%!PS-Adobe-2.0 %%Creator: dvips(k) 2023.1 (TeX Live 2023) Copyright 2023 Radical Eye Software %%Title: arXiv:2502.13115v1 [cs.LG] 18 Feb 2025 %%CreationDate: Wed Feb 19 03:03:18 2025 %%Pages: 45 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: LMRoman17-Regular LMRoman12-Regular LMMono9-Regular %%+ LMRoman9-Bold LMRoman9-Regular LMRoman9-Italic LMMathItalic9-Regular %%+ LMRoman6-Regular LMMathSymbols9-Regular LMMathItalic6-Regular %%+ LMRoman5-Regular LMMathItalic5-Regular LMRoman12-Bold %%+ LMRoman10-Regular LMMathItalic10-Regular LMMathItalic7-Regular %%+ LMMathSymbols10-Regular LMRoman10-Italic LMRoman10-Bold %%+ LMRoman7-Regular LMMathSymbols7-Regular Dingbats %%+ LMMathSymbols5-Regular LMRoman8-Regular LMMathItalic8-Regular %%+ LMRoman8-Italic LMMathSymbols8-Regular LMMathSymbols6-Regular %%+ LMSans10-Regular LMSans8-Regular LMMathItalic10-Bold LMRoman7-Bold %%+ TeX-mathx10 %%DocumentPaperSizes: Letter %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z main.dvi -o %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2025.02.19:0303 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: l3backend-dvips.pro 0 0 %% %% This is file `l3backend-dvips.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% l3backend-header.dtx (with options: `header,dvips') %% %% Copyright (C) 2019-2023 The LaTeX Project %% %% It may be distributed and/or modified under the conditions of %% the LaTeX Project Public License (LPPL), either version 1.3c of %% this license or (at your option) any later version. 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The work is released % under the GUST Font License. See the MANIFEST-Latin-Modern.txt and % README-Latin-Modern.txt files for the details. 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For the most recent version of % this license see http://www.gust.org.pl/fonts/licenses/GUST-FONT-LICENSE.txt % or http://tug.org/fonts/licenses/GUST-FONT-LICENSE.txt /enclmmathsy[ /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional /prime /infinity /element /owner /triangle /triangleinv /negationslash /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection /unionmulti /logicaland /logicalor /turnstileleft /turnstileright /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv /backslash /wreathproduct /radical /coproduct /nabla /integral /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section /dagger /daggerdbl /paragraph /club /diamond /heart /spade /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /lessorequalslant /greaterorequalslant /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: lm-ts1.enc 0 0 % This file belongs to the Latin Modern package. The work is released % under the GUST Font License. See the MANIFEST-Latin-Modern.txt and % README-Latin-Modern.txt files for the details. For the most recent version of % this license see http://www.gust.org.pl/fonts/licenses/GUST-FONT-LICENSE.txt % or http://tug.org/fonts/licenses/GUST-FONT-LICENSE.txt /enclmts1[ /grave /acute /circumflex /tilde /dieresis /hungarumlaut /ring /caron /breve /macron /dotaccent /cedilla /ogonek /quotesinglbase.ts1 /.notdef /.notdef /.notdef /.notdef /quotedblbase.ts1 /.notdef /.notdef /twelveudash /threequartersemdash /cwmcapital /arrowleft /arrowright /tieaccentlowercase /tieaccentcapital /tieaccentlowercase.new /tieaccentcapital.new /.notdef /cwmascender /blanksymbol /.notdef /.notdef /.notdef /dollar /.notdef /.notdef /quotesingle.ts1 /.notdef /.notdef /asteriskmath /.notdef /comma /hyphendbl /period /fraction /zero.taboldstyle /one.taboldstyle /two.taboldstyle /three.taboldstyle /four.taboldstyle /five.taboldstyle /six.taboldstyle /seven.taboldstyle /eight.taboldstyle /nine.taboldstyle /.notdef /.notdef /angleleft /minus /angleright /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /mho /.notdef /bigcircle /.notdef /Orogate /.notdef /.notdef /.notdef /.notdef /.notdef /ohm /.notdef /.notdef /.notdef /dblbracketleft /.notdef /dblbracketright /arrowup /arrowdown /grave.ts1 /.notdef /born /divorced /died /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /leaf /married /musicalnote /.notdef /.notdef /orogate /.notdef /longs /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /tildelow /hyphendbl.alt /breve.ts1 /caron.ts1 /hungarumlaut.ts1 /dblgrave.ts1 /dagger /daggerdbl /dblverticalbar /perthousand /bullet /centigrade /dollar.oldstyle /cent.oldstyle /florin /colonmonetary /won /naira /guarani /peso /lira /recipe /interrobang /gnaborretni /dong /trademark /permyriad /paragraph.alt /baht /numero /discount /estimated /openbullet /servicemark /quillbracketleft /quillbracketright /cent /sterling /currency /yen /brokenbar /section /dieresis.ts1 /copyright /ordfeminine /copyleft /logicalnot /published /registered /macron.ts1 /degree /plusminus /two.superior /three.superior /acute.ts1 /mu /paragraph /periodcentered /referencemark /one.superior /ordmasculine /radical /onequarter /onehalf /threequarters /Euro /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /multiply /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /divide /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginProcSet: special.pro 0 0 %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup} ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N /erasepage{}N/setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{ count ocount sub{pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginProcSet: color.pro 0 0 %! TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll }repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def /TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{ setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{ /currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC /Yellow{0 0 1 0 setcmykcolor}DC/Goldenrod{0 0.10 0.84 0 setcmykcolor}DC /Dandelion{0 0.29 0.84 0 setcmykcolor}DC/Apricot{0 0.32 0.52 0 setcmykcolor}DC/Peach{0 0.50 0.70 0 setcmykcolor}DC/Melon{0 0.46 0.50 0 setcmykcolor}DC/YellowOrange{0 0.42 1 0 setcmykcolor}DC/Orange{0 0.61 0.87 0 setcmykcolor}DC/BurntOrange{0 0.51 1 0 setcmykcolor}DC /Bittersweet{0 0.75 1 0.24 setcmykcolor}DC/RedOrange{0 0.77 0.87 0 setcmykcolor}DC/Mahogany{0 0.85 0.87 0.35 setcmykcolor}DC/Maroon{0 0.87 0.68 0.32 setcmykcolor}DC/BrickRed{0 0.89 0.94 0.28 setcmykcolor}DC/Red{ 0 1 1 0 setcmykcolor}DC/OrangeRed{0 1 0.50 0 setcmykcolor}DC/RubineRed{ 0 1 0.13 0 setcmykcolor}DC/WildStrawberry{0 0.96 0.39 0 setcmykcolor}DC /Salmon{0 0.53 0.38 0 setcmykcolor}DC/CarnationPink{0 0.63 0 0 setcmykcolor}DC/Magenta{0 1 0 0 setcmykcolor}DC/VioletRed{0 0.81 0 0 setcmykcolor}DC/Rhodamine{0 0.82 0 0 setcmykcolor}DC/Mulberry{0.34 0.90 0 0.02 setcmykcolor}DC/RedViolet{0.07 0.90 0 0.34 setcmykcolor}DC /Fuchsia{0.47 0.91 0 0.08 setcmykcolor}DC/Lavender{0 0.48 0 0 setcmykcolor}DC/Thistle{0.12 0.59 0 0 setcmykcolor}DC/Orchid{0.32 0.64 0 0 setcmykcolor}DC/DarkOrchid{0.40 0.80 0.20 0 setcmykcolor}DC/Purple{ 0.45 0.86 0 0 setcmykcolor}DC/Plum{0.50 1 0 0 setcmykcolor}DC/Violet{ 0.79 0.88 0 0 setcmykcolor}DC/RoyalPurple{0.75 0.90 0 0 setcmykcolor}DC /BlueViolet{0.86 0.91 0 0.04 setcmykcolor}DC/Periwinkle{0.57 0.55 0 0 setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC /CornflowerBlue{0.65 0.13 0 0 setcmykcolor}DC/MidnightBlue{0.98 0.13 0 0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1 0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11 0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0 setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0 0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC /Aquamarine{0.82 0 0.30 0 setcmykcolor}DC/BlueGreen{0.85 0 0.33 0 setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0 0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0 1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC /PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0 setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{ 0.26 0 0.76 0 setcmykcolor}DC/OliveGreen{0.64 0 0.95 0.40 setcmykcolor} DC/RawSienna{0 0.72 1 0.45 setcmykcolor}DC/Sepia{0 0.83 1 0.70 setcmykcolor}DC/Brown{0 0.81 1 0.60 setcmykcolor}DC/Tan{0.14 0.42 0.56 0 setcmykcolor}DC/Gray{0 0 0 0.50 setcmykcolor}DC/Black{0 0 0 1 setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end %%EndProcSet %%BeginProcSet: hps.pro 0 0 %! /HPSdict 20 dict dup begin/braindeaddistill 50 def/rfch{dup length 1 sub 1 exch getinterval}bind def/splituri{dup(#)search{exch pop}{()exch} ifelse dup(file:)anchorsearch{pop exch pop 3 -1 roll pop false}{pop 3 -1 roll exch pop true}ifelse}bind def/lookuptarget{exch rfch dup /TargetAnchors where{pop TargetAnchors dup 3 -1 roll known{exch get true }{pop(target unknown:)print == false}ifelse}{pop pop (target dictionary unknown\012)print false}ifelse}bind def/savecount 0 def/stackstopped{count counttomark sub/savecount exch store stopped count savecount sub 1 sub dup 0 gt{{exch pop}repeat}{pop}ifelse}bind def /tempstring 128 string def/targetvalidate{1 index dup length 127 gt exch tempstring cvs dup(/)search{pop pop pop exch pop true exch}{pop}ifelse token{pop length 0 ne}{true}ifelse or not}bind def/targetdump-hook where {pop}{/targetdump-hook{dup mark exch gsave initmat setmatrix{{mark/Dest 4 2 roll targetvalidate{aload pop exch pop/Page 3 1 roll/View exch[exch /FitH exch]/DEST pdfmark}{cleartomark}ifelse}forall}stackstopped pop grestore}bind def}ifelse/baseurl{mark exch 1 dict dup 3 -1 roll/Base exch put/URI exch/DOCVIEW{pdfmark}stackstopped pop}bind def /externalhack systemdict/PDF known def/oldstyle true def/initmat matrix currentmatrix def/actiondict 2 dict dup/Subtype/URI put def /weblinkhandler{dup 3 1 roll mark 4 1 roll/Title 4 1 roll splituri 