# Federated Learning-Based Cell-Free Massive MIMO System for Privacy-
Preserving

Jiayi Zhang, , Jing Zhang, Derrick Wing Kwan Ng, and Bo Ai J. Zhang and J.
Zhang are with the School of Electronic and Information Engineering, Beijing
Jiaotong University, Beijing, China. (email: jiayizhang@bjtu.edu.cn).D. W. K.
Ng is with School of Electrical Engineering and Telecommunications, University
of New South Wales, Sydney, N.S.W., Australia (email: w.k.ng@unsw.edu.au).B.
Ai is with State Key Laboratory of Rail Traffic Control and Safety, Beijing
Jiaotong University, Beijing 100044, China (email: boai@bjtu.edu.cn).

###### Abstract

Cell-free massive MIMO (CF mMIMO) is a promising next generation wireless
architecture to realize federated learning (FL). However, sensitive
information of user equipments (UEs) may be exposed to the involved access
points or the central processing unit in practice. To guarantee data privacy,
effective privacy-preserving mechanisms are defined in this paper. In
particular, we demonstrate and characterize the possibility in exploiting the
inherent quantization error, caused by low-resolution analog-to-digital
converters (ADCs) and digital-to-analog converters (DACs), for privacy-
preserving in a FL CF mMIMO system. Furthermore, to reduce the required uplink
training time in such a system, a stochastic non-convex design problem that
jointly optimizing the transmit power and the data rate is formulated. To
address the problem at hand, we propose a novel power control method by
utilizing the successive convex approximation approach to obtain a suboptimal
solution. Besides, an asynchronous protocol is established for mitigating the
straggler effect to facilitate FL. Numerical results show that compared with
the conventional full power transmission, adopting the proposed power control
method can effectively reduce the uplink training time under various practical
system settings. Also, our results unveil that our proposed asynchronous
approach can reduce the waiting time at the central processing unit for
receiving all user information, as there are no stragglers that requires a
long time to report their local updates.

###### Index Terms:

Cell-free massive MIMO, federated learning, power control, differential
privacy.

##  I Introduction

Massive multiple-input multiple-output (MIMO) is an unprecedented technique to
increase both the spectral efficiency (SE) and energy efficiency (EE) of
communication systems. As a result, massive MIMO has already been utilized in
practical cellular systems [1, 2, 3, 4, 5]. However, some serious concerns
have recently been raised about data privacy. Indeed, mobile devices nowadays
are often equipped with high computing capabilities enabling them to collect
and process large amounts of data [6, 7]. Specifically, numerous applications
perform data preprocessing and classification for predicting possible future
events via using various machine learning technologies [8, 9]. The vast amount
of data of devices is generally collected for numerous private applications
carrying sensation information and thus naturally causes privacy concerns. On
the other hand, it is generally challenging to transmit all the data to a
central processing unit (CPU) for training a deep learning model. Besides, due
to the limited resources of wireless systems, sending a large amount of data
through wireless link is not always possible as it would introduce expensive
communication costs and exceedingly long communication delays.

###  I-A Related Works

In order to address the above challenges, it is necessary to design a novel
machine learning (ML) technology such that each user equipment (UE) can be
trained locally based on the data it collects and collaboratively establish a
shared global learning model. One of the most promising decentralized learning
methods to achieve this goal is FL [10, 11]. In particular, multiple UEs are
allowed to jointly train a global ML model without having to exchange raw data
among them or transfer their data to the CPU [12]. Specifically, the CPU first
broadcasts the latest global model to all the participating UEs. Next, the UEs
calculate the corresponding local update based on the available data and then
send their local models back to the CPU. Repeat these steps until a certain
level of global model accuracy is reached. In this way, only local model
parameters are exchanged, thereby reducing the required communication
signaling overhead.

Despite raw data sharing wireless channel can be avoided via FL, UEs’
sensitive information can still be possibly revealed through any form of the
leaked information. For example, a malicious CPU can perform a model inversion
attack [13] to infer the presence of individual data samples. Moreover, other
adversaries can apply differential attacks to the wireless communication phase
that performs data exchanged between the CPU and distributed access points
(APs) [10]. In the literature, there are three popular techniques for
maintaining privacy, including anonymization, data encryption and differential
privacy (DP) [14, 15, 16], with different drawback. For instance,
anonymization strategies do not guarantee complete level of protection from
adversaries; cryptographic techniques are computationally expensive. In
contrast, differential privacy is easy to implement and provides provable
privacy guarantee. Specifically, DP prevents the sensitive information of UEs
from being easily exposed even if the CPU/adversaries can access the model
parameters and acquire the knowledge of the adopted training mechanism [17].
In fact, one appealing approach to realize DP is via dedicated noise injection
[18]. The main idea of this approach is to deliberately introduce some noises
to the uploaded local model updates such that the CPU/adversaries cannot infer
any sensitive information from exploiting the actual data. Recently,
remarkable efforts have been made to investigate DP mechanisms for wireless FL
through artificial noise injection. For instance, in [19], Gaussian noise was
added to the local updated data and the power control was applied to realize
different levels of DP protection. Besides, the results in [18] and [20]
showed that the inherent channel noise can be exploited for guaranteeing DP
FL. Indeed, by deploying a proper power control, one can harness the channel
noise to achieve privacy for free. Also, in [21], the inherent hardware-
induced distortion was exploited to facilitate local model updates and a power
allocation strategy was proposed to provide guaranteed DP. On the other hand,
the existence of straggler effect also creates a bottleneck in realizing
effective FL in wireless networks [22]. By definition, the CPU needs to wait
until it receives training updates from all UEs before processing any next
steps. Therefore, some straggler UEs with unfavorable links may greatly slow
down the entire FL process and reduce its practicality [22, 23].

###  I-B Contributions

All the aforementioned works, e.g. [18, 19, 20], only assume simple wireless
environments, e.g. Gaussian channels with additive white noise, which does not
consider the fluctuation of practical wireless channels. Also, in practice,
when the number of UEs increases, the required training time could be
significantly prolonged. In such scenarios, to serve a large number of UEs via
the same time/frequency resources, cell-free massive multiple-input multiple-
output (CF mMIMO) systems have been applied for supporting FL [24]. Herein,
multiple distributed APs are connected to the CPU through capacity-unlimited
fronthaul links to serve UEs coherently. Since the large number of APs can
provide a rich macro-diversity gain for ensuring uniform received power
strength, the performance of CF mMIMO supported FL is less prone to UEs with
weak communication links. However, the current literature focuses on cell-free
massive MIMO systems implemented with federate learning is still limited. A
scheme for CF mMIMO networks to support any FL framework was proposed in [25]
for the first time. An optimization problem was also formulated to jointly
optimize the local accuracy, transmit power, data rate, and users¡¯ processing
frequency, and is solved by employing the online successive convex
approximation approach. Also, a UE selection approach was proposed in [26] to
mitigate the straggler effect with UE sampling for FL in CF mMIMO networks. It
selects only a small subset of UEs for participating in one FL process. In
[27], a novel scheme that asynchronously executes the iterations of FL
processes was designed for multicasting downlink and conventional uplink
transmission protocols. However, the privacy-preserving is not considered in
[25, 26, 27]. In [28], the authors developed and analyzed a privacy-preserving
channel estimation schemes in CF mMIMO systems. Yet, the design in [28] only
provides the data privacy protection during the channel estimation phase,
therefore the data information transmitted by the UE in the payload data
transmission phase still with high potential of leakage. Besides, practical
digital CF mMIMO communication systems adopt low-resolution analog-to-digital
converters (ADCs) and digital-to-analog converters (DACs) to reduce associated
power consumption and hardware cost [29]. In fact, these inherent noise can be
exploited to enhance DP that is somewhat overlooked in the literature.

Motivated by the above discussion, we consider a practical CF mMIMO supported
FL framework and demonstrate that the inherent quantization noise caused by
low-resolution ADCs and DACs in CF mMIMO can be exploited as a useful privacy-
preserving mechanism. Our contributions are listed as follows:

  * •

First, we capitalize the quantization noise introduced by the low-resolution
ADCs and DACs to prevent the CPU/adversaries from exploiting the actual local
gradient updates to infer sensitive information and hence, realize privacy-
preserving. Within this proposed framework, we derive an upper-bound to
characterize the privacy violation probability and adopt it for formulating a
privacy preservation condition.

  * •

Then, we provide the closed-from convergence analysis of the DP mechanism,
taking into account the quantization noise, the length of the local updates,
and the total data size. It can be observed from the upper bound of the
average optimality gap that the noise added in the initial iterations is less
damaging to the final optimality gap than that added in later iterations.
Besides, the initial optimality gap decays geometrically as the number of
iteration increases. Note that the quantization noise is exploited for privacy
protection, therefore, different quantization accuracies can realize different
DP protection levels.

  * •

Third, in order to minimize the uplink training time of CF mMIMO-supported-FL,
we formulate a stochastic nonconvex optimization problem that jointly
optimizes the transmit power and the data rate, subject to the practical
constraints on energy consumption of the UEs with different quantization
accuracies of the ADCs and DACs. Our numerical results reveal that the
proposed power control method can effectively reduce the uplink training time
under different privacy protection levels. Besides, our power control method
still performs well over various baseline schemes for different number of APs
and UEs.

  * •

Finally, we propose an asynchronous FL protocol to alleviate the straggler
effect with two simple parameters, lag tolerance and lag percent,
respectively. We also compare the performance of synchronous FL and
asynchronous FL and the empirically analyze the impacts caused by lag
tolerance and lag percent.

