Optimality of Glauber dynamics for general-purpose Ising model sampling and free energy approximation

TL;DR

This paper provides evidence that Glauber dynamics is essentially optimal for sampling from general Ising models and approximating free energy, by linking these problems to computational hardness conjectures.

Contribution

The authors establish a reduction from Wishart matrix hypothesis testing to Ising model sampling and free energy approximation, suggesting Glauber dynamics is near-optimal for these tasks.

Findings

Glauber dynamics is shown to be optimal for general-purpose Ising model sampling.

The paper links sampling hardness to a Wishart matrix hypothesis testing problem.

Evidence suggests Glauber dynamics-based simulated annealing is optimal for free energy approximation.

Abstract

Recently, Eldan, Koehler, and Zeitouni (2020) showed that Glauber dynamics mixes rapidly for general Ising models so long as the difference between the largest and smallest eigenvalues of the coupling matrix is at most $1 - \epsilon$ for any fixed $\epsilon > 0$. We give evidence that Glauber dynamics is in fact optimal for this "general-purpose sampling" task. Namely, we give an average-case reduction from hypothesis testing in a Wishart negatively-spiked matrix model to approximately sampling from the Gibbs measure of a general Ising model for which the difference between the largest and smallest eigenvalues of the coupling matrix is at most $1 + \epsilon$ for any fixed $\epsilon > 0$. Combined with results of Bandeira, Kunisky, and Wein (2019) that analyze low-degree polynomial algorithms to give evidence for the hardness of the former spiked matrix problem, our results in turn give evidence for the hardness of general-purpose sampling improving on Glauber dynamics. We also give a similar reduction to approximating the free energy of general Ising models, and again infer evidence that simulated annealing algorithms based on Glauber dynamics are optimal in the general-purpose setting.