# Optimal Resource Management for Hierarchical Federated Learning over HetNets
with Wireless Energy Transfer

Rami Hamdi, Ahmed Ben Said, Emna Baccour, Aiman Erbad, Amr Mohamed, Mounir
Hamdi, and Mohsen Guizani A preliminary version of this work [1] has appeared
in the proceedings of IEEE Global Communications Conference in 2021.R. Hamdi
is with the AFG College with the University of Aberdeen, Doha, Qatar (email:
rami.hamdi@afg-aberdeen.edu.qa).A. B. Said and A. Mohamed are with the College
of Engineering, Qatar University, Doha, Qatar (email<EMAIL_ADDRESS>amrm@qu.edu.qa).E. Baccour, A. Erbad, and M. Hamdi are with the Division of
Information and Computing Technology, College of Science and Engineering,
Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>mhamdi@hbku.edu.qa).M. Guizani
is with the Machine Learning Department, Mohamed Bin Zayed University of
Artificial Intelligence (MBZUAI), Abu Dhabi, UAE (email: mguizani@ieee.org).

###### Abstract

Remote monitoring systems analyze the environment dynamics in different smart
industrial applications, such as occupational health and safety, and
environmental monitoring. Specifically, in industrial Internet of Things (IoT)
systems, the huge number of devices and the expected performance put pressure
on resources, such as computational, network, and device energy. Distributed
training of Machine and Deep Learning (ML/DL) models for intelligent
industrial IoT applications is very challenging for resource limited devices
over heterogeneous wireless networks (HetNets). Hierarchical Federated
Learning (HFL) performs training at multiple layers offloading the tasks to
nearby Multi-Access Edge Computing (MEC) units. In this paper, we propose a
novel energy-efficient HFL framework enabled by Wireless Energy Transfer (WET)
and designed for heterogeneous networks with massive Multiple-Input Multiple-
Output (MIMO) wireless backhaul. Our energy-efficiency approach is formulated
as a Mixed-Integer Non-Linear Programming (MINLP) problem, where we optimize
the HFL device association and manage the wireless transmitted energy. However
due to its high complexity, we design a Heuristic Resource Management
Algorithm, namely H2RMA, that respects energy, channel quality, and accuracy
constraints, while presenting a low computational complexity. We also improve
the energy consumption of the network using an efficient device scheduling
scheme. Finally, we investigate device mobility and its impact on the HFL
performance. Our extensive experiments confirm the high performance of the
proposed resource management approach in HFL over HetNets, in terms of
training loss and grid energy costs.

###### Index Terms:

HetNets, Hierarchical federated learning, device association, wireless energy
transfer, energy efficiency.

##  I Introduction

Occupational health and safety (OHS) has recently attracted the attention of
governmental agencies in order to provide workers’ protection and minimize
health risks [2, 3, 4]. More generally, remote monitoring systems that capture
the environment dynamics and monitor individuals are used in different
applications to help senior citizens, patients, and factory workers.
Distributed Artificial Intelligence (AI) [5] and communication technologies
allow continuous monitoring for Key Health Indicators (KHIs) of a large number
of workers using relevant industrial IoT devices (smart phones, mobile
devices, wearables, smart cameras, and smart watches). The collected KHIs are
then transmitted to a central or a nearby edge Central Unit (CU) to be
processed by machine and deep learning algorithms and help in the decision
making tasks.

With an increasing number of connected devices, centralized Machine and Deep
Learning (ML/DL) algorithms require more and more memory resources and huge
processing (i.e. CPU) [6]. In a wireless network setting with real-time and
low-latency requirements, a centralized learning scheme becomes inefficient as
it induces delay and may compromise the privacy of exchanged data between
devices. Hence, decentralized learning approaches have emerged, particularly
Federated Learning (FL) [6, 7]. FL enables to train a shared model among
multiple devices, where each device uses only its private local data [7].
Indeed, the devices are training and updating the weights of the global model
they received using their local data samples. Then, they send the new model
weights to a CU for aggregation into a shared learning model. Finally, the CU
broadcasts the aggregated model weights to the FL participants in order to
update their models. This routine is iterated until reaching the desired model
performance. FL addresses the challenge of data privacy, while enhancing the
bandwidth usage as the amount of transferred data to the CU decreases.
However, with the nexus of thousands of devices, several challenges need to be
addressed including communication bottlenecks, increase in energy consumption,
congestion, and and data skewness i.e. non-i.i.d data [10]. To cope with these
issues, Hierarchical Federated Learning (HFL) has been presented [8, 9]. HFL
enables offloading the computation task to the Multi-Access Edge Computing
(MEC) in device vicinity, hence increasing the energy efficiency and enhancing
the response time of the IoT nodes. In this new configuration, the devices
send their local updates to their corresponding MEC for learning and model
aggregation. Then, the MECs transfer their aggregated models to the CU in
order to compute the global aggregation. The most distinctive feature of HFL
is the ability to address and manage diverse data distribution, e.g.,
non-i.i.d property of the data. This is achieved by carefully assigning
devices to MECs, hence speeding up convergence and attaining higher
performance levels [10]. We highlight that if the data is i.i.d, the HFL may
not have a big contribution to the system performance.

In a wireless communication environment with constant inter-device
interference, latency constraint, increase in energy consumption and device
heterogeneity, the implementation of HFL becomes challenging. Therefore, many
research efforts have recently focused on making HFL practical for wireless
communication networks. More precisely, recent works have tackled resources
allocation problems, including wireless resource allocation, edge-device
assignment, device selection and energy management with the objective of
optimizing the performance of such system fully exploiting its potential. On
the other hand, Wireless Energy Transfer (WET) represents a promising
technology whose objective is to address the problem of energy bottlenecks in
wireless networks [11]. The system is based on using electromagnetic radiation
waves transmitted from a power source to deliver energy to a wireless device.
Delivering power using wireless technology satisfies the frequent charging for
mobile devices. Hence, it represents a key technology for the design of
energy-efficient IoT wireless networks [12].

One of the challenges facing the ML/DL distribution is the high-energy
consumption. Thus, to enhance the energy efficiency of HFL processed over
HetNets with wirless backhaul, we propose to incorporate energy harvesting
[13], where each MEC transmits the harvested energy to its related devices via
WET. Compared to the existing schemes in [8, 9, 10, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26], our work is distinctive as it focuses on
designing an energy-efficient HFL architecture assisted by WET. As per our
knowledge, we are the first to consider the implementation of FL over HetNets
with massive MIMO wireless backhaul enabled by WET. Furthermore, a practical
mobility scenario that represents the workers in a construction site is
considered by developing an efficient dynamic resource allocation scheme. The
contributions of our work include the following:

  * •

We propose a novel energy-efficient HFL framework, designed for HetNets with
massive MIMO wireless backhaul and enabled by WET.

  * •

We formulate our energy-efficiency approach as a Mixed-Integer Non-Linear
Programming, where we optimize the HFL device association and manage the
wireless transmitted energy.

  * •

We derive the optimal device association analytically and we use Brute-Force
search to obtain the energy management decisions.

  * •

Due to the high complexity of the optimal solution, we propose a device
association and energy management algorithm that aims to enhance the energy-
efficiency of the proposed HFL framework.

  * •

Some devices with high energy requirements may be deactivated for some frames.
Hence, we propose a device scheduling strategy to decrease the energy
consumption without deteriorating the learning accuracy.

  * •

We develop an efficient dynamic device association scheme, where we consider a
mobility scenario that represents the workers in a construction site.

  * •

Finally, we conduct an extensive evaluation of the proposed system to prove
the energy efficiency and high accuracy performance of our resource
management, device association, and device scheduling schemes under different
network configurations.

We organize the remainder of this article as follows: In section II, we
explore the related works that study one-layer and hierarchical FL. In section
III, we present the system model. The optimization problem is formulated in
Section IV. We investigate the resource management in Section V. The impact of
mobility in FL performance is investigated in Section VI. Evaluation results
are provided in Section VII. Lastly, conclusions are drawn in Section VIII.

TABLE I: List of key Notations. Symbol |  Description |  Symbol |  Description
---|---|---|---
$N_{\text{cu}}$ |  Number of antennas at the CU |  $\mathbf{H}_{m}$ |  Channel matrix between the CU and MEC $m$
$N_{\text{mec}}$ |  Number of antennas at each MEC |  $\mathbf{g}_{m,k}$ |  Channel vector between device $k$ and MEC $m$
$K$ |  Number of devices |  $\beta_{m,k}$ |  Pathloss
$L$ |  Number of frames |  $\sigma^{2}$ |  Noise variance
$M$ |  Number of MECs |  $\mathbf{W}_{m}$ |  Detection matrix at MEC $m$
$T$ |  Duration of each frame |  $E^{\text{dev}}_{m,k}(l)$ |  Energy required by device $k$ assigned to MEC $m$ at frame $l$
$\Lambda_{m}$ |  Set of devices associated with MEC $m$ |  $E^{\text{cir}}$ |  Fixed circuit energy consumption
$K_{m}$ |  Cardinal of $\Lambda_{m}$ |  $E^{\text{ac}}_{m,k}(l)$ |  Required transmit energy of device $k$ at frame $l$
$C$ |  Number of classes |  $Q$ |  Size of the local FL model
$\mathcal{D}_{k}$ |  Dataset of device $k$ |  $B_{max}$ |  Maximal battery capacity
$\bm{x}_{k}$ |  Input of device $k$ |  $A_{m,k}(l)$ |  Amount of harvested energy by device $k$ assigned to MEC $m$ at frame $l$
$\bm{y}_{k}$ |  Output of device $k$ |  $B_{k}(l)$ |  Battery level of device $k$ at frame $l$
$\bm{S}$ |  Data matrix of the devices |  $E^{\text{wet}}_{m}(l)$ |  Transferred energy from MEC $m$ at frame $l$
$S_{c,k}$ |  Number of training data samples of the device $k$ for the class $c$ |  $\alpha_{m}(l)$ |  Grid’s weights
$\bm{w}_{k}(l)$ |  Weight vector of the local ML model of device $k$ at frame $l$ |  $\Delta$ |  Total grid energy consumption cost
$\cup_{j\in\Lambda_{m}}\mathcal{D}_{j}$ |  Virtual dataset available at MEC $m$ |  $\chi_{m,k}$ |  Device association index
$p^{(m)}(i)$ |  Probability of the class $i$ per MEC $m$’s distribution |  $p(\cdot)$ |  Global classes distribution

##  II Related Work

###  II-A Federated Learning

A number of studies have explored resource allocation for FL [27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37] over wireless networks. In [27], an energy
harvesting FL framework was developed by proposing efficient user association
and scheduling schemes. Minimizing the energy costs of FL wireless systems was
investigated, while taking into consideration a limited energy budget by
developing an efficient resource management scheme. In [29], a data
augmentation and adaptive user scheduling scheme was developed to overcome the
challenges related to the imbalanced distribution of data in FL. The
convergence time was theoretically investigated and optimized in [30] by
designing an efficient control scheme. The FL performance was improved in [31]
by proposing an efficient user scheduling policy. Moreover, the authors of
[32] designed an optimal user selection algorithm that minimizes the FL
convergence rate. In [33], a user selection scheme was proposed to decrease
the effect of unreliable users and enhance the accuracy of federated learning
training. To address the issue of battery limitation for FL users, initial
works [34, 35, 36, 37] incorporated WET systems when performing FL tasks. In
[34], the authors assumed that the unmanned aerial vehicles (UAVs) have WET
capability and proposed a suitable FL solution in terms of energy efficiency.
In [35], a heterogeneous mobile architecture is implemented and an optimal
resource allocation solution is proposed that minimizes the energy consumption
of FL devices. In [36], a low-complexity beamforming strategy was proposed to
minimize the mean-square error in FL based systems enabled by WET. The authors
of [37] developed an analytic FL framework that establishes a trade-off
between the model convergence and the power consumption which represents a
guideline to ensure efficient learning tasks.

###  II-B Hierarchical Federated Learning

Others works have tackled the optimization of resource allocation for HFL [9,
10, 19, 14, 17, 8, 15, 21, 18, 22, 23, 24, 25, 26, 16, 20]. More specifically,
the authors in [9] used HFL to design a reliable anomaly detection strategy
for IoT systems. Moreover, to face the challenges caused by non-i.i.d. data,
the authors in [10] designed an edge-device assignment scheme to enhance the
federated averaging performance. Considering the same constraint of non-i.i.d.
data among users, the authors in [19] proposed an efficient resource
allocation and user assignment scheme in HFL for heterogeneous IoT systems.
Meanwhile, the authors of [14] proposed a novel HFL architecture aiming to
minimize the consumed energy and the total training latency under challenging
data distribution. Energy-efficiency was also the focus of the work in [17],
where authors have introduced a power control and client association solution
for HFL. The work in [8] targeted the minimization of the global cost using a
novel resource allocation and device association algorithm. Another extremely
important metric in FL systems is the latency, that was the objective of the
works in [21] and [15]. While the former optimized the association between
multiple servers and devices, the latter designed a HFL framework across
heterogeneous cellular networks. The authors in [18] proposed a flexible
decentralized control over the training process in order to enhance the
privacy of HFL, while the authors of [22] suggested using blockchain to meet
the privacy requirements of internet of vehicles composing the HFL system. In
[23], a novel HFL framework based on clustering is proposed to enhance the
test accuracy. The authors of [24] investigated HFL by proposing a hybrid data
partitioning algorithm. In [25], an efficient HFL algorithm is presented with
an objective to determine the optimal cluster structure. In [26], a resource
allocation heuristic was designed to guarantee the HFL convergence rate
subject to fairness constraint. Finally, the game theory was used in [16] and
[20] to design an efficient resource allocation strategy for HFL.

Unlike the works proposed in the literature, we present in this paper a novel
HFL architecture considering massive MIMO wireless backhaul, i.e. the links
between the MECs and the devices and the links between MECs are wireless.
Moreover, we consider a practical mobility scenario that represents the
workers in a construction site. Additionally, the use of energy harvesting for
HFL was not considered in most of the related works. Hence, we consider in
this work that the MECs have WET capability and the devices are equipped with
energy harvesting batteries, which allows to enhance the energy efficiency.
Taking into account these challenging constraints, we formulate and
investigate a novel resource optimization problem.

##  III System Model and Energy Model

###  III-A Hierarchical Federated Learning Model

Massive MIMO is based on multiplexing a large number of antennas
simultaneously to satisfy the demand of multiple users in the same time-
frequency resource, which contributes to improving energy efficiency, spectral
efficiency, and latency [38]. Furthermore, HetNets, which are based on network
densification in multiple dimensions with different types of cells, allow to
efficiently exploit the wireless resources and improve the QoS. Incorporating
massive MIMO and HetNets in IoT wireless networks is suitable to optimize the
FL performance. Hence, we opt for a HetNet as presented in Fig. 1. Our system
is composed of a massive MIMO Central Unit (CU) equipped with $N_{\text{cu}}$
antennas, and $M$ MECs, each one is endowed with $N_{\text{mec}}$ MIMO
antennas ($N_{\text{cu}}>MN_{\text{mec}}$). Different MECs communicate with
the CU through a wireless backhaul link [39]. Furthermore, each MEC $m$ serves
multiple single-antenna devices. Let $\Lambda_{m}$ be the set of $K_{m}$
devices served by the MEC $m$. We note that the total number of devices in the
network is equal to $K$. We study our system over an interval of time that we
divide into $L$ frames of duration $T$. Table I presents different key
notations introduced in our paper. The device association index is defined as:

$\chi_{m,k}=\left\\{\begin{array}[]{ll}1,&\text{if device }k\text{ is
associated to }\text{MEC}~{}m,\\\ 0,&\text{otherwise}.\end{array}\right.$ (1)

Each device $k$ owns a dataset $\mathcal{D}_{k}=\\{\bm{x}_{k},\bm{y}_{k}\\}$,
where $\bm{x}_{k}$ and $\bm{y}_{k}$ denote the input and output of the device
$k$. These devices train their FL models locally and send the obtained weights
to their related MECs. Then, each MEC aggregates the weights of its devices
and forwards the results to the CU. Finally, the CU aggregates the weights
received from all MECs and shares the updated FL model with all devices. This
process is repeated until reaching a desirable accuracy. Let the Matrix
$\bm{S}\in\mathbb{N}^{C\times K}$ describe the splitting of data among all
devices, where $C$ denotes the total number of classes in the learning
problem. Particularly, each element $S_{c,k}$ represents the number of data
samples owned by the device $k$ and belonging to the class $c$.

Let $\bm{w}_{k}(l)$ represent the vector of FL weights of the local model
trained within the device $k$ at a frame $l$. After aggregating all weights,
first incoming from the devices, then collected from the MECs, the CU is
considered globally fitting the resultant weights by minimizing a particular
loss function. In our work, we opt for the cross-entropy loss function that
can be presented as follows:

$\mathcal{L}\left(\bm{w}\right)=\sum_{c=1}^{C}-p(c)\mathbb{E}_{\bm{x}\sim
q(\bm{x}\mid y=c)}\left[\log d_{c}(\bm{x};\bm{w})\right],$ (2)

where $p(\cdot)$ defines the global distribution of classes,
$q(\cdot\mid\cdot)$ denotes the likelihood function, and
$d_{c}(\bm{x};\bm{w})$ represents the probability that the input $\bm{x}$
belongs to the class $c$, under the parameters $\bm{w}$.
After receiving the weights of the local FL models from the related devices,
each MEC computes the aggregation following the equation in [10]:

$\bm{w}^{\text{mec}}_{m}(l)=\sum_{k\in\Lambda_{m}}\frac{\mid\mathcal{D}_{k}\mid}{|\cup_{j\in\Lambda_{m}}\mathcal{D}_{j}|}\bm{w}_{k}(l),$
(3)

where $\cup_{j\in\Lambda_{m}}\mathcal{D}_{j}$ is the grouping of local
datasets owned by different devices related to the MEC $m$. Accordingly, it
can be considered as a virtual dataset trained at the MEC $m$. The next step
is to aggregate the weights received from all the MECs at the central unit, as
done in [10]:

$\bm{w}^{\text{cu}}(l)=\sum_{m=1}^{M}\frac{|\cup_{k\in\Lambda_{m}}\mathcal{D}_{k}|}{|\cup_{j=1:K}\mathcal{D}_{j}|}\bm{w}^{\text{mec}}_{m}(l)$
(4)

Finally, the aggregated weights are broadcasted to all devices and this
process is iterated until reaching a targeted convergence rate.
Because of the distribution and federation of the learning, a deviation
between the FL and aggregated weights may occur. This divergence is studied in
[10] and quantified as follows:

$\theta=\sum_{m=1}^{M}\frac{|\cup_{k\in\Lambda_{m}}\mathcal{D}_{k}|}{|\cup_{j=1:K}\mathcal{D}_{j}|}\times\left\|D^{(m)}\right\|_{1},$
(5)

where $D^{(m)}=\left\\{\left\|p(i)-p^{(m)}(i)\right\|\right\\}_{i=1}^{C}$, and
$p^{(m)}(i)$ is the probability of the class $i$ per MEC $m$’s distribution.

###  III-B Communication Model

We assume that the channel between the CU and the MEC $m$ undergoes a Rician
fading represented by $\mathbf{H}_{m}\in\mathbb{C}^{M_{s}\times N}$. On the
other hand, we consider a Gaussian i.i.d. small-scale fading channel between
the MEC $m$ and each device $k\in\Lambda_{m}$, denoted as $\mathbf{f}_{m,k}$.
The access link between the MEC $m$ and each device $k\in\Lambda_{m}$ is
denoted by a vector
$\mathbf{g}_{m,k}=\mathbf{\beta}^{\frac{1}{2}}_{m,k}\mathbf{f}_{m,k}\in\mathbb{C}^{N_{\text{mec}}\times
1}$, where $\beta_{m,k}$ represents the pathloss. To this end, the channel
matrix is defined as
$\mathbf{G}_{m}=[\mathbf{g}_{m,k}]_{k=1:K_{m}}\in\mathbb{C}^{N_{\text{mec}}\times
K_{m}}$.
In our system, we assume a perfect Channel State Information (CSI) of all
devices at each MEC. Thus, all MECs can apply a cooperative Zero Forcing (ZF)
scheme to suppress the effect of inter-cell and intra-cell interferences as
done in [40]. Let
$\mathbf{Z}_{m}=\mathbf{G}_{m}(\mathbf{G}^{H}_{m}\mathbf{G}_{m})^{-1}$ be the
decoder matrix of MEC $m$. Accordingly, the rate of a device $k$ connected to
the MEC $m$ can be formulated as:

$r^{\text{dev}}_{m,k}(l)=\lambda\cdot\log_{2}\left(1+\frac{p^{\text{dev}}_{m,k}(l)}{\sigma^{2}\left[(\mathbf{G}^{H}_{m}\mathbf{G}_{m})^{-1}\right]_{k,k}}\right),$
(6)

where $\lambda$ is the bandwidth, and $\sigma^{2}$ represents the variance of
an Additive White Gaussian Noise (AWGN).

When the MECs receive the FL weights from their related devices, they proceed
the aggregation and forward their aggregated weights to the central unit. At
the central unit side, in case of multiple antennas receivers, the Block
Diagonalization (BD) is used to mitigate the multi-device interference. This
approach is known by its high performance in MIMO systems [41]. Therefore, we
assign for each MEC $m$ a detection matrix $\mathbf{W}_{m}$, in order to
distinguish different signals received from different MECs. Following [41],
$\mathbf{W}_{m}$ lies in the null space of matrix $\overline{\mathbf{H}}_{m}$,
which is given by
$\overline{\mathbf{H}}_{m}=[\mathbf{H}^{T}_{1}...\mathbf{H}^{T}_{m-1}\mathbf{H}^{T}_{m+1}...\mathbf{H}^{T}_{M}]^{T}$.
The decoder matrix for MEC $m$ is obtained as
$\mathbf{W}_{m}=\mathbf{V}^{(0)}_{m}\mathbf{V}^{(1)}_{m}$, where
$\mathbf{V}^{(0)}_{m}$ is composed of the last
$N_{\text{cu}}-\overline{J}_{m}$ right singular vectors of
$\overline{\mathbf{H}}_{m}$, $\mathbf{V}^{(1)}_{m}$ represents the first
$J_{m}$ right singular vectors of
$\overline{\mathbf{H}}_{m}\mathbf{V}^{(0)}_{m}$,
$\overline{J}_{m}=\text{rank}(\overline{\mathbf{H}}_{m})$, and
$J_{m}=\text{rank}(\overline{\mathbf{H}}_{m}\mathbf{V}^{(0)}_{m})$. Hence, the
uplink rate of MEC $m$ is given as [42]:

$r^{\text{mec}}_{m}(l)=\lambda\cdot\log_{2}\left(1+\frac{p^{\text{mec}}_{m}(l)\left\|\mathbf{H}_{m}\mathbf{W}_{m}\right\|^{2}_{F}}{\sigma^{2}}\right).$
(7)

Figure 1: WET-enabled HFL over HetNets in battery powered end-devices.

###  III-C Energy Model

In this section, we evaluate the energy needed by the MECs and their related
devices to compute different FL operations. First, the energy required by a
device $k$ connected to an MEC $m$ to complete the FL tasks in the frame $l$
is expressed by:

$E^{\text{dev}}_{m,k}(l)=E^{\text{cir}}+E^{\text{cmp}}_{k}+E^{\text{ac}}_{m,k}(l),$
(8)

where $E^{\text{cir}}$ is a fixed circuit energy consumed by filters, mixers,
and digital to analog converters. $E^{\text{cmp}}_{k}$ is the energy required
by the device $k$ to train its local data, while $E^{\text{ac}}_{m,k}(l)$ is
the energy required to transmit the resulted weights at frame $l$.
The energy needed by the device $k$ to train its data can be calculated as
done in [43]:

$E^{\text{cmp}}_{k}=\varsigma\omega\vartheta^{2}Q,$ (9)

Where $\varsigma$ is an energy coefficient that depends on the device chip,
$\omega$ represents the number of CPU cycles, $\vartheta$ is the CPU clock
frequency, and Q denotes the size of the local model that highly depends on
the deployed FL algorithm. It is worth mentioning that all local and global
models in all devices, MECs, and CU present the same weights’ size.
Next, to send the weights of the local trained model to its assigned MEC $m$
at a frame $l$, the device $k$ requires an energy equal to:

$E^{\text{ac}}_{m,k}(l)=\frac{p^{\text{dev}}_{m,k}(l)Q}{r^{\text{dev}}_{m,k}(l)}.$
(10)

where $p^{\text{dev}}_{m,k}(l)$ is the required transmit power of the device
$k$ connected to the MEC $m$ at a frame $t$. Based on equation (6),
$p^{\text{dev}}_{m,k}(l)$ can be formulated as:

$p^{\text{dev}}_{m,k}(l)=\sigma^{2}\left[(\mathbf{G}^{H}_{m}\mathbf{G}_{m})^{-1}\right]_{k,k}\left(2^{\frac{r^{\text{dev}}_{m,k}(l)}{\lambda}}-1\right).$
(11)

Hence, by replacing $p^{\text{dev}}_{m,k}(l)$ by its expression in (10), we
obtain:

$E^{\text{ac}}_{m,k}(l)=\frac{\sigma^{2}Q\left[(\mathbf{G}^{H}_{m}\mathbf{G}_{m})^{-1}\right]_{k,k}\left(2^{\frac{r^{\text{dev}}_{m,k}(l)}{\lambda}}-1\right)}{r^{\text{dev}}_{m,k}(l)}.$
(12)

Each device in our FL system has a battery with a maximum capacity equal to
$B_{max}$. These on-device batteries are also used to store the wireless
transferred energy. Let $A_{m,k}(l)$ represent the amount of energy
transmitted to a device $k$ connected to the MEC $m$ at a frame of time $l$.
Hence, we define the battery level update of any device $k$ related to the MEC
$m$ as follows:

$B_{k}(l+1)=\min\left(B_{\max},B_{k}(l)-E^{\text{dev}}_{m,k}(l)+A_{m,k}(l)\right).$
(13)

Furthermore, we define the energy consumed by any MEC $m$ using WET to supply
its associated devices at a frame $l$ as follows:

$E^{\text{mec}}_{m}(l)=E^{\text{cir}}+E^{\text{wet}}_{m}(l)+E^{\text{bh}}_{m}(l),$
(14)

where $E^{\text{wet}}_{m}(l)$ denotes the energy transferred by the MEC $m$ to
its related devices using the WET system, at a frame $l$. Meanwhile,
$E^{\text{bh}}_{m}(l)$ represents the energy consumed by the MEC $m$ to
transmit the aggregated FL weights to the CU. We remind that $E^{\text{cir}}$
is a fixed circuit energy consumption.
Next, we adopt an optimal beamformer for a MIMO WET system to calculate the
energy supplied to each device $k$ related to MEC $m$ at a frame $l$. This
energy is expressed as:

$A_{m,k}(l)=E^{\text{wet}}_{m}(l)\beta_{m,k}\xi_{m,k}N_{\text{mec}},$ (15)

where $\xi_{m,k}$ is the beamforming coefficient for device $k$.
The energy required by the MEC $m$ to transmit the aggregated FL weights to
the central unit at a frame $l$ is given by:

$E^{\text{bh}}_{m}(l)=\frac{p^{\text{mec}}_{m}(l)Q}{r^{\text{mec}}_{m}(l)}.$
(16)

The transmit power $p^{\text{mec}}_{m}(l)$ can be derived from (7) as:

$p^{\text{mec}}_{m}(l)=\frac{\sigma^{2}}{\left\|\mathbf{H}_{m}\mathbf{W}_{m}\right\|^{2}_{F}}\left(2^{\frac{r^{\text{mec}}_{m}(l)}{\lambda}}-1\right).$
(17)

Hence, by replacing $p^{\text{mec}}_{m}(l)$ by its expression in (16), we
obtain:

$E^{\text{bh}}_{m}(l)=\frac{\sigma^{2}Q\left(2^{\frac{r^{\text{mec}}_{m}(l)}{\lambda}}-1\right)}{\left\|\mathbf{H}_{m}\mathbf{W}_{m}\right\|^{2}_{F}r^{\text{mec}}_{m}(l)}.$
(18)

We assume that the energy consumed by each MEC $m$ at a frame of time $l$ is
weighted by a parameter $\alpha_{m}(l)$ [44]. Thus, the total cost of the
energy consumed by all MECs is calculated as follows:

$\Delta=\sum_{l=1}^{L}\sum_{m=1}^{M}\alpha_{m}(l)E^{\text{mec}}_{m}(l).$ (19)

##  IV Problem Formulation

Our objective in this work is to design an energy-efficient HFL framework for
heterogeneous networks, where devices are supplied by wireless energy
transfer. More specifically, we aim at minimizing the cost of the total grid
energy consumed by different MECs by optimizing the device association and the
wireless energy, constrained by the divergence between the FL’s weights and
central weights. Since the deviation between the FL and central weights should
not exceed a given threshold, the local data distributions at different
devices should be considered when deriving the device association. On these
bases, the optimization problem is formulated as (20a).
Constraint (20a) ensures that the deviation between the central weights and
the FL’s weights does not exceed a threshold $\theta_{\max}$. Constraint (20b)
guarantees that each device in the network is assigned to only one MEC.
Equation (20c) ensures that the consumed energy by the devices at each frame
is constrained by the available energy at the batteries. Equation (20d)
ensures that we cannot store energy more than the maximum battery capacity.
Finally, the last two constraints (20e) and (20f) guarantee that the
transferred amounts of energy via wireless links are positive, and the device
association decisions are binaries.

$\begin{split}\underset{\underset{m=1,\ldots,M,k=1,\ldots,K,l=1,\ldots,L}{\\{\chi_{m,k},E^{\text{wet}}_{m}(l)\\}}}{\min}\Delta=\sum_{l=1}^{L}\sum_{m=1}^{M}\alpha_{m}(l)E^{\text{mec}}_{m}(l)\end{split}$
(20)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\rm{s.\;t.}\;\scalebox{1.0}{$\theta\leq\theta_{\max},$}$
(20a) (20b) (20c) (20d) (20e) (20f)

Furthermore, we investigate an extended version of the formulated problem (20)
by incorporating device scheduling. Some devices with high amount of wireless
transferred energy may be unscheduled for certain communication rounds in
order to improve the energy-efficiency of the network without deteriorating
the FL training loss. The optimization problem becomes more challenging.
Moreover, we extend the formulated problem (20) by incorporating a mobility
scenario and assuming dynamic device association where the association index
became variable over time as $\chi_{m,k}(l)$.

##  V Resource Management

###  V-A Optimal Solution

In this section, we derive the optimal solution of the problem formulated in
(20). First of all, we can see that our optimization is a Mixed-Integer Non-
Linear Programming (MINLP) due to its combinatorial nature and the non-
linearity of the constraint (20a). Hence, we propose to use the following
theorem to find the optimal energy transferred by each MEC at each time frame,
and consequently simplify the problem.
Theorem 1. For a MEC $m$ and the set $\Lambda_{m}$ of its associated devices,
the optimal transferred energy $E^{\text{wet*}}_{m}(l)$ from MEC $m$ to its
associated devices is given by:

$E^{\text{wet*}}_{m}(l)=\max_{k\in\Lambda_{m}}\frac{E^{\text{dev}}_{m,k}(l)-B_{k}(l)}{\beta_{m,k}\xi_{m,k}N_{\text{mec}}}.$
(21)

###### Proof.

At a frame $l$, each device $k$ related to a MEC $m$ is supplied by the
harvested energy via WET, in addition to its own battery. The total energy of
$k$ is expressed as follows:

$E^{\text{dev}}_{m,k}(l)=B_{k}(l)+A_{m,k}(l).$ (22)

Thus, by replacing $A_{m,k}(l)$ by its expression presented in (15), we can
derive the energy $E^{\text{wet}}_{m}(l)$ transferred by the MEC $m$ to device
$k$:

$E^{\text{wet}}_{m}(l)=\frac{E^{\text{dev}}_{m,k}(l)-B_{k}(l)}{\beta_{m,k}\xi_{m,k}N_{\text{mec}}}.$
(23)

Each MEC serves its related devices based on the highest required energy. More
precisely, the MEC finds the device that requires the highest amount of energy
and serves it along with the others with this same amount. ∎

Using the expression of the transferred energy $E^{\text{wet}}_{m}(l)$
presented in equation (21), our optimization problem (20) can be simplified as
it becomes an integer non-linear problem. This simplified problem that aims
now to find the optimal device association is combinatorial and is known to be
NP-hard [45]. However, it can be solved using Brute-Force Search (BFS).
Although deriving the optimal solution is still highly complex, the results
can be used as a benchmark to our heuristic approach that will be proposed in
what follows. It is also worth mentioning that by neglecting the constraint
(20a), the optimization becomes an integer linear problem that can be solved
using numerical tools such as CVX.

###  V-B Heuristic Approach

Due to the combinatorial complexity of the optimal solution that may prohibit
its online deployment, we propose a heuristic algorithm that solves the device
association and energy management problem with low complexity. Similarly to
the optimization, our algorithm aims to minimize the cost of the consumed grid
energy, while respecting the FL divergence constraints.
Let $R_{m}$ denote the range of the MEC $m$, which is the pathloss of the
farthest associated device. Based on (16), we can see that the device with the
higher pathloss receives the lowest amount of energy via WET. In other words,
any device assigned to the MEC $m$ with a lower pathloss than $R_{m}$, i.e.,
$\beta_{m,k}\leq R_{m}$, may benefit from a higher transferred energy.
Accordingly, each device $k$ tries to find the set of MECs having a higher
range than its pathloss. Next, it selects the one with minimum divergence
$\theta$. This device association process is done one by one. If one of the
devices fails to find a MEC with a suitable range, it will be associated to
the closest one. Our proposed device association heuristic reduces not only
the energy consumption but also the divergence between FL and central weights,
which contributes to minimize the training loss. Finally, for each frame $l$,
we can calculate the amount of energy transmitted by each MEC via WET using
equation (21). Algorithm 1 presents different steps of our proposed HFL
Heuristic Resource Management Algorithm (H2RMA). Moreover, the proposed
algorithm is described in Fig. 2 and we provide an example of the device
association procedure in Fig. 3. We can see in Fig. 3 that Dev 3 is within the
range of MEC 1 and MEC 2. Hence, it chooses to be associated with the MEC that
minimizes the divergence. For Dev 4, it is outside the range of all the MECs.
Hence, it chooses to be associated with the MEC under minimal energy
requirement. Then, the range of MEC 4 is updated following Dev 4.

Figure 2: Diagram of the proposed algorithm H2RMA.

Figure 3: An example of the device association procedure.

We investigate the complexity of H2RMA. The device association process is
presented in lines (3-20) using two loops for with a complexity equal to
$O(MK)$. In the for loop in lines (21-29), the power control is conducted with
a complexity in the order $O(L(2K+M))$. Hence, the complexity of H2RMA can be
expressed as:

$C^{\text{H2RMA}}=O(MK+L(2K+M)).$ (24)

Hence, H2RMA can be executed in polynomial time.

1:Initialization:

2:$\chi_{i_{k},k}\leftarrow 0,~{}m=1..M,~{}k=1..K$,

3:$R_{m}\leftarrow 0,~{}m=1..M$

4:for $k=1:K$ do

5: Initialize $i_{k}\leftarrow 0$

6: Initialize $\theta_{\min}\leftarrow\theta_{\max}$

7: for $m=1:M$ do

8: if $\beta_{m,k}\leq R_{m}$ then, verify if device $k$ is within the range
of MEC $m$.

9: compute $\theta$

10: if $\theta\leq\theta_{\min}$ then

11: $i_{k}\leftarrow m$, select the MEC with the minimum divergence.

12: $\theta_{\min}\leftarrow\theta$

13: end if

14: end if

15: end for

16: if $i_{k}=0$ then, if device $k$ is not within the range of any MEC.

17: $i_{k}\leftarrow\operatorname*{argmin}_{m=1..M}\beta_{m,k}$, associate
with the closest MEC $m*$.

18: $R_{m*}\leftarrow\beta_{m*,k}$, update the range of the MEC $m*$.

19: end if

20: $\chi_{i_{k},k}\leftarrow 1$

21:end for

22:for $l=1:L$ do

23: compute the energy required by each device.

24: compute $E^{\text{wet}}_{m}(l),~{}m=1..M$ using (21).

25: send the weights $\bm{w}_{k}(l)$ to the related MECs and aggregation using
(4).

26:
$B_{k}(l+1)\leftarrow\min\left(B_{\max},B_{k}(l)-E^{\text{dev}}_{m,k}(l)+A_{m,k}(l)\right)$,
battery update

27: send the weights $\bm{w}^{\text{mec}}_{m}(l)$ to the central unit to
compute the aggregation using (5).

28: send the aggregated weights to all devices.

29: different devices update their FL training models.

30:end for

Algorithm 1 HFL Heuristic Resource Management Algorithm (H2RMA)

###  V-C Device Scheduling

We now investigate the device scheduling for our proposed HFL framework. In
fact, at a given frame $l$, some devices may face bad channel conditions and
require high amount of wireless transferred energy, without contributing much
on the HFL procedure. This costs the system unnecessary energy consumption
that can be avoided. In this scenario, a practical idea to save the grid
energy is to deactivate some devices for some frames based on the available
energy and time-varying channel. Moreover, the scheduled devices at each frame
should respect the maximal divergence constraint so as not to degrade of FL
performance.
Including device scheduling to the optimization problem (20) which is MINLP
increases the complexity of finding the optimal solution. This optimal device
scheduling can be obtained with BFS. However, its high computational
complexity prevents it from being conceivable. Hence, we propose an efficient
heuristic device scheduling scheme to save unnecessary grid energy cost. We
start by considering that all devices are selected. The devices to be
deactivated at the current frame are those that require high amount of energy
exceeding a certain threshold $E_{\text{th}}$ given by the set:

$\Omega=\\{k\in\\{1,\ldots,K\\}|E^{\text{ac}}_{m,k}(l)>E_{\text{th}}\\}.$ (25)

We deactivate the devices in $\Omega$ one by one iteratively and we start by
the ones that require higher transmit energy. At each iteration, the device is
deactivated only if the maximal divergence constraint is satisfied so as not
to degrade FL performance.