_Notations:_ Throughout the paper, $\mathbb{R}$ and $\mathbb{C}$ represents
the sets of all real and complex values, respectively. We denote a complex
zero-mean normal distribution with variance $\sigma^{2}$ by ${{\cal
N}_{\mathbb{C}}}\left({0,{\sigma^{2}}}\right)$. The cardinality of a set
$\mathcal{A}$ is denoted by $\left|{\cal A}\right|$. Furthermore, the
difference between two sets is defined as ${\cal A}^{\prime}-{\cal
A}^{\prime\prime}=\left\\{{\left.x\right|x\in{\cal A}^{\prime},x\notin{\cal
A}^{\prime\prime}}\right\\}$. The union of sets ${{\cal A}_{1}},\cdots,{{\cal
A}_{K}}$ is represented by ${\cal A}=\bigcup\nolimits_{k=1}^{K}{{{\cal
A}_{k}}}$. Boldfaced lower-case letters, e.g., $\bf{a}$, represents the
vectors, ${\bf{a}}^{H}$, ${\bf{a}}^{*}$, and $\left\|{\bf{a}}\right\|$ denote
Hermitian transpose, conjugate, and Euclidean norm of $\bf{a}$, respectively.

##  II System Model

Figure 1: Illustrations of a CF mMIMO system supported FL with low-resolutions
ADCs equipped at the APs.

As shown in Fig. 1, we consider a CF mMIMO supported FL system consisting of
$K$ single-antenna UEs and $L$ single-antenna APs [30]. All APs and UEs are
randomly located in an $D\times D$ area. Each UE is served by all the APs over
the same time/frequency resources. A CPU is connected to all APs via ideal
wireless fronthaul. UE $k$, $k\in\left\\{{1,\cdots,K}\right\\}$, is equipped
with its own local dataset ${{\cal
B}_{k}}\buildrel\Delta\over{=}\left\\{{\left({{{\bf{x}}_{kn}},{y_{kn}}}\right)}\right\\}_{n=1}^{{B_{k}}}$,
where $B_{k}$ is the data size and ${\left({{{\bf{x}}_{kn}},{y_{kn}}}\right)}$
is the corresponding $n$th data sample. The objective of FL is to find an
$d\times 1$ optimal model vector ${\bf{w}}$ that minimizes the global loss
function [21]:

$\mathop{\text{minimize}}\limits_{\bf{w}}{\rm{}}F\left({\bf{w}}\right)=\frac{1}{K}\sum\limits_{i=1}^{K}{{F_{i}}}\left({\bf{w}}\right),$
(1)

where ${B_{{\rm{tot}}}}=\sum\limits_{i=1}^{K}{{B_{k}}}$ and
${{F_{k}}\left({\bf{w}}\right)}$ is the local loss function which is given by

${F_{k}}\left({\mathbf{w}}\right)=\frac{1}{{{B_{k}}}}\sum\nolimits_{\left({{{\mathbf{x}}_{k}},{y_{k}}}\right)\in{\mathcal{B}_{k}}}{f\left({{\mathbf{w}},{{\mathbf{x}}_{k}},{y_{k}}}\right)},$
(2)

where ${f\left({{\bf{w}};{{\bf{x}}_{k}},{y_{k}}}\right)}$ is the sample-wise
loss function that quantifies the prediction error of the model ${\bf{w}}$ on
the training samples ${{\bf{x}}_{kn}}$ with respect to the labels $y_{kn}$.

###  II-A Learning Protocol

In order to address problem (1), we apply the distributed stochastic gradient
descent (SGD) [31, 32] at the CPU and the UEs. Note that the CPU and the UEs,
respectively, act as the central server and the clients in the general FL
framework. The APs with CF mMIMO are only used to relay the training updates
between the CPU and the UEs. The specific procedure is summarized as follows.

  * •

Step 1: Downlink communication for model download. The CPU broadcasts the
current model, i.e., ${\bf{w}}^{t}$, to all the UEs, where $t$ represents the
communication round, $t=1,\cdots,T$.

  * •

Step 2: Local computation. Each UE computes the gradient of the local loss
function in (2) via

$\nabla{F_{k}}\left({{{\bf{w}}^{t}}}\right)=\frac{1}{{{B_{k}}}}\sum\limits_{\left({{{\bf{x}}_{k}},{y_{k}}}\right)\in{{\cal
B}_{k}}}\nabla{f\left({{{\bf{w}}^{t}},{{\bf{x}}_{k}},{y_{k}}}\right)},\forall
t,$ (3)

where $\nabla{F_{k}}\left({{{\bf{w}}^{t}}}\right)$ is the gradient of
${F_{k}}\left({{{\bf{w}}^{t}}}\right)$.

  * •

Step 3: Uplink communication for model upload. The UEs send the gradient of
the local loss function to the CPU utilizing the same time and frequency
resources.

  * •

Step 4: Global computation. Based on the received signal, the CPU obtains an
estimated $\widehat{\nabla F}\left({{{\bf{w}}^{t}}}\right)$ of the global
gradient by computing

$\nabla
F\left({{{\bf{w}}^{t}}}\right)=\frac{1}{{{B_{{\rm{tot}}}}}}\sum\limits_{i=1}^{K}{\nabla{F_{i}}\left({{{\bf{w}}^{t}}}\right)}.$
(4)

Then, the CPU updates the current global model as

${{\bf{w}}^{t+1}}={{\bf{w}}^{t}}-\eta\widehat{\nabla
F}\left({{{\bf{w}}^{t}}}\right),$ (5)

where ${\eta}$ denotes the learning rate.

Note that Steps 2 to 4 are repeated until a convergence criterion is met.

###  II-B Communication Model

The channel coefficient between AP $l$ and UE $k$ is denoted as
$h_{kl}\in\mathbb{C}$, which is modeled as
${h_{kl}}=\sqrt{{\beta_{kl}}}{g_{kl}}$, where $\beta\in\mathbb{R}$ represents
the large-scale fading and $g_{kl}\in\mathbb{C}$ represents the small-scale
fading coefficient, respectively [33]. We adopt the block fading model where
$h_{kl}$ is a constant in each time-frequency block. Without loss of
generality, each block contains ${\tau_{c}}$ channel uses, which consists of
$\tau_{p}$ channel uses dedicated for acquiring the channel state information
(CSI) and $\tau_{c}-\tau_{p}$ channel uses for the uncoded transmission of the
$d$-dimensional gradient vector [18]. Besides, we assume that perfect CSI is
available at the APs [18]. In the following, we derive the uplink training
expressions when consider inherent noise induced by the low-resolution ADCs at
the APs or low-resolution DACs at the UEs.

####  II-B1 Low-resolution ADCs equipped at the APs

At communication round $t$ of Step 4, the received signal
${\mathbf{y}}_{l}^{t}$ at AP $l$ is given as

${\mathbf{y}}_{l}^{t}=\sum\limits_{i=1}^{K}{\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}}\right\|}^{2}}}}}{h_{il}}{\mathbf{s}}_{i}^{t}}+{{\mathbf{n}}_{l}},$
(6)

where

${\mathbf{s}}_{i}^{t}=\left|{{{{\mathcal{B}}}_{i}}}\right|\nabla{F_{i}}\left({{{\mathbf{w}}^{t}}}\right),\;\;\;\forall
i\in\left\\{{1,\cdots,K}\right\\},$ (7)

where ${p_{k}^{t}}$ denotes the transmit power for UE $i$,
${{\mathbf{n}}_{l}}$ is the additive noise with independent
${\mathcal{N}_{\mathbb{C}}}\left({0,{\sigma^{2}}{{\mathbf{I}}_{d}}}\right)$,
and ${\sigma^{2}}$ is the noise power per antenna. We adopt the linear
additive quantization noise model [34] to capture the quantization loss and
the noise caused by low-resolution ADCs, which yields

$\mathcal{Q}\left({{\mathbf{y}}_{l}^{t}}\right)=\alpha\sum\limits_{i=1}^{K}{\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}}\right\|}^{2}}}}}{h_{il}}{\mathbf{s}}_{i}^{t}}+\alpha{{\mathbf{n}}_{l}}+{\mathbf{n}}_{l}^{{\text{uq}}},$
(8)

where $\alpha=\frac{{\pi\sqrt{3}}}{2}{2^{-2b}}$ is a linear gain depending on
the number of quantization bits adopted in ADCs, $b$, and
${\mathbf{n}}_{l}^{{\text{uq}}}$ represents the additive Gaussian noise with
covariance matrix111The linear gain $\alpha$ for different quantization bits
can be approximated according to [29]. In general, one can adopt $b=10$ to
mimic the perfect ADCs case. [35]

${{\mathbf{R}}_{{\mathbf{n}}_{l}^{{\text{uq}}}}^{t}}=\alpha\left({1-\alpha}\right)\left({\sum\limits_{i=1}^{K}{{p_{i}^{t}}{\beta_{il}}}+{\sigma^{2}}}\right){{\mathbf{I}}_{d}}.$
(9)

We consider a fully distributed CF mMIMO system, in which the data detection
is performed at the APs [36, 37]. When applying the maximum-ratio combining
(MRC) for low computational complexity, the local processed signal for UE $k$
at AP $l$ at communication round $t$ is given as

