Formulation of reverse diffusion process in DDPM ( 1 - alpha_prod_t = beta_prod_t assumption)

[jadevaibhav]

Describe the bug

I looked into sampling code of DDPM, and I believe there's a mistake:

I believe the code makes assumption that 1 - alpha_prod_t = beta_prod_t, which simply isn't true.

Original sampling algorithm:
original paper

x_0 prediction from sample and predicted noise :

Eqn 15 from paper, as referenced in code snippet as well:

code implementation in diffusers:

diffusers/src/diffusers/schedulers/scheduling_ddpm.py

Lines 446 to 449 in 1e8cf27

	# 2. compute predicted original sample from predicted noise also called
	# "predicted x_0" of formula (15) from https://arxiv.org/pdf/2006.11239.pdf
	if self.config.prediction_type == "epsilon":
	pred_original_sample = (sample - beta_prod_t ** (0.5) * model_output) / alpha_prod_t ** (0.5)

Here, it clearly seems like the 1 - alpha_prod_t = beta_prod_t is being used.

investigated from post by @AlejandroBaron in #9431_

Reproduction

I haven't tested out code, but this seems fundamental formulation issue.

Logs

No response

System Info

0.30.3

Who can help?

@sayakpaul @DN6 @yiy

[jadevaibhav]

It looks like its a unfortunate naming, where beta_prod_t is defined as 1 - alpha_prod_t:

diffusers/src/diffusers/schedulers/scheduling_ddpm.py

Lines 438 to 444 in 1e8cf27

	# 1. compute alphas, betas
	alpha_prod_t = self.alphas_cumprod[t]
	alpha_prod_t_prev = self.alphas_cumprod[prev_t] if prev_t >= 0 else self.one
	beta_prod_t = 1 - alpha_prod_t
	beta_prod_t_prev = 1 - alpha_prod_t_prev
	current_alpha_t = alpha_prod_t / alpha_prod_t_prev
	current_beta_t = 1 - current_alpha_t

I think it should be renamed for clarity.

[asomoza]

cc: @yiyixuxu

[sayakpaul]

Perhaps a comment could be added to clarify the confusion. That seems like the easiest solution to me.

[jadevaibhav]

Thanks! Also, could you please clarify why is this formulation is used for sampling (step function)? The algo in paper sample from the Gaussian(pred_prev_mean, variance). Why do we first convert to the predicted original sample (pred_x_0) and then use following for mean calculation:

diffusers/src/diffusers/schedulers/scheduling_ddpm.py

Lines 468 to 475 in 1e8cf27

	# 4. Compute coefficients for pred_original_sample x_0 and current sample x_t
	# See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
	pred_original_sample_coeff = (alpha_prod_t_prev ** (0.5) * current_beta_t) / beta_prod_t
	current_sample_coeff = current_alpha_t ** (0.5) * beta_prod_t_prev / beta_prod_t
	# 5. Compute predicted previous sample µ_t
	# See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
	pred_prev_sample = pred_original_sample_coeff * pred_original_sample + current_sample_coeff * sample

[github-actions]

This issue has been automatically marked as stale because it has not had recent activity. If you think this still needs to be addressed please comment on this thread.

Please note that issues that do not follow the contributing guidelines are likely to be ignored.