差分プライバシーによる安全な連合学習の実現

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-:$PSQPSBUJPO ࠩ෼ϓϥΠόγʔʹΑΔ ҆શͳ࿈߹ֶशͷ࣮ݱ ∁ڮ ཌྷ c5TVCBTB 5",")"4)* -*/&Ϡϑʔגࣜձࣾ 1SJWBDZ5SVTU5FBN'FEFSBUFE-FBSOJOH5FBN -*/&Ϡϑʔݚڀॴ ্੮ݚڀһ UTVCBTBUBLBIBTIJ!MZDPSQDPKQ

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-:$PSQPSBUJPO ࿈߹ֶशͱϓϥΠόγʔ σʔλ׆༻࣌ͷϓϥΠόγʔϦεΫ %JGGFSFOUJBM1SJWBDZʢࠩ෼ϓϥΠόγʔʣ ࠩ෼ϓϥΠόγʔʹΑΔ҆શͳ࿈߹ֶश ۙ೥ͷൃలతͳ࿩୊ʢ͕࣌ؒ͋Ε͹ʣ -*/&ϠϑʔͰͷݚڀ։ൃͷࣄྫʹ͍ͭͯ΋ॴʑͰ؆ܿʹ঺հ͠·͢ ຊߨٛͷίϯςϯπ

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-:$PSQPSBUJPO ࿈߹ֶश 'FEFSBUFE-FBSOJOH • ΫϥΠΞϯτ܈ͱαʔόʔͱ͕ڠௐ͢Δ෼ࢄܕͷػցֶशʢ$SPTTEFWJDF'-Λ૝ఆʣ • ΫϥΠΞϯτͷσʔλ͔Βܭࢉͨ͠ߋ৽৘ใ ޯ഑ ͚ͩΛαʔόʔʹఏڙ • σʔλ͸ΫϥΠΞϯτ಺ʹཹ·Δ à ϓϥΠόγʔʹ഑ྀ /POQBSUJDJQBOUTPG'- (MPCBM.PEFM

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-:$PSQPSBUJPO ࿈߹ֶशͷಋೖࣄྫ IUUQTBDMBOUIPMPHZPSHBDMJOEVTUSZQEG 'FEFSBUFE-FBSOJOHPG(CPBSE -BOHVBHF.PEFM (PPHMF

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-:$PSQPSBUJPO ʮ-*/&ʯΞϓϦͷελϯϓαδΣετ • ʮ͋Γ͕ͱ͏ʯ౳ͷจࣈΛೖྗͨ͠ࡍʹɺҙຯͷ͍ۙελϯϓΛਪનදࣔ͢Δػೳ • ΑΓߴ͍ਫ਼౓ͰϢʔβʔͷ޷Έʹ͋ͬͨఏҊ͢ΔͨΊɺ࿈߹ֶशͱࠩ෼ϓϥΠόγʔΛಋೖ IUUQTXXXZPVUVCFDPNXBUDI WL5#TIH0C IUUQTUFDIWFSTFNFKBTFTTJPOT IUUQTMJOFDPSQDPNKBTFDVSJUZBSUJDMF

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-:$PSQPSBUJPO ޯ഑͔Βݩͷσʔλ͸෮ݩͰ͖Δ͔ʁ l*OWFSUJOH(SBEJFOUT )PXFBTZJTJUUPCSFBLQSJWBDZJOGFEFSBUFEMFBSOJOH z IUUQTBSYJWPSHBCT :FT (SBEJFOU*OWFSTJPO߈ܸ

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-:$PSQPSBUJPO 1SJWBDZ&OIBODJOH5FDIOPMPHJFT 1&5T 5&& 5SVTUFE&YFDVUJPO&OWJSPONFOU 4.1$ 4FDVSF.VMUJQBSUZ$PNQVUBUJPO 'FEFSBUFE-FBSOJOH %JGGFSFOUJBM1SJWBDZ 4FDVSF$PNQVUJOH

