# An Energy and Carbon Footprint Analysis of Distributed and Federated
Learning

Stefano Savazzi, Vittorio Rampa, Sanaz Kianoush, Mehdi Bennis

###### Abstract

Classical and centralized Artificial Intelligence (AI) methods require moving
data from producers (sensors, machines) to energy hungry data centers, raising
environmental concerns due to computational and communication resource
demands, while violating privacy. Emerging alternatives to mitigate such high
energy costs propose to efficiently distribute, or federate, the learning
tasks across devices, which are typically low-power. This paper proposes a
novel framework for the analysis of energy and carbon footprints in
distributed and federated learning (FL). The proposed framework quantifies
both the energy footprints and the carbon equivalent emissions for vanilla FL
methods and consensus-based fully decentralized approaches. We discuss optimal
bounds and operational points that support green FL designs and underpin their
sustainability assessment. Two case studies from emerging 5G industry
verticals are analyzed: these quantify the environmental footprints of
continual and reinforcement learning setups, where the training process is
repeated periodically for continuous improvements. For all cases,
sustainability of distributed learning relies on the fulfillment of specific
requirements on communication efficiency and learner population size. Energy
and test accuracy should be also traded off considering the model and the data
footprints for the targeted industrial applications.

##  I Introduction

Training deep Machine Learning (ML) models at the network edge has reached
notable gains in terms of accuracy across many tasks, applications and
scenarios. However, such improvements have been acquired at the cost of large
computational and communication resources, as well as significant energy
footprints which are currently overlooked. Vanilla ML requires all training
procedures to be conducted inside data centers [1] that collect data from
producers, such as sensors, machines and personal devices. Centralized based
learning requires high energy costs for networking and data maintenance, while
introduces privacy issues as well. In addition, many mission critical ML tasks
require data centers to learn continuously [2] to track changes in dataset
distributions. Data centers are thus becoming more and more energy-hungry and
responsible of significant CO2 (carbon) emissions. These amount to about $15$%
of the equivalent global Green House Gas (GHG) emissions of the entire
Information and Communication Technology (ICT) ecosystem [3], estimated as
$0.5\div 1$ GtCO2e (gigatonnes of CO2 equivalent emissions) in 2020.
Electricity consumed by global data centers is also estimated to be between
$1.1$% and $1.5$% of the total worldwide electricity use.

Emerging alternatives to centralized big-data analytics enable decisions to be
made at much more granular levels [4]. The Federated Learning (FL) approach
[5] is a recently proposed distributed AI paradigm [6] in line with this
trend. Under FL, the ML model parameters, e.g. the weights and biases
$\mathbf{W}$ of Deep Neural Networks (DNN), are collectively optimized across
several resource-constrained edge/fog devices, that act as _both_ data
producers _and_ local learners. FL distributes the computing tasks across many
devices characterized by low-power consumption profiles, compared with data
centers, and each device has access to small datasets [2] only. With a
judicious system design taking into account both accuracy and energy, FL is
expected to bring significant reduction in terms of energy footprints,
obviating the need for a large centralized infrastructure for cooling or power
delivery.

Figure 1:  From top to bottom: Centralized Learning (CL), Federated Learning
(FL) coordinated by the Parameter Server (PS), namely Federated Averaging
(FA), and Consensus-driven Federated Learning, enabling fully decentralized
implementations (i.e., without PS).

###  I-A Federated, consensus-driven and centralized learning: related works

As shown in Fig. 1, vanilla FL algorithms, such as Federated Averaging (FA)
[4], allow devices to learn a local model under the orchestration of a
Parameter Server (PS). Devices might be either co-located in the same
geographic area or distributed in different zones. Typically, the PS
aggregates the received local models to obtain a _global model_ that is fed
back to the devices. The PS functions are substantially less energy-hungry
compared to centralized learning (CL) and can be implemented at the network
edge. Different FL implementations have emerged in the past few years [7]
targeting several scenarios [2, 8]. The optimization of the population of
learners is critical in FL and is typically explored when power and bandwidth
of devices are limited [9]. Delay and energy tradeoffs between learning and
communication are considered in [10], while the problem of transmission
scheduling in small-scale fading is analyzed in [11]. Quantization and
compression of model parameters, or gradients, is useful to minimize bandwidth
usage and to avoid straggler effects [12, 13, 14]. For a comprehensive survey
on FL please refer to [4].

The above mentioned designs leverage on a server-client architecture [7] where
the PS represents a single-point of failure and penalizes scalability. In
analogy with distributed ledger, consensus-driven learning replaces the PS
functions as it lets the local model parameters be consensually shared and
synchronized across multiple devices via device-to-device (D2D) or sidelink
communications [2]. Such fully decentralized solution relies _solely_ on in-
network processing and a consensus-based federation model, i.e., often based
on distributed weighted averaging [15]-[18].

Regardless of whether the network architecture is server-client or
decentralized via consensus methods, distributed learning requires many
communication rounds for convergence that often consist of UpLink (UL) and
DownLink (DL) communications through a Wireless Wide Area access and core
Network (WWAN). Large energy bills might be therefore observed for
communication [10]. The choice between distributed and centralized learning
targeting sustainable, energy-aware designs is thus expected to be driven by
communication vs computing cost balancing.

###  I-B Contributions

The paper develops a novel framework for the analysis of energy and carbon
footprints in distributed ML, including, for the first time, comparisons and
tradeoff considerations about vanilla, consensus-driven FL and centralized
learning on the data center. Despite initial attempts to assess FL resource
efficiency [8], optimize delay/energy [10, 11] and carbon footprints [3, 20],
quantifying the optimal operating points towards green or sustainable setups
is yet unexplored. To fill this void, the paper dives into the implementations
of selected distributed learning tools, and discusses optimized designs and
operational conditions with respect to energy-efficiency and low carbon
emissions. A general framework is first developed for energy and carbon
footprint assessment: differently from [3, 10], it considers all the system
parameters and components of the learning process, including the PS, required
in vanilla FL, as well as the distributed model averaging, required in
decentralized setups. Next, sustainable regions in the parameter space are
identified to steer the choice between centralized and distributed training in
practical setups: the regions highlight bounds, or necessary conditions on
communication, computing efficiencies, data and model size that make FL more
promising than CL with respect to its carbon footprint. Unlike classical FL
platforms [4, 10] defined on top of edge devices with large computing power
[8], the paper focuses on learning tasks suitable for low-power embedded
wireless devices (e.g., robots, vehicles, drones) typically adopted in
Industrial Internet of Things (IIoT) applications [7]. These setups are
characterized by devices/learners retaining small training datasets, while
running medium/small-sized ML models due to their constrained internal memory.
Devices are also equipped with low-power radio interfaces supporting deep
sleep modes, cellular (e.g., 5G and NB-IoT), as well as direct mode, or
sidelink, communication interfaces.

The proposed co-design of learning and communication is validated using real
world data and a test-bed platform characterized by low-power devices
implementing real-time distributed model training on top of the Message
Queuing Telemetry Transport (MQTT) protocol. The real-time test-bed is used to
measure the training time for different learning policies, while a _what-if_
analysis quantifies the estimated energy and carbon footprints considering the
impact of various communication settings, as well as the influence of the
learner population size.

###  I-C Paper structure and organization

The paper is organized as follows. Section II describes the framework for
energy consumption evaluation of different distributed learning strategies.
Energy modelling is carried out separately for centralized learning
(Sect.II-A), vanilla FA (Sect. II-B), including the novel policy FA-D that
keeps inactive learners in deep-sleep, and decentralized consensus-driven FA
(Sect. II-C). Necessary adaptations of the proposed energy models to
_continual_ and _Reinforcement Learning_ (RL) paradigms [21] are also
discussed in Sect. II-D.

Sect. III considers the carbon footprints (Sect. III-A) and the sustainable
regions (Sect. III-B) that provide necessary requirements to steer the choice
between centralized and distributed learning paradigms. Requirements are
verified first in Sect. IV using the MNIST [22] and the CIFAR10 [23] datasets
(Sect. IV-A) separately and through comparative analysis (Sect. IV-B). Next,
practical design problems are explored in Sect. V targeting two industry
relevant use cases. In particular, the FL network implementation is based on
the MQTT protocol (Sect. V-A) while 4 communication efficiency profiles are
considered (Sect. V-B). Both continual (Sect. V-C) and RL (Sect. V-D)
scenarios are analyzed where the training process must be periodically
repeated using new input data that change frequently to follow either a time-
varying industrial process or the same model under training. These tools are
critical in 5G industry verticals, i.e., Industry 4.0 (I4.0) applications, and
could raise significant energy concerns unless specific actions are taken.
Finally, conclusion are summarized in Sect. VI.

##  II Energy footprint modeling framework

The proposed framework provides insights into how the different components of
the CL (i.e., the datacenter) and FL architectures (i.e. the local learners,
the core network and the PS), contribute to the energy bill and to the carbon
emissions in terms of system accuracy and number of rounds. The learning
system consists of $K$ devices and one data center ($k=0$). Each device $k>0$
has a dataset $\mathcal{E}_{k}$ of examples $(\mathbf{x}_{h},y_{h})$ that are
typically collected independently. In supervised learning, examples
$\mathbf{x}_{h}$ are labelled as $y_{h}$ for training. Unsupervised setups,
i.e., RL, are discussed in Sect. II-D. For all cases, the objective of the
learning system is to train a DNN model $\hat{y}(\mathbf{W};\mathbf{x})$ that
transforms the input data $\mathbf{x}$ into the desired $C$ outputs
$\hat{y}\in\left\\{y_{c}\right\\}_{c=1}^{C}$, i.e., the output classes. Model
parameters are here specified by the matrix $\mathbf{W}$ [4]. The training
system uses the examples in $\mathcal{E}=\bigcup_{k=1}^{K}\mathcal{E}_{k}$ to
minimize a loss function $L$ of the form

$L(\mathcal{E}|\mathbf{W})=\sum_{k=1}^{K}L_{k}(\mathcal{E}_{k}|\mathbf{W}).$
(1)

In what follows, unless stated otherwise, we use the cross-entropy loss
function [4], namely
$L_{k}=-\sum_{h}y_{h}\log\left[\hat{y}(\mathbf{W};\mathbf{x}_{h})\right]$.
Minimization of (1) is typically iterative and gradient-based [5]: it runs
over a pre-defined number ($n$) of learning rounds that depend on a specified
target loss threshold, namely $L(\mathcal{E}|\mathbf{W})\leq\overline{\xi}$,
i.e., corresponding to a required accuracy. For example, using Stochastic
Gradient Descent (SGD)

$\mathbf{W}_{t+1}\leftarrow\mathbf{W}_{t}-\mu_{s}\times\nabla
L_{k}(\mathcal{E}_{k}|\mathbf{W}_{t}),$ (2)

examples $\mathcal{E}_{k}$ are drawn randomly from the full training set
$\mathcal{E}$, while $\mu_{s}$ is the step-size and $\nabla
L_{k}(\mathcal{E}_{k}|\mathbf{W}_{t})=\nabla_{\mathbf{W}_{t}}\left[L(\mathcal{E}_{k}|\mathbf{W}_{t})\right]$
is the gradient of the loss function (1) over the assigned batches of data
$\mathcal{E}_{k}$ given the model $\mathbf{W}_{t}$.

The energy footprint of device $k$, namely the total amount of energy consumed
by the learning process, is broken down into computing and communication
components. Considering one learning round, all the energy costs are modelled
as a function of the computing energy $E_{k}^{(\mathrm{C})}$ required by the
optimizer (2) and the energy $E_{k,h}^{(\mathrm{T})}$ per correctly
received/transmitted bit over the wireless link ($k,h$). The training data or
the DNN model parameters are quantized before transmission into
$b(\mathcal{E}_{k})$ and $b(\mathbf{W})$ bits, respectively. The quantization
scheme assigns a fixed number of bits (here 32 bits) to each parameter of the
DNN model. Compression techniques such as model pruning, sparsification [24],
parameter selection and/or differential transmission schemes [12, 25] are
extremely helpful to scale down the footprint $b(\mathbf{W})$ in large DNNs.
However, due to the great variety of compression techniques, we have
considered here only a simple quantization scheme. The modifications to
include a particular technique can be trivially detailed using the specific
data $b(\mathcal{E}_{k})$ and model size $b(\mathbf{W})$ as the result of the
compression processes.

We define the energy cost for UL communication with the data center (or the
PS), co-located with the Base Station (BS), as $E_{k,0}^{(\mathrm{T})}$.
Similarly, $E_{0,k}^{(\mathrm{T})}$ refers to DL communication, i.e., from the
PS, or the BS, to the _k_ -th device. Communication costs incorporate both
transmission and decoding operations [26]. In contrast to digital
implementations, analog FL designs [9, 17] get over the restrictions of time
scheduled access: rather than spending energy $E_{k,h}^{(\mathrm{T})}$ for
sending bits over orthogonal links, they exploit the superposition property of
wireless transmissions for analog aggregation [27], scaling down the number of
channel uses, and the communication cost. Although not considered in this
paper, comparing analog and digital FL in the light of energy footprints is an
open problem of wide interest. The energy $E_{k}^{(\mathrm{C})}$ for computing
includes the cost of the learning round, namely the local gradient-based
optimizer and the data storage costs. All costs are quantified on average:
notice that routing through the radio access and the core network can vary but
might be assumed as stationary apart from failures or replacements.

###  II-A Centralized Learning

Under CL, the model training is carried out inside the data center $k=0$,
assumed co-located with the BS, and exploits the processing power provided by
racks of CPU (Central Processing Units), GPU (Graphic Processing Units) and
other specialized AI accelerators such as NPU (Neural Processing Units) and
TPU (Tensor Processing Units). Therefore, the energy cost per round
$E_{0}^{(\mathrm{C})}=P_{0}\cdot T_{0}\cdot B$ depends on the power
consumption $P_{0}$ [20] of the aforementioned hardware units, the time span
$T_{0}$ required for processing an individual batch of data, i.e. minimizing
the loss $L(\cdot|\mathbf{W})$, and the number $B$ of batches per round [10].
Data are collected independently by the devices while we neglect here the cost
of initial dataset loading since it is a one-step process only. For $n$
rounds, chosen to satisfy a target loss $\overline{\xi}$, the total energy
$E_{\mathrm{CL}}$ in Joule [J] consists of computing
$E_{\mathrm{CL}}^{(\mathrm{L})}$ and communication
$E_{\mathrm{CL}}^{(\mathrm{C})}$ costs:

$E_{\mathrm{CL}}(n)=\underset{E_{\mathrm{CL}}^{(\mathrm{L})}}{\underbrace{\gamma\cdot
n\cdot
E_{0}^{(\mathrm{C})}}}+\underset{E_{\mathrm{CL}}^{(\mathrm{C})}}{\underbrace{\alpha\cdot\sum_{k=1}^{K}b(\mathcal{E}_{k})\cdot
E_{k,0}^{(\mathrm{T})}}.}$ (3)

The energy cost for computing inside the data center scales as
$E_{\mathrm{CL}}^{(\mathrm{L})}(n)=\gamma\cdot n\cdot E_{0}^{(\mathrm{C})}$
with $\mathrm{\gamma}>1$ being the Power Usage Effectiveness (PUE) of the data
center [31, 32]. The PUE accounts for the additional power consumed for data
storage, power delivery and cooling: $\gamma$ values are in the range $1.1\div
1.8$ [32, 44]. The energy consumed for communication of raw data is
$E_{\mathrm{CL}}^{(\mathrm{C})}=\alpha\sum_{k=1}^{K}b(\mathcal{E}_{k})\cdot
E_{k,0}^{(\mathrm{T})}$ and quantifies the energy for moving
$b(\mathcal{E}_{k})$ bits of the $k$-th local training dataset
$\mathcal{E}_{k}$ for $\alpha$ times and for all the $K$ devices. Parameter
$\alpha$ counts the number of times a new training dataset is uploaded by the
devices while learning is in progress. For example, in one-time learning
processes, the full training dataset is available by devices before the
training starts: therefore, it is $\alpha=1$. On the other hand, in continual
learning applications, devices produce new training data continuously [2, 21]:
data thus needs to be moved while learning is in progress, therefore it is
$1<\alpha\leq n$.

Figure 2:  Distributed learning algorithms: a) Federated Averaging (FA, FA-D),
b) CFA with sidelink communications.

###  II-B Federated learning with server (FA) and deep-sleep (FA-D)

Unlike CL, FL distributes the learning process across a selected, typically
time-varying [4], subset $\mathcal{N}_{t}$ of $K_{a}<K$ _active_ devices as
shown in Fig. 1(a). At each round $t$, the local dataset $\mathcal{E}_{k}$ is
used to train the local model $\mathbf{W}_{k,t}$ using a gradient-based Local
Optimizer (LO) and a global model instance ($\mathbf{W}_{t}$), obtained from
the server (PS). The goal is to minimize the local loss $L_{k}$ as

$\mathbf{W}_{k,t}=\underset{\mathbf{W}}{\mathrm{arg}\mathrm{min}}\thinspace
L_{k}(\cdot|\mathbf{W})+\upsilon\left\|\mathbf{W}-\mathbf{W}_{t}\right\|^{2},$
(4)

where $\upsilon\left\|\mathbf{W}-\mathbf{W}_{t}\right\|^{2}$ is the proximal
term [11, 28] often used when data is non-identically distributed (non-IID).
The local model $\mathbf{W}_{k,t}$ is then forwarded to the PS [4] over the
UL. The PS is co-located with the BS and is in charge of updating the global
model $\mathbf{W}_{t+1}$ for the following round $t+1$ through the aggregation
of the $K_{a}$ received models [5]:

$\mathbf{W}_{t+1}\leftarrow\sum_{k\in\mathcal{N}_{t}}\sigma_{k}\cdot\mathbf{W}_{k,t},$
(5)

with $\sigma_{k}=\tfrac{Q_{k}}{Q}$ being the ratio between the number of local
examples $Q_{k}$ and global $Q=\sum_{k\in\mathcal{N}_{t}}Q_{k}$ ones,
respectively. Notice that for IID data distributions, it is
$\sigma_{k}=\tfrac{1}{K_{a}}$ [6]. The new model $\mathbf{W}_{t+1}$ is finally
sent back to the devices over the DL.

Federated Averaging (FA). In vanilla FL methods, such as FA, the PS chooses
$K_{a}<K$ active devices on each round to communicate their local model.
During this process, the remaining $K-K_{a}$ devices are powered on [5, 6] as
they run the LO (4) and decode the updated global model $\mathbf{W}_{t}$
obtained from the PS. For $n$ rounds, the total end-to-end energy can be
written as

$E_{\mathrm{FA}}(n)=E_{\mathrm{FA}}^{(\mathrm{L})}(n)+E_{\mathrm{FA}}^{(\mathrm{C})}(n),$
(6)

namely, the superposition of the computing $E_{\mathrm{FA}}^{(\mathrm{L})}(n)$
and the communication $E_{\mathrm{FA}}^{(\mathrm{C})}(n)$ costs, considering
both the devices and the PS consumption. In particular:

$\begin{array}[]{c}E_{\mathrm{FA}}^{(\mathrm{L})}(n)=\gamma\cdot
n\cdot\beta\cdot
E_{0}^{(\mathrm{C})}\,+n\cdot\sum_{k=1}^{K}E_{k}^{(\mathrm{C})},\\\
E_{\mathrm{FA}}^{(\mathrm{C})}(n)=b(\mathbf{W})\left[n\cdot\sum_{k=1}^{K}\gamma\cdot
E_{0,k}^{(\mathrm{T})}+\sum_{t=1}^{n}\sum_{k\in\mathcal{N}_{t}}E_{k,0}^{(\mathrm{T})}\right].\end{array}\begin{aligned}
\end{aligned}$ (7)

The energy cost for computing $E_{\mathrm{FA}}^{(\mathrm{L})}(n)$ in (7) is
due to the PS energy $\gamma\cdot n\cdot\beta\cdot E_{0}^{(\mathrm{C})}$
needed for model averaging and the LO,
$n\cdot\sum_{k=1}^{K}E_{k}^{(\mathrm{C})}$ that is implemented by all the $K$
devices [10]. The energy for model averaging $\beta\cdot E_{0}^{(\mathrm{C})}$
is considerably smaller than the gradient-based optimization term
$E_{0}^{(\mathrm{C})}$ on the data center, therefore it is $\beta\ll 1$. In
IoT setups, the devices are also usually equipped with embedded low-
consumption CPUs, SoCs (System on Chips) or $\mu$CU (microcontrollers): thus,
it is reasonable to assume $E_{k}^{(\mathrm{C})}<E_{0}^{(\mathrm{C})}$. The
communication energy $E_{\mathrm{FA}}^{(\mathrm{C})}(n)$ in (7) models the
cost of the global model transmission over the DL, i.e., $n\cdot
b(\mathbf{W})\sum_{k=1}^{K}\gamma\cdot E_{0,k}^{(\mathrm{T})}$, and the UL
communication from selected $K_{a}$ active devices, i.e.,
$b(\mathbf{W})\sum_{t=1}^{n}\sum_{k\in\mathcal{N}_{t}}E_{k,0}^{(\mathrm{T})}$
in the set $\mathcal{N}_{t}$: as depicted in Fig. 2(a), it is assumed that the
PS is co-located with the BS. The model size $b(\mathbf{W})$ quantifies the
size in bits of model parameters to be exchanged111More precisely,
$b(\mathbf{W})$ quantifies the size of the subset of the (trainable) model
layers, or parameters, exchanged on each FL round: however, to simplify the
reasoning, it is herein referred to as “model size”. . As analyzed in the
Sect. III, both model $b(\mathbf{W})$ and local data size $b(\mathcal{E}_{k})$
are critical for sustainable designs. Notice that, as opposed to data,
$b(\mathbf{W})$ is roughly the same for each device, although small changes
might be observed when using lossy compression, sparsification or parameter
selection methods [12, 24, 25].

