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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(1),183-202
PublishedOnlineJanuary2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.121022
ÄuFÝ6ÄÚBBÚ•S“Â KŠ©
ÛhŽ{
©©©www§§§$$$½½½777
∗
B²ŒÆ§êƆÚOÆ§B²B
ÂvFϵ2022c1228F¶¹^Fϵ2023c123F¶uÙFϵ2023c130F
Á‡
©ïÄÛhoe‘kšàK²ºx4z¯K§Ù¥›”¼ê´à¼ê§K‘
•MCP¼ê"JÑÄuFÝ6 ÄÚBarzilar-Borwein(BB)Ú•S“ KŠ©ÛhŽ
{(ISTDP)"Äk§ÄuŽ{zgS“þéFÝV\pdD(§y²TŽ{äk©Ûho
5Ÿ"Ùg§Äu±BBÚ•‰Á&Ú?1‚|¢S“ KŠŽ{§y²TŽ{Œ±Âñu
?¿‰½°Ý"Ïd§ISTDPŽ{´˜«Œ±÷vÛho‡¦ÅìÆS`zŽ{"
'…c
©Ûho§FÝ6ħ KŠŽ{§MCPK
IterativeShrinkThresholdDifferential
PrivacyAlgorithmBasedonGradient
PerturbationandBBStepSize
WenliYuan,DingtaoPeng
∗
∗ÏÕŠö"
©ÙÚ^:©w,$½7.ÄuFÝ6ÄÚBBÚ•S“ KŠ©ÛhŽ{[J].A^êÆ?Ð,2023,12(1):
183-202.DOI:10.12677/aam.2023.121022
©w§$½7
SchoolofMathematicsandStatistics,GuizhouUniversity,GuiyangGuizhou
Received:Dec.28
th
,2022;accepted:Jan.23
rd
,2023;published:Jan.30
th
,2023
Abstract
Inthispaper,westudytheproblemofempiricalriskminimizationwithnonconvex
regularizationunderprivacyprotection,wherethelossfunctionisaconvexfunction
and the regular term is theMCP function.An iterative shrinkage threshold difference
privacyalgorithm(ISTDP)basedongradientperturbationandBarzir-Borwein(BB)
stepsizeisproposed.First,ISTDPalgorithmisprovedtohavethepropertyof
differentialprivacyprotectionsinceGaussnoiseisaddedtothegradientforeach
iterationinthealgorithm.Secondly,itisprovedthattheISTDPalgorithmcan
convergeatanygivenaccuracyduetothefactthatISTDPalgorithmadoptsan
iterative shrinkage thresholdalgorithmtogether withalinesearch beginningwithBB
stepsize.Therefore,ISTDPalgorithmisakindofmachinelearningoptimization
algorithmwhichcansatisfytherequirementofprivacyprotection.
Keywords
DifferentialPrivacyProtection,GradientPerturbation,ShrinkageThreshold
Algorithm,MCPRegularization
