# Mobility in Near-Field 6G: Why Mobile Terahertz Communications MUST Operate
in the Near Field?

V. Petrov1, J. Jornet1, and A. Singh2 1Department of Electrical and Computer
Engineering
Northeastern University, Boston, MA, USA E-mail: {v.petrov,
<EMAIL_ADDRESS>
2Department of Engineering
SUNY Polytechnic Institute, Utica, NY, USA E-mail<EMAIL_ADDRESS>
# Mobility in 6G: Why Non-Stationary Terahertz Communications MUST Operate in
the Near Field

V. Petrov1, J. Jornet1, and A. Singh2 1Department of Electrical and Computer
Engineering
Northeastern University, Boston, MA, USA E-mail: {v.petrov,
<EMAIL_ADDRESS>
2Department of Engineering
SUNY Polytechnic Institute, Utica, NY, USA E-mail<EMAIL_ADDRESS>
###### Abstract

TBD Toward, Network/Networking, THz, Wideband (we talk about bandwidth in the
end) to the title?

##  I Introduction

In recent years, terahertz (THz) band communications have been almost
unanimously accepted as being a key enabling technology for future wireless
standards [piesiewicz2007short, federici2010terahertz, Song2011,
akyildiz2022terahertz, rappaport2019wireless, sengupta2018terahertz]. In part,
the very high bandwidth available at THz frequencies can be utilized to design
very high data rate and low latency networks. With the THz technology gap
closing, the proliferation of THz radios has been utilized in demonstrating
stationary wideband [sen2019experimental], long range [sen2023multi], and high
data rate links. Thz band communications are thus becoming closer and closer
to being deployed in future wireless communication systems.

With the feasibility of THz communications already proven, investigations and
preliminary results in diverse scenarios, especially envisioned for the sixth
generation (6G) of wireless standards, such as in facilitating joint
communications and sensing [akyildiz2022terahertz, sarieddeen2020next], air-
to-air links [afrl_dec_2022], and CubeSat mega-constellations
[alqaraghuli2021performance] are underway. Here, the main motivation is to not
just overcome the huge path losses at THz frequencies but to also facilitate
non-stationary mobile links. More specifically, the link configuration
parameters such as the distance as well as the orientation can be constantly
changing, which can make the system design complex.

A significant complication that arises here is from the fact that, on the one
hand, very high gain antenna apertures are required to overcome the huge path
losses at THz frequencies. On the other hand, such devices would exhibit a
large near-field, which indicates that many use cases can fall under the near-
field paradigm, within which several of the far-field assumptions do not hold
[balanis2016antenna]. These observations manifest independently of the
techniques utilized for realizing the high-gain apertures. Whether the system
is configured with highly directional antennas and lenses [vdi_antennas,
miwave_antennas, fan2017development, llombart2011novel], with commercially
available designs demonstrating gains of as much as 55 dBi, or through antenna
arrays or reconfigurable intelligent surfaces that are still under
investigation [liaskos2019novel, di2019smart].

Primarily, the gain of an aperture depends on its physical size, which also
effects the corresponding near-field distance, or the Fraunhofer zone
[balanis2016antenna]. Thus, as shown in Fig. 1 if a system is configured such
that it provides a minimum performance threshold up to a certain distance
$d_{max}$, it would have a near-field zone that could potentially be breached
in the case on non-stationary links. This is an important observation since a
significant portion of modern wireless communications rely on the empirical
spherical wave model [balanis2016antenna], within which the propagation of a
wave (signal) is assumed to be spherically outward. This model is further
simplified under the far-field assumption through the plane wave approximation
[balanis2016antenna], wherein the curvature of the spherical wave is modeled
as a plane wave.

When a system operated in the near-field, these simplifications are no longer
valid and can lead to significant inefficacies [singh2022wavefront,
dovelos2021electromagnetic, near_field_6g_magazine]. Recognizing this, recent
investigations have explored incorporating the near-field spherical wave
effect through beamfocusing [near_field_6g_magazine, dovelos2021intelligent].
In addition, several new wave models and propagation features, such as non-
diffracting Bessel beams [durnin1987exact, singh2022wavefront] and self-
accelerating Airy beams [siviloglou2007observation] have also been explored
for their attractive use cases in THz communication systems [petrov4mobile,
singh2022bessel, something_from_mittleman]. Nonetheless, each of these
propagation characteristics features complex trade-offs and configurations. In
addition, operating within the near-field would require a novel analysis of
the channel model, security features, MAC protocols, and more.

As such, there is a clear need to evaluate a prospective link from the
fundamental principles of system requirements and derive the exact
applicability and limitations of what would require us to accept and adopt the
near-field zone as a likely case of operation.

In this paper, we intentionally model an ideal, artificial setup where a
mobile THz communication system aims to deliver its best performance when
working exclusively in the THz far field. We then analytically show that the
possible bandwidth that can be utilized in such a system will be limited by a
combination of factors that can be accurately assessed. Finally, we
numerically illustrate that in many practical cases, the resulting value of
bandwidth is small and unattractive for high-rate THz communications, and
conclusively highlight that the near-field is a valid regime of operation of
THz communications that aims to deliver the promises of the neat generation of
wireless standards.

##  II System Model

Our system model is illustrated in Fig. 1. We model a point-to-point non-
stationary THz communication link between a stationary THz Access Point (THz-
AP) and a mobile THz User Equipment (THz-UE). Both sides are equipped with
planar THz antenna arrays with $\lambda/2$ spacing in-between the elements,
where $\lambda$ is the wavelength corresponding to the central frequency $f$,
so $\lambda=c/f$. The THz-AP has an array of $N_{1}\times N_{1}$ elements and
a physical dimensions of $D_{1}\times D_{1}$ cm2, while the corresponding
array at the THz-UE side is of $N_{2}\times N_{2}$ elements and $D_{2}\times
D_{2}$ cm2, respectively.

The THz-AP and the THz-UE are separated by the distance $d$ varying from a
certain minimal distance, $d_{\text{min}}$111In practical deployments, this
value may be determined by i.e., the safety limits or the height of the THz-AP
mounted on a lamppost or a wall versus the height of the THz-UE carried by a
pedestrian outdoors., to a maximum communication range, $d_{\text{max}}$,
determining the coverage of the THz-AP. To simplify the derivations presented
below, we present the case with only the mobility of the THz-UE along the line
$[d_{\text{min}},d_{\text{max}}]$, so the antenna arrays are assumed to always
be in broadsides of each other. The analysis can be extended to the case where
the mutual rotations are also considered. Importantly, while the numerical
results slightly change, the main qualitative conclusions still hold222Real
mobile networks, obviously, have both types of mobility (variable distance and
mutual rotations accounted for with beamsteering). However, modeling the
mobility along the line is already sufficient to prove our main points..

The transmit (Tx) power is $P_{\text{Tx}}$ equally spread over the signal
bandwidth $B$. The receiver (Rx) side is featured by the noise figure
$N_{\text{F}}$. As the resulting communication model in Section III is
symmetric, both uplink and downlink are modeled: in downlink, $P_{\text{Tx}}$
stands for the THz-AP Tx power and $N_{\text{F}}$ – for the THz-UE noise
figure; in uplink, $P_{\text{Tx}}$ stands for the THz-UE Tx power and
$N_{\text{F}}$ – for the THz-AP noise figure. We also introduce two additional
coefficients needed for our analysis: (i) the mobility coefficient
$M=d_{\text{max}}/d_{\text{min}}$; and (ii) the antenna inequality coefficient
$L=D_{1}/D_{2}$.

Figure 1: Non-stationary THz communication system working exclusively in the
THz far field.

##  III Analysis: The Proof from the Opposite

In this study, we utulize proving from the opposite. _Let us assume we design
a mobile THz communication systems but aim to completely avoid working in the
THz near field. What kind of limitations does this impose?_

Key notations is summarized in Table LABEL:tab:notation.

We start by recalling that the near-field to the far-field border is usually
determined by the Fraunhofer distance, $d_{\text{F}}$ [fraunhoferdistance].
For all the distances $d\geq d_{\text{F}}$, the operation is still in the far
field zone, while any $d<d_{\text{F}}$ already belong to the near field
[balanis].

The distance $d_{\text{F}}$ is conventionally determined as
$\frac{2D}{\lambda}$, where $\lambda$ stands for the wavelength of the
transmitted signal and $D$ – for the maximum physical dimension of the
antenna. This particular formula is applicable exclusively to the setup with
the Tx antenna of a size $D$ and a _point_ Rx or, due to the channel
reciprocity, also applicable to the setup with a point transmitter and the Rx
antenna of the maximum size $D$.

However, for prospective THz communication systems, such a setup is quite
unrealistic. Instead, a more practical approach is to assume that both sides
are equipped with directional antenna arrays of a given size, as detailed in
the previous section. Here, the canonical equation for the Fraunhofer distance
is not directly applicable. Therefore, before proceeding with the performance
evaluation, we first extend the Fraunhofer distance equation for the case with
the directional THz antenna arrays used at both the Tx and the Rx:

$\displaystyle
d_{\text{F}}=\frac{4(D_{1}+D_{2})^{2}}{\lambda}=\lambda(N_{1}+N_{2})^{2}.$ (1)

A discussion on how (1) is derived is in Appendix A333Likely, (1) is not
presented for the first time ever (i.e., similar cases are discussed in
[near_field_6g_magazine] and [primer]) We just need an accurate equation to
proceed and haven’t found a reference to cite it in this exact form (for two
square arrays)..

