差分プライバシーによる 安全な連合学習の実現

LINEヤフー株式会社 Privacy & Trust Team / Federated Learning Team
LINEヤフー研究所 上席研究員
髙橋翼 Tsubasa TAKAHASHI

2024/1/24 名古屋大学での講義資料です

Transcript

1. ˜
   -:
   $PSQPSBUJPO ࠩ෼ϓϥΠόγʔʹΑΔ ҆શͳ࿈߹ֶशͷ࣮ݱ ∁ڮ ཌྷ c
   5TVCBTB 5",")"4)* -*/&Ϡϑʔגࣜձࣾ 1SJWBDZ
   
   5SVTU
   5FBN
   
   'FEFSBUFE
   -FBSOJOH
   5FBN -*/&Ϡϑʔݚڀॴ ্੮ݚڀһ UTVCBTBUBLBIBTIJ!MZDPSQDPKQ
2. ˜
   -:
   $PSQPSBUJPO ࿈߹ֶशͱϓϥΠόγʔ σʔλ׆༻࣌ͷϓϥΠόγʔϦεΫ %JGGFSFOUJBM
   1SJWBDZʢࠩ෼ϓϥΠόγʔʣ ࠩ෼ϓϥΠόγʔʹΑΔ҆શͳ࿈߹ֶश ۙ೥ͷൃలతͳ࿩୊ʢ͕࣌ؒ͋Ε͹ʣ      -*/&ϠϑʔͰͷݚڀ։ൃͷࣄྫʹ͍ͭͯ΋ॴʑͰ؆ܿʹ঺հ͠·͢  ຊߨٛͷίϯςϯπ
3. ˜
   -:
   $PSQPSBUJPO  ࿈߹ֶश 'FEFSBUFE
   -FBSOJOH • ΫϥΠΞϯτ܈ͱαʔόʔͱ͕ڠௐ͢Δ෼ࢄܕͷػցֶशʢ$SPTTEFWJDF
   '-Λ૝ఆʣ • ΫϥΠΞϯτͷσʔλ͔Βܭࢉͨ͠ߋ৽৘ใ ޯ഑
   ͚ͩΛαʔόʔʹఏڙ • σʔλ͸ΫϥΠΞϯτ಺ʹཹ·Δ à ϓϥΠόγʔʹ഑ྀ /POQBSUJDJQBOUT
   PG
   '- (MPCBM
   .PEFM
4. ˜
   -:
   $PSQPSBUJPO  ࿈߹ֶशͷಋೖࣄྫ IUUQTBDMBOUIPMPHZPSHBDMJOEVTUSZQEG 'FEFSBUFE
   -FBSOJOH
   PG
   (CPBSE -BOHVBHF
   .PEFM
   (PPHMF
5. ˜
   -:
   $PSQPSBUJPO  ʮ-*/&ʯΞϓϦͷελϯϓαδΣετ • ʮ͋Γ͕ͱ͏ʯ౳ͷจࣈΛೖྗͨ͠ࡍʹɺҙຯͷ͍ۙελϯϓΛਪનදࣔ͢Δػೳ • ΑΓߴ͍ਫ਼౓ͰϢʔβʔͷ޷Έʹ͋ͬͨఏҊ͢ΔͨΊɺ࿈߹ֶशͱࠩ෼ϓϥΠόγʔΛಋೖ IUUQTXXXZPVUVCFDPNXBUDI WL5#TIH0C IUUQTUFDIWFSTFNFKBTFTTJPOT IUUQTMJOFDPSQDPNKBTFDVSJUZBSUJDMF
6. ˜
   -:
   $PSQPSBUJPO  ޯ഑͔Βݩͷσʔλ͸෮ݩͰ͖Δ͔ʁ l*OWFSUJOH
   (SBEJFOUT
    )PX
   FBTZ
   JT
   JU
   UP
   CSFBL
   QSJWBDZ
   JO
   GFEFSBUFE
   MFBSOJOH z IUUQTBSYJWPSHBCT :FT (SBEJFOU
   *OWFSTJPO߈ܸ
7. ˜
   -:
   $PSQPSBUJPO  1SJWBDZ
   &OIBODJOH
   5FDIOPMPHJFT
   1&5T 5&&
   5SVTUFE
   &YFDVUJPO
   &OWJSPONFOU 4.1$
   4FDVSF
   .VMUJQBSUZ
   $PNQVUBUJPO 'FEFSBUFE
   -FBSOJOH %JGGFSFOUJBM
   1SJWBDZ 4FDVSF
   $PNQVUJOH
8. ˜
   -:
   $PSQPSBUJPO  1SJWBDZ
   &OIBODJOH
   5FDIOPMPHJFT
   1&5T 1&5Tʹ͸ɺ
   ҉߸౳ʹΑΔΞΫηε੍ޚٕज़ͱ 
   ৘ใͦͷ΋ͷΛՃ޻͢Δٕज़ ͕͋Δ ࿈߹ֶशɾ ہॴࠩ෼ϓϥΠόγʔ ಗ໊Խɾσʔλ߹੒ ࠩ෼ϓϥΠόγʔ ੺͍఺ઢΑΓࠨଆ͕ϓϥΠόγʔอޢͷൣғ FH ಗ໊Խ΍ࠩ෼ϓϥΠόγʔ FH ࿈߹ֶश΍ൿີܭࢉ
9. ˜
   -:
   $PSQPSBUJPO σʔλ׆༻࣌ͷϓϥΠόγʔϦεΫ 
10. ˜
    -:
    $PSQPSBUJPO  /FUGMJY
    1SJ[Fͷ࠶ࣝผͷ໰୊  • ಗ໊Խ͞Ε͍ͯͳ͍֎෦ͷσʔλϕʔε *.%#
    ͱͷಥ߹ʹΑͬͯɺಗ໊Խ͞Ε͍ͯͨίϝ ϯτͱϢʔβ͕ରԠ෇͍ͯ͠·ͬͨ • ίϝϯτʹ͸੓࣏৴৚͕Θ͔Δ΋ͷ΋͋ͬͨ • ݸͷධՁͱ೔෇ िؒҎ಺ͷਫ਼౓
    ͕෼͔Ε͹ͷਫ਼౓ͰݸਓΛಛఆͰ͖ͨ 6TFSOBNF 5JUMF 3BUF "MJDF """  ###  %%%  #PC """  999  1TFVEP
    *% 5JUMF 3BUF $PNNFOU Y """  ###  %%%  *U
    TFFNT
    YYY B :::  999  "OPOZNJ[FE
    /FUGMJY
    .PWJF
    SBOLJOH
    EBUB /POBOPOZNPVT
    *.%#
    NPWJF
    SBUJOH +0*/
11. ˜
    -:
    $PSQPSBUJPO  ࠩ෼߈ܸ ͋Δσʔλϕʔε͔Βੜ੒ͨ͠ෳ਺ͷ౷ܭ৘ใͷʮࠩ෼ʯ͔Βಛఆͷݸਓʹؔ͢Δ৘ใΛਪఆ BWH
    TBMBSZ 
    ສԁ BWH
    TBMBSZ 
    ສԁ ʲ2VJ[ʳ"MJDFͷ೥ऩ͸ʁ ʜ ʜ ʜ ʜ "MJDF "MJDF͕ୀ৬ 
    FOHJOFFST 
    FOHJOFFST ೥ ೥
12. ˜
    -:
    $PSQPSBUJPO  2VJ[ͷճ౴ BWH
    TBMBSZ 
    ສԁ BWH
    TBMBSZ 
    ສԁ ʲ2VJ[ʳ"MJDFͷ೥ऩ͸ʁ ʜ ʜ ʜ ʜ "MJDF "MJDF͕ୀ৬ 
    FOHJOFFST 
    FOHJOFFST ೥ ೥ 
    Y
    
