Problem in Aerial Federated Learning
Item Type Article
Authors Zhagypar, Ruslan;Kouzayha, Nour Hicham;Elsawy, Hesham;Dahrouj, Hayssam;Al-Naffouri, Tareq Y.
Citation Zhagypar, R., Kouzayha, N., ElSawy, H., Dahrouj, H., & Al-Naffouri, T. Y. (2023). Characterization of the Global Bias Problem in Aerial Federated Learning. IEEE Wireless Communications Letters, 1–1.
https://doi.org/10.1109/lwc.2023.3273318 Eprint version Post-print
DOI 10.1109/lwc.2023.3273318
Publisher Institute of Electrical and Electronics Engineers (IEEE) Journal IEEE Wireless Communications Letters
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Download date 2023-12-06 18:53:45
Link to Item http://hdl.handle.net/10754/686804
Characterization of the Global Bias Problem in Aerial Federated Learning
Ruslan Zhagypar,Student Member, IEEE,Nour Kouzayha, Member, IEEE,Hesham ElSawy, Senior Member, IEEE,Hayssam Dahrouj, Senior Member, IEEE, and Tareq Y. Al-Naffouri, Senior Member, IEEE
Abstract—Unmanned aerial vehicles (UAVs) mobility enables flexible and customized federated learning (FL) at the net- work edge. However, the underlying uncertainties in the aerial- terrestrial wireless channel may lead to a biased FL model.
In particular, the distribution of the global model and the aggregation of the local updates within the FL learning rounds at the UAVs are governed by the reliability of the wireless channel.
This creates an undesirable bias towards the training data of ground devices with better channel conditions, and vice versa.
This paper characterizes the global bias problem of aerial FL in large-scale UAV networks. To this end, the paper proposes a channel-aware distribution and aggregation scheme to enforce equal contribution from all devices in the FL training as a means to resolve the global bias problem. We demonstrate the convergence of the proposed method by experimenting with the MNIST dataset and show its superiority compared to existing methods. The obtained results enable system parameter tuning to relieve the impact of the aerial channel deficiency on the FL convergence rate.
Index Terms—Federated learning, Unmanned aerial vehicle (UAV), stochastic geometry, wireless channel, MCP, scheduling.
I. INTRODUCTION
T
He development of the sixth generation (6G) of wire- less networks enables the joint use of unmanned aerial vehicles (UAVs) and federated learning (FL). The potential of UAVs coupled with the FL algorithm has been recognized in [2] [3]. The FL-based edge computing in a UAV network enhances client quality-of-service and maintains privacy [4].Authors of [5] advocate the role of UAVs in FL over terrestrial base stations (BS) due to UAVs mobility, which allows for customized learning from a specific group of devices. Not only do UAVs offer more extensive coverage, but their on demand deployment also makes them a cost-effective solution [6].
The global bias problem in FL occurs due to the unreliable nature of wireless channels over which local and global param- eters are transmitted. During the FL process, the global param- eters are broadcasted to the devices via downlink, and updated local parameters are sent back to an aggregator through uplink.
If the wireless channel conditions are poor, the transmissions of either local or global parameters may be lost, resulting in a bias towards devices with better connectivity. This bias leads to a drift in the global model away from optimality, resulting in reduced accuracy and slower convergence during the learning process. To address this problem, the paper considers a single- tier network consisting of UAVs as aggregators and ground devices executing local learning. The paper characterizes the
An extended version of this paper can be found on archive [1].
R. Zhagypar, N. Kouzayha, and T. Y. Al-Naffouri are with King Abdullah University of Science and Technology, Thuwal, Saudi Arabia (e-mail: [email protected];
[email protected]; [email protected]).
H. ElSawy is with the School of Computing, Queen’s University, ON, Canada (e-mail:
H. Dahrouj is with the Department of Electrical Engineering, University of Sharjah, United Arab Emirates (e-mail: [email protected]).
global bias problem by deriving the UAV’s download and devices’ upload success probabilities, and by integrating these metrics into the aggregation step of the FL process to ensure unbiased results.
In the existing literature, several studies attempt to account for the wireless channel’s impact on the FL convergence in terrestrial networks [7] [8]. To this end, stochastic geometry is utilized as the main mathematical framework to provide analytical formulations of key performance metrics that assist the FL algorithms. For instance, in [9], the authors highlight the impacts of client update success probability along with resource constraints on the learning performance. Theoretical analysis made in [10] show that the learning latency depends on the number of devices with high uplink (UL) success probability. Authors of [11] apply the framework to explore the effect of dynamic signal-to-interference-plus-noise ratio (SINR) threshold on learning. Although there are studies that optimize resource allocation for faster communication in aerial FL without using a stochastic geometry approach, such as [12]
and [13], these studies neglect to address the issue of bias.
