Classifier-Free Guidance Combination

Problem Statement

Combine a conditional and an unconditional noise prediction into the guided prediction that every text-to-image sampler actually uses.

Background

Classifier-free guidance (Ho & Salimans, 2022) drops the conditioning at random during training so one network learns both $\hat{\varepsilon}_\theta(x_t, t, c)$ and $\hat{\varepsilon}_\theta(x_t, t, \varnothing)$. At sampling time the two are extrapolated:

$\hat{\varepsilon}_{\text{guided}} = \hat{\varepsilon}_\theta(x_t, t, \varnothing) + w\left(\hat{\varepsilon}_\theta(x_t, t, c) - \hat{\varepsilon}_\theta(x_t, t, \varnothing)\right)$

The bracket is (proportional to) the score of the implicit classifier $p(c \mid x_t)$: the direction in noise space that makes the sample more like the prompt. Guidance walks along it.

Read off the boundary cases, because they are the whole intuition: $w = 0$ gives the unconditional model, $w = 1$ gives the plain conditional model (no guidance at all), and $w > 1$ extrapolates past the conditional prediction. The usual Stable Diffusion default of guidance_scale = 7.5 is therefore a 7.5x overshoot, trading diversity for prompt fidelity. Note that w = 1 meaning "off" is exactly why some codebases skip the second forward pass when guidance_scale <= 1.

Your Task

Implement:

def cfg_combine(eps_uncond, eps_cond, w):

Return the guided noise prediction, shaped like the inputs.

Input Format

• eps_uncond, eps_cond: NumPy arrays of matching shape.
• w (float): the guidance scale.

Output Format

A NumPy array shaped like the inputs.

Sample

u = np.array([0.1, -0.2])
c = np.array([0.3, 0.2])
print(np.round(cfg_combine(u, c, 7.5), 4).tolist())

The difference is [0.2, 0.4], so the result is [0.1 + 7.50.2, -0.2 + 7.50.4] = [1.6, 2.8].