論文紹介：Stein Variational Gradient Descent

Stein Variational Gradient Descent 原論文のもっともわかりやすくていねいな日本語解説です（著者の感想）（2020年10月現在）

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1. ࿦จ঺հɿStein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm ి௨େঙ໺ݚ M2 ઒ౡوେ October 19, 2020
2. ໨࣍ 1. ࿦จʹ͍ͭͯ 2. ϕΠζਪ࿦ͱม෼ਪ࿦ 3. Stein’s Identity ͔Β Kernelized Stein Discrepancy ΁ 4. Stein Variational Gradient Descent 5. ࣮ݧ 2
3. ࿦จʹ͍ͭͯ
4. ࿦จ৘ใ Title: Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm Authors: Qinag Liu, Dilin Wangʢൃද࣌ॴଐɿμʔτϚεେֶɼ ݱࡏ͸ͦΕͧΕ UT Austin, Facebookʣ Conference: NeurIPS 2016 3
5. ࿦จ֓ཁ Χʔωϧ๏ & ཻࢠϕʔεͷม෼ਪ࿦๏ Stein Variational Gradient Descent (SVGD) ΛఏҊ X (Kernelized) Stein Discrepancy ʹج͍ͮͯਅͷ෼෍ͱۙࣅ෼ ෍ͷ KL μΠόʔδΣϯεΛ࠶ؼతʹ࠷খԽ X ݁ہ͸͍͔ʹ KL μΠόʔδΣϯε࠷খԽΛղ͔͘ʹؼண X Χʔωϧ๏Λ͏·͘࢖ͬͯ KL μΠόʔδΣϯεͷʢ൚ؔ਺ʣ ޯ഑߱Լͷߋ৽ࣜΛಋग़ X ฏۉ৔ۙࣅ͕ෆཁͳͷͰ௨ৗͷม෼ਪ࿦ͷΑ͏ͳࣄޙ෼෍ʹ ಠཱੑΛԾఆ͠ͳ͍ X ཻࢠ਺ 1 Ͱ MAP ղ͕ಘΒΕΔ X ݪཧతʹଟๆੑʹରͯ͠ϩόετ 1 1࿦จதʹ໌ه͞Ε͍ͯΔΘ͚Ͱ͸ͳ͍ 4
6. ϕΠζਪ࿦ͱม෼ਪ࿦
7. ϕΠζਪ࿦͓͞Β͍ɿϕΠζͷఆཧ ঺հ࿦จͷલఏ஌ࣝͳͷͰ͓͞Β͍ʢ࿦จͷൣғ֎ʣ ϕΠζͷఆཧ D = {Dk}ɿσʔλू߹ɼx ∈ X ⊂ RdɿύϥϝʔλϕΫτϧ ࣄલ෼෍ p0(x) ͷ΋ͱࣄޙ෼෍ p(x) ͸ҎԼͷΑ͏ʹॻ͚Δɿ p(x) ∶= p(x∣D) = ¯ p(x)/Z, ¯ p(x) = p0(x)p(D∣x), Z = p(D) = ∫ ¯ p(x)dx ͳΜͱ͔ࣄޙ෼෍ p(x) or ¯ p(x) ΛٻΊΔͷ͕ϕΠζਪ࿦ͷ໨ඪ ▷ Ұൠʹѻ͍ͮΒ͍ࣄޙ෼෍ΛͲ͏ۙࣅ͢Δ͔ʁ 5
