Optimizing Mean Field Spin Glasses with External Field

Mark Sellke

Published 2021-05-07Version 1

We consider the Hamiltonians of mean-field spin glasses, which are certain random functions $H_N$ defined on high-dimensional cubes or spheres in $\mathbb R^N$. The asymptotic maximum values of these functions were famously obtained by Talagrand and later by Panchenko and by Chen. The landscape of $H_N$ is described by various forms of replica symmetry breaking (RSB) depending on the model parameters. When full RSB occurs, efficient algorithms have recently emerged to compute an approximate maximizer. On the other hand, when full RSB does not hold one typically expects a so-called overlap gap property to prevent efficient algorithms from succeeding at this task. We give an two-phase message pasing algorithm to approximately maximize $H_N$ whenever full RSB holds, generalizing recent works by allowing a non-trivial external field. Moreover we give a branching variant of the algorithm which constructs an ultrametric tree of approximate maxima.