AI 中文总结
该研究提出微分同胚收缩采样器（DCS），解决MCMC采样器处理重尾分布时的梯度消失问题，实验显示其在PosteriorDB基准及高维示例中性能优于现有重尾采样器。
AI 中文摘要
我们引入一类新的一致遍历MCMC算法，称为微分同胚收缩采样器（Diffeomorphic Contraction Sampler，DCS），并为DCS提供快速非渐近混合保证，该算法针对$\boldsymbol{R}^d$上具有任意重多项式尾的分布。DCS解决了MCMC采样器的一个著名问题：MCMC采样器通常难以处理无界高维状态空间与消失梯度的组合问题。DCS将$\boldsymbol{R}^d$上的目标分布拉回至欧氏球$B(R)\boldsymbol{\times}\boldsymbol{R}^d$，随后通过Ball Walk、Hit-and-Run等算法在凸集$B(R)$上对变换后的密度进行采样。选择径向微分同胚收缩使得$B(R)$上的拉回密度有界，这意味着所有具有有限多项式矩的目标分布都满足一致遍历性。DCS的非渐近界需要更强的假设，例如拉回密度的对数凹性；在实践中，这通过变分推断调优的$B(R)$的预处理自同构近似实现。数值模拟测试表明，在PosteriorDB基准中作为真实后验出现的多维重尾目标上，DCS的性能显著优于无回退采样器（No-U-Turns sampler）；在高维示例中，DCS在数值上也优于最近提出的针对重尾目标分布的球投影采样器。
英文摘要
We introduce a new class of uniformly ergodic MCMC algorithms, termed Diffeomorphic Contraction Sampler (DCS), and provide fast non-asymptotic mixing guarantees for DCS targeting distributions on $\R^d$ with arbitrarily heavy polynomial tails. DCS provides a solution to a well-known problem for MCMC samplers, which typically struggle with the combination of unbounded high-dimensional state space and vanishing gradients. The DCS pulls back a target on $\R^d$ onto a Euclidean ball $B(R)\subset\R^d$ and then samples from the transformed density on the convex set $B(R)$ via algorithms such as the Ball Walk, Hit-and-Run and others. A radial diffeomorphic contraction is chosen so that the pull-back density on $B(R)$ is bounded, implying uniform ergodicity for \textit{all} targets with a finite polynomial moment. Non-asymptotic bounds for DCS require stronger assumptions such as log-concavity of the pull-back density. In practice, this is achieved approximately by a preconditioned automorphism of the ball $B(R)$, tuned via Variational Inference. Numerical simulation tests demonstrate that the DCS outperforms significantly the No-U-Turns sampler on multi-dimensional heavy-tailed targets arising as real-world posteriors in PosteriorDB benchmark. DCS also numerically outperforms in high-dimensional examples recently developed spherical projection samplers for heavy-tailed target distributions.
Comments38 pages, 5 figures; for a short YouTube video describing the main algorithm and its properties see https://youtu.be/cOlmRLxeQM0
