非定常な多腕バンディット問題における変化検出アプローチの線形モデルへの拡張/Extension of Change Detection Method for Non-Stationary Linear Multi Armed Bandit

2020.07.10 第50回 情報処理学会 インターネットと運用技術研究会
https://www.iot.ipsj.or.jp/meeting/50-program/

1. ࡾ୐ ༔հ, ܀ྛ ݈ଠ࿠ / Pepabo R&D Institute, GMO Pepabo, Inc. 2020.07.10 ୈ50ճ ৘ใॲཧֶձ Πϯλʔωοτͱӡ༻ٕज़ݚڀձ ඇఆৗͳଟ࿹όϯσΟοτ໰୊ʹ͓͚Δ มԽݕग़ΞϓϩʔνͷઢܗϞσϧ΁ͷ֦ு
2. 1. ͸͡Ίʹ 2. ؔ࿈ݚڀ 3. ఏҊख๏ 4. ධՁ 5. ·ͱΊ 2 ໨࣍
3. 1. ͸͡Ίʹ
4. • WebαʔϏεʹ͓͚Δ৘ใաଟ໰୊Λղܾ͢ΔɺਪનγεςϜͷಋೖ • → ͳΜΒ͔ͷํࡦʹج͖ͮଟ਺ͷબ୒ࢶ͔Βར༻ऀ͕ڵຯΛ࣋ͭ΋ͷΛఏ Ҋ͢ΔγεςϜ 4 എܠ • ঎඼ • ޿ࠂ • ਪનΞϧΰϦζϜ
5. • WebαʔϏεʹ͓͚Δ৘ใաଟ໰୊Λղܾ͢ΔɺਪનγεςϜͷಋೖ • → ͳΜΒ͔ͷํࡦʹج͖ͮଟ਺ͷબ୒ࢶ͔Βར༻ऀ͕ڵຯΛ࣋ͭ΋ͷΛఏ Ҋ͢ΔγεςϜ 5 എܠ • γεςϜʹͱͬͯར༻ऀͷᅂ޷͸ະ஌ʢਪଌͷͨΊͷ৘ใ஝ੵ͕ඞཁʣ • ͋Δ࣌఺Ͱ༗༻ͱߟ͑ΒΕΔ৘ใΛʮ׆༻ʯͨ͠ఏҊΛߦ͍ͭͭɺΑΓ༗༻ͳ ఏҊʹͭͳ͕Δ৘ใͷʮ୳ࡧʯ͕ٻΊΒΕΔʢػձଛࣦͷճආʣ ӡ༻্ͷ՝୊
6. • ͋Δ࣌఺ʹ͓͚Δ࠷ળͳબ୒ͷͨΊɺ׆༻ͱ୳ࡧͷτϨʔυΦϑΛղܾ͍ͨ͠ • → ଟ࿹όϯσΟοτ໰୊ͱΈͳͯ͠ɺಉ໰୊ͷղ๏Λద༻[1] 6 ݚڀͷ໨త <>
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7. • ͋Δ࣌఺ʹ͓͚Δ࠷ળͳબ୒ͷͨΊɺ׆༻ͱ୳ࡧͷτϨʔυΦϑΛղܾ͍ͨ͠ • → ଟ࿹όϯσΟοτ໰୊ͱΈͳͯ͠ɺಉ໰୊ͷղ๏Λద༻ 7 ݚڀͷ໨త • ҰํͰɺैདྷͷಉ໰୊ͷղ๏Ͱ͸ɺར༻ऀͷᅂ޷͕ʮଟ༷͔ͭܧଓతʹมԽʯ ͢ΔΑ͏ͳ؀ڥʹॆ෼ʹରԠɾ௥ै͢Δ͜ͱ͕Ͱ͖ͳ͍ʢؔ࿈ݚڀͰઆ໌ʣ • ͜ͷ؀ڥʹରԠՄೳͳղ๏ΛఏҊ͢Δ͜ͱͰɺར༻ऀͷᅂ޷ͷଟ༷ੑ΍มԽʹ ௥ै͢ΔఏҊΛՄೳʹ͍ͨ͠
8. 2. ؔ࿈ݚڀ
