论文概要
研究领域: ML 作者: Dawei Li, Xiaotian Jiang, Mingyi Hong 发布时间: 2026-07-25 arXiv: 2507.20474
中文摘要
Barzilai-Borwein（BB）方法在连续优化中表现出强大的实际性能，但其收敛动态仍然 poorly understood。特别是，一个核心的未解决问题是BB是否对几乎所有严格凸二次问题和初始点超线性收敛。我们对这个问题给出了否定回答。具体而言，对于每个有限维度n≥4，我们构造了一个非空开集，因此具有正Lebesgue测度，的严格凸二次问题族和初始点，对于这些问题长Barzilai-Borwein方法（BB1）收敛但不能根超线性收敛。更精确地说，以显式常数ρ_min=10^{-6}、ρ_max=0.61，梯度的每个谱分量都被相应的几何序列上下界约束。因此，梯度范数和误差的能量范数满足具有相同速率的双边几何估计，而目标差距满足具有平方速率的相应估计。特别是，这三个量都被几何序列下界约束，排除了超线性收敛。该构造高度非平凡，基于对四维投影化BB动力学的非共振、吸引七周期的计算机辅助证明。
原文摘要
Barzilai-Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension n>=4, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai-Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants rho_min=10^{-6}, rho_max=0.61, every spectral component of the gradient is bounded above and below by the corresponding geometric …
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