A Robust Perceptron Cycling Theorem and Applications

RESUMO

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.