${\mathbf{\hat{s}}}_{kl}^{t}=\alpha\sum\limits_{i=1}^{K}{\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}}\right\|}^{2}}}}}h_{kl}^{*}{h_{il}}{\mathbf{s}}_{i}^{t}}+\alpha
h_{kl}^{*}{{\mathbf{n}}_{l}}+h_{kl}^{*}{\mathbf{n}}_{l}^{{\text{uq}}}.$ (10)

Then, the APs convey the local processed signal to the CPU. The received
signal from all the APs at the CPU is given as

${\mathbf{r}}_{k}^{t}=\sum\limits_{l=1}^{L}{{\mathbf{\hat{s}}}_{kl}^{t}}=\alpha\sum\limits_{l=1}^{L}{\sum\limits_{i=1}^{K}{\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}}\right\|}^{2}}}}}h_{kl}^{*}{h_{il}}{\mathbf{s}}_{i}^{t}}}+{\bm{\omega}}_{k}^{t},$
(11)

where
${\bm{\omega}}_{k}^{t}=\alpha\sum\limits_{l=1}^{L}{h_{kl}^{*}{{\mathbf{n}}_{l}}}+\sum\limits_{l=1}^{L}{h_{kl}^{*}{\mathbf{n}}_{l}^{{\text{uq}}}}$
is the effective noise distributed according to
${\mathcal{N}_{\mathbb{C}}}\left({0,{{\left({m_{k}^{t}}\right)}^{2}}{{\mathbf{I}}_{d}}}\right)$,
with
${{\left({m_{k}^{t}}\right)}^{2}}={{\alpha^{2}}\sum\limits_{l=1}^{L}{{\beta_{kl}}{\sigma^{2}}}++\alpha\left({1-\alpha}\right)\sum\limits_{l=1}^{L}{{\beta_{kl}}}\left({\sum\limits_{i=1}^{K}{{p_{i}}{\beta_{il}}}+{\sigma^{2}}}\right)}$.

Now, we rigorously analyze the performance by using the achievable rates.
According to [30, 38], the achievable rate of UE $k$, ${R_{k}}$, is

$\displaystyle{R_{k}}\leqslant{r_{k}},$ (12)
$\displaystyle{r_{k}}=\left({1-\frac{{{\tau_{p}}}}{{{\tau_{c}}}}}\right)B{\log_{2}}\left({1+{\text{SIN}}{{\text{R}}_{k}}}\right),$
(13)

where $B$ is the bandwidth and

${\text{SIN}}{{\text{R}}_{k}}=\frac{{{p_{k}}{A_{k}}}}{{\sum\nolimits_{i\neq
k}^{K}{{p_{i}}}{B_{ki}}+{C_{k}}+{D_{k}}+\sum\nolimits_{i=1}^{K}{{p_{i}}}{E_{ki}}}},$
(14)

where

$\displaystyle{A_{k}}={\alpha^{2}}{\left|{\sum\limits_{l=1}^{L}{{\beta_{kl}}}}\right|^{2}},\;\;\;{B_{ki}}={\alpha^{2}}\sum\limits_{l=1}^{L}{{\beta_{kl}}{\beta_{il}}},$
$\displaystyle{C_{k}}={\alpha^{2}}{\sigma^{2}}\sum\limits_{l=1}^{L}{{\beta_{kl}}},\;\;\;{D_{k}}=d\alpha\left({1-\alpha}\right)\sum\limits_{l=1}^{L}{{\beta_{kl}}}{\sigma^{2}},$
$\displaystyle{E_{ki}}=d\alpha\left({1-\alpha}\right)\sum\limits_{l=1}^{L}{{\beta_{il}}{\beta_{kl}}}.$
(15)

According to [25], the uplink latency of UE $k$ in each iteration involves the
transmission delay of sending the global uplink training update from it to the
APs and from the APs to the CPU, i.e.,

${t_{k}}=\frac{S}{{{R_{k}}}},\;\;\;{t_{l}}=\frac{{KS}}{{\sum\limits_{k=1}^{K}{{R_{k}}}}},$
(16)

respectively, where $S$ is the data size. Therefore, the total uplink training
time is

$T_{\text{time}}=\mathop{\max}\limits_{k}\left\\{{\frac{ST}{{{R_{k}}}}}\right\\}+\frac{{KST}}{{\sum\limits_{k=1}^{K}{{R_{k}}}}}.$
(17)

####  II-B2 Low-resolution DACs at the UEs

With low-resolution DACs equipped at the UEs, the signal sent by UE $i$,
${{{\bf{s}}_{i}}}$, to the APs is given as

${\cal
Q}\left({{{\bf{s}}_{i}}}\right)={\zeta}{{{\bf{s}}_{i}}}+{\bf{n}}_{i}^{\rm{q}},\forall
i\in\left\\{{1,\cdots,K}\right\\},$ (18)

where $\zeta=\frac{{\pi\sqrt{3}}}{2}{2^{-2b}}$ is a linear gain depending on
the number of quantization bits adopted in DACs, $b$, and the elements of
${{{\bf{n}}_{i}}}^{\rm{q}}$ are i.i.d. ${\cal C}{\cal
N}\left({0,{\zeta}\left({1-{\zeta}}\right){p_{i}}}\right)$ random variables
[39, 40]. Besides, according to (17), in synchronous FL [41, 42], which refers
to the CPU needs to wait for receiving the training updates from all the UEs,
the straggler UEs with unfavorable links may greatly slow down the entire FL
process and reduce its practicality.

On the other hand, in asynchronous FL, even if all local model updates are not
received, the aggregate model can be derived. Therefore, in order to mitigate
the straggler effect, the asynchronous communication mode is considered. Then,
the connection relationship between the APs and the UEs can be expressed as

${d_{il}^{t}}=\left\\{{\begin{array}[]{*{20}{c}}{1,l\in{{\cal M}_{i}}},\\\
{0,l\notin{{\cal M}_{i}}},\end{array}}\right.\;\;\;t=1,\cdots,T,$ (19)

where

$\displaystyle{{\cal
M}_{i}^{t}}=\left\\{{l:{d_{il}^{t}}=1,l\in\left\\{{1,\cdots
L}\right\\}}\right\\}.$ (20)

Note that ${d_{il}^{t}=1}$ if the $l$th AP is allowed to serve UE $i$ at
communication round $t$ and 0 otherwise. Therefore, ${{\cal M}_{i}^{t}}$
denotes the subset of APs that serve UE $i$ at communication round $t$.
Therefore, at communication round $t$, the transmitted signal from UE $k$
processed by the low-precision DAC is given as

$\displaystyle{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{s}}}}_{k}^{t}\\!\\!=\\!\\!{\cal
Q}\left({{\bf{s}}_{k}^{t}}\right)\\!\\!=\\!\\!{\zeta}{\bf{s}}_{k}^{t}+{\left({{\bf{n}}_{k}^{\rm{q}}}\right)^{t}}.$
(21)

The received signal ${\mathbf{y}}_{l}^{t}$ at AP $l$ is

$\displaystyle{\bf{y}}_{l}^{t}$
$\displaystyle=\sum\limits_{i=1}^{K}{d_{il}^{t}}{h_{il}^{t}{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{s}}}}_{i}^{t}}+{\bf{n}}_{l}^{t}=\sum\limits_{i=1}^{K}{{\zeta}{d_{il}^{t}}h_{il}^{t}{\bf{s}}_{i}^{t}}+\sum\limits_{i=1}^{K}{d_{il}^{t}}{h_{il}^{t}{{\left({{\bf{n}}_{k}^{\rm{q}}}\right)}^{t}}}+{\bf{n}}_{l}^{t}=\sum\limits_{i=1}^{K}{{\zeta}{d_{il}^{t}}h_{il}^{t}{\bf{s}}_{i}^{t}}+{\bm{\omega}}_{k}^{t},$
(22)

where
${\bm{\omega}}_{k}^{t}=\sum\limits_{i=1}^{K}{{d_{il}^{t}}h_{il}^{t}{{\left({{\bf{n}}_{k}^{\rm{q}}}\right)}^{t}}}+{\bf{n}}_{l}^{t}$
is the effective noise distributed according to
${\mathcal{N}_{\mathbb{C}}}\left({0,\sigma_{{\rm{eff}}}^{2}{{\bf{I}}_{d}}}\right)$
and
$\sigma_{{\rm{eff}}}^{2}=\sum\limits_{i=1}^{K}{{d_{il}^{t}}\beta_{il}^{t}{\zeta_{i}}}\left({1-{\zeta_{i}}}\right){{\bf{I}}_{d}}+{\sigma^{2}}{{\bf{I}}_{d}}$.

