Noise schedules

The schedule is a design choice

The betas are not learned and not sacred. A schedule is a curve from almost no noise to almost pure noise, and its shape decides how the model spends its capacity. The original linear schedule moves from one in ten thousand to two hundredths over a thousand steps. It works, but it destroys the signal early. By the halfway point alpha_bar is already tiny, so most of the second half is spent denoising pure noise.

A good schedule spends roughly equal effort across the range of signal-to-noise ratios where the model can learn something. That is a vague target. The cosine schedule is a practical approximation. Its alpha_bar follows a cosine squared curve, which keeps more signal for longer and reaches zero smoothly at the end. Stable Diffusion uses a scaled linear schedule with a different start and end, tuned for latents rather than pixels.

Reading a schedule

Three numbers tell you most of what a schedule does. The first is where alpha_bar crosses one half, which is roughly where signal and noise are balanced. The second is how many steps have a signal-to-noise ratio below one. That is the part of training spent on mostly noise. The third is the value at the final step. A schedule that ends above zero has not destroyed the data, and the sampler will have to finish the job itself.

Plot log signal-to-noise ratio against t, and the curve is almost a straight line for a well-behaved schedule. That is not an accident. The model’s difficulty is roughly uniform in log SNR. A schedule that is linear in log SNR therefore gives every difficulty an equal share of steps. The cosine curve is close to that. The linear beta curve is not, and that is exactly its weakness. The linear curve spends its first two hundred steps at a log SNR above four, where almost any denoiser succeeds. It spends its last six hundred below zero, where the target is nearly pure noise.

A cosine schedule

import torch
from diffusers import DDPMScheduler

scheduler = DDPMScheduler(
    num_train_timesteps=1000,
    beta_schedule="squaredcos_cap_v2",
)
print(scheduler.betas[:5])
print(scheduler.alphas_cumprod[[0, 250, 500, 750, 999]])

# The Stable Diffusion schedule is a scaled linear one.
sd = DDPMScheduler(
    num_train_timesteps=1000,
    beta_schedule="scaled_linear",
    beta_start=0.00085,
    beta_end=0.012,
)

Zero terminal SNR and its consequences

Some schedules are built to reach alpha_bar of exactly zero at the last step. That sounds like a detail and is not. If the final distribution is not exactly standard normal, the sampler starts from the wrong distribution. The model also never sees the pure-noise case it is asked to invert. Zero terminal SNR fixes the mismatch, and it makes the model’s job at the last step well defined.

It also breaks the simple noise parameterization at the final step. There the coefficient of x_0 is zero and the coefficient of eps is one, so predicting eps and predicting x_0 are degenerate. That is one reason v-prediction and zero-terminal schedules are often paired. If you change one of these choices, check the others. A schedule, a parameterization and a sampler form a set, and mixing them across a checkpoint is how silent quality loss happens.

Resolution and the schedule

A schedule is defined in terms of variance, not pixels. It transfers between image sizes better than most people expect. What does not transfer is how hard the task is. A 1024 image has more spatial detail, and the same alpha_bar leaves a different amount of visible structure. Models trained at one resolution and sampled at another often need the timestep range shifted. The trick is called dynamic thresholding or trailing timesteps.

The practical rule is to keep the schedule fixed at training. Treat any change at sampling as a guess that must be measured. Fewer steps, a shifted range, a different final alpha: each is a change to the distribution the model was trained on. Some are harmless, some are not. The only way to know is a fixed evaluation set and a metric you trust. A schedule is cheap to change and expensive to get wrong.