Title: Exponential Convergence Guarantees for Iterative Markovian Fitting
Venue: NeurIPS 2025
Accepted: yes
Keywords: Iterative Markovian Fitting

Convergence guarantees

Schrödinger Bridge

Generative Models
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies non-asymptotic convergence guarantees for IMF procedure. The exponential-speed convergence is established under suitable assumptions, of which the key ones are strong or weak log-concavity of the reference distributions. The manuscript comes with remarks which connect theoretical layouts with practical use-cases.\n\nSTRENGTHS AND WEAKNESSES:\nI found the paper to be more-or-less well-written and the considered problem (convergence of IMF) to be interesting. At the same time, from a practical point of view, paper has no practical implications; from a theoretical point of view - it is interesting to study whether the assumptions needed for the convergence hold true could be relaxed - current assumptions (on reference distributions) seems to be rather restrictive. Overall, I have a good impression of the paper.\nMore specific points:\n- The paper has no experiments validating the theoretical conclusions. In particular, it is reasonable to run DSBM on, e.g., gaussian measures/perturbed gaussian measures to see the convergence rates\n- There is a preprint studying the convergence of IMF [Kholkin]. Their analysis seems to be rather restrictive and hold only for gaussians, but anyway they are highly related.\n\n[Kholkin] Kholkin et. al., Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting, 2024\n\nQUESTIONS:\n1. Assumptions H4, H6 - need to be explicitly stated that $(x\\_0, x_T) \\mapsto - \\log p_{0,t,T}^U(x_0, x_t, x_T)$ is [properties from these assumptions] hold for each $x_t$\n\n2. Theorem 1 and Theorem 2: It seems we need $T > \\max{(\\alpha_\\mu^{-1}, \\alpha_\\nu^{-1}, \\alpha^{-1})}$, e.g., for the contraction in Theorem 1 holds true. \n3. A lot of repeated bibtex records: \n  - [AVE22] , [AVE23a] ([AVE23a] does not exists, note that [AVE22] is ICLR-23)\n  - [AVE23b] - missed author Nicholas M. Boffi\n  - [DBTHD21a], [DBTHD21b]\n  - [For40a], [For40b]\n  - [Kul68a], [Kul68b]\n  - [SDBCD23], [SDBCD24] (One from NeuriPS 23, the other from NeurIPS 24 ???)\n  - [VTLL21a], [VTLL21b]\n    \n   Double check bibtex files if you generate bibtex records with llms. \n\n4. Theorem 4: From the proof it seems we do not need the bridge $\\text{bR}^U$ to satisfy $\\text{T2}(\\xi)$\n\n5. Line 645: “the byproduct of conditions [...] leads to” - I think for clarity it is better to be more specific and detailed here, so does in line 656 in the proof of Theorem 6.\n\n6. Formula after line 656: $k\\_{\\overline{h}\\_{t, y}}$ -> $k_{\\nabla \\overline{h}_{t, y}}$.\n\nLIMITATIONS:\nDiscussed above", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-25"}

comments: SUMMARY:
The authors analyze the convergence of the Iterative Markovian Fitting (IMF) algorithm for estimating Schrödinger Bridges. In particular, they develop the first non-asymptotic convergence rates for Iterative Markovian Fitting. An analysis is provided for two important regimes:  strongly and weakly log-concave marginal distributions, where the latter case even captures certain non-convex marginals.

STRENGTHS AND WEAKNESSES:
Strengths:
- The authors provide the first non-asymptotic convergence result for an important algorithm for generative modeling.
- The new contraction results for Markovian projections could be of independent interest.
- Mostly clear structure and exposition, with assumptions are clearly spelled out. Most of them (with one exception, see below) are discussed in detail and settings provided where they hold.