3 -1 roll dup length 0 gt{cvn/Dest exch 4 2 roll}{pop}ifelse{externalhack{ /HTTPFile exch}{actiondict dup 3 -1 roll/URI exch put/Action exch} ifelse}{externalhack{/HTTPFile exch}{/File exch/Action/GoToR}ifelse} ifelse counttomark 2 sub -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll /Color exch oldstyle{/LNK}{/Subtype/Link/ANN}ifelse gsave initmat setmatrix{pdfmark}stackstopped grestore}bind def/externalhandler where{ pop}{/externalhandler{2 copy{weblinkhandler}exec{/externalhack externalhack not store 2 copy{weblinkhandler}exec{/externalhack externalhack not store/oldstyle false store 2 copy{weblinkhandler}exec{ (WARNING: external refs disabled\012)print/externalhandler{pop pop}bind store externalhandler}{pop pop}ifelse}{pop pop/externalhack externalhack not store}ifelse}{pop pop/externalhandler{weblinkhandler pop}bind store} ifelse}bind def}ifelse/pdfmnew{dup type/stringtype eq{externalhandler}{ exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll/Color exch/LNK gsave initmat setmatrix pdfmark grestore}{pop pop}ifelse}ifelse}bind def/pdfmold{dup type/stringtype eq{externalhandler}{exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop pop 0 3 getinterval /Rect 3 1 roll/Border exch/LNK gsave initmat setmatrix pdfmark grestore} {pop pop}ifelse}ifelse}bind def/pdfm where{pop}{/pdfm /currentdistillerparams where{pop currentdistillerparams dup /CoreDistVersion known{/CoreDistVersion get}{0}ifelse dup braindeaddistill le{(WARNING: switching to old pdfm because version =) print ==/pdfmold}{pop/pdfmnew}ifelse load}{/pdfmark where{pop{dup type /stringtype eq{externalhandler}{2 copy mark 3 1 roll{pdfmnew} stackstopped{2 copy mark 3 1 roll{pdfmold}stackstopped{ (WARNING: pdfm disabled\012)print/pdfm{pop pop}store}{ (WARNING: new pdfm failed, switching to old pdfm\012)print/pdfm/pdfmold load store}ifelse}{/pdfm/pdfmnew load store}ifelse pop pop}ifelse}}{{ pop pop}}ifelse}ifelse bind def}ifelse end def %%EndProcSet TeXDict begin @defspecial systemdict /pdfmark known{userdict /?pdfmark systemdict /exec get put}{userdict /?pdfmark systemdict /pop get put userdict /pdfmark systemdict /cleartomark get put}ifelse /DvipsToPDF{72.27 mul Resolution div} def/PDFToDvips{72.27 div Resolution mul} def/BPToDvips{72 div Resolution mul}def product (Ghostscript) search {pop pop pop revision 927 gt}{pop false} ifelse{/BorderArrayPatch{} def}{/BorderArrayPatch{[exch{dup dup type/integertype eq exch type/realtype eq or{BPToDvips}if}forall]}def} ifelse /HyperBorder {1 PDFToDvips} def/H.V {pdf@hoff pdf@voff null} def/H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def/H.S {currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF 72 add /pdf@hoff exch def HyperBorder sub /pdf@llx exch def} def/H.L {2 sub dup/HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def/pdf@urx exch def} def/H.A {H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def} def/H.R {currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def} def /pgfHrgb{/pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfArgb} def /pgfdir { dup 0 moveto dup 5 index lineto } bind def} bind def /pgfVrgb{/pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfArgb} def /pgfdir { dup 0 exch moveto dup 5 index exch lineto } bind def} bind def /pgfArgb{ /pgfdiff 8 index round cvi 8 index round cvi sub 2 mul 1 add def 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div pgfheight 9 index 9 index 9 index 14 index pgfdiff { 3 index 3 index 3 index setrgbcolor pgfdir stroke 4 -1 roll 7 index add 4 -1 roll 6 index add 4 -1 roll 5 index add 4 -1 roll .5 sub } repeat mark 15 1 roll cleartomark exch pop }bind def /pgfR1rgb{ newpath dup dup dup 0 360 arc clip newpath dup /pgfendx exch def /pgfendy exch def 0.875 setlinewidth [] 0 setdash /pgfshade {pgfRrgb} def /pgfstartx exch def /pgfstarty exch def /pgfdiffx pgfendx pgfstartx sub def /pgfdiffy pgfendy pgfstarty sub def dup /pgfdomb exch def }bind def /pgfR2rgb{ newpath 0.5 add pgfcircx pgfcircy 3 2 roll 0 360 arc setrgbcolor fill pop}bind def /pgfRrgb{ /pgfdiff 8 index round cvi 8 index round cvi sub 4 mul 1 add def /pgfcircx pgfstartx 9 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 9 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 8 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 8 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 8 index 8 index 8 index 13 index pgfdiff { 3 index 3 index 3 index setrgbcolor pgfcircx pgfcircy 2 index 0 360 arc closepath stroke 4 -1 roll 6 index add 4 -1 roll 5 index add 4 -1 roll 4 index add 4 -1 roll .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 14 1 roll cleartomark exch pop }bind def /pgfHcmyk{ /pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfAcmyk} def /pgfdir { dup 0 moveto dup 6 index lineto } bind def} bind def /pgfVcmyk{ /pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfAcmyk} def /pgfdir { dup 0 exch moveto dup 6 index exch lineto } bind def} bind def /pgfAcmyk{ /pgfdiff 10 index round cvi 10 index round cvi sub 2 mul 1 add def 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div pgfheight 12 index 12 index 12 index 12 index 18 index pgfdiff { 4 index 4 index 4 index 4 index setcmykcolor pgfdir stroke 5 -1 roll 9 index add 5 -1 roll 8 index add 5 -1 roll 7 index add 5 -1 roll 6 index add 5 -1 roll .5 sub } repeat mark 19 1 roll cleartomark exch pop }bind def /pgfR1cmyk{ newpath dup dup dup 0 360 arc clip newpath dup /pgfendx exch def /pgfendy exch def 0.875 setlinewidth [] 0 setdash /pgfshade {pgfRcmyk} def /pgfstartx exch def /pgfstarty exch def /pgfdiffx pgfendx pgfstartx sub def /pgfdiffy pgfendy pgfstarty sub def dup /pgfdomb exch def }bind def /pgfR2cmyk{ newpath 0.5 add pgfcircx pgfcircy 3 2 roll 0 360 arc setcmykcolor fill pop}bind def /pgfRcmyk{ /pgfdiff 10 index round cvi 10 index round cvi sub 4 mul 1 add def /pgfcircx pgfstartx 11 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 11 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 10 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 10 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 3 index 8 index sub pgfdiff div 11 index 11 index 11 index 11 index 17 index pgfdiff { 4 index 4 index 4 index 4 index setcmykcolor pgfcircx pgfcircy 2 index 0 360 arc closepath stroke 5 -1 roll 8 index add 5 -1 roll 7 index add 5 -1 roll 6 index add 5 -1 roll 5 index add 5 -1 roll .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 18 1 roll cleartomark exch pop }bind def /pgfHgray{ /pgfheight exch 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/pgfcircx pgfstartx 5 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 5 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 4 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 4 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def dup 2 index sub pgfdiff div 2 index 5 index pgfdiff { 1 index setgray pgfcircx pgfcircy 2 index 0 360 arc closepath stroke exch 2 index add exch .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 6 1 roll cleartomark exch pop }bind def /pgfsc{}bind def/pgffc{}bind def/pgfstr{stroke}bind def/pgffill{fill}bind def/pgfeofill{eofill}bind def/pgfe{a dup 0 rlineto exch 0 exch rlineto neg 0 rlineto closepath}bind def/pgfw{setlinewidth}bind def/pgfs{save pgfpd 72 Resolution div 72 VResolution div neg scale magscale{1 DVImag div dup scale}if pgfx neg pgfy neg translate pgffoa .setopacityalpha}bind def/pgfr{pgfsd 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pgfcheckcolorrange 16777215 mul round cvi 1 index 0 2 index -16 bitshift put 1 index 1 2 index 65535 and -8 bitshift put 1 index 2 2 index 255 and put pop } bind def /pgfcheckcolorrange { dup 0.0 lt {pop 0.0} if dup 1.0 gt {pop 1.0} if } bind def /pgfchanneldepthcmyk 8 def /pgfchanneldepthrgb 8 def /pgfchanneldepthgray 24 def /pgfcolorsamplecmyk 4 string def /pgfcolorsamplergb 3 string def /pgfcolorsamplegray 3 string def /pgfrangecmyk [0 1 0 1 0 1 0 1] def /pgfrangergb [0 1 0 1 0 1] def /pgfrangegray [0 1] def /pgf1{gsave exec 1.0 pgfw 2.00002 0.0 moveto -6.00006 4.00005 lineto -3.00003 0.0 lineto -6.00006 -4.00005 lineto pgffill grestore} bind def /pgf2{gsave exec 1.0 pgfw 0.8 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -3.00003 4.00005 moveto -2.75002 2.50002 0.0 0.24998 0.75 0.0 curveto 0.0 -0.24998 -2.75002 -2.50002 -3.00003 -4.00005 curveto pgfstr grestore} bind def /pgf3{gsave exec 1.0 pgfw [ ] 0.0 setdash 0.0 -5.00005 moveto 0.0 5.00005 lineto pgfstr grestore} bind def /pgf4{gsave exec 1.0 pgfw [ ] 0.0 setdash -3.00003 -5.00005 moveto 0.0 -5.00005 lineto 0.0 5.00005 lineto -3.00003 5.00005 lineto pgfstr grestore} bind def /pgf5{gsave exec 1.0 pgfw [ ] 0.0 setdash -2.00002 -5.00005 moveto 1.0 -3.00003 1.0 3.00003 -2.00002 5.00005 curveto pgfstr grestore} bind def /pgf6{gsave exec 1.0 pgfw [ ] 0.0 setdash -4.50003 -5.00005 moveto 0.49998 0.0 lineto -4.50003 5.00005 lineto pgfstr grestore} bind def /pgf7{gsave exec 1.0 pgfw -2.50002 0.0 translate [ ] 0.0 setdash 3.00003 0.0 moveto 3.00003 1.65689 1.65689 3.00003 0.0 3.00003 curveto -1.65689 3.00003 -3.00003 1.65689 -3.00003 0.0 curveto -3.00003 -1.65689 -1.65689 -3.00003 0.0 -3.00003 curveto 1.65689 -3.00003 3.00003 -1.65689 3.00003 0.0 curveto closepath gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath grestore} bind def /pgf8{gsave exec 1.0 pgfw [ ] 0.0 setdash 1.0 0.0 moveto -5.00005 3.00003 lineto -11.00012 0.0 lineto -5.00005 -3.00003 lineto closepath gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath grestore} bind def @fedspecial end %%BeginFont: TeX-mathx10 %!PS-AdobeFont-1.0: TeX-mathx10 001.001 % Filtered by type1fix.pl 0.08 