公平性を保証したAI/機械学習アルゴリズムの最新理論

ެฏੑΛอূͨ͠"*ػցֶश ΞϧΰϦζϜͷ࠷৽ཧ࿦ ୈճ"*ηϛφʔʮػցֶशʗਓ޻஌ೳͷެฏੑʯ ෱஍Ұే ஜ೾େֶཧݚ*"*1 GVLVDIJ!DTUTVLVCBBDKQ

ࣗݾ঺հ w ໊લ෱஍Ұే 'VLVDIJ,B[VUP w ॴଐஜ೾େֶγεςϜ৘ใܥॿڭ w ܦྺ w
ஜ೾େֶγεςϜ৘ใ޻ֶઐ߈Պത࢜ޙظ՝ఔमྃ w ཧݚ"*1ಛผݚڀһ w ݱࡏஜ೾େֶγεςϜ৘ใܥॿڭ w ݱࡏཧݚ"*1٬һݚڀһ w ݚڀڵຯ w ػցֶशʹ͓͚ΔެฏੑɼϓϥΠόγʔͷཧ࿦໘ w ਺ཧ౷ܭɼಛʹɼ൚ؔ਺ਪఆ

ػցֶशք۾Ͱͷެฏੑͷ஫໨ ACM FAT* ࠃࡍձٞ "$.'"5 """*"$."*&4 ࠃࡍϫʔΫγϣοϓ '"5.- ট଴ߨԋ
w*$.- -4XFFOFZ w/*14 ,$SBXGPSE w,%% $%XPSL w,%% +.8JOH "*GPS4PDJBM(PPE $IBMMFOHFTBOE0QQPSUVOJUJFTGPS"* JO'JOBODJBM4FSWJDFT "*&UIJDT84

͸͡Ίʹ w ਆቇઌੜͷެԋ w ެฏͳػցֶशͷجૅతͳ෦෼ w ຊެԋ w ެฏͳػցֶशͷ࠷ۙͷݚڀ੒ՌɾτϐοΫ

ࠓ೔ͷ಺༰ w ෼ྨɼճؼʹ͓͚Δެฏੑͷอূํ๏ w දݱֶशʹ͓͚Δެฏੑͷอূํ๏ w ݸਓެฏੑԼʹ͓͚Δֶशཧ࿦ w όϯσΟοτ໰୊ʹ͓͚Δެฏੑ

ࠓ೔ؚ·ͳ͍಺༰ w ެฏੑΛอূͨ͠ΫϥελϦϯά w ެฏੑΛอূͨ͠ϥϯΩϯά w ҼՌΛݩʹͨ͠ެฏੑ w ػցֶशΞϧΰϦζϜͷಁ໌ੑ΍આ໌Մೳੑ

ެฏͳֶशʹؔ͢Δݚڀ

෼ྨֶशͷ࢓૊Έ ϥϕϧ෇͖σʔλ ֶशΞϧΰϦζϜ ෼ྨث f

෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ

෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ͳΔ΂͘౰ͨΔΑ͏ f Λબ୒͍ͨ͠

෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ f ʹΑΔ෼ྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ

ެฏͳ෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ͳΔ΂͘ ౰ͨΔΑ͏ʹf Λબ୒ ެฏͳf
Λબ୒

ެฏͳ෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ lެฏͳf zͱ͸ʁ f ʹΑΔ෼ྨޡ͕ࠩ
࠷খʹͳΔΑ͏ʹ͢Δ

ઃఆ w ؆୯ͷͨΊʹڭࢣ͋Γ෼ྨ໰୊ͷΈΛߟ͑Δ w ɹɹɹɹɹɹɹɹɹɹֶྺɼ৬ྺɼࢿ֨ͳͲ w ɹɹɹɹɹɹɹɹɹɹੑผɼਓछɼफڭɼ੓࣏ࢤ޲ɼ೥ྸͳͲ w ɹɹɹɹɹɹɹɹɹɹ༧ଌ͍ͨ͠΋ͷ FH
࠾൱ w ɹɹɹɹɹɹɹɹɹɹΞϧΰϦζϜʹΑͬͯ༧ଌ͞Εͨϥϕϧ ೖྗ X ϥϕϧ Y ༧ଌϥϕϧ ̂ Y ผͷೖྗ X S = உੑ S = ঁੑ ೖྗ X ϥϕϧ Y ηϯγςΟϒଐੑ S ༧ଌϥϕϧ ̂ Y ֶश

ެฏੑج४ w ༷ʑͳެฏੑج४ w %FNPHSBQIJDQBSJUZ w &RVBMJ[FEPEET<)BSEU > w $BMJCSBUJPO<1MFJTT
> w *OEJWJEVBMGBJSOFTT<%XPSL >

%FNPHSBQIJDQBSJUZ w ηϯγςΟϒଐੑͰ৚͚݅ͮΒΕͨ༧ଌϥϕϧͷ෼෍͕Ұக w ෼෍Ͱ͸ͳ͘༧ଌਫ਼౓ِཅੑِӄੑͷҰகΛ໨ࢦ͢΋ͷ ΋͋Γ %FNPHSBQIJDQBSJUZ ℙ{ ̂ Y
∈ 𝒜|S = s} = ℙ{ ̂ Y ∈ 𝒜|S = s′ } ೚ҙͷ𝒜, s, s′ ʹ͍ͭͯ ࠾༻ ඇ࠾༻ ࠾༻ ඇ࠾༻ உੑ ঁੑ = ̂ Y|S = உੑ ̂ Y|S = ঁੑ

