不確実性下における目的と手段の統合的探索に向けた連続腕バンディットの応用 / iot70_gp_rff_mab

2025.07.28 第70回インターネットと運用技術研究発表会（IOT70）
https://www.iot.ipsj.or.jp/meeting/70-program/

Transcript

1. ࡾ୐༔հ / Pepabo R&D Institute, GMO Pepabo, Inc. 2025.07.28 ୈ70ճΠϯλʔωοτͱӡ༻ٕज़ݚڀൃදձʢIOT70ʣ ෆ࣮֬ੑԼʹ͓͚Δ ໨తͱखஈͷ౷߹త୳ࡧʹ޲͚ͨ ࿈ଓ࿹όϯσΟοτͷԠ༻
2. 1. ͸͡Ίʹ 2. ໨తͱखஈͷ౷߹త୳ࡧͷ՝୊ 3. ఏҊख๏ 4. ධՁ 5. ͓ΘΓʹ  2 ໨࣍
3. 1. ͸͡Ίʹ
4. • ࣮ӡ༻͞ΕΔ৘ใγεςϜʹ͓͚Δҙࢥܾఆͷෳࡶ͞ • ྫɿEC αΠτʹ͓͚Δ໨తͱखஈઃܭͷଟ༷͞ • ໨తɿԿΛୡ੒͍͔ͨ͠ʢࢦඪʣͱ୭ʹର͔ͯ͠ʢλʔήοτʣ • खஈɿ্هΛͲ͏࣮ݱ͢Δ͔ʢΫʔϙϯදࣔɺਪનվળɺUIมߋͳͲʣ  4 ݚڀͷഎܠʢ1/2ʣ
5. • ࣮ӡ༻͞ΕΔ৘ใγεςϜͰੜ͡Δ՝୊ • ࣮ࡍʹ͸ɺʮ͜ͷλʔήοτʹ͜ͷखஈͰ͜ͷࢦඪΛվળʯͱ໨తΛݻఆ ্ͨ͠ͰࢪࡦΛࢼ͢ͷ͕Ұൠత • → ੒Ռ͕ग़Δ͔Ͳ͏͔͸΍ͬͯΈͳ͍ͱΘ͔Βͳ͍ • ࢼߦʹ͸ίετɾϦεΫ͕൐͏ʢදࣔมߋʹΑΔ཭୤ɺӡ༻ෛՙͳͲʣ • ͞Βʹɺࢪࡦͷ݁Ռͱͯ͠໨తͦͷ΋ͷ͕ݟ௚͞ΕΔ͜ͱ΋͋Δ • ྫɿ૝ఆ͍ͯͨ͠ᅂ޷ͷϢʔβʔΑΓ΋ɺผͷᅂ޷܏޲Λ࣋ͭ૚ͷํ͕ࢪࡦͷޮՌ͕ߴ͔ͬͨ  5 ݚڀͷഎܠʢ2/2ʣ
6. • ࣮ӡ༻͞ΕΔ৘ใγεςϜʹର͢Δཁٻ • ໨తʢࢦඪɾλʔήοτʣ΋खஈ΋ݻఆͤͣʹ୳ࡧ͍ͨ͠ • ࢼߦ݁ՌΛ΋ͱʹɺޮՌͷ͋Γͦ͏ͳ૊߹ͤΛࣗ཯తʹݟग़͍ͨ͠ • ຊݚڀͷఏڙ͢Δ࿮૊Έ • ໨తͱखஈΛڞ௨ͷಛ௃ۭؒʹຒΊࠐΈ • ૊߹ͤʢλʔήοτ × ࢦඪ × खஈʣΛஞ࣍తʹ୳ࡧ͢Δ࢓૊Έ • → ࣗ཯తAIΤʔδΣϯτͷ؀ڥదԠΛࢧ͑Δҙࢥܾఆج൫΁  6 ݚڀͷ໨త
7. 2. ໨తͱखஈͷ౷߹త୳ࡧͷ՝୊
8. • ࣮ӡ༻Ͱ͸ɺ໨తͱखஈ͸ݻఆ͞Εͨ΋ͷͰ͸ͳ͘ɺ૬ޓʹӨڹ͠߹͏ • ࢼߦͷ݁Ռɺ౰ॳͷ໨తʢࢦඪ΍λʔήοτʣ͕ݟ௚͞ΕΔ͜ͱ΋͋Δ • ੒Ռͷग़΍͍͢ର৅΍ධՁ͕࣠ޙ͔Βݟ͔ͭΔ৔߹΋ • ຊݚڀͰ͸ɺ໨తͦͷ΋ͷ΋୳ࡧର৅ͱΈͳ͢  8 ໨తͱखஈͷؔ܎ੑ
