Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-type Samplers

Sitan Chen, Giannis Daras, Alex Dimakis
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:4462-4484, 2023.

Abstract

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov’s theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-chen23e, title = {Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For {DDIM}-type Samplers}, author = {Chen, Sitan and Daras, Giannis and Dimakis, Alex}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {4462--4484}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/chen23e/chen23e.pdf}, url = {https://proceedings.mlr.press/v202/chen23e.html}, abstract = {We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov’s theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.} }
Endnote
%0 Conference Paper %T Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-type Samplers %A Sitan Chen %A Giannis Daras %A Alex Dimakis %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-chen23e %I PMLR %P 4462--4484 %U https://proceedings.mlr.press/v202/chen23e.html %V 202 %X We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov’s theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.
APA
Chen, S., Daras, G. & Dimakis, A.. (2023). Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-type Samplers. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:4462-4484 Available from https://proceedings.mlr.press/v202/chen23e.html.

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