Algorithmic Threshold for Multi-Species Spherical Spin Glasses

Brice Huang, Mark Sellke

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.