9. • ໨తʢࢦඪ΍λʔήοτʣɺखஈ͸ଟ༷Ͱɺ૊߹ͤ͸๲େ • ͢΂ͯͷ૊߹ͤΛݸผʹࢼ͢ͷ͸ݱ࣮తͰͳ͍ • ໨త͝ͱʹ༗ޮͳखஈ͕ҟͳΔ͚ͩͰͳ͘ɺଞͷ໨తʹ΋సҠՄೳͳखஈ ͷ୳ࡧ΋ॏཁ • → ૊߹ۭͤؒશମΛର৅ͱͨ͠ޮ཰తͳ୳ࡧ͕ٻΊΒΕΔ  9 ୳ࡧۭؒͷෳࡶੑ
10. • ໨తͱखஈΛ࿈ଓϕΫτϧۭؒʹຒΊࠐΈɺڞ௨දݱͱ͢Δ • ཭ࢄతͳ૊߹ͤͰ͸ͳ͘ɺ࿈ଓ্ۭؒͰ୳ࡧΛߦ͏ • ҙຯతͳྨࣅੑΛอͬͨ··ɺ൚༻తͳॲཧ͕Մೳ • ݕ౼ର৅ΛϕΫτϧͱͯ͠ѻ͏͜ͱͰɺޮ཰తͳۙ๣୳ࡧ͕Ͱ͖Δ  10 ຒΊࠐΈදݱʹΑΔ௿࣍ݩߏ଄ͷ׆༻
11. • ໨తͱखஈͷ૊߹ۭͤؒΛ࿈ଓ࿹όϯσΟοτͱͯ͠ఆࣜԽ • ใुͱෆ࣮֬ੑΛಉ࣌ʹ༧ଌ • ୳ࡧͱ׆༻ͷόϥϯεΛ੍ޚ • ༧ଌͷෆ࣮֬ੑΛ΋ͱʹɺϦεΫͷߴ͍બ୒ࢶΛճආՄೳ  11 ࿈ଓ࿹όϯσΟοτʹΑΔ୳ࡧͷ࿮૊Έ
12. • ʮ࿹ʯͱݺ͹ΕΔෳ਺ͷީิ͔ΒಘΒΕΔใुΛ࠷େԽ͢Δ໰୊ • ϓϨΠϠʔ͸Ұ౓ͷࢼߦͰ1ͭͷ࿹Λબ୒͠ɺใुΛಘΔ • ͦΕͧΕͷ࿹͸͋Δใु෼෍ʹै͍ใुΛੜ੒ • ͨͩ͠ɺϓϨΠϠʔ͸͜ͷใु෼෍Λࢼߦͷ݁Ռ͔Βਪଌ͢Δඞཁ͕͋Δ  12 ଟ࿹όϯσΟοτ໰୊ ʜ બ୒ ใु ਪଌ ʮ࿹ʯ͸εϩοτϚγʔϯͷʮΞʔϜʢArmʣʯʹ༝དྷ IUUQTJDPOTDPN
13. • ʮ࿹ʯͱݺ͹ΕΔෳ਺ͷީิ͔ΒಘΒΕΔใुΛ࠷େԽ͢Δ໰୊ • ϓϨΠϠʔ͸Ұ౓ͷࢼߦͰ1ͭͷ࿹Λબ୒͠ɺใुΛಘΔ • ͦΕͧΕͷ࿹͸͋Δใु෼෍ʹै͍ใुΛੜ੒ • ͨͩ͠ɺϓϨΠϠʔ͸͜ͷใु෼෍Λࢼߦͷ݁Ռ͔Βਪଌ͢Δඞཁ͕͋Δ  13 ଟ࿹όϯσΟοτ໰୊ • ϓϨΠϠʔ͸͋Δ࣌఺ͷ࿹ͷධՁʹج͖ͮʮ׆༻ʯͱʮ୳ࡧʯΛฒߦͯ͠ߦ͏ • ͜ͷτϨʔυΦϑΛղফ͢ΔͨΊʹ༷ʑͳํࡦ͕ఏҊ͞Ε͍ͯΔ
14. • ࿈ଓ͢Δແݶݸͷʮ࿹ʯͷީิ͔ΒಘΒΕΔใुΛ࠷େԽ͢Δ໰୊ • ϓϨΠϠʔ͸Ұ౓ͷࢼߦͰ1ͭͷ࿹Λબ୒͠ɺใुΛಘΔ • ͦΕͧΕͷ࿹͸͋Δใु෼෍ʹै͍ใुΛੜ੒ • ͨͩ͠ɺϓϨΠϠʔ͸͜ͷใु෼෍Λࢼߦͷ݁Ռ͔Βਪଌ͢Δඞཁ͕͋Δ  14 ࿈ଓ࿹όϯσΟοτ໰୊ બ୒ ใु ਪଌ ʜ ʜ ʜ
15.  15 ໨తͱखஈͷ౷߹త୳ࡧ ໨తू߹ खஈू߹ Encoder Encoder ౷߹ۭؒ ༧ଌͷ෼෍ ༧ଌͷෆ࣮֬ੑ બ୒ ใु ༧ଌͱෆ࣮֬ੑɺ ػձଛࣦΛ ߟྀͨ͠ީิͷબఆ ʢ࿈ଓ࿹όϯσΟοτʣ
16. • Ψ΢εաఔʢGaussian Process, GPʣ͸ɺσʔλ͔Βɺ͋Δؔ਺ͷ෼෍Λ֬཰ աఔͱͯ͠ٻΊΔ • ଟ਺ͷجఈؔ਺  ʹΑΔߴ͍දݱྗ • Χʔωϧ๏ʹΑͬͯجఈؔ਺ͷ໌͕ࣔෆཁ ϕ: ℝD → ℝ  16 Ψ΢εաఔϞσϧͱ࿈ଓ࿹όϯσΟοτ k(p, q) ≜ (p 𝖳 q + c)m, m = 2, p, q ∈ ℝ2 ͜ͷଟ߲ࣜΧʔωϧؔ਺͸࿡ͭͷجఈؔ਺ʹΑΔม׵ͱ಺ੵΛऔͬͨ݁ՌͱҰக
