# The points of space are not points: their wavefunctions remove the
ultraviolet divergences of quantum field theory

Kevin Cahill<EMAIL_ADDRESS>Department of Physics and Astronomy
University of New Mexico
Albuquerque, New Mexico 87106

(September 3, 2024)

###### Abstract

The points $p(x)$ of space are observables that obey quantum mechanics with
average values $\langle p(x)\rangle$ that obey general relativity. In empty
flat 3-space, a point $p(x)$ may be represented as the sum
$\bm{p}(x)=\langle\bm{p}(x)\rangle+\bm{\epsilon}(x)$ of its average value
$\langle\bm{p}(x)\rangle=\bm{x}$ and its quantum remainder $\bm{\epsilon}(x)$.
The ubiquitous Fourier exponential $\exp(i\bm{k}\cdot\bm{p}(x))$ is then
$\exp(i\bm{k}\cdot(\bm{x}+\bm{\epsilon}))$. If the probability distribution of
the quantum remainders $\bm{\epsilon}(x)$ is normal and of width $\sigma$,
then the average of the exponential
$\langle\exp(i\bm{k}(\bm{x}+\bm{\epsilon})\rangle$ contains a gaussian factor
$\exp(-\sigma^{2}\bm{k}^{2}/2)$ which makes self-energies, Feynman diagrams
and dark energy ultraviolet finite. This complex of ideas requires new bosons
and suggests that supersymmetry is softly broken.

##  I Introduction

Einstein’s equations $G_{ik}=8\pi\,G\,T_{ik}$ relate the energy-momentum
tensor $T_{ik}$ to the Einstein tensor $G_{ik}$ which is a sum of products of
the metric $g_{ik}$, its inverse and its derivatives. The metric $g_{ik}(x)$
itself is quadratic in the derivatives

$\displaystyle g_{ik}(x)={}$ $\displaystyle\frac{\partial p(x)}{\partial
x^{i}}\cdot\frac{\partial p(x)}{\partial x^{k}}\equiv{}-\frac{\partial
p^{0}(x)}{\partial x^{i}}\frac{\partial p^{0}(x)}{\partial
x^{k}}+\sum_{\alpha=1}^{s}\frac{\partial p^{\alpha}(x)}{\partial
x^{i}}\frac{\partial p^{\alpha}(x)}{\partial x^{k}}$ (1)

of the point $p(x)$ which lies in a flat semi-euclidian embedding space of one
dimension of time and $s\leq 19$ dimensions of space [1, 2, 3, 4]. Einstein’s
equations therefore relate derivatives of the points $p(x)$ of our
4-dimensional curved space to the energy-momentum tensor $T_{ik}$ which is a
sum of products of hermitian operators. It follows that the points $p(x)$ are
themselves hermitian operators. Their average values $\langle p(x)\rangle$
obey general relativity, and their remainders $\epsilon(x)=p(x)-\langle
p(x)\rangle$ obey quantum mechanics. Their coordinates $x$ are merely labels.

##  II Empty, flat space

In empty flat space $\mathcal{E}$, the average
$\langle\mathcal{E}|\bm{p}(x)|\mathcal{E}\rangle\equiv\langle\bm{p}(x)\rangle$
of the spatial components of a point $\bm{p}(\bm{x})$ is its position
$\langle\bm{p}(x)\rangle=\bm{x}$ in the rest frame of the CMB. A point
$\bm{p}(x)$ of space is the sum of its position $\bm{x}$ and its quantum-
mechanical remainder $\bm{\epsilon}(x)$

$\bm{p}(x)={}\langle\mathcal{E}|\bm{p}(x)|\mathcal{E}\rangle+\bm{\epsilon}(x)=\bm{x}+\bm{\epsilon}(x).$
(2)

In empty flat space $\mathcal{E}$ and in the rest frame of the CMB, the
probability distribution
$P(\bm{\epsilon})=|\langle\bm{\epsilon}|\mathcal{E}\rangle|^{2}$ of the three
components $\bm{\epsilon}(x)$ should depend only upon $\bm{\epsilon}^{2}$ and
not upon $\bm{x}$ and be tightly concentrated about $\bm{\epsilon}=0$. For
simplicity I will assume that the components $\bm{\epsilon}(x)$ are normally
distributed

$\displaystyle P(\bm{\epsilon})\equiv{}$
$\displaystyle|\langle\bm{\epsilon}|\mathcal{E}\rangle|^{2}={}\frac{1}{(2\pi)^{3/2}\sigma^{3}}\,\exp\left(-\frac{\bm{\epsilon}^{2}}{2\sigma^{2}}\right)$
(3)

in which $\sigma$ is a length of the order of $M_{\text{\tiny{Planck}}}^{-1}$.
The present formalism may entail violations of Lorentz invariance on scales of
the order of the inverse of the Planck mass.

The average value of the exponential $\exp(\pm
i\bm{k}\cdot(\bm{x}+\bm{\epsilon}(x)))$ in flat space $\mathcal{E}$ is then

$\displaystyle\langle\mathcal{E}|\,\exp(\pm i\bm{k}$
$\displaystyle\cdot(\bm{x}+\bm{\epsilon}(x)))\,|\mathcal{E}\rangle={}\int
d^{3}\bm{\epsilon}(\bm{x})\,\,P(\bm{\epsilon}(x))\exp(\pm
i\bm{k}\cdot(\bm{x}+\bm{\epsilon}(x)))$ $\displaystyle={}$ $\displaystyle\int
d^{3}\bm{\epsilon}(\bm{x})\,\,|\langle\bm{\epsilon}(\bm{x})|\mathcal{E}\rangle|^{2}\exp(\pm
i\bm{k}\cdot(\bm{x}+\bm{\epsilon}(x)))$ $\displaystyle={}$
$\displaystyle\int\frac{d^{3}\bm{\epsilon}}{\sigma^{3}(2\pi)^{3/2}}\,\,e^{\pm
i\bm{k}\cdot(\bm{x}+\bm{\epsilon})-\bm{\epsilon}^{2}/2\sigma^{2}}={}\exp\big{(}\pm
i\bm{k}\cdot\bm{x}-\sigma^{2}\bm{k}^{2}/2\big{)}$ (4)

which is the original exponential $\exp(\pm i\bm{k}\cdot\bm{x})$ multiplied by
the damping factor $\exp(-\sigma^{2}\bm{k}^{2}/2)$.