Federated Averaging with deep-sleep (FA-D). An alternative to FA, referred to
as FA-D, lets the inactive $K-K_{a}$ devices to turn off their computing
hardware and communication interface, and set into a deep-sleep mode, when not
needed by the PS. Deep sleep mode should be paired with efficient hardware
able to power up and down in fractions of milliseconds: this is also a key
component of 5G [26] and supported by popular interfaces, such as NB-IoT [47].
In contrast to vanilla FA, FA-D limits the consumption to the $K_{a}$ active
devices, while these might change from round to round on a duty-cycle basis.
As verified in the following, FA-D brings significant per-round energy
benefits, in exchange for larger number of rounds
$n_{\mathrm{FA-D}}>n_{\mathrm{FA}}$, for the same target accuracy
$\overline{\xi}$. Considering the duty cycling process, the energy footprint
of FA-D becomes

$E_{\mathrm{FA-D}}(n)=E_{\mathrm{FA-D}}^{(\mathrm{L})}(n)+E_{\mathrm{FA-D}}^{(\mathrm{C})}(n),$
(8)

now with

$\begin{array}[]{c}E_{\mathrm{FA-D}}^{(\mathrm{L})}(n)=\gamma\cdot
n\cdot\beta\cdot
E_{0}^{(\mathrm{C})}\,+\sum_{t=1}^{n}\left[\sum_{k\in\mathcal{N}_{t}}E_{k}^{(\mathrm{C})}+\sum_{k\notin\mathcal{N}_{t}}E_{k,\mathrm{sleep}}^{(\mathrm{C})}\right],\\\
E_{\mathrm{FA-D}}^{(\mathrm{C})}(n)=b(\mathbf{W})\cdot\sum_{t=1}^{n}\sum_{k\in\mathcal{N}_{t}}\left[\gamma\cdot
E_{0,k}^{(\mathrm{T})}+E_{k,0}^{(\mathrm{T})}\right].\end{array}\begin{aligned}
\end{aligned}$ (9)

Notice that the LO cost is now constrained to the active devices,
$\sum_{t=1}^{n}\sum_{k\in\mathcal{N}_{t}}E_{k}^{(\mathrm{C})}$ as with global
model communication,
$b(\mathbf{W})\cdot\sum_{t=1}^{n}\sum_{k\in\mathcal{N}_{t}}\gamma\cdot
E_{0,k}^{(\mathrm{T})}$. Finally, $E_{k,\mathrm{sleep}}^{(\mathrm{C})}$
accounts for the energy consumption of inactive devices in deep-sleep mode (if
not negligible w.r.t. $E_{k}^{(\mathrm{C})}$ ).

###  II-C Consensus-driven Federated Averaging (CFA)

Decentralized FL techniques [16, 17, 18] let the devices mutually exchange
their local models, possibly over peer-to-peer networks. Unlike with FA, the
PS is not needed as it is replaced by a consensus mechanism. In particular, we
propose a consensus-driven FA approach (CFA) where the federated nodes
exchange the local ML model parameters and update them sequentially by
distributed averaging. As shown in Fig. 2(b), devices mutually exchange their
local model parameters $\mathbf{W}_{k,t}$ with an assigned number $N$ of
neighbors [12, 13, 17]. Distributed weighted averaging [16, 18] is used to
combine the received models.

Let $\mathcal{N}_{k,t}$ be the set that contains the $N$ chosen neighbors of
node $k$ at round $t$: on every new round ($t>0$), the device updates the
local model $\mathbf{W}_{k,t}$ using the parameters $\mathbf{W}_{h,t}$
obtained from the neighbor device(s) as

$\mathbf{W}_{k,t+1}\leftarrow\mathbf{W}_{k,t}+\sum_{h\in\mathcal{N}_{k,t}}\sigma_{h,k}\cdot(\mathbf{W}_{h,t}-\mathbf{W}_{k,t}),$
(10)

where the weights $\sigma_{h,k}$ are chosen to guarantee a stable solution as

$\sigma_{h,k}=\frac{Q_{h}}{\sum_{h\in\mathcal{N}_{k,t}}Q_{h}},$ (11)

and for IID data, it is $\sigma_{h,k}=\tfrac{1}{N}$, $\forall h,k$.
Distributed weighted averaging (10) is followed by LO (4) on $\mathcal{E}_{k}$
that can also include proximal regularization [28]. For $K_{a}<K$ active
devices in the set $\mathcal{N}_{t}$ and $n$ rounds, the energy footprint is
again broken down into learning and communication costs as

$E_{\mathrm{CFA}}(n)=E_{\mathrm{CFA}}^{(\mathrm{L})}(n)+E_{\mathrm{CFA}}^{(\mathrm{C})}(n),$
(12)

with

$\begin{array}[]{c}E_{\mathrm{CFA}}^{(\mathrm{L})}(n)=\sum_{t=1}^{n}\left[\sum_{k\in\mathcal{N}_{t}}E_{k}^{(\mathrm{C})}+\sum_{k\notin\mathcal{N}_{t}}E_{k,\mathrm{sleep}}^{(\mathrm{C})}\right],\\\
E_{\mathrm{CFA}}^{(\mathrm{C})}(n)=\sum_{t=1}^{n}\sum_{k\in\mathcal{N}_{t}}\sum_{h\in\mathcal{N}_{k,t}}b(\mathbf{W}_{k,t})\cdot
E_{k,h}^{(\mathrm{T})}.\end{array}\begin{aligned} \end{aligned}$ (13)

The sum $\sum_{h\in\mathcal{N}_{k,t}}b(\mathbf{W}_{k,t})\cdot
E_{k,h}^{(\mathrm{T})}$ now models the total energy spent by the device $k$ to
diffuse the local model parameters to selected neighbors $h$ in the set
$\mathcal{N}_{t}$ at round $t$. Since the PS is not used, CFA is particularly
promising in reducing the energy footprint.

CFA requires sending data over peer-to-peer links ($k,h$): in WWAN, each link
is implemented by UL transmission from the source $k$ to the core network
access point (i.e., router or BS), followed by a DL communication from the
router(s) to the destination device $h$, therefore

$E_{k,h}^{(\mathrm{T})}=E_{k,0}^{(\mathrm{T})}+\gamma\cdot
E_{0,h}^{(\mathrm{T})},$ (14)

where $\gamma$ is the PUE of the BS or router hardware (if any).
Alternatively, CFA can leverage direct, or sidelink, transmissions [29].
Sidelink (SL) features were introduced by 3GPP (PC5 interface) in release $12$
and $13$, namely the proximity service (ProSe). They support direct (D2D)
communications with minor involvement of the BS, or eNB [19, 41]. Since the
signal relay through the BS is not needed, excluding an initial link setup for
resource allocation, sidelink communications are expected to make a
significant step towards green and sustainable networks, serving as enabling
technology for an efficient implementation of CFA. Both communication
architectures will be addressed next.

###  II-D Continual and reinforcement learning paradigms

The energy framework previously discussed is suitable for modelling the
environmental footprints of conventional supervised, or one-time, ML problems.
On the other hand, novel learning paradigms are emerging in industrial IoT
setups. In the following sections, we highlight some necessary adaptations
with a particular focus on continual and RL paradigms. Both scenarios are
further discussed in the case studies of Sect. V-C.

Continual learning. In continual learning paradigms the training process must
be periodically repeated to follow a time-varying industrial process and using
new input data or observations that change frequently. When the input data
changes, a new training process starts using either the most recent global
model instance as an initialization or meta-learning tools [33] that quickly
adapt to model changes or new tasks [35]. Continual learning methods consist
of an initial training stage ($t_{0}$), where the model parameters are
optimized over $n_{0}$ rounds using local training data
$\mathcal{E}_{k}(t_{0})$, followed by periodic tuning/re-training stages at
subsequent time instants $t_{1},...,t_{i}$. Model retraining uses a smaller
amount of (few shot) data [34] as
$b[\mathcal{E}_{k}(t_{i})]<b[\mathcal{E}_{k}(t_{0})]$, and a reduced number of
rounds, namely $n_{i}\ll n_{0},\forall i\geq 1$. Each stage has a cost of
$E^{(t_{i})}(n_{i})$, typically with $E^{(t_{i})}(n_{i})\ll
E^{(t_{0})}(n_{0})$ since fewer rounds are needed.

Reinforcement learning (RL). RL [21] is used to train policies, typically deep
ML models, that map observations of an environment to a set of _actions_ ,
while trying to maximize a long-term _reward_. Similarly as for continual
learning, the training process must be periodically repeated, i.e., at time
instants $t_{0},...,t_{i}$ using new input observations
$\mathcal{E}_{k}(t_{i})$, or _states_ , that change frequently as they are
influenced by the same model under training. In this paper, we resort to the
popular Deep Q-Learning (DQL) method [30]. The goal is to train a state-action
value function $\mathbf{W}$, namely the Q-function, that uses the input states
to predict the expected future rewards for each output action, i.e.,
$\hat{y}\in\left\\{y_{c}\right\\}_{c=1}^{C}$. The Q-function is typically a
deep ML model while training is obtained using gradient-based optimization.
Considering $K$ learning devices, the training process on each round consists
of new actions that generate new states/observations of the environment:
actions could be obtained by maximizing the local Q-function
$\mathbf{W}_{k,t}$ under training (exploitation) or they can be randomly and
independently chosen by devices (exploration). The training data are therefore
determined by the learned Q-function and change at every round [30]. For CL,
the training data is moved to a data center on each round, therefore
$\alpha=n$ in (3). On the other hand, FA, FA-D and CFA algorithms let the
devices exchange the parameters of their local Q-functions, rather than the
observations. All the algorithms proposed in Sects. II-B and II-C can be thus
applied without significant modifications. Finally, it is worth noticing that
since observations are collected on each new round, the energy footprint must
also include the cost of training data collection. A specific example is given
in Sect. V-D.

##  III Design principles for low-carbon emissions

The section discusses the main factors that are expected to steer the choice
between centralized and federated learning paradigms towards sustainable
designs. Sustainability is here measured in terms of equivalent GHG emissions,
referred to as carbon footprints. The goal is to identify the operating
conditions that are necessary for FL policies (FA, FA-D and CFA) to emit lower
carbon than CL. Advantages of FL compared with CL are related to communication
and computing costs, as well as model $b(\mathbf{W})$ and data
$b(\mathcal{E}_{k})$ size. Considering the energy models detailed in Sect. II,
the carbon footprints (Sect. III-A) are obtained for all the FL policies
($\mathrm{C}_{\mathrm{FA}},\mathrm{C}_{\mathrm{FA-D}}$ and
$\mathrm{C}_{\mathrm{CFA}}$) as well as for CL ($\mathrm{C}_{\mathrm{CL}}$).
Sustainable regions (Sect. III-B) highlight necessary conditions on
communication and computing energy costs (energy and computing efficiencies)
as well as on model v. data footprint ratio
$\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}$.

