# Semi-asynchronous Hierarchical Federated Learning for Cooperative
Intelligent Transportation Systems

Qimei Chen, Zehua You, and Hao Jiang Q. Chen, Z. You, and H. Jiang are with
the School of Electronic Information, Wuhan University, Wuhan 430072, China
(e-mail: {chenqimei; youzehua<EMAIL_ADDRESS>
###### Abstract

_Cooperative Intelligent Transport System_ (C-ITS) is a promising network to
provide safety, efficiency, sustainability, and comfortable services for
automated vehicles and road infrastructures by taking advantages from
participants. However, the components of C-ITS usually generate large amounts
of data, which makes it difficult to explore data science. Currently,
federated learning has been proposed as an appealing approach to allow users
to cooperatively reap the benefits from trained participants. Therefore, in
this paper, we propose a novel _Semi-asynchronous Hierarchical Federated
Learning_ (SHFL) framework for C-ITS that enables elastic edge to cloud model
aggregation from data sensing. We further formulate a joint edge node
association and resource allocation problem under the proposed SHFL framework
to prevent personalities of heterogeneous road vehicles and achieve
communication-efficiency. To deal with our proposed _Mixed integer nonlinear
programming_ (MINLP) problem, we introduce a distributed _Alternating
Direction Method of Multipliers_ (ADMM)-_Block Coordinate Update_ (BCU)
algorithm. With this algorithm, a tradeoff between training accuracy and
transmission latency has been derived. Numerical results demonstrate the
advantages of the proposed algorithm in terms of training overhead and model
performance.

###### Index Terms:

C-ITS, federated learning, asynchronous, edge association, resource
allocation.

##  I Introduction

With the rapid developments of automated vehicles and Internet-of-Vehicles,
_Cooperative Intelligent Transport Systems_ (C-ITS) is expected to provide
safety, traffic efficiency, and comfortable infotainment services for vehicles
with an accurate neighborhood view [1]. Specifically, the C-ITS components
including vehicles, road infrastructure units, personal devices, and traffic
command centers can achieve additional traffic management and convenience with
the help of enhanced visions from surrounding participants [2].

Currently, both the _Wireless Access in Vehicular Environments_ (WAVE) and the
_European Telecommunications Standards Institute_ (ETSI) standards have been
proposed consecutively to define spectrum allocation, traffic parameters
setting, and protocol stack to enable _Quality-of-Service_ (QoS) and security
procedure for C-ITS [3]. Several works have also studied the technologies for
C-ITS. The authors in [4] introduce a choreography-based service composition
platform to accelerate the reuse-based development for a choreography-based
urban coordinative application. The authors in [5] proposed a novel
topological graph convolutional network to predict the urban traffic flow and
traffic density.

The authors in [6] analyze the relationship among security, QoS, and safety
for vehicles in C-ITS. In addition, the authors in [7] investigate service-
oriented cooperation models and mechanisms for autonomous vehicles in C-ITS.
Nevertheless, they have neglected to explore the large amounts of traffic data
generated from transportation sensor networks, which is essential for advanced
C-ITS applications like connected and autonomous vehicles, traffic control and
prediction, and road safety. Although some attempts have been made to utilize
data science for C-ITS [8], there still exists various stuff challenges to be
solved, such as privacy protection, computational complexity, and data
heterogeneity.

With the significant advance cloud computing, big data, and machine learning,
intelligent transportation is a trend and requirement. As one of the most
dominant distributed machine learning technologies, _Federated Learning_ (FL)
has been widely studied to deal with the data science recently [9, 10, 11].
The FL technology allows participant devices to collaboratively build a shard
model while preserving privacy data locally [12]. Particularly, the prevalent
FL algorithm, namely federated averaging, allows each device to train a model
locally with its own dataset, and then transmits the model parameters to the
central controller for a global aggregation [13]. Intuitively, FL presents a
great potential for C-ITS to facilitate the large data management. However,
directly applying FL to C-ITS still faces three major deficiencies: 1) limited
wireless resources; 2) high latency; 3) obliterated data diversity.

According to [18, 19], _Federated Learning training at Edge networks_ (FEL)
has been regarded as a solution to facilitate the above limitations through
bringing model training closer to the data produced locally. Compared with the
conventional cloud centric FL approaches, the implementation of FEL can
provide high wireless resources utilization since less information is required
to be transmitted to the cloud. In addition, FEL has a much lower transmission
latency and higher privacy than the conventional FL by making decisions at the
edge nodes. In [15], the authors develop an importance aware joint data
selection and resource allocation algorithm to maximize the resource and
learning efficiencies. Meanwhile, the authors in [16] propose an adaptive
federated learning mechanism in resource constrained edge computing systems.
Along the FEL, the authors in [17] propose a novel _Hierarchical Federated
Edge Learning_ (HFEL) framework, where edge servers deployed with base
stations fixedly and can upload edge aggregation model to the cloud. The above
HFEL enables great potentials in low latency and energy efficiency.

However, the participant C-ITS devices usually have heterogeneous resources,
which lead to _non-independent-identically distributed_ (non-iid) private data
during the communication [20, 21]. The existing federated learning methods
mainly utilize the synchronous model aggregation mechanism, where the central
server needs to wait for the slowest device to complete the training in each
communication round [14, 22]. With heterogeneous data, the transmission
latency of each synchronous model aggregation mechanism is unacceptable for
time-sensitive devices. In this way, several works have proposed asynchronous
model aggregation methods, where only one participant device would update the
global model each time [24, 25, 26]. Nonetheless, the training round under
asynchronous methods is several times than that of synchronous methods.
Moreover, due to the asynchrony, gradient staleness may be difficult to
control [26]. Therefore, the authors in [23] design an $n$-softsync
aggregation model that can significantly reduce training time by combines the
benefits of both synchronous and asynchronous aggregations.

Inspired by the above analyses, we aim to leverage a novel _Semi-asynchronous
Hierarchical Federated Learning_ (SHFL) framework that can provide safety,
efficiency, sustainability, and comfortable services to C-ITS. Specifically,
the proposed SHFL framework consists of both edge and cloud layers, where each
edge node aggregates all of homogeneous local models and the cloud layer
aggregates parts of heterogeneous edge models. These selected nodes would
update the global model once the selected slowest node finishes training,
which combines the merits of both synchronous and asynchronous aggregations.
For further performance enhancement, we formulate a joint edge node
association and resource allocation optimization problem to prevent
heterogeneous edge node personalities as well as ensure communication-
efficient of the whole system. The objective function is a _Mixed Integer
NonLinear Programming_ (MINLP) problem, which has been solved by a distributed
_Alternating Direction Method of Multipliers_ (ADMM)-_Block Coordinate Update_
(BCU) algorithm. It is shown that the proposed algorithm can achieve near
optimal with low computational complexity. To the best of our knowledge, this
is the first work that applying improved FL into the C-ITS.

Overall, the main contributions of this work can be listed as follows.

  * •

We propose a novel SHFL framework by applying the synchronous aggregation
model for local-edge and the semi-asynchronous aggregation model for edge-
cloud to provide safety, traffic efficiency, and comfortable infotainment
services for C-ITS.

  * •

To reserve the personalities of heterogeneous edge nodes, we introduce an
elastic edge model update method based on the distance between the global
model and the edge model.

  * •

We formulate a joint edge node association and resource allocation problem to
achieve communication-efficiency by achieving a tradeoff between training
accuracy and transmission latency. A distributed ADMM-BCU algorithm has been
used to solve the MINLP problem.

The rest of this paper is organized as follows. Section II introduces the
system model and the SHFL learning mechanism. In Section III, we formulate the
communication-efficient problem. A joint edge node association and resource
allocation strategy is presented in Section IV. Section V presents the
numerical results, followed by the conclusions in Section VI.

