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# Deep genenergyprobdoc

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This is explanation about Energy-Based GaN by Bengio

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### Deep genenergyprobdoc

1. 1. Deep Directed Generative Models with Energy-Based Probability Estimation By Yoshua Bengio mabonki0725 ()1 June 16, 2017
2. 2. (1) Alpha SL 64 RL Q- (2) AI (3) (4) VAE GAN Energy-Base ( ) IRL Figure:
3. 3. Eθ(x) -NFAHJ x Eθ(x) x Pθ(x) = 1 Z(θ) exp (−Eθ(x)) Figure: x=0 =1 x= =0 3 / 11
4. 4. EΘ(x) ˜Pθi (x) = 1 1 + exp −WT i x + bi PΘ(x) = 1 ZΘ i ˜Pθi (x) = 1 ZΘ eEΘ(x) EΘ(x) = i log 1 + e−(W T i x+bi) ! Eθ(x) = 1 σ2 xT x − bT x − i log 1 + eW T i x+bi 4 / 11
5. 5. L(Θ, D′ ) = − 1 N N i=1 log Pθ(x(i) ) (5) PositivePhase( ) NegativePhase( ∂L(Θ, D′) ∂Θ = − 1 N N i=1 ∂ log Pθ(x(i)) ∂Θ = − 1 N N i=1 ∂ log exp(Eθ(x(i) )) Zθ ∂Θ (6) = − 1 N N i=1 ∂ log exp(Eθ(x(i))) ∂Θ + ∂ log 1 ZΘ ∂Θ (7) = 1 N N i=1 ∂Eθ(x(i)) ∂Θ − Ex∼Pθ(x) ∂Eθ(x) ∂Θ (8) ≈ Ex+∼PD(x) ∂Eθ(x+) ∂Θ P ositiveP hase − Ex−∼Pθ(x) ∂Eθ(x) ∂Θ NegativeP hase (9) 5 / 11
6. 6. L(Θ, D′) x+ x− Pφ(y = 1|x) = σ(−Eφ(x)) = 1 1 + exp(−Eφ(x)) (10) P(y = 0) = p(y = 1) = 1 2 Positive Phase Negative Phase E(x,y)∼P (x,y) − ∂ log Pφ(y|x) ∂φ = E(x,y)∼P (x,y) − ∂ log Pφ(y = 1|x)yPφ(y = 0|x)(1−y) ∂φ (11) = − 1 2 Ex+∼PD(x) ∂ log Pφ(y = 1|x+) ∂φ + Ex−∼PΘ(x) ∂ log Pφ(y = 0|x−) ∂φ (12) = 1 2 Ex+∼PD(x) Pφ(y = 0|x+ ) ∂Eφ(x+) ∂φ − Ex−∼PΘ(x) Pφ(y = 1|x− ) ∂Eφ(|x−) ∂φ ≈ 1 4 Ex+∼PD(x) ∂Eφ(x+) ∂φ − Ex−∼PΘ(x) ∂Eφ(x−) ∂φ (13) 6 / 11
7. 7. Deep Generative Model with Enegrgy-Based 2 Positive Phase Negative Phase DeepLearning Figure: 7 / 11
8. 8. Deep Energy Model Positive Phase x+ Negative Phase x− Energy Energy Eθ(x) = 1 σ2 xT x − bT x − i log 1 + eW T i fφ(x)+bi (14) fφ(x) (7) ∂L(Θ,D′ ) ∂Θ Deep Energy Model ∂L(Θ, D′) ∂Θ ≈ Ex+∼PD(x) ∂Eθ(x+) ∂Θ P ositiveP hase − Ex−∼Pθ(x) ∂Eθ(x) ∂Θ NegativeP hase (15) 8 / 11
9. 9. Deep Generative Model (Negative Phase ) DKL(Pφ(x)||Pθ(x)) = Ex−∼Pφ(x) − log Pθ(x− ) − H(Pφ(x)) (16) Negativ Phase ∂ ∂φ Ex−∼Pφ(x) − log Pθ(x− ) = ∂ ∂φ Ez−∼P i(z) − log Pθ(Gφ(z)) (17) = Ez∼P (z) ∂Eθ(Gφ(z)) ∂φ (18) ≈ 1 N N i=1 ∂Eθ(Gφ(zi)) ∂φ zi ∼ P(z) (19) Gφ(zi) Deep Network zi ∼ P(z) (1 ∼ −1) Negativ Phase H(Pφ(x)) ≈ αi H(N(µαi , σi)) = αi 1 2 log 2 exp(πσ2 αi ) (20) 9 / 11
10. 10. Figure: Figure: 10 / 11
11. 11. VAE GAN 1 (VAE+GAN VAE GAN / Controlable Text Generation by Salakhutdinov 2 Generator Discrimetor CNN Adversarial Neural Machine Translation Energy-Base GAN Energy 11 / 11