Distributed Conditional Generative Adversarial Networks (GANs) for Data-Driven Millimeter Wave Communications in UAV Networks
11 Distributed Conditional Generative AdversarialNetworks (GANs) for Data-Driven MillimeterWave Communications in UAV Networks
Qianqian Zhang , Aidin Ferdowsi , Walid Saad , and Mehdi Bennis Bradley Department of Electrical and Computer Engineering, Virginia Tech, VA, USA, Emails: { qqz93,aidin,walids } @vt.edu. Centre for Wireless Communications, University of Oulu, Finland, Email: mehdi.bennis@oulu.fi.
Abstract
In this paper, a novel framework is proposed to perform data-driven air-to-ground channel estimationfor millimeter wave (mmWave) communications in an unmanned aerial vehicle (UAV) wireless network.First, an effective channel estimation approach is developed to collect mmWave channel information,allowing each UAV to train a stand-alone channel model via a conditional generative adversarial network(CGAN) along each beamforming direction. Next, in order to expand the application scenarios ofthe trained channel model into a broader spatial-temporal domain, a cooperative framework, basedon a distributed CGAN architecture, is developed, allowing each UAV to collaboratively learn themmWave channel distribution in a fully-distributed manner. To guarantee an efficient learning process,necessary and sufficient conditions for the optimal UAV network topology that maximizes the learningrate for cooperative channel modeling are derived, and the optimal CGAN learning solution per UAV issubsequently characterized, based on the distributed network structure. Simulation results show that theproposed distributed CGAN approach is robust to the local training error at each UAV. Meanwhile, alarger airborne network size requires more communication resources per UAV to guarantee an efficientlearning rate. The results also show that, compared with a stand-alone CGAN without informationsharing and two other distributed schemes, namely: A multi-discriminator CGAN and a federated-learning CGAN method, the proposed distributed CGAN approach yields a higher modeling accuracywhile learning the environment, and it achieves a larger average data rate in the online performance ofUAV downlink mmWave communications.
Index Terms – generative adversarial network; millimeter wave; UAV communications; beyond 5G.
A preliminary version of this work appears in the proceedings of IEEE ICC 2021 [1]. a r X i v : . [ c s . I T ] F e b I. I
NTRODUCTION
Millimeter wave (mmWave) frequency bands are a pillar of next-generation wireless systems asthey will enable ultra-high-speed communications and airborne wireless networks [2]. In order toovercome the fast attenuation of mmWave signals, multiple-input multiple-output (MIMO) tech-nologies with highly-directional beamforming are employed so as to increase the cell throughputand improve the communication reliability. Compared with the sub-6 GHz spectrum, the higherfrequency of mmWave yields a shorter coherence time for the wireless channels. Therefore,mmWave communication links are more time-sensitive and require frequent channel measure-ments [3]. Meanwhile, highly-directional beamforming requires accurate knowledge of angle-of-arrivals (AoAs) and angle-of-departures (AoDs) for the propagation paths in order to achievebeam alignment between transmitter and receiver [4]. However, both the channel estimation andbeam training can incur heavy communication overhead and reduce the spectrum efficiency [5].Therefore, in order to improve the transmission performance, it is essential to have an accuratemodel to characterize a mmWave link and estimate its underlying MIMO channels.Compared with a terrestrial communication network, mmWave channel modeling for an air-borne, drone-based wireless system is more challenging [6]. Airborne wireless networks havebeen increasingly considered as a suitable platform to deploy mmWave communications, due toa high possibility of line-of-sight (LOS) link states. For example, an unmanned aerial vehicle(UAV)-based station (BS) can dynamically adjust its location so as to maintain a LOS channelwith transceivers [7]. However, a mobile UAV BS must often provide connectivity to a muchlarger geographical area compared with a typical terrestrial cellular BS. Moreover, compared toa terrestrial channel, the air-to-ground (A2G) channel includes more model parameters, such asthe 3D location and dynamic orientation of the UAV, which makes the channel modeling processmore challenging. Meanwhile, given that the mmWave channel response is time and locationdependent, the statistical model generated from one communication environment experienced bya UAV, at a given time and spatial coordinate, cannot be flexibly generalized into other temporalor spatial settings [8]. As a result, traditional channel modeling methods, such as ray-tracing,can become very difficult and time-consuming to measure mmWave A2G channels efficiently.Indeed, most current A2G channel models are calibrated at sub-6 GHz frequencies, while astandard-defined A2G model over mmWave bands, as well as the experimental data, has beenvery limited [9]. However, for a UAV BS, it is essential to have an accurate A2G channel model to estimate the mmWave link state, and, thus, save pilot training time and transmit power forefficient communications. In order to address the challenges of mmWave A2G channel modeling,a data-driven approach can be applied, where a UAV BS collects the A2G channel informationduring its cellular service, and then, build a stochastic channel model to estimate the long-term channel parameters. Such a UAV can also collaborate with neighboring UAVs to build ageneralized spatial-temporal map of the mmWave environment.
A. Related Works
In order to capture the stochastic characteristics of mmWave channels, a number of data-drivenmodeling approaches were developed in [2], [5], and [9]–[15]. Traditional methods, such asspatial-temporal correlation [2] and compressed sensing [10], were investigated for characterizingmmWave MIMO transmissions. The authors in [5] developed a neural network approach forcharacterizing mmWave transmissions and estimating MIMO channels. A deep learning datasetis developed in [11] to extract the propagation feature of mmWave-based communication links.However, all of the proposed modeling frameworks in [2], [5], [10], and [11] focus on a terrestrialmmWave transmission scenario, and their results are not applicable for A2G channel modeling,due to distinct propagation environments. To specifically address the characteristics of airbornemmWave communications, some recent works in [9] and [12]–[15] investigate the UAV-relatedmmWave channel from different perspectives. The authors in [12] provided a comprehensivesurvey on A2G propagation channel modeling for both the microwave and mmWave spectrumbands. The authors in [9] developed a generative neural network to predict the mmWave linkstate and model statistical channel parameters between a UAV and a ground BS. An empiricalpropagation loss model is proposed in [13] based on an extensive measurement for UAV-to-UAVcommunications at 60 GHz, and a traditional ray tracing method is applied in [14] to build ageometry-based stochastic model for UAV-to-vehicle communications at 28 GHz. Furthermore,the received signal strength and delay spread of mmWave transmissions is analyzed in [15] toprovide further details for A2G channels. However, all of the prior art in [9] and [12]–[15]studies the characteristics of mmWave channel models based on a single and local dataset, and,thus, the generated channel model is constrained by a limited amount of channel samples and afew dedicated measurement environments. As such, these existing models cannot be used as ageneral and standardized model for A2G mmWave channels.
In order to extend the channel model to large-scale application scenarios, a promising solutionis to use a cooperative modeling approach with multiple, distributed channel datasets. In a recentwork [16], the authors developed a federated learning (FL) framework to train the channel modelfrom distributed data sources. However, the centralized network topology of the FL frameworkrequires a global controller for information aggregation, and, thus, it cannot operate in a fullydistributed network as is the case in an airborne network. The work in [17] characterized a time-varying channel model via continuous data exchange in a distributed wireless system. However,sharing the raw channel data in a real-time manner yields heavy communication overhead.Furthermore, beyond the discriminative models in [5] and [16], a generative machine learningmodel is applied in recent works [8] and [18] to model the wireless channel. The authors in[8] proposed a generative adversarial network (GAN) framework to model the wireless channelbased on massive raw data, and the work in [18] employed a conditional GAN to representunknown channels to enable the encoding and modulation optimization, given the pilot traininginformation. However, all of the prior works in [8] and [16]–[18] do not focus on the characteristicof mmWave frequencies or A2G wireless links. Thus, their results are not applicable to themmWave channel estimation in the UAV communications. Given that a generative model canlearn the application range of the channel model from the temporal-spatial information in thedataset while training the channel features, it provides a better learning framework comparedwith a discriminative approach (e.g., such as those in [5] and [16]). Remarkably, there are nofully distributed generative learning frameworks developed to deal with the problem of data-driven mmWave channel modeling in prior works. As such, to fill this gap, in this work, afully-distributed cooperative generative learning model will be developed to characterize theenvironment of A2G mmWave links.