9. • ʮ࿹ʯͱݺ͹ΕΔෳ਺ͷީิ͔ΒಘΒΕΔใुΛ࠷େԽ͢Δ໰୊ • ϓϨΠϠʔ͸Ұ౓ͷࢼߦͰ1ͭͷ࿹Λબ୒͠ɺใुΛಘΔ • ͦΕͧΕͷ࿹͸͋Δใु෼෍ʹै͍ใुΛੜ੒ • ͨͩ͠ɺϓϨΠϠʔ͸͜ͷใु෼෍Λࢼߦͷ݁Ռ͔Βਪଌ͢Δඞཁ͕͋Δ 9 ଟ࿹όϯσΟοτ໰୊ • ϓϨΠϠʔ͸͋Δ࣌఺ͷ࿹ͷධՁʹج͖ͮʮ׆༻ʯͱʮ୳ࡧʯΛฒߦͯ͠ߦ͏ • ͜ͷτϨʔυΦϑΛղফ͢ΔͨΊʹ༷ʑͳղ๏͕ఏҊ͞Ε͍ͯΔ
10. ଟ࿹όϯσΟοτ໰୊ͷ૝ఆ͢Δ࿹ͷ֬཰෼෍ͷछྨ 10 Arm0 Arm1 Arm2 User(s) System ਪఆͨ֬͠཰෼෍ ਅͷ֬཰෼෍ Recommend Click
11. • ࿹͝ͱͷใु෼෍͸ৗʹಉ͡Ͱ͋Δͱ͍͏Ծఆ • → ঢ়گ΍ଐੑ͝ͱʹใु෼෍͕ҟͳΔͷͰ͸ͳ͍͔ʁ • ྫʣ೥୅͝ͱʹਓؾͷ঎඼͕ҧ͏ɺ࠷ۙɺಉ͡ΧςΰϦͷ঎඼Λങͬͨ • → ࣌ؒͷܦաʹΑͬͯ΋ใु෼෍͕มԽ͢ΔͷͰ͸ͳ͍͔ʁ • ྫʣ৽͍͠χϡʔεهࣄɺҰ࣌తͳྲྀߦ 11 ଟ࿹όϯσΟοτ໰୊ͷ֦ு • ʮઢܗʯ·ͨ͸ʮඇఆৗʯͳଟ࿹όϯσΟοτ໰୊ͱ֦ͯ͠ு͕ߦΘΕ͍ͯΔ
12. • ࿹͝ͱʹෳ਺ͷจ຺͕͋Γɺจ຺ʹԠͯ͡ใु෼෍͕ܾ·Δଟ࿹όϯσΟοτ ໰୊ͷઃఆ • ຊݚڀใࠂͰ͸ɺจ຺͸ɺෳ਺ͷཁҼͷύϥϝʔλͷ૊Έ߹ΘͤͰදݱ͞ Εͨঢ়ଶͷ͜ͱΛࢦ͢ • → ཁҼύϥϝʔλͷ஋͕{0,1}ͷ৔߹ɺจ຺͸ཁҼ਺ ʹରͯ͠ ύλʔϯ d 2d 12 ઢܗͳଟ࿹όϯσΟοτ໰୊ • ઢܗͳଟ࿹όϯσΟοτ໰୊ͷղ๏Ͱ͸ɺจ຺ͷ֬཰෼෍Ͱ͸ͳ͘ɺཁҼ͝ͱ ͷ܎਺ʢઢܗύϥϝʔλʣΛਪఆ͢Δ͜ͱͰ֤จ຺ʹ͓͚ΔใुΛ༧૝͢Δ
13. ઢܗͳଟ࿹όϯσΟοτ໰୊ 13 Arm0 Arm1 Arm2 User(s) System ਪఆͨ֬͠཰෼෍ ਅͷ֬཰෼෍ Recommend Click Context = 0 Context = 0
14. ઢܗͳଟ࿹όϯσΟοτ໰୊ 14 Arm0 Arm1 Arm2 User(s) System ਪఆͨ֬͠཰෼෍ ਅͷ֬཰෼෍ Recommend Click Context = 1 Context = 1
15. • ಉ͡จ຺ʹ͓͍ͯ΋࣌ؒܦաʹΑͬͯใु෼෍͕มԽ͢Δଟ࿹όϯσΟοτ໰ ୊ͷ໰୊ઃఆ • पظతͳมԽͰ͋Ε͹ཁҼͷύϥϝʔλʹؚΊΔ͜ͱͰରԠͰ͖Δ͕ෆن ଇͳ৔߹ʢඇఆৗͳมԽͷ৔߹ʣ͸͜ͷݶΓͰ͸ͳ͍ 15 ඇఆৗͳଟ࿹όϯσΟοτ໰୊ • ඇఆৗͳଟ࿹όϯσΟοτ໰୊ͷղ๏Ͱ͸ɺաڈʹ؍ଌͨ͠ใुʹଊΘΕͣ࿹ ͷධՁΛਝ଎ʹߋ৽͢Δ͜ͱͰ֤จ຺ʹ͓͚ΔใुΛ༧૝͢Δ ׂҾܕΞϓϩʔν มԽݕग़ܕΞϓϩʔν
16. ඇఆৗͳଟ࿹όϯσΟοτ໰୊ 16 Arm0 Arm1 Arm2 User(s) System ਪఆͨ֬͠཰෼෍ ਅͷ֬཰෼෍ Recommend Click t = 0~ t = 100~ t = 0~ t = 100~ t = 99 t = 99