17. • Ψ΢εաఔʢGaussian Process, GPʣ͸ɺσʔλ͔Βɺ͋Δؔ਺ͷ෼෍Λ֬཰ աఔͱͯ͠ٻΊΔ • ଟ਺ͷجఈؔ਺  ʹΑΔߴ͍දݱྗ • Χʔωϧ๏ʹΑͬͯجఈؔ਺ͷ໌͕ࣔෆཁ • ඇઢܗੑͱෆ࣮֬ੑΛѻ͑ΔͨΊଟ࿹όϯσΟοτ໰୊΁ͷ਌࿨ੑ͕ߴ͍ • → ΧʔωϧߦྻͱͦͷٯߦྻͷܭࢉΛؚΉͨΊɺ ɹ ֶशσʔλ਺ʹରֶͯ͠श͕࣌ؒࢦ਺ؔ਺తʹ૿Ճ ϕ: ℝD → ℝ  17 Ψ΢εաఔϞσϧͱ࿈ଓ࿹όϯσΟοτ  K−1  K ∈ ℝN×N  K−1 →  k(xi , xj )
18. • ࿈ଓ࿹ バ ϯ デ Οοτʹ͓͚ΔThompson Sampling で ͸ɺީิ఺ू߹  ʹରͯ͠GPͷ༧ଌ෼෍͔ΒٻΊͨؔ਺஋ͷ͏ͪɺ࠷େ஋Λ༩͑ Δ఺  Λ࣍ͷબ୒࿹ͱ͢Δ • → ߴ࣍ݩۭؒʹ͓͍ͯ͸ɺਫ਼౓ͷ֬อͷͨΊʹީิ఺ू߹਺  Λଟ͘औΔ ɹ ඞཁ͕͋Γɺܭࢉෛՙ͕૿େ͢Δ {x(1) * , …, x(M) * } ˜ x = arg max x(m) * ˜ f(x(m) * ) M  18 Ψ΢εաఔϞσϧͱ࿈ଓ࿹όϯσΟοτ
19. • ᶃ ߴ࣍ݩͳಛ௃ۭؒʹ͓͚ΔϞσϧͷֶशɾਪ࿦ίετͷ૿େ • ᶄ ީิ఺ू߹ͷߏஙʹґଘ͠ͳ͍ޮ཰తͳ࠷దԽͷࠔ೉͞ • ᶅ ߴ࣍ݩੑʹΑΔ༧ଌੑೳͷྼԽͱ൚Խೳྗͷ௿Լ • ᶆ ϋΠύʔύϥϝʔλਪఆʹ͓͚Δஞ࣍ߋ৽࣌ͷܭࢉෛՙ  19 ຊݚڀͷओͳٕज़՝୊
20. 3. ఏҊख๏
21. • ཚ୒ԽϑʔϦΤಛ௃ʢRandom Fourier Features, RFFʣ[5] ͸ɺ͋Δ֬཰෼෍ ͔Βͷ  ݸͷαϯϓϧΛ༻͍ͯΧʔωϧؔ਺  Λ  ͱۙࣅ͢Δख๏ • ͳ͓ɺ  • ֬཰෼෍  ͸Χʔωϧؔ਺ͷछྨʹΑܾͬͯ·Δ p(ω) R′  = R/2 k(xi , xj ) ̂ k(xi , xj ) = z(xi )⊤z(xj ) z(xi ) = 1/R′  (cos(ω⊤ 1 xi ), sin(ω⊤ 1 xi ), …, cos(ω⊤ R′  xi ), sin(ω⊤ R′  xi )) p(ω)  21 ରࡦᶃɿΧʔωϧߦྻͷܭࢉෛՙ΁ͷରॲ [5] Miguel L ́azaro-Gredilla, Joaquin Quinonero-Candela, Carl Edward Rasmussen, and An ́ıbal R Figueiras-Vidal. Sparse spectrum gaussian process regression. The Journal of Machine Learning Research, Vol. 11, pp. 1865– 1881, 2010.  