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.

翻译：在自旋玻璃和无序系统理论中，一个普遍观点是：非平衡动力学无法在物理时间尺度上找到具有局部严格凸性等性质的稳定局部最优解。在Sherrington–Kirkpatrick自旋玻璃模型中，Behrens-Arpino-Kivva-Zdeborová与Minzer-Sah-Sawhney近期提出猜想：尽管整个能量景观中存在指数级数量的此类最优解，但所有高效算法可能本质上都无法克服这一障碍。我们证明该搜索问题对次数$D\leq o(N)$的多项式算法呈现强低度计算困难性：任何此类算法输出稳定局部最优解的概率仅为$o(1)$。据我们所知，这是首个证明即使常数次多项式算法以$o(1)$概率求解无植入结构的随机搜索问题的结果。为证明该结论，我们发展了集成重叠间隙性质的通用强化方法，并作为副产品将先前关于自旋玻璃优化、最大独立集、随机$k$-SAT及伊辛感知器的结果改进至强低度计算困难性。最后对于无外场的球面自旋玻璃，我们证明朗之万动力学无法在维度无关时间内找到稳定局部最优解。