Joint Scheduling and Resource Allocation for Federated Learning in SWIPT-Enabled Micro UAV Swarm Networks

Wanli Wen,Yunjian Jia,Wenchao Xia

1 School of Microelectronics and Communication Engineering,Chongqing University,Chongqing 400044,China

2 National Mobile Communications Research Laboratory,Southeast University,Nanjing 210009,China

3Jiangsu Key Laboratory of Wireless Communications,Nanjing University of Posts and Telecommunications,Nanjing 210003,China

Abstract: Micro-UAV swarms usually generate massive data when performing tasks.These data can be harnessed with various machine learning (ML) algorithms to improve the swarm’s intelligence.To achieve this goal while protecting swarm data privacy, federated learning (FL) has been proposed as a promising enabling technology.During the model training process of FL,the UAV may face an energy scarcity issue due to the limited battery capacity.Fortunately, this issue is potential to be tackled via simultaneous wireless information and power transfer (SWIPT).However, the integration of SWIPT and FL brings new challenges to the system design that have yet to be addressed, which motivates our work.Specifically,in this paper, we consider a micro-UAV swarm network consisting of one base station(BS)and multiple UAVs, where the BS uses FL to train an ML model over the data collected by the swarm.During training,the BS broadcasts the model and energy simultaneously to the UAVs via SWIPT,and each UAV relies on its harvested and battery-stored energy to train the received model and then upload it to the BS for model aggregation.To improve the learning performance,we formulate a problem of maximizing the percentage of scheduled UAVs by jointly optimizing UAV scheduling and wireless resource allocation.The problem is a challenging mixed integer nonlinear programming problem and is NP-hard in general.By exploiting its special structure property,we develop two algorithms to achieve the optimal and suboptimal solutions, respectively.Numerical results show that the suboptimal algorithm achieves a near-optimal performance under various network setups,and significantly outperforms the existing representative baselines.considered.

Keywords: micro unmanned aerial vehicle; federated learning; simultaneous wireless information and power transfer;scheduling;resource allocation

I.INTRODUCTION

In recent years, the micro unmanned aerial vehicle(UAV)swarms have been widely exploited for service provisioning in many fields, such as parcel delivery,agriculture, and surveillance [1-4].While supporting these services, UAVs in the swarms often generate big streams of data traffic.If these data can be harnessed with machine learning(ML)algorithms,the intelligence of the UAV swarm could be significantly improved.However, traditional ML algorithms need to concentrate the training data on a central computing node such as a cellular base station (BS).For the UAV swarm,data privacy may be compromised when transmitted to the central node.To overcome this privacy issue,the concept of federated learning(FL)has been proposed as a promising enabling technology[5].FL can take advantage of the computing power of the UAVs to co-train a common ML model without exposing their data and thus has attracted a lot of attention in recent years.

Due to the limited computation and communication resources, only a portion of clients (e.g., UAV in our paper) can be scheduled to participate in the model training process of FL.Recently,some research efforts have pointed out that the client scheduling of FL will significantly affect the learning performance.For instance,the authors in[6]proved that scheduling more clients at each iteration of FL improves the statistical learning performance.Thus, how to efficiently determine a joint design of client scheduling and resources allocation is critical to the implementation of FL.So far,there are serval representative works have studied this problem.Specifically, based on the principle of over-the-air computation, the authors in [7] proposed a joint client scheduling and beamforming problem to maximize the number of scheduled clients at each iteration of FL.The authors in[8-11]separately examined a joint client scheduling and bandwidth allocation problem to either minimize the training time[8],training loss[9],and energy consumption at scheduled clients[10]or maximize the total number of scheduled clients during entire training process [11].With aiming to reduce the number of iterations required for FL convergence, the work in [12] proposed a joint UAV scheduling and power allocation scheme.In[13],the authors formulated a joint client scheduling, power,and bandwidth allocation problem to minimize the training loss.Note that the micro UAV are in general equipped with capacity-limited batteries.When UAVs are performing FL tasks, especially in harsh environment,it may be unfavorable or even impractical to replace the onboard batteries.Therefore, UAVs participating in FL may face a serious energy dilemma.Note that, such energy scarcity issue may lead to a reduction in the number of UAVs participating in FL,which degrades the learning performance.To facilitate the implementation of FL at the micro-UAV swarm, it is of necessity to take the energy scarcity issue into account,which unfortunately are neglected by the above discussed works.