###  III-A Carbon footprints and model simplifications

The carbon footprints
$\mathrm{C}_{\mathrm{FA}},\mathrm{C}_{\mathrm{FA-D}},\mathrm{C}_{\mathrm{CFA}}$
and $\mathrm{C}_{\mathrm{CL}}$ are summarized in Tab. I for all the proposed
FL algorithms as well as for CL. Based on the energy models (3)-(13), the
estimated emissions are evaluated by multiplying each individual energy
contribution, namely $E_{k}^{(\mathrm{C})}$ and $E_{k,h}^{(\mathrm{T})}$, by
the corresponding carbon intensity ($\mathrm{CI}_{k}$) of the electricity
generation [46]. The $\mathrm{CI}_{k}$ terms depend on the specific
geographical regions where the devices $k=0,...,K$ are installed, and are
measured in kg CO2-equivalent emissions per kWh (kgCO2-eq/kWh): they quantify
how much carbon emissions are produced per kilowatt hour of locally generated
electricity.

Targeting general rules for sustainability assessment, the following
simplifications are applied to carbon footprints in Tab. I. First,
communication costs are quantified on average, in terms of the corresponding
energy efficiencies (EE), standardized by ETSI (European Telecommunications
Standards Institute [38]). These are defined as the ratio between the data
volume originated in DL
($\mathrm{EE}_{\mathrm{D}}=[E_{0,k}^{(\mathrm{T})}]^{-1}$), UL
($\mathrm{EE}_{\mathrm{U}}=[E_{k,0}^{(\mathrm{T})}]^{-1}$) or sidelink
transmissions ($\mathrm{EE}_{\mathrm{S}}=[E_{k,h}^{(\mathrm{T})}]^{-1}$) and
the network energy consumption observed during the period required to deliver
the same data. Efficiency terms are measured here in bit/Joule [bit/J] [39,
40] and we consider different choices of $\mathrm{EE}_{\mathrm{D}}$,
$\mathrm{EE}_{\mathrm{U}}$ and $\mathrm{EE}_{\mathrm{S}}$ depending on the
specific network implementation. In particular, when the sidelink interface is
not available, but WWAN is used222It is worth mentioning that it would be
$[\mathrm{EE}_{\mathrm{S}}]^{-1}=[\mathrm{EE}_{\mathrm{D}}]^{-1}+[\mathrm{EE}_{\mathrm{U}}]^{-1}$
if the PUE $\gamma$ of the Base Station or the router hardware were $\gamma=1$
as mentioned in Sect. II-B. , it is
$[\mathrm{EE}_{\mathrm{S}}]^{-1}\simeq[\mathrm{EE}_{\mathrm{D}}]^{-1}+[\mathrm{EE}_{\mathrm{U}}]^{-1}$.
Considering now the computing costs, we define the computing efficiency of the
data center (or PS) as $\mathrm{EE}_{\mathrm{C}}=[E_{0}^{(\mathrm{C})}]^{-1}$.
It quantifies how much energy per learning round is consumed and it is
measured in terms of number of rounds per Joule [round/J]. The computing
efficiency of the devices $k>0$ are modeled here as
$[E_{k}^{(\mathrm{C})}]^{-1}=\frac{\mathrm{EE_{\mathrm{C}}}}{\varphi_{k}}$
with

$\varphi_{k}=\frac{E_{k}^{(\mathrm{C})}}{E_{0}^{(\mathrm{C})}}=\frac{P_{k}}{P_{0}}\cdot\frac{T_{k}}{T_{0}}.$
(15)

Low-power devices typically experience a much larger local batch time
$T_{k}>T_{0}$ compared with data center $T_{0}$. On the other hand, they use
substantially lower power ($P_{k}\ll P_{0}$).

TABLE I:  Communication and computing carbon footprints. |  Communication $\mathrm{C}^{(\mathrm{C})}$ footprint |  Computing $\mathrm{C}^{(\mathrm{L})}$ footprint
---|---|---
$\mathrm{C}_{\mathrm{CL}}$: | $\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}=\alpha\cdot\sum_{k=1}^{K}b(\mathcal{E}_{k})\cdot\frac{\mathrm{CI}_{k}}{\mathrm{EE_{U}}}$ | $\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}=n_{\mathrm{CL}}\cdot\gamma\cdot\frac{\mathrm{CI}_{0}}{\mathrm{EE_{C}}}$
$\mathrm{C}_{\mathrm{FA}}$: | $\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})}=n_{\mathrm{FA}}b(\mathbf{W})\left(\sum_{k=1}^{K_{a}}\frac{\mathrm{CI}_{k}}{\mathrm{EE_{U}}}+\gamma\cdot K\cdot\frac{\mathrm{CI}_{0}}{\mathrm{EE_{D}}}\right)$ | $\mathrm{C}_{\mathrm{FA}}^{(\mathrm{L})}=n_{\mathrm{FA}}\left(\sum_{k=1}^{K}\frac{\varphi_{k}\cdot\mathrm{CI}_{k}}{\mathrm{EE_{C}}}+\beta\cdot\gamma\cdot\frac{\mathrm{CI}_{0}}{\mathrm{EE_{C}}}\right)$
$\mathrm{C}_{\mathrm{FA-D}}$: | $\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}=n_{\mathrm{FA-D}}b(\mathbf{W})\left(\sum_{k=1}^{K_{a}}\frac{\mathrm{CI}_{k}}{\mathrm{EE_{U}}}+\gamma\cdot K_{a}\cdot\frac{\mathrm{CI}_{0}}{\mathrm{EE_{D}}}\right)$ | $\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{L})}=n_{\mathrm{FA-D}}\left(\sum_{k=1}^{K_{a}}\frac{\varphi_{k}\cdot\mathrm{CI}_{k}}{\mathrm{EE_{C}}}+\beta\cdot\gamma\cdot\frac{\mathrm{CI}_{0}}{\mathrm{EE_{C}}}\right)$
$\mathrm{C}_{\mathrm{CFA}}$: | $\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})}=n_{\mathrm{CFA}}b(\mathbf{W})\left(\sum_{k=1}^{K_{a}}\frac{N\cdot\mathrm{CI}_{k}}{\mathrm{EE_{S}}}\right)$ | $\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{L})}=n_{\mathrm{CFA}}\sum_{k=1}^{K_{a}}\frac{\varphi_{k}\cdot\mathrm{CI}_{k}}{\mathrm{EE_{C}}}$

###  III-B Carbon-aware sustainable regions and requirements

Sustainability of FL depends on the specific operating conditions about
communication ($\mathrm{EE}_{\mathrm{D}}$, $\mathrm{EE}_{\mathrm{U}}$,
$\mathrm{EE}_{\mathrm{S}}$) and computing ($\mathrm{EE}_{\mathrm{S}}$)
efficiency, as well as model $b(\mathbf{W})$ and data $b(\mathcal{E}_{k})$
footprints. In what follows, we dive into such operational points to highlight
practical or necessary requirements for green designs. To simplify the
reasoning, we consider here $\mathrm{CI}_{k}\approx\mathrm{CI},\forall k$,
while the general results with arbitrary carbon intensity values
$\mathrm{CI}_{k}$ are shown in the Appendix. All the requirements analyzed
below are summarized in Tab II.

TABLE II:  Region definitions and requirements. | Region definition | Requirement
---|---|---
$\mathcal{R}_{\mathrm{DU}}$: | $\left\\{\text{$\mathrm{EE}_{\mathrm{D}}$,$\mathrm{EE}_{\mathrm{U}}$}:\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$ | $\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}\left(\frac{\alpha}{n\cdot K_{a}}\cdot\frac{\sum_{k=1}^{K}b(\mathcal{E}_{k})}{b(\mathbf{W})}-1\right)>\gamma\cdot\frac{K}{K_{a}}$
| $\left\\{\text{$\mathrm{EE}_{\mathrm{D}}$,$\mathrm{EE}_{\mathrm{U}}$}:\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$ | $\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}\left(\frac{\alpha}{n\cdot K_{a}}\cdot\frac{\sum_{k=1}^{K}b(\mathcal{E}_{k})}{b(\mathbf{W})}-1\right)>\gamma$
$\mathcal{R}_{\mathrm{SU}}$: | $\left\\{\text{$\mathrm{EE}_{\mathrm{S}}$,$\mathrm{EE}_{\mathrm{U}}$}:\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$ | $\frac{\mathrm{EE_{S}}}{\mathrm{EE}_{\mathrm{U}}}\cdot\frac{\alpha}{n\cdot K_{a}}>N\cdot\frac{b(\mathbf{W})}{\sum_{k=1}^{K}b(\mathcal{E}_{k})}$
$\mathcal{R}_{b(\mathbf{W})}$: | $\left\\{b(\mathbf{W}),b(\mathcal{E}_{k}):\max[\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}]<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$ | $\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}<\frac{\alpha}{n}\cdot\frac{K}{K_{a}}$
$\left\\{b(\mathbf{W}),b(\mathcal{E}_{k}):\max[\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}]<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$ | $\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}<\frac{\alpha}{n}\cdot\frac{K}{N\cdot K_{a}}$
$\mathcal{R}_{\mathrm{CI}}$: | $\left\\{\text{$\mathrm{CI}_{k}$},\forall k:\frac{\mathrm{C}_{\mathrm{FA}}^{(\mathrm{L})}}{n_{\mathrm{FA}}}<\frac{\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}}{n_{\mathrm{CL}}}\right\\}$ | $\sum_{k=1}^{K}\frac{\varphi_{k}}{1-\beta}\mathrm{CI}_{k}<\gamma\mathrm{CI}_{0}$
$\left\\{\text{$\mathrm{CI}_{k}$},\forall k:\frac{\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{L})}}{n_{\mathrm{FA-D}}}<\frac{\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}}{n_{\mathrm{CL}}}\right\\}$ | $\sum_{k=1}^{K_{a}}\frac{\varphi_{k}}{1-\beta}\mathrm{CI}_{k}<\gamma\mathrm{CI}_{0}$
$\left\\{\text{$\mathrm{CI}_{k}$},\forall k:\frac{\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{L})}}{n_{\mathrm{CFA}}}<\frac{\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}}{n_{\mathrm{CL}}}\right\\}$ | $\sum_{k=1}^{K}\varphi_{k}\mathrm{CI}_{k}<\gamma\mathrm{CI}_{0}$

Requirements on uplink and downlink efficiencies in cellular communications.
FL policies make extensive use of UL/DL communications either for local model
parameters upload or global model download. Quantifying necessary requirements
on communication efficiencies
$(\text{$\mathrm{EE}_{\mathrm{D}}$,}\mathrm{EE}_{\mathrm{U}})$ is thus
particularly critical and constitutes a key indicator for the optimal choice
between centralized and distributed learning. The first problem we tackle is
to identify the region of the parameter space

