# Mobility, Communication and Computation Aware Federated Learning for
Internet of Vehicles

Md Ferdous Pervej13, Jianlin Guo1, Kyeong Jin Kim1, Kieran Parsons1, Philip
Orlik1, Stefano Di Cairano1, Marcel Menner1, Karl Berntorp1, Yukimasa Nagai2,
and Huaiyu Dai3
This work was done while Md Ferdous Pervej was working at MERL as an intern.
1Mitsubishi Electric Research Laboratories (MERL), Cambridge, MA, 02139 USA
1Emails:{guo, kkim, parsons, porlik, dicairano, menner<EMAIL_ADDRESS>2Information Technology R&D Center, Mitsubishi Electric Corporation, Kamakura,
Kanagawa 2478501 Japan
3Department of Electrical and Computer Engineering, NC State University,
Raleigh, NC, 27606 USA
3Emails:{mpervej<EMAIL_ADDRESS>
###### Abstract

While privacy concerns entice connected and automated vehicles to incorporate
on-board federated learning (FL) solutions, an integrated vehicle-to-
everything communication with heterogeneous computation power aware learning
platform is urgently necessary to make it a reality. Motivated by this, we
propose a novel mobility, communication and computation aware online FL
platform that uses on-road vehicles as learning agents. Thanks to the advanced
features of modern vehicles, the on-board sensors can collect data as vehicles
travel along their trajectories, while the on-board processors can train
machine learning models using the collected data. To take the high mobility of
vehicles into account, we consider the delay as a learning parameter and
restrict it to be less than a tolerable threshold. To satisfy this threshold,
the central server accepts partially trained models, the distributed roadside
units (a) perform downlink multicast beamforming to minimize global model
distribution delay and (b) allocate optimal uplink radio resources to minimize
local model offloading delay, and the vehicle agents conduct heterogeneous
local model training. Using real-world vehicle trace datasets, we validate our
FL solutions. Simulation shows that the proposed integrated FL platform is
robust and outperforms baseline models. With reasonable local training
episodes, it can effectively satisfy all constraints and deliver near ground
truth multi-horizon velocity and vehicle-specific power predictions.

###### Index Terms:

Internet of Vehicle, online federated learning, vehicle-to-everything
communication, on-board computation power, vehicle-specific power prediction.

##  I Introduction

Modern vehicles are packed with various on-board sensors to sense diversified
data for fulfilling higher automation levels. On the one hand, data collected
by individual vehicles may be non-IID (independent and identically
distributed) and contain imperfections. Training independent a machine
learning (ML) model solely on individual vehicle data may lead to non-
robustness and prediction may not suffice in some strict applications. In
addition, these independently sensed data may not avail predicting a global
task with data features/labels that these sensors have never observed.
Moreover, some vehicles may not have enough computation resources to train a
ML model. On the other hand, sending raw data to a central server is
impractical due to extensive privacy threats and enormous communication
overhead. As such, the above-mentioned issues may very well become a
bottleneck to delivering Society of Automotive Engineers (SAE) level $5$
automation on the road.

The recent advances of privacy-preserving federated learning (FL) can bring a
solution to this bottleneck. FL is an advanced ML technique that allows
training ML models locally based on the trainer’s local data [1]. Therefore,
it ensures user privacy protection and can effectively reduce communication
overhead. Most importantly, FL incorporates data features in collaborative
heterogeneous datasets, which allows robust traffic model training by
eliminating data imperfection contained in individual dataset and ensures a
faster convergence [2]. Robust traffic models can be distributed to vehicles
for their tasks at any location on the road. Leveraging FL, modern vehicles
can not only train traffic models on their on-board processors but can also
keep their privacy unscathed. Therefore, an integrated FL platform with
advanced vehicle-to-everything (V2X) communications seems inevitable for the
rising demand on higher automation.

While FL with vehicular agents can indeed bring manifold benefits, it also
raises new challenges. Perhaps, two major concerns are ($1$) the high mobility
and ($2$) the delay. Due to the high mobility, the time a vehicle connects to
a connection point may be short. As a result, the long delay may not be
acceptable. Even though FL can comparatively reduce the communication
overhead, model distribution (delivering the global model from the central
server to vehicles) and offloading (uploading local models from vehicles to
the central server) still take time. Furthermore, local model training using
the limited computation power of the on-board processors may be a dominant
contributor to the delay. As such, hyper-parameters such as total global
training rounds (communication rounds between the central server and the
vehicle agents), total local training iterations and vehicle agent selection
are critical for swift model training and updating.

Advanced ML techniques are widely used to predict different traffic metrics
such as traffic flow, congestion and trajectory. Nevertheless, independent
learning may not be sufficient due to data heterogeneity, non-IID distribution
and imperfection. Therefore, crowdsource-based collaborative learning such as
FL is luring researchers with its ensured privacy protection and use of
diverse data samples of the learning agents. There exist works that leverage
FL in the vehicular environment. However, most of these studies have either
not considered the mobility of the vehicles or not examined the underlying
heterogeneous computation power of the vehicles. In [3], the authors proposed
a multi-task FL framework to predict average vehicle velocity using stationary
agents. The authors in [4] presented a FL based vehicular cooperative
positioning architecture to model the location of the vehicles. However, this
work does not consider model training delay and computation power
heterogeneity. In addition, the authors assumed data augmentation over
vehicle-to-vehicle (V2V) links that raises privacy concerns and communication
overhead. an agent-edge-cloud FL architecture was proposed in [5] where the
authors intended to minimize communication latency between the server and the
agent. However, the agents are stationary. A similar architecture was used in
[6] for object detection in vehicular environment with a focus on context
instead of mobility, communication and computation.

In a vehicular environment, mobility is a nagging concern. This work proposes
an integrated vehicular FL platform that takes account of the vehicle mobility
and the distributed model training delay including local model training time,
model queuing time, and model transmission time.

The contributions of this paper are summarized as follows:

  * •

We present a two-tier hierarchical vehicular FL platform to address the short
connectivity issue caused by the high mobility to achieve robust traffic model
training.

  * •

Instead of applying traditional FedAvg based FL, we propose an extended
FedProx [7] based FL to incorporate vehicle mobility, communication delay,
queuing delay, model training delay and data heterogeneity.

  * •

In our FL platform, each roadside unit (RSU) solves a multicast beamforming
problem to maximize the minimum sum rate of its associated agents for model
distribution in the downlink. To offload the locally trained models, each RSU
solves a complex combinatorial problem to allocate optimal radio resources to
the learning agents.

  * •

While any ML task can be solved using the proposed online FL platform, we
predict multi-horizon vehicle-specific power (VSP) requirements by exploiting
real-world vehicle datasets.

TABLE I: Table of Notation Symbol |  Description
---|---
$v$ |  A vehicle in the on-road vehicle set $\mathcal{V}$
$b$ |  A roadside unit (RSU) in the distributed RSU set $\mathcal{B}$
$\mathcal{V}_{k}$ |  Set of vehicles selected as learning agents in global training round $k$
$\mathcal{V}_{k}^{b}$ |  Set of vehicle agents associated with $b$ in global training round $k$
$\mathbf{w}$ |  Global machine learning model
$\mathbf{w}_{k}$ |  Global machine learning model in global training round $k$
$\mathbf{w}_{k}^{v}$ |  Machine learning model trained by vehicle $v$ in global round $k$
$\Omega$ |  Total network bandwidth for vehicle-RSU communications
$\Omega_{b}$ |  Network bandwidth assigned to RSU b
$z_{b}$ |  A physical resource block (pRB) in RSU $b$’s pRB set $\mathcal{Z}_{b}$
$\mathbf{g}_{b}^{z}$ |  Downlink multicast beamforming vector of RSU b over pRB $z_{b}$
$\Gamma_{b,z}^{v,\text{dn}}$ |  Downlink SNR from RSU $b$ to vehicle $v$ over pRB $z_{b}$
$\Gamma_{b,z}^{v,\text{up}}$ |  Uplink SNR from vehicle $v$ to RSU $b$ over pRB $z_{b}$
$d_{v}^{\text{down}}$ |  Downlink global model transmission time from RSU $b$ to vehicle $v$
$d_{v}^{\text{up}}$ |  Uplink local model transmission time from vehicle $v$ to RSU $b$
$d_{v}^{\text{cmp}}$ |  The amount of time for vehicle $v$ to locally train model $\mathbf{w}_{k}$
$d_{v}^{q,\text{up}}$ |  Uplink queuing time of model $\mathbf{w}_{k}^{v}$ at vehicle $v$
$\mathrm{d}_{v}^{\text{tot}}$ |  Total delay at vehicle $v$ in a global training round
$\mathrm{d}^{\text{thr}}$ |  Time threshold for all $v\in\mathcal{V}_{k}$ to finish model training & offloading
$L_{v}$ |  Number of local training iterations at vehicle $v$
$\mathcal{D}_{k}^{v}$ |  Dataset of vehicle $v$ in a global training round $k$
$P_{b}^{z}$ |  RSU $b$’s transmission power over pRB $z_{b}$, $P_{b}^{z}\leq P_{b}^{max}$
$P_{v}^{b}$ |  Uplink transmission power from vehicle $v$ to RSU $b$

##  II System Model and V2X Communication Model

This section presents our system model and communication model for the
proposed vehicular FL platform. Moreover, for convenience, we summarize the
main notations in Table I.

