Superlinear Convergence of Cyclic Gradient Methods with Historical Ritz Step Sizes

Abstract

Simplified spectral models predict dimension-dependent superlinear convergence for gradient methods, but the corresponding behavior of actual iterations remains only partially understood. Historical methods introduce an additional difficulty: the gradients defining the step sizes differ from those receiving them. We study this offset for cyclic reuse of historical Ritz step sizes on strictly convex quadratics. Within the existing limited memory steepest descent with retards (LMSDR) family, a rolling cyclic Ritz (RCR) rule reuses $m$ historical Ritz values $\ell$ times per cycle, ending with a complete sweep. At $q=m+1$ active distinct eigenvalues, an exact delayed recurrence gives almost-everywhere doubly exponential convergence with an explicit rate and exceptional weights. Nonempty open basins also exist for $m+1 \binom{q-1}{m}$. Increasing $\ell$ admits larger $q$ for fixed $m$. The guarantee extends to history-dependent choices from finite sampling menus under a bound on the number of admissible sampling sequences. Geometrically spaced decisions retain the SCR condition and rate range. Every update is counted, including sampling; a numerical illustration compares finite-tolerance update counts.