Recently, radio frequency (RF) signal based wireless power transfer (WPT) has been envisioned as a promising solution to remedy the energy scarcity issue of wireless devices[14].By integrating WPT with the communication networks, the researchers further proposed the concept of simultaneous wireless information and power transfer (SWIPT) to enable ubiquitous wireless communications in a self-sustainable manner,where the communication and energy requirements of wireless devices can be met simultaneously[15-17].Thanks to the broadcast nature of RF signals,SWIPT is particularly suitable for powering micro UAVs, making it possible to tackle their energy scarcity issue.The aforementioned discussions stimulate a paradigm shift in the design of SWIPT enabled FL.Nevertheless, the integration of SWIPT and FL brings new challenges to the optimal system design.On the one hand, the client scheduling and resource allocation decisions in FL now rely on the amount of energy harvested by each UAV.On the other hand,intensified competition for the limited communication resources are sparked between energy harvesting and global model broadcasting/uploading in FL.

In this paper, we would like to address the above challenges.We consider an SWIPT-enabled micro-UAV swarm network consisting of one BS and multiple micro UAVs.The BS aims to train an ML model over the data that reside on all UAVs by using FL.At each iteration of FL, the BS employs SWIPT to broadcast both the global model and energy simultaneously to all UAVs, and then each UAV relies on its harvested and battery-stored energy to train the received model and upload the updated local model to the BS for global model aggregation.To improve the statistical learning performance of FL, we maximize the percentage of scheduled UAVs at each iteration under the constraints of computation and communication resources of the UAVs.The main contributions of the paper are summarized as follows:

• We establish the mathematical model of the wireless powered FL system.To improve the learning performance of FL, we formulate a problem of maximizing the percentage of scheduled UAVs at each iteration of FL by jointly optimizing UAV scheduling and sub-channel assignment as well as broadcasting/uploading time and power allocation.The formulated problem is a challenging mixed integer nonlinear programming (MINLP)problem and is NP-hard in general.To render its tractability, we transform it into an equivalent MINLP problem with a special property by using some appropriate transformations.

• With the transformed equivalent problem,we propose an algorithm to obtain its optimal solutionby using the generalized benders decomposition(GBD) technique [18], which, however, suffers from worst-case exponential time complexity.

• To reduce the computational complexity, we devise a low-complexity algorithm to obtain a suboptimal solution by transforming the continuous relaxation of the original problem into a convex problem and rounding the optimal solution of the convex problem in a greedy manner.

Beyond the above contributions, extensive numerical results show that the proposed suboptimal algorithm can achieve almost the same performance as the optimal one under various network setups, and significantly outperforms the other representative baselines considered.

The rest of this paper is organized as follows.Section II introduces the system model.Section III formulates a joint UAV scheduling and resource allocation problem and transforms the original intractable problem into an equivalent MINLP problem with a special property.Sections IV and V propose an optimal solution and a low-complexity suboptimal solution, respectively.Numerical results are presented in Section VI.Finally, conclusion is drawn in Section VII.Unless otherwise specified, the notations used throughout the paper are summarized in Table 1.

Table 1. Notations.

II.SYSTEM MODEL

As shown in Figure 1,we consider an SWIPT-enabled micro UAV swarm network consisting of one singleantenna BS and a swarm ofKsingle-antenna micro UAVs,denoted by setK≜{1,2,··· ,K}.The BS integrates a radio frequency(RF)power transmitter and has a stable power supply.Each UAV in the swarmKis equipped with a hybrid information and power transfer receiver,and thus it can scavenge energy from the received RF signal.We consider that all UAVs are flying around the BS at the same altitude.Particularly,we assume that under the coordination of the BS, each UAV can guarantee constant speed and altitude and avoid collisions between UAVs within the swarm.We refer to the flying area centred on the BS as the patrol area (i.e., the gray area in Figure 1) of the UAV swarm.During flight, the UAV swarm collects data from the patrol area,aiming to perform some ML tasks such as localization,trajectory planning,and target recognition.LetwithNk=|Nk|denote the data set locally collected by UAVk ∈K,where|·|denotes the cardinality of a set,(xnk,ynk) is referred to as a data sample, xnk ∈RDandynk ∈R denote aD-dimensional input vector and the corresponding label,respectively.Further,letdenote the whole data set of all UAVs.