###  II-A System Model

We consider an Internet of Vehicles (IoV) network that consists of a central
server, multiple RSUs and on-road vehicles. Denote the vehicle set and the RSU
set by $\mathcal{V}=\\{v\\}_{v=1}^{V}$ and $\mathcal{B}=\\{b\\}_{b=1}^{B}$,
respectively. At a high level, the learning process progresses as follows. The
central server first selects vehicular agents and distributes a global model,
parameterized by its weight $\mathbf{w}$, to the selected agents. Upon
receiving $\mathbf{w}$, an agent $v$ trains the received model using its local
data and offloads the updated model weight $\mathbf{w}^{(v)}$ to the server.
The server then aggregates all agents’ local models to build an updated global
model. The training process continues until the global model reaches an
expected level of precision.

To perform such distributed model training in a vehicular environment, an
efficient infrastructure-to-vehicle (I$2$V) / vehicle-to-infrastructure
(V$2$I) communication platform is needed for sharing $\mathbf{w}$ and
$\mathbf{w}^{(v)}$s. Therefore, this paper proposes a practical dense
heterogeneous network architecture, where multiple RSUs are deployed over a
considered region of interest (RoI) to connect vehicles as shown in Fig. 1.
These RSUs are connected to a central server with high-speed wired links. The
vehicles travel on the roads and get connected to the network via these RSUs.

The motivation for adopting the proposed system model is three-folded.
Firstly, the on-board computation power of vehicles is limited. It may take a
longer time to finish the local model training. Therefore, vehicles may move
away from the RSUs used by the server to distribute the global model. The
proposed system model enables vehicles to be reasonably covered by other RSUs
while they perform local model training. Secondly, it aims to ensure V$2$I
connectivity in a larger area for comprehensive observations and robust model
training. Thirdly, it may not be possible to establish direct vehicle-server
communication links due to the limited coverage of wireless links. The
proposed system model can not only ensure vehicle data privacy, but also allow
the vehicles to accumulate more observations. It can be used by various
vehicular applications that highly rely on observations of vehicles, e.g., the
adaptive advanced driver-assistance systems (ADAS).

Figure 1: Proposed vehicular FL architecture for IoV

###  II-B Vehicular Communication Model

Note that due to high-speed wired links between the server and RSUs, the
server-RSU communication delay is negligible and ignored in this work. Assume
each vehicle has a single antenna, while each RSU has $n_{b}$ antennas.
Besides, let the total network bandwidth, for the vehicle-RSU communications,
be $\Omega$ Hz. We divide the radio resource to the RSUs so that the
consecutive RSUs have independent radio resources. We reuse the radio
resources by ensuring that the reusing RSUs are far away from each other.
Furthermore, we divide the allocated radio resources into orthogonal physical
resource blocks (pRBs). Without loss of generality, let us denote the radio
resource of RSU $b$ by $\Omega_{b}$ and denote the pRB set of $b$ by
$\mathcal{Z}_{b}=\\{z_{b}\\}_{z_{b}=1}^{Z_{b}}$. We consider that all pRBs
have the same size $\omega$. Moreover, with an appropriate frequency reuse
pattern and orthogonal partitioning of the system bandwidth, there is no
interference in our proposed system model.

The proposed network operates in time division duplex (TDD) mode and can
exploit channel reciprocity. The channel is assumed to be quasi-static block
fading, i.e., it remains static within a block but varies across different
blocks. Denote the wireless channel between the RSU $b$ and vehicle $v$ over
pRB $z_{b}$ by $\mathbf{h}_{b,z}^{v}\in\mathbb{C}^{n_{b}\times 1}$. Moreover,
we model the channel as
$\mathbf{h}_{b,z}^{v}=\sqrt{\zeta}_{b}^{v}\tau_{b}^{v}\breve{\mathbf{h}}_{b,z}^{v}$,
where $\sqrt{\zeta}_{b}^{v}$, $\tau_{b}^{v}$, and
$\breve{\mathbf{h}}_{b,z}^{v}$ are large scale fading, log-Normal shadowing,
and small scale fading channel response, respectively. We model the pathloss
following $3$GPP’s UMa model [8] as suggested in [9]. For simplicity, we
assume RSUs and vehicles can mitigate Doppler spread perfectly.

In 3GPP based IoV network, the RSUs can be connected with each other via the
$\mathrm{X_{n}}$ interface. In addition, we consider a mobility model where
each agent calculates the reference signal received powers (RSRPs) from its
serving RSU and adjacent RSUs. Once the A$3$ event [10] is triggered, the
agent reports the measurements to the serving RSU. The serving RSU then
handovers the agent to the target RSU [10]. As such, an agent is always
associated with only one RSU, i.e., $\sum_{b=1}^{B}a_{v}^{b}=1$, $\forall
v\in\mathcal{V}$, where $a_{v}^{b}\in\\{0,1\\}$ is an indicator function that
takes value $1$ when vehicle $v$ is associated to RSU $b$.

####  II-B1 Downlink Communication Model

In the downlink, as a RSU transmits the same model to all of its associated
agents, each RSU $b$ can multicast the ML model $\mathbf{w}$111In a practical
wireless network, the RSU can use the multicast broadcast single frequency
network (MBSFN) subframe [10] to broadcast the model.. We assume that each RSU
has fixed pRBs to multicast the ML model. Denote the downlink multicast
beamforming vector of RSU $b$ over pRB $z_{b}$ by
$\mathbf{g}_{b}^{z}\in\mathbb{C}^{n_{b}\times 1}$. Then, the received signal
at vehicle $v$ over pRB $z_{b}$ is expressed as

$y_{b,z}^{v,\text{dn}}=\sqrt{P_{b}^{z}}a_{v}^{b}({\mathbf{h}_{b,z}^{v}})^{H}\mathbf{g}_{b}^{z}s_{b}^{v}+\eta,$
(1)

where $P_{b}^{z}$ is $b$’s transmission power over pRB $z_{b}$, $s_{b}^{v}$ is
the unit power transmitted data symbol of $b$ intended for vehicle $v$ and
$({\bm{x}})^{H}$ denotes the conjugate transpose. Moreover, $\eta\sim
CN(0,\sigma^{2})$ is the circularly symmetric complex Gaussian noise with zero
mean and variance $\sigma^{2}$.

To this end, we calculate the downlink signal-to-noise ratio (SNR) over pRB
$z_{b}$ as follows:

$\Gamma_{b,z}^{v,\text{dn}}={P_{b}^{z}|({\mathbf{h}_{b,z}^{v}})^{H}\mathbf{g}_{b}^{z}|^{2}}/{\omega\sigma^{2}}.$
(2)

When the global FL model is distributed from the server, we calculate the
downlink rate at vehicle $v$ from RSU $b$, over all pRB
$z_{b}\in\mathcal{Z}_{b}$, as

$C_{v}^{\text{dn}}=a_{v}^{b}\cdot\omega\sum\nolimits_{z_{b}=1}^{Z_{b}}\mathbb{E}_{\mathbf{h}}\bigl{[}\log_{2}(1+\Gamma_{b,z}^{v,\text{dn}})\bigr{]},$
(3)

where $\mathbb{E}_{\mathbf{h}}[\cdot]$ is the expectation over
$\mathbf{h}_{b,z}^{v}$.

####  II-B2 Uplink Communication Model

For the uplink, assuming linear receiver vector
$\mathbf{g}_{v,z}^{b}\in\mathbb{C}^{1\times n_{b}}$, we calculate the
effective received signal at RSU $b$ from vehicle $v$ as

$y_{v,z}^{b,\text{up}}=\sqrt{P_{v}^{b}}a_{v}^{b}\mathbf{h}_{v,z}^{b}({\mathbf{g}_{v,z}^{b}})^{H}s_{v}^{b}+\eta,$
(4)

where $P_{v}^{b}$ is the uplink transmission power of vehicle $v$ and
$s_{v}^{b}$ is the intended uplink transmitted symbol of $v$. Moreover, this
gives us the following uplink SNR

$\Gamma_{b,z}^{v,\text{up}}={P_{v}^{b}a_{v}^{b}\bigl{|}\mathbf{h}_{v,z}^{b}({\mathbf{g}_{v,z}^{b}})^{H}\bigr{|}^{2}}/{\omega\sigma^{2}}.$
(5)

Similar to $C_{v}^{\text{dn}}$, we calculate the uplink rate as

$C_{v}^{\text{up}}=a_{v}^{b}\cdot\omega\sum\nolimits_{z_{b}=1}^{Z_{b}}\mathbb{I}_{b,z}^{v}\cdot\mathbb{E}_{\mathbf{h}}\bigl{[}\log_{2}(1+\Gamma_{b,z}^{v,\text{up}})\bigr{]},$
(6)

where $\mathbb{I}_{b,z}^{v}\in\\{0,1\\}$ is an indicator function that takes
value $1$ when $z_{b}\in\mathcal{Z}_{b}$ is assigned to vehicle $v$.