##  III Fields in Flat Space

The usual Fourier decomposition of a scalar field in flat space is

$\displaystyle\phi(x)={}$
$\displaystyle\int\frac{d^{3}k}{\sqrt{(2\pi)^{3}2\omega_{k}}}\,\Big{[}e^{ikx}a(k)+e^{-ikx}a^{\dagger}(k)\Big{]}$
(5)

in which $kx=\bm{k}\cdot\bm{x}-\omega_{k}x^{0}$ and
$\omega_{k}=\sqrt{\bm{k}^{2}+m^{2}}$. Replacing the exponentials $e^{\pm ikx}$
by their average values (4) in empty flat space, we find for the averaged
field

$\displaystyle\langle\mathcal{E}|\,\phi(x)\,|\mathcal{E}\rangle={}$
$\displaystyle\int\frac{d^{3}k}{\sqrt{(2\pi)^{3}2\omega_{k}}}\,\Big{[}e^{ikx}a(k)+e^{-ikx}a^{\dagger}(k)\Big{]}\,e^{-\sigma^{2}\bm{k}^{2}/2}.$
(6)

As $\sigma\to 0$, the averaged field $\big{\langle}\phi(x)\big{\rangle}$
becomes the usual field (5), but as long as $\sigma\neq 0$, the factor
$e^{-\sigma^{2}\bm{k}^{2}/2}$ makes Feynman diagrams finite.

##  IV Basic commutation relation

The equal-time commutation relation for a scalar field with $\bm{x}$ and
$\bm{y}$ replaced by $\bm{x}+\bm{\epsilon}(x)$ and $\bm{y}+\bm{\epsilon}(y)$
is

$\begin{split}\bm{[}\phi(t,\bm{x}+\bm{\epsilon}(x)),\pi(t,\bm{y}+\bm{\epsilon}(y))\bm{]}={}&i\,\int\frac{d^{3}k}{(2\pi)^{3}}\exp\Big{(}i\bm{k}\cdot\big{(}\bm{x}+\bm{\epsilon}(x)-\bm{y}-\bm{\epsilon}(y)\big{)}\Big{)}.\end{split}$
(7)

Its average in empty flat space $\mathcal{E}$ is

$\displaystyle\langle\mathcal{E}|\bm{[}\phi(t,\bm{x}),\pi(t,\bm{y})\bm{]}|\mathcal{E}\rangle={}$
$\displaystyle
i\\!\int\frac{d^{3}k\,d^{3}\bm{\epsilon}(x)\,d^{3}\bm{\epsilon}(y)}{(2\pi\sigma)^{6}}e^{i\bm{k}(\bm{x}+\bm{\epsilon}(x)-\bm{y}-\bm{\epsilon}(y))-(\bm{\epsilon}^{2}(x)+\bm{\epsilon}^{2}(y))/2\sigma^{2}}$
$\displaystyle={}$ $\displaystyle
i\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k}\cdot(\bm{x}-\bm{y})-\sigma^{2}\bm{k}^{2}}=\frac{i}{(2\sqrt{\pi}\sigma)^{3}}\,e^{-(\bm{x}-\bm{y})^{2}/4\sigma^{2}}$
(8)

which approaches the usual expression as $\sigma\to 0$ but is finite for all
$\bm{x}$ and $\bm{y}$.

##  V Feynman’s propagator

For scalar fields, the Feynman propagator is

$\displaystyle\Delta_{\text{F}}(x-y)=\int\frac{d^{4}q}{(2\pi)^{4}}\,\,\frac{e^{iq(x-y)}}{q^{2}+m^{2}-i\epsilon}.$
(9)

With $\bm{x}\to\bm{x}+\bm{\epsilon}(x)$ and
$\bm{y}\to\bm{y}+\bm{\epsilon}(y)$, it is

$\displaystyle\Delta_{\text{F}}(x-y)=\int\frac{d^{4}q}{(2\pi)^{4}}\,\,\frac{e^{iq(x+\bm{\epsilon}(x)-y-\bm{\epsilon}(y))}}{q^{2}+m^{2}-i\epsilon}.$
(10)

This expression averaged over the normal distribution (3) is

$\displaystyle\langle\mathcal{E}|\Delta_{\text{F}}(x-y)|\mathcal{E}\rangle={}$
$\displaystyle\int\frac{d^{4}q\,d^{3}\bm{\epsilon}(x)\,d^{3}\bm{\epsilon}(y)}{(2\pi)^{4}}\,\frac{e^{iq(x+\bm{\epsilon}(x)-y-\bm{\epsilon}(y))}}{q^{2}+m^{2}-i\epsilon}\,P(\bm{\epsilon}(x))P(\bm{\epsilon}(y))$
$\displaystyle={}$
$\displaystyle\int\frac{d^{4}q\,d^{3}\bm{\epsilon}_{x}\,d^{3}\bm{\epsilon}_{y}}{(2\pi)^{6}\sigma^{6}}\frac{e^{iq(x+\bm{\epsilon}(x)-y-\bm{\epsilon}(y))-(\bm{\epsilon}^{2}(x)+\bm{\epsilon}^{2}(y))/2\sigma^{2}}}{q^{2}+m^{2}-i\epsilon}$
(11) $\displaystyle={}$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}\,\,\frac{e^{iq(x-y)-\sigma^{2}\bm{q}^{2}}}{q^{2}+m^{2}-i\epsilon}\equiv\Delta_{\text{F},\sigma}(x-y)$

which reduces to the usual propagator as $\sigma\to 0$ but is finite even at
$x=y$.

##  VI Self-energy

In second-order $\phi^{3}$ theory with action density

$\mathcal{L}={}-\frac{1}{2}\partial_{i}\phi\,\partial^{i}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{3!}\phi^{3}$
(12)

($i=0$ to 3), the self-energy amplitude

$\displaystyle A(k,k^{\prime})={}$
$\displaystyle\frac{\lambda^{2}\delta(k-k^{\prime})}{2(2\pi)^{3}\sqrt{2k^{0}2k^{\prime
0}}}\\!\int\\!\frac{d^{4}q}{(q^{2}+m^{2}-i\epsilon)((q-k)^{2}+m^{2}-i\epsilon)}$
(13)

is quadratically divergent.

If the propagators (11) averaged over fluctuations
$\Delta_{\text{F},\sigma}(x-y)$ are used, then the amplitude $A(k,k^{\prime})$
becomes a convergent integral

$\displaystyle A_{\sigma}(k,k^{\prime})={}$
$\displaystyle\frac{\lambda^{2}\,\delta(k-k^{\prime})}{2(2\pi)^{3}\,\sqrt{2k^{0}2k^{\prime
0}}}\\!\int\\!d^{4}q\,\,\frac{e^{-\sigma^{2}(\bm{q}^{2}+(\bm{q}-\bm{k})^{2})}}{(q^{2}+m^{2}-i\epsilon)((q-k)^{2}+m^{2}-i\epsilon)}$
(14)

which at $\sigma=0$ is the usual formula (13) and at $k=0$ is

$\displaystyle A_{\sigma}(0,k^{\prime})={}$
$\displaystyle\frac{\lambda^{2}\,\delta(k^{\prime})}{4(2\pi)^{3}\,m}\\!\int\\!d^{4}q\,\,\frac{e^{-2\,\sigma^{2}\bm{q}^{2}}}{(q^{2}+m^{2}-i\epsilon)^{2}}={}-\frac{im\lambda^{2}\delta(k^{\prime})}{4\pi\sigma^{2}}\,e^{\sigma^{2}m^{2}}K_{1}(\sigma^{2}m^{2}).$
(15)

