Energy-Efficient and Data-Optimized Federated Learning for Distributed on-Device Intelligence

IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 1
Ene gy-E icien and Da a-Op imized Fede a ed
Lea ning o Dis ibu ed On-De ice In elligence
Ioannis P o oge os, Ma ia Diaman i, Membe , IEEE, Dimi ios Spa ha akis,
and Symeon Papa assiliou, Senio Membe , IEEE
Abs ac —The ad en o A i icial In elligence (AI), equi ing
la ge da a om di e se de ices, necessi a es no el a chi ec u es
ha inco po a e edge de ices in he pipeline o AI ope a ions.
Fede a ed Lea ning (FL) is becoming he s a e-o -p ac ice o
dis ibu ed, collabo a i e aining o AI models a he ne wo k
edge, espec ing da a p i acy and anonymi y conce ns. Howe e ,
implemen ing FL on ba e y-powe ed edge de ices wi h limi ed
compu ing capabili ies necessi a es inno a i e app oaches o
balance aining accu acy and ene gy e iciency. In his wo k,
we p opose a amewo k o ene gy-e icien and da a-op imized
FL o acili a e on-de ice in elligence. The p oposed amewo k
join ly op imizes he (a) selec ion o he mos impo an da a
samples om he de ices’ local aining da ase s o main ain
FL model accu acy, and hei (b) compu ing equency and (c)
uplink ansmission powe o minimize he ene gy consump ion
du ing local compu a ion and communica ion phases. A mul i-
p ocesso model is conside ed o each de ice, ex ending da a
selec ion o alloca e da a samples o each de ice’s p ocesso s. The
ini ially o mula ed non-con ex and combina o ial op imiza ion
p oblem is decomposed in o sub-p oblems, each equi alen ly
ans o med in o a con ex o m. An Al e na ing Op imiza ion
(AO) app oach is hen applied o he sub-p oblems, yielding a
close- o-op imal solu ion o he o iginal p oblem. The p oposed
algo i hm is dis inguished in e ms o scalabili y, achie ing
a leas 30% ene gy educ ion on he MNIST and CIFAR10
da ase s compa ed o he second-bes benchma king schemes,
while main aining FL model accu acy a nea ly he same le el.
O e all, he p oposed amewo k p o es scalable and sui able o
deploymen in ealis ic esou ce-cons ained edge en i onmen s.
Index Te ms—Fede a ed lea ning, on-de ice in elligence, da a
selec ion, ene gy e iciency, ansmission powe con ol, compu -
ing equency scaling, al e na ing op imiza ion.
I. INTRODUCTION
The ad en o cu ing-edge A i icial In elligence (AI) appli-
ca ions and La ge Language Models (LLMs) [1] equi es a as
amoun o aining da a o igina ing om di e se de ices a he
ne wo k edge. In his con ex , Fede a ed Lea ning (FL) [2] has
eme ged as an e ec i e echnique o dis ibu ed and collab-
o a i e aining o AI models om mul iple sou ces o in o -
ma ion. Unlike con en ional cloud-based lea ning app oaches,
FL is ypically conduc ed a he ne wo k’s edge, whe e da a
I. P o oge os, M. Diaman i, D. Spa ha akis, and S. Papa assiliou a e
wi h he Ins i u e o Communica ion and Compu e Sys ems (ICCS),
School o Elec ical and Compu e Enginee ing, Na ional Technical
Uni e si y o A hens, Zog a ou, G eece, 15780, e-mail: [email p o ec ed];
[email p o ec ed]; [email p o ec ed]; pa-
pa [email p o ec ed].
This wo k was pa ly suppo ed by P ojec 6G-LEADER unded by he
Eu opean Union suppo ed by Sma Ne wo ks and Se ices Join Unde aking
(SNS JU) (G an 101192080).
Manusc ip ecei ed 14 June, 2025; e ised 18 Augus , 2025.
is gene a ed. This minimizes communica ion delays caused by
cons an da a exchange while p ese ing p i acy by keeping
sensi i e da a on he de ice [3]. Speci ically, he edge de ices
upda e hei local models using hei local da ase s and ans-
mi he model pa ame e s o a cen al en i y (i.e., edge se e ),
whe e he weigh s a e agg ega ed o gene a e a global model.
This no el a chi ec u e pa es he way o a new gene a ion o
In elligen Cybe -Physical Sys ems (ICPSs) [4], [5], cha ac e -
ized by on-de ice AI. Ne e heless, se e al challenges a ise
conce ning he quali y o FL aining while mee ing se e al
key equi emen s, such as esponse ime and ene gy e iciency.
Implemen ing FL o e wi eless ne wo ks exace ba es hese
challenges by ac o s such as spec um sca ci y and he inhe -
en ba e y limi a ions o edge de ices [6]. These issues hinde
he pe o mance o dis ibu ed lea ning by causing la ency
and inc easing in e e ence du ing he ansmission o local
model pa ame e s. Non-O hogonal Mul iple Access (NOMA)
echniques can enhance spec um e iciency by allowing simul-
aneous ansmissions o e he same ime- equency esou ces,
di e en ia ing de ices by hei powe le els. The e o e, powe
con ol solu ions mus be concu en ly conside ed o ensu e
easible mul i-de ice channel sha ing while ope a ing wi hin
he ime limi s imposed by he FL p ocess [7]. Beyond
e icien communica ion, he p ac ical implemen a ion o FL
necessi a es ensu ing he sus ainabili y o edge de ices om
he compu ing iewpoin , especially when conside ing ba e y-
powe ed edge de ices wi h limi ed compu ing capabili ies [8].
Execu ing compu a ionally-in ensi e AI wo kloads in pa allel
wi h equen communica ion can be highly powe -consuming,
esul ing in ba e y d ainage. Consequen ly, i is essen ial o
me iculously design solu ions o ene gy-e icien AI ope a-
ions ha gua an ee consis en pe o mance and eliable model
aining while join ly minimizing he powe consump ion o
he de ices’ compu ing esou ces and da a ansmission.
While se e al seminal wo ks ha e p oposed ene gy-e icien
solu ions o FL h ough combined adio and compu e esou ce
alloca ion [9], [10], he join op imiza ion o aining da a
emains unde explo ed. Op imizing he size and quali y o he
local aining da a hea ily a ec s o e all FL pe o mance in
accu acy and ene gy consump ion, being closely in e wined
wi h compu ing esou ce alloca ion a he de ices. Toge he
wi h adio esou ce alloca ion, signi ican in e dependencies
a e in oduced among he op imiza ion a iables, esul ing in a
pa icula ly in ica e p oblem. A holis ic amewo k ha spans
all phases o he FL pipeline, op imally handling da a, com-
pu ing, and adio esou ces while balancing model accu acy
wi h o al ene gy consump ion, is cu en ly missing.
IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 2
In his wo k, we aim o b idge he gap in he li e a u e
ega ding join ene gy-e icien and da a-op imized FL ope a-
ions. To his end, we ocus on an FL scena io whe e a cen al-
ized edge se e ains a global image classi ica ion model by
agg ega ing pa ame e s om edge de ices. The edge de ices
use mul iple p ocesso s o local aining and communica e
he local model pa ame e s o e he same ime- equency
esou ces using he powe -domain NOMA echnique. In his
mul i-p ocesso and sha ed- adio- esou ce se ing, we join ly
op imize da a selec ion, uplink ansmission powe alloca ion,
and compu ing equency scaling a each de ice and p ocesso .
