Algorithmic Threshold for Multi-Species Spherical Spin Glasses

TL;DR

This paper characterizes the algorithmic threshold for optimizing multi-species spherical spin glasses, introducing new methods that extend previous results to non-convex models and providing explicit formulas for special cases.

Contribution

It develops a novel approach to establish the branching overlap gap property without interpolation, applying to non-convex models and generalizing single-species results.

Findings

Determined the algorithmic threshold for multi-species spherical spin glasses.

Extended results to models with non-convex covariance.

Provided closed-form formulas for pure models matching Kac-Rice results.

Abstract

We study efficient optimization of the Hamiltonians of multi-species spherical spin glasses. Our results characterize the maximum value attained by algorithms that are suitably Lipschitz with respect to the disorder through a variational principle that we study in detail. We rely on the branching overlap gap property introduced in our previous work and develop a new method to establish it that does not require the interpolation method. Consequently our results apply even for models with non-convex covariance, where the Parisi formula for the true ground state remains open. As a special case, we obtain the algorithmic threshold for all single-species spherical spin glasses, which was previously known only for even models. We also obtain closed-form formulas for pure models which coincide with the $E_{\infty}$ value previously determined by the Kac-Rice formula.