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-:$PSQPSBUJPO 1SJWBDZ&OIBODJOH5FDIOPMPHJFT 1&5T 1&5Tʹ͸ɺ ҉߸౳ʹΑΔΞΫηε੍ޚٕज़ͱ ৘ใͦͷ΋ͷΛՃ޻͢Δٕज़ ͕͋Δ ࿈߹ֶशɾ ہॴࠩ෼ϓϥΠόγʔ ಗ໊Խɾσʔλ߹੒ ࠩ෼ϓϥΠόγʔ ੺͍఺ઢΑΓࠨଆ͕ϓϥΠόγʔอޢͷൣғ FH ಗ໊Խ΍ࠩ෼ϓϥΠόγʔ FH ࿈߹ֶश΍ൿີܭࢉ

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-:$PSQPSBUJPO σʔλ׆༻࣌ͷϓϥΠόγʔϦεΫ

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-:$PSQPSBUJPO /FUGMJY1SJ[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 """ ### %%% *UTFFNTYYY B ::: 999 "OPOZNJ[FE/FUGMJY.PWJFSBOLJOHEBUB /POBOPOZNPVT*.%#NPWJFSBUJOH +0*/

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-:$PSQPSBUJPO ࠩ෼߈ܸ ͋Δσʔλϕʔε͔Βੜ੒ͨ͠ෳ਺ͷ౷ܭ৘ใͷʮࠩ෼ʯ͔Βಛఆͷݸਓʹؔ͢Δ৘ใΛਪఆ BWHTBMBSZ ສԁ BWHTBMBSZ ສԁ ʲ2VJ[ʳ"MJDFͷ೥ऩ͸ʁ ʜ ʜ ʜ ʜ "MJDF "MJDF͕ୀ৬ FOHJOFFST FOHJOFFST ೥ ೥

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-:$PSQPSBUJPO 2VJ[ͷճ౴ BWHTBMBSZ ສԁ BWHTBMBSZ ສԁ ʲ2VJ[ʳ"MJDFͷ೥ऩ͸ʁ ʜ ʜ ʜ ʜ "MJDF "MJDF͕ୀ৬ FOHJOFFST FOHJOFFST ೥ ೥ Yr Y ສԁ ౴͑

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-:$PSQPSBUJPO ࠩ෼߈ܸͷࣄྫ 'BDFCPPLͷ1**CBTFE5BSHFUJOH ݱࡏ͸मਖ਼ࡁΈ • ޿ࠂλʔήςΟϯάͷޮՌΛܭଌ͢ΔͨΊͷ౷ܭ෼ੳ"1* • ։੍ࣔޚͷͨΊʹᮢ஋ʹΑΔอޢΛಋೖ͍͕ͯͨ͠ɺόΠύε͞Εͨ • ి࿩൪߸΍ϝʔϧΞυϨεɺ8FCӾཡཤྺ͕࿙Ӯ IUUQTXXXZPVUVCFDPNXBUDI W-Q *X:WY(QL IUUQTXXXGUDHPWTZTUFNGJMFTEPDVNFOUTQVCMJD@FWFO UTQQSJWBDZDPONJTMPWF@QEG

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-:$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

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-:$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 &TUJNBUFBT JOUSBJOEBUB &TUJNBUFBT PVUPGUSBJOEBUB "UUBDL.PEFMͷ܇࿅ʹ͸ɺ5BSHF.PEFMΛ໛฿ͨ͠4IBEPX.PEFMΛར༻ ϝϯόʔγοϓਪ࿦΁ͷରࡦͱͯ͠ɺࠩ෼ϓϥΠόγʔͷอূ͕༗ޮ

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-:$PSQPSBUJPO Ϟσϧ౪༻߈ܸ .PEFM4UFBMJOH λʔήοτϞσϧ͔Βͷग़ྗͱͦͷೖྗΛ༻͍ͯɺڍಈΛ໛฿ͨ͠ϞσϧΛߏங 4ZOUIFUJD *OQVUT 0VUQVUT 5BSHFU.PEFM *NJUBUJPO.PEFM 4IBEPX.PEFM

