Geometric adaptive Monte Carlo in random environment

TL;DR

This paper introduces a geometric adaptive Monte Carlo method that balances local geometric information and computational efficiency, improving sampling effectiveness in complex, high-dimensional target distributions.

Contribution

It proposes a novel adaptive Monte Carlo sampler that switches between geometric and adaptive proposals using an exponential schedule to optimize efficiency.

Findings

Balances geometric information use with computational cost

Achieves higher effective sample size for given computational resources

Flexible complexity setting based on model needs

Abstract

Manifold Markov chain Monte Carlo algorithms have been introduced to sample more effectively from challenging target densities exhibiting multiple modes or strong correlations. Such algorithms exploit the local geometry of the parameter space, thus enabling chains to achieve a faster convergence rate when measured in number of steps. However, acquiring local geometric information can often increase computational complexity per step to the extent that sampling from high-dimensional targets becomes inefficient in terms of total computational time. This paper analyzes the computational complexity of manifold Langevin Monte Carlo and proposes a geometric adaptive Monte Carlo sampler aimed at balancing the benefits of exploiting local geometry with computational cost to achieve a high effective sample size for a given computational cost. The suggested sampler is a discrete-time stochastic process in random environment. The random environment allows to switch between local geometric and adaptive proposal kernels with the help of a schedule. An exponential schedule is put forward that enables more frequent use of geometric information in early transient phases of the chain, while saving computational time in late stationary phases. The average complexity can be manually set depending on the need for geometric exploitation posed by the underlying model.