To always stay in the far field, the minimal communication range
$d_{\text{min}}$ must then fulfill the following condition:

$\displaystyle d_{\text{min}}\geq\frac{4(D_{1}+D_{2})^{2}}{\lambda},$ (2)

which we refer to as _Condition 1_. Hence, $D_{1}+D_{2}\leq\frac{\sqrt{\lambda
d_{\text{min}}}}{2}$.

However, to reliably operate, the system must also satisfy another condition,
_Condition 2_ :

$\displaystyle S(d_{\text{max}})\geq S_{\text{L}},$ (3)

where $S(d_{\text{max}})$ is the signal-to-noise ratio (SNR) at
$d=d_{\text{max}}$, while $S_{\text{L}}$ is the SNR threshold. If the SNR
condition is fulfilled at the cell edge, it is also fulfilled for all other
distances. Hence, maintaining (3) is sufficient for reliable connectivity.

We then recall that the antenna gains $G_{1}$ and $G_{2}$ can be computed for
square arrays as $G_{1}=\pi N^{2}_{1}$ and $G_{2}=\pi N^{2}_{2}$, respectively
[balanis]. Thus, (3) can be reformulated as444We model a line-of-sight (LoS)
channel in the THz transparency window, so the molecular absorption is of
secondary importance comparing to the spreading loss, transmit power, and
antenna gains.:

$\displaystyle\quad{}\quad{}\frac{P_{\text{Tx}}G_{1}G_{2}}{N_{0}}\Big{(}\frac{\lambda}{4\pi
d_{\text{max}}}\Big{)}^{2}\geq S_{\text{L}}\quad{}\Rightarrow$
$\displaystyle\Rightarrow\quad{}\frac{P_{\text{Tx}}\frac{4D^{2}_{1}}{\lambda^{2}}\frac{4D^{2}_{2}}{\lambda^{2}}\frac{\lambda^{2}\pi^{2}}{16\pi^{2}d^{2}_{\text{max}}}}{N_{0}}\geq
S_{\text{L}}\quad{}\Rightarrow$
$\displaystyle\Rightarrow\quad{}\big{(}D_{1}D_{2}\big{)}^{2}\geq\frac{T\lambda^{2}N_{0}d^{2}_{\text{max}}}{P_{\text{Tx}}}\quad{}\Rightarrow$
$\displaystyle\Rightarrow\quad{}D_{1}D_{2}\geq\lambda
d_{\text{max}}\sqrt{\frac{N_{\text{F}}kTB}{P_{\text{Tx}}}}10^{\frac{S_{\text{L},\text{dB}}}{20}},$
(4)

where $k$ is the Boltzmann constant, $T$ is the system temperature in K, $B$
stands for the bandwidth of the transmitted signal in Hz, while
$S_{\text{L},\text{dB}}$ is the SNR threshold $S_{\text{L}}$ in dB scale.

To reliably perform the data exchange while always staying in the THz far
field, the mobile THz system from Section II must then satisfy both _Condition
1_ and _Condition 2_ :

$\displaystyle\begin{cases}D_{1}+D_{2}&\leq\frac{\sqrt{\lambda
d_{\text{min}}}}{2}\\\ \,\,\,D_{1}D_{2}&\geq\lambda
d_{\text{max}}\sqrt{\frac{N_{\text{F}}kTB}{P_{\text{Tx}}}}10^{\frac{S_{\text{L},\text{dB}}}{20}}.\end{cases}$
(5)

We now proceed with solving this system of equations for $D_{1}$ and $D_{2}$.
First, let us set the maximum allowed $D_{2}$ from the first equation as
$D_{2}=\frac{\sqrt{\lambda d_{\text{min}}}}{2}-D_{1}$ and substitute it into
the second equation. This step leads to

$\displaystyle D_{1}\big{(}\sqrt{\lambda d_{\text{min}}}-2D_{1}\big{)}\geq
2\lambda d_{\text{max}}\sqrt{Z}10^{\frac{S_{\text{L},\text{dB}}}{20}},$ (6)

where $Z$ is set to $Z=\frac{N_{\text{F}}kTB}{P_{\text{Tx}}}$ to simplify the
notation.

Then, it can be noted that (6) can be converted into the following monic
quadratic inequation for $D_{1}$:

$\displaystyle D^{2}_{1}-D_{1}\frac{\sqrt{\lambda d_{\text{min}}}}{2}+\lambda
d_{\text{max}}\sqrt{Z}10^{\frac{S_{\text{L},\text{dB}}}{20}}\leq 0$ (7)

that can be rewritten as

$\displaystyle D^{2}_{1}+pD_{1}+q\leq 0,$ (8)

where $p=-\frac{\sqrt{\lambda d_{\text{min}}}}{2}$ and $q=\lambda
d_{\text{max}}\sqrt{Z}10^{\frac{S_{\text{L},\text{dB}}}{20}}$.

We then analyze (8) closer. First, the coefficient for $D^{2}_{1}$ is equal to
$1$, so always positive. Hence, the branches of the parabola are looking up.
Then, the presented quadratic inequation has up to two roots, where its value
equals $0$. Let us define those roots as $x_{1}$ and $x_{2}$, where $x_{1}\leq
x_{2}$. Then, as the expression must be less or equal to $0$, the legitimate
values for $D_{1}$ that satisfy this inequation are $D_{1}\in[x_{1},x_{2}]$.

Here, one may notice that in practice, $D_{1}$ has a certain physical meaning
– the size of the antenna array, so $D_{1}>0$ must be always enforced.
Therefore, the sign of $x_{1}$ (if it exists) becomes of interest to determine
the valid range of the $D_{1}$ values. For this purpose, we call Vieta’s
formulas connecting the roots of the polynomial to its coefficient [???]. For
the quadratic case, if at least one of the roots exists, these give:

$\displaystyle\begin{cases}x_{1}+x_{2}=-p=\frac{\sqrt{\lambda
d_{\text{min}}}}{2}\\\ x_{1}x_{2}=q=\lambda
d_{\text{max}}\sqrt{Z}10^{\frac{S_{\text{L},\text{dB}}}{20}},\end{cases}$ (9)

where $p$ and $q$ are the polynomial coefficients defined above.

Analyzing the right sides of (9), we conclude that both $x_{1}$ and $x_{2}$
(if they exist) must be non-negative555Here, we notice that $q$ is always
positive, as there are no negative (or even zero-value) multipliers in it.
Hence, if at least one of the roots $x_{1}$, $x_{2}$ exist, they are either
both positive or both negative. However, they cannot be both negative as
$x_{1}+x_{2}=-p$ is always non-negative, as $\sqrt{\lambda d_{\text{min}}}/4$
does not have negative multipliers either (here, we say _non-negative_ instead
of _positive_ , as, strictly speaking, $d_{\text{min}}$ may be $0$). So, both
$x_{1}$ and $x_{2}$ are non-negative. Hence, as $x_{1}\geq 0$, $D_{1}>0$ is
always enforces if $D_{1}\in[x_{1},x_{2}]$.. Therefore,
$D_{1}\in[x_{1},x_{2}]$ is sufficient to determine the legitimate values for
$D_{1}$ to satisfy (7).

Now let us proceed with analyzing the conditions, where
$D_{1}\in[x_{1},x_{2}]\neq\emptyset$, so there is at least one value for
$D_{1}$ that fulfils (7). Here, let us recall that for the quadratic
polynomial, the existence of at least one possible root is determined by the
sign of its discriminant $\left(p/2\right)^{2}-q$. _Hence, there is always at
least one value for $D_{1}$ that satisfies both Condition 1 and Condition 2
(so that the modeled mobile THz system is both far-field and reliable) if and
only if $\left(p/2\right)^{2}-q$ is non-negative:_

$\displaystyle\exists D_{1}:\Big{(}D^{2}_{1}-D_{1}\frac{\sqrt{\lambda
d_{\text{min}}}}{2}+\lambda
d_{\text{max}}\sqrt{Z}10^{\frac{S_{\text{L},\text{dB}}}{20}}\Big{)}\leq
0\quad{}\Leftrightarrow$
$\displaystyle\Leftrightarrow\quad{}\Big{(}\frac{\sqrt{\lambda
d_{\text{min}}}}{4}\Big{)}^{2}-\lambda
d_{\text{max}}\sqrt{Z}10^{\frac{S_{\text{L},\text{dB}}}{20}}\geq 0$ (10)

From (10), the following restriction for the signal bandwidth, $B$, is
formulated:

$\displaystyle
B\leq\frac{P_{\text{Tx}}d^{2}_{\text{min}}}{256d^{2}_{\text{max}}10^{\frac{S_{\text{L},\text{dB}}}{10}}N_{\text{F}}kT}.$
(11)

STOP HERE

By substituting the mobility coefficient, $M$, defined as
$M=d_{\text{max}}/d_{\text{min}}$, the previous inequality can be finally
rewritten as:

$\displaystyle B\leq\frac{1}{256M^{2}kT}\cdot
10^{\frac{P_{\text{Tx},\text{dBm}}-S_{\text{L},\text{dB}}-N_{\text{F},\text{dB}}-30}{10}},$
(12)

where $P_{\text{Tx},\text{dBm}}$ is the transmit power in dBm,
$S_{\text{L},\text{dB}}$ is the SNR threshold in dB, and
$N_{\text{F},\text{dB}}$ is the Rx’s noise figure in dB.

Let us analyze (11) now. First, as the right side of (11) is non-negative, it
appears that can be a THz system that satisfies both _Condition 1_ and
_Condition 2_. However, importantly, such a system will have to operate with a
limited bandwidth that satisfies (11). REWRTE BETTER: Hence, from a frequency-
dependent Fraunhofer distance, we arrive at bandwidth-limited restriction
limiting the bandwidth, not the frequency.

