Tutorial on Generative Adversarial Networks youtube: https://www.youtube.com/playlist?list=PLeeHDpwX2Kj5Ugx6c9EfDLDojuQxnmxmU

1. Generative Adversarial Networks
Mark Chang

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

4. Mini-max Two-player Game

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)

18. Global Optimum
D(G(z)) = 0.5
real data
generated
data
G(z)

19. Convergence Proof
• Global Optimum Exists
• Converge to Global Optimum

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)

29. Experiment
• Quantitative Analyses
• Qualitative Analyses

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)

32. Experiment
• Qualitative Analyses :
– Visualization of samples from the model
real datagenerated data
real data
generated data

33. Experiment
• Qualitative Analyses :
– linearly interpolation in z space
prior:z ⇠ pz(z)

34. Further Research
Unsupervised Representation Learning with Deep
Convolutional Generative Adversarial Networks
Alec Radford, Luke Metz, Soumith Chintala

35. Further Research
Adversarial Autoencoders
Alireza Makhzani, Jonathon Shlens, Navdeep
Jaitly, Ian Goodfellow, Brendan Frey

36. Further Research
InfoGAN: Interpretable Representation Learning by
Information Maximizing Generative Adversarial Nets
Xi Chen, Yan Duan, Rein Houthooft, John
Schulman, Ilya Sutskever, Pieter Abbeel

37. Source Code
• Original paper (theano):
– https://github.com/goodfeli/adversarial
• Tensorflow implementation:
– https://github.com/ckmarkoh/GAN-tensorflow