17. ඇఆৗͳଟ࿹όϯσΟοτ໰୊ 17 Arm0 Arm1 Arm2 User(s) System ਪఆͨ֬͠཰෼෍ ਅͷ֬཰෼෍ Recommend Click t = 0~ t = 100~ t = 0~ t = 100~ t = 199 t = 199
18. • ࿹ͷใु෼෍ͷมԽΛલఏʹɺܧଓతʹ࿹ͷධՁΛߋ৽͢ΔΞϓϩʔν • աڈͷࢼߦճ਺ɺใुʹରׂͯ͠Ҿ཰Λಋೖ • → ৽͍͠؍ଌ݁ՌΛॏཁࢹ • UCB1΍Thomson SamplingΛ֦ுͨ͠ղ๏͕ఏҊ͞Ε͍ͯΔ • Discounted UCB, Sliding Window UCB • Dynamic Thompson Sampling, Discounted TS 18 ඇఆৗͳଟ࿹όϯσΟοτ໰୊ͷղ๏ - ׂҾܕ ✓ Trialt Trialt 1 · · · Trialt n Rewardt Rewardt 1 · · · Rewardt n ◆ 0 B B B @ ↵1 ↵2 . . . ↵n 1 C C C A <latexit sha1_base64="sXrCZl7s65TuB2kkSnvZFuw+Fac=">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</latexit>
19. • ࿹ͷใु෼෍ͷมԽΛܖػʹɺ࿹Λ࠶ධՁ͢ΔΞϓϩʔν • ᶃ ؍ଌͨ͠ใुΛ࣌ܥྻσʔλͱΈͳ͠ɺ౎౓ɺมԽݕग़ΛࢼΈΔ • ᶄ มԽΛݕग़ͨ͠৔߹ɺมԽޙͷࢼߦճ਺ɺใुͰ࿹ͷධՁΛߋ৽ • มԽݕग़͸طଘͷख๏Λར༻ՄೳʢPage-Hinkley test, ADWINʣ • UCB1΍Thomson SamplingͱมԽݕग़Λ૊Έ߹Θͤͨղ๏͕ఏҊ͞Ε͍ͯΔ • Adapt-EvE, S-TS-ADWIN 19 ඇఆৗͳଟ࿹όϯσΟοτ໰୊ͷղ๏ - มԽݕग़ܕ ✓ Trialt Trialt 1 · · · Trialt n Rewardt Rewardt 1 · · · Rewardt n ◆ <latexit sha1_base64="U2Q8+JZHKTI1SORJwl0rrVRGhZw=">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</latexit> มԽલ มԽޙ
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21. • ଟ༷͔ͭܧଓతʹมԽ͢Δ؀ڥΛલఏͱ͍ͯ͠ͳ͍ 21 ैདྷख๏ͷ՝୊ ઢܗʢෳ਺ͷจ຺ʣ ඇఆৗ
