Skew Brownian Motion and Complexity of the ALPS Algorithm

TL;DR

This paper proves that a scaled version of the ALPS algorithm converges to skew Brownian motion, providing insights into its mixing times in high-dimensional settings.

Contribution

It establishes the weak convergence of ALPS to skew Brownian motion and analyzes its mixing time complexity under certain assumptions.

Findings

ALPS converges to skew Brownian motion under scaling.

Mixing time is O(d[log(d)]^2) or O(d) depending on the version.

Provides theoretical foundation for ALPS efficiency in multimodal sampling.

Abstract

Simulated tempering is a popular method of allowing MCMC algorithms to move between modes of a multimodal target density {\pi}. The paper [24] introduced the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper, we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that under appropriate assumptions, the ALPS algorithm mixes in time O(d[log(d)]^2 ) or O(d), depending on which version is used.