In this work, we study non-asymptotic bounds on correlation between two time realizations of stable linear systems with isotropic Gaussian noise. Consequently, via sampling from a sub-trajectory and using \emph{Talagrands'} inequality, we show that empirical averages of reward concentrate around steady state (dynamical system mixes to when closed loop system is stable under linear feedback policy ) reward , with high-probability. As opposed to common belief of larger the spectral radius stronger the correlation between samples, \emph{large discrepancy between algebraic and geometric multiplicity of system eigenvalues leads to large invariant subspaces related to system-transition matrix}; once the system enters the large invariant subspace it will travel away from origin for a while before coming close to a unit ball centered at origin where an isotropic Gaussian noise can with high probability allow it to escape the current invariant subspace it resides in, leading to \emph{bottlenecks} between different invariant subspaces that span $\mathbb{R}^{n}$, to be precise : system initiated in a large invariant subspace will be stuck there for a long-time: log-linear in dimension of the invariant subspace and inversely to log of inverse of magnitude of the eigenvalue. In the problem of Ordinary Least Squares estimate of system transition matrix via a single trajectory, this phenomenon is even more evident if spectrum of transition matrix associated to large invariant subspace is explosive and small invariant subspaces correspond to stable eigenvalues. Our analysis provide first interpretable and geometric explanation into intricacies of learning and concentration for random dynamical systems on continuous, high dimensional state space; exposing us to surprises in high dimensions
翻译:在这项研究工作中,我们研究了稳定线性系统的两个时间实现之间的相关性的非渐进界限,该系统带有各向同性的高斯噪声。因此,通过从子轨迹中采样并使用\emph{Talagrand}的不等式,我们展示了关于稳态(当闭环系统在线性反馈策略下稳定时,动态系统混合到该稳态)回报的经验证据平均值的集中化,具有高概率。与常见的认为,大的谱半径意味着样本间的相关性更强不同,\emph{系统特有的本征值代数和几何重数之间的差异导致与系统转移矩阵有关的大不变子空间};一旦该系统进入大的不变子空间,它将会通过一段时间远离原点,然后再接近位于原点处的一个以单位球为中心的地方,此处各向同性高斯噪声可以以高概率让其逃离其当前所在的不变子空间,从而导致跨越覆盖 $\mathbb{R}^{n}$ 的不同不变子空间之间的瓶颈,以确切:系统在大不变子空间中启动将会很长时间停留在那里:大小对于不变子空间的维数是对数级的,而逆向的本征值幅度值的对数则是倒数。在通过单个轨迹的普通最小二乘估计系统转移矩阵的问题中,如果与大不变子空间有关的转移矩阵的频谱具有爆发性,而小不变子空间对应于稳定的本征值,则这种现象会更加明显。我们的分析为解释连续高维状态空间上随机动力系统的学习和集中化提供了首个可解释的几何说明;使我们了解高维度中的惊喜。