    r 
    Y
     
    ສԁ ౴͑
13. ˜
    -:
    $PSQPSBUJPO  ࠩ෼߈ܸͷࣄྫ 'BDFCPPLͷ1**CBTFE
    5BSHFUJOH
    ݱࡏ͸मਖ਼ࡁΈ • ޿ࠂλʔήςΟϯάͷޮՌΛܭଌ͢ΔͨΊͷ౷ܭ෼ੳ"1* • ։੍ࣔޚͷͨΊʹᮢ஋ʹΑΔอޢΛಋೖ͍͕ͯͨ͠ɺόΠύε͞Εͨ • ి࿩൪߸΍ϝʔϧΞυϨεɺ8FCӾཡཤྺ͕࿙Ӯ IUUQTXXXZPVUVCFDPNXBUDI W-Q *X:WY(QL IUUQTXXXGUDHPWTZTUFNGJMFTEPDVNFOUTQVCMJD@FWFO UTQQSJWBDZDPONJTMPWF@QEG
14. ˜
    -:
    $PSQPSBUJPO  ػցֶशϞσϧʹର͢ΔڴҖ ܇࿅ʹ࢖༻ͨ͠ σʔλ ܇࿅ʹະ࢖༻ͷ σʔλ 0.7 0.1 ⋮ 0.05 0.25 0.12 ⋮ 0.45 5BSHFU
    .PEFM .BMJDJPVT *OTJEFSʹΑΔ ܇࿅σʔλ౳ͷ ࡡऔɾ౪ௌ ѱҙͷ͋Δୈࡾऀʹ ΑΔϞσϧߏ଄΍ ܇࿅σʔλͷਪఆ 5SBJOJOH 1IBTF *OGFSFODF 1IBTF
15. ˜
    -:
    $PSQPSBUJPO  ϝϯόʔγοϓਪ࿦߈ܸ ͋Δσʔλ͕Ϟσϧͷ܇࿅ʹར༻͞Ε͔ͨ൱͔Λਪఆ͢Δ߈ܸ ܇࿅ʹ࢖༻ͨ͠ σʔλ ܇࿅ʹະ࢖༻ͷ σʔλ 0.7 0.1 ⋮ 0.05 0.25 0.12 ⋮ 0.45 5BSHFU
    .PEFM "UUBDL
    .PEFM   &TUJNBUF
    BT JO
    USBJO
    EBUB &TUJNBUF
    BT PVU
    PG
    USBJO
    EBUB "UUBDL
    .PEFMͷ܇࿅ʹ͸ɺ5BSHF
    .PEFMΛ໛฿ͨ͠4IBEPX
    .PEFMΛར༻ ϝϯόʔγοϓਪ࿦΁ͷରࡦͱͯ͠ɺࠩ෼ϓϥΠόγʔͷอূ͕༗ޮ
16. ˜
    -:
    $PSQPSBUJPO  Ϟσϧ౪༻߈ܸ .PEFM
    4UFBMJOH λʔήοτϞσϧ͔Βͷग़ྗͱͦͷೖྗΛ༻͍ͯɺڍಈΛ໛฿ͨ͠ϞσϧΛߏங 4ZOUIFUJD
    *OQVUT
    0VUQVUT 5BSHFU
    .PEFM *NJUBUJPO
    .PEFM 4IBEPX
    .PEFM
17. ˜
    -:
    $PSQPSBUJPO  ػցֶशٕज़ʹର͢ΔϓϥΠόγʔอޢٕज़͕ඞཁ TBNQMF
    𝑥 ϓϥΠόγʔอޢ͞Εͨ ػցֶशϞσϧ Ϟσϧ͔Βͷग़ྗ ܇࿅σʔλͷਪఆ͕ࠔ೉ ܇࿅σʔλ TBNQMF
    𝑥 ػցֶशϞσϧ Ϟσϧ͔Βͷग़ྗ ܇࿅σʔλ ʮࠩ෼ϓϥΠόγʔʯʹΑΔ ϓϥΠόγʔอޢػցֶश
18. ˜
    -:
    $PSQPSBUJPO ࠩ෼ϓϥΠόγʔ 
19. ˜
    -:
    $PSQPSBUJPO  ࠩ෼ϓϥΠόγʔͱ͸ʁ ϊΠζʢ ʣ͕Ճࢉ͞Ε͍ͯΔͱϓϥΠόγʔอޢ͞Ε͍ͯΔΑ͏ʹײ͡·ͤΜ͔ʁ ౷ܭత ͳग़ྗ ϊΠζ ΛՃࢉ 𝜖  ∞    TUSPOH XFBL   ʜ ࠩ෼ϓϥΠόγʔ͕ఏڙ͢Δ΋ͷ • ϊΠζͷՃࢉʹର͢Δʢ͋ΔલఏͰͷʣ ཧ࿦తͳϓϥΠόγʔอޢͷई౓ • ॴఆͷϓϥΠόγʔڧ౓ 𝜖 ͷ ୡ੒ʹඞཁͳϊΠζͷಋग़
20. ˜
    -:
    $PSQPSBUJPO  ͭͷओཁͳϓϥΠόγʔϞσϧ $FOUSBM
    %JGGFSFOUJBM
    1SJWBDZ -PDBM
    %JGGFSFOUJBM
    1SJWBDZ ୈࡾऀ΁ͷ ౷ܭ஋ͷެ։ ػඍͳ ౷ܭͷऩू αʔόʔΛ ৴༻͠ͳ͍ αʔόʔΛ ৴༻͢Δ ಋೖࣄྫ • $ISPNFϒϥ΢βͰͷ౷ܭऩू (PPHMF • J04σόΠεͰͷ౷ܭऩू "QQMF • ελϯϓαδΣετͷ࿈߹ֶश -*/&Ϡϑʔ ಋೖࣄྫ • ೥౓ถࠃࠃ੎ௐࠪ 64
    $FOTVT
    • 'VMM
    63-T
    EBUB
    TFU
    .FUB • /FYU
    XPSE
    QSFEJDUJPO
    JO
    (CPBSE (PPHMF • "VEJFODF
    &OHBHFNFOU
    "1*
    -JOLFE*O ·ͣ͸ͪ͜Β͔Β ʢ͝ࢀߟʣ"
    MJTU
    PG
    SFBMXPSME
    VTF
    PG
    EJGGFSFOUJBM
    QSJWBDZ
    IUUQTEFTGPOUBJOFTQSJWBDZSFBMXPSMEEJGGFSFOUJBMQSJWBDZIUNM
21. ˜
    -:
    $PSQPSBUJPO  𝝐, 𝜹 %JGGFSFOUJBM
    1SJWBDZ ϝΧχζϜ ℳ: 𝒟 → 𝒮 ͕ 𝜖, 𝛿 %1
    Λຬͨ͢ͱ͸ɺྡ઀σʔλϕʔε 𝐷, 𝐷) ∈ 𝒟 ͓Αͼ೚ҙͷग़ྗͷू߹ 𝑆 ⊆ 𝒮 ʹରͯ͠ҎԼ͕੒Γཱͭͱ͖Ͱ͋Δ %BUBCBTF
    𝑫 0VUQVU 𝜖  ∞    TUSPOH XFBL   ʜ Pr ℳ 𝐷 ∈ 𝑆 ≤ exp 𝜖 Pr ℳ 𝐷! ∈ 𝑆 + 𝛿 ℳ 𝑫′ɿ BEKBDFOU
    PG
    𝑫 ℳ 𝜖 ≥ 0, 𝛿 ∈ 0,1 ) ೖྗ͕มԽͯ͠΋ग़ྗʹ͸΄ͱΜͲʢexp 𝜖 ఔ౓͔͠ʣӨڹ͕ͳ͍ %XPSL
    %JGGFSFOUJBM
    QSJWBDZ
    *$"-1
    
