15. Score-based Model 3

Controlling the Generation Process

• Can we control the generation process?

• $P(x)$ $\downarrow$

• Inverse distribution: $P(x

  y)$

• Bayes’ Rule:

  • $p(x

    y) = \frac{p(x) \cdot p(y

    x)}{p(y)}$

• Score function decomposition:

  • $\nabla_x \log P(x

    y) = \nabla_x \log P(x) + \nabla_x \log P(y

    x)$

  • $= \nabla_x \log P(x) + \nabla_x \log P(y

    x)$

    • $\approx S_\theta(x)$ (Unconditional Score)

    • Classifier term

• $\Rightarrow$ $p(y

  x)$지만 learn 하면 되는데, 이건 사실 image classifier이다.

• $p(y)$는 training 없이 specify 될 수 있다. 그냥 하면 됨.

• Conclusion: Score-based gen model 하나면 다 된다.

  • Classifier + unconditional gen model $P(x)$ (inverse problem)

Classifier Guidance

• $P(x)$

  • ↳ GAN 성능을 위해 classifier 강화 필요

• DDPM은 unconditional gen만 됨. label 같이 줘도 X.

  • ↳ Adversarial 하지 않으면서 discriminator를 만들자.

• $\nabla_x \log P(y

  x)$ 가 그 역할을 하겠다!

Basic Idea

• Goal: $\nabla_{x_t} \log P(x_t

  y)$ 를 learn하자. (원래는 $\nabla_{x} \log P(x_t) \approx S_\theta(x_t, t)$)

  • ↳ $\nabla \log P(x_t

    y) = \nabla \log (\frac{P(y

    x_t) \cdot P(x_t)}{P(y)})$

  • $\rightarrow$ Classifier guide

  • $= \nabla_x \log P(x_t) + \nabla \log P(y

    x_t)$

  • 그림: 같은 방향으로 guide

Training & Sampling

• Note: $\nabla \log P(x_t

  y) = \nabla \log P(x_t) + \nabla \log P(y

  x_t)$

• Training:

  • ① Score of unconditional diffusion model

  • ② Classifier takes $X_t \rightarrow$ predicts $y$

• Sampling:

  • ↳ Unconditional score function + Gradient of noisy classifier

  • $P(\cdot)$는 share 할 수 있고, $P_\phi(y

    x_t, t)$ 로 처리가능

Guidance Scale

• ↳ Hyper parameter ($\gamma$)

• $\nabla \log P_\gamma(x_t

  y) = \nabla \log P(x_t) + \gamma \nabla \log P(y

  x_t)$

  • $\gamma=0$: $y$ 무시.

  • $\gamma \rightarrow \infty$: $y$와 강력히 연결. (Quality $\uparrow$ Diversity $\downarrow$)

How?

• Train classifier (noise 추가된 거)

Limitation

• Noise-aware classifier (Standard classifier X)

• Unstable gradients

• Computational cost.

Classifier-Free Guidance (CFG)

• ↳ CFG uses a single diffusion model

• Recall: $\nabla \log p(x_t

  y) = \nabla \log P(x_t) + \nabla \log p(y

  x_t)$

• Implicit classifier:

  • $\nabla \log p(y

    x_t) = \nabla \log P(x_t

    y) - \nabla \log P(x_t)$

  • ↳ Conditioning dropout.

• Formula:

  • $\nabla \log p_\gamma(x_t

    y) = \gamma \nabla \log p(x_t

    y) + (1-\gamma) \nabla \log p(x_t)$

  • ↳ Unconditional diffusion conditioning drop out

Conditioning Dropout

• $P(x	y)$를 학습하자. (Conditioning dropout)

  • ↳ $y$를 가끔 없앰 (10~20%.)

  • ↳ $\emptyset$로 대체됨

• $P(x	y)$와 $P(x)$가 모두 기능함.

Guidance Scale Analysis

• $\nabla \log p_\gamma(x_t

  y) = \gamma \nabla \log p(x_t

  y) + (1-\gamma) \nabla \log p(x_t)$

  • $[\gamma=0]$: Unconditional generation

  • $[\gamma=1]$: Standard generation

  • $[\gamma>1]$: Unconditional score 방향 반대로 이동.

    • Reduce the probs of generating samples that do not use conditioning info.

    • (ex. GLIDE)

Steps in CFG

1. Prepare: A collection of (image, caption)

2. Training:

   • Train a unified model $\epsilon_\theta(x_t, c)$

   • ↳ Random caption dropping (10%)

   • ↳ Caption kept (90%)

3. Sample:

   • $X_T$를 뽑고

   • (a) Unconditional $\epsilon_\theta(x_t, \emptyset)$: Baseline prediction

   • (b) Conditional $\epsilon_\theta(x_t, c)$: Conditioning

   • (c) Apply $\gamma$: $\hat{\epsilon}\theta(x_t) = \epsilon{uncond} + \gamma(\epsilon_{cond} - \epsilon_{uncond})$

     • (고양이 쪽 noise) - (noise 없) $\rightarrow$ 고양이 벡터

Modern Architecture

Latent Diffusion Model (LDM)

• ↳ Latent space에서 하기. 대부분의 bits는 perceptual detail임.

• Structure:

  • $x \rightarrow E \mapsto z \rightarrow \text{Diff} \longrightarrow Z_T$

  • $z \rightarrow D \rightarrow x$

  • U-Net uses Condition.

• Advantage:

  • ① Simple denoising + Faster synthesis (Normal distribution 과 유사)

  • ② Expressive design

  • ③ Autoencoder 사용가능.

Diffusion Transformer (DiT)

• ↳ U-Net을 Transformer로 대체.

• 확산 model forward SDE.

• Reverse 에서의 PF-ODE 가능이 증명.

Advanced Models

DDIM

• ↳ Deterministic sampling.

• Predicted:

  • $X_{t-1} = \sqrt{\bar{\alpha}{t-1}} \hat{X}_0 + \sqrt{1-\bar{\alpha}{t-1}} \epsilon$

  • (Note: Formula in note approximates to deterministic update)

• Comparison:

  • 원래는 $X_{t-1} = \mu_\theta(x_t, t) + \beta_t \cdot \epsilon$

• Benefits:

  • ① Higher quality

  • ② Consistency

Consistency Models

• ↳ DDIM 보다 빠르게.

• Concept:

  • All points on latent space $\rightarrow$ same point in data space.

  • $f_\theta(x_t, t) \longrightarrow X_0$ for $\forall t$

  • ODE trajectory ($X_T \rightarrow X_0$)

• Training:

  • ① Directly train $f_\theta$

  • ② Consistency distillation

    • Teacher: Multi-step

    • $\downarrow$

    • Student: Consistency

Flow Matching

• (바로 ODE를 배우기)

• ↳ Learning velocity!

• $\Rightarrow$ Encourages straighter, simpler path: Faster sampling

• How:

  • Noise $\leftrightarrow$ Data

  • Learn Score

• Example: Rectified flow

Examples

• Stable Diffusion 3: MMDiT + Rectified flow

• Flux: Parallel Transformer

• Nano Banana: Autoregressive