8. ϕΠζਪ࿦͓͞Β͍ɿม෼ਪ࿦ ม෼ਪ࿦ p(x)ɿະ஌ͷ෼෍ɼq(x)ɿൺֱతѻ͍΍͍ؔ͢਺ܗͷ෼෍ ֬཰෼෍ಉ࢜ͷྨࣅ౓ΛଌΔద౰ͳࢦඪΛ࠷దԽ͠ɼq Λ p ʹ͚ۙͮΔ ΋ͬͱ΋Α͋͘Δม෼ਪ࿦ͷ࿮૊Έɿ 1. ύϥϝʔλϕΫτϧ x ͷཁૉͷࣄޙ෼෍ʹద౰ʹಠཱੑΛԾ ఆʢฏۉ৔ۙࣅʣ (e.g. x1,x2 x3 ∣ D, x = [x1,x2,x3]⊺) 2. ର਺पล໬౓ ZʢͷԼքʣΛ࠷େԽ͢ΔΑ͏ʹ q Λߋ৽ X ਅͷσʔλ෼෍ʹؔ͢Δ Z ͷظ଴஋͕େ͖͍΄Ͳɼਅͷσʔ λ෼෍ͱϞσϧͷ KL μΠόʔδΣϯε͸খ͘͞ͳΔ 2 2จݙ [1] ͳͲࢀরɽఆ͔ٛΒ͙͢ʹࣔͤΔɽ 6
9. ϕΠζਪ࿦͓͞Β͍ɿม෼ਪ࿦ͷಛ௃ ௨ৗͷม෼ਪ࿦ͷ࿮૊Έͷಛ௃ɿ X MCMC ΑΓऩଋ͕͸΍͍ X ʢ࠷దԽ๏ʹΑΔ͕ʣہॴղʹτϥοϓ͞ΕΔ X ଟๆੑͷ p(x) Λ୯ๆੑͷ q(x) Ͱۙࣅͯ͠΋ͻͱͭͷϞʔυ ͔͠ݟΒΕͳ͍ X ΑΓޮ཰తͰۙࣅੑೳͷྑ͍ม෼ਪ࿦Λߦ͏ʹ͸ɼઃఆͨ͠ Ϟσϧʹର͢Δਂ͍ཧղ͕ඞཁ X ࣄޙ෼෍ͷͲͷ֬཰ม਺ؒʹಠཱੑΛԾఆ͢Δ͔ʁ ▷ ϞσϧʹґଘͤͣҰൠʹ࢖͑Δม෼ਪ࿦๏Λߟ͍͑ͨ 7
10. Stein’s Identity ͔Β Kernelized Stein Discrepancy ΁
11. Stein’s Identity Ծఆͱه߸ X σʔλू߹ D = {Dk} ͕ i.i.d. ʢಠཱಉ෼෍ʣ X ࣄޙ෼෍ p(x) ͸ X ্ͷ࿈ଓ͔ͭඍ෼Մೳͳؔ਺ X ϕ(x) = [ϕ1(x),...,ϕd(x)]⊺ ∈ Rdɿ͋Δਖ਼ଇ৚݅Λຬͨ ͢ϕΫτϧ஋ؔ਺ʢ࣍ϖʔδࢀরʣ Stein’s Identity (ελΠϯͷ౳ࣜʣ Stein Operator Ap ΛҎԼͰఆٛʢจݙ [2] ͳͲʹΑΔఆٛʣ ɿ Apϕ(x) = ∇x log p(x) ⋅ ϕ(x) + ∇x ⋅ ϕ(x) ∈ R. ͜ͷͱ͖ҎԼͷ Stein’s Identity ͕੒Γཱͭɿ Ex∼p[Apϕ(x)] = 0. 8
12. ิ଍ɿStein’s Identity ͷਖ਼ଇੑ৚݅ ϕ(x) ͷਖ਼ଇੑ৚݅ ϕ ͕ҎԼͷ͍ͣΕ͔ͷڥք৚݅Λຬͨ͢ͳΒ Stein’s Identity ͕੒ཱʢ∂X ͸ X ͷڥքʣ 1. p(x)ϕ(x) = 0, ∀x ∈ ∂X (X is compact) 2. limr→∞ ∮Br p(x)ϕ(x) ⋅ n(x)dS = 0 (X = Rd) ͜͜Ͱ Br ͸൒ܘ r ͷ௒ٿɼn(x) ͸ Br ͷ୯Ґ๏ઢϕ Ϋτϧ Ψ΢εͷൃࢄఆཧ ∫ X ∇x ⋅ (p(x)ϕ(x))dx = ∫ ∂X p(x)ϕ(x) ⋅ n(x)dx = 0. ͜ΕΛ༻͍ͯਖ਼ଇੑ৚݅ ⇒ Steins’ Identity Λࣔ͢ 9
13. ิ଍ɿStein’s Identity ͷূ໌ ূ໌ Stein Operator Ap ͷఆٛΑΓ Apϕ(x) = ∇x log p(x) ⋅ ϕ(x) + ∇x ⋅ ϕ(x) = ∇xp(x) ⋅ ϕ(x) + p(x)∇x ⋅ ϕ(x) p(x) = ∇x ⋅ (p(x)ϕ(x)) p(x) . ͕ͨͬͯ͠ Ex∼p[Apϕ(x)] = ∫ X ∇x ⋅ (p(x)ϕ(x)) p(x) p(x)dx = ∫ X ∇x ⋅ (p(x)ϕ(x))dx = 0. ◻ 10