##  VII Vacuum energy

The hamiltonian for a single scalar field is

$\displaystyle H={}$
$\displaystyle\frac{1}{2}\int\Big{(}\dot{\phi}^{2}(x)+(\nabla\phi(x))^{2}+m^{2}\phi^{2}(x)\Big{)}d^{3}x.$
(16)

Its average value in the flat-space vacuum $|\mathcal{E}\rangle$ is

$\displaystyle\langle\mathcal{E}|H|\mathcal{E}\rangle={}$
$\displaystyle\frac{1}{2}\int\langle\mathcal{E}|\dot{\phi}^{2}(x)+(\nabla\phi(x))^{2}+m^{2}\phi^{2}(x)|\mathcal{E}\rangle
d^{3}x$ (17)

in which, for instance, $\phi^{2}(x)$ is the limit of $\phi(x)\,\phi(y)$ as
$y\to x$. Using the averaged field (6), we find for the energy density

$\displaystyle\rho_{m}={}$
$\displaystyle\int\frac{d^{3}k}{2(2\pi)^{3}}\,\sqrt{m^{2}+\bm{k}^{2}}\,\,e^{-\sigma^{2}\bm{k}^{2}}={}\frac{m^{2}}{16\pi^{2}\,\sigma^{2}}\,e^{\sigma^{2}m^{2}/2}\,K_{1}(\sigma^{2}m^{2}/2).$
(18)

The energy density due to a single fermionic mode of mass $m_{f}$ is
${}-\rho_{m_{f}}$.

##  VIII Dark energy, dark matter and supersymmetry

So if there are $B$ bosonic modes each of mass $m_{b}$ and $F$ fermionic modes
each of mass $m_{f}$, then the density of dark energy is an alternating sum of
the densities $\rho_{m_{b}}$ and $\rho_{m_{f}}$

$\displaystyle\rho_{\text{de}}^{\text{th}}={}$
$\displaystyle\frac{1}{16\,\pi^{2}\sigma^{2}}\left(\sum_{b=1}^{B}m_{b}^{2}\,e^{\sigma^{2}m_{b}^{2}/2}K_{1}\big{(}\sigma^{2}m_{b}^{2}/2\big{)}-\sum_{f=1}^{F}m_{f}^{2}\,e^{\sigma^{2}m^{2}_{f}/2}K_{1}\big{(}\sigma^{2}m^{2}_{f}/2\big{)}\right).$
(19)

There are $B=30$ known bosonic modes and $F=96$ known fermionic modes if the
three neutrinos are nearly massless Dirac neutrinos.

The fundamental length $\sigma$ is assumed to be very short, perhaps of the
order of the Planck length, so in the formula (19) for the density of dark
energy $\rho_{\text{de}}^{\text{th}}$, the first two terms in the expansion of
the modified Bessel function

$K_{1}({\textstyle{\frac{1}{2}}}\sigma^{2}m^{2})=\frac{2}{\sigma^{2}m^{2}}+\frac{\sigma^{2}m^{2}}{8}\Big{(}2\gamma-1+2\log({\textstyle{\frac{1}{4}}}\sigma^{2}m^{2})\Big{)}+\dots$
(20)

are the most important. Here $\gamma=0.5772156649$ is the Euler–Mascheroni
constant.

Keeping only the first term in this expansion (20), we get for the density of
dark energy

$\displaystyle\rho_{\text{de}}^{\text{th}}\approx{}$
$\displaystyle\frac{B-F}{8\,\pi^{2}\,\sigma^{4}}={}-\frac{33}{4\,\pi^{2}\,\sigma^{4}}$
(21)

which is negative and too great by many orders of magnitude. So if the
theoretical density of dark energy $\rho_{\text{de}}^{\text{th}}$ is
approximately equal to the observed density of dark energy
$\rho^{\text{obs}}_{\text{de}}$, then we must have $B=F$ as in supersymmetry.
Thus there must be at least 66 bosonic modes. These modes must be very heavy,
dark or both.

If $B=F$, then the estimate (19) for $\rho_{\text{de}}^{\text{th}}$ and the
approximation (20) give

$\displaystyle\rho_{\text{de}}^{\text{th}}\approx{}$
$\displaystyle\frac{1}{16\pi^{2}}\sum_{\alpha}(-1)^{2j_{\alpha}}\Big{[}\frac{m_{\alpha}^{2}}{\sigma^{2}}+\Big{(}\frac{m_{\alpha}^{4}}{8}+\frac{\sigma^{2}m_{\alpha}^{6}}{16}\Big{)}\Big{(}2\gamma-1+2\log({\textstyle{\frac{1}{4}}}\sigma^{2}m_{\alpha}^{2})\Big{)}\Big{]}.$
(22)

In models with softly broken supersymmetry, the alternating sum of squared
masses vanishes $\sum_{\alpha}(-1)^{2j}m^{2}_{\alpha}=0$ at tree level leaving

$\displaystyle\rho_{\text{de}}^{\text{th}}\approx{}$
$\displaystyle\frac{1}{128\pi^{2}}\sum_{\alpha}(-1)^{2j_{\alpha}}\Big{[}m_{\alpha}^{4}\Big{(}2\gamma-1+2\log({\textstyle{\frac{1}{4}}}\sigma^{2}m_{\alpha}^{2})\Big{)}+\sigma^{2}m_{\alpha}^{6}\,\log({\textstyle{\frac{1}{4}}}\sigma^{2}m_{\alpha}^{2})\Big{]}.$
(23)

This estimate $\rho_{\text{de}}^{\text{th}}$ is generically orders of
magnitude greater than the observed dark-energy density
$\rho^{\text{obs}}_{\text{de}}=2.74\times 10^{-47}\,(\text{GeV})^{4}$ unless
the alternating sum of quartic masses $\sum_{\alpha}(-1)^{2j}m^{4}_{\alpha}=0$
vanishes.

##  IX Conclusion

Einstein’s equation $G_{ik}=8\pi\,G\,T_{ik}$ relates derivatives of points
$p(x)$ to the energy-momentum tensor which is a quantum-mechanical operator.
The points $p(x)$ of space therefore are quantum-mechanical operators. Their
average values $\langle p(x)\rangle$ obey general relativity, and their
remainders $\epsilon(x)=p(x)-\langle p(x)\rangle$ obey quantum mechanics.