22. ˜
    -:
    $PSQPSBUJPO  -BQMBDFϝΧχζϜ ࠷΋Α͘஌ΒΕͨϥϯμϜԽॲཧΛ࣮ݱ͢ΔϝΧχζϜʢ𝛿 = 0Λ૝ఆʣ ℳ 𝐷 = 𝑓 𝐷 + Lap 0, Δ- 𝜖 -BQMBDF
    .FDIBOJTN 𝜖 = 10, Δ! = 1 𝜖 = 1, Δ! = 1 ฏۉɿ
    ඪ४ภࠩɿ 2 "! # ͷ ϥϓϥε෼෍͔ΒϊΠζΛαϯϓϦϯά Δ. = sup /,/!∈𝒟 𝑓 𝐷 − 𝑓 𝐷) 3 ℓ𝟏 TFOTJUJWJUZ Δ45678 = 1 Δ9:;7 = 1 𝑛 ˞
    ૝ఆ͢Δྡ઀%#ʹґଘ &Y 4FOTJUJWJUZ
    𝚫𝒇 ೚ҙͷҰϨίʔυͷมԽ˞͕ؔ਺ͷग़ྗʹӨڹΛ༩͑Δ౓߹͍ %XPSL
    %JGGFSFOUJBM
    QSJWBDZ
    *$"-1
    
23. ˜
    -:
    $PSQPSBUJPO  ʢ͝ࢀߟʣ-BQMBDFϝΧχζϜͷূ໌ Pr[𝑀 𝐷 = 𝑦] Pr[𝑀 𝐷) = 𝑦] = Π< 𝑃=;> 𝑦< − 𝑓 𝐷 < Π< 𝑃=;> 𝑦< − 𝑓 𝐷) < = Π< exp 𝑏?3 𝑦< − 𝑓 𝐷 < − 𝑦< − 𝑓 𝐷) < ≤ Π< exp 𝑏?3 𝑓 𝐷 < − 𝑓 𝐷) < = exp 𝑏?3 C < 𝑓 𝐷 < − 𝑓 𝐷) < = exp 𝑏?3 𝑓 𝐷 − 𝑓 𝐷) 3 = exp 𝜖 Δ. 𝑓 𝐷 − 𝑓 𝐷) 3 ≤ exp 𝜖 𝑃=;> 𝑥 = 1 2𝑏 exp(−𝑏?3|𝑥|) 𝑏 = Δ. 𝜖 Δ. ≥ 𝑓 𝐷 − 𝑓 𝐷) 3 𝑥3 − 𝑥@ ≤ |𝑥3 − 𝑥@|
24. ˜
    -:
    $PSQPSBUJPO  ྡ઀σʔλϕʔε ೚ҙͷཁૉ͚͕ͩҟͳΔσʔλϕʔεͷ૊ɻ%1Ͱ૝ఆ͢Δʮࠩ෼ʯΛϞσϧԽ /".& $BODFS "MJDF :FT #PC /P $ZOUIJB /P %BWJE :FT ʜ /".& $BODFS "MJDF :FT #PC /P $ZOUIJB /P %BWJE :FT &WF :FT /".& $BODFS "MJDF :FT $ZOUIJB /P %BWJE :FT /".& $BODFS "MJDF :FT #PC /P $ZOUIJB /P %BWJE :FT 'SBOD /P อޢର৅ͷ%# 𝑑# 𝐷, 𝐷$ = 1 𝑑# ⋅,⋅ ɿϋϛϯάڑ཭ /".& $BODFS "MJDF :FT #PC /P %BWJE :FT /".& $BODFS "MJDF /P #PC /P $ZOUIJB /P %BWJE :FT /".& $BODFS "MJDF :FT #PC /P $ZOUIJB :FT %BWJE :FT /".& $BODFS "MJDF :FT #PC :FT $ZOUIJB /P %BWJE :FT /".& $BODFS "MJDF :FT #PC /P $ZOUIJB /P %BWJE /P ྡ઀%#
    BEESFNPWBM ྡ઀%#
    SFQMBDFNFOU ϊΠζͷେ͖͞͸࠷େͰ BEESFNPWBMͷഒ
25. ˜
    -:
    $PSQPSBUJPO  6TFSMFWFM
    1SJWBDZ
    
