General Framework of GAN
fGAN: General
Framework of GAN
f is convex
f(1) = 0
f-divergence
= 0
Because f
is convex
= 1
If P and Q are the same
distributions,
smallest
f is convex
f(1) = 0
f-divergence
KL
Reverse KL
Chi Square
Fenchel Conjugate
•
Every convex function f has a conjugate function f*
f is convex, f(1) = 0
Fenchel Conjugate
•
Every convex function f has a conjugate function f*
f is convex, f(1) = 0
Fenchel Conjugate
•
Every convex function f has a conjugate function f*
Something like
exponential?
= 1t – 0
Fenchel Conjugate
•
Every convex function f has a conjugate function f*
•
(f*)* = f
Connection with GAN
Connection with GAN
Samples from P
Samples from Q
familiar?
Original GAN has
different V(G,D)
Using the f-divergence
you like
https://arxiv.org/pdf/1606.00709.pdf
Flaw in Optimization?
Modified from Ian
Goodfellow’s tutorial
Mode Collapse
Training with too many iterations ……
Mode Dropping
BEGAN
on CelebA
Generator
at iteration t
Generator
at iteration t+1
Generator
at iteration t+2
Generator switches mode during training
Outlook:
Ensemble
Generator
1
Generator
2
……
……
To generate an image
fGAN is a framework that evaluates the difference between two distributions by utilizing f-divergence, with f being a convex function. This concept can be understood through examples like KL divergence, Reverse KL divergence, and Chi-Square. Additionally, the Fenchel Conjugate method plays a crucial role in optimizing convex functions in GAN settings. Connection with GAN further explores the relationship between functions and their outputs within the context of generative adversarial networks.
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Presentation Transcript
fGAN: General Framework of GAN
? and ? are two distributions. ? ? and q ? are the probability of sampling ?. f-divergence ? ? ? ? ??f is convex f(1) = 0 ???||? evaluates the difference of P and Q ???||? = ? ? ? ? = 1 If ? ? = ? ? for all ? ? ? ? ? ???||? = ? ? ? ?? = 0 smallest ? = 0 ? ? ? ? ???||? = ? ? ? ?? If P and Q are the same distributions, ? ? ? ? ? Because f is convex ? ? ? ?? ???||? has the smallest value, which is 0 ? = ? 1 = 0
f-divergence f is convex ? ? ? ? ???||? = ? ? ? ?? f(1) = 0 ? ? ? = ????? KL ? ? ? ? ? ? ? ? ? ? ? ? ???||? = ? ? ??? ?? = ? ? ??? ?? ? ? ? ? = ???? Reverse KL ? ? ? ? ? ? ? ? ???||? = ? ? ??? ?? = ? ? ??? ?? ? ? ? ? = ? 12 Chi Square 2 2 ? ? ? ? ? ? ? ? ? ? ???||? = ? ? 1 ?? = ?? ? ?
? ? ? ? ???||? = ? ? ? ?? Fenchel Conjugate ? f is convex, f(1) = 0 Every convex function f has a conjugate function f* ? ? = max ? ??? ? ?? ? ? ? ?1 = max ? ??? ? ??1 ? ? ? ?2 = ? ?1 max ? ??? ? ??2 ? ? ?1?1 ? ?1 ? ?2 ?3?2 ? ?3 ?2?1 ? ?2 ?2?2 ? ?2 ?3?1 ? ?3 ?1?2 ? ?1 ?2 ?1 ?
? ? ? ? ???||? = ? ? ? ?? Fenchel Conjugate ? f is convex, f(1) = 0 Every convex function f has a conjugate function f* ? ? = max ? ??? ? ?? ? ? ?1? ? ?1 ? ?1 ? ?2 ?2? ? ?2 ?3? ? ?3 ?2 ?1 ?
Fenchel Conjugate Every convex function f has a conjugate function f* ? ? = max ? ??? ? ?? ? ? 10t 10 log 10 ? ? = ????? Something like exponential? 1t 1log1 = 1t 0 ? ? = ??? ? 1 0.1t 0.1log0.1
Fenchel Conjugate Every convex function f has a conjugate function f* (f*)* = f ? ? = max ? ??? ? ?? ? ? ? ? = ??? ? 1 ? ? = ????? ? ? = max ? ??? ? ?? ????? ? ? = ?? ????? Given t, find x maximizing ? ? ? ???? 1 = 0 ? = ??? ? 1 ? ? = ??? ? 1 ? ??? ? 1 ? 1 = ??? ? 1
Connection with GAN ? ? = ? ??? ? ?? ? ? max ? ??? ? ?? ? ? ? ? = max ? ? ? ? ? ? ? ? ? ? ? ? ???||? = ? ? ? ?? ? ? ? ? ? ? ? ? = ? ? max ?? ? ??? ? ? ? ? ? ? ? ?? max ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ???||? ? ? ?? D is a function whose input is x, and output is t ? ? ? ? ? ? ?? = ? ? ? ? ?? ? ?
Connection with GAN ? ? ? ? ? ?? ???||? max ? ? ? ? ?? D ? ? ??~?? ? ? = max ??~?? ? D Samples from Q Samples from P ??~??? ? ? Original GAN has different V(G,D) ???????||?? = max ??~?????? ? D ? = ???min ???????||?? ? ??~??? ? ? = ???min max ? ??~?????? ? ? familiar? = ???min max ? ? ?,? ?
??~???? ? ???????||?? = max ??~?????? ? D Using the f-divergence you like https://arxiv.org/pdf/1606.00709.pdf
Modified from Ian Goodfellow s tutorial Flaw in Optimization? ?? = ????????????? ?? Reverse ?? = ????? ??????? ?? ?? ????? ????? ?? ?? Minimize KL(??||?????) (reverse KL) Maximum likelihood (minimize KL(?????||??))
Mode Collapse Training with too many iterations : real data : generated data
Mode Dropping Generator switches mode during training Generator at iteration t Generator at iteration t+1 Generator at iteration t+2 BEGAN on CelebA
Train a set of generators: ?1,?2,,?? To generate an image Outlook: Ensemble Random pick a generator ?? Use ??to generate the image Generator 2 Generator 1