When the MRC scheme is employed, we set ${v_{kl}}=d_{kl}^{t}{h_{kl}}$, the
local processed signal of ${\mathbf{s}}_{kl}^{t}$ at AP $l$ is given as

$\displaystyle{\bf{\hat{s}}}_{kl}^{t}=v_{kl}^{*}{{\bf{y}}_{l}^{t}}=v_{kl}^{*}\sum\limits_{i=1}^{K}{{h_{il}}{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{s}}}}_{i}^{t}}+v_{kl}^{*}{{\bf{n}}_{l}}$
$\displaystyle=d_{kl}^{t}h_{kl}^{*}{h_{kl}}{\zeta_{k}}{\bf{s}}_{k}^{t}+d_{kl}^{t}h_{kl}^{*}{h_{kl}}{\left({{\bf{n}}_{k}^{\rm{q}}}\right)^{t}}+d_{kl}^{t}h_{kl}^{*}\sum\limits_{i\neq
k}^{K}{{h_{il}}{\bf{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\smile$}}\over{s}}}}_{i}^{t}}+d_{kl}^{t}\hat{h}_{kl}^{*}{{\bf{n}}_{l}}.$
(23)

Then, the achievable SE for UE $k$ at communication round $t$ can be obtained
in the following closed-form

${\rm{SINR}}_{k}^{t}=\frac{{p_{k}^{t}A_{k}^{t}}}{{\sum\limits_{i=1}^{K}{p_{i}^{t}C_{ki}^{t}}+\sum\limits_{i=1}^{K}{p_{i}^{t}E_{ki}^{t}}+F_{k}^{t}}},$
(24)

where

$\displaystyle
A_{k}^{t}={\left({\zeta^{t}}\right)^{2}}{\left|{\sum\limits_{l=1}^{L}{d_{kl}^{t}{\beta_{kl}}}}\right|^{2}},\;\;\;C_{ki}^{t}={\left({\zeta^{t}}\right)^{2}}\sum\limits_{l=1}^{L}{d_{kl}^{t}{\beta_{kl}}}{\beta_{il}},$
$\displaystyle
E_{k}^{t}=d\sum\limits_{l=1}^{L}{d_{il}^{t}{\beta_{kl}}{\beta_{il}}\zeta^{t}\left({1-\zeta^{t}}\right)},\;\;\;F_{k}^{t}=d\sum\limits_{l=1}^{L}{d_{kl}^{t}}{\beta_{kl}}{\sigma^{2}}.$
(25)

Therefore, the total uplink training time is

$T_{\text{time}}=\sum\limits_{t=1}^{T}{\frac{S}{{R_{k}^{t}}}}+\sum\limits_{t=1}^{T}{\frac{{\left|{{{\cal
K}^{t}}}\right|S}}{{\sum\limits_{k\in{\cal K}}{R_{k}^{t}}}}},$ (26)

where
$\mathcal{K}^{t}={\mathcal{D}_{1}^{t}}\cup{\mathcal{D}_{2}^{t}}\cdots{\mathcal{D}_{L}^{t}}$
and ${\mathcal{D}_{l}^{t}}=\left\\{{i:{d_{il}^{t}}=1,i\in\left\\{{1,\cdots
K}\right\\}}\right\\}$. Compared with (16), it can be observed that choosing
an appropriate serving UEs cluster ${{\cal D}_{i}^{t}}$ in each iteration can
effectively reduce the total uplink training time.

##  III Differential Privacy Analysis

DP is a privacy mechanism to fight against differential attacks and to ensure
that the sensitive information of UEs is not exposed [10]. The standard
definition of DP imposes a point-wise upper bound on the divergence between
the distributions $P\left({\left.{\bf{y}}\right|{\cal B}}\right)$ and
$P\left({\left.{\bf{y}}\right|{\cal B}^{\prime}}\right)$, where $\bf{y}$ is
the received signal and ${\cal B}$ and ${\cal B}^{\prime}$ are two
“neighboring” global data sets which only differ by one sample at one UE.

In this section, we derive the upper-bound of the privacy preservation
condition and provide the convergence analysis in the CF mMIMO-supported FL
with noise injection by using low-resolution ADCs or DACs.

###  III-A Low-resolution ADCs Equipped at the APs

####  III-A1 Privacy Preservation Condition

######  Definition 1.

For two adjacent datasets ${{\mathcal{B}^{{}^{\prime}}_{j}}}$ and
${{\mathcal{B}^{{}^{\prime\prime}}_{j}}}$ with
${\left|{{{\mathcal{B}^{{}^{\prime}}_{j}}}-{{\mathcal{B}^{{}^{\prime\prime}}_{j}}}}\right|}=1$
for some UEs $j$ and
${\left|{{{\mathcal{B}^{{}^{\prime}}_{i}}}-{{\mathcal{B}^{{}^{\prime\prime}}_{i}}}}\right|}=0$
for all $i\neq j$, the communication and learning protocol is
$({\mathbb{\epsilon}},\delta)$-differentially private, where
${\mathbb{\epsilon}}>0$, and $\delta\in\left[{0,1}\right)$, when we have the
following inequality for UE $k$ [18]:

$\Pr\left({\left.{{\mathbf{r}}_{k}^{t}}\right|{{\mathcal{B}^{{}^{\prime}}_{k}}}}\right)\leqslant\exp\left({\mathbb{\epsilon}}\right)\Pr\left({\left.{{\mathbf{r}}_{k}^{t}}\right|{{\mathcal{B}^{{}^{\prime\prime}}_{k}}}}\right)+\delta,$
(27)

where $\Pr$ refers to the probability of a certain event. After $T$
iterations, the $({\mathbb{\epsilon}},\delta)$ DP condition in (27) can be
written as

$\Pr\left({\left|{\ln\left({\prod\limits_{t=1}^{T}{\frac{{P\left({\left.{{\mathbf{r}}_{k}^{t}}\right|{\mathbf{r}}_{k}^{t-1}\cdots{\mathbf{r}}_{k}^{1},{\mathcal{B}_{k}^{{}^{\prime}}}}\right)}}{{P\left({\left.{{\mathbf{r}}_{k}^{t}}\right|{\mathbf{r}}_{k}^{t-1}\cdots{\mathbf{r}}_{k}^{1},{\mathcal{B}_{k}^{{}^{\prime\prime}}}}\right)}}}}\right)}\right|\\!\leqslant\\!{\mathbb{\epsilon}}}\right)\\!\geqslant\\!1-\delta.$
(28)

The $({\mathbb{\epsilon}},\delta)$-DP condition ensures that for all possible
adjacent datasets, the absolute value of the left side of (28) can be bounded
by ${\mathbb{\epsilon}}$ with probability at least $1-\delta$. Note that the
values ${\mathbb{\epsilon}}$ and $\delta$ stand for the similarity of the
result distribution of the random mechanism performed on the data sets
${\mathcal{B}_{k}^{{}^{\prime}}}$ and ${\mathcal{B}_{k}^{{}^{\prime\prime}}}$,
and are interpreted as a privacy level [43]. The lower ${\mathbb{\epsilon}}$
and $\delta$ indicate a higher level of privacy.

The sensitivity $\Delta_{k}^{t}$ of the noiseless received signal
${\mathbf{r}}_{k}^{t}-{\bm{\omega}}_{k}^{t}$ is defined as

$\displaystyle\Delta_{k}^{t}$
$\displaystyle=\mathop{\max}\limits_{\mathcal{B}_{k}^{{}^{\prime}},\mathcal{B}_{k}^{{}^{\prime\prime}}}\left\|{\alpha\sum\limits_{l=1}^{L}{\sum\limits_{i=1}^{K}{\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime}}}\right)}\right\|}^{2}}}}}h_{kl}^{*}{h_{il}}{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime}}}\right)}}}\right.$
$\displaystyle\left.{-\alpha\sum\limits_{l=1}^{L}{\sum\limits_{i=1}^{K}{\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime\prime}}}\right)}\right\|}^{2}}}}}h_{kl}^{*}{h_{il}}{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime\prime}}}\right)}}}\right\|.$
(29)

Equation (III-A1) can be bounded as

$\Delta_{k}^{t}\leqslant\mathop{\max}\limits_{i}2\alpha\sqrt{p_{i}^{t}}\left|{h_{kl}^{*}{h_{il}}}\right|$
(30)

with the help of [19] and the triangular inequality. Then, according to (11),
we can obtain

${\ln\\!\\!\left(\\!{\prod\limits_{t=1}^{T}{\frac{{P\left({\left.{{\mathbf{r}}_{k}^{t}}\right|{\mathbf{r}}_{k}^{t-1}\cdots{\mathbf{r}}_{k}^{1},{\mathcal{B}_{k}^{{}^{\prime}}}}\right)}}{{P\left({\left.{{\mathbf{r}}_{k}^{t}}\right|{\mathbf{r}}_{k}^{t-1}\cdots{\mathbf{r}}_{k}^{1},\mathcal{B}_{k}^{{}^{\prime\prime}}}\right)}}}}\right)}\\!\\!=\\!\\!{\sum\limits_{t=1}^{T}{\ln}\\!\\!\left(\\!\\!{\frac{{\exp\left({\frac{{{{\left\|{{\bm{\omega}}_{k}^{t}}\right\|}^{2}}}}{{2{{\left({m_{k}^{t}}\right)}^{2}}}}}\right)}}{{\exp\left({\frac{{{{\left\|{{\bm{\omega}}_{k}^{t}+{\mathbf{v}}_{k}^{t}}\right\|}^{2}}}}{{2{{\left({m_{k}^{t}}\right)}^{2}}}}}\right)}}}\right)},$
(31)

where

$\displaystyle{\mathbf{v}}_{k}^{t}$
$\displaystyle=\alpha\sum\limits_{l=1}^{L}\sum\limits_{i=1}^{K}\left({\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime\prime}}}\right)}\right\|}^{2}}}}}h_{kl}^{*}{h_{il}}{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime\prime}}}\right)}\right.$
$\displaystyle\left.{-\sqrt{\frac{{p_{i}^{t}}}{{{{\left\|{{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime}}}\right)}\right\|}^{2}}}}}h_{kl}^{*}{h_{il}}{\mathbf{s}}_{i}^{t}\left({\mathcal{B}_{k}^{{}^{\prime}}}\right)}\right),$
(32)

with $\left\|{{\mathbf{v}}_{k}^{t}}\right\|\leqslant\Delta_{k}^{t}$.

