发表机构
The Hong Kong Polytechnic University; Tsinghua University（香港理工大学; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种秩自适应Frank--Wolfe方法，通过非凸谱Oracle和有效秩替代，在更新秩不超过最优秩时实现线性收敛，并保持低计算成本。
AI 中文摘要
对于谱面体上凸优化的Frank--Wolfe (FW)方法，当每次迭代的更新秩不超过（未知的）最优秩$r^*$时，块更新变体能否线性收敛仍然是一个开放问题。现有的块更新和谱FW方法需要更新秩至少为$r^*$——并且通常需要事先知道$r^*$——才能获得线性收敛速率。本文开发了一种秩自适应的FW方法，其更新秩满足每次迭代$\u0052widehat{r}_t\ue005le r_t\ue005le r^*$，并且在二次增长和严格互补性（谱FW分析中常用的两个条件）下，经过有限次预热后线性收敛。该方法基于两个设计。第一个是非凸谱Oracle，其动机来自单纯形与谱面体之间的几何联系；它产生当前迭代的阈值秩$r_t$和闭式低秩解。然而，精确计算$r_t$需要完整的特征分解。第二个引入了当前迭代的有效秩$\u0052widehat{r}_t$，这是一种廉价的替代，继承了谱Oracle的最优性。算法在阈值秩和有效秩之间切换，使得实际的FW更新使用$\u0052widehat{r}_t$，保持每次迭代的成本与标准FW相当，并最终识别出$r^*$。这些结果缩小了谱面体上Frank--Wolfe方法的低秩效率与快速收敛之间的差距。数值实验证明了所提出方法的优势。
英文摘要
For Frank--Wolfe (FW) methods for convex optimization over the spectrahedron, it remains open whether a block-update variant can be linearly convergent when the update rank never exceeds the (unknown) optimal rank $r^*$ at each iteration. Existing block and spectral FW methods require an update rank at least $r^*$---and typically prior knowledge of $r^*$---to obtain a linear rate. This paper develops a rank-adaptive FW method whose update ranks satisfy $\widehat{r}_t\le r_t\le r^*$ at every iteration and which converges linearly after a finite burn-in under quadratic growth and strict complementarity, the two conditions commonly used in spectral FW analyses. The method is built on two designs. The first is a nonconvex spectral oracle, motivated by the geometric connection between the simplex and the spectrahedron; it yields a thresholding rank $r_t$ of the current iterate and a closed-form low-rank solution. Computing $r_t$ exactly, however, requires a full eigendecomposition. The second introduces the efficient rank $\widehat{r}_t$ of the current iterate, a cheap surrogate that inherits the optimality properties of the spectral oracle. The algorithm switches between the thresholding rank and the efficient rank so that the actual FW update uses $\widehat{r}_t$, keeps the per-iteration cost comparable to standard FW, and eventually identifies $r^*$. These results close the gap between low-rank efficiency and fast convergence for Frank--Wolfe methods over the spectrahedron. Numerical experiments demonstrate the advantage of the proposed method.