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-:$PSQPSBUJPO ػցֶशٕज़ʹର͢ΔϓϥΠόγʔอޢٕज़͕ඞཁ TBNQMF𝑥 ϓϥΠόγʔอޢ͞Εͨ ػցֶशϞσϧ Ϟσϧ͔Βͷग़ྗ ܇࿅σʔλͷਪఆ͕ࠔ೉ ܇࿅σʔλ TBNQMF𝑥 ػցֶशϞσϧ Ϟσϧ͔Βͷग़ྗ ܇࿅σʔλ ʮࠩ෼ϓϥΠόγʔʯʹΑΔ ϓϥΠόγʔอޢػցֶश

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-:$PSQPSBUJPO ࠩ෼ϓϥΠόγʔ

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-:$PSQPSBUJPO ࠩ෼ϓϥΠόγʔͱ͸ʁ ϊΠζʢ ʣ͕Ճࢉ͞Ε͍ͯΔͱϓϥΠόγʔอޢ͞Ε͍ͯΔΑ͏ʹײ͡·ͤΜ͔ʁ ౷ܭత ͳग़ྗ ϊΠζ ΛՃࢉ 𝜖 ∞ TUSPOH XFBL ʜ ࠩ෼ϓϥΠόγʔ͕ఏڙ͢Δ΋ͷ • ϊΠζͷՃࢉʹର͢Δʢ͋ΔલఏͰͷʣ ཧ࿦తͳϓϥΠόγʔอޢͷई౓ • ॴఆͷϓϥΠόγʔڧ౓ 𝜖 ͷ ୡ੒ʹඞཁͳϊΠζͷಋग़

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-:$PSQPSBUJPO ͭͷओཁͳϓϥΠόγʔϞσϧ $FOUSBM %JGGFSFOUJBM1SJWBDZ -PDBM%JGGFSFOUJBM1SJWBDZ ୈ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 • 'VMM63-TEBUBTFU .FUB • /FYUXPSEQSFEJDUJPOJO(CPBSE (PPHMF • "VEJFODF&OHBHFNFOU"1* -JOLFE*O ·ͣ͸ͪ͜Β͔Β ʢ͝ࢀߟʣ"MJTUPGSFBMXPSMEVTFPGEJGGFSFOUJBMQSJWBDZIUUQTEFTGPOUBJOFTQSJWBDZSFBMXPSMEEJGGFSFOUJBMQSJWBDZIUNM

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-:$PSQPSBUJPO 𝝐, 𝜹 %JGGFSFOUJBM1SJWBDZ ϝΧχζϜ ℳ: 𝒟 → 𝒮 ͕ 𝜖, 𝛿 %1Λຬͨ͢ͱ͸ɺྡ઀σʔλϕʔε 𝐷, 𝐷) ∈ 𝒟 ͓Αͼ೚ҙͷग़ྗͷू߹ 𝑆 ⊆ 𝒮 ʹରͯ͠ҎԼ͕੒Γཱͭͱ͖Ͱ͋Δ %BUBCBTF𝑫 0VUQVU 𝜖 ∞ TUSPOH XFBL ʜ Pr ℳ 𝐷 ∈ 𝑆 ≤ exp 𝜖 Pr ℳ 𝐷! ∈ 𝑆 + 𝛿 ℳ 𝑫′ɿ BEKBDFOUPG𝑫 ℳ 𝜖 ≥ 0, 𝛿 ∈ 0,1 ) ೖྗ͕มԽͯ͠΋ग़ྗʹ͸΄ͱΜͲʢexp 𝜖 ఔ౓͔͠ʣӨڹ͕ͳ͍ %XPSL%JGGFSFOUJBMQSJWBDZ*$"-1

