Sigmoid Beta Schedule

mediumPROForward Process and Noise SchedulesDiffusion ModelsDiffusion 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.

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Concept primerPREVIEW

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

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