[DL輪読会]GANとエネルギーベースモデル
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This document discusses the connections between generative adversarial networks (GANs) and energy-based models (EBMs). It shows that GAN training can be interpreted as approximating maximum likelihood training of an EBM by replacing the intractable data distribution with a generator distribution. Specifically: 1. GANs train a discriminator to estimate the energy function of an EBM, with the generator minimizing that energy of its samples. 2. EBM training can be seen as alternatively updating the generator and sampling from it, in a manner similar to contrastive divergence for EBMs. 3. This perspective unifies GANs and EBMs, and suggests ways to combine their training procedures to leverage their respective advantages
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[DL輪読会]GANとエネルギーベースモデル
- 1. 1 GANͱΤωϧΪʔϕʔεϞσϧ Shohei Taniguchi, Matsuo Lab (M2)
- 2. ֓ཁ ҎԼͷ3ຊͷจΛϕʔεʹɺGANͱEBMͷؔʹ͍ͭͯ·ͱΊΔ Deep Directed Generative Models with Energy-Based Probability Estimatio n https://arxiv.org/abs/1606.03439 Maximum Entropy Generators for Energy-Based Model s https://arxiv.org/abs/1901.08508 Your GAN is Secretly an Energy-based Model and You Should use Discriminator Driven Latent Samplin g https://arxiv.org/abs/2003.06060
- 3. Outline લఏࣝ • Generative Adversarial Network • Energy-based Model GANͱEBMͷྨࣅ จհ
- 4. Generative Adversarial Network [Goodfellow et al.,2014] ࣝผث ͱੜث ͷϛχϚοΫεήʔϜ ࣝผث Λ࠷େԽ͠ɺੜث Λ࠷খԽ Dθ Gϕ Dθ : ℝdx → [0,1], Gϕ : ℝdz → ℝdx ℒ (θ, ϕ) = 𝔼 p(x) [log Dθ(x)] + 𝔼 p(z) [log (1 − Dθ (Gϕ (z)))] ℒ ℒ
- 5. GANͷֶश GANͷߋ৽ࣜ ℒ (θ, ϕ) = N ∑ i=1 log Dθ(xi) + log (1 − Dθ (Gϕ (zi))) θ ← θ + ηθ ∇θℒ (θ, ϕ) ϕ ← ϕ − ηϕ ∇ϕℒ (θ, ϕ) zi ∼ Normal (0,I)
