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Secure and Private Federated Learning with Diff...

Secure and Private Federated Learning with Differential Privacy

Tutorial on DASFAA 2024 (https://www.dasfaa2024.org/tutorials/#tutorial-5)

This tutorial explores the foundational principles and practical applications of differentially private federated learning, highlighting recent advancements that incorporate secure aggregation techniques.

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Transcript

  1. ˜-:$PSQPSBUJPO  $SPTTEFWJDF 'FEFSBUFE-FBSOJOH 1SPDFEVSFPG'FEFSBUFE-FBSOJOH '-  4FSWFSEJTUSJCVUFTUIFHMPCBMNPEFMUPDMJFOUT  &BDIDMJFOUDSBGUTBOVQEBUFJOGP

    JF HSBEJFOU VTJOH PXOMPDBMEBUBBOEMPDBMNPEFM DPQZPGHMPCBMNPEFM  &BDIDMJFOUTFOETUIFVQEBUFJOGPUPUIFTFSWFS  4FSWFSBHHSFHBUFTBCBUDIPGDPMMFDUFEVQEBUFJOGP  UIFOVQEBUFTUIFHMPCBMNPEFMVTJOHUIFBHHSFHBUF 1SPQFSUJFTPG'- • 3BXEBUBOFWFSMFBWFTDMJFOUT • 4FSWFSEPOPUNBOBHFUIFSBXEBUB • 1FSTPOBMJ[BUJPODBOCFQSPWJEFECZ VTJOHMPDBMNPEFMUSBJOFECZMPDBMEBUB • $PNQVUBUJPOBMSFTPVSDFMJNJUBUJPOBUDMJFOU • $PNNVOJDBUJPOMJNJUBUJPO JF 8J'J POMZ • $POTJEFSDMJFOUESPQPVUT JF QPXFSPGG /POQBSUJDJQBOUTPG'- (MPCBM.PEFM ̍   
  2. ˜-:$PSQPSBUJPO  '-X%JGGFSFOUJBM1SJWBDZJTBTPMVUJPO • %1JTBTPMVUJPOUPNJUJHBUFUIFJOGFSFODFXJUIUIFPSFUJDBMHVBSBOUFFT • $MJFOUTQSFTFSWFUIFJSQSJWBDZCZFOTVSJOH%1UIFNTFMWFTUISPVHISBOEPNJ[BUJPO %JGGFSFOUJBM 1SJWBDZ 'FEFSBUFE-FBSOJOH

    EBUBNJOJNJ[BUJPOXJUIPVUDFOUSBMJ[FE EBUBDPMMFDUJPOBOEQSJWBDZCZEFGBVMU %JGGFSFOUJBM1SJWBDZ EBUBBOPOZNJ[BUJPOQSFWFOUJOHJOGFSFODFT BHBJOTUUIFTFSWFSBOEUIFPUIFSDMJFOUT
  3. ˜-:$PSQPSBUJPO  8IBUJT%JGGFSFOUJBM1SJWBDZ %POUZPVGFFMMJLFZPVSQSJWBDZJTQSPUFDUFEJGUIFOPJTF  JTBEEJUJWF 4UBUJTUJDBM $PNQVUBUJPO /PJTF "EEJUJPO

    𝜖  ∞    TUSPOH XFBL   ʜ %JGGSFOUJBM1SJWBDZQSPWJEFT • 5IFPSFUJDBMQSJWBDZNFBTVSF WJBOPJTFBEEJUJPO • )PXUPEFSJWFUIFOPJTFSFRVJSFEUP BDIJFWFBHJWFOQSJWBDZQBSBNFUFS𝜖
  4. ˜-:$PSQPSBUJPO  5XP1SJWBDZ.PEFMT3FBMXPSME&YBNQMF $FOUSBM %JGGFSFOUJBM1SJWBDZ -PDBM%JGGFSFOUJBM1SJWBDZ 4UBUT3FMFBTF 4UBUT(BUIFSJOH /P3FRVJSFEUP USVTUDVSBUPS

