发表机构
University of Passau（帕绍大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了一个鲁棒的感知机循环定理，适用于非对称正定算子，并导出Frank--Wolfe算法的$O(k^{-1})$最后迭代界，应用于博弈、鞍点及交通分配问题。
AI 中文摘要
Block和Levin的经典感知机循环定理（\n\cite{BlockLevin1970}）限制了校正序列，其选定的更新来自有限集合，且与当前状态的内积非正。我们证明了加性轨迹$z_{k+1}=z_k+u_k$（对所有整数$k\geq0$）的鲁棒变体，其中增量属于有限集合$U\subset E$，这里$E$是有限维实内积空间，且$0\in\conv U$。设$A\colon E\to E$具有正定对称部分，但不要求对称，且$B\geq0$。若每个$u_k$是$u\mapsto\ip{Az_k}{u}$在$U$上的$B$-近似最小化器，则$\sup_{k\geq0}\norm{z_k}\leq C(1+\norm{z_0}+B)$，其中$C=C(E,U,A)$独立于初始状态、$B$及所有允许的更新选择。我们的核心算法结果是：对于多胞形上具有相对内部解的仿射强单调变分不等式，调和顶点返回Frank--Wolfe算法的$O(k^{-1})$最后迭代范数界。该结果将Bach、Lacoste-Julien和Obozinski（\cite[第4.2节]{BachEtAl2012}）的二次Frank--Wolfe/herding保证推广到非对称仿射算子。我们将此界应用于强稳定线性群体博弈中的经验最优响应、二次正则化双线性鞍点问题，以及具有强制性仿射非对称成本映射的交通分配问题。
英文摘要
The classical perceptron cycling theorem of Block and Levin \cite{BlockLevin1970} bounds correction sequences whose selected updates come from a finite set and have nonpositive inner product with the current state. We prove a robust variant for additive trajectories $z_{k+1}=z_k+u_k$, for all integers $k\geq0$, with increments in a finite set $U\subset E$, where $E$ is a finite-dimensional real inner-product space and $0\in\conv U$. Let $A\colon E\to E$ have positive-definite symmetric part, without requiring symmetry, and let $B\geq0$. If each $u_k$ is a $B$-approximate minimizer of $u\mapsto\ip{Az_k}{u}$ over $U$, then $\sup_{k\geq0}\norm{z_k}\leq C(1+\norm{z_0}+B)$, with $C=C(E,U,A)$ independent of the initial state, $B$, and all admissible update choices. We derive two algorithmic consequences. The first is an $O(k^{-1})$ last-iterate norm bound for harmonic vertex-returning Frank--Wolfe for affine strongly monotone variational inequalities on polytopes with relatively interior solutions. It extends the quadratic Frank--Wolfe/herding guarantee of Bach, Lacoste-Julien, and Obozinski \cite[Section~4.2]{BachEtAl2012} to nonsymmetric affine operators. The second consequence concerns oblique relaxation for linear inequalities. Greedy corrections through a fixed matrix with positive-definite symmetric part remain bounded even for inconsistent systems. In particular, for a square matrix $G$ with positive-definite symmetric part, repeatedly increasing the coordinate corresponding to a most-violated inequality of $Gx\geq b$ terminates at an exactly feasible point after finitely many unit corrections. This remains true under bounded additive selection errors, provided the stopping test uses the true inequalities.
Comments44 pages, including a weighted version and the application of correction sequences for linear systems
