开环黎曼Frank--Wolfe方法：在误差界和尺度不等式下的快速收敛速率
Open-Loop Riemannian Frank--Wolfe: Fast Rates under Error Bounds and Scaling Inequalities
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中文总结 AI 辅助
本文针对紧致可行集上的黎曼Frank--Wolfe方法，在不同条件下推导了其快速收敛速率，还分析了标准间隙反馈短步并通过数值实验验证结果。
中文摘要 AI 辅助
本文研究了紧致可行集上光滑测地凸优化问题的黎曼Frank--Wolfe方法的快速收敛性，主要研究背景是在Hadamard流形这类主要场景下的优化问题。在一般完备流形上，分析考虑了所有可行的极小化测地线。本文采用仅使用迭代索引的开环步长ηₖ=a/(k+a)，在局部Hölder误差界和局部长度归一化方向尺度条件下，对于θ∈(0,1/2]，所有a>2均给出最终收敛速率O(k^(-1/(1-θ)))；对于强测地凸目标，内球条件可得到O(k^(-2))的速率。在精确黎曼尺度不等式和梯度范数的一致正下界条件下，所有a≥2在显式阈值索引后给出O(k^(-a))的速率，该速率对最近一半迭代的最小Frank--Wolfe间隙也成立。对于单位球中半径R<π/2的测地球，本文建立了尺度不等式α_R=½cot R，得到O(k^(-a))的原始误差和最近窗口间隙速率。本文还在局部误差界条件下分析了标准间隙反馈短步，得到θ<1/2时的O(k^(-1/(1-2θ)))速率和θ=1/2时的线性速率。数值实验验证了预测的速率并比较了仅迭代和基于反馈的步长选择。
英文摘要
We explore fast convergence of the Riemannian Frank--Wolfe method for smooth geodesically convex optimization over compact feasible sets. Hadamard manifolds are the main setting. On general complete manifolds, the analysis accounts for all feasible minimizing geodesics. We consider the open loop step-size $η_k=a/(k+a)$, which only uses the iteration index. Under a local Hölderian error bound and local length-normalized directional scaling, every $a>2$ gives the eventual rate $O(k^{-1/(1-θ)})$ for $θ\in(0,1/2]$. An interior-ball condition yields $O(k^{-2})$ for strongly geodesically convex objectives. Under an exact Riemannian scaling inequality and a uniform positive lower bound on the gradient norm, every $a\geq2$ gives $O(k^{-a})$ after an explicit threshold index. The same rate holds for the smallest Frank--Wolfe gap over the most recent half of the iterates. For geodesic balls of radius \(R<π/2\) in the unit sphere, we establish the scaling inequality with $α_R=\tfrac12\cot R$, yielding $O(k^{-a})$ primal error and recent-window gap rates. We also analyze the standard gap-feedback short step under the local error-bound conditions, obtaining $O(k^{-1/(1-2θ)})$ for $θ<1/2$ and a linear rate for $θ=1/2$. Numerical experiments illustrate the predicted rates and compare iteration-only and feedback-based step selection.