Con a y o simila s udies ha simpli y speci ic model aspec s
and ea da a selec ion and esou ce alloca ion unila e ally, we
p opose a single, low-complexi y, and scalable amewo k ha
join ly add esses his mul i- a iable and highly non-con ex op-
imiza ion p oblem wi hin he conside ed complex FL se ing.
The main con ibu ions o his wo k a e summa ized as:
•The join p oblem o local aining da a selec ion o each
de ice and i s p ocesso s, uplink ansmission powe al-
loca ion, and equency scaling o each de ice p ocesso
is o mula ed. The aim is o maximize FL e iciency and
minimize ene gy consump ion ac oss all phases o he FL
pipeline, including compu ing and communica ion.
•The ini ially o mula ed highly non-con ex p oblem is de-
composed in o h ee sub-p oblems, concluding close- o-
op imal solu ions o he da a selec ion, compu ing, and
adio esou ce alloca ion sepa a ely. The sub-p oblems
a e i e a i ely sol ed using Al e na ing Op imiza ion
(AO) un il he o iginal objec i e unc ion con e ges. The
p oposed algo i hm has a polynomial complexi y o he
numbe o edge de ices, esul ing in a scalable and
compu a ionally e icien solu ion.
•The o e all p oposed ene gy-e icien and da a-op imized
FL amewo k is e alua ed h ough modeling and simu-
la ion, demons a ing i s scalabili y and ene gy e iciency
in compa ison o benchma k schemes o each indi idual
sub-p oblem. The MNIST and CIFAR10 da ase s a e
conside ed o local aining, in oducing di e si y and
enhancing he alidi y o he p oposed amewo k.
•The p oposed amewo k achie es compa able FL model
accu acy and ene gy consump ion wi h o - he-shel so-
lu ions, while educing eal execu ion ime by 1.5 o de s
o magni ude. The da a selec ion scheme achie es 95%
and 83% sample educ ion o he MNIST and CIFAR10
da ase s, espec i ely. O e all, he amewo k esul s in
a leas 30% ene gy educ ion on bo h he MNIST and
CIFAR10 da ase s, while main aining FL model accu acy
close o he second-bes benchma king schemes.
The emainde o he pape is o ganized as ollows. Sec-
ion II summa izes he mos ele an s udies in he li e a u e.
In Sec ion III, he sys em model is p esen ed along wi h he
o mula ion o he join op imiza ion p oblem. Sec ion IV
discusses he o iginal p oblem decomposi ion, he solu ions
o each speci ic sub-p oblem, and he o e all p oposed AO
amewo k. Sec ion V p esen s he nume ical e alua ion o
he p oposed amewo k, and Sec ion VI concludes he pape
and discusses imp o emen s and ex ensions o his wo k.
II. RELATED WORK
In his sec ion, we discuss he mos ele an and ep esen-
a i e subse o wo ks in he li e a u e on da a selec ion and
esou ce alloca ion in FL sys ems. In e es ed eade s can e e
o he su ey in [11], co e ing se e al aspec s o on-de ice
in elligence, such as deploymen scena ios and model p uning,
o acili a e FL in esou ce-cons ained en i onmen s.
A. Da a Selec ion o FL
T aining da a selec ion on edge de ices is c ucial o balance
FL model accu acy and local compu a ion ene gy consump ion
a , making i an ac i ely e ol ing esea ch a ea, e.g., [12]–[16].
Fo ins ance, he wo k in [12] p esen s one o he i s a emp s
a modeling da a impo ance in FL ne wo ks by p oposing
a da a selec ion and communica ion esou ce alloca ion al-
go i hm ha le e ages he squa ed es ima ed g adien no m
o guide impo ance-awa e da a selec ion, he eby educing
end- o-end la ency and imp o ing lea ning e iciency in FL
sys ems. A he same ime, s ochas ic da a selec ion echniques
exis , such as Me cu y in [13], which uses s ochas ic impo -
ance sampling o ensu e unbiased con e gence o he same
solu ion as s anda d dis ibu ed S ochas ic G adien Descen
(SGD), o e ing heo e ical gua an ees.
O he wo ks ocus on adap i e da a selec ion a each FL
ound o op imize speci ic pe o mance me ics such as accu-
acy, ene gy e iciency, o communica ion o e head, e.g., [14],
[15]. FedOL [14] Unmanned Ae ial Vehicles (UAV) execu ing
an FL p ocess, while dynamically adjus ing he size o local
da ase s du ing aining ounds based on hei eal- ime s a us
and FL deadlines. The algo i hm selec s he mos impo an
samples by p io i izing newly added online da a collec ed
by he UAVs, assigning hem he highes p io i y by se ing
hei loss alue o he maximum among exis ing samples. In
his way, he UAVs ocus on esh da a and a oid edundan
aining. Following a simila a ionale, he au ho s in [15] p o-
pose an adap i e da a selec ion s a egy ha enhances aining
e iciency while educing he communica ion o e head due o
edundan da a ansmissions. The employed selec ion s a egy
adap i ely selec s da a by s a ing wi h a minimum sample
size, g adually inc easing i based on model pe o mance, and
upda ing he model in ba ches un il a p ede ined accu acy
h eshold is eached. This ensu es e icien aining unde
dynamic esou ce condi ions. Fo he in e es ed eade , he
su ey in [16] p esen s a e iew o de ice selec ion me hods
wi h an emphasis on da a selec ion in a comp ehensi e manne .
B. Compu ing and Radio Resou ce Op imiza ion in FL
The pe o mance o FL depends hea ily on he alloca ion
o compu ing and adio esou ces, especially when (i) aining
is pe o med by ba e y-powe ed edge de ices and (ii) model
upda es a e ansmi ed o e wi eless channels. Fo his eason,
he p oblems o compu ing and adio esou ce alloca ion ha e
gained much a en ion, being add essed bo h independen ly
and join ly. Indica i e wo ks ocusing exclusi ely on he
compu ing aspec o FL include [17], [18]. The wo k in [17]
conside s a wo- ie FL sys em, whe e In e ne o Things (IoT)
IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 3
de ices wi h mul i-p ocesso compu ing capabili ies execu e
AI aining asks in pa allel. The aim is o e icien ly dis ibu e
aining asks ac oss p ocesso s o balance he compu a ional
load and minimize ene gy consump ion. Con a y o [17],
he au ho s in [18] explo e he pe o mance ade-o s in FL
be ween di e en compu ing esou ces, namely CPUs and
GPUs, while dynamically alloca ing aining asks ac oss bo h
esou ce ypes, aiming o op imize accu acy and la ency. While
GPUs accele a e AI aining wi h pa allel p ocessing, i is
highligh ed ha CPUs o e mo e ene gy-e icien solu ions.