14. Stein Discrepancy Stein’s Identity ΛԠ༻ͯ͠ 2 ͭͷ෼෍ͷ “ဃ཭౓” Λߟ͍͑ͨ ·ͣ͸ҎԼͷ Stein Discrepancy Λఆٛɿ Stein Discrepancy p(x),q(x)ɿX ্ͷͳΊΒ͔ͳ֬཰෼෍ Fɿద౰ͳؔ਺ू߹ ͜ͷͱ͖ Stein Discrepancy Λmaxϕ∈F Ex∼q[Apϕ(x)]ͱఆٛ Stein Discrepancy ͷղऍɿ ϕ ͰॏΈ෇͚ΒΕͨ p ͱ q ͷείΞؔ ਺ͷࠩͷظ଴஋ 3 ∵ Ex∼q[Aqϕ(x)] = 0 ΑΓ Ex∼q[Apϕ(x)] = Ex∼q[Apϕ(x) − Aqϕ(x)] = Ex∼q[(∇x log p(x) − ∇x log q(x)) ⋅ ϕ(x)]. 3จݙ [4] Lemma 2.3 ࢀর 11
15. Kernelized Stein Discrepancy طଘݚڀͰ͸ɼؔ਺ू߹ F ʹ͸ϦϓγοπϊϧϜͷ্քͳͲͷ ੍໿͕༻͍ΒΕ͖ͯͨ 4 F Λ࠶ੜ֩ώϧϕϧτۭؒʹݶఆͯ͠ Stein Discrepancy Λ ѻ͍΍ͨ͘͢͠ͷ͕ Kernelized Stein Discrepancy (KSD) Kernelized Stein Discrepancy (KSD) [4, 5] Hɿਖ਼ఆ஋Χʔωϧ k(x,x′) ͕ఆΊΔ࠶ੜ֩ώϧϕϧτۭؒ Hdɿf = [f1,...,fd]⊺, fi ∈ H ʹରԠ͢Δؔ਺ۭؒ ͜ͷͱ͖ Kernelized Stein Discrepancy D(q,p)ΛҎԼͰఆٛ ɿ D(q,p) = max ϕ∈Hd {Ex∼q[Apϕ(x)] s.t. ∥ϕ∥Hd ≤ 1} 4จݙ [3] ͳͲ 12
16. ิ଍ɿΧʔωϧؔ਺͕ੜ੒͢Δώϧϕϧτۭؒ ਖ਼ఆ஋Χʔωϧ k(x,x′) ͔Βఆ·Δઢܗۭؒ H0 ΛҎԼͰఆٛɿ H0 = {f(x) = m ∑ i=1 αik(x,xi) ∣ αi ∈ R, m ∈ N, xi ∈ X}. H0 ͷཁૉ f = ∑i αik(⋅,xi), g = ∑j βjk(⋅,xj) ʹର͠ɼ ⟨f,g⟩H0 = ∑i,j αiβjk(xi,xj)͸಺ੵͷఆٛΛຬͨ͢ ಺ੵۭؒ H0 ͷίʔγʔྻ͔ΒͳΔू߹ ˜ H0 ͱॻ͘ɿ ˜ H0 = {{fn} ⊂ H ∣ ∥fn − fm∥ → 0 (n,m → ∞)}. ˜ H0 Λಉ஋ؔ܎ {fn} ∼ {gn} ⇔ ∥fn − gn∥ → 0 (n → ∞) Ͱׂͬͯಘ ۭͨؒ͸͋Δώϧϕϧτۭؒ 5H ͱಉҰࢹͰ͖Δ 6 5H ্ͷ͢΂ͯͷίʔγʔྻ͕ H ಺ʹۃݶΛ΋ͭ 6ৄ͍ٞ͠࿦͸ [6, 7] ͳͲࢀর 13