K  K ∈ ℝN×N  k(xi , xj )  K  ZZ⊤  ≃  K ∈ ℝN×N  Z ∈ ℝN×R  Z⊤Z  Z⊤Z ∈ ℝR×R  ⋙ k(xi , xj ) ≃ z(xi )⊤z(xj ) ݸผͷΧʔωϧؔ਺ʹରͯ͠͸ܭࢉίετ͕ ૿Ճ͢Δ͕ɺجఈؔ਺ͷద༻ͱ಺ੵͷࠞ߹ૢ ࡞ͷ݁ՌΛ෼ղͨ͠ͱݟΔ͜ͱ͕Ͱ͖Δ → ύϥϝʔλͷ࣍ݩ਺Λ  ࣍ݩʹݻఆͰ͖Δ R ϕ(x) = (ϕ1 (x), …, ϕ∞ (x))⊤ ∈ ℝ∞ ϕ(xi )⊤ϕ(xj ) = k(xi , xj ) ≈ z(xi )⊤z(xj )
22.  22 ରࡦᶃɿΧʔωϧߦྻͷܭࢉෛՙ΁ͷରॲ  K′   K′   K′  … z z z … ⋮ x1 x2 xN x1 x1 xN−1 GP learning GP with RFF learning  K′  −1 Inv  K′  −1 Inv  K′  −1 Inv  Z⊤Z + Λ  Z⊤Z + Λ  Z⊤Z + Λ … z z z … ⋮ x1 x2 xN x1 x1 xN−1  (Z⊤Z + Λ)−1 Inv  (Z⊤Z + Λ)−1 Inv  (Z⊤Z + Λ)−1 Inv  K′  ∈ ℝN×N  Z⊤Z ∈ ℝR×R
23. • ީิ఺ू߹ͷߏஙΛհ͞ ず ʹؔ਺αϯ プ ϧΛධՁ͢Δख๏ [10] • ରࡦᶃͷ݁ՌɺGPͷؔ਺αϯϓϧ͸  ͱͯۙ͠ࣅ͞ΕΔ •  Ͱ͋Γɺύϥϝʔλ  ͸ީิ఺ू߹ʹґଘ͠ͳ͍ • ޯ഑  ͕ղੳతʹܭࢉͰ͖Δ • → ࿈ଓ্ۭؒͰͷ࠷దԽʹΑΓީิ఺ू߹Λ ɹ࢖Θͣ࣍ͷબ୒఺ΛܾఆՄೳ ɹ ʢ࠷దԽ͋ͨΓͷ  ͷαϯϓϦϯά͸Ұ౓ͷΈʣ ˜ f(x) = z(x)⊤ ˜ w ˜ w ∼ 𝒩 (μw , Σw ) μw , Σw ∇x ˜ f(x) ˜ w  23 ରࡦᶄɿީิ఺ू߹ߏஙͷճආ [10] Sattar Vakili, Henry Moss, Artem Artemev, Vincent Dutordoir, and Victor Picheny. Scalable thompson sampling using sparse gaussian process models. Advances in neural information processing systems, Vol. 34, pp. 5631– 5643, 2021.
