Convergence of cyclic steepest descent for quadratics with two distinct eigenvalues
Abstract
The cyclic steepest descent method computes the exact steepest descent stepsize at the start of each cycle and reuses it for a fixed number m≥2 of iterations. We study this method for strictly convex quadratics whose Hessian has exactly two distinct eigenvalues, of arbitrary multiplicities, and determine its behavior from every initial point. At the minimizer no update is needed. For nonoptimal initializations, a gradient in one eigenspace gives one-step termination. If both eigenspace components are nonzero and have equal norm, the stepsize never changes and the error contracts by the same constant factor at every iteration. In the remaining case, both components are nonzero and unequal in norm, and the method converges doubly exponentially and hence R-superlinearly. Thus R-superlinear convergence fails exactly on an explicit cone of initial points of measure zero. The proof reduces the method to an exact scalar recurrence for the ratio of the two eigenspace components at the start of each cycle. We also determine the leading term of the logarithm of the error at every iteration within a cycle, which identifies the exact exponent of the doubly exponential decay and shows that this convergence is not Q-superlinear.
Related
- Doubly exponential convergence of the cyclic steepest descent method for strictly convex quadratics in arbitrary dimensions
- Counterexamples to R-superlinear convergence of cyclic steepest descent
- PowerStep: Memory-Efficient Adaptive Optimization via ℓp-Norm Steepest Descent
- R-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics
- Global and Unconditional R-Linear Convergence of Rank-Compressed Weighted LMSD Sweeps for Strictly Convex Quadratics
BibTeX
@misc{li2026convergence,
title = {{Convergence of cyclic steepest descent for quadratics with two distinct eigenvalues}},
author = {Yu Li and Qihang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15335},
eprint = {2610.15335},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15335},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convergence of cyclic steepest descent for quadratics with two distinct eigenvalues
AU - Yu Li
AU - Qihang Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.15335
UR - https://arxiv.org/abs/2610.15335
AB - The cyclic steepest descent method computes the exact steepest descent stepsize at the start of each cycle and reuses it for a fixed number m≥2 of iterations. We study this method for strictly convex quadratics whose Hessian has exactly two distinct eigenvalues, of arbitrary multiplicities, and determine its behavior from every initial point. At the minimizer no update is needed. For nonoptimal initializations, a gradient in one eigenspace gives one-step termination. If both eigenspace components are nonzero and have equal norm, the stepsize never changes and the error contracts by the same constant factor at every iteration. In the remaining case, both components are nonzero and unequal in norm, and the method converges doubly exponentially and hence R-superlinearly. Thus R-superlinear convergence fails exactly on an explicit cone of initial points of measure zero. The proof reduces the method to an exact scalar recurrence for the ratio of the two eigenspace components at the start of each cycle. We also determine the leading term of the logarithm of the error at every iteration within a cycle, which identifies the exact exponent of the doubly exponential decay and shows that this convergence is not Q-superlinear.
ER -