Diffusion Training Objectives (ε/v-pred, Zero-SNR)

Diffusion training objectives share the same underlying posterior mean but differ in numerical behavior across signal-to-noise ratios (SNR). The three main forms:

1. ε-pred (original DDPM): predict the injected noise, $\mathcal{L} = \mathbb{E}\left[\Vert\epsilon - \epsilon_\theta(x_t, t)\Vert^2\right]$. 2. x0-pred: predict the clean image; one-step estimation at sampling, but the target is weak at low SNR. 3. v-pred (v-parameterization, used by SD2/Imagen): define the velocity field $v \equiv \alpha_t\epsilon - \sigma_t x_0$ with $\alpha_t = \sqrt{\bar\alpha_t}$ and $\sigma_t = \sqrt{1-\bar\alpha_t}$, objective $\mathcal{L} = \Vert v - v_\theta(x_t,t)\Vert^2$; from $x_t = \alpha_t x_0 + \sigma_t \epsilon$ one derives $x_0 = \alpha_t x_t - \sigma_t v_\theta(x_t,t)$ and $\epsilon = \sigma_t x_t + \alpha_t v_\theta(x_t,t)$, so all three parameterizations are interchangeable at sampling.

Zero-SNR problem: under the classic linear schedule $\bar\alpha_T \neq 0$, the terminal $x_T$ still contains signal, inconsistent with sampling from pure Gaussian noise — the model must guess the signal at the last step. A Zero-SNR schedule sets $\bar\alpha_T = 0$ ($x_T$ pure noise); then the coefficients $\sqrt{1-\bar\alpha_t}$ and $\sqrt{\bar\alpha_t}$ of ε-pred/x0-pred degenerate or diverge at the terminal step, so v-pred is required: its two coefficients $\alpha_t, \sigma_t$ transition smoothly with SNR, loss weighting is naturally balanced across timesteps, and train/inference boundaries match — markedly improving terminal-step denoising.

CFG dropout: with 10–15% probability the condition $c$ is replaced by the empty token $\emptyset$ during training, giving the network an unconditional branch that enables Classifier-Free Guidance at sampling.

interview favorites — “why did SD2 switch to v-pred?”, “what is Zero-SNR?”, “what CFG dropout rate?”; engineering choices when training diffusion models.

ε-pred is dominated by the noise term at high SNR (small $t$) and loses signal gradient at low SNR (large $t$), unbalancing the loss across timesteps; v-pred + Zero-SNR equalizes loss weighting and fixes the terminal boundary, improving training stability and sample quality — now standard in SD2/SDXL/Flux.

Difference between ε-pred and v-pred; definitions of α_t and σ_t in v = α_t·ε − σ_t·x_0?

What is Zero-SNR, and why does it require v-prediction?

Write the interchange relations between v-pred, x_0 and ε; how to recover x_0 from v_θ?

Typical CFG dropout probability; can CFG work without it?

Why do SD2/SDXL/Flux use v-pred over ε-pred, and how does loss weighting vary across timesteps?