Figure 1. Illustration of an SWIPT-enabled micro UAV swarm network.

Figure 2. Workflow of the proposed optimal solution.

At the BS,the focus is to learn an ML model of sizeLbits over data that reside on theKUAVs,i.e.,the BS needs to fit aD-dimensional vector w∈RDso as to minimize a particular loss function by using the data setN.Mathematically,this task can be expressed as

The above three steps will be repeatedItimes.After that,we set w(I)as a desirable solution of the problem in (1), i.e., w(I)→w∗.Note that the above training process completely embodies the characteristics of FL.In the following subsections, we first mathematically model the above three steps of the training process and then establish the associated constraints on the energy and time consumptions with respect to these steps.1

2.1 Simultaneous Global Model and Power Transfer

where the constraint in (3) ensures that the transmission power of the BS does not exceed its maximum transmission powerAccordingly, the energy harvested by UAVk, denoted by(in Joule),is given by

2.2 Local Model Training

In this step, only a subset of UAVs inKcan be involved in the local learning process.Letβkdenote the scheduling indicator for UAVk,where

Here,βk=1 indicates that UAVkis scheduled for local training,andβk= 0 otherwise.Letβ≜(βk)k∈Kdenote the UAV scheduling policy.Then,the time cost of local training at UAVkis given by

whereZkdenotes the bits of the training data stored at UAVk,Wkindicates the number of central processing unit(CPU)cycles that is needed for computing one bit of the training data stored at UAVk, andFkrepresents the frequency of the CPU clock of UAVk.In the meanwhile,the corresponding energy consumption as regards local training at UAVk, denoted by(βk)(in Joule),is quantified below[13]:

whereµkdenotes the energy consumption coefficient,which is related to the hardware architecture of UAVk[20].

2.3 Local Model Uploading

In this step, each scheduled UAV will upload its updated local model to the BS.Similar to Step 1), we consider OFDMA as the physical layer multiple access technique.Specifically, letρk,sdenote the subchannel assignment indicator for UAVk,where

Here,ρk,s= 1 represents that thesth sub-channel is assigned to UAVkfor uploading its updated local model to the BS, andρk,s= 0 otherwise.Note that,if UAVkhas never been scheduled for local training before(i.e.,βk= 0),then it will not be allocated any sub-channel,namely

In addition, we stipulate that each sub-channel can only be occupied by one UAV and each scheduled UAV should be allocated with one sub-channel at least,i.e.,

Define(in Watt) as the transmission power of UAVkon sub-channels, then the achievable rate is given by(in bps).Lett(in second)denote the time duration of Step 3).In order to successfully upload the updated local model from UAVkto the BS, we have the following constraints:

where the constraints in(14)and(15)ensures that the transmission power of UAVkdoes not exceed its maximum transmission powerAccordingly,the energy consumption at UAVkfor model uploading,denoted byis given by

2.4 Energy and Time Constraints

In this work,we consider that the energy harvested by the UAVs and the energy stored by the onboard batteries will be used to perform local model training and uploading as well as flying.Thus,we have

III.PROBLEM FORMULATION AND TRANSFORMATION

It is explicitly demonstrated by[6]that selecting more UAVs at each global iteration of FL induces an improved statistical learning performance (e.g., training loss and prediction error),but at the cost of communication and computation overhead increment.Inspired by this observation, in this paper, we focus on maximizing the percentage of the scheduled UAVs at each global iteration by jointly optimizing UAV scheduling and sub-channel assignment as well as broadcasting/uploading time and power allocation,subject to the constraints on communication and computation resources.Mathematically,we have the following problem.

Problem 1(Percentage of Scheduled UAVs Maximization).