&RVBMJ[FEPEET<)BSEU > w ɹͱɹΛҰகͤ͞ΔΑ͏ʹֶशͰ͖Δ w %FNPHSBQIJDQBSJUZͰ͸Ͱ͖ͳ͍ &RVBMJ[FEPEET ℙ{ ̂ Y
∈ 𝒜|Y = y, S = s} = ℙ{ ̂ Y ∈ 𝒜|Y = y, S = s′ } ೚ҙͷ𝒜, y, s, s′ ʹ͍ͭͯ Y ̂ Y ࠾༻ ඇ࠾༻ உੑ ࠾༻ ඇ࠾༻ ঁੑ ਅͷ࠾༻෼෍ ֶश ࠾༻ ඇ࠾༻ ࠾༻ ඇ࠾༻ ެฏ ֶशͷ݁ՌมԽͨ͠෦෼ ྘ͷ෦෼ ͷҰக

$BMJCSBUJPO<1MFJTT > w Ͱ͋Δ֬཰ͷ༧ଌ஋Λ ͱ͢Δ w Ͱ͋Δ༧ଌ͕ ͷΈʹΑܾͬͯ·ΔΑ͏ʹ੍໿ Y =
1 ̂ p Y = 1 ̂ p $BMJCSBUJPO ℙ{Y = 1| ̂ p = p, S = s} = p ೚ҙͷp, sʹ͍ͭͯ உੑ ࠾༻ ඇ࠾༻ p ̂ p = p|S = உੑ

*OEJWJEVBMGBJSOFTT<%XPSL > w ੑผҎ֎શ͘ಉ͡ਓ͕͍Ε͹࠾൱΋ಉ͡ʹ͢Δ΂͖ w ࣅͨΑ͏ͳਓ͸ࣅͨ݁ՌΛड͚औΔ΂͖ w ֬཰త༧ଌؔ਺ w -JQTDIJU[QSPQFSUZ
೚ҙͷx, x′ ʹ͍ͭͯ D( f(x), f(x′ )) ≤ d(x, x′ ) ≈ ⟹ f : 𝒳 → Δ(𝒴) ݁Ռͷ෼෍ؒͷ ڑ཭

෼ྨɾճؼʹ͓͚Δެฏੑ

ެฏͳ෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ެฏੑج४Λୡ੒ f ʹΑΔ෼ྨޡ͕ࠩ ࠷খʹͳΔΑ͏ʹ͢Δ

ެฏͳֶशͷجຊઓུ w ࠷దԽ໰୊ͱͯ͠ఆࣜԽ w േଇɾ੍໿Λઃ͚Δ͜ͱͰެฏੑΛ੍ޚ minf Err(f ) + ηUnfair(f
) േଇϕʔε minf Err(f ) TVCUP Unfair(f ) ≤ η ੍໿ϕʔε

3FEVDUJPO"QQSPBDI<"HBSXBM > w ֬཰తͳ෼ྨث Λֶश w ͸෼ྨث ্ͷ෼෍ w 3FEVDUJPO
w ֶश໰୊ΛίετηϯγςΟϒֶशͷܥྻʹஔ͖׵͑ Q Q f minQ 𝔼f∼Q ℙ{f(X) ≠ Y} TVCUP M𝔼f∼Q [μ(f )] ≤ c ࠷దԽ໰୊ ެฏੑج४Λ༗ݶݸͷ੍໿Ͱදݱ ྫ%1 𝔼{f(X)|S = 0} = 𝔼{f(X)} 𝔼{f(X)|S = 1} = 𝔼{f(X)}

3FEVDUJPO"QQSPBDI<"HBSXBM > minQ 𝔼f∼Q ℙ{f(X) ≠ Y} TVCUP M𝔼f∼Q
[μ(f )] ≤ c ݩͷ࠷దԽ໰୊ maxλ∈ℝK + ,∥λ∥≤B minQ 𝔼f∼Q ℙ{f(X) ≠ Y} + λ⊤ (M𝔼f∼Q [μ(f )] − c) ੍໿ແ͠࠷దԽ໰୊ ϥάϥϯδϡ৐਺ ަޓ࠷దԽʹΑͬͯҌ఺Λ୳ࡧ

ίετηϯγςΟϒֶश w ༧ଌͷॏΈ͕ҧ͏ w ϥϕϧ͕ϙδςΟϒωΨςΟϒ͔ w αϯϓϧຖ w ྫʣ࣬ױਪఆ w
࣬ױ͕ͳ͍ͷʹؒҧ͏ΑΓ΋࣬ױ͕͋Δͷʹؒҧ͏͜ͱͷ ํ͕໰୊ minf ∑n i=1 (h(Xi )C1 i + (1 − h(Xi ))C0 i ) ίετηϯγςΟϒֶश

3FEVDUJPO"QQSPBDI*NQMJDBUJPO w ༩͑ΒΕͨϥάϥϯδϡ৐਺ ͷ΋ͱͰ෼෍ ͷ࠷খԽ w ίετηϯγςΟϒֶशͰॻ͖׵͑ΒΕΔ λ Q ͸͜͜ͷظ଴஋
μ

3FEVDUJPO"QQSPBDI"MHPSJUIN Λ࠷దԽ Q Λ࠷దԽ λ $PTUTFOTJUJWFֶश

'BJS3FHSFTTJPO2VBOUJUBUJWF%FpOJUJPOT BOE3FEVDUJPOCBTFE"MHPSJUINT <"HBSXBM > w ճؼʹ͓͚ΔެฏੑΛѻ͑ΔΑ͏ʹ֦ு w %FNPHSBQIJDQBSJUZ͔zશͯͷάϧʔϓͷଛࣦ͕ҰఆҎ Լzͱ͍͏ެฏੑ੍໿͕ѻ͑Δ ࿈ଓม਺ͷີ౓෼෍ͷҰக͸೉͍͠
཭ࢄԽͯ͠ϏϯΛ౰ͯΔ ෼ྨ໰୊ʹॻ͖׵͑