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-:$PSQPSBUJPO -BQMBDFϝΧχζϜ ࠷΋Α͘஌ΒΕͨϥϯμϜԽॲཧΛ࣮ݱ͢ΔϝΧχζϜʢ𝛿 = 0Λ૝ఆʣ ℳ 𝐷 = 𝑓 𝐷 + Lap 0, Δ- 𝜖 -BQMBDF.FDIBOJTN 𝜖 = 10, Δ! = 1 𝜖 = 1, Δ! = 1 ฏۉɿ ඪ४ภࠩɿ 2 "! # ͷ ϥϓϥε෼෍͔ΒϊΠζΛαϯϓϦϯά Δ. = sup /,/!∈𝒟 𝑓 𝐷 − 𝑓 𝐷) 3 ℓ𝟏 TFOTJUJWJUZ Δ45678 = 1 Δ9:;7 = 1 𝑛 ˞૝ఆ͢Δྡ઀%#ʹґଘ &Y 4FOTJUJWJUZ𝚫𝒇 ೚ҙͷҰϨίʔυͷมԽ˞͕ؔ਺ͷग़ྗʹӨڹΛ༩͑Δ౓߹͍ %XPSL%JGGFSFOUJBMQSJWBDZ*$"-1

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-:$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@|

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-:$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

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-:$PSQPSBUJPO 6TFSMFWFM1SJWBDZ&WFOUMFWFM1SJWBDZ 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਺ NBYDPOUSJCVUJPO Λ੍ݶ͢Δ͜ͱ͕͋Δ &WFOU %JTFBTF ౶೘ප ന಺ো ഏ͕Μ ң͕Μ ΠϯϑϧΤϯβ ң௵ᙾ ेೋࢦ௎௵ᙾ ௎ด࠹ ϨίʔυFWFOU 6TFSMFWFM1SJWBDZ &WFOUMFWFM1SJWBDZ

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-:$PSQPSBUJPO ࠩ෼ϓϥΠόγʔͷ࣮૷ ώετάϥϜʹ-BQMBDFϝΧχζϜΛద༻͢Δྫ ϓϥΠόγʔϞσϧ • อޢ୯Ґɿ6TFSMFWFM Ϩίʔυਓ • ྡ઀%#ɿBEESFNPWBM

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-:$PSQPSBUJPO -BQMBDFϝΧχζϜͷڍಈ ϥϯμϜੑΛ࣋ͬͨڍಈΛ͢ΔͨΊɺಉ͡ઃఆͰ΋ຖճग़ྗ͕ҟͳΔ 𝜖 = 1, Δ% = 1 𝜖 = 1, Δ% = 1 𝜖 = 1, Δ% = 1

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-:$PSQPSBUJPO -BQMBDFϝΧχζϜͷڍಈ ϓϥΠόγʔύϥϝʔλ𝜖Λมಈʢ𝜖খ à ϓϥΠόγʔอޢڧʣ 𝜖 = 0.1 𝜖 = 0.5 𝜖 = 2 𝜖 = 0.05 𝜖 = 10 Δ% = 1

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-:$PSQPSBUJPO ࠩ෼ϓϥΠόγʔͷղऍɿ౷ܭతԾઆݕఆͷࢹ఺ ϥϯμϜԽ͞Εͨग़ྗΛ؍ଌͨ͠ͱ͖ʹɺݩͷೖྗ %PS%`Λ౰ͯΒΕΔ͔ à ܦݧతͳ ࠩ෼ϓϥΠόγʔ͸͜ͷਪ࿦ͷਖ਼ޡͷ֬཰͔Βಋग़Ͱ͖Δ 𝜖&'( = max log 1 − 𝛿 − FP FN , log 1 − 𝛿 − FN FP &NQJSJDBMEJGGFSFOUJBMQSJWBDZ Kairouz+. The composition theorem for differential privacy. ICML2015 5SVF*OQVU (VFTT 'BMTF1PTJUJWF 𝐷 𝐷$ 'BMTF/FHBUJWF 𝐷$ 𝐷 ℳ 𝑦 𝐷 PS𝐷) 𝐷 PS 𝐷) 𝐷 𝐷′ ྡ઀%#