##  VI Impact of Mobility

In this section, we investigate an applied case of occupational health and
safety by applying our proposed HFL framework in order to minimize the health
risks of the workers in a construction site. In consequence, we choose a
mobility scenario [46] that reflects the workers equipped with smart devices
in a construction site. The dynamic interactions between workers on the site
makes the construction job one of the most risky activities. The mobility of
workers can be modeled by Hidden Markov Models (HMMs) [46], which are powerful
statistical tools that represent the probability distributions over
observation. We can categorize the workers into three states, i.e., static,
normal, and risky, depending on their speed and turning angle. Hence, the
hidden states of the HMM may be defined depending on the values of these two
parameters, i.e., speed and turning angle. These values are provided in Table
II. Moreover, the step length of the workers is modeled by the Gamma
distribution and the turning angle is modeled by the Von Mises distribution.
Let us divide the network into micro-cells with length $\mu$. The average
number of frames that a normal worker moves from a micro-cell to another is
given by:

$\psi_{\text{nml}}=\lceil\frac{\mu}{TV_{\text{nml}}}\rceil,$ (26)

where $\lceil\cdot\rceil$ is the ceiling function, and $V_{\text{nml}}$ is the
average speed of a normal worker. The average number of frames that a risky
worker moves from a micro-cell to another is given by:

$\psi_{\text{rsk}}=\lceil\frac{\mu}{TV_{\text{rsk}}}\rceil,$ (27)

where $V_{\text{rsk}}$ is the average speed of a risky worker. It is clear
that the risky workers change their micro-cells more frequently than the
normal workers. Also, some workers when they pass by certain places in the
network (like office, meeting room, coffee break …), they remain static for a
certain period.

TABLE II: Values of Hidden States. State |  Speed (steps/mn) |  Turning angle (radians)
---|---|---
Static |  0 |  angle < $\pi/2$
Normal |  0 < steps $\leq$ 84 |  $\pi/2\leq$ angle < $\pi$
Risky |  steps > 84 |  angle $\geq\pi$

Considering this mobility scenario, the mobile devices move around and become
far from their associated MECs. Hence, they require higher amounts of wireless
transferred energy. Indeed, performing fixed device association scheme when
the devices are moving involves an increase in the energy consumption cost.
Therefore, we need to design an adaptive device association scheme taking into
account devices mobility.

1:for $l=1:L$ do

2: if $l\mod\psi_{\text{rsk}}=0$ or $l\mod\psi_{\text{nrm}}=0$ then,

3: Update $\Gamma$ the set of mobile devices that changed their micro-cell
during the current frame.

4: for $k\in\Gamma$ do

5: $i_{k}\leftarrow 0$, initialization

6: $\theta_{\min}\leftarrow\theta_{\max}$, initialization

7: for $m=1:M$ do

8: if $\beta_{m,k}\leq R_{m}$ then, verify if the mobile device $k$ is within
the range of MEC $m$.

9: compute $\theta$

10: if $\theta\leq\theta_{\min}$ then

11: $i_{k}\leftarrow m$, select the MEC that has the minimum divergence.

12: $\theta_{\min}\leftarrow\theta$

13: end if

14: end if

15: end for

16: if $i_{k}=0$ then, if the mobile device $k$ is not within the range of any
MEC.

17: $i_{k}\leftarrow\operatorname*{argmin}_{m=1..M}\beta_{m,k}$, associate
with the closest MEC $m*$.

18: $R_{m*}\leftarrow\beta_{m*,k}$, update the range of the MEC $m*$.

19: end if

20: $\chi_{i_{k},k}\leftarrow 1$

21: end for

22: end if

23:end for

Algorithm 2 Dynamic HFL Device Association (DHDA)

We propose a heuristic dynamic device association strategy for HFL over
HetNets enabled by WET. We start by executing the original device association
algorithm H2RM presented in section V. Then, we change the associated MECs of
the normal workers after each $\psi_{\text{nml}}$ frames and for the risky
workers after each $\psi_{\text{rsk}}$ frames. The mobile device $k$ updates
its new associated MEC as follows. It finds first the new set of MECs that
have a range higher than its pathloss. We already defined the range $R_{m}$ of
a MEC $m$ as the pathloss of the farthest associated device. Hence, being
associated with one of these MECs will not induce any supplementary energy
cost. The mobile device will choose the MEC having the minimum divergence
$\theta$. If this device fails to find a MEC with suitable range, the closest
one will be selected. The proposed Dynamic HFL Device Association Algorithm
(DHDA) is illustrated in Algorithm 2.
To find the complexity of DHDA, we need to multiply the computational
complexity of the first for loop of H2RMA by the number of times that we
repeat association of mobile device $\lceil\frac{L}{\psi_{\text{rsk}}}\rceil$.
Hence, the complexity of DHDA is give by:

$C^{\text{DHDA}}=O(\lceil\frac{L}{\psi_{\text{rsk}}}\rceil MK+L(2K+M)).$ (28)

##  VII Numerical Results

In this section, we investigate the performance of our proposed HFL framework
under different network configurations. The simulation settings are as
follows: we positioned the CU at the center of a circular macro-cell and
uniformly distributed the MECs and devices within the considered coverage.
First, we consider that devices are required to recognize digits between 0 and
9, which can be simulated by training a classification on MNIST dataset. In
our simulation, we used a deep learning model to train on the MNIST dataset.
This model is composed of two convolutional layers of 3x3 filters and we chose
Rectified Linear Unit (ReLU) as an activation function. To downsize the
resulting feature maps, we added a max pooling layer, which output is
flattened to be fed into the subsequent fully connected layer. Finally, the
last layer is designed to generate the probabilities of the 10 classes. We
note that we used the dropout technique [47] to reduce the overfitting and
regularize the model. Furthermore, we incorporated the information about the
class weight in the training process to mitigate the problem of imbalanced
classes, and we used simple Stochastic Gradient Descent (SGD) to train the
model and optimize the cross-entropy loss, throughout 20 training epochs.
Next, we investigate CIFAR-10 (Canadian Institute For Advanced Research) which
is a well-known image dataset that is commonly used to validate the
performance of machine learning and computer vision algorithms. The data
consist of 60000 color images of size 32x32 and 10 classes; each class has
6000 images. The class labels are: horse, frog, dog, cat, deer, bird,
automobile, airplane, truck, and ship. The data does not have overlap between
classes, i.e. the classes are mutually exclusive. We summarize the simulation
parameters used in this evaluation section in Table III.

TABLE III: Configuration of the HFL over HetNet environment in our experiments. Symbol |  Description |  Value
---|---|---
$N_{\text{cu}}$ |  number of antennas at the CU |  128
$M$ |  number of MECs |  8
$N_{\text{mec}}$ |  number of antennas at each MEC |  16
$K$ |  number of devices |  20
$L$ |  number of frames |  50
$C$ |  number of classes |  10
$B_{\textrm{max}}$ |  max battery capacity |  1000 J
$B_{k}(0)$ |  initial battery level |  200 J
$\nu$ |  path loss exponent |  3.7
$\lambda$ |  bandwidth |  20 MHz
$\vartheta$ |  frequency of the CPU clock |  $10^{9}$
$\omega$ |  number of CPU cycles |  40
$\varsigma$ |  consumed energy coefficient |  $10^{-27}$
$\theta_{\text{max}}$ |  maximal divergence |  0.5
|  cell radius |  200 m
|  circuit power per RF chain |  30 dBm
|  noise power spectral density |  -174 dBm/Hz

###  VII-A Fixed Device Association without Scheduling

Figure 4: Grid energy cost while varying the number of devices $K$: comparison
between optimal device association, random association, and H2RMA
($M=2,B_{k}(0)=10~{}\text{J}$).

Figure 5: Grid energy cost while varying the number of devices $K$: comparison
between H2RMA and random device association scheme
($M=8,B_{k}(0)=200~{}\text{J}$).

We start by investigating the performance of the proposed HFL framework
considering a static network and fixed device association over time. In Fig.
4, we show the grid energy cost, when varying the number of devices. Deriving
the optimal device association suffers from a high computation complexity.
Thus, we run our simulation using only two MECs in order to establish the
optimal solution as a baseline. Compared to the optimal solution, we can see
that H2RMA achieves a good performance in terms of energy consumption cost,
while offering a much lower computation complexity. In Fig. 5, we illustrate
the grid energy cost incurred by our heuristic compared to the random device
association scheme, when changing the number of devices $K$. Using a large
number of devices, H2RMA presents a highly lower grid energy cost, due to its
adequate device association method. Fig. 6 depicts the performance of our
heuristic algorithm when varying the number of MECs. When the number of MECs
increases, the cost of grid energy decreases, as different devices in the
system are surrounded by a larger set of MECs; hence, they have a higher
probability to be associated to closer MECs. Moreover, we can see clearly,
that our H2RMA algorithm outperforms the random association scheme, for
different numbers of MECs.

Figure 6: Grid energy cost while varying the number of MECs $M$: comparison
between H2RMA and random device association ($K=20,B_{k}(0)=200~{}\text{J}$).

Figure 7: Test accuracy: comparison between H2RMA and random device
association scheme considering MNIST ($K=15,M=3,C=3$).

Fig. 7 presents the accuracy performance of H2RMA compared to the random
association scheme, when tested using MNIST dataset. We can see that our
algorithm converges faster, owing to its efficient device association process
that minimizes the divergence. In Fig. 8, the accuracy performance of H2RMA is
evaluated, while considering CIFAR-10 dataset. Similar to the MNIST dataset,
our proposed device association scheme performs well and outperforms the
random device association scheme in terms of test accuracy.

Figure 8: Test accuracy: comparison between H2RMA and random device
association scheme considering CIFAR-10 ($K=15,M=3,C=2$).

###  VII-B Fixed Device Association with Scheduling

We investigate now in Fig. 9 the impact of the proposed device scheduling on
the grid energy consumed by our system. It is clear that we can save a
significant amount of energy while deactivating some devices that require a
high amount of wireless transferred energy for a certain number of frames.
Moreover, the proposed device scheduling scheme performs very well compared to
the random scheme, because only far devices are being deactivated, while
respecting the maximal divergence constraint.

Figure 9: Grid energy cost while varying the number of devices $K$: comparison
between the proposed scheduling heuristic, random device association and
system scheduling all devices ($M=8,B_{k}(0)=200~{}\text{J}$).

Fig. 10 depicts the performance of our device scheduling heuristic when
varying the number of MECs. We can save a significant amount of grid energy
specifically when the number of MECs increases thanks to the efficient device
scheduling scheme.

Figure 10: Grid energy cost while varying the number of MECs $M$: comparison
between the proposed scheduling heuristic, random device association and
system scheduling all devices ($K=20,B_{k}(0)=200~{}\text{J}$).

Fig. 11 depicts the performance of our heuristic device scheduling algorithm
in terms of test accuracy considering MNIST. It can be seen that when
deactivating some devices randomly, the test accuracy decreases. However, the
proposed heuristic device scheduling scheme allows to enhance the convergence
rate since only a subset of devices that respect the divergence constraint is
scheduled.

Figure 11: Test accuracy considering MNIST: comparison between the proposed
scheduling heuristic, random device association and system scheduling all
devices ($K=15,M=3,C=3$).

###  VII-C Dynamic Device Association

Finally, we implement a mobility scenario for the devices which is described
in Section V. The mobility parameters used in this simulation are illustrated
in Table II. In Fig. 12, we evaluate the cost of grid energy consumed by our
HFL framework, when considering the mobility scenario. Indeed, the proposed
dynamic device association algorithm allows saving significant amount of
energy because it allows us to efficiently adapt the association of the mobile
devices. Furthermore, the performance gap between dynamic and fixed device
association increases when the number of mobile devices in the network becomes
higher.

Figure 12: Grid energy cost while varying the number of devices $K$
considering a mobility scenario ($M=8,B_{k}(0)=200~{}\text{J}$).

Fig. 13 shows the performance of the proposed HFL framework in terms of
convergence rate considering a mobility scenario. The proposed dynamic device
association scheme ensures that mobile devices respect the maximal divergence
constraints and hence achieve better convergence rate than the fixed device
association.

Figure 13: Test accuracy considering a mobility scenario and MNIST dataset
($K=15,M=3,C=3$).

To summarize, the performance of our HFL framework is evaluated by showing the
energy-efficiency of the proposed resource management approaches, in addition
to the high accuracy achieved under different network configurations and using
two well-known datasets, namely MNIST and CIFAR-10. Moreover, we showed the
efficiency of the proposed dynamic device association considering a mobility
scenario in the network.

##  VIII Conclusion

An energy-efficient HFL over HetNets with massive MIMO wireless backhaul
enabled by WET has been investigated in this work. An optimization problem
involving a grid energy consumption cost constrained by energy availability
and maximal divergence, have been developed. In addition, an optimal energy
management has been derived. To achieve this, we have formulated the main
objective as a device association optimization problem that we solved using
BFS. Because of the high complexity to get the optimal solution, we have
proposed an efficient device association heuristic, with low computational
complexity, namely H2RMA. Moreover, a device scheduling strategy has been
designed to save the grid energy without deteriorating the learning accuracy.
The impact of mobility on FL performance has also been investigated by
developing an efficient dynamic device association scheme. Our extensive
experiments confirm the efficiency of our HFL framework in terms of accuracy,
convergence rate and grid energy consumption, when evaluated under different
network configurations.

As a future work, we plan to develop an efficient clustering strategy based on
reinforcement learning for HFL over HetNets enabled by WET.

## Acknowledgment

This work was supported by National Priorities Research Program-Standard
(NPRP-S) Thirteen (13th) Cycle grant $\\#$ NPRP13S-0205-200265 from the Qatar
National Research Fund (a member of Qatar Foundation). The findings achieved
herein are solely the responsibility of the authors.

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License Attribution 4.0 International (CC BY 4.0).

PhysioCHI: Towards Best Practices for Integrating Physiological Signals in
HCI, May 11, 2024, Honolulu, HI, USA

[1]

<EMAIL_ADDRESS>url=https://alchemy18.github.io/pragyasingh/, ]

<EMAIL_ADDRESS>url=https://www.rit.edu/directory/mjkvcs-mohan-kumar, ]
[1]

<EMAIL_ADDRESS>url=https://www.iiitd.ac.in/pushpendra/, ] [1]

[1]Authors are advisors of this research work.

# CAN WE SAY A CAT IS A CAT? UNDERSTANDING THE CHALLENGES IN ANNOTATING
PHYSIOLOGICAL SIGNAL-BASED EMOTION DATA

Pragya Singh  IIIT-Delhi, Delhi, India  Mohan Kumar  Rochester Institute of
Technology, Rochester, New York, US  Pushpendra Singh  IIIT-Delhi, Delhi,
India

(2024)

###### Abstract

Artificial Intelligence (AI) algorithms, trained on emotion data extracted
from physiological signals, provide a promising approach to monitoring
emotions, affect, and mental well-being. However, the field encounters
challenges because there is a lack of effective methods for collecting high-
quality data in everyday settings that genuinely reflect changes in emotion or
affect. This paper presents a position discussion on the current technique of
annotating physiological signal-based emotion data. Our discourse underscores
the importance of adopting a nuanced understanding of annotation processes,
paving the way for a more insightful exploration of the intricate relationship
between physiological signals and human emotions.

###### keywords:

Physiological Signals Affective computing Data-Centric AI Emotion Recognition
Mental Health

##  1 Introduction

Physiological signals have significant potential for widespread emotion
monitoring, offering numerous applications in emotion-affect-mental-wellbeing
recognition. The increasing prevalence of wearable devices equipped with
sensors capable of continuously collecting physiological signal data has
prompted researchers in the human-computer interaction (HCI) and artificial
intelligence (AI) domains to explore their applicability in emotion-affect-
mental-wellbeing monitoring and emotion-based interfaces. In contrast to
methods relying on video, audio, or text for emotion recognition, leveraging
physiological data provides several advantages. It enables unobtrusive data
collection from individuals and groups over extended periods, offering high
temporal resolution. Furthermore, the use of physiological data reduces
susceptibility to conscious manipulation by subjects.

