We propose CerT-MCMC, a framework that equips learned-transport Markov chain Monte Carlo with automatic, rigorous convergence certificates. A normalising flow maps a Gaussian reference to an approximation of the target posterior; the same flow then serves as both the independence Metropolis-Hastings proposal and the basis for a computable spectral-gap bound. We develop two complementary certificates. The covering certificate bounds the weight-ratio oscillation over the full proposal support via finite-sample covering arguments, yielding full-support spectral-gap bounds when a conservative gradient bound is available; its correction term scales as O(n^{-1/D}), making it rapidly weak and eventually vacuous as dimension increases. We prove a matching Omega(n^{-1/D}) lower bound, establishing that this barrier is intrinsic to pointwise Lipschitz certification. The quantile-core certificate restricts attention to a high-probability residual core on which the oscillation is controlled by one-dimensional empirical quantiles, with a finite-sample probability slack of O(n^{-1/2}), independent of the ambient dimension. On synthetic targets (D=2-20), structural-engineering posteriors (D=6,8), real-data logistic regression on the Heart Disease data set (D=13), and synthetic Bayesian logistic regression (D=20), the quantile-core certificate delivers non-vacuous spectral-gap bounds where the covering certificate is vacuous, and its spectral-gap proxy tracks empirical effective sample sizes within 7%. A negative control experiment confirms that the certificate discriminates flow quality by a factor exceeding 10x, whereas acceptance rates differ by only 1.15x. To our knowledge, the dual-certificate framework is the first to provide automatic, dimension-aware convergence certificates for learned-transport MCMC, distinguishing genuine transport failure from proof-technique limitations.

翻译：我们提出CerT-MCMC框架，为学习传输马尔可夫链蒙特卡洛方法配备自动且严格的收敛性证书。归一化流将高斯参考分布映射为目标后验的近似分布；该流同时作为独立Metropolis-Hastings提议分布和可计算谱隙界的基础。我们开发了两类互补证书：覆盖证书通过有限样本覆盖论证在全提议支撑集上约束权重比振荡，当存在保守梯度界时可获得全支撑谱隙界；其修正项按O(n^{-1/D})尺度衰减，随维度增加迅速失效直至为空。我们证明了匹配的Omega(n^{-1/D})下界，表明该障碍是逐点Lipschitz认证的内在特征。分位数核心证书将注意力限制在高概率残差核心区域，该区域振荡由一维经验分位数控制，具有与环境维度无关的O(n^{-1/2})有限样本概率松弛项。在合成目标（D=2-20）、结构工程后验（D=6,8）、心脏疾病数据集上的真实数据逻辑回归（D=13）以及合成贝叶斯逻辑回归（D=20）实验中，当覆盖证书失效时，分位数核心证书仍能提供非空谱隙界，其谱隙代理与经验有效样本量的偏差在7%以内。阴性对照实验证实，证书区分流质量的因子超过10倍，而接受率差异仅为1.15倍。据我们所知，该对偶证书框架首次为学习传输MCMC提供自动且维度感知的收敛性证书，能够有效区分真实的传输失败与证明技术局限性。