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
3Œêâ†<óœUž“,|„ÚÅÂ8êâ´8&Ež“'…].ùêâ8
DOI:10.12677/aam.2023.121022184A^êÆ?Ð
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ŒU´¯•¿•¹¯a&E,ùÛh&E³é‡<Ûh¤î-%,l¦J
ø˜‡ˆÛhoÓž±êâŒ^5Ž{¤•˜‡-‡¯K[1].2006c‡^úi
DworkJÑ©Ûh(DifferentialPrivacy,{¡DP)Vg[2–4],Ò´•Ûhêâ©Ûþ½›
ÛhVg,Ù8´3|^ÛhêâN&EÓžoz‡‡N&E.äN5`,Ò´3˜g
ÚOÎêâ8¥O\½~˜^P¹,Œ¼AƒÓÑÑ[5].•Ò´`?Û˜^P¹,§
3Ø3êâ8¥,é(JK•ŒÑØO,lÃ{l(J¥„Ñ?Û˜^©P¹.•
y‡<ÛhØ³,Óž,qU¦©ÛhŽ{ÑÑ(J¦þO(.©Ûh.ï
~†²ºx4zEâƒ(Ü,˜„²ºx4z.Xe[6–8]:
min
θ
L(θ;D) :=
1
n
n
X
i=1
`(θ,z
i
)
Ù¥D={z
1
,···,z
n
}••¹n^¯aêâz
i
={x
i
,y
i
}∈R
d+1
êâ8.`(θ,z
i
)´Š^uêâz
i
þ
›”¼ê,L(θ;D): R
d
→RL«Š^u‡êâ8Dþ›”¼ê,§´êâ8Dþ¤kê
âz
i
ƒA›”¼ê`(θ,z
i
)²þŠ,Ø”˜„5,§´Y²k..
•;•þã.L[Ü,˜„I‡\\K‘,=XeKz²ºx4z.:
min
θ
F(θ;D) := L(θ;D)+R(θ).(1)
Ù¥R(θ)´K¼ê,~R(θ) = λkθk
1
½öλkθk
2
2
[9],λ>0´Kzëê.
é.(1),›”¼êÚK¼êþ•à¼êž,÷v©ÛhoŽ{8c~^kn«µ
1˜«´ÄuÑÑ6ÄŽ{[9,10],̇´éêâuÙ¥Î(JÚÅìÆSŽ{ÑÑ(
JV\D(,duÑÑ´3•¹kÛhêâþ,ÏdÑÑBkŒU³^rÛh[1];1
«´Äu8I6ÄŽ{[11],̇´é©ÛhÆSµee8I¼êV\D(,•Œ ±y
Ûh5,‡38I6Äœ¹e¼•`) ´5Ã;1n«´ÄuFÝ6ÄŽ{[12–15],
´•3Ž{S“L§¥éz˜S“ Ú¥FÝV\D(,ïÄL²ÐŽ{Œ±ŽÑÛh³Ú
OŽ¯K[16].
K¼êR(θ)´•“ŽL[ÜÚ\,R(θ)=kθk
1
½ökθk
2
2
K¯K)%´
k ,=OþêÆÏ"ØuOëêý¢Š.Ïd,XÛÀJK¼ê,¦T
Kz¯KQU^u“ŽL[ÜqU¦(J•O(´©•Ä-:.FanÚLi[17]•Ñ˜‡Ð
K¼êA¦)Oþäkeão‡5Ÿ:(1)à 5,(2)DÕ5,(3)ëY5,(4)
Oracle5Ÿ.ïÄö®²y²µ3•¦›”œ¹e,òU](šà)K¼êSCAD[17,18],
MCP[19]ÚCapped-`
1
[20–25])Oþäkþã5Ÿ[26,27].
É[17,19,20,24,25,27–30]šàKéu,©?رe©Ûhoe ‘kMCPK‘
DOI:10.12677/aam.2023.121022185A^êÆ?Ð
©w§$½7
²ºx4z¯K:
min
θ
F(θ;D) := L(θ;D)+R(θ),(2)
Ù¥D={z
1
,...,z
n
}∈Z
n
L«‘kn^P¹êâ8,Ù¥Z⊆R
d+1
¡•êâŠ˜m,
z
i
=(x
i
,y
i
)∈R
d+1
, i∈[n]={1,2,...,n}¡•êâP¹,¿…z˜^P¹éA˜‡êâzö,
…•¹T^r¤k&E,nL«êâ8¥•¹P¹^ê,x
i
∈R
d
¡•êâzö5Ÿ&E,
y
i
∈{0,1}⊂R¡•^rI\.L(θ;D)´1wà¼ê…äkFÝLipschitzëY,=•3˜‡~
êL>0,¦
k∇L(w,D)−∇L(u,D)k≤Lkw−uk,∀w,u∈R
d
.
R(θ)MCP(MinimaxConcavePenalty)K¼ê,=R(θ) =
P
d
i=1
φ(θ
i
),Ù¥
φ(t) :=