B. Contributions
The main contribution of this paper is a novel framework that can perform data-driven channelmodeling for mmWave communications in a distributed UAV network. In particular, a learningapproach, based on a distributed conditional generative adversarial network (CGAN) is proposedfor the UAV network to jointly learn the mmWave A2G channel characteristics from multiple,distributed datasets. In summary, our key contributions are: • First, we develop an effective channel measurement approach to collect real-time A2Gchannel information over mmWave frequencies, allowing each UAV to train a stand-alone channel model via a CGAN at each beamforming direction. • Next, to expand the application scenarios of the trained channel model into a broader spatial-temporal domain, we propose a cooperative learning framework, based on the distributedframework of brainstorming GANs [19]. This distributed generative approach allows eachUAV to learn the channel distribution from other agents in a fully distributed manner,while characterizing an underlying distribution of the mmWave channels based on the entirechannel dataset of all the UAVs. In order to avoid revealing the real measured data or thetrained channel model to other agents, each UAV shares synthetic channel samples that aregenerated from its local channel model in each iteration. The proposed approach does notrequire any control center, and it can accommodate different types of neural networks. • To guarantee an efficient learning process in the distributed UAV system, we analyticallyderive the convergence probability of the distributed CGAN learning for each iteration.Then, we theoretically derive the necessary and sufficient conditions for the optimal UAV-to-UAV communication topology that maximizes the learning rate for cooperative channelmodeling. Finally, based on the structure of the distributed UAV network, we characterizethe optimal CGAN learning solution per UAV.Simulation results show that the proposed distributed CGAN approach is resistant and robustto the local training error of each UAV. When the airborne network size becomes larger, morecommunication resources per UAV are required to guarantee an efficient learning rate. Mean-while, in each iteration, by sharing more generated samples, the learning rate of the distributedCGAN scheme will increase, but the data transmission duration will also be larger. To ensure anefficient data transmission, a better wireless link state or a larger transmit power will be requiredso as to improve the transmission rate. The results also show that, compared with a local CGANwithout information sharing and other distributed schemes, such as the multi-discriminator CGANand the FL-based CGAN methods, the proposed distributed CGAN approach yields a highermodeling accuracy in the learning result, and it achieves a higher average data rate in the onlineperformance of UAV downlink mmWave communications.The rest of this paper is organized as follows. Section II presents the communication modeland data collection. The CGAN learning framework, distributed UAV network, and problemformulation are presented in Section III. The optimal network topology and learning solutionsare derived in Section IV. Simulation results are shown in Section V. Conclusions are drawn inSection VI.
II. C
OMMUNICATION M ODEL AND D ATA C OLLECTION
A. Millimeter Wave Channel Model
Consider an airborne cellular network, in which a set I of UAVs provide mmWave downlinkcommunications to ground user equipment (UE). We assume that each UAV is equipped withan uniform linear array of M antennas, and the steering vector of the UAV’s transmit antennasis given by a t ( φ t ) = [1 , e j πλ sin( φ t ) , · · · , e j ( M − πλ sin( φ t ) ] T , where λ is the carrier wavelength, and φ t ∈ [0 , π ) is the AoD. Meanwhile, each UE is equipped with an uniform linear array of N antennas, and the receiver’s antenna vector is a r ( φ r ) = [1 , e j πλ sin( φ r ) , · · · , e j ( N − πλ sin( φ r ) ] T with φ rk being the AoA. Consequently, the MIMO channel matrix H ∈ C N × M can be given by [10] H = L (cid:88) l =1 α l a r ( φ rl ) a Ht ( φ tl ) , (1)where ( · ) H is the conjugate transpose, L is the number of distinct paths, and α l ∈ C , φ tl , and φ rl are the complex channel gain, AoD, and AoA of path l , respectively. Given the fact that theA2G channel over mmWave is very sensitive to blockage and has few scattering links, the valueof L will be much smaller than M × N . Meanwhile, since a massive MIMO mmWave array canprovide a narrow beam that eliminates much of the multipath [20], we can assume that both theUAV and each UE apply a perfect directional radiation technology for beam training purposes,such that both the transmitter’s and receiver’s antennas have only one main lobe without anyside lobes. Thus, the mmWave channel consists of a single path, i.e. L = 1 , which is eitherline-of-sight (LoS), reflected none-line-of-sight (NLoS), or complete outage. This assumption issupported by experimental results in [15] and [21], where a single path-component is observedfrom the GHz UAV communications in the suburban and rural environments. Meanwhile, thetransceiver design in [22] and other prior works in [23]–[25] can further support the use of asingle-path assumption for general mmWave systems.Now, we consider a UAV located at 3D coordinates x and a UE located at 3D coordinates y .Then, at the service time t , the A2G MIMO channel matrix in (1) can be rewritten as H ( x , y , t, φ t , φ r ) = α ( x , y , t, φ t , φ r ) a r ( φ r ) a Ht ( φ t ) , (2)where the channel gain α is jointly determined by the AoD-AoA pair φ (cid:44) ( φ t , φ r ) of themmWave path, as well as the communication environment ( x , y , t ) . B. Channel Estimation and Data Collection
In order to estimate the A2G mmWave channel, each UAV transmits a pilot symbol withsignal power P . Due to the narrow beam, it is necessary to exploit a pre-determined codebookfor beam alignment between the UAV and the UE. Let K be the length of the codebook, and ( w k , q k ) be the k -th pair of beamforming and combining vectors in the codebook. Then, thereceived pilot signal at the UE for the k -th training is r k = √ P q Hk H k w k + q Hk n , (3)where n ∼ CN ( , σ UE I N ) is the noise vector. Here, we assume that both the UAV and eachUE have a perfect knowledge of their antenna arrays’ radiation patterns, such that given thebeamforming w k and combining q k vectors, the AoD φ tk ( w k ) at the UAV and the AoA φ rk ( q k ) at the UE can be uniquely determined, respectively. This assumption has been adopted in [4] and[26], and supported by the antenna design in [22]. Then, let ⊗ be the Kronecker product, andvec ( · ) be the vectorization of a matrix. Then, the received pilot signal in (3) can be rewritten as r k = √ P ( w Tk ⊗ q Hk ) vec ( H k ) + q Hk n , = √ P ( w Tk ⊗ q Hk )[ a ∗ t ( w k ) ⊗ a r ( q k )] α k ( x , y , t, φ k ) + q Hk n , = β k α k ( x , y , t, φ k ) + q Hk n , (4)where ( · ) T is the transpose, ( · ) ∗ is complex conjugate, and β k = √ P ( w Tk ⊗ q Hk )( a ∗ t,k ⊗ a r,k ) ∈ C .After receiving { r k } k =1 , ··· ,K , each UE will send the pilot training information to the UAV viaa sub- GHz uplink [27]. Note that the beamforming vector w k and the combining vector q k from the pre-determined codebook are known by the BS for beam training purposes, and theantenna steering vectors a t and a r can be estimated directly based on w k and q k . Therefore,according to pilot signals in (4), the channel gain from the UAV located at x towards a UElocated y during time t with an AoD-AoA pair φ k can be estimated via ˜ α k ( x , y , t, φ k ) = r k β − k = α k ( x , y , t, φ k ) + ˜ n k , (5)where ˜ n k = β − k q Hk n is the uncorrelated estimation error.During the airborne cellular service, the channel gain ˜ α k can be measured and collected by eachUAV over a spatial-temporal domain, with K different pairs of antenna steering directions. We de-note the channel dataset of a given UAV i as S i = { s n , φ n } n =1 , ··· ,S i = { x n , y n , t n , ˜ α n , φ n } n =1 , ··· ,S i ,where s n = { x n , y n , t n , ˜ α n } is a channel sample, φ n is the AoA-AoD information associated with s n , and S i = |S i | is the total number of channel samples. Based on S i , each UAV can buildits own model for estimating A2G mmWave links in its dedicated measurement area.However, traditional channel modeling approaches, such as ray tracing and regression, cannotprovide a suitable modeling framework for A2G mmWave links for several reasons. First,different from terrestrial wireless channels, there are limited studies on the characteristics of A2Glinks over the mmWave spectrum (e.g., see [15] and [21], however, those works focus on thesmall-scale temporal and spatial characteristics of mmWave channel, but do not develop tractablemodels). As a result, it is difficult to find a rigorous data-driven channel model to optimizemmWave A2G channel parameters, using well-known regression methods [14]. Meanwhile, theamount of channel samples that each UAV owns is usually not sufficient to build an accuratestochastic model that properly captures the amplitude, phase and directional features of theMIMO channel response over a large spatial-temporal domain. For example, the altitude of aUAV largely determines the LOS probability towards UEs, and, hence, an accurate estimationof the mmWave link state will require a large amount of measurement data, which mandatescooperative learning. Moreover, the mmWave channel characteristics are location and timedependent. In order to build a general and standardized A2G mmWave model, the channel datamust be collected from multiple distinct communication environments, and the correspondingchannel parameters will span a larger set of possible values, which brings more challenges forthe channel modeling accuracy. To address the aforementioned challenges for mmWave A2Gchannel estimation, we will introduce a data-driven deep learning approach with cooperativeinformation sharing, such that an accurate channel model can be developed by each UAV overa large-scale spatial and temporal domain.III. D ISTRIBUTED C HANNEL M ODELING VIA C ONDITIONAL
