G ieshamme , Max; P lug, Lukas; S ingl, Michael; Uihlein, And ian
A icle — Published Ve sion
The con inuous s ochas ic g adien me hod: pa II–
applica ion and nume ics
Compu a ional Op imiza ion and Applica ions
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: G ieshamme , Max; P lug, Lukas; S ingl, Michael; Uihlein, And ian (2023) :
The con inuous s ochas ic g adien me hod: pa II–applica ion and nume ics, Compu a ional
Op imiza ion and Applica ions, ISSN 1573-2894, Sp inge US, New Yo k, NY, Vol. 87, Iss. 3, pp.
977-1008,
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The con inuous s ochas ic g adien me hod: pa
II–applica ion and nume ics
Max G ieshamme 1·Lukas P lug1,2 ·Michael S ingl1·And ian Uihlein1
Recei ed: 7 Feb ua y 2023 / Accep ed: 26 Oc obe 2023 / Published online: 24 No embe 2023
© The Au ho (s) 2023, co ec ed publica ion 2024
Abs ac
In his con ibu ion, we p esen a nume ical analysis o he con inuous s ochas-
ic g adien (CSG) me hod, including applica ions om opology op imiza ion and
con e gence a es. In con as o s anda d s ochas ic g adien op imiza ion schemes,
CSG does no disca d old g adien samples om p e ious i e a ions. Ins ead, design
dependen in eg a ion weigh s a e calcula ed o o m a con ex combina ion as an
app oxima ion o he ue g adien a he cu en design. As he app oxima ion e o
anishes in he cou se o he i e a ions, CSG ep esen s a hyb id app oach, s a ing o
like a pu ely s ochas ic me hod and beha ing like a ull g adien scheme in he limi . In
his wo k, he e iciency o CSG is demons a ed o p ac ically ele an applica ions
om opology op imiza ion. These se ings a e cha ac e ized by bo h, a la ge num-
be o op imiza ion a iables and an objec i e unc ion, whose e alua ion equi es he
nume ical compu a ion o mul iple in eg als conca ena ed in a nonlinea ashion. Such
p oblems could no be sol ed by any exis ing op imiza ion me hod be o e. Las ly, wi h
ega ds o con e gence a es, i s es ima es a e p o ided and con i med wi h he help
o nume ical expe imen s.
Keywo ds S ochas ic g adien scheme ·Con e gence analysis ·S ep size ule ·
Back acking line sea ch ·Cons an s ep size
BAnd ian Uihlein
[email p o ec ed]
Max G ieshamme
[email p o ec ed]
Lukas P lug
[email p o ec ed]
Michael S ingl
[email p o ec ed]
1Depa men o Ma hema ics, Chai o Applied Ma hema ics, F ied ich-Alexande -Uni e si ä
E langen-Nü nbe g (FAU), E langen, Ge many
2FAU Compe ence Cen e Scien i ic Compu ing, F ied ich-Alexande -Uni e si ä E langen-Nü nbe g
(FAU), E langen, Ge many
123
978 M. G ieshamme e al.
Ma hema ics Subjec Classi ica ion 65K05 ·90C06 ·90C15 ·90C30
1 In oduc ion
In his pape , we p esen a nume ical analysis o he Con inuous S ochas ic G adien
(CSG) me hod, which was i s p oposed in [1]. La e , in [2], i was shown ha he
e o in he CSG g adien and objec i e unc ion app oxima ion anishes du ing he
cou se o he i e a ions. This key p ope y o CSG yields s ong con e gence esul s
known om classic g adien me hods, e.g., con e gence o he sequence o i e a es
o cons an s ep sizes, which a e beyond he scope o s anda d s ochas ic app oaches
known om li e a u e, like he S ochas ic G adien (SG) me hod [3], o he S ochas ic
A e age G adien (SAG) me hod [4].
Fu he mo e, he app oxima ion p ope y o CSG signi ican ly inc eases he se
o possible applica ions, allowing o mo e complex s uc u es in he op imiza ion
p oblem han he schemes lis ed be o e. While CSG was shown o pe o m be e
han a ious s ochas ic op imiza ion app oaches on academic examples [2], i emains
o see i his is also he case o mo e in ol ed applica ions. Fo his pu pose, we
conside se e al op imiza ion p oblems a ising in he con ex o op imal nanopa icle
design. These applica ions ocus on op imiza ion wi h espec o he esul ing colo
o a pa icula e p oduc , as i ep esen s one o he mos p ominen ields o esea ch
wi hin his se ing [5–10].
Mo eo e , all con e gence esul s s a ed in [2] p o ide no insigh on he a e o
con e gence. Since his plays a c ucial ole o he p ac icabili y o CSG, i is o
g ea impo ance o u he analyze his quan i y. In his con ibu ion, we conjec u e
es ima ed con e gence a es o he gene al CSG me hod and e i y hem nume ically.
1.1 S uc u e o he pape
Sec ion 2in oduces he applica ion om nanopa icle op ics, men ioned abo e. Two
di e en me hods o model he pa icle, a ying g ea ly in compu a ional e o and
design dimension, a e p esen ed. A e de ailing he se ing and challenges in he
low-dimensional op imiza ion p oblem, we compa e he esul s o he CSG me hod o
di e en app oachesbasedon he mincon algo i hmp o idedbyMATLAB(Sec .2.7).
La e on,weanalyze hehigh-dimensionalp oblem o mula ionpu elywi hin heCSG
