networks is

critically challenging due to limited spectrum resource [2], [3]. Meanwhile, many emerging IoT

applications (e.g., environmental monitoring) only aim to collect a particular function of these

massive data rather than to reconstruct each individual data, which is referred to as wireless

data aggregation (WDA) [4]. To meet these demands, over-the-air computation (AirComp) has

recently been considered as an attractive technique to enable fast WDA among massive devices by

seamlessly integrating communication and computation processes [5]. The principle of AirComp

is to exploit the waveform/signal superposition property of multiple-access channels (MAC) such

that an edge server directly receives a function of concurrently transmitted data. This results in

low transmission delays regardless of the amount of devices, and makes AirComp particularly

appealing to data-intensive and/or latency-critical applications such as consensus control [6],

distributed sensing [7], and distributed machine learning [8], [9].

So far, AirComp has been studied from various aspects in single-cell networks, such as single-

input-single-output (SISO) AirComp [7], [10], multiple-input-single-output (MISO) AirComp

[11], [12], and multiple-input-multiple-output (MIMO) AirComp [13]. In single-cell networks,

to achieve accurate computing, AirComp requires the phase and magnitude of all signals to

be aligned at the receiver side. However, channel heterogeneity across devices makes signal

alignment challenging. To cope with this issue, different transceiver designs have been proposed

to compensate for the non-uniform channel fading and suppress the noise. Speciﬁcally, for SISO

AirComp, the authors in [7], [10] proposed the optimal transmit power control and receive

normalizing factor design. For multi-antenna AirComp systems, beamforming vectors at receiver

and/or transmitter were designed to minimize the computation error in [11], [13]. However, the

AirComp performance may deteriorate when one or more power-constrained devices are in deep

fading. To mitigate the communication bottleneck, the authors in [12] employed a promising

reconﬁgurable intelligent surface (RIS) [14], [15] to jointly design the passive beamforming at

the RIS and transceiver. In addition, we in [16], [17] proposed to deploy an unmanned aerial

vehicle (UAV) as a mobile server and to exploit its mobility to avoid any device being in deep

fade, thereby enhancing the performance of AirComp.

Meanwhile, the IoT networks generally involve different WDA tasks, each of which is char-

acterized by their applications (e.g., classiﬁcation FL task [18], regression FL task, and sensing

tasks [19]), data types (e.g., model parameters in machine learning and velocity in connected

car platooning applications), and computing functions (e.g., sum and mean). Hence, researchers

recently advocated the study of AirComp in a multi-cluster network to simultaneously complete

multiple WDA tasks, which is referred to as multi-cluster AirComp [4]. In multi-cluster Air-

Comp, besides signal alignment and noise suppression, inter-cluster interference management

arises as a new challenge. Different from the orthogonal multiple access (OMA) based multi-

cluster networks [20], each server in the multi-cluster AirComp network not only harnesses

intra-cluster interference for function computation, but also suppresses inter-cluster interference

for computation error reduction. As an initial study, the authors in [19] proposed a signal-

and-interference alignment scheme to simultaneously eliminate inter-cell interference and align

intra-cell signals in a two-cell MIMO AirComp network. Subsequently, the authors in [21]

studied the weighted sum mean-square error (MSE) minimization problem in multi-cell SISO

AirComp networks by optimizing the transmit power of devices. Results in [21] showed that,

the performance of AirComp is limited by the inter-cell interference since the transmit power of

the device needs to balance the tradeoff between combating inter-cell interference and enhancing

signal alignment within the cell. These results are established when the static terrestrial BS is

available. However, in remote and under-developed areas, the terrestrial BSs are usually sparsely

deployed or not available. In these harsh circumstances, it is critical to deploy more ﬂexible BSs

to unleash the potential of multi-cluster AirComp.

As a parallel but complementary study, in this paper, we investigate a novel multi-cluster

AirComp framework with multiple UAVs dispatched as ﬂying BSs to cooperatively perform

diverse AirComp tasks, where no terrestrial BS is available. In fact, multiple UAVs have been

deployed as BSs to assist the terrestrial networks for rate-oriented communications [22], [23],

[24]. For example, the authors in [22] studied multi-UAV cooperative communications to achieve

higher data rate and lower access delay. The authors in [23] considered a multi-UAV enabled

wireless network for data collection. This motivates us to study the use of multiple UAVs in multi-

cluster AirComp by taking into account UAV trajectory planning as well as cluster scheduling

and association for interference management, which has the following advantages. First, multi-

UAV cooperation exploits spectrum sharing to allow multiple clusters to be simultaneously

served, thereby increasing the amount of performed task within the given time duration. Second,

scheduling the clusters that are far from each other can avoid strong co-channel interference

and thus improve computation accuracy. Furthermore, the joint design not only shortens the

communication distances between the UAV and its associated cluster to enhance intra-cluster

signal alignment but also enlarges the communication distances between the UAV and its non-

associated clusters to rein in the inter-cluster interference. This joint design is crucial for in-

terference management in multi-cluster AirComp networks, but has not yet been studied in the

literature to the best of the authors’ knowledge.

