αʔϕΠɿ"EWFSTBSJBM5SBJOJOH ଍ཱߒنɼฏ઒ཌྷɼࢁԼོٛɼ౻٢߂࿱ʢத෦େֶɾػց஌֮ϩϘςΟΫεݚڀάϧʔϓʣ IUUQNQSHKQ

w "EWFSTBSJBM"UUBDLɿఢରత߈ܸ ೖྗը૾ʹมԽΛՃ͑ͯ$//ͷޡ෼ྨΛ༠ൃ ਓ͕ਖ਼͘͠෼ྨͰ͖ΔΑ͏ͳαϯϓϧͰ$//Λ߈ܸ͍ͨ͠ w "EWFSTBSJBM%FGFOTFɿఢରత๷ޚ $//Λ"EWFSTBSJBMBUUBDLͷ֎త߈ܸ͔Β๷ޚ ͋ΒΏΔ߈ܸ͔Β$//ΛकΓ͍ͨ "EWFSTBSJBM"UUBDLWT%FGFOTF 2 %FGFOEFS "UUBDLFS $// $POWPMVUJPOBM/FVSBM/FUXPSLT

"EWFSTBSJBM"UUBDL%FGFOTFͷ܏޲ 3 %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

w ඍখઁಈΛ෇༩ͨ͠αϯϓϧ "&T"EWFSTBSJBMFYBNQMFT 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

w ઁಈΛεςοϓͰಋग़͢Δख๏ ޯ഑ͷූ߸Λར༻ͯ͠ೖྗը૾ʹมԽΛՃ͑Δ 'BTU(SBEJFOU4JHO.FUIPE '(4. <*(PPEGFMMPX BS9JW > 5 *(PPEGFMMPX et al. l&YQMBJOJOHBOE)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

w Λ༻͍ͯ೚ҙͷճ਺൓෮ͯ͠߈ܸ͢Δ͜ͱͰ'(4.ΑΓ΋ڧ͍ઁಈΛ֫ಘ ͷൣғ֎ͷઁಈ͸౤Өؔ਺ ʹΑͬͯҾ͖໭͢ α ≤ ϵ [−ϵ, ϵ] Π ϵ OPSN ℓ∞ 1SPKFDUFE(SBEJFOU%FTDFOU 1(% <".BESZ *$-3 > 6 ".BESZ et al. l5PXBSET%FFQ-FBSOJOH.PEFMT3FTJTUBOUUP"EWFSTBSJBM"UUBDLz *$-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

w "EWFSTBSJBMBUUBDLΛܰݮ͢ΔͨΊͷςΫχοΫ ೖྗը૾Λૢ࡞͢Δ๷ޚ "&TΛ༻ֶ͍ͨशʹΑΔ๷ޚˠ"EWFSTBSJBMUSBJOJOH "EWFSTBSJBMEFGFOTF 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 ݕग़ثϕ $// ϕ(xi ) ≤ T "EWFSTBSJBMTBNQMF ϕ(xi ) > T f(xi ; θ) %FUFDUJPO લஈʹݕग़ثΛઃஔͯ͠ɼ҆શͳαϯϓϧ͚ͩࣝผϞσϧ΁౉͢ $// x′  f(xnat ; θ) y θt+1 ← θt − η ∂L (f(x′  ; θt ), y) ∂θt 1BSBNFUFSVQEBUF "&TΛ༻͍ͨϞσϧͷߋ৽ "EWFSTBSJBMUSBJOJOH "&TΛϕʔεʹύϥϝʔλΛߋ৽ͯ͠ϞσϧຊମΛ߈ܸʹڧ͘͢Δ ύϥϝʔλΛݻఆͯ͠ "&Tͷ࡞੒ "&TΛ༻͍ͯ ύϥϝʔλͷߋ৽

w "&TΛֶश͢Δ͜ͱͰϞσϧࣗମΛؤ݈ʹ͢Δ๷ޚख๏ "&Tͷ࡞੒ *OOFSNBYJNJ[BUJPO ύϥϝʔλߋ৽ 0VUFSNJOJNJ[BUJPO "EWFSTBSJBM5SBJOJOH<".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 (SBEJFOUEFTDFOE 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 = + (SBEJFOUBTDFOE "&TΛجʹύϥϝʔλߋ৽ "&Tͷ࡞੒

"EWFSTBSJBM5SBJOJOHͷݚڀಈ޲ 9 ࣝผڥքʹண໨ ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ αϯϓϧ͝ͱʹ8FJHIUJOH .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ ஌ࣝͷৠཹ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

