Strong Low Degree Hardness for Stable Local Optima in Spin Glasses
Brice Huang, Mark Sellke
TL;DR
This work addresses the computational hardness of locating stable local optima (γ-gapped wells) in high-dimensional mean-field spin glasses. It introduces a robust ensemble overlap gap property (OGP) framework to prove strong low-degree hardness for finding gapped states in the Sherrington–Kirkpatrick model, with corresponding hardness extensions to spherical spin glasses and a suite of mean-field CSPs. The main methodological advance is a correlated ensemble construction and a new positive-correlation object that simultaneously controls success and stability across the ensemble, yielding hardness up to degree $D\le o(N)$ and, in several models, beyond Lipschitz algorithms. Additionally, the paper proves that Langevin dynamics fail to locate wells in spherical spin glasses without external field on dimension-free time scales, illustrating fundamental algorithmic limitations in both discrete and continuous random landscapes. Together, these results illuminate deep computational barriers in disordered systems and provide a unified, quantitative OGP-based view of when efficient algorithms fail to find stable local optima.
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-Zdeborová 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 $D\leq o(N)$: 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.