Conside ing wi eless FL sys ems, adio esou ce alloca ion
has been s udied om a ious iewpoin s, conside ing wi eless
channel eliabili y and key me ics, such as global model
accu acy, aining loss, and ene gy consump ion, e.g., [19]–
[21]. On he one hand, bandwid h o subchannel alloca ion
alongside powe con ol has been he ocus o wo ks such
as [19] and [20], espec i ely, while modeling he p obabili y
o success ul ansmission o e he wi eless medium due o
limi ed bandwid h. A di e en s eam o esea ch ocuses on
non-o hogonal mul iplexing o he edge de ices’ ansmis-
sions o local model upda es o he se e , e.g., [21], allowing
o mo e e icien u iliza ion o he a ailable bandwid h. In
his con ex , powe con ol is c i ical o mi iga e in e e ence
and ensu e e ec i e signal decoding while a ge ing FL- ela ed
me ics and ene gy e iciency.
A ema kable amoun o wo k can also be ound on join
e o s ega ding he alloca ion o compu e and adio esou ces
o FL. The mos widely pu sued op imiza ion objec i e is o
minimize he o al ene gy consump ion o bo h compu ing
and communica ion ope a ions while adhe ing o bandwid h
and la ency cons ain s (e.g, [22]) o lea ning pe o mance
cons ain s like he global FL loss (e.g., [23]). Bo h wo ks
in [22], [23] conside o hogonal sha ing o he a ailable band-
wid h and ackle he p oblem o bandwid h o a e alloca ion
along wi h ansmission powe con ol and compu ing e-
quency alloca ion. A simila ene gy consump ion minimiza ion
objec i e is add essed in [24] wi h he di e ence ha powe -
domain NOMA is assumed o be e esou ce u iliza ion.
Mul i-objec i e op imiza ion is also o in e es , bu i has
been explo ed o a lesse ex en . Indica i ely, he minimiza ion
o global FL loss unc ion and o al ene gy consump ion is
ackled in [25]. Also, he wo k in [26] seeks o minimize
bo h he ime and ene gy o e heads due o compu ing and
communica ion, while op imizing he FL p ocess’s con e -
gence speed. Simila communica ion se ings and op imiza ion
a iables o [22], [23] a e conside ed in [25], [26], neglec ing
o accoun o he sca ci y o bandwid h esou ces ia NOMA
echniques. Mo eo e , none o he wo ks in [22]–[26] models
he mul i-p ocesso case om a compu ing pe spec i e, while
he p oblem o da a selec ion is majo ly o e looked unde join
compu e and adio esou ce op imiza ion wo ks in FL.
III. SYSTEM MODEL
We conside he dis ibu ed ICPS illus a ed in Fig. 1, whe e
a se o edge de ices N={1,2, . . . , N}, wi h cons ained
ene gy and compu ing capabili ies, pe o m FL asks. The
p oposed ICPS op imizes he AI model aining pipeline using
Fig. 1: High-le el A chi ec u e o he ICPS.
FL, by dealing wi h h ee main p oblems namely; (a) he da a
selec ion on each edge de ice (b) he equency scaling o
he mul i-p ocesso s o he de ice, and (c) he powe con ol
o e icien da a ansmission. In his wo k, and o p oo -o -
concep pu poses, we conside a ypical image classi ica ion
FL ask, which has b oad indus ial and p ac ical applica-
ions in he ICPS domain, such as ehicula ne wo ks [4] o
sma -ci y en i onmen al moni o ing sys ems using consume
de ices [5]. The ICPS model can be ex ended o in eg a e
edge de ices wi h in e mi en ene gy ha es ing om ambien
sou ces (e.g., sola , wind, adio equency) [27]. In his case,
ensu ing con inuous FL ope a ions and long- e m sus ainabil-
i y equi es accoun ing o he in e play be ween ha es ed
ene gy and he ene gy consumed o local aining and uplink
ansmission, which u he complica es da a selec ion and
esou ce alloca ion, and is pa o ou u u e wo k.
Each de ice npossesses a labeled aining da ase Dn=
{xn,l, yn,l}Dn
l=1, con aining Dnsamples, whe e xn,l is he l- h
inpu sample and yn,l is he co esponding class label in he
FL ask. In each FL ound i, each edge de ice ains a local
model wi h pa ame e s wn. The goal o local aining is o
minimize he loss be ween he ou pu o he on-de ice neu al
ne wo k and he a ge label yn,l o each inpu sample xn,l.
Le Ψ(xn,l,wn)deno e he ou pu o he on-de ice neu al
ne wo k o inpu xn,l ∈ Bnwi h pa ame e ec o wn. Also,
de ine as ℓ(Ψ(xn,l,w), yn,l) he loss unc ion ha quan i ies
he pe -sample e o a e o wa d p opaga ion.
To educe compu a ional o e head and inc ease lea ning
e iciency, local aining is pe o med on a subse o he local
da ase , Bn⊆ Dn,∀n, which comp ises he mos impo an
samples o de ice n. To de e mine he subse Bn, he impo -
ance o each sample lo de ice nis i s es ima ed using he
squa ed no m o he g adien as ollows [28]:
σn,l =




∂ℓ(Ψ(xn,l,w), yn,l)
∂xL
n,l 




2
2
,(1)
whe e ∥·∥2is he L2 no m and xL
lis he inpu o he las laye ’s
ac i a ion unc ion o sample l. This is a well- ounded heu is-
ic me ic based on he assump ion ha samples wi h la ge
g adien s con ibu e mo e signi ican ly o model upda es. Thus,
a la ge σn,l indica es ha sample lhas a highe impac on he
loss and can con ibu e o as e con e gence. By so ing he
IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 4
samples xn,l ∈ Dnacco ding o hei impo ance, he subse
Bno mos ones can be de i ed. Es ima ing sample impo ance
du ing he o wa d pass allows pe o ming backp opaga ion
only on he selec ed samples. This educes compu a ion since
backp opaga ion ypically equi es app oxima ely wice as
many FLOPs pe sample compa ed o o wa d pass [29].
To u he de e mine he speci ic numbe Bno samples
o each de ice (i.e., he ca dinali y o Bn), we de ine he da a
impo ance unc ion g(Bn)o de ice nas ollows:
gn(Bn) =
Bn
X
i=1
σn,i, Bn≤Dn.(2)
This unc ion cap u es he de ice’s con ibu ion o he global
loss educ ion when selec ing he op Bnsamples om Dn
wi h he highes signi icance alues, conside ing an o de ed
sequence σn,1≥σn,2≥ · · · ≥ σn,Dn, s a ing om he mos
impo an ones. No ably, gn(Bn)is conca e wi h espec o
Bn, as discussed in [12], when Bnis elaxed o a con inuous
a iable. By inco po a ing he unc ion gn(Bn)along wi h he
sys em’s o al ene gy consump ion, we o mula e and sol e
he join p oblem o da a selec ion and ansmission powe
and compu ing equency alloca ion owa d ene gy-e icien
and da a-op imized FL ope a ions, as de ailed in Sec ion IV.
