機械学習とセキュリティ

ػցֶशͱηΩϡϦςΟ ๭௨৴ࣄۀձࣾ ΄͛΄͛։ൃ෦ ઀఺2# !TFUUFO@2# %BUB4DJFODF$BGÉ!৽॓

ػցֶशºηΩϡϦςΟ Ø %PT߈ܸΛػցֶशͰݕ஌ Ø Ϛϧ΢ΣΞΛػցֶशͰݕ஌ ػցֶशΛηΩϡϦςΟʹԠ༻ ػցֶशϞσϧࣗମͷηΩϡϦςΟ ػցֶशͰߏஙͨ͠ϞσϧʢػցֶशϞσϧʣΛ૊ΈࠐΜͩγεςϜͰ͸ ैདྷͷγεςϜͱ͸ҟͳͬͨݻ༗ͷ߈ܸ͕ͳ͞ΕΔϦεΫ͕༗Δ ͦͷͨΊɼػցֶशϞσϧಛ༗ͷηΩϡϦςΟରࡦ͕ඞཁ
2

"EWFSTBSJBM"UUBDL ϞσϧΛὃ͢߈ܸ .PEFM&YUSBDUJPO ϞσϧΛ౪Ή߈ܸ .PEFM*OWFSTJPO ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ

"EWFSTBSJBM"UUBDL ܇࿅͞ΕͨػցֶशϞσϧΛὃ͢߈ܸ ݩͷը૾ ఢରతઁಈ "EWFSTBSJBM&YBNQMF ʢςφΨβϧʣ (PPEGFMMPX FUBM ΑΓҾ༻
5

ఢରతઁಈΛݟ͚ͭΔͨΊʹ͸ argmin " , s.t. + ≠ argmin " ,
s.t. + = # ಛఆͷΫϥεʹޡ෼ྨͤ͞Δ͜ͱΛ໨తͱͨ͠ "EWFSTBSJBM"UUBDL ޡ෼ྨͤ͞ΔΫϥε͸ࢦఆͤͣ ͱʹ͔͘Ͳ͔͜ͷΫϥεʹޡ෼ྨͤ͞Δ͜ͱΛ ໨తͱͨ͠"EWFSTBSJBM"UUBDL 6

୅දతͳ"EWFSTBSJBM"UUBDL (PPEGFMMPX FUBM !"# = + sign ∇ ℓ
, /PUBUJPO ɿઁಈڧ౓ʢͲΕ͙Β͍ઁಈΛڧ͘༩͑Δ͔Λද͢ʣ ℓɿଛࣦؔ਺ 3FNBSL '(4.͸Ұ౓ͷޯ഑্ঢͰ࠷దԽΛऴྃ͢Δ͕ɼ͜ΕΛෳ਺ճʹ֦ுͨ͠ͷ͕#*.Ͱɼ CBMMʹऩ·ΔΑ͏ͳ੍໿ͷ΋ͱͰޯ഑্ঢΛ܁Γฦ͢ɽ '(4.΋#*.΋! ϊϧϜͰͷઁಈΛੜ੒͓ͯ͠Γɼ%FFQGPPM͸͜ͷϊϧϜΛ" ʹɼ $8͸# ʹͨ͠΋ͷͱݟΔ͜ͱ͕Ͱ͖Δɽ 7

8IJUF#PY4FUUJOHʹ͓͚Δఢରతઁಈͷݟ͚ͭํ '(4.BUUBDL (PPEGFMMPX FUBM #*. ,VSBLJO FUBM BUUBDL.*'(4.
%POHFUBM %FFQGPPM .PPTBWJ%F[GPPMJ FUBM /FXUPO'PPM +BOHFUBM +4." 1BQFSOPU FUBM (SBEJFOU #BTFE 0CKFDUJWF 'VODUJPO #BTFE $8 $BSMJOJ FUBM &"% $IFOFUBM 0QU.BSHJO 8BSSFOFUBM ˞8IJUF#PY4FUUJOHɿଛࣦؔ਺ͷޯ഑͕ܭࢉͰ͖ͨΓɼϞσϧͷDPOGJEFODFTDPSFΛ஌Δ͜ͱ͕Ͱ͖Δઃఆ 8