Second, interestingly, the dependencies on both $\lambda$, $D_{1}$, and
$D_{2}$ are all gone, so i.o.w., the maximum signal bandwidth $B$, where the
system satisfies the reliability constraints, _Condition 2_ , while always
operating in the THz far field for all the THz-AP–THz-UE distances,
$d\in[d_{\text{min}},d_{\text{max}}]$, emphCondition 1, depends on neither the
system frequency nor the dimensions of the employed antenna arrays.

Finally, we notice that (12) is subject to neither $d_{\text{min}}$ nor
$d_{\text{max}}$, but only their relation
$M=\frac{d_{\text{max}}}{d_{\text{min}}}$, referred to as the _mobility
coefficient_. I.o.w., the maximum system bandwidth does not depend on the
absolute values of the communication distances, but only on the relative width
of the interval $[d_{\text{min}},d_{\text{max}}]$.

Now, let us determine the actual values of $D_{1}$ imposed by (10) to check
how realistic they are. The widest bandwidth constrained by (12) can be
achieved when the polynomial in (10) has just one root, so
$D_{1}=x_{1}=x_{2}$. We now recall (9) determining that
$x_{1}+x_{2}=\frac{\sqrt{\lambda d_{\text{min}}}}{2}$. Hence, the actual value
of $D_{1}$ leading to the widest achievable bandwidth in (12) is:

$\displaystyle D_{1}=x_{1}=x_{2}=\frac{\sqrt{\lambda d_{\text{min}}}}{4}.$
(13)

Now, we recall that in order to solve (5), we set the maximum value of $D_{2}$
to $D_{2}=\frac{\sqrt{\lambda d_{\text{min}}}}{2}-D_{1}$. Hence, substituting
(13) into this formula, we arrive at the following value of $D_{2}$ that leads
to the widest possible value of $B$ in (12):

$\displaystyle D_{2}=\frac{\sqrt{\lambda
d_{\text{min}}}}{2}-D_{1}=\frac{\sqrt{\lambda d_{\text{min}}}}{4}=D_{1}.$ (14)

Hence, it appears that the best solution for the joint _Condition 1_ and
_Condition 2_ requires that the Tx and Rx antennas are of the same physical
size, $D_{1}=D_{2}$. While looking non-obvious first, after careful
consideration, such a result makes sense, as we de-facto solve an optimization
task in (5) aiming to maximize the multiplication of two values, $D_{1}D_{2}$,
while keeping their sum, $D_{1}+D_{2}$ under a certain limit. It can be shown
that the solution for such a problem statement is always the case, where the
parameters are equal to each other.

As demonstrated in Appendix B, simplifying the setup to a symmetric case with
identical Tx and Rx antennas, $D_{1}=D_{2}=D$, upfront when dealing with (5)
notably simplify the analysis presented above still leading to the very same
equation (12).

It is worth noting that the case, where $D_{1}=D_{2}$ is usually not very
typical in mobile environments, where, by default, the antenna size at the
mobile user is notably smaller than the one employed by the access point,
$D_{2}\ll D_{1}$. In contrast, the symmetric case with equal size antennas is
more relevant to a stationary setup (i.e., a THz wireless backhaul link).
Therefore, the condition in (12) can be further simplified by dropping any
mobility there ($M=1$). That leads to the following equation for the maximum
bandwidth of a stationary THz link, $B^{\text{(max)}}_{\text{stationary}}$,
equipped with two equal-size antenna arrays on both size and a single
communication range $d$ that does not change:

$\displaystyle B^{\text{(max)}}_{\text{Stationary}}=\frac{1}{256kT}\cdot
10^{\frac{P_{\text{Tx},\text{dBm}}-S_{\text{L},\text{dB}}-N_{\text{F},\text{dB}}-30}{10}}.$
(15)

This is an important result demonstrating that it is possible to design a
stationary THz link with a required performance without going into the THz
near field zone if the desired system bandwidth is less than
$B^{\text{(max)}}_{\text{Stationary}}$.

Analyzing (15), we notice that (as $k$ is the constant and $T$ is typically
determined by the environmental conditions) the calculated value is determined
by only the interplay between the transmit power, $P_{\text{Tx},\text{dBm}}$,
the required SNR value at the receiver, $S_{\text{L},\text{dB}}$, and the
receiver’s noise figure, $N_{\text{F},\text{dB}}$. It is important to note
that the maximum achievable bandwidth while satisfying the said constraints is
not impacted by the communication range $d$ or even the system central
wavelength, $\lambda$. In contrast, the actual value of $D_{1}$ to achieve the
desired performance over $B^{\text{(max)}}_{\text{Stationary}}$ depends on
both the wavelength and the range, but does not depend on either of the
$B^{\text{(max)}}_{\text{Stationary}}$ parameters ($P_{\text{Tx},\text{dBm}}$,
$S_{\text{L},\text{dB}}$, and $N_{\text{F},\text{dB}}$), leading to an
interesting separation of the dependencies among the
$B^{\text{(max)}}_{\text{Stationary}}$ and the corresponding $D_{1}$.

VIT: Move these interesting discussions as above to the last subsection in the
analysis or a dedicated section? Likely a subsection… with all the key
formulas and their interpretation and analysis before jumping to the numerical
results

We now proceed with the mobile setup, featured by $D_{2}<D_{1}$ and $M\geq 1$.
In reality, the value of $D_{2}$ is primarily subject to the form factor of
the THz-UE and may be fixed to a certain maximum value
$D_{2}=D^{\text{(max)}}_{2}$for different categories of devices, i.e., a
smartphone, a tablet, or extended reality (XR) glasses
[antenna_size_wearables].

When having the maximum $D_{2}$ to a certain fixed value
$D^{\text{(max)}}_{2}$, we can perform a similar analysis for (5) without the
need to solve any quadratic polynomial inequalities.

First, we would calculate the corresponding $D_{1}$ from the first line of (5)
as $D_{1}=\frac{\sqrt{\lambda d_{\text{min}}}}{2}-D^{\text{(max)}}_{2}$. We
then again substitute this intro to the second line leading to the following
limit for the maximum achievable bandwidth of a mobile THz link, while
satisfying both the far-field operation requirement, _Condition 1_ , and the
reliability constraints, _Condition 2_ :

$\displaystyle
B^{\text{(Fixed)}D_{2}}_{\text{Mobile}}\leq\frac{Q^{2}}{4kT\lambda^{2}d^{2}_{\text{max}}}\cdot
10^{\frac{P_{\text{Tx},\text{dBm}}-S_{\text{L},\text{dB}}-N_{\text{F},\text{dB}}-30}{10}},$
(16)

where $Q=D^{\text{(max)}}_{2}\big{(}\sqrt{\lambda
d_{\text{min}}}-2D^{\text{(max)}}_{2}\big{)}$ is used to simplify the
notation.

The value in (16) presents a correct limit for a mobile case with the size of
the THz-UE antenna limited by a certain value $D^{\text{(max)}}_{2}$. However,
analyzing this equation numerically imposes a challenge, as, in contrast to
(15), the achievable bandwidth is now subject to as many as seven individual
parameters, such as both distances $d_{\text{min}}$ and $d_{\text{max}}$, the
wavelength $\lambda$, all three radio parameters $P_{\text{Tx},\text{dBm}}$,
$S_{\text{L},\text{dB}}$, and $N_{\text{F},\text{dB}}$, and the antenna size
limit $D^{\text{(max)}}_{2}$ itself.

An open question is if it is possible to simplify the analysis for a mobile
THz system somehow. A possible solution here is to reformulate the problem to
a similar one (not identical though), where instead of limiting the THz-UE
antenna size to a certain fixed value, we set it to be $L$ times smaller than
the corresponding size of the antenna at the THz-AP666Strictly speaking, these
two formulations are not equivalent to each other, so the results differ
slightly. Still, the numerical difference is not drastic and is well
compensated by a notably simpler formulation presented below..

By incorporating $L=D_{1}/D_{2}$ into (5), we can solve the system in a
similar manner as above leading to the following limit for the bandwidth of a
mobile THz system:

$\displaystyle B_{\text{Mobile}}\leq\frac{L^{4}}{16kT\lambda
M^{2}(L+1)^{2}}\cdot
10^{\frac{P_{\text{Tx},\text{dBm}}-S_{\text{L},\text{dB}}-N_{\text{F},\text{dB}}-30}{10}},$
(17)

We finally observe a substantial level of similarity between (15) and (17)
leading to the following maximum bandwidth of a mobile THz system that
satisfies both the SNR and the far-field requirements:

$\displaystyle
B^{\text{(max)}}_{\text{Mobile}}=B^{\text{(max)}}_{\text{Stationary}}/\Big{(}M^{2}\frac{(L+1)^{4}}{16L^{2}}\Big{)}.$
(18)

Hence, the resulting maximum bandwidth of a mobile THz system is obtained
similarly to the one of a stationary THz link. However, the bandwidth of a
mobile system is always smaller, affected by two sorts of _penalties_ , where
the first penalty is $M^{2}$ determined by the range in the communication
distances, $M=\frac{d_{\text{max}}}{d_{\text{min}}}$, while the second one is
due to the fact that the THz-UE antenna has to be $L$ times smaller than the
one employed by the THz-AP, $L=\frac{D_{1}}{D_{2}}$.

From (15) and (18), we finally derive the required $P_{\text{Tx}}$ value in
(19) to maintain the desired maximum system bandwidth in mobile THz
communications without having to deal with near-field THz propagation.