amewo k, since a compa ison wi h gene ic de e minis ic op imiza ion schemes is ou
o scope, due o he associa ed compu a ional complexi y.
A e wa ds, Sec .3sho ly co e s echniques o es ima e he g adien app oxima-
ion e o du ing he op imiza ion, be o e we ocus on he con e gence a e o CSG
in Sec .4. While he expec ed a es s a ed he ein a e no p o en, we p esen de ailed
nume ical examples o solidi y ou claims. Fu he mo e, we analyze how he con e -
gence a edependson hedimensiono in eg a ionandhow oa oidslowcon e gence,
i he objec i e unc ion admi s addi ional s uc u e.
123
The con inuous s ochas ic g adien me hod 979
2 Nanopa icle design op imiza ion
Since he design o a nanopa icle, i.e., i s shape, size, ma e ial dis ibu ion, e c.,
hea ily impac s i s op ical p ope ies, he ask o op imizing a nanopa icle design
wi h espec o a speci ic op ical p ope y a ises na u ally [11]. In his sec ion, we a e
in e es ed in using hema i e nanopa icles o op imize he colo o a pain ilm [12].
Thus, we s a by in oducing ou main amewo k o his applica ion.
2.1 Colo spaces
Fi s o , we should explain wha op imal colo means in ou se ing. The e a e se e al
di e en me hods o desc ibe colo ma hema ically, e.g., assigning each colo an RGB
ep esen a ion ec o ∈R3, whe e he h ee componen s o co espond o he ed,
g een and blue alue o he colo . In ou applica ion, we a e in e es ed in he colo o
he pain ilm as i appea s o he human eye. The e o e, he unde lying colo space
should be chosen based on he ollowing p ope y:
I he Euclidean dis ance be ween he ep esen a ion ec o s o wo colo s is
small, he colo s should be almos indis inguishable o he human eye.
As i u ns ou , he RGB colo space is a e y poo choice wi h espec o his ea u e.
Hence, we ins ead choose he CIELAB colo space [13], which was in oduced by he
In e na ional Commission o Illumina ion (Commission In e na ionale de l’Eclai age,
CIE), as i was designed wi h his exac pu pose in mind. The CIELAB ep esen a ion
o a colo consis s o h ee alues L,aand b. He e, Lco esponds o he ligh ness
o a colo and anges om 0 (black) o 100 (whi e). The alues o aand b, ypically
wi hin he ange o ±150, desc ibe he colo s posi ion wi h espec o he opponen
colo pai s g een- ed and blue-yellow. A sho o e iew is gi en in Fig.1.
Ano he colo space, which na u ally a ises om ou se ing, is he CIE 1931 XYZ
colo space [14]. The alues o X, Y and Z can be calcula ed by in eg a ing he
op ical p ope ies o a pa icle o e he spec um o isible ligh (400–700 nm), which
we deno e by . Each o hese in eg a ions is weigh ed by he co esponding colo
ma ching unc ions x,y,z:→R.
Thus, in ou applica ion, we will i s calcula e he CIE 1931 XYZ ep esen a ion
o he esul ing colo and hen use he (nonlinea ) colo space ans o ma ion :
R3→R3wi h (X,Y,Z)=(L,a,b), o wo k in he CIELAB colo space. Fo his
ans o ma ion, we de ine a e e ence whi e poin
⎛
⎝
X
Y
Z ⎞
⎠=⎛
⎝
94.72528492
100
107.13012997⎞
⎠
and deno e he ela i e XYZ alues by
˜
X=X
X ,˜
Y=Y
Y ,and ˜
Z=Z
Z .
123
980 M. G ieshamme e al.
Fig. 1 Resul ing colo o
a ious di e en alues o aand
b. Posi i e alues o a esul in
ed colo s, while colo s
co esponding o nega i e alues
o aappea g een. Simila ly,
posi i e b alues yield yellow
colo s, while nega i e b alues
shi he colo in o he blue
spec um. In his igu e, we ixed
L=50
U ilizing he in ended CIE pa ame e s =216
24389 and κ=24389
27 , he LAB colo alues
a e hen gi en by
L=116 (˜
Y)−16,a=500 (˜
X)− (˜
Y)and b=200 (˜
Y)− (˜
Z),
whe e :R→Ris de ined as
( )=3
√ i >
κ +16
116 o he wise .
2.2 Mie heo y and disc e e dipole app oxima ion
Gi en a nanopa icle shape and ma e ial, we can use he ime-ha monic Maxwell’s
equa ions o calcula e i s op ical p ope ies. Speci ically, in ou se ing, we a e in e -
es ed in he abso p ion (Abs), sca e ing (Sca) and geome y ac o (Geo) [15, Sec ion
2.8]. These p ope ies desc ibe he in e ac ions o a pa icle wi h ligh and a e he e o e
dependen no only on he pa icle’s design, bu also i s o ien a ion w. . . he incoming
ligh wa e as well as he wa eleng h o said ligh . The ime equi ed and p ecision
achie ed in hei nume ical calcula ion a e, o cou se, dependen on ou model o he
nanopa icle and he me hod used o sol e Maxwell’s equa ions. Fo ou se ing, we
choose wo di e en app oaches.
On he one hand, we will use he disc e e dipole app oxima ion (DDA) [16–18], in
which he pa icle is disc e ized in o an equidis an g id o dipole cells. Thus, DDA
allows he analysis o a bi a y pa icle shapes and ma e ial dis ibu ions. The down-
side lies wi hin he compu a ional complexi y o he me hod, which scales wi h he
o al numbe o dipoles and he e o e g ows apidly when inc easing he esolu ion.
123

The con inuous s ochas ic g adien me hod 981
While he CSG me hod is s ill capable o sol ing he esul ing op imiza ion p ob-
lem in ou expe imen s, he emendous compu a ional cos associa ed o he DDA
app oach se e ely impede a de ailed analysis o he p oblem. Especially, he e is no
compu a ionally easible, gene ic op imiza ion scheme o compa e ou esul s wi h.
Howe e , we wan o no e ha op imiza ion in he DDA model has al eady been done
in a sligh ly simple se ing, whe e he ull in eg al o e was eplaced by summa ion
o e a small numbe o di e en wa eleng hs [19].
On he o he hand, Mie heo y [20,21] p o ides a nume ically cheap al e na i e,
a he p ice o a mo e es ic i e se ing. In Mie heo y, one only conside s adially
symme ic pa icles. In his special se ing, i is possible o ind analy ic solu ions
based on se ies expansions o he ime-ha monic Maxwell’s equa ions. The e o e, in
ou i s app oach, we will only conside co e-shell pa icles, as he u iliza ion o Mie
heo y allows o a much deepe analysis o he esul ing op imiza ion p oblem and
compa ison o de e minis ic op imiza ion app oaches, which ely on disc e iza ion o
he in eg als.
2.3 Nanopa icles in pain ilm—Kubelka–Munk heo y
As men ioned abo e, he XYZ colo alues o he pain ilm can be calcula ed by
in eg a ion o he co esponding colo ma ching unc ions x,y,zand he impo an
op ical p ope ies o he nanopa icle. The p ecise me hod o ob ain X, Y and Z is gi en
by he Kubelka–Munk heo y [22], augmen ed by a Saunde son co ec ion [23]. Fo a
pain ilm, in which nanopa icles wi h design ua e o ien ed in di ec ion ν∈S2, ha
is illumina ed by ligh wi h wa eleng h λ∈, he esul ing colo can be exp essed
by he Kand S alue
K(u,λ,ν)=Abs(u,λ,ν) and S(u,λ,ν)=Sca(u,λ,ν)
1−Geo(u,λ,ν)