Following similar steps as in [43, Appendix A], we can then obtain the upper-
bound on the privacy preservation condition

$\displaystyle\Pr\left({\left|{\sum\limits_{t=1}^{T}{\frac{{2{{\left({{\bm{\omega}}_{k}^{t}}\right)}^{T}}{\mathbf{v}}_{k}^{t}+{{\left\|{{\mathbf{v}}_{k}^{t}}\right\|}^{2}}}}{{2{{\left({m_{k}^{t}}\right)}^{2}}}}}}\right|>{\mathbb{\epsilon}}}\right)$
$\displaystyle\mathop{\leqslant}\limits^{\left(i\right)}\Pr\left({\left|{\sum\limits_{t=1}^{T}{\frac{{{{\left({{\bm{\omega}}_{k}^{t}}\right)}^{T}}{\mathbf{v}}_{k}^{t}}}{{{{\left({m_{k}^{t}}\right)}^{2}}}}}}\right|>{\mathbb{\epsilon}}-\sum\limits_{t=1}^{T}{\frac{{{{\left\|{{\mathbf{v}}_{k}^{t}}\right\|}^{2}}}}{{2{{\left({m_{k}^{t}}\right)}^{2}}}}}}\right)$
$\displaystyle=2\Pr\left({\sum\limits_{t=1}^{T}{\frac{{{{\left({{\bm{\omega}}_{k}^{t}}\right)}^{T}}{\mathbf{v}}_{k}^{t}}}{{{{\left({m_{k}^{t}}\right)}^{2}}}}}>{\mathbb{\epsilon}}-\sum\limits_{t=1}^{T}{\frac{{{{\left\|{{\mathbf{v}}_{k}^{t}}\right\|}^{2}}}}{{2{{\left({m_{k}^{t}}\right)}^{2}}}}}}\right)$
$\displaystyle\mathop{\leqslant}\limits^{\left({ii}\right)}2\frac{1}{{\sqrt{2\pi\Lambda}}}\int_{{\mathbb{\epsilon}}-\Lambda}^{\infty}{\frac{x}{{{\mathbb{\epsilon}}-\Lambda}}\exp}\left({-\frac{{{x^{2}}}}{{2\Lambda}}}\right)dx,$
(33)

where $(i)$ follows the inequality
$\Pr\left({\left|{X+a}\right|>{\mathbb{\epsilon}}}\right)\leqslant\Pr\left({\left|X\right|+a>{\mathbb{\epsilon}}}\right)$
for an arbitrary $a\geqslant 0$, and $(ii)$ is due to

$\displaystyle\Pr\left({\sum\limits_{t=1}^{T}{\frac{{{{\left({{\bm{\omega}}_{k}^{t}}\right)}^{T}}{\mathbf{v}}_{k}^{t}}}{{{{\left({m_{k}^{t}}\right)}^{2}}}}}>{\mathbb{\epsilon}}-{\Lambda}}\right)$
$\displaystyle=\frac{1}{{\sqrt{2\pi\Lambda}}}\int_{{\mathbb{\epsilon}}-\Lambda}^{\infty}{\exp}\left({-\frac{{{x^{2}}}}{{2\Lambda}}}\right)dx$
$\displaystyle\leqslant\frac{1}{{\sqrt{2\pi\Lambda}}}\int_{{\mathbb{\epsilon}}-\Lambda}^{\infty}{\frac{x}{{{\mathbb{\epsilon}}-\Lambda}}\exp}\left({-\frac{{{x^{2}}}}{{2\Lambda}}}\right)dx$
$\displaystyle=\frac{{\sqrt{\Lambda}}}{{\sqrt{2\pi}\left({{\mathbb{\epsilon}}-\Lambda}\right)}}\exp\left({-\frac{{{{\left({{\mathbb{\epsilon}}-\Lambda}\right)}^{2}}}}{{2\Lambda}}}\right),$
(34)

where
$\Lambda\triangleq\sum\limits_{t=1}^{T}{{{\left({\frac{{\Delta_{k}^{t}}}{{m_{k}^{t}}}}\right)}^{2}}}$.
Finally, the closed-from $({\mathbb{\epsilon}},\delta)$-DP condition is given
by

$\frac{{\sqrt{2\Lambda}}}{{\sqrt{\pi}\left({{\mathbb{\epsilon}}-\Lambda}\right)}}\exp\left({-\frac{{{{\left({{\mathbb{\epsilon}}-\Lambda}\right)}^{2}}}}{{2\Lambda}}}\right)<\delta.$
(35)

####  III-A2 Convergence Analysis

At the $t$-th iteration, the CPU estimates the scaled local gradient as
${\mathbf{r}}_{k}^{t}$, and then the global gradient is estimated as

$\displaystyle\widehat{\nabla
F}\left({{{\mathbf{w}}^{t-1}}}\right)=\frac{1}{{{B_{{\text{tot}}}}}}\sum\limits_{k=1}^{K}{{\mathbf{r}}_{k}^{t-1}}=\alpha\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)+\frac{\alpha}{{{B_{{\text{tot}}}}}}I^{t-1}+\frac{1}{{{B_{{\text{tot}}}}}}\sum\limits_{k=1}^{K}{{\bm{\omega}}_{k}^{t-1}},$
(36)

where
$I^{t-1}\triangleq\sum\limits_{k=1}^{K}{\sqrt{\frac{{p_{k}^{t-1}}}{{{{\left\|{{\mathbf{s}}_{k}^{t-1}}\right\|}^{2}}}}}{\mathbf{s}}_{k}^{t-1}}\sum\limits_{l=1}^{L}{\sum\limits_{i\neq
k}^{K}{{h_{kl}}h_{il}^{*}}}$. With the help of [18, Assumption 1], we have the
following equality

$\displaystyle F\left({{{\mathbf{w}}^{t}}}\right)$
$\displaystyle\leqslant\\!F\left({{{\mathbf{w}}^{t\\!-\\!1}}}\right)\\!\\!+\\!\\!{\left[{\nabla
F\left({{{\mathbf{w}}^{t\\!-\\!1}}}\right)}\right]^{T}}\left[{{{\mathbf{w}}^{t}}\\!\\!-\\!\\!{{\mathbf{w}}^{t\\!-\\!1}}}\right]\\!\\!+\\!\\!\frac{M}{2}{\left\|{{{\mathbf{w}}^{t}}\\!-\\!{{\mathbf{w}}^{t\\!-\\!1}}}\right\|^{2}}$
$\displaystyle=F\left({{{\mathbf{w}}^{t-1}}}\right)\\!-\\!{\left[{\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)}\right]^{T}}\left[{\alpha\eta\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)+\frac{{\alpha\eta}}{{{B_{{\text{tot}}}}}}{I^{t-1}}+\frac{\eta}{{{B_{{\text{tot}}}}}}\sum\limits_{k=1}^{K}{{\bm{\omega}}_{k}^{t-1}}}\right]$
$\displaystyle+\frac{M}{2}\left\|{\alpha\eta\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)+\frac{{\alpha\eta}}{{{B_{{\text{tot}}}}}}{I^{t-1}}}\right.{\left.{+\frac{\eta}{{{B_{{\text{tot}}}}}}\sum\limits_{k=1}^{K}{{\bm{\omega}}_{k}^{t-1}}}\right\|^{2}},$
(37)

where $M$ is a positive constant. Setting $\eta=\frac{1}{M}$ and taking
expectation over the randomness of additive noise, we have

$\displaystyle\mathbb{E}\left\\{{F\left({{{\mathbf{w}}^{t}}}\right)}\right\\}$
$\displaystyle\leqslant
F\left({{{\mathbf{w}}^{t-1}}}\right)-\frac{\alpha}{M}{\left\|{\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)}\right\|^{2}}\\!\\!-\\!\\!{\left[{\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)}\right]^{T}}{\frac{\alpha}{{M{B_{{\text{tot}}}}}}{I^{t-1}}}$
$\displaystyle+\frac{{{\alpha^{2}}}}{{2M}}{\left\|{\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)}\right\|^{2}}+\left({{{\left[{\nabla
F\left({{{\mathbf{w}}^{t-1}}}\right)}\right]}^{T}}\frac{{{\alpha^{2}}}}{{M{B_{{\text{tot}}}}}}{I^{t-1}}}\right)$
$\displaystyle+\frac{1}{{2M}}{\left\|{\frac{\alpha}{{{B_{{\text{tot}}}}}}{I^{t-1}}}\right\|^{2}}+\frac{1}{{2M}}\frac{d}{{B_{{\text{tot}}}^{2}}}\sum\limits_{k=1}^{K}{{{\left({m_{k}^{t}}\right)}^{2}}}.$
(38)

According to [18, Assumption 2], we obtain

$\displaystyle\mathbb{E}\left\\{{F\left({{{\mathbf{w}}^{t}}}\right)}\right\\}-{F^{*}}$
$\displaystyle\leqslant\frac{{M-\alpha\left({2-\alpha}\right)\mu}}{M}\left({\mathbb{E}\left\\{{F\left({{{\mathbf{w}}^{t-1}}}\right)}\right\\}-{F^{*}}}\right)$
$\displaystyle+\frac{{{\alpha^{2}}}}{{2MB_{{\text{tot}}}^{2}}}{\left\|{\frac{\alpha}{{{B_{{\text{tot}}}}}}{I^{t-1}}}\right\|^{2}}+\frac{1}{{2M}}\frac{d}{{B_{{\text{tot}}}^{2}}}\sum\limits_{k=1}^{K}{{{\left({m_{k}^{t-1}}\right)}^{2}}},$
(39)

where $\mu$ is a positive constant. Therefore, by applying the above
inequality repeatedly through $T$ iterations, the results follow immediately.
Finally, the average optimality gap after $T$ iteration is upper bounded by