"EWFSTBSJBM"UUBDL΁ͷରࡦɿ"EWFSTBSJBM5SBJOJOH Ϟσϧͷֶश࣌ʹBEWFSTBSJBMFYBNQMFTͰl༧๷઀छz͓͚ͯ͠͹ BEWFSTBSJBMFYBNQMFTʹର͢Δϩόετੑ΋্͕ΔΜ͡Όͳ͍͔ʁ 9 ℓ!"# , () ≔ ℓ !"#,
!"# + 1 − ℓ , *EFB 3FTVMU n ࣮ࡍʹBEWFSTBSJBMFYBNQMFTʹର͢Δϩόετੑ͸޲্͢Δ ҰํͰʜ n ֶशʹཁ͢Δ͕࣌ؒ૿͑Δ n "EWFSTBSJBMFYBNQMFT΁ͷϩόετੑ͸BEWFSTBSJBMFYBNQMFTͷ࡞Γํʹґଘ͢Δ n ѱҙͷͳ͍ϊΠζ͕৐ͬͨը૾Λ͏·͘෼ྨͰ͖ͳ͘ͳΔ ͱ͍ͬͨ໰୊΋ൃੜ͢Δ

.PEFM&YUSBDUJPOɿϞσϧΛ౪Ή & "SDIJUFDUVSF %FDJTJPO#PVOEBSZ 'VODUJPOBMJUZ ϨΠϠʔͷ਺΍χϡʔϩϯͷ਺ ͞Βʹ׆ੑԽؔ਺ͳͲͷ Ϟσϧͷߏ଄Λ౪ΉλΠϓ ϞσϧͷܾఆڥքΛ ౪ΉλΠϓ
ϞσϧʹΑΔ ೖྗͱग़ྗͷରԠؔ܎Λ ౪ΉλΠϓ 11

.PEFM&YUSBDUJPOͷओཁͳΞϓϩʔν $ , $ $%& ' : ℝ( → ֶश
), ) )%& * +: ℝ( → ֶश 0SJHJOBM.PEFM 4VCTUJUVUF.PEFM "1*ʹΑͬͯฦ͞ΕΔ Λ໨తม਺ͱͯ͠ར༻͢Δ͜ͱͰ୅ཧϞσϧΛߏங͢Δɽ ୅දతͳݚڀͱͯ͠$PSSFJB4JMWBFUBM ΍0SFLPOEZ FUBM ͕ڍ͛ΒΕΔɽ 12

4VCTUJUVUF.PEFMʹؔ͢Δ߈๷ %FGFODF 0GGFODF "1*ୟ͚Δճ਺Λ੍ݶ গͳ͍Ԡ౴ճ਺Ͱ΋ 4VCTUJUVUF.PEFMΛߏங͢Δํ๏͕ఏҊ͞ΕΔ 0SFLPOEZ FUBM 3PTFOCFSHFUBM
13

ʢ޿ٛͷʣ.PEFM*OWFSTJPO"UUBDL ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ 1SPQFSUZ*OGFSFODF"UUBDL .PEFM*OWFSTJPO"UUBDL ద౰ͳ ͕Ϟσϧͷֶशσʔληοτʹؚ·Ε͍ͯΔ͔ʁ Λ໌Β͔ʹ͢Δ߈ܸ ֶशσʔληοτʹؔ͢Δੑ࣭Λਪଌ͢Δ߈ܸ FH൅͕௕͍ਓ΍υϨεΛண͍ͯΔਓ͕ ੑผ෼ྨͷϞσϧͷֶशσʔλʹؚ·Ε͍ͯΔ͔ʁ
ֶशσʔλΛ෮ݩ͢Δ߈ܸ ˞.PEFM*OWFSTJPO"UUBDLʹ.FNCFSTIJQ*OGFSFODF΍1SPQFSUZ*OGFSFODFΛؚΊΔ͔ʹ͸ॾઆ͋Γ 15

("/Λ༻͍ͨ.PEFM*OWFSTJPO ,VTBOP FUBM ิॿσʔληοτ ) , )%& * ,
, ∼$$( ℱ′ ֶशσʔληοτ $ , $ $%& ' , ∼$$( ℱ ΫΤϦ & ,, … * , Ԡ౴ (& ,), … * , (FOFSBUPS Ͱ ℱ ʢͬΆ͍΋ͷʣΛۙࣅ αϯϓϦϯά (FOFSBUPS͔Β ֶशσʔλͬΆ͍΋ͷΛੜ੒ 16

σʔλͷҰ෦͔Β࢒Γͷ෦෼Λ෮ݩ ֶशσʔλ ∈ ℝ( ͔ΒҰ෦ͷಛ௃ྔ͚ͩΛൈ͖ग़ͯ͠ ࡞੒ͨ͠ϕΫτϧ " ∈ ℝ-, <
͔Β ࢒Γͷಛ௃ྔΛ෮ݩ͢Δ ࿦จͰ͸ " ͸࿙Ӯͯ͠΋໰୊ͳ͍ηϯγςΟϒͰͳ͍ಛ௃ྔͰ ࢒Γͷಛ௃ྔ͸ηϯγςΟϒͳಛ௃ྔͩͱ૝ఆ͍ͯ͠Δ ("/Λ༻͍ͨ.PEFM*OWFSTJPO ;IBOFUBM 17

͍΍ɼϜζ͘Ͷʜʁ ./*45Λֶशσʔλͱͨ͠෼ྨϞσϧʹରͯ͠.PEFM *OWFSTJPO"UUBDLΛߦͬͨ݁Ռ ิॿσʔλ͸खॻ͖จࣈͷ਺ࣈͱΞϧϑΝϕοτ ࣮ࡍʹ෮ݩͯ͠Έͨ݁Ռ 18

ػցֶशϞσϧ΁ͷ߈ܸʹؔ͢Δݚڀ ΍ͬͺΓ"EWFSTBSJBM"UUBDL͕μϯτπͳײ͡ ݚڀ͞ΕͯΔײ ֶशσʔλΛ͢΂ͯ෮ݩ͢ΔλΠϓͷ.PEFM*OWFSTJPO"UUBDL͕Ұ൪೉ͦ͠͏ ʢ࣮ࡍʹ೉͔ͬͨ͠ʣ ߈ܸͷ೉͠͞ n "EWFSTBSJBM"UUBDL͸σʔλ͕ߴ࣍ݩʹͳΔͱෆՄආతʹੜ͡ΔͨΊ ຊ࣭తʹରࡦ͕ࠔ೉ʁ n
.PEFM&YUSBDUJPO"UUBDL͸ΫΤϦ੍ݶ͕༗ޮʹࢥ͑Δ͕ ΫΤϦ੍ݶͷ΋ͱͰ΋͋Δఔ౓ͷϞσϧෳ੡͕ग़དྷ͓ͯΓ ࠓޙͷಈ޲ʹ஫໨ ରࡦͷ೉͠͞ 20

3FGFSFODF (PPEGFMMPX *+ 4IMFOT + 4[FHFEZ $ &YQMBJOJOHBOEIBSOFTTJOHBEWFSTBSJBMFYBNQMFT BS9JW
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

3FGFSFODF 8BSSFO ) #P - %BXO 4 %FDJTJPO#PVOEBSZ"OBMZTJTPG"EWFSTBSJBM&YBNQMFT*OUFSOBUJPOBM $POGFSFODFPG-FBSOJOH3FQSFTFOUBUJPOT
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

機械学習とセキュリティ

機械学習とセキュリティ

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ػցֶशͱηΩϡϦςΟ ๭௨৴ࣄۀձࣾ ΄͛΄͛։ൃ෦ ઀఺2# !TFUUFO@2# %BUB4DJFODF$BGÉ!৽॓

"EWFSTBSJBM"UUBDL ϞσϧΛὃ͢߈ܸ .PEFM&YUSBDUJPO ϞσϧΛ౪Ή߈ܸ .PEFM*OWFSTJPO ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ

"EWFSTBSJBM"UUBDL ܇࿅͞ΕͨػցֶशϞσϧΛὃ͢߈ܸ ݩͷը૾ ఢରతઁಈ "EWFSTBSJBM&YBNQMF ʢςφΨβϧʣ (PPEGFMMPX FUBM ΑΓҾ༻

ఢରతઁಈΛݟ͚ͭΔͨΊʹ͸ argmin " , s.t. + ≠ argmin " ,

୅දతͳ"EWFSTBSJBM"UUBDL (PPEGFMMPX FUBM !"# = + sign ∇ ℓ

8IJUF#PY4FUUJOHʹ͓͚Δఢରతઁಈͷݟ͚ͭํ '(4.BUUBDL (PPEGFMMPX FUBM #. ,VSBLJO FUBM BUUBDL.'(4.

"EWFSTBSJBM"UUBDL΁ͷରࡦɿ"EWFSTBSJBM5SBJOJOH Ϟσϧͷֶश࣌ʹBEWFSTBSJBMFYBNQMFTͰl༧๷઀छz͓͚ͯ͠͹ BEWFSTBSJBMFYBNQMFTʹର͢Δϩόετੑ΋্͕ΔΜ͡Όͳ͍͔ʁ 9 ℓ!"# , () ≔ ℓ !"#,

"EWFSTBSJBM"UUBDL ϞσϧΛὃ͢߈ܸ .PEFM&YUSBDUJPO ϞσϧΛ౪Ή߈ܸ .PEFM*OWFSTJPO ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ

.PEFM&YUSBDUJPOɿϞσϧΛ౪Ή & "SDIJUFDUVSF %FDJTJPO#PVOEBSZ 'VODUJPOBMJUZ ϨΠϠʔͷ਺΍χϡʔϩϯͷ਺ ͞Βʹ׆ੑԽؔ਺ͳͲͷ Ϟσϧͷߏ଄Λ౪ΉλΠϓ ϞσϧͷܾఆڥքΛ ౪ΉλΠϓ

.PEFM&YUSBDUJPOͷओཁͳΞϓϩʔν $ , $ $%& ' : ℝ( → ֶश

4VCTUJUVUF.PEFMʹؔ͢Δ߈๷ %FGFODF 0GGFODF "1*ୟ͚Δճ਺Λ੍ݶ গͳ͍Ԡ౴ճ਺Ͱ΋ 4VCTUJUVUF.PEFMΛߏங͢Δํ๏͕ఏҊ͞ΕΔ 0SFLPOEZ FUBM 3PTFOCFSHFUBM

"EWFSTBSJBM"UUBDL ϞσϧΛὃ͢߈ܸ .PEFM&YUSBDUJPO ϞσϧΛ౪Ή߈ܸ .PEFM*OWFSTJPO ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ

("/Λ༻͍ͨ.PEFMOWFSTJPO ,VTBOP FUBM ิॿσʔληοτ ) , )%& ,

σʔλͷҰ෦͔Β࢒Γͷ෦෼Λ෮ݩ ֶशσʔλ ∈ ℝ( ͔ΒҰ෦ͷಛ௃ྔ͚ͩΛൈ͖ग़ͯ͠ ࡞੒ͨ͠ϕΫτϧ " ∈ ℝ-, <

͍΍ɼϜζ͘Ͷʜʁ ./45Λֶशσʔλͱͨ͠෼ྨϞσϧʹରͯ͠.PEFM OWFSTJPO"UUBDLΛߦͬͨ݁Ռ ิॿσʔλ͸खॻ͖จࣈͷ਺ࣈͱΞϧϑΝϕοτ ࣮ࡍʹ෮ݩͯ͠Έͨ݁Ռ 18

3FGFSFODF (PPEGFMMPX *+ 4IMFOT + 4[FHFEZ $ &YQMBJOJOHBOEIBSOFTTJOHBEWFSTBSJBMFYBNQMFT BS9JW

3FGFSFODF 8BSSFO ) #P - %BXO 4 %FDJTJPO#PVOEBSZ"OBMZTJTPG"EWFSTBSJBM&YBNQMFT*OUFSOBUJPOBM $POGFSFODFPG-FBSOJOH3FQSFTFOUBUJPOT