ia he e lec ance
R∞(u,λ,ν)=1+8
3
K(u,λ,ν)
S(u,λ,ν) −8
3
K(u,λ,ν)
S(u,λ,ν)2
+16
3
K(u,λ,ν)
S(u,λ,ν) .
Now, X, Y and Z can be ob ained by
X(u,ν)=
x(λ)(1−ρ0−ρ1)R∞(u,λ,ν)+ρ0
1−ρ1R∞(u,λ,ν) dλ,
Y(u,ν)=
y(λ)(1−ρ0−ρ1)R∞(u,λ,ν)+ρ0
1−ρ1R∞(u,λ,ν) dλ,
Z(u,ν)=
z(λ)(1−ρ0−ρ1)R∞(u,λ,ν)+ρ0
1−ρ1R∞(u,λ,ν) dλ,
whe e ρ0and ρ1a e ma e ial pa ame e s. In ou se ing, which we in oduce in he
nex sec ion, we ha e ρ0=0.04 and ρ1=0.6. Mo eo e , x,yand za e he colo
ma ching unc ions, as gi en in [24].
123
982 M. G ieshamme e al.
Fig. 2 Radially symme ic
co e-shell nanopa icle. The
inne co e (blue) has adius Rin
he ange o 1–75 nm and
consis s o wa e . The hickness
o he hema i e shell ( ed) is
deno ed by dand anges om 1
o 250nm
2.4 P oblem o mula ion
In ou i s se ing, we conside a adially symme ic co e-shell nanopa icle (see
Fig.2), whe e heinne co e consis so wa e , while he ou e shell ismade o hema i e.
Thus, he design uconsis s o he adius R(1–75 nm) o he co e and he hickness
d(1–250 nm) o he ou e hema i e shell, i.e., we ha e u=(R,d)∈U=[1,75]×
[1,250]. Due o he symme y o he pa icle, i s op ical p ope ies do no depend on
he o ien a ion ν∈S2, which is why we omi i in ou u he analysis o his se ing.
As an addi ional laye o di icul y, we can, in p ac ice, no expec all nanopa icles
p esen in he pain ilm o be iden ical copies o design u. Ins ead, when ying o
p oduce nanopa icles o a speci ic design in la ge quan i ies, one usually ends up wi h
a mix u e o pa icles o di e en designs, ollowing a ce ain p obabili y dis ibu ion
μu, which is dependen on he in ended design u.
We model his aspec by assuming ha , gi en a design u=(R,d), he pa icles
p esen in he pain ilm ollow a unca ed no mal dis ibu ion on he space o eason-
able designs R×D=[10−4,150]×[10−4,500]cen e ed a ound u, i.e.,
˜
R∼NR(R,1
10 R)and ˜
d∼ND(d,1
10d).
T unca ing he no mal dis ibu ion o he space R×Dci cum en s nonphysical pa -
icles appea ing in he design dis ibu ions, like designs wi h nega i e componen s.
F om a nume ical poin o iew, he impac is negligible, as he combined weigh o all
excluded designs is below ypical machine p ecision, since a design componen mus
de ia e om he a e age by mo e han 9 s anda d de ia ions in o de o be ejec ed.
As he pain ilm no longe consis s o iden ical pa icles, he Kand S alues in he
Kubelka–Munk model need o be eplaced by hei a e aged coun e pa s
K(u,λ)=R×D
Abs(˜
R,˜
d,λ)dμu(˜
R,˜
d)
and
S(u,λ)=R×D
Sca(˜
R,˜
d,λ)
1−Geo(˜
R,˜
d,λ)
dμu(˜
R,˜
d),
123
The con inuous s ochas ic g adien me hod 983
be o e calcula ing he e lec ance R∞(u,λ)and in eg a ing i o e .
The objec i e in ou applica ion is o p oduce a pain o b igh ed colo . Thus, he
comple e op imiza ion p oblem eads
max
u∈U
1
20 L(u)+19
20 a(u). (1)
Due o he compac ness o U,Rand D,[2, Assump ion 2.2] is ob iously sa is ied. Fu -
he mo e, he mapping om a design u, wa eleng h λand o ien a ion ν o he op ical
p ope ies Abs, Sca and Geo is smoo h [25, Eqs. 1a, 1b, 1c]. Since e e y admissible
design has a hema i e shell o posi i e hickness, we ob ain a lowe bound on Abs
and Sca. By de ini ion, he geome y ac o is always smalle han 1 in absolu e alue.
Consequen ly, R∞depends smoo hly on Abs, Sca and Geo. Now, by cons uc ion, R∞
admi s alues in [0,1]only. The colo ma ching unc ions x,y,za e gi en poin wise
and can hus be in e pola ed wi h Lipschi z con inuous de i a i e. As a esul , X, Y,
Za eL-smoo h unc ion w. . . all a gumen s. Finally, he unc ion , appea ing in he
de ini ion o he colo ans o ma ion mapping , is cons uc ed in an L-smoo h ash-
ion as well, showing ha [2, Assump ion 2.3] is sa is ied o ou se ing. By choosing
in eg a ion weigh s p esen ed in [2, Sec ion 3], we can also sa is y [2, Assump ion
2.4].
2.5 Challenges
The highly condensed ashion, in which (1) is o mula ed, may obscu e a lo o he
di icul ies ha a ise when ying o sol e i . To ge a be e unde s anding o he
p oblem, le us i s analyze he abs ac s uc u e o he objec i e unc ion J(u)=
1
20 L(u)+19
20 a(u):
⎛
⎝
Abs
Sca
Geo⎞
⎠
in eg a e
R×D
−−−−→K
SKubelka-
Munk
−−−−−→R∞
in eg a e

