Adversarial Training

2022年3月10日のPRMU研究会で使用した発表スライドです．

Transcript

1. αʔϕΠɿ"EWFSTBSJBM
   5SBJOJOH
   ଍ཱߒنɼฏ઒ཌྷɼࢁԼོٛɼ౻٢߂࿱ʢத෦େֶɾػց஌֮ϩϘςΟΫεݚڀάϧʔϓʣ
   IUUQNQSHKQ
2. w "EWFSTBSJBM
   "UUBDLɿఢରత߈ܸ
    ೖྗը૾ʹมԽΛՃ͑ͯ$//ͷޡ෼ྨΛ༠ൃ
    ਓ͕ਖ਼͘͠෼ྨͰ͖ΔΑ͏ͳαϯϓϧͰ$//Λ߈ܸ͍ͨ͠
   w "EWFSTBSJBM
   %FGFOTFɿఢରత๷ޚ
    $//Λ"EWFSTBSJBM
   BUUBDLͷ֎త߈ܸ͔Β๷ޚ
    ͋ΒΏΔ߈ܸ͔Β$//ΛकΓ͍ͨ "EWFSTBSJBM
   "UUBDL
   WT
   %FGFOTF 2 %FGFOEFS "UUBDLFS $//
   $POWPMVUJPOBM
   /FVSBM
   /FUXPSLT
3. "EWFSTBSJBM
   "UUBDL%FGFOTFͷ܏޲ 3 %FGFOTF "UUBDL "5(*'
   <$
   $IFO
   BS9JW`> *"5
   <"
   -BNC
   "*4FD`> .5-"5
   <"
   -BVHSPT
   &$$7`> %"5
   <:
   8BOH
   *$.-`> 'FBUVSF
   TDBUUFS
   <)
   ;IBOH
   BOE
   +
   8BOH
   /FVS*14`> $"5
   <2;
   $BJ
   *+$"*`> 6OJWFSTBM
   "5
   <"
   4IBGBIJ
   """*`> "7NJYVQ
   <4
   -FF
   $713`> ("5
   <(
   4SJSBNBOBO
   /FVS*14`> ("*3"5
   <+
   ;IBOH
   *$-3`> ."*-
   <2
   8BOH
   /FVS*14`> 8..3
   <)
   ;FOH
   """*`> #J-"8
   <$
   )PMU[
   0QFO3FWJFX`> -#("5
   <+
   $VJ
   *$$7`> #("5
   <+
   $VJ
   *$$7`> 1$-$
   <9
   -J
   """*`> 1$
   MPTT
   <"
   .VTUBGB
   *$$7`> $//
   X
   3#'
   LFSOFM
   <4
   "
   5BHIBOBLJ
   $713`> "-1
   <)
   ,BOOBO
   BS9JW`> ."35
   <:
   8BOH
   *$-3`> ..-%"
   <5
   1BOH
   *$.-`> ..$
   <5
   1BOH
   *$-3`> -#'(4
   <$
   4[FHFEZ
   *$-3`> '(4.
   <*
   (PPEGFMMPX
   *$-3`> $8
   </
   $BSMJOJ
   BOE
   %8BHOFS
   41`> 1(%
   <"
   .BESZ
   *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
   UBSHFUFE
   <4
   (PXBM
   BS9JW`> 6OJWFSTBM
   "&T
   <4.
   .PPTBWJ%F[GPPMJ
   $713`> "VUP
   "UUBDL
   <'
   $SPDF
   BOE
   .
   )FJO
   *$.-`> '"#
   <'
   $SPDF
   BOE
   .
   )FJO
   *$.-`> 4RVBSF
   "UUBDL
   <"
   "OESJVTIDIFOLP
   &$$7`> "VUP
   1(% "VUP
   1(%%-3 4"*'(.
   <9
   %POH
   $713`> ("."
   <(
   4SJSBNBOBO
   /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
   ,VSBLJO
   *$-38`> ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 4UBOEBSE
   "5
   <*
   (PPEGFMMPX
   *$-3`> "5
   XJUI
   1(%
   <"
   .BESZ
   *$-3`> ˢ
   /BUVSBMͱ"&Tͷ࢖༻ 1(%
   BUUBDLͷ"&TΛར༻ ˢ
   "&TͷΈ μΠφϛοΫͳ
   εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
   <(
   8
   %JOH
   *$-3`> 53"%&4
   <)
   ;IBOH
   *$.-`> '"5
   <+
   ;IBOH
   *$.-`> )"5
   <3
   3BEF
   *$.-8`> 
4. w ඍখઁಈΛ෇༩ͨ͠αϯϓϧ
   "&T
   "EWFSTBSJBM
   FYBNQMFT Λ༻͍ͨ߈ܸ
    ਓ͕஌֮ࠔ೉ͳมԽ
    ߴ͍֬཰Ͱ$//ͷޡ෼ྨΛ༠ൃՄೳ
   w ߈ܸର৅ͷঢ়ଶʹΑͬͯઁಈ࡞੒ํ๏͕ҟͳΔ
    Ϟσϧͷ৘ใ͕ط஌
   8IJUFCPY ɿϞσϧͷޯ഑Λར༻
    Ϟσϧͷ৘ใ͕ະ஌
   #MBDLCPY ɿผͷϞσϧ͔ΒసҠɼೖग़ྗͷؔ܎ੑ͔Βಋग़ "EWFSTBSJBM
   "UUBDL 4 ඍখͳઁಈ "&T /BUVSBMαϯϓϧ
5. w ઁಈΛεςοϓͰಋग़͢Δख๏
    ޯ഑ͷූ߸Λར༻ͯ͠ೖྗը૾ʹมԽΛՃ͑Δ 'BTU
   (SBEJFOU
   4JHO
   .FUIPE
   '(4.
   <*
   (PPEGFMMPX
   BS9JW
   > 5 *
   (PPEGFMMPX
   et al.
   l&YQMBJOJOH
   BOE
   )BSOFTTJOH
   "EWFSTBSJBM
   &YBNQMFTz
   BS9JW
    ˜ xi = xi + ϵ ⋅ sign (∇x′  i L (f(x′  i ; θ), yi)) ޡࠩΛ࠷େʹ͢Δํ޲ʹ Λ ͚ͩҠಈ xi ϵ x′  i = xi + r s . t . r ∼ U[−ϵ, ϵ] Λ CBMM಺ͷϥϯμϜͳҐஔʹҠಈ xi ϵ ϵ
   OPSN ℓ2 ϵ
   OPSN ℓ∞
   
   σʔλू߹ ͔ΒαϯϓϦϯάͨ͠σʔλ (xi , yi ) ∼ D D
   
   Λத৺ͱͨ͠ CBMMͷ൒ܘ ϵ xi ϵ x′  i x ˜ xi x′  i x ˜ xi
6. w Λ༻͍ͯ೚ҙͷճ਺൓෮ͯ͠߈ܸ͢Δ͜ͱͰ'(4.ΑΓ΋ڧ͍ઁಈΛ֫ಘ
    ͷൣғ֎ͷઁಈ͸౤Өؔ਺ ʹΑͬͯҾ͖໭͢ α ≤ ϵ [−ϵ, ϵ] Π ϵ
   OPSN ℓ∞ 1SPKFDUFE
   (SBEJFOU
   %FTDFOU
   1(%
   <"
   .BESZ
   *$-3
   > 6 "
   .BESZ
   et al.
   l5PXBSET
   %FFQ
   -FBSOJOH
   .PEFMT
   3FTJTUBOU
   UP
   "EWFSTBSJBM
   "UUBDLz
   *$-3
    ˜ x1 i = Π (˜ x0 i + α ⋅ sign (∇˜ x0 i L (f(˜ x0 i ; θ), yi))) ˜ xt+1 i = Π (˜ xt i + α ⋅ sign (∇˜ xt i L (f(˜ xt i ; θ), yi))) ˜ x2 i = Π (˜ x1 i + α ⋅ sign (∇˜ x1 i L (f(˜ x1 i ; θ), yi))) ϵ
   OPSN ℓ2 ˜ x0 i x ˜ xt+1 i
   
   εςοϓαΠζ α
   
   CBMM಺ʹҾ͖໭͢౤Өؔ਺ Π ϵ
   
   ൓෮ճ਺ T ˜ x0 i = Π (xi + r) s . t . r ∼ U[−ϵ, ϵ] Λ CBMM಺ͷϥϯμϜͳҐஔʹҠಈ xi ϵ ൓෮తʹઁಈΛಋग़ ˜ x0 i x ˜ xt+1 i
7. w "EWFSTBSJBM
   BUUBDLΛܰݮ͢ΔͨΊͷςΫχοΫ
    ೖྗը૾Λૢ࡞͢Δ๷ޚ
    "&TΛ༻ֶ͍ͨशʹΑΔ๷ޚˠ"EWFSTBSJBM
   USBJOJOH "EWFSTBSJBM
   EFGFOTF 7
   
   "&Tݕग़ͷͨΊͷᮢ஋ T
   
   ઁಈڧ౓ ϵ
   
   ʹର͢Δڭࢣ৴߸ y xnat
   
   ֶश཰ η
   
   ֶशՄೳͳύϥϝʔλ θ ೖྗը૾Λૢ࡞͢Δ๷ޚ "&TΛ༻ֶ͍ͨशʹΑΔ๷ޚ xi ϊΠζআڈ ˜ xi $// f(˜ xi ; θ) %FOPJTJOH ը૾ੜ੒ϞσϧΛ༻͍ͯೖྗը૾͔ΒϊΠζΛআڈ xi ݕग़ث
   ϕ $// ϕ(xi ) ≤ T "EWFSTBSJBM
   TBNQMF ϕ(xi ) > T f(xi ; θ) %FUFDUJPO લஈʹݕग़ثΛઃஔͯ͠ɼ҆શͳαϯϓϧ͚ͩࣝผϞσϧ΁౉͢ $// x′  f(xnat ; θ) y θt+1 ← θt − η ∂L (f(x′  ; θt ), y) ∂θt 1BSBNFUFS
   VQEBUF "&TΛ༻͍ͨϞσϧͷߋ৽ "EWFSTBSJBM
   USBJOJOH "&TΛϕʔεʹύϥϝʔλΛߋ৽ͯ͠ϞσϧຊମΛ߈ܸʹڧ͘͢Δ ύϥϝʔλΛݻఆͯ͠
   "&Tͷ࡞੒ "&TΛ༻͍ͯ
   ύϥϝʔλͷߋ৽
8. w "&TΛֶश͢Δ͜ͱͰϞσϧࣗମΛؤ݈ʹ͢Δ๷ޚख๏
    "&Tͷ࡞੒
   *OOFSNBYJNJ[BUJPO
    ύϥϝʔλߋ৽
   0VUFSNJOJNJ[BUJPO "EWFSTBSJBM
   5SBJOJOH
   <"
   .BESZ
   *$-3
   > 8 min θ 𝔼 (xi ,yi )∼D [ max ∥δ∥p ≤ ϵ L(xi + δ, yi ; θ) ] *OOFSNBYJNJ[BUJPO
    '(4.1(%
   BUUBDLΛ༻͍ͯઁಈΛಋग़
    جຊతʹɼ߈ܸதͷޡࠩؔ਺͸ΫϩεΤϯτϩϐʔޡࠩ 0VUFSNJOJNJ[BUJPO
    "&TΛ༻͍ͯޡ͕ࠩ࠷খʹͳΔΑ͏ʹύϥϝʔλ Λߋ৽
    ͜͜Ͱ࢖͏ޡࠩؔ਺΍"&TʹҰखؒՃ͑Δ͜ͱ͕ଟ͍ θ $// ˜ x f(˜ x; θ) y θt+1 ← θt − η ∂L (f(˜ x; θt ), y) ∂θt (SBEJFOU
   EFTDFOE 
   ύϥϝʔλߋ৽
   0VUFSNJOJNJ[BUJPO $//
   QBSBNFUFS
   fi Y xnat f(˜ x; θ) y δ := α ⋅ sign (∇xnat L (f(xnat ; θ), y)) 
   "&Tͷ࡞੒
   *OOFSNBYJNJ[BUJPO r ∼ U[−ϵ, ϵ] + ˜ x = + (SBEJFOU
   BTDFOE "&TΛجʹύϥϝʔλߋ৽ "&Tͷ࡞੒
9. "EWFSTBSJBM
   5SBJOJOHͷݚڀಈ޲ 9 ࣝผڥքʹண໨ 
   ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ 
   αϯϓϧ͝ͱʹ8FJHIUJOH 
   .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*'
   <$
   $IFO
   BS9JW`> *"5
   <"
   -BNC
   "*4FD`> .5-"5
   <"
   -BVHSPT
   &$$7`> %"5
   <:
   8BOH
   *$.-`> 'FBUVSF
   TDBUUFS
   <)
   ;IBOH
   BOE
   +
   8BOH
   /FVS*14`> $"5
   <2;
   $BJ
   *+$"*`> 6OJWFSTBM
   "5
   <"
   4IBGBIJ
   """*`> "7NJYVQ
   <4
   -FF
   $713`> ("5
   <(
   4SJSBNBOBO
   /FVS*14`> ("*3"5
   <+
   ;IBOH
   *$-3`> ."*-
   <2
   8BOH
   /FVS*14`> 8..3
   <)
   ;FOH
   """*`> #J-"8
   <$
   )PMU[
   0QFO3FWJFX`> -#("5
   <+
   $VJ
   *$$7`> #("5
   <+
   $VJ
   *$$7`> 1$-$
   <9
   -J
   """*`> 1$
   MPTT
   <"
   .VTUBGB
   *$$7`> $//
   X
   3#'
   LFSOFM
   <4
   "
   5BHIBOBLJ
   $713`> "-1
   <)
   ,BOOBO
   BS9JW`> ."35
   <:
   8BOH
   *$-3`> ..-%"
   <5
   1BOH
   *$.-`> ..$
   <5
   1BOH
   *$-3`> -#'(4
   <$
   4[FHFEZ
   *$-3`> '(4.
   <*
   (PPEGFMMPX
   *$-3`> $8
   </
   $BSMJOJ
   BOE
   %8BHOFS
   41`> 1(%
   <"
   .BESZ
   *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
   UBSHFUFE
   <4
   (PXBM
   BS9JW`> 6OJWFSTBM
   "&T
   <4.
   .PPTBWJ%F[GPPMJ
   $713`> "VUP
   "UUBDL
   <'
   $SPDF
   BOE
   .
   )FJO
   *$.-`> '"#
   <'
   $SPDF
   BOE
   .
   )FJO
   *$.-`> 4RVBSF
   "UUBDL
   <"
   "OESJVTIDIFOLP
   &$$7`> "VUP
   1(% "VUP
   1(%%-3 4"*'(.
   <9
   %POH
   $713`> ("."
   <(
   4SJSBNBOBO
   /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
   ,VSBLJO
   *$-38`> 
   ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 
   Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 
   ஌ࣝͷৠཹ 4UBOEBSE
   "5
   <*
   (PPEGFMMPX
   *$-3`> "5
   XJUI
   1(%
   <"
   .BESZ
   *$-3`> ˢ
   /BUVSBMͱ"&Tͷ࢖༻ 1(%
   BUUBDLͷ"&TΛར༻ ˢ
   "&TͷΈ μΠφϛοΫͳ
   εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
   <(
   8
   %JOH
   *$-3`> 53"%&4
   <)
   ;IBOH
   *$.-`> '"5
   <+
   ;IBOH
   *$.-`> )"5
   <3
   3BEF
   *$.-8`>
10. "EWFSTBSJBM
    USBJOJOHͷݚڀಈ޲ 10 ࣝผڥքʹண໨ 
    ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ 
    αϯϓϧ͝ͱʹ8FJHIUJOH 
    .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*'
    <$
    $IFO
    BS9JW`> *"5
    <"
    -BNC
    "*4FD`> .5-"5
    <"
    -BVHSPT
    &$$7`> %"5
    <:
    8BOH
    *$.-`> 'FBUVSF
    TDBUUFS
    <)
    ;IBOH
    BOE
    +
    8BOH
    /FVS*14`> $"5
    <2;
    $BJ
    *+$"*`> 6OJWFSTBM
    "5
    <"
    4IBGBIJ
    """*`> "7NJYVQ
    <4
    -FF
    $713`> ("5
    <(
    4SJSBNBOBO
    /FVS*14`> ("*3"5
    <+
    ;IBOH
    *$-3`> ."*-
    <2
    8BOH
    /FVS*14`> 8..3
    <)
    ;FOH
    """*`> #J-"8
    <$
    )PMU[
    0QFO3FWJFX`> -#("5
    <+
    $VJ
    *$$7`> #("5
    <+
    $VJ
    *$$7`> 1$-$
    <9
    -J
    """*`> 1$
    MPTT
    <"
    .VTUBGB
    *$$7`> $//
    X
    3#'
    LFSOFM
    <4
    "
    5BHIBOBLJ
    $713`> "-1
    <)
    ,BOOBO
    BS9JW`> ."35
    <:
    8BOH
    *$-3`> ..-%"
    <5
    1BOH
    *$.-`> ..$
    <5
    1BOH
    *$-3`> -#'(4
    <$
    4[FHFEZ
    *$-3`> '(4.
    <*
    (PPEGFMMPX
    *$-3`> $8
    </
    $BSMJOJ
    BOE
    %8BHOFS
    41`> 1(%
    <"
    .BESZ
    *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
    UBSHFUFE
    <4
    (PXBM
    BS9JW`> 6OJWFSTBM
    "&T
    <4.
    .PPTBWJ%F[GPPMJ
    $713`> "VUP
    "UUBDL
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> '"#
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> 4RVBSF
    "UUBDL
    <"
    "OESJVTIDIFOLP
    &$$7`> "VUP
    1(% "VUP
    1(%%-3 4"*'(.
    <9
    %POH
    $713`> ("."
    <(
    4SJSBNBOBO
    /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
    ,VSBLJO
    *$-38`> 
    ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 
    Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 
    ஌ࣝͷৠཹ 4UBOEBSE
    "5
    <*
    (PPEGFMMPX
    *$-3`> "5
    XJUI
    1(%
    <"
    .BESZ
    *$-3`> ˢ
    /BUVSBMͱ"&Tͷ࢖༻ 1(%
    BUUBDLͷ"&TΛར༻ ˢ
    "&TͷΈ μΠφϛοΫͳ
    εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
    <(
    8
    %JOH
    *$-3`> 53"%&4
    <)
    ;IBOH
    *$.-`> '"5
    <+
    ;IBOH
    *$.-`> )"5
    <3
    3BEF
    *$.-8`>
11. w ैདྷͷ"5ʹ͓͚Δ໰୊఺
     1(%ͰٻΊͨઁಈ͸ΫϥεڥքΛେ͖͘ލ͙
     ޡ෼ྨ͢Δ͔൱͔͸ઁಈͷॳظํ޲ʹґଘ
    w ઁಈͷ࡞੒ํ๏ʹண໨ͨ͠ख๏
     'SJFOEMZ
    "EWFSTBSJBM
    5SBJOJOH
    '"5
    <+
    ;IBOH
    *$.-
    >
     (VJEFE
    "EWFSTBSJBM
    5SBJOJOH
    ("5
    <(
    4SJSBNBOBO
    /FVS*14
    >
     "EWFSTBSJBM
    7FSUFY
    NJYVQ
    "7NJYVQ
    <4
    -FF
    $713
    > ̍
    ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ͨ͠ख๏ 11
12. w Ϋϥεڥք෇ۙͷઁಈΛٻΊֶͯश
     FBSMZTUPQQFE
    1(%Λར༻ͨ͠ઁಈಋग़ '"5
    <+
    ;IBOH
    *$.-
    > 12 '"5
    'SJFOEMZ
    "EWFSTBSJBM
    5SBJOJOH +
    ;IBOH
    et al.
    l"UUBDLT
    8IJDI
    %P
    OPU
    ,JMM
    5SBJOJOH
    .BLF
    "EWFSTBSJBM
    -FBSOJOH
    4USPOHFSz
    *$.-
     # PVOEBSZ $MBTT
    DFOUFS $MBTT
    DFOUFS ɿ1(%ͰٻΊͨैདྷͷ"&T ɿ'SJFOEMZ
    "EWFSTBSJBM
    &YBNQMFT ℬ[x] = {˜ x ∈ 𝒳 ∣ ∥x − ˜ x∥∞ ≤ ϵ} ˜ xi = arg max ˜ xi ∈ℬ[xi ] L(˜ xi , yi ; θ) ڭࢣ৴߸ͱͷଛࣦ͕࠷େʹͳΔ"&TΛબ୒ 4UBOEBSE
    "EWFSTBSJBM
    &YBNQMFT s . t . L(˜ x, yi ; θ) − min y∈ 𝒴 L(˜ xi , y; θ) ≥ ρ ˜ xi = arg min ˜ xi ∈ℬ[xi ] L(˜ xi , yi ; θ) ڭࢣ৴߸ͱͷଛࣦ͕࠷খɼ͔ͭޡ෼ྨ͕ੜ͍ͯ͡Δ"&TΛબ୒ 'SJFOEMZ
    "EWFSTBSJBM
    &YBNQMFT
13. w 3FMBYBUJPO߲ͷ௥ՃͰద੾ͳॳظํ޲Λ༻͍ͨ"&Tͷ࡞੒
     ޡࠩۂ໘Λ׈Β͔ʹ͢Δ͜ͱͰద੾ͳํ޲ΛٻΊΔ ("5
    <(
    4SJSBNBOBO
    /FVS*14
    > 13 ("5
    (VJEFE
    "EWFSTBSJBM
    5SBJOJOH (
    4SJSBNBOBO
    et al.
    l(VJEFE
    "EWFSTBSJBM
    "UUBDL
    GPS
    &WBMVBUJOH
    BOE
    &OIBODJOH
    "EWFSTBSJBM
    %FGFOTFTz
    /FVS*14
     ˜ xi = xi + ϵ ⋅ sign (∇xi (L(xi , yi ; θ) + λ ⋅ ∥f(xi + r) − f(xi )∥2 2 )) Ϟσϧશମͷ࠷దԽࣜ r ∼ Bern(−ϵ/2,ϵ/2) "&Tͷಋग़ ɿϋΠύʔύϥϝʔλ λ ޡ෼ྨ͕ੜ͡ͳ͍"&Tͷྫ min θ 1 n n ∑ i=1 (L(xi , yi ; θ) + λ ⋅ ∥f(˜ xi ) − f(xi )∥2 2 ) 3FMBYBUJPO߲ 3FMBYBUJPO߲ 3FMBYBUJPO߲ͷ௥Ճ
14. w NJYVQΛར༻ͯ͠/BUVSBMͷੑೳྼԽΛ๷͗ͭͭؤ݈ੑೳΛ޲্
     "EWFSTBSJBM
    WFSUFYʢ܎਺ഒͨ͠"&Tʣͱ/BUVSBMΛ೚ҙͷൺ཰Ͱ಺ૠ "7NJYVQ
    <4
    -FF
    $713
    > 14 "7NJYVQ
    "EWFSTBSJBM
    7FSUFY
    NJYVQ 4
    -FF
    et al.
    l"EWFSTBSJBM
    7FSUFY
    .JYVQ
    5PXBSE
    #FUUFS
    "EWFSTBSJBMMZ
    3PCVTU
    (FOFSBMJ[BUJPOz
    $713
     JOQVU
    x "EWFSTBSJBM
    FYBNQMF "EWFSTBSJBM
    WFSUFY |δ| γ|δ| xav = x + γ ⋅ δx ˜ x = αx + (1 − α)xav ˜ y = αϕ(y, λ1 ) + (1 − α)ϕ(y, λ2 ) ϕ(y, λ) = λy + 1 − λ k − 1 (1 − y) -BCFM
    TNPPUIJOH ը૾ͷNJYVQ "EWFSTBSJBM
    WFSUFYͷ࡞੒ ڭࢣ৴߸ͷNJYVQ α ∼ Beta(1,1) ɿTNPPUIJOHύϥϝʔλ λ1 , λ2 ˠ
    "&Tͷಛ௃Λ܇࿅σʔλͷͭͱͯ͠ٵऩ ˠ
    ܇࿅σʔληοτͷΧόϨοδΛ֦ு
15. "EWFSTBSJBM
    USBJOJOHͷݚڀಈ޲ 15 ࣝผڥքʹண໨ 
    ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ 
    αϯϓϧ͝ͱʹ8FJHIUJOH 
    .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*'
    <$
    $IFO
    BS9JW`> *"5
    <"
    -BNC
    "*4FD`> .5-"5
    <"
    -BVHSPT
    &$$7`> %"5
    <:
    8BOH
    *$.-`> 'FBUVSF
    TDBUUFS
    <)
    ;IBOH
    BOE
    +
    8BOH
    /FVS*14`> $"5
    <2;
    $BJ
    *+$"*`> 6OJWFSTBM
    "5
    <"
    4IBGBIJ
    """*`> "7NJYVQ
    <4
    -FF
    $713`> ("5
    <(
    4SJSBNBOBO
    /FVS*14`> ("*3"5
    <+
    ;IBOH
    *$-3`> ."*-
    <2
    8BOH
    /FVS*14`> 8..3
    <)
    ;FOH
    """*`> #J-"8
    <$
    )PMU[
    0QFO3FWJFX`> -#("5
    <+
    $VJ
    *$$7`> #("5
    <+
    $VJ
    *$$7`> 1$-$
    <9
    -J
    """*`> 1$
    MPTT
    <"
    .VTUBGB
    *$$7`> $//
    X
    3#'
    LFSOFM
    <4
    "
    5BHIBOBLJ
    $713`> "-1
    <)
    ,BOOBO
    BS9JW`> ."35
    <:
    8BOH
    *$-3`> ..-%"
    <5
    1BOH
    *$.-`> ..$
    <5
    1BOH
    *$-3`> -#'(4
    <$
    4[FHFEZ
    *$-3`> '(4.
    <*
    (PPEGFMMPX
    *$-3`> $8
    </
    $BSMJOJ
    BOE
    %8BHOFS
    41`> 1(%
    <"
    .BESZ
    *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
    UBSHFUFE
    <4
    (PXBM
    BS9JW`> 6OJWFSTBM
    "&T
    <4.
    .PPTBWJ%F[GPPMJ
    $713`> "VUP
    "UUBDL
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> '"#
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> 4RVBSF
    "UUBDL
    <"
    "OESJVTIDIFOLP
    &$$7`> "VUP
    1(% "VUP
    1(%%-3 4"*'(.
    <9
    %POH
    $713`> ("."
    <(
    4SJSBNBOBO
    /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
    ,VSBLJO
    *$-38`> 
    ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 
    Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 
    ஌ࣝͷৠཹ 4UBOEBSE
    "5
    <*
    (PPEGFMMPX
    *$-3`> "5
    XJUI
    1(%
    <"
    .BESZ
    *$-3`> ˢ
    /BUVSBMͱ"&Tͷ࢖༻ 1(%
    BUUBDLͷ"&TΛར༻ ˢ
    "&TͷΈ μΠφϛοΫͳ
    εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
    <(
    8
    %JOH
    *$-3`> 53"%&4
    <)
    ;IBOH
    *$.-`> '"5
    <+
    ;IBOH
    *$.-`> )"5
    <3
    3BEF
    *$.-8`>
16. w ैདྷͷ"5ʹ͓͚Δ໰୊఺
     "&TΛޡ෼ྨ͠ͳ͍Α͏ʹڭࢣ৴߸ͱͷΫϩεΤϯτϩϐʔޡࠩΛ࠷খԽ
    ˠ
    "&Tʹաద߹ͯ͠/BUVSBMαϯϓϧΛޡ෼ྨ
    w Ϟσϧग़ྗʹର͢Δਖ਼ଇԽΛ௥Ճͨ͠ख๏
     "EWFSTBSJBM
    -PHJU
    1BJSJOH
    "-1
    <)
    ,BOOBO
    BS9JW
    >
     53"%&4
    <)
    ;IBOH
    *$.-
    > 
    Ϟσϧͷग़ྗ஋ʹର͢Δਖ਼ଇԽͷ௥Ճ 16
17. w "&Tͱ/BUVSBMαϯϓϧͷग़ྗ஋ͱҰகͤ͞Δ
     "&Tͱϕʔεͷ/BUVSBMαϯϓϧ͸ಉ͡ಛ௃Λଊ͑Δ ͳͥਖ਼ଇԽΛ௥Ճ͢Δ͔ 17 #PVOEBSZ "&T $MBTT
    DFOUFS xi ग़ྗ஋ͷϖΞϦϯά
18. w "&Tͱ/BUVSBMαϯϓϧͷग़ྗ஋ͱҰகͤ͞Δ
     "&Tͱϕʔεͷ/BUVSBMαϯϓϧ͸ಉ͡ಛ௃Λଊ͑Δ ϩδοτʹண໨ͨ͠ख๏ 18 "-1
    <)
    ,BOOBO
    BS9JW
    >
    "EWFSTBSJBM
    -PHJU
    1BJSJOH )
    ,BOOBO
    et al.
    l"EWFSTBSJBM
    -PHJU
    1BJSJOHz
    BS9JW
     w /BUVSBMͱ"&TͷϩδοτΛҰக
     ฏۉೋ৐ޡࠩͷ࠷খԽ #PVOEBSZ xi "&T $MBTT
    DFOUFS L(xi , yi ; θ) = LCE (f(˜ xi ), yi ; θ) + λ k ∥f(xi ) − f(˜ xi )∥2 2 ˢ
    "&Tʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ ˢ
    ϩδοτʹର͢Δਖ਼ଇԽ ग़ྗ஋ͷϖΞϦϯά
19. w "&Tͱ/BUVSBMαϯϓϧͷग़ྗ஋ͱҰகͤ͞Δ
     "&Tͱϕʔεͷ/BUVSBMαϯϓϧ͸ಉ͡ಛ௃Λଊ͑Δ ࣄޙ෼෍ʹண໨ͨ͠ख๏ 19 "-1
    <)
    ,BOOBO
    BS9JW
    >
    "EWFSTBSJBM
    -PHJU
    1BJSJOH )
    ,BOOBO
    et al.
    l"EWFSTBSJBM
    -PHJU
    1BJSJOHz
    BS9JW
     w /BUVSBMͱ"&TͷϩδοτΛҰக
     ฏۉೋ৐ޡࠩͷ࠷খԽ #PVOEBSZ xi "&T $MBTT
    DFOUFS L(xi , yi ; θ) = LCE (f(˜ xi ), yi ; θ) + λ k ∥f(xi ) − f(˜ xi )∥2 2 ˢ
    "&Tʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ ˢ
    ϩδοτʹର͢Δਖ਼ଇԽ w /BUVSBMͱ"&Tͷࣄޙ֬཰ΛҰக
     ,-μΠόʔδΣϯεͷ࠷খԽ 53"%&4
    <)
    ;IBOH
    *$.-
    >
    53BEFP ff JOTQJSFE
    "EWFSTBSJBM
    %&GFOTF
    WJB
    4VSSPHBUF
    NJOJNJ[BUJPO L(xi , yi ; θ) = LCE (f(xi ), yi ; θ) + λ ⋅ DKL [p(xi )∥p(˜ xi )] ˢ
    OBUVSBMʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ )
    ;IBOH
    et al.
    l5IFPSFUJDBMMZ
    1SJODJQMFE
    5SBEFP ff
    CFUXFFO
    3PCVTUOFTT
    BOE
    "DDVSBDZz
    *$.-
     ˢ
    ࣄޙ֬཰ʹର͢Δਖ਼ଇԽ ग़ྗ஋ͷϖΞϦϯά
20. "EWFSTBSJBM
    USBJOJOHͷݚڀಈ޲ 20 ࣝผڥքʹண໨ 
    ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ 
    αϯϓϧ͝ͱʹ8FJHIUJOH 
    .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*'
    <$
    $IFO
    BS9JW`> *"5
    <"
    -BNC
    "*4FD`> .5-"5
    <"
    -BVHSPT
    &$$7`> %"5
    <:
    8BOH
    *$.-`> 'FBUVSF
    TDBUUFS
    <)
    ;IBOH
    BOE
    +
    8BOH
    /FVS*14`> $"5
    <2;
    $BJ
    *+$"*`> 6OJWFSTBM
    "5
    <"
    4IBGBIJ
    """*`> "7NJYVQ
    <4
    -FF
    $713`> ("5
    <(
    4SJSBNBOBO
    /FVS*14`> ("*3"5
    <+
    ;IBOH
    *$-3`> ."*-
    <2
    8BOH
    /FVS*14`> 8..3
    <)
    ;FOH
    """*`> #J-"8
    <$
    )PMU[
    0QFO3FWJFX`> -#("5
    <+
    $VJ
    *$$7`> #("5
    <+
    $VJ
    *$$7`> 1$-$
    <9
    -J
    """*`> 1$
    MPTT
    <"
    .VTUBGB
    *$$7`> $//
    X
    3#'
    LFSOFM
    <4
    "
    5BHIBOBLJ
    $713`> "-1
    <)
    ,BOOBO
    BS9JW`> ."35
    <:
    8BOH
    *$-3`> ..-%"
    <5
    1BOH
    *$.-`> ..$
    <5
    1BOH
    *$-3`> -#'(4
    <$
    4[FHFEZ
    *$-3`> '(4.
    <*
    (PPEGFMMPX
    *$-3`> $8
    </
    $BSMJOJ
    BOE
    %8BHOFS
    41`> 1(%
    <"
    .BESZ
    *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
    UBSHFUFE
    <4
    (PXBM
    BS9JW`> 6OJWFSTBM
    "&T
    <4.
    .PPTBWJ%F[GPPMJ
    $713`> "VUP
    "UUBDL
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> '"#
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> 4RVBSF
    "UUBDL
    <"
    "OESJVTIDIFOLP
    &$$7`> "VUP
    1(% "VUP
    1(%%-3 4"*'(.
    <9
    %POH
    $713`> ("."
    <(
    4SJSBNBOBO
    /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
    ,VSBLJO
    *$-38`> 
    ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 
    Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 
    ஌ࣝͷৠཹ 4UBOEBSE
    "5
    <*
    (PPEGFMMPX
    *$-3`> "5
    XJUI
    1(%
    <"
    .BESZ
    *$-3`> ˢ
    /BUVSBMͱ"&Tͷ࢖༻ 1(%
    BUUBDLͷ"&TΛར༻ ˢ
    "&TͷΈ μΠφϛοΫͳ
    εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
    <(
    8
    %JOH
    *$-3`> 53"%&4
    <)
    ;IBOH
    *$.-`> '"5
    <+
    ;IBOH
    *$.-`> )"5
    <3
    3BEF
    *$.-8`>
21. w ैདྷͷ"5ʹ͓͚Δ໰୊఺
     /BUVSBMαϯϓϧͷ෼ྨ݁Ռʹؔ܎ͳ͘ɼશαϯϓϧʹରͯ͠"&TΛఆֶٛͯ͠श
    ˠ
    ޡ෼ྨ͕ੜ͍ͯ͡Δαϯϓϧ͸"5ʹد༩͠ͳ͍
    w αϯϓϧͷ෼ྨ݁ՌʹԠͨ͡ଛࣦܭࢉΛߦ͏ख๏
     .BY.BSHJO
    "EWFSTBSJBM
    5SBJOJOH
    .."
    <(8
    %JOH
    *$-3
    >
     .JTDMBTTJ fi DBUJPO
    "XBSF
    "EWFSTBSJBM
    5SBJOJOH
    ."35
    <:
    8BOH
    *$-3
    > 
    ޡ෼ྨͨ͠αϯϓϧͷޮՌతͳར༻ 21
22. "EWFSTBSJBM
    USBJOJOHͷݚڀಈ޲ 22 ࣝผڥքʹண໨ 
    ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ 
    αϯϓϧ͝ͱʹ8FJHIUJOH 
    .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*'
    <$
    $IFO
    BS9JW`> *"5
    <"
    -BNC
    "*4FD`> .5-"5
    <"
    -BVHSPT
    &$$7`> %"5
    <:
    8BOH
    *$.-`> 'FBUVSF
    TDBUUFS
    <)
    ;IBOH
    BOE
    +
    8BOH
    /FVS*14`> $"5
    <2;
    $BJ
    *+$"*`> 6OJWFSTBM
    "5
    <"
    4IBGBIJ
    """*`> "7NJYVQ
    <4
    -FF
    $713`> ("5
    <(
    4SJSBNBOBO
    /FVS*14`> ("*3"5
    <+
    ;IBOH
    *$-3`> ."*-
    <2
    8BOH
    /FVS*14`> 8..3
    <)
    ;FOH
    """*`> #J-"8
    <$
    )PMU[
    0QFO3FWJFX`> -#("5
    <+
    $VJ
    *$$7`> #("5
    <+
    $VJ
    *$$7`> 1$-$
    <9
    -J
    """*`> 1$
    MPTT
    <"
    .VTUBGB
    *$$7`> $//
    X
    3#'
    LFSOFM
    <4
    "
    5BHIBOBLJ
    $713`> "-1
    <)
    ,BOOBO
    BS9JW`> ."35
    <:
    8BOH
    *$-3`> ..-%"
    <5
    1BOH
    *$.-`> ..$
    <5
    1BOH
    *$-3`> -#'(4
    <$
    4[FHFEZ
    *$-3`> '(4.
    <*
    (PPEGFMMPX
    *$-3`> $8
    </
    $BSMJOJ
    BOE
    %8BHOFS
    41`> 1(%
    <"
    .BESZ
    *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
    UBSHFUFE
    <4
    (PXBM
    BS9JW`> 6OJWFSTBM
    "&T
    <4.
    .PPTBWJ%F[GPPMJ
    $713`> "VUP
    "UUBDL
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> '"#
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> 4RVBSF
    "UUBDL
    <"
    "OESJVTIDIFOLP
    &$$7`> "VUP
    1(% "VUP
    1(%%-3 4"*'(.
    <9
    %POH
    $713`> ("."
    <(
    4SJSBNBOBO
    /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
    ,VSBLJO
    *$-38`> 
    ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 
    Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 
    ஌ࣝͷৠཹ 4UBOEBSE
    "5
    <*
    (PPEGFMMPX
    *$-3`> "5
    XJUI
    1(%
    <"
    .BESZ
    *$-3`> ˢ
    /BUVSBMͱ"&Tͷ࢖༻ 1(%
    BUUBDLͷ"&TΛར༻ ˢ
    "&TͷΈ μΠφϛοΫͳ
    εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
    <(
    8
    %JOH
    *$-3`> 53"%&4
    <)
    ;IBOH
    *$.-`> '"5
    <+
    ;IBOH
    *$.-`> )"5
    <3
    3BEF
    *$.-8`>
23. w ैདྷͷ"5ʹ͓͚Δ໰୊఺
     ΫϥεڥքͱαϯϓϧͷϚʔδϯΛߟֶྀͯ͠श͕͞Ε͍ͯͳ͍
    ˠ
    ڥքʹ͍ۙαϯϓϧ͸߈ܸͷϦεΫ͕ߴ͍
    w Ϋϥεڥք෇͔ۙΒ෺ཧతʹ཭͢ख๏
     1SPUPUZQF
    $POGPSNJUZ
    MPTT
    1$
    MPTT
    <"
    .VTUBGB
    *$$7
    >
     1SPCBCJMJTUJDBMMZ
    $PNQBDU
    MPTT
    XJUI
    -PHJU
    $POTUSBJOUT
    1$-$
    <9
    -J
    """*
    > 
    ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 23
24. w ֶशՄೳͳΫϥεத৺Λ༻͍ͯಛ௃දݱΛૢ࡞
     ਖ਼ղΫϥεɿಛ௃ྔͱΫϥεத৺ͷڑ཭Λ࠷খԽ͢Δ͜ͱͰίϯύΫτʹදݱ
     ෆਖ਼ղΫϥεɿಛ௃ྔͱΫϥεத৺ͷڑ཭Λ࠷େԽ͢Δ͜ͱͰϫΠυʹදݱ 1$
    MPTT
    <"
    .VTUBGB
    *$$7
    > 24 "
    .VTUBGB
    et al.
    l"EWFSTBSJBM
    %FGFOTF
    CZ
    3FTUSJDUJOH
    UIF
    )JEEFO
    4QBDF
    PG
    %FFQ
    /FVSBM
    /FUXPSLTz
    *$$7
     1$
    MPTT
    1SPUPUZQF
    $POGPSNJUZ
    MPTT ˞
    ͸ ʹؚΊͳͯ͘΋҉໧తʹ཭ΕΔ ∥wc yi − wc yj ∥2 LPC 
    ֶशՄೳͳΫϥεॏ৺ wc 
    αϯϓϧ਺ n 
    ʹؔ͢Δಛ௃ϕΫτϧ fi xi 
    Ϋϥε਺ k 
    ϋΠύʔύϥϝʔλ λ1 , λ2 L(xi , yi ) = LCE (xi , yi ) + LPC (xi , yi ) L(xi , yi ; θ) = LCE (xi , yi ; θ) + λ1 ⋅ ∥f(˜ xi ) − wc yi ∥2 − λ2 k − 1 ⋅ ∑ yj ≠yi (∥f(˜ xi ) − wc yj ∥2 + ∥wc yi − wc yj ∥2) ֶश͢Δଛࣦؔ਺ ˢ
    ਖ਼ղΫϥεத৺ͱͷڑ཭ܭࢉ ˢ
    ෆਖ਼ղΫϥεத৺
    ͱͷڑ཭ܭࢉ ˢ
    Ϋϥεத৺ؒͷڑ཭ܭࢉ fi (˜ xi ) wc yj wc yj wc yi ͚ۙͮΔ ཭ ͢ ཭͢ ཭͢ ཭ ͢
25. w ΫϩεΤϯτϩϐʔޡࠩͷܭࢉͳ͠Ͱ1$
    MPTTͱಉ༷ͷૢ࡞
     ਖ਼ղΫϥεͷ֬཰͕೚ҙͷᮢ஋ΛԼճΔαϯϓϧ͸શͯෆਖ਼ղˠίϯύΫτ͔ͭϫΠυͳදݱ
     ਖ਼ղΫϥεͱྡ઀Ϋϥε֬཰ͷ͕ࠩৗʹਖ਼Λཁ੥ˠΫϥε͕ަࠩ͢Δ͜ͱΛ๷ࢭ 1$-$
    <9
    -J
    """*
    > 25 9
    -J
    et al.
    l*NQSPWJOH
    "EWFSTBSJBM
    3PCVTUOFTT
    WJB
    1SPCBCJMJTUJDBMMZ
    $PNQBDU
    -PTT
    XJUI
    -PHJU
    $POTUSBJOUTz
    """*
     1$-$
    1SPCBCJMJTUJDBMMZ
    $PNQBDU
    MPTT
    XJUI
    -PHJU
    $POTUSBJOU 1$-$ͷଛࣦؔ਺ L(xi , yi ; θ) = max (0,py′  (˜ xi ) + τ − pyi (˜ xi )) + λ (max (0,zyi (˜ xi ) − zy′  (˜ xi )) − C′  ) #PVOEBSZ $MBTT
    DFOUFS $MBTT
    DFOUFS τ ˢ
    ෼ྨڥքΛਖ਼ղΫϥεଆʹ ͚ͩҠಈ τ ˢ
    ྡ઀ΫϥεͱͷަࠩΛ๷ࢭ
26. w ΫϩεΤϯτϩϐʔޡࠩͷܭࢉͳ͠Ͱ1$
    MPTTͱಉ༷ͷૢ࡞
     ਖ਼ղΫϥεͷ֬཰͕೚ҙͷᮢ஋ΛԼճΔαϯϓϧ͸શͯෆਖ਼ղˠίϯύΫτ͔ͭϫΠυͳදݱ
     ਖ਼ղΫϥεͱྡ઀Ϋϥε֬཰ͷ͕ࠩৗʹਖ਼Λཁ੥ˠΫϥε͕ަࠩ͢Δ͜ͱΛ๷ࢭ 1$-$
    <9
    -J
    """*
    > 26 9
    -J
    et al.
    l*NQSPWJOH
    "EWFSTBSJBM
    3PCVTUOFTT
    WJB
    1SPCBCJMJTUJDBMMZ
    $PNQBDU
    -PTT
    XJUI
    -PHJU
    $POTUSBJOUTz
    """*
     1$-$
    1SPCBCJMJTUJDBMMZ
    $PNQBDU
    MPTT
    XJUI
    -PHJU
    $POTUSBJOU 1$-$ͷଛࣦؔ਺ L(xi , yi ; θ) = max (0,py′  (˜ xi ) + τ − pyi (˜ xi )) + λ (max (0,zyi (˜ xi ) − zy′  (˜ xi )) − C′  ) #PVOEBSZ $MBTT
    DFOUFS $MBTT
    DFOUFS τ ˢ
    ෆਖ਼ղαϯϓϧ ˢ
    ෼ྨڥքΛਖ਼ղΫϥεଆʹ ͚ͩҠಈ τ ˢ
    ྡ઀ΫϥεͱͷަࠩΛ๷ࢭ
27. w ΫϩεΤϯτϩϐʔޡࠩͷܭࢉͳ͠Ͱ1$
    MPTTͱಉ༷ͷૢ࡞
     ਖ਼ղΫϥεͷ֬཰͕೚ҙͷᮢ஋ΛԼճΔαϯϓϧ͸શͯෆਖ਼ղˠίϯύΫτ͔ͭϫΠυͳදݱ
     ਖ਼ղΫϥεͱྡ઀Ϋϥε֬཰ͷ͕ࠩৗʹਖ਼Λཁ੥ˠΫϥε͕ަࠩ͢Δ͜ͱΛ๷ࢭ 1$-$
    <9
    -J
    """*
    > 27 9
    -J
    et al.
    l*NQSPWJOH
    "EWFSTBSJBM
    3PCVTUOFTT
    WJB
    1SPCBCJMJTUJDBMMZ
    $PNQBDU
    -PTT
    XJUI
    -PHJU
    $POTUSBJOUTz
    """*
     1$-$
    1SPCBCJMJTUJDBMMZ
    $PNQBDU
    MPTT
    XJUI
    -PHJU
    $POTUSBJOU 1$-$ͷଛࣦؔ਺ L(xi , yi ; θ) = max (0,py′  (˜ xi ) + τ − pyi (˜ xi )) + λ (max (0,zyi (˜ xi ) − zy′  (˜ xi )) − C′  ) #PVOEBSZ $MBTT
    DFOUFS $MBTT
    DFOUFS τ ˢ
    ෆਖ਼ղαϯϓϧ ˢ
    ෼ྨڥքΛਖ਼ղΫϥεଆʹ ͚ͩҠಈ τ ˢ
    ྡ઀ΫϥεͱͷަࠩΛ๷ࢭ
28. "EWFSTBSJBM
    USBJOJOHͷݚڀಈ޲ 28 ࣝผڥքʹண໨ 
    ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ 
    αϯϓϧ͝ͱʹ8FJHIUJOH 
    .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*'
    <$
    $IFO
    BS9JW`> *"5
    <"
    -BNC
    "*4FD`> .5-"5
    <"
    -BVHSPT
    &$$7`> %"5
    <:
    8BOH
    *$.-`> 'FBUVSF
    TDBUUFS
    <)
    ;IBOH
    BOE
    +
    8BOH
    /FVS*14`> $"5
    <2;
    $BJ
    *+$"*`> 6OJWFSTBM
    "5
    <"
    4IBGBIJ
    """*`> "7NJYVQ
    <4
    -FF
    $713`> ("5
    <(
    4SJSBNBOBO
    /FVS*14`> ("*3"5
    <+
    ;IBOH
    *$-3`> ."*-
    <2
    8BOH
    /FVS*14`> 8..3
    <)
    ;FOH
    """*`> #J-"8
    <$
    )PMU[
    0QFO3FWJFX`> -#("5
    <+
    $VJ
    *$$7`> #("5
    <+
    $VJ
    *$$7`> 1$-$
    <9
    -J
    """*`> 1$
    MPTT
    <"
    .VTUBGB
    *$$7`> $//
    X
    3#'
    LFSOFM
    <4
    "
    5BHIBOBLJ
    $713`> "-1
    <)
    ,BOOBO
    BS9JW`> ."35
    <:
    8BOH
    *$-3`> ..-%"
    <5
    1BOH
    *$.-`> ..$
    <5
    1BOH
    *$-3`> -#'(4
    <$
    4[FHFEZ
    *$-3`> '(4.
    <*
    (PPEGFMMPX
    *$-3`> $8
    </
    $BSMJOJ
    BOE
    %8BHOFS
    41`> 1(%
    <"
    .BESZ
    *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
    UBSHFUFE
    <4
    (PXBM
    BS9JW`> 6OJWFSTBM
    "&T
    <4.
    .PPTBWJ%F[GPPMJ
    $713`> "VUP
    "UUBDL
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> '"#
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> 4RVBSF
    "UUBDL
    <"
    "OESJVTIDIFOLP
    &$$7`> "VUP
    1(% "VUP
    1(%%-3 4"*'(.
    <9
    %POH
    $713`> ("."
    <(
    4SJSBNBOBO
    /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
    ,VSBLJO
    *$-38`> 
    ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 
    Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 
    ஌ࣝͷৠཹ 4UBOEBSE
    "5
    <*
    (PPEGFMMPX
    *$-3`> "5
    XJUI
    1(%
    <"
    .BESZ
    *$-3`> ˢ
    /BUVSBMͱ"&Tͷ࢖༻ 1(%
    BUUBDLͷ"&TΛར༻ ˢ
    "&TͷΈ μΠφϛοΫͳ
    εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
    <(
    8
    %JOH
    *$-3`> 53"%&4
    <)
    ;IBOH
    *$.-`> '"5
    <+
    ;IBOH
    *$.-`> )"5
    <3
    3BEF
    *$.-8`>
29. w ैདྷͷ"5ʹ͓͚Δ໰୊఺
     શαϯϓϧΛ౳͘͠ଛࣦܭࢉ͢ΔͨΊɼଛࣦ͕େ͖͍αϯϓϧʹґଘͯ͠ޡ͕ࠩఀ଺
    ˠ
    ϞσϧશମͷੑೳྼԽ
    w ΫϥεڥքͱͷزԿతͳڑ཭ؔ܎͔ΒαϯϓϧͷॏΈ෇͚Λ͢Δख๏
     (FPNFUSZ"XBSF
    *OTUBODF
    3FXFJHIUFE
    "EWFSTBSJBM
    5SBJOJOH
    ("*3"5
    <+
    ;IBOH
    *$-3
    >
     8FJHIUFE
    .JOJ.BY
    3JTL
    8..3
    <)
    ;FOH
    """*
    >
     .BSHJO"XBSF
    *OTUBODF
    3FXFJHIUJOH
    -FBSOJOH
    ."*-
    <2
    8BOH
    /FVS*14
    >
     #JMFWFM
    -FBSOBCMF
    "EWFSTBSJBM
    3FXFJHIUJOH
    #J-"8
    <$
    )PMU[
    0QFO3FWJFX
    > 
    αϯϓϧ͝ͱͷ8FJHIUJOHͷར༻ 29 Ϟσϧશମͷ࠷దԽࣜ ֤αϯϓϧͷଛࣦʹର͢ΔॏΈ෇͚ ω(xi , yi ) min θ 1 n n ∑ i=1 L(f(˜ xi ; θ), yi )
30. w 1(%ͷ߈ܸճ਺ʹج͍ͮͨॏΈఆٛ
     গͳ͍ճ਺Ͱޡ෼ྨ͕ੜͨ͡αϯϓϧ΄Ͳ߈ܸͷϦεΫ͕ߴ͍
     ࢦఆճ਺಺Ͱޡ෼ྨ͕ੜ͡ͳ͍αϯϓϧ͸߈ܸͷϦεΫ͕௿͍
    w /BUVSBMͳঢ়ଶͰޡ෼ྨ͍ͯ͠Δ͔൱͔͸ߟྀ͠ͳ͍ ("*3"5
    <+
    ;IBOH
    *$-3
    > 30 +
    ;IBOH
    et al.
    l(&0.&53:"8"3&
    */45"/$&3&8&*()5&%
    "%7&34"3*"-
    53"*/*/(z
    *$-3
     ("*3"5
    (FPNFUSZ"XBSF
    *OTUBODF3FXFJHIUFE
    "EWFSTBSJBM
    5SBJOJOH 8FJHIUJOHؔ਺ ω(xi , yi ) = (1 − tanh(λ + 5 × (1 − 2 × κ(xi , yi )/K))) 2 ω(xi , yi ) ≥ 0 n ∑ i=1 ω(xi , yi ) = 1 
    ߈ܸʹ੒ޭͨ͠࠷খεςοϓ਺ κ(x, y) 
    1(%ͷ૯εςοϓ਺ K 
    ϋΠύʔύϥϝʔλ λ #PVOEBSZ $MBTT
    DFOUFS $MBTT
    DFOUFS           x1 x2           x3           ڥքͱͷϚʔδϯ
31. w Ϋϥε֬཰Λ༻͍ͯڥք·ͰͷزԿతͳڑ཭Λఆٛ
     ූ߸͕ਖ਼͔ͭϚʔδϯ͕େ͖͍ˠখ͞ͳॏΈɿॏཁ౓͕௿͍
     ූ߸͕ਖ਼͔ͭϚʔδϯ͕খ͍͞ˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍
     ූ߸͕ෛˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ࣄޙ෼෍ΛϕʔεʹϚʔδϯΛܾఆ͢Δख๏ 31 $MBTT
    DFOUFS #PVOEBSZ $MBTT
    DFOUFS p = 0.7 p = 0.3 p = 0.8 p = 0.2
32. w Ϋϥε֬཰Λ༻͍ͯڥք·ͰͷزԿతͳڑ཭Λఆٛ
     ූ߸͕ਖ਼͔ͭϚʔδϯ͕େ͖͍ˠখ͞ͳॏΈɿॏཁ౓͕௿͍
     ූ߸͕ਖ਼͔ͭϚʔδϯ͕খ͍͞ˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍
     ූ߸͕ෛˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ࣄޙ෼෍ΛϕʔεʹϚʔδϯΛܾఆ͢Δख๏ 32 8..3
    <)
    ;FOH
    """*
    >
    8FJHIUFE
    .JOJ.BY
    3JTL w ࢦ਺ؔ਺ʹΑΔॏΈఆٛ
     ޡ෼ྨͨ͠αϯϓϧ΄Ͳେ͖ͳॏΈ͚ͮ $MBTT
    DFOUFS #PVOEBSZ $MBTT
    DFOUFS p = 0.7 p = 0.3 p = 0.8 p = 0.2 ω(xi , yi ) = exp (−α ⋅ margin (˜ xi , yi)) margin (˜ xi , yi) = pyi (˜ xi ) − max j≠yi pj (˜ xi ) 8FJHIUJOHؔ਺ )
    ;FOH
    et al.
    l"SF
    "EWFSTBSJBM
    &YBNQMFT
    $SFBUFE
    &RVBM
    "
    -FBSOBCMF
    8FJHIUFE
    .JOJNBY
    3JTL
    GPS
    3PCVTUOFTT
    VOEFS
    /POVOJGPSN
    "UUBDLTz
    """*
    
33. w Ϋϥε֬཰Λ༻͍ͯڥք·ͰͷزԿతͳڑ཭Λఆٛ
     ූ߸͕ਖ਼͔ͭϚʔδϯ͕େ͖͍ˠখ͞ͳॏΈɿॏཁ౓͕௿͍
     ූ߸͕ਖ਼͔ͭϚʔδϯ͕খ͍͞ˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍
     ූ߸͕ෛˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ࣄޙ෼෍ΛϕʔεʹϚʔδϯΛܾఆ͢Δख๏ 33 8..3
    <)
    ;FOH
    """*
    >
    8FJHIUFE
    .JOJ.BY
    3JTL w ࢦ਺ؔ਺ʹΑΔॏΈఆٛ
     ޡ෼ྨͨ͠αϯϓϧ΄Ͳେ͖ͳॏΈ͚ͮ $MBTT
    DFOUFS #PVOEBSZ $MBTT
    DFOUFS p = 0.7 p = 0.3 p = 0.8 p = 0.2 ω(xi , yi ) = exp (−α ⋅ margin (˜ xi , yi)) margin (˜ xi , yi) = pyi (˜ xi ) − max j≠yi pj (˜ xi ) 8FJHIUJOHؔ਺ )
    ;FOH
    et al.
    l"SF
    "EWFSTBSJBM
    &YBNQMFT
    $SFBUFE
    &RVBM
    "
    -FBSOBCMF
    8FJHIUFE
    .JOJNBY
    3JTL
    GPS
    3PCVTUOFTT
    VOEFS
    /POVOJGPSN
    "UUBDLTz
    """*
     w 4JHNPJEؔ਺ʹΑΔॏΈఆٛ
     ϛχόον಺ͰॏΈΛਖ਼نԽ ."*-
    <2
    8BOH
    /FVS*14
    >
    .BSHJO"XBSF
    *OTUBODF
    3FXFJHIUJOH
    -FBSOJOH ω(xi , yi ) = sigmoid (−γ ⋅ (PM(xi , yi ) − β)) PMnat (xi , yi ) = pyi (xi ) − max j≠yi pj (xi ) PMadv (˜ xi , yi ) = pyi (˜ x′  i ) − max j≠yi pj (˜ xi ) PMdif (x′  i , yi ) = pyi (xi ) − pyi (x′  i ) 8FJHIUJOHؔ਺ 2
    8BOH
    et al.
    l1SPCBCJMJTUJD
    .BSHJOT
    GPS
    *OTUBODF
    3FXFJHIUJOH
    JO
    "EWFSTBSJBM
    5SBJOJOHz
    /FVS*14
    
34. "EWFSTBSJBM
    USBJOJOHͷݚڀಈ޲ 34 ࣝผڥքʹண໨ 
    ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ 
    αϯϓϧ͝ͱʹ8FJHIUJOH 
    .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*'
    <$
    $IFO
    BS9JW`> *"5
    <"
    -BNC
    "*4FD`> .5-"5
    <"
    -BVHSPT
    &$$7`> %"5
    <:
    8BOH
    *$.-`> 'FBUVSF
    TDBUUFS
    <)
    ;IBOH
    BOE
    +
    8BOH
    /FVS*14`> $"5
    <2;
    $BJ
    *+$"*`> 6OJWFSTBM
    "5
    <"
    4IBGBIJ
    """*`> "7NJYVQ
    <4
    -FF
    $713`> ("5
    <(
    4SJSBNBOBO
    /FVS*14`> ("*3"5
    <+
    ;IBOH
    *$-3`> ."*-
    <2
    8BOH
    /FVS*14`> 8..3
    <)
    ;FOH
    """*`> #J-"8
    <$
    )PMU[
    0QFO3FWJFX`> -#("5
    <+
    $VJ
    *$$7`> #("5
    <+
    $VJ
    *$$7`> 1$-$
    <9
    -J
    """*`> 1$
    MPTT
    <"
    .VTUBGB
    *$$7`> $//
    X
    3#'
    LFSOFM
    <4
    "
    5BHIBOBLJ
    $713`> "-1
    <)
    ,BOOBO
    BS9JW`> ."35
    <:
    8BOH
    *$-3`> ..-%"
    <5
    1BOH
    *$.-`> ..$
    <5
    1BOH
    *$-3`> -#'(4
    <$
    4[FHFEZ
    *$-3`> '(4.
    <*
    (PPEGFMMPX
    *$-3`> $8
    </
    $BSMJOJ
    BOE
    %8BHOFS
    41`> 1(%
    <"
    .BESZ
    *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMF
    UBSHFUFE
    <4
    (PXBM
    BS9JW`> 6OJWFSTBM
    "&T
    <4.
    .PPTBWJ%F[GPPMJ
    $713`> "VUP
    "UUBDL
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> '"#
    <'
    $SPDF
    BOE
    .
    )FJO
    *$.-`> 4RVBSF
    "UUBDL
    <"
    "OESJVTIDIFOLP
    &$$7`> "VUP
    1(% "VUP
    1(%%-3 4"*'(.
    <9
    %POH
    $713`> ("."
    <(
    4SJSBNBOBO
    /FVS*14`> .VMUJεςοϓͷ࣌୅ <"
    ,VSBLJO
    *$-38`> 
    ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 
    Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 
    ஌ࣝͷৠཹ 4UBOEBSE
    "5
    <*
    (PPEGFMMPX
    *$-3`> "5
    XJUI
    1(%
    <"
    .BESZ
    *$-3`> ˢ
    /BUVSBMͱ"&Tͷ࢖༻ 1(%
    BUUBDLͷ"&TΛར༻ ˢ
    "&TͷΈ μΠφϛοΫͳ
    εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .."
    <(
    8
    %JOH
    *$-3`> 53"%&4
    <)
    ;IBOH
    *$.-`> '"5
    <+
    ;IBOH
    *$.-`> )"5
    <3
    3BEF
    *$.-8`>
35. w ैདྷͷ"5ʹ͓͚Δ໰୊఺
     ௨ৗͷ"5Ͱ͸"&TʹϑΟοτͨ͠ΫϥεڥքΛ֫ಘ
    ˠ
    /BUVSBMαϯϓϧʹର͢Δޡ෼ྨ͕ଟൃ
    w /BUVSBMαϯϓϧͷΫϥεڥքΛΨΠυʹͨ͠ख๏
     #PVOEBSZ
    (VJEFE
    "EWFSTBSJBM
    5SBJOJOH
    #("5
    <+
    $VJ
    *$$7
    >
     -FBSOBCMF
    #PVOEBSZ
    (VJEFE
    "EWFSTBSJBM
    5SBJOJOH
    -#("5
    <+
    $VJ
    *$$7
    > 
    ஌ࣝͷৠཹ 35 ௨ৗͷֶशΛͨ͠Ϟσϧ "5Λͨ͠Ϟσϧ
36. w /BUVSBMͷΫϥεڥքΛΨΠυʹͨ͠"5
     /BUVSBMͷ஌ࣝΛϩόετͳϞσϧ΁ܧঝ
     "&Tʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ͸ܭࢉ͠ͳ͍ -#("5
    <+
    $VJ
    *$$7
    > 36 +
    $VJ
    et al.
    l-FBSOBCMF
    #PVOEBSZ
    (VJEFE
    "EWFSTBSJBM
    5SBJOJOHz
    *$$7
     -#("5
    -FBSOBCMF
    #PVOEBSZ
    (VJEFE
    "EWFSTBSJBM
    5SBJOJOH min θ 𝔼 (xi ,yi )∈D L (ℳrob (xadv i ), ℳnat (xi )) #PVOEBSZ
    (VJEFE
    "EWFSTBSJBM
    5SBJOJOH 1BSBNFUFS
    fi YFE ℳnat ℳrob ℳnat (xi ) xi xadv i ℳrob (xadv i ) MSE (ℳrob (xi ), ℳnat (xadv i )) ֶशࡁΈͷ ͷΫϥεڥքΛ ʹܧঝ
    ˞
    ΫϩεΤϯτϩϐʔޡࠩ͸࢖༻͠ͳ͍ ℳnat ℳrob min θ,θ* 𝔼 (xi ,yi )∈D L (ℳrob (xadv i ), ℳnat (xi )) +β ⋅ L(xi , yi ; θ) ℳnat ℳrob ℳnat (xi ) xi xadv i ℳrob (xadv i ) MSE (ℳrob (xi ), ℳnat (xadv i )) L (xi , yi ; θ) -FBSOBCMF
    #PVOEBSZ
    (VJEFE
    "EWFSTBSJBM
    5SBJOJOH ͱ Λॳظ஋͔Βڞಉֶश
    ˞
    ʹؔͯ͠ΫϩεΤϯτϩϐʔޡࠩ͸࢖༻͠ͳ͍ ℳnat ℳrob ℳrob
37. w ϞσϧͱσʔληοτΛ౷Ұͯ͠෼ྨਫ਼౓ͱ߈ܸʹର͢Δؤ݈ੑΛൺֱ
     ֶश৚݅͸جຊతʹ࿦จهࡌͷ஋Λར༻
    w ڞ௨ͷ৚݅
     σʔληοτɿ$*'"3
     ϕʔεϞσϧɿ8JEF3FT/FU
    w ֶश৚݅ʢ࿦จʹهࡌ͕ͳ͍৔߹ʣ
     ֶशճ਺ɿ
     όοναΠζɿ
     ࠷దԽख๏ɿNPNFOUVN
    4(%
    ֶश཰ɿ
    ֶश཰ͷݮਰɿ
    
    FQPDIͰ
    NPNFOUVNɿ
     *OOFSNBYɿ1(%
    
    
    
    
    ʣ K K = 10 ϵ = 8/255 α = ϵ/4 ධՁ࣮ݧ 37
38. ධՁࢦඪ 38 Acc = 1 M M ∑ i=1 1 [f(xi ) = yi] 
    Ϋϥε෼ྨਫ਼౓
    
    /BUVSBMͳਫ਼౓ɿਪ࿦σʔλʹର͢ΔධՁ
    
    Ϟσϧͷؤ݈ੑೳɿਪ࿦σʔλΛ߈ܸͨ࣌͠ͷਫ਼౓
    
    ैདྷͷධՁࢦඪ 
    ಛ௃ۭؒͷఆྔతධՁ
    
    
    
    தؒ૚ͷಛ௃ྔʹؔ͢ΔධՁࢦඪ
    
    ɾ$BMJOTLJ
    
    )BSBCBT[ࢦඪ
    
    ɾ4JMIPVFUUFείΞ
    
    ࣄޙ֬཰Λ༻͍ͨධՁࢦඪ
    
    ɾ)PNPHFOFJUZείΞ
    
    ɾ$PNQMFUFOFTTείΞ ˠ
    ෼ྨਫ਼౓͚ͩͰͳ͘ಛ௃ۭؒͷఆྔతධՁ΋ඞཁ Ϟσϧ Ϟσϧ      
39. w Ϋϥελ಺͕ίϯύΫτ͔ͭΫϥελ͕ؒϫΠυͳදݱ
     είΞ͕ߴ͍΄Ͳ༏ΕͨΫϥελ෼ׂ
     είΞ͕ʹ͍ۙ΄Ͳෆద੾ͳΫϥελ෼ׂ $BMJOTLJ
    BOE
    )BSBCBT[ࢦඪ 39 5
    $BMJOTLJ
    BOE
    +
    )BSBCBT[
    l"
    EFOESJUF
    NFUIPE
    GPS
    DMVTUFS
    BOBMZTJTz
    $PNNVOJDBUJPOT
    BOE
    4UBUJDT
     Cal . = tr(Bk ) tr(Wk ) × (nE − k) k − 1 Wk = k ∑ q=1 ∑ x∈Cq ∥x − cq ∥2 2 Bk = k ∑ q=1 nq ∥cq − cE ∥2 2 ɿΫϥε਺ k ɿΫϥε ͷΫϥελ Cq q ɿσʔληοτશମͷத৺ cE ɿΫϥε ͷΫϥελத৺ cq q ɿΫϥε ͷαϯϓϧ਺ nq q cE ɿσʔληοτશମͷαϯϓϧ਺ nE c1 c3 c2 c1 c3 c2 C1 C2 C3 Ϋϥε಺෼ࢄɿΫϥελத৺ͱαϯϓϧͷڑ཭ Ϋϥεؒ෼ࢄɿ֤Ϋϥελத৺ͱશମத৺ͷڑ཭
40. w ೚ҙͷαϯϓϧ͕ద੾ͳΫϥελʹଐͯ͠Δ͔Ͳ͏͔Λද͢ࢦඪ
     είΞ͕ʹ͍ۙ΄Ͳ༏ΕͨΫϥελ෼ׂ
     είΞ͕ʹ͍ۙ΄ͲΫϥελ෼্ׂ͕ख͘Ͱ͖ͯͳ͍ 4JMIPVFUUFείΞ
    40 1
    +
    3PVTTFFVX
    l4JMIPVFUUFT
    B
    (SBQIJDBM
    "JE
    UP
    UIF
    *OUFSQSFUBUJPO
    BOE
    7BMJEBUJPO
    PG
    $MVTUFS
    "OBMZTJTz
    $PNQVUBUJPOBM
    BOE
    "QQMJFE
    .BUIFNBUJDT
     Sil . (i) = b(i) − a(i) max(a(i), b(i)) a(i) = 1 |Cin | − 1 ∑ x( j)∈Cin ∥x(i) − x(j)∥2 b(i) = 1 |Cnear | − 1 ∑ x( j)∈Cnear ∥x(i) − x(j)∥2 ɿ ͕ଐ͢ΔΫϥελ Cin x(i) ɿ ͕ଐ͢ΔΫϥελʹ࠷΋͍ۙΫϥελ Cnear x(i) ɿ ·ͨ͸ ʹؚ·ΕΔ೚ҙͷαϯϓϧ x( j) Cin Cnear x(i) a(i) x(j) Cin Ϋϥελ಺ͷฏۉڑ཭ͷܭࢉ x(i) b(i) x(j) Cnear Ϋϥελؒͷฏۉڑ཭ͷܭࢉ
41. w ΫϥελΛجຊ֓೦ͱͨ͠ࢦඪ
     ֤Ϋϥελʹ͓͍֤ͯσʔλ͕ಉ͡Ϋϥεϥϕϧ͔Λ൑ఆ
     Ϋϥελ಺ʹؚ·ΕΔαϯϓϧͷΫϥε਺͕छྨͷͱ͖είΞ͕ߴ͍ )PNPHFOFJUZείΞ 41 "
    3PTFOCFSH
    BOE
    +
    )JSTDICFSH
    l7.FBTVSF
    "
    $POEJUJPOBM
    &OUSPQZ#BTFE
    &YUFSOBM
    $MVTUFS
    &WBMVBUJPO
    .FBTVSFz
    &./-1$P--
     Ϋϥελ಺ͷαϯϓϧ͕ෳ਺ͷΫϥεΛؚΉ
    ˠ
    )PNPHFOFJUZείΞ͕௿͍ ֤Ϋϥελ಺ͷαϯϓϧ͸Ϋϥε਺͕छྨ
    ˠ
    )PNPHFOFJUZείΞ͕ߴ͍ Homo . = { 1 if H(C, K) = 0 1 − H(C|K) H(C) else H(C|K) = − |K| ∑ k=1 |C| ∑ c=1 ack N log ack ∑ |C| c=1 ack H(C) = − |C| ∑ c=1 ∑ |K| k=1 ack n log ∑ |K| k=1 ack n ɿΫϥεू߹
    C = {ci |i = 1,…, n} ɿΫϥελू߹
    K = {ki |i = 1,…, m} ɿΫϥε ͰΫϥελ ʹؚ·ΕΔσʔλू߹ A = {aij } i j
42. w ֤αϯϓϧʹର͢ΔΫϥεϥϕϧΛجຊ֓೦ͱͨ͠ࢦඪ
     ֤Ϋϥεϥϕϧͷσʔλ͕ಉ͡Ϋϥελ಺ʹ͋Δ͔ΛධՁ
     ͭͷΫϥελʹಉ͡Ϋϥεϥϕϧͷαϯϓϧ͕શؚͯ·ΕΔͱ͖είΞ͕ߴ͍ $PNQMFUFOFTTείΞ 42 Comp . = { 1 if H(K, C) = 0 1 − H(K|C) H(K) else H(K|C) = − |C| ∑ c=1 |K| ∑ k=1 ack N log ack ∑ |K| k=1 ack H(K) = − |K| ∑ k=1 ∑ |C| c=1 ack m log ∑ |C| c=1 ack m "
    3PTFOCFSH
    BOE
    +
    )JSTDICFSH
    l7.FBTVSF
    "
    $POEJUJPOBM
    &OUSPQZ#BTFE
    &YUFSOBM
    $MVTUFS
    &WBMVBUJPO
    .FBTVSFz
    &./-1$P--
     Ϋϥελ಺ʹಉ͡Ϋϥεϥϕϧͷαϯϓϧؚ͕·ΕΔ
    ˠ
    $PNQMFUFOFTTείΞ͕ߴ͍ ಉ͡Ϋϥεϥϕϧͷαϯϓϧ͕ෳ਺ͷΫϥελʹଐ͢Δ
    ˠ
    $PNQMFUFOFTTείΞ͕௿͍ ɿΫϥεू߹
    C = {ci |i = 1,…, n} ɿΫϥελू߹
    K = {ki |i = 1,…, m} ɿΫϥε ͰΫϥελ ʹؚ·ΕΔσʔλू߹ A = {aij } i j
43. w ͲͪΒ͔ͭͰ༏ΕͨΫϥελ෼ׂ͔Λܾఆ͢Δ͜ͱ͸ࠔ೉
     جຊతʹ)PNPHFOFJUZͱ$PNQMFUFOFTT͸૬൓͢Δؔ܎
     ద੾ͳΫϥελ෼ׂ͕Ͱ͖ͨ৔߹ɼͭͷείΞ͕ͱ΋ʹߴ͘ͳΔ )PNPHFOFJUZͱ$PNQMFUFOFTT 43 ֤Ϋϥελ಺ͷαϯϓϧ͸Ϋϥε਺͕छྨ
    
    ಉ͡Ϋϥεϥϕϧͷαϯϓϧ͸ͭͷΫϥελʹશͯଐ͢Δ
    ˠ
    $PNQMFUFOFTTɼ)PNPHFOFJUZͱ΋ʹߴ͍ ˠ
    ͭͷࢦඪΛ߹ΘͤͯධՁ͢Δ͜ͱ͕ඞཁ
44. ੑೳൺֱ 44 Ϋϥε෼ྨੑೳ ಛ௃ۭؒͷఆྔධՁ /BUVSBM '(4. 1(% $8㱣 4JM $BM )PNP $PNQ "5         1$
    MPTT         1$-$         "-1         53"%&4         ."35         ("*3"5         8..3         ."*-         #("5         -#("5         "7NJYVQ         ("5         '"5        
45. ੑೳൺֱɿΫϥε෼ྨੑೳ 45 Ϋϥε෼ྨਫ਼౓Ͱ͸"7NJYVQ͕༏ΕͨੑೳΛൃش Ϋϥε෼ྨੑೳ ಛ௃ۭؒͷఆྔධՁ /BUVSBM '(4. 1(% $8㱣 4JM $BM )PNP $PNQ "5         1$
    MPTT         1$-$         "-1         53"%&4         ."35         ("*3"5         8..3         ."*-         #("5         -#("5         "7NJYVQ         ("5         '"5        
46. ੑೳൺֱɿಛ௃ۭؒͷఆྔධՁ 46 தؒ૚ͷಛ௃ۭؒ͸-#("5͕༏ΕͨΫϥελ෼ׂ
    ˠ
    /BUVSBMαϯϓϧͷಛ௃ྔΛ΋ͱʹֶशͨ͠Ըܙ Ϋϥε෼ྨੑೳ ಛ௃ۭؒͷఆྔධՁ /BUVSBM '(4. 1(% $8㱣 4JM $BM )PNP $PNQ "5         1$
    MPTT         1$-$         "-1         53"%&4         ."35         ("*3"5         8..3         ."*-         #("5         -#("5         "7NJYVQ         ("5         '"5         "5ͷಛ௃ۭؒ
    U4/& -#("5ͷಛ௃ۭؒ
    U4/&
47. 1$
    MPTTʹؔ͢Δߟ࡯ 47 1$
    MPTT͸߈ܸʹର͢Δؤ݈ੑೳ͕௿͍͕ɼ༏Εͨಛ௃දݱΛ࣮ݱ Ϋϥε෼ྨੑೳ ಛ௃ۭؒͷఆྔධՁ /BUVSBM '(4. 1(% $8㱣 4JM $BM )PNP $PNQ "5         1$
    MPTT         1$-$         "-1         53"%&4         ."35         ("*3"5         8..3         ."*-         #("5         -#("5         "7NJYVQ         ("5         '"5         1$
    MPTTͷಛ௃ۭؒ
    U4/&
48. w "EWFSTBSJBM
    BUUBDLʹΑͬͯ༰қʹ$//ͷޡ෼ྨΛ༠ൃՄೳ
     ਂ૚ֶशϕʔεͷΞϓϦέʔγϣϯͷηΩϡϦςΟͷڴҖ
    w "EWFSTBSJBM
    5SBJOJOH
    "5 ͷ܏޲
     /BUVSBMαϯϓϧʹର͢Δੑೳ͕ஶ͘͠ྼԽ
    ˠ
    "5͓͚ΔӬԕͷ՝୊
     Ϋϥεڥքͱͷؔ܎ʹண໨ֶͨ͠शํ๏͕τϨϯυ
    Πϯελϯε͝ͱͷ#PPTUJOH
    ෺ཧతʹΫϥεڥք͔Β཭͢ଛࣦͷ௥Ճ
    ֤αϯϓϧͷ෼ྨ݁ՌΛߟྀͨ͠ଛࣦઃܭ
    w ෼ྨੑೳ͚ͩͰͳ͘ಛ௃ۭؒʹ΋ண໨ͨ͠ख๏ൺֱ
     ߴ͍෼ྨੑೳ
    
    ༏Εͨಛ௃ۭؒ
     ಛ௃ۭؒʹର͢ΔఆྔతͳධՁ΋ඇৗʹॏཁ ≠ ·ͱΊ 48
49. ࣮ݧʹ࢖༻ͨ͠ιʔείʔυ 49 IUUQTHJUIVCDPNNBDIJOFQFSDFQUJPOSPCPUJDTHSPVQ"EWFSTBSJBMUSBJOJOH ˣ࣮ݧʹ࢖༻ͨ͠ίʔυˣ