17. ิ଍ɿώϧϕϧτۭؒͷ࠶ੜੑ H0 ͷ಺ੵʹؔͯ͠ ⟨f,k(x,⋅)⟩H0 = ∑ i αik(xi,x) = f(x) ͕Θ͔Γɼ͜ͷੑ࣭Λ H0 ͷ࠶ੜੑͱݺͿ H0 ͷ࠶ੜੑΛ࢖͏ͱɼਖ਼ఆ஋Χʔωϧ k(⋅,⋅) ͔Βఆ·Δώϧϕϧ τۭؒ H ͷ࠶ੜੑ΋ࣔ͞ΕΔ ▷ H ͸࠶ੜ֩ώϧϕϧτۭؒ (RKHS) ͱݺ͹ΕΔ ࠶ੜੑ͸Χʔωϧ๏ͷཧ࿦͚ͮʹ͓͍ͯඇৗʹॏཁ 7 7ޙʹݟΔΑ͏ʹ KSD ͷ࠷దԽʹ΋࢖͏ 14
18. Kernelized Stein Discrepancy ͷ࠷దղ Kernelized Stein Discrepancy ͷ࠷దղ [4] KSD D(q,p) ʹ͓͚Δ ϕ ∈ Hd ʹؔ͢Δ࠷େԽ໰୊ͷղ͸ ϕ(⋅) = ϕ∗ q,p (⋅) / ∥ϕ∗ q,p ∥Hd where ϕ∗ q,p (⋅) = Ex∼q[∇x log p(x)k(x,⋅) + ∇xk(x,⋅)] ʹ͓͍ͯ༩͑ΒΕɼ͜ͷͱ͖ D(q,p) = ∥ϕ∗ q,p ∥Hd ূ໌͸࣍ϖʔδ͔Β 15
19. Kernelized Stein Discrepancy ͷ࠷దղͷಋग़ (1) D(q,p) = maxϕ∈Hd {Ex∼q[Apϕ(x)] s.t. ∥ϕ∥Hd ≤ 1} ূ໌ ؆୯ͷͨΊείΞؔ਺Λsp(x) = ∇x log p(x)ͱॻ͘ͱ Ex∼q[Apϕ(x)] = Ex∼q[Apϕ(x) − Aqϕ(x)] ∵ Stein’s Identity = Ex∼q[(sp(x) − sq(x)) ⋅ ϕ(x)] = d ∑ l=1 Ex∼q[(sl p (x) − sl q (x))⟨ϕl,k(x,⋅)⟩H] ∵ ࠶ੜੑ = d ∑ l=1 ⟨ϕl,Ex∼q[(sl p (x) − sl q (x))k(x,⋅)]⟩H. ಺ੵͷੑ࣭ΑΓɼ૯࿨಺ͷ֤߲͸ҎԼͰ࠷େԽ͞ΕΔɿ ϕl(⋅) = cEx∼q[(sl p (x) − sl q (x))k(x,⋅)] c ͸ਖ਼نԽఆ਺ 16
20. Kernelized Stein Discrepancy ͷ࠷దղͷಋग़ (2) ϕl(⋅) = cEx∼q[(sl p (x) − sl q (x))k(x,⋅)] ϕl(⋅) = cEx∼q[sl p (x)k(x,⋅)+∇xl k(x,⋅)−(sl q (x)k(x,⋅) + ∇xl k(x,⋅))] ͷԼઢ෦͸ɼಉ࣌෼෍ͷظ଴஋ͷੑ࣭ͱ Stein’s Identity ΑΓ Ex∼q[sl q (x) + ∇xl k(x,⋅)] = Eq(x/l ) [Eq(xl;x/l ) [sl q (x)k(x,⋅) + ∇xl k(x,⋅)]] = Eq(x/l ) [0] = 0. ͕ͨͬͯ͠ɼϕl(⋅) ΛϕΫτϧදࣔ͢Ε͹ ϕ(⋅) = cEx∼q[s(x)k(x,⋅) + ∇xk(x,⋅)] = cϕ∗ q,p (⋅) ͱͳΓɼ·ͨϊϧϜ੍໿ ∥ϕ∥Hd ≤ 1 ͔Β c = 1 / ∥ϕ∗ q,p ∥Hd . ࣍ʹ D(q,p) = ∥ϕ∗ q,p ∥Hd Λࣔ͢ 17