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see copyright of original TeX font) def /FamilyName (TeX mathx10) def /FullName (TeX mathx10 Regular) def /ItalicAngle 0 def /Notice (converted after April 2001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-mathx10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 132 /Dcaron put readonly def /FontBBox {-12 -2972 2309 776} readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8B9CA5870C5A2275FCFF321AE516727784E85F7D8A47E122BCE55A2922C3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark {restore}if %%EndFont %%BeginFont: LMRoman7-Bold %!PS-AdobeFont-1.0: LMRoman7-Bold 2.004 %%CreationDate: 7th October 2009 % Generated by MetaType1 (a MetaPost-based engine) % Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman7-Bold known{/LMRoman7-Bold findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 694 194 0 %%EndComments FontDirectory/LMMathItalic10-Bold known{/LMMathItalic10-Bold findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman10-Bold known{/LMRoman10-Bold findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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%!PS-AdobeFont-1.0: LMMathItalic8-Regular 1.200 %%CreationDate: 16th September 2009 % Generated by MetaType1 (a MetaPost-based engine) % Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 556 156 0 %%EndComments FontDirectory/LMMathItalic8-Regular known{/LMMathItalic8-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 347 97 0 %%EndComments FontDirectory/LMMathSymbols5-Regular known{/LMMathSymbols5-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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cleartomark {restore}if %%EndFont %%BeginFont: Dingbats %!PS-AdobeFont-1.0: Dingbats 001.005 %%CreationDate: Tue Oct 19 1999 % Copyright URW Software, Copyright 1997 by URW % URW Software, Copyright 1997 by URW % See the file COPYING (GNU General Public License) for license conditions. % As a special exception, permission is granted to include this font % program in a Postscript or PDF file that consists of a document that % contains text to be displayed or printed using this font, regardless % of the conditions or license applying to the document itself. 12 dict begin /FontInfo 10 dict dup begin /version (001.005) readonly def /Notice (URW Software, Copyright 1997 by URW. See the file COPYING (GNU General Public License) for license conditions. As a special exception, permission is granted to include this font program in a Postscript or PDF file that consists of a document that contains text to be displayed or printed using this font, regardless of the conditions or license applying to the document itself.) readonly def /Copyright (Copyright URW Software, Copyright 1997 by URW) readonly def /FullName (Dingbats ) readonly def /FamilyName (Dingbats) readonly def /Weight (Regular) readonly def /ItalicAngle 0.0 def /isFixedPitch false def /UnderlinePosition -72 def /UnderlineThickness 36 def end readonly def /FontName /Dingbats def /PaintType 0 def /WMode 0 def /FontBBox {-1 -143 981 819} readonly def /FontType 1 def /FontMatrix [0.001 0.0 0.0 0.001 0.0 0.0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 51 /a19 put dup 55 /a23 put readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LMMathSymbols7-Regular %!PS-AdobeFont-1.0: LMMathSymbols7-Regular 1.200 %%CreationDate: 16th September 2009 % Generated by MetaType1 (a MetaPost-based engine) % Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 486 136 0 %%EndComments FontDirectory/LMMathSymbols7-Regular known{/LMMathSymbols7-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def /FullName(LMMathSymbols7-Regular)readonly def /FamilyName(LMMathSymbols7)readonly def /Weight(Normal)readonly def /isFixedPitch false def /ItalicAngle -14.0362 def /UnderlinePosition -102 def /UnderlineThickness 34 def end readonly def /FontName /LMMathSymbols7-Regular def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /PaintType 0 def /FontType 1 def /StrokeWidth 0 def /FontMatrix[0.001 0 0 0.001 0 0]readonly def %/UniqueID 0 def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0525392EECAC163E584A9104D99AD0BC 1B1844A0E222653FA481B8809B26A46F4C483A5D7E95816EA6582584156CFEDE B994ADCFF4645140E3617E4D7E1B0E4541CB9F562E55829B4DD880AABE2229E9 4A9FA259A734D29BBA91BA1E2055CBEA4339BCBFF98D32CEFF11F296225CAABA DCA10577A5D431B714726C1278D8101ABD1BD8D0BD0174FFF9148F8C61C241D9 2AD360A28616CB4A0670C1BF105BFA1C4BAA966BE78FD2F618C28B35E2A9ABAF 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%!PS-AdobeFont-1.0: LMMathItalic7-Regular 1.200 %%CreationDate: 16th September 2009 % Generated by MetaType1 (a MetaPost-based engine) % Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 486 136 0 %%EndComments FontDirectory/LMMathItalic7-Regular known{/LMMathItalic7-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 694 194 0 %%EndComments FontDirectory/LMMathItalic10-Regular known{/LMMathItalic10-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 347 97 0 %%EndComments FontDirectory/LMMathItalic5-Regular known{/LMMathItalic5-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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%!PS-AdobeFont-1.0: LMRoman5-Regular 2.004 %%CreationDate: 7th October 2009 % Generated by MetaType1 (a MetaPost-based engine) % Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman5-Regular known{/LMRoman5-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman9-Bold known{/LMRoman9-Bold findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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MetaType1 (a MetaPost-based engine) % Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 778 222 0 %%EndComments FontDirectory/LMMono9-Regular known{/LMMono9-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman12-Regular known{/LMRoman12-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 805 195 0 %%EndComments FontDirectory/LMRoman17-Regular known{/LMRoman17-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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2175 a SDict begin [/View [/XYZ H.V]/Dest (cite.foster2022complexity) cvn /DEST pdfmark end -42 2175 a 0 TeXcolorgray 42 w FL(D.)c(J.)g(F)-7 b(oster,)29 b(A.)g(Rakhlin,)g(A.)g(Sekhari,)f(and)h(K.)g(Sridharan.)39 b(On)28 b(the)h(complexit)n(y)g(of)g(adv)n(ersarial)c(decision)k (making.)83 2275 y FF(A)l(dvanc)l(es)h(in)g(Neur)l(al)f(Information)i (Pr)l(o)l(c)l(essing)g(Systems)p FL(,)c(35:35404\02535417,)21 b(2022.)p 0 TeXcolorgray -208 2424 a SDict begin H.S end -208 2424 a -42 2424 a SDict begin 12 H.A end -42 2424 a -42 2424 a SDict begin [/View [/XYZ H.V]/Dest (cite.foster2023tight) cvn /DEST pdfmark end -42 2424 a 0 TeXcolorgray 42 w FL(D.)29 b(J.)e(F)-7 b(oster,)28 b(N.)g(Golo)n(wic)n(h,)f(and)h (Y.)g(Han.)38 b(Tigh)n(t)28 b(guaran)n(tees)e(for)h(in)n(teractiv)n(e)g (decision)h(making)f(with)i(the)f(decision-)83 2524 y(estimation)37 b(co)r(e\036cien)n(t.)67 b(In)38 b FF(The)i(Thirty)g(Sixth)e(A)l(nnual) g(Confer)l(enc)l(e)i(on)f(L)l(e)l(arning)g(The)l(ory)p FL(,)j(pages)36 b(3969\0254043.)83 2623 y(PMLR,)27 b(2023.)p 0 TeXcolorgray -208 2773 a SDict begin H.S end -208 2773 a -42 2773 a SDict begin 12 H.A end -42 2773 a -42 2773 a SDict begin [/View [/XYZ H.V]/Dest (cite.han2021generalized) cvn /DEST pdfmark end -42 2773 a 0 TeXcolorgray 42 w FL(Y.)21 b(Han,)i(Z.)d(Liang,)i(Y.)f(W)-7 b(ang,)22 b(and)f(J.)g(Zhang.)k (Generalized)20 b(linear)g(bandits)h(with)g(lo)r(cal)g(di\033eren)n (tial)f(priv)-5 b(acy)e(.)26 b FF(A)l(dvanc)l(es)83 2873 y(in)k(Neur)l(al)f(Information)i(Pr)l(o)l(c)l(essing)g(Systems)p FL(,)c(34:26511\02526522,)21 b(2021.)p 0 TeXcolorgray -208 3022 a SDict begin H.S end -208 3022 a -42 3022 a SDict begin 12 H.A end -42 3022 a -42 3022 a SDict begin [/View [/XYZ H.V]/Dest (cite.hazan2016volumetric) cvn /DEST pdfmark end -42 3022 a 0 TeXcolorgray 42 w FL(E.)31 b(Hazan)g(and)g(Z.)h(Karnin.) 47 b(V)-7 b(olumetric)32 b(spanners:)43 b(an)32 b(e\036cien)n(t)f (exploration)f(basis)h(for)g(learning.)48 b FF(The)34 b(Journal)f(of)83 3122 y(Machine)f(L)l(e)l(arning)d(R)l(ese)l(ar)l(ch)p FL(,)f(17\(1\):4062\0254095,)23 b(2016.)p 0 TeXcolorgray -208 3271 a SDict begin H.S end -208 3271 a -42 3271 a SDict begin 12 H.A end -42 3271 a -42 3271 a SDict begin [/View [/XYZ H.V]/Dest (cite.he2022reduction) cvn /DEST pdfmark end -42 3271 a 0 TeXcolorgray 42 w FL(J.)28 b(He,)g(J.)g(Zhang,)f(and)g (R.)i(Q.)e(Zhang.)37 b(A)28 b(reduction)g(from)f(linear)g(con)n (textual)h(bandits)f(lo)n(w)n(er)g(b)r(ounds)h(to)g(estimations)83 3371 y(lo)n(w)n(er)e(b)r(ounds.)37 b(In)28 b FF(International)i(Confer) l(enc)l(e)g(on)g(Machine)i(L)l(e)l(arning)p FL(,)c(pages)e (8660\0258677.)e(PMLR,)j(2022.)p 0 TeXcolorgray -208 3520 a SDict begin H.S end -208 3520 a -42 3520 a SDict begin 12 H.A end -42 3520 a -42 3520 a SDict begin [/View [/XYZ H.V]/Dest (cite.karwa2017finite) cvn /DEST pdfmark end -42 3520 a 0 TeXcolorgray 42 w FL(V.)51 b(Karw)n(a)c(and)j(S.)h(V)-7 b(adhan.)103 b(Finite)51 b(sample)f(di\033eren)n(tially)g(priv)-5 b(ate)49 b(con\034dence)h(in)n(terv)-5 b(als.)103 b FF(arXiv)51 b(pr)l(eprint)83 3620 y(arXiv:1711.03908)p FL(,)33 b(2017.)p 0 TeXcolorgray -208 3769 a SDict begin H.S end -208 3769 a -42 3769 a SDict begin 12 H.A end -42 3769 a -42 3769 a SDict begin [/View [/XYZ H.V]/Dest (cite.kasiviswanathan2011can) cvn /DEST pdfmark end -42 3769 a 0 TeXcolorgray 42 w FL(S.)22 b(P)-7 b(.)22 b(Kasivisw)n(anathan,)f(H.)h(K.)g(Lee,)h(K.)f(Nissim,)h (S.)f(Raskho)r(dnik)n(o)n(v)-5 b(a,)21 b(and)h(A.)h(Smith.)28 b(What)23 b(can)e(w)n(e)h(learn)f(priv)-5 b(ately?)83 3869 y FF(SIAM)30 b(Journal)f(on)h(Computing)p