##  III Communication and Computation Aware Vehicular FL Problem

We propose vehicular FL by incorporating heterogeneity in mobility,
communication and computation.

###  III-A Vehicular FL

We assume that each vehicle can use its on-board sensors to collect data, such
as longitude, latitude, velocity, acceleration, weather information, etc.,
that can be used for model training. Without loss of generality, denote the
data sensing interval of the vehicles by $\Delta t$. Denote vehicle $v$’s
dataset at time $t$ by
$\mathcal{D}_{v}^{t}=\\{\mathbf{x}_{v}^{i},y_{v}^{i}\\}_{i=1}^{t}$, where
$\mathbf{x}_{v}^{i}$ and $y_{v}^{i}$ are the $i^{\text{th}}$ feature set and
corresponding label, respectively. The entire dataset available at time $t$ is
denoted as $\mathcal{D}^{t}=\bigcup_{v=1}^{V}\mathcal{D}_{v}^{t}$. As the
central server does not have access to dataset $\mathcal{D}^{t}$, it aims to
solve following optimization problem [1]:

$\displaystyle\underset{\mathbf{w}}{\text{minimize }}$ $\displaystyle
f(\mathbf{w})=\mathbb{E}_{v}\left[f_{v}(\mathbf{w})\right]=\sum\nolimits_{v=1}^{V}p_{v}\cdot
f_{v}(\mathbf{w}),$ (7) s. t.
$\displaystyle\quad\mathbf{w}^{(1)}=\mathbf{w}^{(2)},\dots,=\mathbf{w}^{(V)}=\mathbf{w},$

where $\mathbf{w}$ is the global model parameters,
$f_{v}(\mathbf{w})\coloneqq\mathbb{E}_{\\{\mathbf{x}_{v}^{i},y_{v}^{i}\\}\sim\mathcal{D}_{v}^{t}}\left[f_{v}(\\{\mathbf{x}_{v}^{i},y_{v}^{i}\\},\mathbf{w})\right]$
is the local empirical risk function for agent $v$ and $\sum_{v=1}^{V}p_{v}=1$
with the probability $p_{k}={|\mathcal{D}_{v}^{t}|}/{|\mathcal{D}^{t}|}$.

This FedAvg based FL works well when the agents have IID data distribution and
homogeneous computation power. In our case, the vehicle agents have (a)
diverse on-board sensors that lead to non-IID data distribution and (b)
heterogeneous on-board computation power. Therefore, FedProx based FL [7] is
more suitable because it is designed for agents with different computation
resources. Accordingly, FedProx accepts partial works of the stragglers while
each agent tends to solve its local optimization problem inexactly.

We extend the FedProx based FL to the Internet of Vehicles. At the beginning
of a global round denoted by $k$, the server uniformly selects a set of agents
from the vehicle pool. Denote the selected agent set by
$\mathcal{V}_{k}\subseteq\mathcal{V}$. Each agent $v\in\mathcal{V}_{k}$ then
receives the global model $\mathbf{w}_{k}$ and perform local model training to
minimize the following objective function

$\displaystyle\breve{f}_{v}(\mathbf{w};\mathbf{w}_{k})=f_{v}(\mathbf{w})+(\mu/2)\left\|\mathbf{w}-\mathbf{w}_{k}\right\|^{2},$
(8)

where the proximal term is added to control heterogeneity and $\mu\geq 0$ is
the penalty parameter.

Agent $v\in\mathcal{V}_{k}$ solves problem (8) $\gamma_{v}^{k}$-inexactly for
solution $\mathbf{w}^{(v,*)}_{k}$ such that
$\left\|\nabla\breve{f}_{v}(\mathbf{w}^{(v,*)}_{k};\mathbf{w}_{k})\right\|\leq\gamma_{v}^{k}\left\|\nabla\breve{f}_{v}(\mathbf{w}_{k};\mathbf{w}_{k})\right\|$
[7]. The parameter $\gamma_{v}^{k}$ defines how much local computation to be
performed by agent $v$ for solving its respective local sub-problem.

Upon receiving $\mathbf{w}^{(v,*)}_{k}$s from all learning agents, the central
server averages these model parameters to obtain $\mathbf{w}_{k+1}$. We will
explicitly introduce communication and computation aware FL solutions in
Section IV.

###  III-B Delays in Vehicular FL

We consider three main delays between two global communication rounds as
described in the following.

####  III-B1 Queuing Delay

Queuing delay is the waiting time of a vehicular agent before being scheduled
by the associated RSU. It can be an important delay contributor in wireless
networks. Note that we only consider uplink queuing delay as the RSUs
multicast in the downlink. Denote the uplink queuing delay of agent $v$ by
$d_{v}^{q,\text{up}}$, which is the time difference from the time agent $v$
finishes local model training to the time agent $v$ is scheduled to offload
the trained model.

####  III-B2 Local Model Training Delay

Recall that the vehicles have heterogeneous on-board processing unit power.
Denote agent $v$’s processing power by $\rho_{v}$ cycles per second. If per
sample data requires $\eta_{v}$ cycles for processing, then the delay for one
iteration model training is
$d_{v}^{\text{cmp},l}=\left[\left(\eta_{v}\left|\mathcal{D}_{v}\right|\right)\right]/\rho_{v}$.

####  III-B3 Communication Delay

With vehicular agents, we consider delay from the wireless link. We calculate
the downlink and uplink delays based on downlink rate $C_{v}^{\text{dn}}$,
uplink rate $C_{v}^{\text{up}}$ and payload size. Assuming the FL model
parameter is $d$-dimensional, we calculate the required number of bits as
$S=\sum_{i=1}^{d}FPP_{i}$, where $FPP_{i}$ represents the floating point
precision for element $i$. As such, at the beginning of a global round, we
calculate the model distribution delay to agent $v$ via RSU-vehicle downlink
communication as follows:

$d_{v}^{\text{down}}=\kappa\times\text{min}\\{T:\kappa\times\bigl{(}\sum\nolimits_{\bar{t}=1}^{T}C_{v}^{\text{dn}}(\bar{t})\bigr{)}\geq
S,T\in\mathbb{Z}^{+}\\},$ (9)

where $\kappa$ is the transmission time interval (TTI) and
$C_{v}^{\text{dn}}(\bar{t})$ is the achievable downlink capacity based on the
channel realization at slot $\bar{t}$.

Similarly, we can calculate the delay to offload agent $v$’s trained model as
follows:

$d_{v}^{\text{up}}=\kappa\times\text{min}\\{T:\kappa\times\bigl{(}\sum\nolimits_{\bar{t}=1}^{T}C_{v}^{\text{up}}(\bar{t})\bigr{)}\geq
S,T\in\mathbb{Z}^{+}\\},$ (10)

where $C_{v}^{\text{up}}(\bar{t})$ is the achievable uplink capacity based on
the channel realization at slot $\bar{t}$.

####  III-B4 Total Delay in Each Global Communication Round

To this end, we calculate the cumulative delays between two global rounds. The
total delay is the summation of the global model distribution delay, local
model training delay, uplink queuing delay and local model offloading delay of
the agents. We express the total delay for agent $v$ as follows:

$\mathrm{d}_{v}^{\text{tot}}=d_{v}^{\text{down}}+d_{v}^{\text{cmp}}+d_{v}^{q,\text{up}}+d_{v}^{\text{up}}.$
(11)

In this work, we consider a synchronous agent-server update procedure as
described in Section IV-B. The global round update clock time is fixed and
known to the agents. More specifically, the central server sets a hard
threshold denoted as $\mathrm{d}^{\text{thr}}$, by which the server needs to
distribute the global model, agents need to locally train model and offload
the trained models back to the server. In other words,
$\mathrm{d}_{v}^{\text{tot}}\leq\mathrm{d}^{\text{thr}}$ must hold for this
synchronous update process. Clearly, the underlying wireless communication and
computation aspects affect whether the agents can successfully contribute to
the learning process or not.

##  IV Proposed Mobility, Communication and Computation Aware FL Solution

At a high level, our goal is to optimize the IoV network so that
$\mathrm{d}_{v}^{\text{tot}}\leq\mathrm{d}^{\text{thr}}$. In order to do this,
we first optimize the model distribution delay in the downlink.

###  IV-A Model Distribution Delay Optimization

To optimize the downlink model distribution delay, each RSU aims to maximize
the minimum data rate for all of its associated agents. As such, each RSU
finds the downlink multicasting beamforming vector $\mathbf{g}_{b}^{z}$ for
all pRBs by solving the following optimization problem:

$\displaystyle\underset{\mathbf{g}_{b}^{z},~{}\forall
z_{b}\in\mathcal{Z}_{b}}{\text{ maximize }}\underset{\forall
v\in\mathcal{V}_{k}^{b}}{\text{ min }}$
$\displaystyle\quad\left|({\mathbf{h}_{b,z}^{v}})^{H}\mathbf{g}_{b}^{z}\right|^{2},$
(12) subject to: $\displaystyle\left\|\mathbf{g}_{b}^{z}\right\|^{2}\leq 1,$

where $\mathcal{V}_{k}^{b}\subseteq\mathcal{V}_{k}$ is set of agents
associated with RSU $b$ in global round $k$.