"EWFSTBSJBMUSBJOJOHͷݚڀಈ޲ 10 ࣝผڥքʹண໨ ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ αϯϓϧ͝ͱʹ8FJHIUJOH .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ ஌ࣝͷৠཹ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

w ैདྷͷ"5ʹ͓͚Δ໰୊఺ 1(%ͰٻΊͨઁಈ͸ΫϥεڥքΛେ͖͘ލ͙ ޡ෼ྨ͢Δ͔൱͔͸ઁಈͷॳظํ޲ʹґଘ w ઁಈͷ࡞੒ํ๏ʹண໨ͨ͠ख๏ 'SJFOEMZ"EWFSTBSJBM5SBJOJOH '"5 <+;IBOH *$.- > (VJEFE"EWFSTBSJBM5SBJOJOH ("5 <(4SJSBNBOBO /FVS*14 > "EWFSTBSJBM7FSUFYNJYVQ "7NJYVQ <4-FF $713 > ֶ̍शதͷ"&Tͷ࡞੒ํ๏ʹண໨ͨ͠ख๏ 11

w Ϋϥεڥք෇ۙͷઁಈΛٻΊֶͯश FBSMZTUPQQFE1(%Λར༻ͨ͠ઁಈಋग़ '"5<+;IBOH *$.- > 12 '"5'SJFOEMZ"EWFSTBSJBM5SBJOJOH +;IBOH et al. l"UUBDLT8IJDI%POPU,JMM5SBJOJOH.BLF"EWFSTBSJBM-FBSOJOH4USPOHFSz *$.- # PVOEBSZ $MBTTDFOUFS $MBTTDFOUFS 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

w 3FMBYBUJPO߲ͷ௥ՃͰద੾ͳॳظํ޲Λ༻͍ͨ"&Tͷ࡞੒ ޡࠩۂ໘Λ׈Β͔ʹ͢Δ͜ͱͰద੾ͳํ޲ΛٻΊΔ ("5<(4SJSBNBOBO /FVS*14 > 13 ("5(VJEFE"EWFSTBSJBM5SBJOJOH (4SJSBNBOBO et al. l(VJEFE"EWFSTBSJBM"UUBDLGPS&WBMVBUJOHBOE&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

w NJYVQΛར༻ͯ͠/BUVSBMͷੑೳྼԽΛ๷͗ͭͭؤ݈ੑೳΛ޲্ "EWFSTBSJBMWFSUFYʢ܎਺ഒͨ͠"&Tʣͱ/BUVSBMΛ೚ҙͷൺ཰Ͱ಺ૠ "7NJYVQ<4-FF $713 > 14 "7NJYVQ"EWFSTBSJBM7FSUFYNJYVQ 4-FF et al. l"EWFSTBSJBM7FSUFY.JYVQ5PXBSE#FUUFS"EWFSTBSJBMMZ3PCVTU(FOFSBMJ[BUJPOz $713 JOQVUx "EWFSTBSJBMFYBNQMF "EWFSTBSJBMWFSUFY |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) -BCFMTNPPUIJOH ը૾ͷNJYVQ "EWFSTBSJBMWFSUFYͷ࡞੒ ڭࢣ৴߸ͷNJYVQ α ∼ Beta(1,1) ɿTNPPUIJOHύϥϝʔλ λ1 , λ2 ˠ"&Tͷಛ௃Λ܇࿅σʔλͷͭͱͯ͠ٵऩ ˠ܇࿅σʔληοτͷΧόϨοδΛ֦ு

"EWFSTBSJBMUSBJOJOHͷݚڀಈ޲ 15 ࣝผڥքʹண໨ ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ αϯϓϧ͝ͱʹ8FJHIUJOH .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ ஌ࣝͷৠཹ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

w ैདྷͷ"5ʹ͓͚Δ໰୊఺ "&TΛޡ෼ྨ͠ͳ͍Α͏ʹڭࢣ৴߸ͱͷΫϩεΤϯτϩϐʔޡࠩΛ࠷খԽ ˠ"&Tʹաద߹ͯ͠/BUVSBMαϯϓϧΛޡ෼ྨ w Ϟσϧग़ྗʹର͢Δਖ਼ଇԽΛ௥Ճͨ͠ख๏ "EWFSTBSJBM-PHJU1BJSJOH "-1 <),BOOBO BS9JW > 53"%&4<);IBOH *$.- > Ϟσϧͷग़ྗ஋ʹର͢Δਖ਼ଇԽͷ௥Ճ 16

w "&Tͱ/BUVSBMαϯϓϧͷग़ྗ஋ͱҰகͤ͞Δ "&Tͱϕʔεͷ/BUVSBMαϯϓϧ͸ಉ͡ಛ௃Λଊ͑Δ ͳͥਖ਼ଇԽΛ௥Ճ͢Δ͔ 17 #PVOEBSZ "&T $MBTTDFOUFS xi ग़ྗ஋ͷϖΞϦϯά

w "&Tͱ/BUVSBMαϯϓϧͷग़ྗ஋ͱҰகͤ͞Δ "&Tͱϕʔεͷ/BUVSBMαϯϓϧ͸ಉ͡ಛ௃Λଊ͑Δ ϩδοτʹண໨ͨ͠ख๏ 18 "-1<),BOOBO BS9JW > "EWFSTBSJBM-PHJU1BJSJOH ),BOOBO et al. l"EWFSTBSJBM-PHJU1BJSJOHz BS9JW w /BUVSBMͱ"&TͷϩδοτΛҰக ฏۉೋ৐ޡࠩͷ࠷খԽ #PVOEBSZ xi "&T $MBTTDFOUFS L(xi , yi ; θ) = LCE (f(˜ xi ), yi ; θ) + λ k ∥f(xi ) − f(˜ xi )∥2 2 ˢ "&Tʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ ˢ ϩδοτʹର͢Δਖ਼ଇԽ ग़ྗ஋ͷϖΞϦϯά

w "&Tͱ/BUVSBMαϯϓϧͷग़ྗ஋ͱҰகͤ͞Δ "&Tͱϕʔεͷ/BUVSBMαϯϓϧ͸ಉ͡ಛ௃Λଊ͑Δ ࣄޙ෼෍ʹண໨ͨ͠ख๏ 19 "-1<),BOOBO BS9JW > "EWFSTBSJBM-PHJU1BJSJOH ),BOOBO et al. l"EWFSTBSJBM-PHJU1BJSJOHz BS9JW w /BUVSBMͱ"&TͷϩδοτΛҰக ฏۉೋ৐ޡࠩͷ࠷খԽ #PVOEBSZ xi "&T $MBTTDFOUFS L(xi , yi ; θ) = LCE (f(˜ xi ), yi ; θ) + λ k ∥f(xi ) − f(˜ xi )∥2 2 ˢ "&Tʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ ˢ ϩδοτʹର͢Δਖ਼ଇԽ w /BUVSBMͱ"&Tͷࣄޙ֬཰ΛҰக ,-μΠόʔδΣϯεͷ࠷খԽ 53"%&4<);IBOH *$.- > 53BEFP ff JOTQJSFE"EWFSTBSJBM%&GFOTFWJB4VSSPHBUFNJOJNJ[BUJPO L(xi , yi ; θ) = LCE (f(xi ), yi ; θ) + λ ⋅ DKL [p(xi )∥p(˜ xi )] ˢ OBUVSBMʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ );IBOH et al. l5IFPSFUJDBMMZ1SJODJQMFE5SBEFP ff CFUXFFO3PCVTUOFTTBOE"DDVSBDZz *$.- ˢ ࣄޙ֬཰ʹର͢Δਖ਼ଇԽ ग़ྗ஋ͷϖΞϦϯά

"EWFSTBSJBMUSBJOJOHͷݚڀಈ޲ 20 ࣝผڥքʹண໨ ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ αϯϓϧ͝ͱʹ8FJHIUJOH .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ ஌ࣝͷৠཹ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

w ैདྷͷ"5ʹ͓͚Δ໰୊఺ /BUVSBMαϯϓϧͷ෼ྨ݁Ռʹؔ܎ͳ͘ɼશαϯϓϧʹରͯ͠"&TΛఆֶٛͯ͠श ˠޡ෼ྨ͕ੜ͍ͯ͡Δαϯϓϧ͸"5ʹد༩͠ͳ͍ w αϯϓϧͷ෼ྨ݁ՌʹԠͨ͡ଛࣦܭࢉΛߦ͏ख๏ .BY.BSHJO"EWFSTBSJBM5SBJOJOH .." <(8%JOH *$-3 > .JTDMBTTJ fi DBUJPO"XBSF"EWFSTBSJBM5SBJOJOH ."35 <:8BOH *$-3 > ޡ෼ྨͨ͠αϯϓϧͷޮՌతͳར༻ 21

"EWFSTBSJBMUSBJOJOHͷݚڀಈ޲ 22 ࣝผڥքʹண໨ ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ αϯϓϧ͝ͱʹ8FJHIUJOH .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ ஌ࣝͷৠཹ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

w ैདྷͷ"5ʹ͓͚Δ໰୊఺ ΫϥεڥքͱαϯϓϧͷϚʔδϯΛߟֶྀͯ͠श͕͞Ε͍ͯͳ͍ ˠڥքʹ͍ۙαϯϓϧ͸߈ܸͷϦεΫ͕ߴ͍ w Ϋϥεڥք෇͔ۙΒ෺ཧతʹ཭͢ख๏ 1SPUPUZQF$POGPSNJUZMPTT 1$MPTT <".VTUBGB *$$7 > 1SPCBCJMJTUJDBMMZ$PNQBDUMPTTXJUI-PHJU$POTUSBJOUT 1$-$ <9-J """* > ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ 23

w ֶशՄೳͳΫϥεத৺Λ༻͍ͯಛ௃දݱΛૢ࡞ ਖ਼ղΫϥεɿಛ௃ྔͱΫϥεத৺ͷڑ཭Λ࠷খԽ͢Δ͜ͱͰίϯύΫτʹදݱ ෆਖ਼ղΫϥεɿಛ௃ྔͱΫϥεத৺ͷڑ཭Λ࠷େԽ͢Δ͜ͱͰϫΠυʹදݱ 1$MPTT<".VTUBGB *$$7 > 24 ".VTUBGB et al. l"EWFSTBSJBM%FGFOTFCZ3FTUSJDUJOHUIF)JEEFO4QBDFPG%FFQ/FVSBM/FUXPSLTz *$$7 1$MPTT1SPUPUZQF$POGPSNJUZMPTT ˞ ͸ 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 ཭ ͢ ཭͢ ཭͢ ཭ ͢

w ΫϩεΤϯτϩϐʔޡࠩͷܭࢉͳ͠Ͱ1$MPTTͱಉ༷ͷૢ࡞ ਖ਼ղΫϥεͷ֬཰͕೚ҙͷᮢ஋ΛԼճΔαϯϓϧ͸શͯෆਖ਼ղˠίϯύΫτ͔ͭϫΠυͳදݱ ਖ਼ղΫϥεͱྡ઀Ϋϥε֬཰ͷ͕ࠩৗʹਖ਼Λཁ੥ˠΫϥε͕ަࠩ͢Δ͜ͱΛ๷ࢭ 1$-$<9-J """* > 25 9-J et al. l*NQSPWJOH"EWFSTBSJBM3PCVTUOFTTWJB1SPCBCJMJTUJDBMMZ$PNQBDU-PTTXJUI-PHJU$POTUSBJOUTz """* 1$-$1SPCBCJMJTUJDBMMZ$PNQBDUMPTTXJUI-PHJU$POTUSBJOU 1$-$ͷଛࣦؔ਺ L(xi , yi ; θ) = max (0,py′  (˜ xi ) + τ − pyi (˜ xi )) + λ (max (0,zyi (˜ xi ) − zy′  (˜ xi )) − C′  ) #PVOEBSZ $MBTTDFOUFS $MBTTDFOUFS τ ˢ ෼ྨڥքΛਖ਼ղΫϥεଆʹ ͚ͩҠಈ τ ˢ ྡ઀ΫϥεͱͷަࠩΛ๷ࢭ

w ΫϩεΤϯτϩϐʔޡࠩͷܭࢉͳ͠Ͱ1$MPTTͱಉ༷ͷૢ࡞ ਖ਼ղΫϥεͷ֬཰͕೚ҙͷᮢ஋ΛԼճΔαϯϓϧ͸શͯෆਖ਼ղˠίϯύΫτ͔ͭϫΠυͳදݱ ਖ਼ղΫϥεͱྡ઀Ϋϥε֬཰ͷ͕ࠩৗʹਖ਼Λཁ੥ˠΫϥε͕ަࠩ͢Δ͜ͱΛ๷ࢭ 1$-$<9-J """* > 26 9-J et al. l*NQSPWJOH"EWFSTBSJBM3PCVTUOFTTWJB1SPCBCJMJTUJDBMMZ$PNQBDU-PTTXJUI-PHJU$POTUSBJOUTz """* 1$-$1SPCBCJMJTUJDBMMZ$PNQBDUMPTTXJUI-PHJU$POTUSBJOU 1$-$ͷଛࣦؔ਺ L(xi , yi ; θ) = max (0,py′  (˜ xi ) + τ − pyi (˜ xi )) + λ (max (0,zyi (˜ xi ) − zy′  (˜ xi )) − C′  ) #PVOEBSZ $MBTTDFOUFS $MBTTDFOUFS τ ˢ ෆਖ਼ղαϯϓϧ ˢ ෼ྨڥքΛਖ਼ղΫϥεଆʹ ͚ͩҠಈ τ ˢ ྡ઀ΫϥεͱͷަࠩΛ๷ࢭ

w ΫϩεΤϯτϩϐʔޡࠩͷܭࢉͳ͠Ͱ1$MPTTͱಉ༷ͷૢ࡞ ਖ਼ղΫϥεͷ֬཰͕೚ҙͷᮢ஋ΛԼճΔαϯϓϧ͸શͯෆਖ਼ղˠίϯύΫτ͔ͭϫΠυͳදݱ ਖ਼ղΫϥεͱྡ઀Ϋϥε֬཰ͷ͕ࠩৗʹਖ਼Λཁ੥ˠΫϥε͕ަࠩ͢Δ͜ͱΛ๷ࢭ 1$-$<9-J """* > 27 9-J et al. l*NQSPWJOH"EWFSTBSJBM3PCVTUOFTTWJB1SPCBCJMJTUJDBMMZ$PNQBDU-PTTXJUI-PHJU$POTUSBJOUTz """* 1$-$1SPCBCJMJTUJDBMMZ$PNQBDUMPTTXJUI-PHJU$POTUSBJOU 1$-$ͷଛࣦؔ਺ L(xi , yi ; θ) = max (0,py′  (˜ xi ) + τ − pyi (˜ xi )) + λ (max (0,zyi (˜ xi ) − zy′  (˜ xi )) − C′  ) #PVOEBSZ $MBTTDFOUFS $MBTTDFOUFS τ ˢ ෆਖ਼ղαϯϓϧ ˢ ෼ྨڥքΛਖ਼ղΫϥεଆʹ ͚ͩҠಈ τ ˢ ྡ઀ΫϥεͱͷަࠩΛ๷ࢭ

"EWFSTBSJBMUSBJOJOHͷݚڀಈ޲ 28 ࣝผڥքʹண໨ ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ αϯϓϧ͝ͱʹ8FJHIUJOH .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ ஌ࣝͷৠཹ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

w ैདྷͷ"5ʹ͓͚Δ໰୊఺ શαϯϓϧΛ౳͘͠ଛࣦܭࢉ͢ΔͨΊɼଛࣦ͕େ͖͍αϯϓϧʹґଘͯ͠ޡ͕ࠩఀ଺ ˠϞσϧશମͷੑೳྼԽ w ΫϥεڥքͱͷزԿతͳڑ཭ؔ܎͔ΒαϯϓϧͷॏΈ෇͚Λ͢Δख๏ (FPNFUSZ"XBSF*OTUBODF3FXFJHIUFE"EWFSTBSJBM5SBJOJOH ("*3"5 <+;IBOH *$-3 > 8FJHIUFE.JOJ.BY3JTL 8..3 <);FOH """* > .BSHJO"XBSF*OTUBODF3FXFJHIUJOH-FBSOJOH ."*- <28BOH /FVS*14 > #JMFWFM-FBSOBCMF"EWFSTBSJBM3FXFJHIUJOH #J-"8 <$)PMU[ 0QFO3FWJFX > αϯϓϧ͝ͱͷ8FJHIUJOHͷར༻ 29 Ϟσϧશମͷ࠷దԽࣜ ֤αϯϓϧͷଛࣦʹର͢ΔॏΈ෇͚ ω(xi , yi ) min θ 1 n n ∑ i=1 L(f(˜ xi ; θ), yi )

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"EWFSTBSJBM5SBJOJOH 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 $MBTTDFOUFS $MBTTDFOUFS x1 x2 x3 ڥքͱͷϚʔδϯ

w Ϋϥε֬཰Λ༻͍ͯڥք·ͰͷزԿతͳڑ཭Λఆٛ ූ߸͕ਖ਼͔ͭϚʔδϯ͕େ͖͍ˠখ͞ͳॏΈɿॏཁ౓͕௿͍ ූ߸͕ਖ਼͔ͭϚʔδϯ͕খ͍͞ˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ූ߸͕ෛˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ࣄޙ෼෍ΛϕʔεʹϚʔδϯΛܾఆ͢Δख๏ 31 $MBTTDFOUFS #PVOEBSZ $MBTTDFOUFS p = 0.7 p = 0.3 p = 0.8 p = 0.2

w Ϋϥε֬཰Λ༻͍ͯڥք·ͰͷزԿతͳڑ཭Λఆٛ ූ߸͕ਖ਼͔ͭϚʔδϯ͕େ͖͍ˠখ͞ͳॏΈɿॏཁ౓͕௿͍ ූ߸͕ਖ਼͔ͭϚʔδϯ͕খ͍͞ˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ූ߸͕ෛˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ࣄޙ෼෍ΛϕʔεʹϚʔδϯΛܾఆ͢Δख๏ 32 8..3<);FOH """* > 8FJHIUFE.JOJ.BY3JTL w ࢦ਺ؔ਺ʹΑΔॏΈఆٛ ޡ෼ྨͨ͠αϯϓϧ΄Ͳେ͖ͳॏΈ͚ͮ $MBTTDFOUFS #PVOEBSZ $MBTTDFOUFS p = 0.7 p = 0.3 p = 0.8 p = 0.2 ω(xi , yi ) = exp (−α ⋅ margin (˜ xi , yi)) margin (˜ xi , yi) = pyi (˜ xi ) − max j≠yi pj (˜ xi ) 8FJHIUJOHؔ਺ );FOH et al. l"SF"EWFSTBSJBM&YBNQMFT$SFBUFE&RVBM "-FBSOBCMF8FJHIUFE.JOJNBY3JTLGPS3PCVTUOFTTVOEFS/POVOJGPSN"UUBDLTz """*

w Ϋϥε֬཰Λ༻͍ͯڥք·ͰͷزԿతͳڑ཭Λఆٛ ූ߸͕ਖ਼͔ͭϚʔδϯ͕େ͖͍ˠখ͞ͳॏΈɿॏཁ౓͕௿͍ ූ߸͕ਖ਼͔ͭϚʔδϯ͕খ͍͞ˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ූ߸͕ෛˠେ͖ͳॏΈɿॏཁ౓͕ߴ͍ ࣄޙ෼෍ΛϕʔεʹϚʔδϯΛܾఆ͢Δख๏ 33 8..3<);FOH """* > 8FJHIUFE.JOJ.BY3JTL w ࢦ਺ؔ਺ʹΑΔॏΈఆٛ ޡ෼ྨͨ͠αϯϓϧ΄Ͳେ͖ͳॏΈ͚ͮ $MBTTDFOUFS #PVOEBSZ $MBTTDFOUFS p = 0.7 p = 0.3 p = 0.8 p = 0.2 ω(xi , yi ) = exp (−α ⋅ margin (˜ xi , yi)) margin (˜ xi , yi) = pyi (˜ xi ) − max j≠yi pj (˜ xi ) 8FJHIUJOHؔ਺ );FOH et al. l"SF"EWFSTBSJBM&YBNQMFT$SFBUFE&RVBM "-FBSOBCMF8FJHIUFE.JOJNBY3JTLGPS3PCVTUOFTTVOEFS/POVOJGPSN"UUBDLTz """* w 4JHNPJEؔ਺ʹΑΔॏΈఆٛ ϛχόον಺ͰॏΈΛਖ਼نԽ ."*-<28BOH /FVS*14 > .BSHJO"XBSF*OTUBODF3FXFJHIUJOH-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਺ 28BOH et al. l1SPCBCJMJTUJD.BSHJOTGPS*OTUBODF3FXFJHIUJOHJO"EWFSTBSJBM5SBJOJOHz /FVS*14

"EWFSTBSJBMUSBJOJOHͷݚڀಈ޲ 34 ࣝผڥքʹண໨ ֶशதͷ"&Tͷ࡞੒ํ๏ʹண໨ αϯϓϧ͝ͱʹ8FJHIUJOH .JTMFBEͨ͠αϯϓϧͷ׆༻ ௿࣍ݩಛ௃ۭؒ ೖྗը૾ۭؒ %FGFOTF "UUBDL "5(*' <$$IFO BS9JW`> *"5 <"-BNC "*4FD`> .5-"5 <"-BVHSPT &$$7`> %"5 <:8BOH *$.-`> 'FBUVSFTDBUUFS <);IBOHBOE+8BOH /FVS*14`> $"5 <2;$BJ *+$"*`> 6OJWFSTBM"5 <"4IBGBIJ """*`> "7NJYVQ <4-FF $713`> ("5 <(4SJSBNBOBO /FVS*14`> ("*3"5 <+;IBOH *$-3`> ."*- <28BOH /FVS*14`> 8..3 <);FOH """*`> #J-"8 <$)PMU[ 0QFO3FWJFX`> -#("5 <+$VJ *$$7`> #("5 <+$VJ *$$7`> 1$-$ <9-J """*`> 1$MPTT <".VTUBGB *$$7`> $//X3#'LFSOFM <4"5BHIBOBLJ $713`> "-1 <),BOOBO BS9JW`> ."35 <:8BOH *$-3`> ..-%" <51BOH *$.-`> ..$ <51BOH *$-3`> -#'(4 <$4[FHFEZ *$-3`> '(4. <*(PPEGFMMPX *$-3`> $8 $BSMJOJBOE%8BHOFS 41`> 1(% <".BESZ *$-3`> #*. *-$. ઁಈͷ࠷దԽ .VMUJQMFUBSHFUFE <4(PXBM BS9JW`> 6OJWFSTBM"&T <4..PPTBWJ%F[GPPMJ $713`> "VUP"UUBDL <'$SPDFBOE.)FJO *$.-`> '"# <'$SPDFBOE.)FJO *$.-`> 4RVBSF"UUBDL <""OESJVTIDIFOLP &$$7`> "VUP1(% "VUP1(%%-3 4"*'(. <9%POH $713`> ("." <(4SJSBNBOBO /FVS*14`> .VMUJεςοϓͷ࣌୅ <",VSBLJO *$-38`> ଛࣦؔ਺΍ωοτϫʔΫઃܭͷվળ Ϟσϧग़ྗʹର͢Δਖ਼ଇԽ ܭࢉίετͷ࡟ݮ ࣄޙ֬཰ͷਖ਼ଇԽ ஌ࣝͷৠཹ 4UBOEBSE"5 <*(PPEGFMMPX *$-3`> "5XJUI1(% <".BESZ *$-3`> ˢ /BUVSBMͱ"&Tͷ࢖༻ 1(%BUUBDLͷ"&TΛར༻ ˢ "&TͷΈ μΠφϛοΫͳ εςοϓαΠζͷಋೖ 0VUFSNJOJNJ[BUJPOͷଛࣦܭࢉ .." <(8%JOH *$-3`> 53"%&4 <);IBOH *$.-`> '"5 <+;IBOH *$.-`> )"5 <33BEF *$.-8`>

w ैདྷͷ"5ʹ͓͚Δ໰୊఺ ௨ৗͷ"5Ͱ͸"&TʹϑΟοτͨ͠ΫϥεڥքΛ֫ಘ ˠ/BUVSBMαϯϓϧʹର͢Δޡ෼ྨ͕ଟൃ w /BUVSBMαϯϓϧͷΫϥεڥքΛΨΠυʹͨ͠ख๏ #PVOEBSZ(VJEFE"EWFSTBSJBM5SBJOJOH #("5 <+$VJ *$$7 > -FBSOBCMF#PVOEBSZ(VJEFE"EWFSTBSJBM5SBJOJOH -#("5 <+$VJ *$$7 > ஌ࣝͷৠཹ 35 ௨ৗͷֶशΛͨ͠Ϟσϧ "5Λͨ͠Ϟσϧ

w /BUVSBMͷΫϥεڥքΛΨΠυʹͨ͠"5 /BUVSBMͷ஌ࣝΛϩόετͳϞσϧ΁ܧঝ "&Tʹର͢ΔΫϩεΤϯτϩϐʔޡࠩ͸ܭࢉ͠ͳ͍ -#("5<+$VJ *$$7 > 36 +$VJ et al. l-FBSOBCMF#PVOEBSZ(VJEFE"EWFSTBSJBM5SBJOJOHz *$$7 -#("5-FBSOBCMF#PVOEBSZ(VJEFE"EWFSTBSJBM5SBJOJOH min θ 𝔼 (xi ,yi )∈D L (ℳrob (xadv i ), ℳnat (xi )) #PVOEBSZ(VJEFE"EWFSTBSJBM5SBJOJOH 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"EWFSTBSJBM5SBJOJOH ͱ Λॳظ஋͔Βڞಉֶश ˞ ʹؔͯ͠ΫϩεΤϯτϩϐʔޡࠩ͸࢖༻͠ͳ͍ ℳnat ℳrob ℳrob

w ϞσϧͱσʔληοτΛ౷Ұͯ͠෼ྨਫ਼౓ͱ߈ܸʹର͢Δؤ݈ੑΛൺֱ ֶश৚݅͸جຊతʹ࿦จهࡌͷ஋Λར༻ w ڞ௨ͷ৚݅ σʔληοτɿ$*'"3 ϕʔεϞσϧɿ8JEF3FT/FU w ֶश৚݅ʢ࿦จʹهࡌ͕ͳ͍৔߹ʣ ֶशճ਺ɿ όοναΠζɿ ࠷దԽख๏ɿNPNFOUVN4(% ֶश཰ɿ ֶश཰ͷݮਰɿ FQPDIͰ NPNFOUVNɿ *OOFSNBYɿ1(% ʣ K K = 10 ϵ = 8/255 α = ϵ/4 ධՁ࣮ݧ 37

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

w Ϋϥελ಺͕ίϯύΫτ͔ͭΫϥελ͕ؒϫΠυͳදݱ είΞ͕ߴ͍΄Ͳ༏ΕͨΫϥελ෼ׂ είΞ͕ʹ͍ۙ΄Ͳෆద੾ͳΫϥελ෼ׂ $BMJOTLJBOE)BSBCBT[ࢦඪ 39 5$BMJOTLJBOE+)BSBCBT[ l"EFOESJUFNFUIPEGPSDMVTUFSBOBMZTJTz $PNNVOJDBUJPOTBOE4UBUJDT 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཭

w ೚ҙͷαϯϓϧ͕ద੾ͳΫϥελʹଐͯ͠Δ͔Ͳ͏͔Λද͢ࢦඪ είΞ͕ʹ͍ۙ΄Ͳ༏ΕͨΫϥελ෼ׂ είΞ͕ʹ͍ۙ΄ͲΫϥελ෼্ׂ͕ख͘Ͱ͖ͯͳ͍ 4JMIPVFUUFείΞ 40 1+3PVTTFFVX l4JMIPVFUUFTB(SBQIJDBM"JEUPUIF*OUFSQSFUBUJPOBOE7BMJEBUJPOPG$MVTUFS"OBMZTJTz $PNQVUBUJPOBMBOE"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

w ΫϥελΛجຊ֓೦ͱͨ͠ࢦඪ ֤Ϋϥελʹ͓͍֤ͯσʔλ͕ಉ͡Ϋϥεϥϕϧ͔Λ൑ఆ Ϋϥελ಺ʹؚ·ΕΔαϯϓϧͷΫϥε਺͕छྨͷͱ͖είΞ͕ߴ͍ )PNPHFOFJUZείΞ 41 "3PTFOCFSHBOE+)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

w ֤αϯϓϧʹର͢ΔΫϥεϥϕϧΛجຊ֓೦ͱͨ͠ࢦඪ ֤Ϋϥεϥϕϧͷσʔλ͕ಉ͡Ϋϥελ಺ʹ͋Δ͔ΛධՁ ͭͷΫϥελʹಉ͡Ϋϥεϥϕϧͷαϯϓϧ͕શؚͯ·ΕΔͱ͖είΞ͕ߴ͍ $PNQMFUFOFTTείΞ 42 Comp . = { 1 if H(K, C) = 0 1 − H(K|C) H(K) else H(K|C) = − |C| ∑ c=1 |K| ∑ k=1 ack N log ack ∑ |K| k=1 ack H(K) = − |K| ∑ k=1 ∑ |C| c=1 ack m log ∑ |C| c=1 ack m "3PTFOCFSHBOE+)JSTDICFSH l7.FBTVSF"$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

w ͲͪΒ͔ͭͰ༏ΕͨΫϥελ෼ׂ͔Λܾఆ͢Δ͜ͱ͸ࠔ೉ جຊతʹ)PNPHFOFJUZͱ$PNQMFUFOFTT͸૬൓͢Δؔ܎ ద੾ͳΫϥελ෼ׂ͕Ͱ͖ͨ৔߹ɼͭͷείΞ͕ͱ΋ʹߴ͘ͳΔ )PNPHFOFJUZͱ$PNQMFUFOFTT 43 ֤Ϋϥελ಺ͷαϯϓϧ͸Ϋϥε਺͕छྨಉ͡Ϋϥεϥϕϧͷαϯϓϧ͸ͭͷΫϥελʹશͯଐ͢Δ ˠ$PNQMFUFOFTTɼ)PNPHFOFJUZͱ΋ʹߴ͍ ˠͭͷࢦඪΛ߹ΘͤͯධՁ͢Δ͜ͱ͕ඞཁ

ੑೳ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

ੑೳ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

ੑೳ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/&

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/&

w "EWFSTBSJBMBUUBDLʹΑͬͯ༰қʹ$//ͷޡ෼ྨΛ༠ൃՄೳ ਂ૚ֶशϕʔεͷΞϓϦέʔγϣϯͷηΩϡϦςΟͷڴҖ w "EWFSTBSJBM5SBJOJOH "5 ͷ܏޲ /BUVSBMαϯϓϧʹର͢Δੑೳ͕ஶ͘͠ྼԽˠ"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

࣮ݧʹ࢖༻ͨ͠ιʔείʔυ 49 IUUQTHJUIVCDPNNBDIJOFQFSDFQUJPOSPCPUJDTHSPVQ"EWFSTBSJBMUSBJOJOH ˣ࣮ݧʹ࢖༻ͨ͠ίʔυˣ

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