Generative Adversarial Networks
33 likes10,081 views
Tutorial on Generative Adversarial Networks youtube: https://www.youtube.com/playlist?list=PLeeHDpwX2Kj5Ugx6c9EfDLDojuQxnmxmU
![Generative Adversarial Networks
Generator
Network
G(z)prior
min
G
max
D
V (D, G)
generated
data
z ⇠ pz(z)
real data
x ⇠ pdata(x)
1
0
V (D, G) = Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))]
Discriminator
Network
D(x)
sigmoid
function](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-11-320.jpg)
![Training Discriminator Network
max
D
(Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))])
real data
generated
data
D(x) ⇡ 0 ) logD(x) ⌧ 0
should be 1D(x) should be 0D(G(z))
Discriminator
Network
1
0D(x)
D(G(z)) ⇡ 1 ) log(1 D(G(z))) ⌧ 0](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-12-320.jpg)
![Training Discriminator Network
Discriminator
Network
1
0
real data
generated
data
D(x) ⇡ 1 ) logD(x) ⇡ 0
max
D
(Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))])
should be 1D(x) should be 0D(G(z))
D(x)
D(G(z)) ⇡ 0 ) log(1 D(G(z))) ⇡ 0](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-13-320.jpg)
![Training Generator Network
min
G
(Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))])
) min
G
(Ez⇠pz(z)[log(1 D(G(z))])
) max
G
(Ez⇠pz(z)[log(D(G(z))])
There is no in this term.G
should not be 0 => should be 1D(G(z)) D(G(z))](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-14-320.jpg)
![prior
z ⇠ pz(z)
Training Generator Network
Generator
Network
G(z)
max
G
(Ez⇠pz(z)[log(D(G(z))])
generated
data
real data
1
0
D(G(z)) ⇡ 0 ) logD(G(z)) ⌧ 0
should be 1D(G(z))](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-15-320.jpg)
![Training Generator Network
real data
Generator
Network
prior
z ⇠ pz(z)
max
G
(Ez⇠pz(z)[log(D(G(z))])
1
0
generated
data
G(z)
should be 1D(G(z))
D(G(z)) ⇡ 1 ) logD(G(z)) ⇡ 0](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-16-320.jpg)
![Global Optimum Exists
V (D, G) = Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))]
x = G(z) ) z = G 1
(x) ) dz = (G 1
)0
(x)dx
) pg(x) = pz(G 1
(x))(G 1
)0
(x)
=
Z
x
pdata(x)log(D(x))dx +
Z
x
pz(G 1
(x))log(1 D(x))(G 1
)0
(x)dx
=
Z
x
pdata(x)log(D(x))dx +
Z
x
pg(x)log(1 D(x))dx
=
Z
x
pdata(x)log(D(x)) + pg(x)log(1 D(x))dx
=
Z
x
pdata(x)log(D(x))dx +
Z
z
pz(z)log(1 D(G(z)))dz](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-22-320.jpg)
![Global Optimum Exists
= max
D
Z
x
pdata(x)log(D(x)) + pg(x)log(1 D(x))dx
C(G) = max
D
V (G, D)
=
Z
x
pdata(x)log(D⇤
G(x)) + pg(x)log(1 D⇤
G(x))dx
=
Z
x
pdata(x)log(
pdata(x)
pdata(x) + pg(x)
) + pg(x)log(
pg(x)
pdata(x) + pg(x)
)dx
=
Z
x
pdata(x)log(
pdata(x)
pdata(x)+pg(x)
2
) + pg(x)log(
pg(x)
pdata(x)+pg(x)
2
)dx log(4)
= KL[pdata(x)||
pdata(x) + pg(x)
2
] + KL[pg(x)||
pdata(x) + pg(x)
2
] log(4)](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-25-320.jpg)
![Global Optimum Exists
C(G) = KL[pdata(x)||
pdata(x) + pg(x)
2
] + KL[pg(x)||
pdata(x) + pg(x)
2
] log(4)
0 0
KL[pdata(x)||
pdata(x) + pg(x)
2
] = 0
) pdata(x) = pg(x)
when pdata(x) =
pdata(x) + pg(x)
2
min
G