GAN S A. Conditional GAN for Channel Modeling
Given the channel dataset S i , each UAV i can train a local channel model, based on deepneural networks (DNNs). For example, a discriminative learning model, such as in [5] and [16],can be trained to take a spatial-temporal pair as input and outputs a complex channel gain.This discriminative framework enables a UAV to predict the mmWave channel given any newinput, however, it fails to capture any additional information from S i , other than the channelgain value. Indeed, based on the dataset ( x n , y n , t n ) ∀ n ⊂ S i , we can acquire the geographicarea that UAV i has previously visited and the time interval during which UAV i measures the Fig. 1: The learning framework of a stand-alone CGAN for each UAV i .channel information. The underlying distribution of the spatial-temporal pairs in S i will definethe application range of the trained channel model. In order to jointly capture the applicablespatial-temporal domain from S i while learning the channel model, we propose a generativeapproach for the mmWave A2G channel modeling.Given that the pre-determined codebook defines a dedicated set of AoA-AoD pairs for eachUAV and its downlink UE, the antenna steering information φ can be considered as a priorknowledge that does not depend on channel measurement. Then, in order to model the channeldistribution given different AoA-AoD directions, a conditional generative adversarial network(CGAN) framework [28] is applied, where each UAV i has a condition sampler U , a generator G i , a discriminator D i , and a local dataset S i , as shown in Fig. 1. In each training epoch, thecondition sampler draws an AoA-AoD pair out of K possible directions, following a uniformdistribution φ ∼ U [1 , K ] , which is identical for each UAV, and the sampling result φ will beused as the direction condition in the CGAN training. Next, the generator G i ( z , θ gi | φ ) , which is aDNN with a parameter vector θ gi , maps a random input z to the channel sample space S under thecondition φ , and the discriminator D i ( s , θ di | φ ) , which is another DNN with a parameter vector θ di , takes a channel sample s and the condition φ as an input, and outputs a value between and . If the output of D i is close to , then the input sample s = ( x , y , t, α ) is highly likelyto be a real data sample, which contains the channel gain α that is measured between the UAVlocated at x and the UE located at y during the time interval t with an antenna steering pair φ ; otherwise, a zero output of D i means that the input channel sample is fake. Therefore, thegenerator of each UAV i aims to generate channel samples close to the real measurement data,while the discriminator tries to distinguish the fake data from the real channel samples. Let f i be the channel sample distribution in the dataset S i , f Gi be the learned distributionof the generator for UAV i , and f zi be the sampling distribution of the random input z . Then,the goal of a stand-alone CGAN is to train its generator G i to find the channel distribution f i under each condition φ k , while the discriminator D i is used to quantify the learning accuracyof G i . Hence, we model the interactions between the generator and discriminator of UAV i bya zero-sum game framework with a value function [28]: V i ( D i , G i ) = 1 K K (cid:88) k =1 E s ∼ f i (cid:104) log D i ( s | φ k ) (cid:105) + E z ∼ f zi (cid:104) log(1 − D i ( G i ( z | φ k ))) (cid:105) . (6)For each condition φ k , the first term in (6) forces the discriminator to output one for the real data,and the second term penalizes the generated data samples created by the generator. Therefore,the generator of each UAV i aims at minimizing the value function while its discriminator triesto maximize this value. It has been proven in [29] that this stand-alone CGAN game admits aunique Nash equilibrium (NE) where f Gi = f i and D i = 0 . . At the NE, under each condition,the channel sample distribution of the generator is identical to the distribution of the dataset,and thus, the discriminator cannot distinguish between the generated samples and the real data.However, in practice, each UAV only has a limited number of channel samples. Although astand-alone CGAN can learn the channel representation of a UAV’s local dataset, it can be biasedand only feasible for a limited spatial-temporal domain. Once the UAV moves to an unvisitedarea or a new UE appears at a new location, pilot measurement will again be necessary to acquirethe propagation feature of the new environment and update the channel model. However, boththe data collection and the model update processes are time-demanding and energy-consumingfor a UAV platform. Therefore, to avoid repeated channel estimations within the same space, aUAV can learn the channel information from other UAVs that operated in this region. Here, wenote that raw data exchange in a real-time manner among UAVs can yield heavy communicationoverhead and require a lot of energy and spectrum resource. Meanwhile, sharing the location-timeinformation of served UEs to an unauthorized UAV can raise privacy issues, especially when eachUAV belongs to a different network operator. Thus, the data exchange for mmWave A2G channelmodeling in a distributed UAV network must be communication-efficient and privacy-preserving. B. Distributed CGANs Framework
In order to address the challenge for A2G mmWave channel modeling, we propose a distributedCGAN framework, where a generative channel model is trained by each UAV to create channel Fig. 2: An illustration of the distributed CGANs framework with four UAVs, where O = { } , O = { , } , O = { } , and O = { } .samples from an underlying distribution of the overall dataset, without explicitly revealing thedata distribution or showing real data samples. Note that the use of distributed GANs is anemerging area of research in the machine learning community with only a handful of priorworks [19], [30], and [31], none of which was used in the context of a wireless communicationproblem. Given a set I of I UAVs, we consider that the available data in S = S ∪ · · · ∪ S I follow a distribution f . The local dataset S i of each UAV i is collected from different geographicareas or at different service times. Hence, the distribution f i of each local dataset S i does notspan the entire spatial-temporal space of the real channel distribution.The goal is to train the generator distribution f Gi of each UAV i to find the network-widechannel distribution f , under the constraint that no UAV i sends its real dataset S i . To achievethis goal, we extend the newly introduced concept of distributed brainstorming GANs in [19] tothe context of the studied wireless channel modeling problem with learning conditions. Thus, inour framework, each UAV i only shares the generated samples (not the raw data) from G i andthe AoA-AoD conditions with other UAVs in each training epoch. Fig. 2 illustrated the proposeddistributed CGAN framework, where the input of the discriminator for each UAV i comes fromits local dataset S i , and the generated samples from its own generator G i and the generators of itsneighboring UAVs. In this distributed CGAN framework, the generators collaboratively generatechannel samples to fool all of the discriminators while the discriminators try to distinguishbetween the generated and real channel samples. Let N i be the set of UAVs from whom UAV i receives generated samples, and let O i be the UAV set to whom UAV i sends its generated channel samples. Then, for each UAV i , the interaction between its generator and discriminatorcan be modeled by a game-theoretic framework with a value function: V i ( D i , G i , { G j } j ∈N i ) = 1 K K (cid:88) k =1 E s ∼ f bi (cid:104) log D i ( s | φ k ) (cid:105) + E z ∼ f zi (cid:104) log(1 − D i ( G i ( z | φ k ))) (cid:105) , (7)where f bi = π i f i + (cid:80) j ∈N i π ij f