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-:$PSQPSBUJPO ϓϥΠόγʔ༧ࢉͷ؅ཧ σʔλͷ܁Γฦ͠ར༻ʹ൐͏ྦྷੵతͳϓϥΠόγʔͷফඅΛ 𝜖) ͷ߹੒ʹΑͬͯಋग़ à ࣄલʹఆٛͨ͠ϓϥΠόγʔ༧ࢉͷൣғ಺ͰɺԠ౴Մ൱Λ൑அͰ͖Δ TBUJTGZJOH𝝐𝟏 %1 TBUJTGZJOH𝝐𝒌 %1 ʜ 2VFSZ𝒒𝟏 2VFSZ𝒒𝒌 𝝐𝟏 𝝐𝟏 𝝐𝟐 𝝐𝟏 𝝐𝟐 𝝐𝒌 ʜ ʜ EFDMJOF ྦྷੵϓϥΠόγʔফඅ 2VFSZ EBUBBDDFTT ϓϥΠόγʔͷ߹੒ 1SJWBDZ$PNQPTJUJPO 𝜖*+*,- = ∑𝜖) ΫΤϦԠ౴ γεςϜ ػցֶश

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-:$PSQPSBUJPO 1SJWBDZ$PNQPTJUJPO 4FRVFOUJBM$PNQPTJUJPOʢ௚ྻ߹੒ʣ 𝜖 = B )∈ / 𝜖/ 𝜖 = max 𝜖0, … , 𝜖/ -FUℳ% , … , ℳ& TBUJTGZ𝜖% , … , 𝜖& SFTQFDUJWFMZ 𝐷`TUPUBMQSJWBDZDPOTVNQUJPOCZℳ% 𝐷 , … , ℳ& 𝐷 JT 1BSBMMFM$PNQPTJUJPOʢฒྻ߹੒ʣ -FUℳ%, … , ℳ& TBUJTGZ𝜖%, … , 𝜖& SFTQFDUJWFMZ -FU𝐷 = 𝐷% ∪ ⋯ ∪ 𝐷& XIFSF𝐷' ∩ 𝐷()' = ∅ 𝐷`TUPUBMQSJWBDZDPOTVNQUJPOCZℳ% 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 %JGGFSFOUJBM1SJWBDZ 3%1 3%1΍[$%1͕Α͘ར༻͞Ε͍ͯΔ • ௚ྻ߹੒͸ϧʔζ • ཧ૝తͳ߹੒͸ະ஌ è ༷ʑͳ߹੒ఆཧ 𝜖 DPNQPTJUJPOT 4FRVFOUJBM $PNQPTJUJPO JEFBM 3%1

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-:$PSQPSBUJPO ʲݚڀࣄྫʳ)%17JFX GPS&YQMPSJOH%BUBCBTF $POTUSVDUJPOPGJOUFSNFEJBUFQSJWBUJ[FElWJFXz 1WJFX UPXBSETBDUVBMJ[JOHBOZ RVFSZSFTQPOTFTXJUITNBMMFSOPJTF 3FRVJSFNFOU 2VFSZBHOPTUJD/PJTFSFTJTUBODF4QBDFFGGJDJFOU"OBMZUJDBMMZSFMJBCMF "DDFQUFEBU7-%# *EFOUJUZ 1SJWUSFF )%.. 1SJWCBZFT )%17JFX PVST "33 1.94×10! 7.05 35.34 3.79 𝟏. 𝟎𝟎 "WFSBHF3FMBUJWF&SSPSPWFSEBUBTFUT 4J[FPGQWJFX IUUQTBSYJWPSHBCT 4JNQMF4DBMBCMF 3BOEPN3FDVSTJWF #JTFDUJPO