In empty flat 3-space, a point $p(x)$ may be represented as the sum
$\bm{p}(x)=\langle\bm{p}(x)\rangle+\bm{\epsilon}(x)$ of its average value
$\langle\bm{p}(x)\rangle=\bm{x}$ and its quantum remainder $\bm{\epsilon}(x)$.
The ubiquitous Fourier exponential $\exp(i\bm{k}\cdot\bm{p}(x))$ is then
$\exp(i\bm{k}\cdot(\bm{x}+\bm{\epsilon}))$. If the probability distribution of
the quantum remainders $\bm{\epsilon}(x)$ is normal and of width $\sigma$,
then the average of the exponential
$\langle\exp(i\bm{k}(\bm{x}+\bm{\epsilon})\rangle$ contains a gaussian factor
$\exp(-\sigma^{2}\bm{k}^{2}/2)$ which makes self-energies, Feynman diagrams
and dark energy ultraviolet finite. This complex of ideas requires dark matter
and suggests that supersymmetry is softly broken.

##  X Acknowledgments

I am grateful to Rouzbeh Allahverdi, Michael Grady, David Kaiser and Malcolm
Perry for conversations and email.

## References

  * Greene [1970] R. E. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds (American Mathematical Society, Providence, R.I., 1970) pp. iii+63, memoirs of the American Mathematical Society, No. 97\.
  * Clarke [1970] C. J. S. Clarke, On the global isometric embedding of pseudo-Riemannian manifolds, Proc. Roy. Soc. London Ser. A 314, 417 (1970).
  * Müller and Sánchez [2011] O. Müller and M. Sánchez, Lorentzian manifolds isometrically embeddable in $\mathbb{L}^{N}$, Trans. Amer. Math. Soc. 363, 5367 (2011).
  * Aké  _et al._ [2020] L. Aké, J. L. Flores, and M. Sánchez, Structure of globally hyperbolic spacetimes with timelike boundary, Rev. Matem. Iberoamericana , 45 (2020), arXiv:1808.04412 [gr-qc] .


#  A Credibility-aware Swarm-Federated Deep Learning Framework in Internet of
Vehicles

Zhe Wanga, Xinhang Lia, Tianhao Wu∗a, Chen Xua, Lin Zhangba
aSchool of Artificial Intellegence, Beijing University of Posts and
Telecommunications, Beijing 100876, China
bBeijing Information Science and Technology University, Beijing 100096, China

###### Abstract

Federated Deep Learning (FDL) is helping to realize distributed machine
learning in the Internet of Vehicles (IoV). However, FDL’s global model needs
multiple clients to upload learning model parameters, thus still existing
unavoidable communication overhead and data privacy risks. The recently
proposed Swarm Learning (SL) provides a decentralized machine-learning
approach uniting edge computing and blockchain-based coordination without the
need for a central coordinator. This paper proposes a Swarm-Federated Deep
Learning framework in the IoV system (IoV-SFDL) that integrates SL into the
FDL framework. The IoV-SFDL organizes vehicles to generate local SL models
with adjacent vehicles based on the blockchain empowered SL, then aggregates
the global FDL model among different SL groups with a proposed credibility
weights prediction algorithm. Extensive experimental results demonstrate that
compared with the baseline frameworks, the proposed IoV-SFDL framework
achieves a 16.72% reduction in edge-to-global communication overhead while
improving about 5.02% in model performance with the same training iterations.

###### keywords:

Swarm Learning , Federated Deep Learning , Internet of Vehicles , Privacy ,
Efficiency

Fig. 1: An overview of the proposed IoV-SFDL framework. (1) Vehicles in the
end-client layer form an SL group according to the relative distance and task
type, update the latest global model in the group issued by Road Side Unit
(RSU), collect surrounding information with onboard sensors, and train the
model. (2) The vehicles in the SL group connect through blockchain and
aggregate group model in the SL process. (3) RSU nodes at the edge-server
layer receive the model from SL groups and predict the weights according to
group size and model credibility. (4) RSU nodes upload model parameters and
weights to the central server. (5) The central server generates a global model
by aggregating models from SL groups with different weights, and distributes
the global model to the edge server for subsequent updates. After several
iterations, the final model with good performance is obtained.

##  1 Introduction

With new computing and communication technologies coming up, the communication
network’s rapid development provides the possibility for advanced vehicular
services and applications [1], such as autonomous driving and content
delivery. In this context, the Internet of Vehicles (IoV), as a combination of
the Internet of Things (IoT) and Vehicular networks, has gained considerable
attention from industry and academia. Some works in [2, 3, 4] have shown that
compared with data centralized in traditional machine learning (ML), Federated
Deep Learning (FDL) solves privacy concerns and reduces the cost of data
transmission by distributing the training work to clients. Clients keep their
data locally and send the model parameters to a central server for model
aggregation[5]. In this way, FDL provides a parallel scheme to learn the
global model cooperatively and protect data privacy [6].

As for the FDL application in the IoV, it demands the efficiency of
information exchanging about vehicle statuses and road conditions while
protecting data privacy [7]. Scholars have made several communication
efficiency improvements and achieved sound performance in IoV scenarios. For
instance, [8] proposed a Federated-Learning-Based Cooperative Positioning
Scheme (FedVCP) to fully utilize the potential of collaborative edge computing
while protecting data privacy in the IoV system. [9] proposed a novel
Federated Learning-based framework for Plate Recognition (FedLPR) to identify
traffic signs and participants in low latency and consumption while keeping
pace with preserving data privacy. [10] proposed a two-timescale federated
Deep Reinforcement Learning(DRL)-based algorithm to help obtain robust models
in the IoV scenarios.

[11] integrated homomorphic cryptography into parking space searching and
booking for communication security. [12] designed a data collection and pre-
processing scheme based on deep learning, reducing the amount of data uploaded
to the cloud and effectively protecting the data privacy in the IoV system.
Blockchain technology has been applied to IoV for data privacy in recent years
as a newly emerging technology. For instance, [13] created a local Blockchain
for real-world event message exchange among vehicles within the boundary of
the country, and [14] proposed a DRL and blockchain-empowered Spatial Crowd-
sourcing System (DB-SCS). [15] developed a hybrid blockchain architecture that
consists of the permission blockchain and the local Directed Acyclic Graph to
relieve transmission load and address privacy concerns of data providers.

Based on blockchain technology and FDL, The authors in [16] introduced Swarm
Learning (SL) and achieved outstanding performance in the medical field. The
SL framework dispenses with a dedicated server, shares the parameters via the
Swarm network, and builds the models independently on private data at the
individual sites. However, all the above works don’t consider vehicles’
mobility, unreliable communication connection, and a highly dynamic driving
environment. These factors will bring some new challenges for FDL and
blockchain applications in the IoV system, including overcoming the model’s
ineffective and communication inefficiency caused by clients (such as
malicious and redundant sharing parameters) and ensuring clients’ privacy
security from the third party in the process of data transmission [17].

This paper addresses these issues by proposing a Swarm-Federated Deep Learning
framework in the IoV system (IoV-SFDL) with a credibility weights prediction
algorithm. The contributions of this paper can be summarized as follows:

(1) An IoV-SFDL framework is proposed to ensure data privacy with blockchain
technology while minimizing the total cost of edge-to-global communication
overhead, consisting of a central server, RSU-based edge servers, and
vehicles.

