「強化学習」輪読会資料

by icebee16

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ڧԽֶश ྠಡձ #02, #03 @icebee__ CHAPTER 2 ϓϥϯχϯά

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ڧԽֶश ྠಡձ #02, #03 CHAPTER 2 ABOUT ME @icebee__ •ۚ༥ܥۈΊͷࣾձਓೋ೥໨ •ڵຯͱͯ͠͸σʔλ෼ੳ΍ػցֶशɼ౷ܭؔ࿈ͷ࿩ •Kaggle Expert

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CONTENTS Chapter 2 ϓϥϯχϯά 2.1 ४උ 2.1.1 ໨తؔ਺ 2.1.2 ࠷దՁ஋ؔ਺ 2.2 ಈతܭը๏ 2.2.1 ϕϧϚϯ࡞༻ૉ 2.2.2 ϕϧϚϯ࡞༻ૉͷ୯ௐੑ 2.2.3 ಈతܭը๏ͷ਺ཧ 2.2.4 ࠷దํࡦ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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CONTENTS Chapter 2 ϓϥϯχϯά 2.3 ಈతܭը๏ʹΑΔղ๏ 2.3.1 Ձ஋൓෮๏ 2.3.2 ํࡦ൓෮๏ 2.4 ઢܗܭը๏ʹΑΔղ๏ 2.4.1 ओ໰୊ͷಋग़ 2.4.2* ૒ର໰୊ (ࠓճ͸औΓѻΘͳ͍) 2.4.3* ࠷దղͱ࠷దํࡦ (ࠓճ͸औΓѻΘͳ͍) ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.1 ४උ ▸ ϓϥϯχϯά໰୊ ➡ ؀ڥ͕ط஌ͷ৔߹ͷஞ࣍తҙࢥܾఆ໰୊ ➡ ؀ڥ͕ະ஌ͷ৔߹ʹ͋ͨΔڧԽֶश໰୊ΛऔΓѻ͏্Ͱͷج ૅͱͳΔ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.1.1 ໨తؔ਺ ▸ ҎԼͷ໨తؔ਺Λ༻ҙ͢Δ ▸ ্͔ࣜΒಋ͔ΕΔ ͸ॳظঢ়ଶ ʹґଘ͢ΔՄೳੑ͕ߟ͑ΒΕΔ͕ ޙड़(2.2અ)͢Δ໋୊ΑΓ೚ҙͷॳظঢ়ଶʹ͓͍ͯҎԼ͕੒ཱ͢ Δɽ ▸ ΑͬͯఆৗతͳํࡦͷΈΛߟ͑Δ͜ͱͰ໨తؔ਺Λ࠷େԽग़དྷΔɽ ͨͩ͠ɼ ͸ୈmཁૉ͕1Ͱଞͷཁૉ͸θϩͷn࣍ݩϕΫτϧ Ձ஋ؔ਺ͷॏΈ෇͖࿨ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.1.2 ࠷దՁ஋ؔ਺ ▸ ҎԼͷࣜʹͯ࠷దՁ஋ؔ਺(optimal value function)Λಋೖ ▸ Ձ஋ؔ਺ ͸ Λ༻͍ͯҎԼͷ༷ʹهड़Մೳ ▸ ͜ΕΑΓ ͸Ϛϧίϑํࡦͷे෼ੑʹؔ͢ΔܥΛຬͨ͠ɼ࠷ద Ձ஋ؔ਺ʹ͍ͭͯҎԼͷ͕ࣜ੒Γཱͭɽ ॻ੶ࢀߟϖʔδɿp.22 ಉ࣌पล֬཰ɿ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.1.2 ࠷దՁ஋ؔ਺ ▸ ࠷దՁ஋ؔ਺͸ҎԼͷΑ͏ʹॻ͖ԼͤΔɽ ▸ ͜ΕΑΓϕϧϚϯ࠷దํఔࣜ(Bellman optimality equation)ͱݺ ͹ΕΔҎԼͷ͕ࣜ੒Γཱͭɽ Ϧλʔϯͷ࠶ؼੑɿ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.1.2 ࠷దՁ஋ؔ਺ ▸ ࠷దՁ஋ؔ਺͸ҎԼͷΑ͏ʹॻ͖ԼͤΔɽ ▸ ͜ΕΑΓϕϧϚϯ࠷దํఔࣜ(Bellman optimality equation)ͱݺ ͹ΕΔҎԼͷ͕ࣜ੒Γཱͭɽ Ϧλʔϯͷ࠶ؼੑɿ ભҠ֬཰ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2 ಈతܭը๏ ▸ ಈతܭը๏ ➡ ஞ࣍తҙࢥܾఆͷ࠷దԽ໰୊Λղ͘Ξϓϩʔνͷ͜ͱ ➡ ͨͩ͠ɼ۩ମతͳख๏Λࢦ͢΋ͷͰ͸ͳ͘લड़ͷΑ͏ͳΞϓ ϩʔνͷ૯শ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.1 ϕϧϚϯ࡞༻ૉ ▸ ಈతܭը๏Ͱ͸ϕϧϚϯ࡞༻ૉͱݺ͹ΕΔҎԼͷॲཧ Λؔ਺ ʹ܁Γฦ͠ద༻͢Δ͜ͱͰՁ஋ؔ਺ɼ࠷దՁ஋ؔ਺ʹ͚ۙͮΔɽ ▸ ·ͣɼ࣍ͷ̎ͭͷ೚ҙͷؔ਺ʹର͢ΔϕϧϚϯ࡞༻ૉΛಋೖ͢ Δɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.1 