This is a classical multicasting beamforming problem. Note that
$\left|({\mathbf{h}_{b,z}^{v}})^{H}\mathbf{g}_{b}^{z}\right|^{2}=\operatorname{Tr}\left(\mathbf{g}_{b}^{z}{\mathbf{g}_{b}^{z}}^{H}\mathbf{h}_{b,z}^{v}({\mathbf{h}_{b,z}^{v}})^{H}\right)$.
Denote $\mathbf{H}_{b,z}^{v}=\mathbf{h}_{b,z}^{v}({\mathbf{h}_{b,z}^{v}})^{H}$
and $\mathbf{G}_{b,z}=\mathbf{g}_{b}^{z}({\mathbf{g}_{b}^{z}})^{H}$. Then,
problem (12) can be reformulated as follows:

$\displaystyle\underset{\mathbf{G}_{b,z},~{}\forall
z_{b}\in\mathcal{Z}_{b}}{\text{ maximize }}\quad\underset{\forall
v\in\mathcal{V}_{k}^{b}}{\text{ min
}}\quad\operatorname{Tr}\left(\mathbf{G}_{b,z}\mathbf{H}_{b,z}^{v}\right),$
(13) $\displaystyle\text{subject
to:}\quad\operatorname{Tr}\left(\mathbf{G}_{b,z}\right)=1,\mathbf{G}_{b,z}\succeq
0,\mathrm{rank}(\mathbf{G}_{b,z})=1.$

Note that (13) is non-convex due to the $\mathrm{rank}(\mathbf{G}_{b,z})=1$
constraint. We can relax this constraint to obtain the following relaxed
convex problem.

$\displaystyle\underset{\mathbf{G}_{b,z},~{}\forall
z_{b}\in\mathcal{Z}_{b}}{\text{ maximize }}\quad\underset{\forall
v\in\mathcal{V}_{k}^{b}}{\text{ min }}$
$\displaystyle\quad\operatorname{Tr}\left(\mathbf{G}_{b,z}\mathbf{H}_{b,z}^{v}\right),$
(14) subject to:
$\displaystyle\operatorname{Tr}\left(\mathbf{G}_{b,z}\right)=1,\mathbf{G}_{b,z}\succeq
0.$

Optimization problem (14) is in the well-known semidefinite problem (SDP)
form. Each RSU can solve this downlink multicasting beamforming optimization
problem using widely popular convex optimization solver such as CVX [11]. Each
RSU finds the downlink multicasting beamforming vector and distributes the
model to all associated agents. Note that since the entire bandwidth is used
for this downlink distribution, the $d_{v}^{\text{down}}$ is relatively low.
Moreover, the RSU-agent associations will remain unchanged for this short
time222 For example, if RSU coverage radius is $300$ meters and agent velocity
is $120$ miles/hour, the agent is expected to be within the RSU’s coverage for
$\approx 11.18$ seconds. In wireless networks, usually TTI $\kappa\leq 1$
millisecond..

###  IV-B Local Model Training

This work considers a complete synchronous learning framework, where the
server provides all agents a deadline $\mathrm{d}_{\text{k}}^{\text{cmp}}$ for
their local model training. In other words, during round $k$, the agent
receives the model and performs local model training until
$\mathrm{d}_{\text{k}}^{\text{cmp}}$ expires. Recall that the FL global round
update clock time is known to all agents. Therefore, during global round $k$,
upon receiving the global model $\mathbf{w}_{k}$, each agent
$v\in\mathcal{V}_{k}$ can determine the remaining time budget for its local
model computation as

$d_{v}^{\text{cmp}}=\mathrm{d}_{\text{k}}^{\text{cmp}}-d_{v}^{\text{down}}.$
(15)

Therefore, agent $v$ determines its local model training iterations as

$L_{v}=\left\lfloor d_{v}^{\text{cmp}}/d_{v}^{\text{cmp},l}\right\rfloor,$
(16)

where $\lfloor\cdot\rfloor$ is the floor operation. This essentially means
that agent $v\in\mathcal{V}_{k}$ performs $L_{v}$ local stochastic gradient
decent (SGD) steps to minimize its local objective function defined in (8).
Please note that, unlike FedAvg that considers equal $L_{1}=\dots=L_{V}$ [1],
FedProx allows heterogeneous device participation to utilize agents’ resources
efficiently [7].

Note that FedAvg is a special case of the proposed FedProx based solution if
the server sets an common training iteration $L$ and $\mu$ is set to $0$ in
problem (8).

###  IV-C Local Model Offloading Optimization

Upon finishing the local model training, the agent requests uplink radio
resources from its associated RSU to offload the trained model
$\mathbf{w}_{k}^{v}$. The RSU then allocates the pRB for this uplink
communication. Assume RSUs have perfect CSI. Therefore, each RSU can use
maximal ratio combining (MRC) to model the receiver beamforming vector, i.e.,
$\mathbf{g}_{b,z}^{v}=\breve{\mathbf{h}}_{b,z}^{v}/\|\breve{\mathbf{h}}_{b,z}^{v}\|$.
Moreover, depending on the pRB allocation, the uplink queuing delay
$d_{v}^{\text{up}}$ is known to the associated RSU. In this work, we apply
round-robin scheduling for simplicity. To that end, each RSU allocates its
pRBs to the scheduled agents to maximize the network’s uplink throughput. In
other words, each RSU aims to solve following optimization problem:

$\displaystyle\underset{\mathbb{I}_{b,z}^{v},~{}\forall
v\in\mathcal{V}_{k}^{b}}{\text{ maximize }}$
$\displaystyle\omega\sum\nolimits_{z_{b}=1}^{Z_{b}}\mathbb{I}_{b,z}^{v}\cdot\log_{2}\left(1+\Gamma_{b,z}^{v,\text{up}}\right),$
(17) subject to
$\displaystyle\sum\nolimits_{z_{b}=1}^{\mathcal{Z}_{b}}\mathbb{I}_{b,z}^{v}=1,\sum\nolimits_{v\in\mathcal{V}_{k}^{b}}\mathbb{I}_{b,z}^{v}=1,$
$\displaystyle\sum\nolimits_{z_{b}=1}^{\mathcal{Z}_{b}}\sum\nolimits_{v\in\mathcal{V}_{k}^{b}}\mathbb{I}_{b,z}^{v}=|\mathcal{Z}_{b}|,$

where the first constraint is to allocate only one pRB to each scheduled
agent, while the second constraint is adopted to assign a pRB to only one
agent. Moreover, the last constraint ensures that all pRBs are allocated. Note
that while $|\mathcal{V}_{k}|$ can be greater than $|\mathcal{Z}_{b}|$, the
RSU can only schedule $|\mathcal{Z}_{b}|$ agents in a each scheduling
$\kappa$, i.e., $|\mathcal{V}_{k}^{b}|=|\mathcal{Z}_{b}|$.

Problem (17) is a mixed combinatorial optimization problem and NP-hard. To
solve this problem, we stack the SNRs over all pRBs into a gain matrix
$\mathbf{G}_{b}$ and use the well-known Hungarian algorithm [12]. Note that
this algorithm guarantees optimal pRB allocation with time complexity of
$\mathcal{O}(|\mathcal{Z}_{b}|^{3})$ [12]. Algorithm 1 summarizes the process.

1 Input: $\\{\Gamma_{b,z}^{v,up}\\}$ for all active RSUs and vehicle agents ;

2 for  _$b\in\mathcal{B}$_ do

3 Schedule agents following round-robin scheduling;

4 if  _$\bigl{|}\mathcal{V}_{k}^{b}\bigr{|}\neq 0$_ then

5 Initiate gain matrix,
$\mathbf{G}_{b}=zeros(Z_{b}\times\bigl{|}\mathcal{V}_{k}^{b}\bigr{|})$ ;

6 for  _$v\in\mathcal{V}_{k}^{b}$_ do

7 for  _$z_{b}\in\mathcal{Z}_{b}$_ do

8 $\mathbf{G}_{b}[z_{b},v]\leftarrow\Gamma_{b,z}^{v,\text{up}}$;

9

10 end for

11

12 end for

13 Use $\mathrm{Hugarian}$ algorithm [12] to get optimal
$\mathbb{I}_{b,z}^{v}$s ;

14

15

16 end for

Algorithm 1 Optimal Radio Resource Allocation

1 Input: initial global model $\mathbf{w}$;

2 for  _all global communication round, $k\in K$_ do

/* At the beginning of communication round $k$ */

3 Server selects learning agents $\mathcal{V}_{k}\subseteq\mathcal{V}$ ;

4 Each RSU solves optimization problem (14) and distributes

5 end for

6the global model $\mathbf{w}_{k}$ $\triangleright$ Central server broadcasts
global model $\mathbf{w}_{k}$, agents set $\mathcal{V}_{k}$,
$\mathrm{d}_{\text{k}}^{\text{cmp}}$ and threshold $\mathrm{d}^{thr}$ to RSUs.
RSUs then forward the global model $\mathbf{w}_{k}$ and the updated time
information $d_{v}^{\text{cmp}}$ to their associated $v\in\mathcal{V}_{k}$;