−−−−→⎛
⎝
X
Y
Z⎞
⎠
colo
ans .
−−−−→⎛
⎝
L
a
b⎞
⎠−→ J(u).
Since calcula ing J(u)and ∇J(u) equi es in eg a ing he op ical p ope ies in mul i-
ple dimensions and since e alua ing said p ope ies o any combina ion o ˜
R,˜
dand
λ equi es sol ing he ime-ha monic Maxwell’s equa ions, s anda d de e minis ic
app oaches, e.g., ull g adien me hods, un in o a p edisc e iza ion p oblem.
On he one hand, he numbe o in eg a ion poin s needs o be su icien ly la ge
o ou se ing. In Fig.3, a slice h ough he objec i e unc ion o a ixed alue o R
and se e al di e en amoun s o in eg a ion poin s is shown. While we ac ually do
no ca e oo much abou he app oxima ion e o esul ing om a small numbe o
in eg a ion poin s, he a i icial local maxima in oduced in o he objec i e unc ion
by he disc e iza ion se e ely impac he quali y o he op imiza ion. In o he wo ds,
many solu ions o he disc e ized p oblem a e comple ely un ela ed o solu ions o (1).
We wan o no e ha , e en hough no all o he s a iona y poin s in Fig.3co espond
o s a iona y poin s o (1), he p edisc e iza ion s ill leads o e y la egions in he
123
984 M. G ieshamme e al.
Fig. 3 Objec i e unc ion alues
o ixed co e adius o 3 nm.
Di e en g aphs co espond o
di e en disc e iza ions. The
label o a cu e shows in o how
many poin s he in eg als o e
,Rand Dha e been spli ,
espec i ely. Each o he
disc e iza ions in oduces
a i icial s a iona y poin s in o
he objec i e unc ion
objec i e unc ions, which hinde he pe o mance o many sol e s. In Fig.4, his
e ec is displayed.
On he o he hand, he numbe o in eg a ion poin s is hea ily es ic ed by he
compu a ional cos associa ed o he e alua ion o Abs, Sca and Geo. While medium
esolu ions (253∼15000 poin s in o al) a e s ill nume ically ac able o simple Mie
pa icles, hey a e ou igh impossible o achie e in he mo e gene al DDA se ing,
which we wan o conside la e . Fo compa ison: The op imiza ion in [19] was ca ied
ou using a disc e iza ion consis ing o 20 poin s in o al.
We wan o emphasize ha s anda d SG- ype schemes, o e en he S ochas ic Com-
posi ion G adien Descen (SCGD) me hod [26], which was used o he compa ison
o composi e objec i e unc ions in [2, Sec ion 7.2], a e no capable o sol ing (1).
The eason o his lies in he special s uc u e o J, which consis s o se e al in eg als
nes ed in nonlinea unc ions.
2.6 Disc e iza ion
Fo he easons men ioned abo e, we will only compa e he esul s ob ained by CSG o
gene icde e minis ic op imiza ion schemes o a iouschoiceso disc e iza ion. Since
he in eg a ion o e admi s no special s uc u e, we always choose an equidis an
pa i ion o his dimension o in eg a ion. Howe e , o he in eg a ion o e R×D,
we can use ou knowledge o μu o achie e a be e app oxima ion o he ue in eg al.
Ins ead o di iding R×Din o an equidis an g id, we u ilize he ac ha ˜
Rand ˜
d
ollow unca ed one-dimensional no mal dis ibu ions wi h pa ame e s independen
om each o he . Since, o a no mal dis ibu ion, 99.7% o all weigh is concen a ed
in he 3σ-in e al a ound he mean alue, we may only disc e ize his po ion o he
ull domain in each s ep.
Mo eo e , we know he p ecise densi y unc ion o bo h ˜
Rand ˜
d. Thus,
gi en a design un=(Rn,dn), we will pa i ion Rn−3
10 Rn,Rn+3
10 Rnand
123
The con inuous s ochas ic g adien me hod 991
and
S(u,λ)=1
S2S2Sca(u,λ,ν)
1−Geo(u,λ,ν)
dν.
He e, S2deno es he uni sphe e and he pa icle o ien a ion νisassumed obedis-
ibu ed uni o mly andom o e all possible di ec ions.
The design domain is a ball o 300 nm diame e , disc e ized in o n0=65752 dipole
cells. The design u∈[ε, 1]n0=: Ugi es he ela i e amoun o hema i e o wa e
in each cell, wi h ε=10−4. The op ical p ope ies o in e media e (g ey) ma e ial
u(i)∈(0,1)a e gene a ed by linea in e pola ion be ween he espec i e p ope ies o
wa e and hema i e. Consequen ly, each admissible design con ains a posi i e amoun
o hema i e, esul ing in lowe bounds o Abs and Sca. As s a ed in Sec .2.4,[2,
Assump ions 2.2–2.4] a e sa is ied, since changing om Mie heo y o he DDA model
does no in e e e wi h he smoo hness o Abs, Sca and Geo w. . . (u,λ,ν),see[19,
28].
Gene ally, one would combine il e ing echniques and g eyness penaliza ion o
ob ain a smoo h inal design wi hou in e media e ma e ial (see, e.g., [29]). Howe e ,
we explici ly e ain om doing so o p esen a clea analysis o he CSG pe o mance,
wi hou in e e ence om seconda y laye s o smoo hing echniques.
As men ioned abo e, he change o he DDA model signi ican ly inc eases he com-
pu a ional cos o e alua ing Sca, Abs and Geo o a gi en (u,λ,ν)∈U××S2.
Thus, he de e minis ic app oaches used in he p e ious se ing a e no longe compu-
a ionally easible.
Fig. 11 Rep esen a ion o he ini ial designs ( op ow). Red boxes co espond o cells consis ing pu ely
o hema i e, while g ey boxes indica e an a i icial in e media e ma e ial, consis ing o 50% hema i e and
50% wa e . Fo la e e e ences, we deno e he ini ial designs by pla e (100%),pla e (50%) and sc ewd i e
(50%), espec i ely. The di e en inal designs, ob ained by 5.000 i e a ions o SCIBL-CSG wi h ou e
no m (a) a e shown in he bo om ow. Fo be e isibili y, cells wi h less han 50% hema i e a e conside ed
as pu e wa e and le ou o he isualiza ion. Fo each inal design, he amoun o cells disca ded in his
ashion is less han 100 (less han 0.15% o all cells)
123