$\displaystyle
P_{\text{Tx},\text{dBm}}=54.08+S_{\text{L},\text{dB}}+N_{\text{F},\text{dB}}+10\log_{10}\big{(}kT\big{)}+10\log_{10}\big{(}B^{\text{(max)}}_{\text{Mobile}}\big{)}+20\log_{10}\big{(}M\big{)}+20\log_{10}\bigg{(}\frac{(L+1)^{2}}{4L}\bigg{)}$
(19)

We analyze the dynamics of the obtained restrictions for both mobile and
stationary THz links in realistic conditions in the next section.

.

.

##  IV Numerical Results

In this section, the developed mathematical results get numerically
elaborated.

###  IV-A Stationary THz Wireless Link

We start with analyzing the possible restrictions imposed on the stationary
THz link by the need to operate exclusively in the far-field zone, as
illustrated in Fig. 2. First, Fig. 2(a) presents the maximum achievable
bandwidth, $B^{\text{(max)}}_{\text{Stationary}}$, as a function of the
transmit power, $P_{\text{Tx}}$ and the noise figure at the receiver,
$N_{\text{F}}$. The target SNR value at the receiver, $S_{\text{L},\text{dB}}$
is set to a relatively high value of $30$ dB, which is sufficient to i.e.,
maintain a reliable and high-rate backhaul link [priyangshu_thz_weather].
Here, the dashed curve for $N_{\text{F}}=0$ presents a reference scenario with
idealistic hardware impacted only by the thermal noise. We first observe from
Fig. 2(a) that expectedly higher transmit power leads to greater bandwidth
available for communications. The same advantage can be achieved by designing
a better receiver hardware (featured by lower $N_{\text{F}}$). Most
importantly, we study the absolute achievable bandwidth values to observe that
they are in the order of _hundreds of gigahertz and even several terahertz_
for any practical transmit power over $50$ mW that have already been
experimentally demonstrated [thz_equipment_power]. _Hence, neither transmit
power nor noise figure impose any necessity on the stationary system to work
in the THz near field zone._

We continue our study with Fig. 2(b) exploring the possible limitations on
$B^{\text{(max)}}_{\text{Stationary}}$ from a combination of a stringent
target SNR value and various transmit power levels. For illustration, the
noise figure is set to $N_{\text{F}}=10$ dB. Same as above, it is first noted
that the growth of the SNR threshold, $S_{\text{L},\text{dB}}$, decreases the
maximum system bandwidth, while a growing transmit power has an opposite
effect. Notably, we observe that the absolute values of
$B^{\text{(max)}}_{\text{Stationary}}$ are still very high (_hundreds of
gigahertz and up to terahertz_ for any practical transmit power values of over
$10$ mW777Considering that stationary THz wireless will likely be used for
backhaul-type use cases, socket-powered stationary radios can easily operate
with even higher transmit powers [thz_equipment_power].. _Hence, the SNR
requirement also does not impose any stringent limits on the bandwidth of the
stationary far-field THz system, demanding to work in the THz near field
zone._

We finally present Fig. 2(c) exploring the actual values of the antenna sizes
on the Tx and Rx sides, $D$, of a stationary THz link to achieve the maximum
bandwidth still without entering the THz near-field zone. Here, we first
observe that the required antenna size decreases with the central frequency.
This is due to the fact that higher frequency leads to smaller wavelength,
consequently leading to more antenna array elements packed per cm2 of the
antenna array. The latter contributes to the maximum antenna gain and thus
allows to achieve a similar performance to lower frequencies (i.o.w., operate
comfortably over the same bandwidth still meeting the SNR requirement) by
employing physically smaller antennas. In contrast, while the communication
range, $d$, does not impact the actual achievable bandwidth values presented
in Fig. 2(a) and Fig. 2(b), greater $d$ naturally demands larger antenna
arrays on both the Tx and the Rx sides to compensate for the spreading losses.

Still, studying the absolute values of $D$, we notice that for i.e., a typical
$200$ m–long backhaul link at $300$ GHz only roughly $10$ cm $\times$ $10$ cm
antenna arrays are required and the values decreases with the frequency. Here,
it is important to make two notes. First, the actual values of $D$ may be even
smaller in practice, as the said $10$ cm $\times$ $10$ cm arrays are needed to
maintain the maximum bandwidth, $B^{\text{(max)}}_{\text{Stationary}}$. As
studied earlier, that value is several orders of magnitude higher than may
actually be needed, so the real sizes of the antenna arrays for high-rate THz
backhaul link will be even smaller than $10$ cm. Another factor decreasing $D$
here is the ability to replace phased arrays for stationary THz links with
much smaller horn antennas inherently featured by notably shorter near-field
zone [horn_near_field]. So, the required sizes of the antennas are also
practical. _Hence, the actual required size of the antennas also dose not
impose any notable restrictions on the far-field stationary THz wireless
link._

In summary, we can conclude that in most of the target practical scenarios a
stationary THz wireless link can be optimized to work exclusively in the THz
far field while still meeting the desired performance and reliability
constrains.

###  IV-B Mobile THz Wireless Access

We now continue our analysis with Fig. 3 exploring the possible limitations of
the far-field THz wireless, when serving mobile users. Here, we need to
account for three key limitations of the far-field mobile THz system. The
first limitations comes from the need to now maintain a link over a range of
communication distances $[d_{\text{min}},d_{\text{max}}]$ instead of a single
value of $d$. This is important, as the SNR threshold must be satisfied at a
larger $d_{\text{max}}]$ value, so the antennas must be large enough.
Meanwhile, the antenna sizes of a mobile system are limited by the near-field
border already at much smaller $[d_{\text{min}}$. This is accounted in our
study with the mobility coefficient $M=d_{\text{max}}/d_{\text{min}}$.

Second, a practical mobile system cannot assume the antenna size at a mobile
node $D_{2}$, to be same as the antenna size at a larger base station or
access point, $D_{1}$. This inequality of the antenna sizes is captured in our
analysis with the antenna inequality coefficient $L=D_{1}/D_{2}$. Finally, the
maximum transmit power for a battery-powered mobile node is limited by a lower
value than the corresponding value for a stationary backhaul link, due to
health and safety regulations, inability to design a high-power compact radio,
and last but not least the energy-efficiency constrains888VIT: Rewrite… While
a socket-powered stationary THz radio can operate with higher transmit power
values, the battery-powered mobile node needs to account the power consumption
and thus cannot always transmit in uplink with maximum transmit power. We
account for all these constrains when evaluating a mobile THz wireless system
below.

We proceed with Fig. 3(a) exploring the maximum achievable bandwidth of a far-
field mobile THz system as a function of the mobility coefficient, $M$. For
illustration, we set the uplink transmit power of a mobile node to $23$ dBm
and the receiver noise figure to $10$ dB. Considering a typical range of an
indoor small cell to vary between $0.5$ m and $25$ m ($M=50$) and a typical
range of an outdoor micro cell to be in-between $5$ m and $200$ m ($M=40$),
the practical range of $M$ is in the order of $40$ to $50$. In Fig. 3(a), we
first notice that the maximum achievable bandwidth is expectedly decreasing
with the growth of $M$. Still, even for practical value of $M=50$, the mobile
THz system can operate over _hundreds of gigahertz_ of bandwidth when using
equal-size antenna arrays on both the access point and the mobile node
($L=1$). _Hence, with idealistic value of $L=1$, even practical ranges of $M$
lead to extensive values of available bandwidth_.

A similar observation is made from Fig. 3(b), where a practical range of
antenna inequality, $L$, is in the oder of $20$–$30$999VIT: Rewrite this
accounting for the actual antenna sizes in Fig. 3(c)! Considering the
smartphone antenna size to be limited to $\approx$$1-2$ cm, $L=20$ leads to
the antenna array of $20$–$30$ cm in size, which is comparable to existing
antenna systems used for microwave and mmWave mobile communications. The
corresponding value for wearable electronics, i.e., extended reality (XR)
glasses to be slightly smaller than $1$ cm, leading to the same range of the
antenna sizes at the access point, $20$–$30$ cm, when $L\geq 30$.. Here, a
similar trend to Fig. 3(a) is observed. The available bandwidth without
entering the THz near field zone shrinks with $L$. However, if there is no
mobility at all ($M=1$), _several terahertz of bandwidth_ can be occupied even
when the antenna at one of the size is $L=30$ times smaller than the one on
the other. Also, as illustrated in Fig. 3(c), the corresponding values of the
antenna sizes at the access point $D_{1}$ and $D_{2}$ ($d_{\text{min}}=10$ m)
is still feasible. _Hence, with idealistic value of $M=1$, the effect of
antenna inequality, $L$, cannot severely limit the bandwidth of the THz
wireless link._

###  IV-C Does 6G THz Wireless Have to Operate in Near Field?

We finally present Fig. 4 exploring the required transmit power as a function
of the desired bandwidth for both stationary far-field THz wireless and mobile
far-field THz wireless. For all the curves, the noise figure is set to
$N_{\text{F}}=10$ dB and the target SNR, $S_{\text{L},\text{dB}}$ is set to
$20$ dB101010While the actual value of the cell-edge SNR varies around $3$
dB–$5$ dB, this is also accounting for NLoS propagation, fading, and other
effects. Hence, the idealistic SNR value in LoS without fading should be in
the order of $20$ dB to keep a margin for NLoS, fading, weather, and other
unfavourable propagation effects in practical setups.. For mobile THz systems,
we particularly target two prospective form factors of the mobile node
(smartphone, $L\approx 20$, and XR glasses, $L\approx 30$) in two realistic
deployment scenarios (indoor wireless access, $M\approx 50$, and outdoor THz
small cell, $M\approx 40$).