/* Local model training */

7 for  _all $v\in\mathcal{V}_{k}$ in parallel_ do

8 Prepare the local training dataset $\mathcal{D}_{v}^{(k)}$;

9 Determine local training iterations $L_{v}$ using (16)

10 end for

11;

12 Perform local training to minimize loss function defined in (8);

/* At the end of global round $k$ */

13 All $v\in\mathcal{V}_{k}$ request uplink radio resources from their
associated RSUs to offload their locally trained model $\mathbf{w}_{k}^{v}$ ;

14 Each RSU $b$ solves optimization problem (17) to get the optimal pRB
allocation and schedules its associated agents $v\in\mathcal{V}_{k}^{b}$ in
round-robin manner;

15 Each agent $v\in\mathcal{V}_{k}$ then offloads $\mathbf{w}_{k}^{v}$ to the
central server via the associated RSU $b$;

The central server performs model aggregation to update global model using:
$\mathbf{w}_{k+1}\leftarrow(1/\left|\mathcal{V}_{k}\right|)\sum\nolimits_{v=1}^{\left|\mathcal{V}_{k}\right|}\mathbf{w}_{k}^{v}$.
Output: trained global model $\mathbf{w}^{*}$

Algorithm 2 Vehicular FL Algorithm

###  IV-D Summary of the Proposed FL Solution

The proposed mobility, communication and computation aware FL solution is
summarized in Algorithm 2. In each communication round, server selects agents
to train the model. Server broadcasts the model $\mathbf{w}_{k}$, the agent
set $\mathcal{V}_{k}$, time deadlines $\mathrm{d}_{k}^{\text{cmp}}$ and
$\mathrm{d}^{\text{thr}}$ to RSUs. Each RSU then computes optimal
communication delays for its associated agents and broadcasts the model and
the updated time information to its agents. Based on the updated time
information $d_{v}^{\text{cmp}}$ and its local dataset $\mathcal{D}_{k}^{v}$,
each agent performs independent model training for $L_{v}$ iterations. At
training deadline $\mathrm{d}_{k}^{\text{cmp}}$, each RSU schedules its agents
to offload their local models to satisfy deadline threshold
$\mathrm{d}^{\text{thr}}$.

##  V Performance Evaluation

In this section, we first describe our simulation setup and then present
simulation results and discussions.

###  V-A Simulation Setting and Dataset Description

We use real vehicle traces in the Mobile Century Dataset collected by the
Mobile Millennium project in $2008$ [13]. This dataset contains data collected
for $77$ vehicles between 10:00am and 18:00pm on Interstate $880$ (I-$880$),
CA, USA and has vehicle trajectory traces with an approximate interval of 3
seconds. We use GPS log datasets that have four features in each data sample:
$\\{unixtime,~{}latitude,~{}longitude,~{}speed\\}$. From GPS log data of these
$77$ vehicles, we choose $61$ datasets after initial screening. Note that the
traces contain approximately $20.1$ miles stretch of the I-$880$. To serve
these vehicles, we place RSUs along the I-$880$ for the entire stretch.
Particularly, we assume each RSU has a coverage radius of $500$ meters.
Moreover, the same frequencies are reused after $\approx 2000$ meters to
handle interference issues.

For our learning model, we use $1$ gated recurrent unit (GRU) layer with $16$
hidden layers, SGD (momentum=0.95) optimizer with learning rate $0.001$,
$\mu=0.1$, $l=15$, and $h\in\\{1,\dots,12\\}$. We select $10$ agents with
$\rho_{v}\in Uniform(0.2,0.8)$ GHz and $\eta_{v}\in Uniform(10^{4},40^{4})$
cycles/sample. Furthermore, the agents use $11:00-16:30$ trajectory data
samples as their training dataset. We devise the global round update time in
such a way that at the beginning, there are enough data samples to train the
model, and the training data samples $\mathcal{D}_{k}^{v}$ only use data
samples from $11:00:00$ to the global round update time in a causal manner.
This means that the proposed FL solution is implemented in an online fashion,
in which the vehicles gather data as they travel along their trajectory. Other
key V2X simulation parameters are listed in Table II.

TABLE II: V2I/I2V: Key Simulation Parameters Description |  Value
---|---
Antenna per RSU $n_{b}$ |  $4$
pRB per RSU $Z_{b}$ |  $10$
{$v,b$} antenna heights |  {$3$, $25$} m
{$v,b$} antenna gains |  {$3,8$} dBi
{$v,b$} receiver noise figures |  {$9,5$} dB
Noise power $\sigma^{2}$ |  -$174$ dBm
$P_{b}^{max},P_{v}^{b}$ |  $30,23$ [dBm]
TTI $\kappa$ |  $1$ ms
Carrier frequency |  $2.4$ GHz
pRB bandwidth $\omega$ |  $180$ KHz
$\Omega_{b},\Omega$ |  $1.8,3.6$ [MHz]
Frequency reuse |  $\approx 2000$ m

###  V-B FL Task : Vehicle-Specific Power (VSP) Prediction

While the proposed vehicular FL platform is not task-dependent, we consider a
multi-horizon vehicle-specific power (VSP) prediction task to validate our
vehicular FL solutions. In essence, VSP provides the estimation of the
required power demand for the vehicle. For zero road grade, the VSP can be
calculated as [14]

$P_{v}^{t}=\left(\frac{c_{1}}{c_{2}}\right)\frac{Au_{v}^{t}}{m_{v}}+\left(\frac{c_{1}^{2}}{c_{2}}\right)\frac{B{u_{v}^{t}}^{2}}{m_{v}}+\left(\frac{c_{1}^{3}}{c_{2}}\right)\frac{C{u_{v}^{t}}^{3}}{m_{v}}+c_{1}^{2}u_{v}^{t}a_{v}^{t},$
(18)

where $m_{v}$, $u_{v}^{t}$ and $a_{v}^{t}$ are vehicle $v$’s weight,
instantaneous velocity and instantaneous acceleration, respectively. Moreover,
$A$, $B$, $C$, $c_{1}$ and $c_{2}$ are coefficients described in [14].

The VSP can be predicted in two ways: (i) predicting velocity and then using
the predicted velocity to compute VSP via Eq. (18) and (ii) predicting VSP
directly. The approach (i) does not work well because VSP is expressed as a
third-order polynomial in velocity with additional velocity acceleration
product term. Accordingly, it is very sensitive to velocity variation. Even
with near ground truth velocity prediction as shown in Fig. 3, the VSP
accuracy can be poor as shown in Fig. 3.

Figure 2: Velocity prediction for vehicle $0$

Figure 3: VSP calculation from predicted velocity for vehicle $0$

Therefore, we apply approach (ii) in this work. Note that in order to predict
VSP directly with approach (ii), we need a suitable dataset. However, the
dataset [13] does not have VSP information. As such, we use (18) to calculate
VSP and then use the calculated VSP for model training and evaluation in what
follows.

###  V-C Simulation Results and Discussions

We evaluate the proposed FL solutions in four aspects: (i) delay, (ii)
prediction accuracy, (iii) robustness, and (iv) prediction horizon. It is
worth noting that each agent $v$ randomly samples $50\%$ data samples from
$\mathcal{D}_{k}^{v}$ for its local model training333The percentage of data
samples can be adjusted to meet the deadlines.. Moreover, we choose
$\mathrm{d}^{\text{thr}}=3$ seconds and accordingly set estimated local
iterations $L_{v}$ for all $v\in\mathcal{V}_{k}$ in each communication round
$k$.

####  V-C1 Delay in Distributed Model Training

We first show delays between two global communication rounds using the
simulation parameters in Table II.

Figure 4: Delay between two communication rounds Figure 5: Predicted VSP with
different machine learning models

Given a model training time threshold $\mathrm{d}^{\text{thr}}$, the RSUs
optimizes the multicast beamforming vector to maximize the minimum downlink
data rate at the agents by solving optimization problem (14). As expected, the
downlink model distribution delay $d_{v}^{\text{down}}$ is significantly
lower. As shown in the first subplot of Fig. 4, the maximum distribution time
is at most $7\times\kappa$, i.e., $7$ ms.

However, the uplink model offloading time $d_{v}^{\text{up}}$ is significantly
longer than $d_{v}^{\text{down}}$ as shown in the third subplot of Fig. 4. In
fact, it requires more TTIs to offload local models because agents have
limited radio resources and battery power. This is also observed in our
simulation. More specifically, we observe that it requires $\approx 55$ TTIs
to offload an agent’s model in the uplink. This means that we cannot simply
consider fixed channel $\mathbf{h}_{v,z}^{b}$ conditions to calculate
$d_{v}^{\text{up}}$. Therefore, solving optimal pRB allocation problem (17) is
necessary.

Since the on-board computation power of the agents is limited, the dominant
factor in total delay is the local model training delay444If the model size is
significantly large, the communication overhead may become higher. as shown in
the second subplot of Fig. 4. As the global round increases, agents collect
more data in their datasets $\mathcal{D}_{k}^{v}$s. Therefore, the local
training time increases as the global round increases. As a result, total
delay increase as well as shown in the fourth subplot of Fig. 4.

