A1441
Title: Adaptive stereographic MCMC
Authors: Krzysztof Latuszynski - University of Warwick (United Kingdom) [presenting]
Cameron Bell - University of Warwick (United Kingdom)
Gareth Roberts - University of Warwick (United Kingdom)
Abstract: In order to tackle the problem of sampling from heavy-tailed, high-dimensional distributions via Markov chain Monte Carlo (MCMC) methods, an earlier work introduces Stereographic MCMC samplers. However, its improvement in algorithmic efficiency, as well as the computational cost of the implementation, is significantly impacted by the parameters used in this design. To address these design difficulties, the adaptive versions of three stereographic MCMC algorithms - the stereographic random walk (SRW), the stereographic slice sampler (SSS), and the stereographic bouncy particle sampler (SBPS) - is introduced, which automatically update the parameters of the algorithms as the run progresses. The adaptive setup allows for better exploitation of 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, a novel framework is developed 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 obtaining L2 and almost sure convergence results, and a CLT for the adaptive stereographic algorithms.