Problem 1 is a very challenging non-convex MINLP problem and is in general NP-hard[23].Note that,the challenging mainly stems from two aspects.One is the existence of the non-convex constraints, and the other one is the coupling between the binary variables, i.e., (β,ρ), and the continuous variables, i.e.,Thereby, to obtain an optimal solution of Problem 1, some transformations and simplifications are indispensable.

We first introduce some new variables, i.e.,≜andThen, with the newly introduced variables, the constraints in (2), (3), (15), and (18) can be transformed to the following conditions:

Similarly,we rewrite the constraints in(13)and(14)as

Next,based on the foregoing transformations,we obtain an equivalent problem,which is quantified below:

Problem 2(Equivalent Problem of Problem 1).

s.t.(4),(6),(9),(10),(11),(12),(16),(19),(20),(21),(22),(23),(24),(25),(26).

The relationship between Problem 1 and Problem 2 is shown in the following lemma.

Lemma 1.Problem 1 and Problem 2 are equivalent.

Proof.See Appendix VIII.

Before ending this section, we declare that Problem 2 is still a non-convex MINLP problem.However,compared to Problem 1,Problem 2 has an important property,i.e.,Problem 2 is convex with respect to the continuous variablesfor fixed binary variables(β,ρ).Due to this property, an optimal solution of Problem 2 is attainable by using the generalized benders decomposition(GBD) technique.In the following section, we will detail the GBD procedure for solving Problem 2.

IV.OPTIMAL SOLUTION

The main idea of GBD is to decompose Problem 2 into aprimalproblem and amasterproblem,and solve them iteratively until the convergence is reached,as illustrated in Figure 2.Specifically,the primal problem corresponds to Problem 2 for given binary variables.Solving the primal problem gives the information on the lower bound to Problem 2 as well as the Lagrange multipliers that correspond to the constraints.Based on the obtained Lagrange multipliers,the master problem can be derived by utilizing the nonlinear duality theory.Solving the master problem provides the information on the upper bound to Problem 2 and the binary variables that will be used to generate the primal problem in the next iteration.Letj= 0,1,···denote the iteration index.In the following,we elaborate the primal and master problems at thejth iteration,respectively.

4.1 Primal Problem

Problem 3(Primal Problem at Iterationj).

where f(β(j),ρ(j))denotes the optimal value and the constraints in(27),(28),(29), and(30)together with(31)correspond to the constraints in(19),(23),(24),and(26), respectively.Letdenote an optimal solution of Problem 3.3

Since the constraints in (27)-(31) are affected by both the continuous and the binary variables,implying that not all choices of binary variables can make Problem 3 feasible.Thus, it should be treated differently depending on whether Problem 3 is feasible or not.

4.1.1 Feasible Case

If Problem 3 is feasible, we derive the Lagrange dual problem associated with Problem 3.Letλ≜(λi)i=1,2,···,Δ∈denote the Lagrange multipliers associated with the constraints in(27)-(31),where R+denotes the set of nonnegative real numbers and Δ = 3K+2KSis the number of the constraints in(27)-(31).Then the Lagrange dual problem associated with Problem 3 is given by

whereλ(j)denotes an optimal solution anddenotes the partial Lagrangian associated with Problem 3,given by

4.1.2 Infeasible Case

If Problem 3 is infeasible, by adding a slack variableηfor each of the constraints in(27)-(31),we consider the following problem to characterize the violations of these constraints.

Note that,both

4.2 Master Problem

The master problem at iterationjis used to determine the binary variables that will be used as known conditions passed to next iteration,i.e.,iteration(j+1).To be specific, according to [24], the master problem at iterationjcan be expressed as follows.

Problem 4(Master Problem at Iterationj).

(β,ρ)∈D,

Lemma 2.The optimal value ζ(j)of Problem 4 provides an upper bound to Problem 2.

Proof.See Appendix IX.

