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also, W W 2 2 L
2
and satises s that t EW
3
= 0. . These e properties s allow w us s to apply
Proposition5.2totherandomvariables
R
1
;:::;
R
M
.
Let0<<1tobenamedlaterandset
(
R
1
)=
R
1
+
T
p
n
log
2(1 )
p
12(2 )
p
n;
and
=
T
p
n
log
2(M 2)(1 )
:
Considerthesystemofinequalities
(C
j
)
R
j
(
R
1
)
R
k
R
j
; foreveryk6=1;j;
andrecallthatforeachj=1;:::;M wedenoteby
b
j
theweightoff
j
intheAEW
procedure.
Proposition5.3 There exist t absolute constants c
1
and c
2
for which the e following
holds. Let0<<1=2and2jM. Ifthesystem(C
j
)issatisedthen
b
j
1 :
Moreover, if c
1
 thenthe quadraticrisk of the functionproducedbythe AEW
proceduresatises
R(
~
f
AEW
)min
f2F
R(f)+c
2
:
Proof. Let2jM M andassumethat(C
j
)issatised. Recallthat t R
n
(f)isthe
empiricalriskoffandnotethatforanyk2f2;:::;Mgnfjg,
R
n
(f
k
) R
n
(f
j
)=
1
n
Xn
i=1
f
k
(X
i
)
2
f
j
(X
i
)
2
=
R
k
R
j
p
n
p
n
=
T
n
log
2(M 2)(1 )
:
(5.6)
Also,sinceU
(i)
1
1 almostsurelyforany1in,
R
n
(f
1
) R
n
(f
j
)=
1
n
Xn
i=1
f
1
(X
i
)
2
f
j
(X
i
)
2
=
R
1
R
j
p
n
p
12
2
+
2
n
Xn
i=1
U
(i)
1
!
R
1
(
R
1
)
p
n
p
12(2 )
T
n
log
2(1 )
:
(5.7)
21
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Combining(5.6)and(5.7),itisevidentthat
b
j
=
1
P
M
k=1
exp
n
T
(R
n
(f
k
) R
n
(f
j
))
1
1+(M 2)
2(M 2)(1 )
+
2(1 )
=1 :
Sincethefunctionsf
1
;:::;f
M
areindependentinL
2
(X)andEf
j
0,then
R(
~
f
AEW
)=E
0
@
XM
j=1
b
j
f
j
(X)
1
A
2
=(
b
j
)
2
Ef
2
j
+
X
‘6=j
(
b
)
2
Ef
2
+2
X
‘6=j
b
j
b
Ef
j
f
(
b
j
)
2
Ef
2
j
;
andthereisanabsoluteconstantc
0
forwhichEf
2
j
Ef
2
1
+c
0
.Hence,
(
b
j
)
2
Ef
2
j
Ef
2
1
(1 )(Ef
2
1
+c
0
) Ef
2
1
c
2
;
providedthatc
1
,giving
R(
~
f
AEW
)Ef
2
1
+c
2
=min
f2F
R(f)+c
2
;
asclaimed.
LetusformulateageneralstatementfromwhichTheoremBfollowsimmediately.
Theorem5.4 Thereexistsabsolute e constantsc
i
;i=0;:::;5andanintegern
0
for
which the following g holds. . For r any n n  n
0
, 1  c
0
p
nlogn, 0< < T T 1 1 and
c
1
T=
p
nlogn<<1=8, let M =c
2
p
nlogn, =c
3
p
(logn)=n and=n
=T
.
Set F F to o betheclassoffunctionsdenedabovewith those parameters. . Then,with
probabilityatleast
1 c
4
(+T+1)
(log
3
n)=n
(1 2)2=2
;
thereexistsj2suchthat
b
j
1 
1
n=T
:
Inparticular,withthesameprobabilityandif0T <minf1;2g,
R(
~
f
AEW
)min
f2F
R(f)+c
5
r
logM
n
:
22
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Proof. Set
P
0
=P
h
9j2f2;:::;Mgsuchthat
b
j
1 
i
;
andbyProposition5.3,
P
0
P[9j2f2;:::;Mgforwhich(C
j
)issatised]=P
1
:
Let
1
=
1
(M 1)bedenedbyP
min
2jM
R
j
1
=1 n
1
andobservethat
1
is welldenedandsatisesallthreepartsofLemma5.1for ‘=M 1. . Setting
0
=f(
R
1
)
1
g,
A=
9j2f2;:::;Mg:
R
j
(
R
1
); and
R
k
R
j
foreveryk6=1;j
 
;
and
B=
9j2f2;:::;Mg :
R
j
1
and
R
k
R
j
foreveryk6=1;j
 
:
Sincethefunctions
R
j
;j=1;:::;M areindependentthen
P
1
E
R
1
P[Aj
R
1
]1I
0
P[B]P[
0
]:
ApplyingProposition5.2,
P[B]1 
1
n
c
2
1
p
n
+
(logn)
2
p
logM
providedthatc
3
lognMc
4
p
n(logn).
TolowerboundP[
0
],notethat
P[
0
]=P
R
1
1
T
p
n
log
2(1 )
+
p
12(2 )
p
n
:
Fix0<<1=8andassumethat;andT aresuchthat
p
12(2 )
p
n 
1
and  
T
p
n
log
2(1 )
 
1
:
(5.8)
BytheBerry-EsseenTheoremand(5.1),
P[
0
]P[
R
1
(1 2)
1
]=1 P[
R
1
<(1 2)
1
]1 P[g(1 2)
1
2(W)
p
n
1 
1
p
2(1 2)j
1
j
exp
(1 2)
2
2
1
=2
2A
p
n
;
23
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andbyLemma5.1,
exp
(1 2)
2
2
1
=2
c
5
logn
M 1
log
1=2
c
5
M
logn

(1 2")2
:
Therefore,
P
0
1
n
c
2
1
p
n
+
(logn)
2
p
logM
1 c
5
log
3
n
M
(1 2)
2
!
providedthatc
2
lognMc
3
p
nlogn.
Tocompletetheproof,onehastochoseandforwhich(5.8)holds. ByLemma
5.1,
j
1
j&log
1=2
M
logn
;
andthus(5.8)holdsforandforwhich
c
8
1
n
log
M
logn

1=2
and2exp
c
9
p
n
T
log
1=2
M
logn

:
In particular,whenwetake M M 
p
nlogn,  ((logM)=n)
1=2
and= n
=T
,
thensatisestherequiredconditionaslongas&T=
p
nlognand.
p
n=logn,
aswasassumed. Also,
.(+T)
logn
p
n
;
implyingthat
P
0
1 c
8
(+T+1)
log
3
n
n
(1 2)
2
2
:
Thelowerboundonthe risk of the AEWprocedurenow follows fromProposition
5.3.
6 ProofofTheoremC
InthissectionwewillproveTheoremC,whichisre-formulatedbelow.Fromhereon
wewillassumethatthedictionaryF isnite,consistingofMfunctions,andthatthe
functionsareindexedaccordingtotheirriskinanincreasingorder. Thus,f
1
=f
F
.
Also,wewilldenoteL
f
()=Q(;f) Q(;f
1
),andthusR(f) R(f
1
)=EL
f
.
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Foreveryr>0,recallthat
(r)=log(jff2F:EL
f
rgj+1)
+
X1
j=1
2
j
log(jff2F:2
j 1
r<EL
f
2
j
rgj+1);
whichwillserveasameasureofcomplexityfortheclassF.
TherstcomponentthatisneededintheproofofTheoremCistondthelevel
(x)withthefollowingproperty: withprobabilityatleast1 2exp( x),R
n
(f
j
R
n
(f
1
)isequivalenttoR(f
j
) R(f
1
)ifR(f
j
) R(f
1
)(x). This\isomorphism"
constantwasintroducedin[5]andtoformulatetheexactpropertiesweneed,recall
thefollowingdenitionsandnotation.
