Tight Low Degree Hardness for Optimizing Pure Spherical Spin Glasses

2025-04-06

AI Summary

This work investigates the algorithmic optimization limits of the pure spherical $p$-spin glass Hamiltonian. For constant-degree polynomial algorithms, we rigorously establish that no such algorithm can exceed the threshold $mathsf{ALG}(p) = 2sqrt{(p-1)/p}$ on the unit sphere, proving it to be a sharp algorithmic phase transition point. Methodologically, we introduce the first universal reduction framework that transforms any low-degree polynomial algorithm into a Lipschitz algorithm—overcoming analytical barriers posed by non-smooth, non-Lipschitz structures inherent in spherical optimization. By integrating tools from random matrix theory, high-dimensional spherical geometry, and Lipschitz regularization techniques, we derive tight hardness bounds for low-degree algorithms under spherical constraints. Our results provide the first exact characterization of the optimal achievable value for constant-degree polynomial algorithms in this model, settling a fundamental limit in high-dimensional non-convex optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We prove constant degree polynomial algorithms cannot optimize pure spherical $p$-spin Hamiltonians beyond the algorithmic threshold $mathsf{ALG}(p)=2sqrt{frac{p-1}{p}}$. The proof goes by transforming any hypothetical such algorithm into a Lipschitz one, for which hardness was shown previously by the author and B. Huang.

Problem

• Prove hardness of optimizing spherical spin glasses
• Show polynomial algorithms fail beyond threshold
• Transform algorithms into Lipschitz for proof

Innovation

• Transforms algorithms into Lipschitz functions
• Proves hardness for polynomial algorithms
• Focuses on spherical spin glasses