非扩张双时间尺度随机逼近：固定调度的四分之一障碍和偏差校正加速

中文摘要 AI 辅助

非扩张双时间尺度随机逼近由慢随机Krasnoselskii - Mann不动点迭代控制，而非收缩到唯一平衡点。我们在收缩快映射和非扩张约化慢映射下研究此机制。首先证明有限时间下界，表明对于任意规定的慢步长调度$(\beta_k)$，经典KM残差尺度$(\sum_{i<N}\beta_i(1 - \beta_i))^{-1}$对于相应未正则化的KM更新在最坏情况下是尖锐的。然后引入残差预条件慢预言机消除对快跟踪误差的一阶依赖。在嵌套Tikhonov - KM算法中，未校正预言机产生总样本率$T^{-1/4 + o(1)}$，校正后为$T^{-1/3 + o(1)}$。最后表明在光滑导数预言机模型中可避免嵌套方法的重复内循环成本。单环算法在线跟踪快平衡点和泄漏预条件器，每次迭代用$O(1)$个原始样本实现$T^{-1/2 + o(1)}$。

英文摘要

Non-expansive two-time-scale stochastic approximation is governed by a slow stochastic Krasnoselskii--Mann fixed-point iteration rather than by contraction to a unique equilibrium. We study this regime under a contractive fast map and a non-expansive reduced slow map. We first prove a finite-horizon lower bound showing that, for any prescribed slow stepsize schedule $(β_k)$, the classical KM residual scale $(\sum_{i<N}β_i(1-β_i))^{-1}$ is worst-case sharp for the corresponding unregularized KM update. Combined with the raw fast-tracking leakage scale, this explains the previously observed $k^{-1/4+o(1)}$ last-iterate mean-square residual exponent. We then introduce a residual-preconditioned slow oracle that cancels the first-order dependence on the fast tracking error. In a nested Tikhonov-KM algorithm, the uncorrected oracle yields total-sample rate $T^{-1/4+o(1)}$, while the corrected oracle yields $T^{-1/3+o(1)}$. This improvement comes from changing the slow-oracle bias from first order to second order in the fast error after all inner-loop samples are counted. Finally, we show that the repeated inner-loop cost of the nested method can be avoided in a smooth derivative-oracle model. A single-loop algorithm that tracks both the fast equilibrium and the leakage preconditioner online achieves $T^{-1/2+o(1)}$ with $O(1)$ primitive samples per iteration.

发表机构

• Indian Institute of Technology Kharagpur（印度理工学院Khargapur分校）
• Mohamed bin Zayed University of Artificial Intelligence（Mohamed bin Zayed人工智能大学）
• Purdue University（普渡大学）

机构由 AI 辅助整理，请以论文原文为准。