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Negative Prompt as the Unconditional Branch

MediumDiffusion Models 📚 Diffusion Models 1

Problem Statement

In practice the "unconditional" branch of CFG is replaced by a negative prompt prediction, steering away from unwanted content. Implement guidance with an explicit negative branch and report the resulting noise plus how far it moved from the plain conditional prediction.

Background

With a negative-prompt prediction eps_neg in place of the unconditional one, the guided noise is

$\hat{\varepsilon} = \varepsilon_{\text{neg}} + w\,(\varepsilon_{\text{cond}} - \varepsilon_{\text{neg}})$

The larger the gap between eps_cond and eps_neg, the more the guidance pushes. Report the guided prediction and the L2 distance between it and eps_cond (how much guidance changed the base prediction).

Your Task

Implement:

def negative_prompt_guidance(eps_cond, eps_neg, w):

Return a dict with "eps" (list rounded to 4 decimals) and "shift" (L2 distance from eps_cond, rounded to 4 decimals).

Input Format

eps_cond, eps_neg: lists of equal length D.
w (float): guidance scale.

Output Format

A dict with a list and a float.

Sample

print(negative_prompt_guidance([1.0, 0.0], [0.0, 0.0], 2.0))

Output:

{'eps': [2.0, 0.0], 'shift': 1.0}

Example:

Input:

print(negative_prompt_guidance([1.0, 0.0], [0.0, 0.0], 2.0))

Output:

{'eps': [2.0, 0.0], 'shift': 1.0}

Reasoning:

Compute the guided noise vector $\hat{\varepsilon}$ using the formula $\hat{\varepsilon} = \varepsilon_{\text{neg}} + w(\varepsilon_{\text{cond}} - \varepsilon_{\text{neg}})$ :
- For the first element: $0.0 + 2.0 \times (1.0 - 0.0) = 2.0$
- For the second element: $0.0 + 2.0 \times (0.0 - 0.0) = 0.0$
- Resulting vector: $[2.0, 0.0]$
Calculate the difference between the guided noise and the original conditional prediction to determine the shift:
- $\hat{\varepsilon} - \varepsilon_{\text{cond}} = [2.0 - 1.0, 0.0 - 0.0] = [1.0, 0.0]$
Compute the L2 distance (shift) by taking the square root of the sum of squared differences:
- $\text{shift} = \sqrt{1.0^2 + 0.0^2} = \sqrt{1.0} = 1.0$
Round the results to 4 decimal places as required:
- $\text{eps} = [2.0, 0.0]$
- $\text{shift} = 1.0$
The final output is {'eps': [2.0, 0.0], 'shift': 1.0}

Constraints:

len(eps_cond) == len(eps_neg).
eps = eps_neg + w*(eps_cond - eps_neg).
shift = ||eps - eps_cond||_2; round both to 4 decimals; avoid -0.0.

More from Diffusion Models 1

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1. Background Knowledge

Classifier-Free Guidance (CFG) is the dominant technique for controlling diffusion model outputs without a separate classifier. During training, the model learns to predict noise conditioned on a prompt $\text{cond}$ and unconditionally. At inference, instead of sampling from the conditional distribution alone, we extrapolate: $\hat{\varepsilon} = \varepsilon_{\text{uncond}} + w(\varepsilon_{\text{cond}} - \varepsilon_{\text{uncond}})$ , where $w \geq 1$ is the guidance scale. This linear combination amplifies the direction the conditional prediction points relative to the unconditional baseline, sharpening alignment with the prompt.

In practice, the "unconditional" branch is often replaced by a negative prompt prediction $\varepsilon_{\text{neg}}$ , which encodes what we want to avoid (e.g., "blurry, low quality"). The formula becomes $\hat{\varepsilon} = \varepsilon_{\text{neg}} + w(\varepsilon_{\text{cond}} - \varepsilon_{\text{neg}})$ . When $w = 1$ , the result equals $\varepsilon_{\text{cond}}$ exactly. When $w > 1$ , the output is pushed further along the direction $(\varepsilon_{\text{cond}} - \varepsilon_{\text{neg}})$ , increasing contrast between desired and undesired features.

The shift metric—defined as the L2 distance $\|\hat{\varepsilon} - \varepsilon_{\text{cond}}\|_2$ —quantifies how much the guidance altered the base conditional prediction. A larger shift indicates stronger steering away from the negative prompt. This is useful for diagnosing whether guidance is too aggressive (overshooting) or too weak (insufficient control).

2. Algorithm Approach

This is a vector arithmetic problem with a single post-processing step. The core operation is a scaled linear combination of two vectors, followed by a norm computation. No iterative or recursive logic is needed. The approach is:

Compute the difference vector $\mathbf{d} = \varepsilon_{\text{cond}} - \varepsilon_{\text{neg}}$ element-wise.
Scale it by $w$ and add to $\varepsilon_{\text{neg}}$ to obtain $\hat{\varepsilon}$ .
Compute the L2 norm of $(\hat{\varepsilon} - \varepsilon_{\text{cond}})$ , which simplifies algebraically to $|w - 1| \cdot \|\mathbf{d}\|_2$ .
Round both outputs to 4 decimal places and return as a dictionary.

The algebraic simplification in step 3 is optional but useful for verification: since $\hat{\varepsilon} - \varepsilon_{\text{cond}} = (w-1)(\varepsilon_{\text{cond}} - \varepsilon_{\text{neg}})$ , the shift is simply $|w - 1|$ times the L2 norm of the difference vector.

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