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GAN with Mathematics

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Hyeongmin Lee

“Toward Principled Methods for Training GANs, ICLR 2017, 172회 인용”은 Ian Goodfellow의 GAN에 대해서 근본적인 문제점을 제기합니다. 우리는 그냥 이미지를 잘 생성해 주니까 GAN을 사용하는데, 그 원리에 대해서 깊게 생각해 본 적은 없습니다. Generator와 Discriminator의 수렴에 대해서 관심을 가져본 적도 없죠. 이에 대해서 Distance부터 시작해서 수학적으로 질문을 던지는 논문입니다. 결국엔 Data Distribution이 확률 분포로써 어떻게 작용하는지에 대해서 살펴보고자 합니다. 물론 이에 대한 Solution을 제공한 것은 아닙니다만. 이러한 문제 제기는 GAN의 History에서 아주 큰 흐름을 가져왔습니다 - GAN에 대한 흔한 오해 - Kullback Leibler Divergence와 Jensen Shannon Divergence - GAN 알고리즘의 수학적 분석 - GAN을 Training하는 과정에서 발생하는 치명적인 문제점 - 문제점을 해결하기 위한 시도들 - GAN 테크트리: 그래서 무슨 GAN을 사용할까

Engineering
1 of 36
GAN, with Mathematics 연세대학교 영상 및 비디오 패턴 인식 연구실 이형민
GAN?? GAN에 대한 일반적인 오해
GAN?? MAGIC!! 이미지를 만들어주는 알고리즘 데이터 양을 늘려주는 알고리즘
GAN?? ???
Discovering the Distribution!!
Discovering the Distribution!! Discriminative Model Generative Model Sample New Sample Which Label?? Sample GAN
Kullback – Leibler Divergence 두 Distribution이 얼마나 유사한가?
https://www.countbayesie.com/blog/2017/5/9/kullback-leibler-divergence-explained
VS
정보량이 얼만큼 손실되는가?
𝐼𝑖 = − log 𝑝 𝑥𝑖 𝐻 = 𝐸 𝐼𝑖 = − 𝑖=1 𝑁 𝑝 𝑥𝑖 log 𝑝 𝑥𝑖 정보량 엔트로피
𝐷 𝐾𝐿 𝑝 𝑞 = 𝐸 log 𝑝 𝑥 − log 𝑞 𝑥 = 𝑖=1 𝑁 𝑝 𝑥𝑖 log 𝑝(𝑥𝑖) 𝑞(𝑥𝑖) 손실되는 정보량의 기댓값!! 𝐷 𝐾𝐿 = 0.338 𝐷 𝐾𝐿 = 0.477
But!! KL-Divergence는 비대칭 함수!! Jensen – Shannon Divergence를 쓴다. 𝐽𝑆𝐷(𝑝| 𝑞 = 1 2 𝐾𝐿 𝑝 𝑝 + 𝑞 2 + 𝐾𝐿 𝑞 𝑝 + 𝑞 2
Go back to GAN…
𝒛~𝒑 𝒛(𝒛) 𝒙~𝒑 𝒈(𝒙) 𝒙~𝒑 𝒅𝒂𝒕𝒂(𝒙) G
G를 먼저 고정! D를 Optimize 해보자! 𝑉 𝐺, 𝐷 = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝐷 𝑥 + 𝐸𝑧~𝑝 𝑧(𝑧)[log(1 − 𝐷(𝐺(𝑧)))] = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝐷 𝑥 + 𝐸 𝑥~𝑝 𝑔(𝑥) log(1 − 𝐷(𝑥) (∵ 𝐺 𝑧 ~𝑝 𝑔(𝑥)) = 𝑥 𝑝 𝑑𝑎𝑡𝑎 𝑥 log 𝐷 𝑥 + 𝑝 𝑔 𝑥 log 1 − 𝐷 𝑥 𝑑𝑥 D에 대해 미분!! 𝐷 𝐺 ∗ 𝑥 = 𝑝 𝑑𝑎𝑡𝑎(𝑥) 𝑝 𝑑𝑎𝑡𝑎 𝑥 + 𝑝 𝑔(𝑥)
이제 D를 Optimal로 가정하고 G를 Optimize 해보자!! 𝐶 𝐺 = 𝑉 𝐺, 𝐷∗ = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝐷∗ 𝑥 + 𝐸 𝑥~𝑝 𝑔(𝑥)[log(1 − 𝐷∗ (𝑥))] = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝑝 𝑑𝑎𝑡𝑎(𝑥) 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) + 𝐸 𝑥~𝑝 𝑔(𝑥)[log 𝑝 𝑔(𝑥) 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) ] = − log 4 + 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log(𝑝 𝑑𝑎𝑡𝑎(𝑥)) − log( 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) + 𝐸 𝑥~𝑝 𝑔(𝑥) log(𝑝 𝑔(𝑥)) − log( 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) = − log 4 + 𝐾𝐿( 𝑝 𝑑𝑎𝑡𝑎 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) + 𝐾𝐿( 𝑝 𝑔 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) = − log 4 + 2 × 𝐽𝑆𝐷( 𝑝 𝑑𝑎𝑡𝑎 𝑝 𝑔)