However, a well-labelled dataset is a crucial requirement for training AI
algorithms in a supervised fashion. From past experience, it is evident that
gathering physiological signals with their annotated emotional response is
challenging. Prior work on the collection of emotion-affect-mental well-being
datasets has explored both lab-based settings and field settings with
constraint and daily-life activities [1, 2, 3, 4]. Data collection for
physiological signal-based data remains challenging, limiting the available
datasets’ capabilities for training AI algorithms [5]. The major reason for
these challenges lies in the nature of data. Physiological signal-based data
is often collected by attaching sensors to the human body. However, the
availability of portable wearable devices equipped with sensors is often
limited. Another major challenge is the wearable devices equipped with the
sensors often don’t provide raw data, thus making it challenging to collect
physiological data. Moreover, the nature of annotations in emotion-affect-
mental well-being recognition is often not as straightforward as other
application domains in AI; for example, in the computer vision dataset, we can
label a picture of a cat as a cat with surety. The annotation procedure in
this domain is heavily dependent on psychological models, emotion theories and
human annotators, making the procedure of annotation complex and biased on
various levels. In this work, we present a preliminary discussion on the
challenges in collecting and utilising annotations with respect to
physiological signals for training emotion-affect-mental well-being
recognition algorithms. Furthermore, we shed light on the adverse effect that
present annotations have on overall data quality and model performance. Next,
we present some avenues for HCI and AI researchers to overcome these
challenges.

##  2 Related Work

Emotion recognition, a rapidly growing field within artificial intelligence
and human-computer interaction, focuses on identifying and interpreting human
emotions through various data modalities such as facial expressions, voice
tone, physiological signals, and behavioural patterns. With the availability
of wearable sensors, there is an increase in ongoing work in physiological
signal-based emotion recognition [6, 7, 8]. Researchers have leveraged various
physiological signals and their derived features, such as heart rate (HR),
heart rate variability (HRV) measured using electrocardiogram (ECG),
photoplethysmography (PPG) or blood volume pulse (BVP) signals, skin
conductance measured using electrodermal activity (EDA), also known as
galvanic skin response or resistance (GSR), skin temperature (SKT), and muscle
activity measured using electromyogram (EMG) for quantifying changes in
emotional arousal [2, 1, 3, 4, 9]. Furthermore, in the prior literature,
physiological signal-based emotion datasets have been curated primarily in
three settings: i) Laboratory settings [1, 10], ii) field with constraint
settings [4, 3] and iii) real-life settings [11, 12]. For annotating the
physiological data, prior works have employed techniques such as annotations
by external annotators or experts [13], self-reports using self-assessment
questionnaires such as Self-Assessment Manikin (SAM) or Positive and Negative
Affect Schedule (PANAS) [1], experience sampling [11], continuous annotations
by participants using annotation device [14, 15, 16] and self-reflection [17].
Few other works have used the stimulation task as ground truth; for example,
in the WESAD dataset [1], physiological data collected during the Trier Social
Stress Test (TSST) is labelled as stress. Recently, there has been an
increasing focus on collecting physiological data in everyday settings in
order to capture realistic responses [5]. While we see enough work in
improving the setting of emotion data collection, limited attention is given
to the quality of annotations and their effects on overall data quality and
performance of emotion recognition algorithms. This position paper presents an
argument on the limitations of present practices of annotating physiological
signal-based emotion data.

Figure 1: Presently used procedure of Physiological Signal based Emotion Data
collection.

##  3 Understanding the limitations in Annotation Procedure

In this section, we have outlined the limitations present methods of
annotating physiological signal-based emotion data suffer from and possible
avenues to venture.

###  3.1 Annotating the bare-minimum

Over the past decade, there have been multiple endeavours to recognize
emotion-affect-mental well-being using various physiological signal data and
AI. Although AI algorithms exhibit significant performance in laboratory
settings with small sample sizes, identifying emotions, affect, or mental
well-being in everyday settings remains challenging. Prior works on curating
physiological signal-based emotion have primarily adopted three approaches for
annotating physiological data. These approaches include: i) Self-assessment:
wherein emotions are annotated by the person experiencing the emotions, ii)
Expert-annotations: wherein emotions are annotated by an expert observer iii)
Physiological Correlation: wherein emotions are annotated based on the
patterns or changes in physiological signals themselves. For example, specific
patterns in heart rate variability and skin conductance might be associated
with particular emotional states iv) Stimuli Label: labels are assigned based
on external stimuli or events used to elicit specific emotional responses. In
figure 1, we have illustrated the presently used methodology for physiological
signal-based emotion data collection.

It is important to highlight that each of the above-mentioned methods suffers
from certain limitations. For instance, self-reported annotations are often
perceived emotions that can be based on underlying mood irrespective of the
physiological changes and thus can lead to biased labels (For example, a
person with an underlying sad mood may perceive an otherwise happy situation
as sad due to his overall mood at the instance). Similarly, expert-based
annotations are difficult in the case of real-life or ambulatory data
collection. They may also suffer from expert bias (For example, an expert may
annotate the emotions based on the manipulated expressions of a person while a
person is not feeling them underneath). Further physiological correlations are
often misleading in the case of ambulatory settings. They may also suffer from
artefacts and collection environments (For example, electrodermal activity
data may show peaks in case of a change in temperature). Subsequently, labels
based on stimuli frequently fall short in capturing the subjective perception
of diverse participants. These limitations underscore the necessity of
devising a more nuanced approach to annotation that can delve into a
comprehensive understanding of the larger context.

###  3.2 Are Emotion Scales Enough?

Another important aspect of emotion annotation is the choice of rating scale.
Prior work has utilised scales inspired by either Emotional models or
psychological tests for emotion-affect-mental well-being annotations. In prior
literature, scales such as Self-Assessment-Manikins [18], Geneva emotion wheel
[19], PANAS [20], and others have drawn inspiration from emotion theories.
Similarly, standard scales from psychology and psychiatry such as the
Perceived Stress Scale (PSS) [21], State-Trait Anxiety Inventory (STAI) [22],
General Health Questionnaire (GHQ) [23], have been utilized in the past for
annotating physiological data. While employing standard questionnaires for
annotating emotions proves convenient in controlled laboratory environments,
it’s essential to recognize the challenge of replicating such methods in
everyday data collection settings, where soliciting responses from human
participants can be challenging. Another critical consideration pertains to
the design of current scales. Many of the prevalent scales are crafted to
capture annotations over specific durations. Consequently, these scales are
tailored for discrete annotations, such as those made after observing a
stimulus, at the day’s conclusion, or on a weekly basis, making them
unsuitable for obtaining high-resolution continuous annotations. For example,
a scale rooted in an emotional model like SAM excels at capturing overall
valence and arousal within a given duration. However, in cases where there are
numerous changes within the last hour, the annotation we obtain reflects the
overall perceived emotion rather than providing a detailed, high-resolution
account of those fluctuations. Another common observation is that the scales
often suffer from cultural influence and definitions of human subjects’
emotions in a region. Our observations from past work and our experiences
suggest a need for designing shorter, continuous and self-descriptive scales
for better annotations.

###  3.3 Can we say a Cat is a Cat?

Prior work has highlighted that emotions are subjective, and so are
physiological signals. Each individual has their own physiology and data
range. The physiology of a person is also altered based on several bodily and
environmental changes. The observed variations on multiple levels indicate the
complexity of establishing a direct one-to-one mapping between physiological
changes and emotions, affect, or mental well-being. Unlike in other domains
leveraging AI, the annotation process in this context is not straightforward.
This highlights the imperative to move beyond simple annotations and
underscores the importance of not exclusively interpreting physiological
changes as responses solely to alterations in emotions, affect, and mental
well-being.

##  4 Conclusion

In our paper, we presented a position discussion on the limitations present
techniques of annotating emotion-affect-menatl well-being data suffer from. In
the future, we want to explore these challenges further and develop solutions
to overcome these limitations. In conclusion, comprehending the challenges
associated with annotating physiological signal-based emotion data is
imperative for advancing research in emotion recognition and understanding
human affect. The complexities involved in accurately capturing and
interpreting physiological responses underscore the need for innovative
methodologies and robust annotation strategies. Addressing these challenges
will enhance the reliability of physiological data and contribute to the
development of more effective applications and interventions in fields such as
healthcare, human-computer interaction, and psychology.

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