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Generative Adversarial Networks Mark Chang

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Page 1: Generative Adversarial Networks

GenerativeAdversarialNetworks

MarkChang

Page 2: Generative Adversarial Networks

OriginalPaper

• Title:– GenerativeAdversarialNets

• Authors:– IanJ.Goodfellow,JeanPouget-Abadie,MehdiMirza,BingXu,DavidWarde-Farley,Sherjil Ozair,AaronCourville,Yoshua Bengio

• Organization:– UniversitedeMontreal

• URL:– https://arxiv.org/abs/1406.2661

Page 3: Generative Adversarial Networks

Outlines

• Mini-maxTwo-playerGame• GenerativeModelv.s.DiscriminativeModel• GenerativeAdversarialNetworks• ConvergenceProof• Experiment• FurtherResearch

Page 4: Generative Adversarial Networks

Mini-maxTwo-playerGame

Page 5: Generative Adversarial Networks

Mini-maxTwo-playerGame

• :thecurrentplayer• :theopponent• :theactiontakenbycurrentplayer• :theactiontakenbyopponent• :thevaluefunctionofcurrentplayer

-1

1

1

vs = max

as

min

as�vs(as, as�)

vs

ss�

asas�

as

as�

ss�

Page 6: Generative Adversarial Networks

Mini-maxTwo-playerGame

• :thecurrentplayer• :theopponent• :theactiontakenbycurrentplayer• :theactiontakenbyopponent• :thevaluefunctionofcurrentplayer

1

vs = max

as

min

as�vs(as, as�)

vs

ss�

asas�

as

ss�

-1

1

vs = 1

-1

1

1 as�

1

Page 7: Generative Adversarial Networks

GenerativeModelv.s.DiscriminativeModel

• DiscriminativeModel • GenerativeModel

p(x, y)p(y|x)

y = 0

y = 1

y = 0

y = 1

x

x

Page 8: Generative Adversarial Networks

DiscriminativeModel: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

exampleofotherclass

p(y = 0|x) = p(y = 1|x) = 0.5

Page 9: Generative Adversarial Networks

GenerativeModel:p(x,y)

x

p(x, y = 0)

p(x, y = 1)

generatenewexample

exampleofotherclass

Page 10: Generative Adversarial Networks

GenerativeAdversarialNetworks

• GeneratenewdatabyNeuralNetwork

p(x, z) = p(z)p(x|z)

GeneratorNetwork

p(z)p(x|z)prior

generateddataz ⇠ p(z)

samplingx

Page 11: Generative Adversarial Networks

GenerativeAdversarialNetworks

GeneratorNetworkG(z)prior

min

Gmax

DV (D,G)

generateddata

z ⇠ pz(z)

realdatax ⇠ pdata(x)

10

V (D,G) = Ex⇠pdata(x)[logD(x)] + E

z⇠pz(z)[log(1�D(G(z))]

DiscriminatorNetworkD(x)

sigmoidfunction

Page 12: Generative Adversarial Networks

TrainingDiscriminatorNetworkmax

D

(Ex⇠pdata(x)[logD(x)] + E

z⇠pz(z)[log(1�D(G(z))])

realdata

generateddata

D(x) ⇡ 0 ) logD(x) ⌧ 0

shouldbe1D(x) shouldbe0D(G(z))

DiscriminatorNetwork

10

D(x)

D(G(z)) ⇡ 1 ) log(1�D(G(z))) ⌧ 0

Page 13: Generative Adversarial Networks

TrainingDiscriminatorNetwork

DiscriminatorNetwork

10

realdata

generateddata

D(x) ⇡ 1 ) logD(x) ⇡ 0

max

D

(Ex⇠pdata(x)[logD(x)] + E

z⇠pz(z)[log(1�D(G(z))])

shouldbe1D(x) shouldbe0D(G(z))

D(x)

D(G(z)) ⇡ 0 ) log(1�D(G(z))) ⇡ 0

Page 14: Generative Adversarial Networks

TrainingGeneratorNetwork

min

G

(Ex⇠pdata(x)[logD(x)] + E

z⇠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))])

Thereisnointhisterm.G

shouldnotbe0=>shouldbe1D(G(z)) D(G(z))

Page 15: Generative Adversarial Networks

priorz ⇠ pz(z)

TrainingGeneratorNetwork

GeneratorNetworkG(z)

max

G(Ez⇠pz(z)[log(D(G(z))])

generateddata

realdata

10

D(G(z)) ⇡ 0 ) logD(G(z)) ⌧ 0

shouldbe1D(G(z))

Page 16: Generative Adversarial Networks

TrainingGeneratorNetwork

realdata

GeneratorNetwork

priorz ⇠ pz(z)

max

G(Ez⇠pz(z)[log(D(G(z))])

10

generateddata

G(z)

shouldbe1D(G(z))

D(G(z)) ⇡ 1 ) logD(G(z)) ⇡ 0

Page 17: Generative Adversarial Networks

TrainingGenerativeAdversarialNetworks

minG

V (D,G)

max

DV (D,G)

max

DV (D,G)

min

Gmax

DV (D,G)

minG

V (D,G)

max

DV (D,G)

