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Exponential Convergence Guarantees for Iterative Markovian Fitting

Marta Gentiloni Silveri, Giovanni Conforti, Alain Durmus

TL;DR

This work provides the first non-asymptotic exponential convergence guarantees for Iterative Markovian Fitting (IMF) in the Schrödinger Bridge problem. By proving a contraction property for the Markovian projection under mild assumptions on the reference diffusion and marginals, the authors derive explicit KL-divergence convergence rates in two regimes: strongly log-concave and weakly log-concave. The results establish quantitative finite-iteration guarantees for IMF and support theoretical analysis of related Diffusion Schrödinger Bridge Matching (DSBM). The analysis highlights a trade-off between the time horizon $T$ and marginal convexity in achieving fast convergence, and lays groundwork for future work on finite-time/discretized settings and drift-estimation errors.

Abstract

The Schrödinger Bridge (SB) problem has become a fundamental tool in computational optimal transport and generative modeling. To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed-alongside practical approximations like Diffusion Schrödinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, a quantitative, non-asymptotic understanding remains unknown. In this paper, we provide the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a sufficiently large time horizon. Our results encompass two key regimes: one where the marginals are log-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM.

Exponential Convergence Guarantees for Iterative Markovian Fitting

TL;DR

This work provides the first non-asymptotic exponential convergence guarantees for Iterative Markovian Fitting (IMF) in the Schrödinger Bridge problem. By proving a contraction property for the Markovian projection under mild assumptions on the reference diffusion and marginals, the authors derive explicit KL-divergence convergence rates in two regimes: strongly log-concave and weakly log-concave. The results establish quantitative finite-iteration guarantees for IMF and support theoretical analysis of related Diffusion Schrödinger Bridge Matching (DSBM). The analysis highlights a trade-off between the time horizon and marginal convexity in achieving fast convergence, and lays groundwork for future work on finite-time/discretized settings and drift-estimation errors.

Abstract

The Schrödinger Bridge (SB) problem has become a fundamental tool in computational optimal transport and generative modeling. To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed-alongside practical approximations like Diffusion Schrödinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, a quantitative, non-asymptotic understanding remains unknown. In this paper, we provide the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a sufficiently large time horizon. Our results encompass two key regimes: one where the marginals are log-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM.
Paper Structure (21 sections, 7 theorems, 99 equations, 3 algorithms)

This paper contains 21 sections, 7 theorems, 99 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Outline of the Paper.
  3. Notation.
  4. Schrödinger Bridge Problem and Iterative Markovian Fitting
  5. Schrödinger Bridge problem.
  6. Diffusion Flow Matching Models.
  7. Iterative Markovian Fitting
  8. Main Results
  9. Strongly Log-Concave Setting
  10. Weakly Log-Concave Setting
  11. Related Literature and Our Contribution
  12. Conclusion
  13. Technical Appendices and Supplementary Material
  14. Technical conditions
  15. Geometric Perspective on Iterative Markovian Fitting
  16. ...and 6 more sections

Key Result

Theorem 1

Assume ass:uniq_exist_station_solution_langevin to ass:strongly_log_concave. Let $\{\pi^{(n)}_{0,T}\}_{n\ge 1}$ be the IMF sequence defined in alg:IMF_version_1. If $T>\max\{\alpha_\mu^{-1}, \alpha_\nu^{-1}\}$, then, for any $n\in \mathbb{N}$, it holds where $\alpha_\varphi=\alpha_\mu-T^{-1}$ and $\alpha_\psi=\alpha_\nu-T^{-1}$.

Theorems & Definitions (29)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Theorem 2
  • ...and 19 more