Gj is a mixture distribution of UAV i ’s local dataset S i and receiveddata from all neighboring UAVs in N i . Here, we define π i = S i S i + η (cid:80) j ∈N i S j and π ij = ηS j S i + η (cid:80) j ∈N i S j ,where ηS j is the number of generated channel samples that UAV j sends to UAV i in each epoch,and η > . Analogous to [19], given that the value functions of all UAVs are interdependent,we define the total utility function for the distributed UAV network as follows: V ( { D i } Ii =1 , { G i } Ii =1 ) = I (cid:88) i =1 V i ( D i , G i , { G j } j ∈N i ) , (8)where all generators aim at minimizing the total utility function defined in (8), while all dis-criminators try to maximize this value. Therefore, the optimal discriminators and generators ofthe distributed CGAN learning can be derived as a min-max problem as follows: { D ∗ i } Ii =1 , { G ∗ i } Ii =1 = arg min G , ··· ,G I arg max D , ··· ,D I V. (9)Note that, the optimal discriminators and generators in (9) both depend on the structure of theUAV communication system. However, the prior art in [19] and [30] only defined the generalframework of distributed GANs, but it did not account for the presence of data exchange or thewireless networking optimization for information sharing. Therefore, next, we will first define thestructure of the UAV communication system. Then, based on the optimized network topology,we identify the optimal solution to the learning problem in (9). C. Distributed UAV Communication Network
The communication structure of the UAV network is captured by a directed graph G = ( I , E ) ,where I is the set of UAVs, and E is the set of edges. Each edge e ij ∈ E is an ordered UAVpair that corresponds to an air-to-air (A2A) communication link. For example, for any i, j ∈ I ,if e ij ∈ E , then in each CGAN learning iteration, UAV i will send its generated samples to thediscriminator of UAV j . The number of neighboring UAVs from whom UAV i receives generatedsamples is called the in-degree, where N i = |N i | , and the out-degree of UAV i is O i = |O i | .Meanwhile, for any u, v ∈ I , if we can start from u , follow a set of connected non-repeatededges in E , and finally reach v , then we say that a path E u,v exists from u to v , and the length of the path l u,v equals to the number of edges on E u,v . Moreover, we denote the loop path thatstarts and ends both at u as E u , and denote the loop-path length as l u .In order to efficiently share the generated channel samples, orthogonal frequency-divisionmultiple access (OFDMA) techniques with B available resource blocks (RBs) are used to supportthe A2A communication over sub 6-GHz frequencies [27], where B ≥ I . In order to avoidinterference, we require the number of communication links not to exceed the number of RBs,i.e., |E | ≤ B , which is reasonable for UAV networks. Meanwhile, assuming a fixed hoveringlocation for each UAV during the learning stage, the A2A communication rate from UAV i to j using RB b is given as R ij = w b log (cid:16) P ij h ij σ (cid:17) , where w b is the A2A communicationbandwidth, P ij and h ij are the transmit power and path loss from UAV i to j , and σ is thenoise power. A signal-to-noise ratio (SNR) threshold τ is introduced, such that for any UAV pair ( i, j ) , if the received SNR at UAV j is lower than τ , i.e. P ij h ij /σ < τ , then, no RB will beassigned to this A2A communication link, i.e., e ij / ∈ E . In each iteration, each UAV i sends ηS i generated samples to its neighboring UAVs in O i , and the transmission time for the generatedchannel samples should not exceed t τ .Next, we define the convergence time C of the distributed CGAN approach as the expectednumber of iterations that is required for the learning process to converge, multiplied by theduration of each learning iteration [32]. To facilitate the analysis, we consider a fixed size foreach UAV’s dataset, i.e. S = · · · = S I = S , and a homogeneous UAV communication network,where N = · · · = N I = N , so as to guarantee a synchronous learning speed. Then, theprobability that the learning process converges after iteration T can be derived as follows. Theorem 1.
Given the UAV network structure G and the upper-bound training error (cid:15) of eachlocal CGAN, the probability p G ( T ) that the generator distribution f Gi of each UAV i covers theentire data distribution f after the T -th iteration will be given by: p G ( T ) = < T < l max , [(1 − (cid:15) ) η ] l max (1+ Nη ) l max − T = l max ,p G ( l max ) + (cid:80) Ti = l max +1 (cid:104)(cid:81) i − j = l max (cid:16) − [(1 − (cid:15) ) η ] l max (1+ Nη ) j − (cid:17)(cid:105) [(1 − (cid:15) ) η ] l max (1+ Nη ) i − l max < T < l max + l minloop , and for T ≥ l max + l minloop , p G ( T ) = p G ( l max + l minloop − T (cid:88) i = l max + l minloop i − (cid:89) j = l max + l minloop − − [(1 − (cid:15) ) η ] l max (1 + N η ) j − j (cid:89) k = l max + l minloop − γ ( k ) [(1 − (cid:15) ) η ] l max (1 + N η ) i − i (cid:89) l = l max + l minloop γ ( l ) , where l max = max u,v ∈I l u,v is the length of the maximum shortest-path in G , l minloop is the lengthof the shortest loop-path with the same starting UAV as l max , and γ ( T ) ≥ is an accelerationcoefficient.Proof. See Appendix A.Theorem 1 shows that the number of iterations that is required for learning convergence isalways greater than or equal to the maximum shortest-path length l max in G . Meanwhile, theconvergence probability p G ( T ) will decrease as l max becomes larger. Therefore, to optimize theconvergence rate for sampling sharing in the UAV network, it is necessary to minimize themaximum length of shortest paths among all UAVs. Next, based on Theorem 1, the number ofiterations T G ∈ N + , which is required for the distributed CGAN learning to converge with aconfidence level p τ ∈ (0 , , is given by p G ( T G − < p τ ≤ p G ( T G ) . (10)That is, after T G iterations, the generator distribution of each UAV is guaranteed to cover theentire channel distribution with a probability p τ . Meanwhile, we assume that (cid:15) is the upper-bound of the training error that each UAV can achieve within a fixed duration of t (cid:15) for its localadversarial training between its generator and discriminator. Then, given the network structure G , the overall convergence time of the distributed CGAN learning is [32] C ( G ) = ( t τ + t (cid:15) ) · T G . (11)Consequently, in the distributed UAV network with limited communication resources, the objec-tive for the cooperative mmWave channel modeling is to form an optimal A2A communicationnetwork G , such that the expected convergence time of the distributed CGAN learning is mini-mized, i.e., min G C ( G ) (12a)s.t. (cid:88) e ij ∈E P ij ≤ P max , ∀ i ∈ I , (12b) P ij h ij /σ ≥ τ, ∀ e ij ∈ E , (12c) ηS i ρ/R ij ≤ t τ , ∀ e ij ∈ E , (12d) ∃ E i,j ⊂ E , ∀ i, j ∈ I , (12e) I ≤ |E | ≤ B, (12f) where P max is the maximum transmit power for A2A communications, and ρ is the data sizefor each channel sample. Here, (12b) limits the maximum transmit power P max for each UAV,(12c) and (12d) set thresholds for the received SNR and the transmission time of each A2Acommunication link, (12e) requires a strongly connected network in G such that each localchannel dataset can be learned by all the other UAVs via the distributed CGAN framework, and(12f) avoids the interference over A2A communication links. Here, it is worthy noting that ourgoal is not to find the shortest paths in the graph G , but to identify the optimal topology G ∗ ,such that G ∗ yields a minimal value, among all possible topologies, of the maximum shortest-path between any two UAVs. However, in order to solve (12), a central controller is required tooptimize the communication structure based on path loss between each UAV pair. However, ina distributed UAV network, such a centralized entity is often not available, which makes (12)very challenging to solve. To solve this problem in a distributed manner, we must equivalentlydisassemble (12) into a set of sub-problems for each UAV, which will be detailed in the followingsection. Once (12) is solved, we subsequently derive the NE for the distributed learning in (9).IV. O PTIMAL LEARNING FOR DISTRIBUTED
CGAN S In this section, we aim to jointly solve the optimization problems in (9) and (12) to enable anefficient channel modeling approach using the distributed CGAN framework. First, we derive theoptimal structure G ∗ for the UAV communication network that minimizes the convergence timein (12). Next, given the UAV network topology G ∗ , we analytically derive the optimal distributedCGAN solution ( G ∗ i , D ∗ i ) for each UAV i . A. Optimal network structure for A2A UAV communications
In order to optimally solve (12) in a distributed manner without a central controller, we firstconsider a simple scenario, where constraint (12f) is simplified as I = |E | ≤ B, (13)i.e., the number of A2A communication links equals to the number of UAVs. Then, based onconstraints (12e) and (13), we derive the property of the UAV network structure as follows. Theorem 2.