Prior studies assume guaranteed downlink transmission due to abundant power resources of terrestrial BSs. However, this assumption is not applicable in aerial wireless networks with UAVs’ limited energy capacity. No study has addressed the global bias problem of FL in UAV-assisted networks. To the best of the authors’ knowledge, this is the first work that bridges this gap by optimizing a reliable FL algorithm for aerial networks.
In this work, we propose a UAV-assisted FL algorithm in which UAVs provide an intermediate model aggregation from the sky and communicate with ground devices through unreliable wireless channels. Inspired by the results in [9], we jointly characterize the UAV’s download and clients’ upload success probabilities and integrate them in the aggregation step of the FL to counter the mentioned bias problem. Our main contributions are summarized as follows:
• Development of an analytical framework for the modeling and the analysis of aerial FL: Unlike reference [9] which focuses on the UL success probability, the current paper aims at deriving a tractable expression of the joint DL and UL success probability as a function of the system parameters.
• Reducing bias in FL by including joint success probability in aggregation, thereby leading to improved convergence as compared to existing methods, as validated through experi- ments on the MNIST dataset.
• Anaylyzing the impact of various system parameters such as the UAV height and the environment on the performance of FL in large-scale UAV networks: The provided insights can be utilized to optimize FL performance by adjusting the system parameters.
Fig. 1: Illustration of the system model and an example of MCP withN= 6 in a cluster of radiusR= 100m.
II. SYSTEMMODEL
We consider a single-tier network consisting of UAVs that act as aggregators and ground devices that execute local FL learning, as illustrated in Fig. 1. As devices performing the FL with the same UAV are physically close to each other, we utilize a Matern Cluster Process (MCP) to model the network.
This model also represents the case where a UAV flies up to a cluster of devices and hovers over it to conduct an FL training on their datasets. The UAVs are considered the parent points and their locations follow a homogeneous Poisson Point Process (PPP) Φwith intensity λ. Around each UAV located at u ∈ Φ, a set Du of N devices are independently and identically distributed (iid) forming a cluster of radiusR. An example of the MCP model with a cluster radius ofR= 100m and N = 6 is illustrated in the foreground of Fig. 1. Each UAV orchestrates a separate FL process by aggregating the local updates from the devices within its cluster.
As the aerial links are affected with the blockages in the environment and since the devices are considered at the ground level, we use the approximation of the line-of-sight (LOS) probability proposed in [14]. Specifically, the probability of having a LOS link between the UAV and a device k located at a horizontal distancerk is given by:
PL(rk) = 1
1 +aexp (−b[180π arctan (h/rk)−a]), (1) where aandb are constants that depend on the environment and are given in [15, Table I], andhis the elevation height of the UAV. The non line-of-sight (NLOS) probability is thus given as PN(rk) = 1 −PL(rk). As a result, the set of UAVs is divided into two subsets with intensities PL(rk)λ and PN(rk)λ. We assume different path loss exponents and fading parameters for LOS and NLOS transmissions.
We assume the use of directional beamforming to compen- sate for propagation loss in UAVs and devices. The UAVs and devices use sectored antennas with main lobe gainMsand side lobe gain ms, where s ∈ u, d denotes the UAV and device, respectively. The gain on the desired link is maximized with G0 = MuMd. Interfering links have uniformly distributed beam directions, with directionality gains given in Table I.
The probabilities are functions of the main lobe beamwidth θs for s∈u, d.
TABLE I: Four antenna patterns
i 1 2 3 4
Gi MuMd Mumd muMd mumd
pi (θ2πu)(θ2πd) (θ2πu)(1−θ2πd) (1−θ2πu)(θ2πd) (1−θ2πu)(1−θ2πd)
We account for the impact of the distance-dependent path- loss and small-scale fading modeled by the Nakagami-m distribution. Universal frequency reuse is used across different clusters and M resource blocks (RBs) are available at each UAV. We apply random resource scheduling without replace- ment to devices in each cluster due to the limited resources (M ≤ N), effectively eliminating intra-cluster interference.
We utilize the SINR as a metric to characterize the UL channel used for transmitting local parameters from devices to UAVs.
