Deep Markov Model を数式で追う (+ Pyroでの追試)

Transcript

1. Deep Markov Model Λ਺ࣜͰ௥͏(+ PyroͰͷ௥ࢼ)
   খྛᕣՏ @kajyuuen
   

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2. ࣗݾ঺հ
   খྛ ᕣՏ (Koga Kobayashi)
   GitHub, Twitter ID: @kajyuuen
   ஜ೾େֶେֶӃ म࢜2೥
   ීஈ͸ࣗવݴޠॲཧɺಛʹNERΛݚڀ͍ͯ͠·͢
   

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3. ͜ͷൃදʹ͍ͭͯ
   • PPLॳ৺ऀ͕PyroνϡʔτϦΞϧͷDMMͷ਺ࣜΛৄࡉʹ௥ͬͯΈΔ
   • ࠓճͷSLTͰ࿩͞ͳ͍(࿩ͤͳ͍)͜ͱ
   • ELBOͷಋग़ʹ͍ͭͯ
   • Ϟσϧɺۙࣅ෼෍ͦͷ΋ͷͷ֓೦ʹ͍ͭͯͷઆ໌
   • ֬཰తม෼ਪ࿦(SVI; Stochastic Variational Inference)ʹ͍ͭͯ
   

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4. Hidden Markov Model
   p(X, Z, θ, π, A) =
   N
   ∏
   n=1
   {p(θ)p(π)p(A)p(x(n)
   1
   |z(n)
   1
   , θ)p(z(n)
   1
   |π)
   T
   ∏
   i=2
   p(xn
   |zn
   , θ)p(zn
   |zn−1
   , A)}
   ؍ଌσʔλ ɺજࡏม਺ ʹ͍ͭͯߟ͑Δ
   X = {x(1), ⋯, x(N)} Z = {z(1), ⋯, z(N)}
   z(n)
   1
   z(n)
   2
   x(n)
   2 xT
   ⋯
   z(n)
   T
   x(n)
   T
   π A
   θ
   ⋯
   ⋯
   ⋯
   Transition
   Emitter
   Init state
   z1
   ∼ Cat(z1
   |π)
   π ∼ Dir(π|α)
   zn
   ∼
   K
   ∏
   i=1
   Cat(zn
   |A:,i
   )zn−1,i
   x(n)
   1
   A:,i
   ∼ Dir(A:,i
   , β:,i
   )
   xi
   ∼
   K
   ∏
   k=1
   Bern(xi
   |θk
   )zi,k
   θk
   ∼ Beta(θk
   |a, b)
   

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5. Deep Markov Model
   z(n)
   0
   z(n)
   1
   x(n)
   1
   xT
   ⋯
   Gated Transition
   Emitter
   z(n)
   T
   x(n)
   T
   ҎԼɺ Λলུͯ͠ ͱॻ͘
   z(n) z
   p(X, Z, ψ, ξ) =
   N
   ∏
   n=1
   {p(z(n)
   0
   )
   T
   ∏
   i=1
   p(zi
   |MLPtrans
   (zi−1
   ; ψ)) × p(xi
   |MLPemit
   (zi
   ; ξ))}
   ؍ଌσʔλ ɺજࡏม਺ ʹ͍ͭͯߟ͑Δ
   X = {x(1), ⋯, x(N)} Z = {z(1), ⋯, z(N)}
   : MLP
   DMMͰ͸৚݅෇͖֬཰ʹMLPΛಋೖ͢Δ͜ͱͰ
   HMMʹൺ΂ɺΑΓ๛͔ͳදݱ͕ՄೳʹͳΔ
   

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6. Emitter
   MLPemit
   (zi
   ) = μi
   h(1)
   i
   = ReLU(W(1)zi
   ) + b(1)
   h(2)
   i
   = ReLU(W(2)h(1)
   i
   ) + b(2)
   μi
   = Sigmoid(W(2)h(2)
   i
   ) + b(3)
   xi
   ∼ Bern(xi
   |μi
   )
   zi
   xi
   Emitter
   ൪໨ͷજࡏม਺ ͔Β؍ଌσʔλ Λੜ੒͢Δ֬཰
   i zi
   xi
   ͕ै͏෼෍͸ѻ͏σʔλʹΑͬͯҟͳΔ
   xi
   