Under constraint (13), the strongly connected network must have a ring structure,i.e., N i = O i = 1 , N i ∩ N j = ∅ , and O i ∩ O j = ∅ , ∀ i, j ∈ I and i (cid:54) = j . Proof.
See Appendix B.Theorem 2 shows that, given constraints (12e) and (13), the network structure of the UAVcommunication system must be a ring, where each UAV receives the channel sample fromone UAV, and sends its generated data to another UAV. Based on Theorems 1 and 2, we canequivalently reformulate (12) into a set of distributed optimization problems, such that theobjective of each UAV i is to choose the optimal single UAV O i = { o i } to whom UAV i sends its generated channel samples, so as to minimize the convergence time over its maximumshortest-path while satisfying constraints (12b)-(12d), i.e., min o i ∈I − i l max i ( G + e i,o i ) (14a)s. t. P i,o i ≤ P max , (14b) P i,o i h i,o i /σ ≥ τ, (14c) ηS i ρ/R i,o i ≤ t τ , (14d)where I − i is the set of UAVs except for i , G + e i,o i is the graph structure generated by addingan edge e i,o i to G , and l max i is the maximum shortest-path from UAV i to any other UAVs. Wedefine the set of feasible UAVs to whom UAV i can send its generated channel samples whilesatisfying constraints (14b)-(14d) as J i = { j ∈ I − i | P ij ≤ P max , P ij h ij /σ ≥ τ, ηS i ρ/R ij ≤ t τ } . (15)Then, the necessary condition for a feasible solution to (14) is provided next. Proposition 1 (Necessary condition) . Under constraint (13), a feasible UAV network structure G (cid:48) exists, only if (cid:83) Ii =1 J i = I and ∀ i, J i (cid:54) = ∅ hold.Proof. See Appendix C.Proposition 1 shows that if the union of feasible sets does not cover all UAVs, the UAVnetwork cannot form a strongly connected graph, and then, a feasible solution to (14) does notexist. Based on Proposition 1 and Theorem 2, we derive the sufficient condition for the optimalnetwork structure as follows.
Proposition 2 (Sufficient condition) . Under constraint (13), given that (cid:83) Ii =1 J i = I and J i (cid:54) = ∅ hold for all i ∈ I , the optimal UAV network structure is G (cid:48) = ( I , E ) , where E ⊆ { e ij | i ∈ I , j ∈J i } and l max i ( G (cid:48) ) = I − , ∀ i ∈ I . Proof.
See Appendix D.Consequently, under constraints (12b)-(12e) and (13), the optimal UAV network G (cid:48) thatminimizes the convergence time has a ring structure with a communication link set E ⊆ { e ij | i ∈I , j ∈ J i } . The proof of Proposition 2 in Appendix D has shown that G (cid:48) is not only the optimalsolution to the simplified problem (14), but also a feasible solution to our original problem (12).Note that, the solution in Proposition 2 requires that O i = 1 , while the original constraint (12f)allows the value of O i to be greater than one, i.e., the number of edges can be greater thanor equal to the number of UAVs. Given that the total number of UAV communication edgesis |E | = (cid:80) i ∈I O i , constraint (12f) is equivalent to (cid:80) i ∈I O i ≤ B . Next, we derive the optimalsolution for the UAV network structure given that O i ≥ .To find the optimal topology G ∗ that solves (12), we first start with the feasible solution G (cid:48) ,where the strongly connected property in (12e) is satisfied. Then, more communication edgesneed to be added to G (cid:48) to reduce C ( G ) . Here, similarly to (15), we define another feasible setfor each UAV i that satisfies (12b) - (12d) and (12f) as ˆ J i = j ∈ J i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ ˆ J Oi P ij ≤ P max , P ij h ij /σ ≥ τ, ηS i ρ/R ij ≤ t τ , (16)where ˆ J Oi ⊆ ˆ J i ⊆ J i , and ˆ J Oi is a subset that contains any O i components of J i . Note that,when O i = 1 , J i = ˆ J i = ˆ J Oi . Based on the feasible set in (16), we derive the necessaryconditions for the optimal solution to (12) as follows. Corollary 1 (Necessary condition) . The optimal network structure G ∗ = ( I , E ) that maximizesthe convergence rate in (12) requires that G (cid:48) ⊆ G ∗ , and E = { e ij |∀ i ∈ I , ∀ j ∈ ˆ J Oi } .Proof. See Appendix E.Corollary 1 shows that the optimal network structure G ∗ must include the feasible solution G (cid:48) in its topology. Meanwhile, the edge set of G ∗ consists of the feasible set of each UAV i withexact O i components. Based on Corollary 1, the formation approaches for the edge set ˆ J Oi andthe optimal UAV network G ∗ is summarized in Algorithm 1. The proposed network formationalgorithm has a complexity of O ( I O i ) for each UAV i , where O is the big-O notation. Thiscomplexity is reasonable because the number of available RBs O i in distributed UAV networksis usually small. Meanwhile, our proposed network formation algorithm is distributed, because each UAV only needs to know its A2A channel information to other UAVs, and then, it canoptimize the overall network structure by removing extra UAVs within its own feasible set. B. Optimal learning solution for the distributed CGAN framework
Given the optimal network structure G ∗ , based on [19, Proposition 1 and Theorem 1] , theoptimal generator G ∗ i of each UAV i will follow the distribution f Gi ∗ = f bi = π i f i + (cid:88) j ∈N i π ij f Gj , (17)where under each AoA-AoD condition, the generator’s distribution f Gi ∗ of UAV i equals to themixture of the channel distribution f i from its local dataset and the generator’s distribution f Gj of all incoming UAVs j ∈ N i . In this case, during each learning iteration, the discriminator willreceive two portions of the channel samples with the same information, from its locally generatedsamples of G ∗ i , and from the mixture source of S i and { G j } ∀ j ∈N i . As a result, the discriminatorcannot distinguish the generated samples from the real data, and, thus, it will output and with an equal probability of . . Consequently, given the optimal generator G ∗ i , the output ofthe optimal discriminator will be D ∗ i = f bi f bi + f Gi ∗ = 12 . (18)Once the local adversarial training of each UAV i converges to ( G ∗ i , D ∗ i ) , the distributed infor-mation sharing process in the UAV network converges to a unique NE [19], and the generator ofeach UAV i has learned the entire distribution of mmWave channels, i.e., G ∗ i ∼ f Gi ∗ = f . Next,each UAV i can explicitly obtain the channel distribution f of the trained generative model, basedon the channel samples from its optimal generator G ∗ i . The data-driven approach to identify theoptimal distributed CGAN solution ( G ∗ i , D ∗ i ) has been summarized in Algorithm 1.Here, we note that the proposed distributed CGAN approach has a communication load L = T G (cid:80) i ∈I ηS i ρO i = T G ηSρB , which includes all data transmissions within the UAV networkbefore learning convergence. Given that T G has been minimized in the optimal topology G ∗ ,Algorithm 1 also guarantees a minimal communication load for the proposed learning scheme.We can further adjust the communication overhead of our distributed CGAN scheme, by adaptingthe value of η to meet a large range of wireless transmission constraints. Meanwhile, the Although [19] does not account for the conditional learning, by treating the condition as an additional part of learning input,the analytical results in [19] can be extended to support the convergence solution to our distributed CGAN approach [28]. Algorithm 1