Weaknesses:
- The main weakness and acknowledged limitation is that the analysis applies to an idealized (impractical) version of the Iterative Markovian Fitting algorithm, which assumes exact projections, and ignores effects of discretization. Nevertheless, I think the contribution is still a meaningful first step given the importance of the problem and interest of the machine learning community in the area.

QUESTIONS:
- Can you please elaborate on the dimension-dependence of your analysis (at least for some relevant examples / instances)?
- Can you please elaborate why assumption H2 is reasonable? When is it satisfied?

LIMITATIONS:
yes -- the discussion could be strengthened though by highlighting where the key challenges in the current analysis lie that would need to be overcome to analyze more practical variants of Iterative Markovian Fitting.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors propose a non-ordinary method for proving the exponential convergence rate of the IMF algorithm under specific log-concave assumptions on marginal distributions and reference dynamics. This makes possible to use the duality of the Kantorovich metric and the Talagran inequality to prove contraction of the Markovian projection in IMF by the Kullback-Leibler divergence. The authros derive two convergence rate results, for strongly and weakly log-concave settings. The constant in the resulting inequality requires the time horizon to be large enough.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\nThe paper is clearly written and all the required notions are properly introduced. The authors prove a rather nontrivial result for an important class of marginals and a reasonable family of reference measures. The proof of the main technical result (Theorem 4) seemed to be quite an interesting technique.\n\n**Weaknesses**\n- First, the dependence of the Markov projection's compression constant on the time horizon appears somewhat unclear. The existing analysis might alternatively suggest that the IMF convergence rate depends primarily on noise variance, as compellingly demonstrated in prior work [1] (see Figure 6, \"Dependence on the variance of the prior process\"). It would be particularly insightful if the authors could reformulate their main theorem to express the compression constant in terms of the reference process's noise variation while keeping the time horizon fixed. Such refinement could provide a more intuitive interpretation of the convergence behavior.\n- Second, the requirement for a sufficiently large time horizon needs additional justification. The manuscript would be significantly strengthened by including explicit examples demonstrating the absence of compression properties for shorter time horizons. The absence of such examples might suggest that this condition reflects limitations of the current proof technique rather than fundamental properties of the iterative procedure. \n- Third, the assumptions regarding marginal distributions appear somewhat restrictive. By focusing exclusively on unimodal distributions (log-concave and weakly log-concave families), the current theoretical guarantees exclude potentially important cases of multimodal distributions that frequently arise in practical applications. Addressing this limitation through theoretical extension would enhance the paper's applicability.\n- Finally, the absence of numerical validation represents a notable gap in the current research. Empirical results would be particularly valuable for addressing: (1) the behavior of convergence rates for different time horizons, including shorter ones, and (2) the depended of convergence for the different marginal distributions,  possibly including the ones that do not fall ini described by authors assumptions. Including such experimental validation would substantially strengthen confidence in the theoretical findings.\n\nThese suggested improvements, while not diminishing the paper's theoretical contributions, could help establish a more comprehensive understanding of the method's properties and limitations. The authors' attention to these aspects would likely increase both the depth and practical relevance of their work.\n\n\n**References.**\n\n[1] N. Gushchin, D. Selikhanovych, S. Kholkin, E. Burnaev, A. Korotin. Adversarial Schrödinger Bridge Matching. The Thirty-eighth Annual Conference on Neural Information Processing Systems. 2024\n\nQUESTIONS:\n- Would the authors consider reformulating the theorem in terms of the reference process's variance rather than time horizon? This alternative presentation might enhance interpretability and better align with established physical interpretations of the convergence behavior.\n- What evidence suggests that large time horizons are fundamentally necessary? Could you provide examples where compression fails for smaller horizons to clarify this requirement?