FL(,)e (40\(3\):793\025826,)c(2011.)p 0 TeXcolorgray -208 4018 a SDict begin H.S end -208 4018 a -42 4018 a SDict begin 12 H.A end -42 4018 a -42 4018 a SDict begin [/View [/XYZ H.V]/Dest (cite.lattimore2020exploration) cvn /DEST pdfmark end -42 4018 a 0 TeXcolorgray 42 w FL(T.)39 b(Lattimore)g(and)g(C.)g(Szep)r (esv\341ri.)70 b(Exploration)37 b(b)n(y)i(optimisation)g(in)g(partial)f (monitoring.)70 b(In)40 b FF(Confer)l(enc)l(e)h(on)83 4118 y(L)l(e)l(arning)30 b(The)l(ory)p FL(,)f(pages)e(2488\0252515.)c (PMLR,)k(2020.)p 0 TeXcolorgray -208 4267 a SDict begin H.S end -208 4267 a -42 4267 a SDict begin 12 H.A end -42 4267 a -42 4267 a SDict begin [/View [/XYZ H.V]/Dest (cite.lei2024privacy) cvn /DEST pdfmark end -42 4267 a 0 TeXcolorgray 42 w FL(Y.)22 b(Lei,)h(S.)e(Miao,)i(and)e(R.)h(Momot.)k(Priv)-5 b(acy-preserving)18 b(p)r(ersonalized)j(rev)n(en)n(ue)f(managemen)n(t.) 26 b FF(Management)f(Scienc)l(e)p FL(,)83 4367 y(70\(7\):4875\0254892,) e(2024.)p 0 TeXcolorgray -208 4516 a SDict begin H.S end -208 4516 a -42 4516 a SDict begin 12 H.A end -42 4516 a -42 4516 a SDict begin [/View [/XYZ H.V]/Dest (cite.li2024optimal) cvn /DEST pdfmark end -42 4516 a 0 TeXcolorgray 42 w FL(J.)i(Li,)g(D.)h(Simc)n(hi-Levi,)f(and)f(Y.)i(W)-7 b(ang.)32 b(On)24 b(the)h(optimal)g(regret)f(of)h(lo)r(cally)f(priv)-5 b(ate)24 b(linear)g(con)n(textual)h(bandit.)32 b FF(arXiv)83 4616 y(pr)l(eprint)e(arXiv:2404.09413)p FL(,)k(2024.)p 0 TeXcolorgray -208 4765 a SDict begin H.S end -208 4765 a -42 4765 a SDict begin 12 H.A end -42 4765 a -42 4765 a SDict begin [/View [/XYZ H.V]/Dest (cite.li2019nearly) cvn /DEST pdfmark end -42 4765 a 0 TeXcolorgray 42 w FL(Y.)41 b(Li,)k(Y.)c(W)-7 b(ang,)44 b(and)d(Y.)g(Zhou.)76 b(Nearly)40 b(minimax-optimal)h(regret)e(for)i(linearly)f(parameterized)f(bandits.) 77 b(In)83 4865 y FF(Confer)l(enc)l(e)31 b(on)e(L)l(e)l(arning)h(The)l (ory)p FL(,)g(pages)c(2173\0252174.)d(PMLR,)28 b(2019.)p 0 TeXcolorgray -208 5015 a SDict begin H.S end -208 5015 a -42 5015 a SDict begin 12 H.A end -42 5015 a -42 5015 a SDict begin [/View [/XYZ H.V]/Dest (cite.perchet2016batched) cvn /DEST pdfmark end -42 5015 a 0 TeXcolorgray 42 w FL(V.)g(P)n(erc)n (het,)e(P)-7 b(.)28 b(Rigollet,)f(S.)h(Chassang,)e(and)h(E.)g(Sno)n(wb) r(erg.)36 b(Batc)n(hed)27 b(bandit)h(problems.)36 b(2016.)p 0 TeXcolorgray -208 5164 a SDict begin H.S end -208 5164 a -42 5164 a SDict begin 12 H.A end -42 5164 a -42 5164 a SDict begin [/View [/XYZ H.V]/Dest (cite.shariff2018differentially) cvn /DEST pdfmark end -42 5164 a 0 TeXcolorgray 42 w FL(R.)30 b(Shari\033)f(and)g(O.)g(She\033et.)43 b(Di\033eren)n(tially)29 b(priv)-5 b(ate)29 b(con)n(textual)f(linear)h(bandits.)42 b FF(A)l(dvanc)l(es)31 b(in)h(Neur)l(al)f(Information)83 5264 y(Pr)l(o)l(c)l(essing)f(Systems)p FL(,)e(31,)e(2018.)p 0 TeXcolorgray 1908 5649 a(16)p 0 TeXcolorgray eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray -208 83 a SDict begin H.S end -208 83 a -42 83 a SDict begin 12 H.A end -42 83 a -42 83 a SDict begin [/View [/XYZ H.V]/Dest (cite.simchi2020bypassing) cvn /DEST pdfmark end -42 83 a 0 TeXcolorgray 42 w FL(D.)30 b(Simc)n(hi-Levi)g(and)g(Y.)g(Xu.)44 b(Bypassing)29 b(the)h(monster:)41 b(A)30 b(faster)g(and)f(simpler)h(optimal)g(algorithm)e(for)i(con)n (textual)83 183 y(bandits)e(under)f(realizabilit)n(y)-7 b(.)36 b FF(arXiv)30 b(pr)l(eprint)g(arXiv:2003.12699)p FL(,)k(2020.)p 0 TeXcolorgray -208 332 a SDict begin H.S end -208 332 a -42 332 a SDict begin 12 H.A end -42 332 a -42 332 a SDict begin [/View [/XYZ H.V]/Dest (cite.vershynin2019high) cvn /DEST pdfmark end -42 332 a 0 TeXcolorgray 42 w FL(R.)28 b(V)-7 b(ersh)n(ynin.)36 b(High-dimensional)27 b(probabilit)n(y)-7 b(,)27 b(2019.)p 0 TeXcolorgray -208 482 a SDict begin H.S end -208 482 a -42 482 a SDict begin 12 H.A end -42 482 a -42 482 a SDict begin [/View [/XYZ H.V]/Dest (cite.vietri2020private) cvn /DEST pdfmark end -42 482 a 0 TeXcolorgray 42 w FL(G.)38 b(Vietri,)j(B.)d(Balle,)i(A.)e(Krishnam) n(urth)n(y)-7 b(,)39 b(and)f(S.)g(W)-7 b(u.)68 b(Priv)-5 b(ate)37 b(reinforcemen)n(t)g(learning)g(with)h(pac)g(and)f(regret)83 581 y(guaran)n(tees.)e(In)27 b FF(International)k(Confer)l(enc)l(e)f (on)g(Machine)i(L)l(e)l(arning)p FL(,)27 b(pages)g(9754\0259764.)c (PMLR,)28 b(2020.)p 0 TeXcolorgray -208 731 a SDict begin H.S end -208 731 a -42 731 a SDict begin 12 H.A end -42 731 a -42 731 a SDict begin [/View [/XYZ H.V]/Dest (cite.zheng2020locally) cvn /DEST pdfmark end -42 731 a 0 TeXcolorgray 42 w FL(K.)22 b(Zheng,)g(T.)g(Cai,)h(W.)f(Huang,)h(Z.)f(Li,)h(and)f(L.) g(W)-7 b(ang.)27 b(Lo)r(cally)21 b(di\033eren)n(tially)h(priv)-5 b(ate)21 b(\(con)n(textual\))h(bandits)g(learning.)83 830 y FF(A)l(dvanc)l(es)30 b(in)g(Neur)l(al)f(Information)i(Pr)l(o)l(c) l(essing)g(Systems)p FL(,)c(33:12300\02512310,)21 b(2020.)p 0 TeXcolorgray 1908 5649 a(17)p 0 TeXcolorgray eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.18) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 0 a SDict begin H.S end 0 0 a 0 0 a SDict begin 12 H.A end 0 0 a 0 0 a SDict begin [/View [/XYZ H.V]/Dest (appendix.A) cvn /DEST pdfmark end 0 0 a 83 x FM(A)134 b(T)-11 b(ec)l(hnical)45 b(to)t(ols)0 115 y SDict begin H.S end 0 115 a 0 115 a SDict begin 12 H.A end 0 115 a 0 115 a SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.21) cvn /DEST pdfmark end 0 115 a 0 115 a SDict begin H.S end 0 115 a 0 115 a SDict begin 12 H.A end 0 115 a 0 115 a SDict begin [/View [/XYZ H.V]/Dest (lemma.A.1) cvn /DEST pdfmark end 0 115 a 0 TeXcolorgray 150 x Fp(Lemma)32 b(A.1)f(\(Gaussian)h(concen)m(tration\))p 0 TeXcolorgray 42 w FF(F)-6 b(or)32 b(the)f(r)l(andom)h(ve)l(ctor)f FK(z)f FI(2)c FG(R)2657 235 y FJ(d)2732 265 y FF(with)32 b(i.i.d.)46 b Fo(N)p FH(\(0)p FK(;)14 b(\033)3347 235 y FC(2)3384 265 y FH(\))31 b FF(entries,)h(with)0 364 y(pr)l(ob)l(ability)g(at)e(le)l(ast)f FH(1)18 b FI(\000)g FK(\016)s FF(,)942 532 y FI(k)p FK(z)t FI(k)1067 491 y FC(2)p 942 570 163 4 v 979 646 a FK(\033)1029 622 y FC(2)1138 589 y FI(\024)k FK(d)d FH(+)f(2)1412 514 y FD(p)p 1494 514 352 4 v 1494 589 a FK(d)c FH(log)q(\(1)p FK(=\016)s FH(\))k(+)g(2)c(log)o(\(1)p FK(=\016)s FH(\))23 b FI(\024)g FH(2)p FK(d)18 b FH(+)g(3)c(log)o(\(1)p FK(=\016)s FH(\))p FK(:)0 789 y FF(F)-6 b(or)30 b(the)g(r)l(andom)h(symmetric)f (matrix)g FK(Z)f FI(2)24 b FG(R)1486 759 y FJ(d)p FB(\002)p FJ(d)1647 789 y FF(with)31 b(i.i.d.)41 b(Gaussian)31 b(entries)f FK(Z)2732 801 y FJ(ij)2813 789 y FH(=)24 b FK(Z)2959 801 y FJ(j)s(i)3040 789 y FI(\030)f Fo(N)p FH(\(0)p FK(;)14 b(\033)3348 759 y FC(2)3385 789 y FH(\))31 b(\(1)23 b FI(\024)g FK(i)g FI(\024)g FK(j)28 b FI(\024)0 888 y FK(d)p FH(\))p FF(,)j(with)f(pr)l(ob)l(ability)i(at)e(le)l(ast)f FH(1)18 b FI(\000)g FK(\016)s FF(,)1454 1063 y FI(k)p FK(Z)6 b FI(k)1600 1088 y FC(op)1697 1063 y FI(\024)22 b FK(C)6 b(\033)1899 988 y FD(p)p 1983 988 440 4 v 1983 1063 a FK(d)19 b FH(+)f(log\(1)p FK(=\016)s FH(\))p FK(;)0 1227 y FF(wher)l(e)30 b FK(C)36 b FF(is)31 b(an)e(absolute)i(c)l (onstant.)0 1227 y SDict begin H.S end 0 1227 a 0 1227 a SDict begin 12 H.A end 0 1227 a 0 1227 a SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.22) cvn /DEST pdfmark end 0 1227 a 0 1227 a SDict begin H.S end 0 1227 a 0 1227 a SDict begin 12 H.A end 0 1227 a 0 1227 a SDict begin [/View [/XYZ H.V]/Dest (lemma.A.2) cvn /DEST pdfmark end 0 1227 a 0 TeXcolorgray 150 x Fp(Lemma)h(A.2)f(\(Matrix)h (Bernstein's)e(inequalit)m(y)i(\()p 0 0 0.7 TeXcolorrgb 1854 1377 a SDict begin H.S end 1854 1377 a Fp(V)-8 b(ersh)m(ynin)2272 1377 y SDict begin 12 H.L end 2272 1377 a 2272 1377 a SDict begin [/Subtype /Link/Dest (cite.vershynin2019high) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 2272 1377 a 0 TeXcolorgray 33 w Fp([)p 0 0 0.7 TeXcolorrgb 2332 1377 a SDict begin H.S end 2332 1377 a Fp(2019)2524 1377 y SDict begin 12 H.L end 2524 1377 a 2524 1377 a SDict begin [/Subtype /Link/Dest (cite.vershynin2019high) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 2524 1377 a 0 TeXcolorgray Fp(,)29 b(Theorem)j(5.4.1]\)\))p 0 TeXcolorgray 39 w FF(L)l(et)21 b FK(X)3528 1389 y FC(1)3565 1377 y FK(;)14 b(X)3671 1389 y FC(2)3708 1377 y FK(;)g(:)g(:)g(:)g(;)g(X)3962 1389 y FJ(n)0 1476 y FF(b)l(e)37 b(i.i.d.)62 b(zer)l(o-me)l(an)37 b(symmetric)h(matric)l(es)f(in)g FG(R)1657 1446 y FJ(d)p FB(\002)p FJ(d)1788 1476 y FF(.)60 b(Supp)l(ose)38 b(that)f FI(k)o FK(X)2484 1488 y FC(1)2521 1476 y FI(k)2563 1501 y FC(op)2673 1476 y FI(\024)e FK(R)j FF(almost)f(sur)l(ely,)i(and)f (let)f FK(\033)3762 1446 y FC(2)3835 1476 y FH(=)0 1521 y FD(\015)0 1571 y(\015)46 1592 y FG(E)8 b FH([)q FK(X)195 1561 y FC(2)188 1612 y(1)237 1592 y FH(])260 1521 y FD(\015)260 1571 y(\015)307 1625 y FC(op)381 1592 y FF(.)38 b(Then,)31 b(for)g(al)6 b(l)31 b FK(t)23 b FI(\025)f FH(0)p FF(,)1116 1854 y FG(P)1182 1712 y FD( )1246 1709 y(\015)1246 1758 y(\015)1246 1808 y(\015)1246 1858 y(\015)1246 1908 y(\015)1332 1750 y FJ(n)1292 1775 y FD(X)1298 1952 y FJ(i)p FC(=1)1426 1854 y FK(X)1495 1866 y FJ(i)1522 1709 y FD(\015)1522 1758 y(\015)1522 1808 y(\015)1522 1858 y(\015)1522 1908 y(\015)1592 1854 y FI(\025)g FK(t)1709 1712 y FD(!)1798 1854 y FI(\024)h FH(2)p FK(d)14 b FH(exp)2125 1737 y FD(\022)2186 1854 y FI(\000)2393 1798 y FK(t)2423 1768 y FC(2)2460 1798 y FK(=)p FH(2)p 2261 1835 416 4 v 2261 1911 a FK(n\033)2361 1887 y FC(2)2416 1911 y FH(+)19 b FK(R)q(t=)p FH(3)2686 1737 y FD(\023)2761 1854 y FK(:)0 2025 y SDict begin H.S end 0 2025 a 0 2025 a SDict begin 12 H.A end 0 2025 a 0 2025 a SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.23) cvn /DEST pdfmark end 0 2025 a 0 2025 a SDict begin H.S end 0 2025 a 0 2025 a SDict begin 12 H.A end 0 2025 a 0 2025 a SDict begin [/View [/XYZ H.V]/Dest (lemma.A.3) cvn /DEST pdfmark end 0 2025 a 0 TeXcolorgray 70 x Fp(Lemma)32 b(A.3)p 0 TeXcolorgray 40 w FF(Supp)l(ose)e(that)g FK(V)1060 2107 y FC(1)1098 2095 y FK(;)14 b FI(\001)g(\001)g(\001)27 b FK(;)14 b(V)1344 2107 y FJ(n)1419 2095 y FF(ar)l(e)30 b(i.i.d.)41 b(p)l(ositive)31 b(semi-de\034nite)f(r)l(andom)g(matric)l(es)g(such)g(that)f FG(E)9 b FH([)p FK(V)3673 2107 y FJ(i)3706 2095 y FH(])23 b(=)g(\006)0 2195 y FF(and)30 b FI(k)p FK(V)251 2207 y FJ(i)279 2195 y FI(k)320 2220 y FC(op)417 2195 y FI(\024)23 b FH(1)29 b FF(almost)h(sur)l(ely.)39 b(Then)30 b(for)h(any)f(\034xe)l (d)f(p)l(ar)l(ameter)i FK(c)23 b FI(2)g FH([0)p FK(;)14 b FH(1])p FF(,)30 b(with)g(pr)l(ob)l(ability)i(at)e(le)l(ast)g FH(1)18 b FI(\000)g FK(\016)s FF(:)845 2441 y FH(\(1)g FI(\000)g FK(c)p FH(\)\006)h FI(\000)1260 2385 y FH(4)14 b(log)o(\(2)p FK(d=\016)s FH(\))p 1260 2422 394 4 v 1413 2498 a FK(nc)1663 2441 y FE(I)23 b FI(\026)1824 2385 y FH(1)p 1820 2422 50 4 v 1820 2498 a FK(n)1933 2337 y FJ(n)1893 2362 y FD(X)1900 2539 y FJ(i)p FC(=1)2027 2441 y FK(V)2075 2453 y FJ(i)2126 2441 y FI(\026)g FH(\(1)18 b(+)g FK(c)p FH(\)\006)h(+)2629 2385 y(4)14 b(log)o(\(2)p FK(d=\016)s FH(\))p 2629 2422 394 4 v 2783 2498 a FK(nc)3032 2441 y(:)0 2750 y SDict begin H.S end 0 2750 a 0 2750 a SDict begin 12 H.A end 0 2750 a 0 2750 a SDict begin [/View [/XYZ H.V]/Dest (section*.18) cvn /DEST pdfmark end 0 2750 a Fp(Pro)s(of.)82 b FL(Fix)29 b(a)f FK(\025)c(>)g FH(0)k FL(to)g(b)r(e)g(sp)r(eci\034ed)h(later.)38 b(Let)1730 2729 y FD(e)1720 2750 y FK(V)1768 2762 y FJ(i)1820 2750 y FH(=)24 b(\(\006)19 b(+)f FK(\025)p FE(I)p FH(\))2219 2720 y Fz(\000)p FO(1)p FN(=)p FO(2)2356 2750 y FK(V)2404 2762 y FJ(i)2433 2750 y FH(\(\006)h(+)f FK(\025)p FE(I)p FH(\))2743 2720 y Fz(\000)p FO(1)p FN(=)p FO(2)2880 2750 y FL(.)39 b(Note)28 b(that)3324 2680 y FD(\015)3324 2730 y(\015)3380 2729 y(e)3370 2750 y FK(V)3418 2762 y FJ(i)3446 2680 y FD(\015)3446 2730 y(\015)3492 2784 y FC(op)3590 2750 y FI(\024)3692 2718 y FC(1)p 3689 2732 40 4 v 3689 2779 a FJ(\025)3766 2750 y FL(and)0 2884 y FG(E)60 2863 y FD(e)49 2884 y FK(V)48 b FI(\026)23 b FE(I)p FL(,)28 b(and)f(hence)1596 3081 y FG(E)1656 3060 y FD(e)1645 3081 y FK(V)1718 3047 y FC(2)1694 3102 y FJ(i)1778 3081 y FI(\026)c FG(E)1929 3025 y FH(1)p 1925 3062 49 4 v 1925 3138 a FK(\025)2000 3060 y FD(e)1990 3081 y FK(V)2038 3093 y FJ(i)2089 3081 y FI(\026)2190 3025 y FH(1)p 2186 3062 V 2186 3138 a FK(\025)2245 3081 y FE(I)p FK(:)0 3269 y FL(Then)28 b(b)n(y)p 0 0 0.7 TeXcolorrgb 332 3269 a SDict begin H.S end 332 3269 a 0 0 0.7 TeXcolorrgb FL(Lemma)f(A.2)p 0 0 0.7 TeXcolorrgb 755 3211 a SDict begin H.R end 755 3211 a 755 3269 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.A.2) cvn H.B /ANN pdfmark end 755 3269 a 0 TeXcolorgray FL(,)h(with)g(probabilit)n(y)f(at)g(least)g FH(1)18 b FI(\000)g FK(\016)s FL(,)942 3357 y FD(\015)942 3407 y(\015)942 3457 y(\015)942 3507 y(\015)942 3557 y(\015)1027 3399 y FJ(n)988 3424 y FD(X)994 3601 y FJ(i)p FC(=1)1132 3482 y FD(e)1122 3503 y FK(V)1170 3515 y FJ(i)1216 3503 y FI(\000)g FG(E)1359 3482 y FD(e)1348 3503 y FK(V)1397 3515 y FJ(i)1430 3357 y FD(\015)1430 3407 y(\015)1430 3457 y(\015)1430 3507 y(\015)1430 3557 y(\015)1477 3613 y FC(op)1574 3503 y FI(\024)1661 3373 y FD(r)p 1744 3373 464 4 v 1754 3447 a FH(2)p FK(n)p 1754 3484 92 4 v 1776 3560 a(\025)1869 3503 y FH(log\(2)p FK(d=\016)s FH(\))h(+)2319 3447 y(2)14 b(log)o(\(2)p FK(d=\016)s FH(\))p 2319 3484 394 4 v 2470 3560 a(3)p FK(\025)2745 3503 y FH(=:)23 b FK(c)2892 3515 y FJ(\025)2935 3503 y FK(:)0 3792 y FL(Under)28 b(this)f(ev)n(en)n(t,)h(w)n(e)f(ha)n(v)n(e)f FK(v)1012 3762 y Fz(>)1062 3700 y FD(\020)1111 3730 y(P)1199 3751 y FJ(n)1199 3817 y(i)p FC(=1)1335 3772 y FD(e)1324 3792 y FK(V)1372 3804 y FJ(i)1419 3792 y FI(\000)18 b FG(E)1562 3772 y FD(e)1551 3792 y FK(V)1600 3804 y FJ(i)1633 3700 y FD(\021)1683 3792 y FK(v)26 b FI(\024)d FK(c)1873 3804 y FJ(\025)1930 3792 y FI(k)p FK(v)s FI(k)2056 3751 y FC(2)2121 3792 y FL(for)k(all)h FK(v)e FI(2)d FG(R)2562 3762 y FJ(d)2607 3792 y FL(,)28 b(and)f(hence)1121 4056 y FK(n)p FH(\006)19 b FI(\000)f FK(c)1369 4068 y FJ(\025)1412 4056 y FH(\(\006)h(+)f FK(\025)p FE(I)p FH(\))24 b FI(\026)1873 3952 y FJ(n)1834 3977 y FD(X)1840 4154 y FJ(i)p FC(=1)1967 4056 y FK(V)2015 4068 y FJ(i)2066 4056 y FI(\026)f FK(n)p FH(\006)18 b(+)g FK(c)2401 4068 y FJ(\025)2445 4056 y FH(\(\006)g(+)h FK(\025)p FE(I)p FH(\))p FK(:)0 4319 y FL(Therefore,)26 b(w)n(e)i(c)n(ho)r(ose)e FK(\025)e FH(=)954 4279 y FC(4)12 b(log\(2)p FJ(d=\016)r FC(\))p 954 4300 316 4 v 1060 4348 a FJ(nc)1131 4331 y FO(2)1280 4319 y FL(,)27 b(whic)n(h)h(ensures)f FK(c)1896 4331 y FJ(\025)1962 4319 y FI(\024)c FK(nc)k FL(and)1156 4557 y FH(\(1)18 b FI(\000)g FK(c)p FH(\)\006)g FI(\000)g FK(c\025)p FE(I)24 b FI(\026)1806 4501 y FH(1)p 1802 4538 50 4 v 1802 4614 a FK(n)1915 4454 y FJ(n)1875 4478 y FD(X)1882 4655 y FJ(i)p FC(=1)2009 4557 y FK(V)2057 4569 y FJ(i)2108 4557 y FI(\026)f FH(\(1)18 b(+)g FK(c)p FH(\)\006)h(+)f FK(c\025)p FE(I)p FK(;)0 4794 y FL(with)28 b(probabilit)n(y)f(at)g(least)g FH(1)18 b FI(\000)g FK(\016)s FL(.)2723 b Fd(2)0 4810 y SDict begin H.S end 0 4810 a 0 4810 a SDict begin 12 H.A end 0 4810 a 0 4810 a SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.24) cvn /DEST pdfmark end 0 4810 a 0 4810 a SDict begin H.S end 0 4810 a 0 4810 a SDict begin 12 H.A end 0 4810 a 0 4810 a SDict begin [/View [/XYZ H.V]/Dest (lemma.A.4) cvn /DEST pdfmark end 0 4810 a 0 TeXcolorgray 133 x Fp(Lemma)32 b(A.4)p 0 TeXcolorgray 40 w FF(Supp)l(ose)h(that)g FH(\()p FK(X)1119 4955 y FJ(t)1148 4943 y FH(\))1180 4955 y FJ(t)1242 4943 y FF(is)f(a)h(ve)l(ctor-value)l(d)g(martingale-di\033er)l(enc)l(e)h(se) l(quenc)l(e)e(adapte)l(d)i(to)f(the)f(\034ltr)l(ation)0 5043 y FH(\()p FI(F)92 5055 y FJ(t)121 5043 y FH(\))153 5055 y FJ(t)183 5043 y FF(.)38 b(A)l(ssume)29 b(that)h FI(k)p FK(X)833 5055 y FJ(t)861 5043 y FI(k)903 5055 y FC(2)963 5043 y FI(\024)23 b FK(c)1087 5055 y FJ(t)1146 5043 y FF(almost)30 b(sur)l(ely.)38 b(Then)31 b(it)f(holds)h(that)f FI(8)p FK(x)23 b FI(\025)g FH(0)983 5313 y FG(P)1035 5146 y FD(0)1035 5295 y(@)1106 5168 y(\015)1106 5217 y(\015)1106 5267 y(\015)1106 5317 y(\015)1106 5367 y(\015)1192 5209 y FJ(n)1152 5234 y FD(X)1158 5410 y FJ(t)p FC(=1)1286 5313 y FK(X)1355 5325 y FJ(i)1383 5168 y FD(\015)1383 5217 y(\015)1383 5267 y(\015)1383 5317 y(\015)1383 5367 y(\015)1429 5422 y FC(2)1489 5313 y FI(\025)1577 5143 y FD(v)1577 5189 y(u)1577 5239 y(u)1577 5289 y(t)p 1664 5143 207 4 v 1704 5209 a FJ(n)1664 5234 y FD(X)1669 5410 y FJ(t)p FC(=1)1798 5313 y FK(c)1834 5284 y FC(2)1834 5333 y FJ(t)1890 5313 y FH(+)18 b FK(x)2020 5146 y FD(1)2020 5295 y(A)2116 5313 y FI(\024)k FH(exp)2344 5196 y FD(\022)2405 5313 y FI(\000)2609 5257 y FK(x)2656 5227 y FC(2)p 2480 5294 344 4 v 2480 5370 a FH(2)2536 5308 y FD(P)2623 5329 y FJ(n)2623 5395 y(t)p FC(=1)2750 5370 y FK(c)2786 5342 y FC(2)2786 5391 y FJ(t)2833 5196 y FD(\023)2894 5313 y FK(:)p 0 TeXcolorgray 1908 5649 a FL(18)p 0 TeXcolorgray eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.19) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 83 a SDict begin H.S end 0 83 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FK(Z)3416 195 y FJ(t)3445 183 y FI(j)40 b(\024)g FK(c)3649 195 y FJ(t)p FC(+1)3801 183 y FL(for)0 282 y FH(0)23 b FI(\024)f FK(t)h FI(\024)g FK(n)18 b FI(\000)g FH(1)p FL(.)37 b(Therefore,)26 b(Azuma's)i(inequalit)n(y)f(implies)g(that)1252 513 y FG(P)p FH(\()o FK(Z)1392 525 y FJ(n)1456 513 y FI(\000)18 b FK(Z)1596 525 y FC(0)1656 513 y FI(\025)k FK(x)p FH(\))i FI(\024)f FH(exp)2075 396 y FD(\022)2136 513 y FI(\000)2339 457 y FK(x)2386 427 y FC(2)p 2210 494 344 4 v 2210 571 a FH(2)2266 509 y FD(P)2353 529 y FJ(n)2353 596 y(t)p FC(=1)2480 571 y FK(c)2516 543 y FC(2)2516 592 y FJ(t)2563 396 y FD(\023)2625 513 y FK(:)0 766 y FL(The)28 b(desired)f(inequalit)n (y)g(follo)n(ws)f(from)i(the)g(fact)f(that)h FK(Z)1850 778 y FC(0)1910 766 y FH(=)23 b FG(E)8 b FH([)q FI(k)p FK(S)2164 778 y FJ(n)2214 766 y FI(k)2256 778 y FC(2)2293 766 y FH(])23 b FI(\024)2427 694 y FD(p)p 2510 694 272 4 v 72 x FG(E)8 b FI(k)p FK(S)2652 778 y FJ(n)2703 766 y FI(k)2745 738 y FC(2)2745 788 y(2)2805 766 y FI(\024)2892 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b(norm:)39 b FI(k)o FH(\010)p FI(k)1147 1087 y FC(F)1217 1062 y FI(\024)22 b(k)p FK(\036)p FI(k)p FF(.)0 1277 y SDict begin H.S end 0 1277 a 0 1277 a SDict begin 12 H.A end 0 1277 a 0 1277 a SDict begin [/View [/XYZ H.V]/Dest (section*.20) cvn /DEST pdfmark end 0 1277 a Fp(Pro)s(of.)82 b FL(Denote)28 b FK(U)692 1247 y FC(1)p FJ(=)p FC(2)796 1277 y FK(\036)c FH(=)e(\()p FK(v)1028 1289 y FC(1)1066 1277 y FK(;)14 b FI(\001)g(\001)g(\001)28 b FK(;)14 b(v)1305 1289 y FJ(d)1344 1277 y FH(\))p FL(,)28 b(then)g FH(\010)23 b(=)f FI(k)p FK(U)9 b(\036)p FI(k)1985 1247 y FB(\000)p FC(1)2074 1277 y FH(\()p FK(v)2146 1289 y FJ(i)2174 1277 y FK(v)2214 1289 y FJ(j)2249 1277 y FH(\))2281 1289 y FJ(i;j)2383 1277 y FI(2)23 b FG(R)2515 1247 y FJ(d)p FB(\002)p FJ(d)2646 1277 y FL(,)28 b(and)f(hence)252 1545 y FI(k)p FH(\010)p FI(k)396 1557 y FJ(F)473 1545 y FH(=)c FI(k)p FK(U)9 b(\036)p FI(k)760 1511 y FB(\000)p FC(1)848 1403 y FD( )957 1441 y FJ(d)914 1466 y FD(X)920 1643 y FJ(i)p FC(=1)1048 1545 y 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652 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.B.1) cvn H.B /ANN pdfmark end 1838 652 a 0 TeXcolorgray FL(,)g(w)n(e)h(ha)n(v)n(e)1137 906 y FG(P)1189 872 y FN(M)1242 887 y(\022)1275 872 y(;)p Fn(Alg)1361 906 y FH(\()p FK(E)5 b FH(\))23 b FI(\024)g FH(1)p FK(:)p FH(4)k FG(P)1788 872 y FN(M)1841 884 y FO(0)1872 