이론적인 GAN의 Training 시나리오…  Optimize D  Optimize G  Optimize D  … https://www.youtube.com/watch?v=RlAgB0Ooxaw&list=PLlMkM4tgfjnJhhd4wn5aj8fVTYJwIpWkS&index=49
However…
실제 데이터는 공간상에 넓게 분포하지 않음 ‘집합의 크기’ Ex: 수직선상에서 (0,1)의 Measure = 1 𝑅2 에서 (0,1)×(0,1)의 Measure = 1 But, 𝑅2 에서 {(x, y) : 0<x<1, y=0}의 Measure = 0 간단히 말하면, 전체 집합보다 차원이 작은 영역의 Measure는 0이다.
실제 데이터는 공간상에 넓게 분포하지 않음
실제 데이터는 공간상에 넓게 분포하지 않음 Real, Fake data의 영역 모두 Measure가 Zero  두 영역의 교집합 또한 Measure가 Zero
𝑫 𝒙 = 𝟏 𝛁𝑫 𝒙 = 𝟎 𝑫 𝒙 = 𝟎 𝛁𝑫 𝒙 = 𝟎 그래서, Real과 Fake를 구분하는 완벽한 Discriminator가 항상 존재한다. 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒓) 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒈)
𝛁𝑫 𝒙 = 𝟎 Vanishing Gradient
differentiate = 𝜹
D의 Gradient가 0이면 G의 Gradient도 0이 된다.  학습 불가
Solution 1.
Real Data와 Fake Data에 Noise를 추가한다. 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒓) 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒈) Noise를 추가하므로, 당연히 성능 저하!!
Solution 2.
JSD 말고 다른 Distance를 이용하자. Infimum = Greatest Lower Bound inf 0,1 = 0 inf 0,1 = 0 inf 0,1 = 0 inf 0,1 = 0  프로그래밍적으로 구현 불가능  Wasserstein GAN
Solution 3.
Progressive GAN (a.k.a. 자낳괴) 아직 GAN의 학습 방법론에 대한 연구는 현재 진행형…
GAN/VAE 비교 및 코드 Tensorflow https://github.com/hwalsuklee/ tensorflow-generative-model-collections PyTorch https://github.com/znxlwm/ pytorch-generative-model-collections
GAN Loss Function 계열 (학습 안정화) • GAN (NIPS 2014) • Least Square GAN (LSGAN, ICCV 2017) • Wasserstein GAN (WGAN, ICML 2017) • Improved Training of Wasserstein GANs (WGAN_GP, NIPS 2017) • Improving the Improved Training of Wasserstein GANs (CT-GAN, ICLR 2018)
cGAN / Application 계열 (Conditional Generation) • Conditional GAN (cGAN, 2014) • Image to Image Translation with Conditional Adversarial Networks (Pix2Pix, CVPR 2017) • Learning to Discover Cross-Domain Relations with GAN (DiscoGAN, ICML 2017) • Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks (CycleGAN, ICCV 2017)  LSGAN Loss 사용 • Toward Multimodal Image-to-Image Translation (BiCycleGAN, NIPS 2017)  LSGAN Loss 사용 / Image 변환에 다양성 부여 • High-Resolution Image Synthesis and Semantic Manipulation with Conditional GANs (Pix2PixHD, 2017)  LSGAN Loss 사용 / 고화질 영상 생성 • StarGAN: Unified Generative Adversarial Networks for Multi-Domain Image-to-Image Translation (CVPR 2018)  WGAN_GP Loss 사용 / 여러 Domain Image 변환을 동시에 학습 • Progressive Growing of GANs for Improved Quality, Stability and Variation (ProGAN, ICLR 2018)  WGAN_GP Loss 사용 / 실험적 방법으로 생성 능력 극대화
결론? GAN을 우리가 원하는 이미지를 만들어주는 편리하고 신기한 Tool 정도로 만 이해하면 더 깊은 연구를 하는 데 한계가 있다. 우리 분야에도 굉장히 많은 수학적 Background를 요구하는 분야가 많으 며, 수학을 잘하면 논문이 간지난다.