ϕϧϚϯ࡞༻ૉ ▸ ϕϧϚϯظ଴࡞༻ૉ ▸ ϕϧϚϯ࠷ద࡞༻ૉ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.1 ϕϧϚϯ࡞༻ૉ ▸ ·ͨɼՃ͑ͯҎԼͷ͍͔ͭ͘ͷه๏Λಋೖ͢Δɽ ▸ ؆ศԽ ▸ ஞ࣍తͳద༻ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.1 ϕϧϚϯ࡞༻ૉ ▸ ઌΜͯࣔͨ͡͠ϕϧϚϯ࡞༻ૉ͸ͦΕͧΕϕϧϚϯํఔࣜͱҎԼ ͷؔ܎Λ࣋ͭɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.2 ϕϧϚϯ࡞༻ૉͷ୯ௐੑ ▸ ิ୊2.1ʢ୯ௐੑͷิ୊ʣ ▸ ೚ҙͷؔ਺ͷ૊ʹରͯ͠ϕϧϚϯ࡞༻ૉΛద༻ͯ͠΋ͦͷେখ ؔ܎͸อͨΕΔɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.2 ϕϧϚϯ࡞༻ૉͷ୯ௐੑ ▸ ҎԼͷදهΛఆٛ͢Δɽ ▸ ͜ͷͱ͖ϕϧϚϯ࡞༻ૉͷఆٛΑΓҎԼ͕ࣔͤΔ ▸ ิ୊2.2 ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ໋୊2.3ʢಈతܭը๏ͷऩଋੑʣ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ໋୊2.3ʢಈతܭը๏ͷऩଋੑʣূ໌ϝϞ ▸ a. ~ (2.14) ▸ ઌʹ্͛ͨิ୊Λ͏·͘࢖͏͚ͩɽ༗քͰ͋Δ͜ͱΛར༻ͯ͠ ೚ҙͷఆ਺ͰڬΈɼk>∞Ͱఆ਺Λখ͘͢͞Δ͚ͩɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ໋୊2.3ʢಈతܭը๏ͷऩଋੑʣূ໌ϝϞ ▸ b. ~ (2.16) ▸ Ձ஋ؔ਺Λํࡦͷk൪໨Λڥʹ෼ׂ͢Δ͚ͩɽ ▸ (2.16) ~ (2.17) ▸ ใुؔ਺ͷఆٛ(1.1)ʹै֤ͬͯεςοϓͷใुΛ্͔Β཈͑Δɽ ▸ (2.16)ͷӈลೋ߲໨ʹ͍ͭͯղ͍ͯ୅ೖɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ໋୊2.3ʢಈతܭը๏ͷऩଋੑʣূ໌ϝϞ ▸ (2.17) Լ~ (2.18) ▸ ʢp.49ͷԼࣜE[]಺ೋ߲໨ɼγͱSͷఴࣈ͸ޡ২Ͱ͸ʁt > kʣ ▸ v(s)ͷظ଴஋͸min(v(s))ͱmax(v(s))ͰڬΊΔɽ ▸ ग़དྷͨࣜͱ(2.17)ͷࠩ෼ΛͱΔɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ໋୊2.4ʢϕϧϚϯํఔࣜͷղͷҰҙੑʣ ▸ എཧ๏Ͱ؆୯ʹূ໌ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ิ୊2.5ʢϕϧϚϯ࡞༻ૉͷॖখੑʣ ▸ ূ໌͔Βೖͬͨ΄͏͕ཧղ͕ૣ͍ɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ิ୊2.5ʢϕϧϚϯ࡞༻ૉͷॖখੑʣূ໌ϝϞ ▸ (2.21), (2.22), (2.23) ▸ ิ୊2.2͋ͨΓΛ༻͍ͯڬΜͰࣔ͢ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.3 ಈతܭը๏ͷ਺ཧ ▸ ิ୊2.5ʢϕϧϚϯ࡞༻ૉͷॖখੑʣ͔Β ▸ (2.23)ͷv’ʹରͯ͠V*Λ୅ೖɼϕϧϚϯ࠷దԽ࡞༻ૉͷෆಈ఺͕ V*Ͱ͋Δ͜ͱΛ౿·͑Ε͹ҎԼͷ͕ࣜ੒ཱ ▸ ͜Ε͸ϕϧϚϯ࠷దԽ࡞༻ૉΛ೚ҙͷؔ਺ʹదԠ͢Δ͜ͱͰ࠷ద Ձ஋ؔ਺ͱͦͷؔ਺ͷڑ཭͕γͷൺ཰Ͱࢦ਺ؔ਺తʹখ͘͞ͳͬ ͍ͯ͘͜ͱΛࣔ͢ɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.4 ࠷దํࡦ ▸ ఆٛ2.6ʢ࠷దํࡦʣ ▸ ͋Δঢ়ଶsͷظ଴ϦλʔϯΛ࠷େʹ͢Δ͜ͱ͸Մೳ͕ͩ೚ҙͷঢ় ଶsʹରͯ͜͠ΕΛݴ͏ͷ͸೉͍͠ɼͱ͍͏͚ͩɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.4 ࠷దํࡦ ▸ ໋୊2.7ʢ࠷దํࡦͷଘࡏੑͱඞཁे෼৚݅ʣ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.4 ࠷దํࡦ ▸ ໋୊2.7ʢ࠷దํࡦͷଘࡏੑͱඞཁे෼৚݅ʣূ໌ϝϞ ▸ ඞཁ৚݅ɼे෼৚݅ ▸ ʹରͯ͠ϕϧϚϯظ଴࡞༻ૉΛߟ͑ͨͱ͖ɼ͜ͷෆಈ఺͸ ิ୊2.4͔ΒҰҙʹ ͕ͩɼ(2.28)ΑΓ Ͱ΋͋ΔɽΑͬͯ ͦΕΒ͸Ұக͍ͯ͠Δ ▸ ଘࡏ͢Δ͜ͱΛࣔ͢ ▸ ਖ਼௚Α͘Θ͔͍ͬͯͳ͍ɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.4 ࠷దํࡦ ▸ ໋୊2.7ʢ࠷దํࡦͷଘࡏੑͱඞཁे෼৚݅ʣ͔Β ▸ ͷఆٛʹ ΛೖΕͯ͋͛Ε͹ҎԼ͕ݴ͑Δ ▸ ॏΈ෇͖Ձ஋ؔ਺ͷఆ͔ٛΒҎԼ ▸ ͢ͳΘͪɼ࠷దํࡦͷֶशͱ͸໨తؔ਺ Λ࠷େʹ ͢Δํࡦͷ୳ࡧ໰୊Ͱ͋Δͱ͍͑Δɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.2.4 ࠷దํࡦ ▸ લड़ͨࣜ͠ʹ͍ͭͯ ͱ ͱͳ͍ͬͯΔཧ༝ͱͯ͠͸ɼ = 0ͱ ͳ͍ͬͯΔঢ়ଶɼ͢ͳΘͪભҠ͢ΔՄೳੑͷͳ͍ঢ়ଶؒͷύε· Ͱ࠷దԽͷର৅ʹ͢Δඞཁ͸ͳ͍͔Βɽ ▸ ໋୊2.7ͷ݁Ռ͸ҎԼͷೋ఺ΑΓॏཁ ▸ ࠷దԽର৅ͷํࡦू߹Λߟ͑Δʹ͋ͨͬͯ୳ࡧൣғ͸ఆৗͳܾ ఆతํࡦͷू߹ͷΈͰ໰୊ͳ͍͜ͱΛอূͨ͜͠ͱɽ ▸ ֶश͍ͯ͠ΔՁ஋ؔ਺͕࠷దՁ஋ؔ਺ͱҰக͢Δ͜ͱͰ࠷దํ ࡦ͔൱͔Λ൑ఆͰ͖Δ͜ͱɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3 ಈతܭը๏ʹΑΔղ๏ ▸ ಈతܭը๏ͷ۩ମతͳ࣮૷ͱͯ͠ҎԼͷ2ͭΛ঺հ ➡ Ձ஋൓෮๏ʢvalue iteration algorithmʣ ➡ ํࡦ൓෮๏ʢpolicy iteration algorithmʣ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ΞϧΰϦζϜ2.1 ▸ ऩଋ͕อূ͞Ε͍ͯΔͷͰ୯७ͳ܁Γฦ͠ɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ ▸ ূ໌ͷৄࡉ͸ิ଍A.1.1(p.250) ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ऴྃ৚݅ͷᮢ஋ ͱٻ·ΔՁ஋ؔ਺ɼํࡦͷ࣭ʹ͍ͭͯؔ܎ੑ Λ֬ೝ ▸ ޡࠩݶքͷࣜ(2.32)ΛҎԼͷΑ͏ʹॻ͖׵͑Δɽ ▸ ऴྃ৚݅ͷᮢ஋ʹ͍ͭͯఆٛΛ༻͍Ε͹ҎԼͷΑ͏ʹมܗՄೳ ▸ ࠷దՁ஋ؔ਺ ͱਪఆՁ஋ؔ਺ ͷ࠷େޡࠩ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ࣜ(2.34)ΑΓऴྃ൑ఆʹ༻͍Δ ͸ࢦ਺ݮਰ͠ॳظ஋͕༗քͰ ͋ΔͨΊՁ஋൓෮๏͸ඞͣ༗ݶͷճ਺Ͱऴྃ͢Δɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ํࡦ ʹର͢ΔํࡦޡࠩΛҎԼͱఆٛ ▸ Ձ஋൓෮๏Ͱٻ·Δํࡦ ʹରͯ͠ϕϧϚϯ࡞༻ૉͷఆ͔ٛΒ ҎԼ͕੒Γཱͭɽʢ΄Μͱ͔ʁʣ ▸ ࣜ(2.31)΍(2.36)Λ༻͍ͯม׵͢Δ͜ͱͰҎԼ͕੒Γཱͭɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ษڧձதͰࢀՃऀͷํ͕ࣔͯͩͬͨ͘͠͞(2.40)ͷิ଍ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ΑͬͯํࡦޡࠩΛᮢ஋ Λ༻͍ͯҎԼͷΑ͏ʹ཈͑Δɽ ▸ ্ࣜதͷೋஈ໨͔Βࡾஈ໨͸(2.32)ͱ໋୊2.8 b.ଆͷ(2.32)ͱࣅ ͔ͨࣜΒߦ͚ͦ͏ɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ·ͨՁ஋൓෮๏ͷऴྃ৚݅ΛҎԼͰஔ͖׵͑Δ͜ͱΛߟ͑Δɽ ▸ ্৚݅͸ҎԼͷΑ͏ʹॻ͖׵͑ΒΕΔɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ํࡦޡࠩΛ࣍ͷΑ͏ʹ཈͑Δɽ ▸ ࣜ(2.45)Ͱߋʹ্͔Β཈͑ͯ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ํࡦޡࠩͷҎԼͷࣜͷ ͸ ʁ ▸ p.60 (2.47)ࣜ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ϦλʔϯͷׂҾ཰ ͕ҟͳΔ৔߹ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2 ※ਤதॎͷࠇઢ͸൓෮ऴྃճ਺N

