On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates
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Author(s)
Bruno, Stefano
Zhang, Ying
Lim, Dong-Young
Akyildiz, Omer Deniz
Sabanis, Sotirios
Type
Journal Article
Abstract
We provide full theoretical guarantees for the convergence behaviour of diffusion-based generative models under the assumption of strongly log-concave data distributions while our approximating class of functions used for score estimation is made of Lipschitz continuous functions avoiding any Lipschitzness assumption on the score function. We demonstrate via a motivating example, sampling from a Gaussian distribution with unknown mean, the powerfulness of our approach. In this case, explicit estimates are provided for the associated optimization problem, i.e. score approximation, while these are combined with the corresponding sampling estimates. As a result, we obtain the best known upper bound estimates in terms of key quantities of interest, such as the dimension and rates of convergence, for the Wasserstein-2 distance between the data distribution (Gaussian with unknown mean) and our sampling algorithm.
Beyond the motivating example and in order to allow for the use of a diverse range of stochastic optimizers, we present our results using an -accurate score estimation assumption, which crucially is formed under an expectation with respect to the stochastic optimizer and our novel auxiliary process that uses only known information. This approach yields the best known convergence rate for our sampling algorithm.
Beyond the motivating example and in order to allow for the use of a diverse range of stochastic optimizers, we present our results using an -accurate score estimation assumption, which crucially is formed under an expectation with respect to the stochastic optimizer and our novel auxiliary process that uses only known information. This approach yields the best known convergence rate for our sampling algorithm.
Date Issued
2025-02-14
Date Acceptance
2025-02-14
Citation
Transactions on Machine Learning Research, 2025
ISSN
2835-8856
Journal / Book Title
Transactions on Machine Learning Research
Copyright Statement
© 2025 The Author(s). Available open access on OpenReview under a Creative Commons CC-BY licence (https://creativecommons.org/licenses/by/4.0/)
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Publication Status
Published