992 M. G ieshamme e al.
Fu he mo e, we wan o use his example o analyze he impac o he chosen no m
on U××S2, appea ing in he nea es neighbo calcula ion, which was al eady
men ioned in [2, Sec ion 3.5]. To be p ecise, calcula ing he CSG in eg a ion weigh s
equi es he de ini ion o an ou e no m

(u∗,λ
∗,ν∗)
Ou =cuu∗U+cλλ∗+cνν∗S2,
Fig. 12 Objec i e unc ion
app oxima ion o he
sc ewd i e (50%) design. The
blue and o ange cu e show he
esul s o CSG wi h ixed s ep
size τ=0 and di e en
coe icien s o he ou e no m
·Ou . Fo Mon e Ca lo, each
inne in eg al o e S2was
app oxima ed using 40 andom
di ec ions. The ue objec i e
unc ion alue J∗≈37.84 is
indica ed by he dashed line. The
Mon e Ca lo esul s a e
unca ed o he sake o
eadabili y, as i equi es o e
8.000 e alua ions o each a
good app oxima ion o J∗
Fig. 13 CSG objec i e unc ion
app oxima ions du ing he
op imiza ion p ocess o all
ini ial designs and choice (a) o
·Ou , i.e., cu=1, cλ=100
and cν=100. The dashed lines
indica e he objec i e unc ion
alues o each ini ial design,
espec i ely
123
The con inuous s ochas ic g adien me hod 993
whe e ·U,·and ·S2deno e no ms on he co esponding inne spaces and
cu,cλ,cν>0. In his applica ion, we choose he Euclidean no m ·2 o each inne
space. Addi ionally, we ix cu=1, bu conside di e en coe icien s cλand cν.
Fo he op imiza ion, we conside h ee di e en ini ial designs, which a e shown
in Fig.11, op ow. The objec i e unc ion alue as well as he alues o L,aand b
o hese designs we e compu ed using he CSG me hod wi h ixed design, i.e., wi h
cons an s ep size τ=0, and e i ied by Mon e Ca lo (see, e.g., [30]) in eg a ion.
Fo one o he ini ial designs, he objec i e unc ion alue app oxima ion o CSG
and Mon e Ca lo in eg a ion wi h espec o he numbe o e alua ions and di e en
choices o ·Ou is shown in Fig.12.
Fig. 14 Top le o bo om igh : Design e olu ion du ing he op imiza ion p ocess o he sc ewd i e
(50%) ini ial design and ou e no m (a). The design snapsho s we e aken e e y 200 i e a ions. Red boxes
ep esen design cells consis ing o pu e hema i e. In e media e ma e ial is indica ed ia a colo g adien ,
whe e a cell illed wi h 50% wa e and 50% hema i e is colo ed g ey. Based on his g adien , depending on
he a io o hema i e and wa e in a cell, he cell colo is shi ed o ed (mo e hema i e) o blue (mo e wa e )
123
994 M. G ieshamme e al.
Fig. 15 Euclidean dis ance (a e
di iding by √dim(U) o
scaling) be ween in e media e
designs and he espec i e inal
design du ing he SCIBL-CSG
op imiza ion p ocess, ca ied ou
wi h ou e no m (a)
Each design was op imized wi h SCIBL-CSG, using inexac hyb id weigh s o he
in eg a ion o e S2and exac hyb id weigh s o he in eg a ion o e .Fo ·Ou ,
we conside ed ou di e en choices o he pa ame e s:
(a) cu=1, cλ=100 and cν=100
(b) cu=1, cλ=1 and cν=1
(c) cu=1, cλ=1
100 and cν=1
(d) cu=1, cλ=1
100 and cν=1
100
The esul s in case (a) o all h ee ini ial designs a e p esen ed in Fig.13 and he
espec i e design e olu ion o he ini ial design sc ewd i e (50%), shown in Fig.11
op ow, is depic ed in Fig.14. The co esponding inal designs, ob ained a e 5.000
SCIBL-CSG i e a ions, a e p esen ed in Fig.11, bo om ow. As a second measu e o
con e gence in he design space, he e olu ion o he no m dis ance o he espec i e
inal designs a e shown in Fig.15 o all h ee ini ial designs.
Compa ing Figs.12 and 13, we no ice ha CSG, using an app op ia e ou e no m,
inds an op imized design almos as as as i compu es he objec i e unc ion alue
o a gi en design. In o he wo ds: The ull op imiza ion p ocess is only sligh ly
mo e expensi e ha he simple e alua ion o a single design. Mo eo e , CSG inds
an op imal solu ion o (2) long be o e he Mon e Ca lo app oxima ion o he ini ial
objec i e unc ion alue is con e ged.
I should, o cou se, also be no ed, ha choosing ·Ou should be done wi h cau ion,
as Fig.16 shows. While case (a) is, o he bes o ou knowledge, no op imal by any
means, cases (b) and (c) clea ly show wo se esul s. Choosing ·Ou ex emely poo ly,
i.e., case (d), can e en ha e de as a ing e ec s on he pe o mance, see Fig.17.
This, howe e , could also imply ha he pe o mance migh be signi ican ly
imp o ed, i p oblem speci ic inne and ou e no ms would be chosen. Especially
123
The con inuous s ochas ic g adien me hod 995
Fig. 16 CSG objec i e unc ion
alue app oxima ion du ing he
op imiza ion p ocess o he
pla e (100%) ini ial design. The
dashed line shows he ini al
objec i e unc ion alue,
whe eas he di e en g aphs
co espond o he choices (a), (b)
and (c) o ·Ou
Fig. 17 Resul s o he pla e
(100%) ini ial design p esen ed
in Fig.16, augmen ed by he
CSG objec i e unc ion alue
app oxima ion in he case ha
·Ou was chosen acco ding o
(d)
in e en mo e complex se ings, echniques o ob ain such no ms a p io i, o e en
du ing he op imiza ion p ocess i sel , ep esen one o he mos impo an poin s o
u he esea ch.
123
996 M. G ieshamme e al.
3 Online e o es ima ion
Be o e we go in o heo e ical de ails, we i s collec a ew key p ope ies and esul s
conce ning CSG, which we e shown in [2]. In a i s simple se ing, we conside
op imiza ion p oblems o he o m
min J(u)
s. . u∈U⊂Rdo o some do∈N.(3)
Addi ionally, we assume ha Uis compac , and o some d ∈N, he e exis s an open
an bounded se X⊂Rd and a measu e μwi h supp(μ) ⊂X, such ha Jcan be
w i en as J(u)=Xj(u,x)μ(dx). The de ailed se o assump ions is gi en in [2,
Sec ion 2]. Fo now, i is only impo an ha ∇1j:U×X→Rdois bounded and
Lipschi z con inuous, i.e., he e exis C,Lj>0 wi h
∇1j(u,x)≤C,
∇1j(u1,x1)−∇1j(u2,x2)≤Lju1−u2U+x1−x2X
o all (u,x), (u1,x1), (u2,x2)∈U×X. Due o he ini e dimension o all appea ing
spaces, we can choose a bi a y no ms on U,Xand Rdo, and simply deno e hem by
·
U,·
Xand ·, espec i ely, unless speci ic choices a e made in nume ical
expe imen s.
Du ing he op imiza ion p ocess, CSG compu es design dependen in eg a ion
weigh s αkk=1...,n(c . [2, Sec ion 3]) o build an app oxima ion ˆ
Gn o he ue
objec i e unc ion g adien , based on he a ailable samples om p e ious i e a ions
∇1j(uk,xk)k=1,...,n.Tobep ecise,weha e
∇J(u)=X∇1j(u,x)μ(dx)≈
n

