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Optimizing Mean Field Spin Glasses with External Field

Mark Sellke

TL;DR

This work gives an two-phase message pasing algorithm to approximately maximize $H_N$ when a no overlap-gap condition holds and gives a branching variant of the algorithm which constructs a full ultrametric tree of approximate maxima.

Abstract

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.

Optimizing Mean Field Spin Glasses with External Field

TL;DR

This work gives an two-phase message pasing algorithm to approximately maximize when a no overlap-gap condition holds and gives a branching variant of the algorithm which constructs a full ultrametric tree of approximate maxima.

Abstract

We consider the Hamiltonians of mean-field spin glasses, which are certain random functions defined on high-dimensional cubes or spheres in . 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 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 . We give a two-phase message pasing algorithm to approximately maximize 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.

Paper Structure

This paper contains 26 sections, 60 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Optimizing Ising Spin Glasses
  3. Optimizability and No Overlap Gap
  4. Branching IAMP and Spherical Spin Glasses
  5. Technical Preliminaries
  6. Review of Approximate Message Passing
  7. Initializing AMP
  8. Properties of the Parisi PDE and SDE
  9. The Main Algorithm
  10. Phase 1: Finding the Root
  11. Phase 2: Incremental AMP
  12. Computing the Final Energy
  13. Constructing Many Approximate Maximizers
  14. Spherical Models
  15. Proof of Lemma \ref{['lem:RSsphere']}
  16. ...and 11 more sections

Key Result

Theorem 1

Moreover the minimum is attained at a unique $\gamma^{\mathscrsfs{U}}_*\in\mathscrsfs{U}$.

Theorems & Definitions (107)

  • Theorem 1: talagrand2006parisipanchenko2014parisiauffinger2017parisichen2018energy
  • Theorem 2: ams20
  • Corollary 1.1: ams20
  • Proposition 1.2
  • Definition 1.3
  • Lemma 1.3
  • Theorem 3
  • Lemma 1.3
  • Corollary 1.4
  • Remark 1.5
  • ...and 97 more