\n - Might the authors' methodology be, in principle, extensible to more general classes of marginal distributions? One wonders if there exist implicit conditions under which such generalizations could hold?\n - Could numerical experiments be added at least to: (1) verify convergence behavior at various time horizons, including sorter ones? (2) Test the practical implications of the marginal distribution, including possibly violating log-concavity constraints?\n\nLIMITATIONS:\nThe authors have addressed the limitations of their approach in the conclusion.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper analyses the convergence of the (theoretical) IMF scheme to recover the solution of a Schrödinger Bridge (SB) problem for which no quantitative, non-asymptotical results were yet derived. The key ingredient of the proof is to show that the updates are contractive under the assumptions made in the paper. More specifically, in Theorem 4, it is shown that if the Markov kernel associated with the stochastic interpolant of the optimal coupling (which is in fact the solution to the dynamic problem (2) right?) satisfies a Talagrand inequality, then one step of the IMF is contractive. Then, the paper leverages the recent results of Conforti 2024 to recover weak semi-convexity estimates on the optimal coupling. To handle the non-strongly log concave case, the paper uses the fact that weak semi-convexity implies log-Sobolev which in turn implies Talagrand.\n\nSTRENGTHS AND WEAKNESSES:\nThe strength of the paper is to bring the first quantitative, non-asymptotic guarantees under fairly general assumptions ; remarkably, no uniform log-smoothness assumptions is made. \nWeaknesses:\n- from a technical point of view, the main contribution seems to be Theorem 4 since the weak semi-convexity estimates were proven in Conforti 2024. Since I am quite far from the SB community, I cannot tell how hard it was to derive (I acknowledge theoretical results always seem obvious once they are derived though).\n- because of this proof scheme, that relies on weak-semi-convexity estimates, the quantitative guarantees are in fact quite weak; indeed, weak-log-concavity implies exponentially bad log-Sobolev constants. In particular, how does the exponential bound in Theorem 2 compares to practical settings: does IMF fail when the distributions are not strongly log-concave?\n\nQUESTIONS:\nSee above + why use IMF rather than Sinkhorn?\n\nLIMITATIONS:\nI feel that the guarantees offered by IMF are overall much weaker than Sinkhorn (especially with $\\varepsilon = 1$ as it seems to be the case here) for several reasons:\n- Sinkhorn can be implemented \\emph{exactly} (I do not agree with l. 129-133) while IMF cannot because the function (11) needs to be approximated and no guarantees are provided with this approximation.\n- Sinkhorn converges fast in much larger settings.\nIn my opinion, this comparison should have been made in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work studies the convergence of the IMF method for solving the Schr\\”odinger bridge problem. This method consists in alternating Markovian projection steps with stochastic interpolation steps. This method can sometimes outperform Sinkhorn’s algorithm in practice. The paper shows that under log-concavity assumptions on both end marginals as well as the reference path measure, one achieves exponential convergence with rate depending roughly on $1/(\\alpha_\\mu + \\alpha_\\nu)$. The result is also extended to the case of ``weak convexity’’.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- The result (in the strongly convex case) is extremely elegant and stated under interpretable but generic conditions. In particular, the dependence on the convexity parameters $\\alpha_\\mu, \\alpha_\\nu$ makes sense. \n- As far as I can tell, the proof is also very clean and readable, and may inspire further analysis in this direction. The paper in general is quite well-written.\n- I think this is a strong contribution to the literature. As IMF is a well-known algorithm, this will be of interest both to theoreticians and practitioners.\n\n**Weaknesses**\n\n- The algorithms are idealized, and most discretization errors (SDE discretization, estimating the conditional drift) are ignored.\n- There is an additional requirement that $T> \\alpha^{-1}$ in e.g., Theorem 1, which seems strange. Why would this be necessary?\n\nQUESTIONS:\n**Major**\n\n- I am not particularly familiar with the definition of “weak convexity” used here, so some greater explanation around Theorem 2 would be nice. In particular, is this the expected rate? Can we hope to do better in any meaningful sense?\n- Is the large $T$ condition fundamental, or a limitation of the analysis?\n\n**Minor**\n\nL. 177: Steps $\\to$ steps\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-22"}