- 6. GANͷҰൠతͳղऍ ࣝผີʹثൺਪఆث ࣝผثσʔλ ͱੜαϯϓϧͷ ͷ ີൺਪఆͯ͠ͱثͷׂΛՌͨ͢ i.e., ࣝผ࠷͕ثదͳͱ͖ p (x) pϕ (x) = 𝔼 p(z) [p (Gϕ (z))] D* θ (x) = p (x) p (x) + pϕ (x)
- 7. GANͷҰൠతͳղऍ ੜثͷֶशJS divergenceͷ࠷খԽ ࣝผ࠷͕ثదͳͱ͖ ੜث σʔλͱͷJensen-Shannon divergence࠷খԽʹΑΓֶश͞ΕΔ ℒ (θ, ϕ) = JS (p (x) ∥ pϕ (x)) − 2 log 2 JS (p (x) ∥ pϕ (x)) = 1 2 KL ( p (x) ∥ p (x) + pϕ (x) 2 ) + 1 2 KL ( pϕ (x) ∥ p (x) + pϕ (x) 2 ) Gϕ
- 8. Trick −log D ΦϦδφϧͷϩεͩͱɺޯফࣦ͕͜ىΓ͍͢ͷͰɺ ޙΛҎԼͷΑ͏ʹஔ͖͑ΔτϦοΫ͕Α͘ΘΕΔ ͨͩ͠ɺ͜ͷ߹ີൺਪఆʹͮ͘جղऍΓཱͨͳ͍ ℒ (θ, ϕ) = 𝔼 p(x) [log Dθ(x)] − 𝔼 p(z) [log Dθ (Gϕ (z))]
- 9. GANͷੜ σʔλͱͷڑͷࢦඪΛJSҎ֎ʹม͑Δͱɺ༷ʑͳGANͷੜ࡞͕ܥΕΔ ྫɿWasserstein GAN 1-Ϧϓγοπͳؔʢ ʣ ͜ͷͱ͖ɺੜثͷֶश1-Wasserstein distanceͷ࠷খԽͱͳΔ ℒ (θ, ϕ) = 𝔼 p(x) [Dθ(x)] − 𝔼 p(z) [Dθ (Gϕ (z))] Dθ ℝdx → ℝ
- 10. Energy-based Model ΤωϧΪʔؔ Ͱ֬ϞσϧΛද͢ݱΔ ෛͷର ʹൺྫ Eθ (x) pθ (x) = exp (−Eθ (x)) Z (θ) Z (θ) = ∫ exp (−Eθ (x)) dx Eθ (x) −log pθ (x)
- 11. EBMͷֶश Contrastive Divergence EBMͷରͷޯ ܇࿅σʔλͷΤωϧΪʔΛԼ͛ͯɺϞσϧ͔ΒͷαϯϓϧͷΤωϧΪʔΛ্͛Δ ∇θlog pθ (x) = − ∇θEθ (x) + ∇θlog Z (θ) = − ∇θEθ (x) + 𝔼 x′  ∼pθ(x) [∇θEθ (x′  )]
- 12. EBM͔ΒͷαϯϓϦϯά Langevin dynamics ޯϕʔεͷMCMC ͜ͷߋ৽ࣜͰ܁Γฦ͠αϯϓϦϯά͢Δͱɺαϯϓϧͷ ʹऩଋ͢Δ x ← x − η∇xEθ [x] + ϵ ϵ ∼ Normal (0,2ηI) pθ (x) ޯ߱Լ๏ʹϊΠζ͕ͷͬͨܗ
- 13. EBMͷֶश Contrastive Divergence ·ͱΊΔͱɺEBMͷߋ৽ࣜ ℒ (θ, x′  ) = − N ∑ i=1 Eθ (xi) + Eθ (x′  i) θ ← θ + ηθ ∇θℒ (θ, x′  ) x′  i ← x′  i − ηx′  ∇x′  ℒ (θ, x′  ) + ϵ ϵ ∼ Normal (0,2ηI)
- 14. EBMͷֶश Contrastive Divergence ·ͱΊΔͱɺEBMͷߋ৽ࣜ ℒ (θ, x′  ) = − N ∑ i=1 Eθ (xi) + Eθ (x′  i) θ ← θ + ηθ ∇θℒ (θ, x′  ) x′  i ← x′  i − ηx′  ∇x′  ℒ (θ, x′  ) + ϵ ϵ ∼ Normal (0,2ηI) Α͘ݟΔͱGANͬΆ͍
- 15. GANͷߋ৽ࣜ ℒ (θ, ϕ) = N ∑ i=1 log Dθ(xi) + log (1 − Dθ (Gϕ (zi))) θ ← θ + ηθ ∇θℒ (θ, ϕ) ϕ ← ϕ − ηϕ ∇ϕℒ (θ, ϕ) zi ∼ Normal (0,I)
- 16. GANͷߋ৽ࣜ with -logDτϦοΫ ℒ (θ, ϕ) = N ∑ i=1 log Dθ(xi) − log Dθ (Gϕ (zi)) θ ← θ + ηθ ∇θℒ (θ, ϕ) ϕ ← ϕ − ηϕ ∇ϕℒ (θ, ϕ) zi ∼ Normal (0,I)