We first observe from Fig. 4 that the larger system bandwidth naturally
requires greater transmit power for both stationary and mobile THz systems. We
also notice that for XR glasses featured by smaller antennas it is more
challenging to operate only in the THz far field (greater $P_{\text{Tx}}$
required), as a miniature antenna array at the wearable device has to be
compensated by a larger antenna array at the access point, thus increasing the
THz near field zone.

Most importantly, we study the absolute values of $P_{\text{Tx}}$ contrasting
a stationary THz link to a practical mobile THz link with both $M>1$ and
$L>1$. Here, confirming the results in Fig. 2, a stationary THz link can
harness $100$ GHz of bandwidth with small enough antennas to still operate in
the THz far field with the transmit power of as low as just $-10$ dBm. Hence,
no severe limitations on the stationary THz link are anticipated. In contrast,
a realistic THz mobile link, especially involving wearable devices with
physically small antennas demands at least $30$ dBm transmit power to exploit
$10$ GHz of bandwidth without dealing with near-field THz propagation effects.
The value grows to over $40$ dBm when aiming for $100$ GHz-wide channels.
These levels of uplink transmit power are much higher than the state-of-the-
art THz equipment can offer (even stationary systems) and also notably
challenge the energy-efficiency of prospective mobile THz wireless access
systems, making then de-facto impractical. Therefore, it is numerically
illustrated that the “optimal” sizes of the antenna arrays to keep the system
exclusively far-field are not large enough for mobile THz radio. _Hence,
practical mobile THz wireless communication systems will have to operate with
antennas greater than offered by our far-field model developed in Section III
and featured by larger near field zone._

Therefore, the prospective mobile THz wireless communication systems will
almost inevitably operate in the THz near field, at least for a fraction of
its coverage closer to the minimal distance.

Conclusions:

The truth is: not because they are higher in frequency, but because they
intend to occupy greater bandwidth.

(a) Effect of the transmit power

(b) Effect of the SNR threshold

(c) Effect of the central frequency

Figure 2: The maximum achievable bandwidth for the far-field _stationary_ THz
link and the corresponding antenna sizes.

(a) Effect of mobility

(b) Effect of antenna inequality

(c) Difference in antenna sizes at AP and UE

Figure 3: The maximum achievable bandwidth for the far-field _mobile_ THz link
and the corresponding antenna sizes. Figure 4: The maximum achievable
bandwidth if the 6G radio is not near-field. Contrasting _stationary_ 6G radio
to _mobile_ 6G radio.

.

##  Appendix A Far-field Distance Estimation for Two Square Antenna Arrays

One of the widely-adopted definitions of the far-field zone is the distance
from the transmitter, $d\geq d_{\text{F}}$, where the emitted wave is (almost)
“plane”. Common criteria here, adopted i.e., in the Fraunhoufer distance
equation, is that the phase shift between the signal propagating over the
shortest possible LoS path between the Tx antenna and the Rx antenna and the
signal propagating over the longest possible LoS path between the Tx antenna
and the Rx antenna is not exceeding $\Pi/8$ [???].

The conventional Fraunhofer distance separating the far field from the
radiative near field is then determined by:

$\displaystyle d\geq d_{\text{F}}\textrm{, where
}d_{\text{F}}=\frac{2D^{2}}{\lambda},$ (20)

where $D$ stands for the length of the largest physical dimension of the Tx
antenna111111Here, $D$ stands for the projection of the largest dimension to
the plane perpendicular to the wave propagation path, but this can be
neglected if the path is perpendicular to the antenna array, as in our study.
The same approach can be extended to a general case, by accounting for the
corresponding 3D rotation angles of the Tx and the Rx antenna arrays. and
$\lambda$ stands for the wavelength of the transmitted signal.

It is important to note that the original (20) is only applicable to the setup
with a fixed-size Tx antenna and a point Rx (or, due to the reciprocity of the
wireless channel, to the symmetric setup with a point Tx and a fixed-size Rx
antenna). Hence, this equation cannot be applied to the practical case
considered in the paper, where both Tx and Rx are equipped with phased arrays.

In this Appendix, we illustrated how (20) can be extended to the case of a Tx
equipped with a planar antenna array of a physical $D_{1}\times D_{1}$ and an
Rx equipped with a planar array of $D_{2}\times D_{2}$, as illustrated in Fig.
5121212We do not claim such a derivation to be made for the first time, but
the authors were not able to find a proper reference, where the specific case
of two 2D planar arrays is explicitly considered in the literature..

Figure 5: Calculating the analog for the Fraunhofer far-field distance in case
of two 2D antenna arrays of sizes $D_{1}\times D_{1}$ and $D_{2}\times D_{2}$,
respectively.

In Fig. 5, we first identify the lengths of the shortest and the longest
propagation paths ($d_{\text{S}}$ and $d_{\text{L}}$, respectively) as:

$\displaystyle d_{\text{S}}$ $\displaystyle=d_{\text{F}},$ $\displaystyle
d_{\text{L}}$ $\displaystyle=\sqrt{d^{2}+z^{2}},$ (21)

where (from the setup geometry),
$z=\frac{D^{(r)}_{1}+D^{(r)}_{2}}{2}=\frac{D_{1}+D_{2}}{\sqrt{2}}$.

We then recall that, if the signal propagates only $\Pi/8$ longet in phase
over $d_{\text{L}}$ than over $d_{\text{S}}$, then:

$\displaystyle d_{\text{L}}=d_{\text{S}}+x,$ (22)

where $x=\frac{\lambda}{2\Pi/\left(\frac{\Pi}{8}\right)}=\frac{\lambda}{16}$
(the signal phase shift after traveling $\lambda$ meters is $2\Pi$, so the
signal phase shift is $\Pi/8$ when propagating over $16$ times smaller
distance).

Solving (21) and (22) together for $d_{\text{F}}$, we end up with:

$\displaystyle\frac{d_{\text{F}}\lambda}{8}+\frac{\lambda^{2}}{256}=\frac{\left(D^{(r)}_{1}+D^{(r)}_{2}\right)^{2}}{4},$
(23)

where $\frac{\lambda^{2}}{256}$ is asymptotically smaller than other elements
for high-frequency signals and can thus be omitted, leading to:

$\displaystyle
d_{\text{F}}\approx\frac{2\left(D^{(r)}_{1}+D^{(r)}_{2}\right)^{2}}{\lambda}=\frac{4\left(D_{1}+D_{2}\right)^{2}}{\lambda}.$
(24)

For the case of equal-size antenna arrays at the Tx and the Rx sides
($D_{1}=D_{2}=D$), the previous equation becomes

$\displaystyle d_{\text{F}}\approx\frac{16D^{2}}{\lambda}.$ (25)

We finally recall that $D_{1}=\frac{\lambda}{2}N_{1}$ and
$D_{2}=\frac{\lambda}{2}N_{2}$, thus converting (24) and (25) into:

$\displaystyle d_{\text{F}}$
$\displaystyle\approx\lambda\left(N_{1}+N_{2}\right)^{2},$ (26) $\displaystyle
d_{\text{F}}$ $\displaystyle\approx 4\lambda N^{2}.$ (27)

for (i) $N_{1}\times N_{1}$ plus $N_{2}\times N_{2}$ arrays at the Tx and the
Rx and (ii) equal size $N\times N$ arrays at both sides, respectively.

It is important to note that, as per (24), the far-field distance is now
determined by the physical dimensions of both the Tx and the Rx antenna
arrays.

We rely on these considerations and on (24) when determining the far-field
distance in the main body of the paper.

##  Appendix B Symmetric Case Analysis

In this appendix, we present a shorter solution for (5) in case a simpler
system model is assumed, where $D_{1}=D_{2}=D$.

First, directly from the first condition in (5), the maximum value of $D$ can
be determined as

$\displaystyle D=\frac{\sqrt{\lambda d_{\text{min}}}}{4}.$ (28)

Then, this value can be substituted into the second inequality in (5) leading
to:

$\displaystyle\frac{\lambda d_{\text{min}}}{16}\geq\lambda
d_{\text{max}}\sqrt{\frac{N_{\text{F}}kTB}{P_{\text{Tx}}}}10^{\frac{S_{\text{L},\text{dB}}}{20}}$
(29)

that can be solved for $B$ without the need to analyze any quadratic
polynomials as

$\displaystyle
B\leq\frac{P_{\text{Tx}}d^{2}_{\text{min}}}{256d^{2}_{\text{max}}10^{\frac{S_{\text{L},\text{dB}}}{10}}N_{\text{F}}kT},$
(30)

which is equivalent to the constrain on $B$ in (11) given for the general
case, where $D_{1}$ and $D_{2}$ may be different in the main body of the
paper.


# Federated PAC-Bayesian Learning on Non-IID data

###### Abstract

Existing research has either adapted the Probably Approximately Correct (PAC)
Bayesian framework for federated learning (FL) or used information-theoretic
PAC-Bayesian bounds while introducing their theorems, but few considering the
non-IID challenges in FL. Our work presents the first non-vacuous federated
PAC-Bayesian bound tailored for non-IID local data. This bound assumes unique
prior knowledge for each client and variable aggregation weights. We also
introduce an objective function and an innovative Gibbs-based algorithm for
the optimization of the derived bound. The results are validated on real-world
datasets.

Index Terms—  Federated learning, PAC-Bayesian framework, generalization error

##  1 Introduction

To address privacy concerns in distributed learning, federated learning (FL)
has emerged as a viable solution, enabling multiple local clients to
collaboratively train a model while retaining their private data and without
sharing it [1, 2]. However, in real-world scenarios, data across different
devices is not identically and independently distributed (non-IID), which
poses challenges in model training and convergence [3].