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GAN with Mathematics

  • 1. GAN, with Mathematics 연세대학교 영상 및 비디오 패턴 인식 연구실 이형민
  • 6. Discovering the Distribution!! Discriminative Model Generative Model Sample New Sample Which Label?? Sample GAN
  • 7. Kullback – Leibler Divergence 두 Distribution이 얼마나 유사한가?
  • 9. VS
  • 11. 𝐼𝑖 = − log 𝑝 𝑥𝑖 𝐻 = 𝐸 𝐼𝑖 = − 𝑖=1 𝑁 𝑝 𝑥𝑖 log 𝑝 𝑥𝑖 정보량 엔트로피
  • 12. 𝐷 𝐾𝐿 𝑝 𝑞 = 𝐸 log 𝑝 𝑥 − log 𝑞 𝑥 = 𝑖=1 𝑁 𝑝 𝑥𝑖 log 𝑝(𝑥𝑖) 𝑞(𝑥𝑖) 손실되는 정보량의 기댓값!! 𝐷 𝐾𝐿 = 0.338 𝐷 𝐾𝐿 = 0.477
  • 13. But!! KL-Divergence는 비대칭 함수!! Jensen – Shannon Divergence를 쓴다. 𝐽𝑆𝐷(𝑝| 𝑞 = 1 2 𝐾𝐿 𝑝 𝑝 + 𝑞 2 + 𝐾𝐿 𝑞 𝑝 + 𝑞 2
  • 14. Go back to GAN…
  • 15. 𝒛~𝒑 𝒛(𝒛) 𝒙~𝒑 𝒈(𝒙) 𝒙~𝒑 𝒅𝒂𝒕𝒂(𝒙) G
  • 16. G를 먼저 고정! D를 Optimize 해보자! 𝑉 𝐺, 𝐷 = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝐷 𝑥 + 𝐸𝑧~𝑝 𝑧(𝑧)[log(1 − 𝐷(𝐺(𝑧)))] = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝐷 𝑥 + 𝐸 𝑥~𝑝 𝑔(𝑥) log(1 − 𝐷(𝑥) (∵ 𝐺 𝑧 ~𝑝 𝑔(𝑥)) = 𝑥 𝑝 𝑑𝑎𝑡𝑎 𝑥 log 𝐷 𝑥 + 𝑝 𝑔 𝑥 log 1 − 𝐷 𝑥 𝑑𝑥 D에 대해 미분!! 𝐷 𝐺 ∗ 𝑥 = 𝑝 𝑑𝑎𝑡𝑎(𝑥) 𝑝 𝑑𝑎𝑡𝑎 𝑥 + 𝑝 𝑔(𝑥)
  • 17. 이제 D를 Optimal로 가정하고 G를 Optimize 해보자!! 𝐶 𝐺 = 𝑉 𝐺, 𝐷∗ = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝐷∗ 𝑥 + 𝐸 𝑥~𝑝 𝑔(𝑥)[log(1 − 𝐷∗ (𝑥))] = 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log 𝑝 𝑑𝑎𝑡𝑎(𝑥) 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) + 𝐸 𝑥~𝑝 𝑔(𝑥)[log 𝑝 𝑔(𝑥) 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) ] = − log 4 + 𝐸 𝑥~𝑝 𝑑𝑎𝑡𝑎(𝑥) log(𝑝 𝑑𝑎𝑡𝑎(𝑥)) − log( 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) + 𝐸 𝑥~𝑝 𝑔(𝑥) log(𝑝 𝑔(𝑥)) − log( 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) = − log 4 + 𝐾𝐿( 𝑝 𝑑𝑎𝑡𝑎 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) + 𝐾𝐿( 𝑝 𝑔 𝑝 𝑑𝑎𝑡𝑎 𝑥 +𝑝 𝑔(𝑥) 2 ) = − log 4 + 2 × 𝐽𝑆𝐷( 𝑝 𝑑𝑎𝑡𝑎 𝑝 𝑔)
  • 18. 이론적인 GAN의 Training 시나리오…  Optimize D  Optimize G  Optimize D  … https://www.youtube.com/watch?v=RlAgB0Ooxaw&list=PLlMkM4tgfjnJhhd4wn5aj8fVTYJwIpWkS&index=49
  • 20. 실제 데이터는 공간상에 넓게 분포하지 않음 ‘집합의 크기’ Ex: 수직선상에서 (0,1)의 Measure = 1 𝑅2 에서 (0,1)×(0,1)의 Measure = 1 But, 𝑅2 에서 {(x, y) : 0<x<1, y=0}의 Measure = 0 간단히 말하면, 전체 집합보다 차원이 작은 영역의 Measure는 0이다.