Network formation with distributed CGAN learning for mmWave channel modeling
UAV Network Formation:
1. Each UAV i measures channel information h ij for j ∈ I − i , and broadcasts the feasible UAV set ˆ J i ;2. If (cid:83) Ii =1 ˆ J i = I and | ˆ J i | ≥ O i , go to step 3; otherwise, UAVs adjust their locations, and go back to step 1;3. Start with the network graph G where E = { e ij | i ∈ I , j ∈ ˆ J i } ;4. For each UAV i with | ˆ J i | > O i ,Remove one edge e ij from E where j = min j ∈J i l max i ( G − e ij ) − l max i ( G ) , while guaranteeing ( (cid:83) k ∈I − i ˆ J k ) ∪ ( ˆ J i − j ) = I , and ∃ ˆ J Oi ⊆ ( ˆ J i − j ) , (cid:80) k ∈ ˆ J Oi P ik ≤ P max ; Until | ˆ J i | = O i for all i ∈ I . Distributed CGAN learning:
A. Initialize G i and D i for each UAV i ∈ I ;B. Repeat:
Parallel for all i ∈ I :a. Sample u AoA-AoD conditions: φ (1) , · · · , φ ( u ) ∼ U [1 , K ] , and u random inputs: z (1) , · · · , z ( u ) ∼ f zi ;b. Generate u channel samples G i ( z (1) | φ (1) ) , · · · , G i ( z ( u ) | φ ( u ) ) from the generator of UAV i ;c. Send π oi u generated sample to each outgoing UAV o ∈ O i , and receive N i portions of π ij u data samples { s (1) j | φ (1) } , · · · , { s ( π ij u ) j | φ ( π ij u ) } from incoming UAVs in N i ;c. Sample π i u real channel data from local dataset: { s (1) i | φ (1) } , · · · , { s ( π i u ) i | φ ( π i u ) } ∼ S i ;d. Update θ di via mini-batch stochastic gradient ascent: ∇ θ di V ( D i ( θ di )) = u ∇ θ di (cid:104)(cid:80) π i uk =1 log( D i ( s ( k ) i | φ ( k ) )) + (cid:80) uk =1 log(1 − D i ( G i ( z ( k ) | φ ( k ) ))) + (cid:80) j ∈N i (cid:80) π ij uk =1 log( D i ( s ( k ) j | φ ( k ) )) (cid:105) ;e. Update θ gi via mini-batch stochastic gradient descent: ∇ θ gi V ( G i ( θ gi )) = u ∇ θ gi (cid:80) uk =1 log(1 − D i ( G i ( z ( k ) | φ ( k ) ))) ; Until convergence to the NE. complexity of the local adversarial training for each UAV is similar to the original CGANframework in [28]. Thus, the complexity of our distributed learning approach is around T G -times of the original model. Furthermore, compared with the FL-CGAN scheme [31], whereone central agent averages the parameters of the generator and discriminator of each UAV andsend back the parameter update, our proposed approach supports a more flexible structure thatis fully distributed. Indeed, the use of an FL-GAN scheme requires a central controller whichis not available in a UAV network. The training process of FL-CGAN yields communicationoverhead proportional to the model size, which forbids the use of large-sized models given limitedcommunication resources. However, our proposed approach allows each UAV to employ its ownneural network (NN) architecture that can be different from other UAVs. Meanwhile, differentfrom the prior art in [31] where a multi-discriminators GAN (MD-GAN) framework is developed with one centralized generator and multiple, distributed discriminators, our proposed approachenables a synchronous learning for all UAVs, thus, improving time efficiency. More importantly,the experimental results in [19] have shown that the distributed GAN learning outperformsboth the FL-GAN and MD-GAN systems, in terms of modeling accuracy and communicationefficiency. V. S IMULATION R ESULTS AND A NALYSIS
For our simulations, unless state otherwise, we consider an airborne network with I = 4 UAVs using B = 4 RBs to provide wireless service within a geographic area of × m .Each UAV has a mmWave channel dataset that covers one of the regions without overlap, i.e.,a residential area, a city park [33], an urban environment [21], and a suburban area [15]. Withregards to simulation parameters, we set M = 256 , N = 64 , K = 81 , f = 30 GHz, w b = 2 MHz, P max = 40 dBm, σ = − dBm/Hz, (cid:15) = 0 . , p τ = 99% , τ = 10 dB, t τ = 0 . s, ρ = 11 , η = 0 . , and S i = 1000 for each UAV i . We implement a NN with four convolution layers forthe discriminator, and another NN with four transposed convolution layers for the generator. A. Convergence time of the distributed CGAN learning
Fig. 3 shows the convergence rates of the distributed CGAN learning, for different numbers ofavailable RBs and UAVs, respectively. The probability of NE convergence is used as the metricto evaluate the learning rate of the proposed algorithm. First, as shown in Fig. 3a, given a fixednumber of UAVs I = 4 , as the number of RBs B increases from to , the convergence rate ofthe proposed learning approach increases. When B = 4 , each UAV can send its generated channelsamples in each iteration to only one neighboring UAV. In this case, because of the limitationon the number of A2A communication links available, the efficiency of sharing channel sampleis low, and the distributed learning algorithm requires a long time to converge. As B increases,more A2A communication links are available. When B = 12 , each UAVs can send generatedchannel samples to three UAVs in each iteration, which forms a fully connected system, i.e.,each UAV connects with all the other UAVs for information sharing in the distributed system.Thus, the local data of each UAV can be shared efficiently throughout the learning framework,thus leading to a fast convergence rate of T G = 6 epochs . Moreover, Fig. 3a shows that the The number of iterations is not for the local CGAN training within each UAV, but for the number of times that the generatedchannel samples is transmitted to the neighbors by each UAV in the distributed airborne system. (a) The learning rate increases given more RBs. (b) The learning rate decreases for larger networks. Fig. 3: The proposed distributed CGAN approach yields a higher learning rate, given more RBsfor A2A communication, but this learning rate decrease for a larger network with more UAVs.learning process has three distinct stages. For example, when B = 4 , the maximum shortest-path length for a four-UAVs network is l max = 3 . Thus, during the learning iterations T < , theprobability of convergence is always zero. Meanwhile, the minimum length of the loop path is l minloop = 4 . Therefore, at iteration T = l max + l minloop = 7 , the generated channel samples of each UAVbegin to spread in the loop paths that connect all the UAVs in the distributed network. Thus,the learning rate after T = 7 becomes faster. A similar behavior can be observed for B = 8 when