(2) A credibility weights prediction algorithm is proposed to improve the
model convergence efficiency in the global model aggregation process.

(3) Various experiments in the IoV system are implemented to verify the
performance of the IoV-SFDL framework and credibility weights prediction
algorithm on the Next Generation Simulation (NGSIM) dataset.

The rest of this paper is organized as follows: The system architecture is
present in Section 2. The proposed credibility weights prediction algorithm is
detailed in Section 3. Section 4 provides the NGSIM trajectory dataset,
framework performance evaluation, and analysis. Section 5 concludes this
paper.

##  2 System Architecture

The overview of the proposed IoV-SFDL framework is shown in Figure 1. The
problem formation, data transmission and framewrok detail of the proposed IoV-
SFDL will be introduced in this section.

Table 1: SUMMARY OF MAIN NOTATIONS $F(w)$ | The loss function of global model in cloud server
---|---
$F_{SL}(w)$ | The loss function of the aggregated model in SL group
$f_{i}(w)$ | The loss function of the model in the $i^{th}$ vehicle
$W$ | Aggregated model of central server
$W_{SL}$ | Aggregated model of SL group
$L_{m}(i)$ | Local model parameters of the $i^{th}$ vehicle
$R_{m}(i)$ | Merged model parameters of the $i^{th}$ vehicle
$C_{b}(i)$ | The robustness value of the $i^{th}$ SL group
$C_{e}(i)$ | The credibility value of the $i^{th}$ SL group
$\lambda_{i}$ | The model weights of the $i^{th}$ RSU node
$D_{i}$ | The dataset obtained by onboard sensors of the $i^{th}$ vehicle
$e_{i}$ | The driving environment of the $i^{th}$ vehicle
$w^{i}$ | Onboard model of the $i^{th}$ vehicle
$\gamma^{i}$ | The model weights of the $i^{th}$ vehicle

###  2.1 Problem Formation

This paper assumes that a highway with $N$ vehicles facing different traffic
conditions. The main notations used throughout the paper are listed in Table
1. Because vehicular GPS, radars, cameras, and other onboard sensors can
obtain real-time vehicle trajectory, we believe that each vehicle can be
modeled as a triple $G=(V,P,E)$.

(1) $V=\left\\{v_{i}\mid i\in\\{1,2,\ldots\\}\right\\}$ represents all
vehicles, where $v_{i}$ represents vehicle $i$ on the road. $N=|V|$ represents
the number of vehicles on the road.

(2) $P=\left\\{p_{i}\mid v_{i}\in V\right\\}$ represents the private
information of the vehicle, including the speed, acceleration, orientation,
and position.

(3) $E=\left\\{e_{i}\mid v_{i}\in V\right\\}$ is a matrix, which is used to
store the road environment information around the vehicle, that is,
information of the vehicle from four directions in the visual distance.

The information $e_{i}$ of vehicle $v_{i}$ in road environment matrix is
defined as:

$\displaystyle\mathrm{e}_{i}$
$\displaystyle=\left[\begin{array}[]{cc}I_{r}&I_{l}\\\
I_{p}&I_{f}\end{array}\right]$ (1)
$\displaystyle\mathrm{I}\left(v_{i},v_{j}\right)$
$\displaystyle=\begin{cases}0&d\left(v_{i},v_{j}\right)>R\\\
1-\frac{\left(d\left(v_{i},v_{j}\right)\right)^{2}}{R^{2}}&d\left(v_{i},v_{j}\right)\leq
R\end{cases},$

where $d\left(v_{i},v_{j}\right)$ denotes the Euclidean Distance between
$v_{i}$ and $v_{j}$, $R$ represents the visual range of vehicles.

Fig. 2: The workflow in the proposed IoV-SFDL framework

Back propagation of the loss function is essential for ML training to adjust
weights and parameters. The loss function in Deep Neural Network (DNN) models
quantifies the modeling effect of the algorithm on the training data. The goal
of training is to find the optimal parameters to minimize the loss function.
The loss function in machine learning can be roughly divided into
classification loss and regression loss. The general loss functions for
classification include logarithmic loss, focus loss, Kullback-Leibler(KL)
divergence, exponential loss, hinge loss, etc. This paper uses the Mean Square
Error (MSE) as the loss function of the vehicle onboard model. The expression
of the loss function $f_{i}(w)$ of the vehicle $v_{i}$ is as follows:

$f_{i}(w)=\frac{\sum_{n=1}^{N}\left|{y}_{predict}^{n}-y_{observe}^{n}\right|^{2}}{N},$
(2)

where $y_{predict}^{n}$ and $y_{observe}^{n}$ represent the model prediction
and observation of the sample batch $n$, and $N$ is the number of batches.
Different functions are available for parameter merging as a configuration of
the Swarm Learning, such as average, weighted average, minimum, maximum, or
median functions. In this paper, the weighted average is defined as :

$R_{m}(i)=\frac{\sum_{k=1}^{i}\left(\gamma^{k}\times
L_{m}(k)\right)}{i\times\sum_{k=1}^{i}\gamma^{k}},$ (3)

in which $R_{m}(i)$ is merged parameters of the $i^{th}$ vehicle. $L_{m}(k)$
is model parameters from the $k^{th}$ vehicle, $\gamma$ is the model weight.
In this way, the loss function of an SL group in this paper is defined as:

$F_{SL}(w)=\frac{1}{\left|SL\right|}\sum_{i\in SL}[f_{i}(w)+f_{i-1}(w)],$ (4)

where $|SL|$ means the number of vehicles in an SL group.

Redundant or invalid experience data decrease the efficient implementation of
the model aggregation process. According to the model’s effectiveness, the
weights could be flexibly adjusted by the proposed credibility weights
prediction algorithm. Making the model with better results have a higher
weight in model aggregation helps the model converge faster and achieve higher
accuracy in FDL. To describe the problem of model aggregation in IoV-SFDL, we
introduce ${\lambda_{i}=\lambda_{1},\lambda_{2},\dots,\lambda_{|SL|}}$ as the
weights in global model aggregation. $|SL|$ represents the number of SL groups
on the road. The objective function of global learning optimization is defined
as:

$F(w)=\frac{1}{\left|SL\right|}\sum_{i\in SL}\lambda_{i}F_{{SL}_{i}}(w).$ (5)

As mentioned earlier, we use the minimization of the loss of the global model
for the current training task as the objective function:

$min\,F(w)=min\,[\frac{1}{\left|SL\right|}\sum_{i\in
SL}\lambda_{i}F_{{SL}_{i}}(w)].$ (6)

###  2.2 Swarm-Federated Deep Learning Framework

The IoV-SFDL framework in the IoV system contains three layers: A central
server in the cloud layer, RSU-based edge nodes in the edge layer, Vehicle
clients in the end layer. The three-layer structure constitutes the whole IoV-
SFDL architecture, and the operation is divided into three steps as a whole.
First of all, vehicles in the end layer collect trajectory information and
train a time-sensitive local model. According to the SL process, vehicles with
the same training task will form a blockchain network, with local model
parameters transferred and aggregated. Then, the RSUs receive the model
parameters from the SL group and predict model weights according to the
credibility weights prediction algorithm. At last, the parameters and weights
are transmitted to the central server for a global model aggregation.

The model’s aggregation process can be divided into local SL and global FDL
processes. Each vehicle aggregates the model passed from the previous vehicle
with its onboard model in the local swarm learning process. In this process,
the distributed feature of SL is combined with the point-to-point network
based on blockchain. Compared with the general operation of federated deep
learning, it provides a higher level of data security, central coordinator
elimination, and machine learning model attacks protection. According to the
proposed credibility weights prediction algorithm, each RSU predicts the
model’s weight in the global FDL process and sends the model parameters to the
central server. In this process, the global FDL realizes the global model
aggregation through a central server, which aggregates the value of local
model parameters according to their weights to generate a global model.
Compared with the general operation of FDL, the IoV-SFDL provides less edge-
to-global communication overhead and a more effective training process.

Above all, the data transmission scheme in the framework consists of five
steps: (1) Group forming and model updating; (2) Vehicle group-based Swarm
Learning; (3) Credible weights prediction and comparison; (4) Model parameters
and weights uploading; (5) Global Model Aggregation. After several iterations
of these steps, the final model with good performance is obtained.

##  3 Credibility Weights Prediction Scheme

The key to guaranteeing model security and training efficiency in the IoV
system is detecting ineffective models and scheduling less of them in each
communication round. This work proposes a credibility weights prediction
algorithm in the IoV-SFDL framework to prevent unreliable updates and minus
noise from models with poor effectiveness. With the credibility weights
prediction algorithm, the RSU node will evaluate the model’s effectiveness and
robustness of the received model and global model. The model’s credibility
will be predicted as the weight for the global model aggregation process in
each communication round. The total workflow is shown in Figure 2. The
algorithm can help the central server in FDL distinguish effective models from
ineffective ones, thereby preventing malicious or incorrect updates of the
global aggregate process in FDL.

###  3.1 Credibility Weights Prediction Algorithm

The Credible value $C_{i}$ of model $i$ in this paper can be divided as
$C_{b}$ and $C_{e}$, where $C_{b}$ and $C_{e}$ represent the robustness value
and effectiveness value of model from an SL group. To represent the node’s
robustness, we consider the logarithms’ formulation, a single increasing
function. In ML, we believe that the more datasets in the training process,
the better robustness of the model will be. Therefore, in global model
aggregation, the model trained on more datasets can be given higher weight.
Thus, the robustness value of the model with n vehicles in SL is given by the
following formula:

$C_{b}=\log_{k}n.$ (7)

In this formula, we believe that $n$ is the number of vehicles in the current
SL group, $k$ is the max group size among all SL groups. The specific
experimental parameters will be classified in the next section.

To represent the model’s effectiveness, we use observation $B$ made by the RSU
node to illustrate the effective prediction $P(e_{i})$ of a model. The
effectiveness of the model, $P\left(e_{i}\mid B\right)$ is calculated and
predicted by the RSU node based on the observation, which can be derived as:

$P\left(e_{i}\mid B\right)=\frac{P\left(e_{i}\right)P\left(B\mid
e_{i}\right)}{\sum_{j=1}^{n}P\left(e_{j}\right)P\left(B\mid e_{j}\right)},$
(8)

To facilitate the expression and update of effectiveness value, we use beta
distribution to represent aggregate weights from the effectiveness of an RSU
node, defined as:

$\begin{aligned}
f(x;p,q)=\frac{x^{p-1}(1-x)^{q-1}}{\int_{0}^{1}u^{p-1}(1-u)^{p-1}du}\\\ \\\
=\frac{\Gamma(p+q)}{\Gamma(p)\Gamma(q)}x^{p-1}(1-x)^{q-1}\\\ \\\
=\frac{1}{B(p,q)}x^{p-1}(1-x)^{q-1}\end{aligned},$ (9)

where $p>0\text{ and }q>0$ and p and q respectively represent the number of
times that the receiving model is better than the global model and the number
of times that the global model is better than the receiving model, the
$\Gamma(x)$ is the Gamma function, which could be written as:

$\Gamma(x)=\int_{0}^{+\infty}t^{x-1}e^{-t}\mathrm{~{}d}t(x>0),$ (10)

For example, we assume the $a_{i}$ and $b_{i}$ represent the positive and
negative behaviors currently, while p and q represent the initial positive and
negative behaviors. To update the credibility, it is equivalent to updating
the two parameters p and q, as:

$\left\\{\begin{array}[]{l}p^{\text{new }}=a_{i}+p\\\ q^{\text{new
}}=b_{i}+q\end{array}\right.,$ (11)

When a node is initialized without prior knowledge, the credibility of a node
can be expressed as a uniform distribution on $(0,1)$, defined as:

$P(x)=\operatorname{uni}(0,1)=\operatorname{Beta}(1,1).$ (12)

By comparing the effectiveness of the receiving model and the global model
previous time, the RSU can evaluate the effectiveness of the receiving model,
that is, convert into positive behavior and negative behavior. The evaluation
process of the models is detailed in the next section. Suppose that in $n$
rounds of communication, the SL group has sent the model to RSU $(p+q)$ times.
By comparing the effectiveness of the models, these interactions are described
as $p$ times positive behavior and $q$ times negative behavior. With this
information, the RSU can predict the model’s effectiveness from SL group in
the next event. The validity probability of the model is defined as:

$C_{e}=E[Beta(p,q)]=\frac{p}{p+q}.$ (13)

Central server can normalize all the weights and model parameters sent by the
RSUs, and the weights of model from $i^{th}$ RSU is defined as:

$\lambda_{i}=\frac{C_{i}}{\sum_{j=1}^{n}C_{j}}.$ (14)

###  3.2 Effectiveness Comparison Module

In the model aggregation process of FDL, the credibility comparison module at
RSU uses the test dataset to evaluate and compare the model received from SL
group and global model download from central server. Based on the value of the
loss function derived from the test dataset, we define the model’s credibility
for comparing the effectiveness of different models in model aggregation.

$e_{k}^{t}=\frac{f(w^{t},{x}_{ij},y_{ij})-f(W_{k}^{t},{x}_{ij},y_{ij})}{f({w}^{t},{x}_{ij},y_{ij})},$
(15)

where $\left({x}_{ij},y_{ij}\right)$ is a data sampled from the current RSU
node’s test dataset. We use the difference between the loss function value of
the SL model and global model to evaluate the quality of the received SL model
parameters.