Ha ing concluded he aining subse Bn⊆ Dno Bn
samples, he local aining loss o de ice nis calcula ed as:
Ln(Bn,wn) =
Bn
X
l=1
ℓ(Ψ(xn,l,wn), yn,l),∀n∈ N .(3)
Towa d minimizing he a o emen ioned loss, each de ice
pe o ms κlocal model upda es, indexed by j. In each i e a ion
j, he local model pa ame e s wn o each de ice na e
upda ed using he g adien descen ule wi h lea ning a e
η∈[0,1] as ollows:
wj+1
n=wj
n−η∇Ln(Bn,wj
n).(4)
A e local aining, each de ice communica es i s local
g adien gn=1
Bn∇Ln(Bn,wi
n) o he coo dina ing se e .
The se e hen agg ega es hese local g adien s o p oduce a
global g adien gi+1 used o upda e he global model wi+1
o he nex FL ound i+ 1. Speci ically, he global g adien
is compu ed as a weigh ed a e age o he local g adien s:
gi+1 =1
Pn∈N BnX
n∈N
Bngi
n.(5)
Using he global g adien gi, he global model o he FL
ask is upda ed o FL ound i+ 1 as ollows:
wi+1 =wi−ηg.(6)
In he emainde o Sec ion III, we d op he index i
associa ed wi h he FL ounds o no a ion simplici y.
A. Compu ing Model
Following he wo k in [30], he edge de ices a e equipped
wi h a mul ip ocessing model o CPUs. In pa icula , each
de ice is equipped wi h Qnp ocesso s which de ine a se Qn.
Le q
n[FLOPs/sec] deno e each p ocesso ’s scaling equency
ha ope a es be ween a minimum and maximum alue, i.e.,
q
n∈[ q,min
n, q,max
n]. The edge de ices model aining
is pe o med in pa allel [31], meaning he compu a ional
wo kload is delega ed o he a ailable p ocesso s. We de ine
he se s Bq
n⊆ Bn, which ep esen he se o samples ha
each p ocesso q∈ Qnwill handle. Fo he pa i ioning
o he de ices’ da ase s o hei p ocesso s, i holds ha
∪Qn
q=1Bq
n=Bn. Acco dingly, we de ine he wo kload (in
FLOPs) o each p ocesso as Wq
n=Bq
n·NF LOP S, whe e
NF LOP S is he numbe o FLOPs needed o p ocessing a
sample. The e o e, o de ice n, he compu ing ime equi ed
by each p ocesso o p ocess i s wo kload is exp essed as:
Tcmp
n,q =Wq
n
q
n
,∀q∈ Qn[sec].(7)
As a esul , he maximum ime o he local model upda e o
edge de ice nis:
Tcmp
n=Qn
max
q=1 Wq
n
q
n[sec].(8)
Mo eo e , we de ine he powe consump ion o each p oces-
so ollowing he s anda d Dynamic Vol age F equency Scal-
ing (DVFS) modeling u ilized o CMOS ci cui s o exp ess
he powe as a unc ion o he CPU equency, ollowing [31]:
Pcmp
n,q =Cq
n( q
n)3[Wa ],(9)
whe e Cq
n[Wa /(FLOPs/sec)3] de e mines he e iciency o
each p ocesso , i.e., he powe a e g ow h compa ed o he
inc ease o he compu ing equi emen s. Gi en he du a ion o
local aining o each p ocesso o comple e he associa ed
asks as de ined in Eq. (7), we can exp ess he o al ene gy
consump ion o edge de ice nas ollows:
Ecmp
n=
Qn
X
q=1
Cq
nWq
n( q
n)2[Joule].(10)
I is no ed ha he p oposed compu ing model could be
u he complemen ed wi h in elligen AI op imiza ion ech-
niques, such as ede a ed d opou o p uning o models, ha
dynamically adap model a chi ec u e o educe compu a ional
complexi y and p ese e de ice ene gy [32].
B. Communica ion Model
The use s’ local g adien s a e ansmi ed o he coo dina ing
se e o e he same ime- equency esou ces o o al band-
wid h B[Hz] using he powe -domain Non-O hogonal Mul-
iple Access (NOMA) echnique o ensu e e icien spec um
euse. Le Gndeno e he channel gain be ween use nand
he se e . Wi hou loss o gene ali y, assume ha he channel
gains be ween use s and he se e a e so ed in ascending
o de , G1≤···≤Gn≤···≤GN, and he decoding o
he signals begins om he highes channel gain use using
he Successi e In e e ence Cancella ion (SIC) echnique. The
achie ed uplink da a a e o use n o he se e is calcula ed
as:
Rn=BW log2 1 + Gnpn
Pn−1
n′=1 Gn′pn′+N0BW ![bps],
(11)
IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 5
whe e pn[Wa ] is he uplink ansmission powe o use
nand N0[dBm/Hz] is he powe spec al densi y o ze o-
mean Addi i e Whi e Gaussian Noise (AWGN). Aligned wi h
common assump ions in he li e a u e, we assume pe ec
Channel S a e In o ma ion (CSI) and ideal implemen a ion o
he SIC echnique [24]. In case o impe ec ions, addi ional
a iables should be in oduced o cap u e he unce ain ies in
he channel gains and he decoded in e e ence, espec i ely.
Acco dingly, he ansmission ime o use n o communi-
ca ing i s local g adien gn o he se e is:
T x
n=V(gn)
Rn
[sec],(12)
whe e V(gn)[bi s] is he da a size o he local g adien ec o
gn, which is equal o all use s in he sys em. Mo eo e , he
associa ed ansmission ene gy consump ion is gi en by:
E x
n=V(gn)pn
Rn
[Joule].(13)
C. P oblem Fo mula ion
In his wo k, ou objec i e is o maximize FL e iciency
by op imizing he numbe o local aining samples and
minimizing he ene gy consump ion in bo h local compu a ion
and communica ion o g adien s o he se e . To his end,
we o mula e he join op imiza ion p oblem o he numbe
o local aining samples Bnpe de ice and he numbe
o p ocessed samples Bq
npe de ice p ocesso , he uplink
ansmission powe pno each de ice o he se e , and he
equency scaling q
n o each de ice p ocesso . Speci ically, in
each FL ound, we op imize he ollowing objec i e unc ion
o balance da a selec ion and ene gy e iciency ac oss all edge
de ices:
F(Bn, Bq
n, q
n, pn) =
N
X
n=1
gn(Bn)−y
N
X
n=1 Ecmp
n+E x
n.
(14)
The i s e m measu es da a impo ance, while he second
e m ep esen s he o al compu a ion and communica ion
ene gy consump ion. The cons an pa ame e y∈R+se es
as a scaling ac o o balance he adeo be ween hese wo
e ms.