21. Kernelized Stein Discrepancy ͷ࠷దղͷಋग़ (3) x,x′ ∼ q ΛͦΕͧΕಠཱͳ֬཰ม਺ͱ͢Δͱɼ ∥ϕ∗ q,p ∥2 Hd = ⟨Ex∼q [s(x)k(x,⋅) + ∇x k(x,⋅)],Ex′∼q [s(x′)k(x′,⋅) + ∇x′ k(x′,⋅)]⟩ = Ex,x′∼q [⟨s(x)k(x,⋅) + ∇x k(x,⋅),s(x′)k(x′,⋅) + ∇x′ k(x′,⋅)⟩] = Ex,x′∼q [s(x)⊺⟨k(x,⋅),k(x′,⋅)⟩s(x′) + s(x)⊺⟨k(x,⋅),∇x′ k(x′,⋅)⟩ + ⟨∇x k(x,⋅),k(x′,⋅)⟩s(x′) + ⟨∇x k(x,⋅),∇x′ k(x′,⋅)⟩] = Ex,x′∼q [s(x)⊺k(x,x′)s(x′) + s(x)⊺∇x′ k(x,x′) + ∇x k(x,x′)⊺s(x′) + Tr(∇x,x′ k(x,x′))] ͕ಘΒΕΔɽҰํ ϕ(⋅) = ϕ∗ q,p (⋅) / ∥ϕ∗ q,p ∥Hd ΑΓ Ex∼q [Ap ϕ(x)] = Ex∼q [s(p)⊺ϕ∗ q,p (x) + ∇x ⋅ ϕ∗ q,p (x)] / ∥ϕ∗ q,p ∥Hd = Ex,x′∼q [s(x)⊺k(x,x′)s(x′) + s(x)⊺∇x′ k(x,x′) + ∇x k(x,x′)⊺s(x′) + Tr(∇x,x′ k(x,x′))] / ∥ϕ∗ q,p ∥Hd = ∥ϕ∗ q,p ∥Hd = D(q,p) ◻ 18
22. Kernelized Stein Discrepancy ͷ·ͱΊͱੑ࣭ KSD ͷ·ͱΊ p(x),q(x)ɿX ্ͷͳΊΒ͔ͳ֬཰෼෍ k(⋅,⋅) ∶ X × X → Rɿਖ਼ఆ஋Χʔωϧ ͜ͷͱ͖ KSD D(q,p) ͸ҎԼʹΑͬͯ༩͑ΒΕΔɿ D(q,p) = ∥ϕ∗ q,p ∥Hd ϕ∗ q,p (⋅) = Ex∼q[∇x log p(x)k(x,⋅) + ∇xk(x,⋅)] KSD ͷੑ࣭ X p = q ͷ৔߹ͷΈ D(q,p) = 0 ͱͳΔ X p(x) ͦͷ΋ͷͰͳ͘ ∇x log p(x) ʹ͔͠ґଘ͠ͳ͍ ▷ ਖ਼نԽఆ਺ Z ͸Θ͔Βͳͯ͘΋ OKʂ 19
23. Stein Variational Gradient Descent
24. KL μΠόʔδΣϯεͱม෼ਪ࿦ KL μΠόʔδΣϯε p(x),q(x)ɿX ্ͷͳΊΒ͔ͳ֬཰෼෍ KL μΠόʔδΣϯε KL(q∥p) ͸࣍ࣜͰ༩͑ΒΕΔɿ KL(q∥p) = Ex∼q[log q(x) − log p(x)] = Ex∼q[log q(x) − log ¯ p(x) + log Z] KL μΠόʔδΣϯε͸֬཰෼෍ p,q ͷ “ဃ཭౓” Λද͍ͯͨ͠ ▷ ม෼ਪ࿦ɿਅͷࣄޙ෼෍ p ʹม෼ࣄޙ෼෍ q Λ͚ۙͮΔ 20