##  II System Model

In this work, we aim to design a novel SHFL framework for C-ITS that contains
three layers, namely the cloud layer, the edge layer, and the local layer, as
shown in Fig. 1. Here, we consider the vehicle devices have heterogeneous data
structures, namely the local datasets are non-iid. We let homogeneous devices
with similar data size, network bandwidth, and QoS gather in the same edge
node. Hence, the edge nodes are heterogeneous. A shared _Deep Neural Network_
(DNN) model is distributed over the local devices, which has been trained
collaboratively across the devices under their datasets. Different from
conventional FLs, the proposed SHFL framework allows devices train their data
locally, homogeneous devices report their computed parameters to the same edge
node synchronously, and heterogeneous edge nodes upload their models to the
cloud node semi-asynchronously, which can preserve data privacy as well as
improve communication efficiency. In the proposed framework, we assume there
has a set of $K$ edge nodes $\mathcal{K}=\\{1,...,K\\}$. Any edge node $k$
consists of a set of $N_{k}$ local devices, denoted as
$\mathcal{N}_{k}=\\{L_{k,1},...,L_{k,N_{k}}\\}$. Under edge node
$k\in\mathcal{K}$, local device $n\in\mathcal{N}_{k}$ owns a local data set
$D_{k,n}=\\{(\boldsymbol{x}_{j,k,n},{y}_{j,k,n}):j=1,...,|D_{k,n}|\\}$, where
$\boldsymbol{x}_{j,k,n}$ is the $j$-th input training data sample,
${y}_{j,k,n}$ is the $j$-th corresponding output, and $|D_{k,n}|$ denotes the
cardinality of the data set $D_{k,n}$. For simplicity, we assume the SHFL
algorithm with a single output. However, this work can be extended to the
multiple outputs case. In what follows, we would introduce each part of the
proposed SHFL framework at the $t$-th iteration.

Figure 1: Illustration for the SHFL based C-ITS.

###  II-A Edge Aggregation

The edge aggregation stage contains three processes, including local model
computation, local model transmission, and edge model aggregation. In detail,
local model first trained by local data, then local models respectively
transmit to their associated edge nodes for edge aggregation. The detailed
processes are as follows.

####  II-A1 Local Model Computation

Without loss of generality, we consider a supervised machine learning task on
device $n\in\mathcal{N}_{k}$ associated with edge node $k\in\mathcal{K}$,
which has a learning model of $\boldsymbol{w}_{k,n}$. We further define
$f_{n}(\boldsymbol{x}_{j,k,n},y_{j,k,n},\boldsymbol{w}_{k,n})$ as the loss
function of data sample $j$ that quantifies the prediction error between data
sample $\boldsymbol{x}_{j,k,n}$ and output $y_{j,k,n}$. In this work, we
mainly focus on the logistic regression model for the loss function, i.e.,
$f_{n}(\boldsymbol{x}_{j,k,n},y_{j,k,n},\boldsymbol{w}_{k,n})=-\log\left(1+\exp\left(-y_{j,k,n}\boldsymbol{x}_{j,k,n}^{\mathrm{T}}\boldsymbol{w}_{k,n}\right)\right)$.
Hence, the loss function of device $n\in\mathcal{N}_{k}$ associated with edge
node $k\in\mathcal{K}$ on dataset $D_{k,n}$ can be defined as

$\begin{split}F_{k,n}(\boldsymbol{w}_{k,n})=\frac{1}{|D_{k,n}|}\sum_{j=1}^{|D_{k,n}|}f_{k,n}(\boldsymbol{x}_{j,k,n},y_{j,k,n},\boldsymbol{w}_{k,n}),\\\
~{}\forall k\in\mathcal{K},n\in\mathcal{N}_{k}.\end{split}$ (1)

The local update model of device $n\in\mathcal{N}_{k}$ in edge node
$k\in\mathcal{K}$ can be achieved by

$\begin{split}\boldsymbol{w}_{k,n}^{t}=\boldsymbol{w}_{k,n}^{t-1}-\eta\nabla
F_{k,n}(\boldsymbol{w}_{k,n}^{t-1}),~{}\forall
k\in\mathcal{K},n\in\mathcal{N}_{k},\end{split}$ (2)

where $\eta$ is a predefined learning rate.

Define $C_{k,n}$ as the number of CPU cycles for local device
$n\in\mathcal{N}_{k}$ associated with edge node $k\in\mathcal{K}$ to process
one sample data. Assuming each sample data has the same size, the total CPU
cycles to run one local iteration is $C_{k,n}|D_{k,n}|$. We further let
$f_{k,n}$ be the computation frequency of device $n\in\mathcal{N}_{k}$ in edge
node $k\in\mathcal{K}$. In this way, the related local gradient calculation
latency in one round can be formulated as

$T_{k,n}^{c}=\dfrac{C_{k,n}|D_{k,n}|}{f_{k,n}},~{}\forall
k\in\mathcal{K},n\in\mathcal{N}_{k}.$ (3)

####  II-A2 Local Model Transmission

We adopt the _Orthogonal-Frequency-Division Multiple Access_ (OFDMA) technique
for local uplink transmissions. Define $B_{k,n}$ as the bandwidth allocated to
device $n\in\mathcal{N}_{k}$. Therefore, we have
$\sum_{n=1}^{\mathcal{N}_{k}}B_{k,n}=B_{k}$, where $B_{k}$ is the bandwidth
allocated to edge node $k\in\mathcal{K}$ for the transmission between edge
node $k\in\mathcal{K}$ and the associated local devices. Meanwhile, we have
$\sum_{k=1}^{K}B_{k}\leq B_{e}$, where $B_{e}$ is the total bandwidth
allocated for the communication between edge nodes to the local devices.
Therefore, the achievable local uplink data rate from device
$n\in\mathcal{N}_{k}$ to edge node $k\in\mathcal{K}$ can be formulated as

$r_{k,n}^{u}=B_{k,n}\log_{2}\left({1+\frac{P_{k,n}g_{k,n}}{B_{k,n}N_{0}}}\right)\\!,~{}\forall
k\in\mathcal{K},n\in\mathcal{N}_{k},$ (4)

where $P_{k,n}$ is the uplink transmission power of device
$n\in\mathcal{N}_{k}$ in edge node $k\in\mathcal{K}$, $g_{k,n}$ denotes the
channel gain between local device $n\in\mathcal{N}_{k}$ and edge node
$k\in\mathcal{K}$, and $N_{0}$ means the noise power.

Similarly, the achievable downlink data rate for device $n\in\mathcal{N}_{k}$
associated with edge node $k\in\mathcal{K}$ can be expressed as

$r_{k,n}^{d}=B_{k}\log_{2}\left({1+\frac{P_{k}g_{k,n}}{B_{k}N_{0}}}\right)\\!,~{}\forall
k\in\mathcal{K},n\in\mathcal{N}_{k},$ (5)

where $P_{k}$ is the downlink transmission power of edge node
$k\in\mathcal{K}$.

In this work, we use the same training model for the whole communication
system. Therefore, the number of model parameters in each level of model
transfer has the same size. Denote $Z$ as the data size of the model parameter
bits. The local gradient upload latency of device $n\in\mathcal{N}_{k}$ in
edge node $k\in\mathcal{K}$ can be expressed as

$T_{k,n}^{u}=\frac{Z}{r_{k,n}^{u}}=\frac{Z}{B_{k,n}\log_{2}\left({1+\frac{P_{k,n}g_{k,n}}{B_{k,n}N_{0}}}\right)\\!},~{}\forall
k\in\mathcal{K},n\in\mathcal{N}_{k}.$ (6)

Correspondingly, the edge model download latency of device
$n\in\mathcal{N}_{k}$ in edge node $k\in\mathcal{K}$ can be formulated as

$T_{k,n}^{d}=\dfrac{Z}{r_{k,n}^{d}}=\dfrac{Z}{B_{k}\log_{2}\left({1+\frac{P_{k}g_{k,n}}{B_{k}N_{0}}}\right)\\!},~{}\forall
k\in\mathcal{K},n\in\mathcal{N}_{k}.$ (7)

####  II-A3 Edge Model Aggregation

In this work, each edge node can receive the updated model parameters from its
associated homogeneous devices. Since the devices under one edge node usually
have a similar type, we adopt the synchronous aggregation method to average
these updated models. It means that the edge node would wait for the slowest
node to complete training in each round and collect all the connected devices’
updated model parameters. Therefore, the edge model aggregating equation for
edge node $k\in\mathcal{K}$ can be formulated as

$\boldsymbol{w}_{k}^{t}=\frac{\sum_{n=1}^{\mathcal{N}_{k}}|D_{k,n}|\boldsymbol{w}_{k,n}^{t}}{|D_{k}|},~{}\forall
k\in\mathcal{K},$ (8)

where $|D_{k}|=\sum_{n=1}^{\mathcal{N}_{k}}|D_{k,n}|$ is the total number of
data in edge node $k\in\mathcal{K}$.