Hence, we de ine he ollowing join op imiza ion p oblem:
P: max
{Bn,Bq
n, q
n,pn}F(Bn, Bq
n, q
n, pn)(15a)
s. . Tcmp
n≤Tcmp
max,∀n, (15b)
T x
n≤T x
max,∀n, (15c)
Tcmp
n+T x
n≤Tmax,∀n, (15d)
q,min
n≤ q
n≤ q,max
n,∀n, q, (15e)
0≤pn≤pmax
n,∀n, (15 )
0≤Bn≤Dn,∀n, (15g)
Qn
X
q=1
Bq
n=Bn,∀n. (15h)
Cons ain s (15b) and (15c) ensu e ha he compu ing and
communica ion imes pe de ice emain below hei espec-
i e maximum h esholds. Cons ain (15d) is he o al ime
cons ain o an FL ound pe de ice, calcula ed as he sum
o hei compu ing and communica ion imes. Cons ain (15e)
ensu es ha he equency o each p ocesso ope a es be ween
he minimum and maximum alues. Cons ain (15 ) ensu es
ha he uplink ansmission powe o each edge de ice is
below he maximum le el. Finally, Cons ain s (15g) and (15h)
gua an ee ha he selec ed samples o local model upda ing
will no exceed he samples o he local da ase and will equal
he alloca ed wo kload o he a ailable p ocesso s o each
de ice. I should be no ed ha a con inuous elaxa ion o he
in ege a iables Bn, Bq
n,∀q∈ Qn, n ∈ N is selec ed o
ac ably de i e a solu ion. Howe e , as demons a ed in he
ollowing sec ions, he co esponding decision a iables a e
ul ima ely mapped back o he disc e e space.
P oblem Pis non-con ex due o he non-con exi y o
he objec i e unc ion, while he op imiza ion a iables
Bn, Bq
n, q
na e highly coupled, complica ing he de i a ion o
a ac able solu ion. The e o e, i is challenging o ob ain a
global op imal in polynomial ime. In he ollowing, we ou line
he me hodology o decompose he o iginal highly non-con ex
p oblem in o independen con ex sub-p oblems, he solu ions
o which sepa a ely p o ide sub-op imal solu ions o he join
ene gy-e icien and da a-op imized FL op imiza ion p oblem.
IV. ENERGY-EFFICIENT AND DATA-OPTIMIZED FL
SOLUTION
In his sec ion, he o iginal op imiza ion p oblem Pis
decomposed in o h ee independen sub-p oblems, namely he
on-de ice da a selec ion (Sec ion IV-A), compu ing esou ce
alloca ion (Sec ion IV-B), and adio esou ce alloca ion (Sec-
ion IV-C). Each sub-p oblem is analy ically sol ed while
ea ing he o he s as ixed, as analy ically desc ibed in he
espec i e subsec ions. Las , Sec ion IV-D de ails he i e a i e
app oach based on he p inciples o AO [33], [34], acco ding
o which he solu ions o he sub-p oblems a e applied in an
al e na i e manne un il con e gence o he objec i e unc ion
is achie ed.
A. On-De ice Da a Selec ion
Fi s , we del e in o he p oblem o on-de ice da a selec ion,
aiming o de e mine he op imal numbe o samples Bn o
each de ice and Bq
n o each o i s p ocesso s, while e icien ly
alloca ing he samples o he p ocesso s. By ixing he decision
a iables ega ding he equency scaling o all p ocesso s n,q
and he ansmission powe o each edge de ice pn, p oblem
Pis e o mula ed as ollows:
P1 : max
{Bn,Bq
n}
N
X
n=1
g(Bn)−y
N
X
n=1 Ecmp
n+E x
n(16a)
s. . Qn
max
q=1 Wq
n
q
n≤Tmax
n,∀n, (16b)
0≤Bn≤Dn,∀n, (16c)
Qn
X
q=1
Bq
n=Bn,∀n. (16d)

IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 6
whe e Tmax
n= min Tmax −V(gn)
Rn, Tcmp
maxis he maximum
accep able compu ing ime pe de ice, de i ed by join ly
conside ing cons ain s (15c) and (15d).
Th ough in oducing he auxilia y a iables n,∀n, p oblem
P1is equi alen ly ans o med in o p oblem P1′ o emo e
he max ope a o and ob ain a con inuous exp ession o he
o iginal cons ain (16b). Also, by subs i u ing Wq
n=Bq
n·
NF LOP S, p oblem P1is ew i en as:
P1′: max
{Bn,Bq
n, n}
N
X
n=1
g(Bn)−y
N
X
n=1 Ecmp
n+E x
n(17a)
s. . Bq
nNF LOP S
q
n
≤ n,∀n, q (17b)
n≤Tmax
n,∀n(17c)
0≤Bn≤Dn,∀n, (17d)
Qn
X
q=1
Bq
n=Bn,∀n. (17e)
P oblem P1′is a conca e p oblem, consis ing o a conca e
objec i e unc ion wi h espec o Bn, Bq
n,∀n∈ N,∀q∈ Qn,
and con ex se s o cons ain s. Thus, p oblem P1′can be
e icien ly sol ed using s anda d nume ical me hods such as
he in e io poin me hod. Howe e , in he ollowing, we use
Lag ange dual decomposi ion o de i e closed- o m exp es-
sions gi en he Lag ange mul iplie s, gua an eeing ha he
op imal solu ion is ob ained wi hin polynomial ime [35].
The Lag angian unc ion (see [35], Theo em 2.1) o p oblem
P1′is w i en as:
L=−
N
X
n=1
g(Bn) + y
N
X
n=1 Ecmp
n+E x
n+
N
X
n=1
αn(Bn−Dn)
+
N
X
n=1
Qn
X
q=1
λn,q Bq
n·NF LOP S
q
n
− n+
N
X
n=1
µn( n−Tmax
n)
+
N
X
n=1
βnBn+
N
X
n=1
νn
Qn
X
q=1
(Bq
n−Bn),(18)
whe e αn, λn,q, µn, βn, n≥0a e he Lag angian mul iplie s
associa ed wi h he cons ain s (17b)-(17e).
By calcula ing he KKT condi ions, we ob ain:
∀n:∂L
∂Bn
=−∂g(Bn)
∂Bn
+αn+βn−νn= 0 (19)
∀n:∂L
∂Bq
n
=y∂Ecmp
n
∂Bq
n
+λn,q
NF LOP S
q
n
+νn= 0 (20)
∀n:∂L
∂ n
=−
Qn
X
q=1
λn,q +µn= 0 (21)
∀n:λn,q Bq
nNF LOP S
q
n
− n= 0 (22)
∀n:αn(Bn−Dn)=0 (23)
∀n:µn( n−Tmax
n)=0 (24)
∀n:−βnBn= 0 (25)
Gi en ha p oblem P1′is conca e, he KKT condi ions a e
bo h necessa y and su icien o op imali y.