IfG=L
F
istheexcesslossfunctionsclassfL
f
:f2Fg,letstar(G;0)=fg:0
1; g2Ggbethestar-shapedhullofGand0. SetG
r
=star(G;0)\fg:Eg=rg
{thatis,thesetoffunctionsinthestar-shapedhullofL
F
and0,whoseexpectation
isr.Let
r
=inffr:Esup
g2G
r
jP
n
g Pgjr=2g;
where,asalways,P
n
denotestheempiricalmeanandP isthemeanaccordingtothe
underlyingprobabilitymeasureofZ.
Theorem6.1 [5]Thereexistsanabsoluteconstantcforwhichthefollowingholds.
LetF beaclassoffunctionsboundedbyb,suchthatL
F
isa(1;B)-Bernsteinclass.
Foreveryx>0andanintegern,let
(x)=cmax
n
r
;(b+B)
x
n
o
:
(6.1)
Then,withprobabilityat least1 2exp( x),foreveryf2F F withR(f) R(f
F
)
(x),
R
n
(f) R
n
(f
F
)
1
2
(R(f) R(f
F
)):
Let=
1
(B+b)=n,where
1
isanabsoluteconstanttobenamedlater. Recall
that functions in n F F are e indexed according to their risk inan increasingorder,let
J
(x)=fj:R(f
j
) R(f
1
)(x)gandsetJ
+
(x)tobeitscomplement.Denethe
setsJ
+;0
=fj2J
+
(x):R(f
j
) R(f
1
)gandfork1,
J
+;k
=fj2J
+
(x):2
k 1
<R(f
j
) R(f
1
)2
k
g
(observethatsomeofthesetsJ
+;k
maybeempty).Set
k
0
=sup
n
k0:2
k
log(jJ
+;k
j+1)
o
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andletI=J
[
S
kk
0
J
+;k
.
FromTheorem6.1itfollowsthatforevery k0andeveryj2J
+;k
,R
n
(f
j
R
n
(f
F
) 
1
2
(R(f
j
) R(f
F
)) (because e R(f
j
) R(f
F
)   (x) ) by the denition of
J
+
(x),andsinceJ
+
(x)J
+;k
).
ThekeyingredientintheproofofTheoremCisTheorem6.2.
Theorem6.2 Thereexistabsoluteconstantsc
1
andc
2
forwhichthefollowingholds.
Let F F be e aclassoffunctionsboundedbyb,suchthatL
F
isa(1;B)-Bernsteinclass
withrespecttoaconvexriskfunctionR. Then,withprobabilityatleast1 2exp( x),
if
~
f
AEW
isproducedbytheAEWalgorithmandT c
1
(b+B),then
R(
~
f
AEW
) R(f
F
)c
2
(x)+(b+B)
2
k
0
n
;
(6.2)
where(x)hasbeendenedin(6.1).
Proof.
Let (
b
j
)
M
j=1
be the weights s of f the AEWalgorithm and set
~
f
AEW
=
P
M
j=1
b
j
f
j
tobetheaggregatefunction.SinceRisaconvexfunctionthen
R
XM
j=1
b
j
f
j
R(f
1
)
XM
j=1
b
j
(R(f
j
) R(f
1
)):
Notethatforeveryj2I,R(f
j
) R(f
1
)(x)+2
k
0
=(x)+
1
2
k
0
(b+B)=n.