4.3 Proposed Iterative Algorithm Based on GBD

Before ending this section, we discuss the computational complexity of Algorithm 1.It is worth noting that the GBD technique does not change the“NPhard” property of Problem 2.Nevertheless, after decoupling the continuous and binary variables of Problem 2 using GBD,the primal problem,the feasibilitycheck problem and the master problem (i.e., Problem 3,the problem in(34)and Problem 4)can be efficiently solved by general-purpose optimization toolboxes such as CVX [25].Specifically, both Problem 3 and the problem in(34)are convex, whose optimal solutions can be obtained by using CVX.Note that, CVX is based on the primal-dual interior-point method, which has worst-case polynomial-time complexity.Moreover,Problem 4 is a mixed integer linear programming problem, and similarly, we can obtain its optimal solution by using CVX, but this will have worst-case exponential-time complexity.As a result,Algorithm 1 is an exponential algorithm.

V.LOW-COMPLEXITY SUBOPTIMAL SOLUTION

Algorithm 1. Optimal solution of problem 2(problem 1).1: Input: K, S, B, α, T, L, p max,pmax k , Fk, Wk, Zk,µk,ttrk,k ∈K,γk,s,hk,s,k ∈K,s ∈S.2: Output: (ρ⋆,β⋆,t ⋆,t⋆,x⋆,x⋆,y⋆).3: Initialization: set j ←1, LB(0) ←-∞, UB(0) ←∞, I(0) ←∅, J(0) ←∅, ϵ←10−3, and choose β(1)k ←0,ρ(1)k,s ←0 for all k ∈K and s ∈S.4: repeat 5: if Problem 3 is feasible then 6: Obtain (t(j),t(j),x(j),x(j),y(j)) and λ(j) by solving Problem 3 and the dual problem in (32).7: Set LB(j) ← max{LB(j−1),f(β(j),ρ(j))},I(j) ←I(j−1)∪{j}and J(j) ←J(j−1).8: if■■■LB(j)−f(β(j),ρ(j))LB(j)■■■<ϵthen 9: Set (β⋆,ρ⋆,t ⋆,t⋆,x⋆,x⋆,y⋆) ←(β(j),ρ(j),t(j),t(j),x(j),x(j),y(j)).10: end if 11: else if Problem 3 is infeasible then 12: Obtain (t(j),t(j),x(j),x(j),y(j)) and ˆλ(j) by solving the problem in(34)and the dual problem in(40).13: Set J(j) ←J(j−1)∪{j}and I(j) ←I(j−1).14: end if 15: Obtain(ζ(j),β(j+1),ρ(j+1))by solving Problem 4.16: Set UB(j) ←ζ(j).17: Set j ←j+1.18: until■■■UB(j)−LB(j)UB(j)■■■<ϵ

It is worth pointing out that the complexity of Algorithm 1 is unacceptable whenKorSis relatively large.To address this issue, in this section, we develop a low-complexity algorithm to obtain a suboptimal solution of Problem 2, as illustrated in Figure 3.Specifically, the proposed suboptimal algorithm consists of two consecutive stages:continuous relaxationandgreedy rounding.The first stage is to generate a continuous joint scheduling and sub-channel assignment policy by solving the continuous relaxation of Problem 2.The second stage is to determine a feasible solution of Problem 2 by rounding the continuous joint scheduling and sub-channel assignment policy in a greedy manner.Hereinafter,we first detail these two stages,and then analyze the computational complexity of the algorithm.

Figure 3.Workflow of the developed low-complexity suboptimal solution.

5.1 Continuous Relaxation

In this stage, we obtain the continuous joint scheduling and sub-channel assignment policy by relaxing the binary variables in Problem 2, i.e.,βandρ, into two continuous variables between 0 and 1.However, due to the product termρk,stexisted in the equality constraint in(26),directly relaxingβandρwill result in an intractable non-convex problem,which is still challenging to solve.To remedy this issue, before the relaxation,we first transform the intractable constraint in(26) into some tractable linear inequality constraints.To be specific,by using the McCormick envelope[26],we can equivalently convert(26)to the following constraints,i.e.,

Then, we relax the binary constraints in (6) and(9)into

Consequently,we obtain the following continuous relaxation of Problem 2.

Problem 5(Continuous Relaxation of Problem 2).

5.2 Greedy Rounding

Problem 6(Convex Feasibility Problem of Problem 2 for Given(β†,ρ†)).

