← 研踪 PaperTrailarXiv:2610.35195 · v1
Global $\mathcal{O}\left(k^{-2}\right)$ Convergence of Multilevel Gradient-Regularized Newton Methods
首次提交 2026-10-26 18:47
版本更新 2026-10-26 18:47
雷达观测 2026-10-29 00:36
摘要
We develop and analyze multilevel gradient-regularized Newton methods for unconstrained convex optimization with Lipschitz-continuous Hessians. By computing Newton corrections in lower-dimensional subspaces, the proposed methods aim to reduce computational costs while preserving the global $\mathcal{O}\left(k^{-2}\right)$ convergence rate in function values of the full-space method. To establish this rate for multilevel methods, we introduce a class of row-nested restriction operators that ensures monotonicity of the reduced-gradient norms across levels. Numerical experiments with row-nested operators demonstrate convergence behavior comparable to full-space gradient-regularized Newton, while offering substantial computational savings on large-scale instances.
作者备注：28 pages, 3 figures
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v1 · 2026-10-26 18:47 · 最新观测版本
Global $\mathcal{O}\left(k^{-2}\right)$ Convergence of Multilevel Gradient-Regularized Newton Methods
We develop and analyze multilevel gradient-regularized Newton methods for unconstrained convex optimization with Lipschitz-continuous Hessians. By computing Newton corrections in lower-dimensional subspaces, the proposed methods aim to reduce computational costs while preserving the global $\mathcal{O}\left(k^{-2}\right)$ convergence rate in function values of the full-space method. To establish this rate for multilevel methods, we introduce a class of row-nested restriction operators that ensures monotonicity of the reduced-gradient norms across levels. Numerical experiments with row-nested operators demonstrate convergence behavior comparable to full-space gradient-regularized Newton, while offering substantial computational savings on large-scale instances.
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