TWe establish regret lower bounds for adaptively controlling an unknown linear Gaussian system with quadratic costs. We combine ideas from experiment design, estimation theory and a perturbation bound of certain information matrices to derive regret lower bounds exhibiting scaling on the order of magnitude $\sqrt{T}$ in the time horizon $T$. Our bounds accurately capture the role of control-theoretic parameters and we are able to show that systems that are hard to control are also hard to learn to control; when instantiated to state feedback systems we recover the dimensional dependency of earlier work but with improved scaling with system-theoretic constants such as system costs and Gramians. Furthermore, we extend our results to a class of partially observed systems and demonstrate that systems with poor observability structure also are hard to learn to control.

翻译：针对未知线性高斯系统在二次成本下的自适应控制问题，我们建立了遗憾下界。通过融合实验设计、估计理论以及特定信息矩阵的扰动界等思想，我们推导出在时间范围$T$内呈$\sqrt{T}$量级标度的遗憾下界。该下界精确刻画了控制理论参数的作用，表明难以控制的系统同样难以学习控制；在应用于状态反馈系统时，我们复现了早期工作的维度依赖关系，但改进了系统理论常数（如系统成本与格拉姆矩阵）的标度关系。此外，我们将结果扩展至一类部分可观测系统，并证明观测性结构较差的系统同样难以学习控制。