λ|t|−t
2
/(2α),|t|≤αλ,
αλ
2
/2,|t|>αλ,
λ>0,α>1.
ÅìÆS¥Nõ¯KÑ·^u±þ.,ÏLÀØÓ›”¼êŒ±¢yØÓÆS?Ö,
'X~„Ü6£8[11]!Ø•{[31].
©Ì‡(Xeµ1Ü©¥,©JÑÄuFÝ6ÄÚBarzilai-Borwein(BB)Ú•S“
 KŠ©Ûh(ISTDP)Ž{.1nÜ©é¤JÑISTDPŽ{?1n Ø©Û,y²ISTDPŽ{
Ø=÷v©Ûho‡¦,„Œ±Âñ?¿‰½°Ý.1oÜ©?1{üo(.
ÎÒ`²:±eÎÒ0B ©,∀t∈R,|t|L«týéŠ.kxk
1
L«•þx`
1
‰ê.kxkL«
•þx`
2
‰ê.g
k
= ∇L

θ
k
;D

L«›”¼ê3θ
k
?FÝ.θ
∗
L«8I¼ê•`).PrL«
VÇ.sign(t)•ÎÒ¼ê
sign(t) :=













1,t>0,
0,t= 0,
−1,t<0.
2.Ž{µe
!Jј«Ê·ÛhoŽ{))ÄuFÝ6ÄÚBBÚ•S“ KŠ©ÛhŽ
DOI:10.12677/aam.2023.121022186A^êÆ?Ð
©w§$½7
{(IterativeShrinkageThresholdDifferentialPrivacyAlgorithmbasedonGradientPerturbation
andBarzilar-BorweinStepSize,{¡ISTDP).
3ISTDPŽ{z˜S“Ú,éFÝg
k
V\Ñlpd©ÙN(0,σ
2
I)D(,1kÚS“
ÛhýŽ•ε
k
,Ûhëê•δ
k
ž,D(FÝ
ξ
k
= g
k
+b
k
,
b
k
∼N

0,σ
2
I

.
džŒyD(FÝξ
k
(½L§÷v(ε
k
,δ
k
)−©Ûh(½Â3.3).
ÐÚ••#üÑŒ±ŒŒ~‚|¢s¤žm,éŽ{¯„Âñ–'-‡.Barzilai-
Borwein(BB)Ú•5K[32]®²y²´š~kÚ•üÑ.Ïd,ISTDPŽ{æ^BBÚ•Š•
zgS“Á&Ú,=
α
BB
k
=

d
k
θ

T
d
k
θ

d
k
θ

T
d
k
g
Ù¥
d
k
θ
= θ
k
−θ
k−1
,
d
k
g
= g
k
−g
k−1
.
é.(2),ISTDPŽ{ÏL¦)eãf¯K5•#θ:
θ
k+1
= argmin
θ

Q(α
BB
k
,θ
k
,θ) := L

θ
k
;D

+hξ
k
,θ−θ
k
i+
1
2α
BB
k


θ−θ
k


2
+R(θ)

.(3)
¯K(3)du±eC:¯Kµ
θ
k+1
= argmin
θ

1
2


θ−u
k


2
+α
BB
k
R(θ)

,(4)
Ù¥
u
k
= θ
k
−α
BB
k
ξ
k
.
5¿k·k
2
ÚR(θ)Œ©5,¯K(4)Œ±d/©)•d‡ÕáüCþ`z¯Kµ
θ
k+1
i
= argmin
θ
i
{h(θ
i
,u
k
i
) :=
1
2
(θ
i
−u
k
i
)
2
+α
BB
k
φ
i
(θ
i
)},i= 1,2,···,d.
DOI:10.12677/aam.2023.121022187A^êÆ?Ð
©w§$½7
Ù¥u
k
i
= θ
k
i
−α
BB
k
ξ
k
i
.ùd‡üCþ`z¯KäkƒÓ(,•{zÎÒ,·‚ÏLíØþeI
5n¤XeÚ˜/ªµ
t(s) = argmin
t
{h(t,s) :=
1
2
(t−s)
2
+γφ(t)}.
Šâ[30],þã¯KäkXe4ª)µ
t(s) =





t
1
,h
1
(t
1
,s) ≤h
2
(t
2
,s),
t
2
,h
1
(t
1
,s) >h
2
(t
2
,s).
Ù¥
t
1
= argmin
t