- 17. ͱ͓͘ͱ Eθ (x) = − log Dθ (x) ℒ (θ, ϕ) = − N ∑ i=1 Eθ (xi) + Eθ (Gϕ (zi)) θ ← θ + ηθ ∇θℒ (θ, ϕ) ϕ ← ϕ − ηϕ ∇ϕℒ (θ, ϕ) zi ∼ Normal (0,I) GANͷߋ৽ࣜ with -logDτϦοΫ
- 18. GANͱEBMͷྨࣅੑ ℒ (θ, ϕ) = − N ∑ i=1 Eθ (xi) + Eθ (Gϕ (zi)) θ ← θ + ηθ ∇θℒ (θ, ϕ) ϕ ← ϕ − ηϕ ∇ϕℒ (θ, ϕ) zi ∼ Normal (0,I) ℒ (θ, x′  ) = − N ∑ i=1 Eθ (xi) + Eθ (x′  i) θ ← θ + ηθ ∇θℒ (θ, x′  ) x′  i ← x′  i − ηx′  ∇x′  ℒ (θ, x′  ) + ϵ ϵ ∼ Normal (0,2ηI) GAN with - logD trick EBM ΊͬͪΌࣅͯΔ͚Ͳͪΐͬͱҧ͏
- 19. GANͱEBMͷྨࣅੑ ℒ (θ, ϕ) = − N ∑ i=1 Eθ (xi) + Eθ (Gϕ (zi)) θ ← θ + ηθ ∇θℒ (θ, ϕ) ϕ ← ϕ − ηϕ ∇ϕℒ (θ, ϕ) zi ∼ Normal (0,I) ℒ (θ, x′  ) = − N ∑ i=1 Eθ (xi) + Eθ (x′  i) θ ← θ + ηθ ∇θℒ (θ, x′  ) x′  i ← x′  i − ηx′  ∇x′  ℒ (θ, x′  ) + ϵ ϵ ∼ Normal (0,2ηI) GAN with - logD trick EBM ΊͬͪΌࣅͯΔ͚Ͳͪΐͬͱҧ͏ ߋ৽ʹϊΠζ͕ͷΔ αϯϓϧΛߋ৽͢ΔΘΓʹ ϊΠζ͔ΒαϯϓϧΛੜ͢Δؔ Λ ߋ৽͢Δ Gϕ
- 21. Taesup Kim,Yoshua Bengio (Université de Montréal) Deep Directed Generative Models with Energy-Based Probability Estimation https://arxiv.org/abs/1606.03439
- 22. EBMͷֶश Contrastive Divergence EBMͷରͷޯ ͔ΒͷαϯϓϦϯάΛ ͔ΒͷαϯϓϦϯάͰஔ͖͑Δ ∇θlog pθ (x) = − ∇θEθ (x) + 𝔼 x′  ∼pθ(x) [∇θEθ (x′  )] ≈ − ∇θEθ (x)+ 𝔼 z∼p(z) [∇θEθ (Gϕ (z))] pθ (x) Gϕ (z)
- 23. ੜثͷֶश ͱ͢Δͱɺ ͱͳΕྑ͍ͷͰ ͜ͷ2ͭͷͷKL divergenceΛ࠷খԽ͢Δ͜ͱͰֶश͢Δ pϕ (x) = 𝔼 p(z) [δ (Gϕ (z))] pθ (x) = pϕ (x) KL (pϕ∥ pθ) = 𝔼 pϕ [−log pθ (x)] − H (pϕ) αϯϓϧͷΤωϧΪʔΛ Լ͛Δ αϯϓϧͷΤϯτϩϐʔΛ ্͛Δ
- 24. ੜثͷֶश ͳͥΤϯτϩϐʔ߲͕ඞཁ͔ ͠Τϯτϩϐʔ߲͕ͳ͍ͱɺੜثΤωϧΪʔ͕࠷খʢʹີ͕࠷େʣͷ αϯϓϧͷΈΛੜ͢ΔΑ͏ʹֶशͯ͠͠·͏ ‣ GANͷmode collapseͱࣅͨΑ͏ͳݱ ͜ΕΛ͙ͨΊʹΤϯτϩϐʔ߲͕ඞཁ KL (pϕ∥ pθ) = 𝔼 pϕ [−log pθ (x)] − H (pϕ) αϯϓϧͷΤωϧΪʔΛ Լ͛Δ αϯϓϧͷΤϯτϩϐʔΛ ্͛Δ
- 25. ੜثͷֶश ୈ1߲ͷޯɺҎԼͷΑ͏ʹ؆୯ʹࢉܭՄೳ ∇ϕ 𝔼 pϕ [−log pθ (x)] = 𝔼 z∼p(z) [∇ϕEθ (Gϕ (z))]
- 26. ੜثͷֶश ୈ2߲ͷΤϯτϩϐʔղੳతʹ·ٻΒͳ͍ จͰɺόονਖ਼نԽͷεέʔϧύϥϝʔλΛਖ਼نͷࢄͱΈͳͯ͠ ͦͷΤϯτϩϐʔΛ͢ࢉܭΔ͜ͱͰ༻͍ͯ͠Δ H (pϕ) ≈ ∑ ai H ( 𝒩 (μai , σai)) = ∑ ai 1 2 log (2eπσ2 ai)
- 28. ੜαϯϓϧ
