On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions

Afonso S. Bandeira; Antoine Maillard; Richard Nickl; Sven Wang

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arXiv · 2209.02001

On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions

Afonso S. Bandeira·Antoine Maillard·Richard Nickl·Sven Wang

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Abstract

We exhibit examples of high-dimensional unimodal posterior distributions arising in non-linear regression models with Gaussian process priors for which MCMC methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates. Our results apply to worst-case initialised (`cold start') algorithms that are local in the sense that their step-sizes cannot be too large on average. The counter-examples hold for general MCMC schemes based on gradient or random walk steps, and the theory is illustrated for Metropolis-Hastings adjusted methods such as pCN and MALA.

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Afonso S. Bandeira, Antoine Maillard, Richard Nickl, Sven Wang. 2022-09-05. On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions. https://arxiv.org/abs/2209.02001

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