872 y FN(;)p Fn(Alg)1958 906 y FH(\()p FK(E)5 b FH(\))19 b(+)2200 850 y(1)p 2200 887 42 4 v 2200 963 a(4)2251 906 y FK(;)180 b FI(8)p FK(E)29 b FI(\022)22 b FH(\005)p FK(:)0 1150 y FL(In)32 b(this)h(problem,)f(the)h(decision)e(space)h(is)g FH(\005)f(=)f([)p FI(\000)p FH(1)p FK(;)14 b FH(1])p FL(,)32 b(and)g(the)g(loss)g(function)g(is)g FK(L)p FH(\()p FK(\022)r(;)14 b(\031)s FH(\))31 b(=)f FG(E)3200 1162 y FJ(\036)p FB(\030)p FJ(p)3336 1058 y FD(h)3375 1150 y FI(j)p FK(\036)p FH(\()p FK(\031)23 b FI(\000)18 b FK(\022)r FH(\))p FI(j)3728 1108 y FC(2)3766 1058 y FD(i)3835 1150 y FH(=)0 1285 y FK(\025)48 1255 y FC(2)86 1285 y FI(j)p 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1495 y FN(;)p Fn(Alg)3051 1529 y FH(\()p FK(E)5 b FH(\))p FK(:)0 1735 y FL(Th)n(us,)547 1966 y FH(max)15 b FI(f)p FG(E)808 1932 y FN(M)861 1947 y(\022)901 1932 y(;)p Fn(Alg)987 1966 y FK(L)p FH(\()p FK(\022)r(;)f(\031)s FH(\))p FK(;)g FG(E)1322 1932 y FN(M)1376 1944 y FO(0)1414 1932 y FN(;)p Fn(Alg)1500 1966 y FK(L)p FH(\(0)p FK(;)g(\031)s FH(\))p FI(g)22 b(\025)1916 1910 y FK(\025)1964 1880 y FC(2)2002 1910 y FK(\022)2043 1880 y FC(2)p 1916 1947 V 1978 2023 a FH(8)2090 1966 y(\()q FG(P)2175 1932 y FN(M)2228 1947 y(\022)2260 1932 y(;)p Fn(Alg)2346 1966 y FH(\()p FK(E)2444 1932 y FJ(c)2478 1966 y FH(\))d(+)f FG(P)2664 1932 y FN(M)2717 1944 y FO(0)2748 1932 y FN(;)p Fn(Alg)2834 1966 y FH(\()p FK(E)5 b FH(\)\))1814 2180 y FI(\025)1916 2124 y FK(\025)1964 2094 y FC(2)2002 2124 y FK(\022)2043 2094 y FC(2)p 1916 2161 V 1978 2237 a FH(8)2090 2063 y FD(\022)2152 2180 y FG(P)2204 2146 y FN(M)2257 2161 y(\022)2289 2146 y(;)p Fn(Alg)2375 2180 y FH(\()p FK(E)2473 2146 y FJ(c)2507 2180 y FH(\))19 b(+)f(0)p FK(:)p FH(7)27 b FG(P)2827 2146 y FN(M)2880 2161 y(\022)2912 2146 y(;)p Fn(Alg)2998 2180 y FH(\()p FK(E)5 b FH(\))19 b FI(\000)3240 2124 y FH(1)p 3240 2161 42 4 v 3240 2237 a(5)3292 2063 y FD(\023)1814 2416 y FI(\025)1916 2359 y FK(\025)1964 2329 y FC(2)2002 2359 y FK(\022)2043 2329 y FC(2)p 1916 2397 165 4 v 1957 2473 a FH(16)2113 2416 y FI(\025)k FK(c)2237 2428 y FC(0)2288 2416 y FH(min)2441 2299 y FD(\032)2503 2416 y FK(\025;)2805 2359 y FH(1)p 2598 2397 454 4 v 2598 2473 a FK(\025T)2707 2449 y FC(2)2744 2473 y FH(\()p FK(\013)c FH(+)f FK(\014)t FH(\))3014 2449 y FC(2)3062 2299 y FD(\033)3138 2416 y FK(:)0 2643 y FL(This)28 b(is)f(the)h(desired)f(lo)n(w)n(er)f(b)r(ound.)2672 b Fd(2)0 2718 y SDict begin H.S end 0 2718 a 0 2718 a SDict begin 12 H.A end 0 2718 a 0 2718 a SDict begin [/View [/XYZ H.V]/Dest (appendix.C) cvn /DEST pdfmark end 0 2718 a 200 x FM(C)135 b(Pro)t(ofs)45 b(from)p 0 0 0.7 TeXcolorrgb 976 2918 a SDict begin H.S end 976 2918 a 0 0 0.7 TeXcolorrgb FM(Section)g(4)p 0 0 0.7 TeXcolorrgb 1515 2836 a SDict begin H.R end 1515 2836 a 1515 2918 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 1515 2918 a 0 TeXcolorgray 0 2950 a SDict begin H.S end 0 2950 a 0 2950 a SDict begin 12 H.A end 0 2950 a 0 2950 a SDict begin [/View [/XYZ H.V]/Dest (subsection.C.1) cvn /DEST pdfmark end 0 2950 a 166 x Fl(C.1)112 b(Pro)s(of)37 b(of)p 0 0 0.7 TeXcolorrgb 717 3116 a SDict begin H.S end 717 3116 a 0 0 0.7 TeXcolorrgb Fl(Lemma)i(4.1)p 0 0 0.7 TeXcolorrgb 1257 3048 a SDict begin H.R end 1257 3048 a 1257 3116 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.4.1) cvn H.B /ANN pdfmark end 1257 3116 a 0 TeXcolorgray 0 3269 a FL(By)29 b(de\034nition,)i FI(rL)653 3281 y FJ(U)709 3269 y FH(\()p FK(w)802 3239 y FJ(?)841 3269 y FH(\))c(=)f FK(\025\026)1089 3281 y FJ(\027)1150 3269 y FI(\001)20 b FK(U)9 b(w)1320 3239 y FJ(?)1359 3269 y FL(,)30 b(and)f(hence)h FI(k)o(rL)1975 3281 y FJ(U)2031 3269 y FH(\()p FK(w)2124 3239 y FJ(?)2164 3269 y FH(\))p FI(k)c(\024)g FK(\026)2405 3281 y FJ(\027)2446 3269 y FK(\025)p FL(.)43 b(Then,)31 b(b)r(ecause)e FI(L)3169 3281 y FJ(U)3254 3269 y FL(is)h FH(\()p FK(\026)3422 3281 y FJ(\027)3463 3269 y FK(=)p FH(2\))p FL(-strongly)0 3369 y(con)n(v)n(ex)c(o)n(v)n(er)g FK(w)g FI(2)d(W)7 b FL(,)28 b(w)n(e)f(ha)n(v)n(e)679 3568 y FI(L)736 3580 y FJ(U)792 3568 y FH(\()p FK(w)r FH(\))e FI(\025)d(L)1086 3580 y FJ(U)1142 3568 y FH(\()p FK(w)1235 3534 y FJ(?)1274 3568 y FH(\))d(+)f FI(hrL)1566 3580 y FJ(U)1622 3568 y FH(\()p FK(w)1715 3534 y FJ(?)1754 3568 y FH(\))p FK(;)c(w)22 b FI(\000)c FK(w)2048 3534 y FJ(?)2087 3568 y FI(i)g FH(+)2230 3512 y FK(\026)2280 3524 y FJ(\027)p 2230 3549 92 4 v 2255 3625 a FH(4)2345 3568 y FI(k)p FK(w)j FI(\000)d FK(w)2611 3534 y FJ(?)2650 3568 y FI(k)2692 3527 y FC(2)2743 3568 y FK(;)180 b(w)25 b FI(2)f(W)7 b FK(:)0 3779 y FL(In)28 b(particular,)e(using)h FI(L)784 3791 y FJ(U)840 3779 y FH(\()15 b FD(b)-61 b FK(w)933 3749 y FJ(?)931 3802 y(U)988 3779 y FH(\))23 b FI(\024)g(L)1188 3791 y FJ(U)1243 3779 y FH(\()p FK(w)1336 3749 y FJ(?)1376 3779 y FH(\))p FL(,)28 b(w)n(e)f(ha)n(v)n(e)859 3929 y FK(\026)909 3941 y FJ(\027)p 859 3966 V 884 4042 a FH(4)974 3985 y FI(k)14 b FD(b)-60 b FK(w)1077 3951 y FJ(?)1075 4005 y(U)1149 3985 y FI(\000)18 b FK(w)1293 3951 y FJ(?)1332 3985 y FI(k)1374 3943 y FC(2)1434 3985 y FI(\024)k(\000hrL)1744 3997 y FJ(U)1800 3985 y FH(\()p FK(w)1893 3951 y FJ(?)1933 3985 y FH(\))p FK(;)28 b FD(b)-60 b FK(w)2063 3951 y FJ(?)2061 4005 y(U)2135 3985 y FI(\000)18 b FK(w)2279 3951 y FJ(?)2318 3985 y FI(i)23 b(\024)g FK(\026)2511 3997 y FJ(\027)2552 3985 y FK(\025)14 b FI(k)h FD(b)-61 b FK(w)2717 3951 y FJ(?)2715 4005 y(U)2790 3985 y FI(\000)18 b FK(w)2934 3951 y FJ(?)2973 3985 y FI(k)13 b FK(;)0 4195 y FL(and)27 b(hence)h FI(k)14 b FD(b)-60 b FK(w)495 4165 y FJ(?)493 4218 y(U)567 4195 y FI(\000)18 b FK(w)711 4165 y FJ(?)750 4195 y FI(k)k(\024)h FH(4)p FK(\025)p FL(.)0 4345 y(Next,)28 b(for)f(an)n(y)g(v)n(ector)f FK(v)g FI(2)e FG(R)958 4315 y FJ(d)1003 4345 y FL(,)j(w)n(e)h(ha)n(v)n(e)1055 4576 y FG(E)1105 4588 y FJ(\036)p FB(\030)p FJ(p)1241 4576 y FH([)p FI(jh)p FK(\036;)14 b(U)9 b(v)s FI(ij)q FH(])1593 4534 y FC(2)1654 4576 y FI(\024)22 b FG(E)1791 4588 y FJ(\036)p FB(\030)p FJ(p)1927 4576 y FH([)p FI(k)p FK(U)9 b(\036)p FI(k)o FH(])19 b FI(\001)f FG(E)2281 4588 y FJ(\036)p FB(\030)p FJ(p)2417 4459 y FD(\024)2471 4520 y FI(h)p FK(\036;)c(U)9 b(v)s FI(i)2730 4490 y FC(2)p 2471 4557 298 4 v 2520 4633 a FI(k)p FK(U)g(\036)p FI(k)2778 4459 y FD(\025)2822 4576 y FK(:)0 4843 y FL(Note)24 b(that)f FG(E)422 4855 y FJ(\036)p FB(\030)p FJ(p)558 4750 y FD(h)607 4800 y FJ(U)6 b(\036)t(\036)742 4775 y Fz(>)792 4800 y FJ(U)p 607 4823 237 4 v 643 4871 a FB(k)p FJ(U)g(\036)t FB(k)853 4750 y FD(i)915 4843 y FI(\026)23 b FH(2)p FE(I)p FL(,)h(and)f(hence)h(w)n(e)f(ha)n(v)n(e)f FG(E)1867 4855 y FJ(\036)p FB(\030)p FJ(p)2003 4843 y FH([)p FI(k)p FK(U)9 b(\036)p FI(k)o FH(])24 b FI(\024)e FH(2)p FK(d)h FL(and)h FG(E)2673 4855 y FJ(\036)p FB(\030)p FJ(p)2809 4750 y FD(h)2858 4802 y FB(h)p FJ(\036;U)6 b(v)r FB(i)3058 4777 y FO(2)p 2858 4823 233 4 v 2895 4871 a FB(k)p FJ(U)g(\036)p FB(k)3101 4750 y FD(i)3163 4843 y FI(\024)23 b FH(2)14 b FI(k)o FK(v)s FI(k)3432 4801 y FC(2)3470 4843 y FL(.)35 b(Therefore,)0 4962 y(it)28 b(holds)f(that)1460 5145 y FG(E)1510 5157 y FJ(\036)p FB(\030)p FJ(p)1646 5145 y FH([)p FI(jh)p FK(\036;)14 b(U)9 b(v)s FI(ij)q FH(])23 b FI(\024)g FH(2)2151 5070 y FI(p)p 2220 5070 44 4 v 75 x FK(d)14 b FI(k)o FK(v)s FI(k)g FK(:)0 5327 y FL(Substituting)29 b FK(v)d FH(=)c FK(w)g FI(\000)c FK(w)854 5297 y FJ(?)920 5327 y FL(completes)27 b(the)h(pro)r(of.)2179 b Fd(2)p 0 TeXcolorgray 1908 5649 a FL(20)p 0 TeXcolorgray eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.21) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 0 a SDict begin H.S end 0 0 a 0 0 a SDict begin 12 H.A end 0 0 a 0 0 a SDict begin [/View [/XYZ H.V]/Dest (subsection.C.2) cvn /DEST pdfmark end 0 0 a 83 x Fl(C.2)112 b(Pro)s(of)37 b(of)p 0 0 0.7 TeXcolorrgb 717 83 a SDict begin H.S end 717 83 a 0 0 0.7 TeXcolorrgb Fl(Lemma)i(4.2)p 0 0 0.7 TeXcolorrgb 1257 15 a SDict begin H.R end 1257 15 a 1257 83 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.4.2) cvn H.B /ANN pdfmark end 1257 83 a 0 TeXcolorgray 0 236 a FL(W)-7 b(e)28 b(\034rst)e(note)i(that)f (in)p 0 0 0.7 TeXcolorrgb 773 252 a SDict begin H.S end 773 252 a 0 0 0.7 TeXcolorrgb -16 x FL(Algorithm)g(1)p 0 0 0.7 TeXcolorrgb 1214 179 a SDict begin H.R end 1214 179 a 1214 236 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algorithm.1) cvn H.B /ANN pdfmark end 1214 236 a 0 TeXcolorgray FL(,)g(the)h(dataset)e FI(D)k FL(is)d(split)g(equally)g(as)f FI(D)g FH(=)c FI(D)2686 248 y FO(\(0\))2782 236 y FI(t)17 b(\001)d(\001)g(\001)k(t)g(D)3106 248 y FO(\()p FN(K)s Fz(\000)p FO(1\))3278 236 y FL(,)28 b(and)f(iteration)f(at)0 336 y(the)h FK(k)s FL(th)f(ep)r(o)r(c)n(h)g(can)g(b)r(e)g(regarded)f (as)g(a)h(random)f(function)i FH(\()p FK(U)2015 348 y FO(\()p FN(k)q FO(\))2096 336 y FH(;)14 b FI(D)2197 348 y FO(\()p FN(k)q FO(\))2278 336 y FH(\))24 b FI(7!)f FK(U)2497 348 y FO(\()p FN(k)q FO(+1\))2649 336 y FL(.)36 b(Therefore,)25 b(using)h(the)h(comp)r(osition)0 436 y(prop)r(ert)n(y)40 b(of)h(join)n(t)g(DP)g(\()p 0 0 0.7 TeXcolorrgb 861 436 a SDict begin H.S end 861 436 a 0 0 0.7 TeXcolorrgb FL(Lemma)g(C.1)p 0 0 0.7 TeXcolorrgb 1296 378 a SDict begin H.R end 1296 378 a 1296 436 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.C.1) cvn H.B /ANN pdfmark end 1296 436 a 0 TeXcolorgray FL(\),)k(w)n(e)c(only)f(need)h(to)g(v)n (erify)g(that)g FH(\()p FK(U)2576 448 y FO(\()p FN(k)q FO(\))2657 436 y FH(;)14 b FI(D)2758 448 y FO(\()p FN(k)q FO(\))2840 436 y FH(\))45 b FI(7!)h FK(U)3103 448 y FO(\()p FN(k)q FO(+1\))3296 436 y FL(preserv)n(es)39 b FH(\()p FK(\013;)14 b(\014)t FH(\))p FL(-)0 535 y(JDP)27 b(\(with)i(resp)r(ect) e(to)g FI(D)861 547 y FO(\()p FN(k)q FO(\))943 535 y FL(\).)0 736 y(F)-7 