Note that, given a time threshold $\mathrm{d}^{\text{thr}}$, the choice of
$L_{v}$ and $\mathrm{d}_{\text{k}}^{\text{cmp}}$ are critical to guarantee
$\mathrm{d}_{v}^{\text{tot}}\leq\mathrm{d}^{\text{thr}}$. Particularly, to
choose $\mathrm{d}_{\text{k}}^{\text{cmp}}$, the server should consider the
impact of the uncertain channel condition, available radio resources of the
RSU and corresponding expected queuing delay.

####  V-C2 VSP Prediction Accuracy

In this subsection, we show the performance of different ML models on VSP
prediction task. We compare our proposed FedProx based FL model with FedAvg
based FL model + our communication framework, independent LSTM model and a
widely used statistical time-series baseline, namely the AutoRegressive
Integrated Moving Average (ARIMA) model. Particularly, the statistical ARIMA
works well when the data samples are homogeneous and have a clear trend.
However, our data are essentially the behavior of drivers, i.e., highly
heterogeneous. As such, it is hard to find a clear trend. Therefore, ARIMA is
expected to perform poorly. On the other hand, the independent LSTM model only
uses agent’s own data for training, which works for the homogeneous data
distribution case. However, in the highly heterogeneous data distribution
case, the prediction accuracy of the LSTM model is expected to degrade. Our
proposed FedProx based FL model is exposed to diverse training samples of the
selected agents. It is thus anticipated to perform better than these baseline
models. This tendency is validated in our simulation. Fig. 5 takes vehicle $0$
as an example. Clearly, our FedProx based FL model can deliver near-ground
truth VSP prediction, whereas the independent LSTM model and ARIMA model fail
to capture the temporal dynamics. Furthermore our FedProx based FL model
outperforms FedAvg based FL model because of the contribution from the
proximal term in the objective function. As such, the distributed privacy-
preserving FedProx based FL platform cam make accurate predictions in the
highly dynamic IoV networks and can be immensely useful for ADAS and other IoV
applications.

####  V-C3 Robustness of the Proposed FL Platform

The trained global model can be used by any vehicle to perform their tasks. We
show how accurate the model performs. To that end, we tested our trained
global model using the unseen test data samples of all $61$ vehicles. Figs. 11
\- 11 validate that the proposed distributed FL solution works well for all
vehicles. In addition, our FL solutions ensure that the global model can be
trained in an online fashion. Therefore, our distributed FL solutions are
feasible for real-time IoV applications. Thousands of the on-road vehicles can
benefit from our FL solutions while only a subset of vehicles need to perform
model training.

Figure 6: Predicted VSP for vehicle $1$’ with $h=1$

Figure 7: Predicted VSP for vehicle $1$ with $h=4$

Figure 8: Predicted VSP for vehicle $30$ with $h=1$

Figure 9: Predicted VSP for vehicle $30$ with $h=4$

Figure 10: Predicted VSP for vehicle $60$’ with $h=1$

Figure 11: Predicted VSP for vehicle $60$’ with $h=4$

####  V-C4 Impact of Forecast Window Size

Figure 12: Training loss variation with respect to forecast window size

Finally, we illustrate the impact of the forecast window size $h$ with a fixed
lag of $l=15$. As the $h$ increases, the agents need more time to predict the
future VSP. This essentially means that the probability of making an erroneous
prediction increases. We first take $h=1$ and $h=4$ as examples to show the
behavior of the average training loss with respect to the forecast window
size. As shown in Fig. 12, the average training loss for $h=1$ is less than
that for $h=4$ over all communication rounds. Furthermore, as the global
communication round increases, the model is exposed to more training samples
of the diverse agents. As such, the average training loss exhibits the
decreasing tendency.

TABLE III: Average MSE in All $61$ agents Test Data Forecast Window $h$ |  Proposed FL Solution |  Proposed Platform + FedAvg
---|---|---
$1$ |  2.5784784 |  2.591314118
$2$ |  4.63190712 |  4.670199372
$3$ |  6.85608755 |  7.09665202
$4$ |  9.10875644 |  9.55388634
$5$ |  11.72196015 |  11.89405537
$6$ |  13.57525046 |  13.39627498
$7$ |  14.17268583 |  13.95693221
$8$ |  15.43924312 |  15.44841811
$9$ |  16.07935477 |  16.10373348
$10$ |  16.26966842 |  16.83335492
$11$ |  16.85294033 |  16.56411971
$12$ |  16.27049288 |  16.5502178

We next show average training loss for all forecast window sizes using our
proposed vehicular FL solutions amalgamating with FedProx and FedAvg. Table
III shows that the prediction accuracy degrades as the forecast window size
$h$ increases. Therefore, one should carefully choose the forecast window size
based on the required accuracy by the applications such as velocity and VSP
prediction. Furthermore, FedProx works better than FedAvg. Accordingly,
FedProx based FL solution is more suitable for IoV networks.

##  VI Conclusion

This paper proposes a mobility, communication and computation aware integrated
FL platform for vehicular environment. Leveraging V2X communication, we have
devised the I2V/V2I communication platform to exploit on-road vehicles as
learning agents. To maximize local model training time, the proposed FL
platform optimizes downlink and uplink radio resource allocations to minimize
model transmission delay. Moreover, as vehicles have different computation
power and dataset, our FL solution tolerates partial works of the agents,
i.e., vehicle agents perform heterogeneous local model training, which makes
it more suitable for delay-constrained applications in mobile networks. Using
real-world and causal training datasets, we have shown that the proposed FL
algorithms can be implemented in an online fashion. Our simulation results
show that the proposed FL platform is robust and can deliver near ground truth
velocity and VSP predictions.

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# First integral by means of the uniformization over the torus

Orlando Galdames-Bravo  O. Galdames Bravo
Departamento de Matemáticas
Universidad de Valencia
Doctor Moliner, 50
46100 - Burjassot. Spain<EMAIL_ADDRESS>
(Date: 2nd September 2024)

###### Abstract.

The uniformization of a direction field was defined by Finn (in 1973) for the
classification of certain differential operators. In the present note we
recover the idea of uniformization in order to generalize it and apply it to
the existence of first integrals. Roughly speaking, we say that a plane vector
field $X$ on $D\subset\mathbb{R}^{2}$ is uniformizable over $T^{2}$ (the
torus) if there is a constant vector field $Z$ on $\Delta\subset T^{2}$ and a
diffeomorphism $\psi\colon D\to\Delta\subset T^{2}$ such that $d\psi\circ
X=Z\circ\psi$.

###### Key words and phrases:

vector field, index, uniformization, first integral

###### 2010 Mathematics Subject Classification:

Primary 37B30. Secondary 54H20, 74H20

##  1\. Introduction and terminology

The existence of a first integral for a vector field is a classical problem in
the theory of dynamical systems. One of the main tools to this aim is the
study of the rectification or uniformization of a vector field. Roughly
speaking, we understand as uniformization of a vector field with plane domain,
as the existence of a diffeomorphic map that brings such a vector field to a
constant vector field. Local uniformization is a well known property of any
vector field with mild conditions (see e.g [4, Theorem 1.2]). Regarding global
plane domains, there are several results over time that deal with several
techniques in order to find a first integral or solve a related problem, see
for instance [2, 6, 7, 9, 10, 11].

The technique we introduce in the present note is based in the ideas presented
by Finn in [5] (who studied the behavior of the index of a plane direction
field along a Jordan curve that represent the boundary of the domain) and the
suggestion of Martínez-Alfaro for the generalization of this concept to other
not plane manifolds. In [5], the author studied direction fields which have
index $0$ or $2$ along suitable curves. Thanks to the generalization we give
we can extend this study to the index $1$. In brief, a vector field is
uniformizable the sense of Finn, if it is conjugate with a constant one, both
plane vector fields. In our case we assume that the constant vector field can
be defined over the torus. We will see that this point of view will allows us
to obtain an application to the existence of a first integral for certain
vector fields.

