# Sampling from Rough Energy Landscapes

## TL;DR

This paper investigates the impact of multiscale roughness on sampling algorithms for Boltzmann distributions, showing that certain methods are sensitive to roughness while others are robust, and proposing new strategies to improve sampling performance.

## Contribution

It demonstrates the limitations of the Metropolis Adjusted Langevin Algorithm with rough landscapes and introduces alternative methods that resist fine-scale roughness effects.

## Key findings

- Metropolis Adjusted Langevin Algorithm performance deteriorates with increased roughness
- Random Walk Metropolis remains insensitive to roughness
- Proposed new strategies outperform Random Walk Metropolis in rough landscapes

## Abstract

We examine challenges to sampling from Boltzmann distributions associated with multiscale energy landscapes. The multiscale features, or "roughness," corresponds to highly oscillatory, but bounded, perturbations of a smooth landscape. Through a combination of numerical experiments and analysis we demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can be severely attenuated as the roughness increases. In contrast, we prove that Random Walk Metropolis is insensitive to such roughness. We also formulate two alternative sampling strategies that incorporate large scale features of the energy landscape, while resisting the impact of fine scale roughness; these also outperform Random Walk Metropolis. Numerical experiments on these landscapes are presented that confirm our predictions. Open questions and numerical challenges are also highlighted.