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-:$PSQPSBUJPO ࠩ෼ϓϥΠόγʔʹΑΔ ҆શͳ࿈߹ֶश

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-:$PSQPSBUJPO ػցֶशͱࠩ෼ϓϥΠόγʔ ࿈߹ֶशͱதԝࠩ෼ϓϥΠόγʔ ہॴࠩ෼ϓϥΠόγʔ ࿈߹ֶशͱہॴࠩ෼ϓϥΠόγʔ ࠩ෼ϓϥΠόγʔʹΑΔ҆શͳ࿈߹ֶश

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-:$PSQPSBUJPO 4(%ɿ4UPDIBTUJD(SBEJFOU%FTDFOU ඪ४తͳػցֶशख๏ͷҰͭɻ܇࿅σʔλ͔Βޯ഑ HSBEJFOU Λಋग़ͯ͠Ϟσϧͷߋ৽ʹར༻ 𝐚𝐯𝐠. .JOJCBUDI 4BNQMJOH 1FSTBNQMF (SBEJFOU %BUBCBTF𝐷 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(% NJOJCBUDIUSBJOJOH .PEFM6QEBUF ऩଋ͢Δ·Ͱ൓෮܇࿅

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-:$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 %BUBCBTF𝐷 𝐷 = 𝑛 𝑥%, … , 𝑥5 ∈ 𝐷 𝑔' = ∇𝑓(𝑥'; 𝜃) 𝜋' 𝑔( = 𝑔( ⋅ min 1, 𝐶 𝑔( % (SBEJFOU $MJQQJOH (SBEJOUͷℓ: ϊϧϜΛఆ਺$Ͱ੍ݶ à ηϯγςΟϏςΟΛ$ʹ͢Δ ʢ1PTUQSPDFTTJOHఆཧΑΓʣ IUUQTBSYJWPSHBCT .PEFM6QEBUF 𝜃) = 𝜃)*$ − 𝜂 1 𝑛 P (∈ , 𝜋' 𝑔( ; 𝜃)*$ + 𝑁 0, 𝜎 %14(% ϓϥΠόγʔ༧ࢉͷൣғͰ൓෮܇࿅ "CBEJ%FFQ-FBSOJOHXJUI%JGGFSFOUJBM1SJWBDZ$44

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-:$PSQPSBUJPO ʲݚڀࣄྫʳ1SJWBUF%BUB4ZOUIFTJT σʔλΛ໛฿͢Δੜ੒ϞσϧΛࠩ෼ϓϥΠόγʔΛอূ͠ͳ͕Β܇࿅ à σʔλྲྀ௨Λଅਐ 1(. PVST 𝜖, 𝛿 = (1,107;) IUUQTBSYJWPSHBCT *$%& /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 &ODͷSPVOE܇࿅ %11$" à %FDͷ൓෮܇࿅ %14(% "DDFQUFEBU*$%&

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-:$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 /POQBSUJDJQBOUTPG'- (MPCBM.PEFM ̍

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-:$PSQPSBUJPO 'FE4(%"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.PEFM6QEBUF 0OEFWJDF $PNQVUJOH

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-:$PSQPSBUJPO ࿈߹ֶश ࠩ෼ϓϥΠόγʔ (MPCBM.PEFM (MPCBM.PEFM 3BX (SBEJFOU 3BOEPNJ[FE (SBEJFOU /PJTF*OKFDUJPO $FOUSBM.PEFM -PDBM.PEFM

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-:$PSQPSBUJPO $FOUSBM %1'FE4(%"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 .PEFM6QEBUF ϓϥΠόγʔ༧ࢉͷൣғͰ൓෮܇࿅ .D.BIBO-FBSOJOH%JGGFSFOUJBMMZ1SJWBUF3FDVSSFOU-BOHVBHF.PEFMT*$-3IUUQTBSYJWPSHBCT

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-:$PSQPSBUJPO -PDBM%JGGFSFOUJBM1SJWBDZʢہॴࠩ෼ϓϥΠόγʔʣ ෼ࢄ؀ڥԼͰͷ౷ܭऩू΍࿈߹ֶशͷͨΊͷϓϥΠόγʔϞσϧɻΫϥΠΞϯτͰϥϯμϜԽ 1SJWBDZ.PEFM ػඍͳ ౷ܭͷऩू αʔόʔΛ ৴༻͠ͳ͍ ϝΧχζϜ ℳ: 𝒳 → 𝒮 ͕ 𝜖, 𝛿 %1Λຬͨ͢ͱ͸ɺ ೚ҙͷೖྗͷ૊ 𝑥$ , 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 ྡ઀%#͸ SFQMBDFNFOUΛ૝ఆ

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-:$PSQPSBUJPO L3BOEPNJ[FE3FTQPOTF 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 SBOEPNMZTFMFDU 𝑘: |𝒳| JUFNT

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-:$PSQPSBUJPO L33ʹΑΔ౷ܭऩूʢਪఆʣ ूܭ͢ΔϥϯμϜԽϨϙʔτͷ਺͕ଟ͍΄Ͳਖ਼֬ͳ౷ܭΛਪఆͰ͖Δ 1SJWBDZBUTDBMF 𝑁 = 10,000 𝑁 = 10,000,000

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-:$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[FE3FTQPOTF • %JTDSFUF-BQMBDFϝΧχζϜ • 1PJTTPO#JOPNJOBMϝΧχζϜ ৴པϞσϧ αʔόʔΛ৴པ͢Δ αʔόʔͷ৴པ͸ඞཁͳ͍

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-:$PSQPSBUJPO ࠩ෼ϓϥΠόγʔΛอূ͢Δ҆શͳ࿈߹ֶशʢ-%1ʣ (MPCBM.PEFM 3BOEPNJ[FE (SBEJFOU -%1ͳ$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%14FDVSF"HH౳ ޙड़

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-:$PSQPSBUJPO ࿈߹ֶश޲͚ͷϥϯμϜԽॲཧʢ-%1ʣ 1SJW6OJUϝΧχζϜ • 𝜖< -%1Λอূ • ޯ഑ΛϥϯμϜʹαϯϓϦϯά • ྘ PSനΛ 33Ͱબ୒ • ͞Βʹબ୒ྖҬ಺ʢ྘ PSനʣ͔Β Ұ༷αϯϓϧ (BVTTJBOϝΧχζϜ • $%1޲͚ͷ(BVTTJBOϝΧχζϜͷస༻ • ϊΠζΛ$%1޲͚ख๏ͷഒʹ͢Δ ̀ $%1ͱ-%1ͷྡ઀ੑͷҧ͍ʹΑΓ 33ϕʔεͷख๏ $POUJOVPVT/PJTF %JTDSFUF/PJTF %JTDSFUF-BQMBDFϝΧχζϜ • ಠཱͨͭ͠ͷ֬཰ม਺ d1PJTTPO෼෍ ͷࠩͷ෼෍ 4LFMMBNϝΧχζϜ ˞࠶ੜੑͷ͋Δ෼෍Λ༻͍Δඞཁ͕͋Δ IUUQTFOXJLJQFEJBPSHXJLJ4LFMMBN@EJTUSJCVUJPO 4LFMMBN෼෍ #IPXNJDL1SPUFDUJPO"HBJOTU3FDPOTUSVDUJPOBOE *UT"QQMJDBUJPOTJO1SJWBUF'FEFSBUFE-FBSOJOH IUUQTBSYJWPSHBCT "HBSXBM5IF4LFMMBN .FDIBOJTNGPS%JGGFSFOUJBMMZ1SJWBUF 'FEFSBUFE-FBSOJOH/FVS*14IUUQTBSYJWPSHBCT

