发表机构
Xidian University（西安电子科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出优化几何动力学框架，将优化视为参数轨迹等的耦合演化，分离不变障碍与可改善的几何不匹配，引入动态几何复杂度，分析多种相关流和可观测量，为比较自适应优化器提供理论基准。
AI 中文摘要
大多数基于梯度的优化方法在固定背景几何中移动参数，即便其内部状态隐含定义了长度、曲率和预处理的变化概念。我们引入优化几何动力学，一种基准语言，其中优化是参数轨迹、粒子传输分布和受控时变黎曼度量的耦合演化。该语言将不变障碍与可改善的几何不匹配分离。正度量保留临界点和莫尔斯指数，无法消除全局测地线凸性障碍，但可改变条件、分布传输和远离精确临界点的通量。我们引入动态几何复杂度，即降低可观测优化难度所需的最小几何成本。在具有完全正定度量控制的强凸二次目标的预言机基准模型中，此复杂度恰好是相对对数谱到低条件数集的仿射不变距离。我们还分析了黑塞匹配流、谱昂萨格松弛、离散指数投影更新、规范不变可观测量和固定时间局部莫尔斯鞍点通量。本文仅为理论研究，其论断是有证明的形式化陈述，旨在提供不变量和基准成本，以便在指定可允许度量族、曲率估计和离散化误差后，可将可实现的自适应优化器与之比较。
英文摘要
Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.
Comments25 pages, no figures
