Non-asymptotic Convergence of Discrete-time Diffusion Models: New Approach and Improved Rate
(2402.13901)Abstract
The denoising diffusion model emerges recently as a powerful generative technique that converts noise into data. Theoretical convergence guarantee has been mainly studied for continuous-time diffusion models, and has been obtained for discrete-time diffusion models only for distributions with bounded support in the literature. In this paper, we establish the convergence guarantee for substantially larger classes of distributions under discrete-time diffusion models and further improve the convergence rate for distributions with bounded support. In particular, we first establish the convergence rates for both smooth and general (possibly non-smooth) distributions having finite second moment. We then specialize our results to a number of interesting classes of distributions with explicit parameter dependencies, including distributions with Lipschitz scores, Gaussian mixture distributions, and distributions with bounded support. We further propose a novel accelerated sampler and show that it improves the convergence rates of the corresponding regular sampler by orders of magnitude with respect to all system parameters. For distributions with bounded support, our result improves the dimensional dependence of the previous convergence rate by orders of magnitude. Our study features a novel analysis technique that constructs tilting factor representation of the convergence error and exploits Tweedie's formula for handling Taylor expansion power terms.
Overview
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This paper introduces a novel analytical technique for studying the non-asymptotic convergence of discrete-time diffusion probabilistic models (DDPMs), expanding the understanding and capabilities of these generative models.
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It reveals the limitations of previous analyses that focused on distributions with bounded support and extends guarantees to include a wider class of distributions, including those with unbounded support.
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A new accelerated DDPM sampler is introduced, employing Hessian-based estimators to achieve improved convergence rates, particularly for distributions with bounded support.
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The work utilizes a unique analytical framework to precisely characterize error at each step of the reverse process, incorporating tilting factors and the application of Tweedie’s formula.
Analyzing Non-asymptotic Convergence of Discrete-time Diffusion Models with Improved Rates
Discrete-time diffusion models have recently garnered significant attention due to their powerful generative capabilities, providing a promising alternative to traditional generative models. Despite their empirical success, the theoretical understanding of their convergence properties has largely centered around continuous-time formulations, leaving a gap in our understanding of the discrete-time counterparts. This paper presents a novel analytical technique to address the non-asymptotic convergence of discrete-time diffusion probabilistic models (DDPMs), establishing guarantees for a broader class of distributions and achieving improved convergence rates.
Discrepancy in Theoretical Understanding
The transition from continuous-time to discrete-time diffusion models introduces challenges that have hindered the development of a robust theoretical framework. This difficulty primarily stems from the complex nature of discrete steps in the generative process, which complicates the direct application of continuous-time analysis tools. The only preceding work tackling discrete-time models provided non-asymptotic convergence guarantees under the constraint of distributions with bounded support, leaving open questions regarding distributions with unbounded support and high-dimensional dependencies.
Contributions and Novel Analytical Techniques
This work's principal contribution lies in its novel approach to analyzing discrete-time DDPMs, extending non-asymptotic convergence guarantees to encompass a wider class of distributions, including those with unbounded support. The key highlights include:
- Improved Convergence Bound for Smooth Distributions: A new bound on the convergence rate for smooth distributions indicates that the requirement on the bounded support set in previous analyses is overly restrictive. Through refined analysis techniques, this work establishes polynomial-time convergence guarantees for smooth distributions, illustrating that DDPMs can effectively model a broader range of real-world data distributions.
- Extension to General Distributions: The analysis extends these convergence guarantees to general (possibly non-smooth) distributions by employing a novel representation of the distribution generated at each step of the reverse process. This advancement underscores the flexibility of DDPMs in capturing complex data distribution characteristics.
- Accelerated Convergence via Novel Sampler: A significant breakthrough is the development of a new accelerated DDPM sampler by introducing Hessian-based estimators, which sharpens the convergence rate. This enhancement is particularly notable for distributions with bounded support and highlights the potential for practical improvements in DDPM efficiency.
- Analytical Techniques: At the core of these advancements is the introduction of a novel analytical framework that enables precise error characterization at each step of the reverse process. This includes the development of tilting factors to accurately capture convergence errors and the creative application of Tweedie’s formula to manage higher-order Taylor series terms.
Implications and Future Directions
The findings of this paper have profound theoretical and practical implications, demonstrating that DDPMs can be effectively applied to a broader class of distributions than previously understood. This contributes to closing the gap between the empirical success of DDPMs and their theoretical underpinnings, offering a roadmap for future research in this area. Potential avenues for further investigation include exploring the applicability of these techniques to different families of distributions and developing more efficient samplers based on the insights gained from this analysis.
In summary, this work lays foundational groundwork by providing a robust analytical framework for understanding the non-asymptotic convergence properties of discrete-time DDPMs, marking a significant step forward in the theoretical study of generative models. Through its novel contributions, this paper paves the way for exciting new developments in the field of generative modeling.
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