3FQSFTFOUBUJPO-FBSOJOH ֶशΞϧΰϦζϜ දݱث g( )= ϥϕϧ෇͖σʔλ z f( )=Aࢯ z
෼ྨث z දݱσʔλ ֶशΞϧΰϦζϜ

'BJS3FQSFTFOUBUJPO<;BGBS > w ηϯγςΟϒଐੑʹґଘ͠ͳ͍Α͏ͳදݱͷ֫ಘ g( ) g( ) z

*OIFSFOU5SBEFP⒎TJO-FBSOJOH'BJS 3FQSFTFOUBUJPOT<;IBP > w 'BJS3FQFSFTFOUBUJPO͔Β෼ྨثΛ࡞ͬͨ࣌ʹެฏੑ͕ Ͳ͏ͳΔ͔ղੳ g( ) g( )
z f( )=Aࢯ z ෼ྨث ͜͜·Ͱ͸ެฏ ͜Ε͸ެฏʁ

*OWBSJBOUGFBUVSF w 'BJSSFQSFTFOUBUJPOͱಉ͡ߟ͑ w ಛఆͷཁૉʢηϯγςΟϒଐੑͷมಈʣʹґଘ͠ͳ͍Α͏ ͳಛ௃ྔΛٻΊ͍ͨ g( ) g( )
z

ఢରతֶशΛ࢖ͬͨख๏<9JF > w %1Λݩʹͨ͠ެฏੑ w දݱ͔ΒηϯγςΟϒଐੑ͕ࣝผͰ͖ͳ͍ w ࣮ࡍʹηϯγςΟϒଐੑΛࣝผ͢ΔϞσϧΛ࡞Δ w ͜ͷϞσϧͷࣝผਫ਼౓͕ѱ͘ͳΔΑ͏ʹֶश
g( )= f( )=Aࢯ d( )=ࠇਓ z z z දݱث ෼ྨث ࣝผث ෼ྨث͸ͳΔ΂͘ ౰ͨΔΑ͏ʹ ࣝผث͸ͳΔ΂͘ ֎ΕΔΑ͏ʹ

࣮ݧ݁Ռ<9JF > ެฏੑ 3BOEPNHVFTTͱಉ౳͔͠ ηϯγςΟϒଐੑΛ༧ଌͰ͖ͳ͍ ఏҊख๏ʢӈ୺ʣͷ༧ଌਫ਼౓͕ߴ͍

7"&ͱ*OGPSNBUJPOCPUUMFOFDLΛ ݩʹͨ͠ํ๏<.PZFS > w ఢରతֶशͷ໰୊ɿֶश͕஗͍ w 7BSJBUJPOBM"VUP&ODPEFS 7"& Λݩʹޮ཰తͳֶश minf
Likelihood(f(X), Y) + ηI(f(X), S) ࠷దԽ໰୊ 7"&ͷςΫχοΫΛ࢖্ͬͯهͷ໰୊Λۙࣅ͠ͳ͕Βղ͘

ެฏੑͷֶशཧ࿦

ެฏͳ෼ྨֶशͷ࢓૊Έ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ

൚Խੑೳ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ܇࿅σʔλ ςετσʔλ

൚Խੑೳ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ܇࿅σʔλ ςετσʔλ ςετσʔλͰͷ෼ྨੑೳ͸ʁ ܇࿅σʔλͰ
౰ͨΔΑ͏ʹֶश

൚Խޡࠩެฏੑ ֶशΞϧΰϦζϜ ෼ྨث f( )=Aࢯ ϥϕϧ෇͖σʔλ ܇࿅σʔλ ςετσʔλ ςετσʔλͰͷޡࠩ
܇࿅σʔλͰͷޡࠩ ςετσʔλͰͷෆެฏੑ ܇࿅σʔλͰͷෆެฏੑ ൚Խޡࠩ ൚Խެฏੑ

(SPVQGBJSOFTTͷֶशཧ࿦

-FBSOJOH/PO%JTDSJNJOBUPSZ1SFEJDUPST <8PPEXPSUI > w &RVBMJ[FEPEETΛอূ͢ΔͨΊͷΞϧΰϦζϜ <)BSEU *$.-`> w είΞϕʔεͷ෼ྨثΛֶश w
είΞҎ্ w 30$͕Ұக͢ΔΑ͏ʹείΞͷᮢ஋Λௐઅ w ͜ͷΞϧΰϦζϜ͸TVCPQUJNBM w QPTUIPDͰ͸Τϥʔ͕ͩ &RVBMJ[FEPEET੍໿Λຬͨ͢Τϥʔ͕খ͍͞ ෼ྨث͕ଘࡏ͢Δ໰୊͕࡞ΕΔ ̂ Y = 1

-FBSOJOH/PO%JTDSJNJOBUPSZ1SFEJDUPST "MHPSJUIN "OBMZTJT w ΞϧΰϦζϜ֓ཁɿ w σʔληοτΛ൒෼ʹ෼͚Δ w ยํΛ࢖͍&RVBMJ[FEPEET੍໿ΛՊֶͨ͠शΛߦ͏ w
΋͏ยํΛ࢖͍QPTUIPDͰ੍໿Λຬͨ͢Α͏ʹ͢Δ maxy,s (VC(ℱ) + ln(1/δ))/(nPy,s ) ൚Խޡࠩ maxy,s ln(1/δ)/(nPy,s ) ެฏੑޡࠩ

-FBSOJOH/PO%JTDSJNJOBUPSZ1SFEJDUPST "MHPSJUIN "OBMZTJT w ΞϧΰϦζϜ֓ཁɿ w σʔληοτΛ൒෼ʹ෼͚Δ w ยํΛ࢖͍&RVBMJ[FEPEET੍໿ΛՊֶͨ͠शΛߦ͏ w
΋͏ยํΛ࢖͍QPTUIPDͰ੍໿Λຬͨ͢Α͏ʹ͢Δ maxy,s (VC(ℱ) + ln(1/δ))/(nPy,s ) ൚Խޡࠩ maxy,s ln(1/δ)/(nPy,s ) ެฏੑޡࠩ Ұ൪σʔλ͕গͳ͍ ͷαϯϓϧ਺ (y, s) ൚Խޡࠩ͸7$࣍ݩʹ ґଘ ެฏੑ͸7$࣍ݩʹ ඇґଘ

5SBJOJOH8FMM(FOFSBMJ[JOH$MBTTJpFSTGPS 'BJSOFTT.FUSJDTBOE0UIFS%BUB %FQFOEFOU$POTUSBJOUT<$PUUFS > w <8PPEXPSUI >ͷ͞ΒͳΔҰൠԽ w ໨తؔ਺͕෼ྨޡࠩ w
੍໿͕ެฏੑ w 3FEVDUJPOͱࣅͨઓུͰΞϧΰϦζϜΛߏங w ͨͩ͠<8PPEXPSUI >ͱಉ༷ͷઓུ΋࢖͏ w σʔληοτΛͭʹ෼͚ͯ໨తؔ਺ͱ੍໿ΛผʑʹධՁ minθ 𝔼[ℓ0 (X, θ)]TVCUP𝔼[ℓi (X, θ)] ≤ 0 ର৅ͱ͢Δ໰୊

5SBJOJOH8FMM(FOFSBMJ[JOH$MBTTJpFSTGPS 'BJSOFTT.FUSJDTBOE0UIFS%BUB %FQFOEFOU$POTUSBJOUT"OBMZTJT w ࠷దԽख๏Ͱੜ͡Δޡࠩ w ੍໿਺ w ෼ྨثͷෳࡶ͞ ϵ
m Rn (ℱ) ϵ + Rn (ℱ) + ln(1/δ)/n ໨తؔ਺ͷ൚Խޡࠩ (m ln(1/ϵ) + ln(m/δ))/n ੍໿ͷ൚Խޡࠩ

5SBJOJOH8FMM(FOFSBMJ[JOH$MBTTJpFSTGPS 'BJSOFTT.FUSJDTBOE0UIFS%BUB %FQFOEFOU$POTUSBJOUT"OBMZTJT ϵ + Rn (ℱ) + ln(1/δ)/n ໨తؔ਺ͷ൚Խޡࠩ
(m ln(1/ϵ) + ln(m/δ))/n ੍໿ͷ൚Խޡࠩ ൚Խޡࠩ͸ෳࡶ͞ʹ ґଘ ެฏੑ͸੍໿਺ͷΈʹ ґଘ w ࠷దԽख๏Ͱੜ͡Δޡࠩ w ੍໿਺ w ෼ྨثͷෳࡶ͞ ϵ m Rn (ℱ)

*OEJWJEVBMGBJSOFTTͷֶशཧ࿦

1SPCBCMZ"QQSPYJNBUFMZ.FUSJD'BJS -FBSOJOH<3PUICMVN > w Ͱ͋Δ֬཰Λฦ͢෼ྨث Λֶश w ΰʔϧ w ্هͷ੍໿ͷ΋ͱਅͷϥϕϧͱ
ͷ ଛࣦΛ࠷খԽ ̂ Y = 1 h : 𝒳 → [0,1] h ℓ0 ℙx,x′ {|h(x) − h(x′ )| > d(x, x′ ) + γ} ≤ α ݸਓެฏੑ੍໿ (γ, α)

1SPCBCMZ"QQSPYJNBUFMZ.FUSJD'BJS -FBSOJOH"MHPSJUIN "OBMZTJT maxi,j max(0,|h(x) − h(x′ )| − d(xi
, xj )) ≤ γ ֶश࣌ͷ੍໿ ެฏੑҧ൓ʹേଇ ͳΒ͹ΞϧΰϦζϜ͸ޡࠩ ͷ ެฏͳ෼ྨث Λग़ྗ͢Δɽ m = O(poly(1/ϵα ,1/ϵγ ,1/ϵ)) ϵ (α + ϵα , γ + ϵγ ) h αϯϓϧෳࡶ౓ ۙࣅతͳެฏੑ͕-FBSOBCMFͰ ͋Δ͜ͱΛূ໌ ҉߸తͳPOFXBZؔ਺ͷଘࡏԼͰ͸׬શͳެฏੑ͸ࢦ਺తܭࢉ͕࣌ؒඞཁ

ެฏͳόϯσΟοτ໰୊ڧԽֶश ΦϯϥΠϯֶश

όϯσΟοτ໰୊ w ஞ࣍ҙࢥܾఆ໰୊ w ໨త w ྦྷੵใु ͷ࠷େԽ w ͕༩͑ΒΕͨ࣌
Λ બ୒ w ∑T t=1 r(t) x(t) 1 , . . . , x(t) K i r(t) = fi (x(t) i ) όϯσΟοτ໰୊ ΦϯϥΠϯਪનͳͲͷԠ༻ ݸਓԽ޿ࠂਪન 8FCهࣄਪન 4/4༑ਓਪન Ϣʔβ৘ใ x(t) ޿ࠂ i(t) ϑΟʔυόοΫr(t)

'BJSCBOEJU<+PTFQI > w ใुͷେ͖ͦ͞ͷਓͷೳྗͷߴ͞ w ೳྗ͕ߴ͍ਓҎ֎Λ༏ઌతʹબΜͰ͸͍͚ͳ͍ πi (t) > πj
(t)POMZJGfi (x(t) i ) > fj (x(t) j ) ೳྗओٛతެฏੑ " # $ % & fi (x(t) i )

