📘

Sigmoid Beta Schedule

MediumDiffusion Models 📚 Diffusion Models 1

Problem Statement

The sigmoid schedule (popularized for higher-resolution training) shapes betas with a logistic curve so most of the noising happens in the middle of the trajectory. Build it.

Background

Take T points on a linear grid over [-6, 6], pass them through the logistic sigmoid sigma(x) = 1/(1+e^{-x}), then affinely rescale that curve so its minimum maps to beta_start and its maximum to beta_end:

$s_t = \sigma\!\left(-6 + \frac{12 t}{T-1}\right), \qquad \beta_t = \beta_{\text{start}} + \frac{s_t - s_0}{s_{T-1} - s_0}\big(\beta_{\text{end}} - \beta_{\text{start}}\big)$

so beta_0 = beta_start and beta_{T-1} = beta_end exactly.

Your Task

Implement:

def sigmoid_beta_schedule(T, beta_start, beta_end):

Return a list of T betas rounded to 6 decimals.

Input Format

T (int), T >= 2.
beta_start, beta_end (float): 0 < beta_start < beta_end < 1.

Output Format

A list of T floats rounded to 6 decimals.

Sample

print(sigmoid_beta_schedule(3, 0.0001, 0.02))

Output:

[0.0001, 0.01005, 0.02]

Example:

Input:

print(sigmoid_beta_schedule(3, 0.0001, 0.02))

Output:

[0.0001, 0.01005, 0.02]

Reasoning:

Generate Grid Points: With $T=3$ , we create a linear grid of 3 points spanning the interval $[-6, 6]$ . This yields the input values for the sigmoid function: $x = [-6, 0, 6]$ .
Compute Sigmoid Values: We apply the logistic function $\sigma(x) = \frac{1}{1+e^{-x}}$ $σ (x) = \frac{1}{1 + e ^{- x}}$ to each point.
- For $x=-6$ : $\sigma(-6) \approx 0.002473$
- For $x=0$ : $\sigma(0) = 0.5$
- For $x=6$ : $\sigma(6) \approx 0.997527$
Normalize the Curve: We rescale the sigmoid outputs so the minimum maps to 0 and the maximum maps to 1 using the formula $\frac{s_t - s_0}{s_{T-1} - s_0}$ $\frac{s _{t} - s _{0}}{s _{T - 1} - s _{0}}$ .
- The range is $s_{T-1} - s_0 \approx 0.997527 - 0.002473 = 0.995054$ .
- Normalized values:
  - $t=0$ : $\frac{0.002473 - 0.002473}{0.995054} = 0$
  - $t=1$ : $\frac{0.5 - 0.002473}{0.995054} \approx 0.499999$
  - $t=2$ : $\frac{0.997527 - 0.002473}{0.995054} = 1$
Affine Rescaling to Beta Range: We map the normalized values to the interval $[\beta_{\text{start}}, \beta_{\text{end}}] = [0.0001, 0.02]$ $[β_{start}, β_{end}] = [0.0001, 0.02]$ using $\beta_t = \beta_{\text{start}} + \text{normalized}_t \cdot (\beta_{\text{end}} - \beta_{\text{start}})$ $β_{t} = β_{start} + normalized_{t} \cdot (β_{end} - β_{start})$ . The total range is $0.02 - 0.0001 = 0.0199$ $0.02 - 0.0001 = 0.0199$ .
- $t=0$ : $0.0001 + 0 \cdot 0.0199 = 0.0001$
- $t=1$ : $0.0001 + 0.499999 \cdot 0.0199 \approx 0.01005$
- $t=2$ : $0.0001 + 1 \cdot 0.0199 = 0.02$
The final output is [0.0001, 0.01005, 0.02]

Constraints:

T >= 2, 0 < beta_start < beta_end < 1.
Grid is [-6, 6]; rescale so endpoints hit beta_start/beta_end exactly.
Round every beta to 6 decimals.

More from Diffusion Models 1

Classifier-Free Guidance: Uncond/Cond MixingEasy DDPM Posterior Mean and Variance from x0Medium Adaptive CFG Scheduling over the TrajectoryHard Classifier Gradient Guidance StepMedium Negative Prompt as the Unconditional BranchMedium

1. Background Knowledge

In denoising diffusion probabilistic models (DDPMs), the forward process gradually adds Gaussian noise to a data sample over $T$ discrete timesteps. The amount of noise added at each step is controlled by a variance parameter $\beta_t$ , which must satisfy $0 < \beta_t < 1$ . The choice of how $\beta_t$ varies with $t$ —the noise schedule—significantly affects training stability and sample quality.

A linear schedule simply interpolates $\beta_t$ from $\beta_{\text{start}}$ to $\beta_{\text{end}}$ . However, for higher-resolution images, a sigmoid schedule performs better. It concentrates most of the noising in the middle of the trajectory, mimicking a smooth "S-curve." This is achieved by evaluating the logistic sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$ at evenly spaced points and then affinely rescaling the result so that the first and last values match the desired $\beta_{\text{start}}$ and $\beta_{\text{end}}$ exactly.

The key mathematical operation here is affine normalization: given a sequence $s_t$ with known minimum $s_0$ and maximum $s_{T-1}$ , you map it to a new range $[\beta_{\text{start}}, \beta_{\text{end}}]$ via the formula $\beta_t = \beta_{\text{start}} + \frac{s_t - s_0}{s_{T-1} - s_0}(\beta_{\text{end}} - \beta_{\text{start}})$ . This guarantees the endpoints are exact while preserving the sigmoid shape in between.

2. Algorithm Approach

This is a direct formula evaluation problem. There is no iterative search or optimization. The approach is:

Generate $T$ evenly spaced input points on the interval $[-6, 6]$ .
Apply the sigmoid function to each point to obtain raw scores $s_t$ .
Compute the affine rescaling factor using $s_0$ and $s_{T-1}$ .
Apply the rescaling to every $s_t$ to produce $\beta_t$ .
Round each result to 6 decimal places and return as a list.

The entire computation is a single pass over $T$ values with constant-time arithmetic per element.

3. Step-by-Step Strategy

👑

Continue the full explanation

You're reading the free preview. Unlock the complete walkthrough, the code editor, test runner and reference solution with Premium.

🔒

Editor locked

The code editor is locked for Pro problems. It is only available for free problems. Please upgrade to gain access to the code editor for all problems.

solution.py

Python 3.13

Test Results

0/0

Run code to see test results.