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-:$PSQPSBUJPO -%1'FE4(%"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܇࿅

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-:$PSQPSBUJPO ൃలతͳ࿩୊

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-:$PSQPSBUJPO ۙ೥ͷಈ޲ɿ1SJWBDZ5FDI$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

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-:$PSQPSBUJPO 4IVGGMFNPEFMr BOJOUFSNFEJBUFQSJWBDZNPEFM • *OUFSNFEJBUFUSVTUFEFOUJUZlTIVGGMFSz BOPOZNJ[FTMPDBMVTFST`JEFOUJUZ • &BDIDMJFOUFODSZQUTUIFJSSBOEPNJ[FEDPOUFOUXUIFTFSWFS`TQVCMJDLFZ UIFO TIVGGMFSPOMZNJYFTUIFJSJEFOUJGJFTXPMPPLJOHBUUIFDPOUFOUT Q 𝑥0 Q 𝑥1 Q 𝑥3 R 𝑥0 (R 𝑥0, R 𝑥1, … , R 𝑥3) R 𝑥1 R 𝑥3 4FSWFS 3BOEPNJ[FEX𝝐𝟎 4IVGGMF 4IVGGMFS 4FOEUIFTIVGGMFECBUDI BOPOZNJ[FE

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-:$PSQPSBUJPO 1SJWBDZ"NQMJGJDBUJPOWJB4IVGGMJOH • 4IVGGMFSDBOBNQMJGZEJGGFSFOUJBMQSJWBDZà QPTTJCJMJUZUPEFDSFBTFMPDBMOPJTF • 5IFBNQMJGJDBUJPOPOTIVGGMFSUSBOTMBUFT-%1PODMJFOUTJOUP$%1 𝜖5 = 8 -%1 𝛿 = 1067 𝑘 = 10 CZ)JEJOHBNPOHDMPOFT &YBNQMFJOLSBOEPNJ[FESFTQPOTF 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

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-:$PSQPSBUJPO 4IVGGMFE$IFDL*OTGPS'- ࿈߹ֶशʹ4IVGGMOHΛಋೖ • ΫϥΠΞϯτ܈ʢ𝑛ʣͷҰ෦ʢ𝑚 < 𝑛ʣΛαϯϓϧͯ͠ϞσϧΛ൓෮తʹ܇࿅͍ͨ͠ à Ұ܇࿅ͷࢀՃΫϥΠΞϯτ਺𝑚͕গͳ͍ à 4IVGGMJOH͕ෆे෼ à 4VCTBNQMJOHΛߟྀ • 4VCTBNQMJOH͸ΫϥΠΞϯτͷESPQPVUʹ੬ऑ à ؤ݈ͳϞσϧΛಋೖ͍ͨ͠ )JHI6UJMJUZBU4USPOH1SJWBDZ-FWFM "NQMJGJFE 4IVGGMFE$IFDL*OT 𝛾 = 𝑝 1 − 𝑝$ 𝑝DIFDLJOSBUF 𝑝 𝑝$ESPQPVUSBUF 𝛾FGGFDUJWFDIFDLJOSBUF Q 𝑥0 Q 𝑥8 Q 𝑥3 {Q 𝑥8, Q 𝑥3, … , Q 𝑥0}

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-:$PSQPSBUJPO ʲݚڀࣄྫʳ/FUXPSL4IVGGMJOH %FDFOUSBMJ[FETIVGGMJOHWJBNVMUJSPVOESBOEPNXBMLT POBHSBQI *OFBDISPVOE FWFSZDMJFOUSFMBZTIFSSBOEPNJ[FESFQPSUTUPPOFPGIFSOFJHICPST FH GSJFOETPOBTPDJBMOFUXPSL WJBBOFODSZQUFEDIBOOFM 5IFMBSHFSHSBQIBNQMJGJFTQSJWBDZUIFNPSF "DDFQUFEBU4*(.0% IUUQTBSYJWPSHBCT

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