A widely adopted performance metric for quantifying the AirComp computation error is MSE

between the estimated function value and the ground truth [8], [10], [13]. To promote fairness

among clusters, under the given target MSE requirements, we aim to maximize the minimum

amount of performed WDA tasks among all clusters by jointly designing cluster scheduling and

association, UAVs’ trajectories, and transceiver design in a given time duration. However, such

a non-convex max-min fairness problem presents unique challenges due to their discontinuous

objective functions (as a result of the binary cluster scheduling and association variables), non-

convex MSE constraints (because of the coupling between all optimization variables), and non-

convex trajectory design. It is generally challenging to optimally solve such a mixed-integer

non-convex optimization problem.

A. Contributions

The main contributions of this paper are summarized as follows.

•This paper is one of the early attempts to study the multi-UAV enabled AirComp in

multi-cluster wireless networks, where multiple UAVs are deployed to cooperatively per-

form different AirComp tasks. To ensure fairness among clusters, we aim to maximize

the minimum amount of performed WDA tasks among all clusters by jointly optimizing

scheduling and association for clusters, UAVs’ trajectory planning, and transceiver design,

taking into account the target accuracy of AirComp, practical constraints on UAVs, inter-

cluster interference, as well as the total power budgets at devices.

•To render the resulting non-convex mixed-integer nonlinear programming (MINLP) tractable,

by leveraging the bisection method, the original problem reduces to a sequence of the mini-

mum ratio maximization problems, which enables the development of an iterative algorithm.

After adopting block coordinate descent (BCD) [25], besides both convex normalizing

factors and power optimization subproblems being optimally solved, the optimal cluster

scheduling and association are also obtained by applying the low-complexity Lagrange

duality method. For the non-convex UAV trajectory optimization problem, a suboptimal

solution is obtained by using the successive convex approximation (SCA) method.

•Simulation results are presented to show the effectiveness and superiority of the proposed

design and developed algorithm. In multi-cluster AirComp with a single UAV, the per-

formance achieved by the joint design outperforms other benchmarks and can reach the

upper bound. It is also shown that the max-min task amount of the considered UAV

network increases with the mission duration, revealing a performance-access delay tradeoff

in multi-cluster AirComp. Compared to the single-UAV case, the use of multiple UAVs

with effective cooperative interference management can considerably increase the amount

of tasks performed by each cluster while reducing access delays.

B. Organization

The rest of this paper is organized as follows. We presents the system model and the problem

formulation for multi-cluster AirComp assisted by UAVs in Section II. We develop an iterative

algorithm yielding high-quality solutions to solve the formulated problem based on the Bisection

method, BCD, and SCA in Section III. In addition, numerical results are presented in Section

IV to evaluate the performance of the proposed design. In Section V, we draw the conclusions.

Notations: Scalars, column vectors, and matrices are written in italic letters, boldfaced lower-

case letters, and boldfaced upper-case letters respectively, e.g., a,a,A.RM×Ndenotes the space

of a real-valued matrix with with Mrows and Ncolumns. kak2denotes the Euclidean norm of

vector aand aTrepresents its transpose. |S| denotes the cardinality of the set S.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

As shown in Fig. 1, we consider a multi-cluster wireless network, where each cluster consists

of multiple ground devices and multiple UAVs1are deployed to support multiple WDA tasks

1We consider the scenario that the terrestrial BS is not available, e.g., in wild areas. Under this circumstance, the UAVs are

adopted as an alternative to provide wireless services for ground devices [22].

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1UAV-AssistedMulti-ClusterOver-the-AirComputationMinFu,StudentMember,IEEE,YongZhou,Member,IEEE,YuanmingShi,SeniorMember,IEEE,ChunxiaoJiang,SeniorMember,IEEE,andWeiZhang,Fellow,IEEEAbstractInthispaper,westudyunmannedaerialvehicles(UAVs)assistedwirelessdataaggregation(WDA)inmulti-clusternetworks,where...

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