To ob ain he op imal solu ion o he da a selec ion p ob-
lem, he ollowing cases a e conside ed. Fi s , we in es iga e
he gene al case ha λn,q = 0, meaning ha he ime
cons ain (17b) is inac i e. Then, by subs i u ing s a iona y
condi ion (19) o he s a iona y condi ion (20) yields:
∂g(Bn)
∂Bn
−y∂Ecmp
n
∂Bq
n
−αn−βn= 0.(26)
I he complemen a y slackness condi ion (25) is ac i e, i.e.,
β > 0, hen he co esponding cons ain (17e) mus hold, i.e.,
B∗
n= 0 which is ejec ed. I he condi ion is inac i e, i.e.,
β= 0, hen by he complemen a y slackness condi ion (23)
i αn>0 hen B∗
n=Dn, o he wise he op imal solu ion o
B∗
nis ob ained by sol ing Eq. (26):
∂g(Bn)
∂Bn
=y∂Ecmp
n
∂Bq
n
=y
Qn
X
q=1
Cq
nNF LOP S( q
n)2.(27)
As discussed in Sec ion III, unc ion g(Bn)is conca e and
om Eq. (27) he alue o ∂g(Bn)
∂Bnis known. Subsequen ly,
we can de e mine he disc e e alue o B∗
nby u ilizing he
in e se unc ion g−1
n(Bn)and mapping he esul o he closes
disc e e alue. In his way, he op imal numbe o selec ed
samples B∗
nis de e mined o each de ice n. Fu he mo e, by
sol ing Eq. (22), he op imal numbe o samples Bq∗
n o each
p ocesso q∈ Qno de ice nis de i ed. Ha ing calcula ed
bo h B∗
nand Bq∗
n,∀n, q, he alloca ion o samples o each
de ice’s p ocesso s is hen pe o med by so ing he p ocesso s
acco ding o hei ene gy e iciency, as de ailed in Algo i hm 1.
Algo i hm 1 Da a Alloca ion o he De ice’s P ocesso s
1: Bn= 0 and Q∗
n=∅
2: Compu e B∗
n alue om Eq. (27).
3: Compu e op imal alue ∗
n= min Tmax −V(gn)
Rn, Tcmp
max
4: So p ocesso s o edge de ice nacco ding o Cq
n( q
n)2
alue o o m se Q′
n.
5: o each p ocesso q∈ {1,...,Q′
n}do
6: Sol e Eq. (22) and ob ain Bq∗
n= q
n
NF LOP S ∗
n.
7: Bn+= Bq∗
n
8: Add q o he se o ac i e p ocesso s Q∗
n=Q∗
n∪ {q}.
9: i Bn≥B∗
n hen
10: Bq′∗
n←0 o all q′∈ {q+ 1, . . . , Q′
n}
11: b eak o loop;
12: end i
13: end o
14: e u n B∗
n,{Bq∗
n}, ∗,Q∗
n
The o e all p ocedu e is epea ed o all de ices as de-
sc ibed in Algo i hm 1. Speci ically, a e calcula ing ∗
n(S ep
3), he se Q′
nis cons uc ed by so ing he p ocesso s acco d-
ing o he e ms Cq
n( q
n)2(S ep 4) which indica es hei ene gy
e iciency ega ding hei chosen equency. As a esul , he
algo i hm s a s alloca ing da a samples o he mos e icien
p ocesso un il cons ain (17b) is no iola ed (S ep 6) o each
p ocesso . Mo eo e , he se o ac i e p ocesso s Q∗
nis upda ed
wi h his p ocesso (S ep 8). In case he o al alloca ed da a
Bnon he de ice so a exceeds he op imal alue B∗
n(S ep
9), he es o he p ocesso s a e no assigned any da a samples
IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 7
(S ep 10) and we conside ha hey a e no ope a ional in his
ound. O he wise, he algo i hm p oceeds wi h assigning he
emaining da a o he es o he less e icien p ocesso s. The
algo i hm e u ns he bina y alues o B∗
n, Bq∗
nand he se
o ac i e p ocesso s o each de ice Q∗
n.
The complexi y o Algo i hm 1 is mainly d i en by he
so ing ope a ion o he p ocesso s a S ep 4 and he i e a-
ions o e all p ocesso s a S ep 5. Speci ically, so ing he
p ocesso s o each de ice a S ep 4 has a complexi y o
O(Qnlog(Qn)), using well-known so ing algo i hms, e.g.,
Me ge So . The loop in S ep 5 in ol es algeb aic calcula ions
o O(1) complexi y o e a -wo s Qni e a ions, leading o an
o e all complexi y o O(Qn) o he loop. Consequen ly, he
wo s -case complexi y o Algo i hm 1 is O(Qnlog(Qn)).
Finally, i λn,q >0meaning ha he ime cons ain (17b)
is ac i e, hen Eq. (22) yields Bq∗
n= n q
n
NF LOP S . The e o e,
i µn>0 hen he complemen a y slackness condi ion (24)
yields he op imal solu ion ∗
n=Tmax
n,Bq∗
n=Tmax
n q
n
NF LOP S and
using he p imal easibili y condi ion (17e) we ge B∗
n=
PQn
q=1 Bq∗
n.
B. Compu ing Resou ce Alloca ion
Gi en he da a selec ion, he da a alloca ion o each de ice’s
p ocesso s and he se o ac i e p ocesso s Q∗
n, he equency
alloca ion o he p ocesso s ha will p ocess da a samples is
op imized by sol ing:
P2 : min
{ q
n}
N
X
n=1
Q∗
n
X
q=1 Cq
nBq
nNF LOP ( q
n)2(28a)
s. . Bq
nNF LOP ≤ q
nTmax
n,∀n, q ∈ Q∗
n(28b)
q,min
n≤ q
n≤ q,max
n,∀n, q ∈ Q∗
n,(28c)
whe e Tmax
n= min Tmax −V
Rn
, Tcmp
max. P oblem (28) is
a quad a ic p oblem wi h linea cons ain s, howe e s ic ly
con ex. The objec i e unc ion is inc easing wi h ega ds o
he equencies, and hus ob aining a closed- o m solu ion is
i ial:
q∗
n= max( q,min
n,(Bq
nNF LOP )/T max
n).(29)
C. Radio Resou ce Alloca ion
Gi en he da a selec ion and equency alloca ion solu ions
Bn, Bn,q, n,q o each de ice and i s espec i e p ocesso s,
he uplink ansmission powe o communica ing local g a-
dien s o he se e o each de ice is op imized by sol ing:
P3 : min
{pn}E x
n=
N
X
n=1
pnT x
n(30a)
s. . T x
n≤T x,max
n,∀n, (30b)
0≤pn≤pmax
n,∀n, (30c)
whe e T x,max
n= min (Tmax − n, T x
max)is he maximum
allowable ansmission ime o each de ice, de i ed based
on he op imized compu ing ime nob ained h ough da a
selec ion (Sec ion IV-A), he o al ime cons ain Tmax o
an FL ound, and he p ede ined maximum ansmission ime
T x
max.
P oblem P3is challenging o sol e due o he non-con exi y
o he objec i e unc ion wi h espec o pn,∀n, and he
in e dependence o he de ices’ ansmission powe s s emming
om he sha ed communica ion channel and he esul ing in-
e e ence among hem. To add ess his challenge, we le e age
he ac ha ansmission ime T x
n,∀nis uppe -bounded by
cons ain (31b). Following es ablished app oaches om he
li e a u e [36], we can ix T x
n,∀nand e o mula e he p oblem
as:
P3′: min
{pn}
N
X
n=1
pn(31a)
s. . V(gn)
BW log21 + Gnpn
Pn−1
n′=1 Gn′pn′+N0BW =T x,max
n,∀n,
(31b)
0≤pn≤pmax
n∀n, (31c)
whe e cons ain (31b) eplaces cons ain (30b). This e o -
mula ion con inues o con ibu e o minimizing ansmission
ene gy consump ion, since he ansmission ene gy E x
n=
pnT x
n=pnV(gn)
Rndec eases mo e apidly wi h a educ ion
in pn(linea ela ionship) han wi h an inc ease in Rn(loga-
i hmic ela ionship).
