Optimal Scaling Results for Moreau-Yosida Metropolis-adjusted Langevin Algorithms

TL;DR

This paper investigates the optimal scaling of Moreau-Yosida Metropolis-adjusted Langevin algorithms, providing practical guidelines for their implementation and extending understanding beyond traditional MALA methods.

Contribution

It offers new insights into the optimal scaling of proximity-based MCMC methods, including MALA, and supplies practical guidelines for their effective use.

Findings

Derived optimal scaling results for the class of algorithms

Extended stability analysis to non-differentiable targets

Provided practical implementation guidelines

Abstract

We consider a recently proposed class of MCMC methods which uses proximity maps instead of gradients to build proposal mechanisms which can be employed for both differentiable and non-differentiable targets. These methods have been shown to be stable for a wide class of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms (MALA); and have found wide application in imaging contexts. The wider stability properties are obtained by building the Moreau-Yosida envelope for the target of interest, which depends on a parameter $\lambda$. In this work, we investigate the optimal scaling problem for this class of algorithms, which encompasses MALA, and provide practical guidelines for the implementation of these methods.