发表机构
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University（越南国立大学理科大学数学与计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出黎曼无投影优化的几何框架，用幂型双测地线模量量化可行集几何，导出间隙反馈短步策略的收敛速率$O(t^{-p/(p-1)})$，并证明该指数在几何控制类上是最优的。
AI 中文摘要
无投影方法用线性最小化预言机替代投影，使其收敛性从根本上对可行区域的几何形状敏感。我们为黎曼Frank-Wolfe优化中的这种依赖性建立了一个几何框架。一种幂型双测地线模量量化了可行集的内在几何，一个精确的传递原理表明该模量诱导出一个由黎曼对数映射的微分调制的Frank-Wolfe缩放余量。由此产生的分解将可行集几何与周围黎曼畸变的贡献分开。Rauch比较为该畸变提供了显式的曲率相关证书，而在配备仿射不变度量的正定锥上，对数微分允许用相对特征值散布进行精确的谱表征。结合二次误差界，幂型几何产生间隙反馈短步策略的收敛速率$O(t^{-p/(p-1)})$。对于每个固定的$p>3$，双曲平面中的紧致可行集在同一策略下达到匹配速率$\Theta(t^{-p/(p-1)})$。因此，几何导出的指数$p/(p-1)$对于此处考虑的几何控制类上的间隙反馈短步策略是最优的。
英文摘要
Projection-free methods replace projections by linear minimization oracles, making their convergence fundamentally sensitive to the geometry of the feasible region. We develop a geometric framework for this dependence in Riemannian Frank--Wolfe optimization. A power-type double-geodesic modulus quantifies the intrinsic geometry of the feasible set, and an exact transfer principle shows that this modulus induces a Frank--Wolfe scaling margin modulated by the differential of the Riemannian logarithm map. The resulting factorization separates the contributions of feasible-set geometry and ambient Riemannian distortion. Rauch comparison provides explicit curvature-dependent certificates for this distortion, while on the positive-definite cone equipped with the affine-invariant metric the logarithm differential admits an exact spectral characterization in terms of relative eigenvalue spread. Combined with a quadratic error bound, the power-type geometry yields the convergence rate $O(t^{-p/(p-1)})$ for the gap-feedback short-step policy. For every fixed $p>3$, a compact feasible set in a hyperbolic plane attains the matching rate $Θ(t^{-p/(p-1)})$ under the same policy. The geometry-derived exponent $p/(p-1)$ is therefore sharp for the gap-feedback short-step policy over the geometry-controlled class considered here.
Comments50 pages