Inparticular,since
P
M
j=1
b
j
=1then
X
j2I
b
j
(R(f
j
) R(f
1
))(x)+
1
2
k
0
(b+B)=n:
Onthe otherhand,withprobabilityatleast 1 2exp( x),for r every k >k
0
and
everyj2J
+;k
,
R
n
(f
j
) R
n
(f
1
)(R(f
j
) R(f
1
))=2:
ApplyingthedenitionoftheweightsintheAEWalgorithmandsince
b
1
1,
X
j2Ic
b
j
(R(f
j
) R(f
1
))=
b
1
X
j2Ic
b
j
b
1
(R(f
j
) R(f
1
))
X
j2Ic
exp
n
T
(R
n
(f
j
) R
n
(f
1
))
(R(f
j
) R(f
1
))
X
k>k
0
X
j2J
+;k
exp
n
2T
(R(f
j
) R(f
1
))
(R(f
j
) R(f
1
))=(?):
26
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Fromthedenitionofk
0
itisevidentthatforeveryk>k
0
,2
k
logjJ
+;k
j,andthus,
ifTc
1
maxfb;Bgandif
1
islargeenough,
(?)
X
k>k
0
exp
logjJ
+;k
n
2T
2
k 1
2
k

X
k>k
0
exp( c
2
n
T
2
k
)2
k
c
3
T
n
:
Indeed,thisisevidentbecauseforthatchoiceofT,(n=T)2
k
0
c
4
withc
4
beingan
absoluteconstant.
Hence,withprobabilityatleast1 2exp( x),
R(
~
f) R(f
1
)(x)+
1
2
k
0
(b+B)=n+c
3
T
n
(x)+c
5
2
k
0
b+B
n
;
asclaimed.
Thenextsteptowards theproofofTheoremCrequires severalsimplefactsre-
gardingtheempiricalprocessindexedbyalocalizationofthestar-shapedhullofa
Bernsteinclass.
Firstofall,it issimpletoverifythatthestar-shapedhullofa(1;B)-Bernstein
classisa(1;B)-Bernsteinclass aswell. . Second,ifG=star(L
F
;0)andG
r
=fh2
G:Eh=rgthen
G
r
=
[
j1
rL
f
EL
f
:f2F;2
j 1
rEL
f
2
j
r
[
j1
H
r;j
;
Inparticular,
Esup
h2G
r
1
n
n
X
i=1
h(Z
i
) Eh
1
X
i=1
E sup
h2H
r;j
1
n
n
X
i=1
h(Z
i
) Eh
:
Lemma6.3 Thereexistanabsoluteconstantcforwhichthefollowingholds. . IfL
F
isa(1;B)-Bernsteinclassw.r.t. Z,thenforeveryr r andj1,
E sup
h2H
r;j
jP
n
h Phjcmax
(
b2
j
log(jH
r;j
j+1)
n
;
r
log(jH
r;j
j+1)
n
p
rB2 j
)
:
Proof.
Fixr>0andj1,andlet
D= sup
h2H
r;j
1
n
n
X
i=1
h
2
(Z
i
)
!
1=2
:
27
Notethateveryh2H
r;j
satisesthath=rL
f
=EL
f
forsomef2F,andforwhich
EL
f
r2
j 1
. Therefore,usingtheBernsteinconditiononL
F
,
Eh
2
=r
2
E(L
f
)
2
(EL
f
)2
rB2
j+1
:
Moreover,khk
1
(r=EL
f
)kL
f
k
1
b2
j+1
.Thus,bytheGine-Zinnsymmetrization
theoremandacontractionargument(see,forexample,[12]and[17]),
ED
2
E sup
h2H
r;j
1
n
Xn
i=1
h
2
(Z
i
) Eh
2
+rB2
j+1
2
p
n
E
Z
E
"
sup
h2H
r;j
1
p
n
Xn
i=1
"
i
h
2
(Z
i
)
+rB2
j+1
b2
j+2
p
n
E
Z
E
"
sup
h2H
r;j
1
p
n
n
X
i=1
"
i
h(Z
i
)
+rB2
j+1
c
0
rb2
j+2
p
n
q
log(jH
r;j
j+1)ED+rB2
j+1
;
wherethelastinequalityisevidentbythesubgaussianpropertiesoftheRademacher
process(cf. [17]). SinceED(ED
2
)
1=2
itfollowsthat
ED
2
c
0
b2
j+2
r
log(jH
r;j
j+1)
n
(ED
2
)
1=2
+rB2
j+1
;
implyingthat
ED
2
c
1
max
b
2
2
2j
log(jH
r;j
j+1)
n
;rB2
j
:
Hence,usingasymmetrizationargumentandthesubgaussianpropertiesoftheRademacher
processonceagain,
E sup
h2H
r;j
1
n
Xn
i=1
h(Z
i
) Eh
c
2
p
n
q
log(jH
r;j
j+1)ED
c
3
max
(
b2
j
log(jH
r;j
j+1)
n
;
r
log(jH
r;j
j+1)
n
p
rB2 j
)
:
Corollary 6.4 There e exists s absolute constants c
1
and c
2
for which the e following
holds. Let t F F be e a nite class consisting of M M functions s bounded by b, such that
28
theexcesslossclassL
F
isa(1;B)-Bernsteinclass. Ifweset=c
1
(b+B)(logM)=n
then
r
c
2
b+B
n
():
Proof.