C(G) = 0 + 0 log(4) = log(4)](https://image.slidesharecdn.com/generativeadversarialnetworks-161121164827/85/Generative-Adversarial-Networks-26-320.jpg)
More Related Content
What's hot (20)
Viewers also liked (20)
![[系列活動] 資料探勘速遊](https://cdn.slidesharecdn.com/ss_thumbnails/0114ycchendmquicktour-170110050658-thumbnail.jpg?width=560&fit=bounds)
Similar to Generative Adversarial Networks (9)
More from Mark Chang (20)
Recently uploaded (20)
Generative Adversarial Networks
- 2. Original Paper • Title: – Generative Adversarial Nets • Authors: – Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio • Organization: – Universite ́ de Montre ́al • URL: – https://arxiv.org/abs/1406.2661
- 3. Outlines • Mini-max Two-player Game • Generative Model v.s. Discriminative Model • Generative Adversarial Networks • Convergence Proof • Experiment • Further Research
- 5. Mini-max Two-player Game • : the current player • : the opponent • : the action taken by current player • : the action taken by opponent • : the value function of current player -1 1 1 vs = max as min as vs(as, as ) vs s s as as as as s s
- 6. Mini-max Two-player Game • : the current player • : the opponent • : the action taken by current player • : the action taken by opponent • : the value function of current player 1 vs = max as min as vs(as, as ) vs s s as as as s s -1 1 vs = 1 -1 1 1 as 1
- 7. Generative Model v.s. Discriminative Model • Discriminative Model • Generative Model p(x, y)p(y|x) y = 0 y = 1 y = 0 y = 1 x x
- 8. Discriminative Model : p(y|x) x p(y = 0|x) > p(y = 1|x) p(y = 1|x) > p(y = 0|x) p(y = 0|x) + p(y = 1|x) = 1 y = 1 y = 0 y = 1 example of other class p(y = 0|x) = p(y = 1|x) = 0.5
- 9. Generative Model : p(x, y) x p(x, y = 0) p(x, y = 1) generate new example example of other class
- 10. Generative Adversarial Networks • Generate new data by Neural Network p(x, z) = p(z)p(x|z) Generator Network p(z) p(x|z)prior generated dataz ⇠ p(z) sampling x
- 11. Generative Adversarial Networks Generator Network G(z)prior min G max D V (D, G) generated data z ⇠ pz(z) real data x ⇠ pdata(x) 1 0 V (D, G) = Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))] Discriminator Network D(x) sigmoid function
- 12. Training Discriminator Network max D (Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))]) real data generated data D(x) ⇡ 0 ) logD(x) ⌧ 0 should be 1D(x) should be 0D(G(z)) Discriminator Network 1 0D(x) D(G(z)) ⇡ 1 ) log(1 D(G(z))) ⌧ 0
- 13. Training Discriminator Network Discriminator Network 1 0 real data generated data D(x) ⇡ 1 ) logD(x) ⇡ 0 max D (Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))]) should be 1D(x) should be 0D(G(z)) D(x) D(G(z)) ⇡ 0 ) log(1 D(G(z))) ⇡ 0
- 14. Training Generator Network min G (Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))]) ) min G (Ez⇠pz(z)[log(1 D(G(z))]) ) max G (Ez⇠pz(z)[log(D(G(z))]) There is no in this term.G should not be 0 => should be 1D(G(z)) D(G(z))
- 15. prior z ⇠ pz(z) Training Generator Network Generator Network G(z) max G (Ez⇠pz(z)[log(D(G(z))]) generated data real data 1 0 D(G(z)) ⇡ 0 ) logD(G(z)) ⌧ 0 should be 1D(G(z))
