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

Original: arXiv:2606.01078v1 Announce Type: new 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 (\gamma^\ast = 0.828) at (D = 2) (covering in the original space) and (\gamma^\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) ((\gamma^\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.

Rewrite: arXiv:2606.01078v1 Announce Type: new Abstract: Although Transport MCMC has demonstrated significant empirical success in handling difficult posterior distributions by employing normalizing flows to precondition Metropolis--Hastings proposals, previous research has failed to provide rigorous, non-vacuous spectral-gap bounds for these methods. This paper introduces the first such bounds. Specifically, for the independence Metropolis--Hastings algorithm applied to the banana distribution family, we achieve a certified spectral gap of (\gamma^\ast = 0.828) in dimension (D = 2) within the original space, and (\gamma^\ast \ge 7.6\times 10^{-4}) in dimension (D = 5) within an analytically unwarped Gaussian space. The latter certification relies on a grid-certified gradient bound under the specified numerical Lipschitz constraints, with both results holding at 95% confidence. Our methodology is built upon three core components: (i) spectral normalization combined with reduced-scale clipping reduces the flow’s Lipschitz constant dramatically, from (10^{47}) down to (10^4); (ii) we introduce a data-dependent certificate based on an empirical oscillation bound that covers coverage, thereby supplanting vacuous analytical limits; and (iii) training with oscillation regularization reduces empirical oscillation by 60--90% without compromising density fit, enabling practical certification up to (D = 20) where (\gamma^\ast \ge 1.7\times 10^{-4}). Evaluations on four additional target distributions—including a Gaussian mixture, shear-building model, Neal’s funnel, and Bayesian logistic regression—reveal three specific limitations: boundary curvature, target stiffness, and mismatch in tail coverage. Furthermore, comparing affine and spline architectures reveals that, at equivalent negative log-likelihood (NLL) scores, simpler models produce tighter certificates, effectively reversing the conventional hierarchy based on model expressiveness.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC