Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows

• URL: http://arxiv.org/abs/2606.01078v1
• Date: Sun, 31 May 2026 07:46:48 GMT
• Title: Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows
• Abstract summary: Transport MCMC trains a normalizing flow to precondition Metropolis--Hastings proposals.<n>We establish the first numerically non-vacuous, rigorous spectral-gap bound for such samplers.
• Score: 3.8549131582427303
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Transport MCMC trains a normalizing flow to precondition Metropolis--Hastings proposals, achieving high empirical efficiency on challenging posteriors; yet no prior work produces a numerically non-vacuous, rigorous spectral-gap bound for such samplers. We establish the first such bounds. For independence MH on the banana family we certify (γ^\ast = 0.828) at (D = 2) (covering in the original space) and (γ^\ast \ge 7.6\times 10^{-4}) at (D = 5) (covering in an analytically unwarped Gaussian space with a grid-certified gradient bound under the stated numerical Lipschitz certification), both rigorous at 95% confidence. The framework rests on three pillars: (i) spectral normalization with reduced scale clips constrains the flow Lipschitz constant from (10^{47}) to (10^4); (ii) a coverage-based empirical oscillation bound replaces the vacuous analytical bound with a data-dependent certificate; and (iii) oscillation-regularised training cuts the empirical oscillation by 60--90% at no cost to density fit, extending practical certificates through (D = 20) ((γ^\ast \ge 1.7\times 10^{-4})). Tests on four further targets (Gaussian mixture, shear-building, Neal's funnel, Bayesian logistic regression) identify three precise barriers: boundary curvature, target stiffness, and tail-coverage mismatch. An affine-vs-spline comparison shows that simpler architectures yield tighter certificates at identical NLL, inverting the usual expressiveness hierarchy.

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