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

Xuefeng Gao, Lingjiong Zhu

TL;DR

The paper provides the first non-asymptotic Wasserstein convergence guarantees for a broad class of deterministic probability-flow ODE samplers in diffusion models with general forward schedules. It develops a contraction-based analysis under strong-log-concavity of the data and Lipschitz score dynamics, and decomposes total error into initialization, discretization, and score-matching components, all controlled via an exponential-integrator discretization. The results yield explicit iteration complexities for VE and VP forward processes, showing VP generally outperforms VE and establishing a near-tight lower bound of tilde O(sqrt(d)/ε). The work advances theoretical understanding of Wasserstein convergence for deterministic samplers and informs choice of diffusion schedules in practice.

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.

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

TL;DR

The paper provides the first non-asymptotic Wasserstein convergence guarantees for a broad class of deterministic probability-flow ODE samplers in diffusion models with general forward schedules. It develops a contraction-based analysis under strong-log-concavity of the data and Lipschitz score dynamics, and decomposes total error into initialization, discretization, and score-matching components, all controlled via an exponential-integrator discretization. The results yield explicit iteration complexities for VE and VP forward processes, showing VP generally outperforms VE and establishing a near-tight lower bound of tilde O(sqrt(d)/ε). The work advances theoretical understanding of Wasserstein convergence for deterministic samplers and informs choice of diffusion schedules in practice.

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.
Paper Structure (24 sections, 19 theorems, 158 equations, 2 tables)

This paper contains 24 sections, 19 theorems, 158 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Main Results
  5. Assumptions
  6. Main Result
  7. Examples
  8. Outline of the Proof of Theorem \ref{['thm:discrete:2']}
  9. Conclusion
  10. Key Quantities
  11. Proofs of the Main Results
  12. Proof of Theorem \ref{['thm:discrete:2']}
  13. Proof of Proposition \ref{['thm:1']}
  14. Proof of Proposition \ref{['prop:iterates']}
  15. Proof of Proposition \ref{['prop:lower:bound']}
  16. ...and 9 more sections

Key Result

Theorem 2

Suppose that Assumptions assump:p0, assump:M:1 and assump:M hold and the stepsize $\eta\leq\bar{\eta}$, where $\bar{\eta}>0$ has an explicit formula given in bar:eta in Appendix sec:key:quantities. Then, Here, $\mu(t)$ is given in c:t:defn in Appendix sec:key:quantities and where $\phi_{k,\eta}$ is given in phi:defn, $\psi_{k,\eta}$ is given in psi:defn, $\gamma_{j,\eta}$ is given in gamma:defn,

Theorems & Definitions (24)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Remark 4: Comparison of iteration complexities
  • Remark 5
  • Corollary 6
  • Corollary 7
  • Corollary 8
  • Corollary 9
  • Proposition 10
  • ...and 14 more