$\displaystyle\mathbb{E}\left\\{{F\left({{{\mathbf{w}}^{t}}}\right)}\right\\}-{F^{*}}$
$\displaystyle\leqslant{\left({1-\frac{{\alpha\left({2-\alpha}\right)\mu}}{M}}\right)^{T}}\left({\mathbb{E}\left\\{{F\left({{{\mathbf{w}}^{1}}}\right)}\right\\}-{F^{*}}}\right)$
$\displaystyle+\frac{{{\alpha^{2}}}}{{2MB_{{\text{tot}}}^{2}}}\sum\limits_{t=1}^{T}{{{\left({1-\frac{{\alpha\left({2-\alpha}\right)\mu}}{M}}\right)}^{T-t}}}{\left\|{\frac{\alpha}{{{B_{{\text{tot}}}}}}{I^{t}}}\right\|^{2}}$
$\displaystyle+\frac{1}{{2M}}\frac{d}{{B_{{\text{tot}}}^{2}}}\sum\limits_{t=1}^{T}{{{\left({1-\frac{{\alpha\left({2-\alpha}\right)\mu}}{M}}\right)}^{T-t}}}\sum\limits_{k=1}^{K}{{{\left({m_{k}^{t}}\right)}^{2}}}.$
(40)

The first item in (III-A2) reflects the initial optimality gap
$\left({\mathbb{E}\left\\{{F\left({{{\mathbf{w}}^{1}}}\right)}\right\\}-{F^{*}}}\right)$
increases with the increase of $T$, and the third considers the effect of
effective additional noise power. Interestingly, the bound in (III-A2)
indicates that the quantization noise added in the initial iteration would
enlarge the final optimality gap less than the noise added in the later
iterations. This is because the contribution of the noise added in the
iteration $t$ is affected by a factor
${{{\left({1-\frac{{\alpha\left({2-\alpha}\right)\mu}}{M}}\right)}^{T-t}}}$.

###  III-B Low-resolution DACs at the UEs

In this case, the privacy preservation condition becomes

$\displaystyle{{\cal L}_{{\cal B},{\cal
B}^{\prime}}}\left({{\bf{y}}_{l}^{t}}\right)=\ln\left({\prod\limits_{t=1}^{T}{\frac{{P\left({\left.{{\bf{y}}_{l}^{t}}\right|{\bf{y}}_{l}^{t-1},\cdots{\bf{y}}_{l}^{1},{\cal
B}}\right)}}{{P\left({\left.{{\bf{y}}_{l}^{t}}\right|{\bf{y}}_{l}^{t-1},\cdots{\bf{y}}_{l}^{1},{\cal
B}^{\prime}}\right)}}}}\right)=\sum\limits_{t=1}^{T}{\ln}\left({\frac{{P\left({\left.{{\bf{y}}_{l}^{t}}\right|{\bf{y}}_{l}^{t-1},\cdots{\bf{y}}_{l}^{1},{\cal
B}}\right)}}{{P\left({\left.{{\bf{y}}_{l}^{t}}\right|{\bf{y}}_{l}^{t-1},\cdots{\bf{y}}_{l}^{1},{\cal
B}^{\prime}}\right)}}}\right),$ (41)

where

$\displaystyle
P\left({\left.{{\bf{y}}_{l}^{t}}\right|{\bf{y}}_{l}^{t-1},\cdots{\bf{y}}_{l}^{1},{\cal
B}}\right)=\frac{1}{{\sigma_{{\rm{eff}}}^{2}\sqrt{2\pi}}}{\rm{exp}}\left({-\frac{{{{\left\|{{\bf{y}}_{l}^{t}-\sum\limits_{i=1}^{K}{{\alpha_{i}}d_{il}^{t}h_{il}^{t}{\bf{x}}_{i}^{t}\left({{{\cal
B}_{i}}}\right)}}\right\|}^{2}}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}\right),$
(42)

Then, (41) can be rewritten as

$\displaystyle{{\cal L}_{{\cal B},{\cal
B}^{\prime}}}\left({{\bf{y}}_{l}^{t}}\right)=\sum\limits_{t=1}^{T}{\ln}\left({{{{\rm{exp}}\left({-\frac{{{{\left\|{{\bf{y}}_{l}^{t}-\sum\limits_{i=1}^{K}{{\alpha^{t}}h_{il}^{t}{\bf{s}}_{i}^{t}\left({{{\cal
B}_{i}}}\right)}}\right\|}^{2}}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}\right)}}/{{{\rm{exp}}\left({-\frac{{{{\left\|{{\bf{y}}_{l}^{t}-\sum\limits_{i=1}^{K}{{\alpha^{t}}h_{il}^{t}{\bf{s}}_{i}^{t}\left({{{{\cal
B}^{\prime}}_{i}}}\right)}}\right\|}^{2}}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}\right)}}}\right)$
$\displaystyle=\sum\limits_{t=1}^{T}{\ln}\left({{{{\rm{exp}}\left({-\frac{{{{\left\|{{\bm{\omega}}_{l}^{t}}\right\|}^{2}}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}\right)}}/{{{\rm{exp}}\left({-\frac{{{{\left\|{{\bm{\omega}}_{l}^{t}+{\bf{v}}_{l}^{t}}\right\|}^{2}}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}\right)}}}\right)=\sum\limits_{t=1}^{T}{\frac{{{{\left\|{{\bf{v}}_{l}^{t}}\right\|}^{2}}+2{{\left({{\bm{\omega}}_{l}^{t}}\right)}^{T}}{\bf{v}}_{l}^{t}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}=\Gamma_{l}^{t}.$
(43)

Since the sensitivity $\Delta_{l}^{t}$ is given as

$\displaystyle{\bf{v}}_{l}^{t}=\sum\limits_{i=1}^{K}{{\alpha_{i}}\left|{h_{il}^{t}}\right|}\left({{\bf{x}}_{i}^{t}\left({{{{\cal
B}^{\prime}}_{i}}}\right)-{\bf{x}}_{i}^{t}\left({{{\cal
B}_{i}}}\right)}\right),\left\|{{\bf{v}}_{l}^{t}}\right\|\leq
2\mathop{\max}\limits_{i}\sqrt{p_{i}^{t}}\left|{h_{il}^{t}}\right|=\Delta_{l}^{t},$
(44)

the upper-bound on the privacy preservation condition can be derived as

$\displaystyle\Pr\left({\left|{\Gamma_{l}^{t}}\right|>{\mathbb{\epsilon}}}\right)$
$\displaystyle\leq\Pr\left({\left|{\sum\limits_{t=1}^{T}{\frac{{{{\left({{\bm{\omega}}_{l}^{t}}\right)}^{T}}{\bf{v}}_{l}^{t}}}{{{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}}\right|>{\mathbb{\epsilon}}-\sum\limits_{t=1}^{T}{\frac{{{{\left\|{{\bf{v}}_{l}^{t}}\right\|}^{2}}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}}\right)$
$\displaystyle=2\Pr\left({\sum\limits_{t=1}^{T}{\frac{{{{\left({{\bm{\omega}}_{l}^{t}}\right)}^{T}}{\bf{v}}_{l}^{t}}}{{{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}}\right)>{\mathbb{\epsilon}}-\sum\limits_{t=1}^{T}{\frac{{{{\left\|{{\bf{v}}_{l}^{t}}\right\|}^{2}}}}{{2{{\left({\sigma_{{\rm{eff}}}^{2}}\right)}^{2}}}}}$
$\displaystyle\leq
2\Pr\left({\Lambda>{\mathbb{\epsilon}}-\sum\limits_{t=1}^{T}{\frac{1}{2}{{\left({\frac{{\Delta_{l}^{t}}}{{\sigma_{{\rm{eff}}}^{2}}}}\right)}^{2}}}}\right)$
$\displaystyle=\frac{2}{{\sqrt{2\pi\sum\limits_{t=1}^{T}{{{\left({\frac{{\Delta_{l}^{t}}}{{\sigma_{{\rm{eff}}}^{2}}}}\right)}^{2}}}}}}\int_{b}^{\infty}{\exp\left({-\frac{{{x^{2}}}}{{2\sum\limits_{t=1}^{T}{{{\left({\frac{{\Delta_{l}^{t}}}{{\sigma_{{\rm{eff}}}^{2}}}}\right)}^{2}}}}}}\right)}dx,$
(45)

where

$\displaystyle\begin{array}[]{l}b=-\sum\limits_{t=1}^{T}{{{\left({\frac{{\Delta_{l}^{t}}}{{{\sigma_{{\rm{eff}}}}}}}\right)}^{2}}}>0,\;\;\;\frac{2}{{\sqrt{2\pi\nu}}}\int_{-\frac{\nu}{2}}^{\infty}{\exp\left({-\frac{{{x^{2}}}}{{2\nu}}}\right)}dx<\delta,\;\;\;\nu=\sum\limits_{t=1}^{T}{{{\left({\frac{{\Delta_{l}^{t}}}{{{\sigma_{{\rm{eff}}}}}}}\right)}^{2}}}.\end{array}$
(47)

Besides, the convergence can be proved following similar steps as in Section
III-A-(2).

