提出新分析框架，让无需调参的优化算法在非凸问题上更稳定高效。
Analysis of Schedule-Free Nonconvex Optimization
- 构建鲁棒李雅普诺夫框架，仅需平滑性和下界条件即可分析
- 得到无需依赖总步数T的收敛速率：最坏为1/log T，可能达1/T
- 适用于大规模学习中的自适应优化，适合关注算法稳定性研究者
一阶优化方法支撑着大多数大规模学习算法，但其经典收敛性保证依赖于与总迭代次数T相关的精细步长调度，而T通常事先未知。无调度（Schedule-Free, SF）方法通过融合Polyak-Ruppert平均与动量机制，承诺在不依赖T的超参数下实现最优性能，但其在非凸场景下的分析受限或依赖强全局假设。本文引入一种稳健的李雅普诺夫分析框架，在仅假设L-光滑与函数下有界条件下，将SF分析简化为单步下降不等式。由此导出非凸设置下的时长无关界：恒定步长+PR平均为O(1/log T)，线性增长步长为O(log T/T)，多项式平均则呈现连续的O(T^{-(1-α)})收敛率。我们通过性能估计算法（PEP）实验数值验证了这些速率，并提示原始非凸SF算法的1/log T界可能可收紧至1/T。本工作将SF的无时长依赖保证扩展至平滑非凸优化，并指明未来实现最优非凸速率的方向。
原文摘要 · Abstract (English)
First-order methods underpin most large-scale learning algorithms, yet their classical convergence guarantees hinge on carefully scheduled step-sizes that depend on the total horizon $T$, which is rarely known in advance. The Schedule-Free (SF) method promises optimal performance with hyperparameters that are independent of $T$ by interpolating between Polyak--Ruppert averaging and momentum, but nonconvex analysis of SF has been limited or reliant on strong global assumptions. We introduce a robust Lyapunov framework that, under only $L$-smoothness and lower-boundedness, reduces SF analysis to a single-step descent inequality. This yields horizon-agnostic bounds in the nonconvex setting: $O(1/\log T)$ for constant step + PR averaging, $O(\log T/T)$ for a linearly growing step-size, and a continuum of $O(T^{-(1-α)})$ rates for polynomial averaging. We complement these proofs with Performance Estimation Problem (PEP) experiments that numerically validate our rates and suggest that our $O(1/\log T)$ bound on the original nonconvex SF algorithm may tighten to $O(1/T)$. Our work extends SF's horizon-free guarantees to smooth nonconvex optimization and charts future directions for optimal nonconvex rates.
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