the learning rate starts to increase faster at T = 5 , and for B = 12 , the rate dramaticallyincreases from T = 3 . Next, we show the relationship between the convergence rate and thenumber of UAVs in Fig. 3b, for a fixed number of RBs. For larger network sizes, the convergencerate of the channel modeling process decreases, due to a longer path length in the distributedlearning system. In particular, at the beginning of the learning period, a large UAV network willexperience a long and inefficient data exchange stage, which leads to a slow convergence rate.Meanwhile, by comparing Figs. 3a and 3b, we can see that for I = 4 and B = 4 , the distributedlearning algorithm converges at T G = 19 , while for I = 12 and B = 12 , the convergence timeis T G > . Although in both cases each UAV has one RB for channel information sharing,a large UAV network size results in a much slower learning rate. Therefore, when the numberof UAVs increases, guaranteeing an efficient channel modeling approach requires increasing thetotal number of RBs, as well as the average number of RBs per UAV.Fig. 4 shows the convergence rates of the distributed CGAN, for different sizes of shared datasamples η and for different values of the local CGAN training error (cid:15) at each UAV, respectively. (a) The learning rate increases by sharing more samples. (b) The learning rate decreases for a larger training error. Fig. 4: Given more generated samples and a smaller training error in each iteration, the proposeddistributed CGAN approach yields a higher learning rate.Note that, in each iteration, each UAV i sends ηS i generated channel samples to its neighboringUAVs in O i . As shown in Fig. 4a, when η becomes larger, the number of shared samplesin each iteration increases, and, thus, the convergence rate of the proposed distributed CGANapproach becomes faster. However, a larger size of the generated samples usually yields a longertransmission time. Therefore, in order to guarantee a fixed transmission duration t τ , a larger η will require a better A2A channel state, or a larger A2A transmit power, so as to improvethe transmission rate for A2A communications. Next, in Fig. 4b, we evaluate the effect of theupper-limit error (cid:15) for the local CGAN training per UAV to the overall convergence time ofthe distributed learning framework. As the training error threshold (cid:15) increases from . to . ,the convergence time of the distributed CGAN approach increases from T G = 17 to . Bycomparing with Figs. 3 and 4a, the effect of the local training error (cid:15) on the convergence rateis very limited. Therefore, we can conclude that our proposed learning framework has a hightolerance to the local training error per UAV. Meanwhile, a high tolerance of the training errorenables each UAV to choose different NN structures, based on its own computational abilityand on-board energy, for its local CGAN learning. Thus, our approach supports a very flexibledistributed structure for each learning agent. B. Learning results for A2A mmWave channel modeling
In this section, we evaluate the learning performance of our proposed distributed CGAN ap-proach compared with four baselines: a stand-alone CGAN model per UAV without information (a) The proposed distributed CGAN scheme outperformsother distributed and local baseline methods. (b) The communication overhead per iteration increasesgiven more RBs. Fig. 5: Our proposed distributed CGAN approach yields the highest modeling accuracy, andthe lowest communication load, compared with the distributed and local baseline schemes.sharing, a centralized CGAN scheme based on raw channel data from all UAVs, an FL-CGAN,and an MD-CGAN distributed learning scheme. Jensen-Shannon divergence (JSD) is used asthe performance metric, where a lower value of JSD indicates a higher learning accuracy. Fig.5a shows that the modeling accuracy of the proposed distributed CGAN approach outperformsthe local CGAN, MD-CGAN, and FL-CGAN baselines. First, given more UAVs in the system,each UAV covers a smaller service area, and the local generator distribution only applies to alimited spatial domain. Thus, the modeling accuracy of the local learning scheme decreases fora larger network size. However, using the distributed learning approach, each UAV can learn theA2G channel property over a larger location domain from the generated samples of other UAVs.Thus, the modeling accuracy for all three distributed approaches stays the same for differentnetwork sizes. Given that our proposed distributed CGAN scheme does not require a centralagent to aggregate information, it leads to a more robust structure compared with MD-CGANand FL-CGAN, and it yields a higher modeling accuracy among all the distributed CGANapproaches. However, due to a limited training time and the inevitable training error at eachUAV, the overall distributed CGAN learning of the airborne network may converge to a localoptimum. This explains the performance gap between the proposed distributed learning schemeand the centralized raw data sharing method. To mitigate this gap, a deeper NN framework anda longer training time t (cid:15) are needed to decrease the training error for the local CGAN at eachUAV. However, those would come at the expense of additional delay or computations. Fig. 6: Our proposed distributed CGAN approach yields the highest downlink transmission rate,compared with the distributed and local baseline schemes, in the online performance.In Fig. 5b, we show the relationship between the communication overhead per iteration and thenumber of available RBs in the distributed learning network. Note that, the local CGAN does notinvolve any data sharing, thus it is not shown in the figure. Meanwhile, in the centralized CGANscheme, all of the raw channel samples for each UAV will be shared to all the other agents in thefirst iteration. Thus, the total number of iterations for the centralized CGAN scheme is only one.Fig. 5b shows that our proposed distributed CGAN yields the lowest overhead compared with allbaselines. For the MD-CGAN method, in each iteration, a central agent sends the generated datasample to each UAV, and then, each UAV sends the discriminator results back to the agent. Thistwo-directional communication yields a higher overhead, compared with our proposed schemewhere each UAV only sends its generated samples to a dedicated set of UAVs in one direction.The communication overhead of the FL-CGAN scheme is proportional to the parameter size ofthe NN models in both the generator and discriminator. For channel modeling tasks, it is oftenthe case that the size of the CGAN parameters are larger than the size of generated samples ineach iteration. Thus, the FL-CGAN method results in the highest communication load amongall three distributed schemes.
