$O(d/T)$ Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions (2409.18959v1)

Abstract: Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative tasks. Despite their superior empirical performance, existing theoretical guarantees are often constrained by stringent assumptions or suboptimal convergence rates. In this paper, we establish a fast convergence theory for a popular SDE-based sampler under minimal assumptions. Our analysis shows that, provided $\ell_{2}$-accurate estimates of the score functions, the total variation distance between the target and generated distributions is upper bounded by $O(d/T)$ (ignoring logarithmic factors), where $d$ is the data dimensionality and $T$ is the number of steps. This result holds for any target distribution with finite first-order moment. To our knowledge, this improves upon existing convergence theory for both the SDE-based sampler and another ODE-based sampler, while imposing minimal assumptions on the target data distribution and score estimates. This is achieved through a novel set of analytical tools that provides a fine-grained characterization of how the error propagates at each step of the reverse process.

Analyzing $O(d/T)$ Convergence Theory for Diffusion Probabilistic Models

The paper "O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions" by Gen Li and Yuling Yan presents a notable advancement in the theoretical understanding of score-based diffusion models (SGMs). Diffusion probabilistic models are a class of generative models that have shown empirical success across various tasks, including image, audio, and video generation. However, a more complete theoretical convergence understanding has been lacking, which this paper addresses by offering a robust convergence theory for an SDE-based sampler with minimalistic assumptions. This essay explores the key contributions and implications of the research.

Convergence Rate and Assumptions

The central contribution of the paper is the establishment of a convergence rate of $O(d/T)$ for diffusion models under the total variation distance metric. Here, $d$ represents the data dimensionality and $T$ is the number of steps in the diffusion process. The authors achieve this convergence rate by making sparsely restrictive assumptions, essentially only requiring that the target distribution has a finite first-order moment. Unlike prior works, which demand considerably stringent assumptions, such as log-Sobolev inequality or Lipschitz continuity of the score functions, this paper demonstrates convergence with significantly relaxed conditions.

The work also compares the rate of the presented SDE-based samplers with the ODE-based samplers, showing that this enhanced result matches theoretically and empirically, without the additional requirements usually imposed on ODE-based methods—for instance, the need for Jacobian smoothness of score estimates.

Analytical Techniques

The authors employ a variety of novel analytical techniques to support their theoretical models, addressing the systematic characterization of error propagation through the reverse diffusion process. They formulate bounds derived from forward and reverse stochastic differential equations and demonstrate how convergence guarantees can be preserved even with score function estimation errors. Their analysis entails bounding the total variation distance between the generated and target distributions through a precise calibration of score function accuracy requirements and discernment of discretization errors.

Implications for Future Research

The implications of this paper are broad and influential, both theoretically and practically. The relaxed requirement of assumptions for convergence makes diffusion models more accessible and applicable to a wider range of distributions, including those often seen in real-world applications such as high-dimensional natural image distribution.

The paper also sets a foundational precedent for further exploration toward closing the gap between empirical performance and theoretical guarantees of SGMs. Specifically, the results suggest potential improvements in the conditions required for convergence and motivate a re-evaluation of the score estimation process to ensure it aligns with practical implementations where perfect score functions aren't feasible.

Future Directions

The authors have adopted a complementary approach to ODE-based models by establishing favorable properties of SDEs. Future directions could involve extending these analyses to more complex system settings, potentially incorporating non-Gaussian noise models or devising adaptive learning rate schemes within the diffusion processes that adjust themselves based on data-specific properties.

In summary, this paper significantly advances the theoretical framework of score-based diffusion models by providing convergence guarantees under conditions much more pertinent to practical scenarios. This not only enriches our understanding of diffusion processes in probabilistic generative models but also invites future work to build on and extend these foundational results for broader applicability in complex, high-dimensional data settings.