$\mathcal{R}_{\mathrm{DU}}\triangleq\left\\{\text{$\mathrm{EE}_{\mathrm{D}}$,$\mathrm{EE}_{\mathrm{U}}$}:\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$
(16)

such that when
$\left\\{\mathrm{EE}_{\mathrm{D}},\mathrm{EE}_{\mathrm{U}}\right\\}\in\mathcal{R}_{\mathrm{DU}}$
the FA policy emits lower carbon than CL with respect to communication costs.
Defining $b(\mathcal{E})=\sum_{k=1}^{K}b(\mathcal{E}_{k})$ as the size of the
training data across the deployed devices, condition
$\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
in (16) can be written as

$\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}\left(\frac{\alpha}{n\cdot
K_{a}}\cdot\frac{b(\mathcal{E})}{b(\mathbf{W})}-1\right)>\gamma\cdot\frac{K}{K_{a}},$
(17)

with $\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})}$ and
$\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$ in Tab. I (further details are in
the Appendix). Since training costs $\mathrm{C}^{(\mathrm{L})}$ are
overlooked, the bound (17) gives a necessary (but not sufficient) condition
towards sustainability assessment. As verified experimentally, the bound (17)
is revealed tight enough to serve as a practical operating condition when
communication emits much more carbon than training, often verified in practice
[3, 20]. Considering FA-D, the parameter region (16) becomes
$\mathcal{R}_{\mathrm{DU}}\coloneqq\left\\{\text{$\mathrm{EE}_{\mathrm{D}}$,}\mathrm{EE}_{\mathrm{U}}:\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$
while
$\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
can be written as
$\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}\left(\frac{\alpha}{n\cdot
K_{a}}\cdot\frac{b(\mathcal{E})}{b(\mathbf{W})}-1\right)>\gamma$. Compared
with (17), it gives a less stringent requirement when $K_{a}<K$, namely the
active device population $K_{a}$ is smaller than the full population $K$.

Requirements on direct mode communications (sidelink efficiency). CFA does not
need the PS and exploits direct mode communications, replacing UL/DL
communications with sidelinks (SL). Considering again Tab. I, in analogy to
(16), we now quantify necessary requirements on SL efficiency
$\mathrm{EE_{S}}$. The parameter region

$\mathcal{R}_{\mathrm{SU}}\triangleq\left\\{\text{$\mathrm{EE}_{\mathrm{S}}$,$\mathrm{EE}_{\mathrm{U}}$}:\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}$
(18)

collects the operational points
$\left\\{\mathrm{EE}_{\mathrm{S}},\mathrm{EE}_{\mathrm{U}}\right\\}\in\mathcal{R}_{\mathrm{SU}}$
that make the CFA policy more efficient than CL in terms of communication
costs. Condition
$\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
in (18) can be written as

$\frac{\mathrm{EE_{S}}}{\mathrm{EE}_{\mathrm{U}}}\cdot\frac{\alpha}{n\cdot
K_{a}}>N\cdot\frac{b(\mathbf{W})}{b(\mathcal{E})}.$ (19)

Similarly as (17), the bound (19) gives a necessary condition on SL
efficiency. On the other hand, as verified in Sect. IV, it can be effectively
used for practical assessment when SL communication is the major source of
carbon emissions. As discussed in the Appendix, the condition
$\frac{\mathrm{EE_{S}}}{\mathrm{EE_{U}}}+\gamma\frac{K}{K_{a}}\cdot\frac{\mathrm{EE_{S}}}{\mathrm{EE_{D}}}>N$
guarantees that, for each FL round, the CFA policy leaves a smaller carbon
footprint than FA, or FA-D (where $K$ is replaced with $K_{a}$).

Requirements on data and model size. In analogy to regions
$\mathcal{R}_{\mathrm{DU}}$ and $\mathcal{R}_{\mathrm{SU}}$ that set the
operational points for UL/DL and SL efficiency, the parameter region

$\mathcal{R}_{b(\mathbf{W})}\triangleq\left\\{b(\mathbf{W}),b(\mathcal{E}_{k}):\max[\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}]<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}\right\\}.$
(20)

identify the necessary requirements on model $b(\mathbf{W})$ and data
$b(\mathcal{E}_{k})$ size, now considering all the proposed FL policies. Using
(17) and (19), the condition
$\max[\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}]<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
in $\mathcal{R}_{b(\mathbf{W})}$ can be written and simplified as

$\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}<\frac{\alpha}{n}\cdot\frac{K}{\Gamma\cdot
K_{a}}.$ (21)

by assuming
$\frac{\mathrm{EE_{S}}}{\mathrm{EE}_{\mathrm{U}}},\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}\gg
1$ [46] and $\forall k,\,b(\mathcal{E}_{k})=b(\mathcal{E})/K$ as clarified in
the Appendix. In particular, taking into consideration FA and FA-D policies
only, it is $\Gamma=1$, while $\Gamma=N$ when including CFA. Notice that, for
an assigned model $b(\mathbf{W})$ and dataset $b(\mathcal{E}_{k})$ size, a
critical requirement for (21) to hold is minimizing the active population size
($K_{a}<K$) and keeping the number of learning rounds ($n$) to a minimum,
while satisfying a target accuracy. Furthermore, frequent data updates, as
often observed in continual learning ($\alpha>1$), might favor FL.

Requirements on computing efficiency. Considering now the training costs, the
parameter region

$\mathcal{R}_{\mathrm{CI}}\triangleq\left\\{\text{$\mathrm{CI}_{k}$},\forall
k:\frac{\mathrm{C}_{\mathrm{FA}}^{(\mathrm{L})}}{n_{\mathrm{FA}}}<\frac{\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}}{n_{\mathrm{CL}}}\right\\}$
(22)

sets the requirements on carbon intensities such that, for
$\text{$\mathrm{CI}_{k}$}\in\mathcal{R}_{\mathrm{CI}}$, the FA policy emits
lower carbon than CL with respect to the per-round computing costs. After
straightforward manipulation of $\mathrm{C}_{\mathrm{FA}}^{(\mathrm{L})}$ and
$\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}$ from Tab. I, the bound
$\frac{\mathrm{C}_{\mathrm{FA}}^{(\mathrm{L})}}{n_{\mathrm{FA}}}<\frac{\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}}{n_{\mathrm{CL}}}$
in (22) can be written as

$\sum_{k=1}^{K}\frac{\varphi_{k}}{1-\beta}\mathrm{CI}_{k}<\gamma\mathrm{CI}_{0}.$
(23)

Extending requirement (23) to other FL policies, namely FA-D,
$\frac{\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{L})}}{n_{\mathrm{CFA}}}<\frac{\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}}{n_{\mathrm{CL}}}$,
and CFA,
$\frac{\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{L})}}{n_{\mathrm{CFA}}}<\frac{\mathrm{C}_{\mathrm{CL}}^{(\mathrm{L})}}{n_{\mathrm{CL}}}$,
is straightforward. The corresponding requirements are detailed in the Table
II.

##  IV Carbon footprints: examples in cellular networks

In this section we verify the sustainability conditions (17)-(23) and the
regions in Table II for typical supervised image classification problems
($\alpha=1$), using the popular MNIST [22] and CIFAR10 [23] datasets. In line
with 4G/5G NB-IoT cellular network implementations [26], model parameters and
data exchange are here implemented over a WWAN characterized by UL
$\mathrm{EE_{U}}=10$ kbit/J and DL efficiency $\mathrm{EE_{D}}=50$ kbit/J.
Other setups are considered in Sect. V. Three populations of devices are
analyzed, ranging from $K=100$, of which $K_{a}=50$ are active learners,
$K=60$ (with $K_{a}=40$) and $K=30$ ($K_{a}=20$). For these initial tests,
distributed learning has been simulated on a framework that allows to deploy
virtual devices implemented as independent threads running on the same
machine. Each thread acts as learner and process a fraction of an assigned
data set while communicating with the other threads through a resource sharing
system [16]. The simulations allow to compute the number of rounds $n$ that
are necessary to achieve a target accuracy or loss $\overline{\xi}$: energy
and carbon footprints are then obtained for each setup by evaluating
communication and computing costs in Tab. I. For CL consumption, the data
center machine is equipped with a RTX 3090 GPU hardware (with Thermal Design
Power equal to $350$W) having PUE $\gamma=1.67$. We consider the carbon
intensity $\mathrm{CI}_{k}$ figures reported in EU back in 2019 [37]: in
particular, $\mathrm{CI}_{k}=0.9$ kgCO2-eq/kWh, according to [43]
(mlco2.github.io/impact/).

The following analysis follows a what-if approach to quantify the estimated
carbon emissions under different parameter choices. Actual emissions may be
larger than the estimated ones depending on the specific use case and network
implementation: relative comparisons are however meaningful for green design
assessment. Code examples are available online333Federated Learning code
repository: https://github.com/labRadioVision/federated. Accessed: March
2022..

TABLE III:  FL rounds for varying target losses $\overline{\xi}$, and IID vs
non-IID data distributions with $\mathrm{EE}_{\mathrm{U}}=10$ kbit/J,
$\mathrm{EE}_{\mathrm{D}}=50$ kbit/J. Examples with the MNIST and the CIFAR10
datasets are shown as well. TABLE IV:  Communication/computing energy costs
and corresponding carbon footprints increase/decrease w.r.t. CL for the MNIST
dataset.

###  IV-A Impact of conditions (17)-(23) on MNIST and CIFAR

Handwritten image recognition from the MNIST dataset uses the LeNet-1 model
proposed in [22]. Each device obtains $1000$ MNIST training gray scale images
with average (per device) data footprint $b(\mathcal{E}_{k})=6.2$ Mbit,
$\forall k$. Model size with no compression is $b(\mathbf{W})=180$ kbit, so
that $\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}\simeq 0.03$. Considering the
CIFAR set, devices obtain $1000$ CIFAR10 color images,
$b(\mathcal{E}_{k})=24.5$ Mbit, and use a VGG-1 model with $b(\mathbf{W})=2$
Mbit, so that $\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}\simeq 0.08$. In the
following tests, both IID and non-IID data distributions are considered: in
particular for non-IID data, batches now contain examples for $6$ of the $10$
target classes (MNIST or CIFAR), randomly chosen.

