[Paper Review] Following the ground-states of full-RSB spherical spin glasses

Eliran Subag|arXiv (Cornell University)|Dec 11, 2018

Random Matrices and Applications42 references16 citations

TL;DR

This paper introduces a greedy algorithm that follows the ground-state energy landscape of full-RSB spherical spin glasses by tracking the most negative Hessian eigenvectors at each step. It achieves near-ground-state energy in time complexity $O(N^{ ext{deg}( u)})$ for finite mixtures and reaches the conjectural terminal energy $-E_ u$ for pure models, proving the lower bound in Parisi's formula under the replica symmetric assumption.

ABSTRACT

We focus on spherical spin glasses whose Parisi distribution has support of the form $[0,q]$. For such models we construct paths from the origin to the sphere which consistently remain close to the ground-state energy on the sphere of corresponding radius. The construction uses a greedy strategy, which always follows a direction corresponding to the most negative eigenvalues of the Hessian of the Hamiltonian. For finite mixtures $ν(x)$ it provides an algorithm of time complexity $O(N^{{ m deg}(ν)})$ to find w.h.p. points with the ground-state energy, up to a small error. For the pure spherical models, the same algorithm reaches the energy $-E_{\infty}$, the conjectural terminal energy for gradient descent. Using the TAP formula for the free energy, for full-RSB models with support $[0,q]$, we are able to prove the correct lower bound on the free energy (namely, prove the lower bound from Parisi's formula), assuming the correctness of the Parisi formula only in the replica symmetric case.

Motivation & Objective

• To develop an efficient algorithm for optimizing highly non-convex spherical spin glass Hamiltonians with full Replica Symmetry Breaking (full-RSB).
• To construct paths from the origin to the sphere of radius $\sqrt{N}$ that remain close to the ground-state energy at each radius.
• To prove the lower bound in Parisi's formula for full-RSB models with support $[0,q]$, assuming only the replica symmetric case is correct.
• To understand the geometric structure of ground-state configurations through the lens of ultrametric trees and Hessian dynamics.

Proposed method

• Uses a greedy strategy that, at each point, follows the direction of the most negative Hessian eigenvector in the orthogonal complement of the current position.
• Extends the Hamiltonian into the interior of the sphere to define ground-state configurations at each radius $r < \sqrt{N}$, enabling path construction.
• Employs the TAP formula for free energy to derive bounds, linking the algorithmic path to thermodynamic quantities.
• Applies the Crisanti-Sommers representation of the Parisi functional to analyze the limiting free energy and verify the lower bound.
• Uses concentration of measure and spectral properties of the Hessian to control deviations and ensure high-probability success.
• Leverages the ultrametric tree structure as a geometric model: vertices represent centers of heavy spherical bands, approximating ground states at each radius.

Experimental results

Research questions

• RQ1 Can a polynomial-time algorithm efficiently find configurations near the ground-state energy in full-RSB spherical spin glasses?
• RQ2 How does the ground-state energy evolve as a function of radius in the interior of the sphere?
• RQ3 Can the greedy path following the most negative Hessian directions mimic the hierarchical structure of the ultrametric tree?
• RQ4 Does the algorithm achieve the conjectural terminal energy $-E_\infty$ for pure spherical models?
• RQ5 Can the lower bound in Parisi's formula be proven for full-RSB models assuming only the replica symmetric case is valid?

Key findings

• The greedy algorithm achieves a value within $o(1)$ of the ground-state energy on the sphere of radius $\sqrt{N}$ with high probability.
• For finite mixtures $\nu(x)$, the algorithm runs in time complexity $O(N^{\deg(\nu)})$ and finds points with energy close to the ground state.
• For pure spherical models, the algorithm reaches energy $-E_\infty$, the conjectural terminal energy of gradient descent.
• The TAP formula allows proving the correct lower bound in Parisi's formula for full-RSB models with support $[0,q]$, assuming the replica symmetric case is correct.
• The path constructed via Hessian eigenvector descent remains within $o(1)$ of the ground-state energy at each radius, mimicking the ultrametric tree evolution.
• The analysis confirms that the Hessian's spectral properties control the descent path, with at most $O(1)$ directions violating descent conditions with high probability.