24. • εέʔϧ パ ϥϝʔλ  ʹରͯ͠ର਺ਖ਼ن෼෍Λࣄલ෼෍ͱͯ͠Ծఆ͠ɺ࣍ݩ਺ ͷ૿Ճʹ൐͏൚Խੑೳͷ௿ԼΛ཈੍͢Δਖ਼ଇԽख๏ [11] σ2 k D  24 ରࡦᶅɿ༧ଌੑೳͱ൚Խੑೳͷҡ࣋ [11] Carl Hvarfner, Erik Orm Hellsten, and Luigi Nardi. Vanilla bayesian optimization performs great in high dimensions. arXiv preprint arXiv:2402.02229, 2024.  log p(σ2 k ) = − log σ2 k − 1 2σ2 k0 (log σ2 k − μk0 − 1 2 log D) 2 + DPOTU  ͕ա৒ʹେ͖ͳ஋ΛऔΔ͜ͱΛ཈੍ → ϞσϧͷաֶशΛ๷͙໾ׂ σ2 k  ͕ɺ  ʹۙͮ͘Α͏ଅ͢ޮՌ → ࣍ݩ਺  ʹԠͨ͡ద੾ͳ  ͷબ୒Λ༠ಋ log σ2 k μk0 + 1 2 log D D σ2 k  ͷࣄલ෼෍ͷର਺໬౓ σ2 k  ͝ͱͷෛͷର਺໬౓ͷ  ͷ࠷খ஋ D log σ2 k
25. • Ψ΢εաఔϞσϧͰ͸ɺϋΠύʔύϥϝʔλͷਪఆ͕ੑೳʹେ͖͘Өڹ͢Δ • ͜ͷਪఆ͸ର਺໬౓ͷ࠷దԽͱͯ͠ߦΘΕɺͦͷ܁Γฦ͠ܭࢉʹ͓͍ͯ໬౓ͱ ޯ഑ͷߴ଎ͳධՁ͕ٻΊΒΕΔ • ಛʹɺஞ࣍తͳߋ৽͕ٻΊΒΕΔόϯσΟοτ໰୊ͷઃఆͰ͸ॏཁ • ैདྷख๏Ͱ͸  Χʔωϧߦྻͷٯߦྻ΍ߦྻࣜͷܭࢉ͕ϘτϧωοΫ • ຊใࠂͰ͸ɺ໬౓ͱޯ഑ܭࢉ΋RFFϕʔεͰՄೳͱͳΔߏ଄Λಋೖ͠ɺϋΠ ύʔύϥϝʔλͷߴ଎ͳ࠷దԽΛ࣮ݱ N × N  25 ରࡦᶆɿϋΠύʔύϥϝʔλਪఆͷߴ଎Խ
26. • ؍ଌσʔλ਺  ʹґଘ͢Δେن໛ͳΧʔωϧߦྻ  ͷٯߦྻܭࢉ΍ߦ ྻࣜܭࢉΛճආ • →  ʹґଘͨ͠খ͞ͳߦྻͷܭࢉͰ໬౓Λޮ཰తʹධՁ N K ∈ ℝN×N R  26 ରࡦᶆɿRFFΛ׆༻ͨ͠ର਺໬౓ͷಋग़ log ℒ(θ ∣ y) ∝ − log|K| − y⊤K−1y + 2 log p(σ2 k ) = − N log σ2 ε − log|B| − 1 σ2 ε ( ∥y∥2 − 1 σ2 ε ∥Z⊤y∥2 A−1) − 2 log σ2 k − 1 σ2 k0 (log σ2 k − μk0 − 1 2 log D) 2 A = 1 σ2 w I + 1 σ2 ε Z⊤Z B = σ2 w A = I + σ2 w σ2 ε Z⊤Z
27. • ֤ϋΠύʔύϥϝʔλ  ʹର͢Δର਺໬౓ͷޯ഑ θi ∈ θ = (σ2 k , σ2 w , σ2 ϵ )  27 ରࡦᶆɿRFFΛ׆༻ͨ͠ޯ഑ͷಋग़ ∂ ∂θi log ℒ(θ ∣ y) ∝ − tr ( K−1 ∂K ∂θi ) + y⊤K−1 ∂K ∂θi K−1y + ∂ ∂σ2 k log p(σ2 k )  ʹґଘ K ∈ ℝN×N ʜ