MNIST carbon footprints. Tab. III (left column) quantifies the minimum and
maximum number of rounds ($n$) required for FA, FA-D and CFA, as targeting
different validation loss values $\overline{\xi}$. Communication, computing
energy costs, and carbon footprints are instead reported in Tab. IV. The
carbon footprint is quantified by decrease (negative terms) or increase
(positive terms) with respect to the emissions observed for CL, used here as
reference. The carbon emissions of CL are found as $17.6$ gCO2-eq for $K=100$
devices, $14.1$ gCO2-eq for $K=60$, and $5.6$ gCO2-eq for $K=30$. The
parameter region $\mathcal{R}_{b(\mathbf{W})}$ and the upper bound (21) can be
used to quantify the maximum number of rounds below which distributed learning
tools could be considered as sustainable: for
$\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}\simeq 0.03$, and after
straightforward manipulation, it is $n<50$ for $K=30$ and $K=60$, and $n<66$
for $K=100$. FA and FA-D methods both satisfy the previous condition as they
require $n=13\sim 29$ rounds for $\overline{\xi}=0.1$ and fewer rounds
$n=11\sim 26$ for $\overline{\xi}=0.2$. The region $\mathcal{R}_{\mathrm{DU}}$
and condition (17) for $n=29$ rounds, $K=30$, and $K_{a}=20$ indicates
$\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}>2.8$ for FA and
$\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}>1.9$ for FA-D that are both
satisfied as $\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}=5$. CFA leaves
a smaller carbon footprint than CL for IID data distributions. However, when
data is non-IID, it requires up to $n=70$ rounds with $K=100$ devices that
exceeds the bound (21) and thus cause a larger footprint. Notice that, for
CFA, the observed rounds might differ substantially over varying simulation
runs, as the result of the propagation of model parameters through the
network.

CIFAR carbon footprints. Tab. III (right column) and Tab. V collect the
required rounds, the energy and carbon footprints for the CIFAR set,
respectively. Classification with CIFAR requires much more rounds for
convergence than MNIST and a larger model footprint $b(\mathbf{W})$: according
to the region $\mathcal{R}_{b(\mathbf{W})}$ and the bound (21), FL is less
attractive with respect to centralized designs. For CL, the number of rounds
to achieve $\overline{\xi}=0.6$ (accuracy of $77$%) is in the range
$n_{\mathrm{CL}}=30\sim 40$, while the carbon footprint is found to be $267.4$
gCO2-eq for $K=100$ devices, $161.5$ gCO2-eq for $K=60$, and $81.4$ gCO2-eq
for $K=30$, respectively. Tab. V shows that all FL paradigms leave larger
footprints than CL since the required number of rounds exceeds the bound (21),
that prescribes a maximum of $25$ rounds. A sustainable solution could be
achieved only in exchange for a larger loss: in other words, increasing the
target loss to $\overline{\xi}=1.2$ reduces the required number of FL rounds,
but penalizes the accuracy that scales down to $67\%$. This last setup might
give concrete chances to CFA, when data is IID distributed, considering the
lower communication cost per round as shown in the requirement (19) or region
$\mathcal{R}_{\mathrm{SU}}$. Finally, for $\mathrm{CI}_{k}=0.9$,
$\varphi_{k}=0.14$ and $\beta=0.05$, condition (23) on training costs is not
satisfied: therefore, FL emits much more carbon than CL on each learning round
as $\text{$\mathrm{CI}_{k}$}\notin\mathcal{R}_{\mathrm{CI}}$.

TABLE V:  Communication/computing energy costs and corresponding carbon
footprints increase/decrease w.r.t. CL for the CIFAR10 dataset.

###  IV-B MNIST and CIFAR: comparative analysis

By comparing the MNIST and the CIFAR sets, some common properties emerge:
these are conveniently summarized in Fig. 3, focusing on FA-D and CFA methods.
Each point in the scatter plot corresponds to a simulation run that uses FA-D
or CFA over the MNIST or the CIFAR sets. For each combination, represented by
different markers, we run several simulations by varying the data
distribution, from IID to non-IID444Non-IID data footprint
$b(\mathcal{E}_{k})$ might vary in each simulation run, depending on the
training sample distribution., and we report the required number of rounds,
for accuracy $\overline{\xi}$, as well as the corresponding model/data
footprint ratio $\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}$. The bounds (21)
and (17) are superimposed by black and blue dashed lines, respectively: the
area above each line represents unsustainable designs, namely
$\left\\{b(\mathbf{W}),b(\mathcal{E}_{k})\right\\}\notin\mathcal{R}_{b(\mathbf{W})}$
and
$\left\\{\mathrm{EE}_{\mathrm{D}},\mathrm{EE}_{\mathrm{U}}\right\\}\notin\mathcal{R}_{\mathrm{DU}}$,
for which CL is the preferred choice, with respect to carbon emissions. In
most cases, the simulations confirm the predicted trends: green markers
correspond to sustainable solutions for which FA-D, or CFA, leave a lower
carbon footprint than CL, red markers refer to the opposite case. Small model
footprints $b(\mathbf{W})$ compared with data $b(\mathcal{E}_{k})$ constitute
a critical prerequisite for FL sustainability, while learning rounds ($n$)
must be minimized as much as possible. Notice that the required rounds
increase when the data is non-IID. CFA is generally more sensitive to non-IID
distributions than FA-D; on the other hand, in many cases, CFA gives better
footprints in IID situations. Regions $\mathcal{R}_{\mathrm{DU}}$,
$\mathcal{R}_{b(\mathbf{W})}$ and corresponding requirements (17)-(21) give
more effective indicators of sustainability than region
$\mathcal{R}_{\mathrm{CI}}$ and bound (23) on training costs. This is due to
the fact that, in these examples, communication emits much more carbon than
computing.

Figure 3:  Validation of the bounds (21) in black, and of (17) in blue, with
the CIFAR and the MNIST datasets for the CFA and FA-D methods. Markers in the
scatter plot report FA-D or CFA simulation runs for the MNIST or CIFAR
datasets (IID or non-IID), and give the required rounds for the target loss
$\overline{\xi}$ and the corresponding model/data size
$\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}$. Green markers are sustainable
solutions, namely $\mathrm{C}_{\mathrm{FA-D}}<\mathrm{C}_{\mathrm{CL}}$ or
$\mathrm{C}_{\mathrm{CFA}}<\mathrm{C}_{\mathrm{CL}}$, while red markers refers
to the opposite cases.  Figure 4:  Description of case studies. From left to
right: scenario descriptions, hardware specifications for each industrial
setup and corresponding ML systems.

##  V Case studies in industrial IoT

In what follows, we highlight two critical 5G verticals555These case studies
have been taken from 5G Alliance for Connected Industries and Automation:
https://5g-acia.org/ for Industry 4.0 scenarios, further described in Fig. 4.
The interested reader might refer to [48] for sustainability requirements in
industry processes and transition towards the new Green Deal [49].

The first use case, shown in Fig. 4(a), quantifies the carbon footprint of a
continual supervised learning process targeting passive localization of human
operators in a shared workplace (human-robot co-presence monitoring).
Localization is particularly critical to support various human-robot
interactions processes [50] in advanced manufacturing, where robotic
manipulators continuously share the same space with humans. A deep ML model is
adopted for localization and it is trained continuously to track the
variations of data dynamics caused by changes of the workflow processes
(typically, on a daily base). In the second setup, described in Fig. 4(b),
$K=5$ crawling robots interact with the workplace to learn an optimal sequence
of movements and implement a desired trajectory. Robots train a ML model, now
via DQL tools [30] as introduced in Sect. II-D. Both case studies focus on
industrial setups where AI-based machines/robots are co-located in the same
workplace, so that direct communication is possible.

For all case studies, it is assumed that devices have sufficient computing
power to implement FL and LO, but they run medium/small-sized deep learning
models to allow for real-time operations, i.e., localization or motion
control. Hardware and ML systems are described in Fig. 4. The devices, the PS
(when used), and the data centers are located in the west EU regions
previously defined. In particular, the data center, and the PS are located
outside the workplace so that communication is possible only through WWAN
connectivity. They are equipped with the same high performance GPU already
shown in Sect. IV and have the same PUE namely $\gamma=1.67$. However, a
realistic pool of FL learners is considered: manipulator devices are equipped
with dedicated hardware modules (Nvidia Jetson Nano) equipped with a low-power
GPU (128-core Maxwell architecture) and an ARM Cortex-A57 System-on-Chip
(SoC). Crawling robots mount a low-power ARM Cortex-A72 SoC and thus
experience larger batch time $T_{k}$ but lower power $P_{k}$ when implementing
DQL.

As described in the following, the proposed real-time FL platform adopts MQTT
methods for model parameter exchange and several communication profiles. The
carbon footprints for both case studies are analyzed separately, while
sustainable setups are identified based on the requirements in Table II.

TABLE VI:  Summary of main FL parameters for use cases. | Case (a): Continual Learning | Case (b): Reinforcement Learning
---|---|---
$b\left[\mathcal{E}_{k}(t_{i})\right]$: |  $31$ MB ($t_{0}$), $19$ MB ($t_{i}$), $\forall k,i$ |  $24.6$ MB, $\forall k,i$
$b(\mathbf{W})$: |  $1.08$ MB |  $5.6$ MB
$\alpha$: |  $1$ (1 updates per training, $\forall t_{i}$) |  $n$ ($1$ update per round)
$K,K_{a}:$ | $K=9,K_{a}:1\sim 9$ | $K=5,K_{a}=5$
Consumption | Data center/PS ($k=0$) | Devices ($k\geq 1$) | Data center/PS ($k=0$) | Devices ($k\geq 1$)
$P_{k}$: |  $590\,\mathrm{W}$$\,\,(350\,\mathrm{W}$ GPU) |  $15\,\mathrm{W}$$(\varphi_{k}=0.36)$ |  $590\,\mathrm{W}$$\,\,(350\,\mathrm{W}$ GPU) |  $5.1\,\mathrm{W}$ $(\varphi_{k}=0.17)$
$T_{k}$: |  $10\textrm{ms}$ ($B=3$) |  $140\textrm{ms}$ ($B=3$) |  $20\textrm{ms}$ ($B=3$) |  $400\textrm{ms}$ ($B=3$)
$\beta$: | $0.05$ | n.a. | $0.05$ | n.a.
$E_{k,\mathrm{sleep}}^{(\mathrm{C})}:$ | n.a. |  $0.05$ Wh | n.a. |  $0.05$ Wh
$\mathrm{EE_{C}:}$ |  $0.05$ round/J | $0.14$ |  $\mathrm{EE_{C}}=0.03$ round/J | $0.16$
Peripherals | n.a. |  $3$ Wh (radars) | n.a. |  $6.6$ Wh (servos)

###  V-A Networking and MQTT transport

The networking platform used for validation is characterized by $K$ physical
IoT devices and implements distributed model training in real-time on top of
the MQTT protocol, chosen for low-latency and low-overhead characteristics.
For all FL implementations, the exchange of model parameters is performed
through MQTT-compliant _publish_ and _subscribe_ operations that are
implemented with QoS level 1 [53]. The information included in the MQTT
payload are: _i)_ the local model parameters binary encoded with associated
meta-information (i.e., the DNN layer type); _ii)_ the FL round; _iii)_ the
local loss function used as model quality indicator. The MQTT platform is used
to collect measurements of the required training time: the expected carbon
footprints are then quantified for varying communication energy efficiencies
($\mathrm{EE_{D}},$ $\mathrm{EE_{U}},$ $\mathrm{EE_{S}}$). The code structure
is available in [51]. CL is similarly implemented on top of the same MQTT
architecture: in this case, the $K$ devices publish their data to the MQTT
broker that is co-located with the data center.