'BJSCBOEJU"MHPSJUIN w ৴པ͕۠ؒ൪ฏۉ͕ߴ͍ਓʹ࿈ͳΔूஂ͔ΒҰ༷ʹͱΔ τοϓूஂ ͔ΒҰ༷αϯϓϧ

'BJSCBOEJU3FHSFU w ௨ৗจ຺෇͖όϯσΟοτͰͷϨάϨοτղੳ w ཧ૝తͳใु͔ΒͲΕ͚ͩใुΛऔΓಀ͕͔ͨ͠ K3T ln(Tk/δ) ௨ৗόϯσΟοτͷϨάϨοτ T4/5K6/5d3/5
∨ k3 ln(k/δ) ઢܗจ຺෇͖όϯσΟοτͷϨάϨοτ ͕஌ΒΕ͓ͯΓ ΋ূ໌ Ω( T) Ω( K3 ln(1/δ)) ΄΅λΠτͳ ό΢ϯυ ௨ৗ͸ TKd ln(T)

$BMJCSBUFE'BJSOFTTJO#BOEJUT<-JV > w ݸਓެฏੑ੍໿Λຬͨ͠$BMJCSBUJPOҧ൓Λ࠷খʹ͢Δ πi (t) ≠ ℙ{i = arg
maxj rj } $BMJCSBUJPOҧ൓ D(π(t) i , π(t) j ) ≤ ϵ1 D(ri , rj ) + ϵ2 ݸਓެฏੑ੍໿

$BMJCSBUFE'BJSOFTTJO#BOEJUT"MHPSJUIN 5IPNQTPOTBNQMJOHΛݩͷԼ ΞϧΰϦζϜ ࣄޙฏۉΛ࠷େʹ͢ΔͷͰ͸ͳ͘ ࣄޙ෼෍͔ΒαϯϓϧԼใुΛ࠷େʹ ࿹ΛબΜͩճ਺͕গͳ͍ ৔߹͸Ұ༷αϯϓϧ ݸਓެฏੑͷอূ $BMJCSBUJPOҧ൓ Λগͳ͘

$BMJCSBUFE'BJSOFTTJO#BOEJUT"MHPSJUIN 5IPNQTPOTBNQMJOHΛݩͷԼ ΞϧΰϦζϜ ࣄޙฏۉΛ࠷େʹ͢ΔͷͰ͸ͳ͘ ࣄޙ෼෍͔ΒαϯϓϧԼใुΛ࠷େʹ ࿹ΛબΜͩճ਺͕গͳ͍ ৔߹͸Ұ༷αϯϓϧ ݸਓެฏੑͷอূ $BMJCSBUJPOҧ൓ Λগͳ͘
(KT)2/3 $BMJCSBUJPOҧ൓ͷϨάϨοτ ֬཰ Ͱ 1 − δ D(π(t) i , π(t) j ) ≤ 2D(ri , rj ) + ϵ2 ݸਓެฏੑ੍໿

0OMJOF-FBSOJOHXJUIBO6OLOPXO 'BJSOFTT.FUSJD<(JMMFO > w ઢܗจ຺෇͖όϯσΟοτ໰୊ ݸਓެฏੑ੍໿ w ڑ཭വ਺Λ஌Βͳ͍ w ୅ΘΓʹϑΟʔυόοΫʹڑ཭৘ใؚ͕·ΕΔ
|πi (t) − πj (t)| ≤ d(x(t) i , x(t) j ) ݸਓެฏੑ੍໿ Ϣʔβ৘ใx(t) i ޿ࠂ෼෍π(t) ϑΟʔυόοΫr(t) ڑ཭ΦϥΫϧO(t) ੍໿ҧ൓͍ͯ͠Δ ϖΞΛฦ͢ ࣮ࡍʹ͸ ͙Β͍ͷ ҧ൓Λڐ͢ ϵ

0OMJOF-FBSOJOHXJUIBO6OLOPXO 'BJSOFTT.FUSJD"MHPSJUIN w ใु෼෍ͷਪఆͱڑ཭ͷਪఆ͕ಘΒΕΕ͹͋ͱ͸-1 w ใु෼෍ͷਪఆ w ͔Β௨ৗͷઢܗόϯσΟοτͱಉ࣌Α͏ʹਪఆ w ڑ཭ͷਪఆ
w ڑ཭ΦϥΫϧ͔Βਪఆ r(t) maxπ∈ΔK ∑K i=1 ri πi TVCUP|πi − πj | ≤ dij ղ͘΂͖-1 ใु࠷େ ެฏੑ੍໿

0OMJOF-FBSOJOHXJUIBO6OLOPXO 'BJSOFTT.FUSJD3FHSFU 'BJSOFTT w 'BJSOFTT͸ެฏੑ੍໿Λҧ൓ͨ͠ճ਺ w ·ͱΊΔͱ ͕ ʹൺ΂͋Δఔ౓খ͍͞ͳΒ w
3FHSFU w 'BJSOFTT K, d T d T ln(T/δ) K2d2 ln(TKd) K2d2 ln(kdT/ϵ) + K3ϵT + d T ln(T/δ) 3FHSFU K2d2 ln(d/ϵ) 'BJSOFTT Ͱ ʹؔ ͯ͠͸΄΅࠷ద ϵ = 1/K3T T

ڧԽֶश w ڧԽֶश w ໨త w ׂҾใु ͷ࠷େԽ ∑∞ t=τ
γt−τr(t) ঢ়ଶs(t) ߦಈa(t) ϑΟʔυόοΫr(t) ߦಈʹΑͬͯঢ়ଶભҠ ঢ়ଶʹΑͬͯใु෼෍͕มΘΔ