- 16. Training Generator Network real data Generator Network prior z ⇠ pz(z) max G (Ez⇠pz(z)[log(D(G(z))]) 1 0 generated data G(z) should be 1D(G(z)) D(G(z)) ⇡ 1 ) logD(G(z)) ⇡ 0
- 17. Training Generative Adversarial Networks min G V (D, G) max D V (D, G) max D V (D, G) min G max D V (D, G) min G V (D, G) max D V (D, G) G(z) G(z) G(z) G(z)G(z)G(z)
- 20. Global Optimum Exists after training, reaching global optimum x pdata(x) = pg(x) D(x) = 0.5 x discriminator D(x) real data distribution pdata(x) generated data distribution pg(x) z prior pz(z) global optimum: pdata = pg Generator G(z) before training x pdata(x) pg(x) D(x) G(z)
- 21. x pdata(x) pg(x) D⇤ G(x) Global Optimum Exists • For fixed, the optimal discriminator is: D⇤ G(x) = pdata(x) pdata(x) + pg(x) G D pdata(x) = 0, pg(x) = 0.5 D⇤ G(x) = 0 0.5 + 0 = 0 pdata(x) = 0.5, pg(x) = 0.5 D⇤ G(x) = 0.5 0.5 + 0.5 = 0.5 pdata(x) = 0.5, pg(x) = 0 D⇤ G(x) = 0.5 0.5 + 0 = 1
- 22. Global Optimum Exists V (D, G) = Ex⇠pdata(x)[logD(x)] + Ez⇠pz(z)[log(1 D(G(z))] x = G(z) ) z = G 1 (x) ) dz = (G 1 )0 (x)dx ) pg(x) = pz(G 1 (x))(G 1 )0 (x) = Z x pdata(x)log(D(x))dx + Z x pz(G 1 (x))log(1 D(x))(G 1 )0 (x)dx = Z x pdata(x)log(D(x))dx + Z x pg(x)log(1 D(x))dx = Z x pdata(x)log(D(x)) + pg(x)log(1 D(x))dx = Z x pdata(x)log(D(x))dx + Z z pz(z)log(1 D(G(z)))dz
- 23. Global Optimum Exists max D V (D, G) = max D Z x pdata(x)log(D(x)) + pg(x)log(1 D(x))dx ) pdata(x) D(x) pg(x) 1 D(x) = 0 @ @D(x) (pdata(x)log(D(x)) + pg(x)log(1 D(x))) = 0 ) D(x) = pdata(x) pdata(x) + pg(x)
- 24. Global Optimum Exists • Suppose the discriminator is optimal , the optimal generator makes: D⇤ G(x) ) D⇤ G(x) = pdata(x) pdata(x) + pg(x) = 1 2 pdata(x) = pg(x) pdata(x) = 0.3, pg(x) = 0.3 D⇤ G(x) = 0.3 0.3 + 0.3 = 0.5 pdata(x) = 0.8, pg(x) = 0.8 D⇤ G(x) = 0.8 0.8 + 0.8 = 0.5 x D⇤ G(x) pdata(x) = pg(x)
- 25. Global Optimum Exists = max D Z x pdata(x)log(D(x)) + pg(x)log(1 D(x))dx C(G) = max D V (G, D) = Z x pdata(x)log(D⇤ G(x)) + pg(x)log(1 D⇤ G(x))dx = Z x pdata(x)log( pdata(x) pdata(x) + pg(x) ) + pg(x)log( pg(x) pdata(x) + pg(x) )dx = Z x pdata(x)log( pdata(x) pdata(x)+pg(x) 2 ) + pg(x)log( pg(x) pdata(x)+pg(x) 2 )dx log(4) = KL[pdata(x)|| pdata(x) + pg(x) 2 ] + KL[pg(x)|| pdata(x) + pg(x) 2 ] log(4)
- 26. Global Optimum Exists C(G) = KL[pdata(x)|| pdata(x) + pg(x) 2 ] + KL[pg(x)|| pdata(x) + pg(x) 2 ] log(4) 0 0 KL[pdata(x)|| pdata(x) + pg(x) 2 ] = 0 ) pdata(x) = pg(x) when pdata(x) = pdata(x) + pg(x) 2 min G C(G) = 0 + 0 log(4) = log(4)
- 27. D G Converge to Global Optimum V (D, G) Global Optimum max D V (G, D) Current V (D, G)
- 28. Converge to Global Optimum V (D, G) min G V (G, D) D G Global Optimum Current V (D, G)
- 30. Experiment • Quantitative Analyses – log-likelihood estimation – Parzen window: ˆp(x) = 1 N NX i=1 G (x xi)
- 31. Experiment generated data G(z) testing data xi x ˆp(x) log likelihood ˆp(x) = 1 N NX i=1 G (x xi)
- 37. Source Code • Original paper (theano): – https://github.com/goodfeli/adversarial • Tensorflow implementation: – https://github.com/ckmarkoh/GAN-tensorflow