Significant efforts have been made to improve performance and analyze
convergence in non-IID FL [4], but few have provided theoretical guarantees by
establishing generalization bounds. Most existing FL generalization analyses
rely on the Probably Approximately Correct (PAC) Bayesian theory, first
formulated by McAllester [5, 6]. Building on the McAllester’s bound, these
analyses typically compute local bounds or apply existing PAC-Bayesian bounds
directly, overlooking the non-IID nature of FL. This approach is flawed, as
the PAC-Bayesian framework assumes that each data point is IID, ignoring non-
IID data and directly employing the PAC-Bayesian theory, which potentially
results in inaccurate or overly relaxed bounds. Consequently, techniques
developed for the PAC-Bayesian framework are not directly applicable to non-
IID FL. Therefore, this work aims to advance the theoretical underpinnings of
non-IID FL.

Related works. The PAC-Bayesian framework has been extensively researched in
recent years [7, 8, 9], yielding tighter and non-vacuous bounds. However,
there has been limited exploration in the context of FL. Some studies have
proposed information theoretic-based PAC-Bayesian bounds using Rate-distortion
theory to prove generalization bounds [10, 11], providing an information-
theoretic perspective on enhancing generalization capacity. Others have
followed McAllester’s approach, attempting to directly apply the FL paradigm
to the bound. For example, the authors in [12, 13] applied McAllester’s bound
in a multi-step FL scenario; Omni-Fedge [14] used the PAC-Bayesian learning
framework to construct a weighted sum objective function with a penalty,
considering only a local client bound instead of the entire system, which
precludes obtaining global information; and FedPAC [15] employed PAC learning
to balance utility, privacy, and efficiency in FL. However, these approaches
do not account for the non-IIDness of FL.

Our contributions. First, we derive a federated PAC-Bayesian learning bound
for non-IID local data, providing a unified perspective on federated learning
paradigms. To the best of our knowledge, this is the first non-vacuous bound
for a model averaging FL framework. Specifically, due to the non-IID nature of
clients, we assume that each client has unique prior knowledge rather than a
common one. Additionally, the aggregation weights for non-IID clients vary
instead of being uniform. Based on the derived bound, we define an objective
function that can be computed by each local client rather than on the server
and propose a Gibbs-based algorithm dubbed FedPB for its optimization. This
algorithm not only preserves the privacy of each client but also enhances
efficiency. Finally, we validate our proposed bounds and algorithm on two
real-world datasets, demonstrating the effectiveness of our bounds and
algorithm.

##  2 Problem Setting

In this section, we introduce the federated PAC-Bayesian learning setting. The
whole system comprises $K$ clients, each equipped with its own dataset
$S_{i}=(x_{i},y_{i})_{i=1}^{n}\subseteq(\mathcal{X},\mathcal{Y})^{n}$
consisting of $n$ IID data points. Here $\mathcal{X}$ denotes the input space
and $\mathcal{Y}$ denotes the output space. Each dataset $S_{i}$ is presumed
to be drawn from an unknown data generating distribution $D_{k}^{\bigotimes
n}$. Moreover, let
$\ell:\mathcal{Z}\times\mathcal{W}\rightarrow\mathbb{R}^{+}$ be a given loss
function and $h_{k}\in\mathcal{H}$ is a stochastic estimator on client $k$
where $\mathcal{H}$ is the hypothesis class. In the PAC-Bayesian framework,
each client holds a tailored prior distribution $P_{k}$. The objective of each
client is to furnish a posterior distribution $Q_{k}\in\mathcal{M}$, where
$\mathcal{M}$ denotes the set of distributions over $\mathcal{H}$. We then
define the population risk:

$L(Q_{1},\dots,Q_{K})\triangleq\frac{1}{K}\sum_{k=1}^{K}\underset{h_{k}\sim
Q_{k}}{\mathbb{E}}\underset{(x_{k},y_{k})\sim
D_{k}}{\mathbb{E}}[\ell(h_{k}(x_{k}),y_{k})],$ (1)

and the empirical risk:

$\hat{L}\left(Q_{1},\dots,Q_{K}\right)\triangleq\frac{1}{nK}\sum_{k=1}^{K}\underset{h_{k}\sim
Q_{k}}{\mathbb{E}}\sum_{i=1}^{n}\ell\left(h_{k}\left(x_{k,i}\right),y_{k,i}\right),$
(2)

by averaging over the posterior distribution of each client. In federated
learning, each client will upload their posterior distributions to a central
server, and then the server will aggregate the transmitted model in a weighted
manner:

$\displaystyle\bar{P}=\prod_{k=1}^{K}P_{k}^{p(k)},\quad\bar{Q}=\prod_{k=1}^{K}Q_{k}^{p(k)},$

where $\bar{P}$ and $\bar{Q}$ are the global prior and posterior,
respectively, and the averaging weight $p=(p(1),\dots,p(K))$ be a probability
distribution on $\\{1,\dots,K\\}$. For the sake of generality, we can assume
that $p(k)\in(0,1)$ and $\sum_{k=1}^{K}p(k)=1$. For intuition of this
aggregation, we can see that minimizing the weighted objective function is
actually equivalent to maximizing the logarithm of the corresponding
posterior:
$\min_{h}L(h)=\min_{h}\sum_{k=1}^{K}p(k)L_{k}(h)=\max_{h}\ln\prod_{n=1}^{N}\\\
p\left(h\mid\mathcal{D}_{k}\right)^{p(k)}.$ In addition, we denote the
Kullback-Leibler (KL) divergence as
$D_{KL}(Q\|P)\triangleq\underset{Q}{\mathbb{E}}\left[\log\frac{dQ}{dP}\right]$
if $Q\ll P$ and $D_{KL}(Q\|P)=+\infty$ otherwise.

##  3 Main theorem

In this section, we will present our novel bounds on the non-IID FL scenario.

######  Theorem 1 (Federated PAC-Bayesian learning bound).

For any $\delta\in(0,1]$, assume the loss function $\ell(\cdot,\cdot)$ is
bounded in $[0,C]$, the following inequality holds uniformly for all posterior
distributions $Q$ and for any $\delta\in(0,1)$,

$\underset{S_{1},\dots,S_{K}}{\mathbb{P}}\bigg{\\{}\forall{Q_{1},\dots,Q_{K}},L(Q_{1},\dots,Q_{K})\leq\hat{L}(Q_{1},\dots,Q_{K})\\\
+\frac{\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})+\log\frac{1}{\delta}}{\lambda}+\frac{\lambda
C^{2}}{8Kn}\bigg{\\}}>1-\delta$ (3)

###### Proof.

Let the local generalization error:
$\operatorname{gen}(D_{k},h_{k})=\underset{(x_{k},y_{k})\sim
D_{k}}{\mathbb{E}}[\ell(h_{k}(x_{k}),y_{k})]-\frac{1}{n}\sum_{i=1}^{n}\ell\left(h_{k}\left(x_{k,i}\right),y_{k,i}\right)$,
then the global generalization error:
$\overline{\operatorname{gen}}(D,h)=L(Q_{1},\dots,Q_{K})\\\
-\hat{L}(Q_{1},\dots,Q_{K})=\frac{1}{K}\sum_{k=1}^{K}\underset{h_{k}\sim
Q_{k}}{\mathbb{E}}\operatorname{gen}(D_{k},h_{k})$. For any $\lambda>0$ and
$t>0$, applying the Hoeffding’s lemma to $\mathbb{E}[\ell_{i}]-\ell_{i}$, we
have that, for each client $k$,

$\underset{S_{k}}{\mathbb{E}}\underset{P_{k}}{\mathbb{E}}\left[\mathrm{e}^{\frac{\lambda}{K}\operatorname{gen}(D_{k},h_{k})}\right]\leq\mathrm{e}^{\frac{\lambda^{2}C^{2}}{8K^{2}n}}.$

Since each $S_{k}$ may come from different $D_{k}$. i.e., non-IID, we cannot
directly plug this result to the PAC-Bayesian bound. Note that for each client
$k\in[K]$, $P_{i}$ is independent of $S_{1},\dots,S_{K}$, we have that

$\underset{S_{1}}{{\mathbb{E}}}\underset{P_{1}}{{\mathbb{E}}}\dots\underset{S_{K}}{{\mathbb{E}}}\underset{P_{K}}{{\mathbb{E}}}\left[\mathrm{e}^{\frac{\lambda}{K}\sum_{k=1}^{K}\operatorname{gen}(D_{k},h_{k})}\right]\leq\mathrm{e}^{\frac{\lambda^{2}C^{2}}{8Kn}}.$

And we apply Donsker and Varadhan’s variational formula [16] for
$P_{1},\dots,P_{K}$ to get:

$\underset{S_{1},\dots,S_{K}}{{\mathbb{E}}}\left[e^{\underset{Q_{1},\dots,Q_{K}}{\sup}\lambda\underset{Q_{1}}{{\mathbb{E}}}\dots\underset{Q_{K}}{{\mathbb{E}}}[\frac{1}{K}\sum_{k=1}^{K}\operatorname{gen}(D_{k},h_{k})]}\right.\\\
\left./e^{D_{KL}\left(\prod_{k=1}^{K}Q_{k}^{p(k)}\|\prod_{k=1}^{K}P_{k}^{p(k)}\right)}\right]\leq\mathrm{e}^{\frac{\lambda^{2}C^{2}}{8Kn}}.$
(4)

Recall the definition of the global generalization error:

