SIAM Journal on Computing, Volume 53, Issue 1, Page 1-46, February 2024.
Abstract. We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Such problems arise widely in the theory of random graphs, theoretical computer science, and statistical physics. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising [math]-spin glass model and (b) finding a large independent set in a sparse Erdős–Rényi graph. The following families of algorithms are considered: (a) low-degree polynomials of the input—a general framework that captures many prior algorithms; (b) low-depth Boolean circuits; (c) the Langevin dynamics algorithm, a canonical Monte Carlo analogue of the gradient descent algorithm. We show that these families of algorithms cannot have high success probability. For the case of Boolean circuits, our results improve the state-of-the-art bounds known in circuit complexity theory (although we consider the search problem as opposed to the decision problem). Our proof uses the fact that these models are known to exhibit a variant of the overlap gap property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions whose objectives are above a certain threshold are either close to or far from each other. The crux of our proof is that the classes of algorithms we consider exhibit a form of stability (noise-insensitivity): a small perturbation of the input induces a small perturbation of the output. We show by an interpolation argument that stable algorithms cannot overcome the OGP barrier. The stability of Langevin dynamics is an immediate consequence of the well-posedness of stochastic differential equations. The stability of low-degree polynomials and Boolean circuits is established using tools from Gaussian and Boolean analysis—namely hypercontractivity and total influence, as well as a novel lower bound for random walks avoiding certain subsets, which we expect to be of independent interest. In the case of Boolean circuits, the result also makes use of Linial–Mansour–Nisan’s classical theorem. Our techniques apply more broadly to low influence functions, and we expect that they may apply more generally.

中文翻译：

---

布尔电路、低次多项式和朗之万动力学的随机优化问题的难度

SIAM 计算杂志，第 53 卷，第 1 期，第 1-46 页，2024 年 2 月。
摘要。我们考虑寻找具有随机目标函数的优化问题的近乎最优解的问题。这些问题在随机图理论、理论计算机科学和统计物理学中广泛出现。我们考虑的两个具体问题是（a）优化球面或伊辛自旋玻璃模型的哈密顿量，以及（b）在稀疏的 Erdős-Rényi 图中找到一个大的独立集。考虑以下算法族：（a）输入的低次多项式——捕获许多先前算法的通用框架；(b) 低深度布尔电路；(c) Langevin 动力学算法，梯度下降算法的典型蒙特卡洛模拟。我们证明这些算法族不可能有很高的成功概率。对于布尔电路的情况，我们的结果改进了电路复杂性理论中已知的最先进的边界（尽管我们考虑的是搜索问题而不是决策问题）。我们的证明利用了这样一个事实：已知这些模型表现出接近最优解的重叠间隙属性（OGP）的变体。具体来说，对于这两个模型，每两个目标高于某个阈值的解决方案要么彼此接近，要么彼此远离。我们证明的关键在于，我们考虑的算法类别表现出某种形式的稳定性（噪声不敏感）：输入的小扰动会引起输出的小扰动。我们通过插值论证证明稳定算法无法克服 OGP 障碍。朗之万动力学的稳定性是随机微分方程适定性的直接结果。低次多项式和布尔电路的稳定性是使用高斯和布尔分析的工具建立的，即超收缩性和总影响力，以及避免某些子集的随机游走的新下界，我们希望这些子集具有独立的兴趣。对于布尔电路，结果还利用了 Linial-Mansour-Nisan 的经典定理。我们的技术更广泛地适用于低影响函数，并且我们期望它们可以更广泛地应用。