b(or)38 b(the)h(data)g FH(\()p FK(\036)596 748 y FJ(t)626 736 y FK(;)14 b(y)704 748 y FJ(t)733 736 y FH(\))39 b FL(in)g(the)g FK(k)s FL(th)g(ep)r(o)r(c)n(h,)j(the)d(quan) n(tit)n(y)g FH(\010)2062 748 y FJ(t)2133 736 y FH(=)2249 678 y FJ(U)2300 647 y FO(1)p FN(=)p FO(2)2294 700 y(\()p FN(k)q FO(\))2392 678 y FJ(\036)2432 686 y FN(t)2459 678 y FJ(\036)2499 653 y Fz(>)2499 694 y FN(t)2548 678 y FJ(U)2599 647 y FO(1)p FN(=)p FO(2)2593 700 y(\()p FN(k)q FO(\))p 2249 717 442 4 v 2339 765 a FB(k)p FJ(U)2418 776 y FO(\()p FN(k)q FO(\))2500 765 y FJ(\036)2540 773 y FN(t)2567 765 y FB(k)2740 736 y FL(can)f(b)r(e)i(b)r(ounded)f(as)f FI(k)p FH(\010)3593 748 y FJ(t)3622 736 y FI(k)3663 761 y FC(F)3752 736 y FI(\024)j FH(1)0 857 y FL(\()p 0 0 0.7 TeXcolorrgb 32 857 a SDict begin H.S end 32 857 a 0 0 0.7 TeXcolorrgb FL(Lemma)32 b(A.5)p 0 0 0.7 TeXcolorrgb 460 800 a SDict begin H.R end 460 800 a 460 857 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.A.5) cvn H.B /ANN pdfmark end 460 857 a 0 TeXcolorgray FL(\).)51 b(Th)n(us,)33 b FK(H)874 872 y FC(\()p FJ(k)q FC(\))999 857 y FL(de\034ned)g(in)p 0 0 0.7 TeXcolorrgb 1391 878 a SDict begin H.S end 1391 878 a 0 0 0.7 TeXcolorrgb -21 x FL(Eq.)50 b(\(6\))p 0 0 0.7 TeXcolorrgb 1671 795 a SDict begin H.R end 1671 795 a 1671 857 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.6) cvn H.B /ANN pdfmark end 1671 857 a 0 TeXcolorgray 33 w FL(has)32 b(sensitivit)n(y)f FH(\001)g(=)g(1)p FK(=)-5 b(N)40 b FL(under)32 b(F)-7 b(rob)r(enius)32 b(norm.)50 b(Therefore,)0 976 y(b)n(y)31 b(the)g(priv)-5 b(acy)30 b(guaran)n(tee)f(of)i(Gaussian)e(c)n(hannels)h(\()p 0 0 0.7 TeXcolorrgb 1770 976 a SDict begin H.S end 1770 976 a 0 0 0.7 TeXcolorrgb FL(De\034nition)i(2)p 0 0 0.7 TeXcolorrgb 2202 918 a SDict begin H.R end 2202 918 a 2202 976 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (definition.2) cvn H.B /ANN pdfmark end 2202 976 a 0 TeXcolorgray FL(\),)g(the)f(mec)n (hanism)f FH(\()p FK(U)2954 988 y FO(\()p FN(k)q FO(\))3036 976 y FH(;)14 b FI(D)3137 988 y FO(\()p FN(k)q FO(\))3218 976 y FH(\))29 b FI(7!)3409 955 y FD(e)3390 976 y FK(H)3459 988 y FO(\()p FN(k)q FO(\))3571 976 y FL(preserv)n(es)0 1075 y FH(\()p FK(\013;)14 b(\014)t FH(\))p FL(-JDP)37 b(with)g(resp)r(ect)f(to)h FI(D)1099 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2160 470 4 v 71 x FK(d)c FH(+)f(log\()p FK(K)q(=\016)s FH(\))p 2113 2270 708 4 v 2429 2346 a FK(N)2831 2289 y(;)3752 2289 y SDict begin H.S end 3752 2289 a 3752 2289 a SDict begin 15 H.A end 3752 2289 a 3752 2289 a SDict begin [/View [/XYZ H.V]/Dest (equation.C.11) cvn /DEST pdfmark end 3752 2289 a FL(\(11\))0 2507 y FF(wher)l(e)30 b FK(C)293 2519 y FC(0)361 2507 y FF(is)g(a)g(lar)l(ge)g(absolute)h(c)l(onstant.)38 b(In)29 b(the)h(fol)6 b(lowing,)33 b(we)d(denote)g(this)g(event)g(as)g FI(E)7 b FF(.)0 2656 y FL(Therefore,)26 b(w)n(e)i(can)f(simplify)h(the) g(iterations)f(in)p 0 0 0.7 TeXcolorrgb 1597 2673 a SDict begin H.S end 1597 2673 a 0 0 0.7 TeXcolorrgb -17 x FL(Algorithm)h(1)p 0 0 0.7 TeXcolorrgb 2039 2599 a SDict begin H.R end 2039 2599 a 2039 2656 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algorithm.1) cvn H.B /ANN pdfmark end 2039 2656 a 0 TeXcolorgray 27 w FL(as)f(follo)n(ws:)36 b FK(U)2530 2668 y FO(\(0\))2630 2656 y FH(=)22 b FE(I)p FL(,)28 b(and)g(for)f FK(k)f FH(=)c(0)p FK(;)14 b FH(1)p FK(;)g FI(\001)g(\001)g(\001)27 b FK(;)14 b(K)6 b FL(:)1353 2878 y FD(e)1333 2899 y FK(H)1402 2911 y FO(\()p FN(k)q FO(\))1507 2899 y FH(=)27 b FG(E)1648 2911 y FJ(\036)p FB(\030)p FJ(p)1784 2782 y FD(\024)1838 2841 y FK(U)1904 2802 y FO(1)p FN(=)p FO(2)1895 2857 y(\()p FN(k)q FO(\))1995 2841 y FK(\036)t(\036)2097 2811 y Fz(>)2147 2841 y FK(U)2213 2802 y FO(1)p FN(=)p FO(2)2204 2857 y(\()p FN(k)q FO(\))p 1838 2880 467 4 v 1934 2956 a FI(k)p FK(U)2033 2968 y FO(\()p FN(k)q FO(\))2113 2956 y FK(\036)t FI(k)2314 2782 y FD(\025)2377 2899 y FH(+)18 b FK(E)2521 2911 y FO(\()p FN(k)q FO(\))2602 2899 y FK(;)3752 2899 y SDict begin H.S end 3752 2899 a 3752 2899 a SDict begin 15 H.A end 3752 2899 a 3752 2899 a SDict begin [/View [/XYZ H.V]/Dest (equation.C.12) cvn /DEST pdfmark end 3752 2899 a FL(\(12\))1333 3093 y FE(\006)1402 3105 y FO(\()p FN(k)q FO(\))1507 3093 y FH(=)27 b FK(U)1665 3059 y FO(1)p FN(=)p FO(2)1656 3114 y(\()p FN(k)q FO(\))1775 3072 y FD(e)1756 3093 y FK(H)1825 3105 y FO(\()p FN(k)q FO(\))1906 3093 y FK(U)1972 3059 y FO(1)p FN(=)p FO(2)1963 3114 y(\()p FN(k)q FO(\))2082 3093 y FH(+)18 b FK(\025)2213 3105 y FO(\()p FN(k)q FO(\))2294 3093 y FK(U)2351 3105 y FO(\()p FN(k)q FO(\))2432 3093 y FK(;)3752 3093 y SDict begin H.S end 3752 3093 a 3752 3093 a SDict begin 15 H.A end 3752 3093 a 3752 3093 a SDict begin [/View [/XYZ H.V]/Dest (equation.C.13) cvn /DEST pdfmark end 3752 3093 a FL(\(13\))1275 3227 y FK(U)1332 3239 y FO(\()p FN(k)q FO(+1\))1507 3227 y FH(=)27 b(sym\()p FE(\006)1846 3193 y Fz(\000)p FO(1)p FN(=)p FO(2)1846 3248 y(\()p FN(k)q FO(\))1983 3227 y FK(U)2040 3239 y FO(\()p FN(k)q FO(\))2121 3227 y FH(\))p FK(;)3752 3227 y SDict begin H.S end 3752 3227 a 3752 3227 a SDict begin 15 H.A end 3752 3227 a 3752 3227 a SDict begin [/View [/XYZ H.V]/Dest (equation.C.14) cvn /DEST pdfmark end 3752 3227 a FL(\(14\))0 3410 y(where)22 b FH(sym)q(\()p FK(A)p FH(\))h(=)g(\()p FK(A)713 3380 y Fz(>)763 3410 y FK(A)p FH(\))857 3380 y FO(1)p FN(=)p FO(2)949 3410 y FL(,)g FK(E)1056 3422 y FO(\()p FN(k)q FO(\))1161 3410 y FL(is)f(a)h(symmetric)f(matrix.)35 b(W)-7 b(e)23 b(note)f(that)p 0 0 0.7 TeXcolorrgb 2496 3426 a SDict begin H.S end 2496 3426 a 0 0 0.7 TeXcolorrgb -16 x FL(Algorithm)h(1)p 0 0 0.7 TeXcolorrgb 2933 3353 a SDict begin H.R end 2933 3353 a 2933 3410 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (algorithm.1) cvn H.B /ANN pdfmark end 2933 3410 a 0 TeXcolorgray 22 w FL(do)r(es)f(not)h(actually)f (compute)0 3509 y FH(\()p FK(H)101 3521 y FO(\()p FN(K)s FO(\))201 3509 y FK(;)14 b FE(\006)307 3521 y FO(\()p FN(K)s FO(\))406 3509 y FH(\))p FL(,)28 b(and)f(they)h(only)f(app)r (ear)g(in)h(our)e(analysis)h(\(where)g(w)n(e)g(can)g(regard)f FK(E)2717 3521 y FO(\()p FN(K)s FO(\))2840 3509 y FH(=)c(0)p FL(\).)0 3532 y SDict begin H.S end 0 3532 a 0 3532 a SDict begin 12 H.A end 0 3532 a 0 3532 a SDict begin [/View [/XYZ H.V]/Dest (thmt@dummyctr.dummy.31) cvn /DEST pdfmark end 0 3532 a 0 3532 a SDict begin H.S end 0 3532 a 0 3532 a SDict begin 12 H.A end 0 3532 a 0 3532 a SDict begin [/View [/XYZ H.V]/Dest (proposition.C.3) cvn /DEST pdfmark end 0 3532 a 0 TeXcolorgray 127 x Fp(Prop)s(osition)30 b(C.3)p 0 TeXcolorgray 41 w FF(Supp)l(ose)40 b(that)g(the)g(se)l(quenc)l(e)f (of)i(matric)l(es)f FI(f)o FH(\()p FK(U)2284 3671 y FO(\()p FN(k)q FO(\))2365 3659 y FK(;)14 b(H)2471 3671 y FO(\()p FN(k)q FO(\))2553 3659 y FK(;)g FE(\006)2659 3671 y FO(\()p FN(k)q FO(\))2740 3659 y FH(\))p FI(g)39 b FF(is)h(de\034ne)l(d)g(r)l (e)l(cursively)h(by)p 0 0 0.7 TeXcolorrgb 3780 3680 a SDict begin H.S end 3780 3680 a 0 0 0.7 TeXcolorrgb -21 x FF(Eq.)0 3758 y(\(12\))p 0 0 0.7 TeXcolorrgb 152 3696 a SDict begin H.R end 152 3696 a 152 3758 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.C.12) cvn H.B /ANN pdfmark end 152 3758 a 0 TeXcolorgray 34 w FF(-)p 0 0 0.7 TeXcolorrgb 248 3779 a SDict begin H.S end 248 3779 a 0 0 0.7 TeXcolorrgb -21 x FF(\(14\))p 0 0 0.7 TeXcolorrgb 400 3696 a SDict begin H.R end 400 3696 a 400 3758 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.C.14) cvn H.B /ANN pdfmark end 400 3758 a 0 TeXcolorgray FF(,)35 b(with)e FI(k)p FK(E)746 3770 y FO(\()p FN(k)q FO(\))827 3758 y FI(k)869 3785 y FC(op)971 3758 y FI(\024)28 b FK(")p FF(.)48 b(Supp)l(ose)33 b(that)g FK(\025)1716 3770 y FO(\()p FN(k)q FO(\))1826 3758 y FH(=)28 b(\(2)p FK(k)c FH(+)c(5\))p FK(")p FF(,)33 b(and)h FK(")28 b FI(\024)g FH(0)p FK(:)p FH(1)p FF(.)47 b(Then,)35 b(for)e(any)h FK(k)d FI(\025)d FH(1)p FF(,)34 b(it)f(holds)0 3858 y(that)837 4070 y FK(\025)885 4082 y FC(min)1000 4070 y FH(\()p FE(\006)1101 4082 y FO(\()p FN(k)q FO(\))1183 4070 y FH(\))23 b FI(\025)f FH(exp)1466 3953 y FD(\022)1527 4070 y FI(\000)1602 4014 y FH(log\(1)p FK(=")p FH(\))p 1602 4051 294 4 v 1707 4127 a(2)1749 4103 y FJ(k)1905 3953 y FD(\023)1966 4070 y FK(;)184 b(\025)2221 4082 y FC(max)2348 4070 y FH(\()p FE(\006)2449 4082 y FO(\()p FN(k)q FO(\))2531 4070 y FH(\))23 b FI(\024)g FH(exp)2814 3953 y FD(\022)2885 4014 y FH(12)p 2885 4051 84 4 v 2904 4127 a FK(k)2978 3953 y FD(\023)3040 4070 y FK(:)0 4302 y FF(In)29 b(p)l(articular,)j FK(K)c FI(\025)23 b FH(log)14 b(log\(1)p FK(=")p FH(\))k FI(_)g FH(20)p FF(,)30 b(we)g(have)1169 4479 y FH(1)p 1169 4516 42 4 v 1169 4592 a(2)1220 4535 y FE(I)24 b FI(\026)e FG(E)1417 4547 y FJ(\036)p FB(\030)p FJ(p)1553 4418 y FD(\024)1607 4479 y FK(U)1664 4491 y FO(\()p FN(K)s FO(\))1763 4479 y FK(\036)t(\036)1865 4448 y Fz(>)1915 4479 y FK(U)1972 4491 y FO(\()p FN(K)s FO(\))p 1607 4516 465 4 v 1692 4592 a FI(k)p FK(U)1791 4604 y FO(\()p FN(K)s FO(\))1890 4592 y FK(\036)t FI(k)2081 4418 y FD(\025)2143 4535 y FH(+)c FK(\025)2274 4547 y FO(\()p FN(K)s FO(\))2374 4535 y FK(U)2431 4547 y FO(\()p FN(K)s FO(\))2553 4535 y FI(\026)k FH(2)p FE(I)p FK(:)0 4876 y SDict begin H.S end 0 4876 a 0 4876 a SDict begin 12 H.A end 0 4876 a 0 4876 a SDict begin [/View [/XYZ H.V]/Dest (section*.25) cvn /DEST pdfmark end 0 4876 a Fp(Pro)s(of)f(of)p 0 0 0.7 TeXcolorrgb 351 4892 a SDict begin H.S end 351 4892 a 0 0 0.7 TeXcolorrgb -16 x Fp(Prop)s(osition)f(4.3)p 0 0 0.7 TeXcolorrgb 980 4818 a SDict begin H.R end 980 4818 a 980 4876 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (proposition.4.3) cvn H.B /ANN pdfmark end 980 4876 a 0 TeXcolorgray Fp(.)81 b FL(By)p 0 0 0.7 TeXcolorrgb 1209 4876 a SDict begin