h
1
(t,s) :=
1
2
(t−s)
2
+λγ|t|−
t
2
2α
s.t.|t|≤αλ

,
t
2
= argmin
t

h
2
(t,s) :=
1
2
(t−s)
2
+
α(λγ)
2
2
s.t.|t|≥αλ

.
•äN,ÏLOŽŒ±
t
1
= sign(s)z,
t
2
= sign(s)max(αλ,|s|),
ùpz= argmin
t∈C

1
2
(t−|s|)
2
+λγt−
t
2
2α

,Ù¥
C=






0,αλ,min

αλ,max

0,
α(|s|−γλ)
α−1

,α6= 1ž
{0,αλ},ÄK.
S.þ•¡t(s)•Â KŠ¼ê.
3ISTDPŽ{¥,Ø^BBÚ•‰Á&Ú,ïÆÉÚ•I÷v±eeüY²[30]µ
E[F(θ
k+1
;D)] ≤E[F(θ
k
;D)]−
βα
BB
k
2
E[


θ
k+1
−θ
k


2
],β∈(0,1),(5)
´„,3dÚ•OKƒe,8I¼êŠÏ"´üN4~.
¦)¯K(2)ISTDPŽ{µeXeµ
DOI:10.12677/aam.2023.121022188A^êÆ?Ð
©w§$½7
Ž{1(ISTDPŽ{)
•Ñ\:êâ8D= {z
1
,z
2
,...,z
n
}.
•ÀJëê:T>0,η∈(0,1), β∈(0,1)Ú÷v0 <α
min
<α
max
α
min
,α
max
.
•Ð©z:Àθ
0
,θ
1
∈R
d
,θ
0
6= θ
1
,OŽg
0
= ∇
θ
L(θ
0
;D),k= 1;
•Ì‚: whilek<T,do
1.1.OŽFÝ:g
k
= ∇
θ
L

θ
k
;D

,‰FÝV\D(:ξ
k
= g
k
+b
k
,b
k
∼N(0,σ
2
I);
1.2.OŽBBÚ•:d
k
θ
= θ
k
−θ
k−1
,d
k
g
= g
k
−g
k−1
,
α
BB
k
=

d
k
θ

T
d
k
θ

d
k
θ

T
d
k
g
,α
BB
k
:= min{max{α
BB
k
,α
min
},α
max
};
1.3.SÌ‚:•#θ
k+1
1.3.1.OŽ:
ˆ
θ
k+1
= argmin
θ
{Q(α
BB
k
,θ
k
,θ) = L(θ
k
;D)+hξ
k
,θ−θ
k
i+
1
2α
BB
k
kθ−θ
k
k
2
+R(θ)};
1.3.2.XJ
ˆ
θ
k+1
÷v:
E(F(θ
k
;D)−F(
ˆ
θ
k+1
;D)) ≥
βα
BB
k
2
E

k
ˆ
θ
k+1
−θ
k
k
2

,
Kθ
k+1
=
ˆ
θ
k+1
,ÊŽSÌ‚,=Ú1.4;ÄK,α
BB
k
= ηα
BB
k
,=Ú1.3.1;
1.4.•#S“Ú:k:= k+1,=Ú1.1;
•end while;
•ÑÑ: θ
priv
= θ
T
.
3.nØ©Û
!©ÛISTDPŽ{nØ5Ÿ,òy²§Ø=´˜‡÷v©Ûh‡¦Ž{,Óž•´Â
ñ,§ÑÑ(JŒ±÷v?Û‰½°Ý‡¦.
3.1.Ž{1Ûho5
e¡k0©ÛhoÄVgÚ(Ø,2ïÄŽ{1©Ûho5Ÿ.
½Â3.1[6,33]‰½êâ8D,D
0
∈Z
n
,e|(D\D
0
)∪(D
0
\D)|= 1,K¡D,D
0
∈Z
n
•ƒêâ
8,P•D∼D
0
.
5:eêâ8D,D
0
∈Z
n
´ƒ,Kêâ8D
0
Œ±ÏL3êâ8DþV\½öíؘ^P¹
.
½Â3.2[5,33]‰½?¿ü‡ƒêâ8D,D
0
∈Z
n
,M•3êâ8D,D
0
∈Z
n
þ$1˜‡
‘ÅÅ›,P
M
•M¤kŒUÑÑ(J¤8Ü.eé?¿f8S⊆P
M
÷v
Pr[M(D) ∈S] ≤e
ε
Pr
h
M

