通过振荡控制归一化流实现传输MCMC的非平凡认证
Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows
AI总结：
提出振荡控制归一化流框架，首次为传输MCMC采样器提供严格的非平凡谱隙界，通过谱归一化、基于覆盖的经验振荡界和振荡正则化训练，在多个后验分布上实现可认证的采样效率。
摘要：
传输MCMC训练归一化流以预处理Metropolis-Hastings提议，在具有挑战性的后验分布上实现了高经验效率；然而，先前的工作没有为此类采样器产生数值上非平凡的、严格的谱隙界。我们建立了第一个这样的界。对于香蕉族上的独立MH，我们在D=2时认证了γ^*=0.828（在原始空间中覆盖），在D=5时认证了γ^*≥7.6×10^{-4}（在解析解卷的高斯空间中覆盖，并具有网格认证的梯度界，在所述数值Lipschitz认证下），两者均在95%置信度下严格。该框架基于三个支柱：(i) 具有缩减尺度裁剪的谱归一化将流Lipschitz常数从10^{47}约束到10^4；(ii) 基于覆盖的经验振荡界用数据依赖的证书替代了空洞的分析界；(iii) 振荡正则化训练在不损失密度拟合的情况下将经验振荡减少60-90%，将实用证书扩展到D=20（γ^*≥1.7×10^{-4}）。在另外四个目标（高斯混合、剪切建筑、Neal的漏斗、贝叶斯逻辑回归）上的测试确定了三个精确障碍：边界曲率、目标刚度和尾部覆盖不匹配。仿射与样条比较表明，更简单的架构在相同NLL下产生更紧的证书，颠倒了通常的表达性层次。
英文摘要：
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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