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7章 線形モデル上のバンディット問題
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Takashi Nishibayashi
November 30, 2016
Technology
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7章 線形モデル上のバンディット問題
機械学習プロフェッショナルシリーズ輪読会 ~バンディット問題の理論とアルゴリズム~ の資料です
Takashi Nishibayashi
November 30, 2016
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Transcript
ষ ઢܗϞσϧ্ͷόϯσΟοτ ػցֶशϓϩϑΣογϣφϧγϦʔζྠಡձ dόϯσΟοτͷཧͱΞϧΰϦζϜ<>d !IBHJOP 5BLBTIJ/JTIJCBZBTIJ <>ʮόϯσΟοτͷཧͱΞϧΰϦζϜʯຊଟ३ʗதଜಞɾஶ
ࣗݾհ ID: hagino3000 Name: ྛ (Takashi Nishibayashi) Job: Software
Engineer ݱࡏΞυωοτϫʔΫࣄۀऀʹͯ৴ޮͷ࠷ దԽʹैࣄ (ೖࡳՁ֨ௐϩδοΫɾࠂબϩ δοΫͷઃܭ͔Β࣮·Ͱ)
ઢܗόϯσΟοτ s t e6 e4 e5 e3 e2 e1 ྫ
࢝T͔ΒऴUʹ͋ΔσʔλʹԿΒ͔ͷ ܦ࿏Λ௨ͯ͠ΞΫηε͢ΔͦΕͧΕͷࢬ FKͰະͷ௨৴ԆВK͕ੜ͡Δ͜ͷ࣌ ʹԿσʔλʹΞΫηεͨ࣌͠ʹཁ͠ ͨ௨৴࣌ؒΛ࠷খԽ͢ΔࣄΛߟ͑Δ
௨৴ωοτϫʔΫͷྫ T͔ΒUͷܦ࿏i㱨\ ʜ ^࣍ͷΑ͏ͳϕΫτϧai 㱨\ ^ͰදΘͤΔɽ ܦ࿏ϕΫτϧશମͷू߹ΛA\ai ^i 㱨\
^Ͱ͋ΒΘ͢ɽ ͜ͷ࣌ɼܦ࿏ai 㱨AΛ༻͍ͨ࣌ͷ௨৴࣌ؒВ В В В 5ʹରͯ͠
ΑΓҰൠతʹ ֤εϩοτϚγϯ͕d ࣍ݩͷϕΫτϧai 㱨A㱬RdʹରԠɽͦͷใु PSଛࣦ ͕ظͷޡ߲ࠩЏ t Λ༻͍ͨઢܗϞσϧ Ͱද͞ΕΔઃఆʹ͓͍ͯྦྷੵใुͷ࠷େԽΛࢦ͢ΛઢܗόϯσΟο τ
MJOFBSCBOEJU ͱΑͿɽ dcAcͰ͋Γ a1 , a2 ,,,a|A| ͕୯ҐߦྻͱͳΔ߹લষ·Ͱͷجຊతͳό ϯσΟοτʹରԠ͢Δɽ ઢܗόϯσΟοτͷจ຺Ͱଟ͘ͷ߹ʹ֤ީิߦಈ BDUJPO ͱΑ ΕΔɽ
ઢܗόϯσΟοτͷSFHSFU ใुظ࠷େͷߦಈ Λબ͠ଓ͚Δ߹ͱൺֱͨ࣌͠ͷଛࣦ 6$#ํࡦΛద༻ͨ͠߹ͷϦάϨοτKcAcʹରͯ͠0 KMPHT ɼcAc͕ ඇৗʹେ͖͘ͳΔ߹ʹผͷํࡦΛߟ͑Δඞཁ͕͋Δɽ
