Adaptive Stereographic MCMC

In order to tackle the problem of sampling from heavy tailed, high dimensional distributions via Markov Chain Monte Carlo (MCMC) methods, Yang, Latuszy\'nski, and Roberts (2022) (arXiv:2205.12112) introduces the stereographic projection as a tool to compactify $\mathbb{R}^d$ and transform the problem into sampling from a density on the unit sphere $\mathbb{S}^d$. However, the improvement in algorithmic efficiency, as well as the computational cost of the implementation, are still significantly impacted by the parameters used in this transformation. To address this, we introduce adaptive versions three stereographic MCMC algorithms - the Stereographic Random Walk (SRW), the Stereographic Slice Sampler (SSS), and the Stereographic Bouncy Particle Sampler (SBPS) - which automatically update the parameters of the algorithms as the run progresses. The adaptive setup allows to better exploit the power of the stereographic projection, even when the target distribution is neither centered nor homogeneous. Unlike Hamiltonian Monte Carlo (HMC) and other off-the-shelf MCMC samplers, the resulting algorithms are robust to starting far from the mean in heavy-tailed, high-dimensional settings. To prove convergence properties, we develop a novel framework for the analysis of adaptive MCMC algorithms over collections of simultaneously uniformly ergodic Markov operators, which is applicable to continuous-time processes, such as SBPS. This framework allows us to obtain $\mathcal{L}^2$ and almost sure convergence results, and a CLT for our adaptive stereographic algorithms.