'BJSOFTTJO3FJOGPSDFNFOU-FBSOJOH <+BCCBSJ > w ڧԽֶशʹ͓͍ͯ<+PTFQI >ͱಉ༷ͳެฏੑ੍໿ w ݁Ռ w &YBDUͳެฏੑΛୡ੒͢Δʹ͸ࢦ਺తࢼߦ͕ඞཁ
w ࠷దͳϙϦγʔ͕ಘΒΕΔ·Ͱʹे෼ͳεςοϓ਺͸ w ׂҾ཰ ʹ͍ͭͯ͸ࢦ਺తͰ͜Ε͕࠷ద w ଞͷ߲ʹؔͯ͠͸ଟ߲ࣜ ϵ 1/(1 − γ) πi (t) > πj (t)POMZJGfi (s(t) i ) > fj (s(t) j ) ೳྗओٛతެฏੑ

ެฏͳόϯσΟοτ໰୊ڧԽֶश ΦϯϥΠϯֶश·ͱΊ w όϯσΟοτ໰୊ڧԽֶशΦϯϥΠϯֶश w ஞ࣍ҙࢥܾఆ໰୊ʹ͓͚Δใु࠷େԽ w ެฏͳόϯσΟοτ໰୊ڧԽֶशΦϯϥΠϯֶश w ஞ࣍ҙࢥܾఆ໰୊
બ୒֬཰ʹެฏੑ੍໿ w ༷ʑͳઃఆͰͷཧ࿦ղੳ

࠷ۙ࿩୊ͷτϐοΫ

%FMBZFE&⒎FDU<-JV > w ֶशͱςετͷؒʹ࣌ؒతִͨΓ͕͋Δ w ͦͷؒʹαϯϓϧͷ෼෍͕มԽ͢Δ w %FNPHSBQIJDQBSJUZͷਖ਼౰ੑ ೖࢼ
w ශࠔ૚ͷֶੜΛऔΒͳ͍͜ͱͰকདྷශࠔ͕֦େ͢Δ͜ͱͷ๷ࢭ w %1 &0ͷ੍໿Λ͚ͭͨ࣌༧ଌ࣌ͷੑೳ͸Ͳ͏ͳΔ͔ ࣌ࠁ σʔλऩू ֶश ༧ଌ αϯϓϧͷ෼෍͕มԽ

)VNBOJOUIF-PPQ w ݱ࣮తʹ͸ֶश͸ճͰ͸ऴΘΒͳ͍ w γεςϜΛΞοϓσʔτ͠ଓ͚Δ w Ҏલʹߦͬͨҙࢥܾఆֶ͕शσʔλʹӨڹ͢Δ f

ެฏੑΛ૷͏ٕज़ w ຊ౰͸ෆެฏͰ͋Δͷʹ͔͔ΘΒͣੈؒʹ͸ެฏʹݟ͔ͤ ͚Δ w lෆެฏͳϞσϧz͔Βzެฏͳઆ໌z͕ੜ੒Ͱ͖Δ <"JWPEKJ > w lෆެฏͳϞσϧz͔Βzެฏͳϥϕϧ͚ͮσʔληοτzΛ
ੜ੒Ͱ͖Δ<'VLVDIJ > f ͜Ε͸ෆެฏ આ໌ ϕϯνϚʔΫσʔλ ͜Ε͸ެฏ

·ͱΊ w ެฏੑͷ࠷৽ݚڀΛ঺հ w ࠓ೔ͷ಺༰ w ෼ྨɼճؼʹ͓͚Δެฏੑͷอূํ๏ w 3FEVDUJPOBQQSPBDI w
දݱֶशʹ͓͚Δެฏੑͷอূํ๏ w 'BJSͳදݱɼ*OWBSJBOU'FBUVSF w ެฏੑԼʹ͓͚Δֶशཧ࿦ w όϯσΟοτ໰୊ʹ͓͚Δެฏੑ w ͦͷଞ࿩୊ͷτϐοΫ

3FGFSFODFT • [Hardt+16] Moritz Hardt, Eric Price, and Nathan Srebro.
Equality of Opportunity in Supervised Learning. In: NeurIPS, pp. 3315-3323, 2016. https://arxiv.org/abs/1610.02413 • [Pleiss+17] Geoff Pleiss, Manish Raghavan, Felix Wu, Jon Kleinberg, and Kilian Q. Weinberger. On Fairness and Calibration. In: NeurIPS, pp. 5680-5689, 2017. https://arxiv.org/ abs/1709.02012 • [Dwork+12] Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, Rich Zemel. Fairness Through Awareness. In: the 3rd innovations in theoretical computer science conference, pp. 214-226, 2012. https://arxiv.org/abs/ 1104.3913

3FGFSFODFT • [Agarwal+18] Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John
Langford, and Hanna Wallach. A Reductions Approach to Fair Classification. In: ICML, PMLR 80, pp. 60-69, 2018. https://arxiv.org/abs/1803.02453 • [Agarwal+19] Alekh Agarwal, Miroslav Dudík, and Zhiwei Steven Wu. Fair Regression: Quantitative Definitions and Reduction-based Algorithms. In: ICML, PMLR 97, pp. 120-129, 2019. https://arxiv.org/abs/1905.12843 • [Zafar+13] Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning Fair Representations. In: ICML, PMLR 28, pp. 325-333, 2013.