    3FRVJSFEUP USVTUDVSBUPS &YBNQMF • (PPHMF$ISPNFTUBUTHBUIFSJOH (PPHMF • J04NBD04TUBUTHBUIFSJOH "QQMF • 4UJDLFS4VHHFTU'- -: &YBNQMF • 64$FOTVT 64$FOTVT  • 'VMM63-TEBUBTFU .FUB • /FYUXPSEQSFEJDUJPOJO(CPBSE (PPHMF • "VEJFODF&OHBHFNFOU"1* -JOLFE*O "MJTUPGSFBMXPSMEVTFPGEJGGFSFOUJBMQSJWBDZIUUQTEFTGPOUBJOFTQSJWBDZSFBMXPSMEEJGGFSFOUJBMQSJWBDZIUNM
  5. ˜-:$PSQPSBUJPO  6TFSMFWFM%JGGFSFOUJBM1SJWBDZ 𝝐, 𝜹 %JGGFSFOUJBM1SJWBDZ 5IFEJTUSJCVUJPOPGUIFPVUQVUℳ 𝐷 JT OFBSMZUIFTBNFBTℳ

    𝐷! GPSBMMBEKBDFOUEBUBCBTFT𝐷 BOE𝐷′ EJGGFS CZPOFVTFS ℳ ℳ 0VUQVU 1SPCBCJMJUZ Pr ℳ 𝐷 ∈ 𝑆 ≤ exp 𝜖 Pr ℳ 𝐷" ∈ 𝑆 + 𝛿 𝐷 𝐷! ℳ 𝐷 ℳ 𝐷!
  6. ˜-:$PSQPSBUJPO  4UBUJTUJDBM*OUFSQSFUBUJPOPG%1 • $POTJEFSUIFHBNFPGHVFTTJOHUIFJOQVU𝐷 PS𝐷! GSPNBSBOEPNJ[FEPVUQVU • &NQJSJDBM%1DBOCFEFSJWFEVTJOHUIFJOGFSFODFFSSPSTJOMPUTPGUSJBMT 𝜖"#$

    = max log 1 − 𝛿 − FP FN , log 1 − 𝛿 − FN FP &NQJSJDBMEJGGFSFOUJBMQSJWBDZ [Kairouz+, ICML2015] https://arxiv.org/abs/1311.0776 ℳ GSPN𝐷 PS𝐷! 𝐷 𝐷! 3BOEPNJ[FE "MHPSJUIN 𝜖 = 1.0 𝛿
  7. ˜-:$PSQPSBUJPO  3BOEPNJ[BUJPOHJWJOH%1HVBSBOUFFT • /PJTFTDBMFJTEFTJHOFECBTFEPOB IPXNVDIVTFSDPOUSJCVUJPOTBSFCPVOEFE  BOEC IPXUPBHHSFHBUFTVDICPVOEFEVTFSEBUB •

    6TFSDPOUSJCVUJPOBNBHOJUVEFPGDPOUSJCVUJPOCZTJOHMFVTFS`TEBUBUPPVUQVU • 5IFSBOEPNJ[FENFDIBOJTNFNQMPZTFODPEJOH BHHSFHBUJPO BOEQFSUVSCBUJPO &ODPEJOH "HHSFHBUJPO "MJDF #PC $IBSMJF 1FSUVSCBUJPO . +1: . +1: . +1: . +1: . +1: . +1: . +1: 0 1000 2000 3000 4000 5000 NFBO . +1: #PVOEJOHUIFVTFSDPOUSJCVUJPO BUNPTUDPOTUBOU𝐶 .+1:
  8. ˜-:$PSQPSBUJPO  -BQMBDF.FDIBOJTN .PTUXFMMLOPXOSBOEPNJ[FENFDIBOJTNʢBTTVNJOH𝛿 = 0ʣ ℳ 𝐷 = 𝑓