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2.3.1 Ձ஋൓෮๏ ▸ ໋୊2.8ʢಈతܭը๏ͷޡࠩݶքʣ͔Β ▸ ऴྃ৚͕݅ҟͳΔ৔߹ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2 ※ਤதॎͷࠇઢ͸൓෮ऴྃճ਺N

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2.3.2 ํࡦ൓෮๏ ▸ ΞϧΰϦζϜ2.2 ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.3.2 ํࡦ൓෮๏ ▸ ໋୊2.9ʢํࡦվળͷ୯ௐੑʣ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.4 ઢܗܭը๏ʹΑΔղ๏ ▸ ϓϥϯχϯά໰୊Λղ͘͜ͱ͕ग़དྷΔɽஞ࣍తҙࢥܾఆ໰୊ͷ  ੑ࣭Λௐ΂Δ্Ͱ΋༗༻ɽ ▸ ͳ͓ɼઢܗܭը๏ʹ͓͚Δجຊతͳࣄ߲͸ิ଍A.3(p.266)Λࢀরɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.4.1 ओ໰୊ͷಋग़ ▸ ໋୊2.10ʢ࠷దՁ஋ؔ਺ͷ্քͱԼքʣ ▸ Կ౓΋ద༻͢Δ͜ͱͰۃݶΛऔΔ͚ͩɽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2

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2.4.1 ओ໰୊ͷಋग़ ▸ ໋୊2.10ʢ࠷దՁ஋ؔ਺ͷ্քͱԼքʣ͔Β ▸ ͜ͷ৚݅Λຬͨ͢࠷খͷ Λߟ͑Δ͜ͱͰ࠷దՁ஋ؔ਺ͷ্ քͱͳΔؔ਺Λߟ͑Δ͜ͱ͕ग़དྷΔɽ ▸ ॏΈؔ਺ Λߟ͑ɼҎԼͰઢܗܭը໰୊ʹఆࣜԽ ڧԽֶश ྠಡձ #02, #03 CHAPTER 2