##  IV Power Control for Training Time Minimization and An asynchronous FL
protocol

In this section, an optimization problem is formulated to jointly optimize the
transmit power and data rate under the practical constraints on UEs¡¯ energy
consumption with different quantization accuracies of the ADCs/DACs. Note that
the quantization noise is used for privacy protection, therefore, different
quantization accuracies realizes different DP protection levels. By applying
the successive convex approximation approach, we design a computationally-
efficient algorithm to obtain a suboptimal solution of the power allocation.
Besides, we propose an asynchronous FL protocol to alleviate the staleness,
efficiency and better utilize the progress made by stragglers.

###  IV-A Low-resolution ADCs Equipped at the APs

According to (12) and (17), the problem of FL training time minimization with
low-resolution ADCs in the CF mMIMO system can be formulated as

$\displaystyle\mathop{\text{minimize}}\limits_{{\mathbf{p}},{\mathbf{R}}}\;\;\;T_{\text{time}}$
(47a)
$\displaystyle\;\;{\text{s.}}{\text{t.}}\;\;\;0\leqslant{p_{k}}\leqslant{p_{\max}},$
(47b) $\displaystyle\;\;\;\;\;\;\;\;{R_{k}}\leqslant{r_{k}},$ (47c)

where
${\mathbf{p}}={\left[{{p_{1}},\cdots,{p_{K}}}\right]^{T}},{\mathbf{R}}={\left[{{R_{1}},\cdots,{R_{K}}}\right]^{T}}$.
By introducing slack variables ${u_{k}}$, $x$, $x_{1}$, and $x_{2}$, we
reformulate (47a) as follows:

$\displaystyle\mathop{\text{minimize}}\limits_{{\mathbf{u}},{\mathbf{R}},x}\;\;\;x$
(48a) $\displaystyle\;{\text{s.}}{\text{t.}}\;\;\;x\geqslant{x_{1}}+{x_{2}},$
(48b)
$\displaystyle\;\;\;\;\;\;\;\;{x_{1}}\geqslant\frac{ST}{{{R_{k}}}},\;\;\;\;\;\;\;\;{x_{2}}\geqslant\frac{{KST}}{{\sum\limits_{k=1}^{K}{{R_{k}}}}},$
(48c)
$\displaystyle\;\;\;\;\;\;\;\;{R_{k}}\leqslant\left({1-\frac{{{\tau_{p}}}}{{{\tau_{c}}}}}\right)B\times{\log_{2}}\left({1+\frac{{u_{k}^{2}{A_{k}}}}{{\sum\limits_{i\neq
k}^{L}{u_{i}^{2}}{B_{ki}}+{C_{k}}+{D_{k}}+\sum\limits_{i=1}^{K}{u_{i}^{2}}{E_{ki}}}}}\right),$
(48d)
$\displaystyle\;\;\;\;\;\;\;\;u_{k}^{2}\leqslant{p_{\max}},\;\;\;\;\;\;\;\;{u_{k}}\geqslant
0.$ (48e)

Note that different quantization accuracies realize different DP protection
levels. Specifically, in order to satisfy the
$({\mathbb{\epsilon}},\delta)$-differentially private, according to (35), the
appropriate value of the linear gain $\alpha$ which depends on the number of
quantization bits needs to be selected. At the same time, according to (14),
the rate $R_{k}$ is affected by $\alpha$. However, due to the nonconvex
constraint (48d), (55) is still challenging. To address the problem at hand,
we exploit the fact that a function
$f\left({x,y}\right)=\log_{2}\left({1+\frac{{{{\left|x\right|}^{2}}}}{y}}\right)$
has the following lower bound [25]:

$\displaystyle f\left({x,y}\right)$
$\displaystyle\geqslant\log_{2}\left({1+\frac{{{{\left|{{x^{\left(n\right)}}}\right|}^{2}}}}{{{y^{\left(n\right)}}}}}\right)-\frac{{{{\left|{{x^{\left(n\right)}}}\right|}^{2}}}}{{{y^{\left(n\right)}}}}+2\frac{{{x^{\left(n\right)}}x}}{{{y^{\left(n\right)}}}}-\frac{{{{\left|{{x^{\left(n\right)}}}\right|}^{2}}\left({{{\left|x\right|}^{2}}+y}\right)}}{{{y^{\left(n\right)}}\left({{{\left|{{x^{\left(n\right)}}}\right|}^{2}}+{y^{\left(n\right)}}}\right)}},$
(49)

where $n$ denotes the $n$-th iteration of SCA, $x\in\mathbb{R},y>0$, and
${y^{\left(n\right)}}>0$. Therefore, the concave lower bound in (48d) is

$\displaystyle{R_{k}}\leqslant\left({1-\frac{{{\tau_{p}}}}{{{\tau_{c}}}}}\right)B\left({{{\log}_{2}}\left({1+\frac{{{{\left|{\Upsilon_{k}^{\left(n\right)}}\right|}^{2}}}}{{\Pi_{k}^{\left(n\right)}}}}\right)-\frac{{{{\left|{\Upsilon_{k}^{\left(n\right)}}\right|}^{2}}}}{{\Pi_{k}^{\left(n\right)}}}+2\frac{{\Upsilon_{k}^{\left(n\right)}{\Upsilon_{k}}}}{{\Pi_{k}^{\left(n\right)}}}-\frac{{{{\left|{\Upsilon_{k}^{\left(n\right)}}\right|}^{2}}\left({\Upsilon_{k}^{2}+{\Pi_{k}}}\right)}}{{\Pi_{k}^{\left(n\right)}\left({{{\left|{\Upsilon_{k}^{\left(n\right)}}\right|}^{2}}+\Pi_{k}^{\left(n\right)}}\right)}}}\right),$
(50)

where ${\Upsilon_{k}}={u_{k}}\sqrt{{A_{k}}}$, and
${\Pi_{k}}=\sum\limits_{i\neq
k}^{L}{u_{i}^{2}}{B_{ki}}+{C_{k}}+{D_{k}}+\sum\limits_{i=1}^{K}{{p_{i}}}{E_{ki}}$.

Finally, at iteration $t$, problem (48) can be approximated by the following
convex problem for given point $u_{k}^{\left(n\right)}$:

$\mathop{\min}\limits_{\left\\{{{\mathbf{u}},{\mathbf{R}}}\right\\}\in\mathcal{F}}x,$
(51)

where $\mathcal{F}\triangleq\left\\{{\text{(48b), (48c), (48e),
(50)}}\right\\}$ is a convex feasible set. As a result, it can be solved by
convex optimization. Besides, we can further tighten the bounds in (48)
iteratively, which is suboptimal in Algorithm 1. Note that the proposed
algorithm can achieve a suboptimal solution of (48) and the corresponding
convergence can be proved via a similar approach in [25]. It is noticed that
problem (51) is solving simple convex programs. Therefore, the complexity of
Algorithm 1 to solve the problem (51) in the suboptimal method is in
polynomial time.

Algorithm 1 A Suboptimal Algorithm for (48)

1:Set the maximum transmit power for each UE as ${p_{\max}}$; Large-scale
fading coefficients ${\beta_{lk}},\forall l,k$. Initial values for
$u_{k}^{\left(0\right)},\forall k$, and the tolerance $\varepsilon\geq 0$. Set
up $n=1$.

2:The optimal solutions
$u_{k}^{{\rm{opt}}}=u_{k}^{\left(n\right)},R_{k}^{{\rm{opt}}}=R_{k}^{\left(n\right)},\;\forall
k$.

3:Iteration $n$:

  * •

Solve (51) to get its optimal solution
$\left\\{{{{\mathbf{u}}^{*}},{{\mathbf{R}}^{*}}}\right\\}$

  * •

Update
$\left\\{{{{\mathbf{u}}^{\left(n\right)}},{{\mathbf{R}}^{\left(n\right)}}}\right\\}=\left\\{{{{\mathbf{u}}^{*}},{{\mathbf{R}}^{*}}}\right\\}$.

4:Stop if $\left|{x-{x^{\left(n\right)}}}\right|\leqslant\varepsilon$.
Otherwise, go to Step 5.

5:Set $n=n+1$, then go to Step 3.

###  IV-B Low-resolution DACs at the UEs

The problem of FL training time minimization with low-resolution DACs in the
CF mMIMO system can be formulated as

$\displaystyle\mathop{\min}\limits_{{\bf{R}},{\bf{u}}}\;\;\;w$ (52a)
$\displaystyle\;{\rm{s.}}{\rm{t.}}\;\;\;\;w\geq{\rm{}}{t_{1}}+{t_{2}},$ (52b)
$\displaystyle\;\;\;\;\;\;\;\;\;\;{t_{1}}\geq\sum\limits_{t=1}^{T}{\frac{{{S_{u,k}}}}{{R_{k}^{t}}}},\;\;\;{t_{2}}\geq\sum\limits_{t=1}^{T}{\frac{{\left|{{{\cal
K}^{t}}}\right|{S_{u}}}}{{\sum\limits_{k\in{\cal K}}{R_{k}^{t}}}}},$ (52c)
$\displaystyle\;\;\;\;\;\;\;\;\;\;R_{k}^{t}\leq\left({1-\frac{{{\tau_{p}}}}{{{\tau_{c}}}}}\right)B{\log_{2}}\left({1+{\rm{SINR}}_{k}^{t}}\right),$
(52d)
$\displaystyle\;\;\;\;\;\;\;\;\;{\left({u_{k}^{t}}\right)^{2}}\leq{p_{\max}},$
(52e)

where

${\rm{SINR}}_{k}^{t}=\frac{{p_{k}^{t}A_{k}^{t}}}{{\sum\limits_{i=1}^{K}{p_{i}^{t}B_{ki}^{t}}+\sum\limits_{i=1}^{K}{p_{i}^{t}C_{ki}^{t}}+E_{k}^{t}}},$
(53)

and $A_{k}^{t}$, $B_{ki}^{t}$, $C_{ki}^{t}$, and $E_{k}^{t}$ are defined in
(II-B2). Note that problem (52) can be solved follows the similar steps to
problem (48).