    &WFOUMFWFM
    1SJWBDZ ϓϥΠόγʔอޢର৅ͷ୯ҐʢݸਓΠϕϯτΞΠςϜFUDʣ /".& $BODFS "MJDF :FT #PC /P $ZOUIJB /P %BWJE :FT /".& %JTFBTF "MJDF ౶೘ප "MJDF ന಺ো #PC ഏ͕Μ #PC ң͕Μ $ZOUIJB ΠϯϑϧΤϯβ %BWJE ң௵ᙾ %BWJE ेೋࢦ௎௵ᙾ %BWJE ௎ด࠹ Ϩίʔυਓ ෳ਺Ϩίʔυਓ ࠩ෼ϓϥΠόγʔ͕ ҉໧తʹԾఆ͢Δܗࣜ Ұਓ౰ͨΓͷϨίʔυ਺͕NഒʹͳΔͱɺϊΠζͷେ͖͞΋֓ͶNഒʹͳΔ à ϊΠζ͕େ͖͘ͳΓա͗Δ͜ͱΛ๷͙ͨΊɺ Ұਓ౰ͨΓͷ࠷େϨίʔυ਺ NBY
    DPOUSJCVUJPO
    Λ੍ݶ͢Δ͜ͱ͕͋Δ &WFOU %JTFBTF  ౶೘ප  ന಺ো  ഏ͕Μ  ң͕Μ  ΠϯϑϧΤϯβ  ң௵ᙾ  ेೋࢦ௎௵ᙾ  ௎ด࠹ ϨίʔυFWFOU 6TFSMFWFM
    1SJWBDZ &WFOUMFWFM
    1SJWBDZ
26. ˜
    -:
    $PSQPSBUJPO  ࠩ෼ϓϥΠόγʔͷ࣮૷ ώετάϥϜʹ-BQMBDFϝΧχζϜΛద༻͢Δྫ ϓϥΠόγʔϞσϧ • อޢ୯Ґɿ6TFSMFWFM
    Ϩίʔυਓ • ྡ઀%#ɿBEESFNPWBM
27. ˜
    -:
    $PSQPSBUJPO  -BQMBDFϝΧχζϜͷڍಈ ϥϯμϜੑΛ࣋ͬͨڍಈΛ͢ΔͨΊɺಉ͡ઃఆͰ΋ຖճग़ྗ͕ҟͳΔ 𝜖 = 1, Δ% = 1 𝜖 = 1, Δ% = 1 𝜖 = 1, Δ% = 1
28. ˜
    -:
    $PSQPSBUJPO  -BQMBDFϝΧχζϜͷڍಈ ϓϥΠόγʔύϥϝʔλ𝜖Λมಈʢ𝜖
    খ à ϓϥΠόγʔอޢ
    ڧʣ 𝜖 = 0.1 𝜖 = 0.5 𝜖 = 2 𝜖 = 0.05 𝜖 = 10 Δ% = 1
29. ˜
    -:
    $PSQPSBUJPO  ࠩ෼ϓϥΠόγʔͷղऍɿ౷ܭతԾઆݕఆͷࢹ఺ ϥϯμϜԽ͞Εͨग़ྗΛ؍ଌͨ͠ͱ͖ʹɺݩͷೖྗ %
    PS
    %`
    Λ౰ͯΒΕΔ͔ à ܦݧతͳ
    ࠩ෼ϓϥΠόγʔ͸͜ͷਪ࿦ͷਖ਼ޡͷ֬཰͔Βಋग़Ͱ͖Δ 𝜖&'( = max log 1 − 𝛿 − FP FN , log 1 − 𝛿 − FN FP &NQJSJDBM
    EJGGFSFOUJBM
    QSJWBDZ Kairouz+. The composition theorem for differential privacy. ICML2015 5SVF
    *OQVU (VFTT 'BMTF
    1PTJUJWF 𝐷 𝐷$ 'BMTF
    /FHBUJWF 𝐷$ 𝐷 ℳ 𝑦 𝐷 PS
    𝐷) 𝐷 PS
    𝐷) 𝐷 𝐷′ ྡ઀%#
30. ˜
    -:
    $PSQPSBUJPO  ϓϥΠόγʔ༧ࢉͷ؅ཧ σʔλͷ܁Γฦ͠ར༻ʹ൐͏ྦྷੵతͳϓϥΠόγʔͷফඅΛ 𝜖) ͷ߹੒ʹΑͬͯಋग़ à ࣄલʹఆٛͨ͠ϓϥΠόγʔ༧ࢉͷൣғ಺ͰɺԠ౴Մ൱Λ൑அͰ͖Δ TBUJTGZJOH
    𝝐𝟏 %1 TBUJTGZJOH
    𝝐𝒌 %1 ʜ 2VFSZ
    𝒒𝟏 2VFSZ
    𝒒𝒌 𝝐𝟏 𝝐𝟏 𝝐𝟐 𝝐𝟏 𝝐𝟐 𝝐𝒌 ʜ ʜ EFDMJOF ྦྷੵϓϥΠόγʔফඅ 2VFSZ
    EBUB
    BDDFTT ϓϥΠόγʔͷ߹੒ 1SJWBDZ
    $PNQPTJUJPO 𝜖*+*,- = ∑𝜖) ΫΤϦԠ౴ γεςϜ ػցֶश
31. ˜
    -:
    $PSQPSBUJPO  1SJWBDZ
    $PNQPTJUJPO 4FRVFOUJBM
    $PNQPTJUJPOʢ௚ྻ߹੒ʣ 𝜖 = B )∈ / 𝜖/ 𝜖 = max 𝜖0, … , 𝜖/ -FU
    ℳ% , … , ℳ& TBUJTGZ
    𝜖% , … , 𝜖& SFTQFDUJWFMZ 𝐷`T
    UPUBM
    QSJWBDZ
    DPOTVNQUJPO
    CZ
    ℳ% 𝐷 , … , ℳ& 𝐷 JT 1BSBMMFM
    $PNQPTJUJPOʢฒྻ߹੒ʣ -FU
    ℳ%, … , ℳ& TBUJTGZ
    𝜖%, … , 𝜖& SFTQFDUJWFMZ -FU
    𝐷 = 𝐷% ∪ ⋯ ∪ 𝐷& XIFSF
    𝐷' ∩ 𝐷()' = ∅ 𝐷`T
    UPUBM
    QSJWBDZ
    DPOTVNQUJPO
    CZ
    ℳ% 𝐷% , … , ℳ& 𝐷& JT &YBNQMF ℳ0 ℳ1 𝜖0 = 1 ℳ2 𝜖1 = 0.5 𝜖2 = 1.5 ℳ$ ℳ% ℳ& 1 0.5 1.5 1.5 1 1 2 2.5 46. NBY ൃలతͳϓϥΠόγʔ߹੒ "EWBODFE
    $PNQPTJUJPO [$%1 3ÉOZJ %JGGFSFOUJBM
    1SJWBDZ
    3%1 3%1΍[$%1͕Α͘ར༻͞Ε͍ͯΔ • ௚ྻ߹੒͸ϧʔζ • ཧ૝తͳ߹੒͸ະ஌ è ༷ʑͳ߹੒ఆཧ 𝜖 DPNQPTJUJPOT 4FRVFOUJBM
    $PNQPTJUJPO JEFBM 3%1
32. ˜
    -:
    $PSQPSBUJPO  ʲݚڀࣄྫʳ)%17JFX GPS
    &YQMPSJOH
    %BUBCBTF $POTUSVDUJPO
    PG
    JOUFSNFEJBUF
    QSJWBUJ[FE
    lWJFXz
    1WJFX
    UPXBSET
    BDUVBMJ[JOH
    BOZ
    RVFSZ
    SFTQPOTFT
    XJUI
    TNBMMFS
    OPJTF
    3FRVJSFNFOU 2VFSZ
    BHOPTUJD
    