25. ֬཰ม਺ͷม׵ ֬཰ม਺ͷม׵ T ∶ X → Xɿ֬཰ม਺ʹର͢Δ 1 ର 1 ͔ͭͳΊΒ͔ͳࣸ૾ q(x)ɿX ্ͷѻ͍΍͍֬͢཰෼෍ ֬཰ม਺ x ∼ q(x) Λ T Ͱม׵ͨ͠z = T (x)͸ q[T ] (z) = q(T −1(z))∣det(∇zT −1(z))∣ ʹ͕ͨ͠͏ ఏҊख๏ͷํࡦ ॳظ෼෍ q0 ͔Βੜ੒ཻͨ͠ࢠ {xi} ʹ࠶ؼతʹม਺ม׵Λ ࢪ͠ɼࣄޙ෼෍ p ʹ͚͍ۙͮͯ͘ 21
26. KL μΠόʔδΣϯεͱ Stein Operator ఆཧ 3.1 x ∼ q(x) ʹରͯ͠ T (x) = x + ϵϕ(x) ͱ͍͏ม׵Λߟ͑Δɽ ͜ͷͱ͖ z = T (x) ͕͕ͨ͠͏෼෍ q[T ] (z) ʹؔͯ͠ ∇ϵKL(q[T ] ∥p) ∣ ϵ=0 = −Ex∼q[Apϕ(x)] ͕੒Γཱͭ ิ୊ 3.2 ఆཧ 3.1ʹ͓͍ͯ ϕ ∈ {ϕ ∈ Hd ∣ ∥ϕ∥Hd ≤ D(q,p)} ΛԾఆɽ ͜ͷͱ͖ఆཧ 3.1ͷෛͷޯ഑͸ ϕ∗ q,p (⋅) ͷํ޲Ͱ࠷େʹͳΔ Ҏ্ΑΓɼKSD Λϕʔεʹม਺ม׵Λߟ͑Ε͹ OKʂ 22
27. ఆཧ 3.1 ͓Αͼิ୊ 3.2 ͷূ໌ (1) ఆཧ 3.1 ͷূ໌ ∇ϵKL(q[T ] ∥p) = ∇ϵKL(q∥p[T −1] ) = −Ex∼q[∇ϵ log p[T −1] (x)]. ͜͜Ͱp[T −1] = p(T (x))det(∇xT (x))ʹ஫ҙ͢Δͱɼ ∇ϵ log p[T −1] (x) = ∇ϵ log p(T (x)) + log det(∇xT (x)). ӈลୈೋ߲͸ log det ͷඍ෼ެࣜΑΓʢจݙ [8] ࣜ (43)ʣ log det(∇xT (x)) = Tr((∇xT (x))−1∇ϵ∇xT (x)). ӈลୈҰ߲͸ ∇ϵ log p(T (x)) = ∇ϵp(T (x)) / p(T (x)) = ∇T (x) p(T (x)) / p(T (x)) ⋅ ∇ϵT (x) = ∇zp(z) / p(z) ∣z=T (x) ⋅ ∇ϵT (x) = sp(T (x)) ⋅ ∇ϵT (x). 23
28. ఆཧ 3.1 ͓Αͼิ୊ 3.2 ͷূ໌ (2) ∇ϵ log p[T −1] = sp (T (x)) ⋅ ∇ϵ T (x) + Tr((∇x T (x))−1∇ϵ ∇x T (x)) ͜͜Ͱ T (x) = x + ϵϕ(x), ϵ = 0 ͱ͢Δͱɼ T (x) = x, ∇ϵT (x) = ϕ(x), ∇xT (x) = I, ∇ϵ∇xT (x) = ∇xϕ(x). ͜ΕΒΛ୅ೖ͢Ε͹ɼ ∇ϵKL(q[T ] ∥p) = −Ex∼q[∇ϵ log p[T −1] (x)] = −Ex∼q[sp(T (x)) ⋅ ϕ(x) + ∇x ⋅ ϕ(x)] = −Ex∼q[Apϕ(x)] ◻ ิ୊ 3.2 ͸ఆཧ 3.1ͱKSD ͷ࠷దղ͔Βͨͩͪʹࣔ͞ΕΔ ◻ 24