We have organized the paper in a unique section where we develop the ideas
described above. Let us introduce some terminology for the paper. Let $M$ be a
manifold, we denote by $T(M)$ he bundle of tangent vector spaces of $M$. Let
$D\subseteq\mathbb{R}^{2}$ be an closed plane domain, recall that a vector
field $X\colon D\to T(D)$ is said to be uniformizable if it is conjugate with
a constant vector field $U\colon E\to T(E)$ in the plane, for a certain plane
domain $E$, i.e. if there exists a diffeomorphic map $\varphi\colon D\to E$ so
that $d\varphi\circ X=U\circ\varphi$ (see [5, Definition 3] and [4, Section
1.3]). We denote the torus by $T^{2}$. The definition of the index of a
direction field along a curve can be found in [5, Section 1]. Let us define it
for a vector field. We assume that a Jordan curve is an smooth injective
closed curve. As usual we sometimes identify the parametrization of a curve
with its image. Let $D$ be a plane domain and let $\Gamma\colon[0,2\pi]\to D$
be a Jordan curve on $D$. Let $X_{\Gamma}$ be the vector field defined on
$\Gamma$, and $\delta(X_{\Gamma})$ denotes the change of angle from the vector
$X_{\Gamma}(0)$ to the vector $X_{\Gamma}(2\pi)$. Let $X$ be a vector field
defined on $D$, we define the _index of the vector field $X$ along the curve
$\Gamma$_ as

$\operatorname{ind}_{\Gamma}(X):=\cfrac{1}{2\pi}\,\delta(X_{\Gamma})\,.$

A well known formula to compute this index is:

$\operatorname{ind}_{\Gamma}(X)=\cfrac{1}{2\pi}\int_{0}^{2\pi}\cfrac{\det({X|}_{\Gamma},({X|}_{\Gamma})^{\prime})}{\|{X|}_{\Gamma}\|^{2}}\,dt,$
(1.1)

see for instance [3]. A parametrization of the vector field can be written as
$X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$, where
$P,Q\colon D\to\mathbb{R}$ and $x,y$ are suitable coordinates of $D$. Let
$h\colon D\to\mathbb{R}$ be a differentiable function. We say that $h$ is a
_first integral of $X$_ if $X(h)=Ph_{x}+Qh_{y}=0$, where $h_{x}$ and $h_{y}$
denotes the respective derivatives. We will mention a suitable domain called
_corona_ , which is a set of the form
$\\{(x,y)\in\mathbb{R}^{2}:0<r<\|(x,y)\|_{2}<R<\infty\\}$. We denote by
$S^{n}$ the sphere of dimension $n\in\mathbb{N}$. We will identify $T^{2}$
with $S^{1}\times S^{1}$ and with $\mathbb{R}^{2}/[0,2\pi]^{2}$. The _type_
$(p,q)$ of a Jordan curve on the torus ($S^{1}\times S^{1}$) is the number of
laps around each circumference $S^{1}$. We can see the type as an element of
the homotopy group $\pi_{1}(T^{2})$. We refer to [1, 4] for notation,
definitions and basic results that not appear here.

##  2\. First integrals and uniformization over the torus

In this unique section we introduce the concept of uniformization over the
torus and illustrates by means of an example that such a concept is useful for
the existence of a first integral of a given vector field. Then, we give a
condition for the uniformization over the torus of certain vector fields that
involves the index along the boundary of the domain of these vector fields. We
will assume that a domain is a bounded and connected region delimited by
Jordan curves. We also assume that every function and vector field will be
sufficiently differentiable.

######  Definition 2.1.

Let $r\geq 1$ and $D\subseteq\mathbb{R}^{2}$ be a bounded and connected
domain. Let $X\colon D\to T(D)$ be a vector field. Let $\Delta\subseteq T^{2}$
be a domain. We say that $X$ is _uniformizable over the torus_ if there exists
$\Delta\subseteq T^{2}$ such that $X$ is conjugate to a constant vector field
defined on $\Delta$, i.e. there exists a constant vector field
$Z\colon\Delta\to T(\Delta)$ and a diffeomorphism $\psi\colon D\to\Delta$ such
that $d\psi\circ X=Z\circ\psi$.

In the present paper we just deal with the definition above, over the torus,
but one can easily extend it to any other manifold.

######  Example 2.2.

The constant vector field $X=a\;\partial/\partial x+b\;\partial/\partial y$,
where $a,b\in\mathbb{R}$ so that $a/b\in\mathbb{Q}\setminus\\{0\\}$, defined
in a domain of $\mathbb{R}^{2}$, is a clear example of uniformizable vector
field over the torus and over the cylinder.

Let us give an application to the existence of a first integral. Observe that
we also can apply the following to the Finn uniformization.

######  Proposition 2.3.

Let $D\subseteq\mathbb{R}^{2}$ be a compact connected domain and $X$ be a
vector field on $D$ with Hausdorff flow. Assume that $X$ is uniformizable over
the torus, then there is a first integral for $X$.

###### Proof.

By hypothesis, there is a diffeomorphism $\psi=(f,g)$ through which $X$ is
conjugate to a constant vector field $Z:=a\frac{\partial}{\partial
u}+b\frac{\partial}{\partial v}$ on the torus, where
$a,b\in\mathbb{R}^{r}\setminus\\{0\\}$ satisfy that $a/b$ (or $b/a$) is a
rational number. Moreover, the partial differential equation

$\left\\{\begin{array}[]{l}h_{x}=b\,f_{x}-a\,f_{y}\\\
h_{y}=b\,g_{x}-a\,g_{y}\end{array}\right.$ (2.1)

has solution (see for instance [8]). Let $h$ be such a solution. We claim that
$h$ is a first integral of $X$.

Let us denote by $(u,v)$ the coordinates in $T^{2}$ and by $(x,y)$ the
coordinates in $D$. By hypothesis we know that $d\psi\circ X=Z\circ\psi$. Let
us denote $X=P(x,y)\frac{\partial}{\partial x}+Q(x,y)\frac{\partial}{\partial
y}$, where $\dot{x}=P(x,y)$ and $\dot{y}=Q(x,y)$. On the one hand, since $Z$
is constant, there exists $H$ a first integral of $Z$. We can choose, for
instance, $H(u,v)=bu-av$, since $H_{u}=b$ and $H_{v}=-a$ and
$Z(H)=aH_{u}+bH_{v}=0$. On the other hand we can choose the variables $u$ and
$v$ so that $\psi^{-1}(u,v)=(x,y)$, hence

$d\psi\circ
X\circ\psi^{-1}=(f_{x}\dot{x}+g_{x}\dot{y})\frac{\partial}{\partial
u}+(f_{y}\dot{x}+g_{y}\dot{y})\frac{\partial}{\partial v}.$

Therefore

$\begin{split}d\psi\circ
X\circ\psi^{-1}(H)&=(f_{x}\dot{x}+g_{x}\dot{y})H_{u}+(f_{y}\dot{x}+g_{y}\dot{y})H_{v}\\\
&=\dot{x}(H_{u}f_{x}+H_{v}f_{y})+\dot{y}(H_{u}g_{x}+H_{v}g_{y})\\\
&=\dot{x}(bf_{x}-af_{y})+\dot{y}(bg_{x}-ag_{y})=0.\end{split}$ (2.2)

We know that $h$ is solution of (2.1), hence by (2.2)

$X(h)=\dot{x}h_{x}+\dot{y}h_{y}=\dot{x}(b\,f_{x}-a\,f_{y})+\dot{y}(b\,g_{x}-a\,g_{y})=0.$

That means that $h$ is a first integral for $X$. ∎

######  Example 2.4.

Let $D\subseteq\mathbb{R}^{2}$ be a plane domain and $X\colon D\to T(D)$ be
vector field. Let $(a,b)\in\mathbb{R}^{2}\setminus 0$ such that $a/b$ (or
$b/a$) is rational. Assume that there is a diffeomorphism $\psi\colon
D\to\Delta\subseteq T^{2}$ as the one defined above $\psi:=(f,g)$ and assume
that $X:=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ is such
that

$P:=\dfrac{ag_{y}-bf_{y}}{f_{x}g_{y}-f_{y}g_{x}}$

and

$Q:=\dfrac{bf_{x}-ag_{x}}{f_{x}g_{y}-f_{y}g_{x}},$

where the numerators does not vanishes at the same time in $D$, so the vector
field will be regular. Observe that the denominator is the Jacobian of $\psi$,
hence it not vanishes in $D$. This vector field is uniformizable by means of
the constant vector field $a\frac{\partial}{\partial
u}+b\frac{\partial}{\partial v}$, where $u,v$ denotes the coordinates in the
torus.

Proposition and example above illustrates that uniformization over the torus
could be a tool in order to find out the existence of a first integral, in
consequence it is interesting to find conditions related to such a property.
In the following theorem and corollary we show that there exist certain
uniformizables vector fields such that the index along suitable Jordan curves,
that we can take closer to the the boundary of its domain, is equal to $1$.
Later, we easily deduce a corollary for a more general class of vector fields.

Let $Z\colon T^{2}\to T(T^{2})$ be a constant vector field and $\Gamma$ be a
Jordan curve of type $(p,q)\in\mathbb{Z}^{2}\setminus\\{0\\}$ on the torus.
Clearly $\operatorname{ind}_{\Gamma}Z=0$. Then we consider the set
$T^{2}\setminus\Gamma_{0}$ for a suitable Jordan curve with the same type as
$\Gamma$ and transform it diffeomorphically in a corona on $\mathbb{R}^{2}$.
Next result states that the index of the image of such a constant vector field
along the image of $\Gamma$ is $1$.