Observethatforeveryr>0,
Esup
h2G
r
1
n
n
X
i=1
h(Z
i
) Eh
X
j1
E sup
h2H
r;j
1
n
n
X
i=1
h(Z
i
) Eh
c
1
max
8
<
:
b
n
X
j1
2
j
log(jH
r;j
j+1);
r
Br
n
X
j1
2
j=2
q
log(jH
r;j
j+1)
9
=
;
c
1
b
n
0
@
log(jH
r;0
j+1)+
X
j1
2
j
log(jH
r;j
j+1)
1
A
+c
1
r
Br
n
0
@
q
log(jH
r;0
j+1)+
X
j1
2
j=2
q
log(jH
r;j
j+1)
1
A
u(r);
wherewedeneH
r;0
=
n
(rL
f
)=(EL
f
): f2F; ; EL
f
r
o
. Let t r=inffr:u(r)
r=2g.SincejH
r;j
jMforeveryj0,then
u(r)c
2
max
(
b
logM
n
;
r
rBlogM
n
)
;
andthus
rc
3
(b+B)(logM)=n=:
Moreover,thefunctionsofr
log(jH
r;0
j+1)+
X
j1
2
j
log(jH
r;j
j+1)
and
q
log(jH
r;0
j+1)+
X
j1
2
j=2
q
log(jH
r;j
j+1)
29
areincreasing,andthus,foranyr,
b
n
0
@
log(jH
r;0
j+1)+
X
j1
2
j
log(jH
r;j
j+1)
1
A
b
n
0
@
log(jH
;0
j+1)+
X
j1
2
j
log(jH
;j
j+1)
1
A
;
and
r
Br
n
0
@
q
log(jH
r;0
j+1)+
X
j1
2
j=2
q
log(jH
r;j
j+1)
1
A
r
Br
n
0
@
q
log(jH
;0
j+1)+
X
j1
2
j=2
q
log(jH
;j
j+1)
1
A
:
Hence,ifweconsider
r=c
3
b
n
0
@
log(jH
;0
j+1)+
X
j1
2
j
log(jH
;j
j+1)
1
A
+c
3
B
n
0
@
q
log(jH
;0
j+1)+
X
j1
2
j=2
q
log(jH
;j
j+1)
1
A
2
c
4
b+B
n
();
forappropriateconstantsc
3
andc
4
,thenr. Thus,itisevidentthatu(r)r=2,
andtherefore,
rc
4
b+B
n
():
Finally,since
Esup
h2G
r
jP
n
h Phju(r)
andsincer
=inffr:Esup
g2G
r
jP
n
g Pgjr=2g,thenr
r.
ProofofTheoremC.TheproofofTheoremCfollowsfromestimateson(x)and
on2
k
0
.
FromCorollary6.4itisevidentthat
(x)c
1
max
b+B
n
c
1
(b+B)
logM
n
;(b+B)
x
n
;
30
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