5.3 Complexity

The details of the proposed suboptimal algorithm based on continuous relaxation and greedy grounding are summarized in Algorithm 2.Note that the computation cost of Algorithm 2 is dominated by solving Problem 6 for eachm ∈Kandn ∈Sat Step 9-Step 25.Specifically,the complexity of solving Problem 6 by an interior point method isO(κ3.5log(1/ε))[27],whereκ≜KS+S+2 denotes the total number of variables,κ3.5characterizes the complexity order of dominant Hessian matrix calculation,andεdenotes the accuracy of solving Problem 6.As a result,the total complexity order of Algorithm 2 is given byO(KSκ3.5log(1/ε)).Finally,we remark that the output of Algorithm 2,i.e.,is a feasible solution of Problem 2.Later,we shall see that this feasible solution has a promising performance and can be regarded as a suboptimal solution of Problem 2.

Algorithm 2. Low-complexity suboptimal solution of problem 2(problem 1).1: Input:K,S,B,α,T,L,p max,pmax k ,Fk,Wk,Zk,µk,ttrk,k ∈K,γk,s,hk,s,k ∈K,s ∈S.2: Output: (β†,ρ†,t †,t†,x†,x†,y†).3: Initialization:set β†k ←0,ρ†k,s ←0 for all k ∈K and s ∈S,Sava ←S.4: //Continuous Relaxation Stage 5: Obtain(β‡,ρ‡)by solving Problem 5.6: //Greedy Rounding Stage 7: Obtain {v(m)}m∈K by sorting the sequence{β‡k}k∈K in descending order.8: Obtain {u(k,n)}n∈S by sorting the sequence{ρ‡k,s}s∈S in descending order for each k ∈K.9: for each m ∈K do 10: Set β†v(m) ←1 and flag ←0.11: for each n ∈S do 12: Set Sv(m) ←Sava ∩{u(v(m),i)}ni=1.13: if Sv(m) ̸=∅then 14: Set ρ†v(m),j ←1 for all j ∈Sv(m).15: if Problem 6 is feasible then 16: Obtain (t†,t†,x†,x†) by solving Problem 6.17: Set Sava ←SavaSv(m)and flag ←1.18: Goto Step 22 19: end if 20: end if 21: end for 22: if flag=0 then 23: Set β†v(m) ←0 and ρ†v(m),j ←0 for all j ∈Sv(m).24: end if 25: end for 26: Set y†k,s ←ρ†k,s t†,k ∈K,s ∈S.

VI.NUMERICAL RESULTS

In this section,numerical results are provided to gauge the performance of the proposed Algorithm 1 and Algorithm 2,as compared to the following two practical scheduling schemes in FL[19].

• Baseline 1: At each global iteration, the BS randomly selectsKsUAVs fromKto perform local training.

• Baseline 2: At each global iteration, the BS selectsKsUAVs fromKin a round robin fashion to perform local training.

6.1 Scheduling Performance Evaluation

In this subsection,we evaluate the performance of Algorithm 1,Algorithm 2,Baseline 1,and Baseline 2,as shown in Figure 4.Note that,the numerical results are obtained by averaging over 500 global iterations(each corresponding to a randomized channel realization)in FL.Specifically,Figure 4 illustrates the percentage of the scheduled UAVs versus the number of UAVsKand the number of sub-channelsS,respectively.From Figure 4,we can see that the proposed suboptimal algorithm (i.e., Algorithm 2) achieves almost the same performance as the optimal one (i.e., Algorithm 1),which demonstrates the effectiveness of Algorithm 2.In addition, we also see that the performance of the two baseline schemes is almost the same.This is because at each global iteration of FL, each UAV under the two baselines actually has the same scheduling probability.Furthermore,by comparing these four schemes, we can observe that Algorithm 1 and Algorithm 2 significantly outperform the two baselines,due to the fact that the proposed algorithms can wisely and efficiently adapt to the changes of the wireless channel state so that as many UAVs as possible can be scheduled to participate in the model training.

Figure 4. Performance evaluation of the proposed algorithms and baselines.

