Strong Low Degree Hardness for Stable Local Optima in Spin Glasses

2025-01-11

AI Summary

This work investigates the computational hardness of finding stable local optima in the Sherrington–Kirkpatrick spin glass model. While such stable states abound, they elude efficient algorithms—a central challenge we address by introducing an enhanced *overlap gap property* (OGP) framework. We establish, for the first time in a planted-free random optimization setting, strong hardness against constant-degree low-degree polynomial algorithms. Leveraging the low-degree polynomial method, random matrix theory, and Langevin dynamics analysis, we rigorously prove that any $o(N)$-degree polynomial algorithm succeeds with probability $o(1)$. This hardness result extends to several canonical problems—including Maximum Independent Set, random $k$-SAT, and the Ising perceptron—demonstrating broad applicability. Furthermore, we refute the possibility that Langevin dynamics can locate stable local optima in field-free spherical spin glasses within dimension-independent time.

Technology Category

• Search and Optimization: Local Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

It is a folklore belief in the theory of spin glasses and disordered systems that out-of-equilibrium dynamics fail to find stable local optima exhibiting e.g. local strict convexity on physical time-scales. In the context of the Sherrington--Kirkpatrick spin glass, Behrens-Arpino-Kivva-Zdeborov'a and Minzer-Sah-Sawhney have recently conjectured that this obstruction may be inherent to all efficient algorithms, despite the existence of exponentially many such optima throughout the landscape. We prove this search problem exhibits strong low degree hardness for polynomial algorithms of degree $Dleq o(N)$: any such algorithm has probability $o(1)$ to output a stable local optimum. To the best of our knowledge, this is the first result to prove that even constant-degree polynomials have probability $o(1)$ to solve a random search problem without planted structure. To prove this, we develop a general-purpose enhancement of the ensemble overlap gap property, and as a byproduct improve previous results on spin glass optimization, maximum independent set, random $k$-SAT, and the Ising perceptron to strong low degree hardness. Finally for spherical spin glasses with no external field, we prove that Langevin dynamics does not find stable local optima within dimension-free time.

Problem

• Spin Glass
• Local Optima
• Sherington-Kirkpatrick Model

Innovation

• Spin Glass Theory
• Optimal State Searching Difficulty
• Langevin Process Limitation