######  Theorem 2.5.

Let $\Gamma$ be a Jordan curve on the torus $T^{2}$ with type
$(p,q)\in\mathbb{Z}^{2}\setminus\\{0\\}$. Let $Z\colon T^{2}\to T(T^{2})$ be a
constant vector field. Then, there exist a Jordan curve $\Gamma_{0}$ such that
$\Gamma\cap\Gamma_{0}=\emptyset$, a domain $D\in\mathbb{R}^{2}$ and a
diffeomorphism $\varphi\colon T^{2}\setminus\Gamma_{0}\to D$ such that

$\operatorname{ind}_{\varphi\circ\Gamma}(d\varphi\circ
Z\circ\varphi^{-1})=1\,.$

###### Proof.

We will use the representation of the torus by means of the usual equivalence
relation in $[0,2\pi]^{2}$. Since the type of a curve is invariant by
diffeomorphism, without loss of generality, we can assume that
$\Gamma(t)=(pt,qt)$ for $t\in[0,2\pi]$. Again, without loss of generality we
take the constant vector field $Z:=\partial/\partial x$. Thus, $\Gamma$ has
type $(p,q)$ and the representation of $\Gamma$ in
$\mathbb{R}^{2}/[0,2\pi]^{2}$ is the segment between $(0,0)$ and
$(2p\pi,2q\pi)$ in $\mathbb{R}^{2}$. The angle of this segment with respect to
the positive abscissa is $\arctan(\frac{q}{p})$.

Let the Jordan curve $\Gamma_{0}(t):=(pt+1/2,qt+1/2)$, clearly
$\Gamma\cap\Gamma_{0}=\emptyset$. We define the diffeomorphism $\varphi\colon
T^{2}\setminus\Gamma_{0}\to\mathbb{R}^{2}$ given by

$\varphi:=\sigma\circ\tau\circ\rho_{p,q}\,,$ (2.3)

where

$\rho_{p,q}:=\dfrac{1}{p^{2}+q^{2}}\Big{(}\begin{matrix}p&q\\\
-q&p\end{matrix}\Big{)}$

is the clockwise rotation centered at the origin and angle
$\arctan(\frac{q}{p})$, which move $\Gamma$ to the segment
$[0,2\pi\sqrt{p^{2}+q^{2}}]\times\\{0\\}$; $\tau$ is the translation

$\tau(x,y)=(x,y+2\pi),$

which translate the segment in the plane
$[0,2\pi\sqrt{p^{2}+q^{2}}]\times\\{0\\}$ to the segment
$[0,2\pi\sqrt{p^{2}+q^{2}}]\times\\{2\pi\\}$; and $\sigma$ is the change to
polar variables, that is

$\sigma(x,y)=(y\cos(x),y\sin(x)).$

Therefore

$\varphi(x,y)=\dfrac{1}{p^{2}+q^{2}}\big{(}(py-qx+2\pi)\cos(px+qy)\,,\,(py-
qx+2\pi)\sin(px+qy)\big{)}\,.$

We now use the formula (1.1) in order to obtain the index of the vector field
$d\varphi\circ Z\circ\varphi^{-1}$ along $\varphi\circ\Gamma$ by a direct
computation. On the one hand we have that

$(py-qx)|_{\Gamma(t)}=p(qt)-q(pt)=0\quad\mbox{ and
}\quad(px+qy)|_{\Gamma(t)}=(p^{2}+q^{2})t\,.$ (2.4)

For the aim of brevity we define $H:=p^{2}+q^{2}$, $c:=\cos(Ht)$ and
$s:=\sin(Ht)$. Hence

$\varphi\circ\Gamma=\cfrac{2\pi}{H}(c,s)\,,$

defines a curve on a certain domain $D\subset\mathbb{R}^{2}$. On the other
hand we get the following commutative diagram

$\textstyle{[0,2\pi]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma}$$\scriptstyle{\varphi\circ\Gamma}$$\textstyle{T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Z}$$\scriptstyle{\varphi}$$\textstyle{T(T^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\varphi}$$\textstyle{\quad
D\subset\mathbb{R}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(D)}$

Therefore,

${(d\varphi\circ Z\circ\varphi^{-1})|}_{\varphi\circ\Gamma}={d\varphi\circ
Z|}_{\Gamma}={d\varphi(\partial/\partial
x)|}_{\Gamma}={\dfrac{\partial\varphi}{\partial x}|}_{\Gamma}\,.$ (2.5)

The partial derivative of $\varphi$ with respect to $x$ is

$\begin{split}\dfrac{\partial\varphi}{\partial
x}(x,y)=\cfrac{1}{H}&\big{(}-q\cos(px+qy)-p(py-qx+2\pi)\sin(px+qy),\\\
&-q\sin(px+qy)-p(py-qx+2\pi)\cos(px+qy)\big{)}\,.\end{split}$

Thus, taking into account (2.4) and the given notation, we obtain that

${\tfrac{\partial\varphi}{\partial x}|}_{\Gamma}=\dfrac{1}{H}(-qc-2\pi
ps,-qs-2\pi pc)\,.$

Now, having in mind that $c^{\prime}=-Hs$ and $s^{\prime}=Hc$, we can compute
the derivative

$({\tfrac{\partial\varphi}{\partial x}|}_{\Gamma})^{\prime}=(qs+2\pi
pc,-qc-2\pi ps)\,.$

Finally, we have that the square norm is

${\big{\|}{\tfrac{\partial\varphi}{\partial
x}|}_{\Gamma}\big{\|}}_{2}^{2}=\frac{1}{H}\big{(}(qc+2\pi ps)^{2}+(qs+2\pi
pc)^{2}\big{)}\,,$

in consequence

$\det({\tfrac{\partial\varphi}{\partial
x}|}_{\Gamma},({\tfrac{\partial\varphi}{\partial
x}|}_{\Gamma})^{\prime})=\cfrac{1}{H}\big{(}(qc+2p\pi s)^{2}+(qs+2p\pi
c)^{2}\big{)}={\big{\|}{\tfrac{\partial\varphi}{\partial
x}|}_{\Gamma}\big{\|}}_{2}^{2}\,.$

This equality together with (2.5), implies that

$\operatorname{ind}_{\varphi\circ\Gamma}(d\varphi\circ
Z\circ\varphi^{-1})=\cfrac{1}{2\pi}\int_{0}^{2\pi}\frac{\det({\tfrac{\partial\varphi}{\partial
x}|}_{\Gamma},({\tfrac{\partial\varphi}{\partial
x}|}_{\Gamma})^{\prime})}{{\big{\|}{\tfrac{\partial\varphi}{\partial
x}|}_{\Gamma}\big{\|}}_{2}^{2}}\,dt=1\,,$

which is the claim of the theorem. ∎

######  Remark 2.6.

Observe that the domain $D$ given in the proof of theorem above can be taken
diffeomorphic to the interior of the corona
$\\{(x,y)\in\mathbb{R}^{2}:1\leq\sqrt{x^{2}+y^{2}}\leq 2\\}$. Thus any vector
field defined on a compact domain dipheomorphic to $C$, defined by two
contours $\Gamma_{1}$ and $\Gamma_{2}$, so that it is uniformizable over the
torus by means of the inverse of the map $\varphi$ defined in (2.3) is a good
candidate that could satisfies that $\operatorname{ind}_{\Gamma_{i}}X=1$, for
$i=1,2$.

######  Remark 2.7.

One can see that our approach to the study of the index is quite different to
the one given by Finn in [5]. There are two main reasons that bring us to
change the original point of view: First, the notion of index along a curve
has evolved, in fact, sometimes it is called winding number of a vector field
(see e.g. [3]). Second, some results given in the original paper, although
they were correct, they were based (a little) on the intuition.

In our last result we provide a necessary condition for the uniformization
over the torus for suitable vector fields.

######  Corollary 2.8.

Let $X$ be a vector field on a bounded and connected domain. Assume that $X$
is uniformizable over the torus through a domain in $T^{2}$ which is
diffeomorphic to $T^{2}\setminus\Gamma_{0}$ for a Jordan curve $\Gamma_{0}$.
Then, for each Jordan curve $\Gamma$ with the same type as $\Gamma_{0}$
satisfying that $\Gamma\cap\Gamma_{0}=\emptyset$, there exists a
diffeomorphism $\psi$ such that

$\operatorname{ind}_{\varphi\circ\Gamma}(d\psi\circ X\circ\psi^{-1})=1\,.$

###### Proof.

The proof is easy to deduce taking into account the following diagram

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{X}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\!\\!\\!\\!\\!\alpha}$$\scriptstyle{Z}$$\textstyle{T^{2}\setminus\Gamma_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\beta}$$\scriptstyle{\widehat{Z}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{T(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\varphi}$$\textstyle{T(\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\!\\!\\!\\!\\!d\alpha}$$\textstyle{T(T^{2}\setminus\Gamma_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\;d\beta}$$\textstyle{T(C)\,,}$

where $Z$, $\widehat{Z}$, $\varphi$ and $\alpha$ are the maps defined in the
hypothesis and $U$ and $\beta$ are defined in Theorem 2.5. Observe that we can
choose $\widehat{Z}$ as a constant vector field, since $Z$ is constant. Now,
having in mind that the vector field $\widehat{Z}=d\alpha\circ d\varphi\circ
X\circ\varphi^{-1}\circ\alpha^{-1}$ is constant and applying theorem above for
each $\Gamma$ as in the hypothesis, we obtain the result. ∎

Despite we cannot give a sufficient condition, the condition for the index is
meaningful and provides a good clue for the wanted uniformization.

Acknowledgment. The original idea of a generalization of the uniformization in
the sense of Finn was suggested by Professor José A. Martínez Alfaro around
the year 2010. I am very grateful for his support and encouragement along
those years.

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