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Classifier Gradient Guidance Step

MediumDiffusion Models 📚 Diffusion Models 1

Problem Statement

Before classifier-free guidance, Dhariwal & Nichol used an external classifier's gradient to nudge the noise prediction. Implement that guided-noise update.

Background

Classifier guidance shifts the predicted noise by the gradient of the classifier's log-probability of the target class, scaled by the noise level and a strength s:

$\hat{\varepsilon} = \varepsilon - s\,\sqrt{1 - \bar{\alpha}_t}\; \nabla_{x_t} \log p_\phi(y \mid x_t)$

The sqrt(1 - alpha_bar_t) factor converts a score shift into a noise shift. The gradient is supplied.

Your Task

Implement:

def classifier_guidance(eps, grad_log_p, alpha_bar_t, s):

Return the guided noise as a list rounded to 4 decimals.

Input Format

eps, grad_log_p: lists of equal length D.
alpha_bar_t (float) in (0, 1), s (float): guidance strength.

Output Format

A list of D floats rounded to 4 decimals.

Sample

print(classifier_guidance([0.0, 0.0], [1.0, -1.0], 0.75, 2.0))

Output:

[-1.0, 1.0]

Example:

Input:

print(classifier_guidance([0.0, 0.0], [1.0, -1.0], 0.75, 2.0))

Output:

[-1.0, 1.0]

Reasoning:

Calculate the noise scaling factor derived from the time step, which converts the score shift into a noise shift: $\sqrt{1 - \alpha_{\text{bar}}_t} = \sqrt{1 - 0.75} = \sqrt{0.25} = 0.5$ .
Determine the effective guidance magnitude by multiplying the strength $s$ by the scaling factor: $2.0 \times 0.5 = 1.0$ .
Compute the guided noise for the first dimension by subtracting the scaled gradient from the predicted noise: $0.0 - (1.0 \times 1.0) = -1.0$ .
Compute the guided noise for the second dimension by subtracting the scaled gradient from the predicted noise: $0.0 - (1.0 \times -1.0) = 1.0$ .
The final output is [-1.0, 1.0]

Constraints:

len(eps) == len(grad_log_p), 0 < alpha_bar_t < 1.
eps_guided = eps - s*sqrt(1-alpha_bar_t)*grad.
Round to 4 decimals; avoid -0.0.

More from Diffusion Models 1

Classifier-Free Guidance: Uncond/Cond MixingEasy DDPM Posterior Mean and Variance from x0Medium Adaptive CFG Scheduling over the TrajectoryHard Negative Prompt as the Unconditional BranchMedium Latent Encode-Decode Round TripMedium

1. Background Knowledge

In classifier-guided diffusion, a separate neural network (the classifier) is trained to predict the probability of a class label $y$ given a noisy latent $x_t$ . During sampling, we want the diffusion process to move toward regions of high $p_\phi(y \mid x_t)$ . The direction of this movement is given by the gradient $\nabla_{x_t} \log p_\phi(y \mid x_t)$ , which points toward higher log-probability for the target class.

The raw diffusion model predicts noise $\varepsilon$ at timestep $t$ . To incorporate classifier guidance, we shift this prediction by a scaled gradient term. The scaling factor $\sqrt{1 - \bar{\alpha}_t}$ is critical: it converts a score-space shift (in the $x_t$ domain) into an equivalent noise-space shift. This arises from the relationship between the score function $\nabla_{x_t} \log p_t(x_t)$ and the noise prediction $\varepsilon$ in the forward diffusion process, where $\bar{\alpha}_t$ is the cumulative product of per-step signal retention factors.

The guidance strength $s$ controls how aggressively the classifier steers the generation. When $s = 0$ , the classifier has no effect and the output equals the original noise prediction. As $s$ increases, the generated samples become more class-conditional but may lose diversity or fidelity.

2. Algorithm Approach

This is a pointwise element-wise transformation problem. For each dimension $i$ of the input vectors, you compute:

$\hat{\varepsilon}_i = \varepsilon_i - s \cdot \sqrt{1 - \bar{\alpha}_t} \cdot \nabla_i$

where $\nabla_i$ is the $i$ -th component of the gradient vector. The entire operation reduces to:

Compute the scalar coefficient $c = s \cdot \sqrt{1 - \bar{\alpha}_t}$ .
For each index $i$ , subtract $c \cdot \texttt{grad\_log\_p}[i]$ from $\texttt{eps}[i]$ .
Round each result to 4 decimal places.

No loops over timesteps or batch dimensions are needed—this is a single-step, single-sample update.

3. Step-by-Step Strategy

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solution.py

Python 3.13

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