  • 21. 실제 데이터는 공간상에 넓게 분포하지 않음
  • 22. 실제 데이터는 공간상에 넓게 분포하지 않음 Real, Fake data의 영역 모두 Measure가 Zero  두 영역의 교집합 또한 Measure가 Zero
  • 23. 𝑫 𝒙 = 𝟏 𝛁𝑫 𝒙 = 𝟎 𝑫 𝒙 = 𝟎 𝛁𝑫 𝒙 = 𝟎 그래서, Real과 Fake를 구분하는 완벽한 Discriminator가 항상 존재한다. 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒓) 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒈)
  • 24. 𝛁𝑫 𝒙 = 𝟎 Vanishing Gradient
  • 26. D의 Gradient가 0이면 G의 Gradient도 0이 된다.  학습 불가
  • 28. Real Data와 Fake Data에 Noise를 추가한다. 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒓) 𝑴 = 𝑺𝒖𝒑𝒑(𝑷 𝒈) Noise를 추가하므로, 당연히 성능 저하!!
  • 30. JSD 말고 다른 Distance를 이용하자. Infimum = Greatest Lower Bound inf 0,1 = 0 inf 0,1 = 0 inf 0,1 = 0 inf 0,1 = 0  프로그래밍적으로 구현 불가능  Wasserstein GAN
  • 32. Progressive GAN (a.k.a. 자낳괴) 아직 GAN의 학습 방법론에 대한 연구는 현재 진행형…
  • 33. GAN/VAE 비교 및 코드 Tensorflow https://github.com/hwalsuklee/ tensorflow-generative-model-collections PyTorch https://github.com/znxlwm/ pytorch-generative-model-collections
  • 34. GAN Loss Function 계열 (학습 안정화) • GAN (NIPS 2014) • Least Square GAN (LSGAN, ICCV 2017) • Wasserstein GAN (WGAN, ICML 2017) • Improved Training of Wasserstein GANs (WGAN_GP, NIPS 2017) • Improving the Improved Training of Wasserstein GANs (CT-GAN, ICLR 2018)
  • 35. cGAN / Application 계열 (Conditional Generation) • Conditional GAN (cGAN, 2014) • Image to Image Translation with Conditional Adversarial Networks (Pix2Pix, CVPR 2017) • Learning to Discover Cross-Domain Relations with GAN (DiscoGAN, ICML 2017) • Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks (CycleGAN, ICCV 2017)  LSGAN Loss 사용 • Toward Multimodal Image-to-Image Translation (BiCycleGAN, NIPS 2017)  LSGAN Loss 사용 / Image 변환에 다양성 부여 • High-Resolution Image Synthesis and Semantic Manipulation with Conditional GANs (Pix2PixHD, 2017)  LSGAN Loss 사용 / 고화질 영상 생성 • StarGAN: Unified Generative Adversarial Networks for Multi-Domain Image-to-Image Translation (CVPR 2018)  WGAN_GP Loss 사용 / 여러 Domain Image 변환을 동시에 학습 • Progressive Growing of GANs for Improved Quality, Stability and Variation (ProGAN, ICLR 2018)  WGAN_GP Loss 사용 / 실험적 방법으로 생성 능력 극대화
  • 36. 결론? GAN을 우리가 원하는 이미지를 만들어주는 편리하고 신기한 Tool 정도로 만 이해하면 더 깊은 연구를 하는 데 한계가 있다. 우리 분야에도 굉장히 많은 수학적 Background를 요구하는 분야가 많으 며, 수학을 잘하면 논문이 간지난다.