バンディット問題とベイズ最適化の基本

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1. όϯσΟοτ໰୊ͱϕΠζ࠷దԽͷجຊ ྛ উޛ ژ౎େֶେֶӃ ࣛౡɾࢁాݚڀࣨ 2020/07/21
2. ໨࣍ ੈͷதͷෆ࣮֬ͳ؀ڥͰͷҙࢥܾఆͷଟ͘͸όϯσΟοτ໰୊ͱ͠ ͯఆࣜԽ͢Δ͜ͱ͕Ͱ͖ΔɻຊηϛφʔͰ͸όϯσΟοτ໰୊΍ҙ ࢥܾఆ͕࿈ଓͰ͋ΔϕΠζ࠷దԽͷجຊతͳཧղΛ໨ࢦ͢ɻ • όϯσΟοτ໰୊ • Ԡ༻ྫ • ໰୊ઃఆ • ϦάϨοτ • ΞϧΰϦζϜ • ϕΠζ࠷దԽ • ϒϥοΫϘοΫε࠷దԽ • Ԡ༻ྫ • Ψ΢εաఔ • ΞϧΰϦζϜ ˞ ຊηϛφʔ͸جຊతͳཧղͷͨΊʹ໰୊ઃఆʹ஫ྗΛ౰ͯͨ΋ͷ Ͱ͋Γɺ਺ֶతͳݫີੑ΍ΞϧΰϦζϜɺূ໌ͷৄࡉΛ௥Θͳ͍ɻ 2 / 31
3. όϯσΟοτ໰୊ͱ͸ʁ : ྫ (εϩοτϚγʔϯ) ྫ (εϩοτϚγʔϯ) 3 ͭͷεϩοτϚγʔϯ͕͋ΓɺϨόʔ (࿹ɺΞʔϜ) ΛҾ͚͹ະ஌ ͷ֬཰Ͱ౰ͨΓΛҾ͘ɻ߹ܭ 100 ճεϩοτϚγʔϯͷϨόʔΛҾ ࣌͘ɺ֫ಘίΠϯͷ૯਺Λ࠷େԽ͢ΔͨΊʹͲͷΑ͏ʹϨόʔΛҾ ͘΂͖͔ʁ k=1 80% 20% k=2 50% 50% k=3 20% 80% どれが良い？ 3 / 31
4. όϯσΟοτ໰୊ͱ͸ʁ : ྫ (ਪન) ྫ (ਪન) 3 ͭͷ঎඼ΛωοτͰച͓ͬͯΓɺϢʔβʔʹର͠ਪન͢Δ͜ͱͰ ະ஌ͷ֬཰Ͱߪೖ͞ΕΔɻ߹ܭ 100 ճ঎඼Λਪન͢Δ࣌ɺ߹ܭͷར ӹΛ࠷େԽ͢ΔͨΊʹͲͷ঎඼Λਪન͢΂͖͔ʁ 推薦 k=1 k=2 k=3 どれ 買う？ 80% 20% 購入 非購入 4 / 31
5. όϯσΟοτʹ͓͚Δ୳ࡧͱ׆༻ͷτϨʔυΦϑ બ୒ࢶ k = 1, 2, 3 ʹ͓͍ͯɺ2 ͸ 1 ΑΓ΋ྑ͍͜ͱ͕෼͔͍ͬͯ Δɻ͜ͷͱ͖ɺ 3 ͕ 2 ΑΓ΋ྑ͍͔Ͳ͏͔Λ͔֬ΊΔͨΊʹ 3 ΛҾ͘ʁ ஌ࣝͷ୳ࡧ ͦΕͱ΋ ͦͦ͜͜ྑ͍ͱ෼͔͍ͬͯΔ 2 ΛҾ͖ଓ͚Δʁ ஌ࣝͷ׆༻ ୳ࡧ τϨʔυΦϑ ←→ ׆༻ ͜ͷτϨʔυΦϑ͸؍ଌ͕෦෼తͳ͜ͱʹ༝དྷ͢Δɻ k = 1 k = 2 k = 3 t = 1 t = 2 t = 3 t = 4 外れ 外れ 外れ 当たり ෦෼؍ଌͷ৔߹ k = 1 k = 2 k = 3 t = 1 t = 2 t = 3 t = 4 外れ 外れ 外れ 当たり 外れ 当たり 当たり 外れ 当たり 当たり 当たり 外れ શ؍ଌͷ৔߹ 5 / 31
6. ଟ࿹όϯσΟοτ໰୊: ໰୊ઃఆ ଟ࿹όϯσΟοτ໰୊͸ྦྷੵใुΛ࠷େԽ͢ΔͨΊʹ࿈ଓతʹߦಈ (࿹) Λܾఆ͢ΔํࡦΛ࡞Δ໰୊ɻ ه๏: • A = {1, ..., K}: K ݸͷߦಈू߹ • i(t) ∈ A: ࣌ࠁ t ʹબ୒͞Εͨߦಈ • Yi(t) (t): i(t) ʹର͢Δใु • Dt = {Yi(u) (u)}t u=1 : ࣌ࠁ t ·Ͱͷใुͷཤྺ ೖྗ: ߦಈճ਺ T ग़ྗ: ྦྷੵใु T t=1 Yi(t) (t) Λ࠷େԽ͢ΔΑ͏ͳ࿈ଓతͳߦಈ {i(t)}T t=1 6 / 31
