严格凸二次型上秩压缩加权LMSD扫描的全局无条件$R$-线性收敛性
Global and Unconditional $R$-Linear Convergence of Rank-Compressed Weighted LMSD Sweeps for Strictly Convex Quadratics
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中文总结 AI 辅助
针对严格凸二次优化，研究秩压缩LMSD的全局无条件R-线性收敛性，给出均匀端点估计，并证明加权方法共享衰减因子。
中文摘要 AI 辅助
我们研究了严格凸二次优化中具有精确代数秩压缩的有限记忆最速下降（LMSD）方法。在每个循环中，该方法将梯度历史限制在其列空间中，并在下一次扫描中应用压缩投影的所有Ritz值的倒数。如果块起始梯度至多有$p$个活跃的不同特征值，则延迟扫描有限终止。对于非终止轨迹，完整扫描多项式的行列式和Cauchy-Binet公式在不需满秩历史或逐次归一化条件数界的情况下给出全局收敛性。连续的扫描端点定义了相容状态闭锥上的连续正齐次映射。紧致性随后给出$R$-线性端点估计，对相容初始状态一致，常数仅依赖于$H$和$p$；该估计扩展到内部梯度、迭代误差和目标间隙。每个固定的正谱权重$W=\omega(H)$线性共轭于标准方法。因此，加权方法共享其衰减因子，而欧几里得范数前因子至多增加$\sqrt{\kappa_2(W)}$。这包括调和Ritz LMSD、固定幂权重以及延迟BB1和BB2递推。
英文摘要
We study limited memory steepest descent (LMSD) with exact algebraic rank compression for strictly convex quadratic optimization. At each cycle, the method restricts the gradient history to its column space and applies the reciprocals of all Ritz values of the compressed projection in the next sweep. If the block-start gradient has at most $p$ active distinct eigenvalues, the delayed sweep terminates finitely. For nonterminating trajectories, a determinant and Cauchy--Binet formula for the complete-sweep polynomial yields global convergence without a full-rank history or a run-wise normalized-conditioning bound. Consecutive sweep endpoints define a continuous positively homogeneous map on a closed cone of compatible states. Compactness then gives an $R$-linear endpoint estimate, uniform over compatible initial states, with constants depending only on $H$ and $p$; the estimate extends to inner gradients, iterate errors, and objective gaps. Every fixed positive spectral weight $W=ω(H)$ is linearly conjugate to the standard method. The weighted methods therefore share its decay factor, while the Euclidean-norm prefactor is increased by at most $\sqrt{κ_2(W)}$. This includes harmonic-Ritz LMSD, fixed power weights, and the delayed BB1 and BB2 recurrences.
发表机构
- State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算国家重点实验室）
- University of Chinese Academy of Sciences（中国科学院大学）
- School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