For a tagged RSU node, its contribution to the global model aggregation can be
considered positive or negative. Precisely, if $e_{k}^{t}>0$, we assume that
the effectiveness comparison module will characterize RSU node $k$’s
transmission as positive behavior and vice versa. Moreover, we can also use
$e_{k}^{t}$ to represent the effect gap between the local model and the
receiving model imposed by the RSU.

###  3.3 Vehicle Group-based Swarm Learning

In the process of local model aggregation, vehicle group-based swarm learning
is considered. RSUs, as the participants in the local model aggregation
process, are evenly distributed on both sides of the road. The SL framework
cooperatively optimizes the model inside the vehicle through continuous
iteration of the following three steps:

1) Each vehicle carries out the model training task by collecting the privacy
information of local vehicles, such as geographical location, speed, direction
angle, etc.

2) Vehicles with the same subtask and close to each other can form groups and
aggregate models through blockchain in the SL process.

3) After model aggregation, the vehicle sends the model parameters to the RSU
for the further procession.

Generally, the number of vehicles in different SL groups is different due to
the rapid mobility of vehicles on the road and the uncertainty of driving
purpose. By evaluating the credibility of the SL groups and their internal
LSTM model effectiveness, the best weight in the global FDL aggregation
process could be predicted. The whole training process of vehicle group-based
SL is shown in Algorithm 1.

1 Input: $w_{t}^{k}$

2 Output: $W_{SL}$ Initialize $w_{0}^{k}$ ;

3 for  _each round t = 1, 2, …_ do

4 Update model ${w}_{t}^{k}$ from RSU;

5 $b\leftarrow$ spilt $D_{k}$ into batches of size $\mathcal{B}$;

6 for  _each vehicle k = 1, 2, …_ do

7 $w_{t+1}^{k}\leftarrow w_{t}^{k}-\eta\nabla\ell(w_{t}^{k};b)$;

8 $w_{t+1}^{k}\leftarrow\frac{w_{t+1}^{k}+w_{t+1}^{k-1}}{2}$;

9

10 end for

11

12 end for

13$W_{SL}=w_{t+1}^{k}$;

14 return $W_{SL}$;

Algorithm 1 Vehicle Group-based Swarm Learning. The $K$ vehicles are indexed
by $k$; $D_{k}$ is local dataset of vehicle $k$, $\eta$ is the learning rate,
$B$ is the local mini-batch size, $w_{t}^{k}$ is the onboard model of the
vehicle, and $W_{SL}$ represented the aggregated model of SL group.

###  3.4 Global Federated Deep Learning

In the global model aggregation process, an FDL framework with credibility is
considered. As shown in Figure 1, the FDL enables the local RSUs and central
server to collaboratively optimize the global model by continuously iterating
the following two steps:

1) Each RSU receives the model parameters, predicts the weight by the
credibility weights prediction algorithm, and sends the parameters to the
central server in the up-link.

2) The central server collects the model parameters from RSU, aggregates the
global model according to their weights predicted from RSU, and broadcasts the
new model to all the RSUs in the down-link. Along the training process, each
iteration is generally referred to as a communication round.

After sufficient communication rounds involving parameters exchanges and
training, we can obtain the best global FDL model without sacrificing the
vehicle’s privacy. The whole training process of FDL is shown in Algorithm 2.

1 Input: $w^{k}_{t}$, $\lambda^{k}_{t}$

2 Output: $W$ Initialize $w^{k}_{0}$;

3 for  _each round t = 1, 2, …_ do

4 for  _each RSU k_ do

5 $\lambda_{t+1}^{k}\leftarrow WeightsPrediction({w}^{k}_{t})$;

6

7 end for

8 $W\leftarrow\sum_{k=1}^{K}\frac{1}{K}\lambda_{t+1}^{k}w_{t+1}^{k}$;

9

10 end for

return $W$

Algorithm 2 Global Federated Deep Learning. The K RSUs are indexed by k;
$w^{k}_{t}$ is local model parameters at time t, $\lambda_{t}^{k}$ is model
weights of $w^{k}_{t}$, $W_{t}$ is the global model parameters Fig. 3: Loss
comparison of the IoV-SFDL, the Federated-Avg Learning in [18], and the
Communication-Efficient Learning in [19] on different traffic densities.
(a)High-density traffic: 10 vehicles in one group and 6 vehicles in the other
group. (b)Medium-density traffic: 6 vehicles in one group and 4 vehicles in
the other group. (c)Low-density traffic: 2 vehicles in one group and 3
vehicles in the other group.

##  4 Performance Evaluation

In this section, we introduce the simulation setups and then compare the
performance of the proposed IoV-SFDL with the other two baseline frameworks.
All source code about the framework, dataset used, and proposed algorithm in
this study are provided on GitHub (https://github.com/CoderTylor/IoVSFDL-
Swarm-Federated-Deep-Learning-).

###  4.1 Experiment Setup

The hardware is 32GB RAM, i7-10700 CPU 2.90GHz. The experimental platform is
built based on the PyTorch 1.9 framework, a flexible deep learning tool widely
used in academics. The software adopts Python 3.7.0 in the Ubuntu18.04 system.
Inspired by the performance of Social LSTM [20], an LSTM-centered model is
trained as an onboard vehicle training task to predict the vehicle’s
trajectory in 5s. The complete model parameters are listed in Table 2. For all
LSTM learning models, we used the greedy exploration with linearly annealed
from 1.0 to 0.1 in the whole training process. Adam optimizer is used to learn
all neural network parameters with a learning rate of $10^{-4}$.

Table 2: LSTM Model Parameters CELL | (Input, Output)
---|---
LSTM CELL | (64,32)
Input Embedding Layer | (In Feature=9, Out Feature=32)
Social Tensor Conv1 | (32,16, Kernel size=(5,3), stride(2,1))
Social Tensor Conv2 | (16,8, Kernel size=(5,3), stride(2,1))
Social Tensor Embedded | (In Feature=32, Out Feature=32)
Output Layer | (In Feature=32, Out Feature=5)
Dropout | Dropout(p=0, inplace=False)
Activation Function | ReLU
Time Length | 10
Episode | 50
Learning rate | $10^{-4}$
Optimizer | Adam

The simulation defines three traffic flow densities: high-traffic flow
density, medium-traffic flow density, and low-traffic flow density. The number
of total vehicles ranges from 5 to 16, and the size of the SL group ranges
from 2 to 10. The NGSIM trajectory dataset is used to verify the effectiveness
of the IoV-SFDL framework. The dataset is collected for the next-generation
traffic simulation by Federal Highway Administration. It use videos to capture
real-world traffic information, including vehicle velocity, position,
acceleration, lane, etc. As high-resolution real-world vehicle trajectory
data, NGSIM is widely used to explore the characteristics of the trajectory-
prediction process and calibrate and validate the trajectory-prediction
models. Detailed descriptions of the dataset can be found in [21].