D
0

∈S
i
,
DOI:10.12677/aam.2023.121022189A^êÆ?Ð
©w§$½7
Ù¥ε¡•ÛhýŽ,K¡Å›M÷vX©Ûh,P•ε-DifferentialPrivacy,{¡ε−DP.
©ÛhŒ±yêâÛhôÂöÃ{«©ƒêâ8,?¢y3êâ?nL§¥Ø³?
Ûk'êâ8¥?¿˜‡^r&E8.
½Â3.3[8]‰½?¿ü‡ƒêâ8D,D
0
∈Z
n
,M•3êâ8D,D
0
∈Z
n
þ$1˜‡‘Å
Å›,P
M
•M¤kŒUÑÑ(J¤8Ü.eé?¿f8S⊆P
n
÷v
Pr[M(D) ∈S] ≤e
ε
Pr[M(D
0
) ∈S]+δ,
Ù¥ε¡•ÛhýŽ,δ>0¡•X©Ûh Š‡§Ý(˜„Cu0Š),K¡Å›M÷vCq
©Ûh,P•(ε,δ)-DifferentialPrivacy,{¡(ε,δ)−DP.
Cq©ÛhVg´éX©Ûhí2,…äk•2•A^.
½Â3.4[1]‰½¼êf: Z
n
→R
d
,fL
i
¯aÝ∆
i
(f)(i= 1,2)½ÂXe
∆
i
(f) =max
D∼D
0
kf(D)−f(D
0
)k
i
,
Ù¥D,D
0
∈Z
n
•?¿ƒêâ8.
¯¢þ,¯a5•x¼êfŠ^3?Ûƒêâ8eÑÑŠƒm•ŒCz.3Ø—Úå
·Ïœ¹e,ò¯a5¤∆(f).ïÄö~~Äu¼ê¯ aÝ∆(f)ÚÛhýŽε)¤D(,?
ïáƒA©ÛhoÅ›.
½n3.1‰½Š^uêâ8D⊂Z
n
¼êf: D→R
d
,∀ε∈(0,1), e‘ÅÅ›M÷v
M(D) = f(D)+N

0,σ
2
I

,
Ù¥σ÷v
σ≥
∆
2
(f)
ε
s
2log

1.25
δ

∆
2
(f)•¼êfL
2
¯aÝ,N(0,σ
2
I)L«d‘pd©Ù,KÅ›M÷v(ε,δ)−DP.
y².k?Ød=1œ/.Šâ.(2)Œ•,džé¢Š¼êf:Z
n
→RV\ÎÜIO
©ÙD(.Šâ¯ aÝ½Â3.4,∆(f)=∆
1
(f)=∆
2
(f)=max
D∼D
0
|f(D)−f(D
0
)|.D
(x∼N(0,σ
2
),Šâ©z[33],džÛh›”•
DOI:10.12677/aam.2023.121022190A^êÆ?Ð
©w§$½7





log
e
(
−1/2σ
2
)
x
2
e
(−1/2σ
2
)(x+∆f)
2





=



loge
(
−1/2σ
2
)[
x
2
−(x+∆f)
2
]



=




−
1
2σ
2

x
2
−

x
2
+2x∆f+∆f
2





=




−
1
2σ
2

2x∆f+(∆f)
2





.
|x|<
σ
2
ε
∆f
−
∆f
2
ž,




log
e
(
−1/2σ
2
)
x
2
e
(
−1/2σ
2
)
(x+∆f)
2




≤ε.•(Pr





log
e
(
−1/2σ
2
)
x
2
e
(
−1/2σ
2
)
(x+∆f)
2




≤ε

≥1−δ,I‡
Pr

|x|≥
σ
2
ε
∆f
−
∆f
2

<δ,(6)
lI‡
Pr

x≥
σ
2
ε
∆f
−
∆f
2

<
δ
2
.
0 <ε<1,t=
σ
2
ε
∆f
−
∆f
2
,K
Pr[x≥t] =
Z
∞
t
1
√
2πσ
e
−
x
2
2σ
2
dx≤
σ
√
2πt
e
−
t
2
2σ
2
.
Ïd,•I
σ
√
2π
1
t
e
−
t
2
2σ
2
<
δ
2
,
ùduI‡
σ
t
e
−
t
2
2σ
2
<
√
2πδ
2
⇔
t
σ
e
t
2
2σ
2
>
2
√
2πδ
⇔log(
t
σ
)+
t
2
2σ
2
>log(
2
√
2πδ
).
dut=
σ
2
ε
∆f
−
∆f
2
,þªqduI‡
log