7. ଟ࿹όϯσΟοτ໰୊: ྦྷੵϦάϨοτ ࣌ࠁ T ·Ͱͷྦྷੵใु T t=1 Yi(t) (t) Λ࠷େԽ͍ͨ͠ɻ͔͠͠ɺใ ुͷ஋ࣗମ͸໰୊ґଘͰ͋ΔͨΊɺԿΒ͔ͷҙຯͰ࠷దͳํࡦͱͷ ൺֱͰ͋ΔϦάϨοτͱ͍͏֓೦Λಋೖ͠ɺํࡦͷྑ͞ΛଌΔɻ regret(T) = max i∈A T t=1 Yi(t) − T t=1 Yi(t) (t) regret(T) ≥ 0 Ͱ͋Γɺregret(T) = 0 ʹ͍ͨ͠ɻ ظ଴஋ E[regret(T)] Λߟ͑Δ৔߹΋͋Δɻ 7 / 31
8. ଟ࿹όϯσΟοτ໰୊: ࠷ద࿹ࣝผ ྦྷੵใुͰͳ͘ɺ࠷దͳߦಈ i∗ = argmax i∈A Yi ͷࣝผʹڵຯ͕͋Δ৔ ߹͕͋Δɻ͜ͷΑ͏ʹ׆༻ΛߦΘͣʹ୳ࡧͷΈΛߦ͏໰୊Λ࠷ద࿹ ࣝผͱݺͿɻ ྫ) • A/B ςετ • ϕΠζ࠷దԽ • ೳಈֶश ࠷ద࿹ࣝผͰ͸୳ࡧ͔͠ߦΘΕͳ͍͕ɺྦྷੵใु࠷େԽΑΓ؆୯ͳ Θ͚Ͱ͸ͳ͍ɻ ˞ຊηϛφʔͰ͸جຊతʹྦྷੵใु࠷େԽʹয఺Λ౰ͯΔɻ 8 / 31
9. ଟ࿹όϯσΟοτ໰୊: ํࡦͷྑ͞ Q. ৗʹ i(t) = 1 ΛબͿํࡦ͸ྑ͍ʁ A. No(ͱߟ͑Δ): ৗʹ i∗ = 1 ͳΒ regret(T) = 0 ͕ͩɺৗʹ i∗ ̸= 1 ͳΒ 0 < regret(T) = (Yi∗ − Yi)T ≤ (Yi∗ − Ymin)T ࠷ѱέʔε ͭ·Γɺ࠷దղҎ֎ΛҾ͖ଓ͚Δͱ T ʹઢܗͳϦάϨοτ͕ੜͯ͡ ͠·͏ɻͦ͜Ͱɺର৅ͷશͯͷ໰୊ʹ͓͍ͯ࠷ѱέʔεͰ΋ T ʹؔ ͯ͠ྼઢܗͳϦάϨοτΛୡ੒͢Δ͜ͱΛ࠷௿ݶ໨ඪͱ͢Δɻ lim T→∞ E[regret(T)] T = 0 ͜Ε͸௚ײతʹ͸ߦಈճ਺Λ૿΍ͤ͹ i∗ Λݟ͚ͭΒΕΔͱ͍͏ ͜ͱɻ 9 / 31
10. ΞϧΰϦζϜ: Upper Confidence Bound: UCB 1 ߦಈ i ∈ A Λબ୒ (୳ࡧ) 2 ߦಈ ˆ i∗ = argmax i ˆ µi(t) ׆༻ + log t 2Ni(t) ୳ࡧ Λબ୒ Ni(t) ͸ t ·Ͱʹ i ͕બ୒͞Εͨճ਺ɻ ఆཧ: UCB ͷϦάϨοτ্ݶ E[regret(T)] ∈ O(log T) ϔϑσΟϯάͷෆ౳ࣜΛ࢖ͬͯ༗ҙਫ४ 1/t Ͱ UCB ͕࠷େʹͳΔ ߦಈΛબ୒ɻ P (ˆ µi ≤ µ) ≤ exp −2Ni(t) · (µ − ˆ µi)2 = 1 t 10 / 31
11. ΞϧΰϦζϜ: τϯϓιϯαϯϓϦϯά τϯϓιϯαϯϓϦϯά͸ϕΠζਪఆʹجͮ͘ɿ Yi ∼ Bernoulli(µi)ɺµi ∼ Beta(α, β). ೖྗ: α, β > 0 1 i ∈ A ʹର͠ࣄޙαϯϓϧ ˜ µi ∼ Beta(α + N1 i (t), β + N0 i (t)) Λੜ੒ 2 ߦಈ ˆ i∗ = argmax i ˜ µi Λબ୒ (ใुͷਪఆ͕ߴ͍΋ͷ΄Ͳબ͹ Ε΍͍͢) ͜͜ͰɺN1 i (t), N0 i (t) ͸ i ʹର͠ t ·Ͱʹ 1, 0 ͕ग़ͨճ਺Ͱ͋Δɻ τϯϓιϯαϯϓϦϯά͸େ͖͍֬཰Ͱ׆༻Λߦ͍ɺখ͍֬͞཰Ͱ ୳ࡧ͢Δɻ ఆཧ: τϯϓιϯαϯϓϦϯάͷϦάϨοτ্ݶ E[regret(T)] ∈ O(log T) 11 / 31