To the best of our knowledge, there is no research investigating the SL
application under the IoV system, so two other baseline FDL frameworks are
added in the simulations for better gauging the performance of the proposed
IoV-SFDL framework, which are:

(1) Federated-Average Learning (Federated-Avg): Based on the Federated Average
deep learning algorithm proposed in [18], all vehicles on the road
participating in FDL. After training the model locally, the average
aggregation of model parameters is carried out in the central server.

(2) Communication-Efficient Federated Learning (Communication-Efficient):
Based on the Communication-Efficient federated deep learning algorithm
proposed in [19], we add the LSTM network for trajectory prediction in their
code and set the parameters $Frac$ as 0.8 for client selection.

###  4.2 Indicator

Three indicators are chosen to comprehensively evaluate the performance of the
proposed IoV-SFDL framework, such as model Loss, trajectory Prediction
Accuracy, and Prediction Error. The descent speed of the Loss function is the
first indicator to evaluate the converging rate. The Loss of the LSTM model is
defined as:

${\rm
Loss}=\small\frac{\sum_{i=1}^{N}\left|(x_{predict}^{i},y_{predict}^{i})-(x_{observe}^{i},y_{observe}^{i})\right|^{2}}{N}$
(16)

where $(x_{predict},y_{predict})$ is the vehicle trajectory predicted by LSTM,
and $(x_{observe},y_{observe})$ is the actual trajectory of the vehicle in the
dataset.

Then, the Prediction Error is used to evaluate the average Euclidean distance
between predict trajectory and actual trajectory in the NGSIM dataset under
different traffic densities in meters. The Prediction Error is defined as:

$\displaystyle\rm\tiny Pre$ $\displaystyle\rm diction\ Error=$ (17)
$\displaystyle\small\frac{\sum_{i=1}^{N}\sqrt{(x_{predict}^{i}-x_{observe}^{i})^{2}+(y_{predict}^{i}-y_{observe}^{i})^{2}}}{N}.$

At last, the Predict Accuracy is evaluated to estimate the model performance.
For the trajectory after 5 seconds, we compare the distance difference of each
path point as the basis for whether the prediction is true or false. To be
precise, we define positive prediction $P_{T}$, negative prediction $P_{F}$
and prediction accuracy as:

$\text{P}=\begin{cases}\text{ $P_{F}$ }&{\rm Prediction\ Error}>10m\\\ \text{
$P_{T}$ }&{\rm Prediction\ Error}\leq 10m\end{cases}.$ (18)

Based on equation (18), $N(P=P_{T})$ and $N(P=P_{F})$ is defined as the number
of positive and negative prediction, and the Prediction Accuracy is defined
as:

${\rm Prediction\ Accuracy}=\small\frac{N(P=P_{T})}{N(P=P_{T})+N(P=P_{F})}.$
(19)

Fig. 4: Prediction Error comparison of the IoV-SFDL, the Federated-Avg
Learning in [18], and the Communication-Efficient Learning in [19] on
different traffic densities. (a)High-density traffic: 10 vehicles in one group
and 6 vehicles in the other group. (b)Medium-density traffic: 6 vehicles in
one group and 4 vehicles in the other group. (c)Low-density traffic: 2
vehicles in one group and 3 vehicles in the other group. Fig. 5: Prediction
Accuracy Comparison of the IoV-SFDL, the Federated-Avg Learning in [18], and
the Communication-Efficient Learning in [19] on different traffic densities.
(a)High-density traffic: 10 vehicles in one group and 6 vehicles in the other
group. (b)Medium-density traffic: 6 vehicles in one group and 4 vehicles in
the other group. (c)Low-density traffic: 2 vehicles in one group and 3
vehicles in the other group.

###  4.3 Performance of the IoV-SFDL Framework

In Figure 3, we can see that the loss function change is almost the same in
the first ten episodes among IoV-SFDL and the other two benchmark frameworks.
As shown in Figure 3(b) and 3(c), the IoV-SFDL performs better on the same
computing cost than the others. Although Figure 3(a) shows that the change
speed of the loss function of the three frameworks is almost the same under
the condition of high-density traffic flow, the loss function of IoV-SFDL is
lower than the other two benchmark frames after 20 iterations. This is
because, with the increase of traffic flow, the proposed credibility weight
prediction algorithm can help the cloud server identify the redundant model
update, And give this update a minor weight to ensure the convergence speed of
the model.

Figure 4 shows the Prediction Error between predict trajectory and actual
trajectory under different traffic densities in meters. With the same
experimental parameters settings of the three frameworks, the Prediction Error
decrease with the iteration number increase, indicating that the LSTM learning
agent has learned to predict trajectory more precisely. As shown in Figure
4(a), 4(b), and 4(c), the more vehicle clients participating in the FDL, the
redundant data will affect the model prediction error of the two baseline
frameworks. Federated-Avg Learning is the most vulnerable framework facing
redundant training data. In contrast, the IoV-SFDL can converge with less
Prediction Error quickly under all traffic densities given.

Figure 5 shows the model’s Prediction Accuracy comparison for trajectory under
the IoV-SFDL framework with the other two baseline frameworks. We use equation
18 to classify the positive and negative values of the predicted points so
that the LSTM becomes a binary network. Figure 5(c) illustrates that the
Prediction Accuracy difference between the IoV-SFDL framework and baselines in
Prediction Accuracy is not evident under low-density conditions. However, as
shown in Figure 5(a) and 5(b), with more vehicles clients participating in
global model aggregation, the IoV-SFDL can achieve higher model accuracy with
the same computing cost.

Fig. 6: Communication Links of Different Frameworks

Figure 6 shows the communication links establishment difference between IoV-
SFDL and baseline frameworks. Unlike Fed-Avg Learning and Communication-
Efficient Learning, each client participating in global model aggregation has
to establish a two-way link with the central server. The IoV-SFDL only needs
to establish a blockchain connection between clients in the SL group. Each
group merely requires one two-way communication link with the central server
to realize global model aggregation. Under the conditions of 5, 10, 16
vehicles (Low-density, Medium-density, and High-density) on the road, compared
with the other two baseline frameworks, the IoV-SFDL needs the least
communication links to complete the global model aggregation.

Overall, the proposed IoV-SFDL framework is superior to other baseline
frameworks in edge-to-global communication overhead and convergence times.

##  5 Conclusion

This paper proposed an IoV-SFDL framework for protecting data privacy by
integrating the parallel SL into the FDL process. With the help of the SL
process, the reliability of shared data among FDL is guaranteed by blockchain
verification. Moreover, to achieve the high-level performance of the proposed
framework in the IoV system, a credibility weights prediction algorithm was
designed to speed up the convergence of model training in the IoV scenery.
Simulation results showed that the IoV-SFDL framework has less edge-to-global
communication overhead and higher computing efficiency than other state-of-art
FDL frameworks in the IoV system. Our future work will expand the scale of the
experiment and show how our framework protects vehicle privacy data in the
case of attacks.

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