发表机构
Korea Science Academy of KAIST; Graduate School of AI, KAIST（韩国科学技术院科学高等研究院; 韩国科学技术院人工智能研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对梯度下降的非任意时刻与任意时刻设置，建立了比此前更强的收敛率下界，缩小了与对应上界之间的差距。
AI 中文摘要
在凸优化领域，固定步长梯度下降（GD）的速率最优收敛率对于L-利普希茨光滑凸目标函数，已有公认结果为Θ(N⁻¹)。令人惊讶的是，近期多项研究表明，采用非恒定、非自适应的确定性步长调度可加速普通GD。截至目前，非任意时刻与任意时刻设置下的最佳已知上界分别为O(N⁻¹.²⁷¹)（Altschuler和Parrilo，2025；Grimmer等人，2023）与O(N⁻¹.¹¹⁹)（Zhang等人，2025）；而同期非任意时刻与任意时刻设置下的最佳报道下界（或壁垒）分别为Ω(N⁻¹.⁶³⁵)与Ω(N⁻¹.²⁴¹)（Ye和Liu，2026）。本文通过在两种设置下建立GD收敛率的更强下界缩小上述差距：非任意时刻速率界的下界为Ω(N⁻¹.⁴⁵⁰)，任意时刻速率壁垒的下界为Ω(N⁻¹.¹⁸⁴)。
英文摘要
The rate-optimal convergence rate of gradient descent (GD) with a fixed step-size is well known to be $Θ(N^{-1})$ for $L$-Lipschitz smooth convex objectives in the prior art in convex optimization. Surprisingly, several recent works show that we can accelerate vanilla GD by applying a nonconstant, nonadaptive, deterministic step-size schedule. The best-known upper bounds so far in the non-anytime & anytime setups are $O(N^{-1.271})$ [Altschuler and Parrilo, 2025, Grimmer et al., 2023] and $O(N^{-1.119})$ [Zhang et al., 2025], respectively. On the other hand, the best reported lower bounds (or barriers) up to date in the non-anytime & anytime setups are $Ω(N^{-1.635})$ and $Ω(N^{-1.241})$ [Ye and Liu, 2026], respectively. We narrow these gaps by establishing stronger lower bounds for GD's convergence rate in both settings: $Ω(N^{-1.450})$ for the non-anytime rate bound and $Ω(N^{-1.184})$ for the anytime rate barrier.
Comments38 pages, 2 figures. Minor change in abstract, intro, & Fig. 1