    /PJTF
    SFTJTUBODF
    
    4QBDF
    FGGJDJFOU
    
    "OBMZUJDBMMZ
    SFMJBCMF "DDFQUFE
    BU
    7-%#                                                                                                             *EFOUJUZ 1SJWUSFF )%.. 1SJWCBZFT )%17JFX
    PVST "33 1.94×10! 7.05 35.34 3.79 𝟏. 𝟎𝟎 "WFSBHF
    3FMBUJWF
    &SSPS
    PWFS
    
    EBUBTFUT 4J[F
    PG
    QWJFX IUUQTBSYJWPSHBCT 4JNQMF
    
    4DBMBCMF 3BOEPN
    3FDVSTJWF
    #JTFDUJPO
33. ˜
    -:
    $PSQPSBUJPO ࠩ෼ϓϥΠόγʔʹΑΔ ҆શͳ࿈߹ֶश 
34. ˜
    -:
    $PSQPSBUJPO ػցֶशͱࠩ෼ϓϥΠόγʔ ࿈߹ֶशͱதԝࠩ෼ϓϥΠόγʔ ہॴࠩ෼ϓϥΠόγʔ ࿈߹ֶशͱہॴࠩ෼ϓϥΠόγʔ      ࠩ෼ϓϥΠόγʔʹΑΔ҆શͳ࿈߹ֶश
35. ˜
    -:
    $PSQPSBUJPO  4(%ɿ4UPDIBTUJD
    (SBEJFOU
    %FTDFOU ඪ४తͳػցֶशख๏ͷҰͭɻ܇࿅σʔλ͔Βޯ഑ HSBEJFOU Λಋग़ͯ͠Ϟσϧͷߋ৽ʹར༻ 𝐚𝐯𝐠. .JOJCBUDI 4BNQMJOH 1FSTBNQMF (SBEJFOU %BUBCBTF
    𝐷 𝐷 = 𝑛 𝑥%, … , 𝑥5 ∈ 𝐷 𝑔' = ∇𝑓(𝑥'; 𝜃) 𝑔% = ∇𝑓(𝑥% ; 𝜃) 𝑔5 = ∇𝑓(𝑥5; 𝜃) 𝜃6 = 𝜃67% − 𝜂 1 𝑛 < '∈ 5 ∇𝑓 𝑥'; 𝜃67% 4(% NJOJCBUDI
    USBJOJOH .PEFM
    6QEBUF ऩଋ͢Δ·Ͱ൓෮܇࿅
36. ˜
    -:
    $PSQPSBUJPO  %1ͳػցֶशϑϨʔϜϫʔΫɿ%14(% ϞσϧΛߋ৽͢Δޯ഑ HSBEJFOU ΛϥϯμϜԽ à ܇࿅ޙͷύϥϝʔλ΋(𝜖, 𝛿)%1Λอূ 𝐚𝐯𝐠. (BVTTJBO /PJTF (SBEJFOU $MJQQJOH 𝑔% = ∇𝑓(𝑥% ; 𝜃) 𝝅𝑪 𝒈𝟏 𝑔5 = ∇𝑓(𝑥5 ; 𝜃) 3BOEPN 4BNQMJOH $ 1FSTBNQMF (SBEJFOU %BUBCBTF
    𝐷 𝐷 = 𝑛 𝑥%, … , 𝑥5 ∈ 𝐷 𝑔' = ∇𝑓(𝑥'; 𝜃) 𝜋' 𝑔( = 𝑔( ⋅ min 1, 𝐶 𝑔( % (SBEJFOU $MJQQJOH (SBEJOUͷℓ: ϊϧϜΛఆ਺$Ͱ੍ݶ à ηϯγςΟϏςΟΛ$ʹ͢Δ ʢ1PTUQSPDFTTJOHఆཧΑΓʣ IUUQTBSYJWPSHBCT .PEFM
    6QEBUF 𝜃) = 𝜃)*$ − 𝜂 1 𝑛 P (∈ , 𝜋' 𝑔( ; 𝜃)*$ + 𝑁 0, 𝜎 %14(% ϓϥΠόγʔ༧ࢉͷൣғͰ൓෮܇࿅ "CBEJ 
    %FFQ
    -FBSOJOH
    XJUI
    %JGGFSFOUJBM
    1SJWBDZ
    $44
    
37. ˜
    -:
    $PSQPSBUJPO  ʲݚڀࣄྫʳ1SJWBUF
    %BUB
    4ZOUIFTJT σʔλΛ໛฿͢Δੜ੒ϞσϧΛࠩ෼ϓϥΠόγʔΛอূ͠ͳ͕Β܇࿅ à σʔλྲྀ௨Λଅਐ 1(.
    PVST 𝜖, 𝛿 = (1,107;) IUUQTBSYJWPSHBCT *$%& /BÏWF
    .FUIPE 7"&
    X
    %14(% ՝୊ɿ("/΍7"&౳ͷσʔλ߹੒ΞϧΰϦζϜ͸ෳࡶ౓͕ߴ͘ɺ &ODͱ%FDͷ&OEUP&OEͷ܇࿅ʹ๲େͳ൓෮͕ඞཁ à ॴఆͷϓϥΠόγʔ༧ࢉ಺Ͱ܇࿅͕ࠔ೉ à ϊΠζͷӨڹΛड͚΍͘͢ɺੜ੒σʔλͷ඼࣭͕ѱ͍ ఏҊɿ &ODͷ
    SPVOE܇࿅ %11$"
    à %FDͷ൓෮܇࿅ %14(% "DDFQUFE
    BU
    *$%&
38. ˜
    -:
    $PSQPSBUJPO  ࿈߹ֶश 'FEFSBUFE
    -FBSOJOH ࿈߹ֶशͷखॱ  αʔόʔ͕άϩʔόϧϞσϧΛΫϥΠΞϯτ܈ʹ഑෍  ΫϥΠΞϯτ͕ͦΕͧΕݻ༗ͷϩʔΧϧσʔλͰϞσϧΛධՁɺ ޯ഑ͳͲͷߋ৽৘ใΛܭࢉ  ߋ৽৘ใΛαʔόʔʹૹ෇  ऩूͨ͠ߋ৽৘ใΛू໿ͯ͠άϩʔόϧϞσϧΛߋ৽ ಛ௃ • ϩʔΧϧσʔλ͕ΫϥΠΞϯτʹཹ·Δ • αʔόʔ͕σʔλΛ؅ཧ͠ͳͯ͘Α͍ • ΫϥΠΞϯτݻ༗ͷϩʔΧϧϞσϧʹΑΓ Ϣʔβʔͷᅂ޷౳ͷϩʔΧϧͳಛ௃Λ൓ө • ΫϥΠΞϯτͷܭࢉࢿݯʹΑΔ੍໿ • σʔλ௨৴͕8JGJ઀ଓ࣌౳ʹݶΒΕΔ • ΫϥΠΞϯτͷESPQPVU͕͋ΔʢిݯPGG౳ʣ /POQBSUJDJQBOUT
    PG
    '- (MPCBM
    .PEFM ̍   
39. ˜
    -:
    $PSQPSBUJPO  'FE
    4(%
    
    "WH ΫϥΠΞϯτอ༗ͷσʔλͰޯ഑Λܭࢉ à αʔόʔͰू໿ޙʹάϩʔόϧϞσϧΛߋ৽ 𝐚𝐯𝐠. 𝑔% = ∇𝑓(𝑥% ; 𝜃) 𝑔5 = ∇𝑓(𝑥5 ; 𝜃) 3BOEPN 4BNQMJOH $ 1FSDMJFOU (SBEJFOU (MPCBM
    .PEFM 𝜃 𝑥%, … , 𝑥5 ∈ 𝐷 𝑔' = ∇𝑓(𝑥'; 𝜃) (MPCBM
    .PEFM
    6QEBUF 0OEFWJDF $PNQVUJOH
40. ˜
    -:
    $PSQPSBUJPO  ࿈߹ֶश
    ࠩ෼ϓϥΠόγʔ (MPCBM
    .PEFM (MPCBM
    .PEFM 3BX (SBEJFOU 3BOEPNJ[FE (SBEJFOU /PJTF
    *OKFDUJPO $FOUSBM
    .PEFM -PDBM
    .PEFM
41. ˜
    -:
    $PSQPSBUJPO  $FOUSBM
    %1'FE
    4(%
    
    "WH ΫϥΠΞϯτอ༗ͷσʔλͰޯ഑Λܭࢉ à αʔόʔͰू໿ͱϥϯμϜԽޙʹϞσϧΛߋ৽ 𝐚𝐯𝐠. (SBEJFOU $MJQQJOH 𝑔% = ∇𝑓(𝑥% ; 𝜃) 𝝅𝑪 𝒈𝟏 𝑔5 = ∇𝑓(𝑥5 ; 𝜃) 3BOEPN 4BNQMJOH $ 1FSDMJFOU (SBEJFOU 𝑥%, … , 𝑥5 ∈ 𝐷 𝑔' = ∇𝑓(𝑥'; 𝜃) (MPCBM
    .PEFM 𝜃 (BVTTJBO /PJTF .PEFM
    6QEBUF ϓϥΠόγʔ༧ࢉͷൣғͰ൓෮܇࿅ .D.BIBO 
    -FBSOJOH
    %JGGFSFOUJBMMZ
    1SJWBUF
    3FDVSSFOU
    -BOHVBHF
    .PEFMT
    *$-3
    IUUQTBSYJWPSHBCT
42. ˜
    -:
    $PSQPSBUJPO  -PDBM
    %JGGFSFOUJBM
    1SJWBDZʢہॴࠩ෼ϓϥΠόγʔʣ ෼ࢄ؀ڥԼͰͷ౷ܭऩू΍࿈߹ֶशͷͨΊͷϓϥΠόγʔϞσϧɻΫϥΠΞϯτͰϥϯμϜԽ 1SJWBDZ
    .PEFM ػඍͳ ౷ܭͷऩू αʔόʔΛ ৴༻͠ͳ͍ ϝΧχζϜ ℳ: 𝒳 → 𝒮 ͕ 𝜖, 𝛿 %1
    Λຬͨ͢ͱ͸ɺ ೚ҙͷೖྗͷ૊ 𝑥$ , 𝑥% ∈ 𝒳 ͓Αͼ೚ҙͷग़ྗͷू߹ 𝑆 ⊆ 𝒮 ʹରͯ͠ҎԼ͕੒Γཱͭͱ͖Ͱ͋Δ Pr ℳ 𝑥0 ∈ 𝑆 ≤ exp 𝜖 Pr ℳ 𝑥1 ∈ 𝑆 + 𝛿 𝑥0 𝑥1 ℳ ℳ ݟ෼͚͕͔ͭͳ͍ (𝝐, 𝜹)-PDBM
    %1 ྡ઀%#͸ SFQMBDFNFOU
    Λ૝ఆ
43. ˜
    -:
    $PSQPSBUJPO  L3BOEPNJ[FE
    3FTQPOTF
    L33 ίΠϯΛ౤͛ͯදͳΒਅͷ஋ΛɺཪͳΒͦΕҎ֎ͷ஋ΛϥϯμϜʹԠ౴ 𝒳 ∈ { } 3BOEPNJ[FE 0SJHJOBM 𝑅𝑅 𝑥 = M 𝑥 𝑤. 𝑝. exp 𝜖 exp 𝜖 + 𝑘 − 1 𝑥) ∼ 𝒳 ∖ 𝑥 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ℳ SBOEPNMZ
    TFMFDU 𝑘: |𝒳| JUFNT
44. ˜
    -:
    $PSQPSBUJPO  L33ʹΑΔ౷ܭऩूʢਪఆʣ ूܭ͢ΔϥϯμϜԽϨϙʔτͷ਺͕ଟ͍΄Ͳਖ਼֬ͳ౷ܭΛਪఆͰ͖Δ 1SJWBDZ
    BU
    TDBMF 𝑁 = 10,000 𝑁 = 10,000,000
45. ˜
    -:
    $PSQPSBUJPO  $%1ͱ-%1ͷൺֱ $FOUSBM
    %1 -PDBM
    %1 ྡ઀σʔλϕʔε BEESFNPWBM͕ඪ४త %#ͷαΠζ͸Ҏ্ SFQMBDFNFOU͕ඪ४త %#ͷαΠζ͸ ϥϯμϜԽॲཧd౷ܭॲཧ αʔόʔͰ౷ܭॲཧͨ͠ग़ྗʹ ճ͚ͩϊΠζΛՃࢉ  ΫϥΠΞϯτ͕࡞੒ͨ͠ϥϯμϜ ԽϨϙʔτΛαʔόʔʹૹ৴  ϥϯμϜԽϨϙʔτ܈Λू໿ͯ͠ ౷ܭॲཧ à ૬ରతʹϊΠζɿେ ʢTVNͷ৔߹$%1ൺ 𝑛ഒʣ ஶ໊ͳϥϯμϜԽॲཧ • -BQMBDFϝΧχζϜ • (BVTTJBOϝΧχζϜ • ࢦ਺ϝΧχζϜ • 3BOEPNJ[FE
    3FTQPOTF • %JTDSFUF
    -BQMBDFϝΧχζϜ • 1PJTTPO
    #JOPNJOBMϝΧχζϜ ৴པϞσϧ αʔόʔΛ৴པ͢Δ αʔόʔͷ৴པ͸ඞཁͳ͍
46. ˜
    -:
    $PSQPSBUJPO  ࠩ෼ϓϥΠόγʔΛอূ͢Δ҆શͳ࿈߹ֶशʢ-%1ʣ (MPCBM
    .PEFM 3BOEPNJ[FE (SBEJFOU -%1ͳ$SPTTEFWJDF
    '-ͷಛ௃ • αʔόʔΛ৴༻͠ͳ͍෼ࢄϞσϧ • ΫϥΠΞϯτͰॴఆͷ-%1Λอূͨ͠ ϥϯμϜԽϨϙʔτΛ࡞੒ͯ͠ૹ৴ • ϓϥΠόγʔ༧ࢉ͸ΫϥΠΞϯτຖʹ؅ཧ • ϊΠζ͕େ͖͘ͳΓ͕ͪͳͨΊɺͰ͖Δ͚ͩ ଟ͘ͷϥϯμϜԽϨϙʔτΛฏۉԽͯ͠ ϊΠζͷӨڹΛ௿ݮ͍ͨ͠ ༗ӹͳ-%1
    '-Λ࣮ݱ͢Δʹ͸ • ๲େͳ਺ͷΫϥΠΞϯτͷࢀՃ͕ඞཁ • ϓϥΠόγʔϞσϧͷ؇࿨ͷݕ౼΋ඞཁ • %JTUSJCVUFE
    %1
    
    4FDVSF
    "HH౳ ޙड़
47. ˜
    -:
    $PSQPSBUJPO  ࿈߹ֶश޲͚ͷϥϯμϜԽॲཧʢ-%1ʣ 1SJW6OJUϝΧχζϜ • 𝜖< -%1Λอূ • ޯ഑ΛϥϯμϜʹαϯϓϦϯά • ྘ PS
    നΛ 33Ͱબ୒ • ͞Βʹબ୒ྖҬ಺ʢ྘ PS
    നʣ͔Β Ұ༷αϯϓϧ (BVTTJBOϝΧχζϜ • $%1޲͚ͷ(BVTTJBOϝΧχζϜͷస༻ • ϊΠζΛ$%1޲͚ख๏ͷഒʹ͢Δ ̀ $%1ͱ-%1ͷྡ઀ੑͷҧ͍ʹΑΓ 33ϕʔεͷख๏ $POUJOVPVT
    /PJTF %JTDSFUF
    /PJTF %JTDSFUF
    -BQMBDFϝΧχζϜ • ಠཱͨͭ͠ͷ֬཰ม਺ d
    1PJTTPO෼෍
    ͷࠩͷ෼෍ 4LFMMBNϝΧχζϜ ˞
    ࠶ੜੑͷ͋Δ෼෍Λ༻͍Δඞཁ͕͋Δ IUUQTFOXJLJQFEJBPSHXJLJ4LFMMBN@EJTUSJCVUJPO 4LFMMBN෼෍ #IPXNJDL 
    1SPUFDUJPO
    "HBJOTU
    3FDPOTUSVDUJPO
    BOE
    *UT
    "QQMJDBUJPOT
    JO
    1SJWBUF
    'FEFSBUFE
    -FBSOJOH
    IUUQTBSYJWPSHBCT "HBSXBM 
    5IF
    4LFMMBN .FDIBOJTN
    GPS
    %JGGFSFOUJBMMZ
    1SJWBUF
    'FEFSBUFE
    -FBSOJOH
    /FVS*14
    IUUQTBSYJWPSHBCT
48. ˜
    -:
    $PSQPSBUJPO  -%1'FE
    4(%
    
    "WH ΫϥΠΞϯτͰޯ഑ʢ·ͨ͸ࠩ෼ʣͷܭࢉͱϥϯμϜԽ à αʔόʔͰू໿ޙʹϞσϧΛߋ৽ 𝐚𝐯𝐠. (SBEJFOU $MJQQJOH 𝑔% = ∇𝑓(𝑥% ; 𝜃) 𝝅𝑪 𝒈𝟏 𝑔5 = ∇𝑓(𝑥5 ; 𝜃) 3BOEPN 4BNQMJOH $ 1FSDMJFOU (SBEJFOU (MPCBM
    .PEFM 6QEBUF 𝑥%, … , 𝑥5 ∈ 𝐷 𝑔' = ∇𝑓(𝑥'; 𝜃) (MPCBM
    .PEFM 𝜃 1FSUVSCBUJPO ΫϥΠΞϯτͷϓϥΠόγʔ༧ࢉͷ ൣғͰ൓෮܇࿅
49. ˜
    -:
    $PSQPSBUJPO ൃలతͳ࿩୊ 
50. ˜
    -:
    $PSQPSBUJPO  ۙ೥ͷಈ޲ɿ1SJWBDZ
    5FDI
    $PNCP ϓϥΠόγʔอޢͷਫ४ͱσʔλར׆༻ͷ༗༻ੑͷόϥϯεͷվળΛ໨తͱͯ͠ɺηΩϡΞͳ ϋʔυ΢ΣΞ΍ϓϩτίϧΛલఏͱ͢Δࠩ෼ϓϥΠόγʔͷݚڀ͕׆ൃԽ &/1"ΞʔΩςΫνϟ ൿີ෼ࢄ
    ࠩ෼ϓϥΠόγʔ IUUQTXXXBQQMFDPNKQOFXTSPPN BQQMFBOEHPPHMFQBSUOFSPODPWJEDPOUBDU USBDJOHUFDIOPMPHZ IUUQTDPWJETUBUJDDEO BQQMFDPNBQQMJDBUJPOTDPWJEDVSSFOUTUBUJDDPOUBDU USBDJOHQEG&/1"@8IJUF@1BQFSQEG 'FEFSBUFE
    "OBMZUJDT IUUQTBSYJWPSHBCT
51. ˜
    -:
    $PSQPSBUJPO  4IVGGMF
    NPEFM
    r BO
    JOUFSNFEJBUF
    QSJWBDZ
    NPEFM • *OUFSNFEJBUF
    USVTUFE
    FOUJUZ
    lTIVGGMFSz BOPOZNJ[FT
    MPDBM
    VTFST`
    JEFOUJUZ • &BDI
    DMJFOU
    FODSZQUT
    UIFJS
    SBOEPNJ[FE
    DPOUFOU
    X
    UIF
    TFSWFS`T
    QVCMJD
    LFZ
    UIFO
    TIVGGMFS
    POMZ
    NJYFT
    UIFJS
    JEFOUJGJFT
    XP
    MPPLJOH
    BU
    UIF
    DPOUFOUT Q 𝑥0 Q 𝑥1 Q 𝑥3 R 𝑥0 (R 𝑥0, R 𝑥1, … , R 𝑥3) R 𝑥1 R 𝑥3 4FSWFS 3BOEPNJ[FE
    X
    𝝐𝟎 4IVGGMF 4IVGGMFS 4FOE
    UIF
    TIVGGMFE
    CBUDI BOPOZNJ[FE
52. ˜
    -:
    $PSQPSBUJPO  1SJWBDZ
    "NQMJGJDBUJPO
    WJB
    4IVGGMJOH • 4IVGGMFS
    DBO
    BNQMJGZ
    EJGGFSFOUJBM
    QSJWBDZ
    à QPTTJCJMJUZ
    UP
    EFDSFBTF
    MPDBM
    OPJTF • 5IF
    BNQMJGJDBUJPO
    PO
    TIVGGMFS
    USBOTMBUFT
    -%1
    PO
    DMJFOUT
    JOUP
    $%1 𝜖5 = 8 -%1
    