3FGFSFODFT • [Zhao+19] Han Zhao, Geoffrey J. Gordon. Inherent Tradeoffs
in Learning Fair Representations. In: NeurIPS, 2019, to appear. https://arxiv.org/abs/1906.08386 • [Xie+16] Qizhe Xie, Zihang Dai, Yulun Du, Eduard Hovy, Graham Neubig. Controllable Invariance through Adversarial Feature Learning. In: NeurIPS, pp. 585-596, 2016. https://arxiv.org/abs/1705.11122 • [Moyer+18] Daniel Moyer, Shuyang Gao, Rob Brekelmans, Greg Ver Steeg, and Aram Galstyan. Invariant Representations without Adversarial Training. In: NeurIPS, pp. 9084-9893, 2018. https://arxiv.org/abs/1805.09458

3FGFSFODFT • [Woodworth+18] Blake Woodworth, Suriya Gunasekar, Mesrob I. Ohannessian,
Nathan Srebro. Learning Non-Discriminatory Predictors. In: COLT, pp. 1920-1953, 2017. https://arxiv.org/abs/ 1702.06081 • [Cotter+19] Andrew Cotter, Maya Gupta, Heinrich Jiang, Nathan Srebro, Karthik Sridharan, Serena Wang, Blake Woodworth, Seungil You. Training Well-Generalizing Classifiers for Fairness Metrics and Other Data-Dependent Constraints. In: ICML, PMLR 97, pp. 1397-1405, 2019. https:// arxiv.org/abs/1807.00028 • [Rothblum+18] Guy N. Rothblum, Gal Yona. Probably Approximately Metric-Fair Learning. In: ICML, PMLR 80, pp. 5680-5688, 2018. https://arxiv.org/abs/1803.03242

3FGFSFODFT • [Joseph+16] Matthew Joseph, Michael Kearns, Jamie Morgenstern, Aaron
Roth. Fairness in Learning: Classic and Contextual Bandits. In: NeurIPS, pp. 325-333, 2016. • [Liu+17] Yang Liu, Goran Radanovic, Christos Dimitrakakis, Debmalya Mandal, David C. Parkes. Calibrated Fairness in Bandits. In: 4th Workshop on Fairness, Accountability, and Transparency in Machine Learning (FATML), 2017. https://arxiv.org/abs/1707.01875 • [Gillen+18] Stephen Gillen, Christopher Jung, Michael Kearns, Aaron Roth. Online Learning with an Unknown Fairness Metric. In: NeurIPS, pp. 2600-2609, 2018. https:// arxiv.org/abs/1802.06936

3FGFSFODFT • [Jabbari+17] Shahin Jabbari, Matthew Joseph, Michael Kearns, Jamie
Morgenstern, Aaron Roth. Fairness in Reinforcement Learning. In: ICML, PMLR 70, pp. 1617-1626, 2017. https://arxiv.org/abs/1611.03071 • [Liu+18] Lydia T. Liu, Sarah Dean, Esther Rolf, Max Simchowitz, Moritz Hardt. Delayed Impact of Fair Machine Learning. In: ICML, PMLR 80, pp. 3150-3158, 2018. https://arxiv.org/abs/ 1803.04383 • [Aivodji+19] Ulrich Aïvodji, Hiromi Arai, Olivier Fortineau, Sébastien Gambs, Satoshi Hara, Alain Tapp. Fairwashing: the risk of rationalization. In: ICML, 2019. https://arxiv.org/abs/1901.09749 • [Fukuchi+20] Kazuto Fukuchi, Satoshi Hara, Takanori Maehara. Faking Fairness via Stealthily Biased Sampling. In: AAAI, Special Track on AI for Social Impact (AISI), 2020, to appear. https://arxiv.org/abs/ 1901.08291

*$.-/FVS*14 QBQFST

5SBJOJOH8FMM(FOFSBMJ[JOH$MBTTJpFSTGPS 'BJSOFTT.FUSJDTBOE0UIFS%BUB %FQFOEFOU$POTUSBJOUT<$PUUFS *$.-`> w IUUQTBSYJWPSHBCT w σʔλґଘ੍໿ͷ൚Խੑೳͷղੳ w <8PPEXPSUI
$0-5`>ͷ֦ு w <"HBSXBM *$.-`>Λݩʹղੳ

'BJSOFTTXJUIPVU)BSN%FDPVQMFE $MBTTJpFSTXJUI1SFGFSFODF(VBSBOUFFT <6TUVO *$.-`> w IUUQQSPDFFEJOHTNMSQSFTTWVTUVOB w 1SFGFSFODFCBTFEͳެฏੑΛอূ w SBUJPOBMJUZͱFOWZGSFFOFTTΛ໨ࢦֶͨ͠शख๏
w <;BGBS *$.-`>ͷޙଓ w <%XPSL *$.->ͷ֦ு

'BJSOFTTSJTLNFBTVSFT <8JMMJBNTPO *$.-> w IUUQTBSYJWPSHBCT w TVCHSPVQΛߟྀͨ͠৽͍͠'BJSOFTTTDPSFΛ༩͑Δ w ࣋ͭ΂͖ެཧΛఆ͔ٛͯͦ͜͠ΒείΞΛٻΊΔ w
ͦͷఆٛͷ΋ͱֶशΞϧΰϦζϜΛ༩͑Δ w <;BGBS 888`> <%XPSL *$.-`> <%POJOJ /FVS*14`>ͷޙଓ

0OUIF-POHUFSN*NQBDUPG"MHPSJUINJD%FDJTJPO 1PMJDJFT&⒎PSU6OGBJSOFTTBOE'FBUVSF4FHSFHBUJPO UISPVHI4PDJBM-FBSOJOH<)FJEBSJ *$.-`> w IUUQTBSYJWPSHBCT w F⒎PSUΛಋೖͯ͠ɼ౒ྗͨ͠ྔʹର͢Δใुͷҧ͍ʹΑͬ ͯެฏੑΛఆٛ

0CUBJOJOHGBJSOFTTVTJOHPQUJNBM USBOTQPSUUIFPSZ<#BSSJP *$.-`> w IUUQTBSYJWPSHBCT w 8BTTTFSTUBJOEJTUBODFΛ࢖ͬͯ%JTQBSFUF*NQBDU TDPSFΛ֦ு w ࠷దͳ
BQQSPYJNBUFMZ 'BJSͳσʔλ෼෍ΛٻΊΔΞϧΰ ϦζϜΛ։ൃ w <'FMENBO ,%%`>ͷޙଓ

'BJS3FHSFTTJPO2VBOUJUBUJWF%FpOJUJPOT BOE3FEVDUJPOCBTFE"MHPSJUINT <"HBSXBM `*$.-> w IUUQTBSYJWPSHBCT w ճؼʹ͓͚Δެฏੑ w Ϧϓγοπؔ਺Λ༧ଌ͢Δ໰୊
w %FNPHSBQIJDQBSJUZ͔zશͯͷάϧʔϓͷଛࣦ͕ҰఆҎ Լz͕ެฏੑ੍໿ w ࣗ਎ͷ<"HSBXBM *$.-`>ͷޙଓ

%J⒎FSFOUJBMMZ1SJWBUF'BJS-FBSOJOH <+BHJFMTLJ *$.-`> w IUUQTBSYJWPSHBCT w ࠩ෼ϓϥΠόγʔͱಉ࣌ʹ&RVBMJ[FEPEETΛอূ͢Δํ ๏Λ։ൃ w 3FEVDUJPOBQQSPBDI<"HSBXBM
*$.-`>ͱಉ༷ͳํ๏ ͰެฏੑͷอূΛߦ͏

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ଞͷࢦඪ͸ಠཱ͔ͦ͏Ͱͳ͍͔͔͠Θ͔Βͳ͍

5IFJNQMJDJUGBJSOFTTDSJUFSJPOPG VODPOTUSBJOFEMFBSOJOH<-JV *$.-`> w IUUQTBSYJWPSHBCT w ੍໿Λઃ͚ͳ͍ϦεΫ࠷খԽ͕$BMJCSBUJPOΛୡ੒͢Δ࠷ ΋ྑ͍ํ๏Ͱ͋Δ͜ͱΛূ໌

'MFYJCMZ'BJS3FQSFTFOUBUJPO-FBSOJOHCZ %JTFOUBOHMFNFOU<$SFBHFS *$.-`> w IUUQTBSYJWPSHBCT w 'BJSͳηϯγςΟϒଐੑͷ෼෍มԽʹBEBQUJWFͳදݱֶश ͷํ๏ͷ։ൃ w 7"&Λݩʹͨ͠ख๏

-FBSOJOH0QUJNBM'BJS1PMJDJFT </BCJ *$.-`> w IUUQTBSYJWPSHBCT w DBVTBMϕʔεͷެฏੑఆٛ w ஞ࣍ҙࢥܾఆʹ͓͚Δެฏੑ w
࠷దͳҙࢥܾఆࢦ਑ͱͦΕΛਪఆ͢ΔΞϧΰϦζϜͷ։ൃ w ࣗ਎ͷ</BCJ """*`>ͷ֦ு

5PXBSE$POUSPMMJOH%JTDSJNJOBUJPOJO 0OMJOF"E"VDUJPOT<$FMJT *$.-`> w IUUQTBSYJWPSHBCT w "EBMMPDBUJPOʹ͓͚Δෆެฏʹରॲ w 'BJSDPWFSBHFΛୡ੒͢ΔΑ͏ʹ੍໿ w
SFWFOVFͷ੍໿෇͖OPODPOWFY࠷దԽΛߦ͏͕ɼGBDU DPOWFSHFODFΛূ໌

4UBCMFBOE'BJS$MBTTJpDBUJPO <)VBOH *$.-`> w IUUQTBSYJWPSHBCT w 3FHVMBSJ[FE&3.ʹެฏੑ੍໿PSެฏੑਖ਼ଇԽΛೖΕ ͨ࣌ͷTUBCJMJUZΛղੳ

/PJTFUPMFSBOUGBJSDMBTTJpDBUJPO <-BNZ /FVS*14`> w IUUQTBSYJWPSHBCT w ηϯγςΟϒଐੑʹϊΠζ͕ೖ͍ͬͯΔ৔߹ʹ͓͚Δֶश ΞϧΰϦζϜ w ϊΠζΛߟྀͯ͠ΑΓڧ੍͍໿ΛՊ͢
w ࠩ෼ϓϥΠϕʔτͳηϯγςΟϒଐੑΛ࢖ֶͬͨशͱͯ͠ Έͨͱ͖ͷղੳ΋͋Γ

5IF'BJSOFTTPG3JTL4DPSFT#FZPOE $MBTTJpDBUJPO#JQBSUJUF3BOLJOHBOEUIF Y"6$.FUSJD<,BMMVT /FVS*14`> w IUUQTBSYJWPSHBCT w είΞϕʔεͷϥϯΩϯά༧ଌ໰୊ʹ͓͚Δެฏੑ w ৽͘͠ఆٛͨ͠Y"6$ͷҰகੑʹΑͬͯެฏੑΛఆٛ

1PMJDZ-FBSOJOHGPS'BJSOFTTJO3BOLJOH <4JOHI /FVS*14`> w IUUQTBSYJWPSHBCT w จষηοτͱͦͷSFMFWBODFTDPSF͕༩͑ΒΕΔ w ͦͷ্Ͱެฏੑʹ഑ྀͭͭ͠VUJMJUZΛ࠷େʹ͢Δ֬཰తϥ ϯΩϯάΛٻΊΔ

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