P oblem P3′is a Linea P og amming (LP) p oblem, whose
solu ion is de i ed by sol ing cons ain s (31b) o pn,∀n,
leading o he ollowing se o equa ions:
p1=2V(g1)/(BW ·T x,max
1)−1
G1
·N0BW
p2=2V(g2)/(BW ·T x,mx
2)−1
G2
·(N0BW +G1p1)
.
.
.
pn=2V(gn)/(BW ·T x,max
n)−1
Gn
· N0BW +
n−1
X
n′=1
Gn′pn′!,∀n.
(32)
D. Da a-op imized and Ene gy E icien Algo i hm o FL
The solu ion o he join da a-op imized and ene gy-e icien
FL is analy ically p esen ed in Algo i hm 2. Algo i hm 2 is
execu ed a each FL ound i, op imizing he da a selec ion
and compu ing and adio esou ce alloca ions o maximize FL
e iciency and minimize o al ene gy consump ion. Addi ion-
ally, Algo i hm 2 ope a es in i e a ions indexed by , whe e
he solu ions o he sub-p oblems a e upda ed un il he o iginal
objec i e unc ion Fcon e ges, meaning i s alue emains
unchanged be ween consecu i e algo i hm i e a ions.
To calcula e he complexi y o Algo i hm 2, we i s con-
side he complexi y o de i ing he solu ion o each sub-
p oblem. The complexi y o de e mining he da a selec ion
o all de ices using Algo i hm 1 is O(Qnlog(Qn). The
compu ing equency alloca ion has a closed- o m op imal
solu ion o each de ice, yielding a o al complexi y o O(N)
o all de ices. The complexi y o sol ing he sys em o
linea equa ions o de e mine he uplink ansmission powe s
IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 8
o all de ices is also O(N). Le Tdeno e he numbe o
i e a ions equi ed o he AO app oach o con e ge. Then, he
o e all algo i hm’s complexi y is O(T·(Qnlog(Qn)+N)).
Nume ical esul s ega ding he numbe o AO i e a ions and
eal execu ion ime equi ed o Algo i hm 1 o conclude a
solu ion a e p esen ed in Sec ion V, emphasizing i s ema k-
ably lowe complexi y compa ed o s anda d sol e s a ailable
in op imiza ion oolboxes.
Algo i hm 2 O e all Da a-Op imized and Ene gy-E icien FL
Algo i hm
1: Se AO i e a ion index ←0.
2: Ini ialize a easible solu ion o p oblem P:
{Bn(0), Bq
n(0), q
n(0), pn(0)}∀(n,q).
3: epea
4: = + 1
5: Calcula e {Bn( ), Bq
n( )}∀(n,q)based on Algo i hm 1,
o gi en { q
n( −1), pn( −1)}∀(n,q).
6: Calcula e { q
n( )}∀(n,q)based on Eq. (29), o gi en
{Bn( ), Bq
n( ), pn( −1)}∀(n,q).
7: Calcula e {pq
n( )}∀nby sol ing p oblem (32), o
gi en {Bn( ), Bq
n( ), q
n( )}∀(n,q).
8: un il |F( )−F( −1)|< ϵ, ϵ →0
9: e u n {B∗
n, Bq∗
n, q∗
n, p∗
n}∀(n,q)
V. EVALUATION AND RESULTS
In his sec ion, we e alua e he pe o mance o he join
da a-op imized and ene gy-e icien FL algo i hm and solu ion
ia modeling and simula ion. The simula ion se up and op i-
miza ion pa ame e s a e ini ialized as ollows. We conside
a wi eless FL sys em deployed wi hin a ci cula a ea o
200 m adius. The se e is loca ed a he cen e o he
sys em and N= 10 edge de ices a e uni o mly andomly
dis ibu ed. The channel gain be ween he de ices and he
se e is calcula ed acco ding o he dis ance-based pa h loss
model om 3GPP [22], [24], PL = 128.1 + 37.6 log(d),
whe e dkm is he Euclidean dis ance be ween hem. The
o al sys em bandwid h is BW = 20 MHz. The es o he
communica ion- ela ed pa ame e s a e se as; pmax
n= 24 dBm
and N0=−134 dBm/Hz. Each edge de ice nis equipped
wi h Qn= 4 p ocesso s, conside ing equency scaling wi hin
he ange  q,min
n, q,max
n= [1,3] GHz. The capaci ance
coe icien o each p ocesso qis uni o mly dis ibu ed wi hin
he ange Cq
n∼[0.01,1] W(MFLOPs/s)−3. The maximum
ime h eshold o an FL ound is Tmax = 500 ms, wi h 20%
alloca ed o communica ion (T x
max = 100 ms) and 80% o
compu ing (Tcmp
max = 400 ms).
The FL model is ained on he MNIST da ase [37], unless
o he wise explici ly s a ed. The la ge and mo e complex
CIFAR10 da ase [38] is also adop ed o enhance he alidi y
and applicabili y o he p oposed da a selec ion scheme, while
allowing di ec compa ison and ep oducibili y wi h o he
wo ks in he li e a u e. F om he o al o 60,000 samples o
bo h da ase s, 50,000 a e used o aining and he emaining
10,000 o es ing. The aining samples a e e enly dis ibu ed
among he de ices o pe o m image classi ica ion, i.e., 5,000
samples/de ice. No e ha e ec i ely handling he e ogeneous
da a dis ibu ions would bene i om an addi ional mechanism,
beyond on-de ice da a selec ion using he g adien no m,
in eg a ed in o ou amewo k o cumula i ely e alua e da a
impo ance ac oss all de ices. The FL p ocess is pe o med
un il model accu acy con e ges, while he lea ning a e o he
local aining is se equal o η= 10−3. The numbe o FLOPs
equi ed o p ocessing one MNIST sample is NF LOP s = 106,
while he size o he communica ing g adien s o he se e is
calcula ed as V(gn) = 723.04 Kbi s.
Fo he MNIST da ase , each node’s local model uses a
Con olu ional Neu al Ne wo k (CNN) o image classi ica ion,
consis ing o wo con olu ional laye s wi h 10 and 20 il e s,
espec i ely, each ollowed by a ReLU ac i a ion unc ion and
a2×2max pooling laye . The esul ing ea u e maps a e
la ened and passed h ough a ully connec ed laye wi h 50
neu ons. The ou pu laye con ains 10 neu ons wi h so max
ac i a ion unc ion, equal o he numbe o MNIST classes.
The aining is pe o med using he ca ego ical c oss-en opy
loss unc ion. Fo he CIFAR10 da ase , he designed CNN
consis s o wo con olu ional blocks wi h 32 and 64 il e s,
espec i ely, each ollowed by a ReLU ac i a ion unc ion
and a 2×2max pooling laye . The esul is la ened and
passed h ough a ully connec ed dense laye wi h 128 neu ons
and ReLU ac i a ion. The ou pu laye con ains 10 neu ons
using a So max ac i a ion, co esponding o he classes o
he CIFAR10 da ase .
