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

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 the denoising diffusion probabilistic model (DDPM), a widely used 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. Moreover, we show that with careful coefficient design, the convergence rate improves to $O(k/T)$, where $k$ is the intrinsic dimension of the target data distribution. This highlights the ability of DDPM to automatically adapt to unknown low-dimensional structures, a common feature of natural image distributions. These results are 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.

• The paper demonstrates an improved O(d/T) convergence rate, significantly enhancing previous O(√(d/T)) analyses.
• The paper requires only a finite first-order moment for the target distribution, reducing strict assumptions common in earlier studies.
• The paper uses a refined error propagation framework for score estimation, providing robust guarantees for diffusion probabilistic models.

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

Introduction and Background

The paper "O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions" discusses the theoretical underpinnings of Score-Based Generative Models (SGMs), a subset of generative models that have recently gained prominence due to their ability to effectively learn and sample from complex data distributions. Notable models in this class include Denoising Diffusion Probabilistic Models (DDPM) and Denoising Diffusion Implicit Models (DDIM). These models have been applied successfully across various domains, including image, audio, and video generation as well as in fields like molecular design.

At their core, SGMs use a process of transforming Gaussian noise into a target data distribution via a sequence of diffusion steps. This is achieved through reverse diffusion processes governed by score functions, which estimate the gradient of the log-density of the data. Both SDE-based and ODE-based samplers are popular in this context, yet theoretical guarantees have been largely constrained by demanding assumptions or suboptimal convergence rates.

Theoretical Contributions

This research addresses limitations found in previous work that focused on the convergence of SDE-based and ODE-based samplers. The existing studies primarily fell into one or more of the following issues: requiring stringent data assumptions, establishing slow convergence rates, and imposing additional score estimation requirements. By overcoming these challenges, the paper achieves the sharpest convergence rate known for diffusion probabilistic models under minimal assumptions.

1. Convergence Rate: The paper establishes an $O(d/T)$ convergence rate for SDE-based samplers concerning total variation distance, which marks a significant improvement over the earlier $O(\sqrt{d/T})$ rate. This makes it general compared to previous results for ODE-based samplers, which required stricter conditions.
2. Minimal Assumptions: It requires only the target distribution to have a finite first-order moment. This is notably weaker than assumptions in existing literature, which often demand log-Sobolev inequalities or Lipschitz smoothness for score functions.
3. Score Estimation: The theoretical analysis considers only the $\ell_2$-accuracy of score function estimates, as opposed to needing the Jacobian accuracy which has been a stronger requirement in prior studies.

Analytical Framework and Results

The authors introduce a novel analytical framework to map the error propagation at each reverse process step more finely. This enables sharper conclusions regarding convergence, specifically showing how small errors in the estimation of score functions affect the overall system performance.

The results also incorporate prior efforts in the literature while providing improved guarantees without requiring more restrictive assumptions. This is particularly highlighted by achieving similar convergence rates to ODE-based samplers under conditions where dimensionality $d$ and number of steps $T$ are comparable, a circumstance where previous studies typically falter.

Conclusion

The findings in this paper efficiently bridge the gap in existing theories on diffusion probabilistic models by crafting a convergence theory that accomplishes optimal rates with simplified assumptions. These results are pivotal for expanding the applicability of SDE-based SGMs in more general settings without the burden of impractical assumptions about data and functions. The analytical advancements and error characterizations will likely spur further exploration of diffusion models within the machine learning community, driving both methodological developments and applications across diverse domains.