We omit edge model aggregation time due to its strong computing capability.
Hence, the computation and communication latency between each edge
$k\in\mathcal{K}$ and the related local devices can be derived as

$T_{k}^{edge}=\max_{n\in\mathcal{N}_{k}}\left(T_{k,n}^{c}+T_{k,n}^{u}+T_{k,n}^{d}\right),~{}\forall
k\in\mathcal{K}.$ (9)

###  II-B Cloud Aggregation

Similarly, the cloud aggregation stage contains two processes, i.e., edge
model transmission and cloud model aggregation. Particularly, the selected
edge nodes upload their updated model parameters to the cloud for aggregation.
The detailed processes are as follows.

####  II-B1 Edge Model Transmission

Edge nodes would upload their model parameters to the cloud after edge model
aggregations. To ensure uninterrupted transmission from edge to cloud, we also
adopt the OFDMA technique. Hence, the uplink data rate for edge node
$k\in\mathcal{K}$ can be expressed as

$r_{c,k}^{u}=B_{c,k}\log_{2}\left({1+\frac{P_{c,k}g_{c,k}}{B_{c,k}N_{0}}}\right)\\!,~{}\forall
k\in\mathcal{K},$ (10)

where $B_{c,k}$ is the bandwidth allocated to edge node $k\in\mathcal{K}$
transmits to the cloud node, $P_{c,k}$ is the uplink transmission power of
edge node $k\in\mathcal{K}$ to the cloud node, and $g_{c,k}$ denotes the
channel gain between edge node $k\in\mathcal{K}$ and the cloud node.

Correspondingly, the downlink data rate from the cloud node to edge node
$k\in\mathcal{K}$ can be formulated as

$r_{c,k}^{d}=B_{c}\log_{2}\left({1+\frac{P_{c}g_{c,k}}{B_{c}N_{0}}}\right)\\!,~{}\forall
k\in\mathcal{K},$ (11)

where $P_{c}$ is the downlink transmission power of the cloud node, $B_{c}$ is
the total bandwidth for the transmission between the edge nodes and the cloud.
As we would discuss later, only parts of the edge nodes can be selected in
each round. Therefore, we have the constraint of

$\sum_{k=1}^{K}\alpha_{k}B_{c,k}\leq B_{c},$ (12)

where $\alpha_{k}\in\\{0,1\\}$. Here, $\alpha_{k}=1,\forall k\in\mathcal{K}$
indicates edge node $k$ has been selected, and $\alpha_{k}=0,\forall
k\in\mathcal{K}$ otherwise.

In this way, the upload latency from edge node $k\in\mathcal{K}$ to the cloud
node can be written as

$T_{c,k}^{u}=\frac{Z}{r_{c,k}^{u}}=\frac{Z}{B_{c,k}\log_{2}\left({1+\frac{P_{c,k}g_{c,k}}{B_{c,k}N_{0}}}\right)\\!},~{}\forall
k\in\mathcal{K}.$ (13)

Similarly, the downlink latency from the cloud node to edge node
$k\in\mathcal{K}$ can be expressed as

$T_{c,k}^{d}=\dfrac{Z}{r_{c,k}^{d}}=\dfrac{Z}{B_{c}\log_{2}\left({1+\frac{P_{c}g_{c,k}}{B_{c}N_{0}}}\right)\\!},~{}\forall
k\in\mathcal{K}.$ (14)

####  II-B2 Cloud Model Aggregation

Figure 2: Illustration of the semi-asynchronous process.

Since these edge nodes correspond to heterogeneous local datasets, their model
updated periods various. If we adopt the synchronous aggregation model, the
latency for faster training nodes is unacceptable. On the contrary, the
asynchronous method has shorter round latency, however, it requires several
times of training rounds than the synchronous method. Therefore, in this work,
we propose a flexible semi-asynchronous aggregation method by combining the
merits of both synchronous and asynchronous methods. As shown in Fig. 2, the
cloud node would select $|\mathcal{S}^{t}|=\sum_{k=1}^{K}\alpha_{k}$ edge
nodes with the fastest training round for model aggregation, where the set of
selected edge nodes is denoted as $\mathcal{S}^{t}$. Slow nodes would wait for
the next communication round to upload their models. Hence, under the semi-
asynchronous aggregation method, we can achieve a balance between training
accuracy and communication latency. The semi-asynchronous aggregation method
can be written as

$\boldsymbol{w}_{c}^{t}=\boldsymbol{w}_{c}^{t-1}+\sum_{k\in\mathcal{S}^{t}}\frac{|D_{k}|}{\sum_{k=1}^{K}|D_{k}|}(\boldsymbol{w}_{k}^{t}-\boldsymbol{w}_{c}^{t-1}).$
(15)

Also, we ignore the cloud model aggregation latency due to its strong
computing capability. Therefore, the cloud-edge communication latency can be
derived as

$T_{k}^{cloud}=T_{c,k}^{u}+T_{c,k}^{d},~{}\forall k\in\mathcal{K}.$ (16)

Towards this end, the one-round latency for edge node $k\in\mathcal{K}$ is
given by

$T_{k}=T_{k}^{edge}+T_{k}^{cloud},~{}\forall k\in\mathcal{K}.$ (17)

###  II-C Edge Update Model

From Eq. (2), the local updated models are determined by their own
characteristics. Since the non-iid devices that connected with one edge node
have a similar characteristic, the edge aggregation models are heterogeneous.
Therefore, if we directly use the cloud model to update the edge models, the
personalities among edge models would be eliminated. Meanwhile, the accuracy
of the cloud model would be decreased. Hence, we introduce a new edge update
model based on [28], which defines a weight distance formula to represent the
difference among different weight relatives as

$dist\left(\boldsymbol{w}^{t}_{k},\boldsymbol{w}^{t}_{c}\right)=\frac{\left|\left|\boldsymbol{w}^{t}_{k}-\boldsymbol{w}^{t}_{c}\right|\right|}{\left|\left|\boldsymbol{w}^{t}_{c}\right|\right|},\forall
k\in\mathcal{K}.$ (18)

Intuitively, the larger of
$dist\left(\boldsymbol{w}_{k},\boldsymbol{w}_{c}\right)$, the greater of the
model difference.

Typically, deep learning networks that consist of multiple layers and each
layer contains various amounts of weights can be adopted here. For simplicity,
we use a small dataset to obtain the layer with the most obvious
characteristics, which has been denoted as
$\mathcal{L}=\\{\ell_{1},\ell_{2},\cdots\\}$. Thereafter, we introduce a
parameter $\varepsilon_{k}$ to measure the difference between the cloud model
and edge model $k$, which can be formulated as

$\varepsilon_{k}=\frac{1}{\left|\mathcal{L}\right|}\sum_{\ell\in\mathcal{L}}dist\left(\boldsymbol{w}_{k}^{t,\ell},\boldsymbol{w}_{c}^{t,\ell}\right),\forall
k\in\mathcal{K},$ (19)

where $\boldsymbol{w}_{k}^{t,\ell}$ and $\boldsymbol{w}_{c}^{t,\ell}$
represent the weight of the $\ell$-th layer of edge model
$\boldsymbol{w}^{t}_{k}$ and cloud model $\boldsymbol{w}^{t}_{c}$. Meanwhile,
$\left|\mathcal{L}\right|$ is the cardinality of $\mathcal{L}$.

From Eq. (19), $\varepsilon_{k}$ increases with
$dist\left(\boldsymbol{w}_{k},\boldsymbol{w}_{c}\right)$. To keep the
personalities, the edge updated model can be derived by

$\boldsymbol{w}_{k}^{t}\leftarrow\varepsilon_{k}\boldsymbol{w}_{c}^{t}+(1-\varepsilon_{k})\boldsymbol{w}_{k}^{t},\forall
k\in\mathcal{K}.$ (20)

###  II-D Learning Procedure of the SHFL Model

Figure 3: Learning Procedure of the proposed SHFL model.