- 29. Rithesh Kumar,Sherjil Ozair,Anirudh Goyal,Aaron Courville,Yoshua Bengio (Université de Montréal) Maximum Entropy Generators for Energy-Based Models https://arxiv.org/abs/1901.08508
- 30. Τϯτϩϐʔͷࢉܭ จ1ͰΤϯτϩϐʔ ͷࢉܭΛόονਖ਼نԽͷεέʔϧύϥϝʔλͰ ߦ͍͕ͬͯͨɺώϡʔϦεςΟοΫͰཧతͳଥੑͳ͍ KL (pϕ∥ pθ) = 𝔼 pϕ [−log pθ (x)] − H (pϕ) H (pϕ)
- 31. Τϯτϩϐʔͷࢉܭ જࡏม ͱੜثͷग़ྗ ͷ૬ޓใྔΛߟ͑Δͱ z x = Gϕ (z) I(x, z) = H (x) − H (x ∣ z) = 𝔼 p(z) [H (Gϕ (z)) − H (Gϕ (z) ∣ z)]
- 32. Τϯτϩϐʔͷࢉܭ ͕ܾఆతͳؔͷͱ͖ɺ ͳͷͰ ͭ·ΓɺΤϯτϩϐʔͷΘΓʹɺ૬ޓใྔΛ࠷େԽ͢Εྑ͍ Gϕ H (Gϕ (z) ∣ z) = 0 H (pϕ) = 𝔼 p(z) [H (Gϕ (z))] = I (x, z)
- 33. ૬ޓใྔͷਪఆ ૬ޓใྔͷਪఆํ๏ɺ͍ۙΖ͍ΖఏҊ͞Ε͍ͯΔ͕ɺ ͜͜ͰɺDeep InfoMaxͰఏҊ͞ΕͨJS divergenceʹͮ͘جਪఆ๏Λ༻͍Δ ͔Βͷαϯϓϧͱ ͔ΒͷαϯϓϧΛݟ͚ΔࣝผͰث ಉ࣌ʹֶश͢Δ IJSD (x, z) = sup T∈ 𝒯 𝔼 p(x, z) [−sp (−T (x, z))] − 𝔼 p(x)p(z) [sp (T (x, z))] T p (x, z) p (x) p (z)
- 35. Mode Collapse 1000 (or 10000) ݸͷϞʔυΛͭσʔλʢStackedMNISTʣͰֶशͨ͠ͱ͖ʹɺ ϞʔυΛ͍ͭ͘ଊ͑ΒΕ͍ͯΔ͔Λൺֱ͢Δ࣮ݧ MEG (ఏҊ๏) ͯ͢ͷϞʔυΛଊ͓͑ͯΓɺmode collapse͕͜ىΒͳ͍
- 37. Tong Che,Ruixiang Zhang,Jascha Sohl-Dickstein,Hugo Larochelle,Liam Paull,Yuan Cao,Yoshua Bengio (Université de Montréal,Google Brain) Your GAN is Secretly an Energy-based Model and You Should Use Discriminator Driven Latent Sampling https://arxiv.org/abs/2003.06060
- 38. GANͷҰൠతͳղऍ ࣝผີʹثൺਪఆث ࣝผثσʔλ ͱੜαϯϓϧͷ ͷ ີൺਪఆͯ͠ͱثͷׂΛՌͨ͢ i.e., ࣝผ࠷͕ثదͳͱ͖ p (x) pϕ (x) = 𝔼 p(z) [p (Gϕ (z))] D* θ (x) = p (x) p (x) + pϕ (x)
- 39. GANͷҰൠతͳղऍ ࣝผີʹثൺਪఆث ΛγάϞΠυؔɺ ͱ͢Δͱ σʔλ ੜثͷ ͱ ͷੵʹൺྫ͢Δ ➡ ֶशޙͷGANͰ͜ͷ͔Βαϯϓϧ͢Εɺੜͷ্࣭͕͕ΔͷͰʁ σ ( ⋅ ) dθ (x) = σ−1 (D (x)) Dθ (x) = p (x) p (x) + pϕ (x) ⇒ p (x) ∝ pϕ (x) exp (dθ (x)) p (x) pϕ (x) exp (dθ (x))
- 40. જࡏۭؒͰͷMCMC Discriminator Driven Latent Sampling (DDLS) ͔ΒαϯϓϦϯά͍͕ͨ͠ɺσʔλۭؒͰMCMCΛ͢Δͷ ޮ͕ѱ͍͘͠ ΘΓʹੜث ͷજࡏ্ۭؒͰMCMC (Langevin dynamics)Λߦ͏ pϕ (x) exp (dθ (x)) Gϕ (z) E (z) = − log p (z) − dθ (Gϕ (z)) z ← z − η∇zE (z) + ϵ ϵ ∼ Normal (0,2ηI)