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Exponential Convergence Guarantees for Iterative Markovian Fitting
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Gives the first non-asymptotic exponential KL convergence guarantees for Iterative Markovian Fitting (IMF) to the Schrödinger Bridge, for strongly and weakly log-concave marginals and a sufficiently large horizon T, via a contraction result for the Markovian projection.
Problem
IMF is known to converge asymptotically to the Schrödinger Bridge, but no quantitative non-asymptotic convergence rate was available for this iterative Markovian fitting scheme.
Approach
Derives a contraction estimate for the Markovian projection operator under log-concavity (strong or weak, via convexity profiles) of the marginals and the reference Langevin/Brownian bridge, yielding geometric KL decay of the IMF iterates for large T.
Key results
- Under strong log-concavity and T > max{1/alpha_mu, 1/alpha_nu}, KL(pi^(n) | pi*) decays geometrically at rate (L_U/(T(alpha_phi+alpha_psi+alpha)))^n.
- Under weak log-concavity, an analogous exponential rate holds with an extra damping factor exp(9 max{L_mu,L_nu,L}/(alpha_phi+alpha_psi+alpha)) in the constant.
- The rate is <1 only for sufficiently large horizon T, and approaches 1 as T decreases to the threshold.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply IMF with strongly log-concave marginals and sufficiently large time horizon | IMF · Strongly log-concave marginals | KL divergence to the Schrödinger Bridge | Helped |
| Apply IMF with weakly log-concave marginals and sufficiently large time horizon | IMF · Weakly log-concave marginals | KL divergence to the Schrödinger Bridge | Helped |
| Increase marginal and reference convexity | IMF · Strongly log-concave marginals | KL convergence rate | Helped |
| Increase deviations from strong convexity | IMF · Weakly log-concave marginals | KL convergence rate | Hurt |
Limitations
- Results require a sufficiently large time horizon T, whereas Schrödinger Bridge problems typically fix T.
- Analysis ignores drift estimation error and discretization error present in practical IMF/DSBM implementations.
- Assumes exact Markovian projections and log-concavity-type conditions on marginals and reference bridge.
Experimental setup
Metrics: KL divergence
Builds on: Iterative Markovian Fitting Diffusion Flow Matching Diffusion Schrödinger Bridge Matching Iterative Proportional Fitting
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