k=1
αk∇1j(uk,xk)=: ˆ
Gn.
I was shown in [2, Lemma 4.6], ha
∇J(un)−ˆ
Gn→0 o n→∞almos su ely.
Ca e ully in es iga ing he me hods o ob ain he in eg a ion weigh s, we obse e ha


∇J(un)−ˆ
Gn

=


X∇1j(un,x)μ(dx)−ˆ
Gn



=




n

i=1Mi∇1j(un,x)μ(dx)−
n

i=1∇1j(ui,xi)νn(Mi)




,
123

The con inuous s ochas ic g adien me hod 997
whe e νndeno es he measu e associa ed o one o he measu es lis ed in [2, Sec ion
3.6], depending on he choice o in eg a ion weigh s, and
Mk:= x∈X:un−ukU+x−xkX
<un−ujU+x−xjX o all j∈{1,...,n} {k}.
By cons uc ion, Mkcon ains all poin s x∈X, such ha (un,x)is close o (uk,xk)
han o any o he p e ious poin we e alua ed ∇1ja . Fo exac in eg a ion weigh s,
we ha e νn=μand hus


∇J(un)−ˆ
Gn

=




n

i=1Mi∇1j(un,x)μ(dx)−
n

i=1Mi∇1j(ui,xi)μ(dx)




≤
n

i=1Mi∇1j(un,x)−∇1j(ui,xi)μ(dx)
≤
n

i=1Mi
Lj·sup
x∈Mi
Zn(x)μ(dx)
=Lj
n

i=1
μ(Mi)sup
x∈Mi
Zn(x)
≤Ljsup
x∈X
Zn(x).
He e, Znis gi en by
Zn(x):= min
k∈{1,...,n}un−ukU+x−xkX.
In o he wo ds, he app oxima ion e o can be bounded in e ms o he Lipschi z
cons an o ∇1jand he quan i y Zn, which ela es o he size o Vo onoi cells [31]
wi h posi i e in eg a ion weigh s.
Bo h Ljand supx∈XZn(x)can be e icien ly app oxima ed du ing he op imiza ion
p ocess, e.g. by ini e di e ences o he samples ∇1j(ui,xi)i=1,...,nand by
sup
x∈X
Zn(x)≈max
k=1,...,nZn(xk),
yielding an online e o es ima ion. Such an app oxima ion may, o example, be used
in s opping c i e ia.
4 Con e gence a es
Th oughou his sec ion, we assume [2, Assump ions 2.1–2.4] o be sa is ied. Mo e-
o e , o he en i e sec ion, le (un)n∈Nco espond o he CSG i e a es p oduced o a
123
998 M. G ieshamme e al.
ixed andom sequence (xn)n∈N. Then, wi h p obabili y 1, we ha e

ˆ
Gn−∇J(un)
→0,
see [2, Lemma 4.6]
4.1 Theo e ical backg ound
In he con e gence analysis p esen ed in [2], we ha e al eady seen ha he ashion
in which he g adien app oxima ion ˆ
Gnis calcula ed in CSG is c ucial o ˆ
Gn−
∇J(un)→0 and ha his p ope y o CSG in u n is he key o all ad an ages CSG
o e s in compa ison o classic s ochas ic op imiza ion me hods, like con e gence o
cons an s eps, back acking, mo e in ol ed op imiza ion p oblems, e c.
The p ice we pay o his ea u e lies wi hin he dependency o ˆ
Gnon he pas
i e a es. Fo compa ison, he sea ch di ec ion ˆ
GSG
nin a s ochas ic g adien descen
me hod is gi en by
ˆ
GSG
n=∇
1j(un,xn).
Thus, i is independen o all p e ious s eps and ul ills
EXˆ
GSG
n=EX∇1j(un,·)=∇J(un),
i.e., i is an unbiased sample o he ull g adien . The combina ion o hese p ope ies
allows o a s aigh o wa d con e gence a e analysis, see, e.g., [32].
Incon as , ˆ
Gnisingene alno anunbiasedapp oxima ion o∇J(un)andmo eo e
no independen o ui,xi)i=1,...,n−1. The main p oblem in inding he con e gence
a e o un+1−unU→0 is, ha his quan i y depends on he app oxima ion e o
ˆ
Gn−∇J(un), which, as we ha e seen in Sec .3, depends on Zn. Since Zni sel is
deeply connec ed o minkun−ukU, we un in o a ci cula a gumen .
The e o e,up onow,wea eno able op o econ e gence a es o heCSGi e a es.
We can, howe e , s a e a p edic ion o his a e and p o ide nume ical e idence.
Conjec u e 4.1 We conjec u e ha he CSG me hod, applied o p oblem (3),usinga
cons an s ep size τ<2
Land empi ical in eg a ion weigh s, ul ills
un+1−unU=Oln(n)·n−1
max{2,d }
wi h p obabili y 1.
To mo i a e his claim, no e ha , in he p oo o [2, Lemma 4.6], i was shown ha
he e exis s C>0 such ha


