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Hessian descent for spherical spin glasses with uniform log-Sobolev disorder

Fu-Hsuan Ho

TL;DR

This work extends Hessian descent for spherical spin glasses to non-Gaussian disorder satisfying a uniform logarithmic Sobolev inequality. It establishes uniform spectral-edge control by combining a moment-based semicircular limit for the truncated Hessian with log-Sobolev concentration, and leverages ground-state universality to connect the non-Gaussian model to the Gaussian case. Consequently, in full-RSB models, the Hessian-descent procedure yields near-global-minimizers with high probability, aligning with the Gaussian theory under mild moment assumptions. The results broaden the applicability of Hessian-based optimization in high-dimensional random landscapes and strengthen the link between spectral edge behavior and optimization performance. Overall, the paper provides non-asymptotic, uniform-in-x edge controls and universality principles that support efficient descent on log-Sobolev-disordered spin glasses.

Abstract

The present work concerns spherical spin glass models with disorder satisfying a uniform logarithmic Sobolev inequality. We show that the Hessian descent algorithm introduced by Subag can be extended to this setting, thanks to the abundance of small eigenvalues near the edge of the Hessian spectrum. Combined with the ground state universality recently proven by Sawhney and Sellke, this implies that when the model is in the full-RSB phase, the Hessian descent algorithm can find a near-minimum with high probability. Our proof consists of two main ingredients. First, we show that the empirical spectral distribution of the Hessian converges to a semicircular law via the moment method. Second, we use the logarithmic Sobolev inequality to establish concentration and obtain uniform control of the spectral edge.

Hessian descent for spherical spin glasses with uniform log-Sobolev disorder

TL;DR

This work extends Hessian descent for spherical spin glasses to non-Gaussian disorder satisfying a uniform logarithmic Sobolev inequality. It establishes uniform spectral-edge control by combining a moment-based semicircular limit for the truncated Hessian with log-Sobolev concentration, and leverages ground-state universality to connect the non-Gaussian model to the Gaussian case. Consequently, in full-RSB models, the Hessian-descent procedure yields near-global-minimizers with high probability, aligning with the Gaussian theory under mild moment assumptions. The results broaden the applicability of Hessian-based optimization in high-dimensional random landscapes and strengthen the link between spectral edge behavior and optimization performance. Overall, the paper provides non-asymptotic, uniform-in-x edge controls and universality principles that support efficient descent on log-Sobolev-disordered spin glasses.

Abstract

The present work concerns spherical spin glass models with disorder satisfying a uniform logarithmic Sobolev inequality. We show that the Hessian descent algorithm introduced by Subag can be extended to this setting, thanks to the abundance of small eigenvalues near the edge of the Hessian spectrum. Combined with the ground state universality recently proven by Sawhney and Sellke, this implies that when the model is in the full-RSB phase, the Hessian descent algorithm can find a near-minimum with high probability. Our proof consists of two main ingredients. First, we show that the empirical spectral distribution of the Hessian converges to a semicircular law via the moment method. Second, we use the logarithmic Sobolev inequality to establish concentration and obtain uniform control of the spectral edge.

Paper Structure

This paper contains 28 sections, 18 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Definitions and result
  3. Implications for the Hessian descent algorithm
  4. Proof strategy
  5. Outline.
  6. Notation
  7. Regularity of the Hamiltonian
  8. Proof of Proposition \ref{['prop:Lipschitz']}
  9. Proof of \ref{['eq:p directional']}.
  10. Proof of \ref{['eq:p Hessian Op']}.
  11. Proof of \ref{['eq:p Hessian Lip']}.
  12. Spectral universality for the truncated Hessian
  13. Proof of \ref{['eq:uniform mu bar']} in Proposition \ref{['prop:asymptotic freeness']}
  14. Proof of \ref{['eq:truncated moments']} in Proposition \ref{['prop:asymptotic freeness']}
  15. Proof of eq:H and G are close
  16. ...and 13 more sections

Key Result

Theorem 1.1

Suppose Condition ass:LS holds. Then, for all $\varepsilon\in (0,1)$, there exist $\delta>0$ and $c>0$ such that where $\xi"(z)=\sum_{p=2}^\infty \gamma_p^2p(p-1)z^{p-2}$ and $\rho_{\bm{x}}=\norm{\bm{x}}^2/N$.

Theorems & Definitions (32)

  • Theorem 1.1: Uniform control on the edge
  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • proof : Proof of Lemma \ref{['lem:sup estimate']}
  • Proposition 3.1
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem:first reduction']}
  • ...and 22 more