###  IV-C An Asynchronous FL Protocol

Synchronous FL considers synchronous communication during training round
between the CPU and UEs. For the former, the CPU should wait until getting
response from sufficient UEs. Unfortunately, some UEs may be unresponsive in
the training process due to vulnerable wireless network. Then, the CPU drops
such UEs and proceeds on to the next training process. On the contrary,
asynchronous FL enables all UEs to directly send update information to the CPU
after every local round that is dropped in synchronous FL optimization. In
general, asynchronous FL can achieve faster convergence when wireless links
are vulnerable and heterogeneous across the UEs. Therefore, asynchronous FL
has drawn more interests in recent papers.

It can be observed from problem (52) that $d_{kl}$ would determine the
required training time. Therefore, we propose a simple asynchronous FL
protocol based on the large-scale fading coefficient. Specifically, a lag-
tolerant algorithm which allows some UEs to stay asynchronous with the CPU is
provided. Note that straggler UEs refer to UEs that are slower and are still
training locally based on outdated models. Generally, the UE should start
training based on the latest global model received from the CPU.

At the communication iteration $t$, each AP classifies all UEs into three
categories: synchronous, asynchronous and need-to-be-synchronized,
respectively. First, the synchronous UEs refers to the UEs that are served by
at least one AP and complete its local model update based on the latest global
model transmitted from the CPU. Then, since the channel gains of straggler UEs
are not strong enough, they may dramatically slow down the whole FL process.
In order to mitigate the straggler effect, the local training of straggler UEs
are still based on the last version of the global model. Therefore, the
straggler UEs are also called asynchronous UEs. Besides, we assume that all
the UEs need to update their local training model at least are forced to
synchronize after at most $T_{\text{tol}}$ blocks so that the global model
will not be poisoned by the seriously outdated local models, where
$T_{\text{tol}}$ is called the lag tolerance.

In this model, at iteration $t$, the $l$th AP only serves $K_{0,l}\leq K$ APs
corresponding to the $K_{0,l}$ largest large-scale fading coefficients. The
main question arising immediately is how to choose $K_{0,l}$. Naturally, we
can choose $K_{0,l}$ UEs satisfying

$\sum\limits_{i=1}^{{K_{0,l}}}{\frac{{{{\bar{\beta}}_{il}}}}{{\sum\limits_{i^{\prime}=1}^{K}{{\beta_{i^{\prime}l}}}}}}\geq\nu\%,$
(54)

where $\left\\{{{{\bar{\beta}}_{1l}},\cdots,{{\bar{\beta}}_{Kl}}}\right\\}$ is
the sorted (in descending order) set of the set
$\left\\{{{\beta_{1l}},\cdots,{\beta_{Kl}}}\right\\}$, and $\nu$ is the lag
percent. After choosing ${\cal D}_{1}^{t},\cdots,{\cal D}_{L}^{t}$, where
${\mathcal{D}_{l}^{t}}=\left\\{{i:{d_{il}^{t}}=1,i\in\left\\{{1,\cdots
K}\right\\}}\right\\}$, we can follow the same method as in Section IV-A to an
optimized power control.

##  V Numerical Results

In this section, we first evaluate the performance of the proposed power
control algorithm under different DP protection levels in the CF mMIMO
supported FL network. Note that the DP mechanism is realized by the
quantization noise. Therefore, we adopt ADCs and DACs with different
quantization bits to achieve different DP protection levels. More
specifically, we adopt similar parameters setting as in [25]. Note that we do
not propose a new FL framework but rather apply a existed FL framework in CF
mMIMO systems. We focus on how to reduce the training time using different
quantization bits or under differential privacy levels. Therefore, the
simulation on real datasets to see the effectiveness of the considered FL
framework has already performed in [18] and hence, they are not considered in
this paper. Note that the unit of time is seconds in the following figures.

Figure 2: Uplink training time against the number of quantization bits with
$L=10$, $K=3$, $D=1$ km, and $p_{\max}=200$ mW.

Fig. 2 evaluates the uplink training time as a function of the number of
quantization bits in the synchronized model. We also compare the performance
of applying the proposed power control method and the full power transmission
with the synchronized model. It can be seen from Fig. 2 that the uplink
training time decreases as the number of quantization bits increase since the
quantization error reduces. Specifically, the performance of $b=3$ is close to
that of $b=10$, which we refer to as the perfect ADCs case. Therefore,
reducing the quantization bits from 10 to 3 only has a slight impact on the
performance of uplink training time. Also, the fronthaul load can be relaxed
significantly since the data size is reduced. Furthermore, applying the
proposed power control method leads to a huge reduction in terms of uplink
training time. In particular, the uplink training time reduces 35% and 21%
when $b=1$ and $b=10$, respectively.

Figure 3: Uplink training time against the number of APs with $K=3$, $D=1$ km,
and $p_{\max}=200$ mW.

Fig. 3 shows the uplink training time against the number of APs in the
synchronized model. As expected, the uplink training time decreases along with
the increase of the number of APs, which is resulted from from the higher
macro-diversity gain for enhancing the detection performance. Besides, when
$b=1$, applying the proposed power control method leads to a 42% reduction in
terms of uplink training time compared with the one with simple full power
transmission. Although the performance gap between the proposed power control
and the full power transmission reduces when the number of APs increases, the
advantage of our power control method is still obvious for a reasonable number
of APs. Furthermore, the performance with $b=3$ approaches that of perfect
ADCs. Note that the latter means the quantization error is not dominated and
hence, the DP also no longer exists. This indicates that one can use low-
resolution ADCs to realize privacy-preserving with reduced hardware cost and
fronthaul load without increasing the uplink training time noticeably.

Figure 4: Uplink training time against the number of UEs with $L=30$, $D=1$
km, and $p_{\max}=200$ mW.

Fig. 4 compares the uplink training times as a function of the number of UEs
in the synchronized model. As can be seen, the performance of uplink training
time becomes longer when the system has more UEs. This is because the mutual
interference becomes stronger for a larger number of UEs. However, our
proposed power control method can effectively mitigate the impact of mutual
interference. In particular, the performance of uplink training time is
effectively reduced by 48%. Moreover, the performance gap between $b=3$ and
$b=10$ is also negligible, although the gaps are enlarged with the increases
of the number of UEs. For example, when $K=10$, taking $b=3$ leads to 5%
performance loss.

Figure 5: Uplink training time against the number of quantization bits with
$L=10$, $K=4$, $D=1$ km, and $p_{\max}=200$ mW.

Fig. 5 presents the performance of synchronous mode and synchronous mode as a
function of the number of quantization bits when low-resolution DACs are
equipped at the UEs. It can be observed that utilizing the asynchronous mode
can effectively reduce the training time. By using the asynchronous mode, we
can decrease the number of UEs that each AP needs to be served. Consequently,
the uplink training time reduces sharply. In particular, when $b=1$, the
uplink training time with the asynchronous mode and power control is 2.5 times
shorter than that of the synchronous mode.

Figure 6: Uplink training time against the lag tolerance with $L=10$, $K=4$,
$D=1$ km, and $p_{\max}=200$ mW.

Fig. 6 shows the uplink training time against the lag tolerance with lag
percent $\nu=85$. As expected, the uplink training time first decreases along
with the increase of the number of blocks, which is resulted from that data
exchange is less often. Then, the uplink training time increases as the SE of
UEs are declined. Besides, the efficiency of our power control method is
obvious. When the number of blocks is 8, applying the proposed power control
method leads to a 42% reduction in terms of uplink training time.

Figure 7: Uplink training time against $v$% with $L=10$, $K=4$, $D=1$ km, and
$p_{\max}=200$ mW.

Fig. 7 shows the uplink training time against the lag percent with a lag
tolerant is equal to 4, which reflects the number of UEs served by the APs in
each round of communication. It can be observed from Fig. 7 that when
$\nu<80\%$, the uplink training time decreases as $\nu$ increases. The reason
is that the straggler UEs who significantly slow down the whole FL process is
not served by the APs in every communication iteration. However, when $\nu$
continues to reduce, the uplink training time begins to increase, since the
number of users served in each round of communication decreases, which leads
to the decreases of SE that in turn increases the training time. Therefore,
Fig. 7 shows that choosing an appropriate value of $\nu$ can effectively
reduce the training time.

##  VI Conclusions

In this paper, we studied the DP in wireless FL enabled by CF mMIMO systems
with low-resolution ADCs and DACs. By introducing the quantization noise as
the DP mechanism, we derived an expression of the privacy preservation
condition and provided convergence analysis for the proposed model. Targeting
at the uplink training time minimization, we jointly optimized the transmit
power and data rate under different DP protection levels. The simulation
results showed that our proposed power control method can effectively reduce
the uplink training time in all considered cases. In order to mitigate the
effect of straggler, we proposed an asynchronous FL protocol which
incorporates a UE selection algorithm based on the large-scale fading
coefficient decoupling the CPU and the selected UEs for a reduction of uplink
training time and for tackling the tradeoff between faster convergence and
lower communication overhead. To further improve the system performance, some
future extensions can be considered, e.g., channel allocation.

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