A. E alua ion o O e all Algo i hm’s Con e gence and Pe -
o mance
Fi s , we examine he con e gence beha io o he p oposed
Algo i hm 2 wi h espec o he numbe o AO i e a ions
T equi ed o conclude he solu ion o he join p oblem.
Fig. 2a depic s he o al da a impo ance, calcula ed using he
unc ion in Eq. (2), and o al ene gy consump ion esul ing
om bo h compu ing and communica ion as a unc ion o
he AO i e a ions. The esul s ha e been no malized ela i e
o hei ini ial alues a he s a o Algo i hm 2 o mo e
accu a ely e lec he adeo be ween da a impo ance and
ene gy consump ion. The esul s e eal ha a e app oxi-
ma ely 30 AO i e a ions, he o e all algo i hm con e ges o
he inal solu ion. Mo e impo an ly, hough, i is obse ed
ha ene gy consump ion dec eases mo e apidly han he o al
da a impo ance in he sys em, highligh ing he e ec i eness
o he p oposed solu ion in u ilizing ene gy mo e e icien ly
o handling ewe ye signi ican local da a samples. Fig. 2b
and 2c u he demons a e he con e gence o each de ice’s
ansmission and compu ing ene gy, espec i ely. Bo h igu es
show ha he solu ion con e ges o lowe ene gy consump ion
alues om bo h compu ing and communica ion pe spec i es,
al hough his end is mo e p onounced o compu ing ene gy
due o i s la ge scale and ange o alues compa ed o
ansmission ene gy.
Fu he mo e, o e alua e he e ec i eness o Algo i hm 2,
we compa e i s pe o mance o he T us -Cons algo i hm [39]
used o sol ing complex non-linea cons ained op imiza ion
p oblems. The T us -Cons algo i hm is widely implemen ed
IEEE TRANSACTIONS ON CONSUMER ELECTRONICS 9
10 20 30 40 50
AO I e a ions
0
0.5
1
No malized To al Da a
Impo ance / Ene gy
To al Da a Impo ance
To al Ene gy
(a)
10 20 30 40 50
AO I e a ions
0
0.2
0.4
0.6
T ansmission Ene gy [J]
N=10
(b)
10 20 30 40 50
AO I e a ions
0
0.5
1
1.5
2
2.5
Compu ing Ene gy [J]
N=10
2468
1
2
Fi s 8 AO I e a ions
(c)
Fig. 2: Con e gence analysis o he (a) o al impo ance o selec ed da a and ene gy consump ion, (b) compu ing ene gy pe
de ice, and (c) ansmission ene gy pe de ice, o e AO i e a ions.
5 10 15 20
Numbe o De ices
0.835
0.84
0.845
0.85
0.855
0.86
Accu acy
P oposed
T us -Cons Algo i hm
(a)
5 10 15 20
Numbe o De ices
0
2
4
6
To al Ene gy [J]
P oposed
T us -Cons Algo i hm
(b)
5 10 15 20
Numbe o De ices
0
0.5
1
1.5
P oposed- o-T us -Cons
Objec i e a io
(c)
5 10 15 20
Numbe o De ices
0
50
100
150
Execu ion Time [s]
P oposed
T us -Cons Algo i hm
(d)
Fig. 3: Pe o mance analysis o he P oposed and T us -Cons algo i hms, in e ms o (a) accu acy, (b) o al ene gy consump ion,
(c) p oposed- o- us -cons objec i e a io, and (d) eal execu ion ime, o di e en numbe s o de ices.
by a ious op imiza ion oolboxes, such as Py hon’s scipy, and,
in his pa icula e alua ion scena io, is used o sol e he o ig-
inal p oblem P. The aim o his expe imen is o compa e he
adeo be ween pe o mance and scalabili y achie ed by he
p oposed and T us -Cons algo i hms in e ms o FL model
accu acy, ene gy consump ion ac oss he sys em, and eal
execu ion ime. To his end, we conside an inc easing numbe
o de ices in he sys em om N= 5 o N= 20. Th oughou
his expe imen , we conside ha each de ice possesses 500
MNIST samples, such ha he cumula i e knowledge in he
sys em inc eases wi h he addi ion o mo e de ices. To ensu e
he easibili y o he o al FL ound cons ain wi h double he
maximum numbe o de ices in he sys em, we also doubled
he o al ime h eshold o Tmax = 1000 ms.
Fig. 3a and Fig. 3b illus a e he achie ed FL model
accu acy and o al ene gy consump ion, espec i ely, as a
unc ion o he numbe o de ices o bo h algo i hms. The
esul s e eal ha he wo algo i hms achie e nea ly iden ical
global FL model accu acy. Howe e , he p oposed algo i hm
manages o conclude lowe o al ene gy consump ion o he
sys em while his pe o mance gap be ween he algo i hms
becomes mo e p onounced as he numbe o de ices inc eases.
This ou come is expec ed, as o smalle -scale p oblems, he
sol e can explo e he solu ion space mo e ho oughly wi hin
a easonable ime. Fig. 3c summa izes he pe o mance gain
o he p oposed algo i hm compa ed o he T us -Cons sol e
by depic ing he a io o he objec i e unc ion alue achie ed
by each algo i hm. The pe o mance imp o emen consis en ly
exceeds 25% and g ows as he numbe o de ices inc eases.
No ably, wha pa icula ly dis inguishes he p oposed solu ion
is he ime equi ed o execu e he algo i hm, p esen ed in
Fig. 3d. The eal execu ion ime equi ed by he T us -Cons
algo i hm o con e ge is app oxima ely 1.5o de s o magni-
ude highe han ha o he p oposed Algo i hm 2, ende ing
he o me signi ican ly imp ac ical o eal FL deploymen s.
This end is consis en wi h he de i ed complexi y OT·
(Qnlog(Qn)+N)o Algo i hm 2 (see Sec ion IV-D), whe e
he execu ion ime g ows a mos linea ly wi h N, hus explain-
ing he obse ed nea -cons an o mildly inc easing un ime
wi h N. In con as , he o - he-shel T us -Cons algo i hm
sol es i e a i e quad a ic subp oblems, he size o which g ows
wi h he numbe o op imiza ion a iables and cons ain s
(bo h p opo ional o Nin ou se ing). Summa izing, he
p oposed algo i hm based on con en ional op imiza ion and
AO echniques s ikes a good balance be ween FL e iciency,
o al ene gy consump ion, and eal execu ion ime.
B. E alua ion o On-De ice Da a Selec ion
In his sec ion, we e alua e he pe o mance o da a selec-
ion o each de ice and he alloca ion o he de ice’s espec-
i e p ocesso s using he solu ion desc ibed in Sec ion IV-A
and Algo i hm 1. To his end, we i s examine he pu e
pe o mance o da a selec ion in Fig. 4, ollowed by a compa -
a i e e alua ion agains benchma k da a selec ion schemes in
Fig. 5. Aiming o assess he alidi y and applicabili y o ou
p oposed on-de ice selec ion scheme ac oss a ying le els o
classi ica ion ask complexi y, expe imen s on bo h he MNIST
and CIFAR10 da ase s we e conduc ed.