$\overline{\operatorname{gen}}(D,h)=\frac{1}{K}\sum_{k=1}^{K}\underset{h_{k}\sim
Q_{K}}{{\mathbb{E}}}\operatorname{gen}(D_{k},h_{k}),$

and note that
$D_{KL}\left(\prod_{k=1}^{K}Q_{k}^{p(k)}\|\prod_{k=1}^{K}P_{k}^{p(k)}\right)=\\\
\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})$. Applying the Chernoff bound:

$\displaystyle\underset{S_{1},\dots,S_{K}}{\mathbb{P}}\left[\underset{Q_{1},\dots,Q_{K}}{\sup}\lambda\overline{\operatorname{gen}}(D,h)-\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})-\frac{\lambda^{2}C^{2}}{8Kn}>s\right]$
$\displaystyle\leq\underset{S_{1},\dots,S_{K}}{\mathbb{E}}\left[\mathrm{e}^{\underset{Q_{1},\dots,Q_{K}}{\sup}\lambda\overline{\operatorname{gen}}(D,h)-\sum\limits_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})-\frac{\lambda^{2}C^{2}}{8Kn}}\right]e^{-s}$
$\displaystyle\leq\mathrm{e}^{-s}.$

Let $\delta=e^{-s}$, that is, $s=-\log\delta$. Thus, plug this into the above
result we have that

$\underset{S_{1},\dots,S_{K}}{\mathbb{P}}\bigg{\\{}\exists{Q_{1},\dots,Q_{K}},\overline{\operatorname{gen}}(D,h)>\\\
\frac{1}{\lambda}\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})+\frac{\lambda
C^{2}}{8Kn}+\frac{1}{\lambda}\log\frac{1}{\delta}\bigg{\\}}\leq\delta.$

Therefore, we prove the statement by leveraging the complement of the
probability. ∎

The RHS of Equation 3 comprises two components: the empirical term and the
complexity term. Note that our bound eschews the typical smoothness and
convexity assumptions on the loss often made by other FL frameworks. Moreover,
an intuition can be presented for Equation 3 that the bound will be tighter
with the increasing of clients scales, which is further corroborated by the
evaluation in Section 5.3.

######  Corollary 1 (The choice of $\lambda$).

Suppose $\lambda\in\Xi\triangleq\\{0,\dots,\xi\\}$ and $|\cdot|$ denotes the
cardinality of a set. For any $\delta\in(0,1)$ and a properly chosen
$\lambda$, with probability at least $1-\delta$,

$L(Q_{1},\dots,Q_{K})\leq\hat{L}(Q_{1},\dots,Q_{K})\\\
+C\sqrt{\frac{\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})+\log\frac{|\Xi|}{\delta}}{2Kn}}.$
(5)

###### Proof.

Suppose $\mathcal{S}=S_{1}\cap S_{2}\cap\dots\cap S_{K}$ and
$\mathcal{Q}=Q_{1}\cap Q_{2}\cap\dots\cap Q_{K}$. Since we have the previous
result (4) for some fixed $\lambda$, we can sum (4) over all $\lambda\in\Xi$:

$\underset{\lambda\in\Xi}{\sum}\underset{\mathcal{S}}{\mathbb{E}}\left[\mathrm{e}^{\underset{\mathcal{Q}}{\sup}\lambda\overline{\operatorname{gen}}(D,h)-\sum\limits_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})-\frac{\lambda^{2}C^{2}}{8Kn}}\right]\leq|\Xi|,$

which is equivalent to:

$\underset{\mathcal{S}}{\mathbb{E}}\left[\mathrm{e}^{\underset{\mathcal{Q},\lambda\in\Xi}{\sup}\lambda\overline{\operatorname{gen}}(D,h)-\sum\limits_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})-\frac{\lambda^{2}C^{2}}{8Kn}}\right]\leq|\Xi|.$

Again, from the Chernoff bound:

$\displaystyle\underset{\mathcal{S}}{\mathbb{P}}\left[\underset{\mathcal{Q},\lambda\in\Xi}{\sup}\lambda\overline{\operatorname{gen}}(D,h)-\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})-\frac{\lambda^{2}C^{2}}{8Kn}>s\right]$
$\displaystyle\leq\underset{\mathcal{S}}{\mathbb{E}}\left[\mathrm{e}^{\underset{\mathcal{Q},\lambda\in\Xi}{\sup}\lambda\overline{\operatorname{gen}}(D,h)-\sum\limits_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})-\frac{\lambda^{2}C^{2}}{8Kn}}\right]e^{-s}$
$\displaystyle\leq|\Xi|\mathrm{e}^{-s}.$

Solve $\delta=|\Xi|\mathrm{e}^{-s}$ to get:

$\underset{\mathcal{S}}{\mathbb{P}}\bigg{\\{}\exists{\mathcal{Q},\lambda\in\Xi},\overline{\operatorname{gen}}(D,h)>\\\
\frac{\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})+\log\frac{|\Xi|}{\delta}}{\lambda}+\frac{\lambda
C^{2}}{8Kn}\bigg{\\}}\leq\delta$

Choosing a proper minimizer

$\lambda^{*}=\sqrt{8Kn\left(\sum_{k=1}^{K}p(k)D_{KL}(Q_{k}\|P_{k})+\log\frac{|\Xi|}{\delta}\right)}/C,$

we obtain the bound of this corollary. ∎

Equation 5 generally offers a strategy for selecting the value of parameter
$\lambda$ so that the complexity term can be minimized.

##  4 FedPB: Optimize the upper bound

We denote $\mathcal{L}_{k}=\underset{h_{k}\sim
Q_{k}}{\mathbb{E}}\frac{1}{n}\sum_{i=1}^{n}\ell\left(h_{k}\left(x_{k,i}\right),y_{k,i}\right)$.
Consider the following local objective function:

$\mathcal{J}(Q_{k})=\lambda\mathcal{L}_{k}+p(k)D_{KL}(Q_{k}\|P_{k})$ (6)

In our methodology, we introduce FedPB for a general scenario. It comprises
two phases, designed to iteratively optimize the priors and posteriors for
each client. Notably, in contrast to previous studies, clients are not
required to upload their private prior and posterior distributions to the
server, ensuring their privacy.

Phase 1 (Optimize the posterior). Given a fixed parameter $\lambda>0$ and the
prior $P_{k}^{t}$ during the training epoch $t+1$, we aim to optimize the
posterior $Q_{k}^{t}$ as
$\hat{Q}_{k}^{t+1}=\arg\min_{Q_{k}}\mathcal{J}(Q_{k}),$ yielding the solution:

$\frac{d\hat{Q}_{k}^{t+1}}{dP_{k}^{t}}(h)=\frac{\exp\left(-\lambda\ell\left(h,z_{i}\right)\right)}{\mathbb{E}_{h\sim
P_{k}^{t}}\left[\exp\left(-\lambda\ell\left(h,z_{i}\right)\right)\right]}.$

Phase 2 (Optimize the prior). Having derived the optimal posterior
$\hat{Q}_{k}^{t+1}$, the prior can be updated by
$\hat{P}_{k}^{t+1}={Q}_{k}^{t}$ since it minimizes $D_{KL}(Q_{k}\|P_{k})$.

Link with personalized federated learning. Since the prior
$\hat{P}_{k}^{t+1}={Q}_{k}^{t}$ and ${Q}_{k}^{t}$ is equal to the aggregated
global posterior at epoch $t-1$, the prior can be viewed as the global
knowledge. Optimizing the objective function (6) minimizes the disparity
between the global and local knowledge, which is a a prevalent personalization
strategy in FL [17].

Re-parameterization trick. Utilizing the Bayesian neural network [18] as the
local model aligns with our setting where all parameters are random, and we
are optimizing their posterior distribution. In particular, the prior $P_{k}$
and posterior $Q_{k}$ are defined as follows:

$\begin{aligned}
&P_{k}(w;\vartheta_{P_{k}})=\mathcal{N}\left(w;\mu_{P_{k}},\sigma_{P_{k}}^{2}I_{d}\right)=\prod_{i=1}^{d}\mathcal{N}\left(w_{i};\mu_{P_{k},i},\sigma_{P_{k},i}^{2}\right)\\\
&Q_{k}(w;\vartheta_{Q_{k}})=\mathcal{N}\left(w;\mu_{Q_{k}},\sigma_{Q_{k}}^{2}I_{d}\right)=\prod_{i=1}^{d}\mathcal{N}\left(w_{i};\mu_{Q_{k},i},\sigma_{Q_{k},i}^{2}\right)\end{aligned},$

with every model parameter $w_{i}$ being independent. Computing the Gibbs
posterior directly can be challenging, hence we select the gradient descent as
an alternative way. The update rule of (6) at round $t+1$ is:

$\displaystyle\mu_{Q_{k},i}^{t+1}$
$\displaystyle=\mu_{Q_{k},i}^{t}-\lambda\nabla_{\mu_{Q_{k},i}}\mathcal{L}_{k}-\frac{p(k)(\mu_{Q_{k},i}-\mu_{P_{k},i})}{\sigma_{Q_{k},i}^{2}},$
$\displaystyle\sigma_{Q_{k},i}^{t+1}$
$\displaystyle=\sigma_{Q_{k},i}^{t}-\lambda\nabla_{\sigma_{Q_{k},i}}\mathcal{L}_{k}+\frac{p(k)(\sigma_{P_{k},i}^{2}-\sigma_{Q_{k},i}^{2}+\left(\mu_{P_{k},i}-\mu_{Q_{k},i}\right)^{2})}{\sigma_{Q_{k},i}^{3}},$

where the parameter $\lambda$ can be regarded as the learning rate of the GD
and the KL-divergence is the regularization term. Calculating the gradients
$\nabla_{\mu_{Q_{k},i}}$ and $\nabla_{\mu_{Q_{k},i}}$ directly can be
intricate, but the re-parameterization trick is capable of tackling this
issue. Concretely, we translate the $h\sim Q_{k}$ to
$\varepsilon\sim\mathcal{N}(0,I_{d})$ and then compute the deterministic
function $h=\mu+\sigma\odot\varepsilon$, where $\odot$ signifies an element-
wise multiplication. As a result, we have
$\nabla_{\mu_{Q_{k},i}}\underset{h\sim
Q}{\mathbb{E}}l(h)=\nabla_{\mu_{Q_{k},i}}\underset{\varepsilon\sim
q(\varepsilon)}{\mathbb{E}}l(\epsilon)$, indicating its computability using an
end-to-end framework with back-propagation.