G(z) G(z) G(z)

G(z)G(z)G(z)

Page 18: Generative Adversarial Networks

GlobalOptimum

D(G(z)) = 0.5

realdata

generateddataG(z)

Page 19: Generative Adversarial Networks

ConvergenceProof

• GlobalOptimumExists• ConvergetoGlobalOptimum

Page 20: Generative Adversarial Networks

GlobalOptimumExists

aftertraining,reachingglobaloptimum

x

pdata(x) = pg(x)

D(x) = 0.5

x

discriminatorD(x)

realdatadistributionpdata(x)

generateddatadistribution

pg(x)

z

priorpz(z)

globaloptimum: pdata = pg

GeneratorG(z)

beforetraining

x

pdata(x) pg(x)D(x)

G(z)

Page 21: Generative Adversarial Networks

x

pdata(x)pg(x)

D

⇤G(x)

GlobalOptimumExists

• Forfixed,theoptimaldiscriminatoris:

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

Page 22: Generative Adversarial Networks

GlobalOptimumExists

V (D,G) = Ex⇠pdata(x)[logD(x)] + E

z⇠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

p

data

(x)log(D(x))dx+

Z

x

p

z

(G�1(x))log(1�D(x))(G�1)0(x)dx

=

Z

x

p

data

(x)log(D(x))dx+

Z

x

p

g

(x)log(1�D(x))dx

=

Z

x

p

data

(x)log(D(x)) + p

g

(x)log(1�D(x))dx

=

Z

x

p

data

(x)log(D(x))dx+

Z

z

p

z

(z)log(1�D(G(z)))dz

Page 23: Generative Adversarial Networks

GlobalOptimumExists

max

D

V (D,G) = max

D

Z

x

p

data

(x)log(D(x)) + p

g

(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)

Page 24: Generative Adversarial Networks

GlobalOptimumExists

• Supposethediscriminatorisoptimal,theoptimalgeneratormakes:

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)

Page 25: Generative Adversarial Networks

GlobalOptimumExists

= max

D

Z

x

p

data

(x)log(D(x)) + p

g

(x)log(1�D(x))dx

C(G) = max

DV (G,D)

=

Z

x

p

data

(x)log(D⇤G

(x)) + p

g

(x)log(1�D

⇤G

(x))dx

=

Z

x

p

data

(x)log(p

data

(x)

p

data

(x) + p

g

(x)) + p

g

(x)log(p

g

(x)

p

data

(x) + p

g

(x))dx

=

Z

x

p

data

(x)log(p

data

(x)pdata(x)+pg(x)

2

) + p

g

(x)log(p

g

(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)

Page 26: Generative Adversarial Networks

GlobalOptimumExists

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

minG

C(G) = 0 + 0� log(4) = �log(4)

Page 27: Generative Adversarial Networks

D G

ConvergetoGlobalOptimum

V (D,G)

GlobalOptimummax

DV (G,D)

CurrentV (D,G)

Page 28: Generative Adversarial Networks

ConvergetoGlobalOptimum

V (D,G)

minG

V (G,D)

D G

GlobalOptimumCurrentV (D,G)

Page 29: Generative Adversarial Networks

Experiment

• QuantitativeAnalyses• QualitativeAnalyses

Page 30: Generative Adversarial Networks

Experiment

• QuantitativeAnalyses– log-likelihoodestimation– Parzen window: p(x) =

1

N

NX

i=1

G�(x� xi)

Page 31: Generative Adversarial Networks

Experimentgenerated

dataG(z)

testing data

xix

p(x)

loglikelihood

p(x) =1

N

NX

i=1

G�(x� xi)

Page 32: Generative Adversarial Networks

Experiment

• QualitativeAnalyses:– Visualizationofsamplesfromthemodel

realdatagenerateddatarealdata

generateddata

Page 33: Generative Adversarial Networks

Experiment

• QualitativeAnalyses:– linearlyinterpolationinzspace

prior:z ⇠ pz(z)

Page 34: Generative Adversarial Networks

FurtherResearchUnsupervisedRepresentationLearningwithDeepConvolutionalGenerativeAdversarialNetworks

AlecRadford, LukeMetz,Soumith Chintala

Page 35: Generative Adversarial Networks

FurtherResearchAdversarialAutoencodersAlireza Makhzani,JonathonShlens,Navdeep

Jaitly,IanGoodfellow, BrendanFrey

Page 36: Generative Adversarial Networks

FurtherResearchInfoGAN:InterpretableRepresentationLearningbyInformationMaximizingGenerativeAdversarialNets

XiChen,YanDuan,ReinHouthooft, JohnSchulman, Ilya Sutskever, PieterAbbeel

Page 37: Generative Adversarial Networks

Source Code

• Original paper (theano):– https://github.com/goodfeli/adversarial

• Tensorflow implementation:– https://github.com/ckmarkoh/GAN-tensorflow


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