More specifically, Figure 4a and Figure 4b shows the percentage of the scheduled UAVs versus the number of UAVsKwhen the number of sub-channelsSis small (S= 8) and large (S= 64), respectively.From these figures, we can see that the performance of all schemes decreases withK, due to the limited available sub-channels.In addition, we can also see that ifSis small,the Baselines with smallKsoutperform those with largeKs; ifSis large, the Baselines with largeKsslightly outperform those with smallKsbut this superiority diminishes asKincreases.These observations emphasize the importance of reasonable usage of communication resources.Figure 4c and Figure 4d plot the percentage of the scheduled UAVs versus the number of sub-channelsS, when the number of UAVsKis small (K= 8) and large (K= 64),respectively.From these figures, we observe that all of the four schemes can schedule more UAVs asSincreases,demonstrating that the more communications resources available the more UAVs can be scheduled.

Figure 5. Performance evaluation of FL under different scheduling scheme.

6.2 Learning Performance Evaluation

VII.CONCLUSIONS

In this work, we developed a joint client scheduling and wireless resource allocation of FL in an SWIPTenabled micro UAV swarm network, where the BS transfers both ML model and power simultaneously to the UAVs, and the UAVs depend on the harvested and battery-stored energy to perform ML model training and uploading.Based on the proposed model,we formulated a joint UAV scheduling and wireless resource allocation problem to improve the learning performance of FL.The formulated problem is a challenging MINLP problem.For ease of tractability,the formulated problem is transformed into an equivalent MINLP problem with a special property by using some appropriate transformations.We developed two algorithms to achieve the optimal and low-complexity suboptimal solutions to the transformed problem.Numerical results demonstrated that the suboptimal algorithm can achieve almost the same performance as the optimal one under various network setups,and significantly outperforms the other representative baselines considered.

ACKNOWLEDGEMENT

This work is supported by the National Natural Science Foundation of China (No.61971077), the Natural Science Foundation of Chongqing, China(No.cstc2021jcyj-msxmX0458), the open research fund of National Mobile Communications Research Laboratory, Southeast University (No.2022D06), the Fundamental Research Funds for the Central Universities (No.2020CDCGTX074), the Natural Science Foundation on Frontier Leading Technology Basic Research Project of Jiangsu(No.BK20212001),and the Natural Science Research Project of Jiangsu Higher Education Institutions(No.21KJB510034).

NOTES

1The current analysis and optimization framework is built only for the FL tasks and should be tailored appropriately to embrace the conventional computation tasks such as the multi-media transformation and 3D modeling/rendering tasks[26].

2In this paper,to facilitate analysis and optimization,we consider that all micro UAVs fly relatively slowly,so that the channel between each UAV and the BS remains approximately unchanged within each global iteration of FL and independently changes over different global iterations.

3Note that,the objective function of Problem 3 is independent of the continuous variables,and thus any feasible solution of Problem 3 is an optimal solution.

4Note that,when given the UAV scheduling and sub-channel allocation,the two baselines may be infeasible.In this case,there will be no UAVs scheduled to perform model uploading.

APPENDIX

VIII.PROOF OF LEMMA 1

First,by a change of variableswe can eliminate variableand equivalently convert(2),(3)and(18)to(20),(21)and

wherek ∈K.

Then, due to the equality constraints in (26), the constraints in (50) and (51) are clearly equivalent to the constraints in(24)and(25),respectively.As a result, Problem 1 and Problem 2 are equivalent, which completes the proof.

IX.PROOF OF LEMMA 2

whereλ ∈, (a) follows from that there is no duality gap between Problem 3 and its dual problem in(32)withgiven by(33),and(b)is an equivalent transformation after introducing the variableζ.Since setVis only known implicitly, we need to find a way to represent it explicitly by certain inequality constraints.Fortunately,according to[24],the constraint (β,ρ)∈Vcan be equivalently characterized by

s.t.(54),(55).

Clearly,the problem in(56)is equivalent to Problem 2.Finally, by comparing the problem in (56) with the master problem,i.e.,Problem 4,we see that Problem 4 is just a relaxation of problem in(56),and thus it provides a lower bound to Problem 2.Therefore,we complete the proof of Lemma 2.