C. Communication performance for online deployment
Fig. 6 shows the online performance of the proposed channel modeling approach and threebaseline schemes. We run each method for times, and the average transmission rate of theUAV A2G communications is used as the performance metric, with a downlink bandwidth of MHz [34]. Given more UAVs in the network, the average data rates of all schemes naturally increase, due to an averagely smaller service area for each UAV. Fig. 6 first shows that theproposed distributed CGAN approach improves the average data rate by around and ,compared with the MD-CGAN and FL-CGAN schemes, respectively. Meanwhile, the proposedlearning scheme yields a two-fold increase in the downlink rate, compared with the local CGANmethod. However, due to training errors of mmWave A2G channel models, the proposed methodyields a lower data rate, compared with a perfect channel state information (CSI) scheme.VI. C ONCLUSION
In this paper, we have proposed a novel framework for mmWave channel modeling in a UAVcellular network. First, the channel measurement approach has been developed to collect the real-time information. In order to characterize mmWave A2G links in a large spatial-temporal spacewith different AoA-AoD directions, a cooperative learning framework, based on the distributedCGAN, has been developed for each UAV to learn the mmWave channel distribution from otheragents in a fully-distributed manner. We have derived the necessary and sufficient conditions forthe optimal network topology that maximizes the learning rate, and characterized the learningsolution for the generator and discriminator per UAV. Simulation results have shown that thelearning rate will increase by using more A2A communication RBs and sharing more generatedsamples in each iteration. However, a larger UAV network size and a higher training error willincrease convergence time. The results also show that the proposed CGAN approach yields ahigher learning accuracy and a larger average rate for UAV downlink communications, comparedwith a local CGAN, MD-CGAN and FL-CGAN baseline schemes.A
PPENDIX AP ROOF OF T HEOREM { i, u , · · · , u l max } bethe ordered UAVs on the maximum-length shortest-path. Then, we consider a portion of channel information, whose size equals to exactly the information amount that a single channel samplecan contain. A. Single-time probability
First, we derive the probability that this portion of information can be transmitted from thedataset S i of UAV i to the generator of UAV u l max just at the T -th iteration. T < l max : In each iteration, any channel sample can only be transmitted to the next-hopUAV. Thus, when
T < l max , no information can be successfully delivered from UAV i to UAV u l max through the shortest path, and the probability for successful information delivery is zero. l max ≤ T < l max + l minloop : During the first iteration T = 1 , UAV i will send ηS generatedchannel samples to UAV u . Thus, the probability for transmitting the considered portion ofinformation from i to u equals to the sampling ratio η . Meanwhile, given that the trainingerror for the local generator is (cid:15) , there is only (1 − (cid:15) ) possibility that the information can beaccurately transmitted. Thus, the probability that the considered portion of information can besuccessfully transmitted from i to u p in = (1 − (cid:15) ) η . At the same time, UAV u receives generatedsamples from the other N − neighboring UAVs in N u , and thus, the percentage of UAV i ’sinformation in the dataset of UAV u becomes p out = p in Nη = (1 − (cid:15) ) η Nη . Next, when T = 2 ,UAV u generates ηS samples and sends them to UAV u . Then, the probability that UAV i ’sinformation will be transmitted from u to u becomes p in = ηp out = [(1 − (cid:15) ) η ] Nη . Due to generateddata from other UAVs, the percentage of UAV i ’s information at UAV u will be reduced to p out = p in Nη = [(1 − (cid:15) ) η ] (1+ Nη ) . This process will continue, until the sample information is delivered toUAV u l max at T = l max . In this case, we can find the probability that UAV i ’s information arrivesat UAV u l max with l max hops will be p in l max = [(1 − (cid:15) ) η ] l max (1+ Nη ) l max − . In summary, the rule is that, at eachiteration, the percentage of the previously owned channel samples by each UAV gets reducedby Nη , due to the arrival of generated samples from N neighboring UAVs. Meanwhile, thepercentage of channel samples will be reduced by the sampling ratio and a training error of (1 − (cid:15) ) η for each hop along the path. Consequently, the probability that a portion of UAV i ’schannel information can be successfully delivered to UAV u l max within the T -th learning iterationis p in l max ( T ) = [(1 − (cid:15) ) η ] l max (1+ Nη ) T − . T ≥ l max + l minloop : After T ≥ l minloop , UAV i starts receiving information of its own datadistribution from its neighboring UAVs in N i , due to the existence of looping data flow. In thiscase, the reduction of the information percentage in each iteration will be higher than Nη . Thus, an acceleration coefficient γ ( T ) > , where γ ( T → + ∞ ) = 1 + N η , will be addedin the iteration reduction ratio, and the probability of information delivery becomes p in l max ( T ) = [(1 − (cid:15) ) η ] l max (1+ Nη ) T − (cid:81) Ti = l max + l minloop γ ( i ) for T ≥ l minloop . B. Cumulative probability
Next, we will derive the accumulative probability that the considered portion of UAV i hasbeen successfully delivered to UAV u l max after the T -th iteration. T < l max : Given that p in l max ( T ) = 0 , the accumulative probability p G ( T ) = 0 for T < l max . T = l max : Based on the above analysis, p G ( l max ) = p in l max ( l max ) = [(1 − (cid:15) ) η ] l max (1+ Nη ) l max − . l max < T < l max + l minloop : In this case, the accumulative probability of successful infor-mation delivery can be calculated, based on the chain rule, where p G ( T ) = p in l max ( l max ) + [1 − p in l max ( l max )] p in l max ( l max + 1) + · · · + (cid:81) T − i = l max [1 − p in l max ( i )] p in l max ( T ) . Consequently, we can derive p G ( T ) = p G ( l max ) + T (cid:88) i = l max +1 (cid:34) i − (cid:89) j = l max (cid:18) − [(1 − (cid:15) ) η ] l max (1 + N η ) j − (cid:19)(cid:35) [(1 − (cid:15) ) η ] l max (1 + N η ) i − . T ≥ l max + l minloop : When the loop data flow starts, the probability becomes p G ( T ) = p G ( l max + l minloop − T (cid:88) i = l max + l minloop i − (cid:89) j = l max + l minloop − − [(1 − (cid:15) ) η ] l max (1 + N η ) j − j (cid:89) k = l max + l minloop − γ ( k ) [(1 − (cid:15) ) η ] l max (1 + N η ) i − i (cid:89) l = l max + l minloop γ ( l ) , where γ ( l max + l minloop −
1) = 1 . This concludes our proof of Theorem 1.A
PPENDIX BP ROOF OF T HEOREM N i ≥ and O i ≥ .Meanwhile, given (13), the number of edges equals to the number of UAVs, i.e., (cid:80) Ii =1 N i = I and (cid:80) Ii =1 O i = I . Therefore, we can derive that N i = O i = 1 , ∀ i ∈ I . Based on the observationthat each UAV has only an incoming edge and an outgoing edge, the structure of the UAVnetwork must form a ring, so as to keep the strongly connection property, i.e., N i ∩ N j = ∅ , and O i ∩ O j = ∅ , ∀ i, j ∈ I and i (cid:54) = j . This concludes our proof of Theorem 2. A PPENDIX CP ROOF OF P ROPOSITION i ∈ I , such that J i = ∅ . Then, we have O i = 0 , which contradicts to thestrong connected network requirement where O i ≥ . Next, if (cid:80) Ii =1 J i ⊂ I , then, there exists atleast one UAV i , such that none of the other UAVs sends any generated channel samples to it.Thus, for UAV i , N i = 0 , which again contradicts to the strong connected network requirementwhere N i ≥ . Consequently, (cid:83) Ii =1 J i = I and ∀ i, J i (cid:54) = ∅ must hold to guarantee the existenceof a strongly connected structure for the optimal UAV network G ∗ .A PPENDIX DP ROOF OF P ROPOSITION
E ⊆ { e ij | i ∈ I , j ∈ J i } , all thecommunication edges are formed based on the feasible UAV sets. Thus, constraints (12b)-(12d),as well as (14b)-(14d), naturally hold, according to the definition of the feasible set in (15).Meanwhile, given that l max i ( G ∗ ) = I − hold for i ∈ I , the UAV network must have a ringstructure, which leads to a strongly connected graph that satisfies constraint (12e), and guaranteesan equal number of communication links to the number of UAVs that satisfies constraint (12f).This concludes the proof of a feasible solution. Next, based on Theorem 2, a ring is the onlypossible structure for the UAV network, where l max ( G ∗ ) always has a fixed value of I − . Giventhe fixed value of l max ( G ∗ ) , the convergence rate will be constant, for any ordered set of ringstructures. Therefore, the feasible solution leads to one identical outcome of the convergencetime for (14), and it is the optimal result, as long as the UAV network has a ring structure.A PPENDIX EP ROOF OF C OROLLARY G (cid:48) has been proved in Proposition 1 to meet the constraint of a strongly connectednetwork. Thus, the requirement G (cid:48) ⊆ G ∗ will ensure that G ∗ also satisfies the strongly connectedproperty in constraint (12e). Then, E = { e ij | i ∈ I , j ∈ ˆ J Oi } is a necessary condition forsatisfying all the other constraints in (12), where j ∈ ˆ J i ensures that the communicationconstraints (12b) - (12d) are met according to the definition in (16), and | ˆ J i | = O i supports a homogeneous network structure and minimizes the maximum shortest-path in G ∗ while satisfyingthe edge constraint (12f). R EFERENCES [1] Q. Zhang, A. Ferdowsi, and W. Saad, “Distributed generative adversarial networks for mmWave channel modeling inwireless UAV networks,” in
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