Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models (2306.09251v3)

Abstract: Diffusion models, which convert noise into new data instances by learning to reverse a Markov diffusion process, have become a cornerstone in contemporary generative modeling. While their practical power has now been widely recognized, the theoretical underpinnings remain far from mature. In this work, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functions. For a popular deterministic sampler (based on the probability flow ODE), we establish a convergence rate proportional to $1/T$ (with $T$ the total number of steps), improving upon past results; for another mainstream stochastic sampler (i.e., a type of the denoising diffusion probabilistic model), we derive a convergence rate proportional to $1/\sqrt{T}$, matching the state-of-the-art theory. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results characterize how $\ell_2$ score estimation errors affect the quality of the data generation processes. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without resorting to toolboxes for SDEs and ODEs. Further, we design two accelerated variants, improving the convergence to $1/T^2$ for the ODE-based sampler and $1/T$ for the DDPM-type sampler, which might be of independent theoretical and empirical interest.

Accelerating Convergence in Diffusion-Based Generative Models

Overview

The paper explores the convergence dynamics of diffusion models, a mechanism integral to generative modeling that transitions data into noise through a defined forward process and then reverses this process to generate new samples. This work provides a non-asymptotic theoretical foundation for understanding these models' data generation dynamics in discrete time, incorporating $\ell_2$-accurate approximations of the Stein score functions.

Key Contributions

1. Convergence Guarantees:
   • For deterministic samplers like the probability flow ODE, the paper establishes a convergence rate of $1/T$, enhancing previous findings regarding the rate of convergence for deterministic models.
   • The stochastic sampler variant, DDPM, achieves a $1/\sqrt{T}$ convergence rate, aligning with the most advanced theoretical insights.
2. Influence of Score Estimation Errors:
   • The authors explore how errors in $\ell_2$ score estimation impact data generation, providing quantitative characterizations that highlight the minimal conditions required for optimal convergence. The deterministic sampler relies on both $\ell_2$ score and corresponding Jacobian errors, indicating stability during reverse-time processes.
3. Elementary, Non-Asymptotic Framework:
   • Unlike previous studies relying heavily on SDEs and ODEs, this research offers a versatile approach that directly analyzes discrete-time processes using elementary methods. This simplifies understanding and may reduce obstacles for researchers new to such methodologies.
4. Accelerated Variants:
   • Two accelerated versions of the basic samplers are presented, leveraging higher-order corrections to improve convergence rates to $1/T^2$ for the ODE-based sampler and $1/T$ for DDPM, potentially beneficial for theoretical exploration and practical application.

Implications

The findings provide a robust theoretical basis for generative models employing diffusion techniques, with practical implications for refining sample generation speed and accuracy, pertinent to applications in AI content creation (e.g., Stable Diffusion, DALL·E 2). The spotlight on score estimation errors informs better training strategies and model enhancements. The insights could prompt more efficient designs of score-based generative models, minimizing computational expense while maintaining high fidelity in output.

Future Directions

The paper hints at various potential trajectories for further research. One pivotal question is how to refine convergence rates to diminish dimensional dependencies, aiming for tighter and more feasible error boundaries. Additionally, investigating alternative methodologies limiting information acquisition beyond score functions for acceleration holds promise. Finally, holistic guarantees that encapsulate both score acquisition and generative phases represent significant scope for consequential leaps in understanding and applying diffusion models.