ྫ w 8FCαΠτ࠷దԽ w όϯσΟοτ࠷ద༧ࢉ w ઢܕܭըͷҰ෦ͷ͕ະ w όϯσΟοτઢܗܭը w
ߦಈͷީิcAc㱣
จ຺͖όϯσΟοτ ֤ߦಈͷಛྔ͕ai,t ͱ͍ͬͨܗͰ࣌ࠁʹΑΓҟͳΔΛऔΔ͜ͱΛڐ͢ઃ ఆɽ࠷୯७ͳઃఆͱͯ͠ɼ࣌ࠁUʹԠͨ͡จ຺͕ଘࡏ͢Δͷͷظ࠷େͷ ߦಈ࣌ࠁtʹΑΒͳ͍߹Λߟ͑Δɽ ߦಈͱจ຺ͷަޓ࡞༻ແ͠ˠظ࠷େͷσβΠϯ࣌ࠁtʹґଘ͠ͳ͍ σβΠϯʹΑΔಛྔ ΫϦοΫ ࣌ࠁUʹґଘ͢Δಛྔ ฏPSٳɼϢʔβʔͷྸ
FUDʜ
ߦಈͱจ຺ͷަޓ࡞༻͕͋ΔϞσϧ σβΠϯ͕ύλʔϯ ࣌ࠁUʹґଘ͢ΔϢʔβʔ ͷ͕ύλʔϯͷ߹ 3FHSFU֤࣌ࠁͰৗʹظ࠷େΛͱͬͨ߹ͱͷൺֱ
உ உ σβΠϯ" σβΠϯ# ϑΥϯτ"
ϑΥϯτ# உ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# ঁ ঁ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# ঁ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# ࠂΩϟϯϖʔϯ" PS ࠂΩϟϯϖʔϯ# PS ࠂΩϟϯϖʔϯ$ PS ࠂΩϟϖʔϯ% σβΠϯͱϑΥϯτʹ͍ͭͯੑผͱ૬ޓ࡞༻͕͋Δͱͨ͠Ϟσϧ ྫͱͯ͠ใुࠂΛճදࣔͨ࣌͠ͷऩӹ $1.
உ உ σβΠϯ" σβΠϯ# ϑΥϯτ"
ϑΥϯτ# உ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# ঁ ঁ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# ঁ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# σβΠϯͱϑΥϯτʹ͍ͭͯੑผͱ૬ޓ࡞༻͕͋Δͱͨ͠Ϟσϧ ྫͱͯ͠ใुࠂΛճදࣔͨ࣌͠ͷऩӹ $1. ࠂΩϟϯϖʔϯ" PS ࠂΩϟϯϖʔϯ# PS ࠂΩϟϯϖʔϯ$ PS ࠂΩϟϖʔϯ%
உ உ σβΠϯ" σβΠϯ# ϑΥϯτ"
ϑΥϯτ# உ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# ঁ ঁ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# ঁ σβΠϯ" σβΠϯ# ϑΥϯτ" ϑΥϯτ# σβΠϯͱϑΥϯτʹ͍ͭͯੑผͱ૬ޓ࡞༻͕͋Δͱͨ͠Ϟσϧ ྫͱͯ͠ใुࠂΛճදࣔͨ࣌͠ͷऩӹ $1. ࠂΩϟϯϖʔϯ" PS ࠂΩϟϯϖʔϯ# PS ࠂΩϟϯϖʔϯ$ PS ࠂΩϟϖʔϯ%