29. Stein Variational Gradient Descent ఆཧ 3.1ɼิ୊ 3.2ΑΓɼҎԼͷม෼ਪ࿦ΞϧΰϦζϜ͕ಋ͔ΕΔ Stein Variational Gradient Descent 1. ॳظ෼෍ q0(x) ͔Βཻࢠ {x0 i }n i=1 ⊂ X Λαϯϓϧ 2. m = 0 ͔Βద౰ͳճ਺࣍ͷखॱΛ܁Γฦ͢ 2.1 ֶश཰ ϵm ΛఆΊΔ 2.2 ҎԼΛܭࢉ͠ɼm ← m + 1 ͱ͢Δɿ xm+1 i ← xm i + ϵm ˆ ϕ ∗ q,p (xm i ) ˆ ϕ ∗ q,p (x) = 1 n n ∑ j=1 [∇xm j log p(xm j )k(x,xm j ) + ∇xm j k(x,xm j )] ∇xk(x,x) = 0 Λຬͨ͢೚ҙͷΧʔωϧʹ͍ͭͯɼཻࢠ਺ n = 1 Ͱ log p(x) ͷޯ഑ํ޲ʹߋ৽ ▷ MAP ղʹ޲͔͏ 25
30. ࣮ݧ
31. ࣮ݧ 1. ࣮ݧઃఆ ໨ඪɿ1 ࣍ݩࠞ߹Ψ΢ε෼෍ͷۙࣅΛߦ͏ ϋΠύʔύϥϝʔλͷܾఆ X Χʔωϧؔ਺ k(⋅,⋅)ɿRBF Χʔωϧ X RBF ΧʔωϧͷϋΠύʔύϥϝʔλ͸֤ΠςϨʔγϣ ϯ͝ͱʹ median heuristics ʹΑܾͬͯఆ X k(x,x′) = exp(∥x − x′∥2 2 /h) ʹର͠ h = median({xm i }n i=1 )2/log n X ∑j k(xm i ,xm j ) ≈ 1 ఔ౓ʹͳΓɼ๫ΕͮΒ͘ͳΔ X ֶश཰ ϵm ɿ AdaGrad ʹΑܾͬͯఆ 26
32. ࣮ݧ 1. 1 ࣍ݩࠞ߹Ψ΢ε෼෍ (1) ཻࢠ਺ n = 100 ੺఺ઢɿ໨తͷ෼෍ p(x)ɼ྘࣮ઢɿۙࣅ෼෍ q(x) p(x) = 1 3 N(−2,1) + 2 3 N(2,1) q0(x) = N(10,1) ໨తͷ p(x) ͱॳظ෼෍ q0(x) ͷΦʔόʔϥοϓ͕ͳͯ͘΋ OKʂ ▷ SIR ͳͲϦαϯϓϦϯάʹΑΔۙࣅ๏ͱͷେ͖ͳҧ͍ 27
33. ࣮ݧ 1. 1 ࣍ݩࠞ߹Ψ΢ε෼෍ (2) ཻࢠ਺Λม͑ͯظ଴஋ܭࢉͷਫ਼౓ΛϞϯςΧϧϩ๏ͱൺֱ p(x),q0(x)ɿઌ΄Ͳͱಉ༷ ॎ࣠ɿlog10 MSE ͷ 20 ճฏۉ (ω ∼ N(0,1), b ∼ Uniform([0,2π])) ԣ࣠ɿཻࢠ਺ʢαϯϓϧ਺ʣ 28
34. ࣮ݧ 2. ࣮ݧઃఆ ໨ඪɿϕΠδΞϯϩδεςΟ οΫճؼͷࣄޙ෼෍ΛٻΊΔ Covertype Dataset8 Λ࢖༻ʢσʔλ਺ 581,012, ࣍ݩ਺ 54ʣ ϋΠύʔύϥϝʔλͷܾఆ X Χʔωϧؔ਺ k(⋅,⋅)ɿRBF Χʔωϧ X RBF ΧʔωϧͷϋΠύʔύϥϝʔλ͸ൺֱର৅ͷઌߦ ݚڀʹ߹ΘͤΔ X RBF ΧʔωϧͷϋΠύʔύϥϝʔλɿ h = 0.002 × median({xm i }n i=1 )2 X ֶश཰ɿϵm = a / (m + 1)0.55 a ͸ validation set ͔Βܾఆ X ֶशσʔλ 80%ɼςετσʔλ 20% X ϛχόοναΠζɿ50 29