Thus, the UL SINR is given by:
SINRULz =PdG0|hk|2z(rk2+h2)−αz2
IUL+n20 , (2) wherez∈ {L, N}denotes the LOS and NLOS links,αis the path-loss coefficient,Pdis the transmission power of a device, G0 is the directionality gain of the desired link, |hk|2z is the power of the normalized small-scale Nakagami-m fading,rk
is the distance between projections of a selected UAV and transmitting devicek,n20is the noise power, andIULrepresents the interference from inter-cluster devices.
The DL channel used for transmitting the global parameters from UAVs to devices is also prone to errors due to inter- ference, propagation path-loss, and noise. The SINR is also utilized to characterize the DL channel, which is given by
SINRDLz = PuG0|hk|2z(r2k+h2)−αz2
IDL+n20 , (3) where Pu is the transmission power of a UAV, and IDL is the interference from other UAVs. Please note that exact expressions of IDL and IUL can be found in the Appendix of the extended version [1].
Fig. 1 demonstrates the projections of necessary distances for the characterization of inter-cluster interference. The UAV of the typical cluster, u0 ∈ Φ, is positioned right above the origin at an altitudeh. Thus,g =||d||, d∈Du\u0 represents the Euclidean distance to a device in an interfering cluster.
Further, letq=||u||, whereu∈Φ\u0denotes the distance to an interfering UAV. The conditional distance g|q distribution for MCP is given as [16]:
fQMAT(g|q) = 2g
πR2arccos
g2+q2−R2 2gq
U(g− |R−q|)
×U(R+q−g) + 2g
πR2U(R−q−g), (4) withU(·)is the unit step function.
III. FL ALGORITHM
In the FL scenario, geographically dispersed devices interact with a central server, located at the UAV, to train a common learning model. For learning a statistical model from the distributed data, the central server aims to solve the following optimization problem:
minw F(w) =
N
X
k=1
pkFk(w), (5) where w is the learning model parameter, pk = nk/n rep- resents the weight of device k with nk samples in its local dataset Dk, andnis the total number of data samples.Fk(w) denotes the local loss function at device k, given by:
Fk(w) = 1 nk
X
x∈Dk
f(w, x), (6) wheref(w, x)is the point local loss function for data sample x. It should be noted that the dataset Dk is non-iid across different devices. Since the aggregator cannot directly solve (5), an iterative approach should be applied. Hence, we pro- pose the algorithm shown in Algorithm 1, which solves the FL problem in (5) while considering the unreliable DL global model distribution and UL local models aggregation in aerial setting. The aggregation step is the key feature of Algorithm 1 when compared to existing methods. It is worth noting that the work in [9] assumes perfect DL reception and erroneous UL channel for terrestrial settings. In conventional FedAvg algorithms [17], the global aggregation rule at the server does not consider any channel unreliability. However, if we take the channel unreliability into account, then the aggregation of FedAvg becomes biased and is given by
wt+1=
N
X
k=1 M
X
b=1
pkwt+1k 1(k∈St,
SINRDLk,b> τDL, SINRULk,b> τUL), (7)
wherewkt is the local model parameter of devicekat time t.
Since in the proposed method we allow forEnumber of local stochastic gradient descent (SGD) iterations, the value of wtk takes the following form:
wkt =
(wt ift∈ {E,2E,3E, ...}
vkt otherwise, (8)
where the first case denotes the local initialization stages, and vkt is the locally updated model parameter. Note that the local learning in Algorithm 1 is performed in batches of ξtk on devicek at timet.
In this paper, we apply random scheduling in the following manner: the UAV selects uniformly M out of N devices without replacement, thereby guaranteeing that each device uses at most one RB. Hence, on average, the UAV allocates a RB to devicek with probabilityqk. Thus,qk is defined as
qk=E
"M X
b=1
1(k∈St(b))
#
=M
N, (9)
whereSt(b)represents a set of devices scheduled at timetfor a RB b. The indicator function in Algorithm 1 considers the impact of the wireless channel during UL/DL transmission. To mitigate this impact, the weights of local updates are divided by the RB schedulingqk and the joint success probabilityJk, which is derived in Theorem 1. The device with a higher Jk
would have a lower weight in the aggregation step, promoting fairness among all devices. In practice, the proposed scheme
Algorithm 1 FL for UAV-assisted wireless networks Server Executes:initialize w0
foreach round t= 0,1,2, ...do
St ←(random set of M clients out ofN) Broadcastwt to the clients
foreach clientk∈Stin paralleldo vt+1k ←LocalUpdate(k,wt)
Calculate the joint success probabilityJk using (10) end for
wt+1 = wt+PN k=1
PM b=1
pk
qkJk1(k∈St,SINRDLk,b >
τDL, SINRULk,b> τUL)(vt+1k −wt)//aggregation step end for
LocalUpdate(k, wt):
ξtk ←(splitDk into batches)
foreach local epochifrom 0 toE−1 do wkt+i=wtk−ηt+i∇Fk(wt+ik , ξtk) end for
Transmit wt+Ek to the server
adds an extra step of estimating success probabilities for each device for fair aggregation during FL. Success probabilities are functions of the network parameters and channel statistics along with devices’ locations with respect to the UAV. While the network parameters and channel statistics are common knowledge for all UAVs, the devices’ locations can be either shared by the devices or estimated by the UAV via local- ization/ranging techniques. The proof of convergence of the proposed algorithm is similar to the one shown in [9], and is omitted in this paper due to space limitations.
Theorem 1. In a UAV-assisted wireless network modeled as an MCP, the joint success probability for device k defined as the probability that SINRs for both DL and UL exceed predefined thresholdsτDL andτUL, respectively, is given as:
Jk =Ez∈{L,N}[P[SINRDLz > τDL,SINRULz > τUL|z, k∈St]]
=PL(rk|k∈St)×JkL+PN(rk|k∈St)×JkN,
(10) where
Jkz=
mz
X
j=1
mz
j
(−1)j+1
exp
−jηzn20τDL PuG0(r2k+h2)−αz2
LDL
−jηzτDL PuG0(rk2+h2)−αz2
×
mz
X
j=1
mz
j
(−1)j+1
exp
−jηzn20τUL
PdG0(r2k+h2)−αz2
LUL
−jηzτUL
PdG0(r2k+h2)−αz2
, (11) wherez∈ {L, N},k∈Stindicates that devicekis scheduled for a RB at time t, mz is the Nakagami fading parameter for z link with ηz =mz(mz!)−1/mz, τDL and τUL are SINR thresholds for DL and UL transmissions respectively, and LDL(·)andLUL(·)are the Laplace transforms of DL and UL inteferences, respectively, and are provided in the next two lemmas.
Proof:See Appendix A of our extended report available on archive [1].
TABLE II: System parameter values
N, M=100, 90 R, h=100, 120 m λ= 2/(π1502)UAV/m2 Pd, Pu=0.1, 0.25 W αL, αN= 2.1, 3.6 n20= 4.14×10−6W a, b=9.61, 0.16 mL, mN= 3, 1 τDL, τUL=15, 0 dB Mu=10 dB Md=5 dB mu, md=-1, -3 dB
The Laplace transforms of DL and inter-cluster UL inter- ferences are formulated in the following lemmas:
Lemma 1. In a UAV-assisted wireless network modeled as an MCP, the Laplace transform of the DL interference is given by:
LDL(s) = Y
z∈{L,N}
exp −2πλ Z ∞
0
"
1−
4
X
i=1
pi
1 + sPuGi(g2+h2)−αz2 mz
−mz#
qPz(q)dq
! , Proof: See Appendix B of our extended report available on archive [1].
Lemma 2. The Laplace transform of the inter-cluster UL interference in a UAV-assisted network modeled as MCP is given by:
LUL(s) = Y
z∈{L,N}
exp −2πλ
"
Z R
0
h
1−[Oe1z (s, q)]N¯i q dq
+ Z ∞
R
h
1−[Oe2z (s, q)]N¯i q dq
#!
, where N¯ represents the number of interfering devices in a neighboring cluster, which is N¯ = 1 due to the considered scheduling scheme and
Oze1(s, q) = Z R+q
|q−R|
4
X
i=1
pi
1 + sPdGi(g2+h2)−αz2 mz
−mz
× 2g
πR2arccos
g2+q2−R2 2gq
dq
+ Z R−q
0 4
X
i=1
pi
1 +sPdGi(g2+h2)−αz2 mz
−mz
× 2g R2dq, and
Oze2(s, q) = Z R+q
|q−R|
4
X
i=1
pi
1 + sPdGi(g2+h2)−αz2 mz
−mz
× 2g
πR2arccos
g2+q2−R2 2gq
dq.
Proof: See Appendix C of our extended report available on archive [1].
IV. NUMERICALRESULTS
In this section, we provide analytical results and Monte- Carlo simulations to validate the accuracy of the joint success probability given in Theorem 1. We also present simulation results for the proposed FL algorithm in UAV-assisted net- works. Unless otherwise mentioned, the simulation parameters are presented in Table II. We compare our FL aggregator (i.e., that accounts for the joint DL/UL unreliability) with the traditional FedAvg algorithm and with [9] that considers the UL unreliability only. Fig. 2 plots the coverage probabilities for both joint DL and UL, and UL-only instances as a function
20 40 60 80 100 120 140
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
UAV height, (m)
Coverageprobability
Joint (simulation) Joint (analytic) Uplink (simulation) Uplink (analytic)
Fig. 2: Coverage probability plot for parameter values in Table II.
of the UAV height. It can be seen clearly that the results of the analytic expression match the Monte-Carlo simulations, which proves the validity of the conducted analysis. The figure shows the significant impact of DL on transmission success and an optimal UAV height striking a tradeoff between improved LOS and increased propagation loss.
We evaluate our FL algorithm’s performance with global loss and accuracy. We perform FL with E = 2 local SGD iterations and batch size of ξtk = 64 until convergence.
MNIST data is distributed non-iid across all devices in a cluster. The result in Fig. 3a shows the global loss versus the number of communication rounds between a UAV and the associated devices. It can be noticed that the proposed method drastically reduces the loss compared to the method in UL-only case and the conventional FedAvg. The superiority of the proposed method is also reflected in accuracy plots in Fig. 3b. The proposed method reaches the convergence much faster and provides around 25% improvement as compared to the UL-only scenario. The findings of the study emphasize the significance of taking into account the effects of wireless channels and considering both DL and UL in mitigating the global bias problem of FL.
The effect of changing the number of local iterations is studied and the results are given in Fig. 4. One can observe that there is an optimal value for local iterations which isE= 2.
At this value, both methods using joint and UL-only success probabilities demonstrate the highest testing accuracies. It is noted that the proposed joint method always performs better than the method relying on only UL success probability.
The developed model allows to investigate the impact of different system parameters on the learning performance.
Fig. 5 illustrates that the highest accuracy corresponds to the optimal UAV height of h = 50 m, which is supported by the observations from Fig. 2. The reason for such trend is that the higher coverage probability value, the more devices participate in each round of FL. We plot FL training accuracy in four environments for UAV heights of 25 m and 120 m in Figs. 6a and 6b. The impact of height on FL accuracy varies in different environments. In suburban, higher height leads to negative impact on accuracy since it only increases the path- loss without affecting the LOS probability which is almost 1 even at low heights. In high-rise urban, FL accuracy rises due to the growing LOS probability compensating for increasing
0 50 100 150 200 250 0.6
0.8 1 1.2 1.4 1.6 1.8 2 2.2
Communication rounds
Globalloss JointUL-only
FedAvg
Fig. 3a: Global loss across communication rounds.
0 50 100 150 200 250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Communication rounds
Trainingaccuracy
Joint UL-only FedAvg
Fig. 3b: Training accuracy across communication rounds.
1 2 3 4 5 7 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
E
Testingaccuracy
Joint UL-only
Fig. 4: Testing accuracy for different local SGD iteration values.
0 50 100 150 200 250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Communication rounds
Trainingaccuracy h = 25 mh = 50 m
h = 120 m
Fig. 5: Training accuracy for different UAV heights.
0 50 100 150 200 250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Communication rounds
Trainingaccuracy
Suburban Urban Dense urban High-rise urban
Fig. 6a: Training accuracy for different environments ath= 25m.
0 50 100 150 200 250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Communication rounds
Trainingaccuracy Suburban
Urban Dense urban High-rise urban
Fig. 6b: Training accuracy for different environments ath= 120m.
path-loss. This highlights the need to optimize UAV position for improved FL accuracy in aerial networks.
V. CONCLUSION
In this paper, we examine the performance of the FL algorithm in aerial UAV-assisted networks. The unreliable and resource-constrained wireless channel between aggregat- ing UAVs and devices makes the learning inefficient due to biasing the global model towards devices with better channel conditions. Using tools from stochastic geometry, the joint upload and download success probability for FL in UAV- assisted networks is computed and used to unbias the FL ag- gregation rule. The proposed FL algorithm not only surpasses the existing methods but also enables the adjustment of system parameters for optimum learning performance.
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