ˆ
Gn−∇J(un)

≤CX
Zn(x)μ(dx)+dW(μn,μ)
,
123
The con inuous s ochas ic g adien me hod 999
whe e dWdeno es he Wasse s ein dis ance o he wo measu es μnand μ.By[33,
Theo em 1], he empi ical measu e μnsa is ies
EdW(μn,μ)
≤C(d )·Xx3
Xμ(dx)1
3·⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
√ni d =1,
ln(1+n)
√ni d =2,
n−1
d i d ≥3.
This esul is he main mo i a ion o Conjec u e 4.1. I can be shown ha he a e
n−1/d o d ≥3issha pi μco esponds o a uni o m dis ibu ion on X. Thus, in his
case, i is easonable o assume a uni o m dis ibu ion also co esponds o he wo s -
case a e o XZn(s)μ(dx)→0. Assuming ha he di e ence in designs appea ing
in Znis negligible due o he o e all con e gence o CSG, we ob ain he a e
sup
x∈X
Zn(x)=Oln(n)·n−1
max{2,d }.
To see his, we ill X⊂Rd wi h balls (w. . . he no m ·
X) o adius ε>0
and deno e by N(ε) ∈N he numbe o cells. Due o he dimension o X,weha e
ON(ε)=ε−d . Now, o achie e supx∈XZn(x)<ε, we need each o hese cells o
con ain a leas one o he sample poin s (xi)i=1,...,n. I is well-known ha he expec ed
numbe o samples we need o d aw o his o happen is gi en by
N(ε)
N(ε)

k=1
1
k=O−ε−d ln(ε),
whe e we used
n

k=1
1
k=Oln(n) o n→∞.
In o he wo ds, he con e gence a es o XZn(x)μ(dx)→0 and dW(μn,μ)→0
a e compa able.
Now ha we mo i a ed he a es claimed in Conjec u e 4.1 o he app oxima ion
e o ˆ
Gn−∇J(un), we use he ollowing p oposi ion o show ha he a es o
un+1−unU→0 can no be wo se.
P oposi ion 4.2 Assume ha he app oxima ion e o ˆ
Gn−∇J(un)sa is ies
ˆ
Gn−∇J(un)=Oln(n)·n−1
max{2,d }.
Then, unde he assump ions o Conjec u e 4.1, i holds
un+1−unU=Oln(n)·n−1
max{2,d }.
123
1000 M. G ieshamme e al.
P oo Assume o con adic ion ha his is no he case. Thus, he e exis s N∈N
such ha


∇J(un)−ˆ
Gn

≤1
21
τ−L
2un+1−unU o all n≥N.(4)
By he descen lemma [34, Lemma 5.7], he cha ac e is ic p ope y o he p ojec ion
ope a o [34, Theo em 6.41] and he Cauchy-Schwa z inequali y, we ob ain
J(un+1)−J(un)
≤∇J(un)(un+1−un)+L
2un+1−un2
U
=ˆ
G
n(un+1−un)+L
2un+1−un2
U+∇J(un)−ˆ
Gn(un+1−un)
≤L
2−1
τun+1−un2
U+

∇J(un)−ˆ
Gn

·un+1−unU
=L
2−1
τun+1−unU+

∇J(un)−ˆ
Gn

un+1−unU.
Combining his wi h (4)gi esJ(un+1)≤J(un) o all n≥N, since L
2<1
τ. Thus,
he sequence o objec i e unc ion alues J(un)n∈Nis mono onically dec easing o
all n≥N. By con inui y o Jand compac ness o U,Jis bounded and J(un)→¯
J
o some ¯
J∈R. The e o e,
−∞ <¯
J−J(uN)=∞

n=NJ(un+1−J(un)≤1
2L
2−1
τ∞

n=Nun+1−un2
U.
Hence, he se ies
∞

n=Nun+1−un2
U
con e ges, con adic ing un+1−unU= Oln(n)·n−1
max{2,d }.
4.2 Nume ical e i ica ion
We wan o e i y he p oclaimed a es nume ically. Fo his pu pose, we conside wo
op imiza ion p oblems ha can easily be scaled o high dimensions. The i s p oblem
is gi en by
min
u∈U
1
2X
u−x

2
2dx,(5)
123
The con inuous s ochas ic g adien me hod 1007
Acknowledgemen s The esea ch was unded by he Deu sche Fo schungsgemeinscha (DFG, Ge man
Resea ch Founda ion)—P ojec -ID 416229255—CRC 1411).
Funding Open Access unding enabled and o ganized by P ojek DEAL.
Da a a ailabili y s a emen In ou nume ical expe imen s ela ed o con e gence a es, only simple aca-
demic examples we e used o isualize he heo e ical esul s. These can be ep oduced based on he gi en
algo i hms. Fo he nanopa icle design op imiza ion, he co esponding da a is a ailable a h ps://doi.o g/
10.5281/zenodo.10032613.
Decla a ion
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and ins i u ional a ilia ions.
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