σ
2
ε
∆f
−
∆f
2

1
σ

+
"

σ
2
ε
∆f
−
∆f
2

2
1
2σ
2
#
>log

2
√
2πδ

= log
r
2
π
1
δ
!
.
•d•I‡eãüªÓž¤á=Œ
log

σ
2
ε
∆f
−
∆f
2

1
σ

>0;
"

σ
2
ε
∆f
−
∆f
2

2
1
2σ
2
#
≥log
r
2
π
1
δ
!
.(7)
DOI:10.12677/aam.2023.121022191A^êÆ?Ð
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Pc=
εσ
∆f
,Kσ=
c∆f
ε
.du
1
σ

σ
2
ε
∆f
−
∆f
2

=
1
σ

(c∆f)
2
ε
2
ε
∆f
−
∆f
2

=
1
σ

c
2

∆f
ε

−
∆f
2

=
ε
c∆f

c
2

∆f
ε

−
∆f
2

= c−
ε
2c
,
5¿0 <ε≤1,Ïd,c≥
3
2
ž,c−
ε
2c
≥
7
6
,dž,log

σ
2
ε
∆f
−
∆f
2

1
σ

>0÷v.
,˜•¡,
1
2σ
2

σ
2
ε
∆f
−
∆f
2

2
=
1
2

c−
ε
2c

2
=
1
2

c
2
−ε+
ε
2
4c
2

.
c≥
3
2
…0<ε≤1ž,du

c
2
−ε+
ε
2
4c
2

0
c
>0,¤±c
2
−ε+
ε
2
4c
2
≥c
2
−
8
9
.Ïd,‡
¦
"

σ
2
ε
∆f
−
∆f
2

2
1
2σ
2
#
≥log

q
2
π
1
δ

,•I
c
2
−
8
9
>2log
r
2
π
1
δ
!
,
=•I
c
2
>2log
r
2
π
!
+2log

1
δ

+log

e
8
9

= log

2
π

+log

e
8
9

+2log

1
δ

= log

2
π
e
8
9

+2log

1
δ

,
5¿
r
2
π
e
8
9
<1.25,l•Ic
2
>2log

1.25
δ

,=•Iσ≥
∆
2
(f)
ε
s
2log

1.25
δ

.
-R
1
= {x∈R: |x|≤
c
2
∆f
ε
−
∆f
2
},R
2
= {x∈R: |x|>
c
2
∆f
ε
−
∆f
2
},KR= R
1
∪R
2
.∀S⊆R,
½Â
S
1
= {f(D)+x|x∈R
1
},S
2
= {f(D)+x|x∈R
2
},
DOI:10.12677/aam.2023.121022192A^êÆ?Ð
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K
Pr
x∼N(0,σ
2
)
[f(D)+x∈S] =Pr
x∼N(0,σ
2
)
[f(D)+x∈S
1
]+Pr
x∼N(0,σ
2
)
[f(D)+x∈S
2
]
≤Pr
x∼N(0,σ
2
)
[f(D)+x∈S
1
]+δ
≤e
ε

Pr
x∼N(0,σ
2
)
[f(D
0
)+x∈S
1
]

+δ,
ly阑œ¹epdÅ›÷v(ε,δ)-DP.
2?Ød>1œ/.éuf:D→R
d
Ú?¿ƒêâ8D,D
0
,P∆f=∆
2
(f)Úv=
f(D)−f(D
0
)(§“LV\FÝ6ĤùX&E),K•þv÷vkvk≤∆f.D(•
þx∼N(0,σ
2
I),KŠâ©z[33],Ûh›”•





log
e
(
−1
2σ
2
)
kx−µk
2
e
(
−1
2σ
2
)
kx+v−µk
2





=



loge
(
−1
2σ
2
)[
kx−µk
2
−kx+v−µk
2
]