H.S end 1209 4876 a 0 0 0.7 TeXcolorrgb FL(Lemma)19 b(C.2)p 0 0 0.7 TeXcolorrgb 1622 4819 a SDict begin H.R end 1622 4819 a 1622 4876 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.C.2) cvn H.B /ANN pdfmark end 1622 4876 a 0 TeXcolorgray FL(,)g(under)g(the)f(ev)n(en)n(t)g FI(E)7 b FL(,)21 b(the)e(matrix)e FK(E)2784 4888 y FO(\()p FN(k)q FO(\))2889 4876 y FH(=)2996 4855 y FD(e)2977 4876 y FK(H)3046 4888 y FO(\()p FN(k)q FO(\))3127 4876 y FI(\000)p FG(E)3241 4888 y FJ(\036)p FB(\030)p FJ(p)3377 4759 y FD(\024)3431 4817 y FJ(U)3482 4787 y FO(1)p FN(=)p FO(2)3476 4840 y(\()p FN(k)q FO(\))3574 4817 y FJ(\036)t(\036)3658 4792 y Fz(>)3707 4817 y FJ(U)3758 4787 y FO(1)p FN(=)p FO(2)3752 4840 y(\()p FN(k)q FO(\))p 3431 4857 419 4 v 3521 4905 a FB(k)p FJ(U)3600 4916 y FO(\()p FN(k)q FO(\))3681 4905 y FJ(\036)t FB(k)3859 4759 y FD(\025)0 5025 y FL(is)32 b(b)r(ounded)h(as)f FI(k)p FK(E)642 5037 y FO(\()p FN(k)q FO(\))723 5025 y FI(k)765 5052 y FC(op)870 5025 y FI(\024)f FK(")1005 5037 y FJ(N)1067 5025 y FL(.)52 b(Therefore,)33 b(under)f FI(E)7 b FL(,)34 b(w)n(e)e(can)g(apply)p 0 0 0.7 TeXcolorrgb 2414 5042 a SDict begin H.S end 2414 5042 a 0 0 0.7 TeXcolorrgb -17 x FL(Prop)r(osition)f(C.3)p 0 0 0.7 TeXcolorrgb 2991 4968 a SDict begin H.R end 2991 4968 a 2991 5025 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (proposition.C.3) cvn H.B /ANN pdfmark end 2991 5025 a 0 TeXcolorgray FL(,)i(whic)n(h)f(giv)n(es)g(the)h(desired)0 5125 y(results.)3578 b Fd(2)p 0 TeXcolorgray 1908 5649 a FL(22)p 0 TeXcolorgray eop end end %%Page: 23 23 TeXDict begin HPSdict begin 23 22 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray -8 -316 a SDict begin H.S end -8 -316 a -8 -316 a SDict begin H.R end -8 -316 a -8 -316 a SDict begin [/View [/XYZ H.V]/Dest (page.23) cvn /DEST pdfmark end -8 -316 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 0 0 a SDict begin H.S end 0 0 a 0 0 a SDict begin 12 H.A end 0 0 a 0 0 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.C.3.1) cvn /DEST pdfmark end 0 0 a 83 x Fp(C.3.1)94 b(Pro)s(of)32 b(of)p 0 0 0.7 TeXcolorrgb 685 83 a SDict begin H.S end 685 83 a 0 0 0.7 TeXcolorrgb Fp(Lemma)g(C.2)p 0 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FH(\010)2585 827 y FC(\()p FJ(k)q FC(+1\))p FJ(N)2851 812 y FL(are)30 b(indep)r(enden)n(t)h(\(conditional)0 970 y(on)i FK(U)178 982 y FO(\()p FN(k)q FO(\))259 970 y FL(\).)56 b(F)-7 b(urther,)35 b(b)n(y)p 0 0 0.7 TeXcolorrgb 824 970 a SDict begin H.S end 824 970 a 0 0 0.7 TeXcolorrgb FL(Lemma)e(A.5)p 0 0 0.7 TeXcolorrgb 1253 913 a SDict begin H.R end 1253 913 a 1253 970 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.A.5) cvn H.B /ANN pdfmark end 1253 970 a 0 TeXcolorgray FL(,)i(w)n(e)f(ha)n(v)n(e)e FI(k)p FH(\010)1739 982 y FJ(t)1768 970 y FI(k)1809 995 y FC(F)1890 970 y FI(\024)h FH(1)p FL(,)i(and)e FG(E)8 b FH([)q(\010)2388 982 y FJ(t)2423 970 y FI(j)p FK(U)2503 982 y FO(\()p FN(k)q FO(\))2584 970 y FH(])33 b(=)g FG(E)2788 982 y FJ(\036)p FB(\030)p FJ(p)2924 853 y FD(\024)2978 911 y FJ(U)3029 881 y FO(1)p FN(=)p FO(2)3023 934 y(\()p FN(k)q FO(\))3120 911 y FJ(\036)t(\036)3204 886 y Fz(>)3253 911 y FJ(U)3304 881 y FO(1)p FN(=)p FO(2)3298 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y(\()p FN(k)q FO(\))p 1559 1334 467 4 v 1654 1410 a FI(k)p FK(U)1753 1422 y FO(\()p FN(k)q FO(\))1834 1410 y FK(\036)t FI(k)2035 1236 y FD(\025)2079 1232 y(\015)2079 1282 y(\015)2079 1332 y(\015)2079 1382 y(\015)2125 1437 y FC(F)2195 1353 y FI(\024)2292 1295 y FH(1)g(+)2435 1224 y FD(p)p 2518 1224 351 4 v 71 x FH(2)c(log\(1)p FK(=\016)s FH(\))p 2292 1334 577 4 v 2508 1350 a FI(p)p 2577 1350 76 4 v 71 x FK(N)0 1635 y FL(By)p 0 0 0.7 TeXcolorrgb 130 1635 a SDict begin H.S end 130 1635 a 0 0 0.7 TeXcolorrgb FL(Lemma)27 b(A.1)p 0 0 0.7 TeXcolorrgb 553 1578 a SDict begin H.R end 553 1578 a 553 1635 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lemma.A.1) cvn H.B /ANN pdfmark end 553 1635 a 0 TeXcolorgray FL(,)h(w)n(e)f(also)f(ha)n(v)n(e)g FI(k)p FK(Z)1182 1647 y FO(\()p FN(k)q FO(\))1263 1635 y FI(k)1304 1662 y FC(op)1401 1635 y FI(\024)d FK(C)1564 1586 y FJ(\033)1602 1595 y FN(\013;\014)1699 1530 y FI(p)p 1768 1530 322 4 v 56 x FJ(d)p FC(+log)o(\(1)p FJ(=\016)r FC(\))p 1564 1616 526 4 v 1797 1664 a FJ(N)2127 1635 y FL(with)28 b(probabilit)n(y)e(at)i (least)f FH(1)17 b FI(\000)h FK(\016)s FL(.)37 b(T)-7 b(aking)26 b(the)i(union)0 1755 y(b)r(ound)g(o)n(v)n(er)e FK(k)g FH(=)c(0)p FK(;)14 b FH(1)p FK(;)g FI(\001)g(\001)g(\001)27 b FK(;)14 b(K)23 b FI(\000)18 b FH(1)28 b FL(and)f(rescaling)f FK(\016)g FI( )1869 1722 y FJ(\016)p 1838 1736 93 4 v 1838 1784 a FC(2)p FJ(K)1969 1755 y FL(completes)h(the)h(pro)r(of.)1130 b Fd(2)0 1849 y SDict begin H.S end 0 1849 a 0 1849 a SDict begin 12 H.A end 0 1849 a 0 1849 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.C.3.2) cvn /DEST pdfmark end 0 1849 a 121 x Fp(C.3.2)94 b(Pro)s(of)32 b(of)p 0 0 0.7 TeXcolorrgb 685 1986 a SDict begin H.S end 685 1986 a 0 0 0.7 TeXcolorrgb -16 x Fp(Prop)s(osition)f(C.3)p 0 0 0.7 TeXcolorrgb 1346 1912 a SDict begin H.R end 1346 1912 a 1346 1970 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (proposition.C.3) cvn H.B /ANN pdfmark end 1346 1970 a 0 TeXcolorgray 0 2123 a FL(By)21 b(de\034nition)p 0 0 0.7 TeXcolorrgb 487 2144 a SDict begin H.S end 487 2144 a 0 0 0.7 TeXcolorrgb -21 x FL(\(14\))p 0 0 0.7 TeXcolorrgb 635 2061 a SDict begin H.R end 635 2061 a 635 2123 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.C.14) cvn H.B /ANN pdfmark end 635 2123 a 0 TeXcolorgray FL(,)h(for)f(eac)n(h)g FK(k)26 b FI(\025)c FH(1)p FL(,)h(there)e(exists)g(an)g(orthogonal)e(matrix)i FK(V)2485 2135 y FO(\()p FN(k)q FO(+1\))2659 2123 y FL(suc)n(h)g(that)h FK(U)3071 2135 y FO(\()p FN(k)q FO(+1\))3246 2123 y FH(=)g FK(V)3381 2135 y FO(\()p FN(k)q FO(+1\))3534 2123 y FE(\006)3603 2084 y Fz(\000)p FO(1)p FN(=)p FO(2)3603 2139 y(\()p FN(k)q FO(\))3739 2123 y FK(U)3796 2135 y FO(\()p FN(k)q FO(\))3877 2123 y FL(.)0 2223 y(Therefore,)255 2421 y FE(I)h FH(=)k FK(V)454 2433 y FO(\()p FN(k)q FO(+1\))607 2421 y FK(V)674 2387 y Fz(>)655 2442 y FO(\()p FN(k)q FO(+1\))830 2421 y FH(=)c 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y FJ(d)974 3622 y FL(,)28 b(it)g(holds)f(that)h FI(k)o FK(U)1603 3634 y FO(\()p FN(k)q FO(+1\))1755 3622 y FK(\036)p FI(k)23 b FH(=)1957 3551 y FD(\015)1957 3601 y(\015)2003 3622 y FE(\006)2072 3583 y Fz(\000)p FO(1)p FN(=)p FO(2)2072 3637 y(\()p FN(k)q FO(\))2208 3622 y FK(U)2265 3634 y FO(\()p FN(k)q FO(\))2346 3622 y FK(\036)2395 3551 y FD(\015)2395 3601 y(\015)2442 3622 y FL(,)k(and)h(hence)g(w)n(e)f(ha)n(v)n(e)13 3759 y FD(q)p 96 3759 378 4 v 99 x FK(\025)144 3870 y FC(min)258 3858 y FH(\()p FE(\006)359 3870 y FO(\()p FN(k)q FO(\))441 3858 y FH(\))19 b FI(\001)f FG(E)582 3870 y FJ(\036)p FB(\030)p FJ(p)719 3741 y FD(\024)772 3802 y FK(U)829 3814 y FO(\()p FN(k)q FO(+1\))981 3802 y FK(\036\036)1079 3771 y Fz(>)1129 3802 y FK(U)1186 3814 y FO(\()p FN(k)q FO(+1\))p 772 3839 566 4 v 920 3915 a FI(k)p FK(U)1019 3927 y FO(\()p FN(k)q FO(\))1100 3915 y FK(\036)p FI(k)1348 3741 y FD(\025)1415 3858 y FI(\026)23 b FG(E)1552 3870 y FJ(\036)p FB(\030)p FJ(p)1688 3741 y FD(\024)1742 3802 y 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3919 y FO(\()p FN(k)q FO(\))2589 3907 y FI(\000)e FK(w)2733 3873 y FJ(?)2772 3907 y FI(kg)13 b FK(;)0 4106 y FL(where)30 b(w)n(e)g(denote)h FK(Z)729 4103 y FH(:)752 4106 y(=)d FK(C)6 b(\033)960 4035 y FD(p)p 1044 4035 326 4 v 1044 4106 a FH(log\()p FK(K)q(=\016)s FH(\))p FL(,)31 b FK(C)37 b FL(is)31 b(a)f(large)f(univ)n(ersal)g(constan)n(t.) 45 b(By)30 b(sub-Gaussian)g(concen)n(tration,)f(w)n(e)0 4205 y(kno)n(w)e FG(P)p FH(\()p FI(E)7 b FH(\))22 b FI(\025)h FH(1)18 b FI(\000)g FK(\016)s FL(.)0 4355 y(Under)28 b FI(E)7 b FL(,)28 b(w)n(e)f(can)g(b)r(ound)527 4537 y FI(k)o FK(w)627 4549 y FO(\()p FN(k)q FO(+1\))798 4537 y FI(\000)18 b FK(w)942 4503 y FJ(?)981 4537 y FI(k)1023 4496 y FC(2)1083 4537 y FI(\024)27 b FH(\()q(1)18 b FI(\000)g FK(\021)s(\026)p FH(\))c FI(k)o FK(w)1591 4549 y FO(\()p FN(k)q FO(\))1692 4537 y FI(\000)k FK(w)1836 4503 y FJ(?)1874 4537 y FI(k)1916 4496 y FC(2)1972 4537 y FH(+)g(2)p FK(\021)s(Z)h FI(k)o FK(w)2317 4549 y FO(\()p FN(k)q FO(\))2417 4537 y FI(\000)f 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FJ(\016)1480 3611 y FH(\()p FK(N)9 b FH(\))23 b Fk(.)1731 3469 y FD( )1806 3544 y FK(\033)1856 3514 y FC(2)1853 3567 y FJ(\013;\014)1975 3544 y FH(log\()p FK(N)r(=\016)s FH(\))p 1806 3592 492 4 v 2014 3668 a FK(N)2307 3469 y FD(!)2373 3486 y FC(1)p FJ(=)p FC(6)2477 3611 y FK(:)0 3880 y FL(Therefore,)j(w)n(e)i(c)n(ho)r(ose)e FK(T)846 3850 y FC(\()p FJ(j)s FC(\))955 3880 y FH(=)d(2)1085 3850 y FJ(j)1147 3880 y FL(and)k FK(\016)f FH(=)1476 3847 y FC(1)p 1469 3861 49 4 v 1469 3908 a FJ(T)1527 3880 y FL(,)i(and)p 0 0 0.7 TeXcolorrgb 1739 3880 a SDict begin H.S end 1739 3880 a 0 0 0.7 TeXcolorrgb FL(Theorem)f(E.7)p 0 0 0.7 TeXcolorrgb 2212 3822 a SDict begin H.R end 2212 3822 a 2212 3880 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (theorem.E.7) cvn H.B /ANN pdfmark end 2212 3880 a 0 TeXcolorgray 27 w FL(pro)n(vides)f(the)i(follo)n(wing)e (regret)h(b)r(ound:)1431 4095 y FE(Reg)c FI(\024)1706 4020 y FD(p)p 1789 4020 113 4 v 75 x FI(jAj)c(\001)1980 4074 y FH(~)1961 4095 y FK(O)2027 4003 y FD(\020)2077 4095 y FK(\033)2127 4052 y FC(1)p FJ(=)p FC(3)2124 4120 y FJ(\013;\014)2232 4095 y FK(T)2293 4061 y FC(5)p FJ(=)p FC(6)2396 4003 y FD(\021)2446 4095 y FK(:)0 4298 y FL(This)28 b(is)f(the)h(desired)f(regret)f(b)r(ound.)2621 b Fd(2)p 0 TeXcolorgray 1908 5649 a FL(45)p 0 TeXcolorgray eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF