Abstract:

In this talk I will discuss a new class of uniformly ergodic MCMC algorithms, termed Diffeomorphic Contraction Sampler (DCS), with fast non-asymptotic mixing guarantees for target distributions on $\R^d$ with arbitrarily heavy polynomial tails. DCS solves a well-known problem for the state-of-the-art MCMC algorithms, which struggle with the combination of unbounded high-dimensional state space and vanishing gradients.

The DCS pulls back the target on $\R^d$ onto a Euclidean ball $B(R)$ of the same dimension and samples from a transformed density on the convex set $B(R)$ via algorithms such as Ball Walk, Hit-and-Run and others. The radial diffeomorphic contraction guarantees a bounded density on $B(R)$, 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 density on $B(R)$. In practice, this is achieved via a preconditioned automorphism of the ball $B(R)$, tuned by Variational Inference.

Numerical simulation tests demonstrate that the DCS significantly outperforms the No-U-Turns sampler on high-dimensional heavy-tailed targets arising in well-studied, difficult, real-world data sets exhibiting funnel geometry. DCS also numerically outperforms recently developed spherical projection samplers for heavy-tailed target distributions. This is joint work with M. Bresar.