Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions (2209.11215v3)

Abstract: We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL$\cdot$E 2. Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution. In contrast to prior works, our results (1) hold for an $L^2$-accurate score estimate (rather than $L^\infty$-accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small. We view this as strong theoretical justification for the empirical success of SGMs. We also examine SGMs based on the critically damped Langevin diffusion (CLD). Contrary to conventional wisdom, we provide evidence that the use of the CLD does not reduce the complexity of SGMs.

• The paper provides a rigorous framework linking score estimation with efficient sampling under L2 accuracy assumptions.
• It relaxes strong prior assumptions, enabling sampling from complex, non-log-concave distributions using bounded second moments and Lipschitz scores.
• The results extend to distributions on bounded manifolds, offering theoretical support for advanced generative modeling applications.

Essay on "Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions"

The paper "Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions" presents a comprehensive theoretical framework for understanding the convergence properties of score-based generative models (SGMs), particularly focusing on denoising diffusion probabilistic models (DDPMs). The authors provide rigorous convergence guarantees under minimal assumptions, thereby aligning theoretical insights with the remarkable empirical success observed in practice. This essay provides a critical overview of the paper's main contributions and implications for generative modeling.

Overview of Contributions

The authors introduce a framework wherein SGMs can efficiently sample from complex data distributions if equipped with an accurate score estimate. This result is significant given the role of SGMs in powering large-scale generative applications such as DALL·E 2. They depart from prior analyses by relaxing strong assumptions, such as the need for a log-concave distribution or an $L^\infty$-accurate score estimate. Instead, they operate under the assumptions of an $L^2$-accurate score estimate, a Lipschitz score function, and bounded second moment of the data distribution. Their results are significant because they allow non-log-concave distributions, marking a substantial theoretical understanding of why SGMs work so well empirically.

Theoretical Insights and Results

1. Convergence Guarantees: The central theorem demonstrates that if the score estimation error is bounded in $L^2$, the SGM can sample accurately from the target distribution with an iteration complexity that scales polynomially with the dimension and accuracy parameters. This matches known results for Langevin processes under log-Sobolev inequalities, suggesting the results are near-optimal in their domain.
2. Treatment of Complex Distributions: The findings are noteworthy as they suggest that SGMs equipped with accurate score estimates can handle distributions that display substantial multimodality or non-log-concavity—central to practical generative modeling challenges.
3. Sampling from Arbitrary Distributions: The authors extend their results to settings with distributions supported on bounded manifolds. By careful control of the Wasserstein and total variation metrics, they show that SGMs can still provide meaningful samples, a crucial advancement for real-world data distributions that might not possess densities.

Implications and Speculations for Future Developments

The theoretical development includes a reductionist view wherein the challenge of sampling is equated to learning the score function. This revelation crystallizes the pivotal role score estimation plays, decoupling the learning from sampling complexity, and provides a clean reduction from sampling tasks to score learning tasks.

Furthermore, the exploration of critically damped Langevin diffusion (CLD) suggests this method's potential but reveals no dimension-dependent improvements over traditional DDPMs under current analysis techniques. This indicates avenues for further research, such as identifying conditions under which CLD might be beneficial.

Concluding Thoughts

This paper advances the theoretical underpinnings of SGMs, justifying their empirical effectiveness while opening up new questions about the interplay between score learning and sampling. It suggests that future work should explore the statistical properties of score matching in high dimensions and its implications for model training. Additionally, it leaves open the intriguing possibility that inherent structure within real-world score functions could facilitate efficient learning, further bridging the gap between practice and theory. Such directions could solidify score-based methods as a robust backbone for next-generation machine learning applications.