Considering FA and FA-D methods, the PS serves as MQTT broker until the end of
the training process: on each FL round, the PS thus accepts subscriptions from
the $K_{a}$ active devices that publish their model parameters. For FA method,
all the $K$ devices subscribe to the PS broker service, i.e., to download the
updated global model. On the other hand, for FA-D, inactive devices need to
start a new subscription process every time they wake up from deep-sleep. The
$K_{a}$ learners are chosen by a round robin scheduling table [11] that is
distributed by the PS at training start, via a dedicated MQTT topic. The
broker service might cause an increase of the consumption per round, or a
decrease of the computing efficiency $\mathrm{EE_{C}}$. Similarly,
subscriptions and publishing operations reduce the communication efficiency,
$\mathrm{EE_{D}},$ $\mathrm{EE_{U}}$ depending on the payload-to-overhead
ratio of the MQTT messaging [53]. To implement CFA, the MQTT broker is now
used to orchestrate the consensus operations, while the PS functions are
disabled. Every round, one device is turned on and publishes the local model
parameters to a subset of $K_{a}-1$ subscriber devices. Publisher and
subscribers are again assigned by round robin scheduling. Notice that the MQTT
broker consumption is not considered in this study.

###  V-B Communication efficiencies

The estimated carbon emissions are quantified by considering $4$ communication
efficiency profiles [26] described in the following. The first profile
conforms to a LTE design for macro-cell delivery: we set $\mathrm{EE_{U}}=15$
kbit/J and $\mathrm{EE_{D}}=25$ kbit/J as also verified in [40] for a
throughput of $28$ Mbps and a BS consumption of $1.35$ kW. Notice that larger
efficiency values of $6$ J/Mbit and $10$ J/Mbit for UL and DL, respectively,
could be observed with small bulk size of $10$ kB [46]. The second profile
corresponds to a NB-IoT implementation. In this case, we set a larger DL
$\mathrm{EE_{D}}=50$ kbit/J and UL efficiency of $\mathrm{EE_{U}}=25$
kbit/J666In [47], it is shown that sending $20$ byte using $0.9$ Joule is
possible: efficiency reduces with larger packet sizes. . NB-IoT devices also
consume about $10$ µW (i.e., $E_{k,\mathrm{sleep}}^{(\mathrm{C})}\simeq 36$
mWh) in deep-sleep mode [47]. The third profile complies with IETF 6TiSCH mesh
standard based on IEEE 802.15.4e [52] with a SL efficiency of
$\mathrm{EE_{S}}=20$ kbit/J. For the last profile, we adopt a WiFi IEEE
802.11ac implementation, namely the Intel AC 8265 device supporting sidelink
connectivity based on Neighbor Awareness Network [54]. Typical communication
efficiencies are $2100$ mW/Mbps for receiving and $2500$ mW/Mbps for
transmitting [29]; however, they do not include equipment consumption, nor
WiFi and MQTT overheads: the assumed SL efficiency is thus
$\mathrm{EE_{S}}=100$ kbit/J. Notice that sidelink efficiency is expected to
scale up to at least $10$ times in 6G implementations [26].

###  V-C Human-robot interaction: continual learning

The goal of the training task is to learn a ML model for the detection (i.e.,
classification) of the position of the human operators sharing the workspace,
namely the human-robot distance $d$ and the direction of arrival (DOA)
$\theta$. In particular, we address the detection of a human subject in $C=10$
Region Of Interest (ROI), including the one referring to the subject outside
the monitored area: ROIs are detailed in Fig. 4(a). The proposed training
scenario resorts to a network of $K=9$ physical devices where each one is
equipped with a Time-Division Multiple-Input-Multiple Output (TD-MIMO)
Frequency Modulated Continuous Wave (FMCW) radar working in the $77-81$ GHz
band. For details about the robotic manipulators, the industrial environment
and the radars, the interested reader may also refer to [50]. Radars use a
trained deep learning model to obtain position ($d$, $\theta$) information and
the corresponding ROI. In addition, the subject position can be sent to a
programmable logic controller for robot safety control, for emergency stop or
replanning tasks. The ML model adopted for the classification of the operator
location is the LeNet-4 scheme, proposed in [22] with input adaptations, that
consists of $4$ trainable layers and $28$K parameters. Model footprint is
$b(\mathbf{W})=1.08$ MB, the other parameters are detailed in Tab. VI.

The FL system implements a continual learning task (Sect. II-D): at time
$t_{0}$, each device collects a large dataset $\mathbf{x}_{h}$ of raw range-
azimuth data manually labelled, with size $b[\mathcal{E}_{k}(t_{0})]=31$ MB.
This set is used for initial training of the ML model. Re-training stages,
$t_{i}$, $i>0$, are based on new data, $b[\mathcal{E}_{k}(t_{i})]=19$ MB,
collected on a daily basis. Datasets for initial model training and $2$
subsequent example re-training stages, $t_{1},t_{2}$, are available online in
[51]. To simplify the analysis, in what follows, we compared the results of
the proposed FA-D and CFA implementations, while FA is not considered. For
FA-D, the number of active devices ranges from $K_{a}=1$ to $K_{a}=9$; for
CFA, we have assumed $K_{a}=3$ ($1$ publisher, $2$ subscribers per round) up
to $K_{a}=9$ ($1$ publisher, $8$ subscribers per round).

Figure 5:  Continual learning case study. From left to right: (a) carbon
footprints $\mathrm{C}^{(t_{0})}(n_{0})$ and corresponding number of FL rounds
($n_{0}$) during initial training stage $t_{0}$, for varying population of
active learners $K_{a}=1\sim 9$ and $K=9$ radars; (b) carbon footprints
$\mathrm{C}^{(t_{i})}(n_{i})$ and corresponding rounds $(n_{i})$ averaged over
$2$ subsequent re-training stages ($i=1,2$). For all cases, footprints are
evaluated for CL (dashed line), CFA (cross markers), and FA-D (diamond
markers) as well as $4$ communication efficiency profiles that comply with
LTE, NB-IoT, IEEE 802.15.4e and WiFi IEEE 802.11.ac implementations. Rounds
$(n_{i})$ and learners $(K_{a})$ are compared with bound (21) obtained using
the parameters of Tab. VI.

Selection of the device population ($K_{a}$). The example in Fig. 5 uses the
requirements in Table II as a guideline for the selection of the smallest
population size $K_{a}$, so that FL emits lower carbon than CL during initial
($i=0$) and re-training ($i>0$) stages. The figure highlights the carbon
footprints and the number of rounds for both stages. In particular, Fig. 5(a)
highlights the carbon footprints $\mathrm{C}^{(t_{0})}$ and the number of
learning rounds $n_{0}$ that are measured during the initial training phase
($t_{0}$) for varying number of active learners $K_{a}$. Fig. 5(b) shows the
same results, namely $\mathrm{C}^{(t_{i})}$ and $n_{i}$, averaged over the
subsequent re-training stages, $i=1,2$. The target loss is here
$\overline{\xi}=0.1$ for all cases.

Considering the initial training first, FA-D and CFA implemented over small
population sets, i.e., $K_{a}\leq 3$, need a large number of rounds ($n>150$)
and training time. Bound (21) for region $\mathcal{R}_{b(\mathbf{W})}$ gives
some practical guidelines for optimizing $K_{a}$ population. For
$\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}\simeq 0.035$, since
$b[\mathcal{E}_{k}(t_{0})]=31$ MB, and the other parameters in Tab. VI, the
requirement (21) becomes $K_{a}<\tfrac{257}{n}$. Considering CFA, the minimum
population size that satisfies (21) is $K_{a}=6$ since $n_{\mathrm{CFA}}=40$
rounds. Larger populations, i.e., $K_{a}\geq 8$, increase the cost per round,
with no significant savings in terms of training time. $K_{a}=4$ can be chosen
for FA-D as $n_{\mathrm{FA-D}}=40$ rounds. Using (17) for $K_{a}=4$, the
required DL efficiency should comply with
$\mathrm{\tfrac{EE_{D}}{EE_{U}}>6.2}$: on the other hand, since condition on
computing cost (23) is satisfied for $\mathrm{CI}_{k}=0.97$ and
$\varphi_{k}=0.36$, as $K_{a}\cdot\tfrac{0.36}{0.95}<1.67$, a lower DL
efficiency can be tolerated.

Focusing now on model re-training stage, FA-D and CFA leave a smaller
footprint than CL for all choices of $K_{a}$ as they need few rounds ($<20$)
for model update. With $\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}\simeq 0.057$,
since now $b[\mathcal{E}_{k}(t_{i})]=19$ MB, the requirement (21) for FA-D
becomes $K_{a}<\tfrac{158}{n}$ and it is satisfied for $K_{a}=1$ as
$n_{\mathrm{FA-D}}<12$. For CFA, it is $K_{a}=3$ since $n_{\mathrm{CFA}}<20$.
For example, FA-D with only $K_{a}=1$ learner per round emits $1.1$ gCO2-eq
per re-training stage (over LTE networks), which corresponds to a carbon
emission of $0.4$ kgCO2-eq per year, assuming $1$ re-training/day. For the
same setup, CL would cost $5.4$ gCO2-eq, corresponding to roughly to $2$
kgCO2-eq per year.

To sum up, the results suggest that both FA-D and CFA are sustainable choices
for re-training, as they avoid unnecessary data uploads. On the other hand,
for initial training, CL and FL are both competitive and should be considered
carefully based on the available communication interface and sustainability
conditions (17)-(23). Notice that the availability of low-power sidelink
communications makes CFA the preferred choice.

###  V-D Reinforcement learning for robot motion planning

According to the scenario of Fig. 4(b) and the parameters of Tab. VI, the
considered RL setting features $K=5$ networked robots that collaboratively
learn an optimized sequence of motions, to follow an assigned trajectory,
highlighted in green. Since the goal is to quantify the carbon footprint of
the distributed learning processes against CL in a realistic setup, the motion
control problem is simplified by letting the robots move on a 2D regular grid
space consisting of $40$ landmark points, while the action space consists of
$4$ motions: Forward (F), Backward (B), Left (L), and Right (R). Each robot
explores a different site area collecting new training data in real-time: data
are obtained from two cameras, namely a standard RGB camera and a short-range
Time Of Flight (TOF) one [55]. Exploration of the environment and training
data collection is responsible for additional energy consumption (quantified
here as $6.6$ Wh) that depends on robot/servos hardware.