    𝐷 + Lap 0, Δ3 𝜖 -BQMBDF.FDIBOJTN 𝜖 = 10, Δ! = 1 𝜖 = 1, Δ! = 1 4BNQMFBOPJTFGSPNUIF-BQMBDFEJTUSJCVUJPO IBWJOHNFBOɿ TUEɿ 2 "! # Δ# = sup $,$"∈𝒟 𝑓 𝐷 − 𝑓 𝐷! ( ℓ𝟏 TFOTJUJWJUZ Δ)*+,- = 1 Δ./0, = 𝐶 𝑛 &Y 4FOTJUJWJUZ𝚫𝒇 UIFMBSHFTUEJGGFSFODFPGUIFPVUQVU EJGGFSCZPOFVTFS <%XPSL *$"-1> IUUQTMJOLTQSJOHFSDPNDIBQUFS@
  9. ˜-:$PSQPSBUJPO  1SPPG%14BUJTGBDUJPOCZ-BQMBDF.FDIBOJTN Pr[𝑀 𝐷 = 𝑦] Pr[𝑀 𝐷! =

    𝑦] = Π1 𝑃203 𝑦1 − 𝑓 𝐷 1 Π1 𝑃203 𝑦1 − 𝑓 𝐷! 1 = Π1 exp 𝑏4( 𝑦1 − 𝑓 𝐷 1 − 𝑦1 − 𝑓 𝐷! 1 ≤ Π1 exp 𝑏4( 𝑓 𝐷 1 − 𝑓 𝐷! 1 = exp 𝑏4( C 1 𝑓 𝐷 1 − 𝑓 𝐷! 1 = exp 𝑏4( 𝑓 𝐷 − 𝑓 𝐷! ( = exp 𝜖 Δ# 𝑓 𝐷 − 𝑓 𝐷! ( ≤ exp 𝜖 𝑃203 𝑥 = 1 2𝑏 exp(−𝑏4(|𝑥|) 𝑏 = Δ# 𝜖 Δ# ≥ 𝑓 𝐷 − 𝑓 𝐷! ( 𝑥( − 𝑥5 ≤ |𝑥( − 𝑥5|
  10. ˜-:$PSQPSBUJPO  7BSZJOHQSJWBDZQBSBNFUFS𝝐 • 4USPOHQSJWBDZ BU𝜖 = 0.01 TIPXTFYUSFNFMZSBOEPNJ[FEVTFMFTTPVUQVUT •

    8FBLFSQSJWBDZ BU𝜖 = 10 JTBMNPTUTBNFBTUIFPSJHJOBMIJTUPHSBN 𝜖 = 0.1 𝜖 = 1.0 𝜖 = 0.01 𝜖 = 10 8FBL1SJWBDZ1SPUFDUJPO 4USPOH1SJWBDZ1SPUFDUJPO
  11. ˜-:$PSQPSBUJPO  1SJWBDZ$PNQPTJUJPOJO-FBSOJOH*UFSBUJPO • %1DPOTJEFSTUIBUDPOUJOVBMEBUBVTBHFTBDDVNVMBUFQSJWBDZDPOTVNQUJPOT • 5IFBDDVNVMBUFEQSJWBDZDPOTVNQUJPODBOCFEFSJWFECZDPNQPTJUJPOPGFBDI𝝐 • 5IFQSJWBDZQBSBNFUFS𝜖' GPSRVFSZ𝑞'

    JTBMTPDBMMFEQSJWBDZCVEHFU GPS𝑞' BOEJT BTTJHOFEDPOTJEFSJOHUIFUPUBMQSJWBDZCVEHFU • 4JNQMFTUXBZ4FRVFOUJBM$PNQPTJUJPO • 3FDFOU TPQIJTUJDBUFEXBZT 3ÉOZJ %1 [$%1 𝝐𝟏 𝝐𝟏 𝝐𝟐 𝝐𝟏 𝝐𝟐 𝝐𝒌 ʜ ʜ EFDMJOF QSJWBDZDPOTVNQUJPO CBUDIFT EBUBBDDFTT 1SJWBDZ$ PN QPTJUJPO 𝜖()(*+ = ∑𝜖' 5PUBM 1SJWBDZ #VEHFU TUCBUDI OECBUDI 5UI CBUDI %1 "MHPSJUIN 4FRVFOUJBM$PNQPTJUJPO
  12. ˜-:$PSQPSBUJPO  %14(%ʢ%1-FBSOJOH'SBNFXPSLʣ 6QEBUFNPEFMXJUIUIFOPJTFE DMJQQFE HSBEJFOUXIJMFUIFQSJWBDZCVEHFUSFNBJOT 𝐚𝐯𝐠. (BVTTJBO /PJTF 𝑔$

