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Optimization Algorithms for Multi-Species Spherical Spin Glasses

Brice Huang, Mark Sellke

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.

Optimization Algorithms for Multi-Species Spherical Spin Glasses

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 -species model we construct 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.
Paper Structure (23 sections, 23 theorems, 222 equations)

This paper contains 23 sections, 23 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Problem Description
  3. The Value ${\mathsf{ALG}}$
  4. Notations
  5. Achieving Energy ${\mathsf{ALG}}$
  6. Definition of Pseudo-Maximizer
  7. Review of Approximate Message Passing
  8. Stage ${\rm I}$: Finding the Root of the Ultrametric Tree
  9. Stage ${\rm II}$: Descending the Ultrametric Tree
  10. Extensions
  11. Signed AMP
  12. Gradient Computation and Connection to $E_{\infty}$
  13. Branching IAMP and Exponential Concentration
  14. State Evolution: Proof of Proposition \ref{['prop:state_evolution']}
  15. Further Definitions
  16. ...and 8 more sections

Key Result

Proposition 1.3

If the square matrix $M$ is diagonally signed, then the minimal eigenvalue ${\boldsymbol{\lambda}}_{\min}(M)$ has multiplicity $1$, and the corresponding eigenvector $\vec{v}$ has strictly positive entries. Moreover and the supremum is uniquely attained at $\vec{v}$.

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3: huang2023algorithmic, see also huang2023strong
  • Definition 1.4: Algorithmic Threshold, Super-Solvable Case
  • Definition 1.5: Algorithmic Threshold, Strictly Sub-solvable Case
  • Theorem 1
  • Proposition 1.6: huang2023algorithmic
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • ...and 40 more