用复分析理论让优化器看清全局，自动避开局部最优。
Globally aware optimization with resurgence
- 通过求解配分函数的渐近展开，从发散级数中提取全局结构信息。
- 发现临界点值与Borel平面上奇点一一对应，可定位所有极值点。
- 为梯度下降提供理论指导，适合对收敛性要求高的深度学习任务。
现代优化面临的核心挑战是：基于梯度的局部方法无法获取目标函数 $L$ 的全局结构信息，常导致次优收敛且对初始化敏感。本文提出一种新优化框架，利用复分析中的再生理论（resurgence theory），从发散渐近级数中提取全局结构信息。关键洞见在于：参数空间配分函数 $Z(g) = \int e^{-L(θ)/g} dθ$ 的阶乘发散微扰展开，其Borel变换奇点精确编码了目标函数景观中所有临界值。算法通过计算小耦合 $g\ll 1$ 下的 $Z(g)$，提取渐近系数，并识别对应于临界目标函数值的Borel平面奇点。这些目标值为局部优化器提供全局指引，实现有理论依据的学习率自适应和逃离次优区域的能力。相比启发式自适应方法，该方法的引导作用建立在优化景观几何之上。
原文摘要 · Abstract (English)
Modern optimization faces a fundamental challenge: local gradient-based methods provide no global information about the objective function $L$ landscape, often leading to suboptimal convergence and sensitivity to initialization. We introduce a novel optimization framework that leverages resurgence theory from complex analysis to extract global structural information from divergent asymptotic series. Our key insight is that the factorially divergent perturbative expansions of parameter space partition functions encode precise information about all critical objective function value in the landscape through their Borel transform singularities. The algorithm works by computing the statistical mechanical partition function $Z(g) = \int e^{-L(θ)/g} dθ$ for small coupling $g\ll 1$, extracting its asymptotic series coefficients, and identifying Borel plane singularities that correspond one-to-one with critical objective function values. These target values provide global guidance to local optimizers, enabling principled learning rate adaptation and escape from suboptimal regions. Unlike heuristic adaptive methods, targets are theoretically grounded in the geometry of the optimization landscape.
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