Optimizing Mean Field Spin Glasses with External Field

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 approximate maxima of $H_N$ is described by various forms of replica symmetry breaking exhibiting a broad range of possible behaviors. We study the problem of efficiently computing an approximate maximizer of $H_N$. We give a two-phase message pasing algorithm to approximately maximize $H_N$ when a no overlap-gap condition holds. This generalizes several recent works by allowing a non-trivial external field. For even Ising spin glasses with constant external field, our algorithm succeeds exactly when existing methods fail to rule out approximate maximization for a wide class of algorithms. Moreover we give a branching variant of our algorithm which constructs a full ultrametric tree of approximate maxima.