自适应动力系统基本极限的发散性刻画
Divergence-based characterization of fundamental limitations of adaptive dynamical systems
摘要：
自适应动态系统出现在多种环境中，例如，优化、控制、通信、信号处理和机器学习。因此，准确描述其基本局限性至关重要。在本文中，我们考虑自适应控制和/或识别随机动力系统的一般问题，我们的{ \em先验}知识允许我们将系统的度量空间（不确定性集）的一个子集。我们提出了一个信息理论的元定理，捕捉度量的复杂性（或丰富性）的不确定性集，在控制和观察系统的过程中在线获取的信息量之间的权衡，和剩余的不确定性后，观察已经收集。遵循Zames的方法，我们通过不确定性集合的Kolmogorov（度量）熵来量化{ \em a priori}信息，而在线获取的信息被表示为信息分歧之和。一般的理论是用来推导新的极小极大下界的度量识别误差，以及给一个简单的推导所需的最小时间来稳定一个不确定的随机线性系统。
英文摘要：
Adaptive dynamical systems arise in a multitude of contexts, e.g., optimization, control, communications, signal processing, and machine learning. A precise characterization of their fundamental limitations is therefore of paramount importance. In this paper, we consider the general problem of adaptively controlling and/or identifying a stochastic dynamical system, where our {\em a priori} knowledge allows us to place the system in a subset of a metric space (the uncertainty set). We present an information-theoretic meta-theorem that captures the trade-off between the metric complexity (or richness) of the uncertainty set, the amount of information acquired online in the process of controlling and observing the system, and the residual uncertainty remaining after the observations have been collected. Following the approach of Zames, we quantify {\em a priori} information by the Kolmogorov (metric) entropy of the uncertainty set, while the information acquired online is expressed as a sum of information divergences. The general theory is used to derive new minimax lower bounds on the metric identification error, as well as to give a simple derivation of the minimum time needed to stabilize an uncertain stochastic linear system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