-JO6$#ํࡦ ઢܗϞσϧʹର͢Δ6$#ํࡦͰ͋Δ-JO6$#ํࡦ -JO6$#QPMJDZ ֤ߦಈ͝ͱͷ6$#είΞ ͜͜ͰЋt 0 㲋MPH5 Id d
x dͷ୯ҐߦྻɼA-1ߦྻ ͷٯߦྻɽ ޡ߲ࠩͷࢄɼύϥϝʔλ 2 > 0 2 0 , ↵ > 0
-JO6$#ͷ6$#είΞͷߏ ࣌ࠁUʹ͓͚Δ ߦಈJͷظͷਪఆ ࣌ࠁUʹ͓͚Δ ߦಈJͷظͷඪ४ภࠩ
ઢܗϞσϧ্ͷτϯϓιϯநग़ ਖ਼نϞσϧͰͷࣄޙ֬ͷܭࢉɽ ใुͷޡ߲ࠩЏ U ͕طͷࢄМͷਖ਼نʹै͏߹ Λߟ͑ΔͷВͷࣄޙΛར༻͢ΔͱΞϧΰϦζϜ Вͷࣄޙ͔֬Βαϯϓϧ
ޡ߲͕ࠩਖ਼نͰͳ͍߹ w ΫϦοΫͳͲͷϞσϧͰޡ߲ࠩࢄͱͳΓਖ਼نʹ ैΘͳ͍ w ਖ਼֬ͳࣄޙ͔ΒͷཚੜՄೳ͕ͩܭࢉ͕ඇৗʹෳࡶʹ ͳΔ w τϯϓιϯநग़Ͱਖ਼نΛԾఆͨ͠ΞϧΰϦζϜΛར༻ ͕ͯ͠ԿΒ͔ͷఆ3ʹ͍ͭͯྼΨεతͰ͋Εɼదͳ
МΛऔΔࣄͰখ͞ͳϦάϨοτΛୡͰ͖Δ w ࣮ݧతʹτϯϓιϯநग़-JO6$#ํࡦΑΓΑ͍ੑೳͱͳΔ ߹͕ଟ͍ w ΫϦοΫϞσϧͷΑ͏ʹͰ͋Δ߹ʹྼΨε ͷఆ3Λ༻͍Δͷ͕Ұൠత
࣮ݧͯ͠Έͨ w ࣮ίʔυ w IUUQTOCWJFXFSKVQZUFSPSHHJUIVCIBHJOPOPUFCPPLT CMPCNBTUFS.-1@CBOEJU$IBQJQZOC w ใुਅͷВͱਅͷࢄ͔Βੜͨ͠ w ޡ߲ࠩͷطͷࢄใुΛճ΄Ͳੜͤͯ͞ಘͨ
w ݁Ռ w ࣮ݧͰ5IPNQTPOநग़ͱ-JO6$#ͷ͕ࠩग़ͣ w ͨͩ͠ࠂ৴ใु͕ͦͷͰ؍ଌͰ͖ͳ͍ͷͰ-JO6$#͑ͳ͍ w 5IPNQTPOநग़ใु͕Εͯͳ͠ w ύϥϝʔλνϡʔχϯά w М ˠВͷࣄલ͕Ͱ͋Δͱߟ͑Δࣄʹ૬
ϩδεςΟοΫճؼϞσϧ্ͷ όϯσΟοτ ใु͕ͷ߹ʹΑΓదͨ͠Ϟσϧɽใु9J U 㱨\ ^ͷ֬ ͷใुΛߟ͑ͨ࣌ͷظ࠷େͷߦಈ Вͷࣄޙ͕ཉ͍͠ ˠਖ਼نͷΑ͏ͳดͨ͡ܗͰ͋ΒΘͤͳ͍ɽ ˠ͜ͷຊͰϥϓϥεۙࣅΛར༻
ˠͰۙࣅ
ϩδεςΟοΫճؼϞσϧ্ͷόϯ σΟοτͷੑೳ w ཧอূແ͍͕࣮༻্༗ޮΒ͍͠ w 4-4DPUU.VMUJBSNFECBOEJUFYQFSJNFOUTJOUIF POMJOFTFSWJDFFDPOPNZ
ใु͕ͷ߹ͷ࣮ݧ w ͔֬ʹଞͷΞϧΰϦζϜΑΓϩδεςΟοΫճؼϞσϧͷ 5IPNQTPOநग़ͷํ͕ੑೳ͕ྑ͍ w -JO6$#ɼใुͷޡ͕ࠩਖ਼نΛաఔͨ͠5IPNQTPOநग़ &QTJMPO(SFFEZΑΓѱ͘ͳͬͨ w ݁Ռϊʔτ IUUQTOCWJFXFSKVQZUFSPSHHJUIVCIBHJOPOPUFCPPLT
CMPCNBTUFS.-1@CBOEJU$IBQ@CJOBSZ@SFXBSEJQZOC