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24. • ᶃ ؍ଌ͢΂͖࣌ܥྻͷࢦ਺తͳ૿Ճ • ैདྷख๏Ͱ͸୯Ұͷจ຺Ͱൃੜ͢ΔใुͷมԽΛܭଌ • ཁҼͷύϥϝʔλ਺ʹରͯ͠ࢦ਺తʹ૿Ճ͢Δจ຺͝ͱʹ࣌ܥྻΛܭଌ͢Δ ͜ͱʹΑΔਪનγεςϜͷϝϞϦ࢖༻ྔͷ૿Ճ • ᶄ ௥ैੑʹର͢Δվળͷ༨஍ • ׆༻ͱ୳ࡧͷτϨʔυΦϑͷղফͷ࿮૊Έ͕ैདྷख๏ͷ·· • → ࢼߦճ਺ʹରͯ͠Ұఆ 24 ैདྷख๏Λઢܗ͔ͭඇఆৗ΁֦ு͢Δࡍͷ՝୊
25. • ඇఆৗͳଟ࿹όϯσΟοτ໰୊ͷղ๏Λઢܗʹ֦ு • ैདྷख๏Λ൚༻తʹར༻Ͱ͖ɺׂҾܕΞϓϩʔνͱൺֱͯ͠΋ྑ͍ධՁ͕ಘ ΒΕ͍ͯΔɺมԽݕग़ܕΞϓϩʔνͰ͋ΔS-TS-ADWINͷ֦ு • ᶃͷରࡦ: จ຺ͷύλʔϯ͝ͱͰ͸ͳ͘ɺਪఆͨ͠ઢܗύϥϝʔλͷ࿨͔Β มԽݕग़ΛࢼΈΔ • ᶄͷରࡦ: ୳ࡧϋΠύʔύϥϝʔλͷಈతͳௐ੔Λಋೖ͠ɺੵۃతͳ׆༻ͱ ୳ࡧͷ੾Γସ͑ΛࢼΈΔ 25 ఏҊख๏
26. ઢܗύϥϝʔλͷਪఆͱ มԽݕग़
27. • ࢼߦ࣌ͷίϯςΩετ৘ใ ͱใु ͔Β࿹͝ͱͷઢܗύϥϝʔλΛਪఆ • ίϯςΩετ৘ใͷཁҼ਺ʢ=ઢܗύϥϝʔλͷཁҼ਺ʣ͸ ࣍ݩͱ͢Δ b r d 27 ઢܗύϥϝʔλͷਪఆ Bi = Bi + bbT <latexit sha1_base64="unbR7Yhs6Rkk/aTZ4HTBNw48AxE=">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</latexit> Bi = 0 B B B @ b11 b12 · · · b1d b21 b22 · · · b2d . . . . . . . . . . . . bd1 bd2 · · · bdd 1 C C C A <latexit sha1_base64="EnMuxPCN94aEHMciGdHB9I47X5M=">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</latexit> fi = 0 B B B @ br1 br2 . . . brd 1 C C C A <latexit sha1_base64="IJBFM+VMZa5dmBFfd6Csj18HC0o=">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</latexit> fi = fi + br <latexit sha1_base64="uZ3PwZId0gcRyzSKxEc4GfIgZug=">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</latexit> ✓i = 0 B B B @ ✓1 ✓2 . . . ✓d 1 C C C A <latexit sha1_base64="1YjVOLsXj5FWoj4jVQAS3LB2BCc=">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</latexit> ✓i = B 1 i fi <latexit sha1_base64="g09wD2RSWqw0wHG2RtT4ql0+Bd4=">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</latexit> ର֯੒෼͸ͦͷཁҼͷࢼߦճ਺ɻ
    ͦͷଞͷ஋͸ɺର֯੒෼ͱͦͷཁҼͷ ૊Έ߹ΘͤʹΑΔࢼߦճ਺Λද͢ɻ ͦͷཁҼͷྦྷੵใुΛද͢ ͷٯߦྻͱ ͷੵΑΓཁҼ͝ͱͷઢ ܗύϥϝʔλΛਪఆ
    ʢཁҼ͕ഉଞͰ͋Ε͹ฏۉใुֹʣ B f
28. • ਪఆͨ͠ઢܗύϥϝʔλͷ஋ͷ࿨Λ༻͍ͯඇఆৗͳมԽΛݕग़͢Δ • → ཁҼ਺ʹΑΒͳ͍୯Ұͷ஋ͷΈ͔ΒมԽݕग़ΛࢼΈΔʢ ʣ • ADWINʹΑͬͯมԽΛݕग़ͨ͠৔߹ɺมԽલͷࢼߦճ਺ɺྦྷੵใुΛআڈ 2d → d → 1 28 ઢܗύϥϝʔλͷมԽݕग़ ✓i = B 1 i fi <latexit sha1_base64="g09wD2RSWqw0wHG2RtT4ql0+Bd4=">AAACnnichVHLLgRBFD3a+z3YEJuJCbExqUZCJBIhERvxHEYMne5WQ0W/0l0zCZ2JvR+wsCKxEAlLH2DjByx8gliS2Fi409OJILid6jp16p5bp+oaniUCydhjlVJdU1tX39DY1NzS2tae6OhcC9yCb/KM6VqunzX0gFvC4RkppMWzns9127D4urE/U95fL3I/EK6zKg88vmXru47IC1OXRGmJnpxhhzm5x6Ve0sTktCa2wyG1lNeElkixNIsi+ROoMUghjkU3cYscduDCRAE2OBxIwhZ0BPRtQgWDR9wWQuJ8QiLa5yihibQFyuKUoRO7T/9dWm3GrEPrcs0gUpt0ikXDJ2US/eyBXbIXds+u2BN7/7VWGNUoezmg2ahouae1H3evvP2rsmmW2PtU/elZIo/xyKsg717ElG9hVvTFw5OXlYnl/nCAnbNn8n/GHtkd3cApvpoXS3z59A8/BnmhF6MGqd/b8ROsDafVkfTw0mhqaiNuVQN60YdB6scYpjCHRWSo/hEucI0bJanMKvPKQiVVqYo1XfgSSvYDonKaOQ==</latexit> M = d X (B 1 i fi)d <latexit sha1_base64="c84l6UKA0ACsVJICPHfDqHUtk28=">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</latexit> ᶄ
    ઢܗύϥϝʔλͷ૯࿨ ᶅ
    ࣌ܥྻͱͯ͠஋Λอଘ ᶆ
    ஋ ͷมԽΛݕग़ M ᶇ
    มԽલͷ஋Λআڈ ᶃ
    ઢܗύϥϝʔλͷਪఆ A = {Mt 100, Mt 99, · · · , Mt 1} <latexit sha1_base64="nP26vWdqsH93ruEH6JP3kE7SxaE=">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</latexit>
29. ୳ࡧϋΠύʔύϥϝʔλͷ ಈతͳௐ੔
30. • มԽݕग़࣌ɺ࿹ͷࢼߦճ਺͕Ұ౓ݮগ͢Δ͜ͱʹண໨ • → ࿹ͷࢼߦճ਺ʹ൓ͯ͠ɺ୳ࡧϋΠύʔύϥϝʔλΛ૿ݮ 30 ୳ࡧϋΠύʔύϥϝʔλͷಈతͳௐ੔ τ = ∣A∣ ∑ i β ln ∣ Ai ∣ + 1 ֤࿹ͷ΢Οϯυ΢αΠζࢼߦճ਺ ͔ΒٻΊͨ஋ͷ૯࿨Λ༻͍Δ
31. ैདྷͷઢܗͳղ๏΁ͷద༻
32. • ઢܗͳଟ࿹όϯσΟοτ໰୊ͷղ๏Ͱ͋ΔLinear Thompson Samplingͱ LinUCBʹద༻ • ैདྷͷ࿹બఆͷͨΊͷΞϧΰϦζϜΛมߋ͢Δ͜ͱͳ͘ɺਪఆͨ͠ઢܗύϥ ϝʔλͳΒͼʹ୳ࡧϋΠύʔύϥϝʔλΛར༻Մೳ 32 ैདྷͷઢܗͳղ๏΁ͷద༻ ਪఆͨ͠ઢܗύϥϝʔλ ಈతͳ୳ࡧϋΠύʔύϥϝʔλ
33. 4. ධՁ
34. • ඇఆৗͰઢܗͳଟ࿹όϯσΟοτ໰୊ʹର͢ΔੑೳΛධՁ͢ΔγϛϡϨʔγϣ ϯΛߦ͏ɻ • ઢܗͳ؀ڥͱͯ͠ɺཁҼ਺8͔ΒͳΔ256௨Γͷจ຺ʢ͝ͱͷใु෼෍ʣΛ࣋ ͭ࿹0ͱ࿹1Λ༻ҙ͠ɺ2000࣌఺ܦաޙͷྦྷੵใुͳΒͼʹྦྷੵϦάϨοτΛ ֤ղ๏Ͱൺֱɻ • ඇఆৗͳ؀ڥͱͯ͠ɺ࿹1ͷΈ500࣌఺Ͱใु෼෍ΛมԽͤ͞Δɻ • ൺֱ͢Δղ๏͸ɺThompson SamplingɺUCB1ͷΞϧΰϦζϜʹରͯͦ͠Ε ͧΕɺઢܗɺඇఆৗɺઢܗ͔ͭඇఆৗͷ֦ுΛࢪͨ͠΋ͷΛ༻͍Δɻ 34 ࣮ݧઃఆ
35. ࢼߦճ਺ͱྦྷੵϦάϨοτ 35 • มԽݕग़Λ༻͍ͳ͍ख๏Ͱ͸ྦྷੵϦ άϨοτͷ૿Ճ͕ऩଋ͍ͯ͠ͳ͍ • → աڈσʔλͷӨڹ͕ݪҼ • มԽݕग़Λ༻͍ͯ΋จ຺͝ͱʹ؍ଌ ͢Δ৔߹͸ऩଋ·Ͱ͕͔͔࣌ؒΔ • → ਪఆͷͨΊͷσʔλ෼ࢄ͕ݪҼ • ఏҊख๏Λద༻ʹΑΓଟ࿹όϯσΟο τ໰୊ͷղ๏ͷ௥ैੑೳ͕޲্
36. ୳ࡧϋΠύʔύϥϝʔλͷਪҠ 36 • ఏҊख๏Ͱ͸ɺਅͷઢܗύϥϝʔλ ʹมԽ͕͋ͬͨ࣌఺Ҏ߱ͰҰ࣌తʹ ୳ࡧϋΠύʔύϥϝʔλͷ஋্͕ঢ • ఏҊख๏Ͱ͸ɺมԽ࣌ͷΈҰ࣌తʹ ্ঢͤ͞Δ͜ͱ͕Ͱ͖ΔͨΊɺมԽ ͷͳ͍৔߹͸୳ࡧϋΠύʔύϥϝʔλ ͷ஋Λ௿͘อͭ͜ͱ͕Մೳ • → ػձଛࣦͷ௿ݮ
37. ύϥϝʔλͷਪఆ஋ͱ΢Οϯυ΢αΠζͷਪҠ(࿹0) 37 • ࿹0ʹ͓͍ͯɺઢܗͳଟ࿹όϯσΟο τ໰୊ͷղ๏Β͸ɺ࠷ऴతʹઢܗύ ϥϝʔλͷ஋Λਖ਼͘͠ਪఆ͢Δ͜ͱ ͕Ͱ͖ͨ • ਅͷઢܗύϥϝʔλʹมԽ͕ͳ͍؀ ڥͰ͕͋ͬͨɺఏҊख๏Ͱ͸΢Οϯ υ΢αΠζͷมԽ͕ݟΒΕͨɻ • → ϊΠζΛมԽͱͯ͠ݕग़ͯ͠͠ ·ͬͨͨΊɺࠓޙͷରࡦ͕ඞཁ
38. ύϥϝʔλͷਪఆ஋ͱ΢Οϯυ΢αΠζͷਪҠ(࿹1) 38 • ࿹1ʹ͓͍ͯɺैདྷͷઢܗͳଟ࿹όϯ σΟοτ໰୊ͷղ๏͸ɺաڈσʔλ ͷӨڹʹΑΓਅͷ஋ʹରͯ͠ߴ͍ݟ ੵ΋Γ͕ଓ͍ͨ • ఏҊख๏Ͱ͸ɺ΢Οϯυ΢αΠζͷ มԽʹΑΓաڈσʔλΛऔΓআ͍ͨ ݁Ռɺਝ଎ʹ௥ै͢Δ͜ͱ͕Ͱ͖ͨ
39. ύϥϝʔλͷਪఆ஋ͱ΢Οϯυ΢αΠζͷਪҠ(࿹1) 39 • ҰํͰɺఏҊख๏Ͱ͸มԽޙͷਪఆ ஋͕ਅͷ஋ʹରͯ͠Լճͬͯ͠·ͬͯ ͍Δ • ݪҼᶃ: ࿹1͕ඞͣྼΔ࣮ݧઃఆ • → ࿹ͷબ୒ճ਺͕গͳ͍ • ݪҼᶄ: ΢Οϯυ΢αΠζ͕গͳ͍ • → มԽݕग़ʹར༻͢Δܥྻࣗମ Λॖখ͢ΔͨΊݕग़ͷ҆ఆੑ͕ ௿Լ͍ͯ͠ΔՄೳੑ
40. 5. ·ͱΊ
41. • ਪનγεςϜΛ͸͡Ίͱ͢Δɺଟ༷͔ͭܧଓతʹมԽ͢Δ؀ڥΛ૝ఆͨ͠ଟ࿹ όϯσΟοτ໰୊ͷղ๏ΛఏҊͨ͠ • ैདྷͷઢܗɺඇఆৗͳղ๏Λ୯७ʹ૊Έ߹Θͤͨࡍʹൃੜ͢ΔҎԼͷ՝୊Λղ ܾ͢Δ൚༻తͳ֦ுͱͯ͠ఏҊ • ؍ଌ͢΂͖࣌ܥྻͷࢦ਺తͳ૿Ճ → ਪఆͨ͠ઢܗύϥϝʔλͷ࿨Λ༻͍Δ • ௥ैੑ → ಈతͳ୳ࡧϋΠύʔύϥϝʔλΛ༻͍Δ • ධՁͰ͸ɺඇఆৗ͔ͭઢܗͳ؀ڥʹ͓͍ͯैདྷͷղ๏ʹରͯ͠ྦྷੵใुɺྦྷੵ ϦάϨοτʹ͓͍ͯੑೳͷ޲্Λ֬ೝͨ͠ 41 ·ͱΊ
42. • ਪఆͨ͠ઢܗύϥϝʔλͷ”࿨”͕ଟ࣍ݩͷ࣌ܥྻσʔλͷมԽݕग़ʹ༗ޮͰ͋ Δཧ࿦తͳཪ෇͚͸·ͩߦ͍͑ͯͳ͍ • → ଟมྔ࣌ܥྻʹؔ͢ΔมԽݕग़ͷߋͳΔௐ͕ࠪඞཁ • ADWINʹه࿥͢Δ஋ΛADWINͷ΢Οϯυ΢αΠζʹґଘͨ͠஋͔ΒٻΊͯ ͓ΓɺมԽݕग़ʹؔ͢Δ҆ఆੑͷ௿Լʹͭͳ͕͍ͬͯΔՄೳੑ • → ํࣜͷվળΛݕ౼ 42 ٞ࿦