AI 中文总结
本文通过建立确定性与随机算子诱导不动点迭代的时间尺度分离结果，构建算子理论框架并给出可调参数界，用以证明反馈优化方案在两类场景下的收敛性，关联了时间尺度分离与算子理论领域。
AI 中文摘要
时间尺度分离是分析互联动力学系统的有力工具，而算子理论为研究以不动点迭代形式构建的迭代方法的收敛性提供了通用框架，包括优化、学习与控制领域的相关算法。本文通过为确定性和随机算子诱导的不动点迭代建立时间尺度分离结果，将这两个领域关联起来。与时间尺度分离的常规做法一致，我们的结果包含辅助系统，该系统由原始互联系统在时间尺度参数趋于零时产生，分别捕获慢算子和快算子诱导的动力学行为。所提出的算子理论框架可基于标准算子常数，给出该可调参数明确且易于验证的界。为说明结果的适用性，我们将其用于证明确定性和随机场景下反馈优化方案的收敛特性。
英文摘要
Timescale separation is a powerful tool for analyzing interconnected dynamical systems. Meanwhile, operator theory provides a general framework for studying the convergence of iterative methods formulated as fixed-point iterations, including algorithms arising in optimization, learning, and control. In this paper, we bridge these two areas by establishing timescale separation results for fixed-point iterations induced by both deterministic and stochastic operators. As customary in timescale separation, our results involve auxiliary systems that arise from the original interconnection in the limit as the timescale parameter tends to zero and separately capture the dynamics induced by the slow and fast operators. The proposed operator-theoretic framework yields explicit and readily checkable bounds on this tunable parameter, expressed in terms of standard operator constants. To illustrate the applicability of our results, we employ them to prove the convergence properties of a feedback optimization scheme in both deterministic and stochastic settings.