RL and DeepMind model [21] are used to train a policy that maps observations
of the workplace to a set of actions, namely robot motions, while trying to
maximize a long-term reward777Devices get a larger reward whenever they
approach the desired trajectory: code, data and position-reward lookup table
are described in the repository: https://github.com/labRadioVision/Federated-
DQL. Accessed: Mar. 2022.. Exploration and model exploitation follow the
$\varepsilon$-greedy method while, in this example, exploration takes $20$
robot motions for each learning round. DeepMind model consists of $5$
trainable layers and $1.3$M parameters: model footprint from MQTT payload is
$b(\mathbf{W})=5.6$ MB. Notice that for CL the training data is moved on every
round, therefore $\alpha=n$. CFA and FA are considered here with the setup
highlighted in Sect. II-D, with $K_{a}=K$. Data footprint is
$b(\mathcal{E}_{k})=24.6$ MB, therefore
$\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}\simeq 0.23$: with such setup,
condition (21) for RL is satisfied and requirement for region
$\mathcal{R}_{b(\mathbf{W})}$ becomes
$\frac{b(\mathbf{W})}{b(\mathcal{E}_{k})}<\frac{K}{K_{a}}$.

Figure 6:  Reinforcement Learning for robot navigation: estimated CO2
emissions for each policy optimization vs the running reward. Each reward
value corresponds to an average trajectory error compared with the desired
one, measured by root mean squared error (RMSE). Specific parameters are in
Tab. VI. CL is in black, FA in red, CFA in green and ego learning in blue.
Each point of the scatter plot represents one simulation episode.

Impact of target reward/accuracy. Fig. 6 shows the estimated carbon emissions
versus the obtained reward considering CL (black), FA (red), CFA (green) and
ego learning (blue) policies. For the opportunistic/ego learning approach [2],
the robots disable the radio interface and train their local models using the
training data from on-board sensors only. Each point in the scatter plot of
Fig. 6 corresponds to the carbon footprint measured in one simulation episode.
On each new episode, the observed footprint might vary due to the random
exploration phase. In ego learning, the robots explore the environment for a
longer time, thus training takes around $25$ h corresponding to $4500\sim
5000$ rounds since servos need $1$ s per motion. For comparison, CL takes $90$
minutes on average ($250\sim 300$ rounds), while FA needs $180$ minutes
($500\sim 550$ rounds) and CFA $150$ minutes ($400\div 450$ rounds).

RL generally requires more rounds than continual learning to converge, and
cause large emissions when considering the whole training process. Ego
learning gives the lowest footprint as consumption is only due to the robot
and the LO. However, it experiences lower rewards, from $-10$ to $30$ as the
learned trajectory is far from the desired one, with corresponding Root Mean
Squared Errors888Positions far from the desired trajectory have small rewards:
the smaller the reward the larger the RMSE. (RMSE) between $1.5$ m and $2$ m.
CL converges faster at the cost of a large footprint, about $2$ times larger
than FA/CFA and $6$ times larger than ego learning. However, rewards are close
to $70$, that corresponds to learned trajectories with RMSE $0.5$ m. Compared
with CL, FA and CFA performance are now close since all robots are active;
energy savings are obtained at the cost of rewards penalties, scaling down to
$40\sim 60$, with RMSE between $0.5$ m and $1$ m. Considering ego and CL as
extreme cases, where the former gives low-accuracy and small footprint, while
the latter high-accuracy and large footprint, FL is a promising tradeoff
solution, as trading accuracy (reward) with environmental footprint.

##  VI Conclusions

The article proposed a novel framework for the analysis of energy and carbon
footprints in distributed and federated learning. A tradeoff analysis was
characterized for vanilla FL and decentralized, consensus-driven learning,
compared with centralized training. The analyzed algorithms are suitable for
low-power embedded wireless devices with constrained memory, typically adopted
in industrial IoT processes requiring low latency. The paper also explored
operational conditions applicable to both conventional FL methods and
generalized distributed learning policies targeting carbon-efficiency. Minimal
requirements for FL are obtained analytically and have been applied in
different scenarios to steer the choice towards sustainable designs.

Carbon equivalent emissions have been analyzed for two industry relevant 5G
verticals, using real datasets and a test-bed characterized by physical IoT
devices implementing real-time model training on top of the MQTT transport,
tailored to support various learning processes. For each considered case,
centralized versus distributed training impact is discussed for continual and
reinforcement learning problems. Sustainability of the FL depends in many
cases on model vs data footprints, as well as the population of active
devices, that must be selected to balance energy budget and training data
quality. Downlink vs uplink (or sidelink) communication efficiency and the
amount of available green energy are additional key factors. The proposed CFA
and FA with deep sleep mode (FA-D) methods have been shown to be effective as
they minimize the number of active learners at each learning round. The
selection of the learner population size according to the proposed design
patterns provides significant savings per round ($20\%\sim 30\%$ and higher)
compared with baselines. CFA has an advantage over FA-D provided that an
efficient sidelink communication interface is available: the co-design of
learning and communication is thus of high importance. Finally, the energy
footprints of reinforcement learning need much more careful considerations
than conventional supervised learning, due to the considerable carbon
emissions. Results indicate that FL is a promising solution as it trades off
energy and accuracy compared with the extreme cases of ego (low
rewards/footprint) and centralized training (high rewards/footprint).

## Appendix

### Communication costs: FA and FA-D design constraints

Using the carbon footprints shown in Tab. I, the constraint
$\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
in region $\mathcal{R}_{\mathrm{DU}}$ (16) implies that, after straightforward
manipulation, the following condition has to be met

$\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}\cdot}\left[\mathcal{\mathcal{H}}\left(\frac{b(\mathbf{W})}{b(\mathcal{E})}\right)-\frac{1}{K_{a}}\sum_{k=1}^{K_{a}}\mathrm{CI}_{k}\right]>\gamma\cdot\frac{K}{K_{a}}\mathrm{CI}_{0}$
(24)

with

$\mathcal{\mathcal{H}}\left(\frac{b(\mathbf{W})}{b(\mathcal{E})}\right)=\frac{\alpha}{n\cdot
K_{a}}\frac{b(\mathcal{E})}{b(\mathbf{W})}\sum_{k=1}^{K}\sigma_{k}\mathrm{CI}_{k}$
(25)

where $\sigma_{k}$, defined in (5) with $\sum_{k=1}^{K}\sigma_{k}=1,$ counts
the number of the local ($k$) examples out of the number of the global ones,
respectively. Eq. (17) is then obtained by setting
$\mathrm{CI}_{k}=\mathrm{CI}$, $\forall k$. Considering FA-D, the equation
(24) becomes
$\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}\cdot}\mathcal{\mathcal{H}}\left(\frac{b(\mathbf{W})}{b(\mathcal{E})}\right)>\gamma\mathrm{CI}_{0}$.

For $\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}$ large enough as assumed
for derivation of region $\mathcal{R}_{b(\mathbf{W})}$ in problem (20), the
condition
$\max[\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}]<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
is guaranteed as long as

$\mathcal{\mathcal{H}}\left(\frac{b(\mathbf{W})}{b(\mathcal{E})}\right)>\frac{1}{K_{a}}\sum_{k=1}^{K_{a}}\mathrm{CI}_{k},$
(26)

that corresponds to

$\frac{b(\mathbf{W})}{b(\mathcal{E})}\cdot
K<\frac{\alpha}{n}\cdot\frac{\sum_{k=1}^{K}\sigma_{k}\mathrm{CI}_{k}}{\sum_{k=1}^{K_{a}}\mathrm{CI}_{k}}.$
(27)

Constraint (27) can be further simplified by letting, $\forall k,$
$\sigma_{k}\simeq 1/K,$ namely
$b(\mathcal{E}_{k})\simeq\frac{b(\mathcal{E})}{K}$, and
$\mathrm{CI}_{k}=\mathrm{CI}$, to obtain (21) with $\Gamma=1$.

### Communication costs: CFA design constraints

Using Tab. I and (25), the constraint
$\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})}<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
for region (18) results in

$\frac{\mathrm{EE_{S}}}{\mathrm{EE}_{\mathrm{U}}}\cdot\mathcal{\mathcal{H}}\left(\frac{b(\mathbf{W})}{b(\mathcal{E})}\right)>\frac{N}{K_{a}}\sum_{k=1}^{K_{a}}\mathrm{CI}_{k}$
(28)

that simplifies to (19) when, $\forall k$, it is
$b(\mathcal{E}_{k})\simeq\frac{b(\mathcal{E})}{K}$, and
$\mathrm{CI}_{k}=\mathrm{CI}$. Furthermore,

$\mathrm{\frac{EE_{S}}{\mathrm{EE_{U}}}}+\gamma\cdot
K\cdot\frac{\mathrm{CI}_{0}}{\sum_{k=1}^{K_{a}}\mathrm{CI}_{k}}\cdot\mathrm{\frac{EE_{S}}{\mathrm{EE_{D}}}}>N$
(29)

guarantees lower carbon footprint _per round_ than FA, namely
$\frac{\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})}}{n_{\mathrm{CFA}}}<\frac{\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})}}{n_{\mathrm{FA}}}.$
The comparison of CFA vs FA-D gives the same result shown in (29) but a more
stringent condition as $K$ must be replaced with $K_{a}$. As expected,
considering an individual round, CFA footprint is closer to FA-D as both
strategies target the minimization of the active device population.

In WWAN, or cellular settings, D2D connectivity is replaced by UL and DL
communications, namely
$[\mathrm{EE}_{\mathrm{S}}]^{-1}=[\mathrm{EE}_{\mathrm{D}}]^{-1}+[\mathrm{EE}_{\mathrm{U}}]^{-1}$.
In this case the constraint (28) becomes

$\frac{\mathrm{EE_{D}}}{\mathrm{EE}_{\mathrm{U}}}\cdot\left[\mathcal{\mathcal{H}}\left(\frac{b(\mathbf{W})}{b(\mathcal{E})}\right)-\frac{N}{K_{a}}\sum_{k=1}^{K_{a}}\mathrm{CI}_{k}\right]>\frac{N}{K_{a}}\cdot\sum_{k=1}^{K_{a}}\mathrm{CI}_{k},$
(30)

while for $\mathrm{\frac{EE_{D}}{\mathrm{EE}_{\mathrm{U}}}}$ large enough,
sustainability is guaranteed as long as

$\mathcal{\mathcal{H}}\left(\frac{b(\mathbf{W})}{b(\mathcal{E})}\right)>\frac{N}{K_{a}}\sum_{k=1}^{K_{a}}\mathrm{CI}_{k}.$
(31)

Comparing (26) with (31), the condition
$\max[\mathrm{C}_{\mathrm{CFA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA}}^{(\mathrm{C})},\mathrm{C}_{\mathrm{FA-D}}^{(\mathrm{C})}]<\mathrm{C}_{\mathrm{CL}}^{(\mathrm{C})}$
for the derivation of region $\mathcal{R}_{b(\mathbf{W})}$ in (20), can be
rewritten as in (31). Finally, replacing $\forall k$
$\mathrm{CI}_{k}=\mathrm{CI}$, we obtain (21) now with $\Gamma=N$.

## References

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