Strong Low Degree Hardness for Stable Local Optima in Spin Glasses

TL;DR

This paper proves that polynomial algorithms of degree up to o(N) have negligible probability of finding stable local optima in spin glasses, highlighting inherent computational hardness in these systems.

Contribution

It establishes strong low degree hardness for stable local optima in spin glasses, extending to various optimization problems and dynamics.

Findings

Polynomial algorithms of degree D ≤ o(N) fail to find stable local optima with high probability.

Enhanced the ensemble overlap gap property to prove computational hardness.

Langevin dynamics cannot find stable local optima in spherical spin glasses within dimension-free time.

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 $D\leq o(N)$: 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.