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7. Gated Transition
   MLPσi
   (zi−1
   ) = σi
   zi
   ∼ Normal(μi
   , σi
   )
   MLPμi
   (zi−1
   ) = μi
   h(1)
   i
   = ReLU(W(1)
   h
   zi−1
   ) + b(1)
   h
   μi
   = (1 − g(2)
   i
   ) ⊙ (Wμ
   zi−1
   + bμ
   ) + g(2)
   i
   ⊙ h(2)
   i
   g(1)
   i
   = ReLU(W(1)
   g
   zi−1
   ) + b(1)
   g
   g(2)
   i
   = Sigmoid(W(2)
   g
   g(a)
   i
   ) + b(2)
   g
   h(2)
   i
   = W(2)
   h
   h(1)
   i
   + b(2)
   h
   σi
   = Softplus(Wσ
   ReLU(h(2)
   i
   ) + bσ
   )
   zi−1
   zi
   Gated Transition
   ൪໨ͷજࡏม਺ ͔Β ൪໨ͷજࡏม਺ Λੜ੒͢Δ֬཰
   i − 1 zi−1
   i zi
   

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8. ۙࣅ෼෍: Guide
   જࡏม਺ ͷۙࣅ෼෍ ʹΨ΢ε෼෍ΛԾఆ͢Δɻ
   Z = {z(1), ⋯, z(N)} q( ⋅ )
   q(Z|X, ψ) =
   N
   ∏
   n=1
   Normal(z(n) |x(n) , ψ)
   ঈ٫ਪ࿦Λద༻͠ɺม෼ύϥϝʔλΛؔ਺ ʹΑͬͯճؼͯ͠ٻΊΔɻ
   f( ⋅ )
   q(Z|X, ψ) =
   N
   ∏
   n=1
   Normal(z(n) |f(x(n) ; ψ))
   ͜ͷ ΛCombinerͱݺͿɻ
   f( ⋅ )
   

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9. ۙࣅ෼෍: Guide
   zT
   xT
   (μT
   , σT
   )
   hT
   zq
   0
   z1
   x1
   (μ1
   , σ1
   )
   h1
   z2
   x2
   (μ2
   , σ2
   )
   h2
   ⋯
   ⋯
   ⋯
   ⋯
   Combiners
   RNN
   MLPσi
   (zi−1
   ) = σi
   hcombined
   i
   =
   1
   2
   {tanh(Wh
   zi−1
   + b(1)) + hhmm
   i
   }
   μi
   = Wμ
   hcombined
   i
   + bμ
   zi
   ∼ Normal(μi
   , σi
   )
   MLPμi
   (zi−1
   ) = μi
   σi
   = Softplus(Wσ
   hcombined
   i
   + bσ
   )
   

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10. ར༻ͨ͠σʔλ
    • ෳ਺ͷԻූ͔Βߏ੒͞ΕΔָۂσʔλɻ
    • 4෼Իූ͝ͱʹ۠੾ͬͯɺͦ͜ʹؚ·ΕΔෳ਺ͷԻූ͔Βߏ੒͞ΕΔ
    ϕΫτϧ Λೖྗͱ͢Δɻ
    x ∈ ℝ88
    88伴൫
    ྫ: x = {0,0,1,…,1,0}
    

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11. ࣮ݧ݁Ռ
    

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12. ײ૝
    • જࡏม਺͕࿈ଓ஋ͩͬͨͨΊɺ
    HMMΑΓ͸ΧϧϚϯϑΟϧλʹ͍ۙΑ͏ͳҹ৅Λड͚ͨɻ
    • ࣮ݧࣗମ͸͏·͍ͬͨ͘ײ͕͡͠ͳ͍…
    ಛʹELBO͕΄΅ઢܗʹ্͕͍ͬͯΔͷʹ͸ͳʹ͔ݪҼ͕͋Γͦ͏ɻ
    • PPL͸ͱͯ΋໘ന͍͕ɺ΍͸Γਂ૚ֶशͱ͔ʹൺ΂Δͱ৘ใ͕গͳ͍ɻ
    (Ԡ༻ઌ΍࠷ઌ୺ͷϞσϧʹ͍ͭͯڭ͍͖͍͑ͯͨͩͨͰ͢)
    

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13. ࢀߟจݙ
    [1] ػցֶशελʔτΞοϓγϦʔζ ϕΠζਪ࿦ʹΑΔػցֶशೖ໳, ਢࢁ ರࢤ, ਿࢁ ক
    [2] ػցֶशϓϩϑΣογϣφϧγϦʔζ ϕΠζਂ૚ֶश, ਢࢁ ರࢤ
    [3] ࣗવݴޠॲཧͷͨΊͷਂ૚ֶश, Goldberg, Yoav, ଞ
    [4] Structured Inference Networks for Nonlinear State Space Models,
    Rahul G. Krishnan, Uri Shalit, David Sontag
    [5] Deep Markov Model(http://pyro.ai/examples/dmm.html)
    

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