Cyclic steepest descent is $R$-superlinearly convergent for almost every initialization on two-spectrum quadratics

Yu Li; Qihang Wang

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arXiv · 2610.15335

Cyclic steepest descent is $R$-superlinearly convergent for almost every initialization on two-spectrum quadratics

Yu Li·Qihang Wang

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Abstract

We completely classify cyclic steepest descent (CSD) on strictly convex quadratics whose Hessian has exactly two distinct eigenvalues. At the beginning of each cycle CSD recomputes an exact steepest-descent stepsize and reuses it for $m$ updates. For every $m\ge2$, write $A=aP+bQ$ with $0 0$ is precisely the nonterminating exceptional set: there the convergence is $R$-linear with factor $(b-a)/(b+a)$, whereas every other nonterminating initialization is $R$-superlinear and one-eigenspace initializations terminate in one step. The exceptional cone has Lebesgue measure zero, so $R$-superlinear convergence holds almost everywhere in this two-spectrum class. The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and OpenAI GPT-6 Astra. Further details appear in the disclosure at the end of the paper.

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BibTeX RIS

Yu Li, Qihang Wang. 2026-10-11. Cyclic steepest descent is $R$-superlinearly convergent for almost every initialization on two-spectrum quadratics. https://arxiv.org/abs/2610.15335

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