12. όϯσΟοτ͔ΒϒϥοΫϘοΫε࠷దԽ΁ όϯσΟοτ໰୊͸ҎԼͷΑ͏ʹଊ͑ΒΕΔɻ Figure: όϯσΟοτ໰୊ ͜͜Ͱɺk ∈ A, |A| = K < ∞, f : A → {0, 1}, f(k) = K l=1 Yi · (l = k)ɻ ϒϥοΫϘοΫε࠷దԽͰ͸ɺ࿈ଓͳແݶ࣍ݩͷߦಈ X, e.g., X(= A) = R ͱɺԿΒ͔ͷؔ਺ۭؒʹଐ͢Δؔ਺ f ∈ F, f : X → R Λߟ͑Δɻ f <latexit sha1_base64="HicTCq17VCguahIEOOmqHZHAKkc=">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</latexit> <latexit 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13. 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14. ϕΠζ࠷దԽ: ྫ (ړۀ) ྫ (ړۀ) • ໨త: ͨ͘͞Μͷڕ͕ัΕΔ఺Λݟ͚͍ͭͨ • ೖྗ x: ఺ • ؔ਺ධՁ: ړ֫ • ग़ྗ y: ړ֫ྔ 14 / 31
15. ϕΠζ࠷దԽ: ྫ (Խ߹෺ͷઃܭ) ྫ (Խ߹෺ͷઃܭ) • ໨త: ߴ඼࣭ͳԽ߹෺ (ༀ) ͷ։ൃ • ೖྗ x: Խ߹෺ͷಛ௃ྔ • ؔ਺ධՁ: ඼࣭ݕࠪ • ग़ྗ y: ඼࣭ 〜 〜 品質 検査 化合物設計 高品質 15 / 31
16. ϕΠζ࠷దԽ: ྫ (ϋΠύϥνϡʔχϯά) ྫ (ϋΠύʔύϥϝʔλνϡʔχϯά) • ໨త: ߴਫ਼౓ͳ༧ଌΛߦ͏ϋΠύʔύϥϝʔλͷܾఆ • όϦσʔγϣϯσʔλ H = {(ai, bi)}N i=1 ɺϋΠύʔύϥϝʔλ x Λ࣋ͭ༧ଌث hx • ೖྗ x: ϋΠύʔύϥϝʔλ • ؔ਺ධՁ: όϦσʔγϣϯධՁ f(x | H) = 1/N (ai,bi)∈Hk (bi = h(ai | x)) • ग़ྗ y: ༧ଌਫ਼౓ ハイパーパラメータ 評価 精度 モデル 訓練データ f <latexit sha1_base64="hbhMxJUIRo4lKu0RStfiwjJZoXY=">AAADEnicdVJNbxMxEHWWr7J8tXCEg0UaiUMV7fYA9FbBhWMjkbRSHFVe72xi1R8r20sTrZY/wLX8F8QNceUPcON/cGE2AUTaMJKl0Zv33nhGk5VK+pAk3zvRtes3bt7auh3fuXvv/oPtnYcjbysnYCissu4k4x6UNDAMMig4KR1wnSk4zs5et/Xjd+C8tOZtWJQw0XxqZCEFDwgNitPtbtI/SNKD5ym9mqT9ZBndwyefBj8IIUenO52fLLei0mCCUNz7cZqUYVJzF6RQ0MSs8lByccanMMbUcA1+Ui9/2tAeIjktrMNnAl2i/ypqrr1f6AyZmoeZv1xrwU21cRWKl5NamrIKYMSqUVEpGixtx6a5dCCCWmDChZP4Vypm3HERcDnxWpvWPFir/B4V6JavjVRnmW6Qn0PBRiC6ac10Zuc1y6zKWyXd7aa7TRP3mIFzYbXmJq8Zd1PN503NbAmOB+vatZzLMFNSy+DRBKVO4xanbK9losNmg5Zoy5o5TRF734JsZbJBIc0GBYJ/FXGPXh7d+QILV/Ey6HmD5/LnJuj/k9F+P8V8gHfziqxiizwmT8kzkpIX5JC8IUdkSAQB8oFckI/RRfQ5+hJ9XVGjzm/NI7IW0bdfTOMFIg==</latexit> <latexit 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17. Ψ΢εաఔ: ఆٛ ϕΠζ࠷దԽͰ͸͠͹͠͹ f ʹΨ΢εաఔࣄલ෼෍ΛԾఆ͢Δɻ ఆٛ: Ψ΢εաఔ ༗ݶू߹ X = {x1, ..., xN } ∈ XN ʹର͠ɺ f(X) = (f(x1), ..., f(xN ))⊤ ͕ଟ࣍ݩਖ਼ن෼෍ N(µ0(X), σ2(X, X)) ʹै͏ͱ͖ɺؔ਺ f : X → R ͸Ψ΢εաఔʹ ै͏ f ∼ GP(µ0, σ2 0 ) ͱݴ͏ɻ͜͜Ͱɺµ0 : X → R ͸ฏۉؔ਺Ͱ͋ Γɺσ2 0 : X × X → R ͸ڞ෼ࢄؔ਺Ͱ͋Δɻ 17 / 31
18. 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sha1_base64="3VbnLc9MKILtxhCsO1R2owFiaOM=">AAADFHicbVJLbxMxEHaWV1leLRy5WGwjcaiq3V7KsWovHIsgaaU4irxe78aqHyt7liayln/AtfwZbogrd34LF7xJhMhjJEufvpnvG89o8loKB2n6uxfdu//g4aO9x/GTp8+ev9g/eDl0prGMD5iRxl7n1HEpNB+AAMmva8upyiW/ym8uuvzVZ26dMPoTzGs+VrTSohSMQqA+zicnk/0kPU4XgbdBtgIJWsXl5KD3hxSGNYprYJI6N8rSGsaeWhBM8jYmjeM1ZTe04qMANVXcjf3iry3uB6bApbHhacAL9n+Fp8q5ucpDpaIwdZu5jtyVGzVQvht7oesGuGbLRmUjMRjcDY4LYTkDOQ+AMivCXzGbUksZhPXEa206czBGuiPMgluxNpLPc9WG+oKXZMhZknmicjPzJDey6JT4MMkO2zbuE81vmVGK6sITaitFZ60npuaWgrHdWm4FTKVQAlwwCVKrwhYrctRVBofdBl2hqT2xCgfuS0eSpckOhdA7FIH8p4j7eHN068qQ2OZrULM2nEu2eRzbYHhynAX8IU3OzleHs4deozfoLcrQKTpD79ElGiCGKvQV3aFv0V30PfoR/VyWRr2V5hVai+jXX9GcA0c=</latexit> Figure: άϥϑΟΧϧϞσϧ   f1 f2 f3   ∼ N     0 0 0   ,   k(x1, x1) k(x1, x2) k(x1, x3) k(x2, x1) k(x2, x2) k(x2, x3) k(x3, x1) k(x3x2) k(x3, x3)     = N     0 0 0   ,   1 0.99 0.37 0.99 1 0.44 0.37 0.44 1     ڞ෼ࢄؔ਺ k ͸ 2 ఺ x, x′ ͷ f, f′ ͕Ͳͷఔ౓ࣅ͍ͯΔ͔ΛܾΊΔɻ ͭ·ΓɺΨ΢εաఔ͸ X ্Ͱͷ f ͷ׈Β͔͞ΛΨ΢ε෼෍ʹΑΓ දݱ͢Δɻ 18 / 31
19. Ψ΢εաఔ: ࣄޙ෼෍ ࣄલ෼෍ f ∼ GP(0, k) ͱग़ྗϊΠζ y = f(x) + ϵ, ϵ ∼ N(0, σ2) Λ Ծఆ͢ΔɻN ఺ͷσʔλ D = {(xi, yi)}N i=1 ͕༩͑ΒΕͨͱ͖ɺؔ ਺͸ࣄޙฏۉ µD ͱࣄޙ෼ࢄ σ2 D Λ࣋ͭΨ΢εաఔ GP(µD, σ2 D ) ʹ ै͏ɻ͜͜Ͱɺ µD(x) = k⊤(K + σ2IN )−1y, σ2 D (x) = k⊤(K + σ2IN )−1k, Ͱ͋Γɺ{k}i = k(x, xi), {K}i,j = k(xi, xj)ɺy = (y1, ..., yN )⊤ Ͱ ͋Δɻ Figure: 100 αϯϓϧ Figure: 3 σʔλ఺͕༩͑ΒΕͨͱ͖ ͷ 100 ࣄޙαϯϓϧ 19 / 31
20. Ψ΢εաఔ: ࠷దԽ Ψ΢εաఔϞσϧ͸Χʔωϧؔ਺ͷϋΠύʔύϥϝʔλ θ Λ࣋ͭɻ θ ͸ର਺पล໬౓Λ࠷େԽ͢Δ͜ͱʹΑΓ࠷దԽ͞ΕΔ (λΠϓ 2 ࠷໬๏ɺܦݧϕΠζ๏)ɻ log p(y|x, D, θ) = − 1 2 y⊤ K + σ2IN −1 y − 1 2 log K + σ2IN − N 2 log 2π ͦͷଞͷ࠷దԽख๏: • MCMC Λ࢖ͬͨϕΠζਪఆ • ΫϩεόϦσʔγϣϯ 20 / 31
21. ϕΠζ࠷దԽ: ֫ಘؔ਺ ࣍ͷධՁ఺Λ֫ಘؔ਺ a : X → R ͷ࠷దԽʹΑΓܾఆ͢Δɻ xnext = argmax x∈X a(x) ͭ·Γɺmax f(x) Λ max a(x) ʹஔ͖׵͍͑ͯΔɻ 21 / 31
22. ֫ಘؔ਺: GP-UCB Gaussian Process Upper Confidence Bound (GP-UCB): a(x) = µt(x) ׆༻ + βtσt(x) ୳ࡧ ͜͜Ͱɺβt ͸ t ʹؔͯ͠୯ௐ૿Ճ͢Δؔ਺Ͱ͋Δɻ ఆཧ: GP-UCB ͷϦάϨοτ্ݶ regret(T) ∈ O( dTγT ) ͕ߴ֬཰Ͱ੒ཱɻ͜͜ͰɺγT ͸Χʔωϧؔ਺ݻ༗ͷ߲Ͱ͋ΓɺΨ΢ γΞϯΧʔωϧͰ͸ γT ∈ O((log T)(d+1)) Ͱ͋Δɻ 22 / 31