    = ∇𝑓(𝑥$ ; 𝜃) 𝝅𝑪 𝒈𝟏 𝑔' = ∇𝑓(𝑥' ; 𝜃) $ %BUBCBTF𝐷 𝐷 = 𝑛 6QEBUF.PEFM 3FQFBUVOUJMQSJWBDZCVEHFUSFNBJOT TVCTBNQMJOHIFMQTUPTBWFQSJWBDZDPOTVNQUJPO <"CBEJ $44>IUUQTBSYJWPSHBCT (SBEJFOU$MJQQJOH 3BOEPN 4VCTBNQMJOH 1FSTBNQMF (SBEJFOU 𝑥$ , … , 𝑥' ∈ 𝐷 𝑔( = ∇𝑓(𝑥(; 𝜃) 𝜋$ 𝑔% = 𝑔% ⋅ min 1, 𝐶 𝑔% & (SBEJFOU $MJQQJOH $MJQHSBEJFOUℓ𝟐 OPSNBUDPOTUBOU$à 4FOTJUJWJUZPGWFDUPSNFBOJTCPVOEFEBU$C 𝜃' = 𝜃'() − 𝜂 1 𝑏 F %∈ + 𝜋$ 𝑔% ; 𝜃'() + 𝑁 0, 𝜎 ʜ 𝑥! 𝑥"
  13. ˜-:$PSQPSBUJPO  -JCSBSZGPS%14(% • 5FOTPS'MPX1SJWBDZ IUUQTHJUIVCDPNUFOTPSGMPXQSJWBDZ • 0QBDVT IUUQTHJUIVCDPNQZUPSDIPQBDVT •

    BMJCSBSZUIBUFOBCMFTUSBJOJOH1Z5PSDI NPEFMTXJUIEJGGFSFOUJBMQSJWBDZ • TVQQPSUTUSBJOJOHXJUINJOJNBMDPEFDIBOHFTSFRVJSFEPOUIFDMJFOU • USBDLUIFQSJWBDZCVEHFU FYQFOEFEBUBOZHJWFONPNFOU
  14. ˜-:$PSQPSBUJPO  %15SBJOJOHJTTFOTJUJWFUPOPJTF • 1SJWBUFEBUBTZOUIFTJT5SBJOBEBUBTZOUIFTJTNPEFM UIBUJNJUBUFTPSJHJOBMEBUBTFU • *TTVFUIFJSDPNQMJDBUFEUSBJOJOHQSPDFTT FH ("/

    7"& JTTFOTJUJWFUPOPJTF • "QQSPBDITJNQMJGZUIFQSPDFTTCZVTJOHQSJWBUJ[FEEBUBFNCFEEJOHT 5SBJOXJUI 4ZOUIFTJ[F /BÏWF.FUIPE 1(. PVST 1&"3- PVST <5BLBHJ5BLBIBTIJ  *$%&> 1SJWBUJ[FE(FOFSBUJWF.PEFM %FD%11$" (FO%14(% %FD%14(% (FO%14(% %FD3BOE'FBUVSF (FO"EW5SBJO 𝜖, 𝛿 = 1.0,10(, <5BLBHJ5BLBIBTIJ *$%&>IUUQTBSYJWPSHBCT <-JFX *$-3>IUUQTBSYJWPSHBCT <-JFX *$-3>
  15. ˜-:$PSQPSBUJPO  3FDBQ%15SBJOJOH • "EEUIF (BVTTJBO OPJTFUPDMJQQFEHSBEJFOU • "DDPVOUQSJWBDZDPOTVNQUJPOXIJMFJUFSBUJOHUSBJOJOHDPOTJEFSJOHUIFTBNQMJOH SBUFPGCBUDIFTBOEUIFOPJTFTDBMFPGUIF

    (BVTTJBO OPJTF TUCBUDI OECBUDI 5UI CBUDI %15SBJOJOH "MHPSJUIN 𝝐𝟏 𝝐𝟏 𝝐𝟐 𝝐𝟏 𝝐𝟐 𝝐𝒌 ʜ ʜ EFDMJOF QSJWBDZDPOTVNQUJPO CBUDIFT EBUBBDDFTT 1SJWBDZ$ PN QPTJUJPO 𝜖()(*+ = ∑𝜖' 5PUBM 1SJWBDZ #VEHFU 𝜃' = 𝜃'() − 𝜂 1 𝑏 F %∈ + 𝜋$ 𝑔% ; 𝜃'() + 𝑁 0, 𝜎 %14(%
  16. ˜-:$PSQPSBUJPO  'FEFSBUFE-FBSOJOHXJUI-PDBM%1 • %1JTBTPMVUJPOUPNJUJHBUFUIFJOGFSFODFXJUIUIFPSFUJDBMHVBSBOUFFT • $MJFOUTQSFTFSWFUIFJSQSJWBDZCZFOTVSJOH%1UIFNTFMWFTUISPVHISBOEPNJ[BUJPO %JGGFSFOUJBM 1SJWBDZ 'FEFSBUFE-FBSOJOH

    EBUBNJOJNJ[BUJPOXJUIPVUDFOUSBMJ[FE EBUBDPMMFDUJPOBOEQSJWBDZCZEFGBVMU %JGGFSFOUJBM1SJWBDZ EBUBBOPOZNJ[BUJPOQSFWFOUJOHJOGFSFODFT BHBJOTUUIFTFSWFSBOEUIFPUIFSDMJFOUT
  17. ˜-:$PSQPSBUJPO  -PDBM%1 1SJWBUF4UBUT(BUIFSJOH • -PDBM%1BTTVNFTUIBUFBDIDMJFOUSBOEPNJ[FTPXOPVUQVUCFGPSFSFQPSUJOHJU • 1SJWBUFTUBUTHBUIFSJOHX-%1BMMPXTJOGFSSJOHTUBUJTUJDTBCPVUQPQVMBUJPOT XIJMFQSFTFSWJOHUIFQSJWBDZPGJOEJWJEVBMT XJUIPVUBOZUSVTUFEFOUJUZ

    1SJWBUF4UBUT(BUIFSJOHX-%1 Pr ℳ 𝑥, ∈ 𝑆 ≤ exp 𝜖 Pr ℳ 𝑥- ∈ 𝑆 + 𝛿 𝑥! 𝑥# ℳ ℳ *OEJTUJOHVJTIBCMF (𝝐, 𝜹)-PDBM%1 4UBUT(BUIFSJOH /P3FRVJSFEUP USVTUDVSBUPS
  18. ˜-:$PSQPSBUJPO  -PDBM%1'FE4(%"WH 1SJWBUFWFDUPSNFBOFTUJNBUJPOVOEFS-%1DPOTUSBJOUT • &BDIDMJFOUDPNQVUFTUIFJSSBOEPNJ[FEHSBEJFOUCZUIFJSMPDBMEBUB • 4FSWFSFTUJNBUFTUIFNFBOBNPOHBCBUDIPGSBOEPNJ[FEHSBEJFOUTUPVQEBUF UIFHMPCBMNPEFM 𝐚𝐯𝐠.

    (SBEJFOU $MJQQJOH 𝑔$ = ∇𝑓(𝑥$ ; 𝜃) 𝝅𝑪 𝒈𝟏 𝑔' = ∇𝑓(𝑥'; 𝜃) 3BOEPN 4BNQMJOH $ 1FSDMJFOU (SBEJFOU (MPCBM.PEFM 6QEBUF 𝑥$, … , 𝑥' ∈ 𝐷 𝑔( = ∇𝑓(𝑥( ; 𝜃) (MPCBM.PEFM 𝜃 1FSUVSCBUJPO 1SJWBDZCVEHFUTIPVMECFDPOTJEFSFE XIFOUIFTBNFDMJFOUQBSUJDJQBUFTJO UIF'-QSPDFEVSFNVMUJQMFUJNFT
  19. ˜-:$PSQPSBUJPO  -PDBM3BOEPNJ[FSGPS'FEFSBUFE-FBSOJOH • 3BOEPNJ[FESFTQPOTFJTUIFTUBOEBSEBQQSPBDIUPFOTVSF-%1 • %JTDSFUFOPJTFJTCFDPNJOHQPQVMBSSFDFOUMZTJODFTFDVSFBHHSFHBUJPOSFRVJSFTJU 1SJW6OJU .FDIBOJTN •

    3BOEPNMZTBNQMFBHSBEJFOUGSPN UIFVOJUTQIFSF (BVTTJBO.FDIBOJTN %14(%WBSJBOU $BVUJPOOFFEUPNPEJGZOPJTFTDBMFBHBJOTU$%1 3BOEPNJ[F3FTQPOTF $POUJOVPVT/PJTF %JTDSFUF/PJTF • %JTDSFUF-BQMBDF .FDIBOJTN • %JTUSJCVUFE%JTDSFUF (BVTTJBO.FDIBOJTN • 4LFMMBN .FDIBOJTN IUUQTFOXJLJQFEJBPSHXJLJ4LFMMBN@EJTUSJCVUJPO 4LFMMBN EJTUSJCVUJPO #IPXNJDL 1SPUFDUJPO"HBJOTU3FDPOTUSVDUJPOBOE *UT"QQMJDBUJPOTJO1SJWBUF'FEFSBUFE-FBSOJOH IUUQTBSYJWPSHBCT "HBSXBM 5IF4LFMMBN .FDIBOJTNGPS%JGGFSFOUJBMMZ1SJWBUF 'FEFSBUFE-FBSOJOH/FVS*14IUUQTBSYJWPSHBCT
  20. ˜-:$PSQPSBUJPO  4DBMBCMF%1'-4JNVMBUPS • QGM1ZUIPOGSBNFXPSLGPS1SJWBUF'FEFSBUFE-FBSOJOHTJNVMBUJPOT "QQMF • IUUQTHJUIVCDPNBQQMFQGMSFTFBSDI • 1ZUIPOGSBNFXPSLUPFNQPXFSSFTFBSDIFSTUPSVOFGGJDJFOUTJNVMBUJPOTXJUIQSJWBDZ

    QSFTFSWJOHGFEFSBUFEMFBSOJOH '- BOEEJTTFNJOBUFUIFSFTVMUTPGUIFJSSFTFBSDIJO'- • 4VQQPSUCPUI1Z5PSDI BOE5FOTPS'MPX • *ODMVEJOHDPNNPONFDIBOJTNTGPSMPDBMBOEDFOUSBMEJGGFSFOUJBMQSJWBDZ • '-65&"TDBMBCMFGFEFSBUFEMFBSOJOHTJNVMBUJPOQMBUGPSN .JDSPTPGU3FTFBSDI • IUUQTHJUIVCDPNNJDSPTPGUGMTJNVMBUJPO • l'FEFSBUFE-FBSOJOH6UJMJUJFTBOE5PPMTGPS&YQFSJNFOUBUJPOz '-65& • IJHIQFSGPSNBODFPQFOTPVSDF QMBUGPSNGPSGFEFSBUFEMFBSOJOHSFTFBSDIBOEPGGMJOF TJNVMBUJPOT • JODMVEJOHEJGGFSFOUJBMQSJWBDZ RVBOUJ[BUJPO TDBMJOHBOEBWBSJFUZPGPQUJNJ[BUJPOBOE GFEFSBUJPOBQQSPBDIFT
  21. ˜-:$PSQPSBUJPO  -%1'-OFFETMBSHFBNPVOUPGOPJTF • %VFUPUIFBNPVOUPGOPJTFJTWFSZMBSHFUPTBUJTGZ-%1 BDIJFWJOHHPPEGFEFSBUFE MFBSOJOHJTOPUTPFBTZ 𝜖, 𝛿 

    8,10./ 𝜖, 𝛿  8,10./ <.BFEB %#4++PVSOBM>IUUQTECTKPSHKPVSOBMECTK@KPVSOBM@KECTK@KPVSOBM@WPM@@
  22. ˜-:$PSQPSBUJPO  4FDVSF4IVGGMJOH • *OUFSNFEJBUFUSVTUFEFOUJUZlTIVGGMFSz BOPOZNJ[FTMPDBMVTFST`JEFOUJUJFT • 4IVGGMFSDBOBNQMJGZMPDBM%1HVBSBOUFFTCZJUTBEEJUJPOBMSBOEPNJ[BUJPOFGGPSU • 5IFBNQMJGJDBUJPOEFQFOETPOUIFQPQVMBUJPOUIBUQBSUJDJQBUFTJOUIFTIVGGMJOH

    4IVGGMFS QSJWBDZBNQ WJB BOPOZNJ[BUJPO BOPOZNPVTMPHT 1SJWBDZ"NQMJGJDBUJPO 𝜖0 = 8 -%1 𝛿 = 10.1 1SJWBDZBNQJTDBMDVMBUFEVTJOH <'FMENBO >IUUQTBSYJWPSHBCT
  23. ˜-:$PSQPSBUJPO  4FDVSF"HHSFHBUJPO 4FD"HH • 4FD"HH JTBDMBTTPGTFDVSFNVMUJQBSUZDPNQVUBUJPOEFTJHOFEUPDPNQVUFBO BHHSFHBUFWBMVF TVDIBTTVN XJUIPVUSFWFBMJOHBOZJOGPSNBUJPOUPPOFBOPUIFS

    • 4FD"HH FOBCMFTVTUPHBUIFSMPDBMEBUBXJUIUSVTUà 5SVTUBTTVNQUJPOGPS$%1 • &BDIDMJFOUEJTUSJCVUFTTIBSFTUPUIFPUIFST UIFOBHHSFHBUFTUIFN XEQ OPJTF BHHSFHBUFETUBUT ∑
  24. ˜-:$PSQPSBUJPO  %1'53-$PSSFMBUFEOPJTFPOUSVTUFOUJUZ 6OMJLF%14(%BEETJOEFQFOEFOUOPJTFGPSFBDICBUDI %1'53-JOKFDUTDPSSFMBUFE OPJTFBDDPVOUJOHGPSUIFOPJTFBEEJUJPOIJTUPSZ UIFOSFEVDFTUIFOPJTFGPSMBUFS CBUDIFT 𝜃DEF =

    𝜃D − 𝜂 𝑔D + 𝑧D 𝜃DEF = 𝜃D − 𝜂 𝑔D + ; GHI D 𝛽D,G 𝑧DJG %14(% %1'53- IUUQTSFTFBSDIHPPHMFCMPHGFEFSBUFEMFBSOJOHXJUIGPSNBMEJGGFSFOUJBMQSJWBDZHVBSBOUFFT *OEFQFOEFOU/PJTF $PSSFMBUFE/PJTF '53-'PMMPX5IF3FHVMBSJ[FE-FBEFS <,BJSPV[ *$.->IUUQTBSYJWPSHBCT
  25. ˜-:$PSQPSBUJPO  %JTUSJCVUFE%1PO4FDVSF"HH • 4FDVSFBHHSFHBUJPOFOBCMFTVTUPHBUIFSMPDBMEBUBXJUIUSVTU • %JTUSJCVUFE%1SFEVDFMPDBMOPJTFUPBMFWFMFRVJWBMFOUUP$%1BGUFSBHHSFHBUJPO ∑ "HH 𝐦𝐞𝐚𝐧

    = 𝟏 𝒎 Q 𝒊∈ 𝒎 𝒙𝒊 + 𝒛 𝐦𝐞𝐚𝐧 = 𝟏 𝒎 Q 𝒊∈ 𝒎 𝒙𝒊 + 𝒛𝒊 𝒎 -PDBM%1 4FDVSF"HHGPS%JTUSJCVUFE%1 𝐦𝐞𝐚𝐧 = 𝟏 𝒎 Q 𝒊∈ 𝒎 𝒙𝒊 + 𝒛𝒊 $FOUSBM%1 %JTUSJCVUFE%1 .FBOFTUJNBUJPOXJUIUIF(BVTTJBOOPJTF
  26. ˜-:$PSQPSBUJPO  'FEFSBUFE"OBMZUJDT 4UBUJTUJDBMBOBMZUJDTPO%1XJUI4FDVSF"HHBSFEFQMPZFEJOSFBMXPSME UPJNQSPWF UIFCBMBODFCFUXFFOQSJWBDZQSPUFDUJPOBOEVUJMJUZPGEBUBVTF 4FDVSF"HH XJUI -PDBM3BOEPNJ[FS 4FDSFU4IBSJOH

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