Based on the definition of SHFL model, the training procedure of the SHFL
model at the $t$-th iteration proceeds as follows, which is also shown in Fig.
3.

  1. 1.

_Local model training and update_ : Devices in C-ITS train their learning
model and calculate their local gradient as $\nabla
F_{k,n}(\boldsymbol{w}_{k,n}^{t}),\forall
k\in\mathcal{K},n\in\mathcal{N}_{k}$. After receiving
$\boldsymbol{w}_{k}^{t},\forall k\in\mathcal{S}^{t}$, devices in the selected
edge nodes update their learning model based on Eq. (2).

  2. 2.

_Local model upload_ : Local devices upload their concrete models to related
edge nodes based on the local-edge bandwidth allocation scheme.

  3. 3.

_Edge model aggregation_ : After receiving local models, each edge node
computes the average edge model based on Eq. (8). Since device types among
edge nodes are heterogeneous, their edge model aggregations are semi-
asynchronous.

  4. 4.

_Edge node selection and resource management_ : Based on the reports of edge
nodes, the cloud node selects a subset of edge nodes $\mathcal{S}^{t}$ with
the fastest training round and determines the bandwidth allocation.

  5. 5.

_Selected edge model upload_ : The selected edge nodes upload their aggregated
models to the cloud node.

  6. 6.

_Cloud model aggregation and broadcast_ : The cloud server aggregates the
uploaded edge models, and then broadcasts the current aggregated model
$\boldsymbol{w}_{c}^{t}$ to the selected edge nodes.

  7. 7.

_Edge model update and broadcast_ : The selected edge servers broadcast the
updated model $\boldsymbol{w}_{k}^{t}$ to local related devices.

The procedure starts from $t=1$ and repeats the above steps until convergence.

##  III Problem Formulation

As discussed earlier, there exists a tradeoff between the training accuracy
and the transmission latency. Therefore, our goal in this work is to find a
balance between them to provide safety, efficiency, sustainability, and
comfortable services for the SHFL based C-ITS framework.

According to Eq. (2), the local model _Gradient-Norm-Value_ (GNV) influences
the local model updating, which measures the data importance. The GNV of local
device $n\in\mathcal{N}_{k}$ in edge node $k\in\mathcal{K}$ can be expressed
as

$\begin{split}{\boldsymbol{g}}_{k,n}^{{\boldsymbol{w}},t}&=\nabla
F_{k,n}(\boldsymbol{w}_{k,n}^{t})\\\ &=\sum_{D_{k,n}}\frac{\partial
f_{k,n}(\boldsymbol{x}_{j,k,n},y_{j,k,n},\boldsymbol{w}_{k,n}^{t})}{\partial\boldsymbol{w}_{k,n}^{t}},\forall
k\in\mathcal{K},n\in\mathcal{N}_{k}.\end{split}$ (21)

With out loss of generality, we leverage the norm of GNV to present the
importance, which can be written as

$\sigma_{k,n}^{t}=\left|\left|{\boldsymbol{g}}_{k,n}^{{\boldsymbol{w}},t}\right|\right|^{2},\forall
k\in\mathcal{K},n\in\mathcal{N}_{k}.$ (22)

Since an edge node connects homogeneous local devices, the GNVs among these
local devices are approximately equal. Moreover, local devices in one edge
node also have similar training duration, hence, all of these training models
(GNV) would be uploaded. In this way, the GNV of edge node $k\in\mathcal{K}$
can be defined as

$\sigma_{k}^{t}=\sum_{n=1}^{\mathcal{N}_{k}}\sigma_{k,n}^{t},\forall
k\in\mathcal{K}.$ (23)

On the contrary, the cloud node associates with heterogeneous edge nodes, the
GNVs among them various. Intuitively, edge nodes with significant gradients
have more contributions on model updating and convergence. Therefore, the
cloud would preferentially select impactive edge nodes to upload their
information for cloud model aggregation. Then, the GNV of the cloud model can
be written as

$\sigma^{t}=\sum_{k=1}^{K}\alpha^{t}_{k}\sigma^{t}_{k}.$ (24)

For easy of expression, we remove the iteration $t$ in the following.

Now, we are ready to describe the problem formulation. The goal of this work
is to maximize communication-efficient via joint edge node selection and
resource allocation scheduling for a SHFL based C-ITS network. To accelerate
the learning process, it is desirable to select more edge nodes with larger
data importance. However, to shorten the communication and computation
latency, it is better to upload as fewer edge nodes as possible. As a result,
the objective function that represents the tradeoff between GNVs and
transmission latency can be formulated as

$\displaystyle\hskip
11.00008pt\min_{\boldsymbol{\alpha,B_{k,n},B_{c,k}}}\left(-\rho\sum_{k=1}^{K}\alpha_{k}\sigma_{k}+(1-\rho)\max_{k\in
K}\alpha_{k}T_{k}\right),\hskip 11.00008pt\eqref{mnbs}$

subject to

$\sum_{k=1}^{K}\sum_{n=1}^{\mathcal{N}_{k}}B_{k,n}\leq B_{e},$ (25a)
$\sum_{k=1}^{K}\alpha_{k}B_{c,k}\leq B_{c},$ (25b)
$\alpha_{k}\in\\{0,1\\},\forall k\in\mathcal{K},$ (25c)

where $B_{c}=B-B_{e}$, $B$ is the total bandwidth,
$\boldsymbol{\alpha}=[\alpha_{1},\alpha_{2},\cdots,\alpha_{K}]^{\mathrm{T}}$,
$\boldsymbol{B_{k,n}}=[B_{k,1},B_{k,2},\cdots,B_{k,N_{k}}]^{\mathrm{T}}$,
$\boldsymbol{B_{c,k}}=[B_{c,1},B_{c,2},\cdots,B_{c,K}]^{\mathrm{T}}$, and
$\rho\in[0,1]$ is the weight factor that controls the tradeoff between data
importance and transmission latency.

Obviously, (III) is a MINLP problem, which is NP-hard. In the following, we
would introduce an ADMM-BCU method to find the joint edge node selection and
resource allocation strategy.

##  IV Joint Edge Node Selection and Resource Allocation

As known to us, all of the steps in the learning procedure are independent
with the optimal scheduling decision. Denote
$T_{k}^{{}^{\prime}}=\max_{n\in\mathcal{N}_{k}}(T_{k,n}^{c}+T_{k,n}^{u})+T_{c,k}^{u}$,
the original problem (III) can be rewritten as

$\min_{\boldsymbol{\alpha,B_{k,n},B_{c,k}}}\max_{k\in
K}~{}\left\\{-\rho\sum_{k=1}^{K}\alpha_{k}\sigma_{k}+(1-\rho)\alpha_{k}T_{k}^{{}^{\prime}}\right\\},$
(26)

subject to (25a), (25b), and (25c).

To solve the above min-max problem, we first denote
$X=\max_{n\in\mathcal{N}_{k}}\left(T_{k,n}^{c}+T_{k,n}^{u}\right)$ and
$Y=\max_{k\in
K}(-\rho\sum_{k=1}^{K}\alpha_{k}\sigma_{k}+(1-\rho)\alpha_{k}(X+T_{c,k}^{u}))$.
By applying the parametric method [29], (26) can be transformed into

$\min_{{\boldsymbol{\alpha,B_{k,n},B_{c,k}}},X,Y}Y,$ (27) subject to (25a),
(25b), (25c), and $T_{k,n}^{c}+T_{k,n}^{u}\leq X,$ (27a)
$-\rho\sum_{k=1}^{K}\alpha_{k}\sigma_{k}+(1-\rho)\alpha_{k}(X+T_{c,k}^{u})\leq
Y.$ (27b)

Nevertheless, (27) is still a unsolvable MINLP problem. We then introduce an
auxiliary variable $\boldsymbol{\tilde{\alpha}}$ to deal with the binary
vector $\boldsymbol{\alpha}$ [30]. Thereafter, (27) can be reformulated as

$\min_{{\boldsymbol{\alpha,B_{k,n},B_{c,k}}},X,Y}Y,$ (28) subject to (25a),
(25b), (25c), (27a), (27b), and
$\boldsymbol{\alpha}-\boldsymbol{\tilde{\alpha}}=\boldsymbol{0},$ (28a)
$\alpha_{k}(1-\tilde{\alpha}_{k})=0,$ (28b) $0\leq\alpha_{k}\leq 1.$ (28c)

Problem (26) is transferred as a convex problem with equality constraints now.
Hereinafter, we introduce the Augmented Lagrangian (AL) method to solve
problem (28) through penalizing and dualizing the equality constraints (28a)
and (28b) as

$\min_{\boldsymbol{\alpha},\boldsymbol{\tilde{\alpha}},\boldsymbol{B_{k,n}},\boldsymbol{B_{c,k}},X,Y}Y+\frac{1}{2\nu}\sum_{k=1}^{K}\left[\alpha_{k}\left(1-\tilde{\alpha}_{k}\right)+\nu\lambda_{k}\right]^{2}\\\
+\frac{1}{2\nu}\sum_{k=1}^{K}\left(\alpha_{k}-\tilde{\alpha}_{k}+\nu\tilde{\lambda}_{k}\right)^{2},$
(29)

subject to (25a), (25b), (27a), (27b), and (28c).

Here, $\nu$ is the non-negative penalty parameter, and
$\boldsymbol{\lambda}=[\lambda_{1},\lambda_{2},...,\lambda_{K}]$ and
$\boldsymbol{\widetilde{\lambda}}=[\widetilde{\lambda}_{1},\widetilde{\lambda}_{2},...,\widetilde{\lambda}_{K}]$
denote the dual variable vectors correspond to constraints (28a) and (28b),
respectively.

Nevertheless, (29) is still a coupled problem due to the multiply variables of
$Y$, $\boldsymbol{B_{k,n}}$, and $\boldsymbol{B_{c,k}}$. Therefore, in this
work, we propose a distributed ADMM-BCU algorithm that can iteratively
approach a near optimal stable solution with low computational complexity.
Specifically, during each iteration, (29) is decomposed into edge node
selection and resource allocation subproblems, which aim to solve the blocks
of $\\{\boldsymbol{\alpha},X,Y\\}$ and
$\\{\boldsymbol{\tilde{\alpha}},\boldsymbol{B_{k,n}},\boldsymbol{B_{c,k}}\\}$,
respectively.

##### The optimal edge node selection $\\{\boldsymbol{\alpha},X,Y\\}$

Under the fixed resource allocations block
$\left\\{\boldsymbol{\widetilde{\alpha}},\boldsymbol{B_{k,n}},\boldsymbol{B_{c,k}}\right\\}$,
the edge node selection optimization subproblem over the variable block
$\\{\boldsymbol{\alpha},X,Y\\}$ can be rewritten as

$\min_{\boldsymbol{\alpha},X,Y}Y+\frac{1}{2\nu}\sum_{k=1}^{K}\left[\alpha_{k}\left(1-\tilde{\alpha}_{k}\right)+\nu\lambda_{k}\right]^{2}\\\
+\frac{1}{2\nu}\sum_{k=1}^{K}\left(\alpha_{k}-\tilde{\alpha}_{k}+\nu\tilde{\lambda}_{k}\right)^{2},$
(30)

subject to (25b), (27a), (27b), and (28c).

Obviously, (30) is a convex problem, which can be solved by standard tools,
such as CVX.

In what follows, we provide the closed form expression of the optimal edge
node selection by introducing Lemma 1.

######  Lemma 1.

The optimum edge node selection $\boldsymbol{\alpha^{*}}$ can be expressed as

$\alpha_{k}^{*}=\frac{\nu
M_{k}+\nu\kappa_{k}\left(\rho\sigma_{k}-\left(1-\rho\right)\left(X+T_{c,k}^{u}\right)\right)}{\left(1-\tilde{\alpha}_{k}\right)^{2}+1},\forall
k\in\mathcal{K},$ (31)

where
$M_{k}=\psi_{k}-\xi_{k}-\lambda_{k}\left(1-\tilde{\alpha}_{k}\right)-\tilde{\lambda}_{k}-\phi_{k}B_{c,k}+\frac{1}{\nu}\tilde{\alpha}_{k}$.
$\boldsymbol{\phi}$, $\boldsymbol{\zeta}$, $\boldsymbol{\kappa}$,
$\boldsymbol{\psi}$, and $\boldsymbol{\xi}$ are Lagrangian multipliers
correspond to constraints (25b), (27a), (27b), and (28c), which can be found
by one-dimensional search methods based on the complementary slackness
conditions.

###### Proof.

To find the optimal $\alpha_{k},\forall k\in\mathcal{K}$, we apply for the
Lagrangian dual method, which can rearrange Eq. (30) with respect to
$\alpha_{k},\forall k\in\mathcal{K}$ as

$\small{\begin{split}&L(\boldsymbol{\alpha_{k}},\boldsymbol{\phi_{k}},\boldsymbol{\zeta_{k}},\boldsymbol{\kappa_{k}},\boldsymbol{\psi_{k}})\\\
&=\frac{Y}{\sum_{k=1}^{K}\sigma_{k}\alpha_{k}}+\frac{1}{2\nu}\sum_{k=1}^{K}\left[\alpha_{k}\left(1-\tilde{\alpha}_{k}\right)+\nu\lambda_{k}\right]^{2}\\\
&+\frac{1}{2\nu}\sum_{k=1}^{K}\left(\alpha_{k}-\tilde{\alpha}_{k}+\nu\tilde{\lambda}_{k}\right)^{2}\\\
&+\phi_{k}\left(\sum_{k=1}^{K}\alpha_{k}B_{c,k}-B_{c}\right)-\psi_{k}\alpha_{k}-\xi_{k}\left(1-\alpha_{k}\right)\\\
&+\zeta_{k}\left(T_{k,n}^{c}+T_{k,n}^{u}-X\right)\\\
&+\kappa_{k}\left(-\rho\sum_{k=1}^{K}\alpha_{k}\sigma_{k}+\left(1-\rho\right)\alpha_{k}\left(X+T_{c,k}^{u}\right)-Y\right).\end{split}}$
(32)

Calculate the first-order partial derivatives with respect to
$\alpha_{k},\forall k\in\mathcal{K}$, we derive that

$\begin{split}&\frac{\partial
L(\alpha_{k},\phi_{k},\zeta_{k},\kappa_{k},\psi_{k})}{\partial\alpha_{k}}\\\
&=\frac{1}{\nu}\left[\alpha_{k}\left(1-\tilde{\alpha}_{k}\right)^{2}+\alpha_{k}-\tilde{\alpha}_{k}\right]+\lambda_{k}\left(1-\tilde{\alpha}_{k}\right)\\\
&+\tilde{\lambda}_{k}+\phi_{k}B_{c,k}-\psi_{k}+\xi_{k}\\\
&+\kappa_{k}\left(-\rho\sigma_{k}+(1-\rho)\left[X+T_{c,k}^{u}\right]\right),\forall
k\in\mathcal{K}.\end{split}$ (33)

Setting $\frac{\partial
L(\alpha_{k},\mu_{k},\phi_{k},\zeta_{k})}{\partial\alpha_{k}}=0$, we have

$\alpha_{k}^{*}=\frac{\nu
M_{k}+\nu\kappa_{k}\left(\rho\sigma_{k}-(1-\rho)(X+T_{c,k}^{u})\right)}{\left(1-\tilde{\alpha}_{k}\right)^{2}+1},$
(34)

where
$M_{k}=\psi_{k}-\xi_{k}-\lambda_{k}\left(1-\tilde{\alpha}_{k}\right)-\tilde{\lambda}_{k}-\phi_{k}B_{c,k}+\frac{1}{\nu}\tilde{\alpha}_{k}$.
This ends the proof. ∎

From Lemma 1, we find that the edge node selection is mainly determined by the
edge node importance $\sigma_{k}$ and the uplink transmission latency from
edge $k$ to the cloud $T^{u}_{c,k}$. Intuitively, the cloud preferentially
selects the edge node with either a larger edge node importance or a smaller
uplink transmission latency that can improve the communication-efficiency.

##### The optimal bandwidth allocation
$\left\\{\boldsymbol{\widetilde{\alpha}},\boldsymbol{B_{k,n}},\boldsymbol{B_{c,k}}\right\\}$

Similarly, under the fixed edge node selection block
$\left\\{\boldsymbol{\alpha},X,Y\right\\}$, the resource allocation
optimization subproblem over the block
$\left\\{\boldsymbol{\widetilde{\alpha}},\boldsymbol{B_{k,n}},\boldsymbol{B_{c,k}}\right\\}$
can be rearranged as

$\min_{\boldsymbol{\widetilde{\alpha}},\boldsymbol{B_{k,n}},\boldsymbol{B_{c,k}}}\frac{1}{2\nu}\sum_{k=1}^{K}\left[\alpha_{k}\left(1-\tilde{\alpha}_{k}\right)+\nu\lambda_{k}\right]^{2}\\\
+\frac{1}{2\nu}\sum_{k=1}^{K}\left(\alpha_{k}-\tilde{\alpha}_{k}+\nu\tilde{\lambda}_{k}\right)^{2},$
subject to (25a), (25b), and
$\frac{Z}{B_{k,n}\log_{2}\left(1+\frac{P_{k,n}g_{k,n}}{B_{k,n}N_{0}}\right)}+\dfrac{C_{k,n}N_{k,n}}{f_{k,n}}\leq
X,$ (34a) $-\rho\sum_{k=1}^{K}\alpha_{k}\sigma_{k}+(1-\rho)\alpha_{k}\cdot\\\
\left(X+\frac{Z}{B_{c,k}\log_{2}\left(1+\frac{P_{c,k}g_{c,k}}{B_{c,k}N_{0}}\right)}\right)\leq
Y.$ (34b)

Also, it is easy to observe that (34) is a convex problem. For ease of
analyses, we write this problem under the Lagrangian dual formulation, where
(34) can be rearranged as

$\small{\begin{split}&L\left(\boldsymbol{\tilde{\alpha}_{k}},\boldsymbol{B_{k,n}},\boldsymbol{B_{c,k}},\boldsymbol{\beta_{k}},\boldsymbol{\varkappa_{k}},\boldsymbol{\varphi_{k}},\boldsymbol{\tau_{k}}\right)\\\
&=Y+\frac{1}{2\nu}\sum_{k=1}^{K}\left[\alpha_{k}\left(1-\tilde{\alpha}_{k}\right)+\nu\lambda_{k}\right]^{2}\\\
&+\frac{1}{2\nu}\sum_{k=1}^{K}\left(\alpha_{k}-\tilde{\alpha}_{k}+\nu\tilde{\lambda}_{k}\right)^{2}\\\
&+\beta_{k}\left(\sum_{k=1}^{K}\sum_{n=1}^{\mathcal{N}_{k}}B_{k,n}-B_{e}\right)+\varkappa_{k}\left(\sum_{k=1}^{K}\alpha_{k}B_{c,k}-B_{c}\right)\\\
&+\varphi_{k}\left[\frac{Z}{B_{k,n}\log_{2}\left(1+\frac{P_{k,n}g_{k,n}}{B_{k,n}N_{0}}\right)}+\dfrac{C_{k,n}N_{k,n}}{f_{k,n}}-X\right]\\\
&+\tau_{k}\left(-\rho\sum_{k=1}^{K}\alpha_{k}\sigma_{k}\right.\\\
&\left.+(1-\rho)\alpha_{k}\left[X+\frac{Z}{B_{c,k}\log_{2}\left(1+\frac{P_{c,k}g_{c,k}}{B_{c,k}N_{0}}\right)}\right]-Y\right),\end{split}}$
(35)

where $\boldsymbol{\beta}$, $\boldsymbol{\varkappa}$, $\boldsymbol{\varphi}$,
and $\boldsymbol{\tau}$ are the Lagrangian multipliers corresponding to
constraints (25a), (25b), (34a), and (34b), respectively.

By taking $\frac{\partial
L(\tilde{\alpha}_{k},B_{k,n},B_{c,k},\beta_{k},\varkappa_{k},\varphi_{k},\tau_{k})}{\partial
B_{k,n}}=0$, the optimal local-edge uplink bandwidth allocation $B_{k,n}^{*}$
can be derived by

$\varphi_{k}\frac{Z}{\left(B_{k,n}^{*}\log_{2}\left(1+\frac{P_{k,n}g_{k,n}}{B_{k,n}^{*}N_{0}}\right)\right)^{2}}\left[\log_{2}\left(1+\frac{P_{k,n}g_{k,n}}{B_{k,n}^{*}N_{0}}\right)-\right.\\\
\left.\frac{P_{k,n}g_{k,n}}{\left(P_{k,n}g_{k,n}+B_{k,n}^{*}N_{0}\right)\ln
2}\right]=\beta_{k},~{}\forall k\in\mathcal{K},n\in\mathcal{N}_{k}.$ (36)

From Eq. (36), the optimal local-edge bandwidth allocation $B_{k,n}^{*}$ is
mainly influenced by the related channel conditions
$\frac{P_{k,n}g_{k,n}}{N_{0}}$.

Algorithm 1 Joint edge node selection and resource allocation strategy.

1: Initialize the gradient norm value $\sigma_{k},\forall k\in\mathcal{K}$.

2: Set the minimum successive divergence threshold of the objective function
$\epsilon^{\min}$ and the maximum iteration number $R^{\mathrm{max}}$.

3: Set the iteration number $\jmath=0$.

4: Initialize the auxiliary variables
$\boldsymbol{\tilde{\alpha}^{(\jmath)}}$, $\boldsymbol{\lambda^{(\jmath)}}$,
$\boldsymbol{\tilde{\lambda}^{(\jmath)}}$.

5: While $\epsilon^{(\jmath)}\geq\epsilon^{\min}$ and $\jmath\leq
R^{\mathrm{max}}$ do

6: Calculate the optimal device selection decision
$\alpha_{k}^{*(\jmath)}$according to (34).

7: Calculate the optimal bandwidth $B_{k,n}^{*(\jmath)},B_{c,k}^{*(\jmath)}$
according to (36) and (39).

8: Obtain $\tilde{\alpha}_{k}^{*(\jmath)}$ according to (40).

9: Update $\boldsymbol{\lambda^{(\jmath)}}$ and
$\boldsymbol{\tilde{\lambda}^{(\jmath)}}$ according to
$\boldsymbol{\lambda}^{(\jmath)}=\boldsymbol{\lambda}^{(\jmath-1)}+\frac{1}{\nu}\boldsymbol{\alpha}^{*(\jmath-1)}\left(1-\boldsymbol{\tilde{\alpha}}^{*(\jmath-1)}\right),$
(37)
$\boldsymbol{\tilde{\lambda}}^{(\jmath)}=\boldsymbol{\tilde{\lambda}}^{(\jmath-1)}+\frac{1}{\nu}\left(\boldsymbol{\alpha}^{*(\jmath-1)}-\boldsymbol{\tilde{\alpha}}^{*(\jmath-1)}\right).$
(38)

10: Update $\epsilon^{(\jmath)}$ according to (41).

11: Set $\jmath=\jmath+1$.

12: End while

Alternatively, by taking $\frac{\partial
L(\tilde{\alpha}_{k},B_{k,n},B_{c,k},\beta_{k},\varkappa_{k},\varphi_{k},\tau_{k})}{\partial
B_{c,k}}=0$, the optimal edge-cloud uplink bandwidth allocation $B_{c,k}^{*}$
can be obtained by

$\tau_{k}\frac{\left(1-\rho\right)Z}{\left(B_{c,k}^{*}\log_{2}\left(1+\frac{P_{c,k}g_{c,k}}{B_{c,k}^{*}N_{0}}\right)\right)^{2}}\left[\log_{2}\left(1+\frac{P_{c,k}g_{c,k}}{B_{c,k}^{*}N_{0}}\right)-\right.\\\
\left.\frac{P_{c,k}g_{c,k}}{\left(P_{c,k}g_{c,k}+B_{c,k}^{*}N_{0}\right)\ln
2}\right]=\varkappa_{k},~{}\forall k\in\mathcal{K},n\in\mathcal{N}_{k}.$ (39)

Obviously, the optimal edge-cloud uplink bandwidth allocation $B_{c,k}^{*}$
has a similar rule with $B_{k,n}^{*}$.

Thereafter, by setting $\frac{\partial
L(\tilde{\alpha}_{k},B_{k,n},B_{c,k},\beta_{k},\varkappa_{k},\varphi_{k},\tau_{k})}{\tilde{\alpha}_{k}}=0$,
we can obtain the optimal auxiliary variable $\tilde{\alpha}^{*}_{k}$ as

$\tilde{\alpha}_{k}^{*}=\frac{\alpha_{k}\left(1+\alpha_{k}+\nu\lambda_{k}\right)+\nu\tilde{\lambda}_{k}}{1+\alpha_{k}^{2}},~{}\forall
k\in\mathcal{K}.$ (40)

The detailed procedure for the joint edge node selection and resource
allocation scheduling is presented in Algorithm 1.

In Algorithm 1, $\epsilon^{(\jmath)}$ means the successive divergence of the
objective function at the $\jmath$-th iteration [30], which can be defined as

$\epsilon^{(\jmath)}=F^{(\jmath)}-F^{(\jmath-1)},$ (41)

where
$F=Y+\frac{1}{2\nu}\sum_{k=1}^{K}\left[\alpha_{k}\left(1-\tilde{\alpha}_{k}\right)+\nu\lambda_{k}\right]^{2}+\frac{1}{2\nu}\sum_{k=1}^{K}\left(\alpha_{k}-\tilde{\alpha}_{k}+\nu\tilde{\lambda}_{k}\right)^{2}.$

##  V Numerical Results

In this section, we conduct experiments to evaluate the theoretical analyses
and test the performance of the proposed algorithm.

###  V-A Experiment Settings

_CNN model settings:_ For exposition, we consider the learning task of
training image classifiers, which are implemented on a _Convolutional Neural
Network_ (CNN) model, namely VGGNet $16$ [31]. The corresponding training
dataset is CIFAR-10, which contains $50000$ training images and $10000$
testing images with $10$ categories. To simulate the distributions of
heterogeneous data based mobile devices, all data samples are first sorted by
digital labels, and then divided into $100$ shards of size $500$ and each
local device is assigned with $5$ shards. The batch size of each local device
is set as $50$ and the average quantitative bit number of each parameter is
set as $16$ bits. In addition, we adopt the _Stochastic Gradient Descent_
(SGD) optimizer, and the learning rate for the CNN model is set as $0.1$. The
computation frequency of each local device is randomly set between $2$ GHz to
$4$ GHz.

_Wireless communication settings:_ We consider a hierarchical SHFL
communication network consists of one cloud node and $10$ edge nodes. Each
edge node connects with two local devices. Both edge nodes and local devices
are uniformly distributed under the coverage of the cloud node. The total
bandwidth is set as $20$ MHz. Moreover, the uplink transmission powers of each
local device and edge node are set as $10$ dBm and $24$ dBm, respectively.
Also, the downlink transmission powers of each edge node and the cloud node
are set as $10$ dBm and $24$ dBm, respectively. Furthermore, we utilize the
transmission pass loss model of $128.1+37.6\log(d[\mathrm{km}])$. Meanwhile,
the noise power spectral density is set as $N_{0}=-174$ dBm/Hz.

In the ADMM-BCU algorithm, we set the non-negative penalty parameter $\nu$ as
$1$. The minimum successive divergence threshold $\epsilon^{min}$ is set as
$10^{-4}$. In addition, the maximum iteration number of ADMM-BCU algorithm is
set as $200$.

###  V-B SHFL Performance

In this subsection, we present the convergence performance of the proposed
SHFL model. We first introduce the following baselines.

  * •

_Random selection_ : Under this circumstance, CNN is implemented with random
data selection, where both $5$ and $8$ edge nodes randomly selective
conditions are respectively considered.

  * •

_Full selection_ : Under this circumstance, CNN is implemented by selecting
all of the edge nodes.

  * •

_Normal edge update_ : Edge nodes directly use the broadcast cloud model as
their updated model.

(a) Training accuracy.

(b) Training loss.

Figure 4: Convergence performance of the proposed CNN model under different
algorithms.

For simplicity, we assume the transmissions from the selected edge nodes to
the cloud node are uniformly allocated, totally $5$ MHz. Meanwhile, the
transmission bandwidths from local devices to edge nodes are also set as the
same, totally $15$ MHz. Moreover, we set the weighted factor $\rho$ under the
proposed algorithm as $0.8$. Fig. 4 shows the convergence performance of the
proposed CNN based SHFL model. From this figure, we can find that the VGG-16
network starts to converge at about $70$ communication rounds for both the
random selection scheme with $8$ edge nodes, the Full selection scheme, and
the proposed scheme. However, the random selection scheme with $5$ edge nodes
presents the worst convergence performance. Intuitively, it is because the
more devices to be selected, the larger data information can be provided to
the neural network, and thus faster convergence. Moreover, due to the non-iid
datasets, each node has different contributions. Therefore, the random
selection scheme may play a side effect on the whole model, leading to a
decreasing model accuracy. Overall, the proposed algorithm shows a near to the
full selection scheme convergence and accuracy, which can achieve better
performance than the baselines that would be discussed latter.

(a) Training accuracy.

(b) Training loss.

Figure 5: Performance influence from the edge update model.

Fig. 5 presents the performance influence from the edge update model. From
this figure, we find that either the training accuracy or the training loss
under the elastic edge update model is better than that of the normal edge
update model. The fluctuation of these curves are mainly due to the non-iid
data form. Therefore, we can conclude that the elastic edge update model is
significant to keep the personalities of the edge nodes.

###  V-C The Scheduling Performance

In this subsection, we mainly verify the scheduling performance of the
proposed algorithm.

Figure 6: Convergence performance of the proposed ADMM-BCU algorithm.

In Fig. 6, we shows that the proposed ADMM-BCU algorithm has a fast
convergence and a low computational complexity.

Figure 7: Tradeoff between data importance and delay.

Fig. 7 illustrates that a tradeoff exists between data importance and the
transmission latency. The value of $\rho$ starts from $0.4$ to $0.8$ under the
step of $0.05$. This figure shows that a large value of $\rho$ leads to higher
data importance and longer transmission latency, and vise versa. Thus, the
operators can select a suitable value of $\rho$ according to their specific
requirements.

(a) Data importance and the number of selected edge nodes.

(b) Latency.

Figure 8: Performance among the number of selected edge nodes, data
importance, and latency under different mechanisms.

In Fig. 8, we present the performance among the number of selected edge nodes,
data importance, and latency under various weight factors. Fig. 8(a) shows the
number of selected edge nodes and total data importance in different weight
factors under various algorithms. From this subfigure, the number of selected
edge nodes increases with the weight factor $\rho$. When the value of weight
factor $\rho$ is small, i.e., the associated edge nodes are small, the
proposed algorithm has a lower data importance than the random selection
scheme. With the increment of associated edge nodes, the circumstance changes,
which has been explained in Fig. 4. However, the full selection scheme always
has the highest value of data importance at the cost of higher latency, which
is shown in Fig. 8(b). Fig. 8(b) shows the full selection scheme suffers the
highest latency, and the proposed algorithm has the lowest latency after
scheduling. Intuitively, the transmission latency is much lower than the total
latency, which means the data training time is huge. Moreover, the
transmission latency may not meet the requirements of ultra low latency C-ITS
devices. Under this circumstance, we can enlarge the wireless bandwidth by
some resource management technologies.

##  VI Conclusion

This work proposes a novel SHFL framework that consists of local, edge, and
cloud nodes to provide safety, traffic efficiency, and comfortable
infotainment services for C-ITS. Specifically, homogeneous devices are allowed
to associate with one edge node. Therefore, we adopt the synchronous
aggregation model for edge nodes. On the contrary, for the heterogeneous edge
aggregation models, we introduce a semi-asynchronous aggregation model for the
cloud node, where parts of the fastest training edge models can be uploaded at
each iteration. Moreover, we investigate an effective edge-cloud update method
to keep the personalities of the edge nodes. For communication-efficiency, we
propose a joint edge node association and resource allocation strategy, which
illustrates a tradeoff between training accuracy and transmission latency. A
distributed ADMM-BCU algorithm has been adopted to solve the proposed optimal
MINLP problem. Numerical results show that our proposed scheme can accelerate
the training process and improve the performance for C-ITS.

## References

  * [1] L. Chen and C. Englund, “Cooperative ITS, EU standards to accelerate cooperative mobility,” in Proc. _Int. Conf. Connected Vehicles Expo (ICCVE)_ , pp. 681 - 686, Nov. 2014.
  * [2] F. Qu, F.-Y. Wang, and L. Yang, “Intelligent transportation spaces: Vehicles, traffic, communications, and beyond,” _IEEE Commun. Mag._ , vol. 48, no. 11, pp. 136 ¨C 142, Nov. 2010.
  * [3] E. B. Hamida, H. Noura, and W. Znaidi, “Security of cooperative intelligent transport systems: Standards, threats analysis and cryptographic countermeasures,” _Electronics_ , vol. 4, no. 3, pp. 380 - 423, 2015.
  * [4] M. Autili, L. Chen, C. Englund, C. Pompilio, and M. Tivoli, “Cooperative intelligent transport systems: Choreography-based urban traffic coordination,” _IEEE Trans. Intell. Transp. Syst._ , vol. 22, no. 4, pp. 2088 - 2099, Apr. 2021.
  * [5] H. Qiu, Q. Zheng, M. Msahli, G. Memmi, M. Qiu, and J. Lu, “Topological Graph Convolutional Network-Based Urban Traffic Flow and Density Prediction,” _IEEE Trans. on Intel. Trans. Sys._ , 2020
  * [6] M. A. Javed and E. B. Hamida, “On the interrelation of security, QoS, and safety in cooperative ITS,” _IEEE Trans. Intell. Transp. Syst._ , vol. 18, no. 7, pp. 1943 - 1957, Jul. 2017.
  * [7] K. Zhang, A. Yang, H. Su, ”. L. Fortelle, K. Miao, and Y. Yao, “Service-oriented cooperation models and mechanisms for heterogeneous driverless vehicles at continuous static criticlal sections,” _IEEE Trans. Intell. Transp. Syst._ , vol. 18, no. 7, pp. 1867 - 1881, Jul. 2017.
  * [8] J. Zhang, F.-Y. Wang, K. Wang, W.-H. Lin, X. Xu, and C. Chen, “Data-driven intelligent transportation systems: A survey,” _IEEE Trans. Intell. Transp. Syst._ , vol. 12, no. 4, pp. 1624 - 1639, Dec. 2011.
  * [9] V. Smith, C.-K. Chiang, M. Sanjabi, and A. S. Talwalkar, “Federated multi-task learning,” in _Proc. Adv. Neural Inf. Process. Syst._ , pp. 4424 - 4434, 2017.
  * [10] H. H. Yang, Z. Liu, T. Q. S. Quek, and H. V. Poor, “Scheduling policies for federated learning in wireless networks,” _IEEE Trans. Commun._ , vol. 68, no. 1, pp. 317 - 333, Jan. 2020.
  * [11] J.-H. Ahn, O. Simeone, and J. Kang, “Wireless federated distillation for distributed edge learning with heterogeneous data,” _arXiv preprint arXiv:1907.02745_ , 2019.
  * [12] J. Kone$\breve{c}$n$\grave{y}$, H. B. McMahan, D. Ramage, and P. Richt$\acute{a}$rik, “Federated optimization: Distributed machine learning for on-device intelligence,” _arXiv preprint arXiv:1610.02527_ , 2016.
  * [13] H. B. McMahan, E. Moore,D. Ramage, S. Hampson, and B. Y. A. Arcas, “Communication-efficient learning of deep networks from decentralized data,” _Artif. Intell. Statist._ , vol. 54, pp. 1273 - 1282, Apr. 2017.
  * [14] A. Hard, K. Rao, R. Mathews, S. Ramaswamy, F. Beaufays, S. Augenstein, H. Eichner, C. Kiddon, and D. Ramage, “Federated learning for mobile keyboard prediction,” _arXiv preprint arXiv:1811.03604_ , 2018.
  * [15] Y. He, J. Ren, G. Yu, and J. Yuan, “Importance-aware data selection and resource allocation in federated edge learning system,” _IEEE Trans. Vel. Technol._ , vol. 69, no. 11, pp. 13593 - 13605, Nov. 2020.
  * [16] S. Wang, T. Tuor, T. Salonidis, K. K. Leung, C. Makaya, T. He, and K. Chan, “Adaptive federated learning in resource constrained edge computing systems,” _IEEE J. Sel. Areas Commun._ , vol. 37, no. 6, pp. 1205 - 1221, Jun. 2019.
  * [17] S. Luo, X. Chen, Q. Wu, Z. Zhou, and S. Yu, “HFEL: Joint edge association and resource allocation for cost-efficient hierarchical federated edge learning,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 10, pp. 6535 - 6548, Oct. 2020.
  * [18] W. Y. B. Lim, N. C. Luong, D. T. Hoang, Y. Jiao, Y.-C. Liang, Q. Yang, D. Niyato, and C. Miao, “Federated learning in mobile edge networks: A comprehensive survey,” _IEEE Commun. Surveys Tuts._ , vol. 22, no. 3, pp. 2031 - 2063, Third Quarter, 2020.
  * [19] W. Shi, J. Cao, Q. Zhang, Y. Li, and L. Xu, “Edge computing: Vision and challenges,” _IEEE Internet Things J._ , vol. 3, no. 5, pp. 637 - 646, Oct. 2016.
  * [20] S. Itahara, T. Nishio, Y. Koda, M. Morikura, and K. Yamamoto, “Distillation-based semi-supervised federated learning for communication-efficient collaborative training with non-IID private data,” _IEEE Trans. Mobile Comput._ , early access, 2021.
  * [21] T.-C. Chiu, Y.-Y. Shih, A.-C. Pang, C.-S. Wang, W. Weng, and C.-T. Chou, “Semisupervised distributed learning with non-IID data for AIoT service platform,” _IEEE Inter. Things J._ , vol. 7, no. 10, pp. 9266 - 9277, Oct. 2020.
  * [22] Z. Yang, M. Chen, W. Saad, C. S. Hong, and M. Shikh-Bahaei, “Energy efficient federated learning over wireless communication networks,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 3, pp. 1938 - 1949, March 2021.
  * [23] Y. Gao, L. Liu, X. Zheng, C. Zhang, and H. Ma, “Federated sensing: Edge-cloud elastic collaborative learning for intelligent sensing,” _IEEE Inter. Things J._ , early access, 2021.
  * [24] C. Xie, S. Koyejo, and I. Gupta, “Asynchronous federated optimization,” _arXiv preprint arXiv:1903.03934_ , 2019.
  * [25] M. Chen, B. Mao, and T. Ma, “Efficient and robust asynchronous federated learning with stragglers,” in _Submitted to International Conference on Learning Representations_ , 2019.
  * [26] Y. Chen, Y. Ning, M. Slawski, and H. Rangwala, “Asynchronous online federated learning for edge devices with non-IID data,” in _Proc. IEEE Inter. Confer. Big Data_ , 2020.
  * [27] M. R. Sprague, A. Jalalirad, M. Scavuzzo, C. Capota, M. Neun, L. Do, and M. Kopp, “Asynchronous federated learning for geospatial applications,” in _Joint European Conference on Machine Learning and Knowledge Discovery in Databases,_ Springer, pp. 21 - 28, 2018.
  * [28] Y. Zhao, M. Li, N. Lai, N. Suda, D. Civin, and V. Chandra, “Federated learning with non-iid data.” _arXiv preprint arXiv:1806.00582_ , 2018.
  * [29] S. Geisser and W. M. Johnson, _Modes of Parametric Statistical Inference_. Hoboken, NJ, USA: Wiley, 2006.
  * [30] R. Guo, Y. Cai, M. Zhao, Q. Shi, and B. Champagne, “Joint design of beam selection and precoding matrices for mmWave MU-MIMO systems relying on lens antenna arrays,” _IEEE J. Sel. Topics Signal Process._ , vol. 12, no. 2, May 2018.
  * [31] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _arXiv preprint arXiv: 1409.1556_ , 2015.