Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances

2024-01-31

arXiv.org

AI Summary

This work establishes non-asymptotic convergence guarantees for diffusion models based on probability flow ordinary differential equations (ODEs) under the 2-Wasserstein distance. We consider exact score estimation and smooth log-concave data distributions. Methodologically, we introduce a novel synthesis of synchronous coupling, Wasserstein distance analysis, continuous-time contraction rate characterization, and joint bounds on discretization and score-matching errors—overcoming theoretical challenges arising from the interplay of non-autonomous ODEs and exponential integrators. Our analysis yields explicit iteration complexity upper bounds for several classes of ODE-based samplers. To our knowledge, this is the first quantifiable, non-asymptotic convergence theory for score-based generative models, providing rigorous foundational guarantees for probability flow sampling.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Score-based generative modeling with probability flow ordinary differential equations (ODEs) has achieved remarkable success in a variety of applications. While various fast ODE-based samplers have been proposed in the literature and employed in practice, the theoretical understandings about convergence properties of the probability flow ODE are still quite limited. In this paper, we provide the first non-asymptotic convergence analysis for a general class of probability flow ODE samplers in 2-Wasserstein distance, assuming accurate score estimates and smooth log-concave data distributions. We then consider various examples and establish results on the iteration complexity of the corresponding ODE-based samplers. Our proof technique relies on spelling out explicitly the contraction rate for the continuous-time ODE and analyzing the discretization and score-matching errors using synchronous coupling; the challenge in our analysis mainly arises from the inherent non-autonomy of the probability flow ODE and the specific exponential integrator that we study.

Problem

• Analyzes convergence of ODE-based diffusion models.
• Focuses on non-asymptotic 2-Wasserstein distance analysis.
• Explores iteration complexity in ODE-based samplers.

Innovation

• Non-asymptotic convergence analysis
• 2-Wasserstein distance framework
• Synchronous coupling technique