23. ֫ಘؔ਺: τϯϓιϯαϯϓϦϯά τϯϓιϯαϯϓϦϯά: a(x) = g(x) ͜͜Ͱ g ∼ P(f | D)ɻ τϯϓιϯαϯϓϦϯά͸େ͖ͳ֬཰Ͱ׆༻Λߦ͍ɺখ͍֬͞཰Ͱ ୳ࡧΛߦ͏ɻ ఆཧ: τϯϓιϯαϯϓϦϯάͷϦάϨοτ্ݶ E[regret(T)] ∈ O(d TγT ) 23 / 31
24. ϕΠζ࠷దԽ: f ͷ࠷దԽ GP-UCB: a(x) = µt(x) + 3σt(x) Figure: t = 1 24 / 31
25. ϕΠζ࠷దԽ: f ͷ࠷దԽ GP-UCB: a(x) = µt(x) + 3σt(x) Figure: t = 2 25 / 31
26. ϕΠζ࠷దԽ: f ͷ࠷దԽ GP-UCB: a(x) = µt(x) + 3σt(x) Figure: t = 3 26 / 31
27. ϕΠζ࠷దԽ: f ͷ࠷దԽ GP-UCB: a(x) = µt(x) + 3σt(x) Figure: t = 4 27 / 31
28. ϕΠζ࠷దԽ: f ͷ࠷దԽ GP-UCB: a(x) = µt(x) + 3σt(x) Figure: t = 5 28 / 31
29. ϕΠζ࠷దԽ: ϝϦοτͱσϝϦοτ ϝϦοτ • Ψ΢εաఔʹجͮ͘ཧ࿦ղੳ • ௚ײత σϝϦοτ • ߴ࣍ݩͳ৔߹ (≥ 20) ೉͍͠ • Ψ΢εաఔͷ O(N3) ͷܭࢉίετ • ׈Β͔͕͞ x ʹΑͬͯҟͳΔ (ඇఆৗͳ) ৔߹೉͍͠ 29 / 31
30. ݁࿦ • όϯσΟοτ͸୳ࡧͱ׆༻ͷτϨʔυΦϑΛѻ͏໰୊Ͱ͋Δ • Ψ΢εաఔ͸ؔ਺ͷ׈Β͔͞ΛΨ΢ε෼෍ʹΑΓදݱ͢ΔϞσ ϧͰ͋Δ • ϕΠζ࠷దԽ͸ແݶ࣍ݩͷߦಈΛѻ͏όϯσΟοτ໰୊Ͱ͋Δ 30 / 31
31. ࢀߟจݙ • ຊଟ३໵ɼதଜಞ঵ɼόϯσΟοτ໰୊ͷཧ࿦ͱΞϧΰϦζ Ϝɼߨஊࣾɼ2016ɽ • Tor Lattimore, Csaba Szepesvári, Bandit algorithms, preprint, 2018. • Russo, D. and Roy, B. V. Learning to optimize via posterior sampling.Mathematics of OperationsResearch, 39(4):1221–1243, 2014. • Srinivas, N., Krause, A., Kakade, S. M., and Seeger, M. W. Information-theoretic regret bounds forgaussian process optimization in the bandit setting.IEEE Transactions on Information Theory, 58(5):3250–3265, 2012. • Rasmussen, C. and Williams, C. Gaussian Processes for Machine Learning. MIT Press, Cambridge, 2006. 31 / 31