=




1
2σ
2
(kxk
2
−kx+vk
2
)




,
Ù¥µ=(0,...,0)
T
∈R
d
.-a
1
=
v
kvk
,¿òa
1
*¿¤R
d
¥˜|IOÄa
1
,...,a
d
,K•
3β
1
,···,β
d
∈R…β
i
∼N(0,σ
2
)(i= 1,···,d),¦x=
P
m
i=1
β
i
a
i
.Px
[i]
=β
i
a
i
.5¿,dÄ
5Œhx
[i]
,vi= 0,i= 2,...,d,?h
m
P
i=2
x
[i]
,vi= 0.Ïd,
kxk
2
=
m
X
i=1


x
[i]


2
=


x
[1]


2
+
m
X
i=2


x
[i]


2
,
kx+vk
2
=





x
[1]
+
m
X
i=2
x
[i]
+v





2
=


v+x
[1]


2
+
m
X
i=2


x
[i]


2
.
dx
[1]
= β
1
b
1
= β
1
v
kvk
,


v+x
[1]


2
=

kvk(1+
β
1
kvk
)

2
= (kvk+β
1
)
2
.Ïd,
kx+vk
2
−kxk
2
= kv+x
[1]
k
2
−kx
[1]
k
2
= (kvk+β
1
)
2
−β
2
1
= kvk
2
+2β
1
kvk.
Ïkvk≤∆f,




1
2σ
2
(kxk
2
−kx+vk
2
)




=




1
2σ
2
(kvk
2
+2β
1
kvk
2
)




≤




1
2σ
2
[2β
1
∆f+(∆f)
2
]




.
duβ
1
∼N(0,σ
2
),þªL²Ûh›”•6u˜‘‘ÅCþβ
1
,=q£˜‘œ/.dc㘑
œ/(ØŒ,‘êd>1œ¹epdÅ›•,÷v(ε,δ)-DP.
½n3.2Å›M
1
(·)÷v(,δ)−DP,M
2
´?¿ØžÑÛhýŽŽ{,KM
1
ÚM
2
E
ÜM
2
(M
1
(·))÷v(,δ)−DP.
DOI:10.12677/aam.2023.121022193A^êÆ?Ð
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y².DÚD
0
´?¿ü‡ƒêâ8.-Range(M
1
) = SL«M
1
Š•8Ü.
(i)eS´lÑ8Ü,∀t∈Range(M
2
),k
Pr[M
2
(M
1
(D)) = t)] =
X
s∈S
Pr[M
1
(D) = s]Pr[M
2
(s) = t]
≤
X
s∈S
(e
ε
Pr[M
1
(D
0
) = s]+δ)Pr[M
2
(s) = t]
= e
ε
Pr[M
2
(M
1
(D
0
)) = t]+δ
(ii)eS´ëY8Ü,∀t∈Range(M
2
),
Pr[M
2
(M
1
(D)) = t]=
R
s∈S
Pr[M
1
(D) = s]Pr[M
2
(s) = t]ds,æ^†þãy²aqÜ6,Œy
²(ؤá.
©Ûh˜‡-‡A´§Ûho53õ-EÜeŒ±\È.(ε,δ)-DP\ȽnX
e.
Ún3.1[5]∀ε>0,δ>0,δ
0
>0,K(ε,δ)−DPÅ›3k-g·AEÜ(=Ž{zg ‰1(J
Ñ´dƒc(Jg·A¤)e÷v(
p
2klog(1/δ
0
)ε+kε(e
ε
−1),kδ+δ
0
)−DP.
ù˜5Ÿ¦©Ûh·^u¬zŽ{OÚ©Û:˜‡E,Ž{•¹˜X÷v©
ÛhÚ½ž,Œ±(½TŽ{NE÷v©Ûh5.
½n3.3∇
θ
L(θ;D)3Z
n
þ˜—kþ.U,‰½Û hýŽε9ëêδ,e1kgS“ÛhýŽε
k
=
min{
ε
2
√
2Tlog(2/δ)
,
1
2
p
ε
T
},ëêδ
k
=
δ
2T
,D(Y²σ=
2U
ε
k
r
2log

1.25
δ
k

,KŽ{1÷v(ε,δ)−DP.
y².3Ž{11kÚS“¥,=kD(FÝξ
k
(½I‡^ÛhýŽε
k
9ëêδ
k
.3D(
Ñlpd©ÙN(0,σ
2
I)¥ÀIOσ÷vσ=
2U
ε
k
r
2log

1.25
δ
k

≥
∆
2
(
g
k
)
δ
k
r
2log

1.25
δ
k

,Kd
½n3.1Œ•,D(FÝ)÷v(ε
k
,δ
k
)−DP.
1kÚS“´D(FÝ)¤†S“:•#Ž{EÜ,Šâ½n3.2,1kÚS“•÷
v(ε
k
,δ
k
)−DP.
duŽ{N•õS“TÚ,¿…zÚS“¥D(FÝÑdþ˜ÚS“:5(½,ÏdŽ{÷
vT-g·AEÜ.ŠâÚn3.1,δ
k
=
δ
2T
,δ
0
=
δ
2
,ε
k
= min{
ε
2
√
2Tlog(2/δ)
,
1
2
p
ε
T
}ž,k
p
2Tlog(1/δ
0
)ε
k
+Tε
k
(e
ε
k
−1) ≤
ε
2
+2Tε
2
k
≤
ε
2
+
ε
2
= ε,Tδ
k
+δ
0
= δ,
Ïd,‡Ž{S“L§÷v(ε,δ)−DP.
DOI:10.12677/aam.2023.121022194A^êÆ?Ð
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3.2.Ž{1Âñ5
!©ÛISTDPŽ{Âñ5,òy²,‘XS“gêO\§ÑÑ(JŒ ±÷v?¿‰½
°Ý‡¦.
e¡Äky²éz‡k,SÌ‚1.3k•Ú=Œ÷v.
Ún3.2éz‡k≥0,SÌ‚ÊŽOK(Ú1.3.2Øª)–õ
l
log(βα
2
max
+α
max
L)
−logη
+1
m
ÚÒ¬÷v.
y².dÚ1.3.1Œ•,Q(α
BB
k
,
ˆ
θ
k+1
,θ
k
) ≤Q(α
BB
k
,θ
k
,θ
k
),=
L(θ
k
;D)+hg
k
+b
k
,
ˆ
θ
k+1
−θ
k
i+
1
2α
BB
k
k
ˆ
θ
k+1
−θ
k
k
2
+R(
ˆ
θ
k+1
)
≤L(θ
k
;D)+R(θ
k
).(8)
Ï•∇L(θ;D)isL-LipschitzëY,
L(
ˆ
θ
k+1
;D) ≤L(θ
k
;D)+hg
k
,
ˆ
θ
k+1
−θ
k
i+
L
2
k
ˆ
θ
k+1
−θ
k
k
2
.(9)
d(8)Ú(9),
L(
ˆ
θ
k+1
;D)+R(
ˆ
θ
k+1
)+hb
k
,
ˆ
θ
k+1
−θ
k
i+
1
2
(
1
α
BB
k
−L)k
ˆ
θ
k+1
−θ
k
k
2
≤L(θ
k
;D)+R(θ
k
).(10)
d
F(
ˆ
θ
k+1
;D) = L(
ˆ
θ
k+1
;D)+R(
ˆ
θ
k+1
),
F(θ
k
;D) = L(θ
k
;D)+R(θ
k
),
ÚØª(10),
E(F(θ
k
;D)−F(
ˆ
θ
k+1
;D)) ≥
1
2
(
1
α
BB
k
−L)E

k
ˆ
θ
k+1
−θ
k
k
2

.(11)
Ïd,
1
α
BB
k
−L≥βα
BB
k
(=
1
α
BB
k
−βα
BB
k
≥L)ž,Ú1.3.2÷v.du
1
α
−αβ'uα>0´üN4
~¼ê…lim
α→0+
(
1
α
−αβ)=+∞,Ú1.3.2Ø÷vž,α
BB
k
=ηα
BB
k
´U'~η(0<η<1)Ø
α
BB
k
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ë•©z
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