28. • ֤ϋΠύʔύϥϝʔλ  ʹର͢Δର਺໬౓ͷޯ഑ θi ∈ θ = (σ2 k , σ2 w , σ2 ϵ )  28 ରࡦᶆɿRFFΛ׆༻ͨ͠ޯ഑ͷಋग़ʢ  ʣ σ2 ε ∂ ∂σ2 ε log ℒ(θ ∣ y) = − ( N σ2 ε − σ2 w σ2 ε Tr(B−1Z⊤Z) ) + 1 σ4 ε ( y 2 − 2σ2 w y⊤ZB−1Z⊤y + σ4 w ZB−1Z⊤y 2 ) ∂ ∂θi log ℒ(θ ∣ y) ∝ − tr ( K−1 ∂K ∂θi ) + y⊤K−1 ∂K ∂θi K−1y + ∂ ∂σ2 k log p(σ2 k ) • RFFͷߏ଄Λద༻ͨ͠ϊΠζ෼ࢄ  ͷޯ഑ σ2 ε  ʹґଘ K ∈ ℝN×N ʜ ∂K ∂σ2 ε = I • →  ʹґଘͨ͠খ͞ͳߦྻͷܭࢉͰޯ഑Λޮ཰తʹධՁ R
29. • ֤ϋΠύʔύϥϝʔλ  ʹର͢Δର਺໬౓ͷޯ഑ θi ∈ θ = (σ2 k , σ2 w , σ2 ϵ )  29 ରࡦᶆɿRFFΛ׆༻ͨ͠ޯ഑ͷಋग़ʢ  ʣ σ2 w ∂ ∂θi log ℒ(θ ∣ y) ∝ − tr ( K−1 ∂K ∂θi ) + y⊤K−1 ∂K ∂θi K−1y + ∂ ∂σ2 k log p(σ2 k ) • RFFͷߏ଄Λద༻ͨ͠ॏΈ෼ࢄ  ͷޯ഑ σ2 w  ʹґଘ K ∈ ℝN×N ʜ ∂K ∂σ2 w = K • →  ʹґଘͨ͠খ͞ͳߦྻͷܭࢉͰޯ഑Λޮ཰తʹධՁ R ∂ ∂σ2 w log ℒ(θ ∣ y) = − Tr ( 1 σ2 ε Z⊤Z − σ2 w σ2 ε Z⊤Z ⋅ B−1 ⋅ Z⊤Z ) + 1 σ4 ε ((Z⊤y)⊤Z⊤y − 2σ2 w (Z⊤y)⊤Z⊤ZB−1Z⊤y + σ4 w Z⊤ZB−1Z⊤y 2 )
30. • ֤ϋΠύʔύϥϝʔλ  ʹର͢Δର਺໬౓ͷޯ഑ θi ∈ θ = (σ2 k , σ2 w , σ2 ϵ )  30 ରࡦᶆɿRFFΛ׆༻ͨ͠ޯ഑ͷಋग़ʢ  ʣ σ2 k ∂ ∂θi log ℒ(θ ∣ y) ∝ − tr ( K−1 ∂K ∂θi ) + y⊤K−1 ∂K ∂θi K−1y + ∂ ∂σ2 k log p(σ2 k )  ʹґଘ σ2 k ʜ k(x, x′  ) ≈ z(x)⊤z(x′  ), z(x) = 2 R cos(Ω⊤x + b) Ω:,r ∼ 𝒩 (0, σ−2 k I) ࠷దԽର৅ͷ  ͷมߋ͝ͱʹ ࠶αϯϓϦϯά͕ඞཁ = ࠷దԽ݁Ռ͕҆ఆ͠ͳ͍ σ2 k
31. • ֤ϋΠύʔύϥϝʔλ  ʹର͢Δର਺໬౓ͷޯ഑ θi ∈ θ = (σ2 k , σ2 w , σ2 ϵ )  31 ରࡦᶆɿRFFΛ׆༻ͨ͠ޯ഑ͷಋग़ʢ  ʣ σ2 k ∂ ∂θi log ℒ(θ ∣ y) ∝ − tr ( K−1 ∂K ∂θi ) + y⊤K−1 ∂K ∂θi K−1y + ∂ ∂σ2 k log p(σ2 k )  ʹґଘ͠ͳ͍ σ2 k ʜ k(x, x′  ) ≈ z(x)⊤z(x′  ), z(x) = 2 R cos ( 1 σk ˜ Ωx + b ) ˜ Ω ∼ 𝒩 (0, I) ʢ࠶ύϥϝʔλԽʣ
32. • ֤ϋΠύʔύϥϝʔλ  ʹର͢Δର਺໬౓ͷޯ഑ θi ∈ θ = (σ2 k , σ2 w , σ2 ϵ )  32 ରࡦᶆɿRFFΛ׆༻ͨ͠ޯ഑ͷಋग़ʢ  ʣ σ2 k ∂ ∂θi log ℒ(θ ∣ y) ∝ − tr ( K−1 ∂K ∂θi ) + y⊤K−1 ∂K ∂θi K−1y + ∂ ∂σ2 k log p(σ2 k ) ∂ ∂σ2 k log ℒ(θ ∣ y) = − 2σ2 w 1 σ2 ε Tr ( Z⊤ ∂Z ∂σ2 k ) − σ2 w σ2 ε Tr ( Z⊤Z ⋅ B−1 ⋅ Z⊤ ∂Z ∂σ2 k ) + 2σ2 w (Z⊤α)⊤ ( ∂Z ∂σ2 k ) ⊤ α − 2 σ2 k ( 1 + 1 σ2 k0 (log σ2 k − μk0 − 1 2 log D)) • RFFͷߏ଄Λద༻ͨ͠εέʔϧύϥϝʔλ  ͷޯ഑ σ2 k α = 1 σ2 ε y − σ2 w σ2 ε ZB−1Z⊤y ∂Z ∂σ2 k = 2 R ⋅ sin ( 1 σk X˜ Ω⊤ + b ) ⊙ ( X˜ Ω⊤ 2σ3 k ) • →  ʹґଘͨ͠খ͞ͳߦྻͷܭࢉͰޯ഑Λޮ཰తʹධՁ R ਖ਼ଇԽ߲ ʜ
33. 4. ධՁ
34. • ఏҊख๏ͷ༗ޮੑΛ֬ೝ͢ΔͨΊɺγϛϡϨʔγϣϯ࣮ݧΛ࣮ࢪ • ߴ࣍ݩੑͷӨڹ͕ݱΕΔঢ়گͱͯ͠ɺ32࣍ݩͷ୳ࡧۭؒΛઃఆ • ໨తؔ਺ͱͯ͠shifted sphereؔ਺Λ࢖༻ • ֤࣍ݩͷਅͷ࠷ద஋ͱͷڑ཭ͷೋ৐࿨ΛͱΔߏ଄ • ࿈ଓ࿹όϯσΟοτͷ࿮૊Έʹ͓͚Δ࠷େใुͷ ୳ࡧ໰୊Λɺؔ਺࠷খԽͱͯ͠ѻ͏  34 ධՁ໨తͱ໰୊ઃఆ fr (x) = D ∑ i=1 (xi − ri )2 XIFSF x = (x1 , x2 , …, xD )⊤ ∈ [−3.0,3.0]D, r = (r1 , r2 , …, rD ), ri ∼ 𝒰 (−3.0,3.0)
35. • ᶃ ྦྷੵϦάϨοτɿ֤࣌఺Ͱ͜Ε·Ͱબ͹Εͨީิͷ͏ͪ࠷ྑͷ΋ͷͱɺਅ ͷ࠷దղͱͷࠩͷྦྷੵ • ᶄ ਪ࿦࣌ؒɿ࣍ީิͷબఆʹཁͨ͠ܭࢉ࣌ؒ • ֤ख๏ʹ͍ͭͯ200ճͷީิબఆΛ࣮ࢪ͠ɺੑೳΛධՁ • ॳظঢ়ଶͱͯ͠ɺϥϯμϜʹબ͹Εͨ1600఺෼ͷ؍ଌσʔλΛࣄલ෇༩ • ҟͳΔཚ਺γʔυͰ10ճͷγϛϡϨʔγϣϯΛ࣮ࢪɺ݁Ռ͸ͦͷฏۉΛ࢖༻  35 ධՁࢦඪ
36.  36 ൺֱํࡦ RFF ީิ఺ͷ࠷ద Խ ਖ਼ଇԽ ϋΠύʔύϥϝ ʔλߴ଎ਪఆ ํࡦ ✓ ✓ ✓ ✓ GP-RFF (Proposal) ✓ ✓ ✓ GP-RFF-No-Prior ✓ ✓ ✓ GP-RFF-Naive-Gradient ✓ GP - - - - TPE - - - - Random • γϛϡϨʔγϣϯʹ༻͍Δํࡦ
