🤖 AI Summary
This work addresses the slow mixing and vanishing gradient issues of conventional MCMC methods when sampling heavy-tailed distributions in high-dimensional unbounded spaces. The authors propose a novel approach based on radial diffeomorphic contraction, which maps the target distribution into the unit ball and leverages efficient interior-ball random walk algorithms—such as Ball Walk—on this convex domain. To further enhance efficiency, variational inference is employed to pre-tune a spherical self-diffeomorphism that approximately enforces log-concavity. This framework constitutes the first unified ergodic MCMC sampler capable of handling arbitrary polynomially decaying heavy-tailed distributions, accompanied by non-asymptotic rapid mixing guarantees. Empirical evaluations demonstrate substantial improvements over both the No-U-Turn Sampler and existing spherical projection-based samplers on real-world posterior benchmarks from PosteriorDB and synthetic high-dimensional heavy-tailed targets.
📝 Abstract
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.