35. ࣮ݧ 2. ϕΠδΞϯϩδεςΟ οΫճؼ ࠨਤɿཻࢠ਺ʢαϯϓϧ਺ʣn = 100 ʹ͓͚Δֶशۂઢ ӈਤɿm = 3000 ʹ͓͚Δཻࢠ਺ͱςετਫ਼౓ʢཻࢠϕʔεͷख๏ʣ ςετਫ਼౓ͷධՁ͸ͦΕͧΕ 50 ճͷฏۉ SVGD ͕Ұ൪Αͦ͞͏ 30
36. ࣮ݧ 3. ࣮ݧઃఆ ໨ඪɿϕΠδΞϯχϡʔϥϧωοτͷύϥϝʔλਪఆ ܭ 10 ݸͷσʔληοτͰ Probabilistic Back Propagation ͱൺֱ ϋΠύʔύϥϝʔλͷܾఆ X Χʔωϧؔ਺ɿRBF Χʔωϧ with median heuristics X ֶश཰ɿϞϝϯλϜ͖ͭ AdaGrad X ӅΕ૚਺ɿ1ʢϢχοτ਺ 50ʣ X Protein σʔλͷΈӅΕ૚ͷϢχοτ਺ 100 X ׆ੑԽؔ਺ɿReLU X ֶशσʔλ 90%ɼςετσʔλ 10% X ϛχόοναΠζɿ100 X Year σʔλͷΈ 1000 31
37. ࣮ݧ 3. ϕΠδΞϯχϡʔϥϧωοτ RMSE ͱର਺໬౓ʢSVGD ͸ཻࢠ਺ n = 20ʣ ςετਫ਼౓ͷධՁ͸Ұ෦Λআ͖ͦΕͧΕ 20 ճฏۉ ▷ Protein ͸ 5 ճɼYear ͸ 1 ճ ܭࢉ࣌ؒ΋ੑೳ΋ SVGD ͕ڧ͍ 32
38. References i [1] ౉ล੅෉ɼϕΠζ౷ܭͷཧ࿦ͱํ๏ɼίϩφࣾ (2012)ɽ [2] Gong, W., Li, Y., and Hern´ andez-Lobato, J. M., “Sliced Kernelized Stein Discrepancy,” arXiv:2006.16531 (2020). [3] Gorham, J., and Mackey, L., “Measuring Sample Quality with Stein’s Method,” Advances in Neural Information Processing Systems (2015). [4] Liu, Q., Lee, J., and Jordan, M., “A Kernelized Stein Discrepancy for Goodness-of-fit Tests,” in International Conference on Machine Learning (2016).
39. References ii [5] Chwialkowski, K., Strathmann, H., and Gretton, A., “A kernel test of goodness of fit,” in International Conference on Machine Learning (2016). [6] ෱ਫ݈࣍ɼΧʔωϧ๏ೖ໳ –ਖ਼ఆ஋ΧʔωϧʹΑΔσʔλղ ੳ–ɼே૔ॻళ (2010)ɽ [7] ۚ৿ܟจɼ౷ܭతֶशཧ࿦ɼߨஊࣾ (2015)ɽ [8] Petersen, K., and M. S. Pedersen.m “The matrix cookbook,” Version November 15 (2012).