37. • ࠷΋ྦྷੵϦάϨοτ͕খ͔ͬͨ͞ͷ͸ఏҊख๏ GP-RFFʢྦྷੵ஋ɿ4384.9ʣ • → ਅͷ࠷ద஋ 0 ʹର͠ɺ࠷ྑ఺ͷؔ਺஋͸ 18.3 ·Ͱ઀ۙ • → ఏҊख๏͕ߴ࣍ݩۭؒͰ΋༗ޮͳީิબఆ͕ՄೳͰ͋Δ͜ͱΛ֬ೝ • ಋೖ֤ͨ͠ཁૉٕज़͕༗ޮʹػೳ  37 ݁ՌᶃɿϦάϨοτൺֱ
38. • GP-RFF-Naive-Gradientʢߴ଎Խͳ͠ʣ͸GP-RFFͱੑೳʹࠩ͸ݟΒΕͣ • → ໬౓͓Α び ޯ഑ͷཧ࿦త஋͸ಉҰͰ͋ΔͨΊ • GP-RFF-No-Priorʢਖ਼ଇԽͳ͠ʣ͸ੑೳ͕௿Լ • → ਖ਼ଇԽʹΑΔ୳ࡧੑೳͷ޲্͕ࣔࠦ͞Εͨ • ඪ४తͳGPͷੑೳ͸ݦஶʹ௿Լ • → ߴ࣍ݩۭؒʹ͓͚Δީิ఺ू߹ґଘͷਫ਼౓ݶք  38 ݁ՌᶃɿϦάϨοτൺֱ
39. • ఏҊख๏͸GP-RFF-No-PriorΑΓ΋ྦྷੵϦάϨοτ͕খ͘͞ɺਖ਼ଇԽʹΑΔ୳ ࡧੑೳͷվળ͕֬ೝ͞Εͨ • Χʔωϧͷεέʔϧύϥϝʔλʢ  ʣͷա౓ͳ্ঢ͕཈੍͞Εͨ • → ࣍ݩ਺ʹԠͨ͡ద੾ͳൣғʹऩ·Γɺ ɹ ൚Խੑೳ͕ҡ࣋͞Εͨͱߟ͑ΒΕΔ σ2 k  39 ݁ՌᶄɿϋΠύʔύϥϝʔλਪҠ
40. • ͨͩ͠ɺਖ਼ଇԽͷޮՌ͸ɺࣄલ෼෍ύϥϝʔλʢ  ʣͷઃఆʹڧ͘ґଘ •  → ࠓճͷ݁Ռ •  → ਖ਼ଇԽͷޮՌ͕΄΅ݟΒΕͳ͍ •  → ॳظ୳ࡧ͕཈੍͞Ε͗ͯ͢ੑೳ͕ѱԽ • → ϋΠύʔύϥϝʔλͷઃఆ͸ੑೳʹ ɹ େ͖͘Өڹ͢ΔͨΊɺ৻ॏͳબఆ͕ඞཁ σ2 k0 σ2 k0 = 0.005 σ2 k0 = 1.0 σ2 k0 = 0.001  40 ݁ՌᶄɿϋΠύʔύϥϝʔλਪҠ
41. • ఏҊख๏GP-RFF͓ΑͼGP-RFF-No-Prior͸ɺൺֱత୹͍બఆ࣌ؒΛҡ࣋ • → ਖ਼ଇԽͷ༗ແ͸બఆ࣌ؒʹେ͖ͳӨڹΛ༩͑ͳ͍ • → બఆ࣌ؒ͸ࢼߦճ਺ͷ૿Ճʹରͯ͠΄΅ҰఆͰ͋Γɺ؍ଌσʔλ਺ͷ૿ Ճʹ΋҆ఆͯ͠ରԠͰ͖Δ͜ͱΛ֬ೝ  41 ݁Ռᶅɿਪ࿦࣌ؒൺֱ
42. • ఏҊख๏GP-RFF͓ΑͼGP-RFF-No-Prior͸ɺൺֱత୹͍બఆ࣌ؒΛҡ࣋ • → ਖ਼ଇԽͷ༗ແ͸બఆ࣌ؒʹେ͖ͳӨڹΛ༩͑ͳ͍ • → બఆ࣌ؒ͸ࢼߦճ਺ͷ૿Ճʹରͯ͠΄΅ҰఆͰ͋Γɺ؍ଌσʔλ਺ͷ૿ Ճʹ΋҆ఆͯ͠ରԠͰ͖Δ͜ͱΛ֬ೝ • ҰํɺGP͓ΑͼGP-RFF-Naive-Gradient͸ ؍ଌσʔλ਺ʹൺྫͯ͠ܭࢉෛՙ͕૿େ  42 ݁Ռᶅɿਪ࿦࣌ؒൺֱ
43. 5. ͓ΘΓʹ
44. • ໨తͱखஈͷ૊߹ۭͤؒΛ࿈ଓ࿹όϯσΟοτ໰୊ͱͯ͠ఆࣜԽ͠ɺ౷߹తʹ ୳ࡧ͢Δ࿮૊ΈΛఏҊ • ཚ୒ԽϑʔϦΤಛ௃ʢRFFʣΛಋೖͨ͠Ψ΢εաఔϞσϧʹΑΓɺඇઢܗੑͱ ෆ࣮֬ੑΛཱ྆ͭͭ͠ߴޮ཰ͳҙࢤܾఆΛ࣮ݱ • ϋΠύʔύϥϝʔλਪఆʹ΋RFFΛ׆༻͠ɺ໬౓ͱޯ഑ͷߴ଎ͳධՁʹΑͬͯ ஞ࣍࠷దԽΛޮ཰Խ • ࠓޙ͸ɺߴ࣍ݩ͔ͭෳࡶͳ໨తɾखஈۭؒ΁ͷεέʔϥϏϦςΟରԠΛݕ౼ • ࣮σʔλΛ༻͍ͨݕূʹΑΓɺఏҊख๏ͷ࣮༻ੑΛධՁ͍ͯ͘͠  44 ͓ΘΓʹ