##  5 Evaluation

In this section, we demonstrate our algorithm and our theoretical arguments
with the FL non-IID setting. Specifically, the aggregation weight $p(k)$ is
defined as the sample ratio of client $k$ relative to the entire data size
across all clients. For the global aggregation, the global mean and covariance
are calculated by
$\bar{\mu}=\sum_{k=1}^{K}p(k)\mu_{k}\sigma_{k}^{-2}/\sum_{k=1}^{K}p(k)\sigma_{k}^{-2}$
and $\bar{\sigma}=1/\sum_{k=1}^{K}p(k)\sigma_{k}^{-2}$, respectively.
Furthermore, we utilize two real-world datasets: MedMNIST (medical image
analysis) [19] and CIFAR-10 [20]. For each dataset, we adopt three distinct
data-generating approaches for local clients: 1) Balanced: each client holds
an equal number of samples; 2) Unbalanced: varying sample counts per client
(e.g., $[0.05,0.05,0.05,0.05,0.1,0.1,0.1,0.1,0.2,0.2]$ for 10 clients); 3)
Dirichlet: differing sample counts per client following a Dirichlet
distribution [21]. Besides, the entire FL system encompasses $K=10$ clients,
initializing their posterior models uniformly from a global posterior model.

Additionally, we deploy two versions of Bayesian neural networks: one with 2
convolutional layers for MedMNIST and another with 3 layers for CIFAR-10. The
CrossEntropy loss serves as our loss function, and it is optimized by the Adam
optimizer [22] with a learning rate of 1e-3.

###  5.1 Bound evaluation

To validate our bounds, we set the confidence bound on $1-\delta=95\%$. Our
evaluation underscores a correlation between the generalization error and the
complexity, emphasizing the tightness of our bound. Fig. 1 illustrates an
initial increase in the generalization error and a concurrent decrease in
complexity during the early stages of the training process, attributed to
empirical loss optimization. Subsequently, as neural network training
advances, the KL-divergence stabilizes. Throughout this progression, we can
observe that the generalization error is consistently bounded by the
complexity value.

(a) Gen. for MedMNIST

(b) Acc. for MedMNIST

(c) Gen. for CIFAR-10

(d) Acc. for CIFAR-10

Fig. 1: The results of the generalization error and model performance of FedPB
over the Dirichlet generating method.

###  5.2 Data-dependent prior

Here, we preform the ablation study of data-dependent (trainable) prior
compared with data-independent (fixed, chosen before training) prior and
report the mean ± standard deviation (std) accuracy of the global model in
Table 1, evaluated over multiple experimental seeds. The results demonstrate
the superior efficacy of the data-dependent strategy over both datasets across
all three scenarios. This superiority arises from the data-dependent prior’s
ability to harness more global knowledge, combined with its adaptability
during training.

Table 1: Model accuracy (%) for the data-independent prior and data-dependent
prior in three data-generating scenarios.

Method | MedMNIST | CIFAR10
---|---|---
Balanced | Unbalanced | Dirichlet | Balanced | Unbalanced | Dirichlet
| Data-
---
independent
$53.47\pm 1.12$ | $49.44\pm 1.10$ | $55.24\pm 6.92$ | $50.89\pm 0.62$ | $47.19\pm 0.92$ | $57.93\pm 0.55$
| Data-
---
dependent
$\mathbf{77.10\pm 4.25}$ | $\mathbf{77.34\pm 3.42}$ | $\mathbf{77.48\pm 4.75}$ | $\mathbf{84.41\pm 0.94}$ | $\mathbf{79.39\pm 0.56}$ | $\mathbf{86.11\pm 0.53}$

###  5.3 Different client scales

Lastly, we assess the influence of varying client scales on our complexity
bounds. As depicted in Fig. 2, in the evaluation of both datasets with the
Dirichlet generating method, by increasing the number of clients $K$ from 10
to 20 and 50, we observe a consistent decrease in the value of complexity.
This observation aligns with our analysis in Equation 3.

Fig. 2: The impact of different client scales over FedPB on the value of the
complexity term.

## References

  * [1] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas,  “Communication-efficient learning of deep networks from decentralized data,”  in Artificial intelligence and statistics. PMLR, 2017, pp. 1273–1282.
  * [2] Zihao Zhao, Yuzhu Mao, Yang Liu, Linqi Song, Ye Ouyang, Xinlei Chen, and Wenbo Ding,  “Towards efficient communications in federated learning: A contemporary survey,”  Journal of the Franklin Institute, 2023.
  * [3] Yue Zhao, Meng Li, Liangzhen Lai, Naveen Suda, Damon Civin, and Vikas Chandra,  “Federated learning with non-iid data,”  arXiv preprint arXiv:1806.00582, 2018.
  * [4] Xiang Li, Kaixuan Huang, Wenhao Yang, Shusen Wang, and Zhihua Zhang,  “On the convergence of fedavg on non-iid data,”  arXiv preprint arXiv:1907.02189, 2019.
  * [5] David A McAllester,  “Some pac-bayesian theorems,”  in Proceedings of the eleventh annual conference on Computational learning theory, 1998, pp. 230–234.
  * [6] David A McAllester,  “Pac-bayesian model averaging,”  in Proceedings of the twelfth annual conference on Computational learning theory, 1999, pp. 164–170.
  * [7] Matthias Seeger,  “Pac-bayesian generalisation error bounds for gaussian process classification,”  Journal of machine learning research, vol. 3, no. Oct, pp. 233–269, 2002.
  * [8] Olivier Catoni,  “Pac-bayesian supervised classification: the thermodynamics of statistical learning,”  arXiv preprint arXiv:0712.0248, 2007.
  * [9] Luca Oneto, Michele Donini, Massimiliano Pontil, and John Shawe-Taylor,  “Randomized learning and generalization of fair and private classifiers: From pac-bayes to stability and differential privacy,”  Neurocomputing, vol. 416, pp. 231–243, 2020.
  * [10] Milad Sefidgaran, Romain Chor, and Abdellatif Zaidi,  “Rate-distortion theoretic bounds on generalization error for distributed learning,”  Advances in Neural Information Processing Systems, vol. 35, pp. 19687–19702, 2022.
  * [11] LP Barnes, Alex Dytso, and H Vincent Poor,  “Improved information theoretic generalization bounds for distributed and federated learning,”  in 2022 IEEE International Symposium on Information Theory (ISIT). IEEE, 2022, pp. 1465–1470.
  * [12] Milad Sefidgaran, Romain Chor, Abdellatif Zaidi, and Yijun Wan,  “Federated learning you may communicate less often!,”  arXiv preprint arXiv:2306.05862, 2023.
  * [13] Romain Chor, Milad Sefidgaran, and Abdellatif Zaidi,  “More communication does not result in smaller generalization error in federated learning,”  arXiv preprint arXiv:2304.12216, 2023.
  * [14] Sai Anuroop Kesanapalli and BN Bharath,  “Federated algorithm with bayesian approach: Omni-fedge,”  in ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2021, pp. 3075–3079.
  * [15] Xiaojin Zhang, Anbu Huang, Lixin Fan, Kai Chen, and Qiang Yang,  “Probably approximately correct federated learning,”  arXiv preprint arXiv:2304.04641, 2023.
  * [16] Monroe D Donsker and SR Srinivasa Varadhan,  “On a variational formula for the principal eigenvalue for operators with maximum principle,”  Proceedings of the National Academy of Sciences, vol. 72, no. 3, pp. 780–783, 1975.
  * [17] Xu Zhang, Yinchuan Li, Wenpeng Li, Kaiyang Guo, and Yunfeng Shao,  “Personalized federated learning via variational bayesian inference,”  in International Conference on Machine Learning. PMLR, 2022, pp. 26293–26310.
  * [18] Jost Tobias Springenberg, Aaron Klein, Stefan Falkner, and Frank Hutter,  “Bayesian optimization with robust bayesian neural networks,”  Advances in neural information processing systems, vol. 29, 2016\.
  * [19] Jiancheng Yang, Rui Shi, and Bingbing Ni,  “Medmnist classification decathlon: A lightweight automl benchmark for medical image analysis,”  in IEEE 18th International Symposium on Biomedical Imaging (ISBI), 2021, pp. 191–195.
  * [20] Alex Krizhevsky, Geoffrey Hinton, et al.,  “Learning multiple layers of features from tiny images,”  2009\.
  * [21] Mikhail Yurochkin, Mayank Agarwal, Soumya Ghosh, Kristjan Greenewald, Nghia Hoang, and Yasaman Khazaeni,  “Bayesian nonparametric federated learning of neural networks,”  in International conference on machine learning. PMLR, 2019, pp. 7252–7261.
  * [22] Diederik P Kingma and Jimmy Ba,  “Adam: A method for stochastic optimization,”  arXiv preprint arXiv:1412.6980, 2014.