Optimization Algorithms for Multi-Species Spherical Spin Glasses

TL;DR

This paper introduces approximate message passing algorithms for optimizing multi-species spherical spin glasses, achieving the algorithmic threshold energy and constructing multiple approximate critical points depending on the external field strength.

Contribution

It develops generalized algorithms that produce multiple approximate critical points and analyzes their local Hamiltonian behavior, extending understanding of the energy landscape.

Findings

Achieved the algorithmic threshold energy for multi-species models.

Constructed multiple approximate critical points depending on external field strength.

Analyzed local Hamiltonian behavior around each critical point.

Abstract

This paper develops approximate message passing algorithms to optimize multi-species spherical spin glasses. We first show how to efficiently achieve the algorithmic threshold energy identified in our companion work, thus confirming that the Lipschitz hardness result proved therein is tight. Next we give two generalized algorithms which produce multiple outputs and show all of them are approximate critical points. Namely, in an $r$-species model we construct $2^r$ approximate critical points when the external field is stronger than a "topological trivialization" phase boundary, and exponentially many such points in the complementary regime. We also compute the local behavior of the Hamiltonian around each. These extensions are relevant for another companion work on topological trivialization of the landscape.