    
    𝛿 = 1067 𝑘 = 10 CZ
    )JEJOH
    BNPOH
    DMPOFT &YBNQMF
    JO
    LSBOEPNJ[FE
    SFTQPOTF IUUQTBSYJWPSHBCT 1SJWBDZ
    "NQMJGJDBUJPO A 𝑥% A 𝑥: A 𝑥= B 𝑥% (B 𝑥% , B 𝑥: , … , B 𝑥= ) B 𝑥: B 𝑥= 4IVGGMFS 4FSWFS 𝝐𝟎 -%1 𝝐 < 𝝐𝟎 $%1
53. ˜
    -:
    $PSQPSBUJPO  4IVGGMFE
    $IFDL*OT
    GPS
    '- ࿈߹ֶशʹ4IVGGMOHΛಋೖ • ΫϥΠΞϯτ܈ʢ𝑛ʣͷҰ෦ʢ𝑚 < 𝑛ʣΛαϯϓϧͯ͠ϞσϧΛ൓෮తʹ܇࿅͍ͨ͠ à Ұ܇࿅ͷࢀՃΫϥΠΞϯτ਺𝑚͕গͳ͍ à 4IVGGMJOH͕ෆे෼ à 4VCTBNQMJOHΛߟྀ • 4VCTBNQMJOH͸ΫϥΠΞϯτͷESPQPVUʹ੬ऑ à ؤ݈ͳϞσϧΛಋೖ͍ͨ͠ )JHI
    6UJMJUZ
    BU
    4USPOH
    1SJWBDZ
    -FWFM "NQMJGJFE 4IVGGMFE
    $IFDL*OT 𝛾 = 𝑝 1 − 𝑝$ 𝑝
    DIFDLJO
    SBUF
     𝑝  𝑝$
    ESPQPVU
    SBUF 𝛾
    FGGFDUJWF
    DIFDLJO
    SBUF Q 𝑥0 Q 𝑥8 Q 𝑥3 {Q 𝑥8, Q 𝑥3, … , Q 𝑥0}
54. ˜
    -:
    $PSQPSBUJPO  ʲݚڀࣄྫʳ/FUXPSL
    4IVGGMJOH %FDFOUSBMJ[FE
    TIVGGMJOH
    WJB
    NVMUJSPVOE
    SBOEPN
    XBMLT PO
    B
    HSBQI *O
    FBDI
    SPVOE
    FWFSZ
    DMJFOU
    SFMBZT
    IFS
    SBOEPNJ[FE
    SFQPSUT
    UP
    POF
    PG
    IFS
    OFJHICPST
    FH
    GSJFOET
    PO
    B
    TPDJBM
    OFUXPSL
    WJB
    BO
    FODSZQUFE
    DIBOOFM 5IF
    MBSHFS
    HSBQI
    BNQMJGJFT
    QSJWBDZ
    UIF
    NPSF "DDFQUFE
    BU
    4*(.0% IUUQTBSYJWPSHBCT
55. ˜
    -:
    $PSQPSBUJPO  4FDVSF
    "HHSFHBUJPO "
    NBUVSF
    QSJWBDZ
    UFDI
    DPNCP
    UIBU
    BMMPXT
    VT
    UP
    FNQMPZ
    XFBLFS
    %1
    NPEFM
    UIBU
    SFTVMUT
    JO
    TJHOJGJDBOU
    OPJTF
    SFEVDUJPO
    ℎ𝑖𝑠𝑡/ = B )∈['>] 𝑥) + 𝑧) ℎ𝑖𝑠𝑡/ = B )∈['>] 𝑥) + 𝑧) 𝒎𝒌 -PDBM
    NPEFM %JTUSJCVUFE
    NPEFM ∑ 4FDVSF
    "HH %JTUSJCVUFE
    %1 BOUJNFNPSJ[BUJPO "OPOZNPVT $IBOOFM 0OEFWJDF $PNQVUJOH ʜ ʜ )JHI
    VUJMJUZ WJB SFMBYFE
    %1
    NPEFM VOEFS
    TUSPOH
    FOPVHI QSJWBDZ
    UFDI
    DPNCP
56. ˜
    -:
    $PSQPSBUJPO  %JTUSJCVUFE
    %JGGFSFOUJBM
    1SJWBDZ "O
    JOUFSNFEJBUF
    QSJWBDZ
    NPEFM
    X
    MPDBM
    QSJWBUJ[BUJPO
    BOE
    TFDVSF
    QSPUPDPMT /PJTF
    TDBMF
    -%1
    
    -%14IVGGMF ʾ
    %%1 ʾ
    $%1 ℎ𝑖𝑠𝑡/ = B )∈ '> 𝑥) + 𝑧) ℎ𝑖𝑠𝑡/ = B )∈ '> 𝑥) + 𝑧/ ∑ ∑ ∑ -PDBM
    %1 %JTUSJCVUFE
    %1 $FOUSBM
    %1 )VHF
    BNPVOU
    PG
    OPJTF
    OP
    USVTUFE
    FOUJUJFT • -FTT
    OPJTF
    BHBJOTU
    MPDBM
    EQ FRVJWBMFOU
    UP
    $%1
    BGUFS
    TVNNBUJPO • *OTUBMMJOH
    TFDVSF
    
    USVTUFE
    FOUJUZ 'BJS
    OPJTF
    CVU
    USVTUFE
    DVSBUPS
    JT
    SFRVJSFE ℎ𝑖𝑠𝑡/ = B )∈ '> 𝑥) + 𝑧) 𝑚/
57. ˜
    -:
    $PSQPSBUJPO ·ͱΊ 
58. ˜
    -:
    $PSQPSBUJPO ࿈߹ֶशͱϓϥΠόγʔ σʔλ׆༻࣌ͷϓϥΠόγʔϦεΫ %JGGFSFOUJBM
    1SJWBDZʢࠩ෼ϓϥΠόγʔʣ ࠩ෼ϓϥΠόγʔʹΑΔ҆શͳ࿈߹ֶश ۙ೥ͷൃలతͳ࿩୊ʢ4IVGGMJOH
    
    4FDVSF
    "HHSFHBUJPO
    